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Scheduling job shop problems withsequence-dependent setup timesB. Naderi a , M. Zandieh b & S.M.T. Fatemi Ghomi aa Department of Industrial Engineering , Amirkabir University ofTechnology , Tehran, Iranb Department of Industrial Management , Management andAccounting, Shahid Beheshti University , Tehran, IranPublished online: 13 Aug 2009.

To cite this article: B. Naderi , M. Zandieh & S.M.T. Fatemi Ghomi (2009) Scheduling job shopproblems with sequence-dependent setup times, International Journal of Production Research,47:21, 5959-5976, DOI: 10.1080/00207540802165817

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International Journal of Production ResearchVol. 47, No. 21, 1 November 2009, 5959–5976

Scheduling job shop problems with sequence-dependent setup times

B. Naderia, M. Zandiehb* and S.M.T. Fatemi Ghomia

aDepartment of Industrial Engineering, Amirkabir University of Technology, Tehran, Iran;bDepartment of Industrial Management, Management and Accounting,

Shahid Beheshti University, Tehran, Iran

(Received 17 January 2008; final version received 19 April 2008)

In this work we consider job shop problems where the setup times are sequencedependent under minimisation of the maximum completion time or makespan.We present a genetic algorithm to solve the problem. The genetic algorithm ishybridised with a diversification mechanism, namely the restart phase, and asimple form of local search to enrich the algorithm. Various operators andparameters of the genetic algorithm are reviewed to calibrate the algorithm bymeans of the Taguchi method. For the evaluation of the proposed hybridalgorithm, it is compared against existing algorithms through a benchmark.All the results demonstrate that our hybrid genetic algorithm is very effective forthe problem.

Keywords: job shop; sequence dependent setup times; meta-heuristics; geneticalgorithm; Taguchi method

1. Introduction

Job shop scheduling (JSS) is one of the most typical and complicated manufacturingenvironments in production scheduling problems. A job shop consists of a set of n jobs{1, 2, . . . , n} that have to be processed by at most a set of m machines {1, 2, . . . ,m}. Eachjob i has a specific operation order Oi¼ (Oi1,Oi2, . . . ,Oil), where Oij represents the jthmachine that job i must be processed on, and l�m. Machines are continuously available(i.e. there is no machine breakdown). Each machine can process only one job at a time andeach job can be processed by only one machine at a time. Preemption of jobs is notallowed, which means that the processing of jobs cannot be interrupted.

Considering sequence-dependent setup times (SDST) in scheduling problems is gainingincreasing attention among researchers. Actualising the scheduling problem as well as theenormous savings obtained by explicitly incorporating setup times in scheduling decisionsmotivates researchers to consider SDST. We explore the processing of two consecutivejobs on the same machine, where some setup must be performed that depends on theordering of the two jobs. In many real-life situations such as chemicals, printing,pharmaceuticals, and automobile manufacturing (Zandieh et al. 2006), setup operations,such as cleaning or changing tools, are not only often required between jobs but they arealso strongly dependent on the immediately preceding process on the same machine(Sule 1997, Luh et al. 1998). We also assume that setup is non-anticipatory, meaning that

*Corresponding author. Email: m_zandieh@sbu.ac.ir

ISSN 0020–7543 print/ISSN 1366–588X online

� 2009 Taylor & Francis

DOI: 10.1080/00207540802165817

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we require the job for which the setup is required to be available and the machine on which

the job is to be processed to be idle.In the field of operations research, machine scheduling problems are a prevailing

area of research and numerous algorithms, including mathematical programming, branch-

and-bound, heuristics, meta-heuristics, etc., have been proposed. Since job shop

scheduling is one of the most complex combinational optimisations known to be

intractably NP-hard (Ombuki and Ventresca 2004), no exact method has been presented

that can tackle these problems within a reasonable amount of time. Therefore, proposing

different meta-heuristics for the job shop has developed into a very active field of research

(Watanabe et al. 2005, Chandrasekaran et al. 2006, Heinonen and Pettersson 2007, Zhang

et al. 2008). Many researchers have concluded that the genetic algorithm is not effective for

such NP-hard problems and gives inferior performance compared with meta-heuristic

algorithms; however, it is known that the performances of genetic algorithms strongly

depends on the choice of the operators and parameters (Ruiz et al. 2006).In this work we present a high-performing genetic algorithm (GA) for the problem.

Hence, having reviewed different operators and tuned the GA parameters appropriately,

we explore the impact of the fine-tuning of the operators and parameters on the

performance of the genetic algorithm by means of the Taguchi optimisation technique,

which is widely used in quality engineering to build robustness into an experimental setup

(Eddie et al. 2006). We have been motivated to utilise the Taguchi method because it is

regarded as a cost-effective and labour-saving method. It can investigate simultaneously

several factors and distinguish quickly those that have principal impacts by carrying out a

minimal number of possible experiments (Al-Aomar 2006). This approach has been

applied successfully at the parameter design stage to establish optimum process settings in

many fields (Caprihan and Wadhwa 2005, Cheng and Chang 2007, Luo et al. 2008).To enhance the quality of the genetic algorithm even further, we hybridise the GA with

a simple form of local search (henceforth referred to as HGA), and another feature called

the restart phase, which prevents the HGA from becoming stuck in local minima because

of the low diversity of its population. In the following, in order to prove the efficiency and

effectiveness of our algorithm, the performance of the HGA is compared with some

existing methods.The rest of the paper is organised as follows. Section 2 reviews the literature of the

SDST job shop. Section 3 describes the proposed genetic algorithm. In Section 4, we

elaborate the calibration of the proposed genetic algorithm. In Section 5, the experimental

design and a comparison of the proposed HGA with existing methods are presented.

