a constrained solution update strategy for multiobjective

Research ArticleA Constrained Solution Update Strategy for MultiobjectiveEvolutionary Algorithm Based on Decomposition

Yuchao Su Qiuzhen Lin Jia Wang Jianqiang Li Jianyong Chen and Zhong Ming

College of Computer Science and Software Engineering Shenzhen University Shenzhen China

Correspondence should be addressed to Qiuzhen Lin qiuzhlinszueducn

Received 28 November 2018 Accepted 23 January 2019 Published 8 May 2019

Academic Editor Alex Alexandridis

Copyright copy 2019 Yuchao Su et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper proposes a constrained solution update strategy for multiobjective evolutionary algorithm based on decomposition inwhich each agent aims to optimize one decomposed subproblem Different from the existing approaches that assign one solutionto each agent our approach allocates the closest solutions to each agent and thus the number of solutions in an agent may be zeroand no less than one Regarding the agent with no solution it will be assigned one solution in priority once offspring are generatedclosest to its subproblem To keep the same population size the agent with the largest number of solutions will remove one solutionshowing the worst convergence This improves diversity for one agent while the convergence of other agents is not lowered Onthe agent with no less than one solution offspring assigned to this agent are only allowed to update its original solutions Thusthe convergence of this agent is enhanced while the diversity of other agents will not be affected After a period of evolution ourapproach may gradually reach a stable status for solution assignment ie each agent is only assigned with one solution Whencompared to six competitive multiobjective evolutionary algorithms with different population selection or update strategies theexperiments validated the advantages of our approach on tackling two sets of test problems

1 Introduction

In real-world applications it is often needed to handle mul-tiobjective optimization problems (MOPs) [1] such as rec-ommendation systems [2 3] privacy computing [4] andresource assignment [5ndash7] Due to the conflicts amongdifferent objectives the results of MOPs will output a set ofPareto solutions (PS) and theirmapping in the objective spaceis called Pareto front (PF) [8ndash10] These MOPs may becharacterized with complicated features [11ndash13] which can-not be well solved by traditional mathematical methodsInstead multiobjective evolutionary algorithms (MOEAs)can effectively obtain a set of solutions in one single runwhich have shown a very promising performance in tacklingdifferent kinds of MOPs [14ndash16] and become very popularduring the recent decades

In the design of MOEAs evolution and selection are theirtwo important mechanisms [17ndash19] The first one modifiesindividuals in order to approach the true PF while the secondone selects the most promising individuals to constitute thenew population for next generation Based on the selection

mechanisms most MOEAs can be classified into threetypes ie Pareto-based MOEAs [20ndash23] indicator-basedMOEAs [24ndash29] and decomposition-based MOEAs [30ndash36]Compared to the selection operators used in Pareto-basedand indicator-based MOEAs decomposition-based MOEAsare able to provide more flexibility to balance convergenceand diversity [37] which has been found to provide abetter performance when tackling some complicated MOPsas reported in [38] In this sort of MOEAs the targetMOP is decomposed into a set of subproblems which aresolved simultaneously using a set of cooperative agents Eachagent aims to optimize one subproblem in MOEAD [39]Due to the simplicity and effectiveness of MOEAD thisframework has triggered a considerable amount of researchaiming to improve different components of MOEAD suchas the adjustment and generation of weight vectors [40ndash44]dynamic resource allocation [45ndash47] enhanced evolutionaryoperators [48ndash50] and improved population selection orupdate mechanisms [51ndash57]

Especially regarding the population selection or updatemechanisms for decomposition-based MOEAs the offspring

HindawiComplexityVolume 2019 Article ID 3251349 11 pageshttpsdoiorg10115520193251349

2 Complexity

in MOEAD [39] are allowed to update any solution inpopulation However this method may significantly lowerthe diversity when a very good solution may replace mostof the others in several generations In MOEAD-DE [58]its solution update approach is controlled by two presetprobabilities 120575 and nr which obtains a better balance ofconvergence and diversity The offspring is only allowed toupdate the parent solutions from the neighborhood witha probability 120575 and from the entire population with aprobability (1-120575) Moreover an offspring can only replace atmost nr parent solutions This strategy wasmostly used in thefollowing design of decomposition-based MOEAs [49 54]Different from the decomposition approach in MOEAD-DE MOEAD-M2M [38] separates the search space intomultiple search subspace which simples the solving of MOPsin each subspace and the solution update is constrained byincluding the equal number of solutions in each subspaceThus MOEAD-M2M was shown to be very effective forcomplicated MOPs that strongly emphasize diversity (ieMOP problems [38]) To further find a better match ofsolutions and subproblems a stable matching model wasproposed in MOEAD-STM [59] which associates the solu-tions to subproblems according to their respective prefer-ences In this way MOEAD-STM can maintain a goodconvergence speed and population diversity Similarly animproved interrelationshipmodelwas designed inMOEAD-IR [37] to associate the solutions to subproblems based ontheirmutual-preferences which is an essentially diversity firstand convergence second strategy Moreover two improvedversions [53] forMOEAD-STMwere proposed to embed theconcept of the incomplete preference lists in the stable match-ing model which further strengthens the diversity In [51]an adaptive replacement neighborhood size was proposedto assign an offspring to its most appropriate subproblemsobtaining a better balance of convergence and diversity InMOEAD-ACD [54] an adaptive constrained decompositionapproach was presented in which the update regions ofdecomposition approach are constrained to maintain thediversity Moreover to further enhance the performancein MOPs with more than three objectives decompositionapproach and Pareto domination were simultaneously usedinMOEADD [44] decomposition-based-sorting and angle-based-selection approaches were proposed in MOEAD-SAS[57] and the diversity was preferred in solution update byselecting certain closest subproblems for an offspring in [60]

On the other hand another kind of population selectionor update mechanisms in MOEAD aims to improve theirused decomposition functions InMOEAD [39] three tradi-tional decomposition functions ie the weighted sum (WS)approach the Tchebycheff (TCH) approach and the penalty-based boundary intersection (PBI) approach were employedIn [61 62] a local PBI andWS were respectively designed toconstrain the update regions of decomposition approacheswhich avoid the diversity loss In [63 64] an adaptive Paretofront scalarizing (PaS) and penalty-based boundary inter-section (PaP) decomposition approaches were respectivelyintroduced to match the true PFs with various shapes Twodecomposition approaches were presented in MOEAAD[65] and DECAL [66] to deal with the complicated PF In

MOEAAD two coevolved populations were respectivelyupdated by the two decomposition functions to fit differ-ent PF shapes while two novel decomposition functionswere respectively used to accelerate the convergence speedand enhance the population diversity in DECAL RecentlyMOEAD-LTD [67] was proposed to trace the PF shape inwhich the learning module predicts the PF shape and thedecomposition function is adaptively adjusted to fit its PFshape

Most of the above MOEAs all abide one basic principlethat each agent should be assigned with one solution in orderto find the optimal value for its subproblem However thiskind of solution assignment may not be effective and efficientin decomposition-based MOEAs as the solution assigned tothe agent may be far away from its subproblem In such caseit cannot truly reflect the diversity of each agent and cannotprovide the correct neighboring information in evolutionwhich may slow down the convergence as decomposition-based MOEAs are designed as an essentially collabora-tive evolutionary framework Therefore a constrained solu-tion update (CSU) strategy is designed in this paper fordecomposition-basedMOEAs to alleviate the above problemThe solutions are only assigned to the agent that handlesthe closest subproblem This way the correct neighboringinformation can be provided to guide the evolution and it isstraightforward to show the diversity of each agent In thiscase the number of solutions in each agent may be zero orno less than one To maintain the diversity of each agent theoffspring assigned to one agent are only allowed to renew itsoriginal solutions When the agent has no solution it will beassigned one solution in priority once offspring are generatedclosest to its subproblem To keep the same population sizethe agentwith the largest number of solutionswill remove onesolution showing the worst convergence Thus the diversityof one agent is enhanced while the convergence of otheragents is not affected After a period of evolution a stablestatus for solution assignment is anticipated so that eachagent only has one solution When compared to the existingpopulation selection or update strategies for decomposition-based MOEAs our experiments validate the superiority ofthe proposed approach when tackling two sets of complicatedtest MOPs

The main contributions of this paper are clarified below

(1) Each agent may be assigned with no solution or noless than one solution which is different from theexisting approaches that only assign one solution toeach agent This approach can truly reflect the diver-sity on the agents and provide the correct neighboringinformation in evolution

(2) A CSU strategy is designed for each agent in order tomaintain diversity for all the agents without affectingtheir convergence The agent with no solution will beassigned first while the agent with the largest numberof solutions will remove one solution showing theworst convergence By this way a stable status forsolution assignment may be reached so that eachagent only has one solution which ensures diversityin decomposition-based MOEAs

Complexity 3

(3) When solution assignment is under an unstable statussuch that at least one agent is still not assigned anysolution the mating parents are randomly selectedfrom the best solutions from all the agents as theneighboring agentmay have no solutionThis randomselection of mating parents helps to enhance theexploration ability in our algorithm

The rest of this paper is organized as follows Section 2provides the related background such as MOPs and theused decomposition function in this paper Section 3 intro-duces the details of the proposed algorithm MOEAD-CSUThe experimental results and discussions are provided inSection 4 while the conclusions and some future researchdirections are given in Section 5

2 Related Background

21 Multiobjective Optimization Problems Multiobjectiveoptimization problems often need to optimize several con-flicting objectives which can be modeled by

Minimize 119865 (119909) = 1198911 (119909) 1198912 (119909) 119891119898 (119909) (1)

where 119909 = (1199091 1199092 119909119899) isin Ω is an n dimensional decisionvector in the decision space Ω and m is the number ofobjectives The target of MOP in (1) is to minimize all theobjectives simultaneously

22 The Decomposition Function In this paper the modifiedTchebycheff method [55] is used for decomposing the MOPin (1) which is defined by

119892119879119888ℎ (x | w zlowast) = max1le119894le119898

(1003816100381610038161003816119891119894 (x) minus 119911lowast119894 1003816100381610038161003816119908119894 ) (2)

where w = (1199081 1199082 119908119898) is a preset weight vector with119908119894 ge 0 for each 119894 isin [1119898] and sum119898119894=1119908119894 = 1 whilezlowast = 119911lowast1 119911lowast2 119911lowast119898 is the ideal point by setting 119911lowast119894 =min119891119894(x) | x isin Ω for each 119894 isin [1119898] When using Nuniformly distributed weight vectors in (2) the MOP in (1)is decomposed into a set of N subproblems which can besolved by a set of N collaborative agents The populationselection or update strategies designed in decomposition-based MOEAs will reasonably allocate the solutions to theagents [39] Different from the existing approaches [39 58]that assign one solution to each agent the agent in ourapproach is only allocated by the solutions that are closestto its subproblem resulting in the fact that the number ofsolutions in each agent may be zero or no less than one Toshow (2)more visually a case of updating solution is depictedin Figure 1 where s1 is a solution in current population whiles2 and s3 are two offspring For this case s3 can update thesubproblem but s2 cannot do this because the yellow regionis the improvement domain of s1 by the weight vector and(2) and a solution like s3 falling into the region can updatethe subproblem Actually (2) decides the profile of the region[54]

Weight vector (03 07)

f2

f1

s3

s2

s1

zlowast

Figure 1 Update the subproblem by (2)

3 Our Algorithm MOEA-CSU

Let w1w2 w119873 be N weight vectors and 119860 119894 denote theagent which aims to optimize the subproblem in (2) withthe weight vector w119894 (119894 isin [1119873]) In this paper we classifythe status of solution assignment into two kinds ie a stablestatus (each119860 119894 is assigned only one solution) and an unstablestatus (at least one 119860 119894 is not assigned any solution) (119894 isin[1119873]) Generally an initial population often starts from theunstable status while the purpose of our CSU strategy is toreach the stable status which properlymaintains the diversityof each agent

31 Our CSU Strategy Let P and O respectively denotethe parent population and offspring population At eachgeneration the solution set from P assigned to agent 119860 119894 isdenoted by Λ119901119894 (119894 isin [1119873]) while the solution set fromO assigned to agent 119860 119894 is denoted by Λ119900119894 (119894 isin [1119873]) Inthis paper Λ119901119894 and Λ119900119894 can be obtained using the closestvector angles to the weight vector w119894 of agent 119860 119894 as fol-lows

Λ119901119894 = x isin P | ⟨F (x) minus zlowastw119894⟩ le ⟨F (x) minus zlowastw119895⟩for forall119895 isin [1119873] (3)

Λ119900119894 = x isin O | ⟨F (x) minus zlowastw119894⟩ le ⟨F (x) minus zlowastw119895⟩for forall119895 isin [1119873] (4)

where zlowast = (119911lowast1 119911lowast1 119911lowast119898) (m is the number of objectives) isan utopian objective vector that is approximated by the min-imal objective values from the current parent and offspringpopulations ie 119911lowast119894 = min119891119894(x) x isin PcupO for each 119894 isin [1 119898]

4 Complexity

(1) Get Λ119901119894 and Λ119900119894 respectively from P andO with Eqs (3)-(4)(2) for i=1 to N(3) if |Λ119900119894|=0(4) if |Λ119901119894 |==0(5) find one solution x with the minimal value in Eq (2) from Λ119900119894(6) add x into Λ119901119894(7) find one agent Λ119901119896 with the largest number of solutions(8) remove one solution with the worst value in Eq (2) from Λ119901119896(9) else(10) letU = Λ119900119894 cup Λ119901119894 and set Λ119901119894 as an empty set(11) sort the solutions in U ascendingly using the aggregated values in Eq (2)(12) select the first |Λ119901119894 | solutions from U to compose a new Λ119901119894(13) end if(14) end if(15) end for(16) collect all the Λ119901119894 to compose a new P(17) if each Λ119901119894 is not empty(18) status=True solution assignment is under the stable status(19) end if(20) return [P status]

Algorithm 1 CSU(PO N) constrained solution update

and ⟨F(x) minus zlowastw119894⟩ indicates the acute angle of two vectorsF(x) minus zlowast and w119894 as defined by

⟨F (x) minus zlowastw119894⟩= arccos

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816sum119898119896=1 (119891119896 (x) minus 119911lowast119896 ) sdot 119908119894119896

radicsum119898119896=1 (119891119896 (x) minus 119911lowast119896 )2 sdot radicsum119898119896=1 (119908119894119896)21003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816 (5)

The design principle of our method is simple and effectiveWhen Λ119900119894 is empty Λ119901119894 will not be updated Otherwise theoffspring assigned to each agent is only allowed to renew itsoriginal solutions ie the solutions in Λ119900119894 can only renewΛ119901119894 which will speed up the convergence for 119860 119894 while thediversity of other agents is not affected as the solutions in Λ119900119894are not allowed to update the solutions for other agents Inmore detail two cases for Λ119901119894 are considered when Λ119900119894 is notempty ie |Λ119901119894 | = 0 and |Λ119901119894 | gt 0 where |Λ119901119894 | indicates thesize of Λ119901119894

In the case with |Λ119901119894 | = 0 as the agent 119860 119894 is not assignedwith any solution before one solution in Λ119900119894 with the bestaggregated value using (2) is assigned to 119860 119894 To keep thesame population size the agent 119860119896 (119896 isin [1119873]) with thelargest number of solutions is found and then one solutionin Λ119901119894 having the worst aggregated value in (2) is removedPlease note that if more than one agent has the same largestnumber of solutions one of them is randomly selected toremove one worst solution This way the agent 119860 119894 is assignedone solution to optimize its subproblem which enhances itsdiversity while the convergence for other agents (eg 119860119896) isnot affected

