ORIGINAL ARTICLE

An enhanced Pareto-based artificial bee colony algorithmfor the multi-objective flexible job-shop scheduling

Ling Wang & Gang Zhou & Ye Xu & Min Liu

Received: 4 May 2011 /Accepted: 20 September 2011 /Published online: 2 October 2011# Springer-Verlag London Limited 2011

Abstract In this paper, an enhanced Pareto-based artificialbee colony (EPABC) algorithm is proposed to solve themulti-objective flexible job-shop scheduling problem withthe criteria to minimize the maximum completion time, thetotal workload of machines, and the workload of the criticalmachine simultaneously. First, it uses multiple strategies ina combination way to generate the initial solutions as thefood sources with certain quality and diversity. Second,exploitation search procedures for both the employed beesand the onlooker bees are designed to generate the newneighbor food sources. Third, crossover operators aredesigned for the onlooker bee to exchange useful informa-tion. Meanwhile, it uses a Pareto archive set to record thenondominated solutions that participate in crossover with acertain probability. To enhance the local intensification, alocal search based on critical path is embedded in the onlookerbee phase, and a recombination and select strategy isemployed to determine the survival of the individuals. Inaddition, population is suitably adjusted to maintain diversityin scout bee phase. By using Taguchi method of design ofexperiment, the influence of several key parameters isinvestigated. Simulation results based on the benchmarksand comparisons with some existing algorithms demonstratethe effectiveness of the proposed EPABC algorithm.

Keywords Flexible job-shop scheduling problemMulti-objective optimization . Artificial bee colonyalgorithm .Machine assignment . Operation sequence .

Critical path

1 Introduction

The flexible job-shop scheduling problem (FJSP) is ageneralization of the classical job-shop scheduling problem(JSP) and the parallel machine environment with a strongengineering background in textile industry, automobileassembly process, and semiconductor manufacturingprocess. The FJSP consists of the routing subproblem andthe scheduling subproblem, where the former is to assigneach operation to a machine among a set of given machinesand the latter is to sequence the assigned operations on allmachines to obtain a feasible and satisfactory schedule. TheFJSP is of a larger complexity than the JSP that is NP-hardin determining the assignment of operations to machines aswell as the sequence of all operators. Thus, it is ofsignificant importance in both academic and applicationfields to develop or improve the solution algorithms for theFJSP.

In the first work to address the FJSP [1], a polynomialalgorithm was proposed to solve the FJSP with two jobs.Later, hierarchical approaches based on decomposition [2,3] were proposed for the flexible manufacturing system.During the past decade, metaheuristics especially tabusearch (TS) and genetic algorithm (GA) have gainedincreasing attention. In [4], a TS based on the integratedapproach was proposed in which a new neighborhoodstructure was defined for the FJSP. In [5], two neighbor-hoods were developed to improve the TS in [4] with betterperformances in terms of computation time and solutionquality. In [6], a TS combining with two heuristics wasproposed to solve the FJSP with sequence-dependentsetups. In [7], an integrated approach based on GA wasproposed to solve the FJSP under resource constraints. In[8], a learnable genetic architecture was developed forlearning and evolving solutions for the FJSP. In [9] and

L. Wang (*) :G. Zhou :Y. Xu :M. LiuTsinghua National Laboratory for Information Science andTechnology (TNList), Department of Automation,Tsinghua University,Beijing 100084, Chinae-mail: wangling@tsinghua.edu.cn

Int J Adv Manuf Technol (2012) 60:1111–1123DOI 10.1007/s00170-011-3665-z

[10], a GA hybridized with variable neighborhood searchand a GA integrating different strategies were proposed,respectively. In [11], a multistage operation-based GA wasproposed to deal with the problem from the point of view ofdynamic programming. Besides, a parallel variable neigh-borhood search based on six neighborhoods [12], anefficient cultural algorithm [13], and a hybrid TS with anefficient neighborhood [14] were proposed.

Compared to the mono-objective FJSP, the research onthe multi-objective FJSP (MFJSP) is relatively limited. Themethods to solve the MFJSP can be roughly classified intotwo types: weighting approach and Pareto-based approach.The weighting approach solves the MFJSP by transformingit to a mono-objective one by giving each objective aweight, while it is difficult in assigning proper weights toobjectives. The Pareto-based approach solves the MFJSPbased on the Pareto optimality concept and aims atgenerating the set of Pareto optimal solutions. The existingalgorithms belonging to the first type include the localiza-tion approach to solve the assignment problem [15], hybridparticle swarm optimization (PSO) [16, 17], geneticprogramming [18], hybrid TS [19], and knowledge-basedant colony optimization algorithm [20]. The existingalgorithms belonging to the second type include thePareto-based discrete artificial bee colony (P-DABC) [21],memetic algorithm based on NSGA-II [22], multi-objectivegenetic algorithm, [23] and multi-objective particle swarmoptimization [24].