Finally, Section 6 concludes the paper.

2. Literature review

Research on the SDST job shop, like other manufacturing environments, commences with

a single machine. Coleman (1992) proposed an integer programming model for minimising

earliness and tardiness in a single machine with sequence-dependent setups, similar to the

model presented by Baker (1974). Monma and Potts (1989), Coleman (1992) and

Pinedo (1995) showed that the SDST single machine problem was strongly NP-hard.

Gupta (1986) presented a branch-and-bound algorithm for the SDST job shop. Another

branch-and-bound algorithm was proposed by Brucker and Thiele (1996) for the same

problem.

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Much research in production scheduling with SDST is restricted to the flow shop and

its extensions, such as that of Kurz and Askin (2003, 2004), Riuz et al. (2005), Ruiz

and Maroto (2006) and Zandieh et al. (2006). With regard to job shop scheduling, Zhou

and Egbelu (1989) proposed a heuristic method to minimise the makespan. Cheng et al.

(1999) published a tutorial survey of different GAs and job shop scheduling. Park et al.

(2003) proposed a hybrid GA that started from another algorithm. Watanabe et al. (2005)

investigated job shop scheduling and proposed a GA with modified crossover and search

area adaption. They showed using numerical experiments that their algorithm worked

better than the previous GAs. Goncalves et al. (2005) considered job shop scheduling, and

presented a GA hybridised with a local search. They used a random key representation to

encode a solution.Choi and Korkmaz (1997) studied the job shop problem with separable SDST. They

formulated the problem as a mixed-integer program and proposed a heuristic to optimise

the makespan. They also showed that the heuristic was more effective than that proposed

by Zhou and Egbelu (1989). Schutten (1998) considered a job shop with some practical

aspects, such as release and due date, setup times and transportation times. He then

proposed an extension of the shifting bottleneck procedure.Cheung and Zhou (2001) developed an algorithm based on the hybridisation of a

genetic algorithm and a well-known dispatching rule for a SDST job shop where the setup

times were separable. The first operations for each of the m machines were achieved by the

GA while the subsequent operations on each machine were planned using the SPT rule.

The objective was the makespan, and the GA started from SPTS and MWKR.Vinod and Sridharan (2006) considered a dynamic job shop with SDST, and a discrete

event simulation model of the job shop system developed. Two types of scheduling rules

(ordinary and setup-oriented rules) were applied in the simulation model.

Their experimental results demonstrated that setup-oriented rules performed better than

ordinary rules. This difference arose with an increase in the shop load and setup time ratio.

Zhou et al. (2006) proposed an immune algorithm that certifies the diversity of the

antibody. A complete survey of scheduling problems with setup times was given by

Allahverdi et al. (2008).There is hardly any literature on the SDST job shop; moreover, all quoted papers on

GAs, even for a regular job shop without setup times, do not report on the quality of

different GA operators and parameters.

3. Proposed genetic algorithm

Genetic algorithms (GAs) arose in the 1970s by the work of Holland (1975). They were

intended to tackle industrial problems that were difficult to solve with the methods

available at that time. Nowadays, GAs are considered to be one of the typical meta-

heuristic approaches for tackling both discrete and continuous optimisation problems. The

idea behind GAs comes from Darwin’s ‘survival of the fittest’ concept, meaning that good

parents produce better offspring.The GA searches a problem space with a population of chromosomes, each of which

represents an encoded solution. A fitness value is assigned to each chromosome according

to its performance. The more desirable the chromosome, the larger the fitness value. The

population evolves by a set of operators until some stopping criterion is met. A typical

iteration of a GA, a generation, proceeds as follows. The best chromosomes of the current

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population are copied directly to the next generation (reproduction). A selection mechanism

chooses chromosomes of the current population in such a way that the chromosome with

the higher fitness value has a greater probability of being selected. The selected

chromosomes mate and generate new offspring (crossover). After the mating process,

each offspring might mutate by another mechanism called mutation. The new population is

then evaluated again and the whole process is repeated (Goldberg 1989).It is known that the effectiveness of a GA highly depends on the great choice of the

encoding scheme, and the selection, crossover and mutation operators, as well as their

parameters by which they are applied (Ruiz et al. 2006). We will use the Taguchi method

(Taguchi 1986) to set the parameters and operators of the GA. In our case, the factors will

all be operators and parameters of the GA and the output will be the relative effectiveness

of the GA. In the following subsections we describe all parameters and operators used in

the proposed GA.

3.1 Encoding scheme, initialisation procedure and selection mechanism

Encoding schemes are used to make the solutions recognisable for algorithms. The most

frequently used representation in job shop scheduling is the operation-based representa-

tion by which the relative order of the operations of the jobs on the machines in the shop is

determined (Cheng et al. 1999, Watanabe et al. 2005). Each job has a set of operations that

must be processed on m machines. Each job number occurs as many times as the number

of its operations. By scanning the permutation from left to right, the kth occurrence of a

job number refers to the kth operation in the technological sequence of this job.

