Future Generation Computer Systems 25 (2009) 654–661

Contents lists available at ScienceDirect

Future Generation Computer Systems

journal homepage: www.elsevier.com/locate/fgcs

A variable neighborhood search for job shop scheduling with set-up times tominimize makespanV. Roshanaei a, B. Naderi a,∗, F. Jolai b, M. Khalili ca Department of Industrial Engineering, Amirkabir University of Technology, 424 Hafez Avenue, Tehran, Iranb Department of Industrial Engineering, University of Tehran, Tehran, Iranc Department of Industrial Engineering, Azad University, Science and Research branch, Tehran, Iran

a r t i c l e i n f o

Article history:Received 25 August 2008Received in revised form21 December 2008Accepted 19 January 2009Available online 8 February 2009

Keywords:SchedulingJob shopSequence-dependent set-up timeVariable neighborhood searchVariable neighborhood descentInsertion neighborhood

a b s t r a c t

Production Scheduling problems are typically named bases on the processing routes of their jobs ondifferent processors and also the number of processors in each stage. In this paper, we consider theproblem of scheduling a job shop (JSS) where set-up times are sequence-dependent (SDST) to minimizethe maximum completion times of operations or makespan. Our problem is generally formulated asJ/STsd/Cmax. To tackle such an NP-hard problem, a recent effective metaheuristic algorithm known asvariable neighborhood search (VNS) is employed. VNS algorithmshave shownexcellent capability to solvescheduling problems to optimal or near-optimal schedule. Our proposed VNS is readily intelligible yet is arobust solution technique for the problemof SDST JSS. VNS is categorized as a local search-based algorithmarmedwith systematic neighborhood search structures. Our proposed VNS obviates the notoriousmyopicbehavior of local search-based metaheuristic algorithms by the means of several systematic insertionneighborhood search structures. An experimental design based on Taillard’s benchmark is conducted toevaluate the efficiency and effectiveness of our proposed algorithm against some effective algorithms inthe literature. The obtained results strongly support the high performance of our proposed algorithmwithrespect to other well-known heuristic and metaheuristic algorithms.

© 2009 Elsevier B.V. All rights reserved.

1. Introduction

Scheduling is undoubtedly one of themost useful and successfulfields of operations research. As the total cost of productiongenerally constitutes a sizeable proportion of any enterprize, it isnot surprising that each company seeks to discover an efficientdesign of their shop environment as well as proper sequencing ofjobs on different machines to minimize the times of operations.Multi-stage scheduling problems consist of a set of n jobs

requiring execution on more than one machine. Each of thesejobs has a set of operations needing to be processed on a set ofmachines. Each job visits the machines following a certain orderknown as the processing route. If the processing routes are not givenin advance, and have to be chosen, the scheduling problem iscalled an open shop configuration. In open shop scheduling theprocessing order of operations follows no definite precedence. Aslong as the operations of a job are done, the job is done [1]. If eachjob has a fixed and exclusive processing route, the problem is calledjob shop configuration. In this configuration, the processing route

∗ Corresponding author.E-mail address: bahman.naderi@aut.ac.ir (B. Naderi).

0167-739X/$ – see front matter© 2009 Elsevier B.V. All rights reserved.doi:10.1016/j.future.2009.01.004

of operations is determined in advance but is not identical for allthe operations [2]. If the processing routes are fixed and are identicalfor all the jobs, the problem is called a flow shop [3]. In this paper,we consider job shop scheduling which is one of the most appliedproduction scheduling configuration [4].Job shop scheduling (JSS) is one of the most significant issues

in production planning. JSS is important because it determinesprocess maps and process capabilities for most industries. A jobshop consists of a set of n jobs to be processed by a set ofm processors. Each job i has a specific operation order Oi ={Oi1,Oi2, . . . ,Oim}, where Oij represents j-th processor on whichthe job i must be processed on [2]. In regular job shops, eachjob i might not necessitate being processed by all the processors(i.e. each job i needs only a fraction of all the processors). Aprocessor can process at most one job at a time and a job can beprocessed by at most one processor at a time. Additionally, weassume that all the tasks and jobs are independent and availablefor their process at time 0. The m processors are continuouslyavailable. The process of a job i on a processor j cannot beinterrupted. There are infinite buffers between all processors, if ajob needs a processor that is occupied; it waits indefinitely untilit is available. There is no transportation time between processors.The objective is to find a production sequence of the jobs on theprocessors such that one or some selected criteria are optimized.

