a bee evolutionary guiding nondominated sorting genetic algorithm...

Research ArticleA Bee Evolutionary Guiding Nondominated Sorting GeneticAlgorithm II for Multiobjective Flexible Job-Shop Scheduling

Qianwang Deng Guiliang Gong Xuran Gong Like Zhang Wei Liu and Qinghua Ren

State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body Hunan University Changsha 410082 China

Correspondence should be addressed to Guiliang Gong gong guiliang163com

Received 11 November 2016 Accepted 20 February 2017 Published 28 March 2017

Academic Editor J Alfredo Hernandez-Perez

Copyright copy 2017 Qianwang Deng et alThis is an open access article distributed under theCreativeCommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Flexible job-shop scheduling problem (FJSP) is an NP-hard puzzle which inherits the job-shop scheduling problem (JSP)characteristics This paper presents a bee evolutionary guiding nondominated sorting genetic algorithm II (BEG-NSGA-II) formultiobjective FJSP (MO-FJSP) with the objectives to minimize the maximal completion time the workload of the most loadedmachine and the total workload of all machines It adopts a two-stage optimization mechanism during the optimizing processIn the first stage the NSGA-II algorithm with 119879 iteration times is first used to obtain the initial population 119873 in which a beeevolutionary guiding scheme is presented to exploit the solution space extensively In the second stage the NSGA-II algorithmwith GEN iteration times is used again to obtain the Pareto-optimal solutions In order to enhance the searching ability and avoidthe premature convergence an updating mechanism is employed in this stage More specifically its population consists of threeparts and each of them changes with the iteration times What is more numerical simulations are carried out which are based onsome published benchmark instances Finally the effectiveness of the proposed BEG-NSGA-II algorithm is shown by comparingthe experimental results and the results of some well-known algorithms already existed

1 Introduction

As a part of production scheduling and combinatorial optimi-zation problems job-shop scheduling problem (JSP) attractsmore and more researchers from all walks of life (egmechanical engineering mathematics and computer soft-ware engineering) in the recent decades [1ndash5]

Flexible job-shop scheduling problem (FJSP) inherits thecharacteristics of the JSP in which each operation is allowedto be processed by any machine in a given set rather than onespecified machine and it has been proved that the FJSP isstrong NP-hard [6] FJSP consists of two subproblems one isthe routing subproblem that each operation is assigned to onemachine of a set of machines and the other is the schedulingsubproblem that a feasible schedule is obtained by sequencingthe assigned operations on all machines Therefore the FJSPis more difficult to be solved than the classical JSP because ofits need to determine the assignment of operations in relatedmachines [7]

The FJSP is firstly addressed by Brucker and Schlie [8]And they presented a polynomial algorithm with only twojobs and identical machines Brandimarte [9] proposed ahybrid tabu search (TS) algorithm which was based ondecomposition to solve the FJSP Dauzere-Peres and Paulli[10] provided a TS algorithm which was based on the inte-grated approach and developed a new neighbourhood func-tion [11] for the FJSP in terms of solution quality and com-putation time Gao et al [12] proposed a hybrid genetic algo-rithm to solve the FJSP with nonfixed availability constraintsAnd in order to enhance the inheritability this genetic algo-rithm uses an innovative representation method and appliesgenetic operations to phenotype space Saidi-Mehrabad andFattahi [13] developed a TS algorithm that took the operationsequences and sequence-dependent setups into considerationto solve the FJSP A genetic algorithm (GA) combined with avariable neighbourhood search (VNS) was presented by Gaoet al [14] and a GA with different strategies was proposed byPezzella et al [15] Yazdani et al [16] developed a parallel VNS

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2 Computational Intelligence and Neuroscience

algorithm based on the application of multiple independentsearches which increased the exploration of search spaceXing et al [17] put forth a knowledge-based ant colonyoptimization algorithm Recently a novel artificial bee colony(ABC) algorithm [18] and a discrete harmony search (DHS)algorithm [19] were brought forward to solve the FJSP

As is shown above the single-objective optimization ofFJSP (SO-FJSP) has been extensively studied which generallyminimizes themakespan that is the time required to completeall jobs However many industries (eg aircraft semicon-ductors manufacturing and electronics) have trade-offs intheir scheduling problems in which multiple objectives needto be considered to optimize the overall performance of thesystemTherefore theMO-FJSPmay be closer to the realisticproduction environments and needs to be further studiedIn recent years the MO-FJSP has captured more and moreinterests of numerous domain researchers and a great manyof algorithms have been presented [20] Compared with SO-FJSP MO-FJSP has two problems to be dealt with incom-mensurability between objectives and contradiction betweenobjectives (ie optimizing a single-objective generally resultsin deterioration of another objective) [13]

For the MO-FJSP Schaffer [21] provided a genetic algo-rithm with vector evaluated Jurisch [22] presented a branch-and-bound algorithm and some heuristic algorithms Bycombining the VNSwith particle swarm optimization (PSO)Liu et al [23] presented a hybrid metaheuristic to solve theMO-FJSP A new genetic algorithm (GA) which hybridizedwith a bottleneck shifting procedure was developed by Gao etal [24] to solve theMO-FJSP Zhang et al [25] embedded tabusearch (TS) in PSO as a local search to deal with theMO-FJSPXing et al [26] advanced an efficient search method for theMO-FJSP Gonzalez-Rodrıguez et al [27] proposed a genericmultiobjective model which was based on lexicographicalminimization of expected values for FJSP By introducing sev-eral metrics of the multiobjective evaluation in the MO-FJSPliterature Rahmati et al [28] adopted two multiobjectiveevolutionary algorithms for the MO-FJSP

Recently some studies based on the Pareto dominancerelation have been used to solve the MO-FJSP and they aremore desirablewhen comparedwith the prior linearweightedsummation ones [29] The nondominated sorting geneticalgorithm (NSGA) [30] was one of the first methods used tosolve themultiobjective problemKacemet al [31] brought upa Pareto approach based on the hybridization of fuzzy logic(FL) and evolutionary algorithms to solve the MO-FJSP Hoand Tay [32] integrated a guiding local search procedure andan elitism memory mechanism into the evolutionary algori-thm to solve the MO-FJSP

Deb et al [33] came up with a nondominated sorting-basedmultiobjective evolutionary algorithm (MOEA) callednondominated sorting genetic algorithm II (NSGA-II)Wanget al [34] proposed a multiobjective GA based on immuneand entropy principle for the MO-FJSP By employing simu-lated annealing (SA) algorithm as a local search process Fru-tos et al [35] introduced amemetic algorithm (MA) based onthe NSGA-II Wang et al [36] presented an effective Pareto-based estimation of distribution algorithm (P-EDA) inwhichvarious strategies are integrated to maintain quality and

diversity of solutions Rohaninejad et al [37] advanced anMO-FJSP problem with machines capacity constraints tominimize the makespan and overtime costs of machinesKaplanoglu [38] put forth an object-oriented (OO) approachcombined with SA optimization algorithm to solve the MO-FJSP in which a two-string encoding scheme was used toexpress this problem

Our reviewof the above literatures reveals that theNSGA-II algorithm has been widely used to solve the MO-FJSP forits advantages such as high efficiency to optimize the complexproblems and the ability to gain widespread Pareto-optimalsolutions And the algorithms with a two-stage optimizationscheme have been also widely studied to solve the MO-FJSP for it could fully tap the optimization potentials ofvarious metaheuristic algorithms However we found thatthe NSGA-II algorithm has the disadvantage of prematureconvergence to local solution and the algorithms with a two-stage optimization scheme have the disadvantages of beingunable to gain stable and high quality initial population in thefirst stageHence in this paper we propose a bee evolutionaryguidingNSGA-II (BEG-NSGA-II)with a two-stage optimiza-tion scheme to solve the MO-FJSP which aims to fully playthe respective advantages of NSGA-II algorithm and thealgorithms with a two-stage optimization scheme and toovercome the disadvantages of them In the first stage theNSGA-II algorithm with 119879 iteration times is first used toobtain the initial population119873 which consists of three partschanging with the iteration times In order to extensivelyexploit the solution space an effective local search operator isinvented in this stage In the second stage the NSGA-II algo-rithm with GEN iteration times is used to obtain the Pareto-optimal solutions In order to enhance the searching abilityand avoid the premature convergence an updating mecha-nism and some useful genetic operators were employed inthis stage Four famous benchmarks that include 53 openproblems of FJSP are chosen to estimate the performance ofthe proposed algorithm Moreover by comparing the resultsof our algorithm and some existing well-known algorithmsthe virtues of our algorithm can be clearly demonstrated

The rest of this paper is organized as follows The defini-tion and formalization of the MO-FJSP are given in thenext section In Section 3 NSGA-II is briefly introduced andthen the overview and implementation details of the pro-posed BEG-NSGA-II are presented respectively AfterwardsExperimental Studies and Discussions are made in Section 4Finally Conclusions and Future Studies are described inSection 5

2 Problem Definition

The FJSP is commonly defined as follows There is a set of 119899jobs (119869119894 119894 isin 1 2 119899) and a set of 119898 machines (119872119896 119896 isin1 2 119898) One or more operation(s) (119874119894119895 119895 isin 1 2 119899119894 119899119894 is the total number of operations for job 119869119894) isareallowed to be processed by one machine of 119872119894119895 whichconsists of a set of machines of the 119895th operations for job 119869119894119875119894119895119896 denotes the processing time of the 119895th operation for job 119869119894which is processed bymachine 119896 Generally the FJSP consistsof two subproblems the routing subproblem of assigning

Computational Intelligence and Neuroscience 3

Table 1 The 4 times 5 problem

1198721 1198722 1198723 1198724 1198725

Job 111987411 2 5 4 1 211987412 5 4 5 7 511987413 4 5 5 4 5

Job 211987421 2 5 4 7 811987422 5 6 9 8 511987423 4 5 4 54 5

Job 3

11987431 9 8 6 7 911987432 6 1 2 5 411987433 2 5 4 2 411987434 4 5 2 1 5

Job 4 11987441 1 5 2 4 1211987442 5 1 2 1 2

each operation to a machine among a set of machinesavailable and the scheduling subproblem of sequencing theassigned operations on all machines to obtain a feasibleschedule for optimizing a certain objective function [39 40]

One classical 4 times 5 FJSP is shown in Table 1 In this paperwe aim to minimize the following three objectives

(i) Maximal completion time of machines (makespan)

(ii) Workload of the most loaded machine (MW)

(iii) Total workload of all machines (TW)

Some assumptions are put forward

(i) Each operation cannot be interrupted during process-ing

(ii) Each machine can process at most one operation atany time

(iii) One operation cannot be processed by more than onemachine simultaneously

(iv) Moving time between operations and setting up timeof machines are negligible

(v) Machines are independent of each other

(vi) Jobs are independent of each other

The notations used in the definition of multiobjectiveFJSP in this paper are shown in Notations [25]

The mathematical model could be given as follows [25]Objective functions

min 1198911 = max1le119896le119898

(119862119896)

min 1198912 = max1le119896le119898

(119882119896)

min 1198913 =119898

sum119896=1

119882119896

(1)

Subject to 119862119894119895 minus 119862119894(119895minus1) ge 119875119894119895119896119909119894119895119896 119895 = 2 119899119894 forall119894 119895 (2)

[(119862ℎ119892 minus 119862119894119895 minus 119905ℎ119892119896) 119909ℎ119892119896119909119894119895119896 ge 0]

or [(119862119894119895 minus 119862ℎ119892 minus 119905119894119895119896) 119909ℎ119892119896119909119894119895119896 ge 0] forall (119894 119895) (ℎ 119892) 119896(3)

sum119896isin119872119894119895

119909119894119895119896 = 1 (4)

119909119894119895119896 =

1 if machine 119896 is selected for operation 1198741198941198950 otherwise

(5)

Equation (1) indicates the three optimizing objectivesInequality (2) ensures the operation precedence constraintInequality (3) ensures that each machine processes only oneoperation at each time Equation (4) states that one machinecan be selected from the set of available machines for eachoperation [25] According to whether machine 119896 is selectedto process step 119874119894119895 or not equation (5) is used to determinethe value of 119909119894119895119896

3 The Proposed Algorithm

31 Brief Introduction of NSGA-II The nondominated sort-ing genetic algorithm II (NSGA-II) is a population-basedmultiobjective evolutionary algorithm which is widely usedin the optimization of multiobjective problems The coreprocedure of NSGA-II can be briefly formulated as followsFirst using 119875119905 and 119880119905 presents the current parent and


offspring population respectively The sizes of 119875119905 and 119880119905 areboth 119873 Then a new population 119877119905 = 119875119905 cup 119880119905 (of size 2119873) isformed by combining 119875119905 with 119880119905 Furthermore an operatorcalled nondominated sorting is executed which defines 119877119905 asdifferent nondominated levels (rank1 rank2 etc) to choosethe best 119873 members as the new population named 119875119905+1 forthe next generation In NSGA-II the crowding distance iscomputed by a special operator and then the solutions withlarger crowding distance values are selected More details ofthe NSGA-II could refer to [33]

32 The BEG-NSGA-II

321 Framework In this paper a bee evolutionary guidingnondominated sorting genetic algorithm II (BEG-NSGA-II)with a two-stage optimization scheme is proposed for theFJSP inwhich a bee evolutionary guiding scheme is presentedthat focus on the exploitation of solution space and somemechanisms are used to enhance the searching ability andavoid the premature convergence Its framework is shown inFigure 1 More details of steps are described as follows

The First Stage

Step 1 Generate 119873 individuals as the initial populationrandomly and set 119905 = 1

Step 2 Calculate the fitness value of individuals based onobjective 1 objective 2 and objective 3 then select the bestfitness individuals as the queen 1 queen 2 and queen 3correspondingly if 119905 gt 1 compare the fitness value of theparent queens and offspring queens and select the best onesas the new queens correspondingly

Step 3 Use roulette wheel method to select 119875 individualsaccording to the fitness value of objective 1 objective 2 andobjective 3

Step 4 Randomly generate 119877 individuals and combine theseindividuals with the 119875 individuals generated from Step 3respectively

Step 5 The new queens selected from Step 2 take crossoverand mutation with other corresponding individuals gener-ated from Step 4 to produce the offspring population 1 off-spring population 2 and offspring population 3 respectively

Step 6 Combine the three offspring populations into a com-bining population

Step 7 Fast nondominated sorting and congestion comput-ing

Step 8 Select the top 119873 individuals as the offspring popula-tion

Step 9 Set 119905 = 119905 + 1 if 119905 gt 119879 continue next step otherwisereturn Step 2 and the population is replaced by the selected119873 individuals

The Second Stage

Step 10 Select 119878 top individuals (from the top119873 individuals)as the elite individuals

Step 11 Use a binary tournament selection method to selectcross individuals from offspring population (ie the selected119873 individuals) if 119863(119894 119895) gt 119906 use a precedence operationcrossover otherwise use job-based crossover

Step 12 If 119875119898 lt= 01 and if offspring rank = 1 then chooseswapping mutation else choose two binding mutation orreversemutation else the population does not takemutation

Step 13 Randomly generate 1198771015840 new individuals

Step 14 Combine the three-part individuals which get fromSteps 10 12 and 13

Step 15 Use Step 7 to deal with the combining population togenerate offspring population

Step 16 If 119905 gt 119866119864119873 output the result otherwise 119905 = 119905 + 1and return Step 10

The parameters used in this algorithm are defined asfollows

At the first stage

120572 = 119905119879

119875 = 120572 lowast 1198732

119877 = 119873 (1 minus 120572)2

(6)

At the second stage

1205721015840 = 119905119866119864119873 (7)

1198771015840 =119873(1 minus 1205721015840)

2 (8)

1205941015840 + 119878 + 05119873 gt= 119873 (9)

119863(119894 119895) =119871

sum119897=1

10038161003816100381610038161003816min [max [0 10038161003816100381610038161003816119886119894119897 minus 11988611989511989710038161003816100381610038161003816] minus 1 0]

10038161003816100381610038161003816119871 (10)

119863(119894 119895) le 119906 Precedence operation crossover

119863(119894 119895) gt 119906 Job-based crossover(11)

In (6)ndash(9) where 119873 is the number of individuals inthe initial population 119905 is the current iteration 119879 is theiteration times in the first stage and GEN is the iterationtimes in the second stage Equation (9) ensures that there areenough numbers of individuals to be selected in the followingoffspring and we set the value of 119878 as 03 times 119873 Equation (10)


Initial

No

Yes

Calculate the fitness valueof individuals based on

objective 1

Randomlygenerate Rindividuals

Use roulette wheelmethod to select P individualsaccording the fitness valueof objective 1

Offspring population 1 Offspring population 2 Offspring population 3

Combining population

Fast nondominated sorting and congestion computing

Select the top N individuals

The new queens take crossover and mutation with other corresponding individuals

Select cross individuals bybinary tournament selection

POX crossover JBX crossover

Swampingmutation

TBM or reversemutation

Offspring population

eliteindividuals

Select N top individuals

Randomly

individuals

NoYes

Combining population Fast nondominated sorting and congestion computing


Output result

YesNo

NoYes

Generate N individuals at the first generation randomlyand inherit the offspring populations afterwards


objective 2


objective 3

Parentqueen 3

Offspringqueen 3

Select the best one asthe new queen 3

Parentqueen 2

Offspringqueen 2


Parentqueen 1

Offspringqueen 1


Use roulette wheelmethod to select Pindividuals according to thefitness value of objective 2

Use roulette wheel method to select P individualsaccording to the fitnessvalue of objective 3

NoYes

The first stage

The second stage

t = t + 1

t gt GENgenerate R998400

Offspring rank = 1

D(i j) gt u

t = t + 1

t = 1

t gt T

Pm lt= 01

Remain S

Figure 1 The framework of BEG-NSGA-II for MO-FJSP


Operation sequence vector

3 1 2 3 1 2 4 3 4 1 3 2

Machine assignment vector

4 3 5 2 4 5 1 1 3 4 2 5

O31 O11 O21 O32 O33 O34O13 O23O12 O22 O41 O42

J3 J4J2J1

(a) Chromosome

5 10 15 20 250

6

10

13

9

1

5

Time

1012 17 22

20

15 19

M1

M2

M3

M4

M5

O31

O11

O21

O32

O33

O34

O13 O23

O12

O22

O41

O42

(b) Decoding Gantt chart

Figure 2 An encoding and decoding example of a chromosome of 4 times 5 FJSP

is a difference degree function which is used to compute thesimilarity between chromosome 119894 and chromosome 119895 (thetwo chromosomes that will be crossed) where 119871 is the lengthof a chromosome and 119886119894119897 denotes the 119897th gene in chromosome119894 In (11) 119906 is a parameter set as 07The chromosomes tend toselect themultipoint preservative crossover with the decreaseof 119906

322 Chromosome Encoding and Decoding In this paper weuse the encoding method presented by Gao et al [12] TheFJSP problem includes two subproblems one is operationsequence part and the other is machine assignment partChromosome that corresponds to the solution of the FJSPalso consists of two parts The first one is the operationsequence vector and the second one is the machine assign-ment vector Two different encoding methods are used togenerate the two vectors

In terms of the operation sequence vector the operation-based representation method is used which is composed ofthe jobsrsquo numbers This representation uses an array of unin-terrupted integers from 1 to 119899 where 119899 is the total numberof jobs and each integer appears Oni times where Oni is thetotal number of operations of job 119894 therefore the length of theinitial operation sequence population is equal tosum119899119894=1 119874119899119894 Byscanning the operation sequence from left to right the 119898thoccurrence of a job number expresses the 119898th operation ofthis job The operation sequence vector of every initial indi-vidual of population is generated with the randomly encod-ing principle By using these representation features anypermutation of the operation sequence vector can be decodedto a feasible solution

The machine assignment vector indicates the selectedmachines that are assigned to the corresponding operationsfor all jobs It contains 119899 parts and the length of 119894th part isOni hence the length of this vector also equalssum119899119894=1 119874119899119894 The119894th part of this vector expresses themachine assignment set of119894th job Supposing amachine set 119878119894ℎ = 119898119894ℎ1 119898119894ℎ2 119898119894ℎ119862119894ℎcan be selected to process the ℎth operation of job 119894 a genset 1198921198941 1198921198942 119892119894ℎ 119892119894119874119899119894 denotes the 119894th part of machineassignment vector and 119892119894ℎ is an integer between 1 and 119862119894ℎand this means that the ℎth operation of job 119894 is processed by

the 119892119894ℎth machine 119898119894ℎ119892119894ℎ from 119878119894ℎ The machine assignmentvector of every initial individual of population is generated byselecting the available machine randomly for each operationof every job An example is shown in Figure 2(a) and theoperation and machine sequence are shown in it as follows(119874311198721) (119874111198724) (119874211198722) (119874321198721) (119874121198723) (119874221198724) (119874411198722) (119874331198723) (119874421198725) (119874131198725) (119874341198724)and (119874231198725) then we can get the value of objectives of thework by referring to Table 1

When the chromosome representation is decoded eachoperation starts as soon as possible following the precedenceand machine constraints [31] A schedule generated by usingthis decoding method can be ensured to be active schedule[40] The procedure of decoding is implemented as follows

Step 1 Identify the machine of all operations based on themachine assignment vector

Step 2 Identify the set of machines used to process every job

Step 3 Identify the set of operations for every machine

Step 4Determine the allowable starting time of every opera-tion119860119878119894119895 = 119862119894(119895minus1) where119860119878119894119895 denotes the allowable startingtime of operation 119874119894119895 and 119862119894(119895minus1) is the completion time ofoperation 119874119894(119895minus1) for the same job

Step 5 Calculate the idle time of the machine of operation119874119894119895 and get the idle areas [t start t end] where t start is thestart time of these idle areas and t end is the end time ofthese idle areas Scanning these areas from left to right ifmax (119860119878119894119895 119905 start) + 119905119894119895119896 le 119905 end the earliest starting timeis 119878119894119895 = 119905 start else 119878119894119895 = max(119860119878119894119895 119862119894(119895minus1))

Step 6 Calculate the completion time of every operation119862119894119895 = 119878119894119895 + 119905119894119895119896

Step 7Generate the sets of starting time and completion timefor every operation of every job

By using the above procedure a feasible schedule for theFJSP is obtained Figure 2 shows the examples of encoding


and decodingmethods and the processing time andmachinedate of jobs can be seen from Table 1 This example contains4 jobs and 5machines Job 1 and job 2 both have 3 operationsjob 3 and job 4 contains 4 and 2 operations respectivelyFigure 2(a) shows a chromosome which contains two partsthe operation sequence vector and the machine assignmentvector The operation sequence vector is an unpartitionedpermutation with repetitions of job numbers It containsthree 1s three 2s four 3s and two 4s because there are 4 jobsjob 1 contains 3 operations job 2 contains 3 operations job3 contains 4 operations and job 4 contains 2 operations Itslength is 12Themachine assignment vector consists of 4 partsbecause of 4 jobs Its length is also 12 Each part presents themachines selected for the corresponding operations of jobFor example the first part contains 3 numbers which are 4 3and 5 Number 4 means that machine 4 is selected for opera-tion 1 of job 1 number 3meansmachine 3 is selected for oper-ation 2 of job 1 and number 5meansmachine 5 is selected foroperation 3 of job 1 A Gantt chart of a schedule based on thechromosome in Figure 2(a) is shown in Figure 2(b)

323 Crossover Operators In this paper the proposed algo-rithm contains two optimization stages In order to expandthe searching space and avoid premature of local optimalsolutions we use different crossover operators in these twostages

At the first stage a single-point crossover (SPX) or multi-point crossover (MPX) operator is selected randomly (50)for the operation sequence vector and a two-point crossover(TPX) operator is selected for themachine assignment vector

The procedure of SPX is described as follows (1198751 and 1198752are used to denote two parents1198741 and1198742 are used to denotetwo offspring)

Step 1 A random parameter 119896 that meets the inequality 0 lt119896 lt 119875 (the length of operation sequence vector) is generatedto determine the position of the crossover

Step 2 The elements from 1 to 119896 in 1198751 are duplicated to 1198741in the same positions and the elements from 1 to 119896 in 1198752 areduplicated to 1198742 in the same positions

Step 3 Calculate the total number of each element (in thisexample the total number of 1 2 3 and 4 is three three fourand two resp)

Step 4 The elements in 1198752 are appended to the remainingempty positions in1198741 from left to right until the total numberof each element in 1198741 equals each one in 1198751 The elementsin 1198751 are appended to the remaining empty positions in 1198742from left to right until the total number of each element in1198742equals each one in 1198752

The procedure of MPX is described as follows (1198751 and 1198752are used to denote two parents1198741 and1198742 are used to denotetwo offspring)

Step 1 Two random parameters 1198961 and 1198962 that meet theinequality 0 lt 1198961 lt 1198962 lt 119875 as well as 1198961 = 1198962 are generated todetermine the positions of crossover

3 1 2 4 1 2 4 3 3 1 3 2

3 1 2 4 1 3 2 1 2 4 3 3

1 3 2 1 2 2 4 3 4 1 3 3

1 3 2 1 3 1 2 4 2 4 3 3

3 1 2 4 1 2 4 3 3 1 3 2

O1

O2

P1

P1

P2

Figure 3 SPX crossover

3 1 2 4 1 2 4 3 3 1 3 2

1 2 4 1 3 2 1 2 4 3 3 3

1 3 2 1 2 2 4 3 4 1 3 3

3 2 1 3 1 2 4 1 2 4 3 3

3 1 2 4 1 2 4 3 3 1 3 2

O1

O2

P1

P1

P2

Figure 4 MPX crossover

Step 2The elements from 1198961 to 1198962 in 1198751 are appended to theleftmost positions of1198741 and the elements from 1198961 to 1198962 in 1198752are appended to the leftmost positions of 1198742



The examples of SPX andMPX are respectively shown inFigures 3 and 4

For themachine assignment vector a two-point crossover(TPX) has been adopted here as the crossover operator Theprocedure of it is described as follows (1198751 and 1198752 are usedto denote two parents 1198741 and 1198742 are used to denote twooffspring)


Step 2 Append the elements between 1198961 and 1198962 positions in1198752 to the same positions in1198741 append the elements before 1198961


4 3 5 2 4 5 1 1 3 4 2 5

4 3 5 1 2 5 1 1 3 4 2 5

3 4 5 1 2 4 3 2 5 1 2 3

3 4 5 2 4 4 3 2 5 1 2 3

4 3 5 2 4 5 1 1 3 4 2 5

O1

O2

P1

P1

P2

J3 J4J2J1

Figure 5 TPX crossover

and after 1198962 positions in 1198751 to the same positions in 1198741 anduse the same process to generate 1198742

Based on the above procedure we can obtain feasibleoffspring if the parents are feasible An example of TPXcrossover is shown in Figure 5

At the second stage a precedence operation crossover(POX) or job-based crossover (JBX) which is determined byequation (11) is selected for operation sequence vector andthe TPX is also selected for machine assignment vector

The main procedure of POX is described as follows (1198751and 1198752 are used to denote two parents 1198741 and 1198742 are used todenote two offspring)

Step 1 The Job set is randomly divided into two subsetsJobset1 and Jobset2

Step 2The element(s) which belong(s) to Jobset1 in1198751 is (are)appended to the same position(s) in 1198741 and deleted in 1198751and the element(s) which belong(s) to Jobset2 in 1198752 is (are)appended to the same positions in 1198742 and deleted in 1198752

Step 3Append the elements remaining in1198751 to the remainingempty positions in 1198742 from left to right and append theelements remaining in 1198752 to the remaining empty positionsin 1198741 from left to right

The main procedure of JBX is described as follows (1198751and 1198752 are used to denote two parents 1198741 and 1198742 are usedto denote two offspring)


Step 2The element(s) which belong(s) to Jobset1 in1198751 is (are)appended to the same position(s) in 1198741 and the element(s)which belong(s) to Jobset2 in 1198752 is (are) appended to the samepositions in 1198742

Step 3Theelement(s) which belong(s) to Jobset2 in1198752 is (are)appended to the remained empty positions in 1198741 from left toright and the element(s) which belong(s) to Jobset1 in 1198751 is(are) appended to the remained empty positions in 1198742 fromleft to right

3 1 2 4 1 2 4 3 3 1 3 2

3 1 2 2 1 2 4 3 3 1 3 4

1 3 2 1 2 2 4 3 4 1 3 3

1 3 2 1 4 2 4 3 2 1 3 3

3 1 2 4 1 2 4 3 3 1 3 2

Jobset1 = 1 3 Jobset2 = 2 4

O1

O2

P1

P1

P2

Figure 6 POX crossover

3 1 2 4 1 2 4 3 3 1 3 2

3 1 2 2 1 2 4 3 3 1 3 4

1 3 2 1 2 2 4 3 4 1 3 3

3 1 2 1 2 2 4 3 4 3 1 3

3 1 2 4 1 2 4 3 3 1 3 2


O1

O2

P1

P1

P2

Figure 7 JBX crossover

The examples of POX and JBX are respectively shown inFigures 6 and 7

324 Mutation Operators In this paper we have adopteddifferent mutation operators at two stages for purpose ofexpanding the solution space as well as maintaining the goodsolutions

At the first stage a swapping mutation or reverse muta-tion operator is selected randomly (50) with probability 119901119898(set as 01 in our algorithm) for operation sequence vectorAnd a multipoint mutation (MPM) operator is selected formachine assignment vector

The main procedure of swapping mutation operator isdescribed as follows (1198751 and 1198741 are used to denote a parentand offspring resp)

Step 1 Randomly select two positions in 1198751

Step 2 Swap the elements in the selected positions to generate1198741

The main procedure of reverse mutation operator isdescribed as follows (1198751 and 1198741 are used to denote a parentand offspring resp)


Step 2 Reverse the numbers between the selected two posi-tions to generate 1198741


