Schumann, a modeling framework for supply chain management under uncertainty

European Journal of Operational Research 119 (1999) 14±34
www.elsevier.com/locate/orms

Case Study

Schumann, a modeling framework for supply chain management
under uncertainty
a,b,*
L.F. Escudero , E. Galindo a, G. Garc a, E. Gmez a, V. Sabau
õa o a

a
IBERDROLA Ingenier y Consultor Avda. de Burgos 8b, 28036 Madrid, Spain
õa õa,
b
DEIO, Mathematical Science School, Universidad Complutense de Madrid, Madrid, Spain
Received 1 October 1998; accepted 1 October 1998

Abstract

We present a modeling framework for the optimization of a manufacturing, assembly and distribution (MAD)
supply chain planning problem under uncertainty in product demand and component supplying cost and delivery time,
mainly. The automotive sector has been chosen as the pilot area for this type of multiperiod multiproduct multilevel
problem, but the approach has a far more reaching application. A deterministic treatment of the problem provides
unsatisfactory results. We use a 2-stage scenario analysis based on a partial recourse approach, where MAD supply
chain policy can be implemented for a given set of initial time periods, such that the solution for the other periods does
not need to be anticipated and, then, it depends on the scenario to occur. In any case, it takes into consideration all the
given scenarios. Very useful schemes are used for modeling balance equations and multiperiod linking constraints. A
dual approach splitting variable scheme is been used for dealing with the implementable time periods related variables,
via a redundant circular linking representation. Ó 1999 Elsevier Science B.V. All rights reserved.

Keywords: Supply chain planning; Stochastic parameters; Implementable periods; Scenario analysis; Nonanticipativity
principle; 2-stage decision making

1. Introduction of the ®eld is to build and solve e€ectively realistic
mathematical models of the situation under study,
Decision making is inherent to all aspects of allowing the decision makers to explore a huge
industrial, business and social activities. In all of variety of possible alternatives. As reality is com-
them, dicult tasks must be accomplished. One of plex, many of these models are large (in terms of
the most reliable decision support tools available the number of decision variables), and stochastic
today is Optimization, a ®eld at the con¯uence of (there are parameters whose value cannot be
Mathematics and Computer Science. The purpose controlled by the decision maker and are uncer-
tain). The last fact makes the problems dicult to
tackle, yet its solution is critical for many leading
*
Corresponding author. Tel.: +34 1 383 31 80; fax: +34 1 383 organizations in ®elds such as supply chain plan-
33 11; e-mail: leb@uitesa.es ning among many other areas.

0377-2217/99/$ ± see front matter Ó 1999 Elsevier Science B.V. All rights reserved.
PII: S 0 3 7 7 - 2 2 1 7 ( 9 8 ) 0 0 3 6 6 - X

L.F. Escudero et al. / European Journal of Operational Research 119 (1999) 14±34 15

Manufacturing, Assembly and Distribution available to the decision maker when the decision
(MAD) Supply Chain Management is concerned must be made. MAD supply chain planning ap-
with determining supply, production and stock plications, such as those that this work deals
levels in raw materials, subassemblies at di€erent with, exhibit uncertain product demand as well as
levels of the given Bills of Material (BoM), end uncertain procurement and production availabil-
products and information exchange through ity, supply costing and lag time and others. Ad-
(possibly) a set of factories, depots and dealer ditionally, the problem has a large-scale nature
centres of a given production and service network that makes it dicult, even in its deterministic
to meet ¯uctuating demand requirements. If re- version.
sources can be acquired as needed and plant ca- The aim of this work is to present a novel
pacity is in®nitely expandable and contractible at modeling approach for the MAD supply chain
no cost, then the optimal production schedule planning optimization problem under uncertainty
consists of producing end products according to for very large-scale instances. Although the scheme
the demand schedule, and producing and trans- has been primarily designed for tackling MAD
porting subassemblies exactly when needed as in- supply chain planning problems in the automotive
put to the next assembly process. However, in sector, the approach has a far more reaching ap-
many supply chain systems, the supply of some plication to the very broad supply chain area that
raw materials is tightly constrained, with long deals with multiperiod, multiproduct and multi-
production and/or procurement lead times. The level types of problems in manufacturing, assem-
demand for products ¯uctuates, both in total bly and distribution.
volume and in product mix. As a result, just-in- The paper is organized as follows. Section 2
time production is not usually feasible, and when presents the MAD supply chain planning problem
feasible, may result in poor utilization of the sup- to solve. Section 3 gives the notation and the
ply chain. Four key aspects of this problem are meaning of the main parameters and variables.
identi®ed as time, uncertainty, cost and customer Section 4 presents a concept-oriented mathemati-
service level. In these circumstances, the supply cal representation of the model. Section 5 intro-
chain management optimisation consists of de- duces our modeling framework to treat the
ciding on the best utilization of the available re- uncertainty via scenario analysis. Sections 6 and 7
sources in suppliers, factories, depots and give the parameters and variables as well as the
dealerships given the di€erent scenarios for the implementation-oriented mathematical represen-
stochastic parameters along the planning horizon. tation of the deterministic equivalent model for the
Problems with the characteristics given above stochastic version of the problem.
are transformed into mathematical optimization
models. Often there are tens of thousands of
constraints and variables for a deterministic sit- 2. Problem description
uation. The problems can be modeled as large-
scale linear programs. Given today's Operations 2.1. Current state-of-the-art
Research state-of-the-art tools, deterministic lo-
gistics scheduling optimization problems should A global multinational player (e.g., in the au-
not present major diculties for not very large- tomotive sector) would ideally like to take business
scale problem solving, at least. However, it has decisions which span sourcing, manufacturing,
long been recognized (Beale, 1955; Dantzig, 1955) assembly and distribution. Thus, a company with
that traditional deterministic optimization is not multiple suppliers at di€erent levels of the BoM
suitable for capturing the truly dynamic behavior production plants and multiple markets may seek
of most real-world applications. The main reason to allocate demand quantities to di€erent plants
is that such applications involve data uncertain- over the next month, next quarter or next year
ties which arise because information that will be time horizon. Its objective is to minimize the sum
needed in subsequent decision stages is not of manufacturing, assembly and distribution sup-