A permutation with repetition of job numbers merely expresses the order in which the

operations of the jobs are processed.The procedures of encoding and decoding a candidate solution are illustrated by

applying them to an example. Consider a problem with three jobs and three machines. Job

1 has two operations, and Jobs 2 and 3 consist of three operations and are thereby

repeated three times, so that, for this problem, we have eight operations {1 1 2 2 2 3 3 3}.

For each job, Table 1 lists the production sequence and the processing times required on

each machine. For each operation, we generate a random number from a uniform

distribution between (0, 1). These random keys are then sorted to find the relative order of

the operations (Figure 1). Our purpose of representing the solutions by these random keys

is to make our encoding scheme easily adjustable to any operators; in particular, to the

type of crossover we are going to propose in the following subsection.For example, in Figure 1, after sorting the random keys, the third block implies the

second operation of Job 3 because the number 3 has been repeated twice. If a job number

Table 1. The production sequence and the processing times of anexample with three jobs and three machines.

Job iProductionsequence

Processingtimes

1 2, 1 10, 52 3, 2, 1 3, 6, 53 1, 2, 3 7, 8, 10

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is repeated as the number of its operations, the solution is always feasible.

According to the corresponding relative order of the operations, they are processed at

the earliest time the machine and the job are both available. The Gantt chart for this

solution is shown in Figure 2. For this solution, the completion time of the last job

(makespan) is 26 time units.The significant role of the initial solution on the quality of the final result of a search

procedure has already been recognised and emphasised by many researchers in recent

years. What has been utilised so far by the majority of researchers to generate the initial

solution for their algorithms has been the random generation of the initial solution, which

has led them to poor quality solutions. Therefore, one should give meticulous

consideration to the intelligent selection of their initialisation procedure in order to

acquire a satisfactory level of solution quality for such a hard combinatorial problem.

In this work, one of the individuals in the initial population is generated from the shortest

processing time (SPT) dispatching rule of Sule (1997). The remaining individuals

(popsize–1) are generated randomly.In the SPT of Sule, a table is constructed for each machine to list the jobs requiring that

machine arranged in SPT order. In a nutshell, the procedure of the SPT of Sule is as

follows. Whenever a machine is available, the first job on the list is loaded if the job is

available and has completed all its required prerequisite operations. If no such job is

available, the machine is idle until such a job becomes available. The procedure continues

as long as all the jobs are processed by all the required machines.For the selection of parents, there are two classical selection mechanisms, namely

ranking and tournament selection (Goldberg 1989). The choice of selection mechanism is

made in the calibration section.

3.2 Reproduction and crossover

Chromosomes with smaller makespans are more desirable, therefore the pr% of

the chromosomes with the smallest makespan values are automatically copied to the

J2

J1

J3 J1

J3 J2

J3

J2

3 5 7 15 17 21 25 26

M3

M2

M1

Time

Figure 2. Gantt chart of the example.

Random keysoperations

0.49 0.04 0.52 0.67 0.36 0.91 0.11 0.21

1 1 2 2 2 3 3 3

Sort (random keys)operations order

0.04 0.11 0.21 0.36 0.49 0.52 0.67 0.91

1 3 3 2 1 2 2 3

Figure 1. Representation of a candidate solution in GA.

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next generation. This mechanism is called reproduction. The remaining chromosomes

(1 – pr)% are generated through an operator called crossover.New solutions or offspring are produced by crossing two other sequences or parents

through an operator called crossover. The crossover operators must avoid

generating infeasible solutions. The purpose is to generate ‘better’ offspring, i.e. to

create better sequences after combining the parents. There are a wide variety of

proposed crossover operators suitable for the foregoing encoding scheme in the

literature of job shop scheduling problems. The commonly used crossover operators

reported are:

. the modified crossover of Watanabe et al. (2005) (MCW),

. modification of the PPX crossover (MPPX) proposed by Park et al. (2003),

. OX or order crossover, proposed by Davis (1985),

. OP or one-point crossover (Goldberg 1989), and

. PMX or partial-mapped crossover proposed by Goldberg and Lingle (1985).

The sixth crossover is parameterised uniform crossover (PUC) first proposed by

Norman and Bean (1999) and later used by Kurz and Askin (2004) and Zandieh et al.

(2006) for flexible flow line scheduling, which is an extension of standard

flowshop problems. The effectiveness of PUC is confirmed in those papers.

Therefore, we postulated the use of the PUC operator in job shop scheduling, as to

whether the high performance of the PUC operator is transferable to other scheduling

problems.The PUC operator works through random keys (RKs) of the jobs as follows. For

each operation, a random number is generated. If the value is less than 0.7 (following

Norman and Bean (1999)), the RK of operation Oij from the first parent is copied to

the offspring, otherwise the RK from the second parent is selected. For more

clarification, the PUC procedure is explained through an example. Consider we have a

problem with three jobs and each job has two operations; therefore, we have six

operations and six RKs. Again, suppose two parents, shown in Figure 3(step a), are

selected by the selection mechanism to undergo crossover. We generate six random

numbers for each operation Oij from a uniform distribution over the range (0, 1) as

shown in Figure 3(step b). As seen in Figure 3(step c), the first generated random

number is 0.61 and this means that the random key of operation O11 in the

corresponding offspring is selected from the first parent. The procedure proceeds for all

the subsequent operations. We compare the quality of each of the six above crossovers

in the calibration section.

1 1 2 2 3 3Parent 1

0.23 0.12 0.86 0.45 0.66 0.71

1 1 2 2 3 3Parent 2

0.89 0.54 0.20 0.38 0.73 0.82

step a

step b Crossover 0.61 0.32 0.85 0.43 0.72 0.18

1 1 2 2 3 3step c Offspring

0.23 0.12 0.20 0.45 0.73 0.71

Figure 3. Procedure of parameterized uniform crossover.

