a hybrid particle-swarm tabu search algorithm for solving job shop scheduling problems

2044 IEEE TRANSACTIONS ON INDUSTRIAL INFORMATICS, VOL. 10, NO. 4, NOVEMBER 2014

A Hybrid Particle-Swarm Tabu Search Algorithmfor Solving Job Shop Scheduling Problems

Hao Gao, Sam Kwong, Fellow, IEEE, Baojie Fan, and Ran Wang

Abstract—This paper proposes a method for the job shopscheduling problem (JSSP) based on the hybrid metaheuristicmethod. This method makes use of the merits of an improvedparticle swarm optimization (PSO) and a tabu search (TS)algorithm. In this work, based on scanning a valuable regionthoroughly, a balance strategy is introduced into the PSO forenhancing its exploration ability. Then, the improved PSO couldprovide diverse and elite initial solutions to the TS for makinga better search in the global space. We also present a new localsearch strategy for obtaining better results in JSSP. A real-integer encode and decode scheme for associating a solution incontinuous space to a discrete schedule solution is designed forthe improved PSO and the tabu algorithm to directly applytheir solutions for intensifying the search of better solutions.Experimental comparisons with several traditional metaheuristicmethods demonstrate the effectiveness of the proposed PSO–TSalgorithm.

Index Terms—Global search, job shop scheduling, particleswarm optimization (PSO), tabu search (TS).

I. INTRODUCTION

S CHEDULING is one of the important issues in manycombinatorial and optimization problems. It is difficult

to be resolved as it is influenced by the number of task,the difference of criteria, and the objective function. In fact,the job shop scheduling problem (JSSP) is one of the mostcomplicated scheduling problems, which is yet unable toprovide a good solution so far [1], [2]. Although differentmethods have been applied to JSSP, the lower bounds of someJSSPs are still not reached. JSSP is a common problem existsin industrial cases [3]. In the field of computer science, theefficiency of using computers and network mainly depends onhow to distribute the limited resources (e.g., CPU, memorysize, and bandwidth) to multiusers. In [4] and [5], methodsfor JSSP are adjusted and successfully applied to the realindustrial problems of the manufacturers producing industrial

Manuscript received March 17, 2014; revised June 11, 2014; accepted July07, 2014. Date of publication July 23, 2014; date of current version November04, 2014. This work was supported in part by the Introduction Foundation forthe Talent of Nanjing University of Posts and Telecommunications under GrantNY212025, in part by the National Natural Science Foundation of China underGrant 61203270, and in part by the China Postdoctoral Science Foundationunder Grant SBH14015. Paper no. TII-14-0331.

H. Gao and B. Fan are with the College of Automation, Nanjing Uni-versity of Posts and Telecommunications, Nanjing 210023, China. (e-mail:[email protected]; [email protected]).

S. Kwong and R. Wang are with the Department of Computer Sci-ence, City University of Hong Kong, Hong Kong 999077, China (e-mail:[email protected]; [email protected]).

Color versions of one or more of the figures in this paper are availableonline at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TII.2014.2342378

wheels, brake drum, and castors in rubber. Since Fisher andThomson presented the first benchmark of JSSP in 1963 [6],many researches proposed different approaches to solve theseproblems [7]–[9]. However, these traditional algorithms, ingeneral, require a high computational time with the increasein the problem’s size.

Optimization refers to the process of finding the best pos-sible solution of a problem in finite time, which influencesdifferent types of problems in our life [10]–[13]. Comparedwith the traditional optimization and iterative methods, meta-heuristics make few assumptions on the optimization problemand usually can find good solutions with less computationaleffort [14]. The success of metaheuristic algorithms relies ontheir abilities in providing a balance between the exploration(diversification) and the exploitation (intensification). Accord-ing to their search strategies, metaheuristics can be classifiedinto two categories. One type is single-solution-based localsearch algorithms that include simulated annealing (SA) andtabu search (TS). The other type is population-based searchalgorithms which have a learning component that includesant colony optimization (ACO), particle swarm optimization(PSO), genetic algorithm (GA), and immune algorithm (IA).Generally speaking, basic single-solution-based metaheuristicsshow more local search ability, whereas basic population-based metaheuristics have more global search ability.

Several metaheuristic algorithms have been introduced intoJSSP because of their fast-computing abilities. These intel-ligent optimization algorithms can often find good solutionswith less computational effort and have shown considerablesuccess in solving different kinds of scheduling problems[15]–[18]. As the results shown in [15], an improved tabualgorithm named TSAB is very efficient for solving JSSPdue to its strong local search ability in metaheuristics. Butits performance largely depends on an efficient and diverseinitialization method. It is known that the population-basedmetaheuristics can provide more diverse solutions than thesingle-based metaheuristics. Therefore, choosing a population-based metaheuristic for initialization can give the TSAB moreopportunities to explore in the entire solution space.

