sequence-dependent group scheduling problems in flexible flow shops

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Int. J. Production Economics 102 (2006) 66–86

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Sequence-dependent group scheduling problemsin flexible flow shops

Rasaratnam Logendran�, Paula deSzoeke, Faith Barnard

Department of Industrial & Manufacturing Engineering, Oregon State University, 118 Covell Hall, Corvallis, OR 97331-2407, USA

Received 1 June 2004; accepted 1 February 2005

Available online 29 April 2005

Abstract

Group scheduling within the context of sequence dependent setup times in flexible flow shops is considered in this

paper. Flexible flow shops are becoming very popular in industry practice, primarily because of large workloads

required by jobs within groups on some machine types. The objective is to minimize the makespan required to process

jobs in all groups released on the shop floor. There is clearly a need for efficiently solving large problems that have

industrial merit. To address this need, three different algorithms based on tabu search are developed. Problem sizes

ranging in size from small, medium to large are considered along with three levels of flexibility. The higher the number

of stages and the number of parallel machines in each stage, the higher is the flexibility introduced into the problem.

Three different initial solution (IS) finding mechanisms with varying levels of computational difficulty are proposed to

aid the search algorithms in identifying an IS. Thus, problem size is regarded as the main factor, while flexibility, IS

finding mechanism, and algorithms are considered subplot factors. The makespan which speaks for efficacy and

computation time which speaks for efficiency of the algorithms are considered separately as response variables in the

proposed 34 factorial split-plot design used in the detailed statistical experiment. Based on the results, the search

algorithm that uses short term memory is recommended for problems of all sizes and levels of flexibility. Also, as IS

finding mechanisms are found statistically insignificant with respect to both makespan and computation time, the

mechanism which requires the least amount of computation time is recommended.

r 2005 Elsevier B.V. All rights reserved.

Keywords: Sequence-dependent group scheduling; Flexible flow shops; Heuristics; Split-plot design

1. Introduction

Cellular manufacturing has been viewed as an important manufacturing philosophy that has contri-buted to improving productivity. Notable among these contributions are savings in setup time and

e front matter r 2005 Elsevier B.V. All rights reserved.

e.2005.02.006

ng author. Tel.: +1541 737 5239; fax: +1 541 737 5241.

ss: [email protected] (R. Logendran).

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R. Logendran et al. / Int. J. Production Economics 102 (2006) 66–86 67

work-in-process inventories, simplified flow of parts, centralization of responsibility, and improvedhuman relations (Hyer and Wemmerlov, 1989; Suresh and Kay, 1998). Ideally, when manufacturingcells are setup, parts assigned to a cell would be processed completely on machines assigned to the same cell.But in most industrial-size problems a non-ideal situation prevails, in that one or more parts assignedto a cell would require one or more operations to be performed on machines assigned to a cell otherthan its home cell. Such parts are called bottleneck (exceptional) parts in the literature, and machinesthat cater to their needs are called bottleneck machines. In the long run, it might be advantageousto consider duplicating one or more bottleneck machines in the parts’ home cell or subcontract thebottleneck parts, thus creating disaggregated manufacturing cells (Logendran and Sirikrai, 2000). If perfectdisaggregation were achieved, the operations required of parts that belong to a group (part family)would be completely processed on machines assigned to the same cell. What follows is called the groupscheduling problem, and the objective is to identify the sequence of parts that belong to each group as wellas the sequence of groups themselves to optimize some measure of performance. Thus, unlike inconventional scheduling problems, the issues in group scheduling must be dealt with at two levels. At level1, the sequence of parts that belong to each group is determined, and at level 2, the sequence of groupsthemselves is determined.

The existing literature on scheduling groups of jobs with setup times on single or parallel machines wasreviewed by Liaee and Emmons (1997). They consider scheduling with and without the group technologyassumptions (GTA) under a variety of performance measures. In the former, the jobs in the same groupmust be scheduled contiguously, while in the latter the jobs in the same group need not be scheduledcontiguously. Scheduling with GTA requires that jobs in the same group not to be split into sublots.Consequently, the number of sublots in each job is one, and it is logical to schedule the jobs in the samegroup together. Scheduling without the GTA assumes that jobs in a group can be split into sublots, andtherefore two interrelated decisions concerning the number and size of each sublot would have to be made(Cheng et al., 2000). A setup is incurred on a machine when there is a changeover from processing a sublotof one group to a sublot of another group. Appropriately, scheduling with GTA uses group setup times andthat without GTA uses batch setup times, and is investigated either as a sequence-independent or sequence-dependent setup time problem. The research presented in this paper is on flexible flow shop groupscheduling and, as such, we focus our review on flow shop job scheduling, flow shop group scheduling, andflexible flow shop job scheduling to better position our contribution.

As the jobs that belong to a group are similar, the sequence of operations required of them would be thesame. Consequently, they would adhere to a flow-line arrangement like in conventional flow shops. Forminimizing the makespan in level 1 problem, heuristic methods based on optimal and heuristic decisionrules for solving conventional flow shop scheduling problems have been proposed (Radharamanan, 1986;Al-Qattan, 1988; Logendran and Nudtasomboon, 1991). The single-machine group-scheduling problemwithin the context of minimizing the total tardiness was investigated by Nakamura et al. (1978) and Ozdenet al. (1985). Baker (1988) reported a dynamic programming algorithm for identifying an optimal sequenceof parts that minimizes the mean completion time. Webster and Baker (1995) review the research findings ingroup scheduling with regard to minimizing the total weighted flow time and the maximum lateness,although their review extends the results to cover the areas of group scheduling with batch availability andbatch processing. For job scheduling, Sule (1982) and Proust et al. (1991) have considered the two-machineflow shop scheduling problems with (sequence-independent) setup and removal times separated. Gupta andDarrow (1985, 1986) recognize that even the two machine sequence-dependent makespan minimization job(not group)-scheduling problem is NP-hard, and propose approximate algorithms to solve the problem.

