A MILP formulation for an operating room scheduling problem under sterilizing activities constraints.pdf

CONFERENCE PROCEEDINGS
16th
International Conference on Project
Management and Scheduling
Rome, April 17-20, 2018
pms2018.ing.uniroma2.it

Proceedings of the 16th International Conference on Project Management and
Scheduling
PMS 2018 - April 17-20, 2018 - Rome, Italy
ISBN: 9788894982022
Editors:
Massimiliano Caramia
Lucio Bianco
Stefano Giordani
Address of the Editors:
University of Rome “Tor Vergata”
Dipartimento di Ingegneria dell’Impresa
Via del Politecnico, 1
00133 Roma - Italy
Tel. +39 06 72597360
Fax. +39 06 72597305
email: caramia@dii.uniroma2.it
Published by:
TexMat
Via di Tor Vergata, 93-95
00133 Roma
Tel. +39 06 2023572
www.texmat.it
e-mail: info@texmat.it
Place and Date of Publication:
Rome (Italy), March 31, 2018
i

PMS 2018 Preface
Preface
This volume contains the papers that will be presented at PMS 2018, the 16th
International
Conference on Project Management and Scheduling to be held on April, 17-20 2018 in Rome
- Italy.
The EURO Working Group on Project Management and Scheduling was established by Pro-
fessors Luı́s Valadares Tavares and Jan Weglarz during the EURO VIII Conference, Lisbon, in
September 1986. It was decided to organize a workshop every two years. Gathering the most
promising theoretical and applied advances in Project Management and Scheduling, and assess-
ing both the state-of-the-art of this field and its potential to support management systems are
the main objectives of these workshops.
76 extended abstracts have been submitted to the PMS 2018 Confenrece. These valuable con-
tributions were reviewed by 2 referees who are members of the International Program Committee
and distinguished researchers of the associated fields. The proceedings at hand contain the 65
papers that were finally accepted for presentation at the conference. There papers involves 165
authors from 23 different countries.
The 16th
edition of PMS has four plenary speakers: Professor Jacques Carlier (Univer-
sité de Technologie de Compiègne) will present the talk “Comparing event-node graphs with
nonrenewable resources and activity-node graphs with renewable resources”, Professor Erwin
Pesch (University of Siegen) will discuss on “Optimization problems in intermodal transport”,
Professor Ruben Ruiz (University of Valencia) and Professor Erik Demeulemeester (Katholic
University of Leuven) will delight us talking on “Simple metaheuristics for flowshop scheduling:
all you need is local search” and “On the construction of optimal policies for the RCPSP with
stochastic activity durations”, respectively.
The scientific program and the social events will give to all the participants an opportunity
to share research ideas and debate on recent advances on project managment and scheduling. I
am sure that together we will contribute to make PMS 2018 a great success.
Welcome in Rome and have an enjoyable stay!
Rome, 17th April 2018 Massimiliano Caramia (Conference Chair)
iii

PMS 2018 Committees
Organizing Committee
Massimiliano Caramia (Chairman) Università di Roma “Tor Vergata” (Italy)
Lucio Bianco Università di Roma “Tor Vergata” (Italy)
Stefano Giordani Università di Roma “Tor Vergata” (Italy)
Program Committee
Alessandro Agnetis Università di Siena (Italy)
Ali Allahverdi Kuwait University (Kuwait)
Christian Artigues LAAS-CNRS (France)
Francisco Ballestrin Universitat de València (Spain)
Fayez Boctor Université Laval (Canada)
Jazek Blażewicz Poznań University of Technology (Poland)
Massimiliano Caramia Università di Roma “Tor Vergata” (Italy)
Jacques Carlier Université de Technologie de Compiègne (France)
Erik Demeulemeester Katholieke Universiteit Leuven (Belgium)
Joanna Józefowska Poznań University of Technology (Poland)
Sigrid Knust Universität Osnabrück (Germany)
Rainer Kolisch Technische Universität München (Germany)
Mikhail Kovalyov National Academy of Sciences of Belarus (Belarus)
Wieslaw Kubiak Memorial University (Canada)
Erwin Pesch Universität Siegen (Germany)
Chris Potts University of Southampton (United Kingdom)
Rubén Ruiz Universitat Politècnica de València (Spain)
Avraham Shtub Technion - Israel Institute of Technology (Israel)
Funda Sivrikaya Şerifoğlu Istanbul Bilgi Üniversitesi (Turkey)
Vincent T’Kindt Université François Rabelais Tours (France)
Norbert Trautmann Universität Bern (Switzerland)
Mario Vanhoucke Ghent University (Belgium)
Jan Weglarz Poznań University of Technology (Poland)
Jürgen Zimmermann Technische Universität Clausthal (Germany)
Linet Özdamar Yeditepe Üniversitesi (Turkey)
v

PMS 2018 Table of Contents
Table of Contents
Scheduling energy-consuming jobs on parallel machines with piecewise-linear costs and
storage resources: A lot-sizing and scheduling perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Nabil Absi, Christian Artigues, Safia Kedad-Sidhoum, Sandra Ulrich Ngueveu, Janik
Rannou and Omar Saadi
The truck scheduling problem at cross docking terminals: Formulations and valid
inequalities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
Alessandro Agnetis, Lotte Berghman and Cyril Briand
The Price of Fairness in a Two-Agent Single-Machine Scheduling Problem . . . . . . . . . . . . . . . . 9
Alessandro Agnetis, Bo Chen, Gaia Nicosia and Andrea Pacifici
A MILP formulation for an operating room scheduling problem under sterilizing
activities constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
Hasan Al Hasan, Christelle Guéret, David Lemoine and David Rivreau
Modeling and solving a two-stage assembly scheduling problem with buffers . . . . . . . . . . . . . . 17
Carlos Andrés and Julien Maheut
A new polynomial-time algorithm for calculating upper bounds on resource usage for
RCPSP problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
Dmitry Arkhipov, Olga Battaı̈a and Alexander Lazarev
Assembly Flowshops Scheduling Problem to Minimize Maximum Tardiness with Setup
Times. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
Asiye Aydilek, Harun Aydilek and Ali Allahverdi
No-Wait Flowshop Scheduling Problem to Minimize Total Tardiness Subject to Makespan 32
Harun Aydilek, Asiye Aydilek and Ali Allahverdi
A Robust Optimization Model for the Multi-mode Resource Constrained Project
Scheduling Problem with Uncertain Activity Durations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
Noemie Balouka and Izack Cohen
Scheduling data gathering with limited base station memory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
Joanna Berlinska
A Chance Constrained Optimization Approach for Resource Unconstrained Project
Scheduling with Uncertainty in Activity Execution Intensity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
Lucio Bianco, Massimiliano Caramia and Stefano Giordani
Single-machine capacitated lot-sizing and scheduling with delivery dates and quantities . . . 50
Fayez Boctor
Single machine scheduling with m:n relations between jobs and orders: Minimizing the
sum of completion times and its application in warehousing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
Nils Boysen, Konrad Stephan and Felix Weidinger
A MILP formulation for multi-robot pick-and-place scheduling . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
Cyril Briand, Jeanine Codou Ndiaye and Rémi Parlouar
Minimizing resource management costs in a portfolio with resource transfer possibilities . . 62
Jerome Bridelance and Mario Vanhoucke
vii

Vehicle sequencing at transshipment terminals with handover relations . . . . . . . . . . . . . . . . . . . 66
Dirk Briskorn, Malte Fliedner and Martin Tschöke
Synchronous flow shop scheduling with pliable jobs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
Matthias Bultmann, Sigrid Knust and Stefan Waldherr
Computation of the project completion time distribution in Markovian PERT networks. . . 74
Jeroen Burgelman and Mario Vanhoucke
Comparing event-node graphs with nonrenewable resources and activity-node graphs
with renewable resources (Plenary). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
Jacques Carlier
Synchronizing Heterogeneous Vehicles in a Routing and Scheduling Context . . . . . . . . . . . . . . 79
Marc-Antoine Coindreau, Olivier Gallay and Nicolas Zufferey
On the construction of optimal policies for the RCPSP with stochastic activity durations . 83
Erik Demeulemeester
A B&B Approach to Schedule a No-wait Flow Shop to Minimize the Residual Work
Content Under Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
Simone Dolceamore and Marcello Urgo
On Index Policies in Stochastic Scheduling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
Franziska Eberle, Felix Fischer, Jannik Matuschke and Nicole Megow
Unrelated Parallel Machine Scheduling at a TV Manufacturer . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
Ali Ekici, Okan Ozener and Merve Burcu Sarikaya
A new set of benchmark instances for the Multi-Mode Resource Investment Problem . . . . . 100
Patrick Gerhards
A simheuristic for stochastic permutation flow shop problem considering quantitative
and qualitative decision criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
Eliana Marı́a Gonzalez-Neira and Jairo R. Montoya-Torres
An Algorithm for Schedule Delay Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
Pier Luigi Guida and Giovanni Sacco
Minimizing the total weighted completion time in single machine scheduling with
non-renewable resource constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
Peter Gyorgyi and Tamas Kis
The Cyclic Job Shop Problem with uncertain processing times. . . . . . . . . . . . . . . . . . . . . . . . . . . .119
Idir Hamaz, Laurent Houssin and Sonia Cafieri
Modeling techniques for the eS-graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
Máté Hegyháti
Scheduling Multiple Flexible Projects with Different Variants of Genetic Algorithms . . . . . . 128
Luise-Sophie Hoffmann and Carolin Kellenbrink
A comparison of neighborhoods for the blocking job-shop problem with total tardiness
minimization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
Julia Lange
viii

A parallel machine scheduling problem with equal processing time jobs, release dates
and eligibility constraints to minimize total completion time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
Kangbok Lee and Juntaek Hong
A new grey-box approach to solve challenging workforce planning and activities
scheduling problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
Ludovica Maccarrone and Stefano Lucidi
Scheduling Identical Parallel Machines with Delivery Times to Minimize Total Weighted
Tardiness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
Söhnke Maecker and Liji Shen
Modelling and Solving the Hotspot Problem in Air Traffic Control. . . . . . . . . . . . . . . . . . . . . . . . 149
Carlo Mannino, Giorgio Sartor and Patrick Schittekat
A proactive-reactive approach to schedule an automotive assembly line (Plenary) . . . . . . . . . 152
Massimo Manzini, Erik Demeulemeester and Marcello Urgo
Applying a cost, resource or risk perspective to improve tolerance limits for project
control: an empirical validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
Annelies Martens and Mario Vanhoucke
A Metamodel Approach to Projects Risk Management: outcome of an empirical testing
on a set of similar projects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
Federico Minelle, Franco Stolfi, Roberto Di Gioacchino and Stefano Santini
A column generation scheme for the Periodically Aggregated Resource-Constrained
Project Scheduling Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
Pierre-Antoine Morin, Christian Artigues and Alain Haı̈t
Development of a Schedule Cost Model for a Resource Constrained Project that
incorporates Idleness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
Babatunde Omoniyi Odedairo and Victor Oluwasina Oladokun
Optimization problems in intermodal transport (Plenary) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
Erwin Pesch
The Stakeholder Perspective: how management of KPIs can support value generation to
increase the success rate of complex projects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
Massimo Pirozzi
Multi-skill project scheduling in a nuclear research facility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
Oliver Polo Mejia, Marie-Christine Anselmet, Christian Artigues and Pierre Lopez
Scheduling Vehicles with spatial conflicts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
Atle Riise, Carlo Mannino, Oddvar Kloster and Patrick Schittekat
On some approach to solve a scheduling problem with a continuous doubly-constrained
resource. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
Rafal Rozycki and Grzegorz Waligóra
Simple metaheuristics for Flowshop Scheduling: All you need is local search (Plenary) . . . . 194
Rubén Ruiz
ix

Exact Methods for Large Unrelated Parallel Machine Scheduling Problems with
Sequence Dependent Setup Times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
Rubén Ruiz, Luis Fanjul-Peyró and Federico Perea
Power usage minimization in server problems of scheduling computational jobs on a
single processor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
Rafal Różycki, Grzegorz Waligóra and Jan Weglarz
Scheduling resource-constrained projects with makespan-dependent revenues and costly
overcapacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
Andre Schnabel and Carolin Kellenbrink
On the complexity of scheduling start time dependent asymmetric convex processing times209
Helmut A. Sedding
Resource-constrained project scheduling with alternative project structures . . . . . . . . . . . . . . . 213
Tom Servranckx and Mario Vanhoucke
A New Pre-Processing Procedure for the Multi-Mode Resource-Constrained Project
Scheduling Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
Christian Stürck
An O∗
(1.41n
)-time algorithm for a single machine just-in-time scheduling problem with
common due date and symmetric weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
Vincent T’Kindt, Lei Shang and Federico Della Croce
Finding a specific permutation of jobs for a single machine scheduling problem with
deadlines. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
Thanh Thuy Tien Ta and Jean-Charles Billaut
Minimizing makespan on parallel batch processing machines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
Karim Tamssaouet, Stéphane Dauzère-Pérès and Claude Yugma
Order Acceptance and Scheduling Problem with Batch Delivery . . . . . . . . . . . . . . . . . . . . . . . . . . 233
İstenç Tarhan and Ceyda Oğuz
Energy Conscious Scheduling of Robot Moves in Dual-Gripper Robotic Cells . . . . . . . . . . . . . 237
Nurdan Tatar, Hakan Gültekin and Sinan Gürel
A continuous-time assignment-based MILP formulation for the resource-constrained
project scheduling problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242
Norbert Trautmann, Tom Rihm, Nadine Saner and Adrian Zimmermann
A heuristic procedure to solve the integration of personnel staffing in the project
scheduling problem with discrete time/resource trade-offs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246
Mick Van Den Eeckhout, Mario Vanhoucke and Broos Maenhout
Production and distribution planning for smoothing supply-chain variability . . . . . . . . . . . . . . 250
Marie-Sklaerder Vié, Nicolas Zufferey and Leandro Coelho
Modeling Non-preemptive Parallel Scheduling Problem with Precedence Constraints . . . . . . 255
Tianyu Wang and Odile Bellenguez-Morineau
A Branch-and-Bound Procedure for the Resource-Constrained Project Scheduling
Problem with Partially Renewable Resources and Time Windows. . . . . . . . . . . . . . . . . . . . . . . . . 259
Kai Watermeyer and Jürgen Zimmermann
x

Fixed interval multiagent scheduling problem with rejected costs . . . . . . . . . . . . . . . . . . . . . . . . . 263
Boukhalfa Zahout, Ameur Soukhal and Patrick Martineau
Integrating case-based analysis and fuzzy programming for decision support in project
risk response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267
Yao Zhang, Fei Zuo and Xin Guan
Multi-Level Tabu Search for Job Scheduling in a Variable-Resource Environment . . . . . . . . . 272
Nicolas Zuﬀerey
Scheduling a forge with due dates and die deterioration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .276
Olivér Ősz, Balázs Ferenczi and Máté Hegyháti
xi

PMS 2018 Author Index
Author Index
Absi, Nabil 1
Agnetis, Alessandro 5, 9
Al Hasan, Hasan 13
Allahverdi, Ali 26, 32
Andrés, Carlos 17
Anselmet, Marie-Christine 181
Arkhipov, Dmitry 22
Artigues, Christian 1, 165, 181
Aydilek, Asiye 26, 32
Aydilek, Harun 26, 32
Balouka, Noemie 37
Battaı̈a, Olga 22
Bellenguez-Morineau, Odile 255
Berghman, Lotte 5
Berlinska, Joanna 42
Bianco, Lucio 46
Billaut, Jean-Charles 225
Boctor, Fayez 50
Boysen, Nils 55
Briand, Cyril 5, 58
Bridelance, Jerome 62
Briskorn, Dirk 66
Bultmann, Matthias 70
Burgelman, Jeroen 74
Caﬁeri, Sonia 119
Caramia, Massimiliano 46
Carlier, Jacques 78
Chen, Bo 9
Codou Ndiaye, Jeanine 58
Coelho, Leandro 250
Cohen, Izack 37
Coindreau, Marc-Antoine 79
Dauzère-Pérès, Stéphane 229
Della Croce, Federico 221
Demeulemeester, Erik 83, 152
Di Gioacchino, Roberto 160
Dolceamore, Simone 88
Eberle, Franziska 92
Ekici, Ali 96
xiii

Fanjul-Peyró, Luis 195
Ferenczi, Balázs 276
Fischer, Felix 92
Fliedner, Malte 66
Gallay, Olivier 79
Gerhards, Patrick 100
Giordani, Stefano 46
Gonzalez-Neira, Eliana Marı́a 104
Guan, Xin 267
Guida, Pier Luigi 110
Guéret, Christelle 13
Gyorgyi, Peter 115
Gültekin, Hakan 237
Gürel, Sinan 237
Hamaz, Idir 119
Haı̈t, Alain 165
Hegyháti, Máté 123, 276
Hoﬀmann, Luise-Sophie 128
Hong, Juntaek 136
Houssin, Laurent 119
Kedad-Sidhoum, Saﬁa 1
Kellenbrink, Carolin 128, 205
Kis, Tamas 115
Kloster, Oddvar 185
Knust, Sigrid 70
Lange, Julia 132
Lazarev, Alexander 22
Lee, Kangbok 136
Lemoine, David 13
Lopez, Pierre 181
Lucidi, Stefano 141
Maccarrone, Ludovica 141
Maecker, Söhnke 145
Maenhout, Broos 246
Maheut, Julien 17
Mannino, Carlo 149, 185
Manzini, Massimo 152
Martens, Annelies 156
Martineau, Patrick 263
Matuschke, Jannik 92
Megow, Nicole 92
Minelle, Federico 160
Montoya-Torres, Jairo R. 104
xiv

Morin, Pierre-Antoine 165
Ngueveu, Sandra Ulrich 1
Nicosia, Gaia 9
Odedairo, Babatunde Omoniyi 169
Oladokun, Victor Oluwasina 169
Ozener, Okan 96
Oğuz, Ceyda 233
Paciﬁci, Andrea 9
Parlouar, Rémi 58
Perea, Federico 195
Pesch, Erwin 175
Pirozzi, Massimo 176
Polo Mejia, Oliver 181
Rannou, Janik 1
Rihm, Tom 242
Riise, Atle 185
Rivreau, David 13
Rozycki, Rafal 189
Ruiz, Rubén 194, 195
Różycki, Rafal 199
Saadi, Omar 1
Sacco, Giovanni 110
Saner, Nadine 242
Santini, Stefano 160
Sarikaya, Merve Burcu 96
Sartor, Giorgio 149
Schittekat, Patrick 149, 185
Schnabel, Andre 205
Sedding, Helmut A. 209
Servranckx, Tom 213
Shang, Lei 221
Shen, Liji 145
Soukhal, Ameur 263
Stephan, Konrad 55
Stolﬁ, Franco 160
Stürck, Christian 217
T’Kindt, Vincent 221
Ta, Thanh Thuy Tien 225
Tamssaouet, Karim 229
Tarhan, İstenç 233
Tatar, Nurdan 237
Trautmann, Norbert 242
xv

Tschöke, Martin 66
Urgo, Marcello 88, 152
Van Den Eeckhout, Mick 246
Vanhoucke, Mario 62, 74, 156, 213, 246
Vié, Marie-Sklaerder 250
Waldherr, Stefan 70
Waligóra, Grzegorz 189, 199
Wang, Tianyu 255
Watermeyer, Kai 259
Weidinger, Felix 55
Weglarz, Jan 199
Yugma, Claude 229
Zahout, Boukhalfa 263
Zhang, Yao 267
Zimmermann, Adrian 242
Zimmermann, Jürgen 259
Zuﬀerey, Nicolas 79, 250, 272
Zuo, Fei 267
Ősz, Olivér 276
xvi

PMS 2018 Keyword Index
Keyword Index
Activities scheduling 169
Air traffic control 149
Aircraft assembly 88
Alternative project structure 213
Analytical tolerance limits 156
Approximation algorithm 92
Assembly flowshop 17, 26
Assembly line 152
Assignment and scheduling 58
Bargaining problems 9
Batch delivery 233
Batching 229
Benchmark instances 100
Benders’ reformulation 149
Bicriteria optimization 237
Blocking 17, 132
Branch and bound 259
Budgeted uncertainty set 119
Buffer monitoring 156
Buffer size 17
Carpooling 79
Chance constrained optimization 46
Characterization 225
Claim management 110
Column generation 165, 233
Computational complexity 66, 209
Conditional Value at Risk 88
Conflict resolution 185
Constraint programming 22
Context based risk analysis 160
Continuous resource 189
Continuous time mixed integer linear
programming 242
Crossdocking 5
Cyclic scheduling 119
Data gathering network 42
Deadline 225
Decision support system 267
Decomposition 246
Delivery times 145
Deterioration 276
xvii

Discrete continuous scheduling 189
Discrete time/resource trade-oﬀ 246
Distribution planning 250
Doubly constrained resource 189
Dual gripper 237
Due dates 276
Dynamic programming 1, 136, 233
E-government 160
Earliness 104
Eligibility constraints 136
Energy 199
Energy optimization 237
Equal processing time jobs 136
eS-graph 123
Exact methods 195
Exponential time algorithm 221
Fairness 9
Flexible project 128
Flow shop 42, 70, 152
Forge 276
Fuzzy mathematical programming 267
Genetic algorithm 128, 205
Grey box optimization 141
Handover relation 66
Heuristics 50, 145, 205, 263
Hotspot problem 149
Idleness cost 169
Integer programming 5, 50, 149, 255
Iterated local search 246
Job shop scheduling 132, 185, 272
Just in time 221
KPI 176
Limited memory 42
Linear algebra 74
Linear programming 185
Local search 205, 229
Lot sizing 1
Lot sizing and scheduling 50
Makespan 32
xviii

Maximum tardiness 26
Memetic algorithm 145
Metaheuristics 145, 233
Mixed integer linear programming 1, 13, 58, 141, 145, 165, 242, 263, 276
MMLIB 217
Mode reduction 217
Modeling technique 123
Multi skill 181
Multi-mode resource availability cost
problem 100
Multi-mode resource constrained project
scheduling problem 217
Multi-mode resource investment problem 100
Multi-project experimental outcome 160
Multi-project scheduling 128
Multi-robot pick-and-place 58
Multiagent scheduling 9, 263
New objective function 225
No wait 32
Non-renewable resources 115
Nuclear laboratory 181
Operating rooms in health services 13
Optimal policies 83
Optimization 181
Order acceptance 233
Order consolidation 55
Overcapacity 205
Parallel machine scheduling 5, 96, 145, 189, 195, 229, 255
Pareto optimization 263
Partially renewable resources 259
Periodical aggregation 165
PERT 46, 74
Piecewise linear convex processing time 209
Planning 165
Pliability 70
Polynomial algorithms 22
Portfolio management 62
Power 199
Precedence constraints 255
Preemptive scheduling 181
Preprocessing 217
Proactive reactive scheduling 152
Production planning 250
Project 165
Project control 156
xix

Project management 110, 169
Project network 123
Project planning 22
Project risk management 160, 267
Project scheduling 37, 46, 74, 100, 141, 213, 259
Project staﬃng 246
Propagator 22
Qualitative criteria 104
RCMPSP-PS 128
Release dates 136
Resource availability 62
Resource constrained project scheduling
problem 83, 169, 181, 205, 242
Resource transfers 62
Risk response action 267
Robotic cell scheduling 237
Robust optimization 37, 119
Robustness 104
Schedule delay analysis 110
Scheduling under energy constraints
and costs 1
Semiconductor manufacturing 229
Sequence dependent setup times 96, 233
Server problem 199
Setup times 26, 195
Simulated annealing 26, 132
Single machine scheduling 55, 115, 221, 225, 263
Single processor 199
Stakeholder 176
Sterilization unit 13
Stochastic activity durations 83
Stochastic permutation ﬂow shop 104
Stochastic scheduling 88, 92
Success 176
Supply chain management 250
Synchronization 79
Synchronous movement 70
Tabu search 213, 272
Tardiness 104
Time dependent scheduling 209
Total completion time 92, 136
Total tardiness 32, 132
Total weighted completion time 115
Transshipment terminals 66
xx

Truck scheduling 5
Uncertainty 37
Unrelated machine scheduling 96
Variable resources 272
Vehicle routing and scheduling 79
Vehicle sequencing 66
Warehousing 55
Weighted tardiness 145
Workforce management 141
xxi

Scheduling energy-consuming jobs on parallel machines
with piecewise-linear costs and storage resources
A lot-sizing and scheduling perspective
Nabil Absi1
, Christian Artigues2
, Safia Kedad-Sidhoum3
, Sandra U. Ngueveu2
, Janik
Rannou2
, et Omar Saadi3
1
Mines Saint-Etienne and UMR CNRS 6158 LIMOS, Gardanne, France
absi@emse.fr
2
LAAS-CNRS, Université de Toulouse, CNRS, INP, Toulouse, France
sungueve,artigues@laas.fr
3
Sorbonne Université, UPMC, UMR 7606, LIP6, Paris, F-75005, France
safia.kedad-sidhoum@lip6.fr
Keywords: Scheduling under energy constraints and costs, Mixed-integer programming,
Lot-sizing, Dynamic programming
1 Problem formulation and complexity
In this paper, a scheduling and energy source assignment problem is studied. The
problem is abstracted from several applications including data centers (Guérout et al. 2017),
smart buildings (Desdouits et al. 2016), hybrid vehicles (Caux et al. 2017, Ngueveu et
al. 2017) and manufacturing (Haouassi et al. 2016). A set of jobs J has to be scheduled
on a set of energy consuming machines M. The energy consumed by a machine k has a
fixed part Dk depending on whether the machine is switched on or off and a variable part
depending on the tasks that are currently in process. Each task j requires an amount Djk
of energy at each time period it is processed on machine k. We are interested in optimizing
the total energy cost induced by energy production required to satisfy the total energy
demand of a given schedule over a fixed discrete horizon T = {1, . . . , |T|}. At each time
period, the energy required by the schedule can be supplied by two energy sources. One
source is reversible, such as batteries and super-capacitors. Such a source is able not only
to produce energy but also to retrieve it assuming a limited capacity Q. Such a resource is
equivalent to a continuously-divisible storage resource in the scheduling terminology. The
second source is a non-reversible source, only able to produce energy, such as the external
power grid (assuming here that energy cannot be sold to the network). This source is
assumed of infinite capacity, but its usage comes with a cost (see below). In this paper, we
consider a parallel machine environment, such that pj units of each task j must be scheduled
preemptively inside a time window [rj, dj] and have to be assigned at each time period to
one and only one machine. Job units also require resources from a set R (e.g. CPU, RAM)
on their assigned machines. A job requires a non-negative amount cjr on each resource
r ∈ R. On each machine k, resource r is available in a limited amount Ckr. The energy
cost for a period of the scheduling horizon is a piecewise linear (PWL) function ft, t ∈ T, of
the required amount of energy on the non-reversible source. The PWL function is assumed
to be time dependent. This allows to model time dependent electricity prices as well as
previsions of photovoltaic production since the cost can be zero up to a required amount
of energy corresponding to the expected photovoltaic production on the considered time
period. We introduce variables yjkt ∈ {0, 1} indicating whether one unit of job j is assigned
to machine k at time t and zkt ∈ {0, 1} indicating whether machine k is switched on at time
t. Continuous variables are used for energy amounts: xt ≥ 0 gives the amount of energy
used on the non-reversible source at time t, st ≥ 0 is the level of energy remaining in the
1

non-reversible source at time t (s0 is a constant indicating the initial energy level). Based
on these variables, we define a MILP formulation of the problem aiming at minimizing the
total energy cost.
minimize
P
t∈T ft(xt) (1)
subject to P
k∈M yjkt ≤ 1, j ∈ J , t ∈ T (2)
P
k∈M
P
t∈T yjkt = pj, j ∈ J (3)
P
j∈J cjryjkt ≤ Ckr, k ∈ M, r ∈ R, t ∈ T (4)
zkt − yjkt ≥ 0, k ∈ M, j ∈ J , t ∈ T (5)
xt + st − st−1 −
P
j∈J
P
k∈M Djkyjkt −
P
k∈M Dkzkt = 0, t ∈ T (6)
s|T | − s0 ≥ 0, (7)
st − Q ≤ 0, t ∈ T (8)
st ≥ 0 t ∈ T (9)
xt ≥ 0 t ∈ T (10)
yjkt ∈ {0, 1} j ∈ J , k ∈ M, t ∈ T (11)
zkt ≥ 0 k ∈ M, t ∈ T (12)
The total energy cost minimization objective (1) is considered. Constraints (2) state that
a job may be in process on only one machine at a given time. Constraints (3) enforce each
unit of a job to be scheduled on one machine. Constraints (4) are the resource constraints.
Constraints (5) enforce a machine to be switched on at each time it processes at least one
job. Constraints (6) are the energy balance constraints between the schedule demand, the
energy provided by the non-reversible source (xt) and the energy taken from or provided to
the reversible source (st − st−1, that can be positive or negative). Constraint (7) enforces
the final energy level in the reversible source to be at least the initial one. Constraints
(8) are the reversible source capacity constraints (storage limit). Scheduling preemptive
jobs with PWL energy costs is NP-hard, even with an unlimited number of machines and
single non-reversible source (Ngueveu et al. 2016). Therefore the proposed MILP becomes
intractable as the problem size increases.
2 A lot-sizing and scheduling matheuristic
We propose a natural decomposition of the problem. Let dt =
P
j∈J
P
k∈M Djkyjkt
+
P
k∈M Dkzkt denote the total energy demand of a fixed schedule at time t. Then con-
straints (6) can be rewritten
xt + st − st−1 − dt = 0, t ∈ T (13)
Now observe that for fixed dt, problem LSP: min
P
t∈T ft(xt) s.t. (7–10), (13) is a single-
item (continuous) lot sizing problem with PWL production costs where dt is the demand
for period t, xt is the production variable for period t, and st is the variable giving the
amount of inventory at the end of period t. In Absi et al. (2017), the problem is shown to
be NP-hard but for integer inventory levels, a pseudo-polynomial dynamic programming
(DP) algorithm of complexity O(T2
qd) where d is the average demand and q is the average
number of breakpoints of the PWL functions ft is given, generalizing the results of Shaw
and Wagelmans (1998). On the other hand, if the variables st are fixed, by performing
change of variables xt ← xt − st + st−1 for all t, we obtain
xt =
X
j∈J
X
k∈M
Djkyjkt +
X
k∈M
Dkzkt, t ∈ T (14)
2

and problem MSP: min
P
t∈T f′
t(xt) =
P
t∈T ft(xt − st + st−1) s.t. (2–5), (10–12), (14),
which is a parallel machine scheduling problem with PWL costs and a single non-reversible
source, NP-hard in the strong sense as shown in Ngueveu et al. (2016).
A matheuristic is obtained by solving alternatively MSP and LSP. Starting with initial
reversible source transferred amounts st − st−1 = 0 for all t, MSP is solved and the output
energy demand (dt)t∈T is used as input of LSP. The output inventory levels are used to
update the PWL functions f′
t. Then, MSP is solved again, and so on until no improvement
is observed in the objective function.
We first compare the MILP (solved with Cplex) and the pseudo polynomial DP algo-
rithm with fixed demands (only LSP), see table 1. The merits and the drawback of the
two approaches are illustrated on 4 instances with T = 1000, q = 10 breakpoints, and
varying average maximal capacities Q and demands d. Under a 300s time limit for the
MILP, the CPU times (in seconds) and obtained costs are compared. It appears that no
algorithm dominates the other one in terms of CPU time, while the DP is more impacted
by the maximal available capacity for the reversible source. However the MILP shows a
more erratic and unpredictable behavior. Note that due to the integrity requirement of the
inventory levels, the DP costs are higher than the MILP costs.
Table 1. Comparison of MILP and DP on the lot sizing problem (LSP)
T q Q d MILP cost MILP CPU DP cost DP CPU
1000 10 1000 100 6661 300 6916 14.16
1000 10 10000 100 4396 2.90 4508 279
1000 10 100 2000 975523 0.57 975617 1.31
1000 10 1000 2000 940165 300 937886 15
Finally we compare the MSP/LSP decomposition matheuristic (MH) with the full MILP
to solve the global problem. We also illustrate the cost and CPU time differences on 4
instances with varying horizon, number of machines, number of jobs (see table 2). On 3
instances (marked with a ∗) the full MILP reached the time limit. The matheuristic is
only slightly faster than the MILP except on the last instance, where it is much faster.
The costs can be close to the MILP ones although important gaps can also be observed.
In parenthesis, the maximal CPU time per iteration and the iteration number assigned to
MH is indicated. A closer analysis of the CPU times between MSP and LSP reveals that
90% of the CPU time is spent on solving the scheduling problem MSP.
Table 2. Comparison of full MILP and matheuristic on the global problem
T |M| |J | MILP cost MILP CPU MH cost MH CPU
30 2 50 11280 450∗
11280 154 (150× 3)
60 4 150 25966 2000∗
26183 1893 (1000× 2)
120 2 150 5944 2000∗
6972 1855 (1000× 2)
120 1 150 11255 1353 11265 228 (200× 10)
3

3 Conclusion
We have proposed an original lot sizing and scheduling decomposition approach to
solve an energy management and scheduling problem on parallel machines. The lot sizing
subproblem can be solved considerably faster than the scheduling subproblem and conse-
quently, further research on the problem should focus on improvement of the scheduling
solution procedure. To improve the decomposition heuristic, optimality cuts issued from
lot sizing could be designed for the scheduling problem. Another interesting issue is to
consider a non ideal yield of the reversible source. In practice, due to energy conversion
and losses only a fraction of st − st−1 is available to fulfill the demand and a possibly non
linear efficiency function g(st − st − 1) has to be used to compute the obtained energy. It
remains to know whether efficient lot sizing procedure can be devised with such efficiency
functions.
Acknowledgements
This research benefited from the support of the FMJH Program PGMO and from the
support of EDF, Thales, Orange. It also benefited from funding from the Cellule Energie
of the CNRS.
References
Absi N., C. Artigues, S. Kedad-Sidhoum, S.U. Ngueveu and O. Saadi, 2017, “Lot-sizing models
for energy management”. LAAS report.
Caux S., Y. Gaoua, and P. Lopez, 2017, “A combinatorial optimisation approach to energy man-
agement strategy for a hybrid fuel cell vehicle”. Energy, Vol. 133, pp.219-230.
Desdouits C., M. Alamir, R. Giroudeau and C. Le Pape, 2016, “The Sourcing Problem - Energy
Optimization of a Multisource Elevator”, ICINCO, Vol. 1, pp. 19-30.
Guérout T., Y. Gaoua, C. Artigues, G. Da Costa, P. Lopez and T. Monteil, 2017, “Mixed in-
teger linear programming for quality of service optimization in Clouds” Future Generation
Computer Systems 71: 1–17.
Haouassi M., C. Desdouits, R. Giroudeau and C. Le Pape, 2016, “Production scheduling with a
piecewise-linear energy cost function” IEEE Symposium Series on Computational Intelligence
(SSCI), pp. 1-8.
Ngueveu S. U., C. Artigues and P. Lopez, 2016, “Scheduling under a non-reversible energy source:
An application of piecewise linear bounding of non-linear demand/cost functions”, Discrete
Applied Mathematics, Vol. 208, pp. 98-113.
Ngueveu S.U., S. Caux, F. Messine and M. Guemri, 2017, “Heuristics and lower bound for energy
management in hybrid-electric vehicles”, 4OR, to appear.
Shaw D. X., A. P. Wagelmans, 1998, “An algorithm for single-item capacitated economic lot sizing
with piecewise linear production costs and general holding costs”, Management Science, Vol.
44, pp. 831-838.
4

The truck scheduling problem at cross docking
terminals: Formulations and valid inequalities
A. Agnetis1
, L. Berghman2
and C. Briand3
1
A. Agnetis, Università degli Studi di Siena, DIISM, Siena, Italy
agnetis@diism.unisi.it
2
L. Berghman, Université de Toulouse - Toulouse Business School,
20 BD Lascrosses – BP 7010, 31068 Toulouse Cedex 7, France
l.berghman@tbs-education.fr
3
C. Briand, LAAS-CNRS, Université de Toulouse, UPS, Toulouse, France
briand@laas.fr
Keywords: crossdocking, truck scheduling, parallel machine scheduling, integer linear pro-
gramming.
1 Introduction
Crossdocking is a warehouse management concept in which items delivered to a ware-
house by inbound trucks are immediately sorted out, reorganized based on customer de-
mands and loaded into outbound trucks for delivery to customers, without requiring ex-
cessive inventory at the warehouse (J. van Belle et al. 2012). If any item is held in storage,
it is usually for a brief period of time that is generally less than 24 hours. Advantages
of crossdocking can accrue from faster deliveries, lower inventory costs, and a reduction
of the warehouse space requirement (U.M. Apte and S. Viswanathan 2000, N. Boysen et
al. 2010). Compared to traditional warehousing, the storage as well as the length of the
stay of a product in the warehouse is limited, which requires an appropriate coordination
of inbound and outbound trucks (N. Boysen 2010, W. Yu and P.J. Egbelu 2008).
The truck scheduling problem, which decides on the succession of truck processing at the
dock doors, is especially important to ensure a rapid turnover and on-time deliveries. The
problem studied concerns the operational level: trucks are allocated to the different docks
so as to minimize the storage usage during the product transfer. The internal organization
of the warehouse (scanning, sorting, transporting) is not explicitly taken into consideration.
We also do not model the resources that may be needed to load or unload the trucks, which
implies the assumption that these resources are available in sufficient quantities to ensure
the correct execution of an arbitrary docking schedule.
In this abstract, we present a time-indexed formulation, a network formulation and some
valid inequalities. Experimental results will be presented during the talk at the conference.
2 Detailed problem statement
We examine a crossdocking warehouse where incoming trucks i ∈ I need to be unloaded
and outgoing trucks o ∈ O need to be loaded (where I is the set containing all inbound
trucks while O is the set containing all outbound trucks). The warehouse features n docks
that can be used both for loading and unloading. The processing time of truck j ∈ I ∪ O
equals pj. This processing time includes the loading or unloading but also the transporta-
tion of goods inside the crossdock and other handling operations between dock doors. It
is assumed that there is sufficient workforce to load/unload all docked trucks at the same
time. Hence, a truck assigned to a dock does not wait for the availability of a material
handler.
5

The products on the trucks are packed on unit-size pallets, which move collectively as
a unit: re-packing inside the terminal is to be avoided. Each pallet on an inbound truck
i needs to be loaded on an outbound truck o, which gives rise to a start-start precedence
constraint (i, o) ∈ P ⊂ I × O, with P the set containing all couples of inbound trucks i
and outbound trucks o that share a precedence constraint. Each truck j has a release time
rj (planned arrival time) and a deadline ˜
dj (its latest departure time).
Products can be transshipped directly from an inbound to an outbound truck if the
outbound truck is placed at a dock. Otherwise, the products are temporarily stored and
will be loaded later on. Each couple (i, o) ∈ P has a weight wio, representing the number of
pallets that go from inbound truck i to outbound truck o. The problem aims at determining
time-consistent start times si and so of unload and load tasks i and o so as to minimize
the weighted sum of sojourn times of the pallets stocked in the warehouse. Remark that
the time spent by a pallet in the storage area is equal to the flow time of the pallet: the
difference between the start of loading the outbound trailer and the start of unloading the
inbound trailer (i.e., so − si).
Our problem can be modeled as a parallel machine scheduling problem with release
dates, deadlines, and precedence constraints, denoted by Pm|ri, ˜
di, prec|−. As this problem
is a generalization of the 1|rj, ˜
dj|− problem which is NP-complete (J.K. Lenstra et al. 1977),
even finding a feasible solution for the problem is NP-complete.
3 Time-indexed formulation
A time-indexed formulation discretizes the continuous time space into periods τ ∈
T of a fixed length. Let period τ be the interval [t − 1, t[. It is well known that time-
indexed formulations perform well for scheduling problems because the linear programming
relaxations provide strong lower bounds (M. E. Dyer and L. A. Wolsey 1990).
For all inbound trucks i ∈ I and for all time periods τ ∈ Ti, we have
xiτ =





1 if the unloading of inbound truck i is
started during time period τ,
0 otherwise,
(1)
with Ti = {ri + 1, ri + 2, . . . , ˜
di − pi + 1}, the relevant time window for inbound truck i.
Additionally, for all outbound trucks o ∈ O and for all time periods τ ∈ To, we have
yoτ =





1 if the loading of outbound truck o is
started during time period τ,
0 otherwise,
(2)
with To = {ro + 1, ro + 2, . . . , ˜
do − po + 1}, the relevant time window for outbound truck o.
A time-indexed formulation for the considered truck scheduling problem is the following:
min z =
X
(i,o)∈P
X
τ∈T
wioτ (yoτ − xiτ ) (3)
6

subject to
X
τ∈Ti
xiτ = 1 ∀i ∈ I (4)
X
τ∈To
yoτ = 1 ∀o ∈ O (5)
X
τ∈T
τ (xiτ − yoτ ) ≤ 0 ∀(i, o) ∈ P (6)
X
i∈I
τ
X
u=τ−pi+1
xiu +
X
o∈O
τ
X
u=τ−po+1
you ≤ n ∀τ ∈ T (7)
xiτ ∈ {0, 1} ∀i ∈ I; ∀τ ∈ Ti (8)
yoτ ∈ {0, 1} ∀o ∈ O; ∀τ ∈ To (9)
The objective function (3) minimizes the total weighted usage of the storage area. Con-
straints (4) and (5) demand each truck to be assigned to exactly one gate. Constraints (6)
ensure that if there exists a precedence constraint between inbound truck i and outbound
truck o, then o cannot be processed before i. Constraints (7) enforce the capacity of the
docks for any period τ ∈ T .
4 Network formulation
The formulation below makes use of the well-known concept of a critical set (see e.g. (M.
Lombardi and M. Milano 2012)), i.e., a set of tasks which cannot all be performed in
parallel. We introduce a pair of disjunctive precedence constraints for each task pair (u, v) ∈
(I ∪ O)2
with [ru, du] ∩ [rv, dv] 6= ∅, belonging to a critical set (the set of these task pairs is
further referred as C). We let E be the set of all critical sets. Additionally, we also refer to ek
as a specific critical set of k elements and to Em
⊂ E as the set of all minimal critical sets.
To model the disjunction, binary variables αuv are introduced such that αuv = 1 ≡ u ≺ v.
Our problem can be modelled as follows:
min
X
o∈O
sopo −
X
i∈I
sipi (10)
subject to
so − si ≥ 0 ∀(i, o) ∈ P (11)
sv − su + αuv(Muv − pu) ≥ Muv ∀(u, v) ∈ C (12)
sv − s0 ≥ rv ∀v ∈ I ∪ O (13)
s0 − su ≥ pu − du ∀u ∈ I ∪ O (14)
X
(u,v)∈en+1
αuv ≥ 1 ∀en+1 ∈ Em
(15)
αuv ∈ {0, 1} ∀(u, v) ∈ C (16)
su ∈ R ∀u ∈ I ∪ O (17)
with Muv = pu − du + rv and 0 a dummy vertex, which is introduced to represent the
time origin s0 = 0. Note that obviously αuv + αvu ≤ 1, even though this constraint is not
mandatory for the formulation accuracy.
Remark that constraints (15) express the limited capacity of the crossdocking terminal.
Their number is exponential, as the number of minimal critical sets is exponential. Even
7

though including only minimal critical sets is sufficient, we can also consider the non-
minimal critical sets, generalizing (15) as:
X
(u,v)∈ek
αuv ≥ k − n ∀ek ∈ E (18)
with n + 1 ≤ k ≤ I ∪ O .
We will show that this family of constraints can be strengthened by augmenting the
right-and-side, so that it can be replaced by:
X
(u,v)∈ek
αuv ≥
(k − n)(k − n + 1)
2
∀ek ∈ E (19)
5 Solving methodology framework
Intuitively, only a small number of constraints (19) may be required into the formulation
to obtain a feasible (optimal) solution. Consequently, we consider the following cutting-
plane method which consists in introducing progressively constraints of type (19). First, the
problem is solved without any constraint of type (19) using a MILP solver. Then, violated
constraints of type (19) are added for some k > n involving a critical set ek and the solver
is launched again. Now, each time a feasible solution is found by the solver in course of
the branch-and-cut process, violated constraints of type (19) are added on-the-fly. Note
that if such a solution is feasible with respect to the resource capacity, then it is an upper
bound of the initial problem. When the MILP solver ends up with an optimal solution
also capacity-feasible, it is also optimal. Otherwise, violated constraints of type (19) can
be added again and another MILP is ran. Within various computational time limitation
assumptions, the above methodology will be compared in terms of performance (quality of
the upper and lower bounds) with the time-indexed linear programming approach on a set
of artificial problem instances.
References
U.M. Apte and S. Viswanathan, Effective cross docking for improving distribution efficiencies,
International Journal of Logistics: Research and Applications, 3 (3), 291–302.
N. Boysen, Truck scheduling at zero-inventory cross docking terminals, Computers & Operations
Research, 37, 32–41.
N. Boysen and M. Fliedner and A. Scholl, Scheduling inbound and outbound trucks at cross
docking terminals, OR Spectrum, 32, 135–161.
M. E. Dyer and L. A. Wolsey, Formulating the single machine sequencing problem with release
dates as a mixed integer problem, Discrete Applied Mathematics, 26, 255–270.
J.K. Lenstra and A.H.G. Rinnooy Kan and P. Brucker, Complexity of machine scheduling prob-
lems, Annals of Discrete Mathematics, 1, 343–362.
M. Lombardi and M. Milano, A min-flow algorithm for Minimal Critical Set detection in Resource
Constrained Project Scheduling, Artificial Intelligence, 182-183, 58–67.
J. van Belle and P. Valckenaers and D. Cattrysse, Cross docking: State of the art, Omega,40 (6),
827–846.
W. Yu and P.J. Egbelu, Scheduling of inbound and outbound trucks in cross docking systems with
temporary storage, International Journal of Production Economics„ 184, 377–396.
8

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❆❧❡ss❛♥❞r♦ ❆❣♥❡t✐s1
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✱ ❡t ❆♥❞r❡❛ P❛❝✐✜❝✐4
1
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❛❣♥❡t✐s❅❞✐✐s♠✳✉♥✐s✐✳✐t
2
❯♥✐✈❡rs✐t② ♦❢ ❲❛r✇✐❝❦✱ ❯❑
❇♦✳❈❤❡♥❅✇❜s✳❛❝✳✉❦
3
❯♥✐✈❡rs✐t② ♦❢ ❘♦♠❛ ❚r❡✱ ■t❛❧②
♥✐❝♦s✐❛❅❞✐❛✳✉♥✐r♦♠❛✸✳✐t
4
❯♥✐✈❡rs✐t② ♦❢ ❘♦♠❛ ❚♦r ❱❡r❣❛t❛✱ ■t❛❧②
♣❛❝✐❢✐❝✐❅❞✐s♣✳✉♥✐r♦♠❛✷✳✐t
❑❡②✇♦r❞s✿ ♠✉❧t✐❛❣❡♥t s❝❤❡❞✉❧✐♥❣✱ ❜❛r❣❛✐♥✐♥❣ ♣r♦❜❧❡♠s✱ ❢❛✐r♥❡ss✳
✶ ■♥tr♦❞✉❝t✐♦♥
❋❛✐r♥❡ss ✐ss✉❡s ❛r✐s❡ ✐♥ s❡✈❡r❛❧ r❡❛❧✲✇♦r❧❞ ❝♦♥t❡①ts ❛♥❞ ❛r❡ ✐♥✈❡st✐❣❛t❡❞ ✐♥ ❞✐✛❡r❡♥t
r❡s❡❛r❝❤ ❛r❡❛s ♦❢ ♠❛t❤❡♠❛t✐❝s✱ ❣❛♠❡ t❤❡♦r② ❛♥❞ ♦♣❡r❛t✐♦♥s r❡s❡❛r❝❤✳ ■♥ ❝❧❛ss✐❝❛❧ t✇♦✲♣❧❛②❡r
❜❛r❣❛✐♥✐♥❣ ♣r♦❜❧❡♠s✱ t❤❡ ♥♦t✐♦♥ ♦❢ ❢❛✐r♥❡ss ❤❛s ❜❡❡♥ ✐♥tr♦❞✉❝❡❞ t♦ ❝♦♠♣❛r❡ t❤❡ ✉t✐❧✐t② ♦❢
♦♥❡ ❛❣❡♥t t♦ t❤❡ ♦t❤❡r ❛❣❡♥t✬s✳
❍❡r❡ ✇❡ ❛❞❞r❡ss ❢❛✐r♥❡ss ❝♦♥❝❡♣ts ✐♥ t❤❡ ❝♦♥t❡①t ♦❢ ❝❧❛ss✐❝❛❧ s✐♥❣❧❡✲♠❛❝❤✐♥❡ s❝❤❡❞✉❧✐♥❣✳
❚❤❡r❡ ❛r❡ t✇♦ ❛❣❡♥ts✱ ❝❛❧❧❡❞ A ❛♥❞ B✱ ❡❛❝❤ ♦✇♥✐♥❣ ❛ s❡t ♦❢ ❥♦❜s✱ ✇❤✐❝❤ ♠✉st ❜❡ s❝❤❡❞✉❧❡❞
♦♥ ❛ ❝♦♠♠♦♥ ♣r♦❝❡ss✐♥❣ r❡s♦✉r❝❡✳ ❊❛❝❤ s❝❤❡❞✉❧❡ ✐♠♣❧✐❡s ❛ ❝❡rt❛✐♥ ✉t✐❧✐t② ❢♦r ❡❛❝❤ ❛❣❡♥t✳ ❲❡
❛❞♦♣t t❤❡ s✉♠ ♦❢ t❤❡ ❛❣❡♥ts✬ ✉t✐❧✐t✐❡s ❛s ❛♥ ✐♥❞❡① ♦❢ ❝♦❧❧❡❝t✐✈❡ s❛t✐s❢❛❝t✐♦♥ ✭s②st❡♠ ✉t✐❧✐t②✮
❛♥❞ ✇❡ r❡❢❡r t♦ ❛♥② s♦❧✉t✐♦♥ ♠❛①✐♠✐③✐♥❣ s②st❡♠ ✉t✐❧✐t② ❛s ❛ s②st❡♠ ♦♣t✐♠✉♠✳ ❊✈❡♥ ✐❢ ✐t
♠❛①✐♠✐③❡s s②st❡♠ ✉t✐❧✐t②✱ ❛ s②st❡♠ ♦♣t✐♠✉♠ ♠❛② ✇❡❧❧ ❜❡ ❤✐❣❤❧② ✉♥❜❛❧❛♥❝❡❞ ❛♥❞ t❤❡r❡❢♦r❡
♣♦ss✐❜❧② ✉♥❛❝❝❡♣t❛❜❧❡ ❜② t❤❡ ✇♦rs❡✲♦✛ ❛❣❡♥t✳ ❘❛t❤❡r✱ ❛ s♦❧✉t✐♦♥ t❤❛t ✐♥❝♦r♣♦r❛t❡s s♦♠❡
❝r✐t❡r✐♦♥ ♦❢ ❢❛✐r♥❡ss ♠❛② ❜❡ ♠♦r❡ ❛❝❝❡♣t❛❜❧❡✳ ❚❤❡ ♣r♦❜❧❡♠ ✇❡ ✐♥✈❡st✐❣❛t❡ ✐s ❤♦✇ ♠✉❝❤
s②st❡♠ ✉t✐❧✐t② ♠✉st ❜❡ s❛❝r✐✜❝❡❞ ✐♥ ♦r❞❡r t♦ r❡❛❝❤ ❛ ❢❛✐r s♦❧✉t✐♦♥✳ ❚❤❡ q✉❛♥t✐t② t❤❛t ❝❛♣t✉r❡s
t❤✐s ❝♦♥❝❡♣t ✐s ❦♥♦✇♥ ❛s ♣r✐❝❡ ♦❢ ❢❛✐r♥❡ss ✭PoF✮✳ ●✐✈❡♥ ❛♥ ✐♥st❛♥❝❡ ♦❢ ❛ ❜❛r❣❛✐♥✐♥❣ ♣r♦❜❧❡♠✱
❛♥❞ ❛ ❝❡rt❛✐♥ ❞❡✜♥✐t✐♦♥ ♦❢ ❢❛✐r s♦❧✉t✐♦♥✱ PoF ✐s t❤❡ r❡❧❛t✐✈❡ ❧♦ss ✐♥ ♦✈❡r❛❧❧ ✉t✐❧✐t② ♦❢ ❛ ❢❛✐r
s♦❧✉t✐♦♥ ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ s②st❡♠ ♦♣t✐♠✉♠✳ ❉❡♣❡♥❞✐♥❣ ♦♥ t❤❡ s♣❡❝✐✜❝ ♣r♦❜❧❡♠ s❡tt✐♥❣
❛♥❞ ❛❧s♦ ♦♥ t❤❡ ❛❣❡♥t ♣❡r❝❡♣t✐♦♥ ♦❢ ✇❤❛t ❛ ❢❛✐r s♦❧✉t✐♦♥ ✐s✱ ❛ss♦rt❡❞ ❞❡✜♥✐t✐♦♥s ♦❢ ❢❛✐r
s♦❧✉t✐♦♥ ❝❛♥ ❜❡ ❢♦✉♥❞ ✐♥ t❤❡ s❝✐❡♥t✐✜❝ ❧✐t❡r❛t✉r❡✳ ■♥ ♦✉r st✉❞② ✇❡ ❢♦❝✉s ♦♥ t✇♦ ♦❢ t❤❡ ♠♦st
♣♦♣✉❧❛r ❢❛✐r♥❡ss ❞❡✜♥✐t✐♦♥s✳
❈❛r❛❣✐❛♥♥✐s ❡t ❛❧✳ ✭❈❛r❛❣✐❛♥♥✐s ❡t ❛❧✳ ✷✵✶✷✮ ✐♥tr♦❞✉❝❡❞ t❤❡ ❝♦♥❝❡♣t ♦❢ PoF ✐♥ t❤❡ ❝♦♥t❡①t
♦❢ ❢❛✐r ❛❧❧♦❝❛t✐♦♥ ♣r♦❜❧❡♠s✿ ■♥ ♣❛rt✐❝✉❧❛r✱ t❤❡② ❝♦♠♣❛r❡ t❤❡ ✈❛❧✉❡ ♦❢ t♦t❛❧ ❛❣❡♥ts✬ ✉t✐❧✐t② ✐♥ ❛
s②st❡♠ ♦♣t✐♠✉♠ ✇✐t❤ t❤❡ ♠❛①✐♠✉♠ t♦t❛❧ ✉t✐❧✐t② ♦❜t❛✐♥❡❞ ♦✈❡r ❛❧❧ ❢❛✐r s♦❧✉t✐♦♥s ✭t❤❡② ♠❛❦❡
✉s❡ ♦❢ s❡✈❡r❛❧ ♥♦t✐♦♥s ♦❢ ❢❛✐r♥❡ss✱ ♥❛♠❡❧② ♣r♦♣♦rt✐♦♥❛❧✐t②✱ ❡♥✈②✲❢r❡❡♥❡ss ❛♥❞ ❡q✉✐t❛❜✐❧✐t②✮✳ ■♥
✭❇❡rts✐♠❛s ❡t ❛❧✳ ✷✵✶✶✮✱ ❇❡rts✐♠❛s ❡t ❛❧✳ ❢♦❝✉s ♦♥ ♣r♦♣♦rt✐♦♥❛❧ ❢❛✐r♥❡ss ❛♥❞ ♠❛①✲♠✐♥ ❢❛✐r♥❡ss
❛♥❞ ♣r♦✈✐❞❡ ❛ t✐❣❤t ❝❤❛r❛❝t❡r✐③❛t✐♦♥ ♦❢ PoF ❢♦r ❛ ❜r♦❛❞ ❢❛♠✐❧② ♦❢ ❛❧❧♦❝❛t✐♦♥ ♣r♦❜❧❡♠s ✇✐t❤
❝♦♠♣❛❝t ❛♥❞ ❝♦♥✈❡① ❛❣❡♥ts✬ ✉t✐❧✐t② s❡ts✳ ■♥ ✭◆✐❝♦s✐❛ ❡t ❛❧✳ ✷✵✶✼✮ t❤❡ ❛✉t❤♦rs ♣r♦✈❡ ❛ ♥✉♠❜❡r
♦❢ ♣r♦♣❡rt✐❡s ♦♥ t❤❡ ♣r✐❝❡ ♦❢ ❢❛✐r♥❡ss ✇❤✐❝❤ ❤♦❧❞ ❢♦r ❛♥② ❣❡♥❡r❛❧ ♠✉❧t✐✲❛❣❡♥t ♣r♦❜❧❡♠ ✇✐t❤♦✉t
❛♥② s♣❡❝✐❛❧ ❛ss✉♠♣t✐♦♥ ♦♥ t❤❡ ❛❣❡♥ts✬ ✉t✐❧✐t✐❡s✱ ❢♦❝✉s✐♥❣ ♦♥ ♠❛①✲♠✐♥✱ ❑❛❧❛✐✲❙♠♦r♦❞✐♥s❦✐
❛♥❞ ♣r♦♣♦rt✐♦♥❛❧ ❢❛✐r♥❡ss✳ ❙✐t✉❛t✐♦♥s ✐♥ ✇❤✐❝❤ t❤❡ ❛❣❡♥ts ♣✉rs✉❡ t❤❡ ♠✐♥✐♠✐③❛t✐♦♥ ♦❢ t❤❡✐r
❝♦sts ✭r❛t❤❡r t❤❛♥ t❤❡ ♠❛①✐♠✐③❛t✐♦♥ ♦❢ t❤❡✐r ✉t✐❧✐t②✮ ❤❛✈❡ ❜❡❡♥ ❞❡❛❧t ✇✐t❤ ❜② ❊rt♦❣r❛❧ ❛♥❞
❲✉ ✭❊rt♦❣r❛❧ ❛♥❞ ❲✉ ✷✵✵✵✮✱ ✇❤♦ ❞❡r✐✈❡ ❛ ♠❡❛s✉r❡ ♦❢ ❢❛✐r♥❡ss ❛♠♦♥❣ ❛ s❡t ♦❢ s✉♣♣❧② ❝❤❛✐♥
♠❡♠❜❡rs✳ ❆♥♦t❤❡r ❡①❛♠♣❧❡ ♦❢ ❢❛✐r♥❡ss ✐♥ t❤❡ ♣r❡s❡♥❝❡ ♦❢ ❝♦st ❛❧❧♦❝❛t✐♦♥s ❝❛♥ ❜❡ ❢♦✉♥❞ ✐♥
9

✭❇♦❤♠ ❛♥❞ ▲❛rs❡♥ ✶✾✾✹✮✳ ❖✉r ✈✐❡✇ ♦❢ ❢❛✐r♥❡ss ✐s r❡❧❛t❡❞ t♦ s❝❤❡❞✉❧✐♥❣ ♣r♦❜❧❡♠s✱ ❜✉t ✐t
✐s ✇♦rt❤ ♦❜s❡r✈✐♥❣ t❤❛t ❢❛✐r♥❡ss ✐ss✉❡s ❛r✐s❡ ✐♥ ♦t❤❡r ❝♦♥t❡①ts✱ s✉❝❤ ❛s ❢❛✐r r❡♣r❡s❡♥t❛t✐♦♥
♣r♦❜❧❡♠s ✭❇❛❧✐♥s❦✐ ❛♥❞ ❨♦✉♥❣ ✷✵✵✶✮✱ ♦r t❤❡ ❛♣♣♦rt✐♦♥♠❡♥t ♣r♦❜❧❡♠ ✭▲✉❝❛s ✶✾✽✸✮✳ ❚❤❡s❡
♥❡❡❞ t♦ ❜❡ ❞❡❛❧t ✇✐t❤ ❜② ❞✐✛❡r❡♥t ♠❡t❤♦❞s ❛♥❞ ❛❧❣♦r✐t❤♠s t❤❛♥ t❤♦s❡ ♣r❡s❡♥t❡❞ ✐♥ t❤✐s
t❛❧❦✳
✷ ❙❝❤❡❞✉❧✐♥❣ ❜❛r❣❛✐♥✐♥❣ ♣r♦❜❧❡♠s
❇❛r❣❛✐♥✐♥❣ ♣r♦❜❧❡♠s ❛❞❞r❡ss s✐t✉❛t✐♦♥s ✐♥ ✇❤✐❝❤ t✇♦ ♣❧❛②❡rs ✭❛❣❡♥ts✮ ❛r❡ ❢❛❝❡❞ ✇✐t❤ ❛
s❡t ♦❢ ♣♦ss✐❜❧❡ ❛❣r❡❡♠❡♥ts ✭r❡s♦✉r❝❡ s❡t✮✱ ❛♥❞ ♠✉st r❡❛❝❤ ❛ ❝♦♠♣r♦♠✐s❡ ♦✈❡r ♦♥❡ ♦❢ t❤❡♠✳
❍❡r❡ ✇❡ ❛r❡ ❝♦♥❝❡r♥❡❞ ✇✐t❤ t❤❡ ❢♦❧❧♦✇✐♥❣ s❝❤❡❞✉❧✐♥❣ ❜❛r❣❛✐♥✐♥❣ ♣r♦❜❧❡♠✳ ❚❤❡r❡ ❛r❡ t✇♦
❛❣❡♥ts✱ ♥❛♠❡❧② A ❛♥❞ B✳ ❊❛❝❤ ❛❣❡♥t ❤❛s ❛ s❡t ♦❢ ❥♦❜s✱ ✇❤✐❝❤ ❤❛✈❡ t♦ ❜❡ ♣r♦❝❡ss❡❞ ❜② ❛
s✐♥❣❧❡ ♠❛❝❤✐♥❡✳ ❆❣❡♥ts A ✭B✮ ❤❛s ❥♦❜s JA
1 , . . . , JA
nA
✭JB
1 , . . . , JB
nB
✮✱ ♦❢ ❧❡♥❣t❤ pA
1 , . . . , pA
nA
✭pB
1 , . . . , pB
nB
✮✳ ▲❡t PA
=
PnA
j=1 pA
j ✭PB
=
PnB
j=1 pB
j ✮✳ ❏♦❜s ❝❛♥♥♦t ❜❡ ♣r❡❡♠♣t❡❞✱ ❛♥❞ t❤❡
♠❛❝❤✐♥❡ ❝❛♥ ♦♥❧② ♣r♦❝❡ss ♦♥❡ ❥♦❜ ❛t ❛ t✐♠❡✳ ❲❡ ✉s❡ t❤❡ t❡r♠s A✲❥♦❜s ❛♥❞ B✲❥♦❜s t♦ r❡❢❡r
t♦ t❤❡ t✇♦ ❛❣❡♥ts✬ r❡s♣❡❝t✐✈❡ ❥♦❜s✳
●✐✈❡♥ ❛ ❢❡❛s✐❜❧❡ s❝❤❡❞✉❧❡ σ✱ ✇❡ ❧❡t fA
(σ) ❛♥❞ fB
(σ) ❞❡♥♦t❡ t❤❡ ❝♦st ✈❛❧✉❡s ❢♦r t❤❡
t✇♦ ❛❣❡♥ts✳ ❍❡r❡ ✇❡ ❝♦♥s✐❞❡r ❛s r❡s♦✉r❝❡ s❡t t❤❡ s❡t ΣP ♦❢ P❛r❡t♦ ♦♣t✐♠❛❧ s❝❤❡❞✉❧❡s✱
❛s t❤❡② ✐♥❝❧✉❞❡ ❛❧❧ s❡♥s✐❜❧❡ ❝♦♠♣r♦♠✐s❡ s❝❤❡❞✉❧❡s✳ ❋♦r ❡❛❝❤ σ ∈ ΣP ✱ ✇❡ ✇❛♥t t♦ ❞❡✜♥❡
✉t✐❧✐t② ✈❛❧✉❡s uA
(σ) ❛♥❞ uB
(σ)✱ s♦ t❤❛t✱ ❢♦r i = A, B✱ ui
(σ) ≥ 0 ❛♥❞ ui
(σ) ✐♥❝r❡❛s❡s
❛s fi
(σ) ❞❡❝r❡❛s❡s✳ ❋♦r t❤✐s ♣✉r♣♦s❡✱ ✇❡ ♣r♦♣♦s❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❞❡✜♥✐t✐♦♥ ♦❢ ✉t✐❧✐t②✳ ▲❡t
fi
∞ = max{fi
(σ)|σ ∈ ΣP } ✭❢♦r r❡❣✉❧❛r ❢✉♥❝t✐♦♥s fi
(σ)✱ t❤✐s ✐s t❤❡ ♠✐♥✐♠✉♠ ❝♦st t❤❡ ❛❣❡♥t
i ❜❡❛rs ✐❢ ✐ts ❥♦❜s ❛r❡ s❝❤❡❞✉❧❡❞ ❛❢t❡r ❛❧❧ t❤❡ ❥♦❜s ♦❢ t❤❡ ♦t❤❡r ❛❣❡♥t✮✳ ❚❤❡♥
ui
(σ) = fi
∞ − fi
(σ) i = A, B
■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❝♦♥s✐❞❡r t❤❛t ❛♥ ❛❣❡♥t✬s ✉t✐❧✐t② ✐s t❤❡ s❛✈✐♥❣ ❛❝❤✐❡✈❡❞ ✇✐t❤ r❡s♣❡❝t t♦ t❤❡
✇♦rst s❝❤❡❞✉❧❡ ❢♦r t❤❛t ❛❣❡♥t✳ ❲❡ ❛❧s♦ ❧❡t U(σ) = uA
(σ) + uB
(σ) ❞❡♥♦t❡ t❤❡ ♦✈❡r❛❧❧ ✉t✐❧✐t②
♦❢ s❝❤❡❞✉❧❡ σ ❛♥❞ ❧❡t σ∗
❞❡♥♦t❡ t❤❡ s❝❤❡❞✉❧❡ t❤❛t ♠❛①✐♠✐③❡s U(σ) ✭s②st❡♠ ♦♣t✐♠✉♠✮✱ ✐✳❡✳
U(σ∗
) = max
σ∈ΣP
{U(σ)}.
■♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ✇❡ ❛❧s♦ ❧❡t fi∗
= min{fi
(σ)|σ ∈ ΣP }✳
❆s ❢♦r t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ❛ ❢❛✐r s♦❧✉t✐♦♥✱ ✇❡ ❝♦♥s✐❞❡r t❤❡ ❢♦❧❧♦✇✐♥❣ t✇♦ ❝♦♥❝❡♣ts✳
✶✳ ❑❛❧❛✐✲❙♠♦r♦❞✐♥s❦② s♦❧✉t✐♦♥ ✭❑❛❧❛✐ ❛♥❞ ❙♠♦r♦❞✐♥s❦② ✶✾✼✺✮✳ ●✐✈❡♥ σ ∈ ΣP ✱ ❧❡t
ūi
(σ) =
ui
(σ)
fi
∞ − fi∗
❜❡ t❤❡ ♥♦r♠❛❧✐③❡❞ ✉t✐❧✐t② ♦❢ σ ❢♦r ❛❣❡♥t i✳ ❆ ❑❛❧❛✐✲❙♠♦r♦❞✐♥s❦② s❝❤❡❞✉❧❡ σKS ✭❜r✐❡✢②✱
❑❙ s❝❤❡❞✉❧❡✮ ✐s ❞❡✜♥❡❞ ❛s
σKS = arg max
σ
min
i=A,B
{ūi
(σ)}.
❚❤❡ ✐❞❡❛ ✐s t❤❛t ✐♥ σKS t❤❡ ♥♦r♠❛❧✐③❡❞ ✉t✐❧✐t② ♦❢ t❤❡ ❛❣❡♥t ✇❤♦ ✐s ✇♦rs❡✲♦✛ ✐s ♠❛①✐✲
♠✐③❡❞✳ ❙♦✱ ✐♥ σKS t❤❡ t✇♦ ❛❣❡♥ts✬ ♥♦r♠❛❧✐③❡❞ ✉t✐❧✐t② ✈❛❧✉❡s ❛r❡ t②♣✐❝❛❧❧② q✉✐t❡ ❝❧♦s❡✳
❖❜✈✐♦✉s❧②✱ ✉♥❞❡r t❤✐s ❞❡✜♥✐t✐♦♥ ❛ ❑❙ s❝❤❡❞✉❧❡ ❛❧✇❛②s ❡①✐sts✳ ■♥ t❤❡ ❧✐t❡r❛t✉r❡✱ ❛♥✲
♦t❤❡r ❞❡✜♥✐t✐♦♥ ♦❢ ❢❛✐r s♦❧✉t✐♦♥ ✐s t❤❡ ♠❛①✲♠✐♥ s♦❧✉t✐♦♥ ✭❇❡rts✐♠❛s ❡t ❛❧✳ ✷✵✶✶✮✳ ❑❛❧❛✐✲
❙♠♦r♦❞✐♥s❦② ❛♥❞ ♠❛①✲♠✐♥ s♦❧✉t✐♦♥s ❝♦✐♥❝✐❞❡ ✐❢ fA
∞ − fA∗
= fB
∞ − fB∗
✳
10

✷✳ Pr♦♣♦rt✐♦♥❛❧❧② ❢❛✐r s♦❧✉t✐♦♥✳ ❆ s❝❤❡❞✉❧❡ σP F ✐s ♣r♦♣♦rt✐♦♥❛❧❧② ❢❛✐r ✐❢✱ ❢♦r ❛♥② ♦t❤❡r
P❛r❡t♦ ♦♣t✐♠❛❧ s❝❤❡❞✉❧❡ σ✱ ✐t ❤♦❧❞s

uA
(σ) − uA
(σP F )
uA(σP F )

+

uB
(σ) − uB
(σP F )
uB(σP F )

≤ 0. ✭✶✮
❚❤❡ ✐❞❡❛ ❜❡❤✐♥❞ s✉❝❤ ❞❡✜♥✐t✐♦♥ ✐s t❤❡ ❢♦❧❧♦✇✐♥❣✳ ❲❤❡♥ ❝♦♥s✐❞❡r✐♥❣ ♠♦✈✐♥❣ ❢r♦♠ s❝❤❡❞✲
✉❧❡ σP F t♦ ❛♥② ♦t❤❡r s❝❤❡❞✉❧❡ σ✱ t❤❡ r❡❧❛t✐✈❡ ❜❡♥❡✜t t❤❛t ♦♥❡ ❛❣❡♥t ♠❛② ♦❜t❛✐♥ ✐s
❜❛❧❛♥❝❡❞ ❜② ❛ ♥♦t s♠❛❧❧❡r ✉t✐❧✐t② ❞❡❝r❡❛s❡ ❢♦r t❤❡ ♦t❤❡r ❛❣❡♥t✳ ❚❤✐s ✐s ❛❝t✉❛❧❧② t❤❡
s❛♠❡ r❛t✐♦♥❛❧❡ ❜❡❤✐♥❞ t❤❡ ❝♦♥❝❡♣t ♦❢ ◆❛s❤ s♦❧✉t✐♦♥ ✭◆❛s❤ ✶✾✺✵✮✳ ❍♦✇❡✈❡r✱ t❤❡ ◆❛s❤
s♦❧✉t✐♦♥ ✇❛s ✐♥tr♦❞✉❝❡❞ ♦♥❧② ✇✐t❤ r❡s♣❡❝t t♦ ❝♦♠♣❛❝t ❛♥❞ ❝♦♥✈❡① r❡s♦✉r❝❡ s❡ts✱ ✇❤✐❧❡
❉❡✜♥✐t✐♦♥ ✭✶✮ ✐s ♠♦r❡ ❣❡♥❡r❛❧✳ ■♥ ❢❛❝t✱ ✇❤✐❧❡ ❛ ◆❛s❤ s♦❧✉t✐♦♥ ❛❧✇❛②s ❡①✐sts✱ ❛ ♣r♦♣♦r✲
t✐♦♥❛❧❧② ❢❛✐r s♦❧✉t✐♦♥ ♠❛② ♥♦t ❡①✐st✳ ■❢ ✐t ❞♦❡s ❡①✐st✱ t❤❡♥ ✐t ❝♦✐♥❝✐❞❡s ✇✐t❤ t❤❡ s♦❧✉t✐♦♥
♠❛①✐♠✐③✐♥❣ t❤❡ ♣r♦❞✉❝t ♦❢ ✉t✐❧✐t✐❡s✱ ❛♥❞ ❤❡♥❝❡✱ ✐❢ t❤❡ r❡s♦✉r❝❡ s❡t ✐s ❝♦♠♣❛❝t ❛♥❞
❝♦♥✈❡①✱ ✇✐t❤ t❤❡ ◆❛s❤ s♦❧✉t✐♦♥✳
❖✉r st✉❞② ✐♥✈❡st✐❣❛t❡s ❤♦✇ ♠✉❝❤ ✭❣❧♦❜❛❧✮ ✉t✐❧✐t② s❤♦✉❧❞ ❜❡ ❣✐✈❡♥ ✉♣ ✐♥ ♦r❞❡r t♦ ❤❛✈❡ ❛
❢❛✐r s♦❧✉t✐♦♥✳ ❚❤✐s ✐s ❝❛♣t✉r❡❞ ❜② t❤❡ ♣r✐❝❡ ♦❢ ❢❛✐r♥❡ss✱ ❞❡✜♥❡❞ ❛s t❤❡ r❡❧❛t✐✈❡ ❧♦ss ♦❢ ✉t✐❧✐t②
✐♥ ❛ ❢❛✐r s♦❧✉t✐♦♥ ✇✐t❤ r❡s♣❡❝t t♦ ♠❛①✐♠✉♠ ✉t✐❧✐t②✳ ◆♦t✐❝❡ t❤❛t t❤❡r❡ ♠❛② ❜❡ ♠♦r❡ t❤❛♥
♦♥❡ ❢❛✐r s♦❧✉t✐♦♥✱ ❞✐✛❡r✐♥❣ ✐♥ t❡r♠s ♦❢ ❣❧♦❜❛❧ ✉t✐❧✐t②✳ ❍❡r❡ ✇❡ ❛❞♦♣t t❤❡ s❛♠❡ ✈✐❡✇♣♦✐♥t ❛s
✐♥ ✭❑❛rs✉ ❛♥❞ ▼♦rt♦♥ ✷✵✶✺✱ ◆❛❧❞✐ ❡t ❛❧✳ ✷✵✶✻✮✱ ✐✳❡✳✱ ✇❤❡♥❡✈❡r t❤✐s ♦❝❝✉rs✱ ✇❡ ♠❡❛s✉r❡ t❤❡
♣r✐❝❡ ♦❢ ❢❛✐r♥❡ss ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ❜❡st ❢❛✐r s♦❧✉t✐♦♥✳ ■♥ ❢♦r♠❛❧ t❡r♠s✱ ❧❡tt✐♥❣ I ❞❡♥♦t❡ t❤❡
s❡t ♦❢ ❛❧❧ ✐♥st❛♥❝❡s ♦❢ ❛ ❣✐✈❡♥ ♣r♦❜❧❡♠✱ I ♦♥❡ ♦❢ t❤❡♠✱ σ∗
(I) t❤❡ s②st❡♠ ♦♣t✐♠✉♠✱ ΣF t❤❡
s❡t ♦❢ ❛❧❧ ❢❛✐r s❝❤❡❞✉❧❡s ❛♥❞ σF (I) ♦♥❡ ♦❢ t❤❡♠✱ ✇❡ ❞❡✜♥❡ t❤❡ ♣r✐❝❡ ♦❢ ❢❛✐r♥❡ss ❛s✿
PoF = sup
I∈I

min
σF ∈ΣF

U(σ∗
(I)) − U(σF (I))
U(σ∗(I))

. ✭✷✮
◆♦t✐❝❡ t❤❛t t❤✐s ✐s ❛ s✐♠✐❧❛r ❞❡✜♥✐t✐♦♥ t♦ t❤❡ ✇❡❧❧✲❦♥♦✇♥ Pr✐❝❡ ♦❢ ❙t❛❜✐❧✐t② ✭❆♥s❤❡❧❡✈✐❝❤
❡t ❛❧✳ ✷✵✵✹✮✱ r❡♣❧❛❝✐♥❣ t❤❡ r♦❧❡ ♦❢ ◆❛s❤ ❡q✉✐❧✐❜r✐✉♠ ✇✐t❤ ❢❛✐r♥❡ss✳ ❍❡r❡❛❢t❡r✱ ✇❡ ✐♥❞✐❝❛t❡
✇✐t❤ PoFKS ❛♥❞ PoFP F t❤❡ ♣r✐❝❡ ♦❢ ❑❛❧❛✐✲❙♠♦r♦❞✐♥s❦② ❢❛✐r♥❡ss ❛♥❞ ♣r♦♣♦rt✐♦♥❛❧ ❢❛✐r♥❡ss✱
r❡s♣❡❝t✐✈❡❧②✳
✸ ❙❝❡♥❛r✐♦ ❛❞❞r❡ss❡❞
■♥ t❤✐s t❛❧❦ ✇❡ ❛❞❞r❡ss t❤❡ ✈❛❧✉❡ ♦❢ PoF ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ s❝❤❡❞✉❧✐♥❣ s❝❡♥❛r✐♦✳ ❆❣❡♥t
A ♣✉rs✉❡s t❤❡ ♠✐♥✐♠✐③❛t✐♦♥ ♦❢ t❤❡ s✉♠ ♦❢ ✐ts ❥♦❜s✬ ❝♦♠♣❧❡t✐♦♥ t✐♠❡s✱ ✇❤✐❧❡ ❛❣❡♥t B ✐s
✐♥t❡r❡st❡❞ ✐♥ ♠✐♥✐♠✐③✐♥❣ t❤❡ ♠❛①✐♠✉♠ t❛r❞✐♥❡ss ♦❢ ✐ts ❥♦❜s ✇✐t❤ r❡s♣❡❝t t♦ ❛ ❝♦♠♠♦♥
❞✉❡ ❞❛t❡ d✳ ❋♦❧❧♦✇✐♥❣ t❤❡ ✉s✉❛❧ ●r❛❤❛♠✬s ♥♦t❛t✐♦♥✱ ✇❡ ❞❡♥♦t❡ t❤✐s s❝❡♥❛r✐♦ ❛s 1|dB
j =
d|
P
CA
j , TB
max✳ ◆♦t❡ t❤❛t t❤✐s s❝❡♥❛r✐♦ ✐♥❝❧✉❞❡s t❤❡ ❝❛s❡ ✐♥ ✇❤✐❝❤ B ✇❛♥ts t♦ ♠✐♥✐♠✐③❡
✐ts ❥♦❜s✬ ♠❛❦❡s♣❛♥✱ ♦❜t❛✐♥❡❞ ❢♦r d = 0✳ ◆♦t❡ t❤❛t ✐♥ t❤✐s s❝❡♥❛r✐♦✱ ✐♥ ❛♥② P❛r❡t♦ ♦♣t✐♠❛❧
s♦❧✉t✐♦♥ ❛❧❧ B✲❥♦❜s ❛r❡ s❝❤❡❞✉❧❡❞ ❝♦♥s❡❝✉t✐✈❡❧②✱ s♦ ♦♥❡ ❝❛♥ ❛ss✉♠❡ t❤❛t B ♦✇♥s ❛ s✐♥❣❧❡
❥♦❜ ♦❢ ❧❡♥❣t❤ PB
=
PnB
j=1 pB
j ✳ ■♥ t❤✐s s❝❡♥❛r✐♦✱ t❤❡ ✈❛❧✉❡s fA∗
, fA
∞, fB∗
, fB
∞ ❝❛♥ ❛❧❧ ❜❡
❡❛s✐❧② ❝♦♠♣✉t❡❞✳ ■♥ ❢❛❝t✱ fA∗
✐s t❤❡ t♦t❛❧ ❝♦♠♣❧❡t✐♦♥ t✐♠❡ ♦❢ t❤❡ A✲❥♦❜s ✇❤❡♥ t❤❡② ❛r❡
s❡q✉❡♥❝❡❞ ✐♥ ❙P❚ ♦r❞❡r ❛♥❞ fA
∞ = fA∗
+ nAPB
✱ ✇❤✐❧❡ fB∗
= max{0, PB
− d} ❛♥❞ fB
∞ =
max{0, PA
+ PB
− d}✳
❖✉r ❝♦♥tr✐❜✉t✐♦♥s t♦ t❤✐s s❝❡♥❛r✐♦ ❛r❡ s✉♠♠❛r✐③❡❞ ✐♥ ❚❛❜❧❡ ✶✳ ■♥ ♣❛rt✐❝✉❧❛r✱ ✇❡ s❤♦✇
t❤❛t PoFKS = 2/3 ❛♥❞ PoFP F = 1/2✳ ▼♦r❡♦✈❡r✱ ✇❡ s❤♦✇ t❤❛t✱ ✐❢ t❤❡ A✲❥♦❜s ❛r❡ ❛❧r❡❛❞②
♦r❞❡r❡❞ ❜② ♥♦♥❞❡❝r❡❛s✐♥❣ ❧❡♥❣t❤✱ ✐♥ t✐♠❡ O(log nA) ❛ ♣r♦♣♦rt✐♦♥❛❧❧② ❢❛✐r s♦❧✉t✐♦♥ ❝❛♥ ❜❡
❝♦♠♣✉t❡❞ ♦r ♣r♦✈❡❞ t❤❛t ✐t ❞♦❡s ♥♦t ❡①✐st✳
11

❚❛❜❧❡ ✶✳ ❘❡s✉❧ts ❢♦r s❝❡♥❛r✐♦ 1|dB
j = d|
P
CA
j , TB
max (∗
) ✐❢ A✲❥♦❜s ❛r❡ ❣✐✈❡♥ ✐♥ ❙P❚ ♦r❞❡r✳
Pr♦♣♦rt✐♦♥❛❧❧② ❢❛✐r s♦❧✉t✐♦♥ ❑❛❧❛✐✲❙♠♦r♦❞✐♥s❦② s♦❧✉t✐♦♥
P♦❋ ✈❛❧✉❡ ✶✴✷ ✷✴✸
❊①✐st❡♥❝❡ ❊st❛❜❧✐s❤❡❞ ✐♥ O(log nA)∗
✭❛❧✇❛②s ❡①✐sts✮
❘❡❢❡r❡♥❝❡s
❆♥s❤❡❧❡✈✐❝❤✱ ❊✳✱ ❉❛s❣✉♣t❛✱ ❆✳✱ ❑❧❡✐♥❜❡r❣✱ ❏✳✱ ❚❛r❞ös✱ ❊✳✱ ❲❡①❧❡r✱ ❚✳ ❛♥❞ ❚✳ ❘♦✉❣❤❣❛r❞❡♥ ✭✷✵✵✹✮✱
❚❤❡ Pr✐❝❡ ♦❢ ❙t❛❜✐❧✐t② ❢♦r ◆❡t✇♦r❦ ❉❡s✐❣♥ ✇✐t❤ ❋❛✐r ❈♦st ❆❧❧♦❝❛t✐♦♥✱ ✐♥ ✹✺t❤ ❆♥♥✉❛❧ ■❊❊❊
❙②♠♣♦s✐✉♠ ♦♥ ❋♦✉♥❞❛t✐♦♥s ♦❢ ❈♦♠♣✉t❡r ❙❝✐❡♥❝❡ ✭❋❖❈❙✮✱ ♣❛❣❡s ✺✾✲✼✸✳
❇❛❧✐♥s❦✐✱ ▼✳▲✳✱ ❨♦✉♥❣ ❍✳P✳ ✭✷✵✵✶✮✱ ❋❛✐r ❘❡♣r❡s❡♥t❛t✐♦♥✿ ▼❡❡t✐♥❣ t❤❡ ■❞❡❛❧ ♦❢ ❖♥❡ ▼❛♥✱ ❖♥❡ ❱♦t❡✱
❇r♦♦❦✐♥❣s ■♥st✐t✉t✐♦♥ Pr❡ss✱ ❲❛s❤✐♥❣t♦♥✳
❇❡rts✐♠❛s ❉✳✱ ❱✳ ❋❛r✐❛s✱ ◆✳ ❚r✐❝❤❛❦✐s ✭✷✵✶✶✮✱ ❚❤❡ ♣r✐❝❡ ♦❢ ❢❛✐r♥❡ss✱ ❖♣❡r❛t✐♦♥s ❘❡s❡❛r❝❤✱ ✺✾✭✶✮✱
✶✼✕✸✶✳
❇♦❤♠✱ P✳✱ ❇✳ ▲❛rs❡♥ ✭✶✾✾✹✮✱ ❋❛✐r♥❡ss ✐♥ ❛ tr❛❞❡❛❜❧❡✲♣❡r♠✐t tr❡❛t② ❢♦r ❝❛r❜♦♥ ❡♠✐ss✐♦♥s r❡❞✉❝t✐♦♥s
✐♥ ❊✉r♦♣❡ ❛♥❞ t❤❡ ❢♦r♠❡r ❙♦✈✐❡t ❯♥✐♦♥✱ ❊♥✈✐r♦♥♠❡♥t❛❧ ❛♥❞ ❘❡s♦✉r❝❡ ❊❝♦♥♦♠✐❝s✱ ✹✱ ✷✶✾✕✷✸✾✳
❈❛r❛❣✐❛♥♥✐s ■✳✱ ❈✳ ❑❛❦❧❛♠❛♥✐s✱ P✳ ❑❛♥❡❧❧♦♣♦✉❧♦s✱ ▼✳ ❑②r♦♣♦✉❧♦✉ ✭✷✵✶✷✮✱ ❚❤❡ ❡✣❝✐❡♥❝② ♦❢ ❢❛✐r
❞✐✈✐s✐♦♥✱ ✷✵✶✷✱ ❚❤❡♦r② ♦❢ ❈♦♠♣✉t✐♥❣ ❙②st❡♠s✱ ✺✵✭✹✮✱ ✺✽✾✕✻✶✵✳
❊rt♦❣r❛❧ ❑✳✱ ❉✳ ❲✉ ✭✷✵✵✵✮✱ ✏❆✉❝t✐♦♥✲t❤❡♦r❡t✐❝ ❝♦♦r❞✐♥❛t✐♦♥ ♦❢ ♣r♦❞✉❝t✐♦♥ ♣❧❛♥♥✐♥❣ ✐♥ t❤❡ s✉♣♣❧②
❝❤❛✐♥✑✱ ■■❊ ❚r❛♥s❛❝t✐♦♥s✱ ✸✷✭✶✵✮✱ ✾✸✶✲✾✹✵✳
❑❛❧❛✐ ❊✳✱ ▼✳ ❙♠♦r♦❞✐♥s❦② ✭✶✾✼✺✮✱ ❖t❤❡r s♦❧✉t✐♦♥s t♦ ◆❛s❤ ❜❛r❣❛✐♥✐♥❣ ♣r♦❜❧❡♠✱❊❝♦♥♦♠❡tr✐❝❛✱ ✹✸✱
✺✶✸✕✺✶✽✳
❑❛rs✉ Ö✳✱ ❆✳ ▼♦rt♦♥ ✭✷✵✶✺✮✱ ■♥❡q✉✐t② ❛✈❡rs❡ ♦♣t✐♠✐③❛t✐♦♥ ✐♥ ♦♣❡r❛t✐♦♥❛❧ r❡s❡❛r❝❤✱ ❊✉r♦♣❡❛♥ ❏♦✉r✲
♥❛❧ ♦❢ ❖♣❡r❛t✐♦♥❛❧ ❘❡s❡❛r❝❤✱ ✷✹✺✭✷✮✱ ✸✹✸✕✸✺✾✳
▲✉❝❛s✱ ❲✳❋✳✱ ✭✶✾✽✸✮✱ ❚❤❡ ❆♣♣♦rt✐♦♥♠❡♥t Pr♦❜❧❡♠✱ ✐♥✿ ❇r❛♠s ❙✳❏✳✱ ▲✉❝❛s ❲✳❋✳✱ ❙tr❛✣♥ P✳❉✳
✭❡❞s✮✱ P♦❧✐t✐❝❛❧ ❛♥❞ ❘❡❧❛t❡❞ ▼♦❞❡❧s✱ ▼♦❞✉❧❡s ✐♥ ❆♣♣❧✐❡❞ ▼❛t❤❡♠❛t✐❝s✳ ❙♣r✐♥❣❡r✱ ◆❡✇ ❨♦r❦✱
◆❨✱ ✸✺✽✲✸✾✻✳
◆❛❧❞✐ ▼✳✱ ●✳ ◆✐❝♦s✐❛✱ ❆✳ P❛❝✐✜❝✐✱ ❯✳ P❢❡rs❝❤② ✭✷✵✶✻✮✱ ▼❛①✐♠✐♥ ❋❛✐r♥❡ss ✐♥ Pr♦❥❡❝t ❇✉❞❣❡t ❆❧❧♦✲
❝❛t✐♦♥✱ ❊❧❡❝tr♦♥✐❝ ◆♦t❡s ✐♥ ❉✐s❝r❡t❡ ▼❛t❤❡♠❛t✐❝s✱ ✺✺✱ ✻✺✕✻✽✳
◆❛s❤✱ ❏✳ ✭✶✾✺✵✮✱ ❚❤❡ ❜❛r❣❛✐♥✐♥❣ ♣r♦❜❧❡♠✱ ❊❝♦♥♦♠❡tr✐❝❛✱ ✶✽✭✷✮✱ ♣♣✳ ✶✺✺✕✶✻✷✳
◆✐❝♦s✐❛ ●✳✱ ❆✳ P❛❝✐✜❝✐✱ ❯✳ P❢❡rs❝❤②✱ ✷✵✶✼✱ Pr✐❝❡ ♦❢ ❋❛✐r♥❡ss ❢♦r ❛❧❧♦❝❛t✐♥❣ ❛ ❜♦✉♥❞❡❞ r❡s♦✉r❝❡✱
❊✉r♦♣❡❛♥ ❏♦✉r♥❛❧ ♦❢ ❖♣❡r❛t✐♦♥❛❧ ❘❡s❡❛r❝❤✱ ✷✺✼✱ ♣♣✳ ✾✸✸✲✾✹✸✳
12

❆ ▼■▲P ❢♦r♠✉❧❛t✐♦♥ ❢♦r ❛♥ ♦♣❡r❛t✐♥❣ r♦♦♠ s❝❤❡❞✉❧✐♥❣
♣r♦❜❧❡♠ ✉♥❞❡r st❡r✐❧✐③✐♥❣ ❛❝t✐✈✐t✐❡s ❝♦♥str❛✐♥ts
❍✳ ❆❧ ❍❛s❛♥1,2,3
✱ ❈✳ ●✉ér❡t1
✱ ❉✳ ▲❡♠♦✐♥❡2
❛♥❞ ❉✳ ❘✐✈r❡❛✉3
1
❯♥✐✈❡rs✐té ❞✬❆♥❣❡rs✱ ▲❆❘■❙✱ ❆♥❣❡rs✱ ❋r❛♥❝❡
❝❤r✐st❡❧❧❡✳❣✉❡r❡t❅✉♥✐✈✲❛♥❣❡rs✳❢r
2
■▼❚ ❆t❧❛♥t✐q✉❡✱ ▲❙✷◆✱ ◆❛♥t❡s✱ ❋r❛♥❝❡
❞❛✈✐❞✳❧❡♠♦✐♥❡❅✐♠t✲❛t❧❛♥t✐q✉❡✳❢r
3
❯♥✐✈❡rs✐té ❈❛t❤♦❧✐q✉❡ ❞❡ ❧✬❖✉❡st✱ ▲❆❘■❙✱ ❆♥❣❡rs✱ ❋r❛♥❝❡
❛❧❤❛s❛♥✱ r✐✈r❡❛✉❅✉❝♦✳❢r
❑❡②✇♦r❞s✿ ♦♣❡r❛t✐♥❣ r♦♦♠s ✐♥ ❤❡❛❧t❤ s❡r✈✐❝❡s✱ st❡r✐❧✐③❛t✐♦♥ ✉♥✐t✱ s❝❤❡❞✉❧✐♥❣✱ ▼■▲P
✶ ■♥tr♦❞✉❝t✐♦♥
❖♣❡r❛t✐♥❣ r♦♦♠s ❤❛✈❡ ❜❡❡♥ r❡❝♦❣♥✐③❡❞ t♦ ❜❡ t❤❡ ♠❛✐♥ ✐♥❝♦♠❡ s♦✉r❝❡ ❢♦r ❤♦s♣✐t❛❧s
❛s ✐t ❣❡♥❡r❛t❡s ❛r♦✉♥❞ ✻✵✪ ♦❢ ❤♦s♣✐t❛❧ r❡✈❡♥✉❡s ✭▼❛❝❛r✐♦ ❡t✳ ❛❧✳ ✶✾✾✺✮ ❜✉t ✐t ❝♦✉♥ts ❢♦r
❛r♦✉♥❞ ✹✵✪ ♦❢ ❤♦s♣✐t❛❧ ❝♦sts ✭❏❛❝❦s♦♥ ✷✵✵✷✮ t❤r♦✉❣❤♦✉t t❤❡ ✉s❡ ♦❢ ❢❛❝✐❧✐t✐❡s ✭♦♣❡r❛t✐♥❣
r♦♦♠s✱ ❡t❝✳✮ ❛♥❞ t❤❡ ♣❡rs♦♥♥❡❧ ❝♦sts✳ ❚❤✐s ❤✉❣❡ ✜♥❛♥❝✐❛❧ ❢❛❝t♦r ♠❛❦❡s t❤❡ ♦♣❡r❛t✐♥❣ r♦♦♠s
♠❛♥❛❣❡♠❡♥t ❛ ♣r✐♦r✐t② ❢♦r ❤♦s♣✐t❛❧ ♠❛♥❛❣❡rs ✐♥ ♦r❞❡r t♦ ❛❝❤✐❡✈❡ ❛♥ ❡✣❝✐❡♥t ❛♥❞ ❡✛❡❝t✐✈❡
✉s❡ ♦❢ t❤❡ ♦♣❡r❛t✐♥❣ r♦♦♠s✳ ❊①❤❛✉st✐✈❡ ❧✐t❡r❛t✉r❡ r❡✈✐❡✇s ♦♥ t❤❡ ❙✉r❣✐❝❛❧ ❈❛s❡ ❙❝❤❡❞✉❧✐♥❣
✭❙❈❙✮ ♣r♦❜❧❡♠ ❛r❡ r❡♣♦rt❡❞ ✐♥ ✭❈❛r❞♦❡♥ ❡t✳ ❛❧✳ ✷✵✶✵✱ ●✉❡rr✐❡r♦ ❡t✳ ❛❧✳ ✷✵✶✶✮✳
■♥ t❤✐s ♣❛♣❡r✱ ✇❡ st✉❞② ❛ r❡❛❧ s❝❤❡❞✉❧✐♥❣ ♣r♦❜❧❡♠ ✇❤✐❝❤ ❝♦♥s✐sts ✐♥ s❝❤❡❞✉❧✐♥❣ ❛ s❡t
♦❢ ❡❧❡❝t✐✈❡ s✉r❣✐❝❛❧ ❝❛s❡s ✇❤✐❝❤ r❡q✉✐r❡ s✉r❣✐❝❛❧ ✐♥str✉♠❡♥ts ❛♥❞ t♦♦❧s t♦ s❡✈❡r❛❧ ♦♣❡r❛t✐♥❣
r♦♦♠s ✇✐t❤ t❤❡ ♦❜❥❡❝t✐✈❡ ♦❢ ♠✐♥✐♠✐③✐♥❣ t❤❡ ♦♣❡r❛t✐♥❣ ❝♦sts ✇❤✐❧❡ t❛❦✐♥❣ ✐♥t♦ ❛❝❝♦✉♥t
t❤❡ ❛❝t✐✈✐t✐❡s ♦❢ t❤❡ st❡r✐❧✐③✐♥❣ ✉♥✐t✳ ❚♦ t❤❡ ❜❡st ♦❢ ♦✉r ❦♥♦✇❧❡❞❣❡ t❤❡r❡ ❛r❡ ✈❡r② ❢❡✇
❧✐t❡r❛t✉r❡ ♦♥ s✉❝❤ ♣❛rt✐❝✉❧❛r ♣r♦❜❧❡♠✳ ❋♦r ✐♥st❛♥❝❡ ✭❇❡r♦✉❧❡ ❡t✳ ❛❧✳ ✷✵✶✻✮ st✉❞② ❛♥ ♦♣❡r❛t✐♥❣
r♦♦♠ s❝❤❡❞✉❧✐♥❣ ♣r♦❜❧❡♠ ✐♥❝❧✉❞✐♥❣ ♠❡❞✐❝❛❧ ❞❡✈✐❝❡s st❡r✐❧✐③❛t✐♦♥ ❜✉t ✇✐t❤ t❤❡ ♦❜❥❡❝t✐✈❡ ♦❢
r❡❞✉❝✐♥❣ t❤❡ ♥✉♠❜❡r ♦❢ ♠❡❞✐❝❛❧ ❞❡✈✐❝❡s ♥❡❡❞❡❞ ❛t t❤❡ s❛♠❡ t✐♠❡✳ ❆♥❞ ❛ ❜r❛♥❝❤✲❛♥❞✲♣r✐❝❡
t❡❝❤♥✐q✉❡ ✐s ❛♣♣❧✐❡❞ ✐♥ ✭❈❛r❞♦❡♥ ❡t✳ ❛❧✳ ✷✵✵✾✮ t♦ ✜♥❞ t❤❡ ❜❡st ♦r❞❡r ❢♦r s✉r❣❡r✐❡s ✐♥ ❛ ❞❛②
❝❛r❡ ❝❡♥t❡r ✐♥ ♦r❞❡r t♦ ♦♣t✐♠✐③❡ s❡✈❡r❛❧ ♦❜❥❡❝t✐✈❡s ✭♣❡❛❦ ✉s❡ ♦❢ r❡❝♦✈❡r② ❜❡❞s✱ ♦❝❝✉rr❡♥❝❡
♦❢ r❡❝♦✈❡r② ♦✈❡rt✐♠❡✱ ✳✳✳✮ ✇❤✐❧❡ s❛t✐s❢②✐♥❣ t❤❡ ♠❡❞✐❝❛❧ ❞❡✈✐❝❡s st❡r✐❧✐③✐♥❣ ❝♦♥str❛✐♥ts✳
❚❤✐s r❡s❡❛r❝❤ ✇❛s ♣❡r❢♦r♠❡❞ ✐♥ ❝♦❧❧❛❜♦r❛t✐♦♥ ✇✐t❤ t❤❡ ❯♥✐✈❡rs✐t② ❍♦s♣✐t❛❧ ♦❢ ❆♥❣❡rs
✐♥ ❋r❛♥❝❡ ✭❈❍❯ ❆♥❣❡rs✮✱ ✇❤✐❝❤ ❤❛s ❛❧s♦ ♣r♦✈✐❞❡❞ ❤✐st♦r✐❝❛❧ ❞❛t❛ ❢♦r t❤❡ ❡①♣❡r✐♠❡♥ts✳ ❲❡
♣r♦♣♦s❡ ❛ ♠✐①❡❞ ✐♥t❡❣❡r ❧✐♥❡❛r ♣r♦❣r❛♠♠✐♥❣ ♠♦❞❡❧ ❢♦r t❤❡ ♣r♦❜❧❡♠ ✇❤✐❝❤ ✐s s♦❧✈❡❞ ✐♥
❛ ❧❡①✐❝♦❣r❛♣❤✐❝ ✇❛②✳ ❲❡ s❤♦✇ t❤❛t ♦✉r s♦❧✉t✐♦♥s ♣r♦✈✐❞❡ ❝♦♠♣❡t✐t✐✈❡ r❡s✉❧ts ✐♥ t❡r♠s ♦❢
♥✉♠❜❡r ♦❢ ♦♣❡r❛t✐♥❣ r♦♦♠s✱ ❛♥❞ s✐❣♥✐✜❝❛♥t❧② ✐♠♣r♦✈❡ t❤♦s❡ ♦♣❡r❛t✐♦♥❛❧❧② ✐♠♣❧❡♠❡♥t❡❞ ✐♥
t❡r♠s ♦❢ ♦✈❡rt✐♠❡ ❛♥❞ ❡♠❡r❣❡♥❝✐❡s ❛t t❤❡ st❡r✐❧✐③✐♥❣ ✉♥✐t✳
✷ Pr♦❜❧❡♠ ❞❡s❝r✐♣t✐♦♥
❚❤❡ ❈❍❯ ❆♥❣❡rs ✐♥❝❧✉❞❡s s❡✈❡r❛❧ ❜❧♦❝❦s ❛♥❞ ❛ st❡r✐❧✐③✐♥❣ ✉♥✐t ✇❤✐❝❤ ❝❡♥tr❛❧✐③❡s ❛❧❧
t❤❡ st❡r✐❧✐③✐♥❣ ❛❝t✐✈✐t✐❡s✳ ■♥ t❤✐s st✉❞②✱ ✇❡ ❢♦❝✉s ♦♥❧② ♦♥ t❤❡ ❛❝t✐✈✐t✐❡s ♦❢ t❤❡ ❖rt❤♦♣❡❞✐❝
❙✉r❣❡r② ❇❧♦❝❦ ✭❖❙❇✮ ❛♥❞ t❤❡ ❙t❡r✐❧✐③✐♥❣ ❯♥✐t ✭❙❯✮✳ ❚❤❡r❡ ❛r❡ ❛r♦✉♥❞ ✷✺✵✵ s✉r❣❡r✐❡s t❤❛t
❛r❡ ♣❡r❢♦r♠❡❞ ❛t t❤❡ ❖❙❇ ♣❡r ②❡❛r ✐♥ ✐ts ✸ ♦♣❡r❛t✐♥❣ r♦♦♠s✳ ❚❤❡ ♦♣❡♥✐♥❣ ❤♦✉rs ❢♦r t❤❡s❡
✸ ❖❘s ❛r❡ ❞✐✛❡r❡♥t ❛s✿ r♦♦♠ ✶ ❛♥❞ ✷ ❛r❡ ♦♣❡♥ ✺ ❞❛②s ❛ ✇❡❡❦ ❢r♦♠ ✽✿✶✺ t♦ ✶✼✿✵✵✱ ❛♥❞ r♦♦♠
✸ ✐s ♦♣❡♥ ♦♥❧② ✹ ❞❛②s ❛ ✇❡❡❦ ❢r♦♠ ✽✿✶✺ t♦ ✶✹✿✸✵✳ ❇❡t✇❡❡♥ ✶✵ ❛♥❞ ✶✹ s✉r❣❡♦♥s s❤❛r❡ t❤❡s❡
r♦♦♠s ❛❝❝♦r❞✐♥❣ t♦ ❛ ♣❧❛♥♥✐♥❣ ✐♥❞✐❝❛t✐♥❣ t❤❡ ❞❛②s ✇❤❡♥ t❤❡② ♦♣❡r❛t❡✱ ❛♥❞ t❤❡ ❧✐st ♦❢ r♦♦♠s
t❤❛t ❡❛❝❤ s✉r❣❡♦♥ ❝❛♥ ✉s❡ ❡❛❝❤ ❞❛②✳
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❊❛❝❤ s✉r❣❡♦♥ ❤❛s t♦ ♣❡r❢♦r♠ ❛ ❧✐st ♦❢ s✉r❣❡r✐❡s ♦♥ ❛♥ ❤♦r✐③♦♥ ♦❢ ♦♥❡ ♠♦♥t❤ ✿ s♦♠❡ ♦❢
t❤❡♠ ❝❛♥ ❜❡ s❝❤❡❞✉❧❡❞ ❛♥②t✐♠❡ ❞✉r✐♥❣ t❤❡ ♦♣❡♥✐♥❣ ❤♦✉rs ♦❢ t❤❡ r♦♦♠s✱ ✇❤❡r❡❛s ♦t❤❡rs
✭❛♠❜✉❧❛t♦r② s✉r❣❡r✐❡s✮ ❤❛✈❡ t♦ ❜❡ ❝♦♠♣❧❡t❡❞ ❜❡❢♦r❡ ✶✺✿✵✵ t♦ ❛❧❧♦✇ t❤❡ ♣❛t✐❡♥t t♦ ❣♦ ❤♦♠❡
❛t t❤❡ ❡♥❞ ♦❢ t❤❡ ❞❛②✳ ❊❛❝❤ ♦❢ t❤❡s❡ s✉r❣❡r✐❡s ✐s ❝❤❛r❛❝t❡r✐③❡❞ ❜② ❛♥ ❡st✐♠❛t❡❞ ❞✉r❛t✐♦♥
t✐♠❡ ❛♥❞ r❡q✉✐r❡s ❛ ❧✐st ♦❢ s✉r❣✐❝❛❧ ✐♥str✉♠❡♥ts ✇❤✐❝❤ ❛r❡ ♦r❣❛♥✐③❡❞ ✐♥ s♠❛❧❧ ❜♦①❡s ❝❛❧❧❡❞
❦✐ts✳ ❚❤❡s❡ ❦✐ts ❛r❡ ❛✈❛✐❧❛❜❧❡ ✐♥ ❧✐♠✐t❡❞ q✉❛♥t✐t✐❡s✳ ❆❢t❡r ❡❛❝❤ s✉r❣❡r②✱ t❤❡ ✉s❡❞ ❦✐ts ❛r❡ ❦❡♣t
✐♥t♦ ✇❛t❡r ❢♦r ✸✵ ♠✐♥✉t❡s ❢♦r ♣r❡✲❞✐s✐♥❢❡❝t✐♦♥✳ ❚❤❡♥ t❤❡② ❛r❡ ❝♦❧❧❡❝t❡❞ ❛t t❤❡ ♣r❡❞❡✜♥❡❞
♣❡r✐♦❞s ❣✐✈❡♥ ✐♥ t❛❜❧❡ ✶ ❛♥❞ s❡♥t t♦ t❤❡ ❙❯ ❢♦r st❡r✐❧✐③❛t✐♦♥✳
❚❛❜❧❡ ✶✳ ❙❯✬s ♣✐❝❦✉♣s ❛♥❞ ❞❡❧✐✈❡r✐❡s t♦ t❤❡ ❖❙❇✳
P✐❝❦✲✉♣ ✵✼✿✵✵ ✶✶✿✸✵ ✶✸✿✵✵ ✶✹✿✸✵ ✶✻✿✵✵ ✶✼✿✸✵ ✶✽✿✸✵
❉❡❧✐✈❡r② ✵✼✿✵✵ ✲ ✲ ✶✹✿✸✵ ✲ ✶✼✿✸✵ ✲
❆t t❤❡ ❙❯✱ t❤❡ st❡r✐❧✐③❛t✐♦♥ ♣r♦❝❡ss ✐s ❜❡✐♥❣ ♣❡r❢♦r♠❡❞ ✐♥ s❡✈❡r❛❧ st❡♣s ✿ t❤❡ ✐♥str✉♠❡♥ts
❛r❡ ✜rst ❝❧❡❛♥❡❞ ❜② ❛✉t♦♠❛t✐❝ ✇❛s❤❡rs✱ t❤❡♥ r❡❛ss✐❣♥❡❞ ✐♥ t❤❡✐r ❝♦rr❡s♣♦♥❞✐♥❣ ❦✐t ❜❡❢♦r❡
❜❡✐♥❣ ♣r♦❝❡ss❡❞ t❤r♦✉❣❤ st❡r✐❧✐③❛t✐♦♥ ♠❛❝❤✐♥❡s✳ ❋✐♥❛❧❧②✱ t❤❡ ❦✐ts ❛r❡ ❦❡♣t ❛t t❤❡ ❙❯ t♦ ❝♦♦❧
♦✛ ❜❡❢♦r❡ ❜❡✐♥❣ r❡t✉r♥❡❞ t♦ t❤❡ ❜❧♦❝❦✳ ❖♥ ❛✈❡r❛❣❡✱ ✇❤❡♥ ❛ ❦✐t ❛rr✐✈❡s ❛t t❤❡ ❙❯✱ t❤❡ ✇❤♦❧❡
st❡r✐❧✐③❛t✐♦♥ ♣r♦❝❡ss t❛❦❡s ❛r♦✉♥❞ ✹❤✸✵✳ ❋r♦♠ t❤❡s❡ ❞❡❧✐✈❡r②✴❝♦❧❧❡❝t ❤♦✉rs ✐♥ ❚❛❜❧❡ ✶ ❛♥❞
❢r♦♠ t❤❡ ❛✈❡r❛❣❡ ❦✐ts ♣r♦❝❡ss✐♥❣ t✐♠❡ ❛t t❤❡ ❙❯✱ ✇❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ s✐t✉❛t✐♦♥s✿
✶✳ ❆ ❦✐t ❝♦❧❧❡❝t❡❞ ❛t ✶✶✿✸✵✱ ✶✸✿✵✵ ♦r ✶✹✿✸✵ ♦♥ ❞❛② ✭t✮ ❝❛♥ ❜❡ ✉s❡❞ ✐♥ t❤❡ ♠♦r♥✐♥❣ ♦♥ ❞❛②
✭t + 1✮ ✐❢ ✐t ✐s tr❡❛t❡❞ ❛s ❛ ♣r✐♦r✐t② ❛t t❤❡ ❙❯ ✭♣r✐♦r✐t② ❦✐t✱ ❝❛s❡ ✶✮✳ ■❢ ✐t ✐s ♥♦t tr❡❛t❡❞
❛s ❛ ♣r✐♦r✐t②✱ ✐t ✐s ❝♦♥s✐❞❡r❡❞ t❤❛t ✐t ❝❛♥♥♦t ❜❡ ✉s❡❞ ❜❡❢♦r❡ ✶✹✿✸✵ ♦♥ ❞❛② ✭t + 1✮✳
✷✳ ❆ ❦✐t ❝♦❧❧❡❝t❡❞ ❛t ✶✻✿✵✵ ♦♥ ❞❛② ✭t✮ ❝❛♥ ❜❡ ✉s❡❞ ✐♥ t❤❡ ♠♦r♥✐♥❣ ♦♥ ❞❛② ✭t + 1✮ ✐❢ ✐t ✐s
tr❡❛t❡❞ ✉r❣❡♥t❧② ❛t t❤❡ ❙❯ ✭✉r❣❡♥t ❦✐t✮✳ ■❢ ✐t ✐s ♥♦t tr❡❛t❡❞ ✉r❣❡♥t❧②✱ ✐t ❝❛♥ ❜❡ ✉s❡❞
❢r♦♠ ✶✹✿✸✵ ♦♥ ❞❛② ✭t + 1✮✳
✸✳ ❆ ❦✐t ❝♦❧❧❡❝t❡❞ ❛t ✶✼✿✸✵✱ ✶✽✿✸✵ ♦♥ ❞❛② ✭t✮ ♦r ✼✿✵✵ ♦♥ ❞❛② ✭t + 1✮ ❝❛♥ ❜❡ ✉s❡❞ ♦♥ ❞❛②
✭t + 1✮ ❢r♦♠ ✶✹✿✸✵ ✐❢ ✐t ✐s tr❡❛t❡❞ ❛s ❛ ♣r✐♦r✐t② ❛t t❤❡ ❙❯ ✭♣r✐♦r✐t② ❦✐t✱ ❝❛s❡ ✷✮✳ ■❢ ✐t ✐s
♥♦t tr❡❛t❡❞ ❛s ❛ ♣r✐♦r✐t②✱ ✐t ✇✐❧❧ ❜❡ ❛✈❛✐❧❛❜❧❡ ♦♥ ❞❛② ✭t + 1✮ ❢r♦♠ ✶✼✿✸✵✳
■♥ t❤❡ ❝✉rr❡♥t ❞❡❝✐s✐♦♥ ♣r♦❝❡ss ❛t t❤❡ ❈❍❯✱ ✐t ✐s ♦♥❧② ❝❤❡❝❦❡❞ ✇❤❡t❤❡r t❤❡ s✉r❣❡r✐❡s
s❝❤❡❞✉❧❡❞ ❡❛❝❤ ❞❛② ❛r❡ ❝♦♠♣❛t✐❜❧❡ ✇✐t❤ t❤❡ ♥✉♠❜❡r ♦❢ ❦✐ts ♦✇♥❡❞ ❜② t❤❡ ❜❧♦❝❦ ✭t❤❡ st❡r✲
✐❧✐③✐♥❣ ❝♦✉rs❡s ❢♦r ❦✐ts ❛r❡ ♥❡❣❧❡❝t❡❞ ❞✉r✐♥❣ t❤❡ ❛ss✐❣♥♠❡♥t ♦❢ s✉r❣❡r✐❡s t♦ t❤❡ s❤✐❢ts✮✳
❈♦♥s❡q✉❡♥t❧②✱ t❤❡ ♥✉♠❜❡r ♦❢ ✉r❣❡♥t ❛♥❞ ♣r✐♦r✐t② ❦✐ts ✐♥ ❙❯ t♦ ♣r♦❝❡ss r❡♠❛✐♥s s✉❜st❛♥t✐❛❧✳
❚❤❡ ✐♠♣❛❝t ♦♥ t❤❡ ❛❝t✐✈✐t② ❢♦r t❤❡ ❙❯ ✐s ✐♠♠❡❞✐❛t❡✳ ■♥ ♣❛rt✐❝✉❧❛r✱ ❡❛❝❤ ✉r❣❡♥t ❦✐t ✐♠♣❧✐❡s
t♦ st♦♣ ❛ ♠❛❝❤✐♥❡ ✐♥ ♦r❞❡r t♦ ♣r♦❝❡ss ✐t ✐♠♠❡❞✐❛t❡❧② ✭✐♥❞✉❝✐♥❣ ❛ ♥❡❡❞ t♦ r❡✲♣r♦❝❡ss r❡♠♦✈❡❞
❦✐ts ❛❢t❡r✇❛r❞✮✳ ❖♥ t❤❡ ♦t❤❡r ❤❛♥❞✱ ♣r✐♦r✐t② ❦✐ts ❛r❡ t❛❦❡♥ ✐♥t♦ ❛❝❝♦✉♥t ✉♣♦♥ t❤❡✐r ❛rr✐✈❛❧
❜② ♠♦✈✐♥❣ t❤❡♠ t♦ t❤❡ st❛rt ♦❢ t❤❡ q✉❡✉❡ t♦ ❜❡ tr❡❛t❡❞ ✜rst✱ ❛❣❛✐♥st t❤❡ ✜rst✲✐♥ ✜rst ♦✉t
❝❧❛ss✐❝❛❧ ♣♦❧✐❝② ♦❢ ❙❯✳ ❯❧t✐♠❛t❡❧②✱ ✐♥ r❛r❡ ❝❛s❡s✱ s♦♠❡ ❦✐ts ♠❛② ❡✈❡♥ ❜❡ r❡q✉❡st❡❞ ♦✉ts✐❞❡
t❤❡ ❞❡❧✐✈❡r② ❤♦✉rs ✭✈✐♦❧❛t❡❞ ❦✐ts✮✳ ■♥ t❤❛t ❝❛s❡✱ ✐t ✐s ♥❡❝❡ss❛r② t♦ r❡q✉❡st ❛ s♣❡❝✐❛❧ s❤✉tt❧❡✳
❚❤❡ ♣✉r♣♦s❡ ♦❢ t❤✐s ✇♦r❦ ✐s t♦ s❝❤❡❞✉❧❡ ❛❧❧ s✉r❣❡r✐❡s ❛t t❤❡ ❖❙❇ ✇❤✐❧❡ t❛❦✐♥❣ ✐♥t♦
❛❝❝♦✉♥t t❤❡ st❡r✐❧✐③✐♥❣ ♣r♦❝❡ss ✐♥ ♦r❞❡r t♦ r❡❞✉❝❡ t❤❡ ♣r❡ss✉r❡ ♦♥ t❤❡ ❙❯ st❛✛✳
■♥ t❡r♠s ♦❢ ♦❜❥❡❝t✐✈❡s✱ ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ❈❍❯✱ t❤❡ ✜rst ♣r✐♦r✐t② r❡♠❛✐♥s t♦ s❝❤❡❞✉❧❡ ❛❧❧
s✉r❣❡r✐❡s ✐♥ ♦r❞❡r t♦ ♠✐♥✐♠✐③❡ t❤❡ t♦t❛❧ ♦✈❡rt✐♠❡ ♦❢ t❤❡ st❛✛ ♠❡♠❜❡rs ♦❢ t❤❡ ❖❙❇✳ ❚❤❡
s❡❝♦♥❞ ♣r✐♦r✐t② ❝♦♥s✐sts ✐♥ ♠✐♥✐♠✐③✐♥❣ t❤❡ ♥✉♠❜❡r ♦❢ ✉s❡❞ ♦♣❡r❛t✐♥❣ r♦♦♠s✳ ❋✐♥❛❧❧②✱ t❤❡
t❤✐r❞ ♦❜❥❡❝t✐✈❡ ✐s t♦ ❦❡❡♣ t❤❡ ♥✉♠❜❡r ♦❢ ✉r❣❡♥t ❛♥❞ ♣r✐♦r✐t② ❦✐ts ❛s ❧♦✇ ❛s ♣♦ss✐❜❧❡✳
■♥ t❤❡ ♥❡①t s❡❝t✐♦♥✱ ✇❡ ❜r✐❡✢② s❦❡t❝❤ t❤❡ ❜❛s✐s ♦❢ ❛ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧ ❢♦r s♦❧✈✐♥❣ t❤✐s
✐♥t❡❣r❛t❡❞ ❖❙❇✲❙❯ ♣r♦❜❧❡♠✳
14

✸ ▼❛t❤❡♠❛t✐❝❛❧ ❢♦r♠✉❧❛t✐♦♥
■♥ ♦r❞❡r t♦ ♠♦❞❡❧ t❤✐s ♣r♦❜❧❡♠✱ ✇❡ ♣r♦♣♦s❡ ❛ ♠❛t❤❡♠❛t✐❝❛❧ ❢♦r♠✉❧❛t✐♦♥ ❜❛s❡❞ ♦♥ t❤❡
❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ t❤❡ ❞❛② ✐♥ ❢♦✉r ♣❡r✐♦❞s ✿ ♣❡r✐♦❞ ✶ ❢r♦♠ ✽✿✶✺ t♦ ✶✹✿✵✵✱ ♣❡r✐♦❞ ✷ ❢r♦♠ ✶✹✿✵✵
t♦ ✶✹✿✸✵✱ ♣❡r✐♦❞ ✸ ❢r♦♠ ✶✹✿✸✵ t♦ ✶✺✿✸✵ ❛♥❞ ♣❡r✐♦❞ ✹ ❢r♦♠ ✶✺✿✸✵ t♦ ✶✼✿✵✵✳ ❚❤❡ st❛rt✐♥❣ ❛♥❞
❡♥❞✐♥❣ ❤♦✉rs ♦❢ t❤❡s❡ ♣❡r✐♦❞s ❛r❡ ♦❜t❛✐♥❡❞ ❜② t❤❡ ❝♦♠❜✐♥❛t✐♦♥ ♦❢ t❤❡ ❝r✐t✐❝❛❧ ♣✐❝❦✉♣ ❛♥❞
❞❡❧✐✈❡r② ❤♦✉rs ❛t t❤❡ ❖❙❇✱ t❤❡ ♦♣❡♥✐♥❣ ❛♥❞ ❝❧♦s✐♥❣ ❤♦✉rs ♦❢ t❤❡ ♦♣❡r❛t✐♥❣ r♦♦♠s✱ ❛♥❞ t❤❡
❢❛❝t t❤❛t s✉r❣❡r✐❡s ♠✉st ❡♥❞ ✸✵ ♠✐♥✉t❡s ❜❡❢♦r❡ t❤❡ ❝♦❧❧❡❝t ♦❢ t❤❡✐r ♠❡❞✐❝❛❧ ❞❡✈✐❝❡s✳
❲❡ t❤❡♥ ✐♥tr♦❞✉❝❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❞❡❝✐s✐♦♥ ✈❛r✐❛❜❧❡s✿
witr ❜✐♥❛r② ✈❛r✐❛❜❧❡ ❡q✉❛❧ t♦ ✶ ✐❢ ♦♣❡r❛t✐♦♥ i ✐s s❝❤❡❞✉❧❡❞ ❛t ❞❛② t ✐♥ r♦♦♠ r
xbf
itr ❜✐♥❛r② ✈❛r✐❛❜❧❡ ❡q✉❛❧ t♦ ✶ ✐❢ s✉r❣❡r② i ❜❡❣✐♥s ❛t ♣❡r✐♦❞ b ❛♥❞ ✜♥✐s❤❡s ❛t f✱ ♦♥ ❞❛② t✱ ✐♥ r♦♦♠ r
εtr ✐♥t❡❣❡r ✈❛r✐❛❜❧❡ r❡♣r❡s❡♥t✐♥❣ t❤❡ t♦t❛❧ ♦✈❡rt✐♠❡ ✐♥ r♦♦♠ r ❛t ❞❛② t
Ltr ❜✐♥❛r② ✈❛r✐❛❜❧❡ ❡q✉❛❧ t♦ ✶ ✐❢ r♦♦♠ r ✐s ✉s❡❞ ❛t ❞❛② t
Etk ✐♥t❡❣❡r ✈❛r✐❛❜❧❡ r❡♣r❡s❡♥t✐♥❣ t❤❡ t♦t❛❧ ✉r❣❡♥t ❦✐ts ♦❢ t②♣❡ k ❛t ❞❛② t
Y 1
tk ✭Y 2
tk✮ ✐♥t❡❣❡r ✈❛r✐❛❜❧❡ r❡♣r❡s❡♥t✐♥❣ t❤❡ t♦t❛❧ ♣r✐♦r✐t② ❦✐ts ♦❢ ❝❛s❡ ✶ ✭r❡s♣✳ ❝❛s❡ ✷✮ ♦❢ t②♣❡ k ❛t ❞❛② t
■♥ ❛❞❞✐t✐♦♥ t♦ t❤❡ r❡❣✉❧❛r s❝❤❡❞✉❧✐♥❣ ❝♦♥str❛✐♥ts ✭❛❧❧ s✉r❣❡r✐❡s ❤❛✈❡ t♦ ❜❡ s❝❤❡❞✉❧❡❞ ✐♥ t❤❡
❤♦r✐③♦♥✱ ❡t❝✮✱ ♦✉r ♠♦❞❡❧ ✐♥❝❧✉❞❡s ♦t❤❡r ❝♦♥str❛✐♥ts s✉❝❤ ❛s✿
✕ t❤❡ ❡①♣r❡ss✐♦♥s ♦❢ ✉r❣❡♥t ✭✶✮ ❛♥❞ ♣r✐♦r✐t② ❦✐ts ✭❜♦t❤ ❝❛s❡s✮ ✭✷✮✲✭✸✮✿
PO
i=1
PR
r=1 qik.

PJ
f=2
Pf
b=1 xbf
itr +
P2
b=1
PJ
f=b xbf
i(t+1)r

− Qk ≤ Etk ∀t ∈ {1, .., T}, ∀k ∈ {1, .., K} ✭✶✮
PO
i=1
PR
r=1 qik.

PJ
f=1
Pf
b=1 xbf
itr +
P2
b=1
PJ
f=b xbf
i(t+1)r

− Qk − Etk ≤ Y 1
tk ∀t ∈ {1, . . . , T}, ∀k ∈ {1, . . . , K} ✭✷✮
PO
i=1
PR
r=1 qik.

PJ
f=2
Pf
b=1 xbf
itr +
PJ
b=1
PJ
f=b xbf
i(t+1)r

− Qk − Etk ≤ Y 2
tk ∀t ∈ {1, .., T}, ∀k ∈ {1, .., K} ✭✸✮
✇❤❡r❡ R ✐s t❤❡ t♦t❛❧ ♥✉♠❜❡r ♦❢ ❖❘s✱ qik ✐s t❤❡ t♦t❛❧ ♥✉♠❜❡r ♦❢ ❦✐ts ♦❢ t②♣❡ k ∈ {1..K}
t❤❛t s✉r❣❡r② i ∈ {1..O} r❡q✉✐r❡s✱ J r❡♣r❡s❡♥ts t❤❡ t♦t❛❧ ♥✉♠❜❡r ♦❢ ♣❡r✐♦❞s ✭✹✮✱ Qk ✐s
t❤❡ t♦t❛❧ ❛✈❛✐❧❛❜❧❡ q✉❛♥t✐t② ♦❢ ❦✐t k ❛♥❞ T ✐s t❤❡ t♦t❛❧ ♥✉♠❜❡r ♦❢ ❞❛②s ✐♥ t❤❡ ❤♦r✐③♦♥✳
✕ t❤❡ ❝♦♥tr♦❧ ♦❢ t❤❡ ✇♦r❦❧♦❛❞ ♦❢ s✉r❣❡r✐❡s ❢♦r ❡❛❝❤ ✐♥t❡r✈❛❧ ♦❢ t❤❡ ❞❛② ❛♥❞ ❡❛❝❤ r♦♦♠ ✭✹✮✿
O
X
i=1
γ
X
b=β
γ
X
f=b
pi · xbf
itr +
O
X
i=1
β−1
X
b=1
J
X
f=γ+1
dβγ
rt · xbf
itr ≤ dβγ
rt · Ltr + uγ · εtr
∀β ∈ {1, . . . , J}, ∀γ ∈ {β, . . . , J}, ∀t ∈ {1, . . . , T}, ∀r ∈ {1, . . . , R}
✭✹✮
✇❤❡r❡ pi ✐s t❤❡ ❞✉r❛t✐♦♥ ♦❢ s✉r❣❡r② i✱ dβγ
rt ✐s t❤❡ ❞✉r❛t✐♦♥ ❢r♦♠ ♣❡r✐♦❞ β t♦ γ ✐♥ r♦♦♠
r ♦♥ ❞❛② t✱ uγ ✐s ❛ ❜✐♥❛r② ♣❛r❛♠❡t❡r ❡q✉❛❧ t♦ ✶ ✐❢ γ = J✳
❚❤❡ ♠✉❧t✐♣❧❡ ♦❜❥❡❝t✐✈❡ ❢✉♥❝t✐♦♥s ❛r❡ t❛❦❡♥ ✐♥t♦ ❛❝❝♦✉♥t ❜② ✉s✐♥❣ ❛ ❧❡①✐❝♦❣r❛♣❤✐❝ ♠❡t❤♦❞
✭✺✮ ❛♥❞ ❛❢t❡r ❡❛❝❤ ♦❜❥❡❝t✐✈❡ ✐s s♦❧✈❡❞✱ ✐ts ✈❛❧✉❡ ✐s ❛❞❞❡❞ ❛s ❛♥ ✉♣♣❡r ❜♦✉♥❞ t♦ t❤❡ ♠♦❞❡❧✳
❋✐rst✱ f1 ♠✐♥✐♠✐③❡s t❤❡ t♦t❛❧ ♦✈❡r t✐♠❡ ❛♥❞ t❤❡♥ f2 ♠✐♥✐♠✐③❡s t❤❡ ♥✉♠❜❡r ♦❢ ✉s❡❞ r♦♦♠s
❛♥❞ ✜♥❛❧❧② f3 ♠✐♥✐♠✐③❡s t❤❡ t♦t❛❧ ♣❡♥❛❧t② ❝♦st ♦❢ ✉r❣❡♥t ✭cu✮ ❛♥❞ ♣r✐♦r✐t② ✭cp✮ ❦✐ts✳
▼✐♥✐♠✐③❡ Lex

f1 :
PT
t=1
PR
r=1 εtr ; f2 :
PT
t=1
PR
r=1 Ltr ; f3 :
PT
t=1
PK
k=1

cu · Etk + cp · (Y 1
tk + Y 2
tk)

✭✺✮
✹ ❊①♣❡r✐♠❡♥t❛❧ r❡s✉❧ts
❚♦ t❡st ❛♥❞ ✈❛❧✐❞❛t❡ ♦✉r ♠♦❞❡❧✱ ✇❡ ✉s❡❞ ❛ ✶✵ ✐♥st❛♥❝❡s ❜❡♥❝❤♠❛r❦ ♣r♦✈✐❞❡❞ ❜② t❤❡ ❈❍❯✳
❊❛❝❤ ✐♥st❛♥❝❡ ❝♦rr❡s♣♦♥❞s t♦ t❤❡ ❛❝t✐✈✐t② ♦❢ t❤❡ ❖❙❇ ❞✉r✐♥❣ ♦♥❡ ♠♦♥t❤✳ ❚❤❡ ♥✉♠❜❡r ♦❢
s✉r❣❡r✐❡s ✐♥ t❤❡s❡ ✐♥st❛♥❝❡s ✈❛r✐❡s ❢r♦♠ ✶✻✹ t♦ ✷✷✵✳ ❲❡ ✉s❡❞ ❈P❧❡① ✶✷✳✻✳✶ t♦ s♦❧✈❡ t❤❡
♠♦❞❡❧ ❛♥❞ ❛ t✐♠❡ ❧✐♠✐t ♦❢ ✸✻✵✵ s❡❝♦♥❞s ✇❛s s❡t ❢♦r ❡❛❝❤ ♦❢ t❤❡ t❤r❡❡ ♦❜❥❡❝t✐✈❡ ❢✉♥❝t✐♦♥s✳
❆s s❤♦✇♥ ✐♥ t❛❜❧❡ ✷✱ ✇❤✐❝❤ ❝♦♠♣❛r❡s t❤❡ s❝❤❡❞✉❧❡ ♦❢ t❤❡ ❈❍❯ ✇✐t❤ t❤❡ s❝❤❡❞✉❧❡ ❢r♦♠
t❤❡ ▼■▲P✱ ♦✉r ♠♦❞❡❧ ♠❛♥❛❣❡❞ t♦ ❡❧✐♠✐♥❛t❡ t❤❡ ❛♠❜✉❧❛t♦r② s✉r❣❡r✐❡s t❤❛t ✜♥✐s❤ ❛❢t❡r ✶✺✿✵✵
15

❚❛❜❧❡ ✷✳ ❘❡s✉❧ts ❝♦♠♣❛r✐s♦♥
■♥st❛♥❝❡
❈❍❯ s❝❤❡❞✉❧❡ ▼■▲P s❝❤❡❞✉❧❡
★❧❛t❡
❛♠❜s
★✈✐♦❧❛t❡❞
❦✐ts
♦✈❡r
t✐♠❡
★r♦♦♠s ★✉r❣❡♥t ★♣r✐♦r✐t✐❡s
★❧❛t❡
❛♠❜s
★✈✐♦❧❛t❡❞
❦✐ts
♦✈❡r
t✐♠❡
★r♦♦♠s ★✉r❣❡♥t ★♣r✐♦r✐t✐❡s
✶ ✶✷ ✵ ✶✵✹✸ ✺✾ ✾ ✹✵ ✵ ✵ ✵ ✺✽ ✵ ✷
✷ ✶✵ ✺ ✶✸✸✻ ✺✾ ✹ ✹✹ ✵ ✵ ✹✹✵ ✺✺ ✵ ✸
✸ ✼ ✶✵ ✼✻✻ ✹✽ ✸ ✺✻ ✵ ✵ ✶✸✶ ✹✻ ✵ ✶✷
✹ ✶✵ ✼ ✶✷✵✹ ✹✽ ✶✺ ✻✻ ✵ ✵ ✷✶✶ ✹✽ ✵ ✽
✺ ✶✺ ✺ ✶✶✻✺ ✺✾ ✶✹ ✺✽ ✵ ✵ ✶✹✷ ✺✽ ✵ ✾
✻ ✹ ✸ ✶✶✸✽ ✺✷ ✵ ✻✺ ✵ ✵ ✺✺ ✺✷ ✵ ✾
✼ ✺ ✷✻ ✶✹✼✹ ✺✾ ✶ ✾✺ ✵ ✵ ✹✻✶ ✺✾ ✵ ✶✶
✽ ✾ ✵ ✶✵✶✶ ✹✾ ✷ ✺✶ ✵ ✵ ✷✸✹ ✹✽ ✵ ✵
✾ ✻ ✶✷ ✺✽✵ ✹✻ ✽ ✹✺ ✵ ✵ ✽✻ ✹✸ ✵ ✵
✶✵ ✻ ✸ ✼✶✹ ✻✵ ✺ ✸✶ ✵ ✵ ✶✸✸ ✺✸ ✵ ✵
❆✈❡r❛❣❡ ✽✳✹ ✼✳✶ ✶✵✹✸✳✶ ✺✸✳✾ ✻✳✶ ✺✺✳✶ ✵ ✵ ✶✽✾✳✸ ✺✷ ✵ ✺✳✹
❛♥❞ t❤❡ ♥❡❡❞ ♦❢ ❦✐ts t❤❛t ❝❛♥♥♦t ❜❡ ❞❡❧✐✈❡r❡❞ ✐♥ t❤❡ ♥♦r♠❛❧ ❤♦✉rs ✭✈✐♦❧❛t❡❞ ❦✐ts✮✳ ■♥
❛❞❞✐t✐♦♥✱ ♦✉r ♠♦❞❡❧ ❞❡❝r❡❛s❡❞ t❤❡ ♦✈❡rt✐♠❡ ❜② ❛r♦✉♥❞ ✽✶✳✽✺✪ ✭❢r♦♠ ✶✹❤✶✹ t♦ ✸❤✵✾✮ ♣❡r
♠♦♥t❤ ❛♥❞ ✐t ❝❧♦s❡❞ ❛r♦✉♥❞ ✷ r♦♦♠s ✐♥ ❛✈❡r❛❣❡ ❡❛❝❤ ♠♦♥t❤✳ ❋✐♥❛❧❧②✱ ♦✉r ♠♦❞❡❧ ✇❛s ❛❜❧❡
t♦ ❞❡❝r❡❛s❡ t❤❡ ♥✉♠❜❡r ♦❢ ✉r❣❡♥t ❛♥❞ ♣r✐♦r✐t② ❦✐ts ❜② ❛r♦✉♥❞ ✾✶✳✶✼✪ ✭❢r♦♠ ✻✶✳✷ t♦ ✺✳✹✮
♣❡r ♠♦♥t❤✳
✺ ❈♦♥❝❧✉s✐♦♥s ❛♥❞ ♣❡rs♣❡❝t✐✈❡s
❚❤✐s ✇♦r❦ ❢♦❝✉s❡s ♦♥ ❛ r❡❛❧ s✉r❣✐❝❛❧ ❝❛s❡ s❝❤❡❞✉❧✐♥❣ ✐♥❝❧✉❞✐♥❣ st❡r✐❧✐③❛t✐♦♥ ❛❝t✐✈✐t②
❝♦♥str❛✐♥ts✱ ❛♥❞ t❤r❡❡ ♦❜❥❡❝t✐✈❡ ❢✉♥❝t✐♦♥s✳ ❲❡ ♣r♦♣♦s❡ ❛♥ ▼■▲P ❢♦r♠✉❧❛t✐♦♥ ✇❤✐❝❤ ✐s
s♦❧✈❡❞ ✐♥ ❛ ❧❡①✐❝♦❣r❛♣❤✐❝ ❢❛s❤✐♦♥✳ ❖✉r s♦❧✉t✐♦♥s ♣r♦✈✐❞❡ ❝♦♠♣❡t✐t✐✈❡ r❡s✉❧ts ✐♥ t❡r♠s ♦❢ ✉s❡❞
r♦♦♠s✱ ❛♥❞ s✐❣♥✐✜❝❛♥t❧② ✐♠♣r♦✈❡ t❤♦s❡ ♦♣❡r❛t✐♦♥❛❧❧② ✐♠♣❧❡♠❡♥t❡❞ ✐♥ t❡r♠s ♦❢ ♦✈❡rt✐♠❡ ❛♥❞
✉r❣❡♥t ❛♥❞ ♣r✐♦r✐t② ❦✐ts ❛t t❤❡ ❙❯✳ ❙t✐❧❧ ✐♥ ❧✐♥❡ ✇✐t❤ t❤❡ ♥❡❡❞s ♦❢ t❤❡ ❈❍❯ ❆♥❣❡rs✱ t❤❡ ♥❡①t
st❡♣ ✐s t♦ ❛❞❞r❡ss t❤❡ ♦♥❧✐♥❡ ✈❡rs✐♦♥ ♦❢ t❤❡ ♣r♦❜❧❡♠✳
❆❝❦♥♦✇❧❡❞❣❡♠❡♥ts
❚❤❡ ❛✉t❤♦rs ✇♦✉❧❞ ❧✐❦❡ t♦ t❤❛♥❦ ▲✳ ❍✉❜❡rt✱ ❆✳❱✳ ▲❡❜❡❧❧❡ ❛♥❞ ❆✳ ❘♦❜❡❧❡t ❢r♦♠ t❤❡ ❈❍❯
❢♦r s✉❜♠✐tt✐♥❣ ✉s t❤✐s ❝❤❛❧❧❡♥❣✐♥❣ ♣r♦❜❧❡♠ ❛♥❞ ❦✐♥❞❧② ♣r♦✈✐❞✐♥❣ ✉s ✇✐t❤ r❡❛❧ ✐♥st❛♥❝❡s✳ ❚❤✐s
r❡s❡❛r❝❤ ✐s ♣❛rt✐❛❧❧② ❢♦✉♥❞❡❞ ❜② ❆♥❣❡rs ▲♦✐r❡ ▼❡tr♦♣♦❧❡ ✭❆▲▼✮ ❛♥❞ ■▼❚ ❆t❧❛♥t✐q✉❡✳
❘❡❢❡r❡♥❝❡s
❇❡r♦✉❧❡✱ ❇✳✱ ●r✉♥❞❡r✱ ❖✳✱ ❇❛r❛❦❛t✱ ❖✳✱ ❆✉❥♦✉❧❛t✱ ❖✳✱ ▲✉st✐❣✱ ❍✳✱ ✷✵✶✻✱ ❖♣❡r❛t✐♥❣ r♦♦♠ s❝❤❡❞✉❧✐♥❣
✐♥❝❧✉❞✐♥❣ ♠❡❞✐❝❛❧ ❞❡✈✐❝❡s st❡r✐❧✐③❛t✐♦♥✿ t♦✇❛r❞s ❛ tr❛♥s✈❡rs❡ ❧♦❣✐st✐❝✳ ■❋❆❈✲P❛♣❡rs❖♥▲✐♥❡✱
✹✾✭✶✷✮✱ ♣♣✳✶✶✹✻✲✶✶✺✶✳
❈❛r❞♦❡♥✱ ❇✳✱ ❉❡♠❡✉❧❡♠❡❡st❡r✱ ❊✳✱ ❇❡❧✐☎
♥✱ ❏✳✱ ✷✵✵✾✱ ❙❡q✉❡♥❝✐♥❣ s✉r❣✐❝❛❧ ❝❛s❡s ✐♥ ❛ ❞❛②✲❝❛r❡ ❡♥✲
✈✐r♦♥♠❡♥t✿ ❛♥ ❡①❛❝t ❜r❛♥❝❤✲❛♥❞✲♣r✐❝❡ ❛♣♣r♦❛❝❤✳ ❈♦♠♣✉t❡rs ✫ ❖♣❡r❛t✐♦♥s ❘❡s❡❛r❝❤✱ ✸✻✭✾✮✱
♣♣✳✷✻✻✵✲✷✻✻✾✳
❈❛r❞♦❡♥✱ ❇✳✱ ❉❡♠❡✉❧❡♠❡❡st❡r✱ ❊✳✱ ❇❡❧✐☎
♥✱ ❏✳✱ ✷✵✶✵✱ ❖♣❡r❛t✐♥❣ r♦♦♠ ♣❧❛♥♥✐♥❣ ❛♥❞ s❝❤❡❞✉❧✐♥❣✿ ❆
❧✐t❡r❛t✉r❡ r❡✈✐❡✇✳ ❊✉r♦♣❡❛♥ ❥♦✉r♥❛❧ ♦❢ ♦♣❡r❛t✐♦♥❛❧ r❡s❡❛r❝❤✱ ✷✵✶✭✸✮✱ ♣♣✳✾✷✶✲✾✸✷✳
●✉❡rr✐❡r♦✱ ❋✳✱ ●✉✐❞♦✱ ❘✳✱ ✷✵✶✶✱ ❖♣❡r❛t✐♦♥❛❧ r❡s❡❛r❝❤ ✐♥ t❤❡ ♠❛♥❛❣❡♠❡♥t ♦❢ t❤❡ ♦♣❡r❛t✐♥❣ t❤❡❛tr❡✿
❛ s✉r✈❡②✳ ❍❡❛❧t❤ ❝❛r❡ ♠❛♥❛❣❡♠❡♥t s❝✐❡♥❝❡✱ ✶✹✭✶✮✱ ♣♣✳✽✾✲✶✶✹✳
❏❛❝❦s♦♥✱ ❘✳ ▲✳✱ ✷✵✵✷✱ ✏❚❤❡ ❜✉s✐♥❡ss ♦❢ s✉r❣❡r②✳ ▼❛♥❛❣✐♥❣ t❤❡ ❖❘ ❛s ❛ ♣r♦✜t ❝❡♥t❡r r❡q✉✐r❡s ♠♦r❡
t❤❛♥ ❥✉st ■❚✳ ■t r❡q✉✐r❡s ❛ ♣r♦✜t✲♠❛❦✐♥❣ ♠✐♥❞s❡t✱ t♦♦✧✱ ❍❡❛❧t❤ ♠❛♥❛❣❡♠❡♥t t❡❝❤♥♦❧♦❣②✱ ✷✸✭✼✮✱
✷✵✳
▼❛❝❛r✐♦✱ ❆✳✱ ❱✐t❡③✱ ❚✳✱ ❉✉♥♥✱ ❇✳✱ ▼❝❉♦♥❛❧❞✱ ❚✳✱ ✶✾✾✺✱ ✧❲❤❡r❡ ❛r❡ t❤❡ ❝♦sts ✐♥ ♣❡r✐♦♣❡r❛t✐✈❡
❝❛r❡❄✿ ❆♥❛❧②s✐s ♦❢ ❤♦s♣✐t❛❧ ❝♦sts ❛♥❞ ❝❤❛r❣❡s ❢♦r ✐♥♣❛t✐❡♥t s✉r❣✐❝❛❧ ❝❛r❡✧✱ ❆♥❡st❤❡s✐♦❧♦❣②✿ ❚❤❡
❏♦✉r♥❛❧ ♦❢ t❤❡ ❆♠❡r✐❝❛♥ ❙♦❝✐❡t② ♦❢ ❆♥❡st❤❡s✐♦❧♦❣✐sts✱ ✽✸✭✻✮✱ ♣♣✳✶✶✸✽✲✶✶✹✹✳
16

Modeling and solving a two-stage assembly flowshop
scheduling problem with buffers
Andrés C. and Maheut J.
Universitat Politècnica de València, Spain
candres@omp.upv.es, juma2@upv.es
Keywords: Assembly flowshop, limited buffer, scheduling.
1 Abstract
The two-stage assembly scheduling problem is a well-known problem from the literature
with a lot of practical applications. It consists of a system with two stages where a set of
n jobs must be processed in a given sequence. First stage is composed of m machines, each
one produces a component to be assembled in one single machine at the second stage.
One typical assumption in all the previous literature is the absence of buffer limitations
between both stages but this is too unrealistic from a practical point of view. Under Lean
Manufacturing paradigm, it is interesting to minimize the time that all jobs wait in the
buffers and total machine blocking time. This study presents a mathematical model to two
stage assembly flowshop scheduling problem with finite storage conditions together with
a complete enumeration study for small instances. The aim of the research is to show the
effect of buffer size over the total blocking time plus inventory time.
2 Introduction
Since the seminal paper of Johnson (1954) an extensive amount of papers has been
published related with scheduling problem. Most of them with the unrealistic assumption
of no buffer limitations between machines This is too unrealistic under Lean Manufacturing
paradigm and it has been less studied in the literature. The first paper about limited
buffer scheduling was (Dutta and Cunningham 1975) who studied a flowshop problem with
capacitated buffers using dynamic programming. Later, (Papadimitriou and Kanellakis
1980) probed this problem is NP-hard in the strong sense and developed a relation between
a heuristic developed for the problem and buffer size. Due to the complexity of the problem,
several authors developed heuristic approaches. First one was the paper of Leinsten (1990)
what showed a general framework for scheduling problem with capacitated buffer (limited
buffer size, blocking and no wait problems) and studied several heuristic rules, concluding
the high performance of NEH rule of Nawaz et al. (1983). Later Nowicki (1999) developed
a Tabu Search approach using some job properties from graph representation to accelerate
the local search by eliminating sets of solutions that do not improve the current solution.
Other approaches were Tabu Search of Brucker et al. (2003), Genetic Algorithm of Wang
et al. (2006), Particle Swarm Algorithm of Liu et al. (2008), immune system algorithm of
Hsieh et al. (2009) or the Ant Colony Algorithm of Rossi and Lanzetta (2013).
Simultaneously, other kind of scheduling problems called assembly flowshop has at-
tracted the interest of the researchers. Regarding makespan minimization, Lee et al. (1993)
and Potts et al. (1995) probed this problem is NP-hard even for two and M machines at
the first stage. The best approach up to now to solve the problem with makespan was
proposed by Hariri and Potts, (1997) using Branch and Bound techniques. Regarding to-
tal completion time minimization in assembly flow shops, Framinan and Perez-Gonzalez
17

(2017) proposed a constructive heuristic and a metaheuristic that outperform all the pre-
vious heuristics.
However, there are no previous research about limited storage two stage assembly flow-
shop with buffering and blocking time as objective function. So our aim is to study the
effect of buffer size between both stages. First, a complete enumeration study will be pre-
sented and used to investigate the effect of buffer size changes over small size problems
based on Taillard (1993) instances. Later, a mathematical model will be presented and used
to optimize the sequence for medium instances. The results let us confirm the interest of
this kind of problem and the necessity to develop procedures for study realistic instances.
3 Complete enumeration study
In order to show the effect of buffer size over makespan for small instances (up to
nine jobs), a test based in complete enumeration has been carried out considering that
there is an identical buffer of size b between each component machine and the assembly
one. We used some instances from Taillard’s set where first machine was used to represent
processing time in assembly machine and the rest represents processing time in component
manufacturing machines. All the sequences for the same instance have been computed for
buffer size between 0 and 4.
The following figures represent the results for all the 9! sequences from Ta004 instance
(in our case a shop with one assembly machine plus four component machines) and the
empirical distribution of blocking plus buffering times depending of buffer size. It can be
seen from both figures that there are difference between each solution space depending on
buffer size.
0
10000
20000
30000
40000
50000
60000
70000
80000
90000
900 1400 1900 2400 2900 3400 3900 4400 4900
Absolute
frequency
Total buffering plus blockingtime
b=0 b=1 b=2 b=3
Fig. 1. Distribution for N = 9 jobs and different buffer size.
It can be seen that maximum number of solutions decreases for each objective function
value when buffer size increases. On the other hand, when buffer size increases, distribution
width increases too.
Figure 2, shows that when buffer size increases is more difficult to find a good solution
randomly. For example, for zero buffer size all solutions are under 50% from optimal value
18

0
0,2
0,4
0,6
0,8
1
1,2
0 0,5 1 1,5 2 2,5 3
Cumulative
probability
Relative distance frombestsolution
b=0 b=1 b=2 b=3
Fig. 2. Empirical cumulative distribution for N = 8 jobs.
but for buffer size equal to 3, there are only 2.3% of solutions. That support the idea that it
is “easier” to find a good solution randomly in blocking assembly shops for these objective
function.
4 Mathematical model
A second step to study the problem for medium size instances is to develop a mathe-
matical model. It is necessary the following notation to formalize the model.
Let:
i, index for jobs or components i = 1, . . . , n
j, index for sequence positions j = 1, . . . , n
k, index for machines and buffers k = 1, . . . , m
MCk, component manufacturing machines k = 1, . . . , m. Each machine has a capaci-
tated buffer BCk, k = 1, . . . , m.
pcj,k is the processing time of job in position j in machine k
pj is the processing time of job in position j at assembly machine
bk is the capacity of buffer k
SCjk is the starting time of component to be assembled at position j in machine k
Sj is the starting time of job to be assembled at position j in the assembly machine
xij =
{
1, if component/job j is sequenced on position j
0, otherwise.
Thus, mathematical model can be stated as follows:
min z =
n
∑
j=1
m
∑
k=1
(Sj−b − SCjk − pcjk) +
n
∑
j=1
m
∑
k=2
(Sj − Sj−b)
19

s.t.
SC1k = 0, k = 1, . . . , m
S1 ≥ SC1k +
n
∑
i=1
xi1 pcik, k = 1, . . . , m
SCjk ≥ SCj−1,k +
n
∑
i=1
xi,j−1 pcik, j = 2, . . . , n, k = 1, . . . , m
SCjk ≥ Sj−bck−1,k, j 2, . . . , bck + 1, k = 1, . . . , m
Sj ≥ Sj−1 +
n
∑
i=1
xi,j−1 pi, j = 2, . . . , n
Sj ≥ SCjk +
n
∑
i=1
xij pcik, j = 2, . . . , n, k = 1, . . . , m
n
∑
i=1
xij = 1, j = 1, . . . , n
n
∑
j=1
xij = 1, i = 1, . . . , n
xij ∈ {0, 1}, i, j = 1, . . . , n
Sj, SCjk ≥ 0, j = 1, . . . , n, k = 1, . . . , m.
First term on the objective function represents total blocking time while the second
one computes total buffering time for a given sequence. Set of constraints represents the
relations between starting time of every operations under finite storage assumption.
Mathematical model was tested with some Taillard’s instances (Ta001 to Ta020 and
Ta031 to Ta050) adapting them to the assembly flowshop problem. The results show that
it is possible to solve optimally instances until 20 jobs and 4 component machines. More
results about this study will be presented at the conference.
5 Conclusions and future work
In this paper we presented a study about two stage assembly flowshop scheduling prob-
lem with limited buffers of size b. Instead of classical objective functions like makespan or
total flowtime, we study a composed function of total buffering time plus total blocking
time. A complete enumeration study shows that solution space shape changes with the
size of the buffers and it seems a promising field for researchers due to the relation of ob-
jective function with the improvement in production systems under Lean Manufacturing
paradigm.
Moreover, a new mathematical model is described and some results are presented. Our
aim is to develop competitive heuristic procedures to solve realistic instances and get more
insights about the relationship between buffer size and assembly flowshop performance
under finite storage conditions.
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21

❆ ♥❡✇ ♣♦❧②♥♦♠✐❛❧✲t✐♠❡ ❛❧❣♦r✐t❤♠ ❢♦r ❝❛❧❝✉❧❛t✐♥❣ ✉♣♣❡r
❜♦✉♥❞s ♦♥ r❡s♦✉r❝❡ ✉s❛❣❡ ❢♦r ❘❈P❙P ♣r♦❜❧❡♠
❉♠✐tr② ❆r❦❤✐♣♦✈1,2
✱ ❖❧❣❛ ❇❛tt❛ï❛1
❛♥❞ ❆❧❡①❛♥❞❡r ▲❛③❛r❡✈2,3,4,5
1
❉❡♣❛rt♠❡♥t ♦❢ ❈♦♠♣❧❡① ❙②st❡♠s ❊♥❣✐♥❡❡r✐♥❣✱ ■❙❆❊✲❙❯P❆❊❘❖✱ ❯♥✐✈❡rs✐té ❞❡ ❚♦✉❧♦✉s❡✱
❚♦✉❧♦✉s❡✱ ❋r❛♥❝❡❀
2
❱✳❆✳ ❚r❛♣❡③♥✐❦♦✈ ■♥st✐t✉t❡ ♦❢ ❈♦♥tr♦❧ ❙❝✐❡♥❝❡s ♦❢ ❘✉ss✐❛♥ ❆❝❛❞❡♠② ♦❢ ❙❝✐❡♥❝❡s✱ ▼♦s❝♦✇✱
❘✉ss✐❛♥ ❋❡❞❡r❛t✐♦♥❀
3
▲♦♠♦♥♦s♦✈ ▼♦s❝♦✇ ❙t❛t❡ ❯♥✐✈❡rs✐t②✱ ▼♦s❝♦✇✱ ❘✉ss✐❛♥ ❋❡❞❡r❛t✐♦♥❀
4
▼♦s❝♦✇ ■♥st✐t✉t❡ ♦❢ P❤②s✐❝s ❛♥❞ ❚❡❝❤♥♦❧♦❣②✱ ❉♦❧❣♦♣r✉❞♥②✱ ❘✉ss✐❛♥ ❋❡❞❡r❛t✐♦♥❀
5
■♥t❡r♥❛t✐♦♥❛❧ ▲❛❜♦r❛t♦r② ♦❢ ❉❡❝✐s✐♦♥ ❈❤♦✐❝❡ ❛♥❞ ❆♥❛❧②s✐s✱ ◆❛t✐♦♥❛❧ ❘❡s❡❛r❝❤ ❯♥✐✈❡rs✐t②
❍✐❣❤❡r ❙❝❤♦♦❧ ♦❢ ❊❝♦♥♦♠✐❝s✱ ▼♦s❝♦✇✱ ❘✉ss✐❛♥ ❋❡❞❡r❛t✐♦♥✳
❑❡②✇♦r❞s✿ ♣r♦❥❡❝t ♣❧❛♥♥✐♥❣✱ ♣♦❧②♥♦♠✐❛❧ ❛❧❣♦r✐t❤♠s✱ s❝❤❡❞✉❧✐♥❣✱ ❝♦♥str❛✐♥t ♣r♦❣r❛♠♠✐♥❣✱
♣r♦♣❛❣❛t♦r✳
✶ ■♥tr♦❞✉❝t✐♦♥
❚❤❡ ❘❡s♦✉r❝❡✲❈♦♥str❛✐♥❡❞ Pr♦❥❡❝t ❙❝❤❡❞✉❧✐♥❣ Pr♦❜❧❡♠ ✭❘❈P❙P✮ ✐s ❝♦♥s✐❞❡r❡❞✳ ❚❤✐s
♣r♦❜❧❡♠ ✐s ◆P✲❤❛r❞ ✐♥ str♦♥❣ s❡♥s❡ ✭●❛r❡② ❛♥❞ ❏♦❤♥s♦♥ ✶✾✼✺✮✳ ■♥ t❤✐s ♣❛♣❡r✱ ❛ ♥❡✇
♣♦❧②♥♦♠✐❛❧✲t✐♠❡ ❛♣♣r♦❛❝❤ ✐s ❞❡✈❡❧♦♣❡❞ t♦ ✜♥❞ ❛♥ ✉♣♣❡r ❜♦✉♥❞ ♦♥ r❡s♦✉r❝❡ ❝♦♥s✉♠♣t✐♦♥✳
❚❤✐s ❜♦✉♥❞ ❝❛♥ ❜❡ ❛❧s♦ ✉s❡❞ t♦ ❝❛❧❝✉❧❛t❡ ❛ ❧♦✇❡r ❜♦✉♥❞ ❢♦r ♠❛❦❡s♣❛♥✳ ❋✉rt❤❡r♠♦r❡✱ t❤❡
♣r♦❝❡❞✉r❡ ❤❡❧♣s t♦ ✐♥❝r❡❛s❡ t❤❡ ❡✣❝✐❡♥❝② ♦❢ ❡①✐st✐♥❣ ♣r♦♣❛❣❛t♦rs ❛♥❞ t♦ ✐♠♣r♦✈❡ ❝♦♥str❛✐♥t
♣r♦❣r❛♠♠✐♥❣ ♠♦❞❡❧ ♣❡r❢♦r♠❛♥❝❡s ❜② t✐❣❤t❡♥✐♥❣ ❞❡❝✐s✐♦♥ ✈❛r✐❛❜❧❡s ❞♦♠❛✐♥s✳
❲❡ ❝♦♥s✐❞❡r t❤❡ ❢♦❧❧♦✇✐♥❣ ❢♦r♠✉❧❛t✐♦♥ ♦❢ ❘❈P❙P ✇✐t❤ ❝♦♥t✐♥✉♦✉s t✐♠❡✳ ❚❤❡r❡ ✐s ❛ s❡t
♦❢ t❛s❦s N ❛♥❞ ❛ s❡t ♦❢ r❡♥❡✇❛❜❧❡ r❡s♦✉r❝❡s R✳ ❚❤❡ ❝❛♣❛❝✐t② ♦❢ r❡s♦✉r❝❡ X ∈ R ✐s ❞❡✜♥❡❞
❜② ♥♦♥✲♥❡❣❛t✐✈❡ ♣✐❡❝❡✇✐s❡ ❝♦♥st❛♥t ❢✉♥❝t✐♦♥ cX (t) ✇❤✐❝❤ ❝♦♥s✐sts ♦❢ ❝♦♥st❛♥t ❢✉♥❝t✐♦♥s
c1
X (t) = c1
, ∀t ∈ [0, t1); c2
X (t) = c2
, ∀t ∈ [t1, t2); . . . , cm
X (t) = cm
, ∀t ∈ [tm−1, T]✳ ❋♦r ❛♥②
t❛s❦ j ∈ N✱ t❤❡ ❢♦❧❧♦✇✐♥❣ ♣❛r❛♠❡t❡rs ❛r❡ ❣✐✈❡♥✿ pj ✕ ♣r♦❝❡ss✐♥❣ t✐♠❡ ❛♥❞ ajX ✕ r❡q✉✐r❡❞
❛♠♦✉♥t ♦❢ r❡s♦✉r❝❡ X ∈ R ❢♦r t❛s❦ j✳
Pr❡❝❡❞❡♥❝❡ r❡❧❛t✐♦♥s ❜❡t✇❡❡♥ t❛s❦s ❛r❡ ❣✐✈❡♥ ❜② ❛ ❞✐r❡❝t❡❞ ❛❝②❝❧✐❝ ❣r❛♣❤ G = (N, E)✳
■❢ ❛♥ ❡❞❣❡ eji ∈ E ❡①✐sts✱ ✐t ♠❡❛♥s t❤❛t t❛s❦ j ♠✉st ❜❡ ✜♥✐s❤❡❞ ❜❡❢♦r❡ t❤❡ st❛rt✐♥❣ t✐♠❡ ♦❢
t❛s❦ i ✭j → i✮✳
❚✐♠❡ ❤♦r✐③♦♥ T ✐s ❞❡✜♥❡❞ ❛♥❞ ❢♦r ❡❛❝❤ j ∈ N t❤❡ ♣❛r❛♠❡t❡rs ♦❢ t❛s❦ ♣r♦❝❡ss✐♥❣ ❞♦♠❛✐♥
❛r❡ ❣✐✈❡♥✿ rj ✕ r❡❧❡❛s❡ t✐♠❡✱ t❤❡ ❡❛r❧✐❡st t✐♠❡ ❢r♦♠ ✇❤✐❝❤ t❛s❦ j ❝❛♥ ❜❡ st❛rt❡❞ ❛♥❞ Dj ✕
❞❡❛❞❧✐♥❡✱ t❤❡ ❧❛t❡st t✐♠❡ ❢♦r ✜♥✐s❤✐♥❣ t❛s❦ j✳ ■♥ ❝❛s❡ ✐❢ t❤✐s ♣❛r❛♠❡t❡rs ❛r❡ ♥♦t ❣✐✈❡♥ ✇❡
❝❛♥ s❡t rj = 0✱ Dj = T ❢♦r ❡❛❝❤ j ∈ N✱ ❛♥❞ t❤❡♥ ✉s❡ s♦♠❡ ♣♦❧②♥♦♠✐❛❧✲t✐♠❡ ♣r♦♣❛❣❛t♦rs
t♦ t✐❣❤t❡♥ ❞♦♠❛✐♥s [rj, Dj] ♦❢ t❛s❦ ♣r♦❝❡ss✐♥❣✳
❲❡ ❝♦♥s✐❞❡r t❤❡ ❞❡❝✐s✐♦♥ ✈❡rs✐♦♥ ♦❢ ❘❈P❙P ✇✐t❤♦✉t ♦❜❥❡❝t✐✈❡ ❢✉♥❝t✐♦♥✱ ❜✉t ✇❡ ❤❛✈❡ t♦
✜♥❞ ❛ s❝❤❡❞✉❧❡ ✇❤✐❝❤ s❛t✐s✜❡s ♣r❡❝❡❞❡♥❝❡ r❡❧❛t✐♦♥s ❛♥❞ r❡s♦✉r❝❡ ❝♦♥str❛✐♥ts ✇✐t❤ ♠❛❦❡s♣❛♥
✈❛❧✉❡ ❧♦✇❡r t❤❛♥ T✳
❚❤❡ ♣❛♣❡r ✐s ♦r❣❛♥✐s❡❞ ❛s ❢♦❧❧♦✇s✳ ■♥ s❡❝t✐♦♥ ✷ t❤❡ ♦✈❡r✈✐❡✇ ♦❢ r❡s♦✉r❝❡❞✲❜❛s❡❞ ♣r♦♣❛✲
❣❛t♦rs ✐s ♣r❡s❡♥t❡❞✳ ■♥ t❤❡ s❡❝t✐♦♥ ✸ ✇❡ ❣✐✈❡ t❤❡ ♠❛✐♥ ✐❞❡❛ ♦❢ ♦✉r ❛♣♣r♦❛❝❤ ❛♥❞ t❤❡♦r❡♠s ♦♥
✇❤✐❝❤ ✐t ✐s ❜❛s❡❞ ♦♥✳ ❚❤❡♥✱ ✇❡ ❞✐s❝✉ss s♦♠❡ ❣❡♥❡r❛❧✐③❛t✐♦♥s ♦❢ ❘❈P❙P ♣r♦❜❧❡♠ ❢♦r ✇❤✐❝❤
♣r❡s❡♥t❡❞ ❛♣♣r♦❛❝❤ ✐s ❛♣♣❧✐❝❛❜❧❡ ❛♥❞ ♠❛❦❡ s♦♠❡ ❝♦♥❝❧✉s✐♦♥ r❡♠❛r❦s ✐♥ s❡❝t✐♦♥ ✹✳
✷ ❙t❛t❡ ♦❢ t❤❡ ❛rt
❖✉r r❡s❡❛r❝❤ ✐s ❢♦❝✉s❡❞ ♦♥ ✐♠♣r♦✈✐♥❣ ❝♦♥str❛✐♥t ♣r♦❣r❛♠♠✐♥❣ ♠♦❞❡❧ ❛♥❞ ♣r♦♣❛❣❛t♦rs✱
✇❤✐❝❤ ✉s❡ r❡s♦✉r❝❡ ❝♦♥str❛✐♥ts t♦ ♠❛❦❡ ❞❡❝✐s✐♦♥ ✈❛r✐❛❜❧❡ ✐♥t❡r✈❛❧s t✐❣❤t❡r✳ ❚❤❡r❡ ❛r❡ ❛ ❧♦t ♦❢
22

