research article chaotic multiobjective evolutionary...

Research ArticleChaotic Multiobjective Evolutionary Algorithm Based onDecomposition for Test Task Scheduling Problem

Hui Lu Lijuan Yin Xiaoteng Wang Mengmeng Zhang and Kefei Mao

School of Electronic and Information Engineering Beihang University Beijing 100191 China

Correspondence should be addressed to Hui Lu mluhuibuaaeducn

Received 13 March 2014 Revised 20 June 2014 Accepted 20 June 2014 Published 15 July 2014

Academic Editor Jyh-Hong Chou

Copyright copy 2014 Hui Lu et al This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Test task scheduling problem (TTSP) is a complex optimization problem and has many local optima In this paper a hybrid chaoticmultiobjective evolutionary algorithm based on decomposition (CMOEAD) is presented to avoid becoming trapped in localoptima and to obtain high quality solutions First we propose an improving integrated encoding scheme (IES) to increase theefficiency Then ten chaotic maps are applied into the multiobjective evolutionary algorithm based on decomposition (MOEAD)in three phases that is initial population and crossover and mutation operators To identify a good approach for hybrid MOEADand chaos and indicate the effectiveness of the improving IES several experiments are performedThe Pareto front and the statisticalresults demonstrate that different chaotic maps in different phases have different effects for solving the TTSP especially the circlemap and ICMIC map The similarity degree of distribution between chaotic maps and the problem is a very essential factor forthe application of chaotic maps In addition the experiments of comparisons of CMOEAD and variable neighborhood MOEAD(VNM) indicate that our algorithm has the best performance in solving the TTSP

1 Introduction

Test task scheduling problem (TTSP) is an essential part ofthe automatic test system for improving throughput reducingtime and optimizing resource allocation Similar to otherscheduling problems the TTSP is one kind of combinationoptimization problems It is illustrated to be an NP-hardproblem through the analysis of the nature of the problemcarried out by many researchers [1ndash3] In addition throughthe fitness distance analysis [4] we know that the TTSPhas many local optima The algorithm that has strong spacesearching ability is needed to solve the TTSP

Recently many intelligent methods are used for solvingthe TTSP and other similar scheduling problems basedon the problemsrsquo character All these kinds of researchesfocus on improving the searching ability of the algorithmand obtaining optimal or near-optimal solutions for thescheduling problem There are two basic strategies One isto propose an improvement algorithm based on the originalalgorithm such as variable neighborhood multiobjectiveoptimization algorithm based on decomposition (VNM) forthe multiobjective test task scheduling problem [5] Another

is to adopt a hybrid algorithm using two different kinds ofalgorithms For example Lu et al proposed a hybrid particleswarm optimization and taboo search strategies for the singleobjective TTSP [6] Recently the hybrid method becomes amainstream

Different from the hybrid method using different kindsof algorithms using chaos in the evolutionary process rep-resents its advantages in improving the searching abilityLu et al proposed a chaotic nondominated sorting geneticalgorithm for themultiobjective test task scheduling problemand validated the best performance in convergence anddiversity through the experiment and analysis [1] Donaldet al utilized the chaos-induced discrete self-organizingmigrating algorithm to solve the lot-streaming flow shopscheduling problem with setup time [7] Gavrilova andAhmadian studied an on-demand chaotic neural networkfor the broadcast scheduling problem and found an optimaltime division multiple access (TDMA) frame [8] Jiang et alproposed a chaos-based fuzzy regression approach to modelcustomer satisfaction for product design and used a chaoticoptimization algorithm to generate the polynomial structures

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 640764 25 pageshttpdxdoiorg1011552014640764

2 Mathematical Problems in Engineering

of customer satisfaction models [9] Sun et al studied anovel hysteretic noisy chaotic neural network for broadcastscheduling problems in packet radio networks and exhibiteda stochastic chaotic simulated annealing algorithm [10] Thephenomenon that occurred in these researches illustratedthat the evolutionary algorithm embedded with chaos is aneffective and efficient approach for improving the searchingability of the algorithm However all these researches havethe same defect The authors only used one or several chaoticmaps embedded in algorithms to solve practical problemsHowever there is no detailed discussion and analysis

For the TTSP notice that the previous studies mostlyaim at a single objective and only few papers focus onthe multiobjective problem [1 5] Through the analysis ofour previous related work for the multiobjective TTSP weknow the multiobjective evolutionary algorithm based ondecomposition (MOEAD) exhibited the best performancein solving the TTSP [5] in the aspect of convergence anddiversity Therefore using chaotic maps in MOEAD canfurther enhance the quality of the solution for the TTSP

In this paper we propose a chaotic multiobjective evo-lutionary algorithm based on decomposition (CMOEAD)for the TTSP Ten chaotic maps are embedded in threedifferent phases in the evolutionary processThe aim is to giveguidance for the choice of chaotic maps and phases based onthe framework of MOEAD for the TTSP

First the chromosome-encoding scheme is very impor-tant for the problem description and the operation in theevolutionary process For the TTSP there are different kindsof encoding strategies like task sequencing list (TSL) (orthe operations list coding (OLC)) matrix-encoding andintegrated encoding scheme (IES) TSL andmatrix-encodingare not acceptable if a task can be tested on more thanone set of instruments IES can overcome this problembut it cannot realize the selection operation under equalprobability Therefore in this paper the improved IES isproposed by changing the schemes selectionmethod of everytest task As a result the equal probability is realized

Then ten chaotic maps are embedded in MOEADindependently in three phases The ten chaotic maps arebakerrsquos map cat map circle map cubic map Gauss mapICMIC map logistic map sinusoidal map tent map andZaslavskii mapThree phases are initial population crossoveroperator and mutation operator Four benchmarks of theTTSP are used to evaluate the performance of the proposedalgorithm They are 6 times 8 20 times 8 30 times 12 and 40 times 12 Weuse 119899 times119898 to represent the benchmark Here 119899 is the numberof tasks and119898 is the number of instruments

The performance metrics hypervolume (HV) and 119862 [11]are used to evaluate the role of chaotic maps on MOEADTherefore we can find which kind of chaotic map embeddedalgorithm is the best one for solving the TTSP

From the results of experiments it can be seen that thechaotic map embedded MOEAD has good performance tosolve the TTSP for both small and large scale problemsDifferent kinds of chaotic maps have different performancesin different phases of MOEAD but ICMIC map and circlemap in initial population crossover operator and mutationoperator have the best performance The experiments for

comparisons of CMOEAD and VNM show that our algo-rithmperforms better than theVNMin solving theTTSPTheevidence the chaotic map is an effective and efficient methodfor solving the problem with local optima is validated Thesimilarity degree of distribution between chaotic maps andthe problem is a very essential factor for the application ofchaotic maps

The rest of the paper is organized as follows Section 2gives a summary of related work on applying chaos toimprove evolutionary algorithms Section 3 concludes themathematical model proposed by us in previous work forthe integrity In Section 4 ten chaotic maps including bothone-dimensional maps and two-dimensional maps are intro-duced In Section 5 the proposed CMOEAD is describedin detail for solving the TTSP The detail of the encodingscheme and the phases in which chaos can be embedded inevolutionary algorithms are introduced Experimental resultsand performance comparisons are presented and discussed inSection 6 Finally Section 7 concludes the paper

2 Related Work

Recently chaotic sequences have been integrated in theevolutionary process through two types of operations Oneis using chaotic maps to replace random sequences Anotheris to replace the genetic operations These two kinds ofoperations always appear in the same algorithm at once

In detail all the operations can be divided into sevencases They are population initialization setting crossoverprobability setting crossover position setting crossover oper-ator setting mutation probability setting mutation operatorand increasing chaotic disturbance The performance ofdifferent operations is totally different For example adoptingchaoticmaps in the initialization canmaintain the populationdiversity The aim of using chaotic maps to replace standardmutation operator is to avoid the search being trapped in localoptima

For the scheduling problem the situation is the same asthe above in both the single objective and the multiobjectivescheduling problems For the single objective schedulingproblem Cheng et al used the hybrid genetic algorithm andchaos to optimize the hydropower reservoir operation [12]Two methods were adopted to improve the performance ofGA One was the adoption of chaos for initialization andanother was the annealing chaotic mutation operation Theconclusion was that the proposed approach is feasible andeffective in optimal operations of complex reservoir systemsLiu and Cao [13] proposed a chaotic algorithm for thefuzzy job scheduling problem in the grid environment withuncertaintiesThe authors incorporated logistic map with thestandard genetic algorithm and proposed a chaotic mutationoperator based on the feedback of the fitness functionSingh and Mahapatra [14] proposed a swarm optimizationapproach for the flexible flow shop scheduling problemwith multiprocessor tasks The logistic map was used inthis paper Bahi et al [15] considered a novel chaos-basedscheduling scheme for video surveillance to defeat maliciousintruders The concept of chaotic iterations was investigated

Mathematical Problems in Engineering 3

Yu and Gu [16] proposed an improved transiently chaoticneural network approach for the identical parallel machinescheduling problem

For the multiobjective scheduling problem Niknam etal [17] proposed an improved particle swarm optimization(IPSO) for the multiobjective optimal power flow problemconsidering the cost loss emission and voltage stabilityindex To improve the quality of solutions particularly toavoid being trapped in local optima this study presentedan IPSO that profits from chaos and self-adaptive con-cepts to adjust the particle swarm optimization parametersZhou et al [18] established time expenses resources andquality objective functions and used the chaos particleswarmoptimization to solve the resource-constrained projectscheduling problem Fang [19] proposed a quantum immunealgorithm for the multiobjective parallel machine schedulingproblem in textile manufacturing industry Here a novelmutation operator with a chaos-based rotation gate wasinvestigated We proposed a chaotic nondominated sortinggenetic algorithm (CNSGA) to solve the test task schedulingproblem According to the different capabilities of the logisticand the cat chaotic operators the CNSGA approach usingthe cat population initialization the cat or logistic crossoveroperator and the logistic mutation operator has good perfor-mance [1]

All these researches despite the single objective or themultiobjective problem in these scheduling fields have thesame features The chaotic maps are used for improving thesearching ability of the evolutionary algorithm Howevermost of researches only used one kind of chaotic mapsembedded in special phases of the algorithm and compre-hensive analysis is inefficient In fact different kinds of thescheduling problems have different characters and differentchaotic maps have different effects on the algorithms and theproblems Our work will focus on the analysis and design ofchaoticmultiobjective algorithm for the TTSPWe investigatethe guidance for solving the TTSP

3 Mathematical Model for the TTSP

31 The Problem Description The aim of the TTSP is toorganize the execution of 119899 tasks on 119898 instruments In thisproblem there are a set of tasks 119879 = 119905119895

119899

119895=1and a set

of instruments 119877 = 119903119894119898119894=1 The notifications 119875119894119895 119878

119894119895 and

119862119894119895 present the test time the test start time and the test

completion time of task 119905119895 tested on 119903119894 respectively [1] For theTTSP one task must be tested on one or more instrumentsIn other words some instruments collaborate for one testtask A variable 119874119894119895 is defined to express whether the task119905119895 occupies the instrument 119903119894 A task 119905119895 could have several

alternative schemes to complete the test119882119895 = 119908119896119895 119896119895

119896=1is used

to denote the alternative schemes of task 119905119895 where 119896119895 is thenumber of schemes of 119905119895 119870 = 119896119895

119899

119895=1is the set containing

the numbers of schemes that correspond to every task Each119908119896119895 is a subset of 119877 and can be represented as 119908119896119895 = 119903

119906119895119896119906119895119896

119906=1

Here 119906119895119896 is the number of instruments for 119908119896119895 Obviously

cup 1le119896le119896119895 1le119895le119899119908119896119895 = 119877The notification119875119896119895 = max119903119894isin119908119896119895119875

119894119895 is used

to express the test time of 119905119895 for 119908119896119895

32 Constraint Relationship The TTSP has two types ofconstraints the restriction on resources and the precedenceconstraint between the tasksThe restriction on resources canbe expressed as follows

119883119896119896lowast

119895119895lowast = 1 if 119908119896119895 cap 119908

119896lowast

119895lowast =

0 otherwise(1)

The precedence constraint between the tasks can berepresented as follows

119884119895119895lowast =

0 if 119905119895 and 119905119895lowast have equal priorities+119889 if 119905119895 needs to be tested before 119905119895lowast

with at least 119889 unit time that is 119905119895 ≻ 119905119895lowast minus119889 if 119905119895lowast needs to be tested before 119905119895

with at least 119889 unit time that is 119905119895lowast ≻ 119905119895(2)

where 119889 isin 119877+ In this paper 119889 equals the test time of the highpriority task

33 Objective Function In this study we consider two objec-tive functions The model is defined as follows

minimize

max1le119896le1198961198951le119895le119899

max119903119894isin119908119896119895

1198621198941198951

119876

119899

sum

119895=1

119898

sum

119894=1

119875119894119895119874119894119895

(3)

subject to

119862119894119895 = 119878119894119895 + 119875119894119895 (4)

119874119894119895 =

1 if 119905119895 occupies 1199031198940 otherwise

(5)

The first objective function minimizes the maximal testcompletion time and the second objective function mini-mizes themeanworkload of the instrumentsHere119876 denotesthe parallel steps The initial value of 119876 is 1 Assign theinstruments for all of the tasks if119883119896119896

lowast

119895119895lowast = 1 119876 = 119876 + 1Constraint (4) indicates that the setup time of the instru-

ments and the move time between the tasks are negligibleConstraint (5) defines whether the task 119905119895 occupies theinstrument 119903119894 Here we assume 119875119894119895 = 119875

119896119895 to simplify the

problem

4 Chaotic Maps

Ten chaotic maps including both one-dimensional mapsand two-dimensional maps are introduced in this sectionEach one has specific features and different chaotic mapscombinedwith optimization algorithms have different results(Table 1)


Table 1 The list of chaotic maps

Chaotic map Formula Dimensions Range

Bakerrsquos map 119861 (119909 119910) =

(2119909 2119910) for 0 le 119909 lt 05(2 minus 2119909 1 minus

119910

2) for 05 le 119909 lt 1

2 119909 isin (0 1)

Arnoldrsquos cat map 119909119896+1 = 119909119896 + 119910119896 mod (1)119910119896+1 = 119909119896 + 2119910119896 mod (1)

2 119909119896 isin (0 1)

Circle map 119909119896+1 = 119909119896 + 119887 minus (119886

2120587) sin (2120587119909119896) mod (1) 1 (0 1)

119886 = 05 119887 = 02

Cubic map 119909119896+1 = 120588119909119896 (1 minus 1199092119896) 119909119896 isin (0 1) 1 (0 1)

120588 = 259

Gauss map 119909119896+1 =

0 119909119896 = 0

1

119909119896

mod (1) otherwise1 (0 1)

ICMIC map 119909119896+1 = sin(119886

119909119896

) 119886 isin (0infin) 119909119896 isin (minus1 1) 1 [minus1 1]

119886 = 2

Logistic map 119909119896+1 = 119886119909119896 (1 minus 119909119896) 1 (0 1)119886 = 4

Sinusoidal map 119909119896+1 = sin (120587119909119896) 119909119896 isin (0 1) 1 (0 1)

Tent map 119909119896+1 =

119909119896

07119909119896 lt 07

(10

3) 119909119896 (1 minus 119909119896) otherwise

1 (0 1)

Zaslavskii map119909119896+1 = (119909119896 + V + 119886119910119896+1) mod (1)119910119896+1 = cos (2120587119909119896) + 119890

minus119903119910119896

2 119910119896 isin [minus10512 10512]

V = 400 119903 = 3 119886 = 126695

There are two problems for these chaotic maps One isthat the range of ICMIC and Zaslavskii maps is not (0 1)As a result the generated chaotic sequences need the scaletransformation Another is some maps like tent map havefixed points Therefore jumping out from fixed points isnecessary for maintaining the chaos characteristics

Figure 1 shows the distribution of different chaotic mapsIt reveals that bakerrsquos map bat map and tent map haveuniform distribution while other chaotic maps like circlemap cubic map ICMIC map logistic map and Zaslavskiimap have nonuniform distribution relatively

5 Chaotic Multiobjective EvolutionaryAlgorithm Based on Decomposition

51 The Improving Encoding Method for the TTSP Integratedencoding scheme (IES) proposed by our previous research [1]can use one chromosome to contain the information aboutboth the processing sequence of the tasks and the occupancyof the instruments for each task It can transform a discreteoptimization problem into a continuous optimization prob-lem Therefore the encoding efficiency is improved and thecomplexity of the genetic manipulations is reduced

Here we use an example with four tasks and fourinstruments for illustration of the role of IES The detail is inTable 2

The main concept of the IES is to use the relationshipsbetween the decision variables to express the sequence oftasks and use the values of the variables to represent theoccupancy of the instruments for each task This concept isillustrated in Table 3

The entries in the first row are the decision variableswhich range between 0 and 1 They are sorted in ascendingorder The rank of every variable denotes a test task index inthe sequence Thus the second row (or the task sequence) isobtained On the other hand the instrument assignment canalso be obtained from the decision variables If we want toknow which instruments will be occupied by the task 119905119895 119908

119896119895

should be ascertained In other words we should know thevalue of 119896 which can be calculated by the decision variablecorresponding to 119905119895 The formula is as follows

119896 = [119909119894119895 times 10] mod 119896119895 + 1 (6)

Here 119909119894119895 isin [0 1] represents the decision variable thatcorresponds to 119905119895 and 119896119895 is the number of schemes of 119905119895 Forexample for the task 1199051 the corresponding decision variableis 01270 and the number of schemes is 1198961 = 2 Then thevalue of 119896 can be calculated as follows according to (6) 119896 =[01270 times 10] mod 1198961 + 1 = 1 mod 2 + 1 = 2 Therefore11990821 = 1199032 1199034 is occupiedHowever this encoding scheme has one defect that all

schemes are selected with unequal probability For example


0 05 1

0 05 1

0 05 1 0 05 1

0 05 1 0 05 1

0 05 1

0 05 1

0 05 1 0 05 10

500

1000

1500

0

500

1000

1500

0

500

1000

1500

0

500

1000

1500

2000

2000

0

500

1000

1500

2000

0

500

1000

1500

0

500

1000

1500

0

1000

2000

3000

0

1000

2000

3000

0

1000

2000

3000

Bakers Cat Circle

Cubic Gauss ICMIC

Logistic Sinusoidal Tent

Zaslavskii

Figure 1 Distribution of different chaotic maps

for one task 1199054 the number of schemes is 1198964 = 3 Then thevalue of 119896 can be calculated according to (6) as shown inTable 4

As seen from Table 4 the probability of 119896 = 1 is 410but 310 for 119896 = 2 3 It means all schemes are selectedwith unequal probability Based on the original formula weimproved the encoding strategy as follows

119896 = [119909119894119895119896119895 times 10] mod 119896119895 + 1 (7)

Then the value of 119896 can be calculated according to (7) asshown in Table 5

As seen from Table 5 the equal probability of 119896 = 1 2 3is 13 This encoding method never generates duplication ofa certain task and does not generate unfeasible solutions Inaddition equal probability can maintain impartiality for all

Table 2 A TTSP with four tasks and four instruments

119879 119882119895 119908119896119895 119875

119896119895

1199051

11990811 1199031 1199032 511990821 1199032 1199034 3

1199052

11990812 1199031 411990822 1199033 1

1199053 11990813 1199034 2

1199054

11990814 1199031 1199033 411990824 1199032 1199034 311990834 1199032 1199033 7

schemes It can help the algorithms to match the TTSP withmultiple alternative schemes


Table 3 Example of the integrated encoding scheme

Decision variables 119909119894119895 08147 09058 01270 06324Tast sequence 119905119895 3 4 1 2119896 1 1 2 1119908119896119895 1199034 1199031 1199033 1199032 1199034 1199031

119875119896119895 2 4 3 4

Table 4 The integrated encoding scheme

Decisionvariables 119909119894119895

[00 01)

[03 04)

[06 07)

[09 10)

[01 02)

[04 05)

[07 08)

[02 03)

[05 06)

[08 09)

119896 1 2 3

52 Application of Chaotic Maps in MOEAD The multi-objective evolutionary algorithm based on decomposition isoriginated fromTchebycheff decomposition It decomposes amultiobjective problem into a number of scalar optimizationsubproblems and optimizes them simultaneously Each sub-problem is bound with a weight vector and is optimized byusing the information from its several neighbor subproblems[20]

In this paper chaotic variables are used instead of randomvariables in MOEAD Ten chaotic maps are embedded inMOEAD to replace the random operationThree key phasesin evolutionary algorithms initialization crossover andmutation are chosen to be embedded with chaos Differentchaotic maps have different formulas and characters Herewe use sinusoidal map [21] as an example

(1) Initialization In order to guarantee the diversity of theinitial population the chaos initialization is applied in thispaper

For example we assume119873 individuals in population andone of them can be denoted by

119909119904= 1199091119904 1199092119904 119909

119894119904 119909

119899119904 119904 =1 2 119873 119894 =1 2 119873

(8)

Here the initial population is generated by chaos mapsFor example if the sinusoidal map is used for initialization119909119894+1119904 = sin(120587119909

119894119904)

(2) CrossoverCrossover is themost important step in the pro-cess of the evolution It is directly related to the convergencediversity and other performances of the optimal solutions

In this paper a differential evolution (DE) operator isadopted In the DE operator each child individual 119909119905+1119894 isgenerated as follows

119909119905+1119894 =

119909119905119894 + 119865 times (119909

1199051198941 minus 1199091199051198942) if rand lt CR

119909119905119894 otherwise

(9)

Here CR and119865 are two control parameters1199091199051198941 and1199091199051198942 are

two individuals chosen in the neighborhood of 119909119905119894 Since 119865 isa random number that ranges from 0 to 1 119865 can be generated

by chaotic maps instead of random generation For instanceif the sinusoidal map is used and in the 119894th iteration 119865 = 119865119894then in the (119894 + 1)th iteration 119865119904 = 119865119894+1 = sin(120587119865119894)

(3) MutationMutation operator that prevents solutions frombeing trapped into local optima is indispensable in theprocess of the evolution

In this paper a polynomial mutation operator is adoptedFor a solution 119909119904 the polynomial mutation is described as

119909lowast119904 = 119909119904 + (119909

119906119904 minus 119909119897119904) times 120575119904 (10)

where 119909119906119904 and 119909119897119904 are the upper and lower bounds of 119909119904

Consider

120575119904 = (2119906119904)1(120578119898+1)

minus 1 if 119906119904 lt 051 minus (2 times (1 minus 119906119904))

1(120578119898+1) otherwise(11)

Here 119906119904 is a random number ranging from 0 to 1 120578119898 isthe distribution index for the mutation operator Similar tothe crossover scheme we have 119906119904 = 119906119894+1 = sin(120587119906119894) whenusing the sinusoidal map

6 Experiments

We carry out four types of experiments to illustrate theperformances of the mentioned approaches Experiment 1shows the effectiveness of the improving encoding methodbased on one large scale TTSP Experiment 2 aims to solvea small scale TTSP benchmark to measure the performanceof the evolutionary algorithm using chaotic maps in threephases Experiment 3 is similar to experiment 2 except thatit aims to solve the large scale TTSP In both experiments 2and 3 ten chaotic maps are embedded in three differentphases in the original MOEAD algorithm Each time onlyone parameter is modified The Pareto set (PF) is used toshow the effect firstlyThen the performancemetrics HV and119862 are used to further evaluate the performance of chaoticmaps embedded algorithm and the original algorithm Basedon the results of the above experiments we compare theCMOEAD with the VNM [5] in experiment 4

The parameters for all experiments are shown in Table 6119899iter is the number of iterations 119899pop is the scale of thepopulation 119899var is the number of decision variables CR and119875119898 (equal to the reciprocal of 119899var) are the probabilities ofcrossover and mutation operations

61 Experiment 1 The Performance of the Improving Encod-ing Method This experiment shows the effectiveness ofthe improving encoding method in solving the TTSP Theinstance is based on a large scale TTSP 40 times 12 [4] 50 runsof the same experiment have been performed and the bestrun among the 50 runs is given in Figure 2 Here MOEAD-1 MOEAD-2 and MOEAD-3 represent the algorithm withdifferent encoding method of random IES improving IESseparately

We can find from the Pareto front that the improvingencoding method obtains better convergence of the solutionsof the TTSP The equal probability also helps the algorithm


Table 5 The improving integrated encoding scheme

Decision variables 11990911989411989530[0 1)[3 4)[6 7)[9 10)

[12 13)[15 16)[18 19)

[21 22)[24 25)[27 28)

[1 2)[4 5)[7 8)[10 11)

[13 14)[16 17)[19 20)

[22 23)[25 26)[28 29)

[2 3)[5 6)[8 9)[11 12)

[14 15)[17 18)[20 21)

[23 24)[26 27)[29 30)

119896 1 2 3

Table 6 The setting of parameters

6 times 8 20 times 8 30 times 12 40 times 12

119899iter 250119899pop 100119899var 6 20 30 40CR 09119875119898 16 120 130 140

35 40 45 50 55 60 65 70

Makespan

14

16

18

20

22

24

26

28

30

32

34

Mea

n w

orkl

oad

MOEAD-1MOEAD-2

MOEAD-3

Figure 2 Comparison of different encoding methods in solving theTTSP

to obtain good convergence Therefore the improving IES isused in the following experiments because of the efficiency

62 Experiment 2 The Performance for the Small Scale TTSPThis experiment is carried out to show the effectiveness ofCMOEAD for the small scale TTSP 6 times 8 10 times ofthe same experiment have been performed and the bestresults obtained from original MOEAD and many variantsof CMOEAD for this instance are shown in Figures 3 4 and5

For the convenience the algorithms with differentcombinations of chaotic maps and phases are named asldquoCMOEAD-[phase][chaotic map]rdquo The ten chaotic maps(baker cat circle cubic Gauss ICMIC logistic sinusoidaltent and Zaslavskii) are denoted by 1 2 3 10 in alpha-betical order ldquo119877rdquo represents the original MOEAD ldquo119868rdquorepresents the phase for initial population ldquo119862rdquo represents thephase for the crossover operator ldquo119872rdquo represents the phase forthe mutation operator For example the algorithm for initialpopulation by logistic map is named ldquoCMOEAD-I7rdquo

According to the name role Figure 3 indicates the per-formance of the chaotic maps for crossover for solving theTTSP Figure 4 shows the performance of the chaotic maps

for initialization for solving the TTSP Figure 5 shows theperformance of the chaotic maps for mutation for solving theTTSP

For the small scale TTSP the performance for conver-gence is not very obvious from the Pareto set The solutionsobtained from the original algorithm and the chaos embed-ded algorithm overlap each other However the diversity ofthe solutions obtained from the chaos embedded algorithmis better than the original algorithm

63 Experiment 3 The Performance for the Large Scale TTSPThis experiment is carried out to show the effectiveness ofCMOEAD for three large scale problems TTSP 20 times 830 times 12 and 40 times 12 10 times of the same experiment havebeen performed and the best results obtained from originalMOEAD and many variants of CMOEAD are shown inFigures 6 7 8 9 10 11 12 13 and 14 The name role of thefigures is similar to the small scale instance

For the large scale TTSP both the convergence anddiversity of solutions are improved significantly Almost everychaotic map has good performance for the improvement butthe performance is not stable and positive for some chaoticmaps For example the tent baker and cat maps even havenegative effects for the solutions under some situations

64 Performance Analysis Based on the above experimentswe use the statistical data of the comprehensive metricHV and convergence metric 119862 to indicate the results fromdifferent aspects because the figure of Pareto front canprovide only the primary idea but not the comprehensiveeffectThe conclusion about the guidance of chaotic maps forresolving the TTSP will be investigated based on these data

641 Performance Metrics

(1) Hypervolume (see [11]) This quality indicator calculatesthe volume (in the objective space) covered by members of anondominated set of solutions for problems where all objec-tives are to be minimized Mathematically for each solution119894 isin 119878 a hypercube V119894 is constructed with a reference point119882and the solution 119894 as the diagonal corners of the hypercubeThe reference point can simply be found by constructing avector of worst objective function values Thereafter a unionof all hypercubes is found and its hypervolume is calculatedas follows

HV (119878) = Leb(⋃119894isin119878

V119894) (12)


20 25 30 35 40 45

20 25 30 35 40 45 20 25 30 35 40 45 20 25 30 35 40 45

20 25 30 35 40 45 20 25 30 35 40 45 20 25 30 35 40 45

20 25 30 35 40

20 25 30 35 40

20 25 30 35 40 4510

15

20

25

10

15

20

25

10

15

20

25

10

15

20

25

10

15

20

25

10

15

20

25

10

15

20

25

Mea

n w

orkl

oad

10

15

20

25

10

15

20

25

Mea

n w

orkl

oad

10

15

20

25

Mea

n w

orkl

oad

Mea

n w

orkl

oad

Baker

Makespan

Makespan Makespan Makespan

MakespanMakespanMakespan


Cat Circle

Cubic Gauss ICMIC


Zaslavskii

MOEADCMOEAD-C1

MOEADCMOEAD-C2

MOEADCMOEAD-C3

MOEADCMOEAD-C4

MOEADCMOEAD-C5

MOEADCMOEAD-C6

MOEADCMOEAD-C7

MOEADCMOEAD-C8

MOEADCMOEAD-C9

MOEADCMOEAD-C10

Figure 3 Comparison of different chaotic maps for crossover


20 25 30 35 4010

15

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25

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25Baker

Makespan

Makespan

20 25 30 35 40

Makespan20 25 30 35 40

Makespan

20 25 30 35 40

Makespan

20 25 30 35 40

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MOEADCMOEAD-I1

20 25 30 35 40 45

Makespan20 25 30 35 40 45

Makespan20 25 30 35 40 45

Makespan20 25 30 35 40 45

Makespan20 25 30 35 40 45

Cat

MOEADCMOEAD-I2

Circle

MOEADCMOEAD-I3

Cubic

MOEADCMOEAD-I4

Gauss

MOEADCMOEAD-I5

ICMIC

MOEAD

Logistic

MOEAD

Sinusoidal

MOEAD

Tent

MOEADCMOEAD-I9

Zaslavskii

MOEADCMOEAD-I10

CMOEAD-I6

CMOEAD-I7 CMOEAD-I8

Figure 4 Comparison of different chaotic maps for initialization


10

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25

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20 25 30 35 40 4520 25 30 35 40

Makespan Makespan20 25 30 35 40 45

Makespan

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Makespan20 25 30 35 40 45

Makespan20 25 30 35 40 45

Makespan

20 25 30 35 40 45

Makespan20 25 30 35 40 45

Makespan

20 25 30 35 40 45

Makespan

20 25 30 35 40 45

Makespan

MOEADCMOEAD-M1

MOEADCMOEAD-M2

MOEADCMOEAD-M3

MOEADCMOEAD-M4

MOEADCMOEAD-M5

MOEADCMOEAD-M6

MOEADCMOEAD-M7

MOEADCMOEAD-M8

MOEADCMOEAD-M9

MOEADCMOEAD-M10

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii

Figure 5 Comparison of different chaotic maps for mutation


30 40 50 60 70

30 40 50 60 70

12

14

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18

12

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18

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30 40 50 60 70 80

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30 35 40 45 50

20 40 60 80 100

20 40 60 80 10030 40 50 60 70

30 40 50 60 70

30 40 50 6010

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30

20

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10

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Makespan



Makespan

Makespan Makespan

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii

MOEADCMOEAD-C1

MOEADCMOEAD-C2

MOEADCMOEAD-C3

MOEADCMOEAD-C5

MOEADCMOEAD-C4

MOEADCMOEAD-C6

MOEADCMOEAD-C7

MOEADCMOEAD-C8

MOEADCMOEAD-C9

MOEADCMOEAD-C10

Figure 6 Comparison of different chaotic maps for crossover for 20 times 8


14

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25

30 35 40 45 50 30 40 50 60 70 20 40 60 80 100 120

20 40 60 80 100 30 35 40 45 50 30 40 50 60 70

30 40 50 60 70

30 35 40 45 50 30 40 50 60 70 30 35 40 45 50 55

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Makespan Makespan

Makespan

Makespan

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii

MOEADCMOEAD-I1

MOEADCMOEAD-I2

MOEADCMOEAD-I3

MOEADCMOEAD-I5

MOEADCMOEAD-I4

MOEADCMOEAD-I6

MOEADCMOEAD-I7

MOEADCMOEAD-I9

MOEADCMOEAD-I8

MOEADCMOEAD-I10

Figure 7 Comparison of different chaotic maps for initialization for 20 times 8


30 40 50 60 70 30 40 50 60 70 30 35 40 45 50 55

20 40 60 80 100 30 40 50 60 70 80 30 40 50 60 70

30 35 40 45 50 55 20 30 40 50 60 70 30 40 50 60 70

30 35 40 45 50

MOEADCMOEAD-M1

MOEADCMOEAD-M2

MOEADCMOEAD-M3

MOEADCMOEAD-M4

MOEADCMOEAD-M5

MOEADCMOEAD-M6

MOEADCMOEAD-M7

MOEADCMOEAD-M8

MOEADCMOEAD-M9

MOEADCMOEAD-M10

10

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n w

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Makespan

Makespan

Makespan Makespan

Makespan

Makespan

Makespan Makespan

Makespan Makespan

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii

Figure 8 Comparison of different chaotic maps for mutation for 20 times 8


14

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30 40 50 60 70 80

30 40 50 60 70 30 40 50 60 70

30 40 50 60 70

80 30 40 50 60 70 80

30 40 50 60 70 80 30 40 50 60 70 80

30 40 50 60 70 80

20 40 60 80 100 20 40 60 80 100

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Makespan

MOEADCMOEAD-C1

MOEADCMOEAD-C2

MOEADCMOEAD-C3

MOEADCMOEAD-C4

MOEADCMOEAD-C5

MOEADCMOEAD-C6

MOEADCMOEAD-C7

MOEADCMOEAD-C8

MOEADCMOEAD-C9

MOEADCMOEAD-C10

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii



20 40 60 80 100

20 40 60 80 100

20 40 60 80 100

20 40 60 80 100

20 40 60 80 100 30 40 50 60 70 80

30 40 50 60 70 30 40 50 60 7080

30 40 50 60 70 80 30 40 50 60 70 80

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20

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MOEADCMOEAD-I1

MOEADCMOEAD-I2

MOEADCMOEAD-I3

MOEADCMOEAD-I4

MOEADCMOEAD-I5

MOEADCMOEAD-I6

MOEADCMOEAD-I7

MOEADCMOEAD-I8

MOEADCMOEAD-I9

MOEADCMOEAD-I10

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Makespan Makespan

MakespanMakespan Makespan


Makespan

Makespan

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii



30 40 50 60 70 80 30 40 50 60 70 30 40 50 60 70801015

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MOEADCMOEAD-M2

MOEADCMOEAD-M3

MOEADCMOEAD-M4

MOEADCMOEAD-M5

MOEADCMOEAD-M6

MOEADCMOEAD-M7

MOEADCMOEAD-M8

MOEADCMOEAD-M9

MOEADCMOEAD-M10

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Makespan Makespan

Makespan

Makespan

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii



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40 60 80 100

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MOEADCMOEAD-C2

MOEADCMOEAD-C3

MOEADCMOEAD-C4

MOEADCMOEAD-C5

MOEADCMOEAD-C6

MOEADCMOEAD-C7

MOEADCMOEAD-C8

MOEADCMOEAD-C9

MOEADCMOEAD-C10

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Makespan Makespan

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Cubic Gauss ICMIC


Zaslavskii



40 50 60 70 80 40 50 60 70 80

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90

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Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii

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MOEADCMOEAD-I2

MOEADCMOEAD-I3

MOEADCMOEAD-I4

MOEADCMOEAD-I5

MOEADCMOEAD-I6

MOEADCMOEAD-I7

MOEADCMOEAD-I8

MOEADCMOEAD-I9

MOEADCMOEAD-I10



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Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii

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MOEADCMOEAD-M2

MOEADCMOEAD-M3

MOEADCMOEAD-M5

MOEADCMOEAD-M6

MOEADCMOEAD-M7

MOEADCMOEAD-M9

MOEADCMOEAD-M10

MOEADCMOEAD-M4

MOEADCMOEAD-M8



Algorithm

minus02

0

02

04

06

08

1

TTSP 20 lowast 8

C-m

easu

re

(RC

1)

(C1R

)

(RC

2)

(C2R

)

(RC

3)

(C3R

)

(RC

4)

(C4R

)

(RC

5)

(C5R

)

(RC

6)

(C6R

)

(RC

7)

(C7R

)

(RC

8)

(C8R

)

(RC

9)

(C9R

)

(RC

10)

(C10R

)

Figure 15 The boxplots of 119862 for chaotic maps embedded incrossover

Here Leb denotes the Lebesgue measure Algorithmswith larger values of HV are desirable

(2) Coverage Metric 119862 (see [11]) The metric 119862 can be used tocompare the performances of the two-solution sets Assume119860 and 119861 are two sets of nondominated solutions 119862(119860 119861)represents the proportion of points in set 119861 dominated over119860 in the total points in set 119861 Consider

119862 (119860 119861) =

1003816100381610038161003816119909 isin 119861 | exist119910 isin 119860 119910 dominates 1199091003816100381610038161003816|119861|

(13)

The value 119862(119860 119861) = 1 means that all of the solutionsin 119861 are dominated by solutions in 119860 while 119862(119860 119861) = 0means that no solution in 119861 is dominated by a solution in 119860Note that both the119862(119860 119861) and119862(119861 119860) have to be consideredfor comprehensive dominated information for comparingthe different set obtained from different algorithm because119862(119860 119861) = 1 minus 119862(119861 119860)

642 Experiment Results Theaverage values of performancemetrics HV and 119862 of 10 independent runs for both the smalland the large scale TTSPs are in Tables 7 and 8 respectivelyThe symbol is similar to the above mentioned role In all ofthe cases the best performances are denoted in bold

As shown in Tables 7 and 8 most of the combinations ofchaoticmapswithMOEADhave a positive effect for both thesmall scale and the large scale instances However the largerthe scale is the weaker the chaos effect is

In most cases the best performance in Table 7 is consis-tent with that in Table 8 It means the chaotic maps in thespecific location have better convergence and comprehensiveperformances simultaneously In fact metric HV and metric119862 are different aspects to evaluate the algorithm Thereforesome inconsistences exist also Here we represent the sta-tistical results in an intuitive way If both the convergenceand the comprehensive performances of the algorithm withchaotic maps are better than the original algorithm the valueis replaced by ldquo++rdquo When the situation is opposite blank isused to replace the corresponding value Table 9 shows theresults

Algorithm

minus02

0

02

04

06

08

1

TTSP 20 lowast 8

(RI1)

(I1R

)

(RI2)

(I2R

)

(RI3)

(I3R

)

(RI4)

(I4R

)

(RI5)

(I5R

)

(RI6)

(I6R

)

(RI7)

(I7R

)

(RI8)

(I8R

)

(RI9)

(I9R

)

(RI10)

(I10R

)

+

+

+

C-m

easu

re

Figure 16 The boxplots of 119862 for chaotic maps embedded ininitialization

(RM

1)

(M1R

)

(RM

2)

(M2R

)

(RM

3)

(M3R

)

(RM

4)

(M4R

)

(RM

5)

(M5R

)

(RM

6)

(M6R

)

(RM

7)

(M7R

)

(RM

8)

(M8R

)

(RM

9)

(M9R

)

(RM

10)

(M10R

)

Algorithm

minus02

0

02

04

06

08

1

TTSP 20 lowast 8

+

C-m

easu

re

Figure 17 The boxplots of 119862 for chaotic maps embedded in mut-ation

The results show that circle map and ICMIC map in allphases especially in crossover operator have the best perfor-mance Cubic map and logistic map in mutation operatorGauss map in crossover operator and mutation operatorsinusoidal map in crossover operator and initial populationbakerrsquos map in crossover operator and Zaslavskii map ininitial population have a better effect In addition cat map ininitial population and mutation operator also has a little bitof effect

In order to show the above results in an intuitive waythe boxplots of the performance metric 119862 are also adoptedto illustrate the same conclusion Here we use the boxplotsfor TTSP 20 times 8 as an example The name role is similar tothe above mentioned principle The ten chaotic maps (bakercat circle cubic Gauss ICMIC logistic sinusoidal tentand Zaslavskii) are denoted by 1 2 3 10 in alphabeticalorder In addition ldquo119877rdquo represents the original MOEAD ldquo119868rdquorepresents the phase for initial population ldquo119862rdquo representsthe phase for crossover operator ldquo119872rdquo represents the phasefor mutation operator For example the algorithm for initialpopulation by logistic map is named ldquoI7rdquo Figures 15 16 and17 are the boxplots for chaotic maps embedded in crossoverin initialization and in mutation separately


Table 7 The average value of HV

6 times 8 20 times 8

119877 119862 119877 119868 119877 119872 119877 119862 119877 119868 119877 119872

Baker 03312 03555 03312 03240 03312 03716 07471 07724 07471 07472 07471 07480Cat 03312 03361 03312 03592 03312 03429 07471 07460 07471 07580 07471 07370Circle 03312 03514 03312 03538 03312 03528 07471 07502 07471 07556 07471 07608Cubic 03312 03536 03312 03305 03312 03511 07471 07411 07471 07448 07471 07595Gauss 03312 03470 03312 03324 03312 03568 07471 07538 07471 07294 07471 07548ICMIC 03312 03618 03312 03416 03312 03672 07471 07557 07471 07465 07471 07488Logistic 03312 03450 03312 03302 03312 03394 07471 07579 07471 07560 07471 07539Sinusoidal 03312 03514 03312 03373 03312 03386 07471 07540 07471 07483 07471 07547Tent 03312 03566 03312 03318 03312 03416 07471 07442 07471 07486 07471 07448Zaslavskii 03312 03276 03312 03376 03312 03434 07471 07466 07471 07562 07471 07433


119877 119862 119877 119868 119877 119872 119877 119862 119877 119868 119877 119872


Table 8 The average value of 119862


(119877 119862) (119862 119877) (119877 119868) (119868 119877) (119877119872) (119872 119877) (119877 119862) (119862 119877) (119877 119868) (119868 119877) (119877119872) (119872 119877)



(119877 119862) (119862 119877) (119877 119868) (119868 119877) (119877119872) (119872 119877) (119877 119862) (119862 119877) (119877 119868) (119868 119877) (119877119872) (119872 119877)



Table 9 The visualized result


119862 119868 119872 119862 119868 119872 119862 119868 119872 119862 119868 119872

Baker ++ ++ ++ ++Cat ++ ++ ++ ++Circle ++ ++ ++ ++ ++ ++ ++ ++Cubic ++ ++Gauss ++ ++ ++ ++ICMIC ++ ++ ++ ++ ++ ++ ++Logistic ++ ++ ++ ++ ++Sinusoidal ++ ++ ++ ++Tent ++ ++ ++ ++Zaslavskii ++ ++

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25 30 35 40 45

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Makespan

TTSP 6 lowast 8

Solutions

Figure 18 Exhaustive result for TTSP 6 times 8

Overall chaotic maps for crossover and mutation oper-ators are helpful for preventing the solutions from trappingin the local optima and have significant improvement onthe evolutionary algorithms based on the decomposition forsolving the TTSP Circle map and ICMIC map have the bestperformance in ten maps especially Cubic map logistic mapGauss map and sinusoidal map have better contribution insolving those TTSPs

643 Result Analysis We discuss and explore the reason forthese conclusions based on the above resultsWe focus on thedistribution of solutions of the TTSP

We calculate the feasible solutions of a small scaleTTSP 6 times 8 using the method of enumeration that can-not be used in large scale TTSPs The result is shownin Figure 18 The solutions for the true Pareto front are[(23 703) (28 195) (31 18) (36 146)] out of 103680 solu-tions in objective space We can find that the TTSP hasnonuniformdistribution andmany local optima exist amongall the solutions of the TTSP

The chaotic map has the nature to avoid becomingtrapped in local optima The TTSP has many local optimaAll the experiments illustrate the fact that using chaoticmaps embeddedwith the evolutionary algorithm can help theTTSP to obtain good solutions In addition the process of

crossover and mutation is important for jumping out of localoptima The experiments also validate this fact

Furthermore chaotic maps have a superior effect onescaping from local optima but not all of them are effectiveWe want to find the relationship from the distribution Thedistribution of every chaotic map is shown in Figure 1 Somechaotic maps like circle map cubic map and ICMIC mapare relatively nonuniformly distributed It is very similar tothe distribution of the optimal solution of the TTSP Theabove experiments indicate that these chaotic maps have apositive effect onTTSP Some chaoticmaps like catmap haveuniform distributionThe experiments show that they cannotobtain good effect for solving the TTSP in most situationsIt is natural that the effect of chaotic maps is floatingunder different circumstance because of the ergodicity andstochasticity of chaotic maps However the similarity degreeof the distribution between the chaoticmaps and the problemis a very essential factor for the application of chaotic maps

65 Experiment 4 Comparison of CMOEAD and VNMReferring to Table 9 together with the data in Tables 7 and 8we select a few variants of CMOEAD to compare with VNMVNM has been proved to be more suitable to solve the TTSPthan other methods such as chaotic NSGA-II (CNSGA) [5]Therefore a comparison of CMOEAD and VNM is carriedout to illustrate the performance of our algorithm

We take TTSP 20 times 8 and 40 times 12 as representative testproblems For TTSP 20 times 8 we select the three variantsof CMOEAD They are CMOEAD-C1 with bakerrsquos mapin crossover operator CMOEAD-I9 with tent map in theinitial population and CMOEAD-M3 with circle map inthe mutation operator respectively For TTSP 40 times 12 thethree variants of CMOEAD are CMOEAD-C1 with bakerrsquosmap in crossover operator CMOEAD-I10 with Zaslavskiimap in the initial population and CMOEAD-M9 with tentmap in the mutation operator respectively The results of theperformance metrics HV and 119862 of 10 independent runs arein Tables 10 and 11 separately The best results obtained fromVNMand three variants of CMOEAD for different instancesare shown in Figures 19 and 20 ldquo119881rdquo represents the VNM andthe symbol in the table is similar to the abovementioned role


30 40 50 60 7012

13

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16

17

18

19

Makespan30 40 50 60 70

Makespan Makespan

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20 40 60 80 100 12013

14

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16

17

18

19

12

14

16

18

20

24

22

VNMCMOEAD-C1

VNMCMOEAD-I9

VNMCMOEAD-M3

Figure 19 Comparison of VNM and three variants of CMOEAD for 20 times 8

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20

22

24

26

28

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28

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40 50 60 70 80

Makespan Makespan Makespan40 50 60 70 80 90 40 60 80 100 120 140 160

15

20

25

30

35

40

VNMCMOEAD-C1

VNMCMOEAD-I10

VNMCMOEAD-M9


It shows that the solutions obtained by the CMOEADdominate most of the solutions obtained by the VNM inthe above figures The values of 119862 in Table 11 indicate thatCMOEADhas good convergenceThe results in Table 10 alsoshow that the solutions obtained by CMOEAD are of higher

comprehensive performance Therefore the CMOEAD hasthe best performance completely

A short summary can be obtained according to theabove experiments and analyses The improving encodingmethod is effective for solving the TTSP In addition the


Table 10 The value of HV


119881 119862 119881 119868 119881 119872 119881 119862 119881 119868 119881 119872

1 04067 04928 04067 04582 04067 04642 04016 04718 04016 04576 04016 041892 04018 04844 04018 04756 04018 04746 04148 04511 04148 04449 04148 045213 04552 04776 04552 04582 04552 04706 04729 04729 04729 04937 04729 042224 04258 04907 04258 04789 04258 04643 03881 04853 03881 04521 03881 042865 04283 05055 04283 04607 04283 04755 04010 04598 04010 04674 04010 046666 04492 04690 04492 04738 04492 04889 04155 04389 04155 04496 04155 043547 04560 04540 04560 04612 04560 04871 03900 04521 03900 04688 03900 045788 04306 04765 04306 04688 04306 04789 03968 04426 03968 04703 03968 047509 04460 04872 04460 04632 04460 04740 04235 04585 04235 04500 04235 0446810 04458 04897 04458 04803 04458 04767 04002 04406 04002 04465 04002 04537Average 04345 04827 04345 04679 04345 04755 04104 04574 04104 04601 04104 04457Times 1 9 0 10 0 10 0 9 0 10 1 9

Table 11 The value of 119862


(119881 119862) (119862 119881) (119881 119868) (119868 119881) (119881119872) (119872119881) (119881 119862) (119862 119881) (119881 119868) (119868 119881) (119881119872) (119872119881)

1 00000 10000 00000 08333 00000 10000 00000 10000 00000 10000 00000 090002 00000 10000 00000 10000 00000 10000 00000 10000 00000 10000 00000 100003 00000 09000 03750 05000 01667 07000 00000 10000 00000 10000 00000 100004 00000 10000 00000 10000 00000 10000 00000 10000 00000 10000 00000 100005 00000 10000 00000 09000 00000 10000 00000 10000 00000 10000 00000 100006 00000 01667 00833 01667 00000 10000 00000 08571 00000 10000 01000 085717 04444 00000 00000 02000 00000 10000 00000 10000 00000 10000 00000 100008 00000 10000 00000 10000 00000 10000 00000 10000 00000 10000 00000 100009 00000 10000 00000 10000 00000 10000 00000 10000 00000 07778 00000 0777810 00000 10000 01000 05714 00833 05714 00000 10000 00000 10000 00000 10000Average 00444 08067 00558 07171 00250 09271 00000 09857 00000 09778 00100 09535Times 1 9 0 10 0 10 0 10 0 10 0 10

effectiveness of the multiobjective evolutionary algorithmbased on decomposition using chaotic maps which havenonuniform distributions is illustrated for TTSP Further-more the comparisons of CMOEAD andVNM indicate thatour algorithm has the best performance for solving the TTSPThe fact the chaotic map is an effective and efficient methodfor solving the problemwith local optima is illustrated again

7 Conclusion

The TTSP is a complex combinational optimization problemand has many local optimaThis paper focuses on the chaoticmultiobjective evolutionary algorithm based on decomposi-tion for solving the TTSP The improving encoding methodis proposed to increase the encoding efficiency Ten chaoticmaps are embedded in three phases of MOEAD to solvethe TTSP and the results show that the proposed algorithmcan prevent solutions from falling into local optima Theperformance metrics HV and 119862 are used to analyze thealgorithms with chaotic maps In the experimental resultsalmost all chaotic maps have good effects on improving theperformance of evolutionary algorithms to solve the TTSP

TheCMOEAD approaches using the circle and ICMICmapsin all phases have best performance and are very suitable forsolving the TTSP A comparison of CMOEAD and VNM iscarried out to test the performance of our algorithm and theresults also show that the solutions obtained by CMOEADare of higher comprehensive performance Our work givesguidance on choosing chaotic maps and phases for the TTSPFuture work will focus on more chaotic maps embedded inother algorithms for different kinds of problems and discoverthe reasons for their special properties

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank the anonymous reviewersfor their helpful comments in improving their paper Thisresearch is supported by the National Natural Science Foun-dation of China under Grant no 61101153


References

[1] H Lu R Niu J Liu and Z Zhu ldquoA chaotic non-dominatedsorting genetic algorithm for the multi-objective automatic testtask scheduling problemrdquo Applied Soft Computing Journal vol13 2013

[2] R Xia M Q Xiao and J J Cheng ldquoParallel TPS design andapplication based on software architecture components andpatternsrdquo in IEEE Autotestcon pp 234ndash240 Baltimore MdUSA 2007

[3] D Zhou P Qi and T Liu ldquoAn optimizing algorithm forresources allocation in parallel testrdquo in Proceedings of the IEEEInternational Conference on Control and Automation (ICCArsquo09) pp 1997ndash2002 Christchurch New Zealand December2009

[4] H Lu J Liu R Y Niu and Z Zhu ldquoFitness distance analysis forparallel genetic algorithm in the test task scheduling problemrdquoSoft Computing 2013

[5] H Lu Z Zhu X T Wang and L J Yin ldquoA variableneighborhoodMOEAD formulti-objective test task schedulingproblemrdquo Mathematical Problems in Engineering vol 2014Article ID 423621 14 pages 2014

[6] H Lu X Chen and J Liu ldquoParallel test task scheduling withconstraints based on hybrid particle swarm optimization andtaboo searchrdquo Chinese Journal of Electronics vol 21 no 4 pp615ndash618 2012

[7] D Donald S Roman Z Ivan P Michal and B D MagdalenaldquoUtilising the chaos-induced discrete self organising migratingalgorithm to solve the lot-streaming flowshop scheduling prob-lemwith setup timerdquo Soft Computing vol 18 no 4 pp 669ndash6812014

[8] M Gavrilova and K Ahmadian ldquoOn-demand chaotic neuralnetwork for broadcast scheduling problemrdquo Journal of Super-computing vol 59 no 2 pp 811ndash829 2012

[9] H M Jiang C K Kwong W H Ip and Z Q Chen ldquoChaos-based fuzzy regression approach tomodeling customer satisfac-tion for product designrdquo IEEE Transactions on Fuzzy Systemsvol 21 no 5 pp 926ndash936 2013

[10] M Sun L Zhao W Cao Y Xu X Dai and X Wang ldquoNovelhysteretic noisy chaotic neural network for broadcast schedul-ing problems in packet radio networksrdquo IEEE Transactions onNeural Networks vol 21 no 9 pp 1422ndash1433 2010

[11] E Zitzler and L Thiele ldquoMultiobjective evolutionary algo-rithms a comparative case study and the strength Paretoapproachrdquo IEEETransactions onEvolutionaryComputation vol3 no 4 pp 257ndash271 1999

[12] C Cheng W Wang D Xu and K W Chau ldquoOptimizinghydropower reservoir operation using hybrid genetic algorithmand chaosrdquoWater ResourcesManagement vol 22 no 7 pp 895ndash909 2008

[13] D Liu and Y D Cao ldquoCGA chaotic genetic algorithm forfuzzy job scheduling in grid environmentrdquo in ComputationalIntelligence and Security vol 4456 of Lecture Notes in ComputerScience pp 133ndash143 2007

[14] M R Singh and S S Mahapatra ldquoA swarm optimizationapproach for flexible flow shop scheduling with multiprocessortasksrdquo The International Journal of Advanced ManufacturingTechnology vol 62 no 1ndash4 pp 267ndash277 2012

[15] J M Bahi C Guyeux A Makhoul and C Pham ldquoSecurescheduling of wireless video sensor nodes for surveillanceapplicationsrdquo in Ad Hoc Networks vol 89 of Lecture Notes

of the Institute for Computer Sciences Social Informatics andTelecommunications Engineering pp 1ndash15 2012

[16] A Q Yu and X S Gu ldquoAn improved transiently chaotic neuralnetwork approach for identical parallel machine schedulingrdquo inAdvances in Cognitive Neurodynamics ICCN 2007 pp 909ndash9132007

[17] T Niknam M R Narimani J Aghaei and R Azizipanah-Abarghooee ldquoImproved particle swarm optimisation for multi-objective optimal power flow considering the cost loss emis-sion and voltage stability indexrdquo IET Generation Transmissionand Distribution vol 6 no 6 pp 515ndash527 2012

[18] R Zhou C M Ye and H M Ma ldquoModel research ofmulti-objective and resource-constrained project schedulingproblemrdquo in Proceedings of the 19th International Conferenceon Industrial Engineering and Engineering Anagement pp 991ndash1001 2013

[19] Z M Fang ldquoA quantum immune algorithm for multiobjectiveparallel machine schedulingrdquo inAdvances in Swarm IntelligenceLecture Notes in Computer Science vol 6145 pp 321ndash327 2010

[20] Q Zhang and H Li ldquoMOEAD a multiobjective evolutionaryalgorithm based on decompositionrdquo IEEE Transactions onEvolutionary Computation vol 11 no 6 pp 712ndash731 2007

[21] H Peitgen H Jurgens and D Saupe Chaos and FractalsSpringer Berlin Germany 1992

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


of customer satisfaction models [9] Sun et al studied anovel hysteretic noisy chaotic neural network for broadcastscheduling problems in packet radio networks and exhibiteda stochastic chaotic simulated annealing algorithm [10] Thephenomenon that occurred in these researches illustratedthat the evolutionary algorithm embedded with chaos is aneffective and efficient approach for improving the searchingability of the algorithm However all these researches havethe same defect The authors only used one or several chaoticmaps embedded in algorithms to solve practical problemsHowever there is no detailed discussion and analysis

For the TTSP notice that the previous studies mostlyaim at a single objective and only few papers focus onthe multiobjective problem [1 5] Through the analysis ofour previous related work for the multiobjective TTSP weknow the multiobjective evolutionary algorithm based ondecomposition (MOEAD) exhibited the best performancein solving the TTSP [5] in the aspect of convergence anddiversity Therefore using chaotic maps in MOEAD canfurther enhance the quality of the solution for the TTSP

In this paper we propose a chaotic multiobjective evo-lutionary algorithm based on decomposition (CMOEAD)for the TTSP Ten chaotic maps are embedded in threedifferent phases in the evolutionary processThe aim is to giveguidance for the choice of chaotic maps and phases based onthe framework of MOEAD for the TTSP

First the chromosome-encoding scheme is very impor-tant for the problem description and the operation in theevolutionary process For the TTSP there are different kindsof encoding strategies like task sequencing list (TSL) (orthe operations list coding (OLC)) matrix-encoding andintegrated encoding scheme (IES) TSL andmatrix-encodingare not acceptable if a task can be tested on more thanone set of instruments IES can overcome this problembut it cannot realize the selection operation under equalprobability Therefore in this paper the improved IES isproposed by changing the schemes selectionmethod of everytest task As a result the equal probability is realized

Then ten chaotic maps are embedded in MOEADindependently in three phases The ten chaotic maps arebakerrsquos map cat map circle map cubic map Gauss mapICMIC map logistic map sinusoidal map tent map andZaslavskii mapThree phases are initial population crossoveroperator and mutation operator Four benchmarks of theTTSP are used to evaluate the performance of the proposedalgorithm They are 6 times 8 20 times 8 30 times 12 and 40 times 12 Weuse 119899 times119898 to represent the benchmark Here 119899 is the numberof tasks and119898 is the number of instruments

The performance metrics hypervolume (HV) and 119862 [11]are used to evaluate the role of chaotic maps on MOEADTherefore we can find which kind of chaotic map embeddedalgorithm is the best one for solving the TTSP

From the results of experiments it can be seen that thechaotic map embedded MOEAD has good performance tosolve the TTSP for both small and large scale problemsDifferent kinds of chaotic maps have different performancesin different phases of MOEAD but ICMIC map and circlemap in initial population crossover operator and mutationoperator have the best performance The experiments for

comparisons of CMOEAD and VNM show that our algo-rithmperforms better than theVNMin solving theTTSPTheevidence the chaotic map is an effective and efficient methodfor solving the problem with local optima is validated Thesimilarity degree of distribution between chaotic maps andthe problem is a very essential factor for the application ofchaotic maps

The rest of the paper is organized as follows Section 2gives a summary of related work on applying chaos toimprove evolutionary algorithms Section 3 concludes themathematical model proposed by us in previous work forthe integrity In Section 4 ten chaotic maps including bothone-dimensional maps and two-dimensional maps are intro-duced In Section 5 the proposed CMOEAD is describedin detail for solving the TTSP The detail of the encodingscheme and the phases in which chaos can be embedded inevolutionary algorithms are introduced Experimental resultsand performance comparisons are presented and discussed inSection 6 Finally Section 7 concludes the paper

2 Related Work

Recently chaotic sequences have been integrated in theevolutionary process through two types of operations Oneis using chaotic maps to replace random sequences Anotheris to replace the genetic operations These two kinds ofoperations always appear in the same algorithm at once

In detail all the operations can be divided into sevencases They are population initialization setting crossoverprobability setting crossover position setting crossover oper-ator setting mutation probability setting mutation operatorand increasing chaotic disturbance The performance ofdifferent operations is totally different For example adoptingchaoticmaps in the initialization canmaintain the populationdiversity The aim of using chaotic maps to replace standardmutation operator is to avoid the search being trapped in localoptima

For the scheduling problem the situation is the same asthe above in both the single objective and the multiobjectivescheduling problems For the single objective schedulingproblem Cheng et al used the hybrid genetic algorithm andchaos to optimize the hydropower reservoir operation [12]Two methods were adopted to improve the performance ofGA One was the adoption of chaos for initialization andanother was the annealing chaotic mutation operation Theconclusion was that the proposed approach is feasible andeffective in optimal operations of complex reservoir systemsLiu and Cao [13] proposed a chaotic algorithm for thefuzzy job scheduling problem in the grid environment withuncertaintiesThe authors incorporated logistic map with thestandard genetic algorithm and proposed a chaotic mutationoperator based on the feedback of the fitness functionSingh and Mahapatra [14] proposed a swarm optimizationapproach for the flexible flow shop scheduling problemwith multiprocessor tasks The logistic map was used inthis paper Bahi et al [15] considered a novel chaos-basedscheduling scheme for video surveillance to defeat maliciousintruders The concept of chaotic iterations was investigated







119899

119895=1and a set


119894119895 and




119896=1is used


119899



119906119895119896119906119895119896

119906=1



119894119895 is used



119883119896119896lowast

119895119895lowast = 1 if 119908119896119895 cap 119908

119896lowast

119895lowast =

0 otherwise(1)


119884119895119895lowast =






minimize

max1le119896le1198961198951le119895le119899

max119903119894isin119908119896119895

1198621198941198951

119876

119899

sum

119895=1

119898

sum

119894=1

119875119894119895119874119894119895

(3)

subject to

119862119894119895 = 119878119894119895 + 119875119894119895 (4)

119874119894119895 =


(5)


lowast




problem

4 Chaotic Maps





Bakerrsquos map 119861 (119909 119910) =


119910

2) for 05 le 119909 lt 1

2 119909 isin (0 1)


2 119909119896 isin (0 1)

Circle map 119909119896+1 = 119909119896 + 119887 minus (119886

2120587) sin (2120587119909119896) mod (1) 1 (0 1)

119886 = 05 119887 = 02


120588 = 259

Gauss map 119909119896+1 =

0 119909119896 = 0

1

119909119896


ICMIC map 119909119896+1 = sin(119886

119909119896


119886 = 2



Tent map 119909119896+1 =

119909119896

07119909119896 lt 07

(10

3) 119909119896 (1 minus 119909119896) otherwise

1 (0 1)


minus119903119910119896

2 119910119896 isin [minus10512 10512]

V = 400 119903 = 3 119886 = 126695








119896119895


119896 = [119909119894119895 times 10] mod 119896119895 + 1 (6)




0 05 1

0 05 1

0 05 1 0 05 1

0 05 1 0 05 1

0 05 1

0 05 1

0 05 1 0 05 10

500

1000

1500

0

500

1000

1500

0

500

1000

1500

0

500

1000

1500

2000

2000

0

500

1000

1500

2000

0

500

1000

1500

0

500

1000

1500

0

1000

2000

3000

0

1000

2000

3000

0

1000

2000

3000

Bakers Cat Circle

Cubic Gauss ICMIC


Zaslavskii




119896 = [119909119894119895119896119895 times 10] mod 119896119895 + 1 (7)




119879 119882119895 119908119896119895 119875

119896119895

1199051

11990811 1199031 1199032 511990821 1199032 1199034 3

1199052

11990812 1199031 411990822 1199033 1

1199053 11990813 1199034 2

1199054

11990814 1199031 1199033 411990824 1199032 1199034 311990834 1199032 1199033 7





119875119896119895 2 4 3 4



[00 01)

[03 04)

[06 07)

[09 10)

[01 02)

[04 05)

[07 08)

[02 03)

[05 06)

[08 09)

119896 1 2 3





119909119904= 1199091119904 1199092119904 119909

119894119904 119909

119899119904 119904 =1 2 119873 119894 =1 2 119873

(8)


119894119904)



119909119905+1119894 =

119909119905119894 + 119865 times (119909


119909119905119894 otherwise

(9)






119909lowast119904 = 119909119904 + (119909

119906119904 minus 119909119897119904) times 120575119904 (10)


Consider

120575119904 = (2119906119904)1(120578119898+1)


1(120578119898+1) otherwise(11)


6 Experiments








[12 13)[15 16)[18 19)

[21 22)[24 25)[27 28)

[1 2)[4 5)[7 8)[10 11)

[13 14)[16 17)[19 20)

[22 23)[25 26)[28 29)

[2 3)[5 6)[8 9)[11 12)

[14 15)[17 18)[20 21)

[23 24)[26 27)[29 30)

119896 1 2 3



119899iter 250119899pop 100119899var 6 20 30 40CR 09119875119898 16 120 130 140

35 40 45 50 55 60 65 70

Makespan

14

16

18

20

22

24

26

28

30

32

34

Mea

n w

orkl

oad

MOEAD-1MOEAD-2

MOEAD-3













HV (119878) = Leb(⋃119894isin119878

V119894) (12)


20 25 30 35 40 45

20 25 30 35 40 45 20 25 30 35 40 45 20 25 30 35 40 45

20 25 30 35 40 45 20 25 30 35 40 45 20 25 30 35 40 45

20 25 30 35 40

20 25 30 35 40

20 25 30 35 40 4510

15

20

25

10

15

20

25

10

15

20

25

10

15

20

25

10

15

20

25

10

15

20

25

10

15

20

25

Mea

n w

orkl

oad

10

15

20

25

10

15

20

25

Mea

n w

orkl

oad

10

15

20

25

Mea

n w

orkl

oad

Mea

n w

orkl

oad

Baker

Makespan




Cat Circle

Cubic Gauss ICMIC


Zaslavskii

MOEADCMOEAD-C1

MOEADCMOEAD-C2

MOEADCMOEAD-C3

MOEADCMOEAD-C4

MOEADCMOEAD-C5

MOEADCMOEAD-C6

MOEADCMOEAD-C7

MOEADCMOEAD-C8

MOEADCMOEAD-C9

MOEADCMOEAD-C10



20 25 30 35 4010

15

20

25

10

15

20

25

10

15

20

25Baker

Makespan

Makespan

20 25 30 35 40

Makespan20 25 30 35 40

Makespan

20 25 30 35 40

Makespan

20 25 30 35 40

Makespan

Mea

n w

orkl

oad

10

15

20

25

10

15

20

25

10

15

20

25

10

15

20

25

10

15

20

25

10

15

20

25

10

15

20

25

Mea

n w

orkl

oad

Mea

n w

orkl

oad

Mea

n w

orkl

oad

MOEADCMOEAD-I1

20 25 30 35 40 45

Makespan20 25 30 35 40 45

Makespan20 25 30 35 40 45

Makespan20 25 30 35 40 45

Makespan20 25 30 35 40 45

Cat

MOEADCMOEAD-I2

Circle

MOEADCMOEAD-I3

Cubic

MOEADCMOEAD-I4

Gauss

MOEADCMOEAD-I5

ICMIC

MOEAD

Logistic

MOEAD

Sinusoidal

MOEAD

Tent

MOEADCMOEAD-I9

Zaslavskii

MOEADCMOEAD-I10

CMOEAD-I6

CMOEAD-I7 CMOEAD-I8



10

15

20

25

Mea

n w

orkl

oad

10

15

20

25

Mea

n w

orkl

oad

10

15

20

25

Mea

n w

orkl

oad

10

15

20

25

10

15

20

25

10

15

20

25

Mea

n w

orkl

oad

10

15

20

25

10

15

20

25

10

15

20

25

10

15

20

25

20 25 30 35 40 4520 25 30 35 40


Makespan

20 25 30 35 40 45

Makespan20 25 30 35 40 45

Makespan20 25 30 35 40 45

Makespan

20 25 30 35 40 45

Makespan20 25 30 35 40 45

Makespan

20 25 30 35 40 45

Makespan

20 25 30 35 40 45

Makespan

MOEADCMOEAD-M1

MOEADCMOEAD-M2

MOEADCMOEAD-M3

MOEADCMOEAD-M4

MOEADCMOEAD-M5

MOEADCMOEAD-M6

MOEADCMOEAD-M7

MOEADCMOEAD-M8

MOEADCMOEAD-M9

MOEADCMOEAD-M10

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii



30 40 50 60 70

30 40 50 60 70

12

14

16

18

12

14

16

18

30 40 50 60 70 80

30 40 50 60 70 80

12

14

16

18

20

12

14

16

18

20

12

14

16

18

20

14

16

18

20

30 35 40 45 50

20 40 60 80 100

20 40 60 80 10030 40 50 60 70

30 40 50 60 70

30 40 50 6010

15

20

25

10

15

20

25 40

30

20

10

30

25

20

15

10

Mea

n w

orkl

oad

Mea

n w

orkl

oad

Mea

n w

orkl

oad

Mea

n w

orkl

oad

Makespan



Makespan

Makespan Makespan

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii

MOEADCMOEAD-C1

MOEADCMOEAD-C2

MOEADCMOEAD-C3

MOEADCMOEAD-C5

MOEADCMOEAD-C4

MOEADCMOEAD-C6

MOEADCMOEAD-C7

MOEADCMOEAD-C8

MOEADCMOEAD-C9

MOEADCMOEAD-C10



14

16

18

20

14

16

18

20

14

16

18

20

14

16

18

20

12

14

16

18

20

12

14

16

18

20

12

14

16

18

20

12

14

16

18

20

12

14

16

18

20

10

15

20

25

30 35 40 45 50 30 40 50 60 70 20 40 60 80 100 120

20 40 60 80 100 30 35 40 45 50 30 40 50 60 70

30 40 50 60 70

30 35 40 45 50 30 40 50 60 70 30 35 40 45 50 55

Mea

n w

orkl

oad

Mea

n w

orkl

oad

Mea

n w

orkl

oad

Mea

n w

orkl

oad



Makespan Makespan

Makespan

Makespan

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii

MOEADCMOEAD-I1

MOEADCMOEAD-I2

MOEADCMOEAD-I3

MOEADCMOEAD-I5

MOEADCMOEAD-I4

MOEADCMOEAD-I6

MOEADCMOEAD-I7

MOEADCMOEAD-I9

MOEADCMOEAD-I8

MOEADCMOEAD-I10



30 40 50 60 70 30 40 50 60 70 30 35 40 45 50 55

20 40 60 80 100 30 40 50 60 70 80 30 40 50 60 70

30 35 40 45 50 55 20 30 40 50 60 70 30 40 50 60 70

30 35 40 45 50

MOEADCMOEAD-M1

MOEADCMOEAD-M2

MOEADCMOEAD-M3

MOEADCMOEAD-M4

MOEADCMOEAD-M5

MOEADCMOEAD-M6

MOEADCMOEAD-M7

MOEADCMOEAD-M8

MOEADCMOEAD-M9

MOEADCMOEAD-M10

10

15

20

25

14

16

18

20

12

14

16

18

20

12

14

16

18

20

12

14

16

18

20

12

14

16

18

20

12

14

16

18

20

10

15

20

25

10

15

20

25

10

15

20

25

Mea

n w

orkl

oad

Mea

n w

orkl

oad

Mea

n w

orkl

oad

Mea

n w

orkl

oad

Makespan

Makespan

Makespan Makespan

Makespan

Makespan

Makespan Makespan

Makespan Makespan

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii



14

16

18

20

10

15

20

25

16

18

20

22

16

18

20

22

24

10

15

20

25

16

18

20

22

15

20

25

16

18

20

22

24

15

20

25

15

20

25

30 40 50 60 70 80

30 40 50 60 70 30 40 50 60 70

30 40 50 60 70

80 30 40 50 60 70 80

30 40 50 60 70 80 30 40 50 60 70 80

30 40 50 60 70 80

20 40 60 80 100 20 40 60 80 100

Mea

n w

orkl

oad

Mea

n w

orkl

oad

Mea

n w

orkl

oad

Mea

n w

orkl

oad




Makespan

MOEADCMOEAD-C1

MOEADCMOEAD-C2

MOEADCMOEAD-C3

MOEADCMOEAD-C4

MOEADCMOEAD-C5

MOEADCMOEAD-C6

MOEADCMOEAD-C7

MOEADCMOEAD-C8

MOEADCMOEAD-C9

MOEADCMOEAD-C10

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii



20 40 60 80 100

20 40 60 80 100

20 40 60 80 100

20 40 60 80 100

20 40 60 80 100 30 40 50 60 70 80

30 40 50 60 70 30 40 50 60 7080

30 40 50 60 70 80 30 40 50 60 70 80

16

18

20

22

24

16

18

20

22

24

15

20

25

30

15

20

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MOEADCMOEAD-I1

MOEADCMOEAD-I2

MOEADCMOEAD-I3

MOEADCMOEAD-I4

MOEADCMOEAD-I5

MOEADCMOEAD-I6

MOEADCMOEAD-I7

MOEADCMOEAD-I8

MOEADCMOEAD-I9

MOEADCMOEAD-I10

Mea

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Makespan Makespan



Makespan

Makespan

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii



30 40 50 60 70 80 30 40 50 60 70 30 40 50 60 70801015

20

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30

20 40 60 80 100

20 40 60 80 100

30 40 50 60 70 80

40 50 60 70 80 30 40 50 60 70 80 30 40 50 60 70 80

30 40 50 60 70 80

MOEADCMOEAD-M1

MOEADCMOEAD-M2

MOEADCMOEAD-M3

MOEADCMOEAD-M4

MOEADCMOEAD-M5

MOEADCMOEAD-M6

MOEADCMOEAD-M7

MOEADCMOEAD-M8

MOEADCMOEAD-M9

MOEADCMOEAD-M10

Mea

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Makespan Makespan

Makespan

Makespan

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii



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40 50 60 70 80 40 50 60 70 8040 50 60 70 80 90

