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Heuristic for scheduling in a two-machinebicriteria dynamic flowshop with setup andprocessing times separatedLing-Huey Su & Fuh-Der ChouPublished online: 15 Nov 2010.

To cite this article: Ling-Huey Su & Fuh-Der Chou (2000) Heuristic for scheduling in a two-machine bicriteriadynamic flowshop with setup and processing times separated, Production Planning & Control: The Management ofOperations, 11:8, 806-819, DOI: 10.1080/095372800750038418

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PRODUCTION PLANNING & CONTROL, 2000, VOL. 11, NO. 8, 806 ± 819

Heuristic for scheduling in a two-machinebicriteria dynamic ¯ owshop with setup andprocessing times separated

LING-HUEY SU and FUH-DER CHOU

Keywords two-machine dynamic ¯ owshop, bicriteria, setupseparated

Abstract. This paper addresses the two-machine bicriteriadynamic ¯ owshop problem where setup time of a job is sepa-rated from its processing time and is sequenced independently.The performance considered is the simultaneous minimizationof total ¯ owtime and makespan, which is more eå ective inreducing the total scheduling cost compared to the single objec-tive. A frozen-event procedure is ® rst proposed to transform adynamic scheduling problem into a static one. To solve thetransformed static scheduling problem, an integer program-ming model with N

2 ‡5N variables and 7N constraints is for-mulated. Because the problem is known to be NP-complete, aheuristic algorithm with the complexity of O(N3) is provided. Adecision index is developed as the basis for the heuristic. Experi-mental results show that the proposed heuristic algorithm is

eå ective and eæ cient. The average solution quality of the heur-istic algorithm is above 99%. A 15-job case requires only0.0235 s, on average, to obtain a near or even optimal solution.

1. Introduction

The ¯ owshop scheduling problem has been a keen area ofresearch for over thirty years ever since Johnson (1954)proposed the two-stage scheduling problem with themakespan objective. Minimization of total ¯ ow time(F) or mean ¯ ow time ( -

F) in the two-machine ¯ owshopscheduling environment have received increasing atten-tion and have been proved to be an NP-completeproblem (Gonzalez and Sahni 1978). Ignall and

Authors: Ling-Huey Su, Department of Industrial Engineering, Chung-Yuan ChristianUniversity, Chung-Li, Yaoyuan, Taiwan: and Fuh-Der Chou, Department of IndustrialEgineering and Management, Van Nung Institute of Technology, Chung-Li, Taoyuan, Taiwan.E-mail: linghuey@mail.ie.cycu.edu.tw

L IN-HUEY SU is an Associate Professor of Industrial Engineering at the Chung-Yuan ChristianUniversity. Her research interests include production scheduling, operations research, and produc-tion planning and control. She received her MS in Operations Research from Illinois Institute ofTechnology and PhD in Industrial Engineering from Yuan-Ze University.

FUH-DER CHOU is an Associate Professor in the Department of Industrial Engineering andManagement at Van Nung Institute of Technology. He received his MSD in IndustrialEngineering from Chung-Yuan Christian University and PhD in Industrial Engineering andManagement from National Chio-Tung University. His research interests include productionscheduling, global supply chain management, semiconductor manufacturing management, andgroup technology. He has been a consultant and principal investigator in relevant industry- andgovernment-funded projects in Taiwan.

Production Planning & Control ISSN 0953± 7287 print/ISSN 1366± 5871 online # 2000 Taylor & Francis Ltdhttp://www.tandf.co.uk/journals

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Schrage (1965) ® rst proposed a branch-and-bound pro-cedure to obtain optimal solution for this problem. VanDe Velde (1990), and Hoogeveen and Van De Velde(1993) developed diå erent lower bound determinationformulae to facilitate the eæ ciency of the branch-andbound procedure. Kohler and Steiglitz (1975), andCroce et al. (1996) suggested diå erent heuristicalgorithms to obtain the approximate solution. Onlyone criterion (Cmax ;F ;

-F† was considered in these studies.

