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Scheduling a flow-line manufacturing cell: a tabusearch approachJ. SKORIN-KAPOV a & A. J. VAKHARIA ba W. Averill Harriman School for Management and Policy, State University of New York atStony Brook, Stony Brook, NY, 11794–3775, USAb Department of Management Information Systems, Karl Eller Graduate School ofManagement, The University of Arizona, Tucson, AZ, 85721, USA.Published online: 07 May 2007.

To cite this article: J. SKORIN-KAPOV & A. J. VAKHARIA (1993) Scheduling a flow-line manufacturing cell: a tabu searchapproach, International Journal of Production Research, 31:7, 1721-1734, DOI: 10.1080/00207549308956819

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INT. J. PROD. RES., 1993, VOL. 31, No.7, 1721-1734

Scheduling a flow-line manufacturing cell: a tabu search approach

J. SKORIN-KAPOVt and A. 1. VAKHARIAt*

The performance of two heuristic procedures for the scheduling of a flow-linemanufacturing cell was compared. We propose a procedure based on a com­binatorial search technique known as tabu search.The new procedure is comparedwith a heuristic based on simulated annealing which was proposed in earlierresearch.The scheduling problem addressed here differs from the traditional flow­shop scheduling problem in the sense that we are interested in sequencing partfamilies (i.e. groups of jobs which share a similar setup) as well as individual jobswithineach family. The results reveal that the tabu search heuristic outperforms thesimulated annealing heuristic by generating 'better solutions' in less computationtime.

1. IntroductionManufacturing cells have been implemented by several manufacturers (see Hyer

1984 for a representative list of users) in order to improve the operational efficiency oftheir manufacturing processes. In general, cells have been found to lead to reductions insetup costs, labour costs, tooling costs, scrap/rework costs and WIP inventory costs(Hyer and Wemmerlov 1989, Wemmerlov and Hyer 1989, Askin and Vakharia 1990).Further, this has typically been accompanied by a reduction in throughput times and acorresponding increase in the shipment of on-time deliveries.

A manufacturing cell can be regarded as a group of similar machines located inclose proximity to one another and dedicated to the manufacture of a specific numberof part families. Part families consist of a set of similar jobs in terms of processingrequirements. When scheduling part families and individual jobs within each family ina cell there are usually two distinct types of setups considered (Vakharia and Chang1990, Wemrnerlov and Vakharia 1991): (i) a sequence-independent family setup whichis required to change over from one family to another (referred to as a family setup); and(ii) an individual job setup (a comparatively minor setup which is typically included inthe job processing time).

This paper focuses on the scheduling problem in a pure flow-line manufacturingcell. The major objectives of this research are:

(I) To propose an efficient scheduling procedure for sequencing part families andjobs within each family in a manufacturing cell. This method is based on tabusearch, a combinatorial search technique.

Received October 1992.t W. Averill Harriman School for Management and Policy,State Universityof New York at

Stony Brook, Stony Brook, NY 11794-3775, USA.t Department of Management Information Systems, Karl Eller Graduate School of

Management, The University of Arizona, Tucson, AZ 85721, USA.• To whom correspondence should be addressed.

0020-7543/93 $10·00 © 1993 Taylor & Francis Ltd.

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1722 J. Skorin-Kapou and A. J. Vakharia

(2) To compare the proposed method and the simulated annealing basedprocedure proposed by Vakharia and Chang (1990) in terms of makespan (i.e.the difference between the completion time of the last job on the last machineand the start time of the first job on the first machine) and computation time.

The remainder of this paper is organized as follows. In the next section, the problemaddressed in this study is discussed in more detail. The tabu search scheduling heuristicdeveloped in this research is presented in section 3. Section 4 discusses the experimentaldesign and results of a comparison of the heuristic procedures. Finally, the implicationsand conclusions of this study are presented in section 5.

2. Problem descriptionThe problem considered in this paper is to efficiently schedule a pure flow-line

manufacturing cell. As in the Vakharia and Chang (1990) study where the simulatedannealing scheduling heuristic was proposed, the major assumptions made in thisstudy are:

• the part families to be processed in the cell have been identified;• the equipment in the cell is laid out such that all jobs entering the cell follow the

same sequence of operations (i.e.we are considering a pure flow-line manufactur­ing cell);

• the individual job setup times are known and are included in the job processingtimes;

• a changeover between part families requires an additional setup, referred to as afamily setup;

• the setup times Gob and family) are not sequence dependent; and• there are no prespecified priorities between individual jobs or between families of

jobs.

The assumption offamily setup times has typically been made when group technologycells are identified (see, for example, Burbidge 1975, 1979, Han et al. 1985). Therationale for such an assumption is that the parts processed in cells are grouped intopart families so as to minimize tool changeovers on machines; Thus, it appears that indeveloping scheduling policies for manufacturing cells, there would be some oper­ational advantages in processing parts belonging to the same family as a group. Thisaspect distinguishes the cell scheduling problem from the traditional schedulingproblem addressed in earlier literature (Baker 1974). To address the issue of familysetups in addition to individual job setups, a family sequence as well as a sequence forjobs within each part family is typically developed.

