evaluating multiobjective evolutionary algorithms using...

Research ArticleEvaluating Multiobjective Evolutionary AlgorithmsUsing MCDM Methods

Xiaobing Yu 1234 YiQun Lu4 and Xianrui Yu4

1Collaborative Innovation Center on Forecast and Evaluation of Meteorological DisastersNanjing University of Information Science amp Technology Nanjing 210044 China2Research Center for Prospering Jiangsu Province with Talents Nanjing University of Information Science amp TechnologyNanjing 210044 China3China Institute for Manufacture Developing Nanjing University of Information Science amp Technology Nanjing 210044 China4School of Management Science and Engineering Nanjing University of Information Science amp Technology Nanjing 210044 China

Correspondence should be addressed to Xiaobing Yu yuxb111163com

Received 15 November 2017 Accepted 10 February 2018 Published 19 March 2018

Academic Editor David Bigaud

Copyright copy 2018 Xiaobing Yu et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The evaluation of multiobjective evolutionary algorithms (MOEAs) involves many metrics it can be considered as a multiple-criteria decision making (MCDM) problem A framework is proposed to estimate MOEAs in which six MOEAs five performancemetrics and two MCDMmethods are used An experimental study is designed and thirteen benchmark functions are selected tovalidate the proposed framework The experimental results have indicated that the framework is effective in evaluating MOEAs

1 Introduction

Without a loss of generality the mathematical formula ofmultiobjective problems (MOPs) can be expressed as follows

min119865 (119909) = (1198911 (119909) 1198912 (119909) 119891119898 (119909)) 119909 isin Ω (1)

where 997888rarr119909 = (1199091 1199092 119909119899) is the decision vector in thedecision space Ω 119865(119909) is the objective function Generallyspeaking the objectives contradict each otherWe cannot finda single solution to optimize all the objectives Optimizingone objective often leads to deterioration in at least oneobjective

Over the past two decades many multiobjective evo-lutionary algorithms (MOEAs) have been proposed suchas vector evaluated genetic algorithm (VEGA) [1] Paretoarchived evolution strategy (PAES) [2] strength Paretoevolutionary algorithm (SPEA) [3] SPEA2 [4] Paretoenvelope-based selection algorithm (PESA) [5 6] nondom-inated sorting algorithm (NSGA) [7] NSGAII [8] multi-objective evolutionary algorithm based on decomposition(MOEAD) [9 10] indicator-based evolutionary algorithm(IBEA) [11] epsilon-multiobjective evolutionary algorithm

(epsilon-MOEA) [12] multiobjective particle swarm opti-mizer (MOPSO) [13] speed-constrained multiobjective par-ticle swarm optimizer (SMPSO) [14] generalized differentialevolution (GDE3) [15] ABYSS [16] multiobjective symbioticorganism search (MOSOS) [17] multiobjective differentialevolution algorithm (MODEA) [18] grid-based adaptiveMODE (GAMODE) [19] These algorithms make great con-tributions to the development of evolutionary algorithmsand optimization approaches These methods try to makepopulation move towards the optimal Pareto front region

In the single optimization the algorithm performancecan be evaluated by the difference between119891(119909) and functionoptimal value However the method cannot be adopted inMOPs In order to solve the problem many criteria areproposed to evaluate the performance of MOEAs In fact theexperiment results of almost every algorithm indicate that theproposed algorithm is competitive compared with the state-of-the-art algorithms Nondominated objective space andbox plot are adopted in SPEA2 [4] NSGAII employs conver-gence and diversity metrics to compare with SPEA and PAES[8] The set convergence and inverted generation distance(IGD) are used to evaluate the performance of MOEAD [9]

HindawiMathematical Problems in EngineeringVolume 2018 Article ID 9751783 13 pageshttpsdoiorg10115520189751783

2 Mathematical Problems in Engineering

Six multiobjectiveoptimizationalgorithms

Five performancemeasures

Two MCDMmethods

(1) NSGAII(2) PAES(3) SPEA2(4) MOEAD(5) MOSPO(6) SMPSO

(1) GD(2) IGD(3) HV(4) Space(5) Maximum Pareto front error

(1) TOPSIS(2) VIKOR

Empirical study

Figure 1 Evaluation framework

Epsilon indicator is used in IBEA [11] Convergence mea-surement spread hypervolume and computational time areselected as performance metrics in epsilon-MOEA [12] Tovalidate the proposed MOPSO four quantitative perfor-mance indexes (success counting IGD set coverage two-setdifference hypervolume) and qualitative performance index(plotting the Pareto fronts) are adopted [13] Three qualityindicators additive unary epsilon indicator spread andhypervolume are considered in SMPSO [14] Spacing binarymetrics 119862 and 119881 are used in GD3 [15] Three metrics gen-eration distance (GD) spread and hypervolume are used toestimate ABYSS [16] GD diversity computational time andbox plot are considered as measurement in MOSOS [17] GDand diversity metrics are adopted in MOEDA [18] There arethreemetrics GD IGD and hypervolume inGAMODE [19]

Among these metrics some focus on the convergenceof MOEAs while some pay attention to the diversity ofMOEAs Convergence is to measure the ability to attainglobal Pareto front anddiversity is tomeasure the distributionalong the Pareto front It is observed that every proposedalgorithm often introduces few metrics to estimate theperformance based on the results from benchmarks Theconclusions of these MOEAs are that they are the best andcompetitive However it is unfaithfully to measure MOEAsperformance by one or two metrics Every metric can justdemonstrate some specific qualification of performancewhileneglecting other information For instance the metric GDcan provide information about the convergence of MOEAsbut it cannot evaluate the diversity of MOEAs Thereforethese evaluations are not comprehensive It cannot entirelyestimate the whole performance of MOEAs As evaluationof MOEAs involves many metrics it can be regarded as amultiple-criteria decision making (MCDE) problem MCDEtechniques can be used to cope with the problem In order

to overcome the problem and make fair comparisons aframework using MCDE methods is proposed In the frame-work comprehensive performancemetrics are established inwhich both convergence and diversity are considered TwoMCDE methods are employed to evaluate six MOEAs Theefforts can givemore fair and faithful comparisons than singlemetric

The rest of this paper is organized as follows Section 2proposes the framework in which six algorithms five per-formance metrics and two MCDM methods are brieflyintroduced Experiments are presented in Section 3 andconclusions are illustrated in Section 4

2 Evaluation Framework

A framework is proposed to evaluate multiobjective algo-rithms in Figure 1 SixMOEAs five performancemetrics andtwo MCDMmethods are employed in the framework

21 Six MOEAs

(1) NSGAII [8] NSGAII was proposed to solve the highcomputational complexity lack of elitism and specifying ofthe sharing parameter of NSGA In NSGAII a selectionoperator is designed by creating amating pool to combine theparent population and offspring population Nondominatedsort and crowding distance ranking are also implemented inthe algorithm

(2) PAES [2]The Pareto archived evolution strategy (PAES) isa simple evolutionary algorithmThe algorithm is consideredas a (1 + 1) evolution strategy employing local search from apopulation of one but using a reference archive of previouslyfound solutions in order to identify the approximate domi-nance ranking of the current and candidate solutions vectors

Mathematical Problems in Engineering 3

(3) SPEA2 [4] The strength Pareto evolutionary algorithm(SPEA) was proposed in 1999 by Zitzler Based on the SPEAan improved version namely SPEA2 was proposed whichincorporated a fine-grained fitness assignment a densityestimation technique and an enhanced archive truncationmethod

(4)MOEAD [9]Multiobjective evolutionary algorithmbasedon decomposition (MOEAD) was proposed by Li and ZhangIt decomposes a multiobjective optimization problem into anumber of scalar optimization subproblems and optimizesthem simultaneously Each subproblem is optimized by onlyusing information from neighboring subproblems whichmakes the algorithm effective and efficient It won the out-standing paper award of IEEE Transactions on evolutionarycomputation

(5) MOPSO [13] Multiobjective particle swarm optimizer(MOSPO) is based on Pareto dominance and the use of acrowing factor to filter out the list of available leaders Dif-ferent mutation operators act on different subdivisions of theswarm The epsilon-dominance concept is also incorporatedin the algorithm

(6) SMPSO [14] Speed-constrained multiobjective PSO(SMPSO) was proposed in 2009 It allows generating newparticle position in which velocity is extremely high Theturbulence factor and an external archive are designed tostore the nondominated solutions found during search

22 Performance Metrics Nowadays there are many metricsto measure performance of MOEAs Among them thefollowing five metrics are widely employed They can revealthe convergence and diversity of MOEAs very well Howevermany researches just employ a few of them to evaluatealgorithms and argue that their proposed algorithms arethe best In fact it is unfair to give the conclusion withoutcomprehensive metrics and evaluations Therefore the fivemetrics are selected tomake the comprehensive comparisons

