Research Article A New Multiobjective Evolutionary Algorithm Based on Decomposition of the Objective Space for Multiobjective Optimization

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1 Joural of Applied Mathematics Volume 24, Article ID 9647, 9 pages Research Article A New Multiobjective Evolutioary Algorithm Based o Decompositio of the Objective Space for Multiobjective Optimizatio Cai Dai ad Yupig Wag School of Computer Sciece ad Techology, Xidia Uiversity, Xi a 77, Chia Correspodece should be addressed to Cai Dai; daicai843@hotmail.com Received 6 September 23; Accepted 22 December 23; Published 2 Jauary 24 Academic Editor: Mehmet Sezer Copyright 24 C. Dai ad Y. Wag. This is a ope access article distributed uder the Creative Commos Attributio Licese, which permits urestricted use, distributio, ad reproductio i ay medium, provided the origial work is properly cited. I order to well maitai the diversity of obtaied solutios, a ew multiobjective evolutioary algorithm based o decompositio of the objective space for multiobjective optimizatio problems (MOPs) is desiged. I order to achieve the goal, the objective space of a MOP is decomposed ito a set of subobjective spaces by a set of directio vectors. I the evolutioary process, each subobjective space has a solutio, eve if it is ot a Pareto optimal solutio. I such a way, the diversity of obtaied solutios ca be maitaied, which is critical for solvig some MOPs. I additio, if a solutio is domiated by other solutios, the solutio ca geerate more ew solutios tha those solutios, which makes the solutio of each subobjective space coverge to the optimal solutios as far as possible. Experimetal studies have bee coducted to compare this proposed algorithm with classic ad. Simulatio results o six multiobjective bechmark fuctios show that the proposed algorithm is able to obtai better diversity ad more evely distributed Pareto frot tha the other two algorithms.. Itroductio Sice there are may problems with several optimizatio problems or criteria i real world [], multiobjective optimizatio has become a hot research topic. Ulike sigleobjective optimizatio problem, multiobjective optimizatio problem has a series of oiferior alterative solutios, alsokowasparetooptimalsolutios(thesetofpareto optimal solutios is called Pareto frot [2]), which represet the possible trade-off amog various coflictig objectives. Therefore, multiobjective optimizatio algorithms for MOP should be able to () discover solutios as close to the optimal solutios as possible; (2) fid solutios as uiform as possible i the obtaied odomiated frot; (3) determie solutios to cover the true Pareto frot () as broad as possible. However, achievig these three goals simultaeously is still a challege for multiobjective optimizatio algorithms. Amog various multiobjective optimizatio algorithms, multiobjective evolutioary algorithms (MOEA), which makeuseofthestrategyofthepopulatioevolutioaryto optimize the problems, are a effective method for solvig MOPs. I recet years, may MOEAs have bee proposed for solvig the multiobjective optimizatio problems [3 8]. I the MOEA literatures, Goldberg s populatio categorizatio strategy [9] based o odomiace is importat. May algorithms use the strategy to assig a fitess value based o the odomiace rak of members. For example, the odomiated sortig geetic algorithm II [3]was proposed by Deb et al. i 22, which uses the crowdig measure ad the elitism strategy; Zitzler et al. [2] proposed the stregth Pareto evolutioary algorithm II (SPEA2) which was also based o the elitism strategy; Soylu ad Köksala proposed a favorable weight-based evolutioary algorithm (FWEA) [2]. These algorithms maily rely o Pareto domiace to guide their search, particularly their selectio operators. I cotrast, (multiobjective evolutioary algorithm based o decompositio) [22] makes use of traditioal aggregatio methods to trasform the task of approximatig the Pareto frot () ito a umber of sigle-objective optimizatio subproblems. works well o a wide rage of multiobjective problems with may objectives, discrete decisio variables, ad complicated Pareto sets [23, 24].