As a relatively new member of swarm intelligence,artificial bee colony (ABC) [25] is motivated from thecollective behavior of self-organized systems. The ABCalgorithm was first proposed to optimize multivariable andmultimodal continuous functions [25]. Many comparativestudies showed that ABC was competitive to otherpopulation-based algorithms with the advantage of employingfewer control parameters in the continuous space [26–29].Compared with the research of ABC for continuousoptimization, the study on ABC for combinatorial optimiza-tion is considerably limited, including the generalizedassignment problem [30], the traveling salesman problem[31], and the lot streaming flow shop scheduling [32].

Very recently, a hybrid P-DABC was designed to solvethe MFJSP [21]. In this paper, we develop an enhancedPareto-based artificial bee colony (EPABC) algorithm,which is different from the P-DABC [21] at the followingaspects. First, the EPABC uses multiple strategies in acombination way to generate the initial solutions withcertain quality and diversity. Second, the EPABC appliesthe exploitation search procedures in employed bee phaseto generate new neighbor food sources. Third, the exploi-tation search procedures and crossover operators aredeveloped in onlooker bee phase for evolution, and anexternal Pareto archive set is used to record the non-

dominated solutions that participate in crossover in proba-bility. Fourth, the local search strategy based on criticalpath embedded in onlooker bee phase and a recombination/select strategy is used to determine the survival of theindividuals. Fifth, population is suitably adjusted in scoutbee phase to maintain diversity of population. Besides, theinfluence of key parameters is investigated based onTaguchi method of design of experiment and the suitableparameter setting is suggested. Simulation results withbenchmarks and comparisons with some existingalgorithms demonstrate the effectiveness of the proposedEPABC in solving the MFJSP.

The remainder of the paper is organized as follows. InSection 2, the basic notations of multi-objective optimiza-tion are introduced and the MFJSP is formulated. In Section3, the classic ABC algorithm is introduced briefly. TheEPABC algorithm is proposed in Section 4 for solving theMFJSP. The investigation of parameter setting and thenumerical comparisons are provided in Section 5. Finally,we end the paper with some conclusions in Section 6.

2 Multi-objective flexible job-shop scheduling problem

2.1 Multi-objective optimization

Without loss of generality, a multi-objective optimizationproblem can be described as follows:

Minimize y ¼ f ðxÞ ¼ ðf1ðxÞ; f2ðxÞ; :::; fqðxÞÞwhere x 2 Ω; y 2 Rq ð1Þ

where, x is the decision vector in space Ω, y is the objectivevector with q >1 objectives.

Next, we introduce some notations and techniques formulti-objective optimization.

Pareto dominance A solution a is said to dominatesolution b if and only if ∀i∈{1, 2, …,q}: fi (a) ≤ fi (b)and ∃i∈{1, 2, …,q}: fi (a)<fi (b).

Pareto optimal or nondominated solution A solution a isoptimal in the Pareto sense if there does not exist anysolution b that dominates a.

Pareto front Pareto optimal set is the collection of allPareto optimal solutions, and the corresponding image inthe objective space is called the Pareto optimal front.

Archive set An archive set AS is used to record thenondominated solutions. During the searching process, thearchive set is iteratively updated. Once a new nondomi-nated solution is found, it will be added to AS and then all

1112 Int J Adv Manuf Technol (2012) 60:1111–1123

the members dominated by this solution will beremoved. For the MFJSP, solutions in AS are differentfrom each other in terms of the machine assignment vectoror the operation sequence vector. Different from [21], thearchive set is used not only to record the nondominatedsolutions but also to provide some promising solutions toparticipate in crossover operators in the onlooker beephase.

Nondominated sorting Nondominated sorting is used todivide the solutions into several levels according to thedominance degree. The first front contains all thenondominated solutions in the current population, andthe second front contains all the solutions that aredominated by the individuals in the first front only, andso on. Every individual Si in each front is assigned a rankvalue ri. Individuals in first front are given a rank value of1, and individuals in second front are assigned a rankvalue of 2, and so on. Please refer [33] for the detailedimplementation.

Crowding distance In addition to rank value, crowdingdistance di is calculated for each individual si, which is ameasure of how close an individual is to its neighbors.Large crowding distance will result in better diversity in thepopulation. Once the nondominated sorting is completed,the members of the same rank sequence is in an ascentorder according to each objective, and then the crowdingdistance of each solution is defined as the sum of thenormalized distance between its right and left neighbors inthe sequence. As for the first individual s1 and the lastindividual Slast, their crowing distances are defined asinfinity. In this paper, the crowing distance di by consider-ing three objectives is defined as follows.

di ¼1 if i ¼ 1 or i ¼ LastP3

j¼1

objjðsiþ1Þ�objjðsi�1Þobjmax

j �objminj

; otherwise

8<

:ð2Þ

where, objj(si) is the value of the jth objective of si, and

objmaxj and objmin

j are the maximal and minimal values of

the jth objective known so far. Based on the rank andcrowding distance, solution a is better than b if ra<rb or(ra=rb and da>db).

