Self-Adapting Large Neighborhood Search: Application to single-mode scheduling problems

Self-Adapting Large
Neighborhood Search

Application to single-mode
scheduling problems
P. Laborie and D. Godard
[plaborie,dgodard]@ilog.fr

MISTA 2007 – Paris, France – August 28-31, 2007
2
Overview
 Objective
 Model
 Search: self-adapting LNS for scheduling
 Principles
 Relaxation methods portfolio
 Re-optimization methods portfolio
 Experimental results
 Current and future work

3
Objective
 A robust search method for solving real-world
industrial scheduling problems
 Current focus on:
 Non-preemptive scheduling
 Deterministic scheduling
 Single-mode scheduling (no resource allocation)
 Optimization problems for which computing a
feasible schedule (even of poor quality) is not too
difficult
 Temporal cost functions

4
Model
 Activities
 Calendar constraints
 Temporal constraints
 Resources
 Sequence-dependent setup times
 Temporal cost function

5
Model : Activities
 Non-preemptive activities with variable duration
Activity A
Start time: s(A) End time: e(A)
Duration: d(A) == e(A)-s(A)
t
1 activity  3 integer decision variables

6
Model : Calendar constraints
 Constraint restricting the start/end dates of
activities depending on its position in time
 Breaks/Efficiency:
 Forbidden start/end periods:
 Forbidden overlap periods:


)
(
)
(
)
(
)
(
a
end
a
start
dt
t
a
pt 
)
,
[
)...
,
[
)
,
[
)
( 2
2
1
1 n
n
a
start 




 


  



 )
,
[
)...
,
[
)
,
[
))
(
),
(
[ 2
2
1
1 n
n
a
end
a
start 





]
1
,
0
[
: 



7
Model : Temporal constraints
 Any constraint of one of the forms:
t1 + d ≤ t2 or t1 + d = t2
where t1 and t2 are the start or end time-point
of two activities and d an integer variable (or
constant)
 Expressivity: precedence, min/max delays, etc.

8
Model : Resources
 Resource typology:
 Unary resource: Resource of capacity 1. Two
activities requiring the same unary resource cannot
overlap.
 Discrete resource: Resource of capacity Qmax. The
cumulated resource usage by activities demanding the
resource must be kept in the range [Qmin, Qmax]. Qmin,
Qmax can be some known function of time.
t
Qmax(t)
Qmin(t)

9
Model : Resources
 Resource typology (cont’d):
 Reservoir: Resource of capacity Qmax and initial level
L. Activities may consume or produce the reservoir.
The level of the reservoir over time must be kept in
the range [Qmin, Qmax], 0≤Qmin
 State Resource. Resource with a finite set of
possible states. Activities require the resource be in a
particular (set of) state s. Activities requiring
incompatible state may not overlap.

10
Model : Sequence-dependent setup times
 On unary resources
setup(ai,aj) specified by a setup time matrix or a
user-defined function
)
(
)
,
(
)
(
)
(
)
( j
j
i
i
j
i a
start
a
a
setup
a
end
a
start
a
end 




11
Model : Temporal cost function
 Cost function expressed as sum/max
aggregation of semi-convex piecewise linear
(SCPL) functions on activity start/end/durations
 SCPL function:
z, {x / f(x)≤z} is a convex interval
cost
x
cost
x
cost
x
cost
x
cost
x
z

12
 Why using SCPL functions for cost ?
 They are very expressive (see next slides)
 f(x)≤z can be propagated with arc-consistency
without creating holes in the variables domains
 They can be convexified without loosing too much
information

13
 Some particular SCPL functions:
 I: I(t)=t
 H: H(t)=0 if t<0, H(t)=1 otherwise
 V,, 0≤,: V(t)=-.t if t<0, V(t)=.t otherwise
 -I, -H
 If f and g are SCPL, the following functions are SCPL:
+f, .f if 0≤, shift(f,), max(f,g)
Note that f+g is not SCPL in general