In [16], Nowicki and Smutnick stated that search in thelocal area between two local optima should find a new localoptimum or even a global one in JSSP. With componentsnamed pbest and gbest that are, respectively, contributed bythe best solutions achieved by an individual particle and allparticles, PSO algorithm has the characteristic of “memory theelite” compared with other population-based metaheuristics.According to [16], it is feasible to search in the local area

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GAO et al.: HYBRID PARTICLE-SWARM TS ALGORITHM FOR SOLVING JSSPs 2045

between pbest and gbest for getting a more accurate solutionof JSSP. Furthermore, compared with the other population-based metaheuristics, PSO shows fast convergence rate whichcan shorten the computational time of JSSP. Then, choosingthe PSO algorithm to act as the source of the initializationsolution for the TSAB should be a better choice. However,compared with other population-based metaheuristics, previ-ous study reveals that the PSO algorithm may be trapped intolocal optima on its way to the global optimum especially on thehard problem [17]. Then, if we can find an effective method toenhance its exploration ability, the improved algorithm shouldprovide elite and diverse solutions for the initialization ofTSAB. In this paper, we first present a new PSO algorithmwhich not only enhances its exploration ability but alsomaintains its fast convergence rate. Then by combining themerits of TSAB and the improved PSO, we propose a hybridoptimization algorithm (IPSO–TSAB) in which we can use theTSAB to intensify the search in the space determined by theIPSO process. In the proposed algorithm, a new real-integerscheme is incorporated to transform the solutions between theimproved PSO and TSAB, and a local search method in theimproved PSO is used to make a better search between pbestand gbest regions.

The paper is organized as follows. In Section II, we providea brief outline of related work on applying metaheuristicsto solve JSSP. Section III introduces the formulation of theJSSP models. In Section IV, a hybrid particle swarm withtabu algorithm to JSSP is presented. Section V presents theexperimental results and discussion of the proposed method.Finally, conclusion is given in Section VI.

II. RELATED WORK

Due to its simplicity, fast-computing ability, and not problem-specific, metaheuristics have been broadly applied to JSSP. Inthis section, we briefly review related work on this topic.

With more opportunities to search in the neighborhoodfor the potential solution, some single-solution-based meta-heuristics have been successfully applied to JSSP. By usingan improved sampling space, Song et al. [18] introduced animproved SA algorithm to deal with JSSP. The algorithmproposed by Nowicki and Smutnicki [15], which is namedas TSAB, provided significant progress in the area of JSSP.They introduced a new neighborhood definition into the TS.Then, the new approach helps an individual in the TSAB tomake more precise search near the local optimum. It alsoprovides new upper bounds to many instances of JSSP forwhich the global optimum is still unknown. The same authorsfurther their work (i-TSAB) [16] by adding a new approximatemechanism and a path relinking technique. This new algorithmachieved better results with less computational time than theTSAB algorithm. In [19], Jain revealed that the achievementof the TS algorithm is deeply influenced by its initializationprocedure. Then, if we provide elite and diverse solutionsfor the algorithm to search, it should effectively enhance itsperformance on JSSP.

Recently, with the development of the population-basedmetaheuristics, they have been extensively applied to JSSP.In [20], Qing and Wang proposed a hybrid GA for JSSP.

To enhance the global and local search abilities of GA, amixed selection operator and a local search operator weredesigned, respectively. By introducing crossover and mutationoperators into the PSO algorithm, Lian et al. [21] present anew method to solve the JSSP. The results obtained by theimproved algorithm are more efficacious than the comparedalgorithms. Since the population-based metaheuristics are notwell suited for promoting exploitation ability in the searchspace, they show less contribution to the JSSP than thesingle-based metaheuristics especially for TSAB algorithm.But compared with the local metaheuristics, the population-based metaheuristics show more exploration ability and havemore opportunities to get closer to the global optimum. Inview of the shortcoming of the local and population-basedlearning metaheuristics in JSSP, it is apparent that if the currentbarriers within JSSP are to be overcome, hybrid approachesare worthwhile considering. Zhang and Wu [22] proposeda hybrid immune mechanism incorporated with a simulatedannealing algorithm (ISA) for the JSSP. In the ISA, the globalsearch ability of the algorithm mainly depends on the SAmechanism, and the immune mechanism enables the hybridalgorithm to fine-tune to the global optimum. In [23], Linet al. presented a new hybrid algorithm consists of PSO,SA, and multitype individual enhancement scheme for solvingJSSP, where SA is used to make precise search near thesolution found by the PSO algorithm. However, consideringthe performance of applying SA to JSSP, the local searchability of the hybrid algorithm is still not very powerful. Basedon a differential evolutionary (DE) algorithm for identifyingpotential solution areas in the global solution space [24],Ponsich and Coello Coello used the TS to determine the localoptimum within such regions. A hybrid PSO for JSSP problemis proposed in [25]. To better suit the discrete solution spaceof JSSP, the authors modify the updating equation of PSOand the particle’s position separately. Then, the hybrid PSOachieves a better performance than other compared algorithms.Huang and Liao [26] present a hybrid algorithm by combiningan ACO algorithm with a TS algorithm for the JSSP. Theproposed ACO algorithm introduces a novel decompositionmethod to construct a potential solution. Then the tabu isembedded to obtain further precise search.