40 50 60 70 80 90 40 50 60 70 80 90 40 50 60 70 80 90

40 50 60 70 80 90 40 50 60 70 80 90

40 50 60 70 80 90

40 60 80 100

MOEADCMOEAD-C1

MOEADCMOEAD-C2

MOEADCMOEAD-C3

MOEADCMOEAD-C4

MOEADCMOEAD-C5

MOEADCMOEAD-C6

MOEADCMOEAD-C7

MOEADCMOEAD-C8

MOEADCMOEAD-C9

MOEADCMOEAD-C10

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Makespan Makespan

Makespan

Makespan

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii



40 50 60 70 80 40 50 60 70 80

40 50 60 70 80

90

40 50 60 70 80 90

40 50 60 70 80 90 40 50 60 70 80 90

40 50 60 70 80 90

40 60 80 100

40 60 80 100

40 45 50 55 60 65

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Makespan

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii

MOEADCMOEAD-I1

MOEADCMOEAD-I2

MOEADCMOEAD-I3

MOEADCMOEAD-I4

MOEADCMOEAD-I5

MOEADCMOEAD-I6

MOEADCMOEAD-I7

MOEADCMOEAD-I8

MOEADCMOEAD-I9

MOEADCMOEAD-I10



40 50 60 70 80 90

40 50 60 70 80 90 40 50 60 70 80 90 40 50 60 70 80 90

40 50 60 70 80 9040 50 60 70 80 90

40 50 60 70 80 90 40 50 60 70 80

40 50 60 70 80

40 50 60 70 80

15

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Mea

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Makespan

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii

MOEADCMOEAD-M1

MOEADCMOEAD-M2

MOEADCMOEAD-M3

MOEADCMOEAD-M5

MOEADCMOEAD-M6

MOEADCMOEAD-M7

MOEADCMOEAD-M9

MOEADCMOEAD-M10

MOEADCMOEAD-M4

MOEADCMOEAD-M8



Algorithm

minus02

0

02

04

06

08

1

TTSP 20 lowast 8

C-m

easu

re

(RC

1)

(C1R

)

(RC

2)

(C2R

)

(RC

3)

(C3R

)

(RC

4)

(C4R

)

(RC

5)

(C5R

)

(RC

6)

(C6R

)

(RC

7)

(C7R

)

(RC

8)

(C8R

)

(RC

9)

(C9R

)

(RC

10)

(C10R

)




119862 (119860 119861) =


(13)





Algorithm

minus02

0

02

04

06

08

1

TTSP 20 lowast 8

(RI1)

(I1R

)

(RI2)

(I2R

)

(RI3)

(I3R

)

(RI4)

(I4R

)

(RI5)

(I5R

)

(RI6)

(I6R

)

(RI7)

(I7R

)

(RI8)

(I8R

)

(RI9)

(I9R

)

(RI10)

(I10R

)

+

+

+

C-m

easu

re


(RM

1)

(M1R

)

(RM

2)

(M2R

)

(RM

3)

(M3R

)

(RM

4)

(M4R

)

(RM

5)

(M5R

)

(RM

6)

(M6R

)

(RM

7)

(M7R

)

(RM

8)

(M8R

)

(RM

9)

(M9R

)

(RM

10)

(M10R

)

Algorithm

minus02

0

02

04

06

08

1

TTSP 20 lowast 8

+

C-m

easu

re







119877 119862 119877 119868 119877 119872 119877 119862 119877 119868 119877 119872



119877 119862 119877 119868 119877 119872 119877 119862 119877 119868 119877 119872




(119877 119862) (119862 119877) (119877 119868) (119868 119877) (119877119872) (119872 119877) (119877 119862) (119862 119877) (119877 119868) (119868 119877) (119877119872) (119872 119877)



(119877 119862) (119862 119877) (119877 119868) (119868 119877) (119877119872) (119872 119877) (119877 119862) (119862 119877) (119877 119868) (119868 119877) (119877119872) (119872 119877)





119862 119868 119872 119862 119868 119872 119862 119868 119872 119862 119868 119872


40

35

30

25

20

15

25 30 35 40 45

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Makespan

TTSP 6 lowast 8

Solutions











30 40 50 60 7012

13

14

15

16

17

18

19

Makespan30 40 50 60 70

Makespan Makespan

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Mea

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20 40 60 80 100 12013

14

15

16

17

18

19

12

14

16

18

20

24

22

VNMCMOEAD-C1

VNMCMOEAD-I9

VNMCMOEAD-M3


14

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28

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Mea

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40 50 60 70 80


15

20

25

30

35

40

VNMCMOEAD-C1

VNMCMOEAD-I10

VNMCMOEAD-M9








119881 119862 119881 119868 119881 119872 119881 119862 119881 119868 119881 119872

1 04067 04928 04067 04582 04067 04642 04016 04718 04016 04576 04016 041892 04018 04844 04018 04756 04018 04746 04148 04511 04148 04449 04148 045213 04552 04776 04552 04582 04552 04706 04729 04729 04729 04937 04729 042224 04258 04907 04258 04789 04258 04643 03881 04853 03881 04521 03881 042865 04283 05055 04283 04607 04283 04755 04010 04598 04010 04674 04010 046666 04492 04690 04492 04738 04492 04889 04155 04389 04155 04496 04155 043547 04560 04540 04560 04612 04560 04871 03900 04521 03900 04688 03900 045788 04306 04765 04306 04688 04306 04789 03968 04426 03968 04703 03968 047509 04460 04872 04460 04632 04460 04740 04235 04585 04235 04500 04235 0446810 04458 04897 04458 04803 04458 04767 04002 04406 04002 04465 04002 04537Average 04345 04827 04345 04679 04345 04755 04104 04574 04104 04601 04104 04457Times 1 9 0 10 0 10 0 9 0 10 1 9



(119881 119862) (119862 119881) (119881 119868) (119868 119881) (119881119872) (119872119881) (119881 119862) (119862 119881) (119881 119868) (119868 119881) (119881119872) (119872119881)

1 00000 10000 00000 08333 00000 10000 00000 10000 00000 10000 00000 090002 00000 10000 00000 10000 00000 10000 00000 10000 00000 10000 00000 100003 00000 09000 03750 05000 01667 07000 00000 10000 00000 10000 00000 100004 00000 10000 00000 10000 00000 10000 00000 10000 00000 10000 00000 100005 00000 10000 00000 09000 00000 10000 00000 10000 00000 10000 00000 100006 00000 01667 00833 01667 00000 10000 00000 08571 00000 10000 01000 085717 04444 00000 00000 02000 00000 10000 00000 10000 00000 10000 00000 100008 00000 10000 00000 10000 00000 10000 00000 10000 00000 10000 00000 100009 00000 10000 00000 10000 00000 10000 00000 10000 00000 07778 00000 0777810 00000 10000 01000 05714 00833 05714 00000 10000 00000 10000 00000 10000Average 00444 08067 00558 07171 00250 09271 00000 09857 00000 09778 00100 09535Times 1 9 0 10 0 10 0 10 0 10 0 10


7 Conclusion





Acknowledgments



References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra















119899

119895=1and a set


119894119895 and




119896=1is used


119899



119906119895119896119906119895119896

119906=1



119894119895 is used



119883119896119896lowast

119895119895lowast = 1 if 119908119896119895 cap 119908

119896lowast

119895lowast =

0 otherwise(1)


119884119895119895lowast =






minimize

max1le119896le1198961198951le119895le119899

max119903119894isin119908119896119895

1198621198941198951

119876

119899

sum

119895=1

119898

sum

119894=1

119875119894119895119874119894119895

(3)

subject to

119862119894119895 = 119878119894119895 + 119875119894119895 (4)

119874119894119895 =


(5)


lowast




problem

4 Chaotic Maps





Bakerrsquos map 119861 (119909 119910) =


119910

2) for 05 le 119909 lt 1

2 119909 isin (0 1)


2 119909119896 isin (0 1)

Circle map 119909119896+1 = 119909119896 + 119887 minus (119886

2120587) sin (2120587119909119896) mod (1) 1 (0 1)

119886 = 05 119887 = 02


120588 = 259

Gauss map 119909119896+1 =

0 119909119896 = 0

1

119909119896


ICMIC map 119909119896+1 = sin(119886

119909119896


119886 = 2



Tent map 119909119896+1 =

119909119896

07119909119896 lt 07

(10

3) 119909119896 (1 minus 119909119896) otherwise

1 (0 1)


minus119903119910119896

2 119910119896 isin [minus10512 10512]

V = 400 119903 = 3 119886 = 126695








119896119895


119896 = [119909119894119895 times 10] mod 119896119895 + 1 (6)




0 05 1

0 05 1

0 05 1 0 05 1

0 05 1 0 05 1

0 05 1

0 05 1

0 05 1 0 05 10

500

1000

1500

0

500

1000

1500

0

500

1000

1500

0

500

1000

1500

2000

2000

0

500

1000

1500

2000

0

500

1000

1500

0

500

1000

1500

0

1000

2000

3000

0

1000

2000

3000

0

1000

2000

3000

Bakers Cat Circle

Cubic Gauss ICMIC


Zaslavskii




119896 = [119909119894119895119896119895 times 10] mod 119896119895 + 1 (7)




119879 119882119895 119908119896119895 119875

119896119895

1199051

11990811 1199031 1199032 511990821 1199032 1199034 3

1199052

11990812 1199031 411990822 1199033 1

1199053 11990813 1199034 2

1199054

11990814 1199031 1199033 411990824 1199032 1199034 311990834 1199032 1199033 7





119875119896119895 2 4 3 4



[00 01)

[03 04)

[06 07)

[09 10)

[01 02)

[04 05)

[07 08)

[02 03)

[05 06)

[08 09)

119896 1 2 3





119909119904= 1199091119904 1199092119904 119909

119894119904 119909

119899119904 119904 =1 2 119873 119894 =1 2 119873

(8)


119894119904)



119909119905+1119894 =

119909119905119894 + 119865 times (119909


119909119905119894 otherwise

(9)






119909lowast119904 = 119909119904 + (119909

119906119904 minus 119909119897119904) times 120575119904 (10)


Consider

120575119904 = (2119906119904)1(120578119898+1)


1(120578119898+1) otherwise(11)


6 Experiments








[12 13)[15 16)[18 19)

[21 22)[24 25)[27 28)

[1 2)[4 5)[7 8)[10 11)

[13 14)[16 17)[19 20)

[22 23)[25 26)[28 29)

[2 3)[5 6)[8 9)[11 12)

[14 15)[17 18)[20 21)

[23 24)[26 27)[29 30)

119896 1 2 3



119899iter 250119899pop 100119899var 6 20 30 40CR 09119875119898 16 120 130 140

35 40 45 50 55 60 65 70

Makespan

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34

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MOEAD-1MOEAD-2

MOEAD-3













HV (119878) = Leb(⋃119894isin119878

V119894) (12)


20 25 30 35 40 45

20 25 30 35 40 45 20 25 30 35 40 45 20 25 30 35 40 45

20 25 30 35 40 45 20 25 30 35 40 45 20 25 30 35 40 45

20 25 30 35 40

20 25 30 35 40

20 25 30 35 40 4510

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Baker

Makespan




Cat Circle

Cubic Gauss ICMIC


Zaslavskii

MOEADCMOEAD-C1

MOEADCMOEAD-C2

MOEADCMOEAD-C3

MOEADCMOEAD-C4

MOEADCMOEAD-C5

MOEADCMOEAD-C6

MOEADCMOEAD-C7

MOEADCMOEAD-C8

MOEADCMOEAD-C9

MOEADCMOEAD-C10



20 25 30 35 4010

15

20

25

10

15

20

25

10

15

20

25Baker

Makespan

Makespan

20 25 30 35 40

Makespan20 25 30 35 40

Makespan

20 25 30 35 40

Makespan

20 25 30 35 40

Makespan

Mea

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MOEADCMOEAD-I1

20 25 30 35 40 45

Makespan20 25 30 35 40 45

Makespan20 25 30 35 40 45

Makespan20 25 30 35 40 45

Makespan20 25 30 35 40 45

Cat

MOEADCMOEAD-I2

Circle

MOEADCMOEAD-I3

Cubic

MOEADCMOEAD-I4

Gauss

MOEADCMOEAD-I5

ICMIC

MOEAD

Logistic

MOEAD

Sinusoidal

MOEAD

Tent

MOEADCMOEAD-I9

Zaslavskii

MOEADCMOEAD-I10

CMOEAD-I6

CMOEAD-I7 CMOEAD-I8



10

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Mea

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10

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25

20 25 30 35 40 4520 25 30 35 40


Makespan

20 25 30 35 40 45

Makespan20 25 30 35 40 45

Makespan20 25 30 35 40 45

Makespan

20 25 30 35 40 45

Makespan20 25 30 35 40 45

Makespan

20 25 30 35 40 45

Makespan

20 25 30 35 40 45

Makespan

MOEADCMOEAD-M1

MOEADCMOEAD-M2

MOEADCMOEAD-M3

MOEADCMOEAD-M4

MOEADCMOEAD-M5

MOEADCMOEAD-M6

MOEADCMOEAD-M7

MOEADCMOEAD-M8

MOEADCMOEAD-M9

MOEADCMOEAD-M10

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii



30 40 50 60 70

30 40 50 60 70

12

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18

12

14

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18

30 40 50 60 70 80

30 40 50 60 70 80

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30 35 40 45 50

20 40 60 80 100

20 40 60 80 10030 40 50 60 70

30 40 50 60 70

30 40 50 6010

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30

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10

Mea

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Mea

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Mea

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Mea

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Makespan



Makespan

Makespan Makespan

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii

MOEADCMOEAD-C1

MOEADCMOEAD-C2

MOEADCMOEAD-C3

MOEADCMOEAD-C5

MOEADCMOEAD-C4

MOEADCMOEAD-C6

MOEADCMOEAD-C7

MOEADCMOEAD-C8

MOEADCMOEAD-C9

MOEADCMOEAD-C10



14

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10

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25

30 35 40 45 50 30 40 50 60 70 20 40 60 80 100 120

20 40 60 80 100 30 35 40 45 50 30 40 50 60 70

30 40 50 60 70

30 35 40 45 50 30 40 50 60 70 30 35 40 45 50 55

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Mea

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Mea

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Mea

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Makespan Makespan

Makespan

Makespan

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii

MOEADCMOEAD-I1

MOEADCMOEAD-I2

MOEADCMOEAD-I3

MOEADCMOEAD-I5

MOEADCMOEAD-I4

MOEADCMOEAD-I6

MOEADCMOEAD-I7

MOEADCMOEAD-I9

MOEADCMOEAD-I8

MOEADCMOEAD-I10



30 40 50 60 70 30 40 50 60 70 30 35 40 45 50 55

20 40 60 80 100 30 40 50 60 70 80 30 40 50 60 70

30 35 40 45 50 55 20 30 40 50 60 70 30 40 50 60 70

30 35 40 45 50

MOEADCMOEAD-M1

MOEADCMOEAD-M2

MOEADCMOEAD-M3

MOEADCMOEAD-M4

MOEADCMOEAD-M5

MOEADCMOEAD-M6

MOEADCMOEAD-M7

MOEADCMOEAD-M8

MOEADCMOEAD-M9

MOEADCMOEAD-M10

10

15

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25

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Mea

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Mea

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Mea

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Makespan

Makespan

Makespan Makespan

Makespan

Makespan

Makespan Makespan

Makespan Makespan

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii



14

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25

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30 40 50 60 70 80

30 40 50 60 70 30 40 50 60 70

30 40 50 60 70

80 30 40 50 60 70 80

30 40 50 60 70 80 30 40 50 60 70 80

30 40 50 60 70 80

20 40 60 80 100 20 40 60 80 100

Mea

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Mea

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Mea

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Mea

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Makespan

MOEADCMOEAD-C1

MOEADCMOEAD-C2

MOEADCMOEAD-C3

MOEADCMOEAD-C4

MOEADCMOEAD-C5

MOEADCMOEAD-C6

MOEADCMOEAD-C7

MOEADCMOEAD-C8

MOEADCMOEAD-C9

MOEADCMOEAD-C10

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii



20 40 60 80 100

20 40 60 80 100

20 40 60 80 100

20 40 60 80 100

20 40 60 80 100 30 40 50 60 70 80

30 40 50 60 70 30 40 50 60 7080

30 40 50 60 70 80 30 40 50 60 70 80

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10

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20

25

30

MOEADCMOEAD-I1

MOEADCMOEAD-I2

MOEADCMOEAD-I3

MOEADCMOEAD-I4

MOEADCMOEAD-I5

MOEADCMOEAD-I6

MOEADCMOEAD-I7

MOEADCMOEAD-I8

MOEADCMOEAD-I9

MOEADCMOEAD-I10

Mea

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Mea

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Makespan Makespan



Makespan

Makespan

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii



30 40 50 60 70 80 30 40 50 60 70 30 40 50 60 70801015

20

25

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30

20 40 60 80 100

20 40 60 80 100

30 40 50 60 70 80

40 50 60 70 80 30 40 50 60 70 80 30 40 50 60 70 80

30 40 50 60 70 80

MOEADCMOEAD-M1

MOEADCMOEAD-M2

MOEADCMOEAD-M3

MOEADCMOEAD-M4

MOEADCMOEAD-M5

MOEADCMOEAD-M6

MOEADCMOEAD-M7

MOEADCMOEAD-M8

MOEADCMOEAD-M9

MOEADCMOEAD-M10

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Makespan Makespan

Makespan

Makespan

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii



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40 50 60 70 80 40 50 60 70 8040 50 60 70 80 90

40 50 60 70 80 90 40 50 60 70 80 90 40 50 60 70 80 90

40 50 60 70 80 90 40 50 60 70 80 90

40 50 60 70 80 90

40 60 80 100

MOEADCMOEAD-C1

MOEADCMOEAD-C2

MOEADCMOEAD-C3

MOEADCMOEAD-C4

MOEADCMOEAD-C5

MOEADCMOEAD-C6

MOEADCMOEAD-C7

MOEADCMOEAD-C8

MOEADCMOEAD-C9

MOEADCMOEAD-C10

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Makespan Makespan

Makespan

Makespan

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii



40 50 60 70 80 40 50 60 70 80

40 50 60 70 80

90

40 50 60 70 80 90

40 50 60 70 80 90 40 50 60 70 80 90

40 50 60 70 80 90

40 60 80 100

40 60 80 100

40 45 50 55 60 65

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Makespan

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii

MOEADCMOEAD-I1

MOEADCMOEAD-I2

MOEADCMOEAD-I3

MOEADCMOEAD-I4

MOEADCMOEAD-I5

MOEADCMOEAD-I6

MOEADCMOEAD-I7

MOEADCMOEAD-I8

MOEADCMOEAD-I9

MOEADCMOEAD-I10



40 50 60 70 80 90

40 50 60 70 80 90 40 50 60 70 80 90 40 50 60 70 80 90

40 50 60 70 80 9040 50 60 70 80 90

40 50 60 70 80 90 40 50 60 70 80

40 50 60 70 80

40 50 60 70 80

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Makespan

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii

MOEADCMOEAD-M1

MOEADCMOEAD-M2

MOEADCMOEAD-M3

MOEADCMOEAD-M5

MOEADCMOEAD-M6

MOEADCMOEAD-M7

MOEADCMOEAD-M9

MOEADCMOEAD-M10

MOEADCMOEAD-M4

MOEADCMOEAD-M8



Algorithm

minus02

0

02

04

06

08

1

TTSP 20 lowast 8

C-m

easu

re

(RC

1)

(C1R

)

(RC

2)

(C2R

)

(RC

3)

(C3R

)

(RC

4)

(C4R

)

(RC

5)

(C5R

)

(RC

6)

(C6R

)

(RC

7)

(C7R

)

(RC

8)

(C8R

)

(RC

9)

(C9R

)

(RC

10)

(C10R

)




119862 (119860 119861) =


(13)





Algorithm

minus02

0

02

04

06

08

1

TTSP 20 lowast 8

(RI1)

(I1R

)

(RI2)

(I2R

)

(RI3)

(I3R

)

(RI4)

(I4R

)

(RI5)

(I5R

)

(RI6)

(I6R

)

(RI7)

(I7R

)

(RI8)

(I8R

)

(RI9)

(I9R

)

(RI10)

(I10R

)

+

+

+

C-m

easu

re


(RM

1)

(M1R

)

(RM

2)

(M2R

)

(RM

3)

(M3R

)

(RM

4)

(M4R

)

(RM

5)

(M5R

)

(RM

6)

(M6R

)

(RM

7)

(M7R

)

(RM

8)

(M8R

)

(RM

9)

(M9R

)

(RM

10)

(M10R

)

Algorithm

minus02

0

02

04

06

08

1

TTSP 20 lowast 8

+

C-m

easu

re







119877 119862 119877 119868 119877 119872 119877 119862 119877 119868 119877 119872



119877 119862 119877 119868 119877 119872 119877 119862 119877 119868 119877 119872




(119877 119862) (119862 119877) (119877 119868) (119868 119877) (119877119872) (119872 119877) (119877 119862) (119862 119877) (119877 119868) (119868 119877) (119877119872) (119872 119877)



(119877 119862) (119862 119877) (119877 119868) (119868 119877) (119877119872) (119872 119877) (119877 119862) (119862 119877) (119877 119868) (119868 119877) (119877119872) (119872 119877)





119862 119868 119872 119862 119868 119872 119862 119868 119872 119862 119868 119872


40

35

30

25

20

15

25 30 35 40 45

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Makespan

TTSP 6 lowast 8

Solutions











30 40 50 60 7012

13

14

15

16

17

18

19

Makespan30 40 50 60 70

Makespan Makespan

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20 40 60 80 100 12013

14

15

16

17

18

19

12

14

16

18

20

24

22

VNMCMOEAD-C1

VNMCMOEAD-I9

VNMCMOEAD-M3


14

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28

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40 50 60 70 80


15

20

25

30

35

40

VNMCMOEAD-C1

VNMCMOEAD-I10

VNMCMOEAD-M9








119881 119862 119881 119868 119881 119872 119881 119862 119881 119868 119881 119872

1 04067 04928 04067 04582 04067 04642 04016 04718 04016 04576 04016 041892 04018 04844 04018 04756 04018 04746 04148 04511 04148 04449 04148 045213 04552 04776 04552 04582 04552 04706 04729 04729 04729 04937 04729 042224 04258 04907 04258 04789 04258 04643 03881 04853 03881 04521 03881 042865 04283 05055 04283 04607 04283 04755 04010 04598 04010 04674 04010 046666 04492 04690 04492 04738 04492 04889 04155 04389 04155 04496 04155 043547 04560 04540 04560 04612 04560 04871 03900 04521 03900 04688 03900 045788 04306 04765 04306 04688 04306 04789 03968 04426 03968 04703 03968 047509 04460 04872 04460 04632 04460 04740 04235 04585 04235 04500 04235 0446810 04458 04897 04458 04803 04458 04767 04002 04406 04002 04465 04002 04537Average 04345 04827 04345 04679 04345 04755 04104 04574 04104 04601 04104 04457Times 1 9 0 10 0 10 0 9 0 10 1 9



(119881 119862) (119862 119881) (119881 119868) (119868 119881) (119881119872) (119872119881) (119881 119862) (119862 119881) (119881 119868) (119868 119881) (119881119872) (119872119881)

1 00000 10000 00000 08333 00000 10000 00000 10000 00000 10000 00000 090002 00000 10000 00000 10000 00000 10000 00000 10000 00000 10000 00000 100003 00000 09000 03750 05000 01667 07000 00000 10000 00000 10000 00000 100004 00000 10000 00000 10000 00000 10000 00000 10000 00000 10000 00000 100005 00000 10000 00000 09000 00000 10000 00000 10000 00000 10000 00000 100006 00000 01667 00833 01667 00000 10000 00000 08571 00000 10000 01000 085717 04444 00000 00000 02000 00000 10000 00000 10000 00000 10000 00000 100008 00000 10000 00000 10000 00000 10000 00000 10000 00000 10000 00000 100009 00000 10000 00000 10000 00000 10000 00000 10000 00000 07778 00000 0777810 00000 10000 01000 05714 00833 05714 00000 10000 00000 10000 00000 10000Average 00444 08067 00558 07171 00250 09271 00000 09857 00000 09778 00100 09535Times 1 9 0 10 0 10 0 10 0 10 0 10


7 Conclusion





Acknowledgments



References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












Bakerrsquos map 119861 (119909 119910) =


119910

2) for 05 le 119909 lt 1

2 119909 isin (0 1)


2 119909119896 isin (0 1)

Circle map 119909119896+1 = 119909119896 + 119887 minus (119886

2120587) sin (2120587119909119896) mod (1) 1 (0 1)

119886 = 05 119887 = 02


120588 = 259

Gauss map 119909119896+1 =

0 119909119896 = 0

1

119909119896


ICMIC map 119909119896+1 = sin(119886

119909119896


119886 = 2



Tent map 119909119896+1 =

119909119896

07119909119896 lt 07

(10

3) 119909119896 (1 minus 119909119896) otherwise

1 (0 1)


minus119903119910119896

2 119910119896 isin [minus10512 10512]

V = 400 119903 = 3 119886 = 126695








119896119895


119896 = [119909119894119895 times 10] mod 119896119895 + 1 (6)




0 05 1

0 05 1

0 05 1 0 05 1

0 05 1 0 05 1

0 05 1

0 05 1

0 05 1 0 05 10

500

1000

1500

0

500

1000

1500

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1000

1500

0

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2000

2000

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Bakers Cat Circle

Cubic Gauss ICMIC


Zaslavskii




119896 = [119909119894119895119896119895 times 10] mod 119896119895 + 1 (7)




119879 119882119895 119908119896119895 119875

119896119895

1199051

11990811 1199031 1199032 511990821 1199032 1199034 3

1199052

11990812 1199031 411990822 1199033 1

1199053 11990813 1199034 2

1199054

11990814 1199031 1199033 411990824 1199032 1199034 311990834 1199032 1199033 7





119875119896119895 2 4 3 4



[00 01)

[03 04)

[06 07)

[09 10)

[01 02)

[04 05)

[07 08)

[02 03)

[05 06)

[08 09)

119896 1 2 3





119909119904= 1199091119904 1199092119904 119909

119894119904 119909

119899119904 119904 =1 2 119873 119894 =1 2 119873

(8)


119894119904)



119909119905+1119894 =

119909119905119894 + 119865 times (119909


119909119905119894 otherwise

(9)






119909lowast119904 = 119909119904 + (119909

119906119904 minus 119909119897119904) times 120575119904 (10)


Consider

120575119904 = (2119906119904)1(120578119898+1)


1(120578119898+1) otherwise(11)


6 Experiments








[12 13)[15 16)[18 19)

[21 22)[24 25)[27 28)

[1 2)[4 5)[7 8)[10 11)

[13 14)[16 17)[19 20)

[22 23)[25 26)[28 29)

[2 3)[5 6)[8 9)[11 12)

[14 15)[17 18)[20 21)

[23 24)[26 27)[29 30)

119896 1 2 3



119899iter 250119899pop 100119899var 6 20 30 40CR 09119875119898 16 120 130 140

35 40 45 50 55 60 65 70

Makespan

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34

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MOEAD-1MOEAD-2

MOEAD-3













HV (119878) = Leb(⋃119894isin119878

V119894) (12)


20 25 30 35 40 45

20 25 30 35 40 45 20 25 30 35 40 45 20 25 30 35 40 45

20 25 30 35 40 45 20 25 30 35 40 45 20 25 30 35 40 45

20 25 30 35 40

20 25 30 35 40

20 25 30 35 40 4510

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Baker

Makespan




Cat Circle

Cubic Gauss ICMIC


Zaslavskii

MOEADCMOEAD-C1

MOEADCMOEAD-C2

MOEADCMOEAD-C3

MOEADCMOEAD-C4

MOEADCMOEAD-C5

MOEADCMOEAD-C6

MOEADCMOEAD-C7

MOEADCMOEAD-C8

MOEADCMOEAD-C9

MOEADCMOEAD-C10



20 25 30 35 4010

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20

25

10

15

20

25

10

15

20

25Baker

Makespan

Makespan

20 25 30 35 40

Makespan20 25 30 35 40

Makespan

20 25 30 35 40

Makespan

20 25 30 35 40

Makespan

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MOEADCMOEAD-I1

20 25 30 35 40 45

Makespan20 25 30 35 40 45

Makespan20 25 30 35 40 45

Makespan20 25 30 35 40 45

Makespan20 25 30 35 40 45

Cat

MOEADCMOEAD-I2

Circle

MOEADCMOEAD-I3

Cubic

MOEADCMOEAD-I4

Gauss

MOEADCMOEAD-I5

ICMIC

MOEAD

Logistic

MOEAD

Sinusoidal

MOEAD

Tent

MOEADCMOEAD-I9

Zaslavskii

MOEADCMOEAD-I10

CMOEAD-I6

CMOEAD-I7 CMOEAD-I8



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25

20 25 30 35 40 4520 25 30 35 40


Makespan

20 25 30 35 40 45

Makespan20 25 30 35 40 45

Makespan20 25 30 35 40 45

Makespan

20 25 30 35 40 45

Makespan20 25 30 35 40 45

Makespan

20 25 30 35 40 45

Makespan

20 25 30 35 40 45

Makespan

MOEADCMOEAD-M1

MOEADCMOEAD-M2

MOEADCMOEAD-M3

MOEADCMOEAD-M4

MOEADCMOEAD-M5

MOEADCMOEAD-M6

MOEADCMOEAD-M7

MOEADCMOEAD-M8

MOEADCMOEAD-M9

MOEADCMOEAD-M10

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii



30 40 50 60 70

30 40 50 60 70

12

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30 40 50 60 70 80

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10

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Mea

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Mea

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Makespan



Makespan

Makespan Makespan

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii

MOEADCMOEAD-C1

MOEADCMOEAD-C2

MOEADCMOEAD-C3

MOEADCMOEAD-C5

MOEADCMOEAD-C4

MOEADCMOEAD-C6

MOEADCMOEAD-C7

MOEADCMOEAD-C8

MOEADCMOEAD-C9

MOEADCMOEAD-C10



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25

30 35 40 45 50 30 40 50 60 70 20 40 60 80 100 120

20 40 60 80 100 30 35 40 45 50 30 40 50 60 70

30 40 50 60 70

30 35 40 45 50 30 40 50 60 70 30 35 40 45 50 55

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Mea

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Mea

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Mea

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Makespan Makespan

Makespan

Makespan

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii

MOEADCMOEAD-I1

MOEADCMOEAD-I2

MOEADCMOEAD-I3

MOEADCMOEAD-I5

MOEADCMOEAD-I4

MOEADCMOEAD-I6

MOEADCMOEAD-I7

MOEADCMOEAD-I9

MOEADCMOEAD-I8

MOEADCMOEAD-I10



30 40 50 60 70 30 40 50 60 70 30 35 40 45 50 55

20 40 60 80 100 30 40 50 60 70 80 30 40 50 60 70

30 35 40 45 50 55 20 30 40 50 60 70 30 40 50 60 70

30 35 40 45 50

MOEADCMOEAD-M1

MOEADCMOEAD-M2

MOEADCMOEAD-M3

MOEADCMOEAD-M4

MOEADCMOEAD-M5

MOEADCMOEAD-M6

MOEADCMOEAD-M7

MOEADCMOEAD-M8

MOEADCMOEAD-M9

MOEADCMOEAD-M10

10

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Mea

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Makespan

Makespan

Makespan Makespan

Makespan

Makespan

Makespan Makespan

Makespan Makespan

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii



14

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30 40 50 60 70 80

30 40 50 60 70 30 40 50 60 70

30 40 50 60 70

80 30 40 50 60 70 80

30 40 50 60 70 80 30 40 50 60 70 80

30 40 50 60 70 80

20 40 60 80 100 20 40 60 80 100

Mea

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Mea

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Mea

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Makespan

MOEADCMOEAD-C1

MOEADCMOEAD-C2

MOEADCMOEAD-C3

MOEADCMOEAD-C4

MOEADCMOEAD-C5

MOEADCMOEAD-C6

MOEADCMOEAD-C7

MOEADCMOEAD-C8

MOEADCMOEAD-C9

MOEADCMOEAD-C10

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii



20 40 60 80 100

20 40 60 80 100

20 40 60 80 100

20 40 60 80 100

20 40 60 80 100 30 40 50 60 70 80

30 40 50 60 70 30 40 50 60 7080

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MOEADCMOEAD-I1

MOEADCMOEAD-I2

MOEADCMOEAD-I3

MOEADCMOEAD-I4

MOEADCMOEAD-I5

MOEADCMOEAD-I6

MOEADCMOEAD-I7

MOEADCMOEAD-I8

MOEADCMOEAD-I9

MOEADCMOEAD-I10

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Makespan Makespan



Makespan

Makespan

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii



30 40 50 60 70 80 30 40 50 60 70 30 40 50 60 70801015

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20 40 60 80 100

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40 50 60 70 80 30 40 50 60 70 80 30 40 50 60 70 80