V. Roshanaei et al. / Future Generation Computer Systems 25 (2009) 654–661 655

In many real-life situations such as chemical, printing, pharma-ceutical, and automobile manufacturing [5], the set-up operations,such as cleaning up or changing tools, are not only often requiredbetween jobs but they are also strongly dependent on the imme-diately preceding process on the same processor [6,7] (sequence-dependent). Considering sequence-dependent set-up times (SDST)is gaining increasing attention among researchers in recent years.The motivation behind this assumption is to obtain tremendoussavingswhen set-up times are explicitly included in scheduling de-cisions [8].With respect to the corresponding explanation, we takeinto account that there exist sequence-dependent set-up times oneach processor in our problem.It is remarkable that job shop problems have developed into a

very active and popular field of research in recent decades. Manypapers have addressed a wide variety of practical and realisticindustrial assumptions. Among them, considering sequence-dependent set-up times has become recently popular amongresearchers who intend to investigate the scheduling decisions ina more realistic manner.A one-processor sequence-dependent set-up time scheduling

problem is equivalent to a traveling-salesman problem and is NP-hard [9]. Even for a small system, the complexity of this problemis beyond the exact theories. Job shop problems are significantlymore complex than regular single machine scheduling problem.On the other hand, job shop problems are considered as a class ofcombinational optimization problems known as NP-hard ones [9].Owing to the complexity of the job shop scheduling environ-

ment, exact methods are handicapped to render an optimal sched-ule within a reasonable amount of time. Hence, the employmentof approximate solution techniques (metaheuristics) is justified tofind an optimal or near-optimal schedule for such problems [10].The literature of JSS is filled with algorithms with outstandingand acceptable performance but at the expense of being totallycomplex [11,12]. In this paper, we aim at overcoming this draw-back and proposing a variable neighborhood search (VNS) meta-heuristic that is very effective, yet readily intelligible and easilyimplementable. The primary contributions of this paper are theapplication of the VNS algorithm to the JSS problem for the firsttime and utilization of SDST in our problem.The proposed metaheuristic employs three advanced neigh-

borhood search structures based on insertion neighborhoods inframework of a specific case of VNS, called variable neighborhooddescent (VND).The outline of this paper is organized as follows: Section 2

reviews the literature of a SDST job shop. Section 3 elaborates theproposed VNS. In Section 4, experimental design is carried outand the proposed VNS is compared against some other existingmethods. Finally Section 5 concludes the paper and introducessome directions for future work.

2. Literature review of SDST job shop

Many active researchers in the field of scheduling haverecently perceived the privileged status of considering sequence-dependent set-up times in scheduling problems, for it has manyadvantages when the SDST assumption is incorporated in thescheduling decision making process. Research on a SDST job shop,like the other manufacturing environments, commences with asingle machine. An integer programming model was proposed byColeman [13] for the purpose ofminimizing earliness and tardinessin a single machine with sequence-dependent set-ups, similar tothe model presented by Baker [10]. Naderi et al. [2] addresseda SDST job shop and proposed a genetic algorithm hybridizedwith a local search and restart phase. They showed that theiralgorithm outperformed the algorithms in the literature. In [13],it was shown that a SDST single machine is strongly NP-hard.

Gupta [14] presented a branch-and-bound algorithm for a SDSTjob shop. Another branch-and-bound algorithm was proposed byBrucker and Thiele [15] for the same problem. Most of researchesin production scheduling with SDST are restricted to a flowshopand its extensions, such as a flexible flowshop, a hybrid flowshopand flexible flow line scheduling. Kurz and Askin [16] considereddispatching rules for SDST flexible flow lines. They investigatedthree classes of heuristics based on naive greedy heuristics, theinsertion heuristics and Johnson’s rule. Kurz and Askin [17] thenproposed a genetic algorithm, named the Random Key GeneticAlgorithm (RKGA), for the same problem. Ruiz and Maroto [18]considered a hybrid flowshop with SDST and unrelated machines.They presented a calibrated genetic algorithm for the problem.Zandieh et al. [19] proposed an immune algorithm for SDST flexibleflow lines and demonstrated that the algorithm outperforms theRKGA of Kurz and Askin [17].With regard to job shop scheduling, Zhou and Egbelu [20]

proposed a heuristic method to minimize the makespan. Satakeet al. [21] applied a simulated annealing to solve job shopscheduling. Watanabe et al. [22] investigated job shop schedulingand proposed a GA with modified crossover and search areaadaptation. They showed that their algorithm worked better thanthe previous GAs by numerical experiments. Goncalves et al. [23]considered job shop scheduling, and presented a GA which hadbeen hybridized with a local search. They used random keyrepresentation to encode a solution.Choi and Korkmaz [24] studied the job shop problem with