14
 Max/Sum of SCPL covers classical temporal costs:
 Minimize makespan:
maxi { I o end(ai) }
 Minimize earliness/tardiness cost:
sum/maxi { shift(Vi,i
,i) o end(ai) }
 Minimize weighted number of late jobs:
sumi { shift(i.H,i) o end(ai) }
 Minimize weighted sums of activity durations (WIP):
sumi { i.I o duration(ai) }
 Maximize weighted sums of activity durations (quality):
sumi { i.-I o duration(ai) }

15
Search: Self-adapting LNS for scheduling
 Principles
 LNS: Iterative improvement method based on
sequentially:
 Relaxing a fragment of the current solution
 Re-optimizing the relaxed fragment
 Self-adapting:
 Portfolio of methods for relaxing fragments
 Portfolio of methods for re-optimizing a relaxed fragment
 On-line learning techniques to converge on the most
efficient methods on the instance being solved

16
 Notations:
 s: fully instantiated feasible solution (schedule)
 p: scheduling problem
 Ri
i
: ith
relaxation method parameterized by a parameter
vector i=(i,1,…,i,ni
)
p= Ri
i
(s)
 Oj
j
: jth
re-optimization method parameterized by a
parameter vector j= (j,1,…,j,mj
)
s= Oj
j
(p)

17
 One iteration of LNS:
 Input:
 Current best solution: s
 Current probability distributions for methods and parameters: 
 Select methods according to :
 One relaxation method with a set of parameters: Ri
i
 One re-optimization method with a set of parameters: Oj
j
 Apply selected methods:
 s’= [Oj
j
o Ri
i
] (s)
 c = max(0, cost(s)-cost(s’))
 t = computation time
 Reward selected methods and parameters (i,i,j,j):
 =update(,reward, i,i,j,j) where reward = c/t
 Update current best solution
 If c>0: s=s’
upd

18
 Learning
 = update(, reward, x)
 Use classical machine learning scheme to update the
weight of a parameter value x:
(x)(x)reward
  is the learning rate (=0.1 in our experimental
settings)

19
 Input:
i
j
 s’= [Oj
j
o Ri
i
] (s)
 If c>0: s=s’
Ri
i

20
Search: Relaxation methods portfolio
 Partial Order Schedule (POS) definition
 A temporal network that is sufficient to ensure that
all its solutions satisfy the temporal and resource
constraints of the problem
 All relaxation methods in the portfolio start by
computing a POS from the solution and then,
relax a subset of activities F (fragment) on this
temporal network
Ri
= RelaxFi o POS
POS

21
 POS computation on resources
 Unary resource: easy, O(n logn)
 Discrete resource: [Godard&al 2005], O(n log(Cn))
t
c
R
A
B
C
D F
E
H
G
A C B t A C B
F
B
A
C
G
D
E
H

22
 POS computation on resources (cont’d)
 Reservoir: O(n log(Cn))
 Step 1: constraints to prevent reservoir underflow
t
c
R
A B
D
E
F
G
A B
D
E
F
G
1
1
1
2
1

POSMIN

23
 Step 2: constraints to prevent reservoir overflow
t
c
R
A F
A E
A D
B 
B G
A B
D
E
F
G
1
1
1
2
1


24
 Step 2: constraints to prevent reservoir overflow
t
c
R
A F
A E
A D
B 
B G
A F
A E
A D
B 
B G
Uses algorithm for POS computation on discrete resources
POSMAX

25
 Final POS = POSMIN  POSMAX
A
B
D
E
F
G
t
c
R
A B
D
E
F
G

26
 State resource: O(n2)
A
B
C D
I
E
G
H
F
t
R
A
B
C D
I
E
G
H
F

27
 Global POS
 POS = R POSR
 Redundant edges are removed

28
 Partial Order Schedule (POS) definition
 A temporal network that is sufficient to ensure that
all its solutions satisfy the temporal and resource
constraints of the problem
 All relaxation methods in the portfolio start by
computing a POS from the solution and then,
relax a subset of activities F (fragment) on this
temporal network
Ri
= RelaxFi o POS
Relax