3 1 2 4 1 2 4 3 3 1 3 2

3 1 2 4 3 2 4 1 3 1 3 2O1

P1

Figure 8 Swapping mutation operator

3 1 2 4 1 2 4 3 3 1 3 2

3 1 2 1 4 2 4 3 3 1 3 2O1

P1

Figure 9 Reverse mutation operator

The main procedure of MPM operator is described asfollows (1198751 and 1198741 are used to denote a parent and offspringresp)

Step 1 Randomly select 119896 positions in 1198751 (119896 equals the half ofthe length of the machine assignment vector)

Step 2Change the value of these selected positions accordingto their optional machine sets selected for process thecorresponding operations

At the second stage if the individual is at the forefront ofPareto it will select the swappingmutation otherwise choosethe two binding mutation (TBM) or reverse mutation foroperation sequence vectorTheMPMoperator is also selectedfor the machine assignment vector

The main procedure of TBM is described as follows (1198751and 1198741 are used to denote a parent and offspring resp)

Step 1 A random parameter 119898 (119898 lt the length of theoperation sequence vector subtract 3) is generated in 1198751

Step 2 Exchange the elements 119898 with 119898 + 3 and 119898 + 1 with119898 + 2 in 1198751 to generate 1198741

The examples of swapping mutation operator reversemutation operator TBM operator and MPM operator arerespectively shown in Figures 8 9 10 and 11

4 Experimental Studies and Discussions

The proposed BEG-NSGA-II algorithm was coded in MAT-LAB R2014a and implemented on a computer configuredwith Intel Core i3 CPU with 267GHz frequency and 4GBRAM Four famous benchmarks that include 53 open prob-lems of FJSP are chosen to estimate the proposed algorithmwhich were also used by many researchers to evaluate theirapproaches In order to illustrate the performance of theproposed algorithm we compare our algorithm with otherstate-of-the-art reported ones The computational time usedto solve these benchmarks is also compared to show the goodefficiency of the proposed method Because the computationtime and implementing performance are not only affectedby the algorithm itself but also affected by the computerhardware implementing software and coding skills we also

3 1 2 4 1 2 4 3 3 1 3 2

3 1 2 3 3 2 4 4 1 1 3 2O1

P1

Figure 10 TBMmutation operator

4 3 5 2 4 5 1 1 3 4 2 5

2 3 3 2 5 5 1 3 3 4 1 4

J3 J4J2J1

O1

P1

Figure 11 MPM operator for machine assignment vector

Table 2 Parameters of the BEG-NSGA-II

Parameters ValuePopulation size 100Maximal total generation 300Iteration times of first stage 100Iteration times of second stage 200Crossover probability 1Mutation probability 01

append the information of hardware and software as wellas the original computational time with the correspondingalgorithms All experimental simulations were run 20 timesrespectively for each problem of these benchmarks Theadopted parameters of the BEG-NSGA-II are listed in Table 2

41 Experiment 1 The data of Experiment 1 are taken fromKacem et al [31] It contains 4 representative instances(problem 4 times 5 problem 8 times 8 problem 10 times 10 and problem15times10)The experimental results and comparisonswith otherwell-known algorithms are shown in Table 3 (119899 times 119898 meansthat the problem includes 119899 jobs and119898machines 1198911 1198912 and1198913 mean the optimization objectives of makespan MW andTW resp 119879 means the average computer-independent CPUtimes in minute spent on each problem of these benchmarksthe symbol ldquomdashrdquo means the time has not been given in thepaper) BEG-NSGA-II denotes the proposed algorithm Theresults ofALwhich are taken fromKacemet al [31] PSO+SAwhich are taken from Xia andWu [41] hGA which are takenfrom Gao et al [24] MOGA which are taken from Wanget al [34] hPSO which are taken from Shao et al [42] andOO approaches which are taken from Kaplanoglu [38] areused to make comparison with the proposed algorithm Thebolded results are the new Pareto-optimal solutions found inour algorithms

For the problem 4 times 5 the proposed BEG-NSGA-II algo-rithm obtains not only all best solutions in these comparedalgorithms but also another new Pareto-optimal solution(1198911 = 13 1198912 = 7 and 1198913 = 33) The Gantt chart of this newPareto solution is shown in Figure 12 For the problems 8 times 810times10 and 15times10 the proposed algorithm getsmore Pareto-optimal solutions than other listed algorithms but hPSO and


Table3Re

sults

onEx

perim

ent1

Prob

lem

119899times119898

ALa

PSO+SA

bMOGA

chP

SOd

hGA

eOOapproach

fBE

G-N

SGA-

IIg

119891 1119891 2

119891 3T

119891 1119891 2

119891 3T

119891 1119891 2

119891 3T

119891 1119891 2

119891 3T

119891 1119891 2

119891 3T

119891 1119891 2

119891 3T

119891 1119891 2

119891 3T

4times5

mdashmdash

mdashmdash

mdashmdash

mdashmdash

1110

32010

mdashmdash

mdashmdash

mdashmdash

mdashmdash

1110

32003

1110

32010

119

3411

934

128

3212

832

137

33

8times8

1613

75mdash

1512

75mdash

1511

81016

1412

77014

1512

75001

1613

73004

1412

77012

1613

7315

1275

1512

7515

1275

1512

7516

1373

1611

7716

1177

1613

7316

1373

10times10

75

45mdash

76

44mdash

85

42024

75

43020

75

43001

87

41005

75

43018

85

427

642

76

428

542

76

428

741

87

418

542

77

438

542

75

458

741

87

41

15times10

2311

95mdash

1211

91mdash

1111

9114

611

1191

136

1111

91003

1313

91009

1111

91005

2411

9112

1095

1210

9314

1291

1210

9311

1098

1110

9511

1095

a Thec

ompu

tatio

nconfi

guratio

nisno

tlisted

inAL

b Thec

ompu

tatio

nconfi

guratio

nisno

tlisted

inPS

O+SA

c Th

eCPU

timeo

naP

Cwith

2GHzC

PUand2G

Bof

RAM

mem

oryin

C++

d TheC

PUtim

eonap

ersonalcom

puterw

ith2G

HzC

PUand2G

Bof

RAM

mem

oryin

C++

e TheC

PUtim

eona3

0GHzP

entiu

min

Delp

hi

f TheC

PUtim

eonan

IntelC

orei5267

GHzp

rocessor

with

4GBRA

Mof

mem

oryin

Java

g TheC

PUtim

eonan

IntelC

orei3267

GHzp

rocessor

with

4GBRA

Min

MAT

LABR2

014a


O11

O12

O13

O21

O22

O23

O31

O32

O33 O34

O41

O42

13

1 2 3 4 5 6 7 8 9 10 11 12 130

M1

M2

M3

M4

M5

Figure 12 A new Pareto solution of problem 4 times 5 in Experiment 1

2634

130 260 390 520 650 780 910 1040 1170 1300 1430 1560 1690 1820 1950 2080 2210 2340 2470 26000

M1

M2

M3

M4

M5

M10

M6

M7

M8

M9

Figure 13 Gantt chart of a solution of problem 18a in Experiment 2

although it obtains the same results with hPSO it seems toconsume less computation time than the hPSO does

42 Experiment 2 The data of Experiment 2 are taken fromDauzere-Peres and Paulli [10] It contains 18 problems Theexperimental results and comparisonswith otherwell-knownalgorithms are shown in Table 4 (119899 times 119898 means that theproblem includes 119899 jobs and 119898 machines 1198911 1198912 and 1198913mean the optimization objectives of makespan MW andTW resp 119879 means the average computer-independent CPUtimes inminute spent on each problemof these benchmarks)BEG-NSGA-II denotes the proposed algorithm The resultsof MOGA are adopted from Wang et al [34] The boldedresults are the new Pareto-optimal solutions found in ouralgorithms From Table 4 except the problem 01a we cansee that our algorithm can obtain some new Pareto-optimalsolutions which are better for objectives 1198912 or 1198913 or both of

them but a little worse for the objective 1198911 For the problem01a our algorithm gets the same solutions as MOGA Forall problems our algorithm consumes less computationaltime than MOGA Figure 13 illustrates the Gantt chart of theproblem 18a (1198911 = 2634 1198912 = 2156 and 1198913 = 21005)

43 Experiment 3 The data of Experiment 3 are taken fromBrandimarte [9] It contains 10 problems The experimentalresults and comparisons with other well-known algorithmsare shown in Table 5 (119899 times119898means that the problem includes119899 jobs and119898machines 1198911 1198912 and 1198913 mean the optimizationobjectives ofmakespanMW andTW resp119879means the aver-age computer-independent CPU times in minute spent oneach problem of these benchmarks) BEG-NSGA-II denotesthe proposed algorithm The results of SM and MOGA areadopted from [26 34]The bolded results are the new Pareto-optimal solutions found in our algorithms From Table 5


Table 4 Results on Experiment 2

Problem (119899 times 119898) MOGAa BEG-NSGA-IIb

1198911 1198912 1198913 119879 1198911 1198912 1198913 119879

01a (10 times 5)2568 2505 11137 204 2568 2505 11137 1912572 2568 11137 2572 2568 111372594 2554 11137 2594 2554 11137

02a (10 times 5)2289 2263 11137 256 2487 2237 11137 2132313 2238 11137 3014 2234 11137

2564 2236 11137

03a (10 times 5) 2287 2248 11137 290 2492 2247 11137 252256 2252 11137 2516 2234 11137

04a (10 times 5)

2550 2503 11090 207 2882 2572 11069 2032569 2565 11076 2912 2552 110702579 2552 11080 2904 2626 110673095 2727 11064 3019 2665 11066

2997 2523 110732933 2688 110652953 2668 110662938 2606 110682955 2503 11074

05a (10 times 5)

2292 2252 11077 237 2740 2250 10981 2122293 2242 11091 2846 2212 109862297 2255 11054 2747 2229 109882315 2272 11063 2700 2255 109772343 2298 11050 2915 2231 109812358 2322 11038 2818 2208 109902376 2243 11022 2787 2350 109702904 2620 10941 2759 2320 109712945 2571 10941 2667 2237 109833056 2507 10941 2686 2277 10974

2641 2332 109742687 2221 110002779 2237 109812707 2299 109722702 2233 10989

06a (10 times 5)

2250 2233 11009 309 2673 2194 10893 2862254 2223 10994 2716 2194 108912398 2219 10973 2683 2245 108902437 2280 10988 2666 2248 108932744 2448 10850 2679 2229 108922902 2439 10847 2816 2186 109032967 2840 10839 2898 2186 10899

2713 2234 108892758 2209 108902722 2229 10890

07a (15 times 8)

2450 2413 16485 763 2927 2187 16485 6212457 2299 16485 2910 2213 164852484 2289 16485 2889 2264 16485

2891 2219 164852922 2190 16485

08a (15 times 8)2187 2102 16485 827 2621 2091 16485 6352171 2104 16485 2770 2089 16485

2780 2080 16485


Table 4 Continued


1198911 1198912 1198913 119879 1198911 1198912 1198913 119879

09a (15 times 8)2157 2113 16485 1016 2442 2103 16485 7262144 2119 16485 2470 2074 164852158 2102 16485 2455 2081 16485

10a (15 times 8)

2461 2433 16505 755 2998 2317 16487 6432470 2310 16537 3036 2363 164822478 2330 16533 3091 2491 164732482 2360 16499 2984 2319 164862501 2265 16547 3052 2377 164772501 2312 16528 3098 2491 164722547 2476 16490 3013 2293 164873064 2734 16464 3136 2457 16471

3025 2377 164803041 2351 164812980 2258 164943021 2319 16484

11a (15 times 8)

2182 2170 16449 1014 2551 2116 16229 7252202 2114 16476 2599 2131 162242210 2113 16442 2612 2146 162222337 2185 16377 2573 2058 162312874 2389 16247 2753 2186 162172894 2330 16247 2628 2102 162242962 2312 16247 2702 2098 16223

2658 2089 162242555 2111 162292664 2063 16228

12a (15 times 8)

2161 2107 16295 1192 2624 2049 16055 8222168 2130 16220 2620 2022 160682191 2084 16355 2587 2035 160652210 2103 16331 2614 2057 160512315 2125 16292 2585 2043 160612366 2105 16237 2659 2038 160552493 2297 16124 2617 2024 160652631 2309 16112 2575 2057 160572637 2303 16113 2587 2050 160592683 2397 16104 2582 2098 16052

13a (20 times 10)

2595 2036 160592566 2044 16061

2408 2326 21610 2399 2894 2277 21610 10272928 2233 216102896 2244 216102877 2282 21610

14a (20 times 10)2340 2251 21610 2905 2649 2186 21610 12392334 2258 21610 2664 2183 21610

2641 2207 216102738 2182 21610

15a (20 times 10)

2285 2247 21610 3329 2627 2216 21610 13602287 2218 21610 2681 2196 21610

2645 2215 216102652 2201 216102668 2197 216102682 2178 21610


Table 4 Continued


1198911 1198912 1198913 119879 1198911 1198912 1198913 119879

16a (20 times 10)

2447 2354 21602 2152 2965 2279 21534 11482450 2380 21590 3158 2303 215042487 2454 21584 3184 2400 214902492 2417 21576 3039 2279 215232540 2396 21547 3050 2318 215072550 2492 21545 3071 2286 215132568 2428 21540 3275 2281 215113013 2588 21478 3105 2352 214963106 2548 21478 3175 2344 21497

3180 2293 215073130 2312 215033034 2352 21503

17a (20 times 10)

2322 2240 21433 2847 2806 2195 21096 13922322 2280 21362 2750 2165 211202323 2238 21454 2791 2141 211142343 2224 21420 2716 2157 211292480 2285 21344 2849 2148 211072528 2231 21313 2769 2166 211112789 2448 21198 2856 2183 210982808 2303 21200 2773 2145 211112816 2370 21197 2856 2142 21105

2803 2203 210942759 2186 211062789 2176 211012772 2202 21101

18a (20 times 10)

2267 2235 21483 3301 2634 2156 21005 16232269 2206 21408 2642 2126 210062320 2208 21354 2638 2185 209992437 2221 21311 2638 2148 210022531 2310 21285 2674 2148 210002545 2305 21282 2644 2145 21002

2651 2140 210052641 2157 210002664 2145 210012621 2181 210022659 2140 210032661 2137 210042651 2181 209992669 2126 210052708 2125 21009

aThe CPU time on a PC with 2GHz CPU and 2GB of RAMmemory in C++bThe CPU time on an Intel Core i3 267GHz processor with 4GB RAM in MATLAB R2014a

we can see that our algorithm can obtain some new Pareto-optimal solutions in problems MK01 MK04 MK05 MK08and MK10 compared with SM and MOGA and the samesolutions in MK2 MK3 MK6 MK7 and MK9 as MOGAFor all problems our algorithm consumes less computationaltime than MOGA Figure 14 illustrates the Gantt chart of theproblem MK08 (1198911 = 541 1198912 = 533 and 1198913 = 2516)

44 Experiment 4 The data of Experiment 4 are taken fromBarnes and Chambers [43] It contains 21 problems Theexperimental results and comparisonswith otherwell-knownalgorithms are shown in Table 6 (119899 times 119898 means that theproblem includes 119899 jobs and 119898 machines 1198911 1198912 and 1198913mean the optimization objectives of makespan MW andTW resp 119879 means the average computer-independent CPU



Problem (119899 times 119898) SMa MOGAb BEG-NSGA-IIb

1198911 1198912 1198913 119879 1198911 1198912 1198913 119879 1198911 1198912 1198913 119879

MK01 (10 times 6)

42 42 162 478 42 39 158 049 42 39 158 03644 40 154 44 40 15443 40 155 43 40 15540 36 169 40 36 169

47 36 16747 37 16549 37 16450 38 162

MK02 (10 times 6)

28 28 155 302 26 26 151 075 26 26 151 07027 27 146 27 27 14629 27 145 29 27 14529 29 143 29 29 14331 31 141 31 31 14133 33 140 33 33 140

MK03 (15 times 8)

204 204 852 2614 204 199 855 475 204 199 855 366204 144 871 204 144 871204 135 882 204 135 882204 133 884 204 133 884213 199 850 213 199 850214 210 849 214 210 849221 199 847 221 199 847222 199 848 222 199 848231 188 848 231 188 848230 177 848 230 177 848

MK04 (15 times 8)

68 67 352 1774 66 63 345 176 66 63 345 16365 63 362 65 63 36263 61 371 63 61 37162 61 373 60 59 39061 60 382 74 54 34960 59 390 74 55 34873 55 350 72 72 34274 54 349 78 78 33974 55 348 72 66 34890 76 331 73 72 343

MK05 (15 times 4)

177 177 702 826 173 173 683 234 173 173 683 196175 175 682 175 175 682183 183 677 183 183 677185 185 676 179 179 679179 179 679 199 199 674

193 193 675183 183 677

MK06 (10 times 15)

67 431 1879 62 55 424 193 62 55 424 18565 54 417 65 54 41760 58 441 60 58 44162 60 440 62 60 44076 60 362 76 60 36276 74 356 76 74 35678 60 361 78 60 36173 72 360 73 72 36072 72 361 72 72 361100 90 330 100 90 330


Table 5 Continued


1198911 1198912 1198913 119879 1198911 1198912 1198913 119879 1198911 1198912 1198913 119879

MK07 (20 times 5)

150 150 717 568 139 139 693 492 139 139 693 312140 138 686 140 138 686144 144 673 144 144 673151 151 667 151 151 667157 157 662 157 157 662162 162 659 162 162 659166 166 657 166 166 657

MK08 (20 times 10)

523 523 2524 6767 523 515 2524 1204 523 515 2524 834523 497 2534 523 497 2534524 524 2519 524 524 2519578 578 2489 578 578 2489587 587 2484 587 587 2484

541 533 2516542 542 2511560 560 2505

MK09 (20 times 10)

311 299 2374 7776 311 299 2290 1948 311 299 2290 1015310 299 3514 310 299 3514311 301 2287 311 301 2287314 299 2315 314 299 2315315 299 2283 315 299 2283332 302 2265 332 302 2265329 301 2266 329 301 2266328 308 2259 328 308 2259325 299 2275 325 299 2275

MK10 (20 times 15)

227 221 1989 12252 224 219 1980 1787 224 219 1980 1238225 211 1976 235 225 1895233 214 1919 240 215 1905235 218 1897 246 215 1896235 225 1895 252 224 1884240 215 1905 256 211 1919240 216 1888 260 244 1869242 214 1913 266 254 1864246 215 1896 246 206 1938252 224 1884 246 204 1948256 211 1919 246 205 1940260 244 1869 251 202 1948266 254 1864 253 207 1930268 264 1858 249 203 1947276 256 1857 255 202 1943281 268 1854 260 210 1926217 207 2064 250 205 1935214 204 2082 254 203 1942

244 207 1935aThe CPU time on a Pentium IV 24GHz processor with 10 GB of RAMmemory in MATLABbThe CPU time on a PC with 2GHz CPU and 2GB of RAMmemory in C++cThe CPU time on an Intel Core i3 267GHz processor with 4GB RAM in MATLAB R2014a



Problem (119899 times 119898) HGTSa HAb BEG-NSGA-IIc

1198911 119879 1198911 119879 1198911 1198912 1198913 119879mt10c1 (10 times 11) 927 022 927 020 1092 631 5109 018mt10cc (10 times 12) 908 022 908 016 1005 631 5109 020mt10x (10 times 11) 918 025 918 018 1117 556 5109 018mt10xx (10 times 12) 918 020 918 018 1117 556 5109 018mt10xxx (10 times 13) 918 020 918 018 1100 556 5109 019mt10xy (10 times 12) 905 022 905 019 1061 548 5109 021mt10xyz (10 times 13) 847 030 847 016 925 534 5109 023setb4c9 (15 times 11) 914 027 914 026 1035 857 7727 025setb4cc (15 times 12) 907 025 907 025 1023 857 7727 024setb4x (15 times 11) 925 025 925 021 1030 846 7727 020setb4xx (15 times 12) 925 023 925 009 1030 846 7727 010setb4xxx (15 times 13) 925 025 925 015 1033 846 7727 016setb4xy (15 times 12) 910 032 910 020 1020 845 7727 022setb4xyz (15 times 13) 905 025 905 024 1011 838 7727 023seti5c12 (15 times 16) 1170 068 1170 052 1319 1027 11472 048seti5cc (15 times 17) 1136 057 1136 028 1301 888 11472 029seti5x (15 times 16) 1199 063 1198 046 1368 938 11472 052seti5xx (15 times 17) 1197 057 1197 048 1346 938 11472 055seti5xxx (15 times 18) 1197 052 1197 032 1346 938 11472 037seti5xy (15 times 17) 1136 057 1136 029 1301 888 11472 027seti5xyz (15 times 18) 1125 072 1125 055 1270 835 11472 061aThe CPU time on a Xeon E5520 processor with 24GB of RAMmemory in C++bThe CPU time on an Intel 20 GHz Core (TM) 2 Duo processor with 80GB of RAMmemory in C++cThe CPU time on an Intel Core i3 267GHz processor with 4GB RAM in MATLAB R2014a

25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400 425 450 475 500 5250

M1

M2

M3

M4

M5

M10

M6

M7

M8

M9

541

Figure 14 Gantt chart of a solution of problem MK08 in Experiment 3


127060 120 180 240 300 360 420 480 540 600 660 720 780 840 900 960 1020 1080 1140 1200 12600

M1

M2

M3

M4

M5

M10

M11

M12

M13

M14

M15

M16

M17

M18

M6

M7

M8

M9

Figure 15 Gantt chart of problem seti5xyz in Experiment 4

times inminute spent on each problemof these benchmarks)BEG-NSGA-II denotes the proposed algorithm The resultsof HGTS and HA are adopted from [44 45] and there are noother multiobjective optimization algorithms to compare forthese problems in the up to date literaturesThe bolded resultsare the new multiobjective optimization solution found inour algorithms From Table 6 we can see that our algorithmcan solve these problems with little worse objective 1198911 butsimultaneously get the other two objectives 1198912 and 1198913 withless computation time Figure 15 illustrates the Gantt chart ofthe problem seti5xyz (1198911 = 1270 1198912 = 835 and 1198913 = 11472)

45 Discussions From the simulation results of test examples1ndash3 we can see that our proposed BEG-NSGA-II could obtainthe same or more different Pareto-optimal solutions for themost benchmarks of MFJSP with less computation timeThat means more schemes can be chosen by the productionmanagers when they make scheduling decisions with highefficiency From the simulation results of test example 4we can see that our proposed algorithm of BEG-NSGA-IIworks better in multiobjective optimization as well as single-objective optimization

5 Conclusions and Future Studies

In this paper in order to fully play the respective advantagesof nondominated sorting genetic algorithm II (NSGA-II)algorithm and the algorithms with a two-stage optimizationscheme and to overcome the disadvantages of them wedeveloped a bee evolutionary guiding nondominated sortinggenetic algorithm II (BEG-NSGA-II) for solving the multi-objective flexible job-shop scheduling problem with the opti-mization objectives of minimizing the maximal completiontime the workload of the most loaded machine and thetotal workload of all machines A two-stage optimizationmechanism is constructed in the optimization process In thefirst stage the NSGA-II algorithm with 119879 iteration times isfirst used to obtain the initial population 119873 which consists

of three parts changing with the iteration times In this stagean effective local search operator is invented to extensivelyexploit the solution space In the second stage the NSGA-II algorithm with GEN iteration times is used to obtain thePareto-optimal solutions in which an updating mechanismand some useful genetic operators were employed to enhancethe searching ability and avoid the premature convergenceFrom the simulation results we can get the conclusions thatour proposed algorithm BEG-NSGA-II could obtain moredifferent Pareto-optimal solutions for most benchmarks ofMO-FJSPwith less computation timeHence it could providemore schemes for the production managers to choose whenthey make scheduling decisions This proposed computa-tional intelligence method (BEG-NSGA-II) could be widelyused in flexible job-shop scheduling problems especially themultiobjective optimization problems in scheduling filed

In the future we will concentrate on the dynamic andreal-time scheduling problems which possibly include newlyinserted jobs during the production process Meanwhile theredistribution of job operations and machine breakdownsmay also be taken into consideration

Notations

119899 Total number of jobs119898 Total number of machines119899119894 Total number of operations of job 119894119874119894119895 The 119895th operation of job 119894119872119894119895 The set of available machines for the operation119875119894119895119896 Processing time of 119874119894119895 on machine 119896119905119894119895119896 Starting time of operation 119874119894119895 on machine 119896119862119894119895 Completion time of the operation 119874119894119895119894 ℎ Index of jobs 119894 ℎ = 1 2 119899119896 Index of machines 119896 = 1 2 119898119895 119892 Index of operation sequence 119895 119892 = 1 2 119899119894119862119896 The completion time of119872119896119882119896 The workload of119872119896


Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (Grant no 71473077) National HighTechnology Research and Development Program of China(863 Program) (Grant no 2013AA040206) National KeyTechnology RampD Program of China (2015BAF01B00)

References

[1] J Błazewicz W Domschke and E Pesch ldquoThe job shopscheduling problem conventional and new solution tech-niquesrdquo European Journal of Operational Research vol 93 no1 pp 1ndash33 1996

[2] E Perez M Posada and F Herrera ldquoAnalysis of new nichinggenetic algorithms for findingmultiple solutions in the job shopschedulingrdquo Journal of Intelligent Manufacturing vol 23 no 3pp 341ndash356 2012

[3] X Qiu and H Y K Lau ldquoAn AIS-based hybrid algorithmfor static job shop scheduling problemrdquo Journal of IntelligentManufacturing vol 25 no 3 pp 489ndash503 2014

[4] L Asadzadeh ldquoA local search genetic algorithm for the jobshop scheduling problem with intelligent agentsrdquo Computersand Industrial Engineering vol 85 article no 4004 pp 376ndash3832015

[5] W-Y Ku and J C Beck ldquoMixed Integer Programming modelsfor job shop scheduling a computational analysisrdquo Computersand Operations Research vol 73 pp 165ndash173 2016

[6] M R Garey D S Johnson and R Sethi ldquoThe complexity offlowshop and jobshop schedulingrdquo Mathematics of OperationsResearch vol 1 no 2 pp 117ndash129 1976

[7] S Wang L Wang Y Xu and M Liu ldquoAn effective estimationof distribution algorithm for the flexible job-shop schedulingproblem with fuzzy processing timerdquo International Journal ofProduction Research vol 51 no 12 pp 3778ndash3793 2013

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

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

[10] S Dauzere-Peres and J Paulli ldquoAn integrated approach formodeling and solving the general multiprocessor job-shopscheduling problem using tabu searchrdquo Annals of OperationsResearch vol 70 no 1 pp 281ndash306 1997

[11] 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

[12] J Gao M Gen and L Sun ldquoScheduling jobs and maintenancesin flexible job shop with a hybrid genetic algorithmrdquo Journal ofIntelligent Manufacturing vol 17 no 4 pp 493ndash507 2006

[13] M Saidi-Mehrabad andP Fattahi ldquoFlexible job shop schedulingwith tabu search algorithmsrdquo The International Journal ofAdvanced Manufacturing Technology vol 32 no 5-6 pp 563ndash570 2007

[14] L Gao C Y Peng C Zhou and P G Li ldquoSolving flexiblejob shop scheduling problem using general particle swarm

optimizationrdquo in Proceedings of the 36th CIE Conference onComputers amp Industrial Engineering pp 3018ndash3027 TaipeiTaiwan 2006

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

[16] 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

[17] L-N Xing Y-W Chen P Wang Q-S Zhao and J Xiong ldquoAknowledge-based ant colony optimization for flexible job shopscheduling problemsrdquoApplied Soft Computing vol 10 no 3 pp888ndash896 2010

[18] LWang G Zhou Y Xu SWang andM Liu ldquoAn effective arti-ficial bee colony algorithm for the flexible job-shop schedulingproblemrdquoThe International Journal of AdvancedManufacturingTechnology vol 56 no 1 pp 1ndash8 2012

[19] K Z Gao P N Suganthan Q K Pan andM F Tasgetiren ldquoAneffective discrete harmony search algorithm for flexible job shopscheduling problem with fuzzy processing timerdquo InternationalJournal of Production Research vol 53 no 19 pp 5896ndash59112015

[20] M Rabiee M Zandieh and P Ramezani ldquoBi-objective partialflexible job shop scheduling problemNSGA-II NRGAMOGAand PAES approachesrdquo International Journal of ProductionResearch vol 50 no 24 pp 7327ndash7342 2012

[21] J D Schaffer ldquoMultiple objective optimization with vector eval-uated genetic algorithmsrdquo in Proceedings of the 1st InternationalConference on Genetic Algorithms and Their Applications pp93ndash100 Lawrence ErlbaumAssociates Hillsdale NJ USA 1985

[22] B Jurisch ldquoScheduling jobs in shops with multi-purposemachinesrdquo in Software Technologies for Embedded and Ubiqui-tous Systems pp 114ndash125 Springer Berlin Germany 1992

[23] H Liu A Abraham O Choi and S H Moon ldquoVariableneighborhood particle swarm optimization for multi-objectiveflexible job-shop scheduling problemsrdquo in SEAL 2006 vol 4247of Lecture Notes in Computer Science pp 197ndash204 SpringerBerlin Germany 2006

[24] 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

[25] 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

[26] L-N Xing Y-W Chen and K-W Yang ldquoAn efficient searchmethod for multi-objective flexible job shop scheduling prob-lemsrdquo Journal of Intelligent Manufacturing vol 20 no 3 pp283ndash293 2009

[27] I Gonzalez-Rodrıguez C R Vela and J Puente ldquoA geneticsolution based on lexicographical goal programming for amultiobjective job shop with uncertaintyrdquo Journal of IntelligentManufacturing vol 21 no 1 pp 65ndash73 2010

[28] SHA RahmatiM Zandieh andMYazdani ldquoDeveloping twomulti-objective evolutionary algorithms for the multi-objectiveflexible job shop scheduling problemrdquoThe International Journalof Advanced Manufacturing Technology vol 64 no 5ndash8 pp915ndash932 2013


