Solving multi-objective production scheduling problems using metaheuristics

EOR 5871 No. of Pages 20, DTD = 4.3.1

26 November 2003 Disk usedARTICLE IN PRESS

European Journal of Operational Research xxx (2003) xxx–xxx

www.elsevier.com/locate/dsw

Solving multi-objective production schedulingproblems using metaheuristics

T. Loukil a, J. Teghem b,*, D. Tuyttens b

a Universit�ee de Sfax, F.S.G.E., Route l’a�eerodrome km 4, BP 1088, 30185 Sfax, Tunisiab Facult�ee Polytechnique de Mons, 9 Rue de Houdain, Mons 7000, Belgium

Abstract

Most of research in production scheduling is concerned with the optimization of a single criterion. However the

analysis of the performance of a schedule often involves more than one aspect and therefore requires a multi-objective

treatment. In this paper we first present (Section 1) the general context of multi-objective production scheduling,

analyze briefly the different possible approaches and define the aim of this study i.e. to design a general method able to

approximate the set of all the efficient schedules for a large set of scheduling models. Then we introduce (Section 2) the

models we want to treat––one machine, parallel machines and permutation flow shops––and the corresponding

notations. The method used––called multi-objective simulated annealing––is described in Section 3. Section 4 is devoted

to extensive numerical experiments and their analysis. Conclusions and further directions of research are discussed in

the last section.

� 2003 Elsevier B.V. All rights reserved.

Keywords: Production scheduling; Multi-objective optimization

1. Introduction

Most of research in production scheduling is

concerned with the minimization of a single crite-

rion. However, scheduling problems often involve

more than one aspect and therefore require mul-

tiple criteria analysis.

Despite their importance, scant attention hasbeen given to multiple criteria scheduling prob-

lems, especially in the case of multiple machines.

* Corresponding author. Tel.: +32-65-374-680; fax: +32-65-

374-689.

E-mail address: [email protected] (J. Teghem).

0377-2217/$ - see front matter � 2003 Elsevier B.V. All rights reserv

doi:10.1016/j.ejor.2003.08.029

This is due to the extreme complexity of these

combinatorial problems. A comprehensive survey

of multi-objective scheduling problems is in prep-

aration [9].

For a multi-objective optimization problem (P)

minX2D

zkðX Þ; k ¼ 1; . . . ;K; ðPÞ

where functions zk are the objectives, X the vector

of variables and D the set of feasible solutions (D is

discrete in case of combinatorial problem like

scheduling problems).

We recall the basic notion of efficient solution: afeasible solution X � is efficient if there does not exist

any other X 2 D such that zkðX Þ6 zkðX �Þ 8k with at

least one strict inequality.

ed.

mail to: [email protected]

2 T. Loukil et al. / European Journal of Operational Research xxx (2003) xxx–xxx



In the literature concerning multi-objectivescheduling problems we can distinguish five main

approaches:

(a) Hierarchical approach: the objectives consid-

ered are ranked in a priority order and opti-

mized in this order.

(b) Utility approach: a utility function or weight-

ing function––often a weighted linear combi-nation of the objectives––is used to aggregate

the considered objectives in a single one.

(c) Goal programming (or satisficing approach): all

the objectives are taken into account as con-

straints which express some satisficing levels

(or goals) and the objective is to find a solution

which provides a value as close as possible of

the pre-defined goal for each objective. Some-times one objective is chosen as the main

objective and is optimized under the constraint

related to other objectives.

(d) Simultaneous (or Pareto) approach: the aim is

to generate––or to approximate in case of an

heuristic method––the complete set of efficient

solutions.

(e) Interactive approach: at each step of the proce-dure, the decision maker express his prefer-

ences in regard to one (or several) solutions

proposed so that the method will progressively

converge to a satisfying compromise between

the considered objectives.

Each approach has its own advantages and

drawbacks as described in general literature onmulti-objective optimization: some approaches

require more parameters or a priori information

((a), (b) and (c)), some are more pragmatic ((a) and

(b) for instance) but unable to generate some effi-

cient solution; others are more general or theo-

retical (d) or more oriented to real case studies (c)

and (e);. . .Clearly the approach used depends essentially

of the aim of the study and/or the context of the

application treated.

Independently of the approach used, Hoogeveen

[6] and Chen and Bulfin [1] studied the complexity

of the single machine bicriteria and multiple criteria

problems. They have proved that only problems

including flow time as primary criterion, in a hier-

archical optimization, and problems minimizingflow time and maximum tardiness can be solved in

polynomial time. All other problems are either

shown to be NP hard or remain open as far as

computational complexity is concerned. Obviously

the problems including more than one machine and

two criteria are more difficult.

This is the reason why it appears from the

analysis of the literature that often the methodsproposed are

• very sophisticated (complex branch and bound,

dynamic programming, dominance relations,

etc. . .) and complex to implement;

• only able to solve small size problems and with

two objectives;

• completely dependent of the model treated: themethods are no more valid if a small change in

the constraints or a fortiori a change in the

objectives is introduced.

So there is a need for a general method able to

treat a large class of models––even with large scale

instances––and independent of the considered

objectives. This is the aim of the present paper.Effectively, metaheuristics, like simulated

annealing (SA), tabu search and genetic algo-

rithms have demonstrated their ability to solve

combinatorial problems such as vehicle routing,

production scheduling, time tabling, etc. [14]. So,

some authors suggested to adapt metaheuristics

in order to solve multi-objective combinatorial

(MOCO) problems [3]. In particular Ulungu et al.[22,23], conceived a multi-objective simulated

annealing (MOSA) algorithm for solving combi-

natorial optimization problems and an interactive

version was also designed by Teghem et al. [20].

The MOSA method is thus designed to tackle a

MOCO problem (P). The aim is to generate a good

approximation dEðPÞEðPÞ of the set of efficient solutions

EðP Þ (approach (d) above). The procedure is validfor any number K P 2 of objectives. Similarly to a

single objective heuristic in which a potentially

optimal solution emerges, in the MOSA method

the set dEðPÞEðPÞ will contain potentially efficient solu-

tions.

Since scheduling problems are also combinato-

rial problems, applying the metaheuristics to the

https://www.researchgate.net/publication/222135027_Complexity_of_Single_Machine_Multi-Criteria_Scheduling_Problems?el=1_x_8&enrichId=rgreq-91b359c60509ef391cfb7c4593670918-XXX&enrichSource=Y292ZXJQYWdlOzIyMjQyMjAxODtBUzoxODYxODUyODA0MDE0MTJAMTQyMTQwMTQzODk5MQ==

https://www.researchgate.net/publication/265372574_Single-machine_bicriteria_scheduling?el=1_x_8&enrichId=rgreq-91b359c60509ef391cfb7c4593670918-XXX&enrichSource=Y292ZXJQYWdlOzIyMjQyMjAxODtBUzoxODYxODUyODA0MDE0MTJAMTQyMTQwMTQzODk5MQ==

https://www.researchgate.net/publication/228467980_An_interactive_heuristic_method_for_multi-objective_combinatorial_optimization?el=1_x_8&enrichId=rgreq-91b359c60509ef391cfb7c4593670918-XXX&enrichSource=Y292ZXJQYWdlOzIyMjQyMjAxODtBUzoxODYxODUyODA0MDE0MTJAMTQyMTQwMTQzODk5MQ==

https://www.researchgate.net/publication/226211700_An_Annotated_Bibliography_of_Multiobjective_Combinatorial_Optimization?el=1_x_8&enrichId=rgreq-91b359c60509ef391cfb7c4593670918-XXX&enrichSource=Y292ZXJQYWdlOzIyMjQyMjAxODtBUzoxODYxODUyODA0MDE0MTJAMTQyMTQwMTQzODk5MQ==

https://www.researchgate.net/publication/282059174_A_survey_and_annotated_bibliography_of_multiobjective_combinatorial_optimization?el=1_x_8&enrichId=rgreq-91b359c60509ef391cfb7c4593670918-XXX&enrichSource=Y292ZXJQYWdlOzIyMjQyMjAxODtBUzoxODYxODUyODA0MDE0MTJAMTQyMTQwMTQzODk5MQ==

T. Loukil et al. / European Journal of Operational Research xxx (2003) xxx–xxx 3



production scheduling with multiple criteria issuitable (see for instance Neppali et al. [13]).

