Decomposition-Based Multiobjective Evolutionary...

Research ArticleDecomposition-Based Multiobjective EvolutionaryOptimization with Adaptive Multiple Gaussian Process Models

XunfengWu Shiwen Zhang Zhe Gong Junkai Ji Qiuzhen Lin and Jianyong Chen

College of Computer Science and Software Engineering Shenzhen University Shenzhen China

Correspondence should be addressed to Jianyong Chen jychenszueducn

Received 27 October 2019 Revised 20 December 2019 Accepted 2 January 2020 Published 11 February 2020

Academic Editor Yong Xu

Copyright copy 2020 Xunfeng Wu et al +is 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

In recent years a number of recombination operators have been proposed for multiobjective evolutionary algorithms (MOEAs)One kind of recombination operators is designed based on the Gaussian process model However this approach only uses onestandard Gaussian process model with fixed variance which may not work well for solving various multiobjective optimizationproblems (MOPs) To alleviate this problem this paper introduces a decomposition-based multiobjective evolutionary opti-mization with adaptive multiple Gaussian process models aiming to provide a more effective heuristic search for various MOPsFor selecting a more suitable Gaussian process model an adaptive selection strategy is designed by using the performanceenhancements on a number of decomposed subproblems In this way our proposed algorithm owns more search patterns and isable to produce more diversified solutions +e performance of our algorithm is validated when solving some well-known F UFand WFG test instances and the experiments confirm that our algorithm shows some superiorities over six competitive MOEAs

1 Introduction

Multiobjective optimization problems (MOPs) widely existin the fields of scientific research and engineering applica-tions Since the first multiobjective evolutionary algorithm(MOEA) was reported in 1985 [1] MOEAs have beenstudied sufficiently and become one of the hottest researchdirections in the field of evolutionary computation [2ndash4]Internationally MOEAs represented by NSGA-II [5]SPEA2 [6] PAES [7] HypE [8] MOEAD [9] etc havebeen widely used in many application fields According tothe selection strategy to handle the convergence enhance-ment and diversity maintenance most of existing MOEAscan be roughly divided into the following three categories

11Pareto-BasedMOEAs +e basic idea of such algorithmsrepresented by NSGA-II and SPEA2 is to use Pareto-basedranking scheme for sorting the population into differentconvergence layers and then calculate the density of indi-viduals in the last layer In this way the population can besorted according to the dominance relationship and density

estimation and then the relatively superior individuals areselected to the next generation Crowded distance [5] K-nearest neighbor method [6] ε-domination [10 11] grading[12] and other methods [13ndash15] are often used to estimatethe density of the individuals As Pareto-based MOEAs havesome advantages like simple principle easy understandingand fewer parameters this kind of MOEAs has inducedmany research and extensive applications However theirability to guarantee convergence dramatically degradeswhen the number of objectives is larger than three mainlydue to the loss of selection pressure

12 Decomposition-Based MOEAs Decomposition-basedMOEAs transform a MOP into a set of subproblems andthen solve them simultaneously using a collaborative evo-lutionary search such as MOEAD [9] RVEA [16] NSGA-III [17] RPEA [18] SPEAR [19] and RdEA [20] Note thatmost of these algorithms adopt additional reference infor-mation (reference vectors reference points or weight vec-tors) during the environmental selection which helps tomaintain the diversity of the population Due to the

HindawiComplexityVolume 2020 Article ID 9643273 22 pageshttpsdoiorg10115520209643273

advantage with a better mathematical explanation de-composition-based MOEAs have become very popular inrecent years [21ndash24]

13 Indicator-Based MOEAs Indicator-based MOEAs di-rectly employ a performance indicator like hypervolume(HV) generational distance (GD) and R2 to effectivelyguide the selection of promising solutions for next gener-ation IBEA [25] MOMBI-II [26] HyPE [8] GD-MOEA[27] R2-IBEA [28] and DDE [29] are representatives for theindicator-based MOEAs In these MOEAs performanceindicators are used as selection criterion to rank non-dominated solutions that cannot be distinguished by tra-ditional Pareto dominance However for this kind ofMOEAs high computational complexity is usually requiredfor calculating the performance indicator which is verychallenging especially when the number of objectives islarge

+ese MOEAs usually include two main componentsie variation and selection [6] Selection plays an importantrole in MOEAs to maintain the promising solutions asintroduced above for classifying MOEAs while variation isthe key factor to determine the generation quality of solu-tions In [30] the effects of several variation operators arestudied on some test problems with variable linkagesshowing that variable linkages may cause some difficultiesfor MOEAs Actually there are a number of real-valuedvariation approaches having been proposed during the re-cent decades [31] which can be classified into the three mainkinds ie traditional recombination operators estimationof distribution algorithms and the inverse model methodsfor variation

Traditional recombination operators are generally usedin most existing MOEAs due to their simplicity +is kind ofvariation simulates the binary crossover method to producethe real-valued offspring such as simulated binary crossover(SBX) [32] Laplace crossover [33] parent central crossover[34] blend crossover α [35] unimodal normal distributioncrossover [36] and simple crossover [37] Moreover dif-ferential evolution (DE) [38] is also used as the recombi-nation operator in many MOEAs which samples offspringbased on the difference vectors of parents +e evolutionpaths are used in DE [39] to depict the population move-ment and predict its tendency which could produce po-tential solutions to speed up convergence toward the PSRecently a number of hybridized recombination operatorshave been proposed trying to combine the advantages ofdifferential recombination operators In [40] a cooperativeDE framework is designed for constrained MOPs in whichmultiple DE operators are run in different subpopulations tooptimize its own constrained subproblem In [41] four DEoperators are combined and a sliding window is adopted in[42] to provide the reward for each DE operator according tothe enhancement on subproblems Similarly four DE op-erator pools are presented in [43] including two DE oper-ators with complementary search patterns in each pool toprovide an improved search ability In ACGDE [44] anadaptive cross-generation DE operator is designed by

exploiting information from individuals across generationsto adapt the parameters

+e estimation of distribution algorithms (EDAs) [45]exploit the probabilistic models extracted from the pop-ulationrsquos distribution to run variation [46 47] Unlike theabove traditional recombination operators no crossover ormutation procedures will be run in EDAs Instead theglobally statistical information of the selected parents is usedin EDAs to build a posterior probability distribution modeland then offspring are sampled from this built modelSeveral EDAs are studied for solving continuous MOPs in[48 49] while the Gaussian distribution model mixtureGaussian distribution model or mixed Gaussian with theprincipal component analysis (PCA) model are introducedin [50] for offspring variation Different from the individ-ualrsquos information used in traditional recombination oper-ators the local PCA operator is employed in [45 47] togenerate new offspring Different from [47] the PCA modelwas replaced by the locally linear embedding (LLE) [51]model in [52] In this way this model only considers thedecision space without considering too much about thetarget MOP itself

+e inverse model methods for variation utilize machinelearning approaches to capture the connection from theobjective space to the decision space by exploiting thecharacteristics of the target MOP +e representative algo-rithms like IM-MOEA [53] and E-IM-MOEA [54] use theGaussian process-based inverse model to complete cross-over In another work [55] as inspired by LLE manifoldlearning idea a new LLEmodeling approach is introduced touse the mapping function known in the MOP that thedecision space is considered as the high-dimensional spaceand the objective space is regarded as a low-dimensionalspace +us this new modeling method is no longer to buildthe overall low dimensional space of the sample which isthen reflected back to the high dimensional space but itreplaces directly by constructing new samples in high di-mensional space In this way the model mapping from theobjective space to the decision space is built based on theobtained approximated Pareto set during the evolution Inother words this model can use the probability model orsurrogate model to build a bridge from the objective space tothe decision space

In the above three kinds of variation operators tradi-tional recombination operators are often used in manyMOEAs However as these operators are executed based onthe individuals they only provide a finite number of searchpatterns and are also criticized for lack of mathematic ex-planation For the inverse model methods for variation it isvery changeling for mapping from the objective space intothe decision space when tackling some complicated MOPsEDAs usually adopt one standard Gaussian process modelwith fixed variance which may not work well for variouskinds of MOPs To enhance the search capability of EDAsthis paper introduces a decomposition-based MOEA withadaptive multiple Gaussian process models called MOEAD-AMG which is effective in generating more superioroffspring Based on the performance enhancements on anumber of decomposed subproblems one suitable Gaussian

2 Complexity

process model will be selected accordingly In this way ourmethod is more intelligent and can provide more searchpatterns to produce the diversified solutions After evalu-ating our performance on some well-known F UF andWFG test instances the experimental results have validatedthe superiorities of our algorithm over six competitiveMOEAs (MOEAD-SBX [9] MOEAD-DE [56] MOEAD-GM [57] RM-MEDA [47] IM-MOEA [53] and AMEDA[50])

+e rest of this paper is organized as follows Section 2presents some background information of MOPs Gaussianprocess model and some Gaussian process model-basedMOEAs Section 3 introduces the details of MOEAD-AMGAt last Section 4 provides and discusses the simulationresults while Section 5 gives the conclusions and somefuture work

2 Preliminaries

21 Multiobjective Optimization Problems In real life thereexist a number of engineering problems with multiplecomplicated optimization objectives which are often calledmultiobjective optimizations problems (MOPs) +is paperconsiders to solve continuous MOPs as formulated below

min F(x) f1(x) fm(x)( 1113857T

subject to x isin 1113937n

i1ai bi1113858 1113859

(1)

where n and m are respectively the numbers of decisionvariables and objectives ai and bi are respectively the lowerand upper bounds of ith dimensional variable of x in thedecision space x (x1 xn)T isin Rn is a decision variablevector 1113937

ni1[ai bi] sub Rn is the feasible search space

fi Rn⟶ R i 1 m is a continuous mapping andF(x) consists of m continuous objective functions

In MOPs the conflicts often exist among different ob-jectives ie improvement of one objective results in dete-rioration of another Generally there does not exist anoptimal solution that can minimize all the objectives in (1) atthe same time A set of trade-off solutions among the ob-jectives can be found for solving MOPs which are equallyoptimal when considering all the objectives Suppose thatthere are two vectors p (p1 pm)T andq (q1 qm)T isin Rm where m is the number of objec-tives p is said to dominate q in equation (1) denoted byp≺ q if pile qi for all i 1 m and pne q A solutionxlowast isin Ω where Ω is the feasible search space in equation (1)is called Pareto optimal solution if and only if there is noanother x isin Ω such that F(x)≺F(xlowast) +e collection of allthe Pareto optimal solutions is called Pareto optimal set (PS)and the projection of PS in the objective space is calledPareto optimal front (PF)

22 Gaussian Process Model +e recombination operatorbased on the Gaussian process model belongs to EDADifferent from the classical recombination operators therecombination using Gaussian process model uses the

distribution information from the whole population togenerate offspring which can ensure the diversity of searchpatterns

+e Gaussian model is one of the most widely usedprobability models in scientific research and practical ap-plications [58ndash61] In general a random variablex (x1 x2 xn)T with a Gaussian distribution can beexpressed as

x sim N(μΣ) (2)

where μ is an n dimensional mean vector and 1113936 is thecovariance matrix +e probability density function of therandom variable is expressed as

f(x) (2π)minus (n2)

|Σ|exp minus12(x minus μ)

TΣminus 1(x minus μ)11139671113882 (3)

For a given set of data x1 x2 xK the mean vectorand covariance matrix are respectively obtained by

μ 1K

1113944

K

k1x

k

Σ 1

K minus 11113944

K

k1x

kminus μ1113872 1113873 x

kminus μ1113872 1113873

T

(4)

Hence a new solution x x1 x2 xn can be gener-ated using the Gaussian model which can be divided intothree steps of Algorithm 1 At first decompose the co-variance matrix 1113936 into a lower triangular matrix A by usingthe Cholesky decomposition method [62] in line 1 where1113936 AAT +en generate a vector y (y1 yn)T in line2 of which the element yi y 1 n is sampled from astandard Gaussian distribution N(0 1) After that a newtrial solution x can be yielded by x μ + Ay in line 3Generally to improve the search capability of the Gaussianmodel a small variation is added to mutate the new solutiony by the polynomial mutation operator [2]

23 Gaussian Process Model-Based MOEAs In recent yearsseveral Gaussian process model-based MOEAs have beenproposed +eir details are respectively introduced in thefollowing paragraphs At last the motivation of this paper isclarified

In [63] a mixture-based multiobjective iterated densityestimation evolutionary algorithm (MIDEA) with both dis-crete and continuous representations is proposed +is ap-proach employs clustering analysis to discover the nonlineardistribution structure which validates that a simple model isfeasible to describe a cluster and the population can beadequately represented by a mixture of simple models Afterthat a mixture of Gaussian models is proposed to producenew solutions for continuous real-valued MOPs in MIEDA

Although MIEDA is very effective for solving certainMOPs the regularity property of MOPs is not considered sothat it may perform poorly in some problems +us amultiobjective evolutionary algorithm based on decompo-sition and probability model (MOEAD-MG) is designed in[57] In this approach multivariate Gaussian models are

Complexity 3

embedded into MOEAD [9] for continuous multiobjectiveoptimization Either a local Gaussian distribution model isbuilt around a subproblem based on the neighborhood or aglobal model is constructed based on the whole population+us the population distribution can be captured by all theprobability models working together However MOEAD-MG has to reconstruct a Gaussian model for each sub-problem resulting in a large computational cost in theprocess of building the model Moreover when building theGaussian model for the similar subproblems some individ-uals may be repeatedly used which aggravates the compu-tational resources

To reduce the computational cost for themodeling processan improved MOEAD-MG with high modeling efficiency(MOEAD-MG2) is reported in [64] where the neighboringsubproblems can share the same covariance matrix to buildGaussian model for sampling solutions At first a process ofsorting population is used to adjust the sampling orders of thesubproblems which tries to avoid the insufficient diversitycaused by the reuse of the covariancematrix and can obtain theuniformly distributed offspring set+en in global search onlysome subproblems are selected randomly to construct theGaussian model Although MOEAD-MG2 performs well forsolving some MOPs it just builds the Gaussian model forsolution sampling in the neighborhoods as defined in MOEAD which is only used in the MOEAD paradigm

Moreover an adaptive multiobjective estimation ofdistribution algorithm with a novel Gaussian samplingstrategy (AMEDA) is presented in [50] In this work aclustering analysis approach is adopted to reveal thestructure of the population distribution Based on theseclusters a local multivariate Gaussian model or a globalmodel is built for each solution to sample a new solutionwhich can enhance the accuracy of modeling and thesearching ability of AMEDA Moreover an adaptive updatestrategy of the probability is developed to control thecontributions for two types of Gaussian model

From the above studies it can be observed that theirmodeling differences mainly focus on the methods ofselecting sampling solutions However the above Gaussianmodels in [50 63 64] only adopt the standard one with fixedvariance which cannot adaptively adjust the search step sizesaccording to the different characteristics of MOPs To alle-viate the above problem a decomposition-based MOEA withadaptive multiple Gaussian process models (calledMOEAD-AMG) is proposed in this paper Multiple Gaussian modelsare used in our algorithm by using a set of different variances+en based on the performance enhancements of decom-posed subproblems a suitable Gaussian model will be

adaptively selected which helps to enhance the search ca-pability of MOEAD-AMG and can well handle various kindsof MOPs as validated in the experimental section

3 The Proposed Algorithm

In this section our proposed algorithm MOEAD-AMG isintroduced in detail At first the adaptive multiple Gaussianprocess models are described+en the details of MOEAD-AMG are demonstrated

31 Adaptive Multiple Gaussian Process Models Many ap-proaches have been designed by taking advantages of theregularity in distributions of Pareto optimal solutions inboth the decision and objective spaces to estimate thepopulation distribution Considering the mild conditions itcan be induced from the KarushndashKuhnndashTucker (KTT)condition that the PF is (m-1)-dimensional piecewisecontinuous manifolds [65] for an m-objective optimizationproblem +at is to say the PF of a continuous biobjectiveMOP is a piecewise continuous curve while the PF of acontinuous three-objective MOP is a piecewise continuoussurface +us the Gaussian process model has been widelystudied for both single-objective and multiobjective opti-mization [66ndash72] However a single Gaussian process modelis not so effective for modeling the population distributionwhen tackling some complicated MOEAs as studied in [57]In consequence multiple Gaussian models are used in thispaper which can explicitly exploit the ability of differentGaussian models with various distributions By usingmultiple Gaussian process models with good diversity amore suitable one should be adaptively selected to capturethe population structure for sampling new individuals moreaccurately +us five types of Gaussian models are used inthis paper which have the same mean value 0 with differentstandard deviations ie 06 08 10 12 and 14 +edistributions of five Gaussian models (as respectivelyrepresented by g1 g2 g3 g4 and g5) are plotted in Figure 1

To select a suitable one from multiple Gaussian processmodels an adaptive strategy is proposed in this paper toimprove the comprehensive search capability +e proba-bilities of selecting these different models are dependent ontheir performance to optimize the subproblems Once aGaussian process model is selected it will be used to generatethe Gaussian distribution variable y in line 3 of Algorithm 1Before the adaptive strategy comes into play these pre-defined Gaussian models are selected with an equal prob-ability In our approach a set of offspring solutions will beproduced by using different Gaussian distribution variablesy After analyzing the quality of these new offspring solu-tions the contribution rate of each Gaussian process modelcan be calculated It should be noted that the quality of thenew offspring is determined by their fitness value which canbe calculated by many available methods In our work theTchebycheff approach in [9] is adopted as follows

gi(x) g x λi z

11138681113868111386811138681113872 1113873 max1lejlem

λij fi(x) minus zj

11138681113868111386811138681113868

11138681113868111386811138681113868 (5)

(1) Obtain a lower triangular matrix A throughdecomposing the covariance matrix using Choleskyand 1113936 AAT

(2) Generate a single Gaussian distribution variabley (y1 y2 yn)T y sim N(0 1) j 1 2 n

(3) x μ + Ay

ALGORITHM 1 Sampling model

4 Complexity

where z is a reference point λi is the ith weight vector andmis the number of objectives A smaller fitness value of newoffspring which is generated by Gaussian distribution var-iable indicates a greater contribution for the correspondingGaussian model Hence for each Gaussian model the im-provement of fitness (IoF) value can be obtained as follows

IoFkG FekGminus LP minus FekG FekG lt FekGminus LP

0 otherwise1113896 (6)

where FekG is the fitness of new offspring with the kthGaussian distribution in generation G In order to maximizethe effectiveness of the proposed adaptive strategy thisstrategy is executed by each of LP generations in the wholeprocess of our algorithm Afterwards the contribution rates(Cr) of different Gaussian distributions can be calculated asfollows

CrkG IoFkG

1113936Ki1loFiG

+ ε (7)

where ε is a very small value which works when IoF is zero+en the probabilities (Pr) of different distributions beingselected are updated by the following formula

PrkG CrkG

1113936Ki1CriG

(8)

where K is the total number of Gaussian models As de-scribed above we can adaptively adjust the probability thatthe kth Gaussian distribution is selected by updating thevalue of PrkG

32 5e Details of MOEAD-AMG +e above adaptivemultiple Gaussian process models are just used as the re-combination operator which can be embedded into a state-of-the-art MOEA based on decomposition (MOEAD [9])giving the proposed algorithm MOEAD-AMG In thissection the details of the proposed MOEAD-AMG areintroduced

To clarify the running of MOEAD-AMG its pseudo-code is given in Algorithm 2 and some used parameters areintroduced below

(1) N indicates the number of subproblems and also thesize of population

(2) K is the number of Gaussian process models(3) G is the current generation(4) ξ is the parameter which is applied to control the

balance between exploitation and exploration(5) gss is the number of randomly selected subproblems

for constructing global Gaussian model(6) LP is the parameter to control the frequency of using

the proposed adaptive strategy(7) nr is the maximal number of old solutions which are

allowed to be replaced by a new one(8) perm() randomly shuffles the input values and

rand() produces a random real number in [0 1]

Lines 1-2 of Algorithm 2 describe the initializationprocess by setting the population size N and generating Nweight vectors using the approach in [56] to define Nsubproblems in (5) +en an initial population is randomlygenerated to include N individuals and the neighborhoodsize is set to select T neighboring subproblems for con-structing the Gaussian model +en the reference point z

z1 z2 zm1113864 1113865 can be obtained by including all the minimalvalue of each objective In line 3 multiple Gaussian dis-tributions are defined and the corresponding initial prob-abilities are set to an equal value +en a maximum numberof generations is used as the termination condition in line 4and all the subproblems are randomly selected once at eachgeneration in line 5 After that the main evolutionary loop ofthe proposed algorithm is run in lines 5ndash26 For the sub-problem selected at each generation it will go through threecomponents including model construction model upda-tion and population updation as observed fromAlgorithm 2

Lines 6ndash10 run the process of model construction Ascontrolled by the parameter ξ a set of subproblems B iseither the neighborhood of current subproblem Bi in line 7or gss subproblems selected randomly in line 9 If theneighborhood of current subproblem is used the localmultiple Gaussian models are constructed for running ex-ploitation otherwise the global models are constructed forrunning exploration

Lines 11ndash16 implement the process of model updation+e adaptive strategy described in Section 31 is used toupdate the probability of five different Gaussian models ateach of LP generations as shown in line 12+en a Gaussianmodel can be selected in line 14 and the selected model isused to update y in line 15 At last Algorithm 1 is performedto generate a new offspring in line 16 with the selectedsubproblems B for model construction and the generatedGaussian variable y

Lines 17ndash23 give the process of population updation Foreach individual of B in line 18 if the aggregated functionvalue using equation (5) for subproblem is improved by the

Gaussian curves

0

02

04

06

08

1

21 30 4 5ndash2ndash3ndash4 ndash1ndash5

g1-06g2-08g3-10

g4-12g5-14

Figure 1 +e distributions of five Gaussian models

Complexity 5

new offspring x in line 19 the original individual will bereplaced by x in line 20 +e c value is increased by 1 in line21 to prevent premature convergence such that at most cindividuals in B can be replaced

At last the generation counter G is added by 1 and thereference point z is updated by including the new minimalvalue for each objective If the termination in line 4 is notsatisfied the above evolutionary loop in lines 5ndash26 will berun again otherwise the final population will be outputtedin line 28

4 Experimental Studies

41 Test Instances Twenty-eight unconstrained MOP testinstances are employed here as the benchmark problems forempirical studies To be specific UF1-UF10 are used as thebenchmark problems in CEC2009 MOEA competition andF1ndashF9 are proposed in [48] +ese test instances havecomplicated PS shapes We also consider WFG test suite [49]with different problem characteristics including no separabledeceptive degenerate problems mixed PF shape and variabledependencies+e number of decision variables is set to 30 forUF1-UF10 and F1ndashF9 for WFG1-WFG9 the numbers ofposition and distance-related decision variable are respec-tively set to 2 and 4 while the number of objectives is set to 2

42 Performance Metrics

421 5e Inverted Generational Distance (IGD) [73]Here assume that Plowast is a set of solutions uniformly sampledfrom the true PF and S represents a solution set obtained by aMOEA [73] +e IGD value from Plowast to S will calculate theaverage distance from each point of Plowast to the nearest so-lution of S in the objective space as follows