However, the decision-maker usually needs to considertwo or more criteria at the same time. An optimal sche-dule under a speci® c criterion may be poor underanother criterion. Therefore, the multi-criteria two-machine ¯ owshop scheduling problem is important.Selen and Hott (1986), and Wilson (1989) developedinteger programming models to minimize the weightedsum of total ¯ ow time and makespan. Rajendran (1992)formulated the problem to minimize the total ¯ ow timewhere makespan was optimal. He proposed not only abranch-and-bound algorithm to obtain the optimal sche-dule but also two heuristics for improving the solutioneæ ciency. Nagar et al. (1995) presented a branch-and-bound algorithm to solve the two-machine ¯ owshopscheduling problem, for which the objective functionwas to minimize a weighted sum of total ¯ ow time andmakespan, and proposed a heuristic approach to ® nd theupper bound for the branch-and-bound . In addition,Nagar et al. (1996) developed a heuristic algorithm byintegrating a branch-and-bound procedure and a geneticalgorithm to ® nd the approximate solution, under theobjective function of the weighted sum of mean ¯ owtime and makespan. Serifoglu and Ulusoy (1998) devel-oped three branch-and-bound approaches which diå ermainly in their branching strategies under the objectivefunction of the weighted mean ¯ ow time and makespan.Sayin and Karabati (1999) addressed the problem ofminimizing the makespan and sum of completion timessimultaneously, and developed a branch-and-bound pro-cedure that iteratively solves restricted single-objectivescheduling problems until the set of eæ cient solutions iscompletely enumerated. The above-mentioned studiesbelong to the static scheduling problem domain whichassumes that the job available times and machine readytimes are all at time 0. However, in most real worldmanufacturing environments, dynamic scheduling isnecessary because jobs do not always arrive simul-taneously (Baker 1974, French 1982). Sun and Lin(1994) presented a dynamic scheduling frameworkwhich is carried out by solving a series of static backwardscheduling problems. A rolling time window approachwas adopted to decompose the job shop dynamic sched-uling problem, whose objective function was to minimizethe total weighted ¯ ow time ( å N

iˆ1WiFi) and to satisfy alldue-date constraints, into a series of static backward

scheduling problems. Roy and Zhang (1996) developeda fuzzy logic-based dynamic scheduling algorithm for thejob shop scheduling problem.

While studying the ¯ owshop scheduling problem, it isgenerally assumed that the setup time is included in theprocessing times, or negligible, and is therefore ignored.However, in many practical situations setup time isseparable and can be handled in advance. The setuptime is the time span required to prepare machine j forprocessing job i, e.g. by mounting the necessary tools, jigsand ® xtures. When setup times are considered separatedfrom processing times, the completion time of a job maybe reduced because the setup time of the jobs on a sub-sequent machine may be performed while it is idle. Thisreduction time will not be realized when setup times areconsidered as part of processing times. The separate setuptime problem has received little attention in the litera-ture. The application of this problem has been discussedby Sule and Huang (1983). Some of the works includeYoshida and Hitomi (1979), Dileepan and Sen (1991),Allahverdi (1997), Rajendran and Ziegler (1997), andAldowaisan and Allahverdi (1998). A recent surveypaper by Allahverdi et al. (1999) that surveys schedulingproblems involving setup time is presented. As indicatedby the survey article of Allahverdi et al. (1999), most ofthe works in the area of multiple criteria scheduling con-sists of bicriteria studies of the static case.

In order to increase system performance of a two-machine ¯ owshop, it is important to reduce boththroughput time and work in process (WIP) as muchas possible. The scheduling criterion of makespan mini-mization can eå ectively reduce the throughputtime, while the scheduling criterion of total ¯ ow timeminimization can eå ectively reduce work in process.This paper attempts to minimize the weighted sum ofthese two scheduling criteria in a dynamic ¯ owshopenvironment where setup and processing times separated.A frozen-event procedure is proposed to transform thedynamic scheduling into a static one. An integer pro-gramming model with N 2‡5N variables and 7N con-straints for the static scheduling problem is formulated,and a heuristic algorithm with complexity O(N 3) is pro-vided.

2. Notations and de� nitions

The following notations and de® nitions are used todescribe the scheduling problem. Brackets are used toindicate sequential position, i.e. p‰iŠ1 refers to the pro-cessing time of job i on the ® rst machine in a givensequence.

Two-machine bicriteria dynamic � owshop 807

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2.1. Known variables

N number of jobs,¬ weight for the total ¯ ow time,1 ¡ ¬ weight for the makespan,pi1 processing time of job i on the ® rst machine,

i ˆ 1 ;2 ; . . . ;N ;pi2 processing time of job i on the second machine,

i ˆ 1 ;2 ; . . . ;N ,Si1 setup time of job i on the ® rst machine,

i ˆ 1 ;2 ; . . . ;N ,Si2 setup time of job i on the second machine,

i ˆ 1 ;2 ; . . . ;N ;Ei release time of job i, i ˆ 1 ;2 ; . . . ;N ,R1 available ready time of the ® rst machine,R2 available ready time of the second machine.

2.2. Decision variables

Zik if job i is scheduled at the kth rank to beprocessed, i ˆ 1 ;2 ; . . . ;N ; k ˆ 1 ;2 ; . . . ;N .Zik ˆ 1, job i is scheduled at the kth rank to beprocessed; 5 0, otherwise.

