Research ArticleAn MOEAD-ACO with PBI for Many-Objective Optimization

Tianbai Ling1 and ChenWang 12

1Shanghai Key Laboratory of Intelligent Manufacturing and Robotics Shanghai University Shanghai 200072 China2Dept of Mechanical Engineering Hubei Automotive Industries Institute Shiyan 442002 China

Correspondence should be addressed to Chen Wang 20090011huateducn

Received 1 March 2018 Revised 15 May 2018 Accepted 6 August 2018 Published 30 August 2018

Academic Editor Erik Cuevas

Copyright copy 2018 Tianbai Ling andChenWangThis is an open access article distributed under the Creative CommonsAttributionLicense which permits unrestricted use distribution and reproduction in anymedium provided the originalwork is properly cited

Evolutionary algorithms (EAs) are an important instrument for solving the multiobjective optimization problems (MOPs) It hasbeen observed that the combined ant colony (MOEAD-ACO) based on decomposition is very promising for MOPs However asthe number of optimization objectives increases the selection pressure will be released leading to a significant reduction in theperformance of the algorithm It is a significant problem and challenge in the MOEAD-ACO to maintain the balance betweenconvergence and diversity in many-objective optimization problems (MaOPs) In the proposed algorithm an MOEAD-ACOwith the penalty based boundary intersection distance (PBI) method (MOEAD-ACO-PBI) is intended to solve the MaOPs PBIdecomposes the problems with many single-objective problems a weighted vector adjustment method based on clustering anduses different pheromonematrices to solve different single objectives proposedThen the solutions are constructed and pheromonewas updated Experimental results on both CF1-CF4 and suits of C-DTLZ benchmarks problems demonstrate the superiority ofthe proposed algorithm in comparison with three state-of-the-art algorithms in terms of both convergence and diversity

1 Introduction

Over the past few decades EAs have been successfullyimplemented to various real-world optimization problemswith two or three objectives However EAs face manychallenges when dealing with MOPs which have more thanthree objectives and are commonly referred to as MaOPs[1] The foremost reason is the case that the proportionof nondominated solutions in the population rises sharplywith an increase in the number of objectives [2] The Paretodominate relation cannot provide sufficient selection pressureto ensure that the population evolve [3] In recent yearsmore and more multiobjective evolutionary optimizationalgorithms (MOEAs) have been improved for solvingMaOPs[4]

In recent years as system functions become more andmore complex MaOPs have become a hot topic in thefield of evolutionary multiobjective optimization community[5] It has been found that EAs have succeeded in solvingvarious optimization problems However as the number ofoptimization objectives increases sufficient selection pres-sure will be released resulting in a significant reduction in

the performance of the algorithm It is a challenging majorproblem and challenge in the EAs to maintain the balancebetween convergence and diversity in MaOPs

At present more and more MOEAs have made extraor-dinary improvement in solving MaOPs The most popu-lar method is the Pareto-dominance and modified Pareto-dominance that relaxes the form of Pareto-dominance toincrease the selection pressure towards the Pareto front forexample H Jain and Deb based on the NSGAIII frameworkto solve generic MaOPs [6] and fuzzy Pareto-dominance [7]The disadvantage of the above methods is introduced in oneor more parameters and needs to be adjusted heuristicallyThe indicator-based algorithm is promising method forMaOPs It uses a single indicator value to guide the evolutionof the population such as R2 indicator [8] and hypervolumeindicator [9] The last method is based on decompositionDecomposition strategy and neighbourhood concept areintroduced in this method As one of the popular algorithmsMOPs based ondecomposition (MOEAD)were proposed byZhang andLi [10]TheMOEADuses aggregation function tocompare the solution with the uniformly distributed weightvector to retain solution convergence and diversity [4 11]

HindawiMathematical Problems in EngineeringVolume 2018 Article ID 5414869 9 pageshttpsdoiorg10115520185414869

2 Mathematical Problems in Engineering

MOEAD gets a real good convergence and diversity and lowcomputational complexity and thus has an effective method

ACO originally proposed dealing with single-objectivecombination optimization problems [12] The successfulapplication of ACO in single-objective optimization hasled some researchers to develop ACO algorithms for mul-tiobjective optimization problems Recently a multiobjec-tive evolutionary algorithm for multiobjective combinato-rial optimization problem based on decomposition and antcolony algorithm (MOEAD-ACO) was proposed Ke etal combined ACO and decomposition-based multiobjectiveevolutionary algorithm (MOEADACO) to solve biobjectiveTSP [13] Despite the advance in adapting MOEAs forleading with MaOPs the MOEAD-ACO performed muchbetter than the BicriterionAnt in all the test instances TheMOEAD-ACO algorithm successfully solves the biobjectiveTSP problemHowever its performance in cases with three ormore objectives requires further study Murilo Zangari Souzaet al proposed a parallel implementation of MOEAD-ACOon a graphics processing unit (GPU) using NVIDIA CUDAto improve efficiency and achieve high-quality results withina reasonable execution time [14]

At present the MOEAD-ACO optimization algorithmhas been introduced to solve MaOPs in numerous stud-ies However there are few studies on the further useof MOEAD-ACO to solve the MaOPs In this paper weintroduce the PBI method for MOEAD-ACO to solve theMaOPs Andmany-objective optimization techniques can beincorporated into our method to maintain convergence anddiversity of algorithm

From the discussion above we propose a new algorithmwith MOEAD-ACO-PBI designed to solve the MaOPs Thispaper mainly focuses on the algorithm that promotes theperformance of convergence and diversity in many-objectiveoptimization

This remainder of this paper is organized as follows InSection 2 the related work and background were discussedTheMOEAD-ACO-PBI algorithm is described in Section 3In Section 5 the experimental studies are provided todemonstrate the efficiency of the proposed method as wellas some discussions on this paper Finally Section 6 makes aconclusion and points out some future research directions

2 Related Background

21 Basic Definitions General MaOPs can be define asfollows

minimize 119865 (119909) = (1198911 (119909) 1198912 (119909) (119891119898 (119909)119879))subject to x isin Ω sube Rn

(1)

where x = (x1 xn)T is the n-dimensional decisionvariable vector Ω is the decision space and F Ω 997888rarr Rm

consists of m objective functions fi(x) (i = 1 2 m) m ⩾4Rm is called the objective space

Given two decision vectors xy isin Ω x is said to Paretodominate y (x≺y) if

forall119894 isin 1 2 119898 119891119894 (119909) le 119891119894 (119910)exist119894 isin 1 2 119898 119891119894 (119909) lt 119891119894 (119910) (2)

A solution xlowastisinΩ is Pareto optimal if there is no x≺xlowast f(xlowast) iscalled a Pareto optimal objective vector The set of all Paretooptimal objective vectors is called the Pareto front (PF) andthe set of all Pareto optimal solutions is called the Paretooptimal set (PS)

22 MOEAD MOEAD it is the most popular multiobjec-tive evolutionary algorithm in recent years It first initializesN uniform and widely distributed vectors for each individualevolving in the direction of each vector which is equivalent todividing multiobjective optimization problems The solutionto the N scalar optimization subproblems is through theconstant Evolution can simultaneously obtain the optimalsolution of these N subproblems The algorithm proposes 3weighted methods that can be used weighted sum aggrega-tion Tchebycheff approach and PBI approach

The traditional weighted sum aggregation is the mostdirect method to deal withmultiobjective optimization prob-lems and it is also a method used earlier This methodcombines or aggregates all the optimized subobjectivesinto a single-objective and the multiobjective optimizationproblem is converted into a single-objective optimizationproblem

The expression of weighted sum aggregation is

min119892119908119904 (119909 | 120582) = 119898sum119894=1

120582119894119891119894 (119909) (3)

The weighted sum aggregation method generates a setof different Pareto optimal solutions by generating differentweight vectors in the above scalar optimization problemHowever in the case that the shape of the optimal Paretosurface is nonconvex thismethod cannot obtain every Paretooptimal vector

The expression of Tchebycheff approach is

min119892119905119890 (119909 | 120582 119911lowast) = max1le119894le119898

120582119894 | 119891119894 (119909) minus 119911lowast119894 (4)

For each Pareto optimality x there is always a weightvector y such that the solution of (4) is a Pareto optimalsolution which corresponds to the Pareto optimal solutionof the original multiobjective problem And by modifyingthe weight vector different solutions on the Pareto optimalsurface can be obtained One disadvantage of this approachis that the aggregate method curves are not smooth whendealing with continuous multiobjective problems

The PBI method aims to find the intersection of theuppermost boundary and a set of lines In a sense if thelines are evenly distributed the resulting intersections can beseen as providing a good approximation to the overall Paretooptimal boundary These methods can deal with the Paretooptimal boundary nonconvex problemThe PBI method willbe described in detail later

23MOEAD-ACO Thebasic framework ofMOEAD-ACOfirst decomposes the MOPs into Ns single-objective sub-problems by N weight vectors 1205821 120582N Then MOEAD-ACO utilizes N ants to solve these single-objective sub-problems Ant ia is responsible for subproblems is Finally

Mathematical Problems in Engineering 3

(1) Initialization generate Nini Solutions by weight vectors 120582(2) Decomposition by some aggregation approach(3) Solution construction(4) Update of EP(5) Update of pheromone(6) Check the solutions in the neighbourhood and update the solutions(7) Termination judgment and output EP (a set of high-quality solutions XS)

Algorithm 1 MOEAD-ACO

Ant 1

Ant 2Ant 6 Group 2Group 1

f1

f2

The neighbourhood of ant 6

Pareto optimal vector by 1

Pareto optimal vector by 2

Pareto optimal vector by 6

Figure 1 MOEAD-ACO

MOEAD-ACO uses the concept of neighbourhood andgroup for decomposing the MaOPs Figure 1 illustrates themethod of above

Firstly MOEAD-ACO decomposes the MOP into Nssingle-objective subproblems by choosing Nw weight vectors(1205821 120582Nw) Then the subproblem is was associated withweight vector 120582is and its objective function is denoted as119892 (x | 120582is) MOEAD-ACO utilizes Na ants for solvingthese single-objective subproblems while ant ia is responsiblefor subproblem is Both the weighted sum and Tchebycheffapproaches are used inMOEAD-ACO and are recognized asMOEAD-ACOW and MOEAD-ACOT respectively How-ever recently Deb et al proposed that PBI decompositionmethod is more suitable for solving MOPs

The MOEAD-ACO is shown in Algorithm 1 In Algo-rithm 1 for each generation each ant constructs a solutionin Step (3) In Step (4) the newly constructed solution isupdated in the external archive and an ant updates thepheromone matrix of its group in Step (5) if it has found anew nondominated solution In Step (6) each ant updates itscurrent solution if there is a solution that (1) is better than its

p

F(X)

d2

d1

EF

Attainable objective space

f1

f2

Zlowast

Figure 2 Illustration of PBI approach

current solutions and (2) has not been used for updating anyother old solution The algorithm is terminated in Step (7)24 PBI Method In our paper we use the PBI approachbecause of its promising performance for many-objectiveoptimization problemsThe optimization problem of the PBIapproach is defined as

minimize 119892pbi (x | w zlowast) = d1 + 120579d2subject to x isin Ω (5)

where

1198891 =1003816100381610038161003816(119865 (119909) minus 119911lowast) 1198791199081003816100381610038161003816|119882|

1198892 =1003816100381610038161003816119865 (119909) minus (119911 lowast + 1198891119882)1003816100381610038161003816|119882|

(6)

Zlowast= (z1 zlowastm)T is the ideal objective vector with zilowast ltmin fi(x) x isin Ω i isin 1 m a penalty parameter 120579predefined by the user Figure 2 presents an example toillustrate d1 and d2 of a solution xi In the PBI approach d1is used to measure the convergence of xi towards the EF d2is a kind of measure for the population diversity 119892pbi (x |w zlowast) is a composite measure of x for both diversity andconvergenceThe distance between d1 and d2 is controlled bythe parameter 120579 In our paper we set 120579 = 50 for empiricalstudies

4 Mathematical Problems in Engineering

f1

f2

X3 X5 X2

X6

X1 X7

X4 Zlowast

Figure 3 Cluster-based weight vector adjustment

25 Cluster-BasedWeight Vector Adjustment For theMaOPsproblem this paper proposes a weighted vector adjustmentmethod based on clustering and uses different pheromonematrices to solve different single objectives This scalableapproach can help the ant colony algorithm to better solvethe MaOPs problem

After the multiobjective problem is decomposed intoN single-target subproblems by the PBI method N antsare selected to solve these single-target subproblems Anti solves the subproblem i The subproblem i is associatedwith 120582i The corresponding objective function is 119892(x |120582i) N ants are divided into K groups according to theircorresponding weight distances Each group approaches asmall area of the Pareto frontier The same group of antsshare a pheromone matrix that accommodates the locationinformation of the learned Pareto frontier subregions Eachant looks for a special point on the Pareto frontier for a single-target subproblem Each ant has its ownheuristic informationmatrix to store a priori knowledge of its subproblemsThe antirsquos neighbour B (i) contains T ants whose weights are closestto 120582i in N weights and i isin B (i) and ant i is its own neigh-bour

In each iteration of the algorithm if the solution set in thedominating set remains unchanged for the first time it meansthat the current weight vector can no longer meet the needof guiding population evolution and weight adjustment isneeded In this case nondominated populations are classifiedbased on the clustering method assuming that there are 7individuals in the objective space (x1 x2 x7) They hadthree clusters x3 x5 x2 x6 and x1 x7 x4 Among themx3 x6 and x7 are cluster centres as shown in Figure 3

3 Algorithm Description

31 MOEAD-ACO-PBI In this paper we propose an algo-rithm called MOEAD-ACO-PBI The original MOEAD-ACO used a weighted sum and Tchebycheff method to solvethe problemHowever this is difficult because the correlationbetween the algebraic and geometrical interpretation of theweight sets is not valid [15] Recent studies conducted by[16] have shown that MOEAD-PBI is the better methodfor many-objective optimization problems Therefore weintroduce the PBI approach to MOEAD-ACO known asMOEAD-ACO-PBI The MOEAD-ACO-PBI algorithm ispresented in Algorithm 2

(1) Initialization(2) Decomposition by PBI approach(3) Solution construction(4) Update of EP(5) Update of pheromone(6) Update the solutions(7) Termination

Algorithm 2 MOEAD-ACO-PBI

32 Initialization Procedure Each ant ia has a heuristicinformation value 120578119894119886119896119897 each group j has pheromone trail 120591119896119894119886119895and each individual solution xia is a tour represented bypermutation

The initialization procedure of MOEAD-ACO-PBI con-tains six main steps(1) Setting of Nw and 120582Nw the sets of weight vectors120582Nw = 1205821 120582Nw are generated by a systematic approachdeveloped from Das and Dennisrsquos method [17] Here the sizeof 120582 is Nw

119873119908 = 119862119898minus1119867+119898minus1 (7)

where H gt 0 is the quantity of divisions considered alongeach objective coordinate Therefore a subproblem is has aweight vector 120582is that satisfiessum119898119897=1 120582119894119904119897 = 1 and 120582119894119904119896 ge 0 wherem is the number of objectives(2) Initialize the solution xi to the subproblem is and setFi= F (xi) for i = 1 n(3) Set group the number of groups is J which is avalue parameter that requires to be selected The number ofsubproblems of each group is equally distributed by clusteringthe weights in terms of the Euclidean distance(4) Initialize the pheromone information matrix 120591119896119894119895 = 1for all j = 1 k and k = 1 n The motivation is to encouragethe search to focus on exploration in its early stage(5) Initialize 120578119894119896119897 = 1sum119898119895=1 120582119894119895119888119895119896119897 which is the heuristicinformation matrix to the subproblem is for j = 1 k(6) Initialize EP as the set of the entire high-qualitysolutions in F1 FN EP is initialized to be empty

33 Solution Construction Assume that ant ia is in group jand its current solution xi = (x1

i xni) Ant ia constructs itsnew solution through the following steps(1)The probability of choice of the link is set For k l =1 n set

Φ119896119897 = [Γ119894119886119896119897 + Δ lowast ln (119909119894119886 (119896 119897))]119886 (120578119894119886119896119897)120573 (8)

Φkl is representing the attractiveness of the link betweenthe k and l to ant ia The indicator function in (xia (k l)) isequal to 1 if the link (k l) is in tour xia or 0 if otherwise 120572 120573and Δ are the three control parameters(2) First ant ia chooses an ant randomly as its startingpoint and builds its tour Suppose that its current position is land it has not completed its tour From S (not visited so far)

Mathematical Problems in Engineering 5

Table 1 Test functions utilized in this experiment

function Formulation

CF1 1198911(119909) = 1199091 + 2|1198691| sum119895isin1198691(119909119895 minus 119909105(10+3(119895minus2)(119899minus2)))2

1198912(119909) = 1 minus 1199091 + 2100381610038161003816100381611986921003816100381610038161003816 sum119895isin1198692 (119909119895 minus 119909105(10+3(119895minus2)(119899minus2)))2

CF2 1198911(119909) = 1199091 + 2|1198691| sum119895isin1198691(119909119895 minus sin(61205871199091 + 119895120587119899 ))21198912(119909) = 1 minus radic1199091 + 2100381610038161003816100381611986921003816100381610038161003816 sum119895isin1198692 (119909119895 minus cos(61205871199091 + 119895120587119899 ))

2

CF3 1198911(119909) = 1199091 + 2100381610038161003816100381611986911003816100381610038161003816 (4sum119895isin11986911199102119895 minus 2prod119895isin1198691

cos(20119910119895120587radic119895 ) + 2)

CF41198912(119909) = 1 minus 11990921 + 2100381610038161003816100381611986921003816100381610038161003816 (4sum119895isin1198692119910

2119895 minus 2prod119895isin1198692

cos(20119910119895120587radic119895 ) + 2)1198911 = 1199091 + sum

119895isin1198691

ℎ119895(119910119895)1198911 = 1 minus 1199091 + sum

119895isin1198692

ℎ119895(119910119895)

it selects the ant k to visit next according to the followingprobability by the roulette wheel selection