For multi-objective optimization, the developed algo-rithm should obtain more nondominated solutions thatsatisfy multiple objectives with good proximity anddiversity. In other words, it requires the algorithm to obtainmore nondominated solutions, to obtained solutions on orcloser to the optimal Pareto front, and to obtain solutionsdistributed more diversely [34, 35].

2.2 Formulation of MFJSP

The flexible job-shop scheduling considers n jobs to beprocessed on m machines, where each job i consists of asequence of ni operations Oi,j,j=1, 2, …,ni. The processingtime P, i, j, k of Oi,j performed on machine k is given. Eachrouting has to be determined to complete a job. Theexecution of Oi,j requires one machine out of a set of givenmachines Mi;j � M . Let Ci,j be the completion time ofoperation Oi,j. For the MFJSP, it needs to determine boththe assignment of machines and the sequence of operationson all the machines to optimize multiple schedulingobjectives. In this paper, the following three objectives areto be minimized: (1) the maximal completion time ofmachines, i.e., CM; (2) the total workload of machines, i.e.,WT, which is of interest assigning the machine with relativesmall processing time for each operation to improveeconomic efficiency; and (3) the maximal machine work-load, i.e., WM, which considers the workload balanceamong all machines to prevent too much work beenassigned to a single machine. These objectives are inconflict to some extent and have been widely used in manyliteratures [15–17, 20, 21, 37, 43].

Mathematically, the MFJSP is formulated as follows [19]:

MinimizeCM ¼ max1�i�n

fCi;nig ð3Þ

MinimizeWT ¼Xm

k¼1

Xn

i¼1

Xni

j¼1

pi;j;kxi;j;k ð4Þ

MinimizeWM ¼ max1�k�m

fXn

i¼1

Xni

j¼1

pi;j;kxi;j;kg ð5Þ

Subject to : Ci;j > 0;Ci;j � Ci;j�1 � pi;j;k � xi;j;k ;j ¼ 2; ::::::; ni; 8i; k

ð6Þ

X

k2Mi;j

xi;j;k ¼ 1; 8i; j ð7Þ

xi;j;k ¼1; if machine k is selected for operation Oi;j

0; otherwise:

(

ð8Þ

where, Eq. 6 ensures that the operations belonging to thesame job satisfy the precedence constraints, and Eq. 7 statesthat one machine must be selected from the set of availablemachines for each operation.

Int J Adv Manuf Technol (2012) 60:1111–1123 1113

3 Artificial bee colony algorithm

The ABC algorithm an iterative population-based meta-heuristic inspired by the intelligent foraging behaviors ofhoneybee swarm, which contains three kinds of foragingbees including employed bees, onlooker bees, and scoutbees [25]. It starts by associating all employed bees withrandomly generated food sources. Then every employedbee moves to a new food source in the neighborhood of itscurrently associated food source and then evaluates thenectar amount during iterations. After the employed beescomplete the process, they share the nectar information ofthe food sources with the onlooker bees. The employedbee exhausting food source will become the scout bee,which will performs random exploration search for newfood source. Scout bee can be visualized as performingthe job of exploration, whereas employed and onlookerbees can be visualized as performing the job ofexploitation. The above process iterates until a stoppingcriterion is satisfied. The procedure of the basic ABCalgorithm is shown in Fig. 1 [25].

4 EPABC for MFJSP

4.1 Encoding and decoding

The solution of the MFJSP is a combination of machineassignment and operation scheduling decision encodedby the machine assignment vector and the operationsequence vector, respectively [23]. For the operationsequence vector, the number of genes equals to the totalnumber of operations. The operations of each job aredenoted by the corresponding job number, and the kthoccurrence of a job number refers to the kth operation inthe sequence of this job. For the machine assignmentvector, each number represents the machine assigned foreach operation successively.

Due to the precedence constraint among operations ofthe same job, idle time may exist between operations ona machine. Considering the criterion of makespan, weemploy the left-shift scheme [14] to shift the operationsto the left as compact as possible. Given a time intervalbeginning from tSk and ending at tEk on the machine k toallocate the operation Oi,j, Oi,j can be started only after its

immediate job predecessor Oi,j-1 is completed so that thestarting time of Oi,j can be described as followed.

Si;j ¼ maxftSk ;Ci;j�1g ð9ÞIf the operation has no predecessor, Ci,j-1 is replaced with

0. When operation Oi,j is assigned on machine k, it examinesthe idle time intervals between operations that have alreadybeen scheduled on the machine from left to right to find theearliest available time Si,j. If there is enough time span fromSi,j until tEk to complete Oi,j, i.e., maxftSk ;Ci;j�1g þ pi;j;k � tEk ,

then interval ½tSk ; tEk � is available for Oi,j. That is, Oi,j can beleft shifted. Such a left-shift decoding scheme is used toallocate each operation on its assigned machine from left toright following the operation sequence vector of the solution.Thus, it decodes a solution to a detailed schedule.