2 2 Joural of Applied Mathematics has also bee used as a basic elemet i some hybrid algorithms; for example, with differetial evolutio ad particle swarm optimizatio is proposed by Mashwai [25], Li ad Lada-Silva [26] combie ad simulated aealig to solve MOPs, ad other hybrid algorithms also use to solve MOPs (e.g., [27 29]). Moreover, is used to solve various kids of problems (e.g., [3, 3]). I, weight vectors ad aggregate fuctios play a very importat role. I the literature [32], the uiform desig is used to geerate the weight vectors; Qi et al. [33] desig a adaptive weight vector adjustmet to adaptively geerate uiform distributio weight vectors. Ishibuchi et al. [34] propose a idea of automatically choosig betwee the weighted sum ad the weighted Tchebycheff for each solutio i each geeratio. Ay effective MOEAs must well maitai populatio diversity sice their goal is to approximate a set istead of a sigle poit. Pareto-domiace-based algorithms achieve the goal by maitaiig the diversity of obtaied Pareto optimal solutios. If these algorithms obtaiig Pareto optimal solutios caot cover the etire, this goal caot be achieved. The curret algorithms achieve their populatio diversity via the diversity of their subproblems. A elite diversity is used for their selectio i these algorithms. The aggregatio fuctio values of a ew solutio ad a old solutio completely determied whether the ew solutio replaces the old solutio or ot. I some cases, such replacemet ca cause a severe loss of populatio diversity, which is because the aggregatio fuctio value of a solutio oly reflects the extet of the solutio close to the ideal poit which is determied by the aggregatio fuctio ad the weight vector, but it does ot reflect the space where the objectivevectorofthesolutiolocates.thus,theobjective spacewheretheobjectivevectorofthesolutiolocatesshould be cosidered to maitai the diversity of obtaied Pareto optimal solutios. I this paper, the objective space of a MOP is decomposed ito a set of subobjective spaces by a set of directio vectors. Each subobjective space has a solutio: if a ew solutio will replace the solutio, the ew solutio must domiate the solutio ad its objective vector locates i the subobjective space. I such a way, the diversity of obtaied solutios ca be maitaied. I additio, the crowdig distace [3] is used to calculate the fitess value of a solutio for the selectio operators. I this cause, if a solutio is domiated by other solutios, the solutio is more likely to be selected tha other solutios, so it ca geerate more ew solutios to quickly fid the optimal solutio of thesubobjectivespace,whichmakesthesolutioofeach subobjective space coverge to the optimal solutios as far as possible. Based o these approaches, a ew multiobjective evolutioary algorithm,, is desiged. We show that ca sigificatly outperform ad o a set of test istaces used i our experimetal studies. This paper is orgaized as follows. Sectio 2 itroduces the mai cocepts of the multiobjective optimizatio; Sectio 3 describes the evolutioary algorithm based o decompositio of the objective space, while Sectio4 shows the experimet results of proposed algorithm; fially, Sectio 5 gives the coclusios ad future works. 