A Comparative Study of Constraint-Handling Techniques in Evolutionary Constrained Multiobjective Optimization

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1 A omparatve Stdy of onstrant-handlng Technqes n Evoltonary onstraned Mltobectve Optmzaton Ja-Peng L, Yong Wang, Member, IEEE, Shengxang Yang, Senor Member, IEEE, and Zxng a, Senor Member, IEEE Abstract Solvng constraned mltobectve optmzaton problems s one of the most challengng areas n the evoltonary comptaton research commnty. To solve a constraned mltobectve optmzaton problem, an algorthm shold tackle the obectve fnctons and the constrants smltaneosly. As a reslt, many constrant-handlng technqes have been proposed. However, most of the exstng constrant-handlng technqes are developed to solve test nstances (e.g., TPs) wth low dmenson and large feasble regon. On the other hand, expermental comparsons on dfferent constrant-handlng technqes reman scarce. In vew of these two sses, n ths paper we frst constrct 8 test nstances, each of whch exhbts dfferent propertes. Afterward, we choose three representatve constrant-handlng technqes and combne them wth nondomnated sortng genetc algorthm II to stdy the performance dfference on varos condtons. By the expermental stdes, we pont ot the advantages and dsadvantages of dfferent constrant-handlng technqes. Keywords onstrant-handlng technqes, constraned mltobectve optmzaton problems, evoltonary algorthms, test nstances. I. INTRODUTION Nmeros real-world applcatons nvolve mltple obectves and varos constrants, whch can be modeled as constraned mltobectve optmzaton problems (MOPs). Wthot loss of generalty, a MOP can be formlated as: Mnmze f ) ( f), f2 ),, fm )), x,, xn ) S Sbect to: g ) 0,,..., p h ) 0, p,..., q where x s the decson vector, L x U ( {,, n}) s the th decson varable, L and U are the lower and pper bonds of x, S s the decson space, n and m are the nmber of decson varables and obectve fnctons, f ) ( {,, m}) s the th obectve fncton, g ) s the th neqalty constrant, h ) s the ( p)th eqalty constrant, p s the nmber of neqalty constrants, and ( q p) s the nmber of eqalty constrants. J.-P. L and Z. a are wth the School of Informaton Scence and Engneerng, entral Soth Unversty, hangsha 40083, hna. (e-mal: lpcs@cs.ed.cn; zxca@cs.ed.cn) Y. Wang (orrespondng Athor) s wth the School of Informaton Scence and Engneerng, entral Soth Unversty, hangsha 40083, hna, and also wth the School of ompter Scence and Informatcs, De Montfort Unversty, Lecester LE 9BH, UK. (e-mal: ywang@cs.ed.cn) S. Yang s wth the School of ompter Scence and Informatcs, De Montfort Unversty, Lecester LE 9BH, UK. (e-mal: syang@dm.ac.k) The constrant volaton of ndvdal x on the th constrant s calclated as follows: (0, g )), p ) () (0, h ) ), p q where s a predefned tolerance vale to relax the eqalty constrans to a certan extent. The feasble regon of a MOP s a sbspace of the decson space S, and can be defned as { x S ) 0,,, q}. Evoltonary algorthms (EAs) are poplaton based optmzaton approaches nspred by natre []. Drng the last two decades, mch effort has been made to develop and nderstand EAs for dealng wth mltobectve optmzaton problems [2] [3]. onstrant-handlng for sngle-obectve optmzaton problems has also been extensvely researched [4] [5]. However, few attempts have been made to nvestgate evoltonary constraned mltobectve optmzaton [6], whch s challengng and of practcal nterest. For MOPs, we cannot fnd a sngle solton to optmze all the obectves at the same tme snce the obectves are always n conflct wth each other. Therefore, when sng EAs to deal wth MOPs, we have to balance the obectves and fnd a set of optmal tradeoffs. In addton, t s necessary to note that EAs are nconstraned search methods that need addtonal mechansms to deal wth constrants when solvng MOPs. De to the above attrbtes, the solton of MOPs s very dffclt and the research of MOPs based on EAs s stll n ts nfant stage. When solvng MOPs by EAs, the correspondng methods are called constraned mltobectve EAs (MOEAs). Next, we wll brefly ntrodce some representatve MOEAs. Deb et al. [7] extended the feasblty rle n [4] and proposed a constraned-domnaton prncple (DP) to solve MOPs. Based on the work n [7], Oyama et al. [8] proposed a new constrant-handlng technqe, whch ntrodces the dea of nondomnance and nchng concepts n the obectve space nto the constrant space. hafekar et al. [9] ntrodced a steady state GA to solve MOPs. In [9], two constraned mltobectve optmzaton technqes are appled. One technqe s to rn several sngle-obectve GAs concrrently and each GA s to optmze one obectve. The other technqe s to se a sngle-obectve GA to optmze mltple obectves n a seqental order. Yong [0] presented a blended rank mechansm n whch every solton s assgned two Pareto domnance ranks, and both two ranks are taken nto accont to gve the fnal rank of an ndvdal. Jmenez et al. [] combned the Pareto concept n mltobectve optmzaton

2 wth a novel dversty mechansm to sort the poplaton. Woldesenbet and Yen [6] proposed a self-adaptve penalty fncton to handle constrants based on the feasble rato n the crrent poplaton. Santana-Qntero et al. [2] presented a hybrd MOEA by approxmatng the Pareto front of a MOP wth a hybrd approach. In [2], rogh set theory s mplemented to mantan a good qalty of the approxmaton. By combnng DP [7] wth smlated annealng, Sngh et al. [3] proposed a non-greedy constrant-handlng technqe to solve MOPs. Jao et al. [4] proposed a MOEA based on [6], the man deas of whch are to make se of the nformaton of the nfeasble ndvdals and to constrct some modfed obectve fnctons to gde the evoltonary process. Jan and Khanm [5] proposed two modfed constrant-handlng technqes by combnng Tchebycheff aggregaton fncton wth the famos stochastc rankng [6] and DP [7] to solve MOPs. Very recently, an mmne optmzaton algorthm s proposed by Qan et al. [7] to solve MOPs. Takng both convergence and dversty nto accont, they dvded a poplaton nto two sbpoplatons and appled dfferent reprodcton operators to create the next poplaton. Recognzng that there s no sngle constrant-handlng technqe can otperform all others on dfferent knds of problems, Q and Sganthan [8] proposed an ensemble of constrant-handlng technqes. In [8], three constranthandlng technqes,.e., DP [7], self-adaptve penalty (SP) [6], and ε constraned method [9] are adopted. It s noteworthy that most of the above methods are desgned to solve the test nstances (called TPs) n [20], whch were proposed by Deb et al. ffteen years ago. De to the fact that TPs sally have a large proporton of the feasble regon, t s a very mportant topc to desgn MOPs wth more complex characterstcs. On the other hand, the crrent methods are sally developed to solve TPs wth low dmenson (sch as two dmensons), and there are very few stdes condcted on the performance comparson of dfferent constrant-handlng technqes n the commnty of evoltonary constraned mltobectve optmzaton. Motvated by the above consderatons, n ths paper we frstly constrct 8 test nstances wth dfferent characterstcs based on TPs. Afterward, we select three representatve constrant-handlng technqes (DP [7], SP [6], and adaptve tradeoff model (ATM) [2]). Then we stdy ther performance dfference on varos scenaros. The man contrbtons of ths paper can be smmarzed as follows: We desgn 8 new test nstances (called NTPs) based on TPs. ompared wth the orgnal TPs, NTPs ntrodced n ths paper exhbt more complex characterstcs. Specfcally, n NTPs dfferent test nstances have dfferent shapes of the Pareto front, dfferent dmensons of the search space, and dfferent sze of the feasble regon. Systematc experments have been condcted on the 8 test nstances to stdy the performance of the three representatve constrant-handlng technqes accordng to the shape of the Pareto front, the dmenson of search space, and the sze of the feasble regon. The rest of ths paper s organzed as follows. Secton II brefly descrbes the related defntons. Secton III ntrodces three representatve constrant-handlng technqes. Secton IV presents the constrcted test nstances. Secton V expermentally stdes the performance of the chosen constranthandlng technqes. Secton VI concldes ths paper. II. THE RELATED DEFINITIONS OF MOPS A MOP nclde mltple obectves. In mltobectve optmzaton, the comparson of ndvdals s sally based on Pareto domnance. Next, we wll ntrodce for related defntons. Defnton (Pareto Domnance): onsderng the m obectve fnctons f ) ( f), f2 ),, fm )) and two decson vectors x v and x, f {,..., m}, f v ) f ) and {,..., m}, f v ) f ), then x s sad to v Pareto domnate x, denoted as xv x. Defnton 2 (Pareto Optmal Solton): A solton x s called a Pareto optmal solton of a MOP f and only f, x x. x v v Defnton 3 (Pareto Optmal Set): The Pareto optmal set of a MOP can be defned as POS { x xv, x x v }. Defnton 4 (Pareto Front): The Pareto front of a MOP can be defned as PF { f ) x POS}. III. THREE REPRESENTATIVE ONSTRAINT-HANDLING TEHNIQUES FOR MOPS A. onstraned-domnaton Prncple (DP) DP [7] s a smple and effcent technqe to handle constrans, whch compares parwse ndvdals based on the followng rles: When two feasble soltons are compared, the one Pareto domnatng the other s better. When a feasble solton s compared wth an nfeasble solton, the feasble solton s better. When two nfeasble soltons are compared, the one wth smaller degree of constrant volaton s better. B. Self-adaptve Penalty (SP) Another way to solve MOPs s to penalze the nfeasble ndvdal wth penalty fncton, n whch the penalty term added to the obectve fncton s based on the degree of constrant volaton of the nfeasble ndvdal. Among all penalty fncton based methods, the self-adaptve penalty (SP) proposed n [6] s a representatve one. It has two man components: the dstance vale and the penalty fncton. Frstly, the obectve fnctons and the degree of constrant

3 volaton of each ndvdal x n the poplaton are normalzed as follows: mn f ) f f ), {,, m} (2) mn f f q ) ) (3) q where mn f s the mnmm vale of the th obectve fncton, f s the mm vale of the th obectve fncton, and s the mm volaton of the th constrant. Let r f denote the feasblty rato of the crrent poplaton, whch can be calclated as follows: the nmber of feasble ndvdals rf (4) the poplaton sze Afterward, the th ( {,, m}) dstance vale d ) and th ( {,, m}) penalty vale p ) of x can be expressed as follows: ), f rf 0 d ) 2 2 f ( ) ( ), f rf 0 x x (5) p ) ( r ) X ) r Y ) (6) f f where 0, f rf 0 X ) (7) ), f rf 0 0, f x s feasble Y ) f (8) ( ), f x s nfeasble x Fnally, the ftness of x n the th obectve fncton dmenson s the sm of d ) and p ) : F ) d ) p ), {,, m} (9) Then the poplaton s sorted based on the m ftness fnctons F, F2,, Fm va the nondomnated sortng [7].. Adaptve Tradeoff Model (ATM) ATM ntrodces an mportant dea,.e., dvdng the evoltonary process nto three statons by the feasblty proporton of the crrent poplaton. In dfferent statons, ATM adopts dfferent technqes to cope wth the constrants. ) The nfeasble staton: There s no feasble solton n the crrent poplaton. ATM converts a MOP wth m obectves and q constrants nto a nconstraned MOP wth (m+) obectves, by consderng the constrant volaton as an addtonal obectve. Then the nondomnated sortng [7] s appled, and half of the ndvdals wth less constrant volatons n the frst layer are chosen and removed from the poplaton. Afterward, the remanng ndvdals n the poplaton are mplemented the same process ntl a desrable nmber of ndvdals s obtaned. 2) The sem-feasble staton: There exst both nfeasble and feasble soltons n the crrent poplaton. Assme that Z s the crrent poplaton, and then Z s dvded nto the feasble grop and the nfeasble grop based on eqaton (3): Z { x Z ) 0} (0) Z2 { x Z ) 0} () Frstly, ATM ses the followng transformed obectve fncton to penalze the nfeasble ndvdals: ( ), s feasble ' f x f x f ) { * fb _ ( )* fw _, f )}, f x s nfeasble (2) where denotes the feasblty proporton of the last poplaton, mn ( f ) b _ f x, and ( f ). w xz _ f x xz Then, ATM normalzes the transformed obectve fncton and the constrant volaton: ' ' f ) mn f ) x Z f ), {,, } ' ' m (3) f ) mn f ) xz xz 0, f x Z ) ( ) mn ( ) x x xz (4) 2, f x Z 2 ) mn ) xz2 xz2 The fnal ftness of x n the th obectve fncton dmenson s the sm of f ) and ) : F ) f ) ), {,, m} (5) Lke SP, the m ftness fnctons F, F2,, Fm are sed to sort the poplaton by the nondomnated sortng [7]. 3) The feasble phase: In ths phase, all soltons n the poplaton are feasble. Ths, ATM drectly sorts the poplaton by makng se of the nondomnated sortng [7]. IV. ONSTRUTED TEST INSTANES The wdely sed test nstances n evoltonary constraned mltobectve optmzaton are TPs [20], whch can be dvded n two parts. The frst part s TP, n whch the nmber of constrants s changeable, bt wth the ncrease of constrants the shape of the Pareto front almost remans the same. The second part s TP2-TP7. The shapes of the Pareto front of these test nstances are tnable and controlled by sx parameters, bt these test nstances only have one constrant and the sze of the feasble regon s sally large. To stdy the performance of the three chosen constranthandlng technqes n Secton III on dfferent condtons, the above drawbacks of test nstances shold be addressed. In ths paper, a set of new constraned test nstances (called NTPs) s constrcted based on TP2-TP7. In NTPs, the followng new characterstcs have been added: In most of papers, g ) n the second obectve fncton of TP2-TP7 s often set to (+x 2 ). To ncrease the dffclty, n ths paper g ) has been set to the Ronsenbrock fncton [22]: n g) (00 x ) ( x ) ) (6) 2

4 TABLE I Detaled nformaton of the 8 test nstances, where Type I denotes that the Pareto front s dscontnos (see Fg. ), Type II denotes that the Pareto front conssts of both dscontnos and contnos parts (see Fg. 2), and Type III denotes that the Pareto front s contnos (see Fg. 3). Test nstance Parameter vale onstrant Shape of the Pareto front Propertes Dmenson of the decson vector Sze of the feasble regon NTP- 0.2, a 0.2, b 0, c, d 0.5, e, z 0.5, z2 4 and 2 Type I 0D/30D Small (<0.%) NTP-2 0.2, a 0.75, b 0, c, d 0.5, e, z 0.5, z2 4 and 2 Type I 0D/30D Small (<0.%) NTP-3 0.2, a 2, b 0, c, d 6, e, z 0.5, z2 6 and 2 Type I 0D/30D Small (<0.%) NTP-4 0.2, a 0.2, b 0, c, d 0.5, e, z 0.5 Type I 0D/30D Large (>99.9%) NTP-5 0.2, a 0.75, b 0, c, d 0.5, e, z 0.5 Type I 0D/30D Large (>99.9%) NTP-6 0.2, a 2, b 0, c, d 6, e, z 0.5 Type I 0D/30D Large (>99.9%) NTP-7 0.2, a 0.2, b 0, c, d 0.5, e, z, z2 4 and 2 Type II 0D/30D Small (<0.%) NTP-8 0.2, a 0.75, b 0, c, d 0.5, e, z, z2 4 and 2 Type II 0D/30D Small (<0.%) NTP-9 0.2, a 2, b 0, c, d 6, e, z, z2 6 and 2 Type II 0D/30D Small (<0.%) NTP-0 0.2, a 0.2, b 0, c, d 0.5, e, z Type II 0D/30D Large (>99.9%) NTP- 0.2, a 0.75, b 0, c, d 0.5, e, z Type II 0D/30D Large (>99.9%) NTP-2 0.2, a 2, b 0, c, d 6, e, z Type II 0D/30D Large (>99.9%) NTP-3 0.2, a 0.2, b 0, c, d 0.5, e, z 2, z2 4 and 2 Type III 0D/30D Small (<0.%) NTP-4 0.2, a 0.75, b 0, c, d 0.5, e, z 2.5, z2 4 and 2 Type III 0D/30D Small (<0.%) NTP-5 0.2, a 2, b 0, c, d 6, e, z 4, z2 6 and 2 Type III 0D/30D Small (<0.%) NTP-6 0.2, a 0.2, b 0, c, d 0.5, e, z 2 Type III 0D/30D Large (>99.9%) NTP-7 0.2, a 0.75, b 0, c, d 0.5, e, z 2.5 Type III 0D/30D Large (>99.9%) NTP-8 0.2, a 2, b 0, c, d 6, e, z 4 Type III 0D/30D Large (>99.9%) Meanwhle, the decson space has been changed from [0,] n n to [0,5]. An addtonal constrant ( ) 2 ) has been added to adst the sze of the feasble regon. An addtonal parameter ( z ) has been added to the second obectve fncton to control the proporton of the contnos part of the Pareto front. f ) has been changed to f ) n the second obectve, wth the am of makng the nconstraned Pareto front a crve rather than a lne segment. The proposed test nstances can be descrbed as follows: Mnmze f) x f) Mnmze f2) g)( ) z g) sbect to ) cos( )( f2) e) sn( ) f) c d a sn( b (sn( )( f2) e) cos( ) f)) ) ) f ) 0.73 f ) z (7) where, a, b, c, d, and e are sed to control the topology of the Pareto front [20], z s sed to control the proporton of the contnos part of the Pareto front, and z 2 s sed to control the sze of the feasble regon. Based on the above formlaton, 8 test nstances have been desgned n ths paper to test the performance of the three representatve constrant-handlng technqes. The detals of these test nstances have been presented n Table I, and the obectve spaces of these test nstances have been presented n Fg., Fg. 2, and Fg. 3. A. Expermental Setp V. EXPERIMENTAL STUDY In ths secton, we choose the nondomnated sortng genetc algorthm II (NSGA-II) [7] as the mltobectve optmzaton framework, de to ts hgh robstness and ease of mplementaton. Then the three representatve constrant-handlng technqes (DP, SP, and ATM) ntrodced n Secton III are combned wth NSGA-II to solve the 8 test nstances proposed n Secton IV. For all the experments, the parameters were set as follows: the poplaton sze was 00, the crossover rate was 0.8, the mtaton rate was /n, and the mm nmber of generaton was 500. Besdes, the tornament selecton, smlated bnary crossover, and polynomal mtaton were adopted as the genetc operators n each smlaton. As presented n Table I, ths paper ntends to stdy how the followng three propertes of the 8 test nstances nflence the performance of a constrant-handlng technqe for MOPs. The shape of Pareto front The dmenson of decson vector The sze of feasble regon

5 Fg.. The obectve spaces of sx test nstances n type I, where the shade ndcates the feasble regon n the obectve space, the black crcle lne ndcates the nconstraned Pareto front, and the red dotted lne ndcates the constraned Pareto front. Fg. 2. The obectve spaces of sx test nstances n type II, where the shade ndcates the feasble regon n the obectve space, the black crcle lne ndcates the nconstraned Pareto front, and the red dotted lne ndcates the constraned Pareto front. Fg. 3. The obectve spaces of sx test nstances n type III, where the shade ndcates the feasble regon n the obectve space, the black crcle lne ndcates the nconstraned Pareto front, and the red dotted lne ndcates the constraned Pareto front. In order to stdy the effect of the second property, we tested two scenaros: n=0 and n=30. For each scenaro, 00 ndependent rns were mplemented for each test nstance. Accordng to or observaton, the performance of an algorthm sgnfcantly degenerates wth the ncrease of the dmenson of the decson vector. More mportantly,

6 TABLE II Test nstances nclded n each case: 0D-S ncldes the test nstances wth n=0 and small feasble regons, 30D-S ncldes the test nstances wth n=30 and small feasble regons, 0D-L ncldes the test nstances wth n=0 and large feasble regons, and 30D-L ncldes the test nstances wth n=30 and large feasble regons. Type Type I Type II Type III ase 0D-S 30D-S 0D-L 30D-L 0D-S 30D-S 0D-L 30D-L 0D-S 30D-S 0D-L 30D-L Test nstance NTP- NTP-4 NTP-7 NTP-0 NTP-3 NTP-6 NTP-2 NTP-5 NTP-8 NTP- NTP-4 NTP-7 NTP-3 NTP-6 NTP-9 NTP-2 NTP-5 NTP-8 Fg. 4. The expermental reslts of the for cases n type I Fg. 5. The expermental reslts of the for cases n type II Fg. 6. The expermental reslts of the for cases n type III sometmes an algorthm cannot fnd any feasble solton on the nne test nstances wth a small feasble regon when the evolton halts. The nverted generatonal dstance (IGD) [23] s a wdely sed ndcator to evalate the performance of an algorthm for mltobectve optmzaton. Note that f an algorthm cannot fnd any feasble solton, we cannot obtan the IGD vale. Ths, IGD mght not be stable to assess the performance of an algorthm for MOPs wth a small feasble regon. To ths end, we propose a revsed IGD ndcator, called IGD dstrbton ndcator (IGDD) n ths paper. The IGDD vale of an algorthm s compted as follows: Step ): alclate the IGD vale n each rn. As a reslt, we obtan 00 IGD vales n 00 rns. Step 2): Defne two ranges of the IGD vale: range (0, 0.25] and range2 (0.25, 0.5], compte the nmber (denoted as n ) of the IGD vale belongng to range, and compte the nmber (denoted as n 2 ) of the IGD vale belongng to range. 2 Note that n n2 00. Step 3): alclate the IGDD vale by the followng eqaton: IGDD= n ( ) n (8) 2 where s an coeffcent that s sed to adst the weght of n and n. Snce the IGD vale n 2 range s better than that n range, 2 n shold be pt more emphass. In ths paper, was set to 0.8. The IGDD ndcator can be sed to measre the convergence towards the Pareto front. The hgher the IGDD vale, the better the performance of an algorthm. B. omparatve Stdy For the sake of clarty, the comparatve stdy s based on the three dfferent types of shapes of the Pareto front ntrodced n Table I:

7 Fg fnal Pareto fronts obtaned by ATM, DP, and SP (from left to rght) on NTP-4 n type I, whch belongs to 30D-L. Fg fnal Pareto fronts obtaned by ATM, DP, and SP (from left to rght) on NTP- n type II, whch belongs to 0D-L. Fg fnal Pareto fronts obtaned by ATM, DP, and SP (from left to rght) on NTP-3 n type III, whch belongs to 30D-S. Type I denotes that the Pareto front s dscontnos (NTP--NTP-6). Type II denotes that the Pareto front conssts of both dscontnos part and contnos part (NTP-7- NTP-2). Type III denotes that the Pareto front s contnos (NTP-3-NTP-8). Based on the dmenson of the decson vector and the sze of the feasble regon, each type can be dvded nto for cases (0D-S, 30D-S, 0D-L, and 30D-L) as shown n Table II. The IGDD vales n the same case of each type are added together to represent the overall performance of a constrant-handlng technqe (see Fgs. 4-6). For example, the expermental reslt of ATM n 0D-S of Fg. 4 denotes the sm of the IGDD vales on NTP- wth 0D, NTP-2 wth 0D, and NTP-3 wth 0D. From the expermental reslts of Fgs. 4-6, we can gve the followng comments: Type I: As shown n Fg. 4, DP and SP perform smlarly n all the for cases. For the test nstances wth 0D, DP and SP performs mch better than ATM, regardless of the sze of feasble regon. However, ATM otperforms DP and SP on the test nstances wth 30D. Fg. 7 frther llstrates ths phenomenon. Ths fgre shows that the soltons obtaned by ATM are more close to the Pareto front and have a better dstrbton. ompared wth ATM, some soltons fond by DP are far away from the Pareto front, and the soltons obtaned by SP are not well dstrbted. Type II: Smlar to type I, n type II DP and SP srpass ATM on the test nstances wth low dmenson (0D). Fg. 8 provdes the performance comparson of ATM, DP, and SP on NTP- wth 0D. However, ATM and SP are slghtly better than DP on the test nstances wth 30D and small feasble regons. Moreover, ATM ranks the frst on the test nstances wth 30D and large feasble regons. Type III: In the case of 0D-S, DP s better than SP and ATM. In terms of 30D-S, SP ranks the frst, followed by ATM, whch s frther llstrated n Fg. 9. Wth regard to 0D-L and 30D-L, the three

8 constrant-handlng technqes show comparable performance. Overall, the ncrease of the nmber of decson varables poses a grand challenge to the three constrant-handlng technqes. In comparson wth the IGDD vales of the 0D test nstances, the IGDD vales of the 30D test nstances decrease sgnfcantly. VI. ONLUSION AND FUTURE WORK Accordng to the prelmnary expermental comparson n Secton V, the followng conclsons can be made accordng to the IGDD vales of the three constrant-handlng technqes: DP and SP show comparable performance on all test nstances, and they otperform ATM on test nstances wth 0D. ATM exhbts compettve performance on test nstances wth 30D. It s necessary to note that the above conclsons are obtaned by experments. In the ftre, we wll frther analyze the advantages and dsadvantages of the constrant-handlng technqes on dfferent knds of MOPs n prncple. On the other hand, we wll desgn new constrant-handlng technqes to tackle the 8 test nstances developed n ths paper. The Matlab sorce code can be downloaded from Y. Wang s homepage: AKNOWLEDGMENTS Ths work was spported n part by the Innovaton-drven Plan n entral Soth Unversty (No. 205XS02 and No. 205X007), n part by the Natonal Natral Scence Fondaton of hna nder Grant , n part by the EU Horzon 2020 Mare Sklodowska-re Indvdal Fellowshps (Proect ID: 66327), n part by the Engneerng and Physcal Scences Research oncl of UK nder Grant EP/K0030/, n part by the Hnan Provncal Natral Scence Fnd for Dstngshed Yong Scholars (Grant No. 206JJ08), and n part by the Program for New entry Excellent Talents n Unversty nder Grant NET REFERENES [] T. Bäck, D. B. Fogel, and Z. Mchalewcz, Handbook of Evoltonary omptaton. IOP Pblshng Ltd., 997. [2] K. Deb, Mlt-obectve Optmzaton Usng Evoltonary Algorthms. John Wley & Sons, 200. [3]. A. oello oello, D. A. Van Veldhzen, and G. B. Lamont, Evoltonary Algorthms for Solvng Mlt-Obectve Problems. New York: Klwer Academc, [4] K. 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