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Applied Soft Computing 35 (2015) 681–694

Contents lists available at ScienceDirect

Applied Soft Computing

j ourna l h o mepage: www.elsev ier .com/ locate /asoc

clustering-ranking method for many-objective optimization

ei Caia,∗, Shiru Qua, Yuan Yuanb, Xin Yaoc

Northwestern Polytechnical University, Xi’an 710072, ChinaTsinghua University, Beijing 100084, ChinaUniversity of Birmingham, Birmingham B15 2TT, UK

r t i c l e i n f o

rticle history:eceived 11 October 2014eceived in revised form 25 May 2015ccepted 20 June 2015vailable online 8 July 2015

eywords:any-objective optimization

a b s t r a c t

In evolutionary multi-objective optimization, balancing convergence and diversity remains a challengeand especially for many-objective (three or more objectives) optimization problems (MaOPs). To improveconvergence and diversity for MaOPs, we propose a new approach: clustering-ranking evolutionaryalgorithm (crEA), where the two procedures (clustering and ranking) are implemented sequentially. Clus-tering incorporates the recently proposed non-dominated sorting genetic algorithm III (NSGA-III), usinga series of reference lines as the cluster centroid. The solutions are ranked according to the fitness value,which is considered to be the degree of closeness to the true Pareto front. An environmental selection

lusteringiversityankingonvergence

operation is performed on every cluster to promote both convergence and diversity. The proposed algo-rithm has been tested extensively on nine widely used benchmark problems from the walking fish group(WFG) as well as combinatorial travelling salesman problem (TSP). An extensive comparison with sixstate-of-the-art algorithms indicates that the proposed crEA is capable of finding a better approximatedand distributed solution set.

© 2015 Elsevier B.V. All rights reserved.

. Introduction

Many-objective optimization problems (MaOPs) have attractedreat attention in the last few decades, due to the large number ofeal-world applications that have many objectives [1]. In particular,here are many industrial and engineering design problems thatequire more than three objectives to be maximized or minimized.or example, there are control system design problems with 4–10bjectives [2], and software engineering problems with up to 15bjectives [3].

It is commonly accepted that, compared with multi-objectiveptimization problems (MOPs), MaOPs refer to problems with high-imensional objectives (in general more than three) [4]. However,hey both aim to find a set of Pareto optimal solutions, which areefined as the best trade-off among objectives. When plotted inhe objective space, the set of all the Pareto optimal solutions iseferred to as the Pareto front (PF).

Evolutionary algorithms (EAs) are well suited for MOPs, dueo their population based strategy for achieving an approxima-

ion to the PF. In general, EAs achieve a Pareto approximationet in MOPs via pursing two goals – approximating the wholeF and maximizing the diversity of solutions. Fundamentally, the

∗ Corresponding author. Tel.: +86 15991611412.E-mail address: caileid@gmail.com (L. Cai).

ttp://dx.doi.org/10.1016/j.asoc.2015.06.020568-4946/© 2015 Elsevier B.V. All rights reserved.

balance between convergence and diversity in EAs depends on theselection operator used by the multi-objective evolutionary algo-rithm (MOEA). There is a significant interest in expending MOEAinto MaOPs. These methods can be broadly classified by their selec-tion strategies into three categories: dominance, indicator, anddecomposition-based MOEAs [5].

Dominance-based algorithms, such as non-dominated sortinggenetic algorithm II (NSGA-II) [6], strength Pareto evolutionaryalgorithm 2 (SPEA2) [7] and Pareto envelope based selectionalgorithm II (PESA-II) [8], are very effective for low dimensionalobjectives[48], but lose performance on MaOPs [9]. As empiri-cally shown in the existing literature [1,10,11], the non-dominatedobjective vectors in each population become very large and mostindividuals become non-dominated. Thus, these algorithms fail togenerate sufficient selection pressure to drive solutions towardsthe PF. To overcome the drawbacks of Pareto dominance basedMOEAs and make the approach suitable for MaOPs, two new meth-ods are introduced: (1) new preference relations; (2) new diversitypromotion mechanisms.

To derive new preference relations, it is natural to utilize otherpreference relations to tackle the challenge of too many non-dominated solutions. Some efforts in this direction have been

attempted, such as preference order ranking [12], �-dominance[13], fuzzy Pareto dominance [14] and grid dominance [15]. Theunderlying concept of these methods is to utilize a relaxed formof non-dominated relationship to increase the selective pressure

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82 L. Cai et al. / Applied Soft

owards the PF, and where possible simultaneously improve theiversity of the obtained solutions. Although existing work dependsn several sensitive parameters (e.g. the relaxed parameter � in �-ominance [5]), they can provide some new alternatives for MaOPs.

To develop new diversity mechanisms, control strategies aremplemented to ensure that the obtained solutions spread throughhe entire PF uniformly [16]. Several efforts have been made toncorporate this approach. SPEA2 combines strength values withhe kth nearest neighbour method to distinguish individuals in theopulation [7]. It can emphasize the diversity, but the acquiredolutions may be distant from the true PF. The grid based diver-ity maintenance mechanism (hyperbox) is utilized in PESA-II athe mating and environmental selection step [8]. Li et al. pro-osed a new density estimation method [17], in which a simplehift-based density estimation strategy was used to modify andmprove the performance of the NSGA-II, SPEA2, and PESA-II witharying degrees of success. Recently, NSGA-III [18], which uses therocess of reference line generation, was proposed. The crowdingistance operator from NSGA-II is replaced by a clustering oper-tor, which associates population members with fixed clusteringentroids (reference lines). This is a very promising alternative foraOPs.For decomposition-based methods, a series of easily described

nd well scaled single-objective problems are utilized in place ofn m-objective optimization. The core concept is to decomposehe MOP into a series of scalar objective optimization problems19], based on conventional aggregation approaches. The most typ-cal method is the multi-objective evolutionary algorithm basedn decomposition (MOEA/D) [20,21]. It has not only has a highearch ability for continuous optimization, but also performs welln problems with complex Pareto sets [10]. MOEA/D can alsoe considered an aggregation based algorithm since the MOPbjectives in MOEA/D are aggregated by a scaling function [17].ecently, a number of improved versions of MOEA/D have beenuggested, mainly focusing on neighbourhood structures [22,23],elf-adaptation mechanisms [22], and parallelism.

The indicator-based approach introduces a single indicator as anvaluation of performance to guide the process of evolution, whichs equivalent to finding the Pareto optimal set [24]. The core concept

as originally from Zizler’s indicator based evolutionary algorithmIBEA) [25]. Other attempts based on this concept include the S met-ic selection evolutionary multi-objective algorithm (SMS-EMOA)26] and multi-objective covariance matrix adaptation evolutiontrategy (MO-CMA-ES) [27]. Despite its advantages and clear theo-etical properties, the computational cost increases considerablyith increasing number of objectives. To relieve computational

omplexity, other related methods have been proposed. Somencluded the use of more efficient ways [28] to speed up this process29], while others attempt to introduce new performance indica-ors, e.g. R2 method [30,31], promoting the performance of MOEAs.

The core concept underlying these algorithms is to balance theonvergence and diversity in the whole process of EAs. The algo-ithms try to select solutions which are closer to the true PF andave a high diversity among each other. Based on this principle and

nspired by NSGA-III and decomposition based MOEAs, we intro-uce two operators. One is a clustering operator takeing advantagef the diversity promotion mechanism used in NSGA-III, while theecond is a ranking scheme to push the population towards the trueF. To balance those two aspects of performance, we propose a newelection process that incorporates the clustering and iterativelyelects a suitable solution via the ranking metric. The performancef this method is compared with six state-of-the-art algorithms on

series of test problems.The background of our research, including related work and

otivation, is presented in Section 2. In Section 3, we present theetails of the proposed algorithm. In Section 4, we describe the

ting 35 (2015) 681–694

experimental design to compare the proposed algorithm with peermethods. The comparison results and discussions are in Section 5.Finally, in Section 6 we make our conclusions about the proposedalgorithm.

2. Related work and motivation

The basic purpose of MaOPs is to optimize a problem with mobjective functions F(x) = (f1(x), f2(x), . . ., fm(x)), where x ∈ � is thedecision space, and x = (x1, . . ., xn)T is a candidate solution; F : � →R

m consists of m objective functions, which is a mapping from n-dimensional decision space � to objective space R

m. The dominancerelation is defined by x ≺ y,

x ≺ y :⇔ ∀i ∈ (1, . . ., m) : fi(x) ≤ fi(y); ∃j ∈ (1, . . ., m) : fj(x) < fj(y).

A decision vector, x∗ ∈ �, is Pareto optimal if there is no x ∈ �such that x ≺ x∗. The set of all Pareto optimal points is called a ParetoSet (PS) and the Pareto front (PF) is defined as {f(x) ∈ R

m|x ∈ PS}. Forreference-based algorithms, the ideal point z* is defined as a vectorz∗ = (z∗1, . . ., z∗m)T , i.e., ∀i ∈ (1, . . ., m), z∗

i= inf({fi(x)|x ∈ �}).

As identified earlier, the main task of many-objective optimiza-tion can be divided into two aspects: (1) maximize the diversityof the generated solutions, and (2) minimize the distance fromsolutions to the PF.