For the research problem addressed in this paper, two distinct streams of researchare relevant: flow-shop sequencing research and part family scheduling research. Theinterested reader is referred to Vakharia and Chang (1990) and Wemmerl6v andVakharia (1991) where these contributions are discussed in more detail.

3. Tabu search heuristic (TSH) for part family schedulingTabu search is a meta-strategy developed to improve solvability of hard com­

binatorial optimization problems. Its strategic principles in a broader sense have beenlaid out in Glover (1989,1990). The method basically keeps limited track of a searchtrajectory in order to guide the search out of a local optimum (via the use of the so­called 'short-term memory') and to diversify the search (via the use of the 'long-term

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Flow-line scheduling using tabu search 1723

memory'). We define these and other aspects of tabu search as applied to scheduling aflow-line manufacturing cell. Tabu search strategies have already been successfullyapplied to different machine scheduling problems. For example, see Eck (1989),Gloverand Laguna (1989) and Taillard (1989).

In our context, a feasible schedule !l consists of a sequence of part families and asequence of jobs within each family in a manufacturing cell. Let us assume there are Ffamilies, M machines and N I jobs in family f We will define two differentneighbourhoods for such a feasible solution:

• N I(!l) = {!l':!l' is a schedule obtained from O by exchanging families f and f + I,keeping the order of jobs within each family intact,f= I, ... , F -I. (If1=F, thenexchange the last and the first family in the sequence !lJ);

• Nj(!l) = {!l'r:!l'r is a schedule obtained from O by exchanging jobs Jand J+ I infamily f, 1=1,... , F; J= 1,... , N I - I. (If J=N I' then exchange the last and thefirst job in family Il).

3.1. Moves and tabu listsLet us call by [move (j-move) the transition frorn D to a schedule in NA!l) (Nj(!l)).

The value 01a move is a difference between makes pans after and before the move, andtherefore an improving move has a negative value. An iteration of the search iscompleted when the whole neighbourhood of a current schedule is evaluated, and themove with the smallest move value identified and performed. In order to avoidbacktracking to the nearest local optimum, the so-called tabu list contains informationnecessary to forbid a number of recent moves. This length of the tabu list will be calledthe tabu list size. Reflecting the employment of two different types of moves, ourprocedure will make use of two different tabu lists, namely I_tabu list and J_tabu list.The J-tabu list is a vector containing families that were moved from position 1+ 1 toposition I in one of the I_tabu list size recent iterations. Therefore, ifa family is in the 1­tabu list during a current iteration, it cannot be moved back to position1+ I. Similarly,the J_tabu list contains J_tabu list size pairs (f, j) denoting that job j offamily I cannotbe moved to position j +1 at a current iteration. Both tabu lists are updated circularly(i.e. given a tabu list of size L which contains moves M( I), ... ,M(L), when move M(L+ 1) is generated, it is added to the tabu list and move M(I) is disregarded). Hence,figuratively speaking, as soon as a new move is put on top of a pile of moves, youdisregard the bottom move, in order to preserve the height of the pile.

The procedure will basically iterate between the two types of moves as follows.First, a number of iterations with I_moves will be performed. When the exchange offamilies does not appear to give further improvement, iterations involving j_moves willbe invoked in the attempt to improve the schedule. When this does not provide furtherimprovement, the search will go back to performing the J-moves. Different stoppingcriteria were used. The complete description ofthe procedure is provided in section 3.6.

3.2. Makespan evaluationIn order to speed up calculations, we employed the following evaluation of a

makespan. After the initial schedule has been constructed, the starting times for everyfamily and every job within a family are calculated. The makes pan is then obtained byadding the processing time of the last job of the last family in the sequence to its startingtime.

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1724 J. Skorin-Kapov and A. J. Vakharia

Let startf. j.m(Pf,j.m) denote the starting time (processing time) of job j of family Ion machine m. Also, let sf.m be the setup time of family I on machine m. Thecalculations of starting times are then performed recursively as follows:

(I) The starting time of the first job ofthe first family in the sequence is equal to thefamily setup time:

start I. I,' =SI.1

(2) For 1=1; j=l; m=2, ... ,M:

(3) For 1=1; j=2, ... ,N t; m=l:

start l •j. t =startt,j_t.t +Pt.j-t.t

(4) For 1= I; j=2, ... ,N t ; m=2, ... ,M:

start t. j.m= max tstartt.i_ t,m + Pi.j-t.m; startt.j,m_1 + Pt.j.m- d(5) For 1=2, ... , F; j= 1; m= I:

startf. t. 1 =startf-t,Nt_ .. t +Pf-t,Nt-l.t +sf.t

(6) For 1=2, ... ,F; j= I; m=2, ... ,M:

startj t,m = max {startf-I.Nt_t.m +Pr- t.Nt- ,.m + sf.m; start f. t.m-I +P], I.m-.}

