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A single-machine group schedule with fuzzy setupand processing timesChin-Chia Wu a & Wen-Chiung Lee aa Department of Statistics , Feng Chia University , Taichung , 407 , Taiwan R.O.C.Published online: 18 Jun 2013.

To cite this article: Chin-Chia Wu & Wen-Chiung Lee (2005) A single-machine group schedule with fuzzysetup and processing times, Journal of Information and Optimization Sciences, 26:3, 683-691, DOI:10.1080/02522667.2005.10699671

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A single-machine group schedule with fuzzy setup and processingtimes

Chin-Chia Wu∗

Wen-Chiung Lee

Department of Statistics

Feng Chia University

Taichung 407

Taiwan

R.O.C.

Abstract

Group technology is a popular concept in production activities. It is assumed thatall jobs are classified into several groups and the jobs within a group must be processedcontiguously on the same machines. A sequence-independent setup time is incurred betweentwo consecutive scheduling groups. Traditionally, both group setup times and job processingtimes have been assumed to be crisp numbers. This paper presents a single-machine group-scheduling problem when the group setup times and job processing times are treated asfuzzy numbers. A solution is provided to find a group sequence and a job sequence thatminimizes the centroid values of these jobs. A total fuzzy flow time is then calculated for thedecision makers. An example is given to illustrate this approach.

Keywords : Single machine, group technology, fuzzy setup time, fuzzy flow time.

1. Introduction

The group technology concept involves classifying jobs with similardesigns or machining processes into groups. In scheduling problems, jobsin the same group that must be processed consecutively are called agroup technology assumption. The optimal decisions are made regarding

∗E-mail: cchwu@fcu.edu.tw

——————————–Journal of Information & Optimization SciencesVol. 26 (2005), No. 3, pp. 683–691c© Taru Publications 0252-2667/05 $2.00 + 0.25

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684 C. C. WU AND W. C. LEE

the group sequence and the job sequence in each group, called groupsequence and job sequence. A schedule in which both group and jobsequence are specified is referred to as a group schedule.

In the conventional group-scheduling problems, the setup times andjob processing times have been assumed to be constants [1, 3, 4, 5, 7, 8, 22,23, 25, 26]. However, in a real-life situation, it can often be observed thatsome kinds of information are not necessarily deterministic. McCahon andLee [20] pointed out that the job processing time can be estimated withina certain interval and be represented by a fuzzy number. Thus, fuzzynumbers are ideally suited to represent the setup and processing times. Inthis research, we will investigate the single-machine group-schedulingproblem in which all of the setup and processing times are treated as fuzzynumbers.

In the literature, some papers have addressed scheduling problemswith vague information. While some authors treated the processing times[9, 10, 20] or the due dates [6, 11, 12, 13, 21] as fuzzy numbers, othersconsidered both the processing times and due dates as fuzzy numbers [15,18, 24]. Konno and Ishii [14] studied an open shop problem with fuzzyallowable time and fuzzy resource constraints.

To the best of our knowledge, there has been no paper on group-scheduling problems in a vague environment. Hence, the purpose of thispaper is to utilize Zimmermann’s [27] fuzzy set theory to manage thegroup-scheduling problem when only vague information is available. Theremainder of this paper is organized in four sections. The problem state-ment is presented in Section 2. The solution procedures are described inSection 3. An illustrative example is provided in Section 4. The conclusionis given in the last section.

2. Problem statement

In this section, we first summarize the assumptions for the single-machine scheduling problem. Some assumptions about the fuzzy groupsetup times are modified from the general group-scheduling assumptions[5, 7, 23].

Assumptions:

1. All jobs are ready at time zero, no pre-emption is allowed.

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A SINGLE-MACHINE GROUP SCHEDULE 685

2. A machine does not break down and can only handle jobs one at atime.

3. Jobs can be classified into several groups and jobs within the samegroup are processed in succession.

4. The fuzzy group processing time required for completion of a groupis composed of a fuzzy group setup time and the sum of fuzzyprocessing times contained for jobs in that group.

5. The fuzzy group setup time for a group is independent of thesequence of groups.

6. The fuzzy setup time needed to process a job is independent ofthe group sequences and jobs and it is included in the fuzzy jobprocessing time.

7. No passing of groups and jobs is allowed.

The notations used throughout this paper are presented next. Con-sider a single-machine group-scheduling problem in which n jobs aredivided into N different groups that are readily available for process-ing. Groups and jobs are processed according to a designed schedulingsequence that will not be alternated.

Notations:

Gi denotes group i, i = 1, 2, . . . , N.

ni denotes the number of jobs in group Gi, i = 1, 2, . . . , N.

Ji j denotes job j in group Gi, i = 1, 2, . . . , N, j = 1, 2, . . . , ni.

si denotes the fuzzy setup time for group Gi, i = 1, 2, . . . , N.

ti j denotes the fuzzy processing time for job j in group Gi,

i = 1, 2, . . . , N, j = 1, 2, . . . , ni.