separable SDST. They formulated the problem as a mixed integerprogramming and proposed a heuristic to minimize makespan.Schutten [25] considered a job shop with some practical aspects,such as release and due date, set-up times and transportationtimes. He then proposed an extension of the shifting bottleneckprocedure for the problem. Cheung and Zhou [26] developed ahybrid algorithm based on a hybridization of a genetic algorithmand a well-known dispatching rule for a SDST job shop wherethe set-up times were anticipatory. The first operations for eachof the m processors were achieved by GA while the subsequentoperations on each processor were planned by the SPT rule.The objective was makespan minimization. Artiguies et al. [27]studied a SDST job shop with concentration on a formal definitionof schedule generation schemes (SGSs) based on the semi-active, active, and non-delay schedule categories. They reviewedsome priority rules. Vinod and Sridharan [28] considered adynamic job shop with a SDST, and a discrete event simulationmodel of the job shop system was developed. Two types ofscheduling rule (ordinary and set-up-oriented rules) were appliedin simulation models. Their experimental results demonstratedthat set-up-oriented rules performed better than ordinary rules.This difference rose with the increase in shop load and set-uptime ratio. Zhou et al. [29] proposed an immune algorithm whichcertified the diversity of the antibody.A complete survey of scheduling problems with set-up times

was given by Allahverdi et al. [8]. As far as reviewed, there is farless literature for a SDST job shop than one for pure job shops. Weintend to propose a metaheuristic for a SDST job shop.

3. Variable Neighborhood Search (VNS) metaheuristic

3.1. The background of VNS

VNS, a recently proposed metaheuristic technique, has quicklygained widespread success. Large numbers of papers haveattempted to improve upon solutions by using a relatively largearsenal of local search improvement heuristics, based arounddifferent neighbors. The term ‘‘variable neighborhood search’’refers to all local search-based approaches that are centered on

656 V. Roshanaei et al. / Future Generation Computer Systems 25 (2009) 654–661

the principle of systematically exploring more than one type ofneighborhood structure during the search. VNS has been appliedwith success to other problems including [30–33]. The reason forthe utilization of VNS is that metaheuristics are stuck in localoptima: the move required to improve the solution cannot beperformed and the moves in the neighborhood would lead to adeterioration of the solution quality. Such occurrence takes placemainly due to the myopic behavior of metaheuristic algorithms:no built-in operator exists to diversify the search space. Instead ofrelying on advanced metaheuristics mechanisms such as randomperturbations (iterated local search) or memory structures (tabusearch) or crossover and mutation (evolutionary algorithms) VNSproceeds in this case by using different types of neighborhoods,which might contain the required improving moves.VNS is closely related to Iterated Local Search (ILS): instead

of iterating over one constant type of neighborhood structure(i.e. local search) as done in ILS, VNS iterates in an analogous wayover some neighborhood structures until some stopping criterionis met. The central observations of VNS are: (1) A local minimumone neighborhood structure is not necessarily locallyminimalwithrespect to another neighborhood structure. (2) A global optimumis locally optimal with respect to all neighborhood structures.Since approximately all metaheuristics make use of just one

type of neighborhood structure, there exists a high probabilityfor them to get trapped in local optima after a certain number ofiterations. Therefore, it necessitates developing a strong algorithmenjoying diverse systematic neighborhood search structure andsufficient potential to escape from local optima. The reasonswhy VNS has obtained its acceptability and popularity amongresearchers are due to the utilization of several neighborhoodstructures, easy to implement and the high flexibility and brilliantadaptability of VNS for different problems. A variant of VNSis Variable neighborhood descent (VND) with these two maindifferences: (1) In VND, change of neighborhoods is performed ina deterministic way while in basic VNS, the neighborhoods areexplored randomly. (2) In VND, usually the first improvement isaccepted rather than the best improvement while in basic VNS, itis conversely applied. In a subsequent subsection, we are applyinga VNSwith advanced and powerful neighborhood structures underthe framework of VND.

3.2. Encoding scheme

Encoding schemes are used to make a candidate solutionrecognizable for algorithms. A proper encoding scheme plays akey role in maintaining the search-effectiveness of any algorithm.One of the most extensively used encoding schemes in theliterature is operation-based representation [2]. By making useof operation-based representation, the relative order of theoperations of the jobs on the processors is determined. Since thereare precedence constraints among the operations of each job,not all the permutations of the operations give feasible solutions.With respect to above explanations, Gen et al. [34] proposed analternative scheme as follows: Each job i has a set of ni operations.So in the representation, each job number i occurs as the numberof its operations. By scanning the permutation from left to right,the kth occurrence of a job number refers to the kth operationin the technological sequence of this job. A permutation with arepetition of job numbers merely expresses the order in which theoperations of the jobs are processed. It is clear that so long as ajob number repeats as the number of its operations, a solution isalways feasible.The procedures of encoding and decoding a candidate solution

are illustrated by applying them to an example. Consider a problemwith 4 jobs and 3 processors. Jobs 2 and 4 have two operations andJobs 1, 3 consist of three operations, and are thereby repeated three

Table 1The production sequence and the processing times of an example with 4 jobs and 3machines.