16 L.F. Escudero et al. / European Journal of Operational Research 119 (1999) 14±34

plying costs associated with satisfying customer The Supply Chain Management problems have
demands. Alternatively, if product mix is allowed been cast in the form of deterministic mathe-
to vary, the company may seek to maximize net matical optimisation models and many real in-
revenue and/or market share among others. Very stances have been computationally solved. But
frequently the implication for planning is to min- designing and implementing a sound generic
imize work-in-progress across the supply network. model which closely couples the strategic plan-
The plants and selected suppliers and dealers are ning and tactical logistic decisions as well as
described as particular entities with respect to: its captures the time phasing and the uncertainty
direct, indirect and overhead costs; its resources elements of the supply chain remains a challeng-
including machine capacities, labour and raw ing task. See in Escudero (1994b) and Escudero
materials; and its recipes for producing products and Kamesam (1995) and Escudero et al. (1993) a
from raw materials and other resources. A model previous work on modeling the supply chain
for decision support should capture the ¯ow of management optimisation under uncertainty. It is
products and information from the plants through based on a scenario approach that uses the non-
depots to the markets. anticipativity principle (Rockafellar and Wets,
Traditional MAD Supply Chain Management 1991; Wets, 1989) and it is very amenable for
optimisation models develop production plans decomposition schemes, see Escudero et al. (to
that minimize material procurement, inventory appear), Escudero and Salmern (1998), Vladi-
o
holding and labor costs given time varying de- mirou (to appear) and Vladimirou and Zenios
mands. See in Afentakis et al. (1984), Dzielinski (1997) among others. See also the contribution to
and Gomory (1965), Florian and Klein (1971) and the subject made by Baricelli et al. (1996) and
Lasdon and Terjung (1971) examples of these Escudero (1994a).
models where the demand is assumed to be known Finding the right decision support tools is one
or modelled deterministically; see also Goyal and of the most technologically challenging problems
Gunasekaran (1990) for a good survey and addi- that operators and decision makers face today.
tional references. Several approaches, based on di€erent mathe-
A capacity planning system is presented in matical methods, are being pursued with the same
Kekre and Kekre (1985) to explicitly model the aim of optimizing part of or the full problem of
work-in-progress and lead times and to combine it supply chain planning. Stochastic optimization via
with a discrete time mathematical programming scenario analysis is a powerful methodology that
model with deterministic time varying demands. A we propose for the MAD supply chain problem
tactical planning model has been suggested in solution; see Alvarez et al. (1994), Birge (1985),
Graves and Fine (1988) to evaluate capacity Birge et al. (1996), Birge and Louveaux (1988,
loading under varying demand conditions. The 1997), Dempster and Gassmann (1990), Dempster
interrelations between capacity loading, produc- and Thompson (to appear), Escudero et al. (1993),
tion lead times and work-in-progress have been Escudero and Salmern (1998), Gassmann (1990),
o
highlighting in Karmarkar (1987, 1989). See in Kall and Wallace (1994), Mulvey et al. (1995),
Eppen et al. (1989) an excellent discussion on ca- Mulvey and Vladimirou (1991), Mulvey and
pacity planning based on a scenario approach, but Ruszczynski (1992), Ruszczynski (1993), Van
the emphasis is on longer range decisions regard- Slyke and Wets (1969), Wets (1988, 1989) among
ing facility selection for manufacturing. Models others. At this point in time, we know of no suc-
for global chain management optimisation in cessful system that has been developed to solve the
manufacturing have been presented in Cohen and type of real-life problems as described in this work.
Lee (1989), Cohen and Moon (1991) and Shapiro The available sequentially based alternatives can
(1993). Finally, see in Cheng and Miltenburg solve the deterministic version of the full problem
(submitted) a motivation for a hierarchical ap- for a given scenario, or a stochastic version in-
proach to the production planning of BoM, given volving a very small number of scenarios. Even the
its complexity and large-scale dimensions. deterministic version very frequently does not treat


the problem as a whole, given the dimensions and is the available volume at the end of a given time
complexity of the problem. period.
The aim of this work is the study of the ap- The cycle time (i.e., lead time) of a product is
proach which today appears most promising, the set of (consecutive and integer) time periods
based on stochastic optimisation, since interesting that are required for its completion, from the re-
results have already been obtained, either on a lease of the product in the assembly line until it is
small scale or on parts of the problem, exploiting available for use.
the decomposable nature of the representation of The Bill of Materials (BoM) of a product is the
the stochastic model. In this regard it is important structuring of the set of components that are re-
to note that the mathematical procedures of choice quired for its manufacturing/assembly. Note: A
to handle the uncertainty in the optimisation subassembly is a product that belongs to the BoM
model, namely the use of Augmented Lagrangian of some other product(s). On the contrary, a raw
and Benders Decomposition schemes, are partic- component is not assembled by the network and,
ularly well suited for adaptation to a distributed or then, it is supplied from outside sources only.
parallel computation environment. See Escudero A production period is a time period in the
and Kamesam (1993, 1995), Escudero et al. (to product's cycle time. We assume that each com-
appear), Escudero and Salmern (1998) and Higle
o ponent in a BoM is only required in one speci®c
and Sen (1996) among others. production period (e.g., the ®rst week from a two-
week cycle time).
There are two components' supply modes,
2.2. Problem elements namely, standard and expediting modes. The
supply of raw components by using the standard
A planning horizon is a set of (consecutive and mode has a maximum volume allowed per time
integer) time periods of non-necessarily equal period. If more supply is needed it is possible to
length. An end product is the ®nal output of the use the expediting mode by paying an added pen-
manufacturing network. A subassembly is a alty. Note: Procurement is the standard and ex-
product that is assembled by the manufacturing pediting supply modes for raw components.
network and, together with other components, is Subassemblies can also be supplied either by the
used to produce another product. (External de- standard mode (i.e., in-house production) or by
mand and/or procurement for subassemblies is the expediting mode (e.g., third parties, extra
also allowed). By the term product we will refer shifts). In the second case an added penalty must
to both end products and subassemblies. Its be paid. Note: Production is the standard supply
BoM is a concern of the system decision-making. mode and procurement is the expediting supply
Let us use the term component to describe any mode for subassemblies.
part number (i.e., a raw component or a subas- High-tech products are subject to design and
sembly) that is required for the production. We engineering changes and, then, the set of compo-
will name raw component to a component whose nents used in a product (e.g., its BoM) may change
BoM is not a concern of the system decision- during the planning horizon. An E€ective Periods
making (i.e., the supply is only from outside Segment (EPS) of a component in a given product
sources). A transferable component is a compo- is a set of (consecutive and integer) time periods
nent whose available volume at the end of any de®ned by the earliest period and the latest period
time period can be transferred to the next one where the component can be used in the given
(e.g., materials, subassemblies). A non-transfer- product. Engineering changes (EC) are the most
able (raw) component is a component whose frequent reason for having an EPS that is smaller
volume that is not used in a given time period than the length of the planning horizon. Note that
cannot be transferred to the next one (e.g., en- the avoidance of assembling a product with ob-
ergy, machine and labour time, etc.). The stock solete components does not prevent its use for
of a product or a (transferable) raw component satisfying external demand or as a component in


other BoM's. The using of obsolete products can cation of the alternate components mechanism is
be prevented by forcing a zero stock at the ap- the piecewise representation of convex production
propriate time period. A Mandatory E€ective Pe- cost functions.
riods Segment (MEPS) of a component in a given The model should allow to assign products to
product does not allow to assemble the compo- so-called product groups. An aggregate capacity
nent, nor using the old (obsolete) product. Note 1: constraint (weighted Product Going Rate, PGR)
An EC is mandatory if the product with the old for any group (usually, a manufacturing line) can
technology is not allowed to be used after the EPS, be considered per time period. So-called raw
even if the product was assembled in advance. component groups are also allowed, such that the
Note 2: A mandatory EC's must be performed in total amount that is used per group and time pe-
cascade up to end products. riod can be bounded; as an illustration, raw com-
Let us de®ne backlog of a product at the end of ponents from the same supplier or from the same
a time period as the (non-negative) di€erence be- geographical area can be handled by using this
tween the cumulated demand and shipment up to type of functionality.
that period. Multiple external demand sources for Note that the concept of non-transferable (raw)
a product (either end product or a subassembly) components allows to consider resources such as
are allowed. machine capacity, tool and manpower availability,
A practical model should allow the replacement etc. The following data can be given for each
of some components in a BoM by using other product and resource: unit usage, production pe-
components. Let us term prime component to the riod, EPS (representing, e.g., equipment change-
component listed in a BoM. The other compo- over), alternate resources, etc.
nents that can be used (in the standard mode) will Note that single-level production requires that
be termed alternate components. Multiple alter- the components are assembled sequentially along
nates for a prime component are allowed. The the cycle time of the product. On the contrary, a
alternates for a prime component may depend multilevel production allows that subsets of com-
upon both the component's and the product's ponents to be assembled independently and, then,
costs and availability. Each alternate has its own the production resources can be better utilized.
unit usage, e€ective periods segment, procurement/
production cost and fallout. In any case, a prime
subassembly can be substituted by another sub- 3. Data representation
assembly or by a raw component. A raw compo-
nent can be substituted by another raw component 3.1. Sets
only. Note that in-house production and vendor
sourcing can be modeled by using the alternate T set of time periods in the planning
mechanism. horizon
As an illustration, see that the decision on the J set of products
procurement of a raw component from di€erent JE t set of end products
suppliers, to di€erent depots and the supplying of JS t set of subassemblies (JE ’ JS ˆ £
a subassembly from di€erent depots can be mod- and t ˆ JE ‘ JS)
eled by using the alternate components mecha- JSD JS set of subassemblies with external
nism. Additionally, the utilization of the capacity demand
of non-dedicated manufacturing lines can be PG set of product groups
modeled by using the concept of alternate non- th t set of products that belong to
transferable (raw) components. See that the release group h, for h P PG
of a given product can be diverted to di€erent DSj set of (external) demand sources for
lines, each line has its own capacity (e.g., time product j, for j P JE ‘ JSD, such
availability), and a processing time is given for that hƒj ’ DSjH ˆ £ for j,
each product and line. Another interesting appli- jH P JE ‘ JSD j j Tˆ jH