Xk the idle time on the second machine betweenthe completing setup time and the startingprocessing time for the kth ranked job,k ˆ 1 ;2 ; . . . ;N .

Yk for the kth ranked job, the time between itscompletion time on the ® rst machine and itsstarting processing time on the secondmachine, k ˆ 1 ;2 ; . . . ;N .

Zk the idle time on the second machine betweenthe completion time of the k ¡ 1th ranked job onthe second machine and the starting time ofthe kth ranked job on the ® rst machine,k ˆ 1 ;2 ; . . . ;N .Zk ˆ max…Tk ¡ Ck¡1;2 ;0†.

Tk the starting time for the kth ranked job on the® rst machine, k ˆ 1 ;2 ; . . . ;N .

Ckj the completion time for the kth ranked job onthe jth machine, k ˆ 1;2; . . . ;N ; j ˆ 1 ;2.Note that,

Ck2 ˆ R2 ‡Xk

jˆ1

Zj ‡Xk

jˆ1

S‰jŠ2 ‡Xk

jˆ1

Xj ‡Xk

jˆ1

Bj;

k ˆ 1 ;2 ; . . . ;N

or

C12 ˆ R2 ‡Z1 ‡S‰1Š2 ‡X1 ‡B1 and

Ck2 ˆ Ck¡1 ;2 ‡Zk ‡ S‰kŠ2 ‡Xk ‡Bk;k ˆ 2 ;3 ; . . . ;N

2.3. Auxiliary variables

Ak the kth ranked job’ s processing time on the ® rstmachine

Ak ˆXN

iˆ1

Zikpi1; k ˆ 1 ;2 ; . . . ;N :

Bk the kth ranked job’ s processing time on thesecond machine

Bk ˆXN

iˆ1

Zikpi2; k ˆ 1 ;2 ; . . . ;N :

S‰kŠ j the kth ranked job’ s setup time on the jthmachine

S‰kŠ j ˆXN

iˆ1

ZikSij ; k ˆ 1 ;2 ; . . . ;N ; j ˆ 1 ;2:

Dk the kth ranked job’ s release time in the shop

Dk ˆXN

iˆ1

ZikEl; k ˆ 1 ;2 ; . . . ;N :

Cmax the completion time of all jobs in the shop; i.e.makespan

Cmax ˆ CN2 ˆ R2 ‡XN

jˆ1

Zj ‡XN

jˆ1

S‰ jŠ2

‡XN

jˆ1

Xj ‡XN

jˆ1

Bj

Fk the kth ranked job’ s ¯ ow time in the shop

Fk ˆ Ck2 ¡ Dk; k ˆ 1 ;2 ; . . . ;N :

F the summation of all jobs’ ¯ ow times in theshops.

F ˆXN

kˆ1

…Ck2 ¡ Dk†

AS the set of scheduled jobs.NS the set of unscheduled jobs.³ the number of the scheduled jobs in AS.OV the objective function which is

‰¬F ‡ …1 ¡ ¬†Cmax].OV* the best solution obtained so far.STj the jth machine’ s ready time to process the next

job, j ˆ 1 ;2.RTaj the completion time at the jth machine if the

next job to be processed on the jth machine isjob a, i.e.,RTa1 ˆ …ST1 ;Ta† ‡ Sa1 ‡pa1 ; a 2 NS ;andRTa2 ˆ …ST2 ;Ta† ‡ Sa2 ‡Xa ‡ pa2 ; a 2 NS:

Indexa the decision index if job a is the next scheduledjob.

808 L.-H. Su and F.-D. Chou

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Indexa ˆ ‰¬…N ¡ ³ ¡ 1† ‡1Š…Z³‡1 ‡ Sa2 ‡X³‡1 ‡ pa2†‡‰¬…N ¡ ³ ¡ 2† ‡1Š£ min …Z³‡2 ‡ Sb2 ‡X³‡2 ‡ pb2† ;

a* the job with the smallest Indexa.

3. Integer programming

As discussed in section 1, this study addresses the two-machine ¯ owshop bicriteria dynamic scheduling problemwith setup and processing time separated, using the fro-zen-event procedure to transform the dynamic reschedul-ing problem into a static one. Rescheduling is performedwhen any one of the following three events occurs: a newjob arrives; a job is canceled; or a machine breaks down.The integer programming model is formulated as follows.The objective function is to minimize the weighted sum oftotal ¯ ow time and makespan.

