Genetic Algorithm Combined with Tabu Search in a Holonic MultiagentModel for Flexible Job Shop Scheduling Problem

Houssem Eddine Nouri1, Olfa Belkahla Driss1,2 and Khaled Ghedira11Strategies d’Optimisation et Informatique intelligentE, Higher Institute of Management of Tunis, Bardo, Tunis, Tunisia

2Higher Business School of Tunis, University of Manouba, Manouba, Tunisia

Keywords: Scheduling, Flexible Job Shop, Genetic Algorithm, Tabu search, Holonic Multiagent.

Abstract: The Flexible Job Shop scheduling Problem (FJSP) is an extension of the classical Job Shop scheduling Prob-lem (JSP) presenting an additional difficulty caused by the operation assignment problem on one machine outof a set of alternative machines. The FJSP is an NP-hard problem composed by two complementary problems,which are the assignment and the scheduling problems. In this paper, we propose a combination of a geneticalgorithm with a tabu search in a holonic multiagent model for the FJSP. In fact, firstly, a scheduler agent ap-plies a genetic algorithm for a global exploration of the search space. Then, secondly, a local search techniqueis used by a set of cluster agents to guide the research in promising regions of the search space and to improvethe quality of the final population. To evaluate our approach, numerical tests are made based on two sets ofwell known benchmark instances in the literature of the FJSP: Kacem and Brandimarte. The experimentalresults show that our approach is efficient in comparison to other approaches.

1 INTRODUCTION

Scheduling is a field of investigation which has knowna significant growth these last years. The schedulingproblems appear in all the economic areas, from com-puter engineering to industrial production and manu-facturing. The Job Shop scheduling Problem (JSP),which is among the hardest combinatorial optimiza-tion problems (Sonmez and Baykasoglu, 1998), is abranch of the industrial production scheduling prob-lems. The JSP is known as one of the most popu-lar research topics in the literature due to its potentialto dramatically decrease costs and increase through-put (Jones and Rabelo, 1998). The Flexible Job Shopscheduling Problem (FJSP) is an extension of the clas-sical JSP that allows to process operations on one ma-chine out of a set of alternative machines. Hence,the FJSP is more computationally difficult than theJSP. Furthermore the operation scheduling problem,the FJSP presents an additional difficulty caused bythe operation assignment problem to a set of availablemachines. This problem is known to be strongly NP-Hard even if each job has at most three operations andthere are two machines (Garey et al., 1976).

To solve this problem, standard metaheuristicmethods are used for an approximate resolution and tofind near-optimal solutions for the FJSP with accept-

able computational time. (Brandimarte, 1993) pro-posed a hierarchical algorithm based on Tabu Searchmetaheuristic for routing and scheduling with someknown dispatching rules to solve the FJSP. (Hurinket al., 1994) developed a Tabu Search procedure forthe job shop problem with multi-purpose machines.(Dauzere-Peres and Paulli, 1997) presented a newneighborhood structure for the problem, and a list ofTabu moves was used to prevent the local search fromcycling. (Mastrolilli and Gambardella, 2000) usedTabu Search techniques and presented two neighbor-hood functions allowing an approximate resolutionfor the FJSP. (Bozejko et al., 2010a) presented a TabuSearch approach based on a new golf neighborhoodfor the FJSP, and in the same year, (Bozejko et al.,2010b) proposed another new model of a distributedTabu Search algorithm for the FJSP, using a cluster ar-chitecture consisting of nodes equipped with the GPUunits (multi-GPU) with distributed memory. A newversion of Tabu Search algorithm with a fast Pub-lic Critical Block neighborhood structure (TSPCB)was proposed by (Li et al., 2011) to solve the FJSP.For the Genetic Algorithm, it was adopted by (Chenet al., 1999), where their chromosome representationof solutions for the problem was divided into twoparts. The first part defined the routing policy andthe second part took the sequence of operations on

573Nouri H., Belkahla Driss O. and Ghédira K..Genetic Algorithm Combined with Tabu Search in a Holonic Multiagent Model for Flexible Job Shop Scheduling Problem.DOI: 10.5220/0005348105730584In Proceedings of the 17th International Conference on Enterprise Information Systems (ICEIS-2015), pages 573-584ISBN: 978-989-758-096-3Copyright c 2015 SCITEPRESS (Science and Technology Publications, Lda.)

each machine. (Kacem et al., 2002a) used a GeneticAlgorithm with an approach of localization to solvejointly the assignment and job shop scheduling prob-lems with partial and total flexibility, and a secondhybridization of this evolutionary algorithm with thefuzzy logic was presented in (Kacem et al., 2002b).(Jia et al., 2003) proposed a modified Genetic Algo-rithm for the FJSP, where various scheduling objec-tives can be achieved such as minimizing makespan,cost and weighted multiple criteria. (Ho et al., 2007)developed a new architecture named LEarnable Ge-netic Architecture (LEGA) for learning and evolv-ing solutions for the FJSP, allowing to provide anintegration between evolution and learning in an ef-ficient manner within a random search process. Inaddition, other types of metaheuristics were devel-oped in this last few years, such as (Yazdani et al.,2010) implementing a Parallel Variable Neighbor-hood Search (PVNS) algorithm to solve the FJSP us-ing various neighborhood structures. A Knowledge-Based Ant Colony Optimization (KBACO) algorithmwas presented by (Xing et al., 2010) for the FJSP.Moreover, another branch of metaheuristic optimiza-tion approaches appeared, by combining two or morecomplementary approaches in order to create a supe-rior solution procedure allowing to offer a more effi-cient resolution for complex problems. That is why,(Glover et al., 1995) elaborated a study about the na-ture of connections between the genetic algorithm andtabu search metaheuristics, searching to show the ex-isting opportunities for creating a hybrid approachwith these two standard methods to take advantageof their complementary features and to solve difficultoptimization problems. After this pertinent study, thecombination of these two metaheuristics has becomemore well-known in the literature and is used to solvemany complex problems, such as the FJSP. An im-proved genetic algorithm combined with local searchis proposed by (Zhang et al., 2008) to solve the FJSP.In fact, this combination introduced a time-varyingthe crossover probability for the genetic algorithmand a time-varying maximum step size for the tabusearch, allowing to control the local search processand the convergence to the global optimal solution.After two years, (Zhang et al., 2010) presented a sec-ond combined proposition of genetic algorithm withtabu search to solve the multi-objective FJSP. Theyadopted an additional external memory to save andupdate the non-dominated solutions during the opti-mization process. (Gao et al., 2008) adapted a hybridGenetic Algorithm (GA) and a Variable Neighbor-hood Descent (VND) for FJSP. The GA used two vec-tors to represent a solution and the disjunctive graphto calculate it. Then, a VND was applied to improve