♣r♦♣❛❣❛t♦rs✱ ❜❛s❡❞ ♦♥ r❡s♦✉r❝❡ ❝♦♥s✉♠♣t✐♦♥ ❝♦♥str❛✐♥ts✳ ✭▲❛❤r✐❝❤✐ ✶✾✽✷✮ ✜rst❧② ❝❛❧❝✉❧❛t❡❞
✓r❡s♦✉r❝❡ ❝♦♠♣✉❧s♦r② ♣❛rt✔✱ ✭▲❡ P❛♣❡ ✶✾✽✽✮ ❝r❡❛t❡❞ ✓t✐♠❡ t❛❜❧❡s✔✳ ❚❤❡♥ ✭❋♦① ✶✾✾✵✮ ✐♥tr♦✲
❞✉❝❡❞ t❤❡ t❡r♠ ✓r❡s♦✉r❝❡ ♣r♦✜❧❡✔ ❛♥❞ ✭❈❛s❡❛✉ ❛♥❞ ▲❛❜✉rt❤❡ ✶✾✾✻✮ ♣r❡s❡♥t❡❞ ✓r❡s♦✉r❝❡
❤✐st♦❣r❛♠✔✳ ❙✇❡❡♣ ❛❧❣♦r✐t❤♠ t♦ ❝❛❧❝✉❧❛t❡ r❡s♦✉r❝❡ ♣r♦✜ ❧❡ ✇❛s ♣r❡s❡♥t❡❞ ❜② ✭❇❡❧❞✐❝❡❛♥✉
❛♥❞ ❈❛r❧ss♦♥ ✷✵✵✶✮✳ ❙❡✈❡r❛❧ ❡✣❝✐❡♥t ♣r♦♣❛❣❛t♦rs ❜❛s❡❞ ♦♥ t✐♠❡✲t❛❜❧✐♥❣ ❛❧❣♦r✐t❤♠s ✇❡r❡
❞❡✈❡❧♦♣❡❞ ✐♥ ❧✐t❡r❛t✉r❡ ✭❙❝❤✉tt ❡t✳ ❛❧✳ ✷✵✶✶✮✱ ✭❖✉❡❧❧❡t ❛♥❞ ◗✉✐♠♣❡r ✷✵✶✸✮✳ ❖t❤❡r ♣r♦♣❛❣❛✲
t♦rs ✇❡r❡ ❞✐s❝✉ss❡❞ ✐♥ ✭❇❛♣t✐st❡ ❡t✳ ❛❧✳ ✷✵✵✶✮✱ ✭❱✐❧✐♠ ✷✵✵✼✮ ❛♥❞ ✐♥ ♠❛❦❡s♣❛♥ ❧♦✇❡r ❜♦✉♥❞
s✉r✈❡②s ✭◆❡r♦♥ ❡t✳ ❛❧✳ ✷✵✵✻✮ ❛♥❞ ✭❑♥✉st ✷✵✶✺✮✳
✸ ❈❛❧❝✉❧❛t✐♥❣ ❛♥ ✉♣♣❡r ❜♦✉♥❞ ♦♥ ❤✐❣❤❡st ♣♦ss✐❜❧❡ r❡s♦✉r❝❡ ❝♦♥s✉♠♣t✐♦♥
❋♦r ❡❛❝❤ r❡s♦✉r❝❡ X ∈ R ❛♥❞ ❛♥② t✐♠❡ t ∈ [0, T] ✇❡ ❞❡✜♥❡ ❛♥ ✉♣♣❡r ❜♦✉♥❞ ♦♥ ❤✐❣❤❡st
♣♦ss✐❜❧❡ r❡s♦✉r❝❡ ❝♦♥s✉♠♣t✐♦♥ ✐♥ t✐♠❡ ✐♥t❡r✈❛❧ [0, t) ❜② UX (t)✳ ■♥ ✭❆r❦❤✐♣♦✈ ❡t✳ ❛❧✳ ✷✵✶✼✮
✇❡ ♣r❡s❡♥t❡❞ ❛♥ ❛❧❣♦r✐t❤♠ ❢♦r ❛ ❞✐s❝r❡t❡ ✈❡rs✐♦♥ ♦❢ ❘❈P❙P t♦ ❡st✐♠❛t❡ UX (t) ✐♥ O(n2
r(n+
m + r)T log T) ♦♣❡r❛t✐♦♥s✱ ✇❤❡r❡ n ✕ ♥✉♠❜❡r ♦❢ t❛s❦s✱ r ✕ ♥✉♠❜❡r ♦❢ r❡s♦✉r❝❡s✱ m ✕ t❤❡
❤✐❣❤❡st ♥✉♠❜❡r ♦❢ ❜r❡❛❦♣♦✐♥ts ✐♥ r❡s♦✉r❝❡ ❝❛♣❛❝✐t② ❢✉♥❝t✐♦♥✱ T ✕ t✐♠❡ ❤♦r✐③♦♥✳ ❚❤❡ ♠❛✐♥
✐❞❡❛ ♦❢ t❤✐s ❛❧❣♦r✐t❤♠ ✇❛s ❛s ❢♦❧❧♦✇s✳ ❋✐rst ♦❢ ❛❧❧✱ t❤❡ ❛❧❣♦r✐t❤♠ ✉s❡s s♦♠❡ ♣♦❧②♥♦♠✐❛❧✲t✐♠❡
♣r♦♣❛❣❛t♦rs t♦ t✐❣❤t❡♥ ♣r♦❝❡ss✐♥❣ ✐♥t❡r✈❛❧s [rj, Dj] ❢♦r ❡❛❝❤ j ∈ N✳ ❚❤❡♥✱ ❢♦r ❡❛❝❤ t❛s❦
j ∈ N✱ r❡s♦✉r❝❡ X ∈ R ❛♥❞ t✐♠❡s❧♦t t ∈ 1, . . . , T t❤❡ ❤✐❣❤❡st ♣♦ss✐❜❧❡ ❝♦♥s✉♠♣t✐♦♥ ♦❢ X
❜② j ✐♥ ✐♥t❡r✈❛❧ [0, t] ✐s ❝❛❧❝✉❧❛t❡❞ ❛♥❞ ❞❡✜♥❡❞ ❜② AjX(t)✳
❚❤❡♥ t❤❡ ❘❈P❙P ♣r♦❜❧❡♠ ✐s ❝♦♥s✐❞❡r❡❞ ❢♦r ❡❛❝❤ ♣❛✐r ♦❢ r❡s♦✉r❝❡s X, Y ∈ R ❛♥❞ t❤❡
❢♦❧❧♦✇✐♥❣ t❛s❦ ♣r♦❝❡ss✐♥❣ ❝♦♥str❛✐♥ts✿
✕ ♣r❡❡♠♣t✐♦♥s ♦❢ t❛s❦ ♣r♦❝❡ss✐♥❣ ❛r❡ ❛❧❧♦✇❡❞❀
✕ t❛s❦s ❛r❡ ❛❜❧❡ t♦ ❝♦♥t✐♥✉❡ ♣r♦❝❡ss✐♥❣ ❛❢t❡r ❞❡❛❞❧✐♥❡❀
✕ ❛♠♦✉♥t ♦❢ r❡s♦✉r❝❡s X ❛♥❞ Y ❝♦♥s✉♠❡❞ ❜② t❛s❦ j ✐♥ t✐♠❡s❧♦t [t, t + 1] ❝❛♥ ❜❡ ♥♦t
❡q✉❛❧ t♦ ajX ❛♥❞ ajY r❡s♣❡❝t✐✈❡❧②✱ ❜✉t ❢✉♥❝t✐♦♥s ujX(t) ❛♥❞ ujY (t) ✕ t♦t❛❧ ❛♠♦✉♥ts ♦❢
r❡s♦✉r❝❡s X ❛♥❞ Y ✐♥ t✐♠❡ ✐♥t❡r✈❛❧ [0, t] s❤♦✉❧❞ s❛t✐s❢② t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥str❛✐♥ts✿
ujX(t) ≤ AjX (t),
ujY (t) ≤ AjY (t),
ujX(t)
ujY (t)
=
ajX
ajY
.
❚❤❡ ♦❜❥❡❝t✐✈❡ ✐s t♦ ✜♥❞ t❤❡ ❢✉♥❝t✐♦♥s ujX(t) ❛♥❞ ujY (t) ❢♦r ❛♥② t ∈ [0, T] s✉❜❥❡❝t t♦
♠❛①✐♠✐③✐♥❣ t❤❡ ♦❜❥❡❝t✐✈❡ ❢✉♥❝t✐♦♥s
UX|Y (t) =
X
j∈N
ujX(t),
UY |X(t) =
X
j∈N
ujY (t)
❢♦r ❛♥② t ∈ [0, T]✳
❚❤❡ ▼❛st❡r ❆❧❣♦r✐t❤♠ ✇❤✐❝❤ ✜♥❞s ❛♥ ♦♣t✐♠❛❧ s♦❧✉t✐♦♥ ❢♦r t❤✐s ♣r♦❜❧❡♠ ✐t❡r❛t❡s ♦♥
t✐♠❡s❧♦ts t = 1, . . . , T✱ s♦❧✈✐♥❣ t❤❡ ❢♦❧❧♦✇✐♥❣ ♦♣t✐♠✐③❛t✐♦♥ ♣r♦❜❧❡♠ ❢♦r ❡❛❝❤ ♣❛✐r ♦❢ r❡s♦✉r❝❡s
X, Y ✳
❚✐♠❡s❧♦t ♣r♦❜❧❡♠✳ ❋♦r ❡❛❝❤ j ∈ N ✈❛❧✉❡s ujX(t − 1) ❛♥❞ ujY (t − 1) ❛r❡ ❣✐✈❡♥
❛♥❞ ❢✉♥❝t✐♦♥s AjX(t)✱ AjX(t) ❛r❡ ❞❡✜♥❡❞✳ ❉❡t❡r♠✐♥❡ ujX(t) ≥ ujX(t − 1) ❛♥❞ ujY (t) ≥
ujY (t − 1) ❢♦r ❛❧❧ t❛s❦s j ∈ N s✉❝❤ t❤❛t
max UX (t), UY (t)
23

s✉❜❥❡❝t t♦ r❡s♦✉r❝❡ ❝❛♣❛❝✐t✐❡s
X
j∈N
(ujX(t) − ujX(t − 1)) ≤ cX (t),
X
j∈N
(ujY (t) − ujY (t − 1)) ≤ cY (t)
❛♥❞ ❝♦♥str❛✐♥ts
ujX(t) − ujX(t − 1)
ujY (t) − ujY (t − 1)
=
ajX
ajY
,
ujX(t) ≤ AjX(t), ujY (t) ≤ AjY (t).
■❢ ❢♦r ❛♥② t✐♠❡ s❧♦t t❤❡r❡ ✐s ♠♦r❡ t❤❛♥ ♦♥❡ s♦❧✉t✐♦♥ s❛t✐s❢②✐♥❣ t❤❡s❡ ❝♦♥❞✐t✐♦♥s✱ ❝❤♦♦s❡
t❤❡ ♦♥❡ ✉s✐♥❣ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝r✐t❡r✐♦♥✿
min
X
j∈N
q
(ujX(t) − ujX(t − 1))2 + (ujY (t) − ujY (t − 1))2.
❚❤❡ ❞❡✈❡❧♦♣❡❞ ❣❡♦♠❡tr✐❝ ❛❧❣♦r✐t❤♠ s♦❧✈❡s t❤✐s ♦♣t✐♠✐③❛t✐♦♥ ♣r♦❜❧❡♠ ✐♥ O(n2
) ♦♣❡r❛✲
t✐♦♥s✳
❚❤❡ ♦❜❥❡❝t✐✈❡ ♦❢ t❤✐s ♣❛♣❡r ✐s t♦ s❤♦✇ t❤❛t t❤✐s ❛❧❣♦r✐t❤♠ ❝❛♥ ❜❡ ✉s❡❞ ❢♦r ❘❈P❙P
❢♦r♠✉❧❛t✐♦♥ ✇✐t❤ ❝♦♥t✐♥✉♦✉s t✐♠❡✳ ❲❡ ✇✐❧❧ ❝❛❧❧ ❛ t✐♠❡ ♣♦✐♥t Ti ∈ [0, T] ❛ ❜r❡❛❦♣♦✐♥t ✐❢ Ti
✐s ❛ r❡❧❡❛s❡ t✐♠❡ ♦❢ ❛♥② t❛s❦ j ∈ N✱ ✐✳❡✳ Ti
= rj ♦r Ti ✐s ❛ Ti
= tk ✕ st❛rt ♦r ❡♥❞ ♦❢ ❛♥②
s❡❣♠❡♥t ♦❢ ❢✉♥❝t✐♦♥ cX(t)✳ ❚♦t❛❧ ♥✉♠❜❡r ♦❢ ❜r❡❛❦♣♦✐♥ts ✐s b ≤ n + m✳ ❲❡ ❛ss✉♠❡ t❤❛t
t❤❡ ❜r❡❛❦♣♦✐♥ts ❛r❡ ♦r❞❡r❡❞ ✐♥ ❛s❝❡♥❞✐♥❣ ♦r❞❡r✿ 0 = T1 T2 . . . Tb = T✳ ◆♦t❡ t❤❛t
t❤❡ s✐③❡ ♦❢ ❛ t✐♠❡s❧♦t ❞♦❡s ♥♦t ♠❛tt❡r ❢♦r t❤❡ ▼❛st❡r ❆❧❣♦r✐t❤♠✳ ❚❤❡r❡❢♦r❡✱ ✇❡ ❝♦♥s✐❞❡r
✐♥t❡r✈❛❧s [T1, T2), . . . , [Tb−1, Tb) ❧✐❦❡ t✐♠❡s❧♦ts ❛♥❞ ✉s❡ ▼❛st❡r ❆❧❣♦r✐t❤♠ t♦ ✜♥❞ ❢✉♥❝t✐♦♥s
ujX(t), ujY (t) ❢♦r ❡❛❝❤ j ∈ N ❛♥❞ ❡❛❝❤ t✐♠❡s❧♦t✳ ❆❝❝♦r❞✐♥❣ t♦ t❤❡ t❤❡♦r❡♠ ♣r♦✈❡❞ ✐♥
✭❆r❦❤✐♣♦✈ ❡t✳ ❛❧✳ ✷✵✶✼✮ ♦❜t❛✐♥❡❞ ❢✉♥❝t✐♦♥s UX|Y (t) ❛♥❞ UY |X (t) ✇♦✉❧❞ ❜❡ ❛♥ ✉♣♣❡r ❜♦✉♥❞
♦♥ r❡s♦✉r❝❡ X ❛♥❞ Y ❝♦♥s✉♠♣t✐♦♥ r❡s♣❡❝t✐✈❡❧② ❢♦r ❛♥② t = T1, . . . , Tb ✐❢ ❢♦r ❛♥② t✐♠❡s❧♦t
❛♥❞ ❛♥② t ∈ [Tk, Tk+1] u′
jX(t) = const✳ ❚❤❡ ❢♦❧❧♦✇✐♥❣ ❧❡♠♠❛ ♣r♦✈❡s t❤❛t t❤❡s❡ ❝♦♥❞✐t✐♦♥s
❝❛♥ ❜❡ s✐♠♣❧✐✜❡❞✳
▲❡♠♠❛ ✶✳
❙✉♣♣♦s❡ t❤❡r❡ ✐s ❛ s❡t ♦❢ ❢✉♥❝t✐♦♥s u1X(t), . . . , unX (t) ❞❡✜♥❡❞ ♦♥ t✐♠❡s❧♦t [Tk, Tk+1] ✇❤✐❝❤
s❛t✐s❢② t❤❡ ❝♦♥str❛✐♥ts ♦❢ t❤❡ ❚✐♠❡s❧♦t ♣r♦❜❧❡♠ ❢♦r ❛♥② t ∈ [0, T]✳ ❚❤❡♥ ❢✉♥❝t✐♦♥
uujX(t) =
1
Tk+1 − Tk
Z Tk+1
Tk
ujX(t)dt
❞❡✜♥❡❞ ❢♦r ❛❧❧ j ∈ N s❛t✐s✜❡s t❤❡ ❝♦♥str❛✐♥ts ♦❢ t❤❡ ❚✐♠❡s❧♦t ♣r♦❜❧❡♠ ❛♥❞ uu′
jX(t) = const✳
❊❛❝❤ ❢❡❛s✐❜❧❡ s❝❤❡❞✉❧❡ ❞❡✜♥❡s ❛ s❡t ♦❢ ✐♥t❡❣r❛❜❧❡ ❢✉♥❝t✐♦♥s u1X(t), . . . , unX(t)✳ ❙✐♥❝❡
uujX(Tk) ≤ ujX(Tk)✱ ▲❡♠♠❛ ✶ ✐♠♣❧✐❡s t❤❛t ❢✉♥❝t✐♦♥s UX|Y (t) ❛♥❞ UY |X(t) ♣r♦✈✐❞❡❞ ❜②
t❤❡ ▼❛st❡r ❆❧❣♦r✐t❤♠ ❣✐✈❡ ❛♥ ✉♣♣❡r ❜♦✉♥❞ ♦♥ ❛♠♦✉♥t ♦❢ r❡s♦✉r❝❡s ❝♦♥s✉♠❡❞ ❜② t❤❡ t❛s❦s
❜❡❧♦♥❣✐♥❣ t♦ s❡t N ✐♥ t✐♠❡ ✐♥t❡r✈❛❧ [0, Tk+1)✳ ❚❤❡ ❝♦♠♣❧❡①✐t② ♦❢ t❤❡ ●❡♦♠❡tr✐❝ ❆❧❣♦r✐t❤♠
t♦ s♦❧✈❡ ❚✐♠❡s❧♦t ♣r♦❜❧❡♠ ✐s O(n2
)✳ ❍❡♥❝❡ t❤❡ ▼❛st❡r ❆❧❣♦r✐t❤♠ ❝♦♠♣❧❡①✐t② ❡q✉❛❧s t♦
O(n2
(n + m))✱ ✐✳❡✳ t♦t❛❧ ♥✉♠❜❡r ♦❢ ❝②❝❧❡❞ t✐♠❡s❧♦ts ✐s ♥♦t ♠♦r❡ t❤❛♥ n + m✳ ❚❤❡r❡❢♦r❡
❢✉♥❝t✐♦♥s UX|Y (t) ❛♥❞ UY |X(t) ❝♦✉❧❞ ❜❡ ❢♦✉♥❞ ❢♦r ❛❧❧ ♣❛✐rs ♦❢ r❡s♦✉r❝❡s (X, Y ) ∈ R2
✐♥
O(r2
n2
(n + m)) ♦♣❡r❛t✐♦♥s✱ ✇❤❡r❡ r ✕ ♥✉♠❜❡r ♦❢ r❡s♦✉r❝❡s✳ ❆♥ ✉♣♣❡r ❜♦✉♥❞ ♦♥ r❡s♦✉r❝❡
❝♦♥s✉♠♣t✐♦♥ ✐♥ ✐♥t❡r✈❛❧ [0, t) ❝❛♥ ❜❡ ❞❡✜♥❡❞ ❝♦rr❡❝t❧② ❢♦r ❛♥② t = T1, . . . , Tn :
UX (t) = min
Y ∈R
UX|Y (t).
24

✹ ❆♣♣❧✐❝❛t✐♦♥s ✫ ●❡♥❡r❛❧✐③❛t✐♦♥s
■♥ t❤✐s ♣❛♣❡r✱ ❛ ♣♦❧②♥♦♠✐❛❧ ❛❧❣♦r✐t❤♠ t♦ ❡st✐♠❛t❡ ❛♥ ✉♣♣❡r ❜♦✉♥❞ ♦♥ r❡s♦✉r❝❡ ❛♠♦✉♥t
✉s❡❞ ✐♥ t✐♠❡ ✐♥t❡r✈❛❧ [0, t] ✐s ♣r❡s❡♥t❡❞✳ ❚❤✐s ❛♣♣r♦❛❝❤ ❝❛♥ ❜❡ ❛♣♣❧✐❡❞ ♥♦t ♦♥❧② t♦ t❤❡ ❝❧❛s✲
s✐❝❛❧ ❘❈P❙P ❢♦r♠✉❧❛t✐♦♥ ❜✉t ❢♦r ♦t❤❡r ❘❈P❙P st❛t❡♠❡♥ts ✇✐t❤ s❡❣♠❡♥t✲❝♦♥st❛♥t r❡s♦✉r❝❡
❝❛♣❛❝✐t② ❢✉♥❝t✐♦♥s✱ ✐✳❡✳ ❘❈P❙P✴♠❛①✳
❖❜t❛✐♥❡❞ ❢✉♥❝t✐♦♥s UX(t) ❝❛♥ ❜❡ ✉s❡❞ ✐♥ r❡s♦✉r❝❡✲❜❛s❡❞ ♣r♦♣❛❣❛t♦rs t♦ ❡✈❛❧✉❛t❡ r❡s♦✉r❝❡✲
✉s✐♥❣ ♣r♦✜❧❡s✳ ❖✉r ❢✉t✉r❡ r❡s❡❛r❝❤ ✇✐❧❧ ❜❡ ❢♦❝✉s❡❞ ♦♥ ❞❡✈❡❧♦♣♠❡♥t ♦❢ ❞✐r❡❝t ♠❡t❤♦❞s t♦
✐♠♣r♦✈❡ ❡①✐st✐♥❣ r❡s♦✉r❝❡✲❜❛s❡❞ ♣r♦♣❛❣❛t♦rs ❛♥❞ t♦ ❝r❡❛t❡ ♥❡✇ t❡❝❤♥✐q✉❡s ♦❢ ❜♦✉♥❞✐♥❣
r❡s♦✉r❝❡ ✉s❛❣❡ ❢✉♥❝t✐♦♥✳
❆❝❦♥♦✇❧❡❞❣❡♠❡♥ts
❚❤✐s r❡s❡❛r❝❤ ✇❛s s✉♣♣♦rt❡❞ ❜② ■❙❆❊✲❙❯P❆❊❘❖ ❋♦✉♥❞❛t✐♦♥ ❛♥❞ t❤❡ ❘✉ss✐❛♥ ❙❝✐❡♥❝❡
❋♦✉♥❞❛t✐♦♥ ✭❣r❛♥t ✶✼✲✶✾✲✵✶✻✻✺✮✳
❚❤❡ ❛✉t❤♦rs ❛r❡ ❣r❛t❡❢✉❧ t♦ ❊♠♠❛♥✉❡❧ ❍❡❜r❛r❞ ❛♥❞ P✐❡rr❡ ❋❧❡♥❡r ❢♦r ✉s❡❢✉❧ ❛❞✈✐s❡s ♦♥
r❡s♦✉r❝❡✲❜❛s❡❞ ♣r♦♣❛❣❛t♦rs✳
❘❡❢❡r❡♥❝❡s
❆r❦❤✐♣♦✈ ❉✳✱ ❖✳ ❇❛tt❛✐❛ ❛♥❞ ❆✳ ▲❛③❛r❡✈✱ ✷✵✶✼✱ ✏▲♦♥❣✲t❡r♠ ♣r♦❞✉❝t✐♦♥ ♣❧❛♥♥✐♥❣ ♣r♦❜❧❡♠✿ s❝❤❡❞✉❧✲
✐♥❣✱ ♠❛❦❡s♣❛♥ ❡st✐♠❛t✐♦♥ ❛♥❞ ❜♦tt❧❡♥❡❝❦ ❛♥❛❧②s✐s✧✱ ■❋❆❈✲P❛♣❡rs❖♥▲✐♥❡✱ ❱♦❧✳ ✺✵✱ ■✳ ✶✱ ♣♣✳
✼✾✼✵✕✼✾✼✹✳
❇❛♣t✐st❡ P✳✱ ❈✳ ▲❡ P❛♣❡ ❛♥❞ ❲✳ ◆✉✐❥t❡♥✱ ✷✵✵✶✱ ✏ ❈♦♥str❛✐♥t✲❇❛s❡❞ ❙❝❤❡❞✉❧✐♥❣✧✱ ❑❧✉✇❡r ❆❝❛❞❡♠✐❝
P✉❜❧✐s❤❡r✳
❇❡❧❞✐❝❡❛♥✉ ◆✳✱ ▼✳ ❈❛r❧ss♦♥✱ ✷✵✵✶✱ ✏❙✇❡❡♣ ❛s ❛ ●❡♥❡r✐❝ Pr✉♥✐♥❣ ❚❡❝❤♥✐q✉❡ ❆♣♣❧✐❡❞ t♦ t❤❡ ◆♦♥✲
♦✈❡r❧❛♣♣✐♥❣ ❘❡❝t❛♥❣❧❡s ❈♦♥str❛✐♥t✧✱ Pr♦❝❡❡❞✐♥❣s ♦❢ t❤❡ s❡✈❡♥t❤ ■♥t❡r♥❛t✐♦♥❛❧ ❈♦♥❢❡r❡♥❝❡ ♦♥
Pr✐♥❝✐♣❧❡s ❛♥❞ Pr❛❝t✐❝❡ ♦❢ ❈♦♥str❛✐♥t Pr♦❣r❛♠♠✐♥❣✱ ♣♣✳ ✸✼✼✲✸✾✶✳
❈❛s❡❛✉ ❨✳✱ ❋✳ ▲❛❜✉rt❤❡✱ ✶✾✾✻✱ ✏❈✉♠✉❧❛t✐✈❡ ❙❝❤❡❞✉❧✐♥❣ ✇✐t❤ ❚❛s❦ ■♥t❡r✈❛❧s✧✱ Pr♦❝❡❡❞✐♥❣s ♦❢ t❤❡
❏♦✐♥t ■♥t❡r♥❛t✐♦♥❛❧ ❈♦♥❢❡r❡♥❝❡ ❛♥❞ ❙②♠♣♦s✐✉♠ ♦♥ ▲♦❣✐❝ Pr♦❣r❛♠♠✐♥❣✱ ♣♣✳ ✸✻✸✲✸✼✼✳
❋♦①✱ ❇✳ ❘✳✱ ✶✾✾✶✱ ✏◆♦♥✲❝❤r♦♥♦❧♦❣✐❝❛❧ s❝❤❡❞✉❧✐♥❣✧✱ ■♥✿ Pr♦❝❡❡❞✐♥❣s ♦❢ ❆■✱ ❙✐♠✉❧❛t✐♦♥ ❛♥❞ P❧❛♥♥✐♥❣
✐♥ ❍✐❣❤ ❆✉t♦♥♦♠② ❙②st❡♠s✱ ♣♣✳ ✼✷✲✼✼✳
●❛r❡② ▼✳❘✳✱ ❉✳❙✳ ❏♦❤♥s♦♥✱ ✶✾✼✺✱ ✏❈♦♠♣❧❡①✐t② r❡s✉❧ts ❢♦r ♠✉❧t✐♣r♦❝❡ss♦r s❝❤❡❞✉❧✐♥❣ ✉♥❞❡r r❡s♦✉r❝❡
❝♦♥str❛✐♥ts✧✱ ❙■❆▼ ❏♦✉r♥❛❧ ♦♥ ❈♦♠♣✉t✐♥❣✱ ❱♦❧✳ ✹✱ ♣♣✳ ✸✾✼✕✹✶✶✳
❑♥✉st ❙✳✱ ✷✵✶✺✱ ✏▲♦✇❡r ❇♦✉♥❞s ♦♥ t❤❡ ▼✐♥✐♠✉♠ Pr♦❥❡❝t ❉✉r❛t✐♦♥✧✱ ■♥✿❙❝❤✇✐♥❞t ❈✳✱ ❩✐♠♠❡r♠❛♥♥
❏✳✿ ❍❛♥❞❜♦♦❦ ♦♥ Pr♦❥❡❝t ▼❛♥❛❣❡♠❡♥t ❛♥❞ ❙❝❤❡❞✉❧✐♥❣✱ ❱♦❧✳ ✵✶✱ ♣♣✳ ✸✕✺✻✳
❑♦❧✐s❝❤ ❘✳✱ ❆✳ ❙♣r❡❝❤❡r✱ ✶✾✾✼✱ ✏P❙P▲■❇ ✕ ❛ ♣r♦❥❡❝t s❝❤❡❞✉❧✐♥❣ ♣r♦❜❧❡♠ ❧✐❜r❛r②✧✱ ❊✉r ❏ ❖♣❡r ❘❡s✱
❱♦❧✳ ✾✻✭✶✮✱ ♣♣✳ ✷✵✺✲✷✶✻✱ ❤tt♣✿✴✴✇✇✇✳♦♠✲❞❜✳✇✐✳t✉♠✳❞❡✴♣s♣❧✐❜✴✳
▲❛❤r✐❝❤✐ ❆✳✱ ✶✾✽✷✱ ✏❙❝❤❡❞✉❧✐♥❣✿ t❤❡ ◆♦t✐♦♥s ♦❢ ❍✉♠♣✱ ❈♦♠♣✉❧s♦r② P❛rts ❛♥❞ t❤❡✐r ❯s❡ ✐♥ ❈✉♠✉✲
❧❛t✐✈❡ Pr♦❜❧❡♠s✧✱ ■♥✿ ❈✳❘✳ ❆❝❛❞✳ ❙❝✳ P❛✐rs✱ ♣♣✳ ✷✵✾✲✷✶✶✳
▲❡ P❛♣❡ ❈✳✱ ✶✾✽✽✱ ✏❉❡s s②st❡♠❡s ❞✬♦r❞♦♥♥❛♥❝❡♠❡♥t ❡①✐❜❧❡s ❡t ♦♣♣♦rt✉♥✐st❡s✧✱ P❤❉ t❤❡s✐s✳ ❯♥✐✲
✈❡rs✐t❡ P❛r✐s ❳■✳
◆❡r♦♥ ❊✳✱ ❈✳ ❆rt✐❣✉❡s✱ P✳ ❇❛♣t✐st❡✱ ❏✳ ❈❛r❧✐❡r✱ ❏✳❉❛♠❛②✱ ❙✳❉❡♠❛ss❡② ❛♥❞ P✳ ▲❛❜♦r✐❡ ✷✵✵✻✱ ✏▲♦✇❡r
❜♦✉♥❞s ❢♦r ❘❡s♦✉r❝❡ ❈♦♥str❛✐♥❡❞ Pr♦❥❡❝t ❙❝❤❡❞✉❧✐♥❣ Pr♦❜❧❡♠✧✱ ■♥✿❏♦③❡❢♦✇s❦❛ ❏✳✱ ❲❡❣❧❛r③ ❏✳✿
P❡rs♣❡❝t✐✈❡s ✐♥ ▼♦❞❡r♥ Pr♦❥❡❝t ❙❝❤❡❞✉❧✐♥❣✱ ♣♣✳ ✶✻✼✕✷✵✹✳
❖✉❡❧❧❡t P✳✱ ❈✳●✳ ◗✉✐♠♣❡r✱ ✷✵✶✸✱ ❵❚✐♠❡✲❚❛❜❧❡ ❊①t❡♥❞❡❞✲❊❞❣❡✲❋✐♥❞✐♥❣ ❢♦r t❤❡ ❈✉♠✉❧❛t✐✈❡ ❈♦♥✲
str❛✐♥t✧✱ Pr♦❝❡❡❞✐♥❣s ♦❢ t❤❡ ◆✐♥❡t❡❡♥t❤ ■♥t❡r♥❛t✐♦♥❛❧ ❈♦♥❢❡r❡♥❝❡ ♦♥ Pr✐♥❝✐♣❧❡s ❛♥❞ Pr❛❝t✐❝❡
♦❢ ❈♦♥str❛✐♥t Pr♦❣r❛♠♠✐♥❣✱ ♣♣✳ ✺✻✷✲✺✼✼✳
❙❝❤✉tt ❆✳✱ ❋❡②❞② ❚✳✱ ❙t✉❝❦❡② P✳✱ ❲❛❧❧❛❝❡ ▼✳ ✷✵✶✶✱ ✏❊①♣❧❛✐♥✐♥❣ t❤❡ ❝✉♠✉❧❛t✐✈❡ ♣r♦♣❛❣❛t♦r✧✱ ❈♦♥✲
str❛✐♥t✱ ❱♦❧✳ ✶✻✭✸✮ ♣♣✳ ✷✺✵✕✷✽✷✳
❱✐❧✐♠ P✳✱ ✷✵✵✼✱ ✏●❧♦❜❛❧ ❈♦♥str❛✐♥ts ✐♥ ❙❝❤❡❞✉❧✐♥❣✧✱ P❤❉ t❤❡s✐s✳ ❈❤❛r❧❡s ❯♥✐✈❡rs✐t② ✐♥ Pr❛❣✉❡✱ ❋❛❝✲
✉❧t② ♦❢ ▼❛t❤❡♠❛t✐❝s✱ P❤②s✐❝s✱ ❉❡♣❛rt♠❡♥t ♦❢ ❚❤❡♦r❡t✐❝❛❧ ❈♦♠♣✉t❡r ❙❝✐❡♥❝❡✱ ❛♥❞ ▼❛t❤❡♠❛t✐❝❛❧
▲♦❣✐❝✳
25

Assembly Flowshops Scheduling Problem to Minimize
Maximum Tardiness with Setup Times
Asiye Aydilek1
, Harun Aydilek1
, and Ali Allahverdi2
1
Gulf University for Science and Technology, Kuwait
{aydilek.a,aydilek.h}@gust.edu.kw
2
Kuwait University, Kuwait
ali.allahverdi@ku.edu.kw
Keywords: Assembly flowshop, simulated annealing, maximum tardiness, setup time.
1 Introduction
Manufacturing of a product consisting of different parts, where parts are processed by
different machines in parallel at the first stage, and all the processed parts of the product
are assembled at the second stage can be considered as a two-stage assembly flowshop
scheduling problem.
The problem was addressed with respect to different performance measures, e.g., makespan,
total completion time, and maximum lateness. Moreover, the problem was addressed with
zero and non-zero setup times. There are some applications where assumption of zero
setup times is valid. However, the assumption is not valid for some other applications, e.g.,
Allahverdi (2015).
The problem with total tardiness performance measure was addressed by Allahverdi
and Aydilek (2015) assuming zero setup times. They proposed several heuristics including
a genetic algorithm.
Allahverdi and Al-Anzi (2006) provided a dominance relation and presented a few
heuristics for the problem with respect to maximum lateness performance measure for zero
setup times while Al-Anzi and Allahverdi (2007) presented several heuristics, including
Particle Swarm Optimization (PSO), Tabu search, and Self-Adaptive Deferential Evolution
(SDE), for the problem with setup times. Al-Anzi and Allahverdi (2007) showed that the
SDE heuristic outperforms the others not only in terms of the error but also in terms of
the CPU time.
In this paper, we address the problem with maximum tardiness performance measure
with non-zero setup times. The maximum tardiness performance measure is used in the
scheduling literature. This is because for some applications completing a job after its due
date results in a penalty which increases as the gap between the due date and completion
time widens.
It should be noted that a sequence that minimizes maximum lateness also minimizes
maximum tardiness. As a result, the SDE heuristic of Al-Anzi and Allahverdi (2007) is
not only the best existing heuristic for the problem with maximum lateness performance
measure but also the best existing heuristic with respect to the maximum tardiness perfor-
mance measure. Therefore, we compare the performance of our newly proposed algorithm,
to be developed in this paper, with the SDE algorithm of Al-Anzi and Allahverdi (2007),
which is known to be the best existing algorithm for the problem. We show that perfor-
mance of the newly developed algorithm in this paper significantly outperforms that of the
SDE algorithm of Al-Anzi and Allahverdi (2007).
26

2 The proposed Simulated Maximum Insertion (SMI) algorithm
Al-Anzi and Allahverdi (2007) presented several algorithms and showed that the algo-
rithm of self-adaptive differential evolution (SDE) outperforms the others for the two-stage
assembly flowshop scheduling problem with non-zero setup times to minimize maximum
lateness. The algorithm SDE of Al-Anzi and Allahverdi (2007) is the only benchmark ex-
isting algorithm, in the literature, to compare our algorithm with. The parameter values
of the algorithm SDE are taken as the ones given by Al-Anzi and Allahverdi (2007). How-
ever, for a fair comparison, the number of generations in the SDE is selected so that the
algorithm proposed in this paper and the SDE algorithm have the same computational
time. Our algorithm, called SMI, is explained next.
Simulated Maximum Insertion (SMI) algorithm is a hybrid of simulated annealing
algorithm and maximum insertion algorithm. In the maximum insertion algorithm, for a
given sequence, the job with the maximum tardiness is inserted to certain positions in the
sequence and the sequence is updated if the insertion decreases the maximum tardiness.
It is observed that repeating this procedure decreases the maximum tardiness of a given
sequence significantly. Therefore, we combined the maximum insertion algorithm with the
simulated annealing algorithm in order to obtain the hybrid algorithm. In the hybrid
algorithm, given a sequence and given initial parameters, we apply the swap and insertion
operators to obtain two new sequences and the better one among the two is selected. The
current sequence is updated whenever one of these new sequences, the better one, improves
the objective function. If the objective function is not improved, then the current sequence
is updated with the better one with certain probability. When the temperature is high,
this probability is large and as the temperature decreases the probability of choosing an
inferior sequence decreases. In order not to trap to a local solution, the solution space is
explored for high temperatures and exploited for low temperatures. Then, the maximum
insertion algorithm is applied. At high temperatures, the job with the maximum tardiness
is inserted to every z-th position rather than every position in the sequence and this is
repeated z times. Thus, this gives more chances to explore alternative solutions. As the
temperature decreases, the value of z decreases which helps to exploit the sequence. Once
the temperature drops below the final temperature, the maximum insertion algorithm is
applied such that the job with the maximum tardiness is inserted to every position in
the sequence and the procedure is repeated certain times in order to improve the solution
further. In short, inserting the job with maximum tardiness strengthens the exploration
step of the simulated annealing algorithm at the beginning when the temperature is high
and reinforces the exploitation step of the simulated annealing algorithm towards the end
when the temperature is low. The hybrid algorithm requires an initial sequence, which
affects the performance of the algorithm. We construct some initial sequences as follows.
We first convert the problem to a single machine scheduling problem by aggregating the
processing times and setup times at both stages. The aggregation can be performed in
several ways. Four of the aggregated processing times are
AP0(i) = max{ max
j=1,...,m
(tij + sij), (si + pi)},
AP1(i) = max
j=1,...,m
(tij + sij) + (si + pi),
AP2(i) = max
j=1,...,m
(tij + sij),
AP3(i) = max{( max
j=1,...,m
(tij + sij) + min
j=1,...,m
(tij + sij))/2, (si + pi)}.
27

Then, by applying the shortest processing time (SPT) rule to the aggregated processing
times, we obtain a sequence for each one. In addition to these sequences, we also considered
the sequence obtained from earliest due date (EDD) rule and the best performing sequence
among the five sequences is taken as the initial sequence and denoted as seqb.
Simulated annealing algorithm has parameters, which need to be calibrated for the
problem which are initial temperature, TPi, final temperature, TPf , cooling factor, cf,
and number of repetitions, Nr. The following table presents the values considered for the
calibration and the selected values for the parameters followed by the steps of the SMI
algorithm.
Table 1. Considered and selected values for the parameters of SMI
Parameters Tested values Selected values
Initial temperature (TPi) 0.10, 0.11, 0.12, 0.13, 0.14, 0.15 0.12
Final temperature (TPf ) 0.0001, 0.0005, 0.0010, 0.0020 0.0010
Cooling factor (cf) 0.970, 0.975, 0.980, 0.985, 0.990 0.975
Number of repetitions (Nr) 20, 30, 40 30
3 Algorithm Evaluation
The performances of the existing algorithm SDE and the proposed algorithm SMI are
compared in this section. Computations were executed on a PC with Intel Core i7-3520M
CPU processor of 2.9 GHz with 8 GB RAM.
A uniform distribution U(1, 100) is used to generate processing times on all the machines
including the assembly machine. Similarly, setup times at both stages are generated from
a uniform distribution U(0, k · 100) where the parameter k denotes the expected ratio of
setup times to processing times. Job due dates are generated from a uniform distribution
over the interval of [L(1−T −R/2), L(1−T +R/2)] where L denotes an approximate value
for makespan. The parameter R denotes relative range of due dates while the parameter T
denotes tardiness factor. Therefore, as T increases the due dates become smaller. On the
other hand, the difference between job due dates increases as R increases. The generation
of due dates by using this method is common in the scheduling literature. The values of
T and R are usually taken to be between 0 and 1 in the literature. Therefore, we have
also selected R and T values in the same range. In the experimentations, the following LB
value was first used instead of L where
LB = max
(
max
k=1,...,m
{ n
∑
r=1
(
t[k,r] + s[k,r]
)
}
+ min
j
{pj + sj},
max
k=1,...,m
{
min
r=1,...,n
(
t[k,r] + s[k,r]
)
}
− min
j
{sj} +
n
∑
r=1
(
p[r] + s[r]
)
)
.
Nevertheless, the aforementioned LB may lead to an environment where no job is tardy.
Thus, we have generated n random sequences, and computed the average makespan, which
may be considered as an upper bound, denoted by LU. Subsequently, the average of the
LB and LU is computed to obtain the value of L as an approximate makespan.
The values of parameters utilized in the computational experiments are summarized in
Table 2.
28

Table 2. Parameter values
Parameter Considered values
N 30, 40, 50, 60, 70
M 3, 5, 8
K 0.4, 0.8, 1.2
R 0.3, 0.5, 0.7
T 0.2, 0.4, 0.6
There are a total of 405 combinations of n, m, k, R, and T values. For each combination
of the parameter values, fifty replicates are generated. Therefore, a total of 20,250 problems
are considered.
The existing and proposed algorithms are assessed by using the performance mea-
sure of percentage error (Error). The Error is defined as 100(Tmax of the algorithm −
Tmax of the best algorithm)/Tmax of the best algorithm where Tmax denotes maximum tar-
diness.
Figure 1 indicates the error versus the number of jobs for both the SDE and SMI
algorithms. It is obvious from the figure that the proposed SMI algorithm performs signifi-
cantly better than the existing SDE algorithm. The gap between the performances of SDE
and SMI algorithms monotonically increases as n increases. This is another advantage of
SMI over SDE.
Figure 2 summarizes the errors of SMI and SDE algorithms versus the setup to process-
ing time ratio k. The figure clearly indicates that the SMI algorithm performs significantly
better than the SDE algorithm for k values. The performances of both algorithms SMI
and SDE do not seem to be sensitive to k value.
Given that the CPU times of both algorithms are the same, the error of SMI is negligible
compared to the error of SDE algorithm as the overall average error of the SMI algorithm
is 0.057 while that of the SDE algorithm is 4.17. Therefore, the proposed SMI algorithm
reduces the error of the best existing SDE algorithm by 98.6%.
We also performed statistical tests to verify the conclusions stated above. For example,
Figure 3 shows 95% confidence interval graph for the case of n = 70, m = 8, R = 0.3,
T = 0.6, and k = 0.8 for which the performances of the algorithms are the closest. Even in
this case, the p-value is less than 0.01, which implies that the error of SMI is statistically
less than that of SDE.
4 Conclusion
We investigate a two-stage assembly flowshop scheduling problem where setup times
are considered as separate from processing times. The objective is to minimize maximum
tardiness. The literature reveals that the algorithm of Self-Adaptive Differential Evolution
(SDE) performs as the best for the problem. We propose a new hybrid simulated annealing
and insertion algorithm, called SMI. We compare the performance of the proposed SMI
algorithm with that of the best existing algorithm, SDE. The computational experiments
indicate that the proposed SMI algorithm performs significantly better than the existing
SDE algorithm. More specifically, under the same CPU time, the proposed SMI algorithm,
on average, reduces the error of the best existing SDE algorithm over 90%, which indicates
the superiority of the proposed SMI algorithm.
29

References
Al-Anzi, F.S., Allahverdi, A., 2007, “A self-adaptive diﬀerential evolution heuristic for two-stage
assembly scheduling problem to minimize maximum lateness with setup times”, European
Journal of Operational Research, Vol. 182, pp. 80–94.
Allahverdi, A., 2015, “Third comprehensive survey on scheduling problems with setup
times/costs”, European Journal of Operational Research, Vol. 246, pp. 345–378.
Allahverdi, A, Al-Anzi, F.S., 2006, “A PSO and a Tabu Search Heuristics for Assembly Schedul-
ing Problem of the Two-Stage Distributed Database Application”, Computers Operations
Research, Vol. 33, pp. 1056–1080.
Allahverdi, A., Aydilek, H., 2015, “The two stage assembly ﬂowshop scheduling problem to mini-
mize total tardiness”, Journal of Intelligent Manufacturing, Vol. 26, pp. 225–237.
30 35 40 45 50 55 60 65 70
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Number of jobs
Error
SDE
SMI
Fig. 1. Error versus number of jobs.
30

0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
k values
Error
SDE
SMI
Fig. 2. Error versus k values.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
SMI
SDE
Fig. 3. Mean conﬁdence interval (n = 70, m = 8, R = 0.3, T = 0.6, k = 0.8).
31

No-Wait Flowshop Scheduling Problem to Minimize
Total Tardiness Subject to Makespan
Harun Aydilek1
, Asiye Aydilek1
, and Ali Allahverdi2
1
Gulf University for Science and Technology, Kuwait
{aydilek.h,aydilek.a}@gust.edu.kw
2
Kuwait University, Kuwait
ali.allahverdi@ku.edu.kw
Keywords: Total tardiness, makespan, no-wait, algorithm.
1 Introduction
We address the no-wait flowshop scheduling problem. The no-wait flowshop scheduling
problem is applicable in many industries, such as plastic, chemical, and pharmaceutical,
e.g. Hall and Sriskandarajah (1996) and Allahverdi (2016).
The total tardiness and makespan performance measures are considered in this paper.
Today’s fierce global competition makes the total tardiness performance measure important
since customer satisfaction is affected by the fulfillment of due dates. On the other hand,
the makespan performance measure is directly related to resource utilization as many
resources are scarce and efficient utilization of such scarce resources is important for many
manufacturing firms. Therefore, both performance measures are critical.
The m-machine no-wait flowshop scheduling problem with the makespan performance
measure has been addressed widely in the literature. For example, Tseng and Lin (2010)
presented a hybrid genetic algorithm (GA), which hybridizes a novel local search scheme
and the GA. Tseng and Lin (2010) indicated that their hybrid GA performs better than
the heuristics the earlier algorithms Furthermore, Jarboui et al. (2011) presented another
hybrid GA algorithm where the variable neighborhood search is utilized to further improve
(in the last step) their GA. On the other hand, Lin and Ying (2016) proposed a three-
phase heuristic. In the first phase, two constructive heuristics are used to obtain an initial
sequence. In the second phase, the problem is transformed into an asymmetric traveling
salesman problem and an algorithm is used to improve the initial solution. In the last
phase, a mathematical model is used to further improve the solution.
The m-machine no-wait flowshop scheduling problem with a total tardiness (TT) per-
formance measure has also been addressed in the literature. Aldowaisan and Allahverdi
(2012) presented several dispatching rules for the problem with respect to total tardiness.
They also proposed a simulated annealing (SA) and a genetic algorithm (SA). Furthermore,
Liu et al. (2013) presented dispatching rules and constructive heuristics, including a mod-
ified NEH, for the problem. They indicated that the modified NEH performs better than
the dispatching rules and the constructive heuristics. Moreover, Ding et al. (2015) studied
the problem and explored the objective function evaluation incremental properties. They
presented an accelerated NEH and iterated greedy algorithms based on the incremental
properties. They indicated that the accelerated algorithms perform much faster than the
original algorithms. They further showed that their proposed algorithms perform better
than those of Aldowaisan and Allahverdi (2012) and Liu et al. (2013).
The aforementioned research addressed a single criterion while many scheduling en-
vironments require considering multi criteria. We address m-machine no-wait scheduling
problem to minimize total tardiness subject to the constraint that makespan is less than a
certain value.
32

2 Algorithms
We propose an algorithm and adapt three existing algorithms to our problem. The ex-
isting algorithms are given in the next subsection while the proposed algorithm is presented
in the following subsection.
2.1 Existing algorithms
The m-machine no-wait ﬂowshop scheduling problem to minimize total tardiness was
addressed by Aldowaisan and Allahverdi (2012) who presented an algorithm, called FISA,
which was shown to perform as the best out of the six algorithms they considered. Moreover,
Liu et al. (2013) also considered the same problem and proposed six heuristic approaches
and indicated that the heuristic MNEH is the best. In addition, Ding et al. (2015) provided
three algorithms and indicated that the algorithm AIG1 performs the best. We adapt the
algorithms FISA, MNEH, and AIG1 to our problem, which are denoted by A-FISA, A-
MNEH, and A-AIG1. We propose a new algorithm, which is called Algorithm HA, in the
next subsection and compare our algorithm with the existing best algorithms of FISA,
MNEH, and AIG1.
2.2 The proposed algorithm (HA)
The algorithm HA utilizes both the simulated annealing algorithm and the insertion
algorithm.
Algorithm HA
1. Obtain a C value, and choose an initial sequence si, set the parameters ti, tf , λ, N
and I, set the sequence st = si, and i = 1
2. Set the intermediate temperature tt = ti
3. Generate a sequence by swapping two random jobs of st and call it ss
4. If TT(ss) TT(st) then update st with ss. Otherwise, update st with ss if rand e− D
tt
where D = (TT(ss) − TT(st))/TT(st) and rand is U[0, 1]
5. Update the intermediate temperature tt such that, tt = tt · λ
6. If tt tf , go to Step 7, otherwise go to Step 3
7. If Cmax(st) C, update i = i + 1 and go to Step 15. Otherwise, go to Step 8
8. Set pi = n
9. Set pj = 1
10. Insert the job in position pi of the sequence st to position pj and call the new sequence
sm
11. Evaluate C1 = Cmax(sm), and C2 = Cmax(st). If C1 C, update st with sm, and
update i = i + 1, then go to Step 15. Otherwise, go to Step 12
12. Update pj = pj +1. If C1 C2 then update st with sm. Then go to Step 10 if pj n.
Otherwise, go to Step 13
13. Update pi = pi − 1, and go to Step 9 if pi 0. Otherwise, go to Step 14
14. Update st with si if Cmax(st) C. Update i = i + 1 and go to Step 15
15. If i I, go to Step 2
The parameters of the simulated annealing part of the algorithm are calibrated based
on the values given in the following table. Selected values are 0.14 for ti, 0.001 for tf , 0.98
for λ, and 20 for N.
33

3 Algorithm evaluation
Computations were conducted on a PC with Intel Core i7-3520M CPU processor of
2.9 GHz with 8 GB RAM. An appropriate C value is usually given by the scheduler
as stated earlier. However, there is a need to know the C value for the computational
experiments. First we reduce the m-machine problem into a two-machine problem such
that the processing time of machine one is the sum of the processing times on the first m/2
((m + 1)/2 if m is odd) machines while the processing time of machine two is the sum of
the processing times on the remaining m/2 ((m − 1)/2 if m is odd) machines. Then, we
apply Johnson’s algorithm to the two machine problem to obtain a sequence s. Next, we
take the first job in the sequence s and insert it in all the n positions of the sequence s
which results in n different sequences. We take the minimum Cmax of all the n sequences,
which is the C value.
40 45 50 55 60 65 70 75 80
0
10
20
30
40
50
60
70
80
90
100
Number of jobs
ARDI A-FISI
A-MNEH
A-AIG1
HA
Fig. 1. ARDI values of algorithms with respect to n values.
The uniform distribution of U(1, 99) was used to generate processing times on all the
m machines. The uniform distribution of U[LB(1 − T − R/2), LB(1 − T + R/2)] is used
in generating job due dates where LB denotes an approximate value for makespan. The
parameter R indicates a relative range of due dates while the parameter T denotes a
tardiness factor, i.e., a larger T value results in a smaller due date. In contrast, as the
R value increases, the difference between job due dates increases. The values of T and R
are usually taken to be between 0 and 1 in the literature. Thus, we have also selected R
and T values between 0 and 1. We use a lower bound on makespan LB which is used by
Aldowaisan and Allahverdi (2012). The utilized values of n, m, R, and T are summarized
in Table 1.
34

Parameter Considered values
n 40, 50, 60, 70, 80
m 3, 5, 10, 12
R 0.2, 0.6, 1.0,
T 0.2, 0.4, 0.6,
The performance measure utilized in evaluating the algorithms is the Average Relative
Deviation Index (ARDI) as a percentage, which is
ARDI =
100
Nr
Nr
∑
k=1
TTk − TTbest
TTworst − TTbest
Figure 1 summarizes the ARDI values of the proposed algorithm HA, and the adapted
algorithms of A-FISA, A-MNEH, and A-AIG1 with respect to n. The figure clearly shows
that the algorithms HA performs much better than the others.
The overall average ARDI values of the algorithms A-FISA, A-MNEH, A-AIG1, and
HA are 98.5, 65.1, 17.6, and 4.6, respectively. Therefore, the proposed algorithm HA reduces
the error of the best adapted algorithm A-AIG1 by 74%. It should be noted that the CPU
times (less than two minutes) of the algorithms are same.
The aforementioned conclusions are statistically tested by using the Tukey Honest Sig-
nificant Difference (HSD) test at α = 0.025. Figures 2 shows the results for a combination
of the parameters, which is representative of the vast majority of the combinations. The
statistical results, in general, validate the earlier conclusions.
0 20 40 60 80 100 120
HA
A-AIG1
A- MNEH
A-FISI
3 groups have means significantly different from HA
Fig. 2. Confidence intervals for n = 50, m = 10, R = 0.2, T = 0.6.
35

4 Conclusion
We consider the m-machine no-wait flowshop scheduling problem to minimize total
tardiness subject to the constraint that the makespan is less than a given value. We pro-
pose an algorithm, which is a combination of simulated annealing and insertion algorithm.
Moreover, we adapt three best existing algorithms for minimizing total tardiness to our
problem. We conduct extensive computational experiments to compare the performance of
our proposed algorithm with the three best existing algorithms under the same CPU times.
The computational analysis indicates that the error of algorithm is 74 percent smaller than
that of the best of the three adapted algorithms. All the results are statistically verified.
References
Allahverdi, A., 2016, “A survey of scheduling problems with no-wait in process”, European Journal
of Operational Research, Vol. 255, pp. 665–686.
Aldowaisan, T., Allahverdi, A., 2012, “Minimizing total tardiness in no-wait flowshops”, Founda-
tions of Computing and Decision Sciences, Vol. 37, pp. 149–162.
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tardiness minimization in no-wait flowshops”, Int. Journal of Production Research, Vol. 53,
pp. 1002–1018.
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and no-wait in process”, Operations Research, Vol. 44, pp. 510–525.
Jarboui, B., Eddaly, M., Siarry, P., 2011, “A hybrid genetic algorithm for solving no-wait flowshop
scheduling problems”, Int. Journal of Advanced Manufacturing Technology, Vol. 54, pp. 1129–
1143.
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lems using efficient matheuristics”, Omega, Vol. 64, pp. 115–125.
Liu, G., Song, S., Wu, C., 2013, “Some heuristics for no-wait flowshops with total tardiness crite-
rion”, Computers and Operations Research, Vol. 40, pp. 521–525.
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lem”, Int. Journal of Production Economics, Vol. 128, pp. 144–152.
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gies”, ZOR - Zeitschrift für Operations Research, vol. 29, pp. 65–104, 1985.
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and makespan in resource-constrained project scheduling”, International Journal of Produc-
tion Research, vol. 44, no. 2, pp. 215–236, 2006.
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36

A Robust Optimization Model for the Multi-mode
Resource Constrained Project Scheduling Problem with
Uncertain Activity Durations
Noemie Balouka1
and Izack Cohen2
Technion - Israel Institute of Technology, Haifa, Israel
1
nbalouka@tx.technion.ac.il
2
izikc@technion.ac.il
Keywords: project scheduling, uncertainty, robust optimization.
1 Introduction and Motivation
The multi-mode resource constrained project scheduling problem (MRCPSP) aims to
minimize the makespan by selecting activities’ modes and scheduling project activities
under precedence and resource constraints. It extends the resource constrained project
scheduling problem (RCPSP), by assuming that activities can be performed under one of
several modes, where a mode determines an activity’s duration and its resource require-
ments. RCPSP and MRCPSP are NP-hard (Blazewicz et al., 1983), so real problems are
often solved by heuristic methods. Most researches consider the RCPSP and MRCPSP
parameters as deterministic although some of the research treats uncertainty. For exam-
ple, Herroelen and Leus (2005) review procedures for generating feasible baseline schedules
with respect to a specific objective (e.g. makespan minimization or net present value max-
imization) under uncertainty, and mention four approaches: reactive scheduling, stochastic
optimization, fuzzy project scheduling and robust or proactive scheduling. We focus on
the latter; contrary to reactive scheduling that revises the baseline schedule after an unex-
pected event, proactive scheduling plans robust baseline schedules. This approach origins
from the concept of robust optimization, a relatively recent optimization approach (Ben-
Tal et al., 2009) that aims to construct a solution that is feasible for any realization within
a given uncertainty set. Proactive scheduling develops a robust schedule that anticipates
the variability during project execution. Daniels and Kouvelis (1995) are among the first to
introduce the approach of robust optimization in a scheduling environment. However, they
consider a single-machine where the job’s processing times are uncertain and the objective
is to minimize the total flow time over all jobs. Cohen et al. (2007) apply the robust opti-
mization approach to the stochastic time—cost tradeoff problem and show that the price
of robustness is relatively small when using ellipsoid uncertainty sets. To the best of our
knowledge, there are no papers presenting a robust optimization model for the MRCPSP.
There are two recent papers developing a robust optimization model for the RCPSP. In
the first paper (Artigues et al., 2013), a minimax absolute regret problem is presented
to find a schedule minimizing the maximum absolute regret aver all duration scenarios.
Bruni et al. (2017) develop an adjustable robust optimization problem to find the schedule
that minimizes the worst case makespan over all duration realizations varying through a
polyhedral uncertainty set. A Benders approach is considered and a polynomially solvable
case is identified for a specific uncertainty set. In the present paper, we extend this robust
model to the MRCPSP.
2 Model Description
The project network is modeled through a directed graph defined over the set of nodes
V = 0, ..., n + 1. The dummy nodes 0 and n + 1 represent the start and end of the project,
37

receptively. The other nodes are non-dummy activities. The set of arcs E represents prece-
dence relations between the activities. We assume K types of renewable resources, with
a finite capacity per period denoted by Rk. Each activity j can be performed in one of
|Mj| modes, where each mode mj ∈ Mj is characterized by a duration djm and a resource
requirement of type k, rjmj k. A solution to the MRCPSP is a vector of mode combinations
(m1, ..., mn) and a vector of non negative starting times (S0, ..., Sn+1) which result in a
schedule that satisfies the precedence and resource constraints. Given a mode combination
m = (m1, ..., mn), we define Fm ⊆ V to be any subset of activities without precedence re-
lations between them such that
P
i∈Fm
rimik Rk for at least one k ∈ K. This set is called
forbidden since its activities cannot be performed in parallel because of resource conflicts.
We denote Fm as a minimal forbidden set that corresponds to a mode combination m, such
that each of its subsets is not a forbidden set. The MRCPSP solution can be reduced to an
optimal mode combination m and a optimal selection of the set Xm ⊆ (V × V )E of extra
precedences such that the extended graph G′
(V, E∪Xm) is acyclic and Fm(T(E∪Xm)) = ∅
where T(A) denotes the transitive closure of the set A. We assume that the uncertain data
varies within a so-called uncertainty set. A robust feasible solution guarantees that there are
no violations of constraints for all possible realizations within a considered uncertainty set.
An optimal robust solution is one that solves the robust optimization problem. This new
optimization problem is called the robust counterpart. Tractability of robust counterparts
strongly depends on the uncertainty set’s nature. Ben-Tal et al. (2009) show that a robust
counterpart of an uncertain linear problem is also linear under a polyhedral uncertainty
set. A typical example of a polyhedral set is the case of interval uncertainty, also called a
Box. For a non-polyhedral set, such as the case of ellipsoidal uncertainty, Ben-Tal et al.
(2009) show that a robust counterpart of an uncertain linear problem is quadratic. Since we
formulate our problem with integer variables, we assume that uncertainty sets are polyhe-
dral in order to maintain linear constraints. In our model, uncertain durations are defined
over the polyhedral uncertainty set θ ⊆ Nn×M
. For convenience, we denote the subset θm
as the uncertain duration’s support according to a given mode assignment. Indeed, for a
given mode combination m, the corresponding durations vector is dm = (djm1
, ..., djmn
)
which is included in NJ
. The set of combination modes is denoted by M ⊆ Nn
.
We define the robust multi-mode resource constrained scheduling problem (RMRCPSP) as
a robust optimization problem. The objective is to find a mode assignment and a sufficient
selection that minimizes the worst case makespan under uncertainty:
min
m∈M,Xm∈Xm,S(·)
max
dm∈θm
Sn+1(dm) (1)
S0 = 0 (2)
Sj(dm) − Si(dm) ≥ dimi
, ∀(i, j) ∈ E ∪ Xm , ∀dm ∈ θm (3)
The mode and selection decisions, m ∈ M and Xm ∈ Xm respectively, represent a first-
stage decisions that made before the project’s execution. That is, before activity durations
are known. The second-stage decisions concern the starting times Sj(dm) of each activity
under the duration realization dm ∈ θm. When the uncertainty set is a box, it can be
shown that solving the RMRCPSP is equivalent to solving a deterministic MRCPSP for
the worst-case activity duration vector.
3 Development of an Analytical Solution Approach
The structure of the RMRCPSP encourages us to use a Benders’ solution approach
for solving it (Benders, 1962). Bender’s decomposition algorithm is an iterative algorithm;
at the initial iteration, the lower bound of the objective equals −∞ and its upper bound
equals ∞. At each iteration, we solve a master problem that provides an updated lower
bound and a subproblem that provides an updated upper bound. Once the subproblem
38

is solved, valid cuts are calculated and added to the master problem formulation. The
algorithm stops when the lower bound converges to the upper bound.
3.1 The Master Problem
The master problem determines mode and sufficient selections, and its objective is
to minimize the lower bound of the worst case makespan. The mode selection decisions
variables are binary and denoted by xjmj
(equals to 1 if activity j is executed under mode
mj ∈ Mj). The variables about sufficient selections are modeled by resources flow variables,
fijk, corresponding to the number of resources k units transfered from activity i to activity
j and by binary variables yij, representing all the precedence relations in E∪Xm (including
its transitive closure). In order to improve the computational performance of the master
problem, we incorporate relaxation.
3.2 The Subproblem
After selecting the modes and sequencing activities in the master problem, without the
necessity to consider their durations and the uncertainty set, now, we have to schedule the
activities in the subproblem. The objective is to minimize the worst case makespan when
uncertain durations are defined over a polyhedral uncertainty set. The optimal solution
of the master problem at iteration t determines an acyclic subgraph G′
(V, Ut
) where the
set Ut
is defined as follows: Ut
= {(i, j) ∈ V × V : y∗t
ij = 1}. We accordingly update
Xm∗t , the optimal selections at iteration t, when m∗t
= (m∗t
1 , ..., m∗t
n ) denotes the optimal
mode assignment resulted from the master problem at t. Then, a feasible solution for the
RMRCPSP can be determined by solving the following subproblem:
min
S(·)
max
d∈θm∗t
SJ+1(d) (4)
S0 = 0 (5)
Sj(d) − Si(d) ≥ djmj , ∀(i, j) ∈ Ut
, ∀d ∈ θm∗t (6)
We can rewrite the subproblem as:
max
d∈θm∗t
min
S∈Ω(Xm∗t ,θm∗t )
SJ+1(d) (7)
where: Ω(Xm∗t , θm∗t ) = {S ∈ Rn+2
+ : S0 = 0 , Sj − Si ≥ djm∗t
j
, ∀(i, j) ∈ Ut
}, is a set of
activities’ starting times. Using a strong duality result (Beck and Ben-Tal, 2009), we state
that the optimizing under the worst-case makespan in the primal (7), at a generic iteration
t, is equivalent to optimizing under the best case in the dual. The objective function of the
dual problem is non-linear. Then, we focus on the budgeted uncertainty set inspired by
Bertsimas and Sim (2003). The advantage of this polyhedral set is in its flexibility to adjust
the level of conservatism and robustness through the budget Γ, representing the number
of activities which are allowed to deviate from their nominal durations. The parameter Γ
can vary between 0 and n. We assume that each activity duration j performed in mode mj
has a nominal value, ˆ
djmj
and a maximal deviation denoted by ˜
djmj
. Given an optimal
mode combination m∗t
= (m∗t
1 , ..., m∗t
n ), the uncertainty set is defined as:
θm∗t = {djm∗t
j
|j ∈ V, djm∗t
j
= ˆ
djm∗t
j
+ ξj
˜
djm∗t
j
, 0 ≤ ξj ≤ 1,
X
j∈V
ξj ≤ Γ}.
Under this uncertainty set, we can reformulate subproblem (7) as a mixed-integer linear
problem.
39

3.3 Optimality Cuts
Once the subproblem is solved, two valid cuts are calculated and incorporated to the
master problem.
Proposition 1. Given a finite global lower bound L of the problem (1)-(3), and the optimal
solutions at iteration t, x∗t
, y∗t
, M∗t
, the following constraints are valid optimality cuts.
η ≥ (M∗t
− L) ·
X
(i,j)∈Xm∗t
[1/3 · (yij + xm∗t
i
+ xm∗t
j
) − N · (3 − yij − xm∗t
i
− xm∗t
j
)] (8)
− (M∗t
− L)(|Xm∗t | − 1) + L ,
when N is a large number.
Proof. It always holds that:
P
(i,j)∈Xm∗t
[1/3 · (yij + xm∗t
i
+ xm∗t
j
) − N · (3 − yij − xm∗t
i
− xm∗t
j
)] ≤ |Xm∗t |, with equality
only when x = x∗t
and y = y∗t
.
In this case, we have that:
P
(i,j)∈Xm∗t
[1/3 · (yij + xm∗t
i
+ xm∗t
j
) − N · (3 − yij − xm∗t
i
− xm∗t
j
)] − |Xm∗t | = 0 , and then
the right-hand side takes the value M∗t
.
Otherwise,
P
(i,j)∈Xm∗t
[1/3 · (yij + xm∗t
i
+ xm∗t
j
) − N · (3 − yij − xm∗t
i
− xm∗t
j
)] |Xm∗t |,
and then the right-hand side takes a value less than or equal to L.
Proposition 2. The number of cuts that can be added to the master problem is finite, and
then the procedure is finite.
Proof. Proposition 1 in Laporte and Louveaux (1993). We can apply this result here be-
cause x and y are integer variables.
Summary
This research formulates, for the first time to the best of our knowledge, the robust
MRPCPSP and develops an analytical solution approach.
Acknowledgements
We thank the Israeli Ministry of Science and Technology for supporting our research.
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41

❙❝❤❡❞✉❧✐♥❣ ❞❛t❛ ❣❛t❤❡r✐♥❣ ✇✐t❤ ❧✐♠✐t❡❞ ❜❛s❡ st❛t✐♦♥
♠❡♠♦r②
❏♦❛♥♥❛ ❇❡r❧✐➠s❦❛
❋❛❝✉❧t② ♦❢ ▼❛t❤❡♠❛t✐❝s ❛♥❞ ❈♦♠♣✉t❡r ❙❝✐❡♥❝❡✱ ❆❞❛♠ ▼✐❝❦✐❡✇✐❝③ ❯♥✐✈❡rs✐t② ✐♥ P♦③♥❛➠✱
❯♠✉❧t♦✇s❦❛ ✽✼✱ ✻✶✲✻✶✹ P♦③♥❛➠✱ P♦❧❛♥❞
❏♦❛♥♥❛✳❇❡r❧✐♥s❦❛❅❛♠✉✳❡❞✉✳♣❧
❑❡②✇♦r❞s✿ s❝❤❡❞✉❧✐♥❣✱ ❞❛t❛ ❣❛t❤❡r✐♥❣ ♥❡t✇♦r❦✱ ❧✐♠✐t❡❞ ♠❡♠♦r②✱ ✢♦✇ s❤♦♣✳
✶ ■♥tr♦❞✉❝t✐♦♥
❉❛t❛ ❣❛t❤❡r✐♥❣ ♥❡t✇♦r❦s ❛r❡ ✇✐❞❡❧② ✉s❡❞ ✐♥ ♠❛♥② t②♣❡s ♦❢ ❝♦♥t❡♠♣♦r❛r② ❛♣♣❧✐❝❛t✐♦♥s✳
❉✐str✐❜✉t❡❞ ❝♦♠♣✉t✐♥❣ ✐♥tr♦❞✉❝❡s t❤❡ ♥❡❡❞ ❢♦r ❝♦❧❧❡❝t✐♥❣ t❤❡ r❡s✉❧ts ♦❜t❛✐♥❡❞ ❜② r❡♠♦t❡
✇♦r❦❡rs✳ ❉❛t❛ ❣❛t❤❡r✐♥❣ ✇✐r❡❧❡ss s❡♥s♦r ♥❡t✇♦r❦s ✜♥❞ ❡♥✈✐r♦♥♠❡♥t❛❧✱ ♠✐❧✐t❛r②✱ ❤❡❛❧t❤ ❛♥❞
❤♦♠❡ ❛♣♣❧✐❝❛t✐♦♥s ✭❆❦②✐❧❞✐③ ❡t ❛❧✳ ✷✵✵✷✮✳ ❙❝❤❡❞✉❧✐♥❣ ❛❧❣♦r✐t❤♠s ❢♦r ❞❛t❛ ❣❛t❤❡r✐♥❣ ♥❡t✲
✇♦r❦s ✇❡r❡ ♣r♦♣♦s❡❞✱ ❡✳❣✳✱ ❜② ▼♦❣❡s ❛♥❞ ❘♦❜❡rt❛③③✐ ✭✷✵✵✻✮✱ ❈❤♦✐ ❛♥❞ ❘♦❜❡rt❛③③✐ ✭✷✵✵✽✮✱
❇❡r❧✐➠s❦❛ ✭✷✵✶✹✮ ❛♥❞ ❇❡r❧✐➠s❦❛ ✭✷✵✶✺✮✳
■♥ t❤✐s ✇♦r❦✱ ✇❡ ❛♥❛❧②③❡ ❣❛t❤❡r✐♥❣ ❞❛t❛ ✐♥ ❛ ♥❡t✇♦r❦ ✇✐t❤ ❧✐♠✐t❡❞ ❜❛s❡ st❛t✐♦♥ ♠❡♠♦r②✳
❆ ❞❛t❛s❡t ❜❡✐♥❣ r❡❝❡✐✈❡❞ ♦r ♣r♦❝❡ss❡❞ ❜② t❤❡ ❜❛s❡ st❛t✐♦♥ ♦❝❝✉♣✐❡s ❛ ❜❧♦❝❦ ♦❢ ♠❡♠♦r② ♦❢
❣✐✈❡♥ s✐③❡✳ ❚❤❡ t♦t❛❧ s✐③❡ ♦❢ ❝♦❡①✐st✐♥❣ ♠❡♠♦r② ❜❧♦❝❦s ❝❛♥♥♦t ❡①❝❡❡❞ t❤❡ ❜❛s❡ st❛t✐♦♥ ❜✉✛❡r
❝❛♣❛❝✐t②✳ ❖✉r ❣♦❛❧ ✐s t♦ ❣❛t❤❡r ❛♥❞ ♣r♦❝❡ss ❛❧❧ ❞❛t❛ ✇✐t❤✐♥ t❤❡ ♠✐♥✐♠✉♠ ♣♦ss✐❜❧❡ t✐♠❡✳
✷ Pr♦❜❧❡♠ ❢♦r♠✉❧❛t✐♦♥ ❛♥❞ ❝♦♠♣❧❡①✐t②
❲❡ st✉❞② ❛ ❞❛t❛ ❣❛t❤❡r✐♥❣ ♥❡t✇♦r❦ ❝♦♥s✐st✐♥❣ ♦❢ m ✐❞❡♥t✐❝❛❧ ✇♦r❦❡r ♥♦❞❡s P1, . . . , Pm
❛♥❞ ❛ s✐♥❣❧❡ ❜❛s❡ st❛t✐♦♥✳ ◆♦❞❡ Pi ❤❛s t♦ tr❛♥s❢❡r ❞❛t❛s❡t Di ♦❢ s✐③❡ αi ❞✐r❡❝t❧② t♦ t❤❡ ❜❛s❡
st❛t✐♦♥✳ ❲❤❡♥ ❞❛t❛s❡t Di st❛rts ❜❡✐♥❣ s❡♥t✱ ❛ ♠❡♠♦r② ❜❧♦❝❦ ♦❢ s✐③❡ αi ✐s ❛❧❧♦❝❛t❡❞ ❛t t❤❡
❜❛s❡ st❛t✐♦♥✳ ❚❤❡ ❜❛s❡ st❛t✐♦♥ ❤❛s ❧✐♠✐t❡❞ ♠❡♠♦r② ♦❢ s✐③❡ B ≥ maxm
i=1{αi}✳ ❚❤❡ tr❛♥s❢❡r ♦❢
❞❛t❛s❡t Di ♠❛② st❛rt ♦♥❧② ✐❢ t❤❡ ❛♠♦✉♥t ♦❢ ❛✈❛✐❧❛❜❧❡ ♠❡♠♦r② ✐s ❛t ❧❡❛st αi✳ ❙❡♥❞✐♥❣ ❞❛t❛s❡t
Di t❛❦❡s t✐♠❡ Cαi✱ ✇❤❡r❡ C ✐s t❤❡ ♥❡t✇♦r❦ ❝♦♠♠✉♥✐❝❛t✐♦♥ r❛t❡ ✭✐♥✈❡rs❡ ♦❢ s♣❡❡❞✮✳ ❆❢t❡r
❞❛t❛s❡t Di ✐s tr❛♥s❢❡rr❡❞✱ ✐t ❤❛s t♦ ❜❡ ♣r♦❝❡ss❡❞ ❜② t❤❡ ❜❛s❡ st❛t✐♦♥✳ ❚❤✐s t❛❦❡s t✐♠❡ Aαi✱
✇❤❡r❡ A ✐s t❤❡ ❜❛s❡ st❛t✐♦♥ ❝♦♠♣✉t❛t✐♦♥ r❛t❡✳ ❉❛t❛s❡ts ❛r❡ ♣r♦❝❡ss❡❞ ✐♥ t❤❡ ♦r❞❡r ✐♥ ✇❤✐❝❤
t❤❡② ✇❡r❡ r❡❝❡✐✈❡❞✳ ❆s s♦♦♥ ❛s ♣r♦❝❡ss✐♥❣ ❛ ❞❛t❛s❡t ✜♥✐s❤❡s✱ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ♠❡♠♦r②
❜❧♦❝❦ ✐s r❡❧❡❛s❡❞✳ ■t ✐s ❛ss✉♠❡❞ t❤❛t ❜♦t❤ ❝♦♠♠✉♥✐❝❛t✐♦♥ ❛♥❞ ❝♦♠♣✉t❛t✐♦♥ ♦♥ ❛ ❞❛t❛s❡t
❛r❡ ♥♦♥✲♣r❡❡♠♣t✐✈❡✳ ❚❤❡ ❜❛s❡ st❛t✐♦♥ ❝❛♥ ❝♦♠♠✉♥✐❝❛t❡ ✇✐t❤ ❛t ♠♦st ♦♥❡ ♥♦❞❡ ❛t ❛ t✐♠❡
❛♥❞ ✐t ❝❛♥ ♣r♦❝❡ss ❛t ♠♦st ♦♥❡ ❞❛t❛s❡t ❛t ❛ t✐♠❡✳ ❚❤❡ s❝❤❡❞✉❧✐♥❣ ♣r♦❜❧❡♠ ✐s t♦ ♦r❣❛♥✐③❡
❞❛t❛s❡t tr❛♥s❢❡rs s♦ t❤❛t t❤❡ t♦t❛❧ ❞❛t❛ ❣❛t❤❡r✐♥❣ ❛♥❞ ♣r♦❝❡ss✐♥❣ t✐♠❡ ✐s ♠✐♥✐♠✐③❡❞✳
❲❡ ♣r♦✈❡ t❤❛t t❤❡ ❛❜♦✈❡ ♣r♦❜❧❡♠ ✐s str♦♥❣❧② ◆P✲❤❛r❞ ❡✈❡♥ ✐❢ A = C = 1✱ ✉s✐♥❣ ❛
♣s❡✉❞♦♣♦❧②♥♦♠✐❛❧ r❡❞✉❝t✐♦♥ ❢r♦♠ t❤❡ ❜✐♥ ♣❛❝❦✐♥❣ ♣r♦❜❧❡♠ ✭●❛r❡② ❛♥❞ ❏♦❤♥s♦♥ ✶✾✼✾✮✳
✸ ❘❡❧❛t❡❞ ✇♦r❦
❆s ♦♥❧② ♦♥❡ ♥♦❞❡ ❝❛♥ ❝♦♠♠✉♥✐❝❛t❡ ✇✐t❤ t❤❡ ❜❛s❡ st❛t✐♦♥ ❛t ❛ t✐♠❡✱ ♦✉r ❞❛t❛ ❣❛t❤❡r✐♥❣
♥❡t✇♦r❦ ❝❛♥ ❜❡ s❡❡♥ ❛s ❛ t✇♦✲♠❛❝❤✐♥❡ ✢♦✇ s❤♦♣✱ ✇❤❡r❡ t❤❡ ❝♦♠♠✉♥✐❝❛t✐♦♥ ♥❡t✇♦r❦ ✐s t❤❡
✜rst ♠❛❝❤✐♥❡✱ ❛♥❞ t❤❡ ❜❛s❡ st❛t✐♦♥ ✐s t❤❡ s❡❝♦♥❞ ♠❛❝❤✐♥❡✳ ❏♦❜ i ❝♦♥s✐sts ♦❢ t✇♦ ♦♣❡r❛t✐♦♥s✿
s❡♥❞✐♥❣ ❛♥❞ ♣r♦❝❡ss✐♥❣ ❞❛t❛s❡t Di✱ ❛♥❞ r❡q✉✐r❡s αi ✉♥✐ts ♦❢ ❜❛s❡ st❛t✐♦♥ ♠❡♠♦r② r❡s♦✉r❝❡✳
❚❤✉s✱ ✇❡ s♦❧✈❡ ❛ r❡s♦✉r❝❡✲❝♦♥str❛✐♥❡❞ ✢♦✇ s❤♦♣ s❝❤❡❞✉❧✐♥❣ ♣r♦❜❧❡♠ ✭❇➟❛➺❡✇✐❝③ ❡t ❛❧✳ ✶✾✽✸✮✳
■t ♠❛② s❡❡♠ s✐♠✐❧❛r t♦ t✇♦✲♠❛❝❤✐♥❡ ✢♦✇ s❤♦♣ s❝❤❡❞✉❧✐♥❣ ✇✐t❤ ❧✐♠✐t❡❞ ❜✉✛❡r st♦r❛❣❡ ✭s❡❡✱
42

❡✳❣✳✱ ▲❡✐st❡♥ ✭✶✾✾✵✮✮✱ ❜✉t t❤❡r❡ ❛r❡ s✉❜st❛♥t✐❛❧ ❞✐✛❡r❡♥❝❡s ❜❡t✇❡❡♥ t❤❡♠✳ ■♥ ❛ ✢♦✇ s❤♦♣
✇✐t❤ ❧✐♠✐t❡❞ ❜✉✛❡r st♦r❛❣❡✱ t❤❡ ❜✉✛❡r ❝❛♥ ❤♦❧❞ ❛ ✜①❡❞ ♥✉♠❜❡r ♦❢ ❥♦❜s✱ ❛♥❞ ❛ ❥♦❜ ✐s st♦r❡❞
✐♥ t❤❡ ❜✉✛❡r ✇❤❡♥ ✐t ❤❛s ❛❧r❡❛❞② ❜❡❡♥ ♣r♦❝❡ss❡❞ ♦♥ t❤❡ ✜rst ♠❛❝❤✐♥❡ ❜✉t ♥♦t ②❡t st❛rt❡❞
♦♥ t❤❡ s❡❝♦♥❞ ♠❛❝❤✐♥❡✳ ■♥ ♦✉r ♣r♦❜❧❡♠✱ t❤❡ ❜✉✛❡r ❝❛♥ ❤♦❧❞ ❛ ✜①❡❞ ❛♠♦✉♥t ♦❢ ❞❛t❛ ✭❢♦r
❡①❛♠♣❧❡✱ ♦♥❧② ♦♥❡ ❜✐❣ ❞❛t❛s❡t✱ ❜✉t ✉♣ t♦ t❤r❡❡ s♠❛❧❧ ❞❛t❛s❡ts✮✱ ❛♥❞ t❤❡ ❜✉✛❡r ✐s ♦❝❝✉♣✐❡❞
❜② ❛ ❞❛t❛s❡t ♥♦t ♦♥❧② ❜❡t✇❡❡♥✱ ❜✉t ❛❧s♦ ❞✉r✐♥❣ ✐ts tr❛♥s❢❡r ❛♥❞ ♣r♦❝❡ss✐♥❣✳
▲✐♥ ❛♥❞ ❍✉❛♥❣ ✭✷✵✵✻✮ ♣r♦♣♦s❡❞ ❛ r❡❧♦❝❛t✐♦♥ ♣r♦❜❧❡♠ ✇✐t❤ ❛ s❡❝♦♥❞ ✇♦r❦✐♥❣ ❝r❡✇ ❢♦r
r❡s♦✉r❝❡ r❡❝②❝❧✐♥❣✳ ❊❛❝❤ ❥♦❜ ✇❛s ❡①❡❝✉t❡❞ ♦♥ t✇♦ ♠❛❝❤✐♥❡s ✐♥ ❛ ✢♦✇ s❤♦♣ st②❧❡✳ ❚❤❡ i✲t❤
❥♦❜ r❡q✉✐r❡❞ αi ✉♥✐ts ♦❢ ❛ r❡s♦✉r❝❡✱ ❛♥❞ r❡t✉r♥❡❞ βi ✉♥✐ts ♦❢ t❤✐s r❡s♦✉r❝❡ ♦♥ ❝♦♠♣❧❡t✐♦♥✳
❚❤❡ ❣♦❛❧ ✇❛s t♦ ♠✐♥✐♠✐③❡ t❤❡ ♠❛❦❡s♣❛♥ ✇❤✐❧❡ ♥♦t ❡①❝❡❡❞✐♥❣ t❤❡ ❛✈❛✐❧❛❜❧❡ ❛♠♦✉♥t ♦❢ t❤❡
r❡s♦✉r❝❡✳ ❚❤✐s ♣r♦❜❧❡♠✱ ❞❡♥♦t❡❞ ❜② F2|rp|Cmax✱ ✇❛s s❤♦✇♥ t♦ ❜❡ str♦♥❣❧② ◆P✲❤❛r❞✱ ❛♥❞
❤❡✉r✐st✐❝ ❛❧❣♦r✐t❤♠s ❢♦r s♦❧✈✐♥❣ ✐t ✇❡r❡ ♣r♦♣♦s❡❞✳ ❚❤❡ ♣r♦❜❧❡♠ ✇❛s ❢✉rt❤❡r ❛♥❛❧②③❡❞ ❜②
❈❤❡♥❣ ❡t ❛❧✳ ✭✷✵✶✷✮✱ ✇❤♦ ❢♦r♠✉❧❛t❡❞ ✐t ❛s ❛♥ ✐♥t❡❣❡r ❧✐♥❡❛r ♣r♦❣r❛♠✳ ❈♦♠♣❧❡①✐t② r❡s✉❧ts
❢♦r ❛ ♥✉♠❜❡r ♦❢ s♣❡❝✐❛❧ ❝❛s❡s ♦❢ t❤❡ ♣r♦❜❧❡♠ ✇❡r❡ ❛❧s♦ ♣r❡s❡♥t❡❞✳
❖✉r ❞❛t❛ ❣❛t❤❡r✐♥❣ s❝❤❡❞✉❧✐♥❣ ♣r♦❜❧❡♠ ✐s ❡q✉✐✈❛❧❡♥t t♦ ②❡t ❛♥♦t❤❡r s♣❡❝✐❛❧ ❝❛s❡ ♦❢
♣r♦❜❧❡♠ F2|rp|Cmax✱ ✇❤✐❝❤ ❝❛♥ ❜❡ ❞❡♥♦t❡❞ ❜② F2|rp, pi = Cαi, qi = Aαi, βi = αi|Cmax✱
❛♥❞ ✇❛s ♥♦t st✉❞✐❡❞ ✐♥ t❤❡ ❡❛r❧✐❡r ❧✐t❡r❛t✉r❡✳
✹ ❆❧❣♦r✐t❤♠s
■♥ ♦✉r ♣r♦❜❧❡♠✱ ❛ s❝❤❡❞✉❧❡ ✐s ❞❡t❡r♠✐♥❡❞ ❜② t❤❡ ♦r❞❡r ✐♥ ✇❤✐❝❤ t❤❡ ❞❛t❛s❡ts ❛r❡ tr❛♥s❢❡r✲
r❡❞ t♦ t❤❡ ❜❛s❡ st❛t✐♦♥✳ ❊❛❝❤ ❞❛t❛s❡t ✐s s❡♥t ✇✐t❤♦✉t ✉♥♥❡❝❡ss❛r② ❞❡❧❛②✱ ❛s s♦♦♥ ❛s s✉✣❝✐❡♥t
❛♠♦✉♥t ♦❢ ♠❡♠♦r② ✐s ❛✈❛✐❧❛❜❧❡✳
❲❡ ✜rst ♦❜s❡r✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ s②♠♠❡tr② ♣r♦♣❡rt②✳ ❙✉♣♣♦s❡ t❤❛t A = kC✱ ✇❤❡r❡ k ≥ 1✱
❛♥❞ Σ ✐s ❛ s❝❤❡❞✉❧❡ ♦❢ ❧❡♥❣t❤ T ❢♦r ❣✐✈❡♥ ✈❛❧✉❡s ♦❢ B ❛♥❞ (αi)m
i=1✳ ❚❤❡♥✱ ❜② r❡✈❡rs✐♥❣
s❝❤❡❞✉❧❡ Σ✱ ✇❡ ♦❜t❛✐♥ ❛ s❝❤❡❞✉❧❡ ♦❢ ❧❡♥❣t❤ T ❢♦r t❤❡ s❛♠❡ ✈❛❧✉❡s ♦❢ B ❛♥❞ (αi)m
i=1✱ ❝♦♠✲
♠✉♥✐❝❛t✐♦♥ r❛t❡ C′
= kC ❛♥❞ ❝♦♠♣✉t❛t✐♦♥ r❛t❡ A′
= C✳ ■♥ ❝♦♥s❡q✉❡♥❝❡✱ ✇❡ ❝❛♥ ❛ss✉♠❡
✇✐t❤♦✉t ❧♦ss ♦❢ ❣❡♥❡r❛❧✐t② t❤❛t A ≥ C✳
❲❡ ♣r♦♣♦s❡ t❤r❡❡ ✏s✐♠♣❧❡✑ ❤❡✉r✐st✐❝s✳ ❆❧❣♦r✐t❤♠ ■♥❝ s♦rts t❤❡ ❞❛t❛s❡ts ✐♥ t❤❡ ♦r❞❡r ♦❢
✐♥❝r❡❛s✐♥❣ s✐③❡s✳ ❙✐♥❝❡ A ≥ C✱ t❤✐s ✐s t❤❡ ♦r❞❡r t❤❛t ✇♦✉❧❞ ❜❡ r❡t✉r♥❡❞ ❜② t❤❡ ❏♦❤♥s♦♥✬s
❛❧❣♦r✐t❤♠ ❢♦r ♣r♦❜❧❡♠ F2||Cmax ✭❏♦❤♥s♦♥ ✶✾✺✹✮✱ ❛♥❞ ❤❡♥❝❡✱ ❛❧❣♦r✐t❤♠ ■♥❝ ❞❡❧✐✈❡rs ♦♣t✐✲
♠✉♠ s♦❧✉t✐♦♥s ✐❢ t❤❡ ♠❡♠♦r② ❧✐♠✐t B ✐s ❜✐❣ ❡♥♦✉❣❤✳ ❆❧❣♦r✐t❤♠ ❆❧t❡r st❛rts ✇✐t❤ s❡♥❞✐♥❣
t❤❡ s♠❛❧❧❡st ❞❛t❛s❡t✱ t❤❡♥ t❤❡ ❣r❡❛t❡st ♦♥❡✱ t❤❡ s❡❝♦♥❞ s♠❛❧❧❡st✱ t❤❡ s❡❝♦♥❞ ❣r❡❛t❡st✱ ❡t❝✳✱
t❤✉s ❛❧t❡r♥❛t✐♥❣ ❜✐❣ ❛♥❞ s♠❛❧❧ ❞❛t❛s❡ts✳ ❋✐♥❛❧❧②✱ ❛❧❣♦r✐t❤♠ ❘♥❞ tr❛♥s❢❡rs t❤❡ ❞❛t❛s❡ts ✐♥
❛ r❛♥❞♦♠ ♦r❞❡r✳ ❚❤✐s ❛❧❣♦r✐t❤♠ ✇✐❧❧ ❜❡ ✉s❡❞ t♦ ✈❡r✐❢② t❤❡ q✉❛❧✐t② ♦❢ t❤❡ r❡s✉❧ts ❞❡❧✐✈❡r❡❞
❜② t❤❡ r❡♠❛✐♥✐♥❣ ❤❡✉r✐st✐❝s✳
❚❤❡ s❡❝♦♥❞ ❣r♦✉♣ ♦❢ ❛❧❣♦r✐t❤♠s ❛r❡ ✏❛❞✈❛♥❝❡❞✑ ❤❡✉r✐st✐❝s ■♥❝▲♦❝❛❧✱ ❆❧t❡r▲♦❝❛❧ ❛♥❞
❘♥❞▲♦❝❛❧✳ ❊❛❝❤ ♦❢ t❤❡♠ st❛rts ✇✐t❤ ❣❡♥❡r❛t✐♥❣ ❛ s❝❤❡❞✉❧❡ ✉s✐♥❣ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ s✐♠♣❧❡
❤❡✉r✐st✐❝✱ ❛♥❞ t❤❡♥ ❛♣♣❧✐❡s t❤❡ ❢♦❧❧♦✇✐♥❣ ❧♦❝❛❧ s❡❛r❝❤ ♣r♦❝❡❞✉r❡✳ ❋♦r ❡❛❝❤ ♣❛✐r ♦❢ ❞❛t❛s❡ts✱ ✇❡
❝❤❡❝❦ ✐❢ s✇❛♣♣✐♥❣ t❤❡✐r ♣♦s✐t✐♦♥s ✐♥ t❤❡ ❝✉rr❡♥t s❝❤❡❞✉❧❡ ❧❡❛❞s t♦ ❞❡❝r❡❛s✐♥❣ t❤❡ ♠❛❦❡s♣❛♥✳
❚❤❡ s✇❛♣ t❤❛t r❡s✉❧ts ✐♥ t❤❡ s❤♦rt❡st s❝❤❡❞✉❧❡ ✐s ❡①❡❝✉t❡❞✱ ❛♥❞ t❤❡ s❡❛r❝❤ ✐s ❝♦♥t✐♥✉❡❞
✉♥t✐❧ ♥♦ ❢✉rt❤❡r ✐♠♣r♦✈❡♠❡♥t ✐s ♣♦ss✐❜❧❡✳
◆♦t❡ t❤❛t ♦✉r ❛❧❣♦r✐t❤♠s ❝♦✈❡r t❤❡ t❤r❡❡ ❤❡✉r✐st✐❝s H1, H2, H3 ♣r♦♣♦s❡❞ ❜② ▲✐♥ ❛♥❞
❍✉❛♥❣ ✭✷✵✵✻✮ ❢♦r s♦❧✈✐♥❣ ♣r♦❜❧❡♠ F2|rp|Cmax✳ ■♥ ♦✉r s♣❡❝✐❛❧ ❝❛s❡✱ ❜♦t❤ H1 ❛♥❞ H2 r❡t✉r♥
t❤❡ s❛♠❡ r❡s✉❧ts ❛s ■♥❝▲♦❝❛❧✱ ❛♥❞ H3 ✐s ❡q✉✐✈❛❧❡♥t t♦ ❘♥❞▲♦❝❛❧✳
❚♦ ✜♥✐s❤ t❤✐s s❡❝t✐♦♥✱ ❧❡t ✉s ♦❜s❡r✈❡ t❤❛t t❤❡ ❧❡♥❣t❤ ♦❢ ❛ s❝❤❡❞✉❧❡ ♦❜t❛✐♥❡❞ ❢♦r ❛♥
❛r❜✐tr❛r② ❞❛t❛s❡t s❡q✉❡♥❝❡ ❞♦❡s ♥♦t ❡①❝❡❡❞ (A + C)
Pm
i=1 αi✱ ❛♥❞ A
Pm
i=1 αi ✐s ❛ ❧♦✇❡r
❜♦✉♥❞ ♦♥ ❛ s❝❤❡❞✉❧❡ ❧❡♥❣t❤✳ ❚❤✉s✱ t❤❡ ❛♣♣r♦①✐♠❛t✐♦♥ r❛t✐♦ ♦❢ ❛♥② ❛❧❣♦r✐t❤♠ ❢♦r s♦❧✈✐♥❣
♦✉r ♣r♦❜❧❡♠ ✐s ❛t ♠♦st 1 + C/A✳ ❍❡♥❝❡✱ ✇❡ ❝❛♥ s❛② t❤❛t ♦✉r ♣r♦❜❧❡♠ ❜❡❝♦♠❡s ❡❛s✐❡r t♦
s♦❧✈❡ ✇❤❡♥ A ❣❡ts ❧❛r❣❡ ✐♥ ❝♦♠♣❛r✐s♦♥ t♦ C✳
43

✺ ❊①♣❡r✐♠❡♥t❛❧ r❡s✉❧ts
■♥ t❤✐s s❡❝t✐♦♥✱ ✇❡ ❝♦♠♣❛r❡ t❤❡ q✉❛❧✐t② ♦❢ t❤❡ s♦❧✉t✐♦♥s ❛♥❞ t❤❡ ❝♦♠♣✉t❛t✐♦♥❛❧ ❝♦sts
♦❢ t❤❡ ♣r♦♣♦s❡❞ ❤❡✉r✐st✐❝s✳ ❚❤❡ ❛❧❣♦r✐t❤♠s ✇❡r❡ ✐♠♣❧❡♠❡♥t❡❞ ✐♥ ❈✰✰ ❛♥❞ r✉♥ ♦♥ ❛♥ ■♥t❡❧
❈♦r❡ ✐✺✲✷✺✵✵❑ ❈P❯ ❅ ✸✳✸✵ ●❍③ ✇✐t❤ ✻●❇ ❘❆▼✳ ❚❤❡ t❡st ✐♥st❛♥❝❡s ✇❡r❡ ❝♦♥str✉❝t❡❞ ❛s
❢♦❧❧♦✇s✳ ❚❤❡ ❝♦♠♠✉♥✐❝❛t✐♦♥ r❛t❡ ✇❛s C = 1 ❛♥❞ t❤❡ ❝♦♠♣✉t❛t✐♦♥ r❛t❡ ✇❛s A ∈ {1, 2, 5, 10}✳
❲❡ ❣❡♥❡r❛t❡❞ ✏s♠❛❧❧✑ t❡sts ✇✐t❤ m = 10 ❛♥❞ ✏❜✐❣✑ t❡sts ✇✐t❤ m = 100✳ ❚❤❡ ❞❛t❛s❡t s✐③❡s
αi ✇❡r❡ ❝❤♦s❡♥ r❛♥❞♦♠❧② ❢r♦♠ t❤❡ ✐♥t❡r✈❛❧ [1, 2]✳ ❋♦r ❛ ❣✐✈❡♥ s❡t ♦❢ αi✱ ✇❡ ❝♦♠♣✉t❡❞ t❤❡
♠✐♥✐♠✉♠ ❛♠♦✉♥t ♦❢ ♠❡♠♦r② t❤❛t ❛❧❧♦✇s t♦ ❤♦❧❞ ♠♦r❡ t❤❛♥ ♦♥❡ ❞❛t❛s❡t ✐♥ t❤❡ ❜✉✛❡r✱
Bmin = mini6=j{αi + αj}✳ ❚❤❡♥✱ t❤❡ ♠❡♠♦r② ❧✐♠✐t ✇❛s s❡t t♦ B = δBBmin✱ ✇❤❡r❡ δB =
1 + i/10✱ ❢♦r i = 1, 2, . . . , 7✳ ❋♦r ❡❛❝❤ tr✐♣❧❡ ♦❢ m✱ A ❛♥❞ δB ✈❛❧✉❡s✱ ✸✵ ✐♥st❛♥❝❡s ✇❡r❡
❣❡♥❡r❛t❡❞✳ ❉✉❡ t♦ ❧✐♠✐t❡❞ s♣❛❝❡✱ ✇❡ r❡♣♦rt ❤❡r❡ ♦♥❧② ♦♥ ❛ s♠❛❧❧ s✉❜s❡t ♦❢ t❤❡ ♦❜t❛✐♥❡❞
r❡s✉❧ts✳
❚❤❡ ♠❛❦❡s♣❛♥s r❡t✉r♥❡❞ ❜② t❤❡ ❤❡✉r✐st✐❝s ❢♦r t❤❡ s♠❛❧❧ t❡sts ✇❡r❡ ❝♦♠♣❛r❡❞ t♦ t❤❡
♦♣t✐♠✉♠ ✈❛❧✉❡s ❝♦♠♣✉t❡❞ ✉s✐♥❣ t❤❡ ■▲P ❢♦r♠✉❧❛t✐♦♥ ❢r♦♠ ❈❤❡♥❣ ❡t ❛❧✳ ✭✷✵✶✷✮✳ ■t t✉r♥s
♦✉t t❤❛t t❤❡ ❧♦❝❛❧ s❡❛r❝❤ ♣r♦❝❡❞✉r❡ ✐s ✈❡r② ❡✛❡❝t✐✈❡✱ ❛s ❢♦r ❡❛❝❤ t❡st❡❞ s❡tt✐♥❣✱ t❤❡ ❛✈❡r❛❣❡
r❡❧❛t✐✈❡ ❡rr♦rs ♦❢ ❛❧❧ t❤❡ ❛❞✈❛♥❝❡❞ ❤❡✉r✐st✐❝s ✇❡r❡ ❜❡❧♦✇ ✵✳✺✪✳ ❚❤❡ ❛✈❡r❛❣❡ r❡❧❛t✐✈❡ ❡rr♦rs ♦❢
t❤❡ s✐♠♣❧❡ ❛❧❣♦r✐t❤♠s ✇❡r❡ ❜❡t✇❡❡♥ ✸✪ ❛♥❞ ✷✵✪ ❢♦r t❤❡ ♠♦st ❞✐✣❝✉❧t t❡sts ✭✇✐t❤ A = 1✮✳
❈♦♥str✉❝t✐♥❣ ♦♣t✐♠✉♠ s♦❧✉t✐♦♥s ❢♦r ✐♥st❛♥❝❡s ✇✐t❤ m = 100 ✇❛s ♥♦t ♣♦ss✐❜❧❡ ❜❡❝❛✉s❡
♦❢ t❤❡ ❡①♣♦♥❡♥t✐❛❧ ❝♦♠♣❧❡①✐t② ♦❢ t❤❡ ❡①❛❝t ❛❧❣♦r✐t❤♠✳ ❚❤❡r❡❢♦r❡✱ t❤❡ ♦❜t❛✐♥❡❞ ♠❛❦❡s♣❛♥s
✇❡r❡ ❝♦♠♣❛r❡❞ t♦ t❤❡ ❧♦✇❡r ❜♦✉♥❞ ❝♦♠♣✉t❡❞ ❜② ❞✐sr❡❣❛r❞✐♥❣ t❤❡ ♠❡♠♦r② ❧✐♠✐t ❛♥❞ s♦❧✈✐♥❣
♣r♦❜❧❡♠ F2||Cmax ❢♦r ❣✐✈❡♥ A ❛♥❞ (αi)m
i=1✳ ❚❤❡ r❡s✉❧ts ❛r❡ s✉♠♠❛r✐③❡❞ ✐♥ ❚❛❜❧❡ ✶✳
❚❤❡ s♦❧✉t✐♦♥s ♦❜t❛✐♥❡❞ ❜② t❤❡ ❛❞✈❛♥❝❡❞ ❛❧❣♦r✐t❤♠s ❛r❡ ♠✉❝❤ ❜❡tt❡r t❤❛♥ t❤♦s❡ ♦❢ t❤❡
s✐♠♣❧❡ ❛❧❣♦r✐t❤♠s✳ ❚❤❡ ❞✐✛❡r❡♥❝❡s ❜❡t✇❡❡♥ ❛❧❣♦r✐t❤♠s ■♥❝▲♦❝❛❧ ❛♥❞ ❆❧t❡r▲♦❝❛❧ ❛r❡ ✈❡r②
s♠❛❧❧✱ ✇❤✐❧❡ ❘♥❞▲♦❝❛❧ ♣❡r❢♦r♠s s❧✐❣❤t❧② ✇♦rs❡✳ ❍♦✇❡✈❡r✱ ✐t ✐s ❝❧❡❛r t❤❛t ❢♦r δB = 1.2 t❤❡
❜❡st ❝❤♦✐❝❡ ❛♠♦♥❣ t❤❡ s✐♠♣❧❡ ❤❡✉r✐st✐❝s ✐s t❤❡ ■♥❝ ❛❧❣♦r✐t❤♠✱ ❛♥❞ t❤❡ r❡s✉❧ts ♦❢ ❛❧❣♦r✐t❤♠
❆❧t❡r ❛r❡ ❡✈❡♥ ✇♦rs❡ t❤❛♥ t❤♦s❡ ♦❢ t❤❡ r❛♥❞♦♠ ❛❧❣♦r✐t❤♠✳ ❋♦r δB = 1.5 ✇❡ ❤❛✈❡ t❤❡ r❡✈❡rs❡
s✐t✉❛t✐♦♥✿ ❛❧❣♦r✐t❤♠ ❆❧t❡r ✐s t❤❡ ✇✐♥♥❡r✱ ❛♥❞ ■♥❝ ✐s ❡✈❡♥ ✇♦rs❡ t❤❛♥ ❘♥❞ ✐❢ A 1✳
❚❛❜❧❡ ✶✳ ❆✈❡r❛❣❡ r❡❧❛t✐✈❡ ❞✐st❛♥❝❡ ♦❢ t❤❡ s♦❧✉t✐♦♥s ❢r♦♠ t❤❡ ❧♦✇❡r ❜♦✉♥❞✱ ❢♦r m = 100✳
A δB ■♥❝ ❆❧t❡r ❘♥❞ ■♥❝▲♦❝❛❧ ❆❧t❡r▲♦❝❛❧ ❘♥❞▲♦❝❛❧
✶ ✶✳✷ ✵✳✽✶✹ ✵✳✾✼✹ ✵✳✾✵✼ ✵✳✼✵✷ ✵✳✼✶✶ ✵✳✼✷✼
✶✳✺ ✵✳✺✺✼ ✵✳✹✻✸ ✵✳✺✽✻ ✵✳✷✶✻ ✵✳✷✶✶ ✵✳✷✹✺
✷ ✶✳✷ ✵✳✹✶✶ ✵✳✹✾✺ ✵✳✹✺✻ ✵✳✸✹✵ ✵✳✸✹✼ ✵✳✸✻✶
✶✳✺ ✵✳✷✻✷ ✵✳✵✽✷ ✵✳✷✸✶ ✵✳✵✶✺ ✵✳✵✶✺ ✵✳✵✸✾
✺ ✶✳✷ ✵✳✶✻✻ ✵✳✶✾✽ ✵✳✶✽✸ ✵✳✶✸✺ ✵✳✶✸✼ ✵✳✶✹✸
✶✳✺ ✵✳✶✵✾ ✵✳✵✹✼ ✵✳✵✾✼ ✵✳✵✵✾ ✵✳✵✶✵ ✵✳✵✶✼
✶✵ ✶✳✷ ✵✳✵✽✹ ✵✳✵✾✾ ✵✳✵✾✸ ✵✳✵✼✵ ✵✳✵✼✷ ✵✳✵✼✺
✶✳✺ ✵✳✵✺✺ ✵✳✵✷✵ ✵✳✵✹✾ ✵✳✵✵✺ ✵✳✵✵✹ ✵✳✵✵✾
❲❡ r❡♣♦rt ♦♥ t❤❡ ❡①❡❝✉t✐♦♥ t✐♠❡s ♦❢ t❤❡ ❛❧❣♦r✐t❤♠s ✐♥ ❚❛❜❧❡ ✷✳ ❍❡r❡ ✇❡ ❣r♦✉♣ t❤❡ r❡s✉❧ts
❢♦r ❛❧❧ t❡st❡❞ ✈❛❧✉❡s ♦❢ A t♦❣❡t❤❡r✳ ❚❤❡ r✉♥♥✐♥❣ t✐♠❡ ♦❢ ❛❧❧ s✐♠♣❧❡ ❛❧❣♦r✐t❤♠s ✐s ✈❡r② s❤♦rt✱
❛♥❞ t❤❡ ❛❞✈❛♥❝❡❞ ❛❧❣♦r✐t❤♠s ❛r❡ ✜✈❡ ♦r❞❡rs ♦❢ ♠❛❣♥✐t✉❞❡ s❧♦✇❡r✳ ❚❤❡ s❧♦✇❡st ❤❡✉r✐st✐❝ ✐s
❘♥❞▲♦❝❛❧✱ ❛♥❞ t❤❡ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ ■♥❝▲♦❝❛❧ ❛♥❞ ❆❧t❡r▲♦❝❛❧ ❞❡♣❡♥❞s ♦♥ δB✳ ❋♦r δB = 1.2✱
❛❧❣♦r✐t❤♠ ■♥❝▲♦❝❛❧ ✐s ♠✉❝❤ ❢❛st❡r t❤❛♥ ❆❧t❡r▲♦❝❛❧✱ ❛♥❞ ❢♦r δB = 1.5 ✇❡ ❤❛✈❡ t❤❡ ♦♣♣♦s✐t❡
s✐t✉❛t✐♦♥✳ ❚❤✉s✱ ❝♦♥❢♦r♠✐♥❣ t❤❡ ✭✐♥✐t✐❛❧✮ ❞❛t❛s❡t s❡q✉❡♥❝❡ t♦ δB ✈❛❧✉❡ ❧❡❛❞s t♦ ♦❜t❛✐♥✐♥❣
❜❡tt❡r s❝❤❡❞✉❧❡s ✐♥ t❤❡ ❝❛s❡ ♦❢ t❤❡ s✐♠♣❧❡ ❤❡✉r✐st✐❝s✱ ❛♥❞ t♦ s❤♦rt❡r ❡①❡❝✉t✐♦♥ t✐♠❡ ✐♥ t❤❡
❝❛s❡ ♦❢ t❤❡ ❛❞✈❛♥❝❡❞ ❤❡✉r✐st✐❝s✳
44

❚❛❜❧❡ ✷✳ ❆✈❡r❛❣❡ ❛❧❣♦r✐t❤♠ r✉♥♥✐♥❣ t✐♠❡ ✭✐♥ s❡❝♦♥❞s✮✱ ❢♦r m = 100✳
δB ■♥❝ ❆❧t❡r ❘♥❞ ■♥❝▲♦❝❛❧ ❆❧t❡r▲♦❝❛❧ ❘♥❞▲♦❝❛❧
✶✳✷ ✸✳✶✽❊−3 ✷✳✾✵❊−3 ✸✳✸✸❊−3 ✷✳✵✽❊+2 ✸✳✶✶❊+2 ✸✳✽✾❊+2
✶✳✺ ✷✳✻✽❊−3 ✷✳✺✽❊−3 ✷✳✹✵❊−3 ✸✳✶✶❊+2 ✷✳✵✻❊+2 ✻✳✸✵❊+2
✻ ❈♦♥❝❧✉s✐♦♥s
■♥ t❤✐s ✇♦r❦✱ ✇❡ ❛♥❛❧②③❡❞ s❝❤❡❞✉❧✐♥❣ ❞❛t❛ ❣❛t❤❡r✐♥❣ ✇✐t❤ ❧✐♠✐t❡❞ ❜❛s❡ st❛t✐♦♥ ♠❡♠♦r②✳
❆s ✇❡ s❤♦✇❡❞ t❤❛t t❤✐s ♣r♦❜❧❡♠ ✐s str♦♥❣❧② ◆P✲❤❛r❞✱ ❣r♦✉♣s ♦❢ s✐♠♣❧❡ ❛♥❞ ❛❞✈❛♥❝❡❞
❤❡✉r✐st✐❝s ✇❡r❡ ♣r♦♣♦s❡❞✳ ❚❤❡✐r ♣❡r❢♦r♠❛♥❝❡ ✇❛s t❡st❡❞ ✐♥ ❝♦♠♣✉t❛t✐♦♥❛❧ ❡①♣❡r✐♠❡♥ts✳
❚❤❡ s✐♠♣❧❡ ❛❧❣♦r✐t❤♠s ❛r❡ ✈❡r② ❢❛st✱ ❜✉t t❤❡ r❡s✉❧ts t❤❡② ♦❜t❛✐♥ ❛r❡ ♥♦t ✈❡r② ❣♦♦❞ ✐♥ ♠♦st
❝❛s❡s✳ ❚❤❡ ❛❞✈❛♥❝❡❞ ❤❡✉r✐st✐❝s ♣r♦❞✉❝❡ ❤✐❣❤ q✉❛❧✐t② s❝❤❡❞✉❧❡s✱ ❜✉t t❤❡✐r ❡①❡❝✉t✐♦♥ t✐♠❡s
❛r❡ ❧♦♥❣✳ ❲❡ s❤♦✇❡❞ t❤❛t s♦rt✐♥❣ ❞❛t❛s❡ts ❛❝❝♦r❞✐♥❣ t♦ t❤❡✐r s✐③❡s ✐s ❛ ❣♦♦❞ ✐❞❡❛ ✐❢ t❤❡ ❜❛s❡
st❛t✐♦♥ ♠❡♠♦r② ❧✐♠✐t ✐s r❛t❤❡r s♠❛❧❧✳ ■❢ t❤❡ ❜❛s❡ st❛t✐♦♥ ❜✉✛❡r ✐s ❜✐❣ ❡♥♦✉❣❤ t♦ ❤♦❧❞ t❤❡
s♠❛❧❧❡st ❛♥❞ t❤❡ ❜✐❣❣❡st ❞❛t❛s❡t t♦❣❡t❤❡r✱ t❤❡♥ ❛❧t❡r♥❛t✐♥❣ s♠❛❧❧ ❛♥❞ ❜✐❣ ❞❛t❛s❡ts ❛❧❧♦✇s
t♦ ♦❜t❛✐♥ ❜❡tt❡r r❡s✉❧ts✳
❆❝❦♥♦✇❧❡❞❣❡♠❡♥ts
❚❤✐s r❡s❡❛r❝❤ ❤❛s ❜❡❡♥ ♣❛rt✐❛❧❧② s✉♣♣♦rt❡❞ ❜② t❤❡ ◆❛t✐♦♥❛❧ ❙❝✐❡♥❝❡ ❈❡♥tr❡✱ P♦❧❛♥❞✱
❣r❛♥t ✷✵✶✻✴✷✸✴❉✴❙❚✻✴✵✵✹✶✵✳
❘❡❢❡r❡♥❝❡s
❆❦②✐❧❞✐③ ■✳❋✳✱ ❲✳ ❙✉✱ ❨✳ ❙❛♥❦❛r❛s✉❜r❛♠❛♥✐❛♠ ❛♥❞ ❊✳ ❈❛②✐r❝✐✱ ✷✵✵✷✱ ✏❲✐r❡❧❡ss s❡♥s♦r ♥❡t✇♦r❦s✿ ❛
s✉r✈❡②✑✱ ❈♦♠♣✉t❡r ◆❡t✇♦r❦s✱ ❱♦❧✳ ✸✽✱ ♣♣✳ ✸✾✸✲✹✷✷✳
❇❡r❧✐➠s❦❛ ❏✳✱ ✷✵✶✹✱ ✏❈♦♠♠✉♥✐❝❛t✐♦♥ s❝❤❡❞✉❧✐♥❣ ✐♥ ❞❛t❛ ❣❛t❤❡r✐♥❣ ♥❡t✇♦r❦s ✇✐t❤ ❧✐♠✐t❡❞ ♠❡♠♦r②✑✱
❆♣♣❧✐❡❞ ▼❛t❤❡♠❛t✐❝s ❛♥❞ ❈♦♠♣✉t❛t✐♦♥✱ ❱♦❧✳ ✷✸✺✱ ♣♣✳ ✺✸✵✲✺✸✼✳
❇❡r❧✐➠s❦❛ ❏✳✱ ✷✵✶✺✱ ✏❙❝❤❡❞✉❧✐♥❣ ❢♦r ❞❛t❛ ❣❛t❤❡r✐♥❣ ♥❡t✇♦r❦s ✇✐t❤ ❞❛t❛ ❝♦♠♣r❡ss✐♦♥✑✱ ❊✉r♦♣❡❛♥
❏♦✉r♥❛❧ ♦❢ ❖♣❡r❛t✐♦♥❛❧ ❘❡s❡❛r❝❤✱ ❱♦❧✳ ✷✹✻✱ ♣♣✳ ✼✹✹✲✼✹✾✳
❇➟❛➺❡✇✐❝③ ❏✳✱ ❏✳❑✳ ▲❡♥str❛ ❛♥❞ ❆✳❍✳●✳ ❘✐♥♥♦♦② ❑❛♥✱ ✶✾✽✸✱ ✏❙❝❤❡❞✉❧✐♥❣ s✉❜❥❡❝t t♦ r❡s♦✉r❝❡ ❝♦♥✲
str❛✐♥ts✿ ❝❧❛ss✐✜❝❛t✐♦♥ ❛♥❞ ❝♦♠♣❧❡①✐t②✑✱ ❉✐s❝r❡t❡ ❆♣♣❧✐❡❞ ▼❛t❤❡♠❛t✐❝s✱ ❱♦❧✳ ✺✱ ♣♣✳ ✶✶✲✷✹✳
❈❤❡♥❣ ❚✳❈✳❊✳✱ ❇✳▼✳❚✳ ▲✐♥ ❛♥❞ ❍✳▲✳ ❍✉❛♥❣✱ ✷✵✶✷✱ ✏❘❡s♦✉r❝❡✲❝♦♥str❛✐♥❡❞ ✢♦✇s❤♦♣ s❝❤❡❞✉❧✐♥❣ ✇✐t❤
s❡♣❛r❛t❡ r❡s♦✉r❝❡ r❡❝②❝❧✐♥❣ ♦♣❡r❛t✐♦♥s✑✱ ❈♦♠♣✉t❡rs ✫ ❖♣❡r❛t✐♦♥s ❘❡s❡❛r❝❤✱ ❱♦❧✳ ✸✾✱ ♣♣✳ ✶✷✵✻✲
✶✷✶✷✳
❈❤♦✐ ❑✳✱ ❚✳●✳ ❘♦❜❡rt❛③③✐✱ ✷✵✵✽✱ ✏❉✐✈✐s✐❜❧❡ ▲♦❛❞ ❙❝❤❡❞✉❧✐♥❣ ✐♥ ❲✐r❡❧❡ss ❙❡♥s♦r ◆❡t✇♦r❦s ✇✐t❤
■♥❢♦r♠❛t✐♦♥ ❯t✐❧✐t②✑✱ ■♥✿ ■❊❊❊ ■♥t❡r♥❛t✐♦♥❛❧ P❡r❢♦r♠❛♥❝❡ ❈♦♠♣✉t✐♥❣ ❛♥❞ ❈♦♠♠✉♥✐❝❛t✐♦♥s
❈♦♥❢❡r❡♥❝❡ ✷✵✵✽✿ ■P❈❈❈ ✷✵✵✽✱ ♣♣✳ ✾✲✶✼✳
●❛r❡② ▼✳❘✳✱ ❉✳❙✳ ❏♦❤♥s♦♥✱ ✶✾✼✾✳ ✏❈♦♠♣✉t❡rs ❛♥❞ ✐♥tr❛❝t❛❜✐❧✐t②✿ ❆ ❣✉✐❞❡ t♦ t❤❡ t❤❡♦r② ♦❢ ◆P✲
❝♦♠♣❧❡t❡♥❡ss✑✱ ❋r❡❡♠❛♥✱ ❙❛♥ ❋r❛♥❝✐s❝♦✳
❏♦❤♥s♦♥ ❙✳▼✳✱ ✶✾✺✹✱ ✏❖♣t✐♠❛❧ t✇♦✲ ❛♥❞ t❤r❡❡✲st❛❣❡ ♣r♦❞✉❝t✐♦♥ s❝❤❡❞✉❧❡s ✇✐t❤ s❡t✉♣ t✐♠❡s ✐♥❝❧✉✲
❞❡❞✑✱ ◆❛✈❛❧ ❘❡s❡❛r❝❤ ▲♦❣✐st✐❝s ◗✉❛rt❡r❧②✱ ❱♦❧✳ ✶✱ ♣♣✳ ✻✶✲✻✽✳
▲❡✐st❡♥ ❘✳✱ ✶✾✾✵✱ ✏❋❧♦✇s❤♦♣ s❡q✉❡♥❝✐♥❣ ♣r♦❜❧❡♠s ✇✐t❤ ❧✐♠✐t❡❞ ❜✉✛❡r st♦r❛❣❡✑✱ ■♥t❡r♥❛t✐♦♥❛❧ ❏♦✉r✲
♥❛❧ ♦❢ Pr♦❞✉❝t✐♦♥ ❘❡s❡❛r❝❤✱ ❱♦❧✳ ✷✽✱ ♣♣✳ ✷✵✽✺✲✷✶✵✵✳
▲✐♥ ❇✳▼✳❚✳✱ ❍✳▲✳ ❍✉❛♥❣✱ ✷✵✵✻✱ ✏❖♥ t❤❡ r❡❧♦❝❛t✐♦♥ ♣r♦❜❧❡♠ ✇✐t❤ ❛ s❡❝♦♥❞ ✇♦r❦✐♥❣ ❝r❡✇ ❢♦r r❡s♦✉r❝❡
r❡❝②❝❧✐♥❣✑✱ ■♥t❡r♥❛t✐♦♥❛❧ ❏♦✉r♥❛❧ ♦❢ ❙②st❡♠s ❙❝✐❡♥❝❡✱ ❱♦❧✳ ✸✼✱ ♣♣✳ ✷✼✲✸✹✳
▼♦❣❡s ▼✳✱ ❚✳●✳ ❘♦❜❡rt❛③③✐✱ ✷✵✵✻✱ ✑❲✐r❡❧❡ss ❙❡♥s♦r ◆❡t✇♦r❦s✿ ❙❝❤❡❞✉❧✐♥❣ ❢♦r ▼❡❛s✉r❡♠❡♥t ❛♥❞
❉❛t❛ ❘❡♣♦rt✐♥❣✑✱ ■❊❊❊ ❚r❛♥s❛❝t✐♦♥s ♦♥ ❆❡r♦s♣❛❝❡ ❛♥❞ ❊❧❡❝tr♦♥✐❝ ❙②st❡♠s✱ ❱♦❧✳ ✹✷✱ ♣♣✳ ✸✷✼✲
✸✹✵✳
45

A Chance Constrained Optimization Approach for
Resource Unconstrained Project Scheduling with
Uncertainty in Activity Execution Intensity
Lucio Bianco, Massimiliano Caramia and Stefano Giordani
University of Rome “Tor Vergata”, Italy
{bianco,caramia,giordani}@dii.uniroma2.it
Keywords: Chance constrained optimization, PERT, Project scheduling.
1 Introduction
A project network is a directed graph G = (V, A), where the set of nodes V corresponds
to the set of activities of the project, and the set A of its arcs represents the set of (classical)
finish-to-start precedence relations among pairs of activities, i.e., A = {(i, j) : i, j ∈ V }.
The graph G is acyclic and models the project in the so called Activity-On-Node (AON)
representation; nodes can be topologically numbered assuming a single source dummy node
(activity 1) and a single sink dummy node (activity n). Each activity i ∈ V {1, n} has a
duration di. No resource constraint is taken into account.
The problem of computing the minimum value of the project completion time (project
duration or makespan), under the assumption that di’s are deterministic values, is known to
be polynomially solvable. If activities’ durations are uncertain and are modeled by random
variables, the project network becomes stochastic and the objective changes in determining
the project makespan distribution or some characteristic thereof, e.g., its expectation.
In the literature, stochastic project networks are often referred to as PERT-networks,
since PERT (Project Evaluation and Review Technique) was one of the first techniques to
analyze the stochastic behavior of such networks (see, e.g., Clark, 1962). The originators of
PERT proposed to use three estimates for the duration of each activity i, i.e., an optimistic
value āi, a most likely value m̄i, and a pessimistic estimate b̄i. They modeled each activity
duration as a stochastic variable with an appropriate Beta distribution and proposed a
simple approximate method to calculate its expectation. The assumption of a Beta density
function was a matter of convenience that allowed the derivation of nice approximations for
the expected value and the variance of activity durations, but, in practice, these estimates
may be far form the actual expected value and variance of a Beta distributed stochastic
variable. Besides this, PERT model suffers from several other shortcomings: it assumes that
activity durations are independent stochastic variables; it uses the Central Limit theorem
assuming the number of critical activities being large enough; moreover, it suffers from the
merge event bias problem leading PERT to an optimistically biased estimate of the earliest
expected activity starting times, and then also of the project duration.
In this paper we try to cope with uncertainty in activity durations in a novel way
in order to overcome PERT limitations. In Section 2, we present our approach, while in
Section 3 we compare the latter to the PERT model.
2 The Proposed Chance Constrained Optimization Approach
In the following, we present a new approach where activity durations uncertainty is not
modeled by means of stochastic variables directly associated with activity durations, as
done in most of the stochastic approaches in the literature. We assume that the planning
46

horizon is [0, T) (all the activities have to be completed within T), discretized into T unit
time periods [0, 1), [1, 2), . . . , [T − 1, T), indexed by t = 1, 2, . . . , T. Moreover, for each
(non-dummy) activity i ∈ V {1, n} and for each time period t, we introduce the execution
intensity xit as a stochastic variable, with 0 ≤ xit ≤ 1, representing the fraction of activity
i executed in time period t. This means that activity i is completely executed when the sum
of the variables xit over time is equal to one. Typically, in classical project scheduling, one
assumes that the amount of activity carried out during its execution is flat, i.e., xit = 1
di
.
Let si be a non-negative variable defining the start time of activity i ∈ V . Moreover, let
∆i be an integer non-negative variable representing the number of time periods necessary
to complete activity i with a probability at least equal to θ.
Since the project makespan can be expressed as sn, where n is the dummy sink activity
of the project network, the problem of minimizing the makespan of our stochastic project
network has the following conceptual mathematical program
min sn (1)
Prob
{ si+∆i
∑
t=si+1
xit − 1 ≥ 0
}
≥ θ, ∀i ∈ V (2)
si + ∆i ≤ sj, ∀(i, j) ∈ A (3)
si ≥ 0, ∀i ∈ V (4)
∆i ≥ 0 and integer, ∀i ∈ V {1, n}. (5)
The objective function (1) minimizes the start time of the dummy end activity n and
thus the makespan of the project. The constraints have the following meaning: Constraints
(2) regulate the total amount processed of an activity i ∈ V over time, i.e., the probability
that the summation of the execution fractions of i after ∆i unit time periods from the
start time of i is greater than or equal to 1 must not be less than θ. Constraints (3) model
finish-to-start precedence relations between i and j, ∀(i, j) ∈ A. Constraints (4) and (5)
limit the range of variability of the problem variables.
The above model can be interpreted in terms of Chance constrained programming (see,
e.g., Prekopa, 1995) that is one of the main approaches for dealing with stochastic opti-
mization problems. The latter has the following form
min
y∈Y
f(y), s.t. Prob{G(y, ξ) ≥ 0} ≥ θ, (6)
where Y ⊆ Rr
, ξ is a random vector with probability distribution P supported on a set
χ ⊆ Rq
, f : Rr
→ R is a real valued function, and G(y, ξ) ≥ 0, with G : Rr
× χ → Rs
,
refers to a finite system of inequalities. θ ∈ (0, 1) is called the probability level and it is
chosen by the decision maker in order to model the safety requirements. Sometimes, the
probability level is strictly fixed from the very beginning (e.g., θ = 0.95, . . . , 0.99). Next
we first show how to cope with constraints (2) in order to write the chance constrained
program (1)–(5) as a deterministic (linear) program and then solving the latter.
2.1 Phase 1: Estimating ∆i to cope with constraints (2)
Assume that the durations of the activities are sufficiently large, and consider a generic
(non-dummy) activity i ∈ V {1, n}. Assuming the stochastic variables xit being inde-
pendent and identically distributed, by the Central Limit theorem we can state that
47

X̄i =
∑si+∆i
t=si+1 xit is a Normal stochastic variable. That is, X̄i is such that the (standard-
ized) stochastic variable Zi =
X̄i
∆i
−µi
σi
√
∆i ∼ N(0, 1), where N is the Normal distribution,
and µi and σi are the mean value and the standard deviation of xit, respectively.
In the following, we show how to calculate the minimum ∆i such that Prob{X̄i ≥
1} ≥ θ, i.e., such that activity i being completed with probability at least equal to
θ. The latter can be written as Prob
{
Zi ≥
1
∆i
−µi
σi
√
∆i
}
≥ θ, which means that θ ≤
Prob
{
Zi ≥
1
∆i
−µi
σi
√
∆i
}
= Φ
(
−
1
∆i
−µi
σi
√
∆i
)
where we exploited the fact that the Nor-
mal probability density function is an even function and Φ is the quartile of the Normal
distribution which is the inverse of the repartition function in the case of absolutely con-
tinuous density function, and, therefore Φ−1
(θ) ≤ −
1
∆i
−µi
σi
√
∆i. This inequality can be
rewritten as ∆iµi − Φ−1
(θ)σi
√
∆i − 1 ≥ 0, and by solving it in
√
∆i, considering the latter
being non-negative, we have that
√
∆i ≥
Φ−1
(θ)σi +
√
[Φ−1(θ)]
2
σ2
i + 4µi
2µi
=
√
[Φ−1(θ)]
2
σ2
i
4µ2
i
+
√
[Φ−1(θ)]
2
σ2
i
4µ2
i
+
1
µi
. (7)
Assume now that for each (non-dummy) activity i and for all t = si + 1, . . . , si +
∆i, the stochastic variables xit in [0, 1], are identically distributed with a Beta (prior)
probability density function with parameters αi, βi 1. Let us therefore consider the
additional parameters ai = 0, bi = 1, and mi = αi−1
αi+βi−2 . Parameter ai identifies the
pessimistic value of stochastic variable xit, i.e., an estimate of the minimum fraction of
activity i that can be executed in time period t, while parameter bi identifies its optimistic
value, i.e., an estimate of the maximum fraction of activity i that can be executed in time
period t, and finally parameter mi identifies the most likely value or modal value of xit.
Accordingly, we have that µi = αi
αi+βi
and σ2
i = αiβi
(αi+βi)2(αi+βi+1) .
After some easy calculations and substitutions, equation (7) can be written as,
∆i ≥


√
[Φ−1(θ)]
2
4
(αi − 1)/mi − αi + 2
αi((αi − 1)/mi + 3)
+
√
[Φ−1(θ)]
2
4
(αi − 1)/mi − αi + 2
αi((αi − 1)/mi + 3)
+
(αi − 1)/mi + 2
αi


2
. (8)
Choosing the minimum integer value of ∆i fulfilling inequality (8) guaranties that
activity i will be completed in ∆i time periods with probability not less than θ.
2.2 Phase 2: Solving the resulting deterministic problem
With the above choice for ∆i, for each activity i, constraints (2) are satisfied and hence
it can be removed, along with constraints (5), from the chance constrained program (1)–(5).
The latter therefore reduces to the well known natural-date (linear) problem formulation of
the (deterministic) resource unconstrained project scheduling problem, with finish-to-start
precedence relations and where ∆i assumes the role of the duration of activity i, that can
be solved in linear time with respect the number of precedence relations.
The optimal solution value of the latter program provides the (minimum) project du-
ration assuring that the project itself is completed with probability not less than p = θℓ
≥
θn−2
, where ℓ, with 1 ≤ ℓ ≤ n − 2, is the number of non-dummy activities of the largest
activity chain of the project network. Since typically θ is assumed to be very close to 1, i.e.
θ = 1−ϵ with ϵ 0 being a value sufficiently close to 0, we have that p = (1−ϵ)ℓ
≃ 1−ℓϵ,
hence the project will be completed with probability not less than 1 − ℓϵ.
48

Fig. 1. Example of project network data in the proposed model (a) and in the PERT model (b).
3 Example
In the following example we compare our approach with PERT. In particular we are
interested in comparing the project duration estimated by our approach with that provided
by the PERT, given the probability p of project completion. The comparison is done by
considering, for each non-dummy activity i, the optimistic estimation āi and the pessimistic
estimation b̄i of the duration di equal to the value of ∆i calculated with the approach
described in Section 2.1 with θa = 0.01 and with θb = 0.99, respectively. Finally, given
āi and b̄i and the modal value mi of xit, we calculate the modal value m̄i of di as m̄i =
āi + mi(b̄i − āi). In Figure 1 we depict a project network with (a) the input data for our
model and (b) the corresponding input data for the PERT model evaluated as described
above; nodes 1 and 6 are dummies. By computing the values of ∆i with θ = 0.99 for
every activity i, with i = 2, . . . , 5, we have ∆2 = ⌈3.43⌉, ∆3 = ⌈2.43⌉, ∆4 = ⌈3.16⌉, and
∆5 = ⌈3.16⌉. By solving the deterministic model of Phase 2 with these data we have that
the project duration is 12 with a probability (approximately) equal to 0.97. As for the
PERT model (see part b of Figure 1), we have that the project duration, with the same
probability of 0.97, is 7.79 + 1.88 · 0.81 = 9.31, since its mean value is 7.79 an its standard
deviation is 0.81 (1.88 is the number of standard deviations to be added to the mean
value to get a project duration estimation with probability equal to 0.97). Hence, for this
example, PERT underestimates project duration by more than 22.4% with respect to our
approach. We will also compare the project duration given by our approach and by PERT
with that calculated with Monte Carlo simulation on test problems with diﬀerent sizes.
References
1. Clark, C.E., The PERT model for the distribution of an activity time, Operations Research,
10, 405–406 (1962).
2. Prekopa, A., Stochastic Programming. Kluwer Acad. Publ., Dordrecht, Boston (1995).
49

Single-machine capacitated lot-sizing and scheduling
with delivery dates and quantities
Fayez F. Boctor
Centre interuniversitaire de recherche sur les réseaux d’entreprises, la logistique et le transport
(CIRRELT)
Faculté des sciences de l’administration, Université Laval, Canada G1V 0A6
fayez.boctor@fsa.ulaval.ca
Keywords: Lot-sizing and scheduling, Heuristics, Integer programming.
1 Introduction
The standard Capacitated Single-machine Lot-sizing Problem (CSLP) assumes that the
planning horizon is divided into a number of time periods of equal lengths and makes three
implicit and simplifying assumptions. First, it implicitly assumes that a lot that starts
within a given time period should be finished within this same period. This assumption
is needed to simplify the mathematical expression of the capacity constraints. Second, it
assumes that setups cannot be carried over from one period to the next. Thus if the last
run of a period and the first of the next one process the same product, a setup cost for
each of these two runs are included in the cost function. This overestimates the overall
setup cost as, in most cases, these two lots can be processed with one setup (see Jans et al.
2008). Third, it is assumed that delivery occurs only at the end (or the beginning) of each
time period. However, the objective function does not include the inventory holding cost
between the time a lot is finished and the beginning of the next period. This assumption is
also needed to simplify the mathematical expression of the objective function and to make
the determination of processing dates unnecessary. However this underestimates the real
inventory holding cost.
Some research publications (see Sox et al., 1999; Gopalakrishnan, 2000; Suerie et al.,
2003; Porkka et al., 2003) propose to relax the second assumption and allow setup carryover.
However the resulting formulation still: (1) does not include the inventory holding cost
between the time a lot is finished and the beginning of the next period; (2) does not allow
a lot to be finished beyond the end of its starting period; and (3) does not allow delivery
between the beginning and the end of a time period.
A more realistic version of the problem, called hereafter the single-machine capacitated
lot-sizing and scheduling problem with delivery dates and quantities (SCLSP-DDQ), is stud-
ied in this paper. To the best of my knowledge, this problem is not studied in any previous
publication. In this version each product has a set of delivery dates and the quantity to
deliver at each of these dates is known. It is allowed to start the production of a lot at any
time and finish it at any time later within the finite planning horizon provided that the
required demands are delivered at the required dates. Also it allows setups to be carried
over from one period to the next. Finally, the objective function of this new formulation
includes the inventory holding cost of each produced lot from the end of its processing until
the delivery of all its units.
Arranging delivery dates in their ascending order, denoted, t1, t2, . . . , tL, we consider
that the planning horizon is divided into L time intervals of unequal lengths, where the
length of interval l is the time interval between tl−1 and tl. We also allow a lot that starts
within a given time interval, say interval l, be finished within the same interval or within
a later interval, say interval l + r.
50

2 Mathematical formulation
As mentioned above, the proposed formulation allows for setup carryover, takes into
account the inventory holding cost for each produced lot between the finish time and the
delivery of all its units. Also, it determines the sequence and processing dates of all lots to
process.
2.1 Assumptions:
1. A number of products are to be produced by a single machine (or a production line);
2. Each lot of each product is composed of a number of units of the product;
3. A finite planning horizon and the machine cannot processes more than one product at
a time;
4. For each product there is an upper limit on its lots size;
5. Lots of the same product are not necessarily of same size;
6. Processing time of a lot of a given product is composed of the processing time of its
units plus a known sequence-independent setup time;
7. Processing time of a lot of a given product can be either proportional to the quantity
to produce or constant (e.g. in chemical industries). Both cases are modeled in Boctor
(2016); however this paper model the constant processing time case only;
8. Delivery dates and quantities to deliver at these dates are known and deterministic;
9. No backlogging is allowed;
10. Two cost elements are considered: setup cost and inventory holding cost;
11. For each product, unit inventory holding cost per time unit and setup cost are constant.
2.2 Notation:
N number of different products to produce; indexed i
T set of delivery dates indexed in the ascending order; T = {tl; l = 1, . . . , L}
dil quantity of product i to deliver at tl. This demand is nil if it is not required to deliver
any quantity of product i at tl
Pi processing time of a lot of product i including its setup time
ci setup cost of a lot of product i
hi inventory holding cost of a unit of product i per time unit
Qi upper limit on the size of a lot of product i
Fi the required inventory level of product i at the end of the planning horizon
xipl a binary that takes the value 1 if a lot of product i is in position p among those to
finish between tl−1 and tl (even if it starts before tl−1). Notice that, as we have an
upper limit Qi on the lot size of i, more than one lot of product i may be processed
and finished between tl−1 and tl but in different positions in the sequence
qipl the quantity of product i if produced in the p-th position and finishes between tl−1 and
tl
fipl the finish date of product i if produced in the p-th position and finishes between tl−1
and tl
Iil inventory level of product i at tl just after delivering the demand dil
2.3 The SCLSP-DDQ Model
Find xipl ∈ {0, 1}, qipl ≥ 0, fipl ≥ 0, and Iil ≥ 0; i = 1, . . . , N; p = 1, . . . , N; l =
1, . . . , L, which:
51

Minimize :
N
∑
i=1
L
∑
l=1
hiIi,l−1(tl − tl−1) +
N
∑
i=1
N
∑
p=1
L
∑
l=1
hiqipl(tl − fipl) +
N
∑
i=1
N
∑
p=1
L
∑
l=1
cixipl (1)
Subject to:
N
∑
i=1
xipl ≤ 1, p = 1, . . . , N, l = 1, . . . , L (2)
N
∑
i=1
xipl ≤
N
∑
i=1
xi,p−1,l, p = 2, . . . , N, l = 1, . . . , L (3)
qipl ≤ xiplQi, i = 1, . . . , N, p = 1, . . . , N, l = 1, . . . , L (4)
Iil = Ii,l−1 +
N
∑
p=1
qipl − dil, i = 1, . . . , N, l = 1, . . . , L − 1 (5)
Ii,L−1 +
N
∑
p=1
qipL − diL = Fi, i = 1, . . . , N (6)
fi11 ≥ xi11Pi, i = 1, . . . , N (7)
fi1l ≥ xi1ltl−1, i = 1, . . . , N, l = 2, . . . , L (8)
fipl ≥ xiplfj,p−1,l + Pixiplqipl, i = 1, . . . , N, j = 1, . . . , N, p = 2, . . . , N, l = 1, . . . , L
(9)
N
∑
i=1
N
∑
p=2
Pixipj ≤ tl −
N
∑
i=1
xi1lxi1l, l = 1, . . . , L. (10)
The first term in the objective function (1) gives the inventory holding cost of items
over the time intervals tl−1 and tl. The second term gives the inventory holding cost of the
produced items between the end of their processing and the following delivery date. The
third term gives the setup cost of the processed lots. Constraints (2) require that there
is at most one product in each position of each time interval (i.e., the interval between
two consecutive delivery dates). Constraints (3) make sure that if there is no product in a
position then there are no products in the next position. Constraints (4) make sure that
the produced quantities do not exceed the lot-size upper limit. Constraints (5) and (6)
determine the inventory levels and assure that the demands are fulfilled without backlogs.
Constraints (7), (8) and (9) determines the finish times of the lots to produce. Finally,
constraints (10) make sure that we have enough time to produce the required quantities in
each time interval (capacity constraints).
This model is difficult to solve as it contains a large number of variables and constraints.
It is composed of N2
L binary variables, NL(2N +1)+L continuous variables and NL(2N +
1) + L constraints. Thus if we have 10 products and 20 delivery dates our model has 2000
binary variables, 4220 continuous variables and 4220 constraints. It is also important to
note that the objective functions (1) as well as constraints (9) and (10) are quadratic which
adds to the difficulty of solving the model. Actually, we were not able to solve this model
even for instance including 10 products and 8 delivery dates.
A necessary and sufficient condition for the feasibility of this model is:
N
∑
i=1
(
Pi
⌈∑l
j=1 dil
Qi
⌉)
≤ tl, l = 1, . . . , L (11)
52

3 A decomposition solution approach
A first approach to solve the above introduced problem consists of decomposing the
problem into two sub-problems to be solved consecutively. The first sub-problem is the
one of determining the product lots to be processed before each delivery date without
determining their sequence of production (i.e., without specifying their position p). The
second sub-problem is to determine the sequence for the obtained production lots.
The first sub-problem can be formulated as mixed integer linear program (see Boctor,
2016). The optimal solution of this model gives us the number of lots of each product to
be finished by each date tl. Once this optimal solution is obtained we solve a sequencing
problem to determine the starting dates of the required production lots. This second sub-
problem is also modeled by a mixed integer program. Unfortunately, solving this second
model is very time consuming and has a weak LP relaxation. For more details see Boctor
(2016).
4 A hybrid heuristic
This proposed heuristic is a hybrid one composed of a solution construction procedure
followed by 3 improvement procedures. The construction phase is an iterative, backward-
pass heuristic. To construct a production plan, the heuristic starts by setting t = tL, the
latest delivery date. The main iteration of the heuristic is as follows. At each date t we
list the products to deliver at this date and chose from this list the product k that has the
largest value of hkqk
where qk = min{dkt, Qk}. Then we schedule the processing of a lot
of size qk of the product k to finish at t. If qk = dkt we remove product k from our list
otherwise we reduce its demand by qk. Next, we set t = t − Pk. In other words we put t
equal the starting time of the just scheduled job k, and add to the list all the orders for
which the due date is between t and t + Pk if any. If the resulting list contains more than
one order for a given product, we group them into one order. Now, if the list is not empty,
repeat the above and choose a job to schedule. Otherwise, move backward to the latest
date where we have some orders to deliver. The heuristic stops when there are no more
orders to schedule.
The rational of this construction heuristic is to schedule the production of the orders to
deliver at the latest possible time in order to minimize the sum of inventory holding costs.
This procedure may produce a non-feasible schedule where the starting time of some
jobs is before the beginning of the planning horizon (date 0). Even in such a case we apply
the improvement procedures as they may modify this solution in a way that makes it
feasible.
The first improvement procedure attempts to reduce the number of production lots in
order to reduce the number of setups. The procedure considers one product at a time and
repeat the following until no more gain can be achieved. For each lot of the considered
product, determine if a gain can be made by removing this lot and adding its units to the
preceding lots of the same product. The lot leading to the highest gain is removed and we
repeat the same for the remaining lots. This improvement procedure stops if we cannot
achieve any more gain.
The second improvement procedure attempts to move the remaining production lots to
the latest possible date without causing any backlogs. This may be possible as the previous
improvement procedure may remove some production lots making room for processing
other lots in the time interval originally occupied by some of the removed lots.
The third improvement procedure exchanges the position of pairs of production lots as
long as this can lead to cost reduction without backlogs.
53

5 Computational experiment
To assess its performance, the solutions of the hybrid heuristic were compared to those
obtained by the decomposition approach using 100 randomly generated problem instances.
Unfortunately, it is not possible to obtain the optimal solutions of these instances. In
addition the literature does not provide any solution method that can be used to assess
the performance of the proposed heuristics.
For all the generated instances, the number of products is 10 and the number of delivery
dates is 8. Delivery dates are: 40, 60, 80, 100, 120, 150, 180 and 200. For each instance,
the values of hi, ci, Pi and Qi are randomly drawn from uniform distributions. The limits
for these uniform distributions are respectively from 0.05 to 0.25 for hi, from 80 to 200
for ci, from 1 to 5 for Pi, and from 40 to 80 for Qi. To determine the total demand of a
product, a random value is drawn from a uniform distribution between 120 and 180. For
each product, one to four delivery dates are randomly drawn among the 8 possible dates.
The total demand is then partitioned and a quantity is randomly determined for each
delivery date. Note that all test instances are generated in a way to satisfy the necessary
and sufficient feasibility condition (11).
The proposed hybrid heuristic succeeded to solve all test instances with an average
time of less than 1 second. The 3 improvement procedures reduce the total cost obtained
at the construction step by 8.01% in average with a minimum improvement of 2.12% and
a maximum improvement of 16.76%. In more details, the first improvement procedure, the
grouping procedure, produced an improvement of 4.42% in average while the second and
third improvement procedures yielded 1.05% and 2.73% improvement in average. The hy-
brid heuristic produced a better solution than the decomposition approach for 75 instances
while the decomposition approach produced a better solution for the other 25 instances.
Over the entire 100 instances set, in average, the hybrid heuristic produced solutions with
a total cost of 3.38% less than the decomposition approach.
6 Acknowledgements
This research work was partially supported by grant OPG0036509 from the National
Science and Engineering Research Council of Canada (NSERC). This support is gratefully
acknowledged.
References
Boctor FF., 2016, “The Generalized single-facility capacitated lot-sizing and scheduling problem”.
Document de travail 2016-011; Faculté des sciences de l’administration, Université Laval.
Gopalakrishnan M., 2000, “A modified framework for modeling set-up carryover in the capacitated
lot sizing problem”. International Journal of Production Research, 38: 3421–3424.
Jans R. and Degraeve Z., 2008, “Modeling industrial lot sizing problems: a review”. International
Journal of Production Research, 46: 1619–1643.
Porkka P., Vepsalainen A.P.J., and Kuula M., 2003, “Multi-period production planning carrying
over set-up time”. International Journal of Production Research, 41: 1133–1148.
Sox C.R. and Gao Y., 1999, “The capacitated lot sizing problem with setup carry-over”. IIE
Transactions, 31: 173–181.
Suerie C. and Stadtler H., 2003, “The capacitated lot-sizing problem with linked lot sizes”. Man-
agement Science, 49: 1039–1054.
54

❙✐♥❣❧❡ ♠❛❝❤✐♥❡ s❝❤❡❞✉❧✐♥❣ ✇✐t❤ m✿n r❡❧❛t✐♦♥s ❜❡t✇❡❡♥
❥♦❜s ❛♥❞ ♦r❞❡rs✿ ▼✐♥✐♠✐③✐♥❣ t❤❡ s✉♠ ♦❢ ❝♦♠♣❧❡t✐♦♥
t✐♠❡s ❛♥❞ ✐ts ❛♣♣❧✐❝❛t✐♦♥ ✐♥ ✇❛r❡❤♦✉s✐♥❣
◆✐❧s ❇♦②s❡♥1
✱ ❑♦♥r❛❞ ❙t❡♣❤❛♥1
❛♥❞ ❋❡❧✐① ❲❡✐❞✐♥❣❡r1
❋r✐❡❞r✐❝❤✲❙❝❤✐❧❧❡r✲❯♥✐✈❡rs✐t② ❏❡♥❛✱ ●❡r♠❛♥②
④♥✐❧s✳❜♦②s❡♥✱❦♦♥r❛❞✳st❡♣❤❛♥✱❢❡❧✐①✳✇❡✐❞✐♥❣❡r⑥❅✉♥✐✲❥❡♥❛✳❞❡
❑❡②✇♦r❞s✿ ❙✐♥❣❧❡ ♠❛❝❤✐♥❡ s❝❤❡❞✉❧✐♥❣✱ ❲❛r❡❤♦✉s✐♥❣✱ ❖r❞❡r ❝♦♥s♦❧✐❞❛t✐♦♥✳
✶ ❙✐♥❣❧❡ ♠❛❝❤✐♥❡ s❝❤❡❞✉❧✐♥❣ ✇✐t❤ m✿n ❥♦❜✲♦r❞❡r r❡❧❛t✐♦♥s
■♥ t❤✐s ♣❛♣❡r ✇❡ tr❡❛t ❛♥ ❡❧❡♠❡♥t❛r② ❡①t❡♥s✐♦♥ ♦❢ s✐♥❣❧❡ ♠❛❝❤✐♥❡ s❝❤❡❞✉❧✐♥❣ ♣r♦❜❧❡♠
1||
P
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1|m✿n|
P
Co✱ ✐s ❞❡✜♥❡❞ ❛s ❢♦❧❧♦✇s✿
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♦r❞❡rs✳ ❈♦♠♣❧❡t✐♥❣ ❛♥ ♦r❞❡r o ∈ O r❡q✉✐r❡s t❤❡ ❝♦♠♣❧❡t✐♦♥ ♦❢ ❥♦❜ s✉❜s❡t Jo ⊆ J✳ ❏♦❜ s❡ts
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φ✱ ♣r♦❜❧❡♠ 1|m✿n|
P
Co s❡❡❦s ❛ ❥♦❜ s❡q✉❡♥❝❡ ✇❤✐❝❤ ♠✐♥✐♠✐③❡s t❤❡ s✉♠ ♦❢ ♦r❞❡r ❝♦♠♣❧❡t✐♦♥
t✐♠❡s✱ ✐✳❡✳✱
Z(φ) =
X
o∈O
κ(φ,o)
X
k=1
pφ(k).
❖t❤❡r t❤❛♥ ✐♥ ♦✉r ♣r♦❜❧❡♠ s❡tt✐♥❣✱ tr❛❞✐t✐♦♥❛❧ ♣r♦❜❧❡♠ 1||
P
Cj ♣r❡s✉♣♣♦s❡s ❛ 1✿1
r❡❧❛t✐♦♥ ❛♠♦♥❣ ❥♦❜s ❛♥❞ ♦r❞❡rs ❛♥❞ ✐s ✇❡❧❧ ❦♥♦✇♥ t♦ ❜❡ s♦❧✈❛❜❧❡ ✐♥ ♣♦❧②♥♦♠✐❛❧ t✐♠❡ ❜②
♦r❞❡r✐♥❣ ❥♦❜s ❛❝❝♦r❞✐♥❣ t♦ t❤❡ s❤♦rt❡st✲♣r♦❝❡ss✐♥❣✲t✐♠❡s r✉❧❡ ✭❙♠✐t❤ ✶✾✺✻✮✳ ■♥ t❤✐s ♣❛♣❡r✱
✇❡ s❤♦✇ t❤❛t t❤✐s r❡s✉❧t ♥♦ ❧♦♥❣❡r ❤♦❧❞s ❢♦r 1|m✿n|
P
Co✱ ✇❤✐❝❤ ✐s s❤♦✇♥ t♦ ❜❡ str♦♥❣❧②
◆P✲❤❛r❞ ✐♥ ❙❡❝t✐♦♥ ✷✳ ❙❡❝t✐♦♥ ✸ ❡❧❛❜♦r❛t❡s ♦♥ t❤❡ ❛♣♣❧✐❝❛t✐♦♥ ♦❢ 1|m✿n|
P
Co ✐♥ ❞✐str✐❜✉t✐♦♥
❝❡♥t❡rs ♦❢ ❧❛r❣❡ ♦♥❧✐♥❡ r❡t❛✐❧❡rs✱ ❡✳❣✳✱ ❆♠❛③♦♥ ❛♥❞ ❩❛❧❛♥❞♦✳ ❍❡r❡✱ ❜✐♥s ✭❥♦❜s✮ ❝♦♥t❛✐♥✐♥❣
♠✉❧t✐♣❧❡ ✐t❡♠s ❢♦r ❞✐✛❡r❡♥t ♦r❞❡rs ♥❡❡❞ t♦ ❜❡ ♠❛♥✉❛❧❧② s♦rt❡❞ ✐♥t♦ ❛ r❛❝❦ ✭❞✉❜❜❡❞ t❤❡ ♣✉t
✇❛❧❧✮ ❜② ❛ ❤✉♠❛♥ ❧♦❣✐st✐❝s ✇♦r❦❡r ✭♠❛❝❤✐♥❡✮✱ s✉❝❤ t❤❛t ❤✉♠❛♥ ♣❛❝❦❡rs ♦♥ t❤❡ ♦t❤❡r s✐❞❡
♦❢ t❤❡ ♣✉t ✇❛❧❧ r❡❝❡✐✈❡ ❝✉st♦♠❡r ♦r❞❡rs q✉✐❝❦❧② ❛♥❞ ❞♦ ♥♦t r✉♥ ✐❞❧❡ ✇❤✐❧❡ st♦✇✐♥❣ ♦r❞❡rs
✐♥t♦ ❝❛r❞❜♦❛r❞ ❜♦①❡s✳
✷ ❈♦♠♣✉t❛t✐♦♥❛❧ ❝♦♠♣❧❡①✐t②
■♥ t❤✐s s❡❝t✐♦♥✱ ✇❡ ✐♥✈❡st✐❣❛t❡ t❤❡ ❝♦♠♣❧❡①✐t② st❛t✉s ♦❢ 1|m✿n|
P
Co ❛♥❞ r❡✐t❡r❛t❡ t❤❡
❝♦♠♣❧❡①✐t② ♣r♦♦❢ ✐♥✐t✐❛❧❧② ♣r❡s❡♥t❡❞ ❜② ❇♦②s❡♥ ❡t✳ ❛❧✳ ✭✷✵✶✽✮✳ ❚❤❡ tr❛♥s❢♦r♠❛t✐♦♥ ✐s ❢r♦♠
t❤❡ ❧✐♥❡❛r ❛rr❛♥❣❡♠❡♥t ♣r♦❜❧❡♠ ✭▲❆P✮✱ ✇❤✐❝❤ ✐s ✇❡❧❧✲❦♥♦✇♥ t♦ ❜❡ ◆P✲❝♦♠♣❧❡t❡ ✐♥ t❤❡
str♦♥❣ s❡♥s❡✱ s❡❡ ●❛r❡② ❛♥❞ ❏♦❤♥s♦♥ ✭✶✾✼✾✮✳
▲❆P✿ ●✐✈❡♥ ❛ ❣r❛♣❤ G = (V, E) ❛♥❞ ❛ ♣♦s✐t✐✈❡ ✐♥t❡❣❡r K✳ ■s t❤❡r❡ ❛ ♦♥❡✲t♦✲♦♥❡✲❢✉♥❝t✐♦♥
ϑ : V → {1, 2, . . . , |V |}✱ ✐✳❡✳✱ ❛ ♥✉♠❜❡r✐♥❣ ♦❢ ♥♦❞❡s V ✇✐t❤ ✐♥t❡❣❡r ✈❛❧✉❡s ❢r♦♠ 1 t♦ |V |✱
55

s✉❝❤ t❤❛t
P
(u,v)∈E |ϑ(u) − ϑ(v)| ≤ K❄
❚❤❡♦r❡♠✿ 1|m✿n|
P
Co ✐s str♦♥❣❧② ◆P✲❤❛r❞ ❡✈❡♥ ✐❢ ❛❧❧ ❥♦❜s ❤❛✈❡ ✉♥✐t ♣r♦❝❡ss✐♥❣ t✐♠❡✳
Pr♦♦❢✿ ❲✐t❤✐♥ ♦✉r tr❛♥s❢♦r♠❛t✐♦♥ ♦❢ ▲❆P t♦ 1|m✿n|
P
Co ✇❡ ✐♥tr♦❞✉❝❡ ❛ ❥♦❜ ❢♦r ❡❛❝❤
♥♦❞❡✱ s♦ t❤❛t n = |V |✳ ❚❤❡ ✐♥t❡❣❡r ✈❛❧✉❡ ϑ(u) ❛ss✐❣♥❡❞ t♦ ❛ ♥♦❞❡ u ✇✐t❤✐♥ ▲❆P ❝♦rr❡s♣♦♥❞s
t♦ t❤❡ s❡q✉❡♥❝❡ ♣♦s✐t✐♦♥ φ−1
(i) ❛ss✐❣♥❡❞ t♦ ❥♦❜ i ✇✐t❤✐♥ 1|m✿n|
P
Co✳ ●✐✈❡♥ t❤❡ ♠❛①✐♠✉♠
❞❡❣r❡❡ δ(G) = maxu∈V {|{v ∈ V : (u, v) ∈ E}|} ♦❢ t❤❡ ▲❆P ❣r❛♣❤✱ ✇❡ ✐♥tr♦❞✉❝❡ δ(G) ♦r❞❡rs
❢♦r ❡❛❝❤ ♥♦❞❡ u ∈ V ✿ ❋✐rst✱ ❛♥ ♦r❞❡r {u, v} ✐s ❣❡♥❡r❛t❡❞ ❢♦r ❡❛❝❤ ❛❞❥❛❝❡♥t ♥♦❞❡ v✱ s♦ t❤❛t
❢♦r ❡❛❝❤ ❡❞❣❡ (u, v) ∈ E t✇♦ ♦r❞❡rs {u, v} ❛♥❞ {v, u} ❛r❡ ❣❡♥❡r❛t❡❞✳ ❚❤❡♥✱ ❢♦r ❡❛❝❤ ♥♦❞❡
❤❛✈✐♥❣ ❛ ❞❡❣r❡❡ ❧❡ss t❤❛♥ δ(G) ✇❡ ❡①t❡♥❞ t❤❡ ♦r❞❡r s❡t ❜② ❛❞❞✐t✐♦♥❛❧ s✐♥❣❧❡✲❥♦❜✲♦r❞❡rs {u}
✉♥t✐❧ δ(G) ♦r❞❡rs ♣❡r ♥♦❞❡ ❛r❡ ❣❡♥❡r❛t❡❞✳ ■♥ t♦t❛❧✱ δ(G) · |V | s✐♥❣❧❡✲ ❛♥❞ t✇♦✲❥♦❜✲♦r❞❡rs ❛r❡
✐♥tr♦❞✉❝❡❞✳ ❚❤❡ q✉❡st✐♦♥ ✇❡ ❛s❦ ✐s ✇❤❡t❤❡r ✇❡ ❝❛♥ ✜♥❞ ❛ s♦❧✉t✐♦♥ ❢♦r 1|m✿n|
P
Co ✇✐t❤
♦❜❥❡❝t✐✈❡ ✈❛❧✉❡
Z = δ(G) ·
|V | · (|V | + 1)
2
+ K.
❖❜✈✐♦✉s❧②✱ t❤✐s tr❛♥s❢♦r♠❛t✐♦♥ ❝❛♥ ❜❡ ❞♦♥❡ ✐♥ ♣♦❧②♥♦♠✐❛❧ t✐♠❡✳ ❚❤❡ δ(G) ♦r❞❡rs ❛ss♦✲
❝✐❛t❡❞ ✇✐t❤ ❡❛❝❤ ❥♦❜ u ❛r❡ ❡✐t❤❡r s✐♥❣❧❡✲❥♦❜✲♦r❞❡rs✱ ✇❤✐❝❤ ❤❛✈❡ ❝♦♠♣❧❡t✐♦♥ t✐♠❡ φ−1
(u)✱ ♦r
t✇♦✲❥♦❜✲♦r❞❡rs✳ ❊❛❝❤ ♦r❞❡r {u, v} ♦❢ t❤❡ ❧❛tt❡r ❦✐♥❞ ❛❧✇❛②s ❡①✐sts t✇✐❝❡✱ ❜❡❝❛✉s❡ ✐♥ t❤❡ ♥❛♠❡
♦❢ ❡❛❝❤ ❡❞❣❡ (u, v) t✇♦ ✐❞❡♥t✐❝❛❧ ♦r❞❡rs ❛r❡ ✐♥tr♦❞✉❝❡❞✱ ✐✳❡✳✱ ♦♥❡ ✇❤❡♥ ❣❡♥❡r❛t✐♥❣ t❤❡ δ(G)
♦r❞❡rs ❢♦r ♥♦❞❡ u ❛♥❞ t❤❡ ♦t❤❡r ✇❤❡♥ ❣❡♥❡r❛t✐♥❣ ♦r❞❡rs ❢♦r v✳ ❚❤❡ ✉♥✐t ♣r♦❝❡ss✐♥❣ t✐♠❡s
❛❧❧♦✇ ✉s t♦ ♠❡❛s✉r❡ ❝♦♠♣❧❡t✐♦♥ t✐♠❡ ✐♥ s❡q✉❡♥❝❡ ♣♦s✐t✐♦♥s ♦❢ t❤❡ ❥♦❜ s❡q✉❡♥❝❡✳ ❚❤❡ s✉♠ ♦❢
❝♦♠♣❧❡t✐♦♥ t✐♠❡s ❢♦r ❜♦t❤ ♦❢ t❤❡s❡ ♦r❞❡rs ✐s✱ t❤✉s✱ t✇✐❝❡ t❤❡ s❡q✉❡♥❝❡ ♣♦s✐t✐♦♥ ♦❢ t❤❡ ❧❛t❡r
♦❢ ❜♦t❤ ❥♦❜s u ❛♥❞ v✳ ■❢ φ−1
(u) φ−1
(v)✱ t❤✐s ❛♠♦✉♥ts t♦ 2φ−1
(v)✳ ❉✉❡ t♦ t❤❡ ✐♥❡q✉❛❧✐t②
♦❢ φ−1
(u) ❛♥❞ φ−1
(v)✱ ✇❡ ❝❛♥ r❡❛rr❛♥❣❡ 2φ−1
(v) t♦ φ−1
(v)+φ−1
(u)+(φ−1
(v)−φ−1
(u))✳ ■❢
✇❡ ❛ss✐❣♥ t❤❡ ❢♦r♠❡r t✇♦ t✐♠❡ s♣❛♥s φ−1
(v) ❛♥❞ φ−1
(u) t♦ ❥♦❜s v ❛♥❞ u✱ r❡s♣❡❝t✐✈❡❧②✱ t❤❡♥
✐t ❜❡❝♦♠❡s ♦❜✈✐♦✉s t❤❛t t♦ ❡❛❝❤ s❡q✉❡♥❝❡ ♣♦s✐t✐♦♥ i = 1, . . . , n ❡①❛❝t❧② δ(G) t✐♠❡ s♣❛♥s ❛r❡
❛ss✐❣♥❡❞✳ ❚❤✉s✱ ✇❡ ❤❛✈❡ ❛♥ ✐♥❡✈✐t❛❜❧❡ ❛♠♦✉♥t ♦❢ ❝♦♠♣❧❡t✐♦♥ t✐♠❡✱ ✐✳❡✳✱ ✐♥❞❡♣❡♥❞❡♥t ♦❢ t❤❡
s❡q✉❡♥❝❡ ♣♦s✐t✐♦♥s ♦❢ ❥♦❜s✱ ♦❢
δ(G) ·
X
u∈V
φ−1
(u) = δ(G) ·
|V |
X
i=1
i = δ(G) ·
|V | · (|V | + 1)
2
.
❚❤❡ r❡♠❛✐♥✐♥❣ t✐♠❡ s♣❛♥s φ−1
(v) − φ−1
(u) ✇✐t❤✐♥ 1|m✿n|
P
Co✱ ✇❤✐❝❤ ❛r❡ ❞❡♣❡♥❞❡♥t
♦❢ t❤❡ s❡q✉❡♥❝❡ ♣♦s✐t✐♦♥s ♦❢ ❥♦❜s✱ ❡①❛❝t❧② ❡q✉❛❧ t❤❡ ❞✐✛❡r❡♥❝❡ ✐♥ t❤❡ ♥♦❞❡ ♥✉♠❜❡rs ❛ss✐❣♥❡❞
t♦ ❡❛❝❤ ❡❞❣❡ ✇✐t❤✐♥ ▲❆P✱ s♦ t❤❛t ❜♦t❤ ♣r♦❜❧❡♠s ❛r❡ ❞✐r❡❝t❧② tr❛♥s❢❡r❛❜❧❡ ❢r♦♠ ❡❛❝❤ ♦t❤❡r
❛♥❞ t❤❡ t❤❡♦r❡♠ ❤♦❧❞s✳
✸ ❆♣♣❧✐❝❛t✐♦♥ ✐♥ t❤❡ ✇❛r❡❤♦✉s❡s ♦❢ ♦♥❧✐♥❡ r❡t❛✐❧❡rs
❖♥❧✐♥❡ r❡t❛✐❧❡rs ❧✐❦❡ ❆♠❛③♦♥ ❊✉r♦♣❡ ❛♥❞ ❩❛❧❛♥❞♦✱ t②♣✐❝❛❧❧②✱ str✉❝t✉r❡ t❤❡✐r ♦r❞❡r ❢✉❧✲
✜❧♠❡♥t ♣r♦❝❡ss ✐♥t♦ t❤r❡❡ ❜❛s✐❝ st❡♣s✿
✕ P✐❝❦✐♥❣✿ ❋✐rst✱ t❤❡ ✐t❡♠s ❞❡♠❛♥❞❡❞ ❜② ❝✉st♦♠❡r ♦r❞❡rs ♥❡❡❞ t♦ ❜❡ r❡tr✐❡✈❡❞ ❢r♦♠ t❤❡
s❤❡❧✈❡s ♦❢ ❛ ✇❛r❡❤♦✉s❡✳ ▼♦st ♦♥❧✐♥❡ r❡t❛✐❧❡rs ❛♣♣❧② ❛ ♣✐❝❦❡r✲t♦✲♣❛rts ♦r❞❡r ♣✐❝❦✐♥❣ ✐♥
❛ ❜❛t❝❤✐♥❣ ❛♥❞ ③♦♥✐♥❣ ❡♥✈✐r♦♥♠❡♥t ✇❤❡r❡✱ ❛❞❞✐t✐♦♥❛❧❧②✱ ❛ ♠✐①❡❞✲s❤❡❧✈❡s ♣♦❧✐❝② ✭❛❧s♦
❞❡♥♦t❡❞ ❛s s❝❛tt❡r❡❞ st♦r❛❣❡ ✭❲❡✐❞✐♥❣❡r ❛♥❞ ❇♦②s❡♥ ✷✵✶✽✮✮ ✐s ❛♣♣❧✐❡❞✳ ❯♥❞❡r t❤✐s ♣♦❧✐❝②
✉♥✐t ❧♦❛❞s ♦❢ st♦❝❦ ❦❡❡♣✐♥❣ ✉♥✐ts ✭❙❑❯s✮ ❛r❡ ♣✉r♣♦s❡❢✉❧❧② ❜r♦❦❡♥ ❞♦✇♥ ❛♥❞ s✐♥❣❧❡ ✐t❡♠s
❛r❡ s❝❛tt❡r❡❞ ❛❧❧ ♦✈❡r t❤❡ s❤❡❧✈❡s ♦❢ ❛ ✇❛r❡❤♦✉s❡✳ ■♥ t❤✐s ✇❛②✱ t❤❡r❡ ✐s ❛❧✇❛②s s♦♠❡ ✐t❡♠
♦❢ ❛ ❞❡♠❛♥❞❡❞ ❙❑❯ ❝❧♦s❡ ❜② ✐rr❡s♣❡❝t✐✈❡ ♦❢ t❤❡ ❝✉rr❡♥t ♣✐❝❦❡r ❧♦❝❛t✐♦♥✳ ■♥ s✉❝❤ ❛
56

s❡tt✐♥❣✱ ❧❛r❣❡ ♦♥❧✐♥❡ r❡t❛✐❧❡rs ❛♣♣❧② ❞♦③❡♥s ♦❢ ♣✐❝❦❡rs✱ ✇❤✐❝❤ ❛r❡ t②♣✐❝❛❧❧② ❛ss✐❣♥❡❞ t♦
s♣❡❝✐✜❝ ③♦♥❡s ♦❢ t❤❡ ✇❛r❡❤♦✉s❡✳ ❚❤❡② ♣✐❝❦ ❜❛t❝❤❡❞ ♦r❞❡rs ✐♥ ♣❛r❛❧❧❡❧ ✐♥t♦ ❜✐♥s ❡❛❝❤
✜♥❛❧❧② ❝♦♥t❛✐♥✐♥❣ ♣❛rt✐❛❧ ♦r❞❡rs ❢♦r ♠✉❧t✐♣❧❡ ❝✉st♦♠❡rs✳
✕ ■♥t❡r♠❡❞✐❛t❡ st♦r❛❣❡✿ ❈♦♠♣❧❡t❡❞ ❜✐♥s ❛r❡ ❤❛♥❞❡❞ ♦✈❡r t♦ t❤❡ ❝❡♥tr❛❧ ❝♦♥✈❡②♦r s②st❡♠
✇❤❡r❡ t❤❡② ❛r❡ st♦r❡❞ ✉♥t✐❧ ❛❧❧ ❜✐♥s ❜❡❧♦♥❣✐♥❣ t♦ t❤❡ s❛♠❡ ❜❛t❝❤ ❤❛✈❡ ❛rr✐✈❡❞ ❢r♦♠
t❤❡✐r ③♦♥❡s✳ ❖♥❝❡ ❛ ❜❛t❝❤ ✐s ❝♦♠♣❧❡t❡✱ t❤❡ r❡s♣❡❝t✐✈❡ ❜✐♥s ❛r❡ r❡❧❡❛s❡❞ ❢r♦♠ st♦r❛❣❡
❛♥❞ ❝♦♥✈❡②❡❞ t♦✇❛r❞ t❤❡ ❝♦♥s♦❧✐❞❛t✐♦♥ ❛r❡❛✳
✕ ❖r❞❡r ❝♦♥s♦❧✐❞❛t✐♦♥✿ ❚❤❡ ❜✐♥s ♦❢ ❛ ❜❛t❝❤ s✉❝❝❡ss✐✈❡❧② ❛rr✐✈❡ ❛t ❛ ❝♦♥✈❡②♦r s❡❣♠❡♥t ♦❢
t❤❡ ❝♦♥s♦❧✐❞❛t✐♦♥ ❛r❡❛✳ ❍❡r❡✱ ❛ ❧♦❣✐st✐❝s ✇♦r❦❡r ✇❡ ❝❛❧❧ t❤❡ ♣✉tt❡r r❡s✐❞❡s✳ ❚❤❡ ♣✉tt❡r
s✉❝❝❡ss✐✈❡❧② r❡♠♦✈❡s t❤❡ ✐t❡♠s ❢r♦♠ t❤❡ ❝✉rr❡♥t ❜✐♥ ❛♥❞ ♣✉ts t❤❡♠ ✐♥t♦ t❤❡ ♣✉t ✇❛❧❧✳
❚❤❡ ♣✉t ✇❛❧❧ ✐s ❛ s✐♠♣❧❡ r❡❛❝❤✲tr♦✉❣❤ r❛❝❦ s❡♣❛r❛t❡❞ ✐♥t♦ ♠✉❧t✐♣❧❡ s❤❡❧✈❡s✱ ✇❤✐❝❤ ❛r❡
❛❝❝❡ss✐❜❧❡ ❢r♦♠ ❜♦t❤ s✐❞❡s✳ ❊❛❝❤ s❤❡❧❢ ✐s t❡♠♣♦r❛r✐❧② ❛ss✐❣♥❡❞ t♦ ❛ s❡♣❛r❛t❡ ♦r❞❡r ❛♥❞
♦♥❝❡ t❤❡ ♣✉tt❡r s❝❛♥s t❤❡ ❝✉rr❡♥t ✐t❡♠ ❛ ♣✉t✲t♦✲❧✐❣❤t ♠❡❝❤❛♥✐s♠ ✐♥❞✐❝❛t❡s ✐♥t♦ ✇❤✐❝❤
s❤❡❧❢ t❤❡ ❝✉rr❡♥t ✐t❡♠ ✐s t♦ ❜❡ ♣✉t✳ ■♥ t❤✐s ✇❛②✱ ❜✐♥ ❛❢t❡r ❜✐♥ ✐s s♦rt❡❞ ✐♥t♦ t❤❡ ✇❛❧❧✳
❖♥ t❤❡ ♦t❤❡r s✐❞❡ ♦❢ t❤❡ ✇❛❧❧ r❡s✐❞❡ t❤❡ ♣❛❝❦❡rs✳ ❍❡r❡✱ ❛♥♦t❤❡r ♣✉t✲t♦✲❧✐❣❤t ♠❡❝❤❛♥✐s♠
✐♥❞✐❝❛t❡s ❝♦♠♣❧❡t❡❞ ♦r❞❡rs✱ s♦ t❤❛t ❛ ♣❛❝❦❡r ❝❛♥ ❡♠♣t② ❛♥ ✐♥❞✐❝❛t❡❞ s❤❡❧❢ ❛♥❞ ♣❛❝❦
t❤❡ r❡s♣❡❝t✐✈❡ ✐t❡♠s ✐♥t♦ ❛ ❝❛r❞❜♦❛r❞ ❜♦①✳ P❛❝❦❡❞ ♦r❞❡rs ❛r❡✱ ✜♥❛❧❧②✱ ❤❛♥❞❡❞ ♦✈❡r t♦
❛♥♦t❤❡r ❝♦♥✈❡②♦r s②st❡♠ ❜r✐♥❣✐♥❣ t❤❡♠ t♦✇❛r❞s t❤❡ s❤✐♣♣✐♥❣ ❛r❡❛✳
❖✉r ♣r♦❜❧❡♠ 1|m✿n|
P
Co ❝❛♥ ❞✐r❡❝t❧② ❜❡ ❛♣♣❧✐❡❞ t♦ ❞❡t❡r♠✐♥❡ t❤❡ s❡q✉❡♥❝❡ ♦❢ ❜✐♥s✱
✐♥ ✇❤✐❝❤ ❛ ❜❛t❝❤ ✐s r❡❧❡❛s❡❞ ❢r♦♠ ✐♥t❡r♠❡❞✐❛t❡ st♦r❛❣❡✳ ❏♦❜s ❡q✉❛❧ ❜✐♥s ❛♥❞ t❤❡ ♣r♦❝❡ss✐♥❣
s❡q✉❡♥❝❡ ♦❢ ❥♦❜s ♦♥ t❤❡ s✐♥❣❧❡ ♠❛❝❤✐♥❡ ❝♦rr❡s♣♦♥❞s t♦ t❤❡ r❡❧❡❛s❡ s❡q✉❡♥❝❡ ♦❢ t❤❡ ❜❛t❝❤
❢r♦♠ ✐♥t❡r♠❡❞✐❛t❡ st♦r❛❣❡✱ ✇❤✐❝❤ ✐s ❛❧s♦ t❤❡ s❡q✉❡♥❝❡ ✐♥ ✇❤✐❝❤ ❜✐♥s ❛r❡ s♦rt❡❞ ✐♥t♦ t❤❡ ♣✉t
✇❛❧❧✳ ❚❤❡ ♣r♦❝❡ss✐♥❣ t✐♠❡s pj ❞❡♣❡♥❞ ♦♥ t❤❡ ♥✉♠❜❡r ♦❢ ✐t❡♠s ❝♦♥t❛✐♥❡❞ ✐♥ ❡❛❝❤ ❜✐♥ j✳ ❇②
♠✐♥✐♠✐③✐♥❣ t❤❡ s✉♠ ♦❢ ❝♦♠♣❧❡t✐♦♥ t✐♠❡s ♦r❞❡rs ❛r❡ q✉✐❝❦❧② s♦rt❡❞ ✐♥t♦ t❤❡ ♣✉t ✇❛❧❧ ❜② t❤❡
♣✉tt❡r✱ s♦ t❤❛t t❤❡ ♣❛❝❦❡rs ♦♥ t❤❡ ♦t❤❡r s✐❞❡ ♦❢ t❤❡ ✇❛❧❧ r❡❝❡✐✈❡ ♦r❞❡rs s♦♦♥❡r ❛♥❞ ✐❞❧❡ t✐♠❡s
❛r❡ ❛✈♦✐❞❡❞✳ ❲✐t❤ t❤❡ ❤❡❧♣ ♦❢ ❛ ❝♦♠♣r❡❤❡♥s✐✈❡ s✐♠✉❧❛t✐♦♥ st✉❞② ❇♦②s❡♥ ❡t✳ ❛❧✳ ✭✷✵✶✽✮ s❤♦✇
t❤❛t ♦♣t✐♠✐③❡❞ ❜✐♥ s❡q✉❡♥❝❡s ❝♦♥s✐❞❡r❛❜❧② r❡❞✉❝❡ t❤❡ ♣❛❝❦❡rs✬ ✐❞❧❡ t✐♠❡s✳
❋✉t✉r❡ r❡s❡❛r❝❤ s❤♦✉❧❞ ❝♦♥s✐❞❡r ♦✉r s✐♥❣❧❡ ♠❛❝❤✐♥❡ s❝❤❡❞✉❧✐♥❣ ♣r♦❜❧❡♠ ✇✐t❤ m✿n ❥♦❜✲
♦r❞❡r r❡❧❛t✐♦♥s ❢♦r ♦t❤❡r ♦❜❥❡❝t✐✈❡s✳ ❚❤❡r❡ ♠❛② ❜❡ ♦t❤❡r ❝❛s❡s ✇❤❡r❡ t❤❡ tr❛❞✐t✐♦♥❛❧ s❝❤❡✲
❞✉❧✐♥❣ ♣r♦❜❧❡♠✱ ✐✳❡✳✱ ✇✐t❤ ❛ ✶✿✶ r❡❧❛t✐♦♥ ❛♠♦♥❣ ❥♦❜s ❛♥❞ ♦r❞❡rs✱ ✐s s♦❧✈❛❜❧❡ ✐♥ ♣♦❧②♥♦♠✐❛❧
t✐♠❡✱ ✇❤❡r❡❛s t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ♣r♦❜❧❡♠ ✇✐t❤ m✿n ❥♦❜✲♦r❞❡r r❡❧❛t✐♦♥s t✉r♥s ♦✉t ◆P✲❤❛r❞✳
❆❝❦♥♦✇❧❡❞❣❡♠❡♥ts
❚❤✐s r❡s❡❛r❝❤ ❤❛s ❜❡❡♥ s✉♣♣♦rt❡❞ ❜② t❤❡ ●❡r♠❛♥ ❙❝✐❡♥❝❡ ❋♦✉♥❞❛t✐♦♥ ✭❉❋●✮ t❤r♦✉❣❤
t❤❡ ❣r❛♥t ✏P❧❛♥♥✐♥❣ ❛♥❞ ♦♣❡r❛t✐♥❣ s♦rt❛t✐♦♥ ❝♦♥✈❡②♦r s②st❡♠s✧ ✭❇❖ ✸✶✹✽✴✺✲✶✮✳
❘❡❢❡r❡♥❝❡s
❇♦②s❡♥✱ ◆✳✱ ❑✳ ❙t❡♣❤❛♥ ❛♥❞ ❋✳ ❲❡✐❞✐♥❣❡r✱ ✷✵✶✽✱ ✏▼❛♥✉❛❧ ♦r❞❡r ❝♦♥s♦❧✐❞❛t✐♦♥ ✇✐t❤ ♣✉t ✇❛❧❧s✿ ❚❤❡
❜❛t❝❤❡❞ ♦r❞❡r ❜✐♥ s❡q✉❡♥❝✐♥❣ ♣r♦❜❧❡♠✧✱ ❊❯❘❖ ❏♦✉r♥❛❧ ♦♥ ❚r❛♥s♣♦rt❛t✐♦♥ ❛♥❞ ▲♦❣✐st✐❝s ✭t♦
❛♣♣❡❛r✮✳
●❛r❡②✱ ▼✳❘✳✱ ❉✳❙✳ ❏♦❤♥s♦♠✱ ✶✾✺✻✱ ✏❈♦♠♣✉t❡rs ❛♥❞ ✐♥tr❛❝t❛❜✐❧✐t②✿ ❆ ❣✉✐❞❡ t♦ t❤❡ t❤❡♦r② ♦❢ ◆P✲
❝♦♠♣❧❡t❡♥❡ss✧✱ ❋r❡❡♠❛♥✱ ◆❡✇ ❨♦r❦✳
❙♠✐t❤ ❲✳❊✳✱ ✶✾✺✻✱ ✏❱❛r✐♦✉s ♦♣t✐♠✐③❡rs ❢♦r s✐♥❣❧❡✲st❛❣❡ ♣r♦❞✉❝t✐♦♥✧✱ ◆❛✈❛❧ ❘❡s❡❛r❝❤ ▲♦❣✐st✐❝s ◗✉❛r✲
t❡r❧②✱ ❱♦❧✳ ✵✸✱ ♣♣✳ ✺✾✲✻✻✳
❲❡✐❞✐♥❣❡r✱ ❋✳✱ ◆✳ ❇♦②s❡♥✱ ✷✵✶✽✱ ✏❙❝❛tt❡r❡❞ st♦r❛❣❡✿ ❍♦✇ t♦ ❞✐str✐❜✉t❡ st♦❝❦ ❦❡❡♣✐♥❣ ✉♥✐ts ❛❧❧
❛r♦✉♥❞ ❛ ♠✐①❡❞✲s❤❡❧✈❡s ✇❛r❡❤♦✉s❡✧✱ ❚r❛♥s♣♦rt❛t✐♦♥ ❙❝✐❡♥❝❡ ✭t♦ ❛♣♣❡❛r✮✳
57

