research article a variable neighborhood moea/d...

Research ArticleA Variable Neighborhood MOEAD for Multiobjective Test TaskScheduling Problem

Hui Lu Zheng Zhu Xiaoteng Wang and Lijuan Yin

School of Electronic and Information Engineering Beihang University Beijing 100191 China

Correspondence should be addressed to Hui Lu mluhuibuaaeducn

Received 26 October 2013 Revised 2 January 2014 Accepted 19 February 2014 Published 1 April 2014

Academic Editor Kui Fu Chen

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 typical combinational optimization scheduling problem This paper proposes a variableneighborhood MOEAD (VNM) to solve the multiobjective TTSP Two minimization objectives the maximal completion time(makespan) and the mean workload are considered together In order to make solutions obtained more close to the real ParetoFront variable neighborhood strategy is adopted Variable neighborhood approach is proposed to render the crossover spanreasonable Additionally because the search space of the TTSP is so large that many duplicate solutions and local optima willexist the Starting Mutation is applied to prevent solutions from becoming trapped in local optima It is proved that the solutionsgot by VNM can converge to the global optimum by using Markov Chain and Transition Matrix respectively The experimentsof comparisons of VNM MOEAD and CNSGA (chaotic nondominated sorting genetic algorithm) indicate that VNM performsbetter than theMOEAD and theCNSGA in solving the TTSPThe results demonstrate that proposed algorithmVNM is an efficientapproach to solve the multiobjective TTSP

1 Introduction

During recent decades the manufacturing of electronicdevices has become highly integrated and increasinglycomplex As a result the resource and time consumptionexpended on the test of electronic devices became a crucialproblem in engineering application Therefore the researchfor improving the test efficiency is a topic that has attractedextensive attention To address this situation the objectiveof this research is to solve the test task scheduling problem(TTSP) more efficiently

The goal of the TTSP is to arrange the execution of 119899tasks on 119898 instruments It is a difficult nondeterministicpolynomial (NP) problem [1] for optimization TTSP hassome similarities with flexible job shop scheduling problem(FJSP) [2 3] but the resource configuration of the TTSPis more flexible For example in the TTSP one task canbe performed on more than one instrument at a time Theprecedence relationships in the TTSP resemble a networkOne task can have one or more former or latter tasks inthe TTSP Generally speaking feasible solutions are moredifficult to be obtained in the TTSP than that in the FJSP

TTSP FJSP and most scheduling problems belong tocombinational optimization problems For combinationaloptimization problems the search space is too large that thebest solution cannot be obtained by adopting the methodof enumeration for even small-scale problem Therefore theintelligent algorithms based on integer programming modelare devoted to solving these kinds of problems Take FJSPas the example genetic algorithm (GA) [4ndash7] simulatedannealing (SA) [8ndash10] and the tabu search (TS) [11] havebeen successfully applied in solving scheduling optimizationproblem FJSP receives extensive attention and researchesand many hybrid intelligent algorithms are invented forimproving the performance of the solutions For examplea combination of shuffled frog leaping and fuzzy logic isproposed to solve FJSP [12] A particle swarm optimization(PSO) algorithm and TS algorithm are combined to solve themultiobjective FJSP [13] A biogeography-based optimization(BBO) algorithm [14] was proposed for FJSP for findingoptimum or near-optimum solution Hybrid discrete parti-cle swarm optimization for multiobjective flexible job-shopscheduling problemwas proposed in article [15] especially forlarge-scale problems The objective functions are different in

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 423621 14 pageshttpdxdoiorg1011552014423621

2 Mathematical Problems in Engineering

each literature but themakespan the total tardiness the crit-ical machine workload and the total workload of machinesare frequently considered factors in those researches

Different from the research of FJSP the research of TTSPis relatively few because of the development of automatic testsystem However there are still some achievements in TTSPXia et al [16] proposed a method that combined GA andsimulated SA to optimize the parallel efficiency and speedup ratio of the multi-UUT parallel test Genetic Algorithm-AntColonyAlgorithm (GA-ACA) [17] hybrid particle swarmand tabu search [18] and Ant Colony Algorithm [19] are usedto solve parallel test tasks scheduling to obtain the mini-mum makespan A chaotic nondominated sorting geneticalgorithm is proposed to solve multiobjective TTSP [20]Thechaotic operations are combined with NSGA-II [21] in thisapproachThose algorithms have shown excellent property indecreasing costs and improving efficiencies in automatic testsystem

There are also some intelligent algorithms used to solvepower dispatch problems and other scheduling problemsFor example opposition-based learning is employed inopposition-based gravitational search algorithm (OGSA)to solve optimal reactive power dispatch [22] A fuzzifiedmultiobjective PSO (FMOPSO) algorithm is proposed andimplemented to dispatch the electric power [23] An inter-active artificial bee colony algorithm was proposed for themultiobjective environmentaleconomic dispatch problem[24]

In summary most of the researches of scheduling prob-lems focus on the single-objective problem or adopt weightedsum approach to convert the multiobjective problem into asingle-objective problemHowever theweighting coefficientsare difficult to choose and human factors will greatly impactthe performance of the algorithms In fact there are anothertwo kinds of methods for solving themultiobjective problemOne method is the non-Pareto approach utilizing operatorsfor processing the different objectives in a separated wayAnother is the Pareto approaches which are directly basedon the Pareto optimality concept They aim at satisfying twogoals converging towards the Pareto front and also obtainingdiversified solutions scattered all over the Pareto frontThosetwo kinds of methods mainly rely on the performanceand strategies of the algorithms used in the multiobjectiveproblems

In this paper the method based on Tchebycheff decom-position for multiobjective functions was adopted and thealgorithm named MOEAD is used to solve TTSP MOEADis a typical evolutionary algorithm based on decompositionproposed by Zhang and Li [25] This method decomposes amultiobjective optimization problem into a number of scalaroptimization subproblems and optimizes them simultane-ously The results show that MOEAD has a good perfor-mance for the ZDT and DTLZ test problems MOEAD isvery efficient in solving multiobjective problems Researchon MOEAD has also been performed in recent yearsFor example Tan et al [26] proposed a new version ofMOEAD with a uniform design to deal with the multiob-jective problem in higher-dimensional objective spaces This

method can render the distribution of the weighting vectorsmore uniform especially for problems with high dimensionChen et al [27] introduced a guided mutation operator andpriority update to enhance the ability ofMOEAD Stochasticranking and constraint-domination principle are adopted inMOEAD to improve the ability of the algorithm to dealwith constrained multiobjective optimization problems [28]Although these studies have improved the ability ofMOEADfor solving multi-objective problems MOEAD is mainlyused to solve standard test cases like ZDT DTLZ and F1However MOEAD is rarely used to solve combinationaloptimization problems such as FJSP TTSP Peng et al appliedMOEAD to solve Travelling Salesman Problem (TSP) [29]However there is no special improvement for MOEADaccording to the feature ofMOEAD and the property of TSP

The scheduling problems such as TTSP FJSP and TSPand power dispatching problem are a branch of combina-tional optimization problems Because of the properties of thecombinational optimization problems the final best solutionsonly account for a rather small subset of the search spaceHow to avoid the solutions obtained being trapped in localoptima is the key to improve the ability of algorithms todeal with combinational optimization problems Consideringthe fact that the size of the neighborhood is important inMOEAD [25] too large size will lead to degradation andtoo small size will weaken the effect of evolutionary processMoreover there will be many duplicate solutions due to theinfluence of neighborhood updating of MOEAD [25] Thepopulation diversity will decrease obviously Based on theanalyses above variable neighborhood based on a quadraticcurve is adopted to ensure that the crossover span is morereasonable and Gauss mutation is adopted at the beginningof iteration to maintain the diversity of the populationThese two improvements can efficiently enhance the abilityof MOEAD for avoiding the solutions obtained from beingtrapped in local optima The proposed approach cannot onlysolve TTSP but also deal with other scheduling problemsbecause the feasible solutions of TTSP are more difficult toobtain than most scheduling problems such as FJSP TSP

The organization of this paper is as follows A briefintroduction of TTSP is introduced in Section 2 The newmethod for TTSP variable neighborhood MOEAD (VNM)is proposed in Section 3The convergence analysis of VNM isalso presented in Section 4 A large number of experimentalresults and discussions are covered in Section 5 Conclusionsare given in Section 6

2 The Formulation of TTSP

21 The Mathematical Model for TTSP The goal of the TTSPis to arrange the execution of 119899 tasks on119898 instrumentsThereare three main mathematical models for TTSP One model isbased on Petri netThe second is based onGraph theory Andthe third model is based on integer programming Our workin this paper is mainly based on the integer programmingproposed by us in paper [20]

Mathematical Problems in Engineering 3

P1

P4

P2 P3

r1 r2 r3

t1 [r1 r2 r3]

(1 2 3)

12 3

Figure 1 The Petri net model for one task TTSP

211 The Petri Net Model for TTSP Petri net [30 31] wasproposed in 1962 Petri net focuses on the changes of thesystem the conditions for changes the influence of changesand the relationships between changes We assume that thereis one test task 119905

1in TTSP The instruments occupied for 119905

1

are 1199031 1199032 and 119903

3 The Petri net model for this TTSP can be

shown as Figure 1 In this model there are four places (1199011 1199012

1199013 and 119901

4) one transition (119905

1) three tokens (119903

1 1199032 and 119903

3)

three variables (V1 V2 and V

3) four arc expressions (V

1 V2 V3

and (V1 V2 V3)) and a guard ([119903

1 1199032 1199033]) where V

1 V2 and V

3

are bound to 1199031 1199032 and 119903

3

In Figure 1 at the beginning test resources 1199031 1199032 and 119903

3

are vacant The corresponding tokens for three places 1199011 1199012

and 1199013are 1199031 1199032 and 119903

3 respectively Therefore 119903

1 1199032 and

1199033can be allocated to 119905

1 When the 119905

1is finished the tokens

in 1199011 1199012 and 119901

3will be transferred to place 119901

4 The tokens

in 1199014are V1 V2 and V

3 This means that resources 119903

1 1199032 and

1199033are released The Petri net can describe the relationships

between tasks by the places and transitions but the complexmodels are needed to be establishedThe process will increasethe development cost and extend the development cycle

212TheGraphTheoryModel for TTSP Graph theory [32] isan important branch of mathematics By adopting the Graphtheory the complex project planning and processing can bedescribed using ldquographsrdquo In TTSP the vertexes of the graphrepresent the test tasks and the lines between vertexes meanthat some test instruments are common for these two tasksFor example there are four test tasks (119905

1 1199052 1199053 and 119905

4) and

four test instruments (1199031 1199032 1199033 and 119903

4) The instruments set

needed by 1199051 1199052 1199053 1199054are 1199031 1199032 1199032 1199034 1199033 1199034 and 119903

1 1199033

respectively The graph for this TTSP example is shown inFigure 2

Graph theory model can only be adopted by typicaloptimization methods With the increment of the scale ofTTSP the computation expense will greatly increase but

t1 t2

t3 t4

Figure 2 The Graph model for TTSP

typical optimization methods are not suitable for large-scaleTTSP problem Therefore Graph theory model cannot solvelarge-scale TTSP also

213 The Integer Programming Model for TTSP TTSP isa typical integer programming problem For the integerprogramming model for TTSP the TTSP can be describedas follows [20] assume that 119899 tasks and 119898 instrument areincluded in TTSP There is a task set 119879 = 119905

1 1199052 119905

119895 (1 le

119895 le 119899) and an instrument set 119877 = 1199031 1199032 119903

119894 (1 le 119894 le 119898)

119878119894

119895 119862119894119895 and 119875

119894

119895represent the test start time test finish time

and test consumed time of task 119905119895tested on instrument 119903

119894

respectively In the TTSP one task can be tested on morethan one instrument A judgment matrix is used to expresswhether instrument 119903

119894is needed for 119905

119895 The judgment matrix

is defined as the following

119874119894

119895=

1 if 119905119895occupies 119903

119894

0 others(1)

In general task 119905119895may have several possible test schemes

The set of test schemes for 119905119895is defined as 119882

119895= 1199081

119895 1199082

119895

119908

119896119895

119895 (119896119895is the number of test schemes for 119905

119895) The notation

119875119896

119895= max

119903119894isin119908119896119895119875119894

119895is used to express the test time of 119905

119895for 119908119896119895

The following describes the restriction of resources

119883119896119896lowast

119895119895lowast =

1 if 119908119896

119895cap 119908119896lowast

119895lowast = 0

0 others(2)

Basic hypothesis includes three factors At a given timean instrument can only execute one task each task must becompleted without interruption once it starts Assume 119875

119894

119895=

119875119896

119895 119862119894119895= 119878119894

119895+ 119875119894

119895to simplify the problem

22 The Objective Functions for TTSP The objective func-tions are very important in the study of multiobjectiveoptimization problem The makespan is very importantin scheduling problems such as TTSP and FJSP becausethe completion time is an essential factor for schedulingproblem in product process In additional for TTSP the testinstruments have high integration and the test instrumentshave become increasingly expensive Therefore the demand


for reducing the workload of the instruments and increasingthe service life of the test instruments has great significancein TTSPTherefore our work focuses on twomain objectivesOne is tominimize themaximal test completion time and theother is to minimize the mean workload of the instrumentsThese objectives are represented by 119891

1(119909) and 119891

2(119909)

(1)TheMaximal Test Completion Time 1198911(119909)The notification

119862119896

119895= max

119903119894isin119908119896119895119862119894

119895is the test completion time of 119905

119895for119908119896119895Thus

the maximal test completion time of all tasks can be definedas follows

1198911(119909) = max

1le119896le1198961198951le119895le119899

119862119896

119895 (3)

(2) The Mean Workload of the Instruments 1198912(119909) First a new

notation 119876 is introduced to describe the parallel steps Theinitial value of 119876 is 1 Assign the instruments for all of thetasks if 119883119896119896

lowast

119895119895lowast = 1 119876 = 119876 + 1 Therefore the mean workload

of the instruments can be defined as follows

1198912(119909) =

1

119876

119899

sum

119895=1

119898

sum

119894=1

119875119894

119895119874119894

119895 (4)

3 The Variable NeighborhoodMOEAD Algorithm

In this section we proposed a variable neighborhoodMOEAD algorithm (VNM) To obtain solutions close tothe real Pareto Front (PF) of the TTSP two strategies areadopted The variable neighborhood strategy helps to makethe crossover span more reasonable Moreover Gauss muta-tion is adopted at the beginning of the iteration to maintainthe diversity of the population

31TheMain Strategy of the VNM TheVNM is an evolution-ary algorithm based on decomposition The main strategyof the VNM is to decompose a multiobjective optimizationproblem into a number of scalar optimization subproblemsand optimize these subproblems simultaneouslyThe decom-position method used is the Tchebycheff approach [33] Eachsubproblem is bound with a weight vector and then eachsubproblem is updated by obtaining information from itsneighborhood [25] The neighborhood of each subproblemis determined by its weighting vector

Let 1205821 1205822 120582

119873 be a set of weight vectors and

119911lowast

= (119911lowast

1 119911lowast

2 119911

lowast

119898)119879 is defined as the reference point

The problem of the Pareto Front approximation can bedecomposed into 119873 scalar optimization subproblems usingthe Tchebycheff approach and the objective function of the119895th subproblem is defined as

119892te(119909 | 120582

119895 119911lowast) = max1le119894le119898

120582119895

119894

1003816100381610038161003816119891119894(119909) minus 119911

119894

lowast1003816100381610038161003816 119909 isin Ω (5)

Parameter initialization

Randomly generate Npoints in the search space

Update Update the neighborhood

Update EP

Stop criteria satisfied

Output EP

Yes

No

Staring mutation

Crossover

Mutation

zlowast

Update

Figure 3 The main procedure of the VNM

where Ω is the decision space and 120582119895

= (120582119895

1 120582119895

2 120582

119895

119898)119879

119911lowast

119894= min119891

119894(119909) | 119909 sub Ω for each 119894 = 1 2 119898 It is clear

that the VNM is able to minimize all 119873 objective functionssimultaneously in a single run

The main procedure of the VNM can be described asshown in Figure 3

In the part of parameter setting the iteration number119872 the subproblem number 119873 the size of neighborhood 119879

(which ranges from beginning size 119861 to stopping size 119878) andthe population for saving the optimal solutions EP are set

The crossover operation in VNM is as followsFor each individual 119909119905

119894in generation 119905 the child 119909

119905+1

119894can

be obtained by the following equation

119909119905+1

119894=

119909119905

119894+ 1198651times (119909119905

119894minus 119909119905

1198941) + 1198652times (119909119905

119894minus 119909119905

1198942) rand (1)ltCR

119909119905

119894rand (1)geCR

(6)

CR 1198651 and 119865

2are the three control variables for the

crossover 119909119905

1198941and 119909

119905

1198942are two individuals chosen in the

neighborhood of 119909119905119894This crossovermethod canmake full use

of the information from the neighborhood and render theinformation exchange more sufficient

Themain idea ofVNM is given above Two improvementsare involved in the VNM algorithm Variable neighborhoodstrategy is adopted to make the crossover span more rea-sonable Moreover Starting Mutation is used to enhance thediversity of the population


0 50 100 150 200 2505

10

15

20

25

30

Generations

Nei

ghbo

rhoo

d siz

e

Straight lineMonotone parabolaNonmonotone parabola

Figure 4 Three controlling curves for the neighborhood size

32 Variable Neighborhood In the VNM the size of theneighborhood 119879 has a high impact on the performance ofthe algorithm If 119879 is too large the two solutions chosen (119909119897

and 119909119896) for the genetic operation may be unsuitable for the

subproblem and degradation may occur during the progressof the evolution In contrast if119879 is too small the subproblemsare all similar The child individual will be so similar to itsparents that the crossover operation will have a weak effect

119879 is the neighborhood size which determines thecrossover and neighborhood updating span Too large andtoo small 119879 will both have a negative influence on VNMTherefore 119879 should be large enough at the beginning of theevolution period to ensure sufficient information exchange ofthe solutions and 119879 should be sufficiently small in the latterportion of the evolution period such that degradation can beavoided Motivated by this ideology we designed and testedthree curves to find the best 119879 controlling curve

The three curves are shown in Figure 4 In this figure theabscissa is the number of iterations and the ordinate is thesize of the neighborhood 1198721 1198722 and 1198723 represent thestraight line themonotonic parabolic and the nonmonotonicparabolic curves respectively It is worth noting that in curve1198722 the curvature will be 0 at the end of the evolution periodThis means that the rate of change of curvature for1198722 is thefastest of all of the concave monotone parabolas during theperiod of evolution Because the curvature goes to 0 in theend curve 1198722 is determined Assume that if the number ofiterations is 125 the neighborhood of curves 1198721 1198722 and1198723 are 119910