Φ119896119897sum119904isin119862Φ119904119897 (9)

(3) If the ant has visited all the points it returns its tour

34 Update of Pheromone The pheromone trail value of thelink (k l) for group j is updated as follows

120591119894119886119896119897 fl 120588120591119895119896119897+ sum119909isinΠ

1119892 (119909 | 120582119894119886) (10)

120588 is the persistence rate of the pheromone trail Π is theset of all the new solutions it was constructed by the ants ingroup j in the current iteration and was just added to EP Itcontains the link between the antsrsquo k and l 120591min and 120591max areused to limit the range of the pheromone 120591min is given by

120591min = 120576120591max (11)

where 0 lt 120576 lt 1 is a control parameterThen if 120591119894119886119896119897 lt 120591minset 120591119894119886119896119897 = 120591min and if 120591119894119886119896119897 gt 120591max set 120591119894119886119896119897 = 120591max 120591max is set

120591max = 119861 + 1(1 minus 120588) 119892min(12)

where B is the number of nondominated solutions foundin the current iteration and119892min is the smallest value obtainedfor the objective functions of all the Ns subproblems

4 Simulation Results

In this section we compare the MOEAD-ACO-PBIalgorithm with three state-of-the-art algorithms includingMOEAD-ACOW[13] MOEAD [10] and Multiple Ant

Colony System (MACS) [18] The test problems areintroduced in Section 41 Section 42 describes the qualityindicators Three state-of-the-art algorithms used forcomparison and the corresponding parameter settings arebriefly introduced in Section 43

41 Test Problems The well-known MaOPs test functiondefinitions of CF1ndashCF4 [19] are used to test the proposedalgorithm Table 1 shows the CF1ndashCF4 problems with theCF test suites CF1-CF4 are test functions with biobjectiveproblem CF test suites are used to test the low dimensionalobjective function To verify the ability of algorithms to dealwith more objectives we select C1-DTLZ1 C2-DTLZ2 andC3-DTLZ4 [20] used to test the high dimensional objectivefunction CF suites are popular test problem for MaOPs

C1-DTLZ1 is a MaOPs problem the problem is formu-lated as

119862 (119909) = 1 minus 119891119898 (119909)06 minus 119898minus1sum119894=1

119891119894 (119909)05 ge 0 (13)

C2-DTLZ2 is a MaOPs problem the problem is formu-lated as

119862 (119909) = minusminmin119898119894=1 [[

(119891119894 (119909) minus 1)2 + 119898sum119895=1119895 =119894

119891 2(119909)119895

minus 1199032]] times[ 119898sum119894=1

(119891119894 (119909) minus 1radic119898)2 minus 1199032]

ge 0(14)

C3-DTLZ4 is a MaOPs problem the problem is formu-lated as

119862119894 (119909) = 1198912119894 (119909)4 + 119898sum119895=1119895 =119894

1198912119894 (119909) minus 1 ge 0 (15)

6 Mathematical Problems in Engineering

Table 2The average and standard deviation ofGDvalues obtained forCF1ndashCF4C1 DTLZ1 C2 DTLZ2 andC3 DTLZ4with four algorithmsand different number of objectives Best performance is in bold face

M MOEAD-ACO-PBI MOEAD-ACOW MOEAD MACSCF1 2 10013e-4 (114e-4) 11945e-4 (208e-4) 12524e-4 (212e-2) 34214e-2 (292e-2)CF2 2 23325e-2 (105e-2) 20382e-2 (259e-2) 29194e-2 (124e-2) 45986e-2 (428e-2)CF3 2 22667e-1 (168e-1) 37510e-1 (241e-1) 33477e-1 (319e-1) 42804e-1 (132e-1)CF4 2 44754e-2 (129e-1) 54257e-2 (138e-1) 42232e-2 (719e-2) 21410e-1 (278e-1)

3 30152e-3 (213e-4) 32859e-3 (236e-4) 34831e-3 (332e-4) 44339e-3 (133e-4)

C1 DTLZ1 4 30221-3 (223e-4) 25532e-2 (223e-3) 21234e-2 (134e-3) 24452e-2 (212e-2)5 44264e-3 (325e-4) 33354e-2 (223e-2) 54261e-2 (234e-3) 53634e-2 (284e-2)

C2 DTLZ23 74664e-4 (531e-5) 22772e-3 (429e-4) 95259e-3 (167e-3) 79976e-3 (529e-3)4 27056e-3 (115e-4) 11709e-2 (481e-3) 11867e-2 (128e-3) 27389e-2 (108e-2)5 52944e-3 (299e-4) 25550e-2 (110e-2) 27631e-2 (695e-3) 59090e-2(284e-4)

C3 DTLZ43 40236e-4 (223e-4) 69914e-4 (328e-4) 45365e-4 (265e-4) 3514e-3 (326e-3)4 30885e-3 (232e-3) 22035e-2 (382e-2) 13564e-2 (136e-2) 25658e-2 (125e-4)5 30624e-3 (325e-4) 36412e-2 (221e-2) 37225e-2 (562e-3) 53361e-2 (326e-4)

42 Quality Indicators In our empirical study the followingtwo widely used quality indicators were considered The firstone reflects the convergence of an algorithmwhile the secondone reflects the convergence and diversity of the solutionssimultaneously

421 Generational Distance (GD) Indicator For any kindof algorithm let P be the set of final nondominated pointsobtained from the objective space Let Plowast be a set of pointsuniformly spread over the true PF The GD can indicate onlythe convergence of an algorithm [21] and a smaller valueindicates better quality The GD is computed as

119866119863 (119875 119875lowast) = radicsum119906isin119875 119889 (119906 119875lowast)|119875| (16)

422 Inverted Generational Distance (IGD) Indicator Forany kind of algorithm let P be the set of final nondominatedpoints obtained from the objective space and Plowast be a setof points uniformly spread over the true PF The IGD canindicate both the convergence and diversity [21] and asmaller value indicates better quality The IGD is computedas

119868119866119863 (119875 119875lowast) = sumVisin119875lowast 119889 (V 119875)|119875lowast| (17)

43 Parameter Settings The parameters for the four MOEAsconsidered in this paper are listed as follows(1) Some parameter settings in MOEAD-ACO-PBI setT=8 120572=1 120573=8 120588=095 r=08 and 120579=50(2) Some parameter settings in MOEAD-ACOW andMOEAD T=8 120572=1 120573=8 120588=095 and r=08 The otherparameters can be obtained from [15](3) Some parameter settings in MACS 120572=1 120573=2 120588=09maxFcctor=5 and AntNum=10(4) Number of runs and termination condition allalgorithms were programmed in PlatEMO [22] and each

algorithm was independently run 30 times for each testinstance and stopped after 300 generations All the algorithmswere executed on a desktop PC with a 21-GHz CPU 8-GBRAM and Windows 10

5 Results and Discussion

The GD indicator was utilized to compare the conver-gence capacity between the proposed MOEAD-ACO-PBIalgorithm and three other algorithms (Figure 4) Table 2shows the average and standard deviation of the GD valuesobtained by the four algorithms for test suits with differentnumber of objectives Table 2 shows that the MOEAD-ACO-PBI algorithm could perform well on all the testfunctions especially on CF1 and CF3 problems When thenumber of objective test functions is higher the performanceof MOEAD-ACO-PBI algorithm is better The results ofcomparison of the MOEAD-ACO-PBI algorithm with theother three MOEAs in terms of IGD values using the testsuite are provided in Table 3 It shows the average IGDvalues for the four MOEAs over 30 independent runs andthe best average and standard deviation values From theexperimental results of the test problem theMOEAD-ACO-PBI algorithmperforms onCF2 CF4 C1-DTLZ1 C2-DTLZ2and C3-DTLZ4 test problems It is shown that theMOEAD-ACO-PBI algorithm has better performance of convergenceand diversity In addition performance of MOEAD-ACOW

algorithm is also better andMACS performance is the worstThe conclusions are given below(1)MOEAD-ACO-PBI has the best convergence perfor-

mance as its approximate fronts covermost of those returnedby the other algorithms MOEAD-ACOW can obtain well-spread approximately fronts however some of its solutionswere dominated by MOEAD-ACO-PBI(2) It can be observed in Figure 3 that MOEAD-ACO-PBI dominates in most solutions and that the advantage ofMOEAD-ACO-PBI is more pronounced as the number ofobjectives increases

Mathematical Problems in Engineering 7

C1_DTLZ1 (1)

C2_DTLZ2 (1)

C3_DTLZ4 (1)

(2)CF1 (1) (3) (4)

(2)

CF4 (1)

(3) (4)

(2) (3) (4)

(2) (3) (4)

(2)CF3 (1) (3) (4)

(2)CF2 (1) (3) (4)

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1 15 2 25 3 35 4 45 5 1 15 2 25 3 35 4 45 5

1 15 2 25 3 35 4 45 5 1 15 2 25 3 35 4 45 5

1 15 2 25 3 35 4 45 5 1 15 2 25 3 35 4 45 5

1 15 2 25 3 35 4 45 5 1 15 2 25 3 35 4 45 5

1 15 2 25 3 35 4 45 5 1 15 2 25 3 35 4 45 5 1 15 2 25 3 35 4 45 5 1 15 2 25 3 35 4 45 5

1 15 2 25 3 35 4 45 5 1 15 2 25 3 35 4 45 5

1 15 2 25 3 35 4 45 5 1 15 2 25 3 35 4 45 5

1 15 2 25 3 35 4 45 5 1 15 2 25 3 35 4 45 5

1 15 2 25 3 35 4 45 5 1 15 2 25 3 35 4 45 5

Figure 4 Parallel coordinates figure by four algorithms on the four-objective CF1 CF2 and CF3 (1) MOEAD-ACO-PBI (2) MOEAD-ACOW and (3)MOEAD (4)MACS

8 Mathematical Problems in Engineering

Table 3 The average and standard deviation of IGD values obtained for CF1ndashCF4 C1 DTLZ1 C2 DTLZ2 and C3 DTLZ4 with fouralgorithms and different number of objectives Best performance is in bold face

M MOEAD-ACO-PBI MOEAD-ACOW MOEAD MACSCF1 2 64371e-1 (136e-1) 51532e-1 (127e-1) 65734e-1 (108e-1) 78365e-1 (189e-1)CF2 2 45894e-2 (128e-2) 48905e-2 (252e-2) 62428e2 (610e-2) 24116e-1 (361e-1)CF3 2 37740e-1 (778e-2) 31772e-1 (932e-2) 37877e-1 (177e-1) 37581e-1 (254e-1)CF4 2 16256e-1 (634e-2) 24364e-1 (118e-1) 19728e-1 (717e-2) 24654e-1 (132e-1)

C1 DTLZ1 4 22653e-1 (325e-4) 34629e-1 (236e-2) 25698e-1 326e-3) 37241e-1 (321e-4)5 31651e-1 (413e-4) 36234e-1 (112e-1) 37511e-1 (169e-1) 59283e-1 (387e-4)

C2 DTLZ23 51902e-2 (509e-4) 65049e-2 (146e-3) 59751e-2 (485e-3) 93947e-2 (489e-4)4 15255e-1 (849e-4) 23852e-1 (201e-2) 19833e-1 (552e-3) 47290e-1 (894e-4)5 23283e-1 (429e-4) 29947e-1 (128e-1) 29391e-1 (104e-1) 55282e-1 (494e-4)

C3 DTLZ43 51141e-2 (432e-4) 73349e-2 (232e-3) 53365e-2 (332e-3) 65254e-2 (235e-4)4 11347e-1 (132e-4) 25983e-1 (325e-2) 14651e-1 (222e-3) 24014e-1 (752e-4)5 21698e-1 (324e-4) 27321e-1 (236e-1) 26651e-1 (152e-1) 44652e-1 (532e-4)

6 Conclusions

The major contribution of this paper is the proposal of aMOEAD-ACO-PBI algorithm for MaOPs Due to the factthat the number of model optimization objectives increasesthe PBI aggregate function was selected to decompose theobjectives MOEAD-ACO-PBI algorithm can better ensurethe convergence of algorithms and the balance of diversity soas to obtain a better Pareto solution based on the simulationexperiment it could be seen that the results of the MOEAD-ACO-PBI algorithm are significantly better than those ofthe other algorithms Our future work aims at analysingfurther how to cope with constraint and dynamic constraintprocessing and to optimize the number of objectives moreeffectively

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This research work was supported by the national science andtechnology major projects (NO 2009ZX04001-111) subpro-ject of national science and technologymajor (2013ZX04002-031) key laboratory of automotive power train and elec-tronics (Hubei University of Automotive Technology (NoZDK1201703) and Hubei Provincial Education DepartmentYouth Fund (No Q20181801)

References

[1] B Li J Li K Tang and X Yao ldquoMany-objective evolutionaryalgorithms a surveyrdquo ACM Computing Surveys vol 48 no 1article 13 2015

[2] A Zhou B Y Qu H Li S Z Zhao P N Suganthan and QZhang ldquoMultiobjective evolutionary algorithms a survey of the

state of the artrdquo Swarm Evolutionary Computation vol 1 no 1pp 32ndash49 2011

[3] C Wang Y Wang K Wang Y Dong and Y Yang ldquoAnImproved Hybrid Algorithm Based on BiogeographyComplexand Metropolis for Many-Objective OptimizationrdquoMathemat-ical Problems in Engineering vol 2017 2017

[4] A Trivedi D Srinivasan K Sanyal and A Ghosh ldquoA surveyof multiobjective evolutionary algorithms based on decompo-sitionrdquo IEEE Transactions on Evolutionary Computation vol 21no 3 pp 440ndash462 2017

[5] C Von Lucken B Baran and C Brizuela ldquoA survey onmulti-objective evolutionary algorithms for many-objectiveproblemsrdquo Computational Optimization and Applications vol58 no 3 pp 707ndash756 2014

[6] H Jain and K Deb ldquoAn evolutionary many-objective opti-mization algorithm using reference-point based nondominatedsorting approach Part II Handling constraints and extendingto an adaptive approachrdquo IEEE Transactions on EvolutionaryComputation vol 18 no 4 pp 602ndash622 2014

[7] Z He G G Yen and J Zhang ldquoFuzzy-based pareto optimalityformany-objective evolutionary algorithmsrdquo IEEETransactionson Evolutionary Computation vol 18 no 2 pp 269ndash285 2014

[8] D H Phan and J Suzuki ldquoR2 indicator based multiobjectivememetic optimization for the pickup and delivery problemwithtime windows and demands (PDP-TW-Drdquo in Proceedings ofthe Conference on Bioinspired Information and CommunicationsTechnologies vol 22 pp 43ndash50 2014

[9] J Bader and E Zitzler ldquoHypE an algorithm for fast hypervol-ume-based many-objective optimizationrdquo Evolutionary Com-putation vol 19 no 1 pp 45ndash76 2011

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

[11] H Zhu Z He and Y Jia ldquoAn improved reference point basedmulti-objective optimization by decompositionrdquo InternationalJournal of Machine Learning and Cybernetics vol 7 no 4 pp581ndash595 2016

[12] I Alaya C Solnon and K Ghedira ldquoAnt Colony Optimizationfor Multi-Objective Optimizationrdquo in Proceedings of the Prob-lems vol 1 pp 450ndash457 2017

[13] L Ke Q Zhang and R Battiti ldquoMOEAD-ACO a mul-tiobjective evolutionary algorithm using decomposition and

Mathematical Problems in Engineering 9

AntColonyrdquo IEEE Transactions on Cybernetics vol 43 no 6 pp1845ndash1859 2013

[14] M Z De Souza and A T R Pozo ldquoA GPU implementationof MOEAD-ACO for the multiobjective traveling salesmanproblemrdquo in Proceedings of the 3rd Brazilian Conference onIntelligent Systems BRACIS 2014 pp 324ndash329 Brazil October2014

[15] T Lust and J Teghem ldquoTwo-phase pareto local search for thebiobjective traveling salesman problemrdquo Journal of Heuristicsvol 16 no 3 pp 475ndash510 2010

[16] K Deb and H Jain ldquoAn evolutionary many-objective opti-mization algorithmusing reference-point-based nondominatedsorting approach part I solving problemswith box constraintsrdquoIEEE Transactions on Evolutionary Computation vol 18 no 4pp 577ndash601 2014

[17] I Das and J E Dennis ldquoNormal-boundary intersection a newmethod for generating the Pareto surface in nonlinear multi-criteria optimization problemsrdquo SIAM Journal on Optimizationvol 8 no 3 pp 631ndash657 1998

[18] B Baran and M Schaerer ldquoA multiobjective ant colony systemfor vehicle routing problem with time windowsrdquo in Proceedingsof the 21st IASTED International Multi-Conference on AppliedInformatics pp 97ndash102 Austria February 2003

[19] Q Zhang A Zhou S Zhao P N Suganthan W Liu and STiwari ldquoMultiobjective optimization test instances for the CEC2009 special session and competitionrdquo Tech Rep University ofEssex Colchester UK and Nanyang technological University2008

[20] M Asafuddoula T Ray and R Sarker ldquoA decomposition-basedevolutionary algorithm for many objective optimizationrdquo IEEETransactions on Evolutionary Computation vol 19 no 3 pp445ndash460 2015

[21] CWang YWang KWang Y Yang and Y Tian ldquoAn improvedbiogeographycomplex algorithm based on decomposition formany-objective optimizationrdquo in Proceedings of the Interna-tional Journal of Machine Learning Cybernetics (2 vol 2 p 172017

[22] Y Tian R Cheng X Zhang and Y Jin Platemo a matlabplatform for evolutionary multi-objective optimization arXivpreprint arXiv 170100879 2017

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Mathematical Problems in Engineering 3

(1) Initialization generate Nini Solutions by weight vectors 120582(2) Decomposition by some aggregation approach(3) Solution construction(4) Update of EP(5) Update of pheromone(6) Check the solutions in the neighbourhood and update the solutions(7) Termination judgment and output EP (a set of high-quality solutions XS)