A MILP formulation for multi-robot pick-and-place
scheduling
Briand C.1
, Ndiaye J.C.1
and Parlouar R.2
1
LAAS-CNRS, Université de Toulouse, UPS, Toulouse, France
email: briand@laas.fr, ndiaye@laas.fr
2
NOVALYNX, 35 Bis Route de Bessières, 31240 L’Union, France
email: remi.parlouar@novalynx.fr
Keywords: Assignment and Scheduling, Mixed Integer Linear Programming, Multi-robot
pick-and-place.
1 Introduction
The 21st century is marked by the fourth industrial revolution, which embraces many
technologies and concepts. Among them, robotization is often viewed as the most promising
avenues of progress in the field of automated production. Indeed the use of more and more
sophisticated machines and robots is able to bring improvements in production costs, rates,
quality and operators safety.
Among the various types of robots frequently used in production systems are the han-
dling robots, which basically pick parts somewhere in the shop and place them elsewhere.
More specifically, this study focuses on automated packaging systems involving several
handling robots. A packaging system is generally composed of two conveyor belts convey-
ing products and boxes, respectively. A handling robot picks one or several products on
the former conveyor and places them in a box on the latter. The conveyor belts may have
several possible shapes: parallel, perpendicular or circular. The parallel one is the most
common and product and box flows can go in the same or opposite direction, as illustrated
in Figure 1 (taken from (Blanco Rendon 2013)). A pick/place task can only be carried out
by one robot when the corresponding product/box is present inside the working area of
this robot. The normal speed of each conveyor being assumed known, a time window can
be associated with each robot task. Moreover, the pick/place task duration can vary and
depends either on the product or the robot.
Fig. 1. Conveyor belts configurations
58

The problem considered in this paper, further referred to as the Multi-robot-Pick-and-
Place Scheduling Problem (MPPSP), consists in i) assigning products and boxes to robots
and ii) defining a consistent starting time for each pick/place task, so that the filling rate
is maximized (or equivalently, the number of filled boxes over a given time horizon is
maximized). In the case of a pick-task, the starting-time consistency only requires that the
task is performed by the robot during its time execution window (i.e., when the product
is present inside the robot workspace). In the case of a place-task, the previous condition
should obviously hold and, additionally, there are flow constraints: if k products should
be placed inside each box in a one-shot operation, one has to ensure that k pick tasks
have been achieved before the place task can be carried out. Finally, note that in the case
the conveyor speed can be controlled (which is assumed impossible in the present study),
the filling rate can be further improved, which gives rise to a third MPPSP dimension
consisting in the determination of the optimal conveyor speed profiles.
In many existing systems, a vision system is integrated in front of the conveyor entries to
locate the various parts, which allows predicting the working-area entry or exit events a few
seconds before their occurrence. Moreover, in the context of the industry 4.0, all information
about production and packaging processes may be known in advance so that execution
windows of pick/place tasks could be either predicted earlier. Under the assumption of
predictability of the product/box flows, the MPPSP is studied in its offline version in this
paper and a compact Mixed-Integer Linear Programming (MILP) formulation is proposed.
The paper is structured as follows. First, a brief literature overview is made that par-
ticularly put into evidence some relationships between MPPSP and some other well-known
problems of the scheduling literature. Then, our MILP formulation is established that takes
benefits from specific dominance rules, which allows characterizing all the dominant solu-
tions on a robot within a single master-sequence. Some conclusions are drawn in the last
section.
2 Literature overview
A vast majority of the paper of the literature tackles the online version of the problem,
taking interest in designing efficient rules or cooperation mechanisms between robots that
maximize the filling rate, while balancing the working load between robots, e.g., (Blanco
Rendon 2013, Bouchrit 2016, Huang et al. 2015, Humbert et al. 2015). In the OR literature,
Daoud et al. (2014) took interest in designing pick-and-place robotic systems and propose
fast metaheuristics to determine the best schedule rule to be applied to each robot.
For the offline version of MPPSP, the literature is scarcer. In (A. Bouchrit 2016), a
network-based MILP formulation is proposed to solve the offline MPPSP in the case of
a homogeneous product/box flow (each product/box is separated from the next one on
the conveyor by a constant distance). Products are considered as nodes within a network
and the problem amounts to find for each robot the best path to collect the maximum
possible number of products, which gives a pick-and-place task sequence. Many constraints
are taken into account such as conveyor belt velocities, robot load balance, time windows
and flow constraints. Nevertheless, the implementation of this formulation on commercial
solver does not provide satisfying performances as finding optimal solution turns out to be
too time-demanding.
In the scheduling literature, MPPSP is sharing some similarities with the parallel ma-
chine problem with time windows that aims at minimizing the number of tardy jobs,
(denoted as P|rj|
P
Uj in (Pinedo 2008)). This problem is known to be NP-Hard in the
strong sense even for one single machine. Nevertheless, still under the assumption of a
single machine environment, it is polynomially solvable when execution windows have a
59

staircase structure. As a specific feature of MPPSP, we observe that there are several pos-
sible time windows for the execution of a task (depending on the robot implementing it),
which tends to indicate that MPPSP is also related to the Runway Scheduling Problem
(RSP) (Artiouchine et. al. 2008) that consists in sequencing aircraft landing. Note that
RSP is also NP-hard.
3 MILP formulation
This chapter takes an interest in finding a job sequence that maximizes the number of
filled boxes assuming the product and box flows predictable. We consider three sets B, P
and R of B boxes, P products and R robots, respectively. In the notations used below,
index b (p and r, respectively) refers to a box b ∈ B (a product p ∈ P and a robot r ∈ R,
respectively). The pick and place processing times are denoted Dpr and Dbr, which depend
on robot r. We refer to [Spr, Fpr] and [Sbr, Fbr] as the execution windows of product p (box
b, respectively) on robot r.
In the remainder of this paper, as the conveyor speed is assumed constant, we set
Fpr − Spr = ∆pr, ∀(p, r) ∈ P × R (Fbr − Sbr = ∆br, ∀(b, r) ∈ B × R, resp.). Moreover,
without loss of generality, we assume that ∆pr ∆br (products stay longer in the robot
working area than boxes) but, as explained below, it could be the reverse.
Once an assignment of products and boxes to robots is decided (note that a product/box
can possibly not be assigned), the problem left is to find a pick-and-place sequence on each
robot that i) is time feasible and ii) respects the constraint that k picks should always
precede any place operation. For ensuring time feasibility, following the idea proposed by
Briand and Ourari (2013), a master sequence can be considered that characterizes a set
of dominant sequences. This master sequence uses the notion of a top-job, i.e. a job such
that its execution window does not (strictly) include the execution window of any other
job. In our case, as there are only two kinds of time intervals (the pick and place ones) and
because ∆pr ∆br , any place operation is a top job. Therefore, a master sequence Θr
having the form below can be defined for each robot r.
Θr = σ−
1r 1 σ+
1r σ12 σ−
2r 2 σ+
2r · · · i − 1 σ+
i−1r σi−1,i σ+
ir
| {z }
θi−1r
i σ+
ir+ · · ·
Each place task i has two sets σ−
ir and σ+
ir of pick tasks at its left and its right, respec-
tively. More specifically, σ+
i−1r represents products which intervals overlap place interval
i−1 but not place interval i. Similarly, σ−
ir gathers pick tasks such that their intervals over-
lap box interval i but not box interval i − 1. Eventually, σi−1,i gathers product intervals
which overlap both box intervals i − 1 and i. We refer to θi−1r as the subset of pick tasks
located between place task i − 1 and i, with θ0r (θBr) the subset located at the left (the
right) of box 1 (of box B, resp.). Note that the same pick task can belong to several sets θ
and one has to decide whether the task is performed and, if it is performed, in which set
θ. One advantage of a master sequence lies in the fact that, once the previous decisions
made, the time feasibility of the resulting pick-and-place sequence can easily be assessed.
The following formulation takes benefit of the master sequence notion and introduces
the following binary variables. A box b is filled by robot r if binary variable ybr = 1 (0
otherwise). A product p is picked in subset θbr if xbpr = 1.
max z =
P
b
P
r ybr
60

X
b
X
p∈θbr
xbpr ≤ 1 , ∀p ∀r (1)
X
r
ybr ≤ 1 , ∀b (2)
kybr ≤ −k
X
ib
yir +
X
ib
X
p∈θir
xpir ≤ k , ∀b ∀r (3)
The master sequence Θr is time feasible , ∀r (4)
xbpr ∈ {0, 1} , ∀b ∀p ∀r
ybr ∈ {0, 1} , ∀p ∀r
The formulation aims at maximizing the number of filled boxes. Constraints (1-2) en-
force any product/box to be picked/filled once at the most. Constraints (3) aim at satisfying
the (flow) constraint, i.e. k product at the most should be picked before any place opera-
tion. As in (C. Briand and S. Ourari 2013), high level constraints (4) can be implemented
using a set of big-M linear constraints (not stated here for matter of conciseness) that use
integer variables sbr and fbr. Theses variables refer to as the earliest starting time and the
latest finishing time, respectively, of place task b on robot r (this value linearly depending
on the values of other binary variables), provided that sbr + Dbr ≤ fbr.
4 Conclusion
This paper sketches a formulation for solving the offline MPPSP. This formulation has
been tested and validated using some academic instances. A more systematic experimental
study is currently in progress to assess the efficiency of our approach. The special case
where the processing times of the pick/place tasks are identical (i.e., Dpr = Dpickr and
Dbr = Dplacer) will also be considered.
References
Artiouchine K., P. Baptiste, C. Durr, 2008,“Runway Sequencing with Holding Patterns, European
Journal of Operational Research, Vol. 189(3), pp.1254-1266.
Blanco Rendon D.P., 2013, “Modelling and Simulation of a Scheduling Algorithm for a Pick-and-
Place Packaging System, MastersThesis, Polytechnic of Milan.
Bouchrit A., 2016, “Optimal Scheduling for Robotized Pick and Place Packaging Systems, Mas-
tersThesis, Polytechnic of Milan.
Briand C. and S. Ourari, , 2013, “Minimizing the number of tardy jobs for the single machine
scheduling problem: MIP-based lower and upper bounds, RAIRO - Operations Research,
Vol. 47, pp. 33-46.
Daoud S., H. Chehade, F. Yalaoui and L. Amodeo, 2014, “Efficient metaheuristics for pick and
place robotic systems optimization, Journal of Intelligent Manufacturing, Volume 25, pp.
27-41.
Huang Y., R. Chiba, T. Arai, T. Ueyama, J. Ota, 2015, “Robust multi-robot coordination in pick-
and-place tasks based on part-dispatching rules, Robotics and Autonomous Systems, Volume
64, 2015, pp. 70-83.
Humbert G., M.T. Pham, X. Brun, M. Guillemot and D. Noterman, 2015, “Comparative analysis
of pick place strategies for a multi-robot application, Proc. IEEE 20th Conference on
Emerging Technologies and Factory Automation (ETFA), Luxembourg.
Pinedo, M.L., 2008, “Scheduling: Theory, Algorithm and Systems., 3rd Edition, Springer-Verlag,
New York.
61

Minimizing resource management costs in a portfolio
with resource transfer possibilities
Jerome Bridelance1
, Mario Vanhoucke1,2,3
1
Faculty of Economics and Business Administration, Ghent University, Belgium
jerome.bridelance@ugent.be, mario.vanhoucke@ugent.be
2
Technology and Operations Management Area, Vlerick Business School, Belgium
3
UCL School of Management, University College London, UK
Keywords: Resource availability, Resource transfers, Portfolio management.
1 Introduction
The research presented in this abstract is located in the multi-project environment.
Two approaches can be followed when working with a portfolio of projects, each with
their own methodologies. First of all the different projects can be combined into one large
super-project. This is done by adding additional precedence arcs and one dummy-start and
end-activtiy. In that way the problem is again reduced to a resource constrained project
scheduling problem (RCPSP), consequently this is called the single-project approach. There
is also a second way to deal with those multiple projects, namely the multi-project ap-
proach. Within this method every project remains an entity by itself, with its own critical
path (Kurtulus, I. and Davis, E.W. 1982). This second approach is preferred above the first
one, for multiple reasons. To begin, the first approach is nothing more than solving a single
project and takes distance from the multi-project environment. Secondly, up to now less
research has been done on this topic, which creates more opportunities for improvement
(Herroelen, W. 2005). Finally the second approach is a more realistic view of how multiple
projects are dealt with in practice (Browning, T.R. and Yassine, A.A. 2010). Next to those
two different ways of dealing with the schedule part of a portfolio of projects, there are also
different approaches of how the management of resources can be organized. First there is
the easiest method where the resources are all collected in one large resource pool. Those
can then be freely shared among the activities in the portfolio. This method is called the
resource sharing policy. Secondly, on the opposite end of the spectrum, there is the resource
dedication policy (Besikci, U. et. al. 2013). With this approach resources are dedicated to
a particular project at the beginning of the planning horizon. This method does not allow
to share resources between projects and consequently not between activities of different
projects. The policy is applied when sharing resources between projects is not feasible for
example if those projects are geographically too far distributed (Besikci, U. et. al. 2013).
As already mentioned those two ways are both ends of the resource management spectrum.
In between these two, multiple combinations are possible and are probably more realistic
approaches. An example of such an in between methodology is the dedication of resources
to projects but also allowing the transfer of these resources to other projects. According to
research on this topic, resource transfers should already be included in the scheduling part
(Kruger, D. and Scholl, A. 2009).
2 Problem description
This research deals with multi-project management, more precisely the scheduling and
resource management part. We have chosen to work with the multi-project approach.
Consequently every project is a separate entity and of course this decision also has an
62

influence on the used methodology and its accompanied assumptions. The objective of this
research is minimizing the resource costs, including availability and transfer costs. We are
working with a static number of projects which have to be scheduled and resources have
to be assigned to them in order to be executed. Because not all projects have the same
due date, it is not necessary to start all of them as early as possible. All resources should
be dedicated to a particular project and stay unified with it until the project is completely
finished, after that resources can be transferred to other projects. The general renewable
resource availabilities are positioned as low as possible. To accomplish this, projects are
shifted further in time and resources are transferred between them. All this is done while
taking the precedence relations between projects and the projects´ due dates into account.
As a consequence the following assumptions have to be incorporated into the methodology:
- Resources can be transferred between projects, but only when the first project is fin-
ished and the second project should still be started.
- Transfer time is depending on the two projects between which the resources are trans-
ferred and on the amount transferred.
- Projects can not be interrupted in time.
- The due date of every project should be met.
- Precedence relations between projects have to be satisfied.
- Project activities have fixed durations.
3 Extensions on existing literature
This research idea originates from existing literature and is created as an extension on
the combination of those research topics. Liberatore, M.J. and Pollack-Johnson, B. (2003)
came up with a new methodology to minimize the resource availability costs in a sin-
gle project setting, more precisely solving the resource availability cost problem (RACP).
By doing this the project´s due date and the activities´ resource requirements have to be
satisfied. This methodology obtains the minimum resource availabilities for the different re-
source types by deriving them from the solution of resource-constrained project scheduling
decision problems (RCPSDP). These RCPSDPs are solved with only one or two resource
types, all the others are supposed to have an unlimited availability. Resource dedication is
also an important part of this research topic. The first ones to introduce resource dedica-
tion in a multi-mode and multi-project environment were Besikci, U. et. al. (2013). They
presented two solution approaches to solve this problem, which can be divided into two
sub-problems. First there is the dedication of resource capacities to a particular project,
secondly the activities of the projects itself are scheduled. The first methodology works
with a genetic algorithm in combination with a new local improvement heuristic, namely
combinatorial auction. The second methodology employs a langrangian based heuristic and
a subgradient optimization method to find a solution for the resource dedication problem.
The research presented in this abstract combines, adapts and extends the above intro-
duced research studies. Like in Demeulemeester, E. (1995), also in this paper one of the
goals is to minimize the renewable resource availability costs. With the difference that we
now have a portfolio of projects at our disposal between which resources can be transferred.
That is the reason why numerous solutions for the general resource availabilities are pos-
sible. The solution of this problem is not the summation of the optimal RACP solutions
of every project separately. The general resource levels will be lower because transferring
resources is allowed now. Of course these transfers bring along costs as well and shifting
projects further in time can not be done endlessly because of every project´s due date. Pre-
vious research has already investigated the implementation of resource transfers in project
63

scheduling problems. Like in (Lacomme, P. et. al. 2017) where the resource transfers are
introduced in the scheduling problem via routing operations, with the ultimate goal of
minimizing the overall makespan. Another more practical example is the study of (Froger,
A. et. al. 2017). Here resource transfers are implicitly incorporated in the methodology by
only allowing that employees shift work locations on the same day, if these locations are
compatible. A Location is seen as compatible if the travel time between them is negligible
in comparison to a time unit. With the presented research the added value is the combina-
tion of the resource transfers with the undetermined resource availability levels. In contrary
to Besikci, U. et. al. (2013) renewable resources are not only dedicated to projects at the
beginning of the planning horizon. After a project is terminated, the renewable resources
can be assigned to a new project after they are transferred. Which makes the problem a
trade-off between availability and transferring costs, while still satisfying the projects´ due
dates. In figure 1 a comparison is made between on the left side the method presented in
this abstract and on the right side the methodology when every project is scheduled as
early as possible. The considered transfer times are indicated by the arrows in figure 1.
With this latter approach the portfolio´s cost is not optimized as a whole. Underneath
figure 1 the cost difference between the methods is presented. Assuming a transfer cost of
10 euro/unit and an availability cost of 20 euro/week. Information about the projects can
be found in table 1.
Availability
of
resource
1
0
25
50
75
100
125
150
175
200
Time (weeks)
0 1 2 3 4 5 6 7 8
Project 1
Project 2 Project 5
Project
4
Project 3
Availability
of
resource
1
0
25
50
75
100
125
150
175
200
Time (weeks)
0 1 2 3 4 5 6 7 8 9
Project 1
Project 2 Project 5
Project
4
Project 3
Availability
of
resource
2
0
25
50
75
100
125
150
175
200
Time (weeks)
0 1 2 3 4 5 6 7 8 9
Project 1
Project 2 Project 5 Project
4
Project 3
Availability
of
resource
2
0
25
50
75
100
125
150
175
200
Time (weeks)
0 1 2 3 4 5 6 7 8
Project 1
Project 2 Project 5
Project
4
Project 3
Fig. 1. Example: comparison between methods
Calculations
Left approach
Transfer costs: ((50 + 50 + 25 +50) + (75 + 25 + 50)) * 10 euro/unit = 3250 euro
Availability costs: (100 + 125) units * 7 weeks * 20 euro/week = 31500 euro
Total cost = 34750 euro
Right approach
Transfer costs: ((50 + 50) + (75 + 25)) * 10 euro/unit = 2000 euro
Availability costs: (175 + 175) units * 5,5 weeks * 20 euro/week = 38500 euro
Total cost = 40500 euro
64

Table 1. Project portfolio information
Project Duration Res.1 Res.2 Predecessor Due date
Project 1 3 weeks 50 25 / week 3
Project 2 2 weeks 50 75 / week 3
Project 3 2 weeks 50 50 project 1 week 6
Project 4 1 week 75 50 project 1 week 8
Project 5 2 weeks 50 75 project 2 week 8
4 Methodology
The research papers presented in the previous part were used to come up with this new
research problem and gave inspiration of which different methods can be applied to solve
the specific problem. As a consequence, first a full factorial design is set up to conduct a
complete analysis of multiple priority rules. The test problems used to perform this analysis
are generated with different network-, project- and resource-related characteristics, includ-
ing network complexity, the level of parallelism in the project portfolio and difference in
resource type usage by the projects. All this is done to decide in which situation which
priority rule should be used. Priority rule heuristics stay important for multiple reasons. In
comparison to meta-heuristics the computational complexity is lower, which makes them
interesting for larger problems. Next to this, priority rules are often employed to construct
initial solutions for meta-heuristics. Next to these priority rules, a meta-heuristic is con-
structed to test various experiments and provide some managerial insights. The influence
of the following situations on the objective function value is investigated:
- The ratio between resource availability costs and resource transfer costs.
- The ratio between the range in the projects´ due dates and the mean project duration.
- The diversity in usage of different resource types by the projects.
References
Besikci, U., Bilge, U. and Ulusoy, G., 2013, “Resource dedication problem in a multi-project envi-
ronment, Flexible Services and Manufacturing Journal, Vol. 25(1-2), pp. 206-229.
Browning, T.R. and Yassine, A.A., 2010, “Resource-constrained multi-project scheduling: Priority
rule performance revisited, International Journal of Production Economics, Vol. 126(2), pp.
212-228.
Demeulemeester, E., 1995, “Minimizing resource availability costs in time-limited project net-
works, Management Science, Vol. 41(10), pp. 1590-1598.
Froger, A., Gendreau, M., Mendoza, J.E., Pinson, E. and Rousseau, L.M., 2017, “A branch-and-
check approach for a wind turbine maintenance scheduling problem, Computers Operations
Research, Vol. 88, pp. 117-136.
Herroelen, W., 2005, “Project scheduling-Theory and practice, Production and operations man-
agement, Vol. 14(4), pp. 413-432.
Kruger, D. and Scholl, A., 2009, “A heuristic solution framework for the resource constrained
(multi-) project scheduling problem with sequence-dependent transfer times, European Jour-
nal of Operational Research, Vol. 197(2), pp. 492-508.
Kurtulus, I. and Davis, E.W., 1982, “Multi-project scheduling: Categorization of heuristic rules
performance, Management Science, Vol. 28(2), pp. 161-172.
Lacomme, P., Moukrim, A., Quilliot, A. and Vinot, M., 2017, “A new shortest path algorithm to
solve the resource-constrained project scheduling problem with routing from a flow solution,
Engineering Applications of Artificial Intelligence, Vol. 66, pp. 75-86.
Liberatore, M.J. and Pollack-Johnson, B., 2003, “Factors influencing the usage and selection of
project management software, IEEE transactions on Engineering Management, Vol. 50(2),
pp. 164-174.
65

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❤❛♥❞♦✈❡r r❡❧❛t✐♦♥s
❉✐r❦ ❇r✐s❦♦r♥1
✱ ▼❛❧t❡ ❋❧✐❡❞♥❡r2
❛♥❞ ▼❛rt✐♥ ❚s❝❤ö❦❡2
1
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✶ ■♥tr♦❞✉❝t✐♦♥
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s♣❡❝✐✜❝ tr❛♥ss❤✐♣♠❡♥t ♣r♦❝❡ss❡s ♠❛② ❞✐✛❡r ❝♦♥s✐❞❡r❛❜❧② ✇✐t❤ r❡s♣❡❝t t♦ s♣❡❝✐✜❝ ♣r♦❜❧❡♠
❝❤❛r❛❝t❡r✐st✐❝s✱ ❢♦r ✐♥st❛♥❝❡ ❞✉❡ t♦ t❤❡ ✐♥✈♦❧✈❡❞ t❡❝❤♥♦❧♦❣✐❡s ♦r ♠♦❞❡s ♦❢ tr❛♥s♣♦rt❛t✐♦♥✱
❛t t❤❡ ❝♦r❡ ♦❢ ♠❛♥② ♠♦r❡ ✐♥✈♦❧✈❡❞ tr❛♥ss❤✐♣♠❡♥t ♣r♦❜❧❡♠s ❧✐❡ s♦♠❡ ❢✉♥❞❛♠❡♥t❛❧ ❞❡❝✐s✐♦♥s
t❤❛t ♥❡❡❞ t♦ ❜❡ t❛❦❡♥ ✐rr❡s♣❡❝t✐✈❡ ♦❢ t❤❡ s♣❡❝✐✜❝ ❛♣♣❧✐❝❛t✐♦♥✳
●❡♥❡r❛❧❧② s♣❡❛❦✐♥❣✱ ❛t ❛ tr❛♥ss❤✐♣♠❡♥t ♥♦❞❡ ❝♦♠♠♦❞✐t✐❡s ❛r❡ ❡①❝❤❛♥❣❡❞ ❜❡t✇❡❡♥ ❞✐❢✲
❢❡r❡♥t tr❛♥s♣♦rt r❡❧❛t✐♦♥s ✉s✐♥❣ s♦♠❡ s❡t ♦❢ r❡s♦✉r❝❡s ❢♦r ✭✉♥✮❧♦❛❞✐♥❣ ❛♥❞ tr❛♥s♣♦rt❛t✐♦♥✳
❚②♣✐❝❛❧❧②✱ t❤❡ ❝♦♠♠♦❞✐t② ❡①❝❤❛♥❣❡ ❝❛♥ ❜❡ ♠♦❞❡❧❡❞ ❛s ❛ str✐❝t ❤❛♥❞♦✈❡r r❡❧❛t✐♦♥✱ ✐♥ t❤❡
s❡♥s❡ t❤❛t t❤❡ r❡❝❡✐✈✐♥❣ ✈❡❤✐❝❧❡ ❝❛♥♥♦t ❧❡❛✈❡ t❤❡ s②st❡♠ ❜❡❢♦r❡ t❤❡ ✈❡❤✐❝❧❡ t❤❛t s✉♣♣❧✐❡s t❤❡
❝♦♠♠♦❞✐t② ❤❛s ❛rr✐✈❡❞✳ ■♥ ♦r❞❡r t♦ ❢❛❝✐❧✐t❛t❡ ❛♥ ❡❛s② ❛❝❝❡ss t♦ ✐♥❝♦♠✐♥❣ ✈❡❤✐❝❧❡s✱ tr❛♥ss❤✐♣✲
♠❡♥t ♥♦❞❡s ♦❢t❡♥ ♣r♦✈✐❞❡ ❛ s♣❡❝✐❛❧ s❡t ♦❢ ❞♦❝❦✐♥❣ r❡s♦✉r❝❡s ✇❤❡r❡ ✈❡❤✐❝❧❡s ❛r❡ ♣r♦❝❡ss❡❞✳
❚❤✐s ❝❛♥ ❜❡ r❛✐❧✲tr❛❝❦s ✐♥ r❛✐❧✲r❛✐❧ ♦r r❛✐❧✲r♦❛❞ t❡r♠✐♥❛❧s ✭s❡❡ ✭❇♦②s❡♥ ❡t ❛❧✳ ✷✵✶✶✮✮✱ ❜❡rt❤s ✐♥
s❡❛♣♦rts ✭✭❇✐❡r✇✐rt❤ ❛♥❞ ▼❡✐s❡❧ ✷✵✶✺✮✮✱ ✢✐❣❤t ❣❛t❡s ✐♥ ❛✐r♣♦rt ❤✉❜s ✭✭❉♦r♥❞♦r❢ ❡t ❛❧✳ ✷✵✵✼✮✮
❛♥❞ ❞♦❝❦ ❞♦♦rs ❛t ❝r♦ss ❞♦❝❦s ✭✭❇♦②s❡♥ ❛♥❞ ❋❧✐❡❞♥❡r ✷✵✶✵✮✮✳ ❲❤❡♥❡✈❡r t❤❡s❡ r❡s♦✉r❝❡s ❛r❡
s❝❛r❝❡✱ t❤❡r❡ ✐s ❛ ❢✉♥❞❛♠❡♥t❛❧ ❞❡❝✐s✐♦♥ ♣r♦❜❧❡♠ t♦ ❛ss✐❣♥ ✈❡❤✐❝❧❡s t♦ ❞♦❝❦✐♥❣ r❡s♦✉r❝❡s ♦✈❡r
t✐♠❡✱ s✉❝❤ t❤❛t ❛❧❧ ❤❛♥❞♦✈❡r r❡❧❛t✐♦♥s ❛r❡ s❛t✐s✜❡❞✳ ❚②♣✐❝❛❧❧②✱ t❤❡s❡ ❛ss✐❣♥♠❡♥t ♣r♦❜❧❡♠s
❛r❡ s♦❧✈❡❞ ✉♥❞❡r s♦♠❡ t✐♠❡✲ ♦r ❡✣❝✐❡♥❝②✲♦r✐❡♥t❡❞ ♦❜❥❡❝t✐✈❡ ❢✉♥❝t✐♦♥ ✇❤✐❧❡ ❝♦♥s✐❞❡r✐♥❣ s❡✈✲
❡r❛❧ ❛❞❞✐t✐♦♥❛❧ ❝♦♥str❛✐♥ts ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ❧♦❛❞✐♥❣ ❛♥❞ tr❛♥ss❤✐♣♠❡♥t r❡s♦✉r❝❡s✱ st♦r❛❣❡
str❛t❡❣②✱ ❞✉❡ ❞❛t❡s✱ ❡t❝✳
❖♥❡ ✐♠♣♦rt❛♥t ♦r❣❛♥✐③❛t✐♦♥❛❧ ❝♦♥str❛✐♥t ✐♥ ❝r♦ss✲❞♦❝❦✐♥❣ r❡✢❡❝ts ✇❤❡t❤❡r ❝♦♠♠♦❞✐t✐❡s
❝❛♥ ❜❡ st♦r❡❞ ♦♥ t❤❡ ❞♦❝❦ ✢♦♦r ♦r ❤❛✈❡ t♦ ❜❡ tr❛♥s♣♦rt❡❞ ❞✐r❡❝t❧② t♦ ❢r♦♠ ♦♥❡ tr✉❝❦ t♦
t❤❡ ♥❡①t✳ ❲❤✐❧❡ t❡♠♣♦r❛r② st♦r❛❣❡ ✉♣ t♦ ✷✹❤rs ✐s t②♣✐❝❛❧❧② ♣♦ss✐❜❧❡ ✐♥ ♠♦st ❛♣♣❧✐❝❛t✐♦♥s✱ ✐t
♠✐❣❤t r❡str✐❝t❡❞ t♦ r❡❞✉❝❡ ❞♦✉❜❧❡ ❤❛♥❞❧✐♥❣ ♦r ❡♥s✉r❡ t❤❛t ❝♦♦❧✐♥❣ r❡q✉✐r❡♠❡♥ts ❛r❡ ♠❡t✱ s❡❡
✭❇♦②s❡♥ ✷✵✶✵✮ ❛♥❞ ✭❇♦②s❡♥ ❡t ❛❧✳ ✷✵✶✷✮✳ ❋✉rt❤❡r✱ ✐♥ s♦♠❡ ❝r♦ss ❞♦❝❦s t❤❡ ❧♦❛❞✐♥❣ ♣r♦❝❡ss
♦❢ ❛ tr✉❝❦ ♠❛② ❜❡ ✐♥t❡rr✉♣t❡❞✱ t♦ ❝❧❡❛r t❤❡ ❞♦❝❦ ❞♦♦r ❢♦r ❛♥♦t❤❡r ♠♦r❡ ✉r❣❡♥t ✈❡❤✐❝❧❡✱
❡✳❣✳ s❡❡ ✭❆❧♣❛♥ ❡t ❛❧✳ ✷✵✶✶✮ ❛♥❞ ✭❆❧♣❛♥ ❡t ❛❧✳ ✷✵✶✶❜✮✳ ❋✐♥❛❧❧②✱ t❤❡ ❤❛♥❞♦✈❡r r❡❧❛t✐♦♥s t❤❡♠✲
s❡❧✈❡s ❝❛♥ ❜❡ s✉❜❥❡❝t t♦ str✉❝t✉r❛❧ ❝♦♥str❛✐♥ts✳ ❋♦r ✐♥st❛♥❝❡✱ ✐❢ t❤❡ ❝r♦ss✲❞♦❝❦ ✐s r✉♥ ✐♥ ❛♥
❡①❝❧✉s✐✈❡ s❡r✈✐❝❡ ♠♦❞❡ ❢♦r ✐♥✲ ❛♥❞ ♦✉t❜♦✉♥❞ tr✉❝❦s✱ ❡✳❣✳ s❡❡ ✭❇♦②s❡♥ ❛♥❞ ❋❧✐❡❞♥❡r ✷✵✶✵✮
❛♥❞ ✭❈❤♠✐❡❧❡✇s❦✐ ❡t ❛❧✳ ✷✵✵✾✮✱ ♥♦ ✐♥❜♦✉♥❞ tr✉❝❦ r❡❝❡✐✈❡s ❛♥② ❝♦♠♠♦❞✐t② ❢r♦♠ ♦✉t❜♦✉♥❞
tr✉❝❦s✳ ■♥ ❛❞❞✐t✐♦♥ t♦ t❤❛t✱ ❞♦❝❦ ❞♦♦rs ❛r❡ ♣❛rt✐t✐♦♥❡❞ ✐♥t♦ ❞✐s❥♦✐♥t s❡ts✱ s✉❝❤ t❤❛t ✐♥❜♦✉♥❞
✭♦✉t❜♦✉♥❞✮ tr✉❝❦s ❝❛♥ ♦♥❧② ❜❡ ❞♦❝❦❡❞ t♦ s♣❡❝✐✜❝ ✐♥❜♦✉♥❞ ✭♦✉t❜♦✉♥❞✮ ❞♦♦rs✳ ❙✉❝❤ ❣r♦✉♣✲
✐♥❣ ❝♦♥str❛✐♥ts ❛r❡ ❛❧s♦ ❡♥❝♦✉♥t❡r❡❞ ✐♥ ❛♣♣❧✐❝❛t✐♦♥s ✇❤❡r❡ ❞♦♦rs ❛r❡ ❛ss✐❣♥❡❞ t♦ s♣❡❝✐✜❝
tr❛♥s♣♦rt r❡❧❛t✐♦♥s✱ ❡✳❣✳ ❧♦❝❛❧ ♦r ❧♦♥❣ ❞✐st❛♥❝❡ tr❛♥s♣♦rt r❡❧❛t✐♦♥s✳
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■♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ✇❡ ✇✐❧❧ ❛♥❛❧②③❡ t❤❡ str✉❝t✉r❡ ♦❢ ✜♥❞✐♥❣ ❢❡❛s✐❜❧❡ ❞♦❝❦✐♥❣ ❛ss✐❣♥♠❡♥ts
✉♥❞❡r ❤❛♥❞♦✈❡r r❡❧❛t✐♦♥s ✇❤✐❧❡ ❝♦♥s✐❞❡r✐♥❣ t❤❡ ❛❢♦r❡♠❡♥t✐♦♥❡❞ ♣r♦❜❧❡♠ ❝❤❛r❛❝t❡r✐st✐❝s✳
❚❤✐s ❛ss✐❣♥♠❡♥t ♣r♦❜❧❡♠ t②♣✐❝❛❧❧② ❤❛s t♦ ❜❡ s♦❧✈❡❞ ❛s ❛♥ ✐♥t❡❣r❛❧ ♣❛rt ♦❢ ❛♥② s♦❧✉t✐♦♥
str❛t❡❣② t❤❛t s♦❧✈❡s ❞♦❝❦ ❞♦♦r s❝❤❡❞✉❧✐♥❣ ♣r♦❜❧❡♠s ✉♥❞❡r ♦t❤❡r t✐♠❡✲ ♦r ❡✣❝✐❡♥❝②✲♦r✐❡♥t❡❞
♦❜❥❡❝t✐✈❡✳ ■♥ t❤✐s s❡♥s❡✱ ✇❡ st✉❞② ❛ ❝♦r❡ ❞❡❝✐s✐♦♥ ♣r♦❜❧❡♠ t❤❛t ❝❛♥ ❜❡ r❡s♣♦♥s✐❜❧❡ ❢♦r ♠✉❝❤
♦❢ t❤❡ ❝♦♠♣✉t❛t✐♦♥❛❧ ❝❤❛❧❧❡♥❣❡ t❤❛t ✐s ❡♥❝♦✉♥t❡r❡❞ ✐♥ ✈❛r✐♦✉s ❛♣♣❧✐❝❛t✐♦♥s✳ ❋♦r t❤✐s ♣✉r♣♦s❡✱
✇❡ ✇✐❧❧ ✐♥tr♦❞✉❝❡ ❛ ❢♦r♠❛❧ ❢r❛♠❡✇♦r❦ ❢♦r s✉❝❤ ❞❡❝✐s✐♦♥ ♣r♦❜❧❡♠s ✐♥ ❙❡❝t✐♦♥ ✷✳ ❋✉rt❤❡r✱ ✐♥
❙❡❝t✐♦♥ ✸✱ ✇❡ ♦✉t❧✐♥❡ t❤❡ r❡s✉❧ts ♦❢ ❛ ❝♦♠♣r❡❤❡♥s✐✈❡ ❝♦♠♣❧❡①✐t② ❛♥❛❧②s✐s ♦❢ t❤❡ ❞✐✛❡r❡♥t
♣r♦❜❧❡♠ ✈❡rs✐♦♥s ❝♦✈❡r❡❞ ❜② t❤❡ ❢r❛♠❡✇♦r❦✳
✷ ❋♦r♠❛❧ Pr♦❜❧❡♠ ❉❡✜♥✐t✐♦♥
❲❡ ❝♦♥s✐❞❡r ❛ s❡t D ♦❢ ❞♦♦rs ♣❛rt✐t✐♦♥❡❞ ✐♥t♦ q ❣r♦✉♣s D1, . . . , Dq✳ ❉♦♦rs ✐♥ t❤❡ s❛♠❡
s✉❜s❡t ❛r❡ ❛ss✉♠❡❞ t♦ ❜❡ ✐❞❡♥t✐❝❛❧✳ ▼♦r❡♦✈❡r✱ ✇❡ ❝♦♥s✐❞❡r ❛ s❡t V ♦❢ ✈❡❤✐❝❧❡s ♣❛rt✐t✐♦♥❡❞
✐♥t♦ ❣r♦✉♣s V1, . . . , Vq✳ ❱❡❤✐❝❧❡s ✐♥ Vg✱ g = 1, . . . , q✱ ❝❛♥ ❜❡ ❞♦❝❦❡❞ ♦♥❧② ❛t ❞♦♦rs ✐♥ Dq✳
❲❡ ❞✐st✐♥❣✉✐s❤ ❜❡t✇❡❡♥ ♣r♦❜❧❡♠ s❡tt✐♥❣s ✇❤❡r❡ ❡❛❝❤ ✈❡❤✐❝❧❡ ❝❛♥ ❜❡ ❞♦❝❦❡❞ ♦♥❧② ♦♥❝❡
✭✏♦♥❡✑✮✱ ❡❛❝❤ ✈❡❤✐❝❧❡ ❝❛♥ ❜❡ ❞♦❝❦❡❞ ♠✉❧t✐♣❧❡ t✐♠❡s ❛t t❤❡ s❛♠❡ ❞♦♦r ✭✏✐♥t❡rr✉♣t✑✮✱ ❛♥❞
❡❛❝❤ ✈❡❤✐❝❧❡ ❝❛♥ ❜❡ ❞♦❝❦❡❞ ♠✉❧t✐♣❧❡ t✐♠❡s ❛t ❞✐✛❡r❡♥t ❞♦♦rs ✭✏r❡✈✐s✐ts✑✮✳ ❲❡ r❡❢❡r t♦ t❤✐s
♣❛r❛♠❡t❡r ❛s t❤❡ ❞♦❝❦✐♥❣ str❛t❡❣② ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣✳
▼♦r❡♦✈❡r✱ ✇❡ ❤❛✈❡ ❛ s❡t H ⊆ V × V ♦❢ ❤❛♥❞♦✈❡r r❡❧❛t✐♦♥s ✭❍❘s✮✳ ❍❘ (v, w) ∈ H
r❡♣r❡s❡♥ts ✈❡❤✐❝❧❡ v ❤❛♥❞✐♥❣ ♦✈❡r ✭♣❛rt ♦❢✮ ✐ts ❞❡❧✐✈❡r② t♦ w✳ ❲❡ s❛② t❤❛t v s✉♣♣❧✐❡s w
✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣✳ ❲✐t❤ r❡❣❛r❞ t♦ t❤❡ str✉❝t✉r❡ ♦❢ ❍❘s ✇❡ ❛❞r❡ss s♣❡❝✐✜❝ ♣r♦❜❧❡♠ s❡tt✐♥❣s
✉s✐♥❣ t✇♦ ♣❛r❛♠❡t❡rs✱ ♥❛♠❡❧② t❤❡ ♣❛✐r str✉❝t✉r❡ ❛♥❞ t❤❡ ❣r♦✉♣ str✉❝t✉r❡✳ ❋✐rst✱ ❛❝❝♦r❞✐♥❣
t♦ t❤❡ ♣❛✐r str✉❝t✉r❡ ✇❡ ❞✐st✐♥❣✉✐s❤ ❜❡t✇❡❡♥ s❡tt✐♥❣s ✇❤❡r❡ (v, w) ∈ H ✇❤❡♥❡✈❡r (w, v) ∈ H
✭✏s②♠✑✮✱ ✇❤❡r❡ (w, v) 6∈ H ✇❤❡♥❡✈❡r (v, w) ∈ H ✭✏❛s②♠✑✮✱ ❛♥❞ ✇❤❡r❡ ✇❡ ❤❛✈❡ ♥♦ r❡str✐❝t✐♦♥
♦♥ H ✭✏❣❡♥✑✮✳ ❙❡❝♦♥❞✱ ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ❣r♦✉♣ str✉❝t✉r❡ ✇❡ ❞✐st✐♥❣✉✐s❤ ❜❡t✇❡❡♥ s❡tt✐♥❣s
✇❤❡r❡ (v, w) ∈ H ♦♥❧② ✐❢ v ❛♥❞ w ❛r❡ ✐♥ ❞✐✛❡r❡♥t ❣r♦✉♣s ♦❢ ✈❡❤✐❝❧❡s ✭✏✐♥t❡r✑✮ ❛♥❞ ✇❤❡r❡✱
❛❞❞✐t✐♦♥❛❧❧②✱ (v, w) ∈ H ✇✐t❤ v ❛♥❞ w ✐♥ t❤❡ s❛♠❡ ❣r♦✉♣ ✐s ♣♦ss✐❜❧❡ ✭✏✐♥♥❡r✑✮✳ ◆♦t❡ t❤❛t
r❡str✐❝t✐♥❣ ❍❘s t♦ ♣❛✐rs ♦❢ ✈❡❤✐❝❧❡ ✐♥ t❤❡ s❛♠❡ ❣r♦✉♣ ✇♦✉❧❞ ❜❡ ❛ ♥❛t✉r❛❧ t❤✐r❞ ♦♣t✐♦♥ ❜✉t
②✐❡❧❞s ❛ ♣r♦❜❧❡♠ s❡tt✐♥❣ ✇❤✐❝❤ ❞❡❝♦♠♣♦s❡s ✐♥t♦ ❣r♦✉♣✲s♣❡❝✐✜❝ s✉❜♣r♦❜❧❡♠s✳
■❢ (v, w) ∈ H✱ v ❛♥❞ w ♥❡❡❞ t♦ ❜❡ ❞♦❝❦❡❞ s✉❝❤ t❤❛t t❤❡s❡ ❣♦♦❞s ❝❛♥ ❜❡ ✉♥❧♦❛❞❡❞ ❢r♦♠
v✱ tr❛♥s♣♦rt❡❞ t❤r♦✉❣❤ t❤❡ t❡r♠✐♥❛❧ t♦ t❤❡ ❞♦♦r ✇❤❡r❡ w ✐s ❞♦❝❦❡❞✱ ❛♥❞ ❧♦❛❞❡❞ ♦♥t♦ w✳
■♥ ♦r❞❡r t♦ r❡❞✉❝❡ t❤❡ ♣r♦❜❧❡♠ s❡tt✐♥❣ t♦ t❤❡ ✈❡r② ❝♦r❡ ✇❡ ✐❣♥♦r❡ ❞✉r❛t✐♦♥s ❢♦r ✉♥❧♦❛❞✐♥❣
♦r ❧♦❛❞✐♥❣ ❛♥❞ tr❛♥s♣♦rt❛t✐♦♥ t✐♠❡s✳ ❲❡ ❞✐st✐♥❣✉✐s❤✱ ❤♦✇❡✈❡r✱ ❜❡t✇❡❡♥ st♦r❛❣❡ str❛t❡❣✐❡s
✇❤❡r❡ ❣♦♦❞s ❝❛♥ ❜❡ ✐♥t❡r♠❡❞✐❛t❡❧② st♦r❡❞ ✐♥ t❤❡ t❡r♠✐♥❛❧ ✭✏st♦✑✮ ❛♥❞ ✇❤❡r❡ t❤✐s ✐s ♥♦t
❛❧❧♦✇❡❞ ✭✏♥♦❙t♦✑✮✳
❚❤✐s ❣✐✈❡s ✉s ❛ ❢❛♠✐❧② ♦❢ ✸✻ ❞✐✛❡r❡♥t ♣❛r❛♠❡t❡r s❡tt✐♥❣s✳ ■♥ ❡❛❝❤ ♦❢ t❤❡ r❡s✉❧t✐♥❣ ♣r♦❜❧❡♠
s❡tt✐♥❣s ✇❡ ❛r❡ ✐♥t❡r❡st❡❞ ✐♥ s❡q✉❡♥❝❡s ♦❢ ❞♦❝❦✐♥❣ ♦♣❡r❛t✐♦♥s ✭❉❙✮✳ ❙✉❝❤ ❛♥ ♦♣❡r❛t✐♦♥ (v, d)
✐s s♣❡❝✐✜❡❞ ❜② ✈❡❤✐❝❧❡ v ❛♥❞ t❤❡ ❞♦♦r d ✐♥✈♦❧✈❡❞✳ ❋♦r s✉❝❤ ❛♥ ♦♣❡r❛t✐♦♥ t♦ ❜❡ ❢❡❛s✐❜❧❡ t❤❡r❡
❤❛s t♦ ❜❡ ❛ ❣r♦✉♣ ✐♥❞❡① g = 1, . . . , q s✉❝❤ t❤❛t v ∈ Vg ❛♥❞ d ∈ Dg✳ ❆ ❉❙ ✐s ❢❡❛s✐❜❧❡ ✇✐t❤
r❡s♣❡❝t t♦ t❤❡ ❞♦♦r ❛❧❧♦❝❛t✐♦♥ ✐❢ ❡❛❝❤ ♦♣❡r❛t✐♦♥ ✐s ❢❡❛s✐❜❧❡ ❛♥❞ ✐t r❡♣r❡s❡♥ts t❤❡ ♦r❞❡r ✐♥
✇❤✐❝❤ ❞♦❝❦✐♥❣ ♦♣❡r❛t✐♦♥s ❛r❡ ❝❛rr✐❡❞ ♦✉t✳
▲❡t σ ❜❡ ❛ ❉❙✱ l(σ) ✐ts ❧❡♥❣t❤✱ ❛♥❞ σ(k) t❤❡ kt❤ ♦♣❡r❛t✐♦♥ ✐♥ σ✳ ❲❡ s❛② t❤❛t ♦♣❡r❛t✐♦♥
σ(k) = (v, d)✱ k = 1, . . . , l(σ)✱ ✐s ❛❝t✐✈❡ ✐♥ k ❛♥❞✱ ❢✉rt❤❡r♠♦r❡✱ ✐♥ k′
k ✐❢ ❢♦r ❡❛❝❤
k′′
= k + 1, . . . , k′
✇❡ ❤❛✈❡ σ(k′′
) = (w, d′
) ✇✐t❤ w 6= v ❛♥❞ d′
6= d✳ ❚❤❛t ✐s✱ ❛ ❞♦❝❦✐♥❣
♦♣❡r❛t✐♦♥ (v, d) ✐s ❛❝t✐✈❡ ❛s ❧♦♥❣ ❛s v ✐s ♥♦t ❞♦❝❦❡❞ ❛t ❛♥ ♦t❤❡r ❞♦♦r ❛♥❞ ♥♦ ♦t❤❡r ✈❡❤✐❝❧❡
✐s ❞♦❝❦❡❞ ❛t d✳ ▲❡t e(σ, k) = k′
✐❢ σ(k) ✐s ❛❝t✐✈❡ ✐♥ k′
❛♥❞ ✭✐✮ k′
= l(σ) ♦r ✭✐✐✮ σ(k) ✐s ♥♦t
❛❝t✐✈❡ ✐♥ k′
+ 1✳ ❲❡ s❛② [k, e(σ, k)] ✐s t❤❡ ❛❝t✐✈✐t② ✐♥t❡r✈❛❧ ♦❢ σ(k)✳
❆ ❉❙ σ ✐s ❢❡❛s✐❜❧❡ ✇✐t❤ r❡❣❛r❞ t♦ t❤❡ ❞♦❝❦✐♥❣ str❛t❡❣② ♦♥❧② ✐❢
✕ t❤❡ ❞♦❝❦✐♥❣ str❛t❡❣② ✐s ✏♦♥❡ ✈✐s✐t✑ ❛♥❞ σ ❝♦♥t❛✐♥s ❡①❛❝t❧② ♦♥❡ ♦♣❡r❛t✐♦♥ ❢♦r ❡❛❝❤ ✈❡❤✐❝❧❡✱
67

✕ t❤❡ ❞♦❝❦✐♥❣ str❛t❡❣② ✐s ✏✐♥t❡rr✉♣t✑ ❛♥❞ ❢♦r ❛♥② t✇♦ ♦♣❡r❛t✐♦♥s (v, d) ❛♥❞ (w, d′
) ✐♥ σ
✇❡ ❤❛✈❡ v 6= w ♦r d = d′
✱ ♦r
✕ t❤❡ ❞♦❝❦✐♥❣ str❛t❡❣② ✐s ✏r❡✈✐s✐ts✑✳
❆ ❉❙ σ ✐s ❢❡❛s✐❜❧❡ ✇✐t❤ r❡❣❛r❞ t♦ t❤❡ st♦r❛❣❡ str❛t❡❣② ♦♥❧② ✐❢
✕ t❤❡ st♦r❛❣❡ str❛t❡❣② ✐s ✏st♦r❛❣❡✑ ❛♥❞ ❢♦r ❡❛❝❤ (v, w) ∈ H t❤❡r❡ ❛r❡ ♦♣❡r❛t✐♦♥s σ(k) =
(v, d) ❛♥❞ σ(k′
) = (w, d′
) ✇✐t❤ k ≤ e(σ, k′
) ♦r
✕ t❤❡ st♦r❛❣❡ str❛t❡❣② ✐s ✏♥♦ st♦r❛❣❡✑ ❛♥❞ ❢♦r ❡❛❝❤ (v, w) ∈ H t❤❡r❡ ❛r❡ ♦♣❡r❛t✐♦♥s
σ(k) = (v, d) ❛♥❞ σ(k′
) = (w, d′
) ✇✐t❤ [k, e(σ, k)] ❛♥❞ [k′
, e(σ, k′
)] ♦✈❡r❧❛♣♣✐♥❣✳
❆ ❉❙ σ ✐s ❢❡❛s✐❜❧❡ ✐❢ ✐t ✐s ❢❡❛s✐❜❧❡ ✇✐t❤ r❡❣❛r❞ t♦ t❤❡ ❞♦♦r ❛❧❧♦❝❛t✐♦♥✱ t❤❡ ❞♦❝❦✐♥❣ str❛t❡❣②✱
❛♥❞ t❤❡ st♦r❛❣❡ str❛t❡❣②✳
❉❡✜♥✐t✐♦♥ ✶✳ ●✐✈❡♥ ❛ ❞♦❝❦✐♥❣ str❛t❡❣②✱ ❛ st♦r❛❣❡ str❛t❡❣②✱ D1, . . . , Dq✱ V1, . . . , Vq✱ ❛♥❞
H✱ t❤❡ ❞♦❝❦ ♦♣❡r❛t✐♦♥ s❡q✉❡♥❝✐♥❣ ♣r♦❜❧❡♠ ✭❉❖❙P✮ ✐s t♦ ❞❡t❡r♠✐♥❡ ✇❤❡t❤❡r ❛ ❢❡❛s✐❜❧❡ ❉❙
❡①✐sts✳
❲❡ ✇✐❧❧ r❡❢❡r t♦ ❉❖❙P ✇✐t❤ ❛ s♣❡❝✐✜❝ ♣❛r❛♠❡t❡r s❡tt✐♥❣ ❜② ❛ q✉❛❞r✉♣❧❡t
(❞♦❝❦✐♥❣ str❛t❡❣②|st♦r❛❣❡ str❛t❡❣②|❣r♦✉♣ str✉❝t✉r❡|♣❛✐r str✉❝t✉r❡).
❋♦r ❡①❛♠♣❧❡✱ (✐♥t❡rr✉♣t|st♦|✐♥♥❡r|❛s②♠) r❡❢❡rs t♦ t❤❡ ♣r♦❜❧❡♠ s❡tt✐♥❣ ✇❤❡r❡ ✈❡❤✐❝❧❡s ♠❛②
❛♣♣r♦❛❝❤ t❤❡ s❛♠❡ ❞♦♦r ♠✉❧t✐♣❧❡ t✐♠❡s✱ ❣♦♦❞s ❝❛♥ ❜❡ st♦r❡❞✱ t✇♦ ✈❡❤✐❝❧❡s ❞♦ ♥♦t s✉♣♣❧②
❡❛❝❤ ♦t❤❡r✱ ❛♥❞ ❍❘ ✇✐t❤✐♥ ❛ ❣r♦✉♣ ❛r❡ ♣♦ss✐❜❧❡✳
✸ ❈♦♠♣✉t❛t✐♦♥❛❧ ❈♦♠♣❧❡①✐t②
❲❡ ❣✐✈❡ ❛♥ ♦✈❡r✈✐❡✇ ♦❢ r❡s✉❧ts ✐♥ ❚❛❜❧❡ ✶✳ ■♥ t❤♦s❡ ❝❛s❡s ✇❤❡r❡ ❛♥ ❡♥tr② ✐♥ t❤❡ q✉❛❞r✉♣❧❡t
s♣❡❝✐❢②✐♥❣ ❛ ♣r♦❜❧❡♠ s❡tt✐♥❣ ✐s ♥♦t ❣✐✈❡♥ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ r❡s✉❧t ❤♦❧❞s ❢♦r ❛♥② ♣♦ss✐❜❧❡
❡♥tr②✳ ❍♦r✐③♦♥t❛❧ s♦❧✐❞ ❧✐♥❡s s❡♣❛r❛t❡ ♣r♦❜❧❡♠ s❡tt✐♥❣s ❞✐✛❡r✐♥❣ ✐♥ t❤❡ ❞♦❝❦✐♥❣ str❛t❡❣②✳
◆♦✳ Pr♦❜❧❡♠ ❈♦♠♠❡♥t ❈♦♠♣❧❡①✐t②
✶ (♦♥❡|st♦|✐♥♥❡r|❛s②♠) ❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ ✷ ◆P✲❝♦♠♣❧❡t❡ ❢♦r q = 1
✷ (♦♥❡|st♦|✐♥♥❡r|s②♠) ❡q✉✐✈❛❧❡♥t t♦ P❆❚❍ ❲■❉❚❍ ❢♦r q = 1 ◆P✲❝♦♠♣❧❡t❡ ❢♦r q = 1
✸ (♦♥❡|st♦|✐♥♥❡r|❣❡♥) ❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ ✷ ◆P✲❝♦♠♣❧❡t❡ ❢♦r q = 1
✹ (♦♥❡|st♦|✐♥t❡r|❛s②♠) ❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ ✷ ◆P✲❝♦♠♣❧❡t❡ ❢♦r q = 2
✺ (♦♥❡|st♦|✐♥t❡r|s②♠) ❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ ✷ ◆P✲❝♦♠♣❧❡t❡
✻ (♦♥❡|st♦|✐♥t❡r|❣❡♥) ❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ ✺ ◆P✲❝♦♠♣❧❡t❡
✼✲✾ (♦♥❡|♥♦❙t♦|✐♥♥❡r|−) ❡q✉✐✈❛❧❡♥t t♦ ✷ ◆P✲❝♦♠♣❧❡t❡ ❢♦r q = 1
✶✵✲✶✷ (♦♥❡|♥♦❙t♦|✐♥t❡r|−) ❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ ◆P✲❝♦♠♣❧❡t❡
▼■◆ ❈❯❚ ▲■◆❊❆❘ ❆❘❘❆◆●❊▼❊◆❚
✶✸✲✶✽ (✐♥t❡rr✉♣t|st♦| − |−) q ❞♦♦rs s✉✣❝✐❡♥t✱ ✐♥ P
✶✾✲✷✶ (✐♥t❡rr✉♣t|♥♦❙t♦|✐♥♥❡r|−) ❡q✉✐✈❛❧❡♥t t♦ ❱❊❘❚❊❳ ❈❖▲❖❘■◆● ◆P✲❝♦♠♣❧❡t❡ ❢♦r q = 1
✷✷✲✷✹ (✐♥t❡rr✉♣t|♥♦❙t♦|✐♥t❡r|−) q ❞♦♦rs s✉✣❝✐❡♥t ✐♥ P
✷✺✲✸✵ (r❡✈✐s✐ts|st♦| − |−) q ❞♦♦rs s✉✣❝✐❡♥t✱ ✐♥ P
✸✶✲✸✸ (r❡✈✐s✐ts|♥♦❙t♦|✐♥♥❡r|−) 2q ❞♦♦rs s✉✣❝✐❡♥t ✐♥ P
✸✹✲✸✻ (r❡✈✐s✐ts|♥♦❙t♦|✐♥t❡r|−) q ❞♦♦rs s✉✣❝✐❡♥t ✐♥ P
❚❛❜❧❡ ✶✳ ❈♦♠♣✉t❛t✐♦♥❛❧ ❝♦♠♣❧❡①✐t②
68

❘❡❢❡r❡♥❝❡s
❆❧♣❛♥✱ ●✳❀ ▲❛r❜✐✱ ❘✳❀ P❡♥③✱ ❇✳ ✭✷✵✶✶✮✿ ❆ ❜♦✉♥❞❡❞ ❞②♥❛♠✐❝ ♣r♦❣r❛♠♠✐♥❣ ❛♣♣r♦❛❝❤ t♦ s❝❤❡❞✉❧❡
♦♣❡r❛t✐♦♥s ✐♥ ❛ ❝r♦ss ❞♦❝❦✐♥❣ ♣❧❛t❢♦r♠✳ ❈♦♠♣✉t❡rs ✫ ■♥❞✉str✐❛❧ ❊♥❣✐♥❡❡r✐♥❣ ✻✵✱ ✸✽✺✲✸✾✻✳
❆❧♣❛♥✱ ●✳❀ ▲❛❞✐❡r✱ ❆✳✲▲✳❀ ▲❛r❜✐✱ ❘✳❀ P❡♥③✱ ❇✳ ✭✷✵✶✶✮✿ ❍❡✉r✐st✐❝ s♦❧✉t✐♦♥s ❢♦r tr❛♥ss❤✐♣♠❡♥t ♣r♦❜❧❡♠s
✐♥ ❛ ♠✉❧t✐♣❧❡ ❞♦♦r ❝r♦ss ❞♦❝❦✐♥❣ ✇❛r❡❤♦✉s❡✳ ❈♦♠♣✉t❡rs ✫ ■♥❞✉str✐❛❧ ❊♥❣✐♥❡❡r✐♥❣ ✻✶✱ ✹✵✷✲✹✵✽✳
❇✐❡r✇✐rt❤✱ ❈✳❀ ▼❡✐s❡❧✱ ❋✳ ✭✷✵✶✺✮✿ ❆ ❢♦❧❧♦✇✲✉♣ s✉r✈❡② ♦❢ ❜❡rt❤ ❛❧❧♦❝❛t✐♦♥ ❛♥❞ q✉❛② ❝r❛♥❡ s❝❤❡❞✉❧✐♥❣
♣r♦❜❧❡♠s ✐♥ ❝♦♥t❛✐♥❡r t❡r♠✐♥❛❧s✳ ❊✉r♦♣❡❛♥ ❏♦✉r♥❛❧ ♦❢ ❖♣❡r❛t✐♦♥❛❧ ❘❡s❡❛r❝❤ ✷✹✹✱ ✻✼✺✲✻✽✾✳
❇♦②s❡♥✱ ◆✳❀ ❋❧✐❡❞♥❡r✱ ▼✳ ✭✷✵✶✵✮✿ ❈r♦ss ❞♦❝❦ s❝❤❡❞✉❧✐♥❣✿ ❈❧❛ss✐✜❝❛t✐♦♥✱ ❧✐t❡r❛t✉r❡ r❡✈✐❡✇ ❛♥❞ r❡✲
s❡❛r❝❤ ❛❣❡♥❞❛✳ ❖♠❡❣❛ ✸✽✱ ✹✶✸✲✹✷✷✳
❇♦②s❡♥✱ ◆✳ ✭✷✵✶✵✮✿ ❚r✉❝❦ s❝❤❡❞✉❧✐♥❣ ❛t ③❡r♦✲✐♥✈❡♥t♦r② ❝r♦ss ❞♦❝❦✐♥❣ t❡r♠✐♥❛❧s✳ ❈♦♠♣✉t❡rs ✫
❖♣❡r❛t✐♦♥s ❘❡s❡❛r❝❤ ✸✼✱ ✸✷✲✹✶✳
❇♦②s❡♥✱ ◆✳❀ ❏❛❡❤♥✱ ❋✳❀ P❡s❝❤✱ ❊✳ ✭✷✵✶✶✮✿ ❙❝❤❡❞✉❧✐♥❣ ❋r❡✐❣❤t ❚r❛✐♥s ✐♥ ❘❛✐❧✲❘❛✐❧ ❚r❛♥ss❤✐♣♠❡♥t
❨❛r❞s✱ ❚r❛♥s♣♦rt❛t✐♦♥ ❙❝✐❡♥❝❡ ✹✺✱ ✶✾✾✲✷✶✶✳
❇♦②s❡♥✱ ◆✳❀ ❇r✐s❦♦r♥✱ ❉✳❀ ❚s❝❤ö❦❡✱ ▼✳ ✭✷✵✶✷✮✿ ❚r✉❝❦ s❝❤❡❞✉❧✐♥❣ ✐♥ ❝r♦ss✲❞♦❝❦✐♥❣ t❡r♠✐♥❛❧s ✇✐t❤
✜①❡❞ ♦✉t❜♦✉♥❞ ❞❡♣❛rt✉r❡s✳ ❖❘ ❙♣❡❝tr✉♠ ✸✷✱ ✶✸✺✲✶✻✶✳
❈❤♠✐❡❧❡✇s❦✐✱ ❆✳❀ ◆❛✉❥♦❦s✱ ❇✳❀ ❏❛♥❛s✱ ▼✳❀ ❈❧❛✉s❡♥✱ ❯✳ ✭✷✵✵✾✮✿ ❖♣t✐♠✐③✐♥❣ t❤❡ ❉♦♦r ❆ss✐❣♥♠❡♥t ✐♥
▲❚▲✲❚❡r♠✐♥❛❧s✳ ❚r❛♥s♣♦rt❛t✐♦♥ ❙❝✐❡♥❝❡ ✹✷✱ ✶✾✽✲✷✶✵✳
❉♦r♥❞♦r❢✱ ❯✳❀ ❉r❡①❧✱ ❆✳❀ ◆✐❦✉❧✐♥✱ ❨✳❀ P❡s❝❤✱ ❊✳ ✭✷✵✵✼✮✿ ❋❧✐❣❤t ❣❛t❡ s❝❤❡❞✉❧✐♥❣✿ ❙t❛t❡✲♦❢✲t❤❡✲❛rt ❛♥❞
r❡❝❡♥t ❞❡✈❡❧♦♣♠❡♥ts✳ ❖♠❡❣❛ ✸✺✱ ✸✷✻✲✸✸✹✳
69

Synchronous flow shop scheduling with pliable jobs
Matthias Bultmann1
, Sigrid Knust1
, Stefan Waldherr2
1
University of Osnabrück, Germany
{mbultmann,sknust}@uni-osnabrueck.de
2
Technical University of Munich, Germany
stefan.waldherr@in.tum.de
Keywords: flow shop, synchronous movement, pliability
1 Introduction
In this work we consider synchronous flow shop scheduling problems where the process-
ing times of the operations are not fixed in advance. Instead, for each job a total processing
time is given which can be distributed freely among the machines, respecting some lower
and/or upper bounds on the processing times of the operations.
A synchronous flow shop (also called a “flow shop with synchronous movement”) (cf.
Kouvelis and Karabati (2011)) is a variant of a non-preemptive permutation flow shop
where transfers of jobs from one machine to the next take place at the same time after
the operations on all machines are finished. If the processing time of an operation on one
machine is smaller than the maximum processing time of the operations started on the
other machines at the same time, the corresponding machine is idle until the job may be
transferred to the next machine. In contrast, in a classical flow shop the transfer of jobs is
asynchronous: Jobs may be transferred to the next machine as soon as their processing on
the current machine is completed and processing on the next machine immediately starts
as soon as this machine is available.
The term “pliability” was first introduced in Weiß et al. (2016). Within this model, the
processing times of the individual operations of a job are not fixed in advance but may be
determined with some flexibility. They must respect given lower/upper bounds and add up
to the given total processing time of each job. For example, this allows to model situations
where the processing time of an operation can deviate from a fixed amount by some margin,
defined by the lower and upper bounds. Such a model occurs in practice if several machines
are able to process an operation and it is possible to distribute the processing time of an
operation among these machines. For example, workers at an assembly line might be trained
to not only be able to perform a single, dedicated operation, but to also be skilled enough
to work on additional ones. Then, instead of waiting for the next job to be transported to
them, they may continue working on the current job, which may lead to reduced idle times
and hence a better productivity of the assembly line.
2 Problem formulation
We consider a permutation flow shop with m machines M1, . . . , Mm and n jobs where
job j consists of m operations O1j → O2j → . . . → Omj. Operation Oij has to be processed
without preemption on machine Mi for pij time units. In a feasible schedule each machine
processes at most one operation at any time, each job is processed on at most one machine
at any time, and the operations of each job are processed in the predefined order.
The processing is organized in synchronized cycles where jobs are moved from one
machine to the next by an unpaced synchronous transportation system. This means that
in a cycle all current operations start at the same time on the corresponding machines.
70

Only after all operations have finished processing, all jobs are moved to the next machine
simultaneously. The job processed on the last machine Mm leaves the system, a new job
(if available) is put on the first machine M1. As a consequence, the processing time of a
cycle t (its so-called “cycle time” ct) is determined by the maximum processing time of the
operations contained in it. Furthermore, only permutation schedules are feasible, i.e., the
jobs have to be processed in the same order on all machines.
Let Cj be the completion time of job j, i.e., the time when j has been processed on
all machines and leaves the system. The goal is to find a permutation of the jobs such
that the makespan Cmax = maxj Cj is minimized. With each permutation a corresponding
(left-shifted) schedule is associated in which each operation starts as early as possible.
Huang (2008) introduced the notation “synmv” in the β-field of the well-known α|β|γ
scheduling classification scheme to indicate synchronous movement. Hence, the basic syn-
chronous flow shop problem with the makespan objective is denoted by F|synmv|Cmax.
In this work, the jobs are “pliable” in such a way that instead of a fixed individual
processing time pij for operation Oij on Mi we are only given a total processing time pj
of job j. Then, in addition to finding a job permutation, we also have to determine actual
processing times xij ≥ 0 for operations Oij such that
m
X
i=1
xij = pj (j = 1, . . . , n). (1)
In the unrestricted model, there are no constraints on the actual processing times, i.e., we
only have to fulfill
0 ≤ xij ≤ pj (i = 1, . . . , m; j = 1, . . . , n). (2)
To indicate this situation, we add “plbl” in the β-field of the α|β|γ-notation.
In a more realistic, restricted scenario, additionally lower and upper bounds pij
, pij are
given, and the actual processing times have to satisfy
pij
≤ xij ≤ pij (i = 1, . . . , m; j = 1, . . . , n). (3)
To indicate this situation, we add “plbl(pij
, pij)” in the β-field. We also consider the special
case that only lower bounds pij
are given, indicated by “plbl(pij
)”.
We assume all input data (processing times, lower and upper bounds) to be integer and
usually allow that the actual processing times xij may take arbitrary real values. However,
in some applications, the processing times must also be integer. A similar distinction has
been made for scheduling problems with preemption where usually continuous preemption
is allowed, but in some situations jobs can only be split at integer points in time. For some
special cases it was shown that always an optimal preemptive schedule exists where all
interruptions and all starting/completion times occur at integer time points (cf. Baptiste
et al. (2011)). Dealing with the same question for pliability, in the absence of upper bounds
allowing real-valued processing times does not lead to better schedules since we can show
that for problem F|synmv, plbl(pij
)|Cmax always an optimal schedule with integer-valued
processing times exists. Hence, in this case, when looking for an optimal schedule we may
restrict ourselves to schedules with integer processing times. On the other hand, in the
more general situation F|synmv, plbl(pij
, pij)|Cmax with lower and upper bounds, allowing
non-integer processing times can lead to better solutions.
Concerning complexity, problem F2|synmv, plbl|Cmax without any bounds on the pro-
cessing times is already NP-hard.
71

3 Solution approach
Since the problem is NP-hard, we cannot expect a polynomial time exact algorithm.
In preliminary tests, mixed integer linear programs could only be solved to optimality
for very small instances. To achieve good results, we use a two-stage heuristic. It can
be shown that for a fixed job permutation optimal corresponding processing times can
be obtained in polynomial time by linear programming. The problem is decomposed by
employing a local search procedure using the set of all job permutations as search space. For
each permutation corresponding optimal processing times can be calculated with the LP.
Unfortunately, for larger problem instances solving this LP is quite time-consuming. Since
in the local search approach, usually many permutations should be evaluated, it is more
efficient to use a direct combinatorial algorithm with a better run time than an LP solver.
Depending on the size of the problem, even if no such direct algorithm is known, it may
be more efficient to determine only near-optimal processing times heuristically instead of
solving this subproblem to optimality. Then more neighbors can be evaluated in the same
amount of time.
Thus, the subproblem of determining actual processing times for a fixed job permuta-
tion is of special interest. For problem F|synmv, plbl(pij
)|Cmax without upper bounds, we
propose a polynomial-time direct combinatorial algorithm to obtain optimal actual pro-
cessing times. On the other hand, the situation for problem F|synmv, plbl(pij
, pij)|Cmax is
more involved. While the case where arbitrary real-valued processing times are allowed can
be still solved in polynomial time, the problem becomes NP-hard if all actual processing
times are required to be integer.
In the first stage of the two-stage approach, we use a tabu search procedure with a
simple swap neighborhood. In a tabu list we store pairs of swapped jobs. A move is tabu if
it involves two jobs which are currently in the tabu list. In each iteration of the tabu search,
we consider the whole neighborhood (i.e., we evaluate all possible swaps of two jobs) and
perform the best non-tabu move or the best overall move if it results in a schedule with a
new best objective value (aspiration criterion). As inital solution we used a job permutation
calculated by the NEH heuristic (Nawaz et al. (1983)).
4 Computational results
To evaluate the two-stage approach, we simulated a scenario in which for each job
we are first given a “base” processing time for each operation (which corresponds to the
processing of the operation in the model without pliability) and then introduce flexibility
in such a way that we are allowed to deviate from these processing times on each machine
by some amount as long as the total processing time of the job remains the same. For our
test sets, we randomly generated instances of synchronous flow shops with 2 to 5 machines
and 10 to 150 jobs. For each operation Oij we chose a base processing time pB
ij uniformly
distributed over the interval [0, 100]. Using these base processing times, we generated lower
and upper bounds by defining two real-valued parameters α ≥ 0 and β ≥ 1 and setting
pij
= αpB
ij and pij = βpB
ij for all operations Oij. The total processing time pj of job j was
set to the sum of the base processing times of its operations. For each of the combinations
of n and m we generated five instances. Additionally, for each combination n, m and these
base processing times we generated several instances with different α- and β-values. In
the following, we discuss results for three different parameter sets: (α = 0, β = 2), i.e.,
a configuration in which we are allowed to deviate a lot from the base processing times
and two more restrictive parameter sets, (α = 0.5, β = 1.5) and (α = 0.8, β = 1.2). All
72

computational evaluations were performed on a computer with an Intel Core i3-370M 2.4
GHz processor and 4 GB RAM.
α = 0, β = 2 α = 0.5, β = 1.5 α = 0.8, β = 1.2
Initial Tabu Initial Tabu Initial Tabu
m n Gap Gap Time Gap Gap Time Gap Gap Time
2 10 3.90 0.85 0 6.17 2.36 0 9.93 6.09 0
2 50 0.90 0.02 2 2.01 0.25 2 4.10 1.59 2
2 100 0.65 0.01 11 1.61 0.16 18 3.37 0.72 14
2 150 0.58 0.04 49 1.41 0.21 45 2.84 0.81 80
3 10 11.42 5.40 0 13.31 7.01 0 12.35 10.03 0
3 50 5.20 0.51 3 8.51 2.56 3 13.23 6.67 4
3 100 4.33 0.19 32 7.80 1.47 26 11.37 4.56 31
3 150 3.92 0.26 154 6.06 0.98 115 11.15 3.75 159
5 10 20.38 12.98 0 19.36 16.99 0 27.91 22.20 0
5 50 12.23 3.84 7 19.54 9.25 7 26.16 16.74 7
5 100 10.84 2.16 81 14.84 6.12 46 25.27 13.27 69
5 150 10.56 2.33 383 15.21 5.49 291 24.61 12.31 238
Table 1. Results for problem F|synmv, plbl(pij
, pij)|Cmax
Table 1 shows the results of the two-stage approach for F|synmv, plbl(pij
, pij)|Cmax.
The gaps of the initial solutions as well as the results of the tabu search are reported
relative to a lower bound obtained by an LP relaxation. The computation times for the
tabu search are given in seconds, the time required to obtain an initial solution was below
one second for all instances.
Overall, it can be seen that the two-stage approach leads to a large improvement of the
initial solutions. Especially, in situations with high ﬂexibility we can reach near-optimal
solutions even for larger instances in a reasonable amount of time.
References
Baptiste, P., J. Carlier, A. Kononov, M. Queyranne, S. Sevastyanov, and M. Sviridenko, 2011,
“Properties of optimal schedules in preemptive shop scheduling”, Discr. Appl. Math. 159, pp.
272–280.
Huang, K.-L., 2008, “Flow shop scheduling with synchronous and asynchronous transportation
times”, PhD thesis, The Pennsylvania State University.
Kouvelis, P. and S. Karabati, 1999, “Cyclic scheduling in synchronous production lines”, IIE Trans-
act. 31, pp. 709–719.
Nawaz, M. E.E. Enscore, and I. Ham, 1983, “A heuristic algorithm for the m-machine, n-job
ﬂow-shop sequencing problem”, Omega 11, pp. 91–95.
Weiß, C., S. Knust, N.V. Shakhlevich, and S. Waldherr, 2016, “Flow shop and open shop scheduling
with job splitting” Proceedings of the 15th International Conference on Project Management
and Scheduling, Valencia, Spain, pp. 26–29.
73

Computation of the project completion time
distribution in Markovian PERT networks
Jeroen Burgelman1
1
jeroen.burgelman@ugent.be, mario.vanhoucke@ugent.be
2
3
Keywords: project scheduling, PERT, Linear Algebra.
1 Introduction
Since the introduction of PERT networks (Malcolm et al. 1959) uncertainty in ac-
tivity durations has been increasingly modelled using the PERT methodology (Adlakha
and Kulkarni 1989). Recently, uncertainty in activity durations has modelled using in-
creasingly complex probability distributions (Colin and Vanhoucke 2015). Nevertheless,
computing the exact project makespan distribution for project networks is infeasible in
general (Hagstrom 1988). The special case where activity durations are modelled using
independently distributed exponential random variables has received moderate attention
in the literature (Kulkarni and Adlakha 1986), (Azaron et al. 2006). Moreover, this special
case has frequently been used as a basis to study more involved project scheduling prob-
lems (Azaron et al. 2011, Gutin et al. 2015). Therefore the accurate computation of the
resulting project makespan distribution is of vital importance.
This abstract proposes an integrated approach to validate the applicability, accuracy
and robustness of project completion time distribution computations in Markovian PERT
networks. The applicability of methods in the literature hinges on the theoretical assump-
tions underlying these methods. Given the size of the Markov chain, small rounding errors
in the calculation of the project makespan distribution can propagate and result in inac-
curate distribution functions. Furthermore, the ability of different methods to cope with
changes in the input data is assessed.
Section 2 discusses different approaches to compute the project makespan distribution
in Markovian PERT networks. Section 3 discusses the research and preliminary results.
In this research, a project is represented using an acyclic directed graph G = (N, A)
where N is the set of nodes representing the project activities and A is the set of arcs
representing the precedence relations of the project network. The objective is to compute
the cumulative probability distribution of the project makespan:
F(t) = P(Sn+1 ≤ t) (1)
Where Sn+1 is the random variable representing the project makespan and P(Sn+1 ≤ t)
denotes the probability of the project completing before or at time t. Under the assumption
of exponentially distributed activity durations, the project makespan distribution can be
derived by computing the solution to a set of differential equations resulting from an
underlying Continuous Time Markov chain (Kulkarni and Adlakha 1986, Azaron et al.
74

2006). A solution of the linear system of differential equations is given by:
F(t) = e1
T
· eQt
· en (2)
Here, e1 and en are respectively the first and last column of the n ± ×n identity matrix
In and eQt
is the matrix exponential of the infinitissimal generator of the CTMC defined
by (Kulkarni and Adlakha 1986) and (Azaron et al. 2006).
While several approaches exist to compute the matrix exponential (Moler and Van Loan
1978), the performance of most approaches is inadequate for our purpose. The three main
approaches to compute the matrix exponential are matrix decompositions, approximation
methods, scaling and squaring and Krylov methods.
First, matrix decomposition methods can be used to simplify the computation of eQt
by
decomposing Q into a matrix product form Z ·D·Z−1
. This approach has been advocated
in the project scheduling literature (Azaron et al. 2006). Although matrix decomposition
methods have a good performance on small to medium sized problems, the size of the state
space encountered in project makespan distribution computations can be prohibitive for
these methods. Furthermore, the performance of these methods has only been demonstrated
on numerical examples in the literature (Azaron et al. 2006, Azaron et al. 2011).
Second, the scaling and squaring method relies on padé approximants rm of order m in
combination with a scaling parameter s to produce approximations for eQt
.
eQ
≈ rm 2−s
Q
2s
(3)
Therefore the resulting solutions are obtained numerically, in contrast to the exact expres-
sions obtained by the matrix decomposition methods. This approach is more stable than
the matrix decomposition methods and has less theoretical limitations (Moler and Van
Loan 1978).
Finally, Krylov methods (Moler and Van Loan 2003) do not compute the entire matrix
exponential, but approximate the product eQt
· en without computing eQt
explicitly. The
approximation is achieved by the Arnoldi process, to compute a matrix Qk with orthonor-
mal columns and the resulting approximation is given by
eQ
· en ≈ QkeHk
Qk · en (4)
Where the matrix exponential of the upper Heisenberg matrix Hk is easier to compute
and the large state space dimensions of the CTMC are reduced to dimensions k × k. This
method is especially suited to compute project makespan distribution functions for large
scale problems but can suffer from loss of accuracy if the computation of the matrix Qk is
unstable.
The scaling and squaring and Krylov methods have never been assessed in a project
scheduling setting. Therefore, the performance of algorithms to compute project makespan
distribution functions in Markovian PERT networks has never been assessed with regard
to applicability, accuracy and robustness.
3 Research and preliminary results
In this paper, the advantages and limitations of existing approaches to compute project
makespan distributions are compared. Furthermore, we assess the ability of specialised
techniques from linear algebra to overcome the existing limitations. Based on the resulting
analysis, we provide theoretical and managerial insights in the performance of the differ-
ent algorithms and the extent to which existing limitations can be resolved by adapting
traditional project data generation schemes used in the project scheduling literature.
75

The presented approaches are assessed on three key metrics, applicability, accuracy and
robustness. The assessment based on applicability comprises three parts. First the theoret-
ical limitation of the methods are assessed for several standard datasets from the project
literature (Vanhoucke et al. 2016) in terms of the invertibility of Z. Second, since the goal of
the research is to compute the cumulative distribution of the project makespan, computa-
tional results that do not adhere to the properties of cumulative distribution functions, i.e.
inft F(t) = 0, supt F(t) = 1 and monotonicity, essentially make the corresponding method
inapplicable for our purposes. Finally, the computation of the matrix exponential requires
matrix multiplication operations on matrices of vast dimensions, thus potentially causing
memory problems and an incomplete computation of the project makespan distribution.
The accuracy of an approach is measured in the number of significant digits lost during the
computation of the makespan distribution function. High loss of accuracy can make the
computation of probabilities very inaccurate. The robustness of the approaches is gauged
by perturbing the input data of the infinitissimal generator matrix of the CTMC with a
small factor 0.01 ≤ ǫ ≤ 0.02 and measuring the errors in the computation of the project
makespan distribution function by the euclidean norm at the decile values of the computed
distribution function.
Preliminary experiments show that the matrix decomposition methods advocated in
the project scheduling literature exhibit very limited performance in terms of applicability,
accuracy and robustness, regardless of the project data set on which they were assessed.
Moreover, the scaling and squaring algorithm is more robust to small alternations in the
input data, whereas Krylov methods fail to find stable solutions for project networks with
more than 10 activities. To mitigate the limitations inherent in matrix decomposition
methods, a new dataset is constructed building on the fundamental assumption underlying
the method of (Azaron et al. 2006), namely the existence of a set of |S| independent
eigenvectors, where |S| is the size of the state space of the CTMC. The performance of
all methods on the new dataset was tested and the decomposition method proposed in
the project scheduling literature has comparable performance to the scaling and squaring
algorithm, albeit at a higher computational cost.
References
Adlakha V., V. Kulkarni, 1989, “Classified bibliography of research on stochastic PERT networks:
1966 –1988, INFOR: Information Systems and Operational Research, Vol. 27 (3), pp. 272-296.
Azaron A., B. Fynes and M. Modarres, 2011, “Due date assignment in repetitive projects, Inter-
national Journal of Production Economics, Vol. 129, pp. 79-85.
Azaron A., H. Katagiri, K. Kato and M. Sakawa, 2006, “Longest path analysis in networks of
queues: Dynamic scheduling problems, European Journal of Operational Research, Vol. 174
(1), pp. 132-149.
Colin J., M. Vanhoucke, 2015, “Empirical Perspective on Activity Durations for Project Manage-
ment Simulation Studies, Journal of Construction Engineering and Management, Vol. 142
(1), pp. 04015047.
Gutin E., D. Kuhn, and W. Wiesemann, 2015, “Interdiction games on Markovian PERT networks,
Management Science, Vol. 61 (5), pp. 999-1017.
Hagstrom J., 1988, “Computational Complexity of PERT problems, Networks, Vol. 18 (2), pp.
139-147.
Hartmann S., D. Briskorn, 2010, “A Survey of Variants and Extension of the Resource-Constrained
Project Scheduling Problem, European Journal of Operations Research, Vol. 207, pp. 1-14.
Kulkarni V., V. Adlakha, 1986, “Markov and Markov-regenerative PERT networks, Operations
Research, Vol. 34 (5), pp. 769-781.
Malcolm D., J. Roseboom, C. Clark and W. Fazar, 1959, “Application of a technique for a research
and development program evaluation, Operations Research, Vol. 7 (5), pp. 646-669.
76

Moler C.B., C. F. Van Loan, 1978, “Nineteen dubious ways to compute the exponential of a
matrix, SIAM Review, Vol. 20(4), pp. 801-836.
Moler C.B., C. F. Van Loan, 2003, “Nineteen dubious ways to compute the exponential of a matrix:
Twenty-five years later, SIAM Review, Vol. 45(1), pp. 1-47.
Vanhoucke M., J. Coelho and J. Batselier, 2016, “An overview of project data for integrated project
management and control, Journal of Modern Project Management, Vol. 3 (2), pp. 6-21.
77

Comparing event-node graphs with nonrenewable
resources and activity-node graphs with renewable
resources
Jacques Carlier
Université de Technologie de Compiègne
jacques.carlier@uds.utc.fr
1 Abstract
At the end of the ﬁfties, two main approaches were proposed to manage a large project:
the PERT method and the MPM method. In both approaches the project is modelled
by a graph and one has to compute critical paths. In the PERT graph, an activity is
represented by an arc whenever nodes represent events. In the MPM graph, an activity is
represented by a node whenever arcs represent precedence constraints. The drawback of
both methods is that they do not take into account resources. The speciﬁc drawback of
the event-node graph is its large size. The scheduling literature is essentially devoted to
problems with renewable resources and precedence constraints, modelled by an activity-
node graph. Renewable resources are allocated to activities at their starting times and
released at their completion times. A machine is an example of a renewable resource.
The basic problem is the Resource Constrained Project scheduling Problem (RCPSP).
The aim of this talk is to rehabilitate event-node graph and nonrenewable resources. A
nonrenewable resource is produced or consumed by an activity at its occurrence time.
The money is an example of a nonrenewable resource. Our basic problem is the Extended
Resource Constrained Project Scheduling Problem (ERCPSP). We will present a brief
review of literature on ERCPSP. We will explain that several approaches built for RCPSP
can be adapted to ERCPSP. We will also report some polynomial algorithms. Next we
will introduce several lower bounds and some linear programming models inspired from
RCPSP ones. Finally we will report some computational results and explain why it is
useful to study ERCPSP.
78

Synchronizing Heterogeneous Vehicles in a Routing and
Scheduling Context
Marc-Antoine Coindreau1
, Olivier Gallay1
and Nicolas Zufferey2
1
Department of Operations, HEC – University of Lausanne, Switzerland
marc-antoine.coindreau, olivier.gallay@unil.ch
2
Geneva School of Economics and Management, GSEM – University of Geneva, Switzerland
n.zufferey@unige.ch
Keywords: Vehicle Routing and Scheduling, Synchronization, Carpooling.
1 Introduction
The Vehicle Routing Problem (VRP) aims at defining optimal vehicle routes that visit a
set of jobs spread on a given territory. Depending on the context, a job can be a delivery of
goods, a pick up of components, or a service provided on-site. When scheduling the workers’
routes for on-site services, the systematic use of cars (which is the main hypothesis in the
VRP literature) can be inefficient when only light equipments are transported and when
distances between some jobs could allow light transportation modes (e.g., bikes). Moreover,
using independently light transportation modes, as done in the VRP with heterogeneous
fleet (Baldacci et al. 2008), might not be always envisioned because of the limited range
of such transportation modes (i.e., the maximum allowed distance to travel). Indeed, some
jobs might be too distant from the depot (i.e., exceeding the allowed range or the total
allowed duration of a tour). In such contexts, synchronizing light and heavy transportation
modes could be a promising answer.
We focus here (see Section 2) on formulations involving, jointly, light and heavy re-
sources to serve jobs, where both transportation modes can move independently, and where
the light resources can be embedded in the heavy ones on some parts of their routes. The
heavy resource can be a car, a truck or a van. The light resource can be workers on foot,
on bike, equipped with an electric kick scooter, or whatever light transportation mean that
can be easily embedded into the heavy resource.
A rather scarce literature addresses the problem of synchronizing light and heavy re-
sources. In the home health-care context, a recent contribution considers synchronization
of walking and driving (Fikar and Hirsch 2015). It shows that the number of vehicles can
be reduced by up to 90% when an external company picks up and drops off nurses (who are
also allowed to walk). This reduction comes however with an increase on the total number
of workers employed. In the context of light-goods delivery, where foot couriers can be cou-
pled with vans, Lin (2011) shows that both the average cost and the number of used cars
can be reduced in comparison with the approach where vans are treated as independent
transportation modes. This gain on both dimensions is observed even if that study only
considers coordination during the van outbound or return leg. In parcel delivery, coupling
a drone (the light resource) with a single van (the heavy resource) could lead to a gain up
to 20% on the truck use (Murray and Chu 2015).
The above-mentioned papers successfully show the relevance of synchronizing heteroge-
neous vehicles with different characteristics. In this work, we consider the situation where
the workers have the choice between traveling by car, by using electric kick scooters, or
simply walking. Moreover, carpooling is enabled. The potential gain offered by the syn-
chronization of such transportation modes is measured and discussed (see Section 3).
79

2 Synchronizing workers and vehicles with carpooling
We propose a new formulation that allows the synchronization of cars (heavy resource)
and workers (light resource) potentially equipped with an electric kick scooter. On the one
hand, the light resource is cheaper but limited by its speed and range. On the other hand,
the heavy resource is faster and it can transport multiple workers, but at a larger cost and
pollution impact. If not coupled with the heavy resource, the light resource is restricted
to the exploration of the jobs located close to the depot. As a consequence, the heavy
resources would have to make great detours to visit distant jobs in the same tour.
As electric kick scooters can be easily embedded into a car, coordinating and synchro-
nizing these two types of resources turn out to be a promising approach to overcome the
individual drawbacks of each of these two transportation modes. More precisely, we con-
sider the case where carpooling is enabled, meaning that heavy resources can transport
multiple light resources (with a maximum number of Q = 2 workers equipped with elec-
tric kick scooters per car). Workers are split into two categories, the car drivers and the
passengers. Light and heavy resources can couple and uncouple as many times as required.
Drivers are allowed to serve jobs and to use an electric kick scooter to reach jobs, but the
return path to the car is mandatory. Passengers can be picked up elsewhere than at the
drop-off location, after they have been using the electric kick scooter to travel between
jobs that are located nearby. The considered problem is an extension of the classical VRP
with time windows, in which workers with an individually assigned car must leave and
come back to the depot within the working day, after having served the jobs within their
assigned time windows. We focus here on analyzing the impact offered by the introduction
of electric kick scooters, regarding their speed and range.
We have developed a metaheuristic (MH) based on the ruin and recreate principle
(Pisinger and Ropke 2011). MH aims at improving a solution by sequentially removing
and reinserting jobs. Depending on the search phase, MH can remove up to 30% of the
inserted jobs. In general, the more time the search is trapped in the same local optimum,
the more jobs are removed and reinserted afterwards. A typical output solution is given
in Figure 1. Each worker has a color code: light gray for worker w1, gray for w2, double
line for w3, and black for w4. Heavy (resp. light) resource paths are represented with plain
(resp. dashed) lines. w1 and w4 leave the depot in the same car. w4 is dropped off at job
j30 and uses a light resource to travel to j1, where s/he is picked up by w1. Some drivers
are traveling some sub-routes with a light resource, like w2 on path j4 − j8 − j37 − j19. All
workers and vehicles start and end their working day at the central depot located in the
middle of the grid.
3 Computational experiments
To validate the efficiency of MH, its results are compared to optimal VRP solutions
where only heavy resources are used. The optimal VRP solutions are obtained with the
Branch-And-Price algorithm (BP) proposed in Desaulniers et al. (2008). Allowing the
workers to move without a car while enabling carpooling is expected to help managers
reduce both the number of cars used (fcar) and the total driving distance (fdist). f∗
car and
f∗
dist refer to the optimal values of fcar and fdist found by BP, respectively. Depending on
the instance configuration, replacing a heavy resource by a light one can either reduce or
increase the driving distance. The reduction occurs when detours to carry the light resources
are overcompensated by the pooling of the heavy resources, whereas the augmentation
occurs when too many detours are required to carry the light resources. We consider the
case where managers want to reduce f∗
car without increasing f∗
dist.
80

0 2 4 6 8 10
0
2
4
6
8
10
j26
j25
j24
j20
j12
j33
j31
j28 j23
j21
j15
j5
j39
j19
j37
j8
j4
j35
j34
j9
j38
j17
j29
j2
j6
j16
j27
j7
j36
j14
j22
j13
j10
j0
j32
j1
j18
j11
j3
j30
Fig. 1. Solution exhibiting coordination between light and heavy resources.
We consider 60 instances derived from real data of a large energy provider. The car
speed is set to 30 km/h, whereas the light-resource speed is either 4 km/h (for walking)
or 15 km/h (for the electric kick scooter). A 10-km square grid is considered, representing
an urban configuration. The depot is located at the center of the grid, and Euclidean dis-
tances are considered between two job locations. Instances with n ∈ {20, 30, 40, 50} jobs
were generated. Indeed, lower instance sizes do not exhibit enough potential for carpool-
ing, whereas BP is not able to provide optimal VRP solutions for larger sizes. The job
characteristics (i.e., location, duration, time window) are randomly generated, based on
the uniform distribution. The duration of each job belongs to [15, 34] minutes. There are
three types of instances. First, for the 20 All-Day instances, each job has the same time
window [8:00, 15:00], corresponding to the full planning horizon (i.e., the working day).
Second, for the 20 Half-Day instances, each job has either time window [8:00, 11:30] or
[11:30, 15:00]. Finally, for the 20 Quarter-Day instances, the possible time windows are
[8:00, 9:45], [9:45, 11:30], [11:30, 13:15] and [13:15, 15:00]. These three types of instances
represent three service levels that can be offered to the involved clients. Indeed, the shorter
is the time window, the better it is from the client perspective, as s/he has to block a
shorter time period within which s/he can be served.
Table 1 shows the percentage improvements obtained on fdist and fcar where the follow-
ing features are modified: light resource type (i.e., walking vs electric kick scooter), range
(i.e., 5 km vs 10 km), service level (i.e., All-Day vs Half-Day vs Quarter-Day). Average
results (over the 60 instances) are given in the last line. One can observe that the gain of
only allowing walking and carpooling can help decreasing the driving distance by 5.57%
and the number of cars by 5.76% (see the left double column labeled with 5 km). The
results highlight the importance of increasing the speed and range parameters to magnify
the gain offered by the synchronization of the light and heavy resources. Indeed, both fdist
and fcar can be improved by 9.18% and 14.14%, respectively. Note that additional exper-
iments on these instances have shown that without limiting the driving distance to f∗
dist,
the fcar-gain can be up to 19.90%. Last but not least, it is important to have in mind that
conservative assumptions were considered for generating the instances. Indeed, there are
less than 0.5 job per km2
and the average distance between jobs is around 5.5 km, and
hence only 3% of the edges are eligible to be traveled with a light resource (i.e., when the
distance between two jobs is below 1 km). One can reasonably assume that more favorable
81

cases would occur in other practical situations (especially in urban contexts), which would
lead to the amplification of the gains.
Table 1. Potential gain when workers can move without cars (allowing carpooling).
Light resource Walking worker Electric kick scooter
Range 5 km 10 km 5 km 10 km
Objective fcar fdist fcar fdist fcar fdist fcar fdist
All-Day 11.48% 8.56% 14.75% 10.88% 18.03% 9.87% 22.95% 16.92%
Half-Day 6.25% 5.79% 6.25% 8.36% 10.94% 6.29% 12.50% 9.61%
Quarter-Day 0% 3.58% 1.52% 3.42% 6.06% 3.23% 7.58% 4.12%
Average 5.76 % 5.57 % 7.33% 6.93% 7.91 % 5.92 % 14.14 % 9.18 %
4 Conclusion
In this paper, we highlight the relevance of synchronizing heterogeneous vehicles that
vary in their characteristics, more precisely light and heavy resources that differ in their
speed, range and operational cost. Such a coordinated scheduling helps reducing both the
number of heavy resources needed and the total driving distance. Increasing the speed
of the light resource and its range leads to higher gains, and ultimately the obtained
solutions would be close to those which can be achieved by coordinating truck and drones.
Indeed, in the context of delivery, the next step, after having improved the situation by
replacing walking by electric kick scooters, would be to consider drones as light resources.
Interestingly, drones could even be faster than trucks, but additional constraints such as
capacity and landing eligibility would have to be considered.
Acknowledgements
We would like to thank Prof. Guy Desaulniers for providing the optimal VRP results.
References
Baldacci R., Battarra M., and Vigo D., 2008, “Routing a heterogeneous fleet of vehicles, The
Vehicle Routing Problem: Latest Advances and New Challenges, pages 3–27.
Desaulniers G., Lessard F., and Hadjar A., 2008, “Tabu search, partial elementarity, and general-
ized k-path inequalities for the vehicle routing problem with time windows, Transportation
Science, 42(3):387–404.
Murray C. and Chu G., 2015, “The flying sidekick traveling salesman problem: Optimization
of drone-assisted parcel delivery, Transportation Research Part C: Emerging Technologies,
54:86–109.
Fikar C. and Hirsch P., 2015, “A matheuristic for routing real-world home service transport systems
facilitating walking, Journal of Cleaner Production, 105:300–310.
Lin C. K. Y., 2011, “A vehicle routing problem with pickup and delivery time windows, and
coordination of transportable resources, Computers Operations Research, 38(11):1596–
1609.
Pisinger D. and Ropke S., 2007, “A general heuristic for vehicle routing problems, Computers
Operations Research, 34(8):2403–2435.
82

On the construction of optimal policies for the RCPSP
with stochastic activity durations
Erik Demeulemeester
KU Leuven, Faculty of Economics and Business, Department of Decision Sciences and
Information Management, Leuven (Belgium)
Erik.Demeulemeester@kuleuven.be
Keywords: RCPSP, stochastic activity durations, optimal policies.
1 Abstract
In this paper, we will research for what types of resource-constrained project scheduling
problems (RCPSPs) with stochastic durations an optimal policy can be constructed, which
incorporates an optimal baseline schedule as well as optimal continuations whenever the
realized durations of an activity render the baseline schedule or an already adapted version
of it infeasible.
2 State of the art
Every single day millions of small, medium and large projects are being executed. The
planning of these projects is not a simple endeavor. One often hears about the failure to
complete a project within time, within budget and according to specifications (see Flyvb-
jerg, 2005, for a nice overview). Perfect examples thereof are the building of the interna-
tional airport in Denver (200% overrun of the costs), the building of the Chunnel (80%
overrun of the costs) and the organization of the Olympic Games in Athens (a billion Euro
above budget). It might be obvious that project planning didn’t live up to its promise in
these cases (as in many others). Fundamental research in the field of project planning is
therefore of utmost importance.
The vast majority of the project scheduling efforts over the last forty years have con-
centrated on the development of a workable baseline schedule with the goal of obtaining a
project duration that is as short as possible. One traditionally makes the assumption that
the durations of the activities are known and deterministic and that the resources are fully
available. A realistic project, however, will always be subject to disruptions. Many types
of disruptions have been studied in the literature (Yu and Qi, 2004, Wang, 2005, and Zhu
et al., 2005): activities can take longer than primarily expected, resource requirements or
availabilities may vary, due dates may change during the execution of the project, new
activities may have to be inserted (Artigues and Roubellat, 2000), etc. Research in project
scheduling has focused on the one hand on proactive and reactive procedures to counteract
the effects of these disruptions as much as possible: proactive planning attempts to build
a stable project plan that takes the possible disruptions as much as possible into account,
while the reactive planning procedures are called every time the disruption changes the
baseline schedule such that it cannot be executed anymore as planned.
A typical objective function for the proactive project planning phase is the weighted
sum of the deviations between the planned and the realized starting times of the different
activities in the project. Quite some research (e.g., Leus and Herroelen, 2004, Van de Vonder
et al., 2006, 2007) focused itself on the construction of stable project plans and this mainly
under the assumption of uncertain durations. Typically, the solution procedure consisted
of two phases. In the first phase, a baseline plan is built that is feasible with respect
83

to the precedence relations as well as to the resource constraints and that is based on
previously determined durations and resource requirements for every activity. In a second
phase, this plan is made more stable through the introduction of time buffers before the
activities (even if their predecessors take longer than expected, this doesn’t automatically
lead to a postponement of the corresponding activity) and through the determination of
how the resources are passed along from activity to activity. A disadvantage of such a
two-step procedure obviously lies in the fact that the ultimate results depend heavily upon
the plan that was chosen in the first step (typically an optimal plan for the deterministic
version of the RCPSP). However, very recently Davari and Demeulemeester (2016) have
introduced an integrated proactive and reactive project scheduling problem for the RCPSP
with uncertain durations and developed different Markov Decision Process (MDP) models
to solve this problem. This means that not only a good baseline schedule is determined,
but also all good continuations in case certain combinations of the activity durations occur
that prohibit the baseline schedule or an already adapted schedule from being executed as
planned.
A second strand of literature that solves the underlying problem in a totally differ-
ent way is referred to as the stochastic RCPSP (SRCPSP). Methodologies for stochastic
project scheduling view the scheduling problem as a multi-stage decision process. So-called
scheduling policies are used to decide at each of the stages of a multi-stage decision pro-
cess, that occur serially through time at random decision points, which activities selected
from the set of precedence and resource feasible activities (the so-called admissibility con-
straints) have to be started (Ashtiani et al., 2011, Möhring et al., 1984, 1985, and Stork,
2001). The so-called non-anticipativity constraint requires that scheduling decisions can
only be based on the observed past and a priori knowledge about processing time distribu-
tions. The objective is to minimize the expected project duration. Scheduling policies do
not construct a complete schedule before the initiation of the project, but gradually build
a schedule during the project’s implementation. Because of this characteristic, stochastic
scheduling policies are often referred to as purely reactive or on-line procedures. This also
implies that no baseline schedule is constructed, which is considered as one of the more im-
portant drawbacks of this approach. In this SRCPSP, the duration Di of each non-dummy
activity i is a random variable. The random vector (D2, D3, . . . , Dn−1) is written as D.
According to the definitions given in Igelmund and Radermacher (1983ab) and Möhring et
al. (1984, 1985), a scheduling policy Π makes decisions at the decision points t = 0 (the
start of the project) and at the completion times of activities. A decision at time t is to
start at time t a precedence and resource feasible set of activities, S(t), exploiting only
information that has become available up to time t. As soon as the activities have been
finished, the activity durations are known, yielding a realization (sample, scenario) d of
the random vector D. For a given scenario d and a policy Π, the project duration CΠ
max(d)
is the resulting schedule makespan. The objective of the SRCPSP is to select a policy Π∗
that minimizes E(CΠ
max(d)) within a specific class of scheduling policies.
Various classes of scheduling policies have been proposed in the literature. Stork (2001)
reports promising computational results using so-called preselective policies that have been
introduced by Igelmund and Radermacher (1983a) and three important subclasses of the
class of preselective policies: early-start policies (ES-policies), linear preselective policies
(LIN-policies) and activity-based policies (AB-policies). Ashtiani et al. (2011) introduce
pre-processor policies (PP-policies) which make a number of a-priori sequencing decisions in
a pre-processing phase while the remaining decisions are made dynamically during project
execution. Quite some interesting research has been performed on determining the quality
of the different scheduling policies.
84

3 Methodology
The goal of the research in this paper is to find optimal policies for particular versions
of the RCPSP with uncertain activity durations. We will clarify this goal first by a small
example instance for a standard case of the RCPSP with uncertain activity durations.
Fig. 1. Representation of small project network.
Figure 1 represents a small project network of 8 real activities and 2 dummy activities
(representing the start and end of the project), where the distribution of the activity
durations is shown in the table on the right of the figure and the resource requirements for
each activity are shown above the nodes that indicate the activities (the resource availability
is determined to be 8 units per time unit).
Table 1. The starting times for ten feasible schedules
Sk
S1
S2
S3
S4
S5
S6
S7
S8
S9
S10
Sk
0 0 0 0 0 0 0 0 0 0 0
Sk
1 0 0 0 0 0 0 0 0 0 0
Sk
2 1 1 0 1 5 0 7 4 2 7
Sk
3 3 3 4 4 3 3 3 3 5 5
Sk
4 0 0 4 0 0 7 0 0 0 9
Sk
5 6 6 7 7 7 7 7 7 9 14
Sk
6 6 6 7 7 7 12 5 7 9 14
Sk
7 7 8 7 8 12 12 14 12 11 15
Sk
8 11 13 13 12 15 15 17 15 15 20
Sk
9 13 15 15 15 17 18 19 18 18 23
Table 1 represents the starting times for each activity of 10 schedules that are somehow
created and that are feasible for at least one of the realizations of the durations of the
activities. The optimal policy over these 10 schedules for this problem is then as follows:
the optimal baseline schedule is schedule S9
, for which the planned starting times can be
found in the last but one column of Table 1. However, if at time 2 it becomes clear that
activity 1 takes longer than 2 time periods (a 20% chance, see Figure 1), schedule S9
is no
longer feasible (see upper left schedule of Figure 2 where activities 1, 2 and 4 are scheduled
at the same time, needing 2 + 3 + 4 = 9 resource units whereas only 8 are available). At
that time, the optimal policy indicates that one should switch to schedule S8
, which is
represented in the upper right corner of Figure 2. However, if at time 4 it turns out that
85

activity 4 requires a duration of 5 time units, the current schedule becomes infeasible again
(see lower left corner of Figure 2): at that time the optimal policy indicates that one should
switch to schedule S5
(see lower right corner of Figure 2). Obviously, if more and better
schedules could be generated, the resulting proactive/reactive policy will turn out to be
better. This surely is a very interesting topic for further research. This paper, however, will
analyze for which restricted versions of the RCPSP with stochastic durations true optimal
policies can be constructed that do not depend on the generation of a restricted set of
feasible schedules.
Fig. 2. Representation of the diﬀerent schedules in the optimal policy.
References
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constrained project scheduling problem: Exploring the beneﬁts of preprocessing”, Journal of
Scheduling, 14(2), 157–171, 2011.
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ule with cumulative constraints and multiple modes”, European Journal of Operational Re-
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M. Davari and E. Demeulemeester, “The proactive and reactive resource-constrained project
scheduling problem”, Revise and resubmit for European Journal of Operational Research,
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scheduling problems”, Networks, vol. 13, no. 1, pp. 29–48, 1983b.
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tions, vol. 36, no. 7, pp. 667–682, 2004.
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strategies”, ZOR - Zeitschrift für Operations Research, vol. 28, pp. 193–260, 1984.
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gies”, ZOR - Zeitschrift für Operations Research, vol. 29, pp. 65–104, 1985.
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86

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and makespan in resource-constrained project scheduling”, International Journal of Produc-
tion Research, vol. 44, no. 2, pp. 215–236, 2006.
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reactive project scheduling”, Computers Industrial Engineering, vol. 52, no. 1, pp. 11–28,
2007.
G. Yu and X. Qi, Disruption management - Framework, models and applications, New Jersey:
World Scientiﬁc, 2004.
J. Wang, “Constraint-based schedule repair for product development projects with time-limited
Constraints”, International Journal of Production Economics, 95, 399–414, 2005.
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ing”, Journal of the Operational Research Society, vol. 56, pp. 365–381, 2005.
87

A BB Approach to Schedule a No-wait Flow Shop to
Minimize the Residual Work Content Under
Uncertainty
Simone Dolceamore1
, Marcello Urgo1
Mechanical Engineering Department, Politecnico di Milano, Italy
simone.dolceamore@mail.polimi.it, marcello.urgo@polimi.it
Keywords: Stochastic Scheduling, Conditional Value-at-Risk, Aircraft Assembly.
1 Introduction and problem statement
In recent years, approaches providing robust schedules have been increasing their im-
portance in the production scheduling research area. The pursued objective is to obtain
schedules being insensitive - as much as possible - to disturbing factors, protecting the
decision-maker against the impact of unfavorable uncertain events. In this paper we ad-
dress the scheduling of a set of jobs J in a paced assembly line in presence of uncertainty
affecting the availability of production resources. The proposed approach takes inspiration
from the assembly process in the aircraft manufacturing industry. Each job j has to be
processed in the assembly line made up of M stations. Being paced, the line is characterized
by a cycle time, i.e., at a given time, all the parts move to the next station simultaneously.
Hence, within the cycle time, a given deterministic amount of work has to be accomplished
in each station. The availability of production resources, i.e., the available working hours
of the workers during each cycle time, is modeled as a stochastic variable. The manufac-
turing system described is a permutation flow-shop with no-wait property (Emmons and
Vairaktarakis (2013)). The proposed approach address the definition of a robust schedul-
ing for the assembly line aiming at minimizing the conditional value-at-risk (CV aR) of the
residual work content, i.e. the amount of workload that cannot be completed during the
cycle time in the stations, due to a lack of available resources. A branch bound approach
is developed to solve the described problem to optimality. The objective function used, the
CV aR is a measure of risk widely used in the financial research, e.g. in portfolio optimiza-
tion (Rockafellar and Uryasev (1999), Rockafellar and Uryasev (2002)). This class of risk
measure has been already taken into consideration for scheduling approaches (Tolio, T. et
al. (2011), Sarin, S. C. et al. (2014)). Specifically, the permutation flow-shop scheduling
problem (with or without no-wait property) has been addressed in a considerably large
number of papers, e.g., a branch bound approach is developed by (Kim (1995)) with
the objective of minimizing total tardiness, whereas several mixed integer formulations and
an implicit enumeration approach are proposed in (Samarghandi and Behroozi 2017) and
(Samarghandi and Behroozi (2016)). Nevertheless, the proposed scheduling problem has
not been addressed in previous researches.
2 Description of the approach
The proposed branch bound framework relies on a sequential definition of the sched-
ule. At each level l of the associated tree l ∈ J , a partial solution provides the sequence of
the first l jobs scheduled, while the remaining J − l ∈ J S jobs are the candidates to be
scheduled next in the sequence. Hence, each node of the tree has as many child nodes as
the jobs to schedule, each of them representing a partial solution where a different jobs is
88

added to the partial sequence. The solution tree is explored adopting a depth-first strategy
selecting the most promising branches in terms of the best lower bound. At each node,
a lower and an upper bound on the target performance (the residual work content) are
calculated to determine the most promising branches and prune the dominated ones. The
contribution to the objective function of already scheduled jobs is easily calculated. Being
the system a permutation flow-shop, once a job is scheduled in the first station, the cycle
times where it will be processed by the following stations are automatically determined.
Then, considering a single resource with availability Ac for each cycle time period c, the
sequencing of that job j also entails a resource consumption Rjc. If a job j is scheduled to
enter the first station of the line in period p, its contribution to the objective function is:
RWCS∪{j} = ∗
c
(Ac ∗ Rj,c), ∀ j ∈ J S, c = p, . . . p + M − 1 (1)
where ∗ is the convolution operator.
The lower bound distribution of the residual work content caused by an unscheduled
job i ∈ J S +{j} can be estimated through the scheduling of a dummy job ˜
i1, having the
lowest resource request among the ones of the J − l unscheduled jobs. This contribution
can be estimated according to Eq. 2.
RWCLB
S+{j}∪i = ∗
c
(Ac ∗ R˜
i1,c), ∀ i ∈ S + {j} J , c = p, . . . , p + M − 1 (2)
In an dual way, the upper bound distribution of the residual work content caused by
an unscheduled job i ∈ J S + {j} can be estimated scheduling a dummy job ˜
i2 having
highest among the resource request of the J − l unscheduled jobs (Eq. 3).
RWCUB
S+{j}∪i = ∗
c
(Ac ∗ R˜
i2,c), ∀ i ∈ S + {j} J , c = p, . . . , p + M − 1 (3)
Finally, the lower and upper bounds of the considered node can be calculated as:
RWCLB
= ∗
i
RWCLB
S+{j}∪i ∗ ∗
j
RWCS∪{j}, ∀j ∈ S J , i ∈ S + {j} J (4)
RWCUB
= ∗
i
RWCUB
S+{j}∪i ∗ ∗
j
RWCS∪{j}, ∀j ∈ S J , i ∈ S + {j} J (5)
Grounding on these calculations, the lower and upper bounding distributions for the
residual work content can be calculated in each node. Furthermore, these distributions can
also support the calculation of the lower and upper bound of a function of the risk associated
to the resource consumption, e.g., the CV aR, with the aim at assessing the robustness of
the solution. Notice that, Eq. 4 and 5 provides effective bounds for the CV aR only in
case the resource requirements of the jobs are deterministic. In this particular case, the
convolution operator merely shifts the availability distributions without re-shaping it. This
ensures the conditional value-at-risk of residual work content being a regular objective
function.
Figure 1 further depicts the branching scheme adopted, as well as the computation of
the bounds for the CV aR. Blue and black cumulative distribution functions represent the
lower and upper bound distributions respectively. Nodes with a lower bound of the CV aR
higher than the incumbent CV aR are pruned.
89

Schedule: 3,-,-,-
LB= 107, UB= 110
4
0
1
2
3
7
6
5
8
8
9
Schedule: 0,-,-,-
LB= 51, UB= 123
Schedule: 1,-,-,-
LB= 32, UB= 105
Schedule: 2,-,-,-
LB=98, UB= 158
Schedule: 1,3,-,-
LB= 49, UB= 101
Schedule: 1,2,-,-
LB= 37, UB= 56
Schedule: 1,0,-,-
LB= 67, UB= 89
Fig. 1. Branching scheme and bounds computation
3 Testing and Industrial Application
The developed branch-and-bound approach has been implemented in C++ using the
BoB++ library. Computational experiments have been performed on 8 parallel threads
on an Intel Four-Core i7 Processor 7700-HQ@3.4GHz and 16 GB of DDR4 SDRAM. The
performance of the algorithm has been analyzed in terms of the time to find an optimal
solution and the fraction of nodes explored solving 9-jobs instances sampled from a pool
of 68 real orders. The testing instances have been constructed as follows:
1. the resource requirement of a job j in station m is deterministic. In fact, at the time
the assembling of an aircraft is scheduled, order specifications are known and fixed;
2. the resource availability in station m in time cycle c is a discrete triangular distribution,
whose maximum value matches the planned ideal amount of workforce while minimum
and the mode model the variability caused by absenteeism or other lacks of personnel;
3. the risk level used for the CV aR is set to 10%, this value depends on the risk aversion
of the planner, since it defines the quantile of the tail whose expected value must be
minimized.
The algorithm was able to find the optimal solution in 8264.15 seconds on average,
ranging from a minimum of 7803.20 to a maximum of 8819.61. The average number of
evaluated nodes was 280721 over a total of 623547, with an average pruning efficiency of
about 55%. The main cause of the relatively long computational times is due to the modest
variability in terms of workload requirements among the considered orders, because their
assembly process is composed of more or less 90% of mounting and testing operations for
structural components that are common to all the orders, while customization activities
have a lower impact in terms of equivalent man hours. Nevertheless this is partially due
to the oversimplification of the assembly process to a single type of resource and, hence,
reducing the impact of the uncertainty affecting the availability of specific resources. More-
90

Fig. 2. Distribution of the residual work content obtained with the minimization of the CV aR
(right) and the expected value (left).
over, due to the convolution operations, the amplitude of the support of the distributions
has a strong influence on the time needed to accomplish calculations within a single node.
An additional analysis was carried out to compare the proposed approach against
scheduling to minimize the expected value fo the residual work content (RWC). An ex-
ample is provided in Figure 2 showing the histogram of the RWC in the case of the
minimization of the expected value (left) and the CV aR (right). Although the expected
value in both the cases is almost identical, the CV aR is rather different (0.73 against 0.70)
clearly showing that minimizing the CV aR actually reduce its value in the optimal so-
lution. Moreover the distribution on the right shows a low occurrence probability for the
highest values of the RWC, thus demonstrating the capability of the approach to protect
the schedule against the worst cases.
4 Acknowledgments
This research was supported by the EU projects ProRegio (Grant agreement No.
636966) and ReCaM (Grant agreement No: 680759) funded by the European Commis-
sion in the Horizon 2020 programme.
References
Emmons, H. and Vairaktarakis, G., 2013, Flow Shop Scheduling: Theoretical Results, Algorithms,
and Applications, International Series in Operations Research Management Science, Vol.
182, Springer.
Rockafellar, R. T. and Uryasev, S. 2002, Conditional value-at-risk for general loss distributions,
Journal of Banking Finance, Vol. 26, pp. 1443-1471.
Rockafellar, R. T. and Uryasev, S. 1999, Optimization of Conditional Value-at-Risk, Journal of
Risk, Vol. 2, pp. 21-41.
Tolio, T., Urgo, M., and Váncza, J., 2011, Robust production control against propagation of
disruptions, CIRP Annals - Manufacturing Technology, Vol. 60, pp. 489-492.
Sarin, S. C., Sherali, H. D. and Liao, L., 2014, Minimizing conditional-value-at-risk for stochastic
scheduling problems, Journal of Scheuling, Vol. 17, pp. 5-15.
Kim, Y.-D., 1995, Minimizing total tardiness in permutation flowshops, European Journal of Op-
erational Research, Vol. 85, pp. 541-555.
Samarghandi, H and Behroozi, M., 2016, An Enumeration Algorithm for the No-Wait Flow Shop
Problem with Due Date Constraints, IFAC - PapersOnLine, Vol. 49-12, pp. 1803-1808.
Samarghandi, H and Behroozi, M., 2017, On the exact solution of the no-wait flow shop problem
with due date constraints, Computers and Operations Research, Vol. 81, pp. 141-159.
91

On Index Policies in Stochastic Scheduling
Franziska Eberle1
, Felix Fischer2
, Jannik Matuschke3
and Nicole Megow1
1
University of Bremen, Germany. {feberle,nicole.megow}@uni-bremen.de
2
Queen Mary University of London, UK. felix.fischer@qmul.ac.uk
3
Technical University of Munich, Germany. jannik.matuschke@tum.de
Keywords: stochastic scheduling, total completion time, approximation algorithm.
1 Introduction
We investigate a fundamental stochastic scheduling problem where jobs with uncertain
processing times must be scheduled non-preemptively on m identical parallel machines.
We are given a set J of n jobs, where each job j ∈ J is modeled by a random variable
Pj with known distribution. The actual realization of a processing time becomes known
only by executing a job. More precisely, a job notifies the scheduler when it completes.
The goal is to find a policy Π that decides for any point in time which jobs to schedule
such as to minimize the expected total completion time,
P
j∈J E[ CΠ
j ]. Here CΠ
j denotes
the completion time of job j under policy Π, and we drop the superscript whenever it is
clear from the context. The scheduling problem can be stated in the standard three-field
notation as P||E[
P
j Cj ].
The deterministic version of the problem is well-known to be solved optimally by the
Shortest Processing Time (SPT) rule (Rothkopf 1966). A natural generalization of this rule
to the stochastic setting, the Shortest Expected Processing Time (SEPT) rule, is optimal
when processing times follow exponential distributions (Bruno et al. 1981). For arbitrary
distributions no optimal policy is known, and in the past decade research has focused on
approximative policies. A stochastic scheduling policy Π is an α–approximation, for α ≥ 1,
if for all instances I of the problem at hand it holds that
P
j∈JI
E[ CΠ
j ] ≤ α
P
j∈JI
E[ C∗
j ].
Here, C∗
j denotes the completion time under an optimal stochastic scheduling policy on
the given instance I, assuming a priori knowledge of the set of jobs JI and their processing
time distributions Pj, but not their actual realizations. In particular, the optimal policy
also does not know the realizations, i.e., it is non-clairvoyant.
Several approximation algorithms have been developed with approximation guarantees
that depend either on the parameters m and n (Im et al. 2015) or on the probability
distributions of the processing times (Möhring et al. (1999), Megow et al. (2006), Schulz
(2008), and Skutella et al. (2016)). In the latter case, the approximation guarantee is of
order O(∆) where ∆ is an upper bound on the squared coefficients of variation of the
processing time distributions Pj, that is, Var[Pj]/E[ Pj ]2
≤ ∆ for all jobs j. Interestingly,
there is a 2-approximation algorithm for the preemptive (weighted) variant of our stochas-
tic scheduling problem P|pmtn|E[
P
j Cj ] independently of the distributions (Megow and
Vredeveld 2014).
In this note we rule out distribution-independent approximation factors for simple list
scheduling policies in non-preemptive stochastic scheduling. More precisely we consider
so-called index policies that assign the same priority to jobs with the same probability dis-
tribution and schedule jobs one after the other on the first machine that becomes available.
Job-based index policies do not consider the number of jobs or the number of machines.
We give a lower bound of Ω(∆1/4
) for job-based index policies. Somewhat surprisingly
this lower bound is obtained already for very simple instances with only two types of jobs,
identical deterministic jobs and a set of stochastic jobs that all follow the same Bernoulli
distribution. For this class of instances we also give a policy that is an O(m)-approximation.
92

2 Lower bound for index policies
Theorem 1. Any job-based index policy has approximation factor Ω(∆1/4
) for P||E[
P
j Cj ].
To prove this lower bound we consider a simple class of instances that we call Bernoulli-
type instances. This class consists of two types of jobs, deterministic jobs Jd and stochastic
jobs Js, with jobs of each type following the same distribution. A deterministic job j ∈ Jd
has processing time Pj = p, and a stochastic job j ∈ Js has processing time Pj = 0 with
probability q ∈ (0, 1) and Pj = l with probability 1 − q.
Proof. We define two families of Bernoulli-type instances, I1(∆, m) and I2(∆, m), for the
problem P||E[
P
j Cj ] where ∆ is the upper bound on Var[Pj]/E[ Pj ]2
. The instances differ
only in the number of deterministic and stochastic jobs, nd and ns, but not in the processing
time distributions. We define the processing time for deterministic jobs in Jd to be p = 1,
and for stochastic jobs j ∈ Js we define
Pj =
(
0 with probability 1 − 1/∆
∆3/2
with probability 1/∆.
Note that the squared coefficients of variation are bounded from above by ∆.
For such Bernoulli-type instances there are only two job-based index policies, one where
the deterministic jobs have higher priority, denoted by Jd ≺ Js, and one where the stochas-
tic jobs have higher priority, denoted by Js ≺ Jd. We show that for any fixed ∆ 1, there
exists a value of m such that the cost of the schedule produced by Jd ≺ Js on instance
I1(∆, m) is greater by a factor of Ω(∆1/4
) than the cost of the schedule produced by
Js ≺ Jd, and vice versa for instance I2(∆, m). As the instances I1(∆, m) and I2(∆, m) are
indistinguishable to a job-based index policy, this result implies the lower bound.
The First Instance. Instance I1(∆, m) is defined by nd = ∆3/4
m and ns = 1
2 ∆m. We
distinguish both priority orders.
• Jd ≺ Js : When jobs in Jd are scheduled first, then no job in Js starts before nd/m
(assuming w.l.o.g. that nd/m ∈ Z). Thus,
E
X
j∈J
Cj

≥
nd
m
ns =
1
2
∆7/4
m.
• Js ≺ Jd : Let X be a random variable denoting the number of jobs in Js that turn out
to be long. Note that X ∼ Bin(ns, 1/∆) and E[ X ]= m/2. We distinguish two cases.
◦ X 3
4
m : Every stochastic job starts at time 0. Thus, E
h P
j∈Js
Cj | X 3
4 m
i
≤
3
4 ∆3/2
m. Furthermore, at least 1
4 m machines are free for scheduling deterministic
Jobs, Jd, at total cost bounded by E
h P
j∈Jd
Cj | X 3
4 m
i
≤ nd(nd+1)
1
4 m
≤ 8∆3/2
m.
◦ X ≥ 3
4
m : we get a (very crude) upper bound on the expected cost by assuming
all jobs have processing time ∆3/2
and then scheduling them on a single machine:
E
h P
j∈J Cj | X ≥ 3
4 m
i
1
2 (nd + ns)(nd + ns + 1)∆3/2
≤ 3∆7/2
m2
.
To combine both cases and determine the total expected cost, we use the Chernoff-
Hoeffding bound, which gives P

X ≥ 3
4 m

≤ exp(− m
24 ), and we conclude
E
X
j∈J
Cj

≤ P

X
3
4
m

E
X
j∈J
Cj X
3
4
m

+ P

X ≥
3
4
m

E
X
j∈J
Cj X ≥
3
4
m

≤
3
4
∆3/2
m + 8∆3/2
m + exp

−
m
24

· 3∆7/2
m2
= O(∆3/2
m),
for sufficiently large m.
93

Thus, on sufficiently many machines, the index policy Jd ≺ Js has total cost greater by a
factor of Ω(∆1/4
) than the cost of policy Js ≺ Jd.
The Second Instance. Instance I2(∆, m) is defined by nd = ∆5/4
m and ns = 2∆m. Using
similar arguments as in the previous case, we can show that the index policy Js ≺ Jd yields
expected cost that are worse by a factor Ω(∆1/4
) than the cost of policy Jd ≺ Js. ⊓
⊔
3 Upper bound for Bernoulli-type instances
For the class of Bernoulli-type instances introduced above, we show that taking the
number of machines and jobs into account yields an index policy that is O(m)-approximate.
W.l.o.g. let j ∈ Jd have processing time Pj = p, and j ∈ Js have processing time Pj = 0
with probability 1 − 1
l and Pj = l with probability 1
l for l 1. Observe that the cost
caused by individually scheduling Jd or Js starting at time 0 gives a lower bound on the
cost of an optimal policy. We denote these job set-individual scheduling cost by
P
j∈Jt
E[C0
j ]
where t ∈ {s, d}. Obviously, the sum of both also is a lower bound on the optimum cost.
Firstly, note that in case of few deterministic jobs, Js ≺ Jd is an O(1)-approximation.
Lemma 1. Js ≺ Jd is a 2-approximation for Bernoulli-type instances with nd ≤ m.
Proof. The cost of scheduling Js ≺ Jd is at most the cost of Js and the cost of one
deterministic job per machine starting at the completion of the last stochastic job on that
machine. Then, by linearity of expectation,
X
j∈J
E[Cj] =
X
j∈Js
E[C0
j ] +
X
j∈Jd
E[Sj + p] ≤ 2
X
j∈Js
E[C0
j ] + ndp ≤ 2
X
j∈J
E[C∗
j ].

Moreover, if there are less stochastic jobs than deterministic ones, Jd ≺ Js is O(1)-
approximate.
Lemma 2. Jd ≺ Js is a 5-approximation for Bernoulli-type instances with nd m and
ns ≤ 2nd.
Proof. When scheduling in order Jd ≺ Js, machines start processing jobs in Js no later
than
nd
m

p ≤ 2nd
m p, when all jobs in Jd have completed. Thus, the total cost of Js is
X
j∈Js
E[C0
j ] + ns · 2
nd
m
p ≤
X
j∈Js
E[C0
j ] + 4
X
j∈Jd
E[C0
j ] ,
which follows from the well-known deterministic lower bound by Eastman et al. (1964).
Adding the total cost of the deterministic jobs Jd implies the 5-approximation. ⊓
⊔
To handle the remaining instances, recall X, the random variable counting the number
of actual long stochastic jobs. Formally, X :=
P
j∈Js
Xj with Xj := 1{Pj =l} indicating if
j ∈ Js is long. Furthermore, fix a sequence of the stochastic jobs Js and let Πi denote the
position of the ith long job in that sequence.
Lemma 3. For X and Πi defined as before and 1 ≤ i ≤ λm ≤ ns for λ ∈

1, . . . , ⌊ns
m ⌋ ,
the following holds:
(i) E[Πi | X = λm] = i
λm+1 (ns + 1) and E[Πi | λm ≤ X (λ + 1)m] ≤ i
λm+1 (ns + 1).
(ii) E[ns − Πm | m ≤ X 2m] ≥ ns
4m .
Lemma 4. Js ≺ Jd is an O(m)-approximation for Bernoulli-type instances with ns
2nd 2m.
94

Sketch of proof. We analyze the performance of Js ≺ Jd by conditioning on the number X
of long jobs.
• 0 ≤ X m : There is at least one machine available for scheduling the deterministic
jobs. Hence, we loose at most a factor m w.r.t. an optimal solution using at most m
machines.
• λm ≤ X (λ + 1)m for λ ∈

1, . . . , ⌊ns
m
⌋ : All stochastic jobs are finished at the
latest by (λ + 1)l. Beginning at time (λ + 1)l, all machines process deterministic jobs
only. Hence,
X
j∈J
E[Cj | λm ≤ X (λ + 1)m] ≤
X
j∈J
E[C0
j | λm ≤ X (λ + 1)m] + (λ + 1)lnd. (1)
Note that a non-clairvoyant policy does not know the positions of the long jobs. Thus,
such a policy cannot start any of the stochastic jobs coming after the (k · m)th long
one before time k · l for 1 ≤ k ≤ λ. Thus, ns − Πkm stochastic jobs are delayed
by k ·l. For λ = 1, Lemma 3 (ii) implies that scheduling only Js costs at least l ns
4m , i.e.,
P
j∈Js
E[C0
j | m ≤ X 2m] ≥ l ns
4m . For λ ≥ 2, we can show with Lemma 3 (i) that
P
j∈Js
E[C0
j | λm ≤ X (λ + 1)m] ≥ λlns
4 . This bounds the extra term (λ + 1)lnd in
Equation (1) in terms of the optimum cost.
Combining the results for the different values of X, we obtain
X
j∈J
E[Cj] ≤ (8m + 1)
X
j∈J
E[C∗
j ].