Job i Production sequence Processing times

1 1, 3, 2 8, 9, 52 1, 2 4, 103 2, 1, 3 7, 10, 64 3, 2 11, 4

times, so that for this problem we have 10 operations {1 1 1 2 2 33 3 4 4}. Table 1 lists the production sequence and the processingtimes required on each processor. For each operation, we generatea random number from a uniform distribution between (0, 1).These random keys are then sorted to find relative order of theoperations (Fig. 1). Our purpose to represent the solutions by theserandom keys is to make our encoding scheme easily adjustable toany operators; in particular to the types of neighborhood searchstructures thatwe are going to propose in the following subsection.For example in Fig. 1, after sorting the random keys, the fourth

block implies the second operation of Job 1 because number 1 hasbeen repeated twice. If a job number is repeated as the numberof its operations, the solution is always feasible. According thecorresponding relative order of the operations, they are processedat the earliest time the processor and the job are both available.The makespan becomes 28 time units. Fig. 2 shows the Gantt chartof this solution.

3.3. The proposed VNS

In a nutshell, a variable neighborhood search starts from aninitial solution. The initial solution of our VNS is produced by awell-known dispatching rule, Shortest Processing Times (or SPT)proposed by Sule [1] for job shops. We have adapted SPT so as toconsider the existence of SDST.Since a VNS improves its current solution by the means

of different neighborhood search structures, our proposed VNSutilizes three different types of advanced neighborhood searchstructure (NSS) to comply with the framework of a VND versionof VNS. A systematic switch of one type of NSS to another one isdone to purposefully lead us to maintain the probability of visitingthe better solutions.All three types of NSSs that we are going to define are based

on insertion neighborhood. Many researchers have concluded thatthe insertion neighborhood is superior to the swap or exchangeneighborhoods in the field of scheduling [5,4]. The insertionneighborhood of a sequence comprises all those sequences thatcan be obtained by removing some operations and inserting themin other positions. With regard to the given explanation, in eachinsertion neighborhood, three main decisions should be made:

1. Number of operations to be removed from a complete sequenceπ

2. How to choose these operation(s).3. How to relocate the removed operation(s) to construct a newcomplete sequence π́

We define our first NSS as such: an operation is removedfrom the sequence at random and without repetition, and thenrelocated into another random selected position. Considering theabove definition, the three main decisional factors are adoptedas such: number of removed operation is one. All operations areselected one after another without repetition in a random order.The procedure is illustrated by applying it to the previous example.Suppose the current solution is the sequence in Fig. 3. The firstremoved operation is randomly chosen to be the fourth block(i.e. the first operation of Job 2). We randomly regenerate therandom key RK of the fourth block. Now, we sort the operations

V. Roshanaei et al. / Future Generation Computer Systems 25 (2009) 654–661 657

Fig. 1. Representation of a candidate solution in VNS.

Fig. 2. Gantt chart of the example.

Fig. 3. Procedure of NSS type 1.

according to the RKs again. If we observe the first improvement,the associated sequence is accepted and the procedure restarts. Ifnot, this mechanism repeats for the subsequent blocks. As soonas the first improvement is observed the procedure restates. Theprocedure of NSS type 1 is shown in Fig. 3.The whole procedure repeats so long as no improvement is

obtained through inserting all the operations into a new randomlyselected position. After relocating all the operations, we believethat there is little hope for further improvement just by changingthe position of one operation. Hence, it necessitates presentinganother NSS to escape from this local optimum of the first NSS.According to this research finding, we need to introduce an NSS togenerate farther neighbors than just changing the position of oneoperation.In our second NSS the number of removed operations is

set equal to 2. The manner of choosing these 2 operations isall the combinations of two-out-of-NO (NO is total number ofthe operations) operations. To relocate them, the two jobs arereinserted into two new randomly selected positions. Fig. 4 reportsthe general outline of NSS type 2.Having observed the first improvement, the associated se-

quence is accepted and the procedure restarts from NSS type 1. Ifnot, this mechanism repeats for the subsequent combinations. Bythe same reasoning of switching from NSS type 1 to type 2, NSS