Ham et al. (1985) were the first to propose a polynomial-time algorithm for minimizing the makespan ina two-machine group-scheduling problem with sequence-independent setups. The performance ofcombining a single-pass heuristic by Petrov (PT) (Petrov, 1966) and a multiple-pass heuristic by Campbellet al. (CDS) (1970) for completely solving an m-machine group scheduling problem to minimize the

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R. Logendran et al. / Int. J. Production Economics 102 (2006) 66–8668

makespan was reported by Allison (1990). Using the encouraging performance of the new heuristicalgorithm in (Logendran and Nudtasomboon (LN) (1991) for solving the level 1 problem, Logendran et al.(1995) carried out an investigation to determine how well LN performed in comparison with CDS, if each iscombined with PT to completely solve an m-machine group scheduling problem to minimize the makespan.Several heuristics as well as a branch and bound approach to solve the sequence-dependent flow shop groupscheduling problem have been proposed by Schaller et al. (2000). Reddy and Narendran (2003) usedsimulation experiments to investigate the sequence-dependent group scheduling problems that includeddynamic conditions such as non-availability of all jobs at the beginning of the planning horizon.

The above investigations assume that there exists only a unit of a machine in each stage of a groupscheduling problem. This research focuses on a different class of group scheduling problems called theflexible flow shops. In industry situations, if the workload required of parts assigned to the various groupson a machine type (i.e., stage) is high, it is customary to assign two or more identical units of the machine torepresent that stage, thus resulting in a flexible flow shop. In other words, a flow shop is considered flexibleif it consists of two or more units of a machine type in one or more stages of the multi-stage (m-stage)flexible flow shop problem. Investigations on flexible flow shop scheduling with respect to job scheduling

have been reported in the past. Salvador (1973) first investigated a flexible flow shop problem with no in-process buffers for minimizing the makespan using a branch-and-bound based approach. Brah andHunsucker (1991) later developed a branch-and-bound algorithm for the general flow shop with parallelmachines, known as hybrid flow shops, which included in-process buffers. However, their mixed-integerprogramming formulation turned out to be impractical for solving large problem instances. Sriskandarajahand Sethi (1989) developed heuristics with worst-case performance guarantees for minimizing the makespanin a two-stage flexible flow line. Ding and Kittichartphayak (1994) extended the heuristics for regular flowshop scheduling problems to develop three different heuristics for minimizing the makespan in a flexibleflow line.

A flexible flow line with limited in process buffers was considered by Wittrock (1998) for minimizing bothprimary (makespan) and secondary (work-in-process inventory) objectives. Sawik developed heuristics forminimizing the makespan, namely the route idle time minimization (RITM) for limited intermediate buffers(Sawik, 1993) and RITM-NS (no store) for no intermediate buffers (Sawik, 1995). Lee and Vairaktarakis(1994) recognized that the problem of minimizing the makespan in the multi-stage hybrid flow shop is NP-hard and developed heuristics. Sawik (2000) proposed mixed integer programming formulations forminimizing the makespan in a flexible flow line with limited intermediate buffers to identify an exactsolution using a commercially available software package such as CPLEX. These formulations were laterextended for batch scheduling (Sawik, 2002). An experimental investigation was reported by Kurz andAskin (2003) to compare sequencing rules for minimizing the makespan in flexible flow lines with non-anticipatory sequence-dependent setup time for jobs on machines.

To the best of the authors’ knowledge the paper by Logendran et al. (2005) is the only one that hasattempted to minimize the makespan in group scheduling flexible flow shops with sequence-independent

setups. This research focuses on the development and performance of a heuristic algorithm based on tabusearch for minimizing the makepsan in group scheduling flexible flow shops with sequence-dependent setups.Both sequence-independent and sequence-dependent environments are worthy of consideration, dependingupon the problem investigated. Suppose that the interest is to schedule N groups, denoted byG1;G2;G3; . . . ;GN , on an m-stage flexible flow shop. In sequence-independent setups, the estimated setuptime for any one of the N groups on the machine representing each of the m-stages is assumed independentof the group preceding it in the sequence. Contrastingly, in sequence-dependent setups, the estimated setuptime is strongly influenced by the group preceding the one under consideration, thus resulting in differentsetup times on the same machine.

The remainder of the paper is organized as follows. In the next section, the characteristics of the problemare described. In Section 3 we describe the solution algorithm. Section 4 demonstrates the application of the

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algorithm by way of an example. The factors affecting the problem and experimental design are describedin Section 5. The statistical linear model developed and the results obtained are described in Sections 6 and7, respectively. Finally, in Section 8, we present the conclusions of the research.

2. Problem characteristics

In a conventional flow shop group scheduling problem, each stage has only one machine. Thus, whengroups of jobs complete their processing requirements on a machine representing a preceding stagethey will need to wait their turn to get their processing requirements completed on the machine representingthe next stage. In contrast, one or more stages in a flexible flow shop group scheduling problem canhave two or more machines in that stage. Therefore, in a flexible flow shop jobs that belong to twodifferent groups may be processed simultaneously, yet separately, on two different machines representingthat stage, thus offering more flexibility. We assume that only a single-setup is performed on a machinefor processing the jobs that belong to a group. That is, when sequence-dependent setup is performedon a machine, the entire set of jobs that belong to the group are processed on the same machine andmoved to the next stage. Also, the jobs that belong to the various groups are assumed to be availablefor processing on machines at the beginning of scheduling. In addition to the development of anefficient solution algorithm that is capable of solving this complex problem, several importantresearch questions emerge at this point: (1) In the construct of a problem in flexible flow shops, whatare the important factors to consider? (2) How does one decide on suitable levels for the factors chosen? (3)Are the factors used in the construct of a problem statistically significant? (4) Are the factor interactionsstatistically significant? and (5) Given a problem defined by the level of each factor, which algorithmictechnique shows a statistically significantly better makespan that a practitioner can benefit from using?These issues are addressed comprehensively in this paper. In Section 3, we present the motivation for andconstruct of the solution algorithm.

3. Solution alogorithm

The proposed solution algorithm is based on tabu search for efficiently solving large problemsencountered in industry practice. Tabu search has been found to be remarkably efficient for solving a widevariety of hard combinatorial problems. It has the ability to overcome the limitations of local optimality bysuperimposing it on any method that can be characterized as transforming one solution into anotherthrough a sequence of moves (Glover, 1986). The attractiveness of each move is evaluated by using anevaluation function. The ability to cross boundaries of local optimality is inherited by systematicallyimposing and releasing constraints to permit exploration to the otherwise forbidden regions. Also, memoryfunctions of varying time spans are used to intensify and diversify the search into new regions. The short-term memory or tabu list size is used for the former, while long-term memory (LTM) for the latter. Theconcept behind tabu search is presented in detail in a series of papers (Glover, 1986, 1989, 1990a, b). Withregard to scheduling, it has been proven to be a remarkably efficient approach, including that of FMSscheduling problems (Logendran and Sonthinen, 1997) and unrelated-parallel machine scheduling problemwith job-splitting (Logendran and Subur, 2004).