1 1199102 and 119910

3 respectively in accordance with the

equation 1199101minus 1199102

= 1199102minus 1199103 Thus curve 1198723 can be also

determined Curve 1198723 is a nonmonotonic parabolic curveA series of experiments should be performed to compare theinfluence of the three curves on the algorithm to identify thebest controlling curve

33 StartingMutation TheTTSP represents a typical combi-national optimization problem The final best solutions maybe limited to only several points in the solution space Becauseof the neighborhood updating effect of the VNM there will

be many duplicate solutions so that the crossover operationwill have little effectTherefore how to maintain the diversityof the population is the key question for enhancing thealgorithm effect

Motivated by the ideology above a starting Gauss muta-tion is adopted at the beginning of the iteration For a solution119909119894= (119909119894

1 119909119894

2 119909

119894

119872) (119872 is the number of variables) Gauss

mutation is described as the following

for 119895 = 1 2 119872 119909119894lowast

119895=

normal (119909119894119895 120590) rand (1) lt 119901

119909119894

119895rand (1) ge 119901

(7)

119909119894lowast

= (119909119894lowast

1 119909119894lowast

2 119909119894lowast

119872) represents the individual after muta-

tion 119901 is themutation probability normal (119909119894119895 120590) is a number

that obeys the normal distribution 119909119894119895is the mean value

and 120590 is the variance With Starting Mutation the problemwith the initially invalid crossover operation can be resolvedTherefore we can avoid the solutions from becoming trappedin local optima and thus solutions with higher quality areobtained

4 The Convergence Analysis of VNM

Theconvergence analysis of VNM in this section provides thetheory ground for its application The convergence behaviorof VNM is analyzed according to the Markov Chain and thetransfer matrix respectively

41 Strong and Weak Convergence This section proposes thebasic theories of convergence and proves the strong and weakconvergence of VNM from the perspective of Markov Chain

There is a global optimal solution set 119872 for MOPs(multiobjective problem) 119872 is defined as 119872 = 119883 forall119884 isin

119878 119891(119883) ge 119891(119884) It is assumed that (119899) is the populationin evolutionary algorithms

A detailed demonstration for the convergence of MOEAhas been proposed in paper [34] Based on it the definitionsare described as follows

Theorem 1 120572119899 120573119899 and 119903

119899are defined as

120572119899= 119875 (119899 + 1) cap 119872 = 0 (119899) cap 119872 = 0

120573119899= 119875 (119899 + 1) cap 119872 = 0 (119899) cap 119872 = 0

119903119899= 12057311205732sim 120573119899

(8)

If lim119899rarrinfin

119903119899

= 0 (119899) converges to global optimal solutionweakly It is defined as (119899) rarr 119872(119875119882)

Theorem 2 120572119899 120573119899 and 119903


120572119899= 119875 (119899 + 1) cap 119872

119888= 0(119899) cap 119872

119888= 0

120573119899= 119875 (119899 + 1) cap 119872

119888= 0(119899) cap 119872

119888= 0

119903119899= 12057311205732sim 120573119899

(9)



119903119899

= 0 (119899) converges to global optimal solutionstrongly It is defined as (119899) rarr 119872(119875119878)

Based onTheorems 1 and 2 above the demonstration forthe convergence of VNM is described in the following Herelim119899rarrinfin

120573119899

= 0 lim119899rarrinfin

120573119899

= 0 describe the evolutionarytrend of VNMThere is lim

119899rarrinfin119903119899= 0 lim

119899rarrinfin119903119899= 0

Proof It is defined as 119875(119899) = 119875

997888

119883(119899) cap 119872 = 0Based on Bayesian we have

119875 (119899 + 1) = 119875 (119899 + 1) cap 119872 = 0

= 119875 (119899 + 1) cap 119872 = 0(119899) cap 119872 = 0

sdot 119875 (119899) cap 119872 = 0

+ 119875 (119899 + 1) cap 119872 = 0(119899) cap 119872 = 0

sdot 119875 (119899) cap 119872 = 0

= 120572119899119875 (119899) cap 119872 = 0 + 120573

119899119875 (119899) cap 119872 = 0

(10)

Elitist strategy is adopted in VNM 120572119899= 0 Hence

119875 (119899 + 1) = 120573119899119875 (119899) (11)

Then

lim119899rarrinfin

119875 (119899 + 1) = lim119899rarrinfin

120573119899119875 (119899) = lim

119899rarrinfin119903119899119875 (0) = 0

lim119899rarrinfin

119875 (119899) cap 119872 = 0 = 1

(12)

Therefore we have

(119899) 997888rarr 119872(119875119882) (13)

It means that (119899) converges to global optimal solutionweakly

Similarly it is defined as 119875(119899) = 119875(119899) cap 119872119888

= 0By Bayesian formula we have

119875 (119899 + 1) = 119875 (119899 + 1) cap 119872119888

= 0

= 120572119899119875 (119899) cap 119872

119888= 0 + 120573

119899119875 (119899)

(14)

Elitist strategy is adopted in VNM lim119899rarrinfin

120572119899= 0 Hence

lim119899rarrinfin

119875 (119899 + 1) = lim119899rarrinfin

120573119899119875 (119899) = lim

119899rarrinfin119903119899119875 (0) = 0 (15)

Then

lim119899rarrinfin

119875 (119899) cap 119872119888= 0 = 1

lim119899rarrinfin

119875 (119899) isin 119872 = 1

(16)

Therefore we have

(119899) 997888rarr 119872(119875119878) (17)

It means that (119899) converges to global optimal solutionstrongly

42 Convergence to Global Optimal This part focuses on theelitist strategy and proves that the VNM converges to theglobal optimum from the perspective of transfer matrix

Theorem 3 (see [35]) 119875 = (119862 0

119877 119879) is a reducible stochastic

matrix where 119862 119898 times 119898 is primitive stochastic matrix and119877 119879 = 0 Then

119875infin

= lim119896rarrinfin

119875119896= lim119896rarrinfin

(

119862119896

0

119896minus1

sum

119894=0

119879119894119877119862119896minus119894

119879119896)

= (

119862infin

0

119877infin

0)

(18)

where 119875infin is a stable stochastic matrix with 119875infin

= 11015840119901infin 119901infin =

1199010119875infin is unique regardless of the initial distributionThematrix

119901infin satisfies that 119901infin

119894gt 119901 for 1 le 119894 le 119898 and 119901

infin

119894= 0 for

119898 lt 119894 le 119899

According to the previous description of VNM theextended transition matrices for crossover 119862+ mutation119872

+

1

119872+

2 selection 119878

+ can be written as block diagonal matrix andupgrade matrix 119880 is lower triangular

119862+= (

119862 sdot sdot sdot 0

d


) 119872+

1= (

1198721

sdot sdot sdot 0

d


)

119878+= (


d


) 119872+

2= (

1198722

sdot sdot sdot 0

d


)

119880 = (

11988011

sdot sdot sdot 0

d

11988021198991198971

sdot sdot sdot 11988021198991198972119899119897

)

(19)

119862+ 119872+1 119878+ 119872+

2 and 119880 are with 2

119899119897 square matrices 119862 1198721

1198722 119878 and119880

119886119887(1 le 119886 119887 le 2

119899119897) are all with the size of 119899times 119899 (119899is the number of individuals and 119897 is the number of individualattributes)

119886 119887 in 119880119886119887

represents the populationrsquos state sequencenumber (in the order of the populations of the pros andcons from 1 to 2

119899119897) So 119880 is used to represent populationrsquosselection process Each block matrix 119880

119886119887is a selection of

individuals The details in 119880119886119887

can be described as thereare some individuals to make 119906

119894119895= 1 established in each

row Firstly the first individual is compared with all otherindividuals 119906

1119895= 1 if 119895th individual is optimal (there may

be several optima) or 11990611

= 1 if no one is better than it Then


the second individual is compared with all other individualsexcept the first individualThe best individual119892th individualis chosen set 119906

2119892= 1 if 119892th individual is optimal or 119906

22= 1 if

there is no one better than the second individualThe sortingprocess continues until all individuals are sorted To simplifythe difficulty of the problem assume that the there is only oneglobal optimal solution set Then only 119880

11is a unit matrix

whereas all matrices 119880119886119886with 119886 ge 2 are not unit matrices

In VNM the populations go through Gauss mutation119872+

1 crossover 119862

+ mutation 119872+

2 selection 119878

+ and EPupgradematrix119880 It is worth of noticing that (120583+120582) selectionmode is not used in the evolutionary process of VNMand thenumber of individuals remains unchanged This means that119878+= 119868 The transition matrix 119875

+ for VNM is

119875+= 119872+

1119862+119872+

2119878+119880

= (

1198721

sdot sdot sdot 0

d


)(


d


)(

1198722

sdot sdot sdot 0

d


)

times (


d


)(

11988011

sdot sdot sdot 0

d

11988021198991198971

sdot sdot sdot 11988021198991198972119899119897

)

= (

11987211198621198722

sdot sdot sdot 0

d

0 sdot sdot sdot 11987211198621198722

)(

11988011

sdot sdot sdot 0

d

11988021198991198971

sdot sdot sdot 11988021198991198972119899119897

)

= (

1198721119862119872211988011

0 0

sdot sdot sdot 0

1198721119862119872211988021198991198971

sdot sdot sdot 1198721119862119872211988021198991198972119899119897

)

= (

11987511

sdot sdot sdot 0

d

11987521198991198971

sdot sdot sdot 11987521198991198972119899119897

)

(20)

There is 11987511

gt 0 in the transition matrix 119875+ The submatrices

119875119886119897which is with 119886 ge 2 may be gathered in a rectangular

matrix 119877 = 0 so that Theorem 3 can be used to prove that thecorresponding VNM converges to the global optimum [36]

5 Experimental Results and Analysis

Computational experiments are carried out to compare theapproaches and to evaluate the efficiency of the proposedmethodThere are two objectives to minimize the makespanand the mean workload of the instruments In this sectionthe performance metric coverage metric 119862 is introducedfirst There are two experimental instances adopted in thissectionThey are instances of 30 taskswith 12 instruments and40 tasks with 12 instruments which are real-world examplestaken from a missile system The instance of 40 tasks with12 instruments is displayed in Table 1 The instance of 30

Table 1 The instance of 40 tasks with 12 instrumentsTask Scheme Resource Time

1199051

1199081

11199031 1199037

51199082

11199033 1199035

51199083

11199036 11990310

4

1199052

1199081

21199032 11990311

51199082

21199034 1199039

41199083

21199035 1199036

61199084

21199033 1199037

4

1199053

1199081

31199033

71199082

311990312

5

1199054

1199081

41199039

251199082

411990310

221199055

1199081

511990312

14

1199056

1199081

61199031 1199034

71199082

61199033 1199037

81199083

61199036 1199038

8

1199057

1199081

71199031 1199032

41199082

71199033 1199038

21199083

71199037 11990311

3

1199058

1199081

81199031 1199033

51199082

81199036 11990310

41199083

81199037 11990312

7

1199059

1199081

91199031 1199034

111199082

91199037 1199039

131199083

91199038 11990311

12

11990510

1199081

101199032

91199082

101199034

101199083

1011990310

10

11990511

1199081

111199032 1199037

61199082

111199033 11990312

91199083

111199038 1199039

8

11990512

1199081

121199032

111199082

121199035

131199083

1211990311

15

11990513

1199081

131199032

41199082

131199038

51199083

131199039

7

11990514

1199081

141199033

71199082

1411990311

101199083

1411990312

811990515

1199081

1511990312

2

11990516

1199081

161199032

91199082

161199035

71199083

161199038

6

11990517

1199081

171199031 11990310

101199082

171199035 1199039

121199083

1711990311 11990312

1111990518

1199081

181199036

15

11990519

1199081

191199032

81199082

191199035

71199083

1911990310

71199084

1911990312

6

11990520

1199081

201199033

61199082

201199036

41199083

201199039

5

11990521

1199081

211199031 1199034

21199082

211199033 1199035

51199083

211199036 1199038

3


Table 1 Continued

Task Scheme Resource Time

11990522

1199081

221199032

31199082

221199034

41199083

221199036

31199084

2211990310

4

11990523

1199081

231199033

51199082

2311990312

5

11990524

1199081

241199034

141199082

2411990311

1711990525

1199081

251199037

19

11990526

1199081

261199031 1199034

71199082

261199033 1199037

81199083

261199036 1199038

10

11990527

1199081

271199031 1199032

21199082

271199031 1199037

21199083

271199033 1199038

4

11990528

1199081

281199031 1199033

51199082

281199034 1199035

41199083

281199037 11990312

2

11990529

1199081

291199031 1199034

111199082

291199033 1199034

151199083

291199037 1199038

12

11990530

1199081

301199031

91199082

301199034

121199083

3011990312

10

11990531

1199081

311199032 1199033

61199082

311199035 11990311

81199083

311199036 1199039

8

11990532

1199081

321199032

111199082

321199035

131199083

321199036

17

11990533

1199081

331199032

61199082

331199036

51199083

3311990311

4

11990534

1199081

341199033

71199082

341199037

81199083

3411990312

1011990535

1199081

351199039

2

11990536

1199081

361199032

91199082

361199035

71199083

3611990310

6

11990537

1199081

371199031 1199032

101199082

371199037 11990311

71199083

371199035 11990312

1111990538

1199081

3811990310

15

11990539

1199081

391199034

81199082

391199036

71199083

391199039

71199084

3911990310

6

11990540

1199081

401199033

61199082

401199036

51199083

401199039

5

Table 2 Parameters setting

Population Generation CR 1198651 1198652 119875

100 250 05 1 1 005

tasks with 12 instruments is the first 30 tasks in Table 1 Theexperiment of selection of controlling curve for neighbor-hood size is shown in Section 52 The verification of theimprovements of the algorithm is displayed in Section 53 InSection 53 VNM is comparedwithMOEAD In Section 54the proposed algorithm (VNM) is compared with the varia-tions of CNSGA using real-world TTSP problems All of thealgorithms are executed using 50 independent runs In allof the experiments the better performances are denoted inbold The basic algorithm parameter settings are displayed inTable 2 CR 119865

1 and 119865


crossover 119901 is the mutation probability

51 PerformanceMetric Formultiobjective optimization theconvergence to the Pareto-optimal set is the most importanttarget to be considered There are mainly two metrics toevaluate the convergence One is convergence metric 120574 andthe other is convergence metric 119862 The true set of Pareto-optimal solutions is necessary for the calculation of 120574 How-ever the solutions space of TTSP is so large that the true set ofPareto-optimal solutions cannot be obtained by enumerationThe metric 119862 can be used to compare the performancesof the two solutions sets obtained by different algorithmsThe calculation of 119862 needs only the information of thetwo solutions sets Therefore in this paper the convergencemetric 119862 is used to evaluate the performance of the proposedalgorithm

Assume that 119860 and 119861 are two sets of nondominatedsolutions and 119862(119860 119861) is the ratio of the solutions in 119861 thatare dominated by at least one solution in 119860 Hence

119862 (119860 119861) =

1003816100381610038161003816119909 isin 119861 | exist119910 isin 119860 119910 dominates 119909

1003816100381610038161003816

|119861|

(21)

119862(119860 119861) = 1means that all of the solutions in119861 are dominatedby solutions in 119860 and 119862(119860 119861) = 0 means that there isno solution in 119861 dominated by a solution in 119860 Generallyspeaking if 119862(119860 119861) gt 119862(119861 119860) then solution set 119860 is betterthan solution set 119861

52 The Selection of Controlling Curve In this sectionthree curves are designed and tested to identify the best 119879controlling curve 1198721 1198722 and 1198723 respectively representthe straight line monotonic parabolic and nonmonotonicparabolic curves shown in Figure 4 In curve 1198722 the cur-vature will be 0 at the end of the evolution period Becauseof the influence of neighboring updating in MOEAD manyduplicate solutions will be presented in the final evolutionprocess of MOEAD Therefore Starting Mutation is appliedto the beginning of the next iteration to maintain the


0010203040506070809

1

C(M2M1) C(M1M2) C(M2M3) C(M3M2)

C-m

etric

TTSP 30lowast12

Figure 5 The boxplot of three curves for 30 lowast 12 instance

Table 3 Comparison of influence of three curves for 30 lowast 12

instance

Average Times119862(11987211198722) 02213 13119862(11987221198721) 05196 37119862(11987221198723) 04964 36119862(11987231198722) 02069 14

population diversity Tables 3 and 4 show the comparisonof the influence of the three curves on the algorithm usingtwo instancesThe results show that the monotonic paraboliccurve 1198722 has the best performance This means that themonotonic curve with the fastest rate of change of curvatureis themost useful for the algorithm And the boxplots of threecurves for 30 lowast 12 and 40 lowast 12 instances in Figures 5 and 6also give the same conclusion

53 Experiments for Comparisons of VNM and MOEADIn order to verify the improvement of VNM 30 lowast 12 and40 lowast 12 instances are used to test the performance of VNMandMOEADThemonotonic parabolic curve1198722 is selectedas the controlling curve in VNM The neighborhood size inMOEAD is 20 119881 and 119872 respectively represent VNM andMOEADThe results in Tables 5 and 6 show that the concavecurve with the fastest rate of change of curvature obtainedimprovement for VNM The selected curve renders the sizeof the neighborhood more suitable than before

The results of the two independent experiments forcomparison of VNM and MOEAD are shown in Figures 7and 8 for the 30 lowast 12 and 40 lowast 12 instances respectivelyAs shown in the figures the solutions obtained by theVNM dominate most of the solutions obtained by MOEADVariable neighborhood and Starting Mutation improve theperformance of MOEAD efficiently

Figures 9 and 10 are the boxplots for comparison of VNMand MOEAD It shows that the data distribution of VNMis superior to MOEAD VNM has the better performance

0010203040506070809

1

C(M2M1) C(M1M2) C(M2M3) C(M3M2)

C-m

etric

TTSP 40lowast12



instance

Average Times119862(11987211198722) 0244178 14119862(11987221198721) 0501508 36119862(11987221198723) 0533806 38119862(11987231198722) 0242146 12