In other case with |Λ119901119894 | gt 0 the solutions inΛ119901119894 andΛ119900119894 arecombined into U and they are sorted using the aggregatedvalues in (2) with an ascending order The first |Λ119901119894 | solutions

are selected from U to compose a new Λ119901119894 which keeps thesame number of solutions for the agent 119860 119894 By this way theconvergence for the agent 119860 119894 is enhanced while the diversityfor other agents is not affected as the offspring in Λ119900119894 are notallowed to update them

With the above operations the number of solutionsassigned to each agent will be gradually reduced to only oneonce there exists a solution generated around its subproblemFinally this approach may reach the stable status Note thatthe stable status may be unreachable when solving MOPswith complicated PFs eg disconnected and degeneratedPFs To further clarify the CSU strategy its pseudocode isgiven in Algorithm 1 Please note that Algorithm 1 will returnthe updated population P and the status of solutions assign-ment

32 The Used Recombination Operator In this paper theevolutionary operator [68] in MOEAD-M2M is used whichis respectively defined in (6) and (7) as follows

119910 = 119909 + (119911 minus 119909) 1199031 (1 minus (1199032)120578) (6)

119910 = 119910 + 1199033 (1 minus (1199032)120578) (119906 minus 119897) (7)

where x and z are the decision variables from two parentsand y is that of an offspring while u and l are respectivelythe lower and upper bounds for that decision variable Thecrossover operator is defined in (6) where r1 and r2 are tworandom real numbers respectively generated from (-1 1) and(0 1) and 120578 is an index that is set to -(1-119892119866max)

07 (119866maxand 119892 are respectively the maximum number of generationand the current generation)Themutation operator is definedin (7) where r3 is a random real number produced from

Complexity 5

(1) generateN weight vectors w1w2 w119873 initialize the values of 119892 status anoffspring populationO and generate an initial population P = p1p2 p119873

(2) get Λ119901119894 from P with (3)(3) while 119892 lt 119866max(4) for i=1 to N(5) if status ==0(6) collect the best solution of each Λ119901119894 to form the mating pool(7) else(8) set the neighbors of subproblem i as the mating pool(9) end if(10) generate oi with pi and the parents from the mating pool(11) evaluate the objectives in oi and update zlowast in (2)(12) add oi into the offspring population O(13) end for(14) [P status]= CSU(PO N)(15) set 119892 = 119892 + 1 and initialize O as an empty set(16) end while(17) return P

Algorithm 2 MOEAD-CSU

(-025 025) When y is out of the parameter boundary arepair operation will be executed in (8) and (9) as follows

119910 = 119906 minus 1199034 (119906 minus 119910) if 119910 gt 119906 (8)

119910 = 119897 + 1199034 (119910 minus 119897) if 119910 lt 119897 (9)

where 1199034 is a random real number generated in (-05 05)

33 MOEAD-CSU In this section our CSU strategy isembedded into a general framework of decomposition-basedMOEAs named MOEAD-CSU Its pseudocode is providedin Algorithm 2 In line (1) N weight vectors w1w2 w119873are generated and then the value of generation counter 119892is initialized to 1 the value of status is initialized to False(indicating the unstable status of solution assignment) theoffspring population O is initialized as an empty set and aninitial population P = p1 p2 p119873 is generated randomlyin line (1) In line (2) Λ119901119894 for each agent is obtained fromP by (5) If 119892 is smaller than the preset maximum numberof generations 119866max the following evolution and selectionprocedures in lines (4)-(15) are run For each subproblem iin line (4) it is checked whether the status of the solutionsassignment is stable in line (5) If it is not we collect the bestsolution in each Λ119901119894 to form the mating pool Otherwise weset the neighbors of subproblem i as the mating pool in line(8) based on the Euclidean distance between the used weightvectors Here the neighbor size T in line (8) is dynamicallyadjusted according to the number of generations by using

119879 = lfloor119873 times (119866max minus 119892)119866max125

rfloor + 1 (10)

After that an offspring o119894 is generated using the recombina-tion operators defined in (6)-(9) based on pi and the matingparents in line (10) and is further evaluated to get the objective

values in line (11) which is used to update the approximatelyideal point 119911lowast in (2) In line (12) this offspring oi is addedinto the offspring population O After all the offspring arecollected intoO the CSU strategy (Algorithm 1) is run in line(14) with the inputs P O N to get a new population P Inline (15) the value of 119892 is increased by 1 and the offspringpopulationO is reset to an empty set The above evolutionaryprocess will be terminated when 119892 reaches119866max and the finalpopulation P is reported

4 Experimental Results

41 Benchmark Problems and Parameters Settings In thisstudy two complicated test suites (MOP [38] and IMB[69]) were used to assess the performance of MOEAD-CSU including MOP1-MOP7 and IMB1-IMB10 They havecomplicated mathematical features on the PS shapes Pleasenote that MOP1-MOP5 IMB1-IMB3 and IMB7-IMB9 havetwo optimization objectives while MOP6-MOP7 IMB4-IMB6 and IMB10 include three optimization objectivesThe number of decision variables is set to 10 for all thetest problems Regarding the biobjective and three-objectivetest problems the population sizes were respectively setto 100 and 300 as suggested in [38] while the maximumnumbers of function evaluations were respectively set to3times105 and 9times105 The performance of MOEAD-CSU iscompared to six competitive MOEADs with different pop-ulation selection or update strategies ie MOEAD-M2M[38] MOEAD-STM [59] MOEAD-AGR [51] MOEAD-IR [37] MOEAD-DE [58] and MOEAD-ACD [54] Pleasenote that MOEAD-M2M MOEAD-AGR and MOEAD-CSU are run in Matlab while the rest algorithms are realizedin jMetal [70]The parameters in all the compared algorithmswere set as recommended in their original references Thecrossover mutation probability in our algorithm was set to10 and 1n to run (6) and (7) respectively as suggested in[38]

6 Complexity

Table 1 IGD comparison of results of MOEAD-CSU and six competitors on all theMOP and IMB test problems

MOEAD-M2M

MOEAD-STM

MOEAD-AGR MOEAD-IR MOEAD-DE MOEAD-

ACD MOEAD-CSU

MOP1 Mean(Std)

226E-2 ndash(152E-3)

346E-1 ndash(227E-2)

269E-2 ndash(353E-3)

263E-2 ndash(272E-3)

362E-1 ndash(747E-3)

272E-2 ndash(233E-3)

166E-2(438E-4)

MOP2 Mean(Std)

740E-3 ndash(509E-4)

295E-1 ndash(783E-2)

679E-2 ndash(769E-2)

602E-2 ndash(646E-2)

277E-1 ndash(719E-2)

313E-2 ndash(667E-2)

687E-3(229E-3)

MOP3 Mean(Std)

104E-2 ndash(220E-3)

155E-1 ndash(343E-2)

385E-2 ndash(583E-2)

143E-2 ndash(191E-2)

123E-1 ndash(482E-2)

136E-2 ndash(206E-2)

814E-3(376E-3)

MOP4 Mean(Std)

457E-3 =(339E-4)

300E-1 ndash(299E-2)

352E-2 ndash(323E-2)

107E-1 ndash(846E-2)

276E-1 ndash(324E-2)

548E-2 ndash(574E-2)

447E-3(122E-4)

MOP5 Mean(Std)

200E-2 ndash(691E-4)

312E-1 ndash(278E-2)

231E-2 ndash(304E-3)

216E-2 ndash(241E-3)

316E-1 ndash(802E-3)

239E-2 ndash(231E-3)

151E-2(357E-4)

MOP6 Mean(Std)

496E-2 ndash(158E-3)

290E-1 ndash(224E-2)

490E-2 ndash(236E-3)

484E-2 ndash(280E-3)

290E-1 ndash(256E-2)

502E-2 ndash(239E-3)

330E-2(280E-4)

MOP7 Mean(Std)

793E-2 ndash(469E-3)

351E-1 ndash(893E-8)

177E-1 ndash(316E-2)

192E-1 ndash(186E-2)

338E-1 ndash(227E-2)

233E-1 ndash(247E-2)

465E-2(335E-4)

IMB1 Mean(Std)

104E-2 ndash(611E-4)

105E-1 ndash(878E-2)

123E-2 ndash(133E-3)

118E-2 ndash(196E-3)

176E-1 ndash(714E-2)

128E-2 ndash(952E-4)

753E-3(136E-4)

IMB2 Mean(Std)

119E-2 ndash(546E-4)

162E-1 ndash(116E-2)

470E-2 ndash(179E-2)

664E-2 ndash(246E-2)

167E-1 ndash(111E-2)

585E-2 ndash(184E-2)

892E-3(216E-4)

IMB3 Mean(Std)

178E-2 ndash(766E-4)

282E-1 ndash(355E-2)

256E-2 ndash(348E-3)

237E-2 ndash(273E-3)

284E-1 ndash(142E-2)

226E-2 ndash(188E-3)

119E-2(316E-4)

IMB4 Mean(Std)

405E-2 ndash(159E-3)

133E-1 ndash(692E-3)

253E-2 +(504E-4)

239E-2 +(257E-4)

137E-1 ndash(697E-3)

282E-2 =(545E-4)

279E-2(219E-4)

IMB5 Mean(Std)

567E-2 ndash(992E-3)

840E-2 ndash(247E-5)

839E-2 ndash(137E-5)

790E-2 ndash(461E-3)

912E-2 ndash(574E-5)

741E-2 ndash(750E-3)

328E-2(121E-4)

IMB6 Mean(Std)

385E-2 ndash(524E-3)

462E-2 ndash(124E-4)

461E-2 ndash(212E-4)

470E-2 ndash(145E-4)

522E-2 ndash(628E-4)

250E-2 ndash(310E-4)

237E-2(871E-5)

IMB7 Mean(Std)

108E-2 ndash(589E-4)

298E-2 ndash(228E-4)

249E-2 ndash(830E-3)

276E-2 ndash(276E-2)

299E-2 ndash(195E-4)

285E-2 ndash(695E-3)

792E-3(305E-4)

IMB8 Mean(Std)

124E-2 ndash(530E-4)

345E-2 ndash(977E-4)

261E-2 ndash(106E-2)

334E-2 ndash(535E-3)

338E-2 ndash(358E-3)

352E-2 ndash(419E-3)

940E-3(381E-4)

IMB9 Mean(Std)

135E-2 ndash(662E-4)

384E-2 ndash243E-4

325E-2 ndash(916E-3)

379E-2 ndash(342E-3)

383E-2 ndash(229E-4)

396E-2 ndash(120E-3)

118E-2(618E-4)

IMB10 Mean(Std)

479E-2 ndash(868E-4)

282E-2 +(180E-3)

329E-2 +(961E-4)

295E-2 +(855E-4)

363E-2 ndash(195E-3)

317E-2 +(763E-4)

357E-2(302E-4)

BetterWorseSimilar 1601 1610 1520 1520 1700 1511

42 Performance Measures In this paper in order to providea comprehensive assessment on the performance of all thecompetitors two widely used performance indicators ieinverted generational distance (IGD) [71] and Hypervolume(HV) [71] were adopted to measure the convergence andthe diversity of the final solution set A lower value of IGDand a larger value of HV indicate a better performance toapproach the true PF and to spread solutions uniformly alongthe true PFWhen computing the IGD indicator no less than500 sampling points from the true PF were used For the HVcalculation the reference pointswere set to 11 times the upperbound of the PF ie (11 11) for biobjective problems andto (11 11 11) for three-objective problems as suggested in[71]

All the algorithmswere run 30 times and themean resultsand standard deviations were collected for comparison Inorder to have a statistically sound conclusion Wilcoxonrsquosrank sum test with a 5 significance level was conducted to

compare the significance of statistical difference between theresults obtained by MOEAD-CSU and other competitors

43 Performance Comparisons with Six Competitive MOEADs Table 1 gives all the mean IGD results and standarddeviations on MOP and IMB test problems where the bestmean result for each problem is highlighted in boldface Thelast row ldquoBetterWorseSimilarrdquo in Table 1 summarizes thenumbers of test problems in which MOEAD-CSU respec-tively performed better than worse than and similarly to itscompetitors

From Table 1 it is observed that MOEAD-CSU per-formed best on most of the MOP and IMB test problemsAs these problems were designed with complicated mathe-matical features that require more diversity in the populationMOEADs only emphasizing the convergence will get easilytrapped into local PFs That is the reason why MOEAD-STM and MOEAD-DE had a poor performance obtaining

Complexity 7

Table 2 HV comparison of results of MOEAD-CSU and six competitors on all theMOP and IMB test problems

MOEAD-M2M

MOEAD-STM


ACDMOEAD-

CSU

MOP1 Mean(Std)

695E-1 ndash(178E-3)

263E-1 ndash(361E-2)

692E-1 ndash(423E-3)

693E-1 ndash(283E-3)

238E-1 ndash(137E-2)

692E-1 ndash(244E-3)

703E-1513E-4

MOP2 Mean(Std)

439E-1 =(341E-4)

202E-1 ndash(486E-2)

379E-1 ndash(812E-2)

386E-1 ndash(667E-2)

206E-1 ndash(405E-2)

412E-1 ndash(660E-2)

439E-1360E-3

MOP3 Mean(Std)

340E-1 ndash(321E-3)

215E-1 ndash(290E-2)

311E-1 ndash(575E-2)

332E-1 ndash(264E-2)

241E-1 ndash(398E-2)

333E-1 ndash(268E-2)

341E-1400E-3

MOP4 Mean(Std)

595E-1 =(824E-4)

290E-1 ndash(162E-2)

569E-1 ndash(357E-2)

474E-1 ndash(965E-2)

301E-1 ndash(231E-2)

536E-1 ndash(666E-2)

595E-1248E-4

MOP5 Mean(Std)

696E-1 ndash(113E-3)

404E-1 ndash(208E-2)

694E-1 ndash(487E-3)

696E-1 ndash(274E-3)

400E-1 ndash(103E-17)

694E-1 ndash(272E-3)

703E-1465E-4

MOP6 Mean(Std)

822E-1 ndash(174E-3)

634E-1 ndash(248E-2)

824E-1 ndash(362E-3)

828E-1 ndash(242E-3)

634E-1 ndash(297E-2)

828E-1 ndash(221E-3)

840E-1271E-4

MOP7 Mean(Std)

541E-1 ndash(443E-3)

407E-1 ndash(189E-7)

497E-1 ndash(339E-2)

494E-1 ndash(491E-3)

412E-1 ndash(102E-2)

494E-1 ndash(145E-2)

545E-1526E-4

IMB1 Mean(Std)

711E-1 ndash(741E-4)

615E-1 ndash(887E-2)

709E-1 ndash(156E-3)

710E-1 ndash(204E-3)

543E-1 ndash(695E-2)

709E-1 ndash(100E-3)

715E-1156E-4

IMB2 Mean(Std)

570E-1 ndash(703E-4)

391E-1 ndash(914E-3)

525E-1 ndash(233E-2)

501E-1 ndash(302E-2)

390E-1 ndash(718E-3)

510E-1 ndash(234E-2)

574E-1290E-4

IMB3 Mean(Std)

