The single machine ready time scheduling problem with fuzzy processing times

Fuzzy Sets and Systems 127 (2002) 117–129www.elsevier.com/locate/fss

The single machine ready time scheduling problem with fuzzyprocessing times

Chengyao Wanga ;∗ ;1, Dingwei Wanga, W.H. Ipb, D.W. YuencaDepartment of System Engineering, College of Information Science & Engineering, Northeastern University, P.O. Box 135,

Shenyang Liaoning, 110006, People’s Republic of ChinabDepartment of Manufacturing Engineering, Hong Kong Polytechnic University, Hung Hom, Hong KongcDepartment of Mechanical Engineering, Hong Kong Polytechnic University, Hung Hom, Hong Kong

Received 8 June 1999; received in revised form 1 August 2000; accepted 29 August 2000

Abstract

This paper studies a ready time scheduling problem with fuzzy job processing times. Firstly, we assume that the fuzzyprocessing time can be represented by a triangular fuzzy number. Then we apply the technique of chance constrainedprogramming to construct a job completion likelihood pro3le from the job membership functions so that the likelihoodof completing the queued jobs has a crisp mathematical representation. Some properties of the likelihood pro3le aresubsequently discussed. Utilizing the fuzzy extension principle and the concept of job completion likelihood pro3le, theready time scheduling model that maximizes the common ready time with crisp processing times is extended to one withfuzzy processing times. Optimal solutions for the scheduling model under several special conditions are developed, and anecessary condition that the optimal solution must satisfy when the jobs have di7erent due dates and con3dence levels isalso established. c© 2002 Elsevier Science B.V. All rights reserved.

Keywords: Fuzzy processing time; Membership function; Job completion likelihood pro3le; Con3dence level; Ready time

1. Introduction

In today’s rapidly changing global markets, all companies are under increasing pressure to reduce leadtimes and to maintain a high on-time delivery performance. E7ective scheduling of jobs thus becomes thekey to achieving these goals. In recent decades, researchers have achieved substantial advances in schedulingtechniques. However, since most scheduling problems are NP-hard problems, i.e. the time it takes to reacha solution grows exponentially with the increase in the size of the problem, it remains a di@cult task to3nd an optimal solution e@ciently. Researchers also notice that it is actually not very practical to spend toomuch e7ort on searching for an optimal solution in an industrial context because changes often occur to eitherthe availability of resources or the set of jobs [9] that have to be performed. Consequently, many heuristic

∗ Corresponding author.1 This project was partly supported by National Natural Science Foundation (No. 60084003).

0165-0114/02/$ - see front matter c© 2002 Elsevier Science B.V. All rights reserved.PII: S 0165 -0114(01)00084 -7

118 C. Wang et al. / Fuzzy Sets and Systems 127 (2002) 117–129

algorithms have been developed to solve di7erent scheduling problems [17]. All these algorithms aim to reachnear optimal solutions e@ciently. In this paper, the ready time scheduling problem based on a single machinewith fuzzy processing times is studied.In the past, job processing time was usually treated as a crisp value. However, this assumption is not

realistic in many cases. In any scheduling scheme, the job processing times are usually determined in twoways. First, they are determined according to the processing times used to process the same type of jobs.Second, if this type of job has not been processed before, such as for the case of new products, the jobprocessing times are usually estimated by analyzing the processing times of similar jobs. In reality, jobsprocessing times vary due to occasional machine breakdowns, di7erences in machine capacity and levels ofoperators’ skills. This is especially true in cases of new products where the processing times can only beestimated by human subjective views. Therefore, it is reasonable to consider them as fuzzy variables. Forexample, a certain large electrical appliance company in southern China manufactures a range of products onorder. Thousands of part molding tasks with di7erent due-dates need to be processed on injection moldingmachines every day. The products are then supplied to the assembly line. In order to satisfy customers’special needs and preferences, the company has to develop and modify the characteristics and shapes of theproducts frequently. Thus, skilled workers usually estimate these new products’ processing times using theirexperience. They estimate the minimum job completion time aj and the maximum job completion time Iajrequired for the job j, estimate the job processing time variation in the interval (aj; Iaj) and arrive at a mostlikely job completion time aj. They will then say the job completion time is “approximately aj”. Therefore,the value of “approximately aj” can be described by a fuzzy set A with a membership function fA that hasa maximum at “x= aj” [12]. In other words, fuzzy set theory needs to be applied for solving this schedulingproblem.Fuzzy set theory has been studied extensively in the past 30 years. Since Tanaka et al. [18] and Zimmermann