the GA final individuals. (Thamilselvan and Balasub-ramanie, 2009) developed a new algorithm integratingGenetic Algorithm and Tabu Search methods to solvethe JSP. The proposed algorithm is based on the net-work central-clients node architecture inspired from(Andresen et al., 2002), where the genetic algorithmis run on a central node to generaten initial solutions,to be used byn client nodes as a starting solutionfor the Tabu Search algorithm. (Zhang et al., 2014)proposed a model of low-carbon scheduling in theFJSP considering three factors, the makespan, the ma-chine workload for production and the carbon emis-sion for the environmental influence. A metaheuristichybridization algorithm was proposed combining theoriginal Non-dominated Sorting Genetic Algorithm II(NSGA-II) with a Local Search algorithm based on aneighborhood search technique. Moreover, the Par-ticle Swarm Optimization was implemented by (Xiaand Wu, 2005) in a metaheuristic hybridization ap-proach with the Simulated Annealing for the multi-objective FJSP. A combined Particle Swarm Opti-mization and a Tabu Search algorithm were proposedby (Zhang et al., 2009) to solve the multi-objectiveFJSP. (Moslehi and Mahnam, 2011) presented a meta-heuristic approach based on a hybridization of theParticle Swarm Optimization and Local Search algo-rithm to solve the multi-objective FJSP. Furthermore,a new heuristic was developed by (Ziaee, 2014) forthe FJSP. This heuristic is based on a constructive pro-cedure considering simultaneously many factors hav-ing a great effect on the solution quality. Furthermore,distributed artificial intelligence techniques were usedfor this problem, such as the multiagent model pro-posed by (Ennigrou and Ghedira, 2004) composed bythree classes of agents, job agents, resource agentsand an interface agent. This model is based on a lo-cal search method which is the tabu search to solvethe FJSP. Also, this model was improved in (Enni-grou and Ghedira, 2008) where the optimization roleof the interface agent was distributed among the re-source agents. Similarly, (Azzouz et al., 2012) de-veloped a combination of a tabu search and a ge-netic algorithm based on a multiagent system to solvethe FJSP. This approach used two classes of agents,a first class grouping resource agents responsible ofprocessing a local optimization technique based onthe tabu search of (Ennigrou and Ghedira, 2008) anda second class containing an interface agent responsi-ble of processing a global optimization by a geneticalgorithm. (Henchiri and Ennigrou, 2013) proposeda multiagent model based on a hybridization of twometaheuristics, a local optimization process using thetabu search to get a good exploitation of the good ar-eas and a global optimization process integrating the

ICEIS�2015�-�17th�International�Conference�on�Enterprise�Information�Systems

574

Particle Swarm Optimization (PSO) to diversify thesearch towards unexplored areas.

In this paper, we present a combination of a ge-netic algorithm with a tabu search in a holonic multi-agent model for the flexible job shop scheduling prob-lem. This new combined approach follows two prin-cipal steps. In the first step, a scheduler agent ap-plies a genetic algorithm for a global exploration ofthe search space. Then, in the second step, a localsearch technique is used by a set of cluster agents toimprove the quality of the final population. Numer-ical tests were made to evaluate the performance ofour approach based on two data sets of (Kacem et al.,2002b) and (Brandimarte, 1993) for the FJSP, wherethe experimental results show its efficiency in com-parison to other approaches.

The rest of the paper is organized as follows. Insection 2, we define the formulation of the FJSP withits objective function and a simple problem instance.Then, in section 3, we detail the proposed combinedapproach with its holonic multiagent levels. The ex-perimental and comparison results are provided insection 4. Finally, section 5 rounds up the paper witha conclusion.

2 PROBLEM FORMULATION

The flexible job shop scheduling problem (FJSP)could be formulated as follows. There is a set ofnjobs J = {J1, . . . ,Jn} to be processed on a set ofmmachinesM = {M1, . . . ,Mm}. Each jobJi is formedby a sequence ofni operations{Oi,1,Oi,2, . . . ,Oi,ni}to be performed successively according to the givensequence. For each operationOi, j , there is a set of al-ternative machinesM(Oi, j ) capable of performing it.The main objective of this problem is to find a sched-ule minimizing the end date of the last operation ofthe jobs set which is the makespan. The makespanis defined byCmax in equation (1), whereCi is thecompletion time of a jobJi .

Cmax= max1≤i≤n(Ci) (1)

The FJSP scheduling problem is divided in twosub-problems:

• The operations assignment sub-problem that as-signs each operation to an appropriate machine.

• The operations sequencing sub-problem that de-termines a sequence of operations on all the ma-chines.

Furthermore, the adopted hypotheses in this prob-lem are:

• All the machines are available at time zero;

• All jobs are ready for processing at time zero;

• The order of operations for each job is predefinedand cannot be modified;

• There are no precedence constraints among oper-ations of different jobs;

• The processing time of operations on each ma-chine is defined in advance;

• Each machine can process only one operation at atime;

• Operations belonging to different jobs can be pro-cessed in parallel;

• Each job could be processed more than once onthe same machine;

• The interruption during the process of an opera-tion on a machine is negligible.

To explain the FJSP, a sample problem of threejobs and five machines is shown intable 1, where thenumbers present the processing times and the tags “–” mean that the operation cannot be executed on thecorresponding machine.

Table 1: A simple instance of the FJSP.