29
 Relaxation of a fragment Fi
of the POS
F
B
A
C
G
D
E
H
POS
Fragment Fi
RelaxFi
F
B
A
C
G
D
E
H

30
 Portfolio:
 R1
: Randomized relaxation
 R2
: Time-window relaxation
 R3
: Topological relaxation

31
 R1
: Randomized relaxation
 An activity belongs to the relaxed fragment with
probability (0,1]
  is a self-adapting parameter of the relaxation
method

32
 R2
: Time-window relaxation
 Activities are sorted by increasing start time in the
solution
 The fragment consists of all the activities with index
in [.n, (+).n] in the above array
  and  are self-adapting parameters of the
relaxation method

33
 R2
solution
relaxation method
=0.25, =0.5

34
 R2
solution
relaxation method
=0.25, =0.5

35
 R3
: Topological relaxation
 Similar principle as for previous relaxation but
activities are ordered using a lexicographical
comparison:
 Criterion 1: index of strongly connected component of the
activity in the initial temporal network
 Criterion 2: index of connected component of the activity in
the initial temporal network
 Criterion 3: start time of the activity in the solution
relaxation method

36
 Input:
i
j
 s’= [Oj
j
o Ri
i
] (s)
 If c>0: s=s’
Oj
j

37
Search: Re-optimization methods portfolio
 Currently, a unique optimization method is
used:ScheduleJustInTime
 Explores a search tree with a limited number of
failures .n
  is a self-adapting parameter of the optimization
method
 At the root node, indicative start and end times for
activities are computed using an LP relaxation of the
problem

38
 Temporal relaxation:
 Only consider temporal constraints (including the
ones of the relaxed POS) and (convexified) cost
function
z
z
OR
z
z
z
z
z
b
x
a
D
t
t
D
E
e
E
S
s
S
D
d
D
d
s
e
z
i
i
i
i
k
i
k
i
k
i
i
k
i
j
i
i
j
j
i
i
i
i
i
i
i
i
i
i
i
i
i


























,
,
,
,
,
,
,
,
:
to
subject
)
(
minimize
Duration
Time-bounds
Temporal network
Cost function step
Function cost
Cost aggregation

39
function
F
B
D
H
C
A G
E

40
function
F
B
D
H
C
A G
E

41
 Tree search
 Search considers activities by increasing indicative
start times and tries to schedule them as close as
possible to their indicative dates
 In case of backtrack, the activity is marked
unselectable until constraint propagation removes
from the domain of the activity the dates tried on the
left branch

42
Experimental results
 The approach was implemented as a black-box
(no manual parameter)
 Implemented on top of ILOG CP 1.1
 Use constraint propagation of ILOG CP
 Use ILOG CPLEX for temporal relaxation
 Experimented on 20 scheduling benchmarks

43
Experimental results: benchmarks
Unary
Discrete
State
Makespan E/T Duration
Jobshop
Trolley
Hybrid flowshop
ATFM Quality RCPSP
Jobshop E/T
Flowshop E/T
Single proc. tardiness
Unary+
setup
Semicond. testing
Openshop
RCPSP E/T
Cumul. jobshop RCPSP
Shop setup times Airland
Flowshop w/ buffers
Single-machine E/T
Aircraft assembly
Common due-date
Flowshop
Parallel-machine E/T

44
Experimental results: overview
-15% -10% -5% 0% 5% 10% 15%
Average deviation to best known results or approaches
Job-shop
RCPSP
Cumulative Job-shop
Semiconductor
Flow-shop w/ E/T
RCPSP w/ E/T
Air land
Open-shop
Max. quality RCPSP
Trolley
Job-shop w/ E/T
Single machine w/
Aircraft assembly
Air traffic
Flow-shop
Common due-date
Hybrid flow-shop
Single proc.
Shop w/ set-up times
Flow-shop w/ buffers
Problem
 Self-Adapting LNS: results over 20 benchmarks
Details…