DS ‘ DSj for all j P JE ‘ JSD oiYj o€set of component i in the cycle
I set of components time of product j, for i P sj Y j P t . It
IR s set of raw components, such that gives the number of time periods
IR ’ JS ˆ £ and s ˆ IR ‘ JS before the completion of product j
IT IR set of transferable raw components whose BoM includes component i,
(Note: IR À IT is the set of non- such that oiYj ™j À piYj
transferable raw components) aiYj amount of prime component i that
RG set of raw component groups is needed per unit of product j, for
IRh IR set of raw components that belong i P sj Y j P t
to group h, for h P RG ciYj fallout of prime component i in the
sj s set of prime components in the BoM of product j
BoM of product j, for j P t aiYj net amount of prime component i
s iYj s set of components that are alter- that is needed per unit of product j,
nates to prime component i in the for i P sj j P t , where
Y
BoM of product j, for i P s, j P t . —iYj ˆ aiYj …1 À 0X001ciYj †
ls
i number of time periods to supply
raw component i to its depot by
3.2. Constraint related parameters
using the standard mode, for i P IR
le number of time periods to supply
Demand i
component i to its depot by using
DdYt demand from external source d at time the expediting mode, for i P I (Note
period t, for d P DSY t P „ 1: The order should be performed ls i
MBdYt maximum backlog on demand source d or le time periods in advance to the
i
that is allowed at time period t, for time period where the component is
d P DSY t P „ required; Note 2: It is assumed that
ld delivery lag time, i.e., number of time ls ˆ 0 and le ˆ 0 for i being any
i i
periods to deliver the end product or non-transferable (raw) component)
subassembly with external demand d liYj number of time periods required to
after its completion, for d P DS deliver component i from its depot
dd lost (expected) demand fraction of non- for product j, for i P sj Y j P t and
served cumulated demand at any time i P s f Yj Y f P sj Y j P t
…À† …‡†
period for external demand d, for siYj Y siYj earliest time period and latest time
d P DS period, respectively (i.e., e€ective
qd unit lost demand penalty at any time periods segment), where prime
period for external demand d, for component i or any of its alternates
d P DS can be used in the BoM of product
rdYt unit backlog weight at time period t for j, for i P sj Y j P t
…À† …‡†
demand source d, for d P DSY t P „ (Note: piYj T siYj T siYj T j„ j À oiYj )
Bill of Material (BoM)
MEPSiYj ¯ag for the mandatory e€ective
cj cycle time of product j, for j P t periods segment of prime compo-
piYj production time period in the cycle nent i in the BoM of product j, for
time of product j where prime i P sj Y j P t , such that MEPSiYj ˆ 1
component i is needed, for if the ¯ag is on and, otherwise, it is 0
iYj
i P sj Y j P t (Note 1: 1 T piYj T ™j ; bf amount of component f that is
Note 2: Alternate components have needed per unit of product j pro-
the same production period as the vided that it substitutes prime com-
related prime component) ponent i, for f P s iYj , i P sj Y j P t


uiYj
f fallout of alternate component f (1 À ujYt ) yield of product j provided that it
for prime component i in the is made available at time period t
BoM of product j for 0 ` ujYt T 1, for j P t Y t P „
iYj
˜f net amount of component f that MPhYtHH , mPhYt maximum and minimum release
is needed per unit of product j, capacity that is allowed for
provided that it substitutes prime product group h at time period t,
component i, for for t P „ Y h P PG
f P s iYj Y i P sj Y j P t , where VjYt external input volume for prod-
˜iYj ˆ bf ˆ a…1 À 0X001uf †
f
iYj iYj
uct or (transferable) raw compo-
iYj…À† iYj…‡†
sf Y sf earliest time period and latest nent j, at (the beginning of) time
time period, respectively, where period t, for j P t ‘ s„ Y t P „ .
component f can substitute prime Note: The volume V for the ®rst
component i in the BoM of time periods can represent the
product j, for availability of the product or
f P s iYj Y i P sj Y j P t . (Note: (transferable) raw component
…À† iYj…À† iYj…‡† …‡†
siYj T sf T sf T siYj ) that was a work-in-progress at
the beginning of the time horizon.
Technological constraints On the other hand it can repre-
sent the initial stock for period
ujYh capacity usage of resources allocated to tˆ1
product group h per unit of product j, MSjYtH , mSjYt maximum and minimum volume
for j P th Y h P PG of product or (transferable) raw
riYh unit usage of raw component i in group component j that is allowed to
h, for i P IRh Y h P RG keep in stock at (the end of) time
period t, for j P t ‘ ITY t P „ .
Components availability Note: MSjYsj ˆ mSjYsj ˆ 0 for
…‡†
sj ˆ siYj ‡ oiYj Y i P sj Y j P t and
MXiYtH maximum amount of raw com- MEPSiYj ˆ 1; as a consequence,
ponent i that can be ordered MSlYsl ˆ mslYsl ˆ 0 where l P J
from outside sources at (the is any node in the tree whose root
beginning of) time period t by node is component i and the
using the standard mode, for leaves are the end-products
i P IRY t ˆ 1Y F F F Y j„ j À lsi whose BoM (given by prime
MEiYt maximum amount of compo- components) include (directly or
nent i that can be ordered from indirectly) component
outside sources at (the begin- iY sl ˆ sk ‡ okYl and k is a node in
ning of) time period t by using the tree such that k P sl . Note:
the expediting mode, for The zero stock ®xing process is
i P sY t ˆ 1Y F F F Y j„ j À le
i
…À†
interrupted whenever skYl b sk
MRhYt , mRhYt maximum and minimum usage
that is allowed for raw compo-
3.3. Cost function related parameters
nent group h at time period t,
for h P RGY t P „ PCjYt unit production cost for product j
Production and stock levels and time period t, for j P t , t P „ .
Note: No procurement cost is
MZjYt maximum release volume that is included for the components in its
allowed for product j at time BoM; it is already considered
period t, for j P t Y t P „ below