Job Operation M1 M2 M3 M4 M5

J1 O1,1 2 9 4 5 1O1,2 – 6 – 4 –

J2 O2,1 1 – 5 – 6O2,2 3 8 6 – –O2,3 – 5 9 3 9

J3 O3,1 – 6 6 – –O3,2 3 – – 5 4

3 GENETIC ALGORITHMCOMBINED WITH TABUSEARCH IN A HOLONICMULTIAGENT MODEL

(Glover et al., 1995) elaborated a study about the na-ture of connections between the genetic algorithm andtabu search metaheuristics, searching to show the ex-isting opportunities for creating a hybrid approachwith these two standard methods to take advantageof their complementary features and to solve diffi-cult optimization problems. After this pertinent study,the combination of these two metaheuristics has be-come more well-known in the literature, which hasmotivated many researchers to try the hybridizationof these two methods for the resolution of differentcomplex problems in several areas.

(Ferber, 1999) defined a multiagent system as an

Genetic�Algorithm�Combined�with�Tabu�Search�in�a�Holonic�Multiagent�Model�for�Flexible�Job�Shop�Scheduling�Problem

575

artificial system composed of a population of au-tonomous agents, which cooperate with each other toreach common objectives, while simultaneously eachagent pursues individual objectives. Furthermore, amultiagent system is a computational system wheretwo or more agents interact (cooperate or compete, ora combination of them) to achieve some individual orcollective goals. The achievement of these goals is be-yond the individual capabilities and individual knowl-edge of each agent (Botti and Giret, 2008).

(Koestler, 1967) gave the first definition of theterm “holon” in the literature, by combining the twoGreek words “hol” meaning whole and “on” meaningparticle or part. He said that almost everything is botha whole and a part at the same time. In fact, a holonis recursively decomposed at a lower granularity levelinto a community of other holons to produce a hol-archy (Calabrese, 2011). Moreover, a holon may beviewed as a sort of recursive agent, which is a super-agent composed by a sub-agents set, where each sub-agent has its own behavior as a complementary partof the whole behaviour of the super-agent. Holonsare agents able to show an architectural recursiveness(Giret and Botti, 2004).

In this work, we propose a combined metaheuris-tic approach processing two general steps: a first stepof global exploration using a genetic algorithm to findpromising areas in the search space and a clusteringoperator allowing to regroup them in a set of clusters.In the second step, a tabu search algorithm is appliedto find the best individual solution for each cluster.The global process of the proposed approach is imple-mented in a two hierarchical holonic levels adoptedby a recursive multiagent model, named Genetic Al-gorithm combined with Tabu Search in a HolonicMultiagent model (GATS+HM), seefigure 1. Thefirst holonic level is composed by a Scheduler Agentwhich is the Master/Super-agent, preparing the bestpromising regions of the search space, and the sec-ond holonic level containing a set of Cluster Agentswhich are the Workers/Sub-agents, guiding the searchto the global optimum solution of the problem. Eachholonic level of this model is responsible to process astep of the hybrid metaheuristic algorithm and to co-operate between them to attain the global solution ofthe problem.

In fact, the choice of this new metaheuristic com-bination is justified by that the standard metaheuristicmethods use generally the diversification techniquesto generate and to improve many different solutionsdistributed in the search space, or by using localsearch techniques to generate a more improved set ofneighbourhood solutions from an initial solution. Butthey did not guarantee to attain promising areas with

good fitness converging to the global optimum despitethe repetition of many iterations, that is why they needto be more optimized. So, the novelty of our approachis to launch a genetic algorithm based on a diversifica-tion technique to only explore the search space and toselect the best promising regions by the clustering op-erator. Then, applying the intensification technique ofthe tabu search allowing to relaunch the search froman elite solution of each cluster autonomously to at-tain more dominant solutions of the search space.

The use of a multiagent system gives the opportu-nity for distributed and parallel treatments which arevery complimentary for the second step of the pro-posed approach. Indeed, our combined metaheuris-tic approach follows the paradigm of “Master” and“Workers” which are two recursive hierarchical lev-els adaptable for a holonic multiagent model, wherethe Scheduler Agent is the Master/Super-agent of itssociety and the Cluster Agents are its Workers/Sub-agents.

Figure 1: Genetic algorithm combined with tabu search in aholonic multiagent model.

3.1 Scheduler Agent

The Scheduler Agent (SA) is responsible to processthe first step of the hybrid algorithm by using a geneticalgorithm called NGA (Neighborhood-based GeneticAlgorithm) to identify areas with high average fitnessin the search space. In fact, the goal of using the NGAis only to explore the search space, but not to findthe global solution of the problem. Then, a clusteringoperator is integrated to divide the best identified ar-eas by the NGA in the search space to different parts

ICEIS�2015�-�17th�International�Conference�on�Enterprise�Information�Systems

576

where each part is a clusterCLi ∈CL the set of clus-ters, whereCL = {CL1,CL2, . . . ,CLN}. In addition,this agent plays the role of an interface between theuser and the system (initial parameter inputs and finalresult outputs). According to the number of clustersNobtained after the integration of the clustering opera-tor, the SA createsN Cluster Agents (CAs) preparingthe passage to the next step of the global algorithm.After that, the SA remains in a waiting state until thereception of the best solutions found by the CA foreach cluster. Finally, it finishes the process by dis-playing the final solution of the problem.

3.1.1 Individual’s Solution Presentation

The flexible job shop problem is composed by twosub-problems: the machine assignment problem andthe operation scheduling problem, that is why thechromosome representation is encoded in two parts:Machine Assignment part (MA) and Operation Se-quence part (OS).

Figure 2: The chromosome representation of a schedulingsolution.

The first part MA is a vectorV1 with a lengthL equal to the total number of operation and whereeach index represents the selected machine to pro-cess an operation indicated at positionp, seefig-ure 2 (a). For examplep = 2, V1(2) is the se-lected machineM4 for the operationO1,2. The sec-ond part OS is a vectorV2 having the same lengthof V1 and where each index represents an operationOi, j according to the predefined operations of the

job set, seefigure 2 (b). For example the opera-tion sequence 1−2−1−3−2−3−2 can be trans-lated to: (O1,1,M5) → (O2,1,M1) → (O1,2,M4) →(O3,1,M3)→ (O2,2,M3)→ (O3,2,M1)→ (O2,3,M2).