The aim of this paper is to show how such

methods can be used to solve complex multiple

objectives scheduling problems by generation of a

list of potentially efficient solutions.

The paper is organized as follows: the multiple

objectives production scheduling models treated

are considered in Section 2 and the notations areintroduced. Section 3 gives a description of the

MOSA heuristic for a general MOCO problem.

Section 4 reports some computational results and

their analysis; conclusions and further research

directions are presented in Section 5.

2. Multi-objective scheduling problems

It is well known that the optimal solution of

single objective models can be quite different if the

objective is different (for instance, for the simplest

model of one machine, without any additional

constraint, the rule SPT is optimal to minimize Fbut the rule EDD is optimal to minimize the

maximal tardiness Tmax).In fact, often each particular decision maker

wants to minimize a given criterion. For example

in a company, the commercial manager is inter-

ested by satisfying customers and then minimizing

the tardiness. On the other hand, the production

manager wishes to optimize the use of the machine

by minimizing the makespan or the work in pro-

cess by minimizing the maximum flow time. Andeach of these objectives is valid from a general

point of view. Since these objectives are conflict-

ing, a solution may perform well for one objective,

but giving bad results for others. For this reason,

scheduling problems have often a multi-objective

nature (see [9,10]).

Therefore, any proposed scheduling approach

has to find a compromise between them. Such acompromise must correspond to a more satisfying

solution to the manager. In any case, such a

solution must be efficient: for this reason, it ap-

pears interesting to generate the set of efficient

solutions––or at least, a good approximation of

this set––in which he can choose the ‘‘best com-

promise’’ in regard of his preferences. Sometimes,

the procedure can also ask him to constructinteractively this compromise solution (see [20]).

In the following, we use the classical notations:

Cj: the completion time for job j;dj: the due date for job j;Tj ¼ maxð0;Cj � djÞ the tardiness of job j;Ej ¼ maxð0; dj � CjÞ the earliness of job j.

2.1. The objective functions

In our study, seven possible objective functions

are considered.

• CðwÞ ¼ 1

n

Pj wjCj: the mean weighted completion

time,

• TðwÞ ¼ 1

n

Pj wjTj: the mean weighted tardiness,

• EðwÞ ¼ 1

n

Pj wjEj: the mean weighted earliness,

• Cmax ¼ maxj Cj: the maximum completion time

(makespan),

• Tmax ¼ maxj Tj: the maximum tardiness,

• Emax ¼ maxj Ej: the maximum earliness,

• NT the number of tardy jobs,

where wj is a possible weight associated to job j. Inthe models considered and solved by the MOSA

method (Section 4), any subset of fCðwÞ; T ðwÞ;EðwÞ

;Cmax; Tmax;Emax;NTg can be chosen to define

the objectives.

Remarks

• Clearly, in the three first objectives, the factor 1n

can be avoided, considering thus total weightedobjectives.

• We consider here the more classical objective

functions for scheduling problems, but this is

not a limitation: any other objective function

can also be tackled by the method without any

additional difficulties.

• In case of non-regular objectives––like E and

Emax––we do not consider any empty times forthe machines.

2.2. The models

We only analyze, in a first step, the scheduling

models for which the solutions correspond to a

https://www.researchgate.net/publication/222632236_Genetic_algorithms_for_the_two-stage_bicriteria_flowshop_problem?el=1_x_8&enrichId=rgreq-91b359c60509ef391cfb7c4593670918-XXX&enrichSource=Y292ZXJQYWdlOzIyMjQyMjAxODtBUzoxODYxODUyODA0MDE0MTJAMTQyMTQwMTQzODk5MQ==





permutation of jobs; such permutations are calledsequences; three types of models are considered:

• One machine model.

• Parallel (non-necessary identical) machines

model: in this case, ðm� 1Þ fictitious jobs are

introduced to separate the jobs assigned to the

m machines. (For instance, for a problem with

nine jobs on three machines, the sequence

9; 3; 2; f ; 4; 6; 1; 8; f ; 7; 5

(with ‘‘f ’’ for fictious), means that jobs 9, 3 and

2 are scheduled on machine 1; jobs 4, 6, 1 and 8

on machine 2 and jobs 7 and 5 on machine 3).

• Permutation flow shop: the jobs share the same

processing order and in addition the sequence ofjobs is identical on each machine.

2.3. Sequence neighborhoods

In the SA scheme used (see Section 3) a neigh-

bor sequence of the current sequence will be cho-

sen randomly at each iteration.

Three different ways are used to define aneighbor of a sequence:

• in the first, two jobs i and j are exchanged in the

sequence;

• in the second, one job i is inserted at a different

place k;

• in the third, at each iteration, a random choice is

made between the first two.

The method of Section 3 has been implemented

with the three types of neighborhoods.

3. The MOSA method

This method is designed to tackle a MOCOproblem (P).

The principle idea of MOSA method can be

briefly summarized as follows.

One begins with an initial iterate X0 and initial-

izes the set of potentially efficient points PE to

contain X0. One then samples a point Y in the

neighborhood of the current iterate. But instead

of accepting Y if it is better than the current iter-

ate on an objective, we now accept it if it is not

dominated by any of the points currently in the

set PE. In this case, we make Y the current iter-ate, add it to PE, and throw out any point in PEthat are dominated by Y .

On the other hand, if Y is dominated, we still

make it the current iterate with some probability.

The only complicated aspect of this scheme is

the necessity to use a weighting function for

computing this probability. The introduction

of a weight vector induces a privileged directionof search to the efficient frontier. So, to be able

to cover all the efficient frontier, a diversified set

of weight vectors must be considered.

3.1. Preliminaries

• A weighting function sðzðX Þ; kÞ is chosen, the ef-

fect of this choice on the procedure is small dueto the stochastic character of the method. The

weighted sum is very well known criterion and

it is the easiest function to compute:

sðzðX Þ; kÞ ¼XKk¼1

kkzkðX Þ:

• The three classic parameters of a SA procedure

are initialized: T0: initial temperature (or alternatively an ini-

tial acceptance probability P0);

a (<1): the cooling factor;

Nstep: the length of temperature step in the

cooling schedule.

• A stopping criterion is fixed:

Nstop: the maximum number of iterations

without improvement.• A neighborhood NðxÞ of feasible solutions in the

vicinity of a solution x is defined. This definition

is problem-dependent (see Section 2.3).

3.2. Determination of PEðkðlÞÞ, l 2 L

The method requires, a wide diversified set of

weights to define a family of weighting functions.




Different weight vectors kl, l 2 L, are generatedwhere ðkðlÞk ; k ¼ 1; . . . ;KÞ with kðlÞk > 0 8k andPK

k¼1 kðlÞk ¼ 1 8 l 2 L.

For each l 2 L the following procedure is ap-

plied to determine a list PEðkðlÞÞ of potentially

efficient solutions.

1. Initialization

• Draw at random an initial solution x0.

Fig. 1. Illustration of PEðkÞ.

This step is problem-dependent.

• Evaluate zkðx0Þ 8k.• PEðkðlÞÞ ¼ fx0g; Ncount ¼ n ¼ 0.

2. Iteration n• Draw at random a solution y 2 NðxnÞ.• Evaluate zkðyÞ and determine Dzk ¼ zkðyÞ�

zkðxnÞ 8k.• Calculate Ds ¼ sðzðyÞ; kÞ � sðzðxnÞ; kÞ.