Diversity promotion is important to MOEAs to ensure a goodapproximation to the Pareto optimal solutions. The crowding dis-tance, which serves as a summation of objective-wise distanceamong population, is adopted in the last accepted non-dominatedlevel in NSGA-II [6] for diversity enhancing. The cluster analy-sis operator, first applied by Morse [32], reduces the number ofnon-dominated solutions, and can also be considered as a diver-sity maintenance mechanism. Following Morse’s work, SPEA [33]can be regarded as the most successful algorithm to reduce the PSusing clustering. In the environmental selection of SPEA, when thenumber of solutions in an archive set is excessive, cluster analysispartitions m elements in the archive set into pre-defined numberof groups, n, via the average linkage method [32], and excludes thefurther archived members. In principle, the clustering techniqueused in SPEA implements a truncation method as well as ensur-ing diversity. However, this kind of clustering based method hasbeen replaced by nearest neighbour density estimation in SPEA2[7] since it is not strongly elite preserving and can be very timeconsuming. Although SPEA performs unsatisfactorily due to com-puting complexity, it does indicate that clustering based methodsmay be an outstanding way to promote diversity.

NSGA-III [18] has been recently proposed, partly utilizing theclustering mechanism. Compared with the original NSGA-II [6],NSGA-III has significant modification in the selection mechanism,especially the maintenance of diversity. The main procedure ofNSGA-III is described as follows.

Suppose the size of the parent population, Pt, is N. Initially, aseries of reference points are generated by a systematic approach[34]. Thereafter, similar to NSGA-II, the offspring population, Qt,will be produced from Pt with some operations, including randomselection, simulated binary crossover (SBX), and polynomial muta-tion [35]. A selection mechanism is employed in the combinedpopulation Rt = Pt ∪ Qt to exclude the extra solutions. Specifically,Rt is firstly sorted by the method of fast-non-dominated sor-ting, and then categorized into different non-dominated levels(F0, F1, . . ., Fl). The new population Pt+1 is constructed by selectingindividuals from F0 to Fl , until |Pt+1| = N. It is worth noting that not

all the individuals would be selected into Pt+1 due to |Pt ∪ Qt| = 2N.To further eliminate the extra solutions, another selection mech-anism is also introduced. In NSGA-II, crowding distance was usedto maintain a distribution, and the solutions with larger crowding

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istance values have a higher priority to be chosen. In contrast,SGA-III replaces the crowding distance operator by a clustering-

ike operator. Before that, the objectives and the previous suppliedeference points are first normalized so that they can have an iden-ical range. A perpendicular distance is calculated between everyormalized solution and a reference line. This distance, which cane considered as a metric for similarity, is utilized to associate eachember in population Pt+1 with a reference line.A niche-preservation operation is applied for incremental selec-

ion and operates on the individuals from (Pt+1\Fl). To summarizehis procedure, �j is defined as the niche count for the jth referenceoint, and �j is the number of members in the front, Fl , alreadyssociated with the reference point j. The niche preservation startsrom finding the reference point set Jmin = {j : argmin

j�j} with min-

mum �j, and one j ∈ Jmin is randomly chosen. The details of therocedure are shown below.

If �j = 0, then there is no correspond Rt member associated with

the reference point j. This situation can be divided into two cases:– If �j /= 0, the member with the shortest perpendicular distance

from the reference line is added to Pt+1. The count of �j is thenincreased by one.

– if �j = 0, the current reference point is filtered.If �j ≥ 1, then there is more than one member associated with thereference point in Pt+1\Fl . In that case, a randomly chosen mem-ber from Fl that is associated with the \bar{j}-th reference pointis added to Pt+1, and then the count of �j also needs increasing byone.

Critically, the selection mechanism depends on some points on hyper-plane. The reference line, which is joins from the origino each of these reference points, serves as the centroid of cluster-ng. This clustering scheme not only has satisfactory diversity inbjective space, but also has relatively low computing complexity.he worse-case complexity in one generation is O(N2logm−1N) or(mN2) [18].

Since the set of uniformly distributed reference points are ableo naturally maintain the diversity, it is also employed in ourroposed algorithm, but without the niching procedure. The pre-efined reference weights are reused to maintain convergence.he convergence mechanism of our algorithm arises largely fromOEA/D, which decomposes a MOP into N scalar optimization sub-

roblems simultaneously (where N is the size of the population).hus, the whole PF is transformed into a series of scalar opti-ization problems [20]. The Tchebycheff function, as a weighted

ggregation function, is defined in advance and a MOP can beecomposed into N sub-problems as follows:

te(x|w, z∗) = mmaxi=1{wi|fi(x) − z∗i |}, (1)

here z∗ = (z∗1, z∗2, . . ., z∗m) is the ideal point, and w =w1, w2, . . ., wm)T is the weight vector. This forces sub-problemsith similar weight vectors to have optimal solutions close to

ach other. In MOEA/D, the replacement mechanism is criticalo the whole algorithm. For each neighbouring solution, xu iseplaced by an offspring, y only if gte(y|wu, z*) < gte(xu|wu, z*). Eq.1) indicates the ability of convergence. Since the optimal solutiono Eq. (1) is a Pareto optimal solution to the MOP [36,37], MOEA/Das a strong ability to enhance the selection pressure towardsF.

MOEA/D’s diversity mechanism depends on the nature of itsub-problems, but is too weak to maintain diversity compared withOEA/D’s convergence ability. The clustering mechanism would

nhance the diversity of solutions, but Pareto dominance may not

ting 35 (2015) 681–694 683

strengthen the selection pressure towards the true PF. An ideal wayto deal with MaOPs would be to maintain convergence and diver-sity simultaneously. Motivated by the clustering mechanism andstrong convergence ability of MOEA/D, we propose a clustering-ranking evolutionary algorithm (crEA) for MaOPs, which attemptsto balance diversity and convergence in every generation.

3. Proposed algorithm

3.1. Framework of the proposed clustering-ranking algorithm

The framework of the proposed algorithm is summarized inAlgorithm 1. The reference points � = {�1, �2, . . ., �N}, which areregarded as the preparation for clustering, need to be initialized.Then, following the basic procedure of EAs, the initial population,P0, with N individuals is generated. The ideal point z*, which is nec-essary for normalization, is updated in step 3. Steps 5–10 showthe main evolution procedure for iterative optimization. In steps6 and 7, the conventional method of recombination and mutationare introduced on the individuals to produce N members, whichare considered the input for the UpdateIdealPoint in step 9. We thennormalize all 2N individuals via AdptiveNormalize in step 10. Clus-tering is used to split Rt into Nc clusters C = {C1, C2, . . ., CNc } (Step11). Ranking is then employed to classify Rt into different layersF = {F1, F2, . . ., FNf

}. Finally, after all the solutions in Rt have beenassigned to different layers, the selection operator is employed toselect N new individuals for the next generation.

Algorithm 1. The framework of proposed algorithmRequire: H (The number of divisions)

1: � ← ReferencePoints(H)2: P0← InitializePopulation()3: z*← InitializeIdealPoint(P0)4: t ← 15: whiletermination criterion not fulfilled do6: Pt← Recombination(P0)7: Qt← Mutation(Pt)8: Rt← Pt ∪ Qt

9: z*← UpdateIdealPoint(Qt)10: Rt← AdptiveNormalize(Rt , z*)11: (C, Nf ) ← clustering(Rt, �)12: F ← ranking(C, Nf )13: Pt+1 ← selection(F)14: t ← t + 115: end while

3.2. Reference points generation

To emphasize the diversity of the obtained solutions, a clusteroperator is employed. A structured set of reference points need tobe generated. The basic requirement is to create a set of uniformlydistributed units in the objective space. In this work, Das’s methodis adopted [34], with a user-defined integer H which controls thedivisions along each axis.

m∑i=1

xi = H, xi ∈ N (2)

Geometrically, the formulation could be considered as a hyper-plane in m-dimensional space. The total number of such vectors isN = Cm−1

H+m−1, since xi ∈ N. Finally, the evenly distributed referencepoints, �, are generated according to

�k =xk , k = 1, 2, 3. . ., m (3)

H

The predefined reference points are generated in the normal-ized hyperplane ensuring

∑mi=1fi = 1. Taking the three-dimension

problem as an example, there are C25 = 10 reference points on a

684 L. Cai et al. / Applied Soft Computing 35 (2015) 681–694

00.2

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ig. 1. The reference points (square points) in the three-dimensional objective spaceith H = 3.

riangle if H = 3. Fig. 1 illustrates the reference points for H = 3, andhe details of generator is summarized in Algorithm 2.

lgorithm 2. Reference point generatorRequire: H (The number of divisions)

1: Nr ← Cm−1H+m−1

2: {�1, �2, . . ., �Nr } ← {∅, ∅, . . ., ∅}3: for i = 1 : Nr do4: forj = 1 : m do5: �i[j] ← ReferenceGenerator(j) // Refer to Eq. (3)6: end for7: end for

.3. Adaptive normalization

Normalization would be beneficial to enhance the robustness ofhe algorithm, especially for scaled optimization problems [18,20].n NSGA-III, normalization is based on the extreme point in eachbjective axis. However, in practice, the calculation of the hyper-lane relies on solving a linear system of equations [18], which

s computational expensive. We simplify this procedure using aommon method of normalization. For a solution x, the normalizedbjective value is defined as

j(x) =fj(x) − z∗

j

zmaxj− z∗

j

j = 1, 2, . . ., m, (4)

here zmaxj

is the maximum value for every objective fj in the pop-

lation. Given the current population, Pt, and an ideal point, z*, werst initialize the maximum vector zmax = (zmax

1 , zmax2 , . . ., zmax

m ),hen obtain the normalized objective value fj using z*, zmax, andj(x). Algorithm 3 shows the pseudocode for normalization.

lgorithm 3. Adaptive normalizationRequire: Rt (The current population), z* (The ideal point), m (The number

of objective function)1: zmax← { − ∞ , − ∞ , . . ., − ∞ }2: for each solution x in Rt do3: for j = 1 : m do4: if fj(x) > zmax

jthen

5: zmaxi← fj(x)

6: end if7: end for8: end for9: foreach solution x in Rt do10: forj = 1 : m do

11: fj(x) ← Normalize(fj(x), zmax

j, z∗

j) //Refer to Eq. (4)

12: end for13: end for

Fig. 2. An example of distance for clustering in two-dimensional objective space.d�1,2 is the Euclidean distance from the solution to the reference line.