[19] formulated fuzzy mathematical programming problems, many approaches have been proposed in the 3eldof fuzzy optimization. Several successful applications and implementations of fuzzy set theory in productionmanagement have also been reported.Dubois et al. [6] formulated a fuzzy constraint satisfaction problem. In their model the release and due

dates are fuzzy variables. Based on three basic procedures: consistency enforcing, tree search, and look-aheadanalysis, a solving scheme is presented.Ishii [14] introduced the concept of fuzzy due dates to his fuzzy scheduling model. The membership

function of a fuzzy due date assigned to each job represents the grade of satisfaction of a decision-maker forthe completion time of that job. Ishibuchi [13] and Ham [10] followed him and studied di7erent models ofscheduling problems with fuzzy due dates.Recently, Hong [11] has studied LPT scheduling for fuzzy tasks, i.e. tasks that have a fuzzy processing

time. In his treatment, the membership function of the job processing time is not continuous, but discrete,and the number of possible execution time for each task is assumed to be 3nite. To compare solutions, Hongranked fuzzy numbers using the so-called maximizing method and averaging method. Di7erent from Hong’streatment, in our mathematical model, we regard the membership function of the job processing time as beingcontinuous. We establish a job completion likelihood pro3le based on the membership function of the jobprocessing time, and then convert the scheduling problem to a chance constrained programming problem basedon the job completion likelihood pro3le. Solutions can then be ranked without [email protected] remainder of this paper is organized as follows. In Section 2, we will introduce the ready time schedul-

ing problem on a single machine with crisp processing times. In Section 3, assuming that the processing timesare triangular fuzzy numbers, the job completion likelihood pro3le of the fuzzy number will be constructedto establish a criterion for comparing di7erent scheduling schemes. Some properties of the job completionlikelihood pro3le will be discussed. In Section 4, the model of ready time scheduling with crisp processingtimes will be extended to the fuzzy environment. The model will be described by the technique of chanceconstrained programming with fuzzy processing times. Some theorems that are useful in 3nding an optimal

C. Wang et al. / Fuzzy Sets and Systems 127 (2002) 117–129 119

solution for the problem under several special conditions will be developed. For the general condition ofhaving jobs at di7erent due dates and con3dence levels, a necessary condition for the existence of an optimalsolution will be presented. Finally, the conclusions of this paper are presented in Section 5.

2. The single machine ready time scheduling model with crisp processing times

Single machine scheduling problems arise in practice more often than one may expect. Even in job-shopswith multiple machines, when one machine is found to be the bottleneck, it is rational to reduce the multiplemachines scheduling problem to a single machine problem [1].The ready time scheduling problem on a single machine that has crisp processing times can be described as

follows. There are n jobs to be processed by a single machine. Once a job is completed, a new job will startso the machine has no idle time between jobs. Each job j, has a crisp processing time, pj, and a deterministicdue date, dj. We assume that if we start processing the n jobs immediately, there must exist a minimum ofone job sequence so that each job processed under this sequence can meet its due date, i.e. no job is tardy. Inother words, a solution for the scheduling problem must exist. Our task is then to 3nd an optimal sequencesuch that we can postpone job processing as late as possible and still can meet all due dates with no tardyjob. As the duration from now to the moment we begin processing the n jobs is called the ready time, theproblem is equivalent to maximizing the ready time. The model can be expressed as follows:

r = max{rs; s = 1; 2; : : : ; n!:};

s:t: rks = dks −i∑

j=1

pks k = 1; 2; : : : ; n:

rs = min{rks; k = 1; 2; : : : ; n};

where r means the maximum ready time of all possible job sequences, rs means the minimum ready time ofa job sequence s, and rks means the ready time of a job at queue position k of the job sequence s.The model can be regarded as minimizing the makespan of all n jobs under the constraint that each job

is not tardy. This strategy can be pursued as long as the capacity of the single machine is not yet fullyused up. For example, a company may not be able to keep its machines busy all year round due to seasonalvariation of orders. In winter, it may have fewer orders and some machines can be left idle, and need not bekept busy everyday. The company can adopt this strategy to decrease the inventory, and use the ready timeto carry out machine maintenance or upgrade work that will prepare the machines better for the next peakseason.With crisp job processing times, rs= r1s, it has been found that the optimal sequence can be derived by

the order of earliest due date (EDD). With fuzzy job processing times, the case is entirely di7erent.