The lemmas above imply an O(m)-approximation algorithm based on index policies
taking the number of jobs and machines into account. This result for Bernoulli-type in-
stances can be slightly generalized to arbitrary deterministic jobs, i.e., Pj = pj for j ∈ Jd.
Theorem 2. There exists an O(m)-approximate index policy for Bernoulli-type instances
of P||
P
j E[Cj], where the deterministic jobs may vary in size.
References
Bruno, J.L., P.J. Downey, and G.N. Frederickson, 1981, “Sequencing tasks with exponential service
times to minimize the expected flowtime or makespan, J. ACM, Vol. 28, pp. 100-113.
Eastman, W.L., S. Even, and I.M. Isaacs, 1964, “Bounds for the optimal scheduling of n jobs on
m processors, Management Science, Vol. 11, pp. 268-279.
Im, S., B. Moseley, and K. Pruhs, 2015, “Stochastic Scheduling of Heavy-Tailed Jobs, Proc. of
STACS, Vol. 30, pp. 474-486.
Megow, N., M. Uetz, and T. Vredeveld, 2006, “Models and algorithms for stochastic online schedul-
ing, Math. Oper. Res. Vol. 31.3, pp. 513-525.
Megow, N. and T. Vredeveld, 2014, “A tight 2-approximation or preemptive stochastic scheduling,
2014, Math. Oper. Res. Vol. 39.4, pp. 1297-1310.
Möhring, R.H., A.S. Schulz, and M. Uetz, 1999, “Approximation in stochastic scheduling: the
power of LP-based priority policies, J. ACM Vol. 46, pp. 924-942.
Rothkopf, M.H., 1966, “Scheduling with random service times, Management Science, Vol. 12, pp.
703-713.
Schulz, A.S., 2008 “Stochastic online scheduling, Proc. of COCOA, Vol. 5165, pp. 448-457.
Skutella, M., M. Sviridenko, and M. Uetz, 2016, “Unrelated machine scheduling with stochastic
processing times, Math. Oper. Res., Vol. 41.3, pp. 851-864.
95

Unrelated Parallel Machine Scheduling at a TV
Manufacturer
Merve Burcu Sarkaya, Okan Örsan Özener and Ali Ekici
Department of Industrial Engineering, Ozyegin University, Istanbul, Turkey
burcu.cakiroglu@ozu.edu.tr, orsan.ozener@ozyegin.edu.tr, ali.ekici@ozyegin.edu.tr
Keywords: parallel machine scheduling, unrelated machines, sequence-dependent setups.
1 Introduction
In this study, we analyze the scheduling problem faced by a TV manufacturer. TV
manufacturing is planned based on a make-to-order strategy and mass customization due
to diversied customer demand. The manufacturer utilizes multiple heterogeneous assem-
bly/production lines that are specialized to produce TVs with dierent features. Each
customer order is considered as a separate job, and these jobs are completed on one of the
compatible assembly lines. For a given job, only a subset of assembly lines (called compat-
ible assembly lines) can be used to complete the job, and the total processing time of a
job depends on the assembly line used for that job. A job can only be started after all the
materials (especially cell and cardboard box) are available. Before starting a new job on
an assembly line, a setup time (depending on the previous job processed and the new job
to be processed) is required to make the assembly line ready for production.
Our goal is to determine a production schedule with minimum total tardiness and ear-
liness while considering the job-assembly line compatibility, cell and cardboard box avail-
ability, the sequence-dependent setup times between jobs and the workload balance among
the assembly lines. We propose a sequential heuristic approach to address the problem.
The problem analyzed in this study is a variant of the unrelated parallel machine
scheduling problem which is extensively studied in the literature. Logendran et. al. (2007)
study the unrelated parallel machine scheduling problem with sequence- and machine-
dependent setups and unequal release times for the jobs. They further assume that each
machine has a availability constraint which sets the earliest time a machine can be used
for processing jobs. They look for a minimum weighted tardiness solution. Six dierent
search algorithms based on tabu search are developed to identify the best schedule. Lee
et. al. (2013) also study the unrelated parallel machine setting where jobs have sequence-
and machine-dependent setups. Dierent from Logendran et. al. (2007), they assume that
all the jobs are available at the beginning, and the objective is to minimize total tardi-
ness. The authors propose a tabu search algorithm that incorporates various neighborhood
generation methods. Similarly, Zhu and Heady (2000) and Akyol and Bayhan (2008) con-
sider unrelated parallel machine scheduling problem with sequence-dependent setups and
equal release times. Dierent from the studies above, their objective is to minimize the
total weighted earliness and tardiness. Zhu and Heady (2000) propose a mixed integer pro-
gramming formulation, and Akyol and Bayhan (2008) develop a neural network approach
to address the problem. The main dierences between the above mentioned studies and
the current study are machine-job compatibility restrictions and the workload balance re-
quirement. Finally, Zhang et. al. (2007) consider the unrelated parallel machine setting
with sequence-dependent setup times, unequal release times and machine-job compatibil-
ity restrictions. Their objective is to minimize the total weighted tardiness. They convert
the problem into reinforcement learning problems by constructing a semi-Markov decision
process and then apply the Q-Learning algorithm to nd a solution. Dierent from our
96

setting, they do not consider the workload balance among machines and the earliness in
the objective function.
2 Problem Denition
We have n assembly lines and m jobs to be processed on one of these assembly lines.
We use L (:= {1, 2, . . . , n}) to denote the set of assembly lines and I (:= {1, 2, . . . , m}) to
denote the set of jobs. Job i can only be processed on a subset of assembly lines. We use
Li to denote the set of assembly lines job i can be assigned to and Il to denote the set of
jobs that can be assigned to assembly line l. Processing time (in days) of a job depends
on the assembly line it is assigned to. We denote the processing time of job i on assembly
line l by pil. When job j is processed immediately after job i on the same assembly line,
then a sequence-dependent setup time tij is required to make the assembly line ready for
processing job j. Each job has a certain due date di by which the job has to be nished. Job
i can be started on an assembly line after its release date, and preemption is not allowed.
Moreover, the two critical materials (cells and cardboard boxes) specic to each job have
to be ready before a job can be started. Hence, the earliest time a job can be started is the
maximum of the release time of the job, the available time of the cells and the available
time of the cardboard boxes required for that job. We denote the earliest start time of
job i by ri. Finally, in order to maintain a balance between the workload of the assembly
lines, the manufacturer imposes lower and upper limits on the number of jobs that can be
assigned to an assembly line. We use C1 and C2 to denote these lower and upper limits,
respectively. Our goal is to nd an assignment of the jobs to the assembly lines and the
processing order of the jobs on each assembly line with the objective of minimizing the
total tardiness and earliness.
3 Sequential Heuristic Approach
In the proposed approach, called the Sequential Heuristic Approach (SHA), we decom-
pose the set of decisions to be made into two and make one set of decisions at each stage.
More specically, in the rst phase we assign the jobs to the assembly lines. Then, for each
assembly line we determine processing order of the jobs assigned to it. In each phase, we
make the decisions by solving mathematical models.
In the rst phase, we determine which job is assigned to which assembly line. Our
objective in this phase is to minimize the total processing time of the jobs. We also impose
the lower and upper limits on the number of jobs that can be assigned to an assembly line.
We use the following decision variable:
zil =
{
1, if job i is assigned to assembly line l
0, otherwise.
i ∈ I, l ∈ L
The mathematical model solved in the rst phase is as follows:
MIP-A: Min
∑
l∈L
∑
i∈Il
pilzil (1)
s.t.
∑
l∈Li
zil = 1 ∀i ∈ I (2)
∑
i∈Il
zil ≥ C1 ∀l ∈ L (3)
∑
i∈Il
zil ≤ C2 ∀l ∈ L (4)
97

zil ∈ {0, 1} ∀i ∈ I, l ∈ L (5)
In this model, the objective function minimizes the total processing times of the jobs.
Constraints (2) make sure that each job is assigned to an assembly line. Constraints (3)
and (4) are the lower and upper limits on the number of jobs that can be assigned to an
assembly line. Constraints (5) are the sign restrictions. By solving this model, we determine
a feasible assignment of jobs to the assembly lines. Then, in the second phase we decide
on the order jobs are processed on each assembly line. Let Al be the set of jobs assigned
to assembly line l. For assembly line l, we dene the following decision variables:
yik =
{
1, if job i is processed at the kth order
0, otherwise.
i ∈ Al, k ∈ {1, 2, . . . , |Al|}
xij =
{
1, if job i is the immediate predecessor of job j
0, otherwise.
i, j ∈ Al
si = start time of job i i ∈ Al
fi = completion time of job i i ∈ Al
ui = amount of tardiness for job i i ∈ Al
ei = amount of earliness for job i i ∈ Al
We determine the order of jobs for assembly line l by solving the following model:
MIP-S: Min
∑
i∈Al
(ui + ei) (6)
s.t.
∑
k∈{1,2,...,|Al|}
yik = 1 ∀i ∈ Al (7)
∑
i∈Al
yik ≤ 1 ∀k ∈ {1, 2, . . . , |Al|} (8)
yjk + yi,k−1 − xij ≤ 1 ∀i, j ∈ Al, k ∈ {2, . . . , |Al|} (9)
sj − fi + M(1 − xij) − tijxij ≥ 0 ∀i, j ∈ Al (10)
si ≥ ri ∀i ∈ Al (11)
fi − si − pil ≥ 0 ∀i ∈ Al (12)
fi − ui ≤ di ∀i ∈ Al (13)
ei + fi ≥ di ∀i ∈ Al (14)
yik ∈ {0, 1} ∀i ∈ Al, k ∈ {1, 2, . . . , |Al|} (15)
xij ∈ {0, 1} ∀i, j ∈ Al (16)
si, fi, ui, ei ≥ 0 ∀i ∈ Al (17)
In this model, the objective is to minimize the total tardiness and earliness of the jobs.
Constraints (7) make sure that each job is assigned to one of the assembly lines. Constraints
(8) guarantee that no two jobs can be assigned to the same order of an assembly line.
Constraints (9)-(10) enforce the setup times between consecutive jobs. Constraints (11)
impose the earliest start time restriction. The completion time of a job is determined by
Constraints (12). Constraints (13)-(14) determine the tardiness and earliness of each job.
Finally, Constraints (15-17) impose the nonnegativity and binary restrictions.
4 Computational Results
We test the eectiveness of the proposed solution approach on real-life instances. In the
real-life instances, we have 15 assembly lines dedicated for TV manufacturing and 150 jobs
98

to be processed on one of these assembly lines. Processing times of the jobs (depending on
the assembly line used) vary between 12 minutes and 3 days. In terms of assembly line-job
compatibility, depending on the type of the job, it can be processed on 1 up to 14 assembly
lines. On average, a job can be processed on around 8 out of 15 assembly lines. Finally,
in order to balance the workload between the assembly lines the minimum and maximum
number of jobs that can be assigned to an assembly line are set to 2 and 13, respectively.
Currently, the manufacturer uses an advanced planning and scheduling module inte-
grated with Enterprise Resource Planning (ERP) used at the company. After taking orders
from ERP software, this module provides a visual display of the orders, release dates, due
dates, etc. Then, the user assigns the jobs to the lines manually considering the setup times
and earliness and tardiness. Experience of the user is signicantly important in the current
practice.
We test the proposed approach on these real life instances and compare the solutions
found against the current practice in Table 1. In this table, under SHA column we present
the percentage improvements in the total tardiness and earliness taking the solution found
in the current practice as the reference point. We observe that the Sequential Heuristic Ap-
proach (SHA) provides signicant improvements over the current practice. Total tardiness
and earliness is decreased by 77.89% on average.
Table 1. Comparison between the solutions found by SHA and the current practice
Instance SHA
1 98.27%
2 55.88%
3 91.70%
4 83.39%
5 65.75%
6 75.45%
7 77.95%
8 71.92%
9 78.37%
10 80.25%
Average 77.89%
References
Akyol D.E., G.M. Bayhan, 2008, Multi-machine earliness and tardiness scheduling problem: an
interconnected neural network approach, International Journal of Advanced Manufacturing
Technology, Vol. 37, pp. 576-588.
Lee J.-H., J.-M. Yu, D.-H. Lee, 2013, A tabu search algorithm for unrelated parallel machine
scheduling with sequence- and machine-dependent setups: minimizing total tardiness, Inter-
national Journal of Advanced Manufacturing Technology, Vol. 69, pp. 2081-2089.
Logendran R., B. McDonell, B. Smucker, 2007, Scheduling unrelated parallel machines with
sequence-dependent setups, Computers Operations Research, Vol. 34, pp. 420-438.
Zhang Z., L. Zheng, M.X. Weng, 2007, Dynamic parallel machine scheduling with mean weighted
tardiness objective by Q-Learning, International Journal of Advanced Manufacturing Tech-
nology, Vol. 34, pp. 968-980.
Zhu Z., R.B. Heady, 2000, Minimizing the sum of earliness/tardiness in multi-machine scheduling:
a mixed integer programming approach, Computers Industrial Engineering, Vol. 38, pp.
297-305.
99

A new set of benchmark instances for the Multi-Mode
Resource Investment Problem
Patrick Gerhards1
Helmut Schmidt University Hamburg, Germany
patrick.gerhards@hsu-hh.de
Keywords: Project Scheduling, Multi-Mode Resource Investment Problem, Benchmark
Instances
1 Introduction
In this paper we introduce a new set of benchmark instances for the multi-mode re-
source investment problem (MRIP). The MRIP is a project scheduling problem which has
many practical applications such as construction projects or software development. It is
an extension of the resource investment problem (RIP) also known as the resource avail-
ability cost problem (RACP) where the duration and resource requests of the activities
are fixed and no mode choice is available (Möhring 1984). The goal is to find a sched-
ule minimizing the resource costs while maintaining precedence and resource constraints
as well as adhering to a given deadline. It shares some similarities with the multi-mode
resource-constrained project scheduling problem (MRCPSP) where the available resources
are fixed and the shortest possible makespan is to be determined.
Most of the existing work in the literature tackled the single-mode variant of the prob-
lem (RIP). For a good overview on heuristic and exact procedures we refer to Van Peteghem
and Vanhoucke (2015) and Rodrigues and Yamashita (2015), respectively. For the MRIP,
various heuristic approaches have been provided to tackle the problem. The problem was
introduced by Hsu and Kim (2005) who developed a heuristic that combines two pri-
ority rules to schedule the activities. In Qi et al. (2015) apply modified particle swarm
optimization to the MRIP. Both use problem instances from the PSPLIB (Kolisch and
Sprecher 1997) which were originally designed for the MRCPSP and adapt them to get
MRIP instances.
When considering the single mode case of the problem, i.e. the RIP, most of the exist-
ing work uses benchmark instances for the resource-constrained project scheduling problem
(RCSPSP) such as the PSPLIB. For problems with only one mode it works just fine to
adapt those RCPSP instances but when the multi-mode case is considered adapting MR-
CPSP instances has a major shortcoming: when the due date is set too small it can occur
that many modes of the activities become not executable (further explained in Section
3). Hence, the instances lose some of their complexity since these modes can be omitted
with simple preprocessing techniques. Another reason for proposing a benchmark dataset
for the MRIP is that all of existing approaches use different problem instances in their
computational studies which makes a comparison hard. Hence, we propose a new set of
benchmark instances such that future contributions to this problem can be easily compared
to one another (available at https://riplib.hsu-hh.de).
The MRIP is defined by the following properties: A set of nonpreemptable activities
A = {0, ..., n + 1}, precedence constraints E, a set R of renewable resources and a set Rn
of nonrenewable resources. For each activity i there is a set Mi of modes that can be chosen
100

for the execution of activity i. If mode m ∈ Mi is chosen, activity i has duration di,m ∈ Z+
and it has a resource consumption ri,m,k ∈ Z+
for each resource k ∈ R ∪ Rn
. A due date
D ∈ Z+
for the makespan of the project is given. For each resource k ∈ R∪Rn
the available
capacity of the resource has to be chosen and resource cost factors ck ∈ Z+
are given. The
objective is to find a precedence and resource feasible schedule that minimizes the sum of
resource costs.
min
X
k∈R∪Rn
ck · ak (1)
s.t.
X
m∈Mi
LSi
X
t=ESi
xi,m,t = 1 ∀i ∈ A (2)
X
m∈Mi
LSi
X
t=ESi
xi,m,t · (t + di,m) ≤
X
m∈Mj
LSj
X
t=ESj
xj,m,t · t ∀(i, j) ∈ E (3)
X
i∈A
X
m∈Mi
LSi
X
t=ESi
xi,m,t · ri,m,k ≤ ak ∀k ∈ Rn
(4)
X
i∈A
X
m∈Mi
min(t,LSi)
X
q=max(ESi,t−di,m+1)
xi,m,q · ri,m,k ≤ ak ∀k ∈ R, ∀t ∈ T (5)
ak ≥ 0 ∀k ∈ R ∪ Rn
(6)
xi,m,t ∈ {0, 1} ∀i ∈ A, ∀m ∈ Mi, t = ESi, . . . , LSi (7)
The mathematical model presented in (1)–(7) is an adaptation of a model for the
MRCPSP proposed by Talbot (1982). We define binary decision variables xi,m,t which are
set to 1 if and only if activity i starts in mode m in period t (see (7)) and real-valued
decision variables ak which represent the available capacity of resource k (see (6)). For
each activity i we calculate a lower bound ESi and an upper bound LSi for its possible
starting period using forward and backward calculation (FBC) (Kelley 1963).
The objective function (1) minimizes the sum resource costs. Equation (2) makes sure
that for every activity i exactly one mode and one starting time is assigned. With con-
straint (3) we ensure the precedence constraints. Constraints (4) and (5) model the non-
renewable and renewable resource requirements, respectively. The renewable resource can
represent machines or workers as their available amount replenishes every time period. We
also consider nonrenewable resources. They are a powerful tool for the decision maker to
model the budget of the project or the use of external work force.
3 Instance Generation
We group the benchmark instances in three datasets with instances sharing the same
number of activities, namely MRIP30, MRIP50 and MRIP100. The generated instances
have the following characteristics: number of activities |A|, number of modes per activity
|M|, number of renewable resources |R|, due date factor θ, order strength OS and resource
factor RF. Here, order strength measures the fraction of precedence relations in E com-
pared to the total number of possible relations and, hence, is an indicator if the precedence
structure of the project is more parallel or more serial (Mastor 1970). The resource factor
value is the average of how many resources are actually consumed for every mode of all the
activities. Table 1 displays the values that are used. For every parameter combination we
101

Parameter Values
|A| {30, 50, 100}
|M| {3, 6, 9}
|R| {2, 4, 8}
θ {1.2, 1.4, 1.6, 1.8, 2}
OS {0.25, 0.5, 0.75}
RF {0.5, 1}
generated 5 instances, giving us in total a number of 4,050 instances. As done in the liter-
ature, the parameter θ is used to compute the due date of the project as in the following
equation (activity n + 1 is the dummy end activity that marks the end of the project and
has a duration of 0):
D = Round(θ · ESTn+1) (8)
For smaller values of θ many modes can be infeasible. That means that their earliest
finish time (earliest start plus duration of the mode) is larger than their latest finish time
(w.r.t to the latest start of their successors in order to not violate the due date constraint).
This can happen when the due date is relatively small compared to the earliest start time
of the dummy end activity end the fact that the minimal durations of the activities are
used when calculating the earliest and latest start times with FBC. When, for example, we
use θ = 1 then the durations of modes can not differ for activities on the critical path or
all modes with a duration higher than the minimum duration are infeasible (w.r.t to the
due date constraint). Hence, we use only values greater or equal than 1.2 for θ and apply
a repair mechanism when infeasible modes are encountered.
For every instance we have only one nonrenewable resource since it can be shown that an
instance with multiple nonrenewable resources can be transformed in polynomial time into
an instance with just one nonrenewable resource. An optimal solution for the transformed
instance can be translated into a feasible optimal solution of the original instance and vice
versa (the concept of a polynomial-time reduction will be given in the presentation due to
space limitations).
Next, we describe how we actually computed an instance with the desired properties. We
used the network generator RanGen (Demeulemeester et al. 2003) to generate an activity-
on-the-node network with the desired number of activities and the desired order strength
value. Next, we draw for every activity i and all its modes m ∈ Mi the duration di,m as
a discrete uniform distributed random number U{1, 10}. The resource requirements ri,m,k
for every resource k ∈ R∪Rn
are also drawn from U{1, 10}. If the value of RF = 0.5, then
we set arbitrarily half of the renewable resource requirements of each mode to 0. After all
the resource requirements and the duration for an activity is determined, we check if there
are dominated modes. A mode is dominated if there is another mode with shorter or equal
duration and lower or equal resource requirements for all resources. If a dominated mode
occurs, the duration and resource requirement values of the dominated mode as well as
the other mode that is responsible for the domination get redrawn. This is repeated until
each activity has no dominated modes. Then, we calculate the earliest start times (EST)
with the forward pass technique and the due date D of the project as in (8). With D as
an upper bound for the completion of the project we can use a backward pass to compute
latest finish times (LFT) for every activity. We use the EST and LFT to check for infeasible
modes. A mode m of activity i is infeasible if the following inequality does not hold:
ESTi + di,m ≤ LFTi (9)
102

If an infeasible mode is encountered, the values for this mode get redrawn. Since the minimal
durations can change, we compute EST, D and LFT again and repeat this procedure until
no mode is infeasible.
We choose to set the cost factors ck to be 1 for all resources in this benchmark set.
Setting them to another random number or multiplying the resource requirements for the
respective resource by that random number would basically result in the same outcome.
In this benchmark set we get the randomness for the resource allocation by the resource
consumption and the duration of the modes. For future work it could be interesting to
analyse different cost structures or distributions (e.g., cheap resource types versus expensive
resource types which are also considered in the design of the modes).
4 Computational Experiments
We tested the new instances with a relatively simple iterated local search (ILS) and
implemented the mathematical model displayed in (1)-(7) as a integer program (IP) in
Gurobi. Results are presented at the conference due to space limitations but show that the
proposed instances are quite challenging and need further investigation by means of more
advanced metaheuristic procedures.
5 Conclusion
In this work we argue why benchmark datasets for the multi-mode resource investment
problem are needed and which specific features need to be considered regarding the multi-
mode case. We introduce a procedure to obtain instances with no dominated or infeasible
modes and provide those instances such that future research is easier to compare. First
experiments show that the instances at hand are challenging and need further investigation
by exact and heuristic approaches.
References
Demeulemeester, E., M. Vanhoucke, W. Herroelen, “RanGen: A random network generator for
activity-on-the-node networks”, Journal of Scheduling, Vol 6, No. 1, pp. 17-38.
Hsu, C. C., D. S. Kim, 2005, “A new heuristic for the multi-mode resource investment problem”,
Journal of the Operational Research Society, Vol. 56 No. 4, pp. 406-413.
Kelley, J. E., 1963, “The critical-path method: Resources planning and scheduling”, Industrial
Scheduling, Vol. 13, no. 1, pp. 347-365.
Kolisch, R., A. Sprecher, 1997, “PSPLIB - A project scheduling problem library”, European Journal
of Operational Research, Vol. 96, No. 1, pp. 205-216.
Mastor, A., 1970, “An experimental and comparative evaluation of production line balancing
techniques”, Management Science, Vol. 16, No. 11, pp. 728-746.
Möhring, R. H., 1984, “Minimizing costs of resource requirements in project networks subject to
a fixed completion time”, Operations Research, Vol. 32, No. 1, pp. 89-120.
Rodrigues, S. B., D. S. Yamashita, “Exact methods for the resource availability cost problem”,
In: Handbook on Project Management and Scheduling, editors C. Schwindt, J. Zimmermann,
Vol. 1, pp. 319-338, Springer International Publishing, Cham.
Talbot, F. B., 1982, “Resource-constrained project scheduling with time-resource tradeoffs: The
nonpreemptive case”, Management Science, Vol. 28, No. 10, pp. 1197-1210.
Van Peteghem, V., Mario Vanhoucke, “Heuristic methods for the resource availability cost prob-
lem”, In: Handbook on Project Management and Scheduling, editors C. Schwindt, J. Zimmer-
mann, Vol. 1, pp. 339-359, Springer International Publishing, Cham.
Qi, J. J., Y. J. Liu, P. Jiang, B. Guo, 2015, “Schedule generation scheme for solving multi-mode re-
source availability cost problem by modified particle swarm optimization”, Journal of Schedul-
ing, Vol. 18, No. 3 pp. 285-298.
103

A simheuristic for stochastic permutation flow shop
problem considering quantitative and qualitative
decision criteria
Eliana Maria González-Neira1,2
, Jairo R. Montoya-Torres2
1
Departamento de Ingeniería Industrial, Pontificia Universidad Javeriana, Colombia
eliana.gonzalez@javeriana.edu.co
2
Facultad de Ingeniería, Universidad de La Sabana, Colombia
elianagone@unisabana.edu.co, jairo.montoya@unisabana.edu.co
Keywords: stochastic permutation flow shop, robustness, earliness, tardiness, qualitative
criteria.
1 Introduction
The flow shop problem (FSP) has been one of the most well studied problems in schedul-
ing literature (Vallada et al., 2015). Nevertheless, less research has been conducted for the
stochastic case (González-Neira et al., 2017; Gourgand et al., 2000). Li and Ierapetritou
(2008) have mentioned that the fact of designing systematic ways to take into account
stochasticity is as important as the model itself.
Regarding the objective function, four aspects must be mentioned: two related with
quantitative decision criteria, one with qualitative decision criteria and one with robust-
ness of the solutions obtained. Firstly, the makespan for the deterministic case, and the
expected makespan for the stochastic counterpart, have been the most studied measures
(Gourgand et al., 2000; Vallada et al., 2015). However, other criteria that consider due date
related measures are key objectives in today competitive markets, as they may be measures
of customer service level. Moreover, the adoption of just-in-time (JIT) measures such as
earliness/tardiness has been object of interest in the past two decades, because earliness
may cause obsolescence, more inventory holding costs, requirement for more storage space
(Chandra et al., 2009) among others.
Secondly, the fact of consider various objectives simultaneously is natural in real-life
problems (Yenisey and Yagmahan, 2014). Still, most of the literature considers only single
objective problems.
Thirdly, in the scheduling literature, almost all researches, with very few exceptions,
have considered only quantitative decision criteria but, as in other optimization problems,
qualitative criteria are important and can reduce the gap between theory and practice.
Chang and Lo (2001) and Chang et al. (2008) studied a multi-criteria job shop in which
strategic importance of customers was considered as qualitative criteria. The former hy-
bridized a genetic algorithm, tabu search, analytic hierarchy process (AHP) and fuzzy
theory to solve the problem. The later used a hybridization of ant colony algorithm and
AHP. González-Neira et al. (2016) minimized expected costs of tardiness as quantitative
criteria and strategic customer importance as qualitative criteria in a stochastic hybrid
FSP. This last study employed a method based on stochastic multicriteria acceptability
analysis, hybridized with a GRASP and a Monte Carlo simulation to deal with both type
of criteria.
Fourthly, research on scheduling under uncertainties has taken mainly two approaches:
the stochastic approach, in which parameters are modelled with probability distributions
with the goal of minimizing the expected value of a selected measure, and the robust
104

approach, in which uncertain parameters are modelled with intervals and the schedule
obtained is more stable and suffer less variations under uncertainty. Nonetheless, the com-
bination of both approaches has not been addressed. It is known that if an enterprise
collects all data of their production, in short time, it may have sufficient data to estimate
accurately distribution probability of uncertain parameters. By having this probability dis-
tribution, the robust schedule obtained can be more adjusted than other schedules in which
uncertainties are modelled with intervals.
To the best of our knowledge, there is not a work that includes simultaneously the
analysis of a JIT environment with stochastic parameters, and quantitative and qualita-
tive criteria to obtain robust solutions. Hence, the current work proposes a multicriteria
optimization approach to solve a stochastic PFSP that includes both, quantitative and
qualitative decision criteria. As quantitative objectives, the expected earliness/tardiness
E[E/T] and the standard deviation of earliness/tardiness SD(E/T) are addressed; the lat-
ter to obtain more robust schedules. As qualitative measure, the expected fulfilment of
customer importance (E[CI]) of jobs, that gives priority to the most important jobs for the
company, is considered.
2 Proposed solution approach
The proposed methodology consists of a simheuristic that integrates Monte Carlo simu-
lation into an GRASP metaheuristic (Resende and Ribeiro, 2010), hybridized with pareto
archive strategic evolution algorithm (PAES) (Knowles and Corne, 2000) to deal with
multiple objectives. Additionally, the AHP methodology is integrated to qualify all Pareto
solutions under different weight vectors for the selected criteria.
Special variations of GRASP have been proposed to solve multi-objective problems.
Those are combinations of pure and combined strategies for both, construction and local
search phases (Martí et al. 2015). Pure strategies are those in which only one objective
function guides each construction and the entire local search. In this paper, a GRASP
with a pure strategy for construction stage is used. Local search does not need a strategy
because it integrates the PAES algorithm to construct the Pareto Archive.
Two greedy functions were considered for the construction phase. EDD rule to deal
with earliness/tardiness objective, and a penalization assigned to each job depending on
its customer importance and position in the sequence, to deal with qualitative objective (see
Table 1). The reason for penalizing the accomplishment of customer importance in relation
with the position in the sequence, is because it is desired that a job of a very strategic
and important customer for the company be processed in the first positions rather than
in the final positions of the sequence. Likewise, it is undesirable to schedule a job of a not
very much important customer in the first positions, because it would be stolen a position
that should be taken by a job of a customer of greater importance. Obviously if the job is
not tardy it doesn’t matter which position of the sequence it occupies. Considering these
aspects, the penalization scores were defined with the following criteria: i) a job that is
not tardy has a score of zero; ii) if a job is tardy its penalization is greater if the customer
importance is high, and lower if its customer importance is low; iii) a job penalization
increases if job is taking the place of a job that has greater or lower customer importance.
Table 1 presents an example for an instance of 10 jobs. For our experiments we supposed
that there are 5 degrees of customer importance, where 1 is assigned to the most important
clients and 5 to the worst ones. For the instances tested in this project, a random assignment
of the customer importance for each job was done following the probabilities indicated in
Table 2. Of course this scale from 1 to 5 for importance customer, the and probability of the
importance level were established just for the purpose of testing the methodology. In real
105

cases, the assignment of customer importance will not be probabilistic but deterministic
according with the decision maker.
Table 1. Penalizations for position in the sequence depending on customer importance
Job 5 4 10 1 3 7 2 6 9 8
/ Customer importance 1 2 2 2 3 3 4 4 4 5
Job position in sequence 1 1 5 5 5 7 7 7 7 7 5
2 6 1 1 1 4 4 5 5 5 4
3 6 1 1 1 4 4 5 5 5 4
4 6 1 1 1 4 4 5 5 5 4
5 11 5 5 5 1 1 3 3 3 3
6 11 5 5 5 1 1 3 3 3 3
7 16 9 9 9 4 4 1 1 1 2
8 16 9 9 9 4 4 1 1 1 2
9 16 9 9 9 4 4 1 1 1 2
10 21 13 13 13 7 7 3 3 3 1
Table 2. Probabilities of customer importance occurrence
Customer importance 1 2 3 4 5
Probability of occurrence 8% 12% 20% 28% 32%
The main idea of the construction procedure is alternating the two different greedy
functions at each iteration of GRASP. Therefore, suppose that the procedure begins with
EDD for the first iteration. Next it uses customer importance for iteration 2, and repeats
EDD for iteration 3, and so on. The RCL set is defined as the subset of jobs for which greedy
function values are in the first 10% of the total range of greedy function values. Then, a
job is randomly selected from RCL to form part of the partial solution. The procedure
continues until all jobs have been scheduled and then, the local search begins. Local search
phase consists of 2-optimal interchanges between jobs.
To deal with the stochastic nature of the problem, a Monte-Carlo Simulation is embed-
ded into GRASP. Each sequence obtained in both, construction and local search phases,
is simulated with the required number of runs to give an accurate confidence interval of at
least ±1% around each of the three objective functions, following the procedure proposed
by Framinan and Perez-Gonzalez (2015).
Once these three measures are obtained for each solution, the solution is evaluated to
decide if it should enter in the Pareto Archive or not. If it enters, the other solutions already
saved in the Pareto Archive are evaluated to determine if they remain in the Archive or
not. If the solution does not enter in the Pareto Archive, it is discarded. This is done
according PAES method. A GRASP iteration ends when no interchanges can enter to the
Pareto Archive and then, a new iteration begins. The simheuristic stop time is established
as: number of jobs × number of machines × 1s. Once each Pareto frontier is obtained, we
scored all Pareto sequences with the usage of AHP methodology. We used six different
vectors of criteria weights (Table 3) for the three selected measures. These criteria weights
resulted from an AHP qualification process in which we scored an objective function versus
another, in the scale from 1 to 9, as indicated by AHP procedure. One example of a vector
106

of criteria weights is shown Table 4. Then, from each weight vector we could select the best
solution among the Pareto frontier solutions. In order to compute the matrix of option
scores, for each pair of sequences s1 and s2, we divided the expected earliness/tardiness of s1
by the expected earliness/tardiness of s2, so if the division was 1 the earliness/tardiness
of s1 was worse than the earliness/tardiness of s2 and vice versa. Similar divisions were
done for the other two objective functions (standard deviation of earliness/tardiness and
customer importance).
Table 3. Vectors of criteria weights used for qualification of Pareto Solutions
Weights vector
Objective Function 1 2 3 4 5 6
E[E/T] 66.67% 22.22% 66.67% 22.22% 11.11% 11.11%
SD(E/T) 22.22% 66.67% 11.11% 11.11% 66.67% 22.22%
CI 11.11% 11.11% 22.22% 66.67% 22.22% 66.67%
Table 4. Example of priority vector
AHP qualification
Objective E[E/T] SD(E/T) CI Resultant
Function weight vector
E[E/T] 1 3 6 66.67%
SD(E/T) 1/3 1 2 22.22%
CI 1/6 1/2 1 11.11%
3 Analysis of Results
Two probability distributions and two coefficients of variation were selected to model
both, the stochastic processing and setup times. The first 60 Taillard’ benchmark instances
were taken to test the methodology; this corresponds to 960 Pareto frontiers. With the
application of the AHP method, we selected the best solution for each one of the 6 different
vectors of criteria weights, from each Pareto frontier. That means a total of 5760 solutions,
each of which exhibits an AHP score and a value for the three objectives. Three ANOVAs
were executed to analyse jointly the effect of seven factors in the three selected objective
functions (E[E/T], SD(E/T) and E[CI]). The factors and their levels were: probability
distribution of processing times (PDPT) (lognormal -lgn- and uniform -unf-), coefficient
of variation of processing times (CVPT) (0.25 and 0.50), probability distribution of setup
times (PDST) (lgn and unf), coefficient of variation of setup times (CVST) (lgn and unf),
vectors of criteria weights of AHP (WV) (1 to 6), number of jobs (20 and 50) and number
of machines (5, 10 and 20).
According to the results, all main effects are statistically significant in the three mea-
sures, and at least for one objective function the double interaction effects are also signifi-
cant (P-values 0.05). The Main effects plots can be seen in Figure 1. It shows that the
WV discriminates the Pareto solutions, facilitating to the decision maker the selection of
a solution from the Pareto Frontier. Also, it can be seen that for E[E/T] and SD[E/T],
107

the coefficients of variation of both, setup and processing times, affect the response sub-
stantially by incrementing the three objectives as the coefficients of variation increase. The
same happens with E[CI] but not in the same degree. Additionally, the measures tend to
be greater for lognormal probability distribution than for the uniform distribution. This
shows the importance of making an accurate fitting of probability distribution to obtain
adjusted robust measures.
Future work could be directed to analyze another probability distributions and coef-
ficient of variations. In fact, it should be evaluated the case when the processing time
probability distribution of each job has a different variation coefficient, which is normal in
real cases. Finally, another qualitative criteria should be incorporated in the analysis.
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References
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González-Neira E.M., R.G. García-Cáceres, J.P. Caballero-Villalobos, L.P. Molina-Sánchez, J.R.
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109

An Algorithm for Schedule Delay Analysis
Guida P.L.1
, Sacco G.2
1
PMS Project Management Systems, Rome, Italy
pl.guida@alice.it
2
Engineering Ingegneria Informatica, Rome, Italy
giovanni.sacco@eng.it
Keywords: project management, schedule, critical path method, delay analysis, claim
management.
1 Schedule Delay Analysis
The analysis of the schedule delays is a permanent problem of practical application in
project management. From delays can depend the final outcome and success of projects,
particularly where large sums are at stake and time and cost are very sensitive variables,
often dependent on each other. In client-supplier or owner-contractor relations, the schedule
delays usually represent a very important issue, often undermining the commercial outcome
of the whole project. In fact delay claims are a very well-known issue which is often to
be managed in order to settle lengthy negotiations and even juridical cases, tracing to
responsibilities and monetary compensations.
The present work tackles the problem of schedule delays, linking to literature already
present on the topic, and contributing to solutions already published. We finally imple-
ment an algorithmic method, referenced to as “float banking” which can be of interest to
practitioners and stakeholders in the field of project management.
2 Methods and literature review
Literature on schedule delay analysis is plenty of attempts to produce rational ways to
cope with the problem of sharing delay responsibilities between the project actors, namely
Owner and Contractor(s). Literature on the subject has become to appear in the 90s – e.g.
Alkass (1996) – and is still flourishing.
A conventional way to allocate delaying events encountered on a project is to classify
them according to their origin by the responsible party or event – either Owner, Contractor
or Force Majeure – and whether the same events are excusable, compensable or not; which
also should take into account the so-called “snow ball” effect of delay perturbations. In
summary one finds basic types of delays so defined:
EC: owner-caused excusable compensable
EN: owner-caused excusable but not compensable
NN: contractor-caused neither excusable nor compensable.
In particular such effects and originating events can be the subject of extensive treat-
ment in contractual clauses as well as their “correct” analysis and allocation give often rise
to hard claims and juridical cases between Owner and Contractor, due to their financial and
contractual impacts. The subject among others has originated a specific discipline known
as Forensic Schedule Delay, where the so-called Schedule Delay Analysis (SDA) plays a
major role, in the hands of scheduling (e.g. critical path analysis) arbitration and project
management experts, e.g. Pickavance (2010).
In this framework a number of delay analysis methods have been proposed, such as:
110

- As-planned vs as-built
- Impacted as-planned
- As-planned But for
- Collapsed as built,
- “Windows” analysis, and
- Time Impact Analysis
for which the reader is invited to refer to respective literature, e.g. Davison and Mullen
(2009), Keane and Caletka (2015). Furthermore professional associations like Society of
Construction Law (SCL, 2002) and Association for the Advancement of Cost Engineering
International (AACEI, 2007) have published seminal references on the subject. For the
present discussion we particularly refer to the paper by Braimah (2013), which also provided
us the case study here developed. The topic remains relatively complex to tackle when
confronted with real case problems, though foundation theory has progressed and can be
helpful to assist claimants and defendants in the courts. Among more specific issues arising
on this subject one may recall delay concurrency, float ownership, acceleration and “pacing”
(i.e. the slowing down work activities dependent on another party’s lateness).
The most relevant approaches of analysing schedule delays appear the last two in the
above cited list – Windows and Time Impact analysis – which have most inspired this
paper. The aim is to improve methods which can be used in real-time and are efficient and
convincing in providing logical solutions, also aligned to legal practices.
2.1 Time Impact Analysis
Time Impact Analysis (TIA) method applies re-scheduling at each specific delay or
delaying event, the schedule being updated to a possibly new completion date, including a
new or more critical paths. A picture of the project is developed each time it experiences
a disturbing event, imputing delay responsibilities as soon as they occur, which can also
steer management to undertake timely control actions.
In traditional literature authors report that the method may not be practical due to the
large number of delay events/causes and the laborious re-planning work required. However
the writing authors believe that the technique is now useable thanks to modern scheduling
tools and project management architectures which are taking place in the construction
field applications and yard control offices, where relatively complex projects should not be
dispensed any more. Moreover real-time delay assessment can consolidate project informa-
tion and respective performance either by Owner, Contractor or impacts caused by Force
Majeure (so-called acts of God).
Finally TIA can become standard method for the problem in question, its “algorithmic”
results being less prone to questioning vs. other methods that can provide more approxi-
mate solutions and different results on the same problem, as exemplified in Braimah (2013).
In particular when a TIA analysis is performed following a delaying event, this can im-
pact a large number, theoretically all other project activities, changing their floats and/or
determining a new critical path to the forecasted completion date. In this application we
investigate the effects of changing the single float values of other activities, while tracking
responsibilities of the parties concerned – Owner, Contractor and Force Majeure – and
improving the attention so far dedicated to float management by previous literature.
In our conceptual model an activity float is like an economic reserve or resource which
can be impacted by another activity behaviour, up to being nullified or forced to become
negative, where a project delay is to occur1
. To this aim a model from economics is bor-
rowed. Assuming each activity has an elementary account, where the float plays like reserve
1
Assessment of negative floats and trends is a customary way to analyse schedule delays phe-
nomena, as discussed for instance by Keane and Caletka (2015).
111

funds, these may be decreased or increased by other activities behaviors, representing cred-
its and debts, alike in accounting practice. In particular assuming that floats are owned by
the responsible party, Owner or Contractor, any values changed by the “same” party are
of no charge, while e.g. a decreased float by the other party can represent a future credit.
Total time budget is eventually synthesized on the project account, say completion date or
total delay.
2.2 Proposed algorithm and method
The management of floats for each activity means recording of their evolution during
the project and re-assigning pro-quota their reduction/gain to the respective party. In par-
ticular during the execution of a project, float possessed by an activity may increase or
decrease due to other activities behaviour or external events. For example, some activ-
ity may become critical, so being penalized in future progress without having any direct
responsibility.
The present implementation manages this accounting by introducing an appropriate
data structure, defined as float bank, which is updated during the project dynamics and
trace delays back to their original causes and responsible parties. Any time an activity
duration and hence its float is changed, float banking updates the relevant information,
such as event causing the float change and its responsible party. Besides zeroing, floats can
increase, decrease and become negative. Moreover, following each event, one updates the
history of all floats and activities concerned, can trace the changes of the critical path(s)
and record whether the activities become critical or hypercritical (negative total float),
with additional project delay.
In practice one can evaluate the impacts due to: - Owner (e.g. impoverishing his or some
contractor safety margins); - Contractor (e.g. reducing project efficiency and escalating
costs or liquidated damages); - Force majeure, with no direct responsibility on the project
performance, but accepted as act of God. Therefore at any moment one can have an account
of all integral float values and originating causes, like a bank statement.
2.3 Implementation method
The general logic of the method here implemented can be outlined as follows:
From project start
For each detected delaying event in chronological order:
compute the impact on the activity and all other activities possibly impacted.
determine the event responsibility (Owner, Contractor, FM) of all changes.
These steps apply the CPM scheduling algorithm, recomputing the critical path and
updating the float bank.
Make available the new information to project and contractual management rules.
(Recycle for new delay event until the project end).
The specific algorithmic procedures and required data administration cannot be fully
described here due to space limitation. A more complete paper will be made available on-
line [see ScheDA in References] and is planned to be submitted to a project management
journal.
As already mentioned we only report here the results obtained with the case study by
Braimah (2013) while other cases from other literature on the subject have also positively
been tested for validation.
112

3 Case study
From the referenced case study, where a planning network of 12 activities is defined
and 10 delaying events of various timing impact are injected during the project course,
one obtains the results in Table.1. In this exercise the project original duration of 40 days
was delayed by 11 days, with delays justified (EC, NN) according to classification already
reported.
Table 1. Summary of delay analysis results for the case study according to different methodologies
Delay analysis methodology Delay
EC NN
As-planned vs As Built 9 2
Impacted As-planned 6 8
As-planned But for:
a) Contractor’s point of view 4 7
b) Owner’s point of view 9 2
Collapsed As-built 6 5
Window Analysis 7 4
Time Impact Analysis 6 5
One can see the summary of delay responsibility allocation, shared between EC and NN,
as produced by the different methods, taken from the cited reference, Braimah (2013), with
the additional and last row (bolded italics) obtained by our application. As already said,
different and more heuristics based methods may provide different results; in particular
one method (Impacted as Planned) gives a total delay greater than the actual one.
Besides aligning to the various approaches present in the literature, the method here
developed provides additional focus on the float dynamics and float management modelling,
which, according to our knowledge, is not so explicitly developed in previous papers. The
more recent paper on the subject appears to be Yang and Kao (2012) whose algorithmic
mechanics is however different from ours. Here and by previous authors – particularly
Hehazy and Zhang (2005) – the question of how selecting the rescheduling window is
discussed, in order to be efficient and not losing information. While rescheduling “every
day” may seem more correct and safe, other considerations may induce selecting different
window intervals, e.g. for taking into account “complete” activity influences and better
considering the acceleration and slowing down effects of some activities.
4 Conclusions and future development
In case of project and contract claims, the computer-assisted methods can provide more
efficient, transparent and rational mean to settle disputes than clumsy and difficult ways to
reconstruct the work history from yard journal, records etc. The algorithmic method here
developed can be the core of a more general approach for evaluating the schedule delays
on field project applications. The implementation of the proposed model using examples
form literature and other published cases is providing positive results. More difficult is to
get access, or authorization for publication of real life applications, which often are related
to legal cases and therefore are protected by privacy or difficult to disclose.
In principle floats is a resource that should be given due consideration in contractual
arrangements. Among the relationships that the method of float banking may have with
113

other project management fields of interest, we recall the Critical Chain Method (CCM)
where the concept of buffer management is introduced, see e.g. Leach (2000); in this regard
the float accounting can be considered to improve the concept of buffer control introduced
in CCM.
Once implemented, the system can better support ways to settle claims, arbitration and
other procedures between claimant and defendant, resorting to court as the last chance.
Specifically, we are developing the method as a web-based tool that can be made avail-
able and demonstrated for gaining feed-back from prospective users. This project is nick-
named ScheDA (Scheduling Delay Analysis), which means in Italian language a recording
or reporting sheet, based on some template or standard format. The same term originates
from late Latin “schedula” or strip of paper, and later meaning a “note” or something
to use as reference. Before scheda meant one of the strips forming a papyrus sheet, also
literally in Greek “skhida” (σχέδη), that is piece of wood, table, paper or small notebook.
Acknowledgements
We acknowledge Engineering Ingegneria Informatica Company and particularly Salva-
tore Di Rienzo, who supported a preliminary implementation of the method during the
graduate internship of Giovanni Sacco.
References
Alkass S. et al., 1996, “Construction delay analysis techniques”, J. Constr. Manag. Econ., Vol. 14,
pp. 375–394.
Association for the Advancement of Cost Engineering International (AACEI), 2007, “Recom-
mended Practice No. 29R-03, Forensic Schedule Analysis”.
Braimah N., 2013, “Construction Delay Analysis Techniques”, Buildings, Vol. 3, pp. 506–531.
Davison R.P., Mullen J., 2009, Evaluating Contract Claims, 2nd ed., Wiley-Blackwell.
Hegazy T., Zhang K., 2005, “Daily window delay analysis”, J. Constr. Eng. Manag., ASCE 2005,
Vol. 131, pp. 505–512.
Keane P.J., Caletka A.F., 2015, Delay Analysis in Construction Contracts, 2nd ed., Wiley-
Blackwell.
Leach L.P., 2000, Critical Chain Project Management, Artech House.
Pickavance K., 2010, Delay and Disruption in Construction Contracts, 4th ed., Sweet Maxwell.
ScheDA (Schedule Delay Analysis), www.pmscheda.it.
Society of Construction Law (SCL), 2002, “Protocol for determining extensions of Time and
Compensations for delay and disruption”, SCL, Burbage, UK.
Yang J-B., Kao C-K, 2012, “Critical path effect based delay analysis method for contruction
projects”, Int. Journal of Project Management, Vol. 30, pp. 385–397.
114

Minimizing the total weighted ompletion time in single
ma hine s heduling with non-renewable resour e
onstraints
Péter Györgyi1
, Tamás Kis1
Institute for Computer S ien e and Control, Budapest, Hungary
gyorgyi.petersztaki.mta.hu, kis.tamassztaki.mta.hu
Keywords: single ma hine s heduling, non-renewable resour es, total weighted ompletion
time.
1 Introdu tion
In a ma hine s heduling problem with non-renewable resour es, besides the ma hine(s),
there are non-renewable resour es, like raw materials, energy, or money, onsumed by the
jobs. The non-renewable resour es have some initial sto k, and they are replenished over
time in given quantities. The obje tive fun tion an be any of the widely-used optimization
riteria in ma hine s heduling problems, see e.g., Carlier (1984) or Györgyi and Kis (2017).
Now, we onsider a single ma hine variant with a single non-renewable resour e. For-
mally, there is a single ma hine, a set of n jobs J , and a non-renewable resour e. Ea h job
j has a pro essing time pj 0, a weight wj 0, and resour e requirement aj ≥ 0. The
non-renewable resour e has an initial sto k b̃1 ≥ 0 at time u1 = 0, and it is replenished
at q − 1 distin t supply dates 0 u2 · · · uq in quantities b̃ℓ ≥ 0 for ℓ = 2, . . . , q.
However, the total demand does not ex eed the total supply, i.e.,
P
j∈J aj ≤
Pq
ℓ=1 b̃ℓ. The
umulative supply up to supply date uℓ is bℓ =
Pℓ
k=1 b̃k. A s hedule spe ies the starting
time Sj of ea h job j ∈ J ; it is feasible if (i) no pair jobs overlap in time, i.e., Sj1 +pj1 ≤ Sj2
or Sj2 + pj2 ≤ Sj1 for ea h pair of distin t jobs j1 and j2, and (ii) for ea h time point t,
the total supply until time t is not less than the total onsumption of those jobs starting
not later than t, i.e., if uℓ ≤ t is the last supply date before t, then
P
j∈J:Sj ≤t aj ≤ bℓ.
An example problem along with a feasible s hedule is depi ted in Figure 1. There are 5
jobs represented by 5 re tangles. For ea h job j, the width of the orresponding re tangle
indi ates its pro essing time, while the resour e requirement aj is provided in the re tangle.
Further on, there is an initial supply of b̃1 = 3 at time u1 = 0, and two more supplies at u2
and u3 with supplied quantities b̃2 = 4 and b̃3 = 6, respe tively. In the depi ted s hedule,
job j1 annot start earlier, sin e it requires 2 units from the resour e, but there is only
b̃1 + b̃2 − a2 − a5 − a3 = 1 unit on sto k before the supply arrives at u3.
Fig. 1. A feasible s hedule (n = 5, q = 3)
115

We aim at nding a feasible s hedule S minimizing the total weighted ompletion time
P
j∈J wjCj, where Cj := Sj +pj denotes the ompletion time of job j. Using the standard
α|β|γ notation, we denote our problem by 1|nr = 1|
P
wjCj, where 'nr = 1' indi ates that
we have only one type of non-renewable resour e.
1.1 Previous results
The rst results of the area are from the 1980s. Carlier (1984) presented several omplex-
ity results for variants where the makespan, the maximum lateness, or the total ompletion
time have to be minimized in single and parallel ma hine environments. Slowinski (1984)
examined a preemptive version of the problem for parallel ma hines. Toker et. al. (1991)
and Xie (1997) applied redu tions to the two-ma hine ow shop problem for variants where
the supplies arrive uniformly over time. Grigoriev et. al. (2005) presented easy approxi-
mation algorithms for the makespan and the lateness obje tive. Gafarov et. al. (2011)
proved several omplexity results for various obje tive fun tions. Györgyi and Kis (2014)
presented approximation s hemes for the makespan obje tive in ase of one resour e. This
was extended for a onstant number of resour es by Györgyi and Kis (2015b) and for par-
allel ma hines by Györgyi and Kis (2017) and by Györgyi (2017). Györgyi and Kis (2015a)
proved redu tions between the makespan minimization problem with two supply dates and
variants of the Knapsa k Problem. The most relevant ante edent of this resear h is Kis
(2015), whi h onsidered the same obje tive fun tion and presented an FPTAS for the
problem with q = 2.
1.2 Preliminaries
This paper examines variants with more supplies, where we an state job independent
onne tions among the pro essing times, the resour e requirements and the weights. If
these onne tions are strong enough we an nd easy ordering rules that yield optimal
s hedules, see Table 1. In the next se tions we deal with two other variants.
Table 1. Easy variants of 1|nr = 1|
P
wj Cj.
Variant Optimal s hedule
pj = aj = ā non-in reasing wj order
pj = wj = 1 non-de reasing aj order
aj = wj = 1 SPT order
wj = w̄, pj = aj SPT order
aj = ā, pj = wj LPT order
Noti e that SPT and LPT means that jobs are ordered in in reasing, respe tively,
de reasing pro essing time order. In the orresponding algorithm, jobs are simply s heduled
in in reasing (SPT) / de reasing (LPT) pro essing time order. If the resour e level is below
the requirement of the next job, we simply wait until enough supply arrives.
While the SPT order gives the optimal s hedule for the problem 1||
P
Cj (all job weights
are 1), the LPT order is originally used in a list s heduling algorithm for the parallel
ma hine problem P||Cmax where it yields a 4/3-approximation algorithm.
2 The problem 1|nr = 1, pj = aj = wj|
P
wjCj
Surprisingly, this very restri tive ase is already NP-hard:
116

Theorem 1. The problem 1|nr = 1, q = 2, pj = aj = wj|
P
wjCj is weakly NP-hard, and
1|nr = 1, pj = aj = wj|
P
wjCj is strongly NP-hard.
These omplexity results are new, formerly only the NP-hardness of the variant 1|nr =
1, q = 2|
P
Cj (see Kis (2015)) and that of 1|nr = 1|
P
Cj (Carlier (1984), Kis (2015))
were known.
However, we ould derive a 2-approximation algorithm for it.
Theorem 2. S heduling the jobs in LPT order is a 2-approximation algorithm for 1|nr =
1, pj = aj = wj|
P
wjCj.
3 A PTAS for 1|nr = 1, pj = wj, q = const|
P
wjCj
In this se tion we des ribe an PTAS (polynomial time approximation s heme) for 1|nr =
1, pj = wj, q = const|
P
wjCj. Noti e that the resour e onsumption of the jobs is job-
dependent, but the number of supplies is a onstant, not part of the input. A PTAS is a
family of algorithms {Aε}ε0, su h that for ea h ε 0, Aε is an (1 + ε)-approximation
algorithm for the problem with a omplexity polynomially bounded in the size of the input.
Let Psum :=
P
j pj be the total pro essing time of the jobs. Let ∆ := 1 + (ε/q2
).
We will guess the total pro essing time of those jobs s heduled after uℓ for ℓ = 2, . . . , q,
where a guess is a q − 1 dimensional ve tor of non-in reasing numbers Pg
2 , . . . , Pg
q , i.e.,
Pg
ℓ ≥ Pg
ℓ+1 ≥ 1 for ℓ = 2, . . . , q − 1, and ea h Pg
ℓ is of the form ∆t
for some integer
t ≥ 0 with ∆t
≤ Psum. Also x Pg
1 := Psum. For any guess, dene the set of medium
size jobs Mℓ := {j | pj ≥ (∆ − 1)Pg
ℓ }. Note that Mq ⊇ Mq−1 ⊇ · · · ⊇ M1, sin e
Pg
q ≤ Pg
q−1 ≤ · · · ≤ Pg
1 . Let Sℓ be the omplement of Mℓ, i.e., Sℓ := {j | pj (∆ − 1)Pg
ℓ }.
Clearly, Sq ⊆ Sq−1 ⊆ · · · ⊆ S1. After these preliminaries, the PTAS for 1|nr = 1, pj =
wj, q = const|
P
wjCj onsists of the following steps:
1. Consider ea h possible guess (Pg
2 , . . . , Pg
q ) of the total pro essing time of those jobs
starting after the supply dates u2, . . . , uq, respe tively. For ea h possible guess dene
the sets of jobs Mℓ and Sℓ (see above), and perform the steps 2-5. After pro essing all
the guesses, go to Step 6.
2. For ea h ℓ = 1, . . . , q, hoose at most 1/(∆ − 1) medium size jobs from Mℓ (sin e the
sets Mℓ are not disjoint, are must be taken to hoose ea h job at most on e). For ea h
possible hoi e (T1, . . . , Tq) of the medium size jobs (where Tℓ ⊆ Mℓ), perform steps
3-5. After evaluating all hoi es, ontinue with the next guess in Step 1.
3. Determine a s hedule of the medium jobs. That is, for ℓ = q, . . . , 2, s hedule the jobs
in Tℓ in any order after uℓ ontiguously, and if ne essary, push to the right the jobs in
Sq
ℓ′=ℓ+1 Tℓ′ .
4. Let J u
0 be the set of uns heduled jobs. For ℓ = q, q − 1, . . . , 1, repeat the following. In
a general step with ℓ ≥ 2, pi k jobs from J u
q−ℓ ∩ Sℓ in non-in reasing aj/pj order until
the sele ted subset Kℓ satises p(Kℓ) + p(Tℓ) ≥ Pg
ℓ − (1/∆)Pg
ℓ+1, or if no more jobs left,
Kℓ = J u
q−ℓ ∩ Sℓ. In either ase, insert the jobs of Kℓ in any order after uℓ and after all
the jobs in T1 ∪ · · · ∪ Tℓ−1, and before all the jobs in Tℓ ∪
Sq
ℓ′=ℓ+1(Kℓ′ ∪ Tℓ′ ) (pushing
some of them to the right if ne essary). Let J u
q−ℓ+1 := J u
q−ℓ Kℓ and ontinue with
ℓ − 1 until ℓ = 1 or no more uns heduled jobs are left. For ℓ = 1 just s hedule all the
remaining jobs from time u1 = 0 on (pushing the already s heduled jobs to the right,
if ne essary). If the omplete s hedule obtained satises the resour e onstraints, then
ontinue with Step 5, otherwise with the next hoi e of medium size jobs in Step 2.
5. Compute the obje tive fun tion value of the omplete s hedule obtained in step (4), and
store this s hedule as the best s hedule if it is the rst feasible s hedule or if it is better
than the best feasible s hedule found so far. Continue with next hoi e of medium size
jobs in Step 2.
117

6. Output the best s hedule found in the previous steps.
Theorem 3. The proposed algorithm is an PTAS for 1|nr = 1, pj = wj, q = const|
P
wjCj.
A knowledgements
This work has been supported by the National Resear h, Development and Innova-
tion O e - NKFIH grant K112881, and by the GINOP-2.3.2-15-2016-00002 grant of the
Ministry of National E onomy of Hungary.
Referen es
Carlier J., 1984, Problèmes d'ordonnan ement à ontraintes de ressour es: algorithmes et om-
plexité, Université Paris VI-Pierre et Marie Curie, Institut de programmation
Gafarov E.R., A.A. Lazarev and F. Werner, 2011, Single ma hine s heduling problems with
nan ial resour e onstraints: Some omplexity results and properties, Math. So ial S i.,
Vol. 62, pp. 7-13.
Grigoriev A., M. Holthuijsen and J. van de Klundert, 2005, Basi s heduling problems with raw
material onstraints, Naval Res. Logist., Vol. 52, pp. 527-553.
Györgyi P., 2017, A PTAS for a resour e s heduling problem with arbitrary number of parallel
ma hines, Oper. Res. Lett., Vol. 45, pp. 604-609.
Györgyi P., T. Kis, 2014, Approximation s hemes for single ma hine s heduling with non-
renewable resour e onstraints, J. S hed., Vol. 17, pp. 135-144.
Györgyi P., T. Kis, 2015a, Redu tions between s heduling problems with non-renewable resour es
and knapsa k problems, Theoret. Comput. S i., Vol. 565, pp. 63-76.
Györgyi P., T. Kis, 2015b, Approximability of s heduling problems with resour e onsuming
jobs, Ann. Oper. Res., Vol. 235, pp. 319-336.
Györgyi P., T. Kis, 2017, Approximation s hemes for parallel ma hine s heduling with non-
renewable resour es, European J. Oper. Res., Vol. 258, pp. 113-123.
Kis T., 2015, Approximability of total weighted ompletion time with resour e onsuming jobs,
Oper. Res. Lett., Vol. 43, pp. 595-598.
Slowinski R., 1984, Preemptive s heduling of independent jobs on parallel ma hines subje t to
nan ial onstraints, European J. Oper. Res., Vol. 15, pp. 366-373.
Toker A., S. Kondak i and N. Erkip, 1991, S heduling under a non-renewable resour e onstraint,
J. Oper. Res. So ., Vol. 42, pp. 811-814.
Xie J., 1997, Polynomial algorithms for single ma hine s heduling problems with nan ial on-
straints, Oper. Res. Lett., Vol. 21, pp. 39-42.
118

The Cyclic Job Shop Problem with uncertain
processing times
Idir Hamaz1
, Laurent Houssin1
and Sonia Cafieri2
1
LAAS-CNRS, Universite de Toulouse, CNRS, UPS, Toulouse, France
{ihamaz, lhoussin}@laas.fr
2
ENAC, Universite de Toulouse, F-31055 Toulouse, France
sonia.cafieri@enac.fr
Keywords: Cyclic scheduling, budgeted uncertainty set, robust optimization.
1 Introduction
Most models for scheduling problems assume deterministic parameters. In contrast, real
world scheduling problems are often subject to many sources of uncertainty, for example
activities duration can decrease or increase, machines can break down, new activities can be
incorporated, etc. In this paper, we focus on scheduling problems that are cyclic and where
activity durations are affected by uncertainty. Indeed, the best solution for a deterministic
problem can quickly become the worst one in the presence of uncertainties.
In this paper, we consider the Cyclic Job Shop Problem (CJSP) where processing times
are affected by uncertainty. Several studies were conducted on the deterministic CJSP.
The CJSP with identical parts is studied in (Roundy, R. 1992). The author shows that the
problem is NP-hard and designs a branch and bound algorithm to solve the problem. Hanen
(1994) investigates the general CJSP and presents a branch and bound procedure to tackle
the problem. A general framework for modeling and solving cyclic scheduling problems is
presented in (Brucker, P. and Kampmeyer, T. 2008). The authors present different models
for cyclic versions of the job shop problem. However, a few works consider cyclic scheduling
problems under uncertainty. Che, A. et. al. (2015) investigate the cyclic hoist scheduling
problem with processing time window constraints where the hoist transportation times
are uncertain. The authors define a robustness measure for cyclic hoist schedule and a
bi-objective mixed integer linear program to optimize the cycle time and the robustness.
In order to deal with uncertainty, we use a robust optimization approach. We model
the uncertain parameters by using the idea of uncertainty set proposed by Bertsimas and
Sim (2004). Each task duration belongs to an interval, and the number of parameters that
can deviate from their nominal values is bounded by a positive number called budget of
uncertainty. This parameter allows us to control the degree of conservatism of the resulting
schedule. Finally, we propose a branch and bound procedure that computes the minimum
cycle time for the robust CJSP such that, for each scenario in the uncertainty set, there
exists a feasible cyclic schedule.
2 Problems description
2.1 Basic Cyclic Scheduling Problem (BCSP)
We are given a set of n generic operations T = {1, ..., n}. Each operation i ∈ T is
characterized by a non-negative processing time pi and has to be performed infinitely often
without preemption. We denote i, k the kth
occurrence of the generic operation i and
t(i, k) the starting time of kth
occurrence of the operation i.
119

The operations are subjected to a set of precedence constraints (uniform constraints).
The constraints between the occurrences i, k and j, k + Hij are given by
t(i, k) + pi 6 t(j, k + Hij), ∀ i ∈ T , ∀ k ≥ 1 (1)
where Hij is an integer that represents the depth of the occurrence shift, usually referred
to as height. The Hij parameter is an occurrence shift between the operations i and j.
For instance, for each execution of the occurrence i, k , the next execution of j is the
occurrence j, k + Hij .
A schedule S is an assignment of starting time t(i, k) for each occurrence i, k of
task i ∈ T . Such schedule is called periodic with cycle time α if it satisfies
t(i, k) = t(i, 0) + αk, ∀ i ∈ T , ∀ k ≥ 1 (2)
where α is the cycle time and represents the difference between the stating times of two
successive occurrences of the same task.
Therefore, a schedule S can be entirely defined by the staring times ti = t(i, 0) of the
first occurrences and the cycle time.
In this study, the objective is to minimize the cycle time α while satisfying the prece-
dence constraints between operations. Notice that different objective functions exist for
cyclic scheduling problems, such as work in progress minimization or both cycle time and
work in progress minimization.
A bi-valued directed graph G = (T , U) can be associated with any instance of BCSP. In
this graph, a node (resp. an arc) of G corresponds to a generic operation (resp. constraints)
in the BCSP. Each arc (i, j) of G has two valuations, the length Lij = pi and the height Hij.
These arcs are called uniform arcs and are built by considering the precedence constraints.
For instance, a precedence constraint between task i and task j leads to an arc (i, j) of G
labeled with Lij = pi and Hij. We denote H(c) (resp. L(c)) the height (resp. length) of a
circuit c in graph G the sum of heights (resp. lengths) of the arcs composing the circuit c.
The minimum cycle time is given by the maximum circuit ratio of the graph which is
defined by
α = max
c∈C
∑
(i,j)∈c Lij
∑
(i,j)∈c Hij
where C is the set of all circuits in G.
We call critical circuit the circuit c realizing the maximum circuit ratio. Several algo-
rithms have been proposed for the computation of critical circuits. An experimental study
about maximum circuit ratio algorithms was published in (Dasdan, A. 2004). The author
remarks that, among the several tested algorithms, the most efficient one is the Howard’s
algorithm. Although the algorithm has a pseudo-polynomial complexity, it shows notewor-
thy practical results.
Once the cycle time is determined, the starting times (ti)i∈T can be determined by
computing the longest path in the graph G where each arc (i, j) ∈ U is valued with
pi − αHij.
2.2 Cyclic Job Shop Problem (CJSP)
In the present work, we focus on the cyclic job shop problem (CJSP). The difference
with the problem defined above is that for CJSP the number of machines is lower than the
number of tasks to perform. As a result, the same resource must be shared between different
operations. A CJSP can be considered as a BCSP equipped with resource constraints.
120

Each occurrence of an operation i ∈ T has to be executed, without preemption, on the
machine M(i) ∈ M = {1, ..., m}. Operations are grouped on a set of jobs J , where a job j
represents a sequence of elementary operations that must be executed in order. To avoid
overlapping between the tasks executed on the same machine, for each pair of operations
i and j where M(i) = M(j), the following disjunctive constraint holds
∀ i, j s.t. M(i) = M(j), ∀k, l ∈ N : t(i, k) ≤ t(j, l) ⇒ t(i, k) + pi ≤ t(j, l) (3)
In summary, a cyclic job shop problem is defined by
• a set T of elementary tasks,
• a set M of machines,
• for each task i ∈ T , a processing time pi and a machine M(i) ∈ M on which the task
has to be performed,
• a set P of precedence constraints,
• a set D of disjunctive constraints that occur when two tasks are mapped on the same
machine,
• a set J of jobs corresponding to a production sequence of generic operations. More
precisely, a job J1 defines a sequence J1 = t1,1 . . . t1,k to be executed in that order.
The CJSP can be represented by directed graph G = (V, P ∪ D), called disjunctive
graph. The sequence of operations that belongs to the same job are linked by uniform arcs
in P where the heights are equal to 0. Additionally, for each pair of generic operations i and
j executed on the same machine, a disjunctive pair of arcs (i, j) and (j, i) occurs. These
arcs are labeled respectively with Lij = pi and Hij = Kij, and Lji = pj and Hji = Kji
where Kij is an occurrence shift variable that satisfies Kij + Kji = 1 (Hanen C 1994).
The following bounds on occurrence shift variables Kij have been proposed in (Hanen
C 1994):
K−
ij ≤ Kij ≤ 1 − K−
ij . (4)
with
K−
ij = 1 − min{H(µ) | µ from j to i in G}. (5)
A schedule is an assignment of all the occurrence shifts, i.e., determine precedence
relations on the operation occurrences mapped to the same machine. Note that once the
occurrence shifts are determined the problem is equivalent to the BCSP, therefore, the
minimum cycle time can be obtained by the cited algorithms.
Previous studies have shown that the problem is NP-Hard (Hanen C 1994) for cycle
time minimization.
2.3 Robust Cyclic Job Shop Problem (RCJSP)
In this paper, we investigate the robust version of the CJSP. More precisely, we are
interested in the CJSP where processing times are affected by uncertainty and belong
to a finite uncertainty set U. Based on the budget of uncertainty concept introduced in
(Bertsimas, D. and Sim, M. 2004), the processing time deviations can be modeled trough
the following uncertainty set:
UΓ
=
{
(pi)i∈T ∈ Rn
: pi = p̄i + p̂iξi, ∀ i ∈ T ; ξi ∈ {0, 1};
∑
i∈T
ξi ≤ Γ
}
where p̄i represents the nominal processing time of operation i and p̂i its deviation. The pa-
rameter Γ is a positive integer and represents an upper bound on the number of processing
times deviating from their nominal value.
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The objective of the problem is to find, for a given budget of uncertainty Γ, the mini-
mum cycle time such that, for each p ∈ UΓ
, there exists a vector (t(p)i)i∈T satisfying both
the precedence and disjunctive constraints.
3 Branch and bound procedure for the RCJSP
Recently, an Howard’s algorithm adaptation taking into account the uncertainty set
UΓ
has been presented in (Hamaz, I. et. al. 2017). The computational experiments on the
algorithm show small execution times for robust BCSP instances.
To take into account the uncertainty on the processing times for the RCJSP, we develop
a branch and bound procedure that uses the robust version of the Howard’s algorithm. The
procedure starts by initializing the upper bound on the cycle time to
∑
i∈T pi + pf where
pf is the sum of the first Γ greatest deviations and the lower bound to the optimal cycle
time of G = (T , U) computed by the Howard’s algorithm adaptation.
We use the same branching scheme as in (Fink, M. et. al. 2012). The search tree is
initialized with a node (the root) where the graph G = (T , U) contains only the uniform
arcs U and no fixed disjunctions. Then, the branching is performed on unfixed disjunctions
Kij. For this purpose, a successor node is created for each value on the interval [K−
ij , 1−K−
ij ].
The value of the node is then computed by running the robust version of the Howard’s
algorithm with G = (T , U
′
∪ {(i, j), (j, i)}), where U
′
contains the uniform arcs and a
precedent fixed disjunctive arcs. When all the occurrence shifts are fixed, a feasible schedule
is obtained, then the upper bound can be updated.
Preliminary numerical results show that the branch and bound procedure (implemented
in C++ and executed on an Intel Xeon E5-2695 processor running at 2.30GHz CPU)
delivers promising results. Besides, the algorithm is insensitive regarding the value of the
budget of uncertainty.
Once the optimal cycle time computed by the branch and bound procedure, a periodic
schedule SΓ
= (α, ((t(p)i)i∈T ) can be determined for each p ∈ UΓ
.
4 Conclusion
The RCJSP with budgeted uncertainty set is addressed in this paper. We present a
branch and bound procedure that uses a Howard’s algorithm adaptation. Further investi-
gation will address dominance rules to speed up the branch and bound procedure.
References
Bertsimas, D., Sim, M., 2004, “The price of robustness, Operations research, Vol. 52(1), pp. 35-53.
Brucker, P., Kampmeyer, T., 2008, “A general model for cyclic machine scheduling problems,
Discrete Applied Mathematics, Vol. 156(13), pp. 2561-2572.
Che, A., Feng, J., Chen, H., and Chu, C., 2015, “Robust optimization for the cyclic hoist scheduling
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Dasdan, A., 2004, “Experimental analysis of the fastest optimum cycle ratio and mean algorithms,
ACM Transactions on Design Automation of Electronic Systems, Vol. 9(4), pp. 385-418.
Fink, M., Rahhou, T. B. and Houssin, L., 2012, “A new procedure for the cyclic job shop problem,
IFAC Proceedings Volumes, Vol. 45(6), pp. 69-74.
Hamaz, I., Houssin, L. and Cafieri, S., 2017, “Robust Basic Cyclic Scheduling Problem,
Manuscript submitted for publication.
Hanen, C., 1994, “Study of a NP-hard cyclic scheduling problem: The recurrent job-shop, Euro-
pean journal of operational research, Vol. 72(1), pp. 82-101.
Roundy, R., 1992, “Cyclic schedules for job shops with identical jobs, Mathematics of operations
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122

Modeling techniques for the eS-graph
Hegyháti M.
Department of IT, Széchenyi István University, Hungary
hegyhati@sze.hu
Keywords: scheduling, modeling, eS-graph.
1 Motivation
The scheduling of batch processes has been addressed with a variety of techniques
in the literature: Mixed-Integer Linear Programming models (Floudas and Lin, 2004), S-
graph Framework (Sanmartí et al., 2002), Linear-Priced Timed Automata (LPTA, Panek
et al., 2008), just to name the most frequent ones. Many of the approaches use an internal
representation of the scheduling problem, e.g. a State Task Network (STN, Kondili et al.,
1993), Resource Task Network (RTN, Pantelides, 1993), State Sequence Network (SSN,
Majozi and Zhu, 2001), that is used as a basis for formulating the mathematical model
for the optimization algorithms. An important advantage of the S-graph framework is that
the initial representation and the mathematical model for the algorithms are the same, the
so-called S-graph which is a directed acyclic graph. This transparency makes the approach
easier to understand, and modeling issues (Hegyháti et al., 2009) are easily avoided. This
simple model, however, carries limitations too. To apply it for industrial examples, minor
extensions or alterations are needed in some cases. The recently proposed eS-graph model
(Hegyháti, 2014) aims to extend its modeling power to avoid the need for further extensions,
while keeping the simplicity and transparency of the S-graph. In this work, the modeling
power of the eS-graph framework is presented through several industrial examples.
2 Short introduction of the S-graph framework
The S-graph framework was originally proposed for the short-term scheduling of chem-
ical batch processes. The framework is based on two major components:
• The mathematical model, the S-graph
• The branch-and-bound algorithm to ﬁnd the optimal schedule
The S-graph model is a directed graph with weighted arcs. The nodes represent tasks
and products, which are illustrated by circles in the graphical representation. The circles
have labels attached that are the name of the product for the product nodes, and the
name of the task above the name of the suitable unit in case of task nodes. These nodes are
connected with weighted arcs, which express the production order of the tasks belonging to
the same product. Moreover, the weight of an arc is a lower bound on the necessary timing
diﬀerence between the two connected nodes. This graph, called recipe-graph, is extended
by the algorithm with arcs representing scheduling decisions. The algorithms report the
S-graph model of the optimal solution that can be easily and unambiguously converted to
a common graphical representation, such as a Gantt-chart. An example S-graph is shown
in Figure 1 with 9 tasks and 2 products. The scheduling decisions are expressed by the gray
arcs, e.g., the execution of t9 can not start earlier than the that of t4. The arcs representing
scheduling decisions are zero-weighted by default, which is often omitted in the graphical
representation as well.
123

Since its introduction, the S-graph framework has been applied to numerous case stud-
ies which often required small extensions, alterations of the model to fit and address the
problem-specific constraints. As an example, solving wet-etch scheduling problems required
an extension of the original S-graph model to implement zero-wait policy (Hegyháti et al.,
2014), or a new algorithm has been proposed to solve problems with the objective of
throughput maximization (Majozi and Friedler, 2006). While the individual difficulty of
these separate extensions vary, keeping them compatible is a challenging task on both theo-
retical and software implementation levels. To overcome this issue, the eS-graph framework
was proposed.
4
5
t1
e1
t2
e2
t3
e1
t4
e3
t7
e2
t8
e3
t5
e1
p1
t9
e1
p2
2
3
3
6
1
4
t6
e1
2
Fig. 1. S-graph representing a schedule for two products.
3 The eS-graph model
The eS-graph model can be seen as a generalization of the S-graph model. In the S-
graph framework, there is a one-to-one relation between the nodes of the graph and the
tasks that have to be carried out using the available resources. The eS-graph model relaxes
this connection, and has the following basic building blocks:
• Events – are the atomic building blocks, that represent a single transition of the states
of the system. In the model, events are represented with nodes.
• Subprocesses – are the generalizations of tasks. Any activity that spans over some
events and requires the presence of some resources continuously can be considered as a
subprocess. Formally, any subset of the events can be a subprocess. Subprocesses may
also overlap each other.
Each subprocess may be carried out by a set of resources simultaneously, however,
several of such sets may exist, and it may influences the weights of the arcs within the
subprocess.
The original S-graph model can be seen as a special class of eS-graphs, where
1. The events considered are the starting of the tasks and removal of the products from
the last processing step.
2. Each subprocess spans over the event representing the start of a task, and its out-
neighbors in the recipe graph.
3. The resource sets of each subprocess are singletons.
124

The generalized definition, however, enables implementation of a wider set of practical
considerations without the need for further extensions to the model or the algorithms. The
eS-graph model can be solved to optimality by a slightly modified version of the original
S-graph algorithms, or with a precedence based MILP model.
4 An example eS-graph model
To illustrate the expressive power of the eS-graph framework, parts of the model for
scheduling job cells with automated guided vehicles is presented here. In this practical
study, there are several workstations where the jobs must be processed. The intermediates
between the stations are transported via automated guided vehicles (AGVs) (Zeng et al.,
2014).
The problem entails several specific constraints which cannot be modeled in a straight-
forward way in the S-graph framework, or the STN, RTN, or SSN representations. Two
such constraints are selected here, and used as an illustration for the simple modeling
techniques with the eS-graph model:
1. For each transportation, an arbitrary AGV is needed, which traverses through specific
line segments. While the AGV is transporting something, no other AGV can use the
same line segments. However, each station has its own loading area, thus, the path
segments become free when the AGV arrives to its destination.
2. Loading and unloading of intermediates takes a specific time, for which both the station
and the assigned AGV is required. It is not mandatory however, that the same AGV
is used for subsequent transportations of the same product.
Part of the proposed model is presented in Figure 2., where the events and subprocesses
are the following:
e1: job leaving machine m1.
e2: job arriving to machine m2.
e3: job starting to be processed on machine m2.
e4: job finishing to be processed on machine m2.
e5: job leaving machine m2.
e6: job arriving to machine m3.
sp1: transportation processes between machines m1 and m2.
sp2: transportation processes between machines m2 and m3.
sp3: traversing between machines m1 and m2.
sp4: traversing between machines m2 and m3.
sp5: manufacturing step of the job on machine m2.
As both events e2 and e3 are covered by both sp1 and sp5, both the AGV assigned to
the transportation and the machine m2 are needed for the unloading of the intermediate, as
required by the second statement above. sp2 however, is a completely different subprocess
from sp1, thus, a different AGV may be selected for it.
As for the first statement, the subprocess sp3 is a subset of sp1, and requires the
segments of one of the suitable paths between machines m1 and m2. While the AGV is
moving between the machines, the segments are unavailable for other AGVs, however, they
become free when it arrives to the loading area of m2. The same holds for sp4 and sp2,
however, there is only one possible path between m2 and m3.
125

Unloading
time
Loading
time
sp5: {m2}
sp1: {agv1},{agv2} sp2: {agv1},{agv2}
e3 e4
Processing
time
e1 e2 e1
e5 e6
Transfer
time
Transfer
time
sp3: {p2,p3,p4}, {p1,p5} sp4: {p6,p7}
Fig. 2. Part of an eS-graph for the illustrative example.
5 Results
The modeling power of the eS-graph framework was examined on several industrial case
studies from the literature. The strong and weak points of this modeling technique were
identified, and the modified S-graph algorithms were compared with a proposed precedence
based MILP model on these examples.
6 Conclusions
The eS-graph model allows to model a much wider set of practical constraints arising
in industrial scheduling problems without the need for any extension on the model or the
applied algorithms. There are limitations to this model as well, however, it holds a great
potential. eS-graph can be seen as a middle ground between problem-specific scheduling
models, such as the S-graph, and the very general models like MILP and LPTA. While eS-
graphs are still scheduling specific, they are general enough so that the algorithms working
on them can be developed independently from the problems that are modeled with it. From
each acceleration, however, all of the modeled scheduling problem classes can benefit.
7 Acknowledgements
This research was supported by the National Research, Development and Innovation
Office (NKFIH) K108405 and by the EFOP-3.6.2-16-2017-00015 “HU-MATHS-IN; Inten-
sification of the activity of the Hungarian Industrial Innovation Service Network” grant.
Supported by the ÚNKP-17-4 New National Excellence Program of the Ministry of Human
Capacities.
126

References
Floudas C. A., X. Lin, 2004, “Continuous-time versus discrete-time approaches for scheduling of
chemical processes: a review”, Computers Chemical Engineering, Vol. 28, pp. 2109–2129.
Hegyháti M., 2014, “A combinatorial modeling tool for event-based batch process scheduling: the
eS-graph”, presented at ASCONIKK VOCAL 2014, Veszprem, Hungary.
Hegyháti M., T. Majozi, T. Holczinger, F. Friedler, 2009, “Practical infeasibility of cross-transfer
in batch plants with complex recipes: S-graph vs MILP methods”, Chemical Engineering
Science, Vol. 64, pp. 605–610.
Hegyháti M., O. Osz, B. Kovács, F. Friedler, 2014, “Scheduling of automated wet-etch stations”,
Chemical Engineering Transactions, Vol. 39, pp. 433–438.
Kondili E., C. Pantelides, R. Sargent, 1993, “A general algorithm for short-term scheduling of
batch operations–I. MILP formulation”, Computers Chemical Engineering, Vol. 17, pp.
211–227.
Majozi T, F. Friedler, 2006, “Maximization of throughput in a multipurpose batch plant under
fixed time horizon: S-graph approach”, Industrial Engineering Chemistry Research, Vol. 45,
pp. 6713–6720.
Majozi T, X. X. Zhu, 2001, “A novel continuous-time MILP formulation for multipurpose batch
plants. 1. Short-term scheduling”, Industrial Engineering Chemistry Research, Vol. 40(25),
pp. 5935–5949.
Panek S., S. Engell, S. Subbiah, O. Stursberg, 2008, “Scheduling of multi-product batch plants
based upon timed automata models”, Computers Chemical Engineering, Process Systems
Engineering: Contributions on the State-of-the-Art - Selected extended Papers from ESCAPE
’16/PSE 2006., Vol. 32, pp. 275–291.
Pantelides C., 1993, “Unified frameworks for optimal process planning and scheduling”, Proceedings
of the second international conference on foundations of computer-aided process operations,
pp. 253–274.
Sanmarti E., T. Holczinger, L. Puigjaner, F. Friedler, 2002, “Combinatorial framework for effective
scheduling of multipurpose batch plants”, AIChE Journal, Vol. 48(11), pp. 2557–2570.
Zeng C., J. Tang, C. Yan, 2014, “Scheduling of no buffer job shop cells with blocking constraints
and automated guided vehicles”, Applied Soft Computing, Vol. 24, pp. 1033–1046.
127

Scheduling Multiple Flexible Projects with Different
Variants of Genetic Algorithms
Luise-Sophie Hoffmann and Carolin Kellenbrink
Institute of Production Management, Leibniz Universität Hannover, Germany
luise-sophie.hoffmann, carolin.kellenbrink@prod.uni-hannover.de
Keywords: Multi-Project Scheduling, RCMPSP-PS, Flexible Project, Genetic Algorithm.
1 Introduction
In Kellenbrink and Helber (2015), the resource-constrained project scheduling problem
with a flexible project structure (RCPSP-PS) is introduced. For such flexible projects, the
project structure is not known in advance. Instead, it depends on model-endogenous deci-
sions in which exactly one activity out of a decision set has to be selected. Those activities
can trigger further decisions or cause activities. Therefore, in addition to scheduling the
activities, the project structure has to be chosen.
The regeneration of complex capital goods, like aircraft engines, is an example for such
a flexible project due to different technical repair options. The regeneration is usually con-
ducted by an external service provider. These service providers mostly handle several dif-
ferent projects at the same time which results in a problem setting similar to the scheduling
of multiple projects, cf., e.g., Pritsker et al. (1969). Therefore, the RCPSP-PS is extended
to the resource-constrained multi-project scheduling problem with flexible project struc-
tures (RCMPSP-PS). Furthermore, different variants of genetic algorithms regarding the
representation are presented and evaluated.
2 Problem Setting
A flexible project l ∈ L comprises a set of different activities. These activities can be
divided into sets of mandatory activities j ∈ Vl and optional activities. In each decision
e ∈ El triggered by an activity a(l, e), exactly one of the optional activities j ∈ Wle has
to be chosen for implementation. A decision is triggered, if the triggering activity a(l, e) is
implemented. While a mandatory decision is assumed to be triggered by the start-dummy
job, a non-mandatory decision is triggered by an optional job. Additionally, the decision
for the implementation of an optional activity j ∈ Wle may cause the implementation of
further activities i ∈ Blj.
Due to the different possible project structures, not only renewable resources r ∈ R
but also nonrenewable resources n ∈ N have to be considered. This may lead to infeasible
combinations of project structures. Furthermore, we consider specific due date δl for all
projects. In case this due date is not met, specific delay costs cl for each delayed period
occur. Overall, the aim is to minimize the total delay cost.
Figure 1 shows an example for the given problem setting with two different flexible
projects I and II. These projects consist of eight non-dummy activities each. The project’s
decisions and caused activities, the resource consumption kljr and kljn, as well as the
duration dlj of the activities are given in the figure. While scheduling those two projects,
the capacities of one renewable and one nonrenewable resource are considered.
Project I contains two decisions. The first decision is mandatory and thus triggered by
the start-dummy activity I-1. Hence, the decision between the implementation of activity
I-4 and of activity I-5 has to be made. If activity I-4 is selected in the first decision, it
128

Project I
WI1
a(I,1)=I-1
WI2
a(I,2)=I-4
BI-5
I-1
0
0,0
I-2
3
2,1
I-3
1
2,2
I-4
4
1,2
I-5
2
3,1
I-6
3
2,1
I-7
3
1,3
I-8
4
1,2
I-9
2
2,1
I-10
0
0,0
Project II
WII1
a(II,1)=II-1
BII-5
II-1
0
0,0
II-2
2
1,3
II-3
1
2,2
II-4
2
3,1
II-5
1
1,2
II-6
5
2,2
II-7
4
3,1
II-8
1
2,1
II-9
2
2,2
II-10
0
0,0
Legend
l-i
dli
klir, klin
l-j
dlj
kljr, kljn
decision set caused activities
Fig. 1. Example of multiple flexible projects
triggers the second decision on activities I-7 and I-8. In case activity I-5 is selected in the
first decision, activity I-6 is caused. For scheduling project I, the capacity of the renewable
resource is KIr = 2 and the capacity of the nonrenewable resource is KIn = 9. With
the given capacity of the nonrenewable resource, all three possible project structures are
feasible. The due date for project I is at the end of period seven. The delay costs for each
period equal seven units.
Project II has only one mandatory decision on the three activities II-5, II-6 and II-7.
While activity II-6 and II-7 neither trigger a decision nor cause an activity, activity II-5
causes activity II-9. The resource capacities are KIIr = 3 and KIIn = 10. The capacity of
the nonrenewable resource leads to an infeasible solution in case activity II-5, which causes
activity II-9, is chosen. The due date of project II is at the end of period eight with delay
costs of three units per period.
When scheduling both projects separately, in project I the optional activities I-4 and I-7
are implemented and we get a six period delay resulting in total delay cost of 6·7 = 42 units.
The optimal schedule of project II shows no delay. In this project the optional activity II-7
is selected and scheduled.
For scheduling both projects simultaneously, we assume capacities of Kr = 2 + 3 = 5
and Kn = 9 + 10 = 19. In the optimal schedule, project I is finished without a delay
but project II gets a one period delay. This leads to total delay cost of three units, which
is lower than the result of 42 units for considering separate schedules. Furthermore, the
selected project structures of both projects have changed. In project I activities I-5 and
I-6 are implemented. In project II activities II-5 and II-6 are scheduled. This shows that
for multiple flexible projects not only the scheduling but also the chosen project structures
influence each other.
3 Genetic Algorithms
Many approaches for scheduling multiple projects make use of priority rules, cf. Brown-
ing and Yassine (2010) for an overview. Therefore, in Hoffmann et al. (2017) different
priority rules to solve the RCMPSP-PS were evaluated. However, the numerical results
have not been overly satisfying. Presumably, the interaction of the differently prioritized
129

projects, the project structures and the scheduling is too complex to be represented by a
priority rule. Therefore, the use of a genetic algorithm seems to be more promising. In the
following, we sketch different options to represent an individual and a solution, respectively.
To evaluate the different approaches, we present first numerical results. The test set
of our numerical study contains 1728 PSPLIB-based two-project instances with 15 non-
dummy activities each. However, we excluded 112 instances that could not been solved to
proven optimality by a standard solver as well as 15 infeasible instances and one instance
with total delay cost of zero.
3.1 Selection of the project structure
In addition to the scheduling of activities, for flexible projects the solution representa-
tion has to include the decision on the project structure. Therefore, Kellenbrink and Helber
(2015) use a choice list to indicate the selected activities. Due to the effectiveness of the
random-key representation for scheduling projects, cf., e.g., Gonçalves et al. (2008), we
additionally developed a random-key based representation for the decision on the project
structure. According to our numerical study, both approaches obtain comparable results.
3.2 Scheduling the activities
There are different variants of genetic algorithms known in the literature to schedule
single projects with a fixed project structure. Two important aspects of a genetic algorithm
are the representation of a schedule and its decoding, including the schedule generation
scheme (SGS) applied.
Hartmann (1998) introduces a genetic algorithm to solve the resource-constrained project
scheduling problem using an activity list. Gonçalves et al. (2008) achieve good results by
using a random-key representation to schedule the activities of multiple projects with a
fixed project structure. Our numerical results show that the random-key representation
leads to better results for our problem setting.
For each representation, the fitness can be determined by applying the serial SGS. Hart-
mann (2002) introduces a self-adapting genetic algorithm where the solution representation
includes the decision on using the parallel or the serial SGS. We have refined this approach
by giving information on the SGS used in each scheduling step. In Figure 2, the relative

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VHULDO6*6
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VWHSZLVHFKDQJLQJ6*6
Fig. 2. Consideration of different SGS

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Fig. 3. Consideration of project preferences
130

deviations from the total delay cost of the considered instances are given. After only eight
generations, the stepwise changing SGS outperforms the other approaches.
When computing the numerical results in Figure 2, we did not take the different impor-
tance of the projects into account. However, Hoffmann et al. (2017) show that the inclusion
of project preferences is important for scheduling multiple projects with flexible project
structures.
Thus, we define representations where project priorities are added. The project prior-
ities are represented as random-keys which are then multiplied by the specific delay costs
to itensify the effect. Those project preferences determine for each scheduling step which
project is prioritized. As another option, we use a project list which directly gives the infor-
mation, which project is scheduled next. Figure 3 shows our results for the representation
of project preferences. The usage of project priorities enhances the solutions found. After
30 generations, the mean deviation from the optimum amounts to 1.39%.
4 Outlook
As described above, there are many different possibilities to define the representation
for the different aspects of the given problem setting. Our numerical results show efficient
combinations of representations working together best regarding performance and com-
putational time. We will focus on applying our approach to larger instances containing
more than two projects in future research. Moreover, we will determine the potential of
using forward-backward-improvement while scheduling the activities. In addition, we will
alternate the evolutionary process and evaluate resulting effects.
Acknowledgements
The authors thank the German Research Foundation (DFG) for the financial support
of this research project in the CRC 871 “Regeneration of Complex Capital Goods”.
References
Browning T. R., A. A. Yassine, 2010, “Resource-constrained multi-project scheduling: Priority
rule performance revisited”, International Journal of Production Economics, Vol. 126 (2), pp.
212-228.
Gonçalves J. F., J. J. M. Mendes, M. G. C. Resende, 2008, “A genetic algorithm for the resource
constrained multi-project scheduling problem”, European Journal of Operational Research,
Vol. 189 (3), pp. 1171-1190.
Hartmann S., 1998, “A competitive genetic algorithm for resource-constrained project scheduling”,
Naval Research Logistics, Vol. 45 (7), pp. 733-750.
Hartmann S., 2002, “A self-adapting genetic algorithm for project scheduling under resource con-
straints”, Naval Research Logistics, Vol. 49 (5), pp. 433-448.
Hoffmann L.-S., T. Kuprat, C. Kellenbrink, M. Schmidt, P. Nyhuis, 2017, “Priority Rule-based
Planning Approaches for Regeneration Processes”, Procedia CIRP, Vol. 59, pp. 89-94.
Kellenbrink C., S. Helber, 2015, “Scheduling resource-constrained projects with a flexible project
structure”, European Journal of Operational Research, Vol. 246 (2), pp. 379-391.
Pritsker A. A. B., L. J. Watters, P. M. Wolfe, 1969, “Multiproject Scheduling with Limited Re-
sources: A Zero-One Programming Approach”, Procedia CIRP, Vol. 59, pp. 89-94.
131

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✶ ■♥tr♦❞✉❝t✐♦♥
❚❤❡ ❥♦❜✲s❤♦♣ s❝❤❡❞✉❧✐♥❣ ♣r♦❜❧❡♠ ✐s ♦♥❡ ♦❢ t❤❡ ✇❡❧❧✲st✉❞✐❡❞ ✐ss✉❡s ✐♥ s❝❤❡❞✉❧✐♥❣ r❡s❡❛r❝❤✳
❚❤❡ ✐♥t❡❣r❛t✐♦♥ ♦❢ ❜❧♦❝❦✐♥❣ ❝♦♥str❛✐♥ts ✐s ♠♦t✐✈❛t❡❞ ❜② r❡❛❧✲✇♦r❧❞ ❛♣♣❧✐❝❛t✐♦♥s ❧✐❦❡ t❤❡
s❝❤❡❞✉❧✐♥❣ ♦❢ tr❛✐♥s ✐♥ ❛ ♥❡t✇♦r❦ ❛♥❞ t❤❡ ♣r♦❞✉❝t✐♦♥ ♦❢ ❤✉❣❡ ✐t❡♠s✳ ■t r❡❢❡rs t♦ t❤❡ ❛❜s❡♥❝❡
♦❢ ❜✉✛❡rs ✐♥ t❤❡ ♣❧❛♥♥✐♥❣ s②st❡♠✱ s♦ t❤❛t ❛ ❥♦❜ ❜❧♦❝❦s ❛ ♠❛❝❤✐♥❡ ✉♥t✐❧ ✐ts s✉❜s❡q✉❡♥t
♠❛❝❤✐♥❡ ✐s ✐❞❧❡✳ ❆s ❛ ❝✉st♦♠❡r✲♦r✐❡♥t❡❞ ♦♣t✐♠✐③❛t✐♦♥ ❝r✐t❡r✐♦♥✱ t❤❡ ♠✐♥✐♠✐③❛t✐♦♥ ♦❢ t❤❡
t♦t❛❧ t❛r❞✐♥❡ss ♦❢ ❛❧❧ ❥♦❜s ✇✐t❤ r❡❣❛r❞ t♦ ❣✐✈❡♥ ❞✉❡ ❞❛t❡s ✐s ❝♦♥s✐❞❡r❡❞✳
❉✐✛❡r❡♥t ♠❛t❤❡♠❛t✐❝❛❧ ♣r♦❣r❛♠♠✐♥❣ ❢♦r♠✉❧❛t✐♦♥s ♦❢ t❤❡ ❜❧♦❝❦✐♥❣ ❥♦❜✲s❤♦♣ ♣r♦❜❧❡♠
✭❇❏❙P✮ ✇✐t❤ t♦t❛❧ t❛r❞✐♥❡ss ♠✐♥✐♠✐③❛t✐♦♥ ❛r❡ t❡st❡❞ ❛♥❞ ❞✐s❝✉ss❡❞ ✐♥ ▲❛♥❣❡ ❛♥❞ ❲❡r♥❡r
✭✷✵✶✼✮✳ ❚❤❡ r❡s✉❧ts ♣r♦✈✐❞❡ ❛♥ ✐♥❞✐❝❛t✐♦♥ t♦ t❤❡ ♥❡❝❡ss✐t② ♦❢ ❤❡✉r✐st✐❝ ♠❡t❤♦❞s ❢♦r t❤❡
❇❏❙P✳ ■♥ ❧✐♥❡ ✇✐t❤ t❤✐s ✐❞❡❛✱ s❡✈❡r❛❧ ❛✉t❤♦rs ♣r❡s❡♥t ❤❡✉r✐st✐❝ ❛♣♣r♦❛❝❤❡s t♦ t❛❝❦❧❡ r❡❧❛t❡❞
t②♣❡s ♦❢ ❥♦❜✲s❤♦♣ s❝❤❡❞✉❧✐♥❣ ♣r♦❜❧❡♠s✳ ▼✐♥✐♠✐③✐♥❣ t❤❡ ♠❛❦❡s♣❛♥✱ t❤❡ ❇❏❙P ✐s s♦❧✈❡❞ ❜② ❛
❣❡♥❡t✐❝ ❛❧❣♦r✐t❤♠ ✐♥ ❇r✐③✉❡❧❛ ❡t✳ ❛❧✳ ✭✷✵✵✶✮ ❛♥❞ ❜② ❛ t❛❜✉ s❡❛r❝❤ ❤❡✉r✐st✐❝ ✐♥ ●rö✢✐♥ ❛♥❞
❑❧✐♥❦❡rt ✭✷✵✵✾✮✳ ❉✐✛❡r❡♥t ♥❡✐❣❤❜♦r❤♦♦❞s ❛❞❛♣t❡❞ t♦ ❛ t♦t❛❧ ✇❡✐❣❤t❡❞ t❛r❞✐♥❡ss ♦❜❥❡❝t✐✈❡
❛r❡ ♣r❡s❡♥t❡❞ ❜② ❑✉❤♣❢❛❤❧ ❛♥❞ ❇✐❡r✇✐rt❤ ✭✷✵✶✻✮ ❢♦r t❤❡ ❥♦❜✲s❤♦♣ ♣r♦❜❧❡♠ ✇✐t❤♦✉t ❜❧♦❝❦✐♥❣
❝♦♥str❛✐♥ts✳ ❋✉rt❤❡r♠♦r❡✱ ❇ür❣② ✭✷✵✶✼✮ ❛♣♣❧✐❡s ❛ ❣r❛♣❤✲❜❛s❡❞ t❛❜✉ s❡❛r❝❤ ❛♣♣r♦❛❝❤ t♦ t❤❡
❇❏❙P ❝♦♥s✐❞❡r✐♥❣ ❞✐✛❡r❡♥t r❡❣✉❧❛r ♦♣t✐♠✐③❛t✐♦♥ ❝r✐t❡r✐❛✳ ❇❛s❡❞ ♦♥ ❛ ❣❡♥❡r❛❧✐③❡❞ ❣r❛♣❤ ❢♦r✲
♠✉❧❛t✐♦♥ ❢♦r t❤❡ ❇❏❙P✱ ❛ ❤②❜r✐❞ ❜r❛♥❝❤✲❛♥❞✲❜♦✉♥❞✲♠❡t❤♦❞ ✐s ❛♣♣❧✐❡❞ t♦ ❛ tr❛✐♥ s❝❤❡❞✉❧✐♥❣
♣r♦❜❧❡♠ ✇✐t❤ ❛ t❛r❞✐♥❡ss✲❜❛s❡❞ ♦❜❥❡❝t✐✈❡ ✐♥ ❉✬❆r✐❛♥♦ ❡t✳ ❛❧✳ ✭✷✵✵✼✮✳
■♥ ❝♦♥tr❛st t♦ t❤❡ t❡❝❤♥✐q✉❡s ❣✐✈❡♥ ✐♥ t❤❡ ❧✐t❡r❛t✉r❡✱ t✇♦ ♣❡r♠✉t❛t✐♦♥✲❜❛s❡❞ ❤❡✉r✐st✐❝
❛♣♣r♦❛❝❤❡s ❛r❡ ♣r❡s❡♥t❡❞ ❛♥❞ ❝♦♠♣❛r❡❞ ✐♥ t❤✐s ♣❛♣❡r✳ ❙✐♥❝❡ t♦t❛❧ t❛r❞✐♥❡ss ♠✐♥✐♠✐③❛t✐♦♥
❝♦rr❡s♣♦♥❞s t♦ ❛ r❡❣✉❧❛r ♦♣t✐♠✐③❛t✐♦♥ ❝r✐t❡r✐♦♥✱ ❛ s♦❧✉t✐♦♥ t♦ t❤❡ ♣r♦❜❧❡♠ ✐s ❛ s❝❤❡❞✉❧❡
❞❡✜♥❡❞ ❜② t❤❡ ♦♣❡r❛t✐♦♥ s❡q✉❡♥❝❡s ♦♥ t❤❡ ♠❛❝❤✐♥❡s✳ ❍❡r❡✱ t❤❡s❡ s❡q✉❡♥❝❡s ❛r❡ ❣✐✈❡♥ ❜② ❛
❧✐st ♦r ♣❡r♠✉t❛t✐♦♥ ♦❢ ❛❧❧ ♦♣❡r❛t✐♦♥s✳ ❚✇♦ ✇❡❧❧✲❦♥♦✇♥ str❛t❡❣✐❡s ❛r❡ ✐♠♣❧❡♠❡♥t❡❞ t♦ s❡t ✉♣
❛ ♥❡✐❣❤❜♦r❤♦♦❞✳ ❋✐rst✱ ❛ ♥❡✐❣❤❜♦r ✐s ❞❡t❡r♠✐♥❡❞ ❜② ❛♥ ❛❞❥❛❝❡♥t ♣❛✐r✇✐s❡ ✐♥t❡r❝❤❛♥❣❡ ✭❆P■✮
♦❢ t✇♦ ♦♣❡r❛t✐♦♥s ♦♥ ❛ ♠❛❝❤✐♥❡✳ ❙❡❝♦♥❞✱ ❛ ♥❡✐❣❤❜♦r ✐s ❞❡✜♥❡❞ ❜② ❛ r❛♥❞♦♠ ❧❡❢t✇❛r❞ s❤✐❢t
♦❢ ❛❧❧ ♦♣❡r❛t✐♦♥s ♦❢ ❛ ❥♦❜ ✐♥ t❤❡ ♣❡r♠✉t❛t✐♦♥✳
❲❤✐❧❡ t❤❡s❡ ♥❡✐❣❤❜♦r❤♦♦❞s ❛r❡ s✉❝❝❡ss❢✉❧❧② ❛♣♣❧✐❡❞ t♦ ❥♦❜✲s❤♦♣ ♣r♦❜❧❡♠s ✇✐t❤♦✉t ❛❞✲
❞✐t✐♦♥❛❧ ❝♦♥str❛✐♥ts ❛♥❞ ❝❤❛r❛❝t❡r✐st✐❝s ❧✐❦❡ ❝♦♥♥❡❝t❡❞♥❡ss ❝❛♥ ❜❡ s❤♦✇♥ ❡❛s✐❧②✱ t❤❡r❡ ❛r❡
s✐❣♥✐✜❝❛♥t ❢❡❛s✐❜✐❧✐t② ✐ss✉❡s ♦❝❝✉rr✐♥❣ ✐♥ t❤❡ ❇❏❙P✳ ❙✐♥❝❡ ❛ ❣✐✈❡♥ ♣❡r♠✉t❛t✐♦♥ ❞♦❡s ♥♦t
♥❡❝❡ss❛r✐❧② ❝♦rr❡s♣♦♥❞ t♦ ❛ ❢❡❛s✐❜❧❡ s❝❤❡❞✉❧❡✱ ❝♦♠♣❧❡① ❝♦♥str✉❝t✐♦♥ ❛♥❞ r❡♣❛✐r ♣r♦❝❡❞✉r❡s
❤❛✈❡ t♦ ❜❡ ✉s❡❞ t♦ ❞❡✜♥❡ ❢❡❛s✐❜❧❡ ♥❡✐❣❤❜♦rs✳ ❚❤❡r❡❢♦r❡✱ ♣❡r❢♦r♠❛♥❝❡ ❛♥❞ ❝❤❛r❛❝t❡r✐st✐❝s ♦❢
t❤❡s❡ ♥❡✐❣❤❜♦r❤♦♦❞ str✉❝t✉r❡s ♥❡❡❞ ✐♥✲❞❡♣t❤ ✐♥✈❡st✐❣❛t✐♦♥ ❢♦r t❤❡ ❇❏❙P ✳
■♥ t❤✐s ♣❛♣❡r✱ s♣❡❝✐❛❧ ❡♠♣❤❛s✐s ✐s ❣✐✈❡♥ t♦ t❤❡ ❛❞❥✉st♠❡♥t ♦❢ t❤❡ ♥❡✐❣❤❜♦r❤♦♦❞ t♦ t❤❡
♦♣t✐♠✐③❛t✐♦♥ ❝r✐t❡r✐♦♥✳ ■♥ ❧✐♥❡ ✇✐t❤ t❤❡ ✐❞❡❛ ♦❢ ♦❜s❡r✈✐♥❣ ❛ ❝r✐t✐❝❛❧ ♣❛t❤ ❢♦r ❛ ♠❛❦❡s♣❛♥ ♦❜✲
❥❡❝t✐✈❡✱ ♥❡✐❣❤❜♦rs ❛r❡ ❞❡✜♥❡❞ ❜❛s❡❞ ♦♥ ❝❤♦✐❝❡s ♦❢ ✐♥t❡r❝❤❛♥❣❡s ❛♥❞ s❤✐❢ts ♠❛❞❡ ❢r♦♠ t❤❡ s❡t
♦❢ t❛r❞② ❥♦❜s✳ ●❡♥❡r❛❧ ❛♥❞ t❛r❞✐♥❡ss✲❜❛s❡❞ ♥❡✐❣❤❜♦r❤♦♦❞s ❛r❡ ❞❡s❝r✐❜❡❞ ❛♥❞ ✐♠♣❧❡♠❡♥t❡❞ ✐♥
❛ s✐♠✉❧❛t❡❞ ❛♥♥❡❛❧✐♥❣ ✭❙❆✮ ♠❡t❛❤❡✉r✐st✐❝✳ ❈♦♠♣✉t❛t✐♦♥❛❧ ❡①♣❡r✐♠❡♥ts ❛r❡ ❞♦♥❡ ♦♥ tr❛✐♥
s❝❤❡❞✉❧✐♥❣✲✐♥s♣✐r❡❞ ✐♥st❛♥❝❡s ❛s ✇❡❧❧ ❛s ♦♥ ❜❡♥❝❤♠❛r❦ ✐♥st❛♥❝❡s ❢r♦♠ ▲❛✇r❡♥❝❡ ✭✶✾✽✹✮✳
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❈♦♥❝❧✉s✐♦♥s ❛r❡ ❞r❛✇♥ r❡❣❛r❞✐♥❣ t❤❡ s♦❧✉t✐♦♥ q✉❛❧✐t② ♦❢ t❤❡ ❤❡✉r✐st✐❝ ♠❡t❤♦❞s ❝♦♠♣❛r❡❞
t♦ t❤❡ r❡s✉❧ts ♦❜t❛✐♥❡❞ ❜② s♦❧✈✐♥❣ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ▼■P ❢♦r♠✉❧❛t✐♦♥s✳
✷ Pr♦❜❧❡♠ ❞❡s❝r✐♣t✐♦♥
❆ s❡t ♦❢ ❥♦❜s J = {Ji | i = 1, ..., n} ✐s ❣✐✈❡♥✱ ✇❤❡r❡ ❡❛❝❤ ❥♦❜ ❝♦♥s✐sts ♦❢ ❛ s❡t ♦❢
♦♣❡r❛t✐♦♥s ❛♥❞ Oi,j ❞❡♥♦t❡s t❤❡ j✲t❤ ♦♣❡r❛t✐♦♥ ♦❢ ❥♦❜ Ji✳ ❚❤❡ t❡❝❤♥♦❧♦❣✐❝❛❧ r♦✉t❡ ♦❢ ❛ ❥♦❜
Ji ✐s ❞❡✜♥❡❞ ❜② t❤❡ r❡q✉✐r❡♠❡♥t ♦❢ ❛ ❝❡rt❛✐♥ ♠❛❝❤✐♥❡ Mk ∈ M ❜② ❡❛❝❤ ♦♣❡r❛t✐♦♥✱ ✇❤❡r❡
M ❞❡s❝r✐❜❡s t❤❡ s❡t ♦❢ ♠❛❝❤✐♥❡s✳ ❆❞❞✐t✐♦♥❛❧❧②✱ r❡❧❡❛s❡ ❞❛t❡s ri ❛♥❞ ❞✉❡ ❞❛t❡s di ❛r❡ ❣✐✈❡♥
❢♦r Ji ∈ J ❛♥❞ r❡❝✐r❝✉❧❛t✐♦♥ ✐s ❛❧❧♦✇❡❞✳ ❆♠♦♥❣ ❛❧❧ s❝❤❡❞✉❧❡s✱ ✇❤✐❝❤ ❛r❡ ❢❡❛s✐❜❧❡ ✇✐t❤ r❡❣❛r❞
t♦ t❡❝❤♥♦❧♦❣✐❝❛❧ r♦✉t❡ ❛♥❞ ❜❧♦❝❦✐♥❣ ❝♦♥str❛✐♥ts✱ ❛ s❝❤❡❞✉❧❡ ✇✐t❤ ♠✐♥✐♠❛❧ t♦t❛❧ t❛r❞✐♥❡ss ✐s
t♦ ❜❡ ❢♦✉♥❞✳ ❚❤❡ ❝♦♥s✐❞❡r❡❞ ❇❏❙P ✐s ❝❤❛r❛❝t❡r✐③❡❞ ❜② J | ri, di, block, recr |
P
Ti✳
❆ s❝❤❡❞✉❧❡ ❝❛♥ ❜❡ ❡①♣r❡ss❡❞ ❜② ❛♥ ♦♣❡r❛t✐♦♥✲❜❛s❡❞ r❡♣r❡s❡♥t❛t✐♦♥ sop
❝♦rr❡s♣♦♥❞✐♥❣ t♦
t❤❡ ♣❡r♠✉t❛t✐♦♥ ♦❢ ♦♣❡r❛t✐♦♥s ❛♥❞ ❜② ❛ ♠❛❝❤✐♥❡✲❜❛s❡❞ r❡♣r❡s❡♥t❛t✐♦♥ sma
❝♦rr❡s♣♦♥❞✐♥❣
t♦ t❤❡ ♦♣❡r❛t✐♦♥ s❡q✉❡♥❝❡s ♦♥ t❤❡ ♠❛❝❤✐♥❡s✳ ❚❤✉s✱ ❡✈❡r② ♦♣❡r❛t✐♦♥ Oi,j ✐s ❛ss✐❣♥❡❞ t♦ ❛
❧✐st ✐♥❞❡① lidx(Oi,j) ∈ {1, 2, . . . , nop} ✐♥ sop
✱ ✇❤❡r❡ nop ❞❡♥♦t❡s t❤❡ ♥✉♠❜❡r ♦❢ ♦♣❡r❛t✐♦♥s✱
❛♥❞ t♦ ❛ ♠❛❝❤✐♥❡ ✐♥❞❡① midx(Oi,j) ∈ {1, 2, . . . , Rk} ✐♥ sma
✱ ✇❤❡r❡ Rk ❞❡♥♦t❡s t❤❡ ♥✉♠❜❡r
♦❢ ♦♣❡r❛t✐♦♥s ♦♥ ♠❛❝❤✐♥❡ Mk✳ ❚❤❡s❡ ✐♥❞❡①❡s ❝❛♥ ❛❧s♦ ❜❡ r❡❢❡rr❡❞ t♦ ❛s ♣♦s✐t✐♦♥s ✐♥ t❤❡
♣❡r♠✉t❛t✐♦♥ ❛♥❞ ♦♥ t❤❡ ♠❛❝❤✐♥❡✱ r❡s♣❡❝t✐✈❡❧②✳
✸ P❡r♠✉t❛t✐♦♥✲❜❛s❡❞ ♥❡✐❣❤❜♦r❤♦♦❞s ❢♦r t❤❡ ❜❧♦❝❦✐♥❣ ❥♦❜✲s❤♦♣ ♣r♦❜❧❡♠ ✇✐t❤
t♦t❛❧ t❛r❞✐♥❡ss ♠✐♥✐♠✐③❛t✐♦♥
❉❡✜♥✐♥❣ ♥❡✐❣❤❜♦rs ❜② ❆P■s ✐s ❛ ✇❡❧❧✲❦♥♦✇♥ str❛t❡❣② ✐♥ ❥♦❜✲s❤♦♣ s❝❤❡❞✉❧✐♥❣✳ ❲✳❧✳♦✳❣✳✱
❛ ♣❛✐r ♦❢ ♦♣❡r❛t✐♦♥s ✇✐❧❧ ♦♥❧② ❜❡ ✐♥t❡r❝❤❛♥❣❡❞✱ ✐❢ t❤❡r❡ ✐s ♥♦ ✐❞❧❡ t✐♠❡ ♦♥ t❤❡ ♠❛❝❤✐♥❡
❜❡t✇❡❡♥ t❤❡s❡ ♦♣❡r❛t✐♦♥s✳ ■♥ t❤✐s ♣❛♣❡r✱ ❛ ❣❡♥❡r❛❧ ❆P■ ♥❡✐❣❤❜♦r❤♦♦❞ ✐s s❡t ✉♣ ❜② ❝❤♦♦s✐♥❣
t❤❡ ♥❡✐❣❤❜♦r✲❞❡✜♥✐♥❣ ❆P■ ❢r♦♠ t❤❡ s❡t ♦❢ ❛❧❧ ♣♦ss✐❜❧❡ ♣❛✐rs ♦❢ ♦♣❡r❛t✐♦♥s✳ ❋✉rt❤❡r♠♦r❡✱
t❤❡ ❚❆P■ ♥❡✐❣❤❜♦r❤♦♦❞ ✐s ❞❡s❝r✐❜❡❞ ❜② ❝❤♦♦s✐♥❣ t❤❡ ♥❡✐❣❤❜♦r✲❞❡✜♥✐♥❣ ❆P■ ❢r♦♠ t❤❡ s❡t ♦❢
♣♦ss✐❜❧❡ ♣❛✐rs ♦❢ ♦♣❡r❛t✐♦♥s ❢♦r ✇❤✐❝❤ t❤❡ s❡❝♦♥❞ ✭❧❡❢t✇❛r❞ s❤✐❢t❡❞✮ ♦♣❡r❛t✐♦♥ ❜❡❧♦♥❣s t♦ ❛
t❛r❞② ❥♦❜✳ ❇♦t❤ ♥❡✐❣❤❜♦r❤♦♦❞s ❛r❡ s❡t ✉♣ ✉s✐♥❣ t❤❡ ♠❛❝❤✐♥❡✲❜❛s❡❞ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ t❤❡
s♦❧✉t✐♦♥✳
■♥ ♦r❞❡r t♦ ✐♥✈♦❧✈❡ s♦♠❡ r❛♥❞♦♠♥❡ss ✐♥ t❤❡ ♦♣t✐♠✐③❛t✐♦♥ ♣r♦❝❡ss✱ t❤❡ ❚❏ ♥❡✐❣❤❜♦r❤♦♦❞
✐s ❞❡✜♥❡❞ ❛♥❞ ♦♣❡r❛t❡❞ ♦♥ t❤❡ ♦♣❡r❛t✐♦♥✲❜❛s❡❞ r❡♣r❡s❡♥t❛t✐♦♥✳ ❍❡r❡✱ ❛ ❥♦❜ ✐s r❛♥❞♦♠❧②
❝❤♦s❡♥ ❢r♦♠ t❤❡ s❡t ♦❢ t❛r❞② ❥♦❜s ❛♥❞ ❛❧❧ ✐ts ♦♣❡r❛t✐♦♥s ❛r❡ s❤✐❢t❡❞ t♦ ❛r❜✐tr❛r② ♣♦s✐t✐♦♥s
✇✐t❤ ❧♦✇❡r ❧✐st ✐♥❞❡①❡s ✐♥ t❤❡ ♣❡r♠✉t❛t✐♦♥✳
❆❧❧ t❤r❡❡ ♥❡✐❣❤❜♦r❤♦♦❞s ❛r❡ ❡①❡♠♣❧❛r② ✐❧❧✉str❛t❡❞ ❢♦r ❛♥ ✐♥st❛♥❝❡ ✇✐t❤ ✸ ♠❛❝❤✐♥❡s ❛♥❞
✹ ❥♦❜s ✐♥ ❋✐❣✉r❡ ✸✳ ❆ ❢❡❛s✐❜❧❡ s❝❤❡❞✉❧❡ ✐s ❣✐✈❡♥✱ ✇❤❡r❡ ❥♦❜ J4 ✐s t❛r❞② ❛♥❞ ❥♦❜ J2 ✐s ✜♥✐s❤❡❞
♦♥ t✐♠❡✳ ❆ ♥❡✐❣❤❜♦r ✐♥ t❤❡ ❚❏ ♥❡✐❣❤❜♦r❤♦♦❞ ✐s ❣❡♥❡r❛t❡❞ ❜② s❤✐❢t✐♥❣ ❛❧❧ ♦♣❡r❛t✐♦♥ ♦❢ t❤❡
t❛r❞② ❥♦❜ J4 t♦ ♣♦s✐t✐♦♥s ✇✐t❤ ❧♦✇❡r ❧✐st ✐♥❞❡①❡s✳ ❆s ❛♥ ❡①❛♠♣❧❡ ♦❢ ❛ ❚❆P■ ♥❡✐❣❤❜♦r✱ t❤❡
♣❛✐r O3,1 ❛♥❞ O4,2 ♦♥ ♠❛❝❤✐♥❡ M1 ✐s ❝❤♦s❡♥✱ s✐♥❝❡ ♦♣❡r❛t✐♦♥ O4,2 ❜❡❧♦♥❣s t♦ t❤❡ t❛r❞② ❥♦❜
J4✳ ■♥ t❤❡ ♠♦r❡ ❣❡♥❡r❛❧ ❆P■ ♥❡✐❣❤❜♦r❤♦♦❞✱ ♦♥❡ ♣♦ss✐❜❧❡ ♥❡✐❣❤❜♦r✲❞❡✜♥✐♥❣ ❆P■ r❡✈❡rs❡s t❤❡
♦r❞❡r ♦❢ t❤❡ ♦♣❡r❛t✐♦♥s O3,2 ❛♥❞ O2,2 ♦♥ ♠❛❝❤✐♥❡ M2✳
■♥ ♦♣❡r❛t✐♥❣ t❤❡s❡ t❤r❡❡ ♥❡✐❣❤❜♦r❤♦♦❞s✱ t❤❡ r❡s✉❧t✐♥❣ ♦♣❡r❛t✐♦♥s✲❜❛s❡❞ r❡♣r❡s❡♥t❛t✐♦♥s
❛r❡ ✐♥❢❡❛s✐❜❧❡ ✇✐t❤ r❡❣❛r❞ t♦ ❜❧♦❝❦✐♥❣ ❝♦♥str❛✐♥ts ❢♦r ♠♦st ♦❢ t❤❡ ♥❡✐❣❤❜♦rs✳ ❆ ❝♦♠♣❧❡① r❡✲
♣❛✐r ♣r♦❝❡❞✉r❡ ✐s ❛♣♣❧✐❡❞ t♦ ❝♦♥str✉❝t ❢❡❛s✐❜❧❡ ♥❡✐❣❤❜♦rs ✇❤✐❧❡ t❛❦✐♥❣ t❤❡ ♥❡✐❣❤❜♦r✲❞❡✜♥✐♥❣
❆P■ ❛s ❣✐✈❡♥✳ ❇② ❞♦✐♥❣ s♦✱ ♥❡❝❡ss❛r② ❝❤❛♥❣❡s ✐♥ t❤❡ s❝❤❡❞✉❧❡ ❛r❡ ♠❛❞❡ t♦ r❡❣❛✐♥ ❢❡❛s✐❜✐❧✲
✐t②✳ ●rö✢✐♥ ❛♥❞ ❑❧✐♥❦❡rt ✭✷✵✵✾✮ ♣r❡s❡♥t ❛ ❝♦♥♥❡❝t❡❞ ♥❡✐❣❤❜♦r❤♦♦❞ ❢♦r t❤❡ ❇❏❙P ❜❛s❡❞ ♦♥
❛ ❥♦❜✲✐♥s❡rt✐♦♥ t❡❝❤♥✐q✉❡ ✇✐t❤ ❛♥ ✉♥❞❡r❧②✐♥❣ ❞✐s❥✉♥❝t✐✈❡ ❣r❛♣❤✳ ❚❤❡ ♥❡✐❣❤❜♦r❤♦♦❞s ❝♦♥✲
s✐❞❡r❡❞ ✐♥ t❤✐s ♣❛♣❡r ❛r❡ ♠♦r❡ ❣❡♥❡r❛❧ ❛♥❞ ♦♣❡r❛t❡ ♦♥ ❛ s✐♠♣❧❡ ❧✐st str✉❝t✉r❡✳ ❙✐♥❝❡ t❤❡
❝♦♥♥❡❝t✐✈✐t② ✐s ♥♦t ②❡t s❤♦✇♥✱ ❝♦♠♣✉t❛t✐♦♥❛❧ ❡①♣❡r✐♠❡♥ts ❛r❡ ❛ ❣♦♦❞ ✐♥❞❡① ♦❢ ♣❡r❢♦r♠❛♥❝❡
133

❋✐❣✳ ✶✳ ■❧❧✉str❛t✐♦♥ ♦❢ t❤r❡❡ ❞✐✛❡r❡♥t ♥❡✐❣❤❜♦rs ♦❢ ❛ s❝❤❡❞✉❧❡
❛♥❞ ♣♦t❡♥t✐❛❧✳ ❋✉rt❤❡r♠♦r❡✱ ✐t ✐s ♥♦t ❝❧❡❛r ✇❤❡t❤❡r ❛ r❡str✐❝t✐♦♥ ♦❢ t❤❡ ❆P■ ♥❡✐❣❤❜♦r❤♦♦❞
❜❛s❡❞ ♦♥ t❤❡ ♦♣t✐♠✐③❛t✐♦♥ ❝r✐t❡r✐♦♥ ✐s r❡❛s♦♥❛❜❧❡ ❢♦r t❤❡ ♠✐♥✐♠✐③❛t✐♦♥ ♦❢ t♦t❛❧ t❛r❞✐♥❡ss✳
✹ ❈♦♠♣✉t❛t✐♦♥❛❧ ❘❡s✉❧ts
❚❤❡ ❝♦♠♣✉t❛t✐♦♥❛❧ ❡①♣❡r✐♠❡♥ts ❛r❡ ❞♦♥❡ ♦♥ r❛♥❞♦♠❧② ❣❡♥❡r❛t❡❞ tr❛✐♥ s❝❤❡❞✉❧✐♥❣✲
✐♥s♣✐r❡❞ ✐♥st❛♥❝❡s ✭❚❙✮ ❛s ✇❡❧❧ ❛s ♦♥ ▲❛✇r❡♥❝❡ ✐♥st❛♥❝❡s ✭▲❆✮ ❛❞❛♣t❡❞ ❢♦r t❤❡ ❇❏❙P✳
❚❤❡ r❡❧❡❛s❡ ❞❛t❡s ri ♦❢ t❤❡ ❥♦❜s ❛r❡ ❣❡♥❡r❛t❡❞ s♦ t❤❛t ❥♦❜s ❛r❡ ❢♦r❝❡❞ t♦ ♦✈❡r❧❛♣ ✐♥ t✐♠❡ ❛♥❞
t❤❡ ❞✉❡ ❞❛t❡s ❛r❡ ❞❡t❡r♠✐♥❡❞ ❜② di = δ ·
P
pi,j ✇✐t❤ ❛ t✐❣❤t ❞✉❡ ❞❛t❡ ❢❛❝t♦r ♦❢ δ = 1.2✳ ❚❤❡
s✐③❡ ♦❢ t❤❡ ✐♥st❛♥❝❡s ✐s ❞❡♥♦t❡❞ ❜② (m, n)✱ ✇❤❡r❡ m ❝♦rr❡s♣♦♥❞s t♦ t❤❡ ♥✉♠❜❡r ♦❢ ♠❛❝❤✐♥❡s
❛♥❞ n ✐♥❞✐❝❛t❡s t❤❡ ♥✉♠❜❡r ♦❢ ❥♦❜s✳ ❚❤❡r❡ ❛r❡ ✜✈❡ ❞✐✛❡r❡♥t ✐♥st❛♥❝❡s ❢♦r ❡❛❝❤ ✐♥st❛♥❝❡ s✐③❡✳
❆ s✐♠✉❧❛t❡❞ ❛♥♥❡❛❧✐♥❣ ✭❙❆✮ ✐s ✉s❡❞ t♦ s♦❧✈❡ t❤❡ ❣✐✈❡♥ ♣r♦❜❧❡♠s✱ ✇❤❡r❡ t❤❡ ❚❏ ♥❡✐❣❤✲
❜♦r❤♦♦❞ ✐s ❛♣♣❧✐❡❞ ✇✐t❤ ❛ ♣r♦❜❛❜✐❧✐t② ♦❢ 0.1 ❛♥❞ t❤❡ ❆P■ ❛♥❞ ❚❆P■ ♥❡✐❣❤❜♦r❤♦♦❞s ❛r❡
❝♦♠♣❧❡♠❡♥t❛r② ❛♣♣❧✐❡❞ ✇✐t❤ ❛ ♣r♦❜❛❜✐❧✐t② ♦❢ 0.9✱ r❡s♣❡❝t✐✈❡❧②✳ ❆ ❣❡♦♠❡tr✐❝ ❝♦♦❧✐♥❣ s❝❤❡♠❡
ti+1 = k · ti ✐s ✉s❡❞ ✇✐t❤ k ∈ {0.99, 0.995, 0.999}✳ ❚❤❡ ✐♥✐t✐❛❧ ❛♥❞ t❡r♠✐♥❛❧ t❡♠♣❡r❛t✉r❡✱
t0 ❛♥❞ T✱ ❛r❡ ❝❤♦s❡♥ ✐♥ ❛❝❝♦r❞❛♥❝❡ t♦ t❤❡ r❛♥❣❡ ♦❢ t❤❡ ♦❜❥❡❝t✐✈❡ ❢✉♥❝t✐♦♥ ✈❛❧✉❡s ✇✐t❤
(t0, T) ∈ {(20, 10), (200, 50), (1000, 100)}✳ ❚❤❡ t♦t❛❧ ♥✉♠❜❡r ♦❢ ✐t❡r❛t✐♦♥s ❞♦♥❡ ❜② t❤❡ ❙❆
r❛♥❣❡s ❞❡♣❡♥❞❡♥t ♦♥ t❤❡ ✐♥st❛♥❝❡ s✐③❡ ❜❡t✇❡❡♥ ✶✶✵✵✵ ❛♥❞ ✻✹✵✵✵✳
❚❤❡ ❜❡st ♦✉t ♦❢ t❤❡ ✜✈❡ r✉♥s ❢♦r ❡❛❝❤ ✐♥st❛♥❝❡ ✭✇✳r✳t✳ t❤❡ ♦❜❥❡❝t✐✈❡ ❢✉♥❝t✐♦♥ ✈❛❧✉❡✮ ✐s
❝♦♠♣❛r❡❞ t♦ t❤❡ r❡s✉❧ts ♦❜t❛✐♥❡❞ ❜② s♦❧✈✐♥❣ ❛ ▼■P ❢♦r♠✉❧❛t✐♦♥ ❢♦r t❤❡ ❇❏❙P ✇✐t❤ ■❇▼
■▲❖● ❈P▲❊❳ ✶✷✳✻✳✶✳ ❛s ❣✐✈❡♥ ✐♥ ▲❛♥❣❡ ❛♥❞ ❲❡r♥❡r ✭✷✵✶✼✮✳ ❚❤❡ ❝♦♠♣✉t❛t✐♦♥❛❧ ❡①♣❡r✐♠❡♥ts
✐♥✈♦❧✈✐♥❣ t❤❡ ▼■P s♦❧✈❡r✱ t❤❡ ❙❆ ✇✐t❤ t❛r❞✐♥❡ss✲❜❛s❡❞ ♥❡✐❣❤❜♦r❤♦♦❞s ✭❚❏✱ ❚❆P■✮ ❛♥❞ t❤❡
❙❆ ♠❛✐♥❧② r❡❧②✐♥❣ ♦♥ t❤❡ ❣❡♥❡r❛❧ ♥❡✐❣❤❜♦r❤♦♦❞ ✭❚❏✱ ❆P■✮ ❛r❡ s✉♠♠❛r✐③❡❞ ✐♥ ❚❛❜❧❡ ✶✳ ❋♦r
❡❛❝❤ ✐♥st❛♥❝❡ s✐③❡✱ t❤❡ ♥✉♠❜❡r ♦❢ ✐♥st❛♥❝❡s s♦❧✈❡❞ t♦ ♦♣t✐♠❛❧✐t② ✭♦♣t✮ ❜② t❤❡ ▼■P s♦❧✈❡r
❛♥❞ t❤❡ ♥✉♠❜❡r ♦❢ ✐♥st❛♥❝❡s✱ ❢♦r ✇❤✐❝❤ ❛ ❢❡❛s✐❜❧❡ s♦❧✉t✐♦♥ ✐s ❢♦✉♥❞✱ ❛r❡ ❣✐✈❡♥✳ ❇❡❧♦✇✱ t❤❡
♥✉♠❜❡r ♦❢ ✐♥st❛♥❝❡s ❢♦r ✇❤✐❝❤ t❤❡ ✈❛r✐❛♥ts ♦❢ t❤❡ ❙❆ ♦❜t❛✐♥❡❞ t❤❡ ♦♣t✐♠❛❧ s♦❧✉t✐♦♥ ♦r
✐♠♣r♦✈❡❞ t❤❡ ❜❡st ❦♥♦✇♥ ❢❡❛s✐❜❧❡ s♦❧✉t✐♦♥ ✭♦♣t✴✐♠✮ ✐s st❛t❡❞✳ ❆❞❞✐t✐♦♥❛❧❧②✱ t❤❡ ♥✉♠❜❡r ♦❢
✐♥st❛♥❝❡s ❢♦r ✇❤✐❝❤ t❤❡ ❙❆ ❛♣♣r♦❛❝❤❡s r❡❛❝❤❡❞ ❛ s♦❧✉t✐♦♥ ✇✐t❤✐♥ ❛ ✶✵✪ ❣❛♣ ❝♦♠♣❛r❡❞ t♦
t❤❡ ❜❡st ❦♥♦✇♥ s♦❧✉t✐♦♥ ❢♦✉♥❞ ❜② t❤❡ ▼■P s♦❧✈❡r ✐s ❣✐✈❡♥✳
❚❤❡ ❝♦♠♣❛r✐s♦♥ ♦❢ t❤❡ ♣❡r❢♦r♠❛♥❝❡ ♦❢ t❤❡ ♥❡✐❣❤❜♦r❤♦♦❞s ✐s ❞♦♥❡ ✜rst❧② ❜❛s❡❞ ♦♥ t❤❡
t♦t❛❧ ♥✉♠❜❡r ♦❢ ✐♥st❛♥❝❡s s♦❧✈❡❞ ✇✐t❤ ✭♥❡❛r✲✮♦♣t✐♠❛❧ s♦❧✉t✐♦♥s ❛♥❞ s❡❝♦♥❞❧② r❡❣❛r❞✐♥❣
t❤❡ ♥✉♠❜❡r ♦❢ ✐♥st❛♥❝❡s s♦❧✈❡❞ t♦ ♦♣t✐♠❛❧✐t② ♦r ✐♠♣r♦✈❡♠❡♥t ❢♦r ❡❛❝❤ ✐♥st❛♥❝❡ s✐③❡✳ ❚❤❡
s✉♣❡r✐♦r s❡tt✐♥❣ ✐s ❡♠♣❤❛s✐③❡❞ ✐♥ ❜♦❧❞ ❢❛❝❡✳ ■t ❝❛♥ ❜❡ ♦❜s❡r✈❡❞ t❤❛t t❤❡ ❙❆ ✇✐t❤ t❤❡ ❆P■
♥❡✐❣❤❜♦r❤♦♦❞ ♦❜t❛✐♥s ❜❡tt❡r ♦r ❡q✉✐✈❛❧❡♥t s♦❧✉t✐♦♥s ❢♦r ❡✐❣❤t ♦❢ t❤❡ t❡♥ ❣✐✈❡♥ ✐♥st❛♥❝❡ s✐③❡s✳
❚❤✐s ✐♥❞✐❝❛t❡s t❤❛t ❞✉❡ t♦ t❤❡ t❛r❞✐♥❡ss✲❜❛s❡❞ ♦♣t✐♠✐③❛t✐♦♥ ❝r✐t❡r✐♦♥ ❛♥❞ ❛ ❤✐❣❤ ♥✉♠❜❡r ♦❢
✐♥t❡r❞❡♣❡♥❞❡♥❝✐❡s ❝❛✉s❡❞ ❜② ❜❧♦❝❦✐♥❣ ❝♦♥str❛✐♥ts t❤❡ ✐❞❡❛ ♦❢ ❛♥ ❛❞❛♣t✐♦♥ t♦ t❤❡ ♦❜❥❡❝t✐✈❡
134

❚❛❜❧❡ ✶✳ ❈♦♠♣✉t❛t✐♦♥❛❧ r❡s✉❧ts ♦❢ ❛♣♣❧②✐♥❣ ❛♥ ❙❆ t♦ t❤❡ ❇❏❙P
❚❙ ✐♥st✳ ▲❛✇r❡♥❝❡ ✐♥st❛♥❝❡s
(m, n) (11, 10) (11, 15) (5, 10) (5, 15) (5, 20) (10, 10) (10, 15) (10, 20) (10, 30) (15, 15)
t♦t❛❧ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺
▼■P
♦♣t ✺ ✸ ✺ ✶ ✲ ✺ ✶ ✲ ✲ ✷
❢❡❛s✐❜❧❡ ✲ ✷ ✲ ✹ ✺ ✲ ✹ ✺ ✶ ✸
❙❆ ✲ ✭❚❏✱ ❚❆P■✮
♦♣t✴✐♠ ✹ ✶ ✸ ✷ ✷ ✸ ✶ ✷ ✺ ✲
10% ✶ ✸ ✷ ✶ ✸ ✶ ✶ ✶ ✲ ✲
❙❆ ✲ ✭❚❏✱ ❆P■✮
♦♣t✴✐♠ ✺ ✲ ✹ ✶ ✸ ✸ ✷ ✶ ✺ ✲
10% ✲ ✷ ✶ ✹ ✷ ✶ ✲ ✷ ✲ ✲
❢✉♥❝t✐♦♥ ❞♦❡s ♥♦t ✐♠♣r♦✈❡ t❤❡ ❆P■ ♥❡✐❣❤❜♦r❤♦♦❞ ❢♦r t❤❡ ❇❏❙P✳ ❙t❛t✐st✐❝❛❧ ✐ss✉❡s✱ ✇❤✐❝❤
❛r❡ ♥♦t ♣r❡s❡♥t❡❞ ✐♥ ❞❡t❛✐❧ ❤❡r❡✱ s❤♦✇ t❤❛t t❤❡ ❧❛r❣❡r ♥✉♠❜❡r ♦❢ ♣♦ss✐❜❧❡ ♥❡✐❣❤❜♦rs ❝❛✉s❡s
❛ ❤✐❣❤❡r ❞❡✈✐❛t✐♦♥ ✐♥ t❤❡ ♦❜❥❡❝t✐✈❡ ❢✉♥❝t✐♦♥ ✈❛❧✉❡s ♦❢ t❤❡ ❜❡st s♦❧✉t✐♦♥s ❢♦✉♥❞ ❜✉t ❧❡❛❞s t♦
❜❡tt❡r r❡s✉❧ts ♦♥ ❛✈❡r❛❣❡✳
✺ ❈♦♥❝❧✉s✐♦♥
■♥ t❤✐s ♣❛♣❡r✱ ❣❡♥❡r❛❧ ❛♥❞ t❛r❞✐♥❡ss✲❜❛s❡❞ ♥❡✐❣❤❜♦r❤♦♦❞ str✉❝t✉r❡s ❢♦r t❤❡ ❇❏❙P ❛r❡
❡♠❜❡❞❞❡❞ ✐♥ ❛ ❙❆ ❛♥❞ t❡st❡❞ ✇✐t❤ r❡❣❛r❞ t♦ t❤❡✐r ♣❡r❢♦r♠❛♥❝❡ ❝♦♠♣❛r❡❞ t♦ ▼■P s♦❧✈✐♥❣
t❡❝❤♥✐q✉❡s✳ ❈♦♠♣✉t❛t✐♦♥❛❧ ❡①♣❡r✐♠❡♥ts ❛r❡ ❞♦♥❡ ♦♥ tr❛✐♥✲s❝❤❡❞✉❧✐♥❣ ✐♥s♣✐r❡❞ ❛♥❞ ❜❡♥❝❤✲
♠❛r❦ ✐♥st❛♥❝❡s✳ ❚❤❡ r❡s✉❧ts ❣✐✈❡ ❡✈✐❞❡♥❝❡ t♦ t❤❡ ❢❛❝t t❤❛t ❛♥ ❛❞❛♣t✐♦♥ ♦❢ t❤❡ ❆P■ ♥❡✐❣❤✲
❜♦r❤♦♦❞ t♦ t❤❡ ♦❜❥❡❝t✐✈❡ ❜② ❡①❝❧✉s✐✈❡❧② ❝❤♦♦s✐♥❣ ❧❡❢t✇❛r❞ ✐♥t❡r❝❤❛♥❣❡❞ ♦♣❡r❛t✐♦♥s ♦❢ t❛r❞②
❥♦❜s ❞♦❡s ♥♦t ✐♠♣r♦✈❡ t❤❡ s♦❧✉t✐♦♥ q✉❛❧✐t②✳ ❙✐❣♥✐✜❝❛♥t ❢❡❛s✐❜✐❧✐t② ✐ss✉❡s ✐♥✈♦❧✈❡❞ ❜② t❤❡
❜❧♦❝❦✐♥❣ ❝♦♥str❛✐♥ts s❡❡♠ t♦ ♥❡❝❡ss✐t❛t❡ ❛ ❧❛r❣❡r ♥✉♠❜❡r ♦❢ ♣♦ss✐❜❧❡ ♥❡✐❣❤❜♦rs t♦ ♦❜t❛✐♥
✭♥❡❛r✲✮♦♣t✐♠❛❧ s♦❧✉t✐♦♥s ✇✐t❤ ❤✐❣❤❡r ❢r❡q✉❡♥❝②✳
❘❡❢❡r❡♥❝❡s
❇r✐③✉❡❧❛✱ ❈✳✱ ❨✳ ❩❤❛♦ ❛♥❞ ◆✳ ❙❛♥♥♦♠✐②❛✱ ✷✵✵✶✱ ✬◆♦✲✇❛✐t ❛♥❞ ❇❧♦❝❦✐♥❣ ❏♦❜✲❙❤♦♣s✿ ❈❤❛❧❧❡♥❣✐♥❣
♣r♦❜❧❡♠s ❢♦r ●❆s✬✱ ✐♥ ■❊❊❊ ■♥t❡r♥❛t✐♦♥❛❧ ❈♦♥❢❡r❡♥❝❡ ♦♥ ❙②st❡♠s✱ ▼❛♥ ❛♥❞ ❈②❜❡r♥❡t✐❝s✱ ♣♣✳
✷✸✹✾✕✷✸✺✹✳
❇ür❣②✱ ❘✳✱ ✷✵✶✼✱ ✬❆ ♥❡✐❣❤❜♦r❤♦♦❞ ❢♦r ❝♦♠♣❧❡① ❥♦❜ s❤♦♣ s❝❤❡❞✉❧✐♥❣ ♣r♦❜❧❡♠s ✇✐t❤ r❡❣✉❧❛r ♦❜❥❡❝✲
t✐✈❡s✬✱ ❏♦✉r♥❛❧ ♦❢ ❙❝❤❡❞✉❧✐♥❣✱ ❱♦❧✳ ✷✵✱ ♣♣✳ ✸✾✶✕✹✷✷✳
❉✬❆r✐❛♥♦✱ ❆✳✱ ❉✳ P❛❝❝✐❛r❡❧❧✐ ❛♥❞ ▼✳ Pr❛♥③♦✱ ✷✵✵✼✱ ✬❆ ❜r❛♥❝❤ ❛♥❞ ❜♦✉♥❞ ❛❧❣♦r✐t❤♠ ❢♦r s❝❤❡❞✉❧✐♥❣
tr❛✐♥s ✐♥ ❛ r❛✐❧✇❛② ♥❡t✇♦r❦✬✱ ❊✉r♦♣❡❛♥ ❏♦✉r♥❛❧ ♦❢ ❖♣❡r❛t✐♦♥❛❧ ❘❡s❡❛r❝❤✱ ❱♦❧✳ ✶✽✸✱ ♣♣✳ ✻✹✸✕✻✺✼✳
●rö✢✐♥ ❍✳✱ ❆✳ ❑❧✐♥❦❡rt✱ ✷✵✵✾✱ ✬❆ ♥❡✇ ♥❡✐❣❤❜♦r❤♦♦❞ ❛♥❞ t❛❜✉ s❡❛r❝❤ ❢♦r t❤❡ ❜❧♦❝❦✐♥❣ ❥♦❜ s❤♦♣✬✱
❉✐s❝r❡t❡ ❆♣♣❧✐❡❞ ▼❛t❤❡♠❛t✐❝s✱ ❱♦❧✳ ✶✺✼✱ ♣♣✳ ✸✻✹✸✕✸✻✺✺✳
❑✉❤♣❢❛❤❧✱ ❏✳✱ ❈✳ ❇✐❡r✇✐rt❤✱ ✷✵✶✻✱ ✬❆ st✉❞② ♦♥ ❧♦❝❛❧ s❡❛r❝❤ ♥❡✐❣❤❜♦r❤♦♦❞s ❢♦r t❤❡ ❥♦❜ s❤♦♣ s❝❤❡❞✉❧✲
✐♥❣ ♣r♦❜❧❡♠ ✇✐t❤ t♦t❛❧ ✇❡✐❣❤t❡❞ t❛r❞✐♥❡ss ♦❜❥❡❝t✐✈❡✬✱ ❈♦♠♣✉t❡rs ✫ ❖♣❡r❛t✐♦♥s ❘❡s❡❛r❝❤✱ ❱♦❧✳
✻✻✱ ♣♣✳ ✹✹✕✺✼✳
▲❛♥❣❡✱ ❏✳✱ ❋✳ ❲❡r♥❡r✱ ✷✵✶✼✱ ✬❆♣♣r♦❛❝❤❡s t♦ ♠♦❞❡❧✐♥❣ tr❛✐♥ s❝❤❡❞✉❧✐♥❣ ♣r♦❜❧❡♠s ❛s ❥♦❜✲s❤♦♣ ♣r♦❜✲
❧❡♠s ✇✐t❤ ❜❧♦❝❦✐♥❣ ❝♦♥str❛✐♥ts✬✱ ❏♦✉r♥❛❧ ♦❢ ❙❝❤❡❞✉❧✐♥❣✱ ♣✉❜❧✐s❤❡❞ ♦♥❧✐♥❡✱ ❉❖■ ✶✵✳✶✵✵✼✴s✶✵✾✺✶✲
✵✶✼✲✵✺✷✻✲✵✳
▲❛✇r❡♥❝❡✱ ❙✳✱ ✶✾✽✹✱ ✬❙✉♣♣❧❡♠❡♥t t♦ r❡s♦✉r❝❡ ❝♦♥str❛✐♥❡❞ ♣r♦❥❡❝t s❝❤❡❞✉❧✐♥❣✿ ❛♥ ❡①♣❡r✐♠❡♥t❛❧ ✐♥✲
✈❡st✐❣❛t✐♦♥ ♦❢ ❤❡✉r✐st✐❝ s❝❤❡❞✉❧✐♥❣ t❡❝❤♥✐q✉❡s✬✱ ●❙■❆✱ ❈❛r♥❡❣✐❡ ▼❡❧❧♦♥ ❯♥✐✈❡rs✐t②✱ P✐tts❜✉r❣❤✳
❖❞❞✐✱ ❆✳✱ ❘✳ ❘❛s❝♦♥✐✱ ❆✳ ❈❡st❛ ❛♥❞ ❙✳ ❙♠✐t❤✱ ✷✵✶✷✱ ✬■t❡r❛t✐✈❡ ✐♠♣r♦✈❡♠❡♥t ❛❧❣♦r✐t❤♠s ❢♦r t❤❡
❜❧♦❝❦✐♥❣ ❥♦❜ s❤♦♣✬✱ ✐♥ ■❈❆P❙✳
135

A parallel machine scheduling problem with equal
processing time jobs, release dates and eligibility
constraints to minimize total completion time
Kangbok Lee and Juntaek Hong
Pohang University of Science and Technology, Korea
{kblee,hongjt3000}@postech.ac.kr
Keywords: equal processing time jobs, release dates, eligibility constraints, total comple-
tion time.
1 Introduction
We consider a problem of scheduling n jobs on m identical parallel machines to minimize
the total completion time. Let J = {1, . . . , n} be the set of jobs and M = {1, . . . , m} be the
set of machines. Job j has a given release date rj and a set of machines that can process
job j, which is called the eligible set of job j and denoted by Mj for j ∈ J. All jobs have an
equal processing time, denoted by p. This problem is denoted by P|rj, Mj, pj = p|
∑
Cj.
When the number of machines is considered a fixed constant, the problem is denoted by
Pm|rj, Mj, pj = p|
∑
Cj.
There are a lot of papers on the parallel machine scheduling to minimize the total
completion time. It is known that R||
∑
Cj can be solved in O(n3
) time by assignment
formulation (Bruno et al. 1974, Horn 1973). Note that P|Mj|
∑
Cj is a special case of
R||
∑
Cj because we can consider that pij = pj for i ∈ Mj and pij = ∞ for i ̸∈ Mj. Thus,
P|Mj|
∑
Cj can be solved in O(n3
) time as well.
If release dates are involved, the complexity of the problem changes. While Lenstra
et al. (1977) showed that 1|rj|
∑
Cj is strongly NP-hard, 1|rj, prmt|
∑
Cj can be solved
by Shortest Remaining Processing Time (SRPT) rule optimally. Brucker and Kravchenko
(2004) proved that P|rj, prmt|
∑
Cj is strongly NP-hard.
As for equal processing time jobs cases, the problem P|rj, pj = p, Dj|
∑
Cj where
job j has a deadline Dj can be solved in O(mn2
) time (Simons and Warmuth 1989).
Qm|rj, pj = p|
∑
Cj is solvable in O(mn2m+1
) time (Dessouky et al. 1990). However, the
complexity for Q|rj, pj = p|
∑
Cj with an arbitrary number of machines is still open. For
preemptive case, Brucker and Kravchenko (2005) showed that P|rj, pj = p, prmt|
∑
Cj can
be solved in polynomial time by providing a linear programming formulation. Kravchenko
and Werner (2009) generalized the previous result for the problem Q|rj, pj = p, prmt|
∑
Cj
and provided a more complicated linear programming formulation with O(mn3
) variables
and constraints to solve the problem in polynomial time.
In Section 2, we show that the problem with a fixed m, Pm|rj, Mj, pj = p|
∑
Cj, can
be solved in polynomial time. For an arbitrary m, it is unknown whether the problem is
polynomial solvable or not. Thus, we present an approximation algorithm for the problem
with an arbitrary m along with worst case analysis in Section 3. Section 4 shows the
experimental results with a modified algorithm and Section 5 concludes the paper.
136

2 Dynamic programming algorithm with fixed m
2.1 Preliminary
Let M = {Mj|j ∈ J}, which is the set of all distinct eligible sets of all jobs. Then,
M ⊂ 2M
∅. Let k = |M| and, without loss of generality, M = {M1
, . . . Mk
}. Note that
k 2m
.
Proposition 1. There exists an optimal schedule in which jobs having Me
∈ M as the
eligible set scheduled at each machine are scheduled in Earliest Release Date first (ERD)
order.
From Proposition 1, we define the partition of the set of jobs as Je
= {j|Mj = Me
} for
e = 1, . . . , k. Let ne
= |Je
|. We assume that jobs in Je
are sorted in ERD order. Let re
(j)
denote the release date of j-th job in Je
for j = 1, . . . , ne
.
Proposition 2. There exists an optimal schedule in which the completion time of job j
scheduled at a machine has a form of rj + ap for some j ∈ J and a ∈ ¸{1, . . . , n}.
From Proposition 2, we define the set of candidates for completion times of jobs as
Λ = {t|t = rj + ap, j ∈ J, a ∈ {1, . . . , n}}. Thus, |Λ| = n2
.
2.2 Dynamic programming algorithm
We consider a partial schedule with first be
jobs from Je
for e = 1, . . . , k in which the
latest completion time of machine i is ti for i ∈ Λ. Let α = (ti : b1
i , b2
i , . . . , be
i , . . . , bk
i ) be
the state of machine i where be
i set of jobs among first be
jobs from Je
are scheduled at
machine i for e = 1, . . . , k, where
∑m
i=1 be
i = be
. Note that if i ̸∈ Me
, then be
i = 0.
The state of a partial schedule is a collection of states of all machines and is denoted
by α = (α1, α2, . . . , αm) where αi denotes the state of machine i and αi has the latest
completion time ti and the collection of be
i ’s, the numbers of scheduled jobs from Me
for
e = 1, . . . , k, i.e., αi = (ti : b1
i , . . . , bk
i ), for i ∈ M.
Let V (α) be the total completion time of the current partial schedule.
Restriction
• ti ∈ Λ for i ∈ M and be
i ∈ {0, 1, . . . , ne
} for e ∈ {1, . . . , k}
Boundary conditions
• V (α0
) = 0 where α0
= (α0
1, α0
2, . . . , α0
m) and α0
i = (0 : 0, . . . , 0) for i ∈ {1, . . . , m}
• V (α) = ∞ if there exists i such that ti 0 or there exist e and i such that be
i 0
Recursive relationship
• V (α) = min{V (α̂(h, f)) + th|h ∈ {1, . . . , m}, f ∈ {1, . . . , k}, rf
(bf
) ≤ th − p, th ∈ Λ}
where
◦ α̂(h, f) = (α̂1, α̂2, . . . , α̂m)
where α̂i = αi for i ̸= h and α̂h = (t̂h : b̂1
h, . . . , b̂k
h)
◦ bf
=
∑m
i=1 bf
i
where b̂e
h = be
h for e ̸= f and b̂f
h = bf
h − 1
◦ t̂h = th − p and t̂h ∈ Λ
• If there does not exist a pair of (h, f), then V (α) = ∞.
Optimal condition
137

• min{V (α)|be
= ne
for e ∈ {1, . . . , k}}
The time complexity for an arbitrary eligibility case is O(m2m−1
n2+(1+2m
)m
), so the
proposed Dynamic Programming (DP) algorithm is polynomial for a fixed m. However,
for an arbitrary m, the complexity of the problem is still unknown. Thus, we propose an
approximation algorithm for it in the next section.
3 Approximation algorithm
Let I be an instance of the problem, P|rj, Mj, pj = p|
∑
Cj, and z(I) be the optimal
objective function value of I. Then, we consider two problem instances that can be defined
from I:
• IL: rj is redefined ad ⌊
rj
p ⌋ × p
• IU : rj is redefined ad ⌈
rj
p ⌉ × p
Since the release dates of IL and IU are integer multiples of p, by scaling, these can be
regarded as problem instances of P|rj, Mj, pj = 1|
∑
Cj where rj is a non-negative integer.
P|rj, Mj, pj = 1|
∑
Cj can be solved by assignment formulation in polynomial time.
The optimal solution of IU is feasible to I, and is a 2-approximation solution of I
because z(IL) ≤ z(I) ≤ z(IU ) and z(IU ) − z(IL) ≤ np. This approximation ratio is tight
from the following example. Consider a problem instance with one job with processing time
p and infinitesimal release date ∆. Since ∆ 0, release date of the job in IU is p. The
optimal objective value of IU is 2p, while optimal objective value of I is p + ∆. As ∆ → 0,
the ratio between optimal objective values of IU and I approaches 2.
4 Experiments and Result
For the practical purpose, we can elaborate the algorithm. After solving with IU , we
can keep machine assignment and the job sequence at each machine while we schedule
jobs as early as possible. From this procedure, we can reduce the objective function value
from z(IU ). For more effect, jobs assigned to the same machine m should be ERD-ordered
according to the original release date rj ∈ I. For this to happen, we propose the following
time-indexed MIP formulation.
Parameters
• m: number of machines
• n: number of jobs
• p: processing time of all jobs
• r′
j: modified release date of job j, (r′
j = ⌊
rj
p ⌋p for IL, r′
j = ⌈
rj
p ⌉p for IU )
• δj = r′
j − rj
• T: the set of possible starting times, T = {0, p, 2p, . . . , max{r′
j} + np}
• Pijt: big number P if i ̸∈ Mj or t r′
j for i ∈ M, j ∈ J, t ∈ T, and 0 otherwise
Variables
• xijt: 1 if job j starts at time t by machine i, 0 otherwise
Minimize
m
∑
i=1
n
∑
j=1
∑
t∈T
txijt(1 + ϵδi) + np + Pijtxijt
138

Subject to
m
∑
i=1
∑
t∈T
xijt = 1, for j ∈ J
n
∑
j=1
xijt ≤ 1, for i ∈ M, t ∈ T
xijt ∈ {0, 1}, for i ∈ M, j ∈ J, t ∈ T.
The term δj in objective function enables jobs with earliest original release dates rj to
be processed earlier than those who have the same r′
j but greater rj. The term Pijtxijt
in objective function eliminates the schedules in which any job violates machine eligibility
and release date constraints.
We can apply this idea to IL as well by keeping machine assignment and the job
sequence at each machine while scheduling jobs as early as possible. The total completion
times from the solutions IL and IU obtained this way will be denoted as z′
(IL) and z′
(IU ),
respectively.
We also consider a simple priority-rule based algorithm, denoted as Greedy, as follows:
• First, choose the job with earliest release date(ERD) rule. If more than one job have
the same ERD, choose the one with smallest eligibility |Mj|.
• Second, choose the machine i ∈ Mj with earliest available time. If a tie happens, choose
the one with smallest job eligibility among remaining jobs.
• Assign chosen job to chosen machine’s job queue, iterate until all jobs are assigned.
The total completion time obtained by Greedy algorithm will be denoted as grd(I).
So far we propose three simple algorithms to obtain a feasible schedule. For given
problem instance, the algorithm’s objective value (ALG) is defined as a minimum of z′
(IL),
z′
(IU ), and grd(I). Optimal objective value (OPT) can be calculated by an exact MIP
formulation or dynamic programming algorithm. In order to evaluate the performance of
the proposed algorithm, we conduct an experiment with randomly generated test instances.
Total 30,000 instances are created with following parameters.
• m ∈ {2, 3, 4, 8, 16}
• n ∈ {2m, 3m, 4m, 5m, 6m}
• p ∈ {2, 3, 4, 8, 16}
• rj ∼ U
(
0, np
m · dr
)
, dr ∈ {0.5, 1, 1.5, 2}
• E(|Mj|) = m · dM, dM ∈ {0.4, 0.6, 0.8} subject to |Mj| ≥ 1 ∀j ∈ J
• Replication per each setting: 20
The longest computation time among 30,000 instances is less than 10 seconds. The
result says 28,604 instances (95.3% of all instances) show ALG = OPT and in only 1,396
instances (4.7% of all), ALG was strictly greater than OPT. The worst case ratio of the
ALG/OPT was 19/18 (≈ 1.056). It shows that the proposed algorithm effectively works
in the worst perspective.
The average ALG/OPT trend according to each parameter is shown in graphs below.
The mean of ALG/OPT increased as number of machines m increases until m = 8,
and decreased when m = 16. The algorithm is expected to perform well when number
of machines is large, and the result is as expected. The ratio between number of jobs n
and m seems to have no effect on ALG/OPT because it neither makes the problem easier
nor harder. The algorithm performs better as processing time p increases. It is considered
reasonable because using the sequence of IL(IU ) on original instance I have the same effect
as pushing jobs backward(forward) in the range of p. The algorithm performs better when
139

Fig. 1. Trend graphs of ALG/OPT.
the mean of release dates is smaller, which is similar to problem with no release dates. The
algorithm performs better when the mean of |Mj|/|M| is larger, which is closer to problem
with no eligibility constraints.
5 Conclusions
We propose a polynomial time DP algorithm for Pm|rj, Mj, pj = p|
∑
Cj with fixed
m. With a slight modification, this result can be extended to Qm|rj, Mj, pj = p|
∑
Cj
without increasing the time complexity by defining Λi = {t|t = rj + ap/vi, j ∈ J, a ∈
{1, . . . , n}} with vi being a speed of machine i as a set of candidate completion times
at machine i for i ∈ M. The proposed approximation algorithm is a 2-approximation
algorithm and its practical modification works very well experimentally in both worst and
average perspectives.
References
Brucker, P. and Kravchenko, S.A., 2004, “Complexity of mean flow time scheduling problems with
release dates”, Universität Osnabrück, Fachbereich Mathematik/Informatik, OSM Reihe P,
Heft 251.
Bruno, J., Coffman, E.G. Jr, and Sethi, R., 1974, “Scheduling independent tasks to reduce mean
finishing time”, Communications of the ACM, Vol. 17(7), pp. 382-–387.
Horn, W.A., 1973, “Technical note –minimizing average flow time with parallel machines”, Oper-
ations Research, Vol. 21(3), pp. 846—47.
Kravchenko, S.A., Werner, F., 2009, “Preemptive scheduling on uniform machines to minimize
mean flow time”, Computers Operations Research, Vol. 36(10), pp. 2816—2821.
Lenstra, J.K., Kan, A.R., Brucker, P., 1977, “Complexity of machine scheduling problems”, Annals
of Discrete Mathematics, Vol. 1, pp. 343—362.
Simons, B.B., and Warmuth, M.K., 1989, “A fast algorithm for multiprocessor scheduling of unit-
length jobs”, SIAM Journal on Computing, Vol. 18, pp. 690—710.
140

A new grey-box approach to solve challenging workforce
planning and activities scheduling problems
Stefano Lucidi1,2
and Ludovica Maccarrone1,2
1
Department of Computer, Control, and Management Engineering,
University of Rome La Sapienza , Italy
lucidi, maccarrone@dis.uniroma1.it
2
ACTOR s.r.l., via Nizza 45, 00198 Rome, Italy
stefano.lucidi, ludovica.maccarrone@act-OperationsResearch.com
Keywords: Workforce management, Project scheduling, Grey-box optimization, MILP.
1 Introduction and motivations
Nowadays, a key success factor for many large enterprises is the ability of properly man-
aging labor cost and timetables. This is the reason why workforce planning and scheduling
tools are now getting more and more developed.
Two are the typical issues arising in such applications: the rst is related to the medium
and long-term goal of estimating the amount of workers that the company will require in
future periods. The second, mostly linked to short-term operations, involves the assignment
of human resources to activities in order to meet deadlines and industrial plans.
In practice, to conduct a complete analysis and evaluate the eectiveness of a solution
it is important to take into account both time and nancial objectives, considering not only
the need of reducing durations and delays but also the ability to do so within reasonable
budgets. The result is a trade-o problem looking at the same time at avoiding resource
underutilization and incapacity to comply with due dates.
In the following, we present a new approach to solve the workforce scheduling problem
in complex applicative contexts such as manufacturing and logistics, characterized by the
simultaneous processing of several activities, the occupation of wide areas, the coexistence
of independent workloads, the use of advanced machineries and, above all, the employment
of dierent types of operators, having various abilities and experience levels.
Standard approaches usually address this issue by dening distinct planning, scheduling
and allocation problems. However, within the considered context, the problem of providing
the right number of workers with the right skills at the right time is inherently linked to the
schedule of the activities. For this reason, we rather propose a strategy to tackle all these
aspects at the same time, taking into account a reasonable time horizon. As a result we
obtain a large problem requiring not only a proper representation of processes complexity,
but also a feasible assignment of operators to tasks and an optimized activities scheduling.
In what follows, the structure of the problem is formalized and a specialized simulation-
based decomposition framework is proposed.
Hereinafter, we will consider systems where one or more processes are executed. Rough-
ly, a process can be described in terms of two basic denitions: the skills employed and
the component activities. A skill represents the ability of an operator to perform certain
tasks, thus identifying a worker type. An activity can be any non-interruptible elementary
time-consuming operation requiring skilled operators to be completed.
141

Activities may be linked by some precedence constraints, but have variable starting
times that can be modied in order to create optimal schedules satisfying logical and s-
trategical restrictions. Indeed, activities may be subject to release and deadline constraints,
and are aected by workforce availability limitations.
A basic assumption of our approach is that the number of skills required by each activity
is not xed and therefore there exist many feasible combinations of operators guaranteeing
tasks completion. In particular, allowing to vary the workforce assignments between a lower
and an upper limit, we evidently admit variability to operations processing times. Such
aspect heavily characterizes our procedure. Assuming it is not possible to derive analytic
functions expressing the link between allocated skills and time to complete the activities,
we have based our solution method on the use of a set of ad hoc simulators, having as
input a vector of worker availabilities and as output a duration estimate.
The result is a simulation-optimization problem facing a typical trade-o between dif-
ferent objectives. On the one hand it aims at reducing the employment cost, minimizing
the number of necessary skilled operators, on the other, it encourages an optimal activities
scheduling, trying to parallelize the tasks and decrease the overall completion time.
2.1 Mathematical formulation
In order to introduce the general mathematical formulation we have developed for this
problem, we rst need to list some basic denitions. We will consider m activities and n
dierent skills. Let A = {1, ..., m} be the set of indexes for activities, and S = {1, ..., n}
be the set of indexes for skills. Each activity is non-preemptable and is characterized by
a variable processing time, a release date rj and a due date dj. Both rj and dj are real
parameters and can be set to zero and innity to nullify the associated constraints.
Precedence relations are given by the set Q of ordered index pairs, such that (j1, j2) ∈ Q
means that the execution of activity j2 must start after the end of activity j1. The same
concept can be expressed by an activity-on-node graph whose nodes correspond to activi-
ties, and arcs represent sequence constraints. From this perspective, a necessary condition
to guarantee consistent precedence relations is that the graph contains no cycles.
Our problem formulation involves three main types of decision variables. First, the total
number of operators made available for each skill is represented by a vector y ∈ Nn
, such
that yi denotes the availability of resource i. Second, integer variables xij are required to
indicate the number of workers with skill i assigned to activity j. Finally, the starting-
time continuous variables tj are introduced for each activity j, thus making possible the
scheduling.
Alongside these denitions, two auxiliary variables γjc and θjc are used in our mathe-
matical model, where j and c both belong to A; their meaning will be soon clear.
Ultimately, minding our assumption on the dependence among operators assignments
and time to complete the activities, we can identify the output of the j-th simulator with
the symbol τj = ϕj(x1j, . . . , xij, . . . , xnj), so expressing the processing time of activity j
as an unknown function of the variable skill allocations.
We can therefore formalize the problem in a bilevel programming formulation having
as upper-level and lower-level objectives two generic functions. Their global eect can be
thought of as the combination of two conicting components: the rst accounting for the
workforce cost, the second expressing a time objective. As an example of this trade-o, we
can consider a situation where variables yi are, at the same time, pushed down to lower
salaries expenses, and pushed up to relax resource constraints and obtain better results
in activities scheduling, improving, for example, the overall makespan, the sum of projects
completion times or the average nish time of activities.
We propose the following formulation:
142

min
x,y,τ,t,γ,θ
f1(x, y, τ, t, γ, θ) (1)
s.t. lij ≤ xij ≤ uij i ∈ S, j ∈ A (2)
xij ≤ yi i ∈ S, j ∈ A (3)
yi ≤
∑
j∈A
xij i ∈ S (4)
τj = ϕj(x1j, .., xnj) j ∈ A (5)
yi ∈ N i ∈ S (6)
xij ∈ N i ∈ S, j ∈ A (7)
τj ∈ R+
j ∈ A (8)
(t, γ, θ) ∈ arg min
t,γ,θ
f2(t, γ, θ) (9)
s.t. tj ≥ rj j ∈ A (10)
tj ≤ dj − τj j ∈ A (11)
tȷ̃ ≥ tȷ̂ + τȷ̂ (ȷ̂, ȷ̃) ∈ Q (12)
∑
j∈A
xijγjc ≤ yi i ∈ S, c ∈ A (13)
tc − tj ≥ M(γjc − 1) j ∈ A, c ∈ A (14)
tc − tj ≤ τj + M(1 − γjc) − ε j ∈ A, c ∈ A (15)
tc − tj ≥ −Mθjc + τj −
τj
2
γjc j ∈ A, c ∈ A (16)
tc − tj ≤ M(1 − θjc) +
τj
2
γjc − ε j ∈ A, c ∈ A (17)
tj ∈ R+
j ∈ A (18)
γjc ∈ {0, 1} j ∈ A, c ∈ A (19)
θjc ∈ {0, 1} j ∈ A, c ∈ A (20)
The upper and lower level objective functions are respectively contained in (1) and (9).
In (2) are the bounds for variables xij. Constraints (3) and (4) express two concepts: the
availability of operators with skill i must be (i ) enough to guarantee that each activity
can be independently executed (e.g. if scheduled in sequence with the others), and (ii ) not
more than the total amount of resources that would be needed if all the activities were
parallelized. Relation (5) brings processing time simulations into the problem. Constraints
(10) and (11) give release date and deadline limits, while inequalities (12) describe the
precedence relations between activities.
In order to understand the meaning of constraints from (13) to (17), it is rst necessary
to clarify the role of binary variable γ. For each couple of activities (j, c), we have that γjc
is equal to 1 if j is in progress when c is starting, 0 otherwise. Thus, we make use of the
following double implication, which is guaranteed by inequalities (14)(17) where ε and M
are two appropriate small and large constants:
γjc = 1 ⇔ tj ≤ tc tj + τj
Then, constraints (13) indicate the relation between available and allocated operators,
i.e. the sum of resources simultaneously occupied cannot exceed the total number of work-
ers, for each skill i.
143

Finally, (6)(8) and (18)(20) dene variables domains. Notice that activities durations
τ are black-box values varying on the positive side of the real axis. This assumes a particular
meaning when the structure of lower-level formulation is analyzed: indeed, if we consider
each τj to be externally calculated (once a value for every xij and yi is xed by the upper-
level decision-maker) and f2 to be the overall makespan, we can prove our problem to fall
under the standard denition of Resource Constrained Project Scheduling Problem (see
Artigues, Demassey and Néron (2008)), with additional due date constraints.
However, due to the a priori unknown values of processing times, an appropriate com-
parison of our formulation with existing ones makes sense only by considering analogous
approaches, as those proposed by Artigues, Michelon and Reusser (2003) and Koné, Ar-
tigues, Lopez and Mongeau (2011), that admit continuous starting time variables and do
not recourse to time horizon discretization. In this respect, it is worth making two ob-
servations: the rst is that, similarly to authors just cited, we have developed a MILP
formulation of the problem (that is evident when looking at upper-level variables as con-
stants). The second, instead, captures the dierence between our and previous approaches.
In particular, by exploiting the relations between pairs of activities, we are able to formu-
late the same problem in a new way which diers and in some cases outperforms existing
methods in terms of total amount of variables and constraints.
Anyway, the solution of the RCPSP constituting our lower-level optimization is not the
only source of complexity in our procedure. The presence of black-box values calculated
by simulators is an important issue to be addressed. For this reason, we propose a decom-
position approach modeling the problem from a new grey-box optimization perspective.
3 Simulation-Optimization framework
Our solution framework is composed of three main nested blocks as shown in Figure 1.
The most external one is a black-box optimization formulation working on variables yi and
xij, subject to constraints (2)(4) and (6)(7). Its objective function, denoted by ˜
f, has
the structure of (1) and is calculated every time from the results of inner blocks.
In turn, the second module, represented by the resource constrained scheduling formu-
lation described above, is (approximately) solved at every iteration, immediately after the
execution of the third block, that takes the xij as inputs, runs a parallel simulation for
each activity j, and returns the processing times τj.
Fig. 1. Framework structure
References
Artigues C., Demassey S., Néron E., 2008, Resource-Constrained Project Scheduling: Models,
Algorithms, Extensions and Applications, ISTE, London, UK.
Artigues C.,Michelon P.,Reusser S., 2003, Insertion techniques for static and dynamic resource-
constrained project scheduling, European Journal of Operational Research, Vol.149 pp.249-67.
Koné O., Artigues C., Lopez P., Mongeau M., 2011, Event-based MILP models for resource-
constrained project scheduling problems, Computers Operations Research, Vol.38 pp.3-13.
144

❙❝❤❡❞✉❧✐♥❣ ■❞❡♥t✐❝❛❧ P❛r❛❧❧❡❧ ▼❛❝❤✐♥❡s ✇✐t❤ ❉❡❧✐✈❡r②
❚✐♠❡s t♦ ▼✐♥✐♠✐③❡ ❚♦t❛❧ ❲❡✐❣❤t❡❞ ❚❛r❞✐♥❡ss
❙ö❤♥❦❡ ▼❛❡❝❦❡r1
✱ ▲✐❥✐ ❙❤❡♥2
1
❋❛❝✉❧t② ♦❢ ❇✉s✐♥❡ss ❛♥❞ ❊❝♦♥♦♠✐❝s✱ ❚❡❝❤♥✐s❝❤❡ ❯♥✐✈❡rs✐tät ❉r❡s❞❡♥✱ ✵✶✵✻✷ ❉r❡s❞❡♥✱ ●❡r♠❛♥②
s♦❡❤♥❦❡✳♠❛❡❝❦❡r❅t✉✲❞r❡s❞❡♥✳❞❡
2
❈❤❛✐r ♦❢ ❖♣❡r❛t✐♦♥s ▼❛♥❛❣❡♠❡♥t✱ ❲❍❯ ✲ ❖tt♦ ❇❡✐s❤❡✐♠ ❙❝❤♦♦❧ ♦❢ ▼❛♥❛❣❡♠❡♥t✱ ✺✻✶✼✾
❱❛❧❧❡♥❞❛r✱ ●❡r♠❛♥②
❧✐❥✐✳s❤❡♥❅✇❤✉✳❡❞✉
❑❡②✇♦r❞s✿ s❝❤❡❞✉❧✐♥❣✱ ♣❛r❛❧❧❡❧ ♠❛❝❤✐♥❡s✱ ❞❡❧✐✈❡r② t✐♠❡s✱ ♠❡♠❡t✐❝ ❛❧❣♦r✐t❤♠✳
✶ ■♥tr♦❞✉❝t✐♦♥
■♥ ♠♦st ♠❛♥✉❢❛❝t✉r✐♥❣ ❛♥❞ ❞✐str✐❜✉t✐♦♥ s②st❡♠s✱ s❡♠✐✲✜♥✐s❤❡❞ ❥♦❜s ❛r❡ tr❛♥s❢❡rr❡❞ ❢r♦♠
♦♥❡ ❢❛❝✐❧✐t② t♦ ❛♥♦t❤❡r ❢♦r ❢✉rt❤❡r ♣r♦❝❡ss✐♥❣ ♦r ✜♥✐s❤❡❞ ❥♦❜s ❛r❡ ❞❡❧✐✈❡r❡❞ t♦ t❤❡ ❝✉st♦♠❡r✳
■♥ t❤❡ ❧❛tt❡r ❝❛s❡✱ t❤❡ ❥♦❜ ❝♦♠♣❧❡t✐♦♥ t✐♠❡ ✐s ❞❡✜♥❡❞ ❛s t❤❡ t✐♠❡ ❜② ✇❤✐❝❤ t❤❡ ❥♦❜ ❛rr✐✈❡s
❛t t❤❡ ❝✉st♦♠❡r✳ ❉✐✛❡r❡♥t ♦♣❡r❛t✐♦♥s ♠✉st ❜❡ ❝❛r❡❢✉❧❧② ❝♦♦r❞✐♥❛t❡❞ t♦ ❛❝❤✐❡✈❡ ✐❞❡❛❧ ♦✈❡r❛❧❧
s②st❡♠ ♣❡r❢♦r♠❛♥❝❡ ✭▲❡❡ ❛♥❞ ❈❤❡♥ ✷✵✵✶✮✳
▼✉❧t✐♣❧❡ ❞❡✜♥✐t✐♦♥s ♦❢ ❞❡❧✐✈❡r② t✐♠❡s ❡①✐st ✐♥ t❤❡ ♠❛❝❤✐♥❡ s❝❤❡❞✉❧✐♥❣ ❧✐t❡r❛t✉r❡✳ ▼❛❣❣✉
❛♥❞ ❉❛s ✭✶✾✽✵✮ ✜rst ❝♦♥s✐❞❡r ❥♦❜ tr❛♥s♣♦rt❛t✐♦♥ ✐♥ ❛ t✇♦✲♠❛❝❤✐♥❡ ✢♦✇ s❤♦♣ ♠❛❦❡s♣❛♥
♣r♦❜❧❡♠ ✇❤❡r❡ ❥♦❜✲❞❡♣❡♥❞❛♥t tr❛♥s♣♦rt❛t✐♦♥ t✐♠❡s ♦❝❝✉r ❜❡t✇❡❡♥ t❤❡ ♣r♦❝❡ss✐♥❣ st❛❣❡s
❛♥❞ tr❛♥s♣♦rt❛t✐♦♥ ❝❛♣❛❝✐t② ✐s ✉♥❧✐♠✐t❡❞✳ P♦tts ✭✶✾✽✵✮ st✉❞✐❡s t❤❡ ♣r♦❜❧❡♠ ♦❢ s❝❤❡❞✉❧✐♥❣
❥♦❜s ♦♥ ❛ s✐♥❣❧❡ ♠❛❝❤✐♥❡ ✇✐t❤ r❡❧❡❛s❡ ❞❛t❡s ❛♥❞ ❥♦❜✲❞❡♣❡♥❞❡♥t ❞❡❧✐✈❡r② t✐♠❡s t♦ ♠✐♥✐♠✐③❡
t❤❡ t✐♠❡ ❜② ✇❤✐❝❤ ❛❧❧ ❥♦❜s ❛r❡ ❞❡❧✐✈❡r❡❞✳ ❆ s✐♠✐❧❛r ♣r♦❜❧❡♠ ✇✐t❤ ✐❞❡♥t✐❝❛❧ r❡❧❡❛s❡ ❞❛t❡s ✐s
st✉❞✐❡❞ ❜② ❲♦❡❣✐♥❣❡r ✭✶✾✾✹✮ ❢♦r t❤❡ ♣❛r❛❧❧❡❧ ♠❛❝❤✐♥❡ ❝❛s❡✳ ▲❡❡ ❛♥❞ ❈❤❡♥ ✭✷✵✵✶✮ ❞❡✜♥❡
t✇♦ t②♣❡s ♦❢ s❝❤❡❞✉❧✐♥❣ ♣r♦❜❧❡♠s ✇✐t❤ ❥♦❜ ❞❡❧✐✈❡r② ✇❤❡r❡ t❤❡ tr❛♥s♣♦rt❛t✐♦♥ ❝❛♣❛❝✐t② ✐s
❧✐♠✐t❡❞ ✐♥ t❡r♠s ♦❢ ❜♦t❤ ❛✈❛✐❧❛❜❧❡ ♥✉♠❜❡r ♦❢ ✈❡❤✐❝❧❡s ❛♥❞ ✈❡❤✐❝❧❡ ❝❛♣❛❝✐t②✳ ❚②♣❡✲✶ tr❛♥s✲
♣♦rt❛t✐♦♥ ❝♦♥s✐❞❡rs ❥♦❜ tr❛♥s♣♦rt❛t✐♦♥ ✐♥s✐❞❡ ❛ ♠❛♥✉❢❛❝t✉r✐♥❣ ❢❛❝✐❧✐t② ❜❡t✇❡❡♥ ♣r♦❝❡ss✐♥❣
st❛❣❡s ❛♥❞ t②♣❡✲✷ tr❛♥s♣♦rt❛t✐♦♥ t❛❦❡s ♣❧❛❝❡ ❜❡t✇❡❡♥ t❤❡ ❢❛❝✐❧✐t② ❛♥❞ ❛ ❝✉st♦♠❡r ❛r❡❛✳
■♥ ❜♦t❤ ❝❛s❡s✱ ❥♦❜s s❤❛r❡ ❛ ❝♦♠♠♦♥ ❞❡❧✐✈❡r② t✐♠❡ ❛♥❞ t❤❡ ♦❜❥❡❝t✐✈❡ ✐s t♦ ♠✐♥✐♠✐③❡ t❤❡
♠❛❦❡s♣❛♥✳ ❆ ❝♦♠♣❧❡①✐t② ❛♥❛❧②s✐s ✐s ♣r❡s❡♥t❡❞ ❢♦r s✐♥❣❧❡ ♠❛❝❤✐♥❡✱ ♣❛r❛❧❧❡❧ ♠❛❝❤✐♥❡✱ ❛♥❞
✢♦✇ s❤♦♣ ❡♥✈✐r♦♥♠❡♥ts✳ ❈❤❛♥❣ ❛♥❞ ▲❡❡ ✭✷✵✵✹✮ st✉❞② t②♣❡✲✷ tr❛♥s♣♦rt❛t✐♦♥ ❢♦r t❤❡ s✐♥❣❧❡
❛♥❞ ♣❛r❛❧❧❡❧ ♠❛❝❤✐♥❡ ❝❤❛s❡ ✇✐t❤ ❥♦❜s✱ t❤❛t r❡q✉✐r❡ ❞✐✛❡r❡♥t ❛♠♦✉♥ts ♦❢ s♣❛❝❡ ♦♥ t❤❡ tr❛♥s✲
♣♦rt❛t✐♦♥ ✈❡❤✐❝❧❡✳ ❋✉rt❤❡r♠♦r❡✱ t✇♦ s❡♣❛r❛t❡ ❝✉st♦♠❡r ❛r❡❛s ❡①✐st✳ ❑♦✉❧❛♠❛s ❛♥❞ ❑②♣❛r✐s✐s
✭✷✵✶✵✮ ♣r❡s❡♥t ❛ s✐♥❣❧❡ ♠❛❝❤✐♥❡ ♣r♦❜❧❡♠ ✇❤❡r❡ ❥♦❜s ❤❛✈❡ ♣❛st✲s❡q✉❡♥❝❡✲❞❡♣❡♥❞❡♥t ❞❡❧✐✈✲
❡r② t✐♠❡s ♣r♦♣♦rt✐♦♥❛❧ t♦ t❤❡✐r ✇❛✐t✐♥❣ t✐♠❡ ❜❡❢♦r❡ ♣r♦❝❡ss✐♥❣✳ P♦❧②♥♦♠✐❛❧✲t✐♠❡ ❛❧❣♦r✐t❤♠s
❛r❡ ♣r❡s❡♥t❡❞ ❢♦r ♠✉❧t✐♣❧❡ ♦♣t✐♠✐③❛t✐♦♥ ❝r✐t❡r✐❛✳ ❆♥♦t❤❡r ❞❡✜♥✐t✐♦♥ ♦❢ ❥♦❜ ❞❡❧✐✈❡r② t✐♠❡s ✐s
♣r❡s❡♥t❡❞ ❜② ❈❤❡♥ ❡t✳ ❛❧✳ ✭✷✵✶✻✮ ✇❤♦ ✐♥✈❡st✐❣❛t❡ ❛ ♣❛r❛❧❧❡❧ ♠❛❝❤✐♥❡ ♣r♦❜❧❡♠ ✇❤❡r❡ ❛ s❡t ♦❢
❞❡❧✐✈❡r② t✐♠❡s ❛r❡ ❣✐✈❡♥ ❛♥❞ ❡❛❝❤ ❞❡❧✐✈❡r② t✐♠❡ ♥❡❡❞s t♦ ❜❡ ❛ss✐❣♥❡❞ t♦ ❛♥ ✐♥❞✐✈✐❞✉❛❧ ❥♦❜✳
■♥ t❤✐s ♣❛♣❡r✱ ✇❡ ❝♦♥s✐❞❡r t❤❡ ❞❡❧✐✈❡r② ❛s♣❡❝t ✇❤❡♥ s❝❤❡❞✉❧✐♥❣ ❥♦❜s ♦♥ ✐❞❡♥t✐❝❛❧ ♣❛r❛❧❧❡❧
♠❛❝❤✐♥❡s t♦ ♠✐♥✐♠✐③❡ t❤❡ t♦t❛❧ ✇❡✐❣❤t❡❞ t❛r❞✐♥❡ss ✭❚❲❚✮ ❛♥❞ ❞❡❧✐✈❡r② t✐♠❡s ❛r❡ ♠❛❝❤✐♥❡✲
❞❡♣❡♥❞❡♥t✳ ❲❡ ✜rst ❢♦r♠❛❧❧② ❞❡s❝r✐❜❡ t❤❡ ♣r♦❜❧❡♠ ❛♥❞ ♣r❡s❡♥t ❛ ♠✐①❡❞ ✐♥t❡❣❡r ❧✐♥❡❛r
♣r♦❣r❛♠♠✐♥❣ ❢♦r♠✉❧❛t✐♦♥ ✭▼■▲P✮✳ ❆❢t❡r✇❛r❞s✱ ❛ ♠❡♠❡t✐❝ ❛❧❣♦r✐t❤♠ ✭▼❆✮ ✐s ❞❡✈❡❧♦♣❡❞
❛♥❞ ❝♦♠♣❛r❡❞ t♦ t❤❡ ▼■▲P ❛s ✇❡❧❧ ❛s ♠✉❧t✐♣❧❡ ✇❡❧❧✲❦♥♦✇♥ s❝❤❡❞✉❧✐♥❣ ❤❡✉r✐st✐❝s ♦♥ ❛
❧❛r❣❡ s❡t ♦❢ r❛♥❞♦♠❧② ❣❡♥❡r❛t❡❞ t❡st ♣r♦❜❧❡♠ ✐♥st❛♥❝❡s✳ ■♥ ❛❞❞✐t✐♦♥✱ t❤❡ ✐♠♣❛❝t ♦❢ t❤❡
✐♥st❛♥❝❡ ♣❛r❛♠❡t❡r s❡tt✐♥❣✱ ❡s♣❡❝✐❛❧❧② t❤❡ ❞❡❧✐✈❡r② t✐♠❡s✱ ♦♥ t❤❡ ❛❧❣♦r✐t❤♠ ♣❡r❢♦r♠❛♥❝❡ ✐s
✐♥✈❡st✐❣❛t❡❞✳
145