3.4. Clustering operator

The implementation of our proposed algorithm consists of twomain operators: clustering and ranking, which are implementedsequentially for diversity and convergence. In terms of clustering,after some naturally different � have been initialized, we need toassociate every solution on Rt with a reference line.

Algorithm 4. Clustering operatorRequire: Rt (The current population), � (The reference points set)

1: Nc← |�|, Nf← 02: {C1, C2, . . ., CNc } ← (∅, ∅, . . ., ∅)3: foreach solution x in Rt do4: index ← 05: dmin← + ∞6: n ← 17: for j = 1 : |�| do8: d ← ComputeDistance(f(x), �j) //Refer to Eq. (6)9: if d < dmin then10: dmin← d11: index ← j12: end if13: end for14: Cindex← Cindex ∪ {x}15: if Nf < |Cindex| then16: Nf← |Cindex|17: end if18: end for

Let us consider the perpendicular distance. For clustering, theperpendicular distance, dr,2(x), is defined in Eq. (5) to evaluate the

similarity. Suppose that f(x) = (f1(x), f2(x), . . ., ˜fm(x))T

is the nor-malized objective vector, then

dr,2 =∥∥∥∥f(x) − dr,1(x)

�i

||�i||

∥∥∥∥ , (5)

where dr,1(x) is the distance between origin and the projectionpoint, p; and dr,1(x) = ||f(x) · �i||/||�i||. The illustration of this dis-tance is shown in Fig. 2.

The final distance for clustering is computed from Eq. (6). Theith solution with minimum distance to the jth reference line canbe categorized to the cluster Cj. Algorithm 4 summarizes the mainprocedure of clustering. Each solution in the current population canbe associated with a certain reference line.

Taking Fig. 3 as an example, the distances between solution

E and two reference lines are d

�1,2 = 0.0894 and d�2,2 = 0.4919.

Solution E should associate with �1 since dE�1,2 < dE

�2,2. Similarly,

we can assign solution A and B to the reference �1 with the

L. Cai et al. / Applied Soft Computing 35 (2015) 681–694 685

Fig. 3. An example of the cluster operator in two-dimensional objective space. Thesta

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Fig. 4. An example of ranking procedure on two-dimensional objective spaces. Dif-ferent shapes (circle and triangle points) indicate different clusters. The fitness

olutions A, B, and E have been associated with reference �1 . (For interpretation ofhe references to colour in the text, the reader is referred to the web version of therticle.)

inimum dr,2 value (dA�1,2 = dB

�1,2 = 0.0894, dA�2,2 = 0.5814, and

B�2,2 = 0.3578).

r,2 =∥∥∥∥f(x) − �i||f(x) · �i||

‖�i‖2

∥∥∥∥ (6)

.5. Ranking operator

The second procedure is to rank the solutions in every clustero promote convergence. As we mentioned above, the fitness valueor ranking comes from the aggregation function, which is defineds Eq. (7).

lgorithm 5. Ranking operatorRequire: {C1, C2, . . ., CNc } (The clustered solutions), Nf (The number of layers)

1: {F1, F2, . . ., FNf} ← {∅, ∅, . . ., ∅}

2: for i = 1 : Nc do3: forj = 1 : |Ci| do4: � i← � i ∪ fitness(Ci[j], �i) // Refer to Eq. (7)5: end for6: Ci← AscendingSort(Ci , � i)7: end for8: for i = 1 : Nf do9: for j = 1 : Nc do10: Fj ← Fj ∪ Cj[i]11: end for12: end for

T(x) = mmaxk=1

{1�k

(fk(x))}

(7)

Lower values of FT(x) tend to have a high probability to approx-mate true PF. Geometrically, this metric can serve as a contour linelong with the reference vector. To better understand this value,onsider the example in Fig. 3. For individuals A and B, their fitnessre

TA = max{

0.30.5

,0.81

}= 0.8, and FTB = max

{0.40.5

,0.61

}= 0.8,

espectively, and so FTA = FTB. Moreover the fitness value for indi-idual E is

TE = max{

0.50.5

,0.81

}= 1.

As shown in Fig. 3, with the same fitness value, the solution A

nd B are located at same contour line along with �1 (the secondlue dashed line in Fig. 3). The contour of solution A and B are closero the true PF compared with the contour of solution E (the thirdlue dashed line).

values of solutions are represented by the size of the shape. Those solutions canbe ranked to three layers via their fitness value.

In practical use, FT(x) is regarded as a criterion to rank the solu-tions in each cluster. We first arrange the current solutions in thecluster to different layers using the ranking operator, which sortsthe solutions in ascending order by their fitness. After the first iter-ative ranking step, the first layer of solutions, F1, is composed ofindividuals with the lowest fitness in the cluster Ci. Then, the rank-ing operator continues to construct the next layer in each cluster,and the solution with next-lowest fitness will be labelled with theF2. The main processes of fitness calculation and ranking are shownin Algorithm 5.

3.6. Selection operator

Algorithm 6. Selection operatorRequire: {F1, F2, . . ., FNf

}1: Pt+1← ∅2: for i = 1 : Nf do3: if |Fi| = min(|Fi|, (N − |Pt+1|) then4: Pt+1 ← Pt+1 ∪ Fi

5: else6: Shuffle(Fi)7: Pt+1 ← Pt+1 ∪ Fi[1 : (N − |Pt+1|)]8: end if9: end for

Finally, the selection operator is introduced to the solution set.In principle, solutions from lower levels of a ranked set and locatedat different clusters are preferred choices. In other words, the selec-tion procedure is operated from F1 to higher levels until the numberof selected solutions equals the population size (i.e., |Pt+1| = Nc).Algorithm 6 describes the main steps of the selection operator. It isworth noting that, in the last accepted level, the order of solutionsneed to be shuffled (step 6 in Algorithm 6), which means that if weneed to select the rest of solutions in the last accepted level, wemay just pick them randomly.

To better understand the selection operator, considering the 2-objective minimization instance in Fig. 4 as an example. There aresix individuals in the objective space. The distance between everyindividual solution’s position and reference line is calculated byEq. (6). After comparison, solutions a, b, and c are assigned to onecluster C1 = {a, b, c}, whereas d, e, and f are categorized to anothercluster C2 = {d, e, f}. Fitness is computed for every individual, andsorted (ascending) in each cluster. Based on the fitness, a and d areassigned to layer F ; layer F contains the individuals b and e; and

1 2c and f belong to layer F3. After the cooperation of clustering andranking, solutions labelled with lower layer and divergent clusterswill have high convergence as well as diversity. Following the first

6 Computing 35 (2015) 681–694

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Table 1General parameters.

Algorithms

Population N≈120Maximum number of fitness evaluations 100,000Crossover operator pc=0.9, �c=30Mutation operator pm = 1/n, �m=20

WFGObjectives m = 4:10Position parameter k = m − 1Distance parameter l = 25 − m

Objectives m = 6, 8, 10

86 L. Cai et al. / Applied Soft

terative selection, individuals a and d are selected into the archiveet. The procedure of selection continues until |Pt+1| = N.

.7. Computational complexity analysis

We analyse the computational complexity of the proposed algo-ithm, taking an MOP with m objectives and the algorithm with

individuals as an example. From Algorithm 1, the generic oper-tors in steps 6–8 require O(nN) computations. The updating ofdeal points (step 9) requires O(mN) computations. The normal-zation of the population (Step 10) requires O(mN) computations.he clustering and ranking procedures, which are regarded as theey operators for our proposed method, require O(mN2) and O(N2)omputations, respectively. Thus, the total complexity of our pro-osed crEA approximates to O(mN2), which is the same as that ofOEA/D [20].

. Experimental design

We detail several experiments to analyse the performance of theroposed method compare with other six state of the art MOEAs.

.1. Test sets

To test the effectiveness of the proposed algorithm, we usecalable test functions for simulation. We adopt an artificially con-tructed problems of walking fish group (WFG) [38], which areopular in the study of MOEAs. The WFG test set covers ninecalable, multi or many-objective test problems. They are easy toescribe and implement and suitable to comprehensively evaluatehe performance of optimization algorithms.

A wide range of 9 problems are defined for users to test thebility of a proposed algorithm in different situations. For betterescription and implementation, the exact Pareto-optimal surfacesf this test suit are given. These Pareto optimal fronts are definedy many kinds of shape functions, including linear, mixed, multi-odal, disconnected, and degenerate (refer to [38] for more details

f WFG problems). Before commencing, the parameters of WFGust be assigned. In this work, the position-related parameter, k,as set to m − 1, and the distance-related parameter, l, was set to

5 − m.To test the performance of crEA for solving combinatorial opti-

ization problems, we considered the multi-objective travellingalesman problem (TSP) [39], which is briefly discussed below, as

supplementary method.The basic purpose of TSP is to identify the set of optimal Hamil-

onian cycles on a network L = (V, C), where V = {v1, v2, . . ., vN},nd C = {ck : k ∈ {1, 2, . . ., m}} are sets of K nodes and cost matri-es between nodes, respectively. From [39], the m cost objectivesan be generated. For each matrix, ck, which is constructed by therevious matrix, ck−1,

k(i, j) = TSPcp × ck−1(i, j) + (1 − TSPcp) × rand, (8)

here ck(i, j) is the cost value between the pair of node i and in matrix ck, and TSPcp ∈ (−1, 1) acts as correlation parameter.t is important to highlight that the first matrix, c1, is generatedy assigning each pair of nodes with a uniform random numberetween 0 and 1.