iYj iYj
PAf Yt unit extra production cost for ZAf Yt volume of product j that is made
product j and time period t, due to available at (the end of) time period t,
a substitution of prime component by using the (production) standard
i by alternate component f, for mode, whose prime component i is
f P s iYj Y i P sj Y j P t Y t P „ . Note: substituted by alternate component f,
No procurement cost is included for f P s iYj , i P sj , j P t , t P „ . Note 1:
iYj
for the alternate component; it is ZPtiYj , ZAf Yt do not exist for s iYj ˆ £
considered below where i P sj , j P t . Note 2: ZPiYj , t
HjYt unit holding cost for product or ZAiYj T ZjYt for s iYj Tˆ £, i P sj , j P t ,
f Yt
(transferable) raw component j tP„
and time period t, for j P t ‘ IT, XiYt volume of raw component i that is
t P „ . Note: This element can also ordered at (the beginning of) time
be used for discouraging building- period t by using the (procurement)
ahead in case of tie, by considering standard mode, for i P IR, t P „
HjYt as a stock penalization; in this EiYt volume of (raw or subassembly) com-
case, a typical value can be ponent i that is ordered at (the begin-
10 ´ 10À4 ning of) time period t by using the
SCs Y SCe
iYt iYt unit procurement cost for raw (procurement) expediting mode, for
component i and time period t by i P s, t P „
using the standard and expediting YdYt volume of served demand for external
modes, respectively, for i P IR, source d that is being shipped at (the
tP„ end of) time period t, for d P DS, t P „
SCe
iYt unit procurement cost for subas- LdYt lost demand from external source d at
sembly i and time period t (by time period t, for d P DS, t P „
using the expediting mode), for BdYt backlog volume for external demand
i P JS, t P „ source d at (the end of) time period t,
for d P DS, t P „
SjYt volume of product or (transferable) raw
component j to keep in stock at (the end
3.4. Optimization variables of) time period t, for j P t ‘ IT, t P „

ZjYt volume of product j that is made
Note. For the deterministic model (see Sec-
available at (the end of) time period t,
tion 4), the X- and E-variables may also refer to
by using the (production) standard
the (beginning of) time period where the compo-
mode for shipment, production or
nent is available for depot. However, in the sto-
stock, for j P t, t P „ . Note: The end
chastic setting (see Section 7) the related time
products are made available only for
period is important, since the component's envi-
shipment or stock. The subassemblies
ronment may di€er from the ordering time period
are made available for production,
up to the receiving time period.
shipment (if there is external demand)
or stock.
ZPtiYj volume of product j that is made
available at (the end of) time period t, 3.5. Lag indices and operators
by using the (production) standard
mode, whose prime component i was The X-, E-, Z-, ZP-, ZA- and Y-variables' e€ect
not substituted by any alternate com- on latter time periods to the indexed period can be
ponent, although it could be for taken into account by instrumenting the following
i P sj Y j P t Y t P „ . Note: s iYj Tˆ £ identities to be used for time period s in the model


below, for s P „ . (Note: This e€ect is due to the l- using prime and alternate components in the
lag time interval among other reasons.) products' BoM), and the procurement standard
and expediting costs as well as lost demand pe-
s s s À ls
i Vi P IR nalization.
…procurement standard mode†Y ˆˆ
zÃ ˆ min
1 PCjYs jYs
jPt sP„
e
s sÀ le
i Vi P s ˆˆˆ ˆ
‡ c3 PAf Ys ZAiYj
iYj
f Ys
…procurement expediting mode†Y jPt sP„ iPsj f Ps iYj
ˆˆ ˆ ˆ
r ‡ SCs ˆiYs
iYs ‡ SCe iiYs
iYs
s s À ™j ‡ 1 Vj P t …product release†Y iPIR sˆ1Y„ s iPs sˆ1Y„ e
ˆˆ
sp s ‡ liXj ‡ oiYj Vi P sj Y j P t ‡ qd vdYs Y …1†
dPDS sP„
…prime component supply†Y
where c3 is given in Section 3.5, and
a f Yj
s s ‡ liYj ‡ of Yj Vi P s Y f P sj Y j P t
„ s j„ j À ls Y
i „ e j„ j À le
i …2†
…alternate component supply†Y
subject to Eqs. (5)±(24).
d
s s À ld Vd P DS Option 2: Optimizing the market opportunities
by minimizing the maximum weighted product
…external demand delivery†X backlog.

Let c1 Y c2 Y c3 denote the operators for zÃ ˆ min zaz P rdYs fdYs
2 Vd P DSY s P „ Y …3†
j P t Y s P „ , such that
…À† …‡†
1 for siYj T s ‡ liYj T siYj Y Option 3: Optimizing the market opportunities
c1
0 otherwiseY by minimizing the total weighted product backlog
and lost demand penalization.
for i P sj ,
ˆˆ
f Yj…À† f Yj…‡† zÃ ˆ min
3 ‰rdYs fdYs ‡ qd vdYs Š …4†
c2 1 for si Ts ‡ liYj T si Y dPDS sP„
0 otherwiseY
for i P s f Yj Y f P sj ,
f Yj…À† f Yj…‡†
c3 1 for si T s À of Yj T si Y
4.2. End-product balance equations
0 otherwiseY
for i P s f Yj Y f P sj . The balance equations for the production,
shipment and stock of end products are given
by the constraints (5). These constraints state that,
for given end product and time period, the stock
4. A concept-oriented deterministic model volume at the end of the previous period plus the
external input volume at the beginning of the
4.1. Objective functions current period plus the net production volume that
has been completed at the given period must
Option 1: Optimizing the system resources us- equate the shipment volume plus the stock volume
age by minimizing the total production costs (by at the end of the current period.

ˆ
ƒjYsÀ1 ‡ †jYs ‡ ujYs jYs ˆ ‰dYs ‡ ƒjYs lowing constraint is required for the subassembly's
dPDSj ¯ow at the beginning of the given time period.
Vj P JEY s P „ X …5† ƒiYsÀ1 ‡ †iYs ‡ iiYse
ˆ
Note. ƒjYsÀ1 does not exist for s ˆ 1 given †jYs . À c1 —iYj jYsp
jPt aiPsj Xs iYj ˆ£
ˆ iYj
À c1 —iYj ZPsp
4.3. Subassembly balance equations
jPt aiPsj Xs iYj Tˆ£
ˆ
The balance equations for the production, À c2 ˜f Yj ZAf Yja P 0
i iYs
shipment (internal as well as external demand f Psj YjPt XiPs f Yj

satisfaction) and stock are given by the constraints Vi P JSY s P „ X …7†
(6). These constraints state that, for given subas-
sembly and time period, the stock volume at the
end of the previous period plus the external input
volume at the beginning of the current period plus 4.4. External demand balance equations
the subassembly supply volume at the beginning of
the current period by using the expediting mode The balance equations for the external demand
plus the net production volume at the end of the serviceability are given by the constraints (8).
current period must equate the component's vol- These constraints state that, for given external
ume that is shipped at the beginning of the current demand source and time period, the backlog vol-
period to satisfy internal needs given by the BoM ume at the end of the previous time period plus the
of other products plus the shipment volume at the demand volume at the given period must equate
end of the current period to satisfy external de- the shipment volume to satisfy the external de-
mand plus the stock volume at the end of the mand plus the lost (non-served) demand volume
current period. during the period plus the backlog volume at the
end of the given period.
ƒiYsÀ1 ‡ †iYs ‡ iiYse ‡ uiYs iYs
ˆ ˆ iYj
fdYsÀ1 ‡ hdYs ˆ ‰dYsd ‡ vdYs ‡ fdYs
ˆ c1 —iYj jYs p ‡ c1 —iYj ZPsp
Vd P DSY s P „ Y …8†
jPt aiPsj Xs iYj ˆ£ jPt aiPsj Xs iYj Tˆ£
ˆ ˆ
‡ c2 ˜f Yj ZAf Yja ‡ ‰dYs ‡ ƒiYs where sd is given in Section 3.5, such that the lost
i iYs
f Psj YjPt XiPs f Yj dPDSi demand vdYs can be expressed.
À Á
Vi P JSY s P „ Y …6† vdYs ˆ dd fdYsÀ1 ‡ hdYs À ‰dYsd P 0X …9†
where, se , sp ,sa , c1 , c2 are given in Section 3.5.
Note 1. c1 ˆ 0 does not allow to use prime
component (subassembly) i in the BoM of product 4.5. Raw component balance equations
j. It is the case where time period s does not belong
to the E€ective Periods Segment (EPS) for using The balance equations for the procurement,
subassembly i in the BoM of product j. utilization and stock of (transferable and non-
Note 2. c2 T c1 . See that c2 ˆ 0 does not allow transferable) raw components are given by the
alternate component i to substitute prime com- constraints (10). These constraints state that, for
ponent f in the BoM of product j, due to EPS given component and time period, the stock vol-
reasons. ume (only for transferable components) at the end
of the previous time period plus the external input
The balance equations as given by the con- volume at the beginning of the current period
straints (6) do not consider the synchronisation of (only for transferable components) plus the sup-
the stock balance's input and output. So, the fol- ply volume at the beginning of the current period