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3.3 Mutation

A mutation operator is utilised to slightly change the sequence, i.e. generating a new butsimilar sequence. The main purpose of applying mutation is to avoid convergence toa local optimum and diversify the population. The mutation operator can also be seen asa simple form of local search. Three different mutation operators are widely used in thegenetic algorithm literature.

. SWAP mutation. The RKs of two randomly selected operations are swapped.

. Inversion mutation. The RKs between two randomly selected cut points arereversed.

. Single point mutation. The RK of one randomly selected job is randomlyregenerated.

Further details of these mutation operators can be found in Michalewicz (1996).The probability of mutation (Pm) is defined as the probability that each offspringundergoes a mutation operation. In this paper, this probability is considered as anadaptive parameter, meaning that it might be changed according to the diversity in thecurrent population. The Pm increases twice for generations where the diversity rate is lowerthan the pre-defined threshold �. The diversity value is computed as

Diversity rate ¼fa � fbfb

,

where fa and fb are the average and best makespan of the individuals in the currentgeneration, respectively. Preliminarily instances show that �¼ 0.1 may be a good thresholdfor our problem.

3.4 Restart phase

The population evolves as the GA proceeds. Sometimes the population has a low diversityfor the process to avoid becoming trapped in a local optimum. To overcome this problem,it has recently become popular to apply a mechanism called the restart phase. We use arestart phase based on the ideas of a similar scheme used by Ruiz et al. (2006). This worksas follows. If the best-seen makespan is not promoted for more than a pre-specifiednumber of generations (no_change), the restart phase commences to regenerate thepopulation by the following process:

(1) sort the population in ascending order of makespan;(2) skip the first 25% of individuals from the sorted list (the best individuals); and(3) from the remaining 75% of individuals, 50% are replaced by single-point

mutations of the first 25% best individuals (one single mutation). The 50% areregenerated randomly from the solution space.

With the above processes, we expect to diversify the current population, and as a resultmaintain the chance of finding a better solution.

3.5 Local search

Local search performs a quick exploration around a solution. Making use of thissystematic procedure locates us in a better neighbourhood of the current solution.

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The purpose of the local search is to examine the possibility of finding a solution with

a better objective function. The effectiveness of enhancing a GA by amalgamating it with

a local search (HGA) has frequently been explored by researchers (Ruiz et al. 2005, 2006,

Ruiz and Stutzle 2008).In this work, a local search is applied to the best individual of each generation, but not

to all individuals in the population. The local search that we utilised is very simple and can

be described as follows. The RK of the first operation in the sequence x (x1) is randomly

regenerated. If this new sequence v results in a better objective function, the current

solution x is replaced by the new sequence v.This procedure iterates for all the subsequent operations in the sequence x. The process

of the local search for current solution x terminates if we observe any improvement in

ith5g, where g is the number of operations involved. The local search procedure is

presented in Figure 4. The general outline of the genetic algorithm with all foregoing

features is summarised in Figure 5.

Figure 5. The general outline of HGA.

Figure 4. The procedure of the local search.

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4. Calibration of the parameters and operators of the proposed genetic algorithm

Appropriate design of the parameters and operators has a significant impact on theefficiency of the genetic algorithm. The choice of the GA parameters and operators highlydepends on the type of problem. However, most users often adjust the parameters andoperators manually based on the reference values of previous, similar literature. In thissection, we study the behaviour of the different operators and parameters of the proposedHGA. All different combinations of the aforementioned factors and parameters yieldmany alternative HGAs.

In order to calibrate the algorithms, there are several ways to statistically design theexperimental investigation, but the most frequently used and exhaustive approach is a fullfactorial experiment (Montgomery 2000, Ruiz et al. 2005). This approach is not alwaysefficient because it becomes increasingly difficult to carry out investigations when thenumber of factors becomes significantly large. To reduce the number of required tests,a fractional factorial experiment (FFE) was developed (Cochran and Cox 1992). FFEallows only a portion of the total possible combinations to estimate the main effect of thefactors and some of their interactions. Taguchi (1986) developed a family of FFE matricesthat eventually reduces the number of experiments, but still provides sufficientinformation. In the Taguchi method, orthogonal arrays are used to study a largenumber of decision variables with a small number of experiments.

The traditional method of factorial design is to exploit all possible combinations andconditions in an experiment comprising multi-factors. For example, if the factorial designis implemented for four three-level factors, the total number of trials required would be afull combination of 81 (34) trials, rather than nine trials by the orthogonal array L9(3

4).Taguchi separates the factors into two main groups: controllable and noise factors.

Noise factors are those over which we have no direct control. Since elimination of the noisefactors is impractical and often impossible, the Taguchi method seeks to minimise theeffect of noise and to determine the optimal level of the important controllable factorsbased on the concept of robustness (Tsai et al. 2007). In addition to determining theoptimal levels, Taguchi establishes the relative significance of individual factors in terms oftheir main effects on the objective function.

Taguchi created a transformation of the repetition data to another value which isthe measure of variation. The transformation is the signal-to-noise (S/N) ratio, whichexplains why this type of parameter design is called a robust design (Phadke 1989,Al-Aomar 2006). Here, the term ‘signal’ denotes the desirable value (response variable)and ‘noise’ denotes the undesirable value (standard deviation). So the S/N ratioindicates the amount of variation present in the response variable. The aim is tomaximise the signal-to-noise ratio.