IGD S Plowast

( 1113857 1113936xisinPlowastdist(x s)

Plowast (9)

where dist (x S) returns the minimal distance from onesolution x in Plowast to one solution in S and |Plowast| returns the sizeof Plowast

422 Hypervolume (HV) [74] Here assume that a referencepoint Zr (zr

1 zr2 zr

m) in the objective space is domi-nated by all Pareto-optimal objective vectors [74] +en theHV metric will measure the size of the objective spacedominated by the solutions in S and bounded by Zr asfollows

HV(S) VOL cupxisinS f1(x) zr11113858 1113859 times middot middot middot times fm(x) z

rm1113858 1113859( 1113857

(10)

(1) Initialize N subproblems including weight vector w N individuals and neighborhood size T(2) Initialize the reference point z z1 z2 zm1113864 1113865 j 1 2 m(3) Predefine y sim N(0 σ) N(0 σ) Pk0 1K(4) while not terminate do(5) for i isin perm( x1 x2 xN1113864 1113865) do

Model construction(6) if rand( )lt ξ then(7) Define neighborhood individuals B Bi

(8) else(9) Select gss individuals from x1 x2 xN1113864 1113865) to construct B(10) end

Model updation(11) if mod(G LP) 0 then(12) Update PrkG using equations (5)ndash(8)(13) end(14) Select a Gaussian model according to PrkG

(15) Generate y based on the selected Gaussian model(16) Add B and y into Algorithm 1 to generate a new offspring x

Population updating(17) Set counter c 0(18) for each j isin B do(19) if gi(x)ltgi(xj) and clt nr then(20) xj is replaced by x

(21) Set c c+1(22) end(23) end(24) end(25) GG+ 1(26) Update the reference point z

(27) end(28) Output the final population

ALGORITHM 2 MOEAD-AMG

6 Complexity

where VOL(middot) indicates the Lebesgue measure In ourempirical studies Zr (11 11)T and Zr (11 11 11)T

are respectively set for the biobjective and three-objectivecases in UF and F test problems Due to the different scaledvalues in different objectives Zr (20 40)T is set for WFGtest suite

Both IGD and HV metrics can reflect the convergenceand diversity for the solution set S simultaneously+e lowerIGD value (or the larger HV value) indicates the betterquality of S for approximating the entire true PF In thispaper six competitive MOEAs including MOEAD-SBX[9] MOEAD-DE [56] MOEAD-GM [57] RM-MEDA[47] IM-MOEA [53] and AMEDA [50] are included tovalidate the performance of our proposed MOEAD-AMGAll the comparison results obtained by these MOEAs re-garding IGD and HV are presented in the correspondingtables where the best mean metric values are highlighted inbold and italics In order to have statistically sound con-clusions Wilcoxonrsquos rank sum test at a 5 significance levelis conducted to compare the significance of difference be-tween MOEAD-AMG and each compared algorithm

43 5e Parameter Settings +e parameters of MOEAD-SBX MOEAD-DE MOEAD-GM RM-MEDA IM-MOEA and AMEDA are respectively set according toreferences [9 47 53 56 57 50] All these MOEAs areimplemented in MATLAB +e detailed parameter settingsof all the algorithms are summarized as follows

431 Public Parameters For population size N we setN 300 for biobjective problems and N 600 for three-objective problems For number of runs and terminationcondition each algorithm is independently launched by 20times on each test instance and the termination condition ofan algorithm is the predefined maximal number of functionevaluations which is set to 300000 for UF instances 150000for F instances and 200000 for WFG instances We set theneighborhood size T 20 and the mutation probabilitypm 1n where n is the number of decision variables foreach test problem and its distribution index is μm 20

432 Parameters in MOEAD-SBX +e crossover proba-bility and distribution index are respectively set to 09 and30

433 Parameters in MOEAD-DE +e crossover rateCR 1 and the scaling factor F 05 in DE as recommendedin [9] the maximal number of solution replacement nr 2and the probability of selecting the neighboring subprob-lems δ 09

434 Parameters in MOEAD-GM +e neighborhood sizefor each subproblem K 15 for biobjective problems andK 30 for triobjective problems (this neighborhood size K iscrucial for generating offspring and updating parents inevolutionary process) the parameter to balance the

exploitation and exploration pn 08 and the maximalnumber of old solutions which are allowed to be replaced bya new one C 2

435 Parameters in RM-MEDA +enumber of clustersK isset to 5 (this value indicates the number of disjoint clustersobtained by using local principal component analysis onpopulation) and the maximum number of iterations in localPCA algorithm is set to 50

436 Parameters in IM-MOEA +e number of referencevectors K is set to 10 and the model group size L is set to 3

437 Parameters in AMEDA +e initial control probabilityβ0 09 the history length H 10 and the maximumnumber of clusters K 5 (this value indicates the maximumnumber of local clusters obtained by using hierarchicalclustering analysis approach on population K)

438 Parameters in MOEAD-AMG +e number ofGaussian process models K 5 the initial Gaussian distri-bution is (0 062) (0 082) (0 102) (0 122) and (0 142)LP 10 nr 2 ξ 09 and gss 40

44 Comparison Results and Discussion In this section theproposed MOEAD-AMG is compared with six competitiveMOEAs including MOEAD-SBX MOEAD-DE MOEAD-GM RM-MEDA IM-MOEA and AMEDA +e resultsobtained by these algorithms on the 28 test problems aregiven in Tables 1 and 2 regarding the IGD and HV metricsrespectively When compared to other algorithms theperformance of MOEAD-AMG has a significant im-provement when the multiple Gaussian process models areadaptively used It is observed that the proposed algorithmcan improve the performance ofMOEAD inmost of the testproblems Table 1 summarizes the statistical results in termsof IGD values obtained by these compared algorithmswhere the best result of each test instance is highlighted +eWilcoxon rank sum test is also adopted at a significance levelof 005 where symbols ldquo+rdquo ldquominus rdquo and ldquosimrdquo indicate that resultsobtained by other algorithms are significantly better sig-nificantly worse and no difference to that obtained by ouralgorithm MOEAD-AMG To be specific MOEAD-AMGshows the best results on 12 out of the 28 test instances whilethe other compared algorithms achieve the best results on 33 5 2 2 and 1 out of the 28 problems in Table 1 Whencompared with MOEAD-SBX MOEAD-DE MOEAD-GM RM-MEDA IM-MOEA and AMEDA MOEAD-AMG yields 23 21 18 22 24 and 25 significantly bettermean IGD metric values respectively When compared toRM-MEDA IM-MOEA and AMEDA MOEAD-AMG hasshown an absolute advantage as it outperforms them on 2224 and 25 test problems regarding IGD values respectivelywhereas RM-MEDA outperforms MOEAD-AMG on F8UF3 WFG2 and WFG6 IM-MOEA outperforms MOEAD-AMG on F1 UF10 WFG4 and WFG6 and AMEDA

Complexity 7

Table 1 Performance comparison of MOEAD-SBX MOEAD-DE MOEAD-GM RM-MEDA IM-MOEA AMEDA and MOEAD-AMG in terms of the average IGD values and standard deviation on F UF and WFG problems

Problems MOEAD-SBX MOEAD-DE MOEAD-GM RM-MEDA IM-MOEA AMEDA MOEAD-AMG

F1 343E minus 03(800E minus 04)minus

366E minus 03(194E minus 04)minus

136E minus 03(136E minus 05)sim


127Eminus 03(216Eminus 05)+


135E minus 03(128E minus 05)







285Eminus 03(295Eminus 04)
































138E+ 00(411E minus 01)minus







120E minus 02(514E minus 03)+











UF1 105E minus 01(271E minus 02)minus





















UF4 519E minus 02(361E minus 03)+















156Eminus 01(489Eminus 02)sim


































WFG1 771Eminus 01(532Eminus 02)+

106E+ 00(762E minus 03)+





111E+ 00(117E minus 02)

WFG2 579E minus 02(212E minus 02)minus
























































Total 5023 6121 7318 4222 4024 3025+e best average values obtained by these algorithms for each instance are given in bold and italics

8 Complexity

Table 2 Performance comparison of MOEAD-SBX MOEAD-DE MOEAD-GM RM-MEDA IM-MOEA AMEDA and MOEAD-AMG in terms of the average HV values and standard deviation on F UF and WFG problems































































































































104E+ 00(129Eminus 02)



000E+ 00(000E+ 00)minus

000E+ 00(000E+ 00)minus


000E+ 00(000E+ 00)minus


WFG1 208E+ 00(166Eminus 01)+

129E+ 00(190E minus 02)+





112E+ 00(250E minus 02)

WFG2 441E+ 00(147E minus 02)minus


446E+ 00(676E minus 04)sim

445E+ 00(133E minus 03)sim


445E+ 00(165E minus 02)sim

446E+ 00(870Eminus 04)







398E+ 00(580Eminus 04)

WFG4 170E+ 00(132Eminus 03)+

169E+ 00(306E minus 03)+


150E+ 00(423E minus 03)sim

154E+ 00(210E minus 02)+

162E+ 00(261E minus 02)+

150E+ 00(560E minus 03)

WFG5 145E+ 00(107E minus 02)sim

145E+ 00(725Eminus 03)sim

145E+ 00(320E minus 04)sim



144E+ 00(912E minus 04)sim

145E+ 00(275E minus 04)




168E+ 00(949E minus 04)+

167E+ 00(707E minus 03)+

169E+ 00(152Eminus 03)+

162E+ 00(245E minus 01)


169E+ 00(135E minus 03)sim





170E+ 00(240Eminus 04)







141E+ 00(298Eminus 03)






164E+ 00(386E minus 03)sim

164E+ 00(146E minus 03)


Complexity 9

outperforms MOEAD-AMG onWFG2 WFG4 and WFG6when considering IGD values Regarding the comparisons toMOEAD-SBX and MOEAD-DE MOEAD-AMG alsoshows some advantages as it outperforms them on at least 21test problems Except for UF4-UF5 UF10 WFG1 andWFG4-WFG5 MOEAD-AMG performs better on mostcases regarding the IGD values Since a multivariateGaussian model is used in MOEAD-GM to capture thepopulation distribution from a global view MOEAD-GMcan give the best IGD values on 7 cases of test instancesHowever MOEAD-AMG performs better than MOEAD-AMG on 18 test problems +ese statistical results aboutIGD indicate MOEAD-AMG has the better optimizationperformance when compared with other algorithms onmostof the test instances

For HV metric shown in Table 2 the experimental re-sults also demonstrate the superiority of MOEAD-AMG onthese test problems It is clear from Table 2 that MOEAD-AMG obtains the best results in 13 out of 28 cases whileMOEAD-SBX MOEAD-DE MOEAD-GM RM-MEDAIM-MOEA and AMEDA preform respectively best in 2 35 1 2 and 2 cases As observed from Table 2 considering theF instances except for MOEAD-GM MOEAD-AMG isonly beaten by IM-MOEA on F1 while MOEA-GM is a littlebetter than MOEAD-AMG on F6ndashF8 On UF problemsMOEAD-AMG obtains significantly better results thanMOEAD-SBX and RM-MEDA on all cases in terms of HVMOEAD-DE performs better on UF4 and UF5 andMOEAD-GMperforms better onUF3-UF5 andUF7Whencompared to IM-MEDA and AMEDA they only performbetter than MOEAD-AMG on UF10 and UF6 respectivelyFor WFG instances MOEAD-DE and RM-MEDA onlyobtain statistically similar results with MOEAD-AMG onWFG5 and WFG9 respectively MOEAD-AMG signifi-cantly outperformsMOEAD-GM onWFG1-WFG9 Expectfor WFG1 and WFG4 MOEAD-AMG shows the superiorperformance on most WFG problems AMEDA only has thebetter HV result on WFG6 +erefore it is reasonable todraw a conclusion that the proposed MOEAD-AMGpresents a superior performance over each compared al-gorithm when considering all the test problems

Moreover we also use the R2 indicator to further showthe superior performance of MOEAD-AMG and thesimilar conclusions can be obtained from Table 3

To examine the convergence speed of the seven algo-rithms the mean IGD metric values versus the fitnessevaluation numbers for all the compared algorithms over 20independent runs are plotted in Figures 2ndash4 respectively forsome representative problems from F UF and WFG testsuites It can be observed from these figures that the curves ofthe mean IGD values obtained by MOEAD-AMG reach thelowest positions with the fastest searching speed on mostcases including F2ndashF5 F9 UF1-UF2 UF8-UF9WFG3 andWFG7-WFG8 Even for F1 F6ndashF7 UF6-UF7 and WFG9MOEAD-AMG achieves the second lowest mean IGDvalues in our experiment +e promising convergence speedof the proposed MOEAD-AMG might be attributed to theadaptive strategy used in the multiple Gaussian processmodels

To observe the final PFs Figures 5 and 6 present the finalnondominated fronts with the median IGD values found byeach algorithm over 20 independent runs on F4ndashF5 F9 UF2and UF8-UF9 Figure 5 shows that the final solutions of F4yielded by RM-MEDA IM-MOEA and AMEDA do notreach the PF while MOEAD-GM and MOEAD-AMGhave good approximations to the true PF Nevertheless thesolutions achieved by MOEAD-AMG on the right end ofthe nondominated front have better convergence than thoseachieved by MOEAD-GM In Figure 5 it seems that F5 is ahard instance for all compared algorithms+is might be dueto the fact that the optimal solutions to two neighboringsubproblems are not very close to each other resulting inlittle sense to mate among the solutions to these neighboringproblems+erefore the final nondominated fronts of F5 arenot uniformly distributed over the PF especially in the rightend However the proposed MOEAD-AMG outperformsother algorithms on F5 in terms of both convergence anddiversity For F9 that is plotted in Figure 5 except forMOEAD-GM and MOEAD-AMG other algorithms showthe poor performance to search solutions which can ap-proximate the true PF With respect to UF problems inFigure 6 the final solutions with median IGD obtained byMOEAD-AMG have better convergence and uniformlyspread on the whole PF when compared with the solutionsobtained by other compared algorithms +ese visual com-parison results reveal that MOEAD-AMG has much morestable performance to generate satisfactory final solutionswith better convergence and diversity for the test instances

Based on the above analysis it can be concluded thatMOEAD-AMG performs better than MOEAD-SBXMOEAD-DE MOEAD-GM RM-MEDA IM-MOEA andAMEDA when considering all the F UF and WFG in-stances which is attributed to the use of adaptive strategy toselect a more suitable Gaussian model for running recom-bination+eremay be different causes to let MOEAD-SBXMOEAD-DE MOEAD-GM RM-MEDA IM-MOEA andAMEDA perform not so well on the test suites For RM-MEDA and AMEDA they select individuals to construct themodel by using the clustering method which cannot showgood diversity as that in the MOEAD framework Bothalgorithms using the EDA method firstly classify the indi-viduals and then use all the individuals in the same class assamples for training model Implicitly an individual in thesame class can be considered as a neighboring individualHowever on the boundary of each class there may existindividuals that are far away from each other and there mayexist individuals from other classes that are closer to thisboundary individual which may result in the fact that theirmodel construction is not so accurate For IM-MOEA thatuses reference vectors to partition the objective space itstraining data are too small when dealing with the problemswith low decision space like some test instances used in ourempirical studies

45 Effectiveness of the Proposed Adaptive StrategyAccording to the experimental results in the previous sec-tion MOEAD-AMG with the adaptive strategy shows great

10 Complexity

Table 3 Performance comparison of MOEAD-SBX MOEAD-DE MOEAD-GM RM-MEDA IM-MOEA AMEDA and MOEAD-AMG in terms of the average R2 values and standard deviation on F UF and WFG problems

Problems MOEADminus SBX MOEADminus DE MOEADminus GM RMminus MEDA IMminus MOEA AMEDA MOEADminus AMG



















































































163E minus 01(115E minus 020minus



















































WFG1 698Eminus 01(259Eminus 02)minus














WFG3 452E minus 01(212E minus 04)+














WFG5 602Eminus 01(166Eminus 03)sim































592Eminus 01(675Eminus 04)minus





Complexity 11

F1

IGD

val

ues

10ndash1

10ndash2

10ndash3

0 1 15 205Fitness evaluation numbers

MOEAD-AMGAMEDAIM-MOEARM-MEDA

MOEAD-GMMOEAD-DEMOEAD-SBX

times105

(a)

F2

IGD

val

ues 10ndash1

100

10ndash2

10ndash3



05 1 15 20Fitness evaluation numbers times105

(b)F3

IGD

val

ues

10ndash1

100

10ndash2

10ndash3




(c)

F4

IGD

val

ues

10ndash1

100

10ndash2

10ndash3




(d)F5

IGD

val

ues

10ndash1

100

10ndash2




(e)

F6

IGD

val

ues

10ndash1

100



1 2 30Fitness evaluation numbers times105

(f )

Figure 2 Continued

12 Complexity

F7

IGD

val

ues

10ndash1

100

101

10ndash2

10ndash3




(g)

F9

IGD

val

ues 10ndash1

100

10ndash2

10ndash3




(h)

Figure 2 +e convergence curves of all the compared algorithms on F1ndashF7 and F9

UF1

IGD

val

ues 10ndash1

100

10ndash2

10ndash3




times105

(a)

UF2

IGD

val

ues

10ndash1

100

10ndash2




(b)UF6

IGD

val

ues

10ndash1

100

101




(c)

UF7

IGD

val

ues

10ndash2

10ndash1

100




(d)

Figure 3 Continued

Complexity 13

UF8

IGD

val

ues

10ndash1

100



1 320Fitness evaluation numbers times105

(e)

UF9

IGD

val

ues

10ndash2

10ndash1

100




(f )

Figure 3 +e convergence curves of all the compared algorithms on UF1-UF2 and UF6-UF9

WFG3

IGD

val

ues

10ndash1

100

10ndash2

times105




(a)

WFG7

IGD

val

ues

10ndash1

100

10ndash2

times105




(b)WFG8

IGD

val

ues

10ndash1

100

times105




(c)

WFG9

IGD

val

ues

10ndash1

100

10ndash2

times105




(d)

Figure 4 +e convergence curves of all the compared algorithms on WFG3 and WFG7-WFG9

14 Complexity

MOEAD-SBX-F4

0

02

04

06

08

1

12

02 04 06 080

(a)

MOEAD-DE-F4

0

02

04

06

08

1

12

02 04 06 08 10

(b)

MOEAD-GM-F4

0

02

04

06

08

1

12

05 1 15 20

(c)

RM-MEDA-F4

0

02

04

06

08

1

02 04 06 08 1 120

(d)IM-MOEA-F4

05 1 15 200

02

04

06

08

1

12

14

(e)

AM-EDA-F4

05 1 15 2 25 300

02

04

06

08

1

(f )

MOEAD-AMG-F4

02

0

04

06

08

1

12

04 0602 08 120 1

(g)

Ture-PF-F4

0

02

04

06

08

1

02 04 06 08 10

(h)MOEAD-SBX-F5

0

02

04

06

08

1

12

02 04 06 08 10

(i)

MOEAD-DE-F5

0

02

04

06

08

1

02 04 06 08 1 120

(j)

0

02

04

06

08

1

12

02 04 06 08 1 120

MOEAD-GM-F5

(k)

RM-MEDA-F4

0

02

04

06

08

1

02 04 06 08 1 120

(l)IM-MOEA-F5

02 04 06 08 1 1200

02

04

06

08

1

(m)

AM-EDA-F5

02 04 06 08 100

02

04

06

08

1

(n)

MOEAD-AMG-F5

02 04 06 08 1 1200

02

04

06

08

1

12

(o)

Ture-PF-F5

0

02

04

06

08

1

02 04 06 08 10

(p)MOEAD-SBX-F9

0

02

04

06

08

1

12

02 04 06 08 1 120

(q)

MOEAD-DE-F9

0

02

04

06

08

1

12

02 04 06 08 1 120

(r)

MOEAD-GM-F9

0

02

04

06

08

1

12

02 04 06 08 1 120

(s)

RM-MEDA-F9

0

02

04

06

08

1

12

02 04 06 08 1 120

(t)IM-MOEA-F9

0

1

2

3

4

5

1 2 3 540

(u)

AM-EDA-F9

02 04 06 08 1 1200

02

04

06

08

1

12

(v)

MOEAD-AMG-F9

02 04 06 08 1 1200

02

04

06

08

1

12

(w)

Ture-PF-F9

0

02

04

06

08

1

02 04 06 08 10

(x)

Figure 5 +e final populations obtained by all the compared algorithms on F4 F5 and F9

Complexity 15

MOEAD-SBX-UF2

02

04

06

08

1

12

02 04 06 080

(a)

MOEAD-DE-UF2

0

02

04

06

08

1

12

02 04 06 08 1 120

(b)

MOEAD-GM-UF2

0

02

04

06

08

1

12

02 04 06 08 1 120

(c)

RM-MEDA-UF2

0

02

04

06

08

1

02 04 06 08 10

(d)IM-MOEA-UF2

02 04 06 08 1 1200

02

04

06

08

1

(e)

AM-EDA-UF2

0

02

04

06

08

1

12

02 04 06 08 1 120

(f )

MOEAD-AMG-UF2

02 04 06 08 100

02

04

06

08

1

12

(g)

Ture-PF-UF2

0

02

04

06

08

1

02 04 06 08 10

(h)

2

1

00

12

3

005

115

MOEAD-SBX-UF8

(i)

3

2

1

00 0

2 24 4

MOEAD-DE-UF8

(j)

2

23

1

1

00

MOEAD-GM-UF8

151

050

(k)

1

05

00

24

6

0

12

RM-MEDA-UF8

(l)IM-MOEA-UF8

1

05

00

510 1

050

(m)

AM-EDA-UF8

1

05

00

105

02

4

(n)

MOEAD-AMG-UF8

1

15

15 151 1

05

05 050 00

(o)

Ture-PF-UF8

1

1 1

05

05 05

00 0

(p)MOEAD-SBX-UF9

15

1

05

00 0

1 1

22

3

(q)

MOEAD-DE-UF9

02

46

4

2

00

24

(r)

MOEAD-GM-UF9

0 0

2 2

4 4

15

1

05

0

(s)

RM-MEDA-UF9

15

1

05

00 0

2 2446

(t)RM-MEDA-UF9

15

1

05

00 0

2 2446

(u)

RM-MEDA-UF9

15

1

05

00 0

224

46

(v)

MOEAD-AMG-UF9

1

05

00 0

2 24 4

6 6

(w)

05

Ture-PF-UF9

1

1 105 05

00 0

(x)

Figure 6 +e final populations obtained by all the compared algorithms on UF2 UF8 and UF9

16 Complexity

Table 4 Performance comparison of MOEAD-FMGwith various Gaussian distributions andMOEAD-AMG in terms of the average IGDvalues and standard deviations on F UF and WFG problems

Problems MOEAD-FMG (0062)

MOEAD-FMG (0082)

MOEAD-FMG (0102)

MOEAD-FMG (0122)

MOEAD-FMG (0142) MOEAD-AMG

F1 129Eminus 03 (800Eminus04)+
































470Eminus 02 (765Eminus03)+












































































UF10 619Eminus 01 (823Eminus02)+











111E+ 00(117Eminus 02)