If Ds6 0, we accept the new solution:

xnþ1 y; Ncount ¼ 0:

Else we accept the new solution with a certain

probability p ¼ expð� DsTnÞ:

xnþ1 p y Ncount ¼ 0;

1�pxn Ncount ¼ Ncount þ 1:

(

• If necessary, update the list PEðkðlÞÞ with thesolution y.

• n nþ 1

–– If nðmodNstepÞ ¼ 0, then Tn ¼ aTn�1;

else Tn ¼ Tn�1;

–– If Ncount ¼ Nstop or T < Tstop, then stop;

else iterate.

3.3. Generation of dEðPÞEðPÞ

Because of the use of a weighting function, a

given weight vector kðlÞ induces a privileged

direction on the efficient frontier (see Fig. 1 in case

of a bi-objective problem).

The procedure generates only a good subset of

potentially efficient solutions in that direction and

these solutions are often dominated by somesolutions generated with another weight vector. So

it is thus necessary to consider a wide diversified

set of weights to cover ‘‘all the directions’’. This set

of weights kðlÞk , l 2 L, is generated uniformly:

kðlÞk 2 0;1

r;2

r; . . . ;

r � 1

r; 1

� �

defining jLj ¼ r þ K � 1

K � 1

� �weight vectors in

diversified directions.

The number jLj must be fixed experimentally in

function of the dimensions of problem (P) (see

Section 4); it must be large enough to cover all the

efficient frontier of problem (P). To obtain a good

approximation dEðPÞEðPÞ to EðP Þ it is thus necessary to

filter the set [jLjl¼1PEðkðlÞÞ. This operation is very

simple and consists only in making pairwise

comparisons of all the solutions contained in the

sets PEðkðlÞÞ and removing the dominated solu-

tions. This filtering procedure is denoted by ^ such

that

dEðPÞEðPÞ ¼jLj

l¼1

PEðkðlÞÞ:

A great number of experiments is required todetermine the number jLj of set of weights suffi-

cient to give a good approximation of the whole

efficient frontier.

4. Experimental results

An intensive work of numerical experimenta-tions has been performed in three major steps,

corresponding to Sections 4.2, 4.3 and 4.4:




• comparison with existing single optimization re-

sults;

• comparison with existing multi-objective opti-

mization results;

• treatment of randomly generated multi-objec-

tive instances.

We first present in Section 4.1 how instances

can be randomly generated.

4.1. Generation of instances

There exists in the literature a classical way torandomly generate instances of scheduling problems:

Table 1

Test on unicriterion problems 40/1//T w

R T Problem

number in [2]

Best solution

Crauwels MO

0.2 0.4 6 173.88 17

7 158.10 15

8 171.63 17

0.6 11 436.63 44

12 482.80 48

13 731.40 73

0.8 16 1807.93 181

17 1965.58 196

18 1857.75 186

0.6 0.4 56 52.48 5

57 56.50 6

58 123.40 12

0.6 61 507.03 51

62 335.08 34

63 494.28 50

0.8 66 1634.65 166

67 1643.90 165

68 1961.28 197

1 0.4 106 0

107 12.90 1

108 83.85 8

0.6 111 786.95 82

112 529.23 53

113 676.93 68

0.8 116 1169.25 120

117 1259.10 127

118 636.50 64

• the processing times are uniformly distributed in

the interval ½0; 100�;• the due dates dj are uniformly distributed in the

interval ½P ð1� T � R2Þ; P ð1� T þ R

2Þ� where

• P ¼Pn

j¼1 pj in the case of a single machine,

• P ¼Pn

j¼1 pj=m in the case of m parallel ma-

chine,

• P ¼ ðnþ m� 1Þ�pp with �pp the mean total pro-cessing time of the n jobs in the case of m ma-

chines in series (permutation flow job),

• R, T are two parameters taking their values in

the sets f0:2; 0:6; 1g and f0:4; 0:6; 0:8g respec-

tively.

Deviation (%) from the best solution founded in

literature vs MOSA

SA Deviation (%) Mean Maximum

8.97 2.93 1.06 2.93

8.50 0.25

1.62 0

0.00 0.77 0.76 0.82

6.10 0.68

7.40 0.82

7.75 0.54 0.33 0.54

8.12 0.13

3.72 0.32

3.98 2.87 6.73 13.10

3.90 13.10

8.60 4.21

2.90 1.16 1.63 2.12

0.48 1.61

4.75 2.12

5.12 1.86 1.07 1.86

7.67 0.84

1.50 0.52

0 0 5.65 15.89

4.95 15.89

4.75 1.07

4.25 4.74 2.76 4.74

8.08 1.67

9.58 1.87

3.80 2.95 2.03 2.95

2.00 1.02

9.98 2.12

Table 2


R T Problem

number in [2]

Best solution Deviation (%) from the best solution founded in

literature vs MOSA

Crauwels MOSA Deviation (%) Mean Maximum

0.2 0.4 6 525.52 525.52 0 0.21 0.64

7 228.06 228.06 0

8 169.98 171.06 0.64

0.6 11 870.08 870.08 0 0.08 0.23

12 727.56 729.22 0.23

13 907.66 907.66 0

0.8 16 1758.04 1758.04 0 0.05 0.14

17 1685.20 1687.50 0.14

18 2095.90 2095.90 0

0.6 0.4 56 25.16 25.16 0 0 0

57 73.58 73.58 0

58 50.44 50.44 0

0.6 61 504.24 511.82 1.50 0.60 1.50

62 346.74 347.70 0.28

63 614.58 614.74 0.03

0.8 66 1537.56 1540.32 0.18 0.09 0.18

67 1708.26 1708.56 0.02

68 1855.12 1856.46 0.07

1 0.4 106 0 0 0 1.36 4.08

107 34.40 35.74 4.08

108 0 0 0

0.6 111 546.20 546.20 0 0.52 1.56

112 317.34 317.34 0

113 702.12 713.08 1.56

0.8 116 714.54 714.54 0 0.10 0.29

117 1438.44 1438.44 0

118 1308.66 1312.46 0.29




When T increases, the due dates are morerestrictive; when R increases, the due date are more

diversified.

4.2. Benchmarks of single optimization problems

Although the MOSA method is designed for

multi-objective framework, it is possible to apply it

for single-objective problems. Moreover, for suchproblems, there exist some benchmarks in the lit-

erature. It is also an opportunity to test the influ-

ence of the SA parameters (see Section 3), to fix the

value of these parameters and to validate the SA

approach. After analysis, the following valueshave been chosen:

T0 ¼ 50; a ¼ 0:975; Nstep ¼ 500;

Tstop ¼ 1e� 0:4; Nstop ¼ 2500:

4.2.1. Single machine problems

In Crauwels et al. [2] there exists a large

benchmark for single machine instances with

objective TðwÞ

, giving the optimal value (or the best

known value).For instances with 40, 50 and 100 jobs respec-

tively, we choose three problems for each pair of

https://www.researchgate.net/publication/220669223_Local_Search_Heuristics_for_the_Single_Machine_Total_Weighted_Tardiness_Scheduling_Problem?el=1_x_8&enrichId=rgreq-91b359c60509ef391cfb7c4593670918-XXX&enrichSource=Y292ZXJQYWdlOzIyMjQyMjAxODtBUzoxODYxODUyODA0MDE0MTJAMTQyMTQwMTQzODk5MQ==




parameters ðR; T Þ (see Section 4.1). So 81 problemsare solved.

For each type of problems, Tables 1–3 give the

slack between the value obtained with MOSA for

objective TðwÞ

and the best known value, in terms

of percentages of it. It appears

• that the performance is improved with the

dimension of the instance: in Table 3 (100 jobs)the slack is often zero and almost always less

than 1%;

• for fixed value of parameter T , the performances

decrease when R is increases i.e. when the due

dates are more dispersed.

