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Received 25 September 2006

Accepted 6 May 2008Available online 24 May 2008

Group technology

Single machine

Deteriorating jobs

Makespan

Sum of completion times

g inte

recent years. However, the group technology is relatively unexplored in this eld. In

addition, the group setup times are assumed to be known and xed. In reality, process

setup or preparation often requires more time as food quality deteriorates or a patients

times

enter the rolling machine, drops below a certain level,

required to control a re increases if there is a delay in the

time-stu-stas,d Ng,

2007; Wang and Cheng, 2007; Raut et al., 2007; Cheng

with starting time-dependent processing times were

Contents lists available at ScienceDirect

w.e

Int. J. Productio

ARTICLE IN PRESS

Int. J. Production Economics 115 (2008) 1281330925-5273/$ - see front matter & 2008 Elsevier B.V. All rights reserved.

doi:10.1016/j.ijpe.2008.05.004given by Alidaee and Womer (1999) and Cheng et al.(2004).

On the other hand, the production efciency can beincreased by grouping various parts and products with

Corresponding author. Fax: +886424517250.E-mail address: wclee@fcu.edu.tw (W.-C. Lee).requiring the ingot to be reheated before rolling (Guptaand Gupta, 1988), in re ghting when the time and effort

et al., 2008; Lee et al., 2008a, b; Lee et al.). Comprehensivereviews of different models and problems concerning jobsscheduling and has received increasing attention in recentyears. Examples of increasing processing times occur inmany situations, for instance, in scheduling of steel rollingmills where the temperature of an ingot, while waiting to

its starting time. Since then, related models ofdependent processing times have been extensivelydied from a variety of perspectives (Voutsinas and Co2002; Wu and Lee, 2006; Wang et al., 2006; Kang anbe known and xed from the rst job to be processed untilthe last job to be completed. However, there are manysituations in which a job that is processed later consumesmore time that the same job when processed earlier.Scheduling in this setting is known as deteriorating job

health condition worsens (Mosheiov, 1996).The deteriorating job-scheduling problem was rst

introduced independently by Gupta and Gupta (1988) andBrowne and Yechiali (1990). These authors constructedmodels where the processing time of a job is a function of1. Introduction

For many years, job-processingand job-processing times are lengthened as jobs wait to be processed. Specically, two

single-machine group-scheduling problems are investigated where the group setup

times and the job-processing times are both increasing functions of their starting times.

We rst prove that the makespan minimization problem remains polynomially solvable

when the deterioration is present. We then show that the sum of completion times

problem is polynomially solvable when the numbers of jobs in each group are equal. For

the case of unequal job sizes, a heuristic algorithm is proposed, and the computational

experiments show that the performance of the heuristic is fairly accurately when the

deterioration rate is small.

& 2008 Elsevier B.V. All rights reserved.

are assumed to

start of the re-ghting effort (Kunnathur and Gupta,1990), and in a medical procedure where more extensivemedical treatment might be necessary as a patientsKeywords: condition worsens. Therefore, this paper considers a situation where both setup timesSingle-machine group-scheduling ptimes and job-processing times

Chin-Chia Wu, Wen-Chiung Lee

Department of Statistics, Feng Chia University, Taichung, Taiwan

a r t i c l e i n f o

Article history:

a b s t r a c t

There is a growin

journal homepage: wwblems with deteriorating setup

rest in the research of deteriorating job-scheduling problems in

lsevier.com/locate/ijpe

n Economics

reduced; the variability of tasks is reduced, and hence

ARTICLE IN PRESS

C.-C. Wu, W.-C. Lee / Int. J. Production Economics 115 (2008) 128133 129worker training is simplied (Burbidge, 1979; Websterand Baker, 1995; Ng and Cheng, 2005; Chen et al., 1997;Chiou et al., 2007; Schaller, 2007). However, the grouptechnology has not been considered by most researchersin deteriorating jobs-scheduling problems until recently.Guo and Wang (2005) investigated the makespan mini-mization problem where the actual processing time isgiven by pij(t) pij(a+bt). They showed that the problem ispolynomially solvable under the group technology as-sumption. Under the same model, Xu et al. (2006) provedthat the total weighted completion time minimizationproblem remains polynomially solvable. Moreover, Wanget al. (2007) showed that single-machine group-schedul-ing problems are polynomially solvable where theobjectives are to minimize the makespan and the totalcompletion time under the model that pij(t) aijbijt.However, they all assume that the group setup times areconstant. Since longer setup or preparation might benecessary as food quality deteriorates or a patientscondition worsens, we consider a situation where thesetup time grows and jobs deteriorate as they wait forprocessing. To the best of our knowledge, Wu et al. (2008)were probably the only authors who studied the deterior-ating jobs-scheduling problems under the group technol-ogy with the assumption that both the setup times andthe job-processing times are functions of their startingtimes. They showed that the makespan and the totalcompletion time problems remain polynomially solvableunder the model that si(t) dit and pij(t) aijt. In thispaper, we investigate two single-machine-schedulingproblems in the context of group technology under adifferent model where the setup times and the job-processing times are both linear functions of their startingtimes.