4.2 Population initialization

To guarantee an initial population with a certain quality anddiversity, some different strategies are utilized in a hybridway to generate Psize=population size (PS)×n×m initialsolutions as the food sources, where PS is a parameterrelated to population size Psize.

To generate initial machine assignments, we use threerules. (1) Random rule. Randomly select a machine fromthe candidate machines set for each operation and thenplace it at the position in the machine assignment vector. (2)Global minimum processing time rule [10]. (3) Localminimum processing time rule [14]. In our algorithm, theinitial machine assignments of 60% solutions in thepopulation are generated by rule 2, 30% solutions by rule 3,and 10% solutions by rule 1.

Once the machines are assigned to all the operations, allthe operations will be sequenced. A schedule is feasibleonly when all the precedence constraints among operationsof the same job are not violated. We apply three rules togenerate initial operation sequences. (1) Random rule;randomly generate the sequence of the operations on eachmachine. (2) Most time remaining rule [10]; sequence thejobs in the order of non-increasing remaining time, that is,the job with the most remaining time will be selected first.(3) Most number of operations remaining rule [10]; the jobwith most remaining operations unprocessed has a highpriority to be selected. In our algorithm, the initial operationsequences 20% solutions in the population are generated by

Step 1: Send the employed bees to the food sources and determine the nectar amounts;

Step 2: Calculate the probability value of each food source;

Step 4: Send the onlooker bees to their food sources according to the probability values;

Step 5: Send the scouts to the search area for discovering new food sources randomly;

Step 6: Record the best food source found so far;

Step 7: If a stopping criterion is met, then output the best food source; otherwise, go back to Step 1.

Fig. 1 Procedure of the basicABC algorithm

1114 Int J Adv Manuf Technol (2012) 60:1111–1123

rule 1, 40% solutions by rule 2, and 40% solutions byrule 3.

4.3 Exploitation search

Usually employed bees and onlooker bees use the same searchoperator to perform exploitation search. Since both machineassignment and operation sequence should be considered forMFJSP, we design different exploitation searches to evolvemachine assignment and operation sequence.

4.3.1 Exploitation search for machine assignment

The exploitation search for machine assignment vector ofsolution si is as follows:

Step 1. Randomly generate an integer I from 1 to Z (totalnumber of operations);

Step 2. Randomly select I positions from the machineassignment vector of si;

Step 3. For each selected position, replace the machinewith a different machine randomly chosen fromthe candidate machine set to generate a newsolution snew (no change happens if the set onlyincludes one machine).

Step 4. Evaluate snew. If snew dominates si, replace si withsnew. If they have different objectives and do notdominate each other, then keep si while updatingAS with snew.

4.3.2 Exploitation search for operation sequence

The exploitation search for operation sequence vector ofsolution si is as follows:

Step 1. Randomly select two jobs J1 and J2, and record thepositions of the two jobs (suppose the number ofoperations of J1 is less than or equal to that of J2).

Step 2. Fill J1 into the positions of J2 from left to right andthen fill J2 into the positions of J1 to generate anew solution snew.

Step 3. Evaluate snew. If snew dominates si, replace si withsnew. If they have different objectives and do notdominate each other, then keep si while updatingAS with snew.

Figure 2 illustrates an example for the above procedure,where jobs 1 and 2 are selected.

4.4 Employed bee phase

In the basic ABC algorithm, the employed bees independentlyperform exploration search with multiple different neighbor-hoods for promising food sources around a given food source si.

In this paper, the exploitation search procedures formachine assignment vector and operation sequence vector(Section 4.3) are applied by the employed bees to generatenew neighboring food sources. With the property of the twooperators, better food sources will be reserved for furtherevolution.

4.5 Onlooker bee phase

In this paper, multiple search operators are used in theonlooker bee phase, including the exploitation search,crossover, local search, and recombination and selection.

4.5.1 Exploitation search

In the basic ABC, an onlooker bee chooses a foodsource si according to a certain probability that isproportional to its fitness value. Since it is difficult todetermine the fitness value properly in multi-objectiveoptimization, the tournament selection [32] with size 3 isapplied in this paper.

In particular, randomly select three employed beesolutions from the population, and then determine the bestemployed bee solution according to the rank and crowdingdistance as the food source of the onlooker bee. After allthe onlooker bees have selected their food sources, eachonlooker bee generates a new neighboring solution snew byusing the exploitation search procedures in Section 4.3 formachine assignment vector and operation sequence vectorand then update all the food sources with the bettersolutions to generate population Pt.

4.5.2 Crossover operator

Crossover operator is designed for onlooker bees toexchange information to generate promising solutions. Foreach solution generated by the onlooker bee si using theexploration search, another solution s′ is randomly selectedfrom AS with probability Pas or from the population withprobability 1–Pas by tournament selection. Thus, a tempo-

rary population P»t is generated that contains all the new

solutions produced by the crossover operator. For theMFJSP, crossover operators for machine assignment andoperation sequence are developed as follows.

