# Benchmarks for basic scheduling problems

Éric D. Taillard

^{1}TL;DR: This paper proposes 260 randomly generated scheduling problems whose size is greater than that of the rare examples published, and the objective is the minimization of the makespan.

Abstract: In this paper, we propose 260 randomly generated scheduling problems whose size is greater than that of the rare examples published. Such sizes correspond to real dimensions of industrial problems. The types of problems that we propose are: the permutation flow shop, the job shop and the open shop scheduling problems. We restrict ourselves to basic problems: the processing times are fixed, there are neither set-up times nor due dates nor release dates, etc. Then, the objective is the minimization of the makespan.

Topics: Flow shop scheduling (73%), Job shop scheduling (66%), Open shop (63%), Job shop (62%), Open-shop scheduling (57%)

1

BENCHMARKS

FOR

BASIC SCHEDULING PROBLEMS

E. TAILLARD

ORWP89/21 Dec. 1989

Abstract

In this paper, we propose 260 scheduling problems whose size is greater than that of the rare

examples published. Such sizes correspond to real dimensions of industrial problems.

The types of problems that we propose are : the permutation flow shop, the job shop and the open

shop scheduling problems.

We restrict us to basic problems : the processing times are fixed, there are neither set-up times nor

due dates nor release dates, etc. Then, the objective is the minimization of the makespan.

Keywords : Combinatorial optimization, Scheduling, Benchmarks.

Introduction

The types of problems discussed in this paper (permutation flow shop, job shop and

open shop scheduling problems) have been widely studied in the literature using exact

or heuristic methods, but a common comparison base is missing. We hope that this

paper will fill a gap in this domain.

The three-field nomenclature described in Lawler et al. [5] names these problems

F| |C

max

, J| |C

max

and O| |C

max

respectively. They certainly belong to the most studied

ones among the scheduling problems. Let us describe them briefly.

There are n jobs that have to be performed on m unrelated machines ; in our case,

every job consists of m non preemptible operations. Every operation of a job uses a

different machine during a given time and may wait before being processed.

For the permutation flow shop problem, the operations of every job must be

processed on machines 1 … m in this order. Moreover, the processing order of the jobs

on the machines is the same for every machine. The problem consists in finding a

permutation of the n jobs that minimizes the makespan.

In the case of the job shop problem, any processing order of the jobs on the

machines is allowed. For every job, the operations must be processed in a given order

on the machines, but this order may differ according to the jobs.

For the open shop problem, every operation is assigned to a given machine but the

order of the operations of every job is totally free.

The aim of this paper is to present unsolved problems whose size corresponds to the

one of industrial problems. These problems must be easy to generate.

Published in European Journal of Operational Research, vol.

64, no. 2, pp. 278-285, which should be cited to refer to

this work.

DOI: 10.1016/0377-2217(93)90182-M

2

Generating interesting problems

As we do not know any exact method to solve exactly the problems we want to

propose, we have used heuristic methods to get hopefully good solutions of these

problems. These heuristic methods are based on taboo search techniques. Taboo search

is described very generally in Glover [4] and one can find some of its practical

applications to the flow shop sequencing problem in Taillard [8] and Widmer et al. [10],

and to the job shop scheduling problem in Taillard [9]. Taboo search is very easy to

implement and generally provides excellent results, but it requires a great amount of

CPU time.

In order to propose problems that are as difficult as possible (the most interesting

ones), we have generated many instances of problems that we have “solved” in a

summary way with taboo search. Then, we have chosen the 10 problems that seemed to

be the hardest ones and we have solved them once more, allowing our heuristic method

to perform a higher number of iteration.

Obviously, the choice of the hardest problems is very subjective. We decided that a

problem was interesting if the best makespan we found was far from a lower bound of

the makespans and if many attempts to solve the problem (starting from various initial

solutions) did not provide the same solution. Such a method enabled us to detect the

simplest problems but we may not propose problems that have a local optimum with a

large attraction basin.

The problems

The problems we propose are randomly generated with a good random number

generator proposed in Bratley [1]. We recall its implementation so that this paper is self

contained.

A problem will be entirely defined by the initial value of the seed of the random

generator and by the way of generating it.

For every type of problem, we give a simple manner of computing a lower bound of

the makespan ; in particular, this permits to verify the generation of the problem.

For every size of problem, we give the total number of instances we have generated

(summary resolution), the maximum number of iteration of taboo search that were done

(long resolution) and the proportion of problems that were solved up to the lower

bound, that is to say optimally. For every type and every size of problem, we give 10

instances.

For each instance, we give the initial value of the random generator seed, the best

value of the makespan we have found (i. e. an upper bound of the optimal makespan)

and a lower bound of all the makespans.