30 40 50 60 70 80

MOEADCMOEAD-M1

MOEADCMOEAD-M2

MOEADCMOEAD-M3

MOEADCMOEAD-M4

MOEADCMOEAD-M5

MOEADCMOEAD-M6

MOEADCMOEAD-M7

MOEADCMOEAD-M8

MOEADCMOEAD-M9

MOEADCMOEAD-M10

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Makespan Makespan

Makespan

Makespan

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii



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40 50 60 70 80 40 50 60 70 8040 50 60 70 80 90

40 50 60 70 80 90 40 50 60 70 80 90 40 50 60 70 80 90

40 50 60 70 80 90 40 50 60 70 80 90

40 50 60 70 80 90

40 60 80 100

MOEADCMOEAD-C1

MOEADCMOEAD-C2

MOEADCMOEAD-C3

MOEADCMOEAD-C4

MOEADCMOEAD-C5

MOEADCMOEAD-C6

MOEADCMOEAD-C7

MOEADCMOEAD-C8

MOEADCMOEAD-C9

MOEADCMOEAD-C10

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Makespan Makespan

Makespan

Makespan

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii



40 50 60 70 80 40 50 60 70 80

40 50 60 70 80

90

40 50 60 70 80 90

40 50 60 70 80 90 40 50 60 70 80 90

40 50 60 70 80 90

40 60 80 100

40 60 80 100

40 45 50 55 60 65

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Makespan

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii

MOEADCMOEAD-I1

MOEADCMOEAD-I2

MOEADCMOEAD-I3

MOEADCMOEAD-I4

MOEADCMOEAD-I5

MOEADCMOEAD-I6

MOEADCMOEAD-I7

MOEADCMOEAD-I8

MOEADCMOEAD-I9

MOEADCMOEAD-I10



40 50 60 70 80 90

40 50 60 70 80 90 40 50 60 70 80 90 40 50 60 70 80 90

40 50 60 70 80 9040 50 60 70 80 90

40 50 60 70 80 90 40 50 60 70 80

40 50 60 70 80

40 50 60 70 80

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Makespan

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii

MOEADCMOEAD-M1

MOEADCMOEAD-M2

MOEADCMOEAD-M3

MOEADCMOEAD-M5

MOEADCMOEAD-M6

MOEADCMOEAD-M7

MOEADCMOEAD-M9

MOEADCMOEAD-M10

MOEADCMOEAD-M4

MOEADCMOEAD-M8



Algorithm

minus02

0

02

04

06

08

1

TTSP 20 lowast 8

C-m

easu

re

(RC

1)

(C1R

)

(RC

2)

(C2R

)

(RC

3)

(C3R

)

(RC

4)

(C4R

)

(RC

5)

(C5R

)

(RC

6)

(C6R

)

(RC

7)

(C7R

)

(RC

8)

(C8R

)

(RC

9)

(C9R

)

(RC

10)

(C10R

)




119862 (119860 119861) =


(13)





Algorithm

minus02

0

02

04

06

08

1

TTSP 20 lowast 8

(RI1)

(I1R

)

(RI2)

(I2R

)

(RI3)

(I3R

)

(RI4)

(I4R

)

(RI5)

(I5R

)

(RI6)

(I6R

)

(RI7)

(I7R

)

(RI8)

(I8R

)

(RI9)

(I9R

)

(RI10)

(I10R

)

+

+

+

C-m

easu

re


(RM

1)

(M1R

)

(RM

2)

(M2R

)

(RM

3)

(M3R

)

(RM

4)

(M4R

)

(RM

5)

(M5R

)

(RM

6)

(M6R

)

(RM

7)

(M7R

)

(RM

8)

(M8R

)

(RM

9)

(M9R

)

(RM

10)

(M10R

)

Algorithm

minus02

0

02

04

06

08

1

TTSP 20 lowast 8

+

C-m

easu

re







119877 119862 119877 119868 119877 119872 119877 119862 119877 119868 119877 119872



119877 119862 119877 119868 119877 119872 119877 119862 119877 119868 119877 119872




(119877 119862) (119862 119877) (119877 119868) (119868 119877) (119877119872) (119872 119877) (119877 119862) (119862 119877) (119877 119868) (119868 119877) (119877119872) (119872 119877)



(119877 119862) (119862 119877) (119877 119868) (119868 119877) (119877119872) (119872 119877) (119877 119862) (119862 119877) (119877 119868) (119868 119877) (119877119872) (119872 119877)





119862 119868 119872 119862 119868 119872 119862 119868 119872 119862 119868 119872


40

35

30

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20

15

25 30 35 40 45

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Makespan

TTSP 6 lowast 8

Solutions











30 40 50 60 7012

13

14

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19

Makespan30 40 50 60 70

Makespan Makespan

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20 40 60 80 100 12013

14

15

16

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18

19

12

14

16

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24

22

VNMCMOEAD-C1

VNMCMOEAD-I9

VNMCMOEAD-M3


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40 50 60 70 80


15

20

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30

35

40

VNMCMOEAD-C1

VNMCMOEAD-I10

VNMCMOEAD-M9








119881 119862 119881 119868 119881 119872 119881 119862 119881 119868 119881 119872

1 04067 04928 04067 04582 04067 04642 04016 04718 04016 04576 04016 041892 04018 04844 04018 04756 04018 04746 04148 04511 04148 04449 04148 045213 04552 04776 04552 04582 04552 04706 04729 04729 04729 04937 04729 042224 04258 04907 04258 04789 04258 04643 03881 04853 03881 04521 03881 042865 04283 05055 04283 04607 04283 04755 04010 04598 04010 04674 04010 046666 04492 04690 04492 04738 04492 04889 04155 04389 04155 04496 04155 043547 04560 04540 04560 04612 04560 04871 03900 04521 03900 04688 03900 045788 04306 04765 04306 04688 04306 04789 03968 04426 03968 04703 03968 047509 04460 04872 04460 04632 04460 04740 04235 04585 04235 04500 04235 0446810 04458 04897 04458 04803 04458 04767 04002 04406 04002 04465 04002 04537Average 04345 04827 04345 04679 04345 04755 04104 04574 04104 04601 04104 04457Times 1 9 0 10 0 10 0 9 0 10 1 9



(119881 119862) (119862 119881) (119881 119868) (119868 119881) (119881119872) (119872119881) (119881 119862) (119862 119881) (119881 119868) (119868 119881) (119881119872) (119872119881)

1 00000 10000 00000 08333 00000 10000 00000 10000 00000 10000 00000 090002 00000 10000 00000 10000 00000 10000 00000 10000 00000 10000 00000 100003 00000 09000 03750 05000 01667 07000 00000 10000 00000 10000 00000 100004 00000 10000 00000 10000 00000 10000 00000 10000 00000 10000 00000 100005 00000 10000 00000 09000 00000 10000 00000 10000 00000 10000 00000 100006 00000 01667 00833 01667 00000 10000 00000 08571 00000 10000 01000 085717 04444 00000 00000 02000 00000 10000 00000 10000 00000 10000 00000 100008 00000 10000 00000 10000 00000 10000 00000 10000 00000 10000 00000 100009 00000 10000 00000 10000 00000 10000 00000 10000 00000 07778 00000 0777810 00000 10000 01000 05714 00833 05714 00000 10000 00000 10000 00000 10000Average 00444 08067 00558 07171 00250 09271 00000 09857 00000 09778 00100 09535Times 1 9 0 10 0 10 0 10 0 10 0 10


7 Conclusion





Acknowledgments



References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 05 1

0 05 1

0 05 1 0 05 1

0 05 1 0 05 1

0 05 1

0 05 1

0 05 1 0 05 10

500

1000

1500

0

500

1000

1500

0

500

1000

1500

0

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Bakers Cat Circle

Cubic Gauss ICMIC


Zaslavskii




119896 = [119909119894119895119896119895 times 10] mod 119896119895 + 1 (7)




119879 119882119895 119908119896119895 119875

119896119895

1199051

11990811 1199031 1199032 511990821 1199032 1199034 3

1199052

11990812 1199031 411990822 1199033 1

1199053 11990813 1199034 2

1199054

11990814 1199031 1199033 411990824 1199032 1199034 311990834 1199032 1199033 7





119875119896119895 2 4 3 4



[00 01)

[03 04)

[06 07)

[09 10)

[01 02)

[04 05)

[07 08)

[02 03)

[05 06)

[08 09)

119896 1 2 3





119909119904= 1199091119904 1199092119904 119909

119894119904 119909

119899119904 119904 =1 2 119873 119894 =1 2 119873

(8)


119894119904)



119909119905+1119894 =

119909119905119894 + 119865 times (119909


119909119905119894 otherwise

(9)






119909lowast119904 = 119909119904 + (119909

119906119904 minus 119909119897119904) times 120575119904 (10)


Consider

120575119904 = (2119906119904)1(120578119898+1)


1(120578119898+1) otherwise(11)


6 Experiments








[12 13)[15 16)[18 19)

[21 22)[24 25)[27 28)

[1 2)[4 5)[7 8)[10 11)

[13 14)[16 17)[19 20)

[22 23)[25 26)[28 29)

[2 3)[5 6)[8 9)[11 12)

[14 15)[17 18)[20 21)

[23 24)[26 27)[29 30)

119896 1 2 3



119899iter 250119899pop 100119899var 6 20 30 40CR 09119875119898 16 120 130 140

35 40 45 50 55 60 65 70

Makespan

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MOEAD-1MOEAD-2

MOEAD-3













HV (119878) = Leb(⋃119894isin119878

V119894) (12)


20 25 30 35 40 45

20 25 30 35 40 45 20 25 30 35 40 45 20 25 30 35 40 45

20 25 30 35 40 45 20 25 30 35 40 45 20 25 30 35 40 45

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20 25 30 35 40

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Baker

Makespan




Cat Circle

Cubic Gauss ICMIC


Zaslavskii

MOEADCMOEAD-C1

MOEADCMOEAD-C2

MOEADCMOEAD-C3

MOEADCMOEAD-C4

MOEADCMOEAD-C5

MOEADCMOEAD-C6

MOEADCMOEAD-C7

MOEADCMOEAD-C8

MOEADCMOEAD-C9

MOEADCMOEAD-C10



20 25 30 35 4010

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Makespan

Makespan

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Makespan20 25 30 35 40

Makespan

20 25 30 35 40

Makespan

20 25 30 35 40

Makespan

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MOEADCMOEAD-I1

20 25 30 35 40 45

Makespan20 25 30 35 40 45

Makespan20 25 30 35 40 45

Makespan20 25 30 35 40 45

Makespan20 25 30 35 40 45

Cat

MOEADCMOEAD-I2

Circle

MOEADCMOEAD-I3

Cubic

MOEADCMOEAD-I4

Gauss

MOEADCMOEAD-I5

ICMIC

MOEAD

Logistic

MOEAD

Sinusoidal

MOEAD

Tent

MOEADCMOEAD-I9

Zaslavskii

MOEADCMOEAD-I10

CMOEAD-I6

CMOEAD-I7 CMOEAD-I8



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Makespan

20 25 30 35 40 45

Makespan20 25 30 35 40 45

Makespan20 25 30 35 40 45

Makespan

20 25 30 35 40 45

Makespan20 25 30 35 40 45

Makespan

20 25 30 35 40 45

Makespan

20 25 30 35 40 45

Makespan

MOEADCMOEAD-M1

MOEADCMOEAD-M2

MOEADCMOEAD-M3

MOEADCMOEAD-M4

MOEADCMOEAD-M5

MOEADCMOEAD-M6

MOEADCMOEAD-M7

MOEADCMOEAD-M8

MOEADCMOEAD-M9

MOEADCMOEAD-M10

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii



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30 40 50 60 70

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30 40 50 60 70 80

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Makespan



Makespan

Makespan Makespan

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii

MOEADCMOEAD-C1

MOEADCMOEAD-C2

MOEADCMOEAD-C3

MOEADCMOEAD-C5

MOEADCMOEAD-C4

MOEADCMOEAD-C6

MOEADCMOEAD-C7

MOEADCMOEAD-C8

MOEADCMOEAD-C9

MOEADCMOEAD-C10



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20 40 60 80 100 30 35 40 45 50 30 40 50 60 70

30 40 50 60 70

30 35 40 45 50 30 40 50 60 70 30 35 40 45 50 55

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Makespan Makespan

Makespan

Makespan

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii

MOEADCMOEAD-I1

MOEADCMOEAD-I2

MOEADCMOEAD-I3

MOEADCMOEAD-I5

MOEADCMOEAD-I4

MOEADCMOEAD-I6

MOEADCMOEAD-I7

MOEADCMOEAD-I9

MOEADCMOEAD-I8

MOEADCMOEAD-I10



30 40 50 60 70 30 40 50 60 70 30 35 40 45 50 55

20 40 60 80 100 30 40 50 60 70 80 30 40 50 60 70

30 35 40 45 50 55 20 30 40 50 60 70 30 40 50 60 70

30 35 40 45 50

MOEADCMOEAD-M1

MOEADCMOEAD-M2

MOEADCMOEAD-M3

MOEADCMOEAD-M4

MOEADCMOEAD-M5

MOEADCMOEAD-M6

MOEADCMOEAD-M7

MOEADCMOEAD-M8

MOEADCMOEAD-M9

MOEADCMOEAD-M10

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Makespan

Makespan

Makespan Makespan

Makespan

Makespan

Makespan Makespan

Makespan Makespan

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii



14

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Makespan

MOEADCMOEAD-C1

MOEADCMOEAD-C2

MOEADCMOEAD-C3

MOEADCMOEAD-C4

MOEADCMOEAD-C5

MOEADCMOEAD-C6

MOEADCMOEAD-C7

MOEADCMOEAD-C8

MOEADCMOEAD-C9

MOEADCMOEAD-C10

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii



20 40 60 80 100

20 40 60 80 100

20 40 60 80 100

20 40 60 80 100

20 40 60 80 100 30 40 50 60 70 80

30 40 50 60 70 30 40 50 60 7080

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MOEADCMOEAD-I1

MOEADCMOEAD-I2

MOEADCMOEAD-I3

MOEADCMOEAD-I4

MOEADCMOEAD-I5

MOEADCMOEAD-I6

MOEADCMOEAD-I7

MOEADCMOEAD-I8

MOEADCMOEAD-I9

MOEADCMOEAD-I10

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Mea

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Mea

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Mea

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Makespan Makespan



Makespan

Makespan

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii



30 40 50 60 70 80 30 40 50 60 70 30 40 50 60 70801015

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20 40 60 80 100

20 40 60 80 100

30 40 50 60 70 80

40 50 60 70 80 30 40 50 60 70 80 30 40 50 60 70 80

30 40 50 60 70 80

MOEADCMOEAD-M1

MOEADCMOEAD-M2

MOEADCMOEAD-M3

MOEADCMOEAD-M4

MOEADCMOEAD-M5

MOEADCMOEAD-M6

MOEADCMOEAD-M7

MOEADCMOEAD-M8

MOEADCMOEAD-M9

MOEADCMOEAD-M10

Mea

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Mea

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Makespan Makespan

Makespan

Makespan

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii



14

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40 50 60 70 80 40 50 60 70 8040 50 60 70 80 90

40 50 60 70 80 90 40 50 60 70 80 90 40 50 60 70 80 90

40 50 60 70 80 90 40 50 60 70 80 90

40 50 60 70 80 90

40 60 80 100

MOEADCMOEAD-C1

MOEADCMOEAD-C2

MOEADCMOEAD-C3

MOEADCMOEAD-C4

MOEADCMOEAD-C5

MOEADCMOEAD-C6

MOEADCMOEAD-C7

MOEADCMOEAD-C8

MOEADCMOEAD-C9

MOEADCMOEAD-C10

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Mea

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Makespan Makespan

Makespan

Makespan

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii



40 50 60 70 80 40 50 60 70 80

40 50 60 70 80

90

40 50 60 70 80 90

40 50 60 70 80 90 40 50 60 70 80 90

40 50 60 70 80 90

40 60 80 100

40 60 80 100

40 45 50 55 60 65

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Makespan

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii

MOEADCMOEAD-I1

MOEADCMOEAD-I2

MOEADCMOEAD-I3

MOEADCMOEAD-I4

MOEADCMOEAD-I5

MOEADCMOEAD-I6

MOEADCMOEAD-I7

MOEADCMOEAD-I8

MOEADCMOEAD-I9

MOEADCMOEAD-I10



40 50 60 70 80 90

40 50 60 70 80 90 40 50 60 70 80 90 40 50 60 70 80 90

40 50 60 70 80 9040 50 60 70 80 90

40 50 60 70 80 90 40 50 60 70 80

40 50 60 70 80

40 50 60 70 80

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Makespan

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii

MOEADCMOEAD-M1

MOEADCMOEAD-M2

MOEADCMOEAD-M3

MOEADCMOEAD-M5

MOEADCMOEAD-M6

MOEADCMOEAD-M7

MOEADCMOEAD-M9

MOEADCMOEAD-M10

MOEADCMOEAD-M4

MOEADCMOEAD-M8



Algorithm

minus02

0

02

04

06

08

1

TTSP 20 lowast 8

C-m

easu

re

(RC

1)

(C1R

)

(RC

2)

(C2R

)

(RC

3)

(C3R

)

(RC

4)

(C4R

)

(RC

5)

(C5R

)

(RC

6)

(C6R

)

(RC

7)

(C7R

)

(RC

8)

(C8R

)

(RC

9)

(C9R

)

(RC

10)

(C10R

)




119862 (119860 119861) =


(13)





Algorithm

minus02

0

02

04

06

08

1

TTSP 20 lowast 8

(RI1)

(I1R

)

(RI2)

(I2R

)

(RI3)

(I3R

)

(RI4)

(I4R

)

(RI5)

(I5R

)

(RI6)

(I6R

)

(RI7)

(I7R

)

(RI8)

(I8R

)

(RI9)

(I9R

)

(RI10)

(I10R

)

+

+

+

C-m

easu

re


(RM

1)

(M1R

)

(RM

2)

(M2R

)

(RM

3)

(M3R

)

(RM

4)

(M4R

)

(RM

5)

(M5R

)

(RM

6)

(M6R

)

(RM

7)

(M7R

)

(RM

8)

(M8R

)

(RM

9)

(M9R

)

(RM

10)

(M10R

)

Algorithm

minus02

0

02

04

06

08

1

TTSP 20 lowast 8

+

C-m

easu

re







119877 119862 119877 119868 119877 119872 119877 119862 119877 119868 119877 119872



119877 119862 119877 119868 119877 119872 119877 119862 119877 119868 119877 119872




(119877 119862) (119862 119877) (119877 119868) (119868 119877) (119877119872) (119872 119877) (119877 119862) (119862 119877) (119877 119868) (119868 119877) (119877119872) (119872 119877)



(119877 119862) (119862 119877) (119877 119868) (119868 119877) (119877119872) (119872 119877) (119877 119862) (119862 119877) (119877 119868) (119868 119877) (119877119872) (119872 119877)





119862 119868 119872 119862 119868 119872 119862 119868 119872 119862 119868 119872


40

35

30

25

20

15

25 30 35 40 45

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Makespan

TTSP 6 lowast 8

Solutions











30 40 50 60 7012

13

14

15

16

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18

19

Makespan30 40 50 60 70

Makespan Makespan

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20 40 60 80 100 12013

14

15

16

17

18

19

12

14

16

18

20

24

22

VNMCMOEAD-C1

VNMCMOEAD-I9

VNMCMOEAD-M3


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28

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40 50 60 70 80


15

20

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30

35

40

VNMCMOEAD-C1

VNMCMOEAD-I10

VNMCMOEAD-M9








119881 119862 119881 119868 119881 119872 119881 119862 119881 119868 119881 119872

1 04067 04928 04067 04582 04067 04642 04016 04718 04016 04576 04016 041892 04018 04844 04018 04756 04018 04746 04148 04511 04148 04449 04148 045213 04552 04776 04552 04582 04552 04706 04729 04729 04729 04937 04729 042224 04258 04907 04258 04789 04258 04643 03881 04853 03881 04521 03881 042865 04283 05055 04283 04607 04283 04755 04010 04598 04010 04674 04010 046666 04492 04690 04492 04738 04492 04889 04155 04389 04155 04496 04155 043547 04560 04540 04560 04612 04560 04871 03900 04521 03900 04688 03900 045788 04306 04765 04306 04688 04306 04789 03968 04426 03968 04703 03968 047509 04460 04872 04460 04632 04460 04740 04235 04585 04235 04500 04235 0446810 04458 04897 04458 04803 04458 04767 04002 04406 04002 04465 04002 04537Average 04345 04827 04345 04679 04345 04755 04104 04574 04104 04601 04104 04457Times 1 9 0 10 0 10 0 9 0 10 1 9



(119881 119862) (119862 119881) (119881 119868) (119868 119881) (119881119872) (119872119881) (119881 119862) (119862 119881) (119881 119868) (119868 119881) (119881119872) (119872119881)

1 00000 10000 00000 08333 00000 10000 00000 10000 00000 10000 00000 090002 00000 10000 00000 10000 00000 10000 00000 10000 00000 10000 00000 100003 00000 09000 03750 05000 01667 07000 00000 10000 00000 10000 00000 100004 00000 10000 00000 10000 00000 10000 00000 10000 00000 10000 00000 100005 00000 10000 00000 09000 00000 10000 00000 10000 00000 10000 00000 100006 00000 01667 00833 01667 00000 10000 00000 08571 00000 10000 01000 085717 04444 00000 00000 02000 00000 10000 00000 10000 00000 10000 00000 100008 00000 10000 00000 10000 00000 10000 00000 10000 00000 10000 00000 100009 00000 10000 00000 10000 00000 10000 00000 10000 00000 07778 00000 0777810 00000 10000 01000 05714 00833 05714 00000 10000 00000 10000 00000 10000Average 00444 08067 00558 07171 00250 09271 00000 09857 00000 09778 00100 09535Times 1 9 0 10 0 10 0 10 0 10 0 10


7 Conclusion





Acknowledgments



References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119875119896119895 2 4 3 4



[00 01)

[03 04)

[06 07)

[09 10)

[01 02)

[04 05)

[07 08)

[02 03)

[05 06)

[08 09)

119896 1 2 3





119909119904= 1199091119904 1199092119904 119909

119894119904 119909

119899119904 119904 =1 2 119873 119894 =1 2 119873

(8)


119894119904)



119909119905+1119894 =

119909119905119894 + 119865 times (119909


119909119905119894 otherwise

(9)






119909lowast119904 = 119909119904 + (119909

119906119904 minus 119909119897119904) times 120575119904 (10)


Consider

120575119904 = (2119906119904)1(120578119898+1)


1(120578119898+1) otherwise(11)


6 Experiments








[12 13)[15 16)[18 19)

[21 22)[24 25)[27 28)

[1 2)[4 5)[7 8)[10 11)

[13 14)[16 17)[19 20)

[22 23)[25 26)[28 29)

[2 3)[5 6)[8 9)[11 12)

[14 15)[17 18)[20 21)

[23 24)[26 27)[29 30)

119896 1 2 3



119899iter 250119899pop 100119899var 6 20 30 40CR 09119875119898 16 120 130 140

35 40 45 50 55 60 65 70

Makespan

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MOEAD-1MOEAD-2

MOEAD-3













HV (119878) = Leb(⋃119894isin119878

V119894) (12)


20 25 30 35 40 45

20 25 30 35 40 45 20 25 30 35 40 45 20 25 30 35 40 45

20 25 30 35 40 45 20 25 30 35 40 45 20 25 30 35 40 45

20 25 30 35 40

20 25 30 35 40

20 25 30 35 40 4510

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Baker

Makespan




Cat Circle

Cubic Gauss ICMIC


Zaslavskii

MOEADCMOEAD-C1

MOEADCMOEAD-C2

MOEADCMOEAD-C3

MOEADCMOEAD-C4

MOEADCMOEAD-C5

MOEADCMOEAD-C6

MOEADCMOEAD-C7

MOEADCMOEAD-C8

MOEADCMOEAD-C9

MOEADCMOEAD-C10



20 25 30 35 4010

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10

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10

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25Baker

Makespan

Makespan

20 25 30 35 40

Makespan20 25 30 35 40

Makespan

20 25 30 35 40

Makespan

20 25 30 35 40

Makespan

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MOEADCMOEAD-I1

20 25 30 35 40 45

Makespan20 25 30 35 40 45

Makespan20 25 30 35 40 45

Makespan20 25 30 35 40 45

Makespan20 25 30 35 40 45

Cat

MOEADCMOEAD-I2

Circle

MOEADCMOEAD-I3

Cubic

MOEADCMOEAD-I4

Gauss

MOEADCMOEAD-I5

ICMIC

MOEAD

Logistic

MOEAD

Sinusoidal

MOEAD

Tent

MOEADCMOEAD-I9

Zaslavskii

MOEADCMOEAD-I10

CMOEAD-I6

CMOEAD-I7 CMOEAD-I8



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20 25 30 35 40 4520 25 30 35 40


Makespan

20 25 30 35 40 45

Makespan20 25 30 35 40 45

Makespan20 25 30 35 40 45

Makespan

20 25 30 35 40 45

Makespan20 25 30 35 40 45

Makespan

20 25 30 35 40 45

Makespan

20 25 30 35 40 45

Makespan

MOEADCMOEAD-M1

MOEADCMOEAD-M2

MOEADCMOEAD-M3

MOEADCMOEAD-M4

MOEADCMOEAD-M5

MOEADCMOEAD-M6

MOEADCMOEAD-M7

MOEADCMOEAD-M8

MOEADCMOEAD-M9

MOEADCMOEAD-M10

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii



30 40 50 60 70

30 40 50 60 70

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30 40 50 60 70 80

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30 35 40 45 50

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20 40 60 80 10030 40 50 60 70

30 40 50 60 70

30 40 50 6010

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Mea

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Makespan



Makespan

Makespan Makespan

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii

MOEADCMOEAD-C1

MOEADCMOEAD-C2

MOEADCMOEAD-C3

MOEADCMOEAD-C5

MOEADCMOEAD-C4

MOEADCMOEAD-C6

MOEADCMOEAD-C7

MOEADCMOEAD-C8

MOEADCMOEAD-C9

MOEADCMOEAD-C10



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20 40 60 80 100 30 35 40 45 50 30 40 50 60 70

30 40 50 60 70

30 35 40 45 50 30 40 50 60 70 30 35 40 45 50 55

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Mea

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Makespan Makespan

Makespan

Makespan

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii

MOEADCMOEAD-I1

MOEADCMOEAD-I2

MOEADCMOEAD-I3

MOEADCMOEAD-I5

MOEADCMOEAD-I4

MOEADCMOEAD-I6

MOEADCMOEAD-I7

MOEADCMOEAD-I9

MOEADCMOEAD-I8

MOEADCMOEAD-I10



30 40 50 60 70 30 40 50 60 70 30 35 40 45 50 55

20 40 60 80 100 30 40 50 60 70 80 30 40 50 60 70

30 35 40 45 50 55 20 30 40 50 60 70 30 40 50 60 70

30 35 40 45 50

MOEADCMOEAD-M1

MOEADCMOEAD-M2

MOEADCMOEAD-M3

MOEADCMOEAD-M4

MOEADCMOEAD-M5

MOEADCMOEAD-M6

MOEADCMOEAD-M7

MOEADCMOEAD-M8

MOEADCMOEAD-M9

MOEADCMOEAD-M10

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Mea

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Makespan

Makespan

Makespan Makespan

Makespan

Makespan

Makespan Makespan

Makespan Makespan

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii



14

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30 40 50 60 70 30 40 50 60 70

30 40 50 60 70

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30 40 50 60 70 80

20 40 60 80 100 20 40 60 80 100

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Makespan

MOEADCMOEAD-C1

MOEADCMOEAD-C2

MOEADCMOEAD-C3

MOEADCMOEAD-C4

MOEADCMOEAD-C5

MOEADCMOEAD-C6

MOEADCMOEAD-C7

MOEADCMOEAD-C8

MOEADCMOEAD-C9

MOEADCMOEAD-C10

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii



20 40 60 80 100

20 40 60 80 100

20 40 60 80 100

20 40 60 80 100

20 40 60 80 100 30 40 50 60 70 80

30 40 50 60 70 30 40 50 60 7080

30 40 50 60 70 80 30 40 50 60 70 80

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MOEADCMOEAD-I1

MOEADCMOEAD-I2

MOEADCMOEAD-I3

MOEADCMOEAD-I4

MOEADCMOEAD-I5

MOEADCMOEAD-I6

MOEADCMOEAD-I7

MOEADCMOEAD-I8

MOEADCMOEAD-I9

MOEADCMOEAD-I10

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Mea

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Mea

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Makespan Makespan



Makespan

Makespan

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii



30 40 50 60 70 80 30 40 50 60 70 30 40 50 60 70801015

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30

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30

20 40 60 80 100

20 40 60 80 100

30 40 50 60 70 80

40 50 60 70 80 30 40 50 60 70 80 30 40 50 60 70 80

30 40 50 60 70 80

MOEADCMOEAD-M1

MOEADCMOEAD-M2

MOEADCMOEAD-M3

MOEADCMOEAD-M4

MOEADCMOEAD-M5

MOEADCMOEAD-M6

MOEADCMOEAD-M7

MOEADCMOEAD-M8

MOEADCMOEAD-M9

MOEADCMOEAD-M10

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Mea

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Mea

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Mea

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Makespan Makespan

Makespan

Makespan

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii



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40 50 60 70 80 40 50 60 70 8040 50 60 70 80 90

40 50 60 70 80 90 40 50 60 70 80 90 40 50 60 70 80 90

40 50 60 70 80 90 40 50 60 70 80 90

40 50 60 70 80 90

40 60 80 100

MOEADCMOEAD-C1

MOEADCMOEAD-C2

MOEADCMOEAD-C3

MOEADCMOEAD-C4

MOEADCMOEAD-C5

MOEADCMOEAD-C6

MOEADCMOEAD-C7

MOEADCMOEAD-C8

MOEADCMOEAD-C9

MOEADCMOEAD-C10

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Mea

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Makespan Makespan

Makespan

Makespan

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii



40 50 60 70 80 40 50 60 70 80

40 50 60 70 80

90

40 50 60 70 80 90

40 50 60 70 80 90 40 50 60 70 80 90

40 50 60 70 80 90

40 60 80 100

40 60 80 100

40 45 50 55 60 65

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24

16

18

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24

16

18

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24

15

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15

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15

20

25

15

20

25

14

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18

20

22

14

16

18

20

22

Mea

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oad

Mea

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orkl

oad

Mea

n w

orkl

oad

Mea

n w

orkl

oad




Makespan

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii

MOEADCMOEAD-I1

MOEADCMOEAD-I2

MOEADCMOEAD-I3

MOEADCMOEAD-I4

MOEADCMOEAD-I5

MOEADCMOEAD-I6

MOEADCMOEAD-I7

MOEADCMOEAD-I8

MOEADCMOEAD-I9

MOEADCMOEAD-I10



40 50 60 70 80 90

40 50 60 70 80 90 40 50 60 70 80 90 40 50 60 70 80 90

40 50 60 70 80 9040 50 60 70 80 90

40 50 60 70 80 90 40 50 60 70 80

40 50 60 70 80

40 50 60 70 80

15

20

25

15

20

25

15

20

25

16

18

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24

16

18

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24

16

18

20

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24

16

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22

16

18

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22

24

15

20

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30

15

20

25

30

Mea

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oad

Mea

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oad

Mea

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oad

Mea

n w

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oad




Makespan

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii

MOEADCMOEAD-M1

MOEADCMOEAD-M2

MOEADCMOEAD-M3

MOEADCMOEAD-M5

MOEADCMOEAD-M6

MOEADCMOEAD-M7

MOEADCMOEAD-M9

MOEADCMOEAD-M10

MOEADCMOEAD-M4

MOEADCMOEAD-M8



Algorithm

minus02

0

02

04

06

08

1

TTSP 20 lowast 8

C-m

easu

re

(RC

1)

(C1R

)

(RC

2)

(C2R

)

(RC

3)

(C3R

)

(RC

4)

(C4R

)

(RC

5)

(C5R

)

(RC

6)

(C6R

)

(RC

7)

(C7R

)

(RC

8)

(C8R

)

(RC

9)

(C9R

)

(RC

10)

(C10R

)




119862 (119860 119861) =


(13)





Algorithm

minus02

0

02

04

06

08

1

TTSP 20 lowast 8

(RI1)

(I1R

)

(RI2)

(I2R

)

(RI3)

(I3R

)

(RI4)

(I4R

)

(RI5)

(I5R

)

(RI6)

(I6R

)

(RI7)

(I7R

)

(RI8)

(I8R

)

(RI9)

(I9R

)

(RI10)

(I10R

)

+

+

+

C-m

easu

re


(RM

1)

(M1R

)

(RM

2)

(M2R

)

(RM

3)

(M3R

)

(RM

4)

(M4R

)

(RM

5)

(M5R

)

(RM

6)

(M6R

)

(RM

7)

(M7R

)

(RM

8)

(M8R

)

(RM

9)

(M9R

)

(RM

10)

(M10R

)