The parameter settings of TSP used in this study are the same ashose adopted in [17] except for the objectives. The characteristicsf TSP are shown in Table 1.

.2. Performance metrics

To compare the proposed algorithm with other selected algo-ithms, we adopt the generational distance (GD) rather than

TSP Correlation parameter TSPcp = −0.2, 0, 0.2Number of cities cN = 30

well-known inverted generational distance (IGD) [40]. The calcula-tion of IGD depends on the sampled points on the PF. Although theexact Pareto-optimal surface is given, it is still difficult to samplethose points on the high dimensional objective space and guaran-tee uniform distribution. Furthermore, it is difficult to confirm thesuitable number of sampled points for our evaluation, especially forhigh dimensional objective space. In contrast, the GD metric [41]relies on the minimum distance from solutions to the true PF. It canbe defined as

IGD(S) = 1|S|

√√√√|S|∑

i=1

d2i

(9)

Let S be the obtained solution set approximation to the PF inthe objective space. Then GD provides a metric of how close S isto the true PF, with a lower value preferable. It is relatively sim-ple to implement this metric when the true PF surface is regular,e.g. WFG4 to WFG9 problems. In those problems, the PF surfaceis concave, and we can easily acquire this distance by calculatingthe distance from solution to a concave surface in objective space.Thus, the GD metric will be utilized to evaluate the performance ofconvergence for all algorithms on WFG4-WFG9 test sets.

To obtain further results and make the experiment more com-prehensive, we use the hypervolume (HV) to measure the propertyof the algorithms. HV is a sum of all the areas bounded by referencepoints, whose value expresses both diversity and convergence.

HV(S) = {⋃

volume(v, r)|v ∈ S, r ∈ r}, (10)

where r denotes reference points dominated by all Pareto-optimalobjective vectors. For problems of WFG 1–9, we set the referencepoint as (3, 5, . . ., 2m + 1)T. Since the exact range of the Pareto frontfor TSP is unknown, according to [17], we adopt the point with 22for each objective as the reference point and use it for normaliza-tion. Finally, the value of HV is calculated using While’s method [29].A larger HV is preferable, indicating a better approximation to thetrue PF and better diversity.

4.3. Other algorithms compared

Our proposed crEA is compared with six state-of-the-artMOEAs: NSGA-III, MOEA/D, GrEA, �-MOEA, SPEA2SDE and HypE.We give a briefly review of each algorithm and describe theirparameter settings.

4.3.1. NSGA-IIINSGA-III [18] was developed under the framework of NSGA-II

[6] and is based on the mechanism of reference weights, which

emphasizes population members to a well-distributed state. Theniche-preservation operation was introduced as a selection oper-ator to identify the suitable solutions of the next generation. Itcan find a well-converged and well-diversified set of solutions

Computing 35 (2015) 681–694 687

ro

4

agopo

4

(cgtpdti

4

omdstt�

4

TraewtslnSi

4

apiiw

4

sT

t(ttt

Table 2Population size depends on the number of objectives in different problems.

No. of objectives Divisions (H) Population size (N)

4 7 C310 = 120

5 5 C49 = 126

6 4 C59 = 126

7 3 C69 = 84

8 3 C710 = 120

parameter setting of �div was acquired. These are shown in Table 3,and Fig. 5 illustrates the effects of different divisions for differentinstances. The ideal �div for WFG and TSP lies between 5 and 21.

Table 3Parameters � for WFG problems.

Problems Obj. �div

WFG1 4,5,6,7,8,9,10 5,14,19,19,12,15,15WFG2 4,5,6,7,8,9,10 18,16,14,13,15,17,17WFG3 4,5,6,7,8,9,10 11,14,13,11,12,14,14WFG4 4,5,6,7,8,9,10 11,10,11,8,9,10,11WFG5 4,5,6,7,8,9,10 10,11,11,10,12,12,14WFG6 4,5,6,7,8,9,10 11,11,10,11,11,11,14WFG7 4,5,6,7,8,9,10 11,10,9,8,9,12,14WFG8 4,5,6,7,8,9,10 10,12,11,8,9,10,11

L. Cai et al. / Applied Soft

epeatedly. This is an efficient algorithm to find a set of Pareto-ptimal solutions, and especially suitable for the MaOPs.

.3.2. MOEA/DMOEA/D [20] can be categorized as a decomposition based

pproach. To initialize, a number of uniform weight vectors areenerated, and the original problem is transformed into a seriesf single-objective problems. This mechanism makes MOEA/Derform very well in multi-objective as well as many-objectiveptimization problems [10].

.3.3. GrEAThe recently proposed grid-based evolutionary algorithm

GrEA) [15] is a new type of dominance-based MOEA. The coreoncept of this algorithm is to utilize a grid dominance relation touide the whole process of evolution. Similar to �-dominance rela-ionship, the grid dominance extends the dominance relationship,roviding a higher selection pressure as well as emphasizing theiversity. A fitness adjustment strategy is also proposed, to guaran-ee that gird-dominated individuals have a possibility to be selectednto archive set.

.3.4. �-MOEAAlthough �-MOEA [42] is originally proposed for multi-

bjective optimization, it has been shown that it is suitable forany-objective optimization [5]. �-MOEA first introduces the �-

ominance concept into MOEA, which can divide the objectivepace into a number of hyperboxes. The purpose is to transformhe domination relationship from a traditional one to a domina-ion based on the hyperbox with a size of �. From the core concept,-MOEA can be classified as dominance based MOEA.

.3.5. SPEA2SDESPEA2 [7] is an evolution from the cluster-based SPEA [33].

he fitness assignment scheme and cluster mechanism in SPEA areeplaced by an improved dominate fitness evaluation scheme and

nearest neighbour density estimation technique, respectively. Lit al. proposed a shift-based density estimation strategy (SDE) [17],hich is effective for MaOPs but with a negligibly extra computa-

ional cost. The core concept is to enhance convergence by pushingolutions with poor convergence to crowded regions. Solutionsocated at crowded regions have a high possibility to be elimi-ated during the evolutionary process. Consequently, the originalPEA2 can be applicable for many-objective optimization with thentegration of SDE strategy.

.3.6. HypEUnlike other MOEAs, HypE [43] belongs to the indicator based

pproach. It attempts to use a metric to guide the search in therocess of evolution. However, rather than computing the actual

ndicator values, Monte Carlo simulation is introduced to approx-mate the hypervolume value. The implementation makes HypE

ell suitable for the MaOPs.

.4. Parameter setting

For all the competing algorithms, we use the same parameterettings with common parameters for fair comparison, as shown inable 1.

The simulated binary crossover (SBX) and polynomial muta-ion (PM) [35] are applied for all MOEAs on continuous problems

WFG1-9). The distribution index �c for crossover is set to 30, andhe mutation operator, �m, is set to 20. For the combinatorial TSP,wo point crossover and swap mutation are used. For all problems,he crossover probability pc and mutation probability pm are set to

9 3 C811 = 165

10 3 C912 = 220

0.9 and 1/n (n denotes the number of decision variables), respec-tively. The maximum number of fitness evaluations, which is thetermination criterion, is set to 100,000 per run and every algorithmruns 30 times for each test instance.

For the reference point based (crEA, NSGA-III) and weight based(MOEA/D) methods, the population size N depends on the numberof reference points or weights. In other words, N is controlled by theparameter of divisions, H. For other peer algorithms, the populationsize is set to any positive integer. To ensure fair comparison, H isconfirmed to such that each algorithm has the same N for a giveninstance. Table 2 shows the exact number of the reference pointsfor the problems with different objectives.

In addition to the general parameters of each algorithm, someunique parameters were defined for different algorithms.

For MOEA/D, it was reported [18] that the commonly usedMOEA/D-DE [21] shows poor performance on many-objectiveproblems. Thus, here we use the original version MOEA/D with theTchebycheff scalar function. In terms of generating weight vectors,every weight vector w is created according to the same referencepoints generator in NSGA-III. Furthermore, parameters for MOEA/Dalso include a neighbourhood size, T, which is set to 10% of the pop-ulation size. The maximal number of solutions replaced by a childsolution nr, is set to 2.

For �-MOEA, the value of � is a vital parameter, since an externalarchive size of population depends on � for different instances. Theoverall performance of the algorithm could also be affected by thevalues of �. In this work, we attempt to set an appropriate valuefor different instances to ensure that the archive size is comparablewith the population size.

A critical parameter for SPEA2SDE is an archive population, Pa.This must be properly set to ensure that it is equal to the populationsize (120).

The parameters of GrEA include only a grid division, �div. Differ-ent choices of �div would significantly affect the performance [15].However, a suitable set of �div for WFG test sets was not given in[15]. In this work, following an extra search procedure, a suitable

WFG9 4,5,6,7,8,9,10 9,12,10,11,12,13,15TSP(−0.2) 6,8,10 8,10,12TSP(0) 6,8,10 8,9,8TSP(0.2) 6,8,10 9,8,11

688 L. Cai et al. / Applied Soft Compu

5 10 15 20 25 30 35 40 45 500.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Number of Divisons

Nor

mal

ized

Hyp

ervo

lum

e

4−obj5−obj6−obj7−obj8−obj9−obj10−obj

s

4

W

5

5

TTo

d

TH

T

Fig. 5. HV for GrEA with different number of divisions on WFG6.