Table 5 Comparison of VNM and MOEAD for 30 lowast 12 instance

Average Times119862(119881119872) 04845 35119862(119872119881) 02104 15


Average Times119862(119881119872) 05256 40119862(119872119881) 01949 10

because of application of variable neighborhood and StartingMutation

54 Experiments for Comparisons of VNM and CNSGA Inthis section the VNM is compared with the CNSGA forTTSP CNSGA is based on NSGA-II NSGA-II has beensuccessfully applied to job shop scheduling problems [37]reactive power dispatch problems [38] and many otherapplications CNSGA has successfully been adopted to solveTTSP [20] Therefore a comparison of VNM and CNSGA iscarried out to test the performance of the proposed algorithmVNM

There are two chaotic sequences logistic map and catmap and the chaotic sequences can be applied in threepositions population initialization crossover and mutationTherefore there are six combinations for CNSGA Thenomenclatures for six variants of CNSGA are shown in


40 50 60 70 80 90 10016

18

20

22

24

26

28

30

32

Makespan

Mea

n w

orkl

oad

VNMMOEAD

TTSP 30lowast12

(a)

40 50 60 70 80 9016

17

18

19

20

21

22

23

24

25

Makespan

Mea

n w

orkl

oad

VNMMOEAD

TTSP 30lowast12

(b)

Figure 7 The comparison of VNM and MOEAD for 30 lowast 12 instance

50 55 60 65 70 75 80 85 9017

18

19

20

21

22

23

24

25

26

27

Makespan

Mea

n w

orkl

oad

TTSP

VNMMOEAD

40lowast12

(a)

50 55 60 65 70 75 8017

18

19

20

21

22

23

Makespan

Mea

n w

orkl

oad

VNMMOEAD

TTSP 40lowast12

(b)


Table 7 Tables 8 and 9 show the comparison of VNM andCNSGA for 30 lowast 12 and 40 lowast 12 instances The VNM isrepresented by 119881 All the comparisons between VNM andthe variations of CNSGA are based on 50 independent exper-iments The average of 119862 metric and better computationaltimes are used for the performance analysis The results fromTables 8 and 9 show that VNMprovides the best performancenot only for the average metric 119862 but also in terms of bettercomputational times than CNSGA

Figures 11 and 12 show the comparisons of VNM and the6 variations of CNSGA for 30lowast 12 instanceThe figures show

Table 7 Nomenclature for six variants of the CNSGA

The logistic map The cat mapInitial population 119871

11198621

Crossover operator 1198712

1198622

Mutation operator 1198713

1198623

that the solutions obtained by the VNMdominatemost of thesolutions obtained by the 6 variations of CNSGA Thereforethe VNM obtains the best performance


0010203040506070809

1

C-m

etric

TTSP

C(VM) C(MV)

30lowast12

Figure 9 The boxplot of comparison of VNM and MOEAD for30 lowast 12 instance

0010203040506070809

1

C-m

etric

TTSP

C(VM) C(MV)

40lowast12


The results of the comparison of VNM and CNSGA forthe 40 lowast 12 instance are shown in Figures 13 and 14The resultsalso show that the solutions obtained by VNM have higherquality

In addition box plots are used to display the perfor-mances of the algorithmsThe box plots of119862metric for VNMandCNSGAwith 30lowast12 and 40lowast12 instances are shown fromFigures 15 16 17 and 18

From the box plots of 119862metric it is clear that the medianof VNM is larger than that of the variations of CNSGA inboth the 30 lowast 12 and 40 lowast 12 instances and the datadistribution of VNM ismore reasonable as well Additionallythe average of VNM is also superior The results showthat VNM demonstrates better performance than CNSGAin solving the multiobjective TTSP The performance ofsolutions obtained by VNM is better than that obtained byCNSGA because that the variable neighborhood is adoptedin VNMThe span of information exchange in VNM changesfollowing the evolutionary process but that in CNSGA stays

Table 8 Comparison of VNM and six variations of CNSGA for 30lowast 12 instance

Average Times119862(119881 119871

1) 04206 36

119862(1198711 119881) 02200 14

119862(119881 1198712) 04077 34

119862(1198712 119881) 02648 16

119862(119881 1198713) 04182 36

119862(1198713 119881) 02248 14

119862(119881 1198621) 04638 35

119862(1198621 119881) 02210 15

119862(119881 1198622) 04602 34

119862(1198622 119881) 02288 16

119862(119881 1198623) 04128 35

119862(1198623 119881) 02525 15


Average Times119862(119881 119871

1) 05243 37

119862(1198711 119881) 02264 13

119862(119881 1198712) 05359 36

119862(1198712 119881) 02282 14

119862(119881 1198713) 05218 38

119862(1198713 119881) 02338 12

119862(119881 1198621) 05044 36

119862(1198621 119881) 02138 14

119862(119881 1198622) 04844 35

119862(1198622 119881) 02169 15

119862(119881 1198623) 05116 37

119862(1198623 119881) 02055 13

the sameThe information from the process of evolution helpsVNM get better performance

The variable neighborhood provides a strategy to improvethe performance of the algorithm For different problemswith different scales the controlling curves for the neigh-borhood size will be different The Starting Mutation canbe also applied to solve other optimization problem in theevolution process The strategies proposed in this paper canbe investigated in other scheduling problem similar to TTSP

6 Conclusion

How to improve the test efficiency is more and more impor-tant in modern industry TTSP has important application


40 50 60 70 80 90 100 11012

14

16

18

20

22

24

26

28

Makespan

Mea

n w

orkl

oad

VNM

TTSP

CNSGA-L1

CNSGA-L2

CNSGA-L3

30lowast12

Figure 11 The comparison of VNM and three variations withlogistic map for CNSGA for 30 lowast 12 instance

30 40 50 60 70 80 90 100 11014

16

18

20

22

24

26

Makespan

Mea

n w

orkl

oad

VNM

TTSP

CNSGA-C1

CNSGA-C2

CNSGA-C3

30lowast12

Figure 12 The comparison of VNM and three variations with catmap for CNSGA for 30 lowast 12 instance

value in modern manufacturing process TTSP is combina-tional optimization problem The final best solutions onlyaccount for a rather small subset of the search space Inorder to help the solutions avoid being trapped in localoptima this paper proposed a new genetic evolutionarymultiobjective optimization algorithm (VNM) to solve theTTSP The variable neighborhood and Starting Mutationstrategies are adopted in VNM to make the crossover spanmore suitable and improve the diversity of populationThree controlling curves for neighborhood size are studiedThe experimental results have shown that the monotonic

40 50 60 70 80 90 100 11014

16

18

20

22

24

26

28

30

32

Makespan

Mea

n w

orkl

oad

VNMCNSGA-

CNSGA-CNSGA-

TTSP

L1

L2

L3

40lowast12


40 50 60 70 80 90 100 110 12016

18

20

22

24

26

28

30

Makespan

Mea

n w

orkl

oad

VNM

TTSP

CNSGA-C1

CNSGA-C2

CNSGA-C3

40lowast12


parabolic has the best performance From the experimentconducted for comparison of VNM and MOEAD we seethat the improved algorithm has made great progress insolving the TTSP problem And the experiment conductedfor comparison of VNM and CNSGA also shows that theimproved algorithm is superior to CNSGA in solving TTSPVNM can also be applied to solve other combinationaloptimization problems such as FJSP and TSP Future workwill focus on two objectives the precedence constraint willbe added to the TTSP and information regarding bottlenecktasks will be considered


TTSP

0010203040506070809

1

C-m

etric

C(V L1) C(V L2) C(V L3)C(L1 V) C(L2 V) C(L3 V)

30lowast12

Figure 15 The boxplot of VNM and three variations with logisticmap for CNSGA for 30 lowast 12 instance

0010203040506070809

1TTSP

C-m

etric

C(V C1) C(V C2) C(V C3)C(C1 V) C(C2 V) C(C3 V)

30lowast12

Figure 16 The boxplot of VNM and three variations with cat mapfor CNSGA for TTSP 30 lowast 12 instance

0010203040506070809

1TTSP

C-m

etric


40lowast12

Figure 17 The boxplot of VNM and three variations with logisticmap for CNSGA for 40 lowast 12

TTSP

0010203040506070809

1

C-m

etric


40lowast12

Figure 18 The boxplot of VNM and three variations with cat mapfor CNSGA for 40 lowast 12

Conflict of Interests

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

Acknowledgments

This research is supported by the National Natural Sci-ence Foundation of China under Grant no 61101153 andthe National 863 Hi-Tech R and D Plan under Grant2011AA110101

References

[1] A Radulescu C Nicolescu A J C van Gemund and P PJonker ldquoCPR mixed task and data parallel scheduling for dis-tributed systemsrdquo in Proceeding of 15th Parallel and DistributedProcessing Symposium pp 1ndash9 San Francisco Calif USA April2001

[2] Z Yin J Cui Y Yang and YMa ldquoJob shop scheduling problembased on DNA computingrdquo Journal of Systems Engineering andElectronics vol 17 no 3 pp 654ndash659 2006

[3] Y Zuo H Y Gu and Y G Xi ldquoModified bottleneck-basedheuristic for large-scale job-shop scheduling problem with asingle bottleneckrdquo Journal of Systems Engineering and Electron-ics vol 18 no 3 pp 556ndash565 2007

[4] N Al-Hinai and T Y Elmekkawy ldquoRobust and stable flexiblejob shop scheduling with random machine breakdowns usinga hybrid genetic algorithmrdquo International Journal of ProductionEconomics vol 132 no 2 pp 279ndash291 2011

[5] J C Chen C-C Wu C-W Chen and K-H Chen ldquoFlexiblejob shop scheduling with parallel machines using geneticalgorithm and grouping genetic algorithmrdquo Expert Systems withApplications vol 39 no 11 pp 10016ndash10021 2012

[6] L De Giovanni and F Pezzella ldquoAn improved genetic algorithmfor the distributed and flexible job-shop scheduling problemrdquoEuropean Journal of Operational Research vol 200 no 2 pp395ndash408 2010

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


[8] R Zhang and C Wu ldquoA hybrid immune simulated annealingalgorithm for the job shop scheduling problemrdquo Applied SoftComputing Journal vol 10 no 1 pp 79ndash89 2010

[9] P Damodaran andM C Velez-Gallego ldquoA simulated annealingalgorithm to minimize makespan of parallel batch processingmachines with unequal job ready timesrdquo Expert Systems withApplications vol 39 no 1 pp 1451ndash1458 2012

[10] K Li Y Shi S-L Yang and B-Y Cheng ldquoParallel machinescheduling problem to minimize the makespan with resourcedependent processing timesrdquo Applied Soft Computing Journalvol 11 no 8 pp 5551ndash5557 2011

[11] Q Zhang H Manier and M-A Manier ldquoA genetic algorithmwith tabu search procedure for flexible job shop schedulingwith transportation constraints and bounded processing timesrdquoComputers and Operations Research vol 39 no 7 pp 1713ndash17232012

[12] W Teekeng and A Thammano ldquoA combination of shuffledfrog leaping and fuzzy logic for flexible job-shop schedulingproblemsrdquo Procedia Computer Science vol 6 pp 69ndash75 2011

[13] G Zhang X Shao P Li and LGao ldquoAn effective hybrid particleswarm optimization algorithm for multi-objective flexible job-shop scheduling problemrdquo Computers and Industrial Engineer-ing vol 56 no 4 pp 1309ndash1318 2009

[14] S H A Rahmati and M Zandieh ldquoA new biogeography-based optimization (BBO) algorithm for the flexible job shopscheduling problemrdquo The International Journal of AdvancedManufacturing Technology vol 58 no 9ndash12 pp 1115ndash1129 2012

[15] X Y Shao W Q Liu Q Liu and C Y Zhang ldquoHybriddiscrete particle swarmoptimization formulti-objective flexiblejob-shop scheduling problemrdquo The International Journal ofAdvancedManufacturing Technology vol 67 no 9ndash12 pp 2885ndash2901 2013

[16] R Xia M Xiao J Cheng and X Fu ldquoOptimizing the multi-UUT parallel test task scheduling based on multi-objectiveGASArdquo in Proceedings of the 8th International Conference onElectronicMeasurement and Instruments (ICEMI rsquo07) pp 4839ndash4844 Xirsquoan China August 2007

[17] J Y Fang H H Xue and M Q Xiao ldquoParallel test tasksscheduling and resources configuration based on GA-ACArdquoJournal of Measurement Science and Instrumentation vol 2 no4 pp 321ndash326 2011

[18] 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

[19] X Liang B G Dong H Gao and D S Yan ldquoParallel test taskscheduling of aircraft electrical system based on cost constraintmatrix and ant colony algorithmrdquo in Proceeding of IEEE 10thInternational Conference on Industrial Informatics pp 178ndash183Beijing Chinese July 2012

[20] H Lu R Y Niu J Liu and Z Zhu ldquoA chaotic non-dominatedsorting genetic algorithm for the multi-objective automatic testtask scheduling problemrdquoApplied Soft Computing vol 13 no 5pp 2790ndash2802 2013

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

[22] B Show V Mukherjee and S P Ghoshal ldquoSolution of reactivepower dispatch of power systems by an opposition-basedgravitational search algorithmrdquo Electrical Power and EnergySystems vol 55 pp 29ndash40 2014

[23] L Wang and C Singh ldquoEnvironmentaleconomic power dis-patch using a fuzzifiedmulti-objective particle swarmoptimiza-tion algorithmrdquo Electric Power Systems Research vol 77 no 12pp 1654ndash1664 2007

[24] O Abedinia D Garmarodi R Rahbar and F JavidzadehldquoMulti-objective environmentaleconomic dispatch using inter-active artificial bee colony algorithmrdquo Journal of Basic andApplied Scientific Research vol 2 no 11 pp 11272ndash11281 2012

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

[26] Y-Y Tan Y-C Jiao H Li and X-K Wang ldquoMOEAD+uniform design a new version of MOEAD for optimizationproblems with many objectivesrdquo Computers and OperationsResearch 2012

[27] C-M Chen Y-P Chen and Q Zhang ldquoEnhancing MOEADwith guided mutation and priority update for multi-objectiveoptimizationrdquo in Proceedings of the IEEE Congress on Evolution-ary Computation (CEC rsquo09) pp 209ndash216 Trondheim NorwayMay 2009

[28] M A Jan and R A Khanum ldquoA study of two penalty-parameterless constraint handling techniques in the frameworkofMOEADrdquoApplied Soft Computing vol 13 no 1 pp 128ndash1482012

[29] W Peng Q Zhang and H Li ldquoComparison betweenMOEADand NSGA-II on the multi-objective travelling salesman prob-lemrdquo Studies in Computational Intelligence vol 171 pp 309ndash3242009

[30] P A Carl Communication with Automata Applied DataResearch Princeton NJ USA 1966

[31] T Zhang B Guo and Y Tan ldquoCapacitated stochastic colouredPetri net-based approach for computing two-terminal reliabilityof multi-state networkrdquo Journal of Systems Engineering andElectronics vol 23 no 2 pp 304ndash313 2012

[32] B Bollobas GraphTheory Springer New York NY USA 1979[33] K Miettinen Nonlinear Multiobjective Optimization Kluwer

Academic Boston Mass USA 1999[34] Y-R Zhou H-Q Min X-Y Xu and Y-X Li ldquoMulti-objective

evolutionary algorithm and its convergencerdquo Chinese Journal ofComputers vol 27 no 10 pp 1415ndash1421 2004

[35] M Iosifescu Finite Markov Processes and Their ApplicationsDover 1980

[36] Q Zhang L Dong F Jiang and X J Zhu ldquoConvergence ofmulti-objective evolutionary computation to its pareto optimalsetrdquo Systems Engineering and Electronics vol 22 no 8 pp 17ndash212000

[37] E Moradi S M T Fatemi Ghomi and M Zandieh ldquoBi-objective optimization research on integrated fixed time inter-val preventive maintenance and production for schedulingflexible job-shop problemrdquo Expert Systems with Applicationsvol 38 no 6 pp 7169ndash7178 2011

[38] S Jeyadevi S Baskar C K Babulal and M Willjuice Irutha-yarajan ldquoSolving multiobjective optimal reactive power dis-patch usingmodifiedNSGA-IIrdquo International Journal of Electri-cal Power and Energy Systems vol 33 no 2 pp 219ndash228 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

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each literature but themakespan the total tardiness the crit-ical machine workload and the total workload of machinesare frequently considered factors in those researches

Different from the research of FJSP the research of TTSPis relatively few because of the development of automatic testsystem However there are still some achievements in TTSPXia et al [16] proposed a method that combined GA andsimulated SA to optimize the parallel efficiency and speedup ratio of the multi-UUT parallel test Genetic Algorithm-AntColonyAlgorithm (GA-ACA) [17] hybrid particle swarmand tabu search [18] and Ant Colony Algorithm [19] are usedto solve parallel test tasks scheduling to obtain the mini-mum makespan A chaotic nondominated sorting geneticalgorithm is proposed to solve multiobjective TTSP [20]Thechaotic operations are combined with NSGA-II [21] in thisapproachThose algorithms have shown excellent property indecreasing costs and improving efficiencies in automatic testsystem

There are also some intelligent algorithms used to solvepower dispatch problems and other scheduling problemsFor example opposition-based learning is employed inopposition-based gravitational search algorithm (OGSA)to solve optimal reactive power dispatch [22] A fuzzifiedmultiobjective PSO (FMOPSO) algorithm is proposed andimplemented to dispatch the electric power [23] An inter-active artificial bee colony algorithm was proposed for themultiobjective environmentaleconomic dispatch problem[24]

In summary most of the researches of scheduling prob-lems focus on the single-objective problem or adopt weightedsum approach to convert the multiobjective problem into asingle-objective problemHowever theweighting coefficientsare difficult to choose and human factors will greatly impactthe performance of the algorithms In fact there are anothertwo kinds of methods for solving themultiobjective problemOne method is the non-Pareto approach utilizing operatorsfor processing the different objectives in a separated wayAnother is the Pareto approaches which are directly basedon the Pareto optimality concept They aim at satisfying twogoals converging towards the Pareto front and also obtainingdiversified solutions scattered all over the Pareto frontThosetwo kinds of methods mainly rely on the performanceand strategies of the algorithms used in the multiobjectiveproblems

