Fuzzy optimal portfolio selection based on multi-objective Mean-Variance-Skewness model by using NSGA-II algorithm

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1 Bullet de l Soété Royle des Sees de Lège, Vol. 85, 6,. 9 - Fuzzy otml otfolo seleto sed o mult-ojetve Me-Ve-Sewess model y usg NSGA-II lgothm Sheh ASHRAFZADH, Mehd MORADZADHFARD,*, Feeydou OHADI Fulty of Mgemet d Aoutg, Islm Azd Uvesty, Kj h, I Fulty of Mgemet d Aoutg, Islm Azd Uvesty, Kj h, I Detmet of Idustl geeg, Islm Azd Uvesty, Kj h, I Astt Costutg otml otfolo s tl deso fo vestos. The lss otfolo models geelly osde me d ve of etu te, whh e mostly teded sgle ojetve d hve ee lyzed d studed ude odtos of etty. But the el wold olem of otfolo seleto, s mult-ojetve olem d ddto to the te of me d ve of etu, othe te suh s lqudty s should lso e osdeed. O the othe hd, te we e fed wth vgue ute dt d otfolo seleto olem must e studed ude odtos of fuzzy uetty. I ths e, we hve develoed ew fuzzy mult-ojetve ogmmg model sed o me-ve-sewess model fo otml otfolo seleto ude fuzzy uetty. The ojetve futos lude mxmzg the exeted etu, mxmzg sewess o the he to g exeted etus d mmzg lqudty s. Rtes of etu, tes of tuove, d the mxmum ume of tyes of stos luded the otfolo hve ee osdeed s tgul fuzzy umes. To solve ths olem eltst o-domted sotg geet lgothm NSGA-II hs ee develoed. Veto evluted geet lgothm VGA d o-domted sotg geet lgothm NSGA wee used to ome d evlute the efome of the oosed solvg method. Flly, the esults oted fom the lgothms outut e omed d lyzed, whh dtes hghe effey of the oosed solvg method omed to othe methods. Keywods: otfolo seleto olem, the me-ve-sewess model, NSGA-II lgothm, mult-ojetve ogmmg, fuzzy umes. Itoduto The m gol of modelg otfolo s to hel the vesto otml otfolo seleto odg to hs efeees d teests s well s the deso evomet []. I 95, Hy Mowtz ovde s otfolo model tht hs eome the ss fo mode otfolo theoy. He ws the fst to exess the oet of the otfolo d dvesfto s foml wy. He showed qutttvely why d how otfolo dvesfto edue otfolo s fo vesto []. Mowtz me-ve model s the most fmous d most oul oh to the otfolo seleto olem. The most outstdg ot of teest Mowtz model s to osde vestmet s ot oly sed o the stdd devto of etus of sto ut to osde the s of the set of vestmet [,]. The me-ve model hs sevel smlfyg ssumtos d sevel weesses, whh leds to effey the el-wold olems. The teo of etu ve ts hgh stte hs the ehvo le low etu whh hs udesle ehvo euse the hgh etu lso otutes esg the ve. I ft, whe the olty dstuto of sto etus s symmet, the ve wll e effet mesue of vestmet s. To oveome the weess of ve s mesue of Mowtz model, some esehes hve egu to use othe s te to deteme the s of the otfolos []. Oe of the solutos oosed s the lto of the te of etu dstuto sewess otml otfolo seleto so tht, me-ve-sewess model ws todued [5,6]. O the othe hd, most studes, the s of sset s detfed s dom vle wth olty dstuto of the etus. But el vestmet, etus of sy ssets e usully uet eoom evomet d ves fom tme to tme. So, se thee e ume of uet metes tht e dffeet fom the dom effets of fl mets, suh s eooms, olts, lws d egultos, eole's sujetve ftos, et., t s mossle fo the vesto to ot ute, olty dstuto of etus of sy ssets. I te, eole's sujetve ftos should e osdeed the deso-mg otfolos, d these ftos e ffetg fl mets. To get ul oo d uetty the fl mets tems of ou lultos, fuzzy methods e geelly moe ote th olty methods. So t s vlued to ly fuzzy set theoy to evlute the uetty the fl mets [,5,7]. * Coesodg utho: Mehd Modzdehfd ml: modzdehfd@u.. 9