(1) GD

GD = radicsum119899119894=1 1198892119894119899 (2)

where 119889119894 = min119895119891(119909119894) minus PFtrue(119909119895) is the distance betweennondominated solution 119891(119909119894) and the nearest Pareto frontsolution in objective space It is to measure the closeness ofthe solutions to the real Pareto front If GD is equal to zerothis reveals that all the nondominated solutions generated arelocated in the real Pareto front Therefore the lower value ofGD indicates that the algorithm has better performance [20]

(2) IGD PFtrue is a set of uniformly distributed points in theobjective space119875119860 is the nondominated solution set obtainedby an algorithm and the distance from PFtrue to 119875119860 is definedas

IGD (119875119860PFtrue) = sum120592isinPFtrue 119889 (120592 119875119860)1003816100381610038161003816PFtrue1003816100381610038161003816 (3)

where 119889(120592 119875119860) is the minimumEuclidean distance between Vand the points in 119875119860 Algorithms with smaller IGD values aredesirable [21 22]

(3) Hypervolume This hypervolume metric calculates thevolume (in the objective space) covered by members ofnondominated solutions sets obtained by MOEAs where allobjectives are to be minimized [16] A hypervolume can becalculated as follows

HV = volume(|119875119860|⋃119894=1

V119894) (4)

The larger the HV value is the better the algorithm is

(4) Spacing The metric spacing is to measure how uniformlythe nondominated set is distributed It can be formulated asfollows

119878 = radic 1119899119899sum119894=1

(119889119894 minus 119889)2 (5)

where 119889119894 is the same as the 119889119894 in GD metric 119889 is theaverage value of 119889119894 and n is the number of individuals innondominated set The smaller the spacing is the better thealgorithm performs [23 24]

(5) Maximum Pareto Front Error It is to measure the worstcase and can be formulated as follows

MPFE = max 119889119894 (6)

where 119889119894 is the same as 119889119894 employed in GD MPFE is thelargest distance among these 119889119894The lower the value of MPFEis the better the algorithm is [25]

In order to elaborate the five metrics Figure 2(a) revealsthe distance used in GD space and MPFE Figure 2(b)presents the distance in IGD metric and Figure 2(c) depictsthe HV metric

23 TOPSIS TOPSIS is one of MCDM methods to evaluatealternatives In TOPSIS the best alternative should havetwo characteristics one is the farthest from the negative-ideal solution and the other one is the nearest to positive-ideal solution The negative-ideal solution is a solution thatmaximizes the cost criteria andminimizes the benefit criteriawhich has all the worst values attainable from the criteriaThe positive-ideal solution minimizes the cost criteria andmaximizes the benefit criteria It is consisted of all the bestvalues attainable from the criteria [26 27] TOPSIS consistsof the following steps

Step 1 (obtain decision matrix) If the number of alternativesis 119869 and the number of criteria is 119899 decision matrix with 119899rows and 119869 columns will be obtained as in Table 1

In Table 1 119891119894119895 (119894 = 1 2 119899 119895 = 1 2 119869) is a valueindicating the performance rating of each algorithm jth withrespect to each criterion 119894th


Table 1 The multiple attribute decision matrixes

Algorithm 1 Algorithm 2 Algorithm 119869Criterion 1 11989111 11989112 1198911119869Criterion 2 11989121 11989122 1198912119869 Criterion i 1198911198941 1198911198942 119891119894119869 Criterion 119899 1198911198991 1198911198992 119891119899119869

Real Pareto front

Nondominated solutions

f2

f1

d1

d2

d3

(a) GD space and MPFE

Real Pareto front


f2

f1

d1

d2

d3

d4

(b) IGD

Real Pareto front


f2

f1

(c) HV

Figure 2 The distance and nondominated solutions used in above metrics

Step 2 (normalize decision matrix) According to (7) thenormalized value 119891119894119895 is calculated as follows

119903119894119895 = 119891119894119895radicsum119869119895=1 1198912119894119895 119894 = 1 2 119899 119895 = 1 2 119869 (7)

Step 3 (calculate the weighted normalized decision matrix)The matrix is calculated by multiplying the normalizeddecision matrix and its weights are presented as

120592119894119895 = 119908119894 times 119903119894119895 (8)

where 119908119894 is the weight of the ith criterionsum119899119894=1 119908119894 = 1Step 4 (find the negative-ideal and positive-ideal solutions)

119860minus = 120592minus1 120592minus2 120592minus119899 = (min

119895120592119894119895 | 119894 isin 1198681015840) (max

119895120592119894119895 | 119894 isin 1198681015840)

119860lowast = 120592lowast1 120592lowast2 120592lowast119899 = (max

119895120592119894119895 | 119894 isin 1198681015840) (min

119895120592119894119895 | 119894 isin 11986810158401015840)

(9)

where 1198681015840 is associated with cost criteria and 11986810158401015840 is associatedwith benefit criteria

Step 5 (calculate the 119899-dimensional Euclidean distance) Theseparation of each algorithm from the ideal solution ispresented as follows

119863+119895 = 119899sum119894=1

119889 (120592119894119895 120592lowast119895 ) (10)

The separation of each algorithm from the negative-idealsolution is defined as follows

119863minus119895 = 119899sum119894=1

119889 (120592119894119895 120592minus119895 ) (11)

Step 6 (calculate the relative closeness to the ideal solution)The relative closeness of the algorithm jth is defined as

119862119862lowast119895 = 119863minus119895119863+119895 + 119863minus119895 119894 = 1 2 119869 (12)

Step 7 (rank algorithms order) The 119862119862 is between 0 and 1The larger the 119862119862 is the better the algorithm 119895 is24 VIKORMethod TheVIKORwas proposed by Opricovicand Tzeng [28ndash31] The method is developed to rank andselect from a set of alternatives The multicriteria rankingindex is introduced based on the idea of closeness to the idealsolutions The VIKOR requires the following steps


Step 1 Determine the worst 119891minus119894 values of criterion and thebest 119891lowast119894 value of criterion as follows (119894 = 1 2 119899)

119891minus119894 = min119895

119891119894119895 for benefit criteria

max119895

119891119894119895 for cost criteria119895 = 1 2 119869

119891lowast119894 = max119895


min119895


(13)

where 119891119894119895 is the value of 119894th criterion for alternative 119886119895 119899is the number of criteria and J is the number of alter-natives

Step 2 119878119895 and 119877119895 (119895 = 1 2 119869) can be formulated asfollows

119878119895 = sum119899119894=1 119908119894 (119891lowast119894 minus 119891119894119895)(119891lowast119894 minus 119891minus119894 )119877119895 = max

119894[119908119894 (119891lowast119894 minus 119891119894119895)(119891lowast119894 minus 119891minus119894 ) ]

(14)

where 119908119894 is the weight of 119894th criteria 119878119895 and 119877119895 are employedto measure ranking

Step 3 Compute the values 119876119895 (119895 = 1 2 119869) as follows119876119895 = 120592 (119878119895 minus 119878lowast)(119878minus minus 119878lowast) + (1 minus 120592) (119877119895 minus 119877lowast)(119877minus minus 119877lowast) 119878lowast = min

119895119878119895 119878minus = max

119895119878119895 119877lowast = min

119895119877119895 119877minus = max

119895119877119895

(15)

where the alternative obtained by 119878lowast is with a maximumutility the alternative acquired by 119877lowast is with a minimumindividual regret of the opponent and 120592 is the weight of thestrategy of the majority of criteria and is often set to 05

Step 4 Rank the alternatives in decreasing order Rank thethree measurements respectively 119876 119878 and 119877Step 5 The alternative 119886 is considered as the best if thefollowing two conditions are met

C1119876(1198861015840)minus119876(119886) ge 1(119869minus1) where 1198861015840 is the alternativewith second position in the ranking list by 119876 and 119869 isthe number of alternativesC2 alternative 119886 should be the best ranked by 119878 or 119877

3 Experiments

The experiments are designed to evaluate the above sixalgorithms In order to make fair comparisons thirteen testbenchmark functions are widely used inMOPs and employedin the experiments They can be divided into two groupsZDT suites and WFG suites All of these test suites areminimization of the objective The detailed information isgiven in Table 2 [32 33]

The mathematical forms of WFG can be obtained in [32]and ZDT suites are presented as follows