type 2 terminates and NSS type 3 starts. In the third NSS, the threerandomly selected operations are randomly relocated into threeother randomly selected positions. As can be seen, we randomlychoose three operations as well as randomly relocating the op-erations in three different positions. Generating a new sequencethrough NSS type 3 is done ϕ times. In the NSS type 3, the num-ber of sequences (ϕ) which are supposed to be generated by themeans of NSS type 3 is a parameter that needs to be tuned. Af-ter observing the first improvement, the corresponding sequenceis accepted to move and the procedure restarts from NSS type 1.In addition to the mentioned decisional difference between NSSs,in NSS type 3, algorithm is obliged to accept one move (the bestsequence among ϕ produced sequences), even if it is inferior withrespect to the current solution. This is so because returning to NSStypes 1 and 2 with the former candidate solution most likely doesnot improve the best solution. This mechanism can be consideredas an acceptance criterion of our proposed VNS to avoid stagnationsituations throughout the search. The procedure of NSS type 3 andthe general outline of the proposed VNS are shown in Figs. 5 and 6,respectively.In sum, we proposed a new version of VNS metaheuristic

algorithm with three types of NSSs. Each of these NSS playsa different, yet influential role on the entire performance ofthe algorithm. NSS type one, intensifies the search space while

658 V. Roshanaei et al. / Future Generation Computer Systems 25 (2009) 654–661

Fig. 4. Procedure of NSS type 2.

Fig. 5. Procedure of NSS type 3.

Fig. 6. General outline of the proposed VNS.

NSS types two and three diversify the search space with avery simple mechanism. Many researchers hybridize originalalgorithms with other supplementary tools to simultaneouslyintensify and diversify the search space to achieve an excellentquality solution of their algorithm. These hybridizations havesuccessfully been applied to different problems but at theexpense of being entirely complicated while our proposed VNShybridizes all these mechanisms with a very simple mechanism.Consequently, our proposed VNS enjoys a simple mechanismfor better comprehension of researchers along with a robustperformance during search process.

4. Experimental design

In this section, we intend to evaluate the performance of ourproposed VNS against some other well-known existing methods.We compare the proposed VNS with the genetic algorithm of

Cheung and Zhou [26] (GA_Cheung), the immune algorithm ofZhou et al. [29] (IA_Zhou) and the hybrid genetic algorithmof Naderi et al. [2] (HGA_Naderi) that have shown effectiveperformance in the literature of a SDST job shop. The shortestprocessing time (SPT) of Sule [1] is also brought into thecomparison. Our purpose to utilize SPT is to use it as upper boundfor a given instance to generally assess the metaheuristics.We implement the algorithms in MATLAB 7.0 and run on a

PC with 2.0 GHz Intel Core 2 Duo and 2 GB of RAM memory.The stopping criterion used when testing all instances with thealgorithms is set to a CPU time limit fixed to n2 × mmilliseconds.Wemake use of this stopping criterion because of its flexibility andapplicability. This stopping criterion is a good measure to assessthe strength of search methods because it allows algorithms tomanifest their competitiveness within a fixed time horizon. n andm represent the number of jobs and stages in our stopping criterionand since n is raised to the power of two, this stopping criterion ismore sensitive toward a rise in the number of jobs than the numberof stages.We use relative percentage deviation (RPD) as a common

performance measure to compare the methods [4]. The bestsolutions obtained for each instance (which is named Minsol) arecalculated by any of the algorithms. RPD is obtained by givenformula below:

RPD =Algsol −MinsolMinsol

· 100 (1)

where Algsol is the objective function value obtained for a givenalgorithm and instance. Clearly, lower values of RPD are preferable.

4.1. Data generation

The data required for the problem includes: the number ofjobs, the number of processors, processing times and set-up times.The way in which we generate instances is based on Taillard [35]benchmarks. There are 8 configurations of number of jobs n

V. Roshanaei et al. / Future Generation Computer Systems 25 (2009) 654–661 659

Fig. 7. Means plot and LSD intervals (at the 95% confidence level) for the differentlevels of parameter ϕ.

and number of processors m containing (15 × 15), (20 × 15),(20 × 20), (30 × 15), (30 × 20), (50 × 15), (50 × 20) and(100 × 20). The processing times are generated from a uniformdistribution between (1, 99) similar to Taillard instances. Wehave four levels for SDST, 25%, 50%, 100% and 125% of maximumpossible processing times; therefore, SDST’s are generated fromfour uniform distributions U (1, 25), U (1, 50), U (1, 100) and U (1,125) similar to [2]. We generate 10 instances for each combinationof n,m and SDST, summing up for 320 instances.

4.2. Parameter tuning

One of the advantages of our proposed VNS is that it has onlyone parameter (ϕ) to be tuned. The considered levels are 10,30, 50 and 70. A set of 80 instances including 10 instances foreach combination of n and m is randomly generated. All the 80instances are solved by 4 VNSs produced by the cited levels. Weuse the ANOVA test to analyze the results. The means plot and LSDintervals for different levels of parameter ϕ are shown in Fig. 7. Ascould be seen, ϕ of 50 provides the best results among the levels.