2411424323

2422414313

Fig. 2 Illustration to the procedure of exploitation search foroperation sequence

Int J Adv Manuf Technol (2012) 60:1111–1123 1115

Crossover for machine assignment For machine assign-ment, two crossover operators are used, where the two-point crossover provided in [36] is used with probability Pc

and the following uniform crossover proposed here is usedwith probability 1–Pc. These operators only change themachine assignment but not change the operation sequence.To the two feasible parents, the solution generated by thesecrossover operators is still feasible.

The procedure of the uniform crossover for solutions siand s′ is described as follows, and an example is illustratedin Fig. 3.

Step 1. Uniformly generate a binary string that isconsisted by 0 and 1 with the same length asthe machine assignment vector;

Step 2. The machine assignment of solution snew inheritsthe element of si at positions with bit 1 whileinherits the element of s′ at positions with bit 0.

Crossover for operation sequence For operation sequencevector of onlooker bees with permutation representation,we develop a modified precedence operation crossover(MPOX) according to the POX [17]. In MPOX, only theoperation sequence is changed while the machine assign-ment keeps unchanged. The MPOX for solutions si and s′ isas follows, and an example is illustrated in Fig. 4.

Step 1. Randomly generate a subset of all jobs.Step 2. The operation sequence of solution snew inherits

the element of si at the position if the elementbelongs to the subset; otherwise, it inherits theelements of s′ that do not belong to the subsetfrom left to right.

4.5.3 Local search based on critical path

To further enhance the exploitation capability, a localsearch based on critical path is designed for onlooker beesto generate promising solutions around the temporary

population P»t .

Denote SEi;j as the earliest starting time of operation Oi,j

and SLi;j as the latest starting time without delaying make-

span. Thus, the earliest completion time of operation Oi,j isCEi;j ¼ SEi;j þ pi;j;k , and the latest completion time is

CLi;j ¼ SLi;j þ pi;j;k . Let PMk

i;j be the operation processed on

machine k right before the operation Oi,j and SMki;j be the

operation processed on machine k right after Oi,j. Let PJi,j=Oi,j–1 be the operation of job i that precedes Oi,j and SJi,j=Oi,j+1 be the operation of job i that follows Oi,j. Since themakespan is no shorter than any possible critical path, it canbe improved only by moving the critical operations.

Let Ol (l=1, 2,…, Nc) be the critical operation to bemoved, where Nc is the total number of critical operationsof a solution in the disjunctive graph G. Moving Ol is todelete it from its current position to get a new graph G– andthen to insert it at another feasible position. Clearly, themakespan of G– is no larger than that of G. If Ol is assignedbefore Oi,j on machine k in G–, it can be started as early asCE�ðPMk

i;jÞ and can be finished as late as SL�i;j without

delaying the required makespan in G–. Besides, Ol cannotviolate the precedence relations of the same job. Thus, theassignable idle time interval for Ol can be defined by

maxfCE�ðPMki;jÞ;CE�ðPJlÞg þ pl;k � minfSL�i;j ; SL�ðSJlÞg.

Such a moving process is repeated until all the criticaloperations are moved. Let Nl be the number of positions tomove Ol feasibly, then the total number of moving

neighbors of a solution is Ntotal ¼PNc

l¼1 Nl.In our algorithm, the new solution obtained by moving

critical operations will replace the old one if the new one isnot dominated by the old one. If they are not dominated byeach other while having different objectives, then it updatesAS with the new one. To take advantage of the move ofcritical operations, we propose the following local searchprocedure for the solution s1i generated by the crossoveroperator to obtain si,new, where the depth of local search isset as 3.

Step 1. Let j=0;Step 2. Generate si,new by moving all critical operations of s1i

in the precedence order of operations from the first

is

news

's

2313234114

2333143112

3233143122

0 1 1 0 0 0 1 0 1 1

Fig. 3 Illustration of the uniform crossover

is

news

's

2411424323

2433424121

2443232411

Fig. 4 Illustration to the MPOX

1116 Int J Adv Manuf Technol (2012) 60:1111–1123

machine to the last one, where for each moving thenew solution is used to update AS when possible;

Step 3. Let s1i ¼ si;new and j=j+1. If j=3, then stop; else,go back to step 2.

To compromise quality and efficiency, the above localsearch is performed with probability onlooker bee phase(PL). By using the local search for the temporary population

P»t , a new population Qt is generated.

4.5.4 Recombination and selection

To determine the population for scout bee phase, thefollowing recombination and selection strategy is used.

First, populations Qt and Pt are combined together afterthe local search. Then, sort all the solutions by using thenondominated sorting, where the solutions in the best

Select a solution s’ from the population by tournament selection

Maximum generation is reached?