P
(by using both the standard and expediting ˆ ˆ
modes) must equate the component's volume that mRhYs T R c1 riYh —iYj zjYsp
is shipped at the beginning of the current period iPIRh jPt aiPsj Xs iYj ˆ£
to satisfy internal needs given by the BoM of ˆ iYj
related products plus the stock volume (only for ‡ c1 riYh —iYj ZPsp
jPt aiPsj Xs iYj Tˆ£
transferable components) at the end of the given
Q
period.
ˆ
‡ c2 riYh ˜f Yj ZAf Yja S T MRhYs
i iYs
ƒiYsÀ1 ‡ †iYs ‡ ˆiYss ‡ iiYse f Psj YjPt XiPs f Yj
ˆ
ˆ c1 —iYj jYsp
jPt aiPsj Xs iYj ˆ£
Vh P RGY s P „ Y …12†
ˆ iYj
‡ c1 —iYj ZPsp where, sp , sa , c1 , c2 are given in Section 3.5.
jPt aiPsj Xs iYj Tˆ£ The maximum and minimum release capacity
ˆ
‡ c2 ˜f Yj ZAf Yja ‡ ƒiYs that is allowed for a given product group at a given
i iYs
f Psj YjPt XiPs f Yj time period can be expressed.
Vi P IRY s P „ X …10† ˆ
mPhYsr T ujYh jYs T MPhYsr Vh P PGY s P „ Y
jPth
where ss , se , sp , sa , c1 , c2 are given in Section 3.5.
Note. †iYs Y ƒiYs do not exist and ss ˆ se ˆ s for i P …13†
IR À IT (i.e., non-transferable components).
where sr is given in Section 3.5.

4.6. Prime component substitution balance equa-
4.8. Bounds on the variables
tions
Maximum product release volume
For a product to be made available at a given
time period and each of its prime components with 0 T jYs T MZjYsr Vs ˆ ™j Y F F F Y j„ jY i P t Y …14†
substitution capabilities in the related BoM, these
constraints state that the product volume where where sr is given in Section 3.5.
the prime component is not substituted plus the
…À† …‡†
product volume where it is substituted by other 0 T ZPiYj
s Vs ˆ siYj ‡ oiYj Y F F F Y siYj ‡ oiYj Y
(alternate) components must equate the product
i P sj as iYj Tˆ £Y j P t Y …15†
volume to be made available at the given period.
ˆ iYj
iYj iYj…À† iYj…‡†
jYs ˆ ZPs ‡ c3 ZAf Ys Vs iYj
0 T ZAf Ys Vs ˆ sf ‡ oiYj Y F F F Y sf ‡ oiYj Y
f Ps iYj iYj
…À† …‡†
f P s Y i P sj Y j P t X …16†
ˆ siYj ‡ oiYj Y F F F Y siYj ‡ oiYj Y
i P sj as iYj Tˆ £Y j P t Y …11† Maximum component procurement volume

where c3 is given in Section 3.5. 0 T ˆiYs T MXiYs Vs ˆ 1Y F F F Y j„ j À ls Y i P IT
i
Vs ˆ Xi Y F F F Y Ci Y i P IR À ITY …17†

4.7. Components and products group bounding
0 T iiYs T MEiYs Vs ˆ 1Y F F F Y j„ j À le Y
i
The maximum and minimum usage that is al-
i P JS ‘ IT Vs ˆ Xi Y F F F Y Ci Y
lowed for a given raw component group at a given
time period can be expressed. i P IR À ITY …18†


where 4.10. An implementation-oriented variables reduc-
n o tion scheme
…À†
X1 ˆ min min siYj À liYj Y
jPt aiPsj
Given the dimensions of the deterministic
n o' model (1)±(24) and the type of its constraints (6),
f Yj…À†
min si À liYj Y …19†
f Psj YjPt aiPs f Yj (7), (9)±(11), one can need to reduce the number of
variables. In particular, the variables ƒjYs Vj P
n o iYj
…‡†
t ‘ IT, vdYs Vd P DS and ZPs Vi P sj as iYj Tˆ £Y j P
Ci ˆ max max siYj À liYj Y t for all s P „ are not explicitly required to solve
jPt aiPsj
n o' the model. As an example, let the following generic
f Yj…‡†
max si À liYj X …20† system:
f Psj YjPt aiPs f Yj

Maximum and minimum product and (trans- † 1 ‡ e 1 ˆ1 ˆ ƒ 1 Y
ferable) raw component stock volume † 2 ‡ ƒ 1 ‡ e 2 ˆ2 ˆ ƒ 2 Y
…25†
† 3 ‡ ƒ 2 ‡ e 3 ˆ3 ˆ ƒ 3 Y
mSjYs T ƒjYs T MSjYs Vs P „ Y j P t ‘ ITX …21†
mSt T ƒt T MSt Y t ˆ 1Y 2Y 3Y
Maximum product backlog, and nonnegative
character of the variables where Xt and St are vectors of variables, Vt is a
vector of data, At a conformable constraint matrix
0 T fdYs T MBdYs Vs P „ Y d P DSY …22†
and mSt and MSt are vectors of constants for t ˆ 1,
2, 3. It is easy to see that Eq. (25) can be replaced
0 T ‰dYs Vs ˆ 1Y F F F Y j„ j À ld Y d P DSX …23†
by the system
mS1 À †1 T e1 ˆ1 T MS1 À †1 Y
mS2 À †1 À †2 T e1 ˆ1 ‡ e2 ˆ2 T MS2 À †1 À †2 Y
4.9. Preventing unnecessary product and component mS3 À †1 À †2 À †3 T e1 ˆ1 ‡ e2 ˆ2 ‡ e3 ˆ3 T MS3 À †1 À †2 À †3 Y
stock …26†

The following penalization can be included in and, in general,
the objective functions (1), (3) and (4) to prevent ˆ ˆ
es ˆs T MSt À †s Y t ˆ 1Y 2Y 3Y …27a†
unnecessary build-ahead: sˆ1Yt sˆ1Y„
ˆ ˆ
aˆ rjYs ƒjYs X …24† ˆ ˆ
jPt ‘IT sP„ mSt À †s T es ˆs Y t ˆ 1Y 2Y 3Y …27b†
sˆ1Yt sˆ1Yt
See constraints (5), (6) and (10) and note that
(24) discourages building for stock in case of tie. and, so, the S-variables are not required at the
Note. See that model (1)±(24) is an LP repre- price of increasing the constraint matrix density.
sentation of a multiperiod multilevel multiproduct Note 1. Constraint (27a) for a given t is re-
problem. Given the problem's dimensions, the dundant provided that Ws ˆ t ‡ 1Y F F F Y j„ j such
€
current representation may have a very large-scale that MSt P MSs À 5ˆt‡1Ys †5 for t ˆ 1Y F F F Y
size. Anyway the products linking is due to the j„ j À 1. So, the constraints (27a) for the € of time
set
BoMs communality, and the periods linking is due periods ftg such that t: MSt ` MSs À 5ˆt‡1Ys †5
to the transferable raw components and product for all sat ` s are only required.
stock from one period to the next one, the cycle Note 2. Constraint (27b) for a given t is re-
time for each product, the components delivery lag dundant, provided that Ws ˆ 1Y F F F Y t À 1 such that
€
time for products and the products delivery lag time mSs P mSt À 5ˆs‡1Yt †5 for t ˆ j„ jY F F F Y 2. So, the
to external demand sources (in case the lag time is constraints (27b) for the€ of time periods ftg
set
more than one time period). So, the problem's el- such that t: mSs ` mSt À 5ˆs‡1Yt †5 for all sas ` t
ements are very much time period inter-related. are only required.


