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Journal ArticleDOI

Benchmarks for basic scheduling problems

22 Jan 1993-European Journal of Operational Research (North-Holland)-Vol. 64, Iss: 2, pp 278-285

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%)

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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>DE@EDQG
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 LPMQ:HKDYHJH QHUDWHGW KH YDOXH VRIG
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
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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Metrics
No. of citations received by the Paper in previous years
YearCitations
20228
2021102
2020105
2019115
2018117
2017131