Algorithm

minus02

0

02

04

06

08

1

TTSP 20 lowast 8

+

C-m

easu

re







119877 119862 119877 119868 119877 119872 119877 119862 119877 119868 119877 119872



119877 119862 119877 119868 119877 119872 119877 119862 119877 119868 119877 119872




(119877 119862) (119862 119877) (119877 119868) (119868 119877) (119877119872) (119872 119877) (119877 119862) (119862 119877) (119877 119868) (119868 119877) (119877119872) (119872 119877)



(119877 119862) (119862 119877) (119877 119868) (119868 119877) (119877119872) (119872 119877) (119877 119862) (119862 119877) (119877 119868) (119868 119877) (119877119872) (119872 119877)





119862 119868 119872 119862 119868 119872 119862 119868 119872 119862 119868 119872


40

35

30

25

20

15

25 30 35 40 45

Mea

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oad

Makespan

TTSP 6 lowast 8

Solutions











30 40 50 60 7012

13

14

15

16

17

18

19

Makespan30 40 50 60 70

Makespan Makespan

Mea

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oad

Mea

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oad

Mea

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oad

20 40 60 80 100 12013

14

15

16

17

18

19

12

14

16

18

20

24

22

VNMCMOEAD-C1

VNMCMOEAD-I9

VNMCMOEAD-M3


14

16

18

20

22

24

26

28

Mea

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oad

14

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24

26

28

Mea

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oad

Mea

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oad

40 50 60 70 80


15

20

25

30

35

40

VNMCMOEAD-C1

VNMCMOEAD-I10

VNMCMOEAD-M9








119881 119862 119881 119868 119881 119872 119881 119862 119881 119868 119881 119872

1 04067 04928 04067 04582 04067 04642 04016 04718 04016 04576 04016 041892 04018 04844 04018 04756 04018 04746 04148 04511 04148 04449 04148 045213 04552 04776 04552 04582 04552 04706 04729 04729 04729 04937 04729 042224 04258 04907 04258 04789 04258 04643 03881 04853 03881 04521 03881 042865 04283 05055 04283 04607 04283 04755 04010 04598 04010 04674 04010 046666 04492 04690 04492 04738 04492 04889 04155 04389 04155 04496 04155 043547 04560 04540 04560 04612 04560 04871 03900 04521 03900 04688 03900 045788 04306 04765 04306 04688 04306 04789 03968 04426 03968 04703 03968 047509 04460 04872 04460 04632 04460 04740 04235 04585 04235 04500 04235 0446810 04458 04897 04458 04803 04458 04767 04002 04406 04002 04465 04002 04537Average 04345 04827 04345 04679 04345 04755 04104 04574 04104 04601 04104 04457Times 1 9 0 10 0 10 0 9 0 10 1 9



(119881 119862) (119862 119881) (119881 119868) (119868 119881) (119881119872) (119872119881) (119881 119862) (119862 119881) (119881 119868) (119868 119881) (119881119872) (119872119881)

1 00000 10000 00000 08333 00000 10000 00000 10000 00000 10000 00000 090002 00000 10000 00000 10000 00000 10000 00000 10000 00000 10000 00000 100003 00000 09000 03750 05000 01667 07000 00000 10000 00000 10000 00000 100004 00000 10000 00000 10000 00000 10000 00000 10000 00000 10000 00000 100005 00000 10000 00000 09000 00000 10000 00000 10000 00000 10000 00000 100006 00000 01667 00833 01667 00000 10000 00000 08571 00000 10000 01000 085717 04444 00000 00000 02000 00000 10000 00000 10000 00000 10000 00000 100008 00000 10000 00000 10000 00000 10000 00000 10000 00000 10000 00000 100009 00000 10000 00000 10000 00000 10000 00000 10000 00000 07778 00000 0777810 00000 10000 01000 05714 00833 05714 00000 10000 00000 10000 00000 10000Average 00444 08067 00558 07171 00250 09271 00000 09857 00000 09778 00100 09535Times 1 9 0 10 0 10 0 10 0 10 0 10


7 Conclusion





Acknowledgments



References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












[12 13)[15 16)[18 19)

[21 22)[24 25)[27 28)

[1 2)[4 5)[7 8)[10 11)

[13 14)[16 17)[19 20)

[22 23)[25 26)[28 29)

[2 3)[5 6)[8 9)[11 12)

[14 15)[17 18)[20 21)

[23 24)[26 27)[29 30)

119896 1 2 3



119899iter 250119899pop 100119899var 6 20 30 40CR 09119875119898 16 120 130 140

35 40 45 50 55 60 65 70

Makespan

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30

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34

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MOEAD-1MOEAD-2

MOEAD-3













HV (119878) = Leb(⋃119894isin119878

V119894) (12)


20 25 30 35 40 45

20 25 30 35 40 45 20 25 30 35 40 45 20 25 30 35 40 45

20 25 30 35 40 45 20 25 30 35 40 45 20 25 30 35 40 45

20 25 30 35 40

20 25 30 35 40

20 25 30 35 40 4510

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25

Mea

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Mea

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Baker

Makespan




Cat Circle

Cubic Gauss ICMIC


Zaslavskii

MOEADCMOEAD-C1

MOEADCMOEAD-C2

MOEADCMOEAD-C3

MOEADCMOEAD-C4

MOEADCMOEAD-C5

MOEADCMOEAD-C6

MOEADCMOEAD-C7

MOEADCMOEAD-C8

MOEADCMOEAD-C9

MOEADCMOEAD-C10



20 25 30 35 4010

15

20

25

10

15

20

25

10

15

20

25Baker

Makespan

Makespan

20 25 30 35 40

Makespan20 25 30 35 40

Makespan

20 25 30 35 40

Makespan

20 25 30 35 40

Makespan

Mea

n w

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oad

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25

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25

10

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10

15

20

25

Mea

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oad

Mea

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oad

Mea

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oad

MOEADCMOEAD-I1

20 25 30 35 40 45

Makespan20 25 30 35 40 45

Makespan20 25 30 35 40 45

Makespan20 25 30 35 40 45

Makespan20 25 30 35 40 45

Cat

MOEADCMOEAD-I2

Circle

MOEADCMOEAD-I3

Cubic

MOEADCMOEAD-I4

Gauss

MOEADCMOEAD-I5

ICMIC

MOEAD

Logistic

MOEAD

Sinusoidal

MOEAD

Tent

MOEADCMOEAD-I9

Zaslavskii

MOEADCMOEAD-I10

CMOEAD-I6

CMOEAD-I7 CMOEAD-I8



10

15

20

25

Mea

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10

15

20

25

Mea

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oad

10

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20

25

Mea

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oad

10

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25

10

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Mea

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oad

10

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25

10

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10

15

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25

10

15

20

25

20 25 30 35 40 4520 25 30 35 40


Makespan

20 25 30 35 40 45

Makespan20 25 30 35 40 45

Makespan20 25 30 35 40 45

Makespan

20 25 30 35 40 45

Makespan20 25 30 35 40 45

Makespan

20 25 30 35 40 45

Makespan

20 25 30 35 40 45

Makespan

MOEADCMOEAD-M1

MOEADCMOEAD-M2

MOEADCMOEAD-M3

MOEADCMOEAD-M4

MOEADCMOEAD-M5

MOEADCMOEAD-M6

MOEADCMOEAD-M7

MOEADCMOEAD-M8

MOEADCMOEAD-M9

MOEADCMOEAD-M10

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii



30 40 50 60 70

30 40 50 60 70

12

14

16

18

12

14

16

18

30 40 50 60 70 80

30 40 50 60 70 80

12

14

16

18

20

12

14

16

18

20

12

14

16

18

20

14

16

18

20

30 35 40 45 50

20 40 60 80 100

20 40 60 80 10030 40 50 60 70

30 40 50 60 70

30 40 50 6010

15

20

25

10

15

20

25 40

30

20

10

30

25

20

15

10

Mea

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oad

Mea

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oad

Mea

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oad

Mea

n w

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oad

Makespan



Makespan

Makespan Makespan

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii

MOEADCMOEAD-C1

MOEADCMOEAD-C2

MOEADCMOEAD-C3

MOEADCMOEAD-C5

MOEADCMOEAD-C4

MOEADCMOEAD-C6

MOEADCMOEAD-C7

MOEADCMOEAD-C8

MOEADCMOEAD-C9

MOEADCMOEAD-C10



14

16

18

20

14

16

18

20

14

16

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20

14

16

18

20

12

14

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12

14

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20

12

14

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20

12

14

16

18

20

12

14

16

18

20

10

15

20

25

30 35 40 45 50 30 40 50 60 70 20 40 60 80 100 120

20 40 60 80 100 30 35 40 45 50 30 40 50 60 70

30 40 50 60 70

30 35 40 45 50 30 40 50 60 70 30 35 40 45 50 55

Mea

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oad

Mea

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oad

Mea

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oad

Mea

n w

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oad



Makespan Makespan

Makespan

Makespan

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii

MOEADCMOEAD-I1

MOEADCMOEAD-I2

MOEADCMOEAD-I3

MOEADCMOEAD-I5

MOEADCMOEAD-I4

MOEADCMOEAD-I6

MOEADCMOEAD-I7

MOEADCMOEAD-I9

MOEADCMOEAD-I8

MOEADCMOEAD-I10



30 40 50 60 70 30 40 50 60 70 30 35 40 45 50 55

20 40 60 80 100 30 40 50 60 70 80 30 40 50 60 70

30 35 40 45 50 55 20 30 40 50 60 70 30 40 50 60 70

30 35 40 45 50

MOEADCMOEAD-M1

MOEADCMOEAD-M2

MOEADCMOEAD-M3

MOEADCMOEAD-M4

MOEADCMOEAD-M5

MOEADCMOEAD-M6

MOEADCMOEAD-M7

MOEADCMOEAD-M8

MOEADCMOEAD-M9

MOEADCMOEAD-M10

10

15

20

25

14

16

18

20

12

14

16

18

20

12

14

16

18

20

12

14

16

18

20

12

14

16

18

20

12

14

16

18

20

10

15

20

25

10

15

20

25

10

15

20

25

Mea

n w

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oad

Mea

n w

orkl

oad

Mea

n w

orkl

oad

Mea

n w

orkl

oad

Makespan

Makespan

Makespan Makespan

Makespan

Makespan

Makespan Makespan

Makespan Makespan

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii



14

16

18

20

10

15

20

25

16

18

20

22

16

18

20

22

24

10

15

20

25

16

18

20

22

15

20

25

16

18

20

22

24

15

20

25

15

20

25

30 40 50 60 70 80

30 40 50 60 70 30 40 50 60 70

30 40 50 60 70

80 30 40 50 60 70 80

30 40 50 60 70 80 30 40 50 60 70 80

30 40 50 60 70 80

20 40 60 80 100 20 40 60 80 100

Mea

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oad

Mea

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oad

Mea

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oad

Mea

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oad




Makespan

MOEADCMOEAD-C1

MOEADCMOEAD-C2

MOEADCMOEAD-C3

MOEADCMOEAD-C4

MOEADCMOEAD-C5

MOEADCMOEAD-C6

MOEADCMOEAD-C7

MOEADCMOEAD-C8

MOEADCMOEAD-C9

MOEADCMOEAD-C10

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii



20 40 60 80 100

20 40 60 80 100

20 40 60 80 100

20 40 60 80 100

20 40 60 80 100 30 40 50 60 70 80

30 40 50 60 70 30 40 50 60 7080

30 40 50 60 70 80 30 40 50 60 70 80

16

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24

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30

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30

15

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25

30

15

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25

15

20

25

15

20

25

15

10

20

25

15

20

25

30

MOEADCMOEAD-I1

MOEADCMOEAD-I2

MOEADCMOEAD-I3

MOEADCMOEAD-I4

MOEADCMOEAD-I5

MOEADCMOEAD-I6

MOEADCMOEAD-I7

MOEADCMOEAD-I8

MOEADCMOEAD-I9

MOEADCMOEAD-I10

Mea

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oad

Mea

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oad

Mea

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oad

Mea

n w

orkl

oad

Makespan Makespan



Makespan

Makespan

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii



30 40 50 60 70 80 30 40 50 60 70 30 40 50 60 70801015

20

25

15

20

25

15

20

25

10

15

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25

30

14

16

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20

22

14

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20

14

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20

14

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22

15

20

25

30

20 40 60 80 100

20 40 60 80 100

30 40 50 60 70 80

40 50 60 70 80 30 40 50 60 70 80 30 40 50 60 70 80

30 40 50 60 70 80

MOEADCMOEAD-M1

MOEADCMOEAD-M2

MOEADCMOEAD-M3

MOEADCMOEAD-M4

MOEADCMOEAD-M5

MOEADCMOEAD-M6

MOEADCMOEAD-M7

MOEADCMOEAD-M8

MOEADCMOEAD-M9

MOEADCMOEAD-M10

Mea

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oad

Mea

n w

orkl

oad

Mea

n w

orkl

oad

Mea

n w

orkl

oad



Makespan Makespan

Makespan

Makespan

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii



14

16

18

20

22

16

18

20

22

16

18

20

22

24

16

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24

16

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24

15

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15

20

25

40 50 60 70 80 40 50 60 70 8040 50 60 70 80 90

40 50 60 70 80 90 40 50 60 70 80 90 40 50 60 70 80 90

40 50 60 70 80 90 40 50 60 70 80 90

40 50 60 70 80 90

40 60 80 100

MOEADCMOEAD-C1

MOEADCMOEAD-C2

MOEADCMOEAD-C3

MOEADCMOEAD-C4

MOEADCMOEAD-C5

MOEADCMOEAD-C6

MOEADCMOEAD-C7

MOEADCMOEAD-C8

MOEADCMOEAD-C9

MOEADCMOEAD-C10

Mea

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oad

Mea

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oad

Mea

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oad

Mea

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oad



Makespan Makespan

Makespan

Makespan

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii



40 50 60 70 80 40 50 60 70 80

40 50 60 70 80

90

40 50 60 70 80 90

40 50 60 70 80 90 40 50 60 70 80 90

40 50 60 70 80 90

40 60 80 100

40 60 80 100

40 45 50 55 60 65

16

18

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24

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24

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24

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Mea

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oad

Mea

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oad

Mea

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oad

Mea

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oad




Makespan

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii

MOEADCMOEAD-I1

MOEADCMOEAD-I2

MOEADCMOEAD-I3

MOEADCMOEAD-I4

MOEADCMOEAD-I5

MOEADCMOEAD-I6

MOEADCMOEAD-I7

MOEADCMOEAD-I8

MOEADCMOEAD-I9

MOEADCMOEAD-I10



40 50 60 70 80 90

40 50 60 70 80 90 40 50 60 70 80 90 40 50 60 70 80 90

40 50 60 70 80 9040 50 60 70 80 90

40 50 60 70 80 90 40 50 60 70 80

40 50 60 70 80

40 50 60 70 80

15

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Mea

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Mea

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Mea

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oad

Mea

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oad




Makespan

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii

MOEADCMOEAD-M1

MOEADCMOEAD-M2

MOEADCMOEAD-M3

MOEADCMOEAD-M5

MOEADCMOEAD-M6

MOEADCMOEAD-M7

MOEADCMOEAD-M9

MOEADCMOEAD-M10

MOEADCMOEAD-M4

MOEADCMOEAD-M8



Algorithm

minus02

0

02

04

06

08

1

TTSP 20 lowast 8

C-m

easu

re

(RC

1)

(C1R

)

(RC

2)

(C2R

)

(RC

3)

(C3R

)

(RC

4)

(C4R

)

(RC

5)

(C5R

)

(RC

6)

(C6R

)

(RC

7)

(C7R

)

(RC

8)

(C8R

)

(RC

9)

(C9R

)

(RC

10)

(C10R

)




119862 (119860 119861) =


(13)





Algorithm

minus02

0

02

04

06

08

1

TTSP 20 lowast 8

(RI1)

(I1R

)

(RI2)

(I2R

)

(RI3)

(I3R

)

(RI4)

(I4R

)

(RI5)

(I5R

)

(RI6)

(I6R

)

(RI7)

(I7R

)

(RI8)

(I8R

)

(RI9)

(I9R

)

(RI10)

(I10R

)

+

+

+

C-m

easu

re


(RM

1)

(M1R

)

(RM

2)

(M2R

)

(RM

3)

(M3R

)

(RM

4)

(M4R

)

(RM

5)

(M5R

)

(RM

6)

(M6R

)

(RM

7)

(M7R

)

(RM

8)

(M8R

)

(RM

9)

(M9R

)

(RM

10)

(M10R

)

Algorithm

minus02

0

02

04

06

08

1

TTSP 20 lowast 8

+

C-m

easu

re







119877 119862 119877 119868 119877 119872 119877 119862 119877 119868 119877 119872



119877 119862 119877 119868 119877 119872 119877 119862 119877 119868 119877 119872




(119877 119862) (119862 119877) (119877 119868) (119868 119877) (119877119872) (119872 119877) (119877 119862) (119862 119877) (119877 119868) (119868 119877) (119877119872) (119872 119877)



(119877 119862) (119862 119877) (119877 119868) (119868 119877) (119877119872) (119872 119877) (119877 119862) (119862 119877) (119877 119868) (119868 119877) (119877119872) (119872 119877)





119862 119868 119872 119862 119868 119872 119862 119868 119872 119862 119868 119872


40

35

30

25

20

15

25 30 35 40 45

Mea

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oad

Makespan

TTSP 6 lowast 8

Solutions











30 40 50 60 7012

13

14

15

16

17

18

19

Makespan30 40 50 60 70

Makespan Makespan

Mea

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Mea

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Mea

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oad

20 40 60 80 100 12013

14

15

16

17

18

19

12

14

16

18

20

24

22

VNMCMOEAD-C1

VNMCMOEAD-I9

VNMCMOEAD-M3


14

16

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26

28

Mea

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Mea

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Mea

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40 50 60 70 80


15

20

25

30

35

40

VNMCMOEAD-C1

VNMCMOEAD-I10

VNMCMOEAD-M9








119881 119862 119881 119868 119881 119872 119881 119862 119881 119868 119881 119872

1 04067 04928 04067 04582 04067 04642 04016 04718 04016 04576 04016 041892 04018 04844 04018 04756 04018 04746 04148 04511 04148 04449 04148 045213 04552 04776 04552 04582 04552 04706 04729 04729 04729 04937 04729 042224 04258 04907 04258 04789 04258 04643 03881 04853 03881 04521 03881 042865 04283 05055 04283 04607 04283 04755 04010 04598 04010 04674 04010 046666 04492 04690 04492 04738 04492 04889 04155 04389 04155 04496 04155 043547 04560 04540 04560 04612 04560 04871 03900 04521 03900 04688 03900 045788 04306 04765 04306 04688 04306 04789 03968 04426 03968 04703 03968 047509 04460 04872 04460 04632 04460 04740 04235 04585 04235 04500 04235 0446810 04458 04897 04458 04803 04458 04767 04002 04406 04002 04465 04002 04537Average 04345 04827 04345 04679 04345 04755 04104 04574 04104 04601 04104 04457Times 1 9 0 10 0 10 0 9 0 10 1 9



(119881 119862) (119862 119881) (119881 119868) (119868 119881) (119881119872) (119872119881) (119881 119862) (119862 119881) (119881 119868) (119868 119881) (119881119872) (119872119881)

1 00000 10000 00000 08333 00000 10000 00000 10000 00000 10000 00000 090002 00000 10000 00000 10000 00000 10000 00000 10000 00000 10000 00000 100003 00000 09000 03750 05000 01667 07000 00000 10000 00000 10000 00000 100004 00000 10000 00000 10000 00000 10000 00000 10000 00000 10000 00000 100005 00000 10000 00000 09000 00000 10000 00000 10000 00000 10000 00000 100006 00000 01667 00833 01667 00000 10000 00000 08571 00000 10000 01000 085717 04444 00000 00000 02000 00000 10000 00000 10000 00000 10000 00000 100008 00000 10000 00000 10000 00000 10000 00000 10000 00000 10000 00000 100009 00000 10000 00000 10000 00000 10000 00000 10000 00000 07778 00000 0777810 00000 10000 01000 05714 00833 05714 00000 10000 00000 10000 00000 10000Average 00444 08067 00558 07171 00250 09271 00000 09857 00000 09778 00100 09535Times 1 9 0 10 0 10 0 10 0 10 0 10


7 Conclusion





Acknowledgments



References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










20 25 30 35 40 45

20 25 30 35 40 45 20 25 30 35 40 45 20 25 30 35 40 45

20 25 30 35 40 45 20 25 30 35 40 45 20 25 30 35 40 45

20 25 30 35 40

20 25 30 35 40

20 25 30 35 40 4510

15

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Mea

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Baker

Makespan




Cat Circle

Cubic Gauss ICMIC


Zaslavskii

MOEADCMOEAD-C1

MOEADCMOEAD-C2

MOEADCMOEAD-C3

MOEADCMOEAD-C4

MOEADCMOEAD-C5

MOEADCMOEAD-C6

MOEADCMOEAD-C7

MOEADCMOEAD-C8

MOEADCMOEAD-C9

MOEADCMOEAD-C10



20 25 30 35 4010

15

20

25

10

15

20

25

10

15

20

25Baker

Makespan

Makespan

20 25 30 35 40

Makespan20 25 30 35 40

Makespan

20 25 30 35 40

Makespan

20 25 30 35 40

Makespan

Mea

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oad

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Mea

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Mea

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Mea

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MOEADCMOEAD-I1

20 25 30 35 40 45

Makespan20 25 30 35 40 45

Makespan20 25 30 35 40 45

Makespan20 25 30 35 40 45

Makespan20 25 30 35 40 45

Cat

MOEADCMOEAD-I2

Circle

MOEADCMOEAD-I3

Cubic

MOEADCMOEAD-I4

Gauss

MOEADCMOEAD-I5

ICMIC

MOEAD

Logistic

MOEAD

Sinusoidal

MOEAD

Tent

MOEADCMOEAD-I9

Zaslavskii

MOEADCMOEAD-I10

CMOEAD-I6

CMOEAD-I7 CMOEAD-I8



10

15

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25

Mea

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10

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25

Mea

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Mea

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10

15

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20 25 30 35 40 4520 25 30 35 40


Makespan

20 25 30 35 40 45

Makespan20 25 30 35 40 45

Makespan20 25 30 35 40 45

Makespan

20 25 30 35 40 45

Makespan20 25 30 35 40 45

Makespan

20 25 30 35 40 45

Makespan

20 25 30 35 40 45

Makespan

MOEADCMOEAD-M1

MOEADCMOEAD-M2

MOEADCMOEAD-M3

MOEADCMOEAD-M4

MOEADCMOEAD-M5

MOEADCMOEAD-M6

MOEADCMOEAD-M7

MOEADCMOEAD-M8

MOEADCMOEAD-M9

MOEADCMOEAD-M10

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii



30 40 50 60 70

30 40 50 60 70

12

14

16

18

12

14

16

18

30 40 50 60 70 80

30 40 50 60 70 80

12

14

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20

12

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30 35 40 45 50

20 40 60 80 100

20 40 60 80 10030 40 50 60 70

30 40 50 60 70

30 40 50 6010

15

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10

15

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25 40

30

20

10

30

25

20

15

10

Mea

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oad

Mea

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oad

Mea

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Mea

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oad

Makespan



Makespan

Makespan Makespan

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii

MOEADCMOEAD-C1

MOEADCMOEAD-C2

MOEADCMOEAD-C3

MOEADCMOEAD-C5

MOEADCMOEAD-C4

MOEADCMOEAD-C6

MOEADCMOEAD-C7

MOEADCMOEAD-C8

MOEADCMOEAD-C9

MOEADCMOEAD-C10



14

16

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14

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14

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25

30 35 40 45 50 30 40 50 60 70 20 40 60 80 100 120

20 40 60 80 100 30 35 40 45 50 30 40 50 60 70

30 40 50 60 70

30 35 40 45 50 30 40 50 60 70 30 35 40 45 50 55

Mea

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Mea

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Mea

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Mea

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Makespan Makespan

Makespan

Makespan

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii

MOEADCMOEAD-I1

MOEADCMOEAD-I2

MOEADCMOEAD-I3

MOEADCMOEAD-I5

MOEADCMOEAD-I4

MOEADCMOEAD-I6

MOEADCMOEAD-I7

MOEADCMOEAD-I9

MOEADCMOEAD-I8

MOEADCMOEAD-I10



30 40 50 60 70 30 40 50 60 70 30 35 40 45 50 55

20 40 60 80 100 30 40 50 60 70 80 30 40 50 60 70

30 35 40 45 50 55 20 30 40 50 60 70 30 40 50 60 70

30 35 40 45 50

MOEADCMOEAD-M1

MOEADCMOEAD-M2

MOEADCMOEAD-M3

MOEADCMOEAD-M4

MOEADCMOEAD-M5

MOEADCMOEAD-M6

MOEADCMOEAD-M7

MOEADCMOEAD-M8

MOEADCMOEAD-M9

MOEADCMOEAD-M10

10

15

20

25

14

16

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20

12

14

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Mea

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Mea

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Mea

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oad

Makespan

Makespan

Makespan Makespan

Makespan

Makespan

Makespan Makespan

Makespan Makespan

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii



14

16

18

20

10

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30 40 50 60 70 80

30 40 50 60 70 30 40 50 60 70

30 40 50 60 70

80 30 40 50 60 70 80

30 40 50 60 70 80 30 40 50 60 70 80

30 40 50 60 70 80

20 40 60 80 100 20 40 60 80 100

Mea

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oad

Mea

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Mea

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Mea

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Makespan

MOEADCMOEAD-C1

MOEADCMOEAD-C2

MOEADCMOEAD-C3

MOEADCMOEAD-C4

MOEADCMOEAD-C5

MOEADCMOEAD-C6

MOEADCMOEAD-C7

MOEADCMOEAD-C8

MOEADCMOEAD-C9

MOEADCMOEAD-C10

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii



20 40 60 80 100

20 40 60 80 100

20 40 60 80 100

20 40 60 80 100

20 40 60 80 100 30 40 50 60 70 80

30 40 50 60 70 30 40 50 60 7080

30 40 50 60 70 80 30 40 50 60 70 80

16

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30

15

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20

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25

15

10

20

25

15

20

25

30

MOEADCMOEAD-I1

MOEADCMOEAD-I2

MOEADCMOEAD-I3

MOEADCMOEAD-I4

MOEADCMOEAD-I5

MOEADCMOEAD-I6

MOEADCMOEAD-I7

MOEADCMOEAD-I8

MOEADCMOEAD-I9

MOEADCMOEAD-I10

Mea

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Mea

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oad

Mea

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Mea

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oad

Makespan Makespan



Makespan

Makespan

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii



30 40 50 60 70 80 30 40 50 60 70 30 40 50 60 70801015

20

25

15

20

25

15

20

25

10

15

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25

30

14

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14

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14

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15

20

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30

20 40 60 80 100

20 40 60 80 100

30 40 50 60 70 80

40 50 60 70 80 30 40 50 60 70 80 30 40 50 60 70 80

30 40 50 60 70 80

MOEADCMOEAD-M1

MOEADCMOEAD-M2

MOEADCMOEAD-M3

MOEADCMOEAD-M4

MOEADCMOEAD-M5

MOEADCMOEAD-M6

MOEADCMOEAD-M7

MOEADCMOEAD-M8

MOEADCMOEAD-M9

MOEADCMOEAD-M10

Mea

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Mea

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Mea

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Mea

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Makespan Makespan

Makespan

Makespan

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii



14

16

18

20

22

16

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24

16

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15

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40 50 60 70 80 40 50 60 70 8040 50 60 70 80 90

40 50 60 70 80 90 40 50 60 70 80 90 40 50 60 70 80 90

40 50 60 70 80 90 40 50 60 70 80 90

40 50 60 70 80 90

40 60 80 100

MOEADCMOEAD-C1

MOEADCMOEAD-C2

MOEADCMOEAD-C3

MOEADCMOEAD-C4

MOEADCMOEAD-C5

MOEADCMOEAD-C6

MOEADCMOEAD-C7

MOEADCMOEAD-C8

MOEADCMOEAD-C9

MOEADCMOEAD-C10

Mea

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Mea

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Mea

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Mea

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Makespan Makespan

Makespan

Makespan

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii



40 50 60 70 80 40 50 60 70 80

40 50 60 70 80

90

40 50 60 70 80 90

40 50 60 70 80 90 40 50 60 70 80 90

40 50 60 70 80 90

40 60 80 100

40 60 80 100

40 45 50 55 60 65

16

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Mea

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Mea

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Mea

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Makespan

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii

MOEADCMOEAD-I1

MOEADCMOEAD-I2

MOEADCMOEAD-I3

MOEADCMOEAD-I4

MOEADCMOEAD-I5

MOEADCMOEAD-I6

MOEADCMOEAD-I7

MOEADCMOEAD-I8

MOEADCMOEAD-I9

MOEADCMOEAD-I10



40 50 60 70 80 90

40 50 60 70 80 90 40 50 60 70 80 90 40 50 60 70 80 90

40 50 60 70 80 9040 50 60 70 80 90

40 50 60 70 80 90 40 50 60 70 80

40 50 60 70 80

40 50 60 70 80

15

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Mea

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Mea

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Mea

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Makespan

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii

MOEADCMOEAD-M1

MOEADCMOEAD-M2

MOEADCMOEAD-M3

MOEADCMOEAD-M5

MOEADCMOEAD-M6

MOEADCMOEAD-M7

MOEADCMOEAD-M9

MOEADCMOEAD-M10

MOEADCMOEAD-M4

MOEADCMOEAD-M8



Algorithm

minus02

0

02

04

06

08

1

TTSP 20 lowast 8

C-m

easu

re

(RC

1)

(C1R

)

(RC

2)

(C2R

)

(RC

3)

(C3R

)

(RC

4)

(C4R

)

(RC

5)

(C5R

)

(RC

6)

(C6R

)

(RC

7)

(C7R

)

(RC

8)

(C8R

)

(RC

9)

(C9R

)

(RC

10)

(C10R

)




119862 (119860 119861) =


(13)





Algorithm

minus02

0

02

04

06

08

1

TTSP 20 lowast 8

(RI1)

(I1R

)

(RI2)

(I2R

)

(RI3)

(I3R

)

(RI4)

(I4R

)

(RI5)

(I5R

)

(RI6)

(I6R

)

(RI7)

(I7R

)

(RI8)

(I8R

)

(RI9)

(I9R

)

(RI10)

(I10R

)

+

+

+

C-m

easu

re


(RM

1)

(M1R

)

(RM

2)

(M2R

)

(RM

3)

(M3R

)

(RM

4)

(M4R

)

(RM

5)

(M5R

)

(RM

6)

(M6R

)

(RM

7)

(M7R

)

(RM

8)

(M8R

)

(RM

9)

(M9R

)

(RM

10)

(M10R

)

Algorithm

minus02

0

02

04

06

08

1

TTSP 20 lowast 8

+

C-m

easu

re







119877 119862 119877 119868 119877 119872 119877 119862 119877 119868 119877 119872



119877 119862 119877 119868 119877 119872 119877 119862 119877 119868 119877 119872




(119877 119862) (119862 119877) (119877 119868) (119868 119877) (119877119872) (119872 119877) (119877 119862) (119862 119877) (119877 119868) (119868 119877) (119877119872) (119872 119877)



(119877 119862) (119862 119877) (119877 119868) (119868 119877) (119877119872) (119872 119877) (119877 119862) (119862 119877) (119877 119868) (119868 119877) (119877119872) (119872 119877)





119862 119868 119872 119862 119868 119872 119862 119868 119872 119862 119868 119872


40

35

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25 30 35 40 45

Mea

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Makespan

TTSP 6 lowast 8

Solutions











30 40 50 60 7012

13

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Makespan30 40 50 60 70

Makespan Makespan

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Mea

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Mea

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20 40 60 80 100 12013

14

15

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14

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VNMCMOEAD-C1

VNMCMOEAD-I9

VNMCMOEAD-M3


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Mea

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Mea

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40 50 60 70 80


15

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40

VNMCMOEAD-C1

VNMCMOEAD-I10

VNMCMOEAD-M9








119881 119862 119881 119868 119881 119872 119881 119862 119881 119868 119881 119872

1 04067 04928 04067 04582 04067 04642 04016 04718 04016 04576 04016 041892 04018 04844 04018 04756 04018 04746 04148 04511 04148 04449 04148 045213 04552 04776 04552 04582 04552 04706 04729 04729 04729 04937 04729 042224 04258 04907 04258 04789 04258 04643 03881 04853 03881 04521 03881 042865 04283 05055 04283 04607 04283 04755 04010 04598 04010 04674 04010 046666 04492 04690 04492 04738 04492 04889 04155 04389 04155 04496 04155 043547 04560 04540 04560 04612 04560 04871 03900 04521 03900 04688 03900 045788 04306 04765 04306 04688 04306 04789 03968 04426 03968 04703 03968 047509 04460 04872 04460 04632 04460 04740 04235 04585 04235 04500 04235 0446810 04458 04897 04458 04803 04458 04767 04002 04406 04002 04465 04002 04537Average 04345 04827 04345 04679 04345 04755 04104 04574 04104 04601 04104 04457Times 1 9 0 10 0 10 0 9 0 10 1 9



(119881 119862) (119862 119881) (119881 119868) (119868 119881) (119881119872) (119872119881) (119881 119862) (119862 119881) (119881 119868) (119868 119881) (119881119872) (119872119881)

1 00000 10000 00000 08333 00000 10000 00000 10000 00000 10000 00000 090002 00000 10000 00000 10000 00000 10000 00000 10000 00000 10000 00000 100003 00000 09000 03750 05000 01667 07000 00000 10000 00000 10000 00000 100004 00000 10000 00000 10000 00000 10000 00000 10000 00000 10000 00000 100005 00000 10000 00000 09000 00000 10000 00000 10000 00000 10000 00000 100006 00000 01667 00833 01667 00000 10000 00000 08571 00000 10000 01000 085717 04444 00000 00000 02000 00000 10000 00000 10000 00000 10000 00000 100008 00000 10000 00000 10000 00000 10000 00000 10000 00000 10000 00000 100009 00000 10000 00000 10000 00000 10000 00000 10000 00000 07778 00000 0777810 00000 10000 01000 05714 00833 05714 00000 10000 00000 10000 00000 10000Average 00444 08067 00558 07171 00250 09271 00000 09857 00000 09778 00100 09535Times 1 9 0 10 0 10 0 10 0 10 0 10


7 Conclusion





Acknowledgments



References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










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Cat

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Circle

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Cubic

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Gauss

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ICMIC

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Logistic

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Sinusoidal

MOEAD

Tent

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Zaslavskii

MOEADCMOEAD-I10

CMOEAD-I6

CMOEAD-I7 CMOEAD-I8



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MOEADCMOEAD-M1

MOEADCMOEAD-M2

MOEADCMOEAD-M3

MOEADCMOEAD-M4

MOEADCMOEAD-M5

MOEADCMOEAD-M6

MOEADCMOEAD-M7

MOEADCMOEAD-M8

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MOEADCMOEAD-M10

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii



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Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii

MOEADCMOEAD-C1

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MOEADCMOEAD-C4

MOEADCMOEAD-C6

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MOEADCMOEAD-C10



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Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii

MOEADCMOEAD-I1

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MOEADCMOEAD-I6

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MOEADCMOEAD-I10



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MOEADCMOEAD-M1

MOEADCMOEAD-M2

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MOEADCMOEAD-M4

MOEADCMOEAD-M5

MOEADCMOEAD-M6

MOEADCMOEAD-M7

MOEADCMOEAD-M8

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MOEADCMOEAD-M10

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Makespan Makespan

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii



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MOEADCMOEAD-C1

MOEADCMOEAD-C2

MOEADCMOEAD-C3

MOEADCMOEAD-C4

MOEADCMOEAD-C5

MOEADCMOEAD-C6

MOEADCMOEAD-C7

MOEADCMOEAD-C8

MOEADCMOEAD-C9

MOEADCMOEAD-C10

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii



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MOEADCMOEAD-I2

MOEADCMOEAD-I3

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MOEADCMOEAD-I6

MOEADCMOEAD-I7

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Makespan Makespan



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Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii



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Cubic Gauss ICMIC


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MOEADCMOEAD-C1

MOEADCMOEAD-C2

MOEADCMOEAD-C3

MOEADCMOEAD-C4

MOEADCMOEAD-C5

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MOEADCMOEAD-C8

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MOEADCMOEAD-C10

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Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii



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Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii

MOEADCMOEAD-I1

MOEADCMOEAD-I2

MOEADCMOEAD-I3

MOEADCMOEAD-I4

MOEADCMOEAD-I5

MOEADCMOEAD-I6

MOEADCMOEAD-I7

MOEADCMOEAD-I8

MOEADCMOEAD-I9

MOEADCMOEAD-I10



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Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii

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MOEADCMOEAD-M6

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MOEADCMOEAD-M9

MOEADCMOEAD-M10

MOEADCMOEAD-M4

MOEADCMOEAD-M8



Algorithm

minus02

0

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04

06

08

1

TTSP 20 lowast 8

C-m

easu

re

(RC

1)

(C1R

)

(RC

2)

(C2R

)

(RC

3)

(C3R

)

(RC

4)

(C4R

)

(RC

5)

(C5R

)

(RC

6)

(C6R

)

(RC

7)

(C7R

)

(RC

8)

(C8R

)

(RC

9)

(C9R

)

(RC

10)

(C10R

)




119862 (119860 119861) =


(13)





Algorithm

minus02

0

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1

TTSP 20 lowast 8

(RI1)

(I1R

)

(RI2)

(I2R

)

(RI3)

(I3R

)

(RI4)

(I4R

)

(RI5)

(I5R

)

(RI6)

(I6R

)

(RI7)

(I7R

)

(RI8)

(I8R

)

(RI9)

(I9R

)

(RI10)

(I10R

)

+

+

+

C-m

easu

re


(RM

1)

(M1R

)

(RM

2)

(M2R

)

(RM

3)

(M3R

)

(RM

4)

(M4R

)

(RM

5)

(M5R

)

(RM

6)

(M6R

)

(RM

7)

(M7R

)

(RM

8)

(M8R

)

(RM

9)

(M9R

)

(RM

10)

(M10R

)

Algorithm

minus02

0

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1

TTSP 20 lowast 8

+

C-m

easu

re







119877 119862 119877 119868 119877 119872 119877 119862 119877 119868 119877 119872



119877 119862 119877 119868 119877 119872 119877 119862 119877 119868 119877 119872




(119877 119862) (119862 119877) (119877 119868) (119868 119877) (119877119872) (119872 119877) (119877 119862) (119862 119877) (119877 119868) (119868 119877) (119877119872) (119872 119877)



(119877 119862) (119862 119877) (119877 119868) (119868 119877) (119877119872) (119872 119877) (119877 119862) (119862 119877) (119877 119868) (119868 119877) (119877119872) (119872 119877)





119862 119868 119872 119862 119868 119872 119862 119868 119872 119862 119868 119872


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Solutions











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VNMCMOEAD-I9

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119881 119862 119881 119868 119881 119872 119881 119862 119881 119868 119881 119872

1 04067 04928 04067 04582 04067 04642 04016 04718 04016 04576 04016 041892 04018 04844 04018 04756 04018 04746 04148 04511 04148 04449 04148 045213 04552 04776 04552 04582 04552 04706 04729 04729 04729 04937 04729 042224 04258 04907 04258 04789 04258 04643 03881 04853 03881 04521 03881 042865 04283 05055 04283 04607 04283 04755 04010 04598 04010 04674 04010 046666 04492 04690 04492 04738 04492 04889 04155 04389 04155 04496 04155 043547 04560 04540 04560 04612 04560 04871 03900 04521 03900 04688 03900 045788 04306 04765 04306 04688 04306 04789 03968 04426 03968 04703 03968 047509 04460 04872 04460 04632 04460 04740 04235 04585 04235 04500 04235 0446810 04458 04897 04458 04803 04458 04767 04002 04406 04002 04465 04002 04537Average 04345 04827 04345 04679 04345 04755 04104 04574 04104 04601 04104 04457Times 1 9 0 10 0 10 0 9 0 10 1 9



(119881 119862) (119862 119881) (119881 119868) (119868 119881) (119881119872) (119872119881) (119881 119862) (119862 119881) (119881 119868) (119868 119881) (119881119872) (119872119881)

1 00000 10000 00000 08333 00000 10000 00000 10000 00000 10000 00000 090002 00000 10000 00000 10000 00000 10000 00000 10000 00000 10000 00000 100003 00000 09000 03750 05000 01667 07000 00000 10000 00000 10000 00000 100004 00000 10000 00000 10000 00000 10000 00000 10000 00000 10000 00000 100005 00000 10000 00000 09000 00000 10000 00000 10000 00000 10000 00000 100006 00000 01667 00833 01667 00000 10000 00000 08571 00000 10000 01000 085717 04444 00000 00000 02000 00000 10000 00000 10000 00000 10000 00000 100008 00000 10000 00000 10000 00000 10000 00000 10000 00000 10000 00000 100009 00000 10000 00000 10000 00000 10000 00000 10000 00000 07778 00000 0777810 00000 10000 01000 05714 00833 05714 00000 10000 00000 10000 00000 10000Average 00444 08067 00558 07171 00250 09271 00000 09857 00000 09778 00100 09535Times 1 9 0 10 0 10 0 10 0 10 0 10


7 Conclusion





Acknowledgments



References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










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MOEADCMOEAD-M1

MOEADCMOEAD-M2

MOEADCMOEAD-M3

MOEADCMOEAD-M4

MOEADCMOEAD-M5

MOEADCMOEAD-M6

MOEADCMOEAD-M7

MOEADCMOEAD-M8

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MOEADCMOEAD-M10

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii



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Makespan Makespan

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii

MOEADCMOEAD-C1

MOEADCMOEAD-C2

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MOEADCMOEAD-C5

MOEADCMOEAD-C4

MOEADCMOEAD-C6

MOEADCMOEAD-C7

MOEADCMOEAD-C8

MOEADCMOEAD-C9

MOEADCMOEAD-C10



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30 40 50 60 70

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Makespan Makespan

Makespan

Makespan

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii

MOEADCMOEAD-I1

MOEADCMOEAD-I2

MOEADCMOEAD-I3

MOEADCMOEAD-I5

MOEADCMOEAD-I4

MOEADCMOEAD-I6

MOEADCMOEAD-I7

MOEADCMOEAD-I9

MOEADCMOEAD-I8

MOEADCMOEAD-I10



30 40 50 60 70 30 40 50 60 70 30 35 40 45 50 55

20 40 60 80 100 30 40 50 60 70 80 30 40 50 60 70

30 35 40 45 50 55 20 30 40 50 60 70 30 40 50 60 70

30 35 40 45 50

MOEADCMOEAD-M1

MOEADCMOEAD-M2

MOEADCMOEAD-M3

MOEADCMOEAD-M4

MOEADCMOEAD-M5

MOEADCMOEAD-M6

MOEADCMOEAD-M7

MOEADCMOEAD-M8

MOEADCMOEAD-M9

MOEADCMOEAD-M10

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Makespan

Makespan

Makespan Makespan

Makespan

Makespan

Makespan Makespan

Makespan Makespan

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii



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Makespan

MOEADCMOEAD-C1

MOEADCMOEAD-C2

MOEADCMOEAD-C3

MOEADCMOEAD-C4

MOEADCMOEAD-C5

MOEADCMOEAD-C6

MOEADCMOEAD-C7

MOEADCMOEAD-C8

MOEADCMOEAD-C9

MOEADCMOEAD-C10

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii



20 40 60 80 100

20 40 60 80 100

20 40 60 80 100

20 40 60 80 100

20 40 60 80 100 30 40 50 60 70 80

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MOEADCMOEAD-I1

MOEADCMOEAD-I2

MOEADCMOEAD-I3

MOEADCMOEAD-I4

MOEADCMOEAD-I5

MOEADCMOEAD-I6

MOEADCMOEAD-I7

MOEADCMOEAD-I8

MOEADCMOEAD-I9

MOEADCMOEAD-I10

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Mea

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Makespan Makespan



Makespan

Makespan

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii



30 40 50 60 70 80 30 40 50 60 70 30 40 50 60 70801015

20

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20 40 60 80 100

30 40 50 60 70 80

40 50 60 70 80 30 40 50 60 70 80 30 40 50 60 70 80

30 40 50 60 70 80

MOEADCMOEAD-M1

MOEADCMOEAD-M2

MOEADCMOEAD-M3

MOEADCMOEAD-M4

MOEADCMOEAD-M5

MOEADCMOEAD-M6

MOEADCMOEAD-M7

MOEADCMOEAD-M8

MOEADCMOEAD-M9

MOEADCMOEAD-M10

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Makespan Makespan

Makespan

Makespan

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii



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MOEADCMOEAD-C1

MOEADCMOEAD-C2

MOEADCMOEAD-C3

MOEADCMOEAD-C4

MOEADCMOEAD-C5

MOEADCMOEAD-C6

MOEADCMOEAD-C7

MOEADCMOEAD-C8

MOEADCMOEAD-C9

MOEADCMOEAD-C10

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Makespan Makespan

Makespan

Makespan

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii



40 50 60 70 80 40 50 60 70 80

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90

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40 50 60 70 80 90 40 50 60 70 80 90

40 50 60 70 80 90

40 60 80 100

40 60 80 100

40 45 50 55 60 65

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Makespan

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii

MOEADCMOEAD-I1

MOEADCMOEAD-I2

MOEADCMOEAD-I3

MOEADCMOEAD-I4

MOEADCMOEAD-I5

MOEADCMOEAD-I6

MOEADCMOEAD-I7

MOEADCMOEAD-I8

MOEADCMOEAD-I9

MOEADCMOEAD-I10



40 50 60 70 80 90

40 50 60 70 80 90 40 50 60 70 80 90 40 50 60 70 80 90

40 50 60 70 80 9040 50 60 70 80 90

40 50 60 70 80 90 40 50 60 70 80

40 50 60 70 80

40 50 60 70 80

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Makespan

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii

MOEADCMOEAD-M1

MOEADCMOEAD-M2

MOEADCMOEAD-M3

MOEADCMOEAD-M5

MOEADCMOEAD-M6

MOEADCMOEAD-M7

MOEADCMOEAD-M9

MOEADCMOEAD-M10

MOEADCMOEAD-M4

MOEADCMOEAD-M8



Algorithm

minus02

0

02

04

06

08

1

TTSP 20 lowast 8

C-m

easu

re

(RC

1)

(C1R

)

(RC

2)

(C2R

)

(RC

3)

(C3R

)

(RC

4)

(C4R

)

(RC

5)

(C5R

)

(RC

6)

(C6R

)

(RC

7)

(C7R

)

(RC

8)

(C8R

)

(RC

9)

(C9R

)

(RC

10)

(C10R

)




119862 (119860 119861) =


(13)





Algorithm

minus02

0

02

04

06

08

1

TTSP 20 lowast 8

(RI1)

(I1R

)

(RI2)

(I2R

)

(RI3)

(I3R

)

(RI4)

(I4R

)

(RI5)

(I5R

)

(RI6)

(I6R

)

(RI7)

(I7R

)

(RI8)

(I8R

)

(RI9)

(I9R

)

(RI10)

(I10R

)

+

+

+

C-m

easu

re


(RM

1)

(M1R

)

(RM

2)

(M2R

)

(RM

3)

(M3R

)

(RM

4)

(M4R

)

(RM

5)

(M5R

)

(RM

6)

(M6R

)

(RM

7)

(M7R

)

(RM

8)

(M8R

)

(RM

9)

(M9R

)

(RM

10)

(M10R

)

Algorithm

minus02

0

02

04

06

08

1

TTSP 20 lowast 8

+

C-m

easu

re







119877 119862 119877 119868 119877 119872 119877 119862 119877 119868 119877 119872



119877 119862 119877 119868 119877 119872 119877 119862 119877 119868 119877 119872




(119877 119862) (119862 119877) (119877 119868) (119868 119877) (119877119872) (119872 119877) (119877 119862) (119862 119877) (119877 119868) (119868 119877) (119877119872) (119872 119877)



(119877 119862) (119862 119877) (119877 119868) (119868 119877) (119877119872) (119872 119877) (119877 119862) (119862 119877) (119877 119868) (119868 119877) (119877119872) (119872 119877)





119862 119868 119872 119862 119868 119872 119862 119868 119872 119862 119868 119872


40

35

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20

15

25 30 35 40 45

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Makespan

TTSP 6 lowast 8

Solutions











30 40 50 60 7012

13

14

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19

Makespan30 40 50 60 70

Makespan Makespan

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20 40 60 80 100 12013

14

15

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19

12

14

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VNMCMOEAD-C1

VNMCMOEAD-I9

VNMCMOEAD-M3


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40 50 60 70 80


15

20

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30

35

40

VNMCMOEAD-C1

VNMCMOEAD-I10

VNMCMOEAD-M9








119881 119862 119881 119868 119881 119872 119881 119862 119881 119868 119881 119872

1 04067 04928 04067 04582 04067 04642 04016 04718 04016 04576 04016 041892 04018 04844 04018 04756 04018 04746 04148 04511 04148 04449 04148 045213 04552 04776 04552 04582 04552 04706 04729 04729 04729 04937 04729 042224 04258 04907 04258 04789 04258 04643 03881 04853 03881 04521 03881 042865 04283 05055 04283 04607 04283 04755 04010 04598 04010 04674 04010 046666 04492 04690 04492 04738 04492 04889 04155 04389 04155 04496 04155 043547 04560 04540 04560 04612 04560 04871 03900 04521 03900 04688 03900 045788 04306 04765 04306 04688 04306 04789 03968 04426 03968 04703 03968 047509 04460 04872 04460 04632 04460 04740 04235 04585 04235 04500 04235 0446810 04458 04897 04458 04803 04458 04767 04002 04406 04002 04465 04002 04537Average 04345 04827 04345 04679 04345 04755 04104 04574 04104 04601 04104 04457Times 1 9 0 10 0 10 0 9 0 10 1 9



(119881 119862) (119862 119881) (119881 119868) (119868 119881) (119881119872) (119872119881) (119881 119862) (119862 119881) (119881 119868) (119868 119881) (119881119872) (119872119881)

1 00000 10000 00000 08333 00000 10000 00000 10000 00000 10000 00000 090002 00000 10000 00000 10000 00000 10000 00000 10000 00000 10000 00000 100003 00000 09000 03750 05000 01667 07000 00000 10000 00000 10000 00000 100004 00000 10000 00000 10000 00000 10000 00000 10000 00000 10000 00000 100005 00000 10000 00000 09000 00000 10000 00000 10000 00000 10000 00000 100006 00000 01667 00833 01667 00000 10000 00000 08571 00000 10000 01000 085717 04444 00000 00000 02000 00000 10000 00000 10000 00000 10000 00000 100008 00000 10000 00000 10000 00000 10000 00000 10000 00000 10000 00000 100009 00000 10000 00000 10000 00000 10000 00000 10000 00000 07778 00000 0777810 00000 10000 01000 05714 00833 05714 00000 10000 00000 10000 00000 10000Average 00444 08067 00558 07171 00250 09271 00000 09857 00000 09778 00100 09535Times 1 9 0 10 0 10 0 10 0 10 0 10


7 Conclusion





Acknowledgments



References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










30 40 50 60 70

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Makespan



Makespan

Makespan Makespan

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii

MOEADCMOEAD-C1

MOEADCMOEAD-C2

MOEADCMOEAD-C3

MOEADCMOEAD-C5

MOEADCMOEAD-C4

MOEADCMOEAD-C6

MOEADCMOEAD-C7

MOEADCMOEAD-C8

MOEADCMOEAD-C9

MOEADCMOEAD-C10



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30 40 50 60 70

30 35 40 45 50 30 40 50 60 70 30 35 40 45 50 55

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Makespan Makespan

Makespan

Makespan

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii

MOEADCMOEAD-I1

MOEADCMOEAD-I2

MOEADCMOEAD-I3

MOEADCMOEAD-I5

MOEADCMOEAD-I4

MOEADCMOEAD-I6

MOEADCMOEAD-I7

MOEADCMOEAD-I9

MOEADCMOEAD-I8

MOEADCMOEAD-I10



30 40 50 60 70 30 40 50 60 70 30 35 40 45 50 55

20 40 60 80 100 30 40 50 60 70 80 30 40 50 60 70

30 35 40 45 50 55 20 30 40 50 60 70 30 40 50 60 70

30 35 40 45 50

MOEADCMOEAD-M1

MOEADCMOEAD-M2

MOEADCMOEAD-M3

MOEADCMOEAD-M4

MOEADCMOEAD-M5

MOEADCMOEAD-M6

MOEADCMOEAD-M7

MOEADCMOEAD-M8

MOEADCMOEAD-M9

MOEADCMOEAD-M10

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Makespan

Makespan

Makespan Makespan

Makespan

Makespan

Makespan Makespan

Makespan Makespan

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii



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30 40 50 60 70 30 40 50 60 70

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80 30 40 50 60 70 80

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30 40 50 60 70 80

20 40 60 80 100 20 40 60 80 100

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Makespan

MOEADCMOEAD-C1

MOEADCMOEAD-C2

MOEADCMOEAD-C3

MOEADCMOEAD-C4

MOEADCMOEAD-C5

MOEADCMOEAD-C6

MOEADCMOEAD-C7

MOEADCMOEAD-C8

MOEADCMOEAD-C9

MOEADCMOEAD-C10

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii



20 40 60 80 100

20 40 60 80 100

20 40 60 80 100

20 40 60 80 100

20 40 60 80 100 30 40 50 60 70 80

30 40 50 60 70 30 40 50 60 7080

30 40 50 60 70 80 30 40 50 60 70 80

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30

MOEADCMOEAD-I1

MOEADCMOEAD-I2

MOEADCMOEAD-I3

MOEADCMOEAD-I4

MOEADCMOEAD-I5

MOEADCMOEAD-I6

MOEADCMOEAD-I7

MOEADCMOEAD-I8

MOEADCMOEAD-I9

MOEADCMOEAD-I10

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Makespan Makespan



Makespan

Makespan

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii



30 40 50 60 70 80 30 40 50 60 70 30 40 50 60 70801015

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20 40 60 80 100

30 40 50 60 70 80

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30 40 50 60 70 80

MOEADCMOEAD-M1

MOEADCMOEAD-M2

MOEADCMOEAD-M3

MOEADCMOEAD-M4

MOEADCMOEAD-M5

MOEADCMOEAD-M6

MOEADCMOEAD-M7

MOEADCMOEAD-M8

MOEADCMOEAD-M9

MOEADCMOEAD-M10

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Makespan Makespan

Makespan

Makespan

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii



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MOEADCMOEAD-C1

MOEADCMOEAD-C2

MOEADCMOEAD-C3

MOEADCMOEAD-C4

MOEADCMOEAD-C5

MOEADCMOEAD-C6

MOEADCMOEAD-C7

MOEADCMOEAD-C8

MOEADCMOEAD-C9

MOEADCMOEAD-C10

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Makespan Makespan

Makespan

Makespan

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii



40 50 60 70 80 40 50 60 70 80

40 50 60 70 80

90

40 50 60 70 80 90

40 50 60 70 80 90 40 50 60 70 80 90

40 50 60 70 80 90

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40 60 80 100

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Makespan

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii

MOEADCMOEAD-I1

MOEADCMOEAD-I2

MOEADCMOEAD-I3

MOEADCMOEAD-I4

MOEADCMOEAD-I5

MOEADCMOEAD-I6

MOEADCMOEAD-I7

MOEADCMOEAD-I8

MOEADCMOEAD-I9

MOEADCMOEAD-I10



40 50 60 70 80 90

40 50 60 70 80 90 40 50 60 70 80 90 40 50 60 70 80 90

40 50 60 70 80 9040 50 60 70 80 90

40 50 60 70 80 90 40 50 60 70 80

40 50 60 70 80

40 50 60 70 80

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Mea

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Mea

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Mea

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Makespan

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii

MOEADCMOEAD-M1

MOEADCMOEAD-M2

MOEADCMOEAD-M3

MOEADCMOEAD-M5

MOEADCMOEAD-M6

MOEADCMOEAD-M7

MOEADCMOEAD-M9

MOEADCMOEAD-M10

MOEADCMOEAD-M4

MOEADCMOEAD-M8



Algorithm

minus02

0

02

04

06

08

1

TTSP 20 lowast 8

C-m

easu

re

(RC

1)

(C1R

)

(RC

2)

(C2R

)

(RC

3)

(C3R

)

(RC

4)

(C4R

)

(RC

5)

(C5R

)

(RC

6)

(C6R

)

(RC

7)

(C7R

)

(RC

8)

(C8R

)

(RC

9)

(C9R

)

(RC

10)

(C10R

)




119862 (119860 119861) =


(13)





Algorithm

minus02

0

02

04

06

08

1

TTSP 20 lowast 8

(RI1)

(I1R

)

(RI2)

(I2R

)

(RI3)

(I3R

)

(RI4)

(I4R

)

(RI5)

(I5R

)

(RI6)

(I6R

)

(RI7)

(I7R

)

(RI8)

(I8R

)

(RI9)

(I9R

)

(RI10)

(I10R

)

+

+

+

C-m

easu

re


(RM

1)

(M1R

)

(RM

2)

(M2R

)

(RM

3)

(M3R

)

(RM

4)

(M4R

)

(RM

5)

(M5R

)

(RM

6)

(M6R

)

(RM

7)

(M7R

)

(RM

8)

(M8R

)

(RM

9)

(M9R

)

(RM

10)

(M10R

)

Algorithm

minus02

0

02

04

06

08

1

TTSP 20 lowast 8

+

C-m

easu

re







119877 119862 119877 119868 119877 119872 119877 119862 119877 119868 119877 119872



119877 119862 119877 119868 119877 119872 119877 119862 119877 119868 119877 119872




(119877 119862) (119862 119877) (119877 119868) (119868 119877) (119877119872) (119872 119877) (119877 119862) (119862 119877) (119877 119868) (119868 119877) (119877119872) (119872 119877)



(119877 119862) (119862 119877) (119877 119868) (119868 119877) (119877119872) (119872 119877) (119877 119862) (119862 119877) (119877 119868) (119868 119877) (119877119872) (119872 119877)





119862 119868 119872 119862 119868 119872 119862 119868 119872 119862 119868 119872


40

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25 30 35 40 45

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Makespan

TTSP 6 lowast 8

Solutions











30 40 50 60 7012

13

14

15

16

17

18

19

Makespan30 40 50 60 70

Makespan Makespan

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Mea

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Mea

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oad

20 40 60 80 100 12013

14

15

16

17

18

19

12

14

16

18

20

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22

VNMCMOEAD-C1

VNMCMOEAD-I9

VNMCMOEAD-M3


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Mea

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Mea

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40 50 60 70 80


15

20

25

30

35

40

VNMCMOEAD-C1

VNMCMOEAD-I10

VNMCMOEAD-M9








119881 119862 119881 119868 119881 119872 119881 119862 119881 119868 119881 119872

1 04067 04928 04067 04582 04067 04642 04016 04718 04016 04576 04016 041892 04018 04844 04018 04756 04018 04746 04148 04511 04148 04449 04148 045213 04552 04776 04552 04582 04552 04706 04729 04729 04729 04937 04729 042224 04258 04907 04258 04789 04258 04643 03881 04853 03881 04521 03881 042865 04283 05055 04283 04607 04283 04755 04010 04598 04010 04674 04010 046666 04492 04690 04492 04738 04492 04889 04155 04389 04155 04496 04155 043547 04560 04540 04560 04612 04560 04871 03900 04521 03900 04688 03900 045788 04306 04765 04306 04688 04306 04789 03968 04426 03968 04703 03968 047509 04460 04872 04460 04632 04460 04740 04235 04585 04235 04500 04235 0446810 04458 04897 04458 04803 04458 04767 04002 04406 04002 04465 04002 04537Average 04345 04827 04345 04679 04345 04755 04104 04574 04104 04601 04104 04457Times 1 9 0 10 0 10 0 9 0 10 1 9



(119881 119862) (119862 119881) (119881 119868) (119868 119881) (119881119872) (119872119881) (119881 119862) (119862 119881) (119881 119868) (119868 119881) (119881119872) (119872119881)

1 00000 10000 00000 08333 00000 10000 00000 10000 00000 10000 00000 090002 00000 10000 00000 10000 00000 10000 00000 10000 00000 10000 00000 100003 00000 09000 03750 05000 01667 07000 00000 10000 00000 10000 00000 100004 00000 10000 00000 10000 00000 10000 00000 10000 00000 10000 00000 100005 00000 10000 00000 09000 00000 10000 00000 10000 00000 10000 00000 100006 00000 01667 00833 01667 00000 10000 00000 08571 00000 10000 01000 085717 04444 00000 00000 02000 00000 10000 00000 10000 00000 10000 00000 100008 00000 10000 00000 10000 00000 10000 00000 10000 00000 10000 00000 100009 00000 10000 00000 10000 00000 10000 00000 10000 00000 07778 00000 0777810 00000 10000 01000 05714 00833 05714 00000 10000 00000 10000 00000 10000Average 00444 08067 00558 07171 00250 09271 00000 09857 00000 09778 00100 09535Times 1 9 0 10 0 10 0 10 0 10 0 10


7 Conclusion





Acknowledgments



References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










14

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30 35 40 45 50 30 40 50 60 70 20 40 60 80 100 120

20 40 60 80 100 30 35 40 45 50 30 40 50 60 70

30 40 50 60 70

30 35 40 45 50 30 40 50 60 70 30 35 40 45 50 55

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Mea

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Makespan Makespan

Makespan

Makespan

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii

MOEADCMOEAD-I1

MOEADCMOEAD-I2

MOEADCMOEAD-I3

MOEADCMOEAD-I5

MOEADCMOEAD-I4

MOEADCMOEAD-I6

MOEADCMOEAD-I7

MOEADCMOEAD-I9

MOEADCMOEAD-I8

MOEADCMOEAD-I10



30 40 50 60 70 30 40 50 60 70 30 35 40 45 50 55

20 40 60 80 100 30 40 50 60 70 80 30 40 50 60 70

30 35 40 45 50 55 20 30 40 50 60 70 30 40 50 60 70

30 35 40 45 50

MOEADCMOEAD-M1

MOEADCMOEAD-M2

MOEADCMOEAD-M3

MOEADCMOEAD-M4

MOEADCMOEAD-M5

MOEADCMOEAD-M6

MOEADCMOEAD-M7

MOEADCMOEAD-M8

MOEADCMOEAD-M9

MOEADCMOEAD-M10

10

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25

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Mea

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Mea

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Mea

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Makespan

Makespan

Makespan Makespan

Makespan

Makespan

Makespan Makespan

Makespan Makespan

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii



14

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25

30 40 50 60 70 80

30 40 50 60 70 30 40 50 60 70

30 40 50 60 70

80 30 40 50 60 70 80

30 40 50 60 70 80 30 40 50 60 70 80

30 40 50 60 70 80

20 40 60 80 100 20 40 60 80 100

Mea

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Mea

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Mea

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Mea

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Makespan

MOEADCMOEAD-C1

MOEADCMOEAD-C2

MOEADCMOEAD-C3

MOEADCMOEAD-C4

MOEADCMOEAD-C5

MOEADCMOEAD-C6

MOEADCMOEAD-C7

MOEADCMOEAD-C8

MOEADCMOEAD-C9

MOEADCMOEAD-C10

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii



20 40 60 80 100

20 40 60 80 100

20 40 60 80 100

20 40 60 80 100

20 40 60 80 100 30 40 50 60 70 80

30 40 50 60 70 30 40 50 60 7080

30 40 50 60 70 80 30 40 50 60 70 80

16

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10

20

25

15

20

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30

MOEADCMOEAD-I1

MOEADCMOEAD-I2

MOEADCMOEAD-I3

MOEADCMOEAD-I4

MOEADCMOEAD-I5

MOEADCMOEAD-I6

MOEADCMOEAD-I7

MOEADCMOEAD-I8

MOEADCMOEAD-I9

MOEADCMOEAD-I10

Mea

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Mea

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Mea

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Mea

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oad

Makespan Makespan



Makespan

Makespan

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii



30 40 50 60 70 80 30 40 50 60 70 30 40 50 60 70801015

20

25

15

20

25

15

20

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10

15

20

25

30

14

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20

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22

15

20

25

30

20 40 60 80 100

20 40 60 80 100

30 40 50 60 70 80

40 50 60 70 80 30 40 50 60 70 80 30 40 50 60 70 80

30 40 50 60 70 80

MOEADCMOEAD-M1

MOEADCMOEAD-M2

MOEADCMOEAD-M3

MOEADCMOEAD-M4

MOEADCMOEAD-M5

MOEADCMOEAD-M6

MOEADCMOEAD-M7

MOEADCMOEAD-M8

MOEADCMOEAD-M9

MOEADCMOEAD-M10

Mea

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Mea

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oad

Mea

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Mea

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oad



Makespan Makespan

Makespan

Makespan

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii



14

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40 50 60 70 80 40 50 60 70 8040 50 60 70 80 90

40 50 60 70 80 90 40 50 60 70 80 90 40 50 60 70 80 90

40 50 60 70 80 90 40 50 60 70 80 90

40 50 60 70 80 90

40 60 80 100

MOEADCMOEAD-C1

MOEADCMOEAD-C2

MOEADCMOEAD-C3

MOEADCMOEAD-C4

MOEADCMOEAD-C5

MOEADCMOEAD-C6

MOEADCMOEAD-C7

MOEADCMOEAD-C8

MOEADCMOEAD-C9

MOEADCMOEAD-C10

Mea

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oad

Mea

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oad

Mea

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oad

Mea

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oad



Makespan Makespan

Makespan

Makespan

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii



40 50 60 70 80 40 50 60 70 80

40 50 60 70 80

90

40 50 60 70 80 90

40 50 60 70 80 90 40 50 60 70 80 90

40 50 60 70 80 90

40 60 80 100

40 60 80 100

40 45 50 55 60 65

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Mea

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oad

Mea

n w

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oad

Mea

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oad

Mea

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oad




Makespan

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii

MOEADCMOEAD-I1

MOEADCMOEAD-I2

MOEADCMOEAD-I3

MOEADCMOEAD-I4

MOEADCMOEAD-I5

MOEADCMOEAD-I6

MOEADCMOEAD-I7

MOEADCMOEAD-I8

MOEADCMOEAD-I9

MOEADCMOEAD-I10



40 50 60 70 80 90

40 50 60 70 80 90 40 50 60 70 80 90 40 50 60 70 80 90

40 50 60 70 80 9040 50 60 70 80 90

40 50 60 70 80 90 40 50 60 70 80

40 50 60 70 80

40 50 60 70 80

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Mea

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Mea

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Mea

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oad

Mea

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oad




Makespan

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii

MOEADCMOEAD-M1

MOEADCMOEAD-M2

MOEADCMOEAD-M3

MOEADCMOEAD-M5

MOEADCMOEAD-M6

MOEADCMOEAD-M7

MOEADCMOEAD-M9

MOEADCMOEAD-M10

MOEADCMOEAD-M4

MOEADCMOEAD-M8



Algorithm

minus02

0

02

04

06

08

1

TTSP 20 lowast 8

C-m

easu

re

(RC

1)

(C1R

)

(RC

2)

(C2R

)

(RC

3)

(C3R

)

(RC

4)

(C4R

)

(RC

5)

(C5R

)

(RC

6)

(C6R

)

(RC

7)

(C7R

)

(RC

8)

(C8R

)

(RC

9)

(C9R

)

(RC

10)

(C10R

)




119862 (119860 119861) =


(13)





Algorithm

minus02

0

02

04

06

08

1

TTSP 20 lowast 8

(RI1)

(I1R

)

(RI2)

(I2R

)

(RI3)

(I3R

)

(RI4)

(I4R

)

(RI5)

(I5R

)

(RI6)

(I6R

)

(RI7)

(I7R

)

(RI8)

(I8R

)

(RI9)

(I9R

)

(RI10)

(I10R

)

+

+

+

C-m

easu

re


(RM

1)

(M1R

)

(RM

2)

(M2R

)

(RM

3)

(M3R

)

(RM

4)

(M4R

)

(RM

5)

(M5R

)

(RM

6)

(M6R

)

(RM

7)

(M7R

)

(RM

8)

(M8R

)

(RM

9)

(M9R

)

(RM

10)

(M10R

)

Algorithm

minus02

0

02

04

06

08

1

TTSP 20 lowast 8

+

C-m

easu

re







119877 119862 119877 119868 119877 119872 119877 119862 119877 119868 119877 119872



119877 119862 119877 119868 119877 119872 119877 119862 119877 119868 119877 119872




(119877 119862) (119862 119877) (119877 119868) (119868 119877) (119877119872) (119872 119877) (119877 119862) (119862 119877) (119877 119868) (119868 119877) (119877119872) (119872 119877)



(119877 119862) (119862 119877) (119877 119868) (119868 119877) (119877119872) (119872 119877) (119877 119862) (119862 119877) (119877 119868) (119868 119877) (119877119872) (119872 119877)





119862 119868 119872 119862 119868 119872 119862 119868 119872 119862 119868 119872


40

35

30

25

20

15

25 30 35 40 45

Mea

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oad

Makespan

TTSP 6 lowast 8

Solutions











30 40 50 60 7012

13

14

15

16

17

18

19

Makespan30 40 50 60 70

Makespan Makespan

Mea

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Mea

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Mea

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oad

20 40 60 80 100 12013

14

15

16

17

18

19

12

14

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18

20

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22

VNMCMOEAD-C1

VNMCMOEAD-I9

VNMCMOEAD-M3


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Mea

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40 50 60 70 80


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20

25

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35

40

VNMCMOEAD-C1

VNMCMOEAD-I10

VNMCMOEAD-M9








119881 119862 119881 119868 119881 119872 119881 119862 119881 119868 119881 119872

