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A tabu search approach for group scheduling in buffer-constrained flow shop cellsMaghsud Solimanpur a & Atabak Elmi aa Mechanical Engineering Department, Faculty of Engineering , Urmia University , Urmia,West Azerbaijan Province, IranPublished online: 23 Feb 2011.

To cite this article: Maghsud Solimanpur & Atabak Elmi (2011) A tabu search approach for group scheduling inbuffer-constrained flow shop cells, International Journal of Computer Integrated Manufacturing, 24:3, 257-268, DOI:10.1080/0951192X.2011.552527

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A tabu search approach for group scheduling in buffer-constrained flow shop cells

Maghsud Solimanpur* and Atabak Elmi

Mechanical Engineering Department, Faculty of Engineering, Urmia University, Urmia, West Azerbaijan Province, Iran

(Received 4 May 2010; final version received 2 January 2011)

The scheduling problem in a cellular manufacturing system (CMS) has been named as group scheduling in theliterature. Due to the similarities in the processing route of the parts being in a group, it is mostly referred to as flowshop group scheduling. This problem consists of two interrelated sub-problems, namely intra-group scheduling andinter-group scheduling. On the other hand, mostly there are limited buffers between successive machines in which thework-in-process inventories can be stored. This article investigates the flow shop group scheduling with limitedbuffers to minimise the total completion time (makespan). Regarding the NP-hardness of this problem, two tabusearch algorithms are proposed for solving each sub-problem. The effectiveness of the proposed algorithms isevaluated on 270 randomly generated problems classified under 27 categories. The results of the proposed algorithmare compared with those of the heuristic published by Solimanpur–Vrat–Shankar (SVS). Computational resultsdemonstrate significant reduction in the average makespan over the SVS-algorithm.

Keywords: group technology; cellular manufacturing; heuristics; scheduling

1. Introduction

The philosophy of group technology (GT) is animportant concept in the design of manufacturingcells. GT can be defined as a disciplined approach inidentifying items, such as parts, and machines by theirattributes; analysing those attributes by looking forsimilarities between and among items; grouping theitems into families according to similarities, and finally,increasing the efficiency and effectiveness of managingthe items by taking advantage of the similarities.Briefly, cellular manufacturing (CM) is a productionsystem in which parts are grouped into dedicatedmanufacturing cells, according to a number ofsimilarities in their design and manufacturing features.The major benefits of CM reported in the literatureare: less production cost, less material handling cost,reduction in throughput time, reduction in work-in-process (WIP), simple production control, etc.(Wemmerlov and Hyer, 1989). Design of cellularmanufacturing system (CMS) concerns issues likemachine-cell formation, production planning, schedul-ing, arrangement of machines inside the cells, arrange-ment of cells, etc. Scheduling is one of the issues in theplanning of CMS, and its effective implementationattains more benefits of CMS (Hitomi and Ham,1976). The scheduling problem in a CMS has beennamed as group scheduling in the literature. It isdominantly believed in the literature that parts within a

part family in CMS can be processed in a flow shopway owing to the similarities existing in the design andproduction flow of parts. When there are intercellularmovements between the manufacturing cells, still partscan be processed in a flow shop way by duplicatingbottleneck machines or subcontracting the exceptionalparts (Logendran et al. 1995). Therefore, this problemis mostly referred to as flow shop group scheduling andconsists of two interrelated sub-problems, namelyintra-group scheduling and inter-group scheduling.Intra-group scheduling is related to the sequencing ofparts within groups, and inter-group schedulingdetermines the sequence through which groups are tobe processed. In general, the methods developed forgroup scheduling problem are extensions of themethods proposed for flow shop scheduling problems.Hejazi and Saghafian (2005) provided a review for theclassical flow shop scheduling problem with makespancriterion. To date, mathematical programming, con-structive heuristics and meta-heuristics have beenproposed for flow shop group scheduling, assuminginfinite buffer size between two successive machines.However, in CM systems due to limited room andstorage facilities in cells, using buffer of infinite size iscostly and may be impossible. Therefore, in most CMsystems buffer either is not used or a limited size isconsidered. In such cases, when a machine finishesprocessing of a particular part, that part cannot leave

*Corresponding author. Email: m.solimanpur@urmia.ac.ir

International Journal of Computer Integrated Manufacturing

Vol. 24, No. 3, March 2011, 257–268

ISSN 0951-192X print/ISSN 1362-3052 online

� 2011 Taylor & Francis

DOI: 10.1080/0951192X.2011.552527

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the machine if next machine is busy and the bufferbetween two machines is full. So, the first machine isblocked until at least a buffer unit becomes available.In classical flow shop group scheduling, a part canstart processing on every machine if previous partfinishes processing on it, while in blocking case nextpart has to remain at current machine until the nextmachine becomes available. Evidently, considerationof limited buffers for flow shop group scheduling ismore practical and applicative than the traditionalapproaches on this problem. Many research workshave been done on scheduling problem with limitedbuffers so far. Hall and Siriskandarajah (1996)provided a survey for scheduling problems with limitedbuffers. Khosla (1995) studied a two-stage flow shopscheduling problem with a finite buffer and proposed amixed integer linear programming (ILP) model, basedon which two lower bounds and several heuristics weredeveloped. Nowicki (1999) proposed a tabu search(TS) approach for permutation flow shop schedulingwith finite buffers between successive machines.Brucker et al. (2003) used the classical disjunctivegraph model and extended it to intermediate buffers.They solved this problem using a TS algorithmbased on the extension of the classical blockapproach theorem. Wang et al. (2006) proposed ahybrid genetic algorithm for general flow shopscheduling with finite buffers between successivemachines. Norman (1999) applied TS for flow shopscheduling problem considering both finite buffersand sequence-dependant setup times. Leisten (1990)presented some priority-based heuristics that can beapplied in both permutation and general flow shopscheduling problems with infinite buffers. Smutnicki(1998) discussed application of the NEH heuristic forblocking flow shop in detail. Wardono and Fathi(2004) proposed a TS algorithm for hybrid flow shopscheduling with limited buffers. Sawik (2000) pre-sented a mixed integer programming formulation forthe scheduling of flexible flow line with limitedbuffers. Thornton and Hunsucker (2004) proposed aconstructive heuristic for hybrid flow shop with nobuffer, while Wang and Tang (2009) presented a TSheuristic for a hybrid flow shop scheduling problemwith finite intermediate buffers. In addition, batchscheduling of a two-machine buffer constrained flowshop with sequence independent setup times andremoval times has been attempted by Pranzo (2004).