WFG2 155E minus 02(212E minus 02)sim
























WFG6 513Eminus 03 (129Eminus02)+






WFG7 550Eminus 03 (155Eminus04)sim


















Total 4321 4321 6121 6022 3025+e better average values of these algorithms for each instance are highlighted in bold italics

Complexity 17

advantages when compared with other algorithms In orderto illustrate the effectiveness of the proposed adaptivestrategy we design some experiments for in-depth analysis

In this section an algorithm called MOEAD-FMG isused to run this comparison experiment Note that thedifference between MOEAD-FMG and MOEAD-AMGlies in that MOEAD-FMG adopts a fixed Gaussian distri-bution to control the generation of new offspring withoutadaptive strategy during the whole evolution Here MOEAD-FMGwith five different distributions (0 062) (0 082) (0102) (0 122) and (0 142) are conducted to solve all the FUF and WFG instances To have a fair comparison othercomponents of MOEAD-FMG are kept the same as that inMOEAD-AMG +e statistical results of the IGD metricvalues obtained by MOEAD-AMG and MOEAD-FMGwith different distributions over 20 independent runs arepresented in Table 4 for the used test problems

As observed from Table 4 MOEAD-AMG obtains thebest mean IGD values on 17 out of 28 cases while all theMOEAD-FMG variants totally achieve the best mean IGDvalues on the remaining 11 cases More specifically for fiveMOEAD-FMG variants with the distributions (0 062) (0082) (0 102) (0 122) and (0 142) they respectivelyobtain the best HV results on 4 1 3 2 and 1 out of 28comparisons As observed from the one-to-one comparisonsin the last row of Table 4 MOEAD-AMG performs betterthan five MOEAD-FMG variants with the distributions (0062) (0 082) (0 102) (0 122) and (0 142) respectivelyon 21 21 21 22 and 25 out of 28 comparisons whereasMOEAD-AMG is only beaten by these five MOEAD-FMGvariants on 4 4 6 6 and 3 comparisons respectively+erefore it can be concluded from Table 4 that the pro-posed adaptive strategy for selecting a suitable Gaussianmodel significantly contributes to the superior performanceof MOEAD-AMG on solving MOPs

+e above analysis only considers five different distri-butions with variance from 062 to 142 +e performancewith a Gaussian distribution with bigger variances is furtherstudied here +us several variants of MOEAD-FMG withthree fixed (0 162) (1 182) and (0 202) are used forcomparison Table 5 lists the IGD comparative results on allinstances adopted As observed from the results MOEAD-AMG also shows the obvious advantages as it performs best


Problems MOEAD-FMG (0 162) MOEAD-FMG (0 182) MOEAD-FMG (0 202) MOEAD-AMGF1 216E minus 03 (206E minus 05)minus 221E minus 03 (303E minus 05)minus 220E minus 03 (732E minus 05)minus 135Eminus 03 (128Eminus 05)F2 103E minus 02 (214E minus 04)minus 110E minus 02 (572E minus 03)minus 140E minus 02 (201E minus 03)minus 285Eminus 03 (295Eminus 04)F3 791E minus 03 (231E minus 04)minus 881E minus 03 (281E minus 03)minus 940E minus 03 (386E minus 03)minus 255Eminus 03 (359Eminus 04)F4 993E minus 03 (719E minus 04)minus 104E minus 02 (449E minus 03)minus 143E minus 02 (603E minus 03)minus 293Eminus 03 (723Eminus 04)F5 967E minus 03 (296E minus 04)minus 133E minus 02 (421E minus 03)minus 149E minus 02 (462E minus 03)minus 794Eminus 03 (127Eminus 03)F6 901E minus 01 (993E minus 03)minus 996E minus 02 (289E minus 02)minus 109E minus 01 (397E minus 02)minus 654Eminus 02 (813Eminus 03)F7 123E minus 01 (721E minus 02)minus 984E minus 02 (221E minus 01)minus 129E minus 01 (808E minus 02)minus 421Eminus 02 (784Eminus 02)F8 295E minus 02 (921E minus 03)minus 285E minus 02 (531E minus 03)minus 295E minus 02 (589E minus 03)minus 155Eminus 02 (107Eminus 02)F9 154E minus 02 (241E minus 03)minus 155E minus 02 (234E minus 03)minus 166E minus 02 (211E minus 02)minus 514Eminus 03 (212Eminus 03)UF1 542E minus 03 (512E minus 04)minus 568E minus 03 (357E minus 03)minus 590E minus 03 (620E minus 02)minus 245Eminus 03 (142Eminus 03)UF2 878E minus 03 (535E minus 03)minus 950E minus 03 (555E minus 03)minus 971E minus 03 (277E minus 03)minus 668Eminus 03 (837Eminus 04)UF3 655Eminus 03 (300Eminus 03)+ 660E minus 03 (552E minus 03)+ 175E minus 02 (229E minus 02)sim 174E minus 02 (199E minus 02)UF4 560Eminus 02 (319Eminus 03)+ 703E minus 02 (712E minus 03)minus 713E minus 02 (609E minus 03)minus 698E minus 02 (578E minus 03)UF5 390Eminus 01 (631Eminus 02)+ 456E minus 01 (506E minus 01)minus 585E minus 01 (502E minus 01)minus 408E minus 01 (950E minus 02)UF6 163E minus 01 (210E minus 01)sim 145Eminus 01 (300Eminus 02)+ 184E minus 01 (257E minus 02)minus 162E minus 01 (206E minus 01)UF7 677E minus 03 (422E minus 04)minus 694E minus 03 (402E minus 03)minus 699E minus 03 (317E minus 02)minus 513Eminus 03 (670Eminus 04)UF8 800E minus 02 (454E minus 03)minus 905E minus 02 (404E minus 02)minus 973E minus 02 (308E minus 03)minus 574Eminus 02 (829Eminus 03)UF9 370E minus 02 (299E minus 03)minus 452E minus 02 (320E minus 02)minus 476E minus 02 (445E minus 02)minus 276Eminus 02 (437Eminus 03)UF10 192E+ 00 (271E minus 01)minus 253E+ 00 (109E minus 01)minus 250E+ 00 (494E minus 03)minus 803Eminus 01 (171Eminus 01)WFG1 124E+ 00 (725E minus 03)minus 128E+ 00 (702E minus 03)minus 234E+ 00 (810E minus 02)minus 111E+ 00 (117Eminus 02)WFG2 177E minus 02 (293E minus 04)minus 179E minus 02 (400E minus 03)minus 192E minus 02 (278E minus 03)minus 155Eminus 02 (324Eminus 04)WFG3 460E minus 01 (222E minus 04)minus 469E minus 01 (501E minus 04)minus 497E minus 01 (101E minus 03)minus 445Eminus 03 (142Eminus 04)WFG4 610E minus 02 (329E minus 03)minus 767E minus 02 (428E minus 04)minus 790E minus 02 (201E minus 03)minus 481Eminus 02 (164Eminus 03)WFG5 666E minus 02 (397E minus 05)minus 657E minus 02 (193E minus 04)minus 676E minus 02 (802E minus 03)minus 652Eminus 02 (414Eminus 04)WFG6 847E minus 02 (478E minus 02)minus 144E minus 01 (402E minus 04)minus 192E minus 01 (309E minus 03)minus 162Eminus 02 (502Eminus 02)WFG7 839E minus 03 (493E minus 05)minus 841E minus 03 (293E minus 04)minus 862E minus 03 (808E minus 04)minus 553Eminus 03 (213Eminus 05)WFG8 845E minus 02 (890E minus 04)minus 906E minus 02 (308E minus 03)minus 112E minus 01 (316E minus 03)minus 577Eminus 02 (160Eminus 03)WFG9 354E minus 02 (340E minus 04)minus 371E minus 02 (221E minus 04)minus 377E minus 02 (194E minus 03)minus 152Eminus 02 (292Eminus 04)Total 3124 2026 0127+e better average values of these algorithms for each instance are given in bold and italics

Table 6 +e standard deviations of Gaussian models adopted byeach K value

K Standard deviation3 08 10 125 06 08 10 12 147 06 08 09 10 11 12 1410 05 06 07 08 09 10 11 12 13 14

18 Complexity

on most of the comparisons ie in 24 out of 28 cases forIGD results while other three competitors are not able toobtain a good performance In addition we can also observethat with the increase of variances used in competitors theperformance becomes worse+e reason for this observationmay be that if we adopt a too larger variance in the wholeevolutionary process the offspring will become irregularleading to the poor convergence +erefore it is not sur-prising that the performance with bigger variances willdegrade It should be noted that the results from Table 5 alsoexplain why we only choose Gaussian distributions from (0062) to (0 142) for MOEAD-AMG

46 Parameter Analysis In the above comparison our al-gorithm MOEAD-AMG is initialized with K 5 includingfive types of Gaussianmodels ie (0 062) (0 082) (0 102)(0 122) and (0 142) In this section we test the proposedalgorithm using different K values (3 5 7 and 10) whichwill use K kinds of Gaussian distributions To clarify theexperimental setting the standard deviations of Gaussianmodels adopted by each K value are listed in Table 6 Asshown by the obtained IGD results presented in Table 7 it isclear that K 5 is a preferable number of Gaussian modelsfor MOEAD-AMG since it achieves the best IGD results in16 of 28 cases It also can be found that as K increases the

average IGD values also increase for most of test problems+is is due to the fact that in our algorithmMOEAD-AMGthe population size is often relatively small which makeslittle sense to set a large K value +us a large K value haslittle effect on improving the performance of the algorithmHowever K 3 seems to be too small which is not able totake full advantage of the proposed adaptive multipleGaussian process models According to the above analysiswe recommend that K 5

5 Conclusions and Future Work

In this paper a decomposition-based multiobjective evo-lutionary algorithm with adaptive multiple Gaussian processmodels (called MOEAD-AMG) has been proposed forsolvingMOPs Multiple Gaussian process models are used inMOEAD-AMG which can help to solve various kinds ofMOPs In order to enhance the search capability an adaptivestrategy is developed to select a more suitable Gaussianprocess model which is determined based on the contri-butions to the optimization performance for all thedecomposed subproblems To investigate the performanceof MOEAD-AMG twenty-eight test MOPs with compli-cated PF shapes are adopted and the experiments show thatMOEAD-AMG has superior advantages on most caseswhen compared to six competitive MOEAs In addition

Table 7 Performance comparison of MOEAD-AMG with different values of K in terms of the average IGD values and standard deviationson F UF and WFG problems

Problems K 3 K 5 K 7 K 10F1 136E minus 03 (125E minus 05) 135E minus 03 (128E minus 05) 134E minus 03 (696E minus 06) 133Eminus 03 (101Eminus 05)F2 329E minus 03 (350E minus 04) 285Eminus 03 (295Eminus 04) 348E minus 03 (136E minus 03) 325E minus 03 (755E minus 04)F3 309E minus 03 (270E minus 04) 255Eminus 03 (359Eminus 04) 329E minus 03 (785E minus 04) 357E minus 03 (102E minus 03)F4 318E minus 03 (757E minus 04) 293Eminus 03 (723Eminus 04) 361E minus 03 (148E minus 03) 315E minus 03 (121E minus 03)F5 942E minus 03 (156E minus 03) 794E minus 03 (127E minus 03) 740Eminus 03 (713Eminus 04) 782E minus 03 (137E minus 03)F6 614E minus 02 (757E minus 03) 654E minus 02 (813E minus 03) 599Eminus 02 (870Eminus 03) 636E minus 02 (734E minus 03)F7 671E minus 03 (704E minus 03) 421Eminus 02 (784Eminus 02) 441E minus 02 (607E minus 02) 936E minus 02 (121E minus 01)F8 789E minus 03 (553E minus 03) 155E minus 02 (107E minus 02) 128Eminus 02 (144Eminus 02) 214E minus 02 (246E minus 02)F9 577E minus 03 (184E minus 03) 514E minus 03 (212E minus 03) 554E minus 03 (223E minus 03) 501Eminus 03 (207Eminus 03)UF1 306E minus 03 (124E minus 03) 245E minus 03 (142E minus 03) 212Eminus 03 (392Eminus 04) 215E minus 03 (625E minus 04)UF2 708E minus 03 (127E minus 03) 668Eminus 03 (837Eminus 04) 673E minus 03 (591E minus 04) 819E minus 03 (303E minus 03)UF3 101Eminus 02 (571Eminus 03) 174E minus 02 (199E minus 02) 151E minus 02 (247E minus 02) 130E minus 02 (114E minus 02)UF4 692Eminus 02 (678Eminus 03) 698E minus 02 (578E minus 03) 741E minus 02 (573E minus 03) 738E minus 02 (495E minus 03)UF5 370E minus 01 (727E minus 02) 408E minus 01 (950E minus 02) 359Eminus 01 (642Eminus 02) 312E minus 01 (457E minus 02)UF6 171E minus 01 (232E minus 01) 162E minus 01 (206E minus 01) 113Eminus 01 (506Eminus 02) 234E minus 01 (270E minus 01)UF7 502Eminus 02 (141Eminus 01) 513E minus 03 (670E minus 04) 533E minus 03 (475E minus 04) 526E minus 03 (749E minus 04)UF8 584E minus 02 (625E minus 03) 574Eminus 02 (829Eminus 03) 596E minus 02 (113E minus 02) 559E minus 02 (979E minus 03UF9 329E minus 02 (139E minus 02) 276Eminus 02 (437Eminus 03) 307E minus 02 (823E minus 03) 338E minus 02 (140E minus 02)UF10 145E+ 00 (284E minus 01) 803E minus 01 (171E minus 01) 724E minus 01 (971E minus 02) 610Eminus 01 (781Eminus 02)WFG1 116E+ 00 (793E minus 03) 111E+ 00 (117Eminus 02) 114E+ 00 (525E minus 03) 112E+ 00 (777E minus 03)WFG2 158E minus 02 (426E minus 04) 155Eminus 02 (324Eminus 04) 156E minus 02 (313E minus 04) 156E minus 02 (337E minus 04)WFG3 507E minus 03 (182E minus 04) 445Eminus 03 (142Eminus 04) 474E minus 03 (180E minus 04) 479E minus 03 (168E minus 04)WFG4 573E minus 02 (202E minus 03) 481Eminus 02 (164Eminus 03) 545E minus 02 (164E minus 03) 520E minus 02 (215E minus 03)WFG5 655E minus 02 (428E minus 04) 652Eminus 02 (414Eminus 04) 653E minus 02 (144E minus 04) 653E minus 02 (318E minus 04)WFG6 118E minus 01 (118E minus 01) 162Eminus 02 (502Eminus 02) 511E minus 02 (814E minus 02) 555E minus 02 (932E minus 02)WFG7 592E minus 03 (156E minus 04) 553Eminus 03 (213Eminus 05) 570E minus 03 (629E minus 05) 563E minus 03 (428E minus 05)WFG8 588E minus 02 (904E minus 04) 577Eminus 02 (160Eminus 03) 581E minus 02 (159E minus 03) 608E minus 02 (135E minus 03)WFG9 166E minus 02 (421E minus 04) 152Eminus 02 (292Eminus 04) 161E minus 02 (651E minus 04) 157E minus 02 (472E minus 04)Best 3 16 6 3+e better average values of these algorithms for each instance are given in bold and italics

Complexity 19

other experiments also have verified the effectiveness of theused adaptive strategy to significantly improve the perfor-mance of MOEAD-AMG

When comparing to generic recombination operatorsand other Gaussian process-based recombination operatorsour proposed method based on multiple Gaussian processmodels is effective to solve most of the test problemsHowever in MOEAD-AMG the number of Gaussianmodels built in each generation is the same as the size ofsubproblems which still needs a lot of computational costIn our future work we will try to enhance the computationalefficiency of the proposed MOEAD-AMG and it is alsointeresting to study the potential application of MOEAD-AMG in solving some many-objective optimization prob-lems and engineering problems

Data Availability

+e source code and data are available from the corre-sponding author upon request

Conflicts of Interest

+e authors declare that they have no conflicts of interest

Acknowledgments

+is study was supported by the National Natural ScienceFoundation of China under grant nos 61876110 61836005and 61672358 Joint Funds of the National Natural ScienceFoundation of China under Key Program Grant U1713212Natural Science Foundation of Guangdong Province undergrant no 2017A030313338 and Shenzhen Technology Planunder grant no JCYJ20170817102218122 +is study wasalso supported by the National Engineering Laboratory forBig Data System Computing Technology and the Guang-dong Laboratory of Artificial Intelligence and DigitalEconomy (SZ) Shenzhen University

References

[1] S JD ldquoMultiple objective optimization with vector valuatedgenetic algorithmsrdquo in Proceedings of the First InternationalConference on Genetic Algorithms and their Applicationspp 93ndash100 1985

[2] K Deb Multi-Objective Optimization Using EvolutionaryAlgorithms vol 16 John Wiley amp Sons New York NY USA2001

[3] C A Coello ldquoAn updated survey of GA-based multiobjectiveoptimization techniquesrdquo ACM Computing Surveys vol 32no 2 pp 109ndash143 2000

[4] A Zhou B-Y Qu H Li S-Z Zhao P N Suganthan andQ Zhang ldquoMultiobjective evolutionary algorithms a surveyof the state of the artrdquo Swarm and Evolutionary Computationvol 1 no 1 pp 32ndash49 2011

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

[6] E Zitzler M Laumanns and L +iele ldquoSPEA2 improvingthe strength Pareto evolutionary algorithmrdquo TIK-reportvol 103 2001

[7] J D Knowles and D W Corne ldquoApproximating the non-dominated front using the Pareto archived evolution strat-egyrdquo Evolutionary Computation vol 8 no 2 pp 149ndash1722000

[8] J Bader and E Zitzler ldquoHypE an algorithm for fast hyper-volume-based many-objective optimizationrdquo EvolutionaryComputation vol 19 no 1 pp 45ndash76 2011

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

[10] M Laumanns L +iele K Deb and E Zitzler ldquoCombiningconvergence and diversity in evolutionary multiobjectiveoptimizationrdquo Evolutionary Computation vol 10 no 3pp 263ndash282 2002

[11] D Hadka and P Reed ldquoBorg an auto-adaptive many-ob-jective evolutionary computing frameworkrdquo EvolutionaryComputation vol 21 no 2 pp 231ndash259 2013

[12] S Yang M Li X Liu and J Zheng ldquoA grid-based evolu-tionary algorithm for many-objective optimizationrdquo IEEETransactions on Evolutionary Computation vol 17 no 5pp 721ndash736 2013

[13] G Wang and H Jiang ldquoFuzzy-dominance and its applicationin evolutionary many objective optimizationrdquo in Proceedingsof the International Conference on Computational Intelligenceand Security Workshops pp 195ndash198 Heilongjiang China2007

[14] F di Pierro S-T Khu and D A Savi ldquoAn investigation onpreference order ranking scheme for multiobjective evolu-tionary optimizationrdquo IEEE Transactions on EvolutionaryComputation vol 11 no 1 pp 17ndash45 2007

[15] C Zhu L Xu and E D Goodman ldquoGeneralization of Pareto-optimality for many-objective evolutionary optimizationrdquoIEEE Transactions on Evolutionary Computation vol 20no 2 pp 299ndash315 2016

[16] R Cheng Y Jin M Olhofer and B Sendhoff ldquoA referencevector guided evolutionary algorithm for many-objectiveoptimizationrdquo IEEE Transactions on Evolutionary Compu-tation vol 20 no 5 pp 773ndash791 2016

[17] K Deb and H Jain ldquoAn evolutionary many-objective opti-mization algorithm using reference-point-based non-dominated sorting approach part I solving problems withbox constraintsrdquo IEEE Transactions on Evolutionary Com-putation vol 18 no 4 pp 577ndash601 2014

[18] Y Liu D Gong X Sun and Y Zhang ldquoMany-objectiveevolutionary optimization based on reference pointsrdquoAppliedSoft Computing vol 50 pp 344ndash355 2017

[19] S Jiang and S Yang ldquoA strength Pareto evolutionary algo-rithm based on reference direction for multiobjective andmany-objective optimizationrdquo IEEE Transactions on Evolu-tionary Computation vol 21 no 3 pp 329ndash346 2017

[20] L Pan C He Y Tian Y Su and X Zhang ldquoA region divisionbased diversity maintaining approach for many-objectiveoptimizationrdquo Integrated Computer-Aided Engineeringvol 24 no 3 pp 279ndash296 2017

[21] W Lin Q Lin Z Zhu J Li J Chen and Z Ming ldquoEvo-lutionary search with multiple utopian reference points indecomposition-based multiobjective optimizationrdquo Com-plexity vol 2019 Article ID 7436712 22 pages 2019

[22] C Dai and X Lei ldquoA multiobjective brain storm optimizationalgorithm based on decompositionrdquo Complexity vol 2019p 11 2019

20 Complexity

[23] Y Su Q Lin J Wang J Li J Chen and Z Ming ldquoAconstrained solution update strategy for multiobjective evo-lutionary algorithm based on decompositionrdquo Complexityvol 2019 Article ID 3251349 11 pages 2019

[24] XWang K Xing C Yan andM Zhou ldquoA novelMOEAD formultiobjective scheduling of flexible manufacturing systemsrdquoComplexity vol 2019 Article ID 5734149 14 pages 2019

[25] E Zitzler and S Kunzli ldquoIndicator-based selection in mul-tiobjective searchrdquo Lecture Notes in Computer ScienceSpringer in Proceedings of the International Conference onParallel Problem Solving from Nature pp 832ndash842 SpringerBirmingham UK 2004

[26] R H Gomez and C A C Coello ldquoImproved metaheuristicbased on the R2 indicator for many-objective optimizationrdquoin Proceedings of the 2015 Annual Conference on Genetic andEvolutionary Computation pp 679ndash686 Madrid Spain 2015

[27] A Menchaca-Mendez and C A C Coello ldquoGD-MOEA anew multiobjective evolutionary algorithm based on thegenerational distance indicatorrdquo in Proceedings of the 8thInternational Conference on Evolutionary Multi-CriterionOptimization pp 156ndash170 Springer Guimaratildees Portugal2015

[28] D H Phan and J Suzuki ldquoR2-IBEA R2 indicator basedevolutionary algorithm for multiobjective optimizationrdquo inProceedings of the IEEE Congress on Evolutionary Computa-tion (CEC rsquo13) pp 1836ndash1845 Cancun Mexico June 2013

[29] C A R Villalobos and C A C Coello ldquoA newmulti-objectiveevolutionary algorithm based on a performance assessmentindicatorrdquo in Proceedings of the Fourteenth InternationalConference on Genetic and Evolutionary Computation Con-ference-GECCO lsquo12 pp 505ndash512 ACM Philadelphia PAUSA 2012

[30] K Deb A Sinha and S Kukkonen ldquoMulti-objective testproblems linkages and evolutionary methodologiesrdquo inProceedings of the 8th annual conference on Genetic andevolutionary computation pp 1141ndash1148 ACM Seattle WAUSA 2006