4.3. Experimental results

In this subsection, firstly we analyze the algorithms in termsof selected objective function (makespan). Secondly, effects ofvariable factors, problem size and numbers of stages, on theperformance of them.For appraising the performance of algorithms, we use a RPD

measure. The results of the experiments, averaged for eachcombination of n and m (40 data per average) are reported inTable 1. As can be seen, different algorithms have obtained variousRPDs for different instances. As we expected, the performancesof metaheuristic algorithms against the single heuristic which isSPT are quite remarkable. Among metaheuristic algorithms, ourproposed VNS holds the first rank with a RPD of 1.11%; and theworst performing algorithm in almost all of instances is SPT with aRPD of 14.38%. After our proposed VNS, HGA_Naderi is ranked thesecond position with the praiseworthy RPD of 1.94%. GA_Cheungand IA_Zhou with RPDs of 5.10% and 9.61% hold the 3rd and 4thranks respectively. All algorithms ranging from SPT to GA_Cheunghave shown unsteady behaviors, while our proposed VNS havesustained its robustness in almost all circumstances.For further precise analysis of the results, we carried out an

analysis of variance (ANOVA). It is necessary to indicate that due tothe considerable difference between SPT and the other algorithms;we exclude it from our ANOVA test. It is necessary to note that

Fig. 8. Means plot and LSD intervals (at the 95% confidence level) for the type ofalgorithm factor.

Table 2Average relative percentage deviation (RPD) for the algorithms grouped by n andm.

No. of jobs (n) No of machines (m) AlgorithmGA_Cheung HGA-

NaderiVNS IA_Zhou SPT

15 15 6.49 2.62 0.33 10.48 16.0920 15 6.12 2.13 1.00 12.10 18.57

20 5.17 2.04 1.39 11.02 14.6330 15 6.74 1.93 1.42 11.81 18.22

20 4.55 2.09 1.31 9.04 15.5150 15 6.53 1.75 1.31 9.41 15.26

20 3.95 1.82 1.14 6.88 11.29100 20 1.27 1.17 1.01 6.11 5.50Average 5.10 1.94 1.11 9.61 14.38

for using the ANOVA test, three main hypotheses, normality,homogeneity of variance and independence of residuals, must bechecked.We did that and found no bias for questioning the validityof the experiment. The means plot and LSD intervals (at the 95%confidence level) for the different algorithms are shown in Fig. 8. Asit is shown, our proposed VNS is statistically better than the otheralgorithms (Table 2).To analyze the behavior of different algorithms in different

situations, we plot the average RPDs of the algorithms in differentlevels of the number of jobs and processors in Figs. 9 and 10,respectively. As shown in Fig. 9, in the cases of n = 40, 70, theVNS is better than HGA_Naderi, while in the case of n = 100,HGA_Naderi works better. As could be seen in Fig. 10, VNS showsbetter performances regarding increasing the number of stages.In this subsection we evaluated the proposed VNS against

some other well-known algorithms in the literature. There is astatistically significant difference between the metaheuristics. Asshown in Fig. 8, VNS statistically supersedes the other algorithms.We also explored the effects of different situations such as differentlevels of the number of jobs and stages on the performances of thealgorithms.

5. Conclusion and future studies

In this communication, we proposed a capable and superiormetaheuristic algorithm known as VNS to solve the problem ofscheduling job shop (JSS) where set-up times were sequence-dependent (SDST) on each processor to minimize the maximumcompletion times of operations or makespan. The significantdifference between our proposedVNS and other local search-basedmetaheuristic algorithms was owing to the fact that our proposed

660 V. Roshanaei et al. / Future Generation Computer Systems 25 (2009) 654–661

Fig. 9. Means plot for the type of algorithm factor versus the number of jobs.

Fig. 10. Means plot for the type of algorithm factor versus themagnitude of SDSTs.

VNS was equipped with several systematic neighborhood searchstructures to remedy the deficiency of the infamous myopicbehavior of above algorithms.We conducted an experimental design based on Taillard’s

benchmark to appraise the performance efficiency and effective-ness of our proposed algorithm against some effective algorithmsin the literature. Our strong belief towards the high performanceof our proposedVNSmaterializedwith regard to otherwell-knownalgorithms in the literaturewhilewe analyzed the obtained results.The computational results verified that the VNS not only dom-inated other well-known algorithms in terms of both computa-tional time and quality solution but also sustained its robustnessin all situations. The evaluations place the VNS in a particular po-sition as one of the ever-improving and most efficient algorithmproposed so far for the problem considered while at the same timebeing simple and easy to implement.As a direction for future research, it could be interesting

to apply some other effective hybrid metaheuristic algorithmsfor the problem considered, and compare them with our VNS,or to explore the generalizability of our algorithm in someother complex scheduling problems, such as flexible job shopand open shop to ascertain if the high performance of ourVNS is communicable to other scheduling problems. Another

clue for future research is considering some other realisticindustrial assumptions, such as machine availability constraintsand unrelated machines. Another opportunity for research isconsidering the problem with the other optimization objectives,such as makespan or maximum tardiness, or even multi-objectivecases.