On the other hand, some researches on flow shopgroup scheduling have been reported in the literature.Yoshida and Hitomi (1979) and Baker (1990) investi-gate the two-machine flow shop group scheduling withthe criterion of makespan minimisation. Yang andChern (2000) consider two-machine flow shop groupscheduling problem such that each group requires the

same setup and removal times on both machines. Yangand Liao (1996) consider a group schedulingproblem with two cells and inter-cellular moves sothat, each part needs a maximum of one machine ineach cell. Logendran and Nudtasomboon (1991)proposed a heuristic, called LN-method for solvingthe stage one problem in group scheduling. Ben-Arieh and Dror (1989) considered group schedulingand emphasised for grouping parts according theirbehavioural parameters. Logendran et al. (1995)investigated the performance of combinations ofmethods; viz. PT proposed by Petrov (1966), LNproposed by Logendran and Nudtasomboon (1991),and CDS proposed by Campbell et al. (1970) andreported that LN-PT method (LN-method for intra-group and PT-method for inter-group scheduling)performs superior over the other combinations.Schaller (2001) reports a new lower bound toevaluate the partial sequences in the branch andbound procedure for the flow shop group schedulingproblem. This lower bound is tighter than the oneproposed by Hitomi and Ham (1976). Schaller (2001)develops a heuristic for large size problems in whichat the first stage the sequence of groups isdetermined using a branch and bound techniqueand at the second stage the jobs within each groupare sequenced through an interchange heuristic. Saadet al. (2002) proposed a TS algorithm for loadingand scheduling problem in CMSs. They alsoprovided a goal programming formulation for partsloading problem and reported a practical case study.Solimanpur et al. (2004) proposed a heuristic calledas Solimanpur–Vrat–Shankar (SVS)-algorithm forsolving cell scheduling problem with inter-cellularmoves and reported the superior performance ofSVS-algorithm over the LN-PT method. Logendranet al. (2004) investigated the group schedulingproblem to minimise the makespan in a flexibleflow shop. They compared either single setup ormultiple setups on machines for processing jobs inthe same group and used the LN-PT method tosolve the problem. Logendran et al. (2005) consid-ered the group scheduling within the context ofsequence dependent setup times in flexible flow shopsand used a TS approach to solve the problem andrecommended a TS approach with the short-termmemory. Salmasi et al. (2010) have developed amathematical programming model for minimisingtotal flow time of the flow shop sequence-dependentgroup scheduling problem. They used a TS algorithmand a hybrid ant colony optimisation (HACO) toheuristically solve the problem. To evaluate theperformance of the algorithms, they have used alower bounding (LB) method based on Branch-and-Price algorithm. They reported the effectiveness of

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HACO algorithm over the TS algorithm. Tavakkoli-Moghaddam et al. (2009) discussed around multi-criteria group scheduling problem and developed ascatter search (SS) method to solve the problem.Bi-objective group scheduling in flexible flow shophas been attempted by Karimi et al. (2009). Theyproposed a multi-phase genetic algorithm to optimiseboth makespan and total weighted tardiness.

The literature review conducted in this researchindicates that the flow shop group scheduling problemwith limited buffers between machines has not beenattempted in the literature. This article bridges this gapand presents a mathematical formulation for flow shopgroup scheduling with limited buffers. A TS approachis proposed for solving the formulated problem. Therest of this article is organised as follows. A detaileddescription and mathematical formulation of flowshop group scheduling with limited buffers is discussedin section 2. The TS approach and its elements aredescribed in Section 3. The elements of the proposedTS are synthesised in Section 4. The computationalresults are presented and compared in Section 5.Section 6 includes conclusions.

2. Problem statement

Flow shop group scheduling with limited buffersconsists of two sub-problems, namely intra-groupscheduling and inter-group scheduling. The intra-group scheduling refers to the sequencing of Hg parts(h ¼ {1, 2, . . . , Hg}) to be processed on a set of Mmachines (m ¼ {1, 2, . . . , M}) in a set of G groups(g ¼ {1, 2, . . . , G}) and Pmg is the setup time ofmachine m to process group g. Without loss ofgenerality, assume that the parts of each group areprocessed in the order of indices of machines, and thereare (M71) buffers between machines. Let bmþ1 denotethe size of buffer between machines m and m þ 1. Thesequence of parts for each group can be represented by apermutation pg ¼ {pg(1), pg(2), . . . , pg(Hg)}, wherepg(h) is the hth part in permutation pg The time neededto process part pg(h) on machine m is tm;pgðkÞ . Let s

gh;m

denote the earliest starting time of part pg(h) on machinem in group g. The objective is to findp�g 2

Qg 8g ¼ f1; 2; . . . ;Gg that minimises the make-

span (the completion time of the last part of group g onthe last machine). The makespan associated withsequence pg denoted by Cmax(pg), is calculated as follows:

Cmax ðpgÞ ¼ sgHg;Mþ tM;pgðHgÞ

; 8g ¼ 1; 2; . . . ;G

t0;pgðkÞ ¼ 0 8g ¼ 1; 2; :::G; 8h ¼ 1; 2; . . . ;Hg

sgh;0 ¼ 0 8g ¼ 1; 2; . . . ;G; 8h ¼ 1; 2; . . . ;Hg

sgh;Mþ1 ¼ 0 8g ¼ 1; 2; . . . ;G; 8h ¼ 1; 2; . . . ;Hg

sg1;m ¼ max Pm;g; sg1;m�1 þ tm�1; pgð1Þ

n o

8g ¼ 1; 2; . . . ;G; 8m ¼ 1; 2; . . . ;M

sgh;m ¼ max sgh;m�1 þ tm�1;pgðkÞ ; sgh�1;m þ tm;pgðh�1Þ

n o

8g ¼ 1; 2; . . . ;G;

8h ¼ 2; . . . ; bmþ1 þ 1; 8m ¼ 1; . . . ;M

sgh;m ¼ maxnsgh;m�1 þ tm�1;pgðkÞ ; s

gh�1;m þ tm;pgðh�1Þ ;

sgðh�bmþ1�1Þ;ðmþ1Þ

o8g ¼ 1; 2; . . . ;G;