2. Multiobjective Optimizatio A multiobjective optimizatio problem ca be formulated as follows [35]: mi y=f(x) =(f (x),f 2 (x),...,f m (x)) s.t. g i (x), i=,2,...,q h j (x) =, j=,2,...,p, where x = (x,...,x ) X R is called decisio variable ad X is -dimesioal decisio space. f i (x)(i =,...,m) is the ith objective to be miimized, g j (x)(j =,2...,q) defies jth iequality costrait, ad h j (x)(j =,2,...,p) defies jth equality costrait. Furthermore, all the costraits determie the set of feasible solutios which is deoted by Ω. To be specific, we try to fid a feasible solutio x Ωmiimizig each objective fuctio f i (x)(i =,...,m) i F. I the followig, four importat defiitios [36] for multiobjective problems are give. Defiitio (Pareto domiace). Pareto domiace betwee solutios x, z Ωis defied as follows. If i {,2,...,m} f i (x) f i (z), i {,2,...,m} f i (x) <f i (z) are satisfied, x domiates (Pareto domiate) z (deoted as x z). Defiitio 2 (Pareto optimal). A solutio vector x is said to be Pareto optimal with respect to Ω if z Ω : z x. Defiitio 3 (Pareto optimal set (PS)). The set of Pareto optimal solutios (PS) is defied as () (2) PS = {x Ω z Ω:z x}. (3) Defiitio 4 (Pareto frot). The Pareto optimal frot () is defied as = {F (x) x PS}. (4) 3. A New Multiobjective Evolutioary Algorithm Based o Decompositio of the Objective Space Thisewalgorithmcosistsofthreeparts:solutiosclassificatio, update strategy, ad selectio strategy which will be itroduced oe by oe i this sectio. 3.. Solutios Classificatio. The objective space of a MOP is decomposed ito a set of subobjective spaces by a set of directio vectors, ad the obtaied solutios are classified by these directio vectors to make each subobjective space have a solutio. For a give set of directio vectors (γ,γ 2,...,γ N ),

3 Joural of Applied Mathematics 3 ad the set of curret obtaied solutios beig POP, these solutios will be classified by the followig formula: P i ={x x POP,Δ(F(x),γ i )= max j N {Δ (F (x),γj )}}, Δ(F(x),γ i )= γi (F (x) Z) T γi F (x) Z, i=,...,m, (5) where Z = (Z,...,Z m ) is a referece poit ad Z i = mi{f i (x) x Ω}, Δ(F(x), γ i )} is the cosie of the agle betwee γ i ad F(x) Z. These solutios are divided ito N classes by the formula (5) ad the objective space Ω is divided ito N subobjective spaces Ω,...,Ω N,whereΩ k (k =,...,N)is Ω k ={F(x) x Ω,Δ(F(x),γ k )= max {Δ (F (x) j N,γj )}}. (6) If a P i ( i N)is empty, a solutio is radomly selected from POP ad put to P i. We would like to make the followig commets o the classificatio (decompositio) method. () This method is equivalet i a sese that the s of all thesesubobjectivespacescostitutetheof(). (2)EvewhethePSof() hasaolieargeometric shape,thepsofeachsubobjectivespacecouldbeclose to liear because it is just a small part of the PS of (). Therefore, formulas (5) ad(6) make() simplertha before, at least i terms of PS shapes. (3) This classificatio (decompositio) method does ot require ay aggregatio methods. A user oly eeds to choose a set of directio vectors. To some extet, it requires little huma labor Update Strategy. The elitist strategy is used to update solutios. Whe it meets oe of the followig coditios, a ew solutio will replace the curret solutio of a subobjective space: ()iftheobjectivevectorofthecurretsolutiodoesot locate i this subobjective space, the objective vector of the ew solutio locates i this space or the curret solutio is domiated by the ew solutio; (2) if the objective vector of the curret solutio locates i this subobjective space, the objective vector of the ew solutio also locates i this space ad the curret solutio is domiated by the ew solutio. The first update coditio makes each subobjective spacehaveasolutiowhoseobjectivevectorlocatesithis subobjective space, which ca well maitai the diversity of obtaied solutios. The secod update coditio makes odomiated solutio be kept, which ca make solutios coverge to the Selectio Strategy. I this work, we hope that those solutioswhicharekeptaftertheaboveupdatestrategy ad domiated by other solutios have a stroger ability to survive tha other solutios. I such a way, these solutios aremorelikelytobeselectedtogeerateewsolutios,ad the their subobjective spaces ca quickly fid their optimal solutios. Therefore, the fially obtaied solutio of