To convert the chromosome values to an activeschedule, we used the priority-based decoding of(Gao et al., 2008). This method considers the idletime which may exist between operations on a ma-chinem, and which is caused by the precedence con-straints of operations belonging to the same jobi. LetSi, j is the starting time of an operationOi, j (whichcan only be started after processing its precedent op-erationOi,( j−1)) with its completion timeCi, j . In ad-dition, we have an execution time interval [tS

m, tEm]

starts formtSm and ends attE

m on a machinem to al-locate an operationOi, j . So, if j = 1, Si, j takestS

m,else if j ≥ 2, it takesmax{tS

m,Ci,( j−1)}. In fact, theavailability of the time interval [tS

m, tEm] for an oper-

ationOi, j is validated by verifying if there is a suffi-cient time period to complete the execution timepi jmof this operation, seeequation (2):

i f j = 1, tSm+ pi jm ≤ tE

m (2)

i f j ≥ 2,max{tSm,Ci,( j−1)}+ pi jm ≤ tE

m

The used priority-based decoding method allowsin each case to assign each operation to its reservedmachine following the presented execution order ofthe operation sequence vectorV2. Also, to schedulean operationOi, j on a machinem, the fixed idle timeintervals of the selected machine are verified to findan allowed available period to its execution. So, if aperiod is found, the operationOi, j is executed there,else it is moved to be executed at the end of the ma-chinem.

Noting that the chromosome fitness is calculatedby Fitness(i) which is the fitness function of eachchromosomei and Cmax(i) is its makespan value,wherei ∈ {1, . . . ,P} andP is the total population size,seeequation (3).

Fitness(i) =1

Cmax(i)(3)

3.1.2 Population Initialization

The initial population is generated randomly follow-ing a uniform law and based on a neighborhood pa-rameter to make the individual solutions more diver-sified and distributed in the search space. In fact, eachnew solution should have a predefined distance withall the other solutions to be considered as a new mem-ber of the initial solution. The used method to deter-minate the neighborhood parameter is inspired from

Genetic�Algorithm�Combined�with�Tabu�Search�in�a�Holonic�Multiagent�Model�for�Flexible�Job�Shop�Scheduling�Problem

577

(Bozejko et al., 2010a), which is based on the per-mutation level of operations to obtain the distance be-tween two solutions. In fact, the dissimilarity distanceis calculated by verifying the difference between twochromosomes in terms of the placement of each op-erationOi, j on its alternative machine set in the ma-chine assignment vectorV1 and its execution order inthe operation sequence vectorV2. So, if there is a dif-ference in the vectorV1, the distance is incrementedby M(Oi, j ) (is the number of possiblen placementfor each operation on its machine set, which is thealternative machine number of each operationOi, j )because it is in the order ofO(n). Then, if thereis a difference in the vectorV2, the distance is in-cremented by 1 because it is in the order ofO(1).Let Chrom1(MA1,OS1) andChrom2(MA2,OS2) twochromosomes of two different scheduling solutions,M(Oi, j ) the alternative number of machines of eachoperationOi, j , L is the total number of operations ofall jobs andDist is the dissimilarity distance. Thedistance is calculated firstly by measuring the differ-ence between the machine assignment vectorsMA1andMA2 which is in order ofO(n), then by verify-ing the execution order difference of the operationsequence vectorsOS1 andOS2 which is in order ofO(1), we give here how to proceed:

Noting that Distmax is the maximal dissimilar-ity distance and it is calculated byequation (4), rep-resenting 100% of difference between two chromo-somes.

Distmax=i,ni

∑i,1

[M(Oi, j)]+L (4)

3.1.3 Selection Operator

The selection operator is used to select the best par-ent individuals to prepare them to the crossover step.This operator is based on a fitness parameter allow-ing to analyze the quality of each selected solution.But progressively the fitness values will be similarfor the most individuals. That is why, we integrate

the neighborhood parameter, where we propose a newcombined parent selection operator named Fitness-Neighborhood Selection Operator (FNSO) allowingto add the dissimilarity distance criteria to the fitnessparameter to select the best parents for the crossoverstep. The FNSO chooses in each iteration two parentindividuals until engaging all the population to cre-ate the next generation. The first parent takes succes-sively in each case a solutioni, wherei ∈ {1, . . . ,P}andP is the total population size. The second par-ent obtains its solutionj randomly by the roulettewheel selection method based on the two Fitness andNeighborhood parameters relative to the selected firstparent, wherej ∈ {1, . . . ,P} \ {i} in theP populationand wherej 6= i. In fact, to use this random method,we should calculate the Fitness-Neighborhood totalFN for the population, seeequation (5), the selec-tion probabilityspk for each individualIk, seeequa-tion (6), and the cumulative probabilitycpk, seeequa-tion (7). After that, a random numberr will be gener-ated from the uniform range [0,1]. Ifr ≤ cp1 then thesecond parent takes the first individualI1, else it getsthe kth individual Ik ∈ {I2, . . . , IP} \ {Ii} and wherecpk−1 < r ≤ cpk.

• The Fitness-Neighborhood total for the popula-tion:

FN =P

∑k=1

[1/(Cmax[k]×Neighborhood[i][k])]

(5)

• The selection probabilityspk for each individualIk:

spk =1/(Cmax[k]×Neighborhood[i][k])

FN(6)

• The cumulative probabilitycpk for each individ-ual Ik:

cpk =k

∑h=1

sph (7)

=⇒ For equations (5), (6) and (7), k ={1,2, . . . ,P} \ {i}

3.1.4 Crossover Operator

The crossover operator has an important role in theglobal process, allowing to combine in each case thechromosomes of two parents in order to obtain newindividuals and to attain new better parts in the searchspace. In this work, this operator is applied withtwo different techniques successively for the parent’schromosome vectors MA and OS.