1 04067 04928 04067 04582 04067 04642 04016 04718 04016 04576 04016 041892 04018 04844 04018 04756 04018 04746 04148 04511 04148 04449 04148 045213 04552 04776 04552 04582 04552 04706 04729 04729 04729 04937 04729 042224 04258 04907 04258 04789 04258 04643 03881 04853 03881 04521 03881 042865 04283 05055 04283 04607 04283 04755 04010 04598 04010 04674 04010 046666 04492 04690 04492 04738 04492 04889 04155 04389 04155 04496 04155 043547 04560 04540 04560 04612 04560 04871 03900 04521 03900 04688 03900 045788 04306 04765 04306 04688 04306 04789 03968 04426 03968 04703 03968 047509 04460 04872 04460 04632 04460 04740 04235 04585 04235 04500 04235 0446810 04458 04897 04458 04803 04458 04767 04002 04406 04002 04465 04002 04537Average 04345 04827 04345 04679 04345 04755 04104 04574 04104 04601 04104 04457Times 1 9 0 10 0 10 0 9 0 10 1 9



(119881 119862) (119862 119881) (119881 119868) (119868 119881) (119881119872) (119872119881) (119881 119862) (119862 119881) (119881 119868) (119868 119881) (119881119872) (119872119881)

1 00000 10000 00000 08333 00000 10000 00000 10000 00000 10000 00000 090002 00000 10000 00000 10000 00000 10000 00000 10000 00000 10000 00000 100003 00000 09000 03750 05000 01667 07000 00000 10000 00000 10000 00000 100004 00000 10000 00000 10000 00000 10000 00000 10000 00000 10000 00000 100005 00000 10000 00000 09000 00000 10000 00000 10000 00000 10000 00000 100006 00000 01667 00833 01667 00000 10000 00000 08571 00000 10000 01000 085717 04444 00000 00000 02000 00000 10000 00000 10000 00000 10000 00000 100008 00000 10000 00000 10000 00000 10000 00000 10000 00000 10000 00000 100009 00000 10000 00000 10000 00000 10000 00000 10000 00000 07778 00000 0777810 00000 10000 01000 05714 00833 05714 00000 10000 00000 10000 00000 10000Average 00444 08067 00558 07171 00250 09271 00000 09857 00000 09778 00100 09535Times 1 9 0 10 0 10 0 10 0 10 0 10


7 Conclusion





Acknowledgments



References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










30 40 50 60 70 30 40 50 60 70 30 35 40 45 50 55

20 40 60 80 100 30 40 50 60 70 80 30 40 50 60 70

30 35 40 45 50 55 20 30 40 50 60 70 30 40 50 60 70

30 35 40 45 50

MOEADCMOEAD-M1

MOEADCMOEAD-M2

MOEADCMOEAD-M3

MOEADCMOEAD-M4

MOEADCMOEAD-M5

MOEADCMOEAD-M6

MOEADCMOEAD-M7

MOEADCMOEAD-M8

MOEADCMOEAD-M9

MOEADCMOEAD-M10

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Makespan

Makespan

Makespan Makespan

Makespan

Makespan

Makespan Makespan

Makespan Makespan

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii



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30 40 50 60 70 80

30 40 50 60 70 30 40 50 60 70

30 40 50 60 70

80 30 40 50 60 70 80

30 40 50 60 70 80 30 40 50 60 70 80

30 40 50 60 70 80

20 40 60 80 100 20 40 60 80 100

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Makespan

MOEADCMOEAD-C1

MOEADCMOEAD-C2

MOEADCMOEAD-C3

MOEADCMOEAD-C4

MOEADCMOEAD-C5

MOEADCMOEAD-C6

MOEADCMOEAD-C7

MOEADCMOEAD-C8

MOEADCMOEAD-C9

MOEADCMOEAD-C10

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii



20 40 60 80 100

20 40 60 80 100

20 40 60 80 100

20 40 60 80 100

20 40 60 80 100 30 40 50 60 70 80

30 40 50 60 70 30 40 50 60 7080

30 40 50 60 70 80 30 40 50 60 70 80

16

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15

20

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30

15

20

25

15

20

25

15

20

25

15

10

20

25

15

20

25

30

MOEADCMOEAD-I1

MOEADCMOEAD-I2

MOEADCMOEAD-I3

MOEADCMOEAD-I4

MOEADCMOEAD-I5

MOEADCMOEAD-I6

MOEADCMOEAD-I7

MOEADCMOEAD-I8

MOEADCMOEAD-I9

MOEADCMOEAD-I10

Mea

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Mea

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Mea

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Mea

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Makespan Makespan



Makespan

Makespan

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii



30 40 50 60 70 80 30 40 50 60 70 30 40 50 60 70801015

20

25

15

20

25

15

20

25

10

15

20

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30

14

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14

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14

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16

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15

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30

20 40 60 80 100

20 40 60 80 100

30 40 50 60 70 80

40 50 60 70 80 30 40 50 60 70 80 30 40 50 60 70 80

30 40 50 60 70 80

MOEADCMOEAD-M1

MOEADCMOEAD-M2

MOEADCMOEAD-M3

MOEADCMOEAD-M4

MOEADCMOEAD-M5

MOEADCMOEAD-M6

MOEADCMOEAD-M7

MOEADCMOEAD-M8

MOEADCMOEAD-M9

MOEADCMOEAD-M10

Mea

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Mea

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Mea

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Mea

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Makespan Makespan

Makespan

Makespan

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii



14

16

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40 50 60 70 80 40 50 60 70 8040 50 60 70 80 90

40 50 60 70 80 90 40 50 60 70 80 90 40 50 60 70 80 90

40 50 60 70 80 90 40 50 60 70 80 90

40 50 60 70 80 90

40 60 80 100

MOEADCMOEAD-C1

MOEADCMOEAD-C2

MOEADCMOEAD-C3

MOEADCMOEAD-C4

MOEADCMOEAD-C5

MOEADCMOEAD-C6

MOEADCMOEAD-C7

MOEADCMOEAD-C8

MOEADCMOEAD-C9

MOEADCMOEAD-C10

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Mea

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Mea

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Mea

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Makespan Makespan

Makespan

Makespan

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii



40 50 60 70 80 40 50 60 70 80

40 50 60 70 80

90

40 50 60 70 80 90

40 50 60 70 80 90 40 50 60 70 80 90

40 50 60 70 80 90

40 60 80 100

40 60 80 100

40 45 50 55 60 65

16

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Mea

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Mea

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Mea

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Makespan

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii

MOEADCMOEAD-I1

MOEADCMOEAD-I2

MOEADCMOEAD-I3

MOEADCMOEAD-I4

MOEADCMOEAD-I5

MOEADCMOEAD-I6

MOEADCMOEAD-I7

MOEADCMOEAD-I8

MOEADCMOEAD-I9

MOEADCMOEAD-I10



40 50 60 70 80 90

40 50 60 70 80 90 40 50 60 70 80 90 40 50 60 70 80 90

40 50 60 70 80 9040 50 60 70 80 90

40 50 60 70 80 90 40 50 60 70 80

40 50 60 70 80

40 50 60 70 80

15

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Mea

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Mea

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Mea

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Makespan

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii

MOEADCMOEAD-M1

MOEADCMOEAD-M2

MOEADCMOEAD-M3

MOEADCMOEAD-M5

MOEADCMOEAD-M6

MOEADCMOEAD-M7

MOEADCMOEAD-M9

MOEADCMOEAD-M10

MOEADCMOEAD-M4

MOEADCMOEAD-M8



Algorithm

minus02

0

02

04

06

08

1

TTSP 20 lowast 8

C-m

easu

re

(RC

1)

(C1R

)

(RC

2)

(C2R

)

(RC

3)

(C3R

)

(RC

4)

(C4R

)

(RC

5)

(C5R

)

(RC

6)

(C6R

)

(RC

7)

(C7R

)

(RC

8)

(C8R

)

(RC

9)

(C9R

)

(RC

10)

(C10R

)




119862 (119860 119861) =


(13)





Algorithm

minus02

0

02

04

06

08

1

TTSP 20 lowast 8

(RI1)

(I1R

)

(RI2)

(I2R

)

(RI3)

(I3R

)

(RI4)

(I4R

)

(RI5)

(I5R

)

(RI6)

(I6R

)

(RI7)

(I7R

)

(RI8)

(I8R

)

(RI9)

(I9R

)

(RI10)

(I10R

)

+

+

+

C-m

easu

re


(RM

1)

(M1R

)

(RM

2)

(M2R

)

(RM

3)

(M3R

)

(RM

4)

(M4R

)

(RM

5)

(M5R

)

(RM

6)

(M6R

)

(RM

7)

(M7R

)

(RM

8)

(M8R

)

(RM

9)

(M9R

)

(RM

10)

(M10R

)

Algorithm

minus02

0

02

04

06

08

1

TTSP 20 lowast 8

+

C-m

easu

re







119877 119862 119877 119868 119877 119872 119877 119862 119877 119868 119877 119872



119877 119862 119877 119868 119877 119872 119877 119862 119877 119868 119877 119872




(119877 119862) (119862 119877) (119877 119868) (119868 119877) (119877119872) (119872 119877) (119877 119862) (119862 119877) (119877 119868) (119868 119877) (119877119872) (119872 119877)



(119877 119862) (119862 119877) (119877 119868) (119868 119877) (119877119872) (119872 119877) (119877 119862) (119862 119877) (119877 119868) (119868 119877) (119877119872) (119872 119877)





119862 119868 119872 119862 119868 119872 119862 119868 119872 119862 119868 119872


40

35

30

25

20

15

25 30 35 40 45

Mea

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Makespan

TTSP 6 lowast 8

Solutions











30 40 50 60 7012

13

14

15

16

17

18

19

Makespan30 40 50 60 70

Makespan Makespan

Mea

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Mea

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Mea

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20 40 60 80 100 12013

14

15

16

17

18

19

12

14

16

18

20

24

22

VNMCMOEAD-C1

VNMCMOEAD-I9

VNMCMOEAD-M3


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40 50 60 70 80


15

20

25

30

35

40

VNMCMOEAD-C1

VNMCMOEAD-I10

VNMCMOEAD-M9








119881 119862 119881 119868 119881 119872 119881 119862 119881 119868 119881 119872

1 04067 04928 04067 04582 04067 04642 04016 04718 04016 04576 04016 041892 04018 04844 04018 04756 04018 04746 04148 04511 04148 04449 04148 045213 04552 04776 04552 04582 04552 04706 04729 04729 04729 04937 04729 042224 04258 04907 04258 04789 04258 04643 03881 04853 03881 04521 03881 042865 04283 05055 04283 04607 04283 04755 04010 04598 04010 04674 04010 046666 04492 04690 04492 04738 04492 04889 04155 04389 04155 04496 04155 043547 04560 04540 04560 04612 04560 04871 03900 04521 03900 04688 03900 045788 04306 04765 04306 04688 04306 04789 03968 04426 03968 04703 03968 047509 04460 04872 04460 04632 04460 04740 04235 04585 04235 04500 04235 0446810 04458 04897 04458 04803 04458 04767 04002 04406 04002 04465 04002 04537Average 04345 04827 04345 04679 04345 04755 04104 04574 04104 04601 04104 04457Times 1 9 0 10 0 10 0 9 0 10 1 9



(119881 119862) (119862 119881) (119881 119868) (119868 119881) (119881119872) (119872119881) (119881 119862) (119862 119881) (119881 119868) (119868 119881) (119881119872) (119872119881)

1 00000 10000 00000 08333 00000 10000 00000 10000 00000 10000 00000 090002 00000 10000 00000 10000 00000 10000 00000 10000 00000 10000 00000 100003 00000 09000 03750 05000 01667 07000 00000 10000 00000 10000 00000 100004 00000 10000 00000 10000 00000 10000 00000 10000 00000 10000 00000 100005 00000 10000 00000 09000 00000 10000 00000 10000 00000 10000 00000 100006 00000 01667 00833 01667 00000 10000 00000 08571 00000 10000 01000 085717 04444 00000 00000 02000 00000 10000 00000 10000 00000 10000 00000 100008 00000 10000 00000 10000 00000 10000 00000 10000 00000 10000 00000 100009 00000 10000 00000 10000 00000 10000 00000 10000 00000 07778 00000 0777810 00000 10000 01000 05714 00833 05714 00000 10000 00000 10000 00000 10000Average 00444 08067 00558 07171 00250 09271 00000 09857 00000 09778 00100 09535Times 1 9 0 10 0 10 0 10 0 10 0 10


7 Conclusion





Acknowledgments



References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










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30 40 50 60 70 80

30 40 50 60 70 30 40 50 60 70

30 40 50 60 70

80 30 40 50 60 70 80

30 40 50 60 70 80 30 40 50 60 70 80

30 40 50 60 70 80

20 40 60 80 100 20 40 60 80 100

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Makespan

MOEADCMOEAD-C1

MOEADCMOEAD-C2

MOEADCMOEAD-C3

MOEADCMOEAD-C4

MOEADCMOEAD-C5

MOEADCMOEAD-C6

MOEADCMOEAD-C7

MOEADCMOEAD-C8

MOEADCMOEAD-C9

MOEADCMOEAD-C10

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii



20 40 60 80 100

20 40 60 80 100

20 40 60 80 100

20 40 60 80 100

20 40 60 80 100 30 40 50 60 70 80

30 40 50 60 70 30 40 50 60 7080

30 40 50 60 70 80 30 40 50 60 70 80

16

18

20

22

24

16

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15

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30

15

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15

20

25

30

15

20

25

15

20

25

15

20

25

15

10

20

25

15

20

25

30

MOEADCMOEAD-I1

MOEADCMOEAD-I2

MOEADCMOEAD-I3

MOEADCMOEAD-I4

MOEADCMOEAD-I5

MOEADCMOEAD-I6

MOEADCMOEAD-I7

MOEADCMOEAD-I8

MOEADCMOEAD-I9

MOEADCMOEAD-I10

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Mea

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Mea

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Mea

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Makespan Makespan



Makespan

Makespan

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii



30 40 50 60 70 80 30 40 50 60 70 30 40 50 60 70801015

20

25

15

20

25

15

20

25

10

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20

25

30

14

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14

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20

14

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20

14

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16

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22

15

20

25

30

20 40 60 80 100

20 40 60 80 100

30 40 50 60 70 80

40 50 60 70 80 30 40 50 60 70 80 30 40 50 60 70 80

30 40 50 60 70 80

MOEADCMOEAD-M1

MOEADCMOEAD-M2

MOEADCMOEAD-M3

MOEADCMOEAD-M4

MOEADCMOEAD-M5

MOEADCMOEAD-M6

MOEADCMOEAD-M7

MOEADCMOEAD-M8

MOEADCMOEAD-M9

MOEADCMOEAD-M10

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Mea

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Mea

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Mea

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Makespan Makespan

Makespan

Makespan

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii



14

16

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24

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24

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24

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24

15

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20

25

40 50 60 70 80 40 50 60 70 8040 50 60 70 80 90

40 50 60 70 80 90 40 50 60 70 80 90 40 50 60 70 80 90

40 50 60 70 80 90 40 50 60 70 80 90

40 50 60 70 80 90

40 60 80 100

MOEADCMOEAD-C1

MOEADCMOEAD-C2

MOEADCMOEAD-C3

MOEADCMOEAD-C4

MOEADCMOEAD-C5

MOEADCMOEAD-C6

MOEADCMOEAD-C7

MOEADCMOEAD-C8

MOEADCMOEAD-C9

MOEADCMOEAD-C10

Mea

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Mea

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Mea

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Mea

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Makespan Makespan

Makespan

Makespan

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii



40 50 60 70 80 40 50 60 70 80

40 50 60 70 80

90

40 50 60 70 80 90

40 50 60 70 80 90 40 50 60 70 80 90

40 50 60 70 80 90

40 60 80 100

40 60 80 100

40 45 50 55 60 65

16

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24

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24

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24

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Mea

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Mea

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Mea

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Makespan

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii

MOEADCMOEAD-I1

MOEADCMOEAD-I2

MOEADCMOEAD-I3

MOEADCMOEAD-I4

MOEADCMOEAD-I5

MOEADCMOEAD-I6

MOEADCMOEAD-I7

MOEADCMOEAD-I8

MOEADCMOEAD-I9

MOEADCMOEAD-I10



40 50 60 70 80 90

40 50 60 70 80 90 40 50 60 70 80 90 40 50 60 70 80 90

40 50 60 70 80 9040 50 60 70 80 90

40 50 60 70 80 90 40 50 60 70 80

40 50 60 70 80

40 50 60 70 80

15

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Mea

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Mea

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Mea

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Makespan

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii

MOEADCMOEAD-M1

MOEADCMOEAD-M2

MOEADCMOEAD-M3

MOEADCMOEAD-M5

MOEADCMOEAD-M6

MOEADCMOEAD-M7

MOEADCMOEAD-M9

MOEADCMOEAD-M10

MOEADCMOEAD-M4

MOEADCMOEAD-M8



Algorithm

minus02

0

02

04

06

08

1

TTSP 20 lowast 8

C-m

easu

re

(RC

1)

(C1R

)

(RC

2)

(C2R

)

(RC

3)

(C3R

)

(RC

4)

(C4R

)

(RC

5)

(C5R

)

(RC

6)

(C6R

)

(RC

7)

(C7R

)

(RC

8)

(C8R

)

(RC

9)

(C9R

)

(RC

10)

(C10R

)




119862 (119860 119861) =


(13)





Algorithm

minus02

0

02

04

06

08

1

TTSP 20 lowast 8

(RI1)

(I1R

)

(RI2)

(I2R

)

(RI3)

(I3R

)

(RI4)

(I4R

)

(RI5)

(I5R

)

(RI6)

(I6R

)

(RI7)

(I7R

)

(RI8)

(I8R

)

(RI9)

(I9R

)

(RI10)

(I10R

)

+

+

+

C-m

easu

re


(RM

1)

(M1R

)

(RM

2)

(M2R

)

(RM

3)

(M3R

)

(RM

4)

(M4R

)

(RM

5)

(M5R

)

(RM

6)

(M6R

)

(RM

7)

(M7R

)

(RM

8)

(M8R

)

(RM

9)

(M9R

)

(RM

10)

(M10R

)

Algorithm

minus02

0

02

04

06

08

1

TTSP 20 lowast 8

+

C-m

easu

re







119877 119862 119877 119868 119877 119872 119877 119862 119877 119868 119877 119872



119877 119862 119877 119868 119877 119872 119877 119862 119877 119868 119877 119872




(119877 119862) (119862 119877) (119877 119868) (119868 119877) (119877119872) (119872 119877) (119877 119862) (119862 119877) (119877 119868) (119868 119877) (119877119872) (119872 119877)



(119877 119862) (119862 119877) (119877 119868) (119868 119877) (119877119872) (119872 119877) (119877 119862) (119862 119877) (119877 119868) (119868 119877) (119877119872) (119872 119877)





119862 119868 119872 119862 119868 119872 119862 119868 119872 119862 119868 119872


40

35

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25

20

15

25 30 35 40 45

Mea

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Makespan

TTSP 6 lowast 8

Solutions











30 40 50 60 7012

13

14

15

16

17

18

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Makespan30 40 50 60 70

Makespan Makespan

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Mea

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20 40 60 80 100 12013

14

15

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17

18

19

12

14

16

18

20

24

22

VNMCMOEAD-C1

VNMCMOEAD-I9

VNMCMOEAD-M3


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Mea

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Mea

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Mea

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40 50 60 70 80


15

20

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30

35

40

VNMCMOEAD-C1

VNMCMOEAD-I10

VNMCMOEAD-M9








119881 119862 119881 119868 119881 119872 119881 119862 119881 119868 119881 119872

1 04067 04928 04067 04582 04067 04642 04016 04718 04016 04576 04016 041892 04018 04844 04018 04756 04018 04746 04148 04511 04148 04449 04148 045213 04552 04776 04552 04582 04552 04706 04729 04729 04729 04937 04729 042224 04258 04907 04258 04789 04258 04643 03881 04853 03881 04521 03881 042865 04283 05055 04283 04607 04283 04755 04010 04598 04010 04674 04010 046666 04492 04690 04492 04738 04492 04889 04155 04389 04155 04496 04155 043547 04560 04540 04560 04612 04560 04871 03900 04521 03900 04688 03900 045788 04306 04765 04306 04688 04306 04789 03968 04426 03968 04703 03968 047509 04460 04872 04460 04632 04460 04740 04235 04585 04235 04500 04235 0446810 04458 04897 04458 04803 04458 04767 04002 04406 04002 04465 04002 04537Average 04345 04827 04345 04679 04345 04755 04104 04574 04104 04601 04104 04457Times 1 9 0 10 0 10 0 9 0 10 1 9



(119881 119862) (119862 119881) (119881 119868) (119868 119881) (119881119872) (119872119881) (119881 119862) (119862 119881) (119881 119868) (119868 119881) (119881119872) (119872119881)

1 00000 10000 00000 08333 00000 10000 00000 10000 00000 10000 00000 090002 00000 10000 00000 10000 00000 10000 00000 10000 00000 10000 00000 100003 00000 09000 03750 05000 01667 07000 00000 10000 00000 10000 00000 100004 00000 10000 00000 10000 00000 10000 00000 10000 00000 10000 00000 100005 00000 10000 00000 09000 00000 10000 00000 10000 00000 10000 00000 100006 00000 01667 00833 01667 00000 10000 00000 08571 00000 10000 01000 085717 04444 00000 00000 02000 00000 10000 00000 10000 00000 10000 00000 100008 00000 10000 00000 10000 00000 10000 00000 10000 00000 10000 00000 100009 00000 10000 00000 10000 00000 10000 00000 10000 00000 07778 00000 0777810 00000 10000 01000 05714 00833 05714 00000 10000 00000 10000 00000 10000Average 00444 08067 00558 07171 00250 09271 00000 09857 00000 09778 00100 09535Times 1 9 0 10 0 10 0 10 0 10 0 10


7 Conclusion





Acknowledgments



References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










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MOEADCMOEAD-I1

MOEADCMOEAD-I2

MOEADCMOEAD-I3

MOEADCMOEAD-I4

MOEADCMOEAD-I5

MOEADCMOEAD-I6

MOEADCMOEAD-I7

MOEADCMOEAD-I8

MOEADCMOEAD-I9

MOEADCMOEAD-I10

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Makespan Makespan



Makespan

Makespan

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii



30 40 50 60 70 80 30 40 50 60 70 30 40 50 60 70801015

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MOEADCMOEAD-M1

MOEADCMOEAD-M2

MOEADCMOEAD-M3

MOEADCMOEAD-M4

MOEADCMOEAD-M5

MOEADCMOEAD-M6

MOEADCMOEAD-M7

MOEADCMOEAD-M8

MOEADCMOEAD-M9

MOEADCMOEAD-M10

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Makespan Makespan

Makespan

Makespan

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii



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40 50 60 70 80 90 40 50 60 70 80 90

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MOEADCMOEAD-C1

MOEADCMOEAD-C2

MOEADCMOEAD-C3

MOEADCMOEAD-C4

MOEADCMOEAD-C5

MOEADCMOEAD-C6

MOEADCMOEAD-C7

MOEADCMOEAD-C8

MOEADCMOEAD-C9

MOEADCMOEAD-C10

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Makespan Makespan

Makespan

Makespan

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii



40 50 60 70 80 40 50 60 70 80

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40 50 60 70 80 90 40 50 60 70 80 90

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Makespan

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii

MOEADCMOEAD-I1

MOEADCMOEAD-I2

MOEADCMOEAD-I3

MOEADCMOEAD-I4

MOEADCMOEAD-I5

MOEADCMOEAD-I6

MOEADCMOEAD-I7

MOEADCMOEAD-I8

MOEADCMOEAD-I9

MOEADCMOEAD-I10



40 50 60 70 80 90

40 50 60 70 80 90 40 50 60 70 80 90 40 50 60 70 80 90

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Makespan

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii

MOEADCMOEAD-M1

MOEADCMOEAD-M2

MOEADCMOEAD-M3

MOEADCMOEAD-M5

MOEADCMOEAD-M6

MOEADCMOEAD-M7

MOEADCMOEAD-M9

MOEADCMOEAD-M10

MOEADCMOEAD-M4

MOEADCMOEAD-M8



Algorithm

minus02

0

02

04

06

08

1

TTSP 20 lowast 8

C-m

easu

re

(RC

1)

(C1R

)

(RC

2)

(C2R

)

(RC

3)

(C3R

)

(RC

4)

(C4R

)

(RC

5)

(C5R

)

(RC

6)

(C6R

)

(RC

7)

(C7R

)

(RC

8)

(C8R

)

(RC

9)

(C9R

)

(RC

10)

(C10R

)




119862 (119860 119861) =


(13)





Algorithm

minus02

0

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1

TTSP 20 lowast 8

(RI1)

(I1R

)

(RI2)

(I2R

)

(RI3)

(I3R

)

(RI4)

(I4R

)

(RI5)

(I5R

)

(RI6)

(I6R

)

(RI7)

(I7R

)

(RI8)

(I8R

)

(RI9)

(I9R

)

(RI10)

(I10R

)

+

+

+

C-m

easu

re


(RM

1)

(M1R

)

(RM

2)

(M2R

)

(RM

3)

(M3R

)

(RM

4)

(M4R

)

(RM

5)

(M5R

)

(RM

6)

(M6R

)

(RM

7)

(M7R

)

(RM

8)

(M8R

)

(RM

9)

(M9R

)

(RM

10)

(M10R

)

Algorithm

minus02

0

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1

TTSP 20 lowast 8

+

C-m

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119877 119862 119877 119868 119877 119872 119877 119862 119877 119868 119877 119872



119877 119862 119877 119868 119877 119872 119877 119862 119877 119868 119877 119872




(119877 119862) (119862 119877) (119877 119868) (119868 119877) (119877119872) (119872 119877) (119877 119862) (119862 119877) (119877 119868) (119868 119877) (119877119872) (119872 119877)



(119877 119862) (119862 119877) (119877 119868) (119868 119877) (119877119872) (119872 119877) (119877 119862) (119862 119877) (119877 119868) (119868 119877) (119877119872) (119872 119877)





119862 119868 119872 119862 119868 119872 119862 119868 119872 119862 119868 119872


40

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Makespan

TTSP 6 lowast 8

Solutions











30 40 50 60 7012

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Makespan30 40 50 60 70

Makespan Makespan

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20 40 60 80 100 12013

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VNMCMOEAD-C1

VNMCMOEAD-I9

VNMCMOEAD-M3


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VNMCMOEAD-C1

VNMCMOEAD-I10

VNMCMOEAD-M9








119881 119862 119881 119868 119881 119872 119881 119862 119881 119868 119881 119872

1 04067 04928 04067 04582 04067 04642 04016 04718 04016 04576 04016 041892 04018 04844 04018 04756 04018 04746 04148 04511 04148 04449 04148 045213 04552 04776 04552 04582 04552 04706 04729 04729 04729 04937 04729 042224 04258 04907 04258 04789 04258 04643 03881 04853 03881 04521 03881 042865 04283 05055 04283 04607 04283 04755 04010 04598 04010 04674 04010 046666 04492 04690 04492 04738 04492 04889 04155 04389 04155 04496 04155 043547 04560 04540 04560 04612 04560 04871 03900 04521 03900 04688 03900 045788 04306 04765 04306 04688 04306 04789 03968 04426 03968 04703 03968 047509 04460 04872 04460 04632 04460 04740 04235 04585 04235 04500 04235 0446810 04458 04897 04458 04803 04458 04767 04002 04406 04002 04465 04002 04537Average 04345 04827 04345 04679 04345 04755 04104 04574 04104 04601 04104 04457Times 1 9 0 10 0 10 0 9 0 10 1 9



(119881 119862) (119862 119881) (119881 119868) (119868 119881) (119881119872) (119872119881) (119881 119862) (119862 119881) (119881 119868) (119868 119881) (119881119872) (119872119881)

1 00000 10000 00000 08333 00000 10000 00000 10000 00000 10000 00000 090002 00000 10000 00000 10000 00000 10000 00000 10000 00000 10000 00000 100003 00000 09000 03750 05000 01667 07000 00000 10000 00000 10000 00000 100004 00000 10000 00000 10000 00000 10000 00000 10000 00000 10000 00000 100005 00000 10000 00000 09000 00000 10000 00000 10000 00000 10000 00000 100006 00000 01667 00833 01667 00000 10000 00000 08571 00000 10000 01000 085717 04444 00000 00000 02000 00000 10000 00000 10000 00000 10000 00000 100008 00000 10000 00000 10000 00000 10000 00000 10000 00000 10000 00000 100009 00000 10000 00000 10000 00000 10000 00000 10000 00000 07778 00000 0777810 00000 10000 01000 05714 00833 05714 00000 10000 00000 10000 00000 10000Average 00444 08067 00558 07171 00250 09271 00000 09857 00000 09778 00100 09535Times 1 9 0 10 0 10 0 10 0 10 0 10


7 Conclusion





Acknowledgments



References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










30 40 50 60 70 80 30 40 50 60 70 30 40 50 60 70801015

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20 40 60 80 100

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30 40 50 60 70 80

MOEADCMOEAD-M1

MOEADCMOEAD-M2

MOEADCMOEAD-M3

MOEADCMOEAD-M4

MOEADCMOEAD-M5

MOEADCMOEAD-M6

MOEADCMOEAD-M7

MOEADCMOEAD-M8

MOEADCMOEAD-M9

MOEADCMOEAD-M10

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Makespan Makespan

Makespan

Makespan

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii



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40 50 60 70 80 90 40 50 60 70 80 90 40 50 60 70 80 90

40 50 60 70 80 90 40 50 60 70 80 90

40 50 60 70 80 90

40 60 80 100

MOEADCMOEAD-C1

MOEADCMOEAD-C2

MOEADCMOEAD-C3

MOEADCMOEAD-C4

MOEADCMOEAD-C5

MOEADCMOEAD-C6

MOEADCMOEAD-C7

MOEADCMOEAD-C8

MOEADCMOEAD-C9

MOEADCMOEAD-C10

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Makespan Makespan

Makespan

Makespan

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii



40 50 60 70 80 40 50 60 70 80

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90

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40 50 60 70 80 90 40 50 60 70 80 90

40 50 60 70 80 90

40 60 80 100

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Makespan

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii

MOEADCMOEAD-I1

MOEADCMOEAD-I2

MOEADCMOEAD-I3

MOEADCMOEAD-I4

MOEADCMOEAD-I5

MOEADCMOEAD-I6

MOEADCMOEAD-I7

MOEADCMOEAD-I8

MOEADCMOEAD-I9

MOEADCMOEAD-I10



40 50 60 70 80 90

40 50 60 70 80 90 40 50 60 70 80 90 40 50 60 70 80 90

40 50 60 70 80 9040 50 60 70 80 90

40 50 60 70 80 90 40 50 60 70 80

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Makespan

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii

MOEADCMOEAD-M1

MOEADCMOEAD-M2

MOEADCMOEAD-M3

MOEADCMOEAD-M5

MOEADCMOEAD-M6

MOEADCMOEAD-M7

MOEADCMOEAD-M9

MOEADCMOEAD-M10

MOEADCMOEAD-M4

MOEADCMOEAD-M8



Algorithm

minus02

0

02

04

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08

1

TTSP 20 lowast 8

C-m

easu

re

(RC

1)

(C1R

)

(RC

2)

(C2R

)

(RC

3)

(C3R

)

(RC

4)

(C4R

)

(RC

5)

(C5R

)

(RC

6)

(C6R

)

(RC

7)

(C7R

)

(RC

8)

(C8R

)

(RC

9)

(C9R

)

(RC

10)

(C10R

)




119862 (119860 119861) =


(13)





Algorithm

minus02

0

02

04

06

08

1

TTSP 20 lowast 8

(RI1)

(I1R

)

(RI2)

(I2R

)

(RI3)

(I3R

)

(RI4)

(I4R

)

(RI5)

(I5R

)

(RI6)

(I6R

)

(RI7)

(I7R

)

(RI8)

(I8R

)

(RI9)

(I9R

)

(RI10)

(I10R

)

+

+

+

C-m

easu

re


(RM

1)

(M1R

)

(RM

2)

(M2R

)

(RM

3)

(M3R

)

(RM

4)

(M4R

)

(RM

5)

(M5R

)

(RM

6)

(M6R

)

(RM

7)

(M7R

)

(RM

8)

(M8R

)

(RM

9)

(M9R

)

(RM

10)

(M10R

)

Algorithm

minus02

0

02

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1

TTSP 20 lowast 8

+

C-m

easu

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119877 119862 119877 119868 119877 119872 119877 119862 119877 119868 119877 119872



119877 119862 119877 119868 119877 119872 119877 119862 119877 119868 119877 119872




(119877 119862) (119862 119877) (119877 119868) (119868 119877) (119877119872) (119872 119877) (119877 119862) (119862 119877) (119877 119868) (119868 119877) (119877119872) (119872 119877)



(119877 119862) (119862 119877) (119877 119868) (119868 119877) (119877119872) (119872 119877) (119877 119862) (119862 119877) (119877 119868) (119868 119877) (119877119872) (119872 119877)





119862 119868 119872 119862 119868 119872 119862 119868 119872 119862 119868 119872


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Makespan

TTSP 6 lowast 8

Solutions











30 40 50 60 7012

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20 40 60 80 100 12013

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VNMCMOEAD-C1

VNMCMOEAD-I9

VNMCMOEAD-M3


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VNMCMOEAD-C1

VNMCMOEAD-I10

VNMCMOEAD-M9








119881 119862 119881 119868 119881 119872 119881 119862 119881 119868 119881 119872

1 04067 04928 04067 04582 04067 04642 04016 04718 04016 04576 04016 041892 04018 04844 04018 04756 04018 04746 04148 04511 04148 04449 04148 045213 04552 04776 04552 04582 04552 04706 04729 04729 04729 04937 04729 042224 04258 04907 04258 04789 04258 04643 03881 04853 03881 04521 03881 042865 04283 05055 04283 04607 04283 04755 04010 04598 04010 04674 04010 046666 04492 04690 04492 04738 04492 04889 04155 04389 04155 04496 04155 043547 04560 04540 04560 04612 04560 04871 03900 04521 03900 04688 03900 045788 04306 04765 04306 04688 04306 04789 03968 04426 03968 04703 03968 047509 04460 04872 04460 04632 04460 04740 04235 04585 04235 04500 04235 0446810 04458 04897 04458 04803 04458 04767 04002 04406 04002 04465 04002 04537Average 04345 04827 04345 04679 04345 04755 04104 04574 04104 04601 04104 04457Times 1 9 0 10 0 10 0 9 0 10 1 9



(119881 119862) (119862 119881) (119881 119868) (119868 119881) (119881119872) (119872119881) (119881 119862) (119862 119881) (119881 119868) (119868 119881) (119881119872) (119872119881)