[31] Q Zhu Q Lin Z Du et al ldquoA novel adaptive hybridcrossover operator for multiobjective evolutionary algo-rithmrdquo Information Sciences vol 345 pp 177ndash198 2016

[32] K Deb and R B Agrawal ldquoSimulated binary crossover forcontinuous search spacerdquo Complex Systems vol 9 no 2pp 115ndash148 1995

[33] K Deep and M +akur ldquoA new crossover operator for realcoded genetic algorithmsrdquo Applied Mathematics and Com-putation vol 188 no 1 pp 895ndash911 2007

[34] K Deb A Anand and D Joshi ldquoA computationally efficientevolutionary algorithm for real-parameter optimizationrdquoEvolutionary Computation vol 10 no 4 pp 371ndash395 2002

[35] L J Eshelman and J D Schaffer ldquoReal-coded genetic algo-rithms and interval-schematardquo in Foundations of GeneticAlgorithms vol 2 pp 187ndash202 Elsevier AmsterdamNetherlands 1993

[36] I Ono H Kita and S Kobayashi ldquoA robust real-coded ge-netic algorithm using unimodal normal distribution crossoveraugmented by uniform crossover effects of self-adaptation ofcrossover probabilitiesrdquo in Proceedings of the 1st AnnualConference on Genetic and Evolutionary Computation-Vol-ume 1 vol 1 pp 496ndash503 Morgan Kaufmann Publishers IncOrlando FL USA 1999

[37] S Tsutsui M Yamamura and T Higuchi ldquoMulti-parentrecombination with simplex crossover in real coded geneticalgorithmsrdquo in Proceedings of the 1st Annual Conference onGenetic and Evolutionary Computation-Volume 1 vol 1

pp 657ndash664 Morgan Kaufmann Publishers Inc Orlando FLUSA 1999

[38] R Storn and K Price ldquoDifferential evolutionndasha simple andefficient heuristic for global optimization over continuousspacesrdquo Journal of Global Optimization vol 11 no 4pp 341ndash359 1997

[39] X He Y Zhou and Z Chen ldquoAn evolution path based re-production operator for many-objective optimizationrdquo IEEETransactions on Evolutionary Computation vol 23 no 1pp 29ndash43 2017

[40] J Wang G Liang and J Zhang ldquoCooperative differentialevolution framework for constrained multiobjective optimi-zationrdquo IEEE Transactions on Cybernetics vol 10 no 11pp 1ndash13 2018

[41] K Li and H Tian ldquoAdaptive differential evolution withevolution memory for multiobjective optimizationrdquo IEEEAccess vol 11 no 17 pp 866ndash876 2018

[42] K Li A Fialho S Kwong and Q Zhang ldquoAdaptive operatorselection with bandits for a multiobjective evolutionary al-gorithm based on decompositionrdquo IEEE Transactions onEvolutionary Computation vol 18 no 1 pp 114ndash130 2014

[43] Q Lin Z Liu Q Yan et al ldquoAdaptive composite operatorselection and parameter control for multiobjective evolu-tionary algorithmrdquo Information Sciences vol 339 pp 332ndash352 2016

[44] X Qiu J X Xu K C Tan and H A Abbass ldquoAdaptive cross-generation differential evolution operators for multi-objectiveoptimizationrdquo IEEE Transactions on Evolutionary Compu-tation vol 20 no 2 pp 232ndash244 2015

[45] Y Sun G G Yen and Z Yi ldquoImproved regularity model-based EDA for many-objective optimizationrdquo IEEE Trans-actions on Evolutionary Computation vol 22 no 5pp 662ndash678 2018

[46] A Zhou Q Zhang and Y Jin ldquoApproximating the set ofPareto-optimal solutions in both the decision and objectivespaces by an estimation of distribution algorithmrdquo IEEETransactions on Evolutionary Computation vol 13 no 5pp 1167ndash1189 2009

[47] Q Zhang A Zhou and Y Jin ldquoRM-MEDA a regularitymodel-based multiobjective estimation of distribution algo-rithmrdquo IEEE Transactions on Evolutionary Computationvol 12 no 1 pp 41ndash63 2008

[48] Q Zhang A Zhou S Zhao and P N Suganthan ldquoMulti-objective optimization test instances for the CEC 2009 specialsession and competitionrdquo Special Session on PerformanceAssessment of Multi-Objective Optimization AlgorithmsTechnical report vol 264 University of Essex ColchesterUK 2008

[49] S Huband P Hingston L Barone and LWhile ldquoA review ofmultiobjective test problems and a scalable test problemtoolkitrdquo IEEE Transactions on Evolutionary Computationvol 10 no 5 pp 477ndash506 2006

[50] T Lin H Zhang K Zhang Z Tu and N Cui ldquoAn adaptivemultiobjective estimation of distribution algorithm with anovel Gaussian sampling strategyrdquo Soft Computing vol 21no 20 pp 6043ndash6061 2017

[51] S T Roweis and L K Saul ldquoNonlinear dimensionality re-duction by locally linear embeddingrdquo Science vol 290no 5500 pp 2323ndash2326 2000

[52] Y Z G Dai L Peng and M Wang ldquoHMOEDA_LLE ahybrid multi-objective estimation of distribution algorithmcombining locally linear embeddingrdquo IEEE Congress onEvolutionary Computation pp 707ndash714 2014

Complexity 21

[53] R Cheng Y Jin K Narukawa and B Sendhoff ldquoA multi-objective evolutionary algorithm using Gaussian process-based inverse modelingrdquo IEEE Transactions on EvolutionaryComputation vol 19 no 6 pp 838ndash856 2015

[54] Y Lin H Liu and Q Jiang ldquoDynamic reference vectors andbiased crossover use for inverse model based evolutionarymulti-objective optimization with irregular Pareto frontsrdquoApplied Intelligence vol 48 no 9 pp 3116ndash3142 2018

[55] Q Yuan G Dai and Y Zhang ldquoA novel multi-objectiveevolutionary algorithm based on LLE manifold learningrdquoEngineering with Computers vol 33 no 2 pp 293ndash305 2017

[56] H Li and Q Zhang ldquoMultiobjective optimization problemswith complicated Pareto sets MOEAD and NSGA-IIrdquo IEEETransactions on Evolutionary Computation vol 13 no 2pp 284ndash302 2009

[57] A Zhou Q Zhang and G Zhang ldquoA multiobjective evo-lutionary algorithm based on decomposition and probabilitymodelrdquo in Proceedings of the IEEE Congress on EvolutionaryComputation pp 1ndash8 Brisbane Australia June 2012

[58] C Liu D Zhou Z Wang D Yang and G Song ldquoDamagedetection of refractory based on principle component analysisand Gaussian mixture modelrdquo Complexity vol 2018 ArticleID 7356189 9 pages 2018

[59] M Lungaroni A Murari E Peluso P Gaudio andM GelfusaldquoGeodesic distance on Gaussian manifolds to reduce the sta-tistical errors in the investigation of complex systemsrdquo Com-plexity vol 2019 Article ID 5986562 24 pages 2019

[60] J Li B Zhang G Lu H Ren and D Zhang ldquoVisual clas-sification with multikernel shared Gaussian process latentvariable modelrdquo IEEE Transactions on Cybernetics vol 49no 8 pp 2886ndash2899 2018

[61] Y Zhou J He and H Gu ldquoPartial label learning via Gaussianprocessesrdquo IEEE Transactions on Cybernetics vol 47 no 12pp 4443ndash4450 2017

[62] M C Seiler and F A Seiler ldquoNumerical recipes in C the art ofscientific computingrdquo Risk Analysis vol 9 no 3 pp 415-4161989

[63] P A N Bosman and D +ierens ldquoMulti-objective optimi-zation with diversity preserving mixture-based iterated densityestimation evolutionary algorithmsrdquo International Journal ofApproximate Reasoning vol 31 no 3 pp 259ndash289 2002

[64] A Zhou Q Zhang and G Zhang ldquoMultiobjective evolu-tionary algorithm based on mixture Gaussian modelsrdquoJournal of Software vol 25 no 5 pp 913ndash928 2014

[65] M Dellnitz O Schutze and T Hestermeyer ldquoCoveringPareto sets by multilevel subdivision techniquesrdquo Journal ofOptimization 5eory and Applications vol 124 no 1pp 113ndash136 2005

[66] P Larra1113957naga and J A Lozano Eds Estimation of DistributionAlgorithms A New Tool for Evolutionary ComputationKluwer Academic Publishers vol 2 Boston MA USA 2002

[67] P A N Bosman and D +ierens ldquoContinuous iterateddensity estimation evolutionary algorithms within the ideaframeworkrdquo in Proceedings of the Genetic and EvolutionaryComputation Conference (GECCO 2000) pp 197ndash200 LasVegas NV USA July 2000

[68] N Hansen and A Ostermeier ldquoCompletely derandomizedself-adaptation in evolution strategiesrdquo Evolutionary Com-putation vol 9 no 2 pp 159ndash195 2001

[69] C Igel N Hansen and S Roth ldquoCovariance matrix adap-tation for multi-objective optimizationrdquo Evolutionary Com-putation vol 15 no 1 pp 1ndash28 2007

[70] T Paul and H Iba ldquoReal-coded estimation of distributionalgorithmrdquo in Proceedings of the 5th Metaheuristics Inter-national Conference pp 61ndash66 Kyoto Japan 2003

[71] M R Wagner A Auger and M Schoenauer ldquoEEDA a newrobust estimation of distribution algorithmrdquo Rapport deRecherche (Research Report) RR-5190 Institut National deRecherche en Informatique et en Automatique (INRIA)Rocquencourt France 2004

[72] W Dong and X Yao ldquoUnified eigen analysis on multivariateGaussian based estimation of distribution algorithmsrdquo In-formation Sciences vol 178 no 15 pp 3000ndash3023 2008

[73] P A N Bosman and D +ierens ldquo+e balance betweenproximity and diversity in multiobjective evolutionary algo-rithmsrdquo IEEE Transactions on Evolutionary Computationvol 7 no 2 pp 174ndash188 2003

[74] E Zitzler and L +iele ldquoMultiobjective evolutionary algo-rithms a comparative case study and the strength Paretoapproachrdquo IEEE Transactions on Evolutionary Computationvol 3 no 4 pp 257ndash271 1999

22 Complexity

advantage with a better mathematical explanation de-composition-based MOEAs have become very popular inrecent years [21ndash24]

13 Indicator-Based MOEAs Indicator-based MOEAs di-rectly employ a performance indicator like hypervolume(HV) generational distance (GD) and R2 to effectivelyguide the selection of promising solutions for next gener-ation IBEA [25] MOMBI-II [26] HyPE [8] GD-MOEA[27] R2-IBEA [28] and DDE [29] are representatives for theindicator-based MOEAs In these MOEAs performanceindicators are used as selection criterion to rank non-dominated solutions that cannot be distinguished by tra-ditional Pareto dominance However for this kind ofMOEAs high computational complexity is usually requiredfor calculating the performance indicator which is verychallenging especially when the number of objectives islarge

+ese MOEAs usually include two main componentsie variation and selection [6] Selection plays an importantrole in MOEAs to maintain the promising solutions asintroduced above for classifying MOEAs while variation isthe key factor to determine the generation quality of solu-tions In [30] the effects of several variation operators arestudied on some test problems with variable linkagesshowing that variable linkages may cause some difficultiesfor MOEAs Actually there are a number of real-valuedvariation approaches having been proposed during the re-cent decades [31] which can be classified into the three mainkinds ie traditional recombination operators estimationof distribution algorithms and the inverse model methodsfor variation

Traditional recombination operators are generally usedin most existing MOEAs due to their simplicity +is kind ofvariation simulates the binary crossover method to producethe real-valued offspring such as simulated binary crossover(SBX) [32] Laplace crossover [33] parent central crossover[34] blend crossover α [35] unimodal normal distributioncrossover [36] and simple crossover [37] Moreover dif-ferential evolution (DE) [38] is also used as the recombi-nation operator in many MOEAs which samples offspringbased on the difference vectors of parents +e evolutionpaths are used in DE [39] to depict the population move-ment and predict its tendency which could produce po-tential solutions to speed up convergence toward the PSRecently a number of hybridized recombination operatorshave been proposed trying to combine the advantages ofdifferential recombination operators In [40] a cooperativeDE framework is designed for constrained MOPs in whichmultiple DE operators are run in different subpopulations tooptimize its own constrained subproblem In [41] four DEoperators are combined and a sliding window is adopted in[42] to provide the reward for each DE operator according tothe enhancement on subproblems Similarly four DE op-erator pools are presented in [43] including two DE oper-ators with complementary search patterns in each pool toprovide an improved search ability In ACGDE [44] anadaptive cross-generation DE operator is designed by

exploiting information from individuals across generationsto adapt the parameters

+e estimation of distribution algorithms (EDAs) [45]exploit the probabilistic models extracted from the pop-ulationrsquos distribution to run variation [46 47] Unlike theabove traditional recombination operators no crossover ormutation procedures will be run in EDAs Instead theglobally statistical information of the selected parents is usedin EDAs to build a posterior probability distribution modeland then offspring are sampled from this built modelSeveral EDAs are studied for solving continuous MOPs in[48 49] while the Gaussian distribution model mixtureGaussian distribution model or mixed Gaussian with theprincipal component analysis (PCA) model are introducedin [50] for offspring variation Different from the individ-ualrsquos information used in traditional recombination oper-ators the local PCA operator is employed in [45 47] togenerate new offspring Different from [47] the PCA modelwas replaced by the locally linear embedding (LLE) [51]model in [52] In this way this model only considers thedecision space without considering too much about thetarget MOP itself

+e inverse model methods for variation utilize machinelearning approaches to capture the connection from theobjective space to the decision space by exploiting thecharacteristics of the target MOP +e representative algo-rithms like IM-MOEA [53] and E-IM-MOEA [54] use theGaussian process-based inverse model to complete cross-over In another work [55] as inspired by LLE manifoldlearning idea a new LLEmodeling approach is introduced touse the mapping function known in the MOP that thedecision space is considered as the high-dimensional spaceand the objective space is regarded as a low-dimensionalspace +us this new modeling method is no longer to buildthe overall low dimensional space of the sample which isthen reflected back to the high dimensional space but itreplaces directly by constructing new samples in high di-mensional space In this way the model mapping from theobjective space to the decision space is built based on theobtained approximated Pareto set during the evolution Inother words this model can use the probability model orsurrogate model to build a bridge from the objective space tothe decision space

In the above three kinds of variation operators tradi-tional recombination operators are often used in manyMOEAs However as these operators are executed based onthe individuals they only provide a finite number of searchpatterns and are also criticized for lack of mathematic ex-planation For the inverse model methods for variation it isvery changeling for mapping from the objective space intothe decision space when tackling some complicated MOPsEDAs usually adopt one standard Gaussian process modelwith fixed variance which may not work well for variouskinds of MOPs To enhance the search capability of EDAsthis paper introduces a decomposition-based MOEA withadaptive multiple Gaussian process models called MOEAD-AMG which is effective in generating more superioroffspring Based on the performance enhancements on anumber of decomposed subproblems one suitable Gaussian

2 Complexity



2 Preliminaries


min F(x) f1(x) fm(x)( 1113857T


i1ai bi1113858 1113859

(1)








x sim N(μΣ) (2)




TΣminus 1(x minus μ)11139671113882 (3)


μ 1K

1113944

K

k1x

k

Σ 1

K minus 11113944

K

k1x

kminus μ1113872 1113873 x

kminus μ1113872 1113873

T

(4)





Complexity 3










gi(x) g x λi z

11138681113868111386811138681113872 1113873 max1lejlem

λij fi(x) minus zj

11138681113868111386811138681113868

11138681113868111386811138681113868 (5)



(3) x μ + Ay


4 Complexity





CrkG IoFkG

1113936Ki1loFiG

+ ε (7)


PrkG CrkG

1113936Ki1CriG

(8)
















Gaussian curves

0

02

04

06

08

1


g1-06g2-08g3-10

g4-12g5-14


Complexity 5







IGD S Plowast


Plowast (9)



1 zr2 zr



rm1113858 1113859( 1113857

(10)










6 Complexity















Complexity 7









































































































































106E+ 00(762E minus 03)+





111E+ 00(117E minus 02)


























































8 Complexity
































































































































104E+ 00(129Eminus 02)



000E+ 00(000E+ 00)minus

000E+ 00(000E+ 00)minus


000E+ 00(000E+ 00)minus


WFG1 208E+ 00(166Eminus 01)+

129E+ 00(190E minus 02)+





112E+ 00(250E minus 02)



446E+ 00(676E minus 04)sim

445E+ 00(133E minus 03)sim


445E+ 00(165E minus 02)sim

446E+ 00(870Eminus 04)







398E+ 00(580Eminus 04)

WFG4 170E+ 00(132Eminus 03)+

169E+ 00(306E minus 03)+


150E+ 00(423E minus 03)sim

154E+ 00(210E minus 02)+

162E+ 00(261E minus 02)+

150E+ 00(560E minus 03)



145E+ 00(320E minus 04)sim



144E+ 00(912E minus 04)sim

145E+ 00(275E minus 04)




168E+ 00(949E minus 04)+

167E+ 00(707E minus 03)+

169E+ 00(152Eminus 03)+

162E+ 00(245E minus 01)


169E+ 00(135E minus 03)sim





170E+ 00(240Eminus 04)







141E+ 00(298Eminus 03)






164E+ 00(386E minus 03)sim

164E+ 00(146E minus 03)


Complexity 9








10 Complexity








































































































































































































Complexity 11

F1

IGD

val

ues

10ndash1

10ndash2

10ndash3




times105

(a)

F2

IGD

val

ues 10ndash1

100

10ndash2

10ndash3




(b)F3

IGD

val

ues

10ndash1

100

10ndash2

10ndash3




(c)

F4

IGD

val

ues

10ndash1

100

10ndash2

10ndash3




(d)F5

IGD

val

ues

10ndash1

100

10ndash2




(e)

F6

IGD

val

ues

10ndash1

100




(f )

Figure 2 Continued

12 Complexity

F7

IGD

val

ues

10ndash1

100

101

10ndash2

10ndash3




(g)

F9

IGD

val

ues 10ndash1

100

10ndash2

10ndash3




(h)


UF1

IGD

val

ues 10ndash1

100

10ndash2

10ndash3




times105

(a)

UF2

IGD

val

ues

10ndash1

100

10ndash2




(b)UF6

IGD

val

ues

10ndash1

100

101




(c)

UF7

IGD

val

ues

10ndash2

10ndash1

100




(d)

Figure 3 Continued

Complexity 13

UF8

IGD

val

ues

10ndash1

100




(e)

UF9

IGD

val

ues

10ndash2

10ndash1

100




(f )


WFG3

IGD

val

ues

10ndash1

100

10ndash2

times105




(a)

WFG7

IGD

val

ues

10ndash1

100

10ndash2

times105




(b)WFG8

IGD

val

ues

10ndash1

100

times105




(c)

WFG9

IGD

val

ues

10ndash1

100

10ndash2

times105




(d)


14 Complexity

MOEAD-SBX-F4

0

02

04

06

08

1

12

02 04 06 080

(a)

MOEAD-DE-F4

0

02

04

06

08

1

12

02 04 06 08 10

(b)

MOEAD-GM-F4

0

02

04

06

08

1

12

05 1 15 20

(c)

RM-MEDA-F4

0

02

04

06

08

1

02 04 06 08 1 120

(d)IM-MOEA-F4

05 1 15 200

02

04

06

08

1

12

14

(e)

AM-EDA-F4

05 1 15 2 25 300

02

04

06

08

1

(f )

MOEAD-AMG-F4

02

0

04

06

08

1

12

04 0602 08 120 1

(g)

Ture-PF-F4

0

02

04

06

08

1

02 04 06 08 10

(h)MOEAD-SBX-F5

0

02

04

06

08

1

12

02 04 06 08 10

(i)

MOEAD-DE-F5

0

02

04

06

08

1

02 04 06 08 1 120

(j)

0

02

04

06

08

1

12

02 04 06 08 1 120

MOEAD-GM-F5

(k)

RM-MEDA-F4

0

02

04

06

08

1

02 04 06 08 1 120

(l)IM-MOEA-F5

02 04 06 08 1 1200

02

04

06

08

1

(m)

AM-EDA-F5

02 04 06 08 100

02

04

06

08

1

(n)

MOEAD-AMG-F5

02 04 06 08 1 1200

02

04

06

08

1

12

(o)

Ture-PF-F5

0

02

04

06

08

1

02 04 06 08 10

(p)MOEAD-SBX-F9

0

02

04

06

08

1

12

02 04 06 08 1 120

(q)

MOEAD-DE-F9

0

02

04

06

08

1

12

02 04 06 08 1 120

(r)

MOEAD-GM-F9

0

02

04

06

08

1

12

02 04 06 08 1 120

(s)

RM-MEDA-F9

0

02

04

06

08

1

12

02 04 06 08 1 120

(t)IM-MOEA-F9

0

1

2

3

4

5

1 2 3 540

(u)

AM-EDA-F9

02 04 06 08 1 1200

02

04

06

08

1

12

(v)

MOEAD-AMG-F9

02 04 06 08 1 1200

02

04

06

08

1

12

(w)

Ture-PF-F9

0

02

04

06

08

1

02 04 06 08 10

(x)


Complexity 15

MOEAD-SBX-UF2

02

04

06

08

1

12

02 04 06 080

(a)

MOEAD-DE-UF2

0

02

04

06

08

1

12

02 04 06 08 1 120

(b)

MOEAD-GM-UF2

0

02

04

06

08

1

12

02 04 06 08 1 120

(c)

RM-MEDA-UF2

0

02

04

06

08

1

02 04 06 08 10

(d)IM-MOEA-UF2

02 04 06 08 1 1200

02

04

06

08

1

(e)

AM-EDA-UF2

0

02

04

06

08

1

12

02 04 06 08 1 120

(f )

MOEAD-AMG-UF2

02 04 06 08 100

02

04

06

08

1

12

(g)

Ture-PF-UF2

0

02

04

06

08

1

02 04 06 08 10

(h)

2

1

00

12

3

005

115

MOEAD-SBX-UF8

(i)

3

2

1

00 0

2 24 4

MOEAD-DE-UF8

(j)

2

23

1

1

00

MOEAD-GM-UF8

151

050

(k)

1

05

00

24

6

0

12

RM-MEDA-UF8

(l)IM-MOEA-UF8

1

05

00

510 1

050

(m)

AM-EDA-UF8

1

05

00

105

02

4

(n)

MOEAD-AMG-UF8

1

15

15 151 1

05

05 050 00

(o)

Ture-PF-UF8

1

1 1

05

05 05

00 0

(p)MOEAD-SBX-UF9

15

1

05

00 0

1 1

22

3

(q)

MOEAD-DE-UF9

02

46

4

2

00

24

(r)

MOEAD-GM-UF9

0 0

2 2

4 4

15

1

05

0

(s)

RM-MEDA-UF9

15

1

05

00 0

2 2446

(t)RM-MEDA-UF9

15

1

05

00 0

2 2446

(u)

RM-MEDA-UF9

15

1

05

00 0

224

46

(v)