each subobjective space is as close as possible to the. I order to achieve the goal, the crowdig distace [3] is used to calculate the fitess value of a solutio for the selectio operators. Because these solutios are domiated by other solutios ad the objective vectors of those solutios do ot locate i these subobjective spaces of these solutios, so i the term of the objective vector, these solutios have fewer solutios i their surroud tha other solutios. Thus, by usig the crowdig distace to calculate the fitess value of a solutio, the fitess values of these solutios are better tha those solutios ad these solutios are more likely to be selected to geerate ew solutios Steps of the Proposed Algorithm. Basedotheabove methods, a ew multiobjective evolutioary algorithm () is proposed ad the steps of the algorithm are as follows. Step (iitializatio). Give N directio vectors (γ,γ 2,...,γ N ), radomly geerate a iitial populatio POP(k),aditssizeisN;letk=.SetZ i = mi{f i (x) x POP(k)}, i m. Step 2 (fitess). Solutios of POP(k) are firstly divided ito N classes by the formula (5) adthefitessvalueofeach solutio i POP(k) is calculated by the crowdig distace. The, some better solutios are selected from the populatio POP(k) adputitothepopulatiopop.ithiswork,biary touramet selectio is used. Step 3 (ew solutios). Apply geetic operators to the paret populatio to geerate offsprig. The set of all these offsprig is deoted as O. Step 4 (update). Z is firstly updated. For each j=,...,m,if Z j > mi{f j (x) x O},thesetZ j = mi{f j (x) x O}. The solutios of POP(k) O are firstly classed by the formula (5); the N best solutios are selected by the update strategy of Sectio 3.2 ad put ito POP(k+).Letk=k+. Step 5 (termiatio). If stop coditio is satisfied, stop; otherwise, go to Step Experimetal Study I this sectio, has bee fully compared with a promisig multiobjective evolutioary algorithm based o decompositio () [22],whichwasrakedfirsti the ucostraied MOEA competitio [37], ad odomiated sortig geetic algorithm II () [3]o seve cotiuous test problems.

4 4 Joural of Applied Mathematics Istace F F2 F3 F4 Table : Multiobjective bechmark fuctios. Objective fuctio f (x) = (+g(x)) x f 2 (x) = (+g(x))( x ) t i =x i si (.5πx i ) f (x) = (+g(x)) x f 2 (x) =(+g(x))( x 2 ) t i =x i si (.5πx i ) i=2 i=2 f (x) =(+g(x)) cos ( πx 2 ) f 2 (x) =(+g(x)) si ( πx 2 ) t i =x i si (.5πx i ) i=2 f (x) =(+g(x))x f 2 (x) = (+g(x))( x.5 cos2 (2πx )) t i =x i si (.5πx i ) i=2 mutatio [4] operators are applied directly to real parameter values i three algorithms, that is,,, ad. For crossover operators, simulated biary crossover (SBX [4]) is used i ad ; differetial evolutio (DE) [4] is used i. The parameter settigs i this paper are as follows. () Cotrol parameters i reproductio operators: (a) distributio idex is 2 ad crossover probability is i the SBX operator; (b) crossover rate is. ad scalig factor is.5 i the DE operator; (c) distributio idex is 2 ad mutatio probability is. i mutatio operator. (2) For two-objective ad three-objective istaces, the populatio size is 5 i the three algorithms. Weight vectors are geerated by usig the method which is used i [22]. The directio vectors of are the weight vectors. The umber of weight vectors is the same as the populatio size. The umber of the weight vectors i the eighborhood i is 2 for all test problems. The Tchebycheff approach [22]isused for as a aggregate fuctio. Other cotrol parameters i are the same as i [22]. (3) Each algorithm is ru 2 times idepedetly for each test istace. All the three algorithms stop after geeratios. F5 F6 f (x) = (+g(x)) x x 2 f 2 (x) = (+g(x)) x ( x 2 ) f 3 (x) = (+g(x))( x ) t i =x i x x 2 i=3 f (x) = (+g(x)) cos(.5πx ) cos(.5πx 2 ) f 2 (x) =(+g(x)) cos(.5πx ) si(.5πx 2 ) f 3 (x) =(+g(x)) si(.5πx ) t i =x i x x 2 i=3 4.. Test Problem. The followig modified ZDT [36] ad DTLZ [38]istacesareusedithispaper.g(x) fuctios are modified to icrease the difficulty of these istaces. These strategies for costructig these test problems are used i [39]. These fuctios with complicated Pareto set shapes ca cause difficulties for MOEAs. Their search space is [, ] ad =. All these istaces are for miimizatio ad described i Table Parameter Settigs. Algorithms are implemeted o a persoal computer (Itel Xeo CPU 2.53 GHz, 3.98 G RAM). The idividuals are all coded as the real vectors. Polyomial 4.3. Experimetal Measures. I order to compare the performace of the differet algorithms quatitatively, performace metrics are eeded. I this paper, the followig performace metrics are used to compare the performace of the differet algorithms quatitatively: geeratioal distace (GD) [42], iverted geeratioaldistace (IGD) [42], ad hypervolume idicator (HV) [43]. GD measures how far the kow Pareto frotisawayfromthetrueparetofrot.ifgdisequalto, all poits of the kow belog to the true. GD allows us to observe whether the algorithm ca coverge to some regio i the true. IGD measures how far the true is away from the kow. If IGD is equal to, the kow cotais every poit of the true. IGD shows whether poits of the kow are evely distributed throughout thetrue.here,gdadigdidicatorsareusedsimultaeously to observe whether the solutios are distributed over the etire ad defie a suboptimal solutio as a good solutio. I experimets, we select 5 evely distributed poitsotheforfourtwo-objectivetestistacesad poits for three-objective test istaces. The hypervolume idicator has bee used widely i evolutioary multiobjective optimizatio to evaluate the performace of search algorithms. It computes the volume of the domiated portio of the objective space relative to a referece poit. Higher values of this performace idicator imply more desirable solutios. The hypervolume idicator measures both the covergece to thetrueparetofrotaddiversityoftheobtaiedsolutios. I our experimets, the referece poit is set to (,...,).

5 Joural of Applied Mathematics 5 F F F Fuctio 2 Fuctio 2 Fuctio 2.5 Fuctio.5 Fuctio.5.5 Fuctio (a) (b) (c) F2 F2 F2 Fuctio 2 Fuctio 2 Fuctio 2.2 Fuctio.4 Fuctio.2 Fuctio.4 (d) (e) (f) F3 F3.5 F3 Fuctio 2 Fuctio 2 Fuctio Fuctio.4 Fuctio Fuctio (g) (h) (i) Figure : Solutios obtaied by,, ad o F F Comparisos of with ad. I this sectio, some simulatio results ad comparisos that prove thepotetialofarepresetedadthecomparisos maily focus o three aspects: () the covergece of the obtaied solutios to the true ; (2) the coverage of the obtaied solutios to the true ; ad (3) the diversity oftheobtaiedsolutios.althoughsuchcomparisosare ot sufficiet, they provide a good basis to estimate the performace of. To visually compare the performace of the three algorithms, the solutios obtaied by them o these test problems are show i Figures ad 2.Obviously,bothad caot locate the global o ay istace. I cotrast, ca approximate the s of these istaces quite well. These Figures idicate that the diversity of solutios obtaied by the algorithm is better tha those obtaied by ad o these test problems. Table 2 presets the mea ad stadard deviatio of IGD, GD, ad HV obtaied by,, ad. Best solutios obtaied are highlighted bold i this table. From the table, we ca obtai that the mea values of IGD obtaied by are much smaller tha those obtaied by the other two algorithms for these six test problems, which idicates that for these test problems, the coverage