In this paper the method based on Tchebycheff decom-position for multiobjective functions was adopted and thealgorithm named MOEAD is used to solve TTSP MOEADis a typical evolutionary algorithm based on decompositionproposed by Zhang and Li [25] This method decomposes amultiobjective optimization problem into a number of scalaroptimization subproblems and optimizes them simultane-ously The results show that MOEAD has a good perfor-mance for the ZDT and DTLZ test problems MOEAD isvery efficient in solving multiobjective problems Researchon MOEAD has also been performed in recent yearsFor example Tan et al [26] proposed a new version ofMOEAD with a uniform design to deal with the multiob-jective problem in higher-dimensional objective spaces This

method can render the distribution of the weighting vectorsmore uniform especially for problems with high dimensionChen et al [27] introduced a guided mutation operator andpriority update to enhance the ability ofMOEAD Stochasticranking and constraint-domination principle are adopted inMOEAD to improve the ability of the algorithm to dealwith constrained multiobjective optimization problems [28]Although these studies have improved the ability ofMOEADfor solving multi-objective problems MOEAD is mainlyused to solve standard test cases like ZDT DTLZ and F1However MOEAD is rarely used to solve combinationaloptimization problems such as FJSP TTSP Peng et al appliedMOEAD to solve Travelling Salesman Problem (TSP) [29]However there is no special improvement for MOEADaccording to the feature ofMOEAD and the property of TSP

The scheduling problems such as TTSP FJSP and TSPand power dispatching problem are a branch of combina-tional optimization problems Because of the properties of thecombinational optimization problems the final best solutionsonly account for a rather small subset of the search spaceHow to avoid the solutions obtained being trapped in localoptima is the key to improve the ability of algorithms todeal with combinational optimization problems Consideringthe fact that the size of the neighborhood is important inMOEAD [25] too large size will lead to degradation andtoo small size will weaken the effect of evolutionary processMoreover there will be many duplicate solutions due to theinfluence of neighborhood updating of MOEAD [25] Thepopulation diversity will decrease obviously Based on theanalyses above variable neighborhood based on a quadraticcurve is adopted to ensure that the crossover span is morereasonable and Gauss mutation is adopted at the beginningof iteration to maintain the diversity of the populationThese two improvements can efficiently enhance the abilityof MOEAD for avoiding the solutions obtained from beingtrapped in local optima The proposed approach cannot onlysolve TTSP but also deal with other scheduling problemsbecause the feasible solutions of TTSP are more difficult toobtain than most scheduling problems such as FJSP TSP

The organization of this paper is as follows A briefintroduction of TTSP is introduced in Section 2 The newmethod for TTSP variable neighborhood MOEAD (VNM)is proposed in Section 3The convergence analysis of VNM isalso presented in Section 4 A large number of experimentalresults and discussions are covered in Section 5 Conclusionsare given in Section 6

2 The Formulation of TTSP

21 The Mathematical Model for TTSP The goal of the TTSPis to arrange the execution of 119899 tasks on119898 instrumentsThereare three main mathematical models for TTSP One model isbased on Petri netThe second is based onGraph theory Andthe third model is based on integer programming Our workin this paper is mainly based on the integer programmingproposed by us in paper [20]


P1

P4

P2 P3

r1 r2 r3

t1 [r1 r2 r3]

(1 2 3)

12 3




1

are 1199031 1199032 and 119903



1199013 and 119901



1 1199032 and 119903

3)



1 V2 V3


1 1199032 1199033]) where V

1 V2 and V

3

are bound to 1199031 1199032 and 119903

3


3


and 1199013are 1199031 1199032 and 119903


1 1199032 and


1 When the 119905


in 1199011 1199012 and 119901


4 The tokens



1 1199032 and




1 1199052 1199053 and 119905

4) and



needed by 1199051 1199052 1199053 1199054are 1199031 1199032 1199032 1199034 1199033 1199034 and 119903

1 1199033



t1 t2

t3 t4




1 1199052 119905

119895 (1 le


119894 (1 le 119894 le 119898)

119878119894

119895 119862119894119895 and 119875

119894



119894





119874119894

119895=

1 if 119905119895occupies 119903

119894

0 others(1)



119895= 1199081

119895 1199082

119895

119908

119896119895



119875119896

119895= max

119903119894isin119908119896119895119875119894


119895for 119908119896119895


119883119896119896lowast

119895119895lowast =

1 if 119908119896

119895cap 119908119896lowast

119895lowast = 0

0 others(2)


119894

119895=

119875119896

119895 119862119894119895= 119878119894

119895+ 119875119894





1(119909) and 119891

2(119909)


119862119896

119895= max

119903119894isin119908119896119895119862119894


119895for119908119896119895Thus


1198911(119909) = max

1le119896le1198961198951le119895le119899

119862119896

119895 (3)



lowast



1198912(119909) =

1

119876

119899

sum

119895=1

119898

sum

119894=1

119875119894

119895119874119894

119895 (4)




Let 1205821 1205822 120582


119911lowast

= (119911lowast

1 119911lowast

2 119911

lowast



119892te(119909 | 120582

119895 119911lowast) = max1le119894le119898

120582119895

119894

1003816100381610038161003816119891119894(119909) minus 119911

119894

lowast1003816100381610038161003816 119909 isin Ω (5)




Update EP


Output EP

Yes

No

Staring mutation

Crossover

Mutation

zlowast

Update



= (120582119895

1 120582119895

2 120582

119895

119898)119879

119911lowast

119894= min119891








119905+1

119894can


119909119905+1

119894=

119909119905

119894+ 1198651times (119909119905

119894minus 119909119905

1198941) + 1198652times (119909119905

119894minus 119909119905

1198942) rand (1)ltCR

119909119905

119894rand (1)geCR

(6)

CR 1198651 and 119865



1198941and 119909

119905






0 50 100 150 200 2505

10

15

20

25

30

Generations

Nei

ghbo

rhoo

d siz

e








1 1199102 and 119910








1 119909119894

2 119909

119894



for 119895 = 1 2 119872 119909119894lowast

119895=

normal (119909119894119895 120590) rand (1) lt 119901

119909119894

119895rand (1) ge 119901

(7)

119909119894lowast

= (119909119894lowast

1 119909119894lowast

2 119909119894lowast











Theorem 1 120572119899 120573119899 and 119903


120572119899= 119875 (119899 + 1) cap 119872 = 0 (119899) cap 119872 = 0

120573119899= 119875 (119899 + 1) cap 119872 = 0 (119899) cap 119872 = 0

119903119899= 12057311205732sim 120573119899

(8)


119903119899


Theorem 2 120572119899 120573119899 and 119903


120572119899= 119875 (119899 + 1) cap 119872

119888= 0(119899) cap 119872

119888= 0

120573119899= 119875 (119899 + 1) cap 119872

119888= 0(119899) cap 119872

119888= 0

119903119899= 12057311205732sim 120573119899

(9)



119903119899



120573119899


120573119899


119899rarrinfin119903119899= 0 lim

119899rarrinfin119903119899= 0


997888


119875 (119899 + 1) = 119875 (119899 + 1) cap 119872 = 0

= 119875 (119899 + 1) cap 119872 = 0(119899) cap 119872 = 0

sdot 119875 (119899) cap 119872 = 0

+ 119875 (119899 + 1) cap 119872 = 0(119899) cap 119872 = 0

sdot 119875 (119899) cap 119872 = 0

= 120572119899119875 (119899) cap 119872 = 0 + 120573

119899119875 (119899) cap 119872 = 0

(10)


119875 (119899 + 1) = 120573119899119875 (119899) (11)

Then

lim119899rarrinfin

119875 (119899 + 1) = lim119899rarrinfin

120573119899119875 (119899) = lim

119899rarrinfin119903119899119875 (0) = 0

lim119899rarrinfin

119875 (119899) cap 119872 = 0 = 1

(12)

Therefore we have

(119899) 997888rarr 119872(119875119882) (13)




119875 (119899 + 1) = 119875 (119899 + 1) cap 119872119888

= 0

= 120572119899119875 (119899) cap 119872

119888= 0 + 120573

119899119875 (119899)

(14)


120572119899= 0 Hence

lim119899rarrinfin

119875 (119899 + 1) = lim119899rarrinfin

120573119899119875 (119899) = lim

119899rarrinfin119903119899119875 (0) = 0 (15)

Then

lim119899rarrinfin

119875 (119899) cap 119872119888= 0 = 1

lim119899rarrinfin

119875 (119899) isin 119872 = 1

(16)

Therefore we have

(119899) 997888rarr 119872(119875119878) (17)



Theorem 3 (see [35]) 119875 = (119862 0



119875infin


119875119896= lim119896rarrinfin

(

119862119896

0

119896minus1

sum

119894=0

119879119894119877119862119896minus119894

119879119896)

= (

119862infin

0

119877infin

0)

(18)


= 11015840119901infin 119901infin =



119894gt 119901 for 1 le 119894 le 119898 and 119901

infin

119894= 0 for

119898 lt 119894 le 119899


+

1

119872+

2 selection 119878


119862+= (


d


) 119872+

1= (

1198721

sdot sdot sdot 0

d


)

119878+= (


d


) 119872+

2= (

1198722

sdot sdot sdot 0

d


)

119880 = (

11988011

sdot sdot sdot 0

d

11988021198991198971

sdot sdot sdot 11988021198991198972119899119897

)

(19)

119862+ 119872+1 119878+ 119872+


119899119897 square matrices 119862 1198721

1198722 119878 and119880

119886119887(1 le 119886 119887 le 2


119886 119887 in 119880119886119887














22= 1 if


11is a unit matrix



1 crossover 119862

+ mutation 119872+

2 selection 119878


+ for VNM is

119875+= 119872+

1119862+119872+

2119878+119880

= (

1198721

sdot sdot sdot 0

d


)(


d


)(

1198722

sdot sdot sdot 0

d


)

times (


d


)(

11988011

sdot sdot sdot 0

d

11988021198991198971

sdot sdot sdot 11988021198991198972119899119897

)

= (

11987211198621198722

sdot sdot sdot 0

d

0 sdot sdot sdot 11987211198621198722

)(

11988011

sdot sdot sdot 0

d

11988021198991198971

sdot sdot sdot 11988021198991198972119899119897

)

= (

1198721119862119872211988011

0 0

sdot sdot sdot 0

1198721119862119872211988021198991198971

sdot sdot sdot 1198721119862119872211988021198991198972119899119897

)

= (

11987511

sdot sdot sdot 0

d

11987521198991198971

sdot sdot sdot 11987521198991198972119899119897

)

(20)

There is 11987511







1199051

1199081

11199031 1199037

51199082

11199033 1199035

51199083

11199036 11990310

4

1199052

1199081

21199032 11990311

51199082

21199034 1199039

41199083

21199035 1199036

61199084

21199033 1199037

4

1199053

1199081

31199033

71199082

311990312

5

1199054

1199081

41199039

251199082

411990310

221199055

1199081

511990312

14

1199056

1199081

61199031 1199034

71199082

61199033 1199037

81199083

61199036 1199038

8

1199057

1199081

71199031 1199032

41199082

71199033 1199038

21199083

71199037 11990311

3

1199058

1199081

81199031 1199033

51199082

81199036 11990310

41199083

81199037 11990312

7

1199059

1199081

91199031 1199034

111199082

91199037 1199039

131199083

91199038 11990311

12

11990510

1199081

101199032

91199082

101199034

101199083

1011990310

10

11990511

1199081

111199032 1199037

61199082

111199033 11990312

91199083

111199038 1199039

8

11990512

1199081

121199032

111199082

121199035

131199083

1211990311

15

11990513

1199081

131199032

41199082

131199038

51199083

131199039

7

11990514

1199081

141199033

71199082

1411990311

101199083

1411990312

811990515

1199081

1511990312

2

11990516

1199081

161199032

91199082

161199035

71199083

161199038

6

11990517

1199081

171199031 11990310

101199082

171199035 1199039

121199083

1711990311 11990312

1111990518

1199081

181199036

15

11990519

1199081

191199032

81199082

191199035

71199083

1911990310

71199084

1911990312

6

11990520

1199081

201199033

61199082

201199036

41199083

201199039

5

11990521

1199081

211199031 1199034

21199082

211199033 1199035

51199083

211199036 1199038

3


Table 1 Continued


11990522

1199081

221199032

31199082

221199034

41199083

221199036

31199084

2211990310

4

11990523

1199081

231199033

51199082

2311990312

5

11990524

1199081

241199034

141199082

2411990311

1711990525

1199081

251199037

19

11990526

1199081

261199031 1199034

71199082

261199033 1199037

81199083

261199036 1199038

10

11990527

1199081

271199031 1199032

21199082

271199031 1199037

21199083

271199033 1199038

4

11990528

1199081

281199031 1199033

51199082

281199034 1199035

41199083

281199037 11990312

2

11990529

1199081

291199031 1199034

111199082

291199033 1199034

151199083

291199037 1199038

12

11990530

1199081

301199031

91199082

301199034

121199083

3011990312

10

11990531

1199081

311199032 1199033

61199082

311199035 11990311

81199083

311199036 1199039

8

11990532

1199081

321199032

111199082

321199035

131199083

321199036

17

11990533

1199081

331199032

61199082

331199036

51199083

3311990311

4

11990534

1199081

341199033

71199082

341199037

81199083

3411990312

1011990535

1199081

351199039

2

11990536

1199081

361199032

91199082

361199035

71199083

3611990310

6

11990537

1199081

371199031 1199032

101199082

371199037 11990311

71199083

371199035 11990312

1111990538

1199081

3811990310

15

11990539

1199081

391199034

81199082

391199036

71199083

391199039

71199084

3911990310

6

11990540

1199081

401199033

61199082

401199036

51199083

401199039

5



100 250 05 1 1 005


1 and 119865





119862 (119860 119861) =


1003816100381610038161003816

|119861|

(21)




0010203040506070809

1

C(M2M1) C(M1M2) C(M2M3) C(M3M2)

C-m

etric

TTSP 30lowast12



instance

Average Times119862(11987211198722) 02213 13119862(11987221198721) 05196 37119862(11987221198723) 04964 36119862(11987231198722) 02069 14





0010203040506070809

1

C(M2M1) C(M1M2) C(M2M3) C(M3M2)

C-m

etric

TTSP 40lowast12



instance

Average Times119862(11987211198722) 0244178 14119862(11987221198721) 0501508 36119862(11987221198723) 0533806 38119862(11987231198722) 0242146 12


Average Times119862(119881119872) 04845 35119862(119872119881) 02104 15


Average Times119862(119881119872) 05256 40119862(119872119881) 01949 10





40 50 60 70 80 90 10016

18

20

22

24

26

28

30

32

Makespan

Mea

n w

orkl

oad

VNMMOEAD

TTSP 30lowast12

(a)

40 50 60 70 80 9016

17

18

19

20

21

22

23

24

25

Makespan

Mea

n w

orkl

oad

VNMMOEAD

TTSP 30lowast12

(b)


50 55 60 65 70 75 80 85 9017

18

19

20

21

22

23

24

25

26

27

Makespan

Mea

n w

orkl

oad

TTSP

VNMMOEAD

40lowast12

(a)

50 55 60 65 70 75 8017

18

19

20

21

22

23

Makespan

Mea

n w

orkl

oad

VNMMOEAD

TTSP 40lowast12

(b)






11198621


1198622


1198623



0010203040506070809

1

C-m

etric

TTSP

C(VM) C(MV)

30lowast12


0010203040506070809

1

C-m

etric

TTSP

C(VM) C(MV)

40lowast12






Average Times119862(119881 119871

1) 04206 36

119862(1198711 119881) 02200 14

119862(119881 1198712) 04077 34

119862(1198712 119881) 02648 16

119862(119881 1198713) 04182 36

119862(1198713 119881) 02248 14

119862(119881 1198621) 04638 35

119862(1198621 119881) 02210 15

119862(119881 1198622) 04602 34

119862(1198622 119881) 02288 16

119862(119881 1198623) 04128 35

119862(1198623 119881) 02525 15


Average Times119862(119881 119871

1) 05243 37

119862(1198711 119881) 02264 13

119862(119881 1198712) 05359 36

119862(1198712 119881) 02282 14

119862(119881 1198713) 05218 38

119862(1198713 119881) 02338 12

119862(119881 1198621) 05044 36

119862(1198621 119881) 02138 14

119862(119881 1198622) 04844 35

119862(1198622 119881) 02169 15

119862(119881 1198623) 05116 37

119862(1198623 119881) 02055 13



6 Conclusion



40 50 60 70 80 90 100 11012

14

16

18

20

22

24

26

28

Makespan

Mea

n w

orkl

oad

VNM

TTSP

CNSGA-L1

CNSGA-L2

CNSGA-L3

30lowast12


30 40 50 60 70 80 90 100 11014

16

18

20

22

24

26

Makespan

Mea

n w

orkl

oad

VNM

TTSP

CNSGA-C1

CNSGA-C2

CNSGA-C3

30lowast12



40 50 60 70 80 90 100 11014

16

18

20

22

24

26

28

30

32

Makespan

Mea

n w

orkl

oad

VNMCNSGA-

CNSGA-CNSGA-

TTSP

L1

L2

L3

40lowast12


40 50 60 70 80 90 100 110 12016

18

20

22

24

26

28

30

Makespan

Mea

n w

orkl

oad

VNM

TTSP

CNSGA-C1

CNSGA-C2

CNSGA-C3

40lowast12




TTSP

0010203040506070809

1

C-m

etric


30lowast12


0010203040506070809

1TTSP

C-m

etric


30lowast12


0010203040506070809

1TTSP

C-m

etric


40lowast12


TTSP

0010203040506070809

1

C-m

etric


40lowast12




Acknowledgments


References















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










P1

P4

P2 P3

r1 r2 r3

t1 [r1 r2 r3]

(1 2 3)

12 3




1

are 1199031 1199032 and 119903



1199013 and 119901



1 1199032 and 119903

3)



1 V2 V3


1 1199032 1199033]) where V

1 V2 and V

3

are bound to 1199031 1199032 and 119903

3


3


and 1199013are 1199031 1199032 and 119903


1 1199032 and


1 When the 119905


in 1199011 1199012 and 119901


4 The tokens



1 1199032 and




1 1199052 1199053 and 119905

4) and



needed by 1199051 1199052 1199053 1199054are 1199031 1199032 1199032 1199034 1199033 1199034 and 119903

1 1199033



t1 t2

t3 t4




1 1199052 119905

119895 (1 le


119894 (1 le 119894 le 119898)

119878119894

119895 119862119894119895 and 119875

119894



119894





119874119894

119895=

1 if 119905119895occupies 119903

119894

0 others(1)



119895= 1199081

119895 1199082

119895

119908

119896119895



119875119896

119895= max

119903119894isin119908119896119895119875119894


119895for 119908119896119895


119883119896119896lowast

119895119895lowast =

1 if 119908119896

119895cap 119908119896lowast

119895lowast = 0

0 others(2)


119894

119895=

119875119896

119895 119862119894119895= 119878119894

119895+ 119875119894





1(119909) and 119891

2(119909)