Apply the non-dominated sorting to population, and then update the Pareto archive set AS by using the solutions in the first Pareto level front

Perform exploration search procedures for every employed bee, and update ASwhen possible

Select a food source in population by tournament selection for every onlooker bee

Perform local search procedure based on critical path for each solution in Pt* with

probability PL to generate new population Qt

Perform crossover operators for each onlooker bee and s’ to generate new population Pt

*

Randomly select a solution from AS to perform exploitation search and local search to generate a new solution, and update population when possible for the

next generation

Set parameters and initialize population with multiple strategies

No

Onlooker bee phase

Output ASYes

YesNo

Perform exploration search procedures for all selected food sources to generate population Pt, and update AS when possible

Select a solution s’ from ASrandomly

Employed bee phase

Scout bee phase

Initialization phase

Rand<Pas

Perform recombination and selection for Pt and Qt, and update AS with the solutions in the best front of the combined population

Fig. 5 Framework of EPABCfor MFJSP

Table 1 Combinations of parameter values

Parameters Factor level

1 2 3 4

PS 0.5 1 1.5 2

PL 0.2 0.5 0.8 1

Pas 0.1 0.3 0.6 0.8

Pc 0.1 0.3 0.6 0.8

Int J Adv Manuf Technol (2012) 60:1111–1123 1117

front are used to update AS. Then, select Psize solutionsfrom the best front to the worst one. If selecting all thesolutions in front Fj may result in the total number ofselected solutions exceeds Psize, certain number ofsolutions in front Fj with larger crowding distance willbe selected.

4.6 Scout bee phase

In the classic ABC, the scout bee is used to produce foodsource randomly to replace the worst solution in thepopulation. Since the solution is generated randomly, globalexploration is stressed in scout bee phase and it also mayhelp enhance population diversity to some extent. In theEPABC, the search in scout bee phase is implemented asfollows.

Step 1. Randomly select a nondominated solution fromAS to perform exploitation search only formachine assignment vector to generate solutionstemp;

Step 2. Perform local search based on the critical path forstemp to generate solution sscout;

Step 3. Determine the worst solution in the populationaccording to rank and crowding distance and thenreplace it with sscout if sscout is better.

4.7 Framework of EPABC

Due to the complexity in solving the MFJSP, it is difficult tosolve the problem effectively with a single searching operator.The EPABC fuses the initialization with multiple strategies,exploitation search procedures, crossover operators, localsearch based on critical path, and recombination and selectionstrategy in the framework of the ABC algorithm. It stresses thebalance of the global exploration and local exploitation, and italso stresses the diversity of population during the searchingprocess. Thus, it is expected to achieve good performances insolving the MFJSP.

Straightforwardly, the framework of the proposedEPABC algorithm is illustrated in Fig. 5.

5 Experimental results

To test the performance of the proposed EPABC algorithm,numerical simulations are carried out based on the well-studied benchmarks including five Kacem instances(cases 1–5) [15], Mk09 instances [2], and five instanceswith job release dates [21, 37]. The algorithm is imple-mented in C++ and run on 2.83-GHz PC with 3.21-GBRAM. For each instance, we run the algorithm 20 timesindependently with the maximum 4×n×m generations.

5.1 Investigation of parameters setting

The EPABC contains several key parameters including theparameter related to PS, the probability to perform the localsearch in PL, the probability for solutions in AS toparticipate in crossover, and the probability of two-pointcrossover operator to evolve the machine assignment (Pc).To investigate the influence of these parameters on theperformance of the EPABC, we implement the Taguchimethod of design of experiment (DOE) [38–42] by usingthe problem case 1. We set four-factor levels for eachparameter as shown in Table 1. The orthogonal arrayL16(4

4) is shown in Table 2.Based on the orthogonal array, we run the EPABC to

solve case 1 with each combination of parameters 20 timesindependently. For each combination, the nondominatedsolutions among all the obtained solutions by all the runs

are collected as the reference set RS* with jRS»j solutions.Clearly, the larger jRS»j is, better the combination is.

Table 2 Orthogonal array and Navg(i) values

Combination number Factor Navg(i)

PS PL Pas Pc

1 1 1 1 1 2.85

2 1 2 2 2 2.7

3 1 3 3 3 3.5

4 1 4 4 4 3.45

5 2 1 2 3 3.3

6 2 2 1 4 3.3

7 2 3 4 1 3.45

8 2 4 3 2 3.4

9 3 1 3 4 3.4

10 3 2 4 3 3.45

11 3 3 1 2 3.55

12 3 4 2 1 3.45

13 4 1 4 2 3.4

14 4 2 3 1 3.5

15 4 3 2 4 3.85

16 4 4 1 3 3.8

Table 3 Response value and significance rank

Level PS PL Pas Pc

1 3.125 3.2375 3.375 3.3125

2 3.3625 3.2375 3.325 3.2625

3 3.4625 3.5875 3.45 3.5125

4 3.6375 3.525 3.4375 3.5

Delta 0.5125 0.35 0.125 0.25

Rank 1 2 4 3

1118 Int J Adv Manuf Technol (2012) 60:1111–1123

Denote the solutions obtained by the ith combination at thejth run as RSij. Thus, the average number of nondominatedsolutions Navg(i) for the ith combination in 20 runs can becalculated with Eq. 10. The obtained Navg(i) for all thecombinations are listed in Table 2.