For HypE, the bounds for the reference point and the number ofampling points were set to 200 and 10,000 respectively.

.5. Significance test

To test the difference statistical significance we use theilcoxon signed-rank test [44] at the 5% significance level.

. Experimental results and discussion

.1. Performance comparison on the WFG problems

Table 4 shows the average HV for WFG1–3 problems.ables 5 and 6 show the average GD and HV for problems WFG 4–9.

able 7 shows a summary of the significance test for HV betweenur proposed crEA and other algorithms.

To quantify how well each algorithm performs overall, we intro-uce the performance score [43]. Suppose we have l algorithms

able 4ypervolume for WFG1-3 problems.

Problem Obj. crEA NSGA-III MOEA/D

WFG1

4 0.891907 0.734276a 0.9191385 0.828429 0.670358a 0.8497236 0.736687 0.653196a 0.8583307 0.761396 0.746941 0.8474798 0.722291 0.739105 0.8851789 0.721236 0.753728 0.889067

10 0.703740 0.742918 0.881584

WFG2

4 0.991257 0.988644a 0.9849485 0.996590 0.992230a 0.9926056 0.997397 0.994043a 0.9950097 0.996926 0.995595a 0.9944288 0.997681 0.994196a 0.9958489 0.997645 0.991129a 0.995700

10 0.996851 0.990317a 0.996382

WFG3

4 0.678276 0.673911a 0.6129735 0.659541 0.665182 0.5036836 0.618123 0.648195 0.4626987 0.452529 0.590028 0.3962358 0.486249 0.614352 0.3689439 0.552448 0.615604 0.351212

10 0.572101 0.618876 0.336562

he best value of for each instance in highlighted in boldface.a The outcome is significantly outperformed by crEA.

ting 35 (2015) 681–694

from A1 to Al. Let ıi,j be 1, if Ai is significantly better than Aj, and 0otherwise. For each algorithm Ai, the performance score is

P(Ai) =l∑

j = 1

j /= i

ıi,j. (11)

This indicates how many other algorithms significantly out-perform the corresponding algorithm on the test case considered.Lower values imply a better algorithm. Fig. 6 shows the aver-age performance score for different test problems and numbers ofobjectives.

WFG1-3 problems cover mixed (convex and concave), discon-nected and degenerate PF, which produce irregular geometries.It is difficult to compute a suitable distance from final solutionsto PF, and hence only quality indicators of HV were calculated inthese cases. For WFG1 problems MOEA/D obtains the best HV on allobjects with one exception (ten dimensions). The SPEA2SDE rankssecond on 6–9 objectives. The proposed crEA and NSGA-III obtainsimilar performance and are superior to �-MOEA and HypE.

WFG2 problems have a convex PF with a number of discon-nected regions. Our proposed crEA outperforms all the others.NSGA-III performs as well as MOEA/D in 5–8 objective problemsbut its performance is inferior in higher objective problems. HypEshows a mixed performance.

GrEA achieves the best results on WFG3 problems. The pro-posed crEA achieves comparable performance to NSGA-III on lowerdimensional instances. The typical phenomenon observed from testset shows that the dominance based algorithm such as NSGA-IIIand GrEA are superior to other algorithms on this case. One possi-ble explanation is that the mechanism of selection in NSGA-III andGrEA may be very suitable for solving the WFG-3 type problems,which have a linear and degenerate PF.

Our proposed algorithm, crEA, outperforms all the other inWFG4–6 problems (Table 5). Although MOEA/D obtains the bestGD values for WFG4, indicating the best convergence, the HV

for MOEA/D is inferior compared to the rest of the algorithms.The major reason could be that the decomposition-based MOEA/Dachieves good convergence, but the differences of this sub-problemmay not cause enough diversity. In terms of overall performance,

GrEA SPEA2SDE �-MOEA HypE

0.853297a 0.835723a 0.691042a 0.660421a

0.847107 0.828675 0.601435a 0.605338a

0.813622 0.825136 0.560717a 0.619288a

0.791440 0.847095 0.550687a 0.679876a

0.820346 0.861546 0.486442a 0.722979 0.806564 0.882259 0.567045a 0.748098

0.793258 0.887317 0.610415a 0.779879

a 0.967553a 0.975656a 0.937291a 0.949897a

a 0.973693a 0.976537a 0.925654a 0.962004a

a 0.977694a 0.977444a 0.914464a 0.968049a

a 0.980534a 0.979252a 0.910210a 0.966945a

a 0.985135a 0.980583a 0.919485a 0.978461a

a 0.986597a 0.982157a 0.931269a 0.982014a

0.988290a 0.982894a 0.941051a 0.987025a

a 0.693345 0.662547a 0.657270a 0.630904a

a 0.667505 0.623776a 0.603236a 0.614468a

a 0.658975 0.606992a 0.585516a 0.603871a

a 0.642007 0.596810 0.549017 0.591027a 0.637121 0.584292 0.537927 0.603505a 0.634607 0.582546 0.531563a 0.617060a 0.636980 0.584602 0.511827a 0.633995

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Table 5Hypervolume for WFG4-6 problems.

Problem Obj. GD HV

crEA NSGAIII MOEA/D GrEA SPEA2SDE �-MOEA HypE crEA NSGAIII MOEA/D GrEA SPEA2SDE �-MOEA HpyE

WFG4

4 0.000203 0.001535a 0.000088 0.001623a 0.004319a 0.005831a 0.003848a 0.821261 0.817583a 0.805485a 0.816027a 0.803169a 0.758574a 0.660393a

5 0.000207 0.002157a 0.000072 0.002205a 0.006405a 0.008929a 0.003960a 0.871801 0.867314a 0.845334a 0.857628a 0.839938a 0.723014a 0.662706a

6 0.000293 0.002532a 0.000065 0.005397a 0.008157a 0.011677a 0.004049a 0.907278 0.898703a 0.870475a 0.885569a 0.857073a 0.655037a 0.672413a

7 0.000221 0.001266a 0.000062 0.004461a 0.009925a 0.013654a 0.004501a 0.923023 0.915660a 0.874972a 0.908740a 0.863414a 0.551750a 0.628298a

8 0.000504 0.002480a 0.000066 0.005837a 0.010945a 0.018806a 0.003846a 0.945210 0.933158a 0.891976a 0.908620a 0.871794a 0.527734a 0.671833a

9 0.001064 0.003708a 0.000107 0.003378a 0.011588a 0.022368a 0.003333a 0.957077 0.940009a 0.908985a 0.889240a 0.875192a 0.510414a 0.714787a

10 0.002366 0.005428a 0.000156 0.004269a 0.009014a 0.022385a 0.002840a 0.955584 0.934727a 0.907818a 0.906077a 0.862058a 0.495748a 0.733878a

WFG5

4 0.006667 0.006837a 0.006007 0.006835a 0.008063a 0.008658a 0.005354 0.785559 0.785012a 0.772293a 0.784602a 0.777110a 0.755695a 0.651597a

5 0.006597 0.006935a 0.005814 0.007120a 0.009160a 0.011301a 0.005417 0.834018 0.833170a 0.815477a 0.831368a 0.811278a 0.742078a 0.666262a

6 0.006573 0.006907a 0.005792 0.007160a 0.010139a 0.014399a 0.005549 0.864217 0.863356a 0.839897a 0.863268a 0.831356a 0.674719a 0.677355a

7 0.007769 0.007832a 0.006957 0.008351a 0.011095a 0.018533a 0.006957 0.871933 0.871604a 0.844823a 0.874197 0.842213a 0.579731a 0.646752a

8 0.006702 0.006859a 0.005880 0.007081a 0.012618a 0.023850a 0.005933 0.897651 0.893177a 0.863451a 0.894419a 0.842238a 0.548507a 0.679222a

9 0.005979 0.006430a 0.005025 0.006767a 0.012878a 0.028781a 0.005128 0.909232 0.904481a 0.868681a 0.904232a 0.851351a 0.517371a 0.704275a

10 0.005677 0.006353a 0.004539 0.006634a 0.011958a 0.031021a 0.004502 0.913604 0.908001a 0.862298a 0.903014a 0.837045a 0.488314a 0.734081a

WFG6

4 0.005706 0.006844a 0.004659 0.005889 0.008528a 0.010051a 0.003996 0.788137 0.786021a 0.782921a 0.790196 0.778106a 0.765910a 0.476326a

5 0.005666 0.007498a 0.004976 0.006663a 0.010597a 0.013298a 0.004090 0.836594 0.831188a 0.811368a 0.832166a 0.811344a 0.772921a 0.468481a

6 0.006141 0.007443a 0.004871 0.007254a 0.013230a 0.017111a 0.004295 0.868032 0.861498a 0.831052a 0.855446a 0.825981a 0.724236a 0.473684a

7 0.007688 0.007771 0.007245 0.009816a 0.015653a 0.023108a 0.005521 0.879032 0.872794a 0.827033a 0.873423a 0.833106a 0.579969a 0.451538a

8 0.007131 0.008054a 0.005577 0.007804a 0.018173a 0.027579a 0.004706 0.894857 0.886140a 0.857590a 0.886604 0.835159a 0.546601a 0.459997a

9 0.006661 0.007611a 0.005335 0.007616a 0.019267a 0.033305a 0.004299 0.904440 0.896412a 0.851090a 0.862421a 0.836598a 0.502312a 0.469268a

10 0.006868 0.007893a 0.005188 0.007419a 0.019167a 0.035994a 0.003728 0.903315 0.894945a 0.827259a 0.886456a 0.834964a 0.473709a 0.498463a

The best value of for each instance in highlighted in boldface.a The outcome is significantly outperformed by crEA.