119862119896

119895= max

119903119894isin119908119896119895119862119894


119895for119908119896119895Thus


1198911(119909) = max

1le119896le1198961198951le119895le119899

119862119896

119895 (3)



lowast



1198912(119909) =

1

119876

119899

sum

119895=1

119898

sum

119894=1

119875119894

119895119874119894

119895 (4)




Let 1205821 1205822 120582


119911lowast

= (119911lowast

1 119911lowast

2 119911

lowast



119892te(119909 | 120582

119895 119911lowast) = max1le119894le119898

120582119895

119894

1003816100381610038161003816119891119894(119909) minus 119911

119894

lowast1003816100381610038161003816 119909 isin Ω (5)




Update EP


Output EP

Yes

No

Staring mutation

Crossover

Mutation

zlowast

Update



= (120582119895

1 120582119895

2 120582

119895

119898)119879

119911lowast

119894= min119891








119905+1

119894can


119909119905+1

119894=

119909119905

119894+ 1198651times (119909119905

119894minus 119909119905

1198941) + 1198652times (119909119905

119894minus 119909119905

1198942) rand (1)ltCR

119909119905

119894rand (1)geCR

(6)

CR 1198651 and 119865



1198941and 119909

119905






0 50 100 150 200 2505

10

15

20

25

30

Generations

Nei

ghbo

rhoo

d siz

e








1 1199102 and 119910








1 119909119894

2 119909

119894



for 119895 = 1 2 119872 119909119894lowast

119895=

normal (119909119894119895 120590) rand (1) lt 119901

119909119894

119895rand (1) ge 119901

(7)

119909119894lowast

= (119909119894lowast

1 119909119894lowast

2 119909119894lowast











Theorem 1 120572119899 120573119899 and 119903


120572119899= 119875 (119899 + 1) cap 119872 = 0 (119899) cap 119872 = 0

120573119899= 119875 (119899 + 1) cap 119872 = 0 (119899) cap 119872 = 0

119903119899= 12057311205732sim 120573119899

(8)


119903119899


Theorem 2 120572119899 120573119899 and 119903


120572119899= 119875 (119899 + 1) cap 119872

119888= 0(119899) cap 119872

119888= 0

120573119899= 119875 (119899 + 1) cap 119872

119888= 0(119899) cap 119872

119888= 0

119903119899= 12057311205732sim 120573119899

(9)



119903119899



120573119899


120573119899


119899rarrinfin119903119899= 0 lim

119899rarrinfin119903119899= 0


997888


119875 (119899 + 1) = 119875 (119899 + 1) cap 119872 = 0

= 119875 (119899 + 1) cap 119872 = 0(119899) cap 119872 = 0

sdot 119875 (119899) cap 119872 = 0

+ 119875 (119899 + 1) cap 119872 = 0(119899) cap 119872 = 0

sdot 119875 (119899) cap 119872 = 0

= 120572119899119875 (119899) cap 119872 = 0 + 120573

119899119875 (119899) cap 119872 = 0

(10)


119875 (119899 + 1) = 120573119899119875 (119899) (11)

Then

lim119899rarrinfin

119875 (119899 + 1) = lim119899rarrinfin

120573119899119875 (119899) = lim

119899rarrinfin119903119899119875 (0) = 0

lim119899rarrinfin

119875 (119899) cap 119872 = 0 = 1

(12)

Therefore we have

(119899) 997888rarr 119872(119875119882) (13)




119875 (119899 + 1) = 119875 (119899 + 1) cap 119872119888

= 0

= 120572119899119875 (119899) cap 119872

119888= 0 + 120573

119899119875 (119899)

(14)


120572119899= 0 Hence

lim119899rarrinfin

119875 (119899 + 1) = lim119899rarrinfin

120573119899119875 (119899) = lim

119899rarrinfin119903119899119875 (0) = 0 (15)

Then

lim119899rarrinfin

119875 (119899) cap 119872119888= 0 = 1

lim119899rarrinfin

119875 (119899) isin 119872 = 1

(16)

Therefore we have

(119899) 997888rarr 119872(119875119878) (17)



Theorem 3 (see [35]) 119875 = (119862 0



119875infin


119875119896= lim119896rarrinfin

(

119862119896

0

119896minus1

sum

119894=0

119879119894119877119862119896minus119894

119879119896)

= (

119862infin

0

119877infin

0)

(18)


= 11015840119901infin 119901infin =



119894gt 119901 for 1 le 119894 le 119898 and 119901

infin

119894= 0 for

119898 lt 119894 le 119899


+

1

119872+

2 selection 119878


119862+= (


d


) 119872+

1= (

1198721

sdot sdot sdot 0

d


)

119878+= (


d


) 119872+

2= (

1198722

sdot sdot sdot 0

d


)

119880 = (

11988011

sdot sdot sdot 0

d

11988021198991198971

sdot sdot sdot 11988021198991198972119899119897

)

(19)

119862+ 119872+1 119878+ 119872+


119899119897 square matrices 119862 1198721

1198722 119878 and119880

119886119887(1 le 119886 119887 le 2


119886 119887 in 119880119886119887














22= 1 if


11is a unit matrix



1 crossover 119862

+ mutation 119872+

2 selection 119878


+ for VNM is

119875+= 119872+

1119862+119872+

2119878+119880

= (

1198721

sdot sdot sdot 0

d


)(


d


)(

1198722

sdot sdot sdot 0

d


)

times (


d


)(

11988011

sdot sdot sdot 0

d

11988021198991198971

sdot sdot sdot 11988021198991198972119899119897

)

= (

11987211198621198722

sdot sdot sdot 0

d

0 sdot sdot sdot 11987211198621198722

)(

11988011

sdot sdot sdot 0

d

11988021198991198971

sdot sdot sdot 11988021198991198972119899119897

)

= (

1198721119862119872211988011

0 0

sdot sdot sdot 0

1198721119862119872211988021198991198971

sdot sdot sdot 1198721119862119872211988021198991198972119899119897

)

= (

11987511

sdot sdot sdot 0

d

11987521198991198971

sdot sdot sdot 11987521198991198972119899119897

)

(20)

There is 11987511







1199051

1199081

11199031 1199037

51199082

11199033 1199035

51199083

11199036 11990310

4

1199052

1199081

21199032 11990311

51199082

21199034 1199039

41199083

21199035 1199036

61199084

21199033 1199037

4

1199053

1199081

31199033

71199082

311990312

5

1199054

1199081

41199039

251199082

411990310

221199055

1199081

511990312

14

1199056

1199081

61199031 1199034

71199082

61199033 1199037

81199083

61199036 1199038

8

1199057

1199081

71199031 1199032

41199082

71199033 1199038

21199083

71199037 11990311

3

1199058

1199081

81199031 1199033

51199082

81199036 11990310

41199083

81199037 11990312

7

1199059

1199081

91199031 1199034

111199082

91199037 1199039

131199083

91199038 11990311

12

11990510

1199081

101199032

91199082

101199034

101199083

1011990310

10

11990511

1199081

111199032 1199037

61199082

111199033 11990312

91199083

111199038 1199039

8

11990512

1199081

121199032

111199082

121199035

131199083

1211990311

15

11990513

1199081

131199032

41199082

131199038

51199083

131199039

7

11990514

1199081

141199033

71199082

1411990311

101199083

1411990312

811990515

1199081

1511990312

2

11990516

1199081

161199032

91199082

161199035

71199083

161199038

6

11990517

1199081

171199031 11990310

101199082

171199035 1199039

121199083

1711990311 11990312

1111990518

1199081

181199036

15

11990519

1199081

191199032

81199082

191199035

71199083

1911990310

71199084

1911990312

6

11990520

1199081

201199033

61199082

201199036

41199083

201199039

5

11990521

1199081

211199031 1199034

21199082

211199033 1199035

51199083

211199036 1199038

3


Table 1 Continued


11990522

1199081

221199032

31199082

221199034

41199083

221199036

31199084

2211990310

4

11990523

1199081

231199033

51199082

2311990312

5

11990524

1199081

241199034

141199082

2411990311

1711990525

1199081

251199037

19

11990526

1199081

261199031 1199034

71199082

261199033 1199037

81199083

261199036 1199038

10

11990527

1199081

271199031 1199032

21199082

271199031 1199037

21199083

271199033 1199038

4

11990528

1199081

281199031 1199033

51199082

281199034 1199035

41199083

281199037 11990312

2

11990529

1199081

291199031 1199034

111199082

291199033 1199034

151199083

291199037 1199038

12

11990530

1199081

301199031

91199082

301199034

121199083

3011990312

10

11990531

1199081

311199032 1199033

61199082

311199035 11990311

81199083

311199036 1199039

8

11990532

1199081

321199032

111199082

321199035

131199083

321199036

17

11990533

1199081

331199032

61199082

331199036

51199083

3311990311

4

11990534

1199081

341199033

71199082

341199037

81199083

3411990312

1011990535

1199081

351199039

2

11990536

1199081

361199032

91199082

361199035

71199083

3611990310

6

11990537

1199081

371199031 1199032

101199082

371199037 11990311

71199083

371199035 11990312

1111990538

1199081

3811990310

15

11990539

1199081

391199034

81199082

391199036

71199083

391199039

71199084

3911990310

6

11990540

1199081

401199033

61199082

401199036

51199083

401199039

5



100 250 05 1 1 005


1 and 119865





119862 (119860 119861) =


1003816100381610038161003816

|119861|

(21)




0010203040506070809

1

C(M2M1) C(M1M2) C(M2M3) C(M3M2)

C-m

etric

TTSP 30lowast12



instance

Average Times119862(11987211198722) 02213 13119862(11987221198721) 05196 37119862(11987221198723) 04964 36119862(11987231198722) 02069 14





0010203040506070809

1

C(M2M1) C(M1M2) C(M2M3) C(M3M2)

C-m

etric

TTSP 40lowast12



instance

Average Times119862(11987211198722) 0244178 14119862(11987221198721) 0501508 36119862(11987221198723) 0533806 38119862(11987231198722) 0242146 12


Average Times119862(119881119872) 04845 35119862(119872119881) 02104 15


Average Times119862(119881119872) 05256 40119862(119872119881) 01949 10





40 50 60 70 80 90 10016

18

20

22

24

26

28

30

32

Makespan

Mea

n w

orkl

oad

VNMMOEAD

TTSP 30lowast12

(a)

40 50 60 70 80 9016

17

18

19

20

21

22

23

24

25

Makespan

Mea

n w

orkl

oad

VNMMOEAD

TTSP 30lowast12

(b)


50 55 60 65 70 75 80 85 9017

18

19

20

21

22

23

24

25

26

27

Makespan

Mea

n w

orkl

oad

TTSP

VNMMOEAD

40lowast12

(a)

50 55 60 65 70 75 8017

18

19

20

21

22

23

Makespan

Mea

n w

orkl

oad

VNMMOEAD

TTSP 40lowast12

(b)






11198621


1198622


1198623



0010203040506070809

1

C-m

etric

TTSP

C(VM) C(MV)

30lowast12


0010203040506070809

1

C-m

etric

TTSP

C(VM) C(MV)

40lowast12






Average Times119862(119881 119871

1) 04206 36

119862(1198711 119881) 02200 14

119862(119881 1198712) 04077 34

119862(1198712 119881) 02648 16

119862(119881 1198713) 04182 36

119862(1198713 119881) 02248 14

119862(119881 1198621) 04638 35

119862(1198621 119881) 02210 15

119862(119881 1198622) 04602 34

119862(1198622 119881) 02288 16

119862(119881 1198623) 04128 35

119862(1198623 119881) 02525 15


Average Times119862(119881 119871

1) 05243 37

119862(1198711 119881) 02264 13

119862(119881 1198712) 05359 36

119862(1198712 119881) 02282 14

119862(119881 1198713) 05218 38

119862(1198713 119881) 02338 12

119862(119881 1198621) 05044 36

119862(1198621 119881) 02138 14

119862(119881 1198622) 04844 35

119862(1198622 119881) 02169 15

119862(119881 1198623) 05116 37

119862(1198623 119881) 02055 13



6 Conclusion



40 50 60 70 80 90 100 11012

14

16

18

20

22

24

26

28

Makespan

Mea

n w

orkl

oad

VNM

TTSP

CNSGA-L1

CNSGA-L2

CNSGA-L3

30lowast12


30 40 50 60 70 80 90 100 11014

16

18

20

22

24

26

Makespan

Mea

n w

orkl

oad

VNM

TTSP

CNSGA-C1

CNSGA-C2

CNSGA-C3

30lowast12



40 50 60 70 80 90 100 11014

16

18

20

22

24

26

28

30

32

Makespan

Mea

n w

orkl

oad

VNMCNSGA-

CNSGA-CNSGA-

TTSP

L1

L2

L3

40lowast12


40 50 60 70 80 90 100 110 12016

18

20

22

24

26

28

30

Makespan

Mea

n w

orkl

oad

VNM

TTSP

CNSGA-C1

CNSGA-C2

CNSGA-C3

40lowast12




TTSP

0010203040506070809

1

C-m

etric


30lowast12


0010203040506070809

1TTSP

C-m

etric


30lowast12


0010203040506070809

1TTSP

C-m

etric


40lowast12


TTSP

0010203040506070809

1

C-m

etric


40lowast12




Acknowledgments


References















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1(119909) and 119891

2(119909)


119862119896

119895= max

119903119894isin119908119896119895119862119894


119895for119908119896119895Thus


1198911(119909) = max

1le119896le1198961198951le119895le119899

119862119896

119895 (3)



lowast



1198912(119909) =

1

119876

119899

sum

119895=1

119898

sum

119894=1

119875119894

119895119874119894

119895 (4)




Let 1205821 1205822 120582


119911lowast

= (119911lowast

1 119911lowast

2 119911

lowast



119892te(119909 | 120582

119895 119911lowast) = max1le119894le119898

120582119895

119894

1003816100381610038161003816119891119894(119909) minus 119911

119894

lowast1003816100381610038161003816 119909 isin Ω (5)




Update EP


Output EP

Yes

No

Staring mutation

Crossover

Mutation

zlowast

Update



= (120582119895

1 120582119895

2 120582

119895

119898)119879

119911lowast

119894= min119891








119905+1

119894can


119909119905+1

119894=

119909119905

119894+ 1198651times (119909119905

119894minus 119909119905

1198941) + 1198652times (119909119905

119894minus 119909119905

1198942) rand (1)ltCR

119909119905

119894rand (1)geCR

(6)

CR 1198651 and 119865



1198941and 119909

119905






0 50 100 150 200 2505

10

15

20

25

30

Generations

Nei

ghbo

rhoo

d siz

e








1 1199102 and 119910








1 119909119894

2 119909

119894



for 119895 = 1 2 119872 119909119894lowast

119895=

normal (119909119894119895 120590) rand (1) lt 119901

119909119894

119895rand (1) ge 119901

(7)

119909119894lowast

= (119909119894lowast

1 119909119894lowast

2 119909119894lowast











Theorem 1 120572119899 120573119899 and 119903


120572119899= 119875 (119899 + 1) cap 119872 = 0 (119899) cap 119872 = 0

120573119899= 119875 (119899 + 1) cap 119872 = 0 (119899) cap 119872 = 0

119903119899= 12057311205732sim 120573119899

(8)


119903119899


Theorem 2 120572119899 120573119899 and 119903


120572119899= 119875 (119899 + 1) cap 119872

119888= 0(119899) cap 119872

119888= 0

120573119899= 119875 (119899 + 1) cap 119872

119888= 0(119899) cap 119872

119888= 0

119903119899= 12057311205732sim 120573119899

(9)



119903119899



120573119899


120573119899


119899rarrinfin119903119899= 0 lim

119899rarrinfin119903119899= 0


997888


119875 (119899 + 1) = 119875 (119899 + 1) cap 119872 = 0

= 119875 (119899 + 1) cap 119872 = 0(119899) cap 119872 = 0

sdot 119875 (119899) cap 119872 = 0

+ 119875 (119899 + 1) cap 119872 = 0(119899) cap 119872 = 0

sdot 119875 (119899) cap 119872 = 0

= 120572119899119875 (119899) cap 119872 = 0 + 120573

119899119875 (119899) cap 119872 = 0

(10)


119875 (119899 + 1) = 120573119899119875 (119899) (11)

Then

lim119899rarrinfin

119875 (119899 + 1) = lim119899rarrinfin

120573119899119875 (119899) = lim

119899rarrinfin119903119899119875 (0) = 0

lim119899rarrinfin

119875 (119899) cap 119872 = 0 = 1

(12)

Therefore we have

(119899) 997888rarr 119872(119875119882) (13)




119875 (119899 + 1) = 119875 (119899 + 1) cap 119872119888

= 0

= 120572119899119875 (119899) cap 119872

119888= 0 + 120573

119899119875 (119899)

(14)


120572119899= 0 Hence

lim119899rarrinfin

119875 (119899 + 1) = lim119899rarrinfin

120573119899119875 (119899) = lim

119899rarrinfin119903119899119875 (0) = 0 (15)

Then

lim119899rarrinfin

119875 (119899) cap 119872119888= 0 = 1

lim119899rarrinfin

119875 (119899) isin 119872 = 1

(16)

Therefore we have

(119899) 997888rarr 119872(119875119878) (17)



Theorem 3 (see [35]) 119875 = (119862 0



119875infin


119875119896= lim119896rarrinfin

(

119862119896

0

119896minus1

sum

119894=0

119879119894119877119862119896minus119894

119879119896)

= (

119862infin

0

119877infin

0)

(18)


= 11015840119901infin 119901infin =



119894gt 119901 for 1 le 119894 le 119898 and 119901

infin

119894= 0 for

119898 lt 119894 le 119899


+

1

119872+

2 selection 119878


119862+= (


d


) 119872+

1= (

1198721

sdot sdot sdot 0

d


)

119878+= (


d


) 119872+

2= (

1198722

sdot sdot sdot 0

d


)

119880 = (

11988011

sdot sdot sdot 0

d

11988021198991198971

sdot sdot sdot 11988021198991198972119899119897

)

(19)

119862+ 119872+1 119878+ 119872+


119899119897 square matrices 119862 1198721

1198722 119878 and119880

119886119887(1 le 119886 119887 le 2


119886 119887 in 119880119886119887














22= 1 if


11is a unit matrix



1 crossover 119862

+ mutation 119872+

2 selection 119878


+ for VNM is

119875+= 119872+

1119862+119872+

2119878+119880

= (

1198721

sdot sdot sdot 0

d


)(


d


)(

1198722

sdot sdot sdot 0

d


)

times (


d


)(

11988011

sdot sdot sdot 0

d

11988021198991198971

sdot sdot sdot 11988021198991198972119899119897

)