ZDT11198911 (119909) = 11990911198912 (119909) = 119892 (119909) [1 minus radic1198911 (119909)119892 (119909) ] 119892 (119909) = 1 + 9 (sum119899119894=2 119909119894)(119899 minus 1) 119909 isin [0 1] (16)

ZDT21198911 (119909) = 11990911198912 (119909) = 119892 (119909) [1 minus (1198911 (119909)119892 (119909) )2] 119892 (119909) = 1 + 9 (sum119899119894=2 119909119894)(119899 minus 1) 119909 isin [0 1] (17)

ZDT31198911 (119909) = 11990911198912 (119909) = 119892 (119909) [1 minus radic1198911 (119909)119892 (119909) minus (1198911 (119909)119892 (119909) ) sin (101205871199091)] 119892 (119909) = 1 + 9 (sum119899119894=2 119909119894)(119899 minus 1) 119909 isin [0 1] (18)

ZDT61198911 (119909) = 1 minus exp (minus41199091) sin6 (61205871199091)1198912 (119909) = 119892 (119909) [1 minus (1198911 (119909)119892 (119909) )2] 119892 (119909) = 1 + 9 [(sum119899119894=2 119909119894)(119899 minus 2) ]025 119909 isin [0 1] (19)

The parameters settings of these algorithms are the sameas the original paper The maximum function evaluations

are set to 25000 Each algorithm runs thirty times and theaverage values of performance metrics are obtained


Table 2 Benchmark test functions information

Problem Separable Modality Bias GeometryZDT1 Yes No No ConvexZDT2 Yes No No ConcaveZDT3 Yes Yes No DisconnectedZDT6 Yes Yes Yes ConcaveWFG1 Yes No Polynomial flat Convex mixedWFG2 No No No Convex disconnectedWFG3 No No No Linear degenerateWFG4 Yes Yes No ConcaveWFG5 Yes Yes No ConcaveWFG6 No No No ConcaveWFG7 Yes No Parameter dependent ConcaveWFG8 No No Parameter dependent ConcaveWFG9 No Yes Parameter dependent Concave

Table 3 Five metric results of ZDT1 results

NSGAII PAES SPEA2 MOEAD MOPSO SMPSOGD 227119864 minus 4 485119864 minus 3 173119864 minus 4 432119864 minus 4 795119864 minus 5 117119864 minus 4IGD 489119864 minus 3 660119864 minus 2 392119864 minus 3 641119864 minus 3 522119864 minus 3 370119864 minus 3MPFE 121119864 minus 2 366119864 minus 1 136119864 minus 2 932119864 minus 3 181119864 minus 3 878119864 minus 3Spacing 672119864 minus 3 311119864 minus 2 317119864 minus 3 101119864 minus 2 615119864 minus 3 140119864 minus 3Hypervolume 659119864 minus 1 607119864 minus 1 662119864 minus 1 656119864 minus 1 660119864 minus 1 662119864 minus 1

Table 4 Normalized decision matrix of five performance metrics

NSGAII PAES SPEA2 MOEAD MOPSO SMPSOGD 00465 09939 00355 00885 00163 00240IGD 00731 09864 00586 00958 00780 00553MPFE 00330 09981 00371 00254 00049 00239Spacing 01969 09115 00929 02960 01802 00410Hypervolume 04131 03805 04150 04112 04137 04150

31 Results In order to elaborate the whole calculationprocess the ZDT1 results of five metrics are presented inTable 3 The four metrics GD IGD MPFE and spacing ofSMPSO are the smallest and hypervolume is the biggestPAES is the worst because the performances of five metricsare the worst among the six algorithms Normalized decisionmatrix of five performances metrics is presented in Table 4Suppose that the weight is equal to 15 Thus according toTable 4 positive-ideal and negative-ideal solutions can bedefined as follows

119885+ = 00163 00553 00239 00410 04150 times 15 119885minus = 09939 09864 09981 09115 03805 times 15

(20)

then the distances119863+ and119863minus are calculated according to (10)and (11) demonstrated in Table 5 The global performance ofeach algorithm is determined by 119862119862lowast calculated in (12) and

presented in Table 5 Therefore the ranking of six algorithmsis as follows SMPSO gt SPEA2 gt MOPSO gt NSGAII gtMOEAD gt PAES SMPSO is the best algorithm and PAES isthe worst one for ZDT1

For VIKOR method the Q S and R are calculated andpresented in Table 6 According to the feature of Q S andR SMPSO is the best one while PAES is the worst oneSPEA2 is better than MOEAD However as the condition119876(1198861015840) minus 119876(119886) gt 1(6 minus 1) = 02 cannot be satisfied S value isused to determine the ranking for NSGAII SPEA2MOEADMOPSO Therefore the ranking among six algorithms isSMPSO gt SPEA2 gtMOPSOgt NSGAII gtMOEAD gt PAES

Tables 7 and 8 give the complete rankings of TOPSISand VIKOR methods for all benchmark functions For thethirteen test functions NSGAII wins in one problemWFG1and performs better in ZDT3 WFG2 WFG4 and WFG8SPEA2 wins in five problems ZDT3 WFG2 WFG4 WFG7and WFG8 SPEA2 provides better performance in ZDT1ZDT2WFG1WFG3 andWFG9However it achieves worse


Table 5 The results of119863119863+119863119863minus and 119862119862 from TOPSIS

NSGAII PAES SPEA2 MOEAD MOPSO SMPSO119863119863+ 01623 18889 00641 02689 01411 00205119863119863minus 17818 0 18369 17143 18175 18750119862119862 09165 0 09663 08644 09280 09892

Table 6 The results of 119876 119878 and 119877 from VIKOR

NSGAII PAES SPEA2 MOEAD MOPSO SMPSOQ 02204 20000 00590 03823 01825 0S 00624 10000 00230 01080 00441 00054R 00358 02000 00119 00586 00320 00038

Table 7 The TOPSIS rankings

Functions NSGAII PAES SPEA2 MOEAD MOPSO SMPSOZDT1 4 6 2 5 3 1ZDT2 4 6 2 3 5 1ZDT3 2 6 1 3 4 5ZDT6 4 6 5 1 2 3WFG1 2 4 1 3 6 5WFG2 2 6 1 3 4 5WFG3 3 6 2 5 1 4WFG4 2 4 1 3 6 5WFG5 5 6 4 2 3 1WFG6 2 6 4 5 3 1WFG7 5 6 1 2 4 3WFG8 2 6 1 4 5 3WFG9 3 6 2 5 4 1

Table 8 The VIKOR rankings


result inWFG6MOEADobtains the best result in ZDT6 andbetter performance in WFG5 MOPSO wins in one problemWFG3 and gets better results in ZDT6 and WFG6 SMPSOprovides the best performance in five problems ZDT1 ZDT2WFG5 WFG6 and WFG9

TOPSIS and VKIOR methods achieve same rankings forZDT1 ZDT2 ZDT3 ZDT6 WFG2 WFG3 WFG4 WFG5WFG8 and WFG9

However TOPSIS and VKIOR methods have differentrankings for WFG1 WFG6 and WFG7 Take the WFG1 as

an instance The final values of TOPSIS and VKIOR arepresented in Table 9 As there are six algorithms J is setto six 1(119869 minus 1) = 1(6 minus 1) = 02 It indicates that theQ value difference between two algorithms should be morethan 02 Otherwise the rank between the two algorithms isdetermined by S or R

From Table 9 it can be noticed that this condition cannotbe met between NSGAII and SPEA2 so the values S areused to compare the two algorithms The value of NSGAIIis smaller than that of SPEA2 so NSGAII is better than


Table 9 The results of 119862119862 119876 S and R value from TOPSIS and VIKOR

Method NSGAII PAES SPEA2 MOEAD MOPSO SMPSOCC 09406 05677 09686 06693 00334 02336Q 00290 13579 00062 08686 20000 18874S 00178 04095 00227 02680 08142 07246R 00178 01461 00134 00984 01667 01667

SMPSO

02 04 06 08 10f1

00102030405f

2

06070809

1

ZDT1 true Pareto frontNondominated solutions by SMPSO

02 04 06 08 10f1

00102030405f

2

06070809

1

ZDT1 true Pareto frontNondominated solutions by PAES

PAES

02 04 06 08 10f1

00102030405f

2

06070809

1


PAES

01 02 03 04 05 06 07 08 090f1

minus1minus05

005

115

2

f2

253

354


PAES

00102030405f

2

06070809

1

02 04 06 08 10f1


SMPSO

minus08minus06minus04minus02

00204

f2

0608

112

01 02 03 04 05 06 07 08 090f1

ZDT3 true Pareto frontNondominated solutions by SPEA2

SPEA2

Figure 3 The Pareto front and nondominated solutions of ZDT1 ZDT2 and ZDT3

SPEA2 NSGAII is the first and SPEA2 is the secondHowever TOPSIS directly uses CC as the ranking criteriaThe CC value of SPEA2 is bigger than NSGAII so it canbe ranked the first and NSGAII is the second in TOPSISmethod