Table 3


R T Problem

number in [2]

Best solution

Crauwels MO

0.2 0.4 6 582.58 58

7 509.72 50

8 594.34 59

0.6 11 1816.49 181

12 2341.79 234

13 1788.40 178

0.8 16 4077.03 407

17 3328.04 332

18 5448.38 545

0.6 0.4 56 90.46 9

57 115.39 11

58 163.13 16

0.6 61 867.93 87

62 870.67 87

63 965.63 96

0.8 66 2438.72 244

67 4010.23 401

68 3990.85 399

1 0.4 106 0

107 11.93 1

108 0

0.6 111 1591.23 160

112 1743.67 175

113 911.69 91

0.8 116 3706.14 371

117 3244.37 325

118 2462.37 246

4.2.2. Permutation flow shop problems

Eleven instances, respectively with

• 5 machines: 20, 50 and 100 jobs;

• 10 and 20 machines; 20, 50, 100 and 200 job-

s;have been chosen in the benchmark of Taillard

[19]. These instances concern the minimization

of the makespan Cmax for permutation flow shopproblems.

For each instance, five experiments of MOSA

have been realized and Table 4 presents the mean

and maximal slack obtained in regard to the

optimal value.

Deviation (%) from the best solution founded in

literature vs MOSA

SA Deviation (%) Mean Maximum

2.58 0 0.02 0.07

9.72 0

4.76 0.07

7.11 0.03 0.01 0.03

1.79 0

8.40 0

7.03 0 0.02 0.06

8.04 0

1.81 0.06

4.78 4.78 1.67 4.78

5.39 0

3.53 0.25

3.29 0.62 0.41 0.62

2.17 0.17

9.95 0.45

2.83 0.17 0.13 0.18

7.25 0.18

3.16 0.06

0 0 0 0

1.93 0

0 0

2.04 0.68 0.65 0.83

1.11 0.43

9.29 0.83

4.37 0.22 0.21 0.27

3.24 0.27

5.68 0.13

•

Table 4

Test on unicriterion problems n/mPF//Cmax

Number of machines Number of tasks Deviation (%) from the best solution founded in literature vs MOSA

Mean Maximum

m ¼ 5 20 0.08 0.39

50 0.59 2.97

100 0.05 0.11

m ¼ 10 20 1.19 1.77

50 0.34 0.64

100 0.29 0.60

200 0.2 0.39

m ¼ 20 20 0.69 3.43

50 1.8 2.28

100 0.96 1.63

200 1.01 1.74

Values of parameters

P0 0.50 Tstop 1e)4

Nit 500 Nstop 5000

a 0.975 Vois. 2

T0 50 Weights 25




The performances appear very stable and quitegood with a mean slack almost always less than

1% (see Table 4).

4.3. Existing multi-objective problems in the litera-

ture

We test the method on some small multi-

objective problems found in the literature.Table 5 gives the results of ten experi-

ments corresponding to Refs. [3,5,7,8,12,15–18,

21].

The columns of this table indicate successively:

1. the reference, the type and the dimension of the

instance;

2. the character, exact or heuristic, of the method;3. the objectives considered.

Note that in Shanthikumar�s paper [18], a

hierarchical optimization is made with NT asmain objective; in papers of Selen–Hott [17],

Ho–Chang [5] and Rajendran [15], a linear

aggregated function of the objectives is opti-

mized;

4. the solutions obtained by these authors;

5. the solutions obtained by our SA approach.

Clearly we found identical solutions (and evensometimes better) with our unique method, prov-

ing that it is quite general. There is just one small

difference for the Koksalan�s method [7] which is

also a heuristic: the approximation of the efficient

set is not completely identical.

4.4. Randomly generated instances

4.4.1. Single machine

Instances with 10, 20, 50 and 100 jobs have

been randomly generated with the method de-

scribed in Section 4.1; all 15 pairs of objectives

have been considered (the makespan is con-

stant for this model and thus not taken into ac-

count). Several hundred experiments have been

realized.

(a) First the impact of some parameters has been

analyzed.

Table 6 shows the impact of parameters Rand T , used in the random generation of

the instances, on the number of potential

efficient solutions obtained. Two cases are

compared: (R ¼ 0:6; T ¼ 0:6) and (R ¼ 0:4;T ¼ 0:3).

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Table 5

Test on the problems founded in literature

Problems (source) Methods Criteria Obtained solutions

By authors With MOSA

Nelson et al. [12],

n ¼ 6, m ¼ 1

Exact (F , NT) (130, 1) (103, 2) (98,3) (93,4) Idem author

(NT, Tmax) (3, 50) (2, 95) (1, 185) Idem author

(F , Tmax) (93, 55) (129, 50) Idem author

(F , NT, Tmax) (137, 1, 185) (130, 1, 190)

(129, 3, 50) (116, 2, 95)

(109, 2, 100) (103, 2, 145)

(98, 3, 55) (93, 4, 55)

Idem author

Shanthikumar [18],

n ¼ 8, m ¼ 1

Exact (Tmax, NT) (44, 3) (44, 3) (37, 4) (29, 5)

Van Wassenhove and

Gelders [21], n ¼ 10, m ¼ 1

Exact (F , Tmax) (26.8, 20) (26.9, 12) (27.2, 10)

(27.4, 7) (31.5, 6)

Idem author

Koksalan (1996)

n ¼ 15, m ¼ 1

Heuristic (F , Tmax) (35.6, 65) (35.7, 57) (35.9, 49)

(36.1, 42) (36.4, 35) (36.7, 28)

(37.2, 23) (37.9, 19) (38.7, 18)

(38.8, 16) (39.8, 15) (39.9, 14)

(41.4, 13) (42.7, 12)

(35.6, 65) (35.7, 57) (35.9,

49) (36.1, 42) (36.4, 35)

(37.2, 23) (37.7, 19) (38.7,

18) (38.8, 16) (40.1, 14)

(42.5, 13) (44.1, 12)

Selen et Hott [17], n ¼ 6,

m ¼ 4, Flow shop

Exact (Cmax, F ) (78, 53) Idem author

Ho et Chang [5], n ¼ 5,

m ¼ 4, Flow shop

Heuristic (Cmax, F ) (213, 156) (213, 156) (234, 155)

Rajendran [15], n ¼ 5,

m ¼ 2, Flow shop

Exact (Cmax, F ) (78, 44, 8) (78, 44.8) (79, 44.2)

Rajendran [16], n ¼ 5,

m ¼ 3, Flow shop

Heuristic (Cmax, F ) (52, 29) (51, 31) (51, 29)

Liao et al. [8]a, n ¼ 6,

m ¼ 2, Flow shop

Exact (Cmax, NT) (29, 1) Idem author

(Cmax, T ) (29, 2) Idem author

Hoogeveen [6], n ¼ 4,

m ¼ 1

Exact (C, Tmax) (12.5, 1) (12.25, 2) (11.75, 3)

(11, 4) (10.25, 8) (10, 13)

Idem author

P0 ¼ 0:5, L ¼ 500, a ¼ 0:975, Nstop ¼ 2500, Tstop ¼ 10–0.4, neighborhood¼ 3, five weight sets.a Liao et al. [8] gave the lower bound for Cmax NT and T , given the partial sequence [1,4] always at the end of the sequences. MOSA

give the same results by scheduling the four remaining tasks and adding tasks 1 and 4 at the end of the solutions obtained; nevertheless,

if we release this constraint, we obtain the following results ð29; 0Þ ð28; 1Þ for ðCmax;NTÞ et (29, 0) (28, 1.33) for ðCmax; T Þ.




