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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 dþ1 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.
References
[1] V.A. Armentano, D.P. Ronconi, Tabu search for total tardiness minimization in flow shop
scheduling problems, Computers and Operations Research 26 (1999) 219–235.
[2] M. Asano, H. Ohta, Single machine scheduling using dominance relation to minimize earliness
subject to ready and due times, International Journal of Production Economics 44 (1996) 35–
43.
[3] P. Brucker, Scheduling Algorithms, second ed., Springer, Berlin, 1998.
[4] J.J. Buckley, Y. Hayashi, Fuzzy genetic algorithm and applications, Fuzzy Sets and Systems 61
(1994) 129–136.
[5] J.J. Buckley, T. Feuring, Evolutionary algorithm solution to fuzzy problems: fuzzy linear
programming, Fuzzy Sets and Systems 109 (2000) 35–53.
[6] D. Dubois, H. Fargier, H. Prade, Fuzzy constraints in job-shop scheduling, Journal of
Intelligent Manufacturing 6 (1995) 215–234.
[7] D. Dubois, H. Prade, Possibility Theory: An Approach to Computerized Processing of
Uncertainty, Plenum Press, New York, 1988.
[8] S. Han, H. Ishii, S. Fujii, One machine scheduling problem with fuzzy duedates, European
Journal of Operational Research 79 (1994) 1–12.
[9] M. Hapke, R. Slowinski, Fuzzy priority heuristics for project scheduling, Fuzzy Sets and
Systems 83 (1996) 291–299.
[10] T.-P. Hong, C.-M. Huang, K.-M. Yu, LPT scheduling for fuzzy tasks, Fuzzy Sets and Systems
97 (1998) 277–286.
[11] H. Ishibuchi, N. Yamamoto, T. Murata, H. Tanaka, Genetic algorithms and neighborhood
search algorithms for fuzzy flow shop scheduling problems, Fuzzy Sets and Systems 67 (1994)
81–100.
[12] H. Ishii, M. Tada, T. Masuda, Two scheduling problems with fuzzy due-dates, Fuzzy Sets and
Systems 46 (1992) 339–347.
[13] T. Itoh, H. Ishii, Fuzzy due-date scheduling problem with fuzzy processing time, International
Transactions in Operational Research 6 (1999) 639–647.
[14] M. Kolonko, Some new results on simulated annealing applied to the job shop scheduling
problem, European Journal of Operational Research 113 (1999) 123–136.
[15] T. Konno, H. Ishii, An open shop scheduling problem with fuzzy allowable time and fuzzy
resource constraint, Fuzzy Sets and Systems 109 (2000) 141–147.
[16] M. Kuroda, Z. Wang, Fuzzy job shop scheduling, International Journal of Production
Economics 44 (1996) 45–51.
[17] M. Litoiu, R. Tadei, Real-time task scheduling with fuzzy deadlines and processing times,
Fuzzy Sets and Systems 117 (2001) 35–45.
[18] B. Liu, K. Iwamura, Chance constrained programming with fuzzy parameters, Fuzzy Sets and
Systems 94 (1998) 227–237.
66 J. Peng, B. Liu / Information Sciences 166 (2004) 49–66
[19] B. Liu, K. Iwamura, A note on chance constrained programming with fuzzy coefficients, Fuzzy
Sets and Systems 100 (1998) 229–233.
[20] B. Liu, Uncertain Programming, John Wiley & Sons, New York, 1999.
[21] B. Liu, Dependent-chance programming with fuzzy decisions, IEEE Transactions on Fuzzy
Systems 7 (1999) 354–360.
[22] B. Liu, Dependent-chance programming in fuzzy environments, Fuzzy Sets and Systems 109
(2000) 97–106.
[23] B. Liu, Theory and Practice of Uncertain Programming, Physica-Verlag, Heidelberg, 2002.
[24] B. Liu, Random fuzzy dependent-chance programming and its hybrid intelligent algorithm,
Information Sciences 141 (2002) 259–271.
[25] B. Liu, Y.-K. Liu, Expected value of fuzzy variable and fuzzy expected value models, IEEE
Transactions on Fuzzy Systems 10 (2002) 445–450.
[26] C.S. McCahon, E.S. Lee, Job sequencing with fuzzy processing times, Computers &
Mathematics with Applications 19 (1990) 31–41.
[27] Z. Michalewicz, Genetic Algorithms+Data Structures¼Evolution Programs, third ed.,
Springer-Verlag, New York, 1996.
[28] E.C. €Ozelkan, L. Duckstein, Optimal fuzzy counterparts of scheduling rules, European
Journal of Operational Research 113 (1999) 593–609.
[29] H. Prade, Using fuzzy set theory in a scheduling problem: a case study, Fuzzy Sets and Systems
2 (1979) 153–165.
[30] M. Sakawa, R. Kubota, Fuzzy programming for multiobjective job shop scheduling with fuzzy
processing time and fuzzy duedate through genetic algorithms, European Journal of
Operational Research 120 (2000) 393–407.
[31] L.A. Zadeh, Fuzzy set as a basis for a theory of possibility, Fuzzy Sets and Systems 1 (1978) 3–
28.