32 Discussion In order to further make comparisons thebest and worst performances of the above six algorithmsare selected The nondominated solutions obtained by thetwo kinds of algorithms are depicted in Figures 3ndash5 WFG1NSGAII and SPEA2 achieve the best ranking according to


0010203040506070809

1

05 0803 06 0704 09 102

MOEAD

f2

f1

by MOEAD

ZDT6 true Pareto frontNondominated solutions

05 0803 06 0704 09 12f1ff


0010203040506070809

1

05 0803 06 0704 09 102

f2

f1

PAES

by PAES


05 0803 06 0704 09 12f1ff


WFG1 true Pareto frontNondominated solutions

05 2151 25 30minus05

005

115

225

335

4SPEA2

f2

f2

f1 f1

minus1

0

1

2

3

4

5

05 2151 25 30

NSGAII

by SPEA2 by NSGAII



05 2151 250f1ff f1ff

05 2151 250


f2

f1

minus1

0

1

2

3

4

5

05 2151 25 30

MOPSO

by MOPSO


f1ff

05 2151 250


f2

f2

f1

05 2151 2500

051

152

253

354

SPEA2

by SPEA2


f1ff

05 21510


f1

05 215100

051

152

253

354

45PAES

by PAES


f1ff

05 21510


Figure 4 Continued



3205 1 25150f1

minus2

minus1

012

f2

345 MOPSO

f2


3205 1 25150f1

minus2

minus1

01234 PAES

f2


05 2151 250f1

005

115

225

335

445 SPEA2

f2


05 2151 250f1

005

115

225

335

445

MOPSO

by MOPSO by PAES

by SPEA2 by MOPSO


3205 1 25150f1ff

f2

ff


3205 1 25150f1ff

minus2

minus1

01234


05 2151 250f1ff

SPEA2

f2

ff


05 2151 250f1ff

005

115

225

335

445

MOPSO

by MOPSO by PAES

Figure 4 The Pareto front and nondominated solutions of ZDT6 WFG1-WFG4

VIKOR and TOPSIS methods respectively so the nondomi-nated solutions of two algorithms are both shown in Figure 4

It can be clearly observed that the better algorithmcan achieve nondominated solutions uniformly distributedalong the Pareto front and the worse algorithm obtainsnondominated solutions in which both convergence anddiversity are poor

ZDT1 ZDT2 WFG5 WFG6 and WFG9 have commonfeatures They do not have local Pareto front and their Paretofronts are continuous SMPSO allows new particle positionin which velocity is very high The turbulence factor andan external archive are designed to store the nondominatedsolutions found during search These mechanisms can helppopulation move quickly towards the Pareto front for thistype of problems in which local Pareto front does not existand Pareto front is continuous

ZDT 3 and WFG 4 have discrete Pareto front SPEA2has achieved performance on the two problems Thus if theproblemhas the discrete Pareto front SPEA2 is a good choice

ZDT6 is a biased function as the first objective functionvalue is larger compared to the second one MOEAD obtainssuperior results so if the problem has the feature MOEADshould be chosen

From Tables 7 and 8 the no-free-lunch theorem can alsobe observed any optimization algorithm improves perfor-mance over one class of problems exactly paid for in loss over

another class No algorithm can achieve the best or worstperformance for all test functions

4 Conclusions

There aremanyMOEAsWhen amultiobjective optimizationalgorithm is proposed the experiment results often indicatethat the algorithm is competitive based on one or two perfor-mance metrics Generally these comparisons are unfair andthe results are unfaithful In order to make fair comparisonsand rank these MOEAs a framework is proposed to evaluateMOEAs The framework employs six well-known MOEAsfive performance metrics and two MCDM methods Thesix MOEAs are NSGAII PAES SPEA2 MOEAD MOPSOand SMPSO The five performance metrics GD IGD MPFEspacing and hypervolume are selected in which both con-vergence and diversity of nondominated solutions are fullyconsidered Two methods are TOPSIS and VIKOR

The results have indicated that SPEA2 is the best algo-rithm and PAES is the worst one However SPEA2 cannotperform well on all test functions and PAES also does notachieve the worst performance on all test functions Theexperiment results are consistent with the no-free-lunchtheoremWhat ismore the observation of experiment resultsshows that the ability of MOEAs to solve MOPs depends onboth MOEAs and the features of MOPs


f2

WFG9 true Pareto frontNondominated solutions by SMPSO

SMPSO

05 1 15 2 250f1

005

115

225

335

454

f2

WFG7 true Pareto frontNondominated solutions by SPEA2

SPEA2

05 1 15 2 250f1

005

115

225

335

445

f2


SPEA2

05 1 15 2 250f1

005

115

225

335

445

f2


SMPSO

005

115

225

335

445

05 1 15 2 250f1

WFG5 true Pareto frontNondominated solutions by PAES

PAES

005

115

225

335

445

f2

05 1 15 2 250f1


SMPSO

005

115

2f2 25

335

445

05 1 15 2 250f1


PAES

005

115

225

335

445

f2

05 1 15 2 250f1


PAES

005

115

225

335

445

f2

f2

05 1 15 2 250f1


PAES

005

115

225

335

445

05 1 15 2 250f1


PAES

005

115

225

335

445

f2

05 15 21f1

Figure 5 The Pareto front and nondominated solutions of WFG5-WFG9


Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The authors thank the Natural Science Foundation of China(Grants nos 71503134 91546117 71373131) Key Project ofNational Social and Scientific Fund Program (16ZDA047)Philosophy and Social Sciences in Universities of Jiangsu(Grant no 2016SJB630016)

References

[1] D Schaffer ldquoMultiple Objective Optimization with VectorEvaluated Genetic Algorithmsrdquo in Proceedings of the 1st Inter-national Conference on Genetic Algorithms pp 93ndash100 1985

[2] J Knowles and D Corne ldquoThe pareto archived evolutionstrategy a new baseline algorithm for Pareto multiobjectiveoptimisationrdquo in Proceedings of the Congress on EvolutionaryComputation (CEC rsquo99) vol 1 pp 98ndash105 July 1999

[3] E Zitzler Evolutionary algorithms for multiobjective optimiza-tion methods and applications [PhD Doctoral dissertation]Swiss Federal Institute of Technology Zurich Switzerland 1999ETH 13398

[4] E Zitzler M Laumanns and L Thiele ldquoSPEA2 Improvingthe Strength Pareto Evolutionary Algorithmrdquo TIK-Report 1032001

[5] D W Corne N R Jerram J D Knowles and M J OatesldquoPESA-II Region-based Selection in Evolutionary Multiobjec-tive Optimizationrdquo in Proceedings of the Genetic and Evolution-ary Computation Conference (GECCO rsquo01) pp 283ndash290 2001

[6] D W Corne J D Knowles and M J Oates ldquoThe Pareto-envelope based selection algorithm formultiobjective optimiza-tionrdquo Parallel Problem Solving from Nature PPSN VI pp 869ndash878 2000

[7] N Srinivas and K Deb ldquoMultiobjective function optimizationusing nondominated sorting genetic algorithmsrdquo EvolutionaryComputation vol 2 no 3 pp 221ndash248 1995

[8] 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

[9] H Li and Q Zhang ldquoMultiobjective optimization problemswith complicated pareto sets MOEAD and NSGA-IIrdquo IEEETransactions on Evolutionary Computation vol 13 no 2 pp284ndash302 2009

[10] Q ZhangW Liu andH Li ldquoThe performance of a new versionof MOEAD on CEC09 unconstrained MOP test instancesrdquo inProceedings of the IEEE Congress on Evolutionary Computation(CEC rsquo09) pp 203ndash208 Trondheim Norway May 2009

[11] E Zitzler and S Kunzli ldquoIndicator-based selection in multiob-jective searchrdquo in Parallel Problem Solving from Nature (PPSNVIII) pp 832ndash842 2004

[12] K Deb MMohan and S Mishra ldquoA Fast Multi-Objective Evo-lutionary Algorithm for Finding Well-Spread Pareto-OptimalSolutionsrdquo KanGAL Report 2003002 2003

[13] M R Sierra and C A C Coello ldquoImproving PSO-basedMulti-Objective Optimization using Crowding Mutation andepsilon-Dominancerdquo in Evolutionary Multi-Criterion Opti-mization vol 3410 pp 505ndash519 2005

