Research Article A Modified Biogeography-Based Optimization for the Flexible Job Shop ...downloads.hindawi.com/journals/mpe/2015/184643.pdf · 2019-07-31 · Research Article A Modified

Research ArticleA Modified Biogeography-Based Optimization forthe Flexible Job Shop Scheduling Problem

Yuzhen Yang

School of Electrical Engineering Shanghai Dianji University Shanghai 200240 China

Correspondence should be addressed to Yuzhen Yang young goodstar126com

Received 20 May 2015 Accepted 28 September 2015

Academic Editor George S Dulikravich

Copyright copy 2015 Yuzhen YangThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The flexible job shop scheduling problem (FJSSP) is a practical extension of classical job shop scheduling problem that is known tobe NP-hard In this paper an effective modified biogeography-based optimization (MBBO) algorithmwith machine-based shiftingis proposed to solve FJSSP with makespan minimizationTheMBBO attaches great importance to the balance between explorationand exploitation At the initialization stage different strategies which correspond to two-vector representation are proposed togenerate the initial habitats At global phase different migration and mutation operators are properly designed At local phase amachine-based shifting decoding strategy and a local search based on insertion to the habitat with best makespan are introduced toenhance the exploitation ability A series of experiments on two well-known benchmark instances are performedThe comparisonsbetween MBBO and other famous algorithms as well as BBO variants prove the effectiveness and efficiency of MBBO in solvingFJSSP

1 Introduction

The job shop scheduling problem (JSSP) is one of the mostdifficult combination optimization problems in manufactur-ing systems As an extension of JSSP the flexible job shopscheduling problem (FJSSP) is much closer to the practicalproduction because of the ability of machines to performmore than one type of operations Therefore the problemthat FJSSP should deal with is not only the sequencing ofoperations like JSSP but also the assignments of machinesfor operations It makes solving FJSSP much more difficultthan classical JSSP Hence the study on FJSSP theoreticallyor methodologically would have positive influence in appli-cation field

Since the first study on FJSSP was proposed by Bruckerand Schlie in 1990 [1] a plenty of research has been carried outand most of them were focused on designing metaheuristicsto solve FJSSP with makespan or tardiness minimizationSince FJSSP consists of two subproblems which are operationsequence subproblem and machine assignment subproblemthe conventional method for solving this problem is to treatthese two subproblems separately with different strategiesBy doing this the internal relationship between these two

subproblems has been neglected As research continues theintegrated approaches by considering the two subproblemssimultaneously are proposed with high complexity and bettersolutions Brandimarte [2] used some dispatching rules tosolve the machine assignment problems and tabu search(TS) for the sequencing problems Mastrolilli and Gam-bardella [3] proposed TS based on several neighborhoodstructures Pezzella et al [4] proposed a genetic algorithm(GA) integrating different strategies Ho and Tay did a serialof studies on FJSSPThey first designed genetic programming(GP) to combine dispatching rules [5] and came up withGENACE via incorporating composite dispatching rules(CDRs) and cultural algorithm [6] and then LEGA based onGA and machine learning mechanism [7] Recently Yazdaniet al [8] developed a parallel variable neighborhood search(PVNS) algorithm based on six neighborhood structuresLei designed a coevolutionary GA [9] and swarm-basedneighborhood search algorithm [10] for fuzzy FJSSP Wanget al proposed single population and bipopulation basedestimation of distribution algorithm (EDA [11] and BEDA[12]) for the FJSSP and came up with better results Yuanand Xu [13] presented a hybrid differential evolution (HDE)algorithm with new speed-up strategy on some local search

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 184643 10 pageshttpdxdoiorg1011552015184643

2 Mathematical Problems in Engineering

With regard to the multiobjective FJSSP weighted meth-ods and Pareto-based methods are commonly studiedThe objectives generally are makespan total work load ofmachines and maximum work load of machine Kacem etal [14 15] proposed several effective evolutionary algorithmsHo et al [16] used CDRs GA and machine learning mech-anism to optimize several objectives Gao [17 18] addedshifting bottleneck and variable neighborhood search toGA Zhang et al [19] developed a hybrid particle swarmoptimization (PSO) incorporated with a novel initializationLi et al [20] introduced a hybrid TS algorithm and discreteartificial bee colony (ABC) algorithm for the MJSSP Wang etal [21] further developed Pareto-based EDA (PEDA)

Biogeography-based optimization (BBO) a combinationof biogeography and engineering science was proposed bySimon in 2008 [22] Due to the similar features with otherbiology-based optimization algorithms such as GA and PSOBBO is applicable to many types of problems that GA andPSO are applied for However BBO also has some uniquefeatures from other types of biology-based optimizationTheinitial populations in BBO survive forever and their charac-teristics change gradually as the search process progressedAnd the main parameters of individuals are decided by theirranking in population which would cause no extra settingprocess

So far on count of all these features BBO has beenapplied to many academic and application problems [23ndash25] However there is little research about BBO to solvescheduling problem In this paper we propose a modifiedBBO (MBBO) to solve the FJSSP with makespan minimiza-tion When designing the algorithm we focus on the balancebetween exploration and exploitation Considering two-vector coding different strategies are introduced to generatethe initial solutions A machine-based shifting decodingmethod is proposed to convert the vectors to feasible scheduleat local exploitation phase Different migration and mutationoperators are designed at global exploration phase Particu-larly a local search is applied to the solution with the bestmakespan to speed up the convergence of the algorithmwhilemaintaining the good character of the best solution at localexploitation phase In addition experiments are performedto examine how the parameters affect the performance of thealgorithm Last but not least we compare MBBO with otherexisting famous algorithms and BBO variants on two sets ofbenchmark instances to test the efficiency and effectivenessof BBO in solving FJSSP

The reminder of the paper is organized as followsIn Section 2 we explain the flexible job shop schedulingproblem as well as the concept and structure of basic BBOalgorithm in Section 3 In Section 4 the modified BBO isdeveloped to FJSSP subsequently Section 5 analyzes theperformance results ofMBBOwhen applied to solve commonbenchmark problems in literature At last we come to ourconclusion and some possible future directions

2 Problem Formulation

FJSSP is one of the most practical and hardest combinatorialoptimization problems During past two decades a bunch of

Table 1 Example of FJSSPwith 3 jobs 4machines and 9 operations

FJSSP Processing Time1198721

1198722

1198723

1198724

1198691

O11

3 4 infin 5O12

4 6 5 infin

O13

3 infin infin 6

1198692

O21

4 infin infin 5O22

infin 5 infin 3O23

5 3 5 2O24

3 6 infin infin

1198693

O31

5 3 4 3O32

4 2 1 infin

literature has been published but no efficient algorithm hasbeen presented yet for solving it to optimality in polynomialtime

Suppose we are given 119899 jobs and 119898 machines Eachmachine can handle one job at most at a time Each job 119894consists of a sequence of operations andneeds to be processedduring an uninterrupted time period on one from a set offeasible machines The purpose is to find a scheme that isthe job sequence on the machines as to optimize one ormore performance measurements makespan in our case Anexample of 3 jobs 4 machines and 9 operations is shown inTable 1 whereinfin indicates that the machine can process theoperation however with infinity processing time And theFJSSP could be divided into two types total FJSSP and partialFJSSP based on whether each operation can be processed onall of the machines

The mathematical model of FJSSP with makespan mini-mization can be formulated as [20]

Minimise 119862119872= max1le119894le119899

119862119894119902119894

Subject to 119862119894119895minus 119862119894119895minus1

ge 119901119894119895119896

119909119894119895119896

119895 = 2 3 119902119894 forall119894 119896

sum

119896isin119872119894119895

119909119894119895119896

= 1 forall119894 119895

119909119894119895119896

=

1 if operation 119874119894119895is processed on machine 119896

0 otherwise

119862119894119895ge 0 forall119894 119895

(1)

The indices and variables of the model are enumerated asfollows 119898 is number of machines 119899 is number of jobs 119894 isjob index 119895 is operation number index 119896 is machine index119902119894is number of operations of job 119894 119862

119894119895is finish time of 119895th

operation of job 119894 119901119894119895119896

is processing time of 119895th operation ofjob 119894 on machine 119896 119872

119894119895is set of machines that can process

119895th operation of job 119894

3 Biogeography-Based Optimization

BBO algorithm proposed by Simon in 2008 is inspiredby the mathematics of biogeography and mainly the work