= (

11987211198621198722

sdot sdot sdot 0

d

0 sdot sdot sdot 11987211198621198722

)(

11988011

sdot sdot sdot 0

d

11988021198991198971

sdot sdot sdot 11988021198991198972119899119897

)

= (

1198721119862119872211988011

0 0

sdot sdot sdot 0

1198721119862119872211988021198991198971

sdot sdot sdot 1198721119862119872211988021198991198972119899119897

)

= (

11987511

sdot sdot sdot 0

d

11987521198991198971

sdot sdot sdot 11987521198991198972119899119897

)

(20)

There is 11987511







1199051

1199081

11199031 1199037

51199082

11199033 1199035

51199083

11199036 11990310

4

1199052

1199081

21199032 11990311

51199082

21199034 1199039

41199083

21199035 1199036

61199084

21199033 1199037

4

1199053

1199081

31199033

71199082

311990312

5

1199054

1199081

41199039

251199082

411990310

221199055

1199081

511990312

14

1199056

1199081

61199031 1199034

71199082

61199033 1199037

81199083

61199036 1199038

8

1199057

1199081

71199031 1199032

41199082

71199033 1199038

21199083

71199037 11990311

3

1199058

1199081

81199031 1199033

51199082

81199036 11990310

41199083

81199037 11990312

7

1199059

1199081

91199031 1199034

111199082

91199037 1199039

131199083

91199038 11990311

12

11990510

1199081

101199032

91199082

101199034

101199083

1011990310

10

11990511

1199081

111199032 1199037

61199082

111199033 11990312

91199083

111199038 1199039

8

11990512

1199081

121199032

111199082

121199035

131199083

1211990311

15

11990513

1199081

131199032

41199082

131199038

51199083

131199039

7

11990514

1199081

141199033

71199082

1411990311

101199083

1411990312

811990515

1199081

1511990312

2

11990516

1199081

161199032

91199082

161199035

71199083

161199038

6

11990517

1199081

171199031 11990310

101199082

171199035 1199039

121199083

1711990311 11990312

1111990518

1199081

181199036

15

11990519

1199081

191199032

81199082

191199035

71199083

1911990310

71199084

1911990312

6

11990520

1199081

201199033

61199082

201199036

41199083

201199039

5

11990521

1199081

211199031 1199034

21199082

211199033 1199035

51199083

211199036 1199038

3


Table 1 Continued


11990522

1199081

221199032

31199082

221199034

41199083

221199036

31199084

2211990310

4

11990523

1199081

231199033

51199082

2311990312

5

11990524

1199081

241199034

141199082

2411990311

1711990525

1199081

251199037

19

11990526

1199081

261199031 1199034

71199082

261199033 1199037

81199083

261199036 1199038

10

11990527

1199081

271199031 1199032

21199082

271199031 1199037

21199083

271199033 1199038

4

11990528

1199081

281199031 1199033

51199082

281199034 1199035

41199083

281199037 11990312

2

11990529

1199081

291199031 1199034

111199082

291199033 1199034

151199083

291199037 1199038

12

11990530

1199081

301199031

91199082

301199034

121199083

3011990312

10

11990531

1199081

311199032 1199033

61199082

311199035 11990311

81199083

311199036 1199039

8

11990532

1199081

321199032

111199082

321199035

131199083

321199036

17

11990533

1199081

331199032

61199082

331199036

51199083

3311990311

4

11990534

1199081

341199033

71199082

341199037

81199083

3411990312

1011990535

1199081

351199039

2

11990536

1199081

361199032

91199082

361199035

71199083

3611990310

6

11990537

1199081

371199031 1199032

101199082

371199037 11990311

71199083

371199035 11990312

1111990538

1199081

3811990310

15

11990539

1199081

391199034

81199082

391199036

71199083

391199039

71199084

3911990310

6

11990540

1199081

401199033

61199082

401199036

51199083

401199039

5



100 250 05 1 1 005


1 and 119865





119862 (119860 119861) =


1003816100381610038161003816

|119861|

(21)




0010203040506070809

1

C(M2M1) C(M1M2) C(M2M3) C(M3M2)

C-m

etric

TTSP 30lowast12



instance

Average Times119862(11987211198722) 02213 13119862(11987221198721) 05196 37119862(11987221198723) 04964 36119862(11987231198722) 02069 14





0010203040506070809

1

C(M2M1) C(M1M2) C(M2M3) C(M3M2)

C-m

etric

TTSP 40lowast12



instance

Average Times119862(11987211198722) 0244178 14119862(11987221198721) 0501508 36119862(11987221198723) 0533806 38119862(11987231198722) 0242146 12


Average Times119862(119881119872) 04845 35119862(119872119881) 02104 15


Average Times119862(119881119872) 05256 40119862(119872119881) 01949 10





40 50 60 70 80 90 10016

18

20

22

24

26

28

30

32

Makespan

Mea

n w

orkl

oad

VNMMOEAD

TTSP 30lowast12

(a)

40 50 60 70 80 9016

17

18

19

20

21

22

23

24

25

Makespan

Mea

n w

orkl

oad

VNMMOEAD

TTSP 30lowast12

(b)


50 55 60 65 70 75 80 85 9017

18

19

20

21

22

23

24

25

26

27

Makespan

Mea

n w

orkl

oad

TTSP

VNMMOEAD

40lowast12

(a)

50 55 60 65 70 75 8017

18

19

20

21

22

23

Makespan

Mea

n w

orkl

oad

VNMMOEAD

TTSP 40lowast12

(b)






11198621


1198622


1198623



0010203040506070809

1

C-m

etric

TTSP

C(VM) C(MV)

30lowast12


0010203040506070809

1

C-m

etric

TTSP

C(VM) C(MV)

40lowast12






Average Times119862(119881 119871

1) 04206 36

119862(1198711 119881) 02200 14

119862(119881 1198712) 04077 34

119862(1198712 119881) 02648 16

119862(119881 1198713) 04182 36

119862(1198713 119881) 02248 14

119862(119881 1198621) 04638 35

119862(1198621 119881) 02210 15

119862(119881 1198622) 04602 34

119862(1198622 119881) 02288 16

119862(119881 1198623) 04128 35

119862(1198623 119881) 02525 15


Average Times119862(119881 119871

1) 05243 37

119862(1198711 119881) 02264 13

119862(119881 1198712) 05359 36

119862(1198712 119881) 02282 14

119862(119881 1198713) 05218 38

119862(1198713 119881) 02338 12

119862(119881 1198621) 05044 36

119862(1198621 119881) 02138 14

119862(119881 1198622) 04844 35

119862(1198622 119881) 02169 15

119862(119881 1198623) 05116 37

119862(1198623 119881) 02055 13



6 Conclusion



40 50 60 70 80 90 100 11012

14

16

18

20

22

24

26

28

Makespan

Mea

n w

orkl

oad

VNM

TTSP

CNSGA-L1

CNSGA-L2

CNSGA-L3

30lowast12


30 40 50 60 70 80 90 100 11014

16

18

20

22

24

26

Makespan

Mea

n w

orkl

oad

VNM

TTSP

CNSGA-C1

CNSGA-C2

CNSGA-C3

30lowast12



40 50 60 70 80 90 100 11014

16

18

20

22

24

26

28

30

32

Makespan

Mea

n w

orkl

oad

VNMCNSGA-

CNSGA-CNSGA-

TTSP

L1

L2

L3

40lowast12


40 50 60 70 80 90 100 110 12016

18

20

22

24

26

28

30

Makespan

Mea

n w

orkl

oad

VNM

TTSP

CNSGA-C1

CNSGA-C2

CNSGA-C3

40lowast12




TTSP

0010203040506070809

1

C-m

etric


30lowast12


0010203040506070809

1TTSP

C-m

etric


30lowast12


0010203040506070809

1TTSP

C-m

etric


40lowast12


TTSP

0010203040506070809

1

C-m

etric


40lowast12




Acknowledgments


References















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 50 100 150 200 2505

10

15

20

25

30

Generations

Nei

ghbo

rhoo

d siz

e








1 1199102 and 119910








1 119909119894

2 119909

119894



for 119895 = 1 2 119872 119909119894lowast

119895=

normal (119909119894119895 120590) rand (1) lt 119901

119909119894

119895rand (1) ge 119901

(7)

119909119894lowast

= (119909119894lowast

1 119909119894lowast

2 119909119894lowast











Theorem 1 120572119899 120573119899 and 119903


120572119899= 119875 (119899 + 1) cap 119872 = 0 (119899) cap 119872 = 0

120573119899= 119875 (119899 + 1) cap 119872 = 0 (119899) cap 119872 = 0

119903119899= 12057311205732sim 120573119899

(8)


119903119899


Theorem 2 120572119899 120573119899 and 119903


120572119899= 119875 (119899 + 1) cap 119872

119888= 0(119899) cap 119872

119888= 0

120573119899= 119875 (119899 + 1) cap 119872

119888= 0(119899) cap 119872

119888= 0

119903119899= 12057311205732sim 120573119899

(9)



119903119899



120573119899


120573119899


119899rarrinfin119903119899= 0 lim

119899rarrinfin119903119899= 0


997888


119875 (119899 + 1) = 119875 (119899 + 1) cap 119872 = 0

= 119875 (119899 + 1) cap 119872 = 0(119899) cap 119872 = 0

sdot 119875 (119899) cap 119872 = 0

+ 119875 (119899 + 1) cap 119872 = 0(119899) cap 119872 = 0

sdot 119875 (119899) cap 119872 = 0

= 120572119899119875 (119899) cap 119872 = 0 + 120573

119899119875 (119899) cap 119872 = 0

(10)


119875 (119899 + 1) = 120573119899119875 (119899) (11)

Then

lim119899rarrinfin

119875 (119899 + 1) = lim119899rarrinfin

120573119899119875 (119899) = lim

119899rarrinfin119903119899119875 (0) = 0

lim119899rarrinfin

119875 (119899) cap 119872 = 0 = 1

(12)

Therefore we have

(119899) 997888rarr 119872(119875119882) (13)




119875 (119899 + 1) = 119875 (119899 + 1) cap 119872119888

= 0

= 120572119899119875 (119899) cap 119872

119888= 0 + 120573

119899119875 (119899)

(14)


120572119899= 0 Hence

lim119899rarrinfin

119875 (119899 + 1) = lim119899rarrinfin

120573119899119875 (119899) = lim

119899rarrinfin119903119899119875 (0) = 0 (15)

Then

lim119899rarrinfin

119875 (119899) cap 119872119888= 0 = 1

lim119899rarrinfin

119875 (119899) isin 119872 = 1

(16)

Therefore we have

(119899) 997888rarr 119872(119875119878) (17)



Theorem 3 (see [35]) 119875 = (119862 0



119875infin


119875119896= lim119896rarrinfin

(

119862119896

0

119896minus1

sum

119894=0

119879119894119877119862119896minus119894

119879119896)

= (

119862infin

0

119877infin

0)

(18)


= 11015840119901infin 119901infin =



119894gt 119901 for 1 le 119894 le 119898 and 119901

infin

119894= 0 for

119898 lt 119894 le 119899


+

1

119872+

2 selection 119878


119862+= (


d


) 119872+

1= (

1198721

sdot sdot sdot 0

d


)

119878+= (


d


) 119872+

2= (

1198722

sdot sdot sdot 0

d


)

119880 = (

11988011

sdot sdot sdot 0

d

11988021198991198971

sdot sdot sdot 11988021198991198972119899119897

)

(19)

119862+ 119872+1 119878+ 119872+


119899119897 square matrices 119862 1198721

1198722 119878 and119880

119886119887(1 le 119886 119887 le 2


119886 119887 in 119880119886119887














22= 1 if


11is a unit matrix



1 crossover 119862

+ mutation 119872+

2 selection 119878


+ for VNM is

119875+= 119872+

1119862+119872+

2119878+119880

= (

1198721

sdot sdot sdot 0

d


)(


d


)(

1198722

sdot sdot sdot 0

d


)

times (


d


)(

11988011

sdot sdot sdot 0

d

11988021198991198971

sdot sdot sdot 11988021198991198972119899119897

)

= (

11987211198621198722

sdot sdot sdot 0

d

0 sdot sdot sdot 11987211198621198722

)(

11988011

sdot sdot sdot 0

d

11988021198991198971

sdot sdot sdot 11988021198991198972119899119897

)

= (

1198721119862119872211988011

0 0

sdot sdot sdot 0

1198721119862119872211988021198991198971

sdot sdot sdot 1198721119862119872211988021198991198972119899119897

)

= (

11987511

sdot sdot sdot 0

d

11987521198991198971

sdot sdot sdot 11987521198991198972119899119897

)

(20)

There is 11987511







1199051

1199081

11199031 1199037

51199082

11199033 1199035

51199083

11199036 11990310

4

1199052

1199081

21199032 11990311

51199082

21199034 1199039

41199083

21199035 1199036

61199084

21199033 1199037

4

1199053

1199081

31199033

71199082

311990312

5

1199054

1199081

41199039

251199082

411990310

221199055

1199081

511990312

14

1199056

1199081

61199031 1199034

71199082

61199033 1199037

81199083

61199036 1199038

8

1199057

1199081

71199031 1199032

41199082

71199033 1199038

21199083

71199037 11990311

3

1199058

1199081

81199031 1199033

51199082

81199036 11990310

41199083

81199037 11990312

7

1199059

1199081

91199031 1199034

111199082

91199037 1199039

131199083

91199038 11990311

12

11990510

1199081

101199032

91199082

101199034

101199083

1011990310

10

11990511

1199081

111199032 1199037

61199082

111199033 11990312

91199083

111199038 1199039

8

11990512

1199081

121199032

111199082

121199035

131199083

1211990311

15

11990513

1199081

131199032

41199082

131199038

51199083

131199039

7

11990514

1199081

141199033

71199082

1411990311

101199083

1411990312

811990515

1199081

1511990312

2

11990516

1199081

161199032

91199082

161199035

71199083

161199038

6

11990517

1199081

171199031 11990310

101199082

171199035 1199039

121199083

1711990311 11990312

1111990518

1199081

181199036

15

11990519

1199081

191199032

81199082

191199035

71199083

1911990310

71199084

1911990312

6

11990520

1199081

201199033

61199082

201199036

41199083

201199039

5

11990521

1199081

211199031 1199034

21199082

211199033 1199035

51199083

211199036 1199038

3


Table 1 Continued


11990522

1199081

221199032

31199082

221199034

41199083

221199036

31199084

2211990310

4

11990523

1199081

231199033

51199082

2311990312

5

11990524

1199081

241199034

141199082

2411990311

1711990525

1199081

251199037

19

11990526

1199081

261199031 1199034

71199082

261199033 1199037

81199083

261199036 1199038

10

11990527

1199081

271199031 1199032

21199082

271199031 1199037

21199083

271199033 1199038

4

11990528

1199081

281199031 1199033

51199082

281199034 1199035

41199083

281199037 11990312

2

11990529

1199081

291199031 1199034

111199082

291199033 1199034

151199083

291199037 1199038

12

11990530

1199081

301199031

91199082

301199034

121199083

3011990312

10

11990531

1199081

311199032 1199033

61199082

311199035 11990311

81199083

311199036 1199039

8

11990532

1199081

321199032

111199082

321199035

131199083

321199036

17

11990533

1199081

331199032

61199082

331199036

51199083

3311990311

4

11990534

1199081

341199033

71199082

341199037

81199083

3411990312

1011990535

1199081

351199039

2

11990536

1199081

361199032

91199082

361199035

71199083

3611990310

6

11990537

1199081

371199031 1199032

101199082

371199037 11990311

71199083

371199035 11990312

1111990538

1199081

3811990310

15

11990539

1199081

391199034

81199082

391199036

71199083

391199039

71199084

3911990310

6

11990540

1199081

401199033

61199082

401199036

51199083

401199039

5



100 250 05 1 1 005


1 and 119865





119862 (119860 119861) =


1003816100381610038161003816

|119861|

(21)




0010203040506070809

1

C(M2M1) C(M1M2) C(M2M3) C(M3M2)

C-m

etric

TTSP 30lowast12



instance

Average Times119862(11987211198722) 02213 13119862(11987221198721) 05196 37119862(11987221198723) 04964 36119862(11987231198722) 02069 14





0010203040506070809

1

C(M2M1) C(M1M2) C(M2M3) C(M3M2)

C-m

etric

TTSP 40lowast12



instance

Average Times119862(11987211198722) 0244178 14119862(11987221198721) 0501508 36119862(11987221198723) 0533806 38119862(11987231198722) 0242146 12


Average Times119862(119881119872) 04845 35119862(119872119881) 02104 15


Average Times119862(119881119872) 05256 40119862(119872119881) 01949 10





40 50 60 70 80 90 10016

18

20

22

24

26

28

30

32

Makespan

Mea

n w

orkl

oad

VNMMOEAD

TTSP 30lowast12

(a)

40 50 60 70 80 9016

17

18

19

20

21

22

23

24

25

Makespan

Mea

n w

orkl

oad

VNMMOEAD

TTSP 30lowast12

(b)


50 55 60 65 70 75 80 85 9017

18

19

20

21

22

23

24

25

26

27

Makespan

Mea

n w

orkl

oad

TTSP

VNMMOEAD

40lowast12

(a)

50 55 60 65 70 75 8017

18

19

20

21

22

23

Makespan

Mea

n w

orkl

oad

VNMMOEAD

TTSP 40lowast12

(b)






11198621


1198622


1198623



0010203040506070809

1

C-m

etric

TTSP

C(VM) C(MV)

30lowast12


0010203040506070809

1

C-m

etric

TTSP

C(VM) C(MV)