(7) For 1=2, ... ,F; j=2, ... ,Nf ; m= I:

startf,j, 1 = startf,j_t. 1 +Pr.t- I. I

(8) For 1=2, ... ,F; j=2, ... ,Nf ; m=2, ... ,M:

startf. i.m= max {startf,j_ t,m + Pf,j-I.m; startf. j.m_ 1 + Pf,j.m- dIf there is change in the schedule, then only the affected starting times of jobs andfamilies are recalculated.

3.3. Tabu list sizesWe experimented with fixed and dynamically changing tabu list sizes. A number of

recent studies reported the improvement in performance of the method when variabletabu list sizes were used (Taillard 1989,Skorin-Kapov 1990a, Glover and Laguna 1991,Glover and Hubscher 1991).

Let Z denote the total numberof'jobs (i.e.Z = Ej~ t N f)' For a real number r, let usdenote its integral part by int (r). When we use fixed tabu list sizes, the following arestopping criteria: if there is no improvement in the last 10 x F iterations of I_moves,switch to j_moves. If there is no improvement in the last Z iterations of j_moves, theneither go back to I_moves (in case an improvement occurred while using j_moves), orcall the long-term memory (if the method uses it), otherwise stop the search.

With respect to the dynamically changing tabu list sizes we implemented thefollowing strategy: if there is no improvement in the prescribed number of iterations,decrease the tabu size to intensify the search in the current region. Following that, whenthere is no improvement in.a prescribed number of iterations, increase the tabu size todiversify the search in the current search region. Specifically, we start by performing 1­moves with the initial I_tabu Jist size. If there is no improvement in the last 5 x F

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Flow-line scheduling using tabu search 1725

iterations, we decrease this size. After 2 x F iterations are performed withoutimprovement using the decreased tabu list size, we increase the initial I_tabu list size.After performing 3 x F iterations with the increased tabu list size without improvement,we switch to j_moves. Following int(Z/3) iterations without improvement with theinitial j_tabu list size, we decrease the size. After int (Z/3) iterations with the decreasedtabu list size without improvement, we increase the initial j_tabu list size. Finally, afterint (Z/3) iterations without improvement with this tabu list size, we return to I_movesas described before.

3.4. Long-term memoryIn order to improve the results by diversifying the search, weexperimented with two

different long-term memories. They are both based on frequencies (f, p), denoting thenumber of times family I occupied position p in schedules during the search process.We start our search with a zero F x F frequency matrix. Each time a new currentsolution is constructed, the entries of the frequency matrix corresponding to familiesand their respective positions in the current schedule are increased by one. The long­term memory based on minimal frequencies LTM_MIN (or the long-term memorybased on maximal frequencies LTM_MAX) created a new starting family sequence byusing the following procedure: take the minimal (maximal) entry, say (f" P,), in thefrequency matrix. Fix family I, in position P,. Update the frequency matrix by deletingthe row corresponding to family I, and column corresponding to position P,. Repeatthe procedure until the family sequence has been created. By using the approach basedon minimal frequencies, new initial family schedules will be created in search regionsnot investigated so far. On the other hand, by using the approach based on maximalfrequencies, diverse family schedules will be created in feasible regions considered'good' during the previous search. A similar approach proved useful in the context ofthe quadratic assignment problem (Skorin-Kapov 1990b). The LTM_MIN could bereferred to as 'intensive diversification', while LTM_MAX represents 'diverse intensif­ication'. Following the creation of the family sequence, jobs were randomly scheduledin their respective families. The frequency matrix is updated during the whole searchprocess.

3.5. Aspiration criterionThis refers to the criterion directing when to override the tabu status of a move. We

implemented the commonly used criterion of performing the tabu move if the resultingmakespan is better than the best found previously.

3.6. Description 01 the algorithmThe following is a detailed description of TSH algorithm developed in this study.

(I) InitializationSet the initial I_tabu_size and j_tabu_size. Set the required number of long­term memory restarts, and start the counter: LTM = 0 Go to 2.

(2) Generate starting solutionIf LTM =0 generate a random family sequence, otherwise generate a familysequence using LTM_MIN or LTM_MAX (i.e. frequencies of families andscheduling positions). Complete the feasible schedule by randomly generating asequence for jobs within each part family. Let this be a current solution n° witha makespan of MAKEo. Let n* represent the incumbent solution with

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1726 J. Skorin-Kapov and A. J. Vakharia

makespan MAKE*. Set n*=no, MAKE*=MAKEo, LTM=LTM+I andgo to 3.

(3) Start counting iterations with family exchangesSet f_iter=O and go to 4.