Algorithm 1 MOEAD-ACO

Ant 1

Ant 2Ant 6 Group 2Group 1

f1

f2

The neighbourhood of ant 6

Pareto optimal vector by 1

Pareto optimal vector by 2

Pareto optimal vector by 6

Figure 1 MOEAD-ACO

MOEAD-ACO uses the concept of neighbourhood andgroup for decomposing the MaOPs Figure 1 illustrates themethod of above

Firstly MOEAD-ACO decomposes the MOP into Nssingle-objective subproblems by choosing Nw weight vectors(1205821 120582Nw) Then the subproblem is was associated withweight vector 120582is and its objective function is denoted as119892 (x | 120582is) MOEAD-ACO utilizes Na ants for solvingthese single-objective subproblems while ant ia is responsiblefor subproblem is Both the weighted sum and Tchebycheffapproaches are used inMOEAD-ACO and are recognized asMOEAD-ACOW and MOEAD-ACOT respectively How-ever recently Deb et al proposed that PBI decompositionmethod is more suitable for solving MOPs

The MOEAD-ACO is shown in Algorithm 1 In Algo-rithm 1 for each generation each ant constructs a solutionin Step (3) In Step (4) the newly constructed solution isupdated in the external archive and an ant updates thepheromone matrix of its group in Step (5) if it has found anew nondominated solution In Step (6) each ant updates itscurrent solution if there is a solution that (1) is better than its

p

F(X)

d2

d1

EF

Attainable objective space

f1

f2

Zlowast

Figure 2 Illustration of PBI approach

current solutions and (2) has not been used for updating anyother old solution The algorithm is terminated in Step (7)24 PBI Method In our paper we use the PBI approachbecause of its promising performance for many-objectiveoptimization problemsThe optimization problem of the PBIapproach is defined as

minimize 119892pbi (x | w zlowast) = d1 + 120579d2subject to x isin Ω (5)

where

1198891 =1003816100381610038161003816(119865 (119909) minus 119911lowast) 1198791199081003816100381610038161003816|119882|

1198892 =1003816100381610038161003816119865 (119909) minus (119911 lowast + 1198891119882)1003816100381610038161003816|119882|

(6)

Zlowast= (z1 zlowastm)T is the ideal objective vector with zilowast ltmin fi(x) x isin Ω i isin 1 m a penalty parameter 120579predefined by the user Figure 2 presents an example toillustrate d1 and d2 of a solution xi In the PBI approach d1is used to measure the convergence of xi towards the EF d2is a kind of measure for the population diversity 119892pbi (x |w zlowast) is a composite measure of x for both diversity andconvergenceThe distance between d1 and d2 is controlled bythe parameter 120579 In our paper we set 120579 = 50 for empiricalstudies

4 Mathematical Problems in Engineering

f1

f2

X3 X5 X2

X6

X1 X7

X4 Zlowast

Figure 3 Cluster-based weight vector adjustment

25 Cluster-BasedWeight Vector Adjustment For theMaOPsproblem this paper proposes a weighted vector adjustmentmethod based on clustering and uses different pheromonematrices to solve different single objectives This scalableapproach can help the ant colony algorithm to better solvethe MaOPs problem

After the multiobjective problem is decomposed intoN single-target subproblems by the PBI method N antsare selected to solve these single-target subproblems Anti solves the subproblem i The subproblem i is associatedwith 120582i The corresponding objective function is 119892(x |120582i) N ants are divided into K groups according to theircorresponding weight distances Each group approaches asmall area of the Pareto frontier The same group of antsshare a pheromone matrix that accommodates the locationinformation of the learned Pareto frontier subregions Eachant looks for a special point on the Pareto frontier for a single-target subproblem Each ant has its ownheuristic informationmatrix to store a priori knowledge of its subproblemsThe antirsquos neighbour B (i) contains T ants whose weights are closestto 120582i in N weights and i isin B (i) and ant i is its own neigh-bour

In each iteration of the algorithm if the solution set in thedominating set remains unchanged for the first time it meansthat the current weight vector can no longer meet the needof guiding population evolution and weight adjustment isneeded In this case nondominated populations are classifiedbased on the clustering method assuming that there are 7individuals in the objective space (x1 x2 x7) They hadthree clusters x3 x5 x2 x6 and x1 x7 x4 Among themx3 x6 and x7 are cluster centres as shown in Figure 3

3 Algorithm Description

31 MOEAD-ACO-PBI In this paper we propose an algo-rithm called MOEAD-ACO-PBI The original MOEAD-ACO used a weighted sum and Tchebycheff method to solvethe problemHowever this is difficult because the correlationbetween the algebraic and geometrical interpretation of theweight sets is not valid [15] Recent studies conducted by[16] have shown that MOEAD-PBI is the better methodfor many-objective optimization problems Therefore weintroduce the PBI approach to MOEAD-ACO known asMOEAD-ACO-PBI The MOEAD-ACO-PBI algorithm ispresented in Algorithm 2

(1) Initialization(2) Decomposition by PBI approach(3) Solution construction(4) Update of EP(5) Update of pheromone(6) Update the solutions(7) Termination

Algorithm 2 MOEAD-ACO-PBI

32 Initialization Procedure Each ant ia has a heuristicinformation value 120578119894119886119896119897 each group j has pheromone trail 120591119896119894119886119895and each individual solution xia is a tour represented bypermutation

The initialization procedure of MOEAD-ACO-PBI con-tains six main steps(1) Setting of Nw and 120582Nw the sets of weight vectors120582Nw = 1205821 120582Nw are generated by a systematic approachdeveloped from Das and Dennisrsquos method [17] Here the sizeof 120582 is Nw

119873119908 = 119862119898minus1119867+119898minus1 (7)

where H gt 0 is the quantity of divisions considered alongeach objective coordinate Therefore a subproblem is has aweight vector 120582is that satisfiessum119898119897=1 120582119894119904119897 = 1 and 120582119894119904119896 ge 0 wherem is the number of objectives(2) Initialize the solution xi to the subproblem is and setFi= F (xi) for i = 1 n(3) Set group the number of groups is J which is avalue parameter that requires to be selected The number ofsubproblems of each group is equally distributed by clusteringthe weights in terms of the Euclidean distance(4) Initialize the pheromone information matrix 120591119896119894119895 = 1for all j = 1 k and k = 1 n The motivation is to encouragethe search to focus on exploration in its early stage(5) Initialize 120578119894119896119897 = 1sum119898119895=1 120582119894119895119888119895119896119897 which is the heuristicinformation matrix to the subproblem is for j = 1 k(6) Initialize EP as the set of the entire high-qualitysolutions in F1 FN EP is initialized to be empty

33 Solution Construction Assume that ant ia is in group jand its current solution xi = (x1

i xni) Ant ia constructs itsnew solution through the following steps(1)The probability of choice of the link is set For k l =1 n set

Φ119896119897 = [Γ119894119886119896119897 + Δ lowast ln (119909119894119886 (119896 119897))]119886 (120578119894119886119896119897)120573 (8)

Φkl is representing the attractiveness of the link betweenthe k and l to ant ia The indicator function in (xia (k l)) isequal to 1 if the link (k l) is in tour xia or 0 if otherwise 120572 120573and Δ are the three control parameters(2) First ant ia chooses an ant randomly as its startingpoint and builds its tour Suppose that its current position is land it has not completed its tour From S (not visited so far)

Mathematical Problems in Engineering 5

Table 1 Test functions utilized in this experiment

function Formulation

CF1 1198911(119909) = 1199091 + 2|1198691| sum119895isin1198691(119909119895 minus 119909105(10+3(119895minus2)(119899minus2)))2

1198912(119909) = 1 minus 1199091 + 2100381610038161003816100381611986921003816100381610038161003816 sum119895isin1198692 (119909119895 minus 119909105(10+3(119895minus2)(119899minus2)))2

CF2 1198911(119909) = 1199091 + 2|1198691| sum119895isin1198691(119909119895 minus sin(61205871199091 + 119895120587119899 ))21198912(119909) = 1 minus radic1199091 + 2100381610038161003816100381611986921003816100381610038161003816 sum119895isin1198692 (119909119895 minus cos(61205871199091 + 119895120587119899 ))

2

CF3 1198911(119909) = 1199091 + 2100381610038161003816100381611986911003816100381610038161003816 (4sum119895isin11986911199102119895 minus 2prod119895isin1198691

cos(20119910119895120587radic119895 ) + 2)

CF41198912(119909) = 1 minus 11990921 + 2100381610038161003816100381611986921003816100381610038161003816 (4sum119895isin1198692119910

2119895 minus 2prod119895isin1198692

cos(20119910119895120587radic119895 ) + 2)1198911 = 1199091 + sum

119895isin1198691

ℎ119895(119910119895)1198911 = 1 minus 1199091 + sum

119895isin1198692

ℎ119895(119910119895)

it selects the ant k to visit next according to the followingprobability by the roulette wheel selection

Φ119896119897sum119904isin119862Φ119904119897 (9)

(3) If the ant has visited all the points it returns its tour

34 Update of Pheromone The pheromone trail value of thelink (k l) for group j is updated as follows

120591119894119886119896119897 fl 120588120591119895119896119897+ sum119909isinΠ

1119892 (119909 | 120582119894119886) (10)

120588 is the persistence rate of the pheromone trail Π is theset of all the new solutions it was constructed by the ants ingroup j in the current iteration and was just added to EP Itcontains the link between the antsrsquo k and l 120591min and 120591max areused to limit the range of the pheromone 120591min is given by

120591min = 120576120591max (11)

where 0 lt 120576 lt 1 is a control parameterThen if 120591119894119886119896119897 lt 120591minset 120591119894119886119896119897 = 120591min and if 120591119894119886119896119897 gt 120591max set 120591119894119886119896119897 = 120591max 120591max is set

120591max = 119861 + 1(1 minus 120588) 119892min(12)

where B is the number of nondominated solutions foundin the current iteration and119892min is the smallest value obtainedfor the objective functions of all the Ns subproblems

4 Simulation Results

In this section we compare the MOEAD-ACO-PBIalgorithm with three state-of-the-art algorithms includingMOEAD-ACOW[13] MOEAD [10] and Multiple Ant

Colony System (MACS) [18] The test problems areintroduced in Section 41 Section 42 describes the qualityindicators Three state-of-the-art algorithms used forcomparison and the corresponding parameter settings arebriefly introduced in Section 43

41 Test Problems The well-known MaOPs test functiondefinitions of CF1ndashCF4 [19] are used to test the proposedalgorithm Table 1 shows the CF1ndashCF4 problems with theCF test suites CF1-CF4 are test functions with biobjectiveproblem CF test suites are used to test the low dimensionalobjective function To verify the ability of algorithms to dealwith more objectives we select C1-DTLZ1 C2-DTLZ2 andC3-DTLZ4 [20] used to test the high dimensional objectivefunction CF suites are popular test problem for MaOPs

C1-DTLZ1 is a MaOPs problem the problem is formu-lated as

119862 (119909) = 1 minus 119891119898 (119909)06 minus 119898minus1sum119894=1

119891119894 (119909)05 ge 0 (13)

C2-DTLZ2 is a MaOPs problem the problem is formu-lated as

119862 (119909) = minusminmin119898119894=1 [[

(119891119894 (119909) minus 1)2 + 119898sum119895=1119895 =119894

119891 2(119909)119895

minus 1199032]] times[ 119898sum119894=1

(119891119894 (119909) minus 1radic119898)2 minus 1199032]

ge 0(14)

C3-DTLZ4 is a MaOPs problem the problem is formu-lated as

119862119894 (119909) = 1198912119894 (119909)4 + 119898sum119895=1119895 =119894

1198912119894 (119909) minus 1 ge 0 (15)

6 Mathematical Problems in Engineering

Table 2The average and standard deviation ofGDvalues obtained forCF1ndashCF4C1 DTLZ1 C2 DTLZ2 andC3 DTLZ4with four algorithmsand different number of objectives Best performance is in bold face

M MOEAD-ACO-PBI MOEAD-ACOW MOEAD MACSCF1 2 10013e-4 (114e-4) 11945e-4 (208e-4) 12524e-4 (212e-2) 34214e-2 (292e-2)CF2 2 23325e-2 (105e-2) 20382e-2 (259e-2) 29194e-2 (124e-2) 45986e-2 (428e-2)CF3 2 22667e-1 (168e-1) 37510e-1 (241e-1) 33477e-1 (319e-1) 42804e-1 (132e-1)CF4 2 44754e-2 (129e-1) 54257e-2 (138e-1) 42232e-2 (719e-2) 21410e-1 (278e-1)

3 30152e-3 (213e-4) 32859e-3 (236e-4) 34831e-3 (332e-4) 44339e-3 (133e-4)

C1 DTLZ1 4 30221-3 (223e-4) 25532e-2 (223e-3) 21234e-2 (134e-3) 24452e-2 (212e-2)5 44264e-3 (325e-4) 33354e-2 (223e-2) 54261e-2 (234e-3) 53634e-2 (284e-2)

C2 DTLZ23 74664e-4 (531e-5) 22772e-3 (429e-4) 95259e-3 (167e-3) 79976e-3 (529e-3)4 27056e-3 (115e-4) 11709e-2 (481e-3) 11867e-2 (128e-3) 27389e-2 (108e-2)5 52944e-3 (299e-4) 25550e-2 (110e-2) 27631e-2 (695e-3) 59090e-2(284e-4)

C3 DTLZ43 40236e-4 (223e-4) 69914e-4 (328e-4) 45365e-4 (265e-4) 3514e-3 (326e-3)4 30885e-3 (232e-3) 22035e-2 (382e-2) 13564e-2 (136e-2) 25658e-2 (125e-4)5 30624e-3 (325e-4) 36412e-2 (221e-2) 37225e-2 (562e-3) 53361e-2 (326e-4)

42 Quality Indicators In our empirical study the followingtwo widely used quality indicators were considered The firstone reflects the convergence of an algorithmwhile the secondone reflects the convergence and diversity of the solutionssimultaneously

421 Generational Distance (GD) Indicator For any kindof algorithm let P be the set of final nondominated pointsobtained from the objective space Let Plowast be a set of pointsuniformly spread over the true PF The GD can indicate onlythe convergence of an algorithm [21] and a smaller valueindicates better quality The GD is computed as

119866119863 (119875 119875lowast) = radicsum119906isin119875 119889 (119906 119875lowast)|119875| (16)

422 Inverted Generational Distance (IGD) Indicator Forany kind of algorithm let P be the set of final nondominatedpoints obtained from the objective space and Plowast be a setof points uniformly spread over the true PF The IGD canindicate both the convergence and diversity [21] and asmaller value indicates better quality The IGD is computedas

119868119866119863 (119875 119875lowast) = sumVisin119875lowast 119889 (V 119875)|119875lowast| (17)

43 Parameter Settings The parameters for the four MOEAsconsidered in this paper are listed as follows(1) Some parameter settings in MOEAD-ACO-PBI setT=8 120572=1 120573=8 120588=095 r=08 and 120579=50(2) Some parameter settings in MOEAD-ACOW andMOEAD T=8 120572=1 120573=8 120588=095 and r=08 The otherparameters can be obtained from [15](3) Some parameter settings in MACS 120572=1 120573=2 120588=09maxFcctor=5 and AntNum=10(4) Number of runs and termination condition allalgorithms were programmed in PlatEMO [22] and each

algorithm was independently run 30 times for each testinstance and stopped after 300 generations All the algorithmswere executed on a desktop PC with a 21-GHz CPU 8-GBRAM and Windows 10

5 Results and Discussion

The GD indicator was utilized to compare the conver-gence capacity between the proposed MOEAD-ACO-PBIalgorithm and three other algorithms (Figure 4) Table 2shows the average and standard deviation of the GD valuesobtained by the four algorithms for test suits with differentnumber of objectives Table 2 shows that the MOEAD-ACO-PBI algorithm could perform well on all the testfunctions especially on CF1 and CF3 problems When thenumber of objective test functions is higher the performanceof MOEAD-ACO-PBI algorithm is better The results ofcomparison of the MOEAD-ACO-PBI algorithm with theother three MOEAs in terms of IGD values using the testsuite are provided in Table 3 It shows the average IGDvalues for the four MOEAs over 30 independent runs andthe best average and standard deviation values From theexperimental results of the test problem theMOEAD-ACO-PBI algorithmperforms onCF2 CF4 C1-DTLZ1 C2-DTLZ2and C3-DTLZ4 test problems It is shown that theMOEAD-ACO-PBI algorithm has better performance of convergenceand diversity In addition performance of MOEAD-ACOW

algorithm is also better andMACS performance is the worstThe conclusions are given below(1)MOEAD-ACO-PBI has the best convergence perfor-

mance as its approximate fronts covermost of those returnedby the other algorithms MOEAD-ACOW can obtain well-spread approximately fronts however some of its solutionswere dominated by MOEAD-ACO-PBI(2) It can be observed in Figure 3 that MOEAD-ACO-PBI dominates in most solutions and that the advantage ofMOEAD-ACO-PBI is more pronounced as the number ofobjectives increases

Mathematical Problems in Engineering 7

C1_DTLZ1 (1)

C2_DTLZ2 (1)

C3_DTLZ4 (1)

(2)CF1 (1) (3) (4)

(2)

CF4 (1)

(3) (4)

(2) (3) (4)

(2) (3) (4)

(2)CF3 (1) (3) (4)

(2)CF2 (1) (3) (4)

060504030201

0

060504030201

0

060504030201

0

07060504030201

0

07060504030201

0

354

325

215

105

0

07060504030201

0

07060504030201

0

070809

1

060504030201

0

141210

86420

070809

060504030201

0

070809

060504030201

0

070809

060504030201

0

0708

060504030201

0

060504030201

0

060504030201

0

07060504030201

0

060504030201

0

060504030201

0

060504030201

0

081

1214

060402

0

081

12

060402

0

081

12

060402

0

081

12

060402

0

081

12

060402

0

03503

04504

02502

01501

0050

03503

04505

04

02502

01501

0050

070605040302010

1 15 2 25 3 35 4 45 5

1

1

15

15

2

2

25

25

3

3

35

35

4

4

45

45

5

1 15 2 25 3 35 4 45 5

1 15 2 25 3 35 4 45 5 1 15 2 25 3 35 4 45 5

51 15 2 25 3 35 4 45 5

1 15 2 25 3 35 4 45 5

1 15 2 25 3 35 4 45 5 1 15 2 25 3 35 4 45 5

1 15 2 25 3 35 4 45 5 1 15 2 25 3 35 4 45 5

1 15 2 25 3 35 4 45 5 1 15 2 25 3 35 4 45 5

1 15 2 25 3 35 4 45 5 1 15 2 25 3 35 4 45 5

1 15 2 25 3 35 4 45 5 1 15 2 25 3 35 4 45 5 1 15 2 25 3 35 4 45 5 1 15 2 25 3 35 4 45 5