40lowast12






Average Times119862(119881 119871

1) 04206 36

119862(1198711 119881) 02200 14

119862(119881 1198712) 04077 34

119862(1198712 119881) 02648 16

119862(119881 1198713) 04182 36

119862(1198713 119881) 02248 14

119862(119881 1198621) 04638 35

119862(1198621 119881) 02210 15

119862(119881 1198622) 04602 34

119862(1198622 119881) 02288 16

119862(119881 1198623) 04128 35

119862(1198623 119881) 02525 15


Average Times119862(119881 119871

1) 05243 37

119862(1198711 119881) 02264 13

119862(119881 1198712) 05359 36

119862(1198712 119881) 02282 14

119862(119881 1198713) 05218 38

119862(1198713 119881) 02338 12

119862(119881 1198621) 05044 36

119862(1198621 119881) 02138 14

119862(119881 1198622) 04844 35

119862(1198622 119881) 02169 15

119862(119881 1198623) 05116 37

119862(1198623 119881) 02055 13



6 Conclusion



40 50 60 70 80 90 100 11012

14

16

18

20

22

24

26

28

Makespan

Mea

n w

orkl

oad

VNM

TTSP

CNSGA-L1

CNSGA-L2

CNSGA-L3

30lowast12


30 40 50 60 70 80 90 100 11014

16

18

20

22

24

26

Makespan

Mea

n w

orkl

oad

VNM

TTSP

CNSGA-C1

CNSGA-C2

CNSGA-C3

30lowast12



40 50 60 70 80 90 100 11014

16

18

20

22

24

26

28

30

32

Makespan

Mea

n w

orkl

oad

VNMCNSGA-

CNSGA-CNSGA-

TTSP

L1

L2

L3

40lowast12


40 50 60 70 80 90 100 110 12016

18

20

22

24

26

28

30

Makespan

Mea

n w

orkl

oad

VNM

TTSP

CNSGA-C1

CNSGA-C2

CNSGA-C3

40lowast12




TTSP

0010203040506070809

1

C-m

etric


30lowast12


0010203040506070809

1TTSP

C-m

etric


30lowast12


0010203040506070809

1TTSP

C-m

etric


40lowast12


TTSP

0010203040506070809

1

C-m

etric


40lowast12




Acknowledgments


References















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119903119899



120573119899


120573119899


119899rarrinfin119903119899= 0 lim

119899rarrinfin119903119899= 0


997888


119875 (119899 + 1) = 119875 (119899 + 1) cap 119872 = 0

= 119875 (119899 + 1) cap 119872 = 0(119899) cap 119872 = 0

sdot 119875 (119899) cap 119872 = 0

+ 119875 (119899 + 1) cap 119872 = 0(119899) cap 119872 = 0

sdot 119875 (119899) cap 119872 = 0

= 120572119899119875 (119899) cap 119872 = 0 + 120573

119899119875 (119899) cap 119872 = 0

(10)


119875 (119899 + 1) = 120573119899119875 (119899) (11)

Then

lim119899rarrinfin

119875 (119899 + 1) = lim119899rarrinfin

120573119899119875 (119899) = lim

119899rarrinfin119903119899119875 (0) = 0

lim119899rarrinfin

119875 (119899) cap 119872 = 0 = 1

(12)

Therefore we have

(119899) 997888rarr 119872(119875119882) (13)




119875 (119899 + 1) = 119875 (119899 + 1) cap 119872119888

= 0

= 120572119899119875 (119899) cap 119872

119888= 0 + 120573

119899119875 (119899)

(14)


120572119899= 0 Hence

lim119899rarrinfin

119875 (119899 + 1) = lim119899rarrinfin

120573119899119875 (119899) = lim

119899rarrinfin119903119899119875 (0) = 0 (15)

Then

lim119899rarrinfin

119875 (119899) cap 119872119888= 0 = 1

lim119899rarrinfin

119875 (119899) isin 119872 = 1

(16)

Therefore we have

(119899) 997888rarr 119872(119875119878) (17)



Theorem 3 (see [35]) 119875 = (119862 0



119875infin


119875119896= lim119896rarrinfin

(

119862119896

0

119896minus1

sum

119894=0

119879119894119877119862119896minus119894

119879119896)

= (

119862infin

0

119877infin

0)

(18)


= 11015840119901infin 119901infin =



119894gt 119901 for 1 le 119894 le 119898 and 119901

infin

119894= 0 for

119898 lt 119894 le 119899


+

1

119872+

2 selection 119878


119862+= (


d


) 119872+

1= (

1198721

sdot sdot sdot 0

d


)

119878+= (


d


) 119872+

2= (

1198722

sdot sdot sdot 0

d


)

119880 = (

11988011

sdot sdot sdot 0

d

11988021198991198971

sdot sdot sdot 11988021198991198972119899119897

)

(19)

119862+ 119872+1 119878+ 119872+


119899119897 square matrices 119862 1198721

1198722 119878 and119880

119886119887(1 le 119886 119887 le 2


119886 119887 in 119880119886119887














22= 1 if


11is a unit matrix



1 crossover 119862

+ mutation 119872+

2 selection 119878


+ for VNM is

119875+= 119872+

1119862+119872+

2119878+119880

= (

1198721

sdot sdot sdot 0

d


)(


d


)(

1198722

sdot sdot sdot 0

d


)

times (


d


)(

11988011

sdot sdot sdot 0

d

11988021198991198971

sdot sdot sdot 11988021198991198972119899119897

)

= (

11987211198621198722

sdot sdot sdot 0

d

0 sdot sdot sdot 11987211198621198722

)(

11988011

sdot sdot sdot 0

d

11988021198991198971

sdot sdot sdot 11988021198991198972119899119897

)

= (

1198721119862119872211988011

0 0

sdot sdot sdot 0

1198721119862119872211988021198991198971

sdot sdot sdot 1198721119862119872211988021198991198972119899119897

)

= (

11987511

sdot sdot sdot 0

d

11987521198991198971

sdot sdot sdot 11987521198991198972119899119897

)

(20)

There is 11987511







1199051

1199081

11199031 1199037

51199082

11199033 1199035

51199083

11199036 11990310

4

1199052

1199081

21199032 11990311

51199082

21199034 1199039

41199083

21199035 1199036

61199084

21199033 1199037

4

1199053

1199081

31199033

71199082

311990312

5

1199054

1199081

41199039

251199082

411990310

221199055

1199081

511990312

14

1199056

1199081

61199031 1199034

71199082

61199033 1199037

81199083

61199036 1199038

8

1199057

1199081

71199031 1199032

41199082

71199033 1199038

21199083

71199037 11990311

3

1199058

1199081

81199031 1199033

51199082

81199036 11990310

41199083

81199037 11990312

7

1199059

1199081

91199031 1199034

111199082

91199037 1199039

131199083

91199038 11990311

12

11990510

1199081

101199032

91199082

101199034

101199083

1011990310

10

11990511

1199081

111199032 1199037

61199082

111199033 11990312

91199083

111199038 1199039

8

11990512

1199081

121199032

111199082

121199035

131199083

1211990311

15

11990513

1199081

131199032

41199082

131199038

51199083

131199039

7

11990514

1199081

141199033

71199082

1411990311

101199083

1411990312

811990515

1199081

1511990312

2

11990516

1199081

161199032

91199082

161199035

71199083

161199038

6

11990517

1199081

171199031 11990310

101199082

171199035 1199039

121199083

1711990311 11990312

1111990518

1199081

181199036

15

11990519

1199081

191199032

81199082

191199035

71199083

1911990310

71199084

1911990312

6

11990520

1199081

201199033

61199082

201199036

41199083

201199039

5

11990521

1199081

211199031 1199034

21199082

211199033 1199035

51199083

211199036 1199038

3


Table 1 Continued


11990522

1199081

221199032

31199082

221199034

41199083

221199036

31199084

2211990310

4

11990523

1199081

231199033

51199082

2311990312

5

11990524

1199081

241199034

141199082

2411990311

1711990525

1199081

251199037

19

11990526

1199081

261199031 1199034

71199082

261199033 1199037

81199083

261199036 1199038

10

11990527

1199081

271199031 1199032

21199082

271199031 1199037

21199083

271199033 1199038

4

11990528

1199081

281199031 1199033

51199082

281199034 1199035

41199083

281199037 11990312

2

11990529

1199081

291199031 1199034

111199082

291199033 1199034

151199083

291199037 1199038

12

11990530

1199081

301199031

91199082

301199034

121199083

3011990312

10

11990531

1199081

311199032 1199033

61199082

311199035 11990311

81199083

311199036 1199039

8

11990532

1199081

321199032

111199082

321199035

131199083

321199036

17

11990533

1199081

331199032

61199082

331199036

51199083

3311990311

4

11990534

1199081

341199033

71199082

341199037

81199083

3411990312

1011990535

1199081

351199039

2

11990536

1199081

361199032

91199082

361199035

71199083

3611990310

6

11990537

1199081

371199031 1199032

101199082

371199037 11990311

71199083

371199035 11990312

1111990538

1199081

3811990310

15

11990539

1199081

391199034

81199082

391199036

71199083

391199039

71199084

3911990310

6

11990540

1199081

401199033

61199082

401199036

51199083

401199039

5



100 250 05 1 1 005


1 and 119865





119862 (119860 119861) =


1003816100381610038161003816

|119861|

(21)




0010203040506070809

1

C(M2M1) C(M1M2) C(M2M3) C(M3M2)

C-m

etric

TTSP 30lowast12



instance

Average Times119862(11987211198722) 02213 13119862(11987221198721) 05196 37119862(11987221198723) 04964 36119862(11987231198722) 02069 14





0010203040506070809

1

C(M2M1) C(M1M2) C(M2M3) C(M3M2)

C-m

etric

TTSP 40lowast12



instance

Average Times119862(11987211198722) 0244178 14119862(11987221198721) 0501508 36119862(11987221198723) 0533806 38119862(11987231198722) 0242146 12


Average Times119862(119881119872) 04845 35119862(119872119881) 02104 15


Average Times119862(119881119872) 05256 40119862(119872119881) 01949 10





40 50 60 70 80 90 10016

18

20

22

24

26

28

30

32

Makespan

Mea

n w

orkl

oad

VNMMOEAD

TTSP 30lowast12

(a)

40 50 60 70 80 9016

17

18

19

20

21

22

23

24

25

Makespan

Mea

n w

orkl

oad

VNMMOEAD

TTSP 30lowast12

(b)


50 55 60 65 70 75 80 85 9017

18

19

20

21

22

23

24

25

26

27

Makespan

Mea

n w

orkl

oad

TTSP

VNMMOEAD

40lowast12

(a)

50 55 60 65 70 75 8017

18

19

20

21

22

23

Makespan

Mea

n w

orkl

oad

VNMMOEAD

TTSP 40lowast12

(b)






11198621


1198622


1198623



0010203040506070809

1

C-m

etric

TTSP

C(VM) C(MV)

30lowast12


0010203040506070809

1

C-m

etric

TTSP

C(VM) C(MV)

40lowast12






Average Times119862(119881 119871

1) 04206 36

119862(1198711 119881) 02200 14

119862(119881 1198712) 04077 34

119862(1198712 119881) 02648 16

119862(119881 1198713) 04182 36

119862(1198713 119881) 02248 14

119862(119881 1198621) 04638 35

119862(1198621 119881) 02210 15

119862(119881 1198622) 04602 34

119862(1198622 119881) 02288 16

119862(119881 1198623) 04128 35

119862(1198623 119881) 02525 15


Average Times119862(119881 119871

1) 05243 37

119862(1198711 119881) 02264 13

119862(119881 1198712) 05359 36

119862(1198712 119881) 02282 14

119862(119881 1198713) 05218 38

119862(1198713 119881) 02338 12

119862(119881 1198621) 05044 36

119862(1198621 119881) 02138 14

119862(119881 1198622) 04844 35

119862(1198622 119881) 02169 15

119862(119881 1198623) 05116 37

119862(1198623 119881) 02055 13



6 Conclusion



40 50 60 70 80 90 100 11012

14

16

18

20

22

24

26

28

Makespan

Mea

n w

orkl

oad

VNM

TTSP

CNSGA-L1

CNSGA-L2

CNSGA-L3

30lowast12


30 40 50 60 70 80 90 100 11014

16

18

20

22

24

26

Makespan

Mea

n w

orkl

oad

VNM

TTSP

CNSGA-C1

CNSGA-C2

CNSGA-C3

30lowast12



40 50 60 70 80 90 100 11014

16

18

20

22

24

26

28

30

32

Makespan

Mea

n w

orkl

oad

VNMCNSGA-

CNSGA-CNSGA-

TTSP

L1

L2

L3

40lowast12


40 50 60 70 80 90 100 110 12016

18

20

22

24

26

28

30

Makespan

Mea

n w

orkl

oad

VNM

TTSP

CNSGA-C1

CNSGA-C2

CNSGA-C3

40lowast12




TTSP

0010203040506070809

1

C-m

etric


30lowast12


0010203040506070809

1TTSP

C-m

etric


30lowast12


0010203040506070809

1TTSP

C-m

etric


40lowast12


TTSP

0010203040506070809

1

C-m

etric


40lowast12




Acknowledgments


References















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












22= 1 if


11is a unit matrix



1 crossover 119862

+ mutation 119872+

2 selection 119878


+ for VNM is

119875+= 119872+

1119862+119872+

2119878+119880

= (

1198721

sdot sdot sdot 0

d


)(


d


)(

1198722

sdot sdot sdot 0

d


)

times (


d


)(

11988011

sdot sdot sdot 0

d

11988021198991198971

sdot sdot sdot 11988021198991198972119899119897

)

= (

11987211198621198722

sdot sdot sdot 0

d

0 sdot sdot sdot 11987211198621198722

)(

11988011

sdot sdot sdot 0

d

11988021198991198971

sdot sdot sdot 11988021198991198972119899119897

)

= (

1198721119862119872211988011

0 0

sdot sdot sdot 0

1198721119862119872211988021198991198971

sdot sdot sdot 1198721119862119872211988021198991198972119899119897

)

= (

11987511

sdot sdot sdot 0

d

11987521198991198971

sdot sdot sdot 11987521198991198972119899119897

)

(20)

There is 11987511







1199051

1199081

11199031 1199037

51199082

11199033 1199035

51199083

11199036 11990310

4

1199052

1199081

21199032 11990311

51199082

21199034 1199039

41199083

21199035 1199036

61199084

21199033 1199037

4

1199053

1199081

31199033

71199082

311990312

5

1199054

1199081

41199039

251199082

411990310

221199055

1199081

511990312

14

1199056

1199081

61199031 1199034

71199082

61199033 1199037

81199083

61199036 1199038

8

1199057

1199081

71199031 1199032

41199082

71199033 1199038

21199083

71199037 11990311

3

1199058

1199081

81199031 1199033

51199082

81199036 11990310

41199083

81199037 11990312

7

1199059

1199081

91199031 1199034

111199082

91199037 1199039

131199083

91199038 11990311

12

11990510

1199081

101199032

91199082

101199034

101199083

1011990310

10

11990511

1199081

111199032 1199037

61199082

111199033 11990312

91199083

111199038 1199039

8

11990512

1199081

121199032

111199082

121199035

131199083

1211990311

15

11990513

1199081

131199032

41199082

131199038

51199083

131199039

7

11990514

1199081

141199033

71199082

1411990311

101199083

1411990312

811990515

1199081

1511990312

2

11990516

1199081

161199032

91199082

161199035

71199083

161199038

6

11990517

1199081

171199031 11990310

101199082

171199035 1199039

121199083

1711990311 11990312

1111990518

1199081

181199036

15

11990519

1199081

191199032

81199082

191199035

71199083

1911990310

71199084

1911990312

6

11990520

1199081

201199033

61199082

201199036

41199083

201199039

5

11990521

1199081

211199031 1199034

21199082

211199033 1199035

51199083

211199036 1199038

3


Table 1 Continued


11990522

1199081

221199032

31199082

221199034

41199083

221199036

31199084

2211990310

4

11990523

1199081

231199033

51199082

2311990312

5

11990524

1199081

241199034

141199082

2411990311

1711990525

1199081

251199037

19

11990526

1199081

261199031 1199034

71199082

261199033 1199037

81199083

261199036 1199038

10

11990527

1199081

271199031 1199032

21199082

271199031 1199037

21199083

271199033 1199038

4

11990528

1199081

281199031 1199033

51199082

281199034 1199035

41199083

281199037 11990312

2

11990529

1199081

291199031 1199034

111199082

291199033 1199034

151199083

291199037 1199038

12

11990530

1199081

301199031

91199082

301199034

121199083

3011990312

10

11990531

1199081

311199032 1199033

61199082

311199035 11990311

81199083

311199036 1199039

8

11990532

1199081

321199032

111199082

321199035

131199083

321199036

17

11990533

1199081

331199032

61199082

331199036

51199083

3311990311

4

11990534

1199081

341199033

71199082

341199037

81199083

3411990312

1011990535

1199081

351199039

2

11990536

1199081

361199032

91199082

361199035

71199083

3611990310

6

11990537

1199081

371199031 1199032

101199082

371199037 11990311

71199083

371199035 11990312

1111990538

1199081

3811990310

15

11990539

1199081

391199034

81199082

391199036

71199083

391199039

71199084

3911990310

6

11990540

1199081

401199033

61199082

401199036

51199083

401199039

5



100 250 05 1 1 005


1 and 119865





119862 (119860 119861) =


1003816100381610038161003816

|119861|

(21)




0010203040506070809

1

C(M2M1) C(M1M2) C(M2M3) C(M3M2)

C-m

etric

TTSP 30lowast12



instance

Average Times119862(11987211198722) 02213 13119862(11987221198721) 05196 37119862(11987221198723) 04964 36119862(11987231198722) 02069 14





0010203040506070809

1

C(M2M1) C(M1M2) C(M2M3) C(M3M2)

C-m

etric

TTSP 40lowast12



instance

Average Times119862(11987211198722) 0244178 14119862(11987221198721) 0501508 36119862(11987221198723) 0533806 38119862(11987231198722) 0242146 12


Average Times119862(119881119872) 04845 35119862(119872119881) 02104 15


Average Times119862(119881119872) 05256 40119862(119872119881) 01949 10





40 50 60 70 80 90 10016

18

20

22

24

26

28

30

32

Makespan

Mea

n w

orkl

oad

VNMMOEAD

TTSP 30lowast12

(a)

40 50 60 70 80 9016

17

18

19

20

21

22

23

24

25

Makespan

Mea

n w

orkl

oad

VNMMOEAD

TTSP 30lowast12

(b)


50 55 60 65 70 75 80 85 9017

18

19

20

21

22

23

24

25

26

27

Makespan

Mea

n w

orkl

oad

TTSP

VNMMOEAD

40lowast12

(a)

50 55 60 65 70 75 8017

18

19

20

21

22

23

Makespan

Mea

n w

orkl

oad

VNMMOEAD

TTSP 40lowast12

(b)






11198621


1198622


1198623



0010203040506070809

1

C-m

etric

TTSP

C(VM) C(MV)

30lowast12


0010203040506070809

1

C-m

etric

TTSP

C(VM) C(MV)