ICEIS�2015�-�17th�International�Conference�on�Enterprise�Information�Systems

578

Machine Vector Crossover. A uniform crossoveris used to generate in each case a mixed vectorbetween two machine vector parents, Parent1-MA1and Parent2-MA2, allowing to obtain two new chil-dren, Child1-MA1′ and Child2-MA2′. This uniformcrossover is based on two assignment cases, if thegenerated number is less than 0.5, the first child getsthe current machine value of parent1 and the secondchild takes the current machine value of parent2. Else,the two children change their assignment direction,first child to parent2 and the second child to parent1.

Operation Vector Crossover. An improved prece-dence preserving order-based on crossover (iPOX),inspired from (Lee et al., 1998), is adapted for theparent operation vector OS. This iPOX operator is ap-plied following four steps, a first step is selecting twoparent operation vectors (OS1 andOS2) and generat-ing randomly two job sub-setsJs1/Js2 from all jobs.A second step is allowing to copy any element inOS1/OS2 that belong toJs1/Js2 into child individualOS′1/OS′2 and retain them in the same position. Thenthe third step deletes the elements that are already inthe sub-setJs1/Js2 from OS1/OS2. Finally, fill orderlythe empty position inOS′1/OS′2 with the reminder ele-ments of OS2/OS1 in the fourth step, see the examplein thefigure 3.

Figure 3: An iPOX crossover example.

3.1.5 Mutation Operator

The mutation operator is integrated to promote thechildren generation diversity. In fact, this operator isapplied on the chromosome of the new children gen-erated by the crossover operation. Also, each part of achild chromosome MA and OS has separately its ownmutation technique.

Machine Vector Mutation. This first operator usesa random selection of an index from the machine vec-tor MA. Then, it replaces the machine number in theselected index by another belonging to the same alter-native machine set, seefigure 4.

Figure 4: A machine vector mutation example.

Operation Vector Mutation. This second operatorselects randomly two indexes index1 and index2 fromthe operation vector OS. Next, it changes the positionof the job number in the index1 to the second index2and inversely.

3.1.6 Replacement Operator

The replacement operator has an important role toprepare the remaining surviving population to be con-sidered for the next iterations. This operator replacesin each case a parent by one of its children which hasthe best fitness in its current family.

Figure 5: The final population transformation by applyingthe clustering operator.

3.1.7 Clustering Operator

By finishing the last iteration of the genetic algorithm,the Scheduler Agent applies a clustering operator us-ing the hierarchical clustering algorithm of (Johnson,

Genetic�Algorithm�Combined�with�Tabu�Search�in�a�Holonic�Multiagent�Model�for�Flexible�Job�Shop�Scheduling�Problem

579

1967) to divide the final population intoN Clusters,seefigure 5, to be treated by the Cluster Agents inthe second step of the global process. The clusteringoperator is based on the neighbourhood parameterwhich is the dissimilarity distance between individu-als. The clustering operator starts by assigning eachindividual Indiv(i) to a clusterCLi , so if we havePindividuals, we have nowP clusters containing justone individual in each of them. For each case, wefixe an individualIndiv(i) and we verify successivelyfor each next individualIndiv( j) from the remainingpopulation (wherei and j ∈ {1, . . . ,P}, i 6= j) if thedissimilarity distanceDist between Indiv(i) andIndiv( j) is less than or equal to a fixed thresholdDist f ix (representing a percentage of difference X%relatively to Distmax, see equation (8)) and whereCluster(Indiv(i)) 6= Cluster(Indiv( j)). If it is thecase, Merge(Cluster(Indiv(i)),Cluster(Indiv( j))),else continue the search for new combination withthe remaining individuals. The stopping condition isby browsing all the population individuals, where weobtained at the endN Clusters.

Dist f ix= Distmax×X% (8)

Figure 6: Distribution of the Cluster Agents in the differentclusters of the search space.

3.2 Cluster Agents

Each Cluster AgentCAi is responsible to apply suc-cessively to each clusterCLi a local search techniquewhich is the Tabu Search algorithm to guide the re-search in promising regions of the search space and to

improve the quality of the final population of the ge-netic algorithm. In fact, this local search is executedsimultaneously by the set of the CAs agents, whereeach CA starts the research from an elite solution ofits cluster searching to attain new more dominant in-dividual solutions separately in its assigned clusterCLi , seefigure 6. The used Tabu Search algorithm isbased on an intensification technique allowing to startthe research from an elite solution in a clusterCLi (apromising part in the search space) in order to collectnew scheduling sequence minimizing the makespan.Let E the elite solution of a clusterCLi , E′ ∈ N(E) isa neighbor of the elite solutionE, eachCLi plays therole of the tabu list with a dynamic length andCmaxis the makespan of the obtained solution. So, this al-gorithm applies in each case two steps, amove andinsert first step inspired from (Mastrolilli and Gam-bardella, 2000) to generate new scheduling combina-tion and a second step to save the best found solutionin the tabu list (which isCLi) after each generationcase. The stopping condition is by attaining the max-imum allowed number of neighbors for a solutionEwithout improvement. We give here how to proceed:

By finishing this local search step, the CAs agentsterminate the process by sending their last best solu-tions to the SA agent.

4 EXPERIMENTAL RESULTS

4.1 Experimental Setup

The proposed GATS+HM is implemented in java lan-guage on a 2.10 GHz Intel Core 2 Duo processor and 3Gb of RAM memory, where we use the Integrated De-velopment Environment (IDE)eclipseto code the al-gorithm and the multiagent platformJade(Bellifem-ine et al., 1999) to create the different agents of ourholonic model. To evaluate its efficiency, numericaltests are made based on two sets of well known bench-mark instances in the literature of the FJSP:

ICEIS�2015�-�17th�International�Conference�on�Enterprise�Information�Systems

580

Table 2: Results of the Kacem instances (part 1).