3. The job completion likelihood pro�le and its properties

3.1. Job processing times as triangular fuzzy numbers

Various de3nitions of fuzzy numbers can be found in the literature [7,8,4,15,3]. For convenience, werepresent the job processing times as triangular fuzzy numbers in this paper.


Fig. 1. The membership function of a fuzzy number A.

Assumption. Every job has a fuzzy processing time, and the fuzzy processing time is a triangular fuzzynumber.

Let a triangular fuzzy number A denote the job processing time. The membership function fA can beexpressed as (Fig. 1)

fA =

fLA (x); a6 x6 a;

fRA (x); a6 x6 Ia;

0 otherwise;

(1)

where fLA : [a; a] → [0; 1] and fR

A : [a; Ia] → [0; 1].The function fL

A : [a; a] → [0; 1] and fRA : [a; Ia] → [0; 1] are continuous and strictly increasing. Therefore,

fLA and fR

A have inverse functions and they will be denoted by gLA and gRA , respectively.Since we are comparing fuzzy quantities when we compare the objective of two possible job sequences, we

must 3rst know how to rank two or more fuzzy numbers. In article [8], Dubois only gave some de3nitionson how to compare a real number and a fuzzy interval. He de3ned over the intervals [A;+∞) and ]A;+∞):

f[A;+∞(x) = �A((−∞; x)) = supu6x

fA(u) =

0; x ¡ a;

fLA (u); a6 u6 a;

1; a ¡ u;

f]A;+∞)(x) = NA((−∞; x[) = infu¿x

(1− fA(u)) =

0; x 6 a;

1− fRA (u); a6 u6 Ia;

1; Ia ¡ u;

where �A and NA are the possibility and necessary measures de3ned in term of the distribution fA.Since each fuzzy number is a set, for fuzzy numbers, no 3xed rule exists for ranking fuzzy numbers. In

this paper, the fuzzy numbers are ranked by using the technique of chance constrained programming.

3.2. Chance constrained programming and job completion likelihood pro=le

The chance constrained programming model was developed by Charnes and Cooper [2] for stochasticprogramming. In 1988, Dubois [5] de3ned fuzzy chance constrained programming. In 1998, Liu and Iwamura


[16] extended the chance constrained programming from stochastic to fuzzy environments, and devised atechnique of fuzzy simulation for the chance constraints, which were usually hard to be converted to their crispequivalents. Liu and Iwamura’s chance constrained programming model makes decisions on the assumptionthat the constraints will hold with at least the possibility � and the chance represents the possibility that theconstraints are satis3ed. Using �, chance constrained programming can turn a system of fuzzy equations intoa system of crisp equations. A chance constrained programming with fuzzy parameters may be written asfollows [16]:

max f(x)

s:t:: Pos{�|gi(x; �)6 0; i = 1; 2; : : : ; p}¿ �;

where x is a decision vector, � is a vector of fuzzy parameters, f(x) is the return function, gi(x; �) areconstraint functions, i=1; 2; : : : ; p, � is a predetermined con3dence level, Pos{•} denotes the possibility ofthe event in {•}.To employ the chance constrained programming model in our ready time scheduling problem with every

� mapped to one and only one point, we establish a job completion likelihood �A of a job for any allocatedprocessing time x, based on the fuzzy membership function fA of its job processing time.

De�nition. The job completion likelihood pro3le, �A of a job with fuzzy processing time A, represents thelikelihood of that job to be completed within a certain allocated processing time x. Suppose we put everycompleted job in a set named the completed job set, then �A represents the likelihood of that job belongingto the completed job set within the allocated processing x.

In scheduling, the manager determines the job sequence and allocates the required resources to accomplishthe related set of operations. He allocates a certain processing time to a job which can possibly complete thejob. To him, the likelihood of the fuzzy processing time A equalling the crisp value a is of much less concernthan the likelihood that the job will fall in the completed job set, with an allocated processing time x.

