MULTI-OBJECTIVE PROGRAMMING UNDER UNCERTAINTY TO SELECT PORTFOLIO: A CASE STUDY IN STOCK EXCHANGE MARKET OF IRAN

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1 Mostafa EKHTIARI, MS Faculty of Idustral ad Mechacal Egeerg, Qazv Brach, Islamc Azad Uversty, Qazv, Ira Alreza ALINEZHAD, PhD Faculty of Idustral ad Mechacal Egeerg, Qazv Brach, Islamc Azad Uversty, Qazv, Ira MULTI-OBJECTIVE PROGRAMMING UNDER UNCERTAINTY TO SELECT PORTFOLIO: A CASE STUDY IN STOCK EXCHANGE MARKET OF IRAN Abstract. Recetly, the ecoomc crss has creased ucertaty degree of perso's vestmet facal marets. Therefore, cosderg ucertaty codtos to optmze facal portfolos s very mportace. Ths paper presets a decso support model called chace costraed goal attamet programmg (CCGAP) to optmze mult-obectve portfolo decso problem uder ucertaty evromet. CCGAP s a combato of the well ow classcal approach of chace costraed programmg (CCP) ad A-pror mult-obectve approach of goal attamet programmg (GAP) that s ow to be a eteso of goal programmg (GP). The proposed model s llustrated a real problem of mult-obectve portfolo selecto by Ira stoc maret, where goal values of stochastc obectves ad rate of retur of securtes are cosdered radom ad ormally dstrbuted dfferet scearos. Keywords: Stochastc programmg, Mult-obectve programmg, Chace costraed goal attamet programmg, Portfolo selecto. JELClassfcato: C6, C63, D8, G. Itroducto The Marowtz covarace model (Marowtz, 952), the classcal ad bascally approach to portfolo optmzato, s based o two coflctg optmzato obectves: the rs ad the epected retur. The mea-varace methodology (Marowtz, 952) for portfolo selecto problem has bee cetral to research actvty ad has served as a bass for the developmet of moder facal theory. Yet, the Marowtz model dsadvatages clude:

2 Mostafa Ehtar, Alreza Alezhad () The Marowtz model was geerally crtczed as ot effcet wth aomatc models of prefereces for choce uder rs (Bell et al., 988). () The Marowtz model was a quadratc model whch fally was o-lear. Because hs model was o-lear, so obtaed results were ofte local optmum. () The mea-varace model of Marowtz was the tme eeded to compute the covarace matr from hstorcal data especally whe problems were large scale ad the dffculty of solvg the large scale quadratc programmg problems. I the lterature, some algorthms such as those proposed by Sharpe (967), Elto et al. (976), Koo (990), ad Youg (998) are geerated order to learze ad mprove the effcecy calculato of the Marowtz covarace model (Shg ad Nagasawa, 999). Serba et al. (20) preseted descrpto of effcet froter for a portfolo made of three assets. The orgalty of ther wor was the combato of classfcato theory ad rs estmato theory to determe the best assets. Amr et al. (20) proposed a adr compromse programmg (NCP) for optmzato of mult-obectve portfolo problem Ira stoc maret. Georgescu (20) preseted stylzed facts dsplayed by the Bucharest stoc echage BET de ad the provded a applcato of GARCH modelg approach to predctg BET de mea-retur ad volatlty-retur processes. I real world, usg Marowtz model would ot have desred performace whe parameters are estmated. Because data about evets the past may ot be ow eactly due to errors measurg or dffcultes samplg, whlst data about evets the future may smply ot be ow wth certaty. For eample, the epected retur of a portfolo was used as a appromato because returs are radom, ad t s hardly possble that the vestor ca group a portfolo by attedg to all of ts possble returs (Lu, 999). I may stuatos there s a eed to mae, hopefully a optmal, decso uder ucertaty. Stochastc programmg deals wth a class of optmzato models ad algorthms whch some of the data may be subect to sgfcat ucertaty. Recetly, a few authors studed stochastc portfolo selecto problems. Several techques have bee troduced to solve stochastc programmg problems. The most popular techque s the chace costraed programmg (CCP) developed by Chares ad Cooper (963). The CCP method s a determstc equvalet formulato of a stochastc problem (Chares ad Cooper, 963). Ths techque allows the ucertaty related to several parameters of the problem such as the costrats coeffcets, the obectves coeffcets ad the goals values. The ma dea behd the CCP techque t to allow the decso-maer to geerate the most satsfactory solutos by mag compromses betwee the varous achevemet degrees of the obectves ad the rs assocated wth these obectves. I fact, the CCP attempts to mamze the epected value of the