The effect induced is not systematic and de-

pends of the pair of objectives optimized; nev-

ertheless a tendency exists: with tardiness

objectives ðT ;NT; TmaxÞ the number of solutions

decrease in the second case corresponding to

more large and concentrated due dates and theinverse situation seems true when objective E is

considered.

• Table 7 compares the results obtained respec-

tively with jLj ¼ 5 and 25 sets of weights in

the MOSA procedure.

The interpretation of such table is the following:

• a1 is j dE1ðPÞE1ðPÞj obtained with L ¼ 5;

• a2 is j dE2ðPÞE2ðPÞj obtained with L ¼ 25;

• h is the number of common solutions to these

two sets

h ¼ j dE1ðPÞE1ðPÞ\ dE2ðPÞE2ðPÞj;

• b1ðb2Þ is the number of solutions of dE1ðPÞE1ðPÞð dE2ðPÞE2ðPÞÞ efficient in regard with all the gener-

ated solutions i.e.

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� �

Table 6

Number of potential efficient solutions for different parameters

Criteria 10 tasks 20 tasks 50 tasks 100 tasks

R ¼ 0:6,

T ¼ 0:6

R ¼ 0:4,

T ¼ 0:3

R ¼ 0:6,

T ¼ 0:6

R ¼ 0:4,

T ¼ 0:3

R ¼ 0:6,

T ¼ 0:6

R ¼ 0:4,

T ¼ 0:3

R ¼ 0:6,

T ¼ 0:6

R ¼ 0:4,

T ¼ 0:3

C, T 33 11 57 32 2 19 4 11

C, E 86 529 150 715 164 560 232 823

C, NT 3 3 4 3 7 2 7 4

C, Tmax 10 4 11 11 33 16 12 13

C, Emax 15 6 22 18 52 40 32 12

T , E 1 5 9 17 25 33 75 64

T , NT 5 1 3 1 7 5 7 1

T , Tmax 8 3 10 2 11 5 9 4

T , Emax 1 1 3 2 9 8 10 11

E, NT 5 5 12 5 20 9 35 24

E, Tmax 1 1 4 2 5 3 13 17

E, Emax 1 2 3 3 6 16 9 5

NT, Tmax 5 3 5 2 10 3 12 9

NT, Emax 2 1 6 1 10 2 7 7

Tmax, Emax 1 1 3 2 2 1 11 14

Total 177 576 302 817 363 723 475 1019

(R ¼ 0:6, T ¼ 0:6) and (R ¼ 0:4, T ¼ 0:3), m ¼ 1, 25 weights, Nstop ¼ 2500, neighborhood¼ 3.

Table 7

Number of potential efficient solutions for different set of weights (5 and 25)


a1=b1 h a2=b2 a1=b1 h a2=b2 a1=b1 h a2=b2 a1=b1 h a2=b2

C, T 33/33 33 33/33 50/6 0 57/50 2/1 0 2/2 4/0 0 4/4

C, E 86/86 86 86/86 125/28 0 150/119 135/13 0 164/149 174/45 0 232/183

C, NT 3/3 3 3/3 4/0 0 4/4 5/0 0 7/7 8/1 0 7/6

C, Tmax 10/10 10 10/10 11/2 1 11/11 27/5 0 33/28 13/9 9 12/12

C, Emax 15/15 15 15/15 17/1 1 22/22 34/8 0 52/39 28/12 0 32/24

T , E 1/1 1 1/1 11/0 0 9/9 41/7 0 25/22 61/12 0 75/59

T , NT 5/5 5 5/5 3/0 0 3/3 9/5 0 7/4 6/1 0 7/7

T , Tmax 8/8 8 8/8 6/1 0 10/9 11/2 0 11/10 16/0 0 9/9

T , Emax 1/1 1 1/1 3/3 3 3/3 6/2 0 9/9 10/10 2 10/4

E, NT 5/5 5 5/5 10/2 0 12/9 23/7 0 20/16 27/7 0 35/28

E, Tmax 1/1 1 1/1 7/0 0 4/4 4/0 0 5/5 12/12 12 13/13

E, Emax 1/1 1 1/1 3/3 1 3/1 5/0 0 6/6 6/4 4 9/9

NT, Tmax 5/5 5 5/5 5/2 2 5/5 9/0 0 10/10 12/8 7 12/12

NT, Emax 2/2 2 2/2 7/0 0 6/6 8/0 0 10/10 6/0 0 7/7

Tmax,

Emax

1/1 1 1/1 9/0 0 3/3 7/0 0 2/2 6/0 0 11/11

Total 177/177 177 177/177 271/48 8 302/258 326/50 0 363/319 389/121 34 475/388

m ¼ 1, R ¼ 0:6, T ¼ 0:6, neighborhood¼ 3, Nstop ¼ 2500.




bi ¼ dEiðPÞEiðPÞ\ dE1ðPÞE1ðPÞ

^ dE2ðPÞE2ðPÞ :It appears from this table that the increase of

the number of weight sets does not improve

the results when n ¼ 10 but is wished when

the number of jobs is increased.

• Table 8 does the same for the value of the

stopping criterion Nstop ¼ 2500 or Nstop ¼

Table 8

Number of potential efficient solutions for two values of Nstop (2500 and 5000)



C, T 33/33 33 33/33 57/20 0 71/54 2/0 0 55/55 4/0 0 31/31

C, E 86/86 86 86/86 150/150 150 150/150 164/164 164 164/164 232/232 232 232/232

C, NT 3/3 3 3/3 4/2 1 4/3 7/2 0 13/11 7/3 0 9/7

C, Tmax 10/10 10 10/10 11/6 0 13/8 33/27 13 31/19 12/6 4 20/17

C, Emax 15/15 15 15/15 22/10 1 27/21 52/23 0 45/28 32/6 0 40/37

T , E 1/1 1 1/1 9/8 0 17/2 25/19 7 39/14 75/38 38 61/61

T , NT 5/5 5 5/5 3/1 0 3/3 7/1 0 7/6 7/5 0 17/12

T , Tmax 8/8 8 8/8 10/9 0 12/3 11/10 9 20/19 9/0 0 13/13

T , Emax 1/1 1 1/1 3/0 0 5/5 9/1 0 20/19 10/0 0 10/10

E, NT 5/5 5 5/5 12/12 12 12/12 20/8 1 21/13 35/35 35 35/35

E, Tmax 1/1 1 1/1 4/1 0 4/4 5/0 0 7/7 13/0 0 5/5

E, Emax 1/1 1 1/1 3/0 0 1/1 6/3 0 10/9 9/0 0 16/16

NT, Tmax 5/5 5 5/5 5/0 0 5/5 10/9 3 10/4 12/0 0 17/17

NT, Emax 2/2 2 2/2 6/6 2 7/2 10/6 1 7/5 7/0 0 4/4

Tmax, Emax 1/1 1 1/1 3/0 0 1/1 2/0 0 1/1 11/0 0 1/1

Total 177/177 177 177/177 302/225 166 332/274 363/273 198 450/374 475/325 279 511/498

m ¼ 1, R ¼ 0:6, T ¼ 0:6, neighborhood¼ 3.