MOEAD-AMG-UF9

1

05

00 0

2 24 4

6 6

(w)

05

Ture-PF-UF9

1

1 105 05

00 0

(x)


16 Complexity



MOEAD-FMG (0082)

MOEAD-FMG (0102)

MOEAD-FMG (0122)

























































































































111E+ 00(117Eminus 02)


















































Complexity 17









18 Complexity








Complexity 19



Data Availability




Acknowledgments


References























20 Complexity
































Complexity 21























22 Complexity



2 Preliminaries


min F(x) f1(x) fm(x)( 1113857T


i1ai bi1113858 1113859

(1)








x sim N(μΣ) (2)




TΣminus 1(x minus μ)11139671113882 (3)


μ 1K

1113944

K

k1x

k

Σ 1

K minus 11113944

K

k1x

kminus μ1113872 1113873 x

kminus μ1113872 1113873

T

(4)





Complexity 3










gi(x) g x λi z

11138681113868111386811138681113872 1113873 max1lejlem

λij fi(x) minus zj

11138681113868111386811138681113868

11138681113868111386811138681113868 (5)



(3) x μ + Ay


4 Complexity





CrkG IoFkG

1113936Ki1loFiG

+ ε (7)


PrkG CrkG

1113936Ki1CriG

(8)
















Gaussian curves

0

02

04

06

08

1


g1-06g2-08g3-10

g4-12g5-14


Complexity 5







IGD S Plowast


Plowast (9)



1 zr2 zr



rm1113858 1113859( 1113857

(10)










6 Complexity















Complexity 7









































































































































106E+ 00(762E minus 03)+





111E+ 00(117E minus 02)


























































8 Complexity
































































































































104E+ 00(129Eminus 02)



000E+ 00(000E+ 00)minus

000E+ 00(000E+ 00)minus


000E+ 00(000E+ 00)minus


WFG1 208E+ 00(166Eminus 01)+

129E+ 00(190E minus 02)+





112E+ 00(250E minus 02)



446E+ 00(676E minus 04)sim

445E+ 00(133E minus 03)sim


445E+ 00(165E minus 02)sim

446E+ 00(870Eminus 04)







398E+ 00(580Eminus 04)

WFG4 170E+ 00(132Eminus 03)+

169E+ 00(306E minus 03)+


150E+ 00(423E minus 03)sim

154E+ 00(210E minus 02)+

162E+ 00(261E minus 02)+

150E+ 00(560E minus 03)



145E+ 00(320E minus 04)sim



144E+ 00(912E minus 04)sim

145E+ 00(275E minus 04)




168E+ 00(949E minus 04)+

167E+ 00(707E minus 03)+

169E+ 00(152Eminus 03)+

162E+ 00(245E minus 01)


169E+ 00(135E minus 03)sim





170E+ 00(240Eminus 04)







141E+ 00(298Eminus 03)






164E+ 00(386E minus 03)sim

164E+ 00(146E minus 03)


Complexity 9








10 Complexity








































































































































































































Complexity 11

F1

IGD

val

ues

10ndash1

10ndash2

10ndash3




times105

(a)

F2

IGD

val

ues 10ndash1

100

10ndash2

10ndash3




(b)F3

IGD

val

ues

10ndash1

100

10ndash2

10ndash3




(c)

F4

IGD

val

ues

10ndash1

100

10ndash2

10ndash3




(d)F5

IGD

val

ues

10ndash1

100

10ndash2




(e)

F6

IGD

val

ues

10ndash1

100




(f )

Figure 2 Continued

12 Complexity

F7

IGD

val

ues

10ndash1

100

101

10ndash2

10ndash3




(g)

F9

IGD

val

ues 10ndash1

100

10ndash2

10ndash3




(h)


UF1

IGD

val

ues 10ndash1

100

10ndash2

10ndash3




times105

(a)

UF2

IGD

val

ues

10ndash1

100

10ndash2




(b)UF6

IGD

val

ues

10ndash1

100

101




(c)

UF7

IGD

val

ues

10ndash2

10ndash1

100




(d)

Figure 3 Continued

Complexity 13

UF8

IGD

val

ues

10ndash1

100




(e)

UF9

IGD

val

ues

10ndash2

10ndash1

100




(f )


WFG3

IGD

val

ues

10ndash1

100

10ndash2

times105




(a)

WFG7

IGD

val

ues

10ndash1

100

10ndash2

times105




(b)WFG8

IGD

val

ues

10ndash1

100

times105




(c)

WFG9

IGD

val

ues

10ndash1

100

10ndash2

times105




(d)


14 Complexity

MOEAD-SBX-F4

0

02

04

06

08

1

12

02 04 06 080

(a)

MOEAD-DE-F4

0

02

04

06

08

1

12

02 04 06 08 10

(b)

MOEAD-GM-F4

0

02

04

06

08

1

12

05 1 15 20

(c)

RM-MEDA-F4

0

02

04

06

08

1

02 04 06 08 1 120

(d)IM-MOEA-F4

05 1 15 200

02

04

06

08

1

12

14

(e)

AM-EDA-F4

05 1 15 2 25 300

02

04

06

08

1

(f )

MOEAD-AMG-F4

02

0

04

06

08

1

12

04 0602 08 120 1

(g)

Ture-PF-F4

0

02

04

06

08

1

02 04 06 08 10

(h)MOEAD-SBX-F5

0

02

04

06

08

1

12

02 04 06 08 10

(i)

MOEAD-DE-F5

0

02

04

06

08

1

02 04 06 08 1 120

(j)

0

02

04

06

08

1

12

02 04 06 08 1 120

MOEAD-GM-F5

(k)

RM-MEDA-F4

0

02

04

06

08

1

02 04 06 08 1 120

(l)IM-MOEA-F5

02 04 06 08 1 1200

02

04

06

08

1

(m)

AM-EDA-F5

02 04 06 08 100

02

04

06

08

1

(n)

MOEAD-AMG-F5

02 04 06 08 1 1200

02

04

06

08

1

12

(o)

Ture-PF-F5

0

02

04

06

08

1

02 04 06 08 10

(p)MOEAD-SBX-F9

0

02

04

06

08

1

12

02 04 06 08 1 120

(q)

MOEAD-DE-F9

0

02

04

06

08

1

12

02 04 06 08 1 120

(r)

MOEAD-GM-F9

0

02

04

06

08

1

12

02 04 06 08 1 120

(s)

RM-MEDA-F9

0

02

04

06

08

1

12

02 04 06 08 1 120

(t)IM-MOEA-F9

0

1

2

3

4

5

1 2 3 540

(u)

AM-EDA-F9

02 04 06 08 1 1200

02

04

06

08

1

12

(v)

MOEAD-AMG-F9

02 04 06 08 1 1200

02

04

06

08

1

12

(w)

Ture-PF-F9

0

02

04

06

08

1

02 04 06 08 10

(x)


Complexity 15

MOEAD-SBX-UF2

02

04

06

08

1

12

02 04 06 080

(a)

MOEAD-DE-UF2

0

02

04

06

08

1

12

02 04 06 08 1 120

(b)

MOEAD-GM-UF2

0

02

04

06

08

1

12

02 04 06 08 1 120

(c)

RM-MEDA-UF2

0

02

04

06

08

1

02 04 06 08 10

(d)IM-MOEA-UF2

02 04 06 08 1 1200

02

04

06

08

1

(e)

AM-EDA-UF2

0

02

04

06

08

1

12

02 04 06 08 1 120

(f )

MOEAD-AMG-UF2

02 04 06 08 100

02

04

06

08

1

12

(g)

Ture-PF-UF2

0

02

04

06

08

1

02 04 06 08 10

(h)

2

1

00

12

3

005

115

MOEAD-SBX-UF8

(i)

3

2

1

00 0

2 24 4

MOEAD-DE-UF8

(j)

2

23

1

1

00

MOEAD-GM-UF8

151

050

(k)

1

05

00

24

6

0

12

RM-MEDA-UF8

(l)IM-MOEA-UF8

1

05

00

510 1

050

(m)

AM-EDA-UF8

1

05

00

105

02

4

(n)

MOEAD-AMG-UF8

1

15

15 151 1

05

05 050 00

(o)

Ture-PF-UF8

1

1 1

05

05 05

00 0

(p)MOEAD-SBX-UF9

15

1

05

00 0

1 1

22

3

(q)

MOEAD-DE-UF9

02

46

4

2

00

24

(r)

MOEAD-GM-UF9

0 0

2 2

4 4

15

1

05

0

(s)

RM-MEDA-UF9

15

1

05

00 0

2 2446

(t)RM-MEDA-UF9

15

1

05

00 0

2 2446

(u)

RM-MEDA-UF9

15

1

05

00 0

224

46

(v)

MOEAD-AMG-UF9

1

05

00 0

2 24 4

6 6

(w)

05

Ture-PF-UF9

1

1 105 05

00 0

(x)


16 Complexity



MOEAD-FMG (0082)

MOEAD-FMG (0102)

MOEAD-FMG (0122)

























































































































111E+ 00(117Eminus 02)


















































Complexity 17









18 Complexity








Complexity 19



Data Availability




Acknowledgments


References























20 Complexity
































Complexity 21























22 Complexity










gi(x) g x λi z

11138681113868111386811138681113872 1113873 max1lejlem

λij fi(x) minus zj

11138681113868111386811138681113868

11138681113868111386811138681113868 (5)



(3) x μ + Ay


4 Complexity





CrkG IoFkG

1113936Ki1loFiG

+ ε (7)


PrkG CrkG

1113936Ki1CriG

(8)
















Gaussian curves

0

02

04

06

08

1


g1-06g2-08g3-10

g4-12g5-14


Complexity 5







IGD S Plowast


Plowast (9)



1 zr2 zr



rm1113858 1113859( 1113857

(10)










6 Complexity















Complexity 7









































































































































106E+ 00(762E minus 03)+





111E+ 00(117E minus 02)


























































8 Complexity
































































































































104E+ 00(129Eminus 02)



000E+ 00(000E+ 00)minus

000E+ 00(000E+ 00)minus


000E+ 00(000E+ 00)minus


WFG1 208E+ 00(166Eminus 01)+

129E+ 00(190E minus 02)+





112E+ 00(250E minus 02)



446E+ 00(676E minus 04)sim

445E+ 00(133E minus 03)sim


445E+ 00(165E minus 02)sim

446E+ 00(870Eminus 04)







398E+ 00(580Eminus 04)

WFG4 170E+ 00(132Eminus 03)+

169E+ 00(306E minus 03)+


150E+ 00(423E minus 03)sim

154E+ 00(210E minus 02)+

162E+ 00(261E minus 02)+

150E+ 00(560E minus 03)



145E+ 00(320E minus 04)sim



144E+ 00(912E minus 04)sim

145E+ 00(275E minus 04)




168E+ 00(949E minus 04)+

167E+ 00(707E minus 03)+

169E+ 00(152Eminus 03)+

162E+ 00(245E minus 01)


169E+ 00(135E minus 03)sim





170E+ 00(240Eminus 04)







141E+ 00(298Eminus 03)






164E+ 00(386E minus 03)sim

164E+ 00(146E minus 03)


Complexity 9








10 Complexity








































































































































































































Complexity 11

F1

IGD

val

ues

10ndash1

10ndash2

10ndash3




times105

(a)

F2

IGD

val

ues 10ndash1

100

10ndash2

10ndash3




(b)F3

IGD

val

ues

10ndash1

100

10ndash2

10ndash3




(c)

F4

IGD

val

ues

10ndash1

100

10ndash2

10ndash3




(d)F5

IGD

val

ues

10ndash1

100

10ndash2




(e)

F6

IGD

val

ues

10ndash1

100




(f )

Figure 2 Continued

12 Complexity

F7

IGD

val

ues

10ndash1

100

101

10ndash2

10ndash3




(g)

F9

IGD

val

ues 10ndash1

100

10ndash2

10ndash3




(h)


UF1

IGD

val

ues 10ndash1

100

10ndash2

10ndash3




times105

(a)

UF2

IGD

val

ues

10ndash1

100

10ndash2




(b)UF6

IGD

val

ues

10ndash1

100

101




(c)

UF7

IGD

val

ues

10ndash2

10ndash1

100




(d)

Figure 3 Continued

Complexity 13

UF8

IGD

val

ues

10ndash1

100




(e)

UF9

IGD

val

ues

10ndash2

10ndash1

100




(f )


WFG3

IGD

val

ues

10ndash1

100

10ndash2

times105




(a)

WFG7

IGD

val

ues

10ndash1

100

10ndash2

times105




(b)WFG8

IGD

val

ues

10ndash1

100

times105




(c)

WFG9

IGD

val

ues

10ndash1

100

10ndash2

times105




(d)


14 Complexity

MOEAD-SBX-F4

0

02

04

06

08

1

12

02 04 06 080

(a)

MOEAD-DE-F4

0

02

04

06

08

1

12

02 04 06 08 10

(b)

MOEAD-GM-F4

0

02

04

06

08

1

12

05 1 15 20

(c)

RM-MEDA-F4

0

02

04

06

08

1

02 04 06 08 1 120

(d)IM-MOEA-F4

05 1 15 200

02

04

06

08

1

12

14

(e)

AM-EDA-F4

05 1 15 2 25 300

02

04

06

08

1

(f )

MOEAD-AMG-F4

02

0

04

06

08

1

12

04 0602 08 120 1

(g)

Ture-PF-F4

0

02

04

06

08

1

02 04 06 08 10

(h)MOEAD-SBX-F5

0

02

04

06

08

1

12

02 04 06 08 10

(i)

MOEAD-DE-F5

0

02

04

06

08

1

02 04 06 08 1 120

(j)

0

02

04

06

08

1

12

02 04 06 08 1 120

MOEAD-GM-F5

(k)

RM-MEDA-F4

0

02

04

06

08

1

02 04 06 08 1 120

(l)IM-MOEA-F5

02 04 06 08 1 1200

02

04

06

08

1

(m)

AM-EDA-F5

02 04 06 08 100

02

04

06

08

1

(n)

MOEAD-AMG-F5

02 04 06 08 1 1200

02

04

06

08

1

12

(o)

Ture-PF-F5

0

02

04

06

08

1

02 04 06 08 10

(p)MOEAD-SBX-F9

0

02

04

06

08

1

12

02 04 06 08 1 120

(q)

MOEAD-DE-F9

0

02

04

06

08

1

12

02 04 06 08 1 120

(r)

MOEAD-GM-F9

0

02

04

06

08

1

12

02 04 06 08 1 120

(s)

RM-MEDA-F9

0

02

04

06

08

1

12

02 04 06 08 1 120

(t)IM-MOEA-F9

0

1

2

3

4

5

1 2 3 540

(u)

AM-EDA-F9

02 04 06 08 1 1200

02

04

06

08

1

12

(v)

MOEAD-AMG-F9

02 04 06 08 1 1200

02

04

06

08

1

12

(w)

Ture-PF-F9

0

02

04

06

08

1

02 04 06 08 10

(x)


Complexity 15

MOEAD-SBX-UF2

02

04

06

08

1

12

02 04 06 080

(a)

MOEAD-DE-UF2

0

02

04

06

08

1

12

02 04 06 08 1 120

(b)

MOEAD-GM-UF2

0

02

04

06

08

1

12

02 04 06 08 1 120

(c)

RM-MEDA-UF2

0

02

04

06

08

1

02 04 06 08 10

(d)IM-MOEA-UF2

02 04 06 08 1 1200

02

04

06

08

1

(e)

AM-EDA-UF2

0

02

04

06

08

1

12

02 04 06 08 1 120

(f )

MOEAD-AMG-UF2

02 04 06 08 100

02

04

06

08

1

12

(g)

Ture-PF-UF2

0

02

04

06

08

1

02 04 06 08 10

(h)

2

1

00

12

3

005

115

MOEAD-SBX-UF8

(i)

3

2

1

00 0

2 24 4

MOEAD-DE-UF8

(j)

2

23

1

1

00

MOEAD-GM-UF8

151

050

(k)

1

05

00

24

6

0

12

RM-MEDA-UF8

(l)IM-MOEA-UF8

1

05

00

510 1

050

(m)

AM-EDA-UF8

1

05

00

105

02

4

(n)

MOEAD-AMG-UF8

1

15

15 151 1

05

05 050 00

(o)

Ture-PF-UF8

1

1 1

05

05 05

00 0

(p)MOEAD-SBX-UF9

15

1

05

00 0

1 1

22

3

(q)

MOEAD-DE-UF9

02

46

4

2

00

24

(r)

MOEAD-GM-UF9

0 0

2 2

4 4

15

1

05

0

(s)

RM-MEDA-UF9

15

1

05

00 0

2 2446

(t)RM-MEDA-UF9

15

1

05

00 0

2 2446

(u)

RM-MEDA-UF9

15

1

05

00 0

224

46

(v)

MOEAD-AMG-UF9

1

05

00 0

2 24 4

6 6

(w)

05

Ture-PF-UF9

1

1 105 05

00 0

(x)


16 Complexity



MOEAD-FMG (0082)

MOEAD-FMG (0102)

MOEAD-FMG (0122)

























































































































111E+ 00(117Eminus 02)


















































Complexity 17









18 Complexity








Complexity 19



Data Availability




Acknowledgments


References























20 Complexity
































Complexity 21























22 Complexity





CrkG IoFkG

1113936Ki1loFiG

+ ε (7)


PrkG CrkG

1113936Ki1CriG

(8)
















Gaussian curves

0

02

04

06

08

1


g1-06g2-08g3-10

g4-12g5-14


Complexity 5







IGD S Plowast


Plowast (9)



1 zr2 zr



rm1113858 1113859( 1113857

(10)










6 Complexity















Complexity 7









































































































































106E+ 00(762E minus 03)+





111E+ 00(117E minus 02)


























































8 Complexity
































































































































104E+ 00(129Eminus 02)



000E+ 00(000E+ 00)minus

000E+ 00(000E+ 00)minus


000E+ 00(000E+ 00)minus


WFG1 208E+ 00(166Eminus 01)+

129E+ 00(190E minus 02)+





112E+ 00(250E minus 02)



446E+ 00(676E minus 04)sim

445E+ 00(133E minus 03)sim


445E+ 00(165E minus 02)sim

446E+ 00(870Eminus 04)







398E+ 00(580Eminus 04)

WFG4 170E+ 00(132Eminus 03)+

169E+ 00(306E minus 03)+


150E+ 00(423E minus 03)sim

154E+ 00(210E minus 02)+

162E+ 00(261E minus 02)+

150E+ 00(560E minus 03)



145E+ 00(320E minus 04)sim



144E+ 00(912E minus 04)sim

145E+ 00(275E minus 04)




168E+ 00(949E minus 04)+

167E+ 00(707E minus 03)+

169E+ 00(152Eminus 03)+

162E+ 00(245E minus 01)


169E+ 00(135E minus 03)sim





170E+ 00(240Eminus 04)







141E+ 00(298Eminus 03)






164E+ 00(386E minus 03)sim

164E+ 00(146E minus 03)


Complexity 9








10 Complexity








































































































































































































Complexity 11

F1

IGD

val

ues

10ndash1

10ndash2

10ndash3




times105

(a)

F2

IGD

val

ues 10ndash1

100

10ndash2

10ndash3




(b)F3

IGD

val

ues

10ndash1

100

10ndash2

10ndash3




(c)

F4

IGD

val

ues

10ndash1

100

10ndash2

10ndash3




(d)F5

IGD

val

ues

10ndash1

100

10ndash2




(e)

F6

IGD

val

ues

10ndash1

100




(f )

Figure 2 Continued

12 Complexity

F7

IGD

val

ues

10ndash1

100

101

10ndash2

10ndash3




(g)

F9

IGD

val

ues 10ndash1

100

10ndash2

10ndash3




(h)


UF1

IGD

val

ues 10ndash1

100

10ndash2

10ndash3




times105

(a)

UF2

IGD

val

ues

10ndash1

100

10ndash2




(b)UF6

IGD

val

ues

10ndash1

100

101




(c)

UF7

IGD

val

ues

10ndash2

10ndash1

100




(d)

Figure 3 Continued

Complexity 13

UF8

IGD

val

ues

10ndash1

100




(e)

UF9

IGD

val

ues

10ndash2

10ndash1

100




(f )


WFG3

IGD

val

ues

10ndash1

100

10ndash2

times105




(a)

WFG7

IGD

val

ues

10ndash1

100

10ndash2

times105




(b)WFG8

IGD

val

ues

10ndash1

100

times105




(c)

WFG9

IGD

val

ues

10ndash1

100

10ndash2

times105




(d)


14 Complexity

MOEAD-SBX-F4

0

02

04

06

08

1

12

02 04 06 080

(a)

MOEAD-DE-F4

0

02

04

06

08

1

12

02 04 06 08 10

(b)

MOEAD-GM-F4

0

02

04

06

08

1

12

05 1 15 20

(c)

RM-MEDA-F4

0

02

04

06

08

1

02 04 06 08 1 120

(d)IM-MOEA-F4

05 1 15 200

02

04

06

08

1

12

14

(e)

AM-EDA-F4

05 1 15 2 25 300

02

04

06

08

1

(f )

MOEAD-AMG-F4

02

0

04

06

08

1

12

04 0602 08 120 1

(g)

Ture-PF-F4

0

02

04

06

08

1

02 04 06 08 10

(h)MOEAD-SBX-F5

0

02

04

06

08

1

12

02 04 06 08 10

(i)

MOEAD-DE-F5

0

02

04

06

08

1

02 04 06 08 1 120

(j)

0

02

04

06

08

1

12

02 04 06 08 1 120

MOEAD-GM-F5

(k)

RM-MEDA-F4

0

02

04

06

08

1

02 04 06 08 1 120

(l)IM-MOEA-F5

02 04 06 08 1 1200

02

04

06

08

1

(m)

AM-EDA-F5

02 04 06 08 100

02

04

06

08

1

(n)

MOEAD-AMG-F5

02 04 06 08 1 1200

02

04

06

08

1

12

(o)

Ture-PF-F5

0

02

04

06

08

1

02 04 06 08 10

(p)MOEAD-SBX-F9

0

02

04

06

08

1

12

02 04 06 08 1 120

(q)

MOEAD-DE-F9

0

02

04

06

08

1

12

02 04 06 08 1 120

(r)

MOEAD-GM-F9

0

02

04

06

08

1

12

02 04 06 08 1 120

(s)

RM-MEDA-F9

0

02

04

06

08

1

12

02 04 06 08 1 120

(t)IM-MOEA-F9

0

1

2

3

4

5

1 2 3 540

(u)

AM-EDA-F9

02 04 06 08 1 1200

02

04

06

08

1

12

(v)

MOEAD-AMG-F9

02 04 06 08 1 1200

02

04

06

08

1

12

(w)

Ture-PF-F9

0

02

04

06

08

1

02 04 06 08 10

(x)


Complexity 15

MOEAD-SBX-UF2

02

04

06

08

1

12

02 04 06 080

(a)

MOEAD-DE-UF2

0

02

04

06

08

1

12

02 04 06 08 1 120

(b)