To establish the job completion likelihood pro3le �A, let us consider a job of fuzzy processing time Awhose membership function fA is as de3ned in Section 3.1. If the allocated processing time x is smaller thanthe lower bound a, the job cannot be completed within the processing time x and the likelihood of the jobbelonging to the completed job set is zero (�=0). If the allocated processing time x is larger than the upperbound Ia, the likelihood of the job belonging to the completed job set is one (�=1). If the allocated processingtime x is larger than a and smaller than Ia, the likelihood will vary between zero and one (�∈ [0; 1]). Wemay therefore de3ne the job completion likelihood pro3le as follows (Fig. 2):

�A =f[A;+∞) + f]A;+∞)

2=

0; x6 a;

fLA

2; a; 6 x6 a;

2− fRA

2; a6 x6 Ia;

1; x¿ Ia:

(2)

Since the membership function fLA : [a; a] → [0; 1] is continuous and strictly increasing, and the member-

ship function fRA : [a; Ia] → [0; 1] is continuous and strictly decreasing, the job completion likelihood pro3le

�A : [a; Ia] → [0; 1] is continuous and strictly increasing.As the job completion likelihood pro3le is continuous and strictly increasing in the interval [a; Ia], the

inverse function exists. The inverse function is denoted as �−1A : [0; 1] → [a; Ia]. It is continuous and strictly


Fig. 2. The completion likelihood pro3le of a fuzzy member A.

increasing and it is easy to prove that

�−1A (�) =

{gLA(2�); 06 �6 0:5;

gRA(2− 2�); 0:56 �6 1;(3)

where gLA and gRA are the inverse functions of fLA and fR

A , respectively.

3.3. Properties of the job completion likelihood pro=le

In a scheduling scheme, each job processing time is a fuzzy variable. The job completion likelihood pro3leof a job at position k (k¿1) of a job sequence, is the sum of the job completion likelihood pro3les of all thejobs at position less than or equal to i. It is called the aggregate job completion likelihood pro=le. Based onfuzzy mathematical principles, the aggregate job completion likelihood pro3le has the following properties.

Property 1. The aggregate job completion likelihood pro3le of two jobs J1 and J2 with corresponding fuzzyprocessing times t1 = [a; a; Ia] and t2 = [b; b; Ib] is �t1 ⊕ t2 (z)= �t1 (x0)+�t2 (y0) and z= x0+y0, where �t1 and �t2are job completion likelihood pro3les of jobs J1 and J2, respectively. The aggregate job completion likelihoodpro3le �t1 ⊕ t2 is also a strictly increasing function in the interval [a+ b; Ia+ Ib].

Property 2. There are n jobs (J1; J2; : : : ; Jn) with fuzzy processing times ti, i=1; 2; : : : ; n in a sequence. Theaggregate job completion likelihood pro3le has an inverse function. If the likelihood � of the last job ispredetermined, then the last job processing time tn can be calculated as follows:

tn = �−1[tn](�) =

n∑i=1

�−1ti (�) =

∑n

i=1gLti(2�); 06 �6 0:5;∑n

i=1gRti (2− 2�); 0:56 �6 1;

(4)

where [tn] represents t1⊕ t2⊕ · · · ⊕ tn.

Property 3. The execution order of jobs within a job sequence does not a7ect the aggregate job completionlikelihood pro3le.

That is to say, for two jobs J1 and J2 with corresponding fuzzy processing times t1 and t2, �t1 ⊕ t2 (x)=�t2 ⊕ t1 (x).


4. Ready time scheduling model with fuzzy processing times

The ready time scheduling model with crisp processing time is used to simultaneously determine the jobready time and production sequence, based on each customer-speci3ed due date. The model is extended fromcrisp processing times to fuzzy processing times.

4.1. Assumptions and notations

The following assumptions and notations are used.

Assumptions

• The schedule is non-preemptive, meaning that a job, once started, will not be interrupted before its com-pletion for starting another job.

• The schedule has no idle time between jobs. Once a job is completed, another job will be startedimmediately.

• The schedule will not include any incomplete job. All jobs in the schedule must be completed.

According to the non-preemptive scheduling hypothesis and the fact that a job must be completed, if there isan available time x on the machine which is smaller than a, the job j cannot be completed, so the processingtime x on the machine smaller than a is not considered. If the processing time x is larger than Ia, there willbe an idle time, so the case of a processing time x larger than Ia is not considered. Only the fuzzy processingtimes that vary in the interval [a; Ia] are considered.