The remainder of this paper is organized as follows.Notation and problem formulation are described inSection 2. The solution procedures for the makespanproblem and the sum of completion times problem aredescribed in Sections 3 and 4, respectively. The conclusionis given in Section 5.

2. Notation and problem statement

In this section, the notation that is used throughout thepaper will be introduced rst, followed by the formulationof the problem.

Notation

m the number of groups (mX2)Gi group i, i 1, 2,y, msimilar designs and/or production processes. This phe-nomenon is known as the group technology in theliterature (Ham et al., 1985). Many advantages have beenclaimed through the wide applications of group technol-ogy. For instance, changeover between different parts aresimplied, thereby reducing the costs involved; partsspend less time waiting, which results in less work-in-process inventory; parts tend to move through productionin a direct route, and hence the manufacturing lead time isnj the number of jobs in Gj, j 1, 2,y, mN the total number of jobs (i.e., n1+n2+y+nm N)Jij job j in Gj, i 1, 2,y, m, j 1, 2,y, nidi the basic setup time for Gj, i 1, 2,y, maij the basic processing time for Jij, i 1, 2, y, m,

j 1, 2,y, niai(j) the basic processing time for the jth job in Gi

when jobs are arranged in non-decreasing orderof their basic processing times; that is,ai1pai2p paini, i 1, 2, y, m, j 1, 2, y,ni

b the deterioration rate of jobs, where b40g the deterioration rate of setup, where g40si the actual setup time of Gipij the actual processing time of JijS a schedule of N jobsCij(S) the completion time of Jij under schedule SCmax the makespan of all jobsSCij the sum of completion times

There are N jobs classied into m groups ready to beprocessed on a single machine. All jobs are available attime t0, where t040. Jobs in the same group are processedconsecutively. A setup time precedes the processing ofeach group. The actual job-processing time of Jij is a linearfunction of its starting time t; that is,

pij aij bt; i 1;2; . . . ;m; j 1;2; . . . ;ni

where b is the deterioration rate. Moreover, the actualsetup time of Gi is also a linear function of its starting timet and as follows:

si di gt; i 1;2; . . . ;m

where g is the deterioration rate.Let G indicate that the problem is a group-scheduling

problem. Using the conventional notation, the makespanand the sum of completion times minimization problemsare denoted as 1/G, si di+gt, pij aij+bt/Cmax and 1/G,si di+gt, pij aij+bt/SCij, respectively.

3. The makespan problem

In this section, we consider a single-machine group-scheduling problem with deteriorating values of time.It is assumed that the actual job-processing time is alinear function of its starting time. Moreover, theactual group setup time is also a linear function of itsstarting time. The objective is to nd a schedule thatminimizes the makespan. The following theorem providesthe optimal schedule for the job sequence and the groupsequence.

Theorem 1. For the 1/G, si di+gt, pij aij+bt/Cmax pro-blem, the optimal schedule is obtained if jobs within the samegroup are ordered according to the smallest basic processing

time rst (SPT) rule:

ai1pai2p paini; i 1;2; . . . ;m,

ARTICLE IN PRESS

C.-C. Wu, W.-C. Lee / Int. J. Production Economics 115 (2008) 128133130and the groups are arranged in non-decreasing order of

1 bnidi Pnil1

ail1 bnil

1 g1 bni 1 .

Proof. We prove the theorem by contradiction. First, wewill show that jobs within the same group are orderedaccording to the SPT rule. Consider an optimal schedule Swhere jobs within the same group do not follow the SPTrule. In this schedule, there must be at least one Gi with atleast two adjacent jobs, say Jiu followed by Jiv, such thataiu4aiv. Furthermore, we assume that the starting time forJiu in S is t. We now perform an adjacent pairwiseinterchange of jobs Jiu and Jiv, leaving the remaining jobsin their original positions, to derive a new sequence S0.Under S, we have

CiuS 1 bt aiu,

and

CivS 1 b2t 1 baiu aiv,

whereas under S0, we obtain

CivS0 1 bt aiv,

and

CiuS0 1 b2t 1 baiv aiu.

Thus, we have

CivS CiuS0 baiu aiv40

since aiu4aiv. It implies that job processed after Jiu and Jivunder S has a later starting time than the same job underS. Thus, the makespan of jobs under S is strictly greaterthan that under S0. This contradicts the optimality of S andproves that jobs in the same group are ordered accordingto SPT rule.