Although the aforementioned algorithms achieve favorableresults in JSSP, how to get better results in the solutionspace still remains as a problem. As mentioned in SectionI, combining the merits of the PSO and tabu algorithms is aneffective method to solve JSSP. Since the PSO may be trappedinto poor local optima, many methods have been proposedfor enhancing its global search ability. Shi and Eberhart [27]introduce an inertia weight operator with linear decreasingmethod to the traditional PSO. Since the value of inertiaweight in the improved algorithm is small in the final stage ofiteration, the global search ability of the improved algorithmis still weaker. Zhao et al. [28]proposed an interior pointmethod for improving the achievements of PSO. Due to its lessopportunities to exploit in the local space, the convergence rateof the improved algorithm could be further accelerated. Linget al. [29] introduced a wavelet mutation operator into PSO(HWPSO). However, the wavelet mutation operator generates


only small values in the final stage of iteration. Then, HWPSOhas little chance to jump out of the local optimum. In [30], amoderate search operator is applied to enable particles have amoderate opportunity to jump out of the local optimum. Butthe improved algorithm shows a weaker fine-tuning abilityin small to mid-range regions. Then, it could not explorethe solution space thoroughly. Furthermore, previous worksrelated to the application of PSO on JSSP [21], [23], [25] donot consider enhancing the exploration ability of the traditionalPSO. Hence, a new PSO algorithm with a balance strategyis proposed in this paper for JSSP. Compared with otherimproved algorithm, a leading point using a balance strategyenables a particle to search closely to its own pbest at thebeginning and then gradually down to gbest with the numberof iteration increases. Therefore, the valuable region betweenthese two points in JSSP is exploited thoroughly. A balanceoperator in our algorithm is used for defining the step sizefor the movement of the particle. Differing from the strategiesmentioned above, the operator continuously compensates themomentum of the particle. Then, the solution space of JSSPis more likely explored.

Considering the characteristic of JSSP, the hybrid algorithmin JSSP should solve the issue on how to map a real solutionof metaheuristics into an integer solution of schedule. Thepast related work needs to modify the updating equation ofthe metaheuristics for satisfying the demand of JSSP (e.g.,priority-based representation in [25] and random key encodingscheme in [23]). Then a real solution is mapped into an integersolution by directly applying the updating equation of theoriginal metaheuristics, which needs further research.

III. JOB SHOP SCHEDULING PROBLEM

The JSSP is described as follows: n different jobs are tobe scheduled on m machines, which process one of the moperations in a job. The aim of JSSP is to minimize themakespan, the finishing time Cmax (π) of the last job in anorder π = {j1, j2, . . . , jn}. The major constraints of JSSP aredescribed as follows.

1) All operations must visit each machine only once.2) Each operation of a job has precedence constraints.3) Each machine processes only one operation at a time.4) Each job cannot be processed simultaneously on differ-

ent machines.The basic model with makespan objective is described as

follows:

π∗= arg{Cmax (π)} → min ∀π ∈ Π.

The flow shop scheduling problem is a special case of JSSPwhere all the jobs are processed in the same processing orderand also has the practice importance in the industrial cases[31], [32] (e.g., in chemical or apparel production processes).To illustrate the process of JSSP, we use a schedule system,which consists of two machines, three jobs, and six operations.The processing order and the running time of each operationare described in Table I. The processing time of operation ρ11on machine 1 is 6 and the processing time of operation ρ12 onmachine 2 is 4. The operation ρj1 must be handled before ρj2for a job j. A processing order (ρ21, ρ11, ρ22, ρ31, ρ32, ρ12) is

TABLE IEXAMPLE OF n ∗m = 3 ∗ 2 JSSP

Fig. 1. Gantt chart of a schedule for the n ∗m = 3 ∗ 2 JSSP.

feasible since it satisfies the constraints described above, andthe makespan of this feasible schedule is 16. The resultingGantt chart for the schedule is demonstrated in Fig. 1.