1 00000 10000 00000 08333 00000 10000 00000 10000 00000 10000 00000 090002 00000 10000 00000 10000 00000 10000 00000 10000 00000 10000 00000 100003 00000 09000 03750 05000 01667 07000 00000 10000 00000 10000 00000 100004 00000 10000 00000 10000 00000 10000 00000 10000 00000 10000 00000 100005 00000 10000 00000 09000 00000 10000 00000 10000 00000 10000 00000 100006 00000 01667 00833 01667 00000 10000 00000 08571 00000 10000 01000 085717 04444 00000 00000 02000 00000 10000 00000 10000 00000 10000 00000 100008 00000 10000 00000 10000 00000 10000 00000 10000 00000 10000 00000 100009 00000 10000 00000 10000 00000 10000 00000 10000 00000 07778 00000 0777810 00000 10000 01000 05714 00833 05714 00000 10000 00000 10000 00000 10000Average 00444 08067 00558 07171 00250 09271 00000 09857 00000 09778 00100 09535Times 1 9 0 10 0 10 0 10 0 10 0 10


7 Conclusion





Acknowledgments



References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










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40 50 60 70 80 90 40 50 60 70 80 90 40 50 60 70 80 90

40 50 60 70 80 90 40 50 60 70 80 90

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MOEADCMOEAD-C1

MOEADCMOEAD-C2

MOEADCMOEAD-C3

MOEADCMOEAD-C4

MOEADCMOEAD-C5

MOEADCMOEAD-C6

MOEADCMOEAD-C7

MOEADCMOEAD-C8

MOEADCMOEAD-C9

MOEADCMOEAD-C10

Mea

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Mea

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Mea

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Mea

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Makespan Makespan

Makespan

Makespan

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii



40 50 60 70 80 40 50 60 70 80

40 50 60 70 80

90

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40 50 60 70 80 90 40 50 60 70 80 90

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Makespan

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii

MOEADCMOEAD-I1

MOEADCMOEAD-I2

MOEADCMOEAD-I3

MOEADCMOEAD-I4

MOEADCMOEAD-I5

MOEADCMOEAD-I6

MOEADCMOEAD-I7

MOEADCMOEAD-I8

MOEADCMOEAD-I9

MOEADCMOEAD-I10



40 50 60 70 80 90

40 50 60 70 80 90 40 50 60 70 80 90 40 50 60 70 80 90

40 50 60 70 80 9040 50 60 70 80 90

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Mea

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Mea

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Makespan

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii

MOEADCMOEAD-M1

MOEADCMOEAD-M2

MOEADCMOEAD-M3

MOEADCMOEAD-M5

MOEADCMOEAD-M6

MOEADCMOEAD-M7

MOEADCMOEAD-M9

MOEADCMOEAD-M10

MOEADCMOEAD-M4

MOEADCMOEAD-M8



Algorithm

minus02

0

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TTSP 20 lowast 8

C-m

easu

re

(RC

1)

(C1R

)

(RC

2)

(C2R

)

(RC

3)

(C3R

)

(RC

4)

(C4R

)

(RC

5)

(C5R

)

(RC

6)

(C6R

)

(RC

7)

(C7R

)

(RC

8)

(C8R

)

(RC

9)

(C9R

)

(RC

10)

(C10R

)




119862 (119860 119861) =


(13)





Algorithm

minus02

0

02

04

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08

1

TTSP 20 lowast 8

(RI1)

(I1R

)

(RI2)

(I2R

)

(RI3)

(I3R

)

(RI4)

(I4R

)

(RI5)

(I5R

)

(RI6)

(I6R

)

(RI7)

(I7R

)

(RI8)

(I8R

)

(RI9)

(I9R

)

(RI10)

(I10R

)

+

+

+

C-m

easu

re


(RM

1)

(M1R

)

(RM

2)

(M2R

)

(RM

3)

(M3R

)

(RM

4)

(M4R

)

(RM

5)

(M5R

)

(RM

6)

(M6R

)

(RM

7)

(M7R

)

(RM

8)

(M8R

)

(RM

9)

(M9R

)

(RM

10)

(M10R

)

Algorithm

minus02

0

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TTSP 20 lowast 8

+

C-m

easu

re







119877 119862 119877 119868 119877 119872 119877 119862 119877 119868 119877 119872



119877 119862 119877 119868 119877 119872 119877 119862 119877 119868 119877 119872




(119877 119862) (119862 119877) (119877 119868) (119868 119877) (119877119872) (119872 119877) (119877 119862) (119862 119877) (119877 119868) (119868 119877) (119877119872) (119872 119877)



(119877 119862) (119862 119877) (119877 119868) (119868 119877) (119877119872) (119872 119877) (119877 119862) (119862 119877) (119877 119868) (119868 119877) (119877119872) (119872 119877)





119862 119868 119872 119862 119868 119872 119862 119868 119872 119862 119868 119872


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Makespan

TTSP 6 lowast 8

Solutions











30 40 50 60 7012

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Makespan30 40 50 60 70

Makespan Makespan

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20 40 60 80 100 12013

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VNMCMOEAD-C1

VNMCMOEAD-I9

VNMCMOEAD-M3


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VNMCMOEAD-C1

VNMCMOEAD-I10

VNMCMOEAD-M9








119881 119862 119881 119868 119881 119872 119881 119862 119881 119868 119881 119872

1 04067 04928 04067 04582 04067 04642 04016 04718 04016 04576 04016 041892 04018 04844 04018 04756 04018 04746 04148 04511 04148 04449 04148 045213 04552 04776 04552 04582 04552 04706 04729 04729 04729 04937 04729 042224 04258 04907 04258 04789 04258 04643 03881 04853 03881 04521 03881 042865 04283 05055 04283 04607 04283 04755 04010 04598 04010 04674 04010 046666 04492 04690 04492 04738 04492 04889 04155 04389 04155 04496 04155 043547 04560 04540 04560 04612 04560 04871 03900 04521 03900 04688 03900 045788 04306 04765 04306 04688 04306 04789 03968 04426 03968 04703 03968 047509 04460 04872 04460 04632 04460 04740 04235 04585 04235 04500 04235 0446810 04458 04897 04458 04803 04458 04767 04002 04406 04002 04465 04002 04537Average 04345 04827 04345 04679 04345 04755 04104 04574 04104 04601 04104 04457Times 1 9 0 10 0 10 0 9 0 10 1 9



(119881 119862) (119862 119881) (119881 119868) (119868 119881) (119881119872) (119872119881) (119881 119862) (119862 119881) (119881 119868) (119868 119881) (119881119872) (119872119881)

1 00000 10000 00000 08333 00000 10000 00000 10000 00000 10000 00000 090002 00000 10000 00000 10000 00000 10000 00000 10000 00000 10000 00000 100003 00000 09000 03750 05000 01667 07000 00000 10000 00000 10000 00000 100004 00000 10000 00000 10000 00000 10000 00000 10000 00000 10000 00000 100005 00000 10000 00000 09000 00000 10000 00000 10000 00000 10000 00000 100006 00000 01667 00833 01667 00000 10000 00000 08571 00000 10000 01000 085717 04444 00000 00000 02000 00000 10000 00000 10000 00000 10000 00000 100008 00000 10000 00000 10000 00000 10000 00000 10000 00000 10000 00000 100009 00000 10000 00000 10000 00000 10000 00000 10000 00000 07778 00000 0777810 00000 10000 01000 05714 00833 05714 00000 10000 00000 10000 00000 10000Average 00444 08067 00558 07171 00250 09271 00000 09857 00000 09778 00100 09535Times 1 9 0 10 0 10 0 10 0 10 0 10


7 Conclusion





Acknowledgments



References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










40 50 60 70 80 40 50 60 70 80

40 50 60 70 80

90

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40 60 80 100

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Mea

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Makespan

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii

MOEADCMOEAD-I1

MOEADCMOEAD-I2

MOEADCMOEAD-I3

MOEADCMOEAD-I4

MOEADCMOEAD-I5

MOEADCMOEAD-I6

MOEADCMOEAD-I7

MOEADCMOEAD-I8

MOEADCMOEAD-I9

MOEADCMOEAD-I10



40 50 60 70 80 90

40 50 60 70 80 90 40 50 60 70 80 90 40 50 60 70 80 90

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Makespan

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii

MOEADCMOEAD-M1

MOEADCMOEAD-M2

MOEADCMOEAD-M3

MOEADCMOEAD-M5

MOEADCMOEAD-M6

MOEADCMOEAD-M7

MOEADCMOEAD-M9

MOEADCMOEAD-M10

MOEADCMOEAD-M4

MOEADCMOEAD-M8



Algorithm

minus02

0

02

04

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08

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TTSP 20 lowast 8

C-m

easu

re

(RC

1)

(C1R

)

(RC

2)

(C2R

)

(RC

3)

(C3R

)

(RC

4)

(C4R

)

(RC

5)

(C5R

)

(RC

6)

(C6R

)

(RC

7)

(C7R

)

(RC

8)

(C8R

)

(RC

9)

(C9R

)

(RC

10)

(C10R

)




119862 (119860 119861) =


(13)





Algorithm

minus02

0

02

04

06

08

1

TTSP 20 lowast 8

(RI1)

(I1R

)

(RI2)

(I2R

)

(RI3)

(I3R

)

(RI4)

(I4R

)

(RI5)

(I5R

)

(RI6)

(I6R

)

(RI7)

(I7R

)

(RI8)

(I8R

)

(RI9)

(I9R

)

(RI10)

(I10R

)

+

+

+

C-m

easu

re


(RM

1)

(M1R

)

(RM

2)

(M2R

)

(RM

3)

(M3R

)

(RM

4)

(M4R

)

(RM

5)

(M5R

)

(RM

6)

(M6R

)

(RM

7)

(M7R

)

(RM

8)

(M8R

)

(RM

9)

(M9R

)

(RM

10)

(M10R

)

Algorithm

minus02

0

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TTSP 20 lowast 8

+

C-m

easu

re







119877 119862 119877 119868 119877 119872 119877 119862 119877 119868 119877 119872



119877 119862 119877 119868 119877 119872 119877 119862 119877 119868 119877 119872




(119877 119862) (119862 119877) (119877 119868) (119868 119877) (119877119872) (119872 119877) (119877 119862) (119862 119877) (119877 119868) (119868 119877) (119877119872) (119872 119877)



(119877 119862) (119862 119877) (119877 119868) (119868 119877) (119877119872) (119872 119877) (119877 119862) (119862 119877) (119877 119868) (119868 119877) (119877119872) (119872 119877)





119862 119868 119872 119862 119868 119872 119862 119868 119872 119862 119868 119872


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Makespan

TTSP 6 lowast 8

Solutions











30 40 50 60 7012

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Makespan30 40 50 60 70

Makespan Makespan

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20 40 60 80 100 12013

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VNMCMOEAD-C1

VNMCMOEAD-I9

VNMCMOEAD-M3


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VNMCMOEAD-C1

VNMCMOEAD-I10

VNMCMOEAD-M9








119881 119862 119881 119868 119881 119872 119881 119862 119881 119868 119881 119872

1 04067 04928 04067 04582 04067 04642 04016 04718 04016 04576 04016 041892 04018 04844 04018 04756 04018 04746 04148 04511 04148 04449 04148 045213 04552 04776 04552 04582 04552 04706 04729 04729 04729 04937 04729 042224 04258 04907 04258 04789 04258 04643 03881 04853 03881 04521 03881 042865 04283 05055 04283 04607 04283 04755 04010 04598 04010 04674 04010 046666 04492 04690 04492 04738 04492 04889 04155 04389 04155 04496 04155 043547 04560 04540 04560 04612 04560 04871 03900 04521 03900 04688 03900 045788 04306 04765 04306 04688 04306 04789 03968 04426 03968 04703 03968 047509 04460 04872 04460 04632 04460 04740 04235 04585 04235 04500 04235 0446810 04458 04897 04458 04803 04458 04767 04002 04406 04002 04465 04002 04537Average 04345 04827 04345 04679 04345 04755 04104 04574 04104 04601 04104 04457Times 1 9 0 10 0 10 0 9 0 10 1 9



(119881 119862) (119862 119881) (119881 119868) (119868 119881) (119881119872) (119872119881) (119881 119862) (119862 119881) (119881 119868) (119868 119881) (119881119872) (119872119881)

1 00000 10000 00000 08333 00000 10000 00000 10000 00000 10000 00000 090002 00000 10000 00000 10000 00000 10000 00000 10000 00000 10000 00000 100003 00000 09000 03750 05000 01667 07000 00000 10000 00000 10000 00000 100004 00000 10000 00000 10000 00000 10000 00000 10000 00000 10000 00000 100005 00000 10000 00000 09000 00000 10000 00000 10000 00000 10000 00000 100006 00000 01667 00833 01667 00000 10000 00000 08571 00000 10000 01000 085717 04444 00000 00000 02000 00000 10000 00000 10000 00000 10000 00000 100008 00000 10000 00000 10000 00000 10000 00000 10000 00000 10000 00000 100009 00000 10000 00000 10000 00000 10000 00000 10000 00000 07778 00000 0777810 00000 10000 01000 05714 00833 05714 00000 10000 00000 10000 00000 10000Average 00444 08067 00558 07171 00250 09271 00000 09857 00000 09778 00100 09535Times 1 9 0 10 0 10 0 10 0 10 0 10


7 Conclusion





Acknowledgments



References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










40 50 60 70 80 90

40 50 60 70 80 90 40 50 60 70 80 90 40 50 60 70 80 90

40 50 60 70 80 9040 50 60 70 80 90

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Mea

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Makespan

Baker Cat Circle

Cubic Gauss ICMIC


Zaslavskii

MOEADCMOEAD-M1

MOEADCMOEAD-M2

MOEADCMOEAD-M3

MOEADCMOEAD-M5

MOEADCMOEAD-M6

MOEADCMOEAD-M7

MOEADCMOEAD-M9

MOEADCMOEAD-M10

MOEADCMOEAD-M4

MOEADCMOEAD-M8



Algorithm

minus02

0

02

04

06

08

1

TTSP 20 lowast 8

C-m

easu

re

(RC

1)

(C1R

)

(RC

2)

(C2R

)

(RC

3)

(C3R

)

(RC

4)

(C4R

)

(RC

5)

(C5R

)

(RC

6)

(C6R

)

(RC

7)

(C7R

)

(RC

8)

(C8R

)

(RC

9)

(C9R

)

(RC

10)

(C10R

)




119862 (119860 119861) =


(13)





Algorithm

minus02

0

02

04

06

08

1

TTSP 20 lowast 8

(RI1)

(I1R

)

(RI2)

(I2R

)

(RI3)

(I3R

)

(RI4)

(I4R

)

(RI5)

(I5R

)

(RI6)

(I6R

)

(RI7)

(I7R

)

(RI8)

(I8R

)

(RI9)

(I9R

)

(RI10)

(I10R

)

+

+

+

C-m

easu

re


(RM

1)

(M1R

)

(RM

2)

(M2R

)

(RM

3)

(M3R

)

(RM

4)

(M4R

)

(RM

5)

(M5R

)

(RM

6)

(M6R

)

(RM

7)

(M7R

)

(RM

8)

(M8R

)

(RM

9)

(M9R

)

(RM

10)

(M10R

)

Algorithm

minus02

0

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1

TTSP 20 lowast 8

+

C-m

easu

re







119877 119862 119877 119868 119877 119872 119877 119862 119877 119868 119877 119872



119877 119862 119877 119868 119877 119872 119877 119862 119877 119868 119877 119872




(119877 119862) (119862 119877) (119877 119868) (119868 119877) (119877119872) (119872 119877) (119877 119862) (119862 119877) (119877 119868) (119868 119877) (119877119872) (119872 119877)



(119877 119862) (119862 119877) (119877 119868) (119868 119877) (119877119872) (119872 119877) (119877 119862) (119862 119877) (119877 119868) (119868 119877) (119877119872) (119872 119877)





119862 119868 119872 119862 119868 119872 119862 119868 119872 119862 119868 119872


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Makespan

TTSP 6 lowast 8

Solutions











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Makespan30 40 50 60 70

Makespan Makespan

Mea

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20 40 60 80 100 12013

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VNMCMOEAD-C1

VNMCMOEAD-I9

VNMCMOEAD-M3


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VNMCMOEAD-C1

VNMCMOEAD-I10

VNMCMOEAD-M9








119881 119862 119881 119868 119881 119872 119881 119862 119881 119868 119881 119872

1 04067 04928 04067 04582 04067 04642 04016 04718 04016 04576 04016 041892 04018 04844 04018 04756 04018 04746 04148 04511 04148 04449 04148 045213 04552 04776 04552 04582 04552 04706 04729 04729 04729 04937 04729 042224 04258 04907 04258 04789 04258 04643 03881 04853 03881 04521 03881 042865 04283 05055 04283 04607 04283 04755 04010 04598 04010 04674 04010 046666 04492 04690 04492 04738 04492 04889 04155 04389 04155 04496 04155 043547 04560 04540 04560 04612 04560 04871 03900 04521 03900 04688 03900 045788 04306 04765 04306 04688 04306 04789 03968 04426 03968 04703 03968 047509 04460 04872 04460 04632 04460 04740 04235 04585 04235 04500 04235 0446810 04458 04897 04458 04803 04458 04767 04002 04406 04002 04465 04002 04537Average 04345 04827 04345 04679 04345 04755 04104 04574 04104 04601 04104 04457Times 1 9 0 10 0 10 0 9 0 10 1 9



(119881 119862) (119862 119881) (119881 119868) (119868 119881) (119881119872) (119872119881) (119881 119862) (119862 119881) (119881 119868) (119868 119881) (119881119872) (119872119881)

1 00000 10000 00000 08333 00000 10000 00000 10000 00000 10000 00000 090002 00000 10000 00000 10000 00000 10000 00000 10000 00000 10000 00000 100003 00000 09000 03750 05000 01667 07000 00000 10000 00000 10000 00000 100004 00000 10000 00000 10000 00000 10000 00000 10000 00000 10000 00000 100005 00000 10000 00000 09000 00000 10000 00000 10000 00000 10000 00000 100006 00000 01667 00833 01667 00000 10000 00000 08571 00000 10000 01000 085717 04444 00000 00000 02000 00000 10000 00000 10000 00000 10000 00000 100008 00000 10000 00000 10000 00000 10000 00000 10000 00000 10000 00000 100009 00000 10000 00000 10000 00000 10000 00000 10000 00000 07778 00000 0777810 00000 10000 01000 05714 00833 05714 00000 10000 00000 10000 00000 10000Average 00444 08067 00558 07171 00250 09271 00000 09857 00000 09778 00100 09535Times 1 9 0 10 0 10 0 10 0 10 0 10


7 Conclusion





Acknowledgments



References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Algorithm

minus02

0

02

04

06

08

1

TTSP 20 lowast 8

C-m

easu

re

(RC

1)

(C1R

)

(RC

2)

(C2R

)

(RC

3)

(C3R

)

(RC

4)

(C4R

)

(RC

5)

(C5R

)

(RC

6)

(C6R

)

(RC

7)

(C7R

)

(RC

8)

(C8R

)

(RC

9)

(C9R

)

(RC

10)

(C10R

)




119862 (119860 119861) =


(13)





Algorithm

minus02

0

02

04

06

08

1

TTSP 20 lowast 8

(RI1)

(I1R

)

(RI2)

(I2R

)

(RI3)

(I3R

)

(RI4)

(I4R

)

(RI5)

(I5R

)

(RI6)

(I6R

)

(RI7)

(I7R

)

(RI8)

(I8R

)

(RI9)

(I9R

)

(RI10)

(I10R

)

+

+

+

C-m

easu

re


(RM

1)

(M1R

)

(RM

2)

(M2R

)

(RM

3)

(M3R

)

(RM

4)

(M4R

)

(RM

5)

(M5R

)

(RM

6)

(M6R

)

(RM

7)

(M7R

)

(RM

8)

(M8R

)

(RM

9)

(M9R

)

(RM

10)

(M10R

)

Algorithm

minus02

0

02

04

06

08

1

TTSP 20 lowast 8

+

C-m

easu

re







119877 119862 119877 119868 119877 119872 119877 119862 119877 119868 119877 119872



119877 119862 119877 119868 119877 119872 119877 119862 119877 119868 119877 119872




(119877 119862) (119862 119877) (119877 119868) (119868 119877) (119877119872) (119872 119877) (119877 119862) (119862 119877) (119877 119868) (119868 119877) (119877119872) (119872 119877)



(119877 119862) (119862 119877) (119877 119868) (119868 119877) (119877119872) (119872 119877) (119877 119862) (119862 119877) (119877 119868) (119868 119877) (119877119872) (119872 119877)





119862 119868 119872 119862 119868 119872 119862 119868 119872 119862 119868 119872


40

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Makespan

TTSP 6 lowast 8

Solutions











30 40 50 60 7012

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Makespan30 40 50 60 70

Makespan Makespan

Mea

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Mea

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20 40 60 80 100 12013

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VNMCMOEAD-C1

VNMCMOEAD-I9

VNMCMOEAD-M3


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VNMCMOEAD-C1

VNMCMOEAD-I10

VNMCMOEAD-M9








119881 119862 119881 119868 119881 119872 119881 119862 119881 119868 119881 119872

1 04067 04928 04067 04582 04067 04642 04016 04718 04016 04576 04016 041892 04018 04844 04018 04756 04018 04746 04148 04511 04148 04449 04148 045213 04552 04776 04552 04582 04552 04706 04729 04729 04729 04937 04729 042224 04258 04907 04258 04789 04258 04643 03881 04853 03881 04521 03881 042865 04283 05055 04283 04607 04283 04755 04010 04598 04010 04674 04010 046666 04492 04690 04492 04738 04492 04889 04155 04389 04155 04496 04155 043547 04560 04540 04560 04612 04560 04871 03900 04521 03900 04688 03900 045788 04306 04765 04306 04688 04306 04789 03968 04426 03968 04703 03968 047509 04460 04872 04460 04632 04460 04740 04235 04585 04235 04500 04235 0446810 04458 04897 04458 04803 04458 04767 04002 04406 04002 04465 04002 04537Average 04345 04827 04345 04679 04345 04755 04104 04574 04104 04601 04104 04457Times 1 9 0 10 0 10 0 9 0 10 1 9



(119881 119862) (119862 119881) (119881 119868) (119868 119881) (119881119872) (119872119881) (119881 119862) (119862 119881) (119881 119868) (119868 119881) (119881119872) (119872119881)

1 00000 10000 00000 08333 00000 10000 00000 10000 00000 10000 00000 090002 00000 10000 00000 10000 00000 10000 00000 10000 00000 10000 00000 100003 00000 09000 03750 05000 01667 07000 00000 10000 00000 10000 00000 100004 00000 10000 00000 10000 00000 10000 00000 10000 00000 10000 00000 100005 00000 10000 00000 09000 00000 10000 00000 10000 00000 10000 00000 100006 00000 01667 00833 01667 00000 10000 00000 08571 00000 10000 01000 085717 04444 00000 00000 02000 00000 10000 00000 10000 00000 10000 00000 100008 00000 10000 00000 10000 00000 10000 00000 10000 00000 10000 00000 100009 00000 10000 00000 10000 00000 10000 00000 10000 00000 07778 00000 0777810 00000 10000 01000 05714 00833 05714 00000 10000 00000 10000 00000 10000Average 00444 08067 00558 07171 00250 09271 00000 09857 00000 09778 00100 09535Times 1 9 0 10 0 10 0 10 0 10 0 10


7 Conclusion





Acknowledgments



References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119877 119862 119877 119868 119877 119872 119877 119862 119877 119868 119877 119872



119877 119862 119877 119868 119877 119872 119877 119862 119877 119868 119877 119872




(119877 119862) (119862 119877) (119877 119868) (119868 119877) (119877119872) (119872 119877) (119877 119862) (119862 119877) (119877 119868) (119868 119877) (119877119872) (119872 119877)



(119877 119862) (119862 119877) (119877 119868) (119868 119877) (119877119872) (119872 119877) (119877 119862) (119862 119877) (119877 119868) (119868 119877) (119877119872) (119872 119877)





119862 119868 119872 119862 119868 119872 119862 119868 119872 119862 119868 119872


40

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Makespan

TTSP 6 lowast 8

Solutions











30 40 50 60 7012

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Makespan30 40 50 60 70

Makespan Makespan

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VNMCMOEAD-C1

VNMCMOEAD-I9

VNMCMOEAD-M3


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VNMCMOEAD-C1

VNMCMOEAD-I10

VNMCMOEAD-M9








119881 119862 119881 119868 119881 119872 119881 119862 119881 119868 119881 119872

1 04067 04928 04067 04582 04067 04642 04016 04718 04016 04576 04016 041892 04018 04844 04018 04756 04018 04746 04148 04511 04148 04449 04148 045213 04552 04776 04552 04582 04552 04706 04729 04729 04729 04937 04729 042224 04258 04907 04258 04789 04258 04643 03881 04853 03881 04521 03881 042865 04283 05055 04283 04607 04283 04755 04010 04598 04010 04674 04010 046666 04492 04690 04492 04738 04492 04889 04155 04389 04155 04496 04155 043547 04560 04540 04560 04612 04560 04871 03900 04521 03900 04688 03900 045788 04306 04765 04306 04688 04306 04789 03968 04426 03968 04703 03968 047509 04460 04872 04460 04632 04460 04740 04235 04585 04235 04500 04235 0446810 04458 04897 04458 04803 04458 04767 04002 04406 04002 04465 04002 04537Average 04345 04827 04345 04679 04345 04755 04104 04574 04104 04601 04104 04457Times 1 9 0 10 0 10 0 9 0 10 1 9



(119881 119862) (119862 119881) (119881 119868) (119868 119881) (119881119872) (119872119881) (119881 119862) (119862 119881) (119881 119868) (119868 119881) (119881119872) (119872119881)

1 00000 10000 00000 08333 00000 10000 00000 10000 00000 10000 00000 090002 00000 10000 00000 10000 00000 10000 00000 10000 00000 10000 00000 100003 00000 09000 03750 05000 01667 07000 00000 10000 00000 10000 00000 100004 00000 10000 00000 10000 00000 10000 00000 10000 00000 10000 00000 100005 00000 10000 00000 09000 00000 10000 00000 10000 00000 10000 00000 100006 00000 01667 00833 01667 00000 10000 00000 08571 00000 10000 01000 085717 04444 00000 00000 02000 00000 10000 00000 10000 00000 10000 00000 100008 00000 10000 00000 10000 00000 10000 00000 10000 00000 10000 00000 100009 00000 10000 00000 10000 00000 10000 00000 10000 00000 07778 00000 0777810 00000 10000 01000 05714 00833 05714 00000 10000 00000 10000 00000 10000Average 00444 08067 00558 07171 00250 09271 00000 09857 00000 09778 00100 09535Times 1 9 0 10 0 10 0 10 0 10 0 10


7 Conclusion





Acknowledgments



References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119862 119868 119872 119862 119868 119872 119862 119868 119872 119862 119868 119872


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Makespan

TTSP 6 lowast 8

Solutions











30 40 50 60 7012

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Makespan30 40 50 60 70

Makespan Makespan

Mea

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Mea

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Mea

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20 40 60 80 100 12013

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VNMCMOEAD-C1

VNMCMOEAD-I9

VNMCMOEAD-M3


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VNMCMOEAD-C1

VNMCMOEAD-I10

VNMCMOEAD-M9








119881 119862 119881 119868 119881 119872 119881 119862 119881 119868 119881 119872

1 04067 04928 04067 04582 04067 04642 04016 04718 04016 04576 04016 041892 04018 04844 04018 04756 04018 04746 04148 04511 04148 04449 04148 045213 04552 04776 04552 04582 04552 04706 04729 04729 04729 04937 04729 042224 04258 04907 04258 04789 04258 04643 03881 04853 03881 04521 03881 042865 04283 05055 04283 04607 04283 04755 04010 04598 04010 04674 04010 046666 04492 04690 04492 04738 04492 04889 04155 04389 04155 04496 04155 043547 04560 04540 04560 04612 04560 04871 03900 04521 03900 04688 03900 045788 04306 04765 04306 04688 04306 04789 03968 04426 03968 04703 03968 047509 04460 04872 04460 04632 04460 04740 04235 04585 04235 04500 04235 0446810 04458 04897 04458 04803 04458 04767 04002 04406 04002 04465 04002 04537Average 04345 04827 04345 04679 04345 04755 04104 04574 04104 04601 04104 04457Times 1 9 0 10 0 10 0 9 0 10 1 9



(119881 119862) (119862 119881) (119881 119868) (119868 119881) (119881119872) (119872119881) (119881 119862) (119862 119881) (119881 119868) (119868 119881) (119881119872) (119872119881)

1 00000 10000 00000 08333 00000 10000 00000 10000 00000 10000 00000 090002 00000 10000 00000 10000 00000 10000 00000 10000 00000 10000 00000 100003 00000 09000 03750 05000 01667 07000 00000 10000 00000 10000 00000 100004 00000 10000 00000 10000 00000 10000 00000 10000 00000 10000 00000 100005 00000 10000 00000 09000 00000 10000 00000 10000 00000 10000 00000 100006 00000 01667 00833 01667 00000 10000 00000 08571 00000 10000 01000 085717 04444 00000 00000 02000 00000 10000 00000 10000 00000 10000 00000 100008 00000 10000 00000 10000 00000 10000 00000 10000 00000 10000 00000 100009 00000 10000 00000 10000 00000 10000 00000 10000 00000 07778 00000 0777810 00000 10000 01000 05714 00833 05714 00000 10000 00000 10000 00000 10000Average 00444 08067 00558 07171 00250 09271 00000 09857 00000 09778 00100 09535Times 1 9 0 10 0 10 0 10 0 10 0 10


7 Conclusion





Acknowledgments



References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










30 40 50 60 7012

13

14

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Makespan30 40 50 60 70

Makespan Makespan

Mea

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Mea

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Mea

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20 40 60 80 100 12013

14

15

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12

14

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20

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VNMCMOEAD-C1

VNMCMOEAD-I9

VNMCMOEAD-M3


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40 50 60 70 80


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VNMCMOEAD-C1

VNMCMOEAD-I10

VNMCMOEAD-M9








119881 119862 119881 119868 119881 119872 119881 119862 119881 119868 119881 119872

1 04067 04928 04067 04582 04067 04642 04016 04718 04016 04576 04016 041892 04018 04844 04018 04756 04018 04746 04148 04511 04148 04449 04148 045213 04552 04776 04552 04582 04552 04706 04729 04729 04729 04937 04729 042224 04258 04907 04258 04789 04258 04643 03881 04853 03881 04521 03881 042865 04283 05055 04283 04607 04283 04755 04010 04598 04010 04674 04010 046666 04492 04690 04492 04738 04492 04889 04155 04389 04155 04496 04155 043547 04560 04540 04560 04612 04560 04871 03900 04521 03900 04688 03900 045788 04306 04765 04306 04688 04306 04789 03968 04426 03968 04703 03968 047509 04460 04872 04460 04632 04460 04740 04235 04585 04235 04500 04235 0446810 04458 04897 04458 04803 04458 04767 04002 04406 04002 04465 04002 04537Average 04345 04827 04345 04679 04345 04755 04104 04574 04104 04601 04104 04457Times 1 9 0 10 0 10 0 9 0 10 1 9



(119881 119862) (119862 119881) (119881 119868) (119868 119881) (119881119872) (119872119881) (119881 119862) (119862 119881) (119881 119868) (119868 119881) (119881119872) (119872119881)

1 00000 10000 00000 08333 00000 10000 00000 10000 00000 10000 00000 090002 00000 10000 00000 10000 00000 10000 00000 10000 00000 10000 00000 100003 00000 09000 03750 05000 01667 07000 00000 10000 00000 10000 00000 100004 00000 10000 00000 10000 00000 10000 00000 10000 00000 10000 00000 100005 00000 10000 00000 09000 00000 10000 00000 10000 00000 10000 00000 100006 00000 01667 00833 01667 00000 10000 00000 08571 00000 10000 01000 085717 04444 00000 00000 02000 00000 10000 00000 10000 00000 10000 00000 100008 00000 10000 00000 10000 00000 10000 00000 10000 00000 10000 00000 100009 00000 10000 00000 10000 00000 10000 00000 10000 00000 07778 00000 0777810 00000 10000 01000 05714 00833 05714 00000 10000 00000 10000 00000 10000Average 00444 08067 00558 07171 00250 09271 00000 09857 00000 09778 00100 09535Times 1 9 0 10 0 10 0 10 0 10 0 10


7 Conclusion





Acknowledgments



References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119881 119862 119881 119868 119881 119872 119881 119862 119881 119868 119881 119872

1 04067 04928 04067 04582 04067 04642 04016 04718 04016 04576 04016 041892 04018 04844 04018 04756 04018 04746 04148 04511 04148 04449 04148 045213 04552 04776 04552 04582 04552 04706 04729 04729 04729 04937 04729 042224 04258 04907 04258 04789 04258 04643 03881 04853 03881 04521 03881 042865 04283 05055 04283 04607 04283 04755 04010 04598 04010 04674 04010 046666 04492 04690 04492 04738 04492 04889 04155 04389 04155 04496 04155 043547 04560 04540 04560 04612 04560 04871 03900 04521 03900 04688 03900 045788 04306 04765 04306 04688 04306 04789 03968 04426 03968 04703 03968 047509 04460 04872 04460 04632 04460 04740 04235 04585 04235 04500 04235 0446810 04458 04897 04458 04803 04458 04767 04002 04406 04002 04465 04002 04537Average 04345 04827 04345 04679 04345 04755 04104 04574 04104 04601 04104 04457Times 1 9 0 10 0 10 0 9 0 10 1 9



(119881 119862) (119862 119881) (119881 119868) (119868 119881) (119881119872) (119872119881) (119881 119862) (119862 119881) (119881 119868) (119868 119881) (119881119872) (119872119881)

1 00000 10000 00000 08333 00000 10000 00000 10000 00000 10000 00000 090002 00000 10000 00000 10000 00000 10000 00000 10000 00000 10000 00000 100003 00000 09000 03750 05000 01667 07000 00000 10000 00000 10000 00000 100004 00000 10000 00000 10000 00000 10000 00000 10000 00000 10000 00000 100005 00000 10000 00000 09000 00000 10000 00000 10000 00000 10000 00000 100006 00000 01667 00833 01667 00000 10000 00000 08571 00000 10000 01000 085717 04444 00000 00000 02000 00000 10000 00000 10000 00000 10000 00000 100008 00000 10000 00000 10000 00000 10000 00000 10000 00000 10000 00000 100009 00000 10000 00000 10000 00000 10000 00000 10000 00000 07778 00000 0777810 00000 10000 01000 05714 00833 05714 00000 10000 00000 10000 00000 10000Average 00444 08067 00558 07171 00250 09271 00000 09857 00000 09778 00100 09535Times 1 9 0 10 0 10 0 10 0 10 0 10


7 Conclusion





Acknowledgments



References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article chaotic multiobjective evolutionary...

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