Ta

Nu

M




5000. The interpretation of this table is simi-

lar to the previous one (a1 and b1 correspond

to Nstop ¼ 2500). The increase of Nstop does

not change the results for instances with 10

jobs; but the results are improved with a lar-

ble 9

mber of potential efficient solutions for two different neighborhoods

Criteria 10 tasks 20 tasks

a1=b1 h a2=b2 a1=b1 h a2=b2

C, T 33/33 33 33/33 68/68 66 66/66

C, E 86/86 86 86/86 190/190 134 134/134

C, NT 3/3 3 3/3 4/4 4 4/4

C, Tmax 10/10 10 10/10 19/10 0 19/10

C, Emax 15/15 15 15/15 36/36 0 32/0

T , E 1/1 1 1/1 15/14 0 16/1

T , NT 5/5 5 5/5 4/4 1 5/2

T , Tmax 8/8 8 8/8 10/7 1 43/4

T , Emax 1/1 1 1/1 4/1 1 3/3

E, NT 5/5 5 5/5 12/8 1 12/4

E, Tmax 1/1 1 1/1 3/0 0 2/2

E, Emax 1/1 1 1/1 2/1 0 3/1

NT, Tmax 5/5 5 5/5 5/3 3 5/5

NT, Emax 2/2 2 2/2 6/6 2 6/2

Tmax, Emax 1/1 1 1/1 1/1 1 1/1

Total 177/177 177 177/177 379/353 214 321/239

¼ 1, R ¼ 0:6, T ¼ 0:6, 25 weights, Nstop ¼ 2500.

ger value of Nstop when the number of jobs in-

crease.

• Tables 9–11 compare the use of the three

neighborhoods described in Section 2.3,

respectively for ð1; 2Þ, ð1; 3Þ and ð2; 3Þ.

(2, 1)

50 tasks 100 tasks

a1=b1 h a2=b2 a1=b1 h a2=b2

11/0 0 11/11 6/6 0 8/2

188/188 0 134/0 271/271 0 209/0

7/6 0 8/1 7/6 0 8/1

19/15 0 25/15 30/21 0 27/12

53/53 0 49/1 60/60 0 47/0

35/35 0 27/0 94/94 0 49/0

7/5 0 6/2 5/4 0 8/4

14/14 0 29/10 22/22 0 19/15

17/17 0 13/0 14/14 0 12/0

26/23 0 25/4 33/28 0 26/8

5/0 0 2/12 6/5 0 10/10

5/5 0 5/0 30/30 0 19/1

8/8 0 10/0 14/8 0 12/10

3/3 0 6/0 1/1 0 4/0

2/2 0 4/0 4/0 0 1/1

400/375 0 454/56 475/567 0 459/66

Table 10

Number of potential efficient solutions for two different neighborhoods (3, 1), m ¼ 1



C, T 33/33 33 33/33 57/13 0 66/50 2/2 0 11/5 4/4 0 8/0

C, E 86/86 86 86/86 150/91 0 134/49 164/162 0 134/5 232/231 0 209/5

C, NT 3/3 3 3/3 4/1 1 4/ 4 7/7 0 8/1 7/5 0 8/2

C, Tmax 10/10 10 10/10 11/0 0 19/19 33/29 0 25/2 12/2 0 27/21

C, Emax 15/15 15 15/15 22/5 0 32/26 52/47 0 49/7 32/31 0 47/15

T , E 1/1 1 1/1 9/9 0 16/0 25/25 0 27/1 75/62 0 49/6

T , NT 5/5 5 5/5 3/0 0 5/5 7/4 0 6/3 7/1 0 8/7

T , Tmax 8/8 8 8/8 10/0 0 13/13 11/10 0 29/9 9/0 0 19/19

T , Emax 1/1 1 1/1 3/0 0 3/3 9/8 0 13/1 10/1 0 12/12

E, NT 5/5 5 5/5 12/5 0 12/6 20/18 0 25/3 35/30 0 26/4

E, Tmax 1/1 1 1/1 4/0 0 2/2 5/0 0 2/2 13/0 0 10/10

E, Emax 1/1 1 1/1 3/0 0 3/3 6/6 0 5/0 9/2 0 19/19

NT, Tmax 5/5 5 5/5 5/0 0 5/5 10/9 0 10/1 12/2 0 12/12

NT, Emax 2/2 2 2/2 6/5 3 6/4 10/10 0 6/0 7/4 0 4/2

Tmax, Emax 1/1 1 1/1 3/0 0 1/1 2/2 0 4/0 11/0 0 1/1

Total 177/177 177 177/177 302/129 4 321/190 363/339 0 454/40 475/375 0 459/135

R ¼ 0:6, T ¼ 0:6, 25 weights, Nstop ¼ 2500.

Table 11

Number potential efficient solutions for two different neighborhoods (3, 2)



C, T 33/33 33 33/33 57/15 0 68/54 2/2 0 11/0 4/4 0 6/0

C, E 86/86 86 86/86 150/23 0 190/162 164/20 0 188/166 232/0 0 271/271

C, NT 3/3 3 3/3 4/1 1 4/ 4 7/4 0 7/4 7/1 0 7/6

C, Tmax 10/10 10 10/10 11/0 0 19/19 33/21 0 19/11 12/0 0 30/30

C, Emax 15/15 15 15/15 22/0 0 36/36 52/1 0 53/51 32/0 0 60/60

T , E 1/1 1 1/1 9/9 0 15/3 25/0 0 35/35 75/0 0 94/94

T , NT 5/5 5 5/5 3/0 0 4/4 7/2 0 7/5 7/0 0 5/5

T , Tmax 8/8 8 8/8 10/0 0 10/10 11/7 0 14/9 9/0 0 22/22

T , Emax 1/1 1 1/1 3/0 0 4/4 9/0 0 17/17 10/0 0 14/14

E, NT 5/5 5 5/5 12/6 1 12/7 20/3 0 26/24 35/14 0 33/24

E, Tmax 1/1 1 1/1 4/0 0 3/3 5/5 0 5/0 13/0 0 6/6

E, Emax 1/1 1 1/1 3/0 0 2/2 6/5 0 5/2 9/0 0 30/30

NT, Tmax 5/5 5 5/5 5/0 0 5/5 10/2 1 8/7 12/0 0 14/14

NT, Emax 2/2 2 2/2 6/1 1 6/6 10/0 0 3/3 7/0 0 1/1

Tmax, Emax 1/1 1 1/1 3/0 0 1/1 2/1 1 2/ 2 11/0 0 4/4

Total 177/177 177 177/177 302/55 3 379/320 363/73 2 400/336 475/19 0 597/581

m ¼ 1, R ¼ 0:6, T ¼ 0:6, 25 weights, Nstop ¼ 2500.




The interpretation of these tables is similar tothe two previous ones.

It appears that often-specially when the

dimensions of the instance increase––the

neighborhood 2 gives the best results andneighborhood 1 the worse.

(b) The number of potential efficient solutions and

the CPU time have been analyzed for each pair

of objectives.

100

150

200

250

T max




Tables 12 and 13 give the mean and the maxi-mal value among three randomly generated in-

stances with 50 jobs. These experiments are

realized with the following values:

jLj ¼ 5; Nstop ¼ 2500; neighborhood 3;

R ¼ 0:6; T ¼ 0:6:

Table 12

Problems 50/1/ /C1, C2 sorted according to mean decreasing

CPU

Comb. CPU (seconds)

Mean Maximum

E, NT 36.00 38

C, Emax 34.67 41

C, E 33.00 33

NT, Emax 32.67 37

C, Tmax 29.33 32

T , E 27.00 27

NT, Tmax 26.67 41

T , NT 26.00 34

T , Emax 23.67 35

T , Tmax 20.67 26

E, Tmax 16.00 18

E, Emax 16.00 31

C, NT 14.00 19

Tmax, Emax 13.00 22

C, T 0.67 1

Five weights sets, neighborhood¼ 3, Nstop ¼ 2500, R ¼ 0:6,

T ¼ 0:6.

Table 13

Problems 50/1/ /C1, C2 sorted according to mean decreasing

=PE=

Comb. =PE=

Mean Maximum

C, E 123.00 135

T , E 41.67 46

C, Emax 33.00 36

C, Tmax 24.67 27

E, NT 22.00 23

T , Tmax 12.33 19

NT, Tmax 12.33 18

E, Emax 10.67 20

T , NT 10.33 16

E, Tmax 9.67 13

C, NT 7.00 8

Tmax, Emax 6.00 7

T , Emax 5.33 6

NT, Emax 5.00 8

C, T 2.33 4

Five weights sets, neighborhood¼ 3, Nstop ¼ 2500, R ¼ 0:6,

T ¼ 0:6.