1 15 2 25 3 35 4 45 5 1 15 2 25 3 35 4 45 5

1 15 2 25 3 35 4 45 5 1 15 2 25 3 35 4 45 5

1 15 2 25 3 35 4 45 5 1 15 2 25 3 35 4 45 5

1 15 2 25 3 35 4 45 5 1 15 2 25 3 35 4 45 5

Figure 4 Parallel coordinates figure by four algorithms on the four-objective CF1 CF2 and CF3 (1) MOEAD-ACO-PBI (2) MOEAD-ACOW and (3)MOEAD (4)MACS

8 Mathematical Problems in Engineering

Table 3 The average and standard deviation of IGD values obtained for CF1ndashCF4 C1 DTLZ1 C2 DTLZ2 and C3 DTLZ4 with fouralgorithms and different number of objectives Best performance is in bold face

M MOEAD-ACO-PBI MOEAD-ACOW MOEAD MACSCF1 2 64371e-1 (136e-1) 51532e-1 (127e-1) 65734e-1 (108e-1) 78365e-1 (189e-1)CF2 2 45894e-2 (128e-2) 48905e-2 (252e-2) 62428e2 (610e-2) 24116e-1 (361e-1)CF3 2 37740e-1 (778e-2) 31772e-1 (932e-2) 37877e-1 (177e-1) 37581e-1 (254e-1)CF4 2 16256e-1 (634e-2) 24364e-1 (118e-1) 19728e-1 (717e-2) 24654e-1 (132e-1)

C1 DTLZ1 4 22653e-1 (325e-4) 34629e-1 (236e-2) 25698e-1 326e-3) 37241e-1 (321e-4)5 31651e-1 (413e-4) 36234e-1 (112e-1) 37511e-1 (169e-1) 59283e-1 (387e-4)

C2 DTLZ23 51902e-2 (509e-4) 65049e-2 (146e-3) 59751e-2 (485e-3) 93947e-2 (489e-4)4 15255e-1 (849e-4) 23852e-1 (201e-2) 19833e-1 (552e-3) 47290e-1 (894e-4)5 23283e-1 (429e-4) 29947e-1 (128e-1) 29391e-1 (104e-1) 55282e-1 (494e-4)

C3 DTLZ43 51141e-2 (432e-4) 73349e-2 (232e-3) 53365e-2 (332e-3) 65254e-2 (235e-4)4 11347e-1 (132e-4) 25983e-1 (325e-2) 14651e-1 (222e-3) 24014e-1 (752e-4)5 21698e-1 (324e-4) 27321e-1 (236e-1) 26651e-1 (152e-1) 44652e-1 (532e-4)

6 Conclusions

The major contribution of this paper is the proposal of aMOEAD-ACO-PBI algorithm for MaOPs Due to the factthat the number of model optimization objectives increasesthe PBI aggregate function was selected to decompose theobjectives MOEAD-ACO-PBI algorithm can better ensurethe convergence of algorithms and the balance of diversity soas to obtain a better Pareto solution based on the simulationexperiment it could be seen that the results of the MOEAD-ACO-PBI algorithm are significantly better than those ofthe other algorithms Our future work aims at analysingfurther how to cope with constraint and dynamic constraintprocessing and to optimize the number of objectives moreeffectively

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This research work was supported by the national science andtechnology major projects (NO 2009ZX04001-111) subpro-ject of national science and technologymajor (2013ZX04002-031) key laboratory of automotive power train and elec-tronics (Hubei University of Automotive Technology (NoZDK1201703) and Hubei Provincial Education DepartmentYouth Fund (No Q20181801)

References

[1] B Li J Li K Tang and X Yao ldquoMany-objective evolutionaryalgorithms a surveyrdquo ACM Computing Surveys vol 48 no 1article 13 2015

[2] A Zhou B Y Qu H Li S Z Zhao P N Suganthan and QZhang ldquoMultiobjective evolutionary algorithms a survey of the

state of the artrdquo Swarm Evolutionary Computation vol 1 no 1pp 32ndash49 2011

[3] C Wang Y Wang K Wang Y Dong and Y Yang ldquoAnImproved Hybrid Algorithm Based on BiogeographyComplexand Metropolis for Many-Objective OptimizationrdquoMathemat-ical Problems in Engineering vol 2017 2017

[4] A Trivedi D Srinivasan K Sanyal and A Ghosh ldquoA surveyof multiobjective evolutionary algorithms based on decompo-sitionrdquo IEEE Transactions on Evolutionary Computation vol 21no 3 pp 440ndash462 2017

[5] C Von Lucken B Baran and C Brizuela ldquoA survey onmulti-objective evolutionary algorithms for many-objectiveproblemsrdquo Computational Optimization and Applications vol58 no 3 pp 707ndash756 2014

[6] H Jain and K Deb ldquoAn evolutionary many-objective opti-mization algorithm using reference-point based nondominatedsorting approach Part II Handling constraints and extendingto an adaptive approachrdquo IEEE Transactions on EvolutionaryComputation vol 18 no 4 pp 602ndash622 2014

[7] Z He G G Yen and J Zhang ldquoFuzzy-based pareto optimalityformany-objective evolutionary algorithmsrdquo IEEETransactionson Evolutionary Computation vol 18 no 2 pp 269ndash285 2014

[8] D H Phan and J Suzuki ldquoR2 indicator based multiobjectivememetic optimization for the pickup and delivery problemwithtime windows and demands (PDP-TW-Drdquo in Proceedings ofthe Conference on Bioinspired Information and CommunicationsTechnologies vol 22 pp 43ndash50 2014

[9] J Bader and E Zitzler ldquoHypE an algorithm for fast hypervol-ume-based many-objective optimizationrdquo Evolutionary Com-putation vol 19 no 1 pp 45ndash76 2011

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

[11] H Zhu Z He and Y Jia ldquoAn improved reference point basedmulti-objective optimization by decompositionrdquo InternationalJournal of Machine Learning and Cybernetics vol 7 no 4 pp581ndash595 2016

[12] I Alaya C Solnon and K Ghedira ldquoAnt Colony Optimizationfor Multi-Objective Optimizationrdquo in Proceedings of the Prob-lems vol 1 pp 450ndash457 2017

[13] L Ke Q Zhang and R Battiti ldquoMOEAD-ACO a mul-tiobjective evolutionary algorithm using decomposition and

Mathematical Problems in Engineering 9

AntColonyrdquo IEEE Transactions on Cybernetics vol 43 no 6 pp1845ndash1859 2013

[14] M Z De Souza and A T R Pozo ldquoA GPU implementationof MOEAD-ACO for the multiobjective traveling salesmanproblemrdquo in Proceedings of the 3rd Brazilian Conference onIntelligent Systems BRACIS 2014 pp 324ndash329 Brazil October2014

[15] T Lust and J Teghem ldquoTwo-phase pareto local search for thebiobjective traveling salesman problemrdquo Journal of Heuristicsvol 16 no 3 pp 475ndash510 2010

[16] K Deb and H Jain ldquoAn evolutionary many-objective opti-mization algorithmusing reference-point-based nondominatedsorting approach part I solving problemswith box constraintsrdquoIEEE Transactions on Evolutionary Computation vol 18 no 4pp 577ndash601 2014

[17] I Das and J E Dennis ldquoNormal-boundary intersection a newmethod for generating the Pareto surface in nonlinear multi-criteria optimization problemsrdquo SIAM Journal on Optimizationvol 8 no 3 pp 631ndash657 1998

[18] B Baran and M Schaerer ldquoA multiobjective ant colony systemfor vehicle routing problem with time windowsrdquo in Proceedingsof the 21st IASTED International Multi-Conference on AppliedInformatics pp 97ndash102 Austria February 2003

[19] Q Zhang A Zhou S Zhao P N Suganthan W Liu and STiwari ldquoMultiobjective optimization test instances for the CEC2009 special session and competitionrdquo Tech Rep University ofEssex Colchester UK and Nanyang technological University2008

[20] M Asafuddoula T Ray and R Sarker ldquoA decomposition-basedevolutionary algorithm for many objective optimizationrdquo IEEETransactions on Evolutionary Computation vol 19 no 3 pp445ndash460 2015

[21] CWang YWang KWang Y Yang and Y Tian ldquoAn improvedbiogeographycomplex algorithm based on decomposition formany-objective optimizationrdquo in Proceedings of the Interna-tional Journal of Machine Learning Cybernetics (2 vol 2 p 172017

[22] Y Tian R Cheng X Zhang and Y Jin Platemo a matlabplatform for evolutionary multi-objective optimization arXivpreprint arXiv 170100879 2017

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Mathematical Problems in Engineering

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4 Mathematical Problems in Engineering

f1

f2

X3 X5 X2

X6

X1 X7

X4 Zlowast

Figure 3 Cluster-based weight vector adjustment

25 Cluster-BasedWeight Vector Adjustment For theMaOPsproblem this paper proposes a weighted vector adjustmentmethod based on clustering and uses different pheromonematrices to solve different single objectives This scalableapproach can help the ant colony algorithm to better solvethe MaOPs problem

After the multiobjective problem is decomposed intoN single-target subproblems by the PBI method N antsare selected to solve these single-target subproblems Anti solves the subproblem i The subproblem i is associatedwith 120582i The corresponding objective function is 119892(x |120582i) N ants are divided into K groups according to theircorresponding weight distances Each group approaches asmall area of the Pareto frontier The same group of antsshare a pheromone matrix that accommodates the locationinformation of the learned Pareto frontier subregions Eachant looks for a special point on the Pareto frontier for a single-target subproblem Each ant has its ownheuristic informationmatrix to store a priori knowledge of its subproblemsThe antirsquos neighbour B (i) contains T ants whose weights are closestto 120582i in N weights and i isin B (i) and ant i is its own neigh-bour

In each iteration of the algorithm if the solution set in thedominating set remains unchanged for the first time it meansthat the current weight vector can no longer meet the needof guiding population evolution and weight adjustment isneeded In this case nondominated populations are classifiedbased on the clustering method assuming that there are 7individuals in the objective space (x1 x2 x7) They hadthree clusters x3 x5 x2 x6 and x1 x7 x4 Among themx3 x6 and x7 are cluster centres as shown in Figure 3

3 Algorithm Description

31 MOEAD-ACO-PBI In this paper we propose an algo-rithm called MOEAD-ACO-PBI The original MOEAD-ACO used a weighted sum and Tchebycheff method to solvethe problemHowever this is difficult because the correlationbetween the algebraic and geometrical interpretation of theweight sets is not valid [15] Recent studies conducted by[16] have shown that MOEAD-PBI is the better methodfor many-objective optimization problems Therefore weintroduce the PBI approach to MOEAD-ACO known asMOEAD-ACO-PBI The MOEAD-ACO-PBI algorithm ispresented in Algorithm 2

(1) Initialization(2) Decomposition by PBI approach(3) Solution construction(4) Update of EP(5) Update of pheromone(6) Update the solutions(7) Termination

Algorithm 2 MOEAD-ACO-PBI

32 Initialization Procedure Each ant ia has a heuristicinformation value 120578119894119886119896119897 each group j has pheromone trail 120591119896119894119886119895and each individual solution xia is a tour represented bypermutation

The initialization procedure of MOEAD-ACO-PBI con-tains six main steps(1) Setting of Nw and 120582Nw the sets of weight vectors120582Nw = 1205821 120582Nw are generated by a systematic approachdeveloped from Das and Dennisrsquos method [17] Here the sizeof 120582 is Nw

119873119908 = 119862119898minus1119867+119898minus1 (7)

where H gt 0 is the quantity of divisions considered alongeach objective coordinate Therefore a subproblem is has aweight vector 120582is that satisfiessum119898119897=1 120582119894119904119897 = 1 and 120582119894119904119896 ge 0 wherem is the number of objectives(2) Initialize the solution xi to the subproblem is and setFi= F (xi) for i = 1 n(3) Set group the number of groups is J which is avalue parameter that requires to be selected The number ofsubproblems of each group is equally distributed by clusteringthe weights in terms of the Euclidean distance(4) Initialize the pheromone information matrix 120591119896119894119895 = 1for all j = 1 k and k = 1 n The motivation is to encouragethe search to focus on exploration in its early stage(5) Initialize 120578119894119896119897 = 1sum119898119895=1 120582119894119895119888119895119896119897 which is the heuristicinformation matrix to the subproblem is for j = 1 k(6) Initialize EP as the set of the entire high-qualitysolutions in F1 FN EP is initialized to be empty

33 Solution Construction Assume that ant ia is in group jand its current solution xi = (x1

i xni) Ant ia constructs itsnew solution through the following steps(1)The probability of choice of the link is set For k l =1 n set

Φ119896119897 = [Γ119894119886119896119897 + Δ lowast ln (119909119894119886 (119896 119897))]119886 (120578119894119886119896119897)120573 (8)

Φkl is representing the attractiveness of the link betweenthe k and l to ant ia The indicator function in (xia (k l)) isequal to 1 if the link (k l) is in tour xia or 0 if otherwise 120572 120573and Δ are the three control parameters(2) First ant ia chooses an ant randomly as its startingpoint and builds its tour Suppose that its current position is land it has not completed its tour From S (not visited so far)

Mathematical Problems in Engineering 5

Table 1 Test functions utilized in this experiment

function Formulation

CF1 1198911(119909) = 1199091 + 2|1198691| sum119895isin1198691(119909119895 minus 119909105(10+3(119895minus2)(119899minus2)))2

1198912(119909) = 1 minus 1199091 + 2100381610038161003816100381611986921003816100381610038161003816 sum119895isin1198692 (119909119895 minus 119909105(10+3(119895minus2)(119899minus2)))2

CF2 1198911(119909) = 1199091 + 2|1198691| sum119895isin1198691(119909119895 minus sin(61205871199091 + 119895120587119899 ))21198912(119909) = 1 minus radic1199091 + 2100381610038161003816100381611986921003816100381610038161003816 sum119895isin1198692 (119909119895 minus cos(61205871199091 + 119895120587119899 ))

2

CF3 1198911(119909) = 1199091 + 2100381610038161003816100381611986911003816100381610038161003816 (4sum119895isin11986911199102119895 minus 2prod119895isin1198691

cos(20119910119895120587radic119895 ) + 2)

CF41198912(119909) = 1 minus 11990921 + 2100381610038161003816100381611986921003816100381610038161003816 (4sum119895isin1198692119910

2119895 minus 2prod119895isin1198692

cos(20119910119895120587radic119895 ) + 2)1198911 = 1199091 + sum

119895isin1198691

ℎ119895(119910119895)1198911 = 1 minus 1199091 + sum

119895isin1198692

ℎ119895(119910119895)

it selects the ant k to visit next according to the followingprobability by the roulette wheel selection

Φ119896119897sum119904isin119862Φ119904119897 (9)

(3) If the ant has visited all the points it returns its tour

34 Update of Pheromone The pheromone trail value of thelink (k l) for group j is updated as follows

120591119894119886119896119897 fl 120588120591119895119896119897+ sum119909isinΠ

1119892 (119909 | 120582119894119886) (10)

120588 is the persistence rate of the pheromone trail Π is theset of all the new solutions it was constructed by the ants ingroup j in the current iteration and was just added to EP Itcontains the link between the antsrsquo k and l 120591min and 120591max areused to limit the range of the pheromone 120591min is given by

120591min = 120576120591max (11)

where 0 lt 120576 lt 1 is a control parameterThen if 120591119894119886119896119897 lt 120591minset 120591119894119886119896119897 = 120591min and if 120591119894119886119896119897 gt 120591max set 120591119894119886119896119897 = 120591max 120591max is set

120591max = 119861 + 1(1 minus 120588) 119892min(12)

where B is the number of nondominated solutions foundin the current iteration and119892min is the smallest value obtainedfor the objective functions of all the Ns subproblems

4 Simulation Results

In this section we compare the MOEAD-ACO-PBIalgorithm with three state-of-the-art algorithms includingMOEAD-ACOW[13] MOEAD [10] and Multiple Ant

Colony System (MACS) [18] The test problems areintroduced in Section 41 Section 42 describes the qualityindicators Three state-of-the-art algorithms used forcomparison and the corresponding parameter settings arebriefly introduced in Section 43

41 Test Problems The well-known MaOPs test functiondefinitions of CF1ndashCF4 [19] are used to test the proposedalgorithm Table 1 shows the CF1ndashCF4 problems with theCF test suites CF1-CF4 are test functions with biobjectiveproblem CF test suites are used to test the low dimensionalobjective function To verify the ability of algorithms to dealwith more objectives we select C1-DTLZ1 C2-DTLZ2 andC3-DTLZ4 [20] used to test the high dimensional objectivefunction CF suites are popular test problem for MaOPs

C1-DTLZ1 is a MaOPs problem the problem is formu-lated as

119862 (119909) = 1 minus 119891119898 (119909)06 minus 119898minus1sum119894=1

119891119894 (119909)05 ge 0 (13)

C2-DTLZ2 is a MaOPs problem the problem is formu-lated as

119862 (119909) = minusminmin119898119894=1 [[

(119891119894 (119909) minus 1)2 + 119898sum119895=1119895 =119894

119891 2(119909)119895

minus 1199032]] times[ 119898sum119894=1

(119891119894 (119909) minus 1radic119898)2 minus 1199032]

ge 0(14)

C3-DTLZ4 is a MaOPs problem the problem is formu-lated as

119862119894 (119909) = 1198912119894 (119909)4 + 119898sum119895=1119895 =119894