Assume that et ˆt above can be decomposed as per di€erence between the proposed solution and
the optimal solution value for each scenario. The
et ˆt e1 ˆt1 À e2 ˆt2 Y
t t t ˆ 1Y 2Y 3Y …28† resulting model does not increase the number of
such that variables in the original representation, but now
there are mjqj constraints. Unfortunately, this
†t ‡ ƒtÀ1 À e2 ˆt2 P 0Y
t t ˆ 1Y 2Y 3X …29† representation does not preserve the structure of
It is easy to see that (29) can be presented as fol- the deterministic model (31), and the objective
lows by eliminating St : function is no longer linear; see in Escudero
ˆ ˆ ˆ (1994b) some procedures to overcome this di-
†s ‡ e1 ˆs1 À
s e2 ˆs2 P 0Y t ˆ 1Y 2Y 3X
s culty. Models of this form are known as scenario
sˆ1Yt sˆ1YtÀ1 sˆ1Yt immunization models, or SI models for short, see
…30† Dantzig (1985) and Dembo (1991), Infanger (1994)
and Mulvey et al. (1995) among others.
As an alternative goal, we could minimize the
expected value of the objective function; in this
5. Uncertainty case model (31) becomes
ˆ T
5.1. General approach min wg ™g z
gPq …32†
The model described in the previous section can g
sXtX ez ˆ p Vg P qY z P 0X
be compacted in the following model structuring:
Note that Eq. (32) gives an implementable policy
min ™T z based on the so-called simple recourse scheme.
sXtX ez ˆ pY …31† Note that the whole vector of decision variables is
z P 0Y anticipated at (the beginning of) time period t ˆ 1.
where c is the vector of the objective function co-
ecients, A the m ´ n constraint matrix, p the
right-hand side (r.h.s.) m-vector and z the n-vector 5.2. Non-anticipative policies
of the decision variables to optimise. It must be
extended in order to deal properly with uncer- Let T denote the set of time periods over the
tainty on the values of some parameters. We may time horizon, T1 is the set (so-called ®rst stage) of
employ a technique so-called scenario analysis, the ®rst time periods from set T whose related
where the uncertainty is modelled via a set of parameters are deterministic and „2 ˆ „ À „1 (the
scenarios, say G. Dempster (1988) among others. so-called second stage), where the realizations of
In our MAD Supply Chain problem, the main the parameters related to set T2 are considered via
stochastic parameters are the production and the set of scenarios G; henceforth, this partition
procurement costs as well as the product demand. will be termed a 2-stage time period framework.
It will mean that the vector c of the objective The SI models do anticipate decisions in z that
function coecients and the r.h.s. p in model (31) for the 2-stage environment may not be needed at
should be represented by cg and pg , for, Vg P q, the ®rst stage. Very frequently the decisions for the
respectively. We also introduce a weight, say wg , to ®rst stage (i.e., the MAD supply chain planning
represent the likelihood that the decision maker decisions to be implemented in the time periods
assigns to scenario g, for g P q. from set T1 ) are the only decisions to be made
One way to deal with the uncertainty is to ob- since at the ®rst time period of the second stage
tain the solution x that best tracks each of the (i.e., time period j„1 j ‡ 1) one may realize that
scenarios, while satisfying the constraints for each some of the data has been changed, some scenarios
scenario. This can be achieved by obtaining a so- vanish, etc. In this case, the models will be usually
lution that minimizes a norm of the weighted up- reoptimized in a rolling planning horizon mode.


When only spot decisions (i.e., decisions for the of I is as follows: s ˆ s x ‘ s xy , where Ix will denote
®rst stage) are to be made, the information about the set of constraint blocks where only the x-
future uncertainty is only taken into account for a variables have non-zero coecients, and s xy ˆ s À
better spot decision making. This type of scheme is s x will denote the set of constraint blocks where x-
termed partial recourse for j„1 j b 1; otherwise, it is and y-variables will have non-zero elements. Let ytg
termed full recourse. denote the vector of variables for time period t for
Let zg denote the vector of the variables related
t t P T2 (i.e., the set of time periods with stochastic
to time period t under scenario g for, t P „ , g P q, parameters) under scenario g for g P q, and Tiy
and zg is the set of vectors fzg Vt P „ g.
t denote the set of time periods whose related y-
The so-called non-anticipative principle was in- variables have nonzero coecients in constraint
troduced in Rockafellar and Wets (1991) and Wets block i, for i P s xy , such that Tiy T2 .
(1989) it states that if two di€erent scenarios, say, g Model (34) has a nice structure that we may
and gH are identical up to time period t on the basis exploit. Two approaches can be used to represent
of the information available about them at that the non-anticipativity constraints (33). One ap-
time period, then the values of the z variables must proach is based on a compact representation, where
be identical up to time period t. Eq. (33) is used to eliminate variables in Eq. (34)
In our case, this condition guarantees that the and, so, to reduce model size, such that there is a
solution obtained from the model is not dependent single vector of variables for each time period from
at the ®rst stage on the information that is not yet set T1 , but any special structure of the constraints
available; the time periods from T1 and T2 are in Eq. (31) is destroyed.
termed implementable time periods and non-im- However, given the structure of problem (1)±
plementable time periods, respectively. In order to (24) and the dimensions of its real-life instances,
introduce this condition in our approach, let N the most attractive approach to represent the non-
denote the set of solutions that satisfy the so-called anticipativity constraints (33) is based on a dual
non-anticipativity constraints. That is, splitting variable representation that requires to
n o split the x-variables into the new vector of vari-
H
z P x zg zg ˆ zg VgY gH P qY t P „1 X
t t …33† ables, say, xg Vg P q, such that Eq. (33) can be
t
represented by using the so-called redundant cir-
So, the Deterministic Equivalent Model (DEM) cular linking scheme,
for the partial recourse version of model (31) can
be expressed: xg À xtg‡1 ˆ 0 Vt P „1 Y g P qX …35†
t
ˆ T
min wg ™g zg
(Note: The convention g ‡ 1 ˆ 1 is used for
gPq …34†
g g g
g ˆ jqjX) The splitting variable representation of
sXtX ez ˆ p Vg P qY z P x Y z P 0X model (34) can be expressed as follows:
For the sake of a better analysis of model (34), ˆˆ T ˆˆ T
let us introduce some additional notation. So, let min —g x g ‡
t t ˜g ytg
t
xYy
the z-vector of variables be partitioned into the gPq tP„1
ˆ
gPq tP„2