[29] Y Yuan and H Xu ldquoMultiobjective flexible job shop schedulingusing memetic algorithmsrdquo IEEE Transactions on AutomationScience and Engineering vol 12 no 1 pp 336ndash353 2015

[30] N Srinivas and K Deb ldquoMuiltiobjective optimization usingnondominated sorting in genetic algorithmsrdquo EvolutionaryComputation vol 2 no 3 pp 221ndash248 1994

[31] I Kacem S Hammadi and P Borne ldquoPareto-optimalityapproach for flexible job-shop scheduling problems hybridiza-tion of evolutionary algorithms and fuzzy logicrdquo Mathematicsand Computers in Simulation vol 60 no 3ndash5 pp 245ndash2762002

[32] N B Ho and J C Tay ldquoSolving multiple-objective flexible jobshop problems by evolution and local searchrdquo IEEE Transac-tions on Systems Man and Cybernetics Part C Applications andReviews vol 38 no 5 pp 674ndash685 2008

[33] K Deb A Pratap S Agarwal and T Meyarivan ldquoA fastand elitist multiobjective genetic algorithm NSGA-IIrdquo IEEETransactions on Evolutionary Computation vol 6 no 2 pp 182ndash197 2002

[34] X Wang L Gao C Zhang and X Shao ldquoA multi-objectivegenetic algorithm based on immune and entropy principle forflexible job-shop scheduling problemrdquo International Journal ofAdvanced Manufacturing Technology vol 51 no 5-8 pp 757ndash767 2010

[35] M Frutos A C Olivera and F Tohme ldquoA memetic algorithmbased on a NSGAII scheme for the flexible job-shop schedulingproblemrdquo Annals of Operations Research vol 181 pp 745ndash7652010

[36] L Wang S 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

[37] M Rohaninejad A Kheirkhah P Fattahi and B Vahedi-Nouri ldquoA hybridmulti-objective genetic algorithmbased on theELECTREmethod for a capacitated flexible job shop schedulingproblemrdquoThe International Journal of AdvancedManufacturingTechnology vol 77 no 1ndash4 pp 51ndash66 2015

[38] V Kaplanoglu ldquoAn object-oriented approach for multi-objective flexible job-shop scheduling problemrdquo Expert Systemswith Applications vol 45 pp 71ndash84 2016

[39] I Kacem S Hammadi and P Borne ldquoApproach by localizationand multiobjective evolutionary optimization for flexible job-shop scheduling problemsrdquo IEEE Transactions on Systems Manamp Cybernetics Part C Applications amp Reviews vol 32 no 1 pp1ndash13 2002

[40] R Cheng M Gen and Y Tsujimura ldquoA tutorial survey ofjob-shop scheduling problems using genetic algorithmsmdashIRepresentationrdquo Computers amp Industrial Engineering vol 30no 4 pp 983ndash997 1996

[41] W Xia and Z Wu ldquoAn effective hybrid optimization approachfor multi-objective flexible job-shop scheduling problemsrdquoComputers amp Industrial Engineering vol 48 no 2 pp 409ndash4252005

[42] X Shao W Liu Q Liu and C Zhang ldquoHybrid discrete particleswarm optimization formulti-objective flexible job-shop sched-uling problemrdquo International Journal of Advanced Manufactur-ing Technology vol 67 no 9ndash12 pp 2885ndash2901 2013

[43] J W Barnes and J B Chambers ldquoFlexible job shop schedulingby tabu search Graduate program in operations research andindustrial engineeringrdquo Tech Rep University of Texas AustinTex USA 1996

[44] J J Palacios I Gonzalez-Rodrıguez C R Vela and J PuenteldquoCoevolutionary makespan optimisation through differentranking methods for the fuzzy flexible job shoprdquo Fuzzy Sets andSystems vol 278 pp 81ndash97 2015

[45] X Li andLGao ldquoAn effective hybrid genetic algorithmand tabusearch for flexible job shop scheduling problemrdquo InternationalJournal of Production Economics vol 174 pp 93ndash110 2016

Submit your manuscripts athttpswwwhindawicom

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Applied Computational Intelligence and Soft Computing

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Human-ComputerInteraction

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algorithm based on the application of multiple independentsearches which increased the exploration of search spaceXing et al [17] put forth a knowledge-based ant colonyoptimization algorithm Recently a novel artificial bee colony(ABC) algorithm [18] and a discrete harmony search (DHS)algorithm [19] were brought forward to solve the FJSP

As is shown above the single-objective optimization ofFJSP (SO-FJSP) has been extensively studied which generallyminimizes themakespan that is the time required to completeall jobs However many industries (eg aircraft semicon-ductors manufacturing and electronics) have trade-offs intheir scheduling problems in which multiple objectives needto be considered to optimize the overall performance of thesystemTherefore theMO-FJSPmay be closer to the realisticproduction environments and needs to be further studiedIn recent years the MO-FJSP has captured more and moreinterests of numerous domain researchers and a great manyof algorithms have been presented [20] Compared with SO-FJSP MO-FJSP has two problems to be dealt with incom-mensurability between objectives and contradiction betweenobjectives (ie optimizing a single-objective generally resultsin deterioration of another objective) [13]

For the MO-FJSP Schaffer [21] provided a genetic algo-rithm with vector evaluated Jurisch [22] presented a branch-and-bound algorithm and some heuristic algorithms Bycombining the VNSwith particle swarm optimization (PSO)Liu et al [23] presented a hybrid metaheuristic to solve theMO-FJSP A new genetic algorithm (GA) which hybridizedwith a bottleneck shifting procedure was developed by Gao etal [24] to solve theMO-FJSP Zhang et al [25] embedded tabusearch (TS) in PSO as a local search to deal with theMO-FJSPXing et al [26] advanced an efficient search method for theMO-FJSP Gonzalez-Rodrıguez et al [27] proposed a genericmultiobjective model which was based on lexicographicalminimization of expected values for FJSP By introducing sev-eral metrics of the multiobjective evaluation in the MO-FJSPliterature Rahmati et al [28] adopted two multiobjectiveevolutionary algorithms for the MO-FJSP

Recently some studies based on the Pareto dominancerelation have been used to solve the MO-FJSP and they aremore desirablewhen comparedwith the prior linearweightedsummation ones [29] The nondominated sorting geneticalgorithm (NSGA) [30] was one of the first methods used tosolve themultiobjective problemKacemet al [31] brought upa Pareto approach based on the hybridization of fuzzy logic(FL) and evolutionary algorithms to solve the MO-FJSP Hoand Tay [32] integrated a guiding local search procedure andan elitism memory mechanism into the evolutionary algori-thm to solve the MO-FJSP

Deb et al [33] came up with a nondominated sorting-basedmultiobjective evolutionary algorithm (MOEA) callednondominated sorting genetic algorithm II (NSGA-II)Wanget al [34] proposed a multiobjective GA based on immuneand entropy principle for the MO-FJSP By employing simu-lated annealing (SA) algorithm as a local search process Fru-tos et al [35] introduced amemetic algorithm (MA) based onthe NSGA-II Wang et al [36] presented an effective Pareto-based estimation of distribution algorithm (P-EDA) inwhichvarious strategies are integrated to maintain quality and

diversity of solutions Rohaninejad et al [37] advanced anMO-FJSP problem with machines capacity constraints tominimize the makespan and overtime costs of machinesKaplanoglu [38] put forth an object-oriented (OO) approachcombined with SA optimization algorithm to solve the MO-FJSP in which a two-string encoding scheme was used toexpress this problem

Our reviewof the above literatures reveals that theNSGA-II algorithm has been widely used to solve the MO-FJSP forits advantages such as high efficiency to optimize the complexproblems and the ability to gain widespread Pareto-optimalsolutions And the algorithms with a two-stage optimizationscheme have been also widely studied to solve the MO-FJSP for it could fully tap the optimization potentials ofvarious metaheuristic algorithms However we found thatthe NSGA-II algorithm has the disadvantage of prematureconvergence to local solution and the algorithms with a two-stage optimization scheme have the disadvantages of beingunable to gain stable and high quality initial population in thefirst stageHence in this paper we propose a bee evolutionaryguidingNSGA-II (BEG-NSGA-II)with a two-stage optimiza-tion scheme to solve the MO-FJSP which aims to fully playthe respective advantages of NSGA-II algorithm and thealgorithms with a two-stage optimization scheme and toovercome the disadvantages of them In the first stage theNSGA-II algorithm with 119879 iteration times is first used toobtain the initial population119873 which consists of three partschanging with the iteration times In order to extensivelyexploit the solution space an effective local search operator isinvented in this stage In the second stage the NSGA-II algo-rithm with GEN iteration times is used to obtain the Pareto-optimal solutions In order to enhance the searching abilityand avoid the premature convergence an updating mecha-nism and some useful genetic operators were employed inthis stage Four famous benchmarks that include 53 openproblems of FJSP are chosen to estimate the performance ofthe proposed algorithm Moreover by comparing the resultsof our algorithm and some existing well-known algorithmsthe virtues of our algorithm can be clearly demonstrated

The rest of this paper is organized as follows The defini-tion and formalization of the MO-FJSP are given in thenext section In Section 3 NSGA-II is briefly introduced andthen the overview and implementation details of the pro-posed BEG-NSGA-II are presented respectively AfterwardsExperimental Studies and Discussions are made in Section 4Finally Conclusions and Future Studies are described inSection 5

2 Problem Definition

The FJSP is commonly defined as follows There is a set of 119899jobs (119869119894 119894 isin 1 2 119899) and a set of 119898 machines (119872119896 119896 isin1 2 119898) One or more operation(s) (119874119894119895 119895 isin 1 2 119899119894 119899119894 is the total number of operations for job 119869119894) isareallowed to be processed by one machine of 119872119894119895 whichconsists of a set of machines of the 119895th operations for job 119869119894119875119894119895119896 denotes the processing time of the 119895th operation for job 119869119894which is processed bymachine 119896 Generally the FJSP consistsof two subproblems the routing subproblem of assigning



1198721 1198722 1198723 1198724 1198725

Job 111987411 2 5 4 1 211987412 5 4 5 7 511987413 4 5 5 4 5

Job 211987421 2 5 4 7 811987422 5 6 9 8 511987423 4 5 4 54 5

Job 3

11987431 9 8 6 7 911987432 6 1 2 5 411987433 2 5 4 2 411987434 4 5 2 1 5

Job 4 11987441 1 5 2 4 1211987442 5 1 2 1 2















min 1198911 = max1le119896le119898

(119862119896)

min 1198912 = max1le119896le119898

(119882119896)

min 1198913 =119898

sum119896=1

119882119896

(1)




sum119896isin119872119894119895

119909119894119895119896 = 1 (4)

119909119894119895119896 =


(5)






32 The BEG-NSGA-II


The First Stage










The Second Stage









At the first stage

120572 = 119905119879

119875 = 120572 lowast 1198732

119877 = 119873 (1 minus 120572)2

(6)

At the second stage

1205721015840 = 119905119866119864119873 (7)

1198771015840 =119873(1 minus 1205721015840)

2 (8)

1205941015840 + 119878 + 05119873 gt= 119873 (9)

119863(119894 119895) =119871

sum119897=1


10038161003816100381610038161003816119871 (10)





Initial

No

Yes


objective 1










Swampingmutation



eliteindividuals


Randomly

individuals

NoYes



Output result

YesNo

NoYes



objective 2


objective 3

Parentqueen 3

Offspringqueen 3


Parentqueen 2

Offspringqueen 2


Parentqueen 1

Offspringqueen 1




NoYes

The first stage

The second stage

t = t + 1


Offspring rank = 1

D(i j) gt u

t = t + 1

t = 1

t gt T

Pm lt= 01

Remain S




3 1 2 3 1 2 4 3 4 1 3 2


4 3 5 2 4 5 1 1 3 4 2 5

O31 O11 O21 O32 O33 O34O13 O23O12 O22 O41 O42

J3 J4J2J1

(a) Chromosome

5 10 15 20 250

6

10

13

9

1

5

Time

1012 17 22

20

15 19

M1

M2

M3

M4

M5

O31

O11

O21

O32

O33

O34

O13 O23

O12

O22

O41

O42




























3 1 2 4 1 2 4 3 3 1 3 2

3 1 2 4 1 3 2 1 2 4 3 3

1 3 2 1 2 2 4 3 4 1 3 3

1 3 2 1 3 1 2 4 2 4 3 3

3 1 2 4 1 2 4 3 3 1 3 2

O1

O2

P1

P1

P2


3 1 2 4 1 2 4 3 3 1 3 2

1 2 4 1 3 2 1 2 4 3 3 3

1 3 2 1 2 2 4 3 4 1 3 3

3 2 1 3 1 2 4 1 2 4 3 3

3 1 2 4 1 2 4 3 3 1 3 2

O1

O2

P1

P1

P2










4 3 5 2 4 5 1 1 3 4 2 5

4 3 5 1 2 5 1 1 3 4 2 5

3 4 5 1 2 4 3 2 5 1 2 3

3 4 5 2 4 4 3 2 5 1 2 3

4 3 5 2 4 5 1 1 3 4 2 5

O1

O2

P1

P1

P2

J3 J4J2J1













3 1 2 4 1 2 4 3 3 1 3 2

3 1 2 2 1 2 4 3 3 1 3 4

1 3 2 1 2 2 4 3 4 1 3 3

1 3 2 1 4 2 4 3 2 1 3 3

3 1 2 4 1 2 4 3 3 1 3 2


O1

O2

P1

P1

P2


3 1 2 4 1 2 4 3 3 1 3 2

3 1 2 2 1 2 4 3 3 1 3 4

1 3 2 1 2 2 4 3 4 1 3 3

3 1 2 1 2 2 4 3 4 3 1 3

3 1 2 4 1 2 4 3 3 1 3 2


O1

O2

P1

P1

P2












3 1 2 4 1 2 4 3 3 1 3 2

3 1 2 4 3 2 4 1 3 1 3 2O1

P1


3 1 2 4 1 2 4 3 3 1 3 2

3 1 2 1 4 2 4 3 3 1 3 2O1

P1












3 1 2 4 1 2 4 3 3 1 3 2

3 1 2 3 3 2 4 4 1 1 3 2O1

P1


4 3 5 2 4 5 1 1 3 4 2 5

2 3 3 2 5 5 1 3 3 4 1 4

J3 J4J2J1

O1

P1








Table3Re

sults

onEx

perim

ent1

Prob

lem

119899times119898

ALa

PSO+SA

bMOGA

chP

SOd

hGA

eOOapproach

fBE

G-N

SGA-

IIg

119891 1119891 2

119891 3T

119891 1119891 2

119891 3T

119891 1119891 2

119891 3T

119891 1119891 2

119891 3T

119891 1119891 2

119891 3T

119891 1119891 2

119891 3T

119891 1119891 2

119891 3T

4times5

mdashmdash

mdashmdash

mdashmdash

mdashmdash

1110

32010

mdashmdash

mdashmdash

mdashmdash

mdashmdash

1110

32003

1110

32010

119

3411

934

128

3212

832

137

33

8times8

1613

75mdash

1512

75mdash

1511

81016

1412

77014

1512

75001

1613

73004

1412

77012

1613

7315

1275

1512

7515

1275

1512

7516

1373

1611

7716

1177

1613

7316

1373

10times10

75

45mdash

76

44mdash

85

42024

75

43020

75

43001

87

41005

75

43018

85

427

642

76

428

542

76

428

741

87

418

542

77

438

542

75

458

741

87

41

15times10

2311

95mdash

1211

91mdash

1111

9114

611

1191

136

1111

91003

1313

91009

1111

91005

2411

9112

1095

1210

9314

1291

1210

9311

1098

1110

9511

1095

a Thec

ompu

tatio

nconfi

guratio

nisno

tlisted

inAL

b Thec

ompu

tatio

nconfi

guratio

nisno

tlisted

inPS

O+SA

c Th

eCPU

timeo

naP

Cwith

2GHzC

PUand2G

Bof

RAM

mem

oryin

C++

d TheC

PUtim

eonap

ersonalcom

puterw

ith2G

HzC

PUand2G

Bof

RAM

mem

oryin

C++

e TheC

PUtim

eona3

0GHzP

entiu

min

Delp

hi

f TheC

PUtim

eonan

IntelC

orei5267

GHzp

rocessor

with

4GBRA

Mof

mem

oryin

Java

g TheC

PUtim

eonan

IntelC

orei3267

GHzp

rocessor

with

4GBRA

Min

MAT

LABR2

014a


O11

O12

O13

O21

O22

O23

O31

O32

O33 O34

O41

O42

13

1 2 3 4 5 6 7 8 9 10 11 12 130

M1

M2

M3

M4

M5


2634

130 260 390 520 650 780 910 1040 1170 1300 1430 1560 1690 1820 1950 2080 2210 2340 2470 26000

M1

M2

M3

M4

M5

M10

M6

M7

M8

M9









1198911 1198912 1198913 119879 1198911 1198912 1198913 119879

01a (10 times 5)2568 2505 11137 204 2568 2505 11137 1912572 2568 11137 2572 2568 111372594 2554 11137 2594 2554 11137

02a (10 times 5)2289 2263 11137 256 2487 2237 11137 2132313 2238 11137 3014 2234 11137

2564 2236 11137

03a (10 times 5) 2287 2248 11137 290 2492 2247 11137 252256 2252 11137 2516 2234 11137

04a (10 times 5)

2550 2503 11090 207 2882 2572 11069 2032569 2565 11076 2912 2552 110702579 2552 11080 2904 2626 110673095 2727 11064 3019 2665 11066

2997 2523 110732933 2688 110652953 2668 110662938 2606 110682955 2503 11074

05a (10 times 5)

2292 2252 11077 237 2740 2250 10981 2122293 2242 11091 2846 2212 109862297 2255 11054 2747 2229 109882315 2272 11063 2700 2255 109772343 2298 11050 2915 2231 109812358 2322 11038 2818 2208 109902376 2243 11022 2787 2350 109702904 2620 10941 2759 2320 109712945 2571 10941 2667 2237 109833056 2507 10941 2686 2277 10974

2641 2332 109742687 2221 110002779 2237 109812707 2299 109722702 2233 10989

06a (10 times 5)

2250 2233 11009 309 2673 2194 10893 2862254 2223 10994 2716 2194 108912398 2219 10973 2683 2245 108902437 2280 10988 2666 2248 108932744 2448 10850 2679 2229 108922902 2439 10847 2816 2186 109032967 2840 10839 2898 2186 10899

2713 2234 108892758 2209 108902722 2229 10890

07a (15 times 8)

2450 2413 16485 763 2927 2187 16485 6212457 2299 16485 2910 2213 164852484 2289 16485 2889 2264 16485

2891 2219 164852922 2190 16485

08a (15 times 8)2187 2102 16485 827 2621 2091 16485 6352171 2104 16485 2770 2089 16485

2780 2080 16485


Table 4 Continued


1198911 1198912 1198913 119879 1198911 1198912 1198913 119879

09a (15 times 8)2157 2113 16485 1016 2442 2103 16485 7262144 2119 16485 2470 2074 164852158 2102 16485 2455 2081 16485

10a (15 times 8)

2461 2433 16505 755 2998 2317 16487 6432470 2310 16537 3036 2363 164822478 2330 16533 3091 2491 164732482 2360 16499 2984 2319 164862501 2265 16547 3052 2377 164772501 2312 16528 3098 2491 164722547 2476 16490 3013 2293 164873064 2734 16464 3136 2457 16471

3025 2377 164803041 2351 164812980 2258 164943021 2319 16484

11a (15 times 8)

2182 2170 16449 1014 2551 2116 16229 7252202 2114 16476 2599 2131 162242210 2113 16442 2612 2146 162222337 2185 16377 2573 2058 162312874 2389 16247 2753 2186 162172894 2330 16247 2628 2102 162242962 2312 16247 2702 2098 16223

2658 2089 162242555 2111 162292664 2063 16228

12a (15 times 8)

2161 2107 16295 1192 2624 2049 16055 8222168 2130 16220 2620 2022 160682191 2084 16355 2587 2035 160652210 2103 16331 2614 2057 160512315 2125 16292 2585 2043 160612366 2105 16237 2659 2038 160552493 2297 16124 2617 2024 160652631 2309 16112 2575 2057 160572637 2303 16113 2587 2050 160592683 2397 16104 2582 2098 16052

13a (20 times 10)

2595 2036 160592566 2044 16061

2408 2326 21610 2399 2894 2277 21610 10272928 2233 216102896 2244 216102877 2282 21610

14a (20 times 10)2340 2251 21610 2905 2649 2186 21610 12392334 2258 21610 2664 2183 21610

2641 2207 216102738 2182 21610

15a (20 times 10)

2285 2247 21610 3329 2627 2216 21610 13602287 2218 21610 2681 2196 21610

2645 2215 216102652 2201 216102668 2197 216102682 2178 21610


Table 4 Continued


1198911 1198912 1198913 119879 1198911 1198912 1198913 119879

16a (20 times 10)

2447 2354 21602 2152 2965 2279 21534 11482450 2380 21590 3158 2303 215042487 2454 21584 3184 2400 214902492 2417 21576 3039 2279 215232540 2396 21547 3050 2318 215072550 2492 21545 3071 2286 215132568 2428 21540 3275 2281 215113013 2588 21478 3105 2352 214963106 2548 21478 3175 2344 21497

3180 2293 215073130 2312 215033034 2352 21503

17a (20 times 10)

2322 2240 21433 2847 2806 2195 21096 13922322 2280 21362 2750 2165 211202323 2238 21454 2791 2141 211142343 2224 21420 2716 2157 211292480 2285 21344 2849 2148 211072528 2231 21313 2769 2166 211112789 2448 21198 2856 2183 210982808 2303 21200 2773 2145 211112816 2370 21197 2856 2142 21105

2803 2203 210942759 2186 211062789 2176 211012772 2202 21101

18a (20 times 10)

2267 2235 21483 3301 2634 2156 21005 16232269 2206 21408 2642 2126 210062320 2208 21354 2638 2185 209992437 2221 21311 2638 2148 210022531 2310 21285 2674 2148 210002545 2305 21282 2644 2145 21002

2651 2140 210052641 2157 210002664 2145 210012621 2181 210022659 2140 210032661 2137 210042651 2181 209992669 2126 210052708 2125 21009







1198911 1198912 1198913 119879 1198911 1198912 1198913 119879 1198911 1198912 1198913 119879

MK01 (10 times 6)

42 42 162 478 42 39 158 049 42 39 158 03644 40 154 44 40 15443 40 155 43 40 15540 36 169 40 36 169

47 36 16747 37 16549 37 16450 38 162

MK02 (10 times 6)

28 28 155 302 26 26 151 075 26 26 151 07027 27 146 27 27 14629 27 145 29 27 14529 29 143 29 29 14331 31 141 31 31 14133 33 140 33 33 140

MK03 (15 times 8)

204 204 852 2614 204 199 855 475 204 199 855 366204 144 871 204 144 871204 135 882 204 135 882204 133 884 204 133 884213 199 850 213 199 850214 210 849 214 210 849221 199 847 221 199 847222 199 848 222 199 848231 188 848 231 188 848230 177 848 230 177 848

MK04 (15 times 8)

68 67 352 1774 66 63 345 176 66 63 345 16365 63 362 65 63 36263 61 371 63 61 37162 61 373 60 59 39061 60 382 74 54 34960 59 390 74 55 34873 55 350 72 72 34274 54 349 78 78 33974 55 348 72 66 34890 76 331 73 72 343

MK05 (15 times 4)

177 177 702 826 173 173 683 234 173 173 683 196175 175 682 175 175 682183 183 677 183 183 677185 185 676 179 179 679179 179 679 199 199 674

193 193 675183 183 677

MK06 (10 times 15)

67 431 1879 62 55 424 193 62 55 424 18565 54 417 65 54 41760 58 441 60 58 44162 60 440 62 60 44076 60 362 76 60 36276 74 356 76 74 35678 60 361 78 60 36173 72 360 73 72 36072 72 361 72 72 361100 90 330 100 90 330


Table 5 Continued


1198911 1198912 1198913 119879 1198911 1198912 1198913 119879 1198911 1198912 1198913 119879

MK07 (20 times 5)

150 150 717 568 139 139 693 492 139 139 693 312140 138 686 140 138 686144 144 673 144 144 673151 151 667 151 151 667157 157 662 157 157 662162 162 659 162 162 659166 166 657 166 166 657

MK08 (20 times 10)

523 523 2524 6767 523 515 2524 1204 523 515 2524 834523 497 2534 523 497 2534524 524 2519 524 524 2519578 578 2489 578 578 2489587 587 2484 587 587 2484

541 533 2516542 542 2511560 560 2505

MK09 (20 times 10)

311 299 2374 7776 311 299 2290 1948 311 299 2290 1015310 299 3514 310 299 3514311 301 2287 311 301 2287314 299 2315 314 299 2315315 299 2283 315 299 2283332 302 2265 332 302 2265329 301 2266 329 301 2266328 308 2259 328 308 2259325 299 2275 325 299 2275

MK10 (20 times 15)

227 221 1989 12252 224 219 1980 1787 224 219 1980 1238225 211 1976 235 225 1895233 214 1919 240 215 1905235 218 1897 246 215 1896235 225 1895 252 224 1884240 215 1905 256 211 1919240 216 1888 260 244 1869242 214 1913 266 254 1864246 215 1896 246 206 1938252 224 1884 246 204 1948256 211 1919 246 205 1940260 244 1869 251 202 1948266 254 1864 253 207 1930268 264 1858 249 203 1947276 256 1857 255 202 1943281 268 1854 260 210 1926217 207 2064 250 205 1935214 204 2082 254 203 1942






25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400 425 450 475 500 5250

M1

M2

M3

M4

M5

M10

M6

M7

M8

M9

541



127060 120 180 240 300 360 420 480 540 600 660 720 780 840 900 960 1020 1080 1140 1200 12600

M1

M2

M3

M4

M5

M10

M11

M12

M13

M14

M15

M16

M17

M18

M6

M7

M8

M9








Notations





Acknowledgments


References























































Advances in

FuzzySystems


Volume 2014












Journal of



Advances in

Multimedia


Biomedical Imaging


Advances in


RoboticsJournal of










Advances in





1198721 1198722 1198723 1198724 1198725

Job 111987411 2 5 4 1 211987412 5 4 5 7 511987413 4 5 5 4 5

Job 211987421 2 5 4 7 811987422 5 6 9 8 511987423 4 5 4 54 5

Job 3

11987431 9 8 6 7 911987432 6 1 2 5 411987433 2 5 4 2 411987434 4 5 2 1 5

Job 4 11987441 1 5 2 4 1211987442 5 1 2 1 2















min 1198911 = max1le119896le119898

(119862119896)

min 1198912 = max1le119896le119898

(119882119896)

min 1198913 =119898

sum119896=1

119882119896

(1)




sum119896isin119872119894119895

119909119894119895119896 = 1 (4)

119909119894119895119896 =


(5)






32 The BEG-NSGA-II


The First Stage










The Second Stage









At the first stage

120572 = 119905119879

119875 = 120572 lowast 1198732

119877 = 119873 (1 minus 120572)2

(6)

At the second stage

1205721015840 = 119905119866119864119873 (7)

1198771015840 =119873(1 minus 1205721015840)

2 (8)

1205941015840 + 119878 + 05119873 gt= 119873 (9)

119863(119894 119895) =119871

sum119897=1


10038161003816100381610038161003816119871 (10)