NavgðiÞ ¼X20

j¼1RS

» \ RSij���

���=20 ð10Þ

With the results in Table 2, we figure out the responsevalue of each parameter and analyze the significance rank ofeach parameter as shown in Table 3. Then, we illustrate thetrend of each factor level in Fig. 6. From Table 3, it can beseen that PS is the most significant parameter among the fourparameters. With a fixed maximum generation, population

with larger size may yield more nondominated solutions inglobal sense. The second significant parameter is PL. FromFig. 6, it suggests larger PL to be used which means the localsearch based on critical path is helpful to solve the problem.Besides, Fig. 6 shows that EPABC is relatively robust on Pasand Pc to certain extent while large values are preferable.According to the above analysis, we set PS=2, PL=0.8, Pas=0.6, and Pc=0.6 for EPABC in the following numerical test.

5.2 Test on Kacem instances

Using the five Kacem instances, we compare the EPABCwith the existing algorithms including AL+CGA [15], PSO

Fig. 6 Factor level trend ofparameters in EPABC

Table 4 Results of the five Kacem instances

n×m Obj AIA ALC+GA PSO+SA PSO+TS P-DABC EPABC

S1 S1 S2 S1 S2 S1 S2 S1 S2 S3 S1 S2 S3 S4 CPU (s)

Case 1 (4×5) CM 16 11 11 12 13 11 12 13 11 1.203WT 34 32 32 32 33 32 32 33 34

WM 10 10 10 8 7 10 8 7 9

Case 2 (8×8) CM 14 15 16 15 16 14 15 14 15 16 14 15 16 16 1.484WT 77 79 75 75 73 77 75 77 75 73 77 75 73 77

WM 12 13 13 12 13 12 12 12 12 13 12 12 13 11

Case 3 (10×7) CM 12 11 12 11 12 11 4.937WT 61 63 60 61 60 62

WM 11 11 12 11 12 10

Case 4 (10×10) CM 7 7 7 7 8 7 8 8 7 8 7 6.262WT 43 45 44 43 41 43 42 41 43 42 42

WM 5 5 6 6 7 5 5 7 5 5 6

Case 5 (15×10) CM 11 23 24 12 11 12 11 11 11 18.532WT 93 95 91 91 93 91 93 91 93

WM 11 11 11 11 11 11 11 11 10

Int J Adv Manuf Technol (2012) 60:1111–1123 1119

+SA [16], PSO+TS [17], AIA [43], and P-DABC [21]. Theresults are listed in Table 4, where the results of the existingalgorithms are directly from literature.

From Table 4, it can be seen that EPABC is the best oneamong all the algorithms for solving the five Kaceminstances. Compared with the existing algorithms, EPABCcan obtain more nondominated solutions or better Paretooptimal solutions with the bold values, which meansEPABC is more effective. For example, for case 3 although

both EPABC and D-PABC obtain three nondominatedsolutions, the solutions (12, 61, 11) obtained by P-DABCis dominated by (11, 61, 11) obtained by EPABC, and (11,63, 11) obtained by P-DABC is dominated by (11, 61, 11)and (11, 62, 10) obtained by EPABC. From the results, it isshown that the capability of D-PABC is enhanced byEPABC in sense of multi-objective optimization. In Fig. 7,the Gantt chart of the nondominated solution (11, 62, 10)obtained by EPABC for case 3 is illustrated. In addition, it

Fig. 7 A nondominatedsolution for case 3

Table 5 Results of the Mk09instance Xing P-DABC EPABC

No. (CM,WT,WM) No. (CM,WT,WM) No. (CM,WT,WM) No. (CM,WT,WM) No. (CM,WT,WM)

1 (311,2374,299) 1 (311,2288,299) 1 (309,2300,299) 22 (322,2273,307) 43 (339,2247,334)

2 (313,2286,311) 2 (309,2300,299) 23 (322,2284,301) 44 (341,2249,326)

3 (319,2280,307) 3 (310,2296,310) 24 (323,2269,307) 45 (341,2250,323)

4 (324,2279,307) 4 (311,2291,299) 25 (323,2283,300) 46 (344,2245,334)

5 (411,2265,349) 5 (311,2283,302) 26 (324,2262,316) 47 (345,2248,332)

6 (417,2253,360) 6 (311,2290,301) 27 (324,2264,315) 48 (346,2244,334)

7 (424,2240,386) 7 (314,2289,300) 28 (324,2277,300) 49 (346,2247,332)

8 (431,2230,402) 8 (316,2281,308) 29 (326,2261,316) 50 (347,2248,331)

9 (474,2223,444) 9 (316,2287,300) 30 (326,2263,315) 51 (347,2241,334)