InstanceProblemn×m AL+CGA (2002) LEGA (2007) KBACO (2010)

MOPSO+LS(2011)

BestDev(%) Best

Dev(%) Best

Dev(%) Best

Dev(%)

case 1 4×5 16 13,250 11 0 11 0 16 31,25case 2 8×8 15 6,666 N/A – 14 0 14 0case 3 10×7 15 26,666 11 0 11 0 15 26,666case 4 10×10 7 0 7 0 7 0 7 0case 5 15×10 23 52,173 12 8,333 11 0 11 0

Table 3: Results of the Kacem instances (part 2).

InstanceProblemn×m

Hybrid NSGA-II(2014) Heuristic (2014) GATS+HM

BestDev(%) Best

Dev(%) Best

AvgCmax

Avg CPU(in seconds)

case 1 4×5 11 0 11 0 11 11,00 0,05case 2 8×8 15 6,666 15 6,666 14 14,20 0,36case 3 10×7 N/A – 13 15,384 11 11,40 0,72case 4 10×10 7 0 7 0 7 7,60 1,51case 5 15×10 11 0 12 8,333 11 11,60 29,71

• Kacem Data(Kacem et al., 2002b): The data setconsists of 5 problems considering a number ofjobs ranging from 4 to 15 with a number of op-erations for each job ranging from 2 to 4, and anumber of operations for all jobs ranges from 12to 56, which will be processed on a number ofmachines ranging from 5 to 10.

• Brandimarte Data(Brandimarte, 1993): The dataset consists of 10 problems considering a numberof jobs ranging from 10 to 20 with a number ofoperations for each job ranging from 5 to 15, anda number of operations for all jobs ranges from55 to 240, which will be processed on a numberof machines ranging from 4 to 15.

Due to the non-deterministic nature of the pro-posed algorithm, we run it five independent times foreach case of the two data instances in order to ob-tain significant results. The computational results arepresented by four metrics such as the best makespan(Best), the average of makespan(Avg Cmax), the av-erage of CPU time in seconds(Avg CPU), and thestandard deviation of makespan(Dev%) which is cal-culated byequation (9). The Mko is the makespanobtained byOur algorithm andMkc is the makespanof an algorithm that we chose toCompare to.

Dev= [(Mkc−Mko)/Mkc]×100% (9)

The used parameter settings for our algorithm areadjusted experimentally and presented as follow:

• Crossover probability 1.0

• Mutation probability 1.0

• Maximum number of iterations 1000

• The population size ranged from 15 to 300 de-pending on the complexity of the problem.

4.2 Experimental Comparisons

To show the efficiency of our GATS+HM algorithm,we compare its obtained results from the two previ-ously cited data sets with other well known algorithmsin the literature of the FJSP. The chosen algorithmsare the TS of (Brandimarte, 1993), the AL+CGA of(Kacem et al., 2002b), the LEGA of (Ho et al., 2007),the MATSLO+ of (Ennigrou and Ghedira, 2008), theKBACO of (Xing et al., 2010), the TS3 of (Bozejkoet al., 2010a), the MOPSO+LS of (Moslehi and Mah-nam, 2011), the MATSPSO of (Henchiri and Enni-grou, 2013), the Hybrid NSGA-II of (Zhang et al.,2014) and the Heuristic of (Ziaee, 2014). The differ-ent comparative results are displayed in thetables 2,3, 4 and5, where the first column takes the name ofeach instance, the second column gives the size eachinstance, withn the number of jobs andm the num-ber of machines(n×m), and the remaining columnsdetail the experimental results of the different chosenapproaches in terms of the best Cmax(Best)and thestandard deviation(Dev %). The bold values in thetables signify the best obtained results and theN/Ameans that the result is not available.

By analyzing thetable 2 and table 3, it can be

Genetic�Algorithm�Combined�with�Tabu�Search�in�a�Holonic�Multiagent�Model�for�Flexible�Job�Shop�Scheduling�Problem

581

Table 4: Results of the Brandimarte instances (part 1).

InstanceProblemn×m TS (1993) LEGA (2007)

MATSLO+(2008) KBACO (2010)

BestDev(%) Best

Dev(%) Best

Dev(%) Best

Dev(%)

MK01 10×6 42 4,761 40 0 40 0 39 -2,564MK02 10×6 32 15,625 29 6,896 32 15,625 29 6,896MK03 15×8 211 3,317 N/A – 207 1,449 204 0MK04 15×8 81 20,987 67 4,477 67 4,477 65 1,538MK05 15×4 186 6,989 176 1,704 188 7,978 173 0MK06 10×15 86 24,418 67 2,985 85 23,529 67 2,985MK07 20×5 157 8,280 147 2,040 154 6,493 144 0MK08 20×10 523 0 523 0 523 0 523 0MK09 20×10 369 15,718 320 2,812 437 28,832 311 0MK10 20×15 296 25 229 3,056 380 41,578 229 3,056

Table 5: Results of the Brandimarte instances (part 2).

InstanceProblemn×m TS3 (2010) MATSPSO (2013) Heuristic (2014) GATS+HM

BestDev(%) Best

Dev(%) Best

Dev(%) Best

AvgCmax

Avg CPU (inseconds)

MK01 10×6 40 0 39-

2,564 42 4,761 40 40,80 0,93MK02 10×6 29 6,896 27 0 28 3,571 27 27,80 1,18MK03 15×8 204 0 207 1,449 204 0 204 204,00 1,55MK04 15×8 65 1,538 65 1,538 75 14,666 64 65,60 4,36MK05 15×4 173 0 174 0,574 179 3,351 173 174,80 8,02MK06 10×15 68 4,411 72 9,722 69 5,797 65 67,00 110,01MK07 20×5 144 0 154 6,493 149 3,355 144 144,00 19,73MK08 20×10 523 0 523 0 555 5,765 523 523,00 11,50MK09 20×10 326 4,601 340 8,529 342 9,064 311 311,80 79,68MK10 20×15 227 2,202 299 25,752 242 8,264 222 224,80 185,64

seen that our algorithm GATS+HM is the best onewhich solves the fives instances of Kacem. In fact,the GATS+HM outperforms the AL+CGA in four outof five instances, the LEGA, the MOPSO+LS and theHybrid NSGA-II in two out of five instances, andthe Heuristic in three out of five instances. Also,our algorithm attains the same results obtained bythe KBACO, but the efficiency of GATS+HM canbe noted by the average of the CPU time in sec-onds which is very acceptable for a real manufactur-ing case.