The algorithm consists of two levels of search, outside search (ORS) and inside search. The ORS consistsof searching for the best sequence of groups in the level 2 problem. Given a sequence of groups, the insidesearch focuses on finding the sequence of jobs in each group (i.e., for the level 1 problem). The final solutionis composed of the best group sequence together with its best job sequence that gives the minimummakespan for the sequence-dependent flexible flow shop group scheduling problem.

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3.1. Initial solution finding mechanisms

In order for the search to begin, the solution algorithm must be provided with an initialsolution (IS). Previous research on tabu search (Logendran and Subur, 2004) has alluded to the possibilityof finding a better final/best solution fairly efficiently, if the quality of the IS is deemed better. Weinvestigate this notion in our research by developing three different IS-finding mechanisms, as describedbelow.

3.1.1. Initial solution 1 (IS1)

IS1 is motivated by its simplicity. Observe that the simplest ordering for jobs within each group as well asthe ordering for groups can be established by following the numerical ordering. That is, the group sequencein IS1 can be represented as: G1 � G2 � . . .� GN . The jobs within each group can be represented as:J112J122 . . .2J1n1 for G1, J21 � J22 � . . .2J2n2 for G2; . . . ; and JN1 � JN2 � . . .2JNnN

for GN , wheren1; n2; . . . ; nN are the number of jobs in groups G1;G2; . . . ;GN , respectively.


IS2 keeps the creation of the group sequence simple, meaning that the ordering for groups is stillestablished using the numerical ordering. However, for identifying the sequence of jobs in each group, wedraw from previous research in the identical parallel machines problem for minimizing the makespan. Inthe identical parallel machines problem there is only one stage, and there is no known polynomial timealgorithm to find a provably optimal sequence for minimizing the makespan. However, the longestprocessing time (LPT) heuristic is regarded as an effective algorithm for minimizing the makespan withknown worst-case-bound (Pinedo, 1995). We use this insight in the development of the job sequence in IS2.As a flexible flow shop consists of multiple stages, we first evaluate the cumulative run time (CRT) for eachjob in a group on all m-stages by adding its run time in the respective stages. Following this, the jobsequence for jobs in a group is determined based on the longest cumulative run time (LCRT), meaning thejob having the highest LCRT will be sequenced first. Should there be a tie for two jobs, it is broken in favorof the smallest job number.


IS3 is intended to characterize the advantage of using LPT in the IS found at both levels—jobs within agroup as well as groups. To initiate the process, a key stage (machine) is identified as that whichrequires the longest cumulative processing time (LCPT) for all groups. As this is a sequence-dependentenvironment, we introduce the notion of minimum setup, and use it to evaluate LCPT. Realistically,going into the current planning horizon, the machines would have a setup corresponding to the lastgroup that was processed in the previous planning horizon. This group is referred to as the referencegroup (R). Each of the N groups in the current planning horizon will need at least the minimum setuptime on each of the m-stages of the flexible flow shop, evaluated as the minimum of all the sequencedependent setup times needed to changeover from the remaining (N�1) groups and the reference group (R).A group would use the sequence dependent setup time corresponding to R, should it be processed as thefirst group in the sequence. Thus the minimum setup time based cumulative processing time (CPT) for allgroups on a machine (stage) is evaluated as the sum of the minimum setup time for each group on that stageand the run times for jobs in all groups. Should there be a tie in the determination of LCPT, it is broken infavor of the smallest machine number. Once the key machine is found, the jobs within each group aresequenced based upon the longest run time (LRT) on the key machine, breaking ties in favor of the smallestjob number. The group sequence is then identified based upon LCPT on the key machine for each group,which is the sum of the run time for jobs and the minimum setup time. Ties are broken in favor of thesmallest group number, if necessary.

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3.2. Algorithmic structure

The IS obtained from applying any of the three IS finding mechanisms above abides by theoperational constraints of the original problem. Thus, it is a feasible solution. For the ORS, theneighborhood for the group sequence in IS is obtained by performing pair-wise exchanges of any twoadjacent groups, including the cyclical exchange. The cyclical exchange is performed by exchanging the firstand the last group in the IS, while maintaining the same positions for the remaining groups. Each newgroup sequence in the neighborhood is thus obtained by performing what is typically called the outsideperturbation on the group sequence in IS. The neighborhood for the job sequence in the inside search isobtained in a similar fashion by performing pair-wise exchanges of any two adjacent jobs including thecyclical exchange. To keep it simple, we limit the inside search to namely partitioned perturbations. It meansthat the job perturbations in one group are not considered in conjunction with jobs already perturbed inanother group.

The algorithm starts by completely evaluating the neighborhoods of the group sequence in IS. The insidesearch is then invoked for the group sequence in IS to determine its best/near optimal solution. The bestsolution is evaluated as that which results in the smallest makespan for the problem. The job sequence inthe IS is inserted into the inside candidate list (ICL) for inside search as its first entry. The ICL thus consistsof the configurations of job sequences for perturbations in the future. The partitioned perturbations areperformed on jobs within each group and the makepsan of each is evaluated. The job sequence with thesmallest makespan is inserted into the ICL as its next entry. Should there be a tie, it is broken in favor of thefirst best solution. The pair of jobs that were perturbed to generate this entry into the ICL is regarded astabu, meaning that in the next iteration perturbations of the same pair would not be considered. If themakespan for this solution is better than that for the IS, it is marked with a star (*) to indicate that it hasthe potential for becoming the next local optimum. It will become the next local optimum should thesolution identified by using it as a parent resulted in either equal or inferior makespan. If such were the case,this solution (2nd entry into the ICL) is marked with double stars (**), and inserted into the inside index list(IIL) as the first local optimum. To enhance the computational efficiency of the algorithm, the terminationof the inside search is governed by two parameters: IIL_size and the limit on the number of non-improvingmoves in inside search (IIT_limit). Preliminary experimentation is performed to determine appropriatesettings for these parameters. Based on such experimentation, the following equations are developed toevaluate such parameters and the inside tabu list size (ITL_size) for small, medium, and large probleminstances. The classification of these problems is provided in Section 5.

