A Multi-Objective Gravitational Search Algorithm Based on Non-Dominated Sorting

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1 32 Internatonal Journal of Swarm Intellgence Research, 3(3), 32-49, July-September 2012 A Mult-Obectve Gravtatonal Search Algorthm Based on Non-Domnated Sortng Had Nobahar, Sharf Unversty of Technology, Iran Mahd Nkusokhan, Sharf Unversty of Technology, Iran Patrck Sarry, Unversté Pars-Est Crétel (UPEC), France ABSTRACT Ths paper proposes an extenson of the Gravtatonal Search Algorthm (GSA) to mult-obectve optmzaton problems. The new algorthm, called Non-domnated Sortng GSA (NSGSA), utlzes the non-domnated sortng concept to update the gravtatonal acceleraton of the partcles. An external archve s also used to store the Pareto optmal solutons and to provde some eltsm. It also gudes the search toward the non-crowdng and the extreme regons of the Pareto front. A new crteron s proposed to update the external archve and two new mutaton operators are also proposed to promote the dversty wthn the swarm. Numercal results show that NSGSA can obtan comparable and even better performances as compared to the prevous multobectve varant of GSA and some other mult-obectve optmzaton algorthms. Keywords: Gravtatonal Search Algorthm, Mult-Obectve Optmzaton, Non-Domnated Sortng, Reorderng Mutaton, Sgn Mutaton INTRODUCTION Over the last decades, several meta-heurstcs have been developed to solve complex sngle and mult-obectve optmzaton problems. Many real-world problems nvolve smultaneous optmzaton of several competng obectves. In these problems, there s no sngle optmal soluton, but rather a set of non-domnated solutons, also called Paretooptmal solutons. The use of meta-heurstcs to solve mult-obectve optmzaton problems s DOI: /sr growng fast, because they can handle problems wth concave and dsconnected Pareto fronts. There are many meta-heurstcs such as Smulated Annealng (SA) (Krkpatrck et al., 1983), Tabu Search (TS) (Glover, 1989, 1990), Evolutonary Algorthms (EAs) (Tang et al., 1996), Ant Colony Optmzaton (ACO) (Dorgo et al., 1996), Partcle Swarm Optmzaton (PSO) (Kennedy & Eberhart, 1995), Gravtatonal Search Algorthm (GSA) (Rashed et al., 2009) and so on. These algorthms are able to solve dfferent optmzaton problems. However, there s no specfc algorthm able to fnd the best solutons of all problems n fnte Copyrght 2012, IGI Global. Copyng or dstrbutng n prnt or electronc forms wthout wrtten permsson of IGI Global s prohbted.

2 Internatonal Journal of Swarm Intellgence Research, 3(3), 32-49, July-September teratons and some algorthms show better performances for partcular problems than the others. Hence, searchng for new heurstc optmzaton algorthms s an open problem. Although both sngle and mult-agent metaheurstcs have ther own mult-obectve varants, the mult-agent meta-heurstcs such as EAs, ACO, and PSO have shown better potental to solve mult-obectve optmzaton problems, because n a mult-obectve problem, a populaton of solutons s gong to be generated, n a sngle run. The mult-obectve varants of the sngle agent meta-heurstcs such as SA and TS are often workng based on the defnton of an aggregaton functon n ther structure (Ulungu et al., 1999; Hansen, 1997). The use of aggregaton functons may reduce the search area and provde only a subset of the Pareto front. GSA s a new mult-agent optmzaton algorthm, nspred from the general gravtatonal law (Rashed et al., 2009). The algorthm s based on the movement of some partcles under the effect of the gravtatonal forces, appled by the others. Hassanzadeh and Rohan (2010) have proposed the frst mult-obectve varant of GSA, called Mult-Obectve GSA (MOGSA). MOGSA uses an external archve to store the non-domnated solutons and uses the same dea as n Smple Mult-Obectve PSO (SMOPSO), proposed by Cagnna et al. (2005), to update the external archve. For ths purpose, the nterval of each obectve functon s dvded nto equal dvsons and consequently the entre performance space s dvded nto hyper-rectangles. When the number of archved members exceeds the maxmum archve length, one member of the most crowded hyper-rectangle s randomly chosen and removed. The man problem n proposng a multobectve varant for GSA s updatng the mass of partcles based on the value of the multple obectves. In MOGSA, the mass of all movng partcles s set to one and the mass of archved partcles s updated based on the dstance from the nearest neghbor, wthn the obectve space. However, no equaton has been presented that relates the mass value to the dstance value. Ths technque dstrbutes the archved elements unformly, smlar to the nchng technque. After calculatng the mass of the movng partcles, they move to the new postons by the forces appled from the archved members. In MOGSA, only the archved partcles apply gravtatonal forces to the movng ones and after each movement, the well-known unform mutaton s appled to the new postons. In ths paper, the authors have proposed a new mult-obectve varant of GSA, called NSGSA. In sngle-obectve GSA, proposed by Rashed et al. (2009), the mass of each partcle s taken to be proportonal to ts ftness. As mentoned, the man problem to propose a multobectve GSA s to establsh a relatonshp between the mass of each partcle and ts multple obectves. In ths paper, the non-domnated sortng concept, proposed n Non-domnated Sortng GA (NSGA) (Srnvas & Deb, 1994), s used to dvde the partcles to several layers wthn the performance space. In ths way, the mass of each partcle wll depend to the rank of the layer t belongs to. NSGSA utlzes a lmted length external archve to store the last found non-domnated solutons. To provde some eltsm, specfc members of the external archve are added to the lst of movng partcles and contrbute n applyng gravtatonal forces to the others. A new crteron s ntroduced to update the external archve, based on a new defned spread ndcator. The mutaton (turbulence) phenomenon s also modeled n NSGSA, not consdered n the orgnal GSA. In ths regard, two new mutaton operators, called sgn and reorderng mutatons, are also proposed. The paper s organzed as follows: At frst, GSA s descrbed n detal. Thereafter, the new mult-obectve varant of GSA, called NSGSA, s ntroduced. Next, numercal results are presented. Then, the results of NSGSA are compared wth those of MOGSA, SMOPSO, and NSGA-II (Deb et al., 2002) and ther senstvty to the value of tuned parameters s also nvestgated. Fnally, the conclusons are outlned. Copyrght 2012, IGI Global. Copyng or dstrbutng n prnt or electronc forms wthout wrtten permsson of IGI Global s prohbted.

3 34 Internatonal Journal of Swarm Intellgence Research, 3(3), 32-49, July-September 2012 GRAVITATIONAL SEARCH ALGORITHM R ( t) = x ( t) x ( t) 2 (4) GSA s a new optmzaton algorthm, proposed by Rashed et al. (2009). Ths algorthm has been nspred from the mass nteractons based on the general gravtatonal law. In the proposed algorthm, the search agents are a collecton of masses, nteract wth each other based on the Newtonan laws on gravty and moton. Also, t uses some stochastc operators to ncrease dversty and the probablty of fndng the global optmum. In the followng, the formulaton of GSA s presented n short. The mass of each partcle s calculated accordng to ts ftness value, as follows: ft ( t) worst( t) m ( t) = = 1, 2,..., n best( t) worst( t) mass m ( t) M ( t) = 0 M ( t) < 1 n mass m ( t) = 1 (1) where t s tme (teraton), ft presents ftness value of the -th partcle, M s the normalzed mass, n mass s the number of partcles, and worst(t) and best(t) are defned for a mnmzaton problem as follows: nmass nmass worst( t) = max{ ft ( t)}, best( t) = mn{ ft ( t)} = 1 = 1 (2) The vector of force, appled by mass to mass, f ( t), s calculated as follows: f t G t M ( t ) M ( t ) ( ) = ( ) ( x x ) R ( t) + ε (3) where x s the poston vector of the -th agent, ε s a small threshold, and G(t)= G(G 0, t) s the gravtatonal coeffcent, ntalzed at the begnnng of the algorthm and s reduced wth tme to control the search accuracy, R (t) s the Eucldan dstance between two agents and, defned as follows: Usng the second Newton s law of moton, the gravtatonal acceleraton of the -th agent due to the -th one at tme t s gven as follows: M ( t) a ( t) = G( t) ( x x ) R ( t) + ε (5) Actually, the gravtatonal force between two partcles s nversely proportonal to the square of the relatve dstance (Hollday et al., 1993), and the Newtonan gravtatonal acceleraton can be stated as follows: M ( t) a ( t) = G( t) ( x x ) 3 R ( t) + ε (6) Rashed et al. (2009) used Equaton (5) nstead of Equaton (6), snce t provdes better results, as they have reported from ther experments. Therefore, the total gravtatonal acceleraton of the -th agent, s calculated as follows: n mass M ( t) a ( t) = G( t) rand x x ( ) = 1 R ( t) + ε (7) where rand s a unform random number wthn the nterval [0,1], consdered to add some stochastc behavor to the acceleraton. The velocty and poston of the agents are updated as follows: v ( t + 1) = rand v ( t) + a ( t) x ( t + 1) = x ( t) + v ( t + 1) (8) where rand s another unform random number n the nterval [0,1]. To prevent the partcles go out of the search space, the postons are bounded as follows: Copyrght 2012, IGI Global. Copyng or dstrbutng n prnt or electronc forms wthout wrtten permsson of IGI Global s prohbted.

4 Internatonal Journal of Swarm Intellgence Research, 3(3), 32-49, July-September x d d d d x x < x l l = d d d x x > x u u (9) R R,best < C (11) where d=1,, n shows the dmenson of the d d search space and x l and x u are lower and upper bounds of the search nterval n dmenson d, respectvely. To make a good compromse between exploraton and explotaton, Rashed et al. (2009) proposed to reduce the number of attractve agents n Equaton (7) over tme. Hence, only a set of best agents,.e., the agents wth bgger mass, apply gravtatonal forces to the others. However, ths polcy may reduce the exploraton power and ncrease the explotaton capablty. In order to avod beng trapped n local optma, the algorthm must perform more exploraton at the begnnng. Over tme (teraton), exploraton must be gradually faded out and explotaton must be faded n. Therefore, n GSA, only the k best (t) of the movng agents wll attract the others. k best (t) s a decreasng functon of tme, wth the ntal value k 0 =n mass. It means that at the begnnng, all agents apply gravtatonal forces, and as tme passes, k best (t) s decreased lnearly. At the end, there wll be ust one agent applyng force to the others. Therefore, Equaton (7) could be modfed as follows: M ( t) a ( t) = G( t) rand ( x x ) K ( t) R ( t) + ε best (10) where K best (t) s the set of best k best (t) agents, found so far. Sarafraz et al. (2011) proposed a new operator, called dsrupton to mprove the exploraton and explotaton of GSA. The authors have clamed that ths operator has been nspred from a natural phenomenon n astrophyscs. In ths method, all solutons except the best one, are dsrupted. The dsrupton s occurred, f the rato of the dstance between mass and ts nearest neghbor,, to ts dstance from the best soluton s smaller than a specfc threshold as follows: Where R and R,best are Eucldan dstances between solutons and and between soluton and the best soluton, respectvely, and C s a specfc threshold. The poston of every mass that satsfes the condton wll be changed as follows: x ( new) = x ( old) D R ( rand) R > 1, best D = 1 + ρ( rand) otherwse (12) Agan, rand s a unform random number wthn the nterval [0,1] and ρ s a small number. L et al. (2011) have proposed an mproved varant of GSA, called Improved GSA (IGSA). Ths algorthm s a combnaton of GSA and PSO. In IGSA, the velocty s updated as follows: v ( t + 1) = rand v ( t) + a ( t) 1 + c rand ( p x ) 1 best, 2 + c rand ( g x ) 2 best 3 (13) Where rand 1, rand 2 and rand 3 are unform random numbers wthn the nterval [0,1], c 1 and c 2 are constants, p best, s the best prevous poston of the th partcle and g best s the best prevous poston of all partcles. NON-DOMINATED SORTING GRAVITATIONAL SEARCH ALGORITHM In ths secton the new mult-obectve varant of GSA, called NSGSA, s ntroduced. The new algorthm utlzes the concept of Pareto optmalty condton and non-domnance. A hgh level descrpton of NSGSA s presented Copyrght 2012, IGI Global. Copyng or dstrbutng n prnt or electronc forms wthout wrtten permsson of IGI Global s prohbted.

5 36 Internatonal Journal of Swarm Intellgence Research, 3(3), 32-49, July-September 2012 Fgure 1. Pseudo-code of NSGSA n the followng subsecton, detals of whch are provded n later subsectons. General Settng Out of the Algorthm Fgure 1 shows the general teratve steps of NSGSA n the form of a pseudo code. The man loop starts after ntalzng the partcle postons and veloctes. Durng any teraton, the ftness of partcles s evaluated. An external archve of the Pareto optmal solutons wth lmted sze s generated n the frst teraton. In later teratons, the updated partcles are compared wth the members of the external archve and the archve s updated based on the Pareto optmalty condton. When the external archve s completed and a new member s gong to be added, a spread ndcator s utlzed to decde whch soluton to be replaced. The movng partcles are then ranked n layers usng the non-domnated sortng algorthm (Srnvas & Deb, 1994). To add some eltsm to the algorthm and to promote the unform dstrbuton of the solutons, some specfc members of the external archve are added to the lst of movng partcles and nstead the same numbers of the movng partcles are dscarded from the worst layer(s). As n sngle obectve GSA the mass of each partcle s a functon of tme and depends on ts ftness. But, n a mult-obectve problem, there are more obectves than one. NSGSA utlzes the non-domnated sortng algorthm to assgn each partcle a rank, regardng the layer t belongs to. Then, rank of each partcle s consdered as ts ftness and the mass s calculated usng Equaton (1). The acceleratons, exerted to the partcles, are calculated accordng to the varyng gravtatonal constant, usng the same relatons, proposed n GSA (Rashed et al., 2009). Then, the velocty and poston of partcles are updated accordng to the knematc relatons. The poston of partcles are also mutated to ncrease the dversty of the algorthm and to prevent premature convergences. The mutaton (or turbulence) strength depends on the velocty of partcles. As the gravtatonal constant and consequently the velocty of partcles are decreased over tme, the mutaton strength s decreased, too. Therefore, exploraton s decreased through the teratons whle at the same tme explotaton ncreases and helps to fnd the fnal solutons accurately. In the followng subsectons, the steps, bolded n Fgure 1 are dscussed n detal. Intalzaton The new algorthm has eght control parameters, defned n the next subsectons, whch must be set before the executon of the algorthm. An empty external archve s generated. Intal Copyrght 2012, IGI Global. Copyng or dstrbutng n prnt or electronc forms wthout wrtten permsson of IGI Global s prohbted.

6 Internatonal Journal of Swarm Intellgence Research, 3(3), 32-49, July-September poston of partcles s set randomly wthn the search ntervals and the ntal veloctes are set to zero for all partcles. Updatng the External Archve The external archve s updated based on the Pareto domnance crteron. If a member of the swarm s domnated by any member of the external archve, t wll not be nserted to the archve. In contrast, f t s non-domnated wth respect to all members of the archve or t domnates some members of the archve, t wll be nserted nto the archve and the domnated member(s) wll be removed. If the number of archved members exceeds the maxmum archve length, the most crowded area must be determned to remove a member from that area. A spread ndcator s ntroduced n the paper to control the length of external archve. Ths ndcator s based on the spread of the ponts wthn the Pareto front. Here, we are nterested n gettng a set of solutons that spans the entre Pareto optmal regon as unformly as possble. For ths purpose, a crowdng dstance measure, d c,, smlar to that proposed by Deb et al. (2002), s defned as follows: m d = d c, ( c, ) 2 (14) = 1 ( =1,..,n ) Where archve d = ft ft c, ( =1,..,n, = 1,..,m) archve (15) and ft s the -th ftness of the pont +1 when +1 the members of the archve are sorted versus the -th obectve functon. It s desred to have nearly equal crowdng dstances. To have nearly equal d c,, the devaton of these dstances from ther average, d c, should be mnmzed. For ths purpose a spread ndcator, δ, s defned as follows: n archve δ = d d [ n m ] d c, c archve = 1 E c (16) where n archve s the archve length and E s the set of extreme ponts, the sze of whch s taken equal to m. Deb et al. (2002) use a smlar equaton for measurng the performance of the algorthms, called as spread metrc. To obtan a unform spread of Pareto archve, δ must be decreased. In order to decrease δ, the dfference between max{d c, } and mn{d c, } must be decreased (=1,, n archve ). In other words, the maxmum d c, must be decreased or the mnmum d c, must be ncreased. Insertng a new member to the archve decreases the maxmum d c, or at least does not change t, but ncreasng the mnmum value s only possble through removng a sutable member. For ths purpose, the two nearest neghbors n the obectve space, are determned and named as members 1 and 2. These two members are removed alternatvely to calculate δ 1 and δ 2 correspondng to the removal of members 1 and 2, respectvely. The member, the presence of whch causes bgger δ s then removed, unless t extends the Pareto front, the case of whch the other member should be removed. Updatng the Lst of Movng Partcles Durng any teraton, some members of the external archve should be nserted to the lst of movng partcles for several reasons: Frst, nserton of such members adds some eltsm to the algorthm. Next, f the extreme ponts of the external archve are also nserted, then the algorthm wll hopefully be able to extend the Pareto front. Fnally, nserton of the members, located n the least crowded area, can repar the search gaps wthn the Pareto front. The lst of movng partcles for the next flght s updated as follows: m extreme ponts of the Pareto front (m sngle-obectve optmal solutons of the external archve) are nserted to the lst. Then, m ponts, located n the least Copyrght 2012, IGI Global. Copyng or dstrbutng n prnt or electronc forms wthout wrtten permsson of IGI Global s prohbted.

7 38 Internatonal Journal of Swarm Intellgence Research, 3(3), 32-49, July-September 2012 Fgure 2. Non-domnated sortng and rankng crowded area, are nserted. These ponts are selected usng the crowdng dstance assgnment algorthm, proposed by Deb et al. (2002). A percent of the archve members, P eltsm, s also selected randomly and nserted to the lst of movng partcles. Fnally, the lst length s lmted to n mass by removng some of the worst partcles, startng from the worst layer of the non-domnated sorted partcles. It should be noted that the ntal velocty of the members, nserted from the external archve to the lst of movng partcles, must be set to zero. Updatng the Mass of Movng Partcles As n sngle obectve GSA, the mass of each partcle s a functon of tme (teraton). It depends on the ftness of partcle and s calculated usng Equaton (1). In a mult-obectve optmzaton problem, there are multple obectves and a sngle ftness functon must be defned correspondng to each soluton. NSGSA utlzes the ranks of the layers, generated usng the nondomnated sortng algorthm to determne the ftness of each partcle. In NSGSA, the m members, mported from the extreme regons of the external archve and the m members, mported from the least crowded area of ths archve are ranked as 1, as depcted n Fgure 2. The P eltsm percent of the archve members, chosen randomly and nserted to the lst of movng partcles are ranked as 2. The members, remaned from the last teraton, get ranks 3 and more, regardng the rank of the layers they belong to. The rank of each partcle s then consdered as ts ftness and ts mass s updated usng Equaton (1). It should also be noted that the mported partcles from the archve usually have bgger mass than the others due to ther better ftness. Therefore, accordng to Equaton (10) ther acceleraton s less than the other partcles. Moreover, as mentoned prevously, the ntal velocty of these new mported partcles s set to zero. Therefore, these new partcles move more slowly than the others and provde fne search around the Pareto front. Updatng the Acceleraton of Partcles NSGSA uses the same equaton as GSA to update the acceleraton of partcles,.e., Equaton (10). The most mportant parameter of ths equaton s the gravtatonal constant, G( t). Rashed et al. (2009) suggest G( t) as follows: α t t G 0 max G( t) = e (17) Copyrght 2012, IGI Global. Copyng or dstrbutng n prnt or electronc forms wthout wrtten permsson of IGI Global s prohbted.

8 Internatonal Journal of Swarm Intellgence Research, 3(3), 32-49, July-September where α and G 0 are the functon parameters. It s dffcult to fnd a sutable constant G 0 for varous test functons. In ths paper, G 0 s proposed as follows: G 0 d d = β max ( x x ) (18) d { 1,.., n} u l where β s a parameter of NSGSA. Accordng to Equaton (10), the acceleratons, exerted to the partcles, depend to G(t). Hgh values of G(t) ndcates that the partcles can move n long steps, sutable for exploraton. Smlarly, low values of G(t) ndcates that the partcles can move n short steps, sutable for explotaton. The quck decrease of G(t) n Equaton (17) causes a fast decay of the exploraton. To mprove ths problem NSGSA utlzes a lnear functon of G(t), defned as follows: G( t) = G 0 ( 1 t / t ) (19) max The Use of Mutaton Operator To the authors experence, the orgnal GSA suffers from the premature convergence n the case of complex multmodal problems. To solve ths problem, a mutaton operator s added to the orgnal GSA to decrease the possblty of trappng n local optma. In realty, the close partcles may mpact to each other and consequently the drecton of ther movement may change. Therefore, to smulate the rregulartes, exst n the movement of the real partcles, two new mutaton operators have been proposed, called sgn and reorderng mutaton. NSGSA utlzes a combnaton of these two mutaton operators as ts mutaton (turbulence) operator. The new mutaton operators, proposed by the authors, are defned n the followng. Sgn Mutaton. In sgn mutaton, to update the poston of each partcle, the sgn of velocty vector changes temporally, wth a predefned probablty, P s, as follows: d d d v ( t + 1) = s v ( t + 1) ( d = 1,.., n, = 1,.., n ) mass s = 1 rand < P d s 1 otherwse x ( t + 1) = x ( t) + v ( t + 1) (20) where v d ( t + 1 ) s the mutated velocty by the sgn mutaton operator and rand s a unform random number, generated n the nterval [0,1]. Ho et al. (2005) propose a smlar mutaton operator. But, there are dfferences between these two mutatons. The sgn mutaton, proposed by Ho et al. (2005), changes the sgn of v d ( t ) ; whereas, the sgn mutaton, proposed here, mutate the poston of the partcle, by temporally changng the sgn of v d ( t + 1 ) to update the poston. Then, n the later teraton, the non-mutated velocty, v d ( t + 1 ), s used to calculate v d ( t + 2 ). Reorderng Mutaton. In ths mutaton, some partcles are chosen randomly to be mutated accordng to reorderng mutaton probablty (P r ). Then, elements of the velocty vector are rearranged randomly. In Fgure 3, dfferent possble mutatons of a velocty vector are shown for a two dmenson space. As can be seen, when the sgn and the reorderng mutatons are appled together, the exploraton space s ncreased wth respect to the case when only one of them s appled. Update and Mutate the Poston of Partcles NSGSA utlzes a combnaton of sgn and reorderng mutatons to update the poston of partcles. The mutaton operator of NSGSA works as follows: the velocty of partcle s frst mutated by the sgn and then by the reorderng mutatons. Therefore, n NSGSA the poston of partcles s updated as follows: Copyrght 2012, IGI Global. Copyng or dstrbutng n prnt or electronc forms wthout wrtten permsson of IGI Global s prohbted.

9 40 Internatonal Journal of Swarm Intellgence Research, 3(3), 32-49, July-September 2012 Fgure 3. Sgn and reorderng mutaton operators and ther combnaton v ( t + 1) = w( t) v ( t) + a ( t) v ( t + 1) = Sgn_Mutate( v ( t + 1)) v ( t + 1) = Reorderng_Mutate( v ( t + 1)) x ( t + 1) = x ( t) + v ( t + 1) (21) where w(t) s the tme varyng nerta coeffcent. In orgnal GSA, ths coeffcent s taken a random number between 0 and 1. As n PSO, t s better to use a decreasng nerta weght than a random one to have proper exploraton and explotaton n the frst and the last teratons, respectvely. NSGSA utlzes a tme varyng weghtng coeffcent as follows: t w( t) = w ( w w ) (22) t max It should be noted that the mutaton operator of NSGSA s appled to the poston vector not the velocty vector. NUMERICAL RESULTS To evaluate the performance of NSGSA, the optmzaton results, obtaned for a set of standard benchmarks, are compared wth the well-known NSGA-II, MOGSA and SMOPSO. Deb et al. (2002) used nne benchmark problems, known as SCH, FON, POL, KUR, ZDT1, ZDT2, ZDT3, ZDT4 and ZDT6, to evaluate the performance of NSGA-II. To compare the results of NSGSA wth those of NSGA-II, the same benchmarks are utlzed here. Cagnna et al. (2005) and Hassanzadeh and Rohan (2010) have used three benchmarks, known as Vennet3 (MOP5), Deb (MOP6) and Bnh2 (MOPC1), to evaluate the performance of MOGSA and SMOPSO, respectvely. These benchmarks are also utlzed here, to compare the performance of NSGSA wth that of MOGSA and SMOPSO. Some metrcs are used to measure the performance of mult-obectve algorthms, the examples of whch can be found n Cagnna et al. (2005) and Deb et al. (2002). In ths study, the two metrcs, defned by Deb et al. (2002), are used to compare the performance of NSGSA wth NSGA-II. The frst metrc (γ), measures the convergence of Pareto front, obtaned by the algorthm to a known set of true Pareto optmal solutons. The second metrc (Δ), called the dversty metrc, measures the spread, acheved among the solutons and the extenson of the Pareto front. In the same way, to compare the Copyrght 2012, IGI Global. Copyng or dstrbutng n prnt or electronc forms wthout wrtten permsson of IGI Global s prohbted.

10 Internatonal Journal of Swarm Intellgence Research, 3(3), 32-49, July-September Table 1. Parameters of NSGSA and ther tuned values Descrpton Parameters value Swarm sze n mass 100 Archve length n archve 100/800 Reorderng mutaton probablty P r 0.4 Sgn mutaton probablty P s 0.9 Percent of eltsm P eltsm 0.5 Intal value of nertal coeffcent w Fnal value of nertal coeffcent w Coeffcent of search nterval β 2.5 performance of NSGSA wth that of MOGSA and SMOPSO, the two metrcs, defned by Hassanzadeh and Rohan (2010), are utlzed. These two metrcs, named as Generatonal Dstance (GD) and Spacng (S) are used to measure the convergence and the spread of Pareto front, respectvely. Parameter Settng NSGSA has eght control parameters. These parameters and ther tuned values are lsted n Table 1. To make the results comparable wth those of NSGA-II, the number of functon evaluatons s consdered to be 25000, as n Deb et al. (2002). Therefore, the maxmum number of teratons, t max, s 250. Also, the length of external archve (narchve) s set to 100. Smlarly, to compare results wth MOGSA and SMOPSO, the number of functon evaluatons s consdered to be for MOP5, for MOP6 and for MOPC1, as n Cagnna et al. (2005) and Hassanzadeh and Rohan (2010) and the length of external archve s set to 800. Results and Dscusson The results, presented by Deb et al. (2002), are for real-coded and bnary coded varants of NSGA-II, none of whch outperforms the other Table 2. Mean and varance of the convergence metrc γ, obtaned for NSGSA and both varant of NSGA-II Algorthm SCH FON POL KUR ZDT1 ZDT2 ZDT3 ZDT4 ZDT6 NSGSA γ σ γ NSGA-II (real-coed) γ σ γ NSGA-II (bnarycoded) γ σ γ Copyrght 2012, IGI Global. Copyng or dstrbutng n prnt or electronc forms wthout wrtten permsson of IGI Global s prohbted.

11 42 Internatonal Journal of Swarm Intellgence Research, 3(3), 32-49, July-September 2012 Fgure 4. Varaton of the normalzed convergence metrc, γ, vs. teraton for a sample run Table 3. Mean and varance of the spread metrc Δ, obtaned for NSGSA and both varant of NSGA-II Algorthm SCH FON POL KUR ZDT1 ZDT2 ZDT3 ZDT4 ZDT6 NSGSA σ NSGA-II (realcoed) σ NSGA-II (bnarycoded) σ n all benchmarks. NSGSA s compared wth both varants of NSGA-II. Table 2 compares the mean and varance of the convergence metrc, γ, obtaned usng NSGA-II (real-coded), NSGA-II (bnary-coded) and NSGSA. As can be seen, regardng the mean value of γ, NSGSA has better converge n FON, POL, KUR, ZDT1, ZDT3, and ZDT6 problems than the two other algorthms, whle n ZDT4, the real-coded NSGA-II, and n ZDT2, the bnary-coded varant outperform the others, and n SCH all algorthms have the same convergence rate. Fgure 4 shows the varaton of γ durng a sample run, obtaned for NSGSA. To make the results comparable, the curves have been normalzed between 0 and 1. Table 3 shows the mean and Copyrght 2012, IGI Global. Copyng or dstrbutng n prnt or electronc forms wthout wrtten permsson of IGI Global s prohbted.

12 Internatonal Journal of Swarm Intellgence Research, 3(3), 32-49, July-September Fgure 5. Varaton of the dversty metrc Δ vs. teraton for a sample run Table 4. Mean and varance of the CPU tme obtaned for NSGSA parameter SCH FON POL KUR ZDT1 ZDT2 ZDT3 ZDT4 ZDT6 T (sec) σ T (sec) varance of the dversty metrc, Δ, obtaned usng the three algorthms. The Pareto front, obtaned usng NSGSA has better dversty than the two other algorthms n all benchmarks except n ZDT4, regardng the mean value of the dversty metrc. In ZDT4, the bnary-coded NSGA-II has obtaned a lttle better spread wth respect to the others. Fgure 5 shows the varaton of dversty metrc versus teraton for a sample run, obtaned for NSGSA. As can be seen, the dversty metrc can be deterorated when a new optmal Pareto soluton s found and nserted to the external archve. All experments took place on a 2.13 GHz Intel laptop machne. Table 4 shows the mean and varance of the CPU tme, shown by T and σ T, respectvely. Table 5. Mean of generatonal dstance metrc, obtaned for NSGSA, SMOPSO and MOGSA Algorthm MOP5 MOP6 MOPC1 NSGSA 1e-4 3e SMOPSO e MOGSA e Copyrght 2012, IGI Global. Copyng or dstrbutng n prnt or electronc forms wthout wrtten permsson of IGI Global s prohbted.

13 44 Internatonal Journal of Swarm Intellgence Research, 3(3), 32-49, July-September 2012 Table 6. Mean of spacng metrc, obtaned for NSGSA, SMOPSO and MOGSA Algorthm MOP5 MOP6 MOPC1 NSGSA e SMOPSO MOGSA Table 7. Varaton of the mean performance metrcs vs. the nerta weght Benchmark w0=0.9,w1=0.5 w(t) =0.5 w(t) =0.9 w(t) = rand ZDT3 γ ZDT4 γ ZDT6 γ Table 8. Varaton of the mean performance metrcs vs. the number of partcles Benchmark ZDT3 ZDT4 ZDT6 n mass γ γ γ Table 5 and Table 6 compare the performance of NSGSA wth that of MOGSA and SMOPSO n the sense of the mean generatonal dstance and spacng metrcs, respectvely. As can be seen n Table 5, NSGSA has superor convergence n MOP5 and MOPC1 problems than the two other algorthms, whle MOGSA has a lttle better convergence n MOP6. Moreover, NSGSA has better dversty than the two other algorthms n all benchmarks, as compared n Table 6. There are two mportant reasons that NSGSA has provded better spread n Pareto front. Frst, an enhanced method to prune the external archve wth the lmted length s used. Second, some specfc members of the archve wth hgher rank are nserted nto Copyrght 2012, IGI Global. Copyng or dstrbutng n prnt or electronc forms wthout wrtten permsson of IGI Global s prohbted.

14 Internatonal Journal of Swarm Intellgence Research, 3(3), 32-49, July-September Table 9. Varaton of the mean performance metrcs vs. the probablty of sgn mutaton Benchmark ZDT3 ZDT4 ZDT γ γ γ P s the lst of movng partcles to search the extreme and the least crowded spaces of the Pareto front. Senstvty Analyss In ths secton, the senstvty of NSGSA to ts parameters s analyzed based on the average performance, obtaned over 10 dfferent runs. In ths analyss, three complex benchmarks ZDT3, ZDT4 and ZDT6 are nvestgated. Table 7 presents the performance of NSGSA for dfferent settngs of the nerta weght. As can be seen, the values of 0.9 and 0.5 for w 0 and w 1 are the best for all benchmarks. The large nerta weght motvates exploraton and the smaller values motvate explotaton. It can also be observed that the random nerta weght, proposed n the orgnal GSA, does not provde a better performance than the current settng. Table 8 presents the performance metrcs for dfferent values of n mass. As can be seen, the use of 100 partcles (current settng) provdes the best result. The performance s deterorated by ncreasng n mass. However, t s not so senstve to decreasng n mass. Table 9 presents the performance metrcs for dfferent values of the sgn mutaton probablty (P s ). Accordng to Table 9 the value of 0.7 (current settng) s the best settng for ZDT3 and ZDT6 and the value of 0.5 provdes a lttle better results for ZDT4. Table 10 presents the performance metrcs for dfferent values of the reorderng mutaton probablty (P r ). Accordng to ths table, the current settng (0.4) s the best for ZDT3 and ZDT6, whle the value of 0.6 seems a better settng for ZDT4. Table 11 presents the performance metrcs for dfferent values of eltsm probablty (P eltsm ). Agan, the current settng (0.5) s the best for all benchmarks. Increasng P eltsm motvates the search around the best solutons, found so far and decreases the exploraton. Conversely, decreasng P eltsm decreases the explotaton. The same results can be concluded from Table 12 for the parameter β. Table 13 presents the average performance metrcs for dfferent values of archve length (n archve ). The archve length determnes the requred number of Pareto optmal solutons. Usually, t s not taken as a tunng parameter. The results, presented n Table 13, show that the performance of NSGSA s not so senstve to ths parameter. The senstvty analyss, made n ths secton, shows that NSGSA s not so senstve to the tuned value of some parameters such as n mass and n archve. Moreover, the best value of some other parameters, such as w 0, w 1, P eltsm and β, s not so senstve to the problem at hand. Copyrght 2012, IGI Global. Copyng or dstrbutng n prnt or electronc forms wthout wrtten permsson of IGI Global s prohbted.

15 46 Internatonal Journal of Swarm Intellgence Research, 3(3), 32-49, July-September 2012 Table 10. Varaton of the mean performance metrcs vs. the probablty of re-orderng mutaton Benchmark ZDT3 ZDT4 ZDT γ γ γ P r Table 11. Varaton of the mean performance metrcs vs. the percent of eltsm Benchmark ZDT3 ZDT4 ZDT6 P eltsm γ γ γ CONCLUSION In ths paper, an extenson of the sngle obectve GSA to mult-obectve optmzaton problems was presented. The proposed algorthm, called NSGSA, utlzes the non-domnated sortng approach to update the gravtatonal acceleratons. An external archve of the Pareto optmal solutons was also used to provde some eltsm. For ths purpose, a percent of the archved members are nserted to the lst of movng partcles. Moreover, to extend the Pareto front and to get a unform spread of solutons, the extreme and the least crowdng members of the external archve are also added to the lst of movng partcles, respectvely. The new found non-domnated solutons are contnually added to the external archve. When the length of the external archve s volated, one member s selected to be removed. For ths purpose, a spread ndcator, defned as the devaton of the members crowdng dstances from the average, was suggested to select a member to be removed. The authors propose the use of ths method n other mult-obectve algorthms to prune the external archve. To promote and preserve dversty wthn the movng partcles, two novel mutaton operators, called sgn and reorderng mutatons Copyrght 2012, IGI Global. Copyng or dstrbutng n prnt or electronc forms wthout wrtten permsson of IGI Global s prohbted.

16 Internatonal Journal of Swarm Intellgence Research, 3(3), 32-49, July-September Table 12. Varaton of the mean value of metrcs vs. the coeffcent of search nterval Benchmark ZDT3 ZDT4 ZDT6 β γ γ γ were proposed. These operators can also be used n other swarm optmzers such as PSO to mprove the dversty wthn the agents. Moreover, some modfcatons were appled to the orgnal GSA on the method the gravtatonal constant s updated. The performance of NSGSA was tested over a set of benchmark problems, borrowed from the lterature, to compare ts results wth those of the real-coded and bnary-coded varants of NSGA-II, MOGSA and SMOPSO. The results show that the new mult-obectve varant of GSA, proposed as NSGSA, can obtan comparable and even better performance n the sense of convergence rate and spread of solutons. Fnally, the senstvty of NSGSA to ts parameters was nvestgated. The results show that the performance s not so senstve to the value of some parameters and the best value of some parameters s the best for all problems. Therefore, some parameters can be hard coded, f necessary, n order to reduce the number of parameters to tune. Table 13. Varaton of the mean performance metrcs vs. archve length Benchmark ZDT3 ZDT4 ZDT6 n archve γ γ γ Copyrght 2012, IGI Global. Copyng or dstrbutng n prnt or electronc forms wthout wrtten permsson of IGI Global s prohbted.

17 48 Internatonal Journal of Swarm Intellgence Research, 3(3), 32-49, July-September 2012 ACKNOWLEDGMENTS The authors would lke to thank Dr. Hossen Nezamabad-pour for provdng the code of sngle-obectve GSA that helped the authors develop the mult-obectve varant. REFERENCES Cagnna, L., Esquvel, S., & Coello Coello, C. A. (2005). A partcle swarm optmzer for multobectve optmzaton. Journal of Computer Scence and Technology, 4(5), Deb, K., Pratap, A., Agarwal, S., & Meyarvan, T. (2002). A fast and eltst mult-obectve genetc algorthm: NSGA-II. IEEE Transactons on Evolutonary Computaton, 6(2), do: / Dorgo, M., Manezzo, V., & Colorn, A. (1996). The ant system: Optmzaton by a colony of cooperatng agents. IEEE Transactons on Systems, Man, and Cybernetcs Part B, 26(1), do: / Glover, F. (1989). Tabu search: Part I. ORSA Journal on Computng, 1(3), Glover, F. (1990). Tabu search: Part II. ORSA Journal on Computng, 2(1), do: /oc Hansen, M. P. (1997). Tabu search for mult-obectve optmzaton: MOTS. In Proceedngs of the Internatonal Conference of the Internatonal Socety on Multple Crtera Decson Makng (pp. 1-16). Hassanzadeh, H. R., & Rohan, M. (2010). A mult-obectve gravtatonal search algorthm. In Proceedngs of the Second Internatonal Conference on Computatonal Intellgence, Communcaton Systems and Networks (pp. 7-12). Ho, S. L., Shyou, Y., Guangzheng, N., Lo, E. W. C., & Wong, H. C. (2005). A partcle swarm optmzaton-based method for mult-obectve desgn optmzatons. IEEE Transactons on Magnetcs, 41(5), do: /tmag Hollday, D., Resnk, R., & Walker, J. M. (1993). Fundamentals of physcs. New York, NY: John Wley & Sons. Kennedy, J., & Eberhart, R. C. (1995). Partcle swarm optmzaton. In Proceedngs of the IEEE Internatonal Conference on Neural Networks (Vol. 4, pp ). Krkpatrck, S., Gelatto, C. D., & Vecch, M. P. (1983). Optmzaton by smulated annealng. Scence, 220, do: /scence L, C., & Zhou, J. (2011). Parameter dentfcaton of hydraulc turbne governng system usng mproved gravtatonal search algorthm. Journal of Energy Converson and Management, 52, do: /.enconman Rashed, E., Nezamabad-pour, H., & Saryazd, S. (2009). GSA: A gravtatonal search algorthm. Informaton Scences, 179(13), do: /. ns Sarafraz, S., Nezamabad-pour, H., & Saryazd, S. (2011). Dsrupton: A new operator n gravtatonal search algorthm. Scenta Iranca, Transacton D: Electrcal and Computer Engneerng, 18(3), Srnvas, N., & Deb, K. (1994). Mult-obectve optmzaton usng non-domnated sortng n genetc algorthms. Evolutonary Computaton, 2(3), do: /evco Tang, K. S., Man, K. F., Kwong, S., & He, Q. (1996). Genetc algorthms and ther applcatons. IEEE Sgnal Processng Magazne, 13(6), do: / Ulungu, E., Teghem, J., Fortemps, P., & Tuyttens, D. (1999). MOSA method: A tool for solvng multobectve combnatoral optmzaton problems. Journal of Mult crtera. Decson Analyss, 8, Had Nobahar was born n Iran n He receved the PhD degree n aerospace engneerng from Sharf Unversty of Technology, n Snce 2007 he s an assstant professor n flght dynamcs and control. Hs man research nterests are the applcaton of heurstc algorthms n aerospace, ntellgent gudance and control systems, and cooperatve gudance and navgaton of swarms. Copyrght 2012, IGI Global. Copyng or dstrbutng n prnt or electronc forms wthout wrtten permsson of IGI Global s prohbted.

18 Internatonal Journal of Swarm Intellgence Research, 3(3), 32-49, July-September Mahd Nkusokhan was born n Iran n He receved hs BS and MS degree n Aerospace Engneerng from Sharf Unversty of technology, Iran, n 2002 and 2004 respectvely. He s currently a PhD canddate maored n Aerospace Engneerng n Sharf Unversty of technology, Iran. Hs research nterests are gudance and control of aerospace vehcles and heurstc optmzaton. Patrck Sarry was born n France n He receved the PhD degree from the Unversty Pars 6, n 1986 and the Doctorate of Scences (Habltaton) from the Unversty Pars 11, n He was frst nvolved n the development of analog and dgtal models of nuclear power plants at Electrcté de France (E.D.F.). Snce 1995 he s a professor n automatcs and nformatcs. Hs man research nterests are computer-aded desgn of electronc crcuts, and the applcatons of new stochastc global optmzaton heurstcs to varous engneerng felds. He s also nterested n the fttng of process models to expermental data, the learnng of fuzzy rule bases, and of neural networks. Copyrght 2012, IGI Global. Copyng or dstrbutng n prnt or electronc forms wthout wrtten permsson of IGI Global s prohbted.