Mathematical Problems in Engineering 3

119868 rarr 119867119899HSI119899

While not 119879Ψ = (119898 119899 120582 120583 Ω119872)

end

Pseudocode 1 Pseudocode of BBO for optimization problems

from MacArthur and Wilson [26] Later a large amount oftheoretical methodological and practical studies on BBOhave come into being

The two main concepts of BBO are habitat suitabilityindex (HSI) and suitability index variables (SIVs) Featuresthat correlate with HSI include rainfall diversity of topo-graphic features land area and temperature And SIVsare considered as the independent variables of the habitatGeographical areas that are well suited for species are saidto own a high HSI Considering the optimization algorithma population of candidate solutions can be represented asvectors Each integer in the solution vector is consideredto be an SIV After assessing performance of the solutionsgood solutions are considered to be habitats with a high HSIand poor ones are considered to be habitats with a low HSITherefore HSI is analogous to fitness in other population-based optimization algorithms A BBO algorithm can bedescribed as in Pseudocode 1

A BBO algorithm is a 3-tuple evolutionary algorithm thatproposes a solution to an optimization problem 119868 rarr

119867119899HSI119899 is a function that creates an initial ecosystem

of habitats and computes each corresponding HSI Ψ =

(119898 119899 120582 120583 Ω119872) is a 6-tuple ecosystem transition functionthat modifies the ecosystem from one optimization iterationto the next which includes119898 the number of SIVs 119899 the sizeof habitats 120582 the immigration rate 120583 the emigration rateΩthe migration operator119872 the mutation operator

As it is clear from Pseudocode 1 the two main operatorsof BBO are migration and mutation Here we present a briefintroduction of these two operators and more details can befound in the literature

Migration operator including immigration and emigra-tion bridges the communication of habitats in an ecosystemThe emigration and immigration rates of each solution areused to probabilistically share information between habitatsDeciding whether or not a habitat performs emigration orimmigration is up to its HSI A habitat with high HSImeaning more species has more opportunities to emigrate toneighboring habitats And a habitat with low HSI meaningsparse species need immigration to increase the diversityof its species Therefore habitats with high HSI have largeremigration rates and smaller immigration rates while oneswith low HSI have larger immigration rates and smalleremigration rates A model of immigration and emigrationrate could be set as linear or nonlinear

Since cataclysmic events can drastically change the HSIof a natural habitat mutation operator is designed to modelthese random events Moreover this operator could increasethe diversity of BBO to further help jump out of local optimal

2 1 3 1 2 2 2 1 3

2 3 1 1 4 3 2 2 1

Job 1 Job 2 Job 3

Operation sequence

Machine assignment

O21 O11 O31 O12 O22 O23 O24 O13 O32

Figure 1 The coding strategy for MBBO

to some extentThemutation rate of each habitat here unlikethe one in GA is not set as the same value randomly It isdecided by probability of species numbers of each habitat119875119904(119905) refers to the possibility of a habitat having 119904 speciesThen

the 119875119904(119905 + Δ119905) can be calculated as

119875119904(119905 + Δ119905) = 119875

119904(119905) (1 minus 120582

119904Δ119905 minus 120583

119904Δ119905) + 119875

119904minus1120582119904minus1Δ119905

+ 119875119904+1120583119904+1Δ119905

(2)

Here give (3) derived from (2) by taking Δ119905 rarr 0

1198751015840

119904

=

minus (120582119904+ 120583119904) 119875119904+ 120583119904+1119875119904+1 119904 = 1

minus (120582119904+ 120583119904) 119875119904+ 120582119904minus1119875119904minus1

+ 120583119904+1119875119904+1 1 lt 119904 le 119899 minus 1

minus (120582119904+ 120583119904) 119875119904+ 120582119904minus1119875119904minus1 119904 = 119899

(3)

And then the mutation rate of each habitat is calculatedaccording to its 119875

119904

119898119904= 119898max (1 minus

119875119904

119875max) (4)

where 119898max is a predefined value meaning the maximummutation rate

4 MBBO for FJSSP

In this section we will propose a modified biogeography-based optimization (MBBO) to solve the FJSSP Differentfrom the traditional shifting decoding strategy a novelmachine-based shifting one which performs a kind of localsearch will be presented to decode the representations tofeasible schedules more effectively

41 Habits Representation It is clear that the FJSSP hastwo subproblems which are the allocation of operations onmachines and the sequencing of operations To handle thesesubproblems our MBBO operates on two chromosomeswhich correspond to the two problems respectively whichhas achieved good results by cooperating with GA [9]Each of them is coded as a 119863-dimensional real-parametervector [119909

1 1199092 1199093 119909

119863] where 119863 is the total number of

operations A scheme of this coding strategy is shown inFigure 1

For the operation sequence vector the operations whichbelong to the same job are signed with the same symbol and


the 119896th occurrence of a job number refers to the 119896th operationof the job For the instance in Table 1 119869

1has three operations

11987411 11987412 and 119874

13 1198692has four operations 119874

21 11987422 11987423 and

11987424 1198693has two operations 119874

31and 119874

32 One possible coding

vector of operation sequence [2 1 3 1 2 2 2 1 3] inPseudocode 1 can be translated into a sequence representedas [11987421 11987411 11987431 11987412 11987422 11987423 11987424 11987413 11987432]

And the machine assignment vector presents theassigned machine of each operation successively Forinstance in Table 1 the operations for the three jobs are3 4 and 2 respectively Hence the first three SIVs standfor the assigned machines for 119869

1successively the following

four SIVs for 1198692 and the last two SIVs for 119869

3 Therefore

the two vectors in Pseudocode 1 can be translated as[(119874211198721) (119874111198722) (119874311198722) (119874121198723) (119874221198724) (11987423

1198723) (119874241198722) (119874131198721) (119874321198721)]

This representation has an advantage that any habitatcan be decoded to a feasible schedule There does notexist the special case that processes the schedule operationswhose technological predecessors have not been scheduledyet Moreover it shows another advantage that the searchtemplate of the evolutionary search has no concern withdetails of particular scheduling problems

42 Decoding Vectors to Feasible Schedules After the rep-resentation the coding vectors must be transformed into afeasible schedule and a powerful decoding strategy plays animportant role in improving the final solutions

The most common decoding strategy for FJSSP existingin literatures is called forcing or left shifting [27] which iswidely applied in classical JSSP The decoding allocates eachoperation on its assigned machine one by one in the orderrepresented by the coding When operation 119874

119894119895is scheduled

on machine 119896 the idle time between operations that havealready been processed on that machine is examined fromleft to right to find the earliest one that is not shorter thanthe process time of operation119874

119894119895 If such an interval exists it

is allocated there otherwise it is allocated at the current endof machine 119896

Obviously the left shifting only shifts operations on thepredefined machine Considering the fact that a machine hasthe ability to perform more than one operation in FJSSP amachine-based shifting decoding strategy is proposed wherethe shifting to other capable machines is also examinedThe process works as follows where 120587

1and 120587

2refer to the

operation sequence vector and machine assignment vectorrespectively

(1) For the first operation of operation sequence vector1205871(1) schedule it on the machine 119896 with the shortest

processing time

(2) For each operation 1205871(119897) 119897 = 2 3 119899 the starting

time of corresponding operation 119874119894119895is calculated as

the following situations where 1198711205872(119894119895)

stands for theorder of operation119874

119894119895on themachine determined by

machine assignment vector 1205872 St(119874

119894119895) refers to the

starting time of operation 119874119894119895

2

2

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

3

1

1

Machine 3

Machine 2

Machine 1 3

Machine 4

2

2

1

20191817

Figure 2 The Gantt chart of shifting decoding process for FJSSP

2 2

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

3 1

1

Machine 3

Machine 2

Machine 1

3Machine 4

22 1

20191817

Figure 3The Gantt chart of machine-shifting decoding process forFJSSP

(a) If 1198711205872(119894119895)

= 1amp119895 = 1 set St(119874119894119895) = 0 and

schedule it on the machine 119896 with the shortestprocessing time

(b) If 1198711205872(119894119895)