The process of finding a solution of JSSP is to decidehow to arrange the total operations to time intervals onmachines and it is named as a schedule. Since each possibleschedule is independent, maximum of n!m different solutionsin the potential solution space need to be searched and thecomplexity of the JSSP is n!m. The CPU time will be too longespecially for large numbers of machine and job, and this is thereason to introduce the metaheuristic-based methods into JSSP.

IV. IPSO–TSAB: A HYBRID PARTICLE SWARM TABU

ALGORITHM FOR JSSP

In [33], Nowicki et al. present the positive correlationbetween the makespan value for JSSP and the local optimum.It reveals the existence of big valley (BV) phenomenon inthe search space. BV contains a large number of the high-quality solutions dispersed over BV area. Majority of theseoptima concentrates in a relatively small area of the solutionspace. Then, we want to find an optimization method with thefollowing properties.

1) Perfect exploration, since BV contains a large numberof solutions, the method should have power explorationability.

2) Diverse exploitation, to make a precise search which isclose to the potential solutions and probably gets morefavorable results.


Fig. 2. Illustration of the real-integer encoding approach.

In this section, we present a hybrid improved particleswarm with tabu algorithm (IPSO–TSAB) that satisfies theaforementioned requirements. Specifically, goal (1) is realizedby an improved PSO algorithm which guaranties the hybridalgorithm to have more chance to search in the entire potentialsolution space during iteration. Then, goal (2) is realized byapplying the hybrid IPSO and TSAB algorithm together suchthat the improved PSO offers a promising solution region toTSAB for intensifying search. Simply put, the TSAB makesa precise search around the region, which is generated by theimproved PSO algorithm, for the exploitation of BV.

A. Position Representation and Initialization

The original PSO is designed for solving continuous func-tions. However, JSSP is a combinatorial problem with discretesolution space. In practice, researchers should modify theposition representation of PSO from continuous to discrete forJSSP. Differing from the traditional methods, the position ofparticles in our algorithm still consists of continuous variables.Thus, we will design a mechanism for the mapping betweenthe continuous space of PSO and the discrete space of JSSP.

For a JSSP problem with n jobs on m machines, the searchspace in such a problem consists of n ∗m dimensions. Forsimulating a potential integral solution of JSSP, a real potentialsolution of PSO is changed into an integral potential solutionof JSSP by a real-integer encoding approach. After an iterationof IPSO–TSAB, the optimized integer series is transformedinto a n ∗m real PSO solution by a real-integer decodingapproach. A random key encoding approach to transform realnumber series into integer series is presented in [23], but itdid not address the problem of mapping integer series to realseries. In this paper, a real-integer encode and decode schemeis proposed for associating a real PSO solution to an integerschedule solution. Then, it makes users easier to apply anoptimization algorithm in continuous space into the discreteJSSP problems.

For n jobs on m machines JSSP, the dth dimension of oneparticle is represented as xid, which represents the weightof an operation. In the real-integer encoding scheme, xid

is sorted in ascending order. And we use an integer series(φ1, φ2, . . . , φn∗m) to record the order number. With theconstraints specified in Section II, each job consists of se-quential operations, it is easy to transform the integer series(φ1, φ2, . . . , φn∗m) to the job index by ceil (φk/n), in whichceil function returns an integer vector made of rounded up

elements in φ. By scanning this permutation with job indexfrom the beginning to the end, it is transformed into a fea-sible operation sequence. In the proposed algorithm, after aniteration of the improved PSO, the neighborhood of particles’solution should be intensively searched by TSAB algorithm.Then the feasible operation sequence must be transformed intoa processing order of operations on machines which meets thedemand of TSAB algorithm on JSSP.

We use the JSSP system in Table I to illustrate the procedureof the real-integer encoding approach. The position of theparticle xi is 0.64, 0.94, 0.61, 0.52, 0.17, and 0.23. Then,it is sorted in ascending order and we get an integer seriesP1 = (5, 6, 4, 3, 1, 2). The integers 3 and 5 in P1 indicatethat these two operations belong to jobs 2 and 3, respectively,since ceil (3/2) = 2 and ceil (5/2) = 3. According to the real-integer encoding approach, we obtain an operation sequenceP2 = (3, 3, 2, 2, 1, 1), which consists of job indexes. Inthis sequence, the first “3” represents the first operation ofjob 3 and the second “3” represents the second operationof job 3. By scanning P2 from the beginning to end, weencode this permutation into an operation sequence P3 =(o31, o32, o21, o22, o11, o12). The element oij of P3 repre-sents the jth jobs on i machine. Then, a feasible initial solutionof TSAB named P4 = (5, 6, 3, 4, 1, 2) is obtained by

P4k = (i− 1) ∗m+ j. (1)

An illustration from a solution of PSO to a solution of TSABby using the real-integer encoding approach is shown in Fig. 2.The makespan of this schedule is 20.