(4) Stopping criterion for family exchangesSet f_iter = I_iter + 1.

(a) With fixed tabu sizesIf no improvement in the last lOx F iterations, go to 6, else go to 5.

(b) With variable tabu sizes

(i) If no improvement in the last 5 x F iterations with the initial f_tabu_size, then decrease the f-tabu_size and go to (ii).

(ii] If no improvement in the last 2 x F iterations with the decreased [:tabu.slze, then increase the f_tabu_size and go to (iii).

(iii) If no improvement in the last 3 x F iterations with the increased [:tabu.size, then set f_tabu_size to its initial value and go to 6.

If at any point there is an improvement then go to 5.

(5) Family exchange phase of searchEvaluate completely the neighbourhood NfInO) and select the best exchange offamilies. In an additional attempt to improve the solution, after the best move isperformed, try to improve the solution by evaluating all adjacent pairwiseexchange of jobs in the family that was moved to the right. Denote the newcomplete sequence byn' and its makespan by MAKE'. If MAKE' <MAKE*,then set n*=n, MAKE*=MAKE'. Set nO=n', MAKEo=MAKE' and goto 4.

(6) Start counting iterations with Jab exchangesSet i_iter = 0 and go to 7.

(7) Stopping criterion for Job exchangesJ_iter = J_iter + 1.

(a) With fixed tabu sizesIf no improvement in the last 2 iterations, go to 9, else go to 8.

(b) With variable tabu sizes

(i) Ifno improvement in the last int (2/3) iterations with the initial J_tabu_size, then decrease the J_tabu_size and to to (ii),

(ii) If no improvement in the last int (2/3) iterations with the decreasedJ_tabu_size, then increase the J_tabu_size and go to (iii).

(iii) If no improvement in the last int (2/3) iterations with the increasedJ_tabu.:.size, then set J_tabu_size to its initial value and go to 9.

If at any point there is an improvement go to 8.

(8) Job exchange phase of searchEvaluate completely the neighbourhood Nj(nO) and select the best exchange ofjobs. Denote the new complete sequence by n' and its makespan by MAKE'.If MAKE' <MAKE*, then set n*=n', MAKE*=MAKE'. Set nO=n',MAKEo=MAKE' and go to 7.

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Flow-line scheduling using tabu search 1727

(9) Switch the direction of searchIf the incumbent was changed during the job exchange phase of search (step 8),go to 3. If the required number of the long-term memory restarts has beenperformed STOP the search, else go to 2.

The variable tabu list sizes in steps 4 (b) and 7 (b) of the algorithm were set as follows:

• f_tabu_size

(I) initial size=int(FI2).(2) decreased size= int (F13).(3) increased size=int(FI0·5).

• j_tabu_size

(I) initial size=int(ZIF).(2) decreased size=int(Z/(2 x F)).(3) increased size= int (Z/(0'5 x F)).

When implementing this procedure for the scheduling problem at hand, we wereinterested in examining the impact of:(a) not using long-term memory versus the use oflong-term memory; and (b) the use of fixed versus variable tabu list sizes. Hence, weoperationalized six versions of the TSH algorithm as follows:

• TSH I: The heuristic with no long-term memory and fixed tabu list sizes.• TSH2: The heuristic with 5 invocations of LTM_MIN and fixed tabu list sizes.• TSH3: The heuristic with 5 invocations of LTM_MAX and fixed tabu list sizes.• TSH4: The heuristic with no long-term memory and variable tabu list sizes.• TSH5: The heuristic with 5 invocations of LTM_MIN and variable tabu list

sizes.• TSH6: The heuristic with 5 invocations of LTM_MAX and variable tabu list

sizes.

We now proceed to a comparison of the multiple versions ofthe TSH algorithm and thesimulated annealing procedure.

4. A comparison of scheduling procedures

4.1. Data setsFor carrying out a comparison of the heuristics, we randomly generated multiple

problems of various sizes. The details for these data sets are as follows:

(I) 30 problems for each of the following values of the parameters F, M and Nf

were generated:

• F=3, M=3, Nf=3 "If• F=3, M=4, Nf=5 "If• F=4, M=4, Nf=4 Vf• F=5, M=5, N(=5 "If• F=6, M=5, Nf=4 v].• F=5, M=6, Nf=8 Vf• F=6, M=6, Nf=6 "If• F=8, M=8, Nf=8 "If• F= IO,M=8, Nf=6 "If• F=IO, M=IO, Nf=IO Vf

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1728 J. Skorin-Kapoo and A. J. Vakharia

(2) For each problem, the processing time for a job at each machine was integeruniformly distributed in (1,10).

(3) For each problem, the setup time for a family at each machine was also integeruniformly distributed in the following three sets: (a) (1,20); (b) (1,50); and (c)(1,100). These parameter values were chosen so as to investigate the impact ofalternative family setup to job processing time ratios.