vectors x and y, and let I denote the set of con- sXtX e1 xg
it t ˆ pi0 Vi P s x Y g P qY
straints in the problem, by taking into consider- i
tP„x
ation the structure exhibited by our deterministic ˆ ˆ
e2 xg ‡ fg ytg ˆ pig Vi P s xy Y g P qY
model (4)±(24) with the modi®cation introduced in i
it t it
tP„x tP„yi
Section 4.10. Let xt denote the vector of variables
for time period t for t P T1 (i.e., the set of time xg À xtg‡1 ˆ 0
t Vt P „1 Y g P qY
periods with deterministic parameters), and „xi xg P0 Vt P „1 Y g P qY
t
denote the set of time periods whose related x-
variables have non-zero coecients in constraint ytg P 0 Vt P „2 Y g P qY
block i, for, i P s, such that Tix T1 . The partition …36†


where —g ˆ wg ™0 Vt P „1 Y g P q; ™0 is the vector of
t t t that a given parameter indexed by time period t
the x-related objective function coecients for has the same value for all scenarios for t P T1 .
time period t for t P „1 ; ˜g ˆ wg ™g is the vector of
t t
the y-related objective function coecients for
time period t for t P „2 under secenario g for 6.2. Lag indices and operators
g P q; e1 and e2 are the x-related constraint ma-
it it
trices for time period t and constraint block i for The X- and E-variables' e€ect on later time
t P „xi , i P s x and t P „xi , i P lxy , respectively; fg is
it periods to the indexed period can be taken into
the y-related constraint matrix for time period t account by instrumenting the following identities
and constraint block i for t P „yi , i P s xy under to be used for time period s under scenario g in the
scenario g for g P q; and pi0 and pig are the r.h.s for model below, for s P T, g P q. (Note: This e€ect is
constraint block i for i P s x and i P s xy under sce- due to the l-lag time interval among other rea-
nario g for g P q, respectively. sons.)
See that the dualization of the non-antic-
ipativity constraints (35) allows jqj independent ts t À lisYg Vi P IR
constraint systems, namely, the subsystem from …procurement standard mode†Y
Eq. (36) related to each scenario. See in Section 7 te t À leYg Vi P s
i
the splitting variable based stochastic version of …procurement expediting mode†X
model (1)±(24) with the modi®cation introduced in
Section 4.10. The operators c1 , c2 , c3 for j P t , s P T have in the
stochastic version of the problem the same type of
equivalences as for the deterministic case (see
6. Parameters and variables for the stochastic Section 3.5), but the superindex g for each sce-
approach nario to occur is also required. Note that the
…À† iYj…À† iYj…‡† …‡†
sij Y sf Y sf Y sij parameters are scenario de-
6.1. Constraint and objective functions related pendent elements and, then, the superindex g is
parameters required.

G is set of scenarios, T1 set of implementable
time periods, and T2 ˆ T À T1 set of non-imple- 6.3. Variables
mentable time periods.
The same type of variables used in the deter-
Remark. It is assumed that all parameters are ministic model are to be used in the stochastic
deterministic (i.e., known values) for time period version (by adding the superindex g to show their
set T1 , but the assumption can be very easily relationship with the scenario to occur), but the
removed. ZP-, L- and S-variables are not required, see the
model below.
The uncertain parameters to be considered in The variables, say x, with time period index, say
the model below are related to the production/ t, from the set T1 of implementable time periods
procurement costs and availability, demand vol- will have the same value under all scenarios. So,
ume and lost fraction, prime and alternate com- xg ˆ xg‡1 Vg P qY t P T1 . On the other hand, a
t t
ponents' e€ective periods segment, product and compact representation of the problem may con-
component external input and stock bounding, sider g ˆ 0 for xg Vt P T1 . In any case, the struc-
t
etc. These parameter types are scenario dependent ture of the implementation-oriented model (see in
elements and, then, the superindex g is required; as Section 4.10 the basic ideas) requires that most of
an example, hg gives the demand from external
dt the variables have non-zero elements in constraints
source d at time period t under scenario g, for related to later time periods; so, either the variables
d P DS, t P T, g P q, etc. On the other hand, note related to set T1 have a copy per scenario (in case of

4
a splitting variable representation) or most of these ˆ ˆ ˆÂ Ã
variables are to be included in the appropriate zÃ ˆ min
3 wg rdYs ‡ qd dg fg
d dYs
gPq dPDS sP„
scenario related constraints for set T2 (in case of a 5
compact representation). ˆ ˆ g
À qd dg ‰dYs
d Y …40†
dPDS sˆ1Y„ d

7. An implementable-oriented 2-stage stochastic where Td as given by Eq. (38), subject to
splitting variable model Eqs. (41)±(67).

7.1. Objective functions
7.2. End-product stock bounding
Option 1: Optimizing the system resource usage
by minimizing the expected total production cost The end-product stock bounding (5) and (21) is
(by using prime and alternate components in the represented by the constraints (41), based on the
products' BoM) and the procurement standard expressions (27). They state that the stock volume
and expediting cost as well as the lost demand for given end-product and time period should be
penalization. appropriately lower and upper bounded for each
4 scenario to consider.
ˆ ˆˆ g
zÃ ˆ min
1 wg PCg jYs
jYs
ˆ g ˆ g
ˆ ˆ g
gPq jPt sP„ mSg À
jYt †jYs T ujYsr jYs À ‰dYs
ˆˆˆ ˆ g g sˆ1Yt sˆ1Yt sˆ1Yt dPDSj
‡ cg PAiYj ZAiYj
3 f Ys f Ys ˆ g
jPt sP„ iPsj f Ps iYj T MSg
jYt À †jYs Vj P JEY t P „ Y g P qX
ˆ ˆ ˆˆ sˆ1Yt
sg g eg g
‡ SCiYs ˆiYs ‡ SCiYs iiYs
iPIR sˆ1Y„ s iPs sˆ1Y„ e
…41†
ˆˆ
‡ qd dg fg
d dYs
See the remarks introduced in Section 4.10.
dPDS sˆ„
5
ˆ ˆ g
À qd dg ‰dYs
d Y …37† 7.3. Subassembly stock bounding
dPDS sˆ1Y„ d

where cg is referred to in Section 6.2, and The subassembly stock bounding (6) and (21) is
3
represented by the constraints (42) and (43), based
„ s j„ j À lisYg Y „ e j„ j À leYg Y
i „ d j„ j À ld on the expressions (27) and (30). These constraints
…38† state that the stock volume for given subassembly
and time period should be appropriately lower and
subject to Eqs. (41)±(67). Note that is a dg hg
d d upper bounded for each scenario to consider.
constant.
Option 2: Optimizing the market opportunities ˆ g
by minimizing the expected maximum weighted mSg À
iYt †jYs
sˆ1Yt
product backlog. ˆ ˆ ˆ ˆ
g g
T iiYse ‡ uiYs iYs À cg —iYj jYsp
1
g

zÃ
2 ˆ min zjz P w rdYs f g gdYs sˆ1Yt sˆ1Yt sˆ1Yt jPt aiPsj
ˆ ˆ À Á g
Vd P DSY s P „ Y g P q …39† ‡ cg
2 —iYj À ˜f Yj ZAf Yja
i iYs
…42†
sˆ1Yt f Psj YjPt aiPs f Yj
subject to Eqs. (41)±(67). ˆ ˆ ˆ
g g
Option 3: Optimizing the market opportunities À ‰dYs T MSg À
iYt †jYs
sˆ1Yt dPDSi sˆ1Yt
by minimizing the expected total weighted product
backlog and lost demand penalization. Vi P JSY t P „ Y g P qY