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Table 6Hypervolume for WFG7-9 problems.

Problem Obj. GD HV

crEA NSGAIII MOEA/D GrEA SPEA2SDE �-MOEA HypE crEA NSGAIII MOEA/D GrEA SPEA2SDE �-MOEA HypE

WFG7

4 0.000716 0.003536a 0.002443a 0.000366 0.002159a 0.005059a 0.000411 0.822937 0.817353a 0.768800a 0.823340 0.816239a 0.796893a 0.546537a

5 0.000894 0.004302a 0.000193 0.001004a 0.003975a 0.011650a 0.000470 0.874288 0.870102a 0.824047a 0.873563a 0.849997a 0.776456a 0.530511a

6 0.000097 0.002648a 0.000093 0.001113a 0.004460a 0.016741a 0.000408a 0.907222 0.903883a 0.872925a 0.909045 0.867493a 0.728999a 0.516668a

7 0.000129 0.000740a 0.000099 0.000899a 0.005884a 0.023247a 0.000566a 0.925962 0.919146a 0.886668a 0.924265a 0.883772a 0.611471a 0.490079a

8 0.000102 0.001891a 0.001847a 0.000773a 0.006650a 0.026829a 0.000445a 0.951131 0.941706a 0.903429a 0.942773a 0.897326a 0.585745a 0.508698a

9 0.000282 0.002900a 0.000275 0.000949a 0.007377a 0.030745a 0.000450a 0.964821 0.953470a 0.916649a 0.921359a 0.896929a 0.538178a 0.570578a

10 0.000720 0.004911a 0.000141 0.000954 0.007697a 0.030440a 0.000448 0.971907 0.957077a 0.933936a 0.937228a 0.903422a 0.501922a 0.643555a

WFG8

4 0.011483 0.011551 0.006575 0.011010 0.013554a 0.013220a 0.013103a 0.772422 0.770983a 0.636523a 0.775094 0.757156a 0.742255a 0.498703a

5 0.014645 0.015403a 0.005807 0.015152a 0.019272a 0.018389a 0.017929a 0.814275 0.810451a 0.540679a 0.807212a 0.779923a 0.746010a 0.478235a

6 0.018793 0.018865 0.006536 0.015577 0.025338a 0.025765a 0.023064a 0.834642 0.838481 0.556934a 0.805155a 0.782792a 0.659709a 0.469743a

7 0.028603 0.029878a 0.010539 0.034072a 0.027826 0.036932a 0.013271 0.825207 0.830845 0.536896a 0.817078a 0.787385a 0.544737a 0.320477a

8 0.031410 0.031957a 0.011274 0.031172 0.032392 0.043438a 0.011140 0.835109 0.844686 0.558047a 0.814866a 0.791557a 0.507713a 0.321918a

9 0.034095 0.034407a 0.011384 0.011656 0.034076 0.052368a 0.008259 0.837145 0.848743 0.549898a 0.804706a 0.792548a 0.491422a 0.329590a

10 0.038268 0.034324 0.011529 0.005664 0.038953 0.061436a 0.007031 0.832772 0.851611 0.542844a 0.820805a 0.789032a 0.466381a 0.317266a

WFG9

4 0.008431 0.009660 0.007091 0.004712 0.006226 0.010115a 0.003641 0.771065 0.771676 0.719179a 0.790762 0.776516 0.738129a 0.708495a

5 0.005669 0.011255a 0.009044a 0.004616 0.008682a 0.016863a 0.004699 0.839044 0.812025a 0.681675a 0.839675a 0.804627a 0.710354a 0.746712a

6 0.007493 0.009819a 0.008721 0.006151 0.009320a 0.023177a 0.004753 0.857669 0.846962a 0.631739a 0.860815 0.820984a 0.674068a 0.775860a

7 0.008704 0.011989a 0.009968 0.007541 0.010540a 0.039900a 0.008834 0.864060 0.849048a 0.619004a 0.869726 0.824324a 0.521681a 0.735332a

8 0.007159 0.011109a 0.008116 0.006839 0.013222a 0.041407a 0.006734 0.881874 0.863690a 0.609327a 0.885506 0.812124a 0.502918a 0.780224a

9 0.007131 0.009778a 0.008675 0.006195 0.013767a 0.037482a 0.005794 0.888346 0.878781a 0.606838a 0.886067 0.819613a 0.498504a 0.775606a

10 0.007698 0.009860a 0.009133a 0.003968 0.013674a 0.033963a 0.006231 0.885590 0.878355a 0.581112a 0.861364a 0.800360a 0.495342a 0.757491a

The best value of for each instance in highlighted in boldface.a The outcome is significantly outperformed by crEA.

L. Cai et al. / Applied Soft Computing 35 (2015) 681–694 691

F ore oo

Nbin�l

fbomGfM

dvoNWmaWtciHN

oa

TSa

‘i‘r

HV changes during the evolutionary process. The proposed crEAhas a clear advantage. NSGA-III stably increases HV and achieves asimilar performance to crEA by the end of the evolutionary steps.

0.9

1

e

ig. 6. Ranking of average performance score. (a) shows the average performance scver all test instances for different number of objectives.

SGA-III and GrEA seem to be the only two algorithms compara-le to the proposed crEA. In the WFG5 and WFG6 problems, crEA

s consistently better than MOEA/D. NSGA-III and GrEA obtains theext-best performance in some instances in terms of HV metric.-MOEA and HypE achieve the poorest performance for these prob-ems.

For WFG7–9 problems (Table 6), GrEA ranks first in terms of HVor lower dimension cases (4–6). However, crEA and GrEA performetter than NSGA-III in terms of convergence (smaller GD). For 7–10bjective instances, �-MOEA and HypE still show the worst perfor-ance, although HypE can achieve a good convergence (smallerD). NSGA-III, GrEA, and the proposed crEA have comparable per-

ormance but crEA is slightly better than others. SPEA2SDE andOEA/D rank third and fourth, respectively.For WFG8 and WFG9 problems, none of the algorithms could be

istinguished in terms of GD value, which implies that the con-ergence of these algorithms is similar. In 5 objective instancesf WFG8, crEA shows superior performance in terms of HV, andSGA-III ranks second. In the higher-dimension objective space ofFG8, MOEA/D and HypE show the worst performance (HV) forost instances. NSGA-III is among the top performing algorithms,

nd crEA and GrEA also achieve good results on these problems. TheFG9 problem, has a multi and deceptive modality of PF, for fur-

her testing the overall performance of any algorithm. The proposedrEA and GrEA show clear advantage over all the other algorithmsn all instances, and NSGA-III obtains the next best. Indicator-basedypE performs better than in former problems but still falls behind

SGA-III and SEPA2SDE.

In summary (Table 7), crEA performs significantly better on mostf problem instances compared with any other algorithms (therere 9 × 7 =63 problem instances in all). Our proposed crEA achieves

able 7ummary of the significance test of HV between the proposed algorithm and otherlgorithms.

NSGA-III MOEA/D GrEA SPEA2SDE �-MOEA HypE

crEAvs.

B 47 55 39 52 61 56W 13 6 19 9 2 6E 3 2 5 2 0 1

B’ (‘W’) means that the number of instances on which the results of crEA are signif-cantly better (worse) than other algorithms.E’ means that the number of instances without detected differences between theesults of two algorithms.

ver all dimensions for different test cases. (b) shows the average performance score

the best score on WFG2 and WFG4–7 problems (Fig. 6a), and onlyyields a slightly worse score than NSGA-III and GrEA on problems ofWFG8–9. However, crEA does not show excellent performances forWFG1 and WFG3. MOEA/D performs best for WFG1 problems andGrEA is among the top performing algorithms on WFG3. crEA showsobvious advantages in MaOPs (Fig. 6b) and returns the best scoreexcept for the 8 objective instance, where it takes second place. It isinteresting to note that NSGA-III and GrEA also show competitiveperformance on MaOPs. Conversely, �-MOEA and HypE do not seemto work very well on any WFG problems with higher number ofobjectives.

5.2. More investigations on crEA

Fig. 7 shows the trajectories of HV with generations for the sixalgorithms for the case of 9-objective WFG7. This indicates how

0 100 200 300 400 500 6000.4

0.5

0.6

0.7

0.8

Generation Numbers

Nor

mal

ized

Hyp

ervo

lum

crEANSGA−IIIMOEA/DGrEASPEA2SDE

Fig. 7. Evolutionary trajectories of Hypervolume for the six algorithms on the 9-objective WFG7 problem.

692 L. Cai et al. / Applied Soft Computing 35 (2015) 681–694

0 5 10 15 20 250

5

10

15

20

25

f1

f2

MOEA/DcrEA

(a) crE A vs MOEA/D

0 5 10 15 20 250

5

10

15

20

25

f1

f2

NSGAIIIcrEA

Fig. 8. Comparison between crEA and other reference point based algorithms on the 10-o

F

Su

(riiie

objective space of the 10-objective TSP with TSPcp = −0.2. crEA

TH

T

ig. 9. The average time for six algorithms for the 10-objective WFG9 instance.

PEA2SDE may have a superior HV at first, but shows somewhatnstable performance.

Fig. 9 shows the average time of each algorithm except HypEsince it costs significantly more time than all others). All the algo-ithms were implemented within the jMetal framework [45] andn the same running environment for fair comparison. SPEA2SDE

s much more computationally expensive than others. For somenstances, although SPEA2SDE achieves the best proximity, itsfficiency remains inferior. The proposed crEA exhibits similar

able 8ypervolume results for TSP problems.

Problem Obj. crEA NSGA-III MOEA/D

TSP(-0.2)6 0.056642 0.053327a 0.069911

8 0.022800 0.015693a 0.014558a

10 0.005912 0.003892a 0.002820a

TSP(0)6 0.063747 0.053618a 0.072104

8 0.018754 0.016216a 0.014154a

10 0.004741 0.003510a 0.002849a

TSP(0.2)6 0.062750 0.063936 0.067030

8 0.015456 0.016501 0.012138a

10 0.003925 0.003689 0.002312a

he best value of for each instance in highlighted in boldface.a The outcome is significantly outperformed by crEA.

(b) crE A vs NSGA-III

bjective TSP problem (TSPcp = −0.2) in two-dimensional objective space f1 and f2.

computational complexity to GrEA but with better proximityresults.

Fig. 10 shows the final solutions of six algorithms for the 10-objective WFG7 by parallel coordinates in a single run. All thealgorithms find the upper boundary of the Pareto front. However,�-MOEA shows poor diversity, since most solutions locate at thecentral area of the objective space. Although MOEA/D can approachthe PF very well, it suffers from the loss of population diversity.NSGA-III can achieve a good balance between convergence anddiversity, but the solutions fail to spread over all the objectivespace. The solutions obtained by our proposed crEA and GrEA coverthe whole optimal range and are well distributed. crEA preformsslightly better than GrEA in terms of diversity as well as conver-gence.

5.3. Performance comparison on TSP problems

Table 8 shows the HV values of compared algorithms for dif-ferent TSP. A typical phenomenon observed from these results isthat crEA, NSGA-III, and MOEA/D do not provide the best for TSPproblems, whereas the performance of SPEA2SDE is noticeablyimproved. One possible reason is that the mechanism of objec-tive normalization in the reference point based method may causedeviation on problems which have been normalized. In the scopeof reference point based methods, the crEA outperforms MOEA/Dand NSGA-III.

Fig. 8 shows crEA, MOEA/D, and NSGA-III two-dimensional

achieve better diversity than MOEA/D, since MOEA/D’s solutionsare overcrowded in some regions of space. NSGA-III performs bet-ter than crEA on diversity because it covers wider regions, but its

GrEA SPEA2SDE �-MOEA HypE

0.099402 0.129840 0.097213 0.031920a

0.026443 0.035983 0.013496a 0.005137a

0.005734 0.008572 0.003116a 0.000758a

0.085262 0.111653 0.088865 0.046165a

0.022123 0.030468 0.016256a 0.006518a

0.005490 0.007262 0.003265a 0.000993a

0.076005 0.098061 0.090875 0.050366a

0.020086 0.026929 0.018043 0.011063a

0.004552 0.006571 0.003403a 0.001931a

L. Cai et al. / Applied Soft Computing 35 (2015) 681–694 693

lgorith

cap

6

lcmbmca

pMapopwodpnd

ppbi

roo

Fig. 10. The final solution set of the six a

onvergence is a little inferior to crEA. crEA achieves a better bal-nce between convergence and diversity in the scope of referenceoint based algorithms.

. Conclusion

Balancing convergence and diversity in EAs remains a chal-enge. This paper proposes an algorithm which introduces alustering-ranking strategy to develop a general framework forany-objective optimization. The introduction of a clustering-

ased method can enhance the diversity and the rankingechanism forces the obtained solutions towards the true PF. The

ombination of clustering and ranking considers both convergencend diversity for every individual in the population.

Extensive experiments were performed to compare the pro-osed crEA over widely used WFG problems with state-of-the-artOEAs, which cover Pareto, decomposition, and indicator based

pproaches. In terms of two defined performance indicators, ourroposed algorithm is significantly better than the peer algorithmsn most test problems. Thus, the proposed algorithm is very com-etitive with other algorithms finding a uniformly distributed andell-approximated population. We present some visual evidence

f superior performance in terms of computational complexity andiversity. The computational cost of crEA is lower than most of theeer algorithms except MOEA/D. Moreover, the proposed crEA canot only find the upper boundary of PF, but also obtains a well-istributed solution set.

We performed an extensive comparison for a combinatorial testroblem. The proposed crEA obtains acceptable performance com-ared with the peer algorithms, and it appears that crEA providesetter performance for both convergence and diversity in compar-

son with peer reference point based algorithms.

Although crEA works well on the MaOPs in WFG, it will be

estricted to specific problems by virtue of the ‘no free lunch’ the-rem [46], since multi-objective optimization problems are notutside the no free lunch concerns [47]. We believe it is necessary

ms on the 10-objective WFG 7 instance.

to clearly define the class of problems an algorithm is designed for.In future work we intend to further investigate the performance ofcrEA for a wider range of problems, especially for the real-worldpractical problems with high number of objectives, problems withcomplicated PF and combinatorial optimization problems. It mayalso be worth further investigating the normalization mechanism,since although we have a much simpler normalization procedurein comparison with NSGA-III, our current method is not an elegantmethod to solve different kinds of problems. Moreover, it wouldalso be possible to introduce the advanced learning mechanisms[49,50,51] into crEA to further enhance its performance.

Acknowledgements

The first author would like to acknowledge the support ofa scholarship launched by China Scholarship Council, and wishto thank the Associate Editor and the anonymous reviewers fortheir valuable comments and helpful suggestions which greatlyimproved the paper quality. Part of this work was done while thefirst author visited CERCIA, School of Computer Science, Universityof Birmingham, UK.

References

[1] V. Khare, X. Yao, K. Deb, Performance Scaling of Multi-objective EvolutionaryAlgorithms, In Proc. of the 2nd International Conference on Evolutionary Multi-Criterion Optimization (EMO’03), in: C.M. Fonseca, P.J. Fleming, E. Zitzler, K.Deb, Z. Lothar (Eds.), Lecture Notes in Computer Science, Springer-Verlag, 2003,pp. 376–390.

[2] P. Fleming, R. Purshouse, R. Lygoe, Many-objective optimization: an engineer-ing design perspective, in: C. Coello Coello, A. HernA-ndez Aguirre, E. Zitzler(Eds.), Evolutionary Multi-Criterion Optimization, Vol. 3410 of Lecture Notesin Computer Science, Springer, Berlin/Heidelberg, 2005, pp. 14–32, http://dx.

doi.org/10.1007/978-3-540-31880-4 2

[3] M.W. Mkaouer, M. Kessentini, S. Bechikh, K. Deb, M.Ó. Cinnéide, Highdimensional search-based software engineering: finding tradeoffs among 15objectives for automating software refactoring using NSGA-III, in: Proceedingsof the 2014 Conference on Genetic and Evolutionary Computation, GECCO

6 Compu

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[[

[

[

[

[

[

[

[

[

[

[

[

[

94 L. Cai et al. / Applied Soft

’14, ACM, New York, NY, USA, 2014, pp. 1263–1270, http://dx.doi.org/10.1145/2576768.2598366

[4] M. Farina, P. Amato, On the optimal solution definition for many-criteriaoptimization problems, in: Proceedings of the NAFIPS-FLINT International Con-ference, 2002, pp. 233–238.

[5] B. Li, J. Li, K. Tang, X. Yao, Many-Objective Evolutionary Algorithms: a Survey,ACM Comput. Surv. (2014), http://dx.doi.org/10.1145/2792984, V, N, Article999 (June 2015).

[6] K. Deb, A. Pratap, S. Agarwal, T. Meyarivan, A fast and elitist multiob-jective genetic algorithm: NSGA-II, IEEE Trans. Evol. Comput. 6 (2) (2002)182–197.

[7] E. Zitzler, M. Laumanns, L. Thiele, E. Zitzler, E. Zitzler, L. Thiele, L. Thiele, SPEA2:Improving the Strength Pareto Evolutionary Algorithm, 2001.

[8] D.W. Corne, N.R. Jerram, J.D. Knowles, M.J. Oates, et al., PESA-II: region-basedselection in evolutionary multiobjective optimization, in: Proceedings of theGenetic and Evolutionary Computation Conference (GECCO’2001), Citeseer,2001.

[9] R.C. Purshouse, P.J. Fleming, Evolutionary many-objective optimisation: anexploratory analysis, in: The 2003 Congress on Evolutionary Computation,2003. CEC’03, Vol. 3, IEEE, 2003, pp. 2066–2073.

10] H. Ishibuchi, Y. Sakane, N. Tsukamoto, Y. Nojima, Evolutionary many-objectiveoptimization by NSGA-II and MOEA/D with large populations, in: IEEE Inter-national Conference on Systems, Man and Cybernetics, 2009. SMC 2009, 2009,pp. 1758–1763, http://dx.doi.org/10.1109/ICSMC.2009.5346628

11] J. Knowles, D. Corne, Quantifying the effects of objective space dimension inevolutionary multiobjective optimization, in: S. Obayashi, K. Deb, C. Poloni,T. Hiroyasu, T. Murata (Eds.), Evolutionary Multi-Criterion Optimization, Vol.4403 of Lecture Notes in Computer Science, Springer, Berlin/Heidelberg, 2007,pp. 757–771, http://dx.doi.org/10.1007/978-3-540-70928-2-57

12] F. di Pierro, S.-T. Khu, D.A. Savic, An investigation on preference order rankingscheme for multiobjective evolutionary optimization, IEEE Trans. Evol. Comput.11 (1) (2007) 17–45.

13] M. Laumanns, L. Thiele, K. Deb, E. Zitzler, Combining convergence and diver-sity in evolutionary multiobjective optimization, Evol. Comput. 10 (3) (2002)263–282.

14] M. Farina, P. Amato, A fuzzy definition of “optimality” for many-criteria opti-mization problems, IEEE Trans. Syst. Man Cybern. A: Syst. Hum. 34 (3) (2004)315–326.

15] S. Yang, M. Li, X. Liu, J. Zheng, A grid-based evolutionary algorithm for many-objective optimization, IEEE Trans. Evol. Comput. 17 (5) (2013) 721–736, http://dx.doi.org/10.1109/TEVC.2012.2227145

16] S. Adra, P. Fleming, Diversity management in evolutionary many-objectiveoptimization, IEEE Trans. Evol. Comput. 15 (2) (2011) 183–195, http://dx.doi.org/10.1109/TEVC.2010.2058117

17] M. Li, S. Yang, X. Liu, Shift-based density estimation for pareto-based algo-rithms in many-objective optimization, IEEE Trans. Evol. Comput. 18 (3) (2014)348–365, http://dx.doi.org/10.1109/TEVC.2013.2262178

18] K. Deb, H. Jain, An Evolutionary Many-Objective Optimization AlgorithmUsing Reference-Point Based Non-Dominated Sorting Approach, Part I: SolvingProblems With Box Constraints, 2013, http://dx.doi.org/10.1109/TEVC.2013.2281535

19] A. Zhou, B.-Y. Qu, H. Li, S.-Z. Zhao, P.N. Suganthan, Q. Zhang, Multiobjectiveevolutionary algorithms: a survey of the state of the art, Swarm Evol. Comput.1 (1) (2011) 32–49, http://dx.doi.org/10.1016/j.swevo.2011.03.001

20] Q. Zhang, H. Li, MOEA/D: a multiobjective evolutionary algorithm based ondecomposition, IEEE Trans. Evol. Comput. 11 (6) (2007) 712–731, http://dx.doi.org/10.1109/TEVC.2007.892759

21] H. Li, Q. Zhang, Multiobjective optimization problems with complicated paretosets, MOEA/D and NSGA-II, IEEE Trans. Evol. Comput. 13 (2) (2009) 284–302,http://dx.doi.org/10.1109/TEVC.2008.925798

22] S.-Z. Zhao, P. Suganthan, Q. Zhang, Decomposition-based multiobjective evo-lutionary algorithm with an ensemble of neighborhood sizes, IEEE Trans.Evol. Comput. 16 (3) (2012) 442–446, http://dx.doi.org/10.1109/TEVC.2011.2166159

23] S. Zapotecas Martinez, C.A. Coello Coello, A direct local search mechanism fordecomposition-based multi-objective evolutionary algorithms, in: 2012 IEEECongress on Evolutionary Computation (CEC), 2012, pp. 1–8, http://dx.doi.org/

10.1109/CEC.2012.6252990

24] M. Fleischer, The measure of pareto optima applications to multi-objectivemetaheuristics, in: C. Fonseca, P. Fleming, E. Zitzler, L. Thiele, K. Deb (Eds.), Evo-lutionary Multi-Criterion Optimization, Vol. 2632 of Lecture Notes in ComputerScience, Springer, Berlin/Heidelberg, 2003, pp. 519–533.

[

[

ting 35 (2015) 681–694

25] E. Zitzler, S. Knzli, Indicator-based selection in multiobjective search, in: X.Yao, E. Burke, J. Lozano, J. Smith, J. Merelo-Guervs, J. Bullinaria, J. Rowe, P. Tio,A. Kabn, H.-P. Schwefel (Eds.), Parallel Problem Solving from Nature – PPSN VIII,Vol. 3242 of Lecture Notes in Computer Science, Springer, Berlin/Heidelberg,2004, pp. 832–842.

26] N. Beume, B. Naujoks, M. Emmerich, SMS-EMOA: multiobjective selectionbased on dominated hypervolume, Eur. J. Oper. Res. 181 (3) (2007) 1653–1669,http://dx.doi.org/10.1016/j.ejor.2006.08.008

27] C. Igel, N. Hansen, S. Roth, Covariance matrix adaptation for multi-objectiveoptimization, Evol. Comput. 15 (1) (2007) 1–28.

28] K. Bringmann, T. Friedrich, An efficient algorithm for computing hypervolumecontributions, Evol. Comput. 18 (3) (2010) 383–402.

29] L. While, L. Bradstreet, L. Barone, A fast way of calculating exact hypervolumes,IEEE Trans. Evol. Comput. 16 (1) (2012) 86–95.

30] R.H. Gómez, C.A.C. Coello, MOMBI: a new metaheuristic for many-objectiveoptimization based on the R2 indicator, in: 2013 IEEE Congress on EvolutionaryComputation (CEC 2013), 2013, pp. 2488–2495.

31] H. Trautmann, T. Wagner, D. Brockhoff, R2-EMOA: focused multiobjectivesearch using R2-indicator-based selection, in: Learning and Intelligent Opti-mization, Springer, 2013, pp. 70–74.

32] J.N. Morse, Reducing the size of the nondominated set: pruning by clustering,Comput. Oper. Res. 7 (1) (1980) 55–66.

33] E. Zitzler, L. Thiele, Multiobjective evolutionary algorithms: a comparative casestudy and the strength pareto approach, IEEE Trans. Evol. Comput. 3 (4) (1999)257–271, http://dx.doi.org/10.1109/4235.797969

34] I. Das, J.E. Dennis, Normal-boundary intersection: a new method for generatingthe pareto surface in nonlinear multicriteria optimization problems, SIAM J.Optim. 8 (3) (1998) 631–657.

35] R.B. Agrawal, K. Deb, R.B. Agrawal, Simulated binary crossover for continuoussearch space, 1994.

36] K. Miettinen, Nonlinear Multiobjective Optimization, vol. 12, Springer, 1999.37] Y. Qi, X. Ma, F. Liu, L. Jiao, J. Sun, J. Wu, MOEA/D with adaptive weight adjust-

ment, Evol. Comput. 22 (2) (2014) 231–264.38] S. Huband, P. Hingston, L. Barone, L. While, A review of multiobjective test

problems and a scalable test problem toolkit, IEEE Trans. Evol. Comput. 10 (5)(2006) 477–506, http://dx.doi.org/10.1109/TEVC.2005.861417

39] D.W. Corne, J.D. Knowles, Techniques for highly multiobjective optimisation:some nondominated points are better than others, in: Proceedings of the 9thAnnual Conference on Genetic and Evolutionary Computation, GECCO ’07, ACM,New York, NY, USA, 2007, pp. 773–780, http://dx.doi.org/10.1145/1276958.1277115

40] K. Deb, L. Thiele, M. Laumanns, E. Zitzler, Scalable test problems for evolutionarymultiobjective optimization, in: A. Abraham, L. Jain, R. Goldberg (Eds.), Evolu-tionary Multiobjective Optimization, Advanced Information and KnowledgeProcessing, Springer, London, 2005, pp. 105–145.

41] D.A.V. Veldhuizen, Multiobjective Evolutionary Algorithms: Classifications,Analyses, and New Innovations, Tech. rep., Evolutionary Computation, 1999.

42] K. Deb, M. Mohan, S. Mishra, Evaluating the ε-domination based multi-objectiveevolutionary algorithm for a quick computation of pareto-optimal solutions,Evol. Comput. 13 (4) (2005) 501–525.

43] J. Bader, E. Zitzler, HYPE: an algorithm for fast hypervolume-based many-objective optimization, Evol. Comput. 19 (1) (2011) 45–76.

44] F. Wilcoxon, Individual comparisons by ranking methods, Biom. Bull. 1 (6)(1945) 80–83, URL http://www.jstor.org/stable/3001968

45] J.J. Durillo, A.J. Nebro, jMetal: A java framework for multi-objective optimiza-tion, Adv. Eng. Softw. 42 (10) (2011) 760–771, http://dx.doi.org/10.1016/j.advengsoft.2011.05.014

46] D.H. Wolpert, W.G. Macready, No free lunch theorems for optimization, IEEETrans. Evol. Comput. 1 (1) (1997) 67–82.

47] T.C. Service, A no free lunch theorem for multi-objective optimization, Inf.Process. Lett. 110 (21) (2010) 917–923.

48] Y. Yuan, H. Xu, Multiobjective Flexible Job Shop Scheduling Using MemeticAlgorithms, IEEE Transactions on Automation Science and Engineering 12 (1)(2015) 336–353.

49] Y. Zheng, B. Jeon, D. Xu, W.Q.M. Jonathan, H. Zhang, Image segmentation bygeneralized hierarchical fuzzy C-means algorithm, J. Intell. Fuzzy. Syst. 28 (2)(2015) 961–973.

50] B. Gu, V.S. Sheng, K.Y. Tay, W. Romano, S. Li, Incremental Support Vector Learn-ing for Ordinal Regression, IEEE Transactions on Neural Networks and LearningSystems 26 (7) (2015) 1403–1416.

51] B. Gu, V.S. Sheng, Z. Wang, D. Ho, S. Osman, S. Li, Incremental learning for�-Support Vector Regression, Neural Networks 67 (2015) 140–150.