Initial

No

Yes


objective 1










Swampingmutation



eliteindividuals


Randomly

individuals

NoYes



Output result

YesNo

NoYes



objective 2


objective 3

Parentqueen 3

Offspringqueen 3


Parentqueen 2

Offspringqueen 2


Parentqueen 1

Offspringqueen 1




NoYes

The first stage

The second stage

t = t + 1


Offspring rank = 1

D(i j) gt u

t = t + 1

t = 1

t gt T

Pm lt= 01

Remain S




3 1 2 3 1 2 4 3 4 1 3 2


4 3 5 2 4 5 1 1 3 4 2 5

O31 O11 O21 O32 O33 O34O13 O23O12 O22 O41 O42

J3 J4J2J1

(a) Chromosome

5 10 15 20 250

6

10

13

9

1

5

Time

1012 17 22

20

15 19

M1

M2

M3

M4

M5

O31

O11

O21

O32

O33

O34

O13 O23

O12

O22

O41

O42




























3 1 2 4 1 2 4 3 3 1 3 2

3 1 2 4 1 3 2 1 2 4 3 3

1 3 2 1 2 2 4 3 4 1 3 3

1 3 2 1 3 1 2 4 2 4 3 3

3 1 2 4 1 2 4 3 3 1 3 2

O1

O2

P1

P1

P2


3 1 2 4 1 2 4 3 3 1 3 2

1 2 4 1 3 2 1 2 4 3 3 3

1 3 2 1 2 2 4 3 4 1 3 3

3 2 1 3 1 2 4 1 2 4 3 3

3 1 2 4 1 2 4 3 3 1 3 2

O1

O2

P1

P1

P2










4 3 5 2 4 5 1 1 3 4 2 5

4 3 5 1 2 5 1 1 3 4 2 5

3 4 5 1 2 4 3 2 5 1 2 3

3 4 5 2 4 4 3 2 5 1 2 3

4 3 5 2 4 5 1 1 3 4 2 5

O1

O2

P1

P1

P2

J3 J4J2J1













3 1 2 4 1 2 4 3 3 1 3 2

3 1 2 2 1 2 4 3 3 1 3 4

1 3 2 1 2 2 4 3 4 1 3 3

1 3 2 1 4 2 4 3 2 1 3 3

3 1 2 4 1 2 4 3 3 1 3 2


O1

O2

P1

P1

P2


3 1 2 4 1 2 4 3 3 1 3 2

3 1 2 2 1 2 4 3 3 1 3 4

1 3 2 1 2 2 4 3 4 1 3 3

3 1 2 1 2 2 4 3 4 3 1 3

3 1 2 4 1 2 4 3 3 1 3 2


O1

O2

P1

P1

P2












3 1 2 4 1 2 4 3 3 1 3 2

3 1 2 4 3 2 4 1 3 1 3 2O1

P1


3 1 2 4 1 2 4 3 3 1 3 2

3 1 2 1 4 2 4 3 3 1 3 2O1

P1












3 1 2 4 1 2 4 3 3 1 3 2

3 1 2 3 3 2 4 4 1 1 3 2O1

P1


4 3 5 2 4 5 1 1 3 4 2 5

2 3 3 2 5 5 1 3 3 4 1 4

J3 J4J2J1

O1

P1








Table3Re

sults

onEx

perim

ent1

Prob

lem

119899times119898

ALa

PSO+SA

bMOGA

chP

SOd

hGA

eOOapproach

fBE

G-N

SGA-

IIg

119891 1119891 2

119891 3T

119891 1119891 2

119891 3T

119891 1119891 2

119891 3T

119891 1119891 2

119891 3T

119891 1119891 2

119891 3T

119891 1119891 2

119891 3T

119891 1119891 2

119891 3T

4times5

mdashmdash

mdashmdash

mdashmdash

mdashmdash

1110

32010

mdashmdash

mdashmdash

mdashmdash

mdashmdash

1110

32003

1110

32010

119

3411

934

128

3212

832

137

33

8times8

1613

75mdash

1512

75mdash

1511

81016

1412

77014

1512

75001

1613

73004

1412

77012

1613

7315

1275

1512

7515

1275

1512

7516

1373

1611

7716

1177

1613

7316

1373

10times10

75

45mdash

76

44mdash

85

42024

75

43020

75

43001

87

41005

75

43018

85

427

642

76

428

542

76

428

741

87

418

542

77

438

542

75

458

741

87

41

15times10

2311

95mdash

1211

91mdash

1111

9114

611

1191

136

1111

91003

1313

91009

1111

91005

2411

9112

1095

1210

9314

1291

1210

9311

1098

1110

9511

1095

a Thec

ompu

tatio

nconfi

guratio

nisno

tlisted

inAL

b Thec

ompu

tatio

nconfi

guratio

nisno

tlisted

inPS

O+SA

c Th

eCPU

timeo

naP

Cwith

2GHzC

PUand2G

Bof

RAM

mem

oryin

C++

d TheC

PUtim

eonap

ersonalcom

puterw

ith2G

HzC

PUand2G

Bof

RAM

mem

oryin

C++

e TheC

PUtim

eona3

0GHzP

entiu

min

Delp

hi

f TheC

PUtim

eonan

IntelC

orei5267

GHzp

rocessor

with

4GBRA

Mof

mem

oryin

Java

g TheC

PUtim

eonan

IntelC

orei3267

GHzp

rocessor

with

4GBRA

Min

MAT

LABR2

014a


O11

O12

O13

O21

O22

O23

O31

O32

O33 O34

O41

O42

13

1 2 3 4 5 6 7 8 9 10 11 12 130

M1

M2

M3

M4

M5


2634

130 260 390 520 650 780 910 1040 1170 1300 1430 1560 1690 1820 1950 2080 2210 2340 2470 26000

M1

M2

M3

M4

M5

M10

M6

M7

M8

M9









1198911 1198912 1198913 119879 1198911 1198912 1198913 119879

01a (10 times 5)2568 2505 11137 204 2568 2505 11137 1912572 2568 11137 2572 2568 111372594 2554 11137 2594 2554 11137

02a (10 times 5)2289 2263 11137 256 2487 2237 11137 2132313 2238 11137 3014 2234 11137

2564 2236 11137

03a (10 times 5) 2287 2248 11137 290 2492 2247 11137 252256 2252 11137 2516 2234 11137

04a (10 times 5)

2550 2503 11090 207 2882 2572 11069 2032569 2565 11076 2912 2552 110702579 2552 11080 2904 2626 110673095 2727 11064 3019 2665 11066

2997 2523 110732933 2688 110652953 2668 110662938 2606 110682955 2503 11074

05a (10 times 5)

2292 2252 11077 237 2740 2250 10981 2122293 2242 11091 2846 2212 109862297 2255 11054 2747 2229 109882315 2272 11063 2700 2255 109772343 2298 11050 2915 2231 109812358 2322 11038 2818 2208 109902376 2243 11022 2787 2350 109702904 2620 10941 2759 2320 109712945 2571 10941 2667 2237 109833056 2507 10941 2686 2277 10974

2641 2332 109742687 2221 110002779 2237 109812707 2299 109722702 2233 10989

06a (10 times 5)

2250 2233 11009 309 2673 2194 10893 2862254 2223 10994 2716 2194 108912398 2219 10973 2683 2245 108902437 2280 10988 2666 2248 108932744 2448 10850 2679 2229 108922902 2439 10847 2816 2186 109032967 2840 10839 2898 2186 10899

2713 2234 108892758 2209 108902722 2229 10890

07a (15 times 8)

2450 2413 16485 763 2927 2187 16485 6212457 2299 16485 2910 2213 164852484 2289 16485 2889 2264 16485

2891 2219 164852922 2190 16485

08a (15 times 8)2187 2102 16485 827 2621 2091 16485 6352171 2104 16485 2770 2089 16485

2780 2080 16485


Table 4 Continued


1198911 1198912 1198913 119879 1198911 1198912 1198913 119879

09a (15 times 8)2157 2113 16485 1016 2442 2103 16485 7262144 2119 16485 2470 2074 164852158 2102 16485 2455 2081 16485

10a (15 times 8)

2461 2433 16505 755 2998 2317 16487 6432470 2310 16537 3036 2363 164822478 2330 16533 3091 2491 164732482 2360 16499 2984 2319 164862501 2265 16547 3052 2377 164772501 2312 16528 3098 2491 164722547 2476 16490 3013 2293 164873064 2734 16464 3136 2457 16471

3025 2377 164803041 2351 164812980 2258 164943021 2319 16484

11a (15 times 8)

2182 2170 16449 1014 2551 2116 16229 7252202 2114 16476 2599 2131 162242210 2113 16442 2612 2146 162222337 2185 16377 2573 2058 162312874 2389 16247 2753 2186 162172894 2330 16247 2628 2102 162242962 2312 16247 2702 2098 16223

2658 2089 162242555 2111 162292664 2063 16228

12a (15 times 8)

2161 2107 16295 1192 2624 2049 16055 8222168 2130 16220 2620 2022 160682191 2084 16355 2587 2035 160652210 2103 16331 2614 2057 160512315 2125 16292 2585 2043 160612366 2105 16237 2659 2038 160552493 2297 16124 2617 2024 160652631 2309 16112 2575 2057 160572637 2303 16113 2587 2050 160592683 2397 16104 2582 2098 16052

13a (20 times 10)

2595 2036 160592566 2044 16061

2408 2326 21610 2399 2894 2277 21610 10272928 2233 216102896 2244 216102877 2282 21610

14a (20 times 10)2340 2251 21610 2905 2649 2186 21610 12392334 2258 21610 2664 2183 21610

2641 2207 216102738 2182 21610

15a (20 times 10)

2285 2247 21610 3329 2627 2216 21610 13602287 2218 21610 2681 2196 21610

2645 2215 216102652 2201 216102668 2197 216102682 2178 21610


Table 4 Continued


1198911 1198912 1198913 119879 1198911 1198912 1198913 119879

16a (20 times 10)

2447 2354 21602 2152 2965 2279 21534 11482450 2380 21590 3158 2303 215042487 2454 21584 3184 2400 214902492 2417 21576 3039 2279 215232540 2396 21547 3050 2318 215072550 2492 21545 3071 2286 215132568 2428 21540 3275 2281 215113013 2588 21478 3105 2352 214963106 2548 21478 3175 2344 21497

3180 2293 215073130 2312 215033034 2352 21503

17a (20 times 10)

2322 2240 21433 2847 2806 2195 21096 13922322 2280 21362 2750 2165 211202323 2238 21454 2791 2141 211142343 2224 21420 2716 2157 211292480 2285 21344 2849 2148 211072528 2231 21313 2769 2166 211112789 2448 21198 2856 2183 210982808 2303 21200 2773 2145 211112816 2370 21197 2856 2142 21105

2803 2203 210942759 2186 211062789 2176 211012772 2202 21101

18a (20 times 10)

2267 2235 21483 3301 2634 2156 21005 16232269 2206 21408 2642 2126 210062320 2208 21354 2638 2185 209992437 2221 21311 2638 2148 210022531 2310 21285 2674 2148 210002545 2305 21282 2644 2145 21002

2651 2140 210052641 2157 210002664 2145 210012621 2181 210022659 2140 210032661 2137 210042651 2181 209992669 2126 210052708 2125 21009







1198911 1198912 1198913 119879 1198911 1198912 1198913 119879 1198911 1198912 1198913 119879

MK01 (10 times 6)

42 42 162 478 42 39 158 049 42 39 158 03644 40 154 44 40 15443 40 155 43 40 15540 36 169 40 36 169

47 36 16747 37 16549 37 16450 38 162

MK02 (10 times 6)

28 28 155 302 26 26 151 075 26 26 151 07027 27 146 27 27 14629 27 145 29 27 14529 29 143 29 29 14331 31 141 31 31 14133 33 140 33 33 140

MK03 (15 times 8)

204 204 852 2614 204 199 855 475 204 199 855 366204 144 871 204 144 871204 135 882 204 135 882204 133 884 204 133 884213 199 850 213 199 850214 210 849 214 210 849221 199 847 221 199 847222 199 848 222 199 848231 188 848 231 188 848230 177 848 230 177 848

MK04 (15 times 8)

68 67 352 1774 66 63 345 176 66 63 345 16365 63 362 65 63 36263 61 371 63 61 37162 61 373 60 59 39061 60 382 74 54 34960 59 390 74 55 34873 55 350 72 72 34274 54 349 78 78 33974 55 348 72 66 34890 76 331 73 72 343

MK05 (15 times 4)

177 177 702 826 173 173 683 234 173 173 683 196175 175 682 175 175 682183 183 677 183 183 677185 185 676 179 179 679179 179 679 199 199 674

193 193 675183 183 677

MK06 (10 times 15)

67 431 1879 62 55 424 193 62 55 424 18565 54 417 65 54 41760 58 441 60 58 44162 60 440 62 60 44076 60 362 76 60 36276 74 356 76 74 35678 60 361 78 60 36173 72 360 73 72 36072 72 361 72 72 361100 90 330 100 90 330


Table 5 Continued


1198911 1198912 1198913 119879 1198911 1198912 1198913 119879 1198911 1198912 1198913 119879

MK07 (20 times 5)

150 150 717 568 139 139 693 492 139 139 693 312140 138 686 140 138 686144 144 673 144 144 673151 151 667 151 151 667157 157 662 157 157 662162 162 659 162 162 659166 166 657 166 166 657

MK08 (20 times 10)

523 523 2524 6767 523 515 2524 1204 523 515 2524 834523 497 2534 523 497 2534524 524 2519 524 524 2519578 578 2489 578 578 2489587 587 2484 587 587 2484

541 533 2516542 542 2511560 560 2505

MK09 (20 times 10)

311 299 2374 7776 311 299 2290 1948 311 299 2290 1015310 299 3514 310 299 3514311 301 2287 311 301 2287314 299 2315 314 299 2315315 299 2283 315 299 2283332 302 2265 332 302 2265329 301 2266 329 301 2266328 308 2259 328 308 2259325 299 2275 325 299 2275

MK10 (20 times 15)

227 221 1989 12252 224 219 1980 1787 224 219 1980 1238225 211 1976 235 225 1895233 214 1919 240 215 1905235 218 1897 246 215 1896235 225 1895 252 224 1884240 215 1905 256 211 1919240 216 1888 260 244 1869242 214 1913 266 254 1864246 215 1896 246 206 1938252 224 1884 246 204 1948256 211 1919 246 205 1940260 244 1869 251 202 1948266 254 1864 253 207 1930268 264 1858 249 203 1947276 256 1857 255 202 1943281 268 1854 260 210 1926217 207 2064 250 205 1935214 204 2082 254 203 1942






25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400 425 450 475 500 5250

M1

M2

M3

M4

M5

M10

M6

M7

M8

M9

541



127060 120 180 240 300 360 420 480 540 600 660 720 780 840 900 960 1020 1080 1140 1200 12600

M1

M2

M3

M4

M5

M10

M11

M12

M13

M14

M15

M16

M17

M18

M6

M7

M8

M9








Notations





Acknowledgments


References























































Advances in

FuzzySystems


Volume 2014












Journal of



Advances in

Multimedia


Biomedical Imaging


Advances in


RoboticsJournal of










Advances in





32 The BEG-NSGA-II


The First Stage










The Second Stage









At the first stage

120572 = 119905119879

119875 = 120572 lowast 1198732

119877 = 119873 (1 minus 120572)2

(6)

At the second stage

1205721015840 = 119905119866119864119873 (7)

1198771015840 =119873(1 minus 1205721015840)

2 (8)

1205941015840 + 119878 + 05119873 gt= 119873 (9)

119863(119894 119895) =119871

sum119897=1


10038161003816100381610038161003816119871 (10)





Initial

No

Yes


objective 1










Swampingmutation



eliteindividuals


Randomly

individuals

NoYes



Output result

YesNo

NoYes



objective 2


objective 3

Parentqueen 3

Offspringqueen 3


Parentqueen 2

Offspringqueen 2


Parentqueen 1

Offspringqueen 1




NoYes

The first stage

The second stage

t = t + 1


Offspring rank = 1

D(i j) gt u

t = t + 1

t = 1

t gt T

Pm lt= 01

Remain S




3 1 2 3 1 2 4 3 4 1 3 2


4 3 5 2 4 5 1 1 3 4 2 5

O31 O11 O21 O32 O33 O34O13 O23O12 O22 O41 O42

J3 J4J2J1

(a) Chromosome

5 10 15 20 250

6

10

13

9

1

5

Time

1012 17 22

20

15 19

M1

M2

M3

M4

M5

O31

O11

O21

O32

O33

O34

O13 O23

O12

O22

O41

O42




























3 1 2 4 1 2 4 3 3 1 3 2

3 1 2 4 1 3 2 1 2 4 3 3

1 3 2 1 2 2 4 3 4 1 3 3

1 3 2 1 3 1 2 4 2 4 3 3

3 1 2 4 1 2 4 3 3 1 3 2

O1

O2

P1

P1

P2


3 1 2 4 1 2 4 3 3 1 3 2

1 2 4 1 3 2 1 2 4 3 3 3

1 3 2 1 2 2 4 3 4 1 3 3

3 2 1 3 1 2 4 1 2 4 3 3

3 1 2 4 1 2 4 3 3 1 3 2

O1

O2

P1

P1

P2










4 3 5 2 4 5 1 1 3 4 2 5

4 3 5 1 2 5 1 1 3 4 2 5

3 4 5 1 2 4 3 2 5 1 2 3

3 4 5 2 4 4 3 2 5 1 2 3

4 3 5 2 4 5 1 1 3 4 2 5

O1

O2

P1

P1

P2

J3 J4J2J1













3 1 2 4 1 2 4 3 3 1 3 2

3 1 2 2 1 2 4 3 3 1 3 4

1 3 2 1 2 2 4 3 4 1 3 3

1 3 2 1 4 2 4 3 2 1 3 3

3 1 2 4 1 2 4 3 3 1 3 2


O1

O2

P1

P1

P2


3 1 2 4 1 2 4 3 3 1 3 2

3 1 2 2 1 2 4 3 3 1 3 4

1 3 2 1 2 2 4 3 4 1 3 3

3 1 2 1 2 2 4 3 4 3 1 3

3 1 2 4 1 2 4 3 3 1 3 2


O1

O2

P1

P1

P2












3 1 2 4 1 2 4 3 3 1 3 2

3 1 2 4 3 2 4 1 3 1 3 2O1

P1


3 1 2 4 1 2 4 3 3 1 3 2

3 1 2 1 4 2 4 3 3 1 3 2O1

P1












3 1 2 4 1 2 4 3 3 1 3 2

3 1 2 3 3 2 4 4 1 1 3 2O1

P1


4 3 5 2 4 5 1 1 3 4 2 5

2 3 3 2 5 5 1 3 3 4 1 4

J3 J4J2J1

O1

P1








Table3Re

sults

onEx

perim

ent1

Prob

lem

119899times119898

ALa

PSO+SA

bMOGA

chP

SOd

hGA

eOOapproach

fBE

G-N

SGA-

IIg

119891 1119891 2

119891 3T

119891 1119891 2

119891 3T

119891 1119891 2

119891 3T

119891 1119891 2

119891 3T

119891 1119891 2

119891 3T

119891 1119891 2

119891 3T

119891 1119891 2

119891 3T

4times5

mdashmdash

mdashmdash

mdashmdash

mdashmdash

1110

32010

mdashmdash

mdashmdash

mdashmdash

mdashmdash

1110

32003

1110

32010

119

3411

934

128

3212

832

137

33

8times8

1613

75mdash

1512

75mdash

1511

81016

1412

77014

1512

75001

1613

73004

1412

77012

1613

7315

1275

1512

7515

1275

1512

7516

1373

1611

7716

1177

1613

7316

1373

10times10

75

45mdash

76

44mdash

85

42024

75

43020

75

43001

87

41005

75

43018

85

427

642

76

428

542

76

428

741

87

418

542

77

438

542

75

458

741

87

41

15times10

2311

95mdash

1211

91mdash

1111

9114

611

1191

136

1111

91003

1313

91009

1111

91005

2411

9112

1095

1210

9314

1291

1210

9311

1098

1110

9511

1095

a Thec

ompu

tatio

nconfi

guratio

nisno

tlisted

inAL

b Thec

ompu

tatio

nconfi

guratio

nisno

tlisted

inPS

O+SA

c Th

eCPU

timeo

naP

Cwith

2GHzC

PUand2G

Bof

RAM

mem

oryin

C++

d TheC

PUtim

eonap

ersonalcom

puterw

ith2G

HzC

PUand2G

Bof

RAM

mem

oryin

C++

e TheC

PUtim

eona3

0GHzP

entiu

min

Delp

hi

f TheC

PUtim

eonan

IntelC

orei5267

GHzp

rocessor

with

4GBRA

Mof

mem

oryin

Java

g TheC

PUtim

eonan

IntelC

orei3267

GHzp

rocessor

with

4GBRA

Min

MAT

LABR2

014a


O11

O12

O13

O21

O22

O23

O31

O32

O33 O34

O41

O42

13

1 2 3 4 5 6 7 8 9 10 11 12 130

M1

M2

M3

M4

M5


2634

130 260 390 520 650 780 910 1040 1170 1300 1430 1560 1690 1820 1950 2080 2210 2340 2470 26000

M1

M2

M3

M4

M5

M10

M6

M7

M8

M9









1198911 1198912 1198913 119879 1198911 1198912 1198913 119879

01a (10 times 5)2568 2505 11137 204 2568 2505 11137 1912572 2568 11137 2572 2568 111372594 2554 11137 2594 2554 11137

02a (10 times 5)2289 2263 11137 256 2487 2237 11137 2132313 2238 11137 3014 2234 11137

2564 2236 11137

03a (10 times 5) 2287 2248 11137 290 2492 2247 11137 252256 2252 11137 2516 2234 11137

04a (10 times 5)

2550 2503 11090 207 2882 2572 11069 2032569 2565 11076 2912 2552 110702579 2552 11080 2904 2626 110673095 2727 11064 3019 2665 11066

2997 2523 110732933 2688 110652953 2668 110662938 2606 110682955 2503 11074

05a (10 times 5)

2292 2252 11077 237 2740 2250 10981 2122293 2242 11091 2846 2212 109862297 2255 11054 2747 2229 109882315 2272 11063 2700 2255 109772343 2298 11050 2915 2231 109812358 2322 11038 2818 2208 109902376 2243 11022 2787 2350 109702904 2620 10941 2759 2320 109712945 2571 10941 2667 2237 109833056 2507 10941 2686 2277 10974

2641 2332 109742687 2221 110002779 2237 109812707 2299 109722702 2233 10989

06a (10 times 5)

2250 2233 11009 309 2673 2194 10893 2862254 2223 10994 2716 2194 108912398 2219 10973 2683 2245 108902437 2280 10988 2666 2248 108932744 2448 10850 2679 2229 108922902 2439 10847 2816 2186 109032967 2840 10839 2898 2186 10899

2713 2234 108892758 2209 108902722 2229 10890

07a (15 times 8)

2450 2413 16485 763 2927 2187 16485 6212457 2299 16485 2910 2213 164852484 2289 16485 2889 2264 16485

2891 2219 164852922 2190 16485

08a (15 times 8)2187 2102 16485 827 2621 2091 16485 6352171 2104 16485 2770 2089 16485

2780 2080 16485


Table 4 Continued


1198911 1198912 1198913 119879 1198911 1198912 1198913 119879

09a (15 times 8)2157 2113 16485 1016 2442 2103 16485 7262144 2119 16485 2470 2074 164852158 2102 16485 2455 2081 16485

10a (15 times 8)

2461 2433 16505 755 2998 2317 16487 6432470 2310 16537 3036 2363 164822478 2330 16533 3091 2491 164732482 2360 16499 2984 2319 164862501 2265 16547 3052 2377 164772501 2312 16528 3098 2491 164722547 2476 16490 3013 2293 164873064 2734 16464 3136 2457 16471

3025 2377 164803041 2351 164812980 2258 164943021 2319 16484

11a (15 times 8)

2182 2170 16449 1014 2551 2116 16229 7252202 2114 16476 2599 2131 162242210 2113 16442 2612 2146 162222337 2185 16377 2573 2058 162312874 2389 16247 2753 2186 162172894 2330 16247 2628 2102 162242962 2312 16247 2702 2098 16223

2658 2089 162242555 2111 162292664 2063 16228

12a (15 times 8)

2161 2107 16295 1192 2624 2049 16055 8222168 2130 16220 2620 2022 160682191 2084 16355 2587 2035 160652210 2103 16331 2614 2057 160512315 2125 16292 2585 2043 160612366 2105 16237 2659 2038 160552493 2297 16124 2617 2024 160652631 2309 16112 2575 2057 160572637 2303 16113 2587 2050 160592683 2397 16104 2582 2098 16052

13a (20 times 10)

2595 2036 160592566 2044 16061

2408 2326 21610 2399 2894 2277 21610 10272928 2233 216102896 2244 216102877 2282 21610

14a (20 times 10)2340 2251 21610 2905 2649 2186 21610 12392334 2258 21610 2664 2183 21610

2641 2207 216102738 2182 21610

15a (20 times 10)

2285 2247 21610 3329 2627 2216 21610 13602287 2218 21610 2681 2196 21610

2645 2215 216102652 2201 216102668 2197 216102682 2178 21610


Table 4 Continued


1198911 1198912 1198913 119879 1198911 1198912 1198913 119879

16a (20 times 10)

2447 2354 21602 2152 2965 2279 21534 11482450 2380 21590 3158 2303 215042487 2454 21584 3184 2400 214902492 2417 21576 3039 2279 215232540 2396 21547 3050 2318 215072550 2492 21545 3071 2286 215132568 2428 21540 3275 2281 215113013 2588 21478 3105 2352 214963106 2548 21478 3175 2344 21497

3180 2293 215073130 2312 215033034 2352 21503

17a (20 times 10)

2322 2240 21433 2847 2806 2195 21096 13922322 2280 21362 2750 2165 211202323 2238 21454 2791 2141 211142343 2224 21420 2716 2157 211292480 2285 21344 2849 2148 211072528 2231 21313 2769 2166 211112789 2448 21198 2856 2183 210982808 2303 21200 2773 2145 211112816 2370 21197 2856 2142 21105

2803 2203 210942759 2186 211062789 2176 211012772 2202 21101

18a (20 times 10)

2267 2235 21483 3301 2634 2156 21005 16232269 2206 21408 2642 2126 210062320 2208 21354 2638 2185 209992437 2221 21311 2638 2148 210022531 2310 21285 2674 2148 210002545 2305 21282 2644 2145 21002

2651 2140 210052641 2157 210002664 2145 210012621 2181 210022659 2140 210032661 2137 210042651 2181 209992669 2126 210052708 2125 21009







1198911 1198912 1198913 119879 1198911 1198912 1198913 119879 1198911 1198912 1198913 119879

MK01 (10 times 6)

42 42 162 478 42 39 158 049 42 39 158 03644 40 154 44 40 15443 40 155 43 40 15540 36 169 40 36 169

47 36 16747 37 16549 37 16450 38 162

MK02 (10 times 6)

28 28 155 302 26 26 151 075 26 26 151 07027 27 146 27 27 14629 27 145 29 27 14529 29 143 29 29 14331 31 141 31 31 14133 33 140 33 33 140

MK03 (15 times 8)

204 204 852 2614 204 199 855 475 204 199 855 366204 144 871 204 144 871204 135 882 204 135 882204 133 884 204 133 884213 199 850 213 199 850214 210 849 214 210 849221 199 847 221 199 847222 199 848 222 199 848231 188 848 231 188 848230 177 848 230 177 848

MK04 (15 times 8)

68 67 352 1774 66 63 345 176 66 63 345 16365 63 362 65 63 36263 61 371 63 61 37162 61 373 60 59 39061 60 382 74 54 34960 59 390 74 55 34873 55 350 72 72 34274 54 349 78 78 33974 55 348 72 66 34890 76 331 73 72 343

MK05 (15 times 4)

177 177 702 826 173 173 683 234 173 173 683 196175 175 682 175 175 682183 183 677 183 183 677185 185 676 179 179 679179 179 679 199 199 674

193 193 675183 183 677

MK06 (10 times 15)

67 431 1879 62 55 424 193 62 55 424 18565 54 417 65 54 41760 58 441 60 58 44162 60 440 62 60 44076 60 362 76 60 36276 74 356 76 74 35678 60 361 78 60 36173 72 360 73 72 36072 72 361 72 72 361100 90 330 100 90 330


Table 5 Continued


1198911 1198912 1198913 119879 1198911 1198912 1198913 119879 1198911 1198912 1198913 119879

MK07 (20 times 5)

150 150 717 568 139 139 693 492 139 139 693 312140 138 686 140 138 686144 144 673 144 144 673151 151 667 151 151 667157 157 662 157 157 662162 162 659 162 162 659166 166 657 166 166 657

MK08 (20 times 10)

523 523 2524 6767 523 515 2524 1204 523 515 2524 834523 497 2534 523 497 2534524 524 2519 524 524 2519578 578 2489 578 578 2489587 587 2484 587 587 2484

541 533 2516542 542 2511560 560 2505

MK09 (20 times 10)

311 299 2374 7776 311 299 2290 1948 311 299 2290 1015310 299 3514 310 299 3514311 301 2287 311 301 2287314 299 2315 314 299 2315315 299 2283 315 299 2283332 302 2265 332 302 2265329 301 2266 329 301 2266328 308 2259 328 308 2259325 299 2275 325 299 2275

MK10 (20 times 15)

227 221 1989 12252 224 219 1980 1787 224 219 1980 1238225 211 1976 235 225 1895233 214 1919 240 215 1905235 218 1897 246 215 1896235 225 1895 252 224 1884240 215 1905 256 211 1919240 216 1888 260 244 1869242 214 1913 266 254 1864246 215 1896 246 206 1938252 224 1884 246 204 1948256 211 1919 246 205 1940260 244 1869 251 202 1948266 254 1864 253 207 1930268 264 1858 249 203 1947276 256 1857 255 202 1943281 268 1854 260 210 1926217 207 2064 250 205 1935214 204 2082 254 203 1942






25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400 425 450 475 500 5250

M1

M2

M3

M4

M5

M10

M6

M7

M8

M9

541



127060 120 180 240 300 360 420 480 540 600 660 720 780 840 900 960 1020 1080 1140 1200 12600