8h ¼ bmþ1 þ 2; . . . ;Hg; 8m ¼ 1; . . . ;M

The inter-group scheduling refers to the sequencingof a set of G groups (g ¼ 1,2, . . . , G}) with the set ofHg parts within each group, which were sequencedpreviously in intra-group scheduling. The groups withtheir sequenced parts are to be processed on a set of Mmachines (m ¼ {1,2, . . . , M}), and there are (m71)buffers between each successive pairs of machines mand m þ 1 with the capacity of bmþ1 units. Sequencingof groups can be represented by a permutationd ¼ {d(1), d(2), . . . , d(G)}, where d(a) is the ath elementof the permutation d. Let S denote the set of all suchpermutations. The objective is to find a permutationd*2S that minimises the total makespan (the comple-tion time of the last part of the last group on the lastmachine). The total makespan for each permutation ddenoted by Cmax(d) is calculated as shown below.

Cmax dð Þ ¼ sd Gð ÞHdðGÞ;M

þ tM;pdðGÞðHdðGÞÞ

t0;pdðGÞðkÞ ¼ 0; 8g ¼ 1; 2; . . . ;G; 8h ¼ 1; 2; . . . ;HdðGÞ

sd Gð Þh;0 ¼ 0; 8g ¼ 1; 2; . . . ;G; 8h ¼ 1; 2; . . . ;HdðGÞ

sd Gð Þh;Mþ1 ¼ 0; 8g ¼ 1; 2; . . . ;G; 8h ¼ 1; 2; . . . ;HdðGÞ

sd 1ð Þ1;m ¼ max Pmdð1Þ; s

dð1Þ1;m�1 þ tm�1;pdð1Þð1Þ

n o;

8m ¼ 1; 2; . . . ;M

sd gð Þ1;m ¼ max

nsdðg�1ÞHdðg�1Þ;m

þ tm;pdðg�1ÞðHdðg�1ÞÞþ Pm;dðgÞ; s

dðgÞ1;m�1

þ tðm�1Þ;pdðgÞð1Þ

o

8g ¼ 2; . . . ;G; 8m ¼ 1; 2; . . . ;M

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sd gð Þh;m ¼ max

nsdðgÞh;m�1þ tm�1;pdðgÞ ðkÞ ; s

dðgÞh�1;m

þ tm;pdðgÞðh�1ÞÞ

o; 8g ¼ 1; 2; . . . ;G;

8h ¼ 2; . . . ; bmþ1 þ 1; 8m ¼ 1; . . . ;M

sd gð Þh;m ¼ max

nsdðgÞh;m�1 þ tm�1;pdðgÞ ðkÞ ; s

dðgÞh�1;m þ tm�pdðgÞðh�1Þ ;

sdðgÞðh�bmþ1�1Þ;ðmþ1Þ

o

8g ¼ 1; 2; . . . ;G; 8h ¼ bmþ1 þ 2; . . . ;HdðgÞ;

8m ¼ 1; . . . ;M

An illustrative example (Table 1) adopted fromHitomi and Ham (1976) is used to show application ofthe equations derived above. In this problem, there arefour groups containing 3, 4, 3 and 4 parts, respectivelyto be processed on five machines. In addition, there arefour limited buffers between successive machines withthe capacities 2, 1, 0, and 1, respectively. The first rowof each group in Table 1 shows setup time of eachmachine in that group.

Figure 1 illustrates a graph representation of group4 for the illustrative example given in Table 1. In thisfigure, each circle stands for an operation of the partindicated in the corresponding column to be performedby the machine indicated in the corresponding row.Hence, a horizontal conjunctive arrow pointing to acircle represents release of the previous part from therelevant machine. Also, a vertical conjunctive arrowpointing to a circle represents release of the relevant

part from the previous machine. Disjunctive arrowsstand for buffer size constraint.

Figure 2 shows the graph illustration of partialsequence d ¼ {1,4}. The values alongside the circlesindicate the earliest start time of the correspondingoperation.

We formulated an integer linear mathematicalprogramming model for group scheduling in bufferconstrained flow-line cells based on completion time ofoperations. This proposed model deals with theminimisation of makespan, where buffers betweenmachines are considered as machines with zeroprocessing time on parts. Let I denote the number ofassumptive machines which is calculated as follows.

I ¼MþXM�1m¼1

bmþ1

The ILP model for the presented problem isdiscussed in the following.

A. Indices

G ¼ index for number of part families.Hg ¼ index for number of parts of part family g.I ¼ index for number of machines.

B. Parameters

Rg,i ¼ setup time of ith machine for processinggth part family.

Tg,h,i ¼ processing time of hth part of gth partfamily on machine i.

C. Decision variables

Cg,h,i ¼ completion time of hth part of gth partfamily on machine i.Sg,i ¼ start time of processing gth part family onith machine.

Figure 1. Graph model of illustrative example for group 4and p4 ¼ {13, 12, 14, 11}.

Table 1. Setup and processing times for the problemadopted from Hitomi and Ham (1976).

GroupSetups and

parts

Machines

1 2 3 4 5

Group 1 Setup 30 15 25 30 10Part 1 41 65 39 79 52Part 2 75 75 68 71 61Part 3 32 25 62 73 54

Group 2 Setup 10 20 15 30 25Part 4 50 41 22 41 55Part 5 30 28 41 48 64Part 6 70 20 56 54 62Part 7 48 34 48 29 52

Group 3 Setup 15 25 30 20 10Part 8 29 55 46 37 31Part 9 26 20 37 51 28Part 10 72 66 40 47 62

Group 4 Setup 25 30 10 25 35Part 11 47 71 29 38 24Part 12 27 69 42 75 57Part 13 78 45 73 74 29Part 14 22 42 35 68 17

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Lg,i ¼ finish time of processing gth part family onith machine.Qg,g ¼ binary variable to determine partial se-quence of part families g and g0.Zg,h,h ¼ binary variable to determine partialsequence of parts h and h0 in gth part family.Cmax ¼ makespan.