= 1amp119895 = 1 set St(119874119894119895) = 119862

119894119895minus1and

schedule it on machine 1205872(119874119894119895)

(c) If 1198711205872(119894119895)

= 1amp119895 = 1 set St(119874119894119895) = min(119862

119894119895) minus

119901119894119895119896

and schedule it on machine 119896(d) If 119871

1205872(119894119895)= 1amp119895 = 1 set St(119874

119894119895) = min(119862

119894119895) minus

119901119894119895119896

while St(119874119894119895) gt 119862

119894119895minus1and schedule it on

machine 119896

(3) Modify the two vectors according to the final sched-ule

The schedules generated by shifting decoding andmachine-based shifting decoding of the example shown inTable 1 and Figure 1 are depicted in Figures 2 and 3 respec-tively The makespan obtained by machine-based shifting is13 while the one obtained by shifting strategy is 20 It is clearthat machine-based strategy functions effectively as a kind oflocal search to improve the final schedule

43 Initial Habitats At the beginning ofMBBO two differentmethods which correspond to two vectors are designed togenerate initial habitats

To generate initial operation ones the simple random ruleis served due to the strong decoding strategy in this paperHowever this is not suitable for the machine assignment vec-tors initialization because of the existence of partial flexibilityAt least one operation cannot be processed on somemachineHence random rule might generate infeasible solutionsHere the following 2-tournament strategy is designed for the


For 119894 = 1 NPif Rand(0 1) lt 120582

119894

using Tournament selection to choose119867119895

Migration (119867119894 119867119895)

endend

Pseudocode 2 Selection process of the migration operator

Rate

120582S-immigration

120583S-emigration

Smax

S1 S2

Figure 4 Cosine migration model curve of BBO

machine assignment vectors For each operation two capablemachines are selected randomly Then the one with shorterprocessing time is chosen as the processing machine

44Migration Operator Migration operator is a probabilisticone to share information between habitats based on emi-gration and immigration rates Each habitat 119867

119894should be

checked whether it performs immigration with respect to itsimmigration rate 120582

119894 If so habitat119867

119895is selected as emigration

habitat with respect to its emigration rate 120583119895 The process is

shown in Pseudocode 2And the immigration rate 120582 and emigration rate 120583 are

calculated as (5) and (6) respectively shown in Figure 4 Onehas

120582119904= (

119868

2) lowast (cos(119904120587

119899) + 1) (5)

120583119904= (

119864

2) lowast (minus cos(119904120587

119899) + 1) (6)

where 119904 refers to the species size of the habitat 119899 is themaximum species size 119868 and 119864 are the immigration andemigration coefficient respectively and generally set to be1 Literature [28] proves this advantage of this cosine modelover linear ones

With regard to the two-vector representation of MBBOthe IPOX [29] and 0-1 uniform crossover are applied to theoperation sequence vector and machine assignment vectorrespectively The processes are depicted in Figures 5 and 6By doing so there are no needs for any modification whichavoid burdening the algorithm with extra computation time

3 1 32 11 222

2 1 13 22 231

1 1 12 33 222

Immigrationhabitat 2

Emigration habitat 1 3

Habitat aftermigration

Figure 5 IPOX operator on operation sequence vector

For 119894 = 1 NPusing119898

119894selecting119867

119894amp Rank

119894= 1

Mutation (119867119894)

end

Pseudocode 3 Selection process of the mutation operator

45 Mutation Operator The mutation operator in BBOperforms the same role as the one in GA to increasediversity of the habitats Similarly the mutation operatorshould undergo a selection process Whether or not a habitatperforms mutation operator is determined by its mutationrate 119898

119894calculated by (2) (3) and (4) The selection process

of mutation operator is shown in Pseudocode 3 Note thatan elitism approach is employed to save the features of thehabitat that has the best solution in MBBO process Thehabitat with the best solution has a mutation rate of 0

Likewise mutation operator on operation sequence vec-tor and machine assignment vector should be designed Arandom arrangement on part of the vector is performedon operation sequence vector shown in Figure 7 and nofurther modification is needed With regard to the machineassignment vector random method is not proper becauseof the existence of partial flexible jobs A subsequencereplacement is applied to the machine assignment vector Asubsequence of the vector is selected randomly And thenfor each SIV in the subsequence the Rolette wheel processis performed to select one machine from a set of capablemachines The current machine is replaced by the selectedmachine The detailed description of this process is shown inFigure 8

46 Local Search for the Best Habitat Local search is usedto search the neighbor habitats of the habitat with the bestHSI By doing so the good characters of habitats with betterHSI are reserved and furthermore the convergence speed ofthe search process is greatly accelerated The local search isrealized as follows The habitat with the best HSI is selectedand performed insertion 119899 times at most The operatorinsert(120587 119897 119906 V) refers to insert 119897 variants after position 120583 toposition ] in the vector 120587 Parameter 119897 is randomly generatedamong [1 1198715] where 119871 is length of the vector If currentneighbor habitat has a better HSI than the original habitatterminate the iterations and replace the original habitat withthe neighbor one if not continue the iterations until thetermination criteria are satisfiedThe pseudocode of this localsearch is shown in Pseudocode 4


1 1 12 23 234

1 4 24 31 142

1 4 12 31 244

Emigration habitat

Immigration habitat

Habitat after migration

1 0 11 00 101

Figure 6 Uniform crossover operator on machine assignmentvector

1 1 12 33 222

2 1 12 33 221

Before mutation

Aftermutation

Figure 7 Process of the mutation operator on operation sequencevector

1 4 12 31 244

4 1 12 31 222

Before mutation

Aftermutation

Figure 8 Process of the mutation operator on machine assignmentvector

select119867119894sube Rank

119894= 1

loop = 1do

randomly select 120583 ] amp 120583 = ]po = randperm(ceil(1198715))1198671015840

119894= insert(119867

119894 po(1) 119906 V)

if1198671015840119894lt 119867119894

119867119894=1198671015840119894

breakendloop++

while loop ≦ 119899Return119867

119894

Pseudocode 4 The pseudocode of local search based on insertion

47 Framework of MBBO With the above design the entireprocedure ofMBBO to solve the FJSSP is depicted in Figure 9

5 Computation Results and Comparisons

Computational experiments are carried out to investigate theperformance of our proposed modified BBO algorithm Inorder to evaluate the performance of MBBO we run thealgorithm on a series of widely used benchmark problemsincluding four Kacem instances [14 30] and ten Brandimarteinstances [2]

All the programs in the experiments were written inMatlab and all the experiments were running on platformusing Intel Core 4 Quad 24GHZ CPUwith 2GB RAM Firstwe applied some experiments to examine parameter settingsof MBBO And then we compared MBBO with other well-known evolutionary algorithms for FJSSP Last but not leastseveral BBO variants were compared and discussed

51 Parameter Discussion The parameters have a great influ-ence on the performance of algorithms There are severalparameters that should be set properly in most evolutionaryalgorithms in order to make the algorithms more effectivesuch as population size crossover possibility and mutationpossibility in GA Things are different with BBO Note thatthe three main parameters in BBO are immigration rateemigration rate and mutation rate of each habitat which arecalculated by their rankingsThus there is noneed to set thesethree parametersThe remaining two parameters in BBO thatshould be designed are habitats size and termination ruleSince the trend in metaheuristic methods is to reduce thenumber of design parameters MBBO has the advantage ofscalability than other algorithms

Considering the different sizes of benchmark problemsa series of experiments were performed on each instanceto investigate how habitat sizes and termination rules affectMBBO The results on four Kacem instances are shown inFigures 10ndash13 and the sizes of habitats were set as 10 50 100and 200

It is obvious that the size of habitats has little impact onthe algorithm in Kacem instances In the first three instancesfour sizes of habitats reached the optimal solution within 50generations In accordance with the results of Kacem04 inFigure 13 larger size of habitats accelerated the convergencespeed of MBBO at the expense of a little computation timeincrementThrough overall consideration the size of habitatsand termination generation are set as 50 and 100 separatelyfor all four Kacem instances

Similarly the two parameters for ten BRdata instanceswere tested by experiments The results showed that theoptimal parameters for different instances were not the sameFigure 14 displayed the situation with MK01 under differentsizes of habitats The situation was similar to the one withKacem04 The settings for all the ten BRdata instances werelisted in Table 2

