An Improved Clonal Algorithm in Multiobjective Optimization

Transcription

1 976 JOURNAL OF SOFTWARE, VOL. 4, NO. 9, NOVEMBER 2009 An Improved Clonal Algorthm n Multobjectve Optmzaton Janyong Chen, Quzhen Ln and Qngbn Hu Department of Computer Scence and Technology, Shenzhen Unversty, Shenzhen, P.R. Chna Emal: jychen@szu.edu.cn, quzheng658@163.com, huqb@szu.edu.cn Abstract In ths paper, we develop a novel clonal algorthm for multobjectve optmzaton (NCMO) whch s mproved from three approaches,.e., dynamc mutaton probablty, dynamc smulated bnary crossover (D-SBX) operator and hybrd mutaton operator combnng wth Gaussan and polynomal mutatons (GP-HM operator). Among them, the GP-HM operator s controlled by the dynamc mutaton probablty. These approaches adopt a coolng schedule, reducng the parameters gradually to a mnmal threshold. By ths means, they can enhance exploratory capabltes, and keep a desrable balance between global search and local search, so as to accelerate the convergence speed to the true Pareto-optmal front. When comparng NCMO wth varous state-of-the-art multobjectve optmzaton algorthms developed recently, smulaton results show that NCMO evdently has better performance. Index Terms multobjectve optmzaton; mmune algorthm; clonal selecton; hybrd mutaton I. INTRODUCTION In some real world problems, there exst many multobjectve optmzaton problems (MOPs) n practcal engneerng and scentfc applcatons. Wthout any further nformaton about the relatonshp among the objectves, t s possble to obtan a set of optmal solutons n whch each soluton s equally preferable when regardng all crtera consdered. As a result, n order to provde multple canddate selectons for dfferent applcatons, we am at as many representatve optmal solutons as possble. Evolutonary algorthms have the ablty to process sets of solutons n parallel and explore bg search spaces n reasonable tme. Therefore, Evolutonary algorthms have been recognzed to be well suted to MOPs. The ablty to handle complex problems, nvolvng features such as dscontnutes, multmodalty, dsjont feasble spaces and nosy functon evaluatons, renforces the potental effectveness of evolutonary algorthms n MOPs [1]. In the last few years, numerous competent Evolutonary algorthms have been proposed as the state-of-the-art Manuscrpt receved January 1, The work was supported by the Natonal Natural Scence Foundaton of Chna (Grant No: ). Janyong Chen, Quzhen Ln and Qngbn Hu are wth the Department of Computer Scence and Technology, Shenzhen Unversty, Shenzhen, P.R. Chna (e-mal: jychen@szu.edu.cn; quzheng658@163.com; huqb@sz u.edu.cn) algorthms for MOPs. For example, NSGA-II [2] was proposed wth a fast nondomnated sortng technology, eltsm and crowdng-dstance assgnment. SPEA-II [3] was proposed wth a fne-graned ftness assgnment strategy, an enhanced archve truncaton method and a new densty estmaton technque. The pareto archved evoluton strategy (PAES) [4] was proposed wth smple (1+1) evoluton strategy. All of them tred to desgn effectve and effcent technologes to mprove the abltes of the convergence and the dversty. On the other hand, Artfcal Immune Systems (AIS) have been developed snce 1990s as a new branch n computatonal ntellgence, whch smulate the defense of the human mmune system aganst bactera, vruses and other nvaders [5]. A number of AIS models have been found applcatons n varous felds such as machnelearnng and pattern-recognton tasks [6], network securty [7], schedulng [8], data mnng [9] and others [10, 11]. In recent years, AIS are also appled n MOPs and studes show ther remarkable performances. For example, Coello Coello and Cortes [12] presented a multobjectve mmune system algorthm (MISA) based on the clonal selecton prncple. Fresch and Repetto [13] proposed a vector artfcal mmune system (VAIS) based on the artfcal mmune network. Gong and Jao et al. [14] proposed a nondomnated neghbor-based mmune algorthm (NNIA), by usng a novel nondomnated neghbor-based selecton technque, proportonal clonng, heurstc search operators and eltsm. For MOPs, one of the key ponts s to fnd a unformly dstrbuted approxmaton set that s as close as possble to the Pareto-optmal front. Based on the noton that the preservaton of representatve solutons can effectvely drve the algorthm toward the Pareto-optmal front [15], most of current studes pay much attenton to ftness assgnment and populaton mantenance such as crowdng-dstance [2], grd mechansm [16] and clusterng technology [17]. However, rare studes pay attenton to search operators, whch can obvously accelerate the convergence speed. In ths paper, we propose a novel clonal algorthm for MOPs (NCMO) based on the mprovement of search operators, amng to speed up the convergence. Three novel approaches,.e., dynamc mutaton probablty, dynamc smulated bnary crossover (D-SBX) operator and hybrd mutaton operator combnng wth Gaussan and polynomal mutatons (GP-HM operator), are proposed n ths paper. Smlar to the temperature parameter n smulated annealng, these approaches also do: /jsw

2 JOURNAL OF SOFTWARE, VOL. 4, NO. 9, NOVEMBER adopt a coolng schedule [17], reducng the parameters gradually to a mnmal threshold. When NCMO s compared wth varous multobjectve optmzaton algorthms developed recently, smulaton results llustrate that NCMO has evdently remarkable performance. The remander of the paper s organzed as follows. Secton 2 brefly descrbes the defnton of MOPs and some mportant terms used n MOPs. In secton 3, we descrbe NCMO n detals and ntroduce some mportant operators. Besdes that, complexty analyss s also gven n ths secton. Secton 4 shows the smulatons of NCMO comparng wth varous multobjectve optmzaton algorthms. Fnally, we present some conclusons. II. MULTIOBJECTIVE OPTIMIZATION PROBLEMS In general, takng mnmzaton problems for example, the purpose of multobjectve optmzaton s to fnd a parameter set P that satsfes the followng equaton. Mn P ( ) ( 1( ), 2( ),, ( )) T Ω F x = f x f x L fm x (1) where Ω s set of the decson vector and m s the number of the functon objectves. For better understandng MOPs, the followng four concepts are mportant [18]: Pareto domnance: A vector x = ( x1, x2, L, x n ) s sad to domnate another vector x 1 = ( x , x 2, L, x n ) 0 1 (noted as x f x ) f and only f f x f x f x p f x m (2) : ( ) ( ) : ( ) ( ), [1, ] 2. Pareto-optmal: A soluton x 0 s sad to be Paretooptmal f and only f x Ω : x f x (3) 3. Pareto-optmal set: the set P ncludes all Paretooptmal solutons: = { Ω: f } (4) P x x x x 4. Pareto-optmal front: The set PF ncludes values of all objectve functons correspondng to the solutons n P: T PF = { f ( x) = ( f ( x), f ( x), L, f ( x)) x P} (5) 1 2 Because the Pareto-optmal set may be nfnte, so t s unpractcal to fnd out all solutons n Pareto-optmal set. In general, the am of multobjectve optmzaton algorthms s to fnd a set of some representatve solutons, whch are dstrbuted unformly and approxmate the Pareto-optmal front as close as possble. III. A NOVEL CLONAL ALGORITHM FOR MOPS It s well known that multobjectve optmzaton algorthms have two fundamental goals: one s to mnmze the dstance of the generated solutons to ft the Pareto-optmal set; the other s to maxmze the dversty of the archve Pareto-set approxmaton [19]. In ths study, n order to accelerate the convergence speed, the proposed algorthm uses three novel approaches,.e., dynamc mutaton probablty, D-SBX operator and GP- HM operator. Besdes that, the man operators of NCMO also nclude proportonal clonng operator and populaton selecton operator. Moreover, eltsm mechansm s used here and an archve s used to preserve nondomnated solutons n order to prevent the loss of eltsts. Frstly, we gve the correspondng pseudo-code of NCMO n Fg.1. In ths fgure, our novel approaches are marked wth boldface. m Fgure 1. the pseudo-code of NCMO A. Proportonal Clonng Operator In bologcal mmune system, clonng means that a group of dentcal cells s generated from a sngle common ancestor and only antbodes wth hgh affnty wll be cloned to attack the pathogens. In NCMO, only part of antbodes wth greater crowdng-dstance values s selected to do proportonal clonng and the number of clones s generated based on ther crowdng-dstance values. By ths means, nondomnated antbodes wth greater crowdng-dstance values wll have more clones. Therefore, the less-crowded regons possess more clones, whch encourages explorng the complete Pareto-optmal front. Assumng that A = { a 1, a 2, L, a n } s the populaton that s gong to do clonng. The proportonal clonng operaton s defned as follows: T ( A ) = [ T ( a ), T ( a ), L, T ( a )] T (6) C C 1 C 2 C n

3 978 JOURNAL OF SOFTWARE, VOL. 4, NO. 9, NOVEMBER 2009 where T ( a ) = q a ( = 1,2, L, n). The value of C q ( = 1,2, L, n) s the number of clones for each antbody a ( = 1,2, L, n), wth as follows: q = n C A ζ ( a, A) ζ ( a, ) j 1 j A = where n C s the gven value of the clone populaton sze and ζ ( a, A) s the crowdng-dstance value of the antbody a ( = 1,2, L, n) wth as follows: (7) where m ζ j( a, A) ζ ( a, A) = (8) f max f mn j = 1 j j f max and f mn s the maxmum and the j j mnmum value of the j-th objectve respectvely and the defnton of ζ ( a, A) can be found n (9). j Noted that when calculatng the number of clones for each antbody, t s possble that the crowdng-dstance value s when the antbody s n boundary. In ths case, t s set as double maxmum crowdng-dstance value except the boundary solutons., f( f j( a) == mn{ f j( ak)}) or ( f j( a) == max{ f j( ak)}), k = 1,2... n; ζ j( a, A) = mn{ f j( ak) f j( al)},( k, l = 1, 2... n) and ( k l ), otherwse; (9) B. D-SBX Operator The recombnaton operator has the ablty to escape from local optmal, and share gene segments from parent chromosomes. Smulated bnary crossover (SBX) [20] s one of man recombnaton operators that have been used n varous real-coded multobjectve algorthms [2,3,14]. However, the SBX operator s always used wth fx probablty (usually set as 0.5) of the varables to get crossed. It may destroy good gene segments at the end of algorthm runnng because too many genes get crossed at ths stage. Therefore, n NCMO, dynamc SBX (D-SBX) s proposed as an mproved recombnaton operator. Because the selecton operator selects antbodes accordng to ther Pareto domnance and dversty estmaton measured by crowdng-dstance value. As a result, f t s executed wth more generatons, the antbodes are becomng closer to the Pareto-optmal front. At the begnnng of the algorthm executon, when the generated antbodes are far away from the Paretooptmal front, the dynamc probablty s set wth relatve large value n order to share good gene segments. By ths means, t s easy to generate better antbodes n hgh probablty. Wth the algorthm runnng, the antbodes are becomng closer to the Pareto-optmal front, so the dynamc probablty s becomng small gradually to a mnmal threshold. Otherwse, the good genes segments may be destroyed and the antbodes can t explore closer to the Pareto-optmal front. Ordnarly, SBX has three controllable parameters [15]: (1) pc: the probablty for a par of parent solutons to do recombnaton; (2) η : the magntude of the expected varaton from the parent values; (3) pv: the probablty of the varables to get crossed. In D-SBX, the genes have dynamc probablty pv to do crossover accordng to the number of generatons. The dynamc probablty of genes pv s defned as: ( pvx - pvy) * gen pv = pvx - (10) maxgen where pvx and pvy s the predefned probablty of the varables to get crossed at the begnnng and at the end of the algorthm executon respectvely, gen s the number of teraton, maxgen s the predefned maxmum number of generatons. C. GP-HM Operator Polynomal mutaton has been used n many realcoded multobjectve optmzaton algorthms [2, 3, 14]. Ordnarly, Polynomal mutaton has two controllable parameters: mutaton probablty and dstrbuton ndex. However, wth the fxed dstrbuton parameter, t s not effcent at searchng the global Pareto-optmal front. For example, at large search space, the convergence speed of polynomal mutaton s very slow wth the relatve large dstrbuton parameter and t s easy to stagnate f many local Pareto-optmal fronts exst. In ths paper, we propose GP-HM operator, whch combnes wth Gaussan and polynomal mutatons. For antbody X = ( x1, x2, L, x n ), the GP-HM operator s defned as: x = x + delta ( yu - yd ), = 1,2, L, n (11) where x and x s the -th decson varables after and before mutaton respectvely, yu s the upper bound of the -th decson varables, yd s the lower bound of the - th decson varables. If polynomal mutaton s selected, delta s a small varaton whch s obtaned from polynomal dstrbuton by usng: where: delta 1 [ 2r ] 1 δ η + + 1, f r < 0.5 = 1 [ r ] η 1 + δ + f r 1 2(1 ), 0.5 max( yu x, x yd) δ = (1 2 r ) * ( ) yu yd η + 1 (12) (13) r s an unformly sample random number between (0,1) and η s the mutaton dstrbuton ndex. The plot of delta obtaned from polynomal dstrbuton (assumng η =20, x = ( yu yd)/2) can be seen n left part of Fg 2. It s noted that when x s equal wth yu or yd, x wll

4 JOURNAL OF SOFTWARE, VOL. 4, NO. 9, NOVEMBER be generated randomly between yu and yd. Otherwse, Gaussan mutaton s selected and delta s a small varaton obtaned from Gaussan dstrbuton by usng: delta = 0.1 N(0,1) (14) where N(0,1) s a random Gaussan number wth mean zero and standard devaton one. The plot of delta obtaned from Gaussan dstrbutons can be seen n rght part of Fg 2. Fgure 2. Plot of delta obtaned from the polynomal and Gaussan dstrbutons respectvely. Seen from the Fg 2, Gaussan mutaton has hgher probablty to generate an offsprng further away from ts parent than polynomal mutaton due to ts long flat tals and low hll. In other words, t has a hgher probablty to escape from a local optmum. On the other hand, t also ndcates that polynomal mutaton has stronger fnegraned search ablty than Gaussan mutaton n small regons. In GP-HM, we combne the advantages of two mutatons by a swtchng parameter. The swtchng parameter s used as a threshold value to control the swtchng of two mutatons. When a randomly generated number s smaller than the swtchng parameter, the Gaussan mutaton s deployed. Otherwse, the polynomal mutaton s deployed. The swtchng parameter sp s proposed as: ( spx - spy) * gen dsp = spx - (15) maxgen where spx and spy s the predefned probablty to do Gaussan mutaton at the begnnng and at the end of the algorthm executon respectvely. In general, there s hgh probablty to do Gaussan mutaton at the begnnng of algorthm, whch can make the proposed algorthm converge fast toward the Pareto-optmal front. When nondomnated solutons are approachng the Pareto-optmal front, the probablty wth Gaussan mutaton gradually decreases whle the probablty wth polynomal mutaton gradually ncreases. By ths means, the algorthm gradually performs more fne-graned search. It s an effectve and hgh-speed search operator, whch well keeps the balance of global search and local search. It s noted that the mutaton probablty should be changed dynamcally accordng to the adaptve ablty of the mmune system. In ths study, the mutaton probablty s only changed accordng to the number of generatons n the frst half part of algorthm runnng as follows: gen pm = (1 + p) * mnpm - 2 p * mnpm* ( ) (16) maxgen where mnpm s the predefned mnmal mutaton probablty that guarantees every antbody wth one gene on average to do mutaton and p s a predefned parameter that adjusts the mutaton scale. Whle n the last half part of algorthm runnng, pm s fxed as mnpm. D. Populaton Selecton And Archve Update Reference [21] has shown that eltsm can speed up the performance of GA sgnfcantly and eltsm scheme has been well adopted by many state-of-the-art EAs [2, 3, 14]. The eltsm mechansm s also deployed here. Nondomnated antbodes are preserved n an archve, whch s helpful to prevent the loss of good solutons once they are found. When the number of nondomnated antbodes exceeds the archve sze, a nondomnated neghbor-based selecton mechansm developed by Gong [14] s used to evaluate the nondomnated antbodes. The selecton mechansm can evdently extend the dversty of populaton and manly focuses on lesscrowded regons. The procedures can be descrbed as follows. After the chld populaton s generated from proportonal clonng operator, mutated wth D-SBX and GP-HM, nondomnated antbodes are dentfed n matng pool that contans the archve and the chld populaton. Then, part of nondomnated antbodes wth greater crowdng-dstance values s selected by usng the nondomnated neghbor-based selecton mechansm as the next generaton populaton and the archve s updated wth the new populaton. E. Complexty Analyss The complexty analyss of NCMO s provded n ths secton. Assumng that the populaton sze, the clone populaton sze and the archve sze are N, the number of functon objectves s M. The basc operators and ther worst tme complexty are gven as follows: 1. Populaton ntalzaton: O(M N). 2. Calculate the ftness value: O(M 2N log(2n)). 3. Proportonal clonng operator: O(M N). 4. D-SBX operator and GP-HM operator: O(M N). 5. Procedure to dentfy the nondomnated ndvduals n the matng pool that contans the archve and the chld populaton: O(M (2N) 2 ). 6. New populaton selecton and archve update: O(2N log(2n)). Therefore, the worst total tme complexty s: O(M N 2 ) (17) Note that the worst tme complexty of NSGA-II, PAES, SPEA, SPEA2 and NNIA s O(M N 2 ), O(M N), O(M N 2 ), O(M N 3 ) and O(M N 2 ) respectvely.

5 980 JOURNAL OF SOFTWARE, VOL. 4, NO. 9, NOVEMBER 2009 IV. EXPERIMENTS In ths secton, we make some smulatons to present the performance of NCMO. At frst, we gve the related knowledge of the smulatons such as benchmark functons and performance metrcs used n ths paper. Next, we gve the smulaton results of NCMO comparng wth varous multobjectve optmzaton algorthms. A. Benchmark Functons In ths study, we use ten benchmark functons to valuate the performance of NCMO. The functons nclude fve ZDT functons and fve three-objectve DTLZ functons wthout any nequalty and equalty constrants. Defntons of the ZDT and DTLZ functons can be found n Ref. [21] and Ref. [22] respectvely. It s noted that the test problems are characterzed wth convexty, dscontnuty and non-unformty and some of them have many local Pareto-optmal fronts. By ths means, they are sutable to test the comprehensve performance of multobjectve optmzaton algorthms. B. Performance metrcs As we menton above, MOPs have two goals. In order to compare the performance of NCMO wth varous multobjectve optmzaton algorthms, we adopt the convergence metrc and the dversty metrc suggested by Ref. [2], and the spacng metrc suggested by Ref. [23]. These metrcs are desgned based on the goals. The frst metrc measures the extent of convergence to a known subset of the Pareto-optmal front. The last two metrcs measure the extent of dversty of an approxmaton set. C. Comparson of NCMO wth varous algorthms In the follow sectons, we compare the performance of NCMO wth varous multobjectve optmzaton algorthms,.e., NSGA-II (real-coded), PAES, SPEA2, MOEO [19] and NNIA. The smulaton results are dscussed respectvely n the followng. The parameters settng for NCMO and NNIA are tabulated n Table I. TABLE I. PARAMETERS SETTING parameters NCMO NNIA Crossover probablty pc Dstrbuton ndex for SBX Mutaton probablty pm / 1/n Dstrbuton ndex for polynomal mutaton Populaton sze Clone populaton sze Populaton(selected to clone) sze Besdes that, the values of pvx and pvy n (10) are set as 0.5 and 0.25 respectvely. The values of spx and spy n (15) are set as 0.1 and 0.02 respectvely. The mnmal mutaton probablty mnpm and parameter p n (16) s set as 1/n (n s the number of decson varables) and 0.2 respectvely. These values of parameters have been determned after an ntensve prelmnary test phase of the algorthm on dfferent benchmark functons. The parameters settng of NSGA-II, PAES, SPEA2, and MOEO are found n Ref. [2] and Ref. [19]. The maxmum of generatons s 250. Wth these parameters settng, NCMO has the same smulaton condtons wth the other algorthms. Table II shows the mean and varance of the convergence metrc and the dversty metrc obtaned by usng NCMO, NNIA MOEO, NSGA-II, PAES and SPEA2 n solvng fve ZDT functons. Table III shows the mean and varance of the convergence metrc and the spacng metrc obtaned by usng NCMO and NNIA n solvng fve DTLZ functons. In ths study, all the expermental results of MOEO, NSGA-II, PAES and SPEA-II come from Ref. [1] and Ref. [19], whereas the expermental results of NCMO and NNIA come from our smulatons. The source code of NNIA can be found n author s web ste [24]. All the smulaton results can be found n Table II and Table III. It s noted that the best result for each benchmark functon s marked wth boldface. TABLE II. MEAN (FIRST ROWS) AND VARIANCE (SECOND ROWS) OF THE CONVERGENCE METRIC AND DIVERSITY METRIC Algor thm NCM O Convergence Dversty ZDT1 ZDT2 ZDT3 ZDT4 ZDT6 ZDT1 ZDT2 ZDT3 ZDT4 ZDT NNIA MOE O NSGA -II E PAES SPEA E TABLE III. MEAN (FIRST ROWS) AND VARIANCE (SECOND ROWS) OF THE CONVERGENCE METRIC AND SPACING METRIC Algorthm Convergence Spacng DTLZ1 DTLZ2 DTLZ3 DTLZ4 DTLZ6 DTLZ1 DTLZ2 DTLZ3 DTLZ4 DTLZ6

6 JOURNAL OF SOFTWARE, VOL. 4, NO. 9, NOVEMBER NCMO NNIA ) Comparson wth some state-of-the-art EAs It can be observed from Table II that when regardng convergence metrc and dversty metrc, NCMO s much better than the state-of-the-art EAs.e., NSGA-II, PAES and SPEA2 n all fve ZDT functons. It apparently llustrates the effectveness of NCMO. In addton, n all cases wth NCMO, NCMO s the smallest at the varance of the convergence metrc and the dversty metrc n many test functons n ten runs except PAES. These smulatons also llustrate the robustness of NCMO. 2) Comparson wth MOEO When regardng the dversty metrc, smulatons from Table II show that MOEO performs better than NCMO except n ZDT3. Whle regardng the convergence metrcs, NCMO s better than MOEO except n ZDT6. In addton, NCMO s smaller than MOEO regardng the varance of the convergence metrc and the dversty metrc n many test functons n ten runs. It s very dffcult for the proposed algorthm to obtan better performance n every metrc. It fts the sayng that most currently best multobjectve optmzaton evolutonary algorthms do not outperform each other, but perform smlarly or are preferable than respect to dfferent performance ndcators [15]. Based on the above dscusson, t s concluded that NCMO and MOEO have ther own advantages. NCMO has faster convergence speed and more robust than MOEO, whle MOEO gets more unformly dstrbuted solutons than NCMO. 3) Comparson wth NNIA Observed from Table II and Table III, NCMO s better than NNIA n four ZDT functons and four DTLZ functons when regardng the convergence metrc. Especally n DTLZ3, NCMO performs much better than NNIA. Because DTLZ3 have x (3 M 1) local Paretooptmal fronts, t s very dffcult to be optmzed by many state-of-the-art multobjectve optmzaton algorthms. It s noted that DTLZ1 also has many local Pareto-optmal fronts, but NCMO performs a ltter worse than NNIA. When regardng the dversty of solutons, although both algorthms use the same selecton mechansm, NCMO gets better results than NNIA n fve ZDT functons and three DTLZ functons. These smulaton results evdently show the advantages of our novel approaches n ths paper. From the above comparson, we tabulate the performance results of NCMO comparng wth varous multobjectve algorthms n Table IV. TABLE IV. THE PERFORMANCE RESULTS OF COMPARISON BETWEEN NCMO WITH VARIOUS MULTIOBJECTIVE ALGORITHMS Convergence metrc Dversty metrc Spacng metrc NCMO/NSGA-II Better Better \ NCMO/PAES Better Better \ NCMO/SPEA Better Better \ NCMO/SPEA-2 Better Better \ NCMO/MOEO Better Worse \ NCMO/NNIA Better Better Better In the above table, Better means that the performance of NCMO s generally better than other algorthms n the correspondng metrc. \ ndcates that there s no comparson between NCMO and other algorthms. Worse means that the performance of NCMO s generally worse than other algorthms n the correspondng metrc. In concluson, t s evdent that NCMO has two man advantages wth the three mprovement approaches. Frstly, NCMO can converge fast to the Pareto-optmal front and has the ablty to escape from local Pareto-optmal fronts. Therefore, NCMO s robust when searchng the global Paretooptmal front n bg search space wth many local Paretooptmal fronts. Secondly, NCMO s capable of keepng the desrable dversty of solutons. To graphcally show the performance of NCMO, we show the mages of the smulatons obtaned by NCMO n solvng all benchmark functons respectvely n Fg. 3. It s noted that the black lne s the Pareto-optmal front and the damond symbol s the correspondng front of the approxmaton set obtaned by usng NCMO. From the smulaton mages, t s easy to get the result that NCMO can get the approxmaton set that s dstrbuted unformly and close to the Pareto-optmal front. Zdt1 Zdt2 Zdt3 Zdt4 Zdt6

7 982 JOURNAL OF SOFTWARE, VOL. 4, NO. 9, NOVEMBER 2009 Dtlz1 Dtlz2 Dtlz3 Dtlz4 Dtlz6 Fgure 3. The mages of the smulatons obtaned by NCMO n solvng all benchmark functons respectvely. 4) Smulatons for the novel approaches In order to further nvestgate the performance of the three mprovement approaches n proposed algorthm, we make 30 ndependent runs of algorthm only wth one of three approaches,.e., dynamc mutaton probablty (SBX operator and polynomal mutaton operator), D- SBX operator (statc mutaton probablty and polynomal mutaton operator) or GP-HM operator (statc mutaton probablty and SBX operator) TABLE V. THE AVERAGE VALUE OF CONVERGENCE METRIC OBTAINED BY ALGORITHM WITH ONE OF THREE NOVEL APPROACHES AND NCMO WITH COMBINED APPROACHES Dynamc mutaton probablty D-SBX GP-HM Combned (NCMO) ZDT E E E E-06 ZDT E-05(9) E-05(5) E E-05 ZDT e E e E-05 ZDT (9) (11) ZDT e e e e-04 DTLZ DTLZ DTLZ DTLZ (1) (4) (2) DTLZ Observed from Table V, as far as the convergence metrc s concerned, the algorthm performs best n ZDT3 f only the dynamc mutaton probablty s used. If only the D-SBX operator s used, the algorthm performs best n ZDT 4 and DTLZ6. If only the GP-HM operator s used, the algorthm performs best n ZDT1 and ZDT 6. When all novel approaches are used, the algorthm performs best n ZDT2, DTLZ1, DTLZ2, DTLZ3, DTLZ6 and DTLZ4. Observed from Table VI, as far as the spacng metrc s concerned, the algorthm performs best n DTLZ6 f only the dynamc mutaton probablty s used. If only D- SBX s used, the algorthm performs best n ZDT1, ZDT2 and ZDT3. If the GP-HM s used, the algorthm performs best n DTLZ2 and DTLZ4. If all novel approaches are used, the algorthm performs best n ZDT3, ZDT4, ZDT6, DTLZ1 and DTLZ3. From the above results, t s concluded that the algorthm wth one of three novel approaches has ts own advantage n dfferent test functons, but NCMO wth all novel approaches performs best n most cases, no matter consderng the convergence metrc or the spacng metrc. It llustrates that algorthm wth D-SBX or GP- HM s sutable to solve the test functons wth many local Pareto-optmal fronts. When solvng ZDT3 wth dscreteness features, algorthm wth dynamc mutaton probablty performs best. It s ratonal that whether these approaches can work well or not s relatvely decded by the features of the test functons. Moreover, n solvng ZDT2, ZDT4 and DTLZ4, algorthm wth one of three novel approaches may only converge to sngle Paretooptmal occasonally, whle NCMO can get multple respectvely. The parameters settng are the same wth the above settng except that the crossover probablty s set as 1.0. The results are shown n Table V and Table VI below. For ncreasng the precson of the convergence value, we fnd a set of more than unformly spaced solutons from the Pareto-optmal front to calculate the convergence metrc for every test functons. The number n brackets means tmes the algorthm converges to sngle approxmate Pareto-optmal. TABLE VI. THE AVERAGE VALUE OF SPACING METRIC OBTAINED BY ALGORITHM WITH ONE OF THREE NOVEL APPROACHES AND NCMO WITH COMBINED APPROACHES. Dynamc mutaton probablty D-SBX GP-HM Combned (NCMO) ZDT ZDT (9) (5) ZDT ZDT (9) (11) ZDT DTLZ DTLZ DTLZ DTLZ (1) (4) (2) DTLZ solutons every tmes. It means that these three novel approaches can be well ncorporated to preserve the populaton dversty. Only use one mproved approach may perform better than NCMO when the optmzed functons have some specal features, but n real lfe problems, t s dffcult to know the detals of the optmzed functon. Therefore, t s hard to choose the mproved approaches. However, f NCMO s used to solve the optmzed functons, t can get better performance n general. IV. CONCLUSIONS In ths paper, we proposed a novel clonal algorthm for MOPs, whch uses three novel approaches wth dynamc mutaton probablty, D-SBX operator and GP- HM operator. The noton of these approaches s straghtforward and easy to be mplemented. When comparng wth varous state-of-the-art algorthms and recently proposed algorthms, smulatons show that NCMO wth these approaches s robust. It can not only evdently accelerate the convergence speed, but also keep the desrable dversty. REFERENCES [1] C.M. Fonseca and P.J. Flemng, An overvew of evolutonary algorthms n multobjectve optmzaton, Evolutonary Computaton, vol. 3, pp. 1-16, [2] Kalyanmoy Deb, Amrt Pratap, Sameer Agarwal and T. Meyarvan, A fast and eltst mult-objectve genetc algorthm: NSGA-II, IEEE Transactons on Evolutonary Computaton, vol. 6, no. 2, pp , 2002.

8 JOURNAL OF SOFTWARE, VOL. 4, NO. 9, NOVEMBER [3] Eckart Ztzler, Marco Laumanns and Lothar Thele, SPEA2: Improvng the strength pareto evolutonary algorthm, Techncal Report 103, Computer Engneerng and Networks Laboratory (TIK), Swss Federal Insttute of Technology (ETH), Zurch, Swtzerland, [4] Joshua Knowles and Davd Corne, Approxmatng the non-domnated front usng the pareto archved evoluton strategy, Evolutonary Computaton, vol. 8, No. 2, pp , [5] Dpankar Dasgupta, Advances n Artfcal Immune Systems, IEEE Computatonal Intellgence Magazne, vol. 1, pp , [6] L.N. De Castro and F.J. Von Zuben, Learnng and Optmzaton Usng the Clonal Selecton, IEEE Transactons on Evolutonary Computaton, vol. 6, pp , [7] P.K. Harmer, P.D. Wllams, G.H. Gunsch and G.B. Lamont, An Artfcal Immune System Archtecture for Computer Securty Applcatons, IEEE Transactons on Evolutonary Computaton, vol. 6, no. 3, pp , [8] Hong-We Ge, Lang Sun, Yan-Chun Lang and Feng Qan, An Effectve PSO and AIS-Based Hybrd Intellgent Algorthm for Job-Shop Schedulng Systems, IEEE Transactons on Man and Cybernetcs, vol. 38, no. 2, pp , [9] A.A. Fretas and J. Tmms, Revstng the Foundatons of Artfcal Immune Systems for Data Mnng, IEEE Transactons on Evolutonary Computaton, vol. 11, no. 4, pp , [10] Canova, A., Fresch, F. and Tartagla, M., Multobjectve Optmzaton of Parallel Cable Layout, IEEE Transactons on Magnetcs, vol. 43, no. 10, pp , [11] A.M. Whtbrook, U. Ackeln and J.M. Garbald, Idotypc Immune Networks n Moble-Robot Control, IEEE Transactons on Systems, Man, and Cybernetcs, vol. 37, no. 6, pp , [12] C.A. Coello Coello and N.C. Cortes, Solvng multobjectve optmzaton problems usng an artfcal mmune system, Genetc Programmng and Evolvable Machnes, vol. 6, pp , [13] F. Fresch and M. Repetto, Mult-objectve optmzaton by a modfed artfcal mmune system algorthm, In Proceedngs of the 4 th Internatonal Conference on artfcal mmune systems, 3627 of Lecture Notes n Computer Scence, pp , [14] Maoguo Gong, Lcheng Jao, Hafeng Du and Lefen Bo, Mult-objectve Immune Algorthm wth Nondomnated Neghbor-based Selecton, Evolutonary Computaton (MIT Press), vol. 16, pp , [15] R.C. Purshouse and P.J. Flemng, On the evolutonary optmzaton of many conflctng objectves, Transactons on Evolutonary Computaton, vol. 11, no. 6, pp , [16] D.W. Corne, J.D. Knowles and M.J. Oates, The paretoenvelope based selecton algorthm for mult-objectve optmzaton, In proceedngs of 6-th nternatonal conference Parallel Problem Solvng from Nature (PPSN VI), pp , [17] S. Bandyopadhyay, S. Saha, U. Maulk and K. Deb, A Smulated Annealng-Based Multobjectve Optmzaton Algorthm: AMOSA, IEEE Transactons on Evolutonary Computaton, vol. 12, no. 3, pp , [18] P.A.N. Bosman and D. Therens, The Balance Between Proxmty and Dversty n Multobjectve Evolutonary Algorthms, Transactons on Evolutonary Computaton, vol. 7, no. 2, pp , [19] Chen Mn-Rong and Lu Yong-Za, A novel eltst multobjectve optmzaton algorthm: Multobjectve extremal optmzaton, European Journal of Operatonal Research, Vol. 188, No. 3, pp , [20] Kalyanmoy Deb and Ram Bhusan Agrawal, Smulated bnary crossover for contnuous search space, Complex Systems, Vol. 9, No. 6, pp , [21] Eckart Ztzler, Kalyanmoy Deb and Lothar Thele, Comparson of multobjectve evolutonary algorthms: Emprcal results, Evoluton Computaton, Vol. 8, No. 2, pp , [22] K. Deb, L. Thele, M. Laumanns and E. Ztzler, Scalable mult-objectve optmzaton test problems, Techncal Report 112, Computer Engneerng and Networks Laboratory (TIK), Swss Federal Insttute of Technology (ETH),Zurch, Swtzerland, [23] J.R. Schott, Fault tolerant desgn usng sngle and multcttera gentetc algorthm optmzaton, Master thess, Massachusetts Insttute of Technology, [24] Janyong Chen s assocate professor n Department of Computer Scence and Technology, Shenzhen Unversty. He got hs PhD from Cty Unversty of Hong Kong, Hong Kong, Chna, n He s nterested n Artfcal Intellgence and Informaton Securty. He worked for ZTE Corporaton as senor engneer of network technology from 2003 to After that, he joned the Shenzhen Unversty as assocated professor. He s vce-charman of Internatonal Telecommuncaton Unon- Telecommuncaton (ITU-T) SG17 snce 2004 and edtor of three recommendatons developed n ITU-T SG17. He has publshed more than 10 papers and appled more than 30 patents n the feld of Artfcal Intellgence and Informaton Securty. Quzhen Ln s a graduate student n the college of Informaton Engneerng, Shenzhen Unversty. He got hs bachelor's degree n Zhaoqng Unversty. He s nterested n Artfcal Intellgence and network Securty.