In the following empirical equations:N is the number of groups; NS the number of stages; SumJobs the sum of the jobs per group, and

SumMach the sum of the machines per stage. Note that SumJobs is computed as the total number of jobs inall groups divided by the number of groups, and SumMach is computed as the total number of machines inall stages divided by the number of stages.

Small Problems

ITL_size ¼ SumJobs=ð10 �NSÞ

IIL_size ¼ SumJobs^1:6=ðN^1:6 � ð2 �NSÞÞ

IIT_limit ¼ ðSumJobsþ SumMachþN þNSÞ=11:3

Medium Problems

ITL_size ¼ SumJobs=ð10 �NSÞ

IIL_size ¼ ð13 � SumJobsÞ=ð6 �N �NSÞÞ

IIT_limit ¼ ð0:3 � SumJobsþNÞ=ð0:25 � SumMachþNSÞ

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Large Problems

ITL_Size ¼ SumJobs=ð10 �NSÞ

IIL_size ¼ ðN � SumJobs^0:5Þ=ð2 � ðNS þ SumMachÞÞÞ

IIT_limit ¼ ððSumJobs � 0:5Þ þNÞ=ðð0:5 � SumMachÞ þ 1:5 �NSÞ

Regular rounding is used with ITL_size and IIT_limit, meaning that a value with fractional part less than0.5 is rounded down, while that with greater or equal to 0.5 is rounded up. For IIL_size, all fractionalvalues are rounded up to the next integer value.

The group sequence along with its job sequence is admitted into the outside candidate list (OCL) as itsfirst entry. When the inside search for IS is terminated, the next new group sequence in its outsideneighborhood is considered for perturbation. To maintain consistency in the application of the algorithm,the job sequence for this group sequence is identified by applying the same IS finding mechanism (IS1, IS2,or IS3) used for finding the IS described above. The inside search is applied to each of the new groupsequences identified in the outside neighborhood and the best solution corresponding to each is determined.Using the best-first strategy for breaking ties, the one that corresponds to the smallest makespan is admittedinto the OCL as its 2nd entry. As with the inside search, the pair of groups that were perturbed to obtainthis solution are duly regarded as tabu, and in the next iteration the same pair is not considered forperturbation. Also, to terminate the ORS, both the outside index list size (OIL_size) and the limit on thenumber of non-improving moves in ORS (OIT_limit) are used. As for the inside search, preliminaryexperimentation led to identifying the following empirical equations for the parameters for terminating thesearch as well as outside tabu list size (OTL_size).

Small Problems

OTL_size ¼ SumJobs=ðN þ SumMachþNS^1:3Þ

OIL_size ¼ SumJobs=ð0:9 � ðN þNSÞÞ

OIT_limit ¼ N^3=ððSumMach � 1:4Þ þ ðNS � 2:3Þ þ SumJobs^1:12Þ

Medium Problems

OTL_size ¼ SumJobs=ðN þ SumMach^1:5þNS^1:3Þ

OIL_size ¼ 0:9 � SumJobs=ð0:3 �N �NSÞ

OIT_limit ¼ ð45 �NÞ=ð1:5 � SumMachþ 1:5 �NS þ 0:5 � SumJobsÞ

Large Problems

OTL_Size ¼ SumJobs=ðN þ SumMach^1:5þNS^1:3Þ

OIL_size ¼ SumJobs=ð1:5 �N þNSÞ

OIT_limit ¼ 5 �N^2=ððSumMach � 1:4Þ þ ðNS � 2:3Þ þ SumJobs^1:12Þ

As in the inside search, regular rounding is used with OTL_size and OIT_limit. For OIL_size, allfractional values are rounded up to the next integer value.

With LTM, the ORS is restarted to explore newer and more promising regions in the search space usingeither the LTM based on maximal frequency (LTM-MAX) or LTM based on minimal frequency (LTM-MIN). This is applied to both inside and outside searches simultaneously. For inside search, the frequencyis evaluated by using a frequency matrix to keep track of the tenure of the position of each job in thecandidate list of solutions throughout the search. Should there be a tie in the application of maximal orminimal frequency based searches, the row-wise first best strategy is used to break ties. For ORS, afrequency matrix is used to keep track of the tenure of the position of each group in the candidate list ofsolutions throughout the search. Again, the row-wise first best strategy is used to break ties in the frequency

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matrix. The number of restarts for the ORS is assumed equal to 2, and this has been found to be effectivefor a flexible manufacturing system’s scheduling problem (Logendran and Sonthinen, 1997). When the limiton OIL_size or OIT-limit is reached without LTM (i.e., short-term memory) or on ORS with LTM, theentire search is terminated. We refer to the three algorithms with short-term memory, LTM-MIN, andLTM-MAX as Algorithms 1, 2, and 3, respectively. The flow chart for the algorithms is presented in Fig. 1.In addition, a pseudo-code is provided in Appendix to better describe the steps associated with thealgorithm.

4. Application of the algorithm—an example

We consider a flexible flow shop with four groups and four stages to demonstrate the applicationof the algorithm. To maintain adequate flexibility, two of the four stages (stages 2 and 4) areassumed to have identical-parallel machines. The first stage consists of one unit of machine type 1 ðM11Þ,second stage two units of machine type 2 (M21 and M22), third stage one unit of machine type 3 ðM31Þ, andfinally the fourth stage consists of three units of machine type 4 (M41;M42, and M43). The sequence-dependent group setup times on the four machines are shown in Tables 1–4, respectively, while Table 5presents the run times for jobs in each group on each of the four machine types. As noted before, R in thesetables refer to the reference group, which is the last group processed on the machines in the previousplanning horizon. For example, in Table 1, the sequence dependent setup time to changeover from R toG1;G2;G3, or G4 on M1 is 2, 1, 3, 6, respectively. The time realized would depend on the first group in thesequence. For instance, if the sequence is G2 � G3 � G4 � G1, the changeover is from R to G2, for asequence dependent setup time of 1. In Table 5, the run times, rik, for job k in group i are presented on eachof the four machines. For instance, the run times of job 2 in group 1 are 16, 15, 6, and 10, on M1;M2;M3

and M4, respectively.The identification of the IS is illustrated with IS3, as it is the most involved in being able to characterize