4.2. Simulated annealing based heuristicAs noted earlier, we compare the performance of the multiple versions of the TSH

proposed in this paper to the simulated annealing based heuristic (SAH) proposed byVakharia and Chang (1990). Details of this procedure are given in the appendix. Assuggested by these authors, we operationalized this method with the followingparameter values: X=25, Y=50, APo= 0'50 and GP=O·IO.

4.3. Performance measuresFor each heuristic method, we recorded the value of the best makespan. In

reporting the results, we standardized the value of the average makespan obtainedusing SAH for each problem size to be 100.Then the average makespan for the TSHmethods was computed relative to this value. Hence, if M, and M, represent the averagemakespans obtained using SAH and any TSH, respectively, then the relative makespanfor the latter is computed as {(M,lM,) x 100}. In such a case, a value below 100indicatesthat TSH outperforms SAH and vice versa. In addition to the relative value of themakespan (referred to as RELMAKE from now on), we also kept track of thecomputation time in seconds for each method (referred to as CTS from now on).

4.4. ResultsThe computational experiments were performed on IBM 3090 with a vector facility.

The TSH procedure was coded in FORTRAN 77 and the code for the SAH method wasobtained from the authors (also coded in FORTRAN 77).The complete results for eachproblem size and setup time distribution are reported in Tables I and 2. We use thenotation U(a, b) to denote the uniform distribution in (a, b). These results are discussedbelow.

4.4.1. Comparison of TSH and SAH(a) For the 3 x 3 x 3 problems, both the TSH and SAH procedures provided

identical RELMAKE value across all setup time distributions. However, the TSHmethod requires considerably less computing time than the SAH method (see CTSvalues in Table I).

(b) For the 3 x 4 x 5 and 4 x 4 x 4 problems the SAH procedure outperforms theTSH procedure in terms of RELMAKE values for all setup time distributions. This isalso true for the 5 x 6 x 8 problems with setup distributions of U(I, 20) and U(I, 50).This result is a direct consequence of the fact that the SAH procedure takes almost500% more computing time than the TSH procedure. To validate this, we ran SAHwith the parameters X =15and Y=IS (for these three problems sizes)in order to spenda similar amount of time as TSH in the search process. The results indicate that:

• For the 3 x 5 x 4 problems, TSH3 outperformed the SAH for a family setup timedistribution ofU(I, 20);TSH2, TSH3 and TSH6 outperformed SAH for a familysetup time distribution ofU(I, 50);and TSH3 and TSH6 outperformed SAH for afamily setup time distribution of U(l, 100).

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Table I. A comparison of the TSH and SAH scheduling procedures.

Problem size (F x M x Nf)

3x3x3 3x4x5 4x4x4 5x5x5 6x5x4

Setup time distribution Setup time distribution Setup time distribution Setup time distribution Setup time distribution "'lis"

Heuristic Measure U(l,20) U(l,50) U(I,loo) U(I,20) U(l,50) U(I, 100) U(I,20) U(I,50) U(I,loo) U(l,20) U(l,50) U(l,loo) U(I,20) U(I,50) U(I,loo) 'l:,g:

SAH RELMAKE 100·00 ioooo 100·00 100·00 100·00 100·00 100·00 100·00 10000 ioooo 100·00 100·00 100·00 100·00 100·00 '"'"CTS 0·240 0·237 0·247 0·470 0475 0·492 0·540 0'536 0'554 1'120 HI8 H53 1·080 1·080 1'109 ":0-TSHI RELMAKE 100·63 100·74 100·19 102-46 101·98 101·21 101·49 100·50 100·43 100·76 100·66 100·\7 99·78 99·82 99·58 '"CTS 0·011 0·009 0·010 0·026 0·022 0·022 0·032 0·030 0·033 0·107 0'094 0·085 0·111 0·095 0'098

l'>..s::TSH2 RELMAKE ioooo 10006 100·00 100·64 100·43 100·31 100·74 100'06 100·24 100·30 100·56 100·08 99·73 99·80 99·55 g:

CTS 0·027 0·025 0·024 0·083 0·077 0·075 0·092 0·091 0'091 0·296 0·270 0·265 0·273 0·256 0·263 ""TSH3 RELMAKE 10007 100·00 ioooo 100·35 100·15 100·13 100·28 100{)1 100·18 99·70 100·09 99·87 99-42 99-62 99·34 s::'"CTS 0-026 0·026 0·025 0·085 0·082 0·077 0'096 0·089 0'092 0·323 0·296 0·283 0·290 0·268 0·277 S·

TSH4 RELMAKE 100·43 10058 100·04 102'19 101·26 100·88 101-17 100'47 10051 100·97 100·70 100·18 99·83 99·84 99·75 ""CTS 0·007 0·006 0·006 0·017 0·016 0·016 0·019 0·018 0-018 0{)88 0·086 0·085 0·100 0'093 0·093 Ei