In order to apply the updating equation of the improvedPSO, the real-integer decoding approach should map an opti-mized discrete solution of TSSB to the corresponding PSO’scontinuous solution. The outline of the real-integer decodingapproach is described as follows.

Step 1) Let B be the solution of a schedule and map an ∗m vectorsolution N of the TSAB algorithm into a m ∗ nmatrix M in which element oij in M represents the job indexof the jth operation on machine i. We use F to record thefirst column operation of each machine in M and S to recordthe serial number of operation in a job which matches theone in F .

Step 2) For k = 1 : length (F ).If the predecessor operation of F (k) in a job has been

scheduled, then find the same job index with F (k) in the Fand store them into the vector C.


Fig. 3. Illustration of the real-integer decoding approach.

Step 3) Find the operation d in C with the smallest valuein S, i.e., the one with the highest priority to be proceeded ina job, and append it to B. Delete d from F and remove theelements in C, then append its following operator of machinei to F .

Step 4) Update S based on the updated F .Step 5) isempty (F ) �= 1 go to Step 2).Step 6) Sort the original continuous solution of PSO in

ascending order by using B as the serial number.Fig. 3 demonstrates an example of the real-integer decoding

approach. The makespan of this schedule is 12.

B. Particle Swarm Optimization

In 1995, PSO was first presented by Kennedy and Eberhart[34]. Recently, it has been actively studied and applied forsolving different real optimization problems [35]–[37]. Eachparticle in PSO makes use of position and velocity-updatingequations to decide its position of the next iteration. Theparticle’s movement is influenced by its own and populationexperiences, named as pbest and gbest. Particles in the basicPSO are updated using

vid(t+ 1) = wvid(t) + c1rand1 ∗ (ppid − xid(t))

+ c2rand2 ∗ (pgd − xid(t)) (2)

xid(t+ 1) = xid(t) + vid(t+ 1) (3)

where w is the inertia weight which guaranties the convergenceof particles. c1 and c2 are the two acceleration coefficients.rand1 and rand2 denote two uniformly distributed randomvariables within [0, 1]. vid and xid represent the velocity andposition of individual i on dimension d, respectively. ppid andpgd represent the best position found by particle i and thepopulation separately.

C. An Improved PSO Algorithm

In order to have more opportunities to explore in the globalsearch space and make a precise search in the potentialsolution space, we present an improved PSO algorithm witha balance strategy. The differences between the proposedalgorithm and the works [30], [38], and [39] are listed below.

1) A new attractor using a balance strategy for particles isproposed.

2) A balance operator β is presented for balancing theexploration and exploitation of the new algorithm.

3) A random best position is proposed that each pbest mayprovide its experience for particles during iteration.

Clerc and Kennedy [17] prove that particles are attractedto one point Q = (Q1, Q2, . . . , Qn) during iterations, andeventually converge to this point at the end of iterations. Then,this point can be regarded as the main search direction of PSO

Qd =c1rand1 ∗ ppid + c2rand2 ∗ pgd

c1rand1 + c2rand2. (4)

In [40], Kirkpatric et al. state a classic heuristic: “Anoptimization algorithm should encourage the exploration inthe initial stages and the exploitation ability later.” In view ofthis, we set R as the main search direction of our algorithm

Rd =(1− α) ∗ rand1 ∗ ppid + α ∗ c2rand2 ∗ pgd

(1− α) ∗ rand1 + α ∗ rand2 (5)

in which α = t/MAXITER and t, MAXITER representthe current and the total iterations, respectively. The maindifference between Rd, Q and its improvement [38] is that theformer use linear decreasing and increasing methods to controlthe influences of pbest and gbest, respectively. Thus, particlesled by R will leave from pbest point and then converge to thegbest point gradually. Then the valuable region between thetwo points is exploited thoroughly.