where sp , sa are given in Section 3.5 and se Y cg Y cg
1 2 where sp , sa are given in Section 3.5 and ss , se , cg ,
1
are referred to in Section 6.2. See the remarks in- cg are referred to in Section 6.2. See the remarks
2
troduced in Section 4.10. introduced in Section 4.10.
ˆ g ˆ g
ˆ ˆ g g
À iiYse À uiYs iYs ‡ c1 —iYj jYsp
sˆ1Yt sˆ1YtÀ1 sˆ1Yt jPt aiPsj
ˆ ˆ À Á 7.6. Non-transferable raw component balance equa-
g
À cg
2 —iYj À ˜f Yj ZAf Yja
i iYs
tions
sˆ1Yt f Psj YjPt aiPs f Yj
ˆ ˆ g
ˆ g The balance equations for the procurement and
‡ ‰dYs T †iYs
sˆ1YtÀ1 dPDSi sˆ1Yt
utilization of non-transferable raw components
(10) are given by the constraints (47). These con-
Vi P JSY t P „ Y g P qX …43†
straints state that the supply volume for given
component and time period must equate the
component's volume that is delivered to satisfy
7.4. External demand balance equations internal needs given by the BoM of related periods
for each scenario to consider:
The balance equations (8) for the external de- g g
ˆ g g
mand serviceability with the expression (9) are ˆiYs ‡ iiYs À c1 —iYj jYsp
jPt aiPsj
given by the constraints (44), for given external ˆ À Á g
demand and time period with the bounds (45) for ‡ cg —iYj À ˜f Yj ZAf Yja ˆ 0
2 i iYs
…47†
each scenario to consider. f Psj YjPt aiPs f Yj
À Á À Á g
À 1 À d g fg g g Vi P IR À ITY s P „ Y g P qY
d dYsÀ1 ‡ 1 À dd ‰dYsd ‡ fdYs
À Á g
ˆ 1 À dg hdYs Vd P DSY s P „ Y g P qY …44† where sp , sa are given in Section 3.5 and cg , cg are
1 2
d
referred to in Section 6.2.
where sd is given in Section 3.5, and
‰dYsd À fg
g g
dYsÀ1 T hdYs X …45†
7.7. Alternate components requirements

For a product to be made available at a given
7.5. Transferable raw component stock bounding time period and each of its prime components with
substitution capabilities in the related BoM, the
The raw component stock bounding (10) and following constraint type is required for each sce-
(21) is represented by the constraints (46), based on nario to consider:
the expressions (27). These constraints state that the ˆ g g
stock volume for given transferable raw component g
jYs P c3 ZAiYj
f Ys
and time period should be appropriately lower and f Ps iYj
upper bounded for each scenario to consider. g g
Vs ˆ XiYj Y F F F Y CiYj Y i P sj as iYj Tˆ £Y j P t Y g P qY
ˆ g
…48†
mSg À
iYt †jYs
sˆ1Yt where c3 is referred to in Section 6.2, and
ˆ g
ˆ g
ˆ ˆ d
T ˆiYss ‡ iiYse À cg —iYj jYsp
1
g
n o
g iYj…À†g
sˆ1Yt sˆ1Yt sˆ1Yt jPt aiPsj XiYj ˆ min sf ‡ oiYj Y …49†
ˆ ˆ g
À f Yj
Á f Yjg
…46† iYj
f Ps
‡ c2 —iYj À ˜i ZAiYsa
n o
sˆ1Yt f Psj YjPt aiPs f Yj g iYj…‡†g
ˆ CiYj ˆ max sf
iYj
‡ oiYj X …50†
g f Ps
T MSg À
iYt †jYs Vi P ITY t P „ Y g P qY
sˆ1Yt See the balance equations (11).


g
7.8. Components and products group bounding 0 T iiYs T MEg
iYs

Vs ˆ 1Y F F F Y j„ j À leYg Y i P JS ‘ ITY g P q
i
The maximum and minimum usage (12) that is
allowed for a given raw component group at a Vs ˆ Xg Y F F F Y Cg Y i P IR À ITY g P qY
i i …56†
given time period can be expressed as follows for
where
each scenario to consider:
n o
P …À†g
ˆ ˆ Xg ˆ min
s min siYj À liYj Y
jPt aiPsj
mRhYs T R cg riYh —iYj jYsp
g

iPIRh jPt aiPsj
1
n o'
f Yj…À†g
Q min si À liYj Y …57†
ˆ À Á f Yjg
‡ cg riYh ˜f Yj À —iYj iYsa S T MRhYs
2 i
f Psj YjPt n g o
…‡†
Cg
i ˆ max max siYj À liYj Y
Vh P RGY s P „ Y g P qY …51† jPt aiPsj

n o'
f Yj…‡†g
where sp , sa are given in Section 3.5 and cg , cg are
1 2 max si À liYj X …58†
referred to in Section 6.2.
The maximum and minimum release capacity Maximum product backlog, and non-negative
(13) that is allowed for a given product group at a character of the variables
given time period can be expressed as follows for
each scenario to consider. 0 T fg T MBdYs
dYs Vs P „ Y d P DSY g P qY …59†
ˆ g
mPhYsr T ujYh jYs T MPhYsr
g
jPth 0 T ‰dYs Vs ˆ 1Y F F F Y j„ j À ld Y d P DSY g P qX
Vh P PGY s P „ Y g P qY …52† …60†
where sr is given in Section 3.5.

7.10. Splitting variable constraints
7.9. Bounds on the variables
The Z-, ZA-, X-, E-, Y- and B-variables indexed
Maximum product release volume with time periods from the set T1 of implementable
time periods for the whole set of scenarios, require
g
0 T jYs T MZjYsr Vs ˆ ™j Y F F F Y j„ jY j P t Y g P qY the following non-anticipativity constraints. For
…53† all g P q, t P „1 :

where sr is given in Section 3.5. g g‡1
jYt À jYt ˆ 0 Vj P t Y …61†
g g
iYjg iYj…À† iYj…‡†
0 T ZAf Ys Vs ˆ sf ‡ oiYj Y F F F Y sf ‡ oiYj Y g g‡1
iYj ZAf Yj À ZAf Yj ˆ0 Vi P s f Yj Y f P sj Y j P t Y
f P s Y i P sj Y j P t Y g P qX …54† iYt iYt

…62†
Maximum component procurement volume
g g g‡1
0 T îYs T MXg
iYs
îYt À îYt ˆ 0 Vi P IRY …63†
Vs ˆ 1Y F F F Y j„ j À lisYg Y
i P ITY g P qY
g g g g‡1
Vs ˆ Xi Y F F F Y Ci Y i P IR À ITY g P qY …55† iiYt À iiYt ˆ 0 Vi P sY …64†


g g‡1
‰dYt À ‰dYt ˆ 0 Vd P DSY …65† ever, the DEM for the 2-stage stochastic problem
has still such big dimensions that it is impractical
to solve it without using some type of decompo-
fg À fg‡1 ˆ 0 Vd P DSX
dYt dYt …66† sition approach. An intensive computational ex-
perimentation is underway, via Augmented
Note. By convention it is assumed that g ‡ 1 ˆ Lagrangian-based and Benders-based Decompo-
0 for g ˆ jqj (circular redundant link). sition approaches, by using real-life instances from
the automotive sector for both sequential and
parallel computing implementations.
7.11. Preventing unnecessary product and compo-
nent stock
Acknowledgements
The following penalization can be included in
the objective functions (37), (39) and (40) to pre- This work has been partially supported by the
vent unnecessary build-up: Europe Commission within the ESPRIT program
P HPCN domaine, project ES26267.
ˆ ˆˆ ˆ
aˆ wg R g
riYs ˆiYss
gPq iPIT tP„ sˆ1Yt
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