Notations�j = the con3dence level of the job j,dj = the due date of the job j,�ks = the con3dence level of a job at position k in a scheduling scheme s belonging to the completed

job set,dks = the due date of a job at position k in a scheduling scheme s,�[k]s = the aggregate job completion likelihood pro3le of all the jobs at or before position k in a scheduling

scheme s,�ks(x) = the job completion likelihood pro3le of a job at position k in a scheduling scheme s,�−1[k]s(�ks) = the processing time of the jobs at or before position k in a scheduling scheme s under the

constraint con3dence �ks,�−1ks (�ks) = the processing time of the job at position k in a scheme s under the constraint con3dence level

�ks.

4.2. Model description

We consider n jobs to be processed by a production system. Each job j has a deterministic due date dj anda fuzzy processing time j= [aj; ajaj]. The task to be performed is to sequence jobs and determine a commonready time, r, for the jobs so that the possibility of completing each job j by its due date is greater than orequal to the con3dence level, �j. The objective is to delay the common ready time subject to the completionpossibility constraints. The model of single facility can be expressed as follows.

r = max{rs; s = 1; 2; : : : ; n!}s:t: �[k]s(dks − rks)¿ �ks; k = 1; 2; : : : ; n

rs = min{rks; k = 1; 2; : : : ; n}: (5)


The inverse function of the constraint �[k]s(dks − rks)¿�ks is dks − rks 6 �−1[k]s. Using property 2, we have

rks¿dis −i∑

j=1

�−1js (�is) = dis −

∑i

j=1gLjs(2�is); 06 �is6 0:5;

∑i

j=1gRjs(2− 2�is); 0:56 �is6 1:

Since the inverse of a job completion likelihood pro3le is an increasing function, the maximum rks for eachconstraint as described in (5) only needs to be considered when the equation is satis3ed with equality.The model can also be described as follows.

r = max{rs; s = 1; 2; : : : ; n!}

s:t: rks = dks −

∑k

j=1gLjs(2�ks); 06 �ks6 0:5;

∑k

j=1gRjs(2− 2�ks); 0:56 �ks6 1;

k = 1; 2; : : : ; n;

rs = min{rks; k = 1; 2; : : : ; n}: (6)

The common ready time rs is the minimum of {r1s; r2s; : : : ; rns}. The equation that renders this minimum rsis called the binding equation. The job associated with the binding equation is called the binding job. It is aformidable task to determine the maximum common ready time for a set of n jobs through enumeration as itrequires the inversion of up to n× n! possibility membership functions. In the following sections, the optimalsolutions for some special conditions of the model are presented instead. For the scheduling of n jobs thathave di7erent due dates and con3dence levels, a necessary condition that an optimal solution must satisfy isalso established.

4.3. Common due date and common con=dence level

First, we consider the case where the jobs have a common due date, d, and a con3dence level, �. The con-straint equations yield the following relationship between the ris in a scheduling scheme s, r1s¿r2s¿ · · ·¿rns.The nth constraint is always binding, regardless of the job sequence. As a result, to determine the maximumcommon ready time, one needs only solve the nth constraint for r.

r = d−

∑n

j=1gLj (2�); 06 �6 0:5;

∑n

j=1gRj (2− 2�); 0:56 �6 1:

4.4. DiAerent due dates and common con=dence level

Next, we consider the case where the jobs have a common con3dence level � and di7erent due dates dj.The constraint equation may be generally stated as follows:

rks = dks −

∑k

j=1gLj (2�); 06 �6 0:5;

∑k

j=1gRj (2− 2�); 0:56 �6 1;

k = 1; 2; : : : ; n:

The optimal solution of this situation can be determined as follows.


Theorem 1. The maximum ready time for jobs that have diAerent due dates and a common con=dence levelare determined by sequencing jobs in the earliest due date (EDD) order.