M1

M2

M3

M4

M5

M10

M11

M12

M13

M14

M15

M16

M17

M18

M6

M7

M8

M9








Notations





Acknowledgments


References























































Advances in

FuzzySystems


Volume 2014












Journal of



Advances in

Multimedia


Biomedical Imaging


Advances in


RoboticsJournal of










Advances in




Initial

No

Yes


objective 1










Swampingmutation



eliteindividuals


Randomly

individuals

NoYes



Output result

YesNo

NoYes



objective 2


objective 3

Parentqueen 3

Offspringqueen 3


Parentqueen 2

Offspringqueen 2


Parentqueen 1

Offspringqueen 1




NoYes

The first stage

The second stage

t = t + 1


Offspring rank = 1

D(i j) gt u

t = t + 1

t = 1

t gt T

Pm lt= 01

Remain S




3 1 2 3 1 2 4 3 4 1 3 2


4 3 5 2 4 5 1 1 3 4 2 5

O31 O11 O21 O32 O33 O34O13 O23O12 O22 O41 O42

J3 J4J2J1

(a) Chromosome

5 10 15 20 250

6

10

13

9

1

5

Time

1012 17 22

20

15 19

M1

M2

M3

M4

M5

O31

O11

O21

O32

O33

O34

O13 O23

O12

O22

O41

O42




























3 1 2 4 1 2 4 3 3 1 3 2

3 1 2 4 1 3 2 1 2 4 3 3

1 3 2 1 2 2 4 3 4 1 3 3

1 3 2 1 3 1 2 4 2 4 3 3

3 1 2 4 1 2 4 3 3 1 3 2

O1

O2

P1

P1

P2


3 1 2 4 1 2 4 3 3 1 3 2

1 2 4 1 3 2 1 2 4 3 3 3

1 3 2 1 2 2 4 3 4 1 3 3

3 2 1 3 1 2 4 1 2 4 3 3

3 1 2 4 1 2 4 3 3 1 3 2

O1

O2

P1

P1

P2










4 3 5 2 4 5 1 1 3 4 2 5

4 3 5 1 2 5 1 1 3 4 2 5

3 4 5 1 2 4 3 2 5 1 2 3

3 4 5 2 4 4 3 2 5 1 2 3

4 3 5 2 4 5 1 1 3 4 2 5

O1

O2

P1

P1

P2

J3 J4J2J1













3 1 2 4 1 2 4 3 3 1 3 2

3 1 2 2 1 2 4 3 3 1 3 4

1 3 2 1 2 2 4 3 4 1 3 3

1 3 2 1 4 2 4 3 2 1 3 3

3 1 2 4 1 2 4 3 3 1 3 2


O1

O2

P1

P1

P2


3 1 2 4 1 2 4 3 3 1 3 2

3 1 2 2 1 2 4 3 3 1 3 4

1 3 2 1 2 2 4 3 4 1 3 3

3 1 2 1 2 2 4 3 4 3 1 3

3 1 2 4 1 2 4 3 3 1 3 2


O1

O2

P1

P1

P2












3 1 2 4 1 2 4 3 3 1 3 2

3 1 2 4 3 2 4 1 3 1 3 2O1

P1


3 1 2 4 1 2 4 3 3 1 3 2

3 1 2 1 4 2 4 3 3 1 3 2O1

P1












3 1 2 4 1 2 4 3 3 1 3 2

3 1 2 3 3 2 4 4 1 1 3 2O1

P1


4 3 5 2 4 5 1 1 3 4 2 5

2 3 3 2 5 5 1 3 3 4 1 4

J3 J4J2J1

O1

P1








Table3Re

sults

onEx

perim

ent1

Prob

lem

119899times119898

ALa

PSO+SA

bMOGA

chP

SOd

hGA

eOOapproach

fBE

G-N

SGA-

IIg

119891 1119891 2

119891 3T

119891 1119891 2

119891 3T

119891 1119891 2

119891 3T

119891 1119891 2

119891 3T

119891 1119891 2

119891 3T

119891 1119891 2

119891 3T

119891 1119891 2

119891 3T

4times5

mdashmdash

mdashmdash

mdashmdash

mdashmdash

1110

32010

mdashmdash

mdashmdash

mdashmdash

mdashmdash

1110

32003

1110

32010

119

3411

934

128

3212

832

137

33

8times8

1613

75mdash

1512

75mdash

1511

81016

1412

77014

1512

75001

1613

73004

1412

77012

1613

7315

1275

1512

7515

1275

1512

7516

1373

1611

7716

1177

1613

7316

1373

10times10

75

45mdash

76

44mdash

85

42024

75

43020

75

43001

87

41005

75

43018

85

427

642

76

428

542

76

428

741

87

418

542

77

438

542

75

458

741

87

41

15times10

2311

95mdash

1211

91mdash

1111

9114

611

1191

136

1111

91003

1313

91009

1111

91005

2411

9112

1095

1210

9314

1291

1210

9311

1098

1110

9511

1095

a Thec

ompu

tatio

nconfi

guratio

nisno

tlisted

inAL

b Thec

ompu

tatio

nconfi

guratio

nisno

tlisted

inPS

O+SA

c Th

eCPU

timeo

naP

Cwith

2GHzC

PUand2G

Bof

RAM

mem

oryin

C++

d TheC

PUtim

eonap

ersonalcom

puterw

ith2G

HzC

PUand2G

Bof

RAM

mem

oryin

C++

e TheC

PUtim

eona3

0GHzP

entiu

min

Delp

hi

f TheC

PUtim

eonan

IntelC

orei5267

GHzp

rocessor

with

4GBRA

Mof

mem

oryin

Java

g TheC

PUtim

eonan

IntelC

orei3267

GHzp

rocessor

with

4GBRA

Min

MAT

LABR2

014a


O11

O12

O13

O21

O22

O23

O31

O32

O33 O34

O41

O42

13

1 2 3 4 5 6 7 8 9 10 11 12 130

M1

M2

M3

M4

M5


2634

130 260 390 520 650 780 910 1040 1170 1300 1430 1560 1690 1820 1950 2080 2210 2340 2470 26000

M1

M2

M3

M4

M5

M10

M6

M7

M8

M9









1198911 1198912 1198913 119879 1198911 1198912 1198913 119879

01a (10 times 5)2568 2505 11137 204 2568 2505 11137 1912572 2568 11137 2572 2568 111372594 2554 11137 2594 2554 11137

02a (10 times 5)2289 2263 11137 256 2487 2237 11137 2132313 2238 11137 3014 2234 11137

2564 2236 11137

03a (10 times 5) 2287 2248 11137 290 2492 2247 11137 252256 2252 11137 2516 2234 11137

04a (10 times 5)

2550 2503 11090 207 2882 2572 11069 2032569 2565 11076 2912 2552 110702579 2552 11080 2904 2626 110673095 2727 11064 3019 2665 11066

2997 2523 110732933 2688 110652953 2668 110662938 2606 110682955 2503 11074

05a (10 times 5)

2292 2252 11077 237 2740 2250 10981 2122293 2242 11091 2846 2212 109862297 2255 11054 2747 2229 109882315 2272 11063 2700 2255 109772343 2298 11050 2915 2231 109812358 2322 11038 2818 2208 109902376 2243 11022 2787 2350 109702904 2620 10941 2759 2320 109712945 2571 10941 2667 2237 109833056 2507 10941 2686 2277 10974

2641 2332 109742687 2221 110002779 2237 109812707 2299 109722702 2233 10989

06a (10 times 5)

2250 2233 11009 309 2673 2194 10893 2862254 2223 10994 2716 2194 108912398 2219 10973 2683 2245 108902437 2280 10988 2666 2248 108932744 2448 10850 2679 2229 108922902 2439 10847 2816 2186 109032967 2840 10839 2898 2186 10899

2713 2234 108892758 2209 108902722 2229 10890

07a (15 times 8)

2450 2413 16485 763 2927 2187 16485 6212457 2299 16485 2910 2213 164852484 2289 16485 2889 2264 16485

2891 2219 164852922 2190 16485

08a (15 times 8)2187 2102 16485 827 2621 2091 16485 6352171 2104 16485 2770 2089 16485

2780 2080 16485


Table 4 Continued


1198911 1198912 1198913 119879 1198911 1198912 1198913 119879

09a (15 times 8)2157 2113 16485 1016 2442 2103 16485 7262144 2119 16485 2470 2074 164852158 2102 16485 2455 2081 16485

10a (15 times 8)

2461 2433 16505 755 2998 2317 16487 6432470 2310 16537 3036 2363 164822478 2330 16533 3091 2491 164732482 2360 16499 2984 2319 164862501 2265 16547 3052 2377 164772501 2312 16528 3098 2491 164722547 2476 16490 3013 2293 164873064 2734 16464 3136 2457 16471

3025 2377 164803041 2351 164812980 2258 164943021 2319 16484

11a (15 times 8)

2182 2170 16449 1014 2551 2116 16229 7252202 2114 16476 2599 2131 162242210 2113 16442 2612 2146 162222337 2185 16377 2573 2058 162312874 2389 16247 2753 2186 162172894 2330 16247 2628 2102 162242962 2312 16247 2702 2098 16223

2658 2089 162242555 2111 162292664 2063 16228

12a (15 times 8)

2161 2107 16295 1192 2624 2049 16055 8222168 2130 16220 2620 2022 160682191 2084 16355 2587 2035 160652210 2103 16331 2614 2057 160512315 2125 16292 2585 2043 160612366 2105 16237 2659 2038 160552493 2297 16124 2617 2024 160652631 2309 16112 2575 2057 160572637 2303 16113 2587 2050 160592683 2397 16104 2582 2098 16052

13a (20 times 10)

2595 2036 160592566 2044 16061

2408 2326 21610 2399 2894 2277 21610 10272928 2233 216102896 2244 216102877 2282 21610

14a (20 times 10)2340 2251 21610 2905 2649 2186 21610 12392334 2258 21610 2664 2183 21610

2641 2207 216102738 2182 21610

15a (20 times 10)

2285 2247 21610 3329 2627 2216 21610 13602287 2218 21610 2681 2196 21610

2645 2215 216102652 2201 216102668 2197 216102682 2178 21610


Table 4 Continued


1198911 1198912 1198913 119879 1198911 1198912 1198913 119879

16a (20 times 10)

2447 2354 21602 2152 2965 2279 21534 11482450 2380 21590 3158 2303 215042487 2454 21584 3184 2400 214902492 2417 21576 3039 2279 215232540 2396 21547 3050 2318 215072550 2492 21545 3071 2286 215132568 2428 21540 3275 2281 215113013 2588 21478 3105 2352 214963106 2548 21478 3175 2344 21497

3180 2293 215073130 2312 215033034 2352 21503

17a (20 times 10)

2322 2240 21433 2847 2806 2195 21096 13922322 2280 21362 2750 2165 211202323 2238 21454 2791 2141 211142343 2224 21420 2716 2157 211292480 2285 21344 2849 2148 211072528 2231 21313 2769 2166 211112789 2448 21198 2856 2183 210982808 2303 21200 2773 2145 211112816 2370 21197 2856 2142 21105

2803 2203 210942759 2186 211062789 2176 211012772 2202 21101

18a (20 times 10)

2267 2235 21483 3301 2634 2156 21005 16232269 2206 21408 2642 2126 210062320 2208 21354 2638 2185 209992437 2221 21311 2638 2148 210022531 2310 21285 2674 2148 210002545 2305 21282 2644 2145 21002

2651 2140 210052641 2157 210002664 2145 210012621 2181 210022659 2140 210032661 2137 210042651 2181 209992669 2126 210052708 2125 21009







1198911 1198912 1198913 119879 1198911 1198912 1198913 119879 1198911 1198912 1198913 119879

MK01 (10 times 6)

42 42 162 478 42 39 158 049 42 39 158 03644 40 154 44 40 15443 40 155 43 40 15540 36 169 40 36 169

47 36 16747 37 16549 37 16450 38 162

MK02 (10 times 6)

28 28 155 302 26 26 151 075 26 26 151 07027 27 146 27 27 14629 27 145 29 27 14529 29 143 29 29 14331 31 141 31 31 14133 33 140 33 33 140

MK03 (15 times 8)

204 204 852 2614 204 199 855 475 204 199 855 366204 144 871 204 144 871204 135 882 204 135 882204 133 884 204 133 884213 199 850 213 199 850214 210 849 214 210 849221 199 847 221 199 847222 199 848 222 199 848231 188 848 231 188 848230 177 848 230 177 848

MK04 (15 times 8)

68 67 352 1774 66 63 345 176 66 63 345 16365 63 362 65 63 36263 61 371 63 61 37162 61 373 60 59 39061 60 382 74 54 34960 59 390 74 55 34873 55 350 72 72 34274 54 349 78 78 33974 55 348 72 66 34890 76 331 73 72 343

MK05 (15 times 4)

177 177 702 826 173 173 683 234 173 173 683 196175 175 682 175 175 682183 183 677 183 183 677185 185 676 179 179 679179 179 679 199 199 674

193 193 675183 183 677

MK06 (10 times 15)

67 431 1879 62 55 424 193 62 55 424 18565 54 417 65 54 41760 58 441 60 58 44162 60 440 62 60 44076 60 362 76 60 36276 74 356 76 74 35678 60 361 78 60 36173 72 360 73 72 36072 72 361 72 72 361100 90 330 100 90 330


Table 5 Continued


1198911 1198912 1198913 119879 1198911 1198912 1198913 119879 1198911 1198912 1198913 119879

MK07 (20 times 5)

150 150 717 568 139 139 693 492 139 139 693 312140 138 686 140 138 686144 144 673 144 144 673151 151 667 151 151 667157 157 662 157 157 662162 162 659 162 162 659166 166 657 166 166 657

MK08 (20 times 10)

523 523 2524 6767 523 515 2524 1204 523 515 2524 834523 497 2534 523 497 2534524 524 2519 524 524 2519578 578 2489 578 578 2489587 587 2484 587 587 2484

541 533 2516542 542 2511560 560 2505

MK09 (20 times 10)

311 299 2374 7776 311 299 2290 1948 311 299 2290 1015310 299 3514 310 299 3514311 301 2287 311 301 2287314 299 2315 314 299 2315315 299 2283 315 299 2283332 302 2265 332 302 2265329 301 2266 329 301 2266328 308 2259 328 308 2259325 299 2275 325 299 2275

MK10 (20 times 15)

227 221 1989 12252 224 219 1980 1787 224 219 1980 1238225 211 1976 235 225 1895233 214 1919 240 215 1905235 218 1897 246 215 1896235 225 1895 252 224 1884240 215 1905 256 211 1919240 216 1888 260 244 1869242 214 1913 266 254 1864246 215 1896 246 206 1938252 224 1884 246 204 1948256 211 1919 246 205 1940260 244 1869 251 202 1948266 254 1864 253 207 1930268 264 1858 249 203 1947276 256 1857 255 202 1943281 268 1854 260 210 1926217 207 2064 250 205 1935214 204 2082 254 203 1942






25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400 425 450 475 500 5250

M1

M2

M3

M4

M5

M10

M6

M7

M8

M9

541



127060 120 180 240 300 360 420 480 540 600 660 720 780 840 900 960 1020 1080 1140 1200 12600

M1

M2

M3

M4

M5

M10

M11

M12

M13

M14

M15

M16

M17

M18

M6

M7

M8

M9








Notations





Acknowledgments


References























































Advances in

FuzzySystems


Volume 2014












Journal of



Advances in

Multimedia


Biomedical Imaging


Advances in


RoboticsJournal of










Advances in





3 1 2 3 1 2 4 3 4 1 3 2


4 3 5 2 4 5 1 1 3 4 2 5

O31 O11 O21 O32 O33 O34O13 O23O12 O22 O41 O42

J3 J4J2J1

(a) Chromosome

5 10 15 20 250

6

10

13

9

1

5

Time

1012 17 22

20

15 19

M1

M2

M3

M4

M5

O31

O11

O21

O32

O33

O34

O13 O23

O12

O22

O41

O42




























3 1 2 4 1 2 4 3 3 1 3 2

3 1 2 4 1 3 2 1 2 4 3 3

1 3 2 1 2 2 4 3 4 1 3 3

1 3 2 1 3 1 2 4 2 4 3 3

3 1 2 4 1 2 4 3 3 1 3 2

O1

O2

P1

P1

P2


3 1 2 4 1 2 4 3 3 1 3 2

1 2 4 1 3 2 1 2 4 3 3 3

1 3 2 1 2 2 4 3 4 1 3 3

3 2 1 3 1 2 4 1 2 4 3 3

3 1 2 4 1 2 4 3 3 1 3 2

O1

O2

P1

P1

P2










4 3 5 2 4 5 1 1 3 4 2 5

4 3 5 1 2 5 1 1 3 4 2 5

3 4 5 1 2 4 3 2 5 1 2 3

3 4 5 2 4 4 3 2 5 1 2 3

4 3 5 2 4 5 1 1 3 4 2 5

O1

O2

P1

P1

P2

J3 J4J2J1













3 1 2 4 1 2 4 3 3 1 3 2

3 1 2 2 1 2 4 3 3 1 3 4

1 3 2 1 2 2 4 3 4 1 3 3

1 3 2 1 4 2 4 3 2 1 3 3

3 1 2 4 1 2 4 3 3 1 3 2


O1

O2

P1

P1

P2


3 1 2 4 1 2 4 3 3 1 3 2

3 1 2 2 1 2 4 3 3 1 3 4

1 3 2 1 2 2 4 3 4 1 3 3

3 1 2 1 2 2 4 3 4 3 1 3

3 1 2 4 1 2 4 3 3 1 3 2


O1

O2

P1

P1

P2












3 1 2 4 1 2 4 3 3 1 3 2

3 1 2 4 3 2 4 1 3 1 3 2O1

P1


3 1 2 4 1 2 4 3 3 1 3 2

3 1 2 1 4 2 4 3 3 1 3 2O1

P1












3 1 2 4 1 2 4 3 3 1 3 2

3 1 2 3 3 2 4 4 1 1 3 2O1

P1


4 3 5 2 4 5 1 1 3 4 2 5

2 3 3 2 5 5 1 3 3 4 1 4

J3 J4J2J1

O1

P1








Table3Re

sults

onEx

perim

ent1

Prob

lem

119899times119898

ALa

PSO+SA

bMOGA

chP

SOd

hGA

eOOapproach

fBE

G-N

SGA-

IIg

119891 1119891 2

119891 3T

119891 1119891 2

119891 3T

119891 1119891 2

119891 3T

119891 1119891 2

119891 3T

119891 1119891 2

119891 3T

119891 1119891 2

119891 3T

119891 1119891 2

119891 3T

4times5

mdashmdash

mdashmdash

mdashmdash

mdashmdash

1110

32010

mdashmdash

mdashmdash

mdashmdash

mdashmdash

1110

32003

1110

32010

119

3411

934

128

3212

832

137

33

8times8

1613

75mdash

1512

75mdash

1511

81016

1412

77014

1512

75001

1613

73004

1412

77012

1613

7315

1275

1512

7515

1275

1512

7516

1373

1611

7716

1177

1613

7316

1373

10times10

75

45mdash

76

44mdash

85

42024

75

43020

75

43001

87

41005

75

43018

85

427

642

76

428

542

76

428

741

87

418

542

77

438

542

75

458

741

87

41

15times10

2311

95mdash

1211

91mdash

1111

9114

611

1191

136

1111

91003

1313

91009

1111

91005

2411

9112

1095

1210

9314

1291

1210

9311

1098

1110

9511

1095

a Thec

ompu

tatio

nconfi

guratio

nisno

tlisted

inAL

b Thec

ompu

tatio

nconfi

guratio

nisno

tlisted

inPS

O+SA

c Th

eCPU

timeo

naP

Cwith

2GHzC

PUand2G

Bof

RAM

mem

oryin

C++

d TheC

PUtim

eonap

ersonalcom

puterw

ith2G

HzC

PUand2G

Bof

RAM

mem

oryin

C++

e TheC

PUtim

eona3

0GHzP

entiu

min

Delp

hi

f TheC

PUtim

eonan

IntelC

orei5267

GHzp

rocessor

with

4GBRA

Mof

mem

oryin

Java

g TheC

PUtim

eonan

IntelC

orei3267

GHzp

rocessor

with

4GBRA

Min

MAT

LABR2

014a


O11

O12

O13

O21

O22

O23

O31

O32

O33 O34

O41

O42

13

1 2 3 4 5 6 7 8 9 10 11 12 130

M1

M2

M3

M4

M5


2634

130 260 390 520 650 780 910 1040 1170 1300 1430 1560 1690 1820 1950 2080 2210 2340 2470 26000

M1

M2

M3

M4

M5

M10

M6

M7

M8

M9









1198911 1198912 1198913 119879 1198911 1198912 1198913 119879

01a (10 times 5)2568 2505 11137 204 2568 2505 11137 1912572 2568 11137 2572 2568 111372594 2554 11137 2594 2554 11137

02a (10 times 5)2289 2263 11137 256 2487 2237 11137 2132313 2238 11137 3014 2234 11137

2564 2236 11137

03a (10 times 5) 2287 2248 11137 290 2492 2247 11137 252256 2252 11137 2516 2234 11137

04a (10 times 5)

2550 2503 11090 207 2882 2572 11069 2032569 2565 11076 2912 2552 110702579 2552 11080 2904 2626 110673095 2727 11064 3019 2665 11066

2997 2523 110732933 2688 110652953 2668 110662938 2606 110682955 2503 11074

05a (10 times 5)

2292 2252 11077 237 2740 2250 10981 2122293 2242 11091 2846 2212 109862297 2255 11054 2747 2229 109882315 2272 11063 2700 2255 109772343 2298 11050 2915 2231 109812358 2322 11038 2818 2208 109902376 2243 11022 2787 2350 109702904 2620 10941 2759 2320 109712945 2571 10941 2667 2237 109833056 2507 10941 2686 2277 10974

2641 2332 109742687 2221 110002779 2237 109812707 2299 109722702 2233 10989

06a (10 times 5)

2250 2233 11009 309 2673 2194 10893 2862254 2223 10994 2716 2194 108912398 2219 10973 2683 2245 108902437 2280 10988 2666 2248 108932744 2448 10850 2679 2229 108922902 2439 10847 2816 2186 109032967 2840 10839 2898 2186 10899

2713 2234 108892758 2209 108902722 2229 10890

07a (15 times 8)

2450 2413 16485 763 2927 2187 16485 6212457 2299 16485 2910 2213 164852484 2289 16485 2889 2264 16485

2891 2219 164852922 2190 16485

08a (15 times 8)2187 2102 16485 827 2621 2091 16485 6352171 2104 16485 2770 2089 16485

2780 2080 16485


Table 4 Continued


1198911 1198912 1198913 119879 1198911 1198912 1198913 119879

09a (15 times 8)2157 2113 16485 1016 2442 2103 16485 7262144 2119 16485 2470 2074 164852158 2102 16485 2455 2081 16485

10a (15 times 8)

2461 2433 16505 755 2998 2317 16487 6432470 2310 16537 3036 2363 164822478 2330 16533 3091 2491 164732482 2360 16499 2984 2319 164862501 2265 16547 3052 2377 164772501 2312 16528 3098 2491 164722547 2476 16490 3013 2293 164873064 2734 16464 3136 2457 16471

3025 2377 164803041 2351 164812980 2258 164943021 2319 16484

11a (15 times 8)

2182 2170 16449 1014 2551 2116 16229 7252202 2114 16476 2599 2131 162242210 2113 16442 2612 2146 162222337 2185 16377 2573 2058 162312874 2389 16247 2753 2186 162172894 2330 16247 2628 2102 162242962 2312 16247 2702 2098 16223

2658 2089 162242555 2111 162292664 2063 16228

12a (15 times 8)

2161 2107 16295 1192 2624 2049 16055 8222168 2130 16220 2620 2022 160682191 2084 16355 2587 2035 160652210 2103 16331 2614 2057 160512315 2125 16292 2585 2043 160612366 2105 16237 2659 2038 160552493 2297 16124 2617 2024 160652631 2309 16112 2575 2057 160572637 2303 16113 2587 2050 160592683 2397 16104 2582 2098 16052

13a (20 times 10)

2595 2036 160592566 2044 16061

2408 2326 21610 2399 2894 2277 21610 10272928 2233 216102896 2244 216102877 2282 21610

14a (20 times 10)2340 2251 21610 2905 2649 2186 21610 12392334 2258 21610 2664 2183 21610

2641 2207 216102738 2182 21610

15a (20 times 10)

2285 2247 21610 3329 2627 2216 21610 13602287 2218 21610 2681 2196 21610

2645 2215 216102652 2201 216102668 2197 216102682 2178 21610


Table 4 Continued


1198911 1198912 1198913 119879 1198911 1198912 1198913 119879

16a (20 times 10)

2447 2354 21602 2152 2965 2279 21534 11482450 2380 21590 3158 2303 215042487 2454 21584 3184 2400 214902492 2417 21576 3039 2279 215232540 2396 21547 3050 2318 215072550 2492 21545 3071 2286 215132568 2428 21540 3275 2281 215113013 2588 21478 3105 2352 214963106 2548 21478 3175 2344 21497

3180 2293 215073130 2312 215033034 2352 21503

17a (20 times 10)

2322 2240 21433 2847 2806 2195 21096 13922322 2280 21362 2750 2165 211202323 2238 21454 2791 2141 211142343 2224 21420 2716 2157 211292480 2285 21344 2849 2148 211072528 2231 21313 2769 2166 211112789 2448 21198 2856 2183 210982808 2303 21200 2773 2145 211112816 2370 21197 2856 2142 21105

2803 2203 210942759 2186 211062789 2176 211012772 2202 21101

18a (20 times 10)

2267 2235 21483 3301 2634 2156 21005 16232269 2206 21408 2642 2126 210062320 2208 21354 2638 2185 209992437 2221 21311 2638 2148 210022531 2310 21285 2674 2148 210002545 2305 21282 2644 2145 21002

2651 2140 210052641 2157 210002664 2145 210012621 2181 210022659 2140 210032661 2137 210042651 2181 209992669 2126 210052708 2125 21009







1198911 1198912 1198913 119879 1198911 1198912 1198913 119879 1198911 1198912 1198913 119879

MK01 (10 times 6)

42 42 162 478 42 39 158 049 42 39 158 03644 40 154 44 40 15443 40 155 43 40 15540 36 169 40 36 169

47 36 16747 37 16549 37 16450 38 162

MK02 (10 times 6)

28 28 155 302 26 26 151 075 26 26 151 07027 27 146 27 27 14629 27 145 29 27 14529 29 143 29 29 14331 31 141 31 31 14133 33 140 33 33 140

MK03 (15 times 8)

204 204 852 2614 204 199 855 475 204 199 855 366204 144 871 204 144 871204 135 882 204 135 882204 133 884 204 133 884213 199 850 213 199 850214 210 849 214 210 849221 199 847 221 199 847222 199 848 222 199 848231 188 848 231 188 848230 177 848 230 177 848

MK04 (15 times 8)

68 67 352 1774 66 63 345 176 66 63 345 16365 63 362 65 63 36263 61 371 63 61 37162 61 373 60 59 39061 60 382 74 54 34960 59 390 74 55 34873 55 350 72 72 34274 54 349 78 78 33974 55 348 72 66 34890 76 331 73 72 343

MK05 (15 times 4)

177 177 702 826 173 173 683 234 173 173 683 196175 175 682 175 175 682183 183 677 183 183 677185 185 676 179 179 679179 179 679 199 199 674

193 193 675183 183 677

MK06 (10 times 15)

67 431 1879 62 55 424 193 62 55 424 18565 54 417 65 54 41760 58 441 60 58 44162 60 440 62 60 44076 60 362 76 60 36276 74 356 76 74 35678 60 361 78 60 36173 72 360 73 72 36072 72 361 72 72 361100 90 330 100 90 330


Table 5 Continued


1198911 1198912 1198913 119879 1198911 1198912 1198913 119879 1198911 1198912 1198913 119879

MK07 (20 times 5)

150 150 717 568 139 139 693 492 139 139 693 312140 138 686 140 138 686144 144 673 144 144 673151 151 667 151 151 667157 157 662 157 157 662162 162 659 162 162 659166 166 657 166 166 657

MK08 (20 times 10)

523 523 2524 6767 523 515 2524 1204 523 515 2524 834523 497 2534 523 497 2534524 524 2519 524 524 2519578 578 2489 578 578 2489587 587 2484 587 587 2484

541 533 2516542 542 2511560 560 2505

MK09 (20 times 10)

311 299 2374 7776 311 299 2290 1948 311 299 2290 1015310 299 3514 310 299 3514311 301 2287 311 301 2287314 299 2315 314 299 2315315 299 2283 315 299 2283332 302 2265 332 302 2265329 301 2266 329 301 2266328 308 2259 328 308 2259325 299 2275 325 299 2275

MK10 (20 times 15)

227 221 1989 12252 224 219 1980 1787 224 219 1980 1238225 211 1976 235 225 1895233 214 1919 240 215 1905235 218 1897 246 215 1896235 225 1895 252 224 1884240 215 1905 256 211 1919240 216 1888 260 244 1869242 214 1913 266 254 1864246 215 1896 246 206 1938252 224 1884 246 204 1948256 211 1919 246 205 1940260 244 1869 251 202 1948266 254 1864 253 207 1930268 264 1858 249 203 1947276 256 1857 255 202 1943281 268 1854 260 210 1926217 207 2064 250 205 1935214 204 2082 254 203 1942






25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400 425 450 475 500 5250

M1

M2

M3

M4

M5

M10

M6

M7

M8

M9

541



127060 120 180 240 300 360 420 480 540 600 660 720 780 840 900 960 1020 1080 1140 1200 12600