McCahon and Lee [20] stated that processing times could be moresuitably represented as intervals with the most probable completion timesomewhere near the middle of each interval. A fuzzy number whichis essentially a generalized interval that could represent this processingtime interval exactly and naturally. The definition of a general fuzzynumber and the manipulation of fuzzy numbers are given in detail inZimmermann [27]. Only the manipulation of triangular fuzzy numbersis considered in this paper. For convenience, suppose that a fuzzy number

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686 C. C. WU AND W. C. LEE

could be denoted by x = (a, b, c) , and the membership function, denotedby µA(x) , describes the degree to which x belongs to A and may benormalized as µA(x) ∈ [0, 1] . Kuroda and Wang [15] further pointed outthat a, b , and c could stand for the optimistic time, the time most likelyand the pessimistic time for carrying out the job.

3. Solution procedures

Defuzzification is an important technology in fuzzy scheduling prob-lems. Several effective defuzzification methods have been proposed inthe literature [2, 16, 17, 19, 20]. In this section, we will adopt the normaltriangular fuzzy number centroid value proposed by Lee and Yao [17].

Suppose the membership function of a normal triangular number Ais given in Figure 1 and it can be defined as follows:

µA(x) =

(x− a)/(b− a) if a ≤ x ≤ b

(c− x)/(c− b) if b ≤ x ≤ c

0 otherwise

where 0 < a < b < c .

Figure 1The membership function of µA(x)

The centroid value R(A) of a fuzzy set A is stated as follows;

R(A) =

∫xµA(x)dx

∫µA(x)dx

=13(a + b + c) .

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A SINGLE-MACHINE GROUP SCHEDULE 687

Based on the above definition, the centroid value of J(i)(k) , the k thjob in the i th group, is

R(i)(k) =i−1

∑u=1

(R(s(u)) + R(T(u))) + R(s(i)) +

k

∑v=1

R(t(i)(v))

,

where R(T(u)) =nu∑

j=1R(t(u)( j)) . Thus, the total centroid value (TC) of any

sequence is

TC =N

∑i=1

n(i)

∑k=1

R(i)(k)

=N

∑i=1

n(i)

i−1

∑u=1

(R(s(u)) + R(T(u))) +

N

∑i=1

n(i)R(s(i))

+N

∑i=1

n(i)

∑j=1

j

∑v=1

R(t(i)(v))

, (1)

where the symbol ( ) is used to signify the order of groups or jobs in agroup schedule. In the above equation, the first term is concerned withthe group sequence and is independent of the job sequences becauseR(s(u))+ R(T(u)) is a constant. The second term is a constant. The last termis concerned with the job sequence in each group, and is not influenced bythe group sequence. Thus, the group sequence and the job sequence canbe determined independently of one another.

It is observed that the first term will be minimized by ordering thegroups in the non-decreasing order of (R(si) + R(Ti))/ni , namely,

R(s(1)) + R(T(1))n(1)

≤ R(s(2)) + R(T(2))n(2)

≤ . . . ≤ R(s(N)) + R(T(N))n(N)

. (2)

The last one is minimized by sequencing the jobs in the non-decreasingorder of the fuzzy job processing time centroid value for each group.That is,

R(t(i)(1)) ≤ R(t(i)(2)) ≤ . . . ≤ R(t(i)(ni)) , i = 1, 2, . . . , N . (3)

Although we adopted Lee and Yao’s [17] centroid value method in theprevious discussion, the result is still true as long as the defuzzificationfunction is a linear operation.

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688 C. C. WU AND W. C. LEE

4. An example

Consider a single machine group-scheduling problem with sevenjobs in three groups. The fuzzy processing times for each job and the fuzzysetup times for each group are provided in Table 1.

Table 1The illustrative example

Group G1 G2 G3

Job J11 J12 J13 J21 J22 J31 J32

Job process- (1, 2, 4) (3, 5, 7) (3, 6, 9) (5, 8, 9) (4, 5, 8) (2, 8, 9) (3, 5, 6)ing time

Group setup (1, 3, 4) (1, 2, 3) (1, 4, 5)time

Using a membership function as given in Figure 1, the centroid valueof job J11 is R(t11) = 2.33 .

Similarly, the centroid values of the group setup times and theprocessing times are calculated and listed in Table 2.

Table 2The centroid values for the example

Group G1 G2 G3

Job J11 J12 J13 J21 J22 J31 J32

R(ti j) 2.33 5 6 7.33 5.67 6.33 4.67

[R(Ti) + R(si)]/ni 5.33 7.50 7.16

From Table 2, the group sequence is (G1, G3, G2) owing to equation(2), and its job sequences are (J11, J12, J13) , (J32, J31) , and (J22, J21) forgroups G3 , G1 , and G2 owing to equation (3), respectively. Therefore,the group schedule is (J11, J12, J13, J32, J31, J22, J21) with a total fuzzy flowtime (84, 177, 245) .

5. Conclusion

Group-scheduling problems, where the group setup times and jobprocessing times are formulized as crisp numbers, have been addressed in

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numerous papers. Contrary to the traditional group-scheduling problems,we investigated a single-machine fuzzy group-scheduling problem wherethe group setup times and the job processing times are all fuzzy numbers.A solution based on the centroid of the normal triangular fuzzy numberwas proposed to solve this problem. In this paper, we only paid attentionto the single-machine group-scheduling problem with fuzzy setup timesand fuzzy processing times. In the future, we will attempt to apply thefuzzy set concept to different problems in the scheduling field.

Acknowledgements. This work was supported by National ScienceCouncil of Taiwan, Republic of China, under Grant number NSC94-2213-E-035-020.

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Received April, 2005

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