Proof. Suppose it is known that two jobs i and j are at adjacent positions in a job sequence and di6dj.Hence two possible sequences ! and " exist, which only di7er by the pairing order of adjacent jobs i and j.In the sequence ", the job i is at position N and the job j is at position N + 1. In the sequence !, the jobj is at position N and the job i is at position N + 1. Let us compare the ready time of jobs j and i in thesequences ! and ".In the sequence !, the ready times of the jobs i and j are

rj! = dj −N−1∑k=1

�−1k! (�)− �−1

j (�);

ri! = di −N−1∑k=1

�−1k! (�)− �−1

j (�)− �−1i (�):

Since di6dj and �−1i (�) ¿ 0, we obtain ri!6rj!, and ri!= r! may also hold. Therefore in the sequence !,

only job i may be the binding job while job j is not.In the sequence ", the ready times of the jobs i and j are

ri" = di −N−1∑k=1

�−1k" (�)− �−1

i (�);

rj" = dj −N−1∑k=1

�−1k" (�)− �−1

i (�)− �−1j (�):

From this, we obtain, ri!6min{ri"; rj"}. A switch from " to ! can lead to a reduction in the ready time.A switch from ! to " will not lead to any reduction in the ready time and may actually increase the readytime. Thus if one continues to perform switches of this kind, the resulting optimal sequence would be in theEDD order.

4.5. DiAerent con=dence levels and common due date

Now we consider the jobs with di7erent con3dence levels �j and a common due date d. The constraint asdescribed in (6) can be re-written as follows:

rks = d−

∑k

j=1gLj (2�ks); 06 �ks6 0:5;

∑k

j=1gRj (2− 2�ks); 0:56 �ks6 1;

k = 1; 2; : : : ; n:

The optimal job sequence for this condition can be determined as follows.

Theorem 2. The maximum ready time for jobs that have diAerent con=dence levels and a common due dateis determined by sequencing jobs in decreasing con=dence level (DCL) order.

Proof. Suppose it is known that two jobs i and j are at adjacent positions in a job sequence and the con3dencelevels �i¿�j. We consider the sequences ! and " which di7er only by the pairing order of adjacent jobs iand j and we compare the ready time of jobs i and j.


In the sequence !, the ready times of the jobs i and j are

rj! = d−N−1∑k=1

�−1k! (�j)− �−1

j (�j);

ri! = d−N−1∑k=1

�−1k! (�i)− �−1

j (�i)− �−1i (�i):

Since the inverse of a job completion likelihood pro3le is an increasing function and �i¿�j, we obtainrj!¿ri!. Therefore in the sequence !, only job i may be the binding job and job j is not.In the sequence ", the ready times of the jobs i and j are

ri" = d−N−1∑k=1

�−1k" (�i)− �−1

i (�i);

rj" = d−N−1∑k=1

�−1k" (�j)− �−1

i (�j)− �−1j (�j):

From this we obtain, ri!6min{ri"; rj"}. A switch from " to ! can lead to a reduction in the ready time.A switch from ! to " does not lead to any reduction in the ready time and may actually increase the readytime. Thus if one continues to perform switches of this kind, the resulting optimal sequence would be in theDCL order.

4.6. DiAerent due dates and diAerent con=dence levels

Finally, we consider the case where both the due dates and con3dence levels vary among jobs. In thissituation, if the EDD and DCL sequencing concur, then the resulting sequence yields the maximum readytime.

Theorem 3. For the scheduling of n jobs which have diAerent due dates and con=dence levels; if EDD andDCL sequencing concur; then the resulting sequence yields the maximum ready time.

Proof. Suppose in a certain sequence #, there are two jobs i and j. Job i is at position N and job j is atposition M of the sequence. In addition, it is known that di6dj; �i¿�j. If we exchange the position of jobsi and j, we obtain a new sequence �. The sequences # and � will di7er only in the positions of jobs i and j.Now, let us compare the ready time of the jobs between the positions N and M in the sequences # and �.For any job h at a position L between the positions N and M of the sequence �, the ready times of job h

at position L and job i at position M are

rL� = dh −L∑

k=1

�−1k� (�h);

ri� = di −M−1∑k=1

�−1k� (�i)− �−1

i (�i):

Since dh¿di; �h6�i, we have rL�¿ri�. In the sequence �, between the positions N and M , only the jobi may be the binding job.