Objective function:

Min Z ˆ ¬XN

kˆ1

…Ck2 ¡ Dk† ‡ …1 ¡ ¬†CN2 …1†

Constraints:

XN

iˆ1

Zik ˆ 1 ; k ˆ 1 ;2 ; . . . ;N …2†

XN

kˆ1

Zik ˆ 1; i ˆ 1 ;2 ; . . . N …3†

Tk ¶ Dk ; k ˆ 1 ;2 ; . . . ;N …4†

T1 ¶ R1 ;Tk ¶ …Tk¡1 ‡ S‰k¡1Š;1 ‡Ak¡1† ; k ˆ 2 ;3 ; . . . ;N

…5†

C12 ˆ R2 ‡Z1 ‡ S‰1Š2 ‡X1 ‡B1

Ck2 ˆ Ck¡1;2 ‡Zk ‡S‰kŠ2 ‡ Xk ‡ Bk ;k ˆ 2 ;3 ; . . . ;N …6†

Z1 ˆ T1 ‡S‰1Š1 ‡A1 ‡ Y1 ¡ R2 ¡ S‰1Š2 ¡ X1

Zk ˆ Tk ‡ S‰kŠ1 ‡Ak ‡ Yk ¡ Ck¡1 ;2 ¡ S‰kŠ2 ¡ Xk ;

k ˆ 2 ;3 ; . . . ;N …7†

X1 ˆ T1 ‡ S‰1Š1 ‡A1 ‡Y1 ¡ R2 ¡ Z1 ¡ S‰1Š2

Xk ˆ Tk ‡S‰kŠ1 ‡ Ak ‡Yk ¡ Ck¡1 ;2 ¡ Zk ¡ S‰kŠ2 ;

k ˆ 2 ;3 ; . . . ;N …8†

Constraint (2) speci® es that only one job be scheduledat the kth job priority. Constraint (3) de® nes that eachjob be scheduled only once. Constraint (4) stipulates thatthe starting time of the kth ranked job be greater than orequal to its release time. Constraint (5) ensures that thestarting time of the kth ranked job is greater than orequal to the previous job’ s completion time on the ® rstmachine. Constraint (6) de® nes the completion time ofthe kth ranked job on the second machine. Constraint (7)can be explained using ® gure 1 where the idle time on thesecond machine before its setup operation of kth rankedjob (Zk) equals

…Tk ‡S‰kŠ1 ‡Ak ‡Yk† ¡ …Ck¡1;2 ‡S‰kŠ2 ‡Xk†:

Constraint (8) can be explained the same way as con-straint (7) using ® gure 1. All variables should be greaterthan or equal to zero and Zik is a binary integer.

4. A heuristic scheduling algorithm

Although the integer programming model provides theoptimal solution, variables and constraints increase dras-tically when the number of jobs increases. Therefore, theoptimal solution is not always attainable within theallowable time. Dynamic scheduling environmentrequires a quick response, so a heuristic scheduling algor-ithm is considered. Figure 2 shows the ¯ ow chart of the

Two-machine bicriteria dynamic � owshop 809

Figure 1. The Gantt chart to illustrate constraints (7) and (8).

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heuristic algorithm. The following is a step by step expla-nation of the algorithm.Step 1. Initialization. Let

AS ˆ f¬g; NS ˆ 1 ;2 ;3 ; . . . ;N ;

³ ˆ 0; ST1 ˆ R1; ST2 ˆ R2;

OV* ˆ a very large number ˆ 1031.

Step 2. Calculate Indexa for each job in NS. Take the jobwith the smallest Indexa as job a* and go to step3. If there is more than one job having the smal-lest Indexa then arbitrarily select one as job a¤and push the rest of the jobs as well as the corre-sponding information (current values ofAS ;NS ; ³;ST1 and ST2) into a stack. These jobsand the corresponding information in the

810 L.-H. Su and F.-D. Chou

Figure 2. The ¯ owchart of the heuristic scheduling algorithm.

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stack are control points for further backtrackingprocesses.

Step 3. Append job a* to AS, delete job a* from NS, andlet ³ ˆ ³ ‡ 1. Recalculate

ST1 ˆ Max …ST1 ;Ta¤† ‡Sa¤1 ‡ pa¤1

and

ST2 ˆ Max …ST2 ;Ta¤† ‡ Sa¤2 ‡X³ ‡pa¤2:

Step 4. If ³ < …N ¡ 1†, then repeat step 2± 4; otherwise goto step 5.

Step 5. Assign the only remaining unscheduled job in NS

to the last position of AS and delete it from NS.Recalculate ST1 ;ST2 and OV. If OV < OV*,then OV* ˆ OV .

Step 6. Check if there is any job in the stack. If there arejobs in the stack, then pop up the job at the top ofthe stack as job a*. Start the backtracking processand continue from step 3.

Step 7. The ® nal result, OV*, of the heuristic schedulingalgorithm is obtained and stop.