1198912119894 (119909) minus 1 ge 0 (15)

6 Mathematical Problems in Engineering

Table 2The average and standard deviation ofGDvalues obtained forCF1ndashCF4C1 DTLZ1 C2 DTLZ2 andC3 DTLZ4with four algorithmsand different number of objectives Best performance is in bold face

M MOEAD-ACO-PBI MOEAD-ACOW MOEAD MACSCF1 2 10013e-4 (114e-4) 11945e-4 (208e-4) 12524e-4 (212e-2) 34214e-2 (292e-2)CF2 2 23325e-2 (105e-2) 20382e-2 (259e-2) 29194e-2 (124e-2) 45986e-2 (428e-2)CF3 2 22667e-1 (168e-1) 37510e-1 (241e-1) 33477e-1 (319e-1) 42804e-1 (132e-1)CF4 2 44754e-2 (129e-1) 54257e-2 (138e-1) 42232e-2 (719e-2) 21410e-1 (278e-1)

3 30152e-3 (213e-4) 32859e-3 (236e-4) 34831e-3 (332e-4) 44339e-3 (133e-4)

C1 DTLZ1 4 30221-3 (223e-4) 25532e-2 (223e-3) 21234e-2 (134e-3) 24452e-2 (212e-2)5 44264e-3 (325e-4) 33354e-2 (223e-2) 54261e-2 (234e-3) 53634e-2 (284e-2)

C2 DTLZ23 74664e-4 (531e-5) 22772e-3 (429e-4) 95259e-3 (167e-3) 79976e-3 (529e-3)4 27056e-3 (115e-4) 11709e-2 (481e-3) 11867e-2 (128e-3) 27389e-2 (108e-2)5 52944e-3 (299e-4) 25550e-2 (110e-2) 27631e-2 (695e-3) 59090e-2(284e-4)

C3 DTLZ43 40236e-4 (223e-4) 69914e-4 (328e-4) 45365e-4 (265e-4) 3514e-3 (326e-3)4 30885e-3 (232e-3) 22035e-2 (382e-2) 13564e-2 (136e-2) 25658e-2 (125e-4)5 30624e-3 (325e-4) 36412e-2 (221e-2) 37225e-2 (562e-3) 53361e-2 (326e-4)

42 Quality Indicators In our empirical study the followingtwo widely used quality indicators were considered The firstone reflects the convergence of an algorithmwhile the secondone reflects the convergence and diversity of the solutionssimultaneously

421 Generational Distance (GD) Indicator For any kindof algorithm let P be the set of final nondominated pointsobtained from the objective space Let Plowast be a set of pointsuniformly spread over the true PF The GD can indicate onlythe convergence of an algorithm [21] and a smaller valueindicates better quality The GD is computed as

119866119863 (119875 119875lowast) = radicsum119906isin119875 119889 (119906 119875lowast)|119875| (16)

422 Inverted Generational Distance (IGD) Indicator Forany kind of algorithm let P be the set of final nondominatedpoints obtained from the objective space and Plowast be a setof points uniformly spread over the true PF The IGD canindicate both the convergence and diversity [21] and asmaller value indicates better quality The IGD is computedas

119868119866119863 (119875 119875lowast) = sumVisin119875lowast 119889 (V 119875)|119875lowast| (17)

43 Parameter Settings The parameters for the four MOEAsconsidered in this paper are listed as follows(1) Some parameter settings in MOEAD-ACO-PBI setT=8 120572=1 120573=8 120588=095 r=08 and 120579=50(2) Some parameter settings in MOEAD-ACOW andMOEAD T=8 120572=1 120573=8 120588=095 and r=08 The otherparameters can be obtained from [15](3) Some parameter settings in MACS 120572=1 120573=2 120588=09maxFcctor=5 and AntNum=10(4) Number of runs and termination condition allalgorithms were programmed in PlatEMO [22] and each

algorithm was independently run 30 times for each testinstance and stopped after 300 generations All the algorithmswere executed on a desktop PC with a 21-GHz CPU 8-GBRAM and Windows 10

5 Results and Discussion

The GD indicator was utilized to compare the conver-gence capacity between the proposed MOEAD-ACO-PBIalgorithm and three other algorithms (Figure 4) Table 2shows the average and standard deviation of the GD valuesobtained by the four algorithms for test suits with differentnumber of objectives Table 2 shows that the MOEAD-ACO-PBI algorithm could perform well on all the testfunctions especially on CF1 and CF3 problems When thenumber of objective test functions is higher the performanceof MOEAD-ACO-PBI algorithm is better The results ofcomparison of the MOEAD-ACO-PBI algorithm with theother three MOEAs in terms of IGD values using the testsuite are provided in Table 3 It shows the average IGDvalues for the four MOEAs over 30 independent runs andthe best average and standard deviation values From theexperimental results of the test problem theMOEAD-ACO-PBI algorithmperforms onCF2 CF4 C1-DTLZ1 C2-DTLZ2and C3-DTLZ4 test problems It is shown that theMOEAD-ACO-PBI algorithm has better performance of convergenceand diversity In addition performance of MOEAD-ACOW

algorithm is also better andMACS performance is the worstThe conclusions are given below(1)MOEAD-ACO-PBI has the best convergence perfor-

mance as its approximate fronts covermost of those returnedby the other algorithms MOEAD-ACOW can obtain well-spread approximately fronts however some of its solutionswere dominated by MOEAD-ACO-PBI(2) It can be observed in Figure 3 that MOEAD-ACO-PBI dominates in most solutions and that the advantage ofMOEAD-ACO-PBI is more pronounced as the number ofobjectives increases

Mathematical Problems in Engineering 7

C1_DTLZ1 (1)

C2_DTLZ2 (1)

C3_DTLZ4 (1)

(2)CF1 (1) (3) (4)

(2)

CF4 (1)

(3) (4)

(2) (3) (4)

(2) (3) (4)

(2)CF3 (1) (3) (4)

(2)CF2 (1) (3) (4)

060504030201

0

060504030201

0

060504030201

0

07060504030201

0

07060504030201

0

354

325

215

105

0

07060504030201

0

07060504030201

0

070809

1

060504030201

0

141210

86420

070809

060504030201

0

070809

060504030201

0

070809

060504030201

0

0708

060504030201

0

060504030201

0

060504030201

0

07060504030201

0

060504030201

0

060504030201

0

060504030201

0

081

1214

060402

0

081

12

060402

0

081

12

060402

0

081

12

060402

0

081

12

060402

0

03503

04504

02502

01501

0050

03503

04505

04

02502

01501

0050

070605040302010

1 15 2 25 3 35 4 45 5

1

1

15

15

2

2

25

25

3

3

35

35

4

4

45

45

5

1 15 2 25 3 35 4 45 5

1 15 2 25 3 35 4 45 5 1 15 2 25 3 35 4 45 5

51 15 2 25 3 35 4 45 5

1 15 2 25 3 35 4 45 5

1 15 2 25 3 35 4 45 5 1 15 2 25 3 35 4 45 5

1 15 2 25 3 35 4 45 5 1 15 2 25 3 35 4 45 5

1 15 2 25 3 35 4 45 5 1 15 2 25 3 35 4 45 5

1 15 2 25 3 35 4 45 5 1 15 2 25 3 35 4 45 5

1 15 2 25 3 35 4 45 5 1 15 2 25 3 35 4 45 5 1 15 2 25 3 35 4 45 5 1 15 2 25 3 35 4 45 5

1 15 2 25 3 35 4 45 5 1 15 2 25 3 35 4 45 5

1 15 2 25 3 35 4 45 5 1 15 2 25 3 35 4 45 5

1 15 2 25 3 35 4 45 5 1 15 2 25 3 35 4 45 5

1 15 2 25 3 35 4 45 5 1 15 2 25 3 35 4 45 5

Figure 4 Parallel coordinates figure by four algorithms on the four-objective CF1 CF2 and CF3 (1) MOEAD-ACO-PBI (2) MOEAD-ACOW and (3)MOEAD (4)MACS

8 Mathematical Problems in Engineering

Table 3 The average and standard deviation of IGD values obtained for CF1ndashCF4 C1 DTLZ1 C2 DTLZ2 and C3 DTLZ4 with fouralgorithms and different number of objectives Best performance is in bold face

M MOEAD-ACO-PBI MOEAD-ACOW MOEAD MACSCF1 2 64371e-1 (136e-1) 51532e-1 (127e-1) 65734e-1 (108e-1) 78365e-1 (189e-1)CF2 2 45894e-2 (128e-2) 48905e-2 (252e-2) 62428e2 (610e-2) 24116e-1 (361e-1)CF3 2 37740e-1 (778e-2) 31772e-1 (932e-2) 37877e-1 (177e-1) 37581e-1 (254e-1)CF4 2 16256e-1 (634e-2) 24364e-1 (118e-1) 19728e-1 (717e-2) 24654e-1 (132e-1)

C1 DTLZ1 4 22653e-1 (325e-4) 34629e-1 (236e-2) 25698e-1 326e-3) 37241e-1 (321e-4)5 31651e-1 (413e-4) 36234e-1 (112e-1) 37511e-1 (169e-1) 59283e-1 (387e-4)

C2 DTLZ23 51902e-2 (509e-4) 65049e-2 (146e-3) 59751e-2 (485e-3) 93947e-2 (489e-4)4 15255e-1 (849e-4) 23852e-1 (201e-2) 19833e-1 (552e-3) 47290e-1 (894e-4)5 23283e-1 (429e-4) 29947e-1 (128e-1) 29391e-1 (104e-1) 55282e-1 (494e-4)

C3 DTLZ43 51141e-2 (432e-4) 73349e-2 (232e-3) 53365e-2 (332e-3) 65254e-2 (235e-4)4 11347e-1 (132e-4) 25983e-1 (325e-2) 14651e-1 (222e-3) 24014e-1 (752e-4)5 21698e-1 (324e-4) 27321e-1 (236e-1) 26651e-1 (152e-1) 44652e-1 (532e-4)

6 Conclusions

The major contribution of this paper is the proposal of aMOEAD-ACO-PBI algorithm for MaOPs Due to the factthat the number of model optimization objectives increasesthe PBI aggregate function was selected to decompose theobjectives MOEAD-ACO-PBI algorithm can better ensurethe convergence of algorithms and the balance of diversity soas to obtain a better Pareto solution based on the simulationexperiment it could be seen that the results of the MOEAD-ACO-PBI algorithm are significantly better than those ofthe other algorithms Our future work aims at analysingfurther how to cope with constraint and dynamic constraintprocessing and to optimize the number of objectives moreeffectively

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This research work was supported by the national science andtechnology major projects (NO 2009ZX04001-111) subpro-ject of national science and technologymajor (2013ZX04002-031) key laboratory of automotive power train and elec-tronics (Hubei University of Automotive Technology (NoZDK1201703) and Hubei Provincial Education DepartmentYouth Fund (No Q20181801)

References

[1] B Li J Li K Tang and X Yao ldquoMany-objective evolutionaryalgorithms a surveyrdquo ACM Computing Surveys vol 48 no 1article 13 2015

[2] A Zhou B Y Qu H Li S Z Zhao P N Suganthan and QZhang ldquoMultiobjective evolutionary algorithms a survey of the

state of the artrdquo Swarm Evolutionary Computation vol 1 no 1pp 32ndash49 2011

[3] C Wang Y Wang K Wang Y Dong and Y Yang ldquoAnImproved Hybrid Algorithm Based on BiogeographyComplexand Metropolis for Many-Objective OptimizationrdquoMathemat-ical Problems in Engineering vol 2017 2017

[4] A Trivedi D Srinivasan K Sanyal and A Ghosh ldquoA surveyof multiobjective evolutionary algorithms based on decompo-sitionrdquo IEEE Transactions on Evolutionary Computation vol 21no 3 pp 440ndash462 2017

[5] C Von Lucken B Baran and C Brizuela ldquoA survey onmulti-objective evolutionary algorithms for many-objectiveproblemsrdquo Computational Optimization and Applications vol58 no 3 pp 707ndash756 2014

[6] H Jain and K Deb ldquoAn evolutionary many-objective opti-mization algorithm using reference-point based nondominatedsorting approach Part II Handling constraints and extendingto an adaptive approachrdquo IEEE Transactions on EvolutionaryComputation vol 18 no 4 pp 602ndash622 2014

[7] Z He G G Yen and J Zhang ldquoFuzzy-based pareto optimalityformany-objective evolutionary algorithmsrdquo IEEETransactionson Evolutionary Computation vol 18 no 2 pp 269ndash285 2014

[8] D H Phan and J Suzuki ldquoR2 indicator based multiobjectivememetic optimization for the pickup and delivery problemwithtime windows and demands (PDP-TW-Drdquo in Proceedings ofthe Conference on Bioinspired Information and CommunicationsTechnologies vol 22 pp 43ndash50 2014

[9] J Bader and E Zitzler ldquoHypE an algorithm for fast hypervol-ume-based many-objective optimizationrdquo Evolutionary Com-putation vol 19 no 1 pp 45ndash76 2011

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

[11] H Zhu Z He and Y Jia ldquoAn improved reference point basedmulti-objective optimization by decompositionrdquo InternationalJournal of Machine Learning and Cybernetics vol 7 no 4 pp581ndash595 2016

[12] I Alaya C Solnon and K Ghedira ldquoAnt Colony Optimizationfor Multi-Objective Optimizationrdquo in Proceedings of the Prob-lems vol 1 pp 450ndash457 2017

[13] L Ke Q Zhang and R Battiti ldquoMOEAD-ACO a mul-tiobjective evolutionary algorithm using decomposition and

Mathematical Problems in Engineering 9

AntColonyrdquo IEEE Transactions on Cybernetics vol 43 no 6 pp1845ndash1859 2013

[14] M Z De Souza and A T R Pozo ldquoA GPU implementationof MOEAD-ACO for the multiobjective traveling salesmanproblemrdquo in Proceedings of the 3rd Brazilian Conference onIntelligent Systems BRACIS 2014 pp 324ndash329 Brazil October2014

[15] T Lust and J Teghem ldquoTwo-phase pareto local search for thebiobjective traveling salesman problemrdquo Journal of Heuristicsvol 16 no 3 pp 475ndash510 2010

[16] K Deb and H Jain ldquoAn evolutionary many-objective opti-mization algorithmusing reference-point-based nondominatedsorting approach part I solving problemswith box constraintsrdquoIEEE Transactions on Evolutionary Computation vol 18 no 4pp 577ndash601 2014

[17] I Das and J E Dennis ldquoNormal-boundary intersection a newmethod for generating the Pareto surface in nonlinear multi-criteria optimization problemsrdquo SIAM Journal on Optimizationvol 8 no 3 pp 631ndash657 1998

[18] B Baran and M Schaerer ldquoA multiobjective ant colony systemfor vehicle routing problem with time windowsrdquo in Proceedingsof the 21st IASTED International Multi-Conference on AppliedInformatics pp 97ndash102 Austria February 2003

[19] Q Zhang A Zhou S Zhao P N Suganthan W Liu and STiwari ldquoMultiobjective optimization test instances for the CEC2009 special session and competitionrdquo Tech Rep University ofEssex Colchester UK and Nanyang technological University2008

[20] M Asafuddoula T Ray and R Sarker ldquoA decomposition-basedevolutionary algorithm for many objective optimizationrdquo IEEETransactions on Evolutionary Computation vol 19 no 3 pp445ndash460 2015

[21] CWang YWang KWang Y Yang and Y Tian ldquoAn improvedbiogeographycomplex algorithm based on decomposition formany-objective optimizationrdquo in Proceedings of the Interna-tional Journal of Machine Learning Cybernetics (2 vol 2 p 172017

[22] Y Tian R Cheng X Zhang and Y Jin Platemo a matlabplatform for evolutionary multi-objective optimization arXivpreprint arXiv 170100879 2017

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Mathematical Problems in Engineering

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Submit your manuscripts atwwwhindawicom

Mathematical Problems in Engineering 5

Table 1 Test functions utilized in this experiment

function Formulation

CF1 1198911(119909) = 1199091 + 2|1198691| sum119895isin1198691(119909119895 minus 119909105(10+3(119895minus2)(119899minus2)))2

1198912(119909) = 1 minus 1199091 + 2100381610038161003816100381611986921003816100381610038161003816 sum119895isin1198692 (119909119895 minus 119909105(10+3(119895minus2)(119899minus2)))2

CF2 1198911(119909) = 1199091 + 2|1198691| sum119895isin1198691(119909119895 minus sin(61205871199091 + 119895120587119899 ))21198912(119909) = 1 minus radic1199091 + 2100381610038161003816100381611986921003816100381610038161003816 sum119895isin1198692 (119909119895 minus cos(61205871199091 + 119895120587119899 ))

2

CF3 1198911(119909) = 1199091 + 2100381610038161003816100381611986911003816100381610038161003816 (4sum119895isin11986911199102119895 minus 2prod119895isin1198691

cos(20119910119895120587radic119895 ) + 2)

CF41198912(119909) = 1 minus 11990921 + 2100381610038161003816100381611986921003816100381610038161003816 (4sum119895isin1198692119910

2119895 minus 2prod119895isin1198692

cos(20119910119895120587radic119895 ) + 2)1198911 = 1199091 + sum

119895isin1198691

ℎ119895(119910119895)1198911 = 1 minus 1199091 + sum

119895isin1198692

ℎ119895(119910119895)

it selects the ant k to visit next according to the followingprobability by the roulette wheel selection

Φ119896119897sum119904isin119862Φ119904119897 (9)

(3) If the ant has visited all the points it returns its tour

34 Update of Pheromone The pheromone trail value of thelink (k l) for group j is updated as follows

120591119894119886119896119897 fl 120588120591119895119896119897+ sum119909isinΠ

1119892 (119909 | 120582119894119886) (10)

120588 is the persistence rate of the pheromone trail Π is theset of all the new solutions it was constructed by the ants ingroup j in the current iteration and was just added to EP Itcontains the link between the antsrsquo k and l 120591min and 120591max areused to limit the range of the pheromone 120591min is given by