Taguchi classifies objective functions into three categories: the smaller-the-better type,the larger-the-better type, and the nominal-is-best type. Since almost all objectivefunctions in scheduling are classified in the smaller-the-better type, the corresponding S/Nratio (Phadke 1989) is

S=N ratio ¼�10 log10ðobjective functionÞ2:

As mentioned earlier, in this study, the control factors are: selection mechanism, thepopulation size, the percent reproduction, the probability of mutation, the no_changeparameter in the restart phase, and the mutation and crossover operators. Different levelsof these factors are shown in Table 2.

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The associated degree of freedom for these seven factors is 16. Therefore, the selected

orthogonal array should have a minimum of 17 rows and seven columns to accommodate

the seven factors. From the standard table of orthogonal arrays, L18 is selected as thefittest orthogonal array design that fulfills all our minimum requirements. But this

orthogonal array still entails some modifications to adapt itself to our experimental design.For clarity, the way in which we adjust our orthogonal array is as follows. Since our

problem consists of one two-level factor, five three-level factors and one six-level factor,and L18 is composed of six three-level factors and one six-level factor, our problem has

to be transformed structurally to mold itself to the standard shape of L18. In doing so,

for each factor that lacks a certain number of levels in comparison with the standard L18,we compensate for this lack by the assignment of extra levels in standard L18 to one of the

optionally selected existing levels of our associated factor. For example, in our case,consider factor A, which has two levels and lacks one level with regard to standard L18.

The extra level in standard L18 is offset by the repetition of the second level in this factor.

In other words, the extra level of standard L18 is assigned to level 2 of factor A to eliminatethis gap. The modified orthogonal array L18 is presented in Table 3.

A set of 32 instances was generated as follows. The set of instances comprises eight

combinations of n and m, i.e. n¼ {15, 20, 30, 50, 100} and m¼ {15, 20}, without considering

the combinations (15, 20) and (100, 15). The processing times and setup times weregenerated from uniform distributions between the interval (1, 99) and (1, 50), respectively.

There are four replicates for each combination, thus summing to 32 instances. To yield

more reliable information we tackled each instance three times. Therefore, we have 96results for each trial for the statistical analyses.

Table 2. Factors and factor levels.

Factor Symbol Levels Type

Selection mechanism A 2 A(1)—RankingA(2)—Tournament

Population size B 3 B(1)—30B(2)—50B(3)—80

Percent reproduction C 3 C(1)—30%C(2)—20%C(3)—10%

Probability of mutation D 3 D(1)—0.05D(2)—0.1D(3)—0.2

no_change parameter in the restart phase E 3 E(1)—10E(2)—15E(3)—20

Mutation operator F 3 F(1)—Single pointF(2)—InversionF(3)—Swap

Crossover operator G 6 G(1)—MCWG(2)—PUCG(3)—MPPXG(4)—OXG(5)—OPG(6)—PMX

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In order to conduct the experiments, we implemented HGA in MATLAB 7.0 run on apersonal computer with a 2.0GHz Intel Core 2 Duo processor and 1 GB RAM memory.We use the relative percentage deviation (RPD) for the makespan as a commonperformance measure to compare the methods. The best solutions obtained for eachinstance (denoted Minsol) are calculated. RPD is obtained from the formula

RPD ¼Algsol �Minsol

Minsol� 100, ð1Þ

where Algsol is the makespan obtained for a given algorithm and instance. Clearly, lowervalues of RPD are preferable. The stopping criterion is set to a CPU time limit fixedat n2�m� 0.5 milliseconds. This stopping criterion not only permits more time as thenumber of jobs or machines increases, but is also more sensitive to an increase in thenumber of jobs than the number of machines.

After obtaining the results of the Taguchi experiment for all the trials, RPDs areindividually transformed into S/N ratios. Figure 6 shows the average S/N ratio obtained ateach level. As indicated in Figure 6, the optimal level of the factors A, B, E, F and G clearlybecomes A(2), B(1), E(1), F(2) and G(2), respectively. However, determining the optimallevel for factors C and D necessitates more investigation. In order to do so, we analyse theresults of the experiment using a different measure, the response variable (RPD). Theresults for each level are shown in Figure 7. This analysis strongly supports our decisionwith respect to the optimal level for factors A, B, F, E and G. It finally turns out that C(1)and D(2) are the preferable levels for factors C and D, respectively. To explore the relativesignificance of individual factors in terms of their main effects on the objective function,analysis of variance (ANOVA) was conducted. The results of the analysis are presentedin Table 4. The crossover operator factor has the greatest effect on the quality of the

Table 3. The modified orthogonal array L18.