The Indexa plays the most important role in the above-mentioned procedure. The Indexa determines the jobsequence to be scheduled. Incorrect estimation of thevalue of Indexa aå ords a poor schedule. If the value ofIndexa is not sensitive suæ ciently, excessive, unnecessarybacktracking occurs. Thus, an appropriate formula toestimate Indexa is essential to the heuristic schedulingalgorithm. The objective function is rearranged as fol-lows.

¬XN

kˆ1

…Ck2 ¡ Dk† ‡ …1 ¡ ¬†CN2

ˆ ¬XN

kˆ1

R2 ‡Xk

jˆ1

Zj ‡Xk

jˆ1

S‰ jŠ2 ‡Xk

jˆ1

Xj ‡Xk

jˆ1

Bj

Á !

¡ ¬XN

kˆ1

Dk ‡ …1 ¡ ¬†

£ R2 ‡XN

jˆ1

Z j ‡XN

jˆ1

S‰ jŠ2 ‡XN

jˆ1

Xj ‡XN

jˆ1

Bj

Á !

ˆ ¬NR2 ¡ ¬XN

kˆ1

Dk ‡ …1 ¡ ¬†R2 ‡XN

kˆ1

¬

£Xk

jˆ1

Zj ‡Xk

jˆ1

S‰ jŠ2 ‡Xk

jˆ1

Xj ‡Xk

jˆ1

Bj

Á !‡ …1 ¡ ¬†

£XN

jˆ1

Zj ‡XN

jˆ1

S‰ jŠ2 ‡XN

jˆ1

Xj ‡XN

jˆ1

Bj

Á !

ˆ ¬…Z1 ‡S‰1Š2 ‡X1 ‡B1†

‡ ¬…Z1 ‡ Z2 ‡S‰1Š2 ‡S‰2Š2 ‡ X1 ‡ X2 ‡ B1 ‡B2†

‡ ¢ ¢ ¢ ‡¬…Z1 ‡ Z2 ‡ ¢ ¢ ¢ ‡ZN ‡S‰1Š2 ‡S‰2Š2 ‡ ¢ ¢ ¢

‡ S‰N Š2 ‡X1 ‡X2 ‡ ¢ ¢ ¢ ‡XN ‡B1 ‡ ¢ ¢ ¢ ‡BN†

‡ …1 ¡ ¬†…Z1 ‡Z2 ‡ ¢ ¢ ¢ ‡ZN ‡S‰1Š2 ‡S‰2Š2 ‡ ¢ ¢ ¢ ‡

£ S‰N Š2 ‡X1 ‡X2 ‡ ¢ ¢ ¢

‡ ZN ‡B1 ‡B2 ‡ ¢ ¢ ¢ ‡BN†

‡ ¬NR2 ¡ ¬XN

kˆ1

Dk ‡ …1 ¡ ¬†R2

ˆXN

kˆ1

…¬N ‡1 ¡ k¬†…Zk ‡ S‰kŠ2 ‡Xk ‡Bk†

‡ fconstantg

For minimization, the constant part is neglected.Therefore, the objective function can be represented asPN

kˆ1…¬N ‡ 1 ¡ k¬†…Zk ‡ S‰kŠ2 ‡Xk ‡Bk†. To make thevalue of Indexa accurate and approach the globalminima, Indexa considers the minimization of both theassignment of the current stage (if job a is scheduled) andthat of one stage further.

Indexa ˆ ‰¬…N ¡ ³ ¡ 1† ‡1Š…Z³‡1 ‡ Sa2 ‡X³‡1 ‡pa2†‡‰¬…N ¡ ³ ¡ 2† ‡1Š£ min…Z³‡2 ‡Sb2 ‡ X³‡2 ‡ pb2† ;

where

Z³‡1 ˆ Max …Ta ¡ ST2 ;0†Z³‡2 ˆ Max …Tb ¡ RTa1 ;0†X³‡1 ˆ Max ‰Max …ST1 ;Ta† ‡Sa1 ‡ pa1

¡ …ST2 ‡Z³‡1 ‡Sa2† ;0ŠX³‡2 ˆ Max ‰Max …RTa1 ;Tb† ‡Sb1 ‡pb1

¡ …RTa2 ‡ Z³‡2 ‡Sb2† ;0Ša 2 NS ;b 2 fNS ¡ fagg:

The heuristic scheduling algorithm applies the greedyapproach. At the ® rst stage, N…N ¡ 1† calculations andcomparisons are required. The kth stage requires…N ¡ k ‡ 1†…N ¡ k† calculations and comparisons. Thenumber of backtracking in all is negligible. The abovedescription suggests that there are at least

N…N ¡ 1† ‡ …N ¡ 1†…N ¡ 2† ‡ ¢ ¢ ¢ ‡ 3 £ 2 ‡2 £ 1fˆ 1

3 N…N ¡ 1†…N ‡ 1†g

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calculations and comparisons. Therefore, the complexityof this proposed algorithm is O…N3†.