✷ Pr♦❜❧❡♠ ❋♦r♠✉❧❛t✐♦♥
❚❤❡ ♣r♦❜❧❡♠ ❝❛♥ ❜❡ ❞❡s❝r✐❜❡❞ ❛s ❢♦❧❧♦✇s✿ ●✐✈❡♥ ❛r❡ ❛ s❡t ♦❢ n ❥♦❜s j = 1, . . . , n ❛♥❞
m ✐❞❡♥t✐❝❛❧ ♠❛❝❤✐♥❡s h = 1, . . . , m✳ ❊❛❝❤ ❥♦❜ ♥❡❡❞s t♦ ❜❡ ♣r♦❝❡ss❡❞ ❜② ♦♥❡ ❛♥❞ ♦♥❧② ♦♥❡
♠❛❝❤✐♥❡ ✇✐t❤♦✉t ✐♥t❡rr✉♣t✐♦♥ ✇❤✐❧❡ ❡❛❝❤ ♠❛❝❤✐♥❡ ❝❛♥ ❤❛♥❞❧❡ ❡①❛❝t❧② ♦♥❡ ❥♦❜ ❛t ❛ t✐♠❡✳
❆❧❧ ❥♦❜s ❛r❡ ❛✈❛✐❧❛❜❧❡ ❛t t✐♠❡ ③❡r♦✳ ❊❛❝❤ ❥♦❜ j ❤❛s ❛ s♣❡❝✐✜❝ ♣r♦❝❡ss✐♥❣ t✐♠❡ pj✱ ❞✉❡
❞❛t❡ dj✱ ❛♥❞ ✇❡✐❣❤t wj✳ ❆ ♠❛❝❤✐♥❡✲❞❡♣❡♥❞❡♥t ❞❡❧✐✈❡r② t✐♠❡ qh ♦❝❝✉rs ✐♠♠❡❞✐❛t❡❧② ✉♣♦♥
❝♦♠♣❧❡t✐♥❣ ❛ ❥♦❜ ♦♥ t❤❡ r❡s♣❡❝t✐✈❡ ♠❛❝❤✐♥❡✳ ❲❤✐❧❡ ❛ ❥♦❜ ✐s ❜❡✐♥❣ tr❛♥s❢❡rr❡❞✱ t❤❡ ♠❛❝❤✐♥❡
♠❛② ❛❧r❡❛❞② st❛rt ♣r♦❝❡ss✐♥❣ t❤❡ ♥❡①t ❥♦❜ ✐♥ ❧✐♥❡✳ ❚❤❡ tr❛♥s♣♦rt❛t✐♦♥ ❝❛♣❛❝✐t② ✐s ❛ss✉♠❡❞
t♦ ❜❡ ✉♥❧✐♠✐t❡❞ ✐♥ t❡r♠s ♦❢ ❜♦t❤ ✈❡❤✐❝❧❡ ❛✈❛✐❧❛❜✐❧✐t② ❛♥❞ ✈❡❤✐❝❧❡ ❝❛♣❛❝✐t②✳ ❚❤❡ ♣r♦❜❧❡♠
✐s t♦ ❞❡t❡r♠✐♥❡ ❛ s❝❤❡❞✉❧❡ π t♦ ♠✐♥✐♠✐③❡ t❤❡ ❚❲❚ (
P
wjTj)✳ ❚❤❡ t❛r❞✐♥❡ss ♦❢ ❛ ❥♦❜ Tj
✐s ❞❡✜♥❡❞ ❛s Tj = max{Cj − dj, 0} ✇✐t❤ Cj ❜❡✐♥❣ t❤❡ t✐♠❡ ❥♦❜ j r❡❛❝❤❡s t❤❡ ❝✉st♦♠❡r✳
❋♦❧❧♦✇✐♥❣ t❤❡ ❝♦♥✈❡♥t✐♦♥❛❧ t❤r❡❡✲✜❡❧❞✲♥♦t❛t✐♦♥ ❜② ●r❛❤❛♠ ❡t✳ ❛❧✳ ✭✶✾✼✾✮✱ t❤❡ ♣r♦❜❧❡♠ ❝❛♥
❜❡ ❡①♣r❡ss❡❞ ❛s Pm|qh|
P
wjTj✳ ❙✐♥❝❡ ✐t ✐s ✇❡❧❧ ❦♥♦✇♥ t❤❛t ♣r♦❜❧❡♠ 1||
P
wjTj ✐s NP✲❤❛r❞
✐♥ t❤❡ str♦♥❣ s❡♥s❡ ✭▲❡♥str❛ ❡t✳ ❛❧✳ ✶✾✼✼✮✱ t❤❡ ♣r♦❜❧❡♠ ❝♦♥s✐❞❡r❡❞ ❤❡r❡ ✐s NP✲❤❛r❞ ✐♥ t❤❡
str♦♥❣ s❡♥s❡ ❛s ✇❡❧❧✳
❚♦ ❢♦r♠✉❧❛t❡ t❤❡ ♣r♦❜❧❡♠ ❛s ❛♥ ▼■▲P✱ ✇❡ ✐♥tr♦❞✉❝❡ t✇♦ ❜✐♥❛r② ❞❡❝✐s✐♦♥ ✈❛r✐❛❜❧❡s xij
✇✐t❤
xij =

1, ✐❢ ❥♦❜ j ✐s s❡q✉❡♥❝❡❞ ✐♠♠❡❞✐❛t❡❧② ❛❢t❡r ❥♦❜ i✱
0, ♦t❤❡r✇✐s❡✱
✭✶✮
❛♥❞ yjh ✇✐t❤
yjh =

1, ✐❢ ❥♦❜ j ✐s t❤❡ ✜rst s❡q✉❡♥❝❡❞ ❥♦❜ ♦♥ ♠❛❝❤✐♥❡ h✱
0, ♦t❤❡r✇✐s❡✳
✭✷✮
❋✉rt❤❡r♠♦r❡✱ ❛ ❞✉♠♠② ❥♦❜ j = n + 1 ✐s ✐♥tr♦❞✉❝❡❞ ✇✐t❤ pn+1 = 0 t❤❛t ♠❛r❦s t❤❡ ❡♥❞ ♦❢
t❤❡ s❝❤❡❞✉❧❡ ♦♥ ❡❛❝❤ ♠❛❝❤✐♥❡✳ ❚❤❡ ♣r♦❜❧❡♠ ❝❛♥ ♥♦✇ ❜❡ ❢♦r♠✉❧❛t❡❞ ❛s ❢♦❧❧♦✇s✿
min
n
X
i=1
wjTj ✭✸✮
s✉❜❥❡❝t t♦
n
X
j=1
yjh ≤ 1 h = 1, . . . , m; ✭✹✮
m
X
h=1
yjh ≤ 1 j = 1, . . . , n; ✭✺✮
yjh +
n
X
i=1,i6=j
xij = 1 j = 1, . . . , n; h = 1, . . . , m; ✭✻✮
n+1
X
i=1,i6=j
xji = 1 j = 1, . . . , n; ✭✼✮
Cj ≥ (pj + qh)yjh j = 1, . . . , n; h = 1, . . . m; ✭✽✮
Cj ≥ Ci + pj − H(1 − xij) i, j = 1, . . . , n; i 6= j; ✭✾✮
Tj ≥ Cj − dj j = 1, . . . , n; ✭✶✵✮
❚❤❡ ♦❜❥❡❝t✐✈❡ ✭✸✮ ✐s t♦ ♠✐♥✐♠✐③❡ t❤❡ ❚❲❚✳ ❈♦♥str❛✐♥ts ✭✹✮ ❛♥❞ ✭✺✮ ❡♥s✉r❡ t❤❛t ❛t ♠♦st ♦♥❡
❥♦❜ ✐s ❛ss✐❣♥❡❞ t♦ t❤❡ ✜rst ♣♦s✐t✐♦♥ ♦♥ ❡❛❝❤ ♠❛❝❤✐♥❡✳ ❈♦♥str❛✐♥ts ✭✻✮ ❛♥❞ ✭✼✮ ❞❡t❡r♠✐♥❡ ❥♦❜
s❡q✉❡♥❝❡s✳ ❚❤❡ ❝♦♠♣❧❡t✐♦♥ t✐♠❡ ♦❢ t❤❡ ✜rst ❥♦❜s ♦♥ ♠❛❝❤✐♥❡s ✐s ❝❛❧❝✉❧❛t❡❞ ❜② ✐♥❡q✉❛❧✐t②
✭✽✮ ✇❤✐❧❡ ✭✾✮ ❞❡✜♥❡s t❤❡ ❝♦♠♣❧❡t✐♦♥ t✐♠❡ ❢♦r t❤❡ r❡♠❛✐♥✐♥❣ ❥♦❜s✱ ✇❤❡r❡ H ✐s ❛ s✉✣❝✐❡♥t
❧❛r❣❡ ♥✉♠❜❡r✳ ❚❤❡ t❛r❞✐♥❡ss ♦❢ ❡❛❝❤ ❥♦❜ ✐s ❝❛❧❝✉❧❛t❡❞ ❜② ✐♥❡q✉❛❧✐t② ✭✶✵✮✳
146

✸ ❚❤❡ ▼❡♠❡t✐❝ ❆❧❣♦r✐t❤♠
❖✉r ▼❆ ❛❞♦♣ts t❤❡ ♣❡r♠✉t❛t✐♦♥✲❧✐❦❡ r❡♣r❡s❡♥t❛t✐♦♥ s❝❤❡♠❡ ❜② ❈❤❡♥❣ ❡t✳ ❛❧✳ ✭✶✾✾✺✮
✇❤❡r❡ t❤❡ ❣❡♥♦t②♣❡ ❝♦♥s✐sts ♦❢ ❥♦❜✲ ❛♥❞ ♣❛rt✐t✐♦♥✐♥❣ s②♠❜♦❧s✳ ■♥ t❤❡ r❡♣r♦❞✉❝t✐♦♥ ♣❤❛s❡✱
t❤❡ ♦✛s♣r✐♥❣ ✐s ❣❡♥❡r❛t❡❞ t❤r♦✉❣❤ ❛ s✉❜s❝❤❡❞✉❧❡ ♣r❡s❡r✈❛t✐♦♥ ❝r♦ss♦✈❡r ♦♣❡r❛t♦r ✭❈❤❡♥❣ ❡t✳
❛❧✳ ✶✾✾✺✮ ❛♥❞ ❛ ♠✉t❛t✐♦♥ ♦♣❡r❛t♦r t❤❛t ✉s❡s ❛♥ ✐♥s❡rt✐♦♥ str❛t❡❣②✳ ❚❤❡ ♠✉t❛t✐♦♥ ♦♣❡r❛t♦r
❛❧t❡rs ✐♥❞✐✈✐❞✉❛❧s ❜② r❡♠♦✈✐♥❣ ❛ r❛♥❞♦♠ ❡❧❡♠❡♥t ❢r♦♠ t❤❡ ❝❤r♦♠♦s♦♠❡ ❛♥❞ r❡✐♥s❡rt✐♥❣
✐t ❛t ❛♥♦t❤❡r r❛♥❞♦♠ ♣♦s✐t✐♦♥ t♦ ❢❛❝✐❧✐t❛t❡ ❞✐✈❡rs✐✜❝❛t✐♦♥✳ ❚❤❡ ♥❡✐❣❤❜♦✉r❤♦♦❞ str✉❝t✉r❡ ♦❢
t❤❡ ♠✉t❛t✐♦♥ ♦♣❡r❛t♦r ✐♥❝❧✉❞❡s ❝❤❛♥❣✐♥❣ t❤❡ ♣♦s✐t✐♦♥ ♦❢ ❥♦❜s ♦♥ ❛ ♠❛❝❤✐♥❡✱ r❡✐♥s❡rt✐♥❣ ❥♦❜s
♦♥ ♦t❤❡r ♠❛❝❤✐♥❡s✱ ❛♥❞ ❝❤❛♥❣✐♥❣ t❤❡ ♦✈❡r❛❧❧ ♣❛rt✐t✐♦♥✐♥❣✳
▼♦r❡ ✐♠♣♦rt❛♥t❧②✱ t❤❡ ▼❆ ✐♥❝♦r♣♦r❛t❡s ❛ ❧♦❝❛❧ s❡❛r❝❤ ✭▲❙✮ t♦ ✐♠♣r♦✈❡ ❛❧❧ ♦✛s♣r✐♥❣
s♦❧✉t✐♦♥s ❛❢t❡r r❡♣r♦❞✉❝t✐♦♥✳ ■♥ t❤❡ s✉❜s❡q✉❡♥t ❤✐❧❧✲❝❧✐♠❜✐♥❣ ♣❤❛s❡✱ t❤❡ ▲❙ ✐s ♣❡r❢♦r♠❡❞
♦♥ ❡❛❝❤ ❣❡♥❡r❛t❡❞ ♦✛s♣r✐♥❣ s♦❧✉t✐♦♥✱ t❤❛t s②st❡♠❛t✐❝❛❧❧② ❡①❛♠✐♥❡s ❛❧❧ ♣♦ss✐❜❧❡✱ ♥♦♥✲tr✐✈✐❛❧
❡①❝❤❛♥❣❡s ♦❢ ❡❧❡♠❡♥ts ♦♥ t❤❡ ❝❤r♦♠♦s♦♠❡ ✇❤✐❧❡ t❤❡ ♠❛①✐♠✉♠ ❧❡♥❣t❤ ♦❢ t❤❡ s❡❛r❝❤ ✐s
❧✐♠✐t❡❞✳ ❈♦♥s✐st❡♥t ✇✐t❤ t❤❡ ♠✉t❛t✐♦♥ ♦♣❡r❛t♦r✱ t❤❡ ♥❡✐❣❤❜♦✉r❤♦♦❞ str✉❝t✉r❡ ♦❢ t❤❡ ▲❙
✐♥❝❧✉❞❡s s✇❛♣♣✐♥❣ ❥♦❜s ♦♥ ♦♥❡ ♠❛❝❤✐♥❡✱ s✇❛♣♣✐♥❣ ❥♦❜s ♦♥ ❞✐✛❡r❡♥t ♠❛❝❤✐♥❡s✱ ❛♥❞ ❝❤❛♥❣✐♥❣
t❤❡ ♦✈❡r❛❧❧ ♣❛rt✐t✐♦♥✐♥❣✳ ◆♦t❡ t❤❛t t❤❡ s❡❧❡❝t✐♦♥ ♦❢ ❞✐✈❡r❣✐♥❣ ♥❡✐❣❤❜♦✉r❤♦♦❞s ❢♦r ♠✉t❛t✐♦♥
❛♥❞ ❤✐❧❧✲❝❧✐♠❜✐♥❣ ✐s ❝r✉❝✐❛❧ t♦ t❤❡ s✉❝❝❡ss ♦❢ ♦✉r ▼❆ ✐♥ ♦r❞❡r t♦ ❡①♣❧♦r❡ t❤❡ s♦❧✉t✐♦♥ s♣❛❝❡
❡✣❝✐❡♥t❧②✳ ❚❤❡ ♥❡✇ ❣❡♥❡r❛t✐♦♥ ✐s t❤❡♥ s❡❧❡❝t❡❞ ❜❛s❡❞ ♦♥ ✜t♥❡ss ♦❢ t❤❡ ❢♦r♠❡r ♣♦♣✉❧❛t✐♦♥
❛♥❞ t❤❡ ♦✛s♣r✐♥❣✳ ❚❤❡ ▼❆ t❡r♠✐♥❛t❡s ❛❢t❡r ❛ ♣r❡❞❡✜♥❡❞ ♥✉♠❜❡r ♦❢ ✐t❡r❛t✐♦♥s ✇✐t❤♦✉t
✐♠♣r♦✈❡♠❡♥t✳
✹ ❈♦♠♣✉t❛t✐♦♥❛❧ ❘❡s✉❧ts
■♥ ♦✉r ♣r❡❧✐♠✐♥❛r② t❡sts✱ ✇❡ ✉s❡ t❤❡ ♣r♦♣♦s❡❞ ▼■▲P ❛♥❞ s❡✈❡r❛❧ ❡①✐st✐♥❣ ❤❡✉r✐st✐❝s ❛s
r❡❢❡r❡♥❝❡ ❢♦r ❝♦♠♣❛r✐s♦♥✳ ❚❤❡ ❤❡✉r✐st✐❝s ✐♥❝❧✉❞❡ t❤❡ ❛♣♣❛r❡♥t t❛r❞✐♥❡ss ❝♦st ✭❆❚❈✮ r✉❧❡
✭❱❡♣s❛❧❛✐♥❡♥ ❛♥❞ ▼♦rt♦♥ ✶✾✽✼✮✱ t❤❡ ♠♦❞✐✜❡❞ ❞✉❡ ❞❛t❡ ✭▼❉❉✮ ❛❧❣♦r✐t❤♠ ❜② ❆❧✐❞❛❡❡ ❛♥❞
❘♦s❛ ✭✶✾✾✼✮✱ ❛♥❞ t❤❡ ❑P▼ ❤❡✉r✐st✐❝ ✭❑♦✉❧❛♠❛s ✶✾✾✹✮✳ ❲❡ ✐♠♣❧❡♠❡♥t❡❞ t❤❡ ▼■▲P ✐♥ ■❇▼
■▲❖● ❈P▲❊❳ ❖♣t✐♠✐③❛t✐♦♥ ❙t✉❞✐♦ ✶✷✳✻ ✇✐t❤ ❛ t✐♠❡ ❧✐♠✐t ♦❢ ✸✵ ♠✐♥✉t❡s ❛♥❞ ❛ ♠❛①✐♠✉♠
♦❢ ✽ t❤r❡❛❞s✳ ❚❤❡ ❤❡✉r✐st✐❝s ❛♥❞ t❤❡ ▼❆ ✇❡r❡ ✐♠♣❧❡♠❡♥t❡❞ ✐♥ ❈✰✰✳ ❊①♣❡r✐♠❡♥ts ✇❡r❡
❝♦♥❞✉❝t❡❞ ♦♥ ❛ ♣❡rs♦♥❛❧ ❝♦♠♣✉t❡r ✇✐t❤ ❛♥ ❆▼❉ ❖♣t❡r♦♥ ✻✷✽✷ ❙❊ ♣r♦❝❡ss✐♥❣ ✉♥✐t ✇✐t❤
✷✳✻●❍③ ❛♥❞ ✶✷✽●❇ ❘❆▼✳ ❘❡s✉❧ts ❢♦r t❤❡ ▼❆ ✇❡r❡ ♦❜t❛✐♥❡❞ ❜② ❦❡❡♣✐♥❣ t❤❡ ❜❡st ♦❜❥❡❝t✐✈❡
✈❛❧✉❡ ♦✉t ♦❢ ✜✈❡ ✐♥❞❡♣❡♥❞❡♥t r✉♥s✳
❚❤❡ ♣r♦❜❧❡♠ ✐♥st❛♥❝❡ ❞❛t❛ ✐♥❝❧✉❞✐♥❣ n, m, pj, wj✱ ❛♥❞ dj ✇❡r❡ ❣❡♥❡r❛t❡❞ ❜❛s❡❞ ♦♥ t❤❡
♠❡t❤♦❞ ♣r♦♣♦s❡❞ ❜② P♦tts ❛♥❞ ❱❛♥ ❲❛ss❡♥❤♦✈❡ ✭✶✾✽✷✮✳ ▼♦r❡♦✈❡r✱ ♠❛❝❤✐♥❡ ❞❡❧✐✈❡r② t✐♠❡s
❛r❡ ✉♥✐❢♦r♠❧② ❞✐str✐❜✉t❡❞ ✇✐t❤ qh ∼ [1, 50] ❢♦r s♠❛❧❧ ❞❡❧✐✈❡r② t✐♠❡s ❛♥❞ ✇✐t❤ qh ∼ [101, 300]
❢♦r ❧❛r❣❡ ❞❡❧✐✈❡r② t✐♠❡s✳ ■♥ t♦t❛❧✱ ✇❡ ❤❛✈❡ ✹✵ ❝♦♥✜❣✉r❛t✐♦♥s✳ ❋♦r ❡❛❝❤ ♦❢ t❤❡♠✱ ✜✈❡ ♣r♦❜❧❡♠
✐♥st❛♥❝❡s ❛r❡ ❣❡♥❡r❛t❡❞✱ ✇❤✐❝❤ r❡s✉❧ts ✐♥ ✷✵✵ ✐♥st❛♥❝❡s✳
❚❛❜❧❡ ✶ s❤♦✇s t❤❡ ❛✈❡r❛❣❡ r❡❧❛t✐✈❡ ♣❡r❝❡♥t❛❣❡ ❞❡✈✐❛t✐♦♥ ✭❆❘P❉✮ ❛❝❤✐❡✈❡❞ ❜② ❡❛❝❤
❛♣♣r♦❛❝❤✳ ❚❤❡ ❆❘P❉ ✐s ❞❡✜♥❡❞ ❛s Z−Z∗
Z∗ ×100✱ ✇❤❡r❡ Z ✐s t❤❡ ♦❜❥❡❝t✐✈❡ ✈❛❧✉❡ ♦❜t❛✐♥❡❞ ❜②
t❤❡ r❡s♣❡❝t✐✈❡ s♦❧✉t✐♦♥ ❛♣♣r♦❛❝❤ ❛♥❞ Z∗
✐s t❤❡ ❜❡st ❢♦✉♥❞ s♦❧✉t✐♦♥ ❛♠♦♥❣ ❛❧❧ ❛♣♣r♦❛❝❤❡s✳
◆♦t❡ t❤❛t t❤❡ r❡s✉❧ts ❢♦r ❛❧❧ ✐♥❞✐✈✐❞✉❛❧ ❞✉❡ ❞❛t❡ s❡tt✐♥❣s ✇❡r❡ s✉♠♠❛r✐③❡❞ t♦ str❡ss t❤❡
✐♠♣❛❝t ♦❢ ❞❡❧✐✈❡r② t✐♠❡s✳ ❆❞❞✐t✐♦♥❛❧❧②✱ t❤❡ ❛✈❡r❛❣❡ ❈P❯ t✐♠❡ r❡q✉✐r❡❞ ❜② t❤❡ ▼❆ ✐s ❣✐✈❡♥✳
■t ❝❛♥ ❜❡ s❡❡♥ t❤❛t t❤❡ ▼❆✱ ✐♥ ❣❡♥❡r❛❧✱ ♦✉t♣❡r❢♦r♠s t❤❡ ♦t❤❡r ❛♣♣r♦❛❝❤❡s✳ ❖♥ t❤❡
♦t❤❡r ❤❛♥❞✱ t❤❡ ❝♦♠♣✉t✐♥❣ t✐♠❡ ✐♥❝r❡❛s❡s s✐❣♥✐✜❝❛♥t❧② ✇✐t❤ t❤❡ ♣r♦❜❧❡♠ s✐③❡✳ ❋♦r n = 20✱
t❤❡ ▼■▲P ❢♦✉♥❞ ❢♦r ✸ ♦✉t ♦❢ ✹✵ ✐♥st❛♥❝❡s s❧✐❣❤t❧② ❜❡tt❡r r❡s✉❧ts t❤❛♥ t❤❡ ▼❆✳ ❋♦r n =
100✱ t❤❡ ▼■▲P s♦❧✉t✐♦♥s ❛r❡ ♥♦t ❝♦♠♣❡t✐t✐✈❡ ❛♥❞ ♥♦ ❢❡❛s✐❜❧❡ s♦❧✉t✐♦♥s ✇❡r❡ ❣❡♥❡r❛t❡❞ ❢♦r
n = 200 ✇✐t❤✐♥ t❤❡ ✸✵ ♠✐♥✉t❡s✳ ❚❤❡ ❞❡❧✐✈❡r② t✐♠❡s ❛♣♣❡❛r t♦ ❤❛✈❡ ❛ str♦♥❣ ✐♠♣❛❝t ♦♥
t❤❡ ♣❡r❢♦r♠❛♥❝❡ ♦❢ t❤❡ s♦❧✉t✐♦♥ ❛♣♣r♦❛❝❤❡s s✐♥❝❡ t❤❡ ❤❡✉r✐st✐❝s ♣❡r❢♦r♠✱ ❝♦♠♣❛r❡❞ t♦ t❤❡
▼❆✱ ❝♦♥s✐❞❡r❛❜❧② ✇♦rs❡ ❢♦r ✐♥st❛♥❝❡s ✇✐t❤ s♠❛❧❧ ❞❡❧✐✈❡r② t✐♠❡s✳ ❚❤✐s ❡✛❡❝t ❞❡s❡r✈❡s ❢✉rt❤❡r
✐♥✈❡st✐❣❛t✐♦♥✳
147

❚❛❜❧❡ ✶✳ ❈♦♠♣✉t❛t✐♦♥❛❧ ❘❡s✉❧ts
Pr♦❜❧❡♠ ❙❡tt✐♥❣ ❆❘P❉ ❈P❯ ❚✐♠❡
n m qh ❆❚❈ ❑P▼ ▼❉❉ ▼■▲P ▼❆ [sec]
✷✵ ✺ ❬✶✱✺✵❪ ✶✹✳✶✼ ✶✶✳✶✼ ✹✸✳✸✶ ✵✳✸✷ ✵✳✵✷ ✵✳✹✶
✷✵ ✺ ❬✶✵✶✱✸✵✵❪ ✵✳✻✽ ✶✳✷✶ ✸✳✾✶ ✵✳✸✽ ✵✳✵✵ ✵✳✸✽
✶✵✵ ✺ ❬✶✱✺✵❪ ✷✸✳✶✸ ✷✽✳✻✼ ✶✵✾✳✵✻ ✶✵✷✶✳✾✹ ✵✳✵✵ ✻✽✳✷✸
✶✵✵ ✺ ❬✶✵✶✱✸✵✵❪ ✻✳✾✽ ✷✼✳✵✸ ✻✾✳✽✷ ✸✵✷✳✾✸ ✵✳✵✵ ✶✵✷✳✹✻
✶✵✵ ✷✵ ❬✶✱✺✵❪ ✷✼✳✸✽ ✶✸✳✾✽ ✹✽✳✽✻ ✽✹✳✸✻ ✵✳✵✵ ✻✽✳✼✻
✶✵✵ ✷✵ ❬✶✵✶✱✸✵✵❪ ✵✳✼✽ ✵✳✸✺ ✸✳✸✹ ✸✹✳✾✻ ✵✳✵✵ ✶✵✵✳✵✹
✷✵✵ ✺ ❬✶✱✺✵❪ ✷✶✳✷✾ ✺✺✳✶✷ ✶✺✾✳✺✽ ✕ ✵✳✵✵ ✺✽✼✳✽✻
✷✵✵ ✺ ❬✶✵✶✱✸✵✵❪ ✶✹✳✶✸ ✹✵✳✸✼ ✶✶✶✳✹✼ ✕ ✵✳✵✵ ✻✾✾✳✾✽
✷✵✵ ✷✵ ❬✶✱✺✵❪ ✸✹✳✽✵ ✷✻✳✻✵ ✼✷✳✼✹ ✕ ✵✳✵✵ ✺✾✺✳✼✼
✷✵✵ ✷✵ ❬✶✵✶✱✸✵✵❪ ✸✳✺✾ ✷✳✽✸ ✶✽✳✶✵ ✕ ✵✳✵✵ ✻✽✺✳✹✾
❆✈❣✳ ✶✹✳✻✾ ✷✵✳✼✸ ✻✹✳✵✷ ✷✹✵✳✽✶ ✵✳✵✵
❚♦ ❝♦♥❝❧✉❞❡✱ t❤❡ ▼❆ s❤♦✇s ♣r♦♠✐s✐♥❣ r❡s✉❧ts ✐♥ ♦✉r ♣r❡❧✐♠✐♥❛r② t❡sts✳ ◆♦♥❡t❤❡❧❡ss✱
❡①t❡♥s✐✈❡ t❡sts ❛♥❞ ❝♦♠♣❛r✐s♦♥ ✇✐t❤ ♦t❤❡r ♠❡t❛❤❡✉r✐st✐❝s ❛r❡ ❞❡s✐r❛❜❧❡✳
❘❡❢❡r❡♥❝❡s
❆❧✐❞❛❡❡ ❇✳✱ ❉✳ ❘♦s❛✱ ✶✾✾✼✱ ✏❙❝❤❡❞✉❧✐♥❣ ♣❛r❛❧❧❡❧ ♠❛❝❤✐♥❡s t♦ ♠✐♥✐♠✐③❡ t♦t❛❧ ✇❡✐❣❤t❡❞ ❛♥❞ ✉♥✲
✇❡✐❣❤t❡❞ t❛r❞✐♥❡ss✧✱ ❈♦♠♣✉t❡rs ✫ ❖♣❡r❛t✐♦♥s ❘❡s❡❛r❝❤✱ ❱♦❧✳ ✷✹✱ ♣♣✳ ✼✼✺✲✼✽✽✳
❈❤❛♥❣ ❨✳ ❈✳✱ ❈✳ ❨✳ ▲❡❡ ✷✵✵✹✱ ✏▼❛❝❤✐♥❡ s❝❤❡❞✉❧✐♥❣ ✇✐t❤ ❥♦❜ ❞❡❧✐✈❡r② ❝♦♦r❞✐♥❛t✐♦♥✧✱ ❊✉r♦♣❡❛♥
❏♦✉r♥❛❧ ♦❢ ❖♣❡r❛t✐♦♥❛❧ ❘❡s❡❛r❝❤✱ ❱♦❧✳ ✶✺✽✱ ♣♣✳ ✹✼✵✲✹✽✼✳
❈❤❡♥ ❨✳✱ ▲✳ ▲✉ ❛♥❞ ❏✳ ❨✉❛♥✱ ✷✵✶✻✱ ✏❚✇♦✲st❛❣❡ s❝❤❡❞✉❧✐♥❣ ♦♥ ✐❞❡♥t✐❝❛❧ ♠❛❝❤✐♥❡s ✇✐t❤ ❛ss✐❣♥❛❜❧❡
❞❡❧✐✈❡r② t✐♠❡s t♦ ♠✐♥✐♠✐③❡ t❤❡ ♠❛①✐♠✉♠ ❞❡❧✐✈❡r② ❝♦♠♣❧❡t✐♦♥ t✐♠❡✧✱ ❚❤❡♦r❡t✐❝❛❧ ❈♦♠♣✉t❡r
❙❝✐❡♥❝❡✱ ❱♦❧✳ ✻✷✷✱ ♣♣✳ ✹✺✲✻✺✳
❈❤❡♥❣ ❘✳✱ ▼✳ ●❡♥ ❛♥❞ ❚✳ ❚♦③❛✇❛✱ ✶✾✾✺✱ ✏▼✐♥♠❛① ❡❛r❧✐♥❡ss✴t❛r❞✐♥❡ss s❝❤❡❞✉❧✐♥❣ ✐♥ ✐❞❡♥t✐❝❛❧
♣❛r❛❧❧❡❧ ♠❛❝❤✐♥❡ s②st❡♠ ✉s✐♥❣ ❣❡♥❡t✐❝ ❛❧❣♦r✐t❤♠s✧✱ ❈♦♠♣✉t❡rs ✫ ■♥❞✉str✐❛❧ ❊♥❣✐♥❡❡r✐♥❣✱ ❱♦❧✳
✷✾✱ ♣♣✳ ✺✶✸✲✺✶✼✳
●r❛❤❛♠ ❘✳ ▲✳✱ ❊✳ ▲✳ ▲❛✇❧❡r✱ ❏✳ ❑✳ ▲❡♥str❛ ❛♥❞ ❆✳ ❍✳ ●✳ ❘✐♥♥♦♦② ❑❛♥✱ ✶✾✼✾✱ ✏❖♣t✐♠✐③❛t✐♦♥
❛♥❞ ❛♣♣r♦①✐♠❛t✐♦♥ ✐♥ ❞❡t❡r♠✐♥✐st✐❝ s❡q✉❡♥❝✐♥❣ ❛♥❞ s❝❤❡❞✉❧✐♥❣✿ ❛ s✉r✈❡②✧✱ ❆♥♥❛❧s ♦❢ ❉✐s❝r❡t❡
▼❛t❤❡♠❛t✐❝s✱ ❱♦❧✳ ✺✱ ♣♣✳ ✷✽✼✲✸✷✻✳
❑♦✉❧❛♠❛s ❈✳✱ ✶✾✾✹✱ ✏❚❤❡ t♦t❛❧ t❛r❞✐♥❡ss ♣r♦❜❧❡♠✿ r❡✈✐❡✇ ❛♥❞ ❡①t❡♥s✐♦♥s✧✱ ❖♣❡r❛t✐♦♥s ❘❡s❡❛r❝❤✱
❱♦❧✳ ✹✷✱ ♣♣✳ ✶✵✷✺✲✶✵✹✶✳
❑♦✉❧❛♠❛s ❈✳✱ ●✳ ❏✳ ❑②♣❛r✐s✐s✱ ✷✵✶✵✱ ✏❙✐♥❣❧❡✲♠❛❝❤✐♥❡ s❝❤❡❞✉❧✐♥❣ ♣r♦❜❧❡♠s ✇✐t❤ ♣❛st✲s❡q✉❡♥❝❡✲
❞❡♣❡♥❞❡♥t ❞❡❧✐✈❡r② t✐♠❡s✧✱ ■♥t❡r♥❛t✐♦♥❛❧ ❏♦✉r♥❛❧ ♦❢ Pr♦❞✉❝t✐♦♥ ❊❝♦♥♦♠✐❝s✱ ❱♦❧✳ ✶✷✻✱ ♣♣✳ ✷✻✹✲
✷✻✻✳
▲❡❡ ❈✳ ❨✳✱ ❩✳ ▲✳ ❈❤❡♥✱ ✷✵✵✶✱ ✏▼❛❝❤✐♥❡ s❝❤❡❞✉❧✐♥❣ ✇✐t❤ tr❛♥s♣♦rt❛t✐♦♥ ❝♦♥s✐❞❡r❛t✐♦♥s✧✱ ❏♦✉r♥❛❧ ♦❢
❙❝❤❡❞✉❧✐♥❣✱ ❱♦❧✳ ✹✱ ♣♣✳ ✸✲✷✹✳
▲❡♥str❛ ❏✳ ❑✳✱ ❆✳ ❍✳ ●✳ ❘✐♥♥♦♦② ❑❛♥ ❛♥❞ P✳ ❇r✉❝❦❡r✱ ✶✾✼✼✱ ✏❈♦♠♣❧❡①✐t② ♦❢ ♠❛❝❤✐♥❡ s❝❤❡❞✉❧✐♥❣
♣r♦❜❧❡♠s✧✱ ❆♥♥❛❧s ♦❢ ❉✐s❝r❡t❡ ▼❛t❤❡♠❛t✐❝s✱ ❱♦❧✳ ✶✱ ♣♣✳ ✸✹✸✲✸✻✷✳
▼❛❣❣✉ P✳ ▲✳✱ ●✳ ❉❛s✱ ✶✾✽✵✱ ✏❖♥ ✷ × ♥ s❡q✉❡♥❝✐♥❣ ♣r♦❜❧❡♠ ✇✐t❤ tr❛♥s♣♦rt❛t✐♦♥ t✐♠❡s ♦❢ ❥♦❜s✧✱
P✉r❡ ❛♥❞ ❆♣♣❧✐❡❞ ▼❛t❤❡♠❛t✐❦❛ ❙❝✐❡♥❝❡s✱ ❱♦❧✳ ✶✷✱ ♣♣✳ ✶✲✻✳
P♦tts ❈✳ ◆✳✱ ✶✾✽✵✱ ✏❆♥❛❧②s✐s ♦❢ ❛ ❤❡✉r✐st✐❝ ❢♦r ♦♥❡ ♠❛❝❤✐♥❡ s❡q✉❡♥❝✐♥❣ ✇✐t❤ r❡❧❡❛s❡ ❞❛t❡s ❛♥❞
❞❡❧✐✈❡r② t✐♠❡s✧✱ ❖♣❡r❛t✐♦♥s ❘❡s❡❛r❝❤✱ ❱♦❧✳ ✷✽✱ ♣♣✳ ✶✹✸✻✲✶✹✹✶✳
P♦tts ❈✳ ◆✳✱ ▲✳ ◆✳ ❱❛♥ ❲❛ss❡♥❤♦✈❡✱ ✶✾✽✷✱ ✏❆ ❞❡❝♦♠♣♦s✐t✐♦♥ ❛❧❣♦r✐t❤♠ ❢♦r t❤❡ s✐♥❣❧❡ ♠❛❝❤✐♥❡ t♦t❛❧
t❛r❞✐♥❡ss ♣r♦❜❧❡♠✧✱ ❖♣❡r❛t✐♦♥s ❘❡s❡❛r❝❤ ▲❡tt❡rs✱ ❱♦❧✳ ✶✱ ♣♣✳ ✶✼✼✲✶✽✶✳
❱❡♣s❛❧❛✐♥❡♥ ❆✳ P✳✱ ❚✳ ❊✳ ▼♦rt♦♥✱ ✶✾✽✼✱ ✏Pr✐♦r✐t② r✉❧❡s ❢♦r ❥♦❜ s❤♦♣s ✇✐t❤ ✇❡✐❣❤t❡❞ t❛r❞✐♥❡ss ❝♦sts✧✱
▼❛♥❛❣❡♠❡♥t ❙❝✐❡♥❝❡✱ ❱♦❧✳ ✸✸✱ ♣♣✳ ✶✵✸✺✲✶✵✹✼✳
❲♦❡❣✐♥❣❡r ●✳ ❏✳✱ ✶✾✾✹✱ ✏❍❡✉r✐st✐❝s ❢♦r ♣❛r❛❧❧❡❧ ♠❛❝❤✐♥❡ s❝❤❡❞✉❧✐♥❣ ✇✐t❤ ❞❡❧✐✈❡r② t✐♠❡s✧✱ ❆❝t❛
■♥❢♦r♠❛t✐❝❛✱ ❱♦❧✳ ✸✶✱ ♣♣✳ ✺✵✸✲✺✶✷✳
148

Modelling and Solving the Hotspot Problem in Air
Traffic Control
Patrick Schittekat, Carlo Mannino, and Giorgio Sartor
SINTEF Digital, Norway
carlo.mannino@sintef.no
1 Introduction
Air traffic management involves the coordination of flights in a particular region of the
world with the objective of guaranteeing their safety while possibly reducing delays. There
usually exist different levels of responsibility.
For example, the Air Traffic Control (ATC) provided by an airport handles airplanes
on the ground and in the controlled airspace in the proximity of the airport while the
Air Navigation Service Providers (ANSP) manage the air traffic of a bigger region or an
entire country. For a discussion of the different issues arising in air traffic management see
(Allignol, 2012). One of the critical tasks of an ANSP is to provide the Air Traffic Flow
Management that consists of preventing overcrowded portions of air space while trying
to exploit their maximum capacity. In fact, the air space within a country or a region is
subdivided in sectors which are assigned to specific controllers. Each controller can handle
no more than a given number of airplanes at a time. Consequently, each sector has its own
capacity, that is the maximum number of airplanes that can occupy the sector in a given
time. Note that not only capacity, but even the shape and number of sectors varies from
time to time. For instance, at peak hours the number of sector increases.
The number of airplanes that will occupy each sector in a given time can be forecast
taking into account the timetable and the planned route of the airplanes. A hotspot is a
sector in which the forecast number of airplanes is greater than its maximum capacity in
at least one point in time. The task of an air traffic flow manager is to prevent hotspots
while guaranteeing an efficient utilization of the air space.
The official flight plans of airplanes may already give rise to hotspots. In addition, the
timetable changes when one or more airplanes are delayed and, as a result, more hotspots
might appear.
When this happens, the air traffic flow manager has to modify the flight plans of many
airplanes in order to avoid hotspots and reduce delays. More specifically, the manager can
delay some take-offs, or reduce speeds on certain trip segments for airborne aircraft. This
procedure is usually carried out in a heuristic way and with little software support, leading
most of the time to suboptimal solutions.
2 A MILP model for the hotspot problem
The MILP model for the Hotspot problem resembles very closely the classic job-shop
scheduling problem with blocking and no-wait constraints (Mascis and Pacciarelli, 2002)
exploited in several papers for different transportation problems.
When airborne an airplane f will traverse an ordered sequence of sectors (s1, s2, . . . , sq).
We define a route as an ordered sequence of pairs aircraft-sector, say O(f) = ((f, s1), (f, s2),
. . . , (f, sq)) = (v1, v2, . . . , vq). An element v ∈ O(f) is then a pair (f, s), where f is a flight
and s a sector. With each element v we associate a minimum traversing time λv and a
maximum traversing time Λv. To simplify the notation, if the route of an airplane f starts
149

at an airport a, then we also consider a as a special sector and we have the special pair
(f, a) in O(f). Similarly, the destination airport will be represented as a special sector.
Consider now the set of all flights F, and let O be the set of sector occupations by all
flights in F. With every element v = (f, s) ∈ O we associate a variable tv, representing
the time airplane f enters sector s. Now let u, v ∈ O correspond to the occupation of two
successive sectors in the route of flight f. Then the following constraints must hold:
λu ≤ tv − tu ≤ Λu. (1)
Now, consider a set of distinct flights F̄ = {f1, . . . , fq} traversing a sector s. For each
flight fi, let ai be the time the flight enters s and di the time the flight exits s (that is, it
enters the next sector). Now, assume that the capacity cs of the sector is not enough to
accommodate all flights in F̄, namely cs |F̄|. Then, at least for a pair f, g of flights in
F̄, f and g do not meet in s, namely either f exits s before g enters s or vice-versa. This
can be expressed by the following disjunctive constraint:
(ag − df ≥ 0) ∨ (af − dg ≥ 0). (2)
The above disjunctive constraint can be linearized in standard fashion by introducing
a binary variable yfg for each ordered pair of flights (f, g) ∈ F̄ × F̄, such that yfg = 1 if f
exits s before g enters and yfg = 0 otherwise. Then, for all pairs in F̄, constraint (2) can
be replaced by the following conjunction of linear constraints:
(i) yfg + ygf = 1, {f, g} ⊆ F̄
(ii) ag − df ≥ −M(1 − yfg) (f, g) ∈ F̄ × F̄
(3)
where M is a suitably large positive constant.
Now, for {f, g} ⊆ F̄, we introduce a binary variable xfg and we let xfg = 1 if f and g
meet in s, and xfg = 0 otherwise. Then, if cs |F̄|, we must have:
∑
{f,g}⊆F̄
xfg ≤
(
|F̄|
2
)
− 1 (4)
so that at least a pair of flights in F̄ do not meet in s. Because constraint (2) actually
holds only if f and g do not meet in s, which in turn depends on the value of xfg, we can
suitably modify (3.i) to take this into account:
yfg + ygf = 1 − xfg. (5)
So, a complete formulation for a set of flights F with their routes traversing a set S of
sectors can be obtained by considering now constraints (1) for all routes, and constraints
(3.ii), (4) and (5) for all sectors s ∈ S and all set F(s) ⊆ F of flights exceeding the capacity
cs of s. Let P ⊂ Rp
be the set of points (x, y, t) satisfying all such inequalities, including the
integer stipulation on variables x, y: then our problem reduces to {min f(t) : (x, y, t) ∈ P}.
The objective f(t) may vary form instance to instance, but for our first set of experiments
it will simply be the (weighted) delay at destination.
150

3 Solution approach: sketch
In principle, problem {min f(t) : (x, y, t) ∈ P} could be solved by simply resorting to
some general purpose commercial solver. However, formulation P has two major sources
of complexity which do not allow such a simple approach, already for small-medium size
realistic instances. First, the family of constraints (4) can grow exponentially with F.
This is tackled in a standard fashion by resorting to the so called “lazy constraints” trick.
Namely, constraints are generated dynamically during the search as lazy constraints - i.e.
only if they are violated by the current integer feasible solution.
Next, in order to make the constraint redundant for certain values of the binary vari-
ables, in (3.ii) we make use of a second, infamous trick, namely we include the large coef-
ficient M (the “big-M trick”). In turn, this makes the formulation very weak and prone to
return poor bounds - and thus often intractable search trees.
Our approach to tackle this problem and solve {min f(t) : (x, y, t) ∈ P} extends the
methodology first developed in (Lamorgese and Mannino, 2016). In particular, we exploit
Benders’ decomposition to obtain a (master) problem only in the binary variables - plus a
few continuous variables to represent the objective function. The decomposition allows us
to get rid of big-M coefficients (at the cost of an increased number of linear constraints).
Moreover, the constraints of the reformulated master correspond to basic graph structures,
such as paths, cycles and trees. The new formulation is obtained by strengthening and
lifting the constraints of a classical Benders’ reformulation1
.
Computational experiments. In preparation.
References
Allignol, C., Barnier, N., Flener, P., and Pearson, J., 2012, “Constraint programming for air traffic
management: a survey 1: In memory of pascal brisset”, The Knowledge Engineering Review,
Vol. 27(3), pp. 361–392.
Lamorgese, L. and Mannino, C., 2016, “A non-compact formulation for job-shop scheduling prob-
lems in transportation” (Dagstuhl Seminar 16171), Dagstuhl Reports, Vol. 6(4), p. 151, 2016,
submited to Operations Research, under revision.
Mascis, A. and Pacciarelli, D., 2002, “Job-shop scheduling with blocking and no-wait constraints”,
European Journal of Operational Research,, Vol. 143(3), pp. 498–517.
1
Strengthening and lifting are non-straightforward and both necessary: the master of the classic
Benders’ reformulation would still contain many copies of the despised big-M.
151

A proactive-reactive approach to schedule an
automotive assembly line
Massimo Manzini1
, Erik Demeulemeester2
and Marcello Urgo1
1
Politecnico di Milano, Mechanical Engineering Department, Milano, Italy
massimo.manzini, marcello.urgo@polimi.it
2
KU Leuven, Faculty of Business and Economics, Department of Decision Sciences and
Information Management, Leuven, Belgium
erik.demeulemeester@kuleuven.be
Keywords: flow-shop, proactive-reactive scheduling, assembly.
1 Introduction and problem statement
The assembly of bodywork parts for the automotive sector is operated in dedicated
assembly lines implementing the sequence of assembling operations through specific join-
ing technologies (e.g., spot welding, clinching, hemming, etc.). These assembly lines are
organized as a set of stations executing assembly operations, input/output stations to load
components and unload final parts, and a transportation device moving parts within the
line. The latter is usually a 7-axis robot shared among the stations. In this paper we con-
sider an assembly line where a batch of parts has to be processed. Assembly operations are
executed by automatic devices while load/unload operations are executed manually. The
line has a single transportation robot to be shared among the stations and the proposed
approach aims at scheduling its missions. Due to the manual execution of load/unload
operations, uncertain process times must be considered, thus, the problem under study
is a Stochastic Resource-Constrained Flow-Shop Scheduling Problem to minimize the time
needed to complete a batch of products, i.e., the makespan. In the need to address uncer-
tainty, specific approaches must be adopted. Examples are the ones optimizing the expected
value of the makespan (Fernandez 1995, Igelmund and Radermacher 1983). Nevertheless,
the minimization of the expected value does not protect against rare but very extreme sce-
narios, as discussed in (Alfieri et al. 2012) and (Manzini and Urgo 2015) for Make-to-Order
processes. To this aim, we propose a proactive-reactive approach providing a baseline sched-
ule and looking for the optimal sequence of the robot considering the actual duration of
operations during the execution of the assembly process. Differently from other approaches
of this class, e.g., (Davari and Demeulemeester 2016), the proposed approach identifies
disjunctive constraints without explicitly deciding the starting times of operations.
2 Solution approach
Consider an Activity-on-Node (AoN ) representation of a flow-shop where V = {0, 1, ..., n}
is the set of nodes representing operations and E = (i, j), i, j ∈ V the set of arcs modeling
precedence constraints. Operation durations are modeled through general and indepen-
dent random distributions p̃ = p̃0, . . . , p̃n, pi being a realization of distribution p̃i and
p = p0, . . . , pn a realization of the entire set p̃. Notice that, if an operation is determin-
istic, the described formulation still applies with a single value as support. The flow-shop
under study has a limited availability of the transporter and hence we consider a single
resource with unary availability. We address the scheduling of shared transporter’s mis-
isons through the decisions over a set of disjunctive constraints named EDC (additional
to the ones in E), resolving resource utilization conflicts.The uncertainty embedded in the
152

problem is addressed by adopting a proactive-reactive approach made up of two steps. The
first step provides the baseline schedule as the optimal sequence of the robot considering a
given duration of the uncertain operations (e.g., a quantile can be used). The second one
is supposed to operate while the baseline schedule is being operated, every time an incon-
gruity between the fixed operation duration and the one experienced in the execution of
the schedule occurs. It checks whether the baseline schedule is supposed to remain optimal
and, if needed, reacts by inverting some of the disjunctive constraints previously selected.
The two steps are described in detail in the following.
2.1 Proactive step
The proactive step hypothesizes that the duration of operations is fixed. In case of un-
certain durations this value can be decided by fixing a quantile q obtaining pq
= pq
0, . . . , pq
n,
without considering any anticipation of associated uncertainty. The scheduling problem is
solved using the deterministic approach presented in (Demeulemeester and Herroelen 1992).
The baseline schedule obtained provides the set of additional constraints EDC. In addition
to this, a sensitivity analysis on the solution is also executed. For each precedence con-
straint in EDC, the range of variability of operation durations is calculated such that, if
the durations go outside this range, then the decision taken for the considered disjunctive
constraint is not optimal anymore, and thus the opposite constraint should be considered.
Consider the constraint (i, j) ∈ EDC assuming durations pq
, and the eligible times of op-
erations i and j, Qpq
i and Qpq
j , defined as the instants on which each operation can start
in terms of all the precedence constraints in E, without considering any of those in EDC.
Define ∆pq
i,j = Qpq
i − Qpq
j as the difference between the eligible times of two operations
linked with a disjunctive constraint (i, j). If the decision on this disjunctive constraint
is optimal, the associated makespan is shorter than the one considering the opposite di-
rection, i.e., S
(i,j)
n ≤ S
(j,i)
n , where S
(i,j)
n is the starting time of operation n, considering
disjunctive constraint (i, j). Clearly, this depends on the duration of the operations in pq
.
The makespan takes advantage of an inversion of the disjunctive constraint if and only
if the lateness of i, compared to Qpq
i , is enough to cause a delay of the makespan that
is longer than the delay caused by an inversion without any lateness of i. More formally,
the inversion is effective if there is a difference between the eligible times that is greater
than ∆T
i,j = ∆pq
i,j − (S
(i,j)
n − S
(j,i)
n ). The threshold ∆T
i,j will be used in the reactive step for
evaluating the optimality of the disjunctive constraint (i, j) during the process execution.
2.2 Reactive step
The reactive step considers a vector of realizations p for the durations of the operation
and grounds on the definition of a state space Ω modeling the execution of the operations in
the flow-shop. The execution of the operations can be modeled through a sequence of states
over time t, ω(p, t) = (O, F, S, dO) ∈ Ω. Each state is fully described by the set of operations
in execution O, their starting times S and their durations dO(i), ∀i ∈ O, as well as the set
of completed ones F. Algorithm 1 models the execution of operations starting from t = 0
with initial state ω(p, 0) = (0, ∅, 0, 0) and finishes when all the operations are completed,
i.e., F = V (steps 1-2). Every time an operation is completed, the set F is updated (step 4)
and, if there is an operation i that can start because all its predecessors are completed (step
6), it is put into execution and added to the set of ongoing operations O (step 11). On the
contrary, if its execution is constrained by the completion of another operation k through
a decision on one disjunctive constraint (k, i) ∈ EDC (step 7), then the algorithm checks
whether (k, i) remains optimal in relation to the realizations in p. This evaluation is done
through the estimation of the probability that the actual difference between the eligible
153

Reactive-Procedure
1 ω(p, 0) = (0, ∅, 0, 0)
2 While F ! = V
3 t = t + 1
4 If dO(i) − S(i) = pi, ∀i ∈ O → F = F + i
5 Else dO(i) = dO(i) + 1
6 If i 6∈ O, i 6∈ F and j ∈ F, ∀j ∈ (j, i)
7 If (k, i) ∈ EDC and P(∆p
k,i(t) ∆T
k,i) T
8 EDC = EDC − (k, i) + (i, k)
9 O = O + i, S(i) = t
10 Else
11 O = O + i , S(i) = t
Algorithm 1: Reactive step algorithm.
Operation Mode Min Max
I 6 5 29
T1 13 − −
A 10 − −
T2 9 − −
O 5 4 21
Table 1: Operation duration in
seconds.
times exceeds the threshold previously identified: P[∆p
k,i(t) ∆T
k,i]. If this probability
exceeds a threshold T, the reaction is applied by inverting the constraint (k, i) (steps 8-9).
The P[∆p
k,i(t) ∆T
k,i] is estimated considering the duration of the operations in O preceding
k and their distributions p̃. The probability that ∆p
k,i(t) is greater than ∆T
k,i is equal to
the probability that the difference between the finish time of the last preceding operation
of k and the eligible time of i is greater than ∆T
k,i, conditioned on the ongoing durations
in dO. We are looking at the residual duration probability of the operations preceding k:
P[∆p
k,i(t) ∆T
k,i] = P[maxl∈prec(k)(dF (l)) − Qi ∆T
k,i | dO(l)] = P[maxl∈prec(k)(dF (l) −
dO(l)) ∆T
k,i − Qi], where prec(k) indicates an operation preceding k.
3 Application
The proposed approach is applied on a single product flow-shop assembling a hood
bodywork. The execution of the process is modeled using the AoN representation in Fig-
ure 1. The process consists of five operations, the first and the last ones model the loading
(I) and unloading (O) of the parts, executed manually. In the third operation (A), a re-
inforcement bar is added through a spot welding process, while the second and fourth
operations are handling tasks (T1 and T2 respectively) operated by the 7-axis robot mov-
ing the hood in the line. The two manual operations follow a triangular distribution, while
the others are deterministic (Table 1). The triangular distributions consider an average ex-
ecution duration as the mode, very close to the minimum value, and a worst-case duration
as the maximum value, modeling the occurrence of a problem or a delay. The approach
addresses the conflicts between transport operations in the production of a whole batch.
These conflicts are depicted with dotted arcs in Figure 1 for a single transport of the first
job, only (T21), but are repeated for the whole batch. In addition, we set the threshold T to
0.5, but let the quantile q, used for fixing the duration in the proactive step, vary between
0.1 and 0.9. We evaluate the performances of the approach in terms of the mean square
error compared to the minimum makespan solution obtained with complete knowledge
of the durations of operations using 10000 runs. In addition, we estimate the approach’s
performances without the reactive step and compare the results. Aggregated performances
for different lengths of the batch (from 5 to 50 jobs) are included in Table 2. Grounding on
these results, the proactive-reactive approach always performs as good or better than the
proactive schedule without reaction (PR and P-only in Table 2). Indeed, if the reactive step
154

P-only PR
Quantile 0.1 0.5 0.9 0.1 0.5 0.9
# jobs
5 5.473 5.473 0.917 0.917 0.917 0.917
10 4.963 4.963 0.980 0.980 0.980 0.980
20 7.445 7.445 1.347 1.347 1.347 1.347
50 8.456 8.456 1.865 1.865 1.865 1.865
Table 2: Aggregated results of the application. Fig. 1: AoN process representation.
does not apply any modification, the baseline solution is automatically applied, as depicted
for the 5 jobs and 90th percentile case. The impact of the number of jobs and the percentile
is also analyzed: the percentile impacts on results of the only-P approach, with better per-
formances for high values. On the other hand, this parameter does not affect the reaction’s
performance due to the uncertainty source being limited to the first and last operations.
The performances get worse as the number of jobs increases for both approaches. As a
conclusion, the proactive approach provides a good baseline schedule, nevertheless, the re-
action step improves the performances when used to manage the occurrence of unexpected
events, providing a good support in the line’s real-time management.
4 Conclusions
In this article we propose a proactive-reactive approach to schedule a semi-automatic
assembly system, with a specific focus on the definition of the reaction policy. The approach
has been tested on a five-operation process with good results, demonstrating that the
application of the reactive step significantly improves the performances of the baseline one.
Future developments will address the investigation of (i) completely manual processes or
(ii) tuning the threshold for the reactive step to match user’s aversion to risk and (iii) the
application of additional disjunctive constraints modeling the schedule of machines besides
handling operations.
Acknowledgments
This research has been supported by ReCaM EU project, grant agreement No: 680759.
References
Alfieri, A., Tolio, T. and Urgo, M., 2012, A two-stage stochastic programming project scheduling
approach to production planning, Int J Adv Man Technol, Vol. 62, pp. 279-290.
Davari, M. and Demeulemeester, E., 2016, The proactive and reactive resource-constrained
project scheduling problem, Working paper.
Demeulemeester, E. L. and Herroelen, W. S., 1992, A Branch-and-Bound Procedure for the
Multiple RCPSP, Man Sci, Vol. 38, pp. 1803-1818.
Fernandez, A. A., 1995, The Optimal Solution to the Resource-Constrained Project Scheduling
Problem with Stochastic Task Durations, Unpublished Doctoral Dissertation.
Igelmund, G. and Radermacher, F. J., 1983, Preselective Strategies for the Optimization of
Stochastic Project Networks under Resource Constraints, Networks, Vol. 13, pp. 1-28.
Manzini, M. and Urgo, M., 2015, Makespan estimation of a production process affected by un-
certainty: Application on MTO production of NC machine tools, J Man Syst, Vol. 37, No.
1, pp. 1-16.
155

Applying a cost, resource or risk perspective to
improve tolerance limits for project control: an
empirical validation
Annelies Martens1
and Mario Vanhoucke1,2,3
1
annelies.martens@ugent.be, mario.vanhoucke@ugent.be
2
3
Keywords: project control, buffer monitoring, analytical tolerance limits
1 Introduction
While timely completion is an important factor of project success, projects often exceed
their predefined deadline. In order to protect this deadline, a project buffer can be placed
at the end of the project. Further, during the project control process, the project progress
can be evaluated using tolerance limits that generate warning signals when the project
deadline is expected to be exceeded.
In this study, four methods that use different perspectives to construct tolerance limits
for the schedule progress of projects are empirically validated on the large and diverse
dataset of Batselier Vanhoucke (2015). Each of the used perspectives, namely the time,
cost, resource and risk perspective, consider project-specific information to determine the
allowable buffer consumption during project execution. Based on this allowable buffer con-
sumption, threshold values for the schedule performance can be set for each project phase.
These threshold values generate warning signals when the project deadline is expected to
be exceeded, such that the project manager can take corrective actions to get the project
back on track. The limits using a time, cost and resource perspective have been proposed in
recent literature. Their performance has been evaluated using artificial data. First, the time
perspective to determine the allowable buffer consumption has been introduced by Colin
Vanhoucke (2015). Since this is the most straightforward approach that requires the
least project-specific information, the resulting limits, which are referred to as linear lim-
its, act as a benchmark for the other perspectives. Second, Martens Vanhoucke (2017a)
proposed a cost perspective by setting the allowable buffer consumption based on the cost
accrue of the project and compare the resulting cost limits to the linear benchmark limits.
Further, Martens Vanhoucke (2017b) use the resource availability and requirements in-
formation to determine the allowable buffer consumption using a resource perspective to
construct resource limits. Finally, in this study, we propose a novel approach that employs
a risk perspective to set the allowable buffer consumption and to construct risk limits. For
each type of limits, we evaluate the ease of implementation and performance for real-life
projects. These limits are discussed in greater detail in section 2. In the remainder of this
section, a brief introduction to project control is given.
Since uncertainty and variation during project execution inevitably result in deviations
from the plan, projects often do not finish on time or within budget. In order to protect
the project deadline from these deviations, a project buffer can be placed at the end of
the project. Moreover, the project control phase is an important component of Integrated
Project Management and Control that focuses on detecting problems and/or opportunities
during project execution such that corrective actions can be taken to get the project back
156

on track (Vanhoucke 2014). The project control process consists of three parts, namely
monitoring the project progress, evaluating this progress, and taking corrective actions
when necessary. A well-known technique to monitor the cost and time progress of projects
is Earned Value Management (EVM, Fleming Koppelman (2010)). This methodology
provides a birds-eye view on the project progress by aggregating the activity progress
information on a higher work breakdown structure (WBS) level. Since both the schedule
and cost performance metrics provided by EVM are cost-based metrics, Earned Schedule
(ES, Lipke (2003)) has been developed as an extension that focuses on the time aspect
of projects. In this study, EVM/ES schedule performance metrics are used to monitor the
project progress. Further, project control tolerance limits are a tool to evaluate the project
progress and to decide whether corrective actions are required. For each project phase,
threshold values for the schedule performance are set. When the measured progress is
below this threshold, the project is expected to exceed its deadline and a warning signal is
generated. When a signal is generated by the tolerance limits, the project manager should
take corrective actions to get the project back on track. In section 2, the different types
of tolerance limits are briefly discussed. Further, results of the empirical experiment are
described in section 3.
2 Tolerance limits for project schedule control
The tolerance limits that have been proposed in recent literature can be classified in
three groups, namely static, statistical and analytical tolerance limits. First, static tolerance
limits are constant throughout the entire project life cycle and do not consider any project-
specific or historical information. These limits are determined by applying rules of thumb
and are introduced by Goldratt (1997) and Leach (2005). Further, statistical tolerance
limits apply concepts of Statistical Process Control (SPC, Shewhart (1931)) and require
historical information or Monte Carlo simulations to define the desired state of the progress
at each project phase. The statistical tolerance limits introduced in literature have been
validated using simulation studies (Colin Vanhoucke 2014, Colin Vanhoucke 2015,
Colin, Martens, Vanhoucke Wauters 2015) or empirical data (Aliverdi, Moslemi Naeni
Salehipour 2013, Bauch Chung 2001, Leu Lin 2008, Lipke Vaughn 2000, Wang,
Jiang, Gou, Che Zhang 2006). Finally, analytical tolerance limits require project-specific
information that is readily available during the scheduling phase to determine the threshold
values for each project phase. Since these limits do not require historical data or Monte
Carlo simulations, they are easier to implement than statistical tolerance limits. Moreover,
by including project-specific information, they are more accurate than static tolerance
limits. This type of tolerance limits has been proposed by Colin Vanhoucke (2015),
Hu, Cui, Demeulemeester Bie (2015), Martens Vanhoucke (2017a) and Martens
Vanhoucke (2017b).
The tolerance limits reviewed in this study are analytical tolerance limits, and follow
the same general procedure to be constructed. First, for each project phase, the allowable
buffer consumption is determined. This reflects the amount of buffer that can be consumed
at each project phase during execution without endangering the project deadline. Sec-
ond, the buffered planned progress (BPP) curve is determined. This curve represents the
project progress when, at each project phase, the allowable buffer consumption is entirely
consumed. The construction process for the BPP-curve is illustrated in Figure 1. Finally,
the threshold values are constructed by comparing the BPP to the planned progress. Con-
sequently, when the actual progress is below the BPP, the project is expected to exceed
its deadline and a warning signal is generated. For a more detailed discussion on the con-
157

struction of this type of tolerance limits, the reader is referred to Martens Vanhoucke
(2017a).
time
€
PD
100%
5%
PV-curve
t1
(25% of PD)
78%
t2
(50% of PD)
25%
t3
(75% of PD)
project
buffer
100%
5%
78%
25%
PD
t1
t3
t2
time
€
PD
100%
5%
PV-curve
t1
(25% of PD)
78%
t2
(50% of PD)
25%
t3
(75% of PD)
BPP-curve
DL
allowable
buffer
consumption
Project
buffer size
Fig. 1. Determining the BPP-curve.
The four different perspectives all propose a different approach to determine the allow-
able buffer consumption. The construction of the BPP and the calculation of the threshold
values, on the contrary, do not differ. First, the linear limits assume that the project buffer
can be consumed proportionally with the time, e.g. at x% of the project makespan, x% of
the buffer can be consumed. Since these limits do not consider the amount of work that
has to be completed during each project phase, cost limits have been introduced. These
limits determine the allowable buffer consumption proportionally with the cost of each
phase. Further, resource limits have been proposed to account for the impact of resource
conflicts on project delays. Finally, we introduce the risk limits, which consider the risk of
each project phase to determine the allowable buffer consumption. Two steps have been
implemented to determine the aggregate risk of each project phase. First, a risk value is
assigned to each project activity. This risk value is defined as the product of the activity
duration variability (σ) as estimated by the project manager and, since activity delays may
affect the actual start of successors, the number of succeeding activities (#succ). Second,
the risk of each project phase is determined by aggregating the risk values of the sched-
uled activities at each phase. The allowable buffer consumption at each project phase is
determined by the risk limits proportionally with this aggregated risk.
3 Research study and preliminary results
In this study, we discuss the merits and pitfalls of using artificial and empirical data to
evaluate the performance of project control tolerance limits. Further, a new perspective,
e.g. a risk perspective, is introduced to assign portions of the project buffer to each project
phase based on the risk level of these phases. We determine the aggregated risk level of
each project phase by considering the estimated activity duration variance and the position
of these activities in the baseline schedule. Finally, we compare the performance of the
different perspectives and evaluate their ease of implementation.
The artificial data used in the simulation studies consists of 900 project networks with
varying topological network structures, generated using the project network generator
RanGen (Demeulemeester, Vanhoucke Herroelen 2003). Risk and variability is added
using generalised beta distributions for the activity durations. Further, the empirical data
consists of a wide variety of real-life projects in different industries from the database of
Batselier Vanhoucke (2015). In this database, the baseline schedule, risk analysis and
project control data of the real-life projects are listed.
The empirical experiment conducted in this study confirms the result of previous sim-
ulation studies performed by Martens Vanhoucke (2017a) and Martens Vanhoucke
158

(2017b), e.g. that including project-specific information improves the efficiency of tolerance
limits for project control. However, deploying the cost perspective improves the efficiency
only slightly in our empirical experiment. Further, while deploying the resource perspec-
tive entails additional effort compared to the other perspectives, this effort enhances the
efficiency substantially. Finally, the novel risk perspective improves the efficiency of the
tolerance limits more than the cost perspective, and is hence an appropriate alternative
when projects are not constrained by scarce resources.
In general, this experiment has shown that including project specific information is an
effective approach to improve the project monitoring efficiency. Further, the results of this
study can be used by project managers to determine which perspectives they can deploy
to monitor their projects.
References
Aliverdi, R., Moslemi Naeni, L. Salehipour, A. (2013). Monitoring project duration and cost in
a construction project by applying statistical quality control charts, International Journal of
Project Management 31(3): 411–423.
Batselier, J. Vanhoucke, M. (2015). Construction and evaluation framework for a real-life project
database, International Journal of Project Management 33: 697–710.
Bauch, G. T. Chung, C. A. (2001). A statistical project control tool for engineering managers,
Project Management Journal 32: 37–44.
Colin, J., Martens, A., Vanhoucke, M. Wauters, M. (2015). A multivariate approach for top-
down project control using earned value management, Decision Support Systems 79: 65–76.
Colin, J. Vanhoucke, M. (2014). Setting tolerance limits for statistical project control using
earned value management, Omega The International Journal of Management Science 49: 107–
122.
Colin, J. Vanhoucke, M. (2015). A comparison of the performance of various project control
methods using earned value management systems, Expert Systems with Applications 42: 3159–
3175.
Demeulemeester, E., Vanhoucke, M. Herroelen, W. (2003). Rangen: A random network generator
for activity-on-the-node networks, Journal of Scheduling 6: 17–38.
Fleming, Q. Koppelman, J. (2010). Earned Value Project Management, 3rd edition edn, Project
Management Institute, Newton Square, Pennsylvania.
Goldratt, E. (1997). Critical Chain, North River Press, Great Barrington, MA.
Hu, X., Cui, N., Demeulemeester, E. Bie, L. (2015). Incorporation of activity sensitivity measures
into buffer management to manage project schedule risk, European Journal of Operational
Research 249: 717–727.
Leach, L. P. (2005). Critical chain project management, Vol. 2nd, Artech House.
Leu, S. S. Lin, Y. C. (2008). Project performance evaluation based on statistical process control
techniques, Journal of Construction Engineering and Management 134: 813–819.
Lipke, W. (2003). Schedule is different, The Measurable News Summer, 31–34.
Lipke, W. Vaughn, J. (2000). Statistical process control meets earned value, CrossTalk: The
Journal of Defense Software Engineering June, 16–20,28–29.
Martens, A. Vanhoucke, M. (2017a). A buffer control method for top-down project control,
European Journal Of Operational Research 262: 274–286.
Martens, A. Vanhoucke, M. (2017b). The integration of constrained resources into top-down
project control, Computers Industrial Engineering 110: 277–288.
Shewhart, W. A. (1931). Economic control of quality of manufactured product, Vol. 509, ASQ
Quality Press.
Vanhoucke, M. (2014). Integrated Project Management and Control: First come the theory, then
the practice, Management for Professionals, Springer.
Wang, Q., Jiang, N., Gou, L., Che, M. Zhang, R. (2006). Practical experiences of cost/schedule
measure through earned value management and statistical process control, Lecture Notes in
Computer Science 3966: 348–354.
159

A Metamodel Approach to Projects Risk Management:
outcome of an empirical testing on a set of similar
projects
F. Minelle1
, F. Stolfi2
, Di Gioacchino2
and Santini2
1
Computer Science Dept “Sapienza” University, Rome-Italy
minelle@di.uniroma1.it
2
PRS Planning, Ricerche e Studi, Rome-Italy
stolfi@prsmonitor.it, digioacchino@prsmonitor.it, santini@prsmonitor.it
Keywords: project risk management, e-government, context-based risk analysis, multi-
project experimental outcome.
1 Looking for a metamodel, context-based, approach to project risk manage-
ment
This paper outlines a metamodel approach, context-based, to project risk management
based on McFarlan model (McFarlan, 1981, M. Baldini et al., 2002), built by the authors
analyzing a set of information technology projects along the entire life cycle. The model
considers each project as characterized by a specific risk level, depending on the following
risk factors:
• Size (project/product volume);
• Innovation extent (technology, process, organization, and so on) of project products/solutions
to be implemented;
• General complexity (impact of induced changes on stakeholders organizations and their
relevant operating processes and/or impact on management of contractual constraints
and clauses between customer/owner and supplier/contractor).
Evaluating the risk factors, the metamodel allows to identify:
• The main project risk (or structural risk);
• The strategy of most suitable countermeasures to be implemented in order to re-
strain negative effects on success criteria values and, as a consequence, on project/product
performances;
• The typical countermeasures, more focused on the proper action, can be selected
progressively depending on suitability level of its own management approach;
• The specific countermeasures; considering typical countermeasures generated by
metamodel, project manager can identify specific actions to mitigate each project risk.
2 The e-government program launched by the Italian Public Administration
and Innovation Department
This model was applied to the Italian e-government program, a process of innovation of
Italian local public administrations (Regions, Provinces, Municipalities etc.) initiated and
funded in the mid-2.000s. The Program promoted the implementation of projects from
Local Public Administrations aimed to delivery e-government services and infrastructure
for citizens and firms.
160

One of the main focus of the e-government program was the implementation of projects
not only by individual local public administrations but mainly from a “group of admin-
istrations” with the possibility of direct or indirect participation to the program. Indirect
participation was about the reuse of products and solutions implemented by other local
public administrations.
The funding of the program was €120 million, covering 134 projects (selected out of
approximately 400 submitted projects) with a total value of approximately €500 million.
The program involved 20 Regions (100%), 93 Provinces (90%), more than 170 Mountain
Communities and more than 4,000 Municipalities (49%).
3 The risk survey process on co-financed e-government projects
3.1 Context-based risk analysis and selection of the “most-likely” effective
countermeasures
The model structure has the following components:
a. Summary: Summary report containing a dashboard of indicators whose values come
from the risk analysis carried out at the project;
b. Detection Model: Form for classification of project risk factors (by importance). Most
of the data concerning these factors are carefully extracted from the executive plan
(mandatorily prepared by the proponent entity, according to a predefined standard)
and minimally integrated with further data from the analysis and interpretation of the
project (done by the authors, as program assessors);
c. Countermeasures: Form dedicated to point out the basic countermeasures, suggested
by the authors and/or adopted by the project, according to the specific structural risks.
In particular, the initial indication of suggested countermeasures was completed during
the project implementation, with information concerning their implementation.
The risk analysis process involves the following steps:
• Filling the Detection Model worksheet;
• Filling and checking the Countermeasures Identification and Implementation worksheet;
• Detection of risk indicators through the summary report dashboard.
Detection Model
This form contains a checklist dedicated to detect the project risks; the description of
such type of risks is listed in a worksheet table whose columns have the following meanings:
• Risk factors: list of the factors to be detected for the purposes of the risk analysis;
• Drafting criteria: they represent the evaluation of the correspondent risk factor,
according to the project team leader point of view. Each risk factor was evaluated
according a scale of 3 values (G=big; M=medium; P=small), with the value limit of
each class defined by analyzing the statistical distribution of projects;
• Source: field used to indicate if data are extrapolated in objective way from the ex-
ecutive plan or, alternatively, submitted according to a specific interpretation by the
project team.
Each element of the checklist contributes to define the criticalities of the projects in
terms of Technological Complexity (TC), Organizational Complexity (OC) and Dimension
(DIM). In particular:
• the risk factors evaluation such indicated in Detection Model allows to identify the
most critical situations;
161

• the identification of specific strategies for the risk management (to prevent or to control
them) allows to get information concerning the types of countermeasures more suitable
according the characteristics of the specific project.
The identification of the types of countermeasures has the purpose:
I. To select the prevailing approach, devoted to:
• contain both organizational issues and integration problems with other initia-
tives/projects (IE-External Integration);
• mitigate both organizational issues and management problems which are internal
to the project itself; such problems also include issues caused by the multiplicity
of stakeholders involved in the project (II-Internal Integration);
• ensure formal and rigorous management of the project, either at the initial state
and during its execution (PC-Formal Planning);
• control, in qualitative way, processes and products realized within the project (QC-
Formal Quality Assurance and Control).
II. Selecting the countermeasures mix, to specifically adopt as the best strategy, belonging
to the above mentioned types, for risk mitigation (higher results in equal effort). Such
approach allow to correlate the assessment of risk factors with the structural risk of the
whole project (risk level) and the prominent approach for risk mitigation. In particular:
a. the structural risk of the whole project has been rated on a 5-level qualitative
scale (Very Low, Low, Medium, High, Very High);
b. The evaluation of risk categories was represented on an only 2-level scale (low
and high) with the aim to reduce the potential combinations generated to identify
the strategy for risk mitigation indicated on the risk mitigation approach table;
c. The strategy for risk mitigation was focused taking into account the weighted
configuration of the Risk Factor Assessment and it is expressed on a scale of 3-level
values (Low, Medium, High) for each management approach (External Integration,
Internal Integration, Formal Planning Control).
Component “Countermeasures”
Each Project Manager uses checklist to self-assess project risk factor identifying main
risk and suitability level of each one of the following management approach: IE-External
integration, II-Internal integration, PC-Formal project management and QC-Quality as-
surance or control.
Based on suitability level of each management approach, metamodel allows to identify
the typical most suitable countermeasures to each specific project; Project Manager may
accept or modify or integrate the suggested typical countermeasures.
3.2 Risk analysis summary and countermeasures actual application rate
The last step of metamodel has to do with verifying the actual compliance of suggested
countermeasures and their application rate. In addition to project manager evaluation,
metamodel allows independent assessor evaluation, aimed to mitigate subjective evaluation
of project manager.
Independent assessor evaluation aimed to understand the suitability of risk management
actions identified by project manager. In order to perform that, assessors analyze project
documentation and may modify or integrate selected countermeasures defined from project
manager.
In order to perform an effective audit of applied countermeasures, metamodel pro-
vides a short description for each countermeasure in terms of: [i] Countermeasure, name of
countermeasure; [ii] Meaning, short description of countermeasure; [iii] Objective evidences,
162

examples of objective evidences that should be found to prove countermeasure was actually
applied.
For each selected typical countermeasure (defined from project manager or integrated
from assessor) the metamodel allows to indicate actual level of applied countermeasure: 0
= not applied; 1 = partially applied, 2 = widely applied; 3 = totally applied.
Matching applied countermeasures versus planned ones allows to define application
rate.
3.3 Outcome achieved from more than one hundred similar projects
The metamodel outlined in this paper has been implemented in more than 130 projects;
though all of them were aimed to design e-government services to local public administra-
tion, they were all different in terms of dimension (volume), cost and duration.
From this experience we can infer two order of results: first result is methodological and
it is about a large and coherent application of this model to a large and distributed set of
projects; second result is about impact of model on project management performances.
About the first order, we tried the “easy for use” and applicability of model in all
projects with different contexts and dimensions. The metamodel contributed to spread
risk management culture in project management teams. Moreover, the metamodel counter-
measures database has been enhanced by the results of the most common countermeasures
applied in the projects.
About the second order, the results analysis, ongoing and final, of e-government program
highlighted as projects which applied suitable countermeasures had a positive impact on
time constraint (cost was “out of scope” of assessor control and quality was measured ex-
post in terms of stakeholder benefits on about 45 projects), with less delay to achieve the
intermediate milestones and to complete the entire project.
Picture below, taken from periodic report of e-government program, shows an example
of correlation between suitability of applied countermeasures and projects delay. In that
picture we can see as projects with “suitable” or “most suitable” countermeasures appli-
cation rate have less delay than projects with “not suitable” countermeasure application
rate.
For instance, projects which showed (the minority, at a certain time) a “most suitable”
countermeasures application rate had, as an average, 82% progress and 12 months delay,
while projects which showed (at a certain time) a “not suitable” countermeasure application
rate had 60% progress and 16 months delay.
4 Conclusions and the way forward
The unusual case of a set of similar, contemporary and independent projects (more than
one hundred), was likely to be an empirical proof of the consistent effectiveness of Risk
Management in improving project patterns. While such “experiment” is not easy to be
repeated on a so large number of projects, because projects have the characteristic to be a
“single shot” items, the authors replicated a similar (and someway more sophisticated) ap-
proach while monitoring a large program (in a multi-years, multi-projects, multi-contracts
environment) for the ICT reengineering of a main governmental Institution. Diagnosis was
excellent, but unfortunately therapy (i.e. countermeasures) not always applied: program
stakeholders (owner and contractors) did not “buy” the approach.
Anyway, the proof of evidence about benefit on projects by using the suggested risk
management approach (or anyone in the literature) would encourage all the project man-
agers and their sponsor to consider it a mandatory task in performing the job they were
assigned to.
163

Fig. 1.
Future plans to improve the above-described risk management approach, would consider
the paradigm shift for Project Management 2.0 (Kerzner, 2015), in order to insert in the
model the evaluation of: (i) soft skill competence in the project team, mainly for the
project manager, project team and “sponsor”, (ii) communication plan and its contents
for the various stakeholder clusters, (iii) consistency of expected benefits, both monetary
and not monetary ones. In addition, final correlation between the applied strategy for risk
countermeasures and project performance, including also final success (proven benefit for
stakeholders) should be thoroughly exploited.
5 Acknowledgements
The authors acknowledge the support of CNIPA (now AgID - Italian Agency for Dig-
italization) which appointed this activity to PRS as a part of the monitoring engagement
to the already mentioned e-government program.
References
McFarlan, 1981, “Porfolio approach to information systems.” Harvard Business Review, pp. 142-
150.
M. Baldini, A. Miola, A. Neri, 2002, Project management e processi progettuali, Franco Angeli
(6th edition).
H. Kerzner, 2015, Project Management 2.0: Leveraging Tools, Distributed Collaboration, and Met-
rics for Project Success, Wiley (1st edition).
164

A column generation scheme for the
Periodically Aggregated Resource-Constrained
Project Scheduling Problem
Pierre-Antoine Morin1,2
, Christian Artigues2
and Alain Haït1,2
1
ISAE SUPAERO, University of Toulouse, Toulouse, France
pierre-antoine.morin@isae.fr, alain.hait@isae.fr
2
LAAS CNRS, University of Toulouse, CNRS, Toulouse, France
artigues@laas.fr
Keywords: project, planning, scheduling, periodical aggregation, mixed integer linear pro-
gramming, column generation.
This abstract is focused on the Periodically Aggregated Resource-Constrained Project
Scheduling Problem (PARCPSP) (Morin et. al. 2017b), that can be seen as a continuous-
time variant of a restricted Resource-Constrained Project Scheduling Problem with par-
tially renewable resources (RCPSP/π) (Böttcher et. al. 1999). The purpose of this work is
to compare an existing compact formulation with a new extended formulation.
The PARCPSP is defined as follows. A set A of activities, subject to end-to-start
precedence relations E ⊂ A × A, and a set R of renewable resources are given. During its
processing (duration pi), activity i ∈ A requires ri,k units of resource k ∈ R (capacity bk).
The scheduling horizon is divided uniformly into a set L of L periods of length ∆. The
PARCPSP can be described by the following abstract model:
Minimize : Sn+1 − S0 (1)
s.t. : Sj − Si ≥ pi ∀(i, j) ∈ E (2)
X
i∈A
ri,k
di,ℓ(Si)
∆
≤ bk ∀k ∈ R , ∀ℓ ∈ L (3)
Where Si is the start date of activity i and di,ℓ(t) is the length of the intersection
of the intervals [(ℓ − 1)∆, ℓ∆] and [t, t + pi]. The objective (1) is to minimize the project
duration (activities 0/n+1 are the dummy beginning/end of the project) under precedence
constraints (2) and periodically aggregated resource constraints (3): for every resource, in
every period, the capacity should not be exceeded on average.
1 Compact model
Two formulations based on mixed (continuous and discrete) time frameworks have been
proposed to model the PARCPSP. Although the computation of the values di,ℓ(Si) can be
done by introducing only step binary variables (Morin et. al. 2017b), we focus here on an
alternative scheme based on period partitionning (Morin et. al. 2017a) that requires more
continuous variables, but involves less constraints, all big-M-free, thus yielding a better
linear relaxation.
Two additional functions are considered. Let λi,ℓ(t) be the length of the intersection of
the intervals [(ℓ − 1)∆, ℓ∆] and (−∞, t]; let µi,ℓ(t) be the length of the intersection of the
intervals [(ℓ − 1)∆, ℓ∆] and [t + pi, +∞) (cf. Figure 1).
Notice that it is easier to describe λi,ℓ and µi,ℓ compared to di,ℓ. Moreover, the intervals
whose lengths are measured by these functions form a partition of period ℓ. Therefore:
λi,ℓ(t) + di,ℓ(t) + µi,ℓ(t) = ∆ ∀i ∈ A , ∀ℓ ∈ L , ∀t ∈ R (4)
165

(
ℓ−
1
)
∆
−
p
i
m
i
n
(
ℓ
∆
−
p
i
,
(
ℓ−
1
)
∆
)
m
a
x
(
ℓ
∆
−
p
i
,
(
ℓ−
1
)
∆
)
ℓ
∆
0
min (pi, ∆)
t
di,ℓ(t)
(a) di,ℓ
(
ℓ−
1
)
∆
ℓ
∆
0
∆
t
λi,ℓ(t)
(b) λi,ℓ
(
ℓ−
1
)
∆
−
p
i
ℓ
∆
−
p
i
0
∆
t
µi,ℓ(t)
(c) µi,ℓ
Fig. 1: Piecewise linear functions di,ℓ, λi,ℓ and µi,ℓ
The values di,ℓ(Si), λi,ℓ(Si) and µi,ℓ(Si) are represented as continuous variables Di,ℓ,
Λi,ℓ and Mi,ℓ, respectively. To model the piecewise linear functions λi,ℓ and µi,ℓ, auxiliary
binary variables are introduced; more precisely, to ensure a non-increasing (resp. non-
decreasing) step behavior of the variables Λi,ℓ (resp. Mi,ℓ), step binary variables zλ
i,ℓ
(resp. zµ
i,ℓ) are required.
Minimize : Sn+1 − S0 (5)
s.t. : Sj − Si ≥ pi ∀(i, j) ∈ E (6)
X
i∈A
ri,kDi,ℓ ≤ bk∆ ∀k ∈ R , ∀ℓ ∈ L (7)
Si =
X
ℓ∈L
Λi,ℓ ∀i ∈ A (8)
Di,ℓ = ∆ − Λi,ℓ − Mi,ℓ ∀i ∈ A , ∀ℓ ∈ L (9)
Di,ℓ ≥ 0 ∀i ∈ A , ∀ℓ ∈ L (10)
X
ℓ∈L
Di,ℓ = pi ∀i ∈ A (11)
zλ
i,ℓ+1 ≤
Λi,ℓ
∆
≤ zλ
i,ℓ ∀i ∈ A , ∀ℓ ∈ L (12)
zµ
i,ℓ−1 ≤
Mi,ℓ
∆
≤ zµ
i,ℓ ∀i ∈ A , ∀ℓ ∈ L (13)
zλ
i,ℓ ∈ {0, 1} ∀i ∈ A , ∀ℓ ∈ L (14)
zµ
i,ℓ ∈ {0, 1} ∀i ∈ A , ∀ℓ ∈ L (15)
The objective (5) is to minimize the project duration, under both precedence constraints
(6) and periodically aggregated resource constraints (7). Constraints (8) enable the com-
putation of start dates Si directly from Λi,ℓ variables, while constraints (9), derived from
166

the partition relation (4), enable the computation of Di,ℓ values that cannot be negative
[constraints (10)]. Constraints (11) permit to balance the values of Λi,ℓλ
i
and Mi,ℓµ
i
, where
ℓλ
i (resp. ℓµ
i ) is the period activity i starts (resp. completes) in. Finally, constraints (12)
[resp. (13)] enforce an interdependent non-increasing (resp. non-decreasing) step behavior
of variables Λi,ℓ and zλ
i,ℓ (resp. Mi,ℓ and zµ
i,ℓ) using binary variables [constraints (14) and
(15)]. Therefore, every variable Λi,ℓ (resp. Mi,ℓ) with ℓ 6= ℓλ
i (resp. ℓ 6= ℓµ
i ) is bound either
to 0 or ∆, as shown in Figure 2.
i
Si
pi
ℓ
1 2 3 4 5 6 7 8 9 10 11
ℓλ
i ℓµ
i
0
∆ Λi,ℓ
0
∆ Mi,ℓ
0
∆ Di,ℓ
Fig. 2: Partition-based mixed time framework
2 Dantzig-Wolfe decomposition
We now introduce a new extended formulation that enhances and exploits the com-
binatorial structure of the PARCPSP. On the one hand, the (restricted) master problem
consists in selecting start dates t ∈ Ti for every activity i ∈ A (binary decision variables xi,t
such that Si =
P
t∈Ti
t xi,t, ∀i ∈ A) in such a way that all constraints are satisfied, while
minimizing the project duration. On the other hand, the sub-problem consists in finding
time points t to insert into sets Ti. Notice that, although the start date of an activity can
be any (real) time point in the (continuous) interval [0, L∆ − pi], only a finite number of
them need to be considered, since optimal solutions match extreme points of a polytope
described by a finite number of constraints.
2.1 Master problem
Minimize :
X
t∈Tn+1
t xn+1,t −
X
t∈T0
t x0,t (16)
αi :
X
t∈Ti
xi,t = 1 ∀i ∈ A (17)
βi,j : −
X
t∈Tj
t xj,t +
X
t∈Ti
t xi,t ≤ −pi ∀(i, j) ∈ E (18)
γk,ℓ :
X
i∈A
X
t∈Ti
ri,k di,ℓ(t) xi,t ≤ bk ∆ ∀k ∈ R , ∀ℓ ∈ L (19)
xi,t ∈ {0, 1} ∀i ∈ A , ∀t ∈ Ti (20)
167

The objective (16) is to minimize the project duration, assigning a unique start date
to each activity [constraints (17)], under both precedence constraints (18) and periodically
aggregated resource constraints (19), using binary variables [constraints (20)].
Notice that dual variables βi,j and γk,ℓ are non-negative. The linear relaxation of the
master problem is obtained by replacing constraints (20) with “xi,t ≥ 0”; notice that
constraints αi imply “xi,t ≤ 1”.
2.2 Sub-problem
Minimize : αi +
X
j∈E⊕
i
βi,j t −
X
j∈E⊖
i
βj,i t +
X
k∈R
X
ℓ∈L
γk,ℓ ri,k di,ℓ(t) (21)
ESi ≤ t ≤ LSi (22)
Where, for each activity i ∈ A: E⊕
i = {j ∈ A : (i, j) ∈ E} (set of direct successors of i),
E⊖
i = {j ∈ A : (j, i) ∈ E} (set of direct predecessors of i), ESi and LSi are respectively
the earliest and latest starting time of i (those input values are typically obtained by
computing longest paths in the activity precedence graph).
Given an activity i ∈ A, the sub-problem returns a candidate start t within the horizon
[constraint (22)] such that the new variable xi,t has the least reduced cost [objective (21)].
This returned date t will be inserted in Ti in the restricted master problem only if needed,
i.e., if the reduced cost of xi,t is negative.
Notice that, after the partition relation (4), the reduced cost of xi,t can be transformed
into a sum of continuous monotonic piecewise linear functions of t. Therefore, the sub-
problem can be solved by a forward algorithm, linear in the number of breakpoints, hence
linear in the number of periods.
Computational experiments will be provided by time of the conference. Depending on
the results, it could be interesting to additionally separate either precedence or periodically
aggregated resource constraints. For instance, the framework proposed by Mingozzi et. al.
(1998) for the standard Resource-Constrained Project Scheduling Problem (RCPSP) could
be adapted to the case of the PARCPSP. The precedence constraints are managed by the
master problem, while the resource constraints are managed by the sub-problem. Instead of
using vector columns with binary components indicating whether an activity is processed
in a unit time period, these components should be replaced with real values in the interval
[0, ∆] indicating how much each activity is processed in a period of length ∆.
References
Mingozzi A., Maniezzo V., Ricciardelli S. and Bianco L., 1998, “An Exact Algorithm for the
Resource-Constrained Project Scheduling Problem Based on a New Mathematical Formula-
tion”, Management Science, Vol. 44, pp. 714–729.
Böttcher J., Drexl A., Kolish R. and Salewski F., 1999, “Project scheduling under partially renew-
able constraints”, Management Science, Vol. 45, pp. 543–559.
Morin P.A., Artigues C. and Haït A., 2017a, “A new mixed time framework for the Periodi-
cally Aggregated Resource-Constrained Project Scheduling Problem”, Proceedings of the 13th
Workshop on Models and Algorithms for Project Scheduling Problems (MAPSP 2017), Seeon
Seebruck, Germany.
Morin P.A., Artigues C. and Haït A., 2017b, “Periodically Aggregated Resource-Constrained
Project Scheduling Problem”, European Journal of Industrial Engineering, Vol. 11, No. 6,
pp. 792–817.
168

Development of a Schedule Cost Model for the
Resource Constrained Project that incorporates
Idleness
Babatunde Omoniyi Odedairo and Victor Oluwasina Oladokun
Department of Industrial and Production Engineering, University of Ibadan, Ibadan, Nigeria
{bo.odedairo,vo.oladokun}@ui.edu.ng
Keywords: project management, activities scheduling, resource-constrained, idleness cost.
1 Introduction
Scheduling project activities is a challenging decision-making process because such de-
cisions must cope with physical, technological and resource availability constraints. In the
classical resource-constrained project scheduling problem (RCPSP), the aim is to deter-
mine the start and finish times for all project activities within the specified precedence
relationship and resource constraints such that maximum completion time can be min-
imised. Möhring (1984) termed this problem as ‘problem of scarce resources’.
In practice, resource availability is often faced with conditions arising from the remote-
ness of project location, logistics cost of resource transportation, and costs associated with
hiring and releasing renewable resources (Sears et al., 2008). On the decision to release and
rehire renewable resources, this is practicable with some resources (e.g. unskilled labour)
and it is usually carried out to simultaneously meet daily manpower needs and eliminate
idleness (or waste). On the other hand, for resources which comes at a high hiring rate (or
are capital intensive) and are used from project start to finish not by a single activity but by
several activities, the decision to release becomes more complicated as it may not fit in with
activity resource requirements (Akpan, 1997; Vanhoucke, 2007; Odedairo, 2016). Therefore,
in a resource constrained project management environment; a replacement strategy should
be planned for resources that require uninterrupted usage and are jointly used by a group
of activities. This is necessary in order to forestall the following: (i) some skilled workers
(or machineries) released for another job may not return on time which can cause delay in
job processing, (ii) there is no guarantee that the same set of resources will be hired, and
(iii) the time to engage new resources might not be available. Hence, it becomes necessary
to decide on the minimum level of additional renewable resources to hire and hold (with all
costs implication) throughout a reasonable time period or for the entire project duration.
Also, while being held, the usage of a resource will differ in one or more time intervals due
to precedence constraints among project activities.
With this reality comes resource use-time and idle-time (and associated costs of usage
and idleness). The cost implications of resource usage and idleness times in this research is
assumed to have the same features as the time-dependent costs (TDC) introduced by Gong
(1997) and further elaborated and explained by Goto et al. (2000) and Vanhoucke (2006)
respectively. In this study, the objective is to characterise RCPSP within the context of
idleness cost (IC) arising from the use of additional hired resources (with TDC features)
held throughout the project makespan. Thereafter, a mathematical model that focuses on
the minimisation of the total schedule cost for the resource constrained project scheduling
problem with idleness cost (RCPSP-IC) will be developed to represent the essence of the
decision problem. The remainder of this paper is structured as follows. In section 2, related
work will be discussed. The mathematical models are presented in section 3 while in section
4, preliminary solution approach and results are discussed.
169

2 Related literature
Imreh and Noga (1999) investigated how scheduling problems change when machine
(resource) costs are considered. They argued that resource usage has associated cost, and if
the required resources are not available, then such can be procured or hired. Other studies
have been carried out on the impact of machine/resource cost on scheduling decisions
(Imreh, 2009; Ruiz-Torres et al., 2010). In their study, Ruiz-Torres et al. (2010) identified
two ways in which resource cost can be conceptualised and modeled as components of the
scheduling process. These are (i) using the duration of time required to process an activity
on a resource i.e. resource use-time and (ii) number of resources used.
As stated earlier, in projects; situations often arise when activity processing requires
uninterrupted availability and usage of specialised resources. Such a resource could be said
to be critical (or a bottleneck), in this context; a resource could be critical if it offers
specialised skills/services and its availability is constrained because it is capital intensive.
Furthermore, in their usage; inefficiencies such as resource idle-time may be encountered
due to predefined precedence constraints between activities in the project. In literature,
the problem of idle-time of resources due to processing of repetitive activities from unit to
unit and within-unit has been researched (Harris and Ioannou, 1998; Vanhoucke, 2007).
El-Rayes and Moselhi (1998) as cited by Vanhoucke (2013) define the term “work con-
tinuity constraints” as a way to schedule repetitive units of a project to enable timely
movement of resources from unit to unit to minimise total resource idle time. Vanhoucke
(2007) concluded that the minimisation of resource idle time for a work continuity optimi-
sation involves a trade-off between project completion and cost of idle time. Although, work
continuity constraints is widely known with repetitive projects; Vanhoucke (2007) opined
that in non-repetitive projects, uninterrupted usage of important resources e.g. specialized
consultants, etc. can also pose problem of idle time minimisation.
Therefore, for a RCPSP-IC; the decision on the number of additional resources to hire
and consequently hold to minimise total resource idle time should be considered during
project planning phase. To the best of our knowledge, there is no study available in which
the number of additional TDC resources in a RCPSP-IC is defined as a decision variable.
3 Problem abstraction and mathematical model
The RCPSP-IC can be stated as follows. Consider a set of activities, n, with index
j = 1, . . . , n numbered from a dummy start and end node of 0 and n+1 respectively. Each
activity j has the following information: an activity is to be processed on X renewable
resources (with an index of m = 1, . . . , X); once started, the processing cannot be inter-
rupted. There is a finish-start precedence relationship with zero time-lag between activities
which enforces each activity to be scheduled after all its predecessors are completed. The
precedence relationship between activities is depicted by activity-on-node (AON) network.
For each activity, its processing time is independent of the schedule and can only be exe-
cuted in a single mode composed of a fixed duration and renewable resource requirements.
For renewable resources, each resource has the following characteristics: a resource
cannot process more than one activity at a time; a pre-specified unit of resource m = X
is available for every period of the project horizon. It is assumed that the project will
require additional hired renewable resources (K) with TDC features. Furthermore, the
additional resources are assumed identical, held from project start to finish with service
time equivalent to the project makespan. The identical nature of the TDC resources allows
for the possibility of parallel processing.
170

3.1 Relationship between Cost of Project Schedule and Number of TDC re-
sources
Since activity scheduling constitutes the core of cost minimisation in project manage-
ment, any strategic plan to minimise cost must be centred on determining a good schedule.
Therefore, for a resource, its time-dependent cost is equivalent to the product of cost of hir-
ing per time (hours, days and weeks) unit and its service (usage) time. For a TDC resource
hired and held throughout the project lead time, cost interpretations of such decision is
shown in Figure 1.
Fig. 1. Relationship between cost of project schedule and number of TDC resources.
In Figure 1, let Z1, be the cost attributable to project completion time, and Z2, the cost
attributable to SRIT (now termed resource idleness cost). Two scenarios are possible, the
ﬁrst is the availability of one TDC resource, in this case, the sum of resource idleness time
(SRIT) will be minimum (zero) because the single resource is assumed to be continuously
busy; however, the project completion time (Cmax) will be maximum. The second scenario
involves multiple TDC resources; it is obvious that some or all the resources will be idle
during one or more time intervals of the project execution due to precedence constraints
between activities. In this case, project completion time is minimised while SRIT (k =
1, . . . , K) is assumed to be maximum.
Arising from the two scenarios, their cost implications can be depicted from Figure 1,
the total cost of project schedule (Z) is assumed to be a combination of two independent
components (Z1 and Z2). The relationship between the behaviour of Z1 and Z2 with respect
to available number of TDC resources (k = 1, . . . , K) can be further conceptualised to show
that, Z1 is a function Cmax and invariably a function of schedule (σ) and, for Z2, it is a
function of Cmax and k.
Therefore, for the RCPSP-IC, the minimum total schedule cost (Zmin) can be expressed
as shown in equations (1)–(3).
Z1 = f(Cmax) ≡ f(σ) (1)
Z2 = f(Cmax, k) ≡ f(σ, k) (2)
Zmin = f(σ) + f(σ, k) (3)
In equation (3), the minimum schedule cost for RCPSP-IC is a function of the schedule
and number of TDC resources.
171

3.2 Mathematical model of Resource Idleness Cost (RIC)
To model resource idleness cost (RIC), the schedule (σ) and number of TDC resources
(k) are defined as decision variables as described in equation (2). Before RIC model is pre-
sented, some notations used and their definitions will be explained. pj is activity processing
time (in days); the assignment variable yjk (1 = activity j is being processed on resource
k and 0 = otherwise); Cday is cost per time unit paid during each day of the project;
s = 1, . . . , S is index for schedule; t = 1, . . . , T is index for time periods; RSk
t(use−time) is
resource use-time for k; RSk
t(idle−time) is resource idle-time for k; Rk is per period avail-
ability of resource k; rjk is the resource unit required by activity j being processed by k
in each period; CRk is the cost of using TDC resource k per time unit (i.e. hiring cost per
day).
The cost associated with resource idleness time is presented in equation (4)–(8). As
explained, each TDC resource is expected to be held throughout the project makespan
(either used or idle); hence, each resource will have a duration equivalent to Cmax (in days)
as presented in equation (4). In equation (5), the idle time component of equation (4) is
obtained.
Cmax(σs) = RSk
t(use−time) + RSk
t(idle−time) (4)
RSk
t(idle−time) = Cmax(σs) − RSk
t(use−time) (5)
If activity j (j = 1, . . . , n) with processing time pj can be processed by TDC resource k,
then, RSk
t(use−time) for resource k is given by equation (6).
RSk
t(use−time) =
n
∑
j=1
pjyjk (6)
Therefore for k = 1, . . . , K, the sum of resource idle time (SRIT) in days can be mathe-
matically expressed as shown in equation (7).
SRIT =
K
∑
k=1

Cmax(σs) −
n
∑
j=1
pjyjk

 (7)
From equation (7), RIC can be expressed as shown in equation (8).
RIC =

CRk


K
∑
k=1

Cmax(σs) −
n
∑
j=1
pjyjk





 (8)
3.3 RCPSP-IC model
The total schedule cost for RCPSP-IC is described in equation (9). The first and second
components are the cost attributable to project completion time and to SRIT respectively.
172

TSC(σ,k) = Cmax(σs) · Cday +

CRk


K
∑
k=1

Cmax(σs) −
n
∑
j=1
pjyjk





 (9)
subject to
FTi ≤ FTj − pj, j = 1, . . . , n, ∀i ∈ IPj, i → j (10)
∑
j∈A(t)
rjk ≤ Rk, k = 1, . . . , K, t = 1, . . . , T (11)
Due to the usage of variable of the classical RCPSP (Pritsker et al., 1969), RCPSP-IC
is subjected to all of constraints already established in RCPSP. Two of these constraints
are described in equation (10)–(11). In equation (10), the precedence relations between
activities is enforced (where FT is finish time of activity; i for predecessor and j for
successor). Equation (11) ensure that resource consumption by each activity j = 1, . . . , n
does not exceed the limit per unit time.
4 Preliminary solution approach and results
A Serial Schedule Generation Scheme with latest finish time (LFT) as priority rule
was preliminary used to generate good schedules. The LFT priority is logically feasible
because an activity’s predecessor must have an earlier late finish time and so appears
earlier in the priority list. In addition, an idleness calculator (IDCalc) was incorporated
into the procedure which computes idle-time for the TDC resources at every time interval of
the project horizon. The computer implementation of the procedure was developed using
MATLAB. Data from a real-life project management situation were collected and the
associated problem solved as RCPSP-IC. For each TDC resource level k (k = 8 to 13), the
best schedule (σ) was obtained, keeping σ (σ = 1 to 6) constant, resource level was varied to
reflect levels of resource availability. Thirty-six (36) pairs of (σ, k) were formulated and for
each pair, Total Schedule Cost (TSC), Cost attributable to project completion time (Z1)
and Resource Idleness Cost (Z2) were calculated respectively. A conflicting relationship
exists between Z1 and Z2. Z1 decreased (increase) with increase (decrease) in resource
level. Z2 increased (decrease) with increase (decrease) in resource level. Hence, TSC for
RCPSP-IC was influenced by both the schedule and number of TDC resources.
References
Akpan E.O.P., 1997, “Optimum resource determination for project scheduling”, Journal of Pro-
duction Planning and Control, Vol. 8, pp. 462–469.
El-Rayes, K., and Moselhi, O., 1998, “Resource-driven scheduling of repetitive activities”, Journal
of Construction management and Economics, Vol. 16, pp. 433–446.
Gong, D.,1997, “Optimization of float use in risk analysis-based network scheduling”, International
Journal of Project Management, Vol.15, pp. 187–192.
Goto E., Joko, T., Fujisawa, K., Katoh, N., and Furusaka, S., 2000, “Maximizing net present value
for generalized resource constrained project scheduling problem”, Nomura Research Institute,
Tokyo, Japan.
Harris, R. B., and Ioannou, P. G., 1998, “Scheduling projects with repeating activities”, Journal
of Construction Engineering and Management, Vol. 124, pp. 269–278.
Imreh, Cs., 2009, “Online scheduling with general machine cost function”, Discrete Applied Math-
ematics, Vol. 157, pp. 2070–2077.
173

Imreh, Cs. and Noga, J. 1999, “Scheduling with machine cost”, in: Proceedings of APPROX99:
pp. 168–176.
Möhring, R.H., 1984, “Minimizing costs of resource requirements in project networks subject to a
ﬁxed completion time”, Operations Research, Vol. 32, pp. 89–120.
Odedairo B.O., 2016, “Development of Scheduling heuristic for the Resource Constrained Project
Management Problem with Idleness Cost”, Unpublished Ph.D Thesis, University of Ibadan.
Pritsker, A., Allan. B., Watters, L.J. and Wolfe, P.M. 1969, “Multiproject scheduling with limited
resources: A zero-one programming approach”, Management Science, Vol. 16. pp. 93–108.
Ruiz-Torres, A.J., López, F.J., Wojciechowski P.J. and Ho, J.C., 2010, “Parallel machine scheduling
problems considering regular measures of performance and machine cost”, The Journal of the
Operational Research Society, Vol. 61(5), pp. 849–857.
Sears, S. K., Sears, G.A. and Clough, R. H., 2008, Construction project management: a practical
guide to field construction management. 5th ed. New York: John Wiley Sons.
Vanhoucke, M., 2006, “Work continuity constraints in project scheduling”, Journal of Construction
Engineering and Management, Vol. 132, pp. 14–25.
Vanhoucke, M., 2007, “Work continuity optimization for the Westerscheldetunnel project in the
Netherlands”, Tijdschrift voor Economie en Management, Vol. 52, pp. 435–449.
Vanhoucke, M., 2013, “Project baseline scheduling: An overview of past experiences”, Journal of
Modern Project Management, Vol. 1, pp. 18–27.
174

Optimization problems in intermodal transport
Erwin Pesch
Universität Siegen
erwin.pesch@uni-siegen.de
1 Abstract
In intermodal container transportation, where containers need to be transported be-
tween customers (shippers or receivers) and container terminals (rail or maritime) and
vice versa, transshipment of containers is commonly arranged at the terminals. Attracting
a higher share of freight traffic on rail requires freight handling in railway terminals that
is more efficient, and which includes technical innovations as well as the development of
suitable optimization approaches and decision-support systems. In this talk we will review
some optimization problems of container processing in railway yards, and analyze basic
decision problems and solution approaches for the two most important yard types: conven-
tional rail-road and modern rail-rail transshipment yards. Furthermore, we review some
of the relevant literature and identify open research challenges. Additionally we address
a container dispatching and conflict-free gantry crane routing problem that arises at a
storage container block in an automated, maritime container terminal. A container block
serves as an intermediate buffer for inbound and outbound containers and exchanges of
containers between water- and landside of a maritime terminal. The considered block is
perpendicular to the waterside and employs two rail mounted gantry cranes. Cranes may
have the same or different sizes and therefore either are based at the opposite sides of the
container block or can cross each other. The question arises in which order and by which
crane containers are transported in order to minimize the makespan and prevent crane
conflicts.
175

The Stakeholder Perspective: how management of
KPIs can support value generation to increase the
success rate of complex projects
Massimo Pirozzi
Istituto Italiano di Project Management, Rome, Italy
pirozzi@isipm.org
Keywords: stakeholder, requirements, expectations, satisfaction, value, success, complex-
ity, measures, KPI, CSF, business.
1 Abstract
In today’s world, growing complexity demands that projects, in order to be successful,
have to satisfy not only stakeholder requirements, which refer to cost, time, and deliv-
ered quality, but also stakeholder expectations, which refer directly to the capability of
generating proper business value. Since business value can be measured only after project
completion, there is the need, during project life cycle, to handle value indicators: manage-
ment of proper Key Performance Indicators turns out to be a powerful Project Management
tool, which can be effectively used to increase the success rate of complex projects.
2 Stakeholder, who is this?
The word “stakeholder” dates back to the beginning of the eighteenth century, meaning
the person who was entrusted with the stakes of bettors, and, then, who was holding all the
bets placed on a game or a race, and, moreover who was paying the money to the winners:
therefore, the first stakeholder was a “holder of interests”. In addition, it is believed that
the first modern meaning of stakeholders, which has been attributed (Freeman, 1984) to
an internal memorandum of Stanford University Research Center dated 1963, was “those
groups without whose support the organization would cease to exist”, while in the first
text on the theory of stakeholders (Freeman, 1984), the definition of stakeholder was “a
stakeholder in an organization is any group or individual who can affect or is affected by
the achievement of the organization’s objectives”. Ten years later (Freeman, 1994), the
concept of generated value was added too, and stakeholders were defined as “participants
in the human process of joint value creation”. Furthermore, starting from the second half
of the eighties, the theory of stakeholder management, which was focused on corporate
social responsibility, incorporated an important ethical component into the concept of
stakeholder.
Definitively, a stakeholder, or an interested party, is a person, or a group of persons, or
an organization, that: has some kind of interest in the project; may affect the project, or
may be affected by the project; participates, or would like to participate, in the project;
can bring a value, which could be either positive or negative, to the project; may have
responsibilities towards the project, which, in turn, is supposed to satisfy stakeholders’
requirements and expectations.
Each project could then include a large variety of stakeholders, as, for example, project
manager, project team, sponsor, funders, partners or shareholders, customers, users, busi-
ness partners, suppliers, authorities, regulatory bodies, central and local public adminis-
tration, potential customers and users, participants and candidates to participate in the
176

project, local communities, web communities, associations, trade unions, media, competi-
tors, and so forth.
3 The stakeholder perspective and the value of project stakeholder relations
All the project stakeholders are important, since all the stakeholders are central towards
each project (Pirozzi, 2017): the stakeholders are both the actors, and the beneficiaries,
of the project, and the stakeholders are the critical success factor of the project, since
they are both the realizers of the results, and the validators, at various levels, of their
satisfaction in terms of needs and expectations. In fact, stakeholders, including the project
manager and the project team, are the doers of the project, as well as stakeholders, in-
cluding customers, users, and funders, are the target groups of the project itself: business
is the domain in which various stakeholders (project manager, project team, project man-
agement office, sponsor, board, shareholders, customers, users, suppliers, investors, central
and local public administration, groups of opinion, local communities, and so forth) inter-
act to create and exchange value. The relationships between the project stakeholders are,
then, real and proper business relationships, which are associated with the generation, and
the exchange, of both material and immaterial value: in general, this flow of value, among
the stakeholders, courses through the project with a continuous exchange of resources and
results.
In fact, organizations define strategies, which are based on their own mission and vision,
then select opportunities in accordance with defined strategy, then set business cases up,
and, finally, start projects up. The inputs of a project, and, specifically, to the project
management initiating process group, include business case, contract, and Statement of
Work (The International Organization for Standardization, 2012): generally, of course,
there are different business cases for different stakeholders, as, for instance, providers and
customers are. While business cases, which are the causes of project start-up, are based
on stakeholder business expectations, whose satisfaction correspond to the achievement
of project goals, contract and SOW, which are the references for project development
and delivery, are based on stakeholder requirements, which are, in turn, the conversion of
different stakeholder expectations in a commonly agreed (at least initially) project scope,
and whose fulfilment correspond to the achievement of project objectives.
A project can be considered really successful when its goals are realized, then achieving
those results that correspond to the stakeholder expectations, and which are characterized
by a satisfactory perceived quality; on the other hand, in order to realize the expecta-
tions of stakeholders (project goals), each project must necessarily achieve its objectives,
by realizing those deliverables that fulfill stakeholder requirements, and which are char-
acterized by a proper delivered quality. Effective Stakeholder Management should target
the satisfaction of both stakeholder requirements and expectations, which corresponds to
the achievement of both project goals and objectives (Figure 1): stakeholder satisfaction,
instead of being “a” critical success factor, proves to be “the” critical success factor; in
fact, projects may not succeed their goals, or may fail at all, for various reasons, which
could be technically very different, but, for sure, each project that was not successful had
at least one key stakeholder whose expectations were not satisfied.
In Stakeholder Management, then, effective management of both the domain of “deliv-
erable”, which is based on delivered contents, and of the domain of “perceivable”, which is
based on relations, becomes essential: the realization of the expectations of the stakehold-
ers, which, of course, implies also their acceptance of the deliverables, is therefore a primary
goal of the project, and it coincides with the most important critical success factor (Pirozzi,
2017). In any case, the stakeholder relations are the core of the project value, since they
177

Fig. 1. The Stakeholder Perspective (Pirozzi, 2017).
are a value, which is fundamental to the existence of the project and to its deﬁnition, but
also since they generate value, which is incorporated in the project, and because they allow
the exchange of value, through the project results, among the stakeholders themselves: the
results of a project are, in fact, the results of the relations among its stakeholders. Stake-
holder perspective, ultimately, supports and determines project success: «The emphasis on
Relationship Management is of special importance in today’s world» (Archibald, 2017).
4 Achieving the planned business value: the success factor in complex projects
PMI’s 2017 Global Project Management Survey (Project Management Institute, 2017)
reported that more of 30% of the projects do not meet their original goals and business
intent, i.e. they do not satisfy stakeholder expectations: therefore, the attention to the
satisfaction of stakeholder expectations must be considered as a critical factor, rather than
as a simple warning. In today’s Project Management, Stakeholder Management becomes,
then, the crucial process group, since it targets effectively the project success, by supporting
the generation of that project value which could satisfy both stakeholder requirements and
stakeholder expectations: if we use the perspective of project success, we can distinguish
two cases, the “classical” projects, and the “complex” projects.
In “classical” projects: project is part of customer core business (as, e.g., in internal
or in outsourcing projects), and/or project deliverables are product oriented, and/or are
tangible (as, e.g., in infrastructure projects), and/or, in any case stakeholder requirements
are either well defined (traditional contexts) or are evolutive, but all stakeholders cooperate
effectively (agile contexts); triple constraints (time, cost, quality) are dominant; relations
with stakeholders are important, and periodical. In classical projects, success is based on
the satisfaction of stakeholder requirements: in fact, there is just a small gap between the
satisfaction of requirements and the satisfaction of expectations, and, then, the measures
of the value could be limited to the measures of costs and of consistency/state of progress
of the deliverables, as usually happens in traditional/agile Project Management.
On the other hand, in “complex” projects: project is a support of customer core business
(as, e.g., in the majority of external projects), and/or project deliverables are oriented to
services, and/or are intangible (as, e.g., in software projects), and/or, in any case, stake-
holder requirements are either not well defined or are evolutive, but not all stakeholders
cooperate effectively; competing constraints are dominant, so that value and reputation
overcome triple constraints (Kerzner, 2015); relations with stakeholders are primary, and
can be continuous, fast, interactive (2.0), evolutionary (Kerzner, 2015). In complex projects,
178

success is based on the satisfaction of stakeholder expectations (Figure 1): since there is
a signiﬁcant gap between the satisfaction of requirements and the satisfaction of expecta-
tions, the measures of the value must include the measure of business value, too. Deﬁni-
tively, in each complex context, «Success is not necessarily achieved by completing the
project within the triple constraint. Success is when the planned business value is achieved
within the imposed constraints and assumptions.» (Kerzner and Saladis, 2009).
5 Managing effectively business value by use of Key Performance Indicators
Value management requires measures: during project life cycle, the measure of actual
cost and the assessment of the state of progress of the deliverables are commonly used as
indicators to estimate time and cost of the project completion, while the measures of the
generated business value, which is “future” with respect to project life cycle, could be done,
unfortunately, only after project completion. Therefore, since, during project life cycle, the
measures of business value are not possible, there is the need of the support of indicators,
which could be used to estimate both the current situation and the possible evolution of
the business value: proper Key Performance Indicators (KPIs) are then required. In each
complex project, an effective Stakeholder Management is thus based on measuring, mon-
itoring, and sharing, value-driven specific Key Performance Indicators: KPIs have to be
S.M.A.R.T (Specific, Measurable, Attainable, Realistic, Time Related), but also few, rel-
evant, actionable, and predictive, and can be shared continuously, quickly, and effectively
with stakeholders through dashboards, which can often replace efficiently traditional re-
ports (Kerzner, 2015). Moreover, the use of dashboards can be effective also in several cases
of reluctant, indifferent, and negative/hostile, stakeholders, because dashboards generally
ask only for answers yes/no, and no-answers can be interpreted positively, too.
Since stakeholders are different, they have different behaviour, and they target different
values: while “providers” (Project Manager, project team, etc.) target technical (delivered)
values, which are typical of Project Management, as triple constraints, project objectives,
and revenues, “investors” (top management, funders etc.) target economic values, as costs,
revenues, and business prospects, and “purchasers” (customers, users etc.) target business
values, as customer costs (that correspond to providers/investors revenues), project goals,
and benefits achievement. Therefore, effective KPIs have to target different types of value,
which refer to both Project Management, economic, and business domains. Examples of
Project Management KPIs include Earned Value, Cost Performance Index, Schedule Per-
formance Index, percentages of completed work packages compared to those which have
been planned, percentages of critical work packages which are aligned to the budget and/or
to the schedule, numbers and percentages relating to resources, risks, revisions, to requests
for change and changes etc.; examples of economic KPIs include economic, financial, mar-
keting, CRM, operational, HR, and sustainability indicators; examples of business value
indicators include specific functional and/or quantitative measures, and relevant percent-
ages of completion and/or of deviation from budget and/or schedule, but also measures
and percentages of stakeholder satisfaction (in terms of both requirements and expecta-
tions), measures and percentages of stakeholder engagement, and, definitively, measures
of perceived value (e.g. business value, social value, quality, reputation, business climate,
innovation, sustainability). In any case, while, in value-driven projects, the use, and the
sharing, of Project Management KPIs, of economic KPIs, and, only if they are considered
precisely measurable, of customer satisfaction KPIs too, can be considered well present in
the literature (Kerzner, 2017), in both value driven and complex projects the use, and the
sharing, of the above mentioned “new” KPIs which are relevant to the perceived value, can
be considered innovative. Furthermore, specific KPIs that are relevant to different busi-
179

ness sectors (e.g., referring to some cases that will be shown in the presentation at the
conference, local public transportation, pharmaceutical industry, railway infrastructure,
sustainable smart cities, web marketing, etc.), could be effectively used, as trend indica-
tors, also in project management during project life cycle, and not only as performance
indicators after project completion.
In Project Management, definitively, any measurable value can be effectively used as
a KPI, and the use of an appropriate selection of KPIs is a powerful tool to target the
success of the complex projects, by supporting both the value generation, and the project
goals achievement.
6 Conclusions
Stakeholders, who are central towards both projects and Project Management, define
success in terms of generation of their own business value: proper value indicators (KPIs)
in the domains of project management, of economics, and of business value, can be, then,
measured, shared with stakeholders, and used, in order to effectively confirm/redirect the
action of the project team during life cycle of the complex projects. Stakeholder Perspec-
tive, in this way, allows targeting both project objectives and project goals, then supporting
both the realisation of deliverables and the accomplishment of value generation, so as to
achieve the overall result of a significant increase of the project success rate. During the
presentation at the conference, a case study will be illustrated, in order to show both the
possibility of having unsuccessful projects, in which objectives can be reached, but stake-
holders expectations are not satisfied, and how the different Key Performance Indicators
could be managed, in particular if a conflict among some of them occurs.
References
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tana B, and Trasarti G., foreword by Archibald R.D., FrancoAngeli.
Freeman R.E., 1984, Strategic Management: a Stakeholder Approach, Pitman series in Business
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Freeman, R. E., 1994, “The politics of stakeholder theory: Some future directions”, Business Ethics
Quarterly, Vol. 4, No. 4.
International Organization for Standardization, 2012, “ISO 21500:2012 Guidance on Project Man-
agement”, International Organization for Standardization.
Kerzner H., 2015, Project Management 2.0 - Leveraging Tools, Distributed Collaboration, and
Metrics for Project Success, Wiley.
Kerzner H., 2017, Project Management Metrics, KPIs, and Dashboards, Third Edition, Wiley.
Kerzner H. and Saladis F.P., 2009, “Value-Driven Project Management”, Wiley.
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Management Survey, Project Management Institute.
180

▼✉❧t✐✲s❦✐❧❧ ♣r♦❥❡❝t s❝❤❡❞✉❧✐♥❣ ✐♥ ❛ ♥✉❝❧❡❛r r❡s❡❛r❝❤
❢❛❝✐❧✐t②
❖❧✐✈❡r P♦❧♦ ▼❡❥✐❛1,2
✱ ▼❛r✐❡✲❈❤r✐st✐♥❡ ❆♥s❡❧♠❡t1
✱ ❈❤r✐st✐❛♥ ❆rt✐❣✉❡s2
❛♥❞ P✐❡rr❡ ▲♦♣❡③2
1
❉❊❈✴❙❊❚❈ ✲ ❈❊❆ ❈❛❞❛r❛❝❤❡✱ ❙t P❛✉❧ ❧❡③ ❉✉r❛♥❝❡✱ ❋r❛♥❝❡
2
▲❆❆❙✲❈◆❘❙✱ ❯♥✐✈❡rs✐té ❞❡ ❚♦✉❧♦✉s❡✱ ❈◆❘❙✱ ❚♦✉❧♦✉s❡✱ ❋r❛♥❝❡
♦❧✐✈❡r✳♣♦❧♦♠❡❥✐❛❅❝❡❛✳❢r
❑❡②✇♦r❞s✿ ❘❈P❙P✱ Pr❡❡♠♣t✐✈❡ s❝❤❡❞✉❧✐♥❣✱ ▼✉❧t✐✲s❦✐❧❧✱ ◆✉❝❧❡❛r ❧❛❜♦r❛t♦r②✱ ❖♣t✐♠✐③❛t✐♦♥
✶ ■♥tr♦❞✉❝t✐♦♥
❚❤✐s ♣❛♣❡r ❛❞❞r❡ss❡s t❤❡ ✇❡❡❦❧② s❝❤❡❞✉❧✐♥❣ ♦❢ t❤❡ ❛❝t✐✈✐t✐❡s ✇✐t❤✐♥ ♦♥❡ ♦❢ t❤❡ r❡s❡❛r❝❤
❢❛❝✐❧✐t✐❡s ♦❢ t❤❡ ❋r❡♥❝❤ ❆❧t❡r♥❛t✐✈❡ ❊♥❡r❣✐❡s ❛♥❞ ❆t♦♠✐❝ ❊♥❡r❣② ❈♦♠♠✐ss✐♦♥ ✭❈❊❆ ✐♥ s❤♦rt
❢♦r ❋r❡♥❝❤✮✳ ❆❢t❡r ❛♥❛❧②③✐♥❣ t❤❡ ♦♣❡r❛t✐♦♥s ❛♥❞ ❝❤❛r❛❝t❡r✐st✐❝s ♦❢ t❤❡ st✉❞✐❡❞ ❧❛❜♦r❛t♦r②✱
✇❡ ❝♦♥❝❧✉❞❡ t❤❛t t❤❡ ♣r♦❜❧❡♠ ✉♥❞❡r ❝♦♥s✐❞❡r❛t✐♦♥ ❛♠♦✉♥ts t♦ ❛♥ ❡①t❡♥s✐♦♥ ♦❢ t❤❡ ❝❧❛ss✐❝❛❧
❘❡s♦✉r❝❡✲❈♦♥str❛✐♥❡❞ Pr♦❥❡❝t ❙❝❤❡❞✉❧✐♥❣ Pr♦❜❧❡♠ ✭❘❈P❙P✮✳
❚❤❡ ❘❈P❙P ✐s ❛ ❝♦♠❜✐♥❛t♦r✐❛❧ ♦♣t✐♠✐③❛t✐♦♥ ♣r♦❜❧❡♠ t❤❛t ❝♦✈❡rs ❛ ✇✐❞❡ r❛♥❣❡ ♦❢
s❝❤❡❞✉❧✐♥❣ s✐t✉❛t✐♦♥s✳ ❚❤❡ ♣r♦❜❧❡♠ ❝♦♥s✐sts ✐♥ s❝❤❡❞✉❧✐♥❣ ♥♦♥✲♣r❡❡♠♣t✐✈❡ t❛s❦s ♦♥ ❧✐♠✲
✐t❡❞ r❡♥❡✇❛❜❧❡ r❡s♦✉r❝❡s✳ ❚❤❡s❡ t❛s❦s ❛r❡ ❧✐♥❦❡❞ t♦❣❡t❤❡r ❜② ♣r❡❝❡❞❡♥❝❡ r❡❧❛t✐♦♥s❤✐♣s ✭t❛s❦
i ❝❛♥♥♦t st❛rt ✇❤✐❧❡ t❛s❦ l ✐s ✐♥ ♣r♦❝❡ss✱ ∀(l, i) ∈ E✮✳ ❯s✉❛❧❧②✱ t❤❡ ♦❜❥❡❝t✐✈❡ ✐s t♦ ✜♥❞ ❛
s♦❧✉t✐♦♥ t❤❛t ♠✐♥✐♠✐③❡s t❤❡ ♠❛❦❡s♣❛♥ ♦❢ t❤❡ ♣r♦❥❡❝t✱ ✇❤✐❧❡ ❝♦♠♣❧②✐♥❣ ❜♦t❤ t❤❡ ♣r❡❝❡❞❡♥❝❡
❝♦♥str❛✐♥ts ❛♥❞ t❤❡ r❡s♦✉r❝❡ ❝♦♥str❛✐♥ts✳
❊✈❡♥ ✐❢ t❤❡ st❛♥❞❛r❞ ✈❡rs✐♦♥ ♦❢ t❤❡ ❘❈P❙P ❛❧❧♦✇s t❤❡ ♠♦❞❡❧✐♥❣ ♦❢ ❛ ❜r♦❛❞ s♣❡❝tr✉♠
♦❢ s❝❤❡❞✉❧✐♥❣ ♣r♦❜❧❡♠s✱ ✐t ♠❛② ♥♦t ❝♦✈❡r ❛❧❧ t❤❡ s✐t✉❛t✐♦♥s t❤❛t ❝❛♥ ❜❡ ❢♦✉♥❞ ✐♥ r❡❛❧✲❧✐❢❡
♣r♦❜❧❡♠s✳ ❊①t❡♥❞❡❞ ✈❡rs✐♦♥s ♦❢ t❤❡ ❘❈P❙P ❛r❡ t❤❡♥ ♥❡❝❡ss❛r②✳ ❋♦r ❛ ♠♦r❡ ❡①❤❛✉st✐✈❡
❧❡❝t✉r❡ ❛❜♦✉t t❤❡ ✈❛r✐❛♥ts ❛♥❞ ❡①t❡♥s✐♦♥s ♦❢ t❤❡ r❡s♦✉r❝❡✲❝♦♥str❛✐♥❡❞ ♣r♦❥❡❝t s❝❤❡❞✉❧✐♥❣
♣r♦❜❧❡♠✱ ✇❡ r❡❢❡r t♦ t❤❡ s✉r✈❡② ♦♥ t❤✐s t♦♣✐❝ ♣✉❜❧✐s❤❡❞ ❜② ❖r❥✐ ❛♥❞ ❲❡✐ ✭✷✵✶✸✮✳ ❆♠♦♥❣ ❛❧❧
t❤❡ ❡①✐st✐♥❣ ❡①t❡♥❞❡❞ ✈❡rs✐♦♥s✱ ✇❡ ❞✐st✐♥❣✉✐s❤ t✇♦ t❤❛t ❛r❡ ♦❢ ❣r❡❛t ✐♥t❡r❡st ❢♦r t❤❡ ♠♦❞❡❧✐♥❣
♦❢ t❤❡ st✉❞✐❡❞ ♣r♦❜❧❡♠✿ t❤❡ Pr❡❡♠♣t✐✈❡ ❘❈P❙P ❛♥❞ t❤❡ ▼✉❧t✐✲❙❦✐❧❧ Pr♦❥❡❝t ❙❝❤❡❞✉❧✐♥❣
Pr♦❜❧❡♠ ✭▼❙P❙P✮✳ ❆ ✜rst ❛tt❡♠♣t t♦ ❝♦♠❜✐♥❡ t❤❡s❡ t✇♦ ♠♦❞❡❧s ❢♦r s❝❤❡❞✉❧✐♥❣ r❡s❡❛r❝❤
❛❝t✐✈✐t✐❡s ❝❛♥ ❜❡ ❢♦✉♥❞ ✐♥ P♦❧♦ ▼❡❥✐❛ ❡t ❛❧✳ ✭✷✵✶✼✮✱ ✇❤❡r❡ ❛ ♣✉r❡ ♣r❡❡♠♣t✐✈❡ ▼❙P❙P ✇✐t❤
♠✉❧t✐✲s❦✐❧❧❡❞ r❡s♦✉r❝❡s ✐s ♣r♦♣♦s❡❞✳ ❍♦✇❡✈❡r✱ ❛♥ ✐♥t❡♥s✐✈❡ ❛♥❛❧②s✐s ♦❢ t❤❡ ❧❛❜♦r❛t♦r② ✉♥❞❡r
st✉❞② ❤✐❣❤❧✐❣❤t❡❞ t❤❡ ♥❡❡❞ t♦ ❞❡✈❡❧♦♣ ❛ ♠♦r❡ ❡①t❡♥❞❡❞ ✈❡rs✐♦♥ ✐♥ ♦r❞❡r t♦ ❤❛✈❡ ❛ ❜❡tt❡r
r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ t❤❡ r❡❛❧✐t②✳ ❚❤❛t ✐s ✇❤② ✇❡ ♣r♦♣♦s❡ ✐♥ t❤✐s ♣❛♣❡r ❛ ♥❡✇ ❡①t❡♥❞❡❞ ✈❛r✐❛♥t
♦❢ t❤❡ ❘❈P❙P✿ ▼❙P❙P ✇✐t❤ ♣❛rt✐❛❧ ♣r❡❡♠♣t✐♦♥✳
❚❤❡ r❡♠❛✐♥❞❡r ♦❢ t❤❡ ♣❛♣❡r ✐s ❛s ❢♦❧❧♦✇s✳ ■♥ t❤❡ ♥❡①t s❡❝t✐♦♥✱ ✇❡ ❜r✐❡✢② ❞❡s❝r✐❜❡ t❤❡
♣r♦❜❧❡♠ ✉♥❞❡r ❝♦♥s✐❞❡r❛t✐♦♥✳ ■♥ ❙❡❝t✐♦♥ ✸✱ ✇❡ ♣r❡s❡♥t t❤❡ ♠✐①❡❞ ✐♥t❡❣❡r ❧✐♥❡❛r ♣r♦❣r❛♠♠✐♥❣
♠♦❞❡❧ r❡♣r❡s❡♥t✐♥❣ t❤❡ ♣❛rt✐❛❧❧② ♣r❡❡♠♣t✐✈❡ ▼❙P❙P ❛♥❞ s♦♠❡ ❝♦♠♣✉t❛t✐♦♥❛❧ ❡①♣❡r✐♠❡♥ts
❝❛rr✐❡❞ ♦✉t✳ ❋✐♥❛❧❧②✱ ✐♥ t❤❡ ❧❛st s❡❝t✐♦♥✱ ✇❡ ❝♦♥❝❧✉❞❡ ❛♥❞ ❞✐s❝✉ss ❢✉t✉r❡ r❡s❡❛r❝❤✳
✷ Pr♦❜❧❡♠ ❞❡s❝r✐♣t✐♦♥
❚❤❡ ❝❧❛ss✐❝❛❧ ✈❡rs✐♦♥ ♦❢ t❤❡ ❘❈P❙P ✐s s✉♣♣♦s❡❞ t♦ ❜❡ ♥♦♥✲♣r❡❡♠♣t✐✈❡✱ t❤❛t ♠❡❛♥s✱
♦♥❝❡ st❛rt❡❞ ❛♥ ❛❝t✐✈✐t② ♠✉st r✉♥ ❝♦♥t✐♥✉♦✉s❧② ✉♥t✐❧ ✐ts ❝♦♠♣❧❡t❡♥❡ss✳ ❍♦✇❡✈❡r✱ ✐♥ s♦♠❡
♣r❛❝t✐❝❛❧ ❛♣♣❧✐❝❛t✐♦♥s ❛s ✐♥ t❤❡ ❝❛s❡ ♦❢ s❝❤❡❞✉❧✐♥❣ r❡s❡❛r❝❤ ♦r ❡♥❣✐♥❡❡r✐♥❣ ❛❝t✐✈✐t✐❡s✱ ✐t
♠❛② ❜❡ ✐♥t❡r❡st✐♥❣ t♦ ❛❧❧♦✇ t❤❡ ♣r❡❡♠♣t✐♦♥✳ ❆❧❧♦✇✐♥❣ ♣r❡❡♠♣t✐♦♥ ♠❛② ❧❡❛❞ t♦ ❛ r❡❞✉❝❡❞
♠❛❦❡s♣❛♥ ♦❢ t❤❡ ♣r♦❥❡❝t✱ ❡s♣❡❝✐❛❧❧② ✇❤❡♥ r❡s♦✉r❝❡ ❛✈❛✐❧❛❜✐❧✐t② ✐s ✈❡r② ❧✐♠✐t❡❞✳ ❖♥ t❤❡ ♦t❤❡r
181

❤❛♥❞✱ ✐t ✐♥❝r❡❛s❡s t❤❡ ♥✉♠❜❡r ♦❢ ♣♦ss✐❜❧❡ s♦❧✉t✐♦♥s ❛♥❞ ❝♦♥s❡q✉❡♥t❧② t❤❡ ❝♦♠♣✉t❛t✐♦♥❛❧
❝♦♠♣❧❡①✐t② ♦❢ t❤❡ ♣r♦❜❧❡♠ ✭❍❡rr♦❡❧❡♥ ❡t ❛❧✳ ✶✾✾✽✮✳
❚r❛❞✐t✐♦♥❛❧❧② ✐♥ t❤❡ ♣r❡❡♠♣t✐✈❡ ❘❈P❙P✱ t❤❡ ♣r❡❡♠♣t✐♦♥ ✐s ❛❧❧♦✇❡❞ ❢♦r ❛❧❧ t❤❡ ❛❝t✐✈✐t✐❡s✳
❍♦✇❡✈❡r✱ ❞✉❡ t♦ s♦♠❡ s❛❢❡t② ❛♥❞ ♦♣❡r❛t✐♦♥❛❧ ❝♦♥str❛✐♥ts✱ ♣r♦♣❡r t♦ ♥✉❝❧❡❛r r❡❣✉❧❛t✐♦♥✱ ✇❡
♠✉st ❢♦r❜✐❞ t❤❡ ♣r❡❡♠♣t✐♦♥ ♦❢ ❛ s✉❜s❡t ♦❢ ❛❝t✐✈✐t✐❡s✳ ❆♥♦t❤❡r ❤②♣♦t❤❡s✐s ♦❢ t❤✐s ✈❛r✐❛♥t
✐s t❤❡ r❡❧❡❛s❡ ♦❢ ❛❧❧ r❡s♦✉r❝❡s ❞✉r✐♥❣ t❤❡ ♣r❡❡♠♣t✐♦♥ ♣❡r✐♦❞s✳ ❲❤❡♥ s❝❤❡❞✉❧✐♥❣ r❡s❡❛r❝❤
❛❝t✐✈✐t✐❡s✱ ✇❡ ♠❛② ❜❡ ✐♥t❡r❡st❡❞ ✐♥ ❛✈♦✐❞✐♥❣ t❤❡ r❡❧❡❛s❡ ♦❢ s♦♠❡ ❡q✉✐♣♠❡♥t ♦r r❡s♦✉r❝❡
❤❛✈✐♥❣ ❛♥ ✐♠♣♦rt❛♥t s❡t✉♣ t✐♠❡ ❢♦r s♦♠❡ ❛❝t✐✈✐t✐❡s✳ ❚❤❛t ✐s ✇❤② ✇❡ ♣r♦♣♦s❡ t♦ ✇♦r❦ ✇✐t❤
❛ ✈❛r✐❛♥t ❛❧❧♦✇✐♥❣ t❤❡ ♣❛rt✐❛❧ r❡❧❡❛s❡ ♦❢ r❡s♦✉r❝❡s ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ❝❤❛r❛❝t❡r✐st✐❝s ♦❢ t❤❡
❛❝t✐✈✐t✐❡s✳ ❲❡ ♠✉st ✐♥❞✐❝❛t❡ ❢♦r ❡❛❝❤ ❛❝t✐✈✐t② ✇❤❛t r❡s♦✉r❝❡ ❝❛♥ ❜❡ r❡❧❡❛s❡❞ ❞✉r✐♥❣ t❤❡
♣r❡❡♠♣t✐♦♥ ♣❡r✐♦❞s✳
❖t❤❡r ❛ss✉♠♣t✐♦♥ ♦❢ t❤❡ ❘❈P❙P ✐s t❤❛t ❡❛❝❤ r❡s♦✉r❝❡ ❤❛s s♣❡❝✐✜❝ ❢✉♥❝t✐♦♥s✱ ♦r ✐♥
♦t❤❡r ✇♦r❞s t❤❡ r❡s♦✉r❝❡s ❛r❡ s✉♣♣♦s❡❞ ♠♦♥♦✲s❦✐❧❧❡❞✳ ❚❤✐s ❤②♣♦t❤❡s✐s ❝❛♥ ❜❡❝♦♠❡ ❢❛❧s❡
✇❤❡♥ ✇❡ ❛r❡ ❛❧s♦ st✉❞②✐♥❣ t❤❡ ❛❧❧♦❝❛t✐♦♥ ♦❢ ❤✉♠❛♥ r❡s♦✉r❝❡s ✇♦r❦✐♥❣ ✐♥ t❤❡ ♣r♦❥❡❝t✳ ■♥
♦✉r st✉❞② ❝❛s❡✱ s♦♠❡ r❡s♦✉r❝❡s ❝♦✉❧❞ ♣❡r❢♦r♠ s❡✈❡r❛❧ ❢✉♥❝t✐♦♥s ❧❡❛❞✐♥❣ ✉s t♦ ❛ ♠✉❧t✐✲s❦✐❧❧
❘❈P❙P ✭▼❙P❙P✮✳ ■♥ t❤❡ ▼❙P❙P✱ ❛ r❡s♦✉r❝❡ ✐s t❤❡r❡❢♦r❡ ❝❤❛r❛❝t❡r✐③❡❞ ❜② t❤❡ s❡t ♦❢ s❦✐❧❧s
✐t ♣♦ss❡ss❡s❀ ❛♥❞ ❛ t❛s❦ ✐s ♥♦ ❧♦♥❣❡r ♦♥❧② ❞❡✜♥❡❞ ❜② t❤❡ q✉❛♥t✐t✐❡s r❡q✉✐r❡❞ ♦❢ ❡❛❝❤ r❡s♦✉r❝❡✱
❜✉t ❛❧s♦ ❜② t❤❡ ♥✉♠❜❡r ♦❢ r❡s♦✉r❝❡s ✇✐t❤ ❛ s♣❡❝✐✜❝ ❝♦♠♣❡t❡♥❝❡✳ ❚❤✐s ✈❛r✐❛♥t ❛❝q✉✐r❡s ❣r❡❛t
✐♠♣♦rt❛♥❝❡ ❢♦r s❝❤❡❞✉❧✐♥❣ ❛❝t✐✈✐t✐❡s ✐♥ ✈❡r② s♣❡❝✐✜❝ ✜❡❧❞s✱ s✉❝❤ ❛s ♣❤❛r♠❛❝❡✉t✐❝❛❧✱ ❝❤❡♠✐❝❛❧
❛♥❞ ♥✉❝❧❡❛r✱ ✇❤❡r❡ t❤❡ r❡❣✉❧❛t✐♦♥ r❡q✉✐r❡s t❤❡ ♣r❡s❡♥❝❡ ♦❢ ❛ ❣r♦✉♣ ♦❢ t❡❝❤♥✐❝✐❛♥s ❤❛✈✐♥❣ ❛
s❡t ♦❢ ✇❡❧❧✲❞❡✜♥❡❞ ❝♦♠♣❡t❡♥❝❡s ❢♦r t❤❡ ❡①❡❝✉t✐♦♥ ♦❢ ❛♥ ❛❝t✐✈✐t②✳
■♥ t❤❡ ▼❙P❙P✱ ❛s ❞❡✜♥❡❞ ❜② ▼♦♥t♦②❛ ❡t ❛❧✳ ✭✷✵✶✹✮✱ t❡❝❤♥✐❝✐❛♥s ❝❛♥ ♦♥❧② r❡s♣♦♥❞ t♦
♦♥❡ s❦✐❧❧ r❡q✉✐r❡♠❡♥t ♣❡r ❛❝t✐✈✐t②✳ ❍♦✇❡✈❡r✱ ✐♥ ♦✉r ♣r❛❝t✐❝❛❧ ❝❛s❡✱ t❡❝❤♥✐❝✐❛♥s ♠❛② r❡s♣♦♥❞
t♦ ♠♦r❡ t❤❛♥ ♦♥❡ s❦✐❧❧ r❡q✉✐r❡♠❡♥t ♣❡r ❛❝t✐✈✐t②✳ ❆❞❞✐t✐♦♥❛❧❧②✱ ❞✉❡ t♦ ♦♣❡r❛t✐♦♥❛❧ ❛♥❞ s❛❢❡t②
r❡❛s♦♥s✱ ✇❡ ♥❡❡❞ t♦ ❣✉❛r❛♥t❡❡ ❛ ♠✐♥✐♠❛❧ ♥✉♠❜❡r ♦❢ t❡❝❤♥✐❝✐❛♥s ♣r❡s❡♥t ❞✉r✐♥❣ t❤❡ ❡①❡❝✉t✐♦♥
♦❢ t❤❡ ❛❝t✐✈✐t②✳
❑❡❡♣✐♥❣ ✐♥ ♠✐♥❞ ❛❧❧ t❤❡ ❛❢♦r❡♠❡♥t✐♦♥❡❞ ❝❤❛r❛❝t❡r✐st✐❝s✱ ❛♥❞ ❧♦♦❦✐♥❣ ❢♦r t❤❡ ♠♦st r❡✲
❛❧✐st✐❝ ♠♦❞❡❧✱ ✇❡ ❞❡❝✐❞❡❞ t♦ ❞❡✈❡❧♦♣ ❛♥ ❡①t❡♥❞❡❞ ✈❛r✐❛♥t ♦❢ ❘❈P❙P ❝♦♠❜✐♥✐♥❣ t❤❡ ❝❤❛r✲
❛❝t❡r✐st✐❝s ♦❢ t❤❡ ▼❙P❙P ❛♥❞ t❤❡ ♣r❡❡♠♣t✐✈❡ ❘❈P❙P✳ ■♥ t❤❡ ♣r♦♣♦s❡❞ ✈❛r✐❛♥t✱ t❤❛t ✇❡
❝❛❧❧❡❞ ▼❙P❙P ✇✐t❤ ♣❛rt✐❛❧ ♣r❡❡♠♣t✐♦♥✱ t❤❡ ♦❜❥❡❝t✐✈❡ ✐s t♦ ✜♥❞ t❤❡ ❜❡st s❝❤❡❞✉❧❡ ❢♦r ❛
s❡t ♦❢ ❛❝t✐✈✐t✐❡s ♦♥ r❡♥❡✇❛❜❧❡ ♠✉❧t✐✲s❦✐❧❧❡❞ r❡s♦✉r❝❡s ✇✐t❤ ❧✐♠✐t❡❞ ❝❛♣❛❝✐t②✱ ❜❡✐♥❣ ❛❜❧❡ t♦
r❡s♣♦♥❞ t♦ ♠♦r❡ t❤❛♥ ♦♥❡ s❦✐❧❧ r❡q✉✐r❡♠❡♥t ♣❡r ❛❝t✐✈✐t②✳ ❆♥ ❛❝t✐✈✐t② ✐s ♥♦✇ ❞❡✜♥❡❞ ❜② ✐ts
❞✉r❛t✐♦♥✱ ♣r❡❝❡❞❡♥❝❡ r❡❧❛t✐♦♥s❤✐♣s ❛♥❞ ❝♦♥st❛♥t r❡q✉✐r❡♠❡♥ts ♦❢ ❜♦t❤ r❡s♦✉r❝❡s ❛♥❞ s❦✐❧❧s✳
Pr❡❡♠♣t✐♦♥ ✐s ♥♦✇ ❤❛♥❞❧❡❞ ✐♥ t❤r❡❡ ❧❡✈❡❧s ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ❛❝t✐✈✐t✐❡s ❝❤❛r❛❝t❡r✐st✐❝s✿ ✶✮
◆♦♥✲♣r❡❡♠♣t✐♦♥✱ ❢♦r ❛❝t✐✈✐t✐❡s ✇❤❡r❡ ♥♦♥❡ ♦❢ t❤❡ r❡s♦✉r❝❡s ❝❛♥ ❜❡ ♣r❡❡♠♣t❡❞❀ ✷✮ P❛rt✐❛❧
♣r❡❡♠♣t✐♦♥✱ ❢♦r ❛❝t✐✈✐t✐❡s ✇❤❡r❡ ❛ s✉❜s❡t ♦❢ r❡s♦✉r❝❡s ❝❛♥ ❜❡ ♣r❡❡♠♣t❡❞❀ ❛♥❞ ✸✮ ❋✉❧❧ ♣r❡✲
❡♠♣t✐♦♥✱ ❢♦r ❛❝t✐✈✐t✐❡s ✇❤❡r❡ ❛❧❧ r❡s♦✉r❝❡s ❝❛♥ ❜❡ ♣r❡❡♠♣t❡❞✳ ■♥ ♦✉r ♣r❛❝t✐❝❛❧ ❝❛s❡✱ ❛❝t✐✈✐t✐❡s
♠❛② ❜❡ s✉❜❥❡❝t t♦ ❛ r❡❧❡❛s❡ ❞❛t❡ ❛♥❞ t♦ ❛ ❞❡❛❞❧✐♥❡ ✭❛❝t✐✈✐t✐❡s ✐♥ t❤❡ s✉❜s❡t B✮ ♦r ❞✉❡ ❞❛t❡
✭t❤✐s ✐s ❞❡t❡r♠✐♥❡❞ ❜② t❤❡ ✐♠♣♦rt❛♥❝❡ ♦❢ t❤❡ ❛❝t✐✈✐t②✮✳ ❆❞❞✐t✐♦♥❛❧❧②✱ ❞✉❡ t♦ t❤❡ ❞✉r❛t✐♦♥s
♦❢ s♦♠❡ ❛❝t✐✈✐t✐❡s ✭❧❛r❣❡r t❤❛♥ t❡❝❤♥✐❝✐❛♥s✬ ✇♦r❦ s❤✐❢ts✮✱ ✇❡ ♥❡❡❞ t♦ r❡❧❛① t❤❡ ❝♦♥str❛✐♥t
st❛t✐♥❣ t❤❛t t❤❡ s❛♠❡ t❡❝❤♥✐❝✐❛♥ ❡①❡❝✉t❡ t❤❡ t♦t❛❧✐t② ♦❢ t❤❡ ❛❝t✐✈✐t②✳
❋♦r ❡st❛❜❧✐s❤✐♥❣ t❤❡ ❝♦♠♣❧❡①✐t② ♦❢ t❤❡ ▼❙P❙P ✇✐t❤ ♣❛rt✐❛❧ ♣r❡❡♠♣t✐♦♥✱ ✇❡ ✉s❡ ❛s ❛
st❛rt✐♥❣ ♣♦✐♥t t❤❡ ❝❧❛ss✐❝❛❧ ❘❈P❙P✳ ❋♦r ❡❛❝❤ ✐♥st❛♥❝❡ ♦❢ t❤❡ ❘❈P❙P ✇❡ ❝❛♥ ♠❛t❝❤ ❛♥
✐♥st❛♥❝❡ ♦❢ t❤❡ ▼❙P❙P ✇✐t❤ ♣❛rt✐❛❧ ♣r❡❡♠♣t✐♦♥✱ ✇❤❡r❡ ❛❧❧ r❡s♦✉r❝❡s ❛r❡ ♠♦♥♦✲s❦✐❧❧❡❞ ❛♥❞
♥♦♥❡ ♦❢ t❤❡ r❡s♦✉r❝❡s ❝❛♥ ❜❡ ♣r❡❡♠♣t❡❞✳ ❙♦✱ ✇❡ ❝❛♥ s❡❡ t❤❡ ❘❈P❙P ❛s ❛ ♣❛rt✐❝✉❧❛r ❝❛s❡ ♦❢
t❤❡ ▼❙P❙P ✇✐t❤ ♣❛rt✐❛❧ ♣r❡❡♠♣t✐♦♥✳ ❚❤❡ ❘❈P❙P ❤❛s ❜❡❡♥ ♣r♦✈❡❞ t♦ ❜❡ str♦♥❣❧② ◆P✲❤❛r❞
✭❇❧❛③❡✇✐❝③ ❡t ❛❧✳ ✶✾✽✸✮❀ ✇❡ ❝❛♥ t❤❡r❡❢♦r❡ ✐♥❢❡r t❤❛t t❤❡ ▼❙P❙P ✇✐t❤ ♣❛rt✐❛❧ ♣r❡❡♠♣t✐♦♥ ✐s
❛❧s♦ str♦♥❣❧② ◆P✲❤❛r❞✳ ❖♥❝❡ ❞❡✜♥❡❞ t❤❡ ❝❤❛r❛❝t❡r✐st✐❝s ❛♥❞ t❤❡ ❝♦♠♣❧❡①✐t② ♦❢ t❤❡ ♣r♦♣♦s❡❞
♣r♦❜❧❡♠✱ ✇❡ ♣r♦❝❡❡❞ t♦ ❢♦r♠❛❧✐③❡ t❤❡ ♣r♦❜❧❡♠ ✉s✐♥❣ ❛ ♠✐①❡❞ ✐♥t❡❣❡r ❧✐♥❡❛r ♣r♦❣r❛♠♠✐♥❣
♠♦❞❡❧ t❤❛t ✇❡ ❞✐s❝✉ss ✐♥ t❤❡ ♥❡①t s❡❝t✐♦♥✳
182

✸ ▼♦❞❡❧✐♥❣
❚❤❡ ❘❈P❙P ❝❛♥ ❜❡ ♠♦❞❡❧❡❞ ✉s✐♥❣ ❞✐✛❡r❡♥t ❛♣♣r♦❛❝❤❡s✿ ❝♦♥t✐♥✉♦✉s t✐♠❡✲❜❛s❡❞ ♠♦❞❡❧s
❜❛s❡❞ ♦♥ ✢♦✇s✱ ❞✐s❝r❡t❡✲t✐♠❡ ♠✐①❡❞ ✐♥t❡❣❡r ❧✐♥❡❛r ♣r♦❣r❛♠♠✐♥❣ ✭▼■▲P✮ ❢♦r♠✉❧❛t✐♦♥s✱ ♦r
❡✈❡♥t✲❜❛s❡❞ ▼■▲P ❢♦r♠✉❧❛t✐♦♥s✳ ❆♠♦♥❣ t❤❡ ❞✐s❝r❡t❡✲t✐♠❡ ❢♦r♠✉❧❛t✐♦♥s✱ ♠♦r❡ ♣r❡❝✐s❡❧② t❤❡
t✐♠❡✲✐♥❞❡①❡❞ ❢♦r♠✉❧❛t✐♦♥s✱ ✇❡ ✜♥❞ t❤❡ s♦✲❝❛❧❧❡❞ ♦♥✴♦✛ ❢♦r♠✉❧❛t✐♦♥✳ ❚❤✐s ❢♦r♠✉❧❛t✐♦♥ ✉s❡s
❜✐♥❛r② ✈❛r✐❛❜❧❡s Yi,t✱ ✇❤❡r❡ Yi,t = 1 ✐❢ ❛❝t✐✈✐t② i ✐s ✐♥ ♣r♦❣r❡ss ❛t t✐♠❡ t ❛♥❞ Yi,t = 0
♦t❤❡r✇✐s❡✳ ❚❤✐s ❢♦r♠✉❧❛t✐♦♥✱ ✇❤✐❝❤ s❡❡♠s t♦ ❜❡ t❤❡ ♠♦st s✉✐t❛❜❧❡ ❢♦r t❤❡ ♣r❡❡♠♣t✐✈❡ ❝❛s❡✱
❤❛s ❜❡❡♥ t❤❡ ❜❛s✐❝ ❢♦r♠✉❧❛t✐♦♥ ❢♦r t❤❡ ❝♦♥str✉❝t✐♦♥ ♦❢ t❡st❡❞ ♠♦❞❡❧s✳
■♥ ♦r❞❡r t♦ ❝❤♦♦s❡ ❛♥ ❡✛❡❝t✐✈❡ ♠♦❞❡❧✱ ✇❡ t❡st❡❞ t✇♦ ♠♦❞❡❧s✱ t❤❛t ❛r❡ s✐♠✐❧❛r ✐♥ ❡ss❡♥❝❡✱
❝♦♥str✉❝t❡❞ ✉s✐♥❣ t❤❡ ♦♥✴♦✛ ❢♦r♠✉❧❛t✐♦♥✳ ■♥ ❜♦t❤ ♠♦❞❡❧s✱ ♠♦st r❡str✐❝t✐♦♥s ❛r❡ ♠♦❞❡❧❡❞
✐♥ t❤❡ s❛♠❡ ✇❛②✳ ❚❤❡ ♠❛✐♥ ❞✐✛❡r❡♥❝❡ ❧✐❡s ✐♥ t❤❡ ✇❛② ✐♥ ✇❤✐❝❤ ✇❡ ❤❛♥❞❧❡ t❤❡ ♣r❡❡♠♣t✐♦♥
♣❡r✐♦❞s✳ ❋♦r t❡st✐♥❣ t❤❡s❡ ♠♦❞❡❧s✱ ✇❡ ❣❡♥❡r❛t❡❞ ❛ s❡t ♦❢ ✐♥st❛♥❝❡s ✐♥s♣✐r❡❞ ❜② r❡❛❧ ❞❛t❛
✉s✐♥❣ t❤❡ ♠❡t❤♦❞ ♣r♦♣♦s❡❞ ✐♥ P♦❧♦ ▼❡❥✐❛ ❡t ❛❧✳ ✭✷✵✶✼✮✳ ❆❢t❡r ❝♦♠♣✉t❛t✐♦♥❛❧ ❡①♣❡r✐♠❡♥ts✱
♦♥❡ ♦❢ t❤❡ ♠♦❞❡❧s s❤♦✇❡❞ s✐❣♥✐✜❝❛♥t❧② ❜❡tt❡r r❡s✉❧ts✱ ❛♥❞ ✐t ✐s ♣r❡s❡♥t❡❞ ❜❡❧♦✇✳
■♥ t❤❡ ♠♦❞❡❧ DOj,t ✐s t❤❡ ♦♣❡r❛t♦r✬s ❛✈❛✐❧❛❜✐❧✐t② ♦✈❡r t❤❡ t✐♠❡✳ Bri,k r❡♣r❡s❡♥ts t❤❡
r❡s♦✉r❝❡ r❡q✉✐r❡♠❡♥ts✳ DRk,t ✐♥❞✐❝❛t❡s t❤❡ r❡s♦✉r❝❡ ❝❛♣❛❝✐t✐❡s✳ P❛r❛♠❡t❡r PRi,k ✐♥❞✐❝❛t❡s
✇❤❡t❤❡r t❤❡ r❡s♦✉r❝❡ k ❝❛♥ ❜❡ ♣r❡❡♠♣t❡❞ ✭PRi,k❂✵✮ ♦r ♥♦t ✭PRi,k❂✶✮✳ ❙❦✐❧❧ r❡q✉✐r❡♠❡♥ts
❛r❡ ❣✐✈❡♥ ✐♥ ♣❛r❛♠❡t❡r Bci,c✳ COj,c ✐♥❞✐❝❛t❡s t❤❡ s❡t ♦❢ s❦✐❧❧s ♦❢ t❡❝❤♥✐❝✐❛♥s ✭COj,c =
1 ✐❢ t❡❝❤♥✐❝✐❛♥ j ❤❛s t❤❡ ❝♦♠♣❡t❡♥❝❡ c✱ ✵ ♦t❤❡r✇✐s❡✮✳ P❛r❛♠❡t❡r Pci ✐♥❞✐❝❛t❡s ✇❤❡t❤❡r
t❡❝❤♥✐❝✐❛♥s ❝❛♥ ❜❡ ♣r❡❡♠♣t❡❞ ✭Pci❂✵✮ ♦r ♥♦t ✭Pci❂✶✮✳ ❚❤❡ ♠✐♥✐♠❛❧ ♥✉♠❜❡r ♦❢ r❡q✉✐r❡❞
t❡❝❤♥✐❝✐❛♥s ✐s ❣✐✈❡♥ ✐♥ Nti✳ Di r❡♣r❡s❡♥ts t❤❡ ❞✉r❛t✐♦♥ ♦❢ ❛❝t✐✈✐t✐❡s✳ P❛r❛♠❡t❡rs dli ❛♥❞ ri
❛r❡ t❤❡ ❞❡❛❞❧✐♥❡s ❛♥❞ r❡❧❡❛s❡ ❞❛t❡s✳
✕ Yi,t ∈ {0, 1}✱ Yi,t = 1 ⇐⇒ ❛❝t✐✈✐t② i ✐s ✐♥ ♣r♦❣r❡ss ❛t t✐♠❡ t
✕ Oj,i,t ∈ {0, 1}✱ Oj,i,t = 1 ⇐⇒ t❡❝❤♥✐❝✐❛♥ j ✐s ❛❧❧♦❝❛t❡❞ t♦ ❛❝t✐✈✐t② i ❛t t✐♠❡ t
✕ Zi,t ∈ {0, 1}✱ Zi,t = 1 ⇐⇒ ❛❝t✐✈✐t② i st❛rts ❛t t✐♠❡ t ♦r ❜❡❢♦r❡
✕ Wi,t ∈ {0, 1}✱ Wi,t = 1 ⇐⇒ ❛❝t✐✈✐t② i ❡♥❞s ❛t t✐♠❡ t ♦r ❛❢t❡r
✕ Ppi,t ∈ {0, 1}✱ Ppi,t = 1 ⇐⇒ ❛❝t✐✈✐t② i ✐s ♣r❡❡♠♣t❡❞ ❛t t✐♠❡ t
✕ Tardi ∈ Z≥0 : ❚❛r❞✐♥❡ss ♦❢ ❛❝t✐✈✐t② i
min
P
i Tardi +
P
i
P
t t ∗ Yi,t ✭✶✮
s.t.
P
i Oj,i,t ≤ DOj,t ∀j, ∀t ✭✷✮
P
i((Yi,t + PRi,k ∗ Ppi,k) ∗ Bri,k) ≤ DRk,t ∀t, ∀k ✭✸✮
(Yi,t + Pci ∗ Ppi,t) ∗ Bci,c ≤
P
j(Oj,i,t ∗ COj,c) ∀i, ∀t, ∀c ✭✹✮
P
j Oj,i,t ≥ (Yi,t + Pci ∗ Ppi,t) ∗ Nti ∀t, ∀i ✭✺✮
P
t Yi,t ≥ Di ∀i ✭✻✮
Dl ∗ (1 − Yi,t) ≥
PT
t′=t Yl,t′ ∀(l, i) ∈ E, ∀t ✭✼✮
PT
t=dli+1 Yi,t ≤ 0 ∀i ∈ B ✭✽✮
Pri−1
t=1 Yi,t ≤ 0 ∀i ✭✾✮
Ppi,t ≥ Zi,t + Wi,t − Yi,t − 1 ∀i, ∀t ✭✶✵✮
Zi,t ≥ Yi,t′ ∀i, ∀t, ∀t′
≤ t ✭✶✶✮
Wi,t ≥ Yi,t′ ∀i, ∀t, ∀t′
≥ t ✭✶✷✮
Zi,t ≤
Pt
t′=1 Yi,t′ ∀i, ∀t ✭✶✸✮
Wi,t ≤
PT
t′=t Yi,t′ ∀i, ∀t ✭✶✹✮
Tardi ≥ t ∗ Yi,t − dli ∀i, ∀t ✭✶✺✮
183

