Scheduling a flowline manufacturing cell with sequence dependent family setup times

Theory and Methodology

Scheduling a ¯owline manufacturing cell with sequence dependentfamily setup times

Je�rey E. Schaller a, Jatinder N.D. Gupta b,*, Asoo J. Vakharia c,1

a Department of Business Administration, Eastern Connecticut State University, Willimantic, CT 06226, USAb Department of Management, College of Business, Ball State University, Muncie, IN 47306, USA

c Department of Decision and Information Sciences, University of Florida, Gainesville, FL 32611, USA

Received 1 April 1998; accepted 1 April 1999

Abstract

This paper considers the problem of scheduling part families and jobs within each part family in a ¯owline man-

ufacturing cell where the setup times for each family are sequence dependent and it is desired to minimize the makespan

while processing parts (jobs) in each family together. Lower bounds on the optimal makespan value and e�cient

heuristic algorithms for ®nding permutation schedules are proposed and empirically evaluated as to their e�ectiveness

in ®nding optimal permutation schedules. Ó 2000 Elsevier Science B.V. All rights reserved.

Keywords: Flowshop scheduling; Part family sequence dependent setups; Manufacturing cells; Heuristic algorithms;

Empirical results

1. Introduction

Manufacturing cells usually consist of a groupof machines that are dedicated to producing aspeci®c number of part families. A part family is aset of parts (called jobs in this paper) that havesimilar requirements in terms of tooling, setupsand operations sequences. Often part families are

assigned to a cell based on operation sequences sothat materials ¯ow and scheduling are simpli®edeven though the part families produced by a cellmay require di�erent tooling. This process ofmanufacturing cell formation may result in eachpart family requiring the same set of machineswhere each job (part) is processed on each machinein the same technological order. Such manufac-turing cells are called pure ¯ow-shop manufacturingcells and resemble the traditional ¯owshops exceptfor the existence of multiple part families. Sinceparts are assigned to families based on tooling andsetup requirements, usually a negligible or minorsetup is needed to change from one part to anotherwithin a family and hence can be included in the

European Journal of Operational Research 125 (2000) 324±339www.elsevier.com/locate/dsw

* Corresponding author. Tel.: +1-765-285-5301; fax: +1-765-

285-8024.

E-mail address: [email protected] (J.N.D. Gupta).1 Asoo J. Vakharia's research was partially supported by a

1997 Summer Research Grant from the Warrington College of

Business Administration at the University of Florida.

0377-2217/00/$ - see front matter Ó 2000 Elsevier Science B.V. All rights reserved.

PII: S 0 3 7 7 - 2 2 1 7 ( 9 9 ) 0 0 3 8 7 - 2

processing times of each job. However, a majorsequence dependent setup is needed to changeprocessing jobs in one part family to another partfamily.

Our interest in this problem stems from thefollowing industrial application. In manufacturingprinting circuit boards (PCBs) on HS-180 auto-mated insertion machines, a FORTUNE 100company had developed a classi®cation of di�er-ent PCB types into PCB families. In preparing formanufacturing, chips to be inserted on all thePCBs in a family were preloaded on the insertionmachines. However, when a switch over betweenPCB families occurred, the number of distinctchips to be loaded on the machines for the newfamily depended on two aspects: (a) chips commonbetween families were not changed so thatchangeover time could be minimized; (b) chips notcommon between families needed to be changed.Thus, the changeover time was dependent on thesequence in which families of PCBs were sched-uled. Finally, note that this company used ®veautomated insertion machines all connectedthrough an automated handling system and allPCBs were individually routed through all ®vemachines in the same order. Thus, the schedulingproblem in this context was a sequence dependentfamily scheduling problem for a pure ¯ow line cell.

To formally de®ne the above manufacturingcell scheduling problem, consider that a given setof n jobs, N � f1; 2; . . . ; ng is to be processed on mmachines in the same technological order creatingthe ¯owshop structure. We assume that each jobbelongs to one of K families and at each machine,there is changeover time from one family to an-other, called setup time. Assume that the process-ing time of job i at machine j is given by pij. Let thesetup time for family f processed immediately afterfamily r at machine j be sj

rf where sjff � 0 for all f.

Further, let the jobs be numbered sequentiallysuch that ®rst n1 jobs belong to the ®rst family,next n2 jobs belong to the second family, . . ., nf

belong to the family f, . . . and ®nally last nK jobsbelong to the Kth family. Thus, n � n1 � � � � � nK .Let Nf represent the set of jobs belonging to familyf (Nf � {1; 2; . . . ; nf }) and F � f1; 2 . . . ;Kg bethe set of K families. Then, the completion time ofjob r�i� in a schedule r � r�1�; . . . ; r�n�� at ma-

chine j, C�r�i�; j�, is given by the following re-cursive relationship:

C�r�i�; j� � max C�r�i�; jn

ÿ 1�;

C�r�iÿ 1�; j� � sjqf

o� pr�i�;j; �1�

where r�iÿ 1� 2 Nq, r�i� 2 Nf , and C�/; j� �C�r�i�; 0� � 0 for all i and j.

Thus, the manufacturing cell scheduling prob-lem considered here is one of ®nding a schedule rthat minimizes its makespan, C�r� � C�r�n�;m�,where r ranges over all those permutations of njobs in which all jobs of the same family are pro-cessed together.

When each family contains only one job (i.e.,nf � 1 for all f 2 F ), the problem becomes a tra-ditional ¯owshop with sequence dependent timeswhich is NP-hard in the strong sense for m P 2(Gupta and Darrow, 1986). From this observa-tion, it follows that the manufacturing cell sched-uling problem considered here is NP-hard in thestrong sense.

Ignoring setup times, Johnson's (1954) algo-rithm minimizes the makespan for the two-ma-chine ¯owshop case and can be extended to thethree-machine case under certain conditions. Inview of NP-hard nature of the general ¯owshopproblem, most researchers have focused on devel-oping heuristic procedures which provide goodpermutation schedules (in which the order of jobprocessing is the same on all machines) within areasonable amount of computational time. How-ever, there is no guarantee that a permutationschedule will be optimal when the shop containsfour or more number of machines.

Recent surveys by Allahverdi et al. (1999) andCheng et al. (2000) of scheduling research in-volving separable setups reports that most priorresearch on manufacturing cell scheduling hasassumed sequence independent setup times (e.g.,Vakharia and Chang, 1990; Skorin-Kapov andVakharia, 1993). The one exception is the paperby Hitomi et al. (1977) where a simulation modelwas developed for shops with three ¯ow patterns.In each shop, the impact of eight scheduling rulesand di�erent ratios of mean setup to processingtimes was investigated. They found that the

J.E. Schaller et al. / European Journal of Operational Research 125 (2000) 324±339 325

performance of each shop con®guration wassigni®cantly in¯uenced by the relative ratio ofsetup times to processing times. Further, sched-uling rules which considered sequence dependentsetups outperformed rules which did not explic-itly do so.

This paper considers the problem of schedulinga pure (¯owshop) manufacturing cell with se-quence dependent family setup times and developspolynomially bounded heuristic algorithms to ®ndminimum makespan permutation schedules. Theproposed algorithms are combinations of existingalgorithms for the ¯owshop problems with andwithout setups and are quite e�ective in solvinglarge-sized problems.

The rest of the paper is organized as follows:Section 2 develops lower bounds on the optimalmakespan. These lower bounds are used in Section3 to describe various heuristic algorithms for ob-taining approximate solutions to the problem.Results of computational tests to evaluate theperformance of heuristic algorithms are reportedin Section 4. Finally, Section 5 discusses the mainresults and describes some fruitful directions forfuture research.

2. Lower bounds

We generalize the machine based bound oftraditional ¯owshop scheduling problem to obtainlower bounds on the optimal makespan for themanufacturing cell scheduling problem consideredhere. To develop these lower bounds, let r be anexisting partial family sequence which has d as thelast family scheduled. Let p be the subset of fam-ilies not included in r. For each family f, let thesubset Qf represent the jobs not included in partialschedule r and pf be any arbitrary schedule of alljobs in Qf . For each family f, each machine j, andeach machine pair �r; j� with r < j, we de®ne thefollowing terms:

af �r; j� � mink2Qf

Xjÿ1

q�r

pkq; �2�

bf �r; j� � mink2Qf

Xj

q�r�1

pkq; �3�

cf �j� �Xk2Qf

pkj: �4�

Further, let xf �j� be the makespan of a Johnsonschedule for jobs in subset Qf considering ma-chines j and j� 1 only.

Then, the lower bound on the completion timeof any schedule rpf (obtained by concatenatingthe remaining jobs in family f, pf to partialschedule r) at machine j, LB�rpf ; j� is given by thefollowing expression:

LB�rpf ; j� � max LB1�rpf ; j�; LB2�rpf ; j�;�

LB3�rpf ; j�; �5�

where

LB1�rpf ; j� � max16 r6 j

C�r; r�n

� srdf � af �r; j�

� cf �j�o; �6�

LB2�rpf ; j� � max16 r6 jÿ1

C�r; r�n

� srdf � af �r; jÿ 1�

� xf �jÿ 1�o; �7�

LB3�rpf ; j� � max16 r6 jÿ1

LB�rpf ; r�� bf �r; j�

: �8�

The lower bounds in Eqs. (6) and (8) are obtainedby adding the processing time for the remainingjobs in family f while the lower bound in Eq. (7) isobtained by adding the unavoidable idle time onmachine j as determined by using Johnson's two-machine rule for machines jÿ 1 and j.

The lower bounds in Eqs. (5)±(8) above assumethat the partial schedule r contains all jobs offamily d and no job from family f. However, if thepartial schedule r contains some jobs of family f,then d � f and the setup time sr

df � 0. With thismodi®cation, the above expressions are valid for®nding a lower bound on the completion time ofthe subset of jobs from family f concatenated tothe partial schedule r.

Now for machine j, de®ne the following �u� 1�city traveling salesman problem (TSP) where cities

326 J.E. Schaller et al. / European Journal of Operational Research 125 (2000) 324±339

correspond to the u� 1 families in the unsched-uled set p and the cost from each city (family) i 2 pto each city (family) q 2 pÿ ff g is given by

cjiq � sj

iq � cq�j�: �9�

Further, for each q 2 pÿ ff g, de®ne

cjqf � b�j;m�; �10�

where for each q 2 p,

cjqq � 1: �11�

Then, the lower bound for machine j can be foundby solving the TSP de®ned above. Since this TSPproblem is NP-hard, a lower bound on its valuecan be found by solving the corresponding as-signment problem. Let LBa�f ; j� be the lowerbound obtained by solving the assignment prob-lem with the cost matrix given by Eqs. (9)±(11)above. Then, the lower bound on the completiontime of any schedule with the partial schedule rf ,LB�rpf � is given as follows:

LB�rpf � � max16 j6m

LB�rpf ; j�� LBa�f ; j�

: �12�

The above lower bounds can be used to describe abranch and bound algorithm to solve the problem.However, such an optimization algorithm is notlikely to be useful in solving moderate to large sizeproblems. If the sequence of jobs within eachfamily is known, the above lower bounds can bemade tighter. To do so, let q�f ; 1� and q�f ; 2� bethe jobs at the ®rst and last sequence position infamily f. Now, rede®ne the af �r; j� and bf �r; j� asfollows:

af �r; j� �Xjÿ1

q�r

pq�f ;1�q; �13�

bf �r; j� �Xj

q�r�1

pq�f ;2�q: �14�

In addition, if we do not wish to solve the as-signment problem to ®nd a lower bound for theTSP, we can use the lower bound for the assign-

ment problem to describe another lower bound asfollows:

LB�rpf � � max16 j6m

LB�rpf ; j�� LBb�f ; j�

; �15�

where LBb�f ; j� is the lower bound for the assign-ment problem de®ned by Eqs. (9)±(11).

Since, the partial schedule r and the sequenceof jobs within each family are not known in ad-vance, and the lower bound at the root node is notlikely to be very e�ective, we use a branch andbound procedure to ®nd the lower bound wherethe completion times of partial schedule r in Eqs.(6) and (7) are replaced by their lower bounds andsearch is limited to only those partial schedulescontaining whole families. Thus, each node in thebranch and bound tree represents an initial partialfamily sequence. A lower bound is calculated foreach node. If the lower bound is greater than orequal to the incumbent value the node is fath-omed, otherwise a node is created for each familynot included in the initial partial sequence. If acomplete family sequence is generated and itslower bound is less than the incumbent value thenthe incumbent value is updated. When the branchand bound procedure terminates the incumbentvalue provides the lower bound on the makespanof an optimal permutation sequence.

In view of the use of branch and bound pro-cedure, computational e�ort required to generatethe above lower bound is not polynomiallybounded. To decrease the computational time re-quired to ®nd the lower bounds, the lower boundto the TSP problem was not found by solving anassignment problem; rather a lower bound for theassignment problem was found as illustrated inEq. (15) above. Since our goal is to use the lowerbound in the evaluation of the proposed heuristicalgorithms, simpli®cation is not of much concern.

3. Heuristic algorithms

The solution of the manufacturing cell sched-uling problem considered here requires two as-pects: sequencing jobs within each family andsequencing various families. While there is inter-action between these two aspects, we assume that


these sequences can be developed independent ofeach other. This assumption may produce subop-timal results but saves considerable computationale�ort in ®nding an approximately optimal permu-tation schedule for the problem. Thus, in the heu-ristic algorithms proposed here, we ®rst develop thesequence of jobs within each family and then usethat sequence to develop the sequence of families.

3.1. Scheduling jobs within a family

The jobs in a given family can be scheduledusing one of several existing heuristic algorithmsfor minimizing makespan in a ¯owshop. Since thejob scheduling problem for each family is sequenceindependent, we draw upon existing research onmanufacturing cell scheduling to evaluate two jobbased scheduling algorithms.

3.1.1. Algorithm C: Campbell±Dudek±Smith basedprocedure

Dannenbring's (1977) empirical results showthat the CDS procedure (Campbell et al., 1970)that generates mÿ 1 arti®cial two-machine ¯ow-shop schedules is quite e�ective in ®nding an ap-proximate solution to the m-stage ¯owshopproblem. Therefore, we use the adaptation pro-posed by Vakharia and Chang (1990) to develop asequence of jobs within each family. To do so, foreach family f � f1; 2; . . . ;Kg, let qf be the sched-ule obtained by the CDS algorithm. Then, thedetails of this procedure (called algorithm C) areas follows.

Input: N � f1; 2; . . . ; ng, Nf for f �f1; 2; . . . ;Kg,pij for i � 1; . . . ; n; j � 1; . . . ;m. Let f � 1, k � 1,and makespan Zf � 1 for f � f1; 2; . . . ;Kg.

Step 1. For each job i 2 Nf , calculate ai �Pkj�1 pij and bi �

Pmj�mÿk�1 pij and enter step 2.

Step 2. For jobs i 2 Nf with ai6 bi, form apartial sequence pf such that apf �1�6 � � � 6 apf �n0�,where n0 is the number of jobs i with ai6 bi. Ap-pend the jobs i 2 Nf with ai > bi to pf so thatbpf �n0�1�P � � � P bpf �nf �.

Step 3. If Zf > C�p�, set Z � C�p� and rf � pf .Set k � k � 1. If k P m, enter step 4; otherwise,return to step 1.

Step 4. If f P K, STOP; otherwise, set f �f � 1, k � 1 and return to step 1.

The time requirement for algorithm C isO��mÿ 1�PK

f�1 nf log�nf � � mnf

� ��which,throughglobal implementation of step 2 for each f and k,can be reduced to O �mÿ 1��n log �n� � mn�� .

3.1.2. Algorithm N: The NEH based procedureThe experimental results by Park et al. (1984)

show that the NEH procedure (Nawaz et al.,1983), based on the use of job insertion in an ex-isting partial schedule may provide makespanwhich is less than that obtained by the CDS pro-cedure. Therefore, we use the adaptation proposedby Vakharia and Chang (1990) to develop a se-quence of jobs within each family. Details of thisprocedure (called algorithm N ) are as follows.

Input: N � f1; 2; . . . ; ng, Nf for f �f1; 2; . . . ;Kg, pij for i � 1; . . . ; n; j � 1; . . . ;m. Letf � 1.

Step 1. For the jobs contained in Nf , letJ � �l�1�; . . . ; l�n�� be the schedule of all nf jobscontained in Nf such that

Pmj�1 pl�i�;j PPm

j�1 pl�i�1�;j for all i6 nf ÿ 1. Let i � 2, andp � �l�1�; l�2��. Further, let p � pppiÿp for06 p6 i. Enter step 2.

Step 2. Let k � l�i� 1�. Generate �i� 1� par-tial sequences represented by ppkpiÿp, where 06p6 i. For each of the generated partial sequences,®nd C�ppkpiÿp�. Set x � x1; . . . ;xi�1f g � ppkpiÿp;

�p6 i� 1g; and C�xm� � min16 j6 i�1 fC�xj�g. Setp � xm and enter step 3.

Step 3. If i < nf , let i � i� 1 and return to step2; otherwise set rf � p and enter step 4.

Step 4. If f P K, STOP; otherwise, set f �f � 1 and return to step 1.

The time requirement for algorithm N isO�mPK

f�1 n3f � which may be reduced to

O�mPKf�1 n2

f � using the results discussed in Tail-

lard (1989).

3.2. Scheduling job families

Gupta and Darrow (1986) described dynamicadaptations of Johnson's algorithm to provideapproximate solution procedures for the two-ma-


chine ¯owshop case with sequence dependent set-up times. As in the CDS procedure, we can de®nearti®cial two-stage ¯owshop scheduling problemsto ®nd an approximate solution to the problem ofscheduling families with sequence dependent setuptimes. Thus, our ®rst algorithm to schedulefamilies is an extension of an algorithm for thetwo-machine ¯owshop problem with sequencedependent setup times described by Gupta andDarrow (1986). The steps of this algorithm G aredescribed below.

3.2.1. Algorithm G: Extended Gupta and Darrowprocedure

Input: N � f1; 2; . . . ; ng, pij, and sjiq for i �

1; . . . ; n; j � 1; . . . ;m, k � 1; i; q � 1; 2; . . . ;K, andmakespan Z � 1.

Step 1. Let r be an initial partial family se-quence containing t1 families with last family as dand q be the post partial family schedule con-taining t2 families with e as its ®rst family. Lett � t1 � t2. Let p be the subset of families notincluded in r and q. Initially, set r � q � /(empty), t1 � t2 � 0 and p � f1; 2; . . . ;Kg. Enterstep 2.

Step 2. Using the ®rst job q in each family i 2 p®nd A�i; k� as follows:

A�i; k� � max16 j6 k

sjdi

(�Xk

x�j

pqx

)ÿ sk�1

di : �16�

Find family a such that A � A�a; k� �mini2p A�i; k�.

Step 3. Using the last job r in each family i 2 p®nd B�i; k� as follows:

B�i; k� �Xm

j�mÿk�1

prj: �17�

Find family b such that B � B�b; k� �mini2p B�i; k�.

Step 4. If A6B, let r � ra, t1 � t1 � 1; other-wise let q � bq, t2 � t2 � 1. If t � t1 � t2 < n, up-date p and return to step 2; otherwise enter step 5.

Step 5. If Z > C�rq�, set Z � C�rq� and d � rq.If k P m, STOP; otherwise, set k � k � 1 and re-turn to step 1.

The computational time requirements for al-gorithm G is O �mÿ 1��K2 � mn�� .

3.2.2. Algorithm G0: Modi®ed Gupta and Darrowprocedure

The processing times at the second stage of thekth arti®cial two-stage ¯owshop problem in algo-rithm G did not include any setup times. The in-clusion of these setup times may result in areduction in makespan of the ®nal schedule ob-tained. Thus, a modi®cation of the above algo-rithm, called algorithm G0, de®nes the processingtimes as follows:


sjdi

(�Xk

x�j

pqx

); �18�

B�i; k� � maxmÿk�16 j6m

Xm

x�j

prx

(� sj

ie

): �19�

3.2.3. Algorithm G00: Improved Gupta and Darrowprocedure

The above algorithm can be further improvedby observing that the minimum possible setup timeincurred for family i in any position should not beconsidered as idle time. To do so, de®ne the fol-lowing two terms:

S�i; k � 1� � minf2p;f 6�i

sk�1fi ; �20�

S0�i;mÿ k� � minf2p;f 6�i

smÿkif : �21�

Then, in the second modi®cation, called algorithmG00, the processing times are de®ned as follows:


sjdi

(�Xk

x�j

pqx

)ÿ S�i; k � 1�; �22�


Xm

x�j

prx

(� sj

ie

)ÿ S0�i;mÿ k�:

�23�


3.2.4. Algorithm B: Extended Baker's group sched-uling algorithm

In each of the algorithms above, the processingtimes for the kth arti®cial two-stage problem weredetermined by considering the processing times ofthe ®rst and last job in a family. It is equally pos-sible to de®ne these processing times in an attemptto minimize idle times at the later (early) machinesearly (later) in the schedule. Baker (1990) extendsJohnson's algorithm to the manufacturing cellscheduling problem with two machines and se-quence independent setup times. First the jobswithin each group (family) are ordered accordingto Johnson's rule. Families are then scheduled byusing Johnson's algorithm to solve a speciallyconstructed two-machine ¯owshop problem whereeach family is treated as a job. Extending the con-cepts in Baker (1990) algorithm to schedule groupsof jobs with sequence dependence setup times, theprocessing times for the kth arti®cial two-stageproblem in algorithm B are de®ned as follows:

A�i; k� � max16 j6 k�1

Bij

� � sjdi

ÿ Bik�1 ÿ sk�1di ; �24�

B�i; k� � Fim ÿ Fi;�mÿk�; �25�

where Fij the completion times of family i at ma-chine j considering zero setup times without anyother family. Likewise, Bij is the completion timeof family i on machine j when the machinesnumbers are reversed.

3.2.5. Algorithm B0: Modi®ed Baker's group sched-uling algorithm

The processing times used in algorithm B canbe modi®ed using the concepts for algorithms G0

and G00. Thus, for algorithm B0, the setup time onmachine k � 1 is not subtracted when calculatingA�i; k�. Also, the processing times B�i; k� aremodi®ed to consider the e�ect of all machinesfrom mÿ k � 1 through m. The processing timesfor the arti®cial problem k in algorithm B0 are,therefore, as follows:

A�i; k� � max16 j6 k�1

Bij

� � sjdi

ÿ Bik�1; �26�


Fij

� � sjie

ÿ Fi;�mÿk�: �27�

3.2.6. Algorithm B00: Improved Baker's groupscheduling algorithm

A second modi®cation of algorithm B is ob-tained by subtracting the minimum possible setuptimes for each family i from its processing times.Thus, the processing times for the arti®cial prob-lem k in algorithm B00 are de®ned as follows:

A�i; k� � max16 j6 k�1

Bij

� � sjdi

ÿ Bik�1

ÿ S�i; k � 1�; �28�


Fij

� � Sjie

ÿ Fi;�mÿk�

ÿ S0�i;mÿ k�: �29�

3.2.7. Algorithm M: Modi®ed NEH procedureAll the algorithms described above use the dy-

namic implementation of CDS procedure toschedule families. It is equally possible to use amodi®ed NEH procedure to develop a family se-quence. To do so, de®ne the average setup time forfamily f 2 F at machine j, sj

f as follows:

sjf �

XK

x�1

sjxf

( ),K: �30�

Further, de®ne the e�ective processing time forfamily f 2 F at machine j, Efj as follows:

Efj � sjf �

Xx2Nf

pxj: �31�

Then, the steps of the modi®ed NEH algorithm,called algorithm M, are described as follows:

Input: N � f1; 2; . . . ; ng, Nf and cf for f �f1; 2; . . . ;Kg, pij for i � 1; . . . ; n; j � 1; . . . ;m.

Step 1. Let J � �l�1�; . . . ; l�K�� be the scheduleof all K families such that

Pmj�1 El�i�;j PPm

j�1 El�i�1�;j for all i6 �K ÿ 1�. Let i � 2, andr � �l�1�; l�2��. Further, let r � rpriÿp for06 p6 i. Enter step 2.

Step 2. Let k � l�i� 1�. Generate �i� 1� par-tial sequences represented by rpkriÿp, where 06p6 i. For each of the generated partial sequences,®nd C�rpkriÿp�. Set x � fx1; . . . ;xi�1g � frpkriÿp;p6 i� 1g; and C�xm� � min16 j6 i�1 fC�xj�g. Setr � xm. Enter step 3.


Step 3. If i < K, set i � i� 1 and return to step2; otherwise accept the family schedule r withmakespan C�r� as the solution of the problem.

The computational time requirements for al-gorithm M is O mK2 � mn2� �.

3.2.8. Algorithm T: Approximate TSP-based algo-rithm

Gupta (1986) showed that the multi-stage¯owshop scheduling problem with sequence de-pendent setup times and no-wait between ma-chines can be formulated as a TSP. Similar to thedevelopments in Gupta (1986), we create an arti-®cial family 0 and for each family pair �r; q� 2 K,de®ne the following:

crq � C�qrqq;m� ÿ C�qq;m� ÿ min06 k�K;k 6�q

smkq ÿ cqm:

�32�

Further, for each family q 2 K, we de®ne:

c0q � C�qq;m� ÿ min06 k6K;k 6�q

smkq ÿ cqm; �33�

cq0 � 0; �34�

cqq � 1: �35�

The above equations describe a �K � 1� city TSP.For algorithm T, we solve this TSP using theclosest city heuristic described by Gavett (1965).The steps of this algorithm T are as follows:

Input: crs, sjrq for r; s � f0; 1; . . . ;K; r 6� sg, Nf

and qf for f � f1; 2; . . . ;Kg, pij for i � 1; . . . ; n;j � 1; . . . ;m.

Step 1. Let r � /, p � f1; 2; . . . ;Kg, r � 0 andenter step 2.

Step 2. Let r � rq, where family i is found suchthat crq � mink2p crk. Let p � pÿ fqg, r � q andenter step 3.

Step 3. If p � /, ®nd the makespan of the ®nalschedule and STOP; otherwise return to step 2.

Algorithm T requires O�K2� computationaltime.

3.2.9. Algorithm T 0: Optimized TSP-based proce-dure

In Algorithm T 0, we use an exact optimizationalgorithm (like a branch and bound procedure) tosolve the TSP optimally to develop a sequence offamilies.

In view of the solution of the TSP problem byusing branch and bound algorithm, computationaltime requirement for algorithm T 0 is unboundedfrom above.

Each of our heuristic methods employs an ap-propriate selection of job scheduling algorithmsfrom C and N to couple that selection with anyone of the family scheduling heuristics G, G0, G00,B, B0, B00, M, T, or T 0. Thus, we get a total of 18heuristics. In CM, for example, algorithm C isapplied ®rst to obtain the sequences of jobs in eachfamily which is input into algorithm M to schedulefamilies which results in a heuristic schedule forthe problem.

3.3. An improvement heuristic

We also propose a descent heuristic which at-tempts to improve on some given schedule whichcan be obtained by one of the methods describedearlier. This heuristic systematically explores so-lutions in the neighborhood of the current sched-ule, i.e., schedules that are obtained by somesuitable perturbation. If the makespan of the newschedule is no less than that of the current sched-ule, then the current schedule is retained, and an-other neighbor is considered. On the other hand, ifthe new schedule has a smaller makespan than thatof the current schedule, then the new schedule re-places the current schedule. The search continuesuntil a local optimum is reached, which means thatno further improvement on the makespan can beachieved by this procedure.

We now provide a formal statement of ourdescent heuristic.

3.3.1. Algorithm D: DescentInput: Schedule S, and pqj, sj

if for i; f �1; . . . ;K, j � 1; . . . ; m, and q � 1; . . . ; n. Set d � 1and f � 2.


Step 1. Obtain schedule S0 by interchangingfamilies schedules rd and rf with makespan C�S0�.If C�S0� < C�S�, Set S � S0 and IMPR � 1. Setf � f � 1 and enter step 2.

Step 2. If f 6K return to step 1, otherwise setd � d � 1; f � d � 1. If d < K return to step 1;otherwise let d � 1; i � 1; k � 2 and enter step 3.

Step 3. Obtain schedule S0 by interchangingjobs in family rd at sequence position i and k withmakespan C�S0�. If C�S0� < C�S�, set S � S0 andIMPR � 1. Set k � k � 1. and enter step 4.

Step 4. If k6 nd return to step 3, otherwise seti � i� 1; k � i� 1. If i < nd return to step 3;otherwise let d � d � 1; i � 1; k � 2. If d 6K,return to step 3; otherwise enter step 5.

Step 5. If IMPR � 1, set IMPR � 0, d � 1 andf � 2 and return to step 1; otherwise terminate thesearch and accept S as the solution with makespanC�S�.

There is no polynomial bound on the timecomplexity of algorithm D. In our implementa-tion, we select the schedule generated by algorithmCM described in Section 3.2 to be input as aninitial solution.

4. Computational results

This section describes the computational testswhich are used to evaluate the e�ectiveness ande�ciency of the various heuristic methods in®nding good quality schedules. All algorithmswere coded in Pascal language, and the tests wereperformed on an HP Vectra VL series 4 personalcomputer.

4.1. Test problems

Three classes of test problems were generated.In each class, processing times at each stage arerandom integers from a uniform distribution�1; 10�. Problem hardness is likely to depend onwhether there is a balance between average pro-cessing times and the average setup times. For thisreason, three di�erent classes of problems wereused in the computational experiments. The setup

times were random integers from the followinguniform distributions in the following ranges:· class 1: �1; 20�;· class 2: �1; 50�;· class 3: �1; 100�.

Note that a family setup time distribution ofU �1; 20� implies that the ratio of mean family setuptime to mean job processing time is approximately2:1 (10.5:5.5); a family setup time distribution ofU �1; 50� implies that the ratio of mean family setuptime to mean job processing time is approximately5:1 (25.5:5.5); and a family setup time distributionof U �1; 100� implies that the ratio of mean familysetup time to mean job processing time is ap-proximately 10:1 (50.5:5.5). Problems were gener-ated with number of families varying between 3and 10, number of jobs in each family randomlygenerated between 1 and 10 and number of ma-chines varying between 3 and 10. For each com-bination of problem parameters, 30 probleminstances were generated.

4.2. E�ectiveness of lower bounds

An optimal branch and bound procedure wasused for each problem in the nine small problemsets containing 3 or 4 families and 3 or 4 machines.The optimal procedure was run for a maximum of30 min for each problem. If an optimal solutionwas found, its makespan (opt) was recorded oth-erwise the best makespan was recorded and treatedas a surrogate for an optimal solution. For eachproblem the percentage error of the lower bound�LB� � ��opt ÿ LB�=opt� � 100% is calculated.Table 1 depicts the average and maximum per-centage error based on the solution of 30 problemsof each size. In addition, Table 1 also shows thenumber of times lower bound equals its optimalmakespan and the number of problems that wereoptimally solved by the branch and boundalgorithm.

From Table 1, all nine problem sets have anaverage percentage error of less than 1% while themaximum percentage error is less than 4%. How-ever, as the number of families or the number ofmachines increases, the number of problemsfor which the lower bound equals its optimal


makespan decreases. This indicates that the forlarge sized problems, the proposed lower boundsmay not be really e�ective, especially at the rootnode.

4.3. E�ectiveness of the heuristic methods

Given that we propose nine family basedscheduling heuristics and use two job basedscheduling heuristics developed in prior researchon manufacturing cell scheduling, we would needto evaluate 18 possible family and job combina-tions of heuristic methods. Further, the improve-ment heuristic (algorithm D) could be applied toany one of these 18 combined heuristics resultingin 36 potential scheduling rules for evaluation.However, since the focus of this paper is on se-quence dependent family based scheduling, wedecided to ®rst identify the better (in terms ofaverage performance) job based scheduling heu-ristic for each of the nine family based procedures.This experiment was conducted on a smallernumber of problems of each size. Preliminary re-sults indicated that job based heuristic C (i.e., theCDS based method) performed better than jobbased heuristic N (i.e., the NEH based method) foreight of the nine family based scheduling heuris-tics. The one exception was heuristic M. Therefore,job based heuristic C was used in conjunction withall nine family based scheduling rules while jobbased heuristic N was used only in conjunctionwith heuristic M. Regarding the improvementheuristic D, as noted earlier we choose the job/family combination algorithm CM as an input to

this heuristic. This is referred to as combined al-gorithm CMD. However, we also designed a ran-dom heuristic R as an input to algorithm D. Thisheuristic generated both an initial sequence of jobswithin each family and a family sequence ran-domly and is referred to as algorithm RD. Thus, atotal of 12 combined heuristic algorithms wereevaluated in this paper.

Each problem instance was solved using each ofthe 12 heuristic algorithms. In addition, for eachproblem, the lower bound on its optimal make-span was found using the branch and bound pro-cedure described in Section 2 above. Using theoutput from these programs, percentage relativeerror rate (RER) for each heuristic algorithm h iscomputed as follows:

RER � �Mh ÿ LB� � 100=LB; �36�

where Mh is the makespan for the heuristic algo-rithm h and LB is the lower bound on the make-span for an optimal permutation sequence.

The average, minimum, and maximum RERvalues for each heuristic algorithm are shown inTables 2±4, respectively. In these tables, the nota-tion for the heuristic methods can be interpreted asfollows. For the ®rst 11 algorithms, the ®rst letterin the acronym represents the job based schedulingmethod used while the second represents thefamily based scheduling method employed. Forexample algorithm CG is the CDS job basedscheduling heuristic C used in conjunction theextended Gupta and Darrow procedure G.

Based on the results presented in Tables 2±4, wecan make the following observations:

Table 1

E�ectiveness of lower bounds

Setup type K � m Average% error Maximum% error # LB � Opt. # Solved

Small 3� 3 0.31 2.42 23 30

3� 4 0.87 2.86 13 30

4� 4 0.64 2.82 13 26

Medium 3� 3 0.35 3.26 24 30

3� 4 0.58 2.35 17 30

4� 4 0.61 2.27 09 28

Large 3� 3 0.08 1.12 27 30

3� 4 0.33 2.38 20 29

4� 4 0.27 1.62 19 30


· Algorithm CMD performed best for all problemsets. Use of the descent algorithm D can signif-icantly improve on the solutions generated byalgorithm CM. As problem size becomes larger,the time to process problems using algorithmCMD could become excessive. For small tomedium size problems or when selection of analgorithm is not sensitive to computer process-ing time, algorithm CMD is the best.

· Algorithm CM performed second best for allproblem sets. On data sets with large problem

sizes the performance of algorithms CG0, CG00,CB0, and CB00 were fairly close to algorithmCM.

· Algorithms CG and CB usually performed theworst. For problems with six or more familiesto schedule and with medium or large setup dis-tributions algorithm CG0 usually performed alittle better than algorithm CG00, and algorithmCB0 usually performed a little better than algo-rithm CB00. For problem sets with ®ve or lessfamilies to schedule or using the small setup

Table 2

Average deviation of heuristic from lower bound

Setup type Algorithm Average relative error rate for problem size K�m

3� 3 3� 4 4� 4 5� 5 5� 6 6� 5 6� 6 8� 8 10� 8 10� 10

Small CG 4.67 6.99 7.36 9.35 9.19 9.56 9.29 10.71 13.50 13.20

CG0 4.88 6.03 7.39 7.98 8.48 7.29 8.89 9.41 10.34 11.97

CG00 4.24 5.36 6.37 7.63 9.11 7.23 7.99 9.97 10.14 11.29

CB 3.38 4.68 6.77 7.55 8.69 8.57 8.40 11.24 12.18 12.25

CB0 4.09 4.64 5.65 6.28 7.88 7.73 7.39 9.94 9.65 10.19

CB00 3.47 4.90 5.27 6.62 8.19 7.09 6.77 9.82 9.83 10.73

CM 1.44 3.53 3.38 5.60 6.48 5.20 6.36 7.22 7.86 8.83

NM 3.62 7.26 7.28 7.50 8.73 7.30 9.13 8.59 8.39 10.00

CT 4.56 6.96 8.82 9.38 11.63 10.48 10.97 15.06 15.83 16.72

CT 0 2.34 4.12 5.08 7.39 8.49 7.48 8.46 10.83 10.74 12.65

CMD 0.67 1.85 1.94 3.15 4.02 3.00 4.06 5.62 5.63 6.86

RD 1.10 2.85 2.50 4.22 4.51 4.99 4.39 6.01 6.95 7.48

Medium CG 6.39 5.17 9.80 9.53 10.68 13.51 11.04 14.08 16.15 15.59

CG0 5.07 5.85 6.88 8.81 8.61 6.99 7.64 9.61 9.51 14.02

CG00 4.95 5.48 8.43 8.64 8.96 9.00 9.18 10.44 10.65 16.36

CB 5.68 6.17 9.32 10.53 10.03 14.40 11.92 13.33 15.11 13.91

CB0 5.78 6.59 6.08 7.89 6.59 7.59 6.85 8.30 8.89 9.83

CB00 3.74 7.29 6.04 7.61 8.08 8.89 8.04 8.82 9.46 10.48

CM 1.92 4.52 3.76 5.60 6.59 5.80 7.09 8.25 8.38 8.58

NM 3.74 5.79 4.88 7.50 7.25 7.03 8.41 8.90 8.33 9.47

CT 6.52 8.72 10.47 12.70 11.31 10.21 11.90 5.11 14.97 15.92

CT 0 3.09 3.81 4.65 6.98 6.92 7.14 8.09 10.68 11.83 12.97

CMD 0.92 2.00 1.96 3.10 3.58 3.68 4.59 5.68 6.11 5.73

RD 1.56 1.70 3.89 3.83 5.07 5.92 5.52 5.97 6.90 8.12

Large CG 10.65 7.58 8.09 14.31 9.76 17.57 14.25 17.93 20.15 18.33

CG0 7.30 5.64 8.09 8.25 8.31 11.26 6.88 9.78 10.26 10.38

CG00 6.57 4.97 8.96 7.81 8.45 9.47 8.76 10.49 11.80 11.55

CB 10.69 7.18 9.60 11.33 9.76 16.74 13.74 17.73 19.75 17.01

CB0 6.24 6.26 7.14 7.24 7.27 8.96 8.06 9.34 10.75 9.81

CB00 6.39 5.99 7.94 5.48 8.25 8.72 9.45 10.48 11.48 11.26

CM 1.82 3.19 3.89 5.31 7.41 5.94 6.45 9.04 9.08 9.41

NM 3.03 4.54 4.73 5.73 8.95 7.49 7.05 9.09 10.04 9.74

CT 6.96 6.09 8.67 13.37 12.10 13.55 13.33 17.32 19.88 19.80

CT 0 2.88 3.93 5.33 7.24 7.95 9.18 9.92 12.02 12.87 13.70

CMD 0.91 1.08 1.95 2.49 3.37 3.29 3.03 6.25 6.22 6.30

RD 2.09 1.65 3.99 4.91 4.82 5.09 4.94 6.72 6.82 7.85


distribution the performance of algorithm G0 vsG00 and B0 vs B00 varied.

· Algorithms CB0 and CB0 usually performed a lit-tle better than algorithms CG0 and CG00. Theperformance of algorithms CG0, CG00, CB0, andCB00 deteriorated the least as problem sizeincreased.

· When the number of machines is greater thanthe number of families, algorithms CB0, CG00,CB0 and CB00 may perform better because thenumber of sequences tested using these algo-

rithms is based on the number of machines inthe cell. However, if there are more families thanmachines algorithm CM will most likely per-form better because the number of sequencestested using this algorithm is based on the num-ber of families to be sequenced.

· There is little di�erence in the performance ofalgorithm CG0 vs CB0 and CG00 vs CB00. This indi-cates that including idle time between jobs with-in a family in algorithms to determine goodfamily sequences has a marginal impact. The

Table 3

Maximum deviation of heuristic from lower bound

Setup type Algorithm Maximum relative error rate for problem size K � m

3� 3 3� 4 4� 4 5� 5 5� 6 6� 5 6� 6 8� 8 10� 8 10� 10

Small CG 15.32 22.88 19.70 18.95 14.64 19.51 14.79 14.61 21.10 16.38

CG0 19.26 21.78 23.48 15.44 13.81 14.12 14.33 14.89 14.29 16.54

CG00 12.88 21.78 14.10 14.10 15.49 14.12 12.97 14.64 14.73 15.75

CB 11.11 10.29 16.67 11.48 14.94 14.83 13.31 15.85 16.18 17.20

CB0 14.57 11.30 15.91 10.74 13.81 14.63 11.36 13.61 13.87 14.41

CB00 12.88 11.30 18.18 11.41 13.64 18.90 11.15 13.35 13.02 15.09

CM 4.92 8.46 8.97 12.66 12.80 9.72 10.40 12.16 10.83 13.48

NM 8.70 17.82 15.58 11.65 13.93 13.26 15.60 12.06 12.53 15.09

CT 16.00 19.18 22.50 17.65 20.38 20.73 16.18 25.07 22.82 25.61

CT 0 6.59 9.56 14.00 13.08 14.58 12.54 13.31 17.18 16.82 21.29

CMD 4.13 8.46 8.33 6.61 8.93 6.90 9.16 8.59 8.96 9.11

RD 9.35 9.93 8.67 10.99 9.64 9.58 8.09 11.38 11.26 13.54

Medium CG 20.18 20.93 24.12 21.70 21.55 21.49 23.21 20.81 21.67 20.40

CG0 26.11 16.67 22.60 16.48 16.87 16.33 17.06 15.96 13.06 19.01

CG00 20.18 16.67 22.81 19.54 16.52 18.01 20.82 15.74 18.47 20.51

CB 20.18 20.93 21.53 22.75 19.20 24.30 26.96 20.19 23.62 18.99

CB0 23.78 26.05 21.40 21.68 14.04 15.26 12.66 13.24 13.73 16.07

CB00 18.12 20.93 17.71 16.09 13.57 17.45 15.65 13.87 16.29 15.57

CM 13.02 20.93 15.54 12.34 14.14 11.95 17.35 14.34 13.23 11.88

NM 11.46 15.00 13.79 15.33 20.54 13.71 19.56 15.07 13.74 13.08

CT 18.12 40.12 24.55 24.45 30.21 19.73 25.53 28.29 24.78 21.73

CT 0 18.12 20.93 11.72 22.32 13.86 23.28 15.77 17.23 17.06 21.06

CMD 11.46 16.28 11.11 8.48 13.13 8.88 15.77 12.68 10.83 9.92

RD 18.12 7.69 19.82 13.11 13.13 16.62 17.41 11.26 12.14 14.20

Large CG 57.89 39.47 31.11 36.83 20.52 31.85 31.16 30.90 30.33 28.67

CG0 31.97 39.47 21.02 18.68 20.80 23.63 14.09 18.58 19.40 18.46

CG00 31.97 39.47 22.13 18.68 24.76 18.99 14.73 18.82 20.99 20.07

CB 43.16 39.47 30.00 21.82 22.44 31.71 26.27 25.55 32.27 25.19

CB0 30.25 39.47 18.03 19.55 16.83 20.68 16.41 20.96 19.07 18.81

CB00 30.25 39.47 22.60 14.02 24.76 16.41 20.12 16.13 17.28 16.24

CM 10.18 18.91 15.22 13.75 18.20 13.00 15.22 19.57 13.25 18.35

NM 11.06 19.33 14.44 15.30 19.66 16.14 14.12 16.77 20.50 17.34

CT 39.60 25.48 29.17 27.71 27.56 24.77 28.18 28.45 32.12 32.72

CT 0 18.90 31.38 16.54 19.88 20.59 25.54 19.36 23.16 23.73 21.30

CMD 8.41 16.39 10.27 9.57 17.07 10.02 10.47 18.17 11.25 11.42

RD 32.80 16.39 14.41 12.32 12.66 12.52 17.21 19.80 13.25 15.02


major factors to consider when developingalgorithms to sequence families are idle time be-tween families including the time to change overfrom one family to another.

· Algorithm CT 0 performed third best for prob-lems with four or less families but its perfor-mance deteriorated relative to several otheralgorithms as problem size became larger. Asproblem size increases, algorithms CT and CT 0

perform poorly, relative to other algorithms.The reason for this may be that only one se-

quence is developed using these algorithmswhile the other algorithms test more sequencesas problem size increases.

· Use of algorithm C to ®nd the schedule jobswithin each family is more e�ective than theuse of algorithm N.

· Use of algorithm D drastically improves a givenschedule as is evidenced by the results for algo-rithms CMD and RD. However, use of algo-rithm CM is relatively more e�ective thanrandom heuristic R.

Table 4

Minimum deviation of heuristic from lower bound

Setup type Algorithm Minimum relative error rate for problem size K � m

3� 3 3� 4 4� 4 5� 5 5� 6 6� 5 6� 6 8� 8 10� 8 10� 10

Small CG 0.00 0.00 0.00 1.96 4.07 0.43 4.75 5.91 9.38 9.73

CG0 0.00 0.00 0.96 1.99 3.05 2.73 3.13 3.80 5.78 5.54

CG00 0.00 0.00 0.96 2.20 0.00 2.58 2.50 5.91 7.71 7.41

CB 0.00 0.58 0.00 1.99 3.01 4.44 3.01 7.33 9.81 9.47

CB0 0.00 0.00 0.00 1.96 3.01 3.41 3.52 6.33 5.62 5.54

CB00 0.00 0.00 0.00 1.79 0.00 0.43 3.99 6.45 6.61 7.69

CM 0.00 0.00 0.00 1.99 0.00 1.74 2.56 3.24 4.64 4.98

NM 0.00 0.00 1.28 1.90 3.08 2.73 3.75 6.01 4.63 5.99

CT 0.00 0.00 0.52 2.20 3.70 0.43 3.44 7.65 10.47 9.29

CT 0 0.00 0.00 0.00 2.20 2.31 1.82 3.54 5.38 3.26 7.94

CMD 0.00 0.00 0.00 0.00 0.00 0.00 1.24 3.20 2.58 4.07

RD 0.00 0.00 0.00 0.43 0.00 1.16 1.25 1.91 2.71 4.08

Medium CG 0.00 0.00 0.96 0.00 0.85 5.91 3.38 9.06 8.41 8.71

CG0 0.00 0.00 0.75 0.00 2.07 0.26 2.52 1.35 3.56 9.21

CG00 0.00 0.00 1.34 0.30 2.07 0.26 2.86 1.35 6.09 12.80

CB 0.00 0.00 0.96 0.30 3.32 4.44 3.12 7.26 8.16 6.18

CB0 0.00 0.00 0.70 0.80 0.85 1.89 1.25 2.60 4.85 6.21

CB00 0.00 0.00 1.05 0.30 2.07 1.22 1.25 3.23 5.30 4.79

CM 0.00 0.00 0.00 0.00 0.84 0.68 1.25 4.20 4.27 3.55

NM 0.00 0.44 0.00 0.00 1.67 0.00 2.39 5.81 4.19 4.39

CT 0.00 0.00 0.00 0.00 2.81 1.36 3.99 2.90 10.00 10.50

CT 0 0.00 0.00 0.00 0.00 0.58 0.34 1.04 6.71 6.69 5.47

CMD 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.63 2.89 2.29

RD 0.00 0.00 0.00 0.00 0.28 0.00 0.25 1.79 3.75 3.45

Large CG 0.00 0.00 0.00 1.34 1.35 5.74 4.28 6.77 8.94 12.23

CG0 0.00 0.00 0.29 1.67 0.23 0.65 0.72 2.50 3.44 3.78

CG00 0.00 0.00 0.00 1.66 0.23 1.02 1.30 3.55 3.88 6.56

CB 0.00 0.00 0.25 1.18 0.23 1.52 1.88 10.04 12.13 6.82

CB0 0.00 0.00 0.00 0.45 0.23 1.35 0.72 2.99 5.11 4.69

CB00 0.00 0.00 0.00 0.67 0.23 1.30 0.54 2.51 4.60 7.04

CM 0.00 0.00 0.00 0.00 1.19 0.00 0.97 2.51 2.63 4.44

NM 0.00 0.00 0.00 0.81 2.15 1.23 0.20 2.93 3.75 3.78

CT 0.00 0.00 0.29 1.83 2.19 2.53 1.24 10.31 8.11 12.32

CT 0 0.00 0.00 0.29 0.80 1.19 1.11 0.97 2.50 4.43 5.51

CMD 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.28 0.12 0.53

RD 0.00 0.00 0.00 0.00 0.00 0.00 0.18 1.46 2.79 1.45


Based on this discussion, it may be concludedthat among the polynomially bounded heuristics,algorithm CM is the best heuristic to solve the¯ow-line manufacturing cell scheduling problemwith family setup times. However, use of algorithmD should be considered to improve the ®nal solu-tion obtained by heuristic CM since in mostpractical cases, it will result in signi®cant im-provement with a reasonable amount of compu-tational e�ort. Further, as lower bounds were usedto test the e�ectiveness of various heuristics, the

actual makespan obtained from these heuristicscould be much closer to the optimal makespanthan that indicated by the above results.

4.4. E�ciency of the heuristic methods

Table 5 shows the average CPU time (seconds)required to solve a problem instance of given sizeby each heuristic algorithm. As expected, the CPUtime requirements for heuristic algorithms not

Table 5

Average CPU time rquired to solve a problem

Setup type Algorithm Average CPU time (s) for problem size K � m

3� 3 3� 4 4� 4 5� 5 5� 6 6� 5 6� 6 8� 8 10� 8 10� 10

Small CG 0.01 0.01 0.01 0.02 0.02 0.02 0.02 0.04 0.05 0.07

CG0 0.01 0.01 0.01 0.02 0.02 0.02 0.02 0.07 0.05 0.08

CG00 0.01 0.01 0.01 0.01 0.02 0.02 0.02 0.07 0.06 0.09

CB 0.01 0.01 0.01 0.01 0.01 0.01 0.02 0.06 0.05 0.07

CB0 0.00 0.01 0.01 0.01 0.01 0.01 0.02 0.04 0.06 0.08

CB00 0.01 0.01 0.01 0.01 0.01 0.01 0.02 0.04 0.06 0.08

CM 0.01 0.01 0.01 0.01 0.02 0.03 0.03 0.08 0.14 0.16

NM 0.01 0.01 0.02 0.03 0.03 0.04 0.05 0.11 0.19 0.23

CT 0.01 0.01 0.01 0.01 0.02 0.01 0.01 0.03 0.04 0.05

CT 0 0.01 0.01 0.01 0.01 0.01 0.01 0.02 0.10 1.98 2.84

CMD 0.04 0.05 0.12 0.20 0.25 0.32 0.44 0.91 1.82 2.21

RD 0.05 0.06 0.13 0.24 0.28 0.36 0.47 1.17 2.07 2.57

Medium CG 0.01 0.01 0.01 0.01 0.02 0.02 0.02 0.04 0.05 0.07

CG0 0.01 0.01 0.01 0.02 0.02 0.02 0.02 0.04 0.05 0.08

CG00 0.01 0.01 0.01 0.02 0.02 0.03 0.02 0.05 0.06 0.09

CB 0.01 0.01 0.01 0.01 0.01 0.01 0.02 0.04 0.05 0.07

CB0 0.01 0.01 0.01 0.01 0.01 0.01 0.02 0.04 0.05 0.08

CB00 0.01 0.01 0.01 0.01 0.01 0.01 0.02 0.04 0.06 0.08

CM 0.01 0.01 0.01 0.01 0.02 0.03 0.03 0.10 0.14 0.16

NM 0.01 0.01 0.02 0.03 0.03 0.04 0.05 0.11 0.19 0.23

CT 0.01 0.01 0.01 0.01 0.02 0.01 0.02 0.04 0.04 0.05

CT 0 0.01 0.01 0.01 0.01 0.01 0.02 0.02 0.06 0.49 0.51

CMD 0.04 0.05 0.13 0.19 0.27 0.30 0.40 0.97 1.84 2.37

RD 0.07 0.06 0.15 0.24 0.26 0.35 0.52 1.04 2.31 2.44

Large CG 0.01 0.01 0.01 0.02 0.02 0.02 0.02 0.05 0.05 0.07

CG0 0.01 0.01 0.01 0.02 0.02 0.02 0.02 0.04 0.05 0.08

CG00 0.01 0.01 0.01 0.02 0.02 0.02 0.02 0.05 0.06 0.09

CB 0.01 0.01 0.01 0.01 0.02 0.02 0.02 0.04 0.05 0.08

CB0 0.01 0.01 0.01 0.01 0.02 0.01 0.02 0.04 0.06 0.08

CB0 0.01 0.01 0.01 0.01 0.01 0.01 0.02 0.04 0.06 0.08

CM 0.01 0.01 0.01 0.01 0.02 0.02 0.02 0.08 0.14 0.16

NM 0.01 0.01 0.02 0.03 0.03 0.04 0.05 0.11 0.19 0.23

CT 0.01 0.01 0.01 0.02 0.02 0.01 0.02 0.03 0.04 0.05

CT 0 0.01 0.01 0.01 0.01 0.02 0.02 0.02 0.05 0.34 0.37

CMD 0.04 0.06 0.12 0.21 0.26 0.30 0.44 0.95 1.82 2.37

RD 0.04 0.06 0.14 0.22 0.25 0.38 0.51 1.20 2.44 2.72


based on the TSP problem solution or the use ofthe improvement heuristic do not depend on themagnitude of the setup times. However, the CPUtime requirements for the heuristic algorithms thatuse of the TSP based approaches or the improve-ment heuristic do depend on the magnitude of thesetup times.

The results in Table 5 also show that the com-putational time requirements for the algorithmsare quite reasonable even though the improvementheuristic D requires considerable more time thanany other heuristic algorithm. However, the com-putational time requirements for the combinedalgorithm CMD are less than those for algorithmRD. Therefore, in view of the relatively better so-lution e�ectiveness shown in Tables 2±4, the use ofalgorithm CM to ®nd an initial solution for theimprovement heuristic D is preferable than the useof the random algorithm R. Based on these results,it may be concluded that any of the proposed al-gorithm can be used to ®nd a practical solution tolarge-sized problems.

5. Conclusions

This paper considered the problem of schedul-ing a set of given jobs in a manufacturing cellwhere the setup times for processing various fam-ilies of jobs are sequence dependent and it is de-sired to minimize makespan. In view of thestrongly NP-hard nature of the problem, severalheuristic algorithms are developed to schedule jobswithin various families, and to obtain a familyschedule for ¯ow-line manufacturing cells whichhave sequence dependent setup times. Computa-tional results over a large number of problemsshow that the use of CDS based heuristic performsbest to schedule jobs within each family while theNEH based heuristic provides minimum make-span for scheduling families. Use of an improve-ment heuristic based on neighborhood searchprovide the lowest makespan value. However, theimprovement algorithm also requires the maxi-mum computational e�ort. In many cases, use ofpolynomially bounded algorithms, as those ob-tained by modifying algorithms for the two-ma-chine problems may provide equally good

schedules in a reasonable amount of computa-tional e�ort.

The development of various heuristic algo-rithms in this paper also results in some fruitfuldirections for future research. First, the proposedalgorithms can also be used to solve problems in-volving some precedence constraints and resourcerestrictions. Since the augmentation of jobs is donedynamically, feasibility conditions with respect tothe precedence constraints and resource restric-tions can be checked before augmenting a job to apartial schedule. Second, several improvements arepossible that will enhance the quality of the solu-tion obtained by the proposed algorithm. Third,the proposed algorithm can be adapted to solve thesequence dependent ¯owline problems with otherobjective functions, like the minimization of thetotal ¯ow time. Finally, the solution from theproposed algorithm can be used as an initial seed ina meta-heuristic (like the Tabu Search or Geneticalgorithm) to improve the quality of their solu-tions. Thus, more research is necessary in the areaof scheduling with sequence dependent setup times.

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Scheduling a flowline manufacturing cell with sequence dependent family setup times