120591min = 120576120591max (11)

where 0 lt 120576 lt 1 is a control parameterThen if 120591119894119886119896119897 lt 120591minset 120591119894119886119896119897 = 120591min and if 120591119894119886119896119897 gt 120591max set 120591119894119886119896119897 = 120591max 120591max is set

120591max = 119861 + 1(1 minus 120588) 119892min(12)

where B is the number of nondominated solutions foundin the current iteration and119892min is the smallest value obtainedfor the objective functions of all the Ns subproblems

4 Simulation Results

In this section we compare the MOEAD-ACO-PBIalgorithm with three state-of-the-art algorithms includingMOEAD-ACOW[13] MOEAD [10] and Multiple Ant

Colony System (MACS) [18] The test problems areintroduced in Section 41 Section 42 describes the qualityindicators Three state-of-the-art algorithms used forcomparison and the corresponding parameter settings arebriefly introduced in Section 43

41 Test Problems The well-known MaOPs test functiondefinitions of CF1ndashCF4 [19] are used to test the proposedalgorithm Table 1 shows the CF1ndashCF4 problems with theCF test suites CF1-CF4 are test functions with biobjectiveproblem CF test suites are used to test the low dimensionalobjective function To verify the ability of algorithms to dealwith more objectives we select C1-DTLZ1 C2-DTLZ2 andC3-DTLZ4 [20] used to test the high dimensional objectivefunction CF suites are popular test problem for MaOPs

C1-DTLZ1 is a MaOPs problem the problem is formu-lated as

119862 (119909) = 1 minus 119891119898 (119909)06 minus 119898minus1sum119894=1

119891119894 (119909)05 ge 0 (13)

C2-DTLZ2 is a MaOPs problem the problem is formu-lated as

119862 (119909) = minusminmin119898119894=1 [[

(119891119894 (119909) minus 1)2 + 119898sum119895=1119895 =119894

119891 2(119909)119895

minus 1199032]] times[ 119898sum119894=1

(119891119894 (119909) minus 1radic119898)2 minus 1199032]

ge 0(14)

C3-DTLZ4 is a MaOPs problem the problem is formu-lated as

119862119894 (119909) = 1198912119894 (119909)4 + 119898sum119895=1119895 =119894

1198912119894 (119909) minus 1 ge 0 (15)

6 Mathematical Problems in Engineering

Table 2The average and standard deviation ofGDvalues obtained forCF1ndashCF4C1 DTLZ1 C2 DTLZ2 andC3 DTLZ4with four algorithmsand different number of objectives Best performance is in bold face

M MOEAD-ACO-PBI MOEAD-ACOW MOEAD MACSCF1 2 10013e-4 (114e-4) 11945e-4 (208e-4) 12524e-4 (212e-2) 34214e-2 (292e-2)CF2 2 23325e-2 (105e-2) 20382e-2 (259e-2) 29194e-2 (124e-2) 45986e-2 (428e-2)CF3 2 22667e-1 (168e-1) 37510e-1 (241e-1) 33477e-1 (319e-1) 42804e-1 (132e-1)CF4 2 44754e-2 (129e-1) 54257e-2 (138e-1) 42232e-2 (719e-2) 21410e-1 (278e-1)

3 30152e-3 (213e-4) 32859e-3 (236e-4) 34831e-3 (332e-4) 44339e-3 (133e-4)

C1 DTLZ1 4 30221-3 (223e-4) 25532e-2 (223e-3) 21234e-2 (134e-3) 24452e-2 (212e-2)5 44264e-3 (325e-4) 33354e-2 (223e-2) 54261e-2 (234e-3) 53634e-2 (284e-2)

C2 DTLZ23 74664e-4 (531e-5) 22772e-3 (429e-4) 95259e-3 (167e-3) 79976e-3 (529e-3)4 27056e-3 (115e-4) 11709e-2 (481e-3) 11867e-2 (128e-3) 27389e-2 (108e-2)5 52944e-3 (299e-4) 25550e-2 (110e-2) 27631e-2 (695e-3) 59090e-2(284e-4)

C3 DTLZ43 40236e-4 (223e-4) 69914e-4 (328e-4) 45365e-4 (265e-4) 3514e-3 (326e-3)4 30885e-3 (232e-3) 22035e-2 (382e-2) 13564e-2 (136e-2) 25658e-2 (125e-4)5 30624e-3 (325e-4) 36412e-2 (221e-2) 37225e-2 (562e-3) 53361e-2 (326e-4)

42 Quality Indicators In our empirical study the followingtwo widely used quality indicators were considered The firstone reflects the convergence of an algorithmwhile the secondone reflects the convergence and diversity of the solutionssimultaneously

421 Generational Distance (GD) Indicator For any kindof algorithm let P be the set of final nondominated pointsobtained from the objective space Let Plowast be a set of pointsuniformly spread over the true PF The GD can indicate onlythe convergence of an algorithm [21] and a smaller valueindicates better quality The GD is computed as

119866119863 (119875 119875lowast) = radicsum119906isin119875 119889 (119906 119875lowast)|119875| (16)

422 Inverted Generational Distance (IGD) Indicator Forany kind of algorithm let P be the set of final nondominatedpoints obtained from the objective space and Plowast be a setof points uniformly spread over the true PF The IGD canindicate both the convergence and diversity [21] and asmaller value indicates better quality The IGD is computedas

119868119866119863 (119875 119875lowast) = sumVisin119875lowast 119889 (V 119875)|119875lowast| (17)

43 Parameter Settings The parameters for the four MOEAsconsidered in this paper are listed as follows(1) Some parameter settings in MOEAD-ACO-PBI setT=8 120572=1 120573=8 120588=095 r=08 and 120579=50(2) Some parameter settings in MOEAD-ACOW andMOEAD T=8 120572=1 120573=8 120588=095 and r=08 The otherparameters can be obtained from [15](3) Some parameter settings in MACS 120572=1 120573=2 120588=09maxFcctor=5 and AntNum=10(4) Number of runs and termination condition allalgorithms were programmed in PlatEMO [22] and each

algorithm was independently run 30 times for each testinstance and stopped after 300 generations All the algorithmswere executed on a desktop PC with a 21-GHz CPU 8-GBRAM and Windows 10

5 Results and Discussion

The GD indicator was utilized to compare the conver-gence capacity between the proposed MOEAD-ACO-PBIalgorithm and three other algorithms (Figure 4) Table 2shows the average and standard deviation of the GD valuesobtained by the four algorithms for test suits with differentnumber of objectives Table 2 shows that the MOEAD-ACO-PBI algorithm could perform well on all the testfunctions especially on CF1 and CF3 problems When thenumber of objective test functions is higher the performanceof MOEAD-ACO-PBI algorithm is better The results ofcomparison of the MOEAD-ACO-PBI algorithm with theother three MOEAs in terms of IGD values using the testsuite are provided in Table 3 It shows the average IGDvalues for the four MOEAs over 30 independent runs andthe best average and standard deviation values From theexperimental results of the test problem theMOEAD-ACO-PBI algorithmperforms onCF2 CF4 C1-DTLZ1 C2-DTLZ2and C3-DTLZ4 test problems It is shown that theMOEAD-ACO-PBI algorithm has better performance of convergenceand diversity In addition performance of MOEAD-ACOW

algorithm is also better andMACS performance is the worstThe conclusions are given below(1)MOEAD-ACO-PBI has the best convergence perfor-

mance as its approximate fronts covermost of those returnedby the other algorithms MOEAD-ACOW can obtain well-spread approximately fronts however some of its solutionswere dominated by MOEAD-ACO-PBI(2) It can be observed in Figure 3 that MOEAD-ACO-PBI dominates in most solutions and that the advantage ofMOEAD-ACO-PBI is more pronounced as the number ofobjectives increases

Mathematical Problems in Engineering 7

C1_DTLZ1 (1)

C2_DTLZ2 (1)

C3_DTLZ4 (1)

(2)CF1 (1) (3) (4)

(2)

CF4 (1)

(3) (4)

(2) (3) (4)

(2) (3) (4)

(2)CF3 (1) (3) (4)

(2)CF2 (1) (3) (4)

060504030201

0

060504030201

0

060504030201

0

07060504030201

0

07060504030201

0

354

325

215

105

0

07060504030201

0

07060504030201

0

070809

1

060504030201

0

141210

86420

070809

060504030201

0

070809

060504030201

0

070809

060504030201

0

0708

060504030201

0

060504030201

0

060504030201

0

07060504030201

0

060504030201

0

060504030201

0

060504030201

0

081

1214

060402

0

081

12

060402

0

081

12

060402

0

081

12

060402

0

081

12

060402

0

03503

04504

02502

01501

0050

03503

04505

04

02502

01501

0050

070605040302010

1 15 2 25 3 35 4 45 5

1

1

15

15

2

2

25

25

3

3

35

35

4

4

45

45

5

1 15 2 25 3 35 4 45 5

1 15 2 25 3 35 4 45 5 1 15 2 25 3 35 4 45 5

51 15 2 25 3 35 4 45 5

1 15 2 25 3 35 4 45 5

1 15 2 25 3 35 4 45 5 1 15 2 25 3 35 4 45 5

1 15 2 25 3 35 4 45 5 1 15 2 25 3 35 4 45 5

1 15 2 25 3 35 4 45 5 1 15 2 25 3 35 4 45 5

1 15 2 25 3 35 4 45 5 1 15 2 25 3 35 4 45 5

1 15 2 25 3 35 4 45 5 1 15 2 25 3 35 4 45 5 1 15 2 25 3 35 4 45 5 1 15 2 25 3 35 4 45 5

1 15 2 25 3 35 4 45 5 1 15 2 25 3 35 4 45 5

1 15 2 25 3 35 4 45 5 1 15 2 25 3 35 4 45 5

1 15 2 25 3 35 4 45 5 1 15 2 25 3 35 4 45 5

1 15 2 25 3 35 4 45 5 1 15 2 25 3 35 4 45 5

Figure 4 Parallel coordinates figure by four algorithms on the four-objective CF1 CF2 and CF3 (1) MOEAD-ACO-PBI (2) MOEAD-ACOW and (3)MOEAD (4)MACS

8 Mathematical Problems in Engineering

Table 3 The average and standard deviation of IGD values obtained for CF1ndashCF4 C1 DTLZ1 C2 DTLZ2 and C3 DTLZ4 with fouralgorithms and different number of objectives Best performance is in bold face

M MOEAD-ACO-PBI MOEAD-ACOW MOEAD MACSCF1 2 64371e-1 (136e-1) 51532e-1 (127e-1) 65734e-1 (108e-1) 78365e-1 (189e-1)CF2 2 45894e-2 (128e-2) 48905e-2 (252e-2) 62428e2 (610e-2) 24116e-1 (361e-1)CF3 2 37740e-1 (778e-2) 31772e-1 (932e-2) 37877e-1 (177e-1) 37581e-1 (254e-1)CF4 2 16256e-1 (634e-2) 24364e-1 (118e-1) 19728e-1 (717e-2) 24654e-1 (132e-1)

C1 DTLZ1 4 22653e-1 (325e-4) 34629e-1 (236e-2) 25698e-1 326e-3) 37241e-1 (321e-4)5 31651e-1 (413e-4) 36234e-1 (112e-1) 37511e-1 (169e-1) 59283e-1 (387e-4)

C2 DTLZ23 51902e-2 (509e-4) 65049e-2 (146e-3) 59751e-2 (485e-3) 93947e-2 (489e-4)4 15255e-1 (849e-4) 23852e-1 (201e-2) 19833e-1 (552e-3) 47290e-1 (894e-4)5 23283e-1 (429e-4) 29947e-1 (128e-1) 29391e-1 (104e-1) 55282e-1 (494e-4)

C3 DTLZ43 51141e-2 (432e-4) 73349e-2 (232e-3) 53365e-2 (332e-3) 65254e-2 (235e-4)4 11347e-1 (132e-4) 25983e-1 (325e-2) 14651e-1 (222e-3) 24014e-1 (752e-4)5 21698e-1 (324e-4) 27321e-1 (236e-1) 26651e-1 (152e-1) 44652e-1 (532e-4)

6 Conclusions

The major contribution of this paper is the proposal of aMOEAD-ACO-PBI algorithm for MaOPs Due to the factthat the number of model optimization objectives increasesthe PBI aggregate function was selected to decompose theobjectives MOEAD-ACO-PBI algorithm can better ensurethe convergence of algorithms and the balance of diversity soas to obtain a better Pareto solution based on the simulationexperiment it could be seen that the results of the MOEAD-ACO-PBI algorithm are significantly better than those ofthe other algorithms Our future work aims at analysingfurther how to cope with constraint and dynamic constraintprocessing and to optimize the number of objectives moreeffectively

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This research work was supported by the national science andtechnology major projects (NO 2009ZX04001-111) subpro-ject of national science and technologymajor (2013ZX04002-031) key laboratory of automotive power train and elec-tronics (Hubei University of Automotive Technology (NoZDK1201703) and Hubei Provincial Education DepartmentYouth Fund (No Q20181801)

References

[1] B Li J Li K Tang and X Yao ldquoMany-objective evolutionaryalgorithms a surveyrdquo ACM Computing Surveys vol 48 no 1article 13 2015

[2] A Zhou B Y Qu H Li S Z Zhao P N Suganthan and QZhang ldquoMultiobjective evolutionary algorithms a survey of the

state of the artrdquo Swarm Evolutionary Computation vol 1 no 1pp 32ndash49 2011

[3] C Wang Y Wang K Wang Y Dong and Y Yang ldquoAnImproved Hybrid Algorithm Based on BiogeographyComplexand Metropolis for Many-Objective OptimizationrdquoMathemat-ical Problems in Engineering vol 2017 2017

[4] A Trivedi D Srinivasan K Sanyal and A Ghosh ldquoA surveyof multiobjective evolutionary algorithms based on decompo-sitionrdquo IEEE Transactions on Evolutionary Computation vol 21no 3 pp 440ndash462 2017

[5] C Von Lucken B Baran and C Brizuela ldquoA survey onmulti-objective evolutionary algorithms for many-objectiveproblemsrdquo Computational Optimization and Applications vol58 no 3 pp 707ndash756 2014

[6] H Jain and K Deb ldquoAn evolutionary many-objective opti-mization algorithm using reference-point based nondominatedsorting approach Part II Handling constraints and extendingto an adaptive approachrdquo IEEE Transactions on EvolutionaryComputation vol 18 no 4 pp 602ndash622 2014

[7] Z He G G Yen and J Zhang ldquoFuzzy-based pareto optimalityformany-objective evolutionary algorithmsrdquo IEEETransactionson Evolutionary Computation vol 18 no 2 pp 269ndash285 2014

[8] D H Phan and J Suzuki ldquoR2 indicator based multiobjectivememetic optimization for the pickup and delivery problemwithtime windows and demands (PDP-TW-Drdquo in Proceedings ofthe Conference on Bioinspired Information and CommunicationsTechnologies vol 22 pp 43ndash50 2014

[9] J Bader and E Zitzler ldquoHypE an algorithm for fast hypervol-ume-based many-objective optimizationrdquo Evolutionary Com-putation vol 19 no 1 pp 45ndash76 2011

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

[11] H Zhu Z He and Y Jia ldquoAn improved reference point basedmulti-objective optimization by decompositionrdquo InternationalJournal of Machine Learning and Cybernetics vol 7 no 4 pp581ndash595 2016

[12] I Alaya C Solnon and K Ghedira ldquoAnt Colony Optimizationfor Multi-Objective Optimizationrdquo in Proceedings of the Prob-lems vol 1 pp 450ndash457 2017

[13] L Ke Q Zhang and R Battiti ldquoMOEAD-ACO a mul-tiobjective evolutionary algorithm using decomposition and

Mathematical Problems in Engineering 9

AntColonyrdquo IEEE Transactions on Cybernetics vol 43 no 6 pp1845ndash1859 2013

[14] M Z De Souza and A T R Pozo ldquoA GPU implementationof MOEAD-ACO for the multiobjective traveling salesmanproblemrdquo in Proceedings of the 3rd Brazilian Conference onIntelligent Systems BRACIS 2014 pp 324ndash329 Brazil October2014

[15] T Lust and J Teghem ldquoTwo-phase pareto local search for thebiobjective traveling salesman problemrdquo Journal of Heuristicsvol 16 no 3 pp 475ndash510 2010

[16] K Deb and H Jain ldquoAn evolutionary many-objective opti-mization algorithmusing reference-point-based nondominatedsorting approach part I solving problemswith box constraintsrdquoIEEE Transactions on Evolutionary Computation vol 18 no 4pp 577ndash601 2014

[17] I Das and J E Dennis ldquoNormal-boundary intersection a newmethod for generating the Pareto surface in nonlinear multi-criteria optimization problemsrdquo SIAM Journal on Optimizationvol 8 no 3 pp 631ndash657 1998

[18] B Baran and M Schaerer ldquoA multiobjective ant colony systemfor vehicle routing problem with time windowsrdquo in Proceedingsof the 21st IASTED International Multi-Conference on AppliedInformatics pp 97ndash102 Austria February 2003

[19] Q Zhang A Zhou S Zhao P N Suganthan W Liu and STiwari ldquoMultiobjective optimization test instances for the CEC2009 special session and competitionrdquo Tech Rep University ofEssex Colchester UK and Nanyang technological University2008

[20] M Asafuddoula T Ray and R Sarker ldquoA decomposition-basedevolutionary algorithm for many objective optimizationrdquo IEEETransactions on Evolutionary Computation vol 19 no 3 pp445ndash460 2015

[21] CWang YWang KWang Y Yang and Y Tian ldquoAn improvedbiogeographycomplex algorithm based on decomposition formany-objective optimizationrdquo in Proceedings of the Interna-tional Journal of Machine Learning Cybernetics (2 vol 2 p 172017

[22] Y Tian R Cheng X Zhang and Y Jin Platemo a matlabplatform for evolutionary multi-objective optimization arXivpreprint arXiv 170100879 2017

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

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

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

6 Mathematical Problems in Engineering

Table 2The average and standard deviation ofGDvalues obtained forCF1ndashCF4C1 DTLZ1 C2 DTLZ2 andC3 DTLZ4with four algorithmsand different number of objectives Best performance is in bold face