the use of LPT at both levels. The minimum setup time for each of the four machines (M1;M2;M3; andM4) is first identified. For example, the minimum setup time for G1 on M1 is evaluated as the minimum ofthe four entries (i.e., 2, 5, 8, 6) in the first column of Table 1, which is 2. Extending the same reasoning forthe other groups would result in the minimum setup time for G1;G2;G3, and G4 on M1 as 2, 1, 1, and 1,respectively. Thus, the sum of the minimum setup time for all four groups on M1 is 5. The CPT for M1

based on the minimum setup is 178 (5+173, where 173 is the sum of the run times for all four groups onM1). The CPT on the remaining machines M2;M3; and M4, can be evaluated as 179, 88, and 298,respectively. The key machine (stage) is M4, as it has the longest CPT of 298 (i.e., max (178, 179, 88, 298)).The LRT on M4 is used to identify the job sequence in each group to result in the following sequences:G1 : r11 � r12;G2 : r23 � r21 � r22;G3 : r32 � r31;G4 : r43 � r42 � r41. To identify the group sequence withIS3, the CPT for G1;G2;G3, and G4 on M4 based on the minimum setup time is identified as 22, 163, 54,and 59, respectively. The group sequence is thus G2 � G4 � G3 � G1. The complete sequence for theIS using IS3 is G2ðr23 � r21 � r22Þ � G4ðr43 � r42 � r41Þ � G3ðr32 � r31Þ � G1ðr11 � r12Þ, and the makespanis 224.

The application of the algorithm below is demonstrated by using the short-term memory. Similarreasoning can be extended to cover the LTM with minimum (LTM-MIN) and maximum (LTM-MAX)frequencies. For the initial group sequence, the neighborhood for the ORS consists of the following groupsequences: ðG4 � G2 � G3 � G1Þ; ðG2 � G3 � G4 � G1Þ; ðG2 � G4 � G1 � G3Þ; ðG1 � G4 � G3 � G2Þ. First,the inside search is performed on the job sequence of the IS to produce the following entries in the CLand IL shown in Table 6. As two non-improvement moves (IIT-limit) have been made, the inside search forthe IS is terminated and the search is switched to the outside. The next group sequence (i.e., G4 � G2 �

G3 � G1Þ in the neighborhood is chosen and the inside search is performed. The search is continued in a

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Fig. 1. Flow chart for the algorithms.


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Table 1

Setup times on M1

G1 G2 G3 G4

R 2 1 3 6

G1 — 2 6 1

G2 5 — 3 1

G3 8 7 — 3

G4 6 2 1 —

Table 2

Setup times on M2

G1 G2 G3 G4

R 6 2 5 7

G1 — 8 4 1

G2 4 — 6 3

G3 7 6 — 1

G4 4 3 2 —

Table 3

Setup times on M3

G1 G2 G3 G4

R 2 2 7 4

G1 — 3 6 8

G2 6 — 2 4

G3 5 5 — 3

G4 2 3 8 —

Table 4

Setup times on M4

G1 G2 G3 G4

R 2 5 3 6

G1 — 8 8 6

G2 6 — 1 1

G3 7 3 — 5

G4 4 8 4 —


similar fashion to produce the best solution for each of the group sequences in the neighborhood of the ISas shown in Table 7.

As the best among the best makespan in the neighborhood is 217, the second entry in Table 7 is chosenas the CL entry, representing the next complete solution (i.e., group/job sequence) for the ORS. Table 8shows the results obtained for the ORS, starting with the IS found from using IS3. Notice that as two

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Table 5

Run times for jobs in each group

Machine G1 G2 G3 G4

r11 r12 r21 r22 r23 r31 r32 r41 r42 r43

M1 18 16 15 18 19 15 19 20 16 17

M2 13 15 15 17 19 25 15 15 17 19

M3 10 6 7 10 8 6 8 9 10 5

M4 10 10 50 50 60 25 28 17 20 21

Table 6

Entries into the CL and IL for the job sequence in the initial solution

CL entry/job sequence Makespan/IL**

G2ðr23 � r21 � r22Þ � G4ðr43 � r42 � r41Þ �G3ðr32 � r31Þ � G1ðr11 � r12Þ 224

G2ðr23 � r21 � r22Þ � G4ðr42 � r43 � r41Þ �G3ðr32 � r31Þ � G1ðr11 � r12Þ 219**



Table 7

Best makespan in the neighborhood of the initial solution

Group/job sequence Best makespan


G2ðr23 � r21 � r22Þ � G3ðr32 � r31Þ �G4ðr43 � r42 � r41Þ � G1ðr11 � r12Þ 217


G1ðr11 � r12Þ �G4ðr43 � r42 � r41Þ �G3ðr31 � r32Þ � G2ðr21 � r23 � r22Þ 328

Table 8

Best entries/complete sequences for the outside search with IS3



G2ðr23 � r21 � r22Þ � G3ðr32 � r31Þ �G4ðr43 � r42 � r41Þ � G1ðr11 � r12Þ 217

G3ðr31 � r32Þ �G2ðr21 � r23 � r22Þ �G4ðr43 � r42 � r41Þ � G1ðr11 � r12Þ 236

G1ðr11 � r12Þ �G2ðr21 � r23 � r22Þ �G4ðr43 � r42 � r41Þ � G3ðr31 � r32Þ 237


non-improvement moves (236 and 237) (OIT-limit) have been found after the evaluation of 217 for themakespan, the entire search is terminated. The best makespan determined with IS3 is thus 217.

Tables 9 and 10 present the best entries obtained for the ORS with IS1 and IS2, respectively. Theidentification of two non-improvement moves has resulted in the termination of search in both cases. FromTables 9 and 10, the best solution for the original problem is 235, which is inferior to the best solutionfound with IS3.

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Table 9




G2ðr21 � r22 � r23Þ � G1ðr11 � r12Þ �G3ðr31 � r32Þ � G4ðr43 � r42 � r41Þ 235


Table 10







5. Factors affecting the problem and experimental design

Either completely randomized design or split-plot design can be used in the development of the factorialdesign. The drawback of using complete randomization is that if a lesser makespan is evaluated for aheuristic, it cannot be attributed entirely to the performance of that heuristic as the decision may have beenpartly influenced by the differences between the structure of the problem instances. Thus, a split-plot designis selected in order to perform an objective comparison of the performance of the algorithms (Montgomery,2001). Problem size is an important factor, as larger the problem, the greater the computational effortneeded to completely solve it using an algorithmic technique. The problem size in a flow shop groupscheduling problem is described by three parameters: number of machines, number of groups, and numberof jobs (Logendran et al., 1995). In a flexible flow shop, the representative parameter for number ofmachines is the number of stages, and each stage in turn can have two or more identical parallel machines.Based on previous research (Logendran et al., 1995), three different problems sizes, namely small, medium,and large are selected. Thus, problem size is regarded as the main factor with three levels, namely small,medium, and large, in the experimental design.