40lowast12






Average Times119862(119881 119871

1) 04206 36

119862(1198711 119881) 02200 14

119862(119881 1198712) 04077 34

119862(1198712 119881) 02648 16

119862(119881 1198713) 04182 36

119862(1198713 119881) 02248 14

119862(119881 1198621) 04638 35

119862(1198621 119881) 02210 15

119862(119881 1198622) 04602 34

119862(1198622 119881) 02288 16

119862(119881 1198623) 04128 35

119862(1198623 119881) 02525 15


Average Times119862(119881 119871

1) 05243 37

119862(1198711 119881) 02264 13

119862(119881 1198712) 05359 36

119862(1198712 119881) 02282 14

119862(119881 1198713) 05218 38

119862(1198713 119881) 02338 12

119862(119881 1198621) 05044 36

119862(1198621 119881) 02138 14

119862(119881 1198622) 04844 35

119862(1198622 119881) 02169 15

119862(119881 1198623) 05116 37

119862(1198623 119881) 02055 13



6 Conclusion



40 50 60 70 80 90 100 11012

14

16

18

20

22

24

26

28

Makespan

Mea

n w

orkl

oad

VNM

TTSP

CNSGA-L1

CNSGA-L2

CNSGA-L3

30lowast12


30 40 50 60 70 80 90 100 11014

16

18

20

22

24

26

Makespan

Mea

n w

orkl

oad

VNM

TTSP

CNSGA-C1

CNSGA-C2

CNSGA-C3

30lowast12



40 50 60 70 80 90 100 11014

16

18

20

22

24

26

28

30

32

Makespan

Mea

n w

orkl

oad

VNMCNSGA-

CNSGA-CNSGA-

TTSP

L1

L2

L3

40lowast12


40 50 60 70 80 90 100 110 12016

18

20

22

24

26

28

30

Makespan

Mea

n w

orkl

oad

VNM

TTSP

CNSGA-C1

CNSGA-C2

CNSGA-C3

40lowast12




TTSP

0010203040506070809

1

C-m

etric


30lowast12


0010203040506070809

1TTSP

C-m

etric


30lowast12


0010203040506070809

1TTSP

C-m

etric


40lowast12


TTSP

0010203040506070809

1

C-m

etric


40lowast12




Acknowledgments


References















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Table 1 Continued


11990522

1199081

221199032

31199082

221199034

41199083

221199036

31199084

2211990310

4

11990523

1199081

231199033

51199082

2311990312

5

11990524

1199081

241199034

141199082

2411990311

1711990525

1199081

251199037

19

11990526

1199081

261199031 1199034

71199082

261199033 1199037

81199083

261199036 1199038

10

11990527

1199081

271199031 1199032

21199082

271199031 1199037

21199083

271199033 1199038

4

11990528

1199081

281199031 1199033

51199082

281199034 1199035

41199083

281199037 11990312

2

11990529

1199081

291199031 1199034

111199082

291199033 1199034

151199083

291199037 1199038

12

11990530

1199081

301199031

91199082

301199034

121199083

3011990312

10

11990531

1199081

311199032 1199033

61199082

311199035 11990311

81199083

311199036 1199039

8

11990532

1199081

321199032

111199082

321199035

131199083

321199036

17

11990533

1199081

331199032

61199082

331199036

51199083

3311990311

4

11990534

1199081

341199033

71199082

341199037

81199083

3411990312

1011990535

1199081

351199039

2

11990536

1199081

361199032

91199082

361199035

71199083

3611990310

6

11990537

1199081

371199031 1199032

101199082

371199037 11990311

71199083

371199035 11990312

1111990538

1199081

3811990310

15

11990539

1199081

391199034

81199082

391199036

71199083

391199039

71199084

3911990310

6

11990540

1199081

401199033

61199082

401199036

51199083

401199039

5



100 250 05 1 1 005


1 and 119865





119862 (119860 119861) =


1003816100381610038161003816

|119861|

(21)




0010203040506070809

1

C(M2M1) C(M1M2) C(M2M3) C(M3M2)

C-m

etric

TTSP 30lowast12



instance

Average Times119862(11987211198722) 02213 13119862(11987221198721) 05196 37119862(11987221198723) 04964 36119862(11987231198722) 02069 14





0010203040506070809

1

C(M2M1) C(M1M2) C(M2M3) C(M3M2)

C-m

etric

TTSP 40lowast12



instance

Average Times119862(11987211198722) 0244178 14119862(11987221198721) 0501508 36119862(11987221198723) 0533806 38119862(11987231198722) 0242146 12


Average Times119862(119881119872) 04845 35119862(119872119881) 02104 15


Average Times119862(119881119872) 05256 40119862(119872119881) 01949 10





40 50 60 70 80 90 10016

18

20

22

24

26

28

30

32

Makespan

Mea

n w

orkl

oad

VNMMOEAD

TTSP 30lowast12

(a)

40 50 60 70 80 9016

17

18

19

20

21

22

23

24

25

Makespan

Mea

n w

orkl

oad

VNMMOEAD

TTSP 30lowast12

(b)


50 55 60 65 70 75 80 85 9017

18

19

20

21

22

23

24

25

26

27

Makespan

Mea

n w

orkl

oad

TTSP

VNMMOEAD

40lowast12

(a)

50 55 60 65 70 75 8017

18

19

20

21

22

23

Makespan

Mea

n w

orkl

oad

VNMMOEAD

TTSP 40lowast12

(b)






11198621


1198622


1198623



0010203040506070809

1

C-m

etric

TTSP

C(VM) C(MV)

30lowast12


0010203040506070809

1

C-m

etric

TTSP

C(VM) C(MV)

40lowast12






Average Times119862(119881 119871

1) 04206 36

119862(1198711 119881) 02200 14

119862(119881 1198712) 04077 34

119862(1198712 119881) 02648 16

119862(119881 1198713) 04182 36

119862(1198713 119881) 02248 14

119862(119881 1198621) 04638 35

119862(1198621 119881) 02210 15

119862(119881 1198622) 04602 34

119862(1198622 119881) 02288 16

119862(119881 1198623) 04128 35

119862(1198623 119881) 02525 15


Average Times119862(119881 119871

1) 05243 37

119862(1198711 119881) 02264 13

119862(119881 1198712) 05359 36

119862(1198712 119881) 02282 14

119862(119881 1198713) 05218 38

119862(1198713 119881) 02338 12

119862(119881 1198621) 05044 36

119862(1198621 119881) 02138 14

119862(119881 1198622) 04844 35

119862(1198622 119881) 02169 15

119862(119881 1198623) 05116 37

119862(1198623 119881) 02055 13



6 Conclusion



40 50 60 70 80 90 100 11012

14

16

18

20

22

24

26

28

Makespan

Mea

n w

orkl

oad

VNM

TTSP

CNSGA-L1

CNSGA-L2

CNSGA-L3

30lowast12


30 40 50 60 70 80 90 100 11014

16

18

20

22

24

26

Makespan

Mea

n w

orkl

oad

VNM

TTSP

CNSGA-C1

CNSGA-C2

CNSGA-C3

30lowast12



40 50 60 70 80 90 100 11014

16

18

20

22

24

26

28

30

32

Makespan

Mea

n w

orkl

oad

VNMCNSGA-

CNSGA-CNSGA-

TTSP

L1

L2

L3

40lowast12


40 50 60 70 80 90 100 110 12016

18

20

22

24

26

28

30

Makespan

Mea

n w

orkl

oad

VNM

TTSP

CNSGA-C1

CNSGA-C2

CNSGA-C3

40lowast12




TTSP

0010203040506070809

1

C-m

etric


30lowast12


0010203040506070809

1TTSP

C-m

etric


30lowast12


0010203040506070809

1TTSP

C-m

etric


40lowast12


TTSP

0010203040506070809

1

C-m

etric


40lowast12




Acknowledgments


References















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0010203040506070809

1

C(M2M1) C(M1M2) C(M2M3) C(M3M2)

C-m

etric

TTSP 30lowast12



instance

Average Times119862(11987211198722) 02213 13119862(11987221198721) 05196 37119862(11987221198723) 04964 36119862(11987231198722) 02069 14





0010203040506070809

1

C(M2M1) C(M1M2) C(M2M3) C(M3M2)

C-m

etric

TTSP 40lowast12



instance

Average Times119862(11987211198722) 0244178 14119862(11987221198721) 0501508 36119862(11987221198723) 0533806 38119862(11987231198722) 0242146 12


Average Times119862(119881119872) 04845 35119862(119872119881) 02104 15


Average Times119862(119881119872) 05256 40119862(119872119881) 01949 10





40 50 60 70 80 90 10016

18

20

22

24

26

28

30

32

Makespan

Mea

n w

orkl

oad

VNMMOEAD

TTSP 30lowast12

(a)

40 50 60 70 80 9016

17

18

19

20

21

22

23

24

25

Makespan

Mea

n w

orkl

oad

VNMMOEAD

TTSP 30lowast12

(b)


50 55 60 65 70 75 80 85 9017

18

19

20

21

22

23

24

25

26

27

Makespan

Mea

n w

orkl

oad

TTSP

VNMMOEAD

40lowast12

(a)

50 55 60 65 70 75 8017

18

19

20

21

22

23

Makespan

Mea

n w

orkl

oad

VNMMOEAD

TTSP 40lowast12

(b)






11198621


1198622


1198623



0010203040506070809

1

C-m

etric

TTSP

C(VM) C(MV)

30lowast12


0010203040506070809

1

C-m

etric

TTSP

C(VM) C(MV)

40lowast12






Average Times119862(119881 119871

1) 04206 36

119862(1198711 119881) 02200 14

119862(119881 1198712) 04077 34

119862(1198712 119881) 02648 16

119862(119881 1198713) 04182 36

119862(1198713 119881) 02248 14

119862(119881 1198621) 04638 35

119862(1198621 119881) 02210 15

119862(119881 1198622) 04602 34

119862(1198622 119881) 02288 16

119862(119881 1198623) 04128 35

119862(1198623 119881) 02525 15


Average Times119862(119881 119871

1) 05243 37

119862(1198711 119881) 02264 13

119862(119881 1198712) 05359 36

119862(1198712 119881) 02282 14

119862(119881 1198713) 05218 38

119862(1198713 119881) 02338 12

119862(119881 1198621) 05044 36

119862(1198621 119881) 02138 14

119862(119881 1198622) 04844 35

119862(1198622 119881) 02169 15

119862(119881 1198623) 05116 37

119862(1198623 119881) 02055 13



6 Conclusion



40 50 60 70 80 90 100 11012

14

16

18

20

22

24

26

28

Makespan

Mea

n w

orkl

oad

VNM

TTSP

CNSGA-L1

CNSGA-L2

CNSGA-L3

30lowast12


30 40 50 60 70 80 90 100 11014

16

18

20

22

24

26

Makespan

Mea

n w

orkl

oad

VNM

TTSP

CNSGA-C1

CNSGA-C2

CNSGA-C3

30lowast12



40 50 60 70 80 90 100 11014

16

18

20

22

24

26

28

30

32

Makespan

Mea

n w

orkl

oad

VNMCNSGA-

CNSGA-CNSGA-

TTSP

L1

L2

L3

40lowast12


40 50 60 70 80 90 100 110 12016

18

20

22

24

26

28

30

Makespan

Mea

n w

orkl

oad

VNM

TTSP

CNSGA-C1

CNSGA-C2

CNSGA-C3

40lowast12




TTSP

0010203040506070809

1

C-m

etric


30lowast12


0010203040506070809

1TTSP

C-m

etric


30lowast12


0010203040506070809

1TTSP

C-m

etric


40lowast12


TTSP

0010203040506070809

1

C-m

etric


40lowast12




Acknowledgments


References















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










40 50 60 70 80 90 10016

18

20

22

24

26

28

30

32

Makespan

Mea

n w

orkl

oad

VNMMOEAD

TTSP 30lowast12

(a)

40 50 60 70 80 9016

17

18

19

20

21

22

23

24

25

Makespan

Mea

n w

orkl

oad

VNMMOEAD

TTSP 30lowast12

(b)


50 55 60 65 70 75 80 85 9017

18

19

20

21

22

23

24

25

26

27

Makespan

Mea

n w

orkl

oad

TTSP

VNMMOEAD

40lowast12

(a)

50 55 60 65 70 75 8017

18

19

20

21

22

23

Makespan

Mea

n w

orkl

oad

VNMMOEAD

TTSP 40lowast12

(b)






11198621


1198622


1198623



0010203040506070809

1

C-m

etric

TTSP

C(VM) C(MV)

30lowast12


0010203040506070809

1

C-m

etric

TTSP

C(VM) C(MV)

40lowast12






Average Times119862(119881 119871

1) 04206 36

119862(1198711 119881) 02200 14

119862(119881 1198712) 04077 34

119862(1198712 119881) 02648 16

119862(119881 1198713) 04182 36

119862(1198713 119881) 02248 14

119862(119881 1198621) 04638 35

119862(1198621 119881) 02210 15

119862(119881 1198622) 04602 34

119862(1198622 119881) 02288 16

119862(119881 1198623) 04128 35

119862(1198623 119881) 02525 15


Average Times119862(119881 119871

1) 05243 37

119862(1198711 119881) 02264 13

119862(119881 1198712) 05359 36

119862(1198712 119881) 02282 14

119862(119881 1198713) 05218 38

119862(1198713 119881) 02338 12

119862(119881 1198621) 05044 36

119862(1198621 119881) 02138 14

119862(119881 1198622) 04844 35

119862(1198622 119881) 02169 15

119862(119881 1198623) 05116 37

119862(1198623 119881) 02055 13



6 Conclusion



40 50 60 70 80 90 100 11012

14

16

18

20

22

24

26

28

Makespan

Mea

n w

orkl

oad

VNM

TTSP

CNSGA-L1

CNSGA-L2

CNSGA-L3

30lowast12


30 40 50 60 70 80 90 100 11014

16

18

20

22

24

26

Makespan

Mea

n w

orkl

oad

VNM

TTSP

CNSGA-C1

CNSGA-C2

CNSGA-C3

30lowast12



40 50 60 70 80 90 100 11014

16

18

20

22

24

26

28

30

32

Makespan

Mea

n w

orkl

oad

VNMCNSGA-

CNSGA-CNSGA-

TTSP

L1

L2

L3

40lowast12


40 50 60 70 80 90 100 110 12016

18

20

22

24

26

28

30

Makespan

Mea

n w

orkl

oad

VNM

TTSP

CNSGA-C1

CNSGA-C2

CNSGA-C3

40lowast12




TTSP

0010203040506070809

1

C-m

etric


30lowast12


0010203040506070809

1TTSP

C-m

etric


30lowast12


0010203040506070809

1TTSP

C-m

etric


40lowast12


TTSP

0010203040506070809

1

C-m

etric


40lowast12




Acknowledgments


References















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0010203040506070809

1

C-m

etric

TTSP

C(VM) C(MV)

30lowast12


0010203040506070809

1

C-m

etric

TTSP

C(VM) C(MV)

40lowast12






Average Times119862(119881 119871

1) 04206 36

119862(1198711 119881) 02200 14

119862(119881 1198712) 04077 34

119862(1198712 119881) 02648 16

119862(119881 1198713) 04182 36

119862(1198713 119881) 02248 14

119862(119881 1198621) 04638 35

119862(1198621 119881) 02210 15

119862(119881 1198622) 04602 34

119862(1198622 119881) 02288 16

119862(119881 1198623) 04128 35

119862(1198623 119881) 02525 15


Average Times119862(119881 119871

1) 05243 37

119862(1198711 119881) 02264 13

119862(119881 1198712) 05359 36

119862(1198712 119881) 02282 14

119862(119881 1198713) 05218 38

119862(1198713 119881) 02338 12

119862(119881 1198621) 05044 36

119862(1198621 119881) 02138 14

119862(119881 1198622) 04844 35

119862(1198622 119881) 02169 15

119862(119881 1198623) 05116 37

119862(1198623 119881) 02055 13



6 Conclusion



40 50 60 70 80 90 100 11012

14

16

18

20

22

24

26

28

Makespan

Mea

n w

orkl

oad

VNM

TTSP

CNSGA-L1

CNSGA-L2

CNSGA-L3

30lowast12


30 40 50 60 70 80 90 100 11014

16

18

20

22

24

26

Makespan

Mea

n w

orkl

oad

VNM

TTSP

CNSGA-C1

CNSGA-C2

CNSGA-C3

30lowast12



40 50 60 70 80 90 100 11014

16

18

20

22

24

26

28

30

32

Makespan

Mea

n w

orkl

oad

VNMCNSGA-

CNSGA-CNSGA-

TTSP

L1

L2

L3

40lowast12


40 50 60 70 80 90 100 110 12016

18

20

22

24

26

28

30

Makespan

Mea

n w

orkl

oad

VNM

TTSP

CNSGA-C1

CNSGA-C2

CNSGA-C3

40lowast12




TTSP

0010203040506070809

1

C-m

etric


30lowast12


0010203040506070809

1TTSP

C-m

etric


30lowast12


0010203040506070809

1TTSP

C-m

etric


40lowast12


TTSP

0010203040506070809

1

C-m

etric


40lowast12




Acknowledgments


References















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










40 50 60 70 80 90 100 11012

14

16

18

20

22

24

26

28

Makespan

Mea

n w

orkl

oad

VNM

TTSP

CNSGA-L1

CNSGA-L2

CNSGA-L3

30lowast12


30 40 50 60 70 80 90 100 11014

16

18

20

22

24

26

Makespan

Mea

n w

orkl

oad

VNM

TTSP

CNSGA-C1

CNSGA-C2

CNSGA-C3

30lowast12



40 50 60 70 80 90 100 11014

16

18

20

22

24

26

28

30

32

Makespan

Mea

n w

orkl

oad

VNMCNSGA-

CNSGA-CNSGA-

TTSP

L1

L2

L3

40lowast12


40 50 60 70 80 90 100 110 12016

18

20

22

24

26

28

30

Makespan

Mea

n w

orkl

oad

VNM

TTSP

CNSGA-C1

CNSGA-C2

CNSGA-C3

40lowast12




TTSP

0010203040506070809

1

C-m

etric


30lowast12


0010203040506070809

1TTSP

C-m

etric


30lowast12


0010203040506070809

1TTSP

C-m

etric


40lowast12


TTSP

0010203040506070809

1

C-m

etric


40lowast12




Acknowledgments


References















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










TTSP

0010203040506070809

1

C-m

etric


30lowast12


0010203040506070809

1TTSP

C-m

etric


30lowast12


0010203040506070809

1TTSP

C-m

etric


40lowast12


TTSP

0010203040506070809

1

C-m

etric


40lowast12




Acknowledgments


References















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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