D. Objective function and constraints

Minimize: Cmax

Subject to:

1. Sg,i�Rg,i

2. Sg0,i þ M . Qg,g0 �Lg,i þ Rg0,i

3. Sg,i þ M . (17Qg,g0)�Lg0,i þ Rg,i

4. Cg,h,i7Tg,h,i�Sg,i

h 2 {1,2, . . . , Hg}5. Cg,h,i 07Tg,h,i 0 �Cg,h,i

h 2 {1,2, . . . , Hg}, i04 i

6. Cg,h0,i7Tg,h0,i þ M .Zg,h,h0 �Cg,h,i

7. Cg,h,i7Tg,h,i þ M . (17Zg,h,h0)�Cg,h0,i

h, h0 2 {1,2, . . . , Hg}8. Lg,i�Cg,h,i

h 2 {1,2, . . . , Hg}9. Cmax ¤�Lg,l

Constraints (1, 2, 3) ensure that starting time ofoperations on each machine for each group (Sg,i) islarger than or equal to the completion time ofoperations of previous group on mentioned machine(Lg,i), plus setup time needed for operating next group(Rg,i) and so, processing each group cannot be startedbefore completion of previous group. Constraint (4)guarantees that processing each part on each machinewill be scheduled after the start time of that group onrelated machine. Constraint (5) ensures that starting

each operation of each part takes place after thecompletion of previous operation of that part. Con-straints (6, 7) maintain the operational sequence of theoperations processed on the same machine. Theseconstraints guarantee that no two operations areprocessed on a machine at a time. Constraint (8)ensures that each group on each machine is completedwhen all operations of that group are completed onthat machine. Finally, constraint (9) defines makespanas the completion time of all groups on the lastmachine.

Due to the large number of decision variables andconstraints of the proposed model, it is difficult tosolve this model by exact methods for large-sizedproblems. Hence, an efficient TS approach is proposedin the next section to solve this problem.

3. Tabu search and its elements

Glover (1989, 1990) proposed the TS method to solvecombinatorial optimisation problems. The principlesof the TS method are the neighbourhood searchapproach and the tabu list. This method is concernedwith imposing restrictions to guide the search processand to allow the search to escape from local optima.Hoos and Stutzle (2005) provide more details for TSmethod and its comparison with other stochastic localsearch methods.

TS algorithm is organised to start at some initialsequence and then move successively among neigh-bouring sequences. At each iteration, a move is madeto build a better sequence in the neighbourhood of thecurrent sequence. The method avoids (makes tabu)sequences with certain attributes in order to preventcycling and guides the search process towards theunexplored regions of the solution space. This is doneusing special short and long-term memory functions.

Figure 2. Graph model of partial sequence of groups 1 and 4 as d ¼ {4, 1}.

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An important form of the short-term regency-basedmemory is embodied in a structure called a tabu list.TS includes two kinds of processes called intensifica-tion and diversification. The first one explores theneighbours around the current solution and the secondone explores unexpected neighbourhoods. Both ofthem use a tabu list but each one uses in its specialmanner. In the simplest form, TS requires thefollowing ingredients: initial sequence, a mechanismfor generating the neighbourhood of the currentsequence, tabu list and stopping criteria (Glover1989, 1990).

3.1. Initial solution

An initial solution for flow shop group schedulingproblem must specify sequence of groups and sequenceof parts within the groups. There are many heuristicsin the literature that can be applied to sequence theparts within the groups. Taillard (1990) has found thatthe NEH-algorithm proposed by Nawaz et al. (1983)performs well for flow shop scheduling problems.Thus, this algorithm is adopted in this article toconstruct the initial sequence of parts within thegroups.

The NEH algorithm is presented as follows:

(1) Sort the n parts by decreasing order of sum ofprocessing times on the machines;

(2) Take the first two parts and schedule them inorder to minimise the partial makespan as ifthere were only two parts;

(3) For the kth part, k ¼ (3, . . . ,n} insert it into theplace, among k possible ones, which minimisesthe partial makespan.

Hence, the NEH heuristic consists of two phases.First, the parts are sorted and then a part sequence isconstructed by evaluating the partial schedules origi-nating from the initial sequence obtained in the firstphase. Also, Logendran et al. (1995) reported that thePT method proposed by Petrov (1966) performs betterthan the other heuristics for group sequencing. There-fore the PT method is used to determine the initialsequence of groups.

3.2. Neighbourhood definition

Let us consider the permutation P ¼ {P(1), P(1), . . . ,P(n)} in which P(a) indicates the ath element (part/group) in permutation P. A neighbourhood forpermutation P is defined as the set of all permutationsthat can be created by a certain perturbation of thecurrent permutation P. There are three differentperturbation schemes that have been used in most of

the existing algorithms developed for permutation-involved problems. These can be defined as follows:

Adjacent exchange: exchange the position of theparts currently on positions (a) and (a þ 1).Random exchange (two exchange): exchange theposition of the parts currently in positions (a)and (b).Insertion: remove the part currently in position (a)and insert it into position (b) (Figure 3).

Taillard (1990) has experimentally shown that theinsertion scheme is more effective and efficient thanothers for flow shop scheduling problems. Therefore,we have used the insertion scheme in both intra- andinter-group scheduling problems.

In the illustrative example, permutation of parts ingroup 4 is p4 ¼ {13,12,14,11}. A neighbour solutioncan be obtained for this sequence by the insertionmove (a, b) ¼ (1, 3) as p04 ¼ {12,14,13,11}. Also, forthe permutation of groups d ¼ {2,3,1,4} a newneighbour can be generated by move (a, b) ¼ (2, 4)as d0 ¼ {2,1,4,3}.

3.3. Move strategy

The move strategy determines the manner by which theneighbours are visited. Three move strategies can beaddressed in the literature: best, first-better and sub-neighbourhood. In the best move strategy, all thepossible neighbours are evaluated and the moveresulting in the best neighbour is incorporated. Thebest move strategy may lead to better solutions butevaluating all possible neighbours requires allocatingmemory to keep track of features of all neighbours andincreases the run time. With the first-better strategy,evaluating the neighbours continues until a bettersolution is found and so the corresponding move isdone. In the third strategy, a sub-neighbourhood of thecurrent solution with reasonable size is entirelyevaluated in a reasonable time to determine the moveto be selected. Some advanced move strategies in localsearch algorithms and effective implementation of TSfor different problems can be studied in Hoos andStutzle (2005). Since there are several intra-groupscheduling in the attempted problem, the first-betterstrategy has been used in the proposed algorithm toshorten the computational time.