45
Current and future work
 Extension of the approach to:
 Multi-mode scheduling:
 Optional activities
 Alternative resources
 Alternative routes/recipes
 Resource costs:
 Setup costs
 Resource usage costs
 Inventory costs
 Feasibility problems

47
Experimental results: details
 Trolley
 Hybrid flow-shop
 Job-shop w/ E/T
 Air traffic management
 Max. quality RCPSP
 Flow-shop w/ E/T
 Cumulative job-shop
 Single proc. Tardiness
 Semiconductor testing
 Open-shop
 RCPSP w/ E/T
 RCPSP
 Shop w/ setup times
 Job-shop
 Air land
 Flow-shop w/ buffers
 Flow-shop
 Aircraft assembly
 Single machine w/ E/T
 Common due-date

48
Results: Trolley
 Generalization of the trolley problem described
in [VanHentenryck&al 1999]
 Job-shop + Trolley resource to move items from one
machine to the other (limited capacity + transition
times)
 15 instances
 Comparison with OPL 3.7 default search
 All 15 instances improved
 Mean relative distance: -11.7%

49
Results: Hybrid flow-shop
 20 randomly selected instances among the biggest ones
from:
F. Sivrikaya-Serifolu and G. Ulusoy. Multiprocessor task
scheduling in multistage hybrid flowshops: a genetic
algorithm approach. Journal of the Operational
Research Society, 55(5):504–512, 2004.
 Sizes: 200, 500 and 1000 operations
 Comparison with results of the above paper

50
Results: Job-shop w/ E/T
 All 48 instances with 15 and 20 jobs from:
P. Baptiste, M. Flamini, and F. Sourd.
Lagrangean bounds and lagrangean heuristics
for just in time job-shop scheduling. Tech.
report, Universita degli Studi di Roma Tre,
2005.

51
Results: Air traffic management
 OPL 3.7 distributed example with about 2000
activities
 Comparison with OPL 3.7 default search

52
Results: Max. quality RCPSP
 RCPSP with objective to maximize weighted
sum of activity duration
 3600 instances described in:
N. Policella, X. Wang, S.F. Smith, and A. Oddi.
Exploiting temporal flexibility to obtain high
quality schedules. In Proc. AAAI-2005, 2005.

53
Results: Max. quality RCPSP
Due date = 25 Due date = 30 Due date = 35
Solved Quality Solved Quality Solved Quality
[Policella-al-2005]
C=3 31.2% 47.21% 100% 50.87% 100% 52.01%
C=5 100% 81.60% 100% 81.37% 100% 81.19%
C=7 100% 95.37% 100% 95.28% 100% 95.19%
[This work]
C=3 97.3% 48.95% 100% 52.15% 100% 53.53%
C=5 100% 83.84% 100% 83.92% 100% 83.70%
C=7 100% 96.61% 100% 96.49% 100% 96.34%

54
Results: Flow-shop w/ E/T
 12 instances of Morton&Pentico textbook
 Compared with
E. Danna and L. Perron. Structured vs.
unstructured large neighborhood search: A case
study on job-shop scheduling problems with
earliness and tardiness costs. In Proc. CP 2003,
pages 817–821, 2003.

55
Results: Cumulative job-shop
 Test on x2 open instances of Nuijten
 Comparison with:
D. Godard, P. Laborie, and W. Nuijten.
Randomized Large Neighborhood Search for
Cumulative Scheduling. In Proc. ICAPS-05,
pages 81–89, 2005.
 The difference is due to parameter learning

56
Results: Single proc. tardiness
 20 randomly selected instances (200-500 activities)
from the SMTTP problem library
(http://www.bilkent.edu.tr/~bkara/start.html)
M. Pavlin, H. H. Hoos, and T. Stutzle. Stochastic local
search for multiprocessor scheduling for minimimum
total tardiness. In Proc. of the 16th Canadian
Conference on Artificial Intelligence, pages 96–113,
2003.
 Mean relative distance: 0.2%

57
Results: Semiconductor testing
 18 randomly instances among the biggest ones
described in:
I.M. Ovacik and R. Uzsoy. Decomposition
methods for scheduling semiconductor testing
facilities. International Journal of Flexible
Manufacturing Systems, 8:357–398, 1996.
 7 UBs improved out of the 18 instances

58
Results: Open-shop
 Test on the instances of Brucker (j8*), Taillard
(tai_20x20_*) and Gueret-Prins (gp10*)
 Comparison with optimal solutions or, for open
instances with results of:
C. Blum. Beam-ACO - hybridizing ant colony
optimization with beam search: an application to open-
shop scheduling. Computers and Operations Research,
32(6):1565–1591, 2005.

59
Results: RCPSP w/ E/T
 Selection of 60 instances from the ones of [1]:
M. Vanhoucke, E. Demeulemeester, and W. Herroelen.
An exact procedure for the resource constrained
weighted earliness tardiness project scheduling
problem. Annals of OR, 102(1-4):179–196, 2001.
 30 randomly selected instances for which [1] finds
optimal solution within 30 s and our approach does not
 30 randomly selected instances for which [1] does not
prove optimality

60
Results: RCPSP
 Test on the 600 biggest instances of the PSPLIB
(j120)
 Comparison with best-known UB reported in the
PSPLIB

61
Results: RCPSP
 Average deviation to path-based lower bound:
32.4%
R. Kolisch and S. Hartmann. Experimental
evaluation of state-of-the-art heuristics for the
resource-constrained project scheduling
problem: An update. European Journal of OR,
2006.

62
Results: RCPSP
32.4

63
Results: Shop w/ setup times
 Jobshop with sequence-dependent setup times,
15 instances of Brucker and Thielle.
C. Artigues and D. Feillet. A branch and bound
method for the job-shop problem with
sequence-dependent setup times. Annals of
Operations Research, 2007 [To appear]

64
Results: Job-shop
 Test on 33 instances:
 abz5-9
 swv1-20
 tail1-40
 yam1-4
 Mean relative distance to best-known UB: 2.8%

65
Results: Air land
 8 instances from:
J.E. Beasley, M. Krishnamoorthy, Y.M. Sharaiha,
and D. Abramson. Scheduling aircraft landings -
the static case. Transportation Science, 34:180–
197, 2000.

66
Results: Flow-shop w/ buffers
 Random selection of 30 instances of size 20x5,
20x10, 20x20, 50x5, 50x10 and 100x5, buffer
sizes 0, 1 and 2 from Taillard’s benchmark.
P. Brucker, S. Heitmann, and J. L. Hurink. Flow-
shop problems with intermediate buffers. OR
Spectrum, 25(4):549–574, 2003.
 14 UBs improved out of 30 instances

67
Results: Flow-shop
 Random selection of 120 instances of size in
range 100-1000 from Taillard’s benchmark.
 Comparison with results reported in OR lib.

68
Results: Aircraft assembly
 Benchmark www.neosoft.com/ benchmrx/
 Comparison with results reported in:
J. Crawford. An approach to resource
constrained project scheduling. In Proc. 1996
Artificial Intelligence and Manufacturing
Research Planning Workshop, 1996.
 Relative distance: 8.7%

69
Results: Single machine w/ E/T
 Comparison with [2]:
F. Sourd and S. Kedad-Sidhoum. An efficient algorithm
for the earliness-tardiness scheduling problem. In
Optimization Online, 2005.
 Selection of 20 instances from BKY benchmark and 20
instances from SKS benchmark
 In each benchmark:
 10 instances for which [2] proves optimality
 10 instances for which [2] does not prove optimality (in this
case, comparison with heuristics HEURn)

70
Results: Common due-date
 20 randomly selected instances of size 100 and
200 activities from:
D. Biskup and M. Feldmann. Benchmarks for
scheduling on a single machine against
restrictive and unrestrictive common due dates.
Computers and OR, 28(8):787–801, 2001.

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