M1

M2

M3

M4

M5

M10

M11

M12

M13

M14

M15

M16

M17

M18

M6

M7

M8

M9








Notations





Acknowledgments


References























































Advances in

FuzzySystems


Volume 2014












Journal of



Advances in

Multimedia


Biomedical Imaging


Advances in


RoboticsJournal of










Advances in














3 1 2 4 1 2 4 3 3 1 3 2

3 1 2 4 1 3 2 1 2 4 3 3

1 3 2 1 2 2 4 3 4 1 3 3

1 3 2 1 3 1 2 4 2 4 3 3

3 1 2 4 1 2 4 3 3 1 3 2

O1

O2

P1

P1

P2


3 1 2 4 1 2 4 3 3 1 3 2

1 2 4 1 3 2 1 2 4 3 3 3

1 3 2 1 2 2 4 3 4 1 3 3

3 2 1 3 1 2 4 1 2 4 3 3

3 1 2 4 1 2 4 3 3 1 3 2

O1

O2

P1

P1

P2










4 3 5 2 4 5 1 1 3 4 2 5

4 3 5 1 2 5 1 1 3 4 2 5

3 4 5 1 2 4 3 2 5 1 2 3

3 4 5 2 4 4 3 2 5 1 2 3

4 3 5 2 4 5 1 1 3 4 2 5

O1

O2

P1

P1

P2

J3 J4J2J1













3 1 2 4 1 2 4 3 3 1 3 2

3 1 2 2 1 2 4 3 3 1 3 4

1 3 2 1 2 2 4 3 4 1 3 3

1 3 2 1 4 2 4 3 2 1 3 3

3 1 2 4 1 2 4 3 3 1 3 2


O1

O2

P1

P1

P2


3 1 2 4 1 2 4 3 3 1 3 2

3 1 2 2 1 2 4 3 3 1 3 4

1 3 2 1 2 2 4 3 4 1 3 3

3 1 2 1 2 2 4 3 4 3 1 3

3 1 2 4 1 2 4 3 3 1 3 2


O1

O2

P1

P1

P2












3 1 2 4 1 2 4 3 3 1 3 2

3 1 2 4 3 2 4 1 3 1 3 2O1

P1


3 1 2 4 1 2 4 3 3 1 3 2

3 1 2 1 4 2 4 3 3 1 3 2O1

P1












3 1 2 4 1 2 4 3 3 1 3 2

3 1 2 3 3 2 4 4 1 1 3 2O1

P1


4 3 5 2 4 5 1 1 3 4 2 5

2 3 3 2 5 5 1 3 3 4 1 4

J3 J4J2J1

O1

P1








Table3Re

sults

onEx

perim

ent1

Prob

lem

119899times119898

ALa

PSO+SA

bMOGA

chP

SOd

hGA

eOOapproach

fBE

G-N

SGA-

IIg

119891 1119891 2

119891 3T

119891 1119891 2

119891 3T

119891 1119891 2

119891 3T

119891 1119891 2

119891 3T

119891 1119891 2

119891 3T

119891 1119891 2

119891 3T

119891 1119891 2

119891 3T

4times5

mdashmdash

mdashmdash

mdashmdash

mdashmdash

1110

32010

mdashmdash

mdashmdash

mdashmdash

mdashmdash

1110

32003

1110

32010

119

3411

934

128

3212

832

137

33

8times8

1613

75mdash

1512

75mdash

1511

81016

1412

77014

1512

75001

1613

73004

1412

77012

1613

7315

1275

1512

7515

1275

1512

7516

1373

1611

7716

1177

1613

7316

1373

10times10

75

45mdash

76

44mdash

85

42024

75

43020

75

43001

87

41005

75

43018

85

427

642

76

428

542

76

428

741

87

418

542

77

438

542

75

458

741

87

41

15times10

2311

95mdash

1211

91mdash

1111

9114

611

1191

136

1111

91003

1313

91009

1111

91005

2411

9112

1095

1210

9314

1291

1210

9311

1098

1110

9511

1095

a Thec

ompu

tatio

nconfi

guratio

nisno

tlisted

inAL

b Thec

ompu

tatio

nconfi

guratio

nisno

tlisted

inPS

O+SA

c Th

eCPU

timeo

naP

Cwith

2GHzC

PUand2G

Bof

RAM

mem

oryin

C++

d TheC

PUtim

eonap

ersonalcom

puterw

ith2G

HzC

PUand2G

Bof

RAM

mem

oryin

C++

e TheC

PUtim

eona3

0GHzP

entiu

min

Delp

hi

f TheC

PUtim

eonan

IntelC

orei5267

GHzp

rocessor

with

4GBRA

Mof

mem

oryin

Java

g TheC

PUtim

eonan

IntelC

orei3267

GHzp

rocessor

with

4GBRA

Min

MAT

LABR2

014a


O11

O12

O13

O21

O22

O23

O31

O32

O33 O34

O41

O42

13

1 2 3 4 5 6 7 8 9 10 11 12 130

M1

M2

M3

M4

M5


2634

130 260 390 520 650 780 910 1040 1170 1300 1430 1560 1690 1820 1950 2080 2210 2340 2470 26000

M1

M2

M3

M4

M5

M10

M6

M7

M8

M9









1198911 1198912 1198913 119879 1198911 1198912 1198913 119879

01a (10 times 5)2568 2505 11137 204 2568 2505 11137 1912572 2568 11137 2572 2568 111372594 2554 11137 2594 2554 11137

02a (10 times 5)2289 2263 11137 256 2487 2237 11137 2132313 2238 11137 3014 2234 11137

2564 2236 11137

03a (10 times 5) 2287 2248 11137 290 2492 2247 11137 252256 2252 11137 2516 2234 11137

04a (10 times 5)

2550 2503 11090 207 2882 2572 11069 2032569 2565 11076 2912 2552 110702579 2552 11080 2904 2626 110673095 2727 11064 3019 2665 11066

2997 2523 110732933 2688 110652953 2668 110662938 2606 110682955 2503 11074

05a (10 times 5)

2292 2252 11077 237 2740 2250 10981 2122293 2242 11091 2846 2212 109862297 2255 11054 2747 2229 109882315 2272 11063 2700 2255 109772343 2298 11050 2915 2231 109812358 2322 11038 2818 2208 109902376 2243 11022 2787 2350 109702904 2620 10941 2759 2320 109712945 2571 10941 2667 2237 109833056 2507 10941 2686 2277 10974

2641 2332 109742687 2221 110002779 2237 109812707 2299 109722702 2233 10989

06a (10 times 5)

2250 2233 11009 309 2673 2194 10893 2862254 2223 10994 2716 2194 108912398 2219 10973 2683 2245 108902437 2280 10988 2666 2248 108932744 2448 10850 2679 2229 108922902 2439 10847 2816 2186 109032967 2840 10839 2898 2186 10899

2713 2234 108892758 2209 108902722 2229 10890

07a (15 times 8)

2450 2413 16485 763 2927 2187 16485 6212457 2299 16485 2910 2213 164852484 2289 16485 2889 2264 16485

2891 2219 164852922 2190 16485

08a (15 times 8)2187 2102 16485 827 2621 2091 16485 6352171 2104 16485 2770 2089 16485

2780 2080 16485


Table 4 Continued


1198911 1198912 1198913 119879 1198911 1198912 1198913 119879

09a (15 times 8)2157 2113 16485 1016 2442 2103 16485 7262144 2119 16485 2470 2074 164852158 2102 16485 2455 2081 16485

10a (15 times 8)

2461 2433 16505 755 2998 2317 16487 6432470 2310 16537 3036 2363 164822478 2330 16533 3091 2491 164732482 2360 16499 2984 2319 164862501 2265 16547 3052 2377 164772501 2312 16528 3098 2491 164722547 2476 16490 3013 2293 164873064 2734 16464 3136 2457 16471

3025 2377 164803041 2351 164812980 2258 164943021 2319 16484

11a (15 times 8)

2182 2170 16449 1014 2551 2116 16229 7252202 2114 16476 2599 2131 162242210 2113 16442 2612 2146 162222337 2185 16377 2573 2058 162312874 2389 16247 2753 2186 162172894 2330 16247 2628 2102 162242962 2312 16247 2702 2098 16223

2658 2089 162242555 2111 162292664 2063 16228

12a (15 times 8)

2161 2107 16295 1192 2624 2049 16055 8222168 2130 16220 2620 2022 160682191 2084 16355 2587 2035 160652210 2103 16331 2614 2057 160512315 2125 16292 2585 2043 160612366 2105 16237 2659 2038 160552493 2297 16124 2617 2024 160652631 2309 16112 2575 2057 160572637 2303 16113 2587 2050 160592683 2397 16104 2582 2098 16052

13a (20 times 10)

2595 2036 160592566 2044 16061

2408 2326 21610 2399 2894 2277 21610 10272928 2233 216102896 2244 216102877 2282 21610

14a (20 times 10)2340 2251 21610 2905 2649 2186 21610 12392334 2258 21610 2664 2183 21610

2641 2207 216102738 2182 21610

15a (20 times 10)

2285 2247 21610 3329 2627 2216 21610 13602287 2218 21610 2681 2196 21610

2645 2215 216102652 2201 216102668 2197 216102682 2178 21610


Table 4 Continued


1198911 1198912 1198913 119879 1198911 1198912 1198913 119879

16a (20 times 10)

2447 2354 21602 2152 2965 2279 21534 11482450 2380 21590 3158 2303 215042487 2454 21584 3184 2400 214902492 2417 21576 3039 2279 215232540 2396 21547 3050 2318 215072550 2492 21545 3071 2286 215132568 2428 21540 3275 2281 215113013 2588 21478 3105 2352 214963106 2548 21478 3175 2344 21497

3180 2293 215073130 2312 215033034 2352 21503

17a (20 times 10)

2322 2240 21433 2847 2806 2195 21096 13922322 2280 21362 2750 2165 211202323 2238 21454 2791 2141 211142343 2224 21420 2716 2157 211292480 2285 21344 2849 2148 211072528 2231 21313 2769 2166 211112789 2448 21198 2856 2183 210982808 2303 21200 2773 2145 211112816 2370 21197 2856 2142 21105

2803 2203 210942759 2186 211062789 2176 211012772 2202 21101

18a (20 times 10)

2267 2235 21483 3301 2634 2156 21005 16232269 2206 21408 2642 2126 210062320 2208 21354 2638 2185 209992437 2221 21311 2638 2148 210022531 2310 21285 2674 2148 210002545 2305 21282 2644 2145 21002

2651 2140 210052641 2157 210002664 2145 210012621 2181 210022659 2140 210032661 2137 210042651 2181 209992669 2126 210052708 2125 21009







1198911 1198912 1198913 119879 1198911 1198912 1198913 119879 1198911 1198912 1198913 119879

MK01 (10 times 6)

42 42 162 478 42 39 158 049 42 39 158 03644 40 154 44 40 15443 40 155 43 40 15540 36 169 40 36 169

47 36 16747 37 16549 37 16450 38 162

MK02 (10 times 6)

28 28 155 302 26 26 151 075 26 26 151 07027 27 146 27 27 14629 27 145 29 27 14529 29 143 29 29 14331 31 141 31 31 14133 33 140 33 33 140

MK03 (15 times 8)

204 204 852 2614 204 199 855 475 204 199 855 366204 144 871 204 144 871204 135 882 204 135 882204 133 884 204 133 884213 199 850 213 199 850214 210 849 214 210 849221 199 847 221 199 847222 199 848 222 199 848231 188 848 231 188 848230 177 848 230 177 848

MK04 (15 times 8)

68 67 352 1774 66 63 345 176 66 63 345 16365 63 362 65 63 36263 61 371 63 61 37162 61 373 60 59 39061 60 382 74 54 34960 59 390 74 55 34873 55 350 72 72 34274 54 349 78 78 33974 55 348 72 66 34890 76 331 73 72 343

MK05 (15 times 4)

177 177 702 826 173 173 683 234 173 173 683 196175 175 682 175 175 682183 183 677 183 183 677185 185 676 179 179 679179 179 679 199 199 674

193 193 675183 183 677

MK06 (10 times 15)

67 431 1879 62 55 424 193 62 55 424 18565 54 417 65 54 41760 58 441 60 58 44162 60 440 62 60 44076 60 362 76 60 36276 74 356 76 74 35678 60 361 78 60 36173 72 360 73 72 36072 72 361 72 72 361100 90 330 100 90 330


Table 5 Continued


1198911 1198912 1198913 119879 1198911 1198912 1198913 119879 1198911 1198912 1198913 119879

MK07 (20 times 5)

150 150 717 568 139 139 693 492 139 139 693 312140 138 686 140 138 686144 144 673 144 144 673151 151 667 151 151 667157 157 662 157 157 662162 162 659 162 162 659166 166 657 166 166 657

MK08 (20 times 10)

523 523 2524 6767 523 515 2524 1204 523 515 2524 834523 497 2534 523 497 2534524 524 2519 524 524 2519578 578 2489 578 578 2489587 587 2484 587 587 2484

541 533 2516542 542 2511560 560 2505

MK09 (20 times 10)

311 299 2374 7776 311 299 2290 1948 311 299 2290 1015310 299 3514 310 299 3514311 301 2287 311 301 2287314 299 2315 314 299 2315315 299 2283 315 299 2283332 302 2265 332 302 2265329 301 2266 329 301 2266328 308 2259 328 308 2259325 299 2275 325 299 2275

MK10 (20 times 15)

227 221 1989 12252 224 219 1980 1787 224 219 1980 1238225 211 1976 235 225 1895233 214 1919 240 215 1905235 218 1897 246 215 1896235 225 1895 252 224 1884240 215 1905 256 211 1919240 216 1888 260 244 1869242 214 1913 266 254 1864246 215 1896 246 206 1938252 224 1884 246 204 1948256 211 1919 246 205 1940260 244 1869 251 202 1948266 254 1864 253 207 1930268 264 1858 249 203 1947276 256 1857 255 202 1943281 268 1854 260 210 1926217 207 2064 250 205 1935214 204 2082 254 203 1942






25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400 425 450 475 500 5250

M1

M2

M3

M4

M5

M10

M6

M7

M8

M9

541



127060 120 180 240 300 360 420 480 540 600 660 720 780 840 900 960 1020 1080 1140 1200 12600

M1

M2

M3

M4

M5

M10

M11

M12

M13

M14

M15

M16

M17

M18

M6

M7

M8

M9








Notations





Acknowledgments


References























































Advances in

FuzzySystems


Volume 2014












Journal of



Advances in

Multimedia


Biomedical Imaging


Advances in


RoboticsJournal of










Advances in




4 3 5 2 4 5 1 1 3 4 2 5

4 3 5 1 2 5 1 1 3 4 2 5

3 4 5 1 2 4 3 2 5 1 2 3

3 4 5 2 4 4 3 2 5 1 2 3

4 3 5 2 4 5 1 1 3 4 2 5

O1

O2

P1

P1

P2

J3 J4J2J1













3 1 2 4 1 2 4 3 3 1 3 2

3 1 2 2 1 2 4 3 3 1 3 4

1 3 2 1 2 2 4 3 4 1 3 3

1 3 2 1 4 2 4 3 2 1 3 3

3 1 2 4 1 2 4 3 3 1 3 2


O1

O2

P1

P1

P2


3 1 2 4 1 2 4 3 3 1 3 2

3 1 2 2 1 2 4 3 3 1 3 4

1 3 2 1 2 2 4 3 4 1 3 3

3 1 2 1 2 2 4 3 4 3 1 3

3 1 2 4 1 2 4 3 3 1 3 2


O1

O2

P1

P1

P2












3 1 2 4 1 2 4 3 3 1 3 2

3 1 2 4 3 2 4 1 3 1 3 2O1

P1


3 1 2 4 1 2 4 3 3 1 3 2

3 1 2 1 4 2 4 3 3 1 3 2O1

P1












3 1 2 4 1 2 4 3 3 1 3 2

3 1 2 3 3 2 4 4 1 1 3 2O1

P1


4 3 5 2 4 5 1 1 3 4 2 5

2 3 3 2 5 5 1 3 3 4 1 4

J3 J4J2J1

O1

P1








Table3Re

sults

onEx

perim

ent1

Prob

lem

119899times119898

ALa

PSO+SA

bMOGA

chP

SOd

hGA

eOOapproach

fBE

G-N

SGA-

IIg

119891 1119891 2

119891 3T

119891 1119891 2

119891 3T

119891 1119891 2

119891 3T

119891 1119891 2

119891 3T

119891 1119891 2

119891 3T

119891 1119891 2

119891 3T

119891 1119891 2

119891 3T

4times5

mdashmdash

mdashmdash

mdashmdash

mdashmdash

1110

32010

mdashmdash

mdashmdash

mdashmdash

mdashmdash

1110

32003

1110

32010

119

3411

934

128

3212

832

137

33

8times8

1613

75mdash

1512

75mdash

1511

81016

1412

77014

1512

75001

1613

73004

1412

77012

1613

7315

1275

1512

7515

1275

1512

7516

1373

1611

7716

1177

1613

7316

1373

10times10

75

45mdash

76

44mdash

85

42024

75

43020

75

43001

87

41005

75

43018

85

427

642

76

428

542

76

428

741

87

418

542

77

438

542

75

458

741

87

41

15times10

2311

95mdash

1211

91mdash

1111

9114

611

1191

136

1111

91003

1313

91009

1111

91005

2411

9112

1095

1210

9314

1291

1210

9311

1098

1110

9511

1095

a Thec

ompu

tatio

nconfi

guratio

nisno

tlisted

inAL

b Thec

ompu

tatio

nconfi

guratio

nisno

tlisted

inPS

O+SA

c Th

eCPU

timeo

naP

Cwith

2GHzC

PUand2G

Bof

RAM

mem

oryin

C++

d TheC

PUtim

eonap

ersonalcom

puterw

ith2G

HzC

PUand2G

Bof

RAM

mem

oryin

C++

e TheC

PUtim

eona3

0GHzP

entiu

min

Delp

hi

f TheC

PUtim

eonan

IntelC

orei5267

GHzp

rocessor

with

4GBRA

Mof

mem

oryin

Java

g TheC

PUtim

eonan

IntelC

orei3267

GHzp

rocessor

with

4GBRA

Min

MAT

LABR2

014a


O11

O12

O13

O21

O22

O23

O31

O32

O33 O34

O41

O42

13

1 2 3 4 5 6 7 8 9 10 11 12 130

M1

M2

M3

M4

M5


2634

130 260 390 520 650 780 910 1040 1170 1300 1430 1560 1690 1820 1950 2080 2210 2340 2470 26000

M1

M2

M3

M4

M5

M10

M6

M7

M8

M9









1198911 1198912 1198913 119879 1198911 1198912 1198913 119879

01a (10 times 5)2568 2505 11137 204 2568 2505 11137 1912572 2568 11137 2572 2568 111372594 2554 11137 2594 2554 11137

02a (10 times 5)2289 2263 11137 256 2487 2237 11137 2132313 2238 11137 3014 2234 11137

2564 2236 11137

03a (10 times 5) 2287 2248 11137 290 2492 2247 11137 252256 2252 11137 2516 2234 11137

04a (10 times 5)

2550 2503 11090 207 2882 2572 11069 2032569 2565 11076 2912 2552 110702579 2552 11080 2904 2626 110673095 2727 11064 3019 2665 11066

2997 2523 110732933 2688 110652953 2668 110662938 2606 110682955 2503 11074

05a (10 times 5)

2292 2252 11077 237 2740 2250 10981 2122293 2242 11091 2846 2212 109862297 2255 11054 2747 2229 109882315 2272 11063 2700 2255 109772343 2298 11050 2915 2231 109812358 2322 11038 2818 2208 109902376 2243 11022 2787 2350 109702904 2620 10941 2759 2320 109712945 2571 10941 2667 2237 109833056 2507 10941 2686 2277 10974

2641 2332 109742687 2221 110002779 2237 109812707 2299 109722702 2233 10989

06a (10 times 5)

2250 2233 11009 309 2673 2194 10893 2862254 2223 10994 2716 2194 108912398 2219 10973 2683 2245 108902437 2280 10988 2666 2248 108932744 2448 10850 2679 2229 108922902 2439 10847 2816 2186 109032967 2840 10839 2898 2186 10899

2713 2234 108892758 2209 108902722 2229 10890

07a (15 times 8)

2450 2413 16485 763 2927 2187 16485 6212457 2299 16485 2910 2213 164852484 2289 16485 2889 2264 16485

2891 2219 164852922 2190 16485

08a (15 times 8)2187 2102 16485 827 2621 2091 16485 6352171 2104 16485 2770 2089 16485

2780 2080 16485


Table 4 Continued


1198911 1198912 1198913 119879 1198911 1198912 1198913 119879

09a (15 times 8)2157 2113 16485 1016 2442 2103 16485 7262144 2119 16485 2470 2074 164852158 2102 16485 2455 2081 16485

10a (15 times 8)

2461 2433 16505 755 2998 2317 16487 6432470 2310 16537 3036 2363 164822478 2330 16533 3091 2491 164732482 2360 16499 2984 2319 164862501 2265 16547 3052 2377 164772501 2312 16528 3098 2491 164722547 2476 16490 3013 2293 164873064 2734 16464 3136 2457 16471

3025 2377 164803041 2351 164812980 2258 164943021 2319 16484

11a (15 times 8)

2182 2170 16449 1014 2551 2116 16229 7252202 2114 16476 2599 2131 162242210 2113 16442 2612 2146 162222337 2185 16377 2573 2058 162312874 2389 16247 2753 2186 162172894 2330 16247 2628 2102 162242962 2312 16247 2702 2098 16223

2658 2089 162242555 2111 162292664 2063 16228

12a (15 times 8)

2161 2107 16295 1192 2624 2049 16055 8222168 2130 16220 2620 2022 160682191 2084 16355 2587 2035 160652210 2103 16331 2614 2057 160512315 2125 16292 2585 2043 160612366 2105 16237 2659 2038 160552493 2297 16124 2617 2024 160652631 2309 16112 2575 2057 160572637 2303 16113 2587 2050 160592683 2397 16104 2582 2098 16052

13a (20 times 10)

2595 2036 160592566 2044 16061

2408 2326 21610 2399 2894 2277 21610 10272928 2233 216102896 2244 216102877 2282 21610

14a (20 times 10)2340 2251 21610 2905 2649 2186 21610 12392334 2258 21610 2664 2183 21610

2641 2207 216102738 2182 21610

15a (20 times 10)

2285 2247 21610 3329 2627 2216 21610 13602287 2218 21610 2681 2196 21610

2645 2215 216102652 2201 216102668 2197 216102682 2178 21610


Table 4 Continued


1198911 1198912 1198913 119879 1198911 1198912 1198913 119879

16a (20 times 10)

2447 2354 21602 2152 2965 2279 21534 11482450 2380 21590 3158 2303 215042487 2454 21584 3184 2400 214902492 2417 21576 3039 2279 215232540 2396 21547 3050 2318 215072550 2492 21545 3071 2286 215132568 2428 21540 3275 2281 215113013 2588 21478 3105 2352 214963106 2548 21478 3175 2344 21497

3180 2293 215073130 2312 215033034 2352 21503

17a (20 times 10)

2322 2240 21433 2847 2806 2195 21096 13922322 2280 21362 2750 2165 211202323 2238 21454 2791 2141 211142343 2224 21420 2716 2157 211292480 2285 21344 2849 2148 211072528 2231 21313 2769 2166 211112789 2448 21198 2856 2183 210982808 2303 21200 2773 2145 211112816 2370 21197 2856 2142 21105

2803 2203 210942759 2186 211062789 2176 211012772 2202 21101

18a (20 times 10)

2267 2235 21483 3301 2634 2156 21005 16232269 2206 21408 2642 2126 210062320 2208 21354 2638 2185 209992437 2221 21311 2638 2148 210022531 2310 21285 2674 2148 210002545 2305 21282 2644 2145 21002

2651 2140 210052641 2157 210002664 2145 210012621 2181 210022659 2140 210032661 2137 210042651 2181 209992669 2126 210052708 2125 21009







1198911 1198912 1198913 119879 1198911 1198912 1198913 119879 1198911 1198912 1198913 119879

MK01 (10 times 6)

42 42 162 478 42 39 158 049 42 39 158 03644 40 154 44 40 15443 40 155 43 40 15540 36 169 40 36 169

47 36 16747 37 16549 37 16450 38 162

MK02 (10 times 6)

28 28 155 302 26 26 151 075 26 26 151 07027 27 146 27 27 14629 27 145 29 27 14529 29 143 29 29 14331 31 141 31 31 14133 33 140 33 33 140

MK03 (15 times 8)

204 204 852 2614 204 199 855 475 204 199 855 366204 144 871 204 144 871204 135 882 204 135 882204 133 884 204 133 884213 199 850 213 199 850214 210 849 214 210 849221 199 847 221 199 847222 199 848 222 199 848231 188 848 231 188 848230 177 848 230 177 848

MK04 (15 times 8)

68 67 352 1774 66 63 345 176 66 63 345 16365 63 362 65 63 36263 61 371 63 61 37162 61 373 60 59 39061 60 382 74 54 34960 59 390 74 55 34873 55 350 72 72 34274 54 349 78 78 33974 55 348 72 66 34890 76 331 73 72 343

MK05 (15 times 4)

177 177 702 826 173 173 683 234 173 173 683 196175 175 682 175 175 682183 183 677 183 183 677185 185 676 179 179 679179 179 679 199 199 674

193 193 675183 183 677

MK06 (10 times 15)

67 431 1879 62 55 424 193 62 55 424 18565 54 417 65 54 41760 58 441 60 58 44162 60 440 62 60 44076 60 362 76 60 36276 74 356 76 74 35678 60 361 78 60 36173 72 360 73 72 36072 72 361 72 72 361100 90 330 100 90 330


Table 5 Continued


1198911 1198912 1198913 119879 1198911 1198912 1198913 119879 1198911 1198912 1198913 119879

MK07 (20 times 5)

150 150 717 568 139 139 693 492 139 139 693 312140 138 686 140 138 686144 144 673 144 144 673151 151 667 151 151 667157 157 662 157 157 662162 162 659 162 162 659166 166 657 166 166 657

MK08 (20 times 10)

523 523 2524 6767 523 515 2524 1204 523 515 2524 834523 497 2534 523 497 2534524 524 2519 524 524 2519578 578 2489 578 578 2489587 587 2484 587 587 2484

541 533 2516542 542 2511560 560 2505

MK09 (20 times 10)

311 299 2374 7776 311 299 2290 1948 311 299 2290 1015310 299 3514 310 299 3514311 301 2287 311 301 2287314 299 2315 314 299 2315315 299 2283 315 299 2283332 302 2265 332 302 2265329 301 2266 329 301 2266328 308 2259 328 308 2259325 299 2275 325 299 2275

MK10 (20 times 15)

227 221 1989 12252 224 219 1980 1787 224 219 1980 1238225 211 1976 235 225 1895233 214 1919 240 215 1905235 218 1897 246 215 1896235 225 1895 252 224 1884240 215 1905 256 211 1919240 216 1888 260 244 1869242 214 1913 266 254 1864246 215 1896 246 206 1938252 224 1884 246 204 1948256 211 1919 246 205 1940260 244 1869 251 202 1948266 254 1864 253 207 1930268 264 1858 249 203 1947276 256 1857 255 202 1943281 268 1854 260 210 1926217 207 2064 250 205 1935214 204 2082 254 203 1942






25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400 425 450 475 500 5250

M1

M2

M3

M4

M5

M10

M6

M7

M8

M9

541



127060 120 180 240 300 360 420 480 540 600 660 720 780 840 900 960 1020 1080 1140 1200 12600

M1

M2

M3

M4

M5

M10

M11

M12

M13

M14

M15

M16

M17

M18

M6

M7

M8

M9








Notations





Acknowledgments


References























































Advances in

FuzzySystems


Volume 2014












Journal of



Advances in

Multimedia


Biomedical Imaging


Advances in


RoboticsJournal of










Advances in




3 1 2 4 1 2 4 3 3 1 3 2

3 1 2 4 3 2 4 1 3 1 3 2O1

P1


3 1 2 4 1 2 4 3 3 1 3 2

3 1 2 1 4 2 4 3 3 1 3 2O1

P1












3 1 2 4 1 2 4 3 3 1 3 2

3 1 2 3 3 2 4 4 1 1 3 2O1

P1


4 3 5 2 4 5 1 1 3 4 2 5

2 3 3 2 5 5 1 3 3 4 1 4

J3 J4J2J1

O1

P1








Table3Re

sults

onEx

perim

ent1

Prob

lem

119899times119898

ALa

PSO+SA

bMOGA

chP

SOd

hGA

eOOapproach

fBE

G-N

SGA-

IIg

119891 1119891 2

119891 3T

119891 1119891 2

119891 3T

119891 1119891 2

119891 3T

119891 1119891 2

119891 3T

119891 1119891 2

119891 3T

119891 1119891 2

119891 3T

119891 1119891 2

119891 3T

4times5

mdashmdash

mdashmdash

mdashmdash

mdashmdash

1110

32010

mdashmdash

mdashmdash

mdashmdash

mdashmdash

1110

32003

1110

32010

119

3411

934

128

3212

832

137

33

8times8

1613

75mdash

1512

75mdash

1511

81016

1412

77014

1512

75001

1613

73004

1412

77012

1613

7315

1275

1512

7515

1275

1512

7516

1373

1611

7716

1177

1613

7316

1373

10times10

75

45mdash

76

44mdash

85

42024

75

43020

75

43001

87

41005

75

43018

85

427

642

76

428

542

76

428

741

87

418

542

77

438

542

75

458

741

87

41

15times10

2311

95mdash

1211

91mdash

1111

9114

611

1191

136

1111

91003

1313

91009

1111

91005

2411

9112

1095

1210

9314

1291

1210

9311

1098

1110

9511

1095

a Thec

ompu

tatio

nconfi

guratio

nisno

tlisted

inAL

b Thec

ompu

tatio

nconfi

guratio

nisno

tlisted

inPS

O+SA

c Th

eCPU

timeo

naP

Cwith

2GHzC

PUand2G

Bof

RAM

mem

oryin

C++

d TheC

PUtim

eonap

ersonalcom

puterw

ith2G

HzC

PUand2G

Bof

RAM

mem

oryin

C++

e TheC

PUtim

eona3

0GHzP

entiu

min

Delp

hi

f TheC

PUtim

eonan

IntelC

orei5267

GHzp

rocessor

with

4GBRA

Mof

mem

oryin

Java

g TheC

PUtim

eonan

IntelC

orei3267

GHzp

rocessor

with

4GBRA

Min

MAT

LABR2

014a


O11

O12

O13

O21

O22

O23

O31

O32

O33 O34

O41

O42

13

1 2 3 4 5 6 7 8 9 10 11 12 130

M1

M2

M3

M4

M5


2634

130 260 390 520 650 780 910 1040 1170 1300 1430 1560 1690 1820 1950 2080 2210 2340 2470 26000

M1

M2

M3

M4

M5

M10

M6

M7

M8

M9









1198911 1198912 1198913 119879 1198911 1198912 1198913 119879

01a (10 times 5)2568 2505 11137 204 2568 2505 11137 1912572 2568 11137 2572 2568 111372594 2554 11137 2594 2554 11137

02a (10 times 5)2289 2263 11137 256 2487 2237 11137 2132313 2238 11137 3014 2234 11137

2564 2236 11137

03a (10 times 5) 2287 2248 11137 290 2492 2247 11137 252256 2252 11137 2516 2234 11137

04a (10 times 5)

2550 2503 11090 207 2882 2572 11069 2032569 2565 11076 2912 2552 110702579 2552 11080 2904 2626 110673095 2727 11064 3019 2665 11066

2997 2523 110732933 2688 110652953 2668 110662938 2606 110682955 2503 11074

05a (10 times 5)

2292 2252 11077 237 2740 2250 10981 2122293 2242 11091 2846 2212 109862297 2255 11054 2747 2229 109882315 2272 11063 2700 2255 109772343 2298 11050 2915 2231 109812358 2322 11038 2818 2208 109902376 2243 11022 2787 2350 109702904 2620 10941 2759 2320 109712945 2571 10941 2667 2237 109833056 2507 10941 2686 2277 10974