❚❤❡ ♦❜❥❡❝t✐✈❡ ✐♥ ✭✶✮ r❡♣r❡s❡♥ts t❤❡ ♠✐♥✐♠✐③❛t✐♦♥ ♦❢ t❤❡ t❛r❞✐♥❡ss ❛♥❞ ❛❧s♦ ❡♥s✉r❡s t❤❡
s❝❤❡❞✉❧✐♥❣ ♦❢ ✉♥✐ts ♦❢ ❞✉r❛t✐♦♥ ♦❢ ❡❛❝❤ ❛❝t✐✈✐t② ❛s s♦♦♥ ❛s ♣♦ss✐❜❧❡✳ ❊q✉❛t✐♦♥s ✭✷✮ ❡♥s✉r❡ t❤❛t
♦♣❡r❛t♦r✬s ❝❛♣❛❝✐t✐❡s ❛r❡ s❛t✐s✜❡❞✳ ■♥ ❡q✉❛t✐♦♥s ✭✸✮✱ ✇❡ ❡♥s✉r❡ t❤❛t ❛❧❧ r❡s♦✉r❝❡ r❡q✉✐r❡♠❡♥ts
❛r❡ s❛t✐s✜❡❞ r❡s♣❡❝t✐♥❣ t❤❡ r❡s♦✉r❝❡ ❝❛♣❛❝✐t✐❡s✳ ❊q✉❛t✐♦♥s ✭✹✮ ❡♥s✉r❡ t❤❡ r❡s♣❡❝t ♦❢ s❦✐❧❧
r❡q✉✐r❡♠❡♥ts t❛❦✐♥❣ ✐♥t♦ ❛❝❝♦✉♥t t❤❡ s❡t ♦❢ s❦✐❧❧s ♦❢ t❡❝❤♥✐❝✐❛♥s✳ ❚❤❡ ❝♦♥str❛✐♥ts ❣✐✈❡♥ ✐♥ ✭✺✮
❛♥❞ ✭✻✮ ❡♥s✉r❡ t❤❡ r❡s♣❡❝t ♦❢ t❤❡ ♠✐♥✐♠❛❧ ♥✉♠❜❡r ♦❢ t❡❝❤♥✐❝✐❛♥s ❛♥❞ ❞✉r❛t✐♦♥ ♦❢ ❛❝t✐✈✐t✐❡s✳
Pr❡❝❡❞❡♥❝❡ ❝♦♥str❛✐♥ts ❛r❡ ❣✐✈❡♥ ✐♥ ✭✼✮✳ ■♥❡q✉❛❧✐t✐❡s ✭✽✮ ❛♥❞ ✭✾✮ ❛r❡ t❤❡ ❝♦♥str❛✐♥ts ❢♦r
❞❡❛❞❧✐♥❡s ❛♥❞ r❡❧❡❛s❡ ❞❛t❡s✳ ❊q✉❛t✐♦♥s ✭✶✵✮ ❞❡t❡r♠✐♥❡ ✇❤❡t❤❡r ❛♥ ❛❝t✐✈✐t② ✐s ♣r❡❡♠♣t❡❞
♦r ♥♦t✳ ■♥❡q✉❛❧✐t✐❡s ✭✶✶✮ t♦ ✭✶✹✮ ❛r❡ ❝♦♥str❛✐♥ts ❢♦r ❣❡tt✐♥❣ t❤❡ ✈❛❧✉❡s ♦❢ ✈❛r✐❛❜❧❡s Zi,t ❛♥❞
Wi,t✳ ❋✐♥❛❧❧②✱ ✐♥❡q✉❛❧✐t✐❡s ✭✶✺✮ ❝❛❧❝✉❧❛t❡ t❤❡ t❛r❞✐♥❡ss✳
❯s✐♥❣ ❈P▲❊❳✱ t❤✐s ♠♦❞❡❧ ❛❧❧♦✇s ✉s t♦ s♦❧✈❡ ♦♣t✐♠❛❧❧② ❛ s❡t ♦❢ s♠❛❧❧ ✐♥st❛♥❝❡s ✭✷✵ ❛❝t✐✈✲
✐t✐❡s ✇✐t❤ ❞✉r❛t✐♦♥ ❜❡t✇❡❡♥ ✶ ❛♥❞ ✶✵ ✉♥✐ts ♦❢ t✐♠❡ ❛♥❞ ❛ ♠❡❛♥ ♦❢ ✹ ♣r❡❝❡❞❡♥❝❡ r❡❧❛t✐♦♥s❤✐♣s✱
✶✸ s❦✐❧❧s✮ ✇✐t❤✐♥ ❛ ♠❡❛♥ t✐♠❡ ♦❢ ✼✳✷✸ s❡❝♦♥❞s✳ ❋♦r ❛ s❡t ♦❢ ❧❛r❣❡r ✐♥st❛♥❝❡s ✭✷✵ ❛❝t✐✈✐t✐❡s
✇✐t❤ ❞✉r❛t✐♦♥ ❜❡t✇❡❡♥ ✺ ❛♥❞ ✷✵ ✉♥✐ts ♦❢ t✐♠❡ ❛♥❞ ❛ ♠❡❛♥ ♦❢ ✻ ♣r❡❝❡❞❡♥❝❡ r❡❧❛t✐♦♥s❤✐♣s✱
✶✸ s❦✐❧❧s✮✱ ✇❡ ✇❡r❡ ♥♦t ❛❜❧❡ t♦ s♦❧✈❡ t❤❡♠ ♦♣t✐♠❛❧❧② ❛❢t❡r ✷ ❤♦✉rs ♦❢ ❝♦♠♣✉t✐♥❣ ❤❛✈✐♥❣
✜♥❛❧ ❣❛♣ ❜❡t✇❡❡♥ ✸✲✶✺✪✳ ❇② ❝♦♥❢❡r❡♥❝❡ t✐♠❡✱ ❤❡✉r✐st✐❝ ♠❡t❤♦❞s ❝❛♣❛❜❧❡ ♦❢ ♦❜t❛✐♥✐♥❣ ❣♦♦❞
❛♥s✇❡rs ✐♥ r❡❞✉❝❡❞ t✐♠❡s ❢♦r ❧❛r❣❡ ✐♥st❛♥❝❡s ✇✐❧❧ ❜❡ ♣r❡s❡♥t❡❞✳
✹ ❈♦♥❝❧✉s✐♦♥s
■♥ t❤✐s ♣❛♣❡r ✇❡ s❤♦✇ ❤♦✇ ♦♣❡r❛t✐♦♥s r❡s❡❛r❝❤ t❡❝❤♥✐q✉❡s ❝❛♥ ❜❡ ❛♣♣❧✐❡❞ t♦ s❝❤❡❞✉❧❡
r❡s❡❛r❝❤ ❛❝t✐✈✐t✐❡s ✇✐t❤✐♥ ❛ ♥✉❝❧❡❛r ❢❛❝✐❧✐t②✳ ❘❡❞✉❝✐♥❣ t❤❡ s❝❤❡❞✉❧✐♥❣ ❤♦r✐③♦♥ ❛❧❧♦✇s ✉s t♦
♠❛♥❛❣❡ t❤❡ ✐♥❤❡r❡♥t ✈❛r✐❛❜✐❧✐t② ♦❢ r❡s❡❛r❝❤ ❛❝t✐✈✐t✐❡s ❛♥❞ ❤❡♥❝❡ t♦ tr❡❛t t❤❡ s❝❤❡❞✉❧✐♥❣
♣r♦❜❧❡♠ ❛s ❛ tr❛❞✐t✐♦♥❛❧ ♦♥❡✳ ❚❤❡ ❛♣♣❧✐❝❛t✐♦♥ ♦❢ ♦♣❡r❛t✐♦♥s r❡s❡❛r❝❤ t❡❝❤♥✐q✉❡s t♦ t❤❡
s❝❤❡❞✉❧✐♥❣ ♣r♦❝❡ss ♦❢ r❡s❡❛r❝❤ ❛❝t✐✈✐t✐❡s ❝❛♥ r❡❞✉❝❡ t❤❡ t✐♠❡ s♣❡♥t ❜② r❡s❡❛r❝❤❡rs ✐♥ t❤❡
♣❧❛♥♥✐♥❣ ♦❢ ❛❝t✐✈✐t✐❡s✱ ❣✐✈✐♥❣ t❤❡♠ ♠♦r❡ t✐♠❡ t♦ ❞❡✈♦t❡ t♦ r❡s❡❛r❝❤✳ ❆❞❞✐t✐♦♥❛❧❧②✱ ✉s✐♥❣
t❤❡s❡ t❡❝❤♥✐q✉❡s ✐♥ t❤❡ ♥✉❝❧❡❛r ✜❡❧❞ ✐♥❝r❡❛s❡ t❤❡ s❛❢❡t② ♦♥ t❤❡ ❢❛❝✐❧✐t② ❜② ❡♥s✉r✐♥❣ t❤❡
r❡s♣❡❝t ♦❢ ❛❧❧ t❡❝❤♥✐❝❛❧ ❝♦♥str❛✐♥ts✳
❚❤❡ ❘❈P❙P ❤❛s ❜❡❡♥ s❤♦✇♥ t♦ ❜❡ ❛ ✈❡r② ♣♦✇❡r❢✉❧ ♠♦❞❡❧✱ ❜❡✐♥❣ ❛❜❧❡ t♦ r❡♣r❡s❡♥t ❛
❤✉❣❡ ❛♠♦✉♥t ♦❢ r❡❛❧✲❧✐❢❡ ♣r♦❜❧❡♠s✳ ❍♦✇❡✈❡r✱ ❢♦r s♦♠❡ ❝♦♠♣❧❡① s②st❡♠s✱ t❤❡ ❝❧❛ss✐❝❛❧ ❘❈P❙P
♠❛② ♥♦t t❛❦❡ ✐♥t♦ ❝♦♥s✐❞❡r❛t✐♦♥ s♦♠❡ ✈❡r② ✐♠♣♦rt❛♥t ❛s♣❡❝ts✳ ❲❡ t❤❡♥ ♣r♦♣♦s❡❞ ✐♥ t❤✐s
♣❛♣❡r t❤❡ ♠✉❧t✐✲s❦✐❧❧ ♣r♦❥❡❝t s❝❤❡❞✉❧✐♥❣ ♣r♦❜❧❡♠ ✇✐t❤ ♣❛rt✐❛❧ ♣r❡❡♠♣t✐♦♥ ❛♥❞ ❛♥ ▼■▲P
❢♦r♠✉❧❛t✐♦♥ ❢♦r ❢♦r♠❛❧✐③✐♥❣ t❤❡ ♣r♦❜❧❡♠✳
❆s ❢✉t✉r❡ ✇♦r❦✱ ✇❡ ♠✉st st✉❞② ✇❛②s t♦ ✐♠♣r♦✈❡ t❤❡ ♣r♦♣♦s❡❞ ♠♦❞❡❧ ✐♥ t❡r♠s ♦❢ t❤❡
q✉❛❧✐t② ♦❢ t❤❡ ❧✐♥❡❛r r❡❧❛①❛t✐♦♥ ❛♥❞ t✐♠❡ s♦❧✈✐♥❣✳ ❲❡ ❛❧s♦ ♥❡❡❞ t♦ ❞❡✈❡❧♦♣ ♥❡✇ ❤❡✉r✐st✐❝s
❛❧❧♦✇✐♥❣ ✉s t♦ ❤❛✈❡ ❣♦♦❞ s♦❧✉t✐♦♥s ✐♥ r❡❛s♦♥❛❜❧❡ t✐♠❡s✳ ■♥ ♦r❞❡r t♦ ❞❡✈❡❧♦♣ ❛❧❣♦r✐t❤♠s ❢♦r
❡①❛❝t s♦❧✈✐♥❣✱ ❛♣♣r♦❛❝❤❡s ❢♦r ❝❛❧❝✉❧❛t✐♥❣ ❣♦♦❞ ❧♦✇❡r ❜♦✉♥❞s ✇✐❧❧ ❜❡ st✉❞✐❡❞✳
❘❡❢❡r❡♥❝❡s
❇❧❛③❡✇✐❝③ ❏✳✱ ▲❡♥str❛ ❏✳❑✳ ❛♥❞ ❘✐♥♥♦♦② ❑❛♥ ❆✳❍✳●✳✱ ✶✾✽✸✱ ✏❙❝❤❡❞✉❧✐♥❣ ♣r♦❥❡❝ts s✉❜❥❡❝t t♦ r❡s♦✉r❝❡
❝♦♥str❛✐♥ts✿ ❝❧❛ss✐✜❝❛t✐♦♥ ❛♥❞ ❝♦♠♣❧❡①✐t②✑✱ ❉✐s❝r❡t❡ ❆♣♣❧✐❡❞ ▼❛t❤❡♠❛t✐❝s✱ ❱♦❧✳ ✺✱ ♣♣✳ ✶✶✲✷✹✳
❍❡rr♦❡❧❡♥ ❲✳✱ ❉❡ ❘❡②❝❦ ❇✳ ❛♥❞ ❉❡♠❡✉❧❡♠❡❡st❡r ❊✳✱ ✶✾✾✽✱ ✏❘❡s♦✉r❝❡✲❝♦♥str❛✐♥❡❞ ♣r♦❥❡❝t s❝❤❡❞✉❧✲
✐♥❣✿ ❆ s✉r✈❡② ♦❢ r❡❝❡♥t ❞❡✈❡❧♦♣♠❡♥ts✑✱ ❈♦♠♣✉t❡rs ✫ ❖♣❡r❛t✐♦♥s ❘❡s❡❛r❝❤✱ ❱♦❧✳ ✷✺✱ ♣♣✳ ✷✼✾✲✸✵✷✳
▼♦♥t♦②❛ ❈✳✱ ❇❡❧❧❡♥❣✉❡③✲▼♦r✐♥❡❛✉ ❖✳✱ P✐♥s♦♥ ❊✳ ❛♥❞ ❘✐✈r❡❛✉ ❉✳✱ ✷✵✶✺✱ ✏■♥t❡❣r❛t❡❞ ❝♦❧✉♠♥ ❣❡♥✲
❡r❛t✐♦♥ ❛♥❞ ❧❛❣r❛♥❣✐❛♥ r❡❧❛①❛t✐♦♥ ❛♣♣r♦❛❝❤ ❢♦r t❤❡ ♠✉❧t✐✲s❦✐❧❧ ♣r♦❥❡❝t s❝❤❡❞✉❧✐♥❣ ♣r♦❜❧❡♠✑✱
❍❛♥❞❜♦♦❦ ♦♥ Pr♦❥❡❝t ▼❛♥❛❣❡♠❡♥t ❛♥❞ ❙❝❤❡❞✉❧✐♥❣✱ ❙♣r✐♥❣❡r ■♥t❡r♥❛t✐♦♥❛❧✱ ♣♣✳ ✺✻✺✲✺✽✻✳
❖r❥✐ ■✳▼✳❏✳ ❛♥❞ ❲❡✐ ❙✳✱ ✷✵✶✸✱ ✏Pr♦❥❡❝t s❝❤❡❞✉❧✐♥❣ ✉♥❞❡r r❡s♦✉r❝❡ ❝♦♥str❛✐♥ts✿ ❆ r❡❝❡♥t ❙✉r✈❡②✑✱
■♥t❡r♥❛t✐♦♥❛❧ ❏♦✉r♥❛❧ ♦❢ ❊♥❣✐♥❡❡r✐♥❣ ❘❡s❡❛r❝❤ ❛♥❞ ❚❡❝❤♥♦❧♦❣②✱ ❱♦❧✳ ✵✷✱ ♣♣✳ ✶✲✷✵✳
P♦❧♦ ▼❡❥✐❛ ❖✳✱ ❆♥s❡❧♠❡t ▼✳✲❈✳✱ ❆rt✐❣✉❡s ❈✳✱ ▲♦♣❡③ P✳✱ ✷✵✶✼✱ ✏❆ ♥❡✇ ❘❈P❙P ✈❛r✐❛♥t t♦ s❝❤❡❞✲
✉❧❡ r❡s❡❛r❝❤ ❛❝t✐✈✐t✐❡s ✐♥ ❛ ♥✉❝❧❡❛r ❧❛❜♦r❛t♦r②✑✱ ■♥t❡r♥❛t✐♦♥❛❧ ❈♦♥❢❡r❡♥❝❡ ♦♥ ❈♦♠♣✉t❡rs ✫
■♥❞✉str✐❛❧ ❊♥❣✐♥❡❡r✐♥❣ ✭❈■❊✲✹✼✮✱ ❖❝t♦❜❡r ✶✶✲✶✸✱ ▲✐s❜♦♥✱ P♦rt✉❣❛❧✳
184

Paper: Scheduling Vehicles with spatial conflicts
Oddvar Kloster1
, Carlo Mannino1
, Atle Riise1
and Patrick Schittekat1
SINTEF Digital, Norway
Carlo Mannino, carlo.mannino@sintef.no
Keywords: Job shop scheduling, Conflict resolution, Linear Programming.
1 Introduction
In several important real world applications we find the problem of scheduling the
movement of objects on a network under spatial constraints on their relative position.
Vehicles moving on a transportation network need to fulfil spatial constraints that prevents
them from colliding, or getting too close to each other. Typically, the movement of a vehicle
in a network is represented as a sequence of atomic movements (the route), each requiring
a certain time and corresponding to the occupation of a specific network resource. For
instance, in railway networks a resource correspond to a track segment, for aircraft a
resource can be either an airborne sector or an airport segment, for boats may be channels
regions etc. In this framework, the standard way of modelling spatial conflicts is to sequence
vehicles on shared resources (by satisfying suitable disjunctive constraints). Examples of
this approach in different contexts are in (Mascis and Pacciarelli 2002) for trains, (Boccia
et. al. 2018) for airplanes, (Günther et. al. 2010) for ships, etc. However, in many cases this
approach can be insufficient, both because the vehicles and the network resources may have
complicated spatial shapes - giving rise to conflicts in non-shared resources, and because
two vehicle actually can be on the same network resource if they are not too close to each
other.
a
b
c
d
Fig. 1: Vehicles moving on a network with potential conflicts.
This work is motivated by an application in air traffic management, namely that of
finding a conflict free trajectory solution for taxiing aircraft at an airport. Simply put, for
each allocation of taxi routes to aircraft, the task is to determine a temporal movement of
each aircraft along it route so that no aircraft collides.
In this work we present a mathematical construction, so called Conflict Diagrams,
and demonstrate how this concept is a powerful mechanism for presenting spatial conflicts
185

between two objects moving on a spatial graph. We discuss the conflict diagram’s properties
and show how they can be constructed and extended following simple rules. We then discuss
how the conflict diagram relates to timing variables associated with each vehicle,and how
conflict diagrams can be used to construct feasible schedules.
2 Conflict diagrams
We are considering movements of vehicles on a spatial graph, where each arc corresponds
to some curve in space (IRn
+ with n = 1, 2, 3). For instance, arcs may represent road
segments, rail tracks, airborne sectors, etc. The route of a vehicle is an ordered sequence
of arcs, with the tail of an arc starting at the head of the previous arc. Consider a pair
of vehicles vx and vy with given routes Rx and Ry of length Lx and Ly, respectively. We
can describe the movement of vehicle vx and vehicle vy along their routes as real function
x = x(t) and y = y(t), respectively, with x(t) : IR+ → [0, Lx], and y(t) : IR+ → [0, Ly],
denoting the position in the route as a distance from the origin at time t. Also, we assume
that both x(t) and y(t) are piece-wise linear functions.
t
x
t
y
x
y
x(t)
y(t)
f(x,y)
Lx
Ly Ly
Lx
Fig. 2: Two trajectories and concurrent trajectory
If we now eliminate parameter t, we obtain a function f(x, y) = 0 (the concurrent tra-
jectory) which describes the concurrent positions of the two vehicles along their respective
routes. Namely, f(x̄, ȳ) = 0 if and only if there is a time t̄ ≥ 0, such that x̄ = x(t̄) and
ȳ = y(t̄). Observe that the curve is defined only in the box BL = {(x, y) ∈ IR2
: 0 ≤ x ≤
Lx, 0, ≤ y ≤ Ly} and that both point (0, 0) and point (Lx, Ly) belong to the curve (and
we call them first and last point).
Now, let (x̄, ȳ) ∈ BL represents a point where the two vehicles are too close to each
other. That is, there is some spatial constraint that says that vehicle vx and vehicle vy
cannot be at these respective positions along their routes at the same time. Such point
(x̄, ȳ) cannot belong to any (feasible) concurrent trajectory f(x, y). We call one such point
a conflict point. We denote by C the set (of the unit box) of conflicting points. To simplify
the discussion, from now on we assume C to be the region delimited by a polygon (see
Figure 3). So, a concurrent trajectory f(x, y) = 0 is infeasible (or ’conflicted’) if it intersects
the conflict region C. In Figure 3), f1 is infeasible, whereas f2 is feasible.
The conflict region C must be constructed from a geometrical analysis of the move-
ment of the two vehicles along their respective routes, resulting in the conflict regions C
(illustrated as the dark grey region in Fig. 3).
By exploiting specific knowledge on how vehicles actually move, the conflict region C can
be extended to the infeasible region C̄, namely a set of points which cannot be intersected
by any concurrent trajectory. For instance, we may assume that airplanes can only move
forward in their trajectories. Points in BL which would necessary lead the vehicles to a
186

point in C have to be prevented, even if strictly speaking they are not conflict points.
Similarly, points in BL C which cannot be reached without crossing C can be neglected.
For forward trajectories, We have the following
Lemma 2.1 Suppose vehicles can only move forward in their route, and let (x∗
, y∗
) ∈ BL.
If there exist non-negative quantities δx ≥ 0 and δy ≥ 0 such that both (x∗
+ δx, y∗
) ∈ C
and (x∗
, y∗
+ δy) ∈ C, then (x∗
, y∗
) is infeasible.
Similarly, if there exist non-positive quantities δx ≤ 0 and δy ≤ 0 such that both (x∗
+
δx, y∗
) ∈ C and (x∗
, y∗
+ δy) ∈ C, then (x∗
, y∗
) is infeasible.
The region C̄ ⊆ C which contains C and all the additional infeasible points associated
with C is called the infeasible region and one can show an effective polynomial algorithm
which builds C̄ from C. If C is the region delimited by a polygon, so is C̄.
Fig. 3: A conflict diagram showing the conflict region and its infeasible region.
Observe that the infeasible region C̄ has always a first vertex (the leftest) and a last
vertex (the one most to the right). All other vertices can be classified into lower vertices
(having now infeasible points below) and left vertices (having no infeasible point to the
left).
3 Feasible concurrent trajectories
Lemma 3.1 We can partition the family of feasible concurrent trajectories into two classes
1. vy − wins: Any point on the trajectory lies above any infeasible point with same x-
coordinate.
2. vx − wins: Any point on the trajectory lies below any infeasible point with same x-
coordinate.
In Figure 3, f2 is an x − wins trajectory. We now focus on vx − win trajectories.
A symmetric result applies to the other case. To simplify the notation, we assume that
f(x, y) = 0 in BL is the set of points satisfying y = g(x), for x ∈ Lx. One can show the
following important result
187

Lemma 3.2 vx wins if, for every vertex (x̄, ȳ) of C̄, the point (x̄, g(x̄)) lies in the area
below C and, for any two adjacent lower vertices (x̄1, ȳ1), (x̄2, ȳ2), f(x, y) is linear between
points (x̄1, g(x̄1)) and(x̄1, g(x̄2)).
Let x̄ = x(t̄x), i.e. t̄x = tx(x̄) is the time when x reaches x̄ on its route. Similarly, let
ȳ = y(t̄y), with t̄y = ty(ȳ). Then the above Lemma is equivalent to the following
Lemma 3.3 vx wins if, for every vertex (x̄, ȳ) of C, we have tx(x̄) ≤ ty(ȳ) and f(x, y) is
linear between points associated with successive low vertices as in Lemma 3.2.
4 Scheduling without conflicts.
Consider a vehicle vx, with piecewise linear trajectory x = x(t), and let X = (x1, . . . , xk =
Lx) be the ordered set of breakpoints. A schedule for vx is a vector tx
∈ IRX
+ , where tx
i
specifies when vx is at point xi. The schedule must satisfy time precedence constraints
associated with the breakpoints, i.e. txi+1 −txi
≥ λi,i+1, where λi,i+1 is the minimum time
necessary to vx to run from xi to xi+1. Precedence constraints may involve also variables
associated with different vehicles, for instance tyk
− txj
≥ λxj yk
. Now, consider a second
vehicle vy, its trajectory y = y(t) and list of breakpoints Y = (y1, . . . , yq = Ly). Suppose
we are given for the pair of vehicles vx, vy a conflict diagram C and its infeasible region
C̄. Also, assume vx wins. We assume that the break points X contain also the set of x
coordinates of the lower vertices of C̄, plus the x coordinates of the first and last vertex in
C̄, namely xf and xl. Denote by X̄ the ordered subsets of X between xf and xl (included).
Let Ȳ = {y ∈ [0, Ly] : y = g(x̄), x̄ ∈ X̄}. We now assume that Y ⊇ Ȳ .
Lemma 4.1 Let XY = {(x̄, ȳ) ∈ X̄ × Ȳ : ȳ = g(x̄)}. If tx̄ ≤ tȳ for all (x̄, ȳ) ∈ ¯
XY , then
f(x, y) is feasible.
In this extended abstract we are not focussing on the actual decisions of who wins, which
requires the definition of a suitable disjunctive program (Mascis and Pacciarelli 2002).
We are currently implementing a system to schedule and route airplanes in an airport.
The solution algorithm is based on the solution of large disjunctive programs, and makes
use of conflict diagrams to represent and identify conflicts, and to associate suitable timing
variables with trajectories. Indeed, the standard shared-resource conflict model would not
suffice to represent the different conflicting situation that may occur. The system will be
tested in April in an official test campaign (sponsored by the EU joint undertaking SESAR
2020). The test case will be Budapest airport (a medium size airport). The test campaign
will last for two weeks, involving several airport ground traffic controllers, and will be
carried out with the support of EUROCONTROL’s simulation platform.
References
Avella P., M. Boccia, C. Mannino, I. Vasilev, 2018, “Time-indexed formulations for the Runway
Scheduling Problem, Transportation Science, to appear.
Günther E., M.E. Lübbecke, R.H. Möhring, 2003, “Ship Traffic Optimization for the Kiel Canal,
Seventh Triennial Symposium on Transportation Analysis (TRISTAN 2010).
Mascis A., D. Pacciarelli, 2002, “Job shop scheduling with blocking and no-wait constraints,
European Journal on Operational Research, Vol. 143 (3), pp. 498-517.
188

On some approach to solve a scheduling problem with
a continuous doubly-constrained resource
Różycki R, Waligóra G
Institute of Computing Science, Poznan University of Technology, Poznan, Poland
{rafal.rozycki, grzegorz.waligora}@cs.put.poznan.pl
Keywords: continuous resource, doubly-constrained resource, parallel machines, discrete-
continuous scheduling.
1 Introduction
The nature of scheduling problems where execution time of a job is not set a priori
and depends on the amount of allocated doubly-constrained resource has its characteristic
specificity. Although the length of the optimal schedule depends in this case both on the
available temporary amount of the doubly-constrained resource and its total amount, in
practice for a specific instance of the problem, both of these restrictions are rarely ac-
tive. This fact can be used in the methodology to solve such problems. In this work, we
will demonstrate this approach on the example of a doubly-constrained resource with a
continuous nature, exemplified by power/energy. We will use the well-known model of a
job (Wȩglarz, 1981), in which its speed of execution depends on the temporary amount of
power allocated to it. We consider the problem of scheduling independent preemptable jobs
with the criterion of minimizing the makespan. The general methodology for solving such
problems is known and has been presented in many papers (Józefowska and Wȩglarz, 1998,
Różycki and Wȩglarz, 2014, Różycki and Wȩglarz, 2015). It involves solving a non-linear
mathematical programming problem. Unfortunately, the practice shows that solving such
problems with known numerical methods is extremely difficult. Below, we present an ap-
proach that in some situations allows to find the optimal solution with less computational
effort.
Let there be given a set of n independent, preemptable jobs and a set of m parallel
identical machines. A job requires a certain amount of continuous doubly-constrained re-
source (power/energy) and a machine to be performed. The power usage of all jobs must
not exceed the limit P at the moment. The consumption of energy by all jobs is limited by
the amount E. The temporary speed of job i depends on the current allocation of power
pi(t) and is described by a continuous increasing function (processing speed function), si,
si(0) = 0. Before assigning the power to a jobs, its execution time is unknown. Instead
of that, job i is characterized by size wi. The problem is to find an assignment of jobs to
machines, and simultaneously an allocation of power to jobs which lead to the schedule of
minimal length. In the further part of the paper we will limit our considerations to concave
processing speed functions, for which the considered problem is non-trivial.
3 General methodology
The general methodology of solving the problem is based on the theorem (known from
e.g. Wȩglarz, 1981), which assumes that the execution of jobs is only limited by power and
energy (there is no limit due to the available number of machines). This theorem defines
189

an allocation of power to jobs, which leads to a schedule of the shortest length. It can be
shown that for the considered concave processing speed functions, it is desirable to perform
the jobs (if possible) in parallel. Optimal constant power allocation pi, i = 1, 2, . . . , n, for
jobs executed in parallel, may be calculated basing on the optimal length of their schedule
T∗
, found as a solution of one of the two nonlinear equations:
T
n
∑
i=1
s−1
i (wi/T) = E (1)
n
∑
i=1
s−1
i (wi/T) = P. (2)
The first equation allows to calculate T∗
from the constraint on energy, the second
from the power limit. Of course, in most cases only one of these restrictions is active.
Unfortunately, it is often difficult to evaluate a priori which one. Therefore, it is justified
to utilize a rule where the easier equation is solved first and then it is checked whether
the calculated power allocation does not violate the second limitation. If the second limit
is violated, it means that for the given instance it is active and the optimal length of the
schedule should be calculated from the second equation. In many practical cases, e.g. for
processing speed functions of form:
si(pi) = p
1/αi
i , αi ∈ {2, 3, 4}, i = 1, 2, . . . , n. (3)
Equations (1) and (2) can be solved analytically since they are algebraic equations of an
order not greater than 4. However, equation (1) is of an order less by 1, and thus should
be solved first.
The above approach can be used in the general situation, where, due to the limited
number of machines, all jobs must not be performed in parallel. A potentially optimal
schedule (see Figure 1) of preemptable jobs can be represented by the sequence of r, r =
(n
m
)
, m - element combinations. A single combination Zk, k = 1, 2, . . . , m, represents the
m jobs performed in parallel in a given part of the schedule. Let us denote by wik the part
of size wi of job i performed in k-th part of a schedule (represented by Zk) and by Ek the
portion of energy allocated to Zk. Set Ki contains the indices of combinations where job
i belongs to. In order to find the optimal schedule, the following nonlinear mathematical
problem has to be solved in the general case:
NLP1: minimize T =
r
∑
k=1
T∗
k ({wik}i∈Zk
, Ek, P) (4)
subject to
r
∑
k=1
Ek ≤ E (5)
∑
k∈Ki
wik = wi, i = 1, 2, . . . , n (6)
wik ≥ 0, i = 1, 2, . . . , n, k ∈ Ki (7)
Ek ≥ 0, k = 1, 2, . . . , r (8)
190

where T∗
k (optimal length of k-th part of schedule) are calculated as functions of Ek, P,
and {wik} for i ∈ Zk from (1) or (2). Ek and wik are variables in NLP1, i.e. an optimal
distribution of energy as well as a optimal distribution of job sizes among combinations
have to be found. NLP1 is non-convex and it is very diﬃcult to solve by standard nonlinear
solvers.
Fig. 1. Exemplary optimal schedule with active energy and power constraint.
3.1 Power constraint only
A schedule for the case of the problem, where limit on energy is inactive is illustrated
Figure 2.
Fig. 2. Exemplary optimal schedule of the problem with power constraint only.
The length of k-th combination, T∗
k , depends on the assigned job sizes and the known
amount of P power. In this case, solving the NLP1 problem can be replaced by solving a
simpler problem:
NLP2: minimize T =
r
∑
k=1
T∗
k ({wik}i∈Zk
, P) (9)
subject to (6)–(7)
191

where T∗
k , k = 1, 2, . . . , r,
is the unique positive root of the equation:
∑
i∈Zk
s−1
i (wik/Tk) = P. (10)
As you can see in the problem, it is not needed to distribute energy between combi-
nations. NLP2 is convex, which greatly simpliﬁes the search for an optimal solution by
numerical methods.
3.2 Energy constraint only
A schedule for the case of inactive power constraint is shown in Figure 3.
Fig. 3. Exemplary optimal schedule of the problem with energy constraint only.
In this case NLP3 can be solved instead of NLP1:
NLP3: minimize T =
r
∑
k=1
T∗
k ({wik}i∈Zk
, Ek) (11)
subject to
r
∑
k=1
Ek = E (12)
(6)–(8)
where T∗
k , k = 1, 2, . . . , r,
is the unique positive root of the equation:
Tk ·
∑
i∈Zk
s−1
i (wik/Tk) = Ek. (13)
Similarly to NLP1, the above problem is non-convex, as it also seeks for optimal distri-
bution of energy among combinations. However, the calculation of T∗
k is easier than in the
general case.
A two-layer specialized numerical method can be proposed here. In the lower layer,
only the optimal division of job sizes among combinations is found. In the higher layer for
a given size division, a dynamic programming algorithm is started, in which energy E is
distributed optimally among combinations in order to obtain the minimal makespan.
192

4 Summary
On the basis of the above considerations, the following methodology of solving the
problem is justiﬁed. Start by solving the convex problem NLP2. Next, check whether the
obtained solution violates the restriction on the available amount of energy E. If this
limitation has been violated, NLP3 should be solved. Then check whether the solution of
NLP3 violates the restriction on P in any part of the schedule. Only when such a rare
situation occurs, it is necessary to solve general NLP1. However, to solve NLP1 one can
use the appropriately modiﬁed implementation of the two-layer method indicated in point
3.2.
References
Józefowska J., J. Wȩglarz, 1998, “On a methodology for discrete-continuous scheduling”, European
Journal of Operational Research, Vol. 107(2), pp. 338–353.
Różycki R., J. Wȩglarz, 2014, “Power-aware scheduling of preemptable jobs on identical parallel
processors to minimize makespan”, Annals of Operations Research, Vol. 213(1), pp. 235–252.
Różycki R., J. Wȩglarz, 2015, “Solving a power-aware scheduling problem by grouping jobs with
the same processing characteristic”, Discrete Applied Mathematics, Vol. 182, pp. 150–161.
Wȩglarz J., 1981, “Project scheduling with continuously-divisible doubly constrained resources”,
Management Science, Vol. 27(9), pp. 1040–1053.
193

Simple metaheuristics for Flowshop Scheduling: All
you need is local search
Rubén Ruiz
Universitat Politècnica de València
rruiz@eio.upv.es
1 Abstract
Many scheduling problems are simply too hard to be solved exactly, especially for in-
stances of medium or large size. As a result, the literature on heuristics and metaheuristics
for scheduling is extensive. More often than not, metaheuristics are capable of generating
solutions close optimality or to tight lower bounds for instances of realistic size in a matter
of minutes. Metaheuristics have been refined over the years and there is literally hundreds of
papers published every year with applications to most domains in many different journals.
Most regrettably, some of these methods are complex in the sense that they have many
parameters that affect performance and hence need careful calibration. Furthermore, many
times published results are hard to reproduce due to specific speed-ups being used or com-
plicated software constructs. These complex methods are difficult to transfer to industries
in the case of scheduling problems. Another important concern is the recently recognized
“tsunami” of novel metaheuristics that mimic the most bizarre natural or human processes,
as for example intelligent water drops, harmony search, firefly algorithms and the like. See
K. Sörensen “Metaheuristics - The Metaphor exposed” (2015), ITOR 22(1):3-18. In this
presentation, we review many different flowshop related problems. From the basic flow-
shop problem with makespan minimization to other objectives like flowtime minimization,
tardiness, flowshops with sequence-dependent setup times, no-idle flowshops or other vari-
ants and extensions, all the way up to complex hybrid flexible flowline problems. We will
show how simple Iterated Greedy (IG) algorithms often outperform much more complex
approaches. IG methods are inherently simple with very few parameters. They are easy to
code and results are easy to reproduce. We will show that for all tested problems so far
they show state-of-the-art performance despite their simplicity. As a result, we will defend
the choice of simpler, yet good performing approaches over complicated metaphor-based
algorithms.
194

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❙❝❤❡❞✉❧✐♥❣ Pr♦❜❧❡♠s ✇✐t❤ ❙❡q✉❡♥❝❡ ❉❡♣❡♥❞❡♥t ❙❡t✉♣
❚✐♠❡s
❘✉❜é♥ ❘✉✐③✱ ▲✉✐s ❋❛♥❥✉❧✲P❡②ró ❛♥❞ ❋❡❞❡r✐❝♦ P❡r❡❛
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❑❡②✇♦r❞s✿ P❛r❛❧❧❡❧ ♠❛❝❤✐♥❡✱ s❝❤❡❞✉❧✐♥❣✱ s❡t✉♣ t✐♠❡s✱ ❡①❛❝t ♠❡t❤♦❞s✳
✶ ■♥tr♦❞✉❝t✐♦♥
❆♥ ✐♠♣♦rt❛♥t s❝❤❡❞✉❧✐♥❣ ♣r♦❜❧❡♠ ❡♥t❛✐❧s ❛ s❡t ♦❢ n ❥♦❜s t❤❛t ❤❛✈❡ t♦ ❜❡ ❛ss✐❣♥❡❞
❛♥❞ s❝❤❡❞✉❧❡❞ t♦ ❛ s❡t ♦❢ m ♠❛❝❤✐♥❡s ✐♥ ♣❛r❛❧❧❡❧✳ ❊❛❝❤ ❥♦❜ ♠✉st ❜❡ ♠❛♥✉❢❛❝t✉r❡❞ ❜②
❡①❛❝t❧② ♦♥❡ ♠❛❝❤✐♥❡✳ ◆♦ ♠❛❝❤✐♥❡ ❝❛♥ ♣r♦❝❡ss ♠♦r❡ t❤❛♥ ♦♥❡ ❥♦❜ ❛t ❛ t✐♠❡✳ ■♥ t❤❡ ♠♦st
❣❡♥❡r❛❧ ❝❛s❡ ♠❛❝❤✐♥❡s ❛r❡ s❛✐❞ t♦ ❜❡ ✉♥r❡❧❛t❡❞✱ ♠❡❛♥✐♥❣ t❤❛t t❤❡ t✐♠❡ ♥❡❡❞❡❞ t♦ ♣r♦❝❡ss
❛ ❣✐✈❡♥ ❥♦❜ ❞❡♣❡♥❞s ♦♥ t❤❡ ♠❛❝❤✐♥❡ t♦ ✇❤✐❝❤ ✐t ✐s ❛ss✐❣♥❡❞✳ ❚❤✐s t✐♠❡ ✐s ❞❡♥♦t❡❞ ❛s
pij✱ i ∈ {1, 2, . . . , m}✱ j ∈ {1, 2, . . . , n}✳ ❯♥r❡❧❛t❡❞ P❛r❛❧❧❡❧ ▼❛❝❤✐♥❡s s❝❤❡❞✉❧✐♥❣ ♣r♦❜❧❡♠s
✭❯P▼✮ ♠♦❞❡❧ ❤✐❣❤ ♦✉t♣✉t ♣r♦❞✉❝t✐♦♥ s❤♦♣s ♦r ❡✈❡♥ ❝❡♥tr❛❧ st❛❣❡s ✐♥ ❝❡rt❛✐♥ ♣r♦❞✉❝t✐♦♥
♣r♦❝❡ss❡s✳ Cj ✐s t❤❡ ❝♦♠♣❧❡t✐♦♥ ♦❢ ❥♦❜ j✳ ❚❤❡ ♠♦st ❝♦♠♠♦♥❧② st✉❞✐❡❞ ♦❜❥❡❝t✐✈❡ ✐s t❤❡
♠✐♥✐♠✐③❛t✐♦♥ ♦❢ ♠❛❦❡s♣❛♥ ✭Cmax✮ ❛♥❞ t❤❡ ♣r♦❜❧❡♠ ✐s ❞❡♥♦t❡❞ ❛s R//Cmax✳ ❚❤❡ ❯P▼
✐s ❦♥♦✇♥ t♦ ❜❡ NP✲❍❛r❞✳ Pr❛❝t✐❝❛❧ ✐♥❞✉str✐❛❧ ♣r♦❜❧❡♠s ❝♦♠♠♦♥❧② ✐♥❝❧✉❞❡ s❡t✉♣ t✐♠❡s✳
❚❤✐s ♣❛♣❡r ❝♦♥s✐❞❡rs t❤❡ ❯♥r❡❧❛t❡❞ P❛r❛❧❧❡❧ ▼❛❝❤✐♥❡ s❝❤❡❞✉❧✐♥❣ ♣r♦❜❧❡♠ ✇✐t❤ s❡q✉❡♥❝❡
❞❡♣❡♥❞❡♥t ❙❡t✉♣ t✐♠❡s ✭❯P▼❙✮ ♦r R/sijk/Cmax✱ ✇❤❡r❡ sijk ❞❡♥♦t❡s t❤❡ ❛♠♦✉♥t ♦❢ s❡t✉♣
t✐♠❡ ♥❡❡❞❡❞ ❛t ♠❛❝❤✐♥❡ i ❛❢t❡r ❥♦❜ j ❛♥❞ ❜❡❢♦r❡ ❥♦❜ k✱ j 6= k✳ ❚❤❡ ❯P▼❙ ✐s s✐❣♥✐✜❝❛♥t❧②
♠♦r❡ ❞✐✣❝✉❧t t❤❛♥ t❤❡ ❯P▼✳ ❆s ❛ ♠❛tt❡r ♦❢ ❢❛❝t✱ ❛ s♣❡❝✐❛❧ ❝❛s❡ ♦❢ t❤❡ ❯P▼❙ ✇✐t❤ ❛ s✐♥❣❧❡
♠❛❝❤✐♥❡ ❝❛♥ ❜❡ ♠♦❞❡❧❡❞ ❛s ❛ ♣❛rt✐❝✉❧❛r ❚r❛✈❡❧✐♥❣ ❙❛❧❡s♠❛♥ Pr♦❜❧❡♠ ✭❚❙P✮✳ ❲❤✐❧❡ t❤❡
❧✐t❡r❛t✉r❡ ♦♥ t❤❡ ❯P▼ ✐s ❡①t❡♥s✐✈❡✱ t❤❡ ❯P▼❙ ❤❛s ❜❡❡♥✱ ❝♦♠♣❛r❛t✐✈❡❧② s♣❡❛❦✐♥❣✱ ♠✉❝❤
❧❡ss st✉❞✐❡❞✳ ❋✉rt❤❡r♠♦r❡✱ ♠♦st ❡①✐st✐♥❣ ❧✐t❡r❛t✉r❡ ❞❡❛❧s ✇✐t❤ ❤❡✉r✐st✐❝s ❛♥❞ ♠❡t❛❤❡✉r✐st✐❝s
❛♥❞ ❡①❛❝t ❛♣♣r♦❛❝❤❡s ❛r❡ ♦♥❧② ✈❛❧✐❞ ❢♦r r❡❧❛t✐✈❡❧② s♠❛❧❧ t♦ ♠❡❞✐✉♠ ✐♥st❛♥❝❡s✳ ❖♥❡ ♦❢ t❤❡
❝♦♥tr✐❜✉t✐♦♥s ♦❢ t❤✐s ♣❛♣❡r ✐s ❛ ♥❡✇ ♠❛t❤❡♠❛t✐❝❛❧ r❡❢♦r♠✉❧❛t✐♦♥ ❢♦r t❤❡ ❯P▼❙✳ ❆♥♦t❤❡r
❝♦♥tr✐❜✉t✐♦♥ ♦❢ t❤✐s ♣❛♣❡r ✐s t❤❡ ❞❡s✐❣♥ ♦❢ ❛♥ ❡✣❝✐❡♥t ♠❛t❤❡♠❛t✐❝❛❧ ♣r♦❣r❛♠♠✐♥❣ ❜❛s❡❞
❛❧❣♦r✐t❤♠✳ ❙♦♠❡ ❡①✐st✐♥❣ ▼■▲P ♠♦❞❡❧s ❝❛♥ ❜❡ ❢♦✉♥❞ ✐♥ t❤❡ ♦❧❞❡r ❧✐t❡r❛t✉r❡ ✇❤❡r❡ ♣r♦❜❧❡♠s
♦❢ ✉♣ t♦ ✶✹ ❥♦❜s ❝♦✉❧❞ ❜❡ s♦❧✈❡❞ t♦ ♦♣t✐♠❛❧✐t②✳ ■t ✇❛s ♥♦t ✉♥t✐❧ t❤❡ ❧❛st ❢❡✇ ②❡❛rs t❤❛t
♠✉❝❤ ❧❛r❣❡r ✐♥st❛♥❝❡s ♦❢ t❤❡ ❯P▼❙ ✇❡r❡ s♦❧✈❡❞ t♦ ♦♣t✐♠❛❧✐t②✳ ❆✈❛❧♦s✲❘♦s❛❧❡s ❡t ❛❧✳ ✭✷✵✶✺✮
♣r♦♣♦s❡❞ ❛ ▼■▲P t❤❛t ❡✣❝✐❡♥t❧② s♦❧✈❡❞ s♦♠❡ ✐♥st❛♥❝❡s ♦❢ ✉♣ t♦ ✻✵ ❥♦❜s ❛♥❞ ✽ ♠❛❝❤✐♥❡s✳ ❆
s✐♠✐❧❛r ▼■▲P ♣r❡✈✐♦✉s❧② s❡r✈❡❞ ❛s ❛ ♠❛st❡r ♣r♦❜❧❡♠ ✐♥ s♦♠❡ ✐t❡r❛t✐✈❡ ❛❧❣♦r✐t❤♠s ♣r❡s❡♥t❡❞
✐♥ ❚r❛♥ ❡t ❛❧✳ ✭✷✵✶✻✮✳ ❚❤❡ s✐③❡s ♦❢ t❤❡ ♣r♦❜❧❡♠s s♦❧✈❡❞ ✐♥ t❤❡s❡ ❧❛st ♣❛♣❡rs ❛r❡ ♠✉❝❤ ❧❛r❣❡r✱
❛❧❜❡✐t ♥♦t ❛❧❧ ❛r❡ s♦❧✈❡❞ t♦ ♦♣t✐♠❛❧✐t②✳
✷ Pr♦♣♦s❡❞ ♠♦❞❡❧s ❛♥❞ ❛❧❣♦r✐t❤♠s
❆✈❛❧♦s✲❘♦s❛❧❡s ❡t ❛❧✳ ✭✷✵✶✺✮ ♣r❡s❡♥t❡❞ t❤❡ ❢♦❧❧♦✇✐♥❣ ▼■▲P✱ ❞❡♥♦t❡❞ ❛s ❆❆❆ ✐♥ t❤✐s
♣❛♣❡r✳ ❆❆❆ ✉s❡s t❤❡ ❢♦❧❧♦✇✐♥❣ ✈❛r✐❛❜❧❡s✿ Xijk = 1 ✐❢ k ✐s t❤❡ s✉❝❝❡ss♦r ♦❢ j ♦♥ ♠❛❝❤✐♥❡
i✱ ③❡r♦ ♦t❤❡r✇✐s❡✳ Yij = 1 ✐❢ j ✐s ♣r♦❝❡ss❡❞ ♦♥ ♠❛❝❤✐♥❡ i✱ ③❡r♦ ♦t❤❡r✇✐s❡✳ C̃j ≥ 0 ✐s t❤❡
195

❝♦♠♣❧❡t✐♦♥ t✐♠❡ ♦❢ ❥♦❜ j✳ ❋✐♥❛❧❧②✱ Cmax ✐s t❤❡ ♠❛①✐♠✉♠ ❝♦♠♣❧❡t✐♦♥ t✐♠❡ ✭♠❛❦❡s♣❛♥✮✳
min Cmax ✭✶✮
s✳t✳
X
j∈N0,k∈N,k6=j
sijkXijk +
X
j∈N
pijYij ≤ Cmax, i ∈ M. ✭✷✮
X
k∈N
Xi0k ≤ 1, i ∈ M ✭✸✮
X
i∈M
Yij = 1, j ∈ N ✭✹✮
Yij =
X
k∈N0,j6=k
Xijk, i ∈ M, j ∈ N ✭✺✮
Yik =
X
j∈N0,j6=k
Xijk, i ∈ M, k ∈ N ✭✻✮
C̃k − C̃j + V (1 − Xijk) ≥ sijk + pik, j ∈ N0, k ∈ N, j 6= k, i ∈ M ✭✼✮
C̃0 = 0 ✭✽✮
Cmax ≥ C̃j, j ∈ N ✭✾✮
Xijk ∈ {0, 1}, Yij ≥ 0, C̃j ≥ 0.
❙❡ts N ❛♥❞ M ❞❡♥♦t❡ t❤❡ ❥♦❜s ❛♥❞ ♠❛❝❤✐♥❡s✳ ❙❡t N0 ✐♥❝❧✉❞❡s ❛ ❞✉♠♠② ❥♦❜✳ ❋✉❧❧ ♥♦t❛✲
t✐♦♥ ❞❡t❛✐❧s ❛r❡ ♦♠✐tt❡❞ ❞✉❡ t♦ s♣❛❝❡ ❝♦♥s✐❞❡r❛t✐♦♥s✳ ❈♦♥str❛✐♥ts ✭✷✮ ❞❡✜♥❡ t❤❡ ♠❛❦❡s♣❛♥✳
❈♦♥str❛✐♥ts ✭✸✮ ❡♥s✉r❡ t❤❛t ❛t ♠♦st ♦♥❡ ❥♦❜ ✐s s❝❤❡❞✉❧❡❞ ❛s t❤❡ ✜rst ♦♥ ❡❛❝❤ ♠❛❝❤✐♥❡✳ ❈♦♥✲
str❛✐♥ts ✭✹✮ st❛t❡ t❤❛t ❡❛❝❤ ❥♦❜ ✐s t♦ ❜❡ ♣r♦❝❡ss❡❞ ♦♥ ❡①❛❝t❧② ♦♥❡ ♠❛❝❤✐♥❡✳ ❈♦♥str❛✐♥ts ✭✺✮
❛♥❞ ✭✻✮ ❡♥s✉r❡ t❤❛t ❛❧❧ ❥♦❜s ❤❛✈❡ ♦♥❡ s✉❝❝❡ss♦r ❛♥❞ ♦♥❡ ♣r❡❞❡❝❡ss♦r✳ ❈♦♥str❛✐♥ts ✭✼✮ ♣r♦✈✐❞❡
❛ r✐❣❤t ♣r♦❝❡ss✐♥❣ ♦r❞❡r ❛♥❞ ❜r❡❛❦ s✉❜t♦✉rs✳ ❈♦♥str❛✐♥t ✭✽✮ s❡ts t❤❡ ❝♦♠♣❧❡t✐♦♥ t✐♠❡ ♦❢ t❤❡
❞✉♠♠② ❥♦❜ t♦ ③❡r♦✳ ❈♦♥str❛✐♥ts ✭✾✮ ❛r❡ ❢❡❛s✐❜❧❡ ❝✉ts✳ ❚❤✐s ♠♦❞❡❧ ✇❛s r❡♣♦rt❡❞ t♦ ❡✣❝✐❡♥t❧②
s♦❧✈❡ s♦♠❡ ✐♥st❛♥❝❡s ♦❢ ✉♣ t♦ ✻✵ ❥♦❜s✳
❲❡ ♣r❡s❡♥t ❛ ♠♦❞❡❧✱ ❝❛❧❧❡❞ ▼❚❩✲❆▼✱ ❜❛s❡❞ ♦♥ t❤❡ ❤❡t❡r♦❣❡♥❡♦✉s tr❛✈❡❧✐♥❣ s❛❧❡s♠❛♥
♣r♦❜❧❡♠ ✭❚❙P✮✳ ❚❤❡ ❚❙P ♦❜t❛✐♥s t❤❡ ♠✐♥✐♠✉♠ ❧❡♥❣t❤ r♦✉t❡ t❤❛t ✈✐s✐ts ❛❧❧ ♥♦❞❡s✴❥♦❜s ♦❢
N ❡①❛❝t❧② ♦♥❝❡✳ ❲❤❡♥ ♦♥❡ ❝♦♥s✐❞❡rs t❤❛t ♠♦r❡ t❤❛♥ ♦♥❡ s❛❧❡s♠❛♥ ✐s ❛✈❛✐❧❛❜❧❡✱ ❛♥❞ t❤❛t
❡❛❝❤ ❝✐t② ♠✉st ❜❡ ✈✐s✐t❡❞ ❜② ❡①❛❝t❧② ♦♥❡ s❛❧❡s♠❛♥✱ ✇❡ ❤❛✈❡ ❛ ♠✉❧t✐♣❧❡ tr❛✈❡❧✐♥❣ s❛❧❡s♠❛♥
♣r♦❜❧❡♠ ✭♠✲❚❙P✮✳ ❚❤❡ ❯P▼❙ ✐s ❛ ❤❡t❡r♦❣❡♥❡♦✉s ♠✲❚❙P✱ ✐♥ ✇❤✐❝❤ t❤❡ ❥♦❜s ❝♦rr❡s♣♦♥❞
t♦ ❝✐t✐❡s✱ ❛♥❞ t❤❡ ♠❛❝❤✐♥❡s ❝♦rr❡s♣♦♥❞ t♦ s❛❧❡s♠❡♥✳ ■❢ t✇♦ ❝✐t✐❡s j ❛♥❞ k ❛r❡ ✈✐s✐t❡❞ ♦♥❡
❛❢t❡r t❤❡ ♦t❤❡r ❜② t❤❡ s❛♠❡ s❛❧❡s♠❛♥ i✱ ✐♥ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❯P▼❙ ✇❡ s❛② t❤❛t k ✐s t❤❡
s✉❝❝❡ss♦r ♦❢ j ♦♥ ♠❛❝❤✐♥❡ i✳ ❚❤❡ ❝♦st ❢♦r t❤❡ s❛❧❡s♠❛♥ ✭♠❛❝❤✐♥❡✮ i ♦❢ tr❛✈❡rs✐♥❣ t❤❡ ❛r❝
❧✐♥❦✐♥❣ ❝✐t✐❡s j ❛♥❞ k ✭♦❢ ♣r♦❝❡ss✐♥❣ ❥♦❜ j ❛♥❞ t❤❡♥ k✮ ✐s ❡q✉❛❧ t♦ pij + sijk✳ ▼❚❩✲❆▼
s❤❛r❡s t❤❡ str✉❝t✉r❡ ❞❡✜♥❡❞ ❜② ❡q✉❛t✐♦♥s ✭✶✮ t♦ ✭✻✮✳ ◆♦t❡ t❤❛t ❝♦♥str❛✐♥ts ✭✼✮ ❛r❡ ❜❛s✐❝❛❧❧②
s✉❜t♦✉r ❡❧✐♠✐♥❛t✐♦♥ ❝♦♥str❛✐♥ts ✭❙❊❈✮✳ ❲❡ s✉❜st✐t✉t❡ ✭✼✮✱ ✭✽✮ ❛♥❞ ✭✾✮ ❜② t❤❡ ✇❡❧❧ ❦♥♦✇♥
▼❚❩ s✉❜t♦✉r ❡❧✐♠✐♥❛t✐♦♥ ❝♦♥str❛✐♥ts✿
Uj − Uk + n
X
i∈M
Xijk ≤ n − 1, j, k ∈ N, j 6= k. ✭✶✵✮
❚❤✐s s❡t ♦❢ ❝♦♥str❛✐♥ts ❡♥s✉r❡s t❤❛t✱ ✐❢ k ✐s t❤❡ s✉❝❝❡ss♦r ♦❢ j ♦♥ ❛♥② ♠❛❝❤✐♥❡ ✭❛♥❞ t❤❡r❡❢♦r❡
P
i∈M Xijk = 1✮✱ t❤❡♥ Uk ≥ Uj + 1✳ ❖t❤❡r✇✐s❡ ✭
P
i∈M Xijk = 0✮✳ ◆♦t❡ t❤❛t ✇✐t❤♦✉t
❝♦♥str❛✐♥ts ✭✼✮✱ ✭✽✮ ❛♥❞ ✭✾✮ ✈❛r✐❛❜❧❡s C̃j ❛r❡ ♥♦ ❧♦♥❣❡r ♥❡❡❞❡❞✳ ■♥st❡❛❞✱ ✇❡ ❞❡✜♥❡ t❤❡
❢♦❧❧♦✇✐♥❣ s❡t ♦❢ ✈❛r✐❛❜❧❡s✿ Uj ∈ Z+ ✐s t❤❡ ♥✉♠❜❡r ♦❢ ❥♦❜s ♣r♦❝❡ss❡❞ ❜❡❢♦r❡ j ♦♥ t❤❡ ♠❛❝❤✐♥❡
196

✇❤❡r❡ j ✐s ♣r♦❝❡ss❡❞✳ ❲❡ ❛❞❞ ❛ ✈❛❧✐❞ ✐♥❡q✉❛❧✐t② ❛❞❛♣t❡❞ ❢r♦♠ ♠✲❚❙P ♣r♦❜❧❡♠s✿
Uj + (n − 1)
X
i∈M
Xi0j ≤ n − 1, j ∈ N. ✭✶✶✮
Uj +
X
i∈M
Xi0j ≥ 1, j ∈ N. ✭✶✷✮
❚❤❡s❡ ❝♦♥str❛✐♥ts ✐♠♣♦s❡ t❤❛t ✐❢ ❛ ❥♦❜ j ✐s t❤❡ ✜rst ❥♦❜ ♦♥ ♦♥❡ ♠❛❝❤✐♥❡ t❤❡♥ Uj = 0 r❡✲
❣❛r❞❧❡ss ✇❤❡t❤❡r j ✐s ❛❧s♦ t❤❡ ❧❛st ❥♦❜ ♦r ♥♦t✳
❚r❛♥ ❡t ❛❧✳ ✭✷✵✶✻✮ ❤❛✈❡ ♣✉❜❧✐s❤❡❞ ❛ ❜r❛♥❝❤ ❛♥❞ ❝❤❡❝❦ ❞❡❝♦♠♣♦s✐t✐♦♥ ❛❧❣♦r✐t❤♠✳ ❆ ♠❛s✲
t❡r ♣r♦❜❧❡♠✱ ❜❛s✐❝❛❧❧② ❝♦♥s✐st✐♥❣ ♦❢ ❝♦♥str❛✐♥ts ✭✶✮ t♦ ✭✻✮✱ ✐s s♦❧✈❡❞✱ ♦❜t❛✐♥✐♥❣ ❛ ❢❡❛s✐❜❧❡
❛ss✐❣♥♠❡♥t ♦❢ ❥♦❜s t♦ ♠❛❝❤✐♥❡s✳ ❚❤❡ ❝②❝❧❡s ❝r❡❛t❡❞ ✐♥ t❤❡ s♦❧✉t✐♦♥s ♦❜t❛✐♥❡❞ ❜② t❤✐s ♠❛s✲
t❡r ♣r♦❜❧❡♠ ❛r❡ ❜r♦❦❡♥ ❜② t❤❡ ❈♦♥❝♦r❞❡ ❚❙P s♦❧✈❡r✱ ②✐❡❧❞✐♥❣ ♦♣t✐♠❛❧ s❝❤❡❞✉❧❡s ♦♥ ❡❛❝❤
♠❛❝❤✐♥❡ ❢♦r t❤❡ ❛ss✐❣♥♠❡♥ts ❣✐✈❡♥ ❜② t❤❡ ♠❛st❡r ♣r♦❜❧❡♠✳ ❲❡ ♣r❡s❡♥t ❛♥ ❛❧❣♦r✐t❤♠ ✇❤✐❝❤
t❛❦❡s s♦♠❡ ♦❢ t❤❡s❡ ✐❞❡❛s ❛♥❞ ❝♦♠❜✐♥❡s t❤❡♠ ✇✐t❤ ♦✉r ♣r♦♣♦s❡❞ ▼❚❩✲❆▼ ♠♦❞❡❧✳ ❚❤✐s
❛❧❣♦r✐t❤♠ ✇♦r❦s ✇✐t❤ ❛ s✐♠✐❧❛r ♠❛st❡r ♣r♦❜❧❡♠✿
min Cmax, s✳t✳✿✭✷✮, ✭✸✮, ✭✹✮, ✭✺✮, ✭✻✮, CUTS, Xijk ∈ [0, 1], Yij ∈ {0, 1}.
❋✐rst t❤❡ ♠❛st❡r ♣r♦❜❧❡♠ ✐s s♦❧✈❡❞ ✇✐t❤ CUTS = ∅✱ ❛❧❧♦✇✐♥❣ ❢♦r ❛ ✷✪ ❣❛♣✱ ❧✐❦❡ ✐♥ ❚r❛♥ ❡t
❛❧✳ ✭✷✵✶✻✮✳ ■♥ s✉❜s❡q✉❡♥t ✐t❡r❛t✐♦♥s t❤❡ ♥❡①t ❢❡❛s✐❜❧❡ s♦❧✉t✐♦♥ t♦ t❤❡ ♠❛st❡r ♣r♦❜❧❡♠ ✇✐❧❧
❜❡ ♦❜t❛✐♥❡❞✳ ❚❤❡s❡ s♦❧✉t✐♦♥s ②✐❡❧❞ ❢❡❛s✐❜❧❡ ❥♦❜✲♠❛❝❤✐♥❡ ❛ss✐❣♥♠❡♥ts✱ ❣✐✈❡♥ ❜② t❤❡ ✈❛❧✉❡s ♦❢
✈❛r✐❛❜❧❡ Y ✱ ❞❡♥♦t❡❞ ❜② yM
✳ ❍♦✇❡✈❡r✱ ♥♦ ❢❡❛s✐❜❧❡ s❡q✉❡♥❝❡ ✐s ❣✉❛r❛♥t❡❡❞ ❛s t❤❡ X ✈❛r✐❛❜❧❡s
❛r❡ r❡❧❛①❡❞ ❛♥❞ ♥♦ s✉❜t♦✉r ❡❧✐♠✐♥❛t✐♦♥ ❝♦♥str❛✐♥ts ❛r❡ ✐♥❝❧✉❞❡❞✳ ❋r♦♠ t❤❡ ❛ss✐❣♥♠❡♥ts yM
♦❜t❛✐♥❡❞ ✐♥ t❤❡ ♠❛st❡r ♣r♦❜❧❡♠✱ ❛ ❢❡❛s✐❜❧❡ s❡q✉❡♥❝❡ ✐s ❜✉✐❧t ❜② s♦❧✈✐♥❣ t❤❡ ❝♦♠♣❧❡t❡ ▼■▲P
♠♦❞❡❧ ✐♥ ✇❤✐❝❤ ✇❡ ♠✐♥✐♠✐③❡ t❤❡ s✉♠ ♦❢ t❤❡ ♠❛❝❤✐♥❡ ❝♦♠♣❧❡t✐♦♥ t✐♠❡s✿
min
X
i∈M
X
j∈N0,k∈N,k6=j
sijkXijk +
X
j∈N
pijyM
ij
s✳t✳✿✭✸✮, ✭✺✮, ✭✻✮, ✭✶✵✮, ✭✶✶✮, ✭✶✷✮, Xijk ∈ {0, 1}, Uj ≥ 0.
❆❢t❡r✇❛r❞s✱ ❝✉ts ❛r❡ ❛❞❞❡❞ t♦ t❤❡ ♠❛st❡r ♣r♦❜❧❡♠✳ ■❢ Nh
i ❞❡♥♦t❡s t❤❡ s❡t ♦❢ ❥♦❜s ❛ss✐❣♥❡❞
t♦ ♠❛❝❤✐♥❡ i ✐♥ t❤❡ ♠❛st❡r ♣r♦❜❧❡♠ ♦❢ ✐t❡r❛t✐♦♥ h✱ t❤❡ ♣r♦♣♦s❡❞ ❝✉t ❛t ✐t❡r❛t✐♦♥ h ✭❞❡♥♦t❡❞
❜② CUT(h)✮ ✐s✿ CUT(h) : Cmax ≥ Chi∗
max −
P
j∈Nh
i
(1 − Yij)θhij✳ ❲❡ ❞❡♥♦t❡ t❤✐s ❛❧❣♦r✐t❤♠
❛s ▼P❆ ✭▼❛t❤❡♠❛t✐❝❛❧ Pr♦❣r❛♠♠✐♥❣ ❜❛s❡❞ ❆❧❣♦r✐t❤♠✮✳
✸ ❈♦♠♣✉t❛t✐♦♥❛❧ ❡✈❛❧✉❛t✐♦♥ ❛♥❞ ❝♦♥❝❧✉s✐♦♥s
❲❡ s❤♦✇ r❡s✉❧ts ❢♦r ❧❛r❣❡ ✐♥st❛♥❝❡s ❢r♦♠ ✷✵✵ t♦ ✶✵✵✵ ❥♦❜s✳ ❲❡ r✉♥ ♦✉r ▼❚❩✲❆▼ ♠♦❞❡❧✱
t❤❡ ♠♦❞❡❧ ❆❆❆ ♦❢ ❆✈❛❧♦s✲❘♦s❛❧❡s ❡t ❛❧✳ ✭✷✵✶✺✮✱ ♦✉r ♣r♦♣♦s❡❞ ▼P❆ ❛♥❞ ♦✉r ♦✇♥ ✐♠♣❧❡✲
♠❡♥t❛t✐♦♥ ♦❢ t❤❡ ❜r❛♥❝❤✲❛♥❞✲❝❤❡❝❦ ❛❧❣♦r✐t❤♠ ♦❢ ❚r❛♥ ❡t ❛❧✳ ✭✷✵✶✻✮ ✭❇✫❈✮✳ ■♥ s✉❝❤ ❛♥
✐♠♣❧❡♠❡♥t❛t✐♦♥ ✇❡ s♦❧✈❡ t❤❡ ♠❛st❡r ♣r♦❜❧❡♠ ✇✐t❤ ●✉r♦❜✐✱ ✐♥st❡❛❞ ♦❢ ❙❈■P ❛s t❤❡ ♦r✐❣✐♥❛❧
❛✉t❤♦rs ❞✐❞✱ ❛♥❞ t❤❡ ❚❙P ✐♥ ❡❛❝❤ ♠❛❝❤✐♥❡ ✐s ❛❧s♦ s♦❧✈❡❞ ✇✐t❤ ●✉r♦❜✐ ✭✐♥st❡❛❞ ♦❢ ❈♦♥✲
❝♦r❞❡✬s ❚❙P s♦❧✈❡r✮✳ ❚❤✐s ✐♠♣❧❡♠❡♥t❛t✐♦♥ ♣r♦✈❡❞ ❢❛r s✉♣❡r✐♦r t❤❛♥ t❤❡ ♦r✐❣✐♥❛❧ ♦♥❡✳ ❚❤❡
▼■▲P ♠♦❞❡❧s ✇❡r❡ r✉♥ ❢♦r ❛ ♠❛①✐♠✉♠ ❈P❯ t✐♠❡ ♦❢ t❤r❡❡ ❤♦✉rs✳ ●✉r♦❜✐ ✼✳✵✳✷ ✐s ✉s❡❞
✭❈P▲❊❳ ✶✷✳✼ ♣r♦✈❡❞ t♦ ❜❡ ✐♥❢❡r✐♦r✮✳ ❲❡ ♠❡❛s✉r❡ t❤❡ ❘❡❧❛t✐✈❡ P❡r❝❡♥t❛❣❡ ❉❡✈✐❛t✐♦♥ ❢r♦♠
t❤❡ ♦♣t✐♠✉♠ ♦r ❜❡st ❧♦✇❡r ❜♦✉♥❞ ✭RPD✮✳ ❚❛❜❧❡ ✶ s❤♦✇s r❡s✉❧ts ❜r♦❦❡♥ ❞♦✇♥ ❜② n ✈❛❧✉❡s✳
▼❚❩✲❆▼ ♣❡r❢♦r♠s ❜❡st ✐♥ t❡r♠s ♦❢ ❛✈❡r❛❣❡ RPD✳ ■t ✐s ❢♦r n = 400 t❤❛t ♦✉r ♣r♦♣♦s❡❞ ♠♦❞❡❧
▼❚❩✲❆▼ ❝❧❡❛r❧② ♦✉t♣❡r❢♦r♠s t❤❡ ❆❆❆ ♠♦❞❡❧✳ ❚❛❜❧❡ ✷ s❤♦✇s t❤❡ ❛✈❡r❛❣❡ r❡s✉❧ts ♦✈❡r t❤❡
❧❛r❣❡ ✐♥st❛♥❝❡s ❢♦r t❤❡ ❜❡st t✇♦ ❛❧❣♦r✐t❤♠s t❡st❡❞✿ ♦✉r ✐♠♣❧❡♠❡♥t❛t✐♦♥ ♦❢ t❤❡ ❇r❛♥❝❤✲❆♥❞✲
❈❤❡❝❦ ✐♥ ✭❚r❛♥ ❡t ❛❧✳ ✷✵✶✻✮ ✭❇✫❈✮ ❛♥❞ ♦✉r ▼❛t❤❡♠❛t❤✐❝❛❧✲Pr♦❣r❛♠♠✐♥❣✲❇❛s❡❞ ❛❧❣♦r✐t❤♠
197

❆❆❆ ▼❚❩✲❆▼
n RPD ❚✐♠❡ RPD ❚✐♠❡
✷✵✵ 1.56 7290.98 1.98 8596.62
✹✵✵ 595.69 10 418.89 174.26 9410.34
❆✈❡r❛❣❡ 298.63 8854.94 88.12 9003.48
❚❛❜❧❡ ✶✳ ▼■▲P ♠♦❞❡❧s ❆❆❆ ❛♥❞ ♣r♦♣♦s❡❞ ▼❚❩✲❆▼ ✭t✐♠❡s ✐♥ s❡❝♦♥❞s✮✳
✭▼P❆✮✳ ✏❇❡st✑ ❝♦rr❡s♣♦♥❞s t♦ t❤❡ t✐♠❡ ❛t ✇❤✐❝❤ t❤❡ ❜❡st ❢❡❛s✐❜❧❡ s♦❧✉t✐♦♥ r❡t✉r♥❡❞ ❜② t❤❡
❛❧❣♦r✐t❤♠ ✇❛s ❢♦✉♥❞✳ ✏▼❛st❡r✑ ❛♥❞ ✏❙❝❤❡❞✑ s❤♦✇ t❤❡ ❛✈❡r❛❣❡ ❈P❯ t✐♠❡ ✐♥ s❡❝♦♥❞s s♣❡♥t
s♦❧✈✐♥❣ t❤❡ ♠❛st❡r ♣r♦❜❧❡♠ ❛♥❞ t❤❡ s❡q✉❡♥❝✐♥❣ ♣r♦❜❧❡♠ r❡s♣❡❝t✐✈❡❧②✳ ❲❡ ♥♦t❡ t❤❛t ❜♦t❤
❇✫❈ ▼P❆
n RPD ❚✐♠❡ ❇❡st ▼❛st❡r ❙❝❤❡❞ RPD ❚✐♠❡ ❇❡st ▼❛st❡r ❙❝❤❡❞
✷✵✵ 0.74 6303 2973 6145 158 0.93 6269 1497 6208 61
✹✵✵ 16.67 7488 4206 6290 1198 0.36 7385 2997 7221 164
✻✵✵ 217.78 8496 6698 5373 3122 0.30 7611 4620 7315 294
✽✵✵ 0.37 8623 5494 8017 598
✶✵✵✵ 0.41 9256 6015 8026 1195
❆✈❡r❛❣❡ 78.40 7429 4626 5936 1493 0.47 7829 4125 7358 462
❚❛❜❧❡ ✷✳ ❘❡s✉❧ts ❢♦r t❤❡ ❧❛r❣❡ ✐♥st❛♥❝❡s ❢♦r t❤❡ ❇✫❈ r❡✐♠♣❧❡♠❡♥t❛t✐♦♥ ❛♥❞ t❤❡ ♣r♦♣♦s❡❞ ▼P❆✳
❛❧❣♦r✐t❤♠s ♣❡r❢♦r♠ s✐♠✐❧❛r❧② ❢♦r n = 200✳ ❍♦✇❡✈❡r✱ ❢♦r n = 400, 600 ♦✉r ▼P❆ ♣r♦❞✉❝❡s
♠✉❝❤ ❧♦✇❡r ❛✈❡r❛❣❡ RPD t❤❛♥ ❇✫❈ ✐♥ s❤♦rt❡r ❈P❯ t✐♠❡s✳ RPD ❛r❡ ❝♦♠♣✉t❡❞ ❛❣❛✐♥st
t❤❡ ❜❡st ❧♦✇❡r ❜♦✉♥❞✳ ❇✫❈ ✇❛s ♥♦t ❛❜❧❡ t♦ ❝♦♣❡ ✇✐t❤ ✐♥st❛♥❝❡s ❧❛r❣❡r t❤❛♥ n = 600✳ ❲❡
t❡st❡❞ t❤❡ ♣r♦♣♦s❡❞ ▼P❆ ❛❧❣♦r✐t❤♠ ❢♦r ✐♥st❛♥❝❡s ♦❢ r❡❛❧❧② ❧❛r❣❡ s✐③❡s ✭n = 800, 1000✮✳ ❆s
❛ ❣❡♥❡r❛❧ ❝♦♥❝❧✉s✐♦♥✱ ♦✉r ♣r♦♣♦s❡❞ ▼P❆ ✐s ❛❜❧❡ t♦ ❣❡♥❡r❛t❡ ❛✈❡r❛❣❡ r❡❧❛t✐✈❡ ♣❡r❝❡♥t❛❣❡
❞❡✈✐❛t✐♦♥s ❢r♦♠ ❧♦✇❡r ❜♦✉♥❞s ♦❢ ✵✳✹✶✪ ✐♥ t❤❡ ❧❛r❣❡st ✐♥st❛♥❝❡s ♦❢ ✶✵✵✵ ❥♦❜s ❢♦r t❤❡ ❯▼P❙✳
❚❤✐s s✐❣♥✐✜❝❛♥t❧② ✐♠♣r♦✈❡s ✉♣♦♥ t❤❡ ♣r❡✈✐♦✉s r❡❝❡♥t r❡s✉❧ts ✐♥ t❤❡ ❧✐t❡r❛t✉r❡ ❜② ❚r❛♥ ❡t ❛❧✳
✭✷✵✶✻✮ ♦❢ ❛❜♦✉t ✷✳✹✽✪ RPD ❢♦r n = 120✳
❆❝❦♥♦✇❧❡❞❣♠❡♥ts
❚❤❡ ❛✉t❤♦rs ❛r❡ ♣❛rt✐❛❧❧② s✉♣♣♦rt❡❞ ❜② t❤❡ ❙♣❛♥✐s❤ ▼✐♥✐str② ♦❢ ❊❝♦♥♦♠② ❛♥❞ ❈♦♠✲
♣❡t✐t✐✈❡♥❡ss✱ ✉♥❞❡r t❤❡ ♣r♦❥❡❝t ✏❙❈❍❊❨❆❘❉ ✕ ❖♣t✐♠✐③❛t✐♦♥ ♦❢ ❙❝❤❡❞✉❧✐♥❣ Pr♦❜❧❡♠s ✐♥
❈♦♥t❛✐♥❡r ❨❛r❞s✑ ✭◆♦✳ ❉P■✷✵✶✺✲✻✺✽✾✺✲❘✮ ✜♥❛♥❝❡❞ ❜② ❋❊❉❊❘ ❢✉♥❞s✳ ❙♣❡❝✐❛❧ t❤❛♥❦s ❛r❡
❞✉❡ t♦ ❚♦♥② ❚✳ ❚r❛♥ ❛♥❞ ❝♦❛✉t❤♦rs ❢♦r ❛❧❧ t❤❡ ❤❡❧♣ r❡❝❡✐✈❡❞ ❞✉r✐♥❣ t❤❡ r❡✐♠♣❧❡♠❡♥t❛t✐♦♥
♦❢ t❤❡✐r ❡✣❝✐❡♥t ♠❡t❤♦❞s✳
❘❡❢❡r❡♥❝❡s
❆✈❛❧♦s✲❘♦s❛❧❡s✱ ❖✳✱ ❆♥❣❡❧✲❇❡❧❧♦✱ ❋✳✱ ❛♥❞ ❆❧✈❛r❡③✱ ❆✳ ✷✵✶✺✱ ✏❊✣❝✐❡♥t ♠❡t❛❤❡✉r✐st✐❝ ❛❧❣♦r✐t❤♠ ❛♥❞
r❡✲❢♦r♠✉❧❛t✐♦♥s ❢♦r t❤❡ ✉♥r❡❧❛t❡❞ ♣❛r❛❧❧❡❧ ♠❛❝❤✐♥❡ s❝❤❡❞✉❧✐♥❣ ♣r♦❜❧❡♠ ✇✐t❤ s❡q✉❡♥❝❡ ❛♥❞
♠❛❝❤✐♥❡✲❞❡♣❡♥❞❡♥t s❡t✉♣ t✐♠❡s✑✱ ■♥t❡r♥❛t✐♦♥❛❧ ❏♦✉r♥❛❧ ♦❢ ❆❞✈❛♥❝❡❞ ▼❛♥✉❢❛❝t✉r✐♥❣ ❚❡❝❤♥♦❧✲
♦❣②✱ ❱♦❧✳ ✼✻✱ ◆♦✳ ✾✲✶✷✱ ♣♣✳ ✶✼✵✺✲✶✼✶✽✳
❚r❛♥✱ ❚✳ ❚✳✱ ❆r❛✉❥♦✱ ❆✳✱ ❛♥❞ ❇❡❝❦✱ ❏✳ ❈✳ ✷✵✶✻✱ ✏❉❡❝♦♠♣♦s✐t✐♦♥ ♠❡t❤♦❞s ❢♦r t❤❡ ♣❛r❛❧❧❡❧ ♠❛❝❤✐♥❡
s❝❤❡❞✉❧✐♥❣ ♣r♦❜❧❡♠ ✇✐t❤ s❡t✉♣s✑✱ ■◆❋❖❘▼❙ ❏♦✉r♥❛❧ ♦♥ ❈♦♠♣✉t✐♥❣✱ ❱♦❧✳ ✷✽✱ ◆♦✳ ✶✱ ♣♣✳ ✽✸✲✾✺✳
198

Power usage minimization in server problems of
scheduling computational jobs on a single processor
Różycki R, Waligóra G, and Wȩglarz J
Poznan University of Technology, Poznan, Poland
{rafal.rozycki, grzegorz.waligora, jan.weglarz}@cs.put.poznan.pl
Keywords: scheduling, single processor, power, energy, server problem.
In this work we deal with a scheduling problem from the field of green computing
(Hurson and Memon, 2012, 2013), where the main idea is to find a good balance between
computing performance and consumption of natural resources. Since during the execution
of a computer program (a computational job), energy is consumed as a main resource,
appropriate power management is a basic technique for applying the green computing
principles. We consider a problem of scheduling n preemptable, independent jobs on a
single processor, where each job requires for its execution the processor as well as an
amount (unknown in advance) of power, and it consumes some amount of energy during the
execution. Power/energy is a doubly constrained, continuous resource available in positive
amounts P and E, respectively. Processing speed of a job depends on the amount of power
allotted to this job at a time, according to the following relation (Wȩglarz, 1976):
ẋi(t) =
dxi(t)
dt
= si(pi(t)), xi(0) = 0, xi(Ci) = wi (1)
where
xi(t) is the state of job i at time t;
si(·) is the continuous, increasing processing speed function of job i, such that si(0) = 0;
pi(t) is the amount of power allotted to job i at time t;
Ci is the completion time (unknown in advance) of job i;
wi is the size (final state) of job i.
Completion of job i requires that:
wi =
∫ Ci
0
si(pi(t))dt. (2)
Thus, job i is characterized by both: processing speed function si and size wi. We will
assume that the processing speed function of each job is strictly concave, as an inverse
of a strictly convex power usage function. Power/energy is a doubly constrained resource
available in positive amounts P and E, respectively. As a consequence, a feasible schedule
of length T has to meet the following constraints for any t ∈ [0, T], where T = maxi{Ci}
(Wȩglarz, 1981):
199

n
∑
i=1
pi(t) ≤ P. (3)
n
∑
i=1
∫ T
0
pi(t)dt ≤ E. (4)
The problem is, in general, to find a vector function p(t) = [p1(t), p1(t), . . . , pn(t)],
pi(t) ≥ 0, i = 1, 2, . . . , n, which, under the constraints imposed, optimizes the chosen
scheduling criterion.
The specificity of the model presented above is that it relates time, power, and energy.
In consequence, three possible optimization problems may arise: minimization of a time-
related criterion, power usage, or energy consumption under constraints imposed on the
other two quantities from which only one or both can be active. In this work we consider
a so-called server problem where the power usage is to be minimized assuming a given
level of a computer system performance. The level of performance is expressed by an
assumed deadline D for the completion of the given set of n jobs. Moreover, we analyze
two cases of the problem: when the energy amount available is not limited, as well as
when it is limited by E. In each case we formulate an appropriate nonlinear mathematical
programming (NLP) problem that finds an optimal power allocation, as well as we discuss
the methodology for solving the problem under consideration.
Let us first formulate two important properties of optimal schedules (Różycki and
Wȩglarz, 2014).
Property 1. For each job i characterized by a strictly concave processing speed function si
and for any energy level E, the following condition holds:
lim
T →∞
Ts−1
i (wi/T) E. (5)
Relation (5) can be interpreted so that extending the execution time of a job results
in decreasing the amount of energy consumed by this job. Therefore, from the energy
minimization point of view, it is desirable to extend the schedule as long as possible.
Another property concerning sever problems follows directly from Property 1.
Property 2. In each type of server problem, if a feasible solution exists, the length of an
optimal schedule is equal to D.
Property 2 is important for server problems since it is known that in order to minimize
power usage or energy consumption the schedule has to be completed at the moment of
deadline, under an obvious condition that the resource amounts are sufficient to do so.
In the next two sections power usage minimization server problems of scheduling com-
putational jobs on a single processor will be discussed. Let us firstly notice that since we
consider concave processing speed functions, scheduling on one processor need not be easier
then scheduling on parallel processors. It follows from the fact that for such functions par-
allel schedules lead to optimal solutions, and they are impossible to construct on a single
processor. As a result, some interesting analytical results for the case of one processor can
be obtained. Secondly, it is worth mentioning that since only sequential schedules may be
considered on a single processor, preemptability of jobs can be neglected. It is obvious that
interrupting a job and resuming it later cannot improve the schedule. Thus, each job i,
i = 1, 2, . . . , n, is processed using a constant amount of power pi 0, i = 1, 2, . . . , n, from
its start to its completion. The pi’s are variables in the NLP formulations presented in the
next two sections.
200

2 Power minimization under unlimited energy
In this server problem, the objective is to minimize the power usage under a given
deadline D for the completion of the last job, and assuming that the amount of energy
available for the execution of all jobs is not limited. Based on Property 2, the following
NLP problem can be formulated:
NLP1: minimize P = max
1=1,...,n
{pi} (6)
subject to
n
∑
i=1
wi
si(pi)
= D. (7)
The objective function (6) represents the maximum power usage over the entire set of
jobs, which is to be minimized. Constraint (7) assures that sum of execution times of all
jobs is equal to deadline D, according to Property 2.
By analyzing problem NLP1, we can formulate an important proposition for the con-
sidered case of a server problem.
Proposition 1. In an optimal schedule all jobs are processed using the same fixed power
amount.
Proof. Let us ﬁrst remove the nonlinearity from the objective function in NLP1. The
resulting problem is:
NLP2: minimize p (8)
subject to pi − p ≤ 0, i = 1, 2, . . . , n (9)
n
∑
i=1
wi
si(pi)
= D. (10)
The Lagrange function for problem NLP2 is as follows:
L(pi, λi, ρ) = p +
n
∑
i=1
λi(pi − p) + ρ
( n
∑
i=1
wi
si(pi)
− D
)
. (11)
Gradient conditions take the form:
∂L
∂p
= 1 −
n
∑
i=1
λi = 0 (12)
∂L
∂pi
= λi + ρ
wi(si(pi))′
(si(pi))2
= 0, i = 1, 2, . . . , n (13)
from which it is known that:
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n
∑
i=1
λi = 1 (14)
λi = −ρ
wi(si(pi))′
(si(pi))2
, i = 1, 2, . . . , n. (15)
Orthogonality conditions are as follows:
λi(pi − p) = 0, i = 1, 2, . . . , n (16)
Now, since wi 0, pi 0, si(0) = 0, and all functions si are increasing and strictly
concave, thus:
wi(si(pi))′
(si(pi))2
0, i = 1, 2, . . . , n. (17)
which means that in (15) for any i, λi = 0 if and only if ρ = 0. However, if ρ = 0, then
it follows from (15) that λi = 0 for every i = 1, 2, . . . , n, which is a contradiction to (14).
Consequently, ρ ̸= 0 and therefore λi ̸= 0 for every i = 1, 2, . . . , n. If so, we can conclude
from (16) that pi = p for every i = 1, 2, . . . , n.
An immediate corollary follows:
Corollary 1. The minimum amount of power p∗
sufficient to execute all jobs before given
deadline D can be found as the unique positive root of the equation:
n
∑
i=1
wi
si(p)
= D. (18)
After ﬁnding the optimal value of p, the minimum level of energy can be calculated
from:
Emin = p∗
· D. (19)
3 Power minimization under limited energy
In this server problem, the power usage is to be minimized under an assumed limited
amount E of energy and a required deadline D. We start with the condition for the existence
of a feasible solution.
Lemma 1. A feasible solution to the problem exists if there exists a solution to the system
of inequalities:
202

n
∑
i=1
wi
si(pi)
≤ D, (20)
n
∑
i=1
wipi
si(pi)
≤ E. (21)
If a feasible solution exists, the following NLP problem, using Property 2, finds a min-
imum power allocation:
NLP3: minimize P = max
i=1,...,n
{pi} (22)
subject to
n
∑
i=1
wi
si(pi)
= D (23)
n
∑
i=1
wipi
si(pi)
≤ E. (24)
In problem NLP3 the power usage (22) is minimized subject to the constraints that the
deadline is met (23) as well as the available amount of energy is not exceeded (24). Notice
that now Corollary 1 can be used to make an attempt to find an optimal solution. If p∗
calculated from (18) fulfils constraint (24), it will define the optimum power allocation.
However, if it does not, it means that a power allocation with different pi, i = 1, 2, . . . , n,
leads to optimum and, in such a case, it is necessary to solve problem NLP3.
In any case, after finding an optimum power allocation, the energy consumption in the
obtained schedule can be calculated from formula:
E∗
=
n
∑
i=1
E∗
i =
n
∑
i=1
wip∗
i
si(p∗
i )
. (25)
However, finding the minimum level of energy sufficient to realize the power-optimal
schedule requires a solution of another NLP problem:
NLP4: minimize Emin =
n
∑
i=1
wipi
si(pi)
(26)
subject to
n
∑
i=1
wi
si(pi)
= D (27)
pi ≤ P∗
, i = 1, . . . , n (28)
where P∗
is the optimal solution to problem NLP3.
4 Summary
In this work we consider a problem of minimizing the power usage while scheduling
preemptable, independent jobs on a single processor to meet a schedule deadline. Each
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job uses some amount of power and consumes some amount of energy. We consider two
situations: when energy is not, and when it is limited. For these cases we formulate mathe-
matical programming problems to ﬁnd optimal power allocations. In the case of unlimited
energy, we prove that all jobs are processed using the same power amount. We also show
how to calculate the minimum amount of energy for the power-optimal schedules.
References
Hurson A., A. Memon, 2012, Green and Sustainable Computing: Part I, Elsevier Science, Academic
Press.
Hurson A., A. Memon, 2013, Green and Sustainable Computing: Part II, Elsevier Science, Aca-
demic Press.
Różycki R., J. Wȩglarz, 2014, “Power-aware scheduling of preemptable jobs on identical parallel
processors to minimize makespan”, Annals of Operations Research, Vol. 213(1), pp. 235–252.
Wȩglarz J., 1976, “Time-optimal control of resource allocation in a complex of operations frame-
work”, IEEE Transactions on Systems, Man and Cybernetics, Vol. 6(11), pp. 783–788.
Wȩglarz J., 1981, “Project scheduling with continuously-divisible doubly constrained resources”,
Management Science, Vol. 27(9), pp. 1040–1053.
204

Scheduling resource-constrained projects with
makespan-dependent revenues and costly overcapacity
André Schnabel and Carolin Kellenbrink
Department of Production Management, Leibniz Universität Hannover, Germany
andre.schnabel, carolin.kellenbrink@prod.uni-hannover.de
Keywords: RCPSP, scheduling, heuristics, local-search, genetic algorithm, overcapacity.
1 Introduction
The resource-constrained project scheduling problem (RCPSP), cf. Pritsker (1969), is a
widely researched combinatorial optimization problem. Solving the RCPSP involves find-
ing an assignment of activity starting times which minimizes the project duration without
violating constraints imposed by precedence relations between activities and limited re-
source availabilities. There are given activities j ∈ J including dummy start activity 0
and dummy end activity J + 1. Each activity j has an associated duration dj, resource
consumptions kjr on renewable resources r ∈ R, and may only start after all of its prede-
cessors i ∈ Pj are finished. Each resource r has a constant capacity availability Kr in each
time period t ∈ T . In any period t, the total resource consumption of each resource r from
all activities executed in this period must not exceed the capacity level Kr.
This problem setting itself is widely applicable and general. Many industrial scheduling
problems like machine scheduling are special cases of the RCPSP. However, there are situ-
ations in practice, in which some assumptions of the RCPSP are invalidated or additional
assumptions are required. A good overview of extensions and generalizations of the RCPSP
can be found in Hartmann and Briskorn (2010).
One aspect often found in industrial applications like aircraft engine remanufactur-
ing is the consideration of costly overcapacity in conjunction with makespan-dependent
revenues. This problem setting resembles known generalizations of the RCPSP like the
makespan minimization for exogenous fluctuating capacities, cf. Hartmann (2015), or like
the minimization of endogenous fluctuating capacities for a given deadline, cf. Deckro
(1989). However, the specific combination of simultaneous makespan and flexible capacity
optimization is not yet covered in literature. Therefore, we propose a new extension of the
RCPSP. We also discuss this topic in a working paper submitted to a journal, see Schnabel
et al. (2017).
2 Problem setting
The resource-constrained project scheduling problem with makespan-specific revenues
and option of overcapacity (RCPSP-ROC) extends the well-known RCPSP by allowing an
increase of the freely available capacity levels on a per period basis through the utilization
of costly and bounded overcapacity. This capacity level extension instrument can be inter-
preted either as overcapacity acquired by renting additional machines (or lease workers) or
as overtime of employees. Furthermore, the RCPSP-ROC incorporates customer specific
revenues depending on the time required for project completion, i.e., makespan. These
revenues are assumed to be monotonically decreasing, meaning that a customer is never
willing to increase his payment in case of a delay.
A new trade-off emerges when combining these two aspects in the RCPSP framework:
The planner may either increase revenue through speedups obtained by additional usage
205

of overcapacity or decrease cost through reduction of overcapacity. However, he can never
simultaneously increase revenue while decreasing costs.
2.1 Model formulation
A precise description of the RCPSP-ROC requires three additional parameters: The
payment reserves of a customer ut : T 7→ R, the costs κr per capacity unit and per pe-
riod of overcapacity, and the upper bound for overcapacity zr. The problem can then be
formalized extending any mixed integer programming formulation of the RCPSP. As one
possible option, we chose to modify the binary pulse variable formulation given in Pritsker
(1969). By definition, xjt is set to one if, and only if, activity j finishes in period t.
Model RCPSP-ROC
max F =
LF TJ+1
X
t=EF TJ+1
ut · xJ+1,t −
X
r∈R
X
t∈T
κr · zrt (1)
subject to
LF Tj
X
t=EF Tj
xjt = 1, j ∈ J (2)
LF Ti
X
t=EF Ti
xit · t ≤
LF Tj
X
t=EF Tj
xjt · t − dj, j ∈ J , i ∈ Pj (3)
J
X
j=1
t+dj −1
X
τ=t
kjr · xjτ ≤ Kr + zrt, r ∈ R, t ∈ T (4)
xjt ∈ {0, 1}, j ∈ J , t ∈ {EFTj, . . . , LFTj} (5)
zrt ∈ [0, zr], r ∈ R, t ∈ T (6)
Equation (1) captures the objective of maximizing the profit, which is computed from
the realized project revenue depending on the makespan and the overcapacity costs incurred
by the schedule. These costs for overcapacity are computed using the auxiliary variable zrt,
which is linked by equations (4) to the amount of overcapacity used.
Equations (2) enforce that each activity is executed exactly once. The required order
of activity execution is incorporated through constraints (3). Restrictions (4) then limit
the cumulative demands in the schedule to the freely available fixed resource availabilities
supplemented by the chosen amount of overcapacity. Since the objective of this model
indirectly minimizes zrt, this auxiliary variable will always store the amount of overcapacity
that was actually necessary in order to gain resource feasibility.
The remaining domains of binary primary variable xjt and continuous variable zrt are
specified in equations (5) and (6) respectively, with equations (6) also enforcing an upper
bound and non-negativity for overcapacity. In order to tighten the latest finishing times
LFTj without excluding the optimal solution, as a deadline we apply the makespan of
the schedule generated by the serial schedule generation scheme (SGS) using the canonical
activity list without using any overcapacity at all.
2.2 Properties of the problem setting
The RCPSP is a special case of the RCPSP-ROC, e.g., ut = −t ∀ t, zr = 0 ∀ r.
Therefore, the NP-hardness of the RCPSP implies the impracticality of solving industrial
206

size instances of the more general RCPSP-ROC in acceptable time using exact solution
methods.
Our problem setting combines the minimization of overcapacity (non-regular term) with
the maximization of makespan-dependent revenues (regular term). The non-regular com-
ponent in the objective prevents direct application of heuristics developed for the RCPSP,
cf. Ballestin and Blanco (2015). This motivates the design, implementation, and evaluation
of novel heuristics for the RCPSP-ROC.
3 Solution approaches
The representation or encoding of the solution is a critical core element of many heuris-
tics. A key concern when designing a solution encoding is the following trade-off: The state
space of the encoding should ideally be big enough to contain at least one optimal solution
while simultaneously being as small as possible.
An example for such an efficient encoding of schedules for the RCPSP is the activity
list. Unfortunately, just reusing the activity list as encoding in conjunction with the serial
SGS as decoding procedure is not possible due to the non-regular objective. Therefore,
new solution encodings were developed for the RCPSP-ROC. They include the amount
of overcapacity permitted in certain periods or the information whether one activity is
allowed to use overcapacity or not. These encodings are embedded in two different types
of heuristics.
3.1 Genetic Algorithms
Genetic algorithms are very powerful for heuristically solving the RCPSP and its vari-
ants, see Kolisch and Hartmann (2006). For any encoding, a genetic algorithm can be
obtained by specifying the construction of the initial population and the genetic operators
(crossover, mutation, selection). An individual is simply an encoded solution and its fit-
ness value is the objective value of that solution. The genetic algorithm for the RCPSP
developed by Hartmann (1998) was used as a starting point for developing the genetic
algorithms and adapted to the encodings for the RCPSP-ROC.
3.2 LocalSolver
Furthermore, LocalSolver, a relatively new commercial proprietary black-box heuristic
solver1
, was used for solving the RCPSP-ROC. LocalSolver provides a modeling language
for specifying the objectives and constraints of the model similar to GAMS or OPL. Ad-
ditionally, it offers an API for specifying models. This API then provides so-called “native
functions”, a mechanism that allows inserting arbitrary functions implemented in a general-
purpose programming language into any expression of the model. The decoding procedures
(e.g. activity list 7→ starting times) already implemented for the genetic algorithms as fit-
ness functions were plugged into small LocalSolver models. These models only specify the
structure of the solution encoding.
4 Results and conclusion
The numerical experiments are based on two test sets consisting of a filtered subset
of 270 and 585 project instances from j30 and j120 respectively based on the PSPLIB,
cf. Schnabel et al. (2017) for details on the instances, e. g., the definition of the revenue
1
http://localsolver.com
207

0
0,2
0,4
0,6
0,8
1
0 0,01 0,02 0,03 0,04 0,05 0,06 0,07 0,08 0,09 0,1
Genetic Algorithm LocalSolver Gurobi
(a) ∅ optimality gaps for j30
0
0,1
0,2
0,3
0,4
0 0,05 0,1 0,15 0,2 0,25 0,3 0,35 0,4 0,45 0,5
Genetic Algorithm LocalSolver
(b) ∅ gaps to best-known solutions for j120
Fig. 1: Numerical results for extended PSPLIB test sets
function. Figure 1a shows the progression of average optimality gaps for small instances
with 30 non-dummy activities in the first 0.1 seconds. The optimal reference solutions
were acquired using Gurobi. Figure 1b shows the progression of average gaps to best-
known solutions for larger instances with 120 non-dummy activities in the first 0.5 seconds.
For each instance, the best-known solution is the highest profit schedule of all considered
methods at the end of computation.
In summary, the results show that the developed genetic algorithm is very competitive
and outperforming the other solution procedures evaluated both in the short and in the
long run with small and large instances. Interestingly, utilizing the developed solution
encodings in LocalSolver was also very efficient, although not as efficient as using the
genetic algorithm. Both heuristic approaches were easily able to beat Gurobi on the MIP-
formulation. These results indicate the possibility of constructing efficient methods for
solving this generalized version of the RCPSP by adapting and combining ideas from
scheduling literature for both cost- and time-based objective functions.
Acknowledgements
The authors thank the German Research Foundation (DFG) for financial support of
this research project in the CRC 871 “Regeneration of Complex Capital Goods”.
References
Ballestin, F. and R. Blanco, 2015, “Theoretical and Practical Fundamentals”, Handbook on Project
Management and Scheduling, Vol. 1, pp. 411-427.
Deckro, R. F. and J. E. Hebert, 1989, “Resource constrained project crashing”, Omega, Vol. 17,
No. 1, pp. 69-79.
Hartmann, S., 1998, “A competitive genetic algorithm for resource-constrained project scheduling”,
Wiley Online Library, Vol. 45, No. 7, pp. 733-750.
Hartmann, S. and D. Briskorn, 2010, “A survey of variants and extensions of the resource-
constrained project scheduling problem”, European Journal of Operational Research, Vol. 207,
No. 1, pp. 1-14.
Hartmann, S., 2015, “Time-Varying Resource Requirements and Capacities”, Handbook on Project
Management and Scheduling, Vol. 1, pp. 163-176.
Kolisch, R. and S. Hartmann, 2006, “Experimental investigation of heuristics for resource-
constrained project scheduling: An update”, European Journal of Operational Research, Vol.
174, No. 1, pp. 23-37.
Prisker A., 1969, “Multiproject scheduling with limited resources: A zero-one programming ap-
proach”, Management Science, Vol. 16, pp. 93-108.
Schnabel, A., C. Kellenbrink and S. Helber, 2017, “Profit-oriented scheduling of resource-
constrained projects with flexible capacity constraints”, Diskussionspapiere - Hannover Eco-
nomic Papers (HEP), No. 593. http://diskussionspapiere.wiwi.uni-hannover.de/index.
php?number=593.
208

On the complexity of scheduling start time dependent
asymmetric convex processing times
Helmut A. Sedding
Institute of Theoretical Computer Science, D-89069 Ulm University, Germany
{firstname}.{lastname}@uni-ulm.de
Keywords: Time-dependent scheduling, Piecewise-linear convex processing times
1 Introduction
Time-dependent scheduling concerns with processing times that are a function of start
time (Gawiejnowicz 2008). In this work, we consider a rather generic problem that allows
for nonmonotonous convex processing times. With it, we aim to lay groundwork for this
recent branch of time-dependent scheduling theory. The problem setting surfaced while
automating production planning of continuously moving assembly lines. In it, we consider
sequencing of assembly operations J for one worker at one workpiece. Notably, before an
operation j ∈ J can be performed, necessary parts need to be fetched from a corresponding
container at the line side. For this purpose, the worker leaves the workpiece and walks
along the conveyor. As the conveyor continually moves the workpiece, walking distance
varies over time. It is minimum at time τj, when the moving workpiece just passes the
according container. Else, it increases linearly. Thus, walk time is depicted by a V-shaped,
piecewise linear function of time. After each walk, the worker performs the corresponding
operation in assembly time lj. The objective is to reduce total walk time by permuting the
operations. In scheduling terms, we subsume a walk and an assembly operation by a job
with a time-dependent processing time, and minimize total makespan.
Definition 1 (Problem P) We are given slopes a ∈ [0, 1], b ∈ [0, ∞), and a set of n
jobs J = {1, . . . , n}. Each job j ∈ J is given an assembly time lj ∈ Q≥0 and an ideal
start time τj ∈ Q. We decide on job sequence S : J → {1, . . . , n}, which is a permutation
of the jobs J to assign each job a distinct position. Its inverse is denoted by S−1
. For
each job j ∈ J, we calculate start time tj = CS−1(S(j)−1), iteratively from the global start
time tmin = C0 (usually zero), and completion time Cj = tj + pj(tj) with the start time
dependent processing time pj(t) = lj + max{−a (t − τj), b (t − τj)}. The objective is to find
a job sequence S that minimizes makespan ϕ(S) = CS−1(n) = Cmax.
In three-field notation, the problem is stated as 1 | pj = lj +max{−a (t−τj), b (t−τj)} |
Cmax. The processing time pj = pj(t) of job j ∈ J is shortest if j starts at t = τj, increasing
with slope a for decreasing t τj, else with slope b, thus it is asymmetric. The completion
time Cj = t + pj(t) is increasing with t because a ≤ 1 and b ≥ 0. Therefore, idle time
between jobs may only increase the objective; it is thus excluded by definition. In the
literature, problem setting is introduced with symmetric factors a = b in Sedding and
Jaehn (2014). The variant in Jaehn and Sedding (2016) measures a job’s deviation from
the mid-time, when exactly half of the whole job has been processed. The case with one
common ideal start time and a variable global start time tmin is polynomial (Farahani and
Hosseini 2013). With a fixed, given tmin and asymmetric slopes however, we show in the
following that the decision problem is NP-complete by reduction from Even Odd Partition.
A highlight of our proof is its compactness compared to the approach for the similar and
more specialized problem setting in Jaehn and Sedding (2016). Moreover, we introduce two
important polynomial cases which additionally apply for multiple ideal start times.
209

2 Polynomial Cases
Lemma 1 Given an instance of P and a job sequence S that sorts the jobs nondecreas-
ingly by lj − bτj. If each job starts at or after its ideal start time tj ≥ τj for all j ∈ J,
objective ϕ(S) is minimum and it is expressed by
ϕ(S) =
∑
j∈J
(lj − bτj) (1 + b)
n−S(j)
. (1)
Lemma 2 Given an instance of P. If each job starts at or before its ideal start time tj ≤ τj
for all j ∈ J, the objective is expressed by
ϕ(S) =
∑
j∈J
(lj + aτj) (1 − a)
n−S(j)
. (2)
If S furthermore sorts the jobs nonincreasingly by lj + aτj, then ϕ(S) is minimum.
On the other hand, given an objective value ϕ, the start time tmin = tS−1(1) of the first
job in the sequence S−1
(1) is
tmin = (1 − a)
−n
ϕ −
n
∑
j∈J
(lj + aτj) (1 − a)
−S(j)
. (3)
3 Computational Complexity
We analyze the computational complexity of P using a partition-type NP-hard problem:
Definition 2 (Even Odd Partition Problem (Garey, Tarjan and Wilfong 1988))
We are given a set of n = 2h natural numbers X = {x1, . . . , xn} where xi−1 xi for all
i = 2, . . . , n. The question is whether there exists a partition of X into subsets X1 and
X2 := Y X1 such that
P
x∈X1
x =
P
x∈X2
x, while for each i = 1, . . . , h, set X1 contains
exactly one element of set {x2i−1, x2i}.
We reduce from the Even Odd Partition Problem to the decision version of P, which
asks, for a given threshold Φ ∈ Q, if there exists a sequence S with makespan ϕ(S) ≤ Φ.
Theorem 1. The decision version of P with a common τ = τj ∀j ∈ J is NP-complete.
Proof. We are given an instance of the Even Odd Partition Problem as of Deﬁnition 2.
Let us deﬁne a corresponding instance of P. For this, we choose an arbitrary a ∈ (0, 1),
and set b = (1 − a)
−1
− 1. Then, b ∈ (0, ∞) and (1 + b) = (1 − a)
−1
. Let job set J =
{1, . . . , 2n + 1}, with ln+j = 0 for j = 1, . . . , n, l2n+1 = 2q for q = 1
2
P
i∈X xi, and
l2k−i = x2k−i (1 + b)
k−h−1
for k = 1, . . . , h and i = 0, 1. Hence, lj−1 lj for j = 2, . . . , n,
and ln l2n+1. Moreover, we set the common ideal start time τ = 0 and the global start
time tmin = −q. The decision version of this instance asks if there exists a sequence S
where objective ϕ = Cmax is below threshold Φ = 3q.
Solving the sequencing problem results in an optimum sequence S, which we divide into
three partial sequences for our analysis. Partial sequence S0 consists of jobs n + 1, . . . , 2n,
and we assume this order without loss of generality. The rest is divided into S1, consisting
of jobs that start before 0, and S2 of jobs that start at or after 0. By Lemma 2, S1 is
sorted nonincreasingly by lj, while S2 has nondecreasing lj (Lemma 1). Moreover, S0 is
between S1 and S2 in S as the jobs in S0 have the smallest assembly times. Let Ĉ denote
the completion time of S1 and t̂ ≥ 0 the start time of S2. Hence, sequence S0 starts at Ĉ
210

and completes at t̂. As a 1, if a job j with lj = 0 starts at Ĉ 0, then it completes
at t̂ 0. This contradicts the sorting of the jobs in S1 ∪ S0. Thus, there must be Ĉ 0.
Then, t̂ = Ĉ (1 + b)
n
by Equation 1. As l2n+1 has the longest assembly time, job 2n + 1 is
either the first job in S1 or the last job in S2. However, we show that job 2n + 1 is not in
S1 if Cmax ≤ Φ: for a contradiction, let S(2n + 1) = 1. Then, job 2n + 1 starts at −q and
has a completion time of −q + l2n+1 + aq 0, hence equals Ĉ = q (1 + a). By Lemma 1,
jobs n + 1, . . . , 2n and then jobs 1, . . . , n are appended in nondecreasing order of lj, thus
Cmax = Ĉ (1 + b)
2n
+
∑
j=1,...,2n
lj (1 + b)
2n−(S(j)−1)
= Ĉ (1 + b)
2n
+
∑
j=1,...,n
lj (1 + b)
n−j
= Ĉ (1 + b)
2n
+
∑
k=1,...,h
l2k−1 (1 + b)
n+2−2k
+ l2k (1 + b)
n+1−2k
Ĉ (1 + b)
2n
+
∑
k=1,...,h
(l2k−1 + l2k) (1 + b)
n+1−2k
= Ĉ (1 + b)
2n
+
∑
k=1,...,h
(
x2k−1 (1 + b)
k−h−1
+ x2k (1 + b)
k−h−1
)
(1 + b)
n+1−2k
Ĉ +
∑
k=1,...,h
(x2k−1 + x2k) = Ĉ + 2q = q (1 + a) + 2q = q (3 + a) 3q = Φ.
Hence, job 2n + 1 is the last job in S2 in any optimum S with Cmax ≤ Φ. Moreover we
note as S0 starts at or after 0, partial sequence S1 constains at most n jobs.
Let S be optimum for the given instance and assume Cmax ≤ Φ. Define h1 as the
number of jobs in S1, and define h2 = n − h1. Given t̂ ≥ 0 and Equation 3, there is
tmin = Ĉ (1 − a)
−h1
−
∑
k=1,...,h1
lS−1
1 (k) (1 − a)
−k
= Ĉ (1 + b)
h1
−
∑
k=1,...,h1
lS−1
1 (k) (1 + b)
k
.
As S2 starts at t̂, with Equation 1 there is
Cmax = t̂ (1 + b)
h2+1
+
∑
k=1,...,h2+1
lS−1
2 (k) (1 + b)
h2+1−k
= t̂ (1 + b)
h2+1
+ l2n+1 +
∑
k=1,...,h2
lS−1
2 (k) (1 + b)
h2+1−k
= t̂ (1 + b)
h2+1
+ l2n+1 +
∑
k=1,...,h2
lS−1
2 (h2+1−k) (1 + b)
k
.
Define d = (1 + b)
n+h2+1
− (1 + b)
h1
, and
f1(k) =
{
lS−1
1 (k), 1 ≤ k ≤ h1,
0, else,
f2(k) =
{
lS−1
2 (h2+1−k), 1 ≤ k ≤ h2,
0, else.
Then with t̂ = Ĉ (1 + b)
n
,
Φ − tmin ≥ Cmax − tmin
⇐⇒ 4q ≥ t̂d + l2n+1 +
∑
k=1,...,h2
lS−1
2 (h2+1−k) (1 + b)
k
+
∑
k=1,...,h1
lS−1
1 (k) (1 + b)
k
⇐⇒ 2q ≥ t̂d +
∑
k=1,...,n
(f1(k) + f2(k)) (1 + b)
k
. (4)
As h1 ≤ n and h2 ≥ 0, there is d 0. In the following, we show that the minimum of
P
k=1,...,n (f1(k) + f2(k)) (1 + b)
k
is 2q, hence Equation 4 requires t̂ = 0.
211

– By Hardy, Littlewood and Pólya (1923, Theorem 368, p. 261) and as (1 + b)
k
increases
with k, sum f1(k) + f2(k) decreases with k a optimum S.
– For any i, j = 1, 2 such that i ̸= j, if fi(k) = 0 for some k while fj(k +1) 0, then S is
not optimum: an improved S rather has fi(k) 0 and fj(k +1) = 0. By this argument
and as h1 + h2 = 2h, it follows that h1 = h2 = h for an optimum S.
– Moreover, Hardy et al.’s (1923) theorem implies fi(k − 1) fj(k) for k = 2, . . . , h
and any i, j = 1, 2. This is the case for an optimum S as of Lemma 1 and Lemma 2.
Therefore, S has {S(2k−1), S(2k)} = {h+1−k, h+k} (in any order) for k = 1, . . . , h.
– It follows that an optimum S has
∑
k=1,...,n
(f1(k) + f2(k)) (1 + b)
k
=
∑
k=1,...,h
(l2k−1 + l2k) (1 + b)
k
=
∑
k=1,...,h
(
x2k−1 (1 + b)
−k
+ x2k (1 + b)
−k
)
(1 + b)
k
=
∑
k=1,...,h
x2k−1 + x2k = 2q.
The value of t̂ follows from Equation 2 and S−1
1 (h + 1 − k) ∈ {2k − 1, 2k} for k = 1, . . . , h:
t̂ = −q (1 − a)
h
+
∑
j=1,...,h
lS−1
1 (j) (1 − a)
h−j
= −q (1 − a)
h
+
∑
k=1,...,h
lS−1
1 (h+1−k) (1 − a)
h−(h+1−k)
= −q (1 − a)
h
+
∑
k=1,...,h
(
xS−1
1 (h+1−k) (1 + b)
k−h−1
)
(1 + b)
1−k
= −q (1 − a)
h
+ (1 − a)
h
∑
j=1,...,h
xS−1
1 (j).
Then, t̂ = 0 ⇐⇒
P
j=1,...,h xS−1
1 (j) = q ⇐⇒ {xS−1
1 (j) | j = 1, . . . , h} = X1 where X1 is a
solution for the Even Odd Partition Problem.
Therefore, the Even Odd Partition Problem instance solves the corresponding P in-
stance and vice versa. As the construction is polynomial and as, given a correct partition,
S and ϕ(S) can be obtained in polynomial time, the stated problem is NP-complete. ⊓
⊔
References
Farahani, M. H. and Hosseini, L.: 2013, Minimizing cycle time in single machine scheduling with
start time-dependent processing times, The International Journal of Advanced Manufacturing
Technology 64(9), 1479–1486.
Garey, M. R., Tarjan, R. E. and Wilfong, G. T.: 1988, One-processor scheduling with symmetric
earliness and tardiness penalties, Mathematics of Operations Research 13, 330–348.
Gawiejnowicz, S.: 2008, Time-dependent scheduling, Monographs in Theoretical Computer Science,
Springer, Berlin and Heidelberg.
Hardy, G. H., Littlewood, J. E. and Pólya, G.: 1923, Inequalities, Cambridge University Press.
Jaehn, F. and Sedding, H. A.: 2016, Scheduling with time-dependent discrepancy times, Journal
of Scheduling 19(6), 737–757.
Sedding, H. A. and Jaehn, F.: 2014, Single machine scheduling with nonmonotonic piecewise linear
time dependent processing times, in T. Fliedner, R. Kolisch and A. Naber (eds), Proceedings
of the 14th International Conference on Project Management and Scheduling, TUM School of
Management, pp. 222–225.
212

Resource-constrained project scheduling with
alternative project structures
Tom Servranckx1
and Mario Vanhoucke1,2,3
1
tom.servranckx@ugent.be, mario.vanhoucke@ugent.be
2
3
Keywords: Project scheduling, Alternative project structure, Tabu search.
1 Introduction
Project scheduling is crucial to project success as it provides a point-of-reference for
long-term resource allocation and project scope management. The resource-constrained
project scheduling problem (RCPSP) is a well-known problem in the context of project
scheduling (Brucker et al. 1999). Many research efforts have focused on the development of
various extensions of the basic RCPSP (Hartmann and Briskorn 2010) as well as multiple
(meta)heuristic and exact solution procedures (Kolisch and Hartmann 2006, Hartmann
and Kolisch 2000). However, one assumption that is retained in most scheduling problems
requires that the project structure is deterministic. This implies that the project structure,
which is imposed by the activities and the precedence relations between the activities,
is fixed and completely known prior to the project execution. However, this assumption
has been rendered obsolete in most real-life projects due to the ever-increasing complexity
and uncertainty in the project environment (Wiers V. 1997). Therefore, several researchers
have already considered improving the flexibility in the execution mode of a project. This
would allow certain project elements to be executed in alternative ways in order to respond
to unexpected disruptions. In this regard, we should mention a well-studied extension of
the RCPSP, the so-called multi-mode RCPSP (MRCPSP) (Elmaghraby S. 1977). In recent
years, a more general scheduling problem has been introduced that considers alternative
execution modes at the higher work package (WP) level in the project work breakdown
structure. In the remainder of this abstract, we will refer to this scheduling problem as the
RCPSP with alternative project structures. The most important feature of this problem
formulation is the incorporation of alternative execution modes in the scheduling phase.
These alternatives are necessary in order to model the uncertain project structure in future
stages or are preferred in order to overcome the complex and fast-changing project envi-
ronment. The objective of the research at hand is to construct a (near) optimal schedule
given the alternative project structure. Therefore, the discussed problem shows how to
leverage alternative project structures in order to tackle an uncertain project environment.
Several research efforts on scheduling with alternative structures have been conducted in
various research fields over the past decades (Kis T. 2003, Capacho et al. 2009, Capek et
al. 2012, Kellenbrink and Helber 2015, Vanhoucke and Coelho 2016).
The main contributions of our research are threefold. (1) The existing research efforts
on scheduling with alternative project structures have been developed largely indepen-
dent. Therefore, we propose a comprehensive classification framework to uniquely identify
and define different types of alternative project structures. (2) Since most of the existing
datasets for the proposed problem formulation are small-scale and randomly generated, we
construct a large dataset of artificial problem instance that supports the proposed frame-
work generated using RanGen 2 (Vanhoucke et al. 2008). (3) We develop a metaheuristic
213

solution approach that is tailored to the specific characteristics of the alternative project
structures in the discussed classification framework to solve the data instances in the new
dataset.
In this abstract, we discuss the RCPSP with alternative project structures which ex-
tends the basic RCPSP by defining alternative ways to execute a subset of interrelated
activities in the project. In order to model the alternative execution modes of the WPs,
we distinguish between fixed and alternative activities. Fixed activities should always be
present in the final project schedule and, consequently, the corresponding resource and
precedence constraints should always be satisfied. However, the presence of the alternative
activities in the final project schedule is optional and depends on the selected alternative
project structures. Consequently, the project scheduling problem consists of two subprob-
lems, i.e. the decision and the scheduling subproblem. The objective is to select for each
WP exactly one alternative execution mode such that the resulting precedence, resource
and logical feasible schedule has a minimal project makespan.
3
0
1
2
7
6 9
4
5
8
10
Fig. 1. Illustrative example of project network with alternative project structures
We illustrate the concept of alternative project structures based on the simple project
network (see figure 1) derived from Kellenbrink and Helber (2015). This example shows a
project network with 9 non-dummy activities (i.e. assume activities 0 and 10, respectively,
the dummy start and end activity) and default finish-to-start precedence relations with a
zero time lag. The symbol ’)’ in figure 1 indicates that a choice is triggered between multiple
alternative execution modes of a WP. Therefore, only one of the corresponding precedence
relations should be considered during project scheduling, e.g. a choice is triggered between
the (alternative) activities 1 and 2. The choice for activity 2 will subsequently cause the
implementation of either activity 4 or 5. Consequently, all non-dummy activities in this
example can be classified as alternative activities since their presence in the final schedule
is optional. Note that the choice for one alternative might enforce the implementation of an
activity that also belongs to another alternative. This is represented in figure 1 by means
of a dotted line, e.g. between activities 7 and 8.
3 Methodology
Based on the aforementioned terminology, we have constructed a classification matrix
to unambiguously define projects with alternative project structures based on the relative
number of alternative activities and the type of relations between the alternative activities.
214

Subsequently, we propose a tabu search (TS) procedure (Glover F. 1986) that is tailored to
the characteristics of the alternative project structure based on the presented classification
matrix. In this research, we first test various strategies for the initial solution generation.
Each strategy will assign a weight to the alternative execution modes based on the total
work content (TWC) or sum of durations (SOD), adjusted for the specifications of the
alternative project structure, in order to prioritise alternatives. The improvement proce-
dure of the TS consists of two components: a neighbourhood structure (NH) and a local
search (LS). Given the nature of the overall project scheduling problem, the proposed TS
alternately improves the scheduling and decision subproblem. Where the NH and LS for
the scheduling subproblem are based on best practices in literature, novel heuristic im-
provement techniques (i.e. NH and LS) are introduced for the decision subproblem. For
each of the strategies, the characteristics of the alternative project structures are used to
guide the search procedure to a high-quality final solution. An overview of the procedure is
given in figure 2. The main methodological contributions are threefold. First, the presented
procedure will not tackle both subproblems in a sequential way, rather in an integrated
way. Secondly, the strategies of the TS are adjusted to incorporate the characteristics of
the problem instances. Third, different variants of the building blocks of the TS, which
either focus on the scheduling or selection subproblem, are constructed.
Initial solution generation
Neighbourhood
structure
Local search
scheduling subproblem
Neighbourhood
structure
Local search
decision subproblem
Final solution
Problem instance
Position
in
classification
matrix
Fig. 2. Overview of TS procedure
The aim of the computational experiments is threefold. First, we compare the perfor-
mance of different strategies for the building blocks of the TS. Secondly, we validate the
solution quality, expressed as the overall best project makespan with a stopping criterion
of 5,000 generated schedules, obtained using the TS through a comparison with a multi-
start LS routine. Third, we quantify the impact of the characteristics of alternative project
structures on the solution quality. The computational experiments provide insights on the
above research questions.
1. The impact of the decision subproblem outperforms the impact of the scheduling sub-
problem on the solution quality.
2. The computational results show that the memory structure of the TS pays off for the
problem at hand as it outperforms the multi-start LS routine.
3. According to expectations, an increased relative number of alternatives significantly
improve the solution quality, while the complexity of the alternative project structure,
215

as expressed by the type of relations between the alternatives, has a negative impact
on the project makespan (see the preliminary results in table 1).
Degree of flexibility
Low Medium High
Degree of Low - -6.23 -9.45
complexity Medium 2.52 0.48 -0.63
High 7.06 5.97 4.71
Table 1. The impact of the degree of flexibility and complexity on the project makespan (%)
The extension of the problem formulation to incorporate other concepts discussed in the
literature on project scheduling with alternatives together with a comparison of the com-
putational results are part of future research.
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Vol. 25 (2), pp. 145-153.
216

A New Pre-Pro essing Pro edure for the Multi-Mode
Resour e-Constrained Proje t S heduling Problem
Christian Stür k
Helmut S hmidt University, Hamburg, Germany
hristian.stuer khsu-hh.de
Keywords: Multi-Mode Resour e-Constrained Proje t S heduling Problem (MRCPSP),
Pre-Pro essing Pro edure, Mode Redu tion, MMLIB.
1 Introdu tion
This paper presents a new pre-pro essing pro edure for the Multi-Mode Resour e-
Constrained Proje t S heduling Problem (MRCPSP). The obje tive is nding a feasible
s hedule with minimal makespan. The MRCPSP is a NP-hard problem. Even nding a
feasible mode assignment is NP- omplete if the instan e has more than one non-renewable
resour e (Kolis h and Drexl (1997)).
To redu e the number of variables of this NP-hard problem, dierent pre-pro essing
te hniques have been presented in the literature. The most ited one is the pro edure
of Spre her et al. (1997). It deletes ine ient modes as well as redundant non-renewable
resour es and has shown to be very ee tive on the PSPLIB instan es (Kolis h and Spre her
(1997)). The approa hes of Zhu et al. (1997) and Stür k and Gerhards (2018) are based
on al ulating new earliest starting times whi h redu e the number of variables in time-
indexed models. While the former approa h uses heuristi methods, the latter one uses
mathemati al programming.
Re ently, the new ben hmark data set MMLIB has been presented by Van Peteghem
and Vanhou ke (2014). These instan es are designed in su h way that the pro edure of
Spre her et al. (1997) does not have any impa t at all. As a result no pro edure exists
whi h is able to redu e the number of modes for the MMLIB instan es. Therefore, we
develop a te hnique whi h aims to redu e the number of modes of these instan es. It an
be used as a pre-pro essing pro edure and be embedded in a solution approa h.
2 Problem des ription
The MRCPSP onsists of a set of a tivities A = {0, ..., n + 1}. A tivity 0 and n + 1 a t
as dummy a tivities and denote the start and the end of the proje t. Ea h a tivity i an be
exe uted in a mode m out of the orresponding set of modes Mi. Pre eden e onstraints
E exist among some a tivities. Ea h a tivity has to be assigned to exa tly one mode and
one starting time while minimising the makespan.
Resour e restri tions have to be adhered. Therefore, a set of renewable resour es R
available per time unit is given. This resour e type replenishes on ea h time unit.
Furthermore, a set of non-renewable resour es Rn
exists whi h do not replenish. A ording
to the hosen mode m, a tivity i has a duration di,m ∈ Z+
as well as a resour e onsumption
ri,m,k ∈ Z+
for ea h resour e k ∈ R ∪ Rn
. A mode annot be hanged on e it is exe uted.
In time-indexed models (for example the one of Talbot (1982)) ea h a tivity i has an
earliest starting time ESi and a latest starting time LSi. Alternatively, earliest ompletion
times ECi and latest ompletion times LCi are used. The values of ESi and ECi an be
derived by using the riti al path method (CPM, Kelley (1963)). For the values of LSi and
217

LCi a feasible time horizon T of the proje t is needed. Based on T the values of LSi and
LCi an be omputed with ba kward re ursion.
3 A new pre-pro essing pro edure
The pre-pro essing pro edure that is presented in this work highly depends on the
quality of T . Therefore, it an only be used if the makespan of a feasible solution of an
instan e is known. The pro edure uses the relationship between LCi and T : the value of T
is set to the known makespan of the feasible solution. Then ea h LCi is omputed, starting
with LCn+1 = T .
With all ESi and LCi values known, ea h mode m of ea h a tivity i ∈ A an be
investigated whether the following ondition (1) holds:
LCi ≥ ESi + di,m . (1)
If an a tivity i′
is started at its earliest starting time ECi′ and the duration di′,m′ of a
mode m′
extends the latest ompletion time LCi′ , the time horizon T annot be rea hed any
more. Sin e the obje tive is minimising the makespan of the proje t, using mode m′
would
not be reasonable. If m′
is part of the mode ve tor it is impossible to rea h a makespan
whi h is equal or better than T . Thus, the optimal makespan an never be realised with
the hoi e of this mode. Therefore, these modes are alled non-optimal modes.
A similar idea of dis arding modes has already been proposed in the bran h-and-bound
pro edure of Spre her and Drexel (1998). The idea was used for trun ating the bran h-
and-bound tree. In ontrast, the presented pro edure has the advantage that it an be used
as pre-pro essing pro edure for exa t and heuristi methods.
An example of a non-optimal mode is given in Figure 1. Consider an a tivity i with the
ESi = 2 and LCi = 6. A tivity i has two modes. Using the rst mode, the a tivity has a
duration of di,1 = 2. Exe uted in the se ond mode the duration of i is di,2 = 5. While the
rst mode fulls ondition (1), the se ond mode does not. Exe uted at the earliest starting
time, it still extends the latest ompletion time for a tivity i. Identied as a non-optimal
mode, the se ond mode of a tivity i an be ex luded before the sear h for the optimal
solution starts.
d i,2 = 5
ES i = 2 LC i = 6 d i,m
Fig. 1. Example of a non-optimal mode
A ne essary ondition for the usage of this pre-pro essing pro edure is a known makespan
of a feasible solution whi h an be used for T . This an be a hieved in two ways:
1. Based on a known feasible solution. This ould be a solution from a previous sear h or
the best solution reported to a database.
2. Embedded into an algorithm or a heuristi ea h time a better solution is found.
The impa t of the proposed pro edure on the MMLIB is investigated in the following
se tion.
218

4 Experimental investigation
To test the ee tiveness of the presented pro edure it is applied to the MMLIB in-
stan es. The number of redu ed modes is used as a measure of ee tiveness due to the
fa t that the number of variables de reases with the redu tion of modes. The experiments
were arried out on a PC with an Intel Xeon X5650 CPU at 2.66 GHz. The algorithm is
implemented in C#.
First of all the pro edure of Spre her et al. (1997) was tested on the MMLIB instan es.
We an onrm the statement of Van Peteghem and Vanhou ke (2014) neither a mode
nor a resour e ould be deleted for any MMLIB instan e.
We then tested the presented pre-pro essing pro edure. The best known solutions
(BKS) reported on the data base www.mmlib.eu are used as T for ea h instan e. Based
on T all LCi values of an instan e were omputed. Then ea h mode of ea h a tivity was
tested whether modes exist that extend the latest ompletion time of an a tivity if this
a tivity is started at its earliest starting time. If an a tivity extends its latest ompletion
time in a mode, this mode was identied as non-optimal and therefore deleted.
Table 1. Non-optimal modes of the MMLIB after using the BKS as T
MMLIB50 MMLIB100 MMLIB+
Total number of instan es 540 540 3,240
Numberofinstan eswithatleastonenon-optimal mode 352 347 1,327
Average redu tion of non-optimal modes 17.69% 13.45% 13,52%
Maximal redu tion 44.00% 32.33% 55.78%
Table 1 shows the number of non-optimal modes that an be identied for the MMLIB
instan es. The omputational time is less than one se ond for ea h instan e. Using the best
known solutions of the data base a mode redu tion was possible for 65.19% / 64.26% /
40.96% of the MMLIB50 / MMLIB100 / MMLIB+ instan es, respe tively. For one instan e
of the MMLIB+ 55.78% of the given modes were identied as non-optimal. This leads to
a signi ant redu tion of variables for this instan e.
After the redu tion of the modes a feasible mode assignment was done with the re-
maining modes of the instan es. Using the MIP-based pro edure presented in Gerhards et
al. (in print) a feasible mode assignment was found for ea h instan e. Thus, the deletion
of non-optimal modes does not lead to infeasibility.
To emphasize the impa t of the pre-pro essing pro edure a MIP implementation of
the mathemati al model of Talbot (1982) was tested. IBM ILOG CPLEX 12.6.3 was used
as the mathemati al solver. The experiments were done for all instan es with a maximal
running time of 30 minutes per instan e. The time horizon T of the model was omputed
as the sum of the maximal duration of ea h a tivity. All ESi and LSi (based on T ) values
were omputed using CPM.
Table 2. Improved best known solutions
MMLIB50 MMLIB100 MMLIB+
MIP (Talbot (1982)) 2 2 9
MIP (Talbot (1982)) with pre-pro essing 2 2 23
219

Table 2 summarizes the number of improved best known solutions. In the rst run
the MIP was started without any further pre-pro essing. Although the MIP was not able
to nd a feasible solution for ea h instan e, it improves the best known solutions of 13
instan es. In the se ond run, the presented pre-pro essing pro edure was used. This led
to an additional improvement of the best known solutions for 27 instan es. This indi ates
the ee tiveness of the pro edure. Due to a smaller number of variables the MIP is able to
improve its solution quality. A more detailed overview of these experiments will be given
during the presentation at the onferen e. All improvements are reported to the database
www.mmlib.eu.
5 Con lusions
This work presents a pre-pro essing pro edure for the MRCPSP whi h is based on
a known makespan of a feasible solution. The omputational experiments show that the
pro edure an be integrated into a solution approa h and has a very short omputation
time. A mode redu tion is possible for 2,026 of the 4,320 MMLIB instan es.
To test the ee tiveness of the pro edure a MIP implementation of the mathemati al
model of Talbot (1982) was used. The rst run did not use the pre-pro essing pro edure.
Applying the pre-pro essing te hnique in the se ond run before starting the MIP lead to an
improvement of the best known solutions for 27 instan es. Thus, the presented te hnique
improves the solution approa h. A more detailed investigation of the instan es ontaining
non-optimal modes as well as a more detailed explanation of the omputational experiments
will be presented at the onferen e.
Referen es
Gerhards, P., Stür k, C. and Fink, A. in print, An Adaptive Large Neighborhood Sear h as a
Matheuristi for the Multi-Mode Resour e-Constrained Proje t S heduling Problem, Euro-
pean Journal of Industrial Engineering, to appear.
Kelley, J. E. 1963, The riti al-path method: Resour es planning and s heduling, Industrial
S heduling, Vol. 13, no. 1, pp. 347-365.
Kolis h, R. and Drexl, A. 1997, Lo al sear h for nonpreemptive multi-mode resour e- onstrained
proje t s heduling, IIE Transa tions, Vol. 29, no. 11, pp. 987-999.
Kolis h, R. and Spre her, A. 1997, PSPLIB a proje t s heduling problem library: OR software
ORSEP operations resear h software ex hange program, European Journal of Operational
Resear h, Vol. 96, no. 1, pp. 205-216.
Spre her, A. and Drexl, A. 1998, Multi-mode resour e- onstrained proje t s heduling by a simple,
general and powerful sequen ing algorithm, European Journal of Operational Resear h, Vol.
107, no. 1, pp. 431-450.
Spre her, A., Hartmann, S. and Drexl, A. 1997, An exa t algorithm for proje t s heduling with
multiple modes, Operations-Resear h-Spektrum, Vol. 19, no. 3, pp. 195-203.
Stür k, C. and Gerhards, P. 2018, Providing Lower Bounds for the Multi-Mode Resour e-
Constrained Proje t S heduling Problem, In: Operations Resear h Pro eedings 2016 eds:
Fink, A., Fügens huh, A. and Geiger, M. J., pp. 551-557, Springer(Cham).
Talbot, F. B. 1982, Resour e- onstrained proje t s heduling with time-resour e tradeos: The
nonpreemptive ase, Management S ien e, Vol. 28, no. 10, pp. 1197-1210.
Van Peteghem, V. and Vanhou ke, M. 2014, An experimental investigation of metaheuristi s for
the multi-mode resour e- onstrained proje t s heduling problem on new dataset instan es,
European Journal of Operational Resear h, Vol. 235, no. 1, pp. 62-72.
Zhu, G., Bard, J. F. and Yu, G. 2006, A bran h-and- ut pro edure for the multimode resour e-
onstrained proje t-s heduling problem, INFORMS Journal on Computing, Vol. 18, no. 3,
pp. 377-390.
220

❆♥ O∗
(1.41n
)✲t✐♠❡ ❛❧❣♦r✐t❤♠ ❢♦r ❛ s✐♥❣❧❡ ♠❛❝❤✐♥❡
❥✉st✲✐♥✲t✐♠❡ s❝❤❡❞✉❧✐♥❣ ♣r♦❜❧❡♠ ✇✐t❤ ❝♦♠♠♦♥ ❞✉❡ ❞❛t❡
❛♥❞ s②♠♠❡tr✐❝ ✇❡✐❣❤ts
❱✐♥❝❡♥t ❚✬❦✐♥❞t
1
❛♥❞ ▲❡✐ ❙❤❛♥❣
1
❛♥❞ ❋❡❞❡r✐❝♦ ❉❡❧❧❛ ❈r♦❝❡
2
1
❯♥✐✈❡rs✐t② ♦❢ ❚♦✉rs✱ ▲❛❜♦r❛t♦r② ♦❢ ❈♦♠♣✉t❡r ❙❝✐❡♥❝❡ ✭❊❆ ✻✸✵✵✮✱ ❊❘▲ ❈◆❘❙ ✻✸✵✺✱ ✻✹ ❛✈❡♥✉❡
❏❡❛♥ P♦rt❛❧✐s✱ ✸✼✷✵✵ ❚♦✉rs✱ ❋r❛♥❝❡
④t❦✐♥❞t✱❧❡✐✳s❤❛♥❣⑥❅✉♥✐✈✲t♦✉rs✳❢r
2
P♦❧✐t❡❝♥✐❝♦ ❞✐ ❚♦r✐♥♦✱ ❉■●❊P✱ ❈♦rs♦ ❉✉❝❛ ❞❡❣❧✐ ❆❜r✉③③✐ ✷✹✱ ✶✵✶✷✾ ❚♦r✐♥♦✱ ■t❛❧②
❢❡❞❡r✐❝♦✳❞❡❧❧❛❝r♦❝❡❅♣♦❧✐t♦✳✐t
❑❡②✇♦r❞s✿ ❙✐♥❣❧❡ ♠❛❝❤✐♥❡✱ ❏✉st✲✐♥✲❚✐♠❡✱ ❊①♣♦♥❡♥t✐❛❧✲t✐♠❡ ❛❧❣♦r✐t❤♠✳
✶ Pr♦❜❧❡♠ st❛t❡♠❡♥t ❛♥❞ ❧✐t❡r❛t✉r❡ r❡✈✐❡✇
■♥ t❤✐s ♣❛♣❡r ✇❡ r❡✈✐s✐t ❛ ✇❡❧❧✲❦♥♦✇♥ ❥✉st✲✐♥✲t✐♠❡ s❝❤❡❞✉❧✐♥❣ ♣r♦❜❧❡♠ ✉♥❞❡r t❤❡ ❧✐❣❤t
♦❢ ❡①♣♦♥❡♥t✐❛❧✲t✐♠❡ ❛❧❣♦r✐t❤♠s✳ ❲❡ ❛r❡ ❣✐✈❡♥ ❛ s❡t ♦❢ n ❥♦❜s✱ ❡❛❝❤ ❥♦❜ i ❜❡✐♥❣ ❞❡✜♥❡❞ ❜② ❛
♣r♦❝❡ss✐♥❣ t✐♠❡ pi ❛♥❞ ❛ ✇❡✐❣❤t wi✱ t❤❡ ❧❛tt❡r r❡✢❡❝t✐♥❣ t❤❡ ♣❡♥❛❧t② ✐♥❞✉❝❡❞ ❜② s❝❤❡❞✉❧✐♥❣
✐t ❡❛r❧② ♦r t❛r❞②✳ ❆❧❧ ❥♦❜s s❤❛r❡ t❤❡ s❛♠❡✱ ♥♦♥ r❡str✐❝t✐✈❡✱ ❝♦♠♠♦♥ ❞✉❡ ❞❛t❡ d ≥
P
i pi✳
❚❤❡ ♦❜❥❡❝t✐✈❡ ✐s t♦ ✜♥❞ ❛ s❝❤❡❞✉❧❡ s ♦❢ ❥♦❜s s✉❝❤ t❤❛t
P
i wi(Ei(s) + Ti(s)) ✐s ♠✐♥✐♠✐③❡❞✱
✇✐t❤ Ti(s) = max(Ci(s) − d; 0) ❛♥❞ Ei(s) = max(d − Ci(s); 0)✳ ◆♦t✐❝❡ t❤❛t t❤❡ ♠❡♥t✐♦♥ ♦❢
s❝❤❡❞✉❧❡ s ♠❛② ❜❡ ♦♠✐tt❡❞ ✇❤❡♥❡✈❡r t❤❡r❡ ✐s ♥♦ ❛♠❜✐❣✉✐t②✳ ❋♦❧❧♦✇✐♥❣ t❤❡ st❛♥❞❛r❞ t❤r❡❡✲
✜❡❧❞s ♥♦t❛t✐♦♥✱ t❤✐s ♣r♦❜❧❡♠ ❝❛♥ ❜❡ r❡❢❡rr❡❞ t♦ ❛s 1|di = d ≥
P
i pi|
P
i wi(Ei + Ti)✳
❚❤✐s ♣r♦❜❧❡♠ ❤❛s ❜❡❡♥ s❤♦✇♥ t♦ ❜❡ NP✲❤❛r❞ ❜② ❍❛❧❧ ❛♥❞ P♦s♥❡r ✭✶✾✾✶✮ ❛♥❞ ❛ ❝♦♠✲
♣r❡❤❡♥s✐✈❡ s✉r✈❡② ♦❢ r❡❧❛t❡❞ ♣r♦❜❧❡♠s ❝❛♥ ❜❡ ❢♦✉♥❞ ✐♥ ✭❚✬❦✐♥❞t ❛♥❞ ❇✐❧❧❛✉t ✷✵✵✻✮ ❛♥❞
✭❏♦③❡❢♦✇s❦❛ ✷✵✵✼✮✳ ■♥t❡r❡st✐♥❣❧②✱ s❡✈❡r❛❧ r❡♠❛r❦❛❜❧❡ ♣r♦♣❡rt✐❡s✱ s✉♠♠❛r✐③❡❞ ✐♥ Pr♦♣❡rt②
✶✱ ❤❛✈❡ ❜❡❡♥ ❡st❛❜❧✐s❤❡❞ ❛❧♦♥❣ t❤❡ ②❡❛rs ♦♥ t❤❛t ♣r♦❜❧❡♠✳ ❚❤❡② ♥♦t❛❜❧② ✐♥❞✉❝❡ t❤❛t t❤❡
❤❛r❞♥❡ss ♦❢ t❤❡ ♣r♦❜❧❡♠ ❝♦♠❡s ❢r♦♠ ❞❡❝✐❞✐♥❣ ❢♦r ❡❛❝❤ ❥♦❜ ✐❢ ✐t ✐s ❜❡tt❡r t♦ s❝❤❡❞✉❧❡ ✐t ❡❛r❧②
♦r t❛r❞②✳
Pr♦♣❡rt② ✶✳ ❚❤❡r❡ ❡①✐sts ♦♣t✐♠❛❧ s♦❧✉t✐♦♥s t♦ t❤❡ 1|di = d ≥
P
i pi|
P
i wi(Ei +Ti) ♣r♦❜❧❡♠
s❛t✐s❢②✐♥❣✿
✶✳ t❤❡r❡ ❛r❡ ♥♦ ♠❛❝❤✐♥❡ ✐❞❧❡ t✐♠❡s ❜❡t✇❡❡♥ t✇♦ ❝♦♥s❡❝✉t✐✈❡ ❥♦❜s✱
✷✳ t❤❡ ✜rst s❝❤❡❞✉❧❡❞ ❥♦❜ ❝❛♥ st❛rt ❛t ❛ t✐♠❡ ❣r❡❛t❡r t❤❛♥ ✵✱
✸✳ t❤❡r❡ ❡①✐sts ❛ ❥♦❜ ✇❤✐❝❤ ❡①❛❝t❧② ❝♦♠♣❧❡t❡s ❛t t✐♠❡ d✱
✹✳ t❤❡ ❝❧❛ss ♦❢ ❱✲s❤❛♣❡ s❝❤❡❞✉❧❡s ✐s ❞♦♠✐♥❛♥t✱ ✐✳❡✳ ❛❧❧ ❡❛r❧② ❥♦❜s ❛r❡ s❡q✉❡♥❝❡❞ ❜② ❞❡❝r❡❛s✲
✐♥❣ ✈❛❧✉❡ ♦❢ t❤❡ r❛t✐♦
pi
wi
✭❲▲P❚ r✉❧❡✮ ✇❤✐❧❡ ❛❧❧ t❛r❞② ❥♦❜s ❛r❡ s❡q✉❡♥❝❡❞ ❜② ✐♥❝r❡❛s✐♥❣
✈❛❧✉❡ ♦❢ t❤❡ r❛t✐♦
pi
wi
✭❲❙P❚ r✉❧❡✮✳
❙❡✈❡r❛❧ ❡①❛❝t ❛❧❣♦r✐t❤♠s ❤❛✈❡ ❜❡❡♥ ♣r♦♣♦s❡❞ t♦ s♦❧✈❡ ✐t ❜✉t ✇✐t❤ t❤❡ ❢♦❝✉s ♦❢ ♣r♦♣♦s✐♥❣
❡✣❝✐❡♥t ❛❧❣♦r✐t❤♠s ♦♥ r❛♥❞♦♠❧② ❣❡♥❡r❛t❡❞ ✐♥st❛♥❝❡s✳ ■♥ t❤✐s ✇♦r❦✱ ✇❡ ❢♦❝✉s ♦♥ ❛ ♠♦r❡
t❤❡♦r❡t✐❝❛❧ ❛♣♣r♦❛❝❤ ✇❤✐❝❤ ❝♦♥s✐sts ✐♥ ❞❡s✐❣♥✐♥❣ ❡①❛❝t ❛❧❣♦r✐t❤♠s ❢♦r ✇❤✐❝❤ ❛ ✏❣♦♦❞ ❜❡✲
❤❛✈✐♦✉r✑ ✐♥ t❤❡ ✇♦rst✲❝❛s❡ ✐s s♦✉❣❤t✳ ❚❤✐s r❡❧❛t❡s t♦ t❤❡ ❛r❡❛ ♦❢ ❡①♣♦♥❡♥t✐❛❧✲t✐♠❡ ❛❧❣♦r✐t❤♠s
❛♥❞ t❤❡ r❡❛❞❡r ✐s r❡❢❡rr❡❞ t♦ t❤❡ ❜♦♦❦ ♦❢ ❋♦♠✐♥ ❛♥❞ ❑r❛ts❝❤ ✭✷✵✶✵✮ ❢♦r ❛ ❣♦♦❞ ✐♥tr♦❞✉❝t✐♦♥✳
❲❤❡♥ ❞❡❛❧✐♥❣ ✇✐t❤ ✇♦rst✲❝❛s❡ ❝♦♠♣❧❡①✐t✐❡s ♦❢ ❡①♣♦♥❡♥t✐❛❧ ❛❧❣♦r✐t❤♠s✱ ✇❡ ✉s✉❛❧❧② ♠❛❦❡ ✉s❡
♦❢ t❤❡ O∗
() ♥♦t❛t✐♦♥✿ ❧❡t T(·) ❜❡ ❛ s✉♣❡r♣♦❧②♥♦♠✐❛❧ ❛♥❞ p(·) ❜❡ ❛ ♣♦❧②♥♦♠✐❛❧✱ ❜♦t❤ ♦♥ t❤❡
✐♥st❛♥❝❡ s✐③❡ ✭✉s✉❛❧❧② t❤❡ ♥✉♠❜❡r ♦❢ ❥♦❜s n ❢♦r s❝❤❡❞✉❧✐♥❣✮✳ ❚❤❡♥✱ ✇❡ ❡①♣r❡ss r✉♥♥✐♥❣✲t✐♠❡
❜♦✉♥❞s ♦❢ t❤❡ ❢♦r♠ O(p(n) · T(n))) ❛s O∗
(T(n))✳ ❲✐t❤ r❡s♣❡❝t t♦ s❝❤❡❞✉❧✐♥❣ ❧✐t❡r❛t✉r❡✱
221

▲❡♥té ❡t✳ ❛❧✳ ✭✷✵✶✹✮ ♣r♦♣♦s❡❞ ❛♥ ✐♥tr♦❞✉❝t✐♦♥ ❛♥❞ ❛ ✜rst r❡✈✐❡✇ ♦❢ ❡①✐st✐♥❣ ❡①♣♦♥❡♥t✐❛❧
❛❧❣♦r✐t❤♠s✳ ❆❞❞✐t✐♦♥❛❧ ✇♦r❦s ❤❛✈❡ ❜❡❡♥ ♣✉❜❧✐s❤❡❞ ❜② ❈②❣❛♥ ❡t✳ ❛❧✳ ✭✷✵✶✶✮✱ ▲❡♥té ❡t✳ ❛❧✳
✭✷✵✶✸✮✱ ●❛rr❛✛❛ ❡t✳ ❛❧✳ ✭✷✵✶✼✮✱ ❙❤❛♥❣ ❡t✳ ❛❧✳ ✭✷✵✶✼❛✮ ❛♥❞ ❙❤❛♥❣ ❡t✳ ❛❧✳ ✭✷✵✶✼❜✮✳ ❋♦r t❤❡
1|di = d ≥
P
i pi|
P
i wi(Ei + Ti) ♣r♦❜❧❡♠ ✐t ✐s ❝❧❡❛r t❤❛t ✐ts ♦♣t✐♠❛❧ s♦❧✉t✐♦♥ ❝❛♥ ❜❡ ❝♦♠✲
♣✉t❡❞ ❜② ♠❡❛♥s ♦❢ ❛ ❜r✉t❡✲❢♦r❝❡ ❛❧❣♦r✐t❤♠ ✇❤✐❝❤ ❡♥✉♠❡r❛t❡s ❛❧❧ ♣♦ss✐❜❧❡ ❛ss✐❣♥♠❡♥ts ♦❢
❥♦❜s t♦ t❤❡ s❡ts ♦❢ ❡❛r❧② ❛♥❞ t❛r❞② ❥♦❜s✳ ❚❤❡♥✱ ❡❛❝❤ s❡t ❝❛♥ ❜❡ s♦rt❡❞ ✐♥ ♣♦❧②♥♦♠✐❛❧ t✐♠❡ ❜②
♠❡❛♥s ♦❢ ❡✐t❤❡r ❲❙P❚ ♦r ❲▲P❚ r✉❧❡s✿ t❤❡ ❧❛st ❡❛r❧② ❥♦❜ t❤❡♥ ❝♦♠♣❧❡t❡s ❛t t✐♠❡ d ✇❤✐❧❡
t❤❡ ✜rst t❛r❞② ❥♦❜s st❛rts ❛t t✐♠❡ d✳ ❚❤✐s ❛❧❣♦r✐t❤♠ r❡q✉✐r❡s O∗
(2n
) t✐♠❡ ❛♥❞ ♣♦❧②♥♦♠✐❛❧
s♣❛❝❡ ✐♥ t❤❡ ✇♦rst✲❝❛s❡✳ ❈♦♥s❡q✉❡♥t❧②✱ ✐t ❜❡❝♦♠❡s ♦❢ ✐♥t❡r❡st t♦ s❡❛r❝❤ ❢♦r ❛♥ ❡①❛❝t ❛❧❣♦✲
r✐t❤♠ t❤❛t ✇♦✉❧❞ ❜❡ ♦❢ ❛ ❧♦✇❡r ✇♦rst✲❝❛s❡ t✐♠❡ ❝♦♠♣❧❡①✐t②✳ ❲❡ s❤♦✇ ✐♥ t❤❡ ♥❡①t s❡❝t✐♦♥
t❤❛t ✐t ✐s ♣♦ss✐❜❧❡ t♦ s♦❧✈❡ t❤❡ 1|di = d ≥
P
i pi|
P
i wi(Ei + Ti) ♣r♦❜❧❡♠ ✐♥ O∗
(2
n
2 ) t✐♠❡
❛♥❞ s♣❛❝❡✳
✷ ❆ ❙♦rt ✫ ❙❡❛r❝❤ ❛❧❣♦r✐t❤♠
❆♠♦♥❣ t❤❡ ❦♥♦✇♥ t❡❝❤♥✐q✉❡s t♦ ❞❡r✐✈❡ ❡①♣♦♥❡♥t✐❛❧✲t✐♠❡ ❛❧❣♦r✐t❤♠s ✭❋♦♠✐♥ ❛♥❞ ❑r❛ts❝❤
✷✵✶✵✮✱ t❤❡r❡ ✐s ❙♦rt ✫ ❙❡❛r❝❤ ✐♥✐t✐❛❧❧② ♣r♦♣♦s❡❞ ❜② ❍♦r♦✇✐t③ ❛♥❞ ❙❛❤♥✐ ✭✶✾✼✹✮ t♦ s♦❧✈❡ t❤❡
❦♥❛♣s❛❝❦ ♣r♦❜❧❡♠ ✐♥ O∗
(2
n
2 ) t✐♠❡ ❛♥❞ s♣❛❝❡✳ ▲❛t❡r ♦♥✱ t❤✐s ♠❡t❤♦❞ ❤❛s ❜❡❡♥ ❡①t❡♥❞❡❞
t♦ s♦❧✈❡ ♠✉❧t✐♣❧❡ ❝♦♥str❛✐♥ts ♣r♦❜❧❡♠s ❜② ▲❡♥té ❡t✳ ❛❧✳ ✭✷✵✶✸✮ ✇❤♦ ❛❧s♦ ❛♣♣❧✐❡❞ ✐t t♦ ❛ s❡t
♦❢ s❝❤❡❞✉❧✐♥❣ ♣r♦❜❧❡♠s✳ ❘♦✉❣❤❧② s♣❡❛❦✐♥❣✱ ✐t ❝♦♥s✐sts ✐♥ s❡♣❛r❛t✐♥❣ ❛♥ ✐♥♣✉t ✐♥st❛♥❝❡ ✐♥t♦
t✇♦ ❡q✉❛❧✲s✐③❡ ✐♥st❛♥❝❡s✱ t❤❡♥ ✐♥ ❡♥✉♠❡r❛t✐♥❣ ❛❧❧ ♣❛rt✐❛❧ s♦❧✉t✐♦♥s ❢♦r ❡❛❝❤ s✉❜✲✐♥st❛♥❝❡
❛♥❞ t❤❡♥ ✜♥❞ t❤❡ ♦♣t✐♠❛❧ s♦❧✉t✐♦♥ ♦❢ t❤❡ ✐♥♣✉t ✐♥st❛♥❝❡ ❜② r❡❝♦♠❜✐♥✐♥❣ ✐♥ ❛ s✉✐t❛❜❧❡ ✇❛②
❛❧❧ t❤♦s❡ ♣❛rt✐❛❧ s♦❧✉t✐♦♥s t❛❦✐♥❣ ❡❛❝❤ t✐♠❡ ♦♥❡ ❢r♦♠ ❡❛❝❤ s✉❜✲✐♥st❛♥❝❡✳ ❚❤✐s ✏❝♦♠♣❧❡①✐t②
❜r❡❛❦✐♥❣✑ ✐s ❞♦♥❡ ❛t t❤❡ ❞❡tr✐♠❡♥t ♦❢ t❤❡ s♣❛❝❡ ❝♦♠♣❧❡①✐t② ✇❤✐❝❤ t✉r♥s t♦ ❜❡ ❡①♣♦♥❡♥t✐❛❧✳
❲✐t❤♦✉t ❧♦ss ♦❢ ❣❡♥❡r❛❧✐t②✱ ❛ss✉♠❡ t❤❛t n ✐s ❡✈❡♥ ❛♥❞ t❤❛t ❥♦❜s ❛r❡ ✐♥❞❡①❡❞ s✉❝❤ t❤❛t
p1
w1
≤ p2
w2
≤ . . . ≤ pn
wn
✳ ■♥ t❤❡ r❡♠❛✐♥❞❡r ✇❡ ✐♠♣❧✐❝✐t❧② ♠❛❦❡ ✉s❡ ♦❢ t❤❡ r❡s✉❧ts ✐♥ Pr♦♣✲
❡rt② ✶ t♦ ❡❧❛❜♦r❛t❡ ♦✉r ❙♦rt ✫ ❙❡❛r❝❤ ❛❧❣♦r✐t❤♠✳ ❋♦r ❛♥② ❣✐✈❡♥ ✐♥st❛♥❝❡ I ♦❢ n ❥♦❜s✱
❧❡t ❜❡ I1 = {1, . . . , n
2 } ❛♥❞ I2 = {n
2 + 1, . . . , n} ❛ ❞❡❝♦♠♣♦s✐t✐♦♥ ✐♥t♦ t✇♦ ❡q✉❛❧✲s✐③❡ s✉❜✲
✐♥st❛♥❝❡s✳ ❇② ❡♥✉♠❡r❛t✐♦♥✱ ❞♦♥❡ ✐♥ O∗
(2
n
2 ) t✐♠❡✱ ✇❡ ❝❛♥ ❜✉✐❧❞ s❡t S1 = {s1
j /j = 1, . . . , 2|I1|
}
✭r❡s♣✳ S2 = {s2
k/k = 1, . . . , 2|I2|
}✮ ✇❤✐❝❤ ✐s t❤❡ s❡t ♦❢ ❛❧❧ ♣♦ss✐❜❧❡ s♦❧✉t✐♦♥s ❜✉✐❧t ❢r♦♠ s✉❜✲
✐♥st❛♥❝❡ I1 ✭r❡s♣✳ I2✮✳ ❲❡ ❤❛✈❡ |S1| = |S2| = 2
n
2 ✳ ❋✐❣✉r❡ ✶ s❤♦✇s✱ ❢♦r ❛♥ ✐♥st❛♥❝❡ I✱ ❛
❝♦♠♣❧❡t❡ s❝❤❡❞✉❧❡ s = s1
j //s2
k ❞❡❝♦♠♣♦s❡❞ ✐♥t♦ t✇♦ ♣❛rt✐❛❧ s♦❧✉t✐♦♥s s1
j = {ǫ1
j ; τ1
j } ∈ S1
❛♥❞ s2
k = {ǫ2
k; τ2
k } ∈ S2✱ ✇✐t❤ ǫy
x ✭r❡s♣✳ τy
x ✮ r❡❢❡rr✐♥❣ t♦ ❛ s❝❤❡❞✉❧❡ ♦❢ ❡❛r❧② ❥♦❜s ✭r❡s♣✳ t❛r❞②
❥♦❜s✮✳ ❇❡s✐❞❡s✱ tb
j r❡❢❡rs t♦ t❤❡ ❝♦♠♣❧❡t✐♦♥ t✐♠❡ ♦❢ t❤❡ ❧❛st ❥♦❜ ✐♥ ǫ2
k ✇❤✐❧❡ tf
j r❡❢❡rs t♦ t❤❡
st❛rt✐♥❣ t✐♠❡ ♦❢ t❤❡ ✜rst ❥♦❜ ✐♥ τ2
k ✳
d
τ1
j
ǫ1
j
tb
j tf
j
ǫ2
k τ2
k
❋✐❣✳ ✶✳ ❉❡❝♦♠♣♦s✐t✐♦♥ ♦❢ ❛ ❝♦♠♣❧❡t❡ s❝❤❡❞✉❧❡ ✐♥t♦ t✇♦ s✉❜✲s❝❤❡❞✉❧❡s s1
j ❛♥❞ s2
k
❲❡ ❝❛♥ s❤♦✇ t❤❡ ❢♦❧❧♦✇✐♥❣ r❡s✉❧t✳
Pr♦♣♦s✐t✐♦♥ ✶✳ ▲❡t s = s1
j //s2
k ❜❡ ❛ ❝♦♠♣❧❡t❡ s❝❤❡❞✉❧❡✱ ❛♥❞ ❧❡t fjk =
P
i∈s wi(Ei(s) +
Ti(s)) ❜❡ t❤❡ ✈❛❧✉❡ ♦❢ t❤❡ ♦❜❥❡❝t✐✈❡ ❢✉♥❝t✐♦♥ ❢♦r s❝❤❡❞✉❧❡ s✳ ❲❡ ❤❛✈❡✿
222

fjk = fj + ck + aktb
j✱
✇✐t❤ fj =
P
i∈s1
j
wi(Ei(s1
j ) + Ti(s1
j ))✱ ck =
P
i∈s2
k
wi(Ei(s2
k) + Ti(s2
k)) + d(
P
i∈ǫ2
k
wi −
P
i∈τ2
k
wi) +
P
i∈τ2
k
wi
P
i∈I1
pi✱ ❛♥❞ ak =
P
i∈τ2
k
wi −
P
i∈ǫ2
k
wi✳ ◆♦t✐❝❡ t❤❛t fj ❛♥❞ tb
j ❛r❡
♦♥❧② ❞❡♣❡♥❞❡♥t ♦♥ s1
j ✱ ✇❤✐❧❡ ck ❛♥❞ ak ❛r❡ ♦♥❧② ❞❡♣❡♥❞❡♥t ♦♥ s2
k✳
❋♦r ❛♥② s1
j ♣❛rt✐❛❧ s❝❤❡❞✉❧❡ fj ❛♥❞ tb
j ❝❛♥ ❜❡ ❝♦♠♣✉t❡❞ ✐♥ O(n) t✐♠❡✱ ✇❤✐❝❤ ✐s ❛❧s♦
t❤❡ ❝❛s❡ ❢♦r ck ❛♥❞ ak ✇❤❡♥❡✈❡r ♣❛rt✐❛❧ s❝❤❡❞✉❧❡ s2
k ✐s ❣✐✈❡♥✳ ■♥ t❤❡ r❡♠❛✐♥❞❡r ✇❡ ❛ss✉♠❡
t❤❛t t❤❡s❡ ✈❛❧✉❡s ❛r❡ ❝♦♠♣✉t❡❞ ✇❤❡♥ ❜✉✐❧❞✐♥❣ s❡ts S1 ❛♥❞ S2 ✇❤✐❝❤ ❞♦❡s ♥♦t ❛✛❡❝t t❤❡
O∗
(2
n
2 ) t✐♠❡ ❝♦♠♣❧❡①✐t② r❡q✉✐r❡❞ ❜② t❤❡ ❙♦rt ✫ ❙❡❛r❝❤ ❛❧❣♦r✐t❤♠ t♦ ❜✉✐❧❞ t❤❡s❡ s❡ts✳ ❚❤✐s
❛❧❣♦r✐t❤♠ t❤❡♥ ♣r♦❝❡❡❞s ❜② s♦rt✐♥❣ s❡t S1 ✐♥ O∗
(2
n
2 ) t✐♠❡✱ s♦ t❤❛t ❢♦r ❛♥② s1
j ❛♥❞ s1
j+1 ✇❡
❤❛✈❡ tb
j ≤ tb
j+1✳ ❋♦r ❛ ❣✐✈❡♥ ♣❛rt✐❛❧ s❝❤❡❞✉❧❡ s1
j ∈ S1✱ st❛rt✐♥❣ ❢r♦♠ j = 1 t♦ j = |S1|✱ t❤❡
❛❧❣♦r✐t❤♠ ♥❡❡❞s t♦ ✜♥❞ ❛ s❝❤❡❞✉❧❡ s2
k ∈ S2 s✉❝❤ t❤❛t fjk ✐s ♠✐♥✐♠✉♠✳ ❚❤❡ ♦♣t✐♠❛❧ s♦❧✉t✐♦♥
❛ss♦❝✐❛t❡❞ t♦ ✐♥st❛♥❝❡ I ✐s t❤❡♥ ❣✐✈❡♥ ❛s t❤❡ ❜❡st ❝♦♠♣❧❡t❡ s♦❧✉t✐♦♥ ♦❜t❛✐♥❡❞✳ ◆♦✇✱ ❧❡t ✉s
t✉r♥ t♦ t❤❡ s❡❛r❝❤ ♦❢ t❤❡ ❜❡st s❝❤❡❞✉❧❡ s2
k ✇❤❡♥ s1
j ✐s ✜①❡❞✳
❲❡ s❡♣❛r❛t❡ s❡t S2 ✐♥t♦ s✉❜✲s❡ts S+
2 = {s2
k ∈ S2/ak ≥ 0} ❛♥❞ S−
2 = {s2
k ∈ S2/ak 0} ❛♥❞
t❤❡ s❡❛r❝❤ ❢♦r t❤❡ ❜❡st ♣❛rt✐❛❧ s❝❤❡❞✉❧❡ s2
k ❝♦♠♣❧❡♠❡♥t✐♥❣ s1
j ✐s ❞♦♥❡ ✜rst ✐♥ S+
2 ❛♥❞ ♥❡①t ✐♥
S−
2 ✳ ■♥ t❤✐s ❛❜str❛❝t✱ ✇❡ ♦♥❧② ❞❡t❛✐❧ ❤♦✇ t❤❡ s❡❛r❝❤ ✐♥ S+
2 ❝❛♥ ❜❡ ❞♦♥❡ ✐♥ O∗
(2
n
2 ) t✐♠❡ ❛♥❞
✇❡ ❝❧❛✐♠ t❤❛t t❤❡ s❛♠❡ r❡s✉❧t ❤♦❧❞s ❢♦r s❡❛r❝❤✐♥❣ ✐♥ S−
2 ✳ ❇❡❢♦r❡ ❞♦✐♥❣ t❤❡ s❡❛r❝❤ ✐♥ S+
2 ✇❤❡♥
s1
j ✐s ✜①❡❞✱ ❛♥ ❡①tr❛ ♣r❡♣r♦❝❡ss✐♥❣ ♦♥ S+
2 ✐s ❞♦♥❡✳ ❲❡ ❦♥♦✇ t❤❛t t❤❡ ❧♦✇❡st ♣♦ss✐❜❧❡ ✈❛❧✉❡
♦❢ tb
j ✐s ❡q✉❛❧ t♦ (d−
P
i∈I1
pi)✱ s♦ ❧❡t ♣❛rt✐❛❧ s❡q✉❡♥❝❡s s2
k ∈ S+
2 ❜❡ r❡✲✐♥❞❡①❡❞ ❜② ✐♥❝r❡❛s✐♥❣
✈❛❧✉❡s ♦❢ αk = (ck +ak(d−
P
i∈I1
pi)) =
P
i∈s2
k
wi(Ei(s2
k)+Ti(s2
k))+
P
i∈ǫ2
k
wi
P
i∈I1
pi✳ ❲❡
❛❧s♦ r❡♠♦✈❡ ❛❧❧ s2
k s✉❝❤ t❤❛t αk ≥ αk−1 ❛♥❞ ak ≥ ak−1✳ ❚❤✐s ❝❛♥ ❜❡ ❞♦♥❡✱ ✐♥❞❡♣❡♥❞❡♥t❧②
❢r♦♠ s1
j ✱ ✐♥ O∗
(2
n
2 ) t✐♠❡✳ ❇② t❤❡ ✇❛②✱ ❛❧❧ ❝♦♥tr✐❜✉t✐♦♥s (fjk − fj) ♦❢ s2
k ∈ S+
2 ❞❡♣❡♥❞ ♦♥
tb
j ❛s ♣✐❝t✉r❡❞ ✐♥ ❋✐❣✉r❡ ✷✳ ❇② ❛ ❞❡❞✐❝❛t❡❞ ❛❧❣♦r✐t❤♠ ✭♥♦t ♣r❡s❡♥t❡❞ ❤❡r❡✮ ✐t ✐s ♣♦ss✐❜❧❡ t♦
❝♦♠♣✉t❡ ❝♦✉♣❧❡s (Tℓ, s2
kℓ
) ✐♥ O∗
(2
n
2 ) t✐♠❡✱ ✇✐t❤ t❤❡ ♠❡❛♥✐♥❣ t❤❛t ✇❤❡♥❡✈❡r tb
j ∈ [Tℓ; Tℓ+1[✱
♣❛rt✐❛❧ s❝❤❡❞✉❧❡ s2
kℓ
❧❡❛❞s t♦ t❤❡ ❝♦♠♣❧❡t❡ s❝❤❡❞✉❧❡ s = s1
j //s2
kℓ
✇✐t❤ ♠✐♥✐♠✉♠ ❝♦st✳ ■♥
t❤❡ ✇♦rst✲❝❛s❡ s❝❡♥❛r✐♦ t❤❡r❡ ❛r❡ O(2
n
2 ) ❝♦✉♣❧❡s✱ ❜✉t ✐♥ ♣r❛❝t✐❝❡ t❤❡r❡ ❝❛♥ ❜❡ ❧❡ss ❝♦✉♣❧❡s✱
❞❡♣❡♥❞✐♥❣ ♦♥ t❤❡ ck✬s ❛♥❞ t❤❡ ak✬s✳ ❙❡❛r❝❤✐♥❣ ✐♥ S+
2 ✐s t❤❡♥ ❡q✉✐✈❛❧❡♥t t♦ s❡❛r❝❤ ✐♥ ❛ ❧✐st
♦❢ ❝♦✉♣❧❡s (Tℓ, s2
kℓ
) ✇❤✐❝❤ ✐s ❛ss✉♠❡❞ t♦ ❜❡ s♦rt❡❞ ❜② ✐♥❝r❡❛s✐♥❣ ✈❛❧✉❡s ♦❢ Tℓ✳ ❚❤❡ s❡❛r❝❤ ✐♥
t❤✐s ❧✐st ❝❛♥ ❜❡ ❞♦♥❡✱ ❢♦r ❛ ❣✐✈❡♥ s1
j ∈ S1✱ ❝❛♥ ❜❡ ❞♦♥❡ ✐♥ O(log(|S+
2 |)) = O(n) t✐♠❡✳ ❚❤❡♥✱
✜♥❞✐♥❣ t❤❡ ❜❡st ♣❛rt✐❛❧ s❝❤❡❞✉❧❡ s2
k ❝♦♠♣❧❡♠❡♥t✐♥❣ s1
j ✱ ✐♥ s❡ts S+
2 ❛♥❞ S−
2 ✱ ❝❛♥ ❜❡ ❞♦♥❡ ✐♥
O(n) t✐♠❡✳
❆s ❢♦r ❡❛❝❤ s1
j ∈ S1 ❛ s❡❛r❝❤ st❡♣ ✐♥ O(n) t✐♠❡ ❤❛s t♦ ❜❡ ❞♦♥❡✱ ✇❡ r❡❛❝❤ ❛ t♦t❛❧ t✐♠❡
❝♦♠♣❧❡①✐t② ✐♥ O∗
(2
n
2 ) ❢♦r ✜♥❞✐♥❣ t❤❡ ♦♣t✐♠❛❧ s♦❧✉t✐♦♥ ♦❢ t❤❡ ✐♥st❛♥❝❡ I ✇❤❡♥❡✈❡r ❛❧❧ s❡ts
S1✱ S+
2 ❛♥❞ S−
2 ❤❛✈❡ ❜❡❡♥ ❜✉✐❧t✳ ❚❤✐s ❞❛t❛ ♣r♦❝❡ss✐♥❣ r❡q✉✐r❡s ❛ t♦t❛❧ ♦❢ O∗
(2
n
2 ) t✐♠❡✱
❧❡❛❞✐♥❣ ❜② t❤❡ ✇❛② t♦ ❛ ❙♦rt ✫ ❙❡❛r❝❤ ❛❧❣♦r✐t❤♠ ✇✐t❤ O∗
(2
n
2 ) t✐♠❡ ❛♥❞ s♣❛❝❡ ✇♦rst✲❝❛s❡
❝♦♠♣❧❡①✐t✐❡s✳
✸ ❋✉t✉r❡ ❞✐r❡❝t✐♦♥s
■♥ t❤✐s ❛❜str❛❝t ✇❡ ❤❛✈❡ s❤♦✇♥ ❤♦✇ t♦ ❜✉✐❧❞ ❛♥ ❡①♣♦♥❡♥t✐❛❧ t✐♠❡ ❛❧❣♦r✐t❤♠ ❢♦r t❤❡
1|di = d ≥
P
i pi|
P
i wi(Ei + Ti) ♣r♦❜❧❡♠✱ r✉♥♥✐♥❣ ✐♥ O∗
(2
n
2 ) ≈ O∗
(1.41n
) t✐♠❡ ❛♥❞ s♣❛❝❡
✐♥ t❤❡ ✇♦rst ❝❛s❡✳ ❚❤✐s ❛❧❣♦r✐t❤♠ ✐s ❜❛s❡❞ ♦♥ t❤❡ ❙♦rt ✫ ❙❡❛r❝❤ ♠❡t❤♦❞ ✇❤✐❝❤ ✇♦r❦s ❜②
❛♣♣r♦♣r✐❛t❡ ❞❛t❛ ♣r♦❝❡ss✐♥❣ ❛♥❞ s♦rt✐♥❣ ♣r♦❝❡❞✉r❡s✳ ■♥t❡r❡st✐♥❣❧②✱ t❤❡ s♦rt✐♥❣ ♣r♦❝❡❞✉r❡
✐s ❡❧❛❜♦r❛t❡❞ ♦♥ ♣❛rt✐❛❧ s❡q✉❡♥❝❡ st❛rt✐♥❣ t✐♠❡s ✇❤✐❝❤ ✐s✱ ✜♥❛❧❧②✱ ♥♦t s✉r♣r✐s✐♥❣ s✐♥❝❡ t✐♠✲
✐♥❣ ♣r♦❜❧❡♠s ♣❧❛② ❛ ❝❡♥tr❛❧ r♦❧❡ ✐♥ ❥✉st✲✐♥✲t✐♠❡ s❝❤❡❞✉❧✐♥❣✳ ❇❡s✐❞❡s✱ t❤✐s ✐s t❤❡ ✜rst r❡s✉❧t
❦♥♦✇♥ ❢♦r s✉❝❤ ♣r♦❜❧❡♠s✳
223