[14] A J Nebro J J Durillo G Nieto C A C Coello F Luna andE Alba ldquoSMPSO a new pso-based metaheuristic for multi-objective optimizationrdquo in Proceedings of the IEEE Sympo-sium on Computational Intelligence in Multi-Criteria Decision-Making (MCDM rsquo09) pp 66ndash73 Nashville Tenn USA April2009

[15] S Kukkonen and J Lampinen ldquoGDE3 The Third EvolutionStep of Generalized Differential Evolutionrdquo KanGAL Report2005013 2005

[16] A J Nebro F Luna E Alba B Dorronsoro J J Durillo andA Beham ldquoAbYSS adapting scatter search to multiobjectiveoptimizationrdquo IEEE Transactions on Evolutionary Computationvol 12 no 4 pp 439ndash457 2008

[17] A Panda and S Pani ldquoA SymbioticOrganisms Search algorithmwith adaptive penalty function to solve multi-objective con-strained optimization problemsrdquo Applied Soft Computing vol46 pp 344ndash360 2016

[18] M Ali P Siarry and M Pant ldquoAn efficient Differential Evolu-tion based algorithm for solving multi-objective optimizationproblemsrdquo European Journal of Operational Research vol 217no 2 pp 404ndash416 2012

[19] J Cheng G G Yen and G Zhang ldquoA grid-based adaptivemulti-objective differential evolution algorithmrdquo InformationSciences vol 367-368 pp 890ndash908 2016

[20] D A Van Veldhuizen Multiobjective Evolutionary AlgorithmsClassification Analysis and New Innovations [PhD disser-taion]Wright-PattersonAir Force BaseOhioOhioUSA 1999

[21] Q Zhang A Zhou S Zhao P N Suganthan W Liu andS Tiwari ldquoMultiobjective optimization test instances for theCEC 2009 special session and competitionrdquo Special Sessionon Performance Assessment of Multi-Objective OptimizationAlgorithms University of Essex UK 2008

[22] Y Liu DGong J Sun andY Jin ldquoAMany-Objective Evolution-ary Algorithm Using A One-by-One Selection Strategyrdquo IEEETransactions on Cybernetics vol 47 no 9 pp 2689ndash2702 2017

[23] X Yu ldquoDisaster prediction model based on support vectormachine for regression and improved differential evolutionrdquoNatural Hazards vol 85 no 2 pp 959ndash976 2017

[24] J R Schoot Fault tolerant design using single and multicriteriagenetic algorithm optimizationDepartment of Aeronautics andAstronautics [MS thesis] Massachusetts Institute of Technol-ogy Cambridge UK 1995

[25] G G Yen and Z He ldquoPerformance metric ensemble formultiobjective evolutionary algorithmsrdquo IEEE Transactions onEvolutionary Computation vol 18 no 1 pp 131ndash144 2014

[26] Z Pei ldquoA note on the TOPSISmethod inMADMproblemswithlinguistic evaluationsrdquo Applied Soft Computing vol 36 articleno 3048 pp 24ndash35 2015

[27] X Yu S Guo J Guo and X Huang ldquoRank B2C e-commercewebsites in e-alliance based on AHP and fuzzy TOPSISrdquo ExpertSystems with Applications vol 38 no 4 pp 3550ndash3557 2011

[28] S Opricovic Multi criteria optimization of Civil EngineeringSystems Faculty of Civil Engineering Belgrade Serbia 1998

[29] S Opricovic and G Tzeng ldquoMulticriteria planning of post-earthquake sustainable reconstructionrdquo Computer-Aided Civiland Infrastructure Engineering vol 17 no 3 pp 211ndash220 2002

[30] S Opricovic and G H Tzeng ldquoCompromise solution byMCDM methods a comparative analysis of VIKOR and TOP-SISrdquo European Journal of Operational Research vol 156 no 2pp 445ndash455 2004


[31] S Opricovic and G Tzeng ldquoExtended VIKOR method incomparison with outranking methodsrdquo European Journal ofOperational Research vol 178 no 2 pp 514ndash529 2007

[32] E Zitzler K Deb and LThiele ldquoComparison of multiobjectiveevolutionary algorithms empirical resultsrdquo Evolutionary Com-putation vol 8 no 2 pp 173ndash195 2000

[33] S Huband P Hingston L Barone and L While ldquoA reviewof multiobjective test problems and a scalable test problemtoolkitrdquo IEEE Transactions on Evolutionary Computation vol10 no 5 pp 477ndash506 2006

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of


Mathematical Problems in Engineering

Applied MathematicsJournal of


Probability and StatisticsHindawiwwwhindawicom Volume 2018

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International Journal of Mathematics and Mathematical Sciences


Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018


Decision SciencesAdvances in


AnalysisInternational Journal of


Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom


Six multiobjectiveoptimizationalgorithms

Five performancemeasures

Two MCDMmethods

(1) NSGAII(2) PAES(3) SPEA2(4) MOEAD(5) MOSPO(6) SMPSO

(1) GD(2) IGD(3) HV(4) Space(5) Maximum Pareto front error

(1) TOPSIS(2) VIKOR

Empirical study

Figure 1 Evaluation framework

Epsilon indicator is used in IBEA [11] Convergence mea-surement spread hypervolume and computational time areselected as performance metrics in epsilon-MOEA [12] Tovalidate the proposed MOPSO four quantitative perfor-mance indexes (success counting IGD set coverage two-setdifference hypervolume) and qualitative performance index(plotting the Pareto fronts) are adopted [13] Three qualityindicators additive unary epsilon indicator spread andhypervolume are considered in SMPSO [14] Spacing binarymetrics 119862 and 119881 are used in GD3 [15] Three metrics gen-eration distance (GD) spread and hypervolume are used toestimate ABYSS [16] GD diversity computational time andbox plot are considered as measurement in MOSOS [17] GDand diversity metrics are adopted in MOEDA [18] There arethreemetrics GD IGD and hypervolume inGAMODE [19]

Among these metrics some focus on the convergenceof MOEAs while some pay attention to the diversity ofMOEAs Convergence is to measure the ability to attainglobal Pareto front anddiversity is tomeasure the distributionalong the Pareto front It is observed that every proposedalgorithm often introduces few metrics to estimate theperformance based on the results from benchmarks Theconclusions of these MOEAs are that they are the best andcompetitive However it is unfaithfully to measure MOEAsperformance by one or two metrics Every metric can justdemonstrate some specific qualification of performancewhileneglecting other information For instance the metric GDcan provide information about the convergence of MOEAsbut it cannot evaluate the diversity of MOEAs Thereforethese evaluations are not comprehensive It cannot entirelyestimate the whole performance of MOEAs As evaluationof MOEAs involves many metrics it can be regarded as amultiple-criteria decision making (MCDE) problem MCDEtechniques can be used to cope with the problem In order

to overcome the problem and make fair comparisons aframework using MCDE methods is proposed In the frame-work comprehensive performancemetrics are established inwhich both convergence and diversity are considered TwoMCDE methods are employed to evaluate six MOEAs Theefforts can givemore fair and faithful comparisons than singlemetric

The rest of this paper is organized as follows Section 2proposes the framework in which six algorithms five per-formance metrics and two MCDM methods are brieflyintroduced Experiments are presented in Section 3 andconclusions are illustrated in Section 4

2 Evaluation Framework

A framework is proposed to evaluate multiobjective algo-rithms in Figure 1 SixMOEAs five performancemetrics andtwo MCDMmethods are employed in the framework

21 Six MOEAs

(1) NSGAII [8] NSGAII was proposed to solve the highcomputational complexity lack of elitism and specifying ofthe sharing parameter of NSGA In NSGAII a selectionoperator is designed by creating amating pool to combine theparent population and offspring population Nondominatedsort and crowding distance ranking are also implemented inthe algorithm

(2) PAES [2]The Pareto archived evolution strategy (PAES) isa simple evolutionary algorithmThe algorithm is consideredas a (1 + 1) evolution strategy employing local search from apopulation of one but using a reference archive of previouslyfound solutions in order to identify the approximate domi-nance ranking of the current and candidate solutions vectors







(1) GD

GD = radicsum119899119894=1 1198892119894119899 (2)






HV = volume(|119875119860|⋃119894=1

V119894) (4)



119878 = radic 1119899119899sum119894=1

(119889119894 minus 119889)2 (5)



MPFE = max 119889119894 (6)









Real Pareto front


f2

f1

d1

d2

d3


Real Pareto front


f2

f1

d1

d2

d3

d4

(b) IGD

Real Pareto front


f2

f1

(c) HV



119903119894119895 = 119891119894119895radicsum119869119895=1 1198912119894119895 119894 = 1 2 119899 119895 = 1 2 119869 (7)