The ranges chosen for each parameter representing each problem size are as follows. The number ofstages is varied from 2 to 3, 4 to 6, and 7 to 9, for small, medium, and large problem instances, respectively.The number of groups is varied from 3 to 5, 6 to 9, and 10 to 12, for small, medium, and large probleminstances, while that for the number of jobs is varied from 2 to 5, 6 to 9, and 10 to 12, respectively, for thethree sizes. All of these parameters as well as the run times and setup times described below are generatedusing the ‘rand’ function in combination with ‘srand’ function in C-programming. Srand takes an unsignedinteger as an argument and sets the seed used for generating random numbers. Rand uses this seed as itsstarting point and returns a uniformly distributed random integer between a specified upper and lower limit,noted above for all of the parameters. In the implementation of the algorithm, we chose the current time asan argument for srand. This made the results different each time the program is run, thus ensuringrandomness.

The run times of jobs in each group are varied from 5 to 75, while that for the setup times of groups arevaried from 5 to 25. Care was exercised to guarantee producing a setup time for each group that is different

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for every previous group, thus ensuring that the setup times were indeed sequence dependent. In groupscheduling problems all of the jobs in a group may not use all of the machines considered. This feature isreferred to as machine skipping in the literature. A probability of 0.2 is assigned to describe this feature,based on uniformly distributed random numbers generated between 0 and 1. That is, if the random numbergenerated is less than 0.8, the job is assumed to have a positive run time on that machine, and zerootherwise. If a job were found to have a positive run time, it is generated from an integer uniformdistribution between 5 and 75, as noted above.

The IS finding mechanism, flexibility, and algorithm used for solving the problem are considered factorsthat are of significant importance in this research. They are thus assigned to the subplot and regarded assubplot factors in a split-plot design. The algorithms are described in Sections 3 and 4. In contrast to a flowshop scheduling problem where there is no flexibility, the flexibility in a flexible flow shop is introduced bythe number of identical parallel machines in each stage. The larger the number of stages in a flexible flowshop and larger the number of parallel machines in each stage, the greater the flexibility introduced in theshop. This concept is used to define flexibility in this research, and three levels of flexibility, namely low,medium, and large, are used.

Our preliminary investigations led to determining low flexibility as that being represented by 1/3 of stageshaving parallel machines, medium flexibility by 2/3 of stages having parallel machines, and high flexibilityby all stages being parallel. Should a real value be evaluated for the number of parallel stages, the roundedup integer value is used. For instance, for medium size, medium flexibility, the number of parallel stages is¼ (2/3)�(5) ¼ 3.33. Thus, a rounded up value of 4 parallel stages would be used. Following the evaluationof the number of stages having parallel machines in a problem instance, it would be necessary to determinethe stages that would have the parallel machines. Uniformly distributed integer values are generated for thispurpose, and if the same integer value is generated for the second time it is discarded and a new value isgenerated and used. For example, in the 4-parallel stages case in a 5-stage problem, 4-2-3-5 is an acceptableorder of generating the random numbers representing which stages will have parallel machines, but not 4-2-2-5. To keep the problem instance manageable, if a stage is determined to be flexible (i.e., known to haveparallel machines), the number of machines representing that stage is assumed to be either 2 or 3 with equalprobability of 0.5. Each of the three subplot factors (IS finding mechanism, flexibility, and algorithm) hasthree levels. Finally, to perform a comprehensive analysis, 5 different replicates (blocks) are used.

6. The model

The model we formulate for performing the split-plot design is a linear model comprised of 5 factors,including one for replicates (i). The four remaining factors are problem size (j), IS finding mechanism (k),flexibility (l), and algorithm (m). We investigate both the effectiveness and the efficiency of the algorithmicapproaches and thus two response variables are considered: yijklm for makespan and zijklm for computationtime. The computation time is considered as a second response variable because it can be significant insolving industrial-size problems. The linear models for this design are:

yijklm or zijklm ¼ mþ ti þ bj þ �1 þ gk þ dl þ Zm þ ðbdÞjl þ ðbgÞjk þ ðbZÞjm þ ðgdÞkl þ ðgZÞkm

þ ðdZÞlm þ ðbgdÞjkl þ ðbdZÞjlm þ ðbgZÞjkm þ ðgdZÞklm þ ðbgdZÞjklm þ �2,

where m, is the overall mean; ti, the treatment effect (replicate); bj , the treatment effect (problem size); gk,the treatment effect (IS finding mechanism); dl , the treatment effect (flexibility); Zm, the treatment effect(algorithm), and other terms represent two-, three- and four-factor interactions. �1 and �2 are whole-plotand sub-plot errors, respectively.

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7. Experimental results

All three algorithms are programmed in Microsoft C and implemented on a server with 2.4GHz dualprocessor, 4GB RAM memory, and Windows 2000 operating system. As 5 replicates are solved with eachcombination of main and subplot factors, a total of 405 (34�5) test problems are considered and theobserved value of makespan and computation time (in seconds) are recorded. Following this, the programsnecessary for performing the experimental design are coded in Statistical Analysis System (SAS) (SASRelease, 1999–2001) and run. Figs. 2 and 3 show the normal probability plots of residuals for makespanand computation time, respectively. Although the plot for computation time is slightly more skewed at thetwo extremes than that for makepsan, for practical purposes, both can be assumed to abide by normalityassumptions. Thus, parametric statistics based on analysis of variance (ANOVA) is used to analyze theresults.

Computation Time125

100

75

50

0

−25

−50

−75

25

0.1 1 5 10 25 50

Normal Presentiles

75 90 95 99 99.9

Res

idua

ls

Fig. 3. Normal probability plot of residuals for computation time.

Makespan400

300

200

100

−100

−200

−300

−400

0

0.1 1 5 10 25 50 75 90 95 99 99.9

Res

idua

ls

Normal Presentiles

Fig. 2. Normal probability plot of residuals for makespan.