2641 2332 109742687 2221 110002779 2237 109812707 2299 109722702 2233 10989

06a (10 times 5)

2250 2233 11009 309 2673 2194 10893 2862254 2223 10994 2716 2194 108912398 2219 10973 2683 2245 108902437 2280 10988 2666 2248 108932744 2448 10850 2679 2229 108922902 2439 10847 2816 2186 109032967 2840 10839 2898 2186 10899

2713 2234 108892758 2209 108902722 2229 10890

07a (15 times 8)

2450 2413 16485 763 2927 2187 16485 6212457 2299 16485 2910 2213 164852484 2289 16485 2889 2264 16485

2891 2219 164852922 2190 16485

08a (15 times 8)2187 2102 16485 827 2621 2091 16485 6352171 2104 16485 2770 2089 16485

2780 2080 16485


Table 4 Continued


1198911 1198912 1198913 119879 1198911 1198912 1198913 119879

09a (15 times 8)2157 2113 16485 1016 2442 2103 16485 7262144 2119 16485 2470 2074 164852158 2102 16485 2455 2081 16485

10a (15 times 8)

2461 2433 16505 755 2998 2317 16487 6432470 2310 16537 3036 2363 164822478 2330 16533 3091 2491 164732482 2360 16499 2984 2319 164862501 2265 16547 3052 2377 164772501 2312 16528 3098 2491 164722547 2476 16490 3013 2293 164873064 2734 16464 3136 2457 16471

3025 2377 164803041 2351 164812980 2258 164943021 2319 16484

11a (15 times 8)

2182 2170 16449 1014 2551 2116 16229 7252202 2114 16476 2599 2131 162242210 2113 16442 2612 2146 162222337 2185 16377 2573 2058 162312874 2389 16247 2753 2186 162172894 2330 16247 2628 2102 162242962 2312 16247 2702 2098 16223

2658 2089 162242555 2111 162292664 2063 16228

12a (15 times 8)

2161 2107 16295 1192 2624 2049 16055 8222168 2130 16220 2620 2022 160682191 2084 16355 2587 2035 160652210 2103 16331 2614 2057 160512315 2125 16292 2585 2043 160612366 2105 16237 2659 2038 160552493 2297 16124 2617 2024 160652631 2309 16112 2575 2057 160572637 2303 16113 2587 2050 160592683 2397 16104 2582 2098 16052

13a (20 times 10)

2595 2036 160592566 2044 16061

2408 2326 21610 2399 2894 2277 21610 10272928 2233 216102896 2244 216102877 2282 21610

14a (20 times 10)2340 2251 21610 2905 2649 2186 21610 12392334 2258 21610 2664 2183 21610

2641 2207 216102738 2182 21610

15a (20 times 10)

2285 2247 21610 3329 2627 2216 21610 13602287 2218 21610 2681 2196 21610

2645 2215 216102652 2201 216102668 2197 216102682 2178 21610


Table 4 Continued


1198911 1198912 1198913 119879 1198911 1198912 1198913 119879

16a (20 times 10)

2447 2354 21602 2152 2965 2279 21534 11482450 2380 21590 3158 2303 215042487 2454 21584 3184 2400 214902492 2417 21576 3039 2279 215232540 2396 21547 3050 2318 215072550 2492 21545 3071 2286 215132568 2428 21540 3275 2281 215113013 2588 21478 3105 2352 214963106 2548 21478 3175 2344 21497

3180 2293 215073130 2312 215033034 2352 21503

17a (20 times 10)

2322 2240 21433 2847 2806 2195 21096 13922322 2280 21362 2750 2165 211202323 2238 21454 2791 2141 211142343 2224 21420 2716 2157 211292480 2285 21344 2849 2148 211072528 2231 21313 2769 2166 211112789 2448 21198 2856 2183 210982808 2303 21200 2773 2145 211112816 2370 21197 2856 2142 21105

2803 2203 210942759 2186 211062789 2176 211012772 2202 21101

18a (20 times 10)

2267 2235 21483 3301 2634 2156 21005 16232269 2206 21408 2642 2126 210062320 2208 21354 2638 2185 209992437 2221 21311 2638 2148 210022531 2310 21285 2674 2148 210002545 2305 21282 2644 2145 21002

2651 2140 210052641 2157 210002664 2145 210012621 2181 210022659 2140 210032661 2137 210042651 2181 209992669 2126 210052708 2125 21009







1198911 1198912 1198913 119879 1198911 1198912 1198913 119879 1198911 1198912 1198913 119879

MK01 (10 times 6)

42 42 162 478 42 39 158 049 42 39 158 03644 40 154 44 40 15443 40 155 43 40 15540 36 169 40 36 169

47 36 16747 37 16549 37 16450 38 162

MK02 (10 times 6)

28 28 155 302 26 26 151 075 26 26 151 07027 27 146 27 27 14629 27 145 29 27 14529 29 143 29 29 14331 31 141 31 31 14133 33 140 33 33 140

MK03 (15 times 8)

204 204 852 2614 204 199 855 475 204 199 855 366204 144 871 204 144 871204 135 882 204 135 882204 133 884 204 133 884213 199 850 213 199 850214 210 849 214 210 849221 199 847 221 199 847222 199 848 222 199 848231 188 848 231 188 848230 177 848 230 177 848

MK04 (15 times 8)

68 67 352 1774 66 63 345 176 66 63 345 16365 63 362 65 63 36263 61 371 63 61 37162 61 373 60 59 39061 60 382 74 54 34960 59 390 74 55 34873 55 350 72 72 34274 54 349 78 78 33974 55 348 72 66 34890 76 331 73 72 343

MK05 (15 times 4)

177 177 702 826 173 173 683 234 173 173 683 196175 175 682 175 175 682183 183 677 183 183 677185 185 676 179 179 679179 179 679 199 199 674

193 193 675183 183 677

MK06 (10 times 15)

67 431 1879 62 55 424 193 62 55 424 18565 54 417 65 54 41760 58 441 60 58 44162 60 440 62 60 44076 60 362 76 60 36276 74 356 76 74 35678 60 361 78 60 36173 72 360 73 72 36072 72 361 72 72 361100 90 330 100 90 330


Table 5 Continued


1198911 1198912 1198913 119879 1198911 1198912 1198913 119879 1198911 1198912 1198913 119879

MK07 (20 times 5)

150 150 717 568 139 139 693 492 139 139 693 312140 138 686 140 138 686144 144 673 144 144 673151 151 667 151 151 667157 157 662 157 157 662162 162 659 162 162 659166 166 657 166 166 657

MK08 (20 times 10)

523 523 2524 6767 523 515 2524 1204 523 515 2524 834523 497 2534 523 497 2534524 524 2519 524 524 2519578 578 2489 578 578 2489587 587 2484 587 587 2484

541 533 2516542 542 2511560 560 2505

MK09 (20 times 10)

311 299 2374 7776 311 299 2290 1948 311 299 2290 1015310 299 3514 310 299 3514311 301 2287 311 301 2287314 299 2315 314 299 2315315 299 2283 315 299 2283332 302 2265 332 302 2265329 301 2266 329 301 2266328 308 2259 328 308 2259325 299 2275 325 299 2275

MK10 (20 times 15)

227 221 1989 12252 224 219 1980 1787 224 219 1980 1238225 211 1976 235 225 1895233 214 1919 240 215 1905235 218 1897 246 215 1896235 225 1895 252 224 1884240 215 1905 256 211 1919240 216 1888 260 244 1869242 214 1913 266 254 1864246 215 1896 246 206 1938252 224 1884 246 204 1948256 211 1919 246 205 1940260 244 1869 251 202 1948266 254 1864 253 207 1930268 264 1858 249 203 1947276 256 1857 255 202 1943281 268 1854 260 210 1926217 207 2064 250 205 1935214 204 2082 254 203 1942






25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400 425 450 475 500 5250

M1

M2

M3

M4

M5

M10

M6

M7

M8

M9

541



127060 120 180 240 300 360 420 480 540 600 660 720 780 840 900 960 1020 1080 1140 1200 12600

M1

M2

M3

M4

M5

M10

M11

M12

M13

M14

M15

M16

M17

M18

M6

M7

M8

M9








Notations





Acknowledgments


References























































Advances in

FuzzySystems


Volume 2014












Journal of



Advances in

Multimedia


Biomedical Imaging


Advances in


RoboticsJournal of










Advances in




Table3Re

sults

onEx

perim

ent1

Prob

lem

119899times119898

ALa

PSO+SA

bMOGA

chP

SOd

hGA

eOOapproach

fBE

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SGA-

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119891 1119891 2

119891 3T

119891 1119891 2

119891 3T

119891 1119891 2

119891 3T

119891 1119891 2

119891 3T

119891 1119891 2

119891 3T

119891 1119891 2

119891 3T

119891 1119891 2

119891 3T

4times5

mdashmdash

mdashmdash

mdashmdash

mdashmdash

1110

32010

mdashmdash

mdashmdash

mdashmdash

mdashmdash

1110

32003

1110

32010

119

3411

934

128

3212

832

137

33

8times8

1613

75mdash

1512

75mdash

1511

81016

1412

77014

1512

75001

1613

73004

1412

77012

1613

7315

1275

1512

7515

1275

1512

7516

1373

1611

7716

1177

1613

7316

1373

10times10

75

45mdash

76

44mdash

85

42024

75

43020

75

43001

87

41005

75

43018

85

427

642

76

428

542

76

428

741

87

418

542

77

438

542

75

458

741

87

41

15times10

2311

95mdash

1211

91mdash

1111

9114

611

1191

136

1111

91003

1313

91009

1111

91005

2411

9112

1095

1210

9314

1291

1210

9311

1098

1110

9511

1095

a Thec

ompu

tatio

nconfi

guratio

nisno

tlisted

inAL

b Thec

ompu

tatio

nconfi

guratio

nisno

tlisted

inPS

O+SA

c Th

eCPU

timeo

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2GHzC

PUand2G

Bof

RAM

mem

oryin

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d TheC

PUtim

eonap

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puterw

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HzC

PUand2G

Bof

RAM

mem

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e TheC

PUtim

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0GHzP

entiu

min

Delp

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f TheC

PUtim

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IntelC

orei5267

GHzp

rocessor

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4GBRA

Mof

mem

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g TheC

PUtim

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with

4GBRA

Min

MAT

LABR2

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O11

O12

O13

O21

O22

O23

O31

O32

O33 O34

O41

O42

13

1 2 3 4 5 6 7 8 9 10 11 12 130

M1

M2

M3

M4

M5


2634

130 260 390 520 650 780 910 1040 1170 1300 1430 1560 1690 1820 1950 2080 2210 2340 2470 26000

M1

M2

M3

M4

M5

M10

M6

M7

M8

M9









1198911 1198912 1198913 119879 1198911 1198912 1198913 119879

01a (10 times 5)2568 2505 11137 204 2568 2505 11137 1912572 2568 11137 2572 2568 111372594 2554 11137 2594 2554 11137

02a (10 times 5)2289 2263 11137 256 2487 2237 11137 2132313 2238 11137 3014 2234 11137

2564 2236 11137

03a (10 times 5) 2287 2248 11137 290 2492 2247 11137 252256 2252 11137 2516 2234 11137

04a (10 times 5)

2550 2503 11090 207 2882 2572 11069 2032569 2565 11076 2912 2552 110702579 2552 11080 2904 2626 110673095 2727 11064 3019 2665 11066

2997 2523 110732933 2688 110652953 2668 110662938 2606 110682955 2503 11074

05a (10 times 5)

2292 2252 11077 237 2740 2250 10981 2122293 2242 11091 2846 2212 109862297 2255 11054 2747 2229 109882315 2272 11063 2700 2255 109772343 2298 11050 2915 2231 109812358 2322 11038 2818 2208 109902376 2243 11022 2787 2350 109702904 2620 10941 2759 2320 109712945 2571 10941 2667 2237 109833056 2507 10941 2686 2277 10974

2641 2332 109742687 2221 110002779 2237 109812707 2299 109722702 2233 10989

06a (10 times 5)

2250 2233 11009 309 2673 2194 10893 2862254 2223 10994 2716 2194 108912398 2219 10973 2683 2245 108902437 2280 10988 2666 2248 108932744 2448 10850 2679 2229 108922902 2439 10847 2816 2186 109032967 2840 10839 2898 2186 10899

2713 2234 108892758 2209 108902722 2229 10890

07a (15 times 8)

2450 2413 16485 763 2927 2187 16485 6212457 2299 16485 2910 2213 164852484 2289 16485 2889 2264 16485

2891 2219 164852922 2190 16485

08a (15 times 8)2187 2102 16485 827 2621 2091 16485 6352171 2104 16485 2770 2089 16485

2780 2080 16485


Table 4 Continued


1198911 1198912 1198913 119879 1198911 1198912 1198913 119879

09a (15 times 8)2157 2113 16485 1016 2442 2103 16485 7262144 2119 16485 2470 2074 164852158 2102 16485 2455 2081 16485

10a (15 times 8)

2461 2433 16505 755 2998 2317 16487 6432470 2310 16537 3036 2363 164822478 2330 16533 3091 2491 164732482 2360 16499 2984 2319 164862501 2265 16547 3052 2377 164772501 2312 16528 3098 2491 164722547 2476 16490 3013 2293 164873064 2734 16464 3136 2457 16471

3025 2377 164803041 2351 164812980 2258 164943021 2319 16484

11a (15 times 8)

2182 2170 16449 1014 2551 2116 16229 7252202 2114 16476 2599 2131 162242210 2113 16442 2612 2146 162222337 2185 16377 2573 2058 162312874 2389 16247 2753 2186 162172894 2330 16247 2628 2102 162242962 2312 16247 2702 2098 16223

2658 2089 162242555 2111 162292664 2063 16228

12a (15 times 8)

2161 2107 16295 1192 2624 2049 16055 8222168 2130 16220 2620 2022 160682191 2084 16355 2587 2035 160652210 2103 16331 2614 2057 160512315 2125 16292 2585 2043 160612366 2105 16237 2659 2038 160552493 2297 16124 2617 2024 160652631 2309 16112 2575 2057 160572637 2303 16113 2587 2050 160592683 2397 16104 2582 2098 16052

13a (20 times 10)

2595 2036 160592566 2044 16061

2408 2326 21610 2399 2894 2277 21610 10272928 2233 216102896 2244 216102877 2282 21610

14a (20 times 10)2340 2251 21610 2905 2649 2186 21610 12392334 2258 21610 2664 2183 21610

2641 2207 216102738 2182 21610

15a (20 times 10)

2285 2247 21610 3329 2627 2216 21610 13602287 2218 21610 2681 2196 21610

2645 2215 216102652 2201 216102668 2197 216102682 2178 21610


Table 4 Continued


1198911 1198912 1198913 119879 1198911 1198912 1198913 119879

16a (20 times 10)

2447 2354 21602 2152 2965 2279 21534 11482450 2380 21590 3158 2303 215042487 2454 21584 3184 2400 214902492 2417 21576 3039 2279 215232540 2396 21547 3050 2318 215072550 2492 21545 3071 2286 215132568 2428 21540 3275 2281 215113013 2588 21478 3105 2352 214963106 2548 21478 3175 2344 21497

3180 2293 215073130 2312 215033034 2352 21503

17a (20 times 10)

2322 2240 21433 2847 2806 2195 21096 13922322 2280 21362 2750 2165 211202323 2238 21454 2791 2141 211142343 2224 21420 2716 2157 211292480 2285 21344 2849 2148 211072528 2231 21313 2769 2166 211112789 2448 21198 2856 2183 210982808 2303 21200 2773 2145 211112816 2370 21197 2856 2142 21105

2803 2203 210942759 2186 211062789 2176 211012772 2202 21101

18a (20 times 10)

2267 2235 21483 3301 2634 2156 21005 16232269 2206 21408 2642 2126 210062320 2208 21354 2638 2185 209992437 2221 21311 2638 2148 210022531 2310 21285 2674 2148 210002545 2305 21282 2644 2145 21002

2651 2140 210052641 2157 210002664 2145 210012621 2181 210022659 2140 210032661 2137 210042651 2181 209992669 2126 210052708 2125 21009







1198911 1198912 1198913 119879 1198911 1198912 1198913 119879 1198911 1198912 1198913 119879

MK01 (10 times 6)

42 42 162 478 42 39 158 049 42 39 158 03644 40 154 44 40 15443 40 155 43 40 15540 36 169 40 36 169

47 36 16747 37 16549 37 16450 38 162

MK02 (10 times 6)

28 28 155 302 26 26 151 075 26 26 151 07027 27 146 27 27 14629 27 145 29 27 14529 29 143 29 29 14331 31 141 31 31 14133 33 140 33 33 140

MK03 (15 times 8)

204 204 852 2614 204 199 855 475 204 199 855 366204 144 871 204 144 871204 135 882 204 135 882204 133 884 204 133 884213 199 850 213 199 850214 210 849 214 210 849221 199 847 221 199 847222 199 848 222 199 848231 188 848 231 188 848230 177 848 230 177 848

MK04 (15 times 8)

68 67 352 1774 66 63 345 176 66 63 345 16365 63 362 65 63 36263 61 371 63 61 37162 61 373 60 59 39061 60 382 74 54 34960 59 390 74 55 34873 55 350 72 72 34274 54 349 78 78 33974 55 348 72 66 34890 76 331 73 72 343

MK05 (15 times 4)

177 177 702 826 173 173 683 234 173 173 683 196175 175 682 175 175 682183 183 677 183 183 677185 185 676 179 179 679179 179 679 199 199 674

193 193 675183 183 677

MK06 (10 times 15)

67 431 1879 62 55 424 193 62 55 424 18565 54 417 65 54 41760 58 441 60 58 44162 60 440 62 60 44076 60 362 76 60 36276 74 356 76 74 35678 60 361 78 60 36173 72 360 73 72 36072 72 361 72 72 361100 90 330 100 90 330


Table 5 Continued


1198911 1198912 1198913 119879 1198911 1198912 1198913 119879 1198911 1198912 1198913 119879

MK07 (20 times 5)

150 150 717 568 139 139 693 492 139 139 693 312140 138 686 140 138 686144 144 673 144 144 673151 151 667 151 151 667157 157 662 157 157 662162 162 659 162 162 659166 166 657 166 166 657

MK08 (20 times 10)

523 523 2524 6767 523 515 2524 1204 523 515 2524 834523 497 2534 523 497 2534524 524 2519 524 524 2519578 578 2489 578 578 2489587 587 2484 587 587 2484

541 533 2516542 542 2511560 560 2505

MK09 (20 times 10)

311 299 2374 7776 311 299 2290 1948 311 299 2290 1015310 299 3514 310 299 3514311 301 2287 311 301 2287314 299 2315 314 299 2315315 299 2283 315 299 2283332 302 2265 332 302 2265329 301 2266 329 301 2266328 308 2259 328 308 2259325 299 2275 325 299 2275

MK10 (20 times 15)

227 221 1989 12252 224 219 1980 1787 224 219 1980 1238225 211 1976 235 225 1895233 214 1919 240 215 1905235 218 1897 246 215 1896235 225 1895 252 224 1884240 215 1905 256 211 1919240 216 1888 260 244 1869242 214 1913 266 254 1864246 215 1896 246 206 1938252 224 1884 246 204 1948256 211 1919 246 205 1940260 244 1869 251 202 1948266 254 1864 253 207 1930268 264 1858 249 203 1947276 256 1857 255 202 1943281 268 1854 260 210 1926217 207 2064 250 205 1935214 204 2082 254 203 1942






25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400 425 450 475 500 5250

M1

M2

M3

M4

M5

M10

M6

M7

M8

M9

541



127060 120 180 240 300 360 420 480 540 600 660 720 780 840 900 960 1020 1080 1140 1200 12600

M1

M2

M3

M4

M5

M10

M11

M12

M13

M14

M15

M16

M17

M18

M6

M7

M8

M9








Notations





Acknowledgments


References























































Advances in

FuzzySystems


Volume 2014












Journal of



Advances in

Multimedia


Biomedical Imaging


Advances in


RoboticsJournal of










Advances in




O11

O12

O13

O21

O22

O23

O31

O32

O33 O34

O41

O42

13

1 2 3 4 5 6 7 8 9 10 11 12 130

M1

M2

M3

M4

M5


2634

130 260 390 520 650 780 910 1040 1170 1300 1430 1560 1690 1820 1950 2080 2210 2340 2470 26000

M1

M2

M3

M4

M5

M10

M6

M7

M8

M9









1198911 1198912 1198913 119879 1198911 1198912 1198913 119879

01a (10 times 5)2568 2505 11137 204 2568 2505 11137 1912572 2568 11137 2572 2568 111372594 2554 11137 2594 2554 11137

02a (10 times 5)2289 2263 11137 256 2487 2237 11137 2132313 2238 11137 3014 2234 11137

2564 2236 11137

03a (10 times 5) 2287 2248 11137 290 2492 2247 11137 252256 2252 11137 2516 2234 11137

04a (10 times 5)

2550 2503 11090 207 2882 2572 11069 2032569 2565 11076 2912 2552 110702579 2552 11080 2904 2626 110673095 2727 11064 3019 2665 11066

2997 2523 110732933 2688 110652953 2668 110662938 2606 110682955 2503 11074

05a (10 times 5)

2292 2252 11077 237 2740 2250 10981 2122293 2242 11091 2846 2212 109862297 2255 11054 2747 2229 109882315 2272 11063 2700 2255 109772343 2298 11050 2915 2231 109812358 2322 11038 2818 2208 109902376 2243 11022 2787 2350 109702904 2620 10941 2759 2320 109712945 2571 10941 2667 2237 109833056 2507 10941 2686 2277 10974

2641 2332 109742687 2221 110002779 2237 109812707 2299 109722702 2233 10989

06a (10 times 5)

2250 2233 11009 309 2673 2194 10893 2862254 2223 10994 2716 2194 108912398 2219 10973 2683 2245 108902437 2280 10988 2666 2248 108932744 2448 10850 2679 2229 108922902 2439 10847 2816 2186 109032967 2840 10839 2898 2186 10899

2713 2234 108892758 2209 108902722 2229 10890

07a (15 times 8)

2450 2413 16485 763 2927 2187 16485 6212457 2299 16485 2910 2213 164852484 2289 16485 2889 2264 16485

2891 2219 164852922 2190 16485

08a (15 times 8)2187 2102 16485 827 2621 2091 16485 6352171 2104 16485 2770 2089 16485

2780 2080 16485


Table 4 Continued


1198911 1198912 1198913 119879 1198911 1198912 1198913 119879

09a (15 times 8)2157 2113 16485 1016 2442 2103 16485 7262144 2119 16485 2470 2074 164852158 2102 16485 2455 2081 16485

10a (15 times 8)

2461 2433 16505 755 2998 2317 16487 6432470 2310 16537 3036 2363 164822478 2330 16533 3091 2491 164732482 2360 16499 2984 2319 164862501 2265 16547 3052 2377 164772501 2312 16528 3098 2491 164722547 2476 16490 3013 2293 164873064 2734 16464 3136 2457 16471

3025 2377 164803041 2351 164812980 2258 164943021 2319 16484

11a (15 times 8)

2182 2170 16449 1014 2551 2116 16229 7252202 2114 16476 2599 2131 162242210 2113 16442 2612 2146 162222337 2185 16377 2573 2058 162312874 2389 16247 2753 2186 162172894 2330 16247 2628 2102 162242962 2312 16247 2702 2098 16223

2658 2089 162242555 2111 162292664 2063 16228

12a (15 times 8)

2161 2107 16295 1192 2624 2049 16055 8222168 2130 16220 2620 2022 160682191 2084 16355 2587 2035 160652210 2103 16331 2614 2057 160512315 2125 16292 2585 2043 160612366 2105 16237 2659 2038 160552493 2297 16124 2617 2024 160652631 2309 16112 2575 2057 160572637 2303 16113 2587 2050 160592683 2397 16104 2582 2098 16052

13a (20 times 10)

2595 2036 160592566 2044 16061

2408 2326 21610 2399 2894 2277 21610 10272928 2233 216102896 2244 216102877 2282 21610

14a (20 times 10)2340 2251 21610 2905 2649 2186 21610 12392334 2258 21610 2664 2183 21610

2641 2207 216102738 2182 21610

15a (20 times 10)

2285 2247 21610 3329 2627 2216 21610 13602287 2218 21610 2681 2196 21610

2645 2215 216102652 2201 216102668 2197 216102682 2178 21610


Table 4 Continued


1198911 1198912 1198913 119879 1198911 1198912 1198913 119879

16a (20 times 10)

2447 2354 21602 2152 2965 2279 21534 11482450 2380 21590 3158 2303 215042487 2454 21584 3184 2400 214902492 2417 21576 3039 2279 215232540 2396 21547 3050 2318 215072550 2492 21545 3071 2286 215132568 2428 21540 3275 2281 215113013 2588 21478 3105 2352 214963106 2548 21478 3175 2344 21497

3180 2293 215073130 2312 215033034 2352 21503

17a (20 times 10)

2322 2240 21433 2847 2806 2195 21096 13922322 2280 21362 2750 2165 211202323 2238 21454 2791 2141 211142343 2224 21420 2716 2157 211292480 2285 21344 2849 2148 211072528 2231 21313 2769 2166 211112789 2448 21198 2856 2183 210982808 2303 21200 2773 2145 211112816 2370 21197 2856 2142 21105

2803 2203 210942759 2186 211062789 2176 211012772 2202 21101

18a (20 times 10)

2267 2235 21483 3301 2634 2156 21005 16232269 2206 21408 2642 2126 210062320 2208 21354 2638 2185 209992437 2221 21311 2638 2148 210022531 2310 21285 2674 2148 210002545 2305 21282 2644 2145 21002

2651 2140 210052641 2157 210002664 2145 210012621 2181 210022659 2140 210032661 2137 210042651 2181 209992669 2126 210052708 2125 21009







1198911 1198912 1198913 119879 1198911 1198912 1198913 119879 1198911 1198912 1198913 119879

MK01 (10 times 6)

42 42 162 478 42 39 158 049 42 39 158 03644 40 154 44 40 15443 40 155 43 40 15540 36 169 40 36 169

47 36 16747 37 16549 37 16450 38 162

MK02 (10 times 6)

28 28 155 302 26 26 151 075 26 26 151 07027 27 146 27 27 14629 27 145 29 27 14529 29 143 29 29 14331 31 141 31 31 14133 33 140 33 33 140

MK03 (15 times 8)

204 204 852 2614 204 199 855 475 204 199 855 366204 144 871 204 144 871204 135 882 204 135 882204 133 884 204 133 884213 199 850 213 199 850214 210 849 214 210 849221 199 847 221 199 847222 199 848 222 199 848231 188 848 231 188 848230 177 848 230 177 848

MK04 (15 times 8)

68 67 352 1774 66 63 345 176 66 63 345 16365 63 362 65 63 36263 61 371 63 61 37162 61 373 60 59 39061 60 382 74 54 34960 59 390 74 55 34873 55 350 72 72 34274 54 349 78 78 33974 55 348 72 66 34890 76 331 73 72 343

MK05 (15 times 4)

177 177 702 826 173 173 683 234 173 173 683 196175 175 682 175 175 682183 183 677 183 183 677185 185 676 179 179 679179 179 679 199 199 674

193 193 675183 183 677

MK06 (10 times 15)

67 431 1879 62 55 424 193 62 55 424 18565 54 417 65 54 41760 58 441 60 58 44162 60 440 62 60 44076 60 362 76 60 36276 74 356 76 74 35678 60 361 78 60 36173 72 360 73 72 36072 72 361 72 72 361100 90 330 100 90 330


Table 5 Continued


1198911 1198912 1198913 119879 1198911 1198912 1198913 119879 1198911 1198912 1198913 119879

MK07 (20 times 5)

150 150 717 568 139 139 693 492 139 139 693 312140 138 686 140 138 686144 144 673 144 144 673151 151 667 151 151 667157 157 662 157 157 662162 162 659 162 162 659166 166 657 166 166 657

MK08 (20 times 10)

523 523 2524 6767 523 515 2524 1204 523 515 2524 834523 497 2534 523 497 2534524 524 2519 524 524 2519578 578 2489 578 578 2489587 587 2484 587 587 2484

541 533 2516542 542 2511560 560 2505

MK09 (20 times 10)

311 299 2374 7776 311 299 2290 1948 311 299 2290 1015310 299 3514 310 299 3514311 301 2287 311 301 2287314 299 2315 314 299 2315315 299 2283 315 299 2283332 302 2265 332 302 2265329 301 2266 329 301 2266328 308 2259 328 308 2259325 299 2275 325 299 2275

MK10 (20 times 15)

227 221 1989 12252 224 219 1980 1787 224 219 1980 1238225 211 1976 235 225 1895233 214 1919 240 215 1905235 218 1897 246 215 1896235 225 1895 252 224 1884240 215 1905 256 211 1919240 216 1888 260 244 1869242 214 1913 266 254 1864246 215 1896 246 206 1938252 224 1884 246 204 1948256 211 1919 246 205 1940260 244 1869 251 202 1948266 254 1864 253 207 1930268 264 1858 249 203 1947276 256 1857 255 202 1943281 268 1854 260 210 1926217 207 2064 250 205 1935214 204 2082 254 203 1942






25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400 425 450 475 500 5250

M1

M2

M3

M4

M5

M10

M6

M7

M8

M9

541



127060 120 180 240 300 360 420 480 540 600 660 720 780 840 900 960 1020 1080 1140 1200 12600