References

[1] D.R. Sule, Industrial Scheduling, PWS Publishing Company, USA, 1997.[2] B. Naderi, M. Zandieh, S.M.T. Fatemi Ghomi, Scheduling job shop problemswith sequence-dependent set-up times, International journal of productionresearch (2008) doi:10.1080/00207540802165817.

[3] O. Engi, A. Döyen, A new approach to solve hybrid flow shop schedulingproblems by artificial immune system, Future Generation Computer Systems20 (2004) 1083–1095.

[4] B. Naderi, M. Zandieh, V. Roshanaei, Scheduling hybrid flowshops withsequence-dependent set-up times to minimize makespan and maximumtardiness, International Journal of Advanced Manufacturing Technology(2008) doi:10.1007/s00170-800-1569-3.

[5] R. Ruiz, T. Stützle, An iterated greedy heuristic for the sequence-dependentset-up times flowshop problem with makespan and weighted tardinessobjectives, European Journal of Operational Research 187 (3) (2008)1143–1159.

[6] R.W. Conway,W.L.Maxwell, Theory of Scheduling, Addison-Wesley,MA, 1967.[7] K.R. Baker, Introduction to Sequence and Scheduling, Wiley, NY, 1974.[8] A. Allahverdi, C.T. Ng, T.C.E. Cheng, Y.M. Kovalyov, A survey of schedulingproblemswith set-up times or costs, European Journal of Operational Research187 (3) (2008) 985–1032.

[9] B. Ombuki, M. Ventresca, Local search genetic algorithms for the job shopscheduling problem, Applied Intelligence 21 (2004) 99–109.

[10] R.S. Chang, J.S. Chang, P.S. Lin, An ant algorithm for balanced job scheduling ingrids, Future Generation Computer Systems 25 (1) (2009) 20–27.

[11] G. Vilcot, J.C. Billaut, A tabu search and a genetic algorithm for solving a bi-criteria general job shop scheduling problem, European Journal of OperationalResearch (2007) doi:10.1016/j.ejor.2007.06.039.

[12] J.T. Tsai, T.K. Liu, W.H. Ho, J.H. Chou, An improved genetic algorithmfor job shop scheduling problems using Taguchi-based crossover, Interna-tional Journal of Advanced Manufacturing Technology (2007) doi:10.1007/s00170-007-1142-5.

[13] B.J. Coleman, Technical note: A simple model for optimizing the singlemachine early/tardy problem with sequence-dependent set-ups, Productionand Operations Management 1 (1992) 225–228.

[14] S.K. Gupta, N Jobs and n machines job shop problems with sequence-dependent set-up times, International Journal of Production Research 20 (5)(1986) 643–656.

[15] P. Brucker, O. Thiele, A branch-and-bound method for general shop problemwith sequence-dependent set-up times, OR Spectrum 18 (1996) 145–161.

[16] M.E. Kurz, R.G. Askin, Comparing scheduling rules for flexible flow lines,International Journal of Production Economics 85 (2003) 371–388.

[17] M.E. Kurz, R.G. Askin, Scheduling flexible flow lines with sequence-dependentset-up times, European Journal of Operational Research 159 (2004) 66–82.

[18] R. Ruiz, C. Maroto, A genetic algorithm for hybrid flowshops with sequence-dependent set-up times and machine eligibility, European Journal ofOperational Research 169 (2006) 781–800.

[19] M. Zandieh, S.M.T. Fatemi Ghomi, S.M. Moattar Husseini, An immunealgorithm approach to hybrid flow shops scheduling with sequence-dependent set-up times, Journal of Applied Mathematics and Computation180 (2006) 111–127.

[20] C. Zhou, P.G. Egbelu, Scheduling in a manufacturing shop with sequence-dependent set-ups, Robotics and Computer Integrated Manufacturing 51(1989) 73–81.

[21] T. Satake, K. Morikawa, K. Takahashi, N. Nakamura, Simulated annealingapproach for minimizing the makespan of the general job shop, InternationalJournal of Production Economics 60–61 (1999) 515–522.

[22] M. Watanabe, K. Ida, M. Gen, A genetic algorithm with modified crossoveroperator and search area adaptation for the job shop scheduling problem,Computers & Industrial Engineering 48 (2005) 743–752.

[23] J.F. Goncalves, J.J.D.M. Mendes, M.G.C. Resende, A hybrid genetic algorithm forthe job shop scheduling problem, European Journal of Operational Research167 (2005) 77–95.