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Table 11

Results from ANOVA for makespan

Source Degree of freedom Sum of squares Mean square F Value Pr4F

Model 80 956,573,299 11,957,166.2 163.44 o.0001

Error 324 23,704,226 73,161.2

Corrected total 404 980,277,525

Source Degree of freedom Type I SS Mean square F Value Pr4F

Size 2 813,981,560 406,990,780.1 5562.93 o.0001

Flexibility 2 89,912,361 44,956,180.5 614.48 o.0001

Size�flexibility 4 52,609,391.2 13,152,347.8 179.77 o.0001

Is 2 146.5 73.3 0 0.999

Size�is 4 4995.4 1248.8 0.02 0.9994

Flexibility�is 4 4769.4 1192.3 0.02 0.9995

Size�flexibility�is 8 26,913.4 3364.2 0.05 1

Algorithm 2 15,914.8 7957.4 0.11 0.897

Size�algorithm 4 8410.4 2102.6 0.03 0.9984

Flexibility�algorithm 4 1227.8 306.9 0 1

Size�flexibility�algorithm 8 2645.8 330.7 0 1

is�algorithm 4 353.8 88.5 0 1

Size�is�algorithm 8 1374.6 171.8 0 1

Flexibility�is�algorithm 8 1189.9 148.7 0 1

Size�flexibility�is�algorithm 16 2044.8 127.8 0 1

is ¼ initial solution finding mechanism.


The summary of results obtained for makespan and computation time are presented in Tables 11 and 12,respectively. Overall, the statistical models for makespan and computation time are significant since theprobability to the right of the F-value evaluated is very low (o0.0001). We assume a significance level ðaÞ of5%, as is commonly done in experimental design. Observe that in the SAS outputs the effect of replicatesand main and subplot errors have been combined into one to give an error of 23,704,226 for makespan and234,530.8 for computation time with 324 degrees of freedom for both. For makespan, only factors that arestatistically significant are size and flexibility, and the interaction between them (size�flexibility).Interestingly, neither the IS finding mechanism nor the algorithm is found statistically significant.Although algorithm 2 (long term memory with minimum frequency) had a slight edge over algorithm 1(short term memory only) and algorithm 3 (long term memory with maximum frequency) in being able tofind a better makespan for large problem instances in particular, the numerical difference in makespanbetween them nevertheless is not adequate to produce a statistically significant difference to claimsuperiority. Thus, no further analysis with respect to slicing on any particular factor or multiplecomparisons is performed.

The results obtained for computation time as a response variable is in marked contrast to that obtainedfor makespan. The factors size, flexibility, algorithm, and the interaction terms size�flexibility andsize�algorithm are notably significant at a significance level of 5%. Again, the IS finding mechanism isfound to be insignificant. Although the three factor interaction size�flexibility�algorithm is not significantand each of the two factor interactions (size�flexibility and size�algorithm) that contribute to the makeupof the three factor (size�flexibility�algorithm) is significant, the test of effect slices for computation time isperformed for two and three factor interactions and the results are presented in Table 13. In particular, forlarge size problems the algorithms record a statistically significant difference as noted in the test of effectslices where the probability is substantially lower (0.0001) than the significance level of 5%. Motivated bythese observations the test for multiple comparisons is performed and the results obtained are summarized

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Table 12

Results from ANOVA for computation time

Source Degree of freedom Sum of squares Mean square F Value Pr4F

Model 80 851081.694 10638.521 14.7 o.0001

Error 324 234530.8 723.86

Corrected total 404 1085612.49

Source Degree of freedom Type I SS Mean square F Value Pr4F

Size 2 532093.886 266046.9432 367.54 o.0001

Flexibility 2 6531.7383 3265.8691 4.51 0.0117

Size�flexibility 4 11776.158 2944.0395 4.07 0.0031

Is 2 1615.9457 807.9728 1.12 0.3288

Size�is 4 3126.6617 781.6654 1.08 0.3664

Flexibility�is 4 335.0765 83.7691 0.12 0.9769

Size�flexibility�is 8 660.2272 82.5284 0.11 0.9987

Algorithm 2 99895.3531 49947.6765 69 o.0001

Size�algorithm 4 185589.032 46397.258 64.1 o.0001

Flexibility�algorithm 4 2188.2469 547.0617 0.76 0.5548

Size�flexibility�algorithm 8 3950.3457 493.7932 0.68 0.7073

is�algorithm 4 679.3284 169.8321 0.23 0.9188

Size�is�algorithm 8 1334.2864 166.7858 0.23 0.9851

Flexibility�is�algorithm 8 441.4716 55.184 0.08 0.9997

Size�flexibility�is�algorithm 16 863.9358 53.996 0.07 1

is ¼ initial solution finding mechanism.

Table 13

Test of effect slices for computation time

Effect Size Number of

degrees of

freedom

Denominator

degrees of

freedom

F Value Pr4F

Size�flexibility Small 2 312 0 1

Size�flexibility Medium 2 312 0.04 0.9613

Size�flexibility Large 2 312 27.71 o.0001

Size�algorithm Small 2 312 0 1

Size�algorithm Medium 2 312 0.27 0.7622

Size�algorithm Large 2 312 432.51 o.0001

Effect Size Flexibility Number of

degrees of

freedom

Denominator

degrees of

freedom

F Value Pr4F

Size�flexibility�algorithm Small Low 2 312 0 1

Size�flexibility�algorithm Small Medium 2 312 0 1

Size�flexibility�algorithm Small High 2 312 0 1

Size�flexibility�algorithm Medium Low 2 312 0.17 0.8463

Size�flexibility�algorithm Medium Medium 2 312 0.07 0.9309

Size�flexibility�algorithm Medium High 2 312 0.06 0.9447

Size�flexibility�algorithm Large Low 2 312 196.32 o.0001

Size�flexibility�algorithm Large Medium 2 312 148.52 o.0001

Size�flexibility�algorithm Large High 2 312 96.95 o.0001


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Table 14

Results from multiple comparisons for computation time

Size Flexibility Algorithm Estimate of

computation time

(s)

Significant

difference in

algorithm

performance?

