An Adaptive Hyperplane Approach for Multiple Objective Optimization Problems with Complex Constraints

Transcription

1 An Adaptve Hyperplane Approach for Multple Obectve Optaton Probles wth Cople Constrants Runwe Cheng Departent of Industral and Inforaton Systes Engneerng Ashaga Insttute of Technology Roo 0, Hausan Bunyo-u, Toyo -000, Japan Phone/Fa: +8() Mtsuo Gen Departent of Industral and Inforaton Systes Engneerng Ashaga Insttute of Technology Oae-chao 68, Ashaga , Japan Phone: +8(84) Fa: +8(84)64-07 Shuel S. Oren Departent of Industral Engneerng and Operatons Research Unversty of Calforna Bereley, CA 9470 USA Phone: +(50) Fa: +(50)64-40 Abstract Many real-world decson probles nvolve ultple and conflctng obectves, whch need to be opted sultaneously whle respectng varous cople constrants. In ths paper, we nvestgated how to solve these nds of probles by usng genetc algorths. A new ftness assgnent ethod for ultple obectve optaton probles - the adaptve hyperplane-based ftness assgnent ethod - was proposed n order to gve a genetc algorth the search pressure toward the postve deal pont n the obectve space. An adaptve penalty functon was used together wth the adaptve hyperplane ethod n order to let genetc search eplore the opta through both feasble and nfeasble areas n the soluton space. A Pareto soluton reservng ethod was ncorporated nto noral genetc algorth loop n order to antan a set of Pareto solutons durng the evolutonary process. INTRODUCTION Optaton deals wth the probles of seeng solutons over a set of possble choces to opte certan crtera. If there s only one crteron to consder, t becoes to a sngle obectve optaton proble, a type studed etensvely for the past 50 years. If there s ore than one crteron and they ust be treated sultaneously, we have ultple obectve optaton probles (Steuer, 986). Multple obectve probles arse n the desgn, odelng, and plannng of any cople real-world probles. Alost every portant real-world decson proble nvolves ultple and conflctng obectves that need to be tacled whle respectng varous cople constrants, leadng to overwhelg proble coplety. Multple obectve optaton probles have been recevng growng nterest fro researchers wth varous bacgrounds snce early 960. Durng the last two decades, genetc algorths have receved consderable attenton as a novel approach to ultple obectve optaton probles, nown as genetc ultobectve optatons or evolutonary ultobectve optaton (Fonseca and Fleg, 995). The growng researches on applyng genetc algorths to ultple obectve optaton probles present a fordable theoretcal and practcal challenge to the atheatcal county. In ths paper, we nvestgated how to solve these nds of probles by usng genetc algorths. A new ftness assgnent ethod for ultple obectve optaton probles - the adaptve hyperplane-based ftness assgnent ethod - was proposed n order to gve a genetc algorth the search pressure toward the postve deal pont n the obectve space. An adaptve penalty functon was used together wth the adaptve hyperplane ethod n order to let genetc search eplore the opta through both feasble and nfeasble areas n the soluton space. A Pareto soluton reservng ethod was ncorporated nto noral genetc algorth loop n order to antan a set of Pareto solutons durng the evolutonary process. We have appled the hyperplane approach to several real world probles. The eperental results were very encouragng and showed that t can be easly appled to ost real world ultobectve optaton probles. MULTIOBJECTIVE OPTIMIZATION PROBLEM Wthout loss of generalty, a ultple obectve optaton proble can be represented forally as follows: { = f ( ), = f ( ),..., s.t. g ( ) 0, =,,..., = f ( )

2 n where R s a vector of n decson varables, f ( the th obectve functon, and g ( the th neualty constrant. The functons for an area of feasble solutons denoted by the set S as follows: n S = { R g ( ) 0, =,,...,, 0 The set of ages of all ponts n S fors the feasble regon n crteron space, denoted by the set Z as follows: Z = { R = f ( ), =,,...,, S We soete graph ultple obectve optatons n both decson space and crteron space. There s a specal pont n the crteron space called an deal pont or a postve deal soluton, denoted by * * * * * = (,,..., ), where = sup{ f ( S. The pont s called an deal pont because usually t s not attanable. * Note that, ndvdually, ay be attanable, but to fnd a pont that can e each obectve functon f ( sultaneously s usually very dffcult. In contradcton to the postve deal pont, a negatve deal soluton s also defned to represent the pessstc status n the crteron space, denoted by = (,,..., ), where = sup{ f ( ) S. Slar to the postve deal pont, the negatve deal pont s also usually not attanable. Indvdually, ay be attanable. In prncple, ultple obectve optaton probles are very dfferent fro sngle obectve optaton probles. For the sngle obectve case, one attepts to obtan the best soluton, whch s absolutely superor to all other alternatves. In the case of ultple obectves, there does not necessarly est such a soluton that s the best wth respect to all obectves because of ncoensurablty and conflct aong obectves. A soluton ay be best n one obectve but worst n other obectves. Therefore, there usually ests a set of solutons for the ultple obectve cases, whch cannot be sply copared wth each other. For such solutons, called nondoated solutons or Pareto optal solutons, no proveent n any obectve functon s possble wthout sacrfcng at least one of the other obectve functons. For a gven nondoated pont n the crteron space Z, ts age pont n the decson space S s called effcent or nonnferor. A pont n S s effcent f and only f ts age n Z s nondoated. In real decson ang cases, we are usually ased to select one of those nondoated solutons as a fnal soluton to a gven proble. It s ost lely, however, that we wll be unable to settle for one of those solutons wthout provdng addtonal preferences regardng varous obectves. Therefore, how to ae a fnal choce fro those alternatve solutons essentally depends on one's subectve preferences. Conceptually, the preference ntends to gve an order to the ncoparable solutons wthn the effcent set by usng ones' value udgents on obectves, The preference reflects ether ones' tradeoffs aong obectves or ephass for soe partcular obectves accordng soe pror nowledge to the proble. Wth a gven preference, we can order the alternatve solutons n the nondoated set, and then we can obtan a fnal soluton, whch s the usual outcoe of a decson ang process. Such a fnal soluton s called bestcoprosed soluton. The general epectaton for a decson ang process can be ether to obtan a coprosed or preferred soluton or to dentfy all nondoated solutons. Therefore, there are bascally two nds of technues for consttutng a soluton ethod to ultple obectve optaton probles: () generatng approaches and () preferencebased approaches. The generatng approaches have been developed to dentfy an entre set of Pareto solutons or an approaton. Preference-based approaches attept to obtan a coprsed or preferred soluton. If we have no pror nowledge for preference structure over obectves, we have to adopt the generatng approach to eae all nondoated alternatves. If we have soe deas of the relatve portance of obectves, we can uantfy the preference. Wth the preference nforaton, a coprosed or preferred soluton can be dentfed. Fro the vewpont of soluton technues, ost tradtonal ethods reduce ultple obectves nto a sngle obectve, and then solve the proble wth atheatcal prograg tools. To utle atheatcal prograg tools to solve a ultple obectve proble, we frst need to epress our preference n ters of nubers, so that the larger the nuber, the stronger the preference. By usng scalaraton technues, ultple obectve optaton probles are usually transfored nto a sngle obectve or a seuence of sngle obectve optaton probles; then tradtonal technues can be adapted to solve the altered probles (Morrs and Oren, 980, Arbel and Oren, 999). Faous ethods aong the are utlty functon approach, weghted-sus approach, and coprose approach. FEATURES OF GENETIC SEARCH The nherent characterstcs of genetc algorths deonstrate why genetc search ay be well suted to ultple obectve optaton probles. The basc feature of genetc algorths s ultple drectonal and global searches by antanng a populaton of potental solutons fro generaton to generaton. The populaton-topopulaton approach s useful when eplorng Pareto solutons. Genetc algorths do not have uch atheatcal reureents and can handle all types of obectve functons and constrants. Because of ther evolutonary nature, genetc algorths can be used to search for solutons wthout regard to the specfc nner worngs of probles. Therefore, t s hoped that we can solve any ore cople probles by usng genetc algorths. Because genetc algorths provde us a great fleblty to hybrde wth conventonal ethods nto ther an fraewor, we can eplot the advantages of both genetc algorths and conventonal ethods to establsh uch

3 ore effcent pleentatons to ultple obectve optaton probles. 4 FITNESS ASSIGNMENT MECHA- NISM Genetc algorths are essentally a type of etastrategy of solutons. When applyng genetc algorths to solve a gven proble, t s necessary to refne each of ther aor coponents, such as encodng ethods, recobnaton operators, ftness assgnent, selectons, constrants handlng, and so on, to obtan an effectve pleentaton to the gven proble. Because ultple obectve optaton probles are natural etensons of constraned and cobnatoral optaton probles, any useful ethods developed for constraned and cobnatoral optaton probles durng the past two decades are readly applcable. Therefore, when consderng how to adapt genetc algorths to ultple obectve optaton probles, we ust need to eae soe specal ssues concernng the probles. 4. FITNESS ASSIGNMENT METHODS One of specal ssues arsng n solvng ultple obectve optaton probles by use of genetc algorths s how to detere the ftness value of ndvduals accordng to ultple obectves. The ftness assgnent echans has been studed etensvely durng the past decade and several ethods have been suggested and tested. Roughly, these ethods can be classfed as follows: () vector evaluaton ethod, () Pareto-based ethod, () weghted-su ethod, (4) coprose ethod, and (5) goal prograg ethod. Perhaps, the frst notable wor to etend sple genetc algorths to solve the ultple obectve optaton probles s the vector evaluaton ethod proposed by Schaffer (985). Instead of usng a scalar ftness easure to evaluate each chroosoe, t uses a vector ftness easure to create the net populaton. There are two nds of Pareto-based ethods: Pareto ranng and Pareto tournaent. The Pareto ranngbased ftness assgnent ethod was frst suggested by Goldberg as a eans of achevng eual reproductve potental for all Pareto ndvduals (Goldberg, 989). It ncludes two aor steps: () Sort the populaton based on Pareto ranng. () Assgn selecton probabltes to ndvduals accordng to the ranng. The Pareto tournaent ethod was proposed by Horn, Nafplots, and Goldberg (994). Instead of nondoated sortng and ranng selecton ethod, a nched Pareto concept was used n tournaent, where a Pareto soluton wth least nuber of ndvduals n ts neghbor wns the copetton. The weghted-su ethod taes ts basc deas fro conventonal ultobectve optatons. It assgns weghts to each obectve functon and cobnes the weghted obectves nto a sngle obectve functon. Conceptually, t s sple to understand and easy to copute; but to ae t wor reures only a proper weghtng vector. Theren, however, les the dffculty. Once ebedded n genetc algorths, ts weaness can be copensated for by the powers of populaton-based and evolutonary search. Several weght-adust ethods have been proposed n order to fully utle the power of genetc search: () fed weght approach, () rando weght approach, and () adaptve weghts (Gen and Cheng, 000). Recently, ths approach was cobned wth a spannng treebased genetc algorth appled to a ult-obectve transportaton proble (Gen and L, 999). Cheng and Gen proposed the coprose ethod as a eans to obtan a coprosed soluton nstead of generatng all Pareto solutons (Cheng and Gen, 998). Its basc dea and technues are borrowed fro conventonal ultple obectve optatons. The coprose approach dentfes solutons that are closest to the deal soluton as detered by soe easure of dstance. Goal prograg s one of the powerful technues for solvng the ultobectve optaton probles. Gen, Lu and Ida nvestgated the applcaton of genetc algorths to solve nonlnear goal prograg probles (Gen, Lu and Ida, 996). Because lecographc orderng aong obectves s preferred n goal prograg, ndvduals are sorted on the value of obectves n a lecographc anner n ther genetc algorths. Indvdual ftness values are then assgned by nterpolatng fro the best to the worst accordng to an eponental functon. Accordng to how uch preference nforaton s ncorporated nto the ftness functon, these approaches range fro coplete preference nforaton gven, as when cobnng obectve functons drectly or prortng the, to no preference nforaton gven, as wth Paretobased approaches. In addton, a notable feature s that gven a rough preference, progressve refneent of the preference can be carred out by evolutonary search. The progressve refneent of preferences s le an nteractve procedure often used n ultple obectve optaton, where preferences are odfed at each teraton by decson aers. What aes t unue s the refneent echans: The preference s refned gradually through the evolutonary search by soe adaptve refneent echans, not by the nterventon of decson aers at each teraton. Of course, an nteractve procedure can also be ebedded n genetc search to gude preference refnng. 4. TWO BASIC APPROACHES Fro the vewpont of ethodology, there are two basc approaches to ultple obectve optatons: generatng approach and preference-based approach. Generatng approaches are used to dentfy an entre set of Pareto solutons or an approaton, whereas preference-based approaches attept to obtan a coprsed or preferred soluton. Conceptually, the vector evaluaton approach, the Pareto ranng-based approach, and the rando weghtng approach are desgned as the generatng ethods; the coprose approach, the adaptve weghtng

4 approach, and the goal prograg approach are desgned as the preference-based approaches. In ultobectve optatons, generatng and preference-based ethods both ehbt ther strengths and weanesses. Generatng technues reure decson aers to ae a udgent by selectng fro aong entre Pareto solutons. For probles wth ore than three crtera, ang a choce becoes very coplcated, ncreasng n dffculty approately eponentally wth the nuber of crtera. Coputatonal costs also ncrease rapdly wth the nuber of crtera. In genetc ultobectve optatons, the stuaton essentally does not change. In contrast, preference-based technues see not to put as great a burden on decson aers as used n ultobectve optaton. Because preferences can be refned gradually along wth the evolutonary process, a rough preference can be ade to wor by evolutonary search. A soluton ethod can be desgned as ether a generatng ethod for obtanng an entre set of Pareto solutons or a preference-based ethod for obtanng a preferred or coprosed soluton. In ultobectve optatons, we cannot have a soluton ethod whch pleents the two dstnct deas nto one soluton procedure, but n genetc ultobectve optaton, we can. 4. THE CONCEPT OF PARETO SOLUTION In a strct sense, the ter of Pareto soluton used n genetc algorths has a dfferent eanng as used n a conventonal way. In the orgnal defnton, a pont s sad to be a Pareto soluton f and only f t s a nondoated pont wth respect to all ponts n the crteron space for a gven proble. In genetc algorths, Pareto solutons are dentfed at each generaton. Because a populaton at each generaton contans only partal solutons of the orgnal proble, a Pareto soluton has ts eanng only wth respect to all solutons currently eaed. A nondoated soluton n one generaton ay becoe doated by a new soluton eerged n a later generaton. Therefore, for a gven generaton of genetc algorths, a Pareto soluton obtaned n that generaton ay be a true Pareto soluton to the proble, or t ay not be. There s no guarantee that a genetc algorth certanly produces Pareto solutons to a gven proble. But a genetc algorth wll provde a better approaton of Pareto solutons. How to antan a set of nondoated ndvduals durng the evolutonary process s a specal ssue for ultple obectve optaton probles. Bascally, there are two dfferent ways to handlng Pareto solutons, whch lead to two dfferent overall structures of genetc algorths pleentatons: () preservng Pareto solutons separately fro populaton pool and () wthout preservng echanss. In ost estng ethods, Pareto solutons are dentfed at each generaton and used only to calculate ftness values or rans for each chroosoe. No echans s provded to guarantee that Pareto solutons generated durng the evolutonary process enter the net generaton. In other words, soe Pareto solutons ay get lost durng the evolutonary process. To avod such saplng errors, a preservng echans for Pareto solutons has been suggested by any researchers (Gen and Cheng, 000). A specal pool for preservng Pareto solutons s added onto the basc structure of genetc algorths. At each generaton, the set of Pareto solutons s updated by deletng all doated solutons and addng all newly generated Pareto solutons. The overall structure wth Pareto preservng s gven as follows: Procedure: Pareto genetc algorths begn t = 0 Intale P(t); Obectves P(t); Pareto E(t); Ftness P(t); whle (not teraton condton) do begn Crossover P(t); Mutaton P(t); Obectve P(t); Update Pareto E(t); Ftness P(t); Selecton P(t+) fro P(t); t = t+; end end Wthout a preservng echans, Pareto solutons can be gathered only fro the last generaton. If the ethod used has a tendency of specaton as entoned by Schaffer (985), the entre populaton wll converge toward the ndvdual optu regons after a large nuber of generatons. The preservng echans s, to a certan etent, helpful to e specaton through a Pareto preservng procedure at each generaton. 5 ADAPTIVE HYPERPLANE AP- PROACH Adaptve hyperplane approach belongs to the type of adaptve weght ethod. It constructs a hyperplane by soe specal ponts n obectve space n each generaton and ftness values of ndvduals are then calculated based on the hyperplane. The hyperplane s adusted adaptvely based on the current generaton to obtan a search pressure toward the postve deal pont n the obectve space. 5. ADAPTIVE WEIGHT The basc dea of assgnng weghts to each obectve functon and cobnng the nto a sngle-obectve functon was frstly proposed by Zadeh (Zadeh, 96). The weghted-sus ethod can be represented as follows: ( ) = w f ( ) = The weght w can be nterpreted as the relatve ephass or worth of that obectve when copared to the other

5 obectves. In the other words, the weght can be nterpreted as representng our preference over obectves. Therefore, an optal soluton to a gven proble relates to a partcular preference structure. Moreover, the optal soluton to the proble s a nondoated soluton provded that all the weghts are postve. Because of the nuercal orderng by the weghted-su functon, there s no abguty n preference coparson. For any two ponts, ether one s better, worse or euvalent to another. Eactly one of the three cases ust happen. There are no such thngs as ndefnte sets n the preference structure. The adaptve weghts approach proposed n ths paper adusts weghts adaptvely accordng to the current generaton n order to obtan a search pressure toward to the postve deal pont. Ths approach s desgned for genetc algorths n order to fully utle the power of genetc search. It wors only due to the nature of populatonbased evolutonary search of genetc algorths. Consder the aton proble wth obectves descrbed n Secton. For the solutons eaed n each generaton, we defne + two etree ponts: the u etree pont and the u etree pont n crtera space as follows: + = {, = {, where and are the al value and al value for obectve n current populaton. Let P denote the set of current populaton. For a gven ndvdual, the al value and al value for each obectve are defned as the follows: = { f ( S, = { f( S, =,,..., The hyper-parallelogra defned by the two etree ponts s a al hyper-parallelogra contanng all current solutons. The two etree ponts are renewed at each generaton. The u etree pont wll gradually approate to the postve deal pont. The adaptve weght for obectve s calculated by the followng euaton: w = =,,..., =,,..., For a gven ndvdual, the weghted-su obectve functon s gven by the followng euaton: ( ) = f( w ( ) = = = For the cases that all obectve functons taes only postve value, the euaton can be splfed as follows: ( ) = f( w = = = As the etree ponts are renewed at each generaton, the weghts are renewed accordngly. Euaton above s a hyperplane defned by the followng etree pont n current solutons: { {... {... {,,,, The hyperplane dvdes the crtera space Z nto two halfspaces: one half space contans the postve deal pont, denoted as Z +, and the other half space contans the negatve deal pont, denoted as Z -. All Pareto solutons eaed le n the space Z +, and all ponts lyng n the Z + have larger ftness values than those n the space Z -. As the u etree pont approates to ts possble largest value along wth the evolutonary progress, the hyperplane wll gradually approach to the postve deal pont. Therefore, the adaptve weght ethod can readust ts weghts accordng to the current populaton n order to obtan a search pressure toward to the postve deal pont. Let us see an eaple of bcrtera aton proble gven below: { = f ( ), = f ( ) s.t. g ( ) 0, =,,..., For a gven generaton, two etree ponts are dentfed as: = { ( ), =,,..., pop_ se = { ( ), =,,..., pop_ se = { ( ), =,,..., pop_ se = { ( ), =,,..., pop_ se and the adaptve weghts are calculated as follows: w = and w = The weghted-su obectve functon s then gven by the followng euaton: ( = w + w = w f( + w f( It s an adaptve ovng lne defned by the etree ponts (, ) and (, ) as shown n Fgure. The rectangle defned by the etree ponts (, ) and (, ) s the al rectangle contanng all current solutons.

6 Fgure : Adaptve weghts and adaptve hyperplane for a bcrtera case. Conceptually, the weghted-su approach can be vewed as an etenson of ethods used n ultobectve optaton to genetc algorths. It assgns weghts to each obectve functon and cobnes the weghted obectves nto a sngle obectve functon. In fact, the weghted-su approaches used n genetc algorths are very dfferent n nature fro that n conventonal ultobectve optatons. In ultobectve optaton, the weghted-su approach s used to obtan a coprose soluton. To ae the ethod wor, all that s needed s a good weghtng vector. It s usually very dffcult to detere a set of approprate weghts for a gven proble. In genetc algorths, the weghted-su approach s prarly used to adust genetc search toward to the Pareto fronter. Weghts are readusted adaptvely along wth the evolutonary process. Therefore, a good weghtng vector s not a andatory precondton to ang genetc algorths wor. In addton, the drawbacs ehbted n the ultobectve optaton can be copensated by the powers of populaton-based search and evolutonary search. 5. ADAPTIVE PENALTY FUNCTION Penalty technue s perhaps the ost coon technue used to handle nfeasble solutons n genetc algorths for constraned optaton probles. In essence, ths technue transfors the constraned proble nto an unconstraned proble by penalng nfeasble solutons, n whch a penalty ter s added to the obectve functon for any volaton of the constrants (Gen and Cheng, 997). The basc dea of the penalty technue s borrowed fro conventonal optaton. In conventonal optaton, the penalty technue s used to generate a seuence of nfeasble ponts whose lt s the optal soluton to the orgnal proble. The aor concern s how to choose a proper value of penalty so as to hasten convergence and avod preature teraton. In genetc algorths, the penalty technue s used to eep a certan aount of nfeasble solutons n each generaton so as to enforce genetc search toward an optal soluton fro both sdes of feasble and nfeasble regons. The aor concern s how to detere the penalty ter so as to stre a balance between the nforaton preservaton (eepng soe nfeasble solutons) and the selectve pressure (reectng soe nfeasble solutons), and both under-penalty and over-penalty (Gen and Cheng, 996a). Gven an ndvdual n current populaton P(t), the adaptve penalty functon s constructed as follows: where p( ) = = α b ( b b ( = {0, g ( b b = { ε, b ( P( t) where b ( s the value of volaton for constrant for the th chroosoe, b s the u volaton for constrant aong current populaton, and ε s a sall postve nuber used to avod ero-dvson. For hghly constraned optaton probles, nfeasble solutons tae a relatvely bg porton aong the populaton at each generaton. The penalty approach adusts the rato of penaltes adaptvely at each generaton n order to ae a balance between the preservaton of nforaton and the pressure for nfeasblty to avod over-penalty. Wth the penalty functon, the ftness functon then taes the followng for; eval( = ( p( 6 APPLICATION TO SOME PROB- LEMS 6. INTERVAL PROGRAMMING PROBLEM Let us consder the followng eaple wth an nterval obectve functon: ( = [5, 7] + [5,0] s.t. g ( = + + g ( = + 4 g ( = , =,,. nteger + [0,0] The proble can be transfored nto the followng bcrtera prograg proble: s.t. L ( = C ( = g ( = + + g ( = + 4 g ( = , =,,. nteger

7 The nteger vector was used as the chroosoe representaton. Unfor crossover and rando perturbaton utaton were used as genetc operators. The ftness value of each ndvdual was calculated by the hyperplane ethod. The Pareto solutons found by the proposed ethod are shown n Fgure (Gen and Cheng, 996b). Fgure : Pareto solutons obtaned by our ethod and Anea s ethod Fgure : Pareto Solutons to the Interval Prograg Proble 6. BICRITERIA LINEAR TRANSPORTATION PROBLEM Consder followng bcrtera lnear transportaton proble gven by Anea and Nar (978): = s.t. = = 8 = = 8, =, = = = 9, =, 0, for all and = = + 9 = 4, + 8 = = = BICRITERIA MINIMUM SPANNING TREE PROBLEM Consder the u spannng tree proble, where each edge has two assocated postve real nubers. Then t can be forulated as the followng bcrtera optaton proble (Zhou and Gen, 999). ( = w, ( = w = = s.t. T where s a bnary decson varable defned as:, = 0, f edge e sselected otherwse and T denotes the set of all spannng trees correspondng to a gven proble. The Prüfer nuber was adopted as the tree encodng. It s capable for representng all possble spannng trees. Unfor crossover and perturbaton utaton were used as genetc operatons, and the adaptve hyperplane ethod was used to detere ftness values for each tree. The Pareto solutons found by the proposed ethod are depcted n Fgure 4 (Zhou and Gen, 999). The allocaton atr was used as the chroosoe representaton. The specal crossover operaton and utaton operaton proposed by Vgnau and Mchalewc (99) were adopted. The ftness value for each ndvdual was detered by the adaptve hyperplane ethod. The Pareto solutons found by the proposed ethod are depcted n Fgure (Yang and Gen, 994). Fro t we can now that the proposed ethod s uch ore effectve than the ethod of Anea and Nar. Fgure 4: Pareto solutons found by the proposed ethod for a 50-verte nstance.

8 7 CONCLUSIONS In general, ultple obectve optaton probles are too cople to be solved easly. There are two nds of dffcultes assocated wth the proble solvng that need to be dstngushed: () dffcultes nherent n probles and () dffcultes related to soluton technues. The ost profound drawbac of any conventonal ethods s that they are very senstve toward to the value of weghts, or the prescrbed order of obectves, or the shape of utlty functons. In essence, such nds of dffcultes are caused by soluton technues but not the proble tself. However, n genetc ultobectve optatons, the drawbacs ehbted n conventonal ethods can be copensated by the powers of populaton-based search and evolutonary search. In ths paper, we proposed a new ftness assgnent ethod for ultple obectve optaton probles: the adaptve hyperplane ethod. It s desgned for genetc algorths to fully utle the power of genetc search. In ths ethod, weghts are adusted adaptvely fro generaton to generaton. It provdes a search pressure toward to the Pareto fronter. An adaptve penalty functon was used together wth the adaptve hyperplane ethod n order to let genetc search eplore the opta through both feasble and nfeasble areas n the soluton space. A Pareto soluton reservng ethod was ncorporated nto a noral genetc algorth loop n order to antan a set of Pareto solutons durng the evolutonary process. The proposed approach has been appled to several real world decsonang probles wth ultple conflct obectves and cople constrants. The eperental results are very encouragng and show that the ethod can be easly appled to ultple obectve optaton probles. Acnowledgents Ths research wor was supported by the Internatonal Scentfc Research Progra, the Grant-n-Ad for Scentfc Research (No : ) by the Mnstry of Educaton, Scence and Culture, the Japanese Governent. References A. Arbel and S. Oren (999). Usng approate gradents n developng nteractve nteror pral-dual ultobectve lnear prograg algorth, European Journal of Operatonal Research: 89:0-. Y. Anea and K. Nar (978). Bcrtera transportaton proble. Manageent Scences: 5: R. Cheng and M. Gen (998). Coprose approachbased genetc algorths for bcrteron shortest path probles. Techncal report, Ashaga Insttute of Technology, Japan. C. Fonseca and P. Fleg (995). An overvew of evolutonary algorths n ultobectve optaton. Evolutonary Coputaton ():-6. M. Gen and R. Cheng (996a). A survey of penalty technues n genetc algorths. Proceedngs of 996 IEEE Internatonal Conference on Evolutonary Coputaton: M. Gen and R. Cheng (996b). Interval prograg usng genetc algorths. Intellgent Autoaton and Control: 4: M. Gen and R. Chang (997). Genetc Algorths & Engneerng Desgn. New Yor: John Wley & Sons. M. Gen and R. Chang (000). Genetc Algorths & Engneerng Optaton. New Yor: John Wley & Sons. M. Gen and Y. L (999). Spannng tree-based genetc algorths for bcrtera fed charge transportaton proble. Proceedng of the Congress on Evolutonary Coputaton: M. Gen, B. Lu and K. Ida (996). Evoluton progra for deterstc and stochastc optatons. European Journal of Operatonal Research: 94(): D. Goldberg (989). Genetc Algorths n Search, Optaton & Machne Learnng. Readng: Addson-Wesley. J. Horn, N. Nafplots and D. Goldberg (994). A nched Pareto genetc algorth for ultobectve optaton. Proceedngs of Frst IEEE Conference on Evolutonary Coputaton: P. Morrs and S. Oren (980). Multattrbute Decson ang by seuental resource allocaton, Operatons Research, 8(): -5. J. Schaffer (985). Multple obectve optaton wth vector evaluated genetc algorths. Proceedngs of the Frst Internatonal Conference on Genetc Algorths R. E. Steuer (986). Multple crtera optaton: theory, coputaton, and applcaton. New Yor, John Wley & Sons. G. Vgnau and Z. Mchalewc (99). A Genetc Algorth for the Lnear Transportaton Proble. IEEE Transactons on Systes, Man, and Cybernetcs: : X. Yang and M. Gen (994). Evoluton progra for bcrtera transportaton proble. Proceedngs of the 6th Internatonal Conference on Coputer and Industral Engneerng: L. Zadeh (96). Optalty and non-scalar-valued perforance crtera. IEEE Transactons on Autoatc Control: 8(59). D. Zheng, M. Gen and R. Cheng (999). Multobectve optaton usng genetc algorths. Engneerng Valuaton and Cost Analyss: :0-0. G. Zhou and M. Gen (999). Genetc algorth approach on ult-crtera u spannng tree proble. European Journal of Operatonal Research: 4:4-5.