MOEAD-GM-UF2

0

02

04

06

08

1

12

02 04 06 08 1 120

(c)

RM-MEDA-UF2

0

02

04

06

08

1

02 04 06 08 10

(d)IM-MOEA-UF2

02 04 06 08 1 1200

02

04

06

08

1

(e)

AM-EDA-UF2

0

02

04

06

08

1

12

02 04 06 08 1 120

(f )

MOEAD-AMG-UF2

02 04 06 08 100

02

04

06

08

1

12

(g)

Ture-PF-UF2

0

02

04

06

08

1

02 04 06 08 10

(h)

2

1

00

12

3

005

115

MOEAD-SBX-UF8

(i)

3

2

1

00 0

2 24 4

MOEAD-DE-UF8

(j)

2

23

1

1

00

MOEAD-GM-UF8

151

050

(k)

1

05

00

24

6

0

12

RM-MEDA-UF8

(l)IM-MOEA-UF8

1

05

00

510 1

050

(m)

AM-EDA-UF8

1

05

00

105

02

4

(n)

MOEAD-AMG-UF8

1

15

15 151 1

05

05 050 00

(o)

Ture-PF-UF8

1

1 1

05

05 05

00 0

(p)MOEAD-SBX-UF9

15

1

05

00 0

1 1

22

3

(q)

MOEAD-DE-UF9

02

46

4

2

00

24

(r)

MOEAD-GM-UF9

0 0

2 2

4 4

15

1

05

0

(s)

RM-MEDA-UF9

15

1

05

00 0

2 2446

(t)RM-MEDA-UF9

15

1

05

00 0

2 2446

(u)

RM-MEDA-UF9

15

1

05

00 0

224

46

(v)

MOEAD-AMG-UF9

1

05

00 0

2 24 4

6 6

(w)

05

Ture-PF-UF9

1

1 105 05

00 0

(x)


16 Complexity



MOEAD-FMG (0082)

MOEAD-FMG (0102)

MOEAD-FMG (0122)

























































































































111E+ 00(117Eminus 02)


















































Complexity 17









18 Complexity








Complexity 19



Data Availability




Acknowledgments


References























20 Complexity
































Complexity 21























22 Complexity







IGD S Plowast


Plowast (9)



1 zr2 zr



rm1113858 1113859( 1113857

(10)










6 Complexity















Complexity 7









































































































































106E+ 00(762E minus 03)+





111E+ 00(117E minus 02)


























































8 Complexity
































































































































104E+ 00(129Eminus 02)



000E+ 00(000E+ 00)minus

000E+ 00(000E+ 00)minus


000E+ 00(000E+ 00)minus


WFG1 208E+ 00(166Eminus 01)+

129E+ 00(190E minus 02)+





112E+ 00(250E minus 02)



446E+ 00(676E minus 04)sim

445E+ 00(133E minus 03)sim


445E+ 00(165E minus 02)sim

446E+ 00(870Eminus 04)







398E+ 00(580Eminus 04)

WFG4 170E+ 00(132Eminus 03)+

169E+ 00(306E minus 03)+


150E+ 00(423E minus 03)sim

154E+ 00(210E minus 02)+

162E+ 00(261E minus 02)+

150E+ 00(560E minus 03)



145E+ 00(320E minus 04)sim



144E+ 00(912E minus 04)sim

145E+ 00(275E minus 04)




168E+ 00(949E minus 04)+

167E+ 00(707E minus 03)+

169E+ 00(152Eminus 03)+

162E+ 00(245E minus 01)


169E+ 00(135E minus 03)sim





170E+ 00(240Eminus 04)







141E+ 00(298Eminus 03)






164E+ 00(386E minus 03)sim

164E+ 00(146E minus 03)


Complexity 9








10 Complexity








































































































































































































Complexity 11

F1

IGD

val

ues

10ndash1

10ndash2

10ndash3




times105

(a)

F2

IGD

val

ues 10ndash1

100

10ndash2

10ndash3




(b)F3

IGD

val

ues

10ndash1

100

10ndash2

10ndash3




(c)

F4

IGD

val

ues

10ndash1

100

10ndash2

10ndash3




(d)F5

IGD

val

ues

10ndash1

100

10ndash2




(e)

F6

IGD

val

ues

10ndash1

100




(f )

Figure 2 Continued

12 Complexity

F7

IGD

val

ues

10ndash1

100

101

10ndash2

10ndash3




(g)

F9

IGD

val

ues 10ndash1

100

10ndash2

10ndash3




(h)


UF1

IGD

val

ues 10ndash1

100

10ndash2

10ndash3




times105

(a)

UF2

IGD

val

ues

10ndash1

100

10ndash2




(b)UF6

IGD

val

ues

10ndash1

100

101




(c)

UF7

IGD

val

ues

10ndash2

10ndash1

100




(d)

Figure 3 Continued

Complexity 13

UF8

IGD

val

ues

10ndash1

100




(e)

UF9

IGD

val

ues

10ndash2

10ndash1

100




(f )


WFG3

IGD

val

ues

10ndash1

100

10ndash2

times105




(a)

WFG7

IGD

val

ues

10ndash1

100

10ndash2

times105




(b)WFG8

IGD

val

ues

10ndash1

100

times105




(c)

WFG9

IGD

val

ues

10ndash1

100

10ndash2

times105




(d)


14 Complexity

MOEAD-SBX-F4

0

02

04

06

08

1

12

02 04 06 080

(a)

MOEAD-DE-F4

0

02

04

06

08

1

12

02 04 06 08 10

(b)

MOEAD-GM-F4

0

02

04

06

08

1

12

05 1 15 20

(c)

RM-MEDA-F4

0

02

04

06

08

1

02 04 06 08 1 120

(d)IM-MOEA-F4

05 1 15 200

02

04

06

08

1

12

14

(e)

AM-EDA-F4

05 1 15 2 25 300

02

04

06

08

1

(f )

MOEAD-AMG-F4

02

0

04

06

08

1

12

04 0602 08 120 1

(g)

Ture-PF-F4

0

02

04

06

08

1

02 04 06 08 10

(h)MOEAD-SBX-F5

0

02

04

06

08

1

12

02 04 06 08 10

(i)

MOEAD-DE-F5

0

02

04

06

08

1

02 04 06 08 1 120

(j)

0

02

04

06

08

1

12

02 04 06 08 1 120

MOEAD-GM-F5

(k)

RM-MEDA-F4

0

02

04

06

08

1

02 04 06 08 1 120

(l)IM-MOEA-F5

02 04 06 08 1 1200

02

04

06

08

1

(m)

AM-EDA-F5

02 04 06 08 100

02

04

06

08

1

(n)

MOEAD-AMG-F5

02 04 06 08 1 1200

02

04

06

08

1

12

(o)

Ture-PF-F5

0

02

04

06

08

1

02 04 06 08 10

(p)MOEAD-SBX-F9

0

02

04

06

08

1

12

02 04 06 08 1 120

(q)

MOEAD-DE-F9

0

02

04

06

08

1

12

02 04 06 08 1 120

(r)

MOEAD-GM-F9

0

02

04

06

08

1

12

02 04 06 08 1 120

(s)

RM-MEDA-F9

0

02

04

06

08

1

12

02 04 06 08 1 120

(t)IM-MOEA-F9

0

1

2

3

4

5

1 2 3 540

(u)

AM-EDA-F9

02 04 06 08 1 1200

02

04

06

08

1

12

(v)

MOEAD-AMG-F9

02 04 06 08 1 1200

02

04

06

08

1

12

(w)

Ture-PF-F9

0

02

04

06

08

1

02 04 06 08 10

(x)


Complexity 15

MOEAD-SBX-UF2

02

04

06

08

1

12

02 04 06 080

(a)

MOEAD-DE-UF2

0

02

04

06

08

1

12

02 04 06 08 1 120

(b)

MOEAD-GM-UF2

0

02

04

06

08

1

12

02 04 06 08 1 120

(c)

RM-MEDA-UF2

0

02

04

06

08

1

02 04 06 08 10

(d)IM-MOEA-UF2

02 04 06 08 1 1200

02

04

06

08

1

(e)

AM-EDA-UF2

0

02

04

06

08

1

12

02 04 06 08 1 120

(f )

MOEAD-AMG-UF2

02 04 06 08 100

02

04

06

08

1

12

(g)

Ture-PF-UF2

0

02

04

06

08

1

02 04 06 08 10

(h)

2

1

00

12

3

005

115

MOEAD-SBX-UF8

(i)

3

2

1

00 0

2 24 4

MOEAD-DE-UF8

(j)

2

23

1

1

00

MOEAD-GM-UF8

151

050

(k)

1

05

00

24

6

0

12

RM-MEDA-UF8

(l)IM-MOEA-UF8

1

05

00

510 1

050

(m)

AM-EDA-UF8

1

05

00

105

02

4

(n)

MOEAD-AMG-UF8

1

15

15 151 1

05

05 050 00

(o)

Ture-PF-UF8

1

1 1

05

05 05

00 0

(p)MOEAD-SBX-UF9

15

1

05

00 0

1 1

22

3

(q)

MOEAD-DE-UF9

02

46

4

2

00

24

(r)

MOEAD-GM-UF9

0 0

2 2

4 4

15

1

05

0

(s)

RM-MEDA-UF9

15

1

05

00 0

2 2446

(t)RM-MEDA-UF9

15

1

05

00 0

2 2446

(u)

RM-MEDA-UF9

15

1

05

00 0

224

46

(v)

MOEAD-AMG-UF9

1

05

00 0

2 24 4

6 6

(w)

05

Ture-PF-UF9

1

1 105 05

00 0

(x)


16 Complexity



MOEAD-FMG (0082)

MOEAD-FMG (0102)

MOEAD-FMG (0122)

























































































































111E+ 00(117Eminus 02)


















































Complexity 17









18 Complexity








Complexity 19



Data Availability




Acknowledgments


References























20 Complexity
































Complexity 21























22 Complexity















Complexity 7









































































































































106E+ 00(762E minus 03)+





111E+ 00(117E minus 02)


























































8 Complexity
































































































































104E+ 00(129Eminus 02)



000E+ 00(000E+ 00)minus

000E+ 00(000E+ 00)minus


000E+ 00(000E+ 00)minus


WFG1 208E+ 00(166Eminus 01)+

129E+ 00(190E minus 02)+





112E+ 00(250E minus 02)



446E+ 00(676E minus 04)sim

445E+ 00(133E minus 03)sim


445E+ 00(165E minus 02)sim

446E+ 00(870Eminus 04)







398E+ 00(580Eminus 04)

WFG4 170E+ 00(132Eminus 03)+

169E+ 00(306E minus 03)+


150E+ 00(423E minus 03)sim

154E+ 00(210E minus 02)+

162E+ 00(261E minus 02)+

150E+ 00(560E minus 03)



145E+ 00(320E minus 04)sim



144E+ 00(912E minus 04)sim

145E+ 00(275E minus 04)




168E+ 00(949E minus 04)+

167E+ 00(707E minus 03)+

169E+ 00(152Eminus 03)+

162E+ 00(245E minus 01)


169E+ 00(135E minus 03)sim





170E+ 00(240Eminus 04)







141E+ 00(298Eminus 03)






164E+ 00(386E minus 03)sim

164E+ 00(146E minus 03)


Complexity 9








10 Complexity








































































































































































































Complexity 11

F1

IGD

val

ues

10ndash1

10ndash2

10ndash3




times105

(a)

F2

IGD

val

ues 10ndash1

100

10ndash2

10ndash3




(b)F3

IGD

val

ues

10ndash1

100

10ndash2

10ndash3




(c)

F4

IGD

val

ues

10ndash1

100

10ndash2

10ndash3




(d)F5

IGD

val

ues

10ndash1

100

10ndash2




(e)

F6

IGD

val

ues

10ndash1

100




(f )

Figure 2 Continued

12 Complexity

F7

IGD

val

ues

10ndash1

100

101

10ndash2

10ndash3




(g)

F9

IGD

val

ues 10ndash1

100

10ndash2

10ndash3




(h)


UF1

IGD

val

ues 10ndash1

100

10ndash2

10ndash3




times105

(a)

UF2

IGD

val

ues

10ndash1

100

10ndash2




(b)UF6

IGD

val

ues

10ndash1

100

101




(c)

UF7

IGD

val

ues

10ndash2

10ndash1

100




(d)

Figure 3 Continued

Complexity 13

UF8

IGD

val

ues

10ndash1

100




(e)

UF9

IGD

val

ues

10ndash2

10ndash1

100




(f )


WFG3

IGD

val

ues

10ndash1

100

10ndash2

times105




(a)

WFG7

IGD

val

ues

10ndash1

100

10ndash2

times105




(b)WFG8

IGD

val

ues

10ndash1

100

times105




(c)

WFG9

IGD

val

ues

10ndash1

100

10ndash2

times105




(d)


14 Complexity

MOEAD-SBX-F4

0

02

04

06

08

1

12

02 04 06 080

(a)

MOEAD-DE-F4

0

02

04

06

08

1

12

02 04 06 08 10

(b)

MOEAD-GM-F4

0

02

04

06

08

1

12

05 1 15 20

(c)

RM-MEDA-F4

0

02

04

06

08

1

02 04 06 08 1 120

(d)IM-MOEA-F4

05 1 15 200

02

04

06

08

1

12

14

(e)

AM-EDA-F4

05 1 15 2 25 300

02

04

06

08

1

(f )

MOEAD-AMG-F4

02

0

04

06

08

1

12

04 0602 08 120 1

(g)

Ture-PF-F4

0

02

04

06

08

1

02 04 06 08 10

(h)MOEAD-SBX-F5

0

02

04

06

08

1

12

02 04 06 08 10

(i)

MOEAD-DE-F5

0

02

04

06

08

1

02 04 06 08 1 120

(j)

0

02

04

06

08

1

12

02 04 06 08 1 120

MOEAD-GM-F5

(k)

RM-MEDA-F4

0

02

04

06

08

1

02 04 06 08 1 120

(l)IM-MOEA-F5

02 04 06 08 1 1200

02

04

06

08

1

(m)

AM-EDA-F5

02 04 06 08 100

02

04

06

08

1

(n)

MOEAD-AMG-F5

02 04 06 08 1 1200

02

04

06

08

1

12

(o)

Ture-PF-F5

0

02

04

06

08

1

02 04 06 08 10

(p)MOEAD-SBX-F9

0

02

04

06

08

1

12

02 04 06 08 1 120

(q)

MOEAD-DE-F9

0

02

04

06

08

1

12

02 04 06 08 1 120

(r)

MOEAD-GM-F9

0

02

04

06

08

1

12

02 04 06 08 1 120

(s)

RM-MEDA-F9

0

02

04

06

08

1

12

02 04 06 08 1 120

(t)IM-MOEA-F9

0

1

2

3

4

5

1 2 3 540

(u)

AM-EDA-F9

02 04 06 08 1 1200

02

04

06

08

1

12

(v)

MOEAD-AMG-F9

02 04 06 08 1 1200

02

04

06

08

1

12

(w)

Ture-PF-F9

0

02

04

06

08

1

02 04 06 08 10

(x)


Complexity 15

MOEAD-SBX-UF2

02

04

06

08

1

12

02 04 06 080

(a)

MOEAD-DE-UF2

0

02

04

06

08

1

12

02 04 06 08 1 120

(b)

MOEAD-GM-UF2

0

02

04

06

08

1

12

02 04 06 08 1 120

(c)

RM-MEDA-UF2

0

02

04

06

08

1

02 04 06 08 10

(d)IM-MOEA-UF2

02 04 06 08 1 1200

02

04

06

08

1

(e)

AM-EDA-UF2

0

02

04

06

08

1

12

02 04 06 08 1 120

(f )

MOEAD-AMG-UF2

02 04 06 08 100

02

04

06

08

1

12

(g)

Ture-PF-UF2

0

02

04

06

08

1

02 04 06 08 10

(h)

2

1

00

12

3

005

115

MOEAD-SBX-UF8

(i)

3

2

1

00 0

2 24 4

MOEAD-DE-UF8

(j)

2

23

1

1

00

MOEAD-GM-UF8

151

050

(k)

1

05

00

24

6

0

12

RM-MEDA-UF8

(l)IM-MOEA-UF8

1

05

00

510 1

050

(m)

AM-EDA-UF8

1

05

00

105

02

4

(n)

MOEAD-AMG-UF8

1

15

15 151 1

05

05 050 00

(o)

Ture-PF-UF8

1

1 1

05

05 05

00 0

(p)MOEAD-SBX-UF9

15

1

05

00 0

1 1

22

3

(q)

MOEAD-DE-UF9

02

46

4

2

00

24

(r)

MOEAD-GM-UF9

0 0

2 2

4 4

15

1

05

0

(s)

RM-MEDA-UF9

15

1

05

00 0

2 2446

(t)RM-MEDA-UF9

15

1

05

00 0

2 2446

(u)

RM-MEDA-UF9

15

1

05

00 0

224

46

(v)

MOEAD-AMG-UF9

1

05

00 0

2 24 4

6 6

(w)

05

Ture-PF-UF9

1

1 105 05

00 0

(x)


16 Complexity



MOEAD-FMG (0082)

MOEAD-FMG (0102)

MOEAD-FMG (0122)

























































































































111E+ 00(117Eminus 02)


















































Complexity 17









18 Complexity








Complexity 19



Data Availability




Acknowledgments


References























20 Complexity
































Complexity 21























22 Complexity









































































































































106E+ 00(762E minus 03)+





111E+ 00(117E minus 02)


























































8 Complexity
































































































































104E+ 00(129Eminus 02)



000E+ 00(000E+ 00)minus

000E+ 00(000E+ 00)minus


000E+ 00(000E+ 00)minus


WFG1 208E+ 00(166Eminus 01)+

129E+ 00(190E minus 02)+





112E+ 00(250E minus 02)



446E+ 00(676E minus 04)sim

445E+ 00(133E minus 03)sim


445E+ 00(165E minus 02)sim

446E+ 00(870Eminus 04)







398E+ 00(580Eminus 04)

WFG4 170E+ 00(132Eminus 03)+

169E+ 00(306E minus 03)+


150E+ 00(423E minus 03)sim

154E+ 00(210E minus 02)+

162E+ 00(261E minus 02)+

150E+ 00(560E minus 03)



145E+ 00(320E minus 04)sim



144E+ 00(912E minus 04)sim

145E+ 00(275E minus 04)




168E+ 00(949E minus 04)+

167E+ 00(707E minus 03)+

169E+ 00(152Eminus 03)+

162E+ 00(245E minus 01)


169E+ 00(135E minus 03)sim





170E+ 00(240Eminus 04)







141E+ 00(298Eminus 03)






164E+ 00(386E minus 03)sim

164E+ 00(146E minus 03)


Complexity 9








10 Complexity








































































































































































































Complexity 11

F1

IGD

val

ues

10ndash1

10ndash2

10ndash3




times105

(a)

F2

IGD

val

ues 10ndash1

100

10ndash2

10ndash3




(b)F3

IGD

val

ues

10ndash1

100

10ndash2

10ndash3




(c)

F4

IGD

val

ues

10ndash1

100

10ndash2

10ndash3




(d)F5

IGD

val

ues

10ndash1

100

10ndash2




(e)

F6

IGD

val

ues

10ndash1

100




(f )

Figure 2 Continued

12 Complexity

F7

IGD

val

ues

10ndash1

100

101

10ndash2

10ndash3




(g)

F9

IGD

val

ues 10ndash1

100

10ndash2

10ndash3




(h)


UF1

IGD

val

ues 10ndash1

100

10ndash2

10ndash3




times105

(a)

UF2

IGD

val

ues

10ndash1

100

10ndash2




(b)UF6

IGD

val

ues

10ndash1

100

101




(c)

UF7

IGD

val

ues

10ndash2

10ndash1

100




(d)

Figure 3 Continued

Complexity 13

UF8

IGD

val

ues

10ndash1

100




(e)

UF9

IGD

val

ues

10ndash2

10ndash1

100




(f )


WFG3

IGD

val

ues

10ndash1

100

10ndash2

times105




(a)

WFG7

IGD

val

ues

10ndash1

100

10ndash2

times105




(b)WFG8

IGD

val

ues

10ndash1

100

times105




(c)

WFG9

IGD

val

ues

10ndash1

100

10ndash2

times105




(d)


14 Complexity

MOEAD-SBX-F4

0

02

04

06

08

1

12

02 04 06 080

(a)

MOEAD-DE-F4

0

02

04

06

08

1

12

02 04 06 08 10

(b)

MOEAD-GM-F4

0

02

04

06

08

1

12

05 1 15 20

(c)

RM-MEDA-F4

0

02

04

06

08

1

02 04 06 08 1 120

(d)IM-MOEA-F4

05 1 15 200

02

04

06

08

1

12

14

(e)

AM-EDA-F4

05 1 15 2 25 300

02

04

06

08

1

(f )

MOEAD-AMG-F4

02

0

04

06

08

1

12

04 0602 08 120 1

(g)

Ture-PF-F4

0

02

04

06

08

1

02 04 06 08 10

(h)MOEAD-SBX-F5

0

02

04

06

08

1

12

02 04 06 08 10

(i)

MOEAD-DE-F5

0

02

04

06

08

1

02 04 06 08 1 120

(j)

0

02

04

06

08

1

12

02 04 06 08 1 120

MOEAD-GM-F5

(k)

RM-MEDA-F4

0

02

04

06

08

1

02 04 06 08 1 120

(l)IM-MOEA-F5

02 04 06 08 1 1200

02

04

06

08

1

(m)

AM-EDA-F5

02 04 06 08 100

02

04

06

08

1

(n)

MOEAD-AMG-F5

02 04 06 08 1 1200

02

04

06

08

1

12

(o)

Ture-PF-F5

0

02

04

06

08

1

02 04 06 08 10

(p)MOEAD-SBX-F9

0

02

04

06

08

1

12

02 04 06 08 1 120

(q)

MOEAD-DE-F9

0

02

04

06

08

1

12

02 04 06 08 1 120

(r)

MOEAD-GM-F9

0

02

04

06

08

1

12

02 04 06 08 1 120

(s)

RM-MEDA-F9

0

02

04

06

08

1

12

02 04 06 08 1 120

(t)IM-MOEA-F9

0

1

2

3

4

5

1 2 3 540

(u)

AM-EDA-F9

02 04 06 08 1 1200

02

04

06

08

1

12

(v)

MOEAD-AMG-F9

02 04 06 08 1 1200

02

04

06

08

1

12

(w)

Ture-PF-F9

0

02

04

06

08

1

02 04 06 08 10

(x)


Complexity 15

MOEAD-SBX-UF2

02

04

06

08

1

12

02 04 06 080

(a)

MOEAD-DE-UF2

0

02

04

06

08

1

12

02 04 06 08 1 120

(b)

MOEAD-GM-UF2

0

02

04

06

08

1

12

02 04 06 08 1 120

(c)

RM-MEDA-UF2

0

02

04

06

08

1

02 04 06 08 10

(d)IM-MOEA-UF2

02 04 06 08 1 1200

02

04

06

08

1

(e)

AM-EDA-UF2

0

02

04

06

08

1

12

02 04 06 08 1 120

(f )