For any job h at a position L between the positions N and M of the sequence #:

rL# = dh −L∑

k=1

�−1k# (�h):

It can be easily seen that rL#¿ri� holds for any job h between positions N and M . A switch from # to �may reduce the ready time. Since jobs i and j are speci3ed at arbitrary positions N and M , the sequence #is an optimal sequence.If the sequences of EDD order and DCL order are di7erent, it is di@cult to determine the optimal sequence

directly. However, it may be possible to reduce the number of potentially optimal sequences, using thefollowing theorem with adjacent jobs.

Theorem 4. In an optimal sequence; if two adjacent jobs i and j have a relationship that di6dj; �i¿�j;the job i must precede job j.

Under this condition, it is usually di@cult to 3nd directly an optimal solution to the problem since it isNP-hard. If the job size is small (n6100), the Branch and Bound algorithm may be used to 3nd the optimalsolution. If the job size is large, it will not allow the optimal solution to be found in a short time, and heuristicalgorithms must be employed to 3nd a satisfactory solution to the problem e@ciently.

4.7. Numerical examples

Suppose jobs a, b, c, d are to be processed on a single machine. Each job processing time can be ex-pressed as a triangular fuzzy number, a= [5; 7; 8], b= [6; 6:5; 7], c= [6; 7:5; 8] and d= [7; 7:5; 9], respectively.From previous discussions, the job completion likelihood pro3le of each job can be established as follows,respectively:

�a(x) =

0; x ¡ 5;

x − 54

; 56 x6 7;

x − 62

; 7 ¡ x6 8;

1; x¿ 8;

�b(x) =

0; x ¡ 6;

x − 61

; 66 x6 6:5;

x − 61

; 6:5 ¡ x6 7;

1; x¿ 7;

�c(x) =

0; x ¡ 6;

x − 63

; 66 x6 7:5;

x − 71

; 7:5¡x6 8;

1; x¿ 8;


�d(x) =

0; x¡ 7;

x − 71

; 76 x6 7:5;

x − 63

; 7:5¡x6 9;

1; x¿ 9:

(1) DiAerent due dates and common con=dence levelLet us assume that the job due dates of a, b, c, d are 45; 42; 46, 40 min, respectively. The jobs have a

common con3dence level of 0.5. The objective is to 3nd the optimal sequence of jobs that yields a maximumcommon ready time.According to Theorem 1, the optimal solution is obtained by sequencing in the EDD order. The optimal

sequence is 〈d; b; a; c〉. The common ready time values are

rd = 40− �−1d (0:5) = 32:5;

rb = 42− �−1d (0:5)− �−1

b (0:5) = 29;

ra = 45− �−1d (0:5)− �−1

b (0:5)− �−1a (0:5) = 22;

rc = 46− �−1d (0:5)− �−1

b (0:5)− �−1a (0:5)− �−1

c (0:5) = 17:5;

r = min{ra; rb; rc; rd} = 17:5:

(2) DiAerent con=dence levels and common due dateLet us assume that the jobs a, b, c, and d have a common due date and di7erent con3dence levels of

0:5; 0:6; 0:7; 0:8, respectively. The objective is to 3nd the optimal sequence that yields a maximum commonready time.According to Theorem 2, the optimal sequence is obtained by sequencing in the DCL order. It is 〈d; c; b; a〉.

The common ready time values are

rd = 40− �−1d (0:8) = 31:6;

rc = 40− �−1d (0:7)− �−1

c (0:7) = 24:2;

rb = 40− �−1d (0:6)− �−1

c (0:6)− �−1b (0:6) = 18;

ra = 40− �−1d (0:5)− �−1

c (0:5)− �−1b (0:5)− �−1

a (0:5) = 11:5;

r = min{ra; rb; rc; rd} = 11:5:

(3) DiAerent due dates and diAerent con=dence levelsLet us assume that each job a, b, c, d has a di7erent due date (45; 42; 46; 40) and a di7erent con3dence

level (0:5; 0:6; 0:7; 0:8). The objective is to 3nd the optimal sequence of jobs that yields a maximum commonready time.Since the EDD order and the DCL order are di7erent, the optimal sequence cannot be determined directly.

From Theorem 4, the job d has the smallest due date and the biggest con3dence level, so it must precedeany other job. From Theorem 4, if job a and b are adjacent jobs that b must precede a. Meeting these twoconstraints reduce the solution space to only four possible sequences: 〈d; a; c; b〉, 〈d; b; c; a〉, 〈d; b; a; c〉 and〈d; c; b; a〉. By calculating each sequence’s ready time, the optimal sequence is found to be 〈d; b; c; a〉. Themaximum common ready time is r=16:5.