M MOEAD-ACO-PBI MOEAD-ACOW MOEAD MACSCF1 2 10013e-4 (114e-4) 11945e-4 (208e-4) 12524e-4 (212e-2) 34214e-2 (292e-2)CF2 2 23325e-2 (105e-2) 20382e-2 (259e-2) 29194e-2 (124e-2) 45986e-2 (428e-2)CF3 2 22667e-1 (168e-1) 37510e-1 (241e-1) 33477e-1 (319e-1) 42804e-1 (132e-1)CF4 2 44754e-2 (129e-1) 54257e-2 (138e-1) 42232e-2 (719e-2) 21410e-1 (278e-1)

3 30152e-3 (213e-4) 32859e-3 (236e-4) 34831e-3 (332e-4) 44339e-3 (133e-4)

C1 DTLZ1 4 30221-3 (223e-4) 25532e-2 (223e-3) 21234e-2 (134e-3) 24452e-2 (212e-2)5 44264e-3 (325e-4) 33354e-2 (223e-2) 54261e-2 (234e-3) 53634e-2 (284e-2)

C2 DTLZ23 74664e-4 (531e-5) 22772e-3 (429e-4) 95259e-3 (167e-3) 79976e-3 (529e-3)4 27056e-3 (115e-4) 11709e-2 (481e-3) 11867e-2 (128e-3) 27389e-2 (108e-2)5 52944e-3 (299e-4) 25550e-2 (110e-2) 27631e-2 (695e-3) 59090e-2(284e-4)

C3 DTLZ43 40236e-4 (223e-4) 69914e-4 (328e-4) 45365e-4 (265e-4) 3514e-3 (326e-3)4 30885e-3 (232e-3) 22035e-2 (382e-2) 13564e-2 (136e-2) 25658e-2 (125e-4)5 30624e-3 (325e-4) 36412e-2 (221e-2) 37225e-2 (562e-3) 53361e-2 (326e-4)

42 Quality Indicators In our empirical study the followingtwo widely used quality indicators were considered The firstone reflects the convergence of an algorithmwhile the secondone reflects the convergence and diversity of the solutionssimultaneously

421 Generational Distance (GD) Indicator For any kindof algorithm let P be the set of final nondominated pointsobtained from the objective space Let Plowast be a set of pointsuniformly spread over the true PF The GD can indicate onlythe convergence of an algorithm [21] and a smaller valueindicates better quality The GD is computed as

119866119863 (119875 119875lowast) = radicsum119906isin119875 119889 (119906 119875lowast)|119875| (16)

422 Inverted Generational Distance (IGD) Indicator Forany kind of algorithm let P be the set of final nondominatedpoints obtained from the objective space and Plowast be a setof points uniformly spread over the true PF The IGD canindicate both the convergence and diversity [21] and asmaller value indicates better quality The IGD is computedas

119868119866119863 (119875 119875lowast) = sumVisin119875lowast 119889 (V 119875)|119875lowast| (17)

43 Parameter Settings The parameters for the four MOEAsconsidered in this paper are listed as follows(1) Some parameter settings in MOEAD-ACO-PBI setT=8 120572=1 120573=8 120588=095 r=08 and 120579=50(2) Some parameter settings in MOEAD-ACOW andMOEAD T=8 120572=1 120573=8 120588=095 and r=08 The otherparameters can be obtained from [15](3) Some parameter settings in MACS 120572=1 120573=2 120588=09maxFcctor=5 and AntNum=10(4) Number of runs and termination condition allalgorithms were programmed in PlatEMO [22] and each

algorithm was independently run 30 times for each testinstance and stopped after 300 generations All the algorithmswere executed on a desktop PC with a 21-GHz CPU 8-GBRAM and Windows 10

5 Results and Discussion

The GD indicator was utilized to compare the conver-gence capacity between the proposed MOEAD-ACO-PBIalgorithm and three other algorithms (Figure 4) Table 2shows the average and standard deviation of the GD valuesobtained by the four algorithms for test suits with differentnumber of objectives Table 2 shows that the MOEAD-ACO-PBI algorithm could perform well on all the testfunctions especially on CF1 and CF3 problems When thenumber of objective test functions is higher the performanceof MOEAD-ACO-PBI algorithm is better The results ofcomparison of the MOEAD-ACO-PBI algorithm with theother three MOEAs in terms of IGD values using the testsuite are provided in Table 3 It shows the average IGDvalues for the four MOEAs over 30 independent runs andthe best average and standard deviation values From theexperimental results of the test problem theMOEAD-ACO-PBI algorithmperforms onCF2 CF4 C1-DTLZ1 C2-DTLZ2and C3-DTLZ4 test problems It is shown that theMOEAD-ACO-PBI algorithm has better performance of convergenceand diversity In addition performance of MOEAD-ACOW

algorithm is also better andMACS performance is the worstThe conclusions are given below(1)MOEAD-ACO-PBI has the best convergence perfor-

mance as its approximate fronts covermost of those returnedby the other algorithms MOEAD-ACOW can obtain well-spread approximately fronts however some of its solutionswere dominated by MOEAD-ACO-PBI(2) It can be observed in Figure 3 that MOEAD-ACO-PBI dominates in most solutions and that the advantage ofMOEAD-ACO-PBI is more pronounced as the number ofobjectives increases

Mathematical Problems in Engineering 7

C1_DTLZ1 (1)

C2_DTLZ2 (1)

C3_DTLZ4 (1)

(2)CF1 (1) (3) (4)

(2)

CF4 (1)

(3) (4)

(2) (3) (4)

(2) (3) (4)

(2)CF3 (1) (3) (4)

(2)CF2 (1) (3) (4)

060504030201

0

060504030201

0

060504030201

0

07060504030201

0

07060504030201

0

354

325

215

105

0

07060504030201

0

07060504030201

0

070809

1

060504030201

0

141210

86420

070809

060504030201

0

070809

060504030201

0

070809

060504030201

0

0708

060504030201

0

060504030201

0

060504030201

0

07060504030201

0

060504030201

0

060504030201

0

060504030201

0

081

1214

060402

0

081

12

060402

0

081

12

060402

0

081

12

060402

0

081

12

060402

0

03503

04504

02502

01501

0050

03503

04505

04

02502

01501

0050

070605040302010

1 15 2 25 3 35 4 45 5

1

1

15

15

2

2

25

25

3

3

35

35

4

4

45

45

5

1 15 2 25 3 35 4 45 5

1 15 2 25 3 35 4 45 5 1 15 2 25 3 35 4 45 5

51 15 2 25 3 35 4 45 5

1 15 2 25 3 35 4 45 5

1 15 2 25 3 35 4 45 5 1 15 2 25 3 35 4 45 5

1 15 2 25 3 35 4 45 5 1 15 2 25 3 35 4 45 5

1 15 2 25 3 35 4 45 5 1 15 2 25 3 35 4 45 5

1 15 2 25 3 35 4 45 5 1 15 2 25 3 35 4 45 5

1 15 2 25 3 35 4 45 5 1 15 2 25 3 35 4 45 5 1 15 2 25 3 35 4 45 5 1 15 2 25 3 35 4 45 5

1 15 2 25 3 35 4 45 5 1 15 2 25 3 35 4 45 5

1 15 2 25 3 35 4 45 5 1 15 2 25 3 35 4 45 5

1 15 2 25 3 35 4 45 5 1 15 2 25 3 35 4 45 5

1 15 2 25 3 35 4 45 5 1 15 2 25 3 35 4 45 5

Figure 4 Parallel coordinates figure by four algorithms on the four-objective CF1 CF2 and CF3 (1) MOEAD-ACO-PBI (2) MOEAD-ACOW and (3)MOEAD (4)MACS

8 Mathematical Problems in Engineering

Table 3 The average and standard deviation of IGD values obtained for CF1ndashCF4 C1 DTLZ1 C2 DTLZ2 and C3 DTLZ4 with fouralgorithms and different number of objectives Best performance is in bold face

M MOEAD-ACO-PBI MOEAD-ACOW MOEAD MACSCF1 2 64371e-1 (136e-1) 51532e-1 (127e-1) 65734e-1 (108e-1) 78365e-1 (189e-1)CF2 2 45894e-2 (128e-2) 48905e-2 (252e-2) 62428e2 (610e-2) 24116e-1 (361e-1)CF3 2 37740e-1 (778e-2) 31772e-1 (932e-2) 37877e-1 (177e-1) 37581e-1 (254e-1)CF4 2 16256e-1 (634e-2) 24364e-1 (118e-1) 19728e-1 (717e-2) 24654e-1 (132e-1)

C1 DTLZ1 4 22653e-1 (325e-4) 34629e-1 (236e-2) 25698e-1 326e-3) 37241e-1 (321e-4)5 31651e-1 (413e-4) 36234e-1 (112e-1) 37511e-1 (169e-1) 59283e-1 (387e-4)

C2 DTLZ23 51902e-2 (509e-4) 65049e-2 (146e-3) 59751e-2 (485e-3) 93947e-2 (489e-4)4 15255e-1 (849e-4) 23852e-1 (201e-2) 19833e-1 (552e-3) 47290e-1 (894e-4)5 23283e-1 (429e-4) 29947e-1 (128e-1) 29391e-1 (104e-1) 55282e-1 (494e-4)

C3 DTLZ43 51141e-2 (432e-4) 73349e-2 (232e-3) 53365e-2 (332e-3) 65254e-2 (235e-4)4 11347e-1 (132e-4) 25983e-1 (325e-2) 14651e-1 (222e-3) 24014e-1 (752e-4)5 21698e-1 (324e-4) 27321e-1 (236e-1) 26651e-1 (152e-1) 44652e-1 (532e-4)

6 Conclusions

The major contribution of this paper is the proposal of aMOEAD-ACO-PBI algorithm for MaOPs Due to the factthat the number of model optimization objectives increasesthe PBI aggregate function was selected to decompose theobjectives MOEAD-ACO-PBI algorithm can better ensurethe convergence of algorithms and the balance of diversity soas to obtain a better Pareto solution based on the simulationexperiment it could be seen that the results of the MOEAD-ACO-PBI algorithm are significantly better than those ofthe other algorithms Our future work aims at analysingfurther how to cope with constraint and dynamic constraintprocessing and to optimize the number of objectives moreeffectively

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This research work was supported by the national science andtechnology major projects (NO 2009ZX04001-111) subpro-ject of national science and technologymajor (2013ZX04002-031) key laboratory of automotive power train and elec-tronics (Hubei University of Automotive Technology (NoZDK1201703) and Hubei Provincial Education DepartmentYouth Fund (No Q20181801)

References

[1] B Li J Li K Tang and X Yao ldquoMany-objective evolutionaryalgorithms a surveyrdquo ACM Computing Surveys vol 48 no 1article 13 2015

[2] A Zhou B Y Qu H Li S Z Zhao P N Suganthan and QZhang ldquoMultiobjective evolutionary algorithms a survey of the

state of the artrdquo Swarm Evolutionary Computation vol 1 no 1pp 32ndash49 2011

[3] C Wang Y Wang K Wang Y Dong and Y Yang ldquoAnImproved Hybrid Algorithm Based on BiogeographyComplexand Metropolis for Many-Objective OptimizationrdquoMathemat-ical Problems in Engineering vol 2017 2017

[4] A Trivedi D Srinivasan K Sanyal and A Ghosh ldquoA surveyof multiobjective evolutionary algorithms based on decompo-sitionrdquo IEEE Transactions on Evolutionary Computation vol 21no 3 pp 440ndash462 2017

[5] C Von Lucken B Baran and C Brizuela ldquoA survey onmulti-objective evolutionary algorithms for many-objectiveproblemsrdquo Computational Optimization and Applications vol58 no 3 pp 707ndash756 2014

[6] H Jain and K Deb ldquoAn evolutionary many-objective opti-mization algorithm using reference-point based nondominatedsorting approach Part II Handling constraints and extendingto an adaptive approachrdquo IEEE Transactions on EvolutionaryComputation vol 18 no 4 pp 602ndash622 2014

[7] Z He G G Yen and J Zhang ldquoFuzzy-based pareto optimalityformany-objective evolutionary algorithmsrdquo IEEETransactionson Evolutionary Computation vol 18 no 2 pp 269ndash285 2014

[8] D H Phan and J Suzuki ldquoR2 indicator based multiobjectivememetic optimization for the pickup and delivery problemwithtime windows and demands (PDP-TW-Drdquo in Proceedings ofthe Conference on Bioinspired Information and CommunicationsTechnologies vol 22 pp 43ndash50 2014

[9] J Bader and E Zitzler ldquoHypE an algorithm for fast hypervol-ume-based many-objective optimizationrdquo Evolutionary Com-putation vol 19 no 1 pp 45ndash76 2011

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

[11] H Zhu Z He and Y Jia ldquoAn improved reference point basedmulti-objective optimization by decompositionrdquo InternationalJournal of Machine Learning and Cybernetics vol 7 no 4 pp581ndash595 2016

[12] I Alaya C Solnon and K Ghedira ldquoAnt Colony Optimizationfor Multi-Objective Optimizationrdquo in Proceedings of the Prob-lems vol 1 pp 450ndash457 2017

[13] L Ke Q Zhang and R Battiti ldquoMOEAD-ACO a mul-tiobjective evolutionary algorithm using decomposition and

Mathematical Problems in Engineering 9

AntColonyrdquo IEEE Transactions on Cybernetics vol 43 no 6 pp1845ndash1859 2013

[14] M Z De Souza and A T R Pozo ldquoA GPU implementationof MOEAD-ACO for the multiobjective traveling salesmanproblemrdquo in Proceedings of the 3rd Brazilian Conference onIntelligent Systems BRACIS 2014 pp 324ndash329 Brazil October2014

[15] T Lust and J Teghem ldquoTwo-phase pareto local search for thebiobjective traveling salesman problemrdquo Journal of Heuristicsvol 16 no 3 pp 475ndash510 2010

[16] K Deb and H Jain ldquoAn evolutionary many-objective opti-mization algorithmusing reference-point-based nondominatedsorting approach part I solving problemswith box constraintsrdquoIEEE Transactions on Evolutionary Computation vol 18 no 4pp 577ndash601 2014

[17] I Das and J E Dennis ldquoNormal-boundary intersection a newmethod for generating the Pareto surface in nonlinear multi-criteria optimization problemsrdquo SIAM Journal on Optimizationvol 8 no 3 pp 631ndash657 1998

[18] B Baran and M Schaerer ldquoA multiobjective ant colony systemfor vehicle routing problem with time windowsrdquo in Proceedingsof the 21st IASTED International Multi-Conference on AppliedInformatics pp 97ndash102 Austria February 2003

[19] Q Zhang A Zhou S Zhao P N Suganthan W Liu and STiwari ldquoMultiobjective optimization test instances for the CEC2009 special session and competitionrdquo Tech Rep University ofEssex Colchester UK and Nanyang technological University2008

[20] M Asafuddoula T Ray and R Sarker ldquoA decomposition-basedevolutionary algorithm for many objective optimizationrdquo IEEETransactions on Evolutionary Computation vol 19 no 3 pp445ndash460 2015

[21] CWang YWang KWang Y Yang and Y Tian ldquoAn improvedbiogeographycomplex algorithm based on decomposition formany-objective optimizationrdquo in Proceedings of the Interna-tional Journal of Machine Learning Cybernetics (2 vol 2 p 172017

[22] Y Tian R Cheng X Zhang and Y Jin Platemo a matlabplatform for evolutionary multi-objective optimization arXivpreprint arXiv 170100879 2017

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

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

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Mathematical Problems in Engineering 7

C1_DTLZ1 (1)

C2_DTLZ2 (1)

C3_DTLZ4 (1)

(2)CF1 (1) (3) (4)

(2)

CF4 (1)

(3) (4)

(2) (3) (4)

(2) (3) (4)

(2)CF3 (1) (3) (4)

(2)CF2 (1) (3) (4)

060504030201

0

060504030201

0

060504030201

0

07060504030201

0

07060504030201

0

354

325

215

105

0

07060504030201

0

07060504030201

0

070809

1

060504030201

0

141210

86420

070809

060504030201

0

070809

060504030201

0

070809

060504030201

0

0708

060504030201

0

060504030201

0

060504030201

0

07060504030201

0

060504030201

0

060504030201

0

060504030201

0

081

1214

060402

0

081

12

060402

0

081

12

060402

0

081

12

060402

0

081

12

060402

0

03503

04504

02502

01501

0050

03503

04505

04

02502

01501

0050

070605040302010

1 15 2 25 3 35 4 45 5

1

1

15

15

2

2

25

25

3

3

35

35

4

4

45

45

5

1 15 2 25 3 35 4 45 5

1 15 2 25 3 35 4 45 5 1 15 2 25 3 35 4 45 5

51 15 2 25 3 35 4 45 5

1 15 2 25 3 35 4 45 5

1 15 2 25 3 35 4 45 5 1 15 2 25 3 35 4 45 5

1 15 2 25 3 35 4 45 5 1 15 2 25 3 35 4 45 5

1 15 2 25 3 35 4 45 5 1 15 2 25 3 35 4 45 5

1 15 2 25 3 35 4 45 5 1 15 2 25 3 35 4 45 5

1 15 2 25 3 35 4 45 5 1 15 2 25 3 35 4 45 5 1 15 2 25 3 35 4 45 5 1 15 2 25 3 35 4 45 5

1 15 2 25 3 35 4 45 5 1 15 2 25 3 35 4 45 5

1 15 2 25 3 35 4 45 5 1 15 2 25 3 35 4 45 5

1 15 2 25 3 35 4 45 5 1 15 2 25 3 35 4 45 5

1 15 2 25 3 35 4 45 5 1 15 2 25 3 35 4 45 5

Figure 4 Parallel coordinates figure by four algorithms on the four-objective CF1 CF2 and CF3 (1) MOEAD-ACO-PBI (2) MOEAD-ACOW and (3)MOEAD (4)MACS