329E-1 ndash(763E-4)

133E-1 ndash(220E-2)

324E-1 ndash(197E-3)

323E-1 ndash(262E-3)

128E-1 ndash(718E-3)

324E-1 ndash(169E-3)

335E-1390E-4

IMB4 Mean(Std)

823E-1 ndash(244E-3)

771E-1 ndash(303E-3)

849E-1 +(780E-4)

853E-1 +(454E-4)

764E-1 ndash(310E-3)

846E-1 =(963E-4)

846E-1245E-4

IMB5 Mean(Std)

548E-1 ndash(676E-3)

545E-1 ndash(231E-5)

545E-1 ndash(234E-5)

543E-1 ndash(791E-4)

532E-1 ndash(243E-4)

542E-1 ndash(846E-4)

571E-1271E-4

IMB6 Mean(Std)

831E-1 ndash(307E-3)

841E-1 ndash(353E-5)

840E-1 ndash(428E-5)

840E-1 ndash(144E-4)

832E-1 ndash(125E-3)

852E-1 ndash(358E-4)

853E-1111E-4

IMB7 Mean(Std)

709E-1 ndash(819E-4)

695E-1 ndash(516E-4)

699E-1 ndash(691E-3)

697E-1 ndash(558E-3)

695E-1 ndash(467E-4)

696E-1 ndash(587E-3)

714E-1514E-4

IMB8 Mean(Std)

567E-1 ndash(833E-4)

545E-1 ndash(114E-3)

555E-1 ndash(118E-2)

546E-1 ndash(604E-3)

546E-1 ndash(394E-3)

544E-1 ndash(477E-3)

573E-1541E-4

IMB9 Mean(Std)

331E-1 ndash(114E-3)

313E-1 ndash(888E-4)

318E-1 ndash(682E-3)

314E-1 ndash(249E-3)

314E-1 ndash(783E-4)

311E-1 ndash(202E-3)

334E-1906E-4

IMB10 Mean(Std)

817E-1 ndash(219E-3)

848E-1 +(135E-3)

839E-1 +(123E-3)

846E-1 +(101E-3)

834E-1 ndash(305E-3)

843E-1 +(998E-4)

836E-1335E-4


IGD resultsmostly under an accuracy of 10minus1 Other competi-tors eg MOEAD-M2M MOEAD-AGR MOEAD-ACDand MOEAD-IR were designed to put more emphasis ondiversity and they performed much better obtaining IGDresults mostly with an accuracy of 10minus2 which is still notso close to the true PFs Since the proposed CSU strategywas used in MOEAD-CSU it strongly emphasizes diversitybut impacts the convergence less MOEAD-CSU properlyconverged to the true PFs obtaining IGD results underan accuracy of 10minus3 for half of test problems adopted OnMOP1 to MOP7 MOEAD-CSU gets the all the best resultsParticularly some results are under an accuracy of 10minus3while the competitors cannot converge to the PF well ToIMB test problems the performance of MOEAD-CSU issuperior except for the results on IMB4 and IMB10 OnIMB4 MOEAD-CSU is worse than MOEAD-ARG andMOEAD-IR similar to MOEAD-ACD and better than therest algorithms For IMB10 MOEAD-STM gets the best

result and MOEAD-DE has a pretty good performance Itindicates that the convergence is important on IMB10 Tosummarize the experimental results on Table 1 MOEAD-CSU is superior to the competitors on most of test problemsSeeing the last row ldquoBetterWorseSimilarrdquo when comparedto six competitive MOEAD variants MOEAD-CSU canperform better on at least 15 cases and worse on at most2 cases which indicates our outstanding performance tobalance convergence and diversity for these test problemsadopted Moreover the HV results provided in Table 2 alsoconfirm the advantages of MOEAD-CSU as MOEAD-CSUperforms best on most of the cases

To visually show our performance the best nondom-inated solution sets obtained by MOEAD-CSU from 30runs were plotted in Figure 2 where the circles indicate thesolutions while the lines and grids mean the true PFs on thebiobjective and three-objective test problems respectivelyOn the test problemswith continuous PFs (ieMOP1-MOP3

8 Complexity

0 02 04 06 08 10

02

04

06

08

1MOP1

0 02 04 06 08 10

02

04

06

08

1MOP2

0 02 04 06 08 10

02

04

06

08

1MOP3

0 02 04 06 08 10

02

04

06

08

1MOP4

0 02 04 06 08 10

02

04

06

08

1MOP5

00 0

05z

MOP6

y x

05 05

1

1 1

000

05

05

MOP7

1

051

15

115 0 02 04 06 08 10

02

04

06

08

1IMB1

0 02 04 06 08 10

02

04

06

08

1IMB2

0 02 04 06 08 10

02

04

06

08

1IMB3

000

05

IMB4

z

xy

0505

1

11

00 0

05

IMB5

1

05 05

15

1 1

00 0

05z

IMB6

y x

05 05

1

1 1 0 02 04 06 08 1 120

02

04

06

08

1

12IMB7

0 02 04 06 08 1 120

02

04

06

08

1

12IMB8

0 02 04 06 08 1 120

02

04

06

08

1

12IMB9

00 0

05z

IMB10

y x

05 05

1

1 1

Figure 2 The nondominated solution sets onMOP1-MOP7 and IMB1-IMB10

MOP5-MOP7 and IMB1-IMB10) MOEAD-CSU can reachthe stable status and find all the optimal values for the agentsEven forMOP4which has a disconnected PFMOEAD-CSUcould properly approach all the segments of the true PF Fromthese plots it is reasonable to conclude that our proposedCSU strategy is very effective in tackling complicated testproblems such asMOP and IMB

5 Conclusions and Future Work

In this paper an enhanced decomposition-based MOEAwitha CSU strategy was presented The agent in our approachaims to optimize the subproblem which is only allocatedwith the solutions that are closest to its subproblem Thusthe number of solution in each agent may be zero or no less

than one which helps to reflect the true diversity among theagents and to provide the correct neighboring informationin evolution To ensure diversity the offspring in each agentare only allowed to update its original solutions In thecase that the agent has no solution one solution will beassigned in priority once there are offspring generated closestto its subproblem Another agent with the largest numberof solutions will remove one solution showing the worstconvergence Therefore for each agent this approach mayenhance its diversity or convergence but will not deteriorateeither of them After assessing its performance on twocomplicated test suites (MOP and IMB) the experimentalresults confirmed the superiority of MOEAD-CSU over sixcompetitive MOEADs with other population selection orupdate strategies

Complexity 9

In our future work the performance of this CSU strategywill be further studied to improve the way in which itreaches the stable status One possible path is to embed anadaptive adjustment strategy for generating weight vectors inMOEAD-CSU which can cooperate with the CSU strategyto attain real-diversity when dealing with disconnected orincomplete PFs The application of MOEAD-CSU in somereal-world problems will also be our future research direc-tion

Data Availability

The source code and source data can be provided by contact-ing with the corresponding author

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work was supported by Shenzhen Technology Planunder Grant JCYJ20170817102218122 the National Natu-ral Science Foundation of China under Grants 6187611061836005 and 61402291 the Joint Funds of the NationalNatural Science Foundation of China under Key ProgramGrant U1713212 and the Natural Science Foundation ofGuangdongProvince underGrant 2017A030313338 Also thiswork was supported by the National Engineering Laboratoryfor Big Data System Computing Technology

References

[1] K Miettinen Nonlinear Multiobjective Optimization KluwerAcademic Publishers Norwell Mass USA 1999

[2] Q Lin X Wang B Hu et al ldquoMultiobjective personalizedrecommendation algorithm using extreme point guided evolu-tionary computationrdquoComplexity vol 2018 Article ID 171635218 pages 2018

[3] X Li D Zhou Q Pan Y Tang and J Huang ldquoWeapon-targetassignment problem by multiobjective evolutionary algorithmbased on decompositionrdquo Complexity vol 2018 Article ID8623051 19 pages 2018

[4] M Eskandari Nasab I Maleksaeedi M Mohammadi and NGhadimi ldquoA new multiobjective allocator of capacitor banksanddistributed generations using a new investigated differentialevolutionrdquo Complexity vol 19 no 5 pp 40ndash54 2014

[5] Z Gao X Cui Y Duan Z Jun and Z Peng ldquoUsingMOPSO foroptimizing randomized response schemes in privacy comput-ingrdquo Mathematical Problems in Engineering vol 2018 ArticleID 7846547 16 pages 2018

[6] X Li J Lai andR Tang ldquoA hybrid constraints handling strategyfor multiconstrained multiobjective optimization problem ofmicrogrid economicalenvironmental dispatchrdquo Complexityvol 2017 Article ID 6249432 12 pages 2017

[7] K Deb Multiobjective Optimization Using Evolutionary Algo-rithms Wiley New York NY USA 2001

[8] S Huband L Barone L While and P Hingston ldquoA scalablemulti-objective test problem toolkitrdquo LectureNotes in ComputerScience vol 3410 pp 280ndash295 2005

[9] K Deb L Thiele M Laumanns and E Zitzler ldquoScalable testproblems for evolutionarymultiobjective optimizationrdquo in Evo-lutionary Multiobjective Optimization Advanced Informationand Knowledge Processing Series pp 105ndash145 Springer BerlinGermany 2005

[10] Q Zhang A Zhou S Zhao P Suganthan W Liu and STiwari ldquoMultiobjective optimization test instances for the CEC2009 special session and competitionrdquo Tech Rep CES-887University of Essex and Nanyang Technological UniversityEssex UKSingapore 2008

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

[12] R Cheng Y JinMOlhofer and B sendhoff ldquoTest problems forlarge-scale multiobjective and many-objective optimizationrdquoIEEE Transactions on Cybernetics vol 47 no 12 pp 4108ndash41212017

[13] R ChengM Li Y Tian et al ldquoA benchmark test suite for evolu-tionary many-objective optimizationrdquo Complex and IntelligentSystems vol 3 no 1 pp 67ndash81 2017

[14] S Yang S Jiang and Y Jiang ldquoImproving the multiobjectiveevolutionary algorithm based on decomposition with newpenalty schemesrdquo Soft Computing vol 21 no 16 pp 4677ndash46912017

[15] K Deb L Thiele M Laumanns and E Zitzler ldquoScalable testproblems for evolutionary multiobjective optimizationrdquo Evolu-tionary Multiobjective Optimization pp 105ndash145 2005

[16] S Huband L Barone L While and P Hingston ldquoA scalablemulti-objective test problem toolkitrdquo in Lecture Notes in Com-puter Science vol 3410 pp 280ndash295 Springer Berlin Germany2005

[17] M Elarbi S Bechikh A Gupta L Ben Said and Y-S OngldquoA new decomposition-based nsga-ii for many-objective opti-mizationrdquo IEEE Transactions on Systems Man and CyberneticsSystems vol 48 no 7 pp 1191ndash1210 2018

[18] Q Lin J Chen Z-H Zhan et al ldquoA hybrid evolutionaryimmune algorithm for multiobjective optimization problemsrdquoIEEE Transactions on Evolutionary Computation vol 20 no 5pp 711ndash729 2016

[19] Y Y Tan Y C Jiao H Li and X K Wang ldquoA modificationtoMOEAD-DE for multiobjective optimization problems withcomplicated Pareto setsrdquo Information Sciences vol 213 pp 14ndash38 2012

[20] K Li K Deb Q Zhang and Q Zhang ldquoEfficient non-domination level update method for steady-state evolutionarymultiobjective optimizationrdquo IEEE Transactions on Cyberneticsvol 47 no 9 pp 2838ndash2849 2017

[21] J Knowles and D Corne ldquoThe pareto archived evolutionstrategy a new baseline algorithm for multiobjective optimi-sationrdquo in Proceedings of the 1999 Congress on EvolutionaryComputation-CEC rsquo99 vol 1 pp 98ndash105 WA USA July 1999

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

[23] S Rostami F Neri andM Epitropakis ldquoProgressive preferencearticulation for decision making in multi-objective optimisa-tion problemsrdquo Integrated Computer-Aided Engineering vol 24no 4 pp 315ndash335 2017

[24] S Rostami and F Neri ldquoCovariance matrix adaptation paretoarchived evolution strategy with hypervolume-sorted adaptivegrid algorithmrdquo Integrated Computer-Aided Engineering vol 23no 4 pp 313ndash329 2016

10 Complexity

[25] S Rostami and F Neri ldquoA fast hypervolume driven selectionmechanism for many-objective optimisation problemsrdquo Swarmand Evolutionary Computation vol 34 pp 50ndash67 2017

[26] E Zitzler and S Kunzli ldquoIndicator-based selection in multiob-jective searchrdquo in Parallel Problem Solving from NaturemdashPPSNVIII vol 3242 of Lecture Notes in Computer Science pp 832ndash842 Springer Berlin Germany 2004

[27] D Brockhoff TWagner andH Trautmann ldquoOn the propertiesof the R2 indicatorrdquo inProceedings of the 14thAnnual Conferenceon Genetic and Evolutionary Computation pp 465ndash472 ACMPhiladelphia Pa USA July 2012

[28] K Bringmann andT Friedrich ldquoAn efficient algorithm for com-puting hypervolume contributionsrdquo Evolutionary Computationvol 18 no 3 pp 383ndash402 2010

[29] S Jiang J Zhang Y-S Ong A N Zhang and P S Tan ldquoAsimple and fast hypervolume indicator-based multiobjectiveevolutionary algorithmrdquo IEEE Transactions on Cybernetics vol45 no 10 pp 2202ndash2213 2015

[30] Z Wang Q Zhang and H Li ldquoBalancing convergence anddiversity by using two different reproduction operators inMOEAD some preliminary workrdquo in Proceedings of the 2015IEEE International Conference on Systems Man and Cybernet-ics pp 2849ndash2854 Kowloon Hong Kong October 2015

[31] F Gu and Y-M Cheung ldquoSelf-organizing map-based weightdesign for decomposition-based many-objective evolutionaryalgorithmrdquo IEEE Transactions on Evolutionary Computationvol 22 no 2 pp 211ndash225 2018

[32] L Ke Q Zhang and R Battiti ldquoMOEAD-ACO a mul-tiobjective evolutionary algorithm using decomposition andAntColonyrdquo IEEETransactions on Cybernetics vol 43 no 6 pp1845ndash1859 2013

[33] S Jiang and S Yang ldquoAn improved multiobjective optimizationevolutionary algorithm based on decomposition for complexpareto frontsrdquo IEEE Transactions on Cybernetics vol 46 no 2pp 421ndash437 2015

[34] H Sato ldquoInverted PBI inMOEAD and its impact on the searchperformance on multi and many-objective optimizationrdquo inProceedings of the 2014 Annual Conference on Genetic andEvolutionary Computation pp 645ndash652 Vancouver CanadaJuly 2014

[35] Y Su J Wang L Ma X Wang Q Lin and J Chen ldquoA novelmany-objective optimization algorithm based on the hybridangle-encouragementdecompositionrdquo in LectureNotes in Com-puter Science vol 10956 pp 47ndash53 Springer InternationalPublishing Cham Switzerland 2018

[36] H Li Q Zhang and J Deng ldquoBiased multiobjective opti-mization and decomposition algorithmrdquo IEEE Transactions onCybernetics vol 47 no 1 pp 52ndash66 2017

[37] K Li S Kwong Q Zhang andK Deb ldquoInterrelationship-basedselection for decomposition multiobjective optimizationrdquo IEEETransactions on Cybernetics vol 45 no 10 pp 2076ndash2088 2015

[38] H-L Liu F Gu and Q Zhang ldquoDecomposition of a multi-objective optimization problem into a number of simple mul-tiobjective subproblemsrdquo IEEE Transactions on EvolutionaryComputation vol 18 no 3 pp 450ndash455 2014

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

[40] Y T Qi X L Ma F Liu L C Jiao J Y Sun and J S WuldquoMOEAD with adaptive weight adjustmentrdquo EvolutionaryComputation vol 22 no 2 pp 231ndash264 2014

[41] H-L Liu L ChenQ Zhang andK Deb ldquoAdaptively allocatingsearch effort in challenging many-objective optimization prob-lemsrdquo IEEE Transactions on Evolutionary Computation vol 22no 3 pp 433ndash448 2018

[42] X Cai Z Mei and Z Fan ldquoA decomposition-based many-objective evolutionary algorithm with two types of adjustmentsfor direction vectorsrdquo IEEE Transactions on Cybernetics vol 48no 8 pp 2335ndash2348 2018

[43] M Asafuddoula H K Singh and T Ray ldquoAn enhanceddecomposition-based evolutionary algorithm with adaptivereference vectorsrdquo IEEETransactions on Cybernetics vol 48 no8 pp 2321ndash2334 2018

[44] K Li K Deb Q Zhang and S Kwong ldquoAn evolutionarymany-objective optimization algorithm based on dominanceand decompositionrdquo IEEE Transactions on Evolutionary Com-putation vol 19 no 5 pp 694ndash716 2015

[45] Q Lin G Jin Y Ma et al ldquoA diversity-enhanced resource allo-cation strategy for decomposition-based multiobjective evolu-tionary algorithmrdquo IEEE Transactions on Cybernetics vol 48no 8 pp 2388ndash2501 2018

[46] A Zhou andQ Zhang ldquoAre all the subproblems equally impor-tant Resource allocation in decomposition-based multiobjec-tive evolutionary algorithmsrdquo IEEE Transactions on Evolution-ary Computation vol 20 no 1 pp 52ndash64 2016

[47] Q Zhang W Liu andH Li ldquoThe performance of a new versionof MOEAD on CEC09 unconstrained MOP test instancesrdquoin Proceedings of the 2009 IEEE Congress on EvolutionaryComputation pp 203ndash208 Trondheim Norway May 2009

[48] Q Lin Z Liu Q Yan et al ldquoAdaptive composite operator selec-tion and parameter control for multiobjective evolutionaryalgorithmrdquo Information Sciences vol 339 pp 332ndash352 2016

[49] K Li A Fialho S Kwong and Q Zhang ldquoAdaptive operatorselection with bandits for a multiobjective evolutionary algo-rithm based on decompositionrdquo IEEE Transactions on Evolu-tionary Computation vol 18 no 1 pp 114ndash130 2014

[50] Q Lin C Tang YMa ZDu J Li and J Chen ldquoA novel adaptivecontrol strategy for decomposition-based multiobjective algo-rithmrdquo Computers amp Operations Research vol 78 pp 94ndash1072017

[51] Z Wang Q Zhang A Zhou M Gong and L Jiao ldquoAdaptivereplacement strategies for MOEADrdquo IEEE Transactions onCybernetics vol 46 no 2 pp 474ndash486 2016

[52] R Wang J Xiong H Ishibuchi G Wu and T Zhang ldquoOnthe effect of reference point in MOEAD for multi-objectiveoptimizationrdquo Applied Soft Computing vol 58 pp 25ndash34 2017

[53] M Wu K Li S Kwong Y Zhou and Q Zhang ldquoMatching-based selection with incomplete lists for decomposition multi-objective optimizationrdquo IEEE Transactions on EvolutionaryComputation vol 21 no 5 pp 714ndash730 2017

[54] LWangQZhangAZhouMGong andL Jiao ldquoConstrainedsubproblems in a decomposition-based multiobjective evolu-tionary algorithmrdquo IEEE Transactions on Evolutionary Compu-tation vol 20 no 3 pp 475ndash480 2016

[55] X Ma Q Zhang G Tian J Yang and Z Zhu ldquoOn tchebycheffdecomposition approaches for multiobjective evolutionaryoptimizationrdquo IEEE Transactions on Evolutionary Computationvol 22 no 2 pp 226ndash244 2018

[56] L Cai S Qu and G Cheng ldquoTwo-archive method for aggrega-tion-based many-objective optimizationrdquo Information Sciencesvol 422 pp 305ndash317 2018

Complexity 11

[57] X Cai Z Yang Z Fan and Q Zhang ldquoDecomposition-based-sorting and angle-based-selection for evolutionary multiobjec-tive and many-objective optimizationrdquo IEEE Transactions onCybernetics vol 47 no 9 pp 2824ndash2837 2017

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

[59] K Li Q Zhang S Kwong M Li and R Wang ldquoStablematching-based selection in evolutionary multiobjective opti-mizationrdquo IEEE Transactions on Evolutionary Computation vol18 no 6 pp 909ndash923 2014

[60] Y Yuan H Xu B Wang B Zhang and X Yao ldquoBalancing con-vergence and diversity in decomposition-based many-objectiveoptimizersrdquo IEEE Transactions on Evolutionary Computationvol 20 no 2 pp 180ndash198 2016

[61] R Wang H Ishibuchi Y Zhang X Zheng and T Zhang ldquoOnthe effect of localized PBI method in MOEAD for multiobjec-tive optimizationrdquo in Proceedings of the 2016 IEEE SymposiumSeries on Computational Intelligence pp 645ndash652 AthensGreece 2016

[62] R Wang Z Zhou H Ishibuchi T Liao and T Zhang ldquoLocal-ized weighted sum method for many-objective optimizationrdquoIEEE Transactions on Evolutionary Computation vol 22 no 1pp 3ndash18 2018

[63] R Wang Q Zhang and T Zhang ldquoDecomposition-basedalgorithms using pareto adaptive scalarizing methodsrdquo IEEETransactions on Evolutionary Computation vol 20 no 6 pp821ndash837 2016

[64] M Ming R Wang Y Zha and T Zhang ldquoPareto adap-tive penalty-based boundary intersection method for multi-objective optimizationrdquo Information Sciences vol 414 pp 158ndash174 2017

[65] M Wu K Li S Kwong and Q Zhang ldquoEvolutionary many-objective optimization based on adversarial decompositionrdquoIEEE Transactions on Cybernetics pp 1ndash12 2018

[66] Y Zhang Y Gong T Gu et al ldquoDECAL decomposition-basedcoevolutionary algorithm for many-objective optimizationrdquoIEEE Transactions on Cybernetics vol 49 no 1 pp 27ndash41 2019

[67] M Wu K Li S Kwong Q Zhang and J Zhang ldquoLearning todecompose a paradigm for decomposition-based multiobjec-tive optimizationrdquo IEEE Transactions on Evolutionary Compu-tation p 1 2018

[68] H-L Liu and X Q Li ldquoThe multiobjective evolutionary algo-rithm based on determined weight and sub-regional searchrdquoin Proceedings of the 2009 IEEE Congress on EvolutionaryComputation pp 1928ndash1934 IEEE Trondheim Norway May2009

[69] H Liu L Chen K Deb and E D Goodman ldquoInvestigatingthe effect of imbalance between convergence and diversity inevolutionary multi-objective algorithmsrdquo IEEE Transactions onEvolutionary Computation vol 21 no 3 pp 408ndash425 2017

[70] J J Durillo A J Nebro and E Alba ldquoThe jmetal frameworkfor multi-objective optimization design and architecturerdquo inProceedings of the IEEE Congress on Evolutionary Computationpp 1ndash8 Barcelona Spain 2010

[71] K Li R Wang T Zhang and H Ishibuchi ldquoEvolutionarymany-objective optimization a comparative study of the state-of-the-artrdquo IEEE Access vol 6 pp 26194ndash26214 2018

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2 Complexity

in MOEAD [39] are allowed to update any solution inpopulation However this method may significantly lowerthe diversity when a very good solution may replace mostof the others in several generations In MOEAD-DE [58]its solution update approach is controlled by two presetprobabilities 120575 and nr which obtains a better balance ofconvergence and diversity The offspring is only allowed toupdate the parent solutions from the neighborhood witha probability 120575 and from the entire population with aprobability (1-120575) Moreover an offspring can only replace atmost nr parent solutions This strategy wasmostly used in thefollowing design of decomposition-based MOEAs [49 54]Different from the decomposition approach in MOEAD-DE MOEAD-M2M [38] separates the search space intomultiple search subspace which simples the solving of MOPsin each subspace and the solution update is constrained byincluding the equal number of solutions in each subspaceThus MOEAD-M2M was shown to be very effective forcomplicated MOPs that strongly emphasize diversity (ieMOP problems [38]) To further find a better match ofsolutions and subproblems a stable matching model wasproposed in MOEAD-STM [59] which associates the solu-tions to subproblems according to their respective prefer-ences In this way MOEAD-STM can maintain a goodconvergence speed and population diversity Similarly animproved interrelationshipmodelwas designed inMOEAD-IR [37] to associate the solutions to subproblems based ontheirmutual-preferences which is an essentially diversity firstand convergence second strategy Moreover two improvedversions [53] forMOEAD-STMwere proposed to embed theconcept of the incomplete preference lists in the stable match-ing model which further strengthens the diversity In [51]an adaptive replacement neighborhood size was proposedto assign an offspring to its most appropriate subproblemsobtaining a better balance of convergence and diversity InMOEAD-ACD [54] an adaptive constrained decompositionapproach was presented in which the update regions ofdecomposition approach are constrained to maintain thediversity Moreover to further enhance the performancein MOPs with more than three objectives decompositionapproach and Pareto domination were simultaneously usedinMOEADD [44] decomposition-based-sorting and angle-based-selection approaches were proposed in MOEAD-SAS[57] and the diversity was preferred in solution update byselecting certain closest subproblems for an offspring in [60]

On the other hand another kind of population selectionor update mechanisms in MOEAD aims to improve theirused decomposition functions InMOEAD [39] three tradi-tional decomposition functions ie the weighted sum (WS)approach the Tchebycheff (TCH) approach and the penalty-based boundary intersection (PBI) approach were employedIn [61 62] a local PBI andWS were respectively designed toconstrain the update regions of decomposition approacheswhich avoid the diversity loss In [63 64] an adaptive Paretofront scalarizing (PaS) and penalty-based boundary inter-section (PaP) decomposition approaches were respectivelyintroduced to match the true PFs with various shapes Twodecomposition approaches were presented in MOEAAD[65] and DECAL [66] to deal with the complicated PF In

MOEAAD two coevolved populations were respectivelyupdated by the two decomposition functions to fit differ-ent PF shapes while two novel decomposition functionswere respectively used to accelerate the convergence speedand enhance the population diversity in DECAL RecentlyMOEAD-LTD [67] was proposed to trace the PF shape inwhich the learning module predicts the PF shape and thedecomposition function is adaptively adjusted to fit its PFshape

Most of the above MOEAs all abide one basic principlethat each agent should be assigned with one solution in orderto find the optimal value for its subproblem However thiskind of solution assignment may not be effective and efficientin decomposition-based MOEAs as the solution assigned tothe agent may be far away from its subproblem In such caseit cannot truly reflect the diversity of each agent and cannotprovide the correct neighboring information in evolutionwhich may slow down the convergence as decomposition-based MOEAs are designed as an essentially collabora-tive evolutionary framework Therefore a constrained solu-tion update (CSU) strategy is designed in this paper fordecomposition-basedMOEAs to alleviate the above problemThe solutions are only assigned to the agent that handlesthe closest subproblem This way the correct neighboringinformation can be provided to guide the evolution and it isstraightforward to show the diversity of each agent In thiscase the number of solutions in each agent may be zero orno less than one To maintain the diversity of each agent theoffspring assigned to one agent are only allowed to renew itsoriginal solutions When the agent has no solution it will beassigned one solution in priority once offspring are generatedclosest to its subproblem To keep the same population sizethe agentwith the largest number of solutionswill remove onesolution showing the worst convergence Thus the diversityof one agent is enhanced while the convergence of otheragents is not affected After a period of evolution a stablestatus for solution assignment is anticipated so that eachagent only has one solution When compared to the existingpopulation selection or update strategies for decomposition-based MOEAs our experiments validate the superiority ofthe proposed approach when tackling two sets of complicatedtest MOPs

The main contributions of this paper are clarified below

(1) Each agent may be assigned with no solution or noless than one solution which is different from theexisting approaches that only assign one solution toeach agent This approach can truly reflect the diver-sity on the agents and provide the correct neighboringinformation in evolution

(2) A CSU strategy is designed for each agent in order tomaintain diversity for all the agents without affectingtheir convergence The agent with no solution will beassigned first while the agent with the largest numberof solutions will remove one solution showing theworst convergence By this way a stable status forsolution assignment may be reached so that eachagent only has one solution which ensures diversityin decomposition-based MOEAs

Complexity 3





Minimize 119865 (119909) = 1198911 (119909) 1198912 (119909) 119891119898 (119909) (1)




(1003816100381610038161003816119891119894 (x) minus 119911lowast119894 1003816100381610038161003816119908119894 ) (2)



f2

f1

s3

s2

s1

zlowast








4 Complexity













119910 = 119909 + (119911 minus 119909) 1199031 (1 minus (1199032)120578) (6)

119910 = 119910 + 1199033 (1 minus (1199032)120578) (119906 minus 119897) (7)



Complexity 5





119910 = 119906 minus 1199034 (119906 minus 119910) if 119910 gt 119906 (8)

119910 = 119897 + 1199034 (119910 minus 119897) if 119910 lt 119897 (9)




rfloor + 1 (10)





6 Complexity


MOEAD-M2M

MOEAD-STM


ACD MOEAD-CSU

MOP1 Mean(Std)

226E-2 ndash(152E-3)

346E-1 ndash(227E-2)

269E-2 ndash(353E-3)

263E-2 ndash(272E-3)

362E-1 ndash(747E-3)

272E-2 ndash(233E-3)

166E-2(438E-4)

MOP2 Mean(Std)

740E-3 ndash(509E-4)

295E-1 ndash(783E-2)

679E-2 ndash(769E-2)

602E-2 ndash(646E-2)

277E-1 ndash(719E-2)

313E-2 ndash(667E-2)

687E-3(229E-3)

MOP3 Mean(Std)

104E-2 ndash(220E-3)

155E-1 ndash(343E-2)

385E-2 ndash(583E-2)

143E-2 ndash(191E-2)

123E-1 ndash(482E-2)

136E-2 ndash(206E-2)

814E-3(376E-3)

MOP4 Mean(Std)

457E-3 =(339E-4)

300E-1 ndash(299E-2)

352E-2 ndash(323E-2)

107E-1 ndash(846E-2)

276E-1 ndash(324E-2)

548E-2 ndash(574E-2)

447E-3(122E-4)

MOP5 Mean(Std)

200E-2 ndash(691E-4)

312E-1 ndash(278E-2)

231E-2 ndash(304E-3)

216E-2 ndash(241E-3)

316E-1 ndash(802E-3)

239E-2 ndash(231E-3)

151E-2(357E-4)

MOP6 Mean(Std)

496E-2 ndash(158E-3)

290E-1 ndash(224E-2)

490E-2 ndash(236E-3)

484E-2 ndash(280E-3)

290E-1 ndash(256E-2)

502E-2 ndash(239E-3)

330E-2(280E-4)

MOP7 Mean(Std)

793E-2 ndash(469E-3)

351E-1 ndash(893E-8)

177E-1 ndash(316E-2)

192E-1 ndash(186E-2)

338E-1 ndash(227E-2)

233E-1 ndash(247E-2)

465E-2(335E-4)

IMB1 Mean(Std)

104E-2 ndash(611E-4)

105E-1 ndash(878E-2)

123E-2 ndash(133E-3)

118E-2 ndash(196E-3)

176E-1 ndash(714E-2)

128E-2 ndash(952E-4)

753E-3(136E-4)

IMB2 Mean(Std)

119E-2 ndash(546E-4)

162E-1 ndash(116E-2)

470E-2 ndash(179E-2)

664E-2 ndash(246E-2)

167E-1 ndash(111E-2)

585E-2 ndash(184E-2)

892E-3(216E-4)

IMB3 Mean(Std)

178E-2 ndash(766E-4)

282E-1 ndash(355E-2)

256E-2 ndash(348E-3)

237E-2 ndash(273E-3)

284E-1 ndash(142E-2)

226E-2 ndash(188E-3)

119E-2(316E-4)

IMB4 Mean(Std)

405E-2 ndash(159E-3)

133E-1 ndash(692E-3)

253E-2 +(504E-4)

239E-2 +(257E-4)

137E-1 ndash(697E-3)

282E-2 =(545E-4)

279E-2(219E-4)

IMB5 Mean(Std)

567E-2 ndash(992E-3)

840E-2 ndash(247E-5)

839E-2 ndash(137E-5)

790E-2 ndash(461E-3)

912E-2 ndash(574E-5)

741E-2 ndash(750E-3)

328E-2(121E-4)

IMB6 Mean(Std)

385E-2 ndash(524E-3)

462E-2 ndash(124E-4)

461E-2 ndash(212E-4)

470E-2 ndash(145E-4)

522E-2 ndash(628E-4)

250E-2 ndash(310E-4)

237E-2(871E-5)

IMB7 Mean(Std)

108E-2 ndash(589E-4)

298E-2 ndash(228E-4)

249E-2 ndash(830E-3)

276E-2 ndash(276E-2)

299E-2 ndash(195E-4)

285E-2 ndash(695E-3)

792E-3(305E-4)

IMB8 Mean(Std)

124E-2 ndash(530E-4)

345E-2 ndash(977E-4)

261E-2 ndash(106E-2)

334E-2 ndash(535E-3)

338E-2 ndash(358E-3)

352E-2 ndash(419E-3)

940E-3(381E-4)

IMB9 Mean(Std)

135E-2 ndash(662E-4)

384E-2 ndash243E-4

325E-2 ndash(916E-3)

379E-2 ndash(342E-3)

383E-2 ndash(229E-4)

396E-2 ndash(120E-3)

118E-2(618E-4)

IMB10 Mean(Std)

479E-2 ndash(868E-4)

282E-2 +(180E-3)

329E-2 +(961E-4)

295E-2 +(855E-4)

363E-2 ndash(195E-3)

317E-2 +(763E-4)

357E-2(302E-4)







Complexity 7


MOEAD-M2M

MOEAD-STM


ACDMOEAD-

CSU

MOP1 Mean(Std)

695E-1 ndash(178E-3)

263E-1 ndash(361E-2)

692E-1 ndash(423E-3)

693E-1 ndash(283E-3)

238E-1 ndash(137E-2)

692E-1 ndash(244E-3)

703E-1513E-4

MOP2 Mean(Std)

439E-1 =(341E-4)

202E-1 ndash(486E-2)

379E-1 ndash(812E-2)

386E-1 ndash(667E-2)

206E-1 ndash(405E-2)

412E-1 ndash(660E-2)

439E-1360E-3

MOP3 Mean(Std)

340E-1 ndash(321E-3)

215E-1 ndash(290E-2)

311E-1 ndash(575E-2)

332E-1 ndash(264E-2)

241E-1 ndash(398E-2)

333E-1 ndash(268E-2)

341E-1400E-3

MOP4 Mean(Std)

595E-1 =(824E-4)

290E-1 ndash(162E-2)

569E-1 ndash(357E-2)

474E-1 ndash(965E-2)

301E-1 ndash(231E-2)

536E-1 ndash(666E-2)

595E-1248E-4

MOP5 Mean(Std)

696E-1 ndash(113E-3)

404E-1 ndash(208E-2)

694E-1 ndash(487E-3)

696E-1 ndash(274E-3)

400E-1 ndash(103E-17)

694E-1 ndash(272E-3)

703E-1465E-4

MOP6 Mean(Std)

822E-1 ndash(174E-3)

634E-1 ndash(248E-2)

824E-1 ndash(362E-3)

828E-1 ndash(242E-3)

634E-1 ndash(297E-2)

828E-1 ndash(221E-3)

840E-1271E-4

MOP7 Mean(Std)

541E-1 ndash(443E-3)

407E-1 ndash(189E-7)

497E-1 ndash(339E-2)

494E-1 ndash(491E-3)

412E-1 ndash(102E-2)

494E-1 ndash(145E-2)

545E-1526E-4

IMB1 Mean(Std)

711E-1 ndash(741E-4)

615E-1 ndash(887E-2)

709E-1 ndash(156E-3)

710E-1 ndash(204E-3)

543E-1 ndash(695E-2)

709E-1 ndash(100E-3)

715E-1156E-4

IMB2 Mean(Std)

570E-1 ndash(703E-4)

391E-1 ndash(914E-3)

525E-1 ndash(233E-2)

501E-1 ndash(302E-2)

390E-1 ndash(718E-3)

510E-1 ndash(234E-2)

574E-1290E-4

IMB3 Mean(Std)

329E-1 ndash(763E-4)

133E-1 ndash(220E-2)

324E-1 ndash(197E-3)

323E-1 ndash(262E-3)

128E-1 ndash(718E-3)

324E-1 ndash(169E-3)

335E-1390E-4

IMB4 Mean(Std)

823E-1 ndash(244E-3)

771E-1 ndash(303E-3)

849E-1 +(780E-4)

853E-1 +(454E-4)

764E-1 ndash(310E-3)

846E-1 =(963E-4)

846E-1245E-4

IMB5 Mean(Std)

548E-1 ndash(676E-3)

545E-1 ndash(231E-5)

545E-1 ndash(234E-5)

543E-1 ndash(791E-4)

532E-1 ndash(243E-4)

542E-1 ndash(846E-4)

571E-1271E-4

IMB6 Mean(Std)

831E-1 ndash(307E-3)

841E-1 ndash(353E-5)

840E-1 ndash(428E-5)

840E-1 ndash(144E-4)

832E-1 ndash(125E-3)

852E-1 ndash(358E-4)

853E-1111E-4

IMB7 Mean(Std)

709E-1 ndash(819E-4)

695E-1 ndash(516E-4)

699E-1 ndash(691E-3)

697E-1 ndash(558E-3)

695E-1 ndash(467E-4)

696E-1 ndash(587E-3)

714E-1514E-4

IMB8 Mean(Std)

567E-1 ndash(833E-4)

545E-1 ndash(114E-3)

555E-1 ndash(118E-2)

546E-1 ndash(604E-3)

546E-1 ndash(394E-3)

544E-1 ndash(477E-3)

573E-1541E-4

IMB9 Mean(Std)

331E-1 ndash(114E-3)

313E-1 ndash(888E-4)

318E-1 ndash(682E-3)

314E-1 ndash(249E-3)

314E-1 ndash(783E-4)

311E-1 ndash(202E-3)

334E-1906E-4

IMB10 Mean(Std)

817E-1 ndash(219E-3)

848E-1 +(135E-3)

839E-1 +(123E-3)

846E-1 +(101E-3)

834E-1 ndash(305E-3)

843E-1 +(998E-4)

836E-1335E-4





8 Complexity

0 02 04 06 08 10

02

04

06

08

1MOP1

0 02 04 06 08 10

02

04

06

08

1MOP2

0 02 04 06 08 10

02

04

06

08

1MOP3

0 02 04 06 08 10

02

04

06

08

1MOP4

0 02 04 06 08 10

02

04

06

08

1MOP5

00 0

05z

MOP6

y x

05 05

1

1 1

000

05

05

MOP7

1

051

15

115 0 02 04 06 08 10

02

04

06

08

1IMB1

0 02 04 06 08 10

02

04

06

08

1IMB2

0 02 04 06 08 10

02

04

06

08

1IMB3

000

05

IMB4

z

xy

0505

1

11

00 0

05

IMB5

1

05 05

15

1 1

00 0

05z

IMB6

y x

05 05

1

1 1 0 02 04 06 08 1 120

02

04

06

08

1

12IMB7

0 02 04 06 08 1 120

02

04

06

08

1

12IMB8

0 02 04 06 08 1 120

02

04

06

08

1

12IMB9

00 0

05z

IMB10

y x

05 05

1

1 1






Complexity 9


Data Availability




Acknowledgments


References

























10 Complexity

































Complexity 11























Journal of












Journal of







Volume 2018






Volume 2018








Complexity 3





Minimize 119865 (119909) = 1198911 (119909) 1198912 (119909) 119891119898 (119909) (1)




(1003816100381610038161003816119891119894 (x) minus 119911lowast119894 1003816100381610038161003816119908119894 ) (2)



f2

f1

s3

s2

s1

zlowast








4 Complexity













119910 = 119909 + (119911 minus 119909) 1199031 (1 minus (1199032)120578) (6)

119910 = 119910 + 1199033 (1 minus (1199032)120578) (119906 minus 119897) (7)



Complexity 5





119910 = 119906 minus 1199034 (119906 minus 119910) if 119910 gt 119906 (8)

119910 = 119897 + 1199034 (119910 minus 119897) if 119910 lt 119897 (9)




rfloor + 1 (10)





6 Complexity


MOEAD-M2M

MOEAD-STM


ACD MOEAD-CSU

MOP1 Mean(Std)

226E-2 ndash(152E-3)

346E-1 ndash(227E-2)

269E-2 ndash(353E-3)

263E-2 ndash(272E-3)

362E-1 ndash(747E-3)

272E-2 ndash(233E-3)

166E-2(438E-4)

MOP2 Mean(Std)

740E-3 ndash(509E-4)

295E-1 ndash(783E-2)

679E-2 ndash(769E-2)

602E-2 ndash(646E-2)

277E-1 ndash(719E-2)

313E-2 ndash(667E-2)

687E-3(229E-3)

MOP3 Mean(Std)

104E-2 ndash(220E-3)

155E-1 ndash(343E-2)

385E-2 ndash(583E-2)

143E-2 ndash(191E-2)

123E-1 ndash(482E-2)

136E-2 ndash(206E-2)

814E-3(376E-3)

MOP4 Mean(Std)

457E-3 =(339E-4)

300E-1 ndash(299E-2)

352E-2 ndash(323E-2)

107E-1 ndash(846E-2)

276E-1 ndash(324E-2)

548E-2 ndash(574E-2)

447E-3(122E-4)

MOP5 Mean(Std)

200E-2 ndash(691E-4)

312E-1 ndash(278E-2)

231E-2 ndash(304E-3)

216E-2 ndash(241E-3)

316E-1 ndash(802E-3)

239E-2 ndash(231E-3)

151E-2(357E-4)

MOP6 Mean(Std)

496E-2 ndash(158E-3)

290E-1 ndash(224E-2)

490E-2 ndash(236E-3)

484E-2 ndash(280E-3)

290E-1 ndash(256E-2)

502E-2 ndash(239E-3)

330E-2(280E-4)

MOP7 Mean(Std)

793E-2 ndash(469E-3)

351E-1 ndash(893E-8)

177E-1 ndash(316E-2)

192E-1 ndash(186E-2)

338E-1 ndash(227E-2)

233E-1 ndash(247E-2)

465E-2(335E-4)

IMB1 Mean(Std)

104E-2 ndash(611E-4)

105E-1 ndash(878E-2)

123E-2 ndash(133E-3)

118E-2 ndash(196E-3)

176E-1 ndash(714E-2)

128E-2 ndash(952E-4)

753E-3(136E-4)

IMB2 Mean(Std)

119E-2 ndash(546E-4)

162E-1 ndash(116E-2)

470E-2 ndash(179E-2)

664E-2 ndash(246E-2)

167E-1 ndash(111E-2)

585E-2 ndash(184E-2)

892E-3(216E-4)

IMB3 Mean(Std)

178E-2 ndash(766E-4)

282E-1 ndash(355E-2)

256E-2 ndash(348E-3)

237E-2 ndash(273E-3)

284E-1 ndash(142E-2)

226E-2 ndash(188E-3)

119E-2(316E-4)

IMB4 Mean(Std)

405E-2 ndash(159E-3)

133E-1 ndash(692E-3)

253E-2 +(504E-4)

239E-2 +(257E-4)

137E-1 ndash(697E-3)

282E-2 =(545E-4)

279E-2(219E-4)

IMB5 Mean(Std)

567E-2 ndash(992E-3)

840E-2 ndash(247E-5)

839E-2 ndash(137E-5)

790E-2 ndash(461E-3)

912E-2 ndash(574E-5)

741E-2 ndash(750E-3)

328E-2(121E-4)

IMB6 Mean(Std)

385E-2 ndash(524E-3)

462E-2 ndash(124E-4)

461E-2 ndash(212E-4)

470E-2 ndash(145E-4)

522E-2 ndash(628E-4)

250E-2 ndash(310E-4)

237E-2(871E-5)

IMB7 Mean(Std)

108E-2 ndash(589E-4)

298E-2 ndash(228E-4)

249E-2 ndash(830E-3)

276E-2 ndash(276E-2)

299E-2 ndash(195E-4)

285E-2 ndash(695E-3)

792E-3(305E-4)

IMB8 Mean(Std)

124E-2 ndash(530E-4)

345E-2 ndash(977E-4)

261E-2 ndash(106E-2)

334E-2 ndash(535E-3)

338E-2 ndash(358E-3)

352E-2 ndash(419E-3)

940E-3(381E-4)

IMB9 Mean(Std)

135E-2 ndash(662E-4)

384E-2 ndash243E-4

325E-2 ndash(916E-3)

379E-2 ndash(342E-3)

383E-2 ndash(229E-4)

396E-2 ndash(120E-3)

118E-2(618E-4)

IMB10 Mean(Std)

479E-2 ndash(868E-4)

282E-2 +(180E-3)

329E-2 +(961E-4)

295E-2 +(855E-4)

363E-2 ndash(195E-3)

317E-2 +(763E-4)

357E-2(302E-4)







Complexity 7


MOEAD-M2M

MOEAD-STM


ACDMOEAD-

CSU

MOP1 Mean(Std)

695E-1 ndash(178E-3)

263E-1 ndash(361E-2)

692E-1 ndash(423E-3)

693E-1 ndash(283E-3)

238E-1 ndash(137E-2)

692E-1 ndash(244E-3)

703E-1513E-4

MOP2 Mean(Std)

439E-1 =(341E-4)

202E-1 ndash(486E-2)

379E-1 ndash(812E-2)

386E-1 ndash(667E-2)

206E-1 ndash(405E-2)

412E-1 ndash(660E-2)

439E-1360E-3

MOP3 Mean(Std)

340E-1 ndash(321E-3)

215E-1 ndash(290E-2)

311E-1 ndash(575E-2)

332E-1 ndash(264E-2)

241E-1 ndash(398E-2)

333E-1 ndash(268E-2)

341E-1400E-3

MOP4 Mean(Std)

595E-1 =(824E-4)

290E-1 ndash(162E-2)

569E-1 ndash(357E-2)

474E-1 ndash(965E-2)

301E-1 ndash(231E-2)

536E-1 ndash(666E-2)

595E-1248E-4

MOP5 Mean(Std)

696E-1 ndash(113E-3)

404E-1 ndash(208E-2)

694E-1 ndash(487E-3)

696E-1 ndash(274E-3)

400E-1 ndash(103E-17)

694E-1 ndash(272E-3)

703E-1465E-4

MOP6 Mean(Std)

822E-1 ndash(174E-3)

634E-1 ndash(248E-2)

824E-1 ndash(362E-3)

828E-1 ndash(242E-3)

634E-1 ndash(297E-2)

828E-1 ndash(221E-3)

840E-1271E-4

MOP7 Mean(Std)

541E-1 ndash(443E-3)

407E-1 ndash(189E-7)

497E-1 ndash(339E-2)

494E-1 ndash(491E-3)

412E-1 ndash(102E-2)

494E-1 ndash(145E-2)

545E-1526E-4

IMB1 Mean(Std)

711E-1 ndash(741E-4)

615E-1 ndash(887E-2)

709E-1 ndash(156E-3)

710E-1 ndash(204E-3)

543E-1 ndash(695E-2)

709E-1 ndash(100E-3)

715E-1156E-4

IMB2 Mean(Std)

570E-1 ndash(703E-4)

391E-1 ndash(914E-3)

525E-1 ndash(233E-2)

501E-1 ndash(302E-2)

390E-1 ndash(718E-3)

510E-1 ndash(234E-2)

574E-1290E-4

IMB3 Mean(Std)

329E-1 ndash(763E-4)

133E-1 ndash(220E-2)

324E-1 ndash(197E-3)

323E-1 ndash(262E-3)

128E-1 ndash(718E-3)

324E-1 ndash(169E-3)

335E-1390E-4

IMB4 Mean(Std)

823E-1 ndash(244E-3)

771E-1 ndash(303E-3)

849E-1 +(780E-4)

853E-1 +(454E-4)

764E-1 ndash(310E-3)

846E-1 =(963E-4)

846E-1245E-4

IMB5 Mean(Std)

548E-1 ndash(676E-3)

545E-1 ndash(231E-5)

545E-1 ndash(234E-5)

543E-1 ndash(791E-4)

532E-1 ndash(243E-4)

542E-1 ndash(846E-4)

571E-1271E-4

IMB6 Mean(Std)

831E-1 ndash(307E-3)

841E-1 ndash(353E-5)

840E-1 ndash(428E-5)

840E-1 ndash(144E-4)

832E-1 ndash(125E-3)

852E-1 ndash(358E-4)

853E-1111E-4

IMB7 Mean(Std)

709E-1 ndash(819E-4)

695E-1 ndash(516E-4)

699E-1 ndash(691E-3)

697E-1 ndash(558E-3)

695E-1 ndash(467E-4)

696E-1 ndash(587E-3)

714E-1514E-4

IMB8 Mean(Std)

567E-1 ndash(833E-4)

545E-1 ndash(114E-3)

555E-1 ndash(118E-2)

546E-1 ndash(604E-3)

546E-1 ndash(394E-3)

544E-1 ndash(477E-3)

573E-1541E-4

IMB9 Mean(Std)

331E-1 ndash(114E-3)

313E-1 ndash(888E-4)

318E-1 ndash(682E-3)

314E-1 ndash(249E-3)

314E-1 ndash(783E-4)

311E-1 ndash(202E-3)

334E-1906E-4

IMB10 Mean(Std)

817E-1 ndash(219E-3)

848E-1 +(135E-3)

839E-1 +(123E-3)

846E-1 +(101E-3)

834E-1 ndash(305E-3)

843E-1 +(998E-4)

836E-1335E-4





8 Complexity

0 02 04 06 08 10

02

04

06

08

1MOP1

0 02 04 06 08 10

02

04

06

08

1MOP2

0 02 04 06 08 10

02

04

06

08

1MOP3

0 02 04 06 08 10

02

04

06

08

1MOP4

0 02 04 06 08 10

02

04

06

08

1MOP5

00 0

05z

MOP6

y x

05 05

1

1 1

000

05

05

MOP7

1

051

15

115 0 02 04 06 08 10

02

04

06

08

1IMB1

0 02 04 06 08 10

02

04

06

08

1IMB2

0 02 04 06 08 10

02

04

06

08

1IMB3

000

05

IMB4

z

xy

0505

1

11

00 0

05

IMB5

1

05 05

15

1 1

00 0

05z

IMB6

y x

05 05

1

1 1 0 02 04 06 08 1 120

02

04

06

08

1

12IMB7

0 02 04 06 08 1 120

02

04

06

08

1

12IMB8

0 02 04 06 08 1 120

02

04

06

08

1

12IMB9

00 0

05z

IMB10

y x

05 05

1

1 1






Complexity 9


Data Availability




Acknowledgments


References

























10 Complexity

































Complexity 11























Journal of












Journal of







Volume 2018






Volume 2018








4 Complexity













119910 = 119909 + (119911 minus 119909) 1199031 (1 minus (1199032)120578) (6)

119910 = 119910 + 1199033 (1 minus (1199032)120578) (119906 minus 119897) (7)



Complexity 5





119910 = 119906 minus 1199034 (119906 minus 119910) if 119910 gt 119906 (8)

119910 = 119897 + 1199034 (119910 minus 119897) if 119910 lt 119897 (9)




rfloor + 1 (10)





6 Complexity


MOEAD-M2M

MOEAD-STM


ACD MOEAD-CSU

MOP1 Mean(Std)

226E-2 ndash(152E-3)

346E-1 ndash(227E-2)

269E-2 ndash(353E-3)

263E-2 ndash(272E-3)

362E-1 ndash(747E-3)

272E-2 ndash(233E-3)

166E-2(438E-4)

MOP2 Mean(Std)

740E-3 ndash(509E-4)

295E-1 ndash(783E-2)

679E-2 ndash(769E-2)

602E-2 ndash(646E-2)

277E-1 ndash(719E-2)

313E-2 ndash(667E-2)

687E-3(229E-3)

MOP3 Mean(Std)

104E-2 ndash(220E-3)

155E-1 ndash(343E-2)

385E-2 ndash(583E-2)

143E-2 ndash(191E-2)

123E-1 ndash(482E-2)

136E-2 ndash(206E-2)

814E-3(376E-3)

MOP4 Mean(Std)

457E-3 =(339E-4)

300E-1 ndash(299E-2)

352E-2 ndash(323E-2)

107E-1 ndash(846E-2)

276E-1 ndash(324E-2)

548E-2 ndash(574E-2)

447E-3(122E-4)

MOP5 Mean(Std)

200E-2 ndash(691E-4)

312E-1 ndash(278E-2)

231E-2 ndash(304E-3)

216E-2 ndash(241E-3)

316E-1 ndash(802E-3)

239E-2 ndash(231E-3)

151E-2(357E-4)

MOP6 Mean(Std)

496E-2 ndash(158E-3)

290E-1 ndash(224E-2)

490E-2 ndash(236E-3)

484E-2 ndash(280E-3)

290E-1 ndash(256E-2)

502E-2 ndash(239E-3)

330E-2(280E-4)

MOP7 Mean(Std)

793E-2 ndash(469E-3)

351E-1 ndash(893E-8)

177E-1 ndash(316E-2)

192E-1 ndash(186E-2)

338E-1 ndash(227E-2)

233E-1 ndash(247E-2)

465E-2(335E-4)

IMB1 Mean(Std)

104E-2 ndash(611E-4)

105E-1 ndash(878E-2)

123E-2 ndash(133E-3)

118E-2 ndash(196E-3)

176E-1 ndash(714E-2)

128E-2 ndash(952E-4)

753E-3(136E-4)

IMB2 Mean(Std)

119E-2 ndash(546E-4)

162E-1 ndash(116E-2)

470E-2 ndash(179E-2)

664E-2 ndash(246E-2)

167E-1 ndash(111E-2)

585E-2 ndash(184E-2)

892E-3(216E-4)

IMB3 Mean(Std)

178E-2 ndash(766E-4)

282E-1 ndash(355E-2)

256E-2 ndash(348E-3)

237E-2 ndash(273E-3)

284E-1 ndash(142E-2)

226E-2 ndash(188E-3)

119E-2(316E-4)

IMB4 Mean(Std)

405E-2 ndash(159E-3)

133E-1 ndash(692E-3)

253E-2 +(504E-4)

239E-2 +(257E-4)

137E-1 ndash(697E-3)

282E-2 =(545E-4)

279E-2(219E-4)

IMB5 Mean(Std)

567E-2 ndash(992E-3)

840E-2 ndash(247E-5)

839E-2 ndash(137E-5)

790E-2 ndash(461E-3)

912E-2 ndash(574E-5)

741E-2 ndash(750E-3)

328E-2(121E-4)

IMB6 Mean(Std)

385E-2 ndash(524E-3)

462E-2 ndash(124E-4)

461E-2 ndash(212E-4)

470E-2 ndash(145E-4)

522E-2 ndash(628E-4)

250E-2 ndash(310E-4)

237E-2(871E-5)

IMB7 Mean(Std)

108E-2 ndash(589E-4)

298E-2 ndash(228E-4)

249E-2 ndash(830E-3)

276E-2 ndash(276E-2)

299E-2 ndash(195E-4)

285E-2 ndash(695E-3)

792E-3(305E-4)

IMB8 Mean(Std)

124E-2 ndash(530E-4)

345E-2 ndash(977E-4)

261E-2 ndash(106E-2)

334E-2 ndash(535E-3)

338E-2 ndash(358E-3)

352E-2 ndash(419E-3)

940E-3(381E-4)

IMB9 Mean(Std)

135E-2 ndash(662E-4)

384E-2 ndash243E-4

325E-2 ndash(916E-3)

379E-2 ndash(342E-3)

383E-2 ndash(229E-4)

396E-2 ndash(120E-3)

118E-2(618E-4)

IMB10 Mean(Std)

479E-2 ndash(868E-4)

282E-2 +(180E-3)

329E-2 +(961E-4)

295E-2 +(855E-4)

363E-2 ndash(195E-3)

317E-2 +(763E-4)

357E-2(302E-4)







Complexity 7


MOEAD-M2M

MOEAD-STM


ACDMOEAD-

CSU

MOP1 Mean(Std)

695E-1 ndash(178E-3)

263E-1 ndash(361E-2)

692E-1 ndash(423E-3)

693E-1 ndash(283E-3)

238E-1 ndash(137E-2)

692E-1 ndash(244E-3)

703E-1513E-4

MOP2 Mean(Std)

439E-1 =(341E-4)

202E-1 ndash(486E-2)

379E-1 ndash(812E-2)

386E-1 ndash(667E-2)

206E-1 ndash(405E-2)

412E-1 ndash(660E-2)

439E-1360E-3

MOP3 Mean(Std)

340E-1 ndash(321E-3)

215E-1 ndash(290E-2)

311E-1 ndash(575E-2)

332E-1 ndash(264E-2)

241E-1 ndash(398E-2)

333E-1 ndash(268E-2)

341E-1400E-3

MOP4 Mean(Std)

595E-1 =(824E-4)

290E-1 ndash(162E-2)

569E-1 ndash(357E-2)

474E-1 ndash(965E-2)

301E-1 ndash(231E-2)

536E-1 ndash(666E-2)

595E-1248E-4

MOP5 Mean(Std)

696E-1 ndash(113E-3)

404E-1 ndash(208E-2)

694E-1 ndash(487E-3)

696E-1 ndash(274E-3)

400E-1 ndash(103E-17)

694E-1 ndash(272E-3)

703E-1465E-4

MOP6 Mean(Std)

822E-1 ndash(174E-3)

634E-1 ndash(248E-2)

824E-1 ndash(362E-3)

828E-1 ndash(242E-3)

634E-1 ndash(297E-2)

828E-1 ndash(221E-3)

840E-1271E-4

MOP7 Mean(Std)

541E-1 ndash(443E-3)

407E-1 ndash(189E-7)

497E-1 ndash(339E-2)

494E-1 ndash(491E-3)

412E-1 ndash(102E-2)

494E-1 ndash(145E-2)

545E-1526E-4

IMB1 Mean(Std)

711E-1 ndash(741E-4)

615E-1 ndash(887E-2)

709E-1 ndash(156E-3)

710E-1 ndash(204E-3)

543E-1 ndash(695E-2)

709E-1 ndash(100E-3)

715E-1156E-4

IMB2 Mean(Std)

570E-1 ndash(703E-4)

391E-1 ndash(914E-3)

525E-1 ndash(233E-2)

501E-1 ndash(302E-2)

390E-1 ndash(718E-3)

510E-1 ndash(234E-2)

574E-1290E-4

IMB3 Mean(Std)

329E-1 ndash(763E-4)

133E-1 ndash(220E-2)

324E-1 ndash(197E-3)

323E-1 ndash(262E-3)

128E-1 ndash(718E-3)

324E-1 ndash(169E-3)

335E-1390E-4

IMB4 Mean(Std)

823E-1 ndash(244E-3)

771E-1 ndash(303E-3)

849E-1 +(780E-4)

853E-1 +(454E-4)

764E-1 ndash(310E-3)

846E-1 =(963E-4)

846E-1245E-4

IMB5 Mean(Std)

548E-1 ndash(676E-3)

545E-1 ndash(231E-5)

545E-1 ndash(234E-5)

543E-1 ndash(791E-4)

532E-1 ndash(243E-4)

542E-1 ndash(846E-4)

571E-1271E-4

IMB6 Mean(Std)

831E-1 ndash(307E-3)

841E-1 ndash(353E-5)

840E-1 ndash(428E-5)

840E-1 ndash(144E-4)

832E-1 ndash(125E-3)

852E-1 ndash(358E-4)

853E-1111E-4

IMB7 Mean(Std)

709E-1 ndash(819E-4)

695E-1 ndash(516E-4)

699E-1 ndash(691E-3)

697E-1 ndash(558E-3)

695E-1 ndash(467E-4)

696E-1 ndash(587E-3)

714E-1514E-4

IMB8 Mean(Std)

567E-1 ndash(833E-4)

545E-1 ndash(114E-3)

555E-1 ndash(118E-2)

546E-1 ndash(604E-3)

546E-1 ndash(394E-3)

544E-1 ndash(477E-3)

573E-1541E-4

IMB9 Mean(Std)

331E-1 ndash(114E-3)

313E-1 ndash(888E-4)

318E-1 ndash(682E-3)

314E-1 ndash(249E-3)

314E-1 ndash(783E-4)

311E-1 ndash(202E-3)

334E-1906E-4

IMB10 Mean(Std)

817E-1 ndash(219E-3)

848E-1 +(135E-3)

839E-1 +(123E-3)

846E-1 +(101E-3)

834E-1 ndash(305E-3)

843E-1 +(998E-4)

836E-1335E-4





8 Complexity

0 02 04 06 08 10

02

04

06

08

1MOP1

0 02 04 06 08 10

02

04

06

08

1MOP2

0 02 04 06 08 10

02

04

06

08

1MOP3

0 02 04 06 08 10

02

04

06

08

1MOP4

0 02 04 06 08 10

02

04

06

08

1MOP5

00 0

05z

MOP6

y x

05 05

1

1 1

000

05

05

MOP7

1

051

15

115 0 02 04 06 08 10

02

04

06

08

1IMB1

0 02 04 06 08 10

02

04

06

08

1IMB2

0 02 04 06 08 10

02

04

06

08

1IMB3

000

05

IMB4

z

xy

0505

1

11

00 0

05

IMB5

1

05 05

15

1 1

00 0

05z

IMB6

y x

05 05

1

1 1 0 02 04 06 08 1 120

02

04

06

08

1

12IMB7

0 02 04 06 08 1 120

02

04

06

08

1

12IMB8

0 02 04 06 08 1 120

02

04

06

08

1

12IMB9

00 0

05z

IMB10

y x

05 05

1

1 1






Complexity 9


Data Availability




Acknowledgments


References

























10 Complexity

































Complexity 11























Journal of












Journal of







Volume 2018






Volume 2018








Complexity 5





119910 = 119906 minus 1199034 (119906 minus 119910) if 119910 gt 119906 (8)

119910 = 119897 + 1199034 (119910 minus 119897) if 119910 lt 119897 (9)




rfloor + 1 (10)





6 Complexity


MOEAD-M2M

MOEAD-STM


ACD MOEAD-CSU

MOP1 Mean(Std)

226E-2 ndash(152E-3)

346E-1 ndash(227E-2)

269E-2 ndash(353E-3)

263E-2 ndash(272E-3)

362E-1 ndash(747E-3)

272E-2 ndash(233E-3)

166E-2(438E-4)

MOP2 Mean(Std)

740E-3 ndash(509E-4)

295E-1 ndash(783E-2)

679E-2 ndash(769E-2)

602E-2 ndash(646E-2)

277E-1 ndash(719E-2)

313E-2 ndash(667E-2)

687E-3(229E-3)

MOP3 Mean(Std)

104E-2 ndash(220E-3)

155E-1 ndash(343E-2)

385E-2 ndash(583E-2)

143E-2 ndash(191E-2)

123E-1 ndash(482E-2)

136E-2 ndash(206E-2)

814E-3(376E-3)

MOP4 Mean(Std)

457E-3 =(339E-4)

300E-1 ndash(299E-2)

352E-2 ndash(323E-2)

107E-1 ndash(846E-2)

276E-1 ndash(324E-2)

548E-2 ndash(574E-2)

447E-3(122E-4)

MOP5 Mean(Std)

200E-2 ndash(691E-4)

312E-1 ndash(278E-2)

231E-2 ndash(304E-3)

216E-2 ndash(241E-3)

316E-1 ndash(802E-3)

239E-2 ndash(231E-3)

151E-2(357E-4)

MOP6 Mean(Std)

496E-2 ndash(158E-3)

290E-1 ndash(224E-2)

490E-2 ndash(236E-3)

484E-2 ndash(280E-3)

290E-1 ndash(256E-2)

502E-2 ndash(239E-3)

330E-2(280E-4)

MOP7 Mean(Std)

793E-2 ndash(469E-3)

351E-1 ndash(893E-8)

177E-1 ndash(316E-2)

192E-1 ndash(186E-2)

338E-1 ndash(227E-2)

233E-1 ndash(247E-2)

465E-2(335E-4)

IMB1 Mean(Std)

104E-2 ndash(611E-4)

105E-1 ndash(878E-2)

123E-2 ndash(133E-3)

118E-2 ndash(196E-3)

176E-1 ndash(714E-2)

128E-2 ndash(952E-4)

753E-3(136E-4)

IMB2 Mean(Std)

119E-2 ndash(546E-4)

162E-1 ndash(116E-2)

470E-2 ndash(179E-2)

664E-2 ndash(246E-2)

167E-1 ndash(111E-2)

585E-2 ndash(184E-2)

892E-3(216E-4)

IMB3 Mean(Std)

178E-2 ndash(766E-4)

282E-1 ndash(355E-2)

256E-2 ndash(348E-3)

237E-2 ndash(273E-3)

284E-1 ndash(142E-2)

226E-2 ndash(188E-3)

119E-2(316E-4)

IMB4 Mean(Std)

405E-2 ndash(159E-3)

133E-1 ndash(692E-3)

253E-2 +(504E-4)

239E-2 +(257E-4)

137E-1 ndash(697E-3)

282E-2 =(545E-4)

279E-2(219E-4)

IMB5 Mean(Std)

567E-2 ndash(992E-3)

840E-2 ndash(247E-5)

839E-2 ndash(137E-5)

790E-2 ndash(461E-3)

912E-2 ndash(574E-5)

741E-2 ndash(750E-3)

328E-2(121E-4)

IMB6 Mean(Std)

385E-2 ndash(524E-3)

462E-2 ndash(124E-4)

461E-2 ndash(212E-4)

470E-2 ndash(145E-4)

522E-2 ndash(628E-4)

250E-2 ndash(310E-4)

237E-2(871E-5)

IMB7 Mean(Std)

108E-2 ndash(589E-4)

298E-2 ndash(228E-4)

249E-2 ndash(830E-3)

276E-2 ndash(276E-2)

299E-2 ndash(195E-4)

285E-2 ndash(695E-3)

792E-3(305E-4)

IMB8 Mean(Std)

124E-2 ndash(530E-4)

345E-2 ndash(977E-4)

261E-2 ndash(106E-2)

334E-2 ndash(535E-3)

338E-2 ndash(358E-3)

352E-2 ndash(419E-3)

940E-3(381E-4)

IMB9 Mean(Std)

135E-2 ndash(662E-4)

384E-2 ndash243E-4

325E-2 ndash(916E-3)

379E-2 ndash(342E-3)

383E-2 ndash(229E-4)

396E-2 ndash(120E-3)

118E-2(618E-4)

IMB10 Mean(Std)

479E-2 ndash(868E-4)

282E-2 +(180E-3)

329E-2 +(961E-4)

295E-2 +(855E-4)

363E-2 ndash(195E-3)

317E-2 +(763E-4)

357E-2(302E-4)







Complexity 7


MOEAD-M2M

MOEAD-STM


ACDMOEAD-

CSU

MOP1 Mean(Std)

695E-1 ndash(178E-3)

263E-1 ndash(361E-2)

692E-1 ndash(423E-3)

693E-1 ndash(283E-3)

238E-1 ndash(137E-2)

692E-1 ndash(244E-3)

703E-1513E-4

MOP2 Mean(Std)

439E-1 =(341E-4)

202E-1 ndash(486E-2)

379E-1 ndash(812E-2)

386E-1 ndash(667E-2)

206E-1 ndash(405E-2)

412E-1 ndash(660E-2)

439E-1360E-3

MOP3 Mean(Std)

340E-1 ndash(321E-3)

215E-1 ndash(290E-2)

311E-1 ndash(575E-2)

332E-1 ndash(264E-2)

241E-1 ndash(398E-2)

333E-1 ndash(268E-2)

341E-1400E-3

MOP4 Mean(Std)

595E-1 =(824E-4)

290E-1 ndash(162E-2)

569E-1 ndash(357E-2)

474E-1 ndash(965E-2)

301E-1 ndash(231E-2)

536E-1 ndash(666E-2)

595E-1248E-4

MOP5 Mean(Std)

696E-1 ndash(113E-3)

404E-1 ndash(208E-2)

694E-1 ndash(487E-3)

696E-1 ndash(274E-3)

400E-1 ndash(103E-17)

694E-1 ndash(272E-3)

703E-1465E-4

MOP6 Mean(Std)

822E-1 ndash(174E-3)

634E-1 ndash(248E-2)

824E-1 ndash(362E-3)

828E-1 ndash(242E-3)

634E-1 ndash(297E-2)

828E-1 ndash(221E-3)

840E-1271E-4

MOP7 Mean(Std)

541E-1 ndash(443E-3)

407E-1 ndash(189E-7)

497E-1 ndash(339E-2)

494E-1 ndash(491E-3)

412E-1 ndash(102E-2)

494E-1 ndash(145E-2)

545E-1526E-4

IMB1 Mean(Std)

711E-1 ndash(741E-4)

615E-1 ndash(887E-2)

709E-1 ndash(156E-3)

710E-1 ndash(204E-3)

543E-1 ndash(695E-2)

709E-1 ndash(100E-3)

715E-1156E-4

IMB2 Mean(Std)

570E-1 ndash(703E-4)

391E-1 ndash(914E-3)

525E-1 ndash(233E-2)

501E-1 ndash(302E-2)

390E-1 ndash(718E-3)

510E-1 ndash(234E-2)

574E-1290E-4

IMB3 Mean(Std)

329E-1 ndash(763E-4)

133E-1 ndash(220E-2)

324E-1 ndash(197E-3)

323E-1 ndash(262E-3)

128E-1 ndash(718E-3)

324E-1 ndash(169E-3)

335E-1390E-4

IMB4 Mean(Std)

823E-1 ndash(244E-3)

771E-1 ndash(303E-3)

849E-1 +(780E-4)

853E-1 +(454E-4)

764E-1 ndash(310E-3)

846E-1 =(963E-4)

846E-1245E-4

IMB5 Mean(Std)

548E-1 ndash(676E-3)

545E-1 ndash(231E-5)

545E-1 ndash(234E-5)

543E-1 ndash(791E-4)

532E-1 ndash(243E-4)

542E-1 ndash(846E-4)

571E-1271E-4

IMB6 Mean(Std)

831E-1 ndash(307E-3)

841E-1 ndash(353E-5)

840E-1 ndash(428E-5)

840E-1 ndash(144E-4)

832E-1 ndash(125E-3)

852E-1 ndash(358E-4)

853E-1111E-4

IMB7 Mean(Std)

709E-1 ndash(819E-4)

695E-1 ndash(516E-4)

699E-1 ndash(691E-3)

697E-1 ndash(558E-3)

695E-1 ndash(467E-4)

696E-1 ndash(587E-3)

714E-1514E-4

IMB8 Mean(Std)

567E-1 ndash(833E-4)

545E-1 ndash(114E-3)

555E-1 ndash(118E-2)

546E-1 ndash(604E-3)

546E-1 ndash(394E-3)

544E-1 ndash(477E-3)

573E-1541E-4

IMB9 Mean(Std)

331E-1 ndash(114E-3)

313E-1 ndash(888E-4)

318E-1 ndash(682E-3)

314E-1 ndash(249E-3)

314E-1 ndash(783E-4)

311E-1 ndash(202E-3)

334E-1906E-4

IMB10 Mean(Std)

817E-1 ndash(219E-3)

848E-1 +(135E-3)

839E-1 +(123E-3)

846E-1 +(101E-3)

834E-1 ndash(305E-3)

843E-1 +(998E-4)

836E-1335E-4





8 Complexity

0 02 04 06 08 10

02

04

06

08

1MOP1

0 02 04 06 08 10

02

04

06

08

1MOP2

0 02 04 06 08 10

02

04

06

08

1MOP3

0 02 04 06 08 10

02

04

06

08

1MOP4

0 02 04 06 08 10

02

04

06

08

1MOP5

00 0

05z

MOP6

y x

05 05

1

1 1

000

05

05

MOP7

1

051

15

115 0 02 04 06 08 10

02

04

06

08

1IMB1

0 02 04 06 08 10

02

04

06

08

1IMB2

0 02 04 06 08 10

02

04

06

08

1IMB3

000

05

IMB4

z

xy

0505

1

11

00 0

05

IMB5

1

05 05

15

1 1

00 0

05z

IMB6

y x

05 05

1

1 1 0 02 04 06 08 1 120

02

04

06

08

1

12IMB7

0 02 04 06 08 1 120

02

04

06

08

1

12IMB8

0 02 04 06 08 1 120

02

04

06

08

1

12IMB9

00 0

05z

IMB10

y x

05 05

1

1 1






Complexity 9


Data Availability




Acknowledgments


References

























10 Complexity

































Complexity 11























Journal of












Journal of







Volume 2018






Volume 2018








6 Complexity


MOEAD-M2M

MOEAD-STM


ACD MOEAD-CSU

MOP1 Mean(Std)

226E-2 ndash(152E-3)

346E-1 ndash(227E-2)

269E-2 ndash(353E-3)

263E-2 ndash(272E-3)

362E-1 ndash(747E-3)

272E-2 ndash(233E-3)

166E-2(438E-4)

MOP2 Mean(Std)

740E-3 ndash(509E-4)

295E-1 ndash(783E-2)

679E-2 ndash(769E-2)

602E-2 ndash(646E-2)

277E-1 ndash(719E-2)

313E-2 ndash(667E-2)

687E-3(229E-3)

MOP3 Mean(Std)

104E-2 ndash(220E-3)

155E-1 ndash(343E-2)

385E-2 ndash(583E-2)

143E-2 ndash(191E-2)

123E-1 ndash(482E-2)

136E-2 ndash(206E-2)

814E-3(376E-3)

MOP4 Mean(Std)

457E-3 =(339E-4)

300E-1 ndash(299E-2)

352E-2 ndash(323E-2)

107E-1 ndash(846E-2)

276E-1 ndash(324E-2)

548E-2 ndash(574E-2)

447E-3(122E-4)

MOP5 Mean(Std)

200E-2 ndash(691E-4)

312E-1 ndash(278E-2)

231E-2 ndash(304E-3)

216E-2 ndash(241E-3)

316E-1 ndash(802E-3)

239E-2 ndash(231E-3)

151E-2(357E-4)

MOP6 Mean(Std)

496E-2 ndash(158E-3)

290E-1 ndash(224E-2)

490E-2 ndash(236E-3)

484E-2 ndash(280E-3)

290E-1 ndash(256E-2)

502E-2 ndash(239E-3)

330E-2(280E-4)

MOP7 Mean(Std)

793E-2 ndash(469E-3)

351E-1 ndash(893E-8)

177E-1 ndash(316E-2)

192E-1 ndash(186E-2)

338E-1 ndash(227E-2)

233E-1 ndash(247E-2)

465E-2(335E-4)

IMB1 Mean(Std)

104E-2 ndash(611E-4)

105E-1 ndash(878E-2)

123E-2 ndash(133E-3)

118E-2 ndash(196E-3)

176E-1 ndash(714E-2)

128E-2 ndash(952E-4)

753E-3(136E-4)

IMB2 Mean(Std)

119E-2 ndash(546E-4)

162E-1 ndash(116E-2)

470E-2 ndash(179E-2)

664E-2 ndash(246E-2)

167E-1 ndash(111E-2)

585E-2 ndash(184E-2)

892E-3(216E-4)

IMB3 Mean(Std)

178E-2 ndash(766E-4)

282E-1 ndash(355E-2)

256E-2 ndash(348E-3)

237E-2 ndash(273E-3)

284E-1 ndash(142E-2)

226E-2 ndash(188E-3)

119E-2(316E-4)

IMB4 Mean(Std)

405E-2 ndash(159E-3)

133E-1 ndash(692E-3)

253E-2 +(504E-4)

239E-2 +(257E-4)

137E-1 ndash(697E-3)

282E-2 =(545E-4)

279E-2(219E-4)

IMB5 Mean(Std)

567E-2 ndash(992E-3)

840E-2 ndash(247E-5)

839E-2 ndash(137E-5)

790E-2 ndash(461E-3)

912E-2 ndash(574E-5)

741E-2 ndash(750E-3)

328E-2(121E-4)

IMB6 Mean(Std)

385E-2 ndash(524E-3)

462E-2 ndash(124E-4)

461E-2 ndash(212E-4)

470E-2 ndash(145E-4)

522E-2 ndash(628E-4)

250E-2 ndash(310E-4)

237E-2(871E-5)

IMB7 Mean(Std)

108E-2 ndash(589E-4)

298E-2 ndash(228E-4)

249E-2 ndash(830E-3)

276E-2 ndash(276E-2)

299E-2 ndash(195E-4)

285E-2 ndash(695E-3)

792E-3(305E-4)

IMB8 Mean(Std)

124E-2 ndash(530E-4)

345E-2 ndash(977E-4)

261E-2 ndash(106E-2)

334E-2 ndash(535E-3)

338E-2 ndash(358E-3)

352E-2 ndash(419E-3)

940E-3(381E-4)

IMB9 Mean(Std)

135E-2 ndash(662E-4)

384E-2 ndash243E-4

325E-2 ndash(916E-3)

379E-2 ndash(342E-3)

383E-2 ndash(229E-4)

396E-2 ndash(120E-3)

118E-2(618E-4)

IMB10 Mean(Std)

479E-2 ndash(868E-4)

282E-2 +(180E-3)

329E-2 +(961E-4)

295E-2 +(855E-4)

363E-2 ndash(195E-3)

317E-2 +(763E-4)

357E-2(302E-4)







Complexity 7


MOEAD-M2M

MOEAD-STM


ACDMOEAD-

CSU

MOP1 Mean(Std)

695E-1 ndash(178E-3)

263E-1 ndash(361E-2)

692E-1 ndash(423E-3)

693E-1 ndash(283E-3)

238E-1 ndash(137E-2)

692E-1 ndash(244E-3)

703E-1513E-4

MOP2 Mean(Std)

439E-1 =(341E-4)

202E-1 ndash(486E-2)

379E-1 ndash(812E-2)

386E-1 ndash(667E-2)

206E-1 ndash(405E-2)

412E-1 ndash(660E-2)

439E-1360E-3

MOP3 Mean(Std)

340E-1 ndash(321E-3)

215E-1 ndash(290E-2)

311E-1 ndash(575E-2)

332E-1 ndash(264E-2)

241E-1 ndash(398E-2)

333E-1 ndash(268E-2)

341E-1400E-3

MOP4 Mean(Std)

595E-1 =(824E-4)

290E-1 ndash(162E-2)

569E-1 ndash(357E-2)

474E-1 ndash(965E-2)

301E-1 ndash(231E-2)

536E-1 ndash(666E-2)

595E-1248E-4

MOP5 Mean(Std)

696E-1 ndash(113E-3)

404E-1 ndash(208E-2)

694E-1 ndash(487E-3)

696E-1 ndash(274E-3)

400E-1 ndash(103E-17)

694E-1 ndash(272E-3)

703E-1465E-4

MOP6 Mean(Std)

822E-1 ndash(174E-3)

634E-1 ndash(248E-2)

824E-1 ndash(362E-3)

828E-1 ndash(242E-3)

634E-1 ndash(297E-2)

828E-1 ndash(221E-3)

840E-1271E-4

MOP7 Mean(Std)

541E-1 ndash(443E-3)

407E-1 ndash(189E-7)

497E-1 ndash(339E-2)

494E-1 ndash(491E-3)

412E-1 ndash(102E-2)

494E-1 ndash(145E-2)

545E-1526E-4

IMB1 Mean(Std)

711E-1 ndash(741E-4)

615E-1 ndash(887E-2)

709E-1 ndash(156E-3)

710E-1 ndash(204E-3)

543E-1 ndash(695E-2)

709E-1 ndash(100E-3)

715E-1156E-4

IMB2 Mean(Std)

570E-1 ndash(703E-4)

391E-1 ndash(914E-3)

525E-1 ndash(233E-2)

501E-1 ndash(302E-2)

390E-1 ndash(718E-3)

510E-1 ndash(234E-2)

574E-1290E-4

IMB3 Mean(Std)

329E-1 ndash(763E-4)

133E-1 ndash(220E-2)

324E-1 ndash(197E-3)

323E-1 ndash(262E-3)

128E-1 ndash(718E-3)

324E-1 ndash(169E-3)

335E-1390E-4

IMB4 Mean(Std)

823E-1 ndash(244E-3)

771E-1 ndash(303E-3)

849E-1 +(780E-4)

853E-1 +(454E-4)

764E-1 ndash(310E-3)

846E-1 =(963E-4)

846E-1245E-4

IMB5 Mean(Std)

548E-1 ndash(676E-3)

545E-1 ndash(231E-5)

545E-1 ndash(234E-5)

543E-1 ndash(791E-4)

532E-1 ndash(243E-4)

542E-1 ndash(846E-4)

571E-1271E-4

IMB6 Mean(Std)

831E-1 ndash(307E-3)

841E-1 ndash(353E-5)

840E-1 ndash(428E-5)

840E-1 ndash(144E-4)

832E-1 ndash(125E-3)

852E-1 ndash(358E-4)

853E-1111E-4

IMB7 Mean(Std)

709E-1 ndash(819E-4)

695E-1 ndash(516E-4)

699E-1 ndash(691E-3)

697E-1 ndash(558E-3)

695E-1 ndash(467E-4)

696E-1 ndash(587E-3)

714E-1514E-4

IMB8 Mean(Std)

567E-1 ndash(833E-4)

545E-1 ndash(114E-3)

555E-1 ndash(118E-2)

546E-1 ndash(604E-3)

546E-1 ndash(394E-3)

544E-1 ndash(477E-3)

573E-1541E-4

IMB9 Mean(Std)

331E-1 ndash(114E-3)

313E-1 ndash(888E-4)

318E-1 ndash(682E-3)

314E-1 ndash(249E-3)

314E-1 ndash(783E-4)

311E-1 ndash(202E-3)

334E-1906E-4

IMB10 Mean(Std)

817E-1 ndash(219E-3)

848E-1 +(135E-3)

839E-1 +(123E-3)

846E-1 +(101E-3)

834E-1 ndash(305E-3)

843E-1 +(998E-4)

836E-1335E-4





8 Complexity

0 02 04 06 08 10

02

04

06

08

1MOP1

0 02 04 06 08 10

02

04

06

08

1MOP2

0 02 04 06 08 10

02

04

06

08

1MOP3

0 02 04 06 08 10

02

04

06

08

1MOP4

0 02 04 06 08 10

02

04

06

08

1MOP5

00 0

05z

MOP6

y x

05 05

1

1 1

000

05

05

MOP7

1

051

15

115 0 02 04 06 08 10

02

04

06

08

1IMB1

0 02 04 06 08 10

02

04

06

08

1IMB2

0 02 04 06 08 10

02

04

06

08

1IMB3

000

05

IMB4

z

xy

0505

1

11

00 0

05

IMB5

1

05 05

15

1 1

00 0

05z

IMB6

y x

05 05

1

1 1 0 02 04 06 08 1 120

02

04

06

08

1

12IMB7

0 02 04 06 08 1 120

02

04

06

08

1

12IMB8

0 02 04 06 08 1 120

02

04

06

08

1

12IMB9

00 0

05z

IMB10

y x

05 05

1

1 1






Complexity 9


Data Availability




Acknowledgments


References

























10 Complexity

































Complexity 11























Journal of












Journal of







Volume 2018






Volume 2018








Complexity 7


MOEAD-M2M

MOEAD-STM


ACDMOEAD-

CSU

MOP1 Mean(Std)

695E-1 ndash(178E-3)

263E-1 ndash(361E-2)

692E-1 ndash(423E-3)

693E-1 ndash(283E-3)

238E-1 ndash(137E-2)

692E-1 ndash(244E-3)

703E-1513E-4

MOP2 Mean(Std)

439E-1 =(341E-4)

202E-1 ndash(486E-2)

379E-1 ndash(812E-2)

386E-1 ndash(667E-2)

206E-1 ndash(405E-2)

412E-1 ndash(660E-2)

439E-1360E-3

MOP3 Mean(Std)

340E-1 ndash(321E-3)

215E-1 ndash(290E-2)

311E-1 ndash(575E-2)

332E-1 ndash(264E-2)

241E-1 ndash(398E-2)

333E-1 ndash(268E-2)

341E-1400E-3

MOP4 Mean(Std)

595E-1 =(824E-4)

290E-1 ndash(162E-2)

569E-1 ndash(357E-2)

474E-1 ndash(965E-2)

301E-1 ndash(231E-2)

536E-1 ndash(666E-2)

595E-1248E-4

MOP5 Mean(Std)

696E-1 ndash(113E-3)

404E-1 ndash(208E-2)

694E-1 ndash(487E-3)

696E-1 ndash(274E-3)

400E-1 ndash(103E-17)

694E-1 ndash(272E-3)

703E-1465E-4

MOP6 Mean(Std)

822E-1 ndash(174E-3)

634E-1 ndash(248E-2)

824E-1 ndash(362E-3)

828E-1 ndash(242E-3)

634E-1 ndash(297E-2)

828E-1 ndash(221E-3)

840E-1271E-4

MOP7 Mean(Std)

541E-1 ndash(443E-3)

407E-1 ndash(189E-7)

497E-1 ndash(339E-2)

494E-1 ndash(491E-3)

412E-1 ndash(102E-2)

494E-1 ndash(145E-2)

545E-1526E-4

IMB1 Mean(Std)

711E-1 ndash(741E-4)

615E-1 ndash(887E-2)

709E-1 ndash(156E-3)

710E-1 ndash(204E-3)

543E-1 ndash(695E-2)

709E-1 ndash(100E-3)

715E-1156E-4

IMB2 Mean(Std)

570E-1 ndash(703E-4)

391E-1 ndash(914E-3)

525E-1 ndash(233E-2)

501E-1 ndash(302E-2)

390E-1 ndash(718E-3)

510E-1 ndash(234E-2)

574E-1290E-4

IMB3 Mean(Std)

329E-1 ndash(763E-4)

133E-1 ndash(220E-2)

324E-1 ndash(197E-3)

323E-1 ndash(262E-3)

128E-1 ndash(718E-3)

324E-1 ndash(169E-3)

335E-1390E-4

IMB4 Mean(Std)

823E-1 ndash(244E-3)

771E-1 ndash(303E-3)

849E-1 +(780E-4)

853E-1 +(454E-4)

764E-1 ndash(310E-3)

846E-1 =(963E-4)

846E-1245E-4

IMB5 Mean(Std)

548E-1 ndash(676E-3)

545E-1 ndash(231E-5)

545E-1 ndash(234E-5)

543E-1 ndash(791E-4)

532E-1 ndash(243E-4)

542E-1 ndash(846E-4)

571E-1271E-4

IMB6 Mean(Std)

831E-1 ndash(307E-3)

841E-1 ndash(353E-5)

840E-1 ndash(428E-5)

840E-1 ndash(144E-4)

832E-1 ndash(125E-3)

852E-1 ndash(358E-4)

853E-1111E-4

IMB7 Mean(Std)

709E-1 ndash(819E-4)

695E-1 ndash(516E-4)

699E-1 ndash(691E-3)

697E-1 ndash(558E-3)

695E-1 ndash(467E-4)

696E-1 ndash(587E-3)

714E-1514E-4

IMB8 Mean(Std)

567E-1 ndash(833E-4)

545E-1 ndash(114E-3)

555E-1 ndash(118E-2)

546E-1 ndash(604E-3)

546E-1 ndash(394E-3)

544E-1 ndash(477E-3)

573E-1541E-4

IMB9 Mean(Std)

331E-1 ndash(114E-3)

313E-1 ndash(888E-4)

318E-1 ndash(682E-3)

314E-1 ndash(249E-3)

314E-1 ndash(783E-4)

311E-1 ndash(202E-3)

334E-1906E-4

IMB10 Mean(Std)

817E-1 ndash(219E-3)

848E-1 +(135E-3)

839E-1 +(123E-3)

846E-1 +(101E-3)

834E-1 ndash(305E-3)

843E-1 +(998E-4)

836E-1335E-4





8 Complexity

0 02 04 06 08 10

02

04

06

08

1MOP1

0 02 04 06 08 10

02

04

06

08

1MOP2

0 02 04 06 08 10

02

04

06

08

1MOP3

0 02 04 06 08 10

02

04

06

08

1MOP4

0 02 04 06 08 10

02

04

06

08

1MOP5

00 0

05z

MOP6

y x

05 05

1

1 1

000

05

05

MOP7

1

051

15

115 0 02 04 06 08 10

02

04

06

08

1IMB1

0 02 04 06 08 10

02

04

06

08

1IMB2

0 02 04 06 08 10

02

04

06

08

1IMB3

000

05

IMB4

z

xy

0505

1

11

00 0

05

IMB5

1

05 05

15

1 1

00 0

05z

IMB6

y x

05 05

1

1 1 0 02 04 06 08 1 120

02

04

06

08

1

12IMB7

0 02 04 06 08 1 120

02

04

06

08

1

12IMB8

0 02 04 06 08 1 120

02

04

06

08

1

12IMB9

00 0

05z

IMB10

y x

05 05

1

1 1






Complexity 9


Data Availability




Acknowledgments


References

























10 Complexity

































Complexity 11























Journal of












Journal of







Volume 2018






Volume 2018








8 Complexity

0 02 04 06 08 10

02

04

06

08

1MOP1

0 02 04 06 08 10

02

04

06

08

1MOP2

0 02 04 06 08 10

02

04

06

08

1MOP3

0 02 04 06 08 10

02

04

06

08

1MOP4

0 02 04 06 08 10

02

04

06

08

1MOP5

00 0

05z

MOP6

y x

05 05

1

1 1

000

05

05

MOP7

1

051

15

115 0 02 04 06 08 10

02

04

06

08

1IMB1

0 02 04 06 08 10

02

04

06

08

1IMB2

0 02 04 06 08 10

02

04

06

08

1IMB3

000

05

IMB4

z

xy

0505

1

11

00 0

05

IMB5

1

05 05

15

1 1

00 0

05z

IMB6

y x

05 05

1

1 1 0 02 04 06 08 1 120

02

04

06

08

1

12IMB7

0 02 04 06 08 1 120

02

04

06

08

1

12IMB8

0 02 04 06 08 1 120

02

04

06

08

1

12IMB9

00 0

05z

IMB10

y x

05 05

1

1 1






Complexity 9


Data Availability




Acknowledgments


References

























10 Complexity

































Complexity 11























Journal of












Journal of







Volume 2018






Volume 2018








Complexity 9


Data Availability




Acknowledgments


References

























10 Complexity

































Complexity 11























Journal of












Journal of







Volume 2018






Volume 2018








10 Complexity

































Complexity 11























Journal of












Journal of







Volume 2018






Volume 2018








Complexity 11























Journal of












Journal of







Volume 2018






Volume 2018















Journal of












Journal of







Volume 2018






Volume 2018








a constrained solution update strategy for multiobjective

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