0

50

96 98 100 102 104 106 108Cmoyen

Fig. 2. Efficient frontier.

There is a clear tendency to have a larger

number of solutions with objective E, speciallywhen the other objective is C.

For each instance we can of course visualize

graphically the results in the objective space. Fig. 2

represents the 25 potential non-dominated points

Table 14

Problems 50/P 10/ /C1, C2 sorted according to mean decreasing

CPU

Criteria CPU

Mean Maximum

Cmax, E 49.00 53

E, Tmax 48.67 53

E, NT 47.00 51

C, E 36.33 38

T , E 27.00 31

Tmax, Emax 21.00 28

C, Emax 20.67 36

T , Emax 20.67 28

NT, Emax 19.00 23

Cmax, Emax 16.67 32

E, Emax 9.33 14

C, NT 2.67 8

C, T 0.67 1

C, Tmax 0.33 1

NT, Tmax 0.33 1

Cmax, C 0.33 1

T , NT <1 <1

T , Tmax <1 <1

Cmax, T <1 <1

Cmax, NT <1 <1

Cmax, Tmax <1 <1

m ¼ 10, five weights by criterion, neighborhood¼ 3,

Nstop ¼ 2500, R ¼ 0:6, T ¼ 0:6.




in the space ðC; TmaxÞ for one instance with 50 jobs.In this way we can have an image of the efficient

frontier.

4.4.2. Parallel machines

Instances with 20 or 50 jobs, 3 or 10 machines

have been randomly generated with the method

described in Section 4.1; all the 21 pairs of objec-

tives have been considered.

(a) Tables 14 and 15 give respectively the mean

and maximal CPU times, the mean and maxi-

mal cardinality of dEðPÞEðPÞ among three randomly

generated instances with 50 jobs and 10 ma-

chines. These experiments are realized with

the following values for parameters:

jLj ¼ 5; Nstop ¼ 2500; neighborhood 3;

R ¼ 0:6; T ¼ 0:6:

It appears that the cardinality of dEðPÞEðPÞ is large

only when one of the objectives is E (or Emax) i.e.

Table 15

Problems 50/P 10/ /C1, C2 sorted according to mean decreasing

=PE=

Criteria =PE=

Moy Maximum

C, E 404.67 452

T , E 62.33 75

E, NT 37.33 43

Cmax, E 35.67 41

E, Tmax 23.67 29

C, Emax 16.33 18

Cmax, Emax 12.00 13

NT, Emax 11.67 13

Tmax, Emax 10.67 11

T , Emax 8.67 11

Cmax, C 2.00 2

Cmax, T 1.67 2

Cmax, NT 1.67 2

C, T 1.33 2

C, NT 1.33 2

C, Tmax 1.33 2

T , Tmax 1.33 2

Cmax, Tmax 1.33 2

T , NT 1.00 1

E, Emax 1.00 1

NT, Tmax 1.00 1


Nstop ¼ 2500, R ¼ 0:6, T ¼ 0:6.

when two more contradictory objectives are con-sidered.

As for the case of one machine, the pair ðE;CÞgenerates really more solutions.

Otherwise, very few solutions are obtained.

Consequently, the CPU time is quite different

according the objectives adopted.

(b) Tables 16 and 17 analyze the impact of param-

eters jLj (¼ 5 or 25) and N stop (¼ 2500 or5000) for problems with three machines, 20

and 50 jobs respectively.

Clearly the impact of the increase of jLj is

important: not on the number of solutions, but on

the quality of the approximation of the efficient

frontier.

Nevertheless, this is not the case for Nstop.

4.4.3. Permutation flowshop

Similar experiments are made for permutation

flow shop models.

Tables 18–21 give the corresponding results and

the comments are exactly the same that for the

case of parallel machines.

5. Conclusions and further research directions

(1) Metaheuristics can be adapted to a multi-

objective framework [3]. In particular, MOSA

method appears as an appropriate tool to analyze

MOCO problems.

In case of multi-objective scheduling models,the main advantage of this method is to be able, by

a unique procedure, to solve a broad class of

models, with various objectives, even for relatively

large instances and any number of objectives.

The numerical results of Section 4 prove the

efficiency of the MOSA method. Yet, an important

parameter is the number jLj of weight vectors to be

used; it must be large enough to cover all theefficient frontier and must be fixed experimentally

as a function of the dimensions of the instance.

(2) We think it is possible to extend the class of

scheduling problems treated by the method, for

instance to general flow shops. Effectively, in this

case we can represent a schedule (i.e. a permuta-

tion of the n jobs for each of the m machines) by a


Table 16

Number potential efficient solutions for Nstop and weights

Criteria 20 tasks, Nstop ¼ 2500, 5 weights/25 weights 20 tasks, Nstop ¼ 5000, 5 weights/25 weights

a1=b1 h a2=b2 a1=b1 h a2=b2

C, T 4/2 0 3/3 1/1 0 2/1

C, E 54/13 0 78/61 54/13 0 78/61

C, NT 3/3 3 3/3 3/0 0 2/2

C, Tmax 6/2 0 9/8 7/2 0 7/7

C, Emax 19/4 0 19/16 19/4 0 19/16

T , E 14/7 0 31/16 14/7 0 31/16

T , NT 1/1 1 1/1 1/1 1 1/1

T , Tmax 5/1 1 4/ 4 1/1 1 1/ 4

T , Emax 9/4 1 11/6 9/4 1 11/6

E, NT 14/3 0 14/11 14/3 0 14/11

E, Tmax 10/0 0 9/9 10/0 0 9/9

E, Emax 1/1 1 1/1 1/1 1 1/1

NT, Tmax 3/0 0 1/1 1/1 1 1/1

NT, Emax 8/8 8 8/8 8/1 1 6/6

Tmax, Emax 5/5 5 5/5 5/3 3 4/ 4

Cmax, C 1/1 1 1/1 1/0 0 1/1

Cmax, T 3/0 0 3/3 2/0 0 2/2

Cmax, E 12/2 1 14/12 12/4 0 14/4

Cmax, NT 3/0 0 2/2 1/1 1 1/1

Cmax, Tmax 3/0 0 1/1 2/0 0 2/2

Cmax, Emax 9/1 1 9/9 8/2 1 9/8

Total 187/58 23 227/181 174/49 11 216/164

Three parallel machines; neighborhood¼ 3.

Table 17

Number of potential efficient solutions for Nstop and weights


a1=b1 h a2=b2 a1=b1 h a2=b2

C, T 1/1 1 1/1 1/1 1 1/1

C, E 999/863 705 999/986 999/863 705 999/986

C, NT 1/1 1 1/1 1/1 1 1/1

C, Tmax 1/1 1 1/1 1/1 1 1/1

C, Emax 27/7 6 35/34 27/12 7 31/25

T , E 63/8 4 64/60 63/8 4 64/60

T , NT 1/1 1 1/1 1/1 1 1/1

T , Tmax 1/1 1 1/1 1/1 1 1/1

T , Emax 13/1 0 14/13 15/1 1 13/13

E, NT 35/13 0 36/21 35/13 0 36/21

E, Tmax 44/25 0 43/25 44/16 0 44/36

E, Emax 1/0 0 1/1 1/1 1 1/1

NT, Tmax 1/1 1 1/1 1/1 1 1/1

NT, Emax 14/2 2 7/7 14/2 2 7/7

Tmax, Emax 15/10 10 15/15 15/10 10 15/15

Cmax, C 1/1 1 1/1 1/1 1 1/1

Cmax, T 1/1 1 1/1 1/1 1 1/1

Cmax, E 74/34 25 80/73 74/34 25 80/73

Cmax, NT 1/1 1 1/1 1/1 1 1/1

Cmax, Tmax 1/1 1 1/1 1/1 1 1/1

Cmax, Emax 22/4 4 28/18 23/4 4 28/18

Total 1317/977 766 1332/1263 1320/974 769 1328/1265

Three parallel machines; neighborhood¼ 3.