52 Results of Kacem Instances Kacem instances consistof four problems with the scale ranging from 4 jobs 5machines to 15 jobs 10 machines And our MBBO algorithmwas compared with several famous algorithms for FJSSPincluding LEGA [7] BBO [31] and BEDA [12]The results arelisted in Table 3 For each instance we run MBBO 50 timesindependently and record the best makespan 119862 and averagemakespan AV(119862) The results of other three algorithms aredirectly taken from relative literatures

From Table 3 it is obvious that MBBO solved the Kaceminstances with both effectiveness and stability In all fourinstances MBBO and BEDA reached the recorded optimalvalues in each of the 50 independent runs


Start Initial parameters and generate initial habitats

Calculate the

each habitat

For each habitatperform migration

operator

Choose emigration habitat and perform migration operator

Yes

No

Perform mutation operator

Perform mutationoperator

YesPerform local search to the

habitat with best HSI

No

Termination

No

End

Yes

HIS 120582 and 120583 of

Figure 9 The framework of the MBBO for FJSSP

10811

112114116118

12122124

0 5 10 15 20 25 30 35 40 45 50Generation

Mak

espa

n (K

acem

01)

5010 100

200

Figure 10 The performance of MBBO with different habitat sizeson Kacem01

0 5 10 15 20 25 30 35 40 45 5011

11512

12513

135

Generation

Mak

espa

n (K

acem

02)

5010 100

200

Figure 11The performance ofMBBOwith different habitat sizes onKacem02

0 5 10 15 20 25 30 35 40 45 5068

772747678

88284

Generation

Mak

espa

n (K

acem

03)

5010 100

200


0 10 20 30 40 50 60 70 80 90 10011

115

12

125

13

Generation

Mak

espa

n (K

acem

04)

5010 100

200


0 20 40 60 80 100 120 140 160 180 200404142434445464748

Generation

Mak

espa

n (B

Rdat

a01)

5010 100

200

Figure 14 The performance of MBBO with different habitat sizeson BRdata01

53 Results of BRdata Instances Brandimart data consists often problems with number of jobs ranging from 10 to 20number of machines ranging from 4 to 15 and number ofoperations for each job ranging from 5 to 15 Similarly foreach instance we run MBBO 50 times independently withdifferent seeds The results are presented in Table 4 wherethe results of other three algorithms are directly taken fromrelative literatures

The results in Table 4 showed that MBBO had a distinctadvantage in solving BRdata In MK01 MK02 MK03 MK04 MK08 and MK09 MBBO achieved the known optimalvalues and acquiredmakespanmuch close to the optimal one


Table 2 The parameters settings for BRdata instances

BRdata 119899 lowast 119898 NP 119879

MK01 10 lowast 6 100 100MK02 10 lowast 6 100 200MK03 15 lowast 8 50 100MK04 15 lowast 8 200 300MK05 15 lowast 4 100 200MK06 10 lowast 15 200 200MK07 20 lowast 5 200 200MK08 20 lowast 10 50 100MK09 20 lowast 10 150 100MK10 20 lowast 15 200 300

Table 3The results for several algorithms onKacemwithmakespanminimization

Instance LEGA BBO BEDA MBBO119862 AV(119862) 119862 AV(119862) 119862 AV(119862) 119862 AV(119862)

Kacem

4 lowast 5 11 11 11 11 11 11 11 1110 lowast 7 11 11 11 1134 11 11 11 1110 lowast 10 7 756 7 75 7 7 7 710 lowast 15 12 1204 12 1256 11 11 11 11

Table 4 The results for several algorithms on BRdata with makes-pan minimization


MK01 40 415 40 41 40 4102 40 40MK02 29 291 28 2825 26 2725 26 268MK03 mdash mdash 204 204 204 204 204 204MK04 67 6734 64 66 60 6369 60 62MK05 176 1781 173 1735 172 17338 173 1735MK06 67 6882 66 665 60 6283 61 631MK07 147 1529 144 14425 139 14155 140 14275MK08 523 52334 523 523 523 523 523 523MK09 320 32774 310 31075 307 31035 307 3072MK10 229 23572 230 23275 206 21192 221 22175

in the rest instancesMBBOperformed better than LEGAandBBO in all instances both effectively and stably Comparingwith LEGA best results of MBBO in four instances are alittle larger but the average of results are much better in sixout of ten instances which indicates that MBBO owns theadvantage of stability and is very robust

54 Comparisons between MBBO and Other BBO VariantsTo further verify the effectiveness of the machine-basedshifting and local search in MBBO we compared the MBBOwith several variants on four Kacem instances and ten BRdatainstances BBO stands for the algorithm proposed by Habibet al [32] And the DBBO refers to the BBO with machine-based shifting decoding strategy The habitat sizes for thesethree algorithms are set to 100 for Kacem instances and inaccordance with Table 2 for BRdata instances Figures 15 16

BBODBBOMBBO

0 20 40 60 80 100 120 140 160 180 2001012141618202224262830

Generation

Mak

espa

n (K

acem

02)

Figure 15 The comparison of convergence between BBO DBBOand MBBO on Kacem02

BBODBBOMBBO

0 50 100 150 200 250 30010152025303540

Generation

Mak

espa

n (K

acem

04)


6065707580859095

100105

0 50 100 150 200 250 300Generation

Mak

espa

n (M

K04)

BBODBBOMBBO

Figure 17 The comparison of convergence between BBO DBBOand MBBO on MK04

and 17 depict the convergence curves of the best makespan ofBBO DBBO and MBBO when solving Kacem02 Kacem04and MK04 separately

Obviously from the three figures it can be seen thatMBBO is of faster convergence speed than BBO and DBBOMoreover MBBO is more powerful to jump out of localoptimal and reach the global optimal value It is because thatMBBO applied a local search to the best habitat instead ofmutation operator which keeps the good characters of better


Kace

m01

Kace

m02

Kace

m03

Kace

m04

MK0

1

MK0

2

MK0

3

MK0

4

MK0

5

MK0

6

MK0

7

MK0

8

MK0

9

MK1

0

0100200300400500600

Instances

Com

puta

tion

time (

s)

BBOMBBO

Figure 18 The comparison of computation time between BBO andMBBO on two datasets

habitats as well as the novel machine-based decoding pro-cess The good performance of DBBO indicates effectivenessof machine-based shifting strategy

It is worth noticing that the faster convergence of MBBOis at expense of computation time Figure 18 shows thatunder the same generations the time MBBO spent is alittle more than BBO on both Kacem and BRdata instancesHowever as shown in Figure 16 MBBO reached the optimalmakespan after 30 generations while BBO still did not con-verge to the optimal value 11 after 200 generationsThereforethere is no doubt that MBBO has great superiority over BBOon convergence speed and quality of solutions in solvingFJSSP

6 Conclusions

In this paper a modified biogeography-based optimizationalgorithm with machine-based shifting decoding strategywas proposed for solving the flexible job shop schedulingproblemwithmakespanminimizationTheMBBOalgorithmapplied a double coding scheme with operation sequenceand machine assignment vector And a special machine-based shifting strategy was proposed to decode the vectorsto a scheduling solution Besides several discrete operatorsin BBO such as emigration immigration and mutationwere designed Moreover a special local search insteadof mutation was applied to the best habitat to speed upthe search process while maintaining the good charactersof better habitats The parameters were investigated andchosen by experiments Comparisons on Kacem and BRdatainstances with several existing famous algorithms indicatethe superiority of the proposed MBBO algorithm in terms ofeffectiveness and efficiency

More comprehensive studies can be applied to extendMBBO algorithm Other possible criteria in multiobjectiveoptimization will be considered Furthermore more localsearch methods will be analyzed to integrate to the MBBO

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

Theauthor appreciates the support of scientific research start-up fund provided by Shanghai Dianji University and Climb-ing Peak Discipline Project of Shanghai Dianji University(Project no 15DFXK01)

References

[1] P Brucker and R Schlie ldquoJob-shop scheduling with multi-purposemachinesrdquoComputing vol 45 no 4 pp 369ndash375 1990

[2] P Brandimarte ldquoRouting and scheduling in a flexible job shopby tabu searchrdquoAnnals of Operations Research vol 41 no 3 pp157ndash183 1993