"'"TSH5 RELMAKE 100-00 100-00 100·00 100·76 100'67 100·34 100'49 100-16 100·24 99·94 99·90 99·67 99·23 99'24 99'13 s::CTS 0-032 0·032 0·032 O{}84 0·083 0·084 0·076 0·075 0·076 0·402 0·408 0'409 0-443 0·434 0·439 '"'"TSH6 RELMAKE 100·07 100·00 100·00 100·69 100·54 100·21 100·40 100·01 100·16 99·86 100·05 99·74 99·25 99·21 99·27 I:l...CTS 0·032 0·032 0-032 0·085 0·084 0·083 0·076 0·075 0·076 0·408 0·409 0·410 0·441 0·437 0·436 ":0-

SAH =Simulated annealing heuristic; TSH I = tabu search heuristic with fixed tabu list and no long-term memory; TSH2 = tabu search heuristic with fixed tabu list and long-term memory based on minimal frequencies; TSH3 =tabu search heuristic with fixed tabu list and long-term memory based on maximal frequencies; TSH4 = tabu search heuristicwith variable tabu list and no long-term memory; TSH5 =tabu search heuristic with variable tabu list and long-term memory based on minimal frequencies; TSH6 = tabu searchheuristic with variable tabu list and long-term memory based on maximal frequencies; RELMAKE = relative makespan as compared to that obtained using SAH; CTS =averagecomputation time in seconds per problem.

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Table 2. A comparison of the TSH and SAH scheduling procedures.

Problem size (F x M x Nf)

6x6x6 5x6x8 8x8x8 IOx8x6 lOx lOx 10

Setup time distribution Setup time distribution Setup time distribution Setup time distribution Setup time distribution

Heuristic Measure U(l,20) U(I,lO) U(l,IOO) U(I,20) U(I,50) U(I, 100) U(I,20) U(I,50) U(l,IOO) U(I,20) U(I,50) U(l,I00) U(I,20) U(l,50) U(I,IOO):-."""'"SAH RELMAKE 100·00 100·00 100-00 100-00 100·00 100·00 100·00 100·00 100'00 100'00 100·00 101}00 100-00 100-00 roooo '".,crs 2·200 2'198 2·239 2·560 2'561 2·632 7·150 7·152 7'1]6 6·476 6·442 6·563 18·520 18·]22 18·615:;.,

TSHI RELMAKE 100-13 99·29 99·27 101·07 100·87 100·20 99·02 98'45 98·62 98·36 97·69 96·95 98·4] 97·89 97-00 ~crs 0·247 0·214 0-217 0,]15 0,]02 0·2]] 1·277 H147 1·016 1·286 1-113 0·958 4·513 3-462 3·397 l>

'"TSH2 RELMAKE 99·80 99·22 99·27 100·99 100·65 100'1] 99·02 98·45 98·60 98·36 97·69 96·95 98-43 97·89 97·00 e'"crs 0·789 0·700 0·698 0·957 0'973 0·875 ],970 3·710 3-743 3-606 3-384 3·276 150410 14·180 14'141 l>

TSH3 RELMAKE 99-66 98·94 99·10 100-23 100·03 99·81 98·77 98·1] 98·53 98·08 97·58 96·83 98·40 97·66 96-96 ;:s"'-crs 0-789 0·757 0-725 1·090 1·071 0·924 4·556 4·096 ],870 4·112 3·573 3-474 15·767 16·148 14·480 4

TSH4 RELMAKE 100·32 99·55 99'18 101·23 100·84 100·11 99·20 98·70 98'69 98'64 97·86 97'15 98·72 98·07 97·11crs 0·218 0·205 0-221 0'243 0-234 0·2]6 1·035 1·023 1-016 1-066 1-031 1·038 4·168 3-735 3·779 :-.

TSH5 RELMAKE 99'60 98·88 98·85 100·05 100·05 99·73 98·61 98·22 98·30 97·90 97-42 96·86 98·22 97·58 96·94~crs 1·054 1·031 1·049 1·229 1-186 1'185 5'092 4·921 5·280 5'196 5-053 5-056 20·341 19·751 18·848 "'"TSH6 RELMAKE 99·56 98·95 98·90 100-09 100·26 99·71 98·46 98·30 98·32 97·87 97-20 96·61 98'19 97·58 96-81 ;:s-l>crs 1-053 1·040 1-041 1·220 1·188 1-187 5·166 5·162 5·274 5·053 5·062 5·204 20·526 19·750 19·739 .,!:l'

SAH = simulated annealing heuristic; TSH I = tabu search heuristic with fixed tabu list and no long-term memory; TSH2 = tabu search heuristic with fixed tabu list and long-term memory based on minimal frequencies; TSH3 = tabu search heuristic with fixed tabu list and long-term based on maximal frequencies; TSH4=tabu search heuristic withvariable tabu list and no long-term memory; TSH5=tabu search heuristic with variable tabu list and long-term memory based on minimal frequencies: TSH6=tabu searchheuristic with variable tabu list and long-term memory based on maximal frequencies; RELMAKE = relative makespan as compared to that obtained using SAH; crs = averagecomputation time in seconds per problem.