120592119894119895 = 119908119894 times 119903119894119895 (8)



119895120592119894119895 | 119894 isin 1198681015840) (max

119895120592119894119895 | 119894 isin 1198681015840)


119895120592119894119895 | 119894 isin 1198681015840) (min

119895120592119894119895 | 119894 isin 11986810158401015840)

(9)



119863+119895 = 119899sum119894=1

119889 (120592119894119895 120592lowast119895 ) (10)


119863minus119895 = 119899sum119894=1

119889 (120592119894119895 120592minus119895 ) (11)






119891minus119894 = min119895


max119895


119891lowast119894 = max119895


min119895


(13)





(14)



119895119878119895 119878minus = max

119895119878119895 119877lowast = min

119895119877119895 119877minus = max

119895119877119895

(15)




3 Experiments


















(20)






















SMPSO

02 04 06 08 10f1

00102030405f

2

06070809

1


02 04 06 08 10f1

00102030405f

2

06070809

1


PAES

02 04 06 08 10f1

00102030405f

2

06070809

1


PAES

01 02 03 04 05 06 07 08 090f1

minus1minus05

005

115

2

f2

253

354


PAES

00102030405f

2

06070809

1

02 04 06 08 10f1


SMPSO


00204

f2

0608

112

01 02 03 04 05 06 07 08 090f1


SPEA2





0010203040506070809

1

05 0803 06 0704 09 102

MOEAD

f2

f1

by MOEAD


05 0803 06 0704 09 12f1ff


0010203040506070809

1

05 0803 06 0704 09 102

f2

f1

PAES

by PAES


05 0803 06 0704 09 12f1ff



05 2151 25 30minus05

005

115

225

335

4SPEA2

f2

f2

f1 f1

minus1

0

1

2

3

4

5

05 2151 25 30

NSGAII

by SPEA2 by NSGAII



05 2151 250f1ff f1ff

05 2151 250


f2

f1

minus1

0

1

2

3

4

5

05 2151 25 30

MOPSO

by MOPSO


f1ff

05 2151 250


f2

f2

f1

05 2151 2500

051

152

253

354

SPEA2

by SPEA2


f1ff

05 21510


f1

05 215100

051

152

253

354

45PAES

by PAES


f1ff

05 21510


Figure 4 Continued



3205 1 25150f1

minus2

minus1

012

f2

345 MOPSO

f2


3205 1 25150f1

minus2

minus1

01234 PAES

f2


05 2151 250f1

005

115

225

335

445 SPEA2

f2


05 2151 250f1

005

115

225

335

445

MOPSO

by MOPSO by PAES

by SPEA2 by MOPSO


3205 1 25150f1ff

f2

ff


3205 1 25150f1ff

minus2

minus1

01234


05 2151 250f1ff

SPEA2

f2

ff


05 2151 250f1ff

005

115

225

335

445

MOPSO

by MOPSO by PAES









4 Conclusions




f2


SMPSO

05 1 15 2 250f1

005

115

225

335

454

f2


SPEA2

05 1 15 2 250f1

005

115

225

335

445

f2


SPEA2

05 1 15 2 250f1

005

115

225

335

445

f2


SMPSO

005

115

225

335

445

05 1 15 2 250f1


PAES

005

115

225

335

445

f2

05 1 15 2 250f1


SMPSO

005

115

2f2 25

335

445

05 1 15 2 250f1


PAES

005

115

225

335

445

f2

05 1 15 2 250f1


PAES

005

115

225

335

445

f2

f2

05 1 15 2 250f1


PAES

005

115

225

335

445

05 1 15 2 250f1


PAES

005

115

225

335

445

f2

05 15 21f1





Acknowledgments


References










































Journal of












Journal of







Volume 2018






Volume 2018














(1) GD

GD = radicsum119899119894=1 1198892119894119899 (2)






HV = volume(|119875119860|⋃119894=1

V119894) (4)



119878 = radic 1119899119899sum119894=1

(119889119894 minus 119889)2 (5)



MPFE = max 119889119894 (6)









Real Pareto front


f2

f1

d1

d2

d3


Real Pareto front


f2

f1

d1

d2

d3

d4

(b) IGD

Real Pareto front


f2

f1

(c) HV



119903119894119895 = 119891119894119895radicsum119869119895=1 1198912119894119895 119894 = 1 2 119899 119895 = 1 2 119869 (7)


120592119894119895 = 119908119894 times 119903119894119895 (8)



119895120592119894119895 | 119894 isin 1198681015840) (max

119895120592119894119895 | 119894 isin 1198681015840)


119895120592119894119895 | 119894 isin 1198681015840) (min

119895120592119894119895 | 119894 isin 11986810158401015840)

(9)



119863+119895 = 119899sum119894=1

119889 (120592119894119895 120592lowast119895 ) (10)


119863minus119895 = 119899sum119894=1

119889 (120592119894119895 120592minus119895 ) (11)






119891minus119894 = min119895


max119895


119891lowast119894 = max119895


min119895


(13)





(14)



119895119878119895 119878minus = max

119895119878119895 119877lowast = min

119895119877119895 119877minus = max

119895119877119895

(15)




3 Experiments


















(20)






















SMPSO

02 04 06 08 10f1

00102030405f

2

06070809

1


02 04 06 08 10f1

00102030405f

2

06070809

1


PAES

02 04 06 08 10f1

00102030405f

2

06070809

1


PAES

01 02 03 04 05 06 07 08 090f1

minus1minus05

005

115

2

f2

253

354


PAES

00102030405f

2

06070809

1

02 04 06 08 10f1


SMPSO


00204

f2

0608

112

01 02 03 04 05 06 07 08 090f1


SPEA2





0010203040506070809

1

05 0803 06 0704 09 102

MOEAD

f2

f1

by MOEAD


05 0803 06 0704 09 12f1ff


0010203040506070809

1

05 0803 06 0704 09 102

f2

f1

PAES

by PAES


05 0803 06 0704 09 12f1ff



05 2151 25 30minus05

005

115

225

335

4SPEA2

f2

f2

f1 f1

minus1

0

1

2

3

4

5

05 2151 25 30

NSGAII

by SPEA2 by NSGAII



05 2151 250f1ff f1ff

05 2151 250


f2

f1

minus1

0

1

2

3

4

5

05 2151 25 30

MOPSO

by MOPSO


f1ff

05 2151 250


f2

f2

f1

05 2151 2500

051

152

253

354

SPEA2

by SPEA2


f1ff

05 21510


f1

05 215100

051

152

253

354

45PAES

by PAES


f1ff

05 21510


Figure 4 Continued



3205 1 25150f1

minus2

minus1

012

f2

345 MOPSO

f2


3205 1 25150f1

minus2

minus1

01234 PAES

f2


05 2151 250f1

005

115

225

335

445 SPEA2

f2


05 2151 250f1

005

115

225

335

445

MOPSO

by MOPSO by PAES

by SPEA2 by MOPSO


3205 1 25150f1ff

f2

ff


3205 1 25150f1ff

minus2

minus1

01234


05 2151 250f1ff

SPEA2

f2

ff


05 2151 250f1ff

005

115

225

335

445

MOPSO

by MOPSO by PAES









4 Conclusions




f2


SMPSO

05 1 15 2 250f1

005

115

225

335

454

f2


SPEA2

05 1 15 2 250f1

005

115

225

335

445

f2


SPEA2

05 1 15 2 250f1

005

115

225

335

445

f2


SMPSO

005

115

225

335

445

05 1 15 2 250f1


PAES

005

115

225

335

445

f2

05 1 15 2 250f1


SMPSO

005

115

2f2 25

335

445

05 1 15 2 250f1


PAES

005

115

225

335

445

f2

05 1 15 2 250f1


PAES

005

115

225

335

445

f2

f2

05 1 15 2 250f1


PAES

005

115

225

335

445

05 1 15 2 250f1


PAES

005

115

225

335

445

f2

05 15 21f1





Acknowledgments


References










































Journal of












Journal of







Volume 2018






Volume 2018











Real Pareto front


f2

f1

d1

d2

d3


Real Pareto front


f2

f1

d1

d2

d3

d4

(b) IGD

Real Pareto front


f2

f1

(c) HV



119903119894119895 = 119891119894119895radicsum119869119895=1 1198912119894119895 119894 = 1 2 119899 119895 = 1 2 119869 (7)


120592119894119895 = 119908119894 times 119903119894119895 (8)



119895120592119894119895 | 119894 isin 1198681015840) (max

119895120592119894119895 | 119894 isin 1198681015840)