The reason for the traditional PSO be trapped into localoptima is mainly due to particles having poor global searchabilities especially in the final stage of iteration [30]. Ifwe continuously give larger momentum for particles duringiteration, it will improve the exploration of the PSO. Fur-thermore, enabling the particles to focus on valuable regionsoffered by R is an effective way to exploit in the potentialsolution regions. In view of above, a new operator β isproposed, which seeks a balance between the exploration andexploitation abilities of particles. The updating equation of thenew algorithm is presented as follows:

xid(t+ 1) = Rid(t)± α ∗ abs(β) ∗ (rbestd − xid(t)) (6)

β = ln(t/(MAXITER ∗ rand)) (7)

rbest =

{if rand > 0.5 pbestielse pbestk k �= i

. (8)

where Rid defines the main search direction of particle i ondimension d. The definition of β is described in (7). Fig. 4demonstrates its simulation results of 1000 iterations. Com-paring with the acceleration coefficients used in PSO which


Fig. 4. Distribution of β.

generate values uniformly distributed in [0, c1 = c2 = 1.4], thebalance operator β continuously generates relatively largervalues, which help particles to have more opportunities toexplore in the solution space especially move away from smallto mid-range regions. Furthermore, part of values generated byβ are small which enable the particles to intensively exploitaround the local space of Rd. Then the search abilities ofthe particles are balanced. The role of α in (6) is similar to� in PSO which influences the convergence rate of particles.Experiments reveal that α is better to set to 0.5 on most ofthe benchmark functions. rbest term used in (6) is calledthe random best position which is contributed by each pbest.Compared with the learning component used in the traditionalPSO, a particle used in the proposed algorithm has morechances to learn from the experiences of other particles butnot only from the experiences of the population and itself.Since the diversity of population is high at the beginning ofiteration, the particle may have more opportunities to searchin the global space. On the contrary, it can make more precisesearch at the end of iterations since the difference betweenpbests is small.

The improved PSO algorithm not only suits for JSSP butalso can be applied to other common continuous problems.The original updating equation of the improved PSO algorithmis designed for the system with the data of each dimension inparticles is independent, but actually the change of particles oneach dimension in JSSP may influence the order of a schedule.So, in this work, we randomly selected a dimension in Xi toupdate by using (6).

D. Simple Tabu Algorithm

We briefly describe the simple TSAB algorithm used in ouralgorithm.

1) Initial solution: In our algorithm, the initialization solu-tion of TSAB is provided by the improved PSO.

2) The neighborhood structure: The neighborhood inTSAB defines a processing order of operations by apply-ing a swap operation between two adjacent operationsin a job. The local search ability of TSAB is directlyaffected by the neighborhood structure.

Fig. 5. Potential value of gbest and pbest in PSO.

3) Tabu list: The solutions of TSAB admit to the newneighborhood are determined through a memory struc-ture which is named as the tabu list. A tabu list consistsof solutions maxt that have been visited in the recentpast. If a potential solution appears on the tabu list, itcannot be revisited until it reaches an expiration point.

4) Back-Jump tracking: Since our algorithm uses the im-proved PSO to produce global and efficient solutionsto the TSAB, we do not use the back-jump trackingstrategy.

E. A New Local Search Strategy

In [34], Nowicki et al. state that the local area between twolocal optima contains a large number of new optimum evenglobal one in JSSP. According to the definitions of pbest andgbest in PSO system, they record the best solutions of indi-vidual and population found so far, respectively. Obviously,we infer that the region between pbest and gbest may bevaluable for JSSP.

To show the potential value of the region between pbestand gbest in the PSO system, the trajectory of a particle inone dimension for 100 iterations is recorded. We use a typicalexample of nonlinear multimodal function, Rastrigin function,as the benchmark. An optimization algorithm can easily betrapped into the local optima of the function on its searchway. The final results are shown in Fig. 5, and the PSO isinitialized as follows:{v0 ∈ [−5, 5] , x0 ∈ [−5, 5] , c1 = c2 = 2, global optimum = 0,

� = 0.9 → 0.4, dim = 2, max iteration = 100, popsize = 20.

In Fig. 5, we find that the region between gbest and pbestcontains the global optimum 0 and it attracts particles toexploit it. Then, the particles may find the global optimum.This example proves that the area between gbest and pbestis valuable in PSO.

According to the theory in [34] and the example shownabove, a local search strategy is proposed to focus on theregion between gbest and pbest. This local search strategyis described as follows.

Step 1) The discrete TSAB solution of pbesti, which isnamed as ppi, records the best potential schedule achieved


Fig. 6. Pseudocode for the IPSO–TSAB algorithm.

by the ith individual particle, and the best potential scheduleachieved by the whole particles is recorded as gp.

Step 2) for z = 1 : length (ppi),if ppi(z) ∼= gp(z), then find the position gp(j) == ppi(z).Move the ppi(z) toward gp(j) step-by-step, and for each fea-

sible processing orders, we calculate and record the makespansCmax(π(mz→j)).

end ifend forStep 3) Find the minimum of Cmax(π), then update ppi and

gp. We also use the real-integer decoding scheme to updatethe pbesti and gbest.

In order to avoid using this strategy on the same datarepeatedly, we only adopt the local search strategy under thecondition that pbesti is updated.

The steps of the hybrid IPSO–TSAB algorithm are asfollows (Fig. 6).