The random number generator

Let us recall the implementation of the linear congruential generator we have used

which is based on the recursive formula X

i+1

= (16 807 X

i

) mod (2

31

- 1). This

implementation uses only 32-bit integers and provides a uniformly distributed sequence

of numbers between 0 and 1 (not contained) :

3

0) Initial seed and X

0

(0 < X

0

< 2

31

- 1)

constants : a = 16 807, b = 127 773, c = 2 836, m = 2

31

- 1

1) Modification of k := X

i

/b

the seed : X

i+1

:= a(X

i

mod b) - kc

If X

i+1

< 0 then let X

i+1

:= X

i+1

+ m

2) New value of

the seed : X

i+1

Current value of

the generator : X

i+1

/m

Below, we shall denote by U(0,1) the pseudorandom number that this generator pro-

vides. We have 0 < U(0,1) < 1 for every generated number.

We shall denote by U[a,b] (with a < b, a and b integers) the integer number

a + U(0,1)

.

(b-a+1). For every random integer generated, we have a 8>DE@EDQG

every integer between a and b has the “same” probability of being chosen. In order to

implement the integer random procedure only with 32-bit integers, the problems have

been chosen in such a way that one never has to deal with a seed X such that :

a +

P

DE; )1(

+

−

⋅

≠

a +

)1( +− DEP

;

Flow shop problems

There are in the literature some problems of this type ; let us quote for example eight

small and simple problems proposed in Carlier [2] and solved exactly in this reference.

The flow shop problems are characterized by the processing times d

ij

of job j on

machine i (1 LPMQ:HKDYHJH QHUDWHGW KH YDOXH VRIG

ij

by the following

way :

For i = 1 to m

For j = 1 to n

d

ij

:= U[1,99]

We propose problems with 5, 10 and 20 machines and from 20 to 500 jobs. We

compute the lower bound of the makespan as presented below.

Let b

i

be the minimum amount of time before machine i starts to work and a

i

be the

minimum amount of time that it remains inactive after its work up to the end of the

operations, and let T

i

be its total processing time. We have :

b

i

= min,

j

( Σ,

k = 1

,

i - 1

d

kj

)

a

i

= min,

j

( Σ,

k = i+1

,

m

d

kj

)

T

i

= Σ,

j

= 1

,

n

d

ij

Clearly, the optimal makespan C

max

* is greater than or equal to

4

LB = max,

i

(b

i

+ T

i

+ a

i

) &

*

,

max

This lower bound is easy to compute and we conjecture that :

∞→

/

lim Prob(C

*

,

max

= LB) = 1

For every size of problem we give the following information (Table 1) :

Nb jobs : The number of jobs.

Nb machines : The number of machines.

Nb instances : The total number of problems generated.

LB reached : The proportion of problems for which we found a solution

for which the makespan was equal to the lower bound (or

equal to the lower bound augmented by 2% for the 500-job

20-machine problems).

Nb iterations : The maximum number of iterations performed by taboo

search (long resolution).

Nb resolutions : The number of attempts to solve the problem from various

initial solutions (long resolution).

Nb

Nb

Nb

LB

Nb

Nb

20 5 100 35%

10

4

3

20 10 100 1%

10

4

3

20 20 100 0%

2

⋅

10

4

3

50 5 70 41%

5

⋅

10

3

3

50 10 70 3%

10

4

3

50 20 70 0%

5

⋅

10

4

3

100 5 10 000 54%

2

⋅

10

3

4

100 10 50 6%

2

⋅

10

4

3

100 20 50 0%

10

4

3

200 10 300 28%

2

⋅

10

3

3

200 20 25 0%

2

⋅

10

3

3

500 20 100

14%

*

10

3

3

*

The value reached for this size was less than or equal to 1.02 times the lower bound.

Table 1. Flow shop problems.

Then we give ten instances for every size of problem with the following information

(Table 2) :

Time seed : The initial value of the random generator’s seed.

UB : An upper bound of the optimal makespan (the best value we

got).

LB : A lower bound of the makespans.

As the aim is to give an upper bound as good as possible but not a fast solving

method, the computation time does not have much importance. However, let us mention

5

that an iteration of taboo search needs about 4

.

10

-6.

n

2.

m seconds on a “Silicon

Graphics” personal workstation (10 Mips).

20 jobs,

5 machines Flow shop

Time seed

UB

LB

873654221

1278

1232

379008056

1359

1290

1866992158

1081

1073

216771124

1293

1268

495070989

1236

1198

402959317

1195

1180

1369363414

1239

1226

2021925980

1206

1170

573109518

1230

1206

88325120

1108

1082

20 jobs,

10 machines Flow shop

Time seed UB LB

587595453 1582 1448

1401007982 1659 1479

873136276 1496 1407

268827376 1378 1308

1634173168 1419 1325

691823909 1397 1290

73807235 1484 1388

1273398721 1538 1363

2065119309 1593 1472

1672900551 1591 1356

20 jobs,

20 machines Flow shop

Time seed UB LB

479340445 2297 1911

268827376 2100 1711

1958948863 2326 1844

918272953 2223 1810

555010963 2291 1899

2010851491 2226 1875

1519833303 2273 1875

1748670931 2200 1880

1923497586 2237 1840

1829909967 2178 1900

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