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

M5M2

M0

M4M8

M6M3

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

M6M3

M1

M7

M9M5

M2

M0M4M8

M6M3

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

M6M3M1M7

M9M5

M2M0M4M8

M6M3

M1

M7

M9

M5

M2

M0

M4

M8

Job 1

time

M6M3

M1

M7

M9M5M2

M0M4M8

M6M3

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 ft10Instance 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

Nomalized |

-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[fac. edges in

-backbone]

6x6 instances

7x7 instances

8x8 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[fac. edges in

-backbone]

8x4 instances

10x5 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[fac. edges in

-backbone]

9x3 instances

12x4 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)

€

mM + M logM( )

Orient a random disjunctive edge the

“wrong” wayEnsures makespan >

N+sqrt(N)*log(N) w.h.p.

€

mM + M logM( )

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

J1J3

J4

J7

J0J2J5

J6J8J9

J9J8

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

J0J2

J96J99

J99J96

J2

J98

J0

J3

J1

J97

Machine 1 Machine 2

. . . . .

.

. . .

. . .

J97J98 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]

6x6 instances

7x7 instances

8x8 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]

8x4 instances

10x5 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]

9x3 instances

12x4 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