MOEAD-AMG-UF2

02 04 06 08 100

02

04

06

08

1

12

(g)

Ture-PF-UF2

0

02

04

06

08

1

02 04 06 08 10

(h)

2

1

00

12

3

005

115

MOEAD-SBX-UF8

(i)

3

2

1

00 0

2 24 4

MOEAD-DE-UF8

(j)

2

23

1

1

00

MOEAD-GM-UF8

151

050

(k)

1

05

00

24

6

0

12

RM-MEDA-UF8

(l)IM-MOEA-UF8

1

05

00

510 1

050

(m)

AM-EDA-UF8

1

05

00

105

02

4

(n)

MOEAD-AMG-UF8

1

15

15 151 1

05

05 050 00

(o)

Ture-PF-UF8

1

1 1

05

05 05

00 0

(p)MOEAD-SBX-UF9

15

1

05

00 0

1 1

22

3

(q)

MOEAD-DE-UF9

02

46

4

2

00

24

(r)

MOEAD-GM-UF9

0 0

2 2

4 4

15

1

05

0

(s)

RM-MEDA-UF9

15

1

05

00 0

2 2446

(t)RM-MEDA-UF9

15

1

05

00 0

2 2446

(u)

RM-MEDA-UF9

15

1

05

00 0

224

46

(v)

MOEAD-AMG-UF9

1

05

00 0

2 24 4

6 6

(w)

05

Ture-PF-UF9

1

1 105 05

00 0

(x)


16 Complexity



MOEAD-FMG (0082)

MOEAD-FMG (0102)

MOEAD-FMG (0122)

























































































































111E+ 00(117Eminus 02)


















































Complexity 17









18 Complexity








Complexity 19



Data Availability




Acknowledgments


References























20 Complexity
































Complexity 21























22 Complexity
































































































































104E+ 00(129Eminus 02)



000E+ 00(000E+ 00)minus

000E+ 00(000E+ 00)minus


000E+ 00(000E+ 00)minus


WFG1 208E+ 00(166Eminus 01)+

129E+ 00(190E minus 02)+





112E+ 00(250E minus 02)



446E+ 00(676E minus 04)sim

445E+ 00(133E minus 03)sim


445E+ 00(165E minus 02)sim

446E+ 00(870Eminus 04)







398E+ 00(580Eminus 04)

WFG4 170E+ 00(132Eminus 03)+

169E+ 00(306E minus 03)+


150E+ 00(423E minus 03)sim

154E+ 00(210E minus 02)+

162E+ 00(261E minus 02)+

150E+ 00(560E minus 03)



145E+ 00(320E minus 04)sim



144E+ 00(912E minus 04)sim

145E+ 00(275E minus 04)




168E+ 00(949E minus 04)+

167E+ 00(707E minus 03)+

169E+ 00(152Eminus 03)+

162E+ 00(245E minus 01)


169E+ 00(135E minus 03)sim





170E+ 00(240Eminus 04)







141E+ 00(298Eminus 03)






164E+ 00(386E minus 03)sim

164E+ 00(146E minus 03)


Complexity 9








10 Complexity








































































































































































































Complexity 11

F1

IGD

val

ues

10ndash1

10ndash2

10ndash3




times105

(a)

F2

IGD

val

ues 10ndash1

100

10ndash2

10ndash3




(b)F3

IGD

val

ues

10ndash1

100

10ndash2

10ndash3




(c)

F4

IGD

val

ues

10ndash1

100

10ndash2

10ndash3




(d)F5

IGD

val

ues

10ndash1

100

10ndash2




(e)

F6

IGD

val

ues

10ndash1

100




(f )

Figure 2 Continued

12 Complexity

F7

IGD

val

ues

10ndash1

100

101

10ndash2

10ndash3




(g)

F9

IGD

val

ues 10ndash1

100

10ndash2

10ndash3




(h)


UF1

IGD

val

ues 10ndash1

100

10ndash2

10ndash3




times105

(a)

UF2

IGD

val

ues

10ndash1

100

10ndash2




(b)UF6

IGD

val

ues

10ndash1

100

101




(c)

UF7

IGD

val

ues

10ndash2

10ndash1

100




(d)

Figure 3 Continued

Complexity 13

UF8

IGD

val

ues

10ndash1

100




(e)

UF9

IGD

val

ues

10ndash2

10ndash1

100




(f )


WFG3

IGD

val

ues

10ndash1

100

10ndash2

times105




(a)

WFG7

IGD

val

ues

10ndash1

100

10ndash2

times105




(b)WFG8

IGD

val

ues

10ndash1

100

times105




(c)

WFG9

IGD

val

ues

10ndash1

100

10ndash2

times105




(d)


14 Complexity

MOEAD-SBX-F4

0

02

04

06

08

1

12

02 04 06 080

(a)

MOEAD-DE-F4

0

02

04

06

08

1

12

02 04 06 08 10

(b)

MOEAD-GM-F4

0

02

04

06

08

1

12

05 1 15 20

(c)

RM-MEDA-F4

0

02

04

06

08

1

02 04 06 08 1 120

(d)IM-MOEA-F4

05 1 15 200

02

04

06

08

1

12

14

(e)

AM-EDA-F4

05 1 15 2 25 300

02

04

06

08

1

(f )

MOEAD-AMG-F4

02

0

04

06

08

1

12

04 0602 08 120 1

(g)

Ture-PF-F4

0

02

04

06

08

1

02 04 06 08 10

(h)MOEAD-SBX-F5

0

02

04

06

08

1

12

02 04 06 08 10

(i)

MOEAD-DE-F5

0

02

04

06

08

1

02 04 06 08 1 120

(j)

0

02

04

06

08

1

12

02 04 06 08 1 120

MOEAD-GM-F5

(k)

RM-MEDA-F4

0

02

04

06

08

1

02 04 06 08 1 120

(l)IM-MOEA-F5

02 04 06 08 1 1200

02

04

06

08

1

(m)

AM-EDA-F5

02 04 06 08 100

02

04

06

08

1

(n)

MOEAD-AMG-F5

02 04 06 08 1 1200

02

04

06

08

1

12

(o)

Ture-PF-F5

0

02

04

06

08

1

02 04 06 08 10

(p)MOEAD-SBX-F9

0

02

04

06

08

1

12

02 04 06 08 1 120

(q)

MOEAD-DE-F9

0

02

04

06

08

1

12

02 04 06 08 1 120

(r)

MOEAD-GM-F9

0

02

04

06

08

1

12

02 04 06 08 1 120

(s)

RM-MEDA-F9

0

02

04

06

08

1

12

02 04 06 08 1 120

(t)IM-MOEA-F9

0

1

2

3

4

5

1 2 3 540

(u)

AM-EDA-F9

02 04 06 08 1 1200

02

04

06

08

1

12

(v)

MOEAD-AMG-F9

02 04 06 08 1 1200

02

04

06

08

1

12

(w)

Ture-PF-F9

0

02

04

06

08

1

02 04 06 08 10

(x)


Complexity 15

MOEAD-SBX-UF2

02

04

06

08

1

12

02 04 06 080

(a)

MOEAD-DE-UF2

0

02

04

06

08

1

12

02 04 06 08 1 120

(b)

MOEAD-GM-UF2

0

02

04

06

08

1

12

02 04 06 08 1 120

(c)

RM-MEDA-UF2

0

02

04

06

08

1

02 04 06 08 10

(d)IM-MOEA-UF2

02 04 06 08 1 1200

02

04

06

08

1

(e)

AM-EDA-UF2

0

02

04

06

08

1

12

02 04 06 08 1 120

(f )

MOEAD-AMG-UF2

02 04 06 08 100

02

04

06

08

1

12

(g)

Ture-PF-UF2

0

02

04

06

08

1

02 04 06 08 10

(h)

2

1

00

12

3

005

115

MOEAD-SBX-UF8

(i)

3

2

1

00 0

2 24 4

MOEAD-DE-UF8

(j)

2

23

1

1

00

MOEAD-GM-UF8

151

050

(k)

1

05

00

24

6

0

12

RM-MEDA-UF8

(l)IM-MOEA-UF8

1

05

00

510 1

050

(m)

AM-EDA-UF8

1

05

00

105

02

4

(n)

MOEAD-AMG-UF8

1

15

15 151 1

05

05 050 00

(o)

Ture-PF-UF8

1

1 1

05

05 05

00 0

(p)MOEAD-SBX-UF9

15

1

05

00 0

1 1

22

3

(q)

MOEAD-DE-UF9

02

46

4

2

00

24

(r)

MOEAD-GM-UF9

0 0

2 2

4 4

15

1

05

0

(s)

RM-MEDA-UF9

15

1

05

00 0

2 2446

(t)RM-MEDA-UF9

15

1

05

00 0

2 2446

(u)

RM-MEDA-UF9

15

1

05

00 0

224

46

(v)

MOEAD-AMG-UF9

1

05

00 0

2 24 4

6 6

(w)

05

Ture-PF-UF9

1

1 105 05

00 0

(x)


16 Complexity



MOEAD-FMG (0082)

MOEAD-FMG (0102)

MOEAD-FMG (0122)

























































































































111E+ 00(117Eminus 02)


















































Complexity 17









18 Complexity








Complexity 19



Data Availability




Acknowledgments


References























20 Complexity
































Complexity 21























22 Complexity








10 Complexity








































































































































































































Complexity 11

F1

IGD

val

ues

10ndash1

10ndash2

10ndash3




times105

(a)

F2

IGD

val

ues 10ndash1

100

10ndash2

10ndash3




(b)F3

IGD

val

ues

10ndash1

100

10ndash2

10ndash3




(c)

F4

IGD

val

ues

10ndash1

100

10ndash2

10ndash3




(d)F5

IGD

val

ues

10ndash1

100

10ndash2




(e)

F6

IGD

val

ues

10ndash1

100




(f )

Figure 2 Continued

12 Complexity

F7

IGD

val

ues

10ndash1

100

101

10ndash2

10ndash3




(g)

F9

IGD

val

ues 10ndash1

100

10ndash2

10ndash3




(h)


UF1

IGD

val

ues 10ndash1

100

10ndash2

10ndash3




times105

(a)

UF2

IGD

val

ues

10ndash1

100

10ndash2




(b)UF6

IGD

val

ues

10ndash1

100

101




(c)

UF7

IGD

val

ues

10ndash2

10ndash1

100




(d)

Figure 3 Continued

Complexity 13

UF8

IGD

val

ues

10ndash1

100




(e)

UF9

IGD

val

ues

10ndash2

10ndash1

100




(f )


WFG3

IGD

val

ues

10ndash1

100

10ndash2

times105




(a)

WFG7

IGD

val

ues

10ndash1

100

10ndash2

times105




(b)WFG8

IGD

val

ues

10ndash1

100

times105




(c)

WFG9

IGD

val

ues

10ndash1

100

10ndash2

times105




(d)


14 Complexity

MOEAD-SBX-F4

0

02

04

06

08

1

12

02 04 06 080

(a)

MOEAD-DE-F4

0

02

04

06

08

1

12

02 04 06 08 10

(b)

MOEAD-GM-F4

0

02

04

06

08

1

12

05 1 15 20

(c)

RM-MEDA-F4

0

02

04

06

08

1

02 04 06 08 1 120

(d)IM-MOEA-F4

05 1 15 200

02

04

06

08

1

12

14

(e)

AM-EDA-F4

05 1 15 2 25 300

02

04

06

08

1

(f )

MOEAD-AMG-F4

02

0

04

06

08

1

12

04 0602 08 120 1

(g)

Ture-PF-F4

0

02

04

06

08

1

02 04 06 08 10

(h)MOEAD-SBX-F5

0

02

04

06

08

1

12

02 04 06 08 10

(i)

MOEAD-DE-F5

0

02

04

06

08

1

02 04 06 08 1 120

(j)

0

02

04

06

08

1

12

02 04 06 08 1 120

MOEAD-GM-F5

(k)

RM-MEDA-F4

0

02

04

06

08

1

02 04 06 08 1 120

(l)IM-MOEA-F5

02 04 06 08 1 1200

02

04

06

08

1

(m)

AM-EDA-F5

02 04 06 08 100

02

04

06

08

1

(n)

MOEAD-AMG-F5

02 04 06 08 1 1200

02

04

06

08

1

12

(o)

Ture-PF-F5

0

02

04

06

08

1

02 04 06 08 10

(p)MOEAD-SBX-F9

0

02

04

06

08

1

12

02 04 06 08 1 120

(q)

MOEAD-DE-F9

0

02

04

06

08

1

12

02 04 06 08 1 120

(r)

MOEAD-GM-F9

0

02

04

06

08

1

12

02 04 06 08 1 120

(s)

RM-MEDA-F9

0

02

04

06

08

1

12

02 04 06 08 1 120

(t)IM-MOEA-F9

0

1

2

3

4

5

1 2 3 540

(u)

AM-EDA-F9

02 04 06 08 1 1200

02

04

06

08

1

12

(v)

MOEAD-AMG-F9

02 04 06 08 1 1200

02

04

06

08

1

12

(w)

Ture-PF-F9

0

02

04

06

08

1

02 04 06 08 10

(x)


Complexity 15

MOEAD-SBX-UF2

02

04

06

08

1

12

02 04 06 080

(a)

MOEAD-DE-UF2

0

02

04

06

08

1

12

02 04 06 08 1 120

(b)

MOEAD-GM-UF2

0

02

04

06

08

1

12

02 04 06 08 1 120

(c)

RM-MEDA-UF2

0

02

04

06

08

1

02 04 06 08 10

(d)IM-MOEA-UF2

02 04 06 08 1 1200

02

04

06

08

1

(e)

AM-EDA-UF2

0

02

04

06

08

1

12

02 04 06 08 1 120

(f )

MOEAD-AMG-UF2

02 04 06 08 100

02

04

06

08

1

12

(g)

Ture-PF-UF2

0

02

04

06

08

1

02 04 06 08 10

(h)

2

1

00

12

3

005

115

MOEAD-SBX-UF8

(i)

3

2

1

00 0

2 24 4

MOEAD-DE-UF8

(j)

2

23

1

1

00

MOEAD-GM-UF8

151

050

(k)

1

05

00

24

6

0

12

RM-MEDA-UF8

(l)IM-MOEA-UF8

1

05

00

510 1

050

(m)

AM-EDA-UF8

1

05

00

105

02

4

(n)

MOEAD-AMG-UF8

1

15

15 151 1

05

05 050 00

(o)

Ture-PF-UF8

1

1 1

05

05 05

00 0

(p)MOEAD-SBX-UF9

15

1

05

00 0

1 1

22

3

(q)

MOEAD-DE-UF9

02

46

4

2

00

24

(r)

MOEAD-GM-UF9

0 0

2 2

4 4

15

1

05

0

(s)

RM-MEDA-UF9

15

1

05

00 0

2 2446

(t)RM-MEDA-UF9

15

1

05

00 0

2 2446

(u)

RM-MEDA-UF9

15

1

05

00 0

224

46

(v)

MOEAD-AMG-UF9

1

05

00 0

2 24 4

6 6

(w)

05

Ture-PF-UF9

1

1 105 05

00 0

(x)


16 Complexity



MOEAD-FMG (0082)

MOEAD-FMG (0102)

MOEAD-FMG (0122)

























































































































111E+ 00(117Eminus 02)


















































Complexity 17









18 Complexity








Complexity 19



Data Availability




Acknowledgments


References























20 Complexity
































Complexity 21























22 Complexity








































































































































































































Complexity 11

F1

IGD

val

ues

10ndash1

10ndash2

10ndash3




times105

(a)

F2

IGD

val

ues 10ndash1

100

10ndash2

10ndash3




(b)F3

IGD

val

ues

10ndash1

100

10ndash2

10ndash3




(c)

F4

IGD

val

ues

10ndash1

100

10ndash2

10ndash3




(d)F5

IGD

val

ues

10ndash1

100

10ndash2




(e)

F6

IGD

val

ues

10ndash1

100




(f )

Figure 2 Continued

12 Complexity

F7

IGD

val

ues

10ndash1

100

101

10ndash2

10ndash3




(g)

F9

IGD

val

ues 10ndash1

100

10ndash2

10ndash3




(h)


UF1

IGD

val

ues 10ndash1

100

10ndash2

10ndash3




times105

(a)

UF2

IGD

val

ues

10ndash1

100

10ndash2




(b)UF6

IGD

val

ues

10ndash1

100

101




(c)

UF7

IGD

val

ues

10ndash2

10ndash1

100




(d)

Figure 3 Continued

Complexity 13

UF8

IGD

val

ues

10ndash1

100




(e)

UF9

IGD

val

ues

10ndash2

10ndash1

100




(f )


WFG3

IGD

val

ues

10ndash1

100

10ndash2

times105




(a)

WFG7

IGD

val

ues

10ndash1

100

10ndash2

times105




(b)WFG8

IGD

val

ues

10ndash1

100

times105




(c)

WFG9

IGD

val

ues

10ndash1

100

10ndash2

times105




(d)


14 Complexity

MOEAD-SBX-F4

0

02

04

06

08

1

12

02 04 06 080

(a)

MOEAD-DE-F4

0

02

04

06

08

1

12

02 04 06 08 10

(b)

MOEAD-GM-F4

0

02

04

06

08

1

12

05 1 15 20

(c)

RM-MEDA-F4

0

02

04

06

08

1

02 04 06 08 1 120

(d)IM-MOEA-F4

05 1 15 200

02

04

06

08

1

12

14

(e)

AM-EDA-F4

05 1 15 2 25 300

02

04

06

08

1

(f )

MOEAD-AMG-F4

02

0

04

06

08

1

12

04 0602 08 120 1

(g)

Ture-PF-F4

0

02

04

06

08

1

02 04 06 08 10

(h)MOEAD-SBX-F5

0

02

04

06

08

1

12

02 04 06 08 10

(i)

MOEAD-DE-F5

0

02

04

06

08

1

02 04 06 08 1 120

(j)

0

02

04

06

08

1

12

02 04 06 08 1 120

MOEAD-GM-F5

(k)

RM-MEDA-F4

0

02

04

06

08

1

02 04 06 08 1 120

(l)IM-MOEA-F5

02 04 06 08 1 1200

02

04

06

08

1

(m)

AM-EDA-F5

02 04 06 08 100

02

04

06

08

1

(n)

MOEAD-AMG-F5

02 04 06 08 1 1200

02

04

06

08

1

12

(o)

Ture-PF-F5

0

02

04

06

08

1

02 04 06 08 10

(p)MOEAD-SBX-F9

0

02

04

06

08

1

12

02 04 06 08 1 120

(q)

MOEAD-DE-F9

0

02

04

06

08

1

12

02 04 06 08 1 120

(r)

MOEAD-GM-F9

0

02

04

06

08

1

12

02 04 06 08 1 120

(s)

RM-MEDA-F9

0

02

04

06

08

1

12

02 04 06 08 1 120

(t)IM-MOEA-F9

0

1

2

3

4

5

1 2 3 540

(u)

AM-EDA-F9

02 04 06 08 1 1200

02

04

06

08

1

12

(v)

MOEAD-AMG-F9

02 04 06 08 1 1200

02

04

06

08

1

12

(w)

Ture-PF-F9

0

02

04

06

08

1

02 04 06 08 10

(x)


Complexity 15

MOEAD-SBX-UF2

02

04

06

08

1

12

02 04 06 080

(a)

MOEAD-DE-UF2

0

02

04

06

08

1

12

02 04 06 08 1 120

(b)

MOEAD-GM-UF2

0

02

04

06

08

1

12

02 04 06 08 1 120

(c)

RM-MEDA-UF2

0

02

04

06

08

1

02 04 06 08 10

(d)IM-MOEA-UF2

02 04 06 08 1 1200

02

04

06

08

1

(e)

AM-EDA-UF2

0

02

04

06

08

1

12

02 04 06 08 1 120

(f )

MOEAD-AMG-UF2

02 04 06 08 100

02

04

06

08

1

12

(g)

Ture-PF-UF2

0

02

04

06

08

1

02 04 06 08 10

(h)

2

1

00

12

3

005

115

MOEAD-SBX-UF8

(i)

3

2

1

00 0

2 24 4

MOEAD-DE-UF8

(j)

2

23

1

1

00

MOEAD-GM-UF8

151

050

(k)

1

05

00

24

6

0

12

RM-MEDA-UF8

(l)IM-MOEA-UF8

1

05

00

510 1

050

(m)

AM-EDA-UF8

1

05

00

105

02

4

(n)

MOEAD-AMG-UF8

1

15

15 151 1

05

05 050 00

(o)

Ture-PF-UF8

1

1 1

05

05 05

00 0

(p)MOEAD-SBX-UF9

15

1

05

00 0

1 1

22

3

(q)

MOEAD-DE-UF9

02

46

4

2

00

24

(r)

MOEAD-GM-UF9

0 0

2 2

4 4

15

1

05

0

(s)

RM-MEDA-UF9

15

1

05

00 0

2 2446

(t)RM-MEDA-UF9

15

1

05

00 0

2 2446

(u)

RM-MEDA-UF9

15

1

05

00 0

224

46

(v)

MOEAD-AMG-UF9

1

05

00 0

2 24 4

6 6

(w)

05

Ture-PF-UF9

1

1 105 05

00 0

(x)


16 Complexity



MOEAD-FMG (0082)

MOEAD-FMG (0102)

MOEAD-FMG (0122)

























































































































111E+ 00(117Eminus 02)


















































Complexity 17









18 Complexity








Complexity 19



Data Availability




Acknowledgments


References























20 Complexity
































Complexity 21























22 Complexity

F1

IGD

val

ues

10ndash1

10ndash2

10ndash3




times105

(a)

F2

IGD

val

ues 10ndash1

100

10ndash2

10ndash3




(b)F3

IGD

val

ues

10ndash1

100

10ndash2

10ndash3




(c)

F4

IGD

val

ues

10ndash1

100

10ndash2

10ndash3




(d)F5

IGD

val

ues

10ndash1

100

10ndash2




(e)

F6

IGD

val

ues

10ndash1

100




(f )

Figure 2 Continued

12 Complexity

F7

IGD

val

ues

10ndash1

100

101

10ndash2

10ndash3




(g)

F9

IGD

val

ues 10ndash1

100

10ndash2

10ndash3




(h)


UF1

IGD

val

ues 10ndash1

100

10ndash2

10ndash3




times105

(a)

UF2

IGD

val

ues

10ndash1

100

10ndash2




(b)UF6

IGD

val

ues

10ndash1

100

101




(c)

UF7

IGD

val

ues

10ndash2

10ndash1

100




(d)

Figure 3 Continued

Complexity 13

UF8

IGD

val

ues

10ndash1

100




(e)

UF9

IGD

val

ues

10ndash2

10ndash1

100




(f )


WFG3

IGD

val

ues

10ndash1

100

10ndash2

times105




(a)

WFG7

IGD

val

ues

10ndash1

100

10ndash2

times105




(b)WFG8

IGD

val

ues

10ndash1

100

times105




(c)

WFG9

IGD

val

ues

10ndash1

100

10ndash2

times105




(d)