[3] M Mastrolilli and L M Gambardella ldquoEffective neighbour-hood functions for the flexible job shop problemrdquo Journal ofScheduling vol 3 no 1 pp 3ndash20 2000

[4] F Pezzella GMorganti and G Ciaschetti ldquoA genetic algorithmfor the flexible job-shop scheduling problemrdquo Computers andOperations Research vol 35 no 10 pp 3202ndash3212 2008

[5] N B Ho and J C Tay ldquoEvolving dispatching rules for solvingthe flexible job-shop problemrdquo in Proceedings of the IEEECongress on Evolutionary Computation vol 3 pp 2848ndash2855Edinburgh UK September 2005

[6] N BHo and J C Tay ldquoGENACE an efficient cultural algorithmfor solving the flexible job-shop problemrdquo in Proceedings of theCongress on Evolutionary Computation (CEC rsquo04) vol 2 pp1759ndash1766 IEEE June 2004

[7] N B Ho and J C Tay ldquoLEGA an architecture for learningand evolving flexible job-shop schedulesrdquo in Proceedings of theIEEE Congress on Evolutionary Computation (CEC rsquo05) vol 2pp 1380ndash1387 September 2005

[8] M Yazdani M Amiri and M Zandieh ldquoFlexible job-shopscheduling with parallel variable neighborhood search algo-rithmrdquo Expert Systems with Applications vol 37 no 1 pp 678ndash687 2010

[9] D Lei ldquoCo-evolutionary genetic algorithm for fuzzy flexible jobshop schedulingrdquo Applied Soft Computing Journal vol 12 no 8pp 2237ndash2245 2012

[10] D Lei and X Guo ldquoSwarm-based neighbourhood searchalgorithm for fuzzy flexible job shop schedulingrdquo InternationalJournal of Production Research vol 50 no 6 pp 1639ndash16492012

[11] L Wang and C Fang ldquoAn effective estimation of distributionalgorithm for the multi-mode resource-constrained projectscheduling problemrdquo Computers and Operations Research vol39 no 2 pp 449ndash460 2012

[12] LWang S Wang Y Xu G Zhou andM Liu ldquoA bi-populationbased estimation of distribution algorithm for the flexible job-shop scheduling problemrdquoComputers amp Industrial Engineeringvol 62 no 4 pp 917ndash926 2012

[13] Y Yuan and H Xu ldquoFlexible job shop scheduling using hybriddifferential evolution algorithmsrdquoComputers amp Industrial Engi-neering vol 65 no 2 pp 246ndash260 2013

[14] I Kacem S Hammadi and P Borne ldquoApproach by localizationand multiobjective evolutionary optimization for flexible job-shop scheduling problemsrdquo IEEE Transactions on Systems Manand Cybernetics Part C Applications and Reviews vol 32 no 1pp 1ndash13 2002

[15] I Kacem S Hammadi and P Borne ldquoPareto-optimalityapproach based on uniform design and fuzzy evolutionary


algorithms for flexible job-shop scheduling problems (FJSPs)rdquoin Proceedings of the IEEE International Conference on SystemsMan and Cybernetics vol 7 pp 1ndash7 IEEE Hammamet TunisiaOctober 2002

[16] N B Ho J C Tay and E M-K Lai ldquoAn effective architecturefor learning and evolving flexible job-shop schedulesrdquoEuropeanJournal of Operational Research vol 179 no 2 pp 316ndash333 2007

[17] J Gao M Gen L Sun and X Zhao ldquoA hybrid of geneticalgorithm and bottleneck shifting for multiobjective flexible jobshop scheduling problemsrdquo Computers amp Industrial Engineer-ing vol 53 no 1 pp 149ndash162 2007

[18] J Gao L Sun and M Gen ldquoA hybrid genetic and variableneighborhood descent algorithm for flexible job shop schedul-ing problemsrdquo Computers amp Operations Research vol 35 no 9pp 2892ndash2907 2008

[19] G Zhang X Shao P Li and LGao ldquoAn effective hybrid particleswarm optimization algorithm for multi-objective flexible job-shop scheduling problemrdquoComputers amp Industrial Engineeringvol 56 no 4 pp 1309ndash1318 2009

[20] J-Q Li Q-K Pan and Y-C Liang ldquoAn effective hybridtabu search algorithm for multi-objective flexible job-shopscheduling problemsrdquo Computers amp Industrial Engineering vol59 no 4 pp 647ndash662 2010

[21] L Wang S Y Wang and M Liu ldquoA Pareto-based estimationof distribution algorithm for the multi-objective flexible job-shop scheduling problemrdquo International Journal of ProductionResearch vol 51 no 12 pp 3574ndash3592 2013

[22] D Simon ldquoBiogeography-based optimizationrdquo IEEE Transac-tions on Evolutionary Computation vol 12 no 6 pp 702ndash7132008

[23] H Kundra and M Sood ldquoCross-country path finding usinghybrid approach of PSO and BBOrdquo International Journal ofComputer Applications vol 7 no 6 pp 15ndash19 2010

[24] K Jamuna and K S Swarup ldquoMulti-objective biogeographybased optimization for optimal PMU placementrdquo Applied SoftComputing vol 12 no 5 pp 1503ndash1510 2012

[25] S Singh E Mittal and G Sachdeva ldquoMulti-objective gain-impedance optimization of Yagi-Uda antenna using NSBBOand NSPSOrdquo International Journal of Computer Applicationsvol 56 no 15 pp 1ndash6 2012

[26] R MacArthur and E Wilson The Theory of BiogeographyPrinceton University Press Princeton NJ USA 1967

[27] R Nakano and T Yamada ldquoConventional genetic algorithmfor job shop problemsrdquo in Proceedings of the 4th InternationalConference on Genetic Algorithms (ICGA rsquo91) pp 474ndash479 SanDiego Calif USA July 1991

[28] H P Ma X Li and S D Lin ldquoAnalysis of migratorate modelsfor biogeography-based optimizationrdquo Journal of SoutheastUniversity (Natural Science Edition) vol 39 supplement pp 16ndash21 2009 (Chinese)

[29] C Y Zhang Y Q Rao P G Li and X Shao ldquoBilevelgenetic algorithm for the flexible job-shop scheduling problemrdquoChinese Journal of Mechanical Engineering vol 43 no 4 pp119ndash124 2007 (Chinese)

[30] I Kacem S Hammadi and P Borne ldquoPareto-optimalityapproach based on uniformdesign and fuzzy evolutionary algo-rithms for flexible job-shop scheduling problems (FJSPs)rdquo inProceedings of the IEEE International Conference on SystemsMan and Cybernetics vol 7 pp 1ndash7 Yasmine HammametTunisia October 2002

[31] S H A Rahmati and M Zandieh ldquoA new biogeography-basedoptimization (BBO) algorithm for the flexible job shop schedul-ing problemrdquo International Journal of Advanced ManufacturingTechnology vol 58 no 9ndash12 pp 1115ndash1129 2012

[32] S Habib A Rahmati and M Zandieh ldquoA new biogeography-based optimization (BBO) algorithm for the flexible job shopscheduling problemrdquo International Journal of Advanced Manu-facturing Technology vol 58 no 9-12 pp 1115ndash1129 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


With regard to the multiobjective FJSSP weighted meth-ods and Pareto-based methods are commonly studiedThe objectives generally are makespan total work load ofmachines and maximum work load of machine Kacem etal [14 15] proposed several effective evolutionary algorithmsHo et al [16] used CDRs GA and machine learning mech-anism to optimize several objectives Gao [17 18] addedshifting bottleneck and variable neighborhood search toGA Zhang et al [19] developed a hybrid particle swarmoptimization (PSO) incorporated with a novel initializationLi et al [20] introduced a hybrid TS algorithm and discreteartificial bee colony (ABC) algorithm for the MJSSP Wang etal [21] further developed Pareto-based EDA (PEDA)

Biogeography-based optimization (BBO) a combinationof biogeography and engineering science was proposed bySimon in 2008 [22] Due to the similar features with otherbiology-based optimization algorithms such as GA and PSOBBO is applicable to many types of problems that GA andPSO are applied for However BBO also has some uniquefeatures from other types of biology-based optimizationTheinitial populations in BBO survive forever and their charac-teristics change gradually as the search process progressedAnd the main parameters of individuals are decided by theirranking in population which would cause no extra settingprocess