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Flow-line scheduling using tabu search 1731

• For the 4 x 4 x 4 problems, TSH2, TSH3, TSH5 and TSH6 outperformed theSAH for a family setup time distribution of U(I, 20); and all the TSH versionsoutperformed SAH for family setup time distributions ofU(I, 50)and U(I, 100).

• For the 5 x 8 x 6 problems, all the TSH versions outperformed SAH for familysetup time distributions of U(I, 20) and U(I,50); and TSH3, TSH5 and TSH6outperformed SAH for a family setup time distribution of U(I, 100).

(c) For all other cases, theTSH procedure outperforms the SAH method in terms ofRELMAKE values. Further, the CPU time requirements are substantially lower forthe TSH procedure as compared to the SAH. Note that for the three largest problems,and across all family setup time distributions, all the tabu search variants were superiorto the SAH solution. Moreover, the variants without long-term memory (TSH I andTSH4) required considerably less computing time.

This leads us to conclude that the TSH procedure clearly outperforms the SAHmethod in terms ofRELMAKE values for all problems and in most cases, requires lesscomputation time. Hence, for the scheduling problem being addressed in this paper, theproposed TSH method seems to be preferred.

4.4.2. Use of long-term memory in TSHBased on section 3.6, note that we operationalized multiple versions of the TSH

method. Essentially, TSHI and TSH4 do not use long-term memory; TSH2 and TSH5reflect the use of long-term memory based on minimal frequencies; while TSH3 andTSH6 reflect the use of 'long-term memory based on maximal frequencies. Incomparison go the TSHI and TSH4 variants of tabu search, the use of long-termmemory improved solution quality in terms of RELMAKE. The following interestingresult across all setup time distributions was obtained. The long-term memory basedon maximal frequencies (LTM_MAX) was consistently more efficient than the oneusing minimal frequencies (LTM_MIN). When employing fixed tabu list sizes, LTM_MAX proved more efficient than LTM_MIN in 28 out of 30 problem sets and in onecase both versions were equivalent. Moreover, LTM_MIN did not improve thesolutions to larger problems. With variable tabu list sizes, LTM_MAX performedbetter than LTM_MIN in 16 out of 30 problem sets and in three problem sets, bothversions provided equivalent results. Coupled with variable tabu list sizes, the use ofboth long-term memory versions improved the solutions to all 30 problem sets ascompared to the versions not using the long-term memory.

4.4.3. Fixed versus variable tabu list sizes in TSHNote that TSHI, TSH2 and TSH3 reflect fixed tabu list sizes while TSH4, TSH5

and TSH6 reflect variable tabu list sizes.The use of variable tabu list sizes was generallypreferred in the context of the problem addressed in this paper. This is clearly reflectedby the fact that in 20 of the 30 sets of problems, the best RELMAKE values wereachieved when variable tabu list sizes were employed.

5. Implications and conclusionsIn this study we developed a heuristic tabu search algorithm for scheduling part

families in a flow-line manufacturing cell. A comparison of six different tabu searchstrategies (designed to test different aspects of tabu search) and simulated annealingheuristic proposed by Vakharia and Chang (1990) was carried out using several data

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1732 J. Skorin-Kapou and A. J. Vakharia

sets with alternative ratios of family setup times to job processing times. The majorconclusions of this study are:

• In comparing the tabu search heuristic with simulated annealing, we found thatthe tabu search found comparable solutions for the smaller problems, and bettersolutions for larger problems. Further, tabu search obtained these solutionsusing less computational effort. Thus, the tabu search heuristic appears to bepreferred over the simulated annealing heuristic for the problem addressed inthis paper.

• In general the use of long-term memory improved solution quality but at theexpense of computation time. More specifically, compared with both variants ofthe tabu search heuristic without the use of long-term memory, the long-termmemory based on maximal frequencies offamilies and their respective schedul­ing positions improved the solutions for the problem sets used in this study. Onthe other hand, the long-term memory based on minimal frequencies, when usedwith fixed tabu lists, did not improve the solutions to the larger problem sets.Although it did improve the solutions to all problems when using the variabletabu list sizes, it was outperformed by the use of long-term memory based on'maximal frequencies. We conjecture that a possible explanation is in theneighbourhood structure: restricting the exchanges to adjacent pairwise ex­changes, the transition from a 'very bad' initial schedule to a 'good' one over alimited amount of search, becomes more difficult. Using the LTM_MIN; weactually penalize good schedules and try to construct an initial family scheduleaway from the good region. On the contrary, by employing LTM_MAX we tryto construct a new initial solution, but in a good feasible region, which intensifiesthe search in that region.