V. SIMULATION RESULTS AND COMPARISONS

A. Experimental Setup

Extensive experimental results of applying the IPSO–TSABto tackle 55 notoriously difficult instances are listed in thissection. The proposed algorithm is compared with TSAB [15],i-TSAB [16], HPSO [25], HIA [41], DE-TS [24], ACOFT[26], and MPSO [23]. The former two algorithms are thepopular tabu algorithms for JSSP. The later five algorithmsare the hybrid algorithms which inherit the merits of the singleand population metaheuristics. The results of these algorithms

are obtained from the published paper. All the comparedalgorithms were tested on some difficult JSSPs, i.e., instancesof Lawrence series (LA36–LA40) and Taillard series (1993)(TA01–TA50). The population size of IPSO in this experimentis set as 30, and the hybrid algorithm is terminated after100 iterations or gbest in the IPSO has not been improvedin 10 iterations. The only parameter α in the IPSO is usedas 0.5. The parameters in the TSAB are used as describedin [15].

The program was coded in C mixed with MATLAB lan-guage (the IPSO was written in MATLAB language andthe TSAB algorithm was written in C language) and runon an Intel Core 2 6600 PC 10 times for each of the 55problems.

B. Simulation Results and Comparisons

In OR library, instances LA01–LA35 are commonly consid-ered as trivial problems. In this paper, we only compare thetested algorithms on LA36–40. For the same reason, we onlyshow the comparison of the results on instances TA01–50 inTA class.

Since the IPSO–TSAB algorithm is a randomized optimiza-tion algorithm, different initial solutions of the IPSO–TSABmay give different final results. Therefore, we also focus ourattention on the stability of the tested algorithms by calculatingthe mean relative error (MBE) [24] for each instance in (9)and (10), where UBbest and UBmean are the best and averagemakespan found by the tested algorithms, respectively, and LBis a lower bound

b-MRE = 100 ∗ (UBbest − LB)/LB (9)

av-MRE = 100 ∗ (UBmean − LB)/LB. (10)

According to our test, the IPSO–TSAB obtains the opti-mal solutions for all the instances LA36–40, but the othercompared algorithms failed to find the optimal solutions forsome instances. From this experiment, we can conclude thatthe IPSO–TSAB is quite competitive in solving these basicinstances. The computational results of TA01–50 are listedin Table II. The final results demonstrate that the proposedhybrid IPSO–TSAB is much better than the other algorithms.It gets equal or the closest results to the upper bound of theseinstances. Furthermore, we also compared some algorithmsby using the values of b-MRE and av-MRE in Table III. Thesmaller these values are, the better the algorithm will be. Fromthe results, it can be found that our algorithm gets the smallestb-MRE and av-MRE values. The average time of the IPSO–TSAB first reaches the final best solution in each run is alsopresented in Tables II.

The main reason for the TSAB and the i-TSAB to haveless contribution than the hybrid IPSO–TSAB on JSSP is thatthe two algorithms belong to the single-based metaheuristicand the performances mostly depend on their initial solutionstrategies. Although the HIA and the MPSO mix the single-based and the population-based metaheuristics together, thelocal search ability of the single-based algorithms used inthese two algorithms are weaker than the TSAB algorithm.Thus, they show poorer achievements than the TSAB and the


TABLE IIPERFORMANCE OF THE COMPARED ALGORITHMS ON THE INSTANCES OF TA01–50

i-TSAB. The HPSO, ACOFT, and DE-TS use the differentpopulation-based algorithms to identify the promising regionsin the solution space and the tabu algorithm to determine thelocal optimum within such regions. They get more accurateresults than the TSAB algorithm in most cases. Based onthe merits of the PSO, the IPSO further applies a balancestrategy and a local search strategy to provide elite and diverse

initial solutions for the TSAB. Then, the IPSO–TSAB gets thebest performances when compared with the other algorithms.According to the results shown in Tables II and III, we canalso find that the gap between IPSO–TSAB and the otheralgorithms becomes increasingly larger with increasing size ofthe problems. This proves our algorithm is suitable for JSSPespecially for large size instances.


TABLE IIICOMPUTATIONAL RESULT WITH b-MRE AND ave-MRE OF TA01–50 TEST PROBLEMS

TABLE IVCOMPUTATIONAL RESULT OF TA21–50 TEST PROBLEMS

Fig. 7. Comparison between IPSO and other algorithms.