5. Example illustration

A four-job, two-machine (i ˆ 1 ;2 ;3 ;4; j ˆ 1 ;2) ¯ ow-shop example is used to illustrate the decision process ofthe proposed heuristic algorithm. The objective functionis assumed to be 0:5F ‡0:5Cmax. Table 1 shows thepertinent information.

First of all, system status must be initialized:AS ˆ f¬g ;NS ˆ f1 ;2 ;3 ;4g, and OV* ˆ 1031. TheIndexa for each job in NS is calculated.

Stage 1.AS ˆ f¬g; NS ˆ 1 ;2 ;3 ;4; ST1 ˆ R1 ˆ 4;ST2 ˆ R2 ˆ 7; ³ ˆ 0; OV* ˆ 1031.

Index1 ˆ …2:5†f0 ‡4 ‡ Max ‰Max …4 ;3† ‡3 ‡ 11 ¡ …7 ‡4† ;0Š ‡ 22g ‡…2:0† Min f…0 ‡ 1 ‡0 ‡17† ;…0 ‡2 ‡0 ‡2† ;…0 ‡1 ‡0 ‡9†g ˆ 90:5

Index2 ˆ …2:5†f0 ‡1 ‡ Max ‰Max …4 ;6† ‡6 ‡9 ¡ …7 ‡1† ;0Š ‡17g…2:0† Min f…0 ‡ 4 ‡0 ‡22† ;…0 ‡2 ‡0 ‡2† ;…0 ‡1 ‡0 ‡9†g ˆ 85:5

Index3 ˆ …2:5†f0 ‡2 ‡ Max ‰Max …4 ;4†‡4 ‡3 ¡ …7 ‡2† ;0Š ‡2g ‡…2:0† Min f…0 ‡ 4 ‡8 ‡22† ;…0 ‡1 ‡12 ‡7† ;…0 ‡ 1 ‡12 ‡9†g ˆ 59:0

Index4 ˆ …2:5†f0 ‡1 ‡ Max ‰Max…4;0†‡2 ‡13 ¡ …7 ‡1† ;0Š ‡9g ‡…2:0† Min f…0 ‡4 ‡1 ‡22† ;…0 ‡1 ‡15 ‡17† ;…0 ‡2 ‡ 0 ‡2†g ˆ 60:5

Since Job 3 has the smallest Indexa which is 59.0, a* is 3.After completion of stage 1, AS ;NS ;ST1 ;ST2 ;³ need tobe updated. The Indexa for each job in NS is then recal-culated.

Stage 2.AS ˆ f3g; NS ˆ f1 ;2 ;4g; ST1 ˆ 11; ST2 ˆ 13;³ ˆ 1; OV* ˆ 1031.

Index1 ˆ …2:0†f0 ‡3 ‡ Max ‰Max …11 ;3† ‡3 ‡ 11 ¡ …13 ‡4† ;0Š ‡ 22g…1:5†Min f…0 ‡ 1 ‡0 ‡17† ;…0 ‡1 ‡ 0 ‡9†g ˆ 81:0

Index2 ˆ …2:0†f0 ‡1 ‡ Max ‰Max …11 ;6† ‡6 ‡ 9 ¡ …13 ‡1† ;0Š ‡ 17g ‡…1:5† Min f…0 ‡4 ‡0 ‡22† ;…0 ‡1 ‡0 ‡9†g ˆ 75:0

Index4 ˆ …2:0†f0 ‡1 ‡ Max ‰Max…11 ;0† ‡2 ‡ 13 ¡ …13 ‡1† ;0Š ‡ 9g…1:5† Min f…0 ‡ 4 ‡1 ‡22† ;…0 ‡1 ‡5 ‡17†g ˆ 78:5

Since Job 2 has the smallest Indexa which is 75.0, a* is2.

Stage 3.AS ˆ f3;2g; NS ˆ f1;4g; ST1 ˆ 26;ST2 ˆ 43; ³ ˆ 2; OV* ˆ 1031.

Index1 ˆ …1:5†f0 ‡4 ‡ Max ‰Max…26 ;3† ‡3 ‡ 11 ¡ …43 ‡4† ;0Š ‡ 22g ‡…1:0†…0 ‡ 1 ‡0 ‡9† ˆ 49:0

Index4 ˆ …1:5†f0 ‡1 ‡ Max ‰Max…26 ;0† ‡2 ‡ 13 ¡ …43 ‡1† ;0Š ‡ 9g‡…1:0†…0 ‡ 4 ‡1 ‡22† ˆ 41

Since Job 4 has the smallest Indexa which is 41.0, a* is4.