So far on count of all these features BBO has beenapplied to many academic and application problems [23ndash25] However there is little research about BBO to solvescheduling problem In this paper we propose a modifiedBBO (MBBO) to solve the FJSSP with makespan minimiza-tion When designing the algorithm we focus on the balancebetween exploration and exploitation Considering two-vector coding different strategies are introduced to generatethe initial solutions A machine-based shifting decodingmethod is proposed to convert the vectors to feasible scheduleat local exploitation phase Different migration and mutationoperators are designed at global exploration phase Particu-larly a local search is applied to the solution with the bestmakespan to speed up the convergence of the algorithmwhilemaintaining the good character of the best solution at localexploitation phase In addition experiments are performedto examine how the parameters affect the performance of thealgorithm Last but not least we compare MBBO with otherexisting famous algorithms and BBO variants on two sets ofbenchmark instances to test the efficiency and effectivenessof BBO in solving FJSSP

The reminder of the paper is organized as followsIn Section 2 we explain the flexible job shop schedulingproblem as well as the concept and structure of basic BBOalgorithm in Section 3 In Section 4 the modified BBO isdeveloped to FJSSP subsequently Section 5 analyzes theperformance results ofMBBOwhen applied to solve commonbenchmark problems in literature At last we come to ourconclusion and some possible future directions

2 Problem Formulation

FJSSP is one of the most practical and hardest combinatorialoptimization problems During past two decades a bunch of

Table 1 Example of FJSSPwith 3 jobs 4machines and 9 operations

FJSSP Processing Time1198721

1198722

1198723

1198724

1198691

O11

3 4 infin 5O12

4 6 5 infin

O13

3 infin infin 6

1198692

O21

4 infin infin 5O22

infin 5 infin 3O23

5 3 5 2O24

3 6 infin infin

1198693

O31

5 3 4 3O32

4 2 1 infin

literature has been published but no efficient algorithm hasbeen presented yet for solving it to optimality in polynomialtime

Suppose we are given 119899 jobs and 119898 machines Eachmachine can handle one job at most at a time Each job 119894consists of a sequence of operations andneeds to be processedduring an uninterrupted time period on one from a set offeasible machines The purpose is to find a scheme that isthe job sequence on the machines as to optimize one ormore performance measurements makespan in our case Anexample of 3 jobs 4 machines and 9 operations is shown inTable 1 whereinfin indicates that the machine can process theoperation however with infinity processing time And theFJSSP could be divided into two types total FJSSP and partialFJSSP based on whether each operation can be processed onall of the machines

The mathematical model of FJSSP with makespan mini-mization can be formulated as [20]

Minimise 119862119872= max1le119894le119899

119862119894119902119894

Subject to 119862119894119895minus 119862119894119895minus1

ge 119901119894119895119896

119909119894119895119896

119895 = 2 3 119902119894 forall119894 119896

sum

119896isin119872119894119895

119909119894119895119896

= 1 forall119894 119895

119909119894119895119896

=

1 if operation 119874119894119895is processed on machine 119896

0 otherwise

119862119894119895ge 0 forall119894 119895

(1)

The indices and variables of the model are enumerated asfollows 119898 is number of machines 119899 is number of jobs 119894 isjob index 119895 is operation number index 119896 is machine index119902119894is number of operations of job 119894 119862

119894119895is finish time of 119895th

operation of job 119894 119901119894119895119896

is processing time of 119895th operation ofjob 119894 on machine 119896 119872

119894119895is set of machines that can process

119895th operation of job 119894

3 Biogeography-Based Optimization

BBO algorithm proposed by Simon in 2008 is inspiredby the mathematics of biogeography and mainly the work


119868 rarr 119867119899HSI119899

While not 119879Ψ = (119898 119899 120582 120583 Ω119872)

end











2 1 3 1 2 2 2 1 3

2 3 1 1 4 3 2 2 1

Job 1 Job 2 Job 3

Operation sequence

Machine assignment

O21 O11 O31 O12 O22 O23 O24 O13 O32




119875119904(119905 + Δ119905) = 119875

119904(119905) (1 minus 120582

119904Δ119905 minus 120583

119904Δ119905) + 119875

119904minus1120582119904minus1Δ119905

+ 119875119904+1120583119904+1Δ119905

(2)


1198751015840

119904

=

minus (120582119904+ 120583119904) 119875119904+ 120583119904+1119875119904+1 119904 = 1


+ 120583119904+1119875119904+1 1 lt 119904 le 119899 minus 1


(3)


119904

119898119904= 119898max (1 minus

119875119904

119875max) (4)


4 MBBO for FJSSP



1 1199092 1199093 119909







11987411 11987412 and 119874


21 11987422 11987423 and


31and 119874






3 Therefore


1198723) (119874241198722) (119874131198721) (119874321198721)]









1and 120587

2refer to the



processing time









2

2

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

3

1

1

Machine 3

Machine 2

Machine 1 3

Machine 4

2

2

1

20191817


2 2

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

3 1

1

Machine 3

Machine 2

Machine 1

3Machine 4

22 1

20191817


(a) If 1198711205872(119894119895)

= 1amp119895 = 1 set St(119874119894119895) = 0 and


(b) If 1198711205872(119894119895)

= 1amp119895 = 1 set St(119874119894119895) = 119862

119894119895minus1and


(c) If 1198711205872(119894119895)

= 1amp119895 = 1 set St(119874119894119895) = min(119862

119894119895) minus

119901119894119895119896


1205872(119894119895)= 1amp119895 = 1 set St(119874

119894119895) = min(119862

119894119895) minus

119901119894119895119896

while St(119874119894119895) gt 119862


machine 119896






For 119894 = 1 NPif Rand(0 1) lt 120582

119894


Migration (119867119894 119867119895)

endend


Rate

120582S-immigration

120583S-emigration

Smax

S1 S2




119894should be







120582119904= (

119868

2) lowast (cos(119904120587

119899) + 1) (5)

120583119904= (

119864


119899) + 1) (6)



3 1 32 11 222

2 1 13 22 231

1 1 12 33 222





For 119894 = 1 NPusing119898


119894amp Rank

119894= 1

Mutation (119867119894)

end








1 1 12 23 234

1 4 24 31 142

1 4 12 31 244

Emigration habitat

Immigration habitat


1 0 11 00 101


1 1 12 33 222

2 1 12 33 221

Before mutation

Aftermutation


1 4 12 31 244

4 1 12 31 222

Before mutation

Aftermutation



119894= 1

loop = 1do


119894= insert(119867

119894 po(1) 119906 V)

if1198671015840119894lt 119867119894

119867119894=1198671015840119894

breakendloop++


119894














Calculate the

each habitat


operator


Yes

No





No

Termination

No

End

Yes

HIS 120582 and 120583 of


10811

112114116118

12122124

0 5 10 15 20 25 30 35 40 45 50Generation

Mak

espa

n (K

acem

01)

5010 100

200


0 5 10 15 20 25 30 35 40 45 5011

11512

12513

135

Generation

Mak

espa

n (K

acem

02)

5010 100

200


0 5 10 15 20 25 30 35 40 45 5068

772747678

88284

Generation

Mak

espa

n (K

acem

03)

5010 100

200


0 10 20 30 40 50 60 70 80 90 10011

115

12

125

13

Generation

Mak

espa

n (K

acem

04)

5010 100

200


0 20 40 60 80 100 120 140 160 180 200404142434445464748

Generation

Mak

espa

n (B

Rdat

a01)

5010 100

200






BRdata 119899 lowast 119898 NP 119879




Kacem







BBODBBOMBBO

0 20 40 60 80 100 120 140 160 180 2001012141618202224262830

Generation

Mak

espa

n (K

acem

02)


BBODBBOMBBO

0 50 100 150 200 250 30010152025303540

Generation

Mak

espa

n (K

acem

04)


6065707580859095

100105

0 50 100 150 200 250 300Generation

Mak

espa

n (M

K04)

BBODBBOMBBO





Kace

m01

Kace

m02

Kace

m03

Kace

m04

MK0

1

MK0

2

MK0

3

MK0

4

MK0

5

MK0

6

MK0

7

MK0

8

MK0

9

MK1

0

0100200300400500600

Instances

Com

puta

tion

time (

s)