From table 4 and table 5, the comparison re-sults show that the GATS+HM obtains nine out often best results for the Brandimarte instances. In-deed, our algorithm outperforms the TS in nine outof ten instances. Moreover, our GATS+HM outper-forms the LEGA and the MATSLO+ in eight out often instances. For the comparison with the KBACO,the GATS+HM obtains the best solutions for theMK02, MK04, MK06 and MK10 instances, but it getsslightly worse result for the MK01 instance. Also, ourhybrid approach outperforms the TS3 in five out of ten

instances. Furthermore, the MATSPSO attained thebest result for the MK01 instance, but our algorithmobtains a set of solutions better than it for the remain-ing instances. In addition, our algorithm outperformsthe Heuristic in all the Brandimarte instances. Bysolving this second data set, our GATS+HM attainsthe same results obtained by some approaches suchas the MK01 for LEGA, MATSLO+ and TS3; theMK02 for MATSPSO; the MK03 for KBACO, TS3and Heuristic; the MK05 and MK07 for KBACO andTS3; the MK09 for KBACO; the MK08 for all thealgorithms only it is not the case for the Heuristic.

By analyzing the computational time in secondsand the comparison results of our algorithm in termof makespan, we can distinguish the efficiency of thenew proposed GATS+HM relatively to the literatureof the FJSP. This efficiency is explained by the flex-ible selection of the promising parts of the searchspace by the clustering operator after the genetic al-gorithm process and by applying the intensificationtechnique of the tabu search allowing to start from anelite solution to attain new more dominant solutions.

ICEIS�2015�-�17th�International�Conference�on�Enterprise�Information�Systems

582

5 CONCLUSIONS

In this paper, we present a new combination of a ge-netic algorithm with a tabu search in a holonic multi-agent model, called GATS+HM, for the flexible jobshop scheduling problem (FJSP). In this approach,a Neighborhood-based Genetic Algorithm is adaptedby a Scheduler Agent (SA) for a global explorationof the search space. Then, a local search technique isapplied by a set of Cluster Agents (CAs) to guide theresearch in promising regions of the search space andto improve the quality of the final population. Numer-ical tests are made to measure the performance of theproposed approach using two well known data setsfrom the literature of the FJSP. The experimental re-sults show that the proposed approach is efficient incomparison to others approaches. In the future work,improvements will be done on the tabu search of eachCluster Agent. In fact, if a Cluster AgentCAi findsa new dominant solutionE′ and it does not attain theallowed number of neighbors from its current solu-tion E, it does not stop, but it creates in each case anew Sub-Cluster AgentSCAi, j allowing to relaunchthe search from the new obtained solutionE′. Thenit continues to generate the remaining allowed neigh-bors from its solutionE searching other dominant so-lutions. In other words, each Cluster Agent distributesthe new found solutionsE′ from its elite solutionEon a new sub-set of Sub-Cluster AgentsSCAi, j tolaunch from them their new search processes, whichenhances more the search in the selected promisingareas and shows more the efficient use of the notion ofthe holonic agents (recursive agents) in our proposedsolution for the FJSP problem. Also, we will searchto treat other extensions of the FJSP, such as by inte-grating new transportation resources constraints in theshop process. So, we will make improvements to ourapproach to adapt it to this new transformation andstudy its effects on the makespan.

REFERENCES

Andresen, D., Kota, S., Tera, M., and Bower, T. (2002). Anip-level network monitor and scheduling system forclusters. InPDPTA, pages 789–795. CSREA Press.

Azzouz, A., Ennigrou, M., Jlifi, B., and Ghedira, K. (2012).Combining tabu search and genetic algorithm in amulti-agent system for solving flexible job shop prob-lem. In Proceedings of the 2012 11th Mexican In-ternational Conference on Artificial Intelligence, MI-CAI’12, pages 83–88, Washington, DC, USA. IEEEComputer Society.

Bellifemine, F., Poggi, A., and Rimassa, G. (1999). Jade -a fipa-compliant agent framework. InIn Proceedings

of the fourth International Conference and Exhibitionon The Practical Application of Intelligent Agents andMulti-Agent Technology, pages 97–108.

Botti, V. and Giret, A. (2008). ANEMONA: A Multi-agent Methodology for Holonic Manufacturing Sys-tems. Springer Series in Advanced Manufacturing.Springer-Verlag.

Bozejko, W., Uchronski, M., and Wodecki, M. (2010a). Thenew golf neighborhood for the flexible job shop prob-lem. In In Proceedings of the International Confer-ence on Computational Science, pages 289–296.

Bozejko, W., Uchronski, M., and Wodecki, M. (2010b).Parallel hybrid metaheuristics for the flexible job shopproblem. Computers and Industrial Engineering,59(2):323–333.

Brandimarte, P. (1993). Routing and scheduling in a flex-ible job shop by tabu search.Annals of OperationsResearch, 41(3):157–183.

Calabrese, M. (2011). Hierarchical-granularity holonicmodelling. Doctoral thesis, Universita degli Studi diMilano, Milano, Italy.

Chen, H., Ihlow, J., and Lehmann, C. (1999). A genetic al-gorithm for flexible job-shop scheduling. InIn Pro-ceedings of the IEEE International Conference onRobotics and Automation, pages 1120–1125.

Dauzere-Peres, S. and Paulli, J. (1997). An integrated ap-proach for modeling and solving the general multipro-cessor job-shop scheduling problem using tabu search.Annals of Operations Research, 70(0):281–306.

Ennigrou, M. and Ghedira, K. (2004). Approche multi-agents basee sur la recherche tabou pour le job shopflexible. In In 14eme Congres Francophone AFRIF-AFIA de Reconnaissance des Formes et IntelligenceArtificielle RFIA, pages 28–30.

Ennigrou, M. and Ghedira, K. (2008). New local diversi-fication techniques for the flexible job shop problemwith a multi-agent approach.Autonomous Agents andMulti-Agent Systems, 17(2):270–287.