Control factor level

Trial A B C D E F G

1 A(1) B(1) C(1) D(1) E(1) F(1) G(1)2 A(1) B(1) C(2) D(2) E(3) F(3) G(2)3 A(1) B(2) C(1) D(3) E(3) F(2) G(3)4 A(1) B(2) C(3) D(1) E(2) F(3) G(4)5 A(1) B(3) C(2) D(3) E(2) F(1) G(5)6 A(1) B(3) C(3) D(2) E(1) F(2) G(6)7 A(2) B(1) C(1) D(3) E(2) F(3) G(6)8 A(2) B(1) C(3) D(1) E(3) F(2) G(5)9 A(2) B(2) C(2) D(2) E(2) F(2) G(1)10 A(2) B(2) C(3) D(3) E(1) F(1) G(2)11 A(2) B(3) C(1) D(2) E(3) F(1) G(4)12 A(2) B(3) C(2) D(1) E(1) F(3) G(3)13 A(2) B(1) C(2) D(3) E(1) F(2) G(4)14 A(2) B(1) C(3) D(2) E(2) F(1) G(3)15 A(2) B(2) C(1) D(2) E(1) F(3) G(5)16 A(2) B(2) C(2) D(1) E(3) F(1) G(6)17 A(2) B(3) C(1) D(1) E(2) F(2) G(2)18 A(2) B(3) C(3) D(3) E(3) F(3) G(1)

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algorithm with a relative importance of 32.7%. After the crossover operator factor, the

population size factor, with a relative importance of 22.14%, is placed in the second rank;

factors C, D, and F have the least effect on the performance of our HGA.In this section, we explore the impact of the different operators and parameters on the

performance of the GA. The research finding is that tuning the operators and parameters

with meticulous care can influence the performance of the GA. For example, a good choice

Table 4. ANOVA table for the S/N ratio.

Factor df SS MS FPercent of X

(relative importance) Cumulative

G 5 7.8247 1.5649 8.4 32.7 32.7B 2 5.0406 2.5203 13.6 22.1 54.8A 1 3.2169 3.2169 17.3 14.4 69.2E 2 1.8321 0.9160 4.9 6.9 76.1F 2 1.2659 0.6330 3.4 4.2 80.3D 2 1.1993 0.5996 3.2 3.9 84.2C 2 0.5216 0.2608 1.4 0.7 84.9Error 1 0.1857 0.1857

Total 17 21.0867

Figure 7. The mean RPD plot for each level of the factors.

Figure 6. The mean S/N ratio plot for each level of the factors.

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for the PUC crossover operator results in an RPD that is at least 1.5% lower than other

crossover operators, or a population size of 30 has an RPD that is almost 1% lower than

the other alternatives. In sum, the chosen levels were as follows: selection mechanism,

tournament; population size, 30; percent reproduction, 20%; probability of mutation, 0.1;

no_change parameter in the restart phase, 10; mutation operator, inversion; and crossover

operator, PUC.

5. Experimental evaluation

In this section, we compare our proposed hybrid genetic algorithm (HGA) with other

existing methods. We compare HGA with the genetic algorithm of Cheung and Zhou

(2001) (GA_Cheung) and the immune algorithm of Zhou et al. (2006) (IA_Zhou) as

previous meta-heuristics proposed in the SDST job shop literature, and the shortest

processing time (SPT) of Sule (1997) as a well-known dispatching rule. To evaluate the

effectiveness of hybridisation with local search, a pure GA was also compared.We implement the algorithms in MATLAB 7.0 run on a personal computer with a

2.0GHz Intel Core 2 Duo processor and 1 GB RAMmemory. The stopping criterion used

when testing all instances with the algorithms was n2�m� 0.5 millisecond computational

time. In addition to allowing for more time as the number of jobs or machines increases

and being more sensitive towards an increase in the number of jobs than the number of

machines, initial tests showed that the algorithms usually attained their best solutions

within this amount of time. Figure 8 shows how to improve the makespan over time as an

example. As plotted, with this stopping criterion, we allow the algorithms sufficient time to

explore the search space.

5.1 Data generation

The data required for the problem include the number of jobs, the number of machines,

processing times and setup times. The way in which we generate instances is based on

Taillard (1993) benchmarks. There are eight configurations of the number of jobs n and

Figure 8. How to improve makespan over time.

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the number of machines 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 the Taillard instances. We have four levels forSDST, 25%, 50%, 100% and 125% of the maximum possible processing times; therefore,SDSTs are generated from four uniform distributions, U(1, 25), U(1, 50), U(1, 100) andU(1, 125), similar to Ruiz et al. (2005). We generated 10 instances for each combination ofn, m and SDST, summing to 320 instances.

5.2 Experimental results

The results of the experiments were transformed into the RPD measure (Equation (1)),averaged for each combination of n and m (40 data points per average) and are shown inTable 5. The results demonstrate the high performance of HGA with a RPD of 0.76% withrespect to the other algorithms. The worst performing algorithm is the SPT dispatchingrule with a RPD of 13.66%. GA_Cheung is more competitive than IA_Zhou.

For a further precise analysis of the results, we carried out an analysis of variance(ANOVA). It should be noted that, when using ANOVA, three main hypotheses,normality, homogeneity of variance and independence of the residuals, must be checked.We did that and found no basis for questioning the validity of the experiment. The meansplot and LSD intervals for the different algorithms are shown in Figure 9. As can be seen,HGA statistically supersedes the other algorithms. To assess the robustness of thealgorithms in different situations, we analysed the possible effects of the problem size(number of jobs) and the magnitude of the setup times on the performance of thealgorithms. A means plot for the interaction between the factors, type of method and thenumber of jobs is shown in Figure 10. It is interesting to see that HGA exhibits robustperformance regarding increased number of jobs and outperforms the other algorithms inall cases.

Figure 11 clarifies the interaction between the quality of the algorithm and themagnitude of the SDST. Surprisingly, there is a clear trend that increasing the size of theSDST results in better performance of the HGA. Comparing the behaviour of HGA versusthe pure GA functionally shows that hybridising with the type of local search improvesthe performance of the GA remarkably in environments with long setup times.