Table 18

Problems 50/PF/ /C1, C2 sorted according to mean decreasing

CPU

Criteria CPU (in seconds)

Mean Maximum

Cmax, E 176.33 179

NT, Emax 169.67 189

E, Tmax 156.67 181

E, NT 150.33 159

Tmax, Emax 141.00 182

NT, Tmax 135.00 149

C, Emax 123.00 198

C, E 122.00 129

Cmax, Emax 122.00 156

T , Emax 116.67 138

T , NT 115.67 142

C, Tmax 108.00 141

T , E 106.00 111

Cmax, NT 104.00 131

Cmax, Tmax 98.67 159

Cmax, C 91.00 170

T , Tmax 85.67 141

C, NT 75.00 99

Cmax, T 58.67 74

E, Emax 37.00 97

C, T 3.00 4


Nstop ¼ 2500, R ¼ 0:6, T ¼ 0:6.

Table 19

Problems 50/PF/ /C1, C2 sorted according to mean decreas-

ing=PE=

Criteria =PE=

Mean Maximum

C, E 28.33 32

C, Emax 19.00 29

NT, Tmax 15.67 18

Cmax, T 13.33 15

T , Tmax 13.00 16

E, NT 12.33 13

T , NT 11.67 16

NT, Emax 11.67 15

C, Tmax 10.33 19

T , Emax 10.33 14

Cmax, C 9.67 13

Tmax, Emax 9.33 12

Cmax, Tmax 9.00 11

Cmax, Emax 8.67 19

T , E 8.00 15

Cmax, E 7.67 14

C, NT 7.33 9

Cmax, NT 6.67 9

E, Emax 6.33 16

E, Tmax 3.67 5

C, T 2.00 4


Nstop ¼ 2500, R ¼ 0:6, T ¼ 0:6.

Table 20

Number of potential efficient solutions for different Nstop and weights


a1=b1 h a2=b2 a1=b1 h a2=b2

C, T 4/0 0 4/4 3/1 1 7/7

C;E 19/2 0 34/33 28/1 0 28/28

C, NT 6/0 0 4/4 5/3 0 4/2

C, Tmax 12/3 0 16/6 15/0 0 14/4

C, Emax 10/9 9 13/13 9/0 0 10/10

T , E 16/2 0 10/8 9/1 0 13/12

T , NT 5/1 0 5/ 4 6/3 0 4/3

T , Tmax 6/5 5 7/7 6/0 0 13/13

T , Emax 2/2 2 2/2 2/1 1 2/ 2

E, NT 11/0 0 10/10 11/0 0 10/10

E, Tmax 4/4 4 4/4 4/3 3 4/ 4

E, Emax 1/1 1 1/1 1/1 1 1/1

NT, Tmax 7/7 7 7/7 6/4 2 6/5

NT, Emax 8/8 8 8/8 7/3 2 6/5

Tmax, Emax 3/3 1 1/1 2/0 0 3/3

Cmax;C 4/2 0 5/2 4/2 0 3/0

Cmax; T 4/3 0 4/3 3/0 0 1/1

Cmax;E 3/2 0 3/1 8/0 0 4/4

Cmax, NT 3/2 2 3/3 2/1 1 2/2

Cmax, Tmax 7/2 2 4/4 5/1 1 4/ 4

Cmax, Emax 3/1 1 2/2 2/2 2 2/2

Total 138/59 42 147/127 138/27 14 141/22

Three machines in series; neighborhood¼ 3.




Table 21

Number of potential efficient solutions for different Nstop and weights


a1=b1 h a2=b2 a1=b1 h a2=b2

C, T 4/0 0 3/3 8/4 4 32/32

C, E 68/19 0 85/63 68/19 0 85/63

C, NT 9/0 0 7/7 12/3 0 10/9

C, Tmax 22/0 0 24/24 30/8 0 27/25

C, Emax 26/9 0 40/29 30/3 0 32/30

T , E 25/0 0 23/23 25/0 0 23/23

T , NT 12/2 0 9/8 12/8 0 6/4

T , Tmax 24/13 0 22/14 28/7 0 26/21

T , Emax 10/3 0 7/6 9/3 0 7/5

E, NT 24/7 0 22/16 24/3 0 23/20

E, Tmax 7/3 0 19/16 10/6 1 15/9

E, Emax 6/0 0 7/7 6/0 0 16/16

NT; Tmax 11/1 0 15/14 11/2 1 13/13

NT; Emax 6/3 0 5/5 6/0 0 6/6

Tmax, Emax 4/1 1 3/3 3/1 1 3/3

Cmax, C 4/4 0 2/0 10/0 0 3/3

Cmax, T 5/0 0 1/1 4/0 0 3/3

Cmax, E 12/3 3 9/9 10/4 3 9/9

Cmax, NT 5/0 0 1/1 2/0 0 1/1

Cmax, Tmax 7/3 2 4/ 4 5/3 3 5/5

Cmax, Emax 4/3 3 5/5 4/3 3 5/5

Total 295/74 9 313/258 317/77 16 350/305

Three machines in series; neighborhood¼ 3.




sequence of nm characters (i.e. m times each index iof a job).

To identify the schedule, we first obtain the

permutation of jobs on machine 1, taking the nindexes of the different jobs appearing first time in

the sequence; then the permutation of jobs on

machine 2 will be given by the order of the n in-

dexes of the different jobs appearing the second

time in the sequence; and so on. With such rep-resentation, the neighborhoods defined in Section

2.3 can be used.

Let us illustrate this idea for a problem with five

jobs and three machines: the sequence

1 2 3 1 4 3 5 2 5 3 1 4 5 2 4

will correspond to the three permutations

ð 1 2 3 4 5 Þ; ð 1 3 2 5 4 Þ;ð 3 1 5 2 4 Þ;

respectively for the m machines.

Of course the length of the sequence can bereduced to nðm� 1Þ characters i only if regular

objectives are considered: effectively, with such

objectives, the permutations on the two first ma-

chines are identical (see [4]).

Similar representation is still valid for job shop

models.

Such idea has not yet been tested and is one

direction of future research.(3) Recently we designed for a MOCO problem

[11], a new method called multi-objective tabu

search (MOTAS) exploiting the same idea as

MOSA but this time with a tabu search scheme

([14]).

The main differences are that

• due to too large neighborhoods, a sub-neighbor-hood must be defined: a simple way is for

instance to sample some solutions in neighbor-

hoods as those of Section 2.3;





• at each iteration, an optimization is made inside

a sub-neighborhood of the current iterate, to de-

fine the ‘‘best neighbor solution’’. To do this, a

weighting function must be used so that, as in

MOSA, a diversified set of weight vectors is re-

quired to search potential efficient solutions in

all the directions;

• as usual, a tabu list must be considered to avoidcycling.

Implementation of MOTAS appears more dif-

ficult due to some parameters being difficult to fix

as function of the dimension of the sub-neigh-

borhood and the length of the tabu list. Different

ways to define the ‘‘best solution’’ in the sub-

neighborhood are also possible.After a more detailed design of the MOTAS

procedure, we intend to test this approach

numerically and to compare it with MOSA on

different kinds of MOCO problems, among them

multi-objective scheduling problems.

(4) In any case, for practical problems, it is

not required to generate a large number of po-

tential efficient solutions; it is better to use aninteractive approach (see (e) in Section 1) to

build a good compromise in accordance to the

preferences of the decision maker. In this spirit,

an interactive version of MOSA has been de-

signed ([20]). The present paper proves that it can

produce solutions of high quality for real case

studies.

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