5. Conclusion

This paper has described a ready time scheduling problem in which the job processing times were continuoustriangular fuzzy numbers. By de3ning the job completion likelihood pro3le, the predetermined con3dencelevel will change the fuzzy constraint equations to crisp ones. A scheduling model on a single machinethat determines the maximum common ready time of n jobs with fuzzy processing times is established.After examining various properties of the job completion likelihood pro3le and the aggregate job completionlikelihood pro3le, the solution of the model is presented for several cases. For n jobs with di7erent con3dencelevels but a common due date, the EDD order gives the optimal job sequence. For n jobs with di7erent duedates but a common con3dence level, the DCL order gives the optimal job sequence. For n jobs with di7erentdue dates and con3dence levels, if the EDD and DCL orders concur, then the resulting sequence is optimal.When the EDD and DCL orders do not concur, a necessary condition that the optimal solution must satisfyis also presented. This condition can help to reduce the solution space.

References

[1] A. Bauer, J. Browne, R. Bowden, J. Duggan, Shop Qoor control systems: from design to implementation, Chapman & Hall, London,1994.

[2] A. Charnes, W.W. Cooper, Chance-constrained programming, Management Sci. 6 (1959) 73–79.[3] C.H. Cheng, A new approach for ranking fuzzy numbers by distance method, Fuzzy Sets and Systems 95 (1998) 307–317.[4] C.H. Cheng, D.L. Mon, Fuzzy system reliability analysis by con3dence interval, Fuzzy Sets and Systems 56 (1993) 23–29.[5] D. Dubois, Linear programming with fuzzy data, in: J.C. Bezdek (Ed.), Analysis of Fuzzy Information, Vol. 3, CRC Press, Boca

Raton, FL, 1988, pp. 241–263.[6] D. Dubois, H. Fargier, H. Prade, Fuzzy constraints in job-shop scheduling, J. Intell. Manufact. 6 (1995) 215–234.[7] D. Dubois, H. Prade, Operation on fuzzy numbers, Internat. J. Systems Sci. 9 (1978) 613–626.[8] D. Dubois, H. Prade, Ranking of fuzzy numbers in the setting of possibility theory, Inform. Sci. 30 (1983) 183–224.[9] B. Grabot, L. Geneste, Dispatching rules in scheduling: a fuzzy approach, Internat. J. Prod. Res. 32 (1994) 903–915.[10] S. Han, H. Ishii, S. Fujii, One machine scheduling problem with fuzzy due dates, Eur. J. Oper. Res. 79 (1994) 1–12.[11] T.P. Hong, C.M. Huang, K.M. Yu, LPT scheduling for fuzzy tasks, Fuzzy Sets and Systems 97 (1998) 226–286.[12] A. Irion, Fuzzy rules and fuzzy functions: a combination of logic and arithmetic operations for fuzzy numbers, Fuzzy Sets and

Systems 99 (1998) 49–56.[13] H. Ishibuchi, N. Yamamoto, T. Murata, H. Tanaka, Genetic algorithms and neighborhood search algorithms for fuzzy Qow shop

scheduling problems, Fuzzy Sets and Systems 67 (1994) 81–100.[14] H. Ishii, M. Tada, T. Masuda, Two scheduling problem with fuzzy due dates, Fuzzy Sets and Fuzzy Systems 46 (1992) 339–347.[15] R. Jain, A procedure for multi-aspect decision making using fuzzy sets, Internat. J. Systems Sci. 8 (1978) 227–282.[16] B. Liu, K. Iwamura, Chance constrained programming with fuzzy parameters, Fuzzy Sets and Systems 94 (1998) 227–237.[17] S.S. Panwalker, W. Iskander, A survey of scheduling rules, Oper. Res. 25 (1977) 45–62.[18] H. Tanaka, T. Okuda, K. Asai, On fuzzy mathematical programming, J. Cybernet. 3 (1974) 37–45.[19] H.J. Zimmermann, Description and optimization of fuzzy systems, Internat. J. Gen. Systems 2 (1976) 209–215.

Documents

The single machine ready time scheduling problem with fuzzy processing times