❋✐❣✳ ✷✳ ❈♦♥tr✐❜✉t✐♦♥s ♦❢ ♣❛rt✐❛❧ s❝❤❡❞✉❧❡s s2
k ∈ S+
2
◆♦t✐❝❡ t❤❛t ✐♥ t❤❡ ♣r❡s❡♥❝❡ ♦❢ ♥♦♥ s②♠♠❡tr✐❝ ✇❡✐❣❤ts✱ Pr♦♣♦s✐t✐♦♥ ✶ ❝❛♥ ❜❡ s❧✐❣❤t❧② ♠♦❞✲
✐✜❡❞ ✇❤✐❝❤ ❡♥❛❜❧❡s t♦ ❛❞❛♣t t❤❡ ♣r♦♣♦s❡❞ ❛❧❣♦r✐t❤♠ ❢♦r t❤❡ 1|di = d ≥
P
i pi|
P
i wiEi+viTi
♣r♦❜❧❡♠✳ ❇❡s✐❞❡s✱ t❤❡ ♣r♦♣♦s❡❞ ❛❧❣♦r✐t❤♠ s✉❣❣❡sts ❛❧s♦ t❤❡ ❡①✐st❡♥❝❡ ♦❢ ❛ ❙♦rt ✫ ❙❡❛r❝❤
❛❧❣♦r✐t❤♠ ❢♦r t❤❡ ♣r♦❜❧❡♠ ❜✉t ✇✐t❤ ✐❞❡♥t✐❝❛❧ ♣❛r❛❧❧❡❧ ♠❛❝❤✐♥❡s✱ ❞❡♥♦t❡❞ ❜② P|di = d ≥
P
i pi|
P
i wiEi + viTi✳
❘❡❢❡r❡♥❝❡s
❈②❣❛♥ ▼✳✱ ❛♥❞ P✐❧✐♣❝③✉❦ ▼✳✱ ❛♥❞ P✐❧✐♣❝③✉❦ ▼✳✱ ❛♥❞ ❲♦❥t❛s③❝③②❦ ❏✳❖✳✱ ✷✵✶✶✱ ✏❙❝❤❡❞✉❧✐♥❣ ♣❛rt✐❛❧❧②
♦r❞❡r❡❞ ❥♦❜s ❢❛st❡r t❤❛♥ 2n
✧✱ ■♥✿ ❉❡♠❡tr❡s❝✉ ❈✳✱ ❍❛❧❧❞♦rss♦♥ ▼✳▼✳ ✭❡❞s✮ ❆❧❣♦r✐t❤♠s ✲ ❊❙❆
✷✵✶✶✳ ❊❙❆ ✷✵✶✶✳ ▲❡❝t✉r❡ ◆♦t❡s ✐♥ ❈♦♠♣✉t❡r ❙❝✐❡♥❝❡✱ ✈♦❧ ✻✾✹✷✱ ❙♣r✐♥❣❡r✳
●❛rr❛✛❛ ▼✳✱ ❛♥❞ ❙❤❛♥❣ ▲✳✱ ❛♥❞ ❉❡❧❧❛ ❈r♦❝❡ ❋✳✱ ❛♥❞ ❚✬❦✐♥❞t ❱✳✱ ✷✵✶✼✱ ✏❆♥ ❡①❛❝t ❡①♣♦♥❡♥t✐❛❧
❇r❛♥❝❤✲❛♥❞✲▼❡r❣❡ ❛❧❣♦r✐t❤♠ ❢♦r t❤❡ s✐♥❣❧❡ ♠❛❝❤✐♥❡ t♦t❛❧ t❛r❞✐♥❡ss ♣r♦❜❧❡♠✧✱ ❤❛❧✳❛r❝❤✐✈❡s✲
♦✉✈❡rt❡s✳❢r✴❤❛❧✲✵✶✹✼✼✽✸✺✳
❍❛❧❧ ◆✳●✳✱ P♦s♥❡r ▼✳❊✳✱ ✶✾✾✶✱ ✏❊❛r❧✐♥❡ss✲t❛r❞✐♥❡ss s❝❤❡❞✉❧✐♥❣ ♣r♦❜❧❡♠s✱ ■✿ ❲❡✐❣❤t❡❞ ❞❡✈✐❛t✐♦♥ ♦❢
❝♦♠♣❧❡t✐♦♥ t✐♠❡s ❛❜♦✉t ❛ ❝♦♠♠♦♥ ❞✉❡ ❞❛t❡✧✱ ❖♣❡r❛t✐♦♥s ❘❡s❡❛r❝❤✱ ❱♦❧✳ ✸✾✱ ♣♣✳ ✽✸✻✲✽✹✻✳
❍♦r♦✇✐t③ ❊✳✱ ❛♥❞ ❙❛❤♥✐ ❙✳✱ ✶✾✼✹✱ ✏❈♦♠♣✉t✐♥❣ P❛rt✐t✐♦♥s ✇✐t❤ ❆♣♣❧✐❝❛t✐♦♥s t♦ t❤❡ ❑♥❛♣s❛❝❦ Pr♦❜✲
❧❡♠✧✱ ❏♦✉r♥❛❧ ♦❢ t❤❡ ❆❈▼✱ ❱♦❧✳ ✷✶✱ ♣♣✳ ✷✼✼✲✷✾✷✳
❏♦③❡❢♦✇s❦❛ ❏✳✱ ✷✵✵✼✱ ✏❏✉st✲✐♥✲❚✐♠❡ ❙❝❤❡❞✉❧✐♥❣✿ ▼♦❞❡❧s ❛♥❞ ❛❧❣♦r✐t❤♠s ❢♦r ❝♦♠♣✉t❡r ❛♥❞ ♠❛♥✉✲
❢❛❝t✉r✐♥❣ s②st❡♠s✧✱ ❙♣r✐♥❣❡r✳
❋♦♠✐♥ ❋✳ ❛♥❞ ❑r❛ts❝❤ ❉✳✱ ✷✵✶✵✱ ✏❊①❛❝t ❡①♣♦♥❡♥t✐❛❧ ❛❧❣♦r✐t❤♠s✧✱ ❙♣r✐♥❣❡r✳
▲❡♥té ❈✳ ❛♥❞ ▲✐❡❞❧♦✛ ▼✳ ❛♥❞ ❙♦✉❦❤❛❧ ❆✳ ❛♥❞ ❚✬❦✐♥❞t ❱✳✱ ✷✵✶✸✱ ✏❖♥ ❛♥ ❡①t❡♥s✐♦♥ ♦❢ t❤❡ ❙♦rt ✫
❙❡❛r❝❤ ♠❡t❤♦❞ ✇✐t❤ ❛♣♣❧✐❝❛t✐♦♥ t♦ s❝❤❡❞✉❧✐♥❣ t❤❡♦r②✧✱ ❚❤❡♦r❡t✐❝❛❧ ❈♦♠♣✉t❡r ❙❝✐❡♥❝❡✱ ❱♦❧✳
✺✶✶✱ ♣♣✳ ✶✸✲✷✷✳
▲❡♥té ❈✳ ❛♥❞ ▲✐❡❞❧♦✛ ▼✳ ❛♥❞ ❙♦✉❦❤❛❧ ❆✳ ❛♥❞ ❚✬❦✐♥❞t ❱✳✱ ✷✵✶✹✱ ✏❊①♣♦♥❡♥t✐❛❧ ❛❧❣♦r✐t❤♠s ❢♦r
s❝❤❡❞✉❧✐♥❣ ♣r♦❜❧❡♠s✧✱ ❤❛❧✳❛r❝❤✐✈❡s✲♦✉✈❡rt❡s✳❢r✴❤❛❧✲✵✵✾✹✹✸✽✷✈✶✳
❙❤❛♥❣ ▲✳✱ ❛♥❞ ●❛rr❛✛❛ ▼✳✱ ❛♥❞ ❉❡❧❧❛ ❈r♦❝❡ ❋✳✱ ❛♥❞ ❚✬❦✐♥❞t ❱✳✱ ✷✵✶✼❛✱ ✏▼❡r❣✐♥❣ ♥♦❞❡s ✐♥ s❡❛r❝❤
tr❡❡s✿ ❛♥ ❡①❛❝t ❡①♣♦♥❡♥t✐❛❧ ❛❧❣♦r✐t❤♠ ❢♦r t❤❡ s✐♥❣❧❡ ♠❛❝❤✐♥❡ t♦t❛❧ t❛r❞✐♥❡ss s❝❤❡❞✉❧✐♥❣ ♣r♦❜✲
❧❡♠✧✱ ■♥ ✶✷t❤ ■♥t❡r♥❛t✐♦♥❛❧ ❙②♠♣♦s✐✉♠ ♦♥ P❛r❛♠❡t❡r✐③❡❞ ❛♥❞ ❊①❛❝t ❈♦♠♣✉t❛t✐♦♥ ✭■P❊❈
✷✵✶✼✮✱ ✈♦❧✉♠❡ ✽✾ ♦❢ ▲■P■❝s✱ ♣❛❣❡s ✷✽✿✶✲✷✽✿✶✷✱ ❱✐❡♥♥❛✱ ❆✉str✐❛✳ ❙❝❤❧♦ss ❉❛❣st✉❤❧ ✲ ▲❡✐❜♥✐③✲
❩❡♥tr✉♠ ❢✉❡r ■♥❢♦r♠❛t✐❦
❙❤❛♥❣ ▲✳✱ ❛♥❞ ▲❡♥té ❈✳ ❛♥❞ ▲✐❡❞❧♦✛ ▼✳ ❛♥❞ ❚✬❦✐♥❞t ❱✳✱ ✷✵✶✼❜✱ ✏❊①❛❝t ❡①♣♦♥❡♥t✐❛❧ ❛❧❣♦r✐t❤♠s
❢♦r ✸✲♠❛❝❤✐♥❡ ✢♦✇s❤♦♣ s❝❤❡❞✉❧✐♥❣ ♣r♦❜❧❡♠s✧✱ ❏♦✉r♥❛❧ ♦❢ ❙❝❤❡❞✉❧✐♥❣✱ ❞♦✐✳♦r❣✴✶✵✳✶✵✵✼✴s✶✵✾✺✶✲
✵✶✼✲✵✺✷✹✲✷✳
❚✬❦✐♥❞t ❱✳ ❛♥❞ ❇✐❧❧❛✉t ❏✳✲❈✳✱ ✷✵✵✻✱ ✏▼✉❧t✐❝r✐t❡r✐❛ ❙❝❤❡❞✉❧✐♥❣✿ ❚❤❡♦r②✱ ▼♦❞❡❧s ❛♥❞ ❆❧❣♦r✐t❤♠s✧✱
❙♣r✐♥❣❡r✳
224

❋✐♥❞✐♥❣ ❛ s♣❡❝✐✜❝ ♣❡r♠✉t❛t✐♦♥ ♦❢ ❥♦❜s ❢♦r ❛ s✐♥❣❧❡
♠❛❝❤✐♥❡ s❝❤❡❞✉❧✐♥❣ ♣r♦❜❧❡♠ ✇✐t❤ ❞❡❛❞❧✐♥❡s
❚❆ ❚❤❛♥❤ ❚❤✉② ❚✐❡♥1
❛♥❞ ❇■▲▲❆❯❚ ❏❡❛♥✲❈❤❛r❧❡s1
❯♥✐✈❡rs✐té ❞❡ ❚♦✉rs✱ ❈◆❘❙✱ ▲■❋❆❚✱ ❊❘▲ ❈◆❘❙ ❘❖❖❚ ✻✸✵✺✱ ❚♦✉rs✱ ❋r❛♥❝❡✳
❥❡❛♥✲❝❤❛r❧❡s✳❜✐❧❧❛✉t❅✉♥✐✈✲t♦✉rs✳❢r✱ t❤❛♥❤t❤✉②t✐❡♥✳t❛❅❡t✉✳✉♥✐✈✲t♦✉rs✳❢r
❑❡②✇♦r❞s✿ s✐♥❣❧❡ ♠❛❝❤✐♥❡✱ ❞❡❛❞❧✐♥❡s✱ ❧❛tt✐❝❡✱ ♥❡✇ ♦❜❥❡❝t✐✈❡ ❢✉♥❝t✐♦♥✳
✶ ■♥tr♦❞✉❝t✐♦♥
■t ✐s ✇❡❧❧ ❦♥♦✇♥ t❤❛t ❛ ❧♦t ♦❢ s❝❤❡❞✉❧✐♥❣ ♣r♦❜❧❡♠s ❤❛✈❡ ❛ ❤✉❣❡ ♥✉♠❜❡r ♦❢ ♦♣t✐♠❛❧ s♦❧✉✲
t✐♦♥s✳ ❚❤✐s ✐s ♣❛rt✐❝✉❧❛r❧② tr✉❡ ❢♦r s♦♠❡ ♣♦❧②♥♦♠✐❛❧ ♣r♦❜❧❡♠s s✉❝❤ ❛s 1||Lmax✱ 1|rj|Cmax✱
F2||Cmax✱ ❡t❝✳ ✭❙♠✐t❤✱ ❲✳❊✳ ❡t✳ ❛❧✳ ✶✾✺✻✮✳ ❚❤❡ ♣✉r♣♦s❡ ♦❢ t❤✐s ♣❛♣❡r ✐s t♦ ❝♦♥tr✐❜✉t❡ t♦
t❤❡ ❝❤❛r❛❝t❡r✐③❛t✐♦♥ ♦❢ ❛❧❧ t❤❡ ♦♣t✐♠❛❧ s♦❧✉t✐♦♥s ♦❢ s✉❝❤ ❛ ♣♦❧②♥♦♠✐❛❧ s❝❤❡❞✉❧✐♥❣ ♣r♦❜❧❡♠✳
❆ ❣❡♥❡r❛❧ ❢r❛♠❡✇♦r❦ t♦ ❝❤❛r❛❝t❡r✐③❡ t❤❡ s❡t ♦❢ ♦♣t✐♠❛❧ s♦❧✉t✐♦♥s ❤❛s ❜❡❡♥ ♣r♦♣♦s❡❞ ✐♥
✭❇✐❧❧❛✉t✱ ❏✲❈✳ ❡t✳ ❛❧✳ ✷✵✶✶❜✮ ❛♥❞ ✭❇✐❧❧❛✉t✱ ❏✲❈✳ ❡t✳ ❛❧✳ ✷✵✶✷✮✱ ❜❛s❡❞ ♦♥ t❤❡ ♣r♦♣❡rt✐❡s ♦❢ t❤❡
❧❛tt✐❝❡ ♦❢ ♣❡r♠✉t❛t✐♦♥s ✭❛❧s♦ ❝❛❧❧❡❞ ♣❡r♠✉t♦❤❡❞r♦♥✮✳ ❲❡ ❝♦♥s✐❞❡r ✐♥ t❤✐s ♣❛♣❡r t❤❡ s✐♥❣❧❡
♠❛❝❤✐♥❡ s❝❤❡❞✉❧✐♥❣ ♣r♦❜❧❡♠ ✇✐t❤ ♠❛①✐♠✉♠ ❧❛t❡♥❡ss ♠✐♥✐♠✐③❛t✐♦♥✱ ❞❡♥♦t❡❞ ❜② 1||Lmax
✭❏❛❝❦s♦♥✱ ❏✲❘✳ ❡t✳ ❛❧✳ ✶✾✺✺✮✳ ❲❡ ❛ss✉♠❡ t❤❛t ❛ ♣r❡✲tr❡❛t♠❡♥t ✐♥ O(n log n) ✐s ♣❡r❢♦r♠❡❞
s♦ t❤❛t t❤❡ ❥♦❜s ❛r❡ ♥✉♠❜❡r❡❞ ✐♥ ❊❉❉ ♦r❞❡r ❛♥❞ ❞✉❡ ❞❛t❡s ❛r❡ ♠♦❞✐✜❡❞ ✐♥t♦ ❞❡❛❞❧✐♥❡s s♦
t❤❛t ❛♥② ♦♣t✐♠❛❧ s❡q✉❡♥❝❡ ❤❛s t♦ ❜❡ ❢❡❛s✐❜❧❡ ✇✐t❤ r❡s♣❡❝t t♦ t❤❡s❡ ❞❡❛❞❧✐♥❡s✳
■♥ t❤❡ ❢r❛♠❡✇♦r❦ ❜❛s❡❞ ♦♥ t❤❡ ❧❛tt✐❝❡✱ ♦♥❡ ♣r♦❜❧❡♠ ✐s t♦ ✜♥❞ ❛ ❢❡❛s✐❜❧❡ s❡q✉❡♥❝❡✱ ❛s
❞❡❡♣ ❛s ♣♦ss✐❜❧❡✳ ■♥❞❡❡❞✱ ❛♥② ❢❡❛s✐❜❧❡ s❡q✉❡♥❝❡ ✐♥ t❤❡ ❧❛tt✐❝❡ ✐s s✉❝❤ t❤❛t ❛❧❧ ✐ts ♣r❡❞❡❝❡ss♦rs
❛r❡ ❛❧s♦ ❢❡❛s✐❜❧❡ ✭s✐♠♣❧❡ ♣❛✐r✇✐s❡ ❡①❝❤❛♥❣❡ ❛r❣✉♠❡♥t✮ ❛♥❞ ✐t ✐s ♣♦ss✐❜❧❡ t♦ ❣✐✈❡ ❡❛s✐❧② t❤❡
❝❤❛r❛❝t❡r✐st✐❝s ♦❢ ❛❧❧ t❤❡s❡ ♣r❡❞❡❝❡ss♦rs✳ ❚♦ ❞❡♥♦t❡ t❤❡ ❧❡✈❡❧ ♦❢ ❛ ❢❡❛s✐❜❧❡ s❡q✉❡♥❝❡ ✐♥ t❤❡
❧❛tt✐❝❡✱ ❛ ♥❡✇ ❢✉♥❝t✐♦♥ ❤❛s ❜❡❡♥ ✐♥tr♦❞✉❝❡❞ ❛♥❞ ✇❡ ✇❛♥t t❤✐s ❧❡✈❡❧ t♦ ❜❡ ❛s s♠❛❧❧ ❛s ♣♦ss✐❜❧❡✳
▲❡t r❡♠❡♠❜❡r ✭s❡❡ ✭❇✐❧❧❛✉t✱ ❏✲❈✳ ❡t✳ ❛❧✳ ✷✵✶✷✮✮ t❤❛t t❤❡ t♦♣ s❡q✉❡♥❝❡ ✐s ❊❉❉ ✇✐t❤ ❧❡✈❡❧
1
2 n(n − 1) ❛♥❞ t❤❡ ❜♦tt♦♠ s❡q✉❡♥❝❡ ✐s t❤❡ ✐♥✈❡rs❡ ❊❉❉ s❡q✉❡♥❝❡ ✇✐t❤ ❧❡✈❡❧ ✵✳ ❚②♣✐❝❛❧❧②✱
✐❢ t❤❡ ✐♥✈❡rs❡ ❊❉❉ s❡q✉❡♥❝❡ ✐s ❢❡❛s✐❜❧❡✱ ✐t ♠❡❛♥s t❤❛t ❛❧❧ t❤❡ ♣r❡❞❡❝❡ss♦rs✱ ✐✳❡✳❡ t❤❡ ♥✦
s❡q✉❡♥❝❡s✱ ❛r❡ ❢❡❛s✐❜❧❡✳
❚❤❡ ♥❡✇ ♦❜❥❡❝t✐✈❡ ❢✉♥❝t✐♦♥ ❞❡♥♦t❡❞ ❜②
P
Nj ❤❛s ❧❡❞ t♦ t❤❡ ✐♥tr♦❞✉❝t✐♦♥ ♦❢ s♦♠❡ ♦t❤❡r
♥❡✇ ♦❜❥❡❝t✐✈❡ ❢✉♥❝t✐♦♥s✱ ❜❛s❡❞ ♦♥ t❤❡ ♣♦s✐t✐♦♥ ♦❢ t❤❡ ❥♦❜s ✐♥ t❤❡ s❡q✉❡♥❝❡✱ ✇❤✐❝❤ ❤❛✈❡ ❜❡❡♥
st✉❞✐❡❞ ✐♥ ✭❚❛✱ ❚✳❚✳❚✐❡♥ ❡t✳ ❛❧✳ ✷✵✶✼❛✮ ❛♥❞ ✭❚❛✱ ❚✳❚✳❚✐❡♥ ❡t✳ ❛❧✳ ✷✵✶✼❜✮✳
✷ ❉❡✜♥✐t✐♦♥ ♦❢ ❢✉♥❝t✐♦♥
P
Nj ❛♥❞ ✜rst r❡s✉❧ts
❲❡ ❝♦♥s✐❞❡r ❛ s❡t ♦❢ n ❥♦❜s t♦ s❝❤❡❞✉❧❡✳ ❚♦ ❡❛❝❤ ❥♦❜ Jj✱ 1 ≤ j ≤ n✱ ✐s ❛ss♦❝✐❛t❡❞
❛ ♣r♦❝❡ss✐♥❣ t✐♠❡ pj ❛♥❞ ❛ ❞❡❛❞❧✐♥❡ ˜
dj✳ ❲✐t❤♦✉t ❧♦ss ♦❢ ❣❡♥❡r❛❧✐t②✱ ✐t ✐s ❛ss✉♠❡❞ t❤❛t
˜
d1 ≤ ˜
d2 ≤ ... ≤ ˜
dn ❛♥❞ t❤❛t s❡q✉❡♥❝❡ EDD = (J1, J2, ..., Jn) ✐s ❢❡❛s✐❜❧❡✳
▲❡t σ ❜❡ ❛ s❡q✉❡♥❝❡✳ ❚❤❡ ❧❡✈❡❧ ♦❢ σ ✐♥ t❤❡ ❧❛tt✐❝❡ ✐s t❤❡ ♥✉♠❜❡r ♦❢ ❝♦✉♣❧❡s (Jj, Jk) s♦
t❤❛t j k ❛♥❞ Jj ♣r❡❝❡❞❡s Jk✳ ❚❤❡r❡❢♦r❡✱ t❤❡ ❝♦♥tr✐❜✉t✐♦♥ ♦❢ Jj t♦ t❤✐s ♦❜❥❡❝t✐✈❡ ❢✉♥❝t✐♦♥
✐s t❤❡ ♥✉♠❜❡r ♦❢ ❥♦❜s ❛❢t❡r Jj ✇✐t❤ ❛♥ ✐♥❞❡① ❣r❡❛t❡r t❤❛♥ j✳ ❲❡ ❞❡♥♦t❡ t❤✐s ♥✉♠❜❡r ❜② Nj✳
▲❡t s✉♣♣♦s❡ t❤❛t xj,k ✐s ❛ ❜✐♥❛r② ✈❛r✐❛❜❧❡ ❡q✉❛❧ t♦ ✶ ✐❢ Jj ✐s ✐♥ ♣♦s✐t✐♦♥ k✳ ❲❡ ❤❛✈❡✿
Nj =
Pn
i=j+1
Pn
h=k+1 xi,h✳
❚❤✐s ♦❜❥❡❝t✐✈❡ ❢✉♥❝t✐♦♥ ❤❛s ♦t❤❡r ❞❡♥♦♠✐♥❛t✐♦♥s ✐♥ t❤❡ ❧✐tt❡r❛t✉r❡✿ t❤❡ ❑❡♥❞❛❧❧✬s t❛✉
❞✐st❛♥❝❡ ✭❝♦✉♥ts t❤❡ ♥✉♠❜❡r ♦❢ ♣❛✐r✇✐s❡ ❞✐s❛❣r❡❡♠❡♥ts ❜❡t✇❡❡♥ t✇♦ r❛♥❦✐♥❣ ❧✐sts✮ ❛♥❞ t❤❡
225

❝r♦ss✐♥❣ ♥✉♠❜❡r ❜❡t✇❡❡♥ t❤❡ ❝♦♥s✐❞❡r❡❞ s❡q✉❡♥❝❡ ❛♥❞ t❤❡ ✐♥✈❡rs❡ ♥✉♠❜❡r✐♥❣ s❡q✉❡♥❝❡✳
◆♦t✐❝❡ t❤❛t ❛ ♣r♦❜❧❡♠✱ ♣r❡s❡♥t✐♥❣ s✐♠✐❧❛r✐t✐❡s ✇✐t❤ ♦✉r ♣r♦❜❧❡♠✱ ✐s ♣r♦✈❡❞ ◆P✲❤❛r❞ ✐♥
✭❇✐❡❞❧✱ ❚✳ ❡t✳ ❛❧✳ ✷✵✵✺✮✳
❲❡ ❝❛♥ ♥♦t✐❝❡ t❤❛t t❤✐s ♦❜❥❡❝t✐✈❡ ❢✉♥❝t✐♦♥ ❞♦❡s ♥♦t ❞❡♣❡♥❞ ♦♥ t❤❡ ❥♦❜s ❝♦♠♣❧❡t✐♦♥
t✐♠❡s✱ ✇❤✐❝❤ ✐s ✉♥✉s✉❛❧ ✐♥ s❝❤❡❞✉❧✐♥❣✳ ❚❤✐s r❡♠❛r❦ ❧❡❛❞s t♦ s♦♠❡ ✜rst ✭s✐♠♣❧❡✮ r❡s✉❧ts✳
• Pr♦❜❧❡♠ 1||
P
Nj
Pr♦❜❧❡♠ 1||
P
Nj ✭✇✐t❤♦✉t ❞✉❡ ❞❛t❡ ♦r ❞❡❛❞❧✐♥❡s✮ ✐s tr✐✈✐❛❧✳ ❙❝❤❡❞✉❧✐♥❣ t❤❡ ❥♦❜s ✐♥ t❤❡
r❡✈❡rs❡ ♦r❞❡r ♦❢ t❤❡✐r ♥✉♠❜❡r✐♥❣ ❧❡❛❞s t♦ ❛ s♦❧✉t✐♦♥ ✇✐t❤
P
Nj = 0✳
• Pr♦❜❧❡♠ 1|pj = p, e
dj|
P
Nj
▲❡t ❝♦♥s✐❞❡r ✜rst t❤❡ 1|pj = 1, e
dj|
P
Nj ♣r♦❜❧❡♠ ❛♥❞ ❝♦♥s✐❞❡r t❤❡ ❢♦❧❧♦✇✐♥❣ ❇❛❝❦✇❛r❞
❛❧❣♦r✐t❤♠ ✭❆❧❣✳ ✶✮✿ s❝❤❡❞✉❧❡ st❛rt✐♥❣ ❜② t❤❡ ❡♥❞ t❤❡ ❢❡❛s✐❜❧❡ ❥♦❜ ✇✐t❤ ♠✐♥✐♠✉♠ ✐♥❞❡①✳ ❚❤✐s
❛❧❣♦r✐t❤♠ s♦❧✈❡s ♣r♦❜❧❡♠ 1|pj = 1, e
dj|
P
Nj t♦ ♦♣t✐♠❛❧✐t② ✭t❤❡ ♣r♦♦❢ ✐s ❛❞♠✐tt❡❞ ❤❡r❡✮✳
■t ✐s ❡❛s② t♦ s❡❡ t❤❛t t❤✐s ❛❧❣♦r✐t❤♠ ❝❛♥ ❛❧s♦ s♦❧✈❡ ♣r♦❜❧❡♠ 1|pj = p, e
dj|
P
Nj✳
✸ Pr♦♣❡rt✐❡s ❛♥❞ r❡s♦❧✉t✐♦♥ ♠❡t❤♦❞s ❢♦r 1|e
dj|
P
Nj
Pr♦♣❡rt② ✶✿ ❆♥ ♦♣t✐♠❛❧ s♦❧✉t✐♦♥ ❝❛♥ ❛❧✇❛②s ❜❡ ❞❡❝♦♠♣♦s❡❞ ✐♥ ❛ s✉❝❝❡ss✐♦♥ ♦❢ ❜❛t❝❤❡s
❞❡✜♥❡❞ ❛s ❢♦❧❧♦✇s✿ t❤❡ ✧❤❡❛❞✧ ♦❢ t❤❡ ❜❛t❝❤ ✐s t❤❡ ❧❛st ❥♦❜ ♦❢ t❤❡ ❜❛t❝❤ ❀ t❤❡ ❥♦❜s ✐♥ t❤❡ ❜❛t❝❤
❛r❡ ✐♥ ❞❡❝r❡❛s✐♥❣ ♥✉♠❜❡r✐♥❣ ♦r❞❡r ❛♥❞ ❤❛✈❡ ❛♥ ✐♥❞❡① ❣r❡❛t❡r t❤❛♥ t❤❡ ❤❡❛❞✳ ❚❤❡r❡❢♦r❡✱ t❤❡
✐♥❞❡① ♦❢ t❤❡ ❤❡❛❞s ❛r❡ ✐♥❝r❡❛s✐♥❣✱ st❛rt✐♥❣ ✇✐t❤ ✐♥❞❡① ✶✳
Pr♦♦❢✳ ❛❞♠✐tt❡❞✳
❊①❛❝t r❡s♦❧✉t✐♦♥ ♠❡t❤♦❞s
❋♦r ❡①❛❝t r❡s♦❧✉t✐♦♥✱ t✇♦ ▼■▲P ♠♦❞❡❧s ✇❡r❡ ♣r❡s❡♥t❡❞ ✐♥ ✭❇✐❧❧❛✉t✱ ❏✲❈✳ ❡t✳ ❛❧✳ ✷✵✶✷✮✳
❚❤❡ ✜rst ♠♦❞❡❧ ✉s❡s ♣♦s✐t✐♦♥❛❧ ✈❛r✐❛❜❧❡s✱ t❤❡ s❡❝♦♥❞ ♠♦❞❡❧ ✉s❡s r❡❧❛t✐✈❡ ♣♦s✐t✐♦♥ ✈❛r✐❛❜❧❡s✳
■♥ t❤✐s ♣❛♣❡r✱ ❛ ❜r❛♥❝❤✲❛♥❞✲❜♦✉♥❞ ❛❧❣♦r✐t❤♠ ✐s ♣r♦♣♦s❡❞ ✇✐t❤ s♦♠❡ ❞♦♠✐♥❛♥❝❡ r✉❧❡s✳
❚❤❡ BB ♠❡t❤♦❞ ❢♦r
P
Nj ❤❛s t❤❡ ❢♦❧❧♦✇✐♥❣ ❝❤❛r❛❝t❡r✐st✐❝s✳ ❆ ♥♦❞❡ ✐s ❞❡✜♥❡❞ ❜② ❛
♣❛rt✐❛❧ s❡q✉❡♥❝❡ S ♦❢ k ❥♦❜s st❛rt✐♥❣ ❜② t❤❡ ❡♥❞ ♦❢ t❤❡ s❝❤❡❞✉❧❡✱ ❛ s❡t ♦❢ n − k ✉♥s❝❤❡❞✉❧❡❞
❥♦❜s S̄✱ ❛ ❧♦✇❡r ❜♦✉♥❞ LB(S)✱ t❤❡ ✐♥❞❡① idx ♦❢ t❤❡ ❤❡❛❞ ♦❢ t❤❡ ❝✉rr❡♥t ❜❛t❝❤ ❛♥❞ t t❤❡
st❛rt✐♥❣ t✐♠❡ ♦❢ t❤❡ ❥♦❜s ✐♥ S✿ t =
P
Jj ∈S̄ pj✳
❆t t❤❡ r♦♦t ♥♦❞❡✱ t❤❡ ✉♥s❝❤❡❞✉❧❡❞ ❥♦❜s ❛r❡ {Jn, Jn−1, ..., J1}✳ ❚❤❡ ✐♥✐t✐❛❧ ✉♣♣❡r ❜♦✉♥❞
UB ✐s ❣✐✈❡♥ ❜② ❛ ❇❛❝❦✇❛r❞ ❛❧❣♦r✐t❤♠ ♦❢ t❤❡ s❛♠❡ t②♣❡ ❛s ❆❧❣✳ ✶✳ ❚❤❡ str❛t❡❣② ♦❢ ❜r❛♥❝❤✐♥❣
❝♦♥s✐sts ✐♥ ❛❞❞✐♥❣ ❛ ❥♦❜ ♦❢ S̄ ✐♥ ✜rst ♣♦s✐t✐♦♥ ♦❢ S✱ r❡s♣❡❝t✐♥❣ t❤❡ ❞❡❛❞❧✐♥❡s✱ ❛♥❞ t❤❡
❡①♣❧♦r❛t✐♦♥ ✐s ❞♦♥❡ ❜② depth − first ✭t❤❡ ❧✐st ♦❢ ♥♦❞❡s ✐s ♠❛♥❛❣❡❞ ❛s ❛ ▲■❋❖ ❧✐st✮✳
❙♦♠❡ ❞♦♠✐♥❛♥❝❡ r✉❧❡s ❛r❡ ✉s❡❞ ❢♦r t❤✐s ♠❡t❤♦❞✳ ▲❡t ❝♦♥s✐❞❡r ❛ ❝✉rr❡♥t ♥♦❞❡ ❛♥❞ ❧❡t
✉s ❞❡♥♦t❡ ❜② Jℓ t❤❡ ✜rst ❥♦❜ ✐♥ S ❛♥❞ ❜② Jh t❤❡ ❥♦❜ ✐♥ S̄ t♦ s❝❤❡❞✉❧❡ ❜❡❢♦r❡ Jℓ✳ ❚❤❡ ❝❤✐❧❞
♥♦❞❡ ✐s ❝r❡❛t❡❞ ♦♥❧② ✐❢ ˜
dh ≥ t✳ ❋✉rt❤❡r♠♦r❡✱ ✐❢ h ℓ ❛♥❞ h idx✱ t❤❡ ♥♦❞❡ ✐s ♥♦t ❝r❡❛t❡❞
✭s❡❡ Pr♦♣❡rt② ✶✮✳ ■❢ h ℓ ❛♥❞ h idx✱ t❤❡ idx ♦❢ t❤❡ ❝❤✐❧❞ ♥♦❞❡ ✐s s❡t t♦ h✳ ■❢ h = 1✱ t❤❡
s❡q✉❡♥❝❡ ✐s ❝♦♠♣❧❡t❡❞ ❜② t❤❡ ❥♦❜s ✐♥ S̄ ✐♥ t❤❡✐r ✐♥✈❡rs❡ ♥✉♠❜❡r✐♥❣ ♦r❞❡r ❛♥❞ t❤✐s ♥♦❞❡ ✐s
❝♦♥s✐❞❡r❡❞ ✐♠♠❡❞✐❛t❡❧② ❛s ❛ ❧❡❛❢ ♦❢ t❤❡ tr❡❡ ✭s❡❡ Pr♦♣❡rt② ✶✮✳
❚❤❡ ❧♦✇❡r ❜♦✉♥❞ ✇♦r❦s ❛s ❢♦❧❧♦✇s✿ ❛ ❞✉♠♠② s❡q✉❡♥❝❡ ✐s ❜✉✐❧t ✇✐t❤ t❤❡ ❥♦❜s ✐♥ S̄ ✐♥
r❡✈❡rs❡ ♥✉♠❜❡r ♦r❞❡r✐♥❣✱ ♣❧✉s t❤❡ ❥♦❜s ✐♥ S✳ ❚❤❡ ❡✈❛❧✉❛t✐♦♥ ♦❢ t❤✐s ❛ ♣r✐♦r✐ ♥♦♥ ❢❡❛s✐❜❧❡
s❡q✉❡♥❝❡ ✐s t❤❡ ❧♦✇❡r ❜♦✉♥❞✳ ❍♦✇❡✈❡r✱ ✐❢ t❤❡ s❡t ♦❢ ✉♥s❝❤❡❞✉❧❡❞ ❥♦❜s ✐s (Jn, Jn−1, ..., J1) ✐♥
t❤✐s ♦r❞❡r✱ ✐t ✐s ♣♦ss✐❜❧❡ t♦ ❝♦♠♣✉t❡ t❤❡ ❧♦✇❡r ❜♦✉♥❞ ✐♥ O(1) t✐♠❡✳
❍❡✉r✐st✐❝ ❛♥❞ ♠❡t❛❤❡✉r✐st✐❝ ♠❡t❤♦❞s
❚✇♦ ♣♦❧②♥♦♠✐❛❧ t✐♠❡ ❤❡✉r✐st✐❝ ♠❡t❤♦❞s ❛r❡ ♣r♦♣♦s❡❞✿ ❛ ❇❛❝❦✇❛r❞ ❛❧❣♦r✐t❤♠ ✭❞❡♥♦t❡❞
BW✱ ❆❧❣✳ ✶✮ ❛♥❞ ❛ ❋♦r✇❛r❞ ❛❧❣♦r✐t❤♠ ✭❞❡♥♦t❡❞ FW✮✳ BW ❜✉✐❧❞s ❛ s♦❧✉t✐♦♥ ❜② t❤❡ ❡♥❞✱
♣✉tt✐♥❣ ✐♥ ❧❛st ♣♦s✐t✐♦♥ t❤❡ ❢❡❛s✐❜❧❡ ❥♦❜ ✇✐t❤ t❤❡ s♠❛❧❧❡st ✐♥❞❡①❀ FW t❛❦❡s t❤❡ ❥♦❜s ✐♥ ❊❉❉
226

♦r❞❡r✱ ♣✉t ❡❛❝❤ ❥♦❜ ❛s ❧❛t❡ ❛s ♣♦ss✐❜❧❡ ❛♥❞ ✐♥s❡rt t❤❡ ❢❡❛s✐❜❧❡ ❥♦❜ ✇✐t❤ t❤❡ ❜✐❣❣❡st ✐♥❞❡①
❜❡❢♦r❡ ✐t✳
❚✇♦ ♠❡t❛❤❡✉r✐st✐❝ ♠❡t❤♦❞s ❛r❡ ♣r♦♣♦s❡❞✿ ❛ ❚❛❜✉ s❡❛r❝❤ ✭❞❡♥♦t❡❞ TS✮ ❛♥❞ ❛ ❙✐♠✉❧❛t❡❞
❆♥♥❡❛❧✐♥❣ ✭SA✮✱ ✇✐t❤ s❡✈❡r❛❧ ✭❝♦♠♠♦♥✮ ♥❡✐❣❤❜♦r❤♦♦❞s ♦♣❡r❛t♦rs✳ ❚❤❡ ✐♥✐t✐❛❧ s♦❧✉t✐♦♥ ♦❢
TS ❛♥❞ SA ✐s t❤❡ ❜❡st s♦❧✉t✐♦♥ ♦❢ BW ❛♥❞ FW✳
✹ ❈♦♠♣✉t❛t✐♦♥❛❧ ❡①♣❡r✐♠❡♥ts
❆❢t❡r ❛ st✉❞② ❛❜♦✉t ❛ r❡❧❛t❡❞ ♣r♦❜❧❡♠ ❜❛s❡❞ ♦♥ ❥♦❜s ♣♦s✐t✐♦♥s✱ ✇❤✐❝❤ ✇❛s ♣r♦✈❡❞ t♦ ❜❡
str♦♥❣❧② ◆P✲❤❛r❞ ✭✭❚❛ ❚✳❚✳❚✐❡♥✱ ❡t✳ ❛❧✳ ✷✵✶✼❛✮✱ ✭❚❛✱ ❚✳❚✳❚✐❡♥✱ ❡t✳ ❛❧✳ ✷✵✶✼❜✮✮✱ t✇♦ t②♣❡s
♦❢ ✐♥st❛♥❝❡s ✇❡r❡ ❣❡♥❡r❛t❡❞✳ ❖♥❡ t②♣❡ ♦❢ ♣✉r❡ r❛♥❞♦♠ ✐♥st❛♥❝❡s✱ ❛♥❞ ♦♥❡ t②♣❡ ♦❢ ✧❞✐✣❝✉❧t✧
✐♥st❛♥❝❡s✳ ❊✈❡♥ ✐❢ t❤❡ ♣r♦❜❧❡♠s ❛r❡ ♥♦t t❤❡ s❛♠❡✱ ✇❡ ❦❡♣t t❤❡s❡ ❞❛t❛ ❢♦r ♦✉r ❝♦♠♣✉t❛t✐♦♥❛❧
❡①♣❡r✐♠❡♥ts✳
❉❛t❛ s❡ts ❋♦r ❡❛❝❤ t②♣❡ ♦❢ ✐♥st❛♥❝❡✱ ✸✵ ✐♥st❛♥❝❡s ❤❛✈❡ ❜❡❡♥ ❣❡♥❡r❛t❡❞ ❢♦r ❡❛❝❤ ✈❛❧✉❡ ♦❢
n✱ ✇✐t❤ n ∈ {10, 20, ..., 100}✳
• ❋♦r t❤❡ ✐♥st❛♥❝❡s ♦❢ t②♣❡ ■✱ r❛♥❞♦♠ ❞❛t❛ s❡ts ❤❛✈❡ ❜❡❡♥ ❣❡♥❡r❛t❡❞ ❛s ❢♦❧❧♦✇s✿ pj ∈
[1, 100]✱ wj ∈ [1, 100] ✱ dj ∈ [(α − β/2)P, (α + β/2)P]✱ ✇✐t❤ P =
P
pj✱ α = 0.75 ❛♥❞
β = 0.25✳
❚❤❡s❡ ✐♥st❛♥❝❡s r❡❝❡✐✈❡ ❛ ♣r❡✲tr❡❛t♠❡♥t✿ ✭✶✮ ❊❉❉ r✉❧❡ ✐s ❛♣♣❧✐❡❞✱ ❣✐✈✐♥❣ L∗
max✳ ❚❤❡♥✱
✭✷✮ ❞✉❡ ❞❛t❡s ❛r❡ ♠♦❞✐✜❡❞ t♦ ❣✐✈❡ ❞❡❛❞❧✐♥❡s✿ ˜
dj = dj + L∗
max✱ ❢♦r ❛♥② j ∈ {1, 2, ..., n}✱
❧✐♠✐t✐♥❣ t❤❡ ❞❡❛❞❧✐♥❡s t♦
P
pj✳ ❋✐♥❛❧❧②✱ ✭✸✮ t❤❡ ❥♦❜s ❛r❡ r❡♥✉♠❜❡r❡❞ ✐♥ ❊❉❉ ♦r❞❡r✳
• ❋♦r t❤❡ ✐♥st❛♥❝❡s ♦❢ t②♣❡ ■■✱ r❛♥❞♦♠ ❞❛t❛ s❡ts ❤❛✈❡ ❜❡❡♥ ❣❡♥❡r❛t❡❞ ❛s ❢♦❧❧♦✇s✿
❋♦r n′
= ⌊n/4⌋ ❥♦❜s✿ pj = 1❀ wj = 0❀ ˜
dj = 4jP/n
❋♦r t❤❡ (n − n′
) r❡♠❛✐♥✐♥❣ ❥♦❜s✿ pj ∈ [1, 100]✱ wj = w0j + P✱ ✇✐t❤ w0j ∈ [1, 100] ❛♥❞
P =
P
pj❀ ˜
dj = P + ⌊n/4⌋
❚❤❡s❡ ✐♥st❛♥❝❡s ❞♦ ♥♦t ♥❡❡❞ t❤❡ ♣r❡✲tr❡❛t♠❡♥t✳
❘❡s✉❧ts ❚❤❡ ❝♦♠♣✉t❛t✐♦♥❛❧ ❡①♣❡r✐♠❡♥ts ❤❛✈❡ ❜❡❡♥ r✉♥ ♦♥ ❛ ❍P Pr♦❇♦♦❦✱ ■♥t❡❧✭❘✮
❈♦r❡✭❚▼✮ ✐✺✲✻✸✵✵ ❈P❯ ❅ ✷✳✹✵●❍③ ✷✳✺✵ ●❍③✱ ❘❆▼ ✶✻✱✵●♦✱ ❙②st❡♠ st②❧❡ ✻✹ ❜✐t✳ ❚❤❡
▼■▲P ♠♦❞❡❧s ❤❛✈❡ ❜❡❡♥ s♦❧✈❡❞ ❜② ■❇▼ ■▲❖● ❈P▲❊❳ ✶✷✳✻✳ ❚❤❡ ❈P❯ t✐♠❡ t♦ s♦❧✈❡ ❡❛❝❤
✐♥st❛♥❝❡ ❤❛s ❜❡❡♥ ❧✐♠✐t❡❞ t♦ ✶✽✵ s❡❝♦♥❞s ❢♦r ❛❧❧ t❤❡ r❡s♦❧✉t✐♦♥ ♠❡t❤♦❞s✳ ❘❡s✉❧ts ❢♦r ✐♥✲
st❛♥❝❡s ♦❢ t②♣❡ ■ ❛♥❞ ■■ ❛r❡ ♣r❡s❡♥t❡❞ ✐♥ ❚❛❜❧❡ ✶✳ ❈♦❧✉♠♥s ▼■▲P✶✱ ▼■▲P✷ ❛♥❞ BB ❝♦♥✲
❝❡r♥ t❤❡ ❡①❛❝t ♠❡t❤♦❞s✱ ✬❝♣✉✬ ✐♥❞✐❝❛t❡s t❤❡ ❛✈❡r❛❣❡ ❝♦♠♣✉t❛t✐♦♥ t✐♠❡ ❛♥❞ ✬♦♣t✬ ✐♥❞✐❝❛t❡s
t❤❡ ♥✉♠❜❡r ♦❢ ✐♥st❛♥❝❡s s♦❧✈❡❞ t♦ ♦♣t✐♠❛❧✐t② ✐♥ ❧❡ss t❤❛♥ ✶✽✵ s❡❝♦♥❞s✳ ❚❤❡ ♦t❤❡r ❝♦❧✉♠♥s
❝♦♥❝❡r♥ t❤❡ ❤❡✉r✐st✐❝ ♠❡t❤♦❞s✳ ❈♦❧✉♠♥s ✬N◦
❇✬ ✐♥❞✐❝❛t❡ t❤❡ ♥✉♠❜❡r ♦❢ t✐♠❡s t❤❡ ♠❡t❤♦❞
✐s t❤❡ ❜❡st ❛♠♦♥❣ ❛❧❧ t❤❡ ♠❡t❤♦❞s✱ ❛♥❞ ∆B1 ✐s ❛ r❡❧❛t✐✈❡ ❞❡✈✐❛t✐♦♥ ❞❡✜♥❡❞ ❜②✿ MIN =
min(MIP1, MIP2, BB, BW, FW) ❛♥❞ ∆B1(H) = H−MIN
H , ∀H ∈ {BW, FW, TS, SA}
❋♦r ❚②♣❡ ■ ✐♥st❛♥❝❡s✱ ♦♥❡ ❝❛♥ s❡❡ t❤❛t ▼■P✶ ✐s ❜❡tt❡r t❤❛♥ ▼■P✷ ❢♦r s♠❛❧❧ ✐♥st❛♥❝❡s✱
❜✉t ❇✫❇ ✐s t❤❡ ❜❡st ❡①❛❝t ♠❡t❤♦❞✱ s♦❧✈✐♥❣ q✉✐t❡ ❛❧❧ ✐♥st❛♥❝❡s ✉♣ t♦ ✼✵ ❥♦❜s✳ ❲✐t❤ ✾✵ ❥♦❜s
t❤❡ BB r❡♠❛✐♥s ✐♥t❡r❡st✐♥❣ ❜✉t ❢♦r ❧❛r❣❡r ✐♥st❛♥❝❡s✱ t❤❡ ❜❡st ♠❡t❤♦❞ ✐s t❤❡ ❙✐♠✉❧❛t❡❞
❆♥♥❡❛❧✐♥❣ ❛❧❣♦r✐t❤♠✳ ❋♦r ❚②♣❡ ■■ ✐♥st❛♥❝❡s✱ ♦♥❡ ❝❛♥ s❡❡ t❤❛t t❤❡ ❡①❛❝t ♠❡t❤♦❞s ❛r❡ ❧✐♠✐t❡❞
t♦ ✐♥st❛♥❝❡s ✇✐t❤ ✉♣ t♦ ✷✵ ❥♦❜s✳ ❆♠♦♥❣ t❤❡ ❤❡✉r✐st✐❝ ❛❧❣♦r✐t❤♠s✱ ❇❲ ✐s t❤❡ ❜❡st ♠❡t❤♦❞
❛♥❞ t❤❡ ❚❛❜✉ ❙❡❛r❝❤ ❛♥❞ t❤❡ ❙✐♠✉❧❛t❡❞ ❆♥♥❡❛❧✐♥❣ ❛r❡ ♥♦t ❛❜❧❡✱ ✐♥ t❤❡ ❧✐♠✐t❡❞ ❝♦♠♣✉t❛t✐♦♥
t✐♠❡ ♦❢ ✶✽✵ s❡❝♦♥❞s✱ t♦ ✐♠♣r♦✈❡ t❤❡ ✐♥✐t✐❛❧ s♦❧✉t✐♦♥✳
227

❚❛❜❧❡ ✶✳ ❘❡s✉❧ts ♦❢ ❚②♣❡ ■ ✫ ■■ ✐♥st❛♥❝❡s
▼■P✶ ▼■P✷ ❇✫❇ BW FW TS SA
n ❝♣✉ ♦♣t ❝♣✉ ♦♣t ❝♣✉ ♦♣t N◦
❇ ∆B1 N◦
❇ ∆B1 N◦
❇ ∆B1 N◦
❇ ∆B1
✭s✮ ✭s✮ ✭s✮ ✭✪✮ ✭✪✮ ✭✪✮ ✭✪✮
❘❡s✉❧ts ♦❢ ❚②♣❡ ■ ✐♥st❛♥❝❡s
✶✵ ✵✱✷✻ ✸✵ ✵✱✷✼ ✸✵ 3.10−5
✸✵ ✷✷ ✷✱✵✵ ✷✽ ✵✱✹✸ ✸✵ ✵ ✸✵ ✵
✷✵ ✹✼✱✷ ✸✵ ✶✵✺ ✷✵ 4.10−4
✸✵ ✺ ✶✺✱✸✻ ✶✸ ✹✱✷✶ ✷✻ ✵✱✸✻ ✷✵ ✵✱✻✺
✸✵ ✶✽✵ ✵ ✶✽✵ ✵ 5.10−3
✸✵ ✶ ✷✵✱✻✻ ✺ ✻✱✺✵ ✶✸ ✷✱✸✾ ✶✶ ✶✱✵✸
✹✵ ✶✽✵ ✵ ✶✽✵ ✵ ✵✱✵✶✹ ✸✵ ✵ ✷✹✱✵✾ ✶ ✽✱✷✸ ✶✸ ✹✱✵✹ ✼ ✶✱✺✺
✺✵ ✶✽✵ ✵ ✶✽✵ ✵ ✵✱✵✼✼ ✸✵ ✵ ✷✽✱✾✷ ✵ ✼✱✹✹ ✺ ✸✱✻✽ ✵ ✶✱✼✷
✻✵ ✶✽✵ ✵ ✶✽✵ ✵ ✷✱✸✾✶ ✸✵ ✵ ✷✽✱✾✵ ✵ ✼✱✶✶ ✶✵ ✷✱✹✷ ✵ ✶✱✷✽
✼✵ ✶✽✵ ✵ ✶✽✵ ✵ ✷✸✱✼✾ ✷✾ ✵ ✷✽✱✹✾ ✵ ✼✱✻✹ ✸ ✸✱✺✾ ✶ ✶✱✸✼
✽✵ ✶✽✵ ✵ ✶✽✵ ✵ ✶✷✼✱✸ ✶✺ ✵ ✸✶✱✸✸ ✵ ✼✱✵✼ ✼ ✷✱✾✵ ✼ ✵✱✹✹
✾✵ ✶✽✵ ✵ ✶✽✵ ✵ ✶✼✹✱✺ ✶ ✵ ✷✺✱✽✺ ✵ ✶✱✼✻ ✶✹ ✲✷✱✹✷ ✶✸ ✲✸✱✸✵
✶✵✵ ✶✽✵ ✵ ✶✽✵ ✵ ✶✽✵ ✵ ✵ ✷✽✱✼✾ ✵ ✵✱✸✵ ✶✷ ✲✶✱✻✷ ✶✾ ✲✸✱✶✵
❘❡s✉❧ts ♦❢ ❚②♣❡ ■■ ✐♥st❛♥❝❡s
✶✵ ✵✱✵✵✶ ✸✵ ✵✱✶ ✸✵ 2.10−3
✸✵ ✷✽ ✵✱✺✽ ✻ ✶✸✱✸✾ ✸✵ ✵ ✸✵ ✵
✷✵ ✵✳✺✵✹ ✸✵ ✶✶✶ ✷✵ ✸✸✱✷✶ ✷✾ ✸✵ ✵ ✵ ✷✻✱✶✹ ✸✵ ✵ ✸✵ ✵
✸✵✳✳✶✵✵ ✶✽✵ ✵ ✶✽✵ ✵ ✶✽✵ ✵ ✸✵ ✵ ✵ ≃✷✼✪ ✸✵ ✵ ✸✵ ✵
✺ ❈♦♥❝❧✉s✐♦♥s ❛♥❞ P❡rs♣❡❝t✐✈❡s
■♥ t❤✐s ♣❛♣❡r✱ ✇❡ ❤❛✈❡ ✐❞❡♥t✐✜❡❞ ❛ ♥❡✇ ❝❛t❡❣♦r② ♦❢ s❝❤❡❞✉❧✐♥❣ ♣r♦❜❧❡♠s✱ ✇✐t❤ t❤❡
❞❡✜♥✐t✐♦♥ ♦❢ ❛ ♥❡✇ ♦❜❥❡❝t✐✈❡ ❢✉♥❝t✐♦♥✳ ❙♦♠❡ tr✐✈✐❛❧ ♣r♦❜❧❡♠s ❛r❡ ✐❞❡♥t✐✜❡❞ ❜✉t t❤❡ ❣❡♥❡r❛❧
♣r♦❜❧❡♠ ✇✐t❤ ❞❡❛❞❧✐♥❡s r❡♠❛✐♥s ♦♣❡♥✳ ❲❡ ♣r♦♣♦s❡ s♦♠❡ ❡①❛❝t ❛♥❞ ❡①♣♦♥❡♥t✐❛❧ ♠❡t❤♦❞s✱
❛s ✇❡❧❧ ❛s ❤❡✉r✐st✐❝ ❛♥❞ ♠❡t❛✲❤❡✉r✐st✐❝ ❛❧❣♦r✐t❤♠s✳ ❚❤❡s❡ ♠❡t❤♦❞s ❛r❡ ❡✈❛❧✉❛t❡❞ ❜② s♦♠❡
❝♦♠♣✉t❛t✐♦♥❛❧ ❡①♣❡r✐♠❡♥ts ♦♥ r❛♥❞♦♠❧② ❣❡♥❡r❛t❡❞ ✐♥st❛♥❝❡s✳ ■♥ t❤❡ ❢✉t✉r❡✱ ✇❡ ✇✐❧❧ ❝♦♥t✐♥✉❡
t♦ ✐♠♣r♦✈❡ t❤❡ ❡①❛❝t ♠❡t❤♦❞s ❜② ✐♥tr♦❞✉❝✐♥❣ ❝✉ts ❛♥❞ ♠♦r❡ ❞♦♠✐♥❛♥❝❡ ❝♦♥❞✐t✐♦♥s✱ ❜✉t
t❤❡ ♠♦st ✐♠♣♦rt❛♥t ♣♦✐♥t ✐s t♦ ✐♥✈❡st✐❣❛t❡ t❤❡ ❝♦♠♣❧❡①✐t② ♦❢ t❤❡ ❣❡♥❡r❛❧ ♣r♦❜❧❡♠✳
❘❡❢❡r❡♥❝❡s
❇✐❡❞❧✱ ❚✳✱ ❇r❛♥❞❡♥❜✉r❣✱ ❋r❛♥③ ❏✳✱ ❉❡♥❣ ❳✳✱ ✧❈r♦ss✐♥❣s ❛♥❞ P❡r♠✉t❛t✐♦♥s✧✱ ■♥✿ ❍❡❛❧② P✳✱ ◆✐❦♦❧♦✈
◆✳❙✳ ✭❡❞s✮ ●r❛♣❤ ❉r❛✇✐♥❣✳ ●❉ ✷✵✵✺✳ ▲❡❝t✉r❡ ◆♦t❡s ✐♥ ❈♦♠♣✉t❡r ❙❝✐❡♥❝❡✱ ✈♦❧ ✸✽✹✸✳ ❙♣r✐♥❣❡r✱
❇❡r❧✐♥✱ ❍❡✐❞❡❧❜❡r❣✱ ✷✵✵✺✳
❇✐❧❧❛✉t✱ ❏✲❈✳✱ ▲♦♣❡③✱ P✳ ✷✵✶✶❜✱ ✧❈❤❛r❛❝t❡r✐③❛t✐♦♥ ♦❢ ❛❧❧ ρ−❛♣♣r♦①✐♠❛t❡❞ s❡q✉❡♥❝❡s ❢♦r s♦♠❡
s❝❤❡❞✉❧✐♥❣ ♣r♦❜❧❡♠s✧✱ ■❊❊❊ ■♥t❡r♥❛t✐♦♥❛❧ ❈♦♥❢❡r❡♥❝❡ ♦♥ ❊♠❡r❣✐♥❣ ❚❡❝❤♥♦❧♦❣✐❡s ❛♥❞ ❋❛❝t♦r②
❆✉t♦♠❛t✐♦♥✱ ❊❚❋❆✱ ❛rt✳ ✳ ◆♦✳ ✻✵✺✾✵✷✻✳
❇✐❧❧❛✉t✱ ❏✲❈✳✱ ❍❡❜r❛r❞✱ ❊✳ ❛♥❞ ▲♦♣❡③✱ P✳ ✷✵✶✷✱ ✧❈♦♠♣❧❡t❡ ❈❤❛r❛❝t❡r✐③❛t✐♦♥ ♦❢ ◆❡❛r✲❖♣t✐♠❛❧ ❙❡✲
q✉❡♥❝❡s ❢♦r t❤❡ ❚✇♦ ▼❛❝❤✐♥❡ ❋❧♦✇ ❙❤♦♣ ❙❝❤❡❞✉❧✐♥❣ Pr♦❜❧❡♠✧✱ ◆✐♥t❤ ■♥t❡r♥❛t✐♦♥❛❧ ❈♦♥❢❡r❡♥❝❡
♦♥ ■♥t❡❣r❛t✐♦♥ ♦❢ ❆rt✐✜❝✐❛❧ ■♥t❡❧❧✐❣❡♥❝❡ ❛♥❞ ❖♣❡r❛t✐♦♥s ❘❡s❡❛r❝❤ ❚❡❝❤♥✐q✉❡s ✐♥ ❈♦♥str❛✐♥t
Pr♦❣r❛♠♠✐♥❣ ✭❈P❆■❖❘✬✷✵✶✷✮✱ ◆❛♥t❡s✱ ❏✉♥❡ ✷✵✶✷✳
❏❛❝❦s♦♥✱ ❏✲❘✳✱ ✶✾✺✺ ✧❙❝❤❡❞✉❧✐♥❣ ❛ ♣r♦❞✉❝t✐♦♥ ❧✐♥❡ t♦ ♠✐♥✐♠✐③❡ ♠❛①✐♠✉♠ t❛r❞✐♥❡ss✧✱ ❘❡s❡❛r❝❤
r❡♣♦rt ✹✸✱ ▼❛♥❛❣❡♠❡♥t ❙❝✐❡♥❝❡ ❘❡s❡❛r❝❤ Pr♦❥❡❝t✱ ❯♥✐✈❡rs✐t② ♦❢ ❈❛❧✐❢♦r♥✐❛✱ ▲♦s ❆♥❣❡❧❡s✱ ✶✾✺✺✳
❙♠✐t❤✱ ❲✳❊✳ ✶✾✺✻ ✧❱❛r✐♦✉s ♦♣t✐♠✐③❡rs ❢♦r s✐♥❣❧❡ st❛❣❡ ♣r♦❞✉❝t✐♦♥✧✱ ◆❛✈❛❧ ❘❡s❡❛r❝❤ ▲♦❣✐st✐❝s ◗✉❛r✲
t❡r❧②✱ ✸✭✶✲✷✮✿✺✾✲✻✻✱ ✶✾✺✻✳
❚❛✱ ❚✳❚✳❚✐❡♥✱ ❇✐❧❧❛✉t✱ ❏✲❈✳✱ ❋❡❜✉❛r② ✷✵✶✼ ✧❈❤❛r❛❝t❡r✐③❛t✐♦♥ ♦❢ ♦♣t✐♠❛❧ s♦❧✉t✐♦♥s ❢♦r s♦♠❡ ♣r♦❜❧❡♠s
s❝❤❡❞✉❧✐♥❣✧✱ ❘❖❆❉❊❋ ❈♦♥❢❡r❡♥❝❡✱ ▼❡t③✱ ❋r❛♥❝❡✳
❚❛✱ ❚✳❚✳❚✐❡♥✱ ❘✐♥❣❡❛r❞✱ ❑✳✱ ❇✐❧❧❛✉t✱ ❏✲❈✳✱ ❖❝t♦❜❡r ✷✵✶✼ ✧◆❡✇ ♦❜❥❡❝t✐✈❡ ❢✉♥❝t✐♦♥s ❜❛s❡❞ ♦♥ ❥♦❜s
♣♦s✐t✐♦♥s ❢♦r s✐♥❣❧❡ ♠❛❝❤✐♥❡ s❝❤❡❞✉❧✐♥❣ ✇✐t❤ ❞❡❛❞❧✐♥❡s ✧✱ ✼t❤ ■❊❙▼ ❈♦♥❢❡r❡♥❝❡✱ ❙❛❛r❜r✉❝❦❡♥✱
●❡r♠❛♥②✳
228

Minimizing makespan on parallel batch processing
machines
Karim Tamssaouet1,2
, Stéphane Dauzère-Pérès1
, Claude Yugma1
and Jacques Pinaton2
1
Ecole des Mines de Saint-Etienne, Department of Manufacturing Sciences and Logistics
CNRS UMR 6158 LIMOS, Gardanne, France
karim.tamssaouet,dauzere-peres,yugma@emse.fr
2
STMicroelectronics Rousset, Rousset, France
jacques.pinaton@st.com
Keywords: Scheduling, Parallel machines, Batching, Local Search, Semiconductor
Manufacturing
1 Introduction and Problem Description
Batch scheduling problems arise in many industries, such as semiconductor manufac-
turing, aircraft manufacturing, steel casting, transportation, material handling, packaging,
and storage systems. A batch is defined as a group of jobs that can be processed jointly
(Brucker et al. (1998)). Batch scheduling problems are a combination of assignment and se-
quencing problems. The two main decisions are: Forming batches (assigning jobs to batches)
and scheduling the batches on the machines.
This work is motivated by semiconductor manufacturing. Our goal is to optimize
scheduling decisions in the diffusion area which is a complex and critical part of front-
end processing in semiconductor manufacturing (eg. Mehta and Uzsoy (1998); Mathirajan
and Sivakumar (2006); Mönch et al. (2012)). The processes in this area are performed
on two types of equipment: Cleaning machines and furnaces (Yugma et al. (2012)).These
machines can process several lots simultaneously. Moreover, the processing durations can
be very long compared to other operations in a front-end semiconductor manufacturing
facility (fab).
As a starting point, we adopt the novel approach recently proposed by Knopp et al.
(2017) for complex job-shop scheduling problems with batching machines. In this approach,
an adaptation of the classical conjunctive graph is introduced to model batches through
arc labels. Using this new representation, called batch-oblivious graph, schedules are con-
structed and improved during the graph traversal. As this representation does not differ
from the conjunctive graph representation for the flexible job-shop scheduling problem, it
is possible to directly apply the move proposed in Dauzère-Pérès and Paulli (1997) which
integrates the resequencing and reassignment of operations. However, the batch-oblivious
approach in Knopp et al. (2017) considers all operations as candidates for the move while
in Dauzère-Pérès and Paulli (1997) only critical operations are considered.
The contribution of this work is to improve the efficiency of the batch-oblivious ap-
proach when it is dealing with scheduling problems on parallel batch processing machines.
Within the context of a neighborhood-based heuristic, which is the case of the batch-
oblivious approach, the efficiency can be reached by reducing the size of the neighborhood,
i.e. by reducing the set of candidate operations to move. After bringing out that the origi-
nal batch-oblivious graph lacks fundamental properties that underlie efficient neighborhood
structures for scheduling problems without batching, a new construction algorithm is pro-
posed to remedy this. Using this new algorithm, we propose two efficient neighborhood
functions that improve the results obtained by the original batch-oblivious approach.
229

We consider n jobs that arrive dynamically and have to be processed on m identical
parallel machines. The maximum batch size of each machine is B jobs. The jobs belong to
F incompatible families. Only jobs of the same family can be processed together in a batch
due to the chemical nature of the processes. All jobs of family f have the same processing
time pf . Job j has a family denoted by f(j) and a release date denoted by rj . We are
interested in minimizing the makespan Cmax. Using the (α|β|γ) notation of Graham et al.
(1979), the scheduling problem can be denoted by: P|p − batch, incompatible, rj|Cmax.
2 Batch Oblivious Approach
Existing disjunctive graph approaches for scheduling problems with batching rely on
the introduction of dedicated nodes and arcs to explicitly represent batches. To facili-
tate modifications of the graph, the batch-oblivious approach reduces this complexity by
encoding batching decisions into edge weights and keep unchanged the structure of the orig-
inal graph. Using this new representation, an original construction algorithm that takes
batching decisions and improves the schedules on the fly is proposed. As it computes the
processing start dates of the operations while it traverses the graph in the topological or-
der, the proposed construction algorithm changes dynamically the graph in order to fill
up the underutilized batches by bringing backward suitable nodes. This algorithm is com-
plemented by the integrated move of Dauzère-Pérès and Paulli (1997) to resequence and
reassign operations. This combination yields a rich neighborhood that is applied within a
Simulated Annealing (SA) metaheuristic. The latter is embedded in a Greedy Randomized
Adaptive Search Procedure (GRASP).
3 New Construction Algorithm
While we adopt most of the batch-oblivious approach, we propose a new construction
algorithm. In the original algorithm, the graph is traversed in topological order to compute
the processing start dates and constitute batches. If a batch is incomplete, the algorithm
searches for a node of a compatible job that can complete the batch among the set of nodes
that have not been yet assigned a processing start date. If such compatible job is found, it
is required to be available before the already decided start date of the incomplete batch.
If it is the case, the job is moved and inserted at the end of the batch sequence.
The study of the resulting graph shows it lacks a fundamental property of efficient
neighborhood functions (Van Laarhoven et al. (1992)): The removal of an operation from
a machine sequence cannot increase start times. It is easier to construct an example of a
batch-oblivious graph when the removal of an operation degrades the solution. The new
construction algorithm proposed in this work then modifies the graph in a way that deleting
an operation cannot increase start times. This algorithm uses Property 1.
Property 1. If all batch operations are sequenced in the non-increasing order of the job
release dates, no operation deletion can degrade the solution quality.
So, instead of inserting an operation at the end of the batch sequence, it will be inserted
in the position that leads to the satisfaction of the condition in Property 1. Note that
the new construction algorithm leads to the same solution and only changes the solution
representation.
4 New Neighborhood Functions
The new algorithm thus uses a solution representation where removing an operation
does not increase start times. This leads to the direct use of the classical move where
230

only critical operations are candidates. This restriction to critical operations is justified
by Property 2. This new neighborhood function can quickly lead to a good solution as it
discards uninteresting moves and only focuses on promising ones.
Property 2. If a move of a non-critical operation can improve the solution, there is always
a move of a critical operation that leads to a solution with the same or a better quality.
Based on the same solution representation, the size of the neighborhood of a solution
that can be reached through the move of critical operations can also be reduced. This
additional reduction uses Property 3. Before stating the property, two types of operation
are given in Definition 1 and Definition 2. This classification is based on the position of an
operation in the sequence of its batch.
Definition 1 (First Batched Operation). It is a first operation in a batch sequence in
the batch oblivious conjunctive graph representation.
Definition 2 (Internal Batched Operation). It is an operation in a batch which is not
the First Batched Operation.
Property 3. If a move of a critical Internal Batched Operation can improve the solution,
there is always a move of a critical First Batched Operation that leads to a solution with
the same or a better quality.
As the constructed solutions respect the condition in Property 1, the first operation in
the batch sequence is in fact the last available among all the operations belonging to the
batch. Property 3 then helps reducing the candidate set of operations to move to those
that are critical and the last available in their batch.
5 Computational Results
To assess the efficiency of the two proposed neighborhood functions, the instances of
Mönch et al. (2005) are used, except that the makespan is considered instead of the total
weighted tardiness. Three implementations are compared:
1. (OI): Original implementation of Knopp et al. (2017).
2. (NC): The construction algorithm is modified so that Property 1 is satisfied. Moreover,
only critical operations are moved instead of any operation.
3. (NCB): Similar to (NC), except that only critical first batched operations are consid-
ered instead of any critical operation.
Table 1 shows the results of the three implementations on the 160 instances using the
same experimentation parameters. Table 1 shows that the ideas presented in this work
improve the performance of the original batch-oblivious approach. For example, (NCB)
obtains the best solution for 70% of the instances while the original approach finds the
best solution for 43%. The dominance of (NC) and (NCB) over (OI) can also be observed
when analyzing the average gap (AverageGap) and the maximum gap (MaxGap) to the
best solution. When comparing (NC) and (NCB), even if the improvement is not significant,
the interest of improving the local search efficiency is confirmed.
231

Table 1. Results for instances of Mönch et al. (2005) with makespan objective
Implementation % Found Best AverageGap MaxGap
(OI) 43 % 0.83 % 6.27 %
(NC) 65 % 0.24 % 3.00 %
(NCB) 70 % 0.20 % 3.00 %
6 Conclusion
In this work, we study the scheduling problem of minimizing makespan on parallel
identical batching machines with dynamic job arrivals and incompatible families. Based
on the batch-oblivious approach presented in Knopp et al. (2017), two new neighborhood
functions are proposed. These two functions improve the search performance by reducing
the size of the neighborhood to explore. Numerical experiments on academic instances
show that the local search efficiency is improved. An important perspective of this work is
to extend the analysis to the case where routing precedence constraints are considered.
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232

Order Acceptance and Scheduling Problem with Batch
Delivery
İstenç Tarhan and Ceyda Oğuz
Koç University, Turkey
itarhan15@ku.edu.tr, coguz@ku.edu.tr
Keywords: Order acceptance, scheduling, batch delivery, sequence dependent setup times,
metaheuristics, dynamic programming
1 Introduction
Scheduling problems are highly diverse though the principal objective of them is similar,
which is satisfying the orders while utilizing the scarce resources as efficiently as possible.
However, as the firms operating on a make-to-order basis, satisfaction of the entire demand
may not be possible due the capacity limitations and tight delivery time requirements faced
by the firm. This necessitates selecting only part of customer orders to maximize the total
revenue, which gives rise to the order acceptance and scheduling (OAS) problems. We
consider a make-to-order production system, where limited production capacity and order
delivery requirements necessitate selective acceptance of the orders. It is often assumed that
each order is delivered individually (i.e. immediately after their completion). However,
orders may be sent in batches to decrease the transportation cost. Therefore, batching
decisions along with order acceptance and scheduling decisions should be considered. Since
batching decisions directly affect the tardiness, all decisions should be taken jointly. Herein,
we study the problem called as the order acceptance and scheduling problems with batching
(OASB) and present an iterated local search algorithm (ILS) to solve it.
2 Literature review
Charnsirisakskul et. al. (2004) define an order acceptance problem in which the cus-
tomer does not place an order if the manufacturer cannot complete the order by the latest
acceptable due date. Oguz et. al. (2010) study a different version of the problem where
sequence dependent setup times and release times are included. They develop a simulated
annealing based heuristic and two constructive heuristics. Cesaret et. al. (2012) propose
a tabu search algorithm for the same problem that improves the best solution of many
test instances. Chaurasia and Singh (2016) develop two hybrid metaheuristic approaches,
a hybrid steady-state genetic algorithm and a hybrid evolutionary algorithm and improved
the best solutions further.
The timing issue and package delivery is determined first by Cheng and Kahlbacher
(1993). Although there have been studies regarding batch delivery in the following years
such as Potts and Kovalyov (2000), Hall and Potts (2005) and Cakici et. al. (2014), batch
delivery in the OAS problem is addressed for the first time by Khalili et. al. (2016). Authors
propose an imperialist competitive algorithm for which the gap between the best solution
found by CPLEX solver in 3600s and the solution found by the proposed algorithm can be
up to 150%. Computational results of this study show that better solution methodologies
can be developed for this problem.
233

3 Problem definition
In a single machine environment, we are given a set of independent orders O at the
beginning of the planning period. For each order i ∈ O, we have data on its customer
qi, qi ∈ Q, its release time ri, processing time pi, due date di, deadline ¯
di such that
di ≤ ¯
di, sequence dependent setup times where each element sti,j is the time that has to
be incurred before order j is processed, if order i immediately precedes order j , revenue ei
which denotes the maximum gain from order i, and unit tardiness penalty cost wi. Orders
are delivered in batches to the customers (it is assumed that there are infinite number of
uncapacitated vehicles for delivery). Each batch can only contain the orders belonging to
a single customer and it can be delivered only if all orders in that batch are completed.
Therefore, the delivery time of an order i, deli, is the completion time of the latest order in
batch ki, ki ∈ K, that includes order i, c̄ki
; deli = c̄ki
and c̄ki
= maxj:kj =ki
cj where cj is
the completion time of order j. The manufacturer may deliver order i until its deadline ¯
di,
but for each time unit beyond its due date, she incurs a tardiness penalty cost. Accordingly,
tardiness of order i, Ti is equal to max{0, deli − di}.
Given a sequence σs of the selected orders S ⊆ O and the number of batches nb
(in another sense, deliveries) of corresponding sequence, revenue generated from order i,
denoted by Ri(σs), is calculated as Ri(σs) = max{0, ei − Tiwi}. Consequently, the total
revenue gained from processing orders in S in sequence σs is TR(σs) =
P
i∈S Ri(σs) and
net revenue is TR − nbf where f is the fixed cost of a delivery. Hence the OASB problem
is to find the set S, the sequence σs and its batching configuration so that the net revenue
is maximized. We are not presenting the corresponding MILP here due to page limitation.
In a nutshell, MILP consists of three groups of constraints regarding i) sequence of orders,
ii) batch configuration of orders and iii) computation of tardiness, completion and delivery
times.
An extensive set of test instances with 10, 15, 25 and 50 orders is solved by CPLEX
using the MILP formulation. Results show that MILP can handle all instances with 10
orders and some instances with 15 orders. However, none of the instances with 25 and 50
orders can be solved to optimality by MILP in one hour. Therefore, we propose an iterated
search algorithm that is capable of solving the majority of the instances with 10 orders to
optimality and for the instances with higher number of orders, providing better solutions
than CPLEX in much shorter time.
4 Proposed metaheuristic algorithm
The general steps of the proposed metaheuristic algorithm ILS are given in Algorithm 1.
Solution x is encoded as a sequence including all orders. The proposed algorithm is founded
on two types of neighborhoods: Swap and Insertion. Swap neighborhood of a solution x
includes all solutions that can be obtained by swapping any two orders of it. Insertion
neighborhood of a solution x includes all solutions that can be obtained by shifting a single
order or consecutive two orders in the schedule of solution x. In the OASB problem, even
though the corresponding neighborhoods may not include a solution that is better than
the incumbent solution, they are likely to include a solution having the same objective
value with the incumbent solution. Thus, it is promising to employ moves, which have the
same objective value as the incumbent solution, simultaneously. To this end, SwapXSwap,
SwapXInsertion and InsertionXInsertion neighborhoods include all solutions generated
by simultaneously employing promising two swap moves, swap and insertion move, and two
insertion moves, respectively. For each neighborhood defined, there is a function returning
the best solution in the corresponding neighborhood and they are employed in variable
neighborhood search manner.
234

If the best solution cannot be improved through these moves, Perturbation(x∗
) is called
to perturb the best solution. Perturbation(x∗
) is performed by first dividing solution x∗
into a set of blocks and then resequencing these blocks. The first part, that is dividing
solution x∗
into a set of unconnected blocks of orders, is realized by ejecting a set of orders
chosen randomly. The second part, that is forming solution x again, is accomplished by
obtaining the optimal sequence of these blocks via a dynamic programming algorithm. To
avoid cycling, that is to ensure that the perturbed solution is different from x∗
, the ejected
orders are not allowed to be appended to the end of blocks that precede them in solution
x∗
.
The algorithm returns to Step 1 after the perturbation and terminates after a certain
number of perturbations.
Algorithm 1 Pseudocode of the proposed algorithm
Input: Current solution x, Objective value of x is f(x), best solution x∗
, perturbation number
p = 0, maximum number of perturbations maxp
Update_best_solution(x′
) {x ← x′
, x∗
← x′
, go to line 2}
1: while p maxp do
2: x′
:= Swap(x)
3: if f(x′
) f(x) then Update_best_solution(x′
) end if
4: x′
:= Insertion(x)
5: if f(x′
) end if
6: x′
:= SwapXSwap(x)
7: if f(x′
) end if
8: x′
:= SwapXInsertion(x)
9: if f(x′
) end if
10: x′
:= InsertionXInsertion(x)
11: if f(x′
) end if
12: x := Perturbation(x∗
)
13: p + +
14: end while
5 Computational results
The proposed algorithm ILS is coded in C++ and the MILP model of the OASB
problem is solved by CPLEX 12.5.1 for comparison purposes. All computations are executed
in an Intel Core i7 with 2.60 GHz and 8 GB of RAM running Windows 7. The proposed
algorithm is tested with the benchmark instances suggested in Cesaret et. al. (2012). Test-
problems have four different sizes (number of orders), more specifically, n = 10, 15, 25 and
50. Two additional parameters are used to create instances with varying characteristics,
namely τ and R. The first is a tardiness factor, while the second is a due date range; both
parameters were chosen from 3 possible values: 0.1, 0.5 and 0.9. For each combination of
these parameters, ten instances are solved and average results are provided in Table 1.
CPLEX is set to terminate in at most one hour and under this limitation, it is unable
to provide tight upper bounds when the the number of orders is higher than 10. Thus, the
optimality gap of the proposed algorithm seems high. However, it outperforms CPLEX in
terms of both solution quality and time when n is larger than 10. While CPLEX could
solve all instances with 10 orders to the optimality, the proposed algorithm can solve the
majority of these instances to the optimality in much shorter time.
235

n R τ Optimality gap (%) Solution time (s)
CPLEX ILS CPLEX ILS
10 0,1 0,1 0,00 0,00 563,73 0,03
0,5 0,00 0,14 158,72 0,02
0,9 0,00 0,00 1,05 0,01
0,5 0,1 0,00 0,22 882,63 0,03
0,5 0,00 0,00 145,61 0,02
0,9 0,00 0,68 2,50 0,01
0,9 0,1 0,00 0,00 662,07 0,04
0,5 0,00 0,08 144,50 0,02
0,9 0,00 0,00 4,09 0,01
15 0,1 0,1 8,81 7,91 3600,00 0,06
0,5 23,35 23,17 3600,00 0,04
0,9 0,00 0,00 48,19 0,02
0,5 0,1 9,24 9,19 3600,00 0,07
0,5 24,35 23,09 3600,00 0,04
0,9 2,34 2,34 689,15 0,02
0,9 0,1 10,99 10,21 3600,00 0,06
0,5 19,94 19,20 3600,00 0,04
0,9 1,02 1,02 791,98 0,02
25 0,1 0,1 13,30 10,11 3600,00 0,60
0,5 25,46 18,65 3600,00 0,28
0,9 13,84 11,94 3600,00 0,12
0,5 0,1 13,39 8,49 3600,00 0,62
0,5 20,59 16,14 3600,00 0,31
0,9 15,28 12,49 3600,00 0,12
0,9 0,1 15,39 9,39 3600,00 0,56
0,5 16,36 13,92 3600,00 0,26
0,9 10,19 10,19 3600,00 0,13
50 0,1 0,1 18,12 7,53 3600,00 2,73
0,5 34,34 15,04 3600,00 1,14
0,9 * * * *
0,5 0,1 20,66 7,10 3600,00 2,96
0,5 30,11 13,93 3600,00 0,99
0,9 * * * *
0,9 0,1 18,18 6,19 3600,00 2,74
0,5 11,24 6,24 3600,00 0,92
0,9 * * * *
Table 1: Computational comparison of the proposed algorithm and CPLEX
*Instances with asterisk cannot be compared since CPLEX runs out of memory.
6 Conclusion
We provide a competitive iterated local search algorithm ILS for the order acceptance
and scheduling problem with batch delivery in a single machine environment. ILS includes a
variable neighborhood search and a dynamic programming algorithm to perturb a solution
if the algorithm is stuck at a local optima. Computational results show that the proposed
algorithm can ﬁnd the optimal solutions for the majority of the instances with 10 orders
and can ﬁnd better solutions than CPLEX for the instances with higher number of orders
in much shorter time.
References
Cakici E., S. J. Mason, H. N. Geisman and J. W. Fowler, 2014, “Scheduling parallel machines with
single vehicle delivery, Journal of Heuristics, Vol. 20, pp. 511-537.
Cesaret B., C. Oguz and F. S. Salman, 2012, “A tabu search algorithm for order acceptance and
scheduling, Computers and Operations Research, Vol. 39, pp. 1197-1205.
Charnsirisakskul K., P. M. Griffin and P. Keskinocak, 2004, “Order selection and scheduling with
leadtime flexibility, IIE Transactions, Vol. 36, pp. 194-205.
Chaurasia S. N., A. Singh, 2016, “Hybrid evolutionary approaches for the single machine order
acceptance and scheduling problem, Applied Soft Computing, Vol. 177, pp. 2033-2049.
Cheng T. C. E., H. G. Kahlbacher, 1993, “Scheduling with delivery and earliness penalties, Asia-
Pacific Journal of Operational Research, Vol. 10, pp. 145-152.
Hall N. G., C. N. Potts, 2005, “Scheduling with batching: A review, European Journal of Opera-
tional Research, Vol. 120, pp. 228-249.
Khalili M., M. Esmailpour and B. Naderi, 2016, “The production-distribution problem with order
acceptance and package delivery: Models and algorithm, Manufacturing Review, Vol. 3, pp.
194-205.
Oguz C., F. S. Salman and Z. B. Yalcin, 2010, “Order acceptance and scheduling decisions in
make-to-order systems, International Journal of Production Economics, Vol. 125, pp. 200-
211.
Potts C. N., M. Y. Kovalyov, 2000, “Scheduling with batching: A review, European Journal of
Operational Research, Vol. 120, pp. 228-249.
236

Energy Conscious Scheduling of Robot Moves in
Dual-Gripper Robotic Cells
Nurdan Tatar1
, Hakan Gültekin1
, Sinan Gürel2
1
TOBB University of Economy and Technology, Ankara, Turkey
{nurdantatar,hakan.gultekin}@gmail.com
2
Middle East Technical University, Ankara, Turkey
gsinan@metu.edu.tr
Keywords: Robotic cell scheduling, Dual gripper, energy optimization, bicriteria opti-
mization.
1 Introduction
In cellular manufacturing systems, one of the most used material handling devices is
an industrial robot. Among the interrelated issues to be considered in using robotic cells,
the scheduling of robot moves is one of the most significant ones. In such a robotic cell,
the processing of the parts are performed by the machines and the transportation of the
parts between the machines and the loading/unloading of the machines are performed by
a material handling robot.
The robotic cells consist of an input device, a series of processing stages, an output
device, and robots for material handling within the cell. Most robotic cells examined in
the literature assume one of the two layouts: linear or circular (Dawande et al. 2005). On
the other hand, two types of robots are discussed in the literature: A single gripper robot,
which can hold only one part at a time, and in contrast, a dual-gripper robot, that can
hold two parts simultaneously. If the robotic cell produces identical parts, we refer to it
as a single part-type. The identical parts cyclic scheduling problem is then to find the
shortest cyclic schedule for the robot; i.e., a sequence of robot moves that can be repeated
indefinitely.
From an optimization point of view, the most widely used objective in the literature is
that of maximizing the throughput; i.e. minimizing the cycle time. To achieve the maximum
throughput (minimum cycle time) it is assumed that the robot performs the operations at
its maximal speed. However, this may not be the most efficient energy consumption policy;
at its maximum speed, the energy consumption is also at its maximum. On the other hand,
the robot may need to wait in front of some of the machines to unload them because of
reaching them earlier than necessary (before the processing is completed). Accordingly,
there is a considerable potential for energy saving (Drobouchevitch et al. 2004).
In this study, we consider an m-machine robotic cell, where a dual-gripper robot is
used, as illustrated in Figure 1. The robot moves linearly along a track (linear layout) and
the system follows the flow shop assumption which means that each part goes through the
same sequence of machines (M1 → M2 → . . . → Mm). However, the sequence of robot
activities may be different. One of the decisions is to determine the optimal sequence of
these robot activities. However, this is a complex task, because even in a two-machine
dual-gripper robotic cells there is a total of 52 robot activity sequences that produce one
part (1-unit cycles) (Sethi et al. 2001).
The other decision is to determine the robot’s speeds in each of its moves to minimize
the total energy consumption. Therefore, we consider a bicriteria problem. There are very
few studies that consider bicriteria robotic cell scheduling problem. Gültekin et al. (2008,
2010) assumed that the processing times on the machines are controllable and considered
237

Fig. 1. A Dual Gripper Robotic Cell.
the problem of minimizing the cycle time and total manufacturing cost in a single grip-
per robotic cell. This study is the first one to consider the energy consumption of the
robot together with the cycle time minimization objective in a dual gripper robotic cell.
For this problem, we adapted the epsilon-constraint method and moved the cycle time
objective to constraints by including an upper bound on it and developed a mixed integer
nonlinear mathematical programming formulation (MINLP). This MINLP is solved using
the BARON solver in GAMS. To improve its performance the nonlinear terms are refor-
mulated to build conic quadratic representation. The new transformed model is a mixed
integer quadratic conic programming (MIQCP) and solved with CPLEX 12.6.2.
In the next section, we define the problem and introduce the notation, in Section 3 we
provide the solution methodology. Section 4 concludes the study.
2 Problem definition and notations
In this section, we develop a notational and mathematical modelling framework for the
given problem. Being consistent with the existing studies in the literature, the following
notation is used to describe the problem.
2.1 Parameters
ϵ: Loading/unloading time of machine
dij: Distance between machine Mi and Mj
vij: Traveling speed of the robot between machine Mi and Mj
δij: Traveling time of the robot between machine Mi and Mj, δij = dij/vij
θ: The time for switching the robot grippers
2.2 Robot states
The following notation is used for our analysis of the robot states:
(g1, g2): where gi ∈ ¸{0, 1, . . . , m + 1}: represents the state of the grippers. For instance, (2,4)
means the first gripper (g1) has a part that requires processing next on machine 2 and
the second gripper (g2) has a part that requires processing next on machine 4, where
gi = 0 (e.g. state (0,0)) means that there is no part on the gripper i.
238

Li: The robot activity that indicates a loading operation onto Mi. Just after loading a part
on this machine and it has no part on the corresponding gripper. Therefore, just after
this operation the corresponding gripper states are either (0, g2) or (g1, 0).
Ui: The robot activity that indicates an unloading operation from Mi. Just after perform-
ing this operation the robot has at least one part in one of its grippers that require
processing next on machine Mi+1. Therefore the corresponding gripper states are either
((i + 1), g2) or (g1, (i + 1)).
In order to understand better, consider the following example of a dual gripper robot
sequence for a two machine case: U0(1, 0) → U1(1, 2) → L1(0, 2) → U2(3, 2) → L2(3, 0) →
L3(0, 0). As it can be seen, there are 2(m + 1) = 6 robot activities in this sequence in
which U0(0, 1) is the first. After unloading a part from the input buffer, the robot unloads
another part with the other gripper from the first machine. Now, both grippers are full
and the robot is in front of machine 1. Since both grippers are not empty (g1, g2 ̸= 0), the
next robot activity must be a loading one. That is why we have a L1 as the third activity.
It means that the robot loads the part that it transported from the input buffer to M1.
Similar sequence of unloading and loading activities are performed on the second machine
and finally the unloaded part from M2 is transported to the output buffer.
Our mathematical programming formulation sequence the unloading and loading activ-
ities of the robot while satisfying the feasibility of this sequence. A feasible sequence must
load and unload each machine once, must not try to load an already loaded machine, un-
load an already empty machine, unload a machine with a loaded gripper, or load a machine
with an empty gripper. The model considers all such feasibility constraints, determines the
starting times of all the sequenced activities together with robot move speeds.
3 Solution methodology
The minimization of the cycle time and the minimization of the energy consumption
are conflicting objectives. In other words, improving one of them will sacrifice the other one
and further achievement on the cycle time (energy consumption) can only be accomplished
at the expense of higher energy consumption (cycle time).
To handle this bicriteria problem, we used the epsilon-constraint method, in which one
of the objectives is written as a constraint with an upper bound on its value. By utilizing
different upper bounds, different non-dominated solutions are generated. In this study, we
considered the cycle time objective as a constraint. Therefore, the problem becomes the
minimization of the total energy consumption subject to a given upper bound on the cycle
time.
Figure 2 shows a set of Pareto efficient solutions for a 2-machine problem instance. It
depicts the rate of change in the energy consumption and the change of optimal robot
sequence when different cycle time upper bounds are used. In this figure Solution (1∗
)
corresponds to the problem where the robot’s speeds are at their upper limits. If a machine
has not completed the processing of a part when the robot arrives to unload it, the robot
waits in front of the machine. In such cases, instead of waiting, the robot can make its
previous moves slower. This situation corresponds to solution (2∗
) in Figure 2. Which
results in the same cycle time with Solution 1∗
with a 28% less energy consumption.
Increasing the cycle time upper bound by 5 units, changes the optimal robot activity
sequence and leads to decrease in energy consumption as depicted in Solution (3∗
). When
the cycle time upper bound is large enough, robot’s speeds of all moves becomes equivalent
to their lower bounds. In this case, the second gripper is never used as depicted in Solution
(4∗
).
239

Fig. 2. Cycle Time vs. Energy Consumption.
Table 1 shows the elapsed times to attain the above solutions with MINLP and MIQCP
formulations. To get all 8 non-dominated solutions took 66.7 seconds with MIQCP and
11529.4 with MINLP for this problem instance.
Table 1. Performance of MIQCP and MINLP for a 2-machine problem
MIQCP MINLP
CT Time (s) Time (s)
38.8 10.1 3646.2
43.8 9.0 3183.1
48.8 5.9 2222.4
53.8 10.8 1571.0
58.8 10.9 274.4
63.8 5.0 384.8
66.01 8.4 162.3
112 6.6 85.3
Total 66.7 11529.4
To test the performance of MIQCP and MINLP models, 100 instances of the prob-
lem are generated with different parameter values. In these test problems, the number of
machines varies between 2 and 6. For two-machine instances, all non-dominated solutions
are evaluated optimally with both MIQCP and MINLP. However, for 3 and 4 machine in-
stances, MINLP could not find any solution within the given time limit of 3 hours, whereas
MIQCP was able to find each non-nominated solution in 23.3 and 231.7 seconds for 3 and
4 machines, respectively. However, it was not possible to solve the instances with 5 and
6 machines with MIQCP with three-hour time limit. When robotic cells with speed con-
trol is compared with robotic cells without any speed control, our results reveal that the
controllability of robot speeds yields 30% energy savings on the average.
240

4 Conclusion
This study, addresses a flow shop robotic cell scheduling problem consisting of m-
machines, each of which performs a different operation on the parts, plus an input and
output buffer, and a dual gripper robot that moves linearly along a track to transport the
parts between the machines. We deal with a bicriteria scheduling problem for optimizing
the cycle time and energy consumption of the robot at the same time.
We developed two mathematical models; a mixed integer nonlinear mathematical pro-
gramming formulation and a mixed integer quadratic conic programming formulation. Both
are evaluated with the same data sets and it is shown that MIQCP is much more efficient
than the MINLP in terms of the solution time. However, for larger number of machines,
the MIQCP formulation also fails to find solutions in reasonable times.
By means of the proposed approach of this study, which utilizes the controllability
of the robot speeds, we can get not only an economic return but also an environmental
benefit through reducing carbon emissions by decreasing the need for electric power across
the manufacturing sector.
For further studies, we plan to develop a heuristic algorithm to solve large problem
instances where the MIQCP formulation is not sufficient. Also, multiple part-type case can
be considered instead of producing identical parts.
Acknowledgements
This research is supported by the Scientific and Technological Council of Turkey (TUBİTAK)
under grant number 215M845.
References
Dawande, M., Geismar, H., Sethi, S., and Sriskandarajah, C., 2005, “Sequencing and Scheduling
in Robotic Cells: Recent Developments”, Journal of Scheduling, Vol. 8(5), pp. 387–426.
Drobouchevitch, I.G., Sethi, S., Sidney, J., and Sriskandarajah, C., 2004, “A note on scheduling
multiple parts in two-machine dual gripper robotic cell: Heuristic algorithm and performance
guarantee”, International Journal of Operations and Quantitative Management, Vol. 10(4),
pp. 297–314.
Gültekin, H., Akturk, M.S., Karasan, O.E., 2008, “Bicriteria robotic cell scheduling”, Journal of
Scheduling, Vol. 11(6), pp. 457–473.
Gültekin, H., Akturk, M.S., Karasan, O.E., 2010, “Bicriteria robotic operation allocation in a
flexible manufacturing cell”,”, Computers Operations Research, Vol. 40(2), pp. 639–653.
Sethi, S., Sidney, J., and Sriskandarajah, C., 2001, “Scheduling in dual gripper robotic cells for
productivity gains”, IEEE Transactions on Robotics and Automation, Vol. 17(3), pp. 324–341.
241

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t✐♠❡ ❛♥❞ s❝❛r❝❡ r❡s♦✉r❝❡s ❢♦r ❡①❡❝✉t✐♦♥❀ s♦✉❣❤t ✐s ❛ ✈❡❝t♦r ♦❢ st❛rt t✐♠❡s ❢♦r t❤❡ ❛❝t✐✈✐t✐❡s
s✉❝❤ t❤❛t ❛❧❧ ♣r❡❝❡❞❡♥❝❡ r❡❧❛t✐♦♥s❤✐♣s ❛r❡ r❡s♣❡❝t❡❞✱ t❤❡ t♦t❛❧ r❡q✉✐r❡❞ q✉❛♥t✐t② ♦❢ ❡❛❝❤
r❡s♦✉r❝❡ ♥❡✈❡r ❡①❝❡❡❞s ✐ts ♣r❡s❝r✐❜❡❞ ❝❛♣❛❝✐t②✱ ❛♥❞ t❤❡ t♦t❛❧ ♣r♦❥❡❝t ❞✉r❛t✐♦♥ ✐s ♠✐♥✐♠✐③❡❞✳
❚❤❡ ❘❈P❙P ♣♦s❡s ❛ ❝❤❛❧❧❡♥❣✐♥❣ ❝♦♠❜✐♥❛t♦r✐❛❧ ♦♣t✐♠✐③❛t✐♦♥ ♣r♦❜❧❡♠❀ ✐♥ ❛❞❞✐t✐♦♥ t♦ ♠❛♥②
♣r♦❜❧❡♠✲s♣❡❝✐✜❝ s♦❧✉t✐♦♥ ❛♣♣r♦❛❝❤❡s✱ ✈❛r✐♦✉s t②♣❡s ♦❢ ♠✐①❡❞✲✐♥t❡❣❡r ❧✐♥❡❛r ♣r♦❣r❛♠♠✐♥❣
✭▼■▲P✮ ♠♦❞❡❧s ❤❛✈❡ ❜❡❡♥ ♣r♦♣♦s❡❞✱ ✇❤✐❝❤ ♥♦✇ r❡❝❡✐✈❡ ✐♥❝r❡❛s❡❞ ❛tt❡♥t✐♦♥ ❞✉❡ t♦ t❤❡
✐♠♣r♦✈❡❞ ♣❡r❢♦r♠❛♥❝❡ ♦❢ ▼■▲P s♦❧✈❡rs ❛♥❞ ❝♦♠♣✉t❡r ❤❛r❞✇❛r❡✳
❚✇♦ ❝❧❛ss❡s ♦❢ ♠♦❞❡❧s ❡①✐st ✭❝❢✳ ❆rt✐❣✉❡s ❡t ❛❧✳ ✷✵✶✺✮✿ ❞✐s❝r❡t❡✲t✐♠❡ ✭❉❚✮ ♠♦❞❡❧s ❛♥❞
❝♦♥t✐♥✉♦✉s✲t✐♠❡ ✭❈❚✮ ♠♦❞❡❧s✳ ■♥ ❉❚ ♠♦❞❡❧s✱ t❤❡ ♣❧❛♥♥✐♥❣ ❤♦r✐③♦♥ ✐s ❞✐✈✐❞❡❞ ✐♥t♦ ❛ s❡t ♦❢
❡q✉❛❧✲❧❡♥❣t❤ t✐♠❡ ✐♥t❡r✈❛❧s✱ ❛♥❞ ❛❝t✐✈✐t✐❡s ❝❛♥ st❛rt ♦♥❧② ❛t t❤❡ ❜❡❣✐♥♥✐♥❣ ♦❢ ❡❛❝❤ ♦❢ t❤❡s❡
✐♥t❡r✈❛❧s❀ ❜② ❝♦♥tr❛st✱ ✐♥ ❈❚ ♠♦❞❡❧s✱ ❛❝t✐✈✐t✐❡s ❝❛♥ st❛rt ❛t ❛♥② ♣♦✐♥t ✐♥ t✐♠❡ ♦✈❡r t❤❡
♣❧❛♥♥✐♥❣ ❤♦r✐③♦♥✳ ■♥ ❣❡♥❡r❛❧✱ ❉❚ ♠♦❞❡❧s ✐♥✈♦❧✈❡ t✐♠❡✲✐♥❞❡①❡❞ ❜✐♥❛r② ✈❛r✐❛❜❧❡s✱ ❡✳❣✳✱ ♣✉❧s❡
✈❛r✐❛❜❧❡s✱ ❝❢✳ Pr✐ts❦❡r ❡t ❛❧✳ ✭✶✾✻✾✮ ❛♥❞ ❈❤r✐st♦✜❞❡s ❡t ❛❧✳ ✭✶✾✽✼✮❀ st❡♣ ✈❛r✐❛❜❧❡s✱ ❝❢✳ ❑❛♣❧❛♥
✭✶✾✽✽✮ ❛♥❞ ❑❧❡✐♥ ✭✷✵✵✵✮❀ st❡♣ ✈❛r✐❛❜❧❡s ❛♥❞ ♣❡r❝❡♥t❛❣❡✲♦❢✲❝♦♠♣❧❡t✐♦♥ ✈❛r✐❛❜❧❡s✱ ❝❢✳ ❇✐❛♥❝♦
❛♥❞ ❈❛r❛♠✐❛ ✭✷✵✶✸✮❀ ♦r ♦♥✴♦✛ ✈❛r✐❛❜❧❡s✱ ❝❢✳ ❑♦♣❛♥♦s ❡t ❛❧✳ ✭✷✵✶✹✮✳ ■♥ ❛❧❧ t❤❡s❡ ♠♦❞❡❧s✱
t❤❡ ♥✉♠❜❡r ♦❢ ❜✐♥❛r② ✈❛r✐❛❜❧❡s ❣r♦✇s ✇✐t❤ t❤❡ ♥✉♠❜❡r ♦❢ t✐♠❡ ✐♥t❡r✈❛❧s ❝♦♥s✐❞❡r❡❞✱ ✇❤✐❝❤
✐s ❞✐s❛❞✈❛♥t❛❣❡♦✉s ✐♥ t❤❡ ❝❛s❡ ♦❢ ❧♦♥❣ ❛❝t✐✈✐t② ❞✉r❛t✐♦♥s✳ ■♥ t❤❡ ✇❡❧❧✲❦♥♦✇♥ ❈❚ ♠♦❞❡❧ ♦❢
❆rt✐❣✉❡s ❡t ❛❧✳ ✭✷✵✵✸✮✱ r❡s♦✉r❝❡ ✢♦✇ ✈❛r✐❛❜❧❡s ❛r❡ ✉s❡❞ t♦ ♠♦❞❡❧ t❤❡ r❡s♦✉r❝❡ ❝♦♥str❛✐♥ts✳
❆❝❝♦r❞✐♥❣ t♦ ❑♦♥é ❡t ❛❧✳ ✭✷✵✶✶✮✱ ❢♦r ✐♥st❛♥❝❡s ✇✐t❤ s❤♦rt ♣❧❛♥♥✐♥❣ ❤♦r✐③♦♥s✱ ❉❚ ♠♦❞❡❧s
❡①❤✐❜✐t ❛ ❜❡tt❡r ♣❡r❢♦r♠❛♥❝❡ t❤❛♥ ❈❚ ♠♦❞❡❧s❀ ❢♦r ✐♥st❛♥❝❡s ✇✐t❤ r❡❧❛t✐✈❡❧② ❧♦♥❣ ♣❧❛♥♥✐♥❣
❤♦r✐③♦♥s✱ ❤♦✇❡✈❡r✱ ❈❚ ♠♦❞❡❧s ♣r♦✈✐❞❡ ❜❡tt❡r r❡s✉❧ts t❤❛♥ ❉❚ ♠♦❞❡❧s✳
■♥ t❤✐s ♣❛♣❡r✱ ✇❡ ♣r❡s❡♥t ❛ ♥♦✈❡❧ ❈❚ ♠♦❞❡❧ ❢♦r t❤❡ ❘❈P❙P❀ ❛ ♣r❡❧✐♠✐♥❛r② ✈❡rs✐♦♥ ♦❢
t❤❡ ♠♦❞❡❧✱ ✇✐t❤ s♦♠❡ r❡❞✉♥❞❛♥t ❝♦♥str❛✐♥ts✱ ❤❛s ❜❡❡♥ ♣✉❜❧✐s❤❡❞ ✐♥ ❘✐❤♠ ❛♥❞ ❚r❛✉t♠❛♥♥
✭✷✵✶✼✮✳ ❚♦ ♠♦❞❡❧ t❤❡ r❡s♦✉r❝❡ ❝♦♥str❛✐♥ts✱ ✇❡ ✉s❡ t✇♦ t②♣❡s ♦❢ ❜✐♥❛r② ✈❛r✐❛❜❧❡s✿ ❛ss✐❣♥♠❡♥t
✈❛r✐❛❜❧❡s s♣❡❝✐❢② ✇❤✐❝❤ ✐♥❞✐✈✐❞✉❛❧ r❡s♦✉r❝❡ ✉♥✐ts ❛r❡ ✉s❡❞ ❢♦r t❤❡ ❡①❡❝✉t✐♦♥ ♦❢ ❡❛❝❤ ❛❝t✐✈✐t②✱
❛♥❞ s❡q✉❡♥❝✐♥❣ ✈❛r✐❛❜❧❡s s♣❡❝✐❢② t❤❡ ♦r❞❡r ✐♥ ✇❤✐❝❤ ♣❛✐rs ♦❢ ❛❝t✐✈✐t✐❡s t❤❛t ❛r❡ ❛ss✐❣♥❡❞ t♦
t❤❡ s❛♠❡ r❡s♦✉r❝❡ ✉♥✐t ❛r❡ ♣r♦❝❡ss❡❞✳ ❚♦ ❡♥❤❛♥❝❡ t❤❡ ♣❡r❢♦r♠❛♥❝❡ ♦❢ t❤❡ ♠♦❞❡❧✱ ✇❡ ♠♦❞✐❢②
t❤❡ s❡q✉❡♥❝✐♥❣ ❝♦♥str❛✐♥ts ❢♦r ♣❛✐rs ❛♥❞ tr✐♣❧❡ts ♦❢ ❛❝t✐✈✐t✐❡s t❤❛t ❝❛♥♥♦t ❜❡ ♣r♦❝❡ss❡❞ ✐♥
♣❛r❛❧❧❡❧✱ ❛♥❞ ✇❡ ❡❧✐♠✐♥❛t❡ s♦♠❡ s②♠♠❡tr✐❝ s♦❧✉t✐♦♥s ❢r♦♠ t❤❡ s❡❛r❝❤ s♣❛❝❡✳ ■♥ ❛ ❝♦♠♣❛r❛t✐✈❡
❛♥❛❧②s✐s✱ ✇❡ ❤❛✈❡ ❛♣♣❧✐❡❞ t❤❡ ♥❡✇ ♠♦❞❡❧ t♦ t✇♦ st❛♥❞❛r❞ t❡st s❡ts ❢r♦♠ t❤❡ ❧✐t❡r❛t✉r❡✳ ❖✉r
❝♦♠♣✉t❛t✐♦♥❛❧ r❡s✉❧ts ✐♥❞✐❝❛t❡ t❤❛t t❤❡ ♠♦❞❡❧ ♣❡r❢♦r♠s ♣❛rt✐❝✉❧❛r❧② ✇❡❧❧ ✇❤❡♥ r❡s♦✉r❝❡s
❛r❡ ✈❡r② s❝❛r❝❡ ♦r ✇❤❡♥ t❤❡ r❛♥❣❡ ♦❢ ❛❝t✐✈✐t② ♣r♦❝❡ss✐♥❣ t✐♠❡s ✐s r❛t❤❡r ❤✐❣❤✳
❚❤❡ r❡♠❛✐♥❞❡r ♦❢ t❤✐s ♣❛♣❡r ✐s str✉❝t✉r❡❞ ❛s ❢♦❧❧♦✇s✳ ■♥ ❙❡❝t✐♦♥ ✷✱ ✇❡ ❞❡s❝r✐❜❡ t❤❡ ♥♦✈❡❧
▼■▲P ♠♦❞❡❧✳ ■♥ ❙❡❝t✐♦♥ ✸✱ ✇❡ r❡♣♦rt ♦♥ t❤❡ ❝♦♠♣✉t❛t✐♦♥❛❧ r❡s✉❧ts✳ ■♥ ❙❡❝t✐♦♥ ✹✱ ✇❡ ♣r❡s❡♥t
s♦♠❡ ❝♦♥❝❧✉❞✐♥❣ r❡♠❛r❦s ❛♥❞ ❛♥ ♦✉t❧♦♦❦ ♦♥ ❢✉t✉r❡ r❡s❡❛r❝❤✳
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✷ ◆♦✈❡❧ ▼■▲P ❢♦r♠✉❧❛t✐♦♥
❚❛❜❧❡ ✶ ♣r♦✈✐❞❡s t❤❡ ♥♦♠❡♥❝❧❛t✉r❡❀ ❛❝t✐✈✐t✐❡s 0 ❛♥❞ n + 1 ❛r❡ ✜❝t✐t✐♦✉s ❛❝t✐✈✐t✐❡s r❡♣r❡✲
s❡♥t✐♥❣ t❤❡ st❛rt ❛♥❞ t❤❡ ❝♦♠♣❧❡t✐♦♥ ♦❢ t❤❡ ♣r♦❥❡❝t✱ r❡s♣❡❝t✐✈❡❧②✳ ❚❤❡ ❜❛s✐❝ ♥♦✈❡❧ ❝♦♥t✐♥✉♦✉s✲
t✐♠❡ ❛ss✐❣♥♠❡♥t✲❜❛s❡❞ ▼■▲P ❢♦r♠✉❧❛t✐♦♥✱ ❝❛❧❧❡❞ ❈❚❆❇ ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣✱ r❡❛❞s ❛s ❢♦❧❧♦✇s✳
(❈❚❆❇)
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▼✐♥✳ Sn+1 (1)
s✳t✳
PRk
l=1 rk
li = rik (i ∈ V, k ∈ R) (2)
rk
li + rk
lj ≤ 1 + yij + yji (i, j ∈ V, k ∈ R, l = 1, . . . , Rk : i j,
(i, j) 6∈ TE) (3)
Si + pi ≤ Sj ((i, j) ∈ E) (4)
Si + pi ≤ Sj + (LSi + pi)(1 − yij) (i, j ∈ V : i 6= j, (i, j) 6∈ TE) (5)
ESi ≤ Si ≤ LSi (i ∈ V ) (6)
yij ∈ {0, 1} (i, j ∈ V : i 6= j, (i, j) 6∈ TE) (7)
rk
li ∈ {0, 1} (i ∈ V, k ∈ R, l = 1, . . . , Rk) (8)
❚❤❡ ♦❜❥❡❝t✐✈❡ ✭✶✮ ✐s t♦ ♠✐♥✐♠✐③❡ t❤❡ ♠❛❦❡s♣❛♥✳ ❈♦♥str❛✐♥ts ✭✷✮ ❡♥s✉r❡ t❤❛t t❤❡ r❡q✉✐r❡❞
♥✉♠❜❡r ♦❢ r❡s♦✉r❝❡ ✉♥✐ts ✐s ❛ss✐❣♥❡❞ t♦ ❡❛❝❤ ❛❝t✐✈✐t②✳ ❈♦♥str❛✐♥ts ✭✸✮ ❧✐♥❦ t❤❡ r❡s♦✉r❝❡✲
❛ss✐❣♥♠❡♥t ✈❛r✐❛❜❧❡s t♦ t❤❡ s❡q✉❡♥❝✐♥❣ ✈❛r✐❛❜❧❡s✿ ✐❢ t❤❡ s❛♠❡ r❡s♦✉r❝❡ ✉♥✐t ✐s ❛ss✐❣♥❡❞
t♦ t✇♦ ❛❝t✐✈✐t✐❡s i ❛♥❞ j✱ t❤❡♥ ❛ s❡q✉❡♥❝❡ ❜❡t✇❡❡♥ t❤❡s❡ t✇♦ ❛❝t✐✈✐t✐❡s ✐s ❡♥❢♦r❝❡❞✳ ❚❤❡
s❡q✉❡♥❝✐♥❣ ✈❛r✐❛❜❧❡s yij ❛r❡ ♥♦t ❞❡✜♥❡❞ ❢♦r t❤❡ ♣❛✐rs ♦❢ ❛❝t✐✈✐t✐❡s (i, j) ❢♦r ✇❤✐❝❤ t❤❡r❡ ✐s
❛ ♣❛t❤ ❢r♦♠ i t♦ j ✐♥ t❤❡ ❛❝t✐✈✐t②✲♦♥✲♥♦❞❡ ❞✐❣r❛♣❤ G = (V, E)❀ t❤❡ tr❛♥s✐t✐✈❡ ❝❧♦s✉r❡ TE
♦❢ E ❝♦♥s✐sts ♦❢ ❛❧❧ t❤❡s❡ ♣❛✐rs ♦❢ ❛❝t✐✈✐t✐❡s✳ ❈♦♥str❛✐♥ts ✭✹✮ ♠♦❞❡❧ t❤❡ ❝♦♠♣❧❡t✐♦♥✲st❛rt
♣r❡❝❡❞❡♥❝❡ r❡❧❛t✐♦♥s❤✐♣s ❛♠♦♥❣ t❤❡ ❛❝t✐✈✐t✐❡s✳ ❈♦♥str❛✐♥ts ✭✺✮ ❧✐♥❦ t❤❡ st❛rt t✐♠❡ ✈❛r✐❛❜❧❡s
t♦ t❤❡ s❡q✉❡♥❝✐♥❣ ✈❛r✐❛❜❧❡s✳ ❈♦♥str❛✐♥ts ✭✻✮ ❡♥s✉r❡ t❤❛t ❡❛❝❤ ❛❝t✐✈✐t② st❛rts ❜❡t✇❡❡♥ ✐ts
❡❛r❧✐❡st ❛♥❞ ✐ts ❧❛t❡st st❛rt t✐♠❡✳
❚♦ ❡♥❤❛♥❝❡ t❤❡ ♣❡r❢♦r♠❛♥❝❡ ♦❢ ♠♦❞❡❧ ❈❚❆❇✱ ✇❡ ✐♥❝❧✉❞❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❡①t❡♥s✐♦♥s✿
✶✳ ❋♦r ❛❧❧ ♣❛✐rs ♦❢ ❛❝t✐✈✐t✐❡s i ❛♥❞ j t❤❛t ❝❛♥♥♦t ❜❡ ♣r♦❝❡ss❡❞ ✐♥ ♣❛r❛❧❧❡❧✱ ✐✳❡✳✱ rik +rjk Rk
❢♦r s♦♠❡ r❡s♦✉r❝❡ k ∈ R✱ ✇❡ ❛❞❞ t❤❡ ❝♦♥str❛✐♥t yij +yji = 1✳ ❆♥❛❧♦❣♦✉s❧②✱ ❢♦r ❛❧❧ tr✐♣❧❡ts
♦❢ ❛❝t✐✈✐t✐❡s i✱ j✱ ❛♥❞ m t❤❛t ❝❛♥♥♦t ❜❡ ♣r♦❝❡ss❡❞ ✐♥ ♣❛r❛❧❧❡❧✱ ✐✳❡✳✱ rik + rjk + rmk Rk
❢♦r s♦♠❡ r❡s♦✉r❝❡ k ∈ R✱ ✇❡ ❛❞❞ t❤❡ ❝♦♥str❛✐♥t yij + yji + yim + ymi + yjm + ymj ≥ 1✳
✷✳ ❆❧❧ ✉♥✐ts ♦❢ ❡❛❝❤ r❡s♦✉r❝❡ ❛r❡ ✐❞❡♥t✐❝❛❧✳ ❚❤❡r❡❢♦r❡✱ t♦ ❡❧✐♠✐♥❛t❡ s♦♠❡ s②♠♠❡tr✐❝ s♦❧✉✲
t✐♦♥s ❢r♦♠ t❤❡ s❡❛r❝❤ s♣❛❝❡ ✇✳❧✳♦✳❣✳✱ ❢♦r ❡❛❝❤ r❡s♦✉r❝❡ k ∈ R✱ ✇❡ s❡❧❡❝t ❛♥ ❛❝t✐✈✐t② i
✇✐t❤ ❧❛r❣❡st r❡q✉✐r❡♠❡♥t rik ❢♦r t❤✐s r❡s♦✉r❝❡ ❛♥❞ ❛ss✐❣♥ t❤❡ ✜rst rik r❡s♦✉r❝❡ ✉♥✐ts t♦
✐ts ❡①❡❝✉t✐♦♥ ❜② ♣r❡s❝r✐❜✐♥❣ rk
li = 1 ❢♦r l = 1, . . . , rik✳
✸ ❈♦♠♣✉t❛t✐♦♥❛❧ st✉❞②
■♥ t❤✐s s❡❝t✐♦♥✱ ✇❡ ❝♦♠♣❛r❡ t❤❡ ♣❡r❢♦r♠❛♥❝❡ ♦❢ t❤❡ ▼■▲P ♠♦❞❡❧ ♣r♦♣♦s❡❞ ✐♥ ❙❡❝t✐♦♥ ✷ t♦
t❤❡ ♣❡r❢♦r♠❛♥❝❡ ♦❢ t❤❡ ✇❡❧❧✲❦♥♦✇♥ ♠♦❞❡❧s ♣r❡s❡♥t❡❞ ✐♥ Pr✐ts❦❡r ❡t ❛❧✳ ✭✶✾✻✾✮✱ ❈❤r✐st♦✜❞❡s ❡t
❛❧✳ ✭✶✾✽✼✮✱ ❑❧❡✐♥ ✭✷✵✵✵✮✱ ❑♦♣❛♥♦s ❡t ❛❧✳ ✭✷✵✶✹✮✱ ❛♥❞ ❆rt✐❣✉❡s ❡t ❛❧✳ ✭✷✵✵✸✮✳ ❲❡ ✐♠♣❧❡♠❡♥t❡❞
t❤❡ ▼■▲P ♠♦❞❡❧s ✐♥ ❆▼P▲✱ ❛♥❞ ✇❡ ✉s❡❞ t❤❡ ●✉r♦❜✐ ❖♣t✐♠✐③❡r ✼✳✺ ✇✐t❤ t❤❡ ❞❡❢❛✉❧t s♦❧✈❡r
s❡tt✐♥❣s t♦ s♦❧✈❡ t❤❡ ♠♦❞❡❧s✳ ❲❡ ♣❡r❢♦r♠❡❞ ❛❧❧ ❝♦♠♣✉t❛t✐♦♥s ♦♥ ❛ ✇♦r❦st❛t✐♦♥ ❡q✉✐♣♣❡❞
✇✐t❤ t✇♦ ✻✲❝♦r❡ ■♥t❡❧ ❳❡♦♥ ❳✺✻✺✵ ❈P❯s ✭✷✳✻✻ ●❍③✱ ✷✹ ●❇ ❘❆▼✮✳ ❲❡ s❡t t❤❡ s♦❧✈❡r t✐♠❡
❧✐♠✐t t♦ ✺✵✵ s❡❝♦♥❞s ♣❡r ✐♥st❛♥❝❡ ❛♥❞ ❧✐♠✐t❡❞ t❤❡ ♥✉♠❜❡r ♦❢ ✉s❡❞ t❤r❡❛❞s t♦ ✹✳ ❋♦r t❤❡
❝♦♠♣❛r❛t✐✈❡ ❛♥❛❧②s✐s✱ ✇❡ ✉s❡❞ t❤❡ ❥✸✵ s❡t ✭✹✽✵ ✐♥st❛♥❝❡s✮ ❢r♦♠ t❤❡ P❙P▲■❇ ✭❝❢✳ ❑♦❧✐s❝❤
❛♥❞ ❙♣r❡❝❤❡r ✶✾✾✻✮ ❛♥❞ t❤❡ P❛❝❦❴❞ s❡t ✭✺✺ ✐♥st❛♥❝❡s✮ ❣❡♥❡r❛t❡❞ ❜② ❑♦♥é ❡t ❛❧✳ ✭✷✵✶✶✮✳
❚❛❜❧❡s ✷✕✹ s✉♠♠❛r✐③❡ t❤❡ r❡s✉❧ts ❢♦r t❡st s❡t ❥✸✵✱ ❢♦r t❤❡ ✐♥st❛♥❝❡s ♦❢ t❡st s❡t ❥✸✵ ✇✐t❤
r❡s♦✉r❝❡ str❡♥❣t❤ ✵✳✷ ✭✐✳❡✳ ✇✐t❤ ✈❡r② s❝❛r❝❡ r❡s♦✉r❝❡s✮✱ ❛♥❞ ❢♦r t❡st s❡t P❛❝❦❴❞✱ r❡s♣❡❝t✐✈❡❧②✳
243

❋♦r t❤❡ ✐♥st❛♥❝❡s ♦❢ s❡t ❥✸✵✱ t❤❡ r❡s✉❧ts ❝❛♥ ❜❡ s✉♠♠❛r✐③❡❞ ❛s ❢♦❧❧♦✇s✳ ❆❧❧ ♠♦❞❡❧s ❡①❝❡♣t
t❤❡ ♠♦❞❡❧ ♦❢ ❑♦♣❛♥♦s ❡t ❛❧✳ ✭✷✵✶✹✮ ♣r♦✈✐❞❡ ❛ ❢❡❛s✐❜❧❡ s♦❧✉t✐♦♥ t♦ ❡❛❝❤ ✐♥st❛♥❝❡ ✭❝♦❧✉♠♥
★ ❋❡❛s✮✳ ❲✐t❤ r❡s♣❡❝t t♦ t❤❡ ♥✉♠❜❡r ♦❢ ✐♥st❛♥❝❡s ❢♦r ✇❤✐❝❤ ❛♥ ♦♣t✐♠❛❧ s♦❧✉t✐♦♥ ✐s ❢♦✉♥❞ ❛♥❞
♦♣t✐♠❛❧✐t② ✐s ♣r♦✈❡♥ ❜② t❤❡ s♦❧✈❡r ✇✐t❤✐♥ t❤❡ t✐♠❡ ❧✐♠✐t ✭★ ❖♣t✮✱ t❤❡ ♠♦❞❡❧s ♦❢ Pr✐ts❦❡r ❡t
❛❧✳ ✭✶✾✻✾✮✱ ❈❤r✐st♦✜❞❡s ❡t ❛❧✳ ✭✶✾✽✼✮ ❛♥❞ ❑♦♣❛♥♦s ❡t ❛❧✳ ✭✷✵✶✹✮ ♣❡r❢♦r♠ ❜❡st❀ t❤❡ s❛♠❡ ❤♦❧❞s
❢♦r t❤❡ ♥✉♠❜❡r ♦❢ ✐♥st❛♥❝❡s ❢♦r ✇❤✐❝❤✱ ❛♠♦♥❣ ❛❧❧ ♠♦❞❡❧s✱ ❛ ❜❡st s♦❧✉t✐♦♥ ✐s ♦❜t❛✐♥❡❞ ✭★
❇❡st✮✳ ❚❤❡ ❛✈❡r❛❣❡ r❡❧❛t✐✈❡ ❞❡✈✐❛t✐♦♥ ✭●❛♣❜❜✮ ❜❡t✇❡❡♥ t❤❡ ♦❜❥❡❝t✐✈❡ ❢✉♥❝t✐♦♥ ✈❛❧✉❡ ✭❖❋❱✮
❛♥❞ t❤❡ ❧♦✇❡r ❜♦✉♥❞ ✭▲❇✮ ♣r♦✈✐❞❡❞ ❜② t❤❡ s♦❧✈❡r✱ ✐✳❡✳ (OFV − LB)/LB✱ ✐s t❤❡ ❧♦✇❡st
❢♦r t❤❡ ♠♦❞❡❧s ♦❢ Pr✐ts❦❡r ❡t ❛❧✳ ✭✶✾✻✾✮ ❛♥❞ ❈❤r✐st♦✜❞❡s ❡t ❛❧✳ ✭✶✾✽✼✮✳ ❚❤❡ ❧♦✇❡st ❛✈❡r❛❣❡
r❡❧❛t✐✈❡ ❞❡✈✐❛t✐♦♥ ❜❡t✇❡❡♥ t❤❡ ❖❋❱ ❛♥❞ t❤❡ ❝r✐t✐❝❛❧✲♣❛t❤ ❜❛s❡❞ ❧♦✇❡r ❜♦✉♥❞ ✭●❛♣❈P▼✮
❛♥❞ t❤❡ ❜❡st ❖❋❱ r❡t✉r♥❡❞ ❜② ❛♥② ♦❢ t❤❡ ♠♦❞❡❧s ✭●❛♣❜❡st✮✱ r❡s♣❡❝t✐✈❡❧②✱ ✐s ♦❜t❛✐♥❡❞ ❜②
t❤❡ ♠♦❞❡❧ ♦❢ Pr✐ts❦❡r ❡t ❛❧✳ ✭✶✾✻✾✮ ❛♥❞ t❤❡ ♠♦❞❡❧ ♣r❡s❡♥t❡❞ ✐♥ t❤✐s ♣❛♣❡r✳ ❋♦r t❤❡ ✐♥st❛♥❝❡s
✇✐t❤ r❡s♦✉r❝❡ str❡♥❣t❤ ✵✳✷✱ t❤❡ ❡①t❡♥❞❡❞ ♠♦❞❡❧ ♣r❡s❡♥t❡❞ ✐♥ t❤✐s ♣❛♣❡r ❡✈❡♥ ♦✉t♣❡r❢♦r♠s
t❤❡ ♦t❤❡r ♠♦❞❡❧s✳ ❋♦r t❤❡ ✐♥st❛♥❝❡s ♦❢ s❡t P❛❝❦❴❞✱ ✇❤✐❝❤ ❤❛✈❡ ❝♦♥s✐❞❡r❛❜❧② ❧♦♥❣❡r ❛❝t✐✈✐t②
❞✉r❛t✐♦♥s t❤❛♥ t❤❡ ✐♥st❛♥❝❡s ♦❢ s❡t ❥✸✵✱ ❛ ❢❡❛s✐❜❧❡ s♦❧✉t✐♦♥ ❢♦r ❛❧❧ ✐♥st❛♥❝❡s ❤❛s ❜❡❡♥ ♦❜t❛✐♥❡❞
♦♥❧② ❜② t❤❡ ♠♦❞❡❧ ♦❢ ❑❧❡✐♥ ✭✷✵✵✵✮ ❛♥❞ t❤❡ ♠♦❞❡❧ ♣r❡s❡♥t❡❞ ✐♥ t❤✐s ♣❛♣❡r❀ ❤♦✇❡✈❡r✱ t❤❡
❞❡✈✐❛t✐♦♥ ❢r♦♠ t❤❡ ❧♦✇❡r ❜♦✉♥❞s ✐s ♥♦t❡❞❧② ❧❛r❣❡r ❢♦r t❤❡ ♠♦❞❡❧ ♦❢ ❑❧❡✐♥ ✭✷✵✵✵✮ t❤❛♥ ❢♦r
t❤❡ ♠♦❞❡❧ ♣r❡s❡♥t❡❞ ✐♥ t❤✐s ♣❛♣❡r✳
✹ ❈♦♥❝❧✉s✐♦♥s
■♥ t❤✐s ♣❛♣❡r✱ ✇❡ ❤❛✈❡ ♣r♦♣♦s❡❞ ❛ ♥♦✈❡❧ ❝♦♥t✐♥✉♦✉s✲t✐♠❡ ▼■▲P ♠♦❞❡❧ ❢♦r t❤❡ ❘❈P❙P
✇❤✐❝❤ ✐s ❜❛s❡❞ ♦♥ ❜✐♥❛r② ✈❛r✐❛❜❧❡s t❤❛t r❡♣r❡s❡♥t t❤❡ ❛ss✐❣♥♠❡♥t ♦❢ t❤❡ ♣r♦❥❡❝t ❛❝t✐✈✐t✐❡s
t♦ ✐♥❞✐✈✐❞✉❛❧ r❡s♦✉r❝❡ ✉♥✐ts ❛♥❞ t❤❡ s❡q✉❡♥t✐❛❧ r❡❧❛t✐♦♥s❤✐♣s ❜❡t✇❡❡♥ ❛❝t✐✈✐t✐❡s t❤❛t ❛r❡
❛ss✐❣♥❡❞ t♦ ❛t ❧❡❛st ♦♥❡ ✐❞❡♥t✐❝❛❧ r❡s♦✉r❝❡ ✉♥✐t✳ ■♥ ❢✉t✉r❡ r❡s❡❛r❝❤✱ ❢✉rt❤❡r ♣♦ss✐❜✐❧✐t✐❡s t♦
❡❧✐♠✐♥❛t❡ s♦♠❡ s②♠♠❡tr✐❝ s♦❧✉t✐♦♥s ❢r♦♠ t❤❡ s❡❛r❝❤ s♣❛❝❡ s❤♦✉❧❞ ❜❡ ❡①♣❧♦✐t❡❞✱ ❛♥❞ t❤❡
♥♦✈❡❧ ♠♦❞❡❧ s❤♦✉❧❞ ❜❡ ❝♦♠♣❛r❡❞ ❛❣❛✐♥st ♦t❤❡r ♠♦❞❡❧s ❦♥♦✇♥ ❢r♦♠ t❤❡ ❧✐t❡r❛t✉r❡✳
❘❡❢❡r❡♥❝❡s
❆rt✐❣✉❡s✱ ❈✳✱ ❑♦♥é✱ ❖✳✱ ▲♦♣❡③✱ P✳✱ ▼♦♥❣❡❛✉✱ ▼✳✱ ✷✵✶✺✱ ✧▼✐①❡❞✲✐♥t❡❣❡r ❧✐♥❡❛r ♣r♦❣r❛♠♠✐♥❣ ❢♦r♠✉✲
❧❛t✐♦♥s✧✱ ✐♥ ❈✳ ❙❝❤✇✐♥❞t ❛♥❞ ❏✳ ❩✐♠♠❡r♠❛♥♥ ✭❡❞s✮✱ ❍❛♥❞❜♦♦❦ ♦♥ Pr♦❥❡❝t ▼❛♥❛❣❡♠❡♥t ❛♥❞
❙❝❤❡❞✉❧✐♥❣ ❱♦❧✳ ✶✱ ❈❤❛♠✿ ❙♣r✐♥❣❡r✱ ♣♣✳ ✶✼✕✹✶✳
❆rt✐❣✉❡s✱ ❈✳✱ ▼✐❝❤❡❧♦♥✱ P✳✱ ❘❡✉ss❡r✱ ❙✳✱ ✷✵✵✸✱ ✧■♥s❡rt✐♦♥ t❡❝❤♥✐q✉❡s ❢♦r st❛t✐❝ ❛♥❞ ❞②♥❛♠✐❝ r❡s♦✉r❝❡✲
❝♦♥str❛✐♥❡❞ ♣r♦❥❡❝t s❝❤❡❞✉❧✐♥❣✧✱ ❊✉r ❏ ❖♣❡r ❘❡s✱ ❱♦❧✳ ✶✹✾✱ ◆♦✳ ✷✱ ♣♣✳ ✷✹✾✕✷✻✼✳
❇✐❛♥❝♦✱ ▲✳✱ ❈❛r❛♠✐❛✱ ▼✳✱ ✷✵✶✸✱ ✧❆ ♥❡✇ ❢♦r♠✉❧❛t✐♦♥ ❢♦r t❤❡ ♣r♦❥❡❝t s❝❤❡❞✉❧✐♥❣ ♣r♦❜❧❡♠ ✉♥❞❡r
❧✐♠✐t❡❞ r❡s♦✉r❝❡s✧✱ ❋❧❡① ❙❡r✈ ▼❛♥✉ ❏✱ ❱♦❧✳ ✷✺✱ ◆♦✳ ✶✕✷✱ ♣♣✳ ✻✕✷✹✳
❈❤r✐st♦✜❞❡s✱ ◆✳✱ ❆❧✈❛r❡③✲❱❛❧❞és✱ ❘✳✱ ❚❛♠❛r✐t✱ ❏✳ ▼✳✱ ✶✾✽✼✱ ✧Pr♦❥❡❝t s❝❤❡❞✉❧✐♥❣ ✇✐t❤ r❡s♦✉r❝❡ ❝♦♥✲
str❛✐♥ts✿ ❛ ❜r❛♥❝❤ ❛♥❞ ❜♦✉♥❞ ❛♣♣r♦❛❝❤✧✱ ❊✉r ❏ ❖♣❡r ❘❡s✱ ❱♦❧✳ ✷✾✱ ◆♦✳ ✸✱ ♣♣✳ ✷✻✷✕✷✼✸✳
❑❛♣❧❛♥✱ ▲✳✱ ✶✾✽✽✱ ✧❘❡s♦✉r❝❡✲❝♦♥str❛✐♥❡❞ ♣r♦❥❡❝t s❝❤❡❞✉❧✐♥❣ ✇✐t❤ ♣r❡❡♠♣t✐♦♥ ♦❢ ❥♦❜s✧✱ P❤❉ t❤❡s✐s✱
❯♥✐✈❡rs✐t② ♦❢ ▼✐❝❤✐❣❛♥✳
❑❧❡✐♥✱ ❘✳✱ ✷✵✵✵✱ ✧❙❝❤❡❞✉❧✐♥❣ ♦❢ r❡s♦✉r❝❡✲❝♦♥str❛✐♥❡❞ ♣r♦❥❡❝ts✧✱ ❇♦st♦♥✿ ❑❧✉✇❡r✳
❑♦❧✐s❝❤✱ ❘✳✱ ❙♣r❡❝❤❡r✱ ❆✳✱ ✶✾✾✻✱ ✧P❙P▲■❇✲❛ ♣r♦❥❡❝t s❝❤❡❞✉❧✐♥❣ ♣r♦❜❧❡♠ ❧✐❜r❛r②✧✱ ❊✉r ❏ ❖♣❡r ❘❡s✱
❱♦❧✳ ✾✻✱ ◆♦✳ ✶✱ ♣♣✳ ✷✵✺✕✷✶✻✳
❑♦♥é✱ ❖✳✱ ❆rt✐❣✉❡s✱ ❈✳✱ ▲♦♣❡③✱ P✳✱ ▼♦♥❣❡❛✉✱ ▼✳✱ ✷✵✶✹✱ ✧❊✈❡♥t✲❜❛s❡❞ ▼■▲P ♠♦❞❡❧s ❢♦r r❡s♦✉r❝❡✲
❝♦♥str❛✐♥❡❞ ♣r♦❥❡❝t s❝❤❡❞✉❧✐♥❣ ♣r♦❜❧❡♠s✧✱ ❈♦♠♣ ❖♣❡r ❘❡s✱ ❱♦❧✳ ✸✽✱ ◆♦✳ ✶✱ ♣♣✳ ✸✕✶✸✳
❑♦♣❛♥♦s✱ ●✳ ▼✳✱ ❑②r✐❛❦✐❞✐s✱ ❚✳ ❙✳✱ ●❡♦r❣✐❛❞✐s✱ ▼✳ ❈✳✱ ✷✵✶✹✱ ✧◆❡✇ ❝♦♥t✐♥✉♦✉s✲t✐♠❡ ❛♥❞ ❞✐s❝r❡t❡✲
t✐♠❡ ♠❛t❤❡♠❛t✐❝❛❧ ❢♦r♠✉❧❛t✐♦♥s ❢♦r r❡s♦✉r❝❡✲❝♦♥str❛✐♥❡❞ ♣r♦❥❡❝t s❝❤❡❞✉❧✐♥❣ ♣r♦❜❧❡♠s✧✱ ❈♦♠✲
♣✉t ❈❤❡♠ ❊♥❣✱ ❱♦❧✳ ✻✽✱ ♣♣✳ ✾✻✕✶✵✻✳
Pr✐ts❦❡r✱ ❆✳ ❆✳ ❇✳✱ ❲❛✐t❡rs✱ ▲✳ ❏✳✱ ❲♦❧❢❡✱ P✳ ▼✳✱ ✶✾✻✾✱ ✧▼✉❧t✐♣r♦❥❡❝t s❝❤❡❞✉❧✐♥❣ ✇✐t❤ ❧✐♠✐t❡❞
r❡s♦✉r❝❡s✿ ❛ ③❡r♦✲♦♥❡ ♣r♦❣r❛♠♠✐♥❣ ❛♣♣r♦❛❝❤✧✱ ▼❛♥❛❣❡ ❙❝✐✱ ❱♦❧✳ ✶✻✱ ◆♦✳ ✶✱ ♣♣✳ ✾✸✕✶✵✽✳
❘✐❤♠✱ ❚✳✱ ❚r❛✉t♠❛♥♥✱ ◆✳✱ ✷✵✶✼✱ ✧❆♥ ❛ss✐❣♥♠❡♥t✲❜❛s❡❞ ❝♦♥t✐♥✉♦✉s✲t✐♠❡ ▼■▲P ♠♦❞❡❧ ❢♦r t❤❡
r❡s♦✉r❝❡✲❝♦♥str❛✐♥❡❞ ♣r♦❥❡❝t s❝❤❡❞✉❧✐♥❣ ♣r♦❜❧❡♠✧✱ ✐♥ ❞❡ ▼❡②❡r✱ ❆✳✱ ❈❤❛✐✱ ❑✳❍✳✱ ❏✐❛♦✱ ❘✳✱
❈❤❡♥✱ ◆✳✱ ❳✐❡✱ ▼✳ ✭❡❞s✮✱ Pr♦❝ ✷✵✶✼ ■❊❊❊ ■♥t ❈♦♥❢ ♦♥ ■♥❞ ❊♥❣ ❊♥❣ ▼❣♠t✱ ❙✐♥❣❛♣♦r❡✱ ✸✺✕✺✾
244

❚❛❜❧❡ ✶✳ ◆♦♠❡♥❝❧❛t✉r❡ ♦❢ t❤❡ ♥♦✈❡❧ ▼■▲P ❢♦r♠✉❧❛t✐♦♥
V ❙❡t ♦❢ ❛❧❧ ❛❝t✐✈✐t✐❡s ✭V := {0, . . . , n + 1}✮
pi Pr♦❝❡ss✐♥❣ t✐♠❡ ♦❢ ❛❝t✐✈✐t② i ∈ V
E ❙❡t ♦❢ ❛❧❧ ♣r❡❝❡❞❡♥❝❡ r❡❧❛t✐♦♥s❤✐♣s
TE ❚r❛♥s✐t✐✈❡ ❝❧♦s✉r❡ ♦❢ ❊
ESi ❊❛r❧✐❡st st❛rt t✐♠❡ ♦❢ ❛❝t✐✈✐t② i ∈ V
LSi ▲❛t❡st st❛rt t✐♠❡ ♦❢ ❛❝t✐✈✐t② i ∈ V
R ❙❡t ♦❢ ❛❧❧ r❡♥❡✇❛❜❧❡ r❡s♦✉r❝❡s
Rk ❈❛♣❛❝✐t② ♦❢ r❡s♦✉r❝❡ k ∈ R
rik ❘❡q✉✐r❡♠❡♥t ♦❢ r❡s♦✉r❝❡ k ∈ R ♣❡r t✐♠❡ ♣❡r✐♦❞ ❢♦r ❡①❡❝✉t✐♦♥ ♦❢ ❛❝t✐✈✐t② i ∈ V
∗ Si ❙t❛rt t✐♠❡ ♦❢ ❛❝t✐✈✐t② i
∗ yij

= 1, ✐❢ ❛❝t✐✈✐t② i ♠✉st ❜❡ ❝♦♠♣❧❡t❡❞ ❜❡❢♦r❡ t❤❡ st❛rt ♦❢ j❀
= 0, ♦t❤❡r✇✐s❡✳
∗ rk
li

= 1, ✐❢ ❛❝t✐✈✐t② i ✐s ♣r♦❝❡ss❡❞ ♦♥ ✉♥✐t l ♦❢ r❡s♦✉r❝❡ k❀
= 0, ♦t❤❡r✇✐s❡✳
❚❛❜❧❡ ✷✳ ❖✈❡r❛❧❧ r❡s✉❧ts ❢♦r t❡st s❡t ❥✸✵ ✭✹✽✵ ✐♥st❛♥❝❡s✮
❋♦r♠✉❧❛t✐♦♥ ★ ❋❡❛s ★ ❖♣t ★ ❇❡st ●❛♣❜❜ ✭✪✮ ●❛♣❈P▼ ✭✪✮ ●❛♣❜❡st ✭✪✮
Pr✐❲❛✐❲♦❧✻✾ ✹✽✵ ✹✹✸ ✹✺✶ ✶✳✵ ✶✸✳✽ ✵✳✷
❈❤r❆❧✈❚❛♠✽✼ ✹✽✵ ✹✹✻ ✹✺✸ ✵✳✾ ✶✸✳✾ ✵✳✸
❑❧❡✵✵ ✹✽✵ ✹✸✷ ✹✹✼ ✶✳✽ ✶✹✳✶ ✵✳✹
❑♦♣❑②r●❡♦✶✹ ✹✼✽ ✹✹✻ ✹✺✺ ✶✳✸ ✶✹✳✵ ✵✳✹
❆rt▼✐❝❘❡✉✵✸ ✹✽✵ ✸✺✹ ✸✽✾ ✶✵✳✽ ✶✻✳✷ ✶✳✼
❈❚❆❇ ❜❛s✐❝ ✹✽✵ ✸✼✹ ✹✵✻ ✸✳✻ ✶✹✳✹ ✵✳✼
❈❚❆❇ ❡①t❡♥❞❡❞ ✹✽✵ ✹✶✼ ✹✹✹ ✶✳✽ ✶✸✳✽ ✵✳✷
❚❛❜❧❡ ✸✳ ❘❡s✉❧ts ❢♦r ❥✸✵ ✐♥st❛♥❝❡s ✇✐t❤ r❡s♦✉r❝❡ str❡♥❣t❤ ✵✳✷ ✭✶✷✵ ✐♥st❛♥❝❡s✮
❋♦r♠✉❧❛t✐♦♥ ★ ❋❡❛s ★ ❖♣t ★ ❇❡st ●❛♣❜❜ ✭✪✮ ●❛♣❈P▼ ✭✪✮ ●❛♣❜❡st ✭✪✮
Pr✐❲❛✐❲♦❧✻✾ ✶✷✵ ✽✸ ✾✶ ✸✳✽ ✹✺✳✼ ✵✳✼
❈❤r❆❧✈❚❛♠✽✼ ✶✷✵ ✽✻ ✾✸ ✸✳✺ ✹✻✳✸ ✶✳✵
❑❧❡✵✵ ✶✷✵ ✼✷ ✽✼ ✼✳✷ ✹✻✳✾ ✶✳✹
❑♦♣❑②r●❡♦✶✹ ✶✶✽ ✽✻ ✾✺ ✺✳✶ ✹✻✳✾ ✶✳✼
❆rt▼✐❝❘❡✉✵✸ ✶✷✵ ✸✾ ✺✸ ✹✵✳✵ ✺✹✳✼ ✻✳✸
❈❚❆❇ ❜❛s✐❝ ✶✷✵ ✼✾ ✾✵ ✾✳✻ ✹✻✳✸ ✶✳✷
❈❚❆❇ ❡①t❡♥❞❡❞ ✶✷✵ ✶✵✵ ✶✶✷ ✹✳✻ ✹✹✳✽ ✵✳✷
❚❛❜❧❡ ✹✳ ❖✈❡r❛❧❧ r❡s✉❧ts ❢♦r t❡st s❡t P❛❝❦❴❞ ✭✺✺ ✐♥st❛♥❝❡s✮
❋♦r♠✉❧❛t✐♦♥ ★ ❋❡❛s ★ ❖♣t ★ ❇❡st ●❛♣❜❜ ✭✪✮ ●❛♣❈P▼ ✭✪✮ ●❛♣❜❡st ✭✪✮
Pr✐❲❛✐❲♦❧✻✾ ✹✽ ✻ ✻ ✾✾✳✷ ✷✶✽✳✺ ✺✾✳✷
❈❤r❆❧✈❚❛♠✽✼ ✵ ✵ ✵ ✲ ✲ ✲
❑❧❡✵✵ ✺✺ ✹ ✹ ✷✾✷✳✶ ✷✾✽✳✼ ✾✺✳✹
❑♦♣❑②r●❡♦✶✹ ✸ ✶ ✶ ✺✸✳✷ ✺✺✳✶ ✸✶✳✻
❆rt▼✐❝❘❡✉✵✸ ✹✽ ✺ ✶✻ ✶✵✸✳✾ ✶✵✸✳✾ ✹✳✼
❈❚❆❇ ❜❛s✐❝ ✺✺ ✶✼ ✹✵ ✶✾✳✸ ✶✶✶✳✶ ✵✳✷
❈❚❆❇ ❡①t❡♥❞❡❞ ✺✺ ✶✾ ✺✸ ✶✷✳✹ ✶✶✵✳✼ ✵✳✶
245

A heuristic procedure to solve the integration of
personnel staffing in the project scheduling problem
with discrete time/resource trade-offs
Mick Van Den Eeckhout1
and Broos Maenhout1
1
Faculty of Economics and Business Administration, Ghent University,Tweekerkenstraat 2,
9000 Gent (Belgium)
mick.vandeneeckhout@ugent.be,mario.vanhoucke@ugent.be,broos.maenhout@ugent.be
2
3
Keywords: Project scheduling, staffing, discrete time/resource trade-offs, decomposition.
1 Introduction
Project scheduling and personnel staffing are two complementary optimisation problems.
In project scheduling, activities are scheduled given precedence relations between these
activities and a constant resource availability. Hartmann and Briskorn (2010) discuss the
characteristics of the resource-constrained project scheduling problem (RCPSP) and give
an overview of different extensions. Personnel resources are one of the most important
resources in project planning, accounting for 30-50% of the total project cost (Adrian 1987)
and therefore it is essential to determine the personnel budget to carry out a project. The
personnel budget results from the composition of a staffing plan and is based on the staffing
requirements generated by the project schedule. However, in personnel staffing time-related
constraints are imposed on the scheduling of individual workers, which complicate the
scheduling of the resources (see Van den Bergh et. al. (2013) for a full overview).
Tackling these two problems sequentially leads to sub-optimal outcomes. On the one hand,
the scheduling of activities determines the staffing requirements and should thus be in line
with the personnel staffing. On the other hand, personnel supply is an important factor
when activities need to be scheduled. When integrating these two-interrelated problems,
benefits can thus arise in both directions. First, additional flexibility is provided for the
project manager if resource scheduling is included. Second, demand management can be
applied to improve the resource utilisation.
2 Problem Definition
As stated above, integrating personnel staffing in project planning increases the schedule
flexibility since resource availabilities can be adapted to the project scheduling requirements.
On top of that, additional schedule flexibility is considered by incorporating different
modes for each activity. Each activity mode is defined by a trade-off between duration
and resource demand, where a longer duration will lead to a smaller resource demand.
Only one type of (renewable) resources is considered, namely personnel resources, which
are divided in regular and temporary workers. The scheduling of the regular workers implies
a manpower days-off scheduling problem with time-related constraints (Van den Bergh et.
al. 2013), whereas temporary workers are hired for a single day. The imposed time-related
246

constraints limit the minimum and maximum weekly assignments, the minimum and max-
imum consecutive days-on and the minimum and maximum consecutive days off for a
single worker. Given the incorporation of multiple modes for each activity and the use of
personnel as only resource, the problem lies in line with the discrete time/resource trade-off
problem in project management (Ranjbar et. al. 2009).
When integrating the two presented problems, an accurate objective function should be
chosen. In project planning, makespan minimisation is the most common objective, whereas
cost minimisation is the primary focus of resource scheduling. These two objectives are thus
conflicting since a short makespan will lead to higher personnel costs and vice versa. To
overcome this issue, a fixed deadline is proposed resulting in strategic budgeting problem
which determines the size of the personnel staff. The objective function makes a trade-off
between the number of regular and temporary workers, since a regular worker should be
paid the entire planning period and a temporary workers has a higher daily cost.
3 Methodology
An iterative heuristic solution procedure was developed to solve the integrated project
scheduling and personnel staffing problem. This procedure combines a heuristic framework,
namely an iterated local search (ILS), with optimal solution procedures in the local search
step. A generic framework of an iterated local search is presented in algorithm 1, where
the different steps will be explained below.
Algorithm 1 Iterated Local Search (Lourenço et. al. 2010)
1: s0 = Generate Initial Solution()
2: s∗
= Local Search (s0)
3: repeat
4: s′
= Perturbation (s∗
, history)
5: s′∗
= Local Search (s′
)
6: s∗
= Acceptance Criterion (s∗
, s′∗
, history)
7: until termination condition met
Due to the observation that the initial solution of the local search is of great importance
for the performance of the algorithm (Lourenço et. al. 2010), different methods were
developed to create this initial solution. Instead of generating one solution, a pool of
solutions was created wherefrom the best solution was selected. This pool can be created
by creating random projects or by incorporating information from the linear programming
(LP) relaxation. In the first case, the assignment of a certain mode and a certain start
time to an activity is based on an uniform distribution. In the second case, the probability
distribution is biased by the fractional decision variable values of the LP relaxation.
The local search step is based on the two types of variables in our problem definition, namely
project and personnel variables, resulting in two types of decomposition strategies, activity-
and personnel-based decomposition. In activity-based decomposition, the complexity of
the project scheduling problem is reduced by fixing a large set of activities and thus
rescheduling only a small set of activities in a (limited) time period. In personnel-based
decomposition, the complexity of the personnel scheduling problem is reduced by fixing
certain staffing assignments or by fixing days-off/days-on in the underlying personnel
247

patterns. The iterated local search takes the solution quality of the local search and
randomisation into account to perform a perturbation move.
Different subsets of 30 instances, taking into account the different network topology mea-
sures (Vanhoucke et. al. 2008) and each subset having a different number of activities,
were selected from the multi-mode Project Scheduling Problem LIBrary (Kolisch and
Sprecher 1997) to test the quality of the solution procedure. Since only one type of
renewable resources is needed, the original modes become inefficient and a random mode
generation was invoked. Due to the integration of personnel scheduling, the daily resource
availability is determined based on the personnel schedule instead of the defined constant
resource availability. The deadline is set at the middle between the shortest and longest
path, unless otherwise stated.
We compare our procedure with a branch-and-price procedure, a branch-and-bound method
and a multi-start heuristic. The branch-and-price is based on Maenhout and Vanhoucke
(2016) and includes an additional layer to branch on the activity modes. The branch-
and-bound method only considers a limited set of personnel patterns, and is also used
as initial upper bound in the previous mentioned paper. Both the branch-and-price and
the branch-and-bound are truncated after 3600 seconds. To evaluate the ILS framework, a
multi-start heuristics was programmed where in each iteration, the local search is applied
on a random schedule. A stopping criteria of 100 iterations was imposed on the ILS and
multi-start heuristic.
When the number of activities is limited to 10 and the deadline is set to the critical path,
the expanded version of the branch-and-price finds the optimal solution for all considered
instances. The solutions obtained by the ILS lie very close to the optimum, leading to the
conclusion that the presented procedure can find near optimal solutions for small instances.
The branch-and-bound does not perform well on these instances, which can be explained
by the limited number of considered personnel patterns.
When the number of activities or the deadline increases, results indicate that the per-
formance of the branch-and-price procedure deteriorates quickly. Certainly when the number
of activities is high, the branch-and-price is unsuitable to find good solutions. Even the
branch-and-bound procedure with a limited set of patterns has a better performance, given
the time limit of one hour. The presented ILS framework outperforms the branch-and-
bound, meaning that better results are obtained in smaller timeframes. Moreover, the
proposed procedure outperforms the multi-start procedure, which advocates the use of the
iterated local search as heuristic framework.
5 Conclusion
Integrating personnel staffing with project planning when discrete time/resource trade-
offs are considered, is a challenging endeavour due to the high complexity. A heuristic
procedure was developed which is based on iterated local search. The local search consists
of decomposing the problem in smaller subproblems by applying different activity-based
and personnel-based decomposition strategies. By relying on randomisation and solution
quality to perform a perturbation move, the algorithm is able to reach good solutions in
relatively small time frames. When comparing the presented algorithm with other optimal
248

or heuristic procedures, it is clear that the presented procedure outperforms the benchmarks
on time and solution quality. Furthermore, for small instances, the proposed procedure leads
to solutions close to the optimum.
References
Adrian, J., 1987, Construction productivity improvement, Elsevier
Hartmann, S. and Briskorn, D., 2010, A survey of variants and extensions of the resource-
constrained project scheduling problem, European Journal of Operations Research
Kolisch, R. and Sprecher, A., 1997, PSPLIB-a project scheduling problem library, European
Journal of Operations Research
Lourenço, H. R., Martin, O.C. and Stützle, T., 2010, Iterated Local Search: Framework and
applications, Handbook of metaheuristics
Maenhout, B. and Vanhoucke, M., 2016, An exact algorithm for an integrated project staffing
problem with a homogeneous workforce, Journal of Scheduling
Ranjbar, M., De Reyck, B. and Kianfar, F., 2009, A hybrid scatter-search for the discrete
time/resource trade-off problem in project scheduling, European Journal of Operations Re-
search
Van den Bergh, J., Beliën, J., De Bruecker, P., Demeulemeester, E. and De Boeck, L, 2013,
Personnel scheduling: A literature review, European Journal of Operations Research
Vanhoucke, M., Coelho, J., Debels, D., Maenhout, B. and Tavares, L.V., 2008, An evaluation of
the adequacy of project network generators with systematically sampled networks, European
Journal of Operations Research
249