10 (484,2210,454) 10 (316,2278,314) 31 (327,2258,316) 52 (352,2240,345)

11 (317,2279,307) 32 (327,2260,315) 53 (353,2240,334)

12 (318,2286,299) 33 (328,2257,326) 54 (362,2239,353)

13 (318,2227,307) 34 (330,2259,315) 55 (407,2230,394)

14 (319,2274,314) 35 (330,2255,326) 56 (422,2226,412)

15 (320,2273,314) 36 (331,2254,323) 57 (432,2224,432)

16 (320,2271,315) 37 (331,2256,316) 58 (434,2225,426)

17 (320,2275,307) 38 (332,2252,322) 59 (435,2218,426)

18 (321,2269,315) 39 (332,2254,321) 60 (443,2223,442)

19 (321,2285,301) 40 (333,2250,328) 61 (449,2217,436)

20 (321,2273,308) 41 (337,2263,310) 62 (454,2210,454)

21 (322,2265,312) 42 (338,2258,315)

1120 Int J Adv Manuf Technol (2012) 60:1111–1123

also shows that EPABC is efficient since the centralprocessing unit (CPU) time cost is very short.

5.3 Test on MK09 instance

Next, we use Mk09 with a relatively larger scale (20×10)for testing. The results of EPABC and the existingalgorithms including Xing [20] and P-DABC [21] are listedin Table 5.

From Table 5, it can be seen that EPABC obtains 62nondominated solutions. In [20], a linear weighted summa-tion was used to aggregate three objectives with tendifferent weight sets, while the obtained solution isdominated by three solutions obtained by EPABC. As forthe P-DABC [21], its ten solutions except (311, 2288, 299)are all dominated by some of solutions obtained by

EPABC. In brief, EPABC can obtain more Pareto optimalor nondominated solutions. Once again, it verifies that theproposed EPABC is of superior performances to theexisting P-DABC in terms of quality and number of theresulted nondominated solutions.

5.4 The instances with job release dates

Finally, we use five instances with job release dates [21, 37]to compare EPABC with Multi-Objective EvolutionaryAlgorithm and Guided Local Search (MOEA-GLS) [37]and P-DABC [21]. The results are listed in Table 6.

From Table 6, it can be seen that EPABC is better thanor competitive to MOEA-GLS and P-DABC. For the 8×8 instance, EPABC obtains one more Pareto optimalsolution (19, 80, 15), whose Gantt chart is illustrated in

Fig. 8 A nondominatedsolution for 8×8 instance

Table 6 Comparison of the fiveinstances with release dates n×m Obj MOEA-GLS P-DABC EPABC

S1 S2 S3 S1 S2 S3 S1 S2 S3 S4 CPU (s)

4×5 CM 16 16 16 16 16 16 1.188WT 32 33 32 33 32 33

WM 8 7 8 7 8 7

8×8 CM 20 20 20 20 20 20 19 2.063WT 73 75 77 73 75 77 80

WM 13 12 11 13 12 11 15

10×7 CM 15 16 15 15 16 15 15 16 15 4.172WT 61 60 62 61 60 62 61 60 62

WM 11 12 10 11 12 10 11 12 10

10×10 CM 12 13 13 12 13 13 6.281WT 43 41 42 43 41 42

WM 6 7 5 6 7 5

15×10 CM 23 23 23 23 23 23 23.219WT 91 93 91 93 91 93

WM 11 10 11 10 11 10

Int J Adv Manuf Technol (2012) 60:1111–1123 1121

Fig. 8. Besides, it also shows that EPABC is efficient sinceonly small amount of CPU time is required.

6 Conclusion

In this paper, an enhanced Pareto-based artificial bee colonyalgorithm has been proposed to solve the multi-objectiveflexible job-shop scheduling problem. By designing effec-tive hybrid initialization scheme, exploitation search pro-cedures, crossover operators for machine assignment andoperation sequence, local search based on critical path,recombination, and selection strategy, the capability ofABC has been enhanced for solving the MFJSP. Theinfluence of key parameters has also been investigatedbased on Taguchi method of DOE test. It has beendemonstrated by simulation results based on the widelystudied benchmarks and comparisons with the existingalgorithms that the proposed EPABC is better than theexisting the algorithms including P-DABC in terms ofquality and number of the obtained nondominatedsolutions. The future work is to study linking othermetaheuristics like bacteria-foraging algorithm withABC to further enhance the performance and to applythe algorithm to solve the scheduling problems in theuncertain environment.

Acknowledgments The authors would like to thank the editor andthe anonymous referees for their valuable comments to improve thispaper. This research is partially supported by National ScienceFoundation of China (61174189, 61025018, 70871065, and60834004), Program for New Century Excellent Talents in University(NCET-10-0505), Doctoral Program Foundation of Institutions ofHigher Education of China (20100002110014), the National KeyBasic Research and Development Program of China (no.2009CB320602), and National Science and Technology Major Projectof China (no. 2011ZX02504-008).

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