Next, we will show that the group order follows the

specied rule. Consider an optimal schedule Q in which

the group order does not follow that rule. In this schedule,

there must be at least two adjacent groups, Gi followed by

Gj, such that

1 bnidi Pnil1

ail1 bnil

1 g1 bni 1

4

1 bnjdj Pnjl1

ajl1 bnjl

1 g1 bnj 1 (1)

Furthermore, we assume that the starting time to pro-

cess Gi in Q is t. Again, we perform an adjacent pairwise

interchange of Gi and Gj, leaving the remaining groups in

their original order, to derive a new group sequence Q0.Under Q, the completion time for the last job in Gj is

Cjnj Q 1 g1 bnjCini Q 1 bnjdj

Xnjl1

ajl1 bnjl. (2)In contrast, under Q0 the completion time for the last jobin Gi is

Cini Q 0 1 g1 bniCjnj Q 0 1 bnidi

Xnil1

ail1 bnil. (3)

From Eqs. (1)(3), it is derived that

Cjnj Q Cini Q 0 1 g1 bnj 1

1 bnidi Xnil1

ail1 bnil" #

1 g1 bni 1

1 bnjdj Xnjl1

ajl1 bnjl" #

40.

It follows that the completion time of the last job in Gjunder Q is strictly greater than that of Gi under Q

0. Thus, itimplies that groups after Gi and Gj under Q are processed

later than the same group under Q0. This contradicts theoptimality of Q and proves that groups are arranged in

non-decreasing order of

1 bnidi Pnil1

ail1 bnil

1 g1 bni 1Therefore, the theorem is proved. &

A simple algorithm based on the result of Theorem 1 todetermine an optimal schedule for the 1/G, si di+gt,pij aij+bt/Cmax problem is described as follows.

Algorithm 1:Step 1. Arrange the jobs of each group in non-

decreasing order of their basic processing times aij, i.e.,ai1pai2p paini, i 1, 2,y, m.

Step 2. Arrange the groups in non-decreasing order of

1 bnidi Pnil1

ail1 bnil

1 g1 bni 1

It is seen that the complexity of obtaining the optimaljob sequence within Gi is O(ni logni) and that of obtainingthe optimal group sequence is O(m logm). Hence, thecomplexity of Algorithm 1 is at most O(N logN). Anillustrative example (Example 1) is given in Appendix.

4. The sum of completion times problem

In this section, we deal with another single-machinegroup-scheduling problem when the deterioration isconsidered. The objective is to nd a schedule thatminimizes the sum of completion times of all jobs. Onceagain, the actual job-processing times and the actualgroup setup times are assumed as linear functions of theirstarting times. In addition, we rst assume that all thegroups have the same job size. That is, n1 ? nm n.We then provide the optimal job sequence and groupsequence under this condition. Finally, we propose aheuristic algorithm for the case of unequal job sizes.

ARTICLE IN PRESS

C.-C. Wu, W.-C. Lee / Int. J. Production Economics 115 (2008) 128133 131Theorem 2. For the 1/G, si di+gt, pij aij+bt/SCij problem,if all the groups have the same number of jobs, then the

optimal schedule is obtained if jobs within the same group

are ordered according to the smallest basic processing time

rst (SPT) rule:

ai1pai2p pain; i 1;2; . . . ;m,and the groups are arranged in non-decreasing order of

(1+b)ndi+Sl 1n ai(l)(1+b)

nl.

Proof. We prove the theorem by contradiction. First, wewill show that jobs within the same group are orderedaccording to the SPT rule. Consider an optimal schedule Swhere the jobs in the same group do not follow the SPT rule.In this schedule, there must be at least one Gi with at leasttwo adjacent jobs, say Jiu followed by Jiv, such that aiu4aiv.Furthermore, we assume that t is the starting time for Jiu inS. We now perform an adjacent pairwise interchange of Jiuand Jiv, leaving the remaining jobs in their original positions,to derive a new sequence S0. Under S,

CivS 1 b2t 1 baiu aiv,whereas under S0,

CiuS0 1 b2t 1 baiv aiu.Furthermore, the completion times of jobs processed beforeJiu and Jiv are the same since they are processed in the sameorder. In addition, we have

CivS CjuS0 baiu aiv;40,since aiu4aiv. It implies that job processed after Jiu and Jivunder S has a later starting time than the same job under S0.Thus,X

CijS X

CijS0XCiuS CivS CivS0 CiuS0 1 baiu aiv40.This contradicts the optimality of S and proves that jobs

within the same group are ordered according to SPT rule.

Next, we will show that the group order follows the

specied rule. Consider an optimal schedule Q in which

the group order does not follow the rule. In this schedule,

there must be at least two adjacent groups, Gi followed by

Gj, such that

1 bndi Xnl1

ail1 bnl41 bndj

Xnl1

ajl1 bnl. (4)

Furthermore, we assume that the starting time to pro-

cess Gi in Q is t. Again, we perform an adjacent pairwise

interchange of Gi and Gj, leaving the remaining groups in

their original order, to derive a new group sequence Q0.Under Q, the completion time for the kth job in Gj is

CikQ t1 g1 bk 1 bkdi

Xkl1

ail1 bkl (5)and the completion time for the kth job in Gj...