Ferber, J. (1999).Multi-Agent Systems: An Introductionto Distributed Artificial Intelligence. Addison-WesleyLongman Publishing Co., Inc., Boston, MA, USA, 1stedition.

Gao, J., Sun, L., and Gen, M. (2008). A hybrid genetic andvariable neighborhood descent algorithm for flexiblejob shop scheduling problems.Computers and Oper-ations Research, 35(9):2892–2907.

Garey, M. R., Johnson, D. S., and Sethi, R. (1976). Thecomplexity of flow shop and job shop scheduling.Mathematics of Operations Research, 1(2):117–129.

Giret, A. and Botti, V. (2004). Holons and agents.Journalof Intelligent Manufacturing, 15(5):645–659.

Glover, F., Kelly, J. P., and Laguna, M. (1995). Geneticalgorithms and tabu search: Hybrids for optimization.Computers & Operations Research, 22(1):111–134.

Henchiri, A. and Ennigrou, M. (2013). Particle swarm op-timization combined with tabu search in a multi-agentmodel for flexible job shop problem. InIn Proceed-ings of the 4th International Conference on SwarmIntelligence, Advances in Swarm Intelligence, pages385–394.

Genetic�Algorithm�Combined�with�Tabu�Search�in�a�Holonic�Multiagent�Model�for�Flexible�Job�Shop�Scheduling�Problem

583

Ho, N. B., Tay, J. C., and Lai, E. M. K. (2007). An ef-fective architecture for learning and evolving flexiblejob-shop schedules.European Journal of OperationalResearch, 179(2):316–333.

Hurink, J., Jurisch, B., and Thole, M. (1994). Tabusearch for the job-shop scheduling problem withmulti-purpose machines.Operations Research Spek-trum, 15(4):205–215.

Jia, H., Nee, A., Fuh, J., and Zhang, Y. (2003). A modifiedgenetic algorithm for distributed scheduling problems.Journal of Intelligent Manufacturing, 14(3):351–362.

Johnson, S. C. (1967). Hierarchical clustering schemes.Psychometrika, 32(3):241–254.

Jones, A. and Rabelo, L. C. (1998). Survey of job shopscheduling techniques. InNational Institute of Stan-dards and Technology, Gaithersburg, MD.

Kacem, I., Hammadi, S., and Borne, P. (2002a). Approachby localization and multiobjective evolutionary opti-mization for flexible job-shop scheduling problems.IEEE Transactions on Systems, Man, and Cybernet-ics, 32(1):1–13.

Kacem, I., Hammadi, S., and Borne, P. (2002b). Pareto-optimality approach for flexible job-shop schedulingproblems: Hybridization of evolutionary algorithmsand fuzzy logic.Mathematics and Computers in Sim-ulation, 60(3-5):245–276.

Koestler, A. (1967).The Ghost in the Machine. Hutchinson,London, United Kingdom, 1st edition.

Lee, K., Yamakawa, T., and Lee, K. M. (1998). A ge-netic algorithm for general machine scheduling prob-lems. In In Proceedings of the second IEEE inter-national Conference on Knowledge-Based IntelligentElectronic Systems, pages 60–66.

Li, J., Pan, Q., Suganthan, P., and Chua, T. (2011). A hybridtabu search algorithm with an efficient neighborhoodstructure for the flexible job shop scheduling problem.The International Journal of Advanced Manufactur-ing Technology, 52(5):683–697.

Mastrolilli, M. and Gambardella, L. (2000). Effectiveneighbourhood functions for the flexible job shopproblem.Journal of Scheduling, 3(1):3–20.

Moslehi, G. and Mahnam, M. (2011). A pareto ap-proach to multi-objective flexible job-shop schedul-ing problem using particle swarm optimization and lo-cal search.International Journal of Production Eco-nomics, 129(1):14–22.

Sonmez, A. I. and Baykasoglu, A. (1998). A new dynamicprogramming formulation of (nm) flow shop sequenc-ing problems with due dates.International Journal ofProduction Research, 36(8):2269–2283.

Thamilselvan, R. and Balasubramanie, P. (2009). Integrat-ing genetic algorithm, tabu search approach for jobshop scheduling.International Journal of ComputerScience and Information Security, 2(1):1–6.

Xia, W. and Wu, Z. (2005). An effective hybrid op-timization approach for multiobjective flexible job-shop scheduling problems.Computers and IndustrialEngineering, 48(2):409–425.

Xing, L., Chen, Y., Wang, P., Zhao, Q., and Xiong, J.(2010). A knowledge-based ant colony optimization

for flexible job shop scheduling problems.AppliedSoft Computing, 10(3):888–896.

Yazdani, M., Amiri, M., and Zandieh, M. (2010). Flexiblejob-shop scheduling with parallel variable neighbor-hood search algorithm.Expert Systems with Applica-tions, 37(1):678–687.

Zhang, C., Gu, P., and Jiang, P. (2014). Low-carbonscheduling and estimating for a flexible job shopbased on carbon footprint and carbon efficiency ofmulti-job processing. Journal of Engineering Man-ufacture, 39(32):1–15.

Zhang, G., Gao, L., and Shi, Y. (2010). A genetic algorithmand tabu search for multi objective flexible job shopscheduling problems. InInternational Conference onComputing, Control and Industrial Engineering, vol-ume 1, pages 251–254.

Zhang, G., Shao, X., Li, P., and Gao, L. (2009). An effec-tive hybrid particle swarm optimization algorithm formulti-objective flexible job-shop scheduling problem.Computers and Industrial Engineering, 56(4):1309–1318.

Zhang, G., Shi, Y., and Gao, L. (2008). A genetic algorithmand tabu search for solving flexible job shop sched-ules. International Symposium on Computational In-telligence and Design, 1:369–372.

Ziaee, M. (2014). A heuristic algorithm for solving flexiblejob shop scheduling problem.The International Jour-nal of Advanced Manufacturing Technology, 71(1-4):519–528.

ICEIS�2015�-�17th�International�Conference�on�Enterprise�Information�Systems

584