Stage 4.AS ˆ f3;2;4g; NS ˆ f1g;ST1 ˆ 41; ST2 ˆ 53; ³ ˆ 3; OV* ˆ 1031.

Cmax ˆ 79; F ˆ 175; OV ˆ 127; OV* ˆ OV ˆ 127.

In stage 4, the values of Cmax ;F and OV are calculated.Because no backtracking is necessary, the heuristic algor-ithm stops here. The ® nal schedule is J3 ¡ J2 ¡ J4 ¡ J1

with Cmax ˆ 79; F ˆ 175; OV* ˆ 127. The solution is thesame as the optimal solution obtained from the integerprogramming model.

6. Experimental results

Experimental results are divided into two parts. The® rst part gives the optimal solution or the lower boundvalue, if the allowable pivoting limit is reached, as thebasis to prove the eå ectiveness of the heuristic schedulingalgorithm. Another part provides the solution obtainedusing the heuristic scheduling algorithm. All experi-mental tests are run on a personal computer withPentium 133 MHz CPU. The machine ready time andjob release time are both uniformly distributed between 0and 10 [U(0, 10)]. The job processing time is uniformly

812 L.-H. Su and F.-D. Chou

Table 1. Pertinent information for the example case.

pi1 pi2 Si1 Si2 Ei

Job 1 (J1) 11 22 3 4 3Job 2 (J2) 9 17 6 1 6Job 3 (J3) 3 2 4 2 4Job 4 (J4) 13 9 2 1 0

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distributed between 1 and 25 [U(1, 25)]. The setup timeis uniformly distributed between 0 and 10 [U(0, 10)].The number of jobs ranges from 2 to 15. Thirty sets oftesting examples are experimented for each job number.Five sets of (¬;1 ¡ ¬) values, (1.0, 0.0), (0.7, 0.3), (0.5,0.5), (0.3, 0.7) and (0.0, 1.0) for the objective function ofeach testing example are calculated. The following for-mula is applied to determine the solution quality of theheuristic scheduling algorithm.

Solution quality ˆ ‰…2 £ optimal or lower bound†¡heuristicŠ=‰…optimal or lower bound†Š£ 100%

The integer programming model does not provideoptimal solutions for some testing examples, within theallowable pivoting limit, so the solutions are substitutedwith lower bound values. Therefore, the actual values ofsolution quality of the proposed heuristic algorithm areslightly higher than those shown in Tables 2 and 3.

The integer programming model solves the problemusing LINDO V5.01 software package. The allowablepivoting limit is set to 900000 so that the computingtime falls within about 3000 CPU seconds. The lowerbound value instead of the optimal solution will beused when the pivoting limit is reached. The heuristicscheduling algorithm is programmed in the C language.Initial data are stored in a ® le. The I/O processing timesproduce some discrepancies. To estimate the executiontime accurately, each testing example is processed con-tinuously 20 times and the average CPU time is calcu-lated.

As table 2 shows, the average computing time of theinteger programming model drastically increases as thenumber of jobs increases. When the number of jobs isgreater than 12, some of the testing examples cannot besolved optimally within the allowable pivoting limit.Therefore, the average execution time of the integer pro-gramming model is somewhat underestimated and theactual execution time is higher than that shown intable 2. The average execution time of the heuristicscheduling algorithm will slowly increase when the num-ber of jobs increases. For example, when ¬ ˆ 0:5 and1 ¡ ¬ ˆ 0:5, the curves for the number of jobs versusthe corresponding average execution time by log functionfor both integer programming model and heuristic algor-ithm are shown in ® gure 3. When the number of jobsincreases to 15, the integer programming model requiresat least 1806.970 s to produce a solution. However, for 61out of 150 sets of the experimental examples, lower boundsolutions are obtained within the allowable pivotinglimit. The average execution time of the heuristic algor-ithm, however, is only 0.0235 seconds. The solution qual-ity is very good which can be seen from table 2. The

complexity of the heuristic scheduling algorithm is fairlylow, but the algorithm can quickly obtain a near oroptimal schedule which satis® es the quick responserequirement of the dynamic rescheduling environment.

Table 2 summarizes the solution quality of the sched-uling algorithm. When optimal solution is obtained usingthe integer programming model, the average solutionquality for each job number is above 98%. When thelower bound value is used because the pivoting limit isreached, the average solution quality for each job num-ber is slightly lower, but still above 90%. The solutionquality is underestimated in this situation. In summary,the average solution quality decreases as the number ofjobs increases, because some testing examples cannotyield optimal solutions under the allowable pivotinglimit. Although the underestimation of the solution qual-ity dilutes the overall solution quality and lowers theaverage solution quality for the heuristic algorithm, theoverall average solution quality for each job number isabove 97.5% (table 2). The overall average solutionquality for 2100 (5*30*(15 ¡ 2 1 1)) sets of testing ex-amples is 99%. Therefore, we can assure that the resultof the heuristic scheduling algorithm is accurate and ispragmatic, in the dynamic rescheduling environment.

Table 3 shows the solution quality for a 15-job case,and ® gure 4 depicts the result. In these 150 sets of testingexamples, 89 sets can obtain optimal solutions within theallowable pivoting limit, while the remaining 61 sets can-not generate the optimal schedules through the integerprogramming model. Although 25 out of 150 optimalschedules are generated using the heuristic algorithm,the overall average solution quality is above 98% forthe 15-job case. In contrast to the slow integer program-ming model, the heuristic scheduling algorithm is suc-cessful at producing ultimate schedules and is quick.

7. Conclusion

To schedule a two-machine ¯ owshop environment,most studies developed optimization procedures for singlecriterion or static scheduling problems. This paperattempts to solve a two-machine bicriteria dynamicscheduling problem with setup and processing time sepa-rated, in which the objective function is to minimize aweighted sum of total ¯ ow time and makespan. A frozen-event procedure is proposed to transform a dynamicscheduling problem into a static one. An integer pro-gramming model and a heuristic scheduling algorithmwith the complexity of O…N3) are formulated.Experimental results show that the proposed heuristicalgorithm solves this problem quickly and accurately.The overall average solution quality of the heuristicalgorithm is 99%. Processing of the 15-job case requires

Two-machine bicriteria dynamic � owshop 813

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814 L.-H. Su and F.-D. ChouT

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0.02

299

98.9

7895

94.1

8712

99.2

0123

97.1

4050

98.9

8141

1423

854.

817

0.02

8641

99.5

2970

97.0

1131

99.2

2087

97.6

9141

99.5

1121

1516

1033

.808

0.03

607

99.8

4137

99.0

0312

99.5

1314

98.6

0121

99.7

8500

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Two-machine bicriteria dynamic � owshop 817

Table 3. The experimental results for the 15-job case.

¬ ˆ 1:0 ¬ ˆ 0:7 ¬ ˆ 0:5 ¬ ˆ 0:3 ¬ ˆ 0:0

01 98.93657 98.97145 98.98951 98.51440 99.2614302 95.71217* 95.72914* 96.25931* 96.98719* 100.0000003 95.49185* 95.72758* 95.93594* 96.01813* 98.60164*04 99.93142 99.93571 99.93817 99.94904 100.0000005 98.11414 98.03195 96.54345* 97.82691* 100.0000006 97.49187* 97.50914* 97.63606* 97.94216* 99.1566107 98.01589* 98.17210* 98.23141* 98.13426* 100.0000008 100.00000 99.51914 100.00000 100.00000 100.0000009 98.21413 98.31499 98.38757 97.31416 100.0000010 99.64312 99.65117 99.69172 99.71742 99.0466011 95.96063* 96.19102* 95.08143* 95.89830* 100.0000012 98.51453 98.55312 98.56721 98.97192 100.0000013 98.86414 98.91426 99.01845 97.79810 100.0000014 97.85120* 97.92134* 98.24949* 98.78314* 100.0000015 94.71321* 94.53472* 94.21952* 94.46134* 99.41011*16 98.83121 98.90412 98.93104 98.10412 99.0416317 95.71411* 95.76917* 96.31585* 96.05660 100.0000018 95.87989* 95.32972* 95.75312* 95.52831* 100.0000019 99.38112 99.49161 99.6168 99.66712 100.0000020 98.72415* 98.65118* 99.95110 98.99868* 100.0000021 99.38414 99.16129 99.47379 98.82071 100.0000022 98.37151 96.72141 94.31253* 95.76181* 99.0100223 99.87407 99.67157 99.44123 99.02162 100.0000024 97.18901* 97.32146* 97.65573* 98.01321* 100.0000025 96.61417* 96.81062* 96.83280* 97.01050* 100.0000026 96.86705* 96.68819* 96.71318* 95.39114* 100.0000027 99.51407 99.54930 99.17012 99.95217 100.0000028 97.16206* 97.18385* 97.45318* 97.76331* 99.5614229 99.28730 99.39145 99.61206 97.24410* 100.0000030 99.10494 99.25143 99.31612 99.61502 100.00000

* cannot obtain the optimal solution within the allowable pivoting limit.

Figure 3. Average execution times by log function at ¬ ˆ 0:5 and 1 ¡ ¬ ˆ 0:5.

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only 0.0235 s on average to obtain an ultimate or evenoptimal solution. The heuristic scheduling algorithm ismore practical to solve real world applications than theinteger programming model.

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Figure 4. The solution quality for 15 jobs.

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