M1

M2

M3

M4

M5

M10

M11

M12

M13

M14

M15

M16

M17

M18

M6

M7

M8

M9








Notations





Acknowledgments


References























































Advances in

FuzzySystems


Volume 2014












Journal of



Advances in

Multimedia


Biomedical Imaging


Advances in


RoboticsJournal of










Advances in






1198911 1198912 1198913 119879 1198911 1198912 1198913 119879

01a (10 times 5)2568 2505 11137 204 2568 2505 11137 1912572 2568 11137 2572 2568 111372594 2554 11137 2594 2554 11137

02a (10 times 5)2289 2263 11137 256 2487 2237 11137 2132313 2238 11137 3014 2234 11137

2564 2236 11137

03a (10 times 5) 2287 2248 11137 290 2492 2247 11137 252256 2252 11137 2516 2234 11137

04a (10 times 5)

2550 2503 11090 207 2882 2572 11069 2032569 2565 11076 2912 2552 110702579 2552 11080 2904 2626 110673095 2727 11064 3019 2665 11066

2997 2523 110732933 2688 110652953 2668 110662938 2606 110682955 2503 11074

05a (10 times 5)

2292 2252 11077 237 2740 2250 10981 2122293 2242 11091 2846 2212 109862297 2255 11054 2747 2229 109882315 2272 11063 2700 2255 109772343 2298 11050 2915 2231 109812358 2322 11038 2818 2208 109902376 2243 11022 2787 2350 109702904 2620 10941 2759 2320 109712945 2571 10941 2667 2237 109833056 2507 10941 2686 2277 10974

2641 2332 109742687 2221 110002779 2237 109812707 2299 109722702 2233 10989

06a (10 times 5)

2250 2233 11009 309 2673 2194 10893 2862254 2223 10994 2716 2194 108912398 2219 10973 2683 2245 108902437 2280 10988 2666 2248 108932744 2448 10850 2679 2229 108922902 2439 10847 2816 2186 109032967 2840 10839 2898 2186 10899

2713 2234 108892758 2209 108902722 2229 10890

07a (15 times 8)

2450 2413 16485 763 2927 2187 16485 6212457 2299 16485 2910 2213 164852484 2289 16485 2889 2264 16485

2891 2219 164852922 2190 16485

08a (15 times 8)2187 2102 16485 827 2621 2091 16485 6352171 2104 16485 2770 2089 16485

2780 2080 16485


Table 4 Continued


1198911 1198912 1198913 119879 1198911 1198912 1198913 119879

09a (15 times 8)2157 2113 16485 1016 2442 2103 16485 7262144 2119 16485 2470 2074 164852158 2102 16485 2455 2081 16485

10a (15 times 8)

2461 2433 16505 755 2998 2317 16487 6432470 2310 16537 3036 2363 164822478 2330 16533 3091 2491 164732482 2360 16499 2984 2319 164862501 2265 16547 3052 2377 164772501 2312 16528 3098 2491 164722547 2476 16490 3013 2293 164873064 2734 16464 3136 2457 16471

3025 2377 164803041 2351 164812980 2258 164943021 2319 16484

11a (15 times 8)

2182 2170 16449 1014 2551 2116 16229 7252202 2114 16476 2599 2131 162242210 2113 16442 2612 2146 162222337 2185 16377 2573 2058 162312874 2389 16247 2753 2186 162172894 2330 16247 2628 2102 162242962 2312 16247 2702 2098 16223

2658 2089 162242555 2111 162292664 2063 16228

12a (15 times 8)

2161 2107 16295 1192 2624 2049 16055 8222168 2130 16220 2620 2022 160682191 2084 16355 2587 2035 160652210 2103 16331 2614 2057 160512315 2125 16292 2585 2043 160612366 2105 16237 2659 2038 160552493 2297 16124 2617 2024 160652631 2309 16112 2575 2057 160572637 2303 16113 2587 2050 160592683 2397 16104 2582 2098 16052

13a (20 times 10)

2595 2036 160592566 2044 16061

2408 2326 21610 2399 2894 2277 21610 10272928 2233 216102896 2244 216102877 2282 21610

14a (20 times 10)2340 2251 21610 2905 2649 2186 21610 12392334 2258 21610 2664 2183 21610

2641 2207 216102738 2182 21610

15a (20 times 10)

2285 2247 21610 3329 2627 2216 21610 13602287 2218 21610 2681 2196 21610

2645 2215 216102652 2201 216102668 2197 216102682 2178 21610


Table 4 Continued


1198911 1198912 1198913 119879 1198911 1198912 1198913 119879

16a (20 times 10)

2447 2354 21602 2152 2965 2279 21534 11482450 2380 21590 3158 2303 215042487 2454 21584 3184 2400 214902492 2417 21576 3039 2279 215232540 2396 21547 3050 2318 215072550 2492 21545 3071 2286 215132568 2428 21540 3275 2281 215113013 2588 21478 3105 2352 214963106 2548 21478 3175 2344 21497

3180 2293 215073130 2312 215033034 2352 21503

17a (20 times 10)

2322 2240 21433 2847 2806 2195 21096 13922322 2280 21362 2750 2165 211202323 2238 21454 2791 2141 211142343 2224 21420 2716 2157 211292480 2285 21344 2849 2148 211072528 2231 21313 2769 2166 211112789 2448 21198 2856 2183 210982808 2303 21200 2773 2145 211112816 2370 21197 2856 2142 21105

2803 2203 210942759 2186 211062789 2176 211012772 2202 21101

18a (20 times 10)

2267 2235 21483 3301 2634 2156 21005 16232269 2206 21408 2642 2126 210062320 2208 21354 2638 2185 209992437 2221 21311 2638 2148 210022531 2310 21285 2674 2148 210002545 2305 21282 2644 2145 21002

2651 2140 210052641 2157 210002664 2145 210012621 2181 210022659 2140 210032661 2137 210042651 2181 209992669 2126 210052708 2125 21009







1198911 1198912 1198913 119879 1198911 1198912 1198913 119879 1198911 1198912 1198913 119879

MK01 (10 times 6)

42 42 162 478 42 39 158 049 42 39 158 03644 40 154 44 40 15443 40 155 43 40 15540 36 169 40 36 169

47 36 16747 37 16549 37 16450 38 162

MK02 (10 times 6)

28 28 155 302 26 26 151 075 26 26 151 07027 27 146 27 27 14629 27 145 29 27 14529 29 143 29 29 14331 31 141 31 31 14133 33 140 33 33 140

MK03 (15 times 8)

204 204 852 2614 204 199 855 475 204 199 855 366204 144 871 204 144 871204 135 882 204 135 882204 133 884 204 133 884213 199 850 213 199 850214 210 849 214 210 849221 199 847 221 199 847222 199 848 222 199 848231 188 848 231 188 848230 177 848 230 177 848

MK04 (15 times 8)

68 67 352 1774 66 63 345 176 66 63 345 16365 63 362 65 63 36263 61 371 63 61 37162 61 373 60 59 39061 60 382 74 54 34960 59 390 74 55 34873 55 350 72 72 34274 54 349 78 78 33974 55 348 72 66 34890 76 331 73 72 343

MK05 (15 times 4)

177 177 702 826 173 173 683 234 173 173 683 196175 175 682 175 175 682183 183 677 183 183 677185 185 676 179 179 679179 179 679 199 199 674

193 193 675183 183 677

MK06 (10 times 15)

67 431 1879 62 55 424 193 62 55 424 18565 54 417 65 54 41760 58 441 60 58 44162 60 440 62 60 44076 60 362 76 60 36276 74 356 76 74 35678 60 361 78 60 36173 72 360 73 72 36072 72 361 72 72 361100 90 330 100 90 330


Table 5 Continued


1198911 1198912 1198913 119879 1198911 1198912 1198913 119879 1198911 1198912 1198913 119879

MK07 (20 times 5)

150 150 717 568 139 139 693 492 139 139 693 312140 138 686 140 138 686144 144 673 144 144 673151 151 667 151 151 667157 157 662 157 157 662162 162 659 162 162 659166 166 657 166 166 657

MK08 (20 times 10)

523 523 2524 6767 523 515 2524 1204 523 515 2524 834523 497 2534 523 497 2534524 524 2519 524 524 2519578 578 2489 578 578 2489587 587 2484 587 587 2484

541 533 2516542 542 2511560 560 2505

MK09 (20 times 10)

311 299 2374 7776 311 299 2290 1948 311 299 2290 1015310 299 3514 310 299 3514311 301 2287 311 301 2287314 299 2315 314 299 2315315 299 2283 315 299 2283332 302 2265 332 302 2265329 301 2266 329 301 2266328 308 2259 328 308 2259325 299 2275 325 299 2275

MK10 (20 times 15)

227 221 1989 12252 224 219 1980 1787 224 219 1980 1238225 211 1976 235 225 1895233 214 1919 240 215 1905235 218 1897 246 215 1896235 225 1895 252 224 1884240 215 1905 256 211 1919240 216 1888 260 244 1869242 214 1913 266 254 1864246 215 1896 246 206 1938252 224 1884 246 204 1948256 211 1919 246 205 1940260 244 1869 251 202 1948266 254 1864 253 207 1930268 264 1858 249 203 1947276 256 1857 255 202 1943281 268 1854 260 210 1926217 207 2064 250 205 1935214 204 2082 254 203 1942






25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400 425 450 475 500 5250

M1

M2

M3

M4

M5

M10

M6

M7

M8

M9

541



127060 120 180 240 300 360 420 480 540 600 660 720 780 840 900 960 1020 1080 1140 1200 12600

M1

M2

M3

M4

M5

M10

M11

M12

M13

M14

M15

M16

M17

M18

M6

M7

M8

M9








Notations





Acknowledgments


References























































Advances in

FuzzySystems


Volume 2014












Journal of



Advances in

Multimedia


Biomedical Imaging


Advances in


RoboticsJournal of










Advances in




Table 4 Continued


1198911 1198912 1198913 119879 1198911 1198912 1198913 119879

09a (15 times 8)2157 2113 16485 1016 2442 2103 16485 7262144 2119 16485 2470 2074 164852158 2102 16485 2455 2081 16485

10a (15 times 8)

2461 2433 16505 755 2998 2317 16487 6432470 2310 16537 3036 2363 164822478 2330 16533 3091 2491 164732482 2360 16499 2984 2319 164862501 2265 16547 3052 2377 164772501 2312 16528 3098 2491 164722547 2476 16490 3013 2293 164873064 2734 16464 3136 2457 16471

3025 2377 164803041 2351 164812980 2258 164943021 2319 16484

11a (15 times 8)

2182 2170 16449 1014 2551 2116 16229 7252202 2114 16476 2599 2131 162242210 2113 16442 2612 2146 162222337 2185 16377 2573 2058 162312874 2389 16247 2753 2186 162172894 2330 16247 2628 2102 162242962 2312 16247 2702 2098 16223

2658 2089 162242555 2111 162292664 2063 16228

12a (15 times 8)

2161 2107 16295 1192 2624 2049 16055 8222168 2130 16220 2620 2022 160682191 2084 16355 2587 2035 160652210 2103 16331 2614 2057 160512315 2125 16292 2585 2043 160612366 2105 16237 2659 2038 160552493 2297 16124 2617 2024 160652631 2309 16112 2575 2057 160572637 2303 16113 2587 2050 160592683 2397 16104 2582 2098 16052

13a (20 times 10)

2595 2036 160592566 2044 16061

2408 2326 21610 2399 2894 2277 21610 10272928 2233 216102896 2244 216102877 2282 21610

14a (20 times 10)2340 2251 21610 2905 2649 2186 21610 12392334 2258 21610 2664 2183 21610

2641 2207 216102738 2182 21610

15a (20 times 10)

2285 2247 21610 3329 2627 2216 21610 13602287 2218 21610 2681 2196 21610

2645 2215 216102652 2201 216102668 2197 216102682 2178 21610


Table 4 Continued


1198911 1198912 1198913 119879 1198911 1198912 1198913 119879

16a (20 times 10)

2447 2354 21602 2152 2965 2279 21534 11482450 2380 21590 3158 2303 215042487 2454 21584 3184 2400 214902492 2417 21576 3039 2279 215232540 2396 21547 3050 2318 215072550 2492 21545 3071 2286 215132568 2428 21540 3275 2281 215113013 2588 21478 3105 2352 214963106 2548 21478 3175 2344 21497

3180 2293 215073130 2312 215033034 2352 21503

17a (20 times 10)

2322 2240 21433 2847 2806 2195 21096 13922322 2280 21362 2750 2165 211202323 2238 21454 2791 2141 211142343 2224 21420 2716 2157 211292480 2285 21344 2849 2148 211072528 2231 21313 2769 2166 211112789 2448 21198 2856 2183 210982808 2303 21200 2773 2145 211112816 2370 21197 2856 2142 21105

2803 2203 210942759 2186 211062789 2176 211012772 2202 21101

18a (20 times 10)

2267 2235 21483 3301 2634 2156 21005 16232269 2206 21408 2642 2126 210062320 2208 21354 2638 2185 209992437 2221 21311 2638 2148 210022531 2310 21285 2674 2148 210002545 2305 21282 2644 2145 21002

2651 2140 210052641 2157 210002664 2145 210012621 2181 210022659 2140 210032661 2137 210042651 2181 209992669 2126 210052708 2125 21009







1198911 1198912 1198913 119879 1198911 1198912 1198913 119879 1198911 1198912 1198913 119879

MK01 (10 times 6)

42 42 162 478 42 39 158 049 42 39 158 03644 40 154 44 40 15443 40 155 43 40 15540 36 169 40 36 169

47 36 16747 37 16549 37 16450 38 162

MK02 (10 times 6)

28 28 155 302 26 26 151 075 26 26 151 07027 27 146 27 27 14629 27 145 29 27 14529 29 143 29 29 14331 31 141 31 31 14133 33 140 33 33 140

MK03 (15 times 8)

204 204 852 2614 204 199 855 475 204 199 855 366204 144 871 204 144 871204 135 882 204 135 882204 133 884 204 133 884213 199 850 213 199 850214 210 849 214 210 849221 199 847 221 199 847222 199 848 222 199 848231 188 848 231 188 848230 177 848 230 177 848

MK04 (15 times 8)

68 67 352 1774 66 63 345 176 66 63 345 16365 63 362 65 63 36263 61 371 63 61 37162 61 373 60 59 39061 60 382 74 54 34960 59 390 74 55 34873 55 350 72 72 34274 54 349 78 78 33974 55 348 72 66 34890 76 331 73 72 343

MK05 (15 times 4)

177 177 702 826 173 173 683 234 173 173 683 196175 175 682 175 175 682183 183 677 183 183 677185 185 676 179 179 679179 179 679 199 199 674

193 193 675183 183 677

MK06 (10 times 15)

67 431 1879 62 55 424 193 62 55 424 18565 54 417 65 54 41760 58 441 60 58 44162 60 440 62 60 44076 60 362 76 60 36276 74 356 76 74 35678 60 361 78 60 36173 72 360 73 72 36072 72 361 72 72 361100 90 330 100 90 330


Table 5 Continued


1198911 1198912 1198913 119879 1198911 1198912 1198913 119879 1198911 1198912 1198913 119879

MK07 (20 times 5)

150 150 717 568 139 139 693 492 139 139 693 312140 138 686 140 138 686144 144 673 144 144 673151 151 667 151 151 667157 157 662 157 157 662162 162 659 162 162 659166 166 657 166 166 657

MK08 (20 times 10)

523 523 2524 6767 523 515 2524 1204 523 515 2524 834523 497 2534 523 497 2534524 524 2519 524 524 2519578 578 2489 578 578 2489587 587 2484 587 587 2484

541 533 2516542 542 2511560 560 2505

MK09 (20 times 10)

311 299 2374 7776 311 299 2290 1948 311 299 2290 1015310 299 3514 310 299 3514311 301 2287 311 301 2287314 299 2315 314 299 2315315 299 2283 315 299 2283332 302 2265 332 302 2265329 301 2266 329 301 2266328 308 2259 328 308 2259325 299 2275 325 299 2275

MK10 (20 times 15)

227 221 1989 12252 224 219 1980 1787 224 219 1980 1238225 211 1976 235 225 1895233 214 1919 240 215 1905235 218 1897 246 215 1896235 225 1895 252 224 1884240 215 1905 256 211 1919240 216 1888 260 244 1869242 214 1913 266 254 1864246 215 1896 246 206 1938252 224 1884 246 204 1948256 211 1919 246 205 1940260 244 1869 251 202 1948266 254 1864 253 207 1930268 264 1858 249 203 1947276 256 1857 255 202 1943281 268 1854 260 210 1926217 207 2064 250 205 1935214 204 2082 254 203 1942






25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400 425 450 475 500 5250

M1

M2

M3

M4

M5

M10

M6

M7

M8

M9

541



127060 120 180 240 300 360 420 480 540 600 660 720 780 840 900 960 1020 1080 1140 1200 12600

M1

M2

M3

M4

M5

M10

M11

M12

M13

M14

M15

M16

M17

M18

M6

M7

M8

M9








Notations





Acknowledgments


References























































Advances in

FuzzySystems


Volume 2014












Journal of



Advances in

Multimedia


Biomedical Imaging


Advances in


RoboticsJournal of










Advances in




Table 4 Continued


1198911 1198912 1198913 119879 1198911 1198912 1198913 119879

16a (20 times 10)

2447 2354 21602 2152 2965 2279 21534 11482450 2380 21590 3158 2303 215042487 2454 21584 3184 2400 214902492 2417 21576 3039 2279 215232540 2396 21547 3050 2318 215072550 2492 21545 3071 2286 215132568 2428 21540 3275 2281 215113013 2588 21478 3105 2352 214963106 2548 21478 3175 2344 21497

3180 2293 215073130 2312 215033034 2352 21503

17a (20 times 10)

2322 2240 21433 2847 2806 2195 21096 13922322 2280 21362 2750 2165 211202323 2238 21454 2791 2141 211142343 2224 21420 2716 2157 211292480 2285 21344 2849 2148 211072528 2231 21313 2769 2166 211112789 2448 21198 2856 2183 210982808 2303 21200 2773 2145 211112816 2370 21197 2856 2142 21105

2803 2203 210942759 2186 211062789 2176 211012772 2202 21101

18a (20 times 10)

2267 2235 21483 3301 2634 2156 21005 16232269 2206 21408 2642 2126 210062320 2208 21354 2638 2185 209992437 2221 21311 2638 2148 210022531 2310 21285 2674 2148 210002545 2305 21282 2644 2145 21002

2651 2140 210052641 2157 210002664 2145 210012621 2181 210022659 2140 210032661 2137 210042651 2181 209992669 2126 210052708 2125 21009







1198911 1198912 1198913 119879 1198911 1198912 1198913 119879 1198911 1198912 1198913 119879

MK01 (10 times 6)

42 42 162 478 42 39 158 049 42 39 158 03644 40 154 44 40 15443 40 155 43 40 15540 36 169 40 36 169

47 36 16747 37 16549 37 16450 38 162

MK02 (10 times 6)

28 28 155 302 26 26 151 075 26 26 151 07027 27 146 27 27 14629 27 145 29 27 14529 29 143 29 29 14331 31 141 31 31 14133 33 140 33 33 140

MK03 (15 times 8)

204 204 852 2614 204 199 855 475 204 199 855 366204 144 871 204 144 871204 135 882 204 135 882204 133 884 204 133 884213 199 850 213 199 850214 210 849 214 210 849221 199 847 221 199 847222 199 848 222 199 848231 188 848 231 188 848230 177 848 230 177 848

MK04 (15 times 8)

68 67 352 1774 66 63 345 176 66 63 345 16365 63 362 65 63 36263 61 371 63 61 37162 61 373 60 59 39061 60 382 74 54 34960 59 390 74 55 34873 55 350 72 72 34274 54 349 78 78 33974 55 348 72 66 34890 76 331 73 72 343

MK05 (15 times 4)

177 177 702 826 173 173 683 234 173 173 683 196175 175 682 175 175 682183 183 677 183 183 677185 185 676 179 179 679179 179 679 199 199 674

193 193 675183 183 677

MK06 (10 times 15)

67 431 1879 62 55 424 193 62 55 424 18565 54 417 65 54 41760 58 441 60 58 44162 60 440 62 60 44076 60 362 76 60 36276 74 356 76 74 35678 60 361 78 60 36173 72 360 73 72 36072 72 361 72 72 361100 90 330 100 90 330


Table 5 Continued


1198911 1198912 1198913 119879 1198911 1198912 1198913 119879 1198911 1198912 1198913 119879

MK07 (20 times 5)

150 150 717 568 139 139 693 492 139 139 693 312140 138 686 140 138 686144 144 673 144 144 673151 151 667 151 151 667157 157 662 157 157 662162 162 659 162 162 659166 166 657 166 166 657

MK08 (20 times 10)

523 523 2524 6767 523 515 2524 1204 523 515 2524 834523 497 2534 523 497 2534524 524 2519 524 524 2519578 578 2489 578 578 2489587 587 2484 587 587 2484

541 533 2516542 542 2511560 560 2505

MK09 (20 times 10)

311 299 2374 7776 311 299 2290 1948 311 299 2290 1015310 299 3514 310 299 3514311 301 2287 311 301 2287314 299 2315 314 299 2315315 299 2283 315 299 2283332 302 2265 332 302 2265329 301 2266 329 301 2266328 308 2259 328 308 2259325 299 2275 325 299 2275

MK10 (20 times 15)

227 221 1989 12252 224 219 1980 1787 224 219 1980 1238225 211 1976 235 225 1895233 214 1919 240 215 1905235 218 1897 246 215 1896235 225 1895 252 224 1884240 215 1905 256 211 1919240 216 1888 260 244 1869242 214 1913 266 254 1864246 215 1896 246 206 1938252 224 1884 246 204 1948256 211 1919 246 205 1940260 244 1869 251 202 1948266 254 1864 253 207 1930268 264 1858 249 203 1947276 256 1857 255 202 1943281 268 1854 260 210 1926217 207 2064 250 205 1935214 204 2082 254 203 1942






25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400 425 450 475 500 5250

M1

M2

M3

M4

M5

M10

M6

M7

M8

M9

541



127060 120 180 240 300 360 420 480 540 600 660 720 780 840 900 960 1020 1080 1140 1200 12600

M1

M2

M3

M4

M5

M10

M11

M12

M13

M14

M15

M16

M17

M18

M6

M7

M8

M9








Notations





Acknowledgments


References























































Advances in

FuzzySystems


Volume 2014












Journal of



Advances in

Multimedia


Biomedical Imaging


Advances in


RoboticsJournal of










Advances in






1198911 1198912 1198913 119879 1198911 1198912 1198913 119879 1198911 1198912 1198913 119879

MK01 (10 times 6)

42 42 162 478 42 39 158 049 42 39 158 03644 40 154 44 40 15443 40 155 43 40 15540 36 169 40 36 169

47 36 16747 37 16549 37 16450 38 162

MK02 (10 times 6)

28 28 155 302 26 26 151 075 26 26 151 07027 27 146 27 27 14629 27 145 29 27 14529 29 143 29 29 14331 31 141 31 31 14133 33 140 33 33 140

MK03 (15 times 8)

204 204 852 2614 204 199 855 475 204 199 855 366204 144 871 204 144 871204 135 882 204 135 882204 133 884 204 133 884213 199 850 213 199 850214 210 849 214 210 849221 199 847 221 199 847222 199 848 222 199 848231 188 848 231 188 848230 177 848 230 177 848

MK04 (15 times 8)

68 67 352 1774 66 63 345 176 66 63 345 16365 63 362 65 63 36263 61 371 63 61 37162 61 373 60 59 39061 60 382 74 54 34960 59 390 74 55 34873 55 350 72 72 34274 54 349 78 78 33974 55 348 72 66 34890 76 331 73 72 343

MK05 (15 times 4)

177 177 702 826 173 173 683 234 173 173 683 196175 175 682 175 175 682183 183 677 183 183 677185 185 676 179 179 679179 179 679 199 199 674

193 193 675183 183 677

MK06 (10 times 15)

67 431 1879 62 55 424 193 62 55 424 18565 54 417 65 54 41760 58 441 60 58 44162 60 440 62 60 44076 60 362 76 60 36276 74 356 76 74 35678 60 361 78 60 36173 72 360 73 72 36072 72 361 72 72 361100 90 330 100 90 330


Table 5 Continued


1198911 1198912 1198913 119879 1198911 1198912 1198913 119879 1198911 1198912 1198913 119879

MK07 (20 times 5)

150 150 717 568 139 139 693 492 139 139 693 312140 138 686 140 138 686144 144 673 144 144 673151 151 667 151 151 667157 157 662 157 157 662162 162 659 162 162 659166 166 657 166 166 657

MK08 (20 times 10)

523 523 2524 6767 523 515 2524 1204 523 515 2524 834523 497 2534 523 497 2534524 524 2519 524 524 2519578 578 2489 578 578 2489587 587 2484 587 587 2484

541 533 2516542 542 2511560 560 2505

MK09 (20 times 10)

311 299 2374 7776 311 299 2290 1948 311 299 2290 1015310 299 3514 310 299 3514311 301 2287 311 301 2287314 299 2315 314 299 2315315 299 2283 315 299 2283332 302 2265 332 302 2265329 301 2266 329 301 2266328 308 2259 328 308 2259325 299 2275 325 299 2275

MK10 (20 times 15)

227 221 1989 12252 224 219 1980 1787 224 219 1980 1238225 211 1976 235 225 1895233 214 1919 240 215 1905235 218 1897 246 215 1896235 225 1895 252 224 1884240 215 1905 256 211 1919240 216 1888 260 244 1869242 214 1913 266 254 1864246 215 1896 246 206 1938252 224 1884 246 204 1948256 211 1919 246 205 1940260 244 1869 251 202 1948266 254 1864 253 207 1930268 264 1858 249 203 1947276 256 1857 255 202 1943281 268 1854 260 210 1926217 207 2064 250 205 1935214 204 2082 254 203 1942






25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400 425 450 475 500 5250

M1

M2

M3

M4

M5

M10

M6

M7

M8

M9

541



127060 120 180 240 300 360 420 480 540 600 660 720 780 840 900 960 1020 1080 1140 1200 12600

M1

M2

M3

M4

M5

M10

M11

M12

M13

M14

M15

M16

M17

M18

M6

M7

M8

M9








Notations





Acknowledgments


References























































Advances in

FuzzySystems


Volume 2014












Journal of



Advances in

Multimedia


Biomedical Imaging


Advances in


RoboticsJournal of










Advances in




Table 5 Continued


1198911 1198912 1198913 119879 1198911 1198912 1198913 119879 1198911 1198912 1198913 119879

MK07 (20 times 5)

150 150 717 568 139 139 693 492 139 139 693 312140 138 686 140 138 686144 144 673 144 144 673151 151 667 151 151 667157 157 662 157 157 662162 162 659 162 162 659166 166 657 166 166 657

MK08 (20 times 10)

523 523 2524 6767 523 515 2524 1204 523 515 2524 834523 497 2534 523 497 2534524 524 2519 524 524 2519578 578 2489 578 578 2489587 587 2484 587 587 2484

541 533 2516542 542 2511560 560 2505

MK09 (20 times 10)

311 299 2374 7776 311 299 2290 1948 311 299 2290 1015310 299 3514 310 299 3514311 301 2287 311 301 2287314 299 2315 314 299 2315315 299 2283 315 299 2283332 302 2265 332 302 2265329 301 2266 329 301 2266328 308 2259 328 308 2259325 299 2275 325 299 2275

MK10 (20 times 15)

227 221 1989 12252 224 219 1980 1787 224 219 1980 1238225 211 1976 235 225 1895233 214 1919 240 215 1905235 218 1897 246 215 1896235 225 1895 252 224 1884240 215 1905 256 211 1919240 216 1888 260 244 1869242 214 1913 266 254 1864246 215 1896 246 206 1938252 224 1884 246 204 1948256 211 1919 246 205 1940260 244 1869 251 202 1948266 254 1864 253 207 1930268 264 1858 249 203 1947276 256 1857 255 202 1943281 268 1854 260 210 1926217 207 2064 250 205 1935214 204 2082 254 203 1942






25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400 425 450 475 500 5250

M1

M2

M3

M4

M5

M10

M6

M7

M8

M9

541



127060 120 180 240 300 360 420 480 540 600 660 720 780 840 900 960 1020 1080 1140 1200 12600

M1

M2

M3

M4

M5

M10

M11

M12

M13

M14

M15

M16

M17

M18

M6

M7

M8

M9








Notations





Acknowledgments


References























































Advances in

FuzzySystems


Volume 2014












Journal of



Advances in

Multimedia


Biomedical Imaging


Advances in


RoboticsJournal of










Advances in







25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400 425 450 475 500 5250

M1

M2

M3

M4

M5

M10

M6

M7

M8

M9

541



127060 120 180 240 300 360 420 480 540 600 660 720 780 840 900 960 1020 1080 1140 1200 12600

M1

M2

M3

M4

M5

M10

M11

M12

M13

M14

M15

M16

M17

M18

M6

M7

M8

M9








Notations





Acknowledgments


References























































Advances in

FuzzySystems


Volume 2014












Journal of



Advances in

Multimedia


Biomedical Imaging


Advances in


RoboticsJournal of










Advances in




127060 120 180 240 300 360 420 480 540 600 660 720 780 840 900 960 1020 1080 1140 1200 12600

M1

M2

M3

M4

M5

M10

M11

M12

M13

M14

M15

M16

M17

M18

M6

M7

M8

M9








Notations





Acknowledgments


References























































Advances in

FuzzySystems


Volume 2014












Journal of



Advances in

Multimedia


Biomedical Imaging


Advances in


RoboticsJournal of










Advances in






Acknowledgments


References























































Advances in

FuzzySystems


Volume 2014












Journal of



Advances in

Multimedia


Biomedical Imaging


Advances in


RoboticsJournal of










Advances in




























Advances in

FuzzySystems


Volume 2014












Journal of



Advances in

Multimedia


Biomedical Imaging


Advances in


RoboticsJournal of










Advances in



a bee evolutionary guiding nondominated sorting genetic algorithm...

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