Figure 3. Insertion scheme.

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3.4. Tabu list

The tabu list is a mechanism to prevent cycling andguide the search process towards the unexploredregions of the solution space. Glover (1989, 1990)provides some general methods of tabu list implemen-tations. There are also several implementations of tabulist in the literature proposed for permutation involvedproblems. A dynamic tabu list is used for both intra-and inter-group scheduling problems in the proposedalgorithm. In the intra-group scheduling, the tabu listsize (LS) depends on group size (Hg) and number ofiterations (iter) and changes over the iterations ofalgorithm as shown in Figure 4.

The tabu LS in the inter-group scheduling changesduring the iterations of algorithm similar to thedynamics shown in Figure 4 except that Hg is replacedby G. As seen in Figure 4, the tabu LS is kept smallduring the intensification phase to make an in-depthexploration of solution space. It is then enlargedduring the diversification phase to avoid rapidconvergence of the algorithm and its entrapment in alocal optimum solution.

4. Description of the algorithm

The abovementioned elements are combined to createthe proposed TS algorithm. The proposed algorithm isbased on the fact that if the tabu LS is selectedappropriately large, diversification of the searchprocess will be increased and if it is selected small,the intensification of the search process will beincreased (Glover, 1989). That is, a short list allows athorough examination of the neighbourhood of a goodsolution while a long list facilitates the escape from alocal optimum to explore new regions of the searchspace. From this point, our adaptive tabu scheme goesfurther in direction of robustness by proposing asimple mechanism for adapting the LS to the status of

the search process. The intensification and diversifica-tion processes can be described as follows:

(1) Intensification process: after updating thecurrent best known solution, the tabu LS isset to an appropriate lower bound value andremains unchanged during the certain numberof iterations. By means of this process, thesolutions being in the close neighbourhood ofthe current best-known solution are searched tofind the possible local optimum.

(2) Diversification process: after the inten-sification process, if the current best-knownsolution is not improved, diversification processstarts and the tabu LS changes to a largervalue. During the diversification process, if thecurrent best-known solution is improved, theintensification process is called to find possiblelocal optimum and the tabu LS is reset to theinitial size.

For the intra-group scheduling, TS-algorithm startsby initialising the parameters and generating an initialsolution for the first group. Next, using the intensifica-tion process, TS algorithm explores the neighbours ofthe current solution pg. Suppose p0g has the first-bettermakespan among the neighbours of the currentsolution pg. The algorithm moves to this neighboursolution, updates the tabu list, and starts to search anew better neighbour solution. After a certain numberof iterations, if the objective did not improve,diversification process explores the solution space untilfinding a promising neighbourhood. Then, the intensi-fication process is applied once again. This processrepeats for the next group after terminating the searchprocess for the previous group and finally, thepreferred solution for each group is obtained. Nowwe have a solution for sequencing the related parts ofeach group and it is the time to find a solution forgroup sequencing, which is referred to as inter-groupscheduling. The algorithm initialises the parametersand generates an initial solution d for group sequen-cing. The intensification process explores the neigh-bourhood of the current solution. Suppose d0 has thefirst-better makespan in the neighbourhood of thecurrent solution. The algorithm moves to d0, updatesthe tabu list and starts to search a new betterneighbour solution. If the last best solution does notimprove for a certain number of iterations, TS-algorithm applies diversification process to find anew promising solution. If such a solution found, theintensification is applied once again. The algorithmterminates when the number of iterations reaches apredetermined value. The pseudo code of the algo-rithm described above is presented in the following:

Figure 4. Dynamic list-size for intra-group scheduling ofparts in group g.

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Begin

//Input: pg7the initial sequence of parts in group g,d7the initial sequence of groups;

Output: p�g� the best sequence of parts ever foundfor group g,d*7the best sequence of groups ever found;

Parameters: MaxCounter7maximum iterations for in-tra-group scheduling, MaxGCounter�maximum iterations for inter-groupschedulingk7parameter to dynamically controlsize of tabu list//

p�g pg, d* d //Save initial solution as the bestsolution yet obtained//

Repeat

initialise unimpiter 0, LS Hg, Counter 0Repeat

(1) Let p0g denote the first-better solution deter-mined in the neighbourhood of the currentsolution (pg).

(2) pg p0g and update Tabu list //replace thecurrent solution by the new one//

(3) If Cmax pgð Þ < Cmax p�gð Þ then

p�g pg;unimpiter 0,LS Hg, //save the bestsolution ever found//Else unimpiter (unimpiter þ 1)

(4) If unimpiter ¼ ¼ k then LS 2Hg

3

(5) If unimpiter ¼ ¼ 2k then LS 2Hg,(6) Counter (Counter þ 1),

Until Counter reaches MaxCounterUntil sequence of parts within all groups is determined

set unimpiter 0, LS G, Counter 0.Repeat

(1) Let d0 denote the first-better solution deter-mined in the neighbourhood of the currentsolution d.

(2) d d0 and update Tabu list //replace thecurrent solution by the new one//

(3) If Cmax(d) 5 Cmax(d*) then

d * d, unimpiter 0,LS G//save thebest solution ever found//Else unimpiter (unimpiter þ 1)

(4) If unimpiter ¼ ¼ k then LS 2G3

(5) If unimpiter ¼ ¼ 2k then LS 2G,(6) Counter (Counter þ 1),

Until Counter reaches MaxGCounter

Return d� and p�g for g ¼ 1; . . . ;GEnd

5. Experimental and computational results

In many production systems the workload flow isvariable due to the inconsistent processing capacity ofmachines, the failure or maintenance of machines,unbalanced job arrivals and fluctuations in processingtimes. Therefore, buffers are employed in these systemsto achieve a monotonic flow of workload. Whileapplication of buffers is a must in most of manufactur-ing environments, the relevant direct and indirectproduction costs will increase when the buffer capa-cities grow. Also, larger buffer storage raises thethroughput of the system at the expense of moreWIP inventory (So, 1990). However, low ratios of WIPto throughput are necessary to maintain a competitiveproduction (Conway et al. 1988). Thus, there havebeen necessities and importance to analyse bufferresources to obtain an optimal buffer capacity foreach machine under certain constraints. In this article,we assume that the buffer capacity is the same betweendifferent machines. Numerical experiments are carriedout in five levels of buffer size: B ¼ 0, 1, 2, 4 andinfinity, respectively.

