Characterizing the Distribution of Low- Makespan Schedules in the Job Shop Scheduling Problem Matthew J. Streeter Stephen 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 M4 M8 M6 M3 M1 M7 M9 M5 M2 M0 M4 M8 Job 1 Job 2 time

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 M4 M8 M6 M3 M1 M7 M9 M5 M2 M0 M4 M8 Job 1 time M6 M3 M1 M7 M9 M5 M2 M0 M4 M8 M6 M3 M1 M7 M9 M5 M2 M0 M4 M8 Job 2 Source 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 opt o  o’ be the optimum makespan among schedules that start o before o’ If o and o’ use the same machine, then min(opt o  o’, opt o’  o ) = [optimal makespan] An edge e = {o,o’} is in  -backbone iff. max(opt o  o’, opt o’  o ) >  *[optimal makespan] Can determine opt o  o’ using branch and bound

|  -backbone| for instance ft10

|  -backbone| for random 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) Orient a random disjunctive edge the “wrong” way Ensures makespan > N+sqrt(N)*log(N) w.h.p. time

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 J8 J9 J8 J2 J7 J6 J0 J3 J1 J5 J4 Machine 1Machine 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 J96 J99 J96 J2 J98 J0 J3 J1 J97 Machine 1Machine 2... J97 J98 J J J’ 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

Results: neighborhood exactness

Neighborhood exactness N r (S) = set of schedules that differ from S on at most r edges exactness(N r ) = probability that a random local optimum under N r is globally optimal A way to quantify “ruggedness” of search landscape

Neighborhood exactness

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?