Table 5. Average relative percentage deviation (RPD) for the algorithms grouped by n and m.

Algorithm

n m SPT GA_Cheung IA_Zhou HGA GA

15 15 14.39 4.93 8.47 1.10 1.9020 15 17.12 4.84 9.96 0.87 3.62

20 13.36 4.00 9.47 0.30 3.3030 15 17.06 5.67 9.04 0.44 3.78

20 14.89 3.99 6.43 1.14 4.0150 15 14.91 6.20 7.07 0.55 4.28

20 11.39 4.05 4.94 1.12 4.03100 20 6.16 2.16 5.66 0.55 2.32

Average 13.66 4.48 7.63 0.76 3.41

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Figure 9. Means plot and LSD intervals (at the 95% confidence level) for the type of algorithmfactor.

Figure 11. Means plot for the interaction between algorithm type and magnitude of SDST.

Figure 10. Means plot for the interaction between algorithms type and number of jobs.

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In the previous evaluation, our proposed HGA was compared with two other meta-heuristics from the literature, the SDST job shop and the SPT dispatching rule. It isinteresting to note that all the tests and analyses support the high performance of ourproposed HGA.

6. Conclusion and future research

In this paper, we have proposed a new genetic algorithm for job shop scheduling withsequence-dependent setup times to minimise the makespan. Our proposed geneticalgorithm was hybridised with two additional features including a simple form of localsearch and a diversification mechanism called the restart phase. We exhaustivelyexplored the different operators and parameters of the genetic algorithm by means ofthe Taguchi method. We adapted the operation-based encoding scheme for a veryeffective crossover, namely parameterised uniform crossover. To evaluate theeffectiveness and robustness of the proposed GA, we conducted a comparison betweenour HGA and two other meta-heuristics in the literature, the SDST job shop and awell-recognised dispatching rule. The results of our benchmark demonstrated thatHGA outperforms the other algorithms.

As a direction for future research, it would be interesting to work on other meta-heuristics, such as particle swarm optimisation and electromagnetic-like, and comparethem with our HGA, or to examine the performance of our algorithm in other complexscheduling problems, such as the flexible job shop and an open shop, to see whether thehigh performance of our HGA is transferable to other scheduling problems. Anotherdirection for future research is to consider other realistic assumptions, such as machineavailability constraints and transportation times between stages. Another opportunity forresearch is to consider the problems with other optimisation objectives, such as totalweighted tardiness or total completion time, or even multi-objective cases.

References

Al-Aomar, R., 2006. Incorporating robustness into Genetic Algorithm search of stochastic

simulation outputs. Simulation Modeling Practice and Theory, 14, 201–223.Allahverdi, A., et al., 2008. A survey of scheduling problems with setup times or costs. European

Journal of Operational Research, 187 (3), 985–1032.

Baker, K.R., 1974. Introduction to sequencing and scheduling. New York: Wiley.Brucker, P. and Thiele, O., 1996. A branch-and-bound method for general shop problem with

sequence-dependent setup times. OR Spectrum, 18, 145–161.Caprihan, R. and Wadhwa, S., 2005. Scheduling of FMSs with information delays: a simulation

study. International Journal of Flexible Manufacturing System, 17, 39–65.Chandrasekaran, M., et al., 2006. Solving job shop scheduling problems using artificial immune

system. International Journal of Advanced Manufacturing Technology, 31, 580–593.

Cheng, B.W. and Chang, C.L., 2007. A study on flowshop scheduling problem combining Taguchi

experimental design and genetic algorithm. Expert Systems with Applications, 32, 415–421.Cheng, R., Gen, M., and Tsujimura, Y., 1999. A tutorial survey of job-shop scheduling problems

using genetic algorithms, part II: hybrid genetic search strategies. Computers & Industrial

Engineering, 36, 343–364.Cheung, W. and Zhou, H., 2001. Using genetic algorithms and heuristics for job shop scheduling

with sequence-dependent setup times. Annals of Operations Research, 107, 65–81.

5974 B. Naderi et al.

Dow

nloa

ded

by [

Upp

sala

uni

vers

itets

bibl

iote

k] a

t 01:

31 0

7 O

ctob

er 2

014

Choi, I. and Korkmaz, O., 1997. Job shop scheduling with separable sequence-dependent setups.

Annals of Operations Research, 70, 155–170.Cochran, W.G. and Cox, G.M., 1992. Experimental designs. 2nd ed. New York: Wiley.Coleman, B.J., 1992. Technical note: a simple model for optimizing the single machine early/tardy

problem with sequence-dependent setups. Production and Operations Management, 1, 225–228.Davis, L., 1985. Applying adaptive algorithms to epistatic domains. In: Proceedings of the

international joint conference on artificial intelligence, (IJCAI) Los Angeles, 162–164.

Eddie, Y.K., Ng, E.W., and Kee, N., 2006. Parametric study of the biopotential equation for breast

tumour identification using ANOVA and Taguchi method. International Federation for

Medical and Biological Engineering, 44, 131–139.Goldberg, D.E. and Lingle, R., 1985. Alleles, loci and the traveling salesman problem. In: J.J.

Grefenstette, ed. In: Proceedings of the first international conference on genetic algorithms.

Hillsdale, NJ: Lawrence Erlbaum Associates, 154–159.Goldberg, D.E., 1989. Genetic algorithms in search, optimisation and machine learning. Reading, MA:

Addison-Wesley.