To test the effectiveness of the proposed algorithm,the computational experiments were carried out on 27categories. Table 2 illustrates the size of problems ateach category. To reduce the effect of randomlygenerated data, 10 random data sets were generatedfrom a uniform distribution in [0,100]. Thus, there are270 problem data sets in 27 categories and eachcategory is determined by three parameters, viz.number of groups, number of parts in each groupand number of machines.

The computational results reported by Solimanpuret al. (2004) indicate that the SVS-algorithm is moreeffective and efficient than the LN-PT methodproposed by Logendran et al. (1995). Moreover,LN-PT method performs better than the otherheuristics, such as PT-LN, PT-CDS and CDS-PT(Logendran et al. 1995). Therefore, the results of theproposed TS-algorithm are compared to those ob-tained by SVS-algorithm. For a fair comparison, bothTS and SVS algorithms were coded in Borland Cþþand run on a Pentium 4, 2.00 GHz PC with 1.0 GBmemory.

Table 3 presents the average makespan of eachcategory obtained by both algorithms. The percentageof reduction (PR) in average makespan obtained byTS-algorithm over the SVS-algorithm is shown inTable 3 for each level of buffer size.

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The sample data reported for PR in Table 3 is usedto compute 100(17a)% confidence interval for thisparameter. Based on central limit theorem, a normaldistribution is considered for the average of these dataand hence a 95% confidence interval for PR can beobtained by PR � 1:96 Sffiffiffi

Np where PR is the sample

mean, S is the standard deviation and n ¼ 27 is thesample size. Table 4 shows the 95% confidence intervalfor the PR (PR) in makespan for different levels ofbuffer size. As seen in Table 4, when for example thereis no buffer between machines, with a 95% confidence,the PR in makespan achieved by the proposed TS overthe SVS-algorithm is at least 0.39 and it is 0.63 at most.As seen, the maximum reduction is attained whenB¼ 4.

Figure 5 compares the main effect of differentparameters on the PR in makespan. The points in thisfigure are the means of the response (PR) at the various

levels of each factor (number of groups, parts,machines, and buffers). As seen in this figure, the PRin makespan increases when the number of groups andmachines increases. In contrary, the value of PRdecreases by the increment in the number of parts. Thisfigure also indicates that the maximum reduction inmakespan is achieved for B¼ 4 and it will not befurther improved for larger buffers.

Figure 6 illustrates the percentage of differencesbetween the best makespan found for different buffersizes including an infinite buffer size. Our computa-tional experience reveals that the critical buffer size upto 20 parts and 10 machines is almost 4. Adoption of alarger buffer size will not improve performance of themanufacturing system and even may increase theproduction costs.

6. Conclusions

This article concerns the scheduling problem in a CMenvironment. An ILP model is proposed to minimisethe makespan in the flow shop group schedulingproblem with limited buffers. NP-hardness of theformulated problem makes it impossible to be solvedby exact methods in a reasonable time. Therefore, a TSalgorithm is developed to solve it heuristically.

Performance of the proposed TS-algorithm iscompared to that of the SVS-algorithm proposed bySolimanpur et al. (2004). The computational resultsindicate that the proposed TS-algorithm outperformsthe SVS-algorithm in terms of minimum makespan.Based on numerical experimentations, favourability ofthe proposed TS over SVS-algorithm enhances inproblems with larger number of machines and groups.Increment in the number of parts deteriorates effec-tiveness of the proposed TS though it performs stillbetter than SVS-algorithm.

It is found that the impact of limited buffers on theperformance of traditional heuristics for flow shopgroup scheduling is significant only for small buffersizes and diminishes rapidly as buffer size increases.This indicates that the buffer capacity can be treated asinfinite once the buffer size reaches a certain criticalsize.

Table 2. Size of test problems.

Number

Category Groups Parts Machines

Ctg 1 3 5 3Ctg 2 3 5 5Ctg 3 3 5 10Ctg 4 3 10 3Ctg 5 3 10 5Ctg 6 3 10 10Ctg 7 3 20 3Ctg 8 3 20 5Ctg 9 5 20 10Ctg 10 5 5 3Ctg 11 5 5 5Ctg 12 5 5 10Ctg 13 5 10 3Ctg 14 5 10 5Ctg 15 5 10 10Ctg 16 5 20 3Ctg 17 5 20 5Ctg 18 5 20 10Ctg 19 7 5 3Ctg 20 7 5 5Ctg 21 7 5 10Ctg 22 7 10 3Ctg 23 7 10 5Ctg 24 7 10 10Ctg 25 7 20 3Ctg 26 7 20 5Ctg 27 7 20 10

PR ¼ Average makespan obtained by SVS algorithm � average makespan obtained by TS algorithm

Average makespan obtained by SVS algorithm

� �� 100

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Table

3.

Computationalresultsunder

differentbuffer

sizes.