3 Mult-obectve Programmg uder Ucertaty to Select Portfolo: A Case Study Stoc Echage Maret of Ira obectves whle assurg a certa probablty of realzato of the dfferet costrats (Aou et al., 2005). Usg mult-obectve stochastc models for stochastc portfolo selecto problems has bee cosdered by may researches ad DMs. For eample, model proposed by Shg ad Nagasawa (999) the mea ad varace of retur of securtes have several scearos wth ow probabltes. Muhlema et al. (978) developed a multobectve stochastc lear programmg formulato of portfolo selecto problem uder ucertaty. Ballestero (200) proposed a formulato of stochastc goal programmg based o utlty fucto ad Mea-Varace model. Tag et al. (200) proposed a chacecostraed problem of portfolo selecto to choose a portfolo wth mmzed stadard devato uder the codto that the probablty that the portfolo's rate of retur s greater tha a epected rate of retur s o less tha a cofdece level. Ballestero (2005) mmzed portfolo sem varace as the obectve fucto stochastc programmg model subect to stadard parametrc costrats whch led to the mea-sem varace effcet froter. Caaoglu ad Ozec (2009) cosdered ths problem a multple perod settg where the vestor mamzes the epected utlty of the termal wealth a stochastc maret. Xu ad Zhag (202) cosdered a stochastc programmg model where the obectve fucto was the varace of a radom fucto ad the costrat fucto was the epected value of the radom fucto. Tamz et al. (996) proposed a two-stage goal programmg model for portfolo selecto. Aou et al. (2005) eplctly troduced the DM s prefereces ad adapted CCP for the SGP model. They llustrated ther formulato through a portfolo selecto eample where the goal values assocated wth each obectve are cosdered ormally dstrbuted. The GP model was proposed by Chares et al. (955). I the GP there s o clear relato betwee the weghts ad the soluto foud as you mght ed up ew vertces both the soluto space, as well as the utopa set. Hece t seems approprate to use a method that besdes havg advatages of GP method, ca apply DM s deas optmzato process of stochastc problem. The goal attamet programmg (GAP) (Gembc, 974) s of methods pror category that ca be cosdered as a specal d of the GP method. Compared to the GP method, the GAP method has the followg advatages: () The GAP has fewer varables. If a mult-obectve problem has K obectve fuctos, m costrats ad varables, the compared to GAP method, the GP method wll has 2K more varables, so that by creasg the dmesos of problem, ths dfferece would be uderstood easer, (2) Whe dmesos of mult-obectve

4 Mostafa Ehtar, Alreza Alezhad problems are creased, the GAP method has hgher speed for optmzg such problems. If a lear programmg problem has varables ad m costrats, the mamum umber of eeded teratos for optmzg ths problem wll be equal to C m!/ m!( m)!(taha, 976), (3) The GAP method allows optmzato of olear mult-obectve problems. Gembc ad Hames (975) showed that the GAP ca optmze o-lear mult-obectve problems too. Ths shows capablty of smultaeous optmzato of lear ad o-lear mult-obectve problems by the GAP method, ad (4) There s a specfc relato betwee obectves preferece weghts ad optmal values of obectves the GAP method (Adersso, 2000) (see Secto 2 for more detals). Therefore our ma motve ths paper s to combe CCP approach ad GAP method ad preset chace costraed goal attamet programmg (CCGAP) method whch ca be used for optmzg mult-obectve stochastc problems. Besde the GAP method s advatages, deas of DM ca be used CCGAP method for decso mag uder ucertaty. I order to llustrate the proposed model, we test t by a mult-obectve problem of portfolo selecto from the Ira stoc echage maret. I ths paper, addto to parameters of rate of retur, goal values of stochastc obectves are cosdered radom too stochastc problem of mult-obectve portfolo selecto. The rest of the paper s orgazed as follows: I Secto 2 we troduce the GAP method ad epla the stochastc programmg ad the CCP approach. I Secto 3, we propose a chace costraed goal attamet programmg (CCGAP) model, whch combe the GAP model ad the CCP approach. The CCGAP allows DM to cosder several coflctg obectves wth radom parameters. I Secto 4, a real case s gve to llustrate the CCGAP model about problem of mult-obectve portfolo selecto. Meawhle we coclude ths paper Secto Portfolo selecto problem Cosder a mult-obectve problem of portfolo selecto as follows: ma f : c, =,..., K = = m f : c, r = K +,..., R subect to : r = r a b, =,..., m = =, 0, =,...,, ()

5 Mult-obectve Programmg uder Ucertaty to Select Portfolo: A Case Study Stoc Echage Maret of Ira where f =,, K are postve obectves such as lqudty, retur, f r r = K+,, R are egatve obectves such as rs, cost ad a, =,..., = b m are costrats of the problem. The s the decso varable of proporto to be vested the securty ad the sum of the proportos vested securtes s equal to. Utl ow, dfferet methods have bee troduced for solvg mult-obectve problems, e.g. Program (). Oe of these methods s GAP (Gembc, 974) whch, a vector of decso mag goals g = (g,, g t ) s assocated wth a set of obectves f = (f,, f t ). The relatve degree of uder-or over-achevemet of the goals s cotrolled by a vector of preferece weghts of obectves w = (w,, w t ). I the case that the more of the obectve s better, the fal model of GAP ca be obtaed as follows: m subect to : f w y g, =,..., K a b, =,..., m = = =, 0, y 0, =,...,, y where w > 0 (for =,, K) ad K (2) w. The term w y troduces a elemet of surplus to Program (2), whch otherwse mposes that the goals be rgdly met. I Program (2), g s goal related to obectve. If the case that less of the obectve s better, the fal model of GAP ca be obtaed as follows: m y subect to : f w y g, r = K +,..., R r r r = a b, =,..., m = =, 0, y 0, =,...,. (3)

6 Mostafa Ehtar, Alreza Alezhad The term w r y troduces a elemet of slacess to Program (3), whch otherwse mposes that the goals be rgdly met. The weghtg vector (w) eables the DM to epress a measure of the relatve trade-offs betwee the obectves. For stace, settg the vector w equal to the tal goals dcates that the same percetage uderor over-attamet of the goals (g) s acheved. The GAP method provdes a coveet tutve terpretato of the programmg problem whch s solvable usg stadard optmzato procedures. Oe of advatages of the GAP method s estece of a specfc relato betwee preferece weghts ad optmal values of each obectve (Adersso, 2000). See Fgure for better uderstadg. f 2 S g 2 Fgure. Represetato of relato betwee w ad f (for =, 2) values the GAP method f Fgure shows mmzato of a two obectves problem soluto space S. The dotted rego shows the optmal ad feasble Pareto froter of soluto space S. I order to optmze ths problem by the GAP method, the cotued vector g shows goal values vector of each obectve whch are specfed by DM. Also the vectors () ad (2) show two dfferet vectors of w values. I vector (): w < w 2 ad vector (2): w > w 2. O ths bass cosderg vectors () ad (2), t ca be sad f < f 2 ad f > f 2, respectvely. For eample, whe value of w decrease, value of f wll decrease too. Ths property allows DM to select a proper preferece weght vector for decso mag order to get more desred results mmum possble tme. I a real case, DMs do ot have eact ad complete formato o decso obectves ad costrats. For portfolo-selecto problems, the collected data does ot behave crsply ad they are typcally ucerta ther ature. Hece stochastc problem related to Program () ca be stated as follows:

7 Mult-obectve Programmg uder Ucertaty to Select Portfolo: A Case Study Stoc Echage Maret of Ira ma f : c, =,..., K = = m f : c, r = K +,..., R subect to : r = r a b, =,..., m = =, 0, =,...,, where c, c r, a ad b are radom parameters. I the CCP approach, the Program (4) wth cosderg postve obectves s coverted to a determstc equvalet program as follows (Préopa, 995): (4) ma E c, =,..., K subect to : = Pr a b α, =,..., m = (5) = =, 0, =,...,, where E c = s the epected value of the obectves wth regards to radom codtos c ad α are threshold values of costrats that are specfed by the DM. 3. Chace costraed goal attamet programmg I ths secto, we propose our determstc equvalet program for the Program (4) based o CCP approach ad GAP method. We call the resultg approach chace costraed goal attamet programmg (CCGAP). I the followg, we preset how to trasform the radom obectves ad the radom costrats. 3.. Radom obectves c are radom ad ormally dstrbuted parameters. The goal g ca be radom parameter where the DM does ot ow ts value wth certaty. Thus we assume that

8 Mostafa Ehtar, Alreza Alezhad 2 the goals of stochastc obectves g N( g, g )where ad (for =,,K) g g 2 are ow (Aou et al., 2005). The obectve s mamzg c ( for,..., K ) ad the: c g,,..., K = = By cosderg the GAP method, t ca be sad whch the obectve s mmzg y (y 0) subect to: P c y g -,,..., K = where s the threshold value of obectve. Let = P g c y -,,..., K = h () = g c, h ()s ormally dstrbuted ad let Var( h ( )) be respectvely the mea ad the varace. ad t ca be sad: P h ( ) y -,,..., K h ( ) E( h ( )) y E( h ( )) P -,,..., K Var( h ( )) Var( h ( )) E( h ( )) ad where ( z) = P N( 0,) z represets the probablty dstrbuto fucto of a stadard ormal dstrbuto. y E( h ( )) Var( h ( )) ( - ),,..., K y E( h ( )) ( - ) ( h ( )) K + Var,,..., - + Var -,,..., = = y E ( g c ) ( - ) ( g c ) K At fal t ca be sad: ( )- ( - ) ( - ) ( ) = = E c Var g c y E g,,..., K.

9 Mult-obectve Programmg uder Ucertaty to Select Portfolo: A Case Study Stoc Echage Maret of Ira 3.2. Radom costrats Radom costrats are hadled as the CCP approach: ormally dstrbuted parameters. The related chace costrat to P a -, =,..., = b m a ad b are radom ad a = b s: Letl() a b, l() s ormally dstrbuted ad let E( l( )) ad = Var( l ( )) be respectvely the mea ad the varace. Thus we have: ad the P l () 0 -, =,..., m l( ) E( l( )) E( l( )) P -, =,..., m Var( l ( )) Var( l ( )) where ( z) = P N( 0,) z represets the probablty dstrbuto fucto of a stadard ormal dstrbuto. E( l ( )) At fal t ca be sad: Var( l( )) ( - ), =,..., m E( l ( )) ( ) ( l ( )) m + - Var 0, =,..., + - Var, =,..., = = E ( a ) ( ) ( a b ) E ( b ) m. The CCGAP whch s equvalet to Program (4) ca be formulated as: m y subect to : = = = E ( c ) ( - ) Var ( g c ) w y E ( g ),,..., K + Var, =,..., = - = E ( a ) ( ) ( a b ) w y E ( b ) m =, 0, y 0, =,...,, (6)

10 where Mostafa Ehtar, Alreza Alezhad K m w w (w, w > 0, for =,, K ad =,, m). = = The Program (6) allows DM s deas about radom parameters, coflctg obectves ad values of threshold be gathered pror to optmzato process. I et secto, we wll llustrate our proposed model for a mult-obectve stochastc problem about optmal portfolo selecto Ira stoc maret. 4. Case study I ths secto we wll llustrate our developed model through a portfolo selecto real problem where the goals of stochastc obectves ad rate of retur of securtes are radom parameters. We cosder a sample of 5 stocs from the Ira stoc echage maret. The data ad observatos (from March 2002 to March 20) of the -sample perod are used as the trag set to determe the models parameters ad specfcatos. For the llustrato purpose, we cosder four selected obectves. These obectves are: The frst obectve s retur stochastc obectve fucto. The rate of retur r ( P P D ) / P ) measures the proftablty of the stoc where the come (,t,t,t,t ca be the form of radom captal ga ad dvded. Here D,t P,t s the prce of the stoc at tme t ad s the dvded receved durg the perod [t, t]. r =,, 5 are radom ad ormally dstrbuted wth ow mea μ ad varace σ 2. Ths stochastc obectve s to be mamzed. The secod obectve s Beta rs obectve fucto. β Cov( r, rm ) / Var( r m ), where r, =,, 5 s the rate of retur of stoc ad r m s the rate of maret retur. Ths obectve dcates the relace of stoc s retur o maret. Lower correlato wth the maret dcates the stoc performace o ts ow rather tha by the movemets of the maret. The am s to choose a dversfed portfolo wth small β. The thrd obectve s tal cost of vestmet obectve fucto. I real world, may people suffer because they have ot eough moey for secure vestmets. Thus the am ths s whch they sped less moey whle wll obta ther favorte results from other obectves. P s the prce of stoc (wth ow formal currecy) the last uder study day. Let N be total umber of estet securtes (stocs) the optmum portfolo. Therefore the tal cost of vestmet obectve fucto ca be obtaed wthout cosderg the value N as follows:

11 Mult-obectve Programmg uder Ucertaty to Select Portfolo: A Case Study Stoc Echage Maret of Ira Z = P ( N ) P ( N )... P ( N ) Z = N ( P P... P ) Z N ( P ) = Z 5 Z = N ( P ) = f 3 P = N N N Fally optmum value of cost for selecto ad allocato of optmum portfolo s equal to Z = f 3 N. We cosder prce of the last day uder study term (P ) to purchase stoc. Ths obectve s to be mmzed. The fourth obectve s purchase rato obectve fucto. PR PN,s / PN T (for =,, 5) s purchase rato of stoc whe the maret s ope. PN,s beg the umber of purchasers of stoc ad PN T beg the total umber of purchases uder study term. The am s to select a portfolo whose stocs are more attractg to buyers. Ths obectve has to be mamzed. The system costrats ca be defed as follows: 5 (a) The sum of the proportos vested stocs s equal to : (Marowtz, 952). (b) Allocated costrats: I order to dversfy the selected portfolos ad mamum utlzato from the all estet capactes of vestmet, DM proposes to vest 25% automotve dustry (for stocs =, 2, 3, 5), bag ad leasg (for stocs = 5, 6, 7, 8), vestmet sectors (for stocs = 4, 3, 4) ad aother sectors (for stocs = 9, 0,, 2). I fact summato of these costrats s equal to the costrat of part (a). Settg a lower ad a upper boud for each stoc order to dversfy the portfolo, 0 0., for =,..., 5, where the s the proporto to be vested the stoc. The ma portfolo selecto problem ca be formulated as follows: 5

12 Mostafa Ehtar, Alreza Alezhad ma f = m f = m f = ma f = subect to : = 5 3 = 5 4 = , , , = , , =,...,5. r P PR (7) The Program (7) s trasformed to a CCGAP as Program (8): m y subect to : 5 = 5 = 5 = 5-5 = = E ( r ) ( α ) Var( g r ) + w y E ( g ), β -w y g P -w y g 3 3 PR + w y g 9 0 2,, , , , , 0 0., y 0, =,...,5,, (8) where 4 = w (w > 0, for =,, 4). I Program (8), the parameters r (for =,, 5) ad g are cosdered radom ad ormally dstrbuted wth ow mea ad varace. Table, presets the mea ad varace values of the radom parameter g. Also Table 2 presets the goal values

13 Mult-obectve Programmg uder Ucertaty to Select Portfolo: A Case Study Stoc Echage Maret of Ira of determstc obectves. The goal value of Beta obectve s equal to (Lee ad Chesser, 980). The other parameters of Program (8) are ow wth certaty. The obectves cosdered ths eample are the rate of retur, the rs β, the tal cost of vestmet ad the purchase rato whch oly the retur obectve s stochastc ad other obectves are determstc. Table. Goal value of stochastc obectve (for = ) Radom goal μ σ 2 g Table 2. Goal values of determstc obectves (for = 2, 3, 4) Obectves ( f ) Determstc goals (g ) f 2 f f I Table 3, we preset data cocerg the 5 dfferet stocs of the Ira stoc maret for the years 2002 to 20. The s colums of the Table 3 are umber, the stocs, the stoc prce the last echaged day, the rs β, the epected rate of retur of each securty, ad the purchase rato of each securty, respectvely. From DM s vewpot, dfferet scearos are proposed for threshold α ad vector of preferece weghts w. Program (8) was appled separately for each scearo by Lgo software pacage ad Table 4 shows optmal portfolo of each scearo ad optmal values of each obectve. Table 3. Data uder study Stoc Stoc prce the last echaged day (P ) Beta rs (β ) Epected rate of retur (μ ) Purchase rato (PR ) PARS AUTO MEH IRAN AUTO SAIPA RAY SAIPA INV

14 Mostafa Ehtar, Alreza Alezhad 5 PERSIAN BANK KAR AFR BANK IRAN LEAS IND & MIN LEAS PARS ALU ALUMTAK IRAN BEHNUSH PARS MINOO OIL IND INV SEPAH INV SAIPA DIESEL Table 4. Optmum values of portfolos wth regard to dfferece scearos α = 0.0 w (0.2,0.2,0.4,0.2) (0.2,0.5,0.,0.2) (0.25,0.25,0.25,0.25) (0.3,0.2,0.2,0.3) f f 2 f 3 f α = w (0.2,0.2,0.4,0.2) (0.2,0.5,0.,0.2) (0.25,0.25,0.25,0.25) (0.3,0.2,0.2,0.3)

15 Mult-obectve Programmg uder Ucertaty to Select Portfolo: A Case Study Stoc Echage Maret of Ira f f 2 f 3 f α = 0.05 w (0.2,0.2,0.4,0.2) (0.2,0.5,0.,0.2) (0.25,0.25,0.25,0.25) (0.3,0.2,0.2,0.3) f f 2 f

16 Mostafa Ehtar, Alreza Alezhad f α = 0. w (0.2,0.2,0.4,0.2) (0.2,0.5,0.,0.2) (0.25,0.25,0.25,0.25) (0.3,0.2,0.2,0.3) f f 2 f f The obtaed results from dfferet scearos Table 4 show some d close tradeoffs betwee obectves. We cosder that r (for =,, 5) ad g are ormally dstrbuted wth ow mea ad varace. I Table 4 the all portfolos were obtaed wth cosderg ucertaty. We otce that the results all portfolos are close to each other, where oly the proportos vested some stocs are chaged. What s uderstood from Table 4 s that by creasg α each vector w, f wll decrease. I other words crease of α ca result worst codto of access to epected rate of retur uder ucertaty. About f 2, t should be sad that vestmet rs ay level of ucertaty s equal to. The results dcate that chages α each vector w have ot fluece o f 3, so that uder ucertaty ad at each level α, more amout from the allocated budget s eeded. By crease of α each vector w, f 4 wll have a decreasg procedure. I the other words, by crease of acceptace probablty of ucertaty, umber of purchasers wll decrease. It seems vestmet portfolo uder ucertaty has less attractveess for purchasers ad vestors. At fal, t seems that crease of α each scearo w has udesred effects o problem s obectves ad

17 Mult-obectve Programmg uder Ucertaty to Select Portfolo: A Case Study Stoc Echage Maret of Ira deterorato procedure of obectves ca be see. Of course ths deterorato procedure of optmal values of obectves s evdet ad ratoal, because t provdes a better ad more tagble represet of realty of vestmet Ira stoc maret uder ucertaty for DM. 5. Cocluso Ivestmet mult-obectve portfolos s mportat from two aspects of level of use of DM s deas ad cosderg all or some of radom parameters. I the begg of ths paper, the GAP method was troduced as a method pror category ad advatages of ths method compared to the GP method was preseted. I the other secto, the fal shape of stochastc programmg problems ad ther optmzato by CCP approach were troduced, so that by combg CCP approach ad GAP method, we proposed the CCGAP method whch ca optmze mult-obectve stochastc problems. At the ed of ths paper, o the bass of dfferet scearos, the proposed model was appled for a real problem of mult-obectve portfolo selecto by four obectves Ira stoc echage maret. I ths problem, the rate of retur of securtes ad the goal value of retur obectve were cosdered as radom parameters wth ormal dstrbuto ad ow mea ad varace. At fal, the obtaed results from optmzato of stochastc problem of mult-obectve portfolo selecto by the CCGAP method dcated that by crease of threshold value each scearo, obectves mprovemet of stochastc problem wll decrease. ACKNOWLEDGEMENT The authors would le to tha Mr. A. Aref from Departmet of Facal Egeerg at the orgazato of Tehra Bourse Echages for hs support collectg the data set. REFERENCES [] Amr, M., Ehtar, M., Yazda, M. (20), Nadr Compromse Programmg: A Model for Optmzato of Mult-obectve Portfolo Problem. Epert Systems wth Applcatos, 38, ; [2] Adersso, J. (2000), A Survey of Mult-obectve Optmzato Egeerg Desg. Departmet of Mechacal Egeerg, Löpg Uversty, Swede;

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