18
Parallel machine scheduling models with fuzzy processing times Jin Peng * , Baoding Liu * Uncertainty Theory Laboratory, Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China Received 8 May 2002; received in revised form 10 March 2003; accepted 8 May 2003 Abstract The purpose of this research is to develop a methodology for modeling parallel machine scheduling problems with fuzzy processing times. Three novel types of fuzzy scheduling models are presented. A hybrid intelligent algorithm is also designed for solving these models. Finally, some numerical examples are provided to demonstrate the computational efficiency of the proposed algorithm. Ó 2003 Elsevier Inc. All rights reserved. Keywords: Scheduling; Fuzzy programming; Genetic algorithm 1. Introduction In the conventional scheduling problem, the parameters such as job pro- cessing times, ready times, due-dates have been assumed to be deterministic [3]. However, in the real-world situations, these parameters are often encountered with uncertainties. Accordingly, scheduling problems have been mainly bran- ched into two categories: deterministic scheduling and uncertain (stochastic, fuzzy, etc.) scheduling. In fact, various factors involved in the scheduling problems are often im- precise or uncertain in nature when we formulate scheduling problems in the * Corresponding authors. Tel.: +86-10-6277-7897; fax: +86-10-6278-1785. E-mail addresses: [email protected] (J. Peng), [email protected] (B. Liu). 0020-0255/$ - see front matter Ó 2003 Elsevier Inc. All rights reserved. doi:10.1016/j.ins.2003.05.012 Information Sciences 166 (2004) 49–66 www.elsevier.com/locate/ins

Parallel machine scheduling models with fuzzy processing times

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Information Sciences 166 (2004) 49–66

www.elsevier.com/locate/ins

Parallel machine scheduling models withfuzzy processing times

Jin Peng *, Baoding Liu *

Uncertainty Theory Laboratory, Department of Mathematical Sciences,

Tsinghua University, Beijing 100084, China

Received 8 May 2002; received in revised form 10 March 2003; accepted 8 May 2003

Abstract

The purpose of this research is to develop a methodology for modeling parallel

machine scheduling problems with fuzzy processing times. Three novel types of fuzzy

scheduling models are presented. A hybrid intelligent algorithm is also designed for

solving these models. Finally, some numerical examples are provided to demonstrate the

computational efficiency of the proposed algorithm.

� 2003 Elsevier Inc. All rights reserved.

Keywords: Scheduling; Fuzzy programming; Genetic algorithm

1. Introduction

In the conventional scheduling problem, the parameters such as job pro-

cessing times, ready times, due-dates have been assumed to be deterministic [3].

However, in the real-world situations, these parameters are often encountered

with uncertainties. Accordingly, scheduling problems have been mainly bran-ched into two categories: deterministic scheduling and uncertain (stochastic,

fuzzy, etc.) scheduling.

In fact, various factors involved in the scheduling problems are often im-

precise or uncertain in nature when we formulate scheduling problems in the

* Corresponding authors. Tel.: +86-10-6277-7897; fax: +86-10-6278-1785.

E-mail addresses: [email protected] (J. Peng), [email protected] (B. Liu).

0020-0255/$ - see front matter � 2003 Elsevier Inc. All rights reserved.

doi:10.1016/j.ins.2003.05.012

50 J. Peng, B. Liu / Information Sciences 166 (2004) 49–66

real-world. This is especially true when human-made factors are incorporated

into the problems. In these cases, it seems more appropriate to consider fuzzy

processing times, fuzzy due-dates, and so on.

So far, much of research work has been performed on fuzzy schedulingproblems. The earliest paper in fuzzy scheduling appeared in 1979 [29]. Ishii

et al. [12] first investigated scheduling problems with fuzzy due-dates. Han et al.

[8] considered single-machine scheduling problem with fuzzy due-dates.

Ishibuchi et al. [11] studied flow shop scheduling with fuzzy processing times.

The fuzzy job shop scheduling problem was analyzed by Kuroda and Wang

[16]. Konno and Ishii [15] discussed an open shop scheduling problem with

fuzzy allowable time and fuzzy resource constraint. Dubois et al. [6] formulated

a simple mathematical model of job-shop scheduling under preference anduncertainty and outlined a combinatorial search method to solve the model.

McCahon and Lee [26] modified Johnson’s algorithm and Ignall and Schrage’s

branch and bound algorithm to accept triangular and trapezoidal fuzzy pro-

cessing times. Hong et al. [10] utilized fuzzy set concept in the largest pro-

cessing time (LPT) algorithm to schedule fuzzy tasks. Recently, Itoh and Ishii

[13] proposed a single machine scheduling model dealing with fuzzy processing

times and due-dates by the possibility measure. Litoiu and Tadei [17] presented

some new models for real-time task scheduling with fuzzy deadlines and pro-cessing times.