14 Complexity

MOEAD-SBX-F4

0

02

04

06

08

1

12

02 04 06 080

(a)

MOEAD-DE-F4

0

02

04

06

08

1

12

02 04 06 08 10

(b)

MOEAD-GM-F4

0

02

04

06

08

1

12

05 1 15 20

(c)

RM-MEDA-F4

0

02

04

06

08

1

02 04 06 08 1 120

(d)IM-MOEA-F4

05 1 15 200

02

04

06

08

1

12

14

(e)

AM-EDA-F4

05 1 15 2 25 300

02

04

06

08

1

(f )

MOEAD-AMG-F4

02

0

04

06

08

1

12

04 0602 08 120 1

(g)

Ture-PF-F4

0

02

04

06

08

1

02 04 06 08 10

(h)MOEAD-SBX-F5

0

02

04

06

08

1

12

02 04 06 08 10

(i)

MOEAD-DE-F5

0

02

04

06

08

1

02 04 06 08 1 120

(j)

0

02

04

06

08

1

12

02 04 06 08 1 120

MOEAD-GM-F5

(k)

RM-MEDA-F4

0

02

04

06

08

1

02 04 06 08 1 120

(l)IM-MOEA-F5

02 04 06 08 1 1200

02

04

06

08

1

(m)

AM-EDA-F5

02 04 06 08 100

02

04

06

08

1

(n)

MOEAD-AMG-F5

02 04 06 08 1 1200

02

04

06

08

1

12

(o)

Ture-PF-F5

0

02

04

06

08

1

02 04 06 08 10

(p)MOEAD-SBX-F9

0

02

04

06

08

1

12

02 04 06 08 1 120

(q)

MOEAD-DE-F9

0

02

04

06

08

1

12

02 04 06 08 1 120

(r)

MOEAD-GM-F9

0

02

04

06

08

1

12

02 04 06 08 1 120

(s)

RM-MEDA-F9

0

02

04

06

08

1

12

02 04 06 08 1 120

(t)IM-MOEA-F9

0

1

2

3

4

5

1 2 3 540

(u)

AM-EDA-F9

02 04 06 08 1 1200

02

04

06

08

1

12

(v)

MOEAD-AMG-F9

02 04 06 08 1 1200

02

04

06

08

1

12

(w)

Ture-PF-F9

0

02

04

06

08

1

02 04 06 08 10

(x)


Complexity 15

MOEAD-SBX-UF2

02

04

06

08

1

12

02 04 06 080

(a)

MOEAD-DE-UF2

0

02

04

06

08

1

12

02 04 06 08 1 120

(b)

MOEAD-GM-UF2

0

02

04

06

08

1

12

02 04 06 08 1 120

(c)

RM-MEDA-UF2

0

02

04

06

08

1

02 04 06 08 10

(d)IM-MOEA-UF2

02 04 06 08 1 1200

02

04

06

08

1

(e)

AM-EDA-UF2

0

02

04

06

08

1

12

02 04 06 08 1 120

(f )

MOEAD-AMG-UF2

02 04 06 08 100

02

04

06

08

1

12

(g)

Ture-PF-UF2

0

02

04

06

08

1

02 04 06 08 10

(h)

2

1

00

12

3

005

115

MOEAD-SBX-UF8

(i)

3

2

1

00 0

2 24 4

MOEAD-DE-UF8

(j)

2

23

1

1

00

MOEAD-GM-UF8

151

050

(k)

1

05

00

24

6

0

12

RM-MEDA-UF8

(l)IM-MOEA-UF8

1

05

00

510 1

050

(m)

AM-EDA-UF8

1

05

00

105

02

4

(n)

MOEAD-AMG-UF8

1

15

15 151 1

05

05 050 00

(o)

Ture-PF-UF8

1

1 1

05

05 05

00 0

(p)MOEAD-SBX-UF9

15

1

05

00 0

1 1

22

3

(q)

MOEAD-DE-UF9

02

46

4

2

00

24

(r)

MOEAD-GM-UF9

0 0

2 2

4 4

15

1

05

0

(s)

RM-MEDA-UF9

15

1

05

00 0

2 2446

(t)RM-MEDA-UF9

15

1

05

00 0

2 2446

(u)

RM-MEDA-UF9

15

1

05

00 0

224

46

(v)

MOEAD-AMG-UF9

1

05

00 0

2 24 4

6 6

(w)

05

Ture-PF-UF9

1

1 105 05

00 0

(x)


16 Complexity



MOEAD-FMG (0082)

MOEAD-FMG (0102)

MOEAD-FMG (0122)

























































































































111E+ 00(117Eminus 02)


















































Complexity 17









18 Complexity








Complexity 19



Data Availability




Acknowledgments


References























20 Complexity
































Complexity 21























22 Complexity

F7

IGD

val

ues

10ndash1

100

101

10ndash2

10ndash3




(g)

F9

IGD

val

ues 10ndash1

100

10ndash2

10ndash3




(h)


UF1

IGD

val

ues 10ndash1

100

10ndash2

10ndash3




times105

(a)

UF2

IGD

val

ues

10ndash1

100

10ndash2




(b)UF6

IGD

val

ues

10ndash1

100

101




(c)

UF7

IGD

val

ues

10ndash2

10ndash1

100




(d)

Figure 3 Continued

Complexity 13

UF8

IGD

val

ues

10ndash1

100




(e)

UF9

IGD

val

ues

10ndash2

10ndash1

100




(f )


WFG3

IGD

val

ues

10ndash1

100

10ndash2

times105




(a)

WFG7

IGD

val

ues

10ndash1

100

10ndash2

times105




(b)WFG8

IGD

val

ues

10ndash1

100

times105




(c)

WFG9

IGD

val

ues

10ndash1

100

10ndash2

times105




(d)


14 Complexity

MOEAD-SBX-F4

0

02

04

06

08

1

12

02 04 06 080

(a)

MOEAD-DE-F4

0

02

04

06

08

1

12

02 04 06 08 10

(b)

MOEAD-GM-F4

0

02

04

06

08

1

12

05 1 15 20

(c)

RM-MEDA-F4

0

02

04

06

08

1

02 04 06 08 1 120

(d)IM-MOEA-F4

05 1 15 200

02

04

06

08

1

12

14

(e)

AM-EDA-F4

05 1 15 2 25 300

02

04

06

08

1

(f )

MOEAD-AMG-F4

02

0

04

06

08

1

12

04 0602 08 120 1

(g)

Ture-PF-F4

0

02

04

06

08

1

02 04 06 08 10

(h)MOEAD-SBX-F5

0

02

04

06

08

1

12

02 04 06 08 10

(i)

MOEAD-DE-F5

0

02

04

06

08

1

02 04 06 08 1 120

(j)

0

02

04

06

08

1

12

02 04 06 08 1 120

MOEAD-GM-F5

(k)

RM-MEDA-F4

0

02

04

06

08

1

02 04 06 08 1 120

(l)IM-MOEA-F5

02 04 06 08 1 1200

02

04

06

08

1

(m)

AM-EDA-F5

02 04 06 08 100

02

04

06

08

1

(n)

MOEAD-AMG-F5

02 04 06 08 1 1200

02

04

06

08

1

12

(o)

Ture-PF-F5

0

02

04

06

08

1

02 04 06 08 10

(p)MOEAD-SBX-F9

0

02

04

06

08

1

12

02 04 06 08 1 120

(q)

MOEAD-DE-F9

0

02

04

06

08

1

12

02 04 06 08 1 120

(r)

MOEAD-GM-F9

0

02

04

06

08

1

12

02 04 06 08 1 120

(s)

RM-MEDA-F9

0

02

04

06

08

1

12

02 04 06 08 1 120

(t)IM-MOEA-F9

0

1

2

3

4

5

1 2 3 540

(u)

AM-EDA-F9

02 04 06 08 1 1200

02

04

06

08

1

12

(v)

MOEAD-AMG-F9

02 04 06 08 1 1200

02

04

06

08

1

12

(w)

Ture-PF-F9

0

02

04

06

08

1

02 04 06 08 10

(x)


Complexity 15

MOEAD-SBX-UF2

02

04

06

08

1

12

02 04 06 080

(a)

MOEAD-DE-UF2

0

02

04

06

08

1

12

02 04 06 08 1 120

(b)

MOEAD-GM-UF2

0

02

04

06

08

1

12

02 04 06 08 1 120

(c)

RM-MEDA-UF2

0

02

04

06

08

1

02 04 06 08 10

(d)IM-MOEA-UF2

02 04 06 08 1 1200

02

04

06

08

1

(e)

AM-EDA-UF2

0

02

04

06

08

1

12

02 04 06 08 1 120

(f )

MOEAD-AMG-UF2

02 04 06 08 100

02

04

06

08

1

12

(g)

Ture-PF-UF2

0

02

04

06

08

1

02 04 06 08 10

(h)

2

1

00

12

3

005

115

MOEAD-SBX-UF8

(i)

3

2

1

00 0

2 24 4

MOEAD-DE-UF8

(j)

2

23

1

1

00

MOEAD-GM-UF8

151

050

(k)

1

05

00

24

6

0

12

RM-MEDA-UF8

(l)IM-MOEA-UF8

1

05

00

510 1

050

(m)

AM-EDA-UF8

1

05

00

105

02

4

(n)

MOEAD-AMG-UF8

1

15

15 151 1

05

05 050 00

(o)

Ture-PF-UF8

1

1 1

05

05 05

00 0

(p)MOEAD-SBX-UF9

15

1

05

00 0

1 1

22

3

(q)

MOEAD-DE-UF9

02

46

4

2

00

24

(r)

MOEAD-GM-UF9

0 0

2 2

4 4

15

1

05

0

(s)

RM-MEDA-UF9

15

1

05

00 0

2 2446

(t)RM-MEDA-UF9

15

1

05

00 0

2 2446

(u)

RM-MEDA-UF9

15

1

05

00 0

224

46

(v)

MOEAD-AMG-UF9

1

05

00 0

2 24 4

6 6

(w)

05

Ture-PF-UF9

1

1 105 05

00 0

(x)


16 Complexity



MOEAD-FMG (0082)

MOEAD-FMG (0102)

MOEAD-FMG (0122)

























































































































111E+ 00(117Eminus 02)


















































Complexity 17









18 Complexity








Complexity 19



Data Availability




Acknowledgments


References























20 Complexity
































Complexity 21























22 Complexity

UF8

IGD

val

ues

10ndash1

100




(e)

UF9

IGD

val

ues

10ndash2

10ndash1

100




(f )


WFG3

IGD

val

ues

10ndash1

100

10ndash2

times105




(a)

WFG7

IGD

val

ues

10ndash1

100

10ndash2

times105




(b)WFG8

IGD

val

ues

10ndash1

100

times105




(c)

WFG9

IGD

val

ues

10ndash1

100

10ndash2

times105




(d)


14 Complexity

MOEAD-SBX-F4

0

02

04

06

08

1

12

02 04 06 080

(a)

MOEAD-DE-F4

0

02

04

06

08

1

12

02 04 06 08 10

(b)

MOEAD-GM-F4

0

02

04

06

08

1

12

05 1 15 20

(c)

RM-MEDA-F4

0

02

04

06

08

1

02 04 06 08 1 120

(d)IM-MOEA-F4

05 1 15 200

02

04

06

08

1

12

14

(e)

AM-EDA-F4

05 1 15 2 25 300

02

04

06

08

1

(f )

MOEAD-AMG-F4

02

0

04

06

08

1

12

04 0602 08 120 1

(g)

Ture-PF-F4

0

02

04

06

08

1

02 04 06 08 10

(h)MOEAD-SBX-F5

0

02

04

06

08

1

12

02 04 06 08 10

(i)

MOEAD-DE-F5

0

02

04

06

08

1

02 04 06 08 1 120

(j)

0

02

04

06

08

1

12

02 04 06 08 1 120

MOEAD-GM-F5

(k)

RM-MEDA-F4

0

02

04

06

08

1

02 04 06 08 1 120

(l)IM-MOEA-F5

02 04 06 08 1 1200

02

04

06

08

1

(m)

AM-EDA-F5

02 04 06 08 100

02

04

06

08

1

(n)

MOEAD-AMG-F5

02 04 06 08 1 1200

02

04

06

08

1

12

(o)

Ture-PF-F5

0

02

04

06

08

1

02 04 06 08 10

(p)MOEAD-SBX-F9

0

02

04

06

08

1

12

02 04 06 08 1 120

(q)

MOEAD-DE-F9

0

02

04

06

08

1

12

02 04 06 08 1 120

(r)

MOEAD-GM-F9

0

02

04

06

08

1

12

02 04 06 08 1 120

(s)

RM-MEDA-F9

0

02

04

06

08

1

12

02 04 06 08 1 120

(t)IM-MOEA-F9

0

1

2

3

4

5

1 2 3 540

(u)

AM-EDA-F9

02 04 06 08 1 1200

02

04

06

08

1

12

(v)

MOEAD-AMG-F9

02 04 06 08 1 1200

02

04

06

08

1

12

(w)

Ture-PF-F9

0

02

04

06

08

1

02 04 06 08 10

(x)


Complexity 15

MOEAD-SBX-UF2

02

04

06

08

1

12

02 04 06 080

(a)

MOEAD-DE-UF2

0

02

04

06

08

1

12

02 04 06 08 1 120

(b)

MOEAD-GM-UF2

0

02

04

06

08

1

12

02 04 06 08 1 120

(c)

RM-MEDA-UF2

0

02

04

06

08

1

02 04 06 08 10

(d)IM-MOEA-UF2

02 04 06 08 1 1200

02

04

06

08

1

(e)

AM-EDA-UF2

0

02

04

06

08

1

12

02 04 06 08 1 120

(f )

MOEAD-AMG-UF2

02 04 06 08 100

02

04

06

08

1

12

(g)

Ture-PF-UF2

0

02

04

06

08

1

02 04 06 08 10

(h)

2

1

00

12

3

005

115

MOEAD-SBX-UF8

(i)

3

2

1

00 0

2 24 4

MOEAD-DE-UF8

(j)

2

23

1

1

00

MOEAD-GM-UF8

151

050

(k)

1

05

00

24

6

0

12

RM-MEDA-UF8

(l)IM-MOEA-UF8

1

05

00

510 1

050

(m)

AM-EDA-UF8

1

05

00

105

02

4

(n)

MOEAD-AMG-UF8

1

15

15 151 1

05

05 050 00

(o)

Ture-PF-UF8

1

1 1

05

05 05

00 0

(p)MOEAD-SBX-UF9

15

1

05

00 0

1 1

22

3

(q)

MOEAD-DE-UF9

02

46

4

2

00

24

(r)

MOEAD-GM-UF9

0 0

2 2

4 4

15

1

05

0

(s)

RM-MEDA-UF9

15

1

05

00 0

2 2446

(t)RM-MEDA-UF9

15

1

05

00 0

2 2446

(u)

RM-MEDA-UF9

15

1

05

00 0

224

46

(v)

MOEAD-AMG-UF9

1

05

00 0

2 24 4

6 6

(w)

05

Ture-PF-UF9

1

1 105 05

00 0

(x)


16 Complexity



MOEAD-FMG (0082)

MOEAD-FMG (0102)

MOEAD-FMG (0122)

























































































































111E+ 00(117Eminus 02)


















































Complexity 17









18 Complexity








Complexity 19



Data Availability




Acknowledgments


References























20 Complexity
































Complexity 21























22 Complexity

MOEAD-SBX-F4

0

02

04

06

08

1

12

02 04 06 080

(a)

MOEAD-DE-F4

0

02

04

06

08

1

12

02 04 06 08 10

(b)

MOEAD-GM-F4

0

02

04

06

08

1

12

05 1 15 20

(c)

RM-MEDA-F4

0

02

04

06

08

1

02 04 06 08 1 120

(d)IM-MOEA-F4

05 1 15 200

02

04

06

08

1

12

14

(e)

AM-EDA-F4

05 1 15 2 25 300

02

04

06

08

1

(f )

MOEAD-AMG-F4

02

0

04

06

08

1

12

04 0602 08 120 1

(g)

Ture-PF-F4

0

02

04

06

08

1

02 04 06 08 10

(h)MOEAD-SBX-F5

0

02

04

06

08

1

12

02 04 06 08 10

(i)

MOEAD-DE-F5

0

02

04

06

08

1

02 04 06 08 1 120

(j)

0

02

04

06

08

1

12

02 04 06 08 1 120

MOEAD-GM-F5

(k)

RM-MEDA-F4

0

02

04

06

08

1

02 04 06 08 1 120

(l)IM-MOEA-F5

02 04 06 08 1 1200

02

04

06

08

1

(m)

AM-EDA-F5

02 04 06 08 100

02

04

06

08

1

(n)

MOEAD-AMG-F5

02 04 06 08 1 1200

02

04

06

08

1

12

(o)

Ture-PF-F5

0

02

04

06

08

1

02 04 06 08 10

(p)MOEAD-SBX-F9

0

02

04

06

08

1

12

02 04 06 08 1 120

(q)

MOEAD-DE-F9

0

02

04

06

08

1

12

02 04 06 08 1 120

(r)

MOEAD-GM-F9

0

02

04

06

08

1

12

02 04 06 08 1 120

(s)

RM-MEDA-F9

0

02

04

06

08

1

12

02 04 06 08 1 120

(t)IM-MOEA-F9

0

1

2

3

4

5

1 2 3 540

(u)

AM-EDA-F9

02 04 06 08 1 1200

02

04

06

08

1

12

(v)

MOEAD-AMG-F9

02 04 06 08 1 1200

02

04

06

08

1

12

(w)

Ture-PF-F9

0

02

04

06

08

1

02 04 06 08 10

(x)


Complexity 15

MOEAD-SBX-UF2

02

04

06

08

1

12

02 04 06 080

(a)

MOEAD-DE-UF2

0

02

04

06

08

1

12

02 04 06 08 1 120

(b)

MOEAD-GM-UF2

0

02

04

06

08

1

12

02 04 06 08 1 120

(c)

RM-MEDA-UF2

0

02

04

06

08

1

02 04 06 08 10

(d)IM-MOEA-UF2

02 04 06 08 1 1200

02

04

06

08

1

(e)

AM-EDA-UF2

0

02

04

06

08

1

12

02 04 06 08 1 120

(f )

MOEAD-AMG-UF2

02 04 06 08 100

02

04

06

08

1

12

(g)

Ture-PF-UF2

0

02

04

06

08

1

02 04 06 08 10

(h)

2

1

00

12

3

005

115

MOEAD-SBX-UF8

(i)

3

2

1

00 0

2 24 4

MOEAD-DE-UF8

(j)

2

23

1

1

00

MOEAD-GM-UF8

151

050

(k)

1

05

00

24

6

0

12

RM-MEDA-UF8

(l)IM-MOEA-UF8

1

05

00

510 1

050

(m)

AM-EDA-UF8

1

05

00

105

02

4

(n)

MOEAD-AMG-UF8

1

15

15 151 1

05

05 050 00

(o)

Ture-PF-UF8

1

1 1

05

05 05

00 0

(p)MOEAD-SBX-UF9

15

1

05

00 0

1 1

22

3

(q)

MOEAD-DE-UF9

02

46

4

2

00

24

(r)

MOEAD-GM-UF9

0 0

2 2

4 4

15

1

05

0

(s)

RM-MEDA-UF9

15

1

05

00 0

2 2446

(t)RM-MEDA-UF9

15

1

05

00 0

2 2446

(u)

RM-MEDA-UF9

15

1

05

00 0

224

46

(v)

MOEAD-AMG-UF9

1

05

00 0

2 24 4

6 6

(w)

05

Ture-PF-UF9

1

1 105 05

00 0

(x)


16 Complexity



MOEAD-FMG (0082)

MOEAD-FMG (0102)

MOEAD-FMG (0122)

























































































































111E+ 00(117Eminus 02)


















































Complexity 17









18 Complexity








Complexity 19



Data Availability




Acknowledgments


References























20 Complexity
































Complexity 21























22 Complexity

MOEAD-SBX-UF2

02

04

06

08

1

12

02 04 06 080

(a)

MOEAD-DE-UF2

0

02

04

06

08

1

12

02 04 06 08 1 120

(b)

MOEAD-GM-UF2

0

02

04

06

08

1

12

02 04 06 08 1 120

(c)

RM-MEDA-UF2

0

02

04

06

08

1

02 04 06 08 10

(d)IM-MOEA-UF2

02 04 06 08 1 1200

02

04

06

08

1

(e)

AM-EDA-UF2

0

02

04

06

08

1

12

02 04 06 08 1 120

(f )

MOEAD-AMG-UF2

02 04 06 08 100

02

04

06

08

1

12

(g)

Ture-PF-UF2

0

02

04

06

08

1

02 04 06 08 10

(h)

2

1

00

12

3

005

115

MOEAD-SBX-UF8

(i)

3

2

1

00 0

2 24 4

MOEAD-DE-UF8

(j)

2

23

1

1

00

MOEAD-GM-UF8

151

050

(k)

1

05

00

24

6

0

12

RM-MEDA-UF8

(l)IM-MOEA-UF8

1

05

00

510 1

050

(m)

AM-EDA-UF8

1

05

00

105

02

4

(n)

MOEAD-AMG-UF8

1

15

15 151 1

05

05 050 00

(o)

Ture-PF-UF8

1

1 1

05

05 05

00 0

(p)MOEAD-SBX-UF9

15

1

05

00 0

1 1

22

3

(q)

MOEAD-DE-UF9

02

46

4

2

00

24

(r)

MOEAD-GM-UF9

0 0

2 2

4 4

15

1

05

0

(s)

RM-MEDA-UF9

15

1

05

00 0

2 2446

(t)RM-MEDA-UF9

15

1

05

00 0

2 2446

(u)

RM-MEDA-UF9

15

1

05

00 0

224

46

(v)

MOEAD-AMG-UF9

1

05

00 0

2 24 4

6 6

(w)

05

Ture-PF-UF9

1

1 105 05

00 0

(x)


16 Complexity



MOEAD-FMG (0082)

MOEAD-FMG (0102)

MOEAD-FMG (0122)

























































































































111E+ 00(117Eminus 02)


















































Complexity 17









18 Complexity








Complexity 19



Data Availability




Acknowledgments


References























20 Complexity
































Complexity 21























22 Complexity



MOEAD-FMG (0082)

MOEAD-FMG (0102)

MOEAD-FMG (0122)

























































































































111E+ 00(117Eminus 02)


















































Complexity 17









18 Complexity








Complexity 19



Data Availability




Acknowledgments


References























20 Complexity
































Complexity 21























22 Complexity









18 Complexity








Complexity 19



Data Availability




Acknowledgments


References























20 Complexity
































Complexity 21























22 Complexity








Complexity 19



Data Availability




Acknowledgments


References























20 Complexity
































Complexity 21























22 Complexity



Data Availability




Acknowledgments


References























20 Complexity
































Complexity 21























22 Complexity
































Complexity 21























22 Complexity























22 Complexity

Date post:	24-Jul-2020
Category:	Documents
Upload:	others
View:	5 times
Download:	0 times

Decomposition-Based Multiobjective Evolutionary...

Documents