B¼

0B¼

1B¼

2B¼

4B¼

Infinite

Category

SVS

TS

mSVS

TS

PR

SVS

TS

PR

SVS

TS

PR

SVS

TS

PR

Ctg-1

1,205.3

1,191

1.18

1,099.1

1,067

2.92

1,075.6

1,042.4

3.08

1,070.9

1,037.2

3.14

1,070.9

1,037.2

3.14

Ctg-2

1,384.9

1,372.5

0.89

1,261.7

1,235.2

2.1

1,249

1,218.6

2.43

1,243.2

1,210.9

2.59

1,243.2

1,210.9

2.59

Ctg-3

1,810.3

1,786.6

1.3

1,745.8

1,679.2

3.81

1,744.3

1,671.4

4.17

1,744.3

1,665.9

4.49

1,744.3

1,665.9

4.49

CTg-4

2,121.2

2,105.2

0.75

1,904

1,887.7

0.85

1,846.3

1,827.5

1.01

1,825.4

1,782.4

2.35

1,825.4

1,782.4

2.35

Ctg-5

2,373.6

2,367.7

0.24

2,104.4

2,078

1.25

2,031.8

2,007.3

1.2

2,002.8

1,992.7

0.15

2,002.8

1,992.7

0.15

Ctg-6

2,895.2

2,872.7

0.77

2,604

2,559.5

1.7

2,581.2

2,522.8

2.26

2,577.8

2,529.8

1.86

2,577.8

2,529.8

1.86

Ctg-7

4,022.5

4,011.9

0.26

3,544.5

3,529.9

0.41

3,375.4

3,355.9

0.57

3,492.7

3,466.3

0.75

3,467.9

3,441.4

0.76

Ctg-8

4,293.1

4,286.2

0.16

3,761.2

3,736.5

0.65

3,656.2

3,603.9

1.43

3,590.5

3,541.3

1.37

3,582.8

3,539.3

1.21

Ctg-9

5,033.5

5,015.2

0.36

4,352.8

4,337.2

0.35

4,276.4

4,241.8

0.8

4,265.6

4,212.6

1.24

4,265.6

4,212.6

1.24

Ctg-10

2,064.9

2,054.9

0.48

1,823.9

1,803.7

1.1

1,774.1

1,704.5

3.92

1,760.2

1,700.2

3.4

1,760.1

1,700.2

3.4

Ctg-11

2,282.5

2,267.2

0.67

1,992.3

1,962.7

1.48

1,968.9

1,869.1

5.06

1,965.1

1,866.4

5.02

1,965.1

1,866.4

5.02

Ctg-12

2,707

2,683.8

0.85

2,472.5

2,429.1

1.75

2,444.8

2,338.1

4.36

2,442.2

2,337.7

4.27

2,442.2

2,337.7

4.27

Ctg-13

3,537.4

3,527.5

0.27

3,193.2

3,153.7

1.23

3,094.6

3,016.9

2.51

3,023.1

2,958.6

2.13

3,007.7

2,955.9

1.72

Ctg-14

3,891.1

3,873.1

0.46

3,409.2

3,344.4

1.93

3,309.5

3,183.4

3.81

3,352.1

3,123

6.83

3,252.1

3,121.8

4Ctg-15

4,594

4,579

0.32

4,027.3

3,962.3

1.61

3,937.4

3,814.4

3.12

3,922.7

3,776.8

3.71

3,922.7

3,776.8

3.71

Ctg-16

6,570.6

6,560.2

0.15

5,962.3

5,939.4

0.38

5,849.8

5,826.8

0.39

5,704.3

5,640.3

1.12

5,600.2

5,488

2Ctg-17

7,200.5

7,182.6

0.24

6,249.6

6,172.9

1.22

6,064.8

5,933.7

2.16

5,936.3

5,791.4

2.44

5,892.3

5,749

2.43

Ctg-18

8,187

8,150.9

0.44

6,922.6

6,854.6

0.98

6,681.9

6,565.3

1.74

6,583.1

6,437.3

2.21

6,583.1

6,437.3

2.21

Ctg-19

2,750.1

2,734.5

0.56

2,409.5

2,364.4

1.87

2,317.5

2,208.3

4.71

2,264.5

2,169.8

4.18

2,255.6

2,169.8

3.8

Ctg-20

3,097.6

3,090.6

0.22

2,684.5

2,638.1

1.72

2,590.4

2,496.4

3.62

2,558.5

2,441.5

4.57

2,551.9

2,441.5

4.32

Ctg-21

3,602.6

3,568.2

0.95

3,174.3

3,092

2.59

3,100.5

2,969.3

4.23

3,079.8

2,945.9

4.34

3,079.8

2,945.9

4.34

Ctg-22

4,938.9

4,929.2

0.19

4,424.8

4,362.1

1.41

4,293.8

4,185.4

2.52

4,197.7

4,079.9

2.8

4,155.3

4,077.8

1.86

Ctg-23

5,438.4

5,400.5

0.69

4,770.7

4,677.7

1.94

4,605.4

4,439.7

3.59

4,495.8

4,278.2

4.84

4,477.2

4,256.4

4.93

Ctg-24

6,255.4

6,215.4

0.63

5,396.8

5,315.2

1.51

5,218.8

5,055.5

3.12

5,175.2

4,961.3

4.13

5,175.2

4,961.3

4.13

Ctg-25

9,137.8

9,125.3

0.13

8,281.1

8,212.7

0.82

8,178.9

8,085.9

1.13

8,057.1

7,902.5

1.91

7,846.4

7,714.1

1.68

Ctg-26

10,140.7

10,114.7

0.25

8,763

8,673.2

1.02

8,434

8,297.4

1.61

8,203.5

8,019.4

2.24

8,138.6

7,897.3

2.96

Ctg-27

11,503.9

11,454.4

0.43

9,618.2

9,475.2

1.48

9,156.6

9,088

0.74

9,038.1

8,747.4

3.21

8,999.2

8,705.3

3.26

Total

1,23,040

1,22,521

13.89

1,07,953.3

1,06,543.6

40.08

1,04,857.9

1,02,569.7

69.29

1,03,572.2

1,00,616.7

81.29

1,02,003.4

1,00,014.7

77.83

Avg

0.5

1.48

2.56

3.01

2.88

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Applicability of the proposed approach can beenriched by considering backtracking within manufac-turing cells, intercellular movements and breakdowns.The group scheduling problem can be considered inhybrid flow shop cells with multiple constraints asunrelated parallel machines, machine eligibility andsequence-dependant setup time of parts. Also, multi-objective formulation and treatment of the groupscheduling problem can be attempted in futureresearches.

Acknowledgement

The authors express their cordial thankfulness and sincereappreciations to the valuable time and constructive com-ments of the reviewers, which have enhanced the quality ofthe article over its initial version.

References

Baker, K.R., 1990. Scheduling groups of jobs in the twomachine flow shop. Mathematical and Computer Model-ing, 13, 29–36.

Ben-Arieh, D. and Dror, M., 1989. Group scheduling usinggroup technology concepts. International Journal ofComputer Integrated Manufacturing, 2 (4), 186–193.

Brucker, P., Heitmann, S., and Hurink, J., 2003. Flow-shopproblems with intermediate buffers. OR Spectrum, 25,549–574.

Campbell, H.G., Dudek, R.A., and Smith, M.L., 1970. Aheuristic algorithm for the n job, m machine sequencingproblem. Management Science, 16, B630–B637.

Conway, R., et al., 1988. The role of work-in-process inven-tory in serial production lines. Operations Research, 36,229–241.

Glover, F., 1989. Tabu search: Part I. ORSA Journal ofComputing, 1, 190–206.