C. Role of IPSO and Local Search Strategy in IPSO–TSAB

As described in Section IV, we present an IPSO algorithmwith a local search strategy. In order to demonstrate itseffectiveness, we compare the IPSO algorithm with other threealgorithms, i.e., Random Search algorithm (RS), traditionalPSO, and IPSO without the local search strategy (IPSOW).For showing the merit of the population-based metaheuristicsin JSSP, we present the RS algorithm in which a randomoperator within [0, 1] instead of IPSO is used to serve asthe initial solution of the TSAB. The difference between theIPSOW and the traditional PSO is that the former uses abalance strategy. We test them on instances TA21–50. For fair

comparison, the running time is set as 1000 s for all the testedalgorithms. In this comparison, the final best achievements ofeach algorithm are shown in Table IV and the convergence rateof these algorithms on TA 25 and 45 are compared in Fig. 7.

Table IV shows the best JSSP values obtained by eachalgorithm. Depending on the balance strategy and the localsearch method, the IPSO shows the best performance withinlimited time. As the RS algorithm uses the random operatorwithin [0, 1] to serve as the initial solution of the TSAB, itprovides different but not accurate results for the TSAB tomake further search. Thus, it shows the worst achievements inthe tested algorithm. The reason for the PSO-based algorithm


to get more competitive results than the Radom-based algo-rithm in most cases is that the learning mechanism used in thePSO directs particles to search in the potential solution space.Then, it offers more accurate results for the TSAB algorithm tosearch. As compared with the other algorithms except IPSO,the IPSO-W uses the balance strategy which helps particlesto have more chances to explore in the solution space in theinitial stage and exploit in the local area in the final stage. Asa result, IPSOW gets more favorable achievements in mostcases. The reason IPSOW shows worse results than IPSO isthat the later further uses a local search strategy to improvethe quality of solution. The results shown in Fig. 7 also provethat the IPSO algorithm has a faster convergence rate than theother algorithms.

VI. CONCLUSION

In this paper, we present a hybrid PSO and tabu algo-rithm for JSSP. The aim is to allow the improved PSOto provide diverse and elite initial solutions for the tabualgorithm. We first present a novel PSO with a balance strategywhich makes particles have more opportunities to explorein the first stage and exploit in the final stage of iteration.Then, the decoding and encoding schemes are employed toexchange the solutions between PSO and tabu algorithmsdirectly. Finally, we use a local search strategy to get moreprecise solution of JSSP. The computational results obtainedby the proposed algorithm are superior to the other comparedalgorithms especially on large dimension instances. Further,extending the hybrid algorithm to solve more complex prob-lems and finding more favorable results in the JSSP merits ourresearch.

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Hao Gao received the M.S. and Ph.D. degrees incomputer science from Jiangnan University, Wuxi,China, in 2006 and 2009, respectively.

Currently, he is an Associate Professor with theCollege of Automation, Nanjing University of Postsand Telecommunications, Nanjing, China. His re-search interests include evolutionary algorithms andcomputer vision.

Sam Kwong (M’93–SM’04–F’14) received the B.S.degree from the State University of New York,Buffalo, NY, USA, in 1983; the M.S. degree fromthe University of Waterloo, Waterloo, ON, Canada,in 1985; and the Ph.D. degree from the Universityof Hagen, Hagen, Germany, in 1996, all in electricalengineering.

From 1985 to 1987, he was a Diagnostic Engineerwith Control Data Canada, Mississauga, Canada. Hejoined as a Member of Scientific Staff with BellNorthern Research, Ottawa, Canada. In 1990, he

became a Lecturer with the Department of Electronic Engineering, CityUniversity of Hong Kong, Hong Kong, China, where he is currently aProfessor with the Department of Computer Science. His research interestsinclude video and image coding, and evolutionary algorithms.

Baojie Fan received the B.S. and M.S. degreesin automation from Qufu Normal University, Qufu,China, and Northwest University, Xi’an, China, in2006 and 2008, respectively, and the Ph.D. degreein pattern recognition and intelligent system fromthe State Key Laboratory of Robotics, ShenyangInstitute Automation, Chinese Academy of Sciences,Shenyang, China, in 2012.

Currently, he is an Assistant Professor with theCollege of Automation, Nanjing University of Postsand Telecommunications, Nanjing, China. His re-

search interests include unmanned aerial vehicle (UAV) vision system, spacerobot, object tracking, and pattern recognition.

Ran Wang received the B.Sc. degree in computerscience from the College of Information Science andTechnology, Beijing Forestry University, Beijing,China, in 2009, and the Ph.D. degree in computerscience from the City University of Hong Kong,Hong Kong, China, in 2014.

Currently, she is a Senior Research Associate withthe Department of Computer Science, City Univer-sity of Hong Kong. Her current research interestsinclude pattern recognition, machine learning, fuzzysets and fuzzy logic, and their related applications.

a hybrid particle-swarm tabu search algorithm for solving job shop scheduling problems

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