BBOMBBO




6 Conclusions





Acknowledgments


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










119868 rarr 119867119899HSI119899

While not 119879Ψ = (119898 119899 120582 120583 Ω119872)

end











2 1 3 1 2 2 2 1 3

2 3 1 1 4 3 2 2 1

Job 1 Job 2 Job 3

Operation sequence

Machine assignment

O21 O11 O31 O12 O22 O23 O24 O13 O32




119875119904(119905 + Δ119905) = 119875

119904(119905) (1 minus 120582

119904Δ119905 minus 120583

119904Δ119905) + 119875

119904minus1120582119904minus1Δ119905

+ 119875119904+1120583119904+1Δ119905

(2)


1198751015840

119904

=

minus (120582119904+ 120583119904) 119875119904+ 120583119904+1119875119904+1 119904 = 1


+ 120583119904+1119875119904+1 1 lt 119904 le 119899 minus 1


(3)


119904

119898119904= 119898max (1 minus

119875119904

119875max) (4)


4 MBBO for FJSSP



1 1199092 1199093 119909







11987411 11987412 and 119874


21 11987422 11987423 and


31and 119874






3 Therefore


1198723) (119874241198722) (119874131198721) (119874321198721)]









1and 120587

2refer to the



processing time









2

2

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

3

1

1

Machine 3

Machine 2

Machine 1 3

Machine 4

2

2

1

20191817


2 2

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

3 1

1

Machine 3

Machine 2

Machine 1

3Machine 4

22 1

20191817


(a) If 1198711205872(119894119895)

= 1amp119895 = 1 set St(119874119894119895) = 0 and


(b) If 1198711205872(119894119895)

= 1amp119895 = 1 set St(119874119894119895) = 119862

119894119895minus1and


(c) If 1198711205872(119894119895)

= 1amp119895 = 1 set St(119874119894119895) = min(119862

119894119895) minus

119901119894119895119896


1205872(119894119895)= 1amp119895 = 1 set St(119874

119894119895) = min(119862

119894119895) minus

119901119894119895119896

while St(119874119894119895) gt 119862


machine 119896






For 119894 = 1 NPif Rand(0 1) lt 120582

119894


Migration (119867119894 119867119895)

endend


Rate

120582S-immigration

120583S-emigration

Smax

S1 S2




119894should be







120582119904= (

119868

2) lowast (cos(119904120587

119899) + 1) (5)

120583119904= (

119864


119899) + 1) (6)



3 1 32 11 222

2 1 13 22 231

1 1 12 33 222





For 119894 = 1 NPusing119898


119894amp Rank

119894= 1

Mutation (119867119894)

end








1 1 12 23 234

1 4 24 31 142

1 4 12 31 244

Emigration habitat

Immigration habitat


1 0 11 00 101


1 1 12 33 222

2 1 12 33 221

Before mutation

Aftermutation


1 4 12 31 244

4 1 12 31 222

Before mutation

Aftermutation



119894= 1

loop = 1do


119894= insert(119867

119894 po(1) 119906 V)

if1198671015840119894lt 119867119894

119867119894=1198671015840119894

breakendloop++


119894














Calculate the

each habitat


operator


Yes

No





No

Termination

No

End

Yes

HIS 120582 and 120583 of


10811

112114116118

12122124

0 5 10 15 20 25 30 35 40 45 50Generation

Mak

espa

n (K

acem

01)

5010 100

200


0 5 10 15 20 25 30 35 40 45 5011

11512

12513

135

Generation

Mak

espa

n (K

acem

02)

5010 100

200


0 5 10 15 20 25 30 35 40 45 5068

772747678

88284

Generation

Mak

espa

n (K

acem

03)

5010 100

200


0 10 20 30 40 50 60 70 80 90 10011

115

12

125

13

Generation

Mak

espa

n (K

acem

04)

5010 100

200


0 20 40 60 80 100 120 140 160 180 200404142434445464748

Generation

Mak

espa

n (B

Rdat

a01)

5010 100

200






BRdata 119899 lowast 119898 NP 119879




Kacem







BBODBBOMBBO

0 20 40 60 80 100 120 140 160 180 2001012141618202224262830

Generation

Mak

espa

n (K

acem

02)


BBODBBOMBBO

0 50 100 150 200 250 30010152025303540

Generation

Mak

espa

n (K

acem

04)


6065707580859095

100105

0 50 100 150 200 250 300Generation

Mak

espa

n (M

K04)

BBODBBOMBBO





Kace

m01

Kace

m02

Kace

m03

Kace

m04

MK0

1

MK0

2

MK0

3

MK0

4

MK0

5

MK0

6

MK0

7

MK0

8

MK0

9

MK1

0

0100200300400500600

Instances

Com

puta

tion

time (

s)

BBOMBBO




6 Conclusions





Acknowledgments


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












11987411 11987412 and 119874


21 11987422 11987423 and


31and 119874






3 Therefore


1198723) (119874241198722) (119874131198721) (119874321198721)]









1and 120587

2refer to the



processing time









2

2

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

3

1

1

Machine 3

Machine 2

Machine 1 3

Machine 4

2

2

1

20191817


2 2

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

3 1

1

Machine 3

Machine 2

Machine 1

3Machine 4

22 1

20191817


(a) If 1198711205872(119894119895)

= 1amp119895 = 1 set St(119874119894119895) = 0 and


(b) If 1198711205872(119894119895)

= 1amp119895 = 1 set St(119874119894119895) = 119862

119894119895minus1and


(c) If 1198711205872(119894119895)

= 1amp119895 = 1 set St(119874119894119895) = min(119862

119894119895) minus

119901119894119895119896


1205872(119894119895)= 1amp119895 = 1 set St(119874

119894119895) = min(119862

119894119895) minus

119901119894119895119896

while St(119874119894119895) gt 119862


machine 119896






For 119894 = 1 NPif Rand(0 1) lt 120582

119894


Migration (119867119894 119867119895)

endend


Rate

120582S-immigration

120583S-emigration

Smax

S1 S2




119894should be







120582119904= (

119868

2) lowast (cos(119904120587

119899) + 1) (5)

120583119904= (

119864


119899) + 1) (6)



3 1 32 11 222

2 1 13 22 231

1 1 12 33 222





For 119894 = 1 NPusing119898


119894amp Rank

119894= 1

Mutation (119867119894)

end








1 1 12 23 234

1 4 24 31 142

1 4 12 31 244

Emigration habitat

Immigration habitat


1 0 11 00 101


1 1 12 33 222

2 1 12 33 221

Before mutation

Aftermutation


1 4 12 31 244

4 1 12 31 222

Before mutation

Aftermutation



119894= 1

loop = 1do


119894= insert(119867

119894 po(1) 119906 V)

if1198671015840119894lt 119867119894

119867119894=1198671015840119894

breakendloop++


119894














Calculate the

each habitat


operator


Yes

No





No

Termination

No

End

Yes

HIS 120582 and 120583 of


10811

112114116118

12122124

0 5 10 15 20 25 30 35 40 45 50Generation

Mak

espa

n (K

acem

01)

5010 100

200


0 5 10 15 20 25 30 35 40 45 5011

11512

12513

135

Generation

Mak

espa

n (K

acem

02)

5010 100

200


0 5 10 15 20 25 30 35 40 45 5068

772747678

88284

Generation

Mak

espa

n (K

acem

03)

5010 100

200


0 10 20 30 40 50 60 70 80 90 10011

115

12

125

13

Generation

Mak

espa

n (K

acem

04)

5010 100

200


0 20 40 60 80 100 120 140 160 180 200404142434445464748

Generation

Mak

espa

n (B

Rdat

a01)

5010 100

200






BRdata 119899 lowast 119898 NP 119879




Kacem







BBODBBOMBBO

0 20 40 60 80 100 120 140 160 180 2001012141618202224262830

Generation

Mak

espa

n (K

acem

02)


BBODBBOMBBO

0 50 100 150 200 250 30010152025303540

Generation

Mak

espa

n (K

acem

04)


6065707580859095

100105

0 50 100 150 200 250 300Generation

Mak

espa

n (M

K04)