Glover, F., 1990. Tabu search: part II. ORSA Journal ofComputing, 2, 4–32.

Hall, N.G. and Siriskandarajah, C., 1996. A survey ofmachine scheduling problems with blocking and no-waitin process. Operations Research, 44, 510–525.

Table 4. The 95% confidence interval for the percentage of reduction in makespan.

B ¼ 0 B ¼ 1 B ¼ 2 B ¼ 4 B ¼ infinity

PR 0.512593 1.484444 2.566296 3.010741 2.882593S 0.323224 0.780248 1.393466 1.537815 1.336403Confidence interval 0.39–0.63 1.19–1.78 2.04–3.09 2.43–3.59 2.38–3.39

Figure 5. Main effect of different factors in PR.

Figure 6. Percentage of difference in makespan for differentbuffer sizes.

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rsity

Lib

rari

es]

at 2

3:16

12

Aug

ust 2

014

Hejazi, R. and Saghafian, S., 2005. Flowshop schedulingproblems with makespan criterion: a review. Interna-tional Journal of Production Research, 43, 2895–2929.

Hitomi, K. and Ham, I., 1976. Operations scheduling forgroup technology applications. Annals CIRP, 25,419–422.

Hoos, H.H. and Stutzle, T., 2005. Stochastic local search,San Francisco, CA: Morgan Kaufmann Publishers.

Karimi, N., Zandieh, M., and Kar-Amooz, R., 2009. Bi-objective group scheduling in hybrid flexible flowshop: amulti phase approach. Expert Systems with Applications,37, 4024–4032.

Khosla, I., 1995. The scheduling problem where multiplemachines compete for a common local buffer. EuropeanJournal of Operational Research, 84, 330–342.

Leisten, R., 1990. Flowshop sequencing problems withlimited buffer storage. International Journal of ProductionResearch, 28, 2085–2100.

Logendran, R., Carson, S., and Hanson, E., 2004. Groupscheduling in flexible flow shops. International Journal ofProduction Economics, 96, 143–155.

Logendran, R., de Szoeke, P., and Barnard, F., 2005.Sequence-dependant group scheduling problems in flex-ible flow shops. International Journal of ProductionEconomics, 102, 66–86.

Logendran, R., Mai, L., and Talkington, D., 1995.Combined heuristics for bi-level group schedulingproblems. International Journal of Production Economics,38, 133–145.

Logendran, R. and Nudtasomboon, N., 1991. Minimizingthe makespan of a group scheduling problem: a newheuristic. International Journal of Production Economics,22, 217–230.

Nawaz, M., Enscore, E., and Ham, I., 1983. A heuristicalgorithm for the m-machine, n-job flow shop sequencingproblem. OMEGA: International Journal of ManagementScience, 11, 91–95.

Norman, B.A., 1999. Scheduling flowshops with finite buffersand sequence-dependent setup times. Computers andIndustrial Engineering, 36, 163–177.

Nowicki, E., 1999. The permutation flow shop with buffers: atabu search approach. European Journal of OperationalResearch, 116, 205–219.

Petrov, V.A., 1966. Flow line group production planning.London: Business Pub. Ltd.

Pranzo, M., 2004. Batch scheduling in a two-machine flowshop with limited buffer and sequence independent setuptimes and removal times. European Journal of Opera-tional Research, 153, 581–592.

Saad Sameh, M., Baykasoglu, A., and Gindy Nabil, N.Z.,2002. A new integrated system for loading and schedul-ing in cellular manufacturing. International Journal ofComputer Integrated Manufacturing, 15 (1), 37–49.

Salmasi, N., Logendran, R., and Skandari, M., 2010. Totalflow time minimization in a flow shop sequence-dependent group scheduling problem. Computers andOperations Research, 37, 199–212.

Sawik, T., 2000. Mixed integer programming for schedulingflexible lines with limited intermediate buffers. Mathe-matical and Computer Modeling, 31, 39–52.

Schaller, J., 2001. A new lower bound for the flow shopgroup scheduling problem. Computers and IndustrialEngineering, 41, 151–161.

Smutnicki, C., 1998. A two-machine permutation flow-shopscheduling with buffers. OR Spectrum, 20, 229–235.

So, K.C., 1990. The impact of buffering strategies on theperformance of production line systems. InternationalJournal of Production Research, 26, 2293–2307.

Solimanpur, M., Vrat, P., and Shankar, R., 2004. A heuristicto minimize makespan of cell scheduling problem.International Journal of Production Economics, 88,231–241.

Taillard, E., 1990. Some efficient heuristic methods for theflow shop sequencing problem. European Journal ofOperational Research, 47, 65–74.

Tavakkoli-Moghaddam, R., et al., 2009. Design of a scattersearch method for a novel multi-criteria group schedulingproblem in a cellular manufacturing system. ExpertSystems with Applications, 37, 2661–2669.

Thornton, H.W. and Hunsucker, J.L., 2004. A new heuristicfor minimal makespan in flow shops with multipleprocessors and no intermediate storage. EuropeanJournal of Operational Research, 152, 96–114.

Wang, L., Zhang, L., and Zheng Da-Z., 2006. An effectivehybrid genetic algorithm for flow shop scheduling withlimited buffers. Computers and Operations Research, 33,2960–2971.

Wang, X. and Tang, L., 2009. A tabu search heuristic for thehybrid flowshop scheduling with finite intermediatebuffers. Computers and Operations Research, 36, 907–918.

Wardono, B. and Fathi, Y., 2004. A tabu search algorithmfor the multi-stage parallel machine problems withlimited buffer capacities. European Journal of OperationalResearch, 155, 380–401.

Wemmerlov, U. and Hyer, N.L., 1989. Cellular manufactur-ing in the U.S. industry: A survey of users. InternationalJournal of Production Research, 27, 1511–1530.

Yang, D.-L. and Chern, M.-S., 2000. Two-machine flowshop group scheduling problem. Computers and Opera-tions Research, 27, 975–985.

Yang, W.H. and Liao, C.-J., 1996. Group scheduling on twocells with inter-cell movement. Computers and OperationsResearch, 23, 997–1006.

Yoshida, T. and Hitomi, K., 1979. Optimal two-stageproduction scheduling with setup times separated. AIIETransaction, 11, 261–263.

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