Algorithm

comparisons

P Value

Small Low 1 0.00 No N/Aa N/A

2 0.00

3 0.00

Small Medium 1 0.00 No N/A N/A

2 0.00

3 0.00

Small High 1 0.00 No N/A N/A

2 0.00

3 0.00

Medium Low 1 0.40 No N/A N/A

2 4.20

3 2.73

Medium Medium 1 0.20 No N/A N/A

2 2.47

3 2.27

Medium High 1 0.13 No N/A N/A

2 2.13

3 2.00

Large Low 1 16.13 Yes 1–2 o.0001

2 136.07 1–3 o.0001

3 122.60 2–3 o.9574

Large Medium 1 12.73 Yes 1–2 o.0001

2 114.33 1–3 o.0001

3 108.87 2–3 o1.000

Large High 1 10.33 Yes 1–2 o.0001

2 95.93 1–3 o.0001

3 83.13 2–3 o.9763

aN/A ¼ not applicable.


in Table 14. Notice that as the algorithms are implemented on a dual 2.4GHz processor server with 4GBRAM, negligible computation times are recorded for small problem instances. The results reveal that forlarge size problems there is indeed a statistically significant difference in computation time at levels offlexibility. This difference is attributed to Algorithm 1 being significantly better than both Algorithms 2 and3. No significant difference between Algorithms 2 and 3 is observed. This, coupled with the fact that nostatistically significant differences are observed for makespan with problems of all sizes, enablesrecommending Algorithm 1 with only short term memory for solving problems of all sizes and levels offlexibility. As the IS finding mechanisms did not have any significant impact on makespan and computationtime, the use of IS finding mechanism 1, which is simplest requiring the least amount of computation time,is also recommended.

8. Conclusions

A sequence-dependent group scheduling problem in flexible flow shops is investigated in this paper.Three different algorithms based on tabu search have been developed to solve problems ranging in size

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from small, medium to large. In addition, three levels of flexibility are considered. In order to aid thesearch algorithms with a better IS, three different IS finding mechanisms that require varyinglevels of computational effort to find an IS are also considered. A detailed statistical experimentbased on split-plot design is performed to analyze both makespan and computation time as twoseparate response variables. The former speaks for the efficacy of the algorithms, while the latterfor the efficiency. Overall, the models representing makespan and computation time are foundsignificant. For makespan however, neither the algorithm nor the IS finding mechanism were foundsignificant for all sizes of problems. But problem size, flexibility and the interaction between thetwo are found significant. For computation time, problem size, flexibility, and algorithm, and the twofactor interaction between size and flexibility and size and algorithm are found significant. Further analysisreveals that for large size problems Algorithm 1 is computationally efficient than Algorithms 2 and 3 at alllevels of flexibility. Thus, Algorithm 1 is recommended for solving the proposed research problems.Further, as the IS finding mechanisms did not have a statistically significant impact on both makepsan andcomputation time, IS finding mechanism 1 that requires the least amount of computational effort is alsorecommended.

Acknowledgments

This research is funded in part by the National Science Foundation (USA) Research Experiences forUndergraduates (REU) Grant No. DMI-0010118. Their support is gratefully acknowledged.

Appendix. Pseudo-code for tabu search

�
Read problem inputs in from file—# stages (NS), # groups (N), # jobs per group, # machines per stage,run time for each job on each stage, set up times for each group on each stage, group sequence, jobsequence. � Find parameters–Outside index list size (OIL_size), Outside # iterations w/o improvements (OIT_limit),
Outside tabu list size (OTL_size), Inside index list size (IIL_size), Inside # iterations w/o improvements(IIT_limit), and Inside tabu list size (ITL_size).
� ORS—performs pair-wise exchanges on groups, investigating job sequences within each group search,
until stopping criteria met.J EVALUATE MAKESPAN time of initial sequence (IS)—insert into OCL, OIL and OTL.J If long term memory (LTM-MIN or LTM-MAX)—loop through designated restart amount

(2 restarts, 3 times through). If short-term memory, do not loop (0 restarts, 1 time through)� Loop while stopping criteria unmet� Cycle through groups 12N

J Exchange groups pair-wise including cyclical exchange (1 and N)J If sequence has been previously investigated, move on to next sequenceJ If exchange is tabu, remember sequenceJ If exchange is not tabu and sequence has not been investigated, perform INSIDE SEARCHJ Save the time, sequence and tabu parameters if better than previous best� If best time is not better than previous OCL entry, investigate sequences saved as tabu. Only save

as best if evaluated as better than previous OCL entry.� Using best time, sequence and tabu parameters, update OTL, OCL and OIL if necessary.

Increment OIT_limit or set to zero as necessary.� If LTM-MIN or LTM-MAX—update long term memory matrix

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� Close stopping criteria loop� If LTM-MIN or LTM-MAX—switch appropriate jobs

J Close restart loopJ Return best time and sequence
� INSIDE SEARCH—performs pair-wise exchanges on jobs within all groups, evaluating makespan time
of each new sequence, until stopping criteria met.J Evaluate makespan time of IS—insert into ICL, IIL and ITL.J If LTM-MIN or LTM-MAX—loop through designated restart amount (2 restarts, 3 times through).

If short-term memory, do not loop (0 restarts, 1 time through)� Loop while stopping criteria unmet� Cycle through groups 12N

J Cycle through jobs within each group� Exchange jobs pair-wise including cyclical exchange (1 and n)� If sequence has been previously investigated, move on to next sequence� If exchange is tabu, remember sequence� If exchange is not tabu and sequence has not been investigated, EVALUATE MAKESPAN

time� Save the time, sequence and tabu parameters if better than previous best

� If best time is not better than previous ICL entry, investigate sequences saved as tabu. Only saveas best if evaluated as better than previous ICL entry.� Using best time, sequence and tabu parameters, update ITL, ICL and IIL if necessary. Increment

IIT_limit or set to zero as necessary.� If LTM-MIN or LTM-MAX—update long term memory matrix� Close stopping criteria loop� If LTM-MIN or LTM-MAX—switch appropriate jobs

J Close restart loopJ Return best time and sequence
� EVALUATE MAKESPAN–returns the makespan time for a given group and job sequence
J Cycle through groups� Cycle through stages� Find the machine within the stage with the least completion time� Assign the current group to that machine on the current stage� Add the appropriate setup time depending on the previous group on that machine� Cycle through the jobs� Cycle through the stages

J Calculate the finish time of each job on each stageJ Return the maximum finish time on the final stage

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