BBODBBOMBBO





Kace

m01

Kace

m02

Kace

m03

Kace

m04

MK0

1

MK0

2

MK0

3

MK0

4

MK0

5

MK0

6

MK0

7

MK0

8

MK0

9

MK1

0

0100200300400500600

Instances

Com

puta

tion

time (

s)

BBOMBBO




6 Conclusions





Acknowledgments


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










For 119894 = 1 NPif Rand(0 1) lt 120582

119894


Migration (119867119894 119867119895)

endend


Rate

120582S-immigration

120583S-emigration

Smax

S1 S2




119894should be







120582119904= (

119868

2) lowast (cos(119904120587

119899) + 1) (5)

120583119904= (

119864


119899) + 1) (6)



3 1 32 11 222

2 1 13 22 231

1 1 12 33 222





For 119894 = 1 NPusing119898


119894amp Rank

119894= 1

Mutation (119867119894)

end








1 1 12 23 234

1 4 24 31 142

1 4 12 31 244

Emigration habitat

Immigration habitat


1 0 11 00 101


1 1 12 33 222

2 1 12 33 221

Before mutation

Aftermutation


1 4 12 31 244

4 1 12 31 222

Before mutation

Aftermutation



119894= 1

loop = 1do


119894= insert(119867

119894 po(1) 119906 V)

if1198671015840119894lt 119867119894

119867119894=1198671015840119894

breakendloop++


119894














Calculate the

each habitat


operator


Yes

No





No

Termination

No

End

Yes

HIS 120582 and 120583 of


10811

112114116118

12122124

0 5 10 15 20 25 30 35 40 45 50Generation

Mak

espa

n (K

acem

01)

5010 100

200


0 5 10 15 20 25 30 35 40 45 5011

11512

12513

135

Generation

Mak

espa

n (K

acem

02)

5010 100

200


0 5 10 15 20 25 30 35 40 45 5068

772747678

88284

Generation

Mak

espa

n (K

acem

03)

5010 100

200


0 10 20 30 40 50 60 70 80 90 10011

115

12

125

13

Generation

Mak

espa

n (K

acem

04)

5010 100

200


0 20 40 60 80 100 120 140 160 180 200404142434445464748

Generation

Mak

espa

n (B

Rdat

a01)

5010 100

200






BRdata 119899 lowast 119898 NP 119879




Kacem







BBODBBOMBBO

0 20 40 60 80 100 120 140 160 180 2001012141618202224262830

Generation

Mak

espa

n (K

acem

02)


BBODBBOMBBO

0 50 100 150 200 250 30010152025303540

Generation

Mak

espa

n (K

acem

04)


6065707580859095

100105

0 50 100 150 200 250 300Generation

Mak

espa

n (M

K04)

BBODBBOMBBO





Kace

m01

Kace

m02

Kace

m03

Kace

m04

MK0

1

MK0

2

MK0

3

MK0

4

MK0

5

MK0

6

MK0

7

MK0

8

MK0

9

MK1

0

0100200300400500600

Instances

Com

puta

tion

time (

s)

BBOMBBO




6 Conclusions





Acknowledgments


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1 1 12 23 234

1 4 24 31 142

1 4 12 31 244

Emigration habitat

Immigration habitat


1 0 11 00 101


1 1 12 33 222

2 1 12 33 221

Before mutation

Aftermutation


1 4 12 31 244

4 1 12 31 222

Before mutation

Aftermutation



119894= 1

loop = 1do


119894= insert(119867

119894 po(1) 119906 V)

if1198671015840119894lt 119867119894

119867119894=1198671015840119894

breakendloop++


119894














Calculate the

each habitat


operator


Yes

No





No

Termination

No

End

Yes

HIS 120582 and 120583 of


10811

112114116118

12122124

0 5 10 15 20 25 30 35 40 45 50Generation

Mak

espa

n (K

acem

01)

5010 100

200


0 5 10 15 20 25 30 35 40 45 5011

11512

12513

135

Generation

Mak

espa

n (K

acem

02)

5010 100

200


0 5 10 15 20 25 30 35 40 45 5068

772747678

88284

Generation

Mak

espa

n (K

acem

03)

5010 100

200


0 10 20 30 40 50 60 70 80 90 10011

115

12

125

13

Generation

Mak

espa

n (K

acem

04)

5010 100

200


0 20 40 60 80 100 120 140 160 180 200404142434445464748

Generation

Mak

espa

n (B

Rdat

a01)

5010 100

200






BRdata 119899 lowast 119898 NP 119879




Kacem







BBODBBOMBBO

0 20 40 60 80 100 120 140 160 180 2001012141618202224262830

Generation

Mak

espa

n (K

acem

02)


BBODBBOMBBO

0 50 100 150 200 250 30010152025303540

Generation

Mak

espa

n (K

acem

04)


6065707580859095

100105

0 50 100 150 200 250 300Generation

Mak

espa

n (M

K04)

BBODBBOMBBO





Kace

m01

Kace

m02

Kace

m03

Kace

m04

MK0

1

MK0

2

MK0

3

MK0

4

MK0

5

MK0

6

MK0

7

MK0

8

MK0

9

MK1

0

0100200300400500600

Instances

Com

puta

tion

time (

s)

BBOMBBO




6 Conclusions





Acknowledgments


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Calculate the

each habitat


operator


Yes

No





No

Termination

No

End

Yes

HIS 120582 and 120583 of


10811

112114116118

12122124

0 5 10 15 20 25 30 35 40 45 50Generation

Mak

espa

n (K

acem

01)

5010 100

200


0 5 10 15 20 25 30 35 40 45 5011

11512

12513

135

Generation

Mak

espa

n (K

acem

02)

5010 100

200


0 5 10 15 20 25 30 35 40 45 5068

772747678

88284

Generation

Mak

espa

n (K

acem

03)

5010 100

200


0 10 20 30 40 50 60 70 80 90 10011

115

12

125

13

Generation

Mak

espa

n (K

acem

04)

5010 100

200


0 20 40 60 80 100 120 140 160 180 200404142434445464748

Generation

Mak

espa

n (B

Rdat

a01)

5010 100

200






BRdata 119899 lowast 119898 NP 119879




Kacem







BBODBBOMBBO

0 20 40 60 80 100 120 140 160 180 2001012141618202224262830

Generation

Mak

espa

n (K

acem

02)


BBODBBOMBBO

0 50 100 150 200 250 30010152025303540

Generation

Mak

espa

n (K

acem

04)


6065707580859095

100105

0 50 100 150 200 250 300Generation

Mak

espa

n (M

K04)

BBODBBOMBBO





Kace

m01

Kace

m02

Kace

m03

Kace

m04

MK0

1

MK0

2

MK0

3

MK0

4

MK0

5

MK0

6

MK0

7

MK0

8

MK0

9

MK1

0

0100200300400500600

Instances

Com

puta

tion

time (

s)

BBOMBBO




6 Conclusions





Acknowledgments


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











BRdata 119899 lowast 119898 NP 119879




Kacem







BBODBBOMBBO

0 20 40 60 80 100 120 140 160 180 2001012141618202224262830

Generation

Mak

espa

n (K

acem

02)


BBODBBOMBBO

0 50 100 150 200 250 30010152025303540

Generation

Mak

espa

n (K

acem

04)


6065707580859095

100105

0 50 100 150 200 250 300Generation

Mak

espa

n (M

K04)

BBODBBOMBBO





Kace

m01

Kace

m02

Kace

m03

Kace

m04

MK0

1

MK0

2

MK0

3

MK0

4

MK0

5

MK0

6

MK0

7

MK0

8

MK0

9

MK1

0

0100200300400500600

Instances

Com

puta

tion

time (

s)

BBOMBBO




6 Conclusions





Acknowledgments


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Kace

m01

Kace

m02

Kace

m03

Kace

m04

MK0

1

MK0

2

MK0

3

MK0

4

MK0

5

MK0

6

MK0

7

MK0

8

MK0

9

MK1

0

0100200300400500600

Instances

Com

puta

tion

time (

s)

BBOMBBO




6 Conclusions





Acknowledgments


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

Research Article A Modified Biogeography-Based Optimization for the Flexible Job Shop ...downloads.hindawi.com/journals/mpe/2015/184643.pdf · 2019-07-31 · Research Article A Modified