[24] I.C. Choi, O. Korkmaz, Job shop schedulingwith separable sequence-dependentset-ups, Annals of Operations research 70 (1997) 155–170.

[25] J.M.J. Schutten, Practical job shop scheduling, Annals of Operations Research83 (1998) 161–178.

[26] W. Cheung, H. Zhou, Using genetic algorithms and heuristics for job shopscheduling with sequence-dependent set-up times, Annals of OperationsResearch 107 (2001) 65–81.

[27] C. Artiguies, P. Lopez, P.D. Ayache, Schedule generation schemes for the jobshop problem with sequence dependent set-up times: Dominance propertiesand computational analysis, Annals of Operations Research 138 (2005) 21–52.

[28] V. Viond, R. Sridharan, Dynamic job shop scheduling with sequence-dependent set-up times: Simulation modeling and analysis, Interna-tional Journal of Advanced Manufacturing Technology (2006) doi:10.1007/s00170-006-0836-4.

V. Roshanaei et al. / Future Generation Computer Systems 25 (2009) 654–661 661

[29] Y. Zhou, L. Beizhi, J. Yang, Study on job shop scheduling with sequence-dependent set-up times using biological immune algorithm, InternationalJournal of Advanced Manufacturing Technology 30 (2006) 105–111.

[30] K. Flesza, K.S. Hindi, Solving the resource-constrained project schedulingproblem by a variable neighborhood search, European Journal of OperationalResearch 155 (2004) 402–413.

[31] C.J. Liao, C.C. Cheng, A variable neighborhood search for minimizing singlemachine weighted earliness and tardiness with common due date, Computersand Industrial Engineering 52 (2007) 404–413.

[32] J. Gao, L. Sun, M. Gen, A hybrid genetic and variable neighborhooddescent algorithm for flexible job shop scheduling problems, Computers andOperations Research 35 (9) (2008) 2892–2907.

[33] B. Naderi, M. Zandieh, S.M.T. Fatemi Ghomi, A study on integratingsequence-dependent set-up time flexible flow lines and preventive mainte-nance scheduling, Journal of Intelligent Manufacturing (2008) doi:10.1007/s10845-008-0157-6.

[34] M. Gen, Y. Tsujimura, E. Kubota, Solving job-shop scheduling problem usinggenetic algorithms, in: Proceedings of the 16th International Conference onComputer and Industrial Engineering, Ashikaga, Japan, 1994, pp. 576–579.

[35] E. Taillard, Benchmarks for basic scheduling problems, European Journal ofOperational Research 64 (1993) 278–285.

V. Roshanaei accomplished his B.A in Industrial Man-agement at Azad University of Tehran (2001–2004).Next, he pursued his scientific career in M.Sc. in Indus-trial Engineering at Amirkabir University of Technology(2006–2008). As research interests, he is exploring theapplications of Multi-Criteria Decision Making (MCDM)and Artificial Intelligence (AI) techniques in the areasof manufacturing systems design (Flow-shops, Job-shopsand Open-shops), supply chain management (SCM) andportfolio optimization. Recently, he has initiated doing re-search in the areas of risk assessment methodologies and

Business Continuity Planning (BCP). He has been working for Bank Mellat since2000. Presently, he is a Systems Engineering expert in Bank Mellat IT Departmentin Tehran.

B. Naderi completed his B.Sc. in Industrial Eng. at Mazan-daran University of Science and Technology (2001–2005),and M.Sc. in Industrial Eng. at Amirkabir University ofTechnology (2005–2007). Currently, he is Ph.D. studentin Industrial Eng. at Amirkabir University of Technology(Start year: 2007). His research interests are the appli-cations of Operations Research (OR) in the productionscheduling, supply chain and portfoliomanagement. Morespecifically, he has been working on Modeling and So-lution Methods including Artificial Intelligence (Heuris-tic and Metaheuristic algorithms) and mathematical ap-

proaches regarding foregoing problems.

F. Jolai completed his B.Sc. and M.Sc. in IndustrialEng. at Amirkabir University of Technology, Tehran, Iran(1986–1992). He obtained his Ph.D. in Industrial Eng. fromINPG, France (1998). Currently, he is anAssociate Professorat Industrial Eng. Department, University of Tehran,Iran. His research interests are project and productionscheduling, supply chain management.

M. Khalili completed his B.Sc. in Industrial Eng. at Mazan-daran University of Science and Technology (2000–2005),andM.Sc. in Industrial Eng. fromAzadUniversity of Tehran(2006–2008). He currently is a Ph.D. student in IndustrialEng. at Azad University, science and research branch (Startyear: 2008). His research interests are project and supplychain management and production scheduling.