Pr♦❞✉❝t✐♦♥ ❛♥❞ ❞✐str✐❜✉t✐♦♥ ♣❧❛♥♥✐♥❣
❢♦r s♠♦♦t❤✐♥❣ s✉♣♣❧②✲❝❤❛✐♥ ✈❛r✐❛❜✐❧✐t②
▼❛r✐❡✲❙❦❧❛❡r❞❡r ❱✐é1
✱ ◆✐❝♦❧❛s ❩✉✛❡r❡②1,2
❛♥❞ ▲❡❛♥❞r♦ ❈♦❡❧❤♦2,3
1
●❡♥❡✈❛ ❙❝❤♦♦❧ ♦❢ ❊❝♦♥♦♠✐❝s ❛♥❞ ▼❛♥❛❣❡♠❡♥t✱ ●❙❊▼✱ ❯♥✐✈❡rs✐t② ♦❢ ●❡♥❡✈❛✱ ❙✇✐t③❡r❧❛♥❞
♠❛r✐❡✲s❦❧❛❡r❞❡r✳✈✐❡❅✉♥✐❣❡✳❝❤✱ ♥✳③✉❢❢❡r❡②❅✉♥✐❣❡✳❝❤
2
❈❡♥tr❡ ■♥t❡r✉♥✐✈❡rs✐t❛✐r❡ ❞❡ ❘❡❝❤❡r❝❤❡ s✉r ❧❡s ❘és❡❛✉① ❞✬❊♥tr❡♣r✐s❡✱ ❧❛ ▲♦❣✐st✐q✉❡ ❡t ❧❡
❚r❛♥s♣♦rt✱ ❈■❘❘❊▲❚✱ ◗✉é❜❡❝✱ ❈❛♥❛❞❛
3
❈❛♥❛❞❛ ❘❡s❡❛r❝❤ ❈❤❛✐r ✐♥ ■♥t❡❣r❛t❡❞ ▲♦❣✐st✐❝s✱ ❯♥✐✈❡rs✐té ▲❛✈❛❧✱ ❈❛♥❛❞❛
❧❡❛♥❞r♦✳❝♦❡❧❤♦❅❝✐rr❡❧t✳❝❛
❑❡②✇♦r❞s✿ ♣r♦❞✉❝t✐♦♥ ♣❧❛♥♥✐♥❣✱ ❞✐str✐❜✉t✐♦♥ ♣❧❛♥♥✐♥❣✱ s✉♣♣❧② ❝❤❛✐♥ ♠❛♥❛❣❡♠❡♥t✱ s✐♠✉✲
❧❛t✐♦♥✱ ♦♣t✐♠✐③❛t✐♦♥✳
✶ ■♥tr♦❞✉❝t✐♦♥
❚❤✐s st✉❞② ✐s ♠♦t✐✈❛t❡❞ ❜② ❛ ❢❛st✲♠♦✈✐♥❣ ❝♦♥s✉♠❡r ❣♦♦❞s ❝♦♠♣❛♥②✱ ❞❡♥♦t❡❞ ❛s ❆❇❈✳ ■t
❤❛s ✐ts ❊✉r♦♣❡❛♥ ❤❡❛❞q✉❛rt❡rs ✐♥ ●❡♥❡✈❛ ❛♥❞ ✐t ❝❛♥♥♦t ❜❡ ♥❛♠❡❞ ❞✉❡ t♦ ❛ ♥♦♥✲❞✐s❝❧♦s✉r❡
❛❣r❡❡♠❡♥t✳ ❲❡ ❝♦♥s✐❞❡r ❛ ✢❛❣s❤✐♣ ♣r♦❞✉❝t ✇✐❞❡❧② ✉s❡❞ ✐♥ t❤❡ ✇♦r❧❞✳ ❈✉rr❡♥t❧②✱ t❤❡ s✉♣♣❧②
❝❤❛✐♥ ✐s ❝❧❛ss✐❝❛❧❧② ♠❛♥❛❣❡❞ ✇✐t❤ ❛ ❞❡❝❡♥tr❛❧✐③❡❞ ♣✉❧❧ ❛♣♣r♦❛❝❤ ❜❛s❡❞ ♦♥ ❛ r❡♦r❞❡r✲♣♦✐♥t
♣♦❧✐❝②✿ ❡✈❡r② ❡❝❤❡❧♦♥ ♦r❞❡rs t♦ t❤❡ ✉♣str❡❛♠ ❡❝❤❡❧♦♥ ✇❤❡♥❡✈❡r t❤❡ ❛✈❛✐❧❛❜❧❡ ✐♥✈❡♥t♦r② ❧❡✈❡❧
r❡❛❝❤❡s t❤❡ r❡♦r❞❡r ♣♦✐♥t✳ ❚❤❡ ♣❧❛♥t ❛❞❥✉sts ✐ts ♣r♦❞✉❝t✐♦♥ t♦ t❤❡s❡ ♦r❞❡rs t♦ ♠✐♥✐♠✐③❡
st♦r❛❣❡✳ ■♥ ❡❛❝❤ ❡❝❤❡❧♦♥✱ ❊❝♦♥♦♠✐❝ ❇❛t❝❤ ◗✉❛♥t✐t✐❡s ✭❊❇◗s✮ ❛r❡ ✉s❡❞ ❛❝❝♦r❞✐♥❣ t♦ ❤♦✇
t❤❡ ♣r♦❞✉❝ts ❛r❡ tr❛♥s♣♦rt❡❞ ✭❡✳❣✳✱ r♦✉♥❞✐♥❣ ✉♣ t♦ ♣❛❧❧❡ts✴❧❛②❡rs✴❝❛s❡s✮✳ ❙✉❝❤ ❛ ❞❡❝❡♥tr❛❧✲
✐③❡❞ ✐♥✈❡♥t♦r②✲♠❛♥❛❣❡♠❡♥t ❛♣♣r♦❛❝❤ ❧❡❛❞s t♦ ❛ s✐❣♥✐✜❝❛♥t ✈♦❧❛t✐❧✐t② ♦❢ t❤❡ ♦r❞❡rs✱ ❛♥❞
t♦ ❛♥ ❡✈❡♥ str♦♥❣❡r ✈♦❧❛t✐❧✐t② ♦❢ t❤❡ ♣r♦❞✉❝t✐♦♥✳ ❚❤✐s s♦✲❝❛❧❧❡❞ ❜✉❧❧✇❤✐♣ ❡✛❡❝t ❤❛s ❜❡❡♥
✇✐❞❡❧② st✉❞✐❡❞ ✐♥ t❤❡ ❧✐t❡r❛t✉r❡ ✭❲❛♥❣ ❛♥❞ ❉✐s♥❡② ✷✵✶✻✮✳ ■t ❝❛♥ ❜❡ ❝❛✉s❡❞ ❜② t❤❡ ❢♦❧❧♦✇✐♥❣
❡❧❡♠❡♥ts✿ ✭✶✮ ❡❛❝❤ ♦r❞❡r❡❞ q✉❛♥t✐t② ✐s r♦✉♥❞❡❞ ✉♣ t♦ ❛♥ ❊❇◗❀ ✭✷✮ ♦✈❡r❡st✐♠❛t✐♦♥ ♦❢ t❤❡
❞❡♠❛♥❞❀ ✭✸✮ ❧❡❛❞✲t✐♠❡ ✉♥❝❡rt❛✐♥t✐❡s ❛❧♦♥❣ t❤❡ s✉♣♣❧② ❝❤❛✐♥✳ ❈♦♥s❡q✉❡♥t❧②✱ ❛ s♠❛❧❧ ❞❡♠❛♥❞
❝❛♥ ❝r❡❛t❡ ❛ ❧❛r❣❡ ♣r♦❞✉❝t✐♦♥ ♦r❞❡r ❜❡❝❛✉s❡ ♦❢ t❤❡s❡ ❛♠♣❧✐✜❝❛t✐♦♥ ❡✛❡❝ts✳ ❆ ❤✐st♦r✐❝❛❧ r❡✲
✈✐❡✇ ♦❢ ❝✉rr❡♥t ♣r❛❝t✐❝❡s ❝r❡❛t✐♥❣ t❤❡ ❜✉❧❧✇❤✐♣ ❡✛❡❝t ❝❛♥ ❜❡ ❢♦✉♥❞ ✐♥ ●❡❛r② ❡t✳ ❛❧✳ ✭✷✵✵✻✮✳ ■t
s❤♦✇s t❤❛t ♠♦st ♦❢ t❤❡ r❡s❡❛r❝❤ ♦♥ t❤❡ t♦♣✐❝ ❡✐t❤❡r ❞❡✈❡❧♦♣s ❡♠♣✐r✐❝❛❧✴❡①♣❡r✐♠❡♥t❛❧ st✉❞✐❡s
t❤❛t ❛♥❛❧②③❡ ❤✐st♦r✐❝❛❧ ❞❛t❛✱ ♦r ✐t ♣r♦♣♦s❡s ♠❛♥❛❣❡♠❡♥t ❣❛♠❡s✱ ♦r ✐t s❡❡❦s ❢♦r ❛ ♠❛t❤❡✲
♠❛t✐❝❛❧ ♠♦❞❡❧ t♦ ❡①♣❧❛✐♥ t❤❡ ❡✛❡❝t ❛♥❞ ❞❡t❡r♠✐♥❡ ✐ts ❢❛❝t♦rs✳ ❙✉r♣r✐s✐♥❣❧②✱ ♦♥❧② ❢❡✇ st✉❞✐❡s
t❛❝❦❧❡ r❡❛❧ ♣r♦❜❧❡♠s ✇✐t❤ s✐♠✉❧❛t✐♦♥✱ ❛♥❞ ♥♦♥❡ ✇✐t❤ t❤❡ ✐♥t❡❣r❛t❡❞ ❛♣♣r♦❛❝❤ ♣r♦♣♦s❡❞
✐♥ t❤✐s ♣❛♣❡r✳ ❍♦✇❡✈❡r✱ ♠❛♥② ❡①❛♠♣❧❡s s❤♦✇ t❤❛t ✐♥t❡❣r❛t❡❞ ♦♣t✐♠✐③❛t✐♦♥ ♦✉t♣❡r❢♦r♠s
s❡q✉❡♥t✐❛❧ ♦♣t✐♠✐③❛t✐♦♥ ✭❉❛r✈✐s❤ ❛♥❞ ❈♦❡❧❤♦ ✷✵✶✼✱ ❚❤❡✈❡♥✐♥ ❡t✳ ❛❧✳ ✷✵✶✼✮✳
❚❤❡ r❡❛❞❡r ✐s r❡❢❡rr❡❞ t♦ ❆①sät❡r ✭✷✵✶✺✮ ❢♦r ❛♥ ♦✈❡r✈✐❡✇ ♦♥ ✐♥✈❡♥t♦r② ♠❛♥❛❣❡♠❡♥t✳
❆ ❜❛s✐❝ ✐♥✈❡♥t♦r②✲♠❛♥❛❣❡♠❡♥t ❛♣♣r♦❛❝❤ t❤❛t ❛✈♦✐❞s t❤❡ ❜✉❧❧✇❤✐♣ ❡✛❡❝t ❝♦♥s✐sts ✐♥ ❞❛✐❧②
♣r♦❞✉❝✐♥❣ t❤❡ s❛♠❡ ❛♠♦✉♥t✱ ❝♦♠♣✉t❡❞ ❛s t❤❡ ❛✈❡r❛❣❡ ❢♦r❡❝❛st❡❞ ❞❡♠❛♥❞ ♦✈❡r ❛ ❧♦♥❣
❤♦r✐③♦♥✳ ❚❤❡ ♣r♦❞✉❝t✐♦♥ ✐s t❤❡♥ ♣✉s❤❡❞ ❛❧♦♥❣ t❤❡ s✉♣♣❧② ❝❤❛✐♥ ❞♦✇♥ t♦ t❤❡ s❤♦♣s✳ ❍♦✇❡✈❡r✱
❡✈❡♥ ✐❢ t❤✐s ♣✉s❤ ❛♣♣r♦❛❝❤ ♣❡r❢❡❝t❧② s♠♦♦t❤❡s t❤❡ ♣r♦❞✉❝t✐♦♥✱ ✐t ✐s ❢❛r ❢r♦♠ ♦♣t✐♠❛❧✐t② ✇✐❤t
r❡s♣❡❝t t♦ s❤♦rt❛❣❡ ❛♥❞ ✐♥✈❡♥t♦r② ❝♦sts✳ ❍❡♥❝❡✱ ❛ ❜❛❧❛♥❝❡ ♥❡❡❞s t♦ ❜❡ ❢♦✉♥❞ ❜❡t✇❡❡♥ t❤❡s❡
♣✉❧❧ ❛♥❞ ♣✉s❤ ♠❡t❤♦❞s✳ ❚❤❡ ♠❛✐♥ ❝♦♥tr✐❜✉t✐♦♥ ♦❢ t❤✐s ✇♦r❦ ✐s t❤❡ ❞❡s✐❣♥ ♦❢ ❛♥ ✐♥t❡❣r❛t❡❞
♣❧❛♥♥✐♥❣✲❜②✲r❛♥❣❡ ✭P❇❘✮ ❛♣♣r♦❛❝❤ ❢♦r ❛ r❡❛❧✲✇♦r❧❞ ♣r♦❜❧❡♠ (P) ❞❡s❝r✐❜❡❞ ✐♥ ❙❡❝t✐♦♥ ✷
✭r❡❧②✐♥❣ ♦♥ ❛♥ ❡✣❝✐❡♥t s✐♠✉❧❛t✐♦♥✲♦♣t✐♠✐③❛t✐♦♥ ❛❧❣♦r✐t❤♠✮✱ ❛❧♦♥❣ ✇✐t❤ t❤❡ ❛❞❛♣t❛t✐♦♥s ♦❢
t❤❡ ❝❧❛ss✐❝❛❧ ♣✉❧❧ ❛♥❞ ♣✉s❤ ❛♣♣r♦❛❝❤❡s ❢♦r (P)✳ ❚❤❡s❡ ♠❡t❤♦❞s ❛r❡ ♥✉♠❡r✐❝❛❧❧② ❝♦♠♣❛r❡❞ ✐♥
❙❡❝t✐♦♥ ✸✳ ❆s ❤✐❣❤❧✐❣❤t❡❞ ✐♥ t❤❡ ❝♦♥❝❧✉s✐♦♥ ✭❙❡❝t✐♦♥ ✹✮✱ P❇❘ ❝❛♥ ❜❡ ❡❛s✐❧② ❛❞❛♣t❡❞ t♦ ♦t❤❡r
250

s✉♣♣❧② ❝❤❛✐♥ ♥❡t✇♦r❦s✱ ❢♦r ✇❤✐❝❤ r❡❞✉❝✐♥❣ t❤❡ ❢♦❧❧♦✇✐♥❣ ❢❡❛t✉r❡s ✐s ✐♠♣♦rt❛♥t✿ s❤♦rt❛❣❡✱
❜✉❧❧✇❤✐♣ ❡✛❡❝t ❛♥❞ ✐♥✈❡♥t♦r② ❧❡✈❡❧✳
✷ Pr❡s❡♥t❛t✐♦♥ ♦❢ t❤❡ ♣r♦❜❧❡♠ (P) ❛♥❞ ♦❢ t❤❡ ♣r♦♣♦s❡❞ P❇❘ ❛♣♣r♦❛❝❤
❚❤❡ ❝♦♥s✐❞❡r❡❞ ✸✲❡❝❤❡❧♦♥ s✉♣♣❧② ❝❤❛✐♥ ✐s ♠❛❞❡ ♦❢ ♦♥❡ ♣❧❛♥t✱ ♦♥❡ ❞✐str✐❜✉t✐♦♥ ❝❡♥t❡r
✭❉❈✮✱ ❛♥❞ ❞♦③❡♥s ♦❢ s❤♦♣s✳ ❊❇◗ ❝♦♥str❛✐♥ts ❤❛✈❡ t♦ ❜❡ s❛t✐s✜❡❞✿ ❢♦r ❡❛❝❤ ❞❛② t✱ t❤❡
s❤✐♣♠❡♥ts st
P,DC ❢r♦♠ t❤❡ ♣❧❛♥t t♦ t❤❡ ❉❈ ✭r❡s♣✳ st
DC,x ❢r♦♠ t❤❡ ❉❈ t♦ ❡❛❝❤ s❤♦♣ x✮
❤❛✈❡ t♦ ❜❡ ✐♥ ♥✉♠❜❡r ♦❢ ❧❛②❡rs ✭r❡s♣✳ ❝❛s❡s✮✳ ❚❤❡ ♥✉♠❜❡r ♦❢ ❝❛s❡s ♣❡r ❧❛②❡r ✭r❡s♣✳ ✐t❡♠
♣❡r ❝❛s❡s✮ ✐s nl ✭r❡s♣✳ nc✮✳ ❙t♦r❛❣❡ ✐s ❛❧❧♦✇❡❞ ❛t t❤❡ ❉❈ ✭r❡s♣✱ ✐♥ ❡❛❝❤ s❤♦♣ x✮✱ ❛♥❞ t❤❡
❝✉rr❡♥t ✐♥✈❡♥t♦r② ❢♦r ❡❛❝❤ ❞❛② t ✐s ❞❡♥♦t❡❞ it
DC ✭r❡s♣✳ it
x✮✳ ❆t t❤❡ ♣❧❛♥t ❧❡✈❡❧✱ st♦r❛❣❡ ✐s
♥♦t ❛❧❧♦✇❡❞✱ ❛♥❞ ❢♦r ❡❛❝❤ ❞❛② t✱ t❤❡ ♣r♦❞✉❝t✐♦♥ pt
♠✉st ❜❡ ✐♥ t❤❡ ✐❞❡❛❧ ♣r♦❞✉❝t✐♦♥ r❛♥❣❡
[Pmin, Pmax] ✭❣✐✈❡♥ ✐♥ ♥✉♠❜❡r ♦❢ ❧❛②❡rs ♣❡r ❞❛②✮✳ ❚❤✐s r❛♥❣❡ ❛✐♠s t♦ ♠✐t✐❣❛t❡ t❤❡ ❜✉❧❧✇❤✐♣
❡✛❡❝t✳ Pr♦❞✉❝✐♥❣ ♦✉t ♦❢ t❤❡s❡ ❧✐♠✐ts ✐s ♣❡♥❛❧✐③❡❞✳ ❋♦r ❆❇❈✱ t❤❡ ❞❛✐❧② r❛♥❣❡ ✐s s❡t t♦ ±20%
♦❢ t❤❡ ❛✈❡r❛❣❡ ❞❛✐❧② ❞❡♠❛♥❞ ❝♦♠♣✉t❡❞ ♦✈❡r ❛ ♣❧❛♥♥✐♥❣ ❤♦r✐③♦♥ ♦❢ ✶✵✵ ❞❛②s ✭❛ss✉♠✐♥❣ t❤❡
❞❛✐❧② ❞❡♠❛♥❞ ❢♦❧❧♦✇s ❛ ♥♦r♠❛❧ ❞✐str✐❜✉t✐♦♥✮✳ ❚❤❡ ❝♦♥s✐❞❡r❡❞ ❧❡❛❞✲t✐♠❡s ✭✐♥ ❞❛②s✮ ❛r❡ t❤❡
❢♦❧❧♦✇✐♥❣✿ L(P) = 1 ❜❡t✇❡❡♥ ♣r♦❞✉❝t✐♦♥ ❛♥❞ ❛✈❛✐❧❛❜✐❧✐t② ✭❛t t❤❡ ♣❧❛♥t ❧❡✈❡❧✮ ❢♦r s❤✐♣♠❡♥t
t♦ ❉❈✱ L(P, DC) = 2 ❢r♦♠ t❤❡ ♣❧❛♥t t♦ t❤❡ ❉❈✱ L(DC) = 1 ❢♦r ❝r♦ss✲❞♦❝❦✐♥❣ t❤r♦✉❣❤ t❤❡
❉❈✱ L(DC, x) = 1 ❢r♦♠ t❤❡ ❉❈ t♦ ❛♥② s❤♦♣ x✳ ❋✐♥❛❧❧②✱ t❤❡ ❡①♣❡❝t❡❞ ❞❡♠❛♥❞ ✭r❡s♣✳ t❤❡
❝♦rr❡s♣♦♥❞✐♥❣ ❧♦st s❛❧❡s✮ ❢♦r ❡❛❝❤ ❞❛② t ❛♥❞ ❡❛❝❤ s❤♦♣ x ✐s ❞❡♥♦t❡❞ dt
x ✭r❡s♣✳ lt
x✮✳
❚❤❡ ♦♥❧② str✉❝t✉r❛❧ ❝♦♥str❛✐♥ts ❢♦r t❤❡ s✉♣♣❧② ❝❤❛✐♥ ❛r❡ t❤❡ ♠❛t❡r✐❛❧ ✢♦✇ ❝♦♥s❡r✈❛t✐♦♥
✐♥ ❡❛❝❤ ❡❝❤❡❧♦♥ ✭✜rst ✐♥ t❤❡ ♣❧❛♥t✱ s❡❝♦♥❞ ✐♥ t❤❡ ❉❈✱ ❛♥❞ t❤✐r❞ ✐♥ ❡❛❝❤ s❤♦♣✮✿





st
P,DC = pt+L(P )
it
DC = it−1
DC + nl · s
t+L(P,DC)
P,DC −
P
x st
DC,x
it
x = it−1
x + nc · s
t+L(DC)+L(DC,x)
DC,x − dt
x + lt
x
✭C✮
■♥st❡❛❞ ♦❢ t❤❡ ✉s✉❛❧ r❡♦r❞❡r✲♣♦✐♥t ❛♣♣r♦❛❝❤✱ ✇❡ ♣r♦♣♦s❡ t♦ ❛ss♦❝✐❛t❡ ❛ ❞❛✐❧② ▼■◆✲▼❆❳
r❛♥❣❡ S(x) ✇✐t❤ ❡❛❝❤ s❤♦♣ x✳ ❚❤❡ ▼■◆ ✐s ✇❤❛t x ♥❡❡❞s t♦ s❛t✐s❢② ✐ts ❞❛✐❧② ❞❡♠❛♥❞✱ ✇❤❡r❡❛s
t❤❡ ▼❆❳ ✐s t❤❡ ❧❛r❣❡st ❞❡s✐r❡❞ ❞❛✐❧② ✐♥✈❡♥t♦r② ❢♦r t❤❡ ❝♦♥s✐❞❡r❡❞ ♣r♦❞✉❝t✳ ▼❆❳ ✐s ❞❡✜♥❡❞
❛s t❤❡ ❛✈❛✐❧❛❜❧❡ ♣❛rt ♦❢ t✇♦ q✉❛♥t✐t✐❡s✿ t❤❡ s❤❡❧❢ ❝❛♣❛❝✐t② ✭✇❤✐❝❤ ✐s ✉s❡❞ ✜rst✮ ♣❧✉s t❤❡ ❜❛❝❦✲
r♦♦♠ ❝❛♣❛❝✐t② ❛ss✐❣♥❡❞ t♦ t❤❡ ♣r♦❞✉❝t ✭✐t ✐s ♥♦t ❛ ❤❛r❞ ❝♦♥str❛✐♥t✱ ❛s ❡❛❝❤ s❤♦♣ ❤❛s ♦t❤❡r
♣r♦❞✉❝ts ❛♥❞ t❤❡r❡❢♦r❡ ❝❛♥ ✜♥❞ ❛ ♣❧❛❝❡ ✐♥ t❤❡ ❜❛❝❦✲r♦♦♠ ✐❢ t❤❡r❡ ✐s t♦♦ ♠✉❝❤ ✐♥✈❡♥t♦r② ♦❢
t❤❡ ❝♦♥s✐❞❡r❡❞ ♣r♦❞✉❝t✮✳ ❚❤❡ s✉♠ ♦❢ t❤❡s❡ t✇♦ q✉❛♥t✐t✐❡s ✐s ❞❡♥♦t❡❞ M(x) ❢♦r ❡❛❝❤ s❤♦♣ x✳
❈♦♥s❡q✉❡♥t❧②✱ t❤❡ ♣r♦❞✉❝t✐♦♥ ❝❛♥ ❜❡ ♣❧❛♥♥❡❞ ❜❛s❡❞ ♦♥ t❤❡ ❛❣❣r❡❣❛t✐♦♥ ♦❢ t❤❡ S(x) r❛♥❣❡s✱
❛♥❞ t❤❡♥ ♣✉s❤❡❞ ❞♦✇♥ t♦ t❤❡ ❉❈ ✭✇❤❡r❡ t❤❡ ♣r♦❞✉❝t ❝❛♥ ❜❡ t❡♠♣♦r❛r✐❧② st♦r❡❞ ❜✉t ❛
st♦r❛❣❡ ♣❡♥❛❧t② ✐s ❞✉❡✮ ❛♥❞ ✜♥❛❧❧② t♦ t❤❡ s❤♦♣s✳ ❚❤❡ ♣r♦❞✉❝t✐♦♥ ♣❧❛♥ ✐s ❢✉rt❤❡r ❝❛❧✐❜r❛t❡❞
✐♥ ♦r❞❡r t♦ ♣❡r❢❡❝t❧② s❛t✐s❢② t❤❡ ❊❇◗ ❝♦♥str❛✐♥ts✳ ❚❤✐s ❛♣♣r♦❛❝❤ s❤♦✉❧❞ ❛❧❧♦✇ ✭✶✮ t❤❡ ❉❈
❤❛✈✐♥❣ ❛ ♠✉❝❤ ♠♦r❡ st❛❜❧❡ r❡s♣♦♥s❡ ✭❡✈❡♥ ✇❤✐❧❡ ❦❡❡♣✐♥❣ ❛ ❧♦✇ ✐♥✈❡♥t♦r②✮✱ ❛♥❞ ✭✷✮ t❤❡ ♣❧❛♥t
s♠♦♦t❤✐♥❣ ✐ts ♣❡❛❦s ♦❢ ♣r♦❞✉❝t✐♦♥ ✭♠✐t✐❣❛t✐♥❣ t❤❡ ❜✉❧❧✇❤✐♣ ❡✛❡❝t✮✳ ❖♥ t❤❡ ♦♥❡ ❤❛♥❞✱ P❇❘
✐s ✐♥♥♦✈❛t✐✈❡ ✐♥ t❤❡ s❡♥s❡ t❤❛t ❢r♦♠ t❤❡ s❤♦♣ ♣❡rs♣❡❝t✐✈❡✱ ❛♥② st♦❝❦ ✇✐t❤✐♥ ✐ts ❛ss♦❝✐❛t❡❞
r❛♥❣❡ S(x) ✐s ♥♦t ♣❡♥❛❧✐③❡❞ ✭✐♥ ❝♦♥tr❛st ✇✐t❤ t❤❡ ✉s✉❛❧ ✉♥✐t st♦r❛❣❡ ❝♦sts✮✳ ❖♥ t❤❡ ♦t❤❡r
❤❛♥❞✱ ❢r♦♠ t❤❡ ♣❧❛♥t ♣❡rs♣❡❝t✐✈❡✱ t❤❡ ♣r♦❞✉❝t✐♦♥ ✈❛r✐❛❜✐❧✐t② ✇✐t❤✐♥ t❤❡ r❛♥❣❡ [Pmin, Pmax]
✐s ♥♦t ♣❡♥❛❧✐③❡❞✳
❚❤r❡❡ ❞✐✛❡r❡♥t ♦❜❥❡❝t✐✈❡ ❢✉♥❝t✐♦♥s ❛r❡ ❝♦♥s✐❞❡r❡❞✱ r❛♥❦❡❞ ✐♥ ❛ ❧❡①✐❝♦❣r❛♣❤✐❝ ♦r❞❡r ❢r♦♠
f1 t♦ f3 ✭✐✳❡✳✱ ❛ ❤✐❣❤❡r✲❧❡✈❡❧ ♦❜❥❡❝t✐✈❡ fi ✐s ✐♥✜♥✐t❡❧② ♠♦r❡ ✐♠♣♦rt❛♥t t❤❛♥ ❛ ❧♦✇❡r✲❧❡✈❡❧
♦❜❥❡❝t✐✈❡ fi+1✮✳ ❚❤❡ ❧❡①✐❝♦❣r❛♣❤✐❝ ❛♣♣r♦❛❝❤ f1 f2 f3 ✇❛s ✈❛❧✐❞❛t❡❞ ❜② ❆❇❈✳ ❚❤❡
t❤r❡❡ ♦❜❥❡❝t✐✈❡ ❢✉♥❝t✐♦♥s ❛r❡✱ ❢♦r ❡❛❝❤ ❞❛② t✿
✕ f1(t) =
P
x lt
x✿ s❤♦rt❛❣❡ ❛t t❤❡ s❤♦♣ ❧❡✈❡❧ ✭✐✳❡✳✱ ❧❡ss t❤❛♥ ▼■◆✮❀
✕ f2(t) = max{Pmin − pt
, 0} + max{pt
− Pmax, 0}✿ ♣r♦❞✉❝t✐♦♥ ♦✉t ♦❢ t❤❡ ✐❞❡❛❧ r❛♥❣❡❀
251

✕ f3(t) = it
+
P
x max{it
x − M(x), 0}✿ ❡①❝❡ss ♦❢ ✐♥✈❡♥t♦r② ✐♥ t❤❡ s❤♦♣s ❛♥❞ ❛t t❤❡ ❉❈
✭✐✳❡✳✱ ♠♦r❡ t❤❛♥ ▼❆❳ ✐♥ t❤❡ s❤♦♣s✱ ❛♥❞ ♠♦r❡ t❤❛♥ ③❡r♦ st♦❝❦ ✐♥ t❤❡ ❉❈✮✳
❆s t❤❡ ❞❡♠❛♥❞ ✐s ♥♦♥✲❞❡t❡r♠✐♥✐st✐❝✱ s✐♠✉❧❛t✐♦♥ ✐♥✈♦❧✈✐♥❣ ❛ r♦❧❧✐♥❣ ❤♦r✐③♦♥ H ✐s ✉s❡❞
✭♦✈❡r ❛ ♣❧❛♥♥✐♥❣ ❤♦r✐③♦♥ ♦❢ ✶✵✵ ❞❛②s✮✳ ❚❤❡ s✐③❡ ♦❢ H ✐s ✜①❡❞ t♦ L(P)+L(P, DC)+L(DC)+
maxx L(DC, x) + 1 ✭✇❤✐❝❤ r❡s✉❧ts ✐♥ ✻ ❞❛②s ❢♦r ❆❇❈✮✳ ❚❤✐s ✇❛②✱ t✇♦ ♣r♦❞✉❝t✐♦♥ ❞❡❝✐s✐♦♥s
✭✐✳❡✳✱ ✐♥✈♦❧✈✐♥❣ t❤❡ ✜rst t✇♦ ❞❛②s ♦❢ H✮ ❝❛♥ r❡❛❝❤ t❤❡ s❤♦♣ ❧❡✈❡❧ ✇✐t❤✐♥ H✳ ◆♦t❡ t❤❛t
❡❛❝❤ ❞❡❝✐s✐♦♥ t❤❛t ❝❛♥♥♦t ✐♠♣❛❝t t❤❡ st♦❝❦s ✐♥ t❤❡ s❤♦♣s ✇✐t❤✐♥ H ❛r❡ s❡t t♦ ③❡r♦✳ ❲❤❡♥
s✐♠✉❧❛t✐♥❣✱ ❢♦r ❡❛❝❤ s❤♦♣ x✱ ♦♥❧② t❤❡ ❞❡♠❛♥❞ ♦❢ t♦❞❛② dt0
x ❛♥❞ t❤♦s❡ ♦❢ t❤❡ ♣r❡✈✐♦✉s ❞❛②s
❛r❡ ❦♥♦✇♥✱ ❛♥❞ t❤❡ ❢♦r❡❝❛st ❢♦r t❤❡ ♥❡①t ❞❛② dt
x ✐s s✐♠♣❧② t❤❡ ❛✈❡r❛❣❡ ❞❛✐❧② ❞❡♠❛♥❞ ♣❧✉s ✐ts
st❛♥❞❛r❞ ❞❡✈✐❛t✐♦♥ ✭❛s ♥❡✐t❤❡r ❛ tr❡♥❞ ♥♦r ❛ s❡❛s♦♥❛❧✐t② ❝❤❛r❛❝t❡r✐③❡ t❤❡ ❝♦♥s✐❞❡r❡❞ ✢❛❣s❤✐♣
♣r♦❞✉❝t✮✳ ❖♥ ❡❛❝❤ ❞❛② ♦❢ t❤❡ ♣❧❛♥♥✐♥❣ ❤♦r✐③♦♥✱ ❛ ✸✲st❡♣ ♦♣t✐♠✐③❛t✐♦♥ ✐s ♣❡r❢♦r♠❡❞✱ ❡❛❝❤
st❡♣ ❜❡✐♥❣ s♦❧✈❡❞ ❜② ❈P▲❊❳✳ ❚❤❡ r❡s✉❧t✐♥❣ P❇❘ ❛♣♣r♦❛❝❤ ✐s s✉♠♠❛r✐③❡❞ ✐♥ ❆❧❣♦r✐t❤♠ ✶✳
❆❧❣♦r✐t❤♠ ✶ P❧❛♥♥✐♥❣✲❜②✲r❛♥❣❡ ❛♣♣r♦❛❝❤
❙❡t t0 ← 1
❲❤✐❧❡ t0 + |H| ≤ 100✱ ❞♦✿
✕ ❢♦r ❡❛❝❤ s❤♦♣ x✱ s❡t dt0
x ❛s t❤❡ ❛❝t✉❛❧ ❦♥♦✇♥ ❞❡♠❛♥❞ ✭✐♥st❡❛❞ ♦❢ ❢♦r❡❝❛st✮❀
✶✳ ♠✐♥✐♠✐③❡ F1 =
P
t∈[t0,t0+|H|] f1(t) ✇❤✐❧❡ s❛t✐s❢②✐♥❣ t❤❡ ❝♦♥str❛✐♥t s❡t (C)❀
❧❡t F⋆
1 ❜❡ t❤❡ ♦❜t❛✐♥❡❞ ♠✐♥✐♠✉♠❀
✷✳ ♠✐♥✐♠✐③❡ F2 =
P
t∈[t0,t0+|H|] f2(t) ✇❤✐❧❡ s❛t✐s❢②✐♥❣ (C)✱ ❛♥❞ s✉❝❤ ❛s F1 = F⋆
1 ❀
❧❡t F⋆
2 ❜❡ t❤❡ ♦❜t❛✐♥❡❞ ♠✐♥✐♠✉♠❀
✸✳ ♠✐♥✐♠✐③❡ F3 =
P
t∈[t0,t0+|H|] f3(t) ✇❤✐❧❡ s❛t✐s❢②✐♥❣ (C)✱ ❛♥❞ s✉❝❤ ❛s (F1, F2) = (F⋆
1 , F⋆
2 )❀
✕ ❢r❡❡③❡ s❤✐♣♠❡♥ts ❛♥❞ ♣r♦❞✉❝t✐♦♥ ♦❢ ❞❛② t0❀
✕ s❡t t0 ← t0 + 1✳
✸ ❘❡s✉❧ts
P❇❘ ✐s ❝♦♠♣❛r❡❞ ✇✐t❤ t❤❡ t✇♦ st❛♥❞❛r❞ s✉♣♣❧②✲❝❤❛✐♥✲♠❛♥❛❣❡♠❡♥t ♠❡t❤♦❞s ♣✉❧❧ ❛♥❞
♣✉s❤ ❞❡s❝r✐❜❡❞ ✐♥ ❙❡❝t✐♦♥ ✶✳ ❖♥ t❤❡ ♦♥❡ ❤❛♥❞✱ ♣✉❧❧ ♠✐♥✐♠✐③❡s t❤❡ r❡t❛✐❧❡r✬s ❝♦sts ✭✐✳❡✳✱ s❤♦rt✲
❛❣❡ ❛♥❞ ✐♥✈❡♥t♦r② ✐♥ t❤❡ s❤♦♣s✮✳ ❖♥ t❤❡ ♦t❤❡r ❤❛♥❞✱ ♣✉s❤ ♠✐♥✐♠✐③❡s t❤❡ ♠❛♥✉❢❛❝t✉r❡r✬s
❝♦sts ✭✐✳❡✳✱ ✐rr❡❣✉❧❛r ♣r♦❞✉❝t✐♦♥✱ ❛♥❞ ✐♥✈❡♥t♦r② ✐♥ t❤❡ ❉❈✮✳
▼❡t❤♦❞ ♣✉❧❧ ✉s❡s r❡♦r❞❡r ♣♦✐♥ts ❢♦r ❜♦t❤ t❤❡ ❉❈ ❛♥❞ t❤❡ s❤♦♣s✳ ❊❛❝❤ t✐♠❡ t❤❡ ❛✈❛✐❧❛❜❧❡
✐♥✈❡♥t♦r② ❧❡✈❡❧ ✐s ❜❡❧♦✇ ✐ts r❡♦r❞❡r ♣♦✐♥t✱ ❛♥ ♦r❞❡r ✐s ♣❧❛❝❡❞ t♦ t❤❡ ✉♣str❡❛♠ ❧❡✈❡❧✳ ❆
♣r♦❞✉❝t✐♦♥ ❜❛t❝❤ ✐s ❧❛✉♥❝❤❡❞ ❡❛❝❤ t✐♠❡ ❛♥ ♦r❞❡r ❝♦♠❡s ❢r♦♠ t❤❡ ❉❈✱ ❛s t❤❡ ♣❧❛♥t ❞♦❡s
♥♦t ❤♦❧❞ ✐♥✈❡♥t♦r②✳ ❚❤❡ r❡♦r❞❡r ♣♦✐♥t ✐s s❡t ❡q✉❛❧ t♦ (D + σ) · L✱ ✇❤❡r❡ D ✐s t❤❡ ❛✈❡r❛❣❡
❞❛✐❧② ❞❡♠❛♥❞ ❢r♦♠ t❤❡ ❞♦✇♥str❡❛♠ ❡❝❤❡❧♦♥✱ σ ✐s t❤❡ ❞❛✐❧② st❛♥❞❛r❞ ❞❡✈✐❛t✐♦♥ ♦❢ D✱ ❛♥❞ L
✐s t❤❡ t♦t❛❧ ❧❡❛❞✲t✐♠❡ ❢r♦♠ t❤❡ ✉♣str❡❛♠ ❡❝❤❡❧♦♥ ✭✐✳❡✳✱ L(P) + L(P, DC) ❢♦r t❤❡ ❉❈✱ ❛♥❞
L(DC) + L(DC, x) ❢♦r ❛♥② s❤♦♣ x✮✳
▼❡t❤♦❞ ♣✉s❤ ✜rst ❝♦♠♣✉t❡s ✐ts ✐❞❡❛❧ ♣r♦❞✉❝t✐♦♥ r❛t❡ p ✭✐♥ ❧❛②❡rs✮✱ ✇❤✐❝❤ ✐s t❤❡ ❛✈❡r❛❣❡
❞❛✐❧② ❞❡♠❛♥❞ ♦✈❡r t❤❡ ✇❤♦❧❡ ♣❧❛♥♥✐♥❣ ❤♦r✐③♦♥ ♦❢ ✶✵✵ ❞❛②s✳ ■t ✐s ✉s✉❛❧❧② ♥♦t ❛♥ ✐♥t❡❣❡r✳ ❚♦
s❛t✐s❢② t❤❡ ❊❇◗ ❝♦♥str❛✐♥t ✇❤✐❧❡ ❤❛✈✐♥❣ ❛♥ ❛❧♠♦st ❝♦♥st❛♥t ♣r♦❞✉❝t✐♦♥ r❛t❡ ✭✐✳❡✳✱ ❛r♦✉♥❞
p✮✱ ❡❛❝❤ ❞❛② t ✭❢r♦♠ t = 1 t♦ t = 100✮✱ ⌈p · t − Q⌉ ❧❛②❡rs ❛r❡ ♣r♦❞✉❝❡❞✱ ✇❤❡r❡ Q ✐s t❤❡
♥✉♠❜❡r ♦❢ ❧❛②❡rs ♣r♦❞✉❝❡❞ ✉♥t✐❧ ❞❛② t✳ ❚❤❡ ❞❛✐❧② ♣r♦❞✉❝❡❞ q✉❛♥t✐t② ✐s s❤✐♣♣❡❞ t♦ t❤❡ ❉❈ ❛s
s♦♦♥ ❛s ♣♦ss✐❜❧❡✱ ❛♥❞ ✇❤❡♥❡✈❡r t❤❡ ❉❈ ❤❛s ❛♥② ✐♥✈❡♥t♦r②✱ ✐t s❤✐♣s ✐t t♦ t❤❡ s❤♦♣s ✇✐t❤♦✉t
❡①❝❡❡❞✐♥❣ t❤❡✐r ❞❡s✐r❡❞ ✐♥✈❡♥t♦r✐❡s ✭❛s ♣r❡✈✐♦✉s❧② ❞❡✜♥❡❞✮✳
252

❚❤❡ t❤r❡❡ ♠❡t❤♦❞s ❛r❡ ❝♦♠♣❛r❡❞ ❢♦r ✷✵ ✐♥st❛♥❝❡s I1 t♦ I20✱ ❣❡♥❡r❛t❡❞ r❛♥❞♦♠❧② ❜❛s❡❞
♦♥ t❤❡ r❡❛❧ ❞❛t❛ ♣r♦✈✐❞❡❞ ❜② ❆❇❈✳ ❊❛❝❤ ✐♥st❛♥❝❡ ✐s ♠❛❞❡ ♦❢ N s❤♦♣s✳ ■♥st❛♥❝❡s I1 t♦ I10
❤❛✈❡ N = 20 s❤♦♣s✱ ✇✐t❤ ❛ ❧❛r❣❡ ❞❛✐❧② ❛✈❡r❛❣❡ ❞❡♠❛♥❞ ♣❡r s❤♦♣ ✭✐♥ ❬✻✱ ✶✷❪ ❝❛s❡s✮✱ ❛♥❞ ❛
❞❡s✐r❡❞ ✐♥✈❡♥t♦r② ♣❡r s❤♦♣ ♦❢ ✻ ❝❛s❡s ✭✷ ❢♦r t❤❡ s❤❡❧❢ ❛♥❞ ✹ ❢♦r t❤❡ ❜❛❝❦✲r♦♦♠✮✳ ■♥st❛♥❝❡s
I11 t♦ I20 ❤❛✈❡ N = 50 s❤♦♣s✱ ✇✐t❤ ❛ s♠❛❧❧ ❞❛✐❧② ❛✈❡r❛❣❡ ❞❡♠❛♥❞ ♣❡r s❤♦♣ ✭✐♥ ❬✶✱ ✹❪ ❝❛s❡s✮✱
❛♥❞ ❛ ❞❡s✐r❡❞ ✐♥✈❡♥t♦r② ♣❡r s❤♦♣ ♦❢ ✸ ❝❛s❡s ✭✶ ❢♦r t❤❡ s❤❡❧❢ ❛♥❞ ✷ ❢♦r t❤❡ ❜❛❝❦✲r♦♦♠✮✳ ❋♦r
✐♥st❛♥❝❡s I1 t♦ I5 ❛♥❞ I11 t♦ I15✱ ✇❡ ❤❛✈❡ σ ∈ [50, 100]% ♦❢ t❤❡ ❛✈❡r❛❣❡ ❞❛✐❧② ❞❡♠❛♥❞✳
■♥ ❝♦♥tr❛st✱ σ ∈ [100, 150]% ❢♦r t❤❡ ♦t❤❡r ✐♥st❛♥❝❡s✳ ❋♦r ❡❛❝❤ ❣r♦✉♣ ♦❢ ✜✈❡ ✐♥st❛♥❝❡s✱ t❤❡
♥✉♠❜❡r ♦❢ ✐t❡♠s ♣❡r ❝❛s❡✴❧❛②❡r ❞✐✛❡rs✳ ▼♦r❡ ♣r❡❝✐s❡❧②✱ t❤❡ ♥✉♠❜❡rs ♦❢ ✐t❡♠s ♣❡r ❝❛s❡ ❛r❡
(6, 8, 12, 16, 20)✱ ✇❤❡r❡❛s t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ♥✉♠❜❡rs ♦❢ ❝❛s❡s ♣❡r ❧❛②❡r ✐s (14, 10, 14, 10, 12)✱
t❤♦s❡ ♥✉♠❜❡rs ❜❡✐♥❣ r❡❛❧ ❞❛t❛ ❢r♦♠ ✜✈❡ ❞✐✛❡r❡♥t ♣❛❝❦ ♠❛t❡r✐❛❧s ♦❢ ❆❇❈✳ ❚❛❜❧❡ ✶ ♣r❡s❡♥ts
t❤❡ r❡s✉❧ts ❢♦r t❤❡ ❝♦♥s✐❞❡r❡❞ ♠❡t❤♦❞s✳ ❚❤❡ ❢♦❧❧♦✇✐♥❣ ❛✈❡r❛❣❡❞ ✐♥❞✐❝❛t♦rs ✭♦✈❡r t❤❡ ♣❧❛♥♥✐♥❣
❤♦r✐③♦♥ ♦❢ ✶✵✵ ❞❛②s✮ ❛r❡ ❣✐✈❡♥ t♦ ❝❛♣t✉r❡ t❤❡ ♣❡r❢♦r♠❛♥❝❡ ♦♥ f1, f2 ❛♥❞ f3✱ r❡s♣❡❝t✐✈❡❧②✳
F1 ✐s t❤❡ s❤♦rt❛❣❡ ♣❡r❝❡♥t❛❣❡ ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ❛✈❡r❛❣❡ ❞❛✐❧② ❞❡♠❛♥❞ ♦✈❡r ❛❧❧ t❤❡ s❤♦♣s✳
F2 ✐s t❤❡ ♦✉t✲♦❢✲r❛♥❣❡ ♣r♦❞✉❝t✐♦♥ ♣❡r❝❡♥t❛❣❡✳ F3 ✐s t❤❡ ♣❡r❝❡♥t❛❣❡ ♦❢ ❡①❝❡❡❞✐♥❣ st♦❝❦ ✇✐t❤
r❡s♣❡❝t t♦ t❤❡ t♦t❛❧ st♦r❛❣❡ ❝❛♣❛❝✐t② t❤❛t ✐s ♥♦t ♣❡♥❛❧✐③❡❞✳ ❚❤❡ ❧❛tt❡r ✐s t❤❡ s✉♠ ♦❢ ❛❧❧ t❤❡
❞❡s✐r❡❞ ✐♥✈❡♥t♦r✐❡s ✐♥ t❤❡ s❤♦♣s ✭❦♥♦✇✐♥❣ t❤❛t t❤❡ ❞❡s✐r❡❞ ✐♥✈❡♥t♦r② ✐♥ t❤❡ ❉❈ ✐s ❛❧✇❛②s
③❡r♦✮✳
❆❧❧ t❤❡ ❛❧❣♦r✐t❤♠s ✇❡r❡ ❝♦❞❡❞ ✇✐t❤ ❈✰✰ ✉♥❞❡r ▲✐♥✉①✱ ❛♥❞ r✉♥ ♦♥ ✸✳✹ ●❍③ ■♥t❡❧ ◗✉❛❞✲
❝♦r❡ ✐✼ ♣r♦❝❡ss♦r ✇✐t❤ ✽ ●❇ ♦❢ ❉❉❘✸ ❘❆▼✳ ❋♦r ❡❛❝❤ t✐♠❡ ✇✐♥❞♦✇ H✱ ❡❛❝❤ ♠❡t❤♦❞ ✐s
❛❜❧❡ t♦ ✜♥❞ ✐ts s♦❧✉t✐♦♥ ✇✐t❤✐♥ s❡❝♦♥❞s✱ ✐♥❝❧✉❞✐♥❣ ❈P▲❊❳ t❤❛t ❛❧✇❛②s ♣r♦✈✐❞❡s ❛♥ ♦♣t✐♠❛❧
s♦❧✉t✐♦♥✳ ❈P▲❊❳ ✐s ❜❛s❡❞ ♦♥ t❤❡ ✸✲st❡♣ ❛❧❣♦r✐t❤♠ ♣r❡s❡♥t❡❞ ✐♥ ❙❡❝t✐♦♥ ✷✱ ❛♥❞ ✐t ❤❛s ❛
t✐♠❡ ❧✐♠✐t ♦❢ ♦♥❡ ♠✐♥✉t❡ ♣❡r ♦❜❥❡❝t✐✈❡✳ ❋♦r ❛❧❧ t❤r❡❡ ♠❡t❤♦❞s✱ t❤❡ s❤♦rt❛❣❡ ♣❡♥❛❧t② ✐s
❧♦❣✐❝❛❧❧② s♠❛❧❧❡r ❢♦r t❤❡ ✐♥st❛♥❝❡s ✇✐t❤ ❛ s♠❛❧❧❡r σ ✭✐✳❡✳✱ ✐♥st❛♥❝❡s I1 t♦ I5✱ ❛♥❞ I11 t♦ I15✮✳
❆s ❡①♣❡❝t❡❞✱ ♣✉❧❧ s❤♦✇s ❛ ✈❡r② s♠❛❧❧ s❤♦rt❛❣❡ ✭♦♥ ❛✈❡r❛❣❡✱ ✵✳✼✻✪ ♦❢ t❤❡ ❞❡♠❛♥❞✮✱ ❜✉t
❛ ✈❡r② ✐rr❡❣✉❧❛r ♣r♦❞✉❝t✐♦♥ ♣❛tt❡r♥ ✭♦♥ ❛✈❡r❛❣❡✱ ✻✶✳✵✸✪ ♦❢ t❤❡ ♣r♦❞✉❝t✐♦♥ ✐s ♦✉t ♦❢ t❤❡
✐❞❡❛❧ r❛♥❣❡✮✳ ❚❤❡ r❡❧❛t✐✈❡❧② ❜✐❣ ✐♥✈❡♥t♦r② ✭♦♥ ❛✈❡r❛❣❡✱ ✷✸✳✵✼✪ ♦❢ t❤❡ ❢r❡❡✲♦❢✲❝♦st ❝❛♣❛❝✐t②✮
✐s ❞✉❡ t♦ st♦r❛❣❡ ❛t t❤❡ ❉❈✳ ❯♥s✉r♣r✐s✐♥❣❧②✱ ♣✉s❤ ❤❛s ♥♦ ♦✉t✲♦❢✲r❛♥❣❡ ♣r♦❞✉❝t✐♦♥✱ ❜✉t
✐t ❤❛s t❤❡ ❜✐❣❣❡st s❤♦rt❛❣❡ ✭♦♥ ❛✈❡r❛❣❡✱ ✶✷✳✹✺✪ ♦❢ t❤❡ ❞❡♠❛♥❞✮ ❛s ✐t ❞♦❡s ♥♦t ❛❞❛♣t ✐ts
♣r♦❞✉❝t✐♦♥ t♦ t❤❡ ❞❡♠❛♥❞ ♣❛tt❡r♥✳ ■♥t❡r❡st✐♥❣❧②✱ ♣✉s❤ s❤♦✇s ♠✉❝❤ ❜❡tt❡r r❡s✉❧ts ✇✐t❤ ❛
❧❛r❣❡r ♥✉♠❜❡r ♦❢ s❤♦♣s✿ t❤❡ s❤♦rt❛❣❡ ✐♥❞✐❝❛t♦r r♦✉❣❤❧② ❣♦❡s ❞♦✇♥ ❢r♦♠ ✷✺✪ ✭✇✐t❤ N = 20✮
t♦ ✺✪ ✭✇✐t❤ N = 50✮✳ ■♥❞❡❡❞✱ t❤❡ ♠♦r❡ s❤♦♣s t❤❡r❡ ❛r❡✱ t❤❡ ♠♦r❡ ♣♦ss✐❜✐❧✐t✐❡s ❛ ✜①❡❞
♣r♦❞✉❝t✐♦♥ ❤❛s t♦ ❜❡ ♣✉s❤❡❞ ❞♦✇♥ t♦ t❤❡ s❤♦♣s✱ ❛♥❞ ❤❡♥❝❡ t❤❡ ❜❡tt❡r t❤❡ ❞✐s♣❛t❝❤✐♥❣ ♦❢
t❤❡ ♣r♦❞✉❝t✐♦♥ ❜❡t✇❡❡♥ t❤❡ s❤♦♣s ❝❛♥ ❜❡✳ P❇❘ ♦✛❡rs t❤❡ ❜❡st r❡s✉❧ts✿ t❤❡ ❛✈❡r❛❣❡ s❤♦rt❛❣❡
✐s ♦♥❧② ✵✳✸✷✪ ♦❢ t❤❡ ❞❡♠❛♥❞✱ t❤❡ ❛✈❡r❛❣❡ ♦✉t✲♦❢✲r❛♥❣❡ ♣r♦❞✉❝t✐♦♥ ✐s ❧✐♠✐t❡❞ t♦ ✵✳✵✷✪✱
❛♥❞ t❤❡ ❛✈❡r❛❣❡ ❝♦st❧② ✐♥✈❡♥t♦r② ✐s ♦♥❧② ✵✳✷✻✪ ♦❢ t❤❡ ❢r❡❡✲♦❢✲❝❤❛r❣❡ ❝❛♣❛❝✐t②✳ ❘❡♠❛r❦❛❜❧②✱
♥♦♥❡ ♦❢ t❤❡ P❇❘ ♣❡r❢♦r♠❛♥❝❡ ✐♥❞✐❝❛t♦r ❡①❝❡❡❞s ✶✪✱ ❛♥❞ ❡✈❡♥ ✵✳✶✶✪ ❢♦r t❤❡ ♦✉t✲♦❢✲r❛♥❣❡
♣r♦❞✉❝t✐♦♥✳ ❚❤❡ ❜✉❧❧✇❤✐♣ ❡✛❡❝t ✐s t❤✉s r❡♠♦✈❡❞✱ ✇❤✐❧❡ ❛❧♠♦st ❛❧✇❛②s ❛✈♦✐❞✐♥❣ s❤♦rt❛❣❡
❛♥❞ ✐♥✈❡♥t♦r② ♣❡♥❛❧t✐❡s✳
253

❚❛❜❧❡ ✶✳ ❈♦♠♣❛r✐s♦♥ ♦❢ ♣✉❧❧✱ ♣✉s❤ ❛♥❞ P❇❘ ❛♣♣r♦❛❝❤❡s ❢♦r r❡❛❧✐st✐❝ ✐♥st❛♥❝❡s✳
P✉❧❧ P✉s❤ P❇❘
■♥st❛♥❝❡ F1 F2 F3 F1 F2 F3 F1 F2 F3
I1 ✵✳✺✺ ✼✺✳✵✵ ✶✽✳✶✷ ✶✽✳✽✷ ✵✳✵✵ ✶✶✳✺✵ ✵✳✵✾ ✵✳✵✶ ✵✳✺✵
I2 ✵✳✻✸ ✺✾✳✵✵ ✸✵✳✾✵ ✶✾✳✶✾ ✵✳✵✵ ✶✼✳✶✷ ✵✳✶✼ ✵✳✵✵ ✵✳✵✵
I3 ✵✳✹✹ ✹✻✳✻✼ ✻✽✳✵✽ ✷✷✳✽✶ ✵✳✵✵ ✹✻✳✶✸ ✵✳✵✼ ✵✳✵✵ ✵✳✵✵
I4 ✵✳✺✺ ✺✹✳✺✵ ✶✽✳✼✾ ✶✻✳✷✽ ✵✳✵✵ ✶✵✳✸✼ ✵✳✵✼ ✵✳✵✶ ✵✳✶✼
I5 ✵✳✼✸ ✷✹✳✵✵ ✶✹✳✼✷ ✶✳✹✻ ✵✳✵✵ ✵✳✵✾ ✵✳✶✶ ✵✳✵✵ ✵✳✵✵
I6 ✶✳✷✽ ✼✵✳✺✵ ✷✻✳✽✶ ✷✼✳✾✷ ✵✳✵✵ ✶✺✳✾✺ ✵✳✺✶ ✵✳✵✶ ✵✳✺✵
I7 ✶✳✵✸ ✼✵✳✵✵ ✹✸✳✼✶ ✷✼✳✾✼ ✵✳✵✵ ✷✽✳✻✾ ✵✳✷✵ ✵✳✵✷ ✵✳✽✷
I8 ✶✳✺✸ ✼✼✳✵✵ ✽✾✳✷✷ ✷✽✳✼✶ ✵✳✵✵ ✺✸✳✻✶ ✵✳✼✹ ✵✳✵✹ ✵✳✽✷
I9 ✶✳✷✹ ✹✷✳✸✸ ✸✻✳✵✺ ✸✵✳✻✾ ✵✳✵✵ ✷✸✳✾✷ ✵✳✻✹ ✵✳✵✵ ✵✳✶✼
I10 ✵✳✽✽ ✽✳✵✵ ✷✺✳✼✻ ✹✳✻✻ ✵✳✵✵ ✸✳✶✶ ✵✳✵✷ ✵✳✵✵ ✵✳✵✵
I11 ✵✳✺✾ ✹✾✳✺✵ ✺✳✷✼ ✹✳✶✼ ✵✳✵✵ ✵✳✵✵ ✵✳✹✸ ✵✳✵✶ ✵✳✸✵
I12 ✵✳✸✺ ✶✶✵✳✵✵ ✼✳✺✵ ✹✳✽✵ ✵✳✵✵ ✵✳✵✵ ✵✳✷✸ ✵✳✵✹ ✵✳✹✷
I13 ✵✳✹✸ ✶✵✺✳✵✵ ✶✹✳✹✺ ✹✳✻✼ ✵✳✵✵ ✵✳✵✵ ✵✳✷✽ ✵✳✶✶ ✵✳✻✸
I14 ✵✳✹✷ ✶✶✸✳✵✵ ✺✳✺✶ ✺✳✶✹ ✵✳✵✵ ✵✳✵✵ ✵✳✸✶ ✵✳✵✸ ✵✳✷✺
I15 ✵✳✸✹ ✶✺✳✵✵ ✹✳✽✸ ✻✳✽✺ ✵✳✵✵ ✵✳✵✵ ✵✳✵✵ ✵✳✵✵ ✵✳✵✵
I16 ✵✳✼✾ ✺✼✳✺✵ ✻✳✺✻ ✹✳✷✵ ✵✳✵✵ ✵✳✵✵ ✵✳✺✻ ✵✳✵✵ ✵✳✶✷
I17 ✵✳✽✵ ✶✷✻✳✵✵ ✶✵✳✵✸ ✹✳✼✺ ✵✳✵✵ ✵✳✵✵ ✵✳✻✼ ✵✳✵✷ ✵✳✷✶
I18 ✶✳✶✽ ✹✼✳✺✵ ✷✵✳✾✹ ✺✳✾✹ ✵✳✵✵ ✵✳✵✵ ✵✳✽✶ ✵✳✵✷ ✵✳✷✶
I19 ✵✳✾✾ ✹✽✳✵✵ ✽✳✵✹ ✹✳✼✺ ✵✳✵✵ ✵✳✵✵ ✵✳✹✽ ✵✳✵✵ ✵✳✵✵
I20 ✵✳✹✻ ✷✷✳✵✵ ✻✳✶✸ ✺✳✷✷ ✵✳✵✵ ✵✳✵✵ ✵✳✵✸ ✵✳✵✵ ✵✳✵✵
❆✈❡r❛❣❡ ✵✳✼✻ ✻✶✳✵✸ ✷✸✳✵✼ ✶✷✳✹✺ ✵✳✵✵ ✶✵✳✺✷ ✵✳✸✷ ✵✳✵✷ ✵✳✷✻
✹ ❈♦♥❝❧✉s✐♦♥
■♥ t❤✐s ✇♦r❦✱ ❛ ♣❧❛♥♥✐♥❣✲❜②✲r❛♥❣❡ ✭P❇❘✮ ❛♣♣r♦❛❝❤ ✐s ♣r♦♣♦s❡❞ ❢♦r t❤❡ ♣r♦❞✉❝t✐♦♥ ❛♥❞
❞✐str✐❜✉t✐♦♥ ♦❢ ❛ ✢❛❣s❤✐♣ ♣r♦❞✉❝t ♦❢ ❛ r❡❛❧ ❝♦♠♣❛♥②✳ P❇❘ ✐s s♣❡❝✐✜❝❛❧❧② ✇❡❧❧ ❛❞❛♣t❡❞
❢♦r ❝♦♥tr♦❧❧✐♥❣ t❤❡ ❜✉❧❧✇❤✐♣ ❡✛❡❝t✳ ■♥ t❤✐s ❝♦♥t❡①t✱ ❛ ❧❡①✐❝♦❣r❛♣❤✐❝ ♦♣t✐♠✐③❛t✐♦♥ ♣r♦❜❧❡♠ ✐s
❞❡s✐❣♥❡❞ ❢♦r ♠✐♥✐♠✐③✐♥❣✿ ✭✶✮ s❤♦rt❛❣❡✱ ✭✷✮ ♦✉t✲♦❢✲r❛♥❣❡ ♣r♦❞✉❝t✐♦♥✱ ✭✸✮ ✉♥❞❡s✐r❡❞ ✐♥✈❡♥t♦r②✳
P❇❘ ✇❛s t❡st❡❞ ❢♦r ✷✵ r❡❛❧✐st✐❝ ✐♥st❛♥❝❡s ❛♥❞ ✈❡r② ❢❛✈♦r❛❜❧② ❝♦♠♣❛r❡❞ ✇✐t❤ t❤❡ ✇❡❧❧✲
❦♥♦✇♥ ♣✉❧❧ ❛♥❞ ♣✉s❤ ❛♣♣r♦❛❝❤❡s✳ ❋✉t✉r❡ ✇♦r❦s ✐♥❝❧✉❞❡ t❤❡ ✐♥✈❡st✐❣❛t✐♦♥ ♦❢ ♠♦r❡ ❝♦♠♣❧❡①
s✐t✉❛t✐♦♥s ✭❡✳❣✳✱ ♠✉❧t✐♣❧❡ ❉❈s✱ ✈❛r✐❛❜❧❡ ❧❡❛❞✲t✐♠❡s✱ ♣r♦♠♦t✐♦♥❛❧ ✇❡❡❦s✮✳
❘❡❢❡r❡♥❝❡s
❆①sät❡r✱ ❙✳✱ ✷✵✶✺✱ ✏■♥✈❡♥t♦r② ❈♦♥tr♦❧✧✱ ❙♣r✐♥❣❡r✳
❉❛r✈✐s❤ ▼✳ ❛♥❞ ▲✳❈✳ ❈♦❡❧❤♦✱ ✷✵✶✼✱ ✏❙❡q✉❡♥t✐❛❧ ✈❡rs✉s ■♥t❡❣r❛t❡❞ ❖♣t✐♠✐③❛t✐♦♥✿ ▲♦t ❙✐③✐♥❣✱ ■♥✈❡♥✲
t♦r② ❈♦♥tr♦❧ ❛♥❞ ❉✐str✐❜✉t✐♦♥✧✱ ❊✉r♦♣❡❛♥ ❏♦✉r♥❛❧ ♦❢ ❖♣❡r❛t✐♦♥❛❧ ❘❡s❡❛r❝❤✱ ❢♦rt❤❝♦♠✐♥❣✳
●❡❛r② ❙✳✱ ❙✳▼✳ ❉✐s♥❡② ❛♥❞ ❉✳❘✳ ❚♦✇✐❧❧✱ ✷✵✵✻✱ ✏❖♥ ❜✉❧❧✇❤✐♣ ✐♥ s✉♣♣❧② ❝❤❛✐♥s ✲ ❍✐st♦r✐❝❛❧ r❡✈✐❡✇✱
♣r❡s❡♥t ♣r❛❝t✐❝❡ ❛♥❞ ❡①♣❡❝t❡❞ ❢✉t✉r❡ ✐♠♣❛❝t✧✱ ■♥t❡r♥❛t✐♦♥❛❧ ❏♦✉r♥❛❧ ♦❢ Pr♦❞✉❝t✐♦♥ ❊❝♦♥♦♠✐❝s✱
❱♦❧✳ ✶✵✶✱ ♣♣✳ ✷✲✶✽✳
❚❤❡✈❡♥✐♥ ❙✳✱ ◆✳ ❩✉✛❡r❡② ❛♥❞ ❘✳ ●❧❛r❞♦♥✱ ✷✵✶✼✱ ✏▼♦❞❡❧ ❛♥❞ ▼❡t❛❤❡✉r✐st✐❝s ❢♦r ❛ ❙❝❤❡❞✉❧✐♥❣
Pr♦❜❧❡♠ ■♥t❡❣r❛t✐♥❣ Pr♦❝✉r❡♠❡♥t✱ ❙❛❧❡ ❛♥❞ ❉✐str✐❜✉t✐♦♥✧✱ ❆♥♥❛❧s ♦❢ ❖♣❡r❛t✐♦♥s ❘❡s❡❛r❝❤✱
❱♦❧✳ ✷✺✾✭✶✮✱ ♣♣✳ ✹✸✼✲✹✻✵✳
❲❛♥❣ ❳✳✱ ❛♥❞ ❙✳▼✳ ❉✐s♥❡②✱ ✷✵✶✻✱ ✏❚❤❡ ❜✉❧❧✇❤✐♣ ❡✛❡❝t✿ Pr♦❣r❡ss✱ tr❡♥❞s ❛♥❞ ❞✐r❡❝t✐♦♥s✧✱ ❊✉r♦♣❡❛♥
❏♦✉r♥❛❧ ♦❢ ❖♣❡r❛t✐♦♥❛❧ ❘❡s❡❛r❝❤✱ ❱♦❧✳ ✷✺✵✱ ♣♣✳ ✻✾✶✲✼✵✶✳
254

▼♦❞❡❧✐♥❣ ◆♦♥✲♣r❡❡♠♣t✐✈❡ P❛r❛❧❧❡❧ ❙❝❤❡❞✉❧✐♥❣ Pr♦❜❧❡♠
✇✐t❤ Pr❡❝❡❞❡♥❝❡ ❈♦♥str❛✐♥ts
❚✐❛♥②✉ ❲❛♥❣1
❛♥❞ ❖❞✐❧❡ ❇❡❧❧❡♥❣✉❡③✲▼♦r✐♥❡❛✉1
▲❙✷◆✭▲❡ ▲❛❜♦r❛t♦✐r❡ ❞❡s ❙❝✐❡♥❝❡s ❞✉ ◆✉♠❡r✐q✉❡ ❞❡ ◆❛♥t❡s✮✱ ❋r❛♥❝❡
t✐❛♥②✉✳✇❛♥❣❅❧s✷♥✳❢r❀ ♦❞✐❧❡✳❜❡❧❧❡♥❣✉❡③❅✐♠t✲❛t❧❛♥t✐q✉❡✳❢r
❑❡②✇♦r❞s✿ ♣❛r❛❧❧❡❧ s❝❤❡❞✉❧✐♥❣✱ ♠♦❞❡❧✐♥❣✱ ♣r❡❝❡❞❡♥❝❡ ❝♦♥str❛✐♥ts✳
✶ ■♥tr♦❞✉❝t✐♦♥
❚❤❡ s❝❤❡❞✉❧✐♥❣ ♣r♦❜❧❡♠ ✇❡ st✉❞② ❞❡❛❧s ✇✐t❤ n ❥♦❜s t♦ ❜❡ ♣r♦❝❡ss❡❞ ♦♥ m ♠❛❝❤✐♥❡s ✇❤✐❧❡
s❛t✐s❢②✐♥❣ t❤❡ ♣r❡❝❡❞❡♥❝❡ ❝♦♥str❛✐♥ts✳ ❈♦♥s✐❞❡r✐♥❣ t❤❡ ♠❛❦❡s♣❛♥✱ t❤✐s ♣r♦❜❧❡♠ ✐s NP✲❤❛r❞
❡✈❡♥ ✇✐t❤ ♥♦ ♣r❡❝❡❞❡♥❝❡ ❝♦♥str❛✐♥ts ❛♥❞ t✇♦ ♠❛❝❤✐♥❡s✭▲❡♥str❛ ❡t ❛❧✳ ✶✾✼✼✮✳ ❚♦ t❤❡ ❜❡st ♦❢
♦✉r ❦♥♦✇❧❡❞❣❡✱ t❤❡r❡ ✐s ♥♦ ❡①❛❝t ♠❡t❤♦❞ ❢♦r t❤✐s ♣r♦❜❧❡♠ ❡✈❡♥ ✇❤❡♥ m ✐s ✜①❡❞✳
■♥ t❤✐s ♣❛♣❡r✱ ✇❡ ❛❞❛♣t s♦♠❡ ♠♦❞❡❧s t♦ ♣❛r❛❧❧❡❧ s❝❤❡❞✉❧✐♥❣ ♣r♦❜❧❡♠s ❛♥❞ ♣r♦♣♦s❡ ❛ ♥❡✇
♦♥❡✳ ❚❤❡♥✱ ✇❡ ❝♦♠♣❛r❡ t❤❡✐r ♣❡r❢♦r♠❛♥❝❡ ❜② t❡st✐♥❣ t❤❡♠ ♦♥ ❜❡♥❝❤♠❛r❦s ✇✐t❤ ♣r❡❝❡❞❡♥❝❡
❝♦♥str❛✐♥ts ❢r♦♠ P❙P▲■❇✳
✷ ▼❛t❤❡♠❛t✐❝❛❧ ▼♦❞❡❧s
■♥ t❤✐s s❡❝t✐♦♥✱ ✇❡ ♣r❡s❡♥t ❞✐✛❡r❡♥t ♠♦❞❡❧s✳ ❊❛❝❤ ♠♦❞❡❧ ✉s❡s t❤❡ ✈❛r✐❛❜❧❡s Cj ❛♥❞ Sj
❛s st❛rt✐♥❣ t✐♠❡ ❛♥❞ ❝♦♠♣❧❡t✐♦♥ t✐♠❡ ♦❢ j✳ ❚❤❡ ♦❜❥❡❝t✐✈❡ ✐s Cmax✱ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥str❛✐♥ts
❤♦❧❞ ❢♦r ❛❧❧ ♠♦❞❡❧s✱ ❛♥❞ t❤❡② ❛r❡ ♦♠✐tt❡❞ ❤❡r❡❛❢t❡r✿
Cj = Sj + pj✱ ∀j ∈ J
Cmax ≥ Cj✱ ∀j ∈ J
Sj ≥ Ci✱ ∀i, j ∈ J ❛♥❞ i ≺ j
✇❤❡r❡ pj ✐s t❤❡ ♣r♦❝❡ss✐♥❣ t✐♠❡ ♦❢ ❥♦❜ j ❛♥❞ i ≺ j ♠❡❛♥s i ♣r❡❝❡❞❡s j✳ J ❛♥❞ M ❛r❡
s❡t ♦❢ ❛❧❧ ❥♦❜s ❛♥❞ ♠❛❝❤✐♥❡s✳ ■♥ t❤❡ ❢♦❧❧♦✇✐♥❣ s❡❝t✐♦♥s✱ i, j ∈ J ❛♥❞ k ∈ M✳ M r❡♣r❡s❡♥ts
❛ ❧❛r❣❡ ♥✉♠❜❡r✱ ✇❤✐❝❤ ❝❛♥ ❜❡ ❞❡✜♥❡❞ ❛s
P
pj✳
✷✳✶ ❘❡❧❛t✐✈❡✲❖r❞❡r✲■♥❞❡①❡❞ ▼♦❞❡❧✶ ✭❘❖■▼✶✮
❚❤✐s ♠♦❞❡❧ ✉s❡s ❜✐♥❛r✐❡s yk
i,j ❛♥❞ zk
i,j ❛s ❞❡❝✐s✐♦♥ ✈❛r✐❛❜❧❡s✳ yk
i,j = 1 ✐❢ i ✐s ❡①❡❝✉t❡❞
✐♠♠❡❞✐❛t❡❧② ❜❡❢♦r❡ j ♦♥ k❀ zk
i,j = 1 ✐❢ i ✐s ❡①❡❝✉t❡❞ ❜❡❢♦r❡ j ♦♥ k✳ ❉✐✛❡r❡♥t ❢♦r♠✉❧❛t✐♦♥s ♦❢
t❤✐s ♠♦❞❡❧ ❝❛♥ ❜❡ s❡❡♥ ✐♥ ❇❧❛③❡✇✐❝③ ❡t ❛❧✳ ✭✶✾✾✶✮ ❛♥❞ ❯♥❧✉ ❛♥❞ ▼❛s♦♥ ✭✷✵✶✵✮ ❢♦r ♣r♦❜❧❡♠s
✇✐t❤♦✉t ♣r❡❝❡❞❡♥❝❡ ❝♦♥str❛✐♥ts✳ ❲❡ ✐♥tr♦❞✉❝❡ Sj ❛♥❞ Cj ❢♦r ♣r❡❝❡❞❡♥❝❡ ❝♦♥str❛✐♥ts ❜②
❛❞❞✐♥❣ ✭✶❜✮✱ ❛♥❞ ✇❡ ♣r♦♣♦s❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❢♦r♠✉❧❛t✐♦♥✿
✭✶❛✮
yk
i,j ≤ zk
i,j✱ ∀i, j ∈ J , k ∈ M
✭✶❜✮
M(1 − zk
i,j) + Sj ≥ Ci✱ ∀i, j ∈ J , k ∈ M
✭✶❝✮
M(1 − zk
i,j) ≥
X
k′6=k
X
q
(zk′
i,j + zk′
q,j + zk′
q,i + zk′
j,q + zk′
i,q)✱ ∀i, j ∈ J , k ∈ M
✭✶❞✮
X
k
(yk
i,j + yk
j,i) ≤ 1✱ ∀i, j ∈ J
✭✶❡✮
X
k
X
i
yk
i,j = 1✱ ∀j ∈ J
255

✭✶❢✮
X
k
X
j
yk
i,j = 1✱ ∀i ∈ J
✭✶❛✮ ❡♥s✉r❡s t❤❛t zk
i,j = 1 ✐❢ yk
i,j = 1✳ ✭✶❝✮✱ ✭✶❞✮✱ ✭✶❡✮ ❛♥❞ ✭✶❢✮ ❢♦r❝❡ ❡❛❝❤ ❥♦❜ ❤❛s ❡①❛❝t❧②
♦♥❡ ♣r❡❞❡❝❡ss♦r ❛♥❞ ♦♥❡ s✉❝❝❡ss♦r✳ ◆♦t✐❝❡ t❤❛t ✐♥ ✭✶❡✮ ❛♥❞ ✭✶❢✮✱ ❡❛❝❤ ❥♦❜ ❤❛s t♦ ❜❡ ❡①❡❝✉t❡❞
❜❡❢♦r❡✭❛❢t❡r✮ s♦♠❡ ♦t❤❡r ❥♦❜✳ ■♥ ♣r❛❝t✐❝❡✱ s♦♠❡ ❞✉♠♠② ❥♦❜s ❛r❡ ❝r❡❛t❡❞ t♦ r❡♣r❡s❡♥t ❥♦❜s
❛❢t❡r✭❜❡❢♦r❡✮ t❤❡ ❧❛st✭✜rst✮ ❡①❡❝✉t❡❞ ❥♦❜s ♦♥ ❡❛❝❤ ♠❛❝❤✐♥❡✳
■♥ ❢❛❝t✱ ✭✶❡✮ ❛♥❞ ✭✶❢✮ ❝♦♥✈❡② t❤❡ s❛♠❡ ♠❡❛♥✐♥❣✿ ✐❢ ❡✈❡r② ❥♦❜ ❢♦❧❧♦✇s ❛♥♦t❤❡r✭❡①❝❡♣t
t❤❡ ✜rst ♦♥❡✮✱ t❤❡♥ ❡✈❡r② ❥♦❜ ❤❛s ❛ ❢♦❧❧♦✇❡r✭❡①❝❡♣t t❤❡ ❧❛st ♦♥❡✮✳ ■❢ ✇❡ r❡♠♦✈❡ ♦♥❡ ♦❢ ✭✶❡✮
❛♥❞ ✭✶❢✮✱ t❤❡ ♠♦❞❡❧ st✐❧❧ ✇♦r❦s ✇❡❧❧✱ ❜✉t ✭✶✻✪ ✐♥ ♦✉r ❡①♣❡r✐♠❡♥t✮ s❧♦✇❡r✳ ❲❡ ❝❛❧❧ t❤✐s ❛
r❡❞✉♥❞❛♥t ❝♦♥str❛✐♥t✳
✷✳✷ ❘❡❧❛t✐✈❡✲❖r❞❡r✲■♥❞❡①❡❞ ▼♦❞❡❧✷ ✭❘❖■▼✷✮
❚❤✐s ♠♦❞❡❧ ✉s❡s ❜✐♥❛r✐❡s zk
i,j ❛♥❞ xk
i ❛s ❞❡❝✐s✐♦♥ ✈❛r✐❛❜❧❡✳ xk
j = 1 ✐❢ j ✐s ♦♥ k✳ ❚♦ ❛ss♦❝✐❛t❡
t❤❡s❡ t✇♦ ✈❛r✐❛❜❧❡s✱ ❛ ♥♦♥✲❧✐♥❡❛r ❝♦♥str❛✐♥t✱ xk
i xk
j = zk
i,j + zk
j,i✱ ∀i, j, k✱ ✐s ❣✐✈❡♥ ✐♥ ▲♦✇
❡t ❛❧✳ ✭✷✵✵✻✮ ❛♥❞ ●❛♦ ❡t ❛❧✳ ✭✷✵✵✻✮ ❆ ❧✐♥❡❛r ✈❡rs✐♦♥ ✐♥ Ö③❣ü✈❡♥ ❡t ❛❧✳ ✭✷✵✶✵✮ ✐♥tr♦❞✉❝❡s
♥❡✇ ✐♥t❡❣❡r ✈❛r✐❛❜❧❡s✳
❲❡ ♣r♦♣♦s❡❞ ❛ ♥❡✇ ❢♦r♠✉❧❛t✐♦♥ ✇❤✐❝❤ ✐s s✉♣❡r✐♦r t♦ t❤❡ ♦t❤❡rs ❜♦t❤ t❤❡♦r❡t✐❝❛❧❧② ❛♥❞
✐♥ ♦✉r ♣r❛❝t✐❝❡ ❛s ✇❡❧❧✿
✭✷❛✮
X
k
xk
j = 1✱ ∀j ∈ J
✭✷❜✮
Ci ≤ Sj + M(1 − zk
i,j)✱ ∀i, j ∈ J , k ∈ M
✭✷❝✮
Mxk
i ≥
X
j
(zk
i,j + zk
j,i)✱ ∀i ∈ J , k ∈ M
✭✷❞✮
X
j
(zk
i,j + zk
j,i) + M(1 − xk
i ) ≥ 1✱ ∀i ∈ J , k ∈ M
✭✷❡✮
M(zk
i,j + zk
j,i) ≥ xk
i + xk
j − 1✱ ∀i, j ∈ J , k ∈ M
✭✷❛✮ ❢♦r❝❡s ❡❛❝❤ ❥♦❜ t♦ ❜❡ ❡①❡❝✉t❡❞ ♦♥❝❡✳ ✭✷❜✮ ❡♥s✉r❡s t❤❛t Ci ≤ Sj ✐❢ zi,j = 1✳ ✭✷❝✮ ❛♥❞
✭✷❞✮ ❝♦♥s✐❞❡r t✇♦ ❝❛s❡s✿ xk
i = 0 t❤❡♥ ∀j ∈ J ✱ zk
i,j = 0❀ xk
i = 1 t❤❡♥ ∃j✱ zk
i,j ∨ zk
j,i = 1✳ ✭✷❡✮
✇♦r❦s ✇❤❡♥ ❜♦t❤ i, j ❛r❡ ♦♥ k✱ ❛♥❞ ❢♦r❝❡s ♦♥❡ t♦ ♣r❡❝❡❞❡ t❤❡ ♦t❤❡r✳
✷✳✸ ❆❜s♦❧✉t❡✲❖r❞❡r✲■♥❞❡①❡❞ ▼♦❞❡❧✭❆❖■▼✮
❚❤✐s ♠♦❞❡❧ ✉s❡s βl
k,j✱ ✇❤✐❝❤ ❡q✉❛❧s 1 ✐❢ j ✐s t❤❡ lt❤ ❥♦❜ ♦♥ k✱ ❛s ❛ ♣r✐♥❝✐♣❛❧ ✈❛r✐❛❜❧❡✳ ■t
✇❛s ♦r✐❣✐♥❛❧❧② ❞❡s✐❣♥❡❞ ❢♦r ♣❛r❛❧❧❡❧ s❝❤❡❞✉❧✐♥❣ ♣r♦❜❧❡♠ ✇✐t❤♦✉t ♣r❡❝❡❞❡♥❝❡ ❝♦♥str❛✐♥ts ❜②
❇❧❛③❡✇✐❝③ ❡t ❛❧✳ ✭✶✾✾✶✮✳ ❚♦ ❛❞❞ ♣r❡❝❡❞❡♥❝❡ ❝♦♥str❛✐♥ts✱ ❉❡♠✐r ❛♥❞ ➑➩❧❡②❡♥ ✭✷✵✶✸✮ ✐♥tr♦❞✉❝❡s
Tl
k t♦ t❤❡ ♠♦❞❡❧✱ ✇❤✐❝❤ ✐s t❤❡ st❛rt✐♥❣ t✐♠❡ ♦❢ t❤❡ lt❤ ❥♦❜ ♦❢ k✳
❍❡r❡✱ ✇❡ ♣r♦♣♦s❡ ❛ s✐♠✐❧❛r ❢♦r♠✉❧❛t✐♦♥✱ ✇❤✐❝❤ ✉s❡s ❢❡✇❡r ✈❛r✐❛❜❧❡s✿
✭✸❛✮
Tl+1
k − Tl
k ≥ pjβl
k,j✱ ∀l ≤ n, j ∈ J , k ∈ M
✭✸❜✮
Tl
k + M(1 − βl
k,j) ≥ Sj✱ ∀l ≤ n, j ∈ J , k ∈ M
✭✸❝✮
Tl
k ≤ M(1 − βl
k,j) + Sj✱ ∀l ≤ n, j ∈ J , k ∈ M
✭✸❞✮
X
j
βl
k,j ≤ 1✱ ∀l ≤ n, k ∈ M
✭✸❡✮
X
k
X
l
βl
k,j = 1✱ ∀j ∈ J
256

✭✸❡✮ ❢♦r❝❡s ❡❛❝❤ ❥♦❜ t♦ ❜❡ ❡①❡❝✉t❡❞ ♦♥❝❡✳ ✭✸❞✮ ❡♥s✉r❡s t❤❛t ♦♥❧② ♦♥❡ ❥♦❜ ❝❛♥ ❜❡ ❡①❡❝✉t❡❞
❛s t❤❡ lt❤ ❥♦❜ ♦♥ k✳ ✭✸❝✮✱ ✭✸❜✮ ❛♥❞ ✭✸❛✮ ✇♦r❦s ✇❤❡♥ βl
k,j = 1✱ t❤❡② ❣✉❛r❛♥t❡❡ Tl
k = Sj ❛♥❞
Tl+1
k − Tl
k = pj✳
✷✳✹ ❈♦♠♣❛❝t ▼♦❞❡❧✭❈▼✮
❲❡ ♣r♦♣♦s❡ ❛ ♥❡✇ ♦r❞❡r✲✐♥❞❡①❡❞ ♠♦❞❡❧ ❤❡r❡✳ ■t ✉s❡s δi,j✱ ✇❤✐❝❤ ❡q✉❛❧s 1 ✐❢ i ✐s ❡①❡❝✉t❡❞
❜❡❢♦r❡ j ♦♥ t❤❡ s❛♠❡ ♠❛❝❤✐♥❡✱ ❛♥❞ xk
j ❛s ❞❡❝✐s✐♦♥ ✈❛r✐❛❜❧❡s✳
✭✹❛✮
Ci − Sj ≤ M(1 − δi,j)✱ ∀i, j ∈ J
✭✹❜✮
M(2 − xk
i − xk
j ) + δi,j + δj,i ≥ 1✱ ∀i, j ∈ J , k ∈ M
✭✹❝✮
M(2 − xk1
i − xk2
j ) ≥ δi,j + δj,i✱ ∀i, j ∈ J , k1, k2 ∈ M ❛♥❞ k1 6= k2
✭✹❞✮
X
k
xk
j = 1✱ ∀j ∈ J
✭✹❛✮ ❝♦♥♥❡❝ts δi,j✱ Ci ❛♥❞ Sj✭✐❢ δi,j = 1 t❤❡♥ Ci ≤ Sj✮✳ ❲❤❡♥ ❜♦t❤ i, j ❛r❡ ♦♥ k✱ ✭✹❜✮
❢♦r❝❡s ♦♥❡ ♣r❡❝❡❞❡s t❤❡ ♦t❤❡r✳ ✭✹❞✮ ♠❛❦❡ ❡❛❝❤ ❥♦❜ ❜❡ ❡①❡❝✉t❡❞ ♦♥❝❡✳ ✭✹❝✮ s❡ts δi,j ❛♥❞ δj,i
❛s 0 ✇❤❡♥ i, j ❛r❡ ♦♥ ❞✐✛❡r❡♥t ♠❛❝❤✐♥❡s✳
◆♦t✐❝❡ t❤❛t ✇❤❡♥ ♠✐♥✐♠✐③✐♥❣ Cmax✱ ✭✹❝✮ ✐s ✉♥♥❡❝❡ss❛r② ❜❡❝❛✉s❡ ✇❤❡♥ i, j ❛r❡ ♦♥ ❞✐✛❡r❡♥t
♠❛❝❤✐♥❡s✱ δi,j = 1 ♦r δj,i = 1 ❝❛♥ ♥♦t r❡❞✉❝❡ Cmax✳ ❍♦✇❡✈❡r✱ ✐t ❤❡❧♣s t♦ ❣✐✈❡ δi,j ❛
❝♦♠♣r❡❤❡♥s✐❜❧❡ ♠❡❛♥✐♥❣✭δi,j = 1 ✐❢ ❛♥❞ ♦♥❧② ✐❢ i ♣r❡❝❡❞❡s j ♦♥ t❤❡ s❛♠❡ ♠❛❝❤✐♥❡✮ ❛♥❞ ❤❛s
❛ ♣♦s✐t✐✈❡ ✐♠♣❛❝t ♦♥ t❤❡ ♠♦❞❡❧ ✭✇❤✐❝❤ ✐s ✐♠♣r♦✈❡❞ ❜② ✶✾✪ ❛❝❝♦r❞✐♥❣ t♦ ♦✉r t❡sts✮✳
✸ ❚❡st ❘❡s✉❧t ❛♥❞ ❆♥❛❧②s✐s
❲❡ t❡st❡❞ t❤❡ ♠♦❞❡❧s ✇✐t❤ ❜❡♥❝❤♠❛r❦s ✇❡ ❜✉✐❧t ❢r♦♠ ♣r❡❝❡❞❡♥❝❡ ❝♦♥str❛✐♥ts ✐♥ P❙P▲■❇✳
❚❤❡ ♣❧❛t❢♦r♠ ✇❡ ✉s❡❞ ✐s✿ ■❇▼ ■▲❖● ❈P▲❊❳ ❖♣t✐♠✐③❛t✐♦♥ ❙t✉❞✐♦ ❱✶✷✳✻✳✵ ♦♥ ■♥t❡❧ ❈♦r❡
✐✼✲✹✻✵✵❯ ❅✷✳✶✵●❍③✳ ❲❡ ❝♦♠♣❛r❡❞ t❤❡✐r ❛✈❡r❛❣❡ t✐♠❡ ❝♦♥s✉♠❡❞ t♦ s♦❧✈❡ ✐♥st❛♥❝❡s ✇✐t❤
❞✐✛❡r❡♥t s❝❛❧❡s ♦❢ ❥♦❜s ♦♥ m = 4 ♠❛❝❤✐♥❡s ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ t❛❜❧❡✿
❚❛❜❧❡ ✶✳ ▼♦❞❡❧s✬ ♣❡r❢♦r♠❛♥❝❡✱ ✇❤❡r❡ ❇❱✴■❱✴❈ ♠❡❛♥s ♥✉♠❜❡r ♦❢ ❜✐♥❛r② ✈❛r✐❛❜❧❡s✴✐♥t❡❣❡r ✈❛r✐✲
❛❜❧❡s✴❝♦♥str❛✐♥ts❀ ❚❈ ♠❡❛♥s ❛✈❡r❛❣❡ t✐♠❡ ❝♦♥s✉♠❡❞ t♦ s♦❧✈❡ t❤❡ ✐♥st❛♥❝❡s❀ ✬✲✬ ♠❡❛♥s t❤❡ ♠♦❞❡❧
❞✐❞ ♥♦t s♦❧✈❡ ❛♥② ✐♥st❛♥❝❡ ♦♥ t❤✐s s❝❛❧❡ ✇✐t❤✐♥ ✻✵✵✵s❡❝
▼♦❞❡❧ ❇❱ ■❱ ❈ ❚❈ ❚❈ ❚❈
n = 15 n = 30 n = 60
❈▼ ✷✼✵ ✸✶ ✶✶✷✺ ✵✳✾✷ ✶✳✷✷ ✶✸✳✸✷
❘❖■▼✷ ✾✵✵ ✸✶ ✶✽✼✺ ✶✳✼✼ ✻✳✹✶ ✾✼✳✷✸
❆❖■▼ ✾✵✵ ✾✶ ✷✼✼✺ ✸✳✶✾ ✶✹✳✻✻ ✕
❘❖■▼✶ ✶✾✷✵ ✸✶ ✷✽✷✽ ✶✺✳✹✶ ✼✾✳✷✸ ✕
❚■▼ ✹✾✷✵ ✸✶ ✺✸✸✽ ✹✽✳✸✾ ✕ ✕
❆s ❝❛♥ ❜❡ s❡❡♥✱ ❈▼ st❛②s ❛❤❡❛❞ ♦❢ t❤❡ ♦t❤❡rs ❛♥❞ r❡q✉✐r❡s ❧❡ss s♣❛❝❡✳ ❚❤❡ ❞✐✛❡r❡♥t
s♣❡❡❞ ♦❢ ♠♦❞❡❧s r❡s✉❧ts ♠♦st❧② ❢r♦♠ t❤❡ ❞✐✛❡r❡♥t ❞❡❝✐s✐♦♥ ✈❛r✐❛❜❧❡s ✉s❡❞✳
■♥s♣✐r❡❞ ❜② t❤❡ ❚✐♠❡✲■♥❞❡①❡❞ ▼♦❞❡❧ ✭❚■▼✮ ❜② ❚❤♦♠❛❧❧❛ ✭✷✵✵✶✮✱ ✇❡ ❛❧s♦ ❢♦r♠✉❧❛t❡❞
❛ ✈❡rs✐♦♥ ♦❢ ❚■▼ ❢♦r ♣❛r❛❧❧❡❧ s❝❤❡❞✉❧✐♥❣ ♣r♦❜❧❡♠✳ ■t ✉s❡s ❜✐♥❛r② ✈❛r✐❛❜❧❡ xk
t,j✱ ✇❤✐❝❤ ✐s ✶
✐❢ j st❛rts ♦♥ k ❛t t✳ ■t ✇❛s ❝♦♠♣❛r❡❞ ✇✐t❤ ♦t❤❡r ♠♦❞❡❧s✱ ❤♦✇❡✈❡r✱ ♥♦t ❞❡✈❡❧♦♣❡❞ ✐♥ t❤❡
♣r❡✈✐♦✉s s❡❝t✐♦♥ ❞✉❡ t♦ ✐ts ♣♦♦r ♣❡r❢♦r♠❛♥❝❡✳ ■t r❡q✉✐r❡s ❛♥ ❡st✐♠❛t✐♦♥ ♦❢ ❛♥ ✉♣♣❡r ❜♦✉♥❞
♦❢ Cmax✳ ■t ✐s s❡t ❛s
P
pj ❢♦r t❤❡ ✇♦rst ❝❛s❡✭s✐♥❣❧❡ ♠❛❝❤✐♥❡✮ ✐♥ ♣r❛❝t✐❝❡✱ ✇❤✐❝❤ ❧❡❛❞s t♦
257

❧❛r❣❡ ❛♠♦✉♥ts ♦❢ ✈❛r✐❛❜❧❡s✳ ■♥ ❢❛❝t✱ t❤❡ ♥✉♠❜❡r ♦❢ ✈❛r✐❛❜❧❡s ❝♦✉❧❞ ❜❡ ❡①tr❡♠❡❧② ❧❛r❣❡ ✐❢ pj ✐s
♥♦t ❛♥ ✐♥t❡❣❡r✳ ❍♦✇❡✈❡r✱ ❚■▼ ✐s st✐❧❧ t❤❡ s❧♦✇❡st ❡✈❡♥ ❢♦r ✐♥st❛♥❝❡s ♦❢ ✉♥✐t✲♣r♦❝❡ss✐♥❣✲t✐♠❡
❥♦❜s✳ ❇♦t❤ ❘❖■▼✷ ❛♥❞ ❘❖■▼✷ ✉s❡ O(n2
m) ❜✐♥❛r② ✈❛r✐❛❜❧❡s✿ ❘❖■▼✶ ✉s❡s t✇♦ ✸✲❞✐♠❡♥s✐♦♥
✈❛r✐❛❜❧❡s✱ ✇❤✐❧❡ ❘❖■▼✷ ❛♥❞ ❆❖■▼ ✉s❡ ♦♥❧② ♦♥❡✳ ❆❖■▼ ✐s t❤❡ ♦♥❧② ♦♥❡ ✇❤♦ r❡q✉✐r❡s ❡①tr❛
✐♥t❡❣❡r ✈❛r✐❛❜❧❡✳ ❚❤❡ ❢❛st❡st ♠♦❞❡❧ ❈▼ ✉s❡s ♦♥❧② ✷✲❞✐♠❡♥s✐♦♥ ❜✐♥❛r② ✈❛r✐❛❜❧❡s ❛♥❞ r❡q✉✐r❡s
❢❡✇❡st ✈❛r✐❛❜❧❡s✱ ✇❤✐❝❤ ♠❛② ❜❡ t❤❡ ♣r✐♥❝✐♣❛❧ ❛❞✈❛♥t❛❣❡ ♦❢ ❈▼✳
❲❡ t❡st❡❞ ❛❧s♦ ✐♥st❛♥❝❡s ✇✐t❤ ❞✐✛❡r❡♥t ♥✉♠❜❡r ♦❢ ♠❛❝❤✐♥❡s✳ ❲❤❡♥ ❥✉❞❣✐♥❣ t❤❡ ♠♦❞❡❧s✱
❝♦♠♣❛r✐s♦♥ r❡s✉❧ts ❛r❡ s✐♠✐❧❛r ❛s ❚❛❜❧❡ ✶✳ ❆❞❞✐t✐♦♥❛❧❧②✱ ✇❡ ✜♥❞ t❤❛t ✇❤❡♥ m ✐s s❡t ❛s ✹✱
t❤❡ ♠♦❞❡❧s t♦♦❦ ❧♦♥❣❡st t✐♠❡ ❢♦r s♦❧✈✐♥❣✳
✹ ❈♦♥❝❧✉s✐♦♥ ❛♥❞ P❡rs♣❡❝t✐✈❡
■♥ t❤✐s ♣❛♣❡r✱ ✇❡ ❛❞❛♣t❡❞ ♠♦❞❡❧s t♦ t❤❡ ♣❛r❛❧❧❡❧ s❝❤❡❞✉❧✐♥❣ ♣r♦❜❧❡♠ ✇✐t❤ ♣r❡❝❡❞❡♥❝❡
❝♦♥str❛✐♥ts ❛♥❞ ♣r♦♣♦s❡❞ ❛ ♥❡✇ ♦♥❡ ✇❤✐❝❤ ♦✉t♣❡r❢♦r♠s t❤❡ ♦t❤❡rs ❜② ♦✉r t❡st✳
■♥ ❛❞❞✐t✐♦♥ ♦❢ r❡❞✉♥❞❛♥t ❝♦♥str❛✐♥ts✱ ✇❡ t❡st❡❞ ❛♥❞ ✜♥❞ t❤❛t s♦♠❡t✐♠❡s t❤❡ r❡❞✉♥❞❛♥t
✈❛r✐❛❜❧❡s✱ s✉❝❤ ❛s Cj ✇❤✐❝❤ ❝♦✉❧❞ t♦t❛❧❧② ❜❡ r❡♣❧❛❝❡❞ ❜② Sj + pj✱ ❛♠❡❧✐♦r❛t❡s t❤❡ ♠♦❞❡❧s✳
❍♦✇ ❛♥ r❡❞✉♥❞❛♥t ❝♦♥str❛✐♥t ♦r ✈❛r✐❛❜❧❡ ❛✛❡❝ts t❤❡ ♠♦❞❡❧ ✐s ✇♦rt❤② ♦❢ ❜❡✐♥❣ ❢✉rt❤❡r
❞✐s❝✉ss❡❞✳
❇❡s✐❞❡s✱ t❤❡ s♦❧✈✐♥❣ t✐♠❡ ❞♦❡s ♥♦t ❞❡♣❡♥❞ ♠❡r❡❧② ♦♥ ✐♥st❛♥❝❡✬s s❝❛❧❡✱ ❜✉t ❛❧s♦ ♦♥ ♥✉♠❜❡r
♦❢ ♠❛❝❤✐♥❡s✱ t❤❡ ♣r♦❝❡ss✐♥❣ t✐♠❡ ♦❢ ❥♦❜s✱ ❛♥❞ t❤❡ s❤❛♣❡ ♦❢ ♣r❡❝❡❞❡♥❝❡ ❝♦♥str❛✐♥ts✳ ❋♦r
❡①❛♠♣❧❡✱ ❛ s✉❜♣r♦❜❧❡♠ ♦❢ s❝❤❡❞✉❧✐♥❣ ❡q✉❛❧✲♣r♦❝❡ss✐♥❣✲❥♦❜s ✇✐t❤ ✐♥✲tr❡❡ ♣r❡❝❡❞❡♥❝❡ ❣r❛♣❤s
❝❛♥ ❜❡ ✜♥✐s❤❡❞ ✐♥ ♣♦❧②♥♦♠✐❛❧ t✐♠❡ ❍✉ ✭✶✾✻✶✮✳ ❚❤❡ ♠♦❞❡❧ ✐s ❤❡❧♣❢✉❧ t♦ st✉❞② ❡①♣❡r✐♠❡♥t❛❧❧②
❤♦✇ t❤❡② ✐♠♣❛❝t t❤❡ s♦❧✈✐♥❣ t✐♠❡✳ ❖✉r ♥❡①t ✇♦r❦ ❞✐r❡❝t✐♦♥ ❢♦❧❧♦✇s t❤✐s ❛♣♣r♦❛❝❤✳
❇✐❜❧✐♦❣r❛♣❤②
❇❧❛③❡✇✐❝③✱ ❏✳✱ ❉r♦r✱ ▼✳ ❛♥❞ ❲❡❣❧❛r③✱ ❏✳✿ ✶✾✾✶✱ ▼❛t❤❡♠❛t✐❝❛❧ ♣r♦❣r❛♠♠✐♥❣ ❢♦r♠✉❧❛t✐♦♥s ❢♦r
♠❛❝❤✐♥❡ s❝❤❡❞✉❧✐♥❣✿ ❆ s✉r✈❡②✱ ❊✉r♦♣❡❛♥ ❏♦✉r♥❛❧ ♦❢ ❖♣❡r❛t✐♦♥❛❧ ❘❡s❡❛r❝❤ ✺✶✭✸✮✱ ✷✽✸✕✸✵✵✳
❉❡♠✐r✱ ❨✳ ❛♥❞ ➑➩❧❡②❡♥✱ ❙✳ ❑✳✿ ✷✵✶✸✱ ❊✈❛❧✉❛t✐♦♥ ♦❢ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧s ❢♦r ✢❡①✐❜❧❡ ❥♦❜✲s❤♦♣
s❝❤❡❞✉❧✐♥❣ ♣r♦❜❧❡♠s✱ ❆♣♣❧✐❡❞ ▼❛t❤❡♠❛t✐❝❛❧ ▼♦❞❡❧❧✐♥❣ ✸✼✭✸✮✱ ✾✼✼✕✾✽✽✳
●❛♦✱ ❏✳✱ ●❡♥✱ ▼✳ ❛♥❞ ❙✉♥✱ ▲✳✿ ✷✵✵✻✱ ❙❝❤❡❞✉❧✐♥❣ ❥♦❜s ❛♥❞ ♠❛✐♥t❡♥❛♥❝❡s ✐♥ ✢❡①✐❜❧❡ ❥♦❜ s❤♦♣
✇✐t❤ ❛ ❤②❜r✐❞ ❣❡♥❡t✐❝ ❛❧❣♦r✐t❤♠✱ ❏♦✉r♥❛❧ ♦❢ ■♥t❡❧❧✐❣❡♥t ▼❛♥✉❢❛❝t✉r✐♥❣ ✶✼✭✹✮✱ ✹✾✸✕✺✵✼✳
❍✉✱ ❚✳ ❈✳✿ ✶✾✻✶✱ P❛r❛❧❧❡❧ s❡q✉❡♥❝✐♥❣ ❛♥❞ ❛ss❡♠❜❧② ❧✐♥❡ ♣r♦❜❧❡♠s✱ ❖♣❡r❛t✐♦♥s r❡s❡❛r❝❤
✾✭✻✮✱ ✽✹✶✕✽✹✽✳
▲❡♥str❛✱ ❏✳ ❑✳✱ ❑❛♥✱ ❆✳ ❘✳ ❛♥❞ ❇r✉❝❦❡r✱ P✳✿ ✶✾✼✼✱ ❈♦♠♣❧❡①✐t② ♦❢ ♠❛❝❤✐♥❡ s❝❤❡❞✉❧✐♥❣ ♣r♦❜✲
❧❡♠s✱ ❆♥♥❛❧s ♦❢ ❞✐s❝r❡t❡ ♠❛t❤❡♠❛t✐❝s ✶✱ ✸✹✸✕✸✻✷✳
▲♦✇✱ ❈✳✱ ❨✐♣✱ ❨✳ ❛♥❞ ❲✉✱ ❚✳✲❍✳✿ ✷✵✵✻✱ ▼♦❞❡❧❧✐♥❣ ❛♥❞ ❤❡✉r✐st✐❝s ♦❢ ❢♠s s❝❤❡❞✉❧✐♥❣ ✇✐t❤
♠✉❧t✐♣❧❡ ♦❜❥❡❝t✐✈❡s✱ ❈♦♠♣✉t❡rs ✫ ♦♣❡r❛t✐♦♥s r❡s❡❛r❝❤ ✸✸✭✸✮✱ ✻✼✹✕✻✾✹✳
Ö③❣ü✈❡♥✱ ❈✳✱ Ö③❜❛❦✙r✱ ▲✳ ❛♥❞ ❨❛✈✉③✱ ❨✳✿ ✷✵✶✵✱ ▼❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧s ❢♦r ❥♦❜✲s❤♦♣ s❝❤❡❞✉❧✲
✐♥❣ ♣r♦❜❧❡♠s ✇✐t❤ r♦✉t✐♥❣ ❛♥❞ ♣r♦❝❡ss ♣❧❛♥ ✢❡①✐❜✐❧✐t②✱ ❆♣♣❧✐❡❞ ▼❛t❤❡♠❛t✐❝❛❧ ▼♦❞❡❧❧✐♥❣
✸✹✭✻✮✱ ✶✺✸✾✕✶✺✹✽✳
❚❤♦♠❛❧❧❛✱ ❈✳ ❙✳✿ ✷✵✵✶✱ ❏♦❜ s❤♦♣ s❝❤❡❞✉❧✐♥❣ ✇✐t❤ ❛❧t❡r♥❛t✐✈❡ ♣r♦❝❡ss ♣❧❛♥s✱ ■♥t❡r♥❛t✐♦♥❛❧
❏♦✉r♥❛❧ ♦❢ Pr♦❞✉❝t✐♦♥ ❊❝♦♥♦♠✐❝s ✼✹✭✶✮✱ ✶✷✺✕✶✸✹✳
❯♥❧✉✱ ❨✳ ❛♥❞ ▼❛s♦♥✱ ❙✳ ❏✳✿ ✷✵✶✵✱ ❊✈❛❧✉❛t✐♦♥ ♦❢ ♠✐①❡❞ ✐♥t❡❣❡r ♣r♦❣r❛♠♠✐♥❣ ❢♦r♠✉❧❛t✐♦♥s
❢♦r ♥♦♥✲♣r❡❡♠♣t✐✈❡ ♣❛r❛❧❧❡❧ ♠❛❝❤✐♥❡ s❝❤❡❞✉❧✐♥❣ ♣r♦❜❧❡♠s✱ ❈♦♠♣✉t❡rs ✫ ■♥❞✉str✐❛❧ ❊♥✲
❣✐♥❡❡r✐♥❣ ✺✽✭✹✮✱ ✼✽✺✕✽✵✵✳
258

A Branch-and-Bound Procedure for the
Resource-Constrained Project Scheduling Problem
with Partially Renewable Resources and Time
Windows
Kai Watermeyer and Jürgen Zimmermann
Clausthal University of Technology, Germany
kai.watermeyer, juergen.zimmermann@tu-clausthal.de
Keywords: Project scheduling, Partially renewable resources, Branch-and-bound.
1 Introduction
In this paper we present a branch-and-bound procedure for the resource-constrained
project scheduling problem with partially renewable resources and time windows (RCPSP/
max,π). For the first time the concept of partially renewable resources is embedded in the
context of projects with general temporal constraints.
Partially renewable resources were introduced by Böttcher et al. (1996) and have just
been considered for projects restricted to precedence constraints (RCPSP/π). For each par-
tially renewable resource a resource capacity for a subset of time periods of the planning
horizon is given. In this way timetabling and complex labor regulation problems can be
modeled as project scheduling problems (Álvarez-Valdés et al. 2006). For the RCPSP/π a
branch-and-bound procedure has been developed in Böttcher et al. (1999) and also approx-
imation procedures in Schirmer (1999) and Álvarez-Valdés et al. (2006, 2008) have been
investigated.
In Section 2 the RCPSP/max,π is described formally. Section 3 presents the enumera-
tion scheme the developed branch-and-bound procedure is based on and in Section 4 the
branch-and-bound procedure is outlined. Finally, in Section 5 the results of a computa-
tional study are presented where we compared the performance of our branch-and-bound
procedure with the outcome of the mixed-integer linear programming solver IBM CPLEX.
The resource-constrained project scheduling problem with time windows and partially
renewable resources (RCPSP/max,π) can be modeled as an activity-on-node network where
the nodes correspond to all activities of the project V = {0, 1, . . . , n + 1} with n real
activities and the fictitious activities 0 and n + 1 representing the start and end of the
project, respectively. Each activity i ∈ V is assigned a non-interuptible processing time
pi ∈ Z≥0 and a resource demand rd
ik ∈ Z≥0 for each partially renewable resource k ∈ R
considered in the project. The arcs of the network given by the set E ⊆ V ×V represent the
temporal constraints between the activities where the arc weight δij ∈ Z for arc hi, ji ∈ E
implicates a minimal time lag between the start times of activity i and activity j which
has to be fulfilled. For each resource k ∈ R a resource capacity Rk and a subset of time
periods of the whole planning horizon Πk ⊆ {1, 2, . . . , ¯
d} is given with ¯
d as a given maximal
project duration. It is assumed that an activity i just consumes a resource k with rd
ik units
in each time period of Πk activity i is in execution where the start times of all activities
are restricted to integer values. The number of time periods an activity i with start time
point Si is in execution during the defined time periods of resource k is given by the so
259

called resource usage ru
ik(Si) := |{Si +1, Si +2, . . . , Si +pi}∩Πk| so that the corresponding
resource consumption can be determined by rc
ik(Si) := ru
ik(Si) · rd
ik.
The objective of the problem is to assign each activity i ∈ V a start time Si so that
all time and resource constraints are fulfilled and the project duration is minimized. In the
following a sequence of start times of all activities S = (S0, S1, . . . , Sn+1) with S0 := 0
is called a schedule where it is said to be time-feasible, resource-feasible or feasible if it
fulfills all temporal constraints, all resource constraints or all constraints, respectively. The
problem RCPSP/max,π can be stated as follows:
Minimize f(S) = Sn+1
subject to Sj − Si ≥ δij (hi, ji ∈ E)
S0 = 0
∑
i∈V
rc
ik(Si) ≤ Rk (k ∈ R)
Si ∈ Z≥0 (i ∈ V )
3 Enumeration scheme
The enumeration scheme of the developed branch-and-bound procedure is based on
a stepwise restriction of the allowed resource usages of the activities of the project. The
procedure starts with the determination of the earliest possible start times ESi of all
activities i ∈ V for the resource-relaxation of RCPSP/max,π. If this schedule is resource-
feasible the optimal solution is already found. Otherwise there is at least one resource k
whose resource capacity Rk is exceeded so that the resource usage of at least one activity
consuming resource k have to be decreased to get a feasible schedule. The enumeration
scheme makes use of the start time dependency of the resource usage ru
ik(·) of all activities
i ∈ V for resource k. It is easy to see that for a feasible schedule S the resource usage
of at least one activity i ∈ V has to be lower than the resource usage of the resource-
infeasible schedule ES, i.e., ru
ik(Si) ≤ ru
ik(ESi) − 1. So we preserve all feasible schedules
by branching the resource-relaxation in subproblems where each subproblem restricts the
resource usage of an activity i with ru
ik(ESi) 0 to ru
ik(ESi) − 1. The resource usage
restriction of activity i for resource k is achieved by permitting only start time points t with
ru
ik(t) ≤ ru
ik(ESi)−1. In order to save these permitted start time points for all activities in
the enumeration process a so called start time restriction Wi for each activity is introduced.
This is set to Wi := {ESi, ESi+1, . . . , LSi} for each activity at the beginning of the process
with LSi as the latest possible start time point of activity i for the resource-relaxation of
RCPSP/max,π. For the subproblem in which the resource usage of activity i is restricted
the start time restriction is set to Wi := Wi ∩ {t ∈ {0, 1, . . . , ¯
d} | ru
ik(t) ≤ ru
ik(ESi) − 1}
so that the resource usage of activity i of resource k is lower or equal to ru
ik(ESi) − 1 if
activity i starts at time point t ∈ Wi. For each achieved subproblem the earliest possible
start time points of all activities have to be determined so that all temporal constraints of
the RCPSP/max,π are fulfilled and also Si ∈ Wi for each i ∈ V is satisfied. This can be
done by a modified label correcting algorithm which determines the earliest possible start
time points denoted by ESi(W) of all activities i ∈ V with a worst-case time complexity
of O(|V ||E| (1 + B)) with B as the number of interruptions of consecutive time points in
Wi over all activities i ∈ V . If all determined and all following subproblems are tackled
like described for the resource-relaxation of the RCPSP/max,π it can be shown that the
procedure determines after a finite number of iterations an optimal schedule or shows the
infeasibility if there is no optimal schedule.
260

4 Branch-and-bound procedure
The enumeration scheme describes the decomposition of the currently considered part of
the solution space in one or more components for a chosen conflict resource, i.e., a resource
whose capacity is exceeded. The strategy to decide which of the conflict resources is used
next to decompose the solution space is called branching strategy. The way to determine
which node in the enumeration tree is considered next is called search strategy. For both
strategies different approaches have been investigated on benchmark test sets.
Before the branch-and-bound procedure is started a preprocessing phase is conducted.
In this step start time points of activities are eliminated for which it can be shown that
they cannot be part of any of the feasible schedules. For this a start time point of an
activity is eliminated if the resource consumption of the activity started at this time point
and the sum of the minimal resource consumptions of all other activities over all start time
points satisfying the temporal constraints to the considered activity exceeds the capacity
of at least one resource.
Furthermore, for each node in the search tree two lower bounds for the project duration
are determined to be able to prune this node and the following parts of the enumeration
tree if one of these lower bounds is greater or equal to the project duration of the best found
solution so far. The first lower bound is given by the minimal possible project duration
taking the start time restrictions of all activities into consideration. The second lower bound
is equal to the minimal project duration for which at least one resource-feasible schedule
in the currently considered part of the search tree exists so that all temporal constraints
to the start and the end of the project are satisfied.
To reduce the search tree even further a dominance rule is used in addition. For this
an unexplored node is called dominated by another node if the restrictions of the resource
usages over all activities and resources are lower or equal to the resource usage restrictions
of the other node. In this case the unexplored node is pruned from the search tree.
5 Performance analysis
In order to evaluate the performance of our branch-and-bound (BnB) procedure we have
compared the obtained results with the outcome of the mixed-integer linear programming
(MILP) solver IBM CPLEX in the latest version 12.7.1. The computational study was
conducted on a PC with Intel Core i7-3820 CPU with 3.6 GHz and 32 GB RAM under
Windows 7. The BnB procedure was coded in C++ and compiled with the 64-bit Visual
Studio 2015 C++-Compiler. The instance sets we have used are adaptions of the well-
known benchmark test set UBO (Schwindt 1998) where we replaced the included renewable
resources by 30 partially renewable resources using the generation procedure described in
Schirmer (1999). Note that there is no instance with a project network containing a cycle
of positive length. In this manner we have generated 729 instances with 10, 20, 50, 100,
and 200 activities, respectively. For the computational study we set the runtime limit to
60 seconds and used an adaption of the MILP given in Böttcher et al. (1999) for the IBM
CPLEX solver. The mathematical program is given as follows:
Minimize
∑
t∈Tn+1
t · xn+1,t
subject to
∑
t∈Ti
xit = 1 (i ∈ V )
∑
t∈Tj
t · xjt ≥
∑
t∈Ti
t · xit + δij (hi, ji ∈ E)
∑
i∈V
rd
ik
∑
v∈Πk
∑
τ∈Qi,(v−1)∩Ti
xiτ ≤ Rk (k ∈ R)
xit ∈ {0, 1} (i ∈ V, t ∈ Ti)
261

The MILP is a time-indexed formulation with binary variables xit for each activity
i ∈ V and each start time point t of the activity in the set Ti := {ESi, ESi + 1, . . . , LSi}.
The binary variable xit takes the value 1 exactly if activity i starts at time point t, i.e.,
t = Si. The set Qit contains all time points activity i could be started so that activity i
would be in execution at time point t, i.e., Qit := {t − pi + 1, . . . , t}.
Table 1. Results of the computational study
UBO10π
UBO20π
UBO50π
UBO100π
UBO200π
BnB CPLEX BnB CPLEX BnB CPLEX BnB CPLEX BnB CPLEX
#opt 511 565 288 391 116 113 58 34 53 5
#feas 55 1 259 160 352 65 333 6 312 1
#infeas 129 132 30 57 0 19 0 3 0 0
#noSol 3 0 34 3 59 330 93 441 101 460
#trivial 31 31 118 118 202 202 245 245 263 263
∅CPU
opt 1.60 0.56 2.51 6.78 1.72 4.89 1.76 15.41 6.21 40.51
∅CPU
infeas 0.44 0.03 2.31 0.45 – 2.44 – 14.90 – –
The results of the computational study are given in Tab. 1 where for each test set,
for instance UBO10π
with 10 activities, the results of the BnB procedure and the IBM
CPLEX solver (CPLEX) are listed. The term #opt stands for the number of optimal solved
instances for which the schedule ES is not optimal, term #feas describes the number of
instances the solution procedure was able to find a solution which could not be proofed
to be optimal and #infeas gives the number of instances the procedure could proof the
infeasibility for. In the following two rows, the number of instances the solution procedure
was not able to find any feasible solution (#noSol) and the number of so called trivial
instances for which the schedule ES is already optimal (#trivial) are given. Finally, the
last rows show the average used CPU time in seconds over all optimal solved (∅CPU
opt ) and
over all instances for which the infeasibility could be proofed (∅CPU
infeas).
In Tab. 1 it can be seen that the IBM CPLEX solver dominates the developed BnB
procedure for the instance sets UBO10π
and UBO20π
. In contrast, the BnB procedure is
able to obtain optimal and feasible solutions for more instances of the test sets UBO50π
,
UBO100π
and UBO200π
.
References
Álvarez-Valdés R., E. Crespo, J.M. Tamarit and F. Villa, 2006, “A scatter search algorithm for
project scheduling under partially renewable resources”, Journal of Heuristics, Vol. 12, pp. 95-
113.
Álvarez-Valdés R., E. Crespo, J.M. Tamarit and F. Villa, 2008, “GRASP and path relinking
for project scheduling under partially renewable resources”, European Journal of Operational
Research, Vol. 189, pp. 1153-1170.
Böttcher J., A. Drexl, R. Kolisch, F. Salewski, 1996, “Project scheduling under partially renewable
resource constraints”, Technical Report, Manuskripte aus den Instituten für Betriebswirt-
schaftslehre 398, University of Kiel.
Böttcher J., A. Drexl, R. Kolisch and F. Salewski, 1999, “Project scheduling under partially
renewable resource constraints”, Management Science, Vol. 45, pp. 544-559.
Schirmer A., 1999, “Project scheduling with scarce resources: models, methods and applications”,
Dr. Kovač, Hamburg.
Schwindt C., “Generation of resource-constrained project scheduling problems subject to temporal
constraints”, Technical Report WIOR-543, University of Karlsruhe.
262

❋✐①❡❞ ✐♥t❡r✈❛❧ ♠✉❧t✐❛❣❡♥t s❝❤❡❞✉❧✐♥❣ ♣r♦❜❧❡♠ ✇✐t❤
r❡❥❡❝t❡❞ ❝♦sts
❇✳ ❩❛❤♦✉t✱ ❆✳ ❙♦✉❦❤❛❧ ❛♥❞ P✳ ▼❛rt✐♥❡❛✉
❯♥✐✈❡rs✐té ❞❡ ❚♦✉rs✱ ❋r❛♥❝❡
▲■❋❆❚ ❊❆ ✻✸✵✵✱ ❈◆❘❙✱ ❘❖❖❚ ❊❘▲ ❈◆❘❙ ✼✵✵✷
❜♦✉❦❤❛❧❢❛✳③❛❤♦✉t✱❛♠❡✉r✳s♦✉❦❤❛❧✱♣❛tr✐❝❦✳♠❛rt✐♥❡❛✉❅✉♥✐✈✲t♦✉rs✳❢r
❑❡②✇♦r❞s✿ ▼✉❧t✐❛❣❡♥t s❝❤❡❞✉❧✐♥❣✱ s✐♥❣❧❡ ♠❛❝❤✐♥❡✱ P❛r❡t♦ ♦♣t✐♠✐③❛t✐♦♥✱ ▼■▲P✱ ❧✐♥❡❛r ❝♦♠✲
❜✐♥❛t✐♦♥ ♦❢ ❝r✐t❡r✐❛✱ ε✲❝♦♥str❛✐♥t✱ ❤❡✉r✐st✐❝s✳
✶ ■♥tr♦❞✉❝t✐♦♥
✑▼✉❧t✐❛❣❡♥t s❝❤❡❞✉❧✐♥❣ ♣r♦❜❧❡♠s✑ ❝♦♥s✐❞❡r t❤❛t s❡✈❡r❛❧ ❛❣❡♥ts ❛r❡ ❝♦♠♣❡t✐♥❣ ❢♦r t❤❡ ✉s❡
♦❢ t❤❡ s❛♠❡ r❡s♦✉r❝❡s✳ ❊❛❝❤ ❛❣❡♥t ✐s r❡s♣♦♥s✐❜❧❡ ❢♦r ❛ s❡t ♦❢ ❥♦❜s✱ ❛♥❞ ❛✐♠s ❛t ♠✐♥✐♠✐③✐♥❣
♦♥❡ ♦❜❥❡❝t✐✈❡ ❢✉♥❝t✐♦♥ t❤❛t ❞❡♣❡♥❞s ♦♥❧② ♦♥ t❤❡ ❝♦♠♣❧❡t✐♦♥ t✐♠❡s ♦❢ ✐ts ❛ss♦❝✐❛t❡❞ s✉❜s❡t
♦❢ ❥♦❜s✳ ❲❤❡♥ t❤❡ s✉❜s❡ts ♦❢ ❥♦❜s ❛r❡ ❞✐s❥♦✐♥t✱ t❤❡ ♣r♦❜❧❡♠s ❛r❡ ❝❛❧❧❡❞ ❈♦♠♣❡t✐♥❣ s❝❤❡❞✉❧✐♥❣
♣r♦❜❧❡♠s ✭❆❣♥❡t✐s✱ ▼✐r❝❤❛♥❞❛♥✐✱ P❛❝❝✐❛r❡❧❧✐ ❛♥❞ P❛❝✐✜❝✐ ✷✵✵✹✮✳ ❙✉❝❤ ♣r♦❜❧❡♠s ❝♦rr❡s♣♦♥❞s
t♦ s♦♠❡ r❡❛❧ ✇♦r❧❞ s✐t✉❛t✐♦♥s ❛s ✐♥tr♦❞✉❝❡❞ ✐♥ ✭❆❣♥❡t✐s✱ ❇✐❧❧❛✉t✱ ●❛✇✐❡❥♥♦✇✐❝③✱ P❛❝❝✐❛r❡❧❧✐
❛♥❞ ❙♦✉❦❤❛❧ ✷✵✶✹✮✳
■♥ t❤✐s st✉❞②✱ ✇❡ ❝♦♥s✐❞❡r t✇♦ ❛❣❡♥ts A ❛♥❞ B✳ ❆❣❡♥t A ✭r❡s♣✳ B✮ ✐s ❛ss♦❝✐❛t❡❞
✇✐t❤ t❤❡ s❡t ♦❢ nA ✭r❡s♣✳ nB✮ ❥♦❜s✱ ❞❡♥♦t❡❞ ❜② NA
= {J1, J2, ..., JnA
} ✭r❡s♣✳ NB
=
{JnA+1, JnA+2, ..., Jn}✮✱ ✇❤❡r❡ n = nA + nB✳
❚❤❡ n ✐♥❞❡♣❡♥❞❡♥t ❥♦❜s s❤♦✉❧❞ ❜❡ s❝❤❡❞✉❧❡❞ ✇✐t❤♦✉t ♣r❡❡♠♣t✐♦♥ ♦♥ ❛ s✐♥❣❧❡ ♠❛❝❤✐♥❡✳
❆❞❞✐t✐♦♥❛❧ r❡♥❡✇❛❜❧❡ r❡s♦✉r❝❡s ❛r❡ ❤♦✇❡✈❡r ♥❡❝❡ss❛r② t♦ ♣r♦❝❡ss ❡❛❝❤ ❥♦❜✳ ❙❡✈❡r❛❧ t②♣❡s ♦❢
s✉❝❤ r❡s♦✉r❝❡s ❛r❡ ♥❡❡❞❡❞✱ ❞❡♥♦t❡❞ Rj, j = 1 . . . k✳ ❍❡♥❝❡✱ ❛t ❡①❡❝✉t✐♦♥ t✐♠❡ ♦❢ ❥♦❜ i✱ rij
✉♥✐ts ♦❢ ❛✈❛✐❧❛❜❧❡ r❡s♦✉r❝❡ ❛r❡ r❡q✉✐r❡❞✳ ❋♦r ❡❛❝❤ ❥♦❜ i✱ t❤❡ st❛rt t✐♠❡ si ❛♥❞ ✐ts ✜♥✐s❤❡❞
t✐♠❡ fi ✭i = 1, . . . , n✮ ❛r❡ ✜①❡❞ ✇❤❡r❡ ✐ts ♣r♦❝❡ss✐♥❣ t✐♠❡ pi = fi − si✳ wi ✐s t❤❡ ✇❡✐❣❤t ♦❢
❥♦❜ i✳ ❉❡❛❧✐♥❣ ✇✐t❤ ❡❛❝❤ t②♣❡ ♦❢ r❡s♦✉r❝❡s✱ t❤❡ ♠❛❝❤✐♥❡ ❝❛♥ ♣r♦❝❡ss ♠♦r❡ t❤❛♥ ♦♥❡ ❥♦❜ ❛t
❛ t✐♠❡ ♣r♦✈✐❞❡❞ t❤❡ r❡s♦✉r❝❡ ❝♦♥s✉♠♣t✐♦♥ ❞♦❡s ♥♦t ❡①❝❡❡❞ ❛ ❣✐✈❡♥ ✈❛❧✉❡ Rj ✭j = 1 . . . k✮✳
❚❤✐s ♠❛❝❤✐♥❡ ✐s ❝♦♥t✐♥✉♦✉s❧② ❛✈❛✐❧❛❜❧❡ ❞✉r✐♥❣ t✐♠❡ ✐♥t❡r✈❛❧ [0, ∞)✳ ❆❧❧ ❞❛t❛ ❛r❡ ❛ss✉♠❡❞
♣♦s✐t✐✈❡ ✐♥t❡❣❡rs✳ ❚❤❡ ♣r♦❝❡ss✐♥❣ t✐♠❡s ♦❢ ❥♦❜s ✐s ❢♦r♠❛tt❡❞ ✐♥ s❧♦tt❡❞ ✇✐♥❞♦✇s✳ ❚❤❡ t♦t❛❧
t✐♠❡ ♣❡r✐♦❞ [0, T] ✐s ♣❛rt✐t✐♦♥❡❞ ✐♥t♦ ❡q✉❛❧ ❧❡♥❣t❤ s❧♦ts (l0) ✇✐t❤ T = maxi,i=1,...,n(fi)✳ ❲❡
s✉♣♣♦s❡ t❤❛t✿ si fi ❛♥❞ ri,j ≤ Rj ❢♦r ❛❧❧ i = 1, . . . , n ❛♥❞ j = 1 . . . k✳ ❚❤❡ ♦❜❥❡❝t✐✈❡ ♦❢
❡❛❝❤ ❛❣❡♥t ✐s t♦ ♠✐♥✐♠✐③❡ ✐ts t♦t❛❧ r❡❥❡❝t❡❞ ❝♦sts✳ ▲❡t xi ❜❡ t❤❡ ❜✐♥❛r② ❞❡❝✐s✐♦♥ ✈❛r✐❛❜❧❡
✇❤❡r❡ xi = 1 ✐❢ ❥♦❜ i ✐s r❡❥❡❝t❡❞❀ ✵ ♦t❤❡r✇✐s❡✳ ❲❡ ❞❡♥♦t❡ t❤❡ r❡❥❡❝t❡❞ ❝♦st ♦❢ ❛❣❡♥ts A
❛♥❞ B ❜② ZA
=
PnA
i=1 wixi ❛♥❞ ZB
=
Pn
i=nA+1 wixi✱ r❡s♣❡❝t✐✈❡❧②✳ ■♥ t❤✐s st✉❞②✱ ❜♦t❤
❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥ ♦❢ ❝r✐t❡r✐❛ ❛♣♣r♦❛❝❤ ❛♥❞ ε✲❝♦♥str❛✐♥t ❛♣♣r♦❛❝❤ ❛r❡ ✉s❡❞ t♦ ❞❡t❡r♠✐♥❡
♦♥❡ P❛r❡t♦ ♦♣t✐♠❛❧ s♦❧✉t✐♦♥✳
❆❝❝♦r❞✐♥❣ t♦ t❤❡ t❤r❡❡✲✜❡❧❞ ♥♦t❛t✐♦♥ ♦❢ ♠✉❧t✐❛❣❡♥t s❝❤❡❞✉❧✐♥❣ ♣r♦❜❧❡♠s ✐♥tr♦❞✉❝❡❞ ✐♥
✭❆❣♥❡t✐s✱ ❇✐❧❧❛✉t✱ ●❛✇✐❡❥♥♦✇✐❝③✱ P❛❝❝✐❛r❡❧❧✐ ❛♥❞ ❙♦✉❦❤❛❧ ✷✵✶✹✮✱ ♣r♦❜❧❡♠s ✇❡ ❛❞❞r❡ss ❛r❡
❞❡♥♦t❡❞ ❜②✿ 1|CO|Fℓ(ZA
, ZB
) ✇✐t❤ Fℓ = λZA
+ (1 − λ)ZB
❀ ❆♥❞ 1|CO|ε(ZB
/ZA
)✳ ❚❤❡s❡
♣r♦❜❧❡♠s ❛r❡ ❛❧❧ ◆P✲❤❛r❞ ❡✈❡♥ ✐❢ ♦♥❧② ♦♥❡ ❛❣❡♥t ✐s ❝♦♥s✐❞❡r❡❞ ✭♠♦♥♦❝r✐t❡r✐❛♥ ❝❛s❡✮ ✭❩❛❤♦✉t✱
❙♦✉❦❤❛❧ ❛♥❞ ▼❛rt✐♥❡❛✉ ✷✵✶✼✮✳
❚❤❡ ❛❞❞r❡ss❡❞ ♣r♦❜❧❡♠ ❝❛♥ ❜❡ ♠❡t ✐♥ ❛ ❞❛t❛ ❝❡♥t❡r ✇❤❡r❡ t❤❡ ♦❜❥❡❝t✐✈❡ ✐s t♦ ♦♣t✐♠✐③❡
t❤❡ ♦❜❥❡❝t✐✈❡ ❢✉♥❝t✐♦♥ ♦❢ ❡❛❝❤ ✉s❡r ✭❛❣❡♥t✮✳ ❱✐rt✉❛❧ ▼❛❝❤✐♥❡s ❱▼s ✭❥♦❜s✮ s✉❜♠✐tt❡❞ ❜② t❤❡
✉s❡rs s❤♦✉❧❞ ❜❡ ❡①❡❝✉t❡❞ ♦♥ t❤❡ s❛♠❡ ❝❧✉st❡r ✭♦♥❧② ♦♥❡ ❝❧✉st❡r ✐s ❝♦♥s✐❞❡r❡❞ ✐♥ t❤✐s st✉❞②✮✳
❋♦r ❡①❛♠♣❧❡✱ t❤✐s ❝❧✉st❡r ♦✇♥s t❤r❡❡ ❧✐♠✐t❡❞ t②♣❡s ♦❢ r❡♥❡✇❛❜❧❡ r❡s♦✉r❝❡s ❈P❯✱ ▼❊▼❖❘❨
❛♥❞ ❙❚❖❘❆●❊ ✇✐t❤ ❝❛♣❛❝✐t✐❡s ❡q✉❛❧ t♦ Q1 ❈P❯✱ ❛ ❝❡rt❛✐♥ q✉❛♥t✐t② ♦❢ ♠❡♠♦r② Q2 ❛♥❞
263

❛ ❝❡rt❛✐♥ st♦r❛❣❡ ❝❛♣❛❝✐t② Q3✳ ■♥ t❤✐s ❝❛s❡✱ t♦ ❡①❡❝✉t❡ V Mi✱ ❛ ♥✉♠❜❡r ♦❢ ✈✐rt✉❛❧ ❈P❯s
ri1✱ ✈✐rt✉❛❧ ♠❡♠♦r② ri2 ❛♥❞ ❤❛r❞ ❞r✐✈❡s ri3 ❛r❡ ♥❡❡❞❡❞✳ ❚❤❡ ♠♦♥♦❝r✐t❡r✐❛♥ ❝❛s❡ ❤❛s ❜❡❡♥
❛❞❞r❡ss❡❞ ✐♥ ✭❆♥❣❡❧❡❧❧✐✱ ❇✐❛♥❝❤❡ss✐ ❛♥❞ ❋✐❧✐♣♣✐ ✷✵✶✹✮ ✇❤❡r❡ t❤❡ ❛✉t❤♦rs ❝♦♥s✐❞❡r ♦♥❧② ♦♥❡
❛❞❞✐t✐♦♥❛❧ r❡s♦✉r❝❡ ✭♠❡♠♦r②✮ ❛♥❞ ❞❡✈❡❧♦♣ ♠❡t❤♦❞s t♦ ❞❡t❡r♠✐♥❡ ♦♥❡ ❢❡❛s✐❜❧❡ s♦❧✉t✐♦♥✳
■♥ t❤❡ ❝♦♥t❡①t ♦❢ ❣r✐❞ ❝♦♠♣✉t✐♥❣✱ ✭❈♦r❞❡✐r♦✱ ❉✉t♦t✱ ▼♦✉♥✐é ❛♥❞ ❚r②str❛♠ ✷✵✶✶✮ ❝♦♥✲
s✐❞❡rs ♦r❣❛♥✐③❛t✐♦♥s t❤❛t s❤❛r❡ ❝❧✉st❡rs t♦ ❞✐str✐❜✉t❡ ♣❡❛❦ ✇♦r❦❧♦❛❞s ❛♠♦♥❣ ❛❧❧ t❤❡ ♣❛r✲
t✐❝✐♣❛♥ts✳ ❊❛❝❤ ❝❧✉st❡r ✐s ❛ss♦❝✐❛t❡❞ ✇✐t❤ ♦♥❡ ❛❣❡♥t ❛♥❞ t❤❡ ❣❧♦❜❛❧ ♦❜❥❡❝t✐✈❡ ❢✉♥❝t✐♦♥ ✐s
t♦ ♠✐♥✐♠✐③❡ t❤❡ ♠❛❦❡s♣❛♥✳ ❚❤❡ ❛✉t❤♦rs ♣r♦♣♦s❡ ❛ ✷✲❛♣♣r♦①✐♠❛t✐♦♥ ❛❧❣♦r✐t❤♠ ❢♦r ✜♥❞✐♥❣
❝♦❧❧❛❜♦r❛t✐✈❡ s♦❧✉t✐♦♥s✳
✷ ❊①❛❝t ♠❡t❤♦❞s
✷✳✶ ▲✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥ ♦❢ ❝r✐t❡r✐❛
❈♦♥s✐❞❡r t❤❡ ❝❧❛ss✐❝❛❧ s❝❤❡❞✉❧✐♥❣ ♣r♦❜❧❡♠ 1||Z ✇❤❡r❡ Z =
P
1≤i≤n w′
ixi✳ ❚❤❡ t✇♦ ❢♦❧✲
❧♦✇✐♥❣ s❝❤❡❞✉❧✐♥❣ ♣r♦❜❧❡♠s ❛r❡ ❡q✉✐✈❛❧❡♥t✿ 1|CO|Fℓ(ZA
, ZB
) ❛♥❞ 1||Z✳ ■♥ ❢❛❝t✱ ✇❡ s❡t
w′
i = λwi ❢♦r ❛❧❧ Ji ∈ NA
❛♥❞ w′
i = (1 − λ)wi ❢♦r ❛❧❧ Ji ∈ NB
✳ ❍❡♥❝❡✱ ✇❡ ♣r♦♣♦s❡ t❤❡
❢♦❧❧♦✇✐♥❣ t✐♠❡ ✐♥❞❡①❡❞ ✐♥t❡❣❡r ❧✐♥❡❛r ♣r♦❣r❛♠♠✐♥❣ ❢♦r♠✉❧❛t✐♦♥ ✭■▲P✮ ✇❤❡r❡✿ xi ✐s ❛ ❜✐♥❛r②
✈❛r✐❛❜❧❡ ❡q✉❛❧ t♦ ✶ ✐❢ ❥♦❜ Ji ✐s r❡❥❡❝t❡❞✱ ✵ ♦t❤❡r✇✐s❡❀ ❆♥❞ yit ✐s ❛ ❜✐♥❛r② ✈❛r✐❛❜❧❡ ❡q✉❛❧ t♦
✶ ✐❢ ❥♦❜ Ji ✐s ❡①❡❝✉t❡❞ ❛t t✐♠❡ t✱ ❛♥❞ ✵ ♦t❤❡r✇✐s❡✳
▼✐♥✐♠✐③❡✿
X
i∈N
w′
ixi
s✉❜❥❡❝t t♦✿
fi−1
X
t=si
yit = (fi − si) ∗ (1 − xi) ∀i ∈ N (1)
X
i∈N
yit ∗ rij ≤ Rj ∀j ∈ R ; ∀t ∈ [0, T] (2)
xi ∈ {0, 1}, yit ∈ {0, 1} , ∀i ∈ N , ∀t ∈ [0, T].
❚❤❡ ❝♦♥str❛✐♥ts ✭✶✮ ❡♥s✉r❡ t❤❛t ✐❢ ❥♦❜ Ji ✐s ♥♦t r❡❥❡❝t❡❞ t❤❡♥ ✐t ✐s s❝❤❡❞✉❧❡❞ ❞✉r✐♥❣ ✐ts
t✐♠❡ ✐♥t❡r✈❛❧✳ ❚❤❡ ❝♦♥str❛✐♥ts ✭✷✮ ❡♥s✉r❡ t❤❛t ♥♦ ♠♦r❡ t❤❛♥ Rj q✉❛♥t✐t✐❡s ♦❢ t❤❡ r❡q✉✐r❡❞
r❡s♦✉r❝❡s ❛r❡ ❝♦♥s✉♠❡❞ ❛t t✐♠❡ t✳
✷✳✷ ε✲❝♦♥str❛✐♥t ❛♣♣r♦❛❝❤
❚♦ ❞❡t❡r♠✐♥❡ ❛ ♥♦♥✲❞♦♠✐♥❛t❡❞ s♦❧✉t✐♦♥✱ ✇❡ ♣r♦♣♦s❡ t♦ ✉s❡ ♣r❡✈✐♦✉s ■▲P ✇❤❡r❡ t❤❡
♦❜❥❡❝t✐✈❡ ❢✉♥❝t✐♦♥ ✐s ♥♦✇✿ ▼✐♥✐♠✐③❡ ZB
=
P
i∈N B wixi✳ ❚❤❡♥ t♦ t❤❡ t✇♦ ✐♥❡q✉❛❧✐t✐❡s ✭✶✮
❛♥❞ ✭✷✮✱ ✇❡ ❛❞❞ ❢♦❧❧♦✇✐♥❣ ♥❡✇ ❝♦♥str❛✐♥t✿ ZA
≤ QA✳ ■t ♠❡❛♥s t❤❛t t❤❡ t♦t❛❧ r❡❥❡❝t❡❞ ❝♦st
❞❡✜♥❡❞ ❜② ♥♦♥✲s❝❤❡❞✉❧❡❞ ❥♦❜s ♦❢ ❛❣❡♥t A ❞♦❡s ♥♦t ❡①❝❡❡❞ ❛ ❣✐✈❡♥ ✈❛❧✉❡ QA✳
❚❤✐s ■▲P ✐s ❛❧s♦ ✉s❡❞ t♦ ❝♦♠♣✉t❡ t❤❡ ♦♣t✐♠❛❧ P❛r❡t♦ ❢r♦♥t✳
✸ ●r❡❡❞② ❤❡✉r✐st✐❝s
❚❤❡ s❝❤❡❞✉❧✐♥❣ ♣r♦❜❧❡♠s ✉♥❞❡r t❤✐s st✉❞② ❤❛✈❡ ♠❛♥② ❛♣♣❧✐❝❛t✐♦♥s✱ ❜✉t t❤❡② ✇❡r❡ ♠♦t✐✲
✈❛t❡❞ ❜② r❡s❡❛r❝❤ ✐♥t♦ ♦♥✲❧✐♥❡ s②st❡♠ ❛♥❞ ✐♥t❡❣r❛t❡❞✲s❡r✈✐❝❡s ♥❡t✇♦r❦s✱ ✇❤❡r❡ t❤❡ ♥✉♠❜❡r ♦❢
❥♦❜s t♦ ❜❡ ♣r♦❝❡ss❡❞ ❝❛♥ ❜❡ ❡①tr❡♠❡❧② ❧❛r❣❡✱ s♦ ❧♦✇ ❝♦♠♣✉t❛t✐♦♥❛❧ r✉♥♥✐♥❣ t✐♠❡ ✐s ❡ss❡♥t✐❛❧✳
❍❡♥❝❡✱ t❤❡ r❡s♦❧✉t✐♦♥ ♠❡t❤♦❞s ♠✉st ❤❛✈❡ ❧♦✇ ❝♦♠♣❧❡①✐t②✱ ♥♦t ❥✉st ♣♦❧②♥♦♠✐❛❧ ❝♦♠♣❧❡①✐t②✳
■♥ t✐s s❡❝t✐♦♥✱ ✇❡ ♣r❡s❡♥t❡ ❧♦✇✲❝♦♠♣❧❡①✐t② ✭O(nlogn)✮ ❣r❡❡❞② ❛❧❣♦r✐t❤♠s✳ ❘♦✉❣❤❧②✱ t❤✐s
❛❧❣♦r✐t❤♠ ✇♦r❦s ❛s ❢♦❧❧♦✇✳ ■❢ ε✲❝♦♥str❛✐♥t ❛♣♣r♦❛❝❤ ✐s ✉s❡❞✱ ❥♦❜s ♦❢ ❡❛❝❤ ❛❣❡♥t ❛r❡ s♦rt❡❞
❛❝❝♦r❞✐♥❣ t♦ ❛ ❣✐✈❡♥ ♣r✐♦r✐t② r✉❧❡✳ ❆t ✜rst✱ ✇❡ tr② t♦ s❝❤❡❞✉❧❡ ❥♦❜s ♦❢ ❛❣❡♥t A ✇✐t❤ r❡s♣❡❝t
♦❢ ✐ts ♦❜❥❡❝t✐✈❡ ✭✐✳❡✳ ZA
≤ QA✮✳ ❏♦❜s ❛r❡ t❛❦❡♥ ❛❝❝♦r❞✐♥❣ t♦ t❤❡✐r ♣r✐♦r✐t② ♦r❞❡r✳ ❏♦❜ ✐s
264

r❡❥❡❝t❡❞ ✐❢ ✐t ❝❛♥ ♥♦t ❜❡ s❝❤❡❞✉❧❡❞✳ ❚❤❡♥✱ ✇✐t❤✐♥ t❤❡ ♦❜t❛✐♥❡❞ s♦❧✉t✐♦♥✱ ✇❡ tr② t♦ s❝❤❡❞✉❧❡
❥♦❜s ♦❢ ❛❣❡♥t B ♠✐♥✐♠✐③✐♥❣ ✐ts ♦❜❥❡❝t✐✈❡ ❢✉♥❝t✐♦♥ ZB
✳
❲❤❡♥ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥ ♦❢ ❝r✐t❡r✐❛ ❛♣♣r♦❛❝❤ ✐s ✉s❡❞✱ ✇❤♦❧❡ ❥♦❜s ❛r❡ s♦rt❡❞ ❛❝❝♦r❞✐♥❣ t♦
❛ ❣✐✈❡♥ ♣r✐♦r✐t② r✉❧❡✳ ❚❤❡♥✱ ✇❡ s♦❧✈❡ t❤❡ s❝❤❡❞✉❧✐♥❣ ♣r♦❜❧❡♠ 1||Z ✇❤❡r❡ Z =
P
1≤i≤n w′
ixi✳
■t ♠❡❛♥s t❤❛t ❥♦❜s ✇✐t❤ ❤✐❣❤❡r ♣r✐♦r✐t② ❛r❡ s❝❤❡❞✉❧❡❞ ✜rst✱ ✐❢ ♣♦ss✐❜❧❡✳
✸✳✶ Pr✐♦r✐t② r✉❧❡s
✶✳ ❲❡✐❣❤t❡❞ ❙❤♦rt❡st Pr♦❝❡ss✐♥❣ ❚✐♠❡ ❋✐rst ✭WSPT✮✿ ❏♦❜s ❛r❡ s♦rt❡❞ ✐♥ ♥♦♥✲
❞❡❝r❡❛s✐♥❣ ♦r❞❡r ♦❢ (fi −si)/wi✱ ✐♥ ❝❛s❡ ♦❢ t✐❡s✱ ❥♦❜ ✇✐t❤ t❤❡ s♠❛❧❧❡st ✜♥✐s❤❡❞ t✐♠❡ ❝♦♠❡
✜rst✱ ♦t❤❡r✇✐s❡ ❧❡①✐❝♦❣r❛♣❤✐❝❛❧ ♦r❞❡r ✐s ❝♦♥s✐❞❡r❡❞✳ ❚❤✐s WSPT r✉❧❡ ❛❧❧♦✇s r❡s♦✉r❝❡s
t♦ ❜❡ r❡❧❡❛s❡❞ ❛s s♦♦♥ ❛s ♣♦ss✐❜❧❡✳
✷✳ ❲❡✐❣❤t❡❞ ❈❛♣❛❝✐t②✲▼❛❦❡s♣❛♥ ✭WCM✮✿ ❏♦❜s ❛r❡ s♦rt❡❞ ✐♥ ♥♦♥✲❞❡❝r❡❛s✐♥❣ ♦r❞❡r
♦❢ t❤❡✐r ♦❝❝✉♣✐❡❞ s♣❛❝❡ ❞✐✈✐❞❡❞ ❜② wi ❣✐✈❡♥ ❜② t❤❡ ❢♦❧❧♦✇✐♥❣ ❢♦r♠✉❧❛✿ (
P
j∈R rij ∗ (fi −
si))/wi✱ ✐♥ ❝❛s❡ ♦❢ t✐❡s✱ t❤❡ ❥♦❜ ✇✐t❤ t❤❡ s♠❛❧❧❡st ✜♥✐s❤❡❞ t✐♠❡ ❝♦♠❡ ✜rst✱ ♦t❤❡r✇✐s❡
❧❡①✐❝♦❣r❛♣❤✐❝❛❧ ♦r❞❡r ✐s ❝♦♥s✐❞❡r❡❞✳ ❚❤❡ ✐❞❡❛ ♦❢ ✉s✐♥❣ WCM r✉❧❡ ✐s t♦ ♠✐♥✐♠✐③❡ t❤❡
s♣❛❝❡ ♦❝❝✉♣✐❡❞ ❜② ❥♦❜s ❞❡✜♥❡❞ ❜② ♣r♦❝❡ss✐♥❣ t✐♠❡ ♣❡r q✉❛♥t✐t✐❡s ♦❢ ❝♦♥s✉♠❡❞ r❡s♦✉r❝❡s✳
✹ ❈♦♠♣✉t❛t✐♦♥❛❧ ❡①♣❡r✐♠❡♥ts
❲❡ ✐♠♣❧❡♠❡♥t❡❞ ♦✉r ❛❧❣♦r✐t❤♠s ✐♥ C✰✰ ❧❛♥❣✉❛❣❡ ❛♥❞ ❡①❡❝✉t❡❞ ❡①♣❡r✐♠❡♥ts ♦♥ ❛
✇♦r❦st❛t✐♦♥ ✇✐t❤ ❛ ✷✳✽ ●❤③ ■♥t❡❧ ❈♦r❡ ✐✼ ♣r♦❝❡ss♦r ❛♥❞ ✽ ●❇ ♦❢ ♠❡♠♦r②✳ ❲❡ ✉s❡❞ ■❇▼
■▲❖● ❈P▲❊❳ ❖♣t✐♠✐③❛t✐♦♥ ❙t✉❞✐♦ ✈❡rs✐♦♥ ✶✷✳✻✳✸ t♦ s♦❧✈❡ t❤❡ ■▲P ♠♦❞❡❧s✳
❲❡ ❛ss❡ss❡❞ t❤❡ ♣❡r❢♦r♠❛♥❝❡ ♦❢ t❤❡ ❛❧❣♦r✐t❤♠s ♦♥ ✺✵ ✐♥st❛♥❝❡s✱ ✇✐t❤ ❛ ♥✉♠❜❡r ♦❢ ❥♦❜s
n ∈ {20, 40, . . . , 100} ✇❤❡r❡ 30% ♦❢ n ❛r❡ ❥♦❜s ♦❢ ❛❣❡♥t A ✭✶✵ ✐♥st❛♥❝❡s ❛r❡ ❣❡♥❡r❛t❡❞ ♣❡r
n✮✳ ❲❡ ❣❡♥❡r❛t❡❞ t❤❡ ❥♦❜✲st❛rt✐♥❣ t✐♠❡s si ✉s✐♥❣ ❛ ❞✐s❝r❡t❡ ✉♥✐❢♦r♠ ❞✐str✐❜✉t✐♦♥ ❜❡t✇❡❡♥
1♠♥ ❛♥❞ 1400♠♥✳ ❙✐♠✐❧❛r❧②✱ ✇❡ ❣❡♥❡r❛t❡❞ t❤❡ ❥♦❜✲✜♥✐s❤✐♥❣ t✐♠❡ ♦❢ ❡❛❝❤ ❥♦❜ Ji ❤❛s ✉s✐♥❣
❛ ❞✐s❝r❡t❡ ✉♥✐❢♦r♠ ❞✐str✐❜✉t✐♦♥ ❜❡t✇❡❡♥ (si + 1)♠♥ ❛♥❞ (1440 − si)♠♥✳ ❲❡ ❝♦♥s✐❞❡r❡❞
t❤r❡❡ t②♣❡s ♦❢ r❡s♦✉r❝❡s✳ ❲✐t❤♦✉t ❧♦st ♦❢ ❣❡♥❡r❛❧✐t②✱ ✇❡ ♥♦r♠❛❧✐③❡ t❤❡ ✉♥✐ts ♦❢ ❛ r❡♥❡✇❛❜❧❡
r❡s♦✉r❝❡ t♦ ✶✵✵✵✳ ❍❡♥❝❡✱ Rj = 1000, j = 1, 2, 3✳ ❋♦r ❡❛❝❤ ❥♦❜ Ji✱ rij r❛♥❞♦♠❧② ❣❡♥❡r❛t❡❞ ✐♥
[1, 1000]✱ i = 1, . . . , n ❛♥❞ j = 1, 2, 3✳
❚❤❡ ❡①♣❡r✐♠❡♥t❛❧ r❡s✉❧ts ❛r❡ s✉♠♠❛r✐③❡❞ ✐♥ ❚❛❜❧❡ ✶✳
❚❛❜❧❡ ✶✳ ❈♦♠♣✉t❛t✐♦♥❛❧ r❡s✉❧ts ❢♦r ♣r♦❜❧❡♠ 1|CO|ε(ZA
, ZB
) ✇✐t❤ nA = 30%n
n ILP WSPT WCM
CPUs |S∗
| CPUs |S| GD %S %wS CPUs |S| GD %S %wS
✷✵ ✵✱✽ ✹✱✶ ✵ ✸✱✺ ✶✱✶✻ ✹✼ ✹✵ ✵ ✸✱✻ ✶✱✵✺ ✹✼ ✹✸
✹✵ ✸✱✵ ✻✱✺ ✵ ✹✱✵ ✸✱✷✵ ✸✹ ✷✺ ✵ ✹✱✵ ✸✱✷✵ ✸✷ ✹✶
✻✵ ✽✱✵ ✶✵✱✷ ✵ ✺✱✸ ✽✱✽✻ ✶✼ ✹✵ ✵ ✺✱✸ ✽✱✼✼ ✶✺ ✹✾
✽✵ ✶✺✱✻ ✶✷✱✽ ✵ ✺✱✸ ✶✸✱✵✵ ✶✵ ✶✷ ✵ ✹✱✾ ✶✵✱✵✵ ✶✹ ✶✸
✶✵✵ ✸✺✱✶ ✷✶✱✾ ✵ ✻✱✻ ✷✶✱✵✵ ✷ ✸✼ ✵ ✻✱✾ ✶✾✱✵✵ ✸ ✹✵
❚❤❡ ✜rst ❝♦❧✉♠♥ ✐♥ ❚❛❜❧❡ ✶ s❤♦✇s t❤❡ s✐③❡ ♦❢ t❤❡ ✐♥st❛♥❝❡ ✭♥✉♠❜❡r ♦❢ ❥♦❜s✮✳ ❲❡ ❝♦♠✲
♣✉t❡❞ t❤❡ ❛✈❡r❛❣❡ ❝♦♠♣✉t❛t✐♦♥ t✐♠❡ ✐♥ s❡❝♦♥❞s r❡q✉✐r❡❞ t♦ ♦❜t❛✐♥ t❤❡ P❛r❡t♦ ❢r♦♥t ❢♦r ❡❛❝❤
♠❡t❤♦❞✿ ILP ♠♦❞❡❧✱ ❲❡✐❣❤t❡❞ ❙❤♦rt❡st Pr♦❝❡ss✐♥❣ ❚✐♠❡ ❋✐rst ✭WSPT✮ ❛♥❞ ❲❡✐❣❤t❡❞
❈❛♣❛❝✐t②✲▼❛❦❡s♣❛♥ ✭WCM✮✳ ❚❤✐s ❝♦♠♣✉t❛t✐♦♥ t✐♠❡ ✐s ❞❡♥♦t❡❞ ❜② CPUs✳ ❚❤❡ s✐③❡ ♦❢
❡①❛❝t ✭r❡s♣✳ ❛♣♣r♦①✐♠❛t❡✮ P❛r❡t♦ ❢r♦♥t ✐s ❞❡♥♦t❡❞ ❜② |S∗
| ✭r❡s♣✳ |S|✮✳
265

❋♦r ❡❛❝❤ ✐♥st❛♥❝❡✱ ✇❡ ❝♦♠♣❛r❡ t❤❡ ❡①❛❝t ❢r♦♥t S∗
❣❡♥❡r❛t❡❞ ❜② ILP ♠♦❞❡❧ ✇✐t❤
t❤❡ P❛r❡t♦ ❢r♦♥t S ❣❡♥❡r❛t❡❞ ❜② ❤❡✉r✐st✐❝s ✭WCM ❛♥❞ WSPT✮✳ ❉✐✛❡r❡♥t ♣❡r❢♦r♠❛♥❝❡
♠❡❛s✉r❡s ♦❢ ❤❡✉r✐st✐❝s ❛r❡ ✉s❡❞ ❛♥❞ ❞❡s❝r✐❜❡❞ ❛s ❢♦❧❧♦✇✳ ●✐✈❡♥ S∗
= {a1, . . . , a|S∗|} ❛♥❞
S = {b1, . . . , b|S|}✱ ✇❡ ❝❛t❡❣♦r✐③❡ t❤❡s❡ ♠❡❛s✉r❡s ✐♥ t✇♦ ❝❧❛ss❡s✿
✕ ❈❛r❞✐♥❛❧✐t② ♠❡❛s✉r❡✿ ✇❡ ❝❛❧❝✉❧❛t❡ t❤❡ s✐③❡ ♦❢ t❤❡ ♦♣t✐♠❛❧ P❛r❡t♦ ❢r♦♥t |S∗
|✱ ❛♥❞ t❤❡
❛♣♣r♦①✐♠❛t❡❞ P❛r❡t♦ ❢r♦♥t |S|✳ ❲❡ t❤❡♥ ❝♦♠❜✐♥❡ t❤❡s❡ ♠❡tr✐❝s t♦ ♦❜t❛✐♥ t❤❡ ♣❡r❝❡♥t❛❣❡
♦❢ str✐❝t ♥♦♥✲❞♦♠✐♥❛t❡❞ s♦❧✉t✐♦♥s ❣❡♥❡r❛t❡❞ ❜② WCM ❛♥❞ WSPT✳
%S =
|S ∩ S∗
|
|S|
∗ 100
❛♥❞ %wS✱ t❤❡ ♣❡r❝❡♥t❛❣❡ ♦❢ ✇❡❛❦ ♥♦♥✲❞♦♠✐♥❛t❡❞ s♦❧✉t✐♦♥s ❣❡♥❡r❛t❡❞ ❜② WCM ❛♥❞
WSPT✳
✕ ❆✈❡r❛❣❡ ♠✐♥✐♠✉♠ ❊✉❝❧✐❞✐❛♥ ❞✐st❛♥❝❡ GD✿ ❧❡t di ❜❡ t❤❡ ♠✐♥✐♠✉♠ ❊✉❝❧✐❞✐❛♥ ❞✐st❛♥❝❡
❜❡t✇❡❡♥ t❤❡ ❡❧❡♠❡♥t bi ∈ S ❛♥❞ s♦♠❡ ❡❧❡♠❡♥t ♦❢ S∗
✳ GD ✐s ❣✐✈❡♥ ❜②✿
GD =
1
|S|
(
|S|
X
i=1
di)
❚❤❡ ✜rst r❡s✉❧t ❝♦♥❝❡r♥s t❤❡ ♣❡r❢♦r♠❛♥❝❡ ♦❢ t❤❡ ♣r♦♣♦s❡❞ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧ ILP✳
❈P▲❊❳ ❞❡❧✐✈❡rs ♦♣t✐♠❛❧ s♦❧✉t✐♦♥s ❢♦r t❤❡ ✺✵ ✐♥st❛♥❝❡s✳ ❚❤❡ r❡q✉✐r❡❞ ❛✈❡r❛❣❡ ❝♦♠♣✉t❛t✐♦♥
t✐♠❡ ♣❡r ✐♥st❛♥❝❡ ✐s ❧❡ss t❤❛♥ ✸✺ s❡❝♦♥❞s ✭✶s ❢♦r ✐♥st❛♥❝❡s ♦❢ ✷✵ ❥♦❜s✮✳ ■♥ ❛❞❞✐t✐♦♥✱ ✇❡
❝♦♥❞✉❝t❡❞ ❛❞❞✐t✐♦♥❛❧ t❡sts t♦ ❛♥❛❧②③❡ t❤❡ ♣❡r❢♦r♠❛♥❝❡ ♦❢ ■▲P ♦♥ ❧❛r❣❡ s✐③❡ ✐♥st❛♥❝❡s ✭✉♣
t♦ ✺✵✵ ❥♦❜s✮✳ ❋♦r ❡①❛♠♣❧❡✱ t❤❡ ♠❛①✐♠✉♠ ❝♦♠♣✉t❛t✐♦♥ t✐♠❡ ♥❡❡❞❡❞ ❜② ■▲P t♦ ❝❛❧❝✉❧❛t❡ t❤❡
❡①❛❝t P❛r❡t♦ ❢r♦♥t ✇✐t❤ ✐♥st❛♥❝❡s ♦❢ ✺✵✵ ❥♦❜s ✐s ✹✺ ♠✐♥✉t❡s✳
❆❝❝♦r❞✐♥❣ t♦ t❤❡ ❞❛t❛ ❞✐s♣❧❛②❡❞ ♦♥ t❤❡ ❚❛❜❧❡ ✶ ❛♥❞ ❛❝❝♦r❞✐♥❣ t♦ ❝❛r❞✐♥❛❧✐t② ♠❡❛s✉r❡✱ t❤❡
♥✉♠❜❡r ♦❢ P❛r❡t♦ s♦❧✉t✐♦♥s ✐♥❝r❡❛s❡s ✇✐t❤ ✐♥❝r❡❛s✐♥❣ ♥✉♠❜❡r ♦❢ ❥♦❜s✳ ❖✈❡r ❛❧❧ ✺✵ ✐♥st❛♥❝❡s✱
✷✷✪ ♦❢ P❛r❡t♦ s♦❧✉t✐♦♥s ❣❡♥❡r❛t❡❞ ❜② ❤❡✉r✐st✐❝ WCM ♦r WSPT ❛r❡ ♦♣t✐♠❛❧ s♦❧✉t✐♦♥s✱
✇❤✐❧❡ ✸✼✪ ♦❢ s♦❧✉t✐♦♥s ❛r❡ ✇❡❛❦❧② P❛r❡t♦ s♦❧✉t✐♦♥s ❣❡♥❡r❛t❡❞ ❜② WCM ✇❤❡♥ WSPT ✜♥❞s
✸✶✪✳ ❉❡❛❧✐♥❣ ✇✐t❤ GD ♠❡❛s✉r❡✱ t❤❡ ❛✈❡r❛❣❡ ♠✐♥✐♠✉♠ ❊✉❝❧✐❞❡❛♥ ❞✐st❛♥❝❡ ❣✐✈❡♥ ❜② WCM
✭r❡s♣✳ WSPT✮ ✐s ✶✹✱✺ ✭r❡s♣✳ ✶✼✱✵✮✳
❲❡ ♦❜s❡r✈❡ t❤❛t ✇❤❡♥ t❤❡ ♥✉♠❜❡r ♦❢ ❥♦❜s ✐♥❝r❡❛s❡s✱ t❤❡ ❛✈❡r❛❣❡ ♣❡r❝❡♥t❛❣❡ ♦❢ t❤❡
❡①❛❝t✴✇❡❛❦❧② P❛r❡t♦ s♦❧✉t✐♦♥s ❣❡♥❡r❛t❡❞ ❜② ❤❡✉r✐st✐❝s WCM ❛♥❞ WSPT ❞❡❝r❡❛s❡s✳ ❋♦r
❡①❛♠♣❧❡✱ ♦♥ t❤❡ ✷✵✲❥♦❜ ✭r❡s♣✳✶✵✵✲❥♦❜✮ ✐♥st❛♥❝❡s✱ ✾✵✪ ✭r❡s♣✳ ✹✸✪✮ ♦❢ t❤❡ P❛r❡t♦ s♦❧✉t✐♦♥s
❣❡♥❡r❛t❡❞ ❜② ❤❡✉r✐st✐❝ WCM ❛r❡ ❡①❛❝t ♦r ✇❡❛❦❧② s♦❧✉t✐♦♥s✳ ❚❤❡r❡❢♦r❡✱ GD ✐♥❝r❡❛s❡s t♦
❛♥ ❛✈❡r❛❣❡ ♦❢ ✶✳✵✺ ✭r❡s♣✳ ✶✾✮✳ ❲❡ ♦❜s❡r✈❡ ❛❧♠♦st t❤❡ s❛♠❡ ♣❡r❢♦r♠❛♥❝❡s ✇✐t❤ WSPT✳ ■♥
❢❛❝t✱ ❞❡❛❧✐♥❣ ✇✐t❤ %wS ♠❡❛s✉r❡ WCM ♦❜t❛✐♥s s♦♠❡ ❛❞✈❛♥t❛❣❡✳
❍♦✇❡✈❡r✱ t❤❡s❡ ♠❡t❤♦❞s ❛r❡ ✈❡r② ✉s❡❢✉❧ t♦ s♦❧✈❡ st✉❞✐❡❞ ♣r♦❜❧❡♠ ❣✐✈❡♥ ✐ts ❝♦♠♣❧❡①✐t②✳
❘❡❢❡r❡♥❝❡s
❆❣♥❡t✐s✱ ❆✳✱ P✳ ▼✐r❝❤❛♥❞❛♥✐✱ ❉✳ P❛❝❝✐❛r❡❧❧✐✱ ❆✳ P❛❝✐✜❝✐✱ ✷✵✵✹✱ ✏❙❝❤❡❞✉❧✐♥❣ ♣r♦❜❧❡♠s ✇✐t❤ t✇♦
❝♦♠♣❡t✐♥❣ ❛❣❡♥ts✧✱ ❖♣❡r❛t✐♦♥s ❘❡s❡❛r❝❤✱ ❱♦❧✳ ✺✷✱ ♣♣✳ ✷✷✾✲✷✹✷✳
❆❣♥❡t✐s ❆✳✱ ❏✳✲❈✳ ❇✐❧❧❛✉t✱ ❙✳ ●❛✇✐❡❥♥♦✇✐❝③✱ ❉✳ P❛❝❝✐❛r❡❧❧✐✱ ❆✳ ❙♦✉❦❤❛❧✱ ✷✵✶✹✱ ✏▼✉❧t✐❛❣❡♥t ❙❝❤❡❞✉❧✲
✐♥❣✱ ▼♦❞❡❧s ❛♥❞ ❆❧❣♦r✐t❤♠s✧✱ ❙♣r✐♥❣❡r✲❱❡r❧❛❣✱ ❇❡r❧✐♥ ❍❡✐❧❞❡❧❜❡r❣ ◆❡✇ ❨♦r❦✳
❆♥❣❡❧❡❧❧✐ ❊✳✱ ◆✳ ❇✐❛♥❝❤❡ss✐✱ ❈✳ ❋✐❧✐♣♣✐✱ ✷✵✶✹✱ ✏❖♣t✐♠❛❧ ✐♥t❡r✈❛❧ s❝❤❡❞✉❧✐♥❣ ✇✐t❤ ❛ r❡s♦✉r❝❡ ❝♦♥✲
str❛✐♥t✑✱ ❈♦♠♣✉t❡rs ✫ ❖♣❡r❛t✐♦♥s ❘❡s❡❛r❝❤✱ ❱♦❧✳ ✺✶✱ ♣♣✳ ✷✻✽✲✷✽✶✳
❈♦r❞❡✐r♦ ❉✳✱ P✳✲❋✳ ❉✉t♦t✱ ●✳ ▼♦✉♥✐é✱ ❉✳ ❚r②str❛♠✱ ✷✵✶✶✱ ✏❚✐❣❤t ❆♥❛❧②s✐s ♦❢ ❘❡❧❛①❡❞ ▼✉❧t✐✲
❖r❣❛♥✐③❛t✐♦♥ ❙❝❤❡❞✉❧✐♥❣ ❆❧❣♦r✐t❤♠s✧✱ ■♥ Pr♦❝❡❡❞✐♥❣s ♦❢ t❤❡ ✷✺t❤ ■❊❊❊ ■♥t❡r♥❛t✐♦♥❛❧ P❛r✲
❛❧❧❡❧ ✫ ❉✐str✐❜✉t❡❞ Pr♦❝❡ss✐♥❣ ❙②♠♣♦s✐✉♠ ✭■P❉P❙✮✱ ■❊❊❊ ❈♦♠♣✉t❡r ❙♦❝✐❡t②✱ ❆♥❝❤♦r❛❣❡✱ ❆▲✱
❯❙❆✱ ♣♣✳ ✶✶✼✼✲✶✶✽✻✳
❩❛❤♦✉t ❊✳✱ ◆✳ ❙♦✉❦❤❛❧✱ ❈✳ ▼❛rt✐♥❡❛✉✱ ✷✵✶✼✱ ✏❋✐①❡❞ ❥♦❜s s❝❤❡❞✉❧✐♥❣ ♦♥ ❛ s✐♥❣❧❡ ♠❛❝❤✐♥❡ ✇✐t❤
r❡♥❡✇❛❜❧❡ r❡s♦✉r❝❡s✧✱ ▼■❙❚❆✬✷✵✶✼✱ ❑✉❛❧❛ ▲✉♠♣✉r✱ ▼❛❧❛②s✐❛✱ ♣♣✳ ✶✲✾✳
266

Integrating case-based analysis and fuzzy programming
for decision support in project risk response
Yao Zhang, Fei Zuo, and Xin Guan
Department of Operations and Logistics Management, School of Business Administration,
Northeastern University, Shenyang, China
yzhang@mail.neu.edu.cn, zfsy30@163.com, guanxin1016@126.com
Keywords: project risk management, Case-based, risk response action (RRA), fuzzy math-
ematical programming, decision support system.
1 Introduction
Project performance is constantly affected by risks, and thus the operation of effective
project risk management (PRM) is significant for the success of the whole project. In
general, PRM consists of three phases: risk identification, risk assessment and risk response.
Though the three processes are of equal importance to the success of PRM, risk response
is always considered to have dir

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