119895120592119894119895 | 119894 isin 1198681015840) (min

119895120592119894119895 | 119894 isin 11986810158401015840)

(9)



119863+119895 = 119899sum119894=1

119889 (120592119894119895 120592lowast119895 ) (10)


119863minus119895 = 119899sum119894=1

119889 (120592119894119895 120592minus119895 ) (11)






119891minus119894 = min119895


max119895


119891lowast119894 = max119895


min119895


(13)





(14)



119895119878119895 119878minus = max

119895119878119895 119877lowast = min

119895119877119895 119877minus = max

119895119877119895

(15)




3 Experiments


















(20)






















SMPSO

02 04 06 08 10f1

00102030405f

2

06070809

1


02 04 06 08 10f1

00102030405f

2

06070809

1


PAES

02 04 06 08 10f1

00102030405f

2

06070809

1


PAES

01 02 03 04 05 06 07 08 090f1

minus1minus05

005

115

2

f2

253

354


PAES

00102030405f

2

06070809

1

02 04 06 08 10f1


SMPSO


00204

f2

0608

112

01 02 03 04 05 06 07 08 090f1


SPEA2





0010203040506070809

1

05 0803 06 0704 09 102

MOEAD

f2

f1

by MOEAD


05 0803 06 0704 09 12f1ff


0010203040506070809

1

05 0803 06 0704 09 102

f2

f1

PAES

by PAES


05 0803 06 0704 09 12f1ff



05 2151 25 30minus05

005

115

225

335

4SPEA2

f2

f2

f1 f1

minus1

0

1

2

3

4

5

05 2151 25 30

NSGAII

by SPEA2 by NSGAII



05 2151 250f1ff f1ff

05 2151 250


f2

f1

minus1

0

1

2

3

4

5

05 2151 25 30

MOPSO

by MOPSO


f1ff

05 2151 250


f2

f2

f1

05 2151 2500

051

152

253

354

SPEA2

by SPEA2


f1ff

05 21510


f1

05 215100

051

152

253

354

45PAES

by PAES


f1ff

05 21510


Figure 4 Continued



3205 1 25150f1

minus2

minus1

012

f2

345 MOPSO

f2


3205 1 25150f1

minus2

minus1

01234 PAES

f2


05 2151 250f1

005

115

225

335

445 SPEA2

f2


05 2151 250f1

005

115

225

335

445

MOPSO

by MOPSO by PAES

by SPEA2 by MOPSO


3205 1 25150f1ff

f2

ff


3205 1 25150f1ff

minus2

minus1

01234


05 2151 250f1ff

SPEA2

f2

ff


05 2151 250f1ff

005

115

225

335

445

MOPSO

by MOPSO by PAES









4 Conclusions




f2


SMPSO

05 1 15 2 250f1

005

115

225

335

454

f2


SPEA2

05 1 15 2 250f1

005

115

225

335

445

f2


SPEA2

05 1 15 2 250f1

005

115

225

335

445

f2


SMPSO

005

115

225

335

445

05 1 15 2 250f1


PAES

005

115

225

335

445

f2

05 1 15 2 250f1


SMPSO

005

115

2f2 25

335

445

05 1 15 2 250f1


PAES

005

115

225

335

445

f2

05 1 15 2 250f1


PAES

005

115

225

335

445

f2

f2

05 1 15 2 250f1


PAES

005

115

225

335

445

05 1 15 2 250f1


PAES

005

115

225

335

445

f2

05 15 21f1





Acknowledgments


References










































Journal of












Journal of







Volume 2018






Volume 2018










119891minus119894 = min119895


max119895


119891lowast119894 = max119895


min119895


(13)





(14)



119895119878119895 119878minus = max

119895119878119895 119877lowast = min

119895119877119895 119877minus = max

119895119877119895

(15)




3 Experiments


















(20)






















SMPSO

02 04 06 08 10f1

00102030405f

2

06070809

1


02 04 06 08 10f1

00102030405f

2

06070809

1


PAES

02 04 06 08 10f1

00102030405f

2

06070809

1


PAES

01 02 03 04 05 06 07 08 090f1

minus1minus05

005

115

2

f2

253

354


PAES

00102030405f

2

06070809

1

02 04 06 08 10f1


SMPSO


00204

f2

0608

112

01 02 03 04 05 06 07 08 090f1


SPEA2





0010203040506070809

1

05 0803 06 0704 09 102

MOEAD

f2

f1

by MOEAD


05 0803 06 0704 09 12f1ff


0010203040506070809

1

05 0803 06 0704 09 102

f2

f1

PAES

by PAES


05 0803 06 0704 09 12f1ff



05 2151 25 30minus05

005

115

225

335

4SPEA2

f2

f2

f1 f1

minus1

0

1

2

3

4

5

05 2151 25 30

NSGAII

by SPEA2 by NSGAII



05 2151 250f1ff f1ff

05 2151 250


f2

f1

minus1

0

1

2

3

4

5

05 2151 25 30

MOPSO

by MOPSO


f1ff

05 2151 250


f2

f2

f1

05 2151 2500

051

152

253

354

SPEA2

by SPEA2


f1ff

05 21510


f1

05 215100

051

152

253

354

45PAES

by PAES


f1ff

05 21510


Figure 4 Continued



3205 1 25150f1

minus2

minus1

012

f2

345 MOPSO

f2


3205 1 25150f1

minus2

minus1

01234 PAES

f2


05 2151 250f1

005

115

225

335

445 SPEA2

f2


05 2151 250f1

005

115

225

335

445

MOPSO

by MOPSO by PAES

by SPEA2 by MOPSO


3205 1 25150f1ff

f2

ff


3205 1 25150f1ff

minus2

minus1

01234


05 2151 250f1ff

SPEA2

f2

ff


05 2151 250f1ff

005

115

225

335

445

MOPSO

by MOPSO by PAES









4 Conclusions




f2


SMPSO

05 1 15 2 250f1

005

115

225

335

454

f2


SPEA2

05 1 15 2 250f1

005

115

225

335

445

f2


SPEA2

05 1 15 2 250f1

005

115

225

335

445

f2


SMPSO

005

115

225

335

445

05 1 15 2 250f1


PAES

005

115

225

335

445

f2

05 1 15 2 250f1


SMPSO

005

115

2f2 25

335

445

05 1 15 2 250f1


PAES

005

115

225

335

445

f2

05 1 15 2 250f1


PAES

005

115

225

335

445

f2

f2

05 1 15 2 250f1


PAES

005

115

225

335

445

05 1 15 2 250f1


PAES

005

115

225

335

445

f2

05 15 21f1





Acknowledgments


References










































Journal of












Journal of







Volume 2018






Volume 2018

















(20)






