Goncalves, J.F., Mendes, J.J.D.M., and Resende, M.G.C., 2005. A hybrid genetic algorithm for the

job shop scheduling problem. European Journal of Operational Research, 167, 77–95.Gupta, S.K., 1986. N Jobs and nmachines job-shop problems with sequence-dependent set-up times.

International Journal of Production Research, 20 (5), 643–656.Heinonen, J. and Pettersson, F., 2007. Hybrid ant colony optimisation and visibility studies applied

to a job-shop scheduling problem. Applied Mathematics and Computation, 187, 989–998.

Holland, J., 1975. Adaptation in natural and artificial systems. Ann Arbor: University of Michigan

Press.Kurz, M.E. and Askin, R.G., 2003. Comparing scheduling rules for flexible flow lines. International

Journal of Production Economics, 85, 371–388.Kurz, M.E. and Askin, R.G., 2004. Scheduling flexible flow lines with sequence-dependent setup

times. European Journal of Operational Research, 159, 66–82.

Luh, P.B., et al., 1998. Job shop scheduling with group dependent setups, finite buffers, and long

time horizon. Annual of Operation Research, 76, 233–259.Luo, F., et al., 2008. Application of Taguchi’s method in the optimisation of bridging efficiency

between confluent and fresh micro carriers in bead-to-bead transfer of Vero cells.

Biotechnology Letters, 30, 645–649.Michalewicz, Z., 1996. Genetic algorithmsþ data structures¼ evolution programs. 3rd ed. Berlin:

Springer.Monma, C.L. and Potts, C.N., 1989. On the complexity of scheduling with batch setup times.

Operations Research, 37, 798–804.

Montgomery, D.C., 2000. Design and analysis of experiments. 5th ed. New York: Wiley.Norman, B.A. and Bean, J.C., 1999. A genetic algorithm methodology for complex scheduling

problems. Naval Research Logistics, 46, 199–211.Ombuki, B. and Ventresca, M., 2004. Local search genetic algorithms for the job shop scheduling

problem. Applied Intelligence, 21, 99–109.

Park, B.J., Choi, H.R., and Kim, H.S., 2003. A hybrid genetic algorithm for the job shop scheduling

problems. Computers & Industrial Engineering, 45, 597–613.Phadke, M.S., 1989. Quality engineering using robust design. Engelwood Cliffs, NJ: Prentice-Hall.Pinedo, M., 1995. Scheduling: theory, algorithms and systems. Engelwood Cliffs, NJ: Prentice-Hall.

Ruiz, R., Maroto, C., and Alcaraz, J., 2005. Solving the flowshop scheduling problem with sequence

dependent setup times using advanced metaheuristics. European Journal of Operational

Research, 165, 34–54.Ruiz, R. and Maroto, C., 2006. A genetic algorithm for hybrid flowshops with sequence dependent

setup times and machine eligibility. European Journal of Operational Research, 169, 781–800.Ruiz, R., Maroto, C., and Alcaraz, J., 2006. Two new robust genetic algorithms for the flowshop

scheduling problem. Omega, 34, 461–476.

International Journal of Production Research 5975

Dow

nloa

ded

by [

Upp

sala

uni

vers

itets

bibl

iote

k] a

t 01:

31 0

7 O

ctob

er 2

014

Ruiz, R. and Stutzle, T., 2008. An iterated greedy heuristic for the sequence dependent setup timesflowshop problem with makespan and weighted tardiness objectives. European Journal ofOperational Research, 187 (3), 1143–1159.

Schutten, J.M.J., 1998. Practical job shop scheduling. Annals of Operations Research, 83, 161–178.

Sule, D.R., 1997. Industrial scheduling. New York: PWS Publishing Company.Taguchi, G., 1986. Introduction to quality engineering. White Plains: Asian Productivity

Organization/UNIPUB.

Taillard, E., 1993. Benchmarks for basic scheduling problems. European Journal of OperationalResearch, 64, 278–285.

Tsai, J.T., et al., 2007. Improved immune algorithm for global numerical optimisation and job-shop

scheduling problems. Applied Mathematics and Computation, 194, 406–424.Viond, V. and Sridharan, R., 2008. Dynamic job-shop scheduling with sequence-dependent setup

times: simulation modeling and analysis. International Journal of Advanced Manufacturing

Technology, 36, 355–372.Watanabe, M., Ida, K., and Gen, M., 2005. A genetic algorithm with modified crossover operator

and search area adaptation for the job-shop scheduling problem. Computers & IndustrialEngineering, 48, 743–752.

Zandieh, M., Fatemi Ghomi, S.M.T., and Moattar Husseini, S.M., 2006. An immune algorithmapproach to hybrid flow shops scheduling with sequence-dependent setup times. Journal ofApplied Mathematics and Computation, 180, 111–127.

Zhang, C.Y., et al., 2008. Very fast TS/SA algorithm for the job shop scheduling problem.Computers & Operations Research, 35, 282–294.

Zhou, C. and Egbelu, P.G., 1989. Scheduling in a manufacturing shop with sequence-dependent

setups. Robotics and Computer Integrated Manufacturing, 51, 73–81.Zhou, Y., Beizhi, Li., and Yang, J., 2006. Study on job shop scheduling with sequence-dependent

setup times using biological immune algorithm. International Journal of AdvancedManufacturing Technology, 30, 105–111.

5976 B. Naderi et al.

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Upp

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