6 6 Joural of Applied Mathematics Fuctio 2 F4.2.4 Fuctio Fuctio 2 F4 Fuctio Fuctio 2 F4.5.5 Fuctio (a) (b) (c) F5 F5 F5 Fuctio 3.5 Fuctio Fuctio Fuctio Fuctio 2.5 Fuctio Fuctio 3.5 Fuctio Fuctio (d) (e) (f) F6 F6 F6 Fuctio Fuctio 2.5 Fuctio Fuctio Fuctio 2.5 Fuctio Fuctio Fuctio 2.5 Fuctio (g) (h) (i) Figure 2: Solutios obtaied by,, ad o F4 F6. of solutios obtaied by to the true is better tha those obtaied by ad. I additio, stadard deviatios of IGD obtaied by are very small for these test problems, which show that the performace of is well stable o these problems. We ca also see that the mea values of GD obtaied by are smaller tha those obtaied by the other two algorithms for test problems F ad F3, which idicate that the covergece of solutios obtaied by to the true is better tha those obtaied by ad. For problems F2adF4,themeavaluesofGDobtaiedbyare slightly larger tha those obtaied by but smaller tha those obtaied by. For problems F5 ad F6, themeavaluesofgdobtaiedbyarelargertha those obtaied by ad. For these four test problems, the covergece of solutios obtaied by to the true is worse tha that obtaied by, which is because the solutios obtaied by ad are cocetrated i certai regios of the, but solutios obtaied by are distributed throughout the ad covergetothe.fromthemeavaluesofgdobtaied by, we ca obtai that solutios obtaied by ca wellcovergetothetrueforthesesixtestproblems.i termsofthemeavalueofhv,thevaluesofhvobtaied by are much bigger tha those obtaied by other two algorithms, which illustrate that the diversities of solutios obtaied by are better tha those obtaied by ad.

7 Joural of Applied Mathematics 7 Table 2: IGD, GD, ad HV obtaied by,, ad o F F6. IGD GD HV Mea SD Mea SD Mea SD F F F F F F Coclusio I this paper, a ew evolutioary algorithm based o decompositio of the objective space is desiged to well maitai the diversity of obtaied solutios. I order to achieve the goal, the objective space of a MOP is decomposed ito a umber of subobjective spaces, ad the obtaied solutios are classed to make each subobjective space have a solutio. For each subobjective space, if a ew solutio will replace its curret solutio whose objective vector locates i this sub-regio, the ew solutio must domiate the curret solutio ad its objective vector locates i the sub-regio; if the objective vector of the curret solutio does ot locate i this sub-regio, the ew solutio domiate the curret solutio or its objective vector locates i the sub-regio. I such a way, good populatio diversity ca be achieved, which is essetial for solvig some MOPs. I additio, i order to improve the covergece of all obtaied solutios, thecrowdigdistaceisusedtocalculatethefitessvalue of a solutio for the selectio operators, which ca make domiated solutios be more likely to be selected to geerate ew solutios. Experimetal studies o six test istaces have implied that this proposed algorithm ca sigificatly outperform ad o these test problems. The future work icludes combiatio of this algorithm ad other evolutioary search techiques ad ivestigatio of its performace o other hard multiobjective optimizatio problems. Coflict of Iterests The authors declare that there is o coflict of iterests regardig the publicatio of this paper. Ackowledgmet This work was supported by the Natioal Natural Sciece Foudatio of Chia (os ad ). Refereces [] K.C.Ta,T.H.Lee,adE.F.Khor, Evolutioaryalgorithms with dyamic populatio size ad local exploratio for multiobjective optimizatio, IEEE Trasactios o Evolutioary Computatio,vol.5,o.6,pp ,2. [2] C.A.CoelloCoello,D.A.vaVeldhuize,adG.B.Lamot, Evolutioary Algorithms for Solvig Multiobjective Problems, Kluwer Academic Publishers, New York, NY, USA, 22. [3] K. Deb, A. Pratap, S. Agarwal, ad T. Meyariva, A fast ad elitist multiobjective geetic algorithm: NSGA-II, IEEE Trasactios o Evolutioary Computatio,vol.6,o.2,pp.82 97, 22.

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