SMPSO

02 04 06 08 10f1

00102030405f

2

06070809

1


02 04 06 08 10f1

00102030405f

2

06070809

1


PAES

02 04 06 08 10f1

00102030405f

2

06070809

1


PAES

01 02 03 04 05 06 07 08 090f1

minus1minus05

005

115

2

f2

253

354


PAES

00102030405f

2

06070809

1

02 04 06 08 10f1


SMPSO


00204

f2

0608

112

01 02 03 04 05 06 07 08 090f1


SPEA2





0010203040506070809

1

05 0803 06 0704 09 102

MOEAD

f2

f1

by MOEAD


05 0803 06 0704 09 12f1ff


0010203040506070809

1

05 0803 06 0704 09 102

f2

f1

PAES

by PAES


05 0803 06 0704 09 12f1ff



05 2151 25 30minus05

005

115

225

335

4SPEA2

f2

f2

f1 f1

minus1

0

1

2

3

4

5

05 2151 25 30

NSGAII

by SPEA2 by NSGAII



05 2151 250f1ff f1ff

05 2151 250


f2

f1

minus1

0

1

2

3

4

5

05 2151 25 30

MOPSO

by MOPSO


f1ff

05 2151 250


f2

f2

f1

05 2151 2500

051

152

253

354

SPEA2

by SPEA2


f1ff

05 21510


f1

05 215100

051

152

253

354

45PAES

by PAES


f1ff

05 21510


Figure 4 Continued



3205 1 25150f1

minus2

minus1

012

f2

345 MOPSO

f2


3205 1 25150f1

minus2

minus1

01234 PAES

f2


05 2151 250f1

005

115

225

335

445 SPEA2

f2


05 2151 250f1

005

115

225

335

445

MOPSO

by MOPSO by PAES

by SPEA2 by MOPSO


3205 1 25150f1ff

f2

ff


3205 1 25150f1ff

minus2

minus1

01234


05 2151 250f1ff

SPEA2

f2

ff


05 2151 250f1ff

005

115

225

335

445

MOPSO

by MOPSO by PAES









4 Conclusions




f2


SMPSO

05 1 15 2 250f1

005

115

225

335

454

f2


SPEA2

05 1 15 2 250f1

005

115

225

335

445

f2


SPEA2

05 1 15 2 250f1

005

115

225

335

445

f2


SMPSO

005

115

225

335

445

05 1 15 2 250f1


PAES

005

115

225

335

445

f2

05 1 15 2 250f1


SMPSO

005

115

2f2 25

335

445

05 1 15 2 250f1


PAES

005

115

225

335

445

f2

05 1 15 2 250f1


PAES

005

115

225

335

445

f2

f2

05 1 15 2 250f1


PAES

005

115

225

335

445

05 1 15 2 250f1


PAES

005

115

225

335

445

f2

05 15 21f1





Acknowledgments


References










































Journal of












Journal of







Volume 2018






Volume 2018

























SMPSO

02 04 06 08 10f1

00102030405f

2

06070809

1


02 04 06 08 10f1

00102030405f

2

06070809

1


PAES

02 04 06 08 10f1

00102030405f

2

06070809

1


PAES

01 02 03 04 05 06 07 08 090f1

minus1minus05

005

115

2

f2

253

354


PAES

00102030405f

2

06070809

1

02 04 06 08 10f1


SMPSO


00204

f2

0608

112

01 02 03 04 05 06 07 08 090f1


SPEA2





0010203040506070809

1

05 0803 06 0704 09 102

MOEAD

f2

f1

by MOEAD


05 0803 06 0704 09 12f1ff


0010203040506070809

1

05 0803 06 0704 09 102

f2

f1

PAES

by PAES


05 0803 06 0704 09 12f1ff



05 2151 25 30minus05

005

115

225

335

4SPEA2

f2

f2

f1 f1

minus1

0

1

2

3

4

5

05 2151 25 30

NSGAII

by SPEA2 by NSGAII



05 2151 250f1ff f1ff

05 2151 250


f2

f1

minus1

0

1

2

3

4

5

05 2151 25 30

MOPSO

by MOPSO


f1ff

05 2151 250


f2

f2

f1

05 2151 2500

051

152

253

354

SPEA2

by SPEA2


f1ff

05 21510


f1

05 215100

051

152

253

354

45PAES

by PAES


f1ff

05 21510


Figure 4 Continued



3205 1 25150f1

minus2

minus1

012

f2

345 MOPSO

f2


3205 1 25150f1

minus2

minus1

01234 PAES

f2


05 2151 250f1

005

115

225

335

445 SPEA2

f2


05 2151 250f1

005

115

225

335

445

MOPSO

by MOPSO by PAES

by SPEA2 by MOPSO


3205 1 25150f1ff

f2

ff


3205 1 25150f1ff

minus2

minus1

01234


05 2151 250f1ff

SPEA2

f2

ff


05 2151 250f1ff

005

115

225

335

445

MOPSO

by MOPSO by PAES









4 Conclusions




f2


SMPSO

05 1 15 2 250f1

005

115

225

335

454

f2


SPEA2

05 1 15 2 250f1

005

115

225

335

445

f2


SPEA2

05 1 15 2 250f1

005

115

225

335

445

f2


SMPSO

005

115

225

335

445

05 1 15 2 250f1


PAES

005

115

225

335

445

f2

05 1 15 2 250f1


SMPSO

005

115

2f2 25

335

445

05 1 15 2 250f1


PAES

005

115

225

335

445

f2

05 1 15 2 250f1


PAES

005

115

225

335

445

f2

f2

05 1 15 2 250f1


PAES

005

115

225

335

445

05 1 15 2 250f1


PAES

005

115

225

335

445

f2

05 15 21f1





Acknowledgments


References










































Journal of












Journal of







Volume 2018






Volume 2018









0010203040506070809

1

05 0803 06 0704 09 102

MOEAD

f2

f1

by MOEAD


05 0803 06 0704 09 12f1ff


0010203040506070809

1

05 0803 06 0704 09 102

f2

f1

PAES

by PAES


05 0803 06 0704 09 12f1ff



05 2151 25 30minus05

005

115

225

335

4SPEA2

f2

f2

f1 f1

minus1

0

1

2

3

4

5

05 2151 25 30

NSGAII

by SPEA2 by NSGAII



05 2151 250f1ff f1ff

05 2151 250


f2

f1

minus1

0

1

2

3

4

5

05 2151 25 30

MOPSO

by MOPSO


f1ff

05 2151 250


f2

f2

f1

05 2151 2500

051

152

253

354

SPEA2

by SPEA2


f1ff

05 21510


f1

05 215100

051

152

253

354

45PAES

by PAES


f1ff

05 21510


Figure 4 Continued



3205 1 25150f1

minus2

minus1

012

f2

345 MOPSO

f2


3205 1 25150f1

minus2

minus1

01234 PAES

f2


05 2151 250f1

005

115

225

335

445 SPEA2

f2


05 2151 250f1

005

115

225

335

445

MOPSO

by MOPSO by PAES

by SPEA2 by MOPSO


3205 1 25150f1ff

f2

ff


3205 1 25150f1ff

minus2

minus1

01234


05 2151 250f1ff

SPEA2

f2

ff


05 2151 250f1ff

005

115

225

335

445

MOPSO

by MOPSO by PAES









4 Conclusions




f2


SMPSO

05 1 15 2 250f1

005

115

225

335

454

f2


SPEA2

05 1 15 2 250f1

005

115

225

335

445

f2


SPEA2

05 1 15 2 250f1

005

115

225

335

445

f2


SMPSO

005

115

225

335

445

05 1 15 2 250f1


PAES

005

115

225

335

445

f2

05 1 15 2 250f1


SMPSO

005

115

2f2 25

335

445

05 1 15 2 250f1


PAES

005

115

225

335

445

f2

05 1 15 2 250f1


PAES

005

115

225

335

445

f2

f2

05 1 15 2 250f1


PAES

005

115

225

335

445

05 1 15 2 250f1


PAES

005

115

225

335

445

f2

05 15 21f1





Acknowledgments


References










































Journal of












Journal of







Volume 2018






Volume 2018










3205 1 25150f1

minus2

minus1

012

f2

345 MOPSO

f2


3205 1 25150f1

minus2

minus1

01234 PAES

f2


05 2151 250f1

005

115

225

335

445 SPEA2

f2


05 2151 250f1

005

115

225

335

445

MOPSO

by MOPSO by PAES

by SPEA2 by MOPSO


3205 1 25150f1ff

f2

ff


3205 1 25150f1ff

minus2

minus1

01234


05 2151 250f1ff

SPEA2

f2

ff


05 2151 250f1ff

005

115

225

335

445

MOPSO

by MOPSO by PAES









4 Conclusions




f2


SMPSO

05 1 15 2 250f1

005

115

225

335

454

f2


SPEA2

05 1 15 2 250f1

005

115

225

335

445

f2


SPEA2

05 1 15 2 250f1

005

115

225

335

445

f2


SMPSO

005

115

225

335

445

05 1 15 2 250f1


PAES

005

115

225

335

445

f2

05 1 15 2 250f1


SMPSO

005

115

2f2 25

335

445

05 1 15 2 250f1


PAES

005

115

225

335

445

f2

05 1 15 2 250f1


PAES

005

115

225

335

445

f2

f2

05 1 15 2 250f1


PAES

005

115

225

335

445

05 1 15 2 250f1


PAES

005

115

225

335

445

f2

05 15 21f1





Acknowledgments


References










































Journal of












Journal of







Volume 2018






Volume 2018









f2


SMPSO

05 1 15 2 250f1

005

115

225

335

454

f2


SPEA2

05 1 15 2 250f1

005

115

225

335

445

f2


SPEA2

05 1 15 2 250f1

005

115

225

335

445

f2


SMPSO

005

115

225

335

445

05 1 15 2 250f1


PAES

005

115

225

335

445

f2

05 1 15 2 250f1


SMPSO

005

115

2f2 25

335

445

05 1 15 2 250f1


PAES

005

115

225

335

445

f2

05 1 15 2 250f1


PAES

005

115

225

335

445

f2

f2

05 1 15 2 250f1


PAES

005

115

225

335

445

05 1 15 2 250f1


PAES

005

115

225

335

445

f2

05 15 21f1





Acknowledgments


References










































Journal of












Journal of







Volume 2018






Volume 2018











Acknowledgments


References










































Journal of












Journal of







Volume 2018






Volume 2018



















Journal of












Journal of







Volume 2018






Volume 2018








evaluating multiobjective evolutionary algorithms using...

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