There are three main approaches reported in the literature for the fuzzy

scheduling problems: fuzzifying directly the classical dispatching rules [28],

using fuzzy ranking [26] and fuzzy dominance relation methods [2], and solving

mathematical programming models to determine the optimal schedules by

heuristic approximation methods [9] including genetic algorithm (GA) [30],

simulated annealing [14], tabu search [1], etc.

A limited amount of literature has been devoted to fuzzy parallel machinescheduling problems (FPMSPs). In this paper, as a practical application, we

focus on the FPMSPs with fuzzy processing times. In addition to single-ob-

jective scheduling models, the multiobjective FPMSPs are also considered and

formulated as three-objective models which not only minimize the fuzzy

maximum tardiness, but also minimize the fuzzy maximum completion time

(makespan) and the fuzzy maximum idleness. We design a hybrid intelligent

algorithm to solve the formulated scheduling models. Effectiveness of the

proposed algorithm is demonstrated through some numerical experiments.The outline of this paper is organized as follows. First, in Section 2 we

briefly review the concepts of possibility, necessity, credibility and expected

value operator for fuzzy variable. Then we describe the assumptions and no-

tations for FPMSP in Section 3. After that, three new types of fuzzy scheduling

models––expectation scheduling model, a-scheduling model, and the most

credible scheduling model––for FPMSP are formulated in Section 4. In Section

5, a hybrid intelligent algorithm designed for solving the models is described in

J. Peng, B. Liu / Information Sciences 166 (2004) 49–66 51

detail. In Section 6, some numerical examples and computational results are

provided to show the effectiveness of the proposed algorithm. Finally, we

concludes this paper with a summary.

2. Preliminaries

Fuzzy set theory and possibility theory [31] provide an alternative and

convenient framework for modeling of real-world fuzzy decision systems

mathematically. In this section, we are going to briefly review some concepts of

possibility, necessity, credibility and expected value for fuzzy variable.

Dubois and Prade [7] developed the possibility measure and necessitymeasure as follows. Let n be a fuzzy variable with membership function lðxÞ,and let r be a real number. The possibility, necessity, and credibility measure of

fn6 rg are defined as

Posfn6 rg ¼ supx6 r

lðxÞ; ð1Þ

Necfn6 rg ¼ 1� supx>r

lðxÞ; ð2Þ

Crfn6 rg ¼ 1

2ðPosfn6 rg þNecfn6 rgÞ; ð3Þ

respectively. Furthermore, the expected value of a fuzzy variable n is defined on

the basis of the credibility measure by Liu and Liu [25] as follows:

E½n� ¼Z 1

0

CrfnP rgdr �Z 0

�1Crfn6 rgdr ð4Þ

provided that at least one of the above two integrals is finite.

3. Assumptions and notations

Assumptions and notations for FPMSP in this paper are described as fol-

lows:

Assumptions

(i) each job has only one operation and can be processed on any machine;

(ii) each machine can process only one job at a time;

(iii) all jobs are available for machine processing simultaneously at time zero;

(iv) the processing times are assumed to be fuzzy variables.

Notationsi ¼ 1; 2; . . . ; n: the jobs to be scheduled;

k ¼ 1; 2; . . . ;m: the machines;

52 J. Peng, B. Liu / Information Sciences 166 (2004) 49–66

tik: the fuzzy processing time of job i on machine k (i ¼ 1; 2; . . . ; n;k ¼ 1; 2; . . . ;m);n: the fuzzy vector (t11; t12; . . . ; t1m; . . . ; tn1; tn2; . . . ; tnm);Di: the due-date of job i (i ¼ 1; 2; . . . ; n);(x,y): the decision vector in which

x ¼ ðx1; x2; . . . ; xnÞ: integer decision variables representing n jobs with

16 xi 6 n and xi 6¼ xj for all i 6¼ j, i; j ¼ 1; 2; . . . ; n;y ¼ ðy1; y2; . . . ; ym�1Þ: integer decision variables with y0 � 06 y1 6 y2 6 � � � 6

ym�1 6 n � ym.

We note that the schedule is fully determined by the decision variables x and

y in the following way. For each k ð16 k6mÞ, if yk ¼ yk�1, then machine k isnot used; if yk > yk�1, then machine k is used and processes the following jobs in

turn: xyk�1þ1 ! xyk�1þ2 ! � � � ! xyk .Let Ciðx; y; nÞ be the completion times of jobs i, i ¼ 1; 2; . . . ; n, respectively.

Then the completion times can be calculated by the following equations:

Cxyk�1þ1ðx; y; nÞ ¼ txyk�1þ1k ð5Þ

and

Cxyk�1þjðx; y; nÞ ¼ Cxyk�1þj�1ðx; y; nÞ þ txyk�1þjk ð6Þ

for 26 j6 yk � yk�1 and k ¼ 1; 2; . . . ;m.It follows that the fuzzy maximum tardiness (called fuzzy tardiness for short

hereafter) f1ðx; y; nÞ, fuzzy makespan f2ðx; y; nÞ, and fuzzy maximum idleness

(called fuzzy idleness for short hereafter) f3ðx; y; nÞ of the schedule ðx; yÞ can be

calculated by

f1ðx; y; nÞ ¼ max16 i6 n

fCiðx; y; nÞ � Dig _ 0; ð7Þ

f2ðx; y; nÞ ¼ max16 k6m

Cxykðx; y; nÞ; ð8Þ

f3ðx; y; nÞ ¼ max16 k6m

f2ðx; y; nÞ(

�Xyk

i¼yk�1þ1

tik

); ð9Þ

where _ denotes the maximum operator.

4. Fuzzy scheduling models

4.1. Expectation scheduling models

In fuzzy scheduling environment, we naturally follow a similar modeling

philosophy in stochastic scheduling environment which optimizes some

J. Peng, B. Liu / Information Sciences 166 (2004) 49–66 53

expected objective functions subject to some expected constraints. Taking full

advantage of the concept of expected value for fuzzy variable mentioned in

Section 2, we can reasonably provide expectation scheduling models for fuzzy

scheduling problems.If we want to minimize the expected tardiness E½f1ðx; y; nÞ�, then we can

employ the following expected tardiness model:

min E½f1ðx; y; nÞ�subject to

16 xi 6 n; i ¼ 1; 2; . . . ; n;xi 6¼ xj; i 6¼ j; i; j ¼ 1; 2; . . . ; n;06 y1 6 y2 � � � 6 ym�1 6 n;xi; yj; i ¼ 1; 2; . . . ; n; j ¼ 1; 2; . . . ;m� 1; integers:

8>>>>>><>>>>>>:

ð10Þ

Similarly, we have the expected makespan model and the expected idleness

model if f1ðx; y; nÞ is replaced with f2ðx; y; nÞ and f3ðx; y; nÞ, respectively.We can also formulate a fuzzy scheduling problem as a fuzzy goal pro-

gramming model according to the priority structure and target levels set by the

decision makers:

Priority 1: The expected tardiness E½f1ðx; y; nÞ� should not exceed the target

value b1, i.e., we have a goal constraint

E½f1ðx; y; nÞ� þ d�1 � dþ

1 ¼ b1

in which dþ

1 will be minimized.

Priority 2: The expected makespan E½f2ðx; y; nÞ� should not exceed the target

value b2, i.e., we have a goal constraint

E½f2ðx; y; nÞ� þ d�2 � dþ

2 ¼ b2

in which dþ2 will be minimized.

Priority 3: The expected idleness E½f3ðx; y; nÞ� should not exceed the target

value b3, i.e., we have a goal constraint

E½f3ðx; y; nÞ� þ d�3 � dþ

3 ¼ b3

in which dþ3 will be minimized.

Thus we have the following fuzzy scheduling model:

lexmin fdþ1 ; d

þ2 ; d

þ3 g

subject to

E½fiðx; y; nÞ� þ d�i � dþ

i ¼ bi; i ¼ 1; 2; 3;16 xi 6 n; i ¼ 1; 2; . . . ; n;xi 6¼ xj; i 6¼ j; i; j ¼ 1; 2; . . . ; n;06 y1 6 y2 � � � 6 ym�1 6 n;xi; yj; i ¼ 1; 2; . . . ; n; j ¼ 1; 2; . . . ;m� 1; integers;dþi ; d

�i P 0; i ¼ 1; 2; 3;

8>>>>>>>>>><>>>>>>>>>>:

ð11Þ

where lexmin represents lexicographically minimizing the objective vector.

54 J. Peng, B. Liu / Information Sciences 166 (2004) 49–66

4.2. a-Scheduling models

Fuzzy chance-constrained programming, suggested by Liu and Iwamura

[18,19], offers another effective method of modeling fuzzy decision systems.Applying fuzzy chance-constrained programming in scheduling problem, we

can build a series of a-scheduling models.

Let us introduce the following definitions of a-tardiness, a-makespan, and a-idleness.

Definition 1. For a given confidence level a 2 ð0; 1�, the values

c1 ¼ inffcjCrff1ðx; y; nÞ6 cgP ag;c2 ¼ inffcjCrff2ðx; y; nÞ6 cgP ag;c3 ¼ inffcjCrff3ðx; y; nÞ6 cgP ag

ð12Þ

are called the a-tardiness, a-makespan, and a-idleness of the schedule (x; y),respectively.

If we want to minimize the a-makespan, then we have the following a-makespan model,

min csubject to

Crff2ðx; y; nÞ6 cgP a;16 xi 6 n; i ¼ 1; 2; . . . ; n;xi 6¼ xj; i 6¼ j; i; j ¼ 1; 2; . . . ; n;06 y1 6 y2 � � � 6 ym�1 6 n;xi; yj; i ¼ 1; 2; . . . ; n; j ¼ 1; 2; . . . ;m� 1; integers:

8>>>>>>>><>>>>>>>>:

ð13Þ

Similarly, if we want to minimize the a-tardiness or a-idleness, then we have

the a-tardiness model or a-idleness model.

In addition, the credibility measure ‘‘Cr’’ may be replaced with the possi-

bility measure ‘‘Pos’’ or necessity measure ‘‘Nec’’. As we know from Dubois

and Prade [7], the necessity measure is the dual of possibility measure. But

neither possibility measure nor necessity measure is self-dual. Fortunately, the

credibility measure is self-dual [23]. In this aspect, the credibility measureshares some properties like the probability measure. Recall that a random

event must hold if its probability is 1, and fail if its probability is 0. Com-

paratively, a fuzzy event may fail even though its possibility is 1, and hold even

though its necessity is 0. However, a fuzzy event must hold if its credibility is 1,

and fail if its credibility is 0. In this way, credibility measure weights possibility

and necessity measures harmoniously. Although many researchers make efforts

to set up fuzzy programming models via possibility or necessity measure suc-

cessfully, it seems more reasonable that we employ credibility measure to set upthe fuzzy scheduling models.

J. Peng, B. Liu / Information Sciences 166 (2004) 49–66 55

We can reformulate a scheduling problem as a fuzzy chance-constrained

goal programming model. Suppose that the management goals have the fol-

lowing priority structures:

At the first priority level, the fuzzy tardiness f1ðx; y; nÞ should not exceed thetarget value b1 with a confidence level a1. Thus we have a goal constraint

Crff1ðx; y; nÞ � b1 6 dþ1 gP a1

in which dþ1 will be minimized.

At the second priority level, the fuzzy makespan f2ðx; y; nÞ should not exceed

the target value b2 with a confidence level a2. Thus we have a goal constraint

Crff2ðx; y; nÞ � b2 6 dþ2 gP a2

in which dþ2 will be minimized.

At the third priority level, the fuzzy idleness f3ðx; y; nÞ should not exceed thetarget value b3 with a confidence level a3. Thus we have a goal constraint

Crff3ðx; y; nÞ � b3 6 dþ3 gP a3

in which dþ3 will be minimized.

Then we have the following fuzzy scheduling model:

lexmin fdþ1 ; d

þ2 ; d

þ3 g

subject to

Crffiðx; y; nÞ � bi 6 dþi gP ai; i ¼ 1; 2; 3;

16 xi 6 n; i ¼ 1; 2; . . . ; n;xi 6¼ xj; i 6¼ j; i; j ¼ 1; 2; . . . ; n;06 y1 6 y2 � � � 6 ym�1 6 n;xi; yj; i ¼ 1; 2; . . . ; n; j ¼ 1; 2; . . . ;m� 1; integers;dþi P 0; i ¼ 1; 2; 3:

8>>>>>>>>>><>>>>>>>>>>:

ð14Þ

4.3. The most credible scheduling models

In practice, there usually exist multiple events in a complex fuzzy scheduling

system. Fuzzy dependent-chance programming, initialized by Liu [20–22],

provides an effective method of modeling fuzzy decision systems in which theunderlying is based on selecting the decision with maximal chances to meet the

events. In other words, dependent-chance programming is related to maxi-

mizing the chance functions of some events in an uncertain environments. We

may construct the most credible scheduling models for FPMSP based on de-

pendent-chance programming.

Definition 2. For a given predetermined time c and a given schedule (x�; y�), if

Crff2ðx�; y�; nÞ6 cgPCrff2ðx; y; nÞ6 cg ð15Þ

56 J. Peng, B. Liu / Information Sciences 166 (2004) 49–66

for any schedule (x; y), then (x�; y�) is called the most credible makespan

schedule.

If we want to maximize the credibility that the fuzzy makespan f2ðx�; y�; nÞdoes not exceed the predetermined time c subject to the constraints, then we

have the most credible makespan model as follows:

max Crff2ðx; y; nÞ6 cgsubject to

16 xi 6 n; i ¼ 1; 2; . . . ; n;

xi 6¼ xj; i 6¼ j; i; j ¼ 1; 2; . . . ; n;

06 y1 6 y2 � � � 6 ym�1 6 n;

xi; yj; i ¼ 1; 2; . . . ; n; j ¼ 1; 2; . . . ;m� 1; integers:

8>>>>>>>><>>>>>>>>:

ð16Þ

Naturally, we may employ the most credible tardiness model and the most

credible idleness model if f2ðx; y; nÞ is replaced with f1ðx; y; nÞ and f3ðx; y; nÞ,respectively.

We may formulate the scheduling problem by fuzzy dependent-chance goal

programming. Suppose that the management goals have the following priority

structures:

At the first priority level, the credibility that the fuzzy tardiness f1ðx; y; nÞdoes not exceed the given time c1 should achieve a confidence level a1. Then we

have a goal constraint

Crff1ðx; y; nÞ6 c1g þ d�1 � dþ

1 ¼ a1

in which d�1 will be minimized.

At the second priority level, the credibility that the fuzzy makespan

f2ðx; y; nÞ does not exceed a given time c2 should achieve a confidence level a2.Then we have a goal constraint

Crff2ðx; y; nÞ6 c2g þ d�2 � dþ

2 ¼ a2

in which d�2 will be minimized.

At the third priority level, the credibility that the fuzzy idleness f3ðx; y; nÞdoes not exceed a given time c3 should achieve a confidence level a3. Then we

have a goal constraint

Crff3ðx; y; nÞ6 c3g þ d�3 � dþ

3 ¼ a3

in which d�3 will be minimized.

J. Peng, B. Liu / Information Sciences 166 (2004) 49–66 57

Thus we have the following fuzzy scheduling model:

lexmin fd�1 ; d

�2 ; d

�3 g

subject to

Crffiðx; y; nÞ6 cig þ d�i � dþ

i ¼ ai; i ¼ 1; 2; 3;16 xi 6 n; i ¼ 1; 2; . . . ; n;xi 6¼ xj; i 6¼ j; i; j ¼ 1; 2; . . . ; n;06 y1 6 y2 � � � 6 ym�1 6 n;xi; yj; i ¼ 1; 2; . . . ; n; j ¼ 1; 2; . . . ;m� 1; integers;dþi ; d

�i P 0; i ¼ 1; 2; 3:

8>>>>>>>>>><>>>>>>>>>>:

ð17Þ

5. Hybrid intelligent algorithm

GAs have demonstrated considerable success in providing good solutions to

many complex optimization problems. They have been well documented in the

literature, such as in [27], and have been applied to a wide variety of optimi-zation problems. In particular, Liu and Iwamura [18,19], Liu [21,22], Buckley

and Hayashi [4], and Buckley and Feuring [5] designed a spectrum of GAs to

solve fuzzy programming models. For detailed expositions, the readers may

consult Liu [20,23] in which numerous GAs have been suggested for solving

uncertain programming models.

In this section, we present a hybrid intelligent algorithm designed to solve

the proposed scheduling models. The hybrid intelligent algorithm here is much

different from the existing algorithms (such as Liu [24]) for general uncertainprogramming models in many aspects. For instance, we have to find a specific

coding method to represent the schedule on parallel machine. Thus, some new

ways of genetic operators have to be designed for this type of problems. The

hybrid intelligent algorithm is in detail explained as follows.

Representation structure: For an FPMSP with n jobs and m machines, the

corresponding decision variables are x ¼ ðx1; x2; . . . ; xnÞ and y ¼ ðy1; y2; . . . ;ym�1Þ, where x represents n jobs with 16 xi 6 n and xi 6¼ xj for all i 6¼ j,i; j ¼ 1; 2; . . . ; n and y represents an assignment of n jobs to m machines withy0 � 06 y1 6 y2 6 � � � 6 ym�1 6 n � ym. Thus, we adopt a chromosome V ¼ðx; yÞ to stand for a schedule.

Initialization process: The pop size chromosomes in the initial population are

randomly generated. For gene section x ¼ ðx1; x2; . . . ; xnÞ, it can be obtained by

rearranging the sequence f1; 2; . . . ; ng randomly. As for a gene section y, it canbe generated by the following way: for each i with 16 i6m� 1, we firstly set yias a random integer between 0 and n. Then we rearrange the sequence

fy1; y2; . . . ; ym�1g from small to large and thus obtain a gene sectiony ¼ ðy1; y2; . . . ; ym�1Þ. We may repeat the above process pop size times until the

58 J. Peng, B. Liu / Information Sciences 166 (2004) 49–66

pop size initial chromosomes V1; V2; . . . ; Vpop size are generated. With this tech-

nique, the initial population was accordingly created to ensure the feasibility of

each chromosome.

Evaluation process: To begin with, we calculate the objective values for allchromosomes by fuzzy simulations (and the fuzzy simulations for computing

the expected value, a-value, and credibility are described in Appendix Afor the

sake of shortness). Then, we rearrange these chromosomes from good to bad

according to the calculated objective values. Here, we adopt the following

rank-based evaluation function:

EvalðViÞ ¼ að1� aÞi�1; i ¼ 1; 2; . . . ; pop size; ð18Þ

where parameter a 2 ð0; 1Þ, i ¼ 1 means the best individual, and i ¼ pop sizethe worst individual.

Selection process: The selection method is based on spinning the roulette

wheel pop size times. We select a single chromosome for a new population each

time until pop size copies of chromosomes are finally obtained.

Crossover process: Crossover operation is an operator that produces two

children out of the parents. Based on the character of the chromosome in thispaper, we adopt the following crossover method to create one child from two

parent chromosomes.

Without loss of generality, let us illustrate the two parental chromosomes by

V1 ¼ ðx1; y1Þ and V2 ¼ ðx2; y2Þ, respectively. Then two children V 01 and V 0

2 are

generated by the proposed crossover operator as follows: V 01 ¼ ðx1; y2Þ and

V 02 ¼ ðx2; y1Þ. It should be noted that the obtained offsprings V 0

1 ¼ ðx1; y2Þ andV 02 ¼ ðx2; y1Þ in this crossover method not only maintain the dimension of the

decision vector but also inherit the feasibility.Mutation process: Mutation operation is an operator that carries out local

modification of the chromosomes. Firstly, we repeat the following steps from

i ¼ 1 to pop size to determine the parents for mutation operator: Generate a

random real number r from the interval ½0; 1�. The chromosome Vi is selected as

a parent for mutation if r < Pm. Secondly, for each selected parent, denoted by

V ¼ ðx; yÞ, it is mutated in the following way: randomly generate two mutation

positions n1 and n2 between 1 and n for the gene section x, and rearrange the

sequence fxn1 ; xn1þ1; . . . ; xn2g at random to form a new sequencefx0n1 ; x

0n1þ1; . . . ; x

0n2g, thus we obtain a new gene section

x0 ¼ ðx1; . . . ; xn1�1; x0n1 ; x0n1þ1; . . . ; x

0n2; xn2þ1; . . . ; xnÞ:

Thirdly, we similarly generate two random mutation positions m1 and m2

between 1 and m� 1 for gene section y, and pick over the subsection

fym1; ym1þ1; . . . ; ym2

g. After that, we set yi as a random integer number y0i be-tween 0 and n for i ¼ m1;m1 þ 1; . . . ;m2. Then rearrange the whole sequencey1; . . . ; ym1�1; y0m1

; y0m1þ1; . . . ; y0m2; ym2þ1; . . . ; ym�1 from small to large and obtain a

J. Peng, B. Liu / Information Sciences 166 (2004) 49–66 59

new gene section y0. Finally, we replace the parent V with the offspring

V 0 ¼ ðx0; y0Þ whose feasibility is always guaranteed.

GA procedure: Following crossover, mutation, evaluation and selection, the

new population is ready for its next evaluation (the fuzzy simulations areembedded in the evaluation process). The GA will terminate after a given

number of cyclic repetitions and output the best chromosome as the optimal

schedule.

6. Numerical examples

In this section, we give three examples to illustrate the effectiveness of the

proposed hybrid intelligent algorithms, which have been coded with C++

language and run on a personal computer. Here the parameters are set as

follows: the population size is 100, the probability of crossover Pc is 0.6, the

probability of mutation Pm is 0.3, and the parameter a in the rank-basedevaluation function is 0.05.

Let us consider an FPMSP with 10 jobs and 3 machines. The processing

times for each job on Machine 1, Machine 2, and Machine 3 are trapezoidal

fuzzy variables. The processing times and the due-dates of the jobs are listed in

Table 1.

Example 1. Assume that the decision makers in the FPMSP have the following

priority structures:At the first priority level, the expected tardiness E½f1ðx; y; nÞ� should be as

little as possible. Then we have a goal constraint

Table

Fuzzy

Jobs

1

2

3

4

5

6

7

8

9

10

E½f1ðx; y; nÞ� þ d�1 � dþ

1 ¼ 0

in which dþ1 will be minimized.

1

processing times and due-dates

Fuzzy processing times Due-dates

Machine 1 Machine 2 Machine 3

(7,9,11,13) (6,9,11,14) (6,9,12,15) 20

(5,7,9,11) (4,7,9,12) (4,7,10,13) 15

(4,6,8,10) (3,6,8,11) (3,6,9,12) 25

(2,4,6,8) (1,4,6,9) (1,4,7,10) 15

(1,3,5,7) (1,3,5,8) (2,5,8,11) 15

(1,3,5,7) (0,3,5,8) (1,4,7,10) 20

(0,1,3,4) (0,2,4,6) (1,3,5,7) 30

(0,1,3,4) (0,1,3,4) (1,3,5,7) 30

(0,1,3,4) (0,1,3,4) (0,2,4,6) 25

(0,1,3,4) (0,1,3,4) (1,2,3,4) 25

60 J. Peng, B. Liu / Information Sciences 166 (2004) 49–66

At the second priority level, the expected makespan E½f2ðx; y; nÞ� should not

exceed the target 18. Thus we have a goal constraint

E½f2ðx; y; nÞ� þ d�2 � dþ

2 ¼ 18

in which dþ2 will be minimized.

At the third priority level, the expected idleness E½f3ðx; y; nÞ� should not

exceed the target 2. Thus we have a goal constraint

E½f3ðx; y; nÞ� þ d�3 � dþ

3 ¼ 2

in which dþ3 will be minimized.

Then we have the following fuzzy scheduling model:

lexmin fdþ1 ; d

þ2 ; d

þ3 g

subject to

E½f1ðx; y; nÞ� þ d�1 � dþ

1 ¼ 0;

E½f2ðx; y; nÞ� þ d�2 � dþ

2 ¼ 18;

E½f3ðx; y; nÞ� þ dþ3 � dþ

3 ¼ 2;

16 xi 6 10; i ¼ 1; 2; . . . ; 10;

xi 6¼ xj; i 6¼ j; i; j ¼ 1; 2; . . . ; 10;

06 y1 6 y2 6 10;

xi; yj; i ¼ 1; 2; . . . ; 10; j ¼ 1; 2; integers;

dþi ; d

�i P 0; i ¼ 1; 2; 3:

8>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>:

ð19Þ

A run of the hybrid intelligent algorithm (5000 cycles in fuzzy simulation,500 generations in GA) shows that the optimal schedule is

Machine 1: 4fi5fi3;

Machine 2: 2 fi6fi9fi7;

Machine 3: 10 fi1 fi8

which can satisfy the first two goals, but the third objective is 2.34.

Example 2. Now let us consider the above mentioned 10-job and 3-machine

FPMSP in another way. If the decision makers want to minimize the 0.95-

makespan subject to the constraints, then we have the following single-objec-

tive 0.95-makespan model:

min csubject to

Crff2ðx; y; nÞ6 cgP 0:95;16 xi 6 10; i ¼ 1; 2; . . . ; 10;xi 6¼ xj; i 6¼ j; i; j ¼ 1; 2; . . . ; 10;06 y1 6 y2 6 10;xi; yj; i ¼ 1; 2; . . . ; 10; j ¼ 1; 2; integers:

8>>>>>>>><>>>>>>>>:

ð20Þ

J. Peng, B. Liu / Information Sciences 166 (2004) 49–66 61

A run of the hybrid intelligent algorithm (5000 cycles in fuzzy simulation, 500

generations in GA) shows that the optimal schedule is

Machine 1: 5fi6fi7fi4;Machine 2: 1fi9fi8 fi10;

Machine 3: 2fi3.

In fact, we have

Crff2ðx�; y�; nÞ6 25:45g � 0:95:

For the optimal schedule (x�; y�), the membership function of f2ðx�; y�; nÞcan be obtained from those of completion times of jobs which are lastly pro-

cessed on each machine. Fig. 1 shows that the simulation computation agrees

with the analytic calculation. In addition, the relation between the a-makespan

and the confidence level a is shown in Fig. 2.

Fig. 1. Membership functions and 0.95-makespan.

Fig. 2. Relation between a-makespan c and confidence level a.

62 J. Peng, B. Liu / Information Sciences 166 (2004) 49–66

Example 3. Let us consider the above mentioned 10-job and 3-machine

FPMSP again. Assume that the decision makers have two management goals

with the following priority structures:

Firstly, we require that the credibility that no tardy jobs occur shouldachieve the confidence level 0.95, then we have a goal constraint

Crff1ðx; y; nÞ6 0g þ d�1 � dþ

1 ¼ 0:95

in which d�1 will be minimized.

Secondly, we hope that the credibility that the makespan does not exceed a

given time 20 should achieve the confidence level 0.85, then we have a goal

constraint

Crff2ðx; y; nÞ6 20g þ d�2 � dþ

2 ¼ 0:85

in which d�2 will be minimized.

As a result, the problem can be formulated as follows:

lexmin fd�1 ; d

�2 g

subject to

Crff1ðx; y; nÞ6 0g þ d�1 � dþ

1 ¼ 0:95;Crff2ðx; y; nÞ6 20g þ d�

2 � dþ2 ¼ 0:85;

16 xi 6 10; i ¼ 1; 2; . . . ; 10;xi 6¼ xj; i 6¼ j; i; j ¼ 1; 2; . . . ; 10;06 y1 6 y2 6 10;xi; yj; i ¼ 1; 2; . . . ; 10; j ¼ 1; 2; integers;dþi ; d

�i P 0; i ¼ 1; 2:

8>>>>>>>>>>>><>>>>>>>>>>>>:

ð21Þ

A run of the hybrid intelligent algorithm (5000 cycles in fuzzy simulation,

600 generations in GA) shows that the optimal schedule is

Machine 1: 4fi5fi3fi7;

Machine 2: 2 fi6fi8;Machine 3: 10 fi1 fi9

which can satisfy the first goal, but the second objective is 0.15. In fact, we also

have

Crff1ðx�; y�; nÞ6 0g � 0:96;

Crff2ðx�; y�; nÞ6 20g � 0:70:

The result of the evolution process is shown in Fig. 3.

Fig. 3. An evolution process.

J. Peng, B. Liu / Information Sciences 166 (2004) 49–66 63

7. Conclusions

In this paper, we have demonstrated three types of fuzzy scheduling modelson parallel machines by means of the newly developed credibility measure.

Since credibility measure weights possibility and necessity measures harmoni-

ously, it seems more reasonable that the credibility measure is employed to set

up the fuzzy scheduling models. Meanwhile, the goal programming models are

also considered in the scheduling problem. We have also presented an effective

hybrid intelligent algorithm to solve the proposed models and provide effi-

ciently computational studies. It may be concluded that the modeling methods

can be extended to other fuzzy scheduling problems.

Acknowledgements

This work was supported by the National Natural Science Foundation ofChina grant no. 60174049, the Sino-French Joint Laboratory for Research in

Computer Science, Control and Applied Mathematics (LIAMA), and the Key

Project of Education Department of Hubei Province, China.

Appendix A

In this Appendix, we present the fuzzy simulation methods related to the

solutions of the proposed models. The general method of fuzzy simulation for

computing the expected value of a fuzzy variable was provided by Liu and Liu

[25]. The general method of fuzzy simulation for computing the possibility of afuzzy event was proposed by Liu [20].

64 J. Peng, B. Liu / Information Sciences 166 (2004) 49–66

A.1. Fuzzy simulation for expected value E[f (x; y; )]

The fuzzy simulation for general expected value E½f ðx; y; nÞ� can be de-

scribed as follows:

Step 1. Uniformly select N sample points si ði ¼ 1; 2; . . . ;NÞ from the e-levelset of n, where e is an appropriate nonnegative number.

Step 2. Compute the values li ¼ lðsiÞ and ai ¼ f ðx; y; siÞ for i ¼ 1; 2; . . . ;N .

Step 3. Rearrange a1; a2; . . . ; aN (correspondingly l1; l2; . . . ; lN ) such that

a1 6 a2 6 � � � 6 aN .Step 4. Calculate the weights wi for i ¼ 1; 2; . . . ;N ; where

w1 ¼1

2ðl1 þ max

16 j6Nlj � max

1<j6NljÞ;

wi ¼1

2ðmax16 j6 i

lj � max16 j<i

lj þ maxi6 j6N

lj � maxi<j6N

ljÞ; 26 i6N � 1;

wN ¼ 1

2ð max16 j6N

lj � max16 j<N

lj þ lNÞ:

Step 5. Calculate the expected value E½f ðx; y; nÞ� ¼PN

i¼1 wiai.

Appendix B. Fuzzy simulation for credibility Cr{ f (x; y; n)6c}

In order to estimate credibility Crff ðx; y; nÞ6 cg for a given decision x and

y, we design the following fuzzy simulation procedure:

Step 1. Set Posff ðx; y; nÞ6 cg ¼ 0 and Posff ðx; y; nÞ > cg ¼ 0.

Step 2. Generate two points s1 and s2 from the e-level set of n, where e is anappropriate nonnegative number.

Step 3. Calculate f ðx; y; s1Þ and f ðx; y; s2Þ.Step 4. Set Posff ðx; y; nÞ6 cg ¼ lðs1Þ if Posff ðx; y; nÞ6 cg < lðs1Þ and

Posff ðx; y; nÞ > cg ¼ lðs2Þ if Posff ðx; y; nÞ > cg < lðs2Þ.Step 5. Repeat Steps 2–4 for N times.

Step 6. Return Crff ðx; y; nÞ6 cg ¼ 12ð1þ Posff ðx; y; nÞ6 cg � Pos ff ðx; y; nÞ>

cgÞ.

Appendix C. Fuzzy simulation for a-value c

In order to find the a-value c, i.e., the minimal value c such that the in-

equality Crff ðx; y; nÞ6 cgP a holds for a given decision vector x and y, wedesign the fuzzy simulation as follows:

J. Peng, B. Liu / Information Sciences 166 (2004) 49–66 65

Step 1. Set c ¼ þ1.

Step 2. Generate a sample point s from the e-level set of n, where e is an

appropriate nonnegative number.

Step 3. If c > f ðx; y; sÞ and Crff ðx; y; nÞ6 f ðx; y; sÞgP a, then set c ¼f ðx;y; sÞ.

Step 4. Repeat Steps 2 and 3 for N times.

Step 5. Return c.

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