8 Mathematical Problems in Engineering

Table 3 The average and standard deviation of IGD values obtained for CF1ndashCF4 C1 DTLZ1 C2 DTLZ2 and C3 DTLZ4 with fouralgorithms and different number of objectives Best performance is in bold face

M MOEAD-ACO-PBI MOEAD-ACOW MOEAD MACSCF1 2 64371e-1 (136e-1) 51532e-1 (127e-1) 65734e-1 (108e-1) 78365e-1 (189e-1)CF2 2 45894e-2 (128e-2) 48905e-2 (252e-2) 62428e2 (610e-2) 24116e-1 (361e-1)CF3 2 37740e-1 (778e-2) 31772e-1 (932e-2) 37877e-1 (177e-1) 37581e-1 (254e-1)CF4 2 16256e-1 (634e-2) 24364e-1 (118e-1) 19728e-1 (717e-2) 24654e-1 (132e-1)

C1 DTLZ1 4 22653e-1 (325e-4) 34629e-1 (236e-2) 25698e-1 326e-3) 37241e-1 (321e-4)5 31651e-1 (413e-4) 36234e-1 (112e-1) 37511e-1 (169e-1) 59283e-1 (387e-4)

C2 DTLZ23 51902e-2 (509e-4) 65049e-2 (146e-3) 59751e-2 (485e-3) 93947e-2 (489e-4)4 15255e-1 (849e-4) 23852e-1 (201e-2) 19833e-1 (552e-3) 47290e-1 (894e-4)5 23283e-1 (429e-4) 29947e-1 (128e-1) 29391e-1 (104e-1) 55282e-1 (494e-4)

C3 DTLZ43 51141e-2 (432e-4) 73349e-2 (232e-3) 53365e-2 (332e-3) 65254e-2 (235e-4)4 11347e-1 (132e-4) 25983e-1 (325e-2) 14651e-1 (222e-3) 24014e-1 (752e-4)5 21698e-1 (324e-4) 27321e-1 (236e-1) 26651e-1 (152e-1) 44652e-1 (532e-4)

6 Conclusions

The major contribution of this paper is the proposal of aMOEAD-ACO-PBI algorithm for MaOPs Due to the factthat the number of model optimization objectives increasesthe PBI aggregate function was selected to decompose theobjectives MOEAD-ACO-PBI algorithm can better ensurethe convergence of algorithms and the balance of diversity soas to obtain a better Pareto solution based on the simulationexperiment it could be seen that the results of the MOEAD-ACO-PBI algorithm are significantly better than those ofthe other algorithms Our future work aims at analysingfurther how to cope with constraint and dynamic constraintprocessing and to optimize the number of objectives moreeffectively

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This research work was supported by the national science andtechnology major projects (NO 2009ZX04001-111) subpro-ject of national science and technologymajor (2013ZX04002-031) key laboratory of automotive power train and elec-tronics (Hubei University of Automotive Technology (NoZDK1201703) and Hubei Provincial Education DepartmentYouth Fund (No Q20181801)

References

[1] B Li J Li K Tang and X Yao ldquoMany-objective evolutionaryalgorithms a surveyrdquo ACM Computing Surveys vol 48 no 1article 13 2015

[2] A Zhou B Y Qu H Li S Z Zhao P N Suganthan and QZhang ldquoMultiobjective evolutionary algorithms a survey of the

state of the artrdquo Swarm Evolutionary Computation vol 1 no 1pp 32ndash49 2011

[3] C Wang Y Wang K Wang Y Dong and Y Yang ldquoAnImproved Hybrid Algorithm Based on BiogeographyComplexand Metropolis for Many-Objective OptimizationrdquoMathemat-ical Problems in Engineering vol 2017 2017

[4] A Trivedi D Srinivasan K Sanyal and A Ghosh ldquoA surveyof multiobjective evolutionary algorithms based on decompo-sitionrdquo IEEE Transactions on Evolutionary Computation vol 21no 3 pp 440ndash462 2017

[5] C Von Lucken B Baran and C Brizuela ldquoA survey onmulti-objective evolutionary algorithms for many-objectiveproblemsrdquo Computational Optimization and Applications vol58 no 3 pp 707ndash756 2014

[6] H Jain and K Deb ldquoAn evolutionary many-objective opti-mization algorithm using reference-point based nondominatedsorting approach Part II Handling constraints and extendingto an adaptive approachrdquo IEEE Transactions on EvolutionaryComputation vol 18 no 4 pp 602ndash622 2014

[7] Z He G G Yen and J Zhang ldquoFuzzy-based pareto optimalityformany-objective evolutionary algorithmsrdquo IEEETransactionson Evolutionary Computation vol 18 no 2 pp 269ndash285 2014

[8] D H Phan and J Suzuki ldquoR2 indicator based multiobjectivememetic optimization for the pickup and delivery problemwithtime windows and demands (PDP-TW-Drdquo in Proceedings ofthe Conference on Bioinspired Information and CommunicationsTechnologies vol 22 pp 43ndash50 2014

[9] J Bader and E Zitzler ldquoHypE an algorithm for fast hypervol-ume-based many-objective optimizationrdquo Evolutionary Com-putation vol 19 no 1 pp 45ndash76 2011

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

[11] H Zhu Z He and Y Jia ldquoAn improved reference point basedmulti-objective optimization by decompositionrdquo InternationalJournal of Machine Learning and Cybernetics vol 7 no 4 pp581ndash595 2016

[12] I Alaya C Solnon and K Ghedira ldquoAnt Colony Optimizationfor Multi-Objective Optimizationrdquo in Proceedings of the Prob-lems vol 1 pp 450ndash457 2017

[13] L Ke Q Zhang and R Battiti ldquoMOEAD-ACO a mul-tiobjective evolutionary algorithm using decomposition and

Mathematical Problems in Engineering 9

AntColonyrdquo IEEE Transactions on Cybernetics vol 43 no 6 pp1845ndash1859 2013

[14] M Z De Souza and A T R Pozo ldquoA GPU implementationof MOEAD-ACO for the multiobjective traveling salesmanproblemrdquo in Proceedings of the 3rd Brazilian Conference onIntelligent Systems BRACIS 2014 pp 324ndash329 Brazil October2014

[15] T Lust and J Teghem ldquoTwo-phase pareto local search for thebiobjective traveling salesman problemrdquo Journal of Heuristicsvol 16 no 3 pp 475ndash510 2010

[16] K Deb and H Jain ldquoAn evolutionary many-objective opti-mization algorithmusing reference-point-based nondominatedsorting approach part I solving problemswith box constraintsrdquoIEEE Transactions on Evolutionary Computation vol 18 no 4pp 577ndash601 2014

[17] I Das and J E Dennis ldquoNormal-boundary intersection a newmethod for generating the Pareto surface in nonlinear multi-criteria optimization problemsrdquo SIAM Journal on Optimizationvol 8 no 3 pp 631ndash657 1998

[18] B Baran and M Schaerer ldquoA multiobjective ant colony systemfor vehicle routing problem with time windowsrdquo in Proceedingsof the 21st IASTED International Multi-Conference on AppliedInformatics pp 97ndash102 Austria February 2003

[19] Q Zhang A Zhou S Zhao P N Suganthan W Liu and STiwari ldquoMultiobjective optimization test instances for the CEC2009 special session and competitionrdquo Tech Rep University ofEssex Colchester UK and Nanyang technological University2008

[20] M Asafuddoula T Ray and R Sarker ldquoA decomposition-basedevolutionary algorithm for many objective optimizationrdquo IEEETransactions on Evolutionary Computation vol 19 no 3 pp445ndash460 2015

[21] CWang YWang KWang Y Yang and Y Tian ldquoAn improvedbiogeographycomplex algorithm based on decomposition formany-objective optimizationrdquo in Proceedings of the Interna-tional Journal of Machine Learning Cybernetics (2 vol 2 p 172017

[22] Y Tian R Cheng X Zhang and Y Jin Platemo a matlabplatform for evolutionary multi-objective optimization arXivpreprint arXiv 170100879 2017

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

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

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

8 Mathematical Problems in Engineering

Table 3 The average and standard deviation of IGD values obtained for CF1ndashCF4 C1 DTLZ1 C2 DTLZ2 and C3 DTLZ4 with fouralgorithms and different number of objectives Best performance is in bold face

M MOEAD-ACO-PBI MOEAD-ACOW MOEAD MACSCF1 2 64371e-1 (136e-1) 51532e-1 (127e-1) 65734e-1 (108e-1) 78365e-1 (189e-1)CF2 2 45894e-2 (128e-2) 48905e-2 (252e-2) 62428e2 (610e-2) 24116e-1 (361e-1)CF3 2 37740e-1 (778e-2) 31772e-1 (932e-2) 37877e-1 (177e-1) 37581e-1 (254e-1)CF4 2 16256e-1 (634e-2) 24364e-1 (118e-1) 19728e-1 (717e-2) 24654e-1 (132e-1)

C1 DTLZ1 4 22653e-1 (325e-4) 34629e-1 (236e-2) 25698e-1 326e-3) 37241e-1 (321e-4)5 31651e-1 (413e-4) 36234e-1 (112e-1) 37511e-1 (169e-1) 59283e-1 (387e-4)

C2 DTLZ23 51902e-2 (509e-4) 65049e-2 (146e-3) 59751e-2 (485e-3) 93947e-2 (489e-4)4 15255e-1 (849e-4) 23852e-1 (201e-2) 19833e-1 (552e-3) 47290e-1 (894e-4)5 23283e-1 (429e-4) 29947e-1 (128e-1) 29391e-1 (104e-1) 55282e-1 (494e-4)

C3 DTLZ43 51141e-2 (432e-4) 73349e-2 (232e-3) 53365e-2 (332e-3) 65254e-2 (235e-4)4 11347e-1 (132e-4) 25983e-1 (325e-2) 14651e-1 (222e-3) 24014e-1 (752e-4)5 21698e-1 (324e-4) 27321e-1 (236e-1) 26651e-1 (152e-1) 44652e-1 (532e-4)

6 Conclusions

The major contribution of this paper is the proposal of aMOEAD-ACO-PBI algorithm for MaOPs Due to the factthat the number of model optimization objectives increasesthe PBI aggregate function was selected to decompose theobjectives MOEAD-ACO-PBI algorithm can better ensurethe convergence of algorithms and the balance of diversity soas to obtain a better Pareto solution based on the simulationexperiment it could be seen that the results of the MOEAD-ACO-PBI algorithm are significantly better than those ofthe other algorithms Our future work aims at analysingfurther how to cope with constraint and dynamic constraintprocessing and to optimize the number of objectives moreeffectively

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This research work was supported by the national science andtechnology major projects (NO 2009ZX04001-111) subpro-ject of national science and technologymajor (2013ZX04002-031) key laboratory of automotive power train and elec-tronics (Hubei University of Automotive Technology (NoZDK1201703) and Hubei Provincial Education DepartmentYouth Fund (No Q20181801)

References

[1] B Li J Li K Tang and X Yao ldquoMany-objective evolutionaryalgorithms a surveyrdquo ACM Computing Surveys vol 48 no 1article 13 2015

[2] A Zhou B Y Qu H Li S Z Zhao P N Suganthan and QZhang ldquoMultiobjective evolutionary algorithms a survey of the

state of the artrdquo Swarm Evolutionary Computation vol 1 no 1pp 32ndash49 2011

[3] C Wang Y Wang K Wang Y Dong and Y Yang ldquoAnImproved Hybrid Algorithm Based on BiogeographyComplexand Metropolis for Many-Objective OptimizationrdquoMathemat-ical Problems in Engineering vol 2017 2017

[4] A Trivedi D Srinivasan K Sanyal and A Ghosh ldquoA surveyof multiobjective evolutionary algorithms based on decompo-sitionrdquo IEEE Transactions on Evolutionary Computation vol 21no 3 pp 440ndash462 2017

[5] C Von Lucken B Baran and C Brizuela ldquoA survey onmulti-objective evolutionary algorithms for many-objectiveproblemsrdquo Computational Optimization and Applications vol58 no 3 pp 707ndash756 2014

[6] H Jain and K Deb ldquoAn evolutionary many-objective opti-mization algorithm using reference-point based nondominatedsorting approach Part II Handling constraints and extendingto an adaptive approachrdquo IEEE Transactions on EvolutionaryComputation vol 18 no 4 pp 602ndash622 2014

[7] Z He G G Yen and J Zhang ldquoFuzzy-based pareto optimalityformany-objective evolutionary algorithmsrdquo IEEETransactionson Evolutionary Computation vol 18 no 2 pp 269ndash285 2014

[8] D H Phan and J Suzuki ldquoR2 indicator based multiobjectivememetic optimization for the pickup and delivery problemwithtime windows and demands (PDP-TW-Drdquo in Proceedings ofthe Conference on Bioinspired Information and CommunicationsTechnologies vol 22 pp 43ndash50 2014

[9] J Bader and E Zitzler ldquoHypE an algorithm for fast hypervol-ume-based many-objective optimizationrdquo Evolutionary Com-putation vol 19 no 1 pp 45ndash76 2011

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

[11] H Zhu Z He and Y Jia ldquoAn improved reference point basedmulti-objective optimization by decompositionrdquo InternationalJournal of Machine Learning and Cybernetics vol 7 no 4 pp581ndash595 2016

[12] I Alaya C Solnon and K Ghedira ldquoAnt Colony Optimizationfor Multi-Objective Optimizationrdquo in Proceedings of the Prob-lems vol 1 pp 450ndash457 2017

[13] L Ke Q Zhang and R Battiti ldquoMOEAD-ACO a mul-tiobjective evolutionary algorithm using decomposition and

Mathematical Problems in Engineering 9

AntColonyrdquo IEEE Transactions on Cybernetics vol 43 no 6 pp1845ndash1859 2013

[14] M Z De Souza and A T R Pozo ldquoA GPU implementationof MOEAD-ACO for the multiobjective traveling salesmanproblemrdquo in Proceedings of the 3rd Brazilian Conference onIntelligent Systems BRACIS 2014 pp 324ndash329 Brazil October2014

[15] T Lust and J Teghem ldquoTwo-phase pareto local search for thebiobjective traveling salesman problemrdquo Journal of Heuristicsvol 16 no 3 pp 475ndash510 2010

[16] K Deb and H Jain ldquoAn evolutionary many-objective opti-mization algorithmusing reference-point-based nondominatedsorting approach part I solving problemswith box constraintsrdquoIEEE Transactions on Evolutionary Computation vol 18 no 4pp 577ndash601 2014

[17] I Das and J E Dennis ldquoNormal-boundary intersection a newmethod for generating the Pareto surface in nonlinear multi-criteria optimization problemsrdquo SIAM Journal on Optimizationvol 8 no 3 pp 631ndash657 1998

[18] B Baran and M Schaerer ldquoA multiobjective ant colony systemfor vehicle routing problem with time windowsrdquo in Proceedingsof the 21st IASTED International Multi-Conference on AppliedInformatics pp 97ndash102 Austria February 2003

[19] Q Zhang A Zhou S Zhao P N Suganthan W Liu and STiwari ldquoMultiobjective optimization test instances for the CEC2009 special session and competitionrdquo Tech Rep University ofEssex Colchester UK and Nanyang technological University2008

[20] M Asafuddoula T Ray and R Sarker ldquoA decomposition-basedevolutionary algorithm for many objective optimizationrdquo IEEETransactions on Evolutionary Computation vol 19 no 3 pp445ndash460 2015

[21] CWang YWang KWang Y Yang and Y Tian ldquoAn improvedbiogeographycomplex algorithm based on decomposition formany-objective optimizationrdquo in Proceedings of the Interna-tional Journal of Machine Learning Cybernetics (2 vol 2 p 172017

[22] Y Tian R Cheng X Zhang and Y Jin Platemo a matlabplatform for evolutionary multi-objective optimization arXivpreprint arXiv 170100879 2017

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

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

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Mathematical Problems in Engineering 9

AntColonyrdquo IEEE Transactions on Cybernetics vol 43 no 6 pp1845ndash1859 2013

[14] M Z De Souza and A T R Pozo ldquoA GPU implementationof MOEAD-ACO for the multiobjective traveling salesmanproblemrdquo in Proceedings of the 3rd Brazilian Conference onIntelligent Systems BRACIS 2014 pp 324ndash329 Brazil October2014

[15] T Lust and J Teghem ldquoTwo-phase pareto local search for thebiobjective traveling salesman problemrdquo Journal of Heuristicsvol 16 no 3 pp 475ndash510 2010

[16] K Deb and H Jain ldquoAn evolutionary many-objective opti-mization algorithmusing reference-point-based nondominatedsorting approach part I solving problemswith box constraintsrdquoIEEE Transactions on Evolutionary Computation vol 18 no 4pp 577ndash601 2014

[17] I Das and J E Dennis ldquoNormal-boundary intersection a newmethod for generating the Pareto surface in nonlinear multi-criteria optimization problemsrdquo SIAM Journal on Optimizationvol 8 no 3 pp 631ndash657 1998

[18] B Baran and M Schaerer ldquoA multiobjective ant colony systemfor vehicle routing problem with time windowsrdquo in Proceedingsof the 21st IASTED International Multi-Conference on AppliedInformatics pp 97ndash102 Austria February 2003

[19] Q Zhang A Zhou S Zhao P N Suganthan W Liu and STiwari ldquoMultiobjective optimization test instances for the CEC2009 special session and competitionrdquo Tech Rep University ofEssex Colchester UK and Nanyang technological University2008

[20] M Asafuddoula T Ray and R Sarker ldquoA decomposition-basedevolutionary algorithm for many objective optimizationrdquo IEEETransactions on Evolutionary Computation vol 19 no 3 pp445ndash460 2015

[21] CWang YWang KWang Y Yang and Y Tian ldquoAn improvedbiogeographycomplex algorithm based on decomposition formany-objective optimizationrdquo in Proceedings of the Interna-tional Journal of Machine Learning Cybernetics (2 vol 2 p 172017

[22] Y Tian R Cheng X Zhang and Y Jin Platemo a matlabplatform for evolutionary multi-objective optimization arXivpreprint arXiv 170100879 2017

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

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

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

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

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom