Multiobjective Simulated Annealing: A Comparative Study to Evolutionary Algorithms

Transcription

1 Multobjectve Smulated Annealng: A Comparatve Study to Evolutonary Algorthms Dongyung Nam and Cheol Hoon Par Abstract As multobjectve optmzaton problems have many solutons, evolutonary algorthms have been wdely used for complex multobjectve problems nstead of smulated annealng. However, smulated annealng also has favorable characterstcs n the multmodal search. We developed several smulated annealng schemes for the multobjectve optmzaton based on ths fact. Smulated annealng and evolutonary algorthms are compared n multobjectve NK model. The prelmnary results of the smulated annealng developed show that smulated annealng method performs well and sometmes better than evolutonary algorthms. More systematcal analyses to the varous problems are dscussed as further researches. Keywords: Multobjectve Optmzaton, Evoluto- nary Algorthms, Smulated Annealng, Pareto Optmalty, NK model 1. Introducton The multobjectve optmzaton problem has a rather dfferent aspect to scalar-objectve one. Instead of fndng one global optmum, whch s a general am n scalar-objectve optmzaton, multobjectve optmzaton must fnd a set of solutons, whch s called the Pareto set, or Pareto optmal fronter, as all the Pareto solutons are equvalently mportant and all of them are the global optmal solutons. As many engneerng and economcal problems are often complex and have ths multple objectves characterstc, whch must be optmzed smultaneously, conventonal optmzaton technques, such as the steepest-descent method, conventonal smplex method, many conventonal evolutonary algorthms, and the smulated annealng method, have dffcultes n extendng themselves to the multobjectve case because they are not orgnally desgned to fnd multple solutons. Typcally multobjectve problems are often solved wth conventonal sngle-objectve optmzaton methods by Correspondng Author: Cheol Hoon Par s wth Department of Electrcal Engneerng, KAIST 373-1, Kusong-dong, Yusong-gu, Taejon , Republc of Korea E-mal: chpar@ee.ast.ac.r usng penalty or weghted sum methods [4,13,22,33,36]. However, the penalty and weghted sum methods also have dffcultes n selectng proper penalty functons and weghtng factors respectvely. The other problem of usng the weghted sum method s t cannot fnd a soluton n a concave regon [6]. To solve ths problem, many researches for multobjectve optmzatons have been suggested and new concepts ntroduced [9,10]. One of these concepts, Pareto optmalty, s wdely used n many multobjectve optmzaton algorthms ncludng evolutonary algorthms. Evolutonary algorthms (EAs) have many nterestng propertes and have been wdely used n varous optmzaton problems from combnatoral problems such as job shop schedulng to real valued parameter optmzaton [2,3]. Also many evolutonary algorthms for solvng the multobjectve problem have been suggested [19,20]. The success of evolutonary approaches n multobjectve optmzaton s manly based on the populaton concept wth the ablty of fndng multple optma smultaneously, whch matches the dea of multobjectve optmzaton. However, the smulated annealng method, whch s reported to gve good performance a many sngle-objectve problems, has been seldom used for the multple objectves problems. The man reason s that smulated annealng usually fnds only one soluton nstead of set of solutons and ths s a crtcal handcap n multobjectve optmzaton [30]. There are four mportant propertes for a good algorthm n multobjectve optmzaton. 1) Searchng precson. The algorthm must fnd the Pareto optmal solutons, whch are global optma n multobjectve optmzaton. When ths s hard to acheve because of problem complexty, t must fnd the possble near solutons to the optmal solutons set. 2) Searchng tme. It must fnd the optmal set effcently. 3) Unform probablty dstrbuton over the optmal set. The solutons found must be wdely spread, or unformly dstrbuted over the real Pareto optmal set nstead of convergng to one pont because every soluton s mportant n multobjectve optmzaton. 4) Informaton about Pareto fronter. The algorthm must gve as much nformaton as possble about the Pareto fronter. Smulated annealng has been appled for multobjectve optmzaton by usng the weght sum method n lmted applcatons. Whdborne used the smulated annealng to solve a problem formulated as the

2 method of nequaltes (MOI) [38]. The objectve of ths paper s to construct a smulated annealng method to fnd all the Pareto solutons, whch satsfes the above propertes. Frst, smulated annealng method, whch s suggested n ths paper, uses the concepts of Pareto optmalty and domnaton to acheve hgh searchng precson. The man drawbac of smulated annealng s searchng-tme, as t s generally nown that smulated annealng taes long tme to fnd the optmum. Though t s also reported that long searchng-tme s not always needed, the second property remans as a man problem of smulated annealng. Smulated annealng has an nterestng advantage n ts unform probablty dstrbuton property as t s mathematcally proved that t can fnd each of the global optma wth the same probablty n a scalar fnte-state problem [12,29]. Consderng that evolutonary algorthms generally use addtonal algorthms such as ftness sharng, nche nducton for spreadng the solutons, smulated annealng can have a more smple and compact structure. The last property comes from the dfference between the propertes of scalar-objectve and multobjectve optmzaton. In solvng scalar-objectve problems, there s no need to fnd all the global optma except some specal cases because every global optmum has the same value. The only thng needed s the optmal cost and parameters wth whch the cost s evaluated. However, the stuaton s dfferent n multobjectve case. As all the Pareto solutons have dfferent cost vectors that have a trade-off relatonshp, a human or a decsonmaer must select a proper soluton from the found Pareto soluton set or sometmes by nterpolatng the found solutons. The rest of ths paper s organzed as follows. Secton 2 formulates the multobjectve optmzaton problem ncludng the concept of Pareto optmalty and domnaton, and descrbes some prevous wors about evolutonary algorthms and the smulated annealng method. Secton 3 shows the dea of multobjectve smulated annealng method and prelmnary results from t. Comparson results to an evolutonary approach are presented. Further research drectons are dscussed n the secton 4 and secton 5 summarzes the smulated annealng method n multobjectve optmzaton and dscusses the comparsons wth the evolutonary approach. 2. Survey of Stochastc Multobjectve Optmzaton Algorthms 2.1 Multobjectve optmzatons For most multobjectve problems, there exsts a set of non-domnated solutons that have a trade off relatonshp each other, and one of the multple objectves of each soluton cannot be mproved wthout sacrfcng any of others. Ths concept s nown as the Pareto optmalty [29]. Defnton 1 Consder, wthout loss of generalty, the mnmzaton of the n components f, = 1,, n, of a vector functon f of a vector varable x n a unverse, where f( x) = ( f1( x), K, f n ( x )). Then a decson vector x u s sad to be Pareto optmal f and only f there s no x v for whch v= f( x v) = ( v1, K, vn) domnates u= f( x u) = ( u1, K, un), that s, there s no x v such that v u {1, K,} n and v < u {1, K, n} f (a) functon graph w.r.t. parameter Pareto optmal fronter 1.0 Domnated solutons (b) functon space graph Fgure 1. The concept of the Pareto optmal fronter The set of all Pareto-optmal decson vectors s called the Pareto optmal set, effcent set, admssble set or the Pareto fronter of the problem. The correspondng set of objectve vectors s called the non-domnated set. In practce, however, t s not unusual for these two terms to be used nterchangeably to descrbe solutons of a multobjectve optmzaton problem. The noton of Pareto optmalty s only a frst step towards the practcal soluton of a multobjectve problem, whch usually nvolves the choce of a sngle compromse soluton from the non-domnated set accordng to some preference nformaton. Fgure 1 shows the concept of the Pareto optmal set clearly. Consderng the specfed f 1

3 multobjectve optmzaton, fgure 1 (a) shows each functon value wth respect to the parameter x and fgure 1 (b) s a plot n whch the x-axs s f 1 and the y- axs s f 2. The mddle sold curved segment of fgure 1(b) s the Pareto optmal fronter - non-domnated set and the two outer dashed curved segments are domnated solutons. The smultaneous optmzaton of multple, possbly competng, objectve functons devates from scalarobjectve optmzaton. Instead of fndng one perfect soluton, multobjectve optmzaton problems tend to be characterzed by a famly of alternatves that must be consdered equvalent n the absence of nformaton concernng the relevance of each objectve relatve to the others. Therefore, the frst objectve n multobjectve optmzaton s to fnd the Pareto set, and the next s to select a proper soluton from the found Pareto soluton set. 2.2 Evolutonary algorthms Many multobjectve optmzaton problems have been successfully solved usng tradtonal mathematcal optmzaton procedures, such as lnear programmng, nteger programmng, and nonlnear programmng. However, many real-world problems nvolve complex and nonlnear propertes that do not ft readly nto one of these tradtonal framewors. Recently, non-gradent, stochastc based search technques such as smulated annealng and evolutonary algorthms have been successfully employed to solve real-world optmzaton problems. There have been four mportant multobjectve search crtera n the hstory of evolutonary algorthms [9]. Plan aggregatng approaches: as conventonal evolutonary algorthms can solve the problem when t has sngle objectve, t s requred to mae scalar ftness functons on whch to wor. In most problems, where no global crteron drectly emerges from the problem formulaton, scalarzaton of the objectve functon has been acheved by aggregatng the multple objectves wth weghtng factors. Several applcatons of evolutonary algorthms n the optmzaton of aggregatng functons have been reported from the begnnng of 1990s, and almost every multobjectve problem used ths method n the frst era of the multobjectve evolutonary hstory [13,22,36]. There are two advantages of usng ths method. The frst s, of course, the smplcty of ths method. There s no need to change the algorthm tself except mang a sngle objectve by usng the weghted sum method. The second s, there s no need for post-processng such as decsonmang because there s only one soluton already. Even though ths algorthm s stll wdely used, the dffculty n selectng proper weghts, and the nablty to fnd solutons n a concave Pareto regon are the man drawbacs of ths algorthm. Populaton-based non-pareto approaches: Schaffer was probably the frst to recognze the possblty of explotng populatons to treat multple, conflctng objectves separately and search for multple nondomnated solutons concurrently n a sngle run [33]. In ths case, hs algorthm uses the concept of specaton nstead of Pareto optmalty. The entre populaton s dvded nto several sub-populatons (specaton) and the dvded sub-populaton was selected usng a selecton mechansm whch consdered only one objectve functon for each sub-populaton. The selected specaton maes a new populaton (next generaton) whch s dvded nto sub-populatons agan after mutaton and crossover operatons. Schaffer suggested ths algorthm Vector Evaluated Genetc Algorthm (VEGA) and hs smulatons showed good results n multobjectve optmzaton. As ths algorthm s also very smple, uses the concept of populaton well, and s able to fnd Pareto solutons ncludng the concave regon just n one run, there have been many researches based on the populaton concept [11,17,25]. However, there are also weanesses n ths algorthm. One s the basng phenomenon: fnal solutons have a tendency to be located on the edge of the Pareto fronter. Also the performance of ths algorthm s severely affected by the objectve values because selecton s determned accordng to one of the values n the objectve vector not the domnaton relatonshp. Pareto-based approaches: Goldberg suggested a multobjectve optmzaton algorthm usng the concept of Pareto optmalty n 1989 [13]. Ths search algorthm, whch consders all the objectves smultaneously and selects the non-domnated solutons wth a hgh probablty, can fnd a good Pareto optmal fronter by the Pareto ranng technque. Soon many research results about the algorthms based on the Pareto concept were publshed [10,19,35]. The advantage of ths algorthm s ts ablty to fnd the Pareto optmal fronter ndfferent to the parameter value, that s, ths algorthm wors well when there s large dfference of average or varance between objectves. The second possble advantage of ths Pareto ranng approach s that solutons that exhbt good performance n many objectve dmensons are more lely to be produced by recombnaton. Nche nducton technques: ths algorthm uses the nche and sharng concepts to spread the searchng agents unformly over the Pareto optmal fronter. Also ths method has a tendency to prohbt the genetc drft phenomenon by forcng the searchng agents not to converge to one pont from the begnnng of the search. Though t s very helpful for the decson-mang to spread out the soluton unformly, ths algorthm has a weaness also. As the sharng technque s affected by the scale dfference severely, spreadng out the soluton s generally domnated by the objectve functon wth the largest varaton. Ths property seems to be opposte to the phlosophy of Pareto optmalty and domnaton. Therefore, t s necessary to control the scales of each parameter before search but t s generally dffcult. There

4 have been many promsng results from ths algorthm by the many researchers n the 1990s [10,17,19,35]. 2.3 Smulated annealng Smulated annealng (SA) s one of the stochastc search algorthms, whch s desgned usng a spn glass model by the Krpatrc [24]. It has been used n wde areas from the combnatoral problems to the real world problems because t performs well on most of optmzaton problems, especally on complex problems [1,26,30]. The powerfulness of SA orgnates n the good selecton scheme and annealng technque. Generally SA used two nds of selecton scheme. One s the Metropols algorthm and the other s the logstc selecton algorthm [27]. Orgnally any nd of selecton that satsfes the detaled balance equaton can be used as a selecton scheme because the detaled balance equaton guarantees the convergence of SA [29]. Another reason why SA performs well s annealng, that s, the gradual temperature reducng technque. As the temperature and the cost dfference manly determne the amount of mutaton n generatng the next searchng pont, SA can do local fne-tunng towards the end of the search to gve fner results. The dsadvantage of SA s, as s well nown, the long annealng tme. There are, of course, many algorthms to compensate for ths such as fast smulated annealng (FSA), very fast smulated re-annealng (VFSR), new smulated annealng (NSA) [21,37,41]. However, there s lttle research nto usng the smulated annealng method for multobjectve optmzaton. The frst and most sgnfcant problem s that SA uses only one search agent. As solvng multobjectve problem generally requres fndng all the solutons at the same tme, usng many search agents wll be effectve n general. Though SA was desgned orgnally to use only one search agent, there have been also many technques for usng mult search agents or for parallelzaton [1]. To use a populaton n SA, however, has a possblty to lose the merts of SA a lttle because those nds of methods usually ental redundant search. 3. Multobjectve Smulated Annealng EAs have been wdely used n varous statc optmzaton problems from combnatoral optmzaton to real parameter optmzaton as a powerful and robust optmzaton technque. There have been a lot of researches showng that EAs are good optmzaton methods, whch has resulted n fast enlargement of ther applcaton areas [7,8,18,32,34]. Many EA researchers have been tryng to characterze EAs mechansms and landscape. One result of ths research s smulaton results wth Royal road functons [28]. Though the Royal road functon was desgned n favor of crossover operatons, evolutonary search do not always outperform varatons of the hll clmbng method and a well-desgned hll clmbng method shows better performance than evolutonary algorthms. In 1995, Wolpert and Macready publshed the No Free Lunch theorem and the theorem showed mathematcally that all algorthms perform equally well over all the functons n the fnte search space [39,40]. Accordng to ths theorem, dscusson about the performance between dfferent algorthms can be meanngless as they perform equally from an average pont of vew. However, the stuaton s dfferent n treatng real world problems, as there are general tendences n ordnary problems. Droste, Jansen and Wegener showed that a partcular algorthm performs better over a subset of the entre functon set n ther paper [5]. It means that there can be a better algorthm to solve restrcted problems. Many researchers also have found that EAs are very promsng algorthms for solvng multobjectve optmzaton problem as they can fnd many good solutons (the Pareto set) n one smulaton. However the SA algorthm has been hardly used for multobjectve optmzaton because SA was orgnally constructed to use only one searchng agent. Ths s nown to be a crtcal weaness of SA as t betrays the phlosophy of multobjectve optmzaton searchng for all the Pareto solutons nstead of only one soluton. As the result of ths weaness, SA has remaned as one of the mproper or not favorable algorthms for multobjectve optmzaton. It s, however, a queston whether SA cannot be used at all for multobjectve optmzaton though t performs well and sometmes better than EAs n solvng sngle objectve optmzaton problems [31]. In ths secton, a possble method for SA s suggested to solve multobjectve optmzaton and ts advantages and dsadvantages are shown by smulaton results. 3.1 Extenson from Sngle-objectve to Multobjectve Multobjectve SA (MOSA) uses the domnaton concept and the annealng scheme for effcent search. The man obstacle for SA n multobjectve optmzaton s ts nablty to fnd multple solutons. However, SA can do the same wor by repeatng the trals as t converges to the global optma wth a unform probablty dstrbuton n the sngle objectve optmzaton. Fgure 2 shows ths characterstc of SA. When there are two global optma, t s proved that SA can fnd each optmum wth probablty 0.5 [29]. When ths fact s also true n multobjectve optmzaton, SA has advantages over EAs because t does not need large memory to eep the populaton; nor does t use addtonal algorthms to spread the solutons over the Pareto fronter. Addtonally MOSA can fnd a small group of Pareto solutons n a short tme wth the demand of urgent smulaton and then fnd more solutons by repeatng the trals for detaled nformaton about the Pareto fronter. The mathematcal tass of showng the unform convergence to the Pareto fronter of MOSA s not completed and remans as a future wor. In ths paper, smulaton results on a smple test-bed wll be presented to show ths property.

5 Functon Global optmum Global optmum (a) State probablty 1.0 Fgure 3. Pseudo-code of multobjectve smulated annealng General scheme: The general SA algorthm nvolves the followng three steps. Frst, the objectve functon correspondng to the energy functon must be dentfed. Second, one must select a proper annealng scheme consstng of decreasng temperature wth ncreasng of teratons. Thrd, a method of generatng a neghbor near the current search poston s needed. In sngle objectve optmzaton problems, the transton probablty scheme s generally selected by the Metropols and logstc algorthms [27,29]. However, the stuaton s dfferent n multobjectve optmzaton and choosng a proper transton probablty s dffcult. Ths problem wll be treated detal n the transton probablty paragraph. The algorthmc descrpton of the MOSA s outlned n fgure 3 where s represents the current search poston (or the current state n a fnte state search problem) and T s the temperature parameter, whch s gradually decreased as tme goes on. A new search poston s s generated by the N(s) functon, ts cost s evaluated and compared wth the prevous cost. When t s determned to be a good soluton by the domnaton test, the new state s accepted. Even when the new poston s not proper (meanng the new poston s domnated by the current state), t s accepted wth some acceptance probablty. When there s no superorty between the current state and the next state, the new state s accepted nstead of the current one because movng n the nondomnated stuaton helps ncrease the spread (b) Fgure 2. Unform dstrbuton property n SA a) the graph of the objectve functon; b) State probablty of the Marov chan as tme goes to nfnty. As there are only two global optma, SA fnds each global optmum wth the same probablty 0.5 s=s 0 T=T 0 Repeat Generate a neghbor s =N(s) If C(s ) domnates C(s) move to s else f C(s) domnates C(s ) move to s wth transton probablty P t (C(s), C(s ), T) else f C(s) and C(s ) do not domnate each other move to s endf T=annealng(T) Endrepeat (untl the termnaton are satsfed) 0.5 Global optmum Global optmum performance and evade local optma. Ths fact wll be shown wth smulaton results. When whether to move or stay s determned, the algorthm repeats ts loop wth lower temperature untl termnaton condtons are satsfed. Neghbor generatng and annealng: In fnte state problems le combnatoral problems (TSP, QAP, NK-model), a general neghbor generatng method s the permute operaton (or the bt flp operaton n a bnary problem), whch must satsfy the reachablty and symmetry condtons. The followng geometrc coolng s wdely employed for the annealng scheme n ths nd of problem. T = α T0 (1) where 0< α < 1 s a coolng rate. For combnatoral problems (ncludng the NK-model), t s usual to generate a neghbor by flppng one bt at a random poston and use the geometrc annealng scheme. Transton probablty: General transton rules such as the Metropols or logstc method cannot be appled drectly to the multobjectve problems because they support only a scalar cost functon. The suggested transton rule n ths paper s very smlar to the Metropols method except that they used a dfferent cost crteron for the multobjectve cost functon. The transton probablty from state to j s, P(, j) = mn{exp( cj (, )/ T),0} (2) t where cj (, ) s the cost crteron for transton from state to j, and T s the annealng temperature. Sx crtera for MOSA are suggested and evaluated. The schemes are as follows: Mnmum cost crteron cj (, ) = mn( c () j c ()) (3) where c () s th cost value n the objectve vector of th state. Maxmum cost crteron cj (, ) = max( c ( j) c ()) (4) Random cost crteron D = 1 cj (, ) = α( c() j c ()) (5) where D s the dmenson of the objectve vector and α s a random varable wth unform probablty dstrbuton. Self cost crteron D cj (, ) = c () (6) = 1 Average cost crteron cj (, ) = D = 1 ( c () j c ()) D (7)

6 Fxed cost crteron cj= (, ) fxed value (8) We tested the above sx crtera on the smple test-beds and found that the random, average, fxed crtera generally show good performance. The performances of the mnmum, maxmum, self cost crtera change greatly dependent on the test-beds. In the followng smulatons, we used the average cost crteron. The man problem wth usng the weghted sum method the nablty to fnd a concave regon does not occur n the suggested MOSA as t uses the domnaton test frst. Move or Stay n non-domnated stuaton: When the new state s the same level of value as the current state, there can exst two schemes move to the new state or stay n the current state. The analyss of ths problem shows that the move scheme s better than the stay one. Wth the stay scheme, search wll end on both edges of the Pareto fronter not enterng the mddle of the fronter. However, wth the move scheme, search wll be contnue nto the mddle part of the fronter, move freely between non-domnated states le a random wal when the temperature s low and eventually wll be dstrbuted unformly over the Pareto fronter as tme goes to nfnty. 3.2 EA technques the nche nducton algorthm The specfcs of the Nche Pareto algorthm are localzed to the mplementaton of selecton - the use of Pareto domnaton tournaments, where two canddates for selecton are compared aganst each ndvdual n the comparson set. In tournament selecton a set of ndvduals s randomly chosen from the current populaton and the best of ths subset s chosen to be represented n the next populaton. In order to obtan a Pareto optmal surface, tournament selecton must be altered to use multple objectves. Selecton pressure s manly determned by the sze of the comparson set t dom ; f the sze of the comparson set s large, there s hgh selecton pressure whch possbly lead to local optma n many cases, f the sze s small, there s low pressure whch mae the convergence of populaton slow. The Pareto ran s the number of elements n the comparson set domnated by the canddate. For example, f the canddate domnates three elements of the comparson set of sze ten, the Pareto ran of the canddate s 3 [13]. Although the Pareto ran scheme encourages the exploraton n the drecton of non-domnated ndvduals, they have a tendency to converge on one pont as tme goes on and wll suffer from populaton drft because ths s the property of most conventonal evolutonary algorthms. To fnd unformly dstrbuted solutons along the Pareto fronter, Goldberg and Rchardson, Deb, Goldberg have ncorporated ftness sharng method by the nche scheme [14,15,16]. Ftness sharng degrades the ndvdual ftness by a sharng functon as f/ m, dvdng the objectve ftness f, by the nche count m, whch reflects the neghborhood crowdng around an ndvdual. In ths paper, ths scheme was adopted n a smplfed form. Instead of recalculatng the ftness functon by the sharng method, the nche count of one canddate s drectly compared to the nche count of the other. As the sharng functon encourages the canddate that s located n the sparse space, gradually all the solutons of the populaton become unformly dstrbuted as the algorthm goes on. 3.3 Comparson on the NK ftness model In ths secton we dscuss whether or not smulated annealng s a promsng tool for solvng to solve hard optmzaton problems by comparng ts performance wth evolutonary algorthms on the multdmensonal verson of Kauffman s NK ftness landscape model [23]. The NK-model of ftness landscapes can be regarded as combnatoral optmzaton problems defned on the bnary space {0,1} N, where N s the length of bnary strng. The ftness functon, f:{0,1} N R s defned by the average of ftness contrbutons of all bts as shown n equaton (9) f 1 N f N = 1 = (9) where the ftness contrbuton f of the -th bt s determned by a random number drawn from unform dstrbuton n the nterval [0,1], dependng on the values ( 1) of tself and K other bts. That s, f has 2 K+ dfferent random numbers. In the NK model, K s the most mportant parameter that nfluences the statstcal property of the NK ftness landscape. K s used to tune the ruggedness of the landscape. For example, when K = 0, the landscape has a unque global optmum but as K ncreases (up to N-1), t becomes more rugged wth an ncreasng number of local optma. As the NK model was orgnally desgned to construct to sngle objectve landscape, we extend t to a multobjectve one. In multobjectve NK model (NKD model), there s one more parameter D that determnes the dmenson sze of multple objectves. In what follows, the parameters of smulated annealng are descrbed: Intal temperature value: The ntal temperature s chosen to be 500 by heurstcs from smulaton results. Annealng scheme: For the NKD model, the geometrc annealng method s used. T+ 1 = αt ( α = 0.995). Neghbor generaton: Generatng a new search poston s done by flppng one bt of parameter strng. Chan length: The chan length represents the number of allowable transtons before the temperature changes ts value. The length of the strng (N) s used for the chan length. Termnaton condton: The algorthm fnshed ts calculaton after pre-defned teraton. In ths smulaton, we set ts value to 5000.

7 Agent number: We used 100 ndependent searchng agents smultaneously. That s, 100 agents search for the Pareto optmal wthout exchangng of nformaton between them. The parameters for the evolutonary algorthm are as follows: Populaton sze: The populaton sze s set to 100 for all the smulatons. Genetc operaton: Conventonal one pont crossover s used wth crossover probablty 0.1 and standard mutaton per bt s used wth the mutaton rate 0.3. Selecton: Pareto based tournament selecton s used wth a comparson set sze of 5 (5% of the populaton). As ths parameter determnes the selecton pressure, t must be chosen carefully. However, as there s no systematc method for choosng ths by consderng the landscape, ths value s chosen based on the emprcal smulaton results. It s an open problem to choose a proper selecton power accordng to each problem and remans as future wor. Nche sze: The nche sze that determnes the sze of the hyper sphere around the canddate s set to 0.1. Termnaton condton: The evolutonary algorthm was desgned to use the same number of teratons for comparson wth smulated annealng. The smulaton runs over 5000 teratons. The frst four tests are conducted on the small sze landscape model where the exact Pareto fronter can be found by exhaustve search. The length of NKD model s 10, the epstatc parameter K changes ts value to 0, 2, 4, 8 and the objectve dmenson s 2. The fgure 4 shows a (pseudo) Pareto optmal fronter that each algorthm found. As t s dffcult to show the performance of multobjectve optmzaton except by showng the Pareto fronter, a randomly chosen typcal graph s presented as an example. In the small sze landscape model, the smulated annealng method can fnd the Pareto fronter more precsely at many tmes, and each soluton spreads wdely over the Pareto fronter n spte of the fact that the smulated annealng method does not use any sharng method. However, t s true that the evolutonary approach wth the sharng technque has a tendency of spreadng more than smulated annealng. We can see ths tendency more easly wth the large sze landscape model. We conducted the second tests to examne the performance of the two algorthms n a large landscape. By changng N to 20, 40, 80 wth K, to 2, 8, the smulated annealng and evolutonary algorthms have been smulated and compared. Fgure 5 shows the comparson results of smulaton. The concluson s that the evolutonary algorthm shows better performance as the sze of landscape becomes large wth better searchng ablty and better spreadng characterstcs. However, smulated annealng also showed satsfactory results from another pont of vew when consderng that smulated annealng does not use any addtonal algorthm and t can be used ndependently. (a) N=10, K=0, D=2 (Low epstatc problem) (b) N=10, K=8, D=2 (Hghly epstatc problem) Fgure 4. Smulaton results on the small sze NKD-model 4. Dscusson and Future Wor 4.1 Nche nducton smulated annealng Though the suggested smulated annealng method gves satsfactory smulaton results n multobjectve optmzaton over a fnte state optmzaton, the NKD model, t s sometmes observed that smulated annealng has dffculty n searchng the Pareto optmal wth unform dstrbuton. That s, multobjectve smulated annealng can fnd the solutons n the easer and noncomplex problem, but the performance s degraded n complex problems wth much randomness le hghly epstatc NKD models. One possble approach for ncreasng the performance of smulated annealng s to use the populaton nformaton effcently. The nche nducton method was reported as a powerful technque n multobjectve evolutonary algorthms [19]. However, usng the nformaton of the populaton le nche nducton should be desgned carefully because t may harm the advantages of the smulated annealng method. 4.2 Performance measures for multobjectve optmzaton

8 One dffculty n comparng the algorthms n the multobjectve test-beds s that there s no systematc crteron to measure the performance of each algorthm. Ths s manly due to the fact that n multobjectve optmzaton, the objectve value tself does not have a sgnfcant meanng. Instead, the confguraton of objectve values s more mportant. Therefore, the conventonal measure s only the plottng of the Pareto set, but t s mpossble to draw the graph when the dmensons of objectves are larger than three. (Even for three-dmensonal graph t s not so easy to determne whch s the better Pareto set) Even f t s possble to plot the graph for more than three objectves, t s not a good measure as there s no quanttatve nformaton. A good performance measure for comparson must have the followng propertes. 1) It must measure the closeness to the real Pareto fronter n numerc value. 2) The unformty of the dstrbuton of solutons over the Pareto fronter must be measured. 3) Addtonal nformaton, e.g. separated fronters number, must be also measured. a) N=20, K=2, D=2 d) N=40, K=8, D=2 b) N=20, K=8, D=2 e) N=80, K=2, D=2 c) N=40, K=2, D=2 Fgure 5. Smulaton results on the large sze NKD-model f) N=80, K=8, D=2

9 4.3 Mathematcal Analyss The most favorable property of smulated annealng s that there s a complete convergence proof for t. The deal annealng and neghbor generatng schemes are deduced from the mathematcal analyss of convergence. Even though these deal schemes have lttle meanng from a practcal pont of vew, for example, the conventonal smulated annealng uses the log-le annealng scheme from the mathematcal result and t taes enormous smulaton tme for the algorthm to converge, guaranteeng the convergence s a fundamental step for constructng an algorthm. Unfortunately, mathematcal analyses about smulated annealng n multobjectve optmzaton have seldom been studed and reman as an open problem. There are two man propertes to be consdered mathematcally: one s the convergence proof and the other s unformty of dstrbuton. As the conventonal smulated annealng method satsfes the detaled balance condton, t s guaranteed to have pseudo-statonary probablty and the global convergence probablty s represented as a smple equaton [29]. However, n the multobjectve case, fndng a proper acceptance probablty crteron, whch satsfes the detaled balance condton, s dffcult. Therefore, even though the pseudo-statonary probablty exsts, t s not easy to fnd the probablty as an equaton form. It s also unclear whether the ndependent smulated annealng algorthm gves unformly dstrbuted solutons over the Pareto fronter or not. Though provng unform dstrbuton over the connected Pareto set s clear and easly explaned by the random wal property, the problem s not so easy when the Pareto set s not a connected one. Ths problem also remans as further wor. 5. Concluson There have been many researches nto usng evolutonary algorthms to solve multobjectve problems and many effcent algorthms have been developed. However, though smulated annealng s also a very powerful searchng algorthm and has gven many good results n varous optmzaton felds, t has been seldom used for the multobjectve optmzaton because t conventonally uses only one search agent, whch maes the search for all solutons n the Pareto set dffcult. Wth the dea that smulated annealng has a unform probablty dstrbuton over global optma, a multobjectve smulated annealng method s suggested. The prelmnary results of the developed algorthms are compared wth an evolutonary algorthm and show that smulated annealng also has good propertes n multobjectve optmzaton. The frst test wth fnte state test-beds shows that ndependent smulated annealng have a tendency of fndng the solutons n the Pareto set wth unform probablty. Ths property was tested over a more complex combnatoral problem the multdmensonal NK model. When the sze of problem s small, smulated annealng showed good performance compared to the evolutonary algorthm. However, the evolutonary algorthm outperforms smulated annealng when the problem sze and the epstatss become large. Expermental results suggest that smulated annealng has much potental n the multobjectve optmzaton feld also. Parallelzng technques and usng populaton nformaton wll be good approaches for ncreasng the performance of MOSA. Also, fndng effcent parallelzng technques and performance measures for multobjectve optmzaton remans as future wor. References [1] R. Azencott, Smulated Annealng: Parallelzaton Technques, Readng, John-Wley & Sons, Inc., [2] T. Bac, U. Hammel and H.-P. Schwefel, Evolutonary Computaton: Comments on the Hstory and Current State, IEEE Trans. on Evolutonary Computaton, pp.3-17, [3] T. Bac, F. Hoffmester, and H. Schwefel, A Survey of Evolutonary Strateges, In Proceedngs of the Fourth ICGA. pp.1-10, Morgan Kauffman, San Mateo, CA, [4] L. Davs, and M. Steenstrup, Genetc Algorthms and Smulated Annealng: an Overvew, In L. Davs (Ed.) Genetc Algorthms and Smulated Annealng, Research Notes n Artfcal Intellgence, pp.1-11, London:Ptman. [5] S. Droste, T. Hansen, and I. Wegener, Perhaps not a Free Lunch but at least a Free Appetzer, Proceedngs of the Genetc and Evolutonary Computatons Conference, pp , [6] P. J. Flemng, and A. P. Pashevch, Computer Aded Control System Desgn Usng a Multobjectve Optmzaton Approach, Proceedngs of the IEE Control 85 Conference, pp , London, UK: IEE, [7] D. B. Fogel, An Introducton to Smulated Evolutonary Optmzaton, IEEE Trans. on Neural Networs, vol.5, pp.3-14, [8] D. B. Fogel, Evolutonary Computaton, IEEE Press, [9] C. M. Fonseca and P. J. Flemng, An Overvew of Evolutonary Algorthms n Multojectve Optmzaton, Evolutonary Computaton, vol.3, no.1, pp.1-16, [10] C. M. Fonseca and P. J. Flemng, Genetc Algorthm for Multobjectve Optmzaton: Formulaton, Dscusson and Generalzaton, Proceedngs of the Ffth Internatonal Conference on Genetc Algorthms, pp , [11] M. P. Fourman, Compacton of Symbolc Layout Usng Genetc Algorthms, In J. J. Grefenstette (Ed.), Genetc Algorthms and Ther Applcatons: Proceedngs of the Frst Internatonal Conference

10 on Genetc Algorthms, pp , Hllsdale, NJ:Lawrence Erlbaum, [12] S. Geman and D. Geman, Stochastc Relaxaton, Gbbs Dstrbtons, and the Bayesan Restoraton of Images, IEEE Trans. on Pattern Analyss and Machne Intellgence, vol.pami-6, no.6, November [13] D. E. Goldberg, Genetc Algorthms n Search, Optmzaton and Machne Learnng. Readng, MA: Addson-Wesley, [14] D. E. Goldberg, K. Deb, and J. Horn, Massve Multmodalty, Decepton, and Genetc Algorthms, IllGAL Report No Urbana-Champagn, IL: Department of General Engneerng, Unversty of Illnos at Urbana-Champagn, [15] D. E. Goldberg, K. Deb, and J. Horn, Massve Multmodalty, Decepton, and Genetc Algorthms, Parallel Problem Solvng From Nature, vol.2, pp.37-46, [16] D. E. Goldberg and J. Rchardson, Genetc Algorthms wth Sharng for Multmodal Functon Optmzaton, Proceedngs of the Second Internatonal Conference on Genetc Algorthms, pp.41-49, [17] P. Hajela, and C. -Y. Ln, Genetc Search Strateges n Multcrteron Optmal Desgn, Structural Optmzaton, vol.4, pp , [18] J. Holland, Adaptaton n Natural and Artfcal systems. MI: Unversty of Mchgan Press, [19] J. Horn and N. Nafplots, Multobjectve Optmzaton Usng the Nched Pareto Genetc Algorthm, IllGAL Report No Urbana- Champagn, IL: Department of General Engneerng, Unversty of Illnos at Urbana-Champagn, [20] J. Horn, N. Nafplots and D. E. Goldberg, A Nched Pareto Genetc Algorthm for Multobjectve Optmzaton, In Proceedngs of the Frst IEEE Conference on Evolutonary Computaton, IEEE World Congress on Computatonal Intellgence, pp.82-87, Pscataway, NJ. IEEE Servce Center, [21] L. Ingber, Very Fast Smulated Re-annealng, Mathematcal and Computer Modellng, vol.16, no.11, pp , [22] G. Jones, R. D. Brown, D. E. Clar, P. Wllet and R. C. Glen, Searchng Databases of Two-Dmensonal and Three-Dmensonal Chemcal Structures Usng Gentc Algorthms, In S. Forrest (Ed.) Genetc Algorthms: Proceedngs of the Ffth Internatonal Conference, pp , San Mateo, CA:Morgan Kaufmann, [23] S. Kauffman, Adaptaton on Rugged Ftness Landscapes, n Lecture n the scences of complexty (D. L. Sten, ed.), pp [24] S. Krpatrc, C. D. Gelatt, and M. P. Vecch, Optmzaton by Smulated Annealng, Scence, vol.220, pp , [25] F. Kursawe, A Varant of Evoluton Strateges for Vector Optmzaton, In H,-P, Schewefel and R. Männer (Eds.), Parallel problem solvng from nature, 1 st worshop, proceedngs, volume 496 of Lecture notes n computer scence, pp , Berln: Sprnger-Verlag, [26] P. J. M. V. Laarhoven and E. H. L. Aarts, Smulated Annealng: Theory and Applcatons. Dordrecht, The Netherlands, D. Redel, [27] N. Metropols, A. W. Rosenbluth, M. N. Rosenbloth, A. H. Teller, and E. Teller, Equaton of State Calculaton by Fast Computng Machnes, Journal of Chemcal Physcs, vol.21, pp , [28] M. Mtchell, An Introducton to Genetc Algorthms, Readng: MIT press, [29] D. Mtra, F. Romeo, and A. Sangovann-Vncentell, Convergence and Fnte-tme Behavor of Smulated Annealng, Advanced Appled Probablty, vol.18, pp , [30] L. J. Par, Hybrd Evolutonary Algorthms wth Heurstc Operators for Combnatoral Optmzaton Problems, Ph. D Thess, [31] L. J. Par, C. H. Par, C. Par, and T. Lee, Applcaton of Genetc Algorthms to Parameter Estmaton of Boprocesses, Med. & Bol. Eng. & Comput., vol.35, pp.1-3, Jan [32] I. Rechenberg, Evolutons Stratege: Optmerung Technscher Systeme Nach Prnzpen der Bologschen Evoluton. Stuttgart Frommann- Holzboog, [33] J. D. Schaffer, Multple Objectve Optmzaton wth Vector Evaluated Genetc Algorthms, In J. J. Grefenstette, edtor, Proceedngs of the Frst Internatonal Conference on Genetc Algorthms, pp Lawrence Earlbaum, [34] J. D. Schaffer, D. Whteley, and L. J. Eshelman, Combnaton of Genetc Algorthms and Neural Networs: A Survey of the State of Art, Proceedng of COGANN-92, pp.1-37, [35] N. Srnvas and K. Deb, Multobjectve Optmzaton Usng Nondomnated Sortng n Genetc Algorthms, Evolutonary Computatons, vol.2, no.3, pp , [36] G. Sysweda, and J. Palmucc, The Applcaton of Genetc Algorthms to Resource Schedulng, In R. K. Belew and L. B. Booer (Eds.) Genetc Algorthms: Proceedngs of the Fourth Internatonal Conference, pp , San Mateo, CA:Morgan Kaufmann, [37] H. H. Szu, and R. L. Hartley, Fast Smulated Annealng, Physcs Letters A, vol.122, pp , [38] J. F. Whdborne, D.-W. Gu, and I. Postlethwate, Smulated Annealng for Multobjectve Control System Desgn, IEE Proceedngs of Control Theory Applcaton, vol.144, no.6, November [39] D. H. Wolpert and W. G. Macready, No Free Lunch Theorems for Search, Techncal Report SFI-TR , Santa Fe Insttute, Santa Fe, 1995.

11 [40] D. H. Wolpert and W. G. Macready, No Free Lunch Theorems for Optmzaton, IEEE Trans. on Evolutonary Computaton, vol.1, no.1, pp.67-82, [41] X. Yao, A New Smulated Annealng Algorthm, Internatonal Journal of Computer Mathematcs, vol.56, pp , Dongyung Nam receved the B.S. degree n 1994, and M.S. degree n 1996 from the Seoul Natonal Unversty, Seoul, Korea, all n electrcal engneerng. He s currently pursung the Ph.D. degree n Electrcal Engneerng at Korea Advanced Insttute of Scence and Technology. Hs research nterests nclude evolutonary algorthm, artfcal neural networ, artfcal lfe, and multobjectve optmzaton. Cheol Hoon Par He receved the B.S. degree n Electroncs Engneerng wth the best student award from Seoul Natonal Unversty, Seoul, Korea, n 1984 and the M.S. and Ph.D. degrees n Electrcal Engneerng from Calforna Insttute of Technology, Pasadena, Calforna, n 1985 and 1990, respectvely. He joned the Department of Electrcal Engneerng at the Korea Advanced Insttute of Scence and Technology n 1991, where he s currently an Assocate Professor. Hs current research nterests are n the area of ntellgent systems ncludng ntellgence, neural networs, fuzzy logc, evolutonary algorthms, and applcaton to recognton, nformaton processng, ntellgent control, and optmzaton. He s a member of IEEE, INNS, and KITE.