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Characterizing the Distribution of Low-Makespan Schedules in the
Job Shop Scheduling Problem
Matthew J. StreeterStephen F. Smith
Carnegie Mellon University
Outline
• Introduction & related work
• Definitions
• Results– Backbone size– Clustering of low-makespan schedules– Neighborhood exactness
• Conclusions
Introduction
• Goal: to determine how optimal schedules are geographically distributed in the search space in random instances of the JSSP, as a function of the job:machine ratio
• A first step toward understanding the success of heuristics (e.g., path relinking) for the JSSP
Related work: the ‘big valley’
• Boese et al. (1994) generated random locally optimal TSP tours, measured two correlations:– correlation between cost of tour and avg. distance to
other tours– correlation between cost of tour and distance to best tour
• Both correlations were suprisingly high; suggests ‘big valley’ distribution of local optima
• Similar correlations found for job shop (Nowicki & Smutnicki 2001) and permutation flow shop (Watson et al. 2002) problems
Related work
• Statistical mechanical analyses of TSP (Mézard and Parisi 1986)
• Empirical studies of backbone size in JSSP, SAT, & other problems (Slaney & Walsh 2001; Watson et al. 2001)
Definitions
JSSP instance
• Set of N jobs, each a sequence of M operations
• Each operation has specific machine, duration
• Each job uses each machine exactly once
M6
M3
M1
M7
M9
M5
M2
M0
M4
M8
M6
M3
M1
M7
M9
M5
M2
M0
M4
M8
Job 1 Job 2
JSSP schedule
• Assigns start time to each operation
• Feasible if– no machine is scheduled to
perform two operations simultaneously, and
– operation i of a job does not start until operation i-1 completes
• Makespan = maximum operation completion time
M6
M3
M1
M7
M9
M5
M2
M0
M4M8
M6M3
M1
M7
M9
M5
M2
M0
M4
M8
Job 1 Job 2
tim
e
Disjunctive Graphs
• Weighted, directed graph representation of JSSP schedule
• Makespan = length of longest weighted path from source to sink
M6
M3
M1
M7
M9
M5
M2
M0
M4M8
M6M3
M1
M7
M9
M5
M2
M0
M4
M8
Job 1
tim
e
M6
M3
M1
M7
M9
M5
M2
M0
M4M8
M6
M3
M1
M7
M9
M5
M2
M0
M4
M8
Job 2Source
Sink
-Backbone
• Set of disjunctive edges that have the same orientation in all schedules whose makespan is within a factor of optimal
Random JSSP instances
• Fixed N and M.
• Each job uses the machines in a random order
• Operation durations are i.i.d.
Results: backbone size
Determining |-backbone| empirically
• Let optoo’ be the optimum makespan among schedules that start o before o’
• If o and o’ use the same machine, thenmin(optoo’, opto’o) = [optimal makespan]
• An edge e = {o,o’} is in -backbone iff. max(optoo’, opto’o) > *[optimal makespan]
• Can determine optoo’ using branch and bound
|-backbone| for instance ft10
Instance ft10
0
0.2
0.4
0.6
0.8
1
1.00 1.02 1.04 1.06 1.09 1.11 1.13
|Normalized
- |backbone
|-backbone| for random instances(A) Job:machine ratio 1:1
0
0.2
0.4
0.6
0.8
1
1 1.1 1.2 1.3 1.4 1.5
[ . E frac edges in
- ]backbone
6 6 x instances
7 7 x instances
8 8 x instances
(B) Job:machine ratio 2:1
0
0.2
0.4
0.6
0.8
1
1 1.1 1.2 1.3 1.4 1.5
[ . E frac edges in
- ]backbone
8 4 x instances
10 5 x instances
(C) Job:machine ratio 3:1
0
0.2
0.4
0.6
0.8
1
1 1.1 1.2 1.3 1.4 1.5
[ . E frac edges in
- ]backbone
9 3 x instances
12 4 x instances
Claim
• For fixed N, a random disjunctive edge is in the 1-backbone w.h.p. (as M )
Proof idea
Schedule, ignoring resource constraints
makespan < N+sqrt(N)*log(N)
w.h.p.
Resolve resource conflicts randomly
Expected increase in makespan is O(1)
€
μM + M log M( )
Orient a random disjunctive edge the
“wrong” way
Ensures makespan > N+sqrt(N)*log(N)
w.h.p.
€
μM + M log M( )
tim
e
Claim
• For fixed M, a random disjunctive edge is not in the 1-backbone w.h.p. (as N )
Proof idea
• Lemma: w.h.p. we can build a schedule with no idle time on any machine
J1
J3
J4
J7
J0
J2
J5
J6
J8J9
J9
J8
J2
J7
J6
J0
J3
J1
J5
J4
Machine 1 Machine 2
Proof idea
• Let e = {o,o’} connect operations of jobs J and J’.
• Remove J and J’ from instance, and use Lemma to build schedule
• Schedule J and J’ during “long” period when all but one machine is idle J1
J3
J0
J2
J96J99
J99
J96
J2
J98
J0
J3
J1
J97
Machine 1 Machine 2
. .
.
. .
.
. .
.
. .
.
J97
J98 J
J
J’
J’
€
N
O(1)
Results: clustering of low-makespan schedules
Clustering of low-makespan schedules
• Use simulated annealing to generate “random” schedules within a given factor of optimal
• For each , compute average distance between random schedules, where distance = # of disjunctive edges with different orientation
Clustering of low-makespan schedules
(A) Job:machine ratio 1:1
0
0.1
0.2
0.3
0.4
0.5
1 1.2 1.4 1.6 1.8 2
[ . ]E dist between schedules
6 6 x instances
7 7 x instances
8 8 x instances
(B) Job:machine ratio 2:1
0
0.1
0.2
0.3
0.4
0.5
1 1.2 1.4 1.6 1.8 2
[ . ]E dist between schedules
8 4 x instances
10 5 x instances
(C) Job:machine ratio 3:1
0
0.1
0.2
0.3
0.4
0.5
1 1.2 1.4 1.6 1.8 2
[ . ]E dist between schedules
9 3 x instances
12 4 x instances
Results: neighborhood exactness
Neighborhood exactness
• Nr(S) = set of schedules that differ from S on at most r edges
• exactness(Nr) = probability that a random local optimum under Nr is globally optimal
• A way to quantify “ruggedness” of search landscape
Neighborhood exactness(A) Job:machine ratio 1:5
0
0.2
0.4
0.6
0.8
1
0 0.1 0.2 0.3
Normalized radius
E[exactness]
3x15 instances
4x20 instances
5x25 instances
(B) Job:machine ratio 1:1
0
0.2
0.4
0.6
0.8
1
0 0.1 0.2 0.3
Normalized radius
E[exactness]
6x6 instances
7x7 instances
8x8 instances
(C) Job:machine ratio 5:1
0
0.2
0.4
0.6
0.8
1
0 0.1 0.2 0.3
Normalized radius
E[exactness]
15x3 instances
20x4 instances
Conclusions
Summary of contributions
• Two analytical results on backbone size in JSSP
• Experimental investigation of clustering of low-makespan schedules as a function of job:machine ratio
• Tool (neighborhood exactness) to quantify ‘ruggedness’ of search landscape
An intuitive picture
• many global optima,
• far apart,• ‘smooth’ search
space
• few global optima,• close together,• ‘smooth’ search
space
0 1
• big valley?
€
NM