2 Bullet de l Soété Royle des Sees de Lège, Vol. 85, 6,. 9 - Also, dsegdg the otfolo lqudty s d tsto osts, modelg d solvg otfolo seleto olems fo sgle-ojetve o ovetg mult-ojetve otfolo seleto model to sgle-ojetve olem d solvg t t lte stge, hve led to models d methods tht e f fom vestmet ssues the el wold d hve lost the effetveess [8,9]. Reetly, some esehes hve ted to edue vuleltes of me-ve Mowtz model. Amm d Khlf [], the tle usg fuzzy tevls, modeled the otfolo otmzto me tht mmzes vestmet s so tht, etu moe th sefed mout wll e detemed. They solved the ole ogmmg model y usg the Lgge multles. The esult s the lloto of fuds to eh vestmet osdeg the udget esttos. Hug [,] defed fuzzy-sem-ve s the m vle of otfolo s d odgly, eseted me-sem-ve fuzzy model. Lu d Zhg [] develoed mult-ojetve otfolo otmzto model sed o fuzzy me-sem ve model fo otfolo seleto. They oosed two me-sem-ve olty model wth the el ostts. Fo solvg oosed models fuzzy mult-ojetve ogmmg method s used to ovet them to sgle ojetve model d the geet lgothm hs ee desged to solve t. Khlf d Zeld [8] eseted fuzzy ogmmg model fo mult-ojetve otfolo seleto model. They osdeed fuzzy umes s ojetve futo oeffets d oosed α uttg tehque d smulted elg met-heust methods to solve the olem. Shh Mohmmd et l. [5] modeled otfolo seleto olem sed o the me-ve-sewess model, whee sto etus e ssumed to e fuzzy vles d to solve ths ole model, geet lgothm wth the eul etwo ws used. Sht l-hmd et l. [] used Geet Algothm NSGA-II detemst multojetve otfolo otmzto olem d demostted ths soluto hs hghe effey omed to othe methods fo solvg mult-ojetve olems. Delv et l. [] doted mult-ojetve geet lgothm to selet the otml otfolo of fltes of Mell s Isfh ove. The esults show tht the otml otfolo of fltes tht hs ee heved y the mult-ojetve geet lgothm s dffeet fom the 's uet otfolo d s oveg dffeet oles d esttos evlg ledg d s moe effet. The m of ths e s to develo ew fuzzy mult-ojetve ogmmg model sed o Me-vesewess model fo otml otfolo seleto ude odtos of fuzzy uetty. The ojetve futos lude mxmzg the exeted etu, mxmzg sewess o he to ot exeted etus d mmzg lqudty s. Rtes of etu d tes of tuove, d the mxmum ume of tyes of stos luded the otfolo hve ee osdeed fo tgul fuzzy umes. To solve ths olem, geet lgothm NSGA-II hs ee develoed d ts efome ws omed d lyzed y geet lgothms VGA d NSGA. Thus smlfyg ssumtos modelg d solvg otfolo seleto olem s edued d oosed model s moe effet el-wold olems. The e s ogzed s follows. I seto the ext defto of the olem d ts mthemtl model hve ee eseted. The methodology of olem-solvg d mult-ojetve geet lgothms hve ee develoed d desed Seto. I seto, the omuttol esults d omsos hve ee exessed ode to demostte the llty d effey of the model d the solvg method. The olusos d detos of futue wo e ovded Seto 5.. Polem Fomulto. xeted vlue, ve d sewess of fuzzy vles Pofesso Lotf A. Zdeh 965 fo the fst tme wth the toduto of fuzzy set theoy ovded etos of ute fomto modelg d oxmte esog wth mthemtl equtos tht tself ought eomous