• Finally, a comparison of the use of fixed versus variable tabu list sizes revealedthat the use of a variable tabu list size is preferred.

Based on these conclusions, we recommend the use of the tabu search heuristic using avariable tabu list and long-term memory incorporated using the LTM_MAX approach(i.e. TSH6 as operationalized in this paper) for obtaining 'good' solutions to thescheduling problem addressed in this paper. At the expense of increased computationtime, it is also plausible that a combination of the two long-term memory variants withvariable tabu list sizes, could prove more efficient.

AcknowledgmentsThe authors would like to acknowledge the support of NSF under grant number

DDM 8909206 (I.S-K.), and the Karl Eller Graduate School of Management at theUniversity of Arizona (Summer Research Grant, AJ.V.).

Appendix

Simulated annealing based heuristic (SAH)In the Vakharia and Chang (1990) study, the SAH algorithm for part family

scheduling was operationalized as follows.

(I) Set X (number of iterations), Y (number of searches at each iteration), APo(acceptance probability), GP (the probability to carry out a pairwise exchangeof families, i.e. I - GP is the probability of carrying out a pairwise interchangeof jobs with a family), £=APo/X.

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Flow-line scheduling using tabu search 1733

(2) Generate a random sequence. This includes a complete sequence for all jobs(nO), a family sequence (r) and a sequence for jobs within each part family (Ilf;f == I, ... , F) and let these be the current solutions with a makespan of MAKEo.Let n* represent the incumbent solution with makespan MAKE*. Set n* ==no,MAKE*==MAKEo and x==O.

(3) Set x==x+ I. If x z-X, STOP else set y==O and continue.

(4) Set y== y+ I. If y> Y, then set APx==APx - £ and to to 3 else continue.

(5) Generate a random number v (0"; v"; 1). If v>GP go to 7, else continue.

(6) In carrying out this step the order ofjobs within each family will not change (i.e.Ilf stays the same '<If). Generate a random number v, (1";v, ~F). Interchangefamily in position v, with that of the family in position v, + I (If v, == F,interchange positions of family in position F with that of family in position 1)and generate sequence r '. Based on interchange of families, specify a newcomplete sequence n' and let the makespan of the new sequence be MAKE'.

(a) If MAKE' ~MAKE*,go to (b) else let the complete sequence n' replacesequence n* in the incumbent solution, set MAKE*==MAKE' and go to(b).

(b) IfMAKE' ~ MAKEo, go to (e) else let the new family sequence,' replace rand let the complete sequence n' replace sequence n° in the currentsolution. Set MAKEo==MAKE' and go to 4.

(c) Generate a random number V 2 (0"; V 2 .,; 1). If V 2 ~ AP x' go to 8 else let thecomplete sequence n' replace sequence n° in the current solution. SetMAKEo==MAKE' and go to 4.

(7) In carrying out this step the sequence for part families stays the same (i.e. r staysthe same). Generate a random number v, (I ";v, ~Z; where Z is the totalnumber of jobs in all families). Let f, be the family in which job v, is included.Interchange the job in position VI with that of the job in position VI + I (ifv, isthe last job in family f" interchange positions of job in position V, with that ofjob in position I for the same family f,) in sequence n°. Let the new sequence beIl}, for family f. and the new complete sequence be n' with associatedmakespan MAKE'.

(a) If MAKE' ~MAKE*,go to (b) else let the complete sequence n' replacesequence n* in the incumbent solution, set MAKE*==MAKE' and go to(b).

(b) If MAKE' ~MAKEo,go to (e) else let the new job sequence for family f,(i.e., Il},) replace Ilf,' and let the complete sequence n' replace sequence n°in the current solution. Set MAKEo==MAKE' and go to 4.

(c) Generate a random number V 2 (0~ V 2 ~ I). If V 2 ~ AP x, go to 8 else let thecomplete sequence n' replace sequence n° and the job sequence for familyf, (i.e. Il}.) replace Ilf' in the current solution. Set MAKEo ==M AKE' andgo to 4.

(8) Carry out a complete pairwise adjacent interchange of the jobs within eachfamily in sequence n°. Let the best makespan of such a search be MAKE' andthe associated complete sequence be n', the family sequence be r and thesequence for jobs within each family be Ilh

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1734 Flow-line scheduling using tabu search

(a) If MAKE 1 ~MAKE*, go to (b) else let the complete sequence 0 1 replacesequence 0* as the incumbent solution, set M AKE* = M AKE1 and go to(b).

(b) If MAKEI~MAKEo, go to 4 else let the complete sequence 0 1 replacesequence 0°, family sequence r l replace family sequence r and jobsequences Jlf. replace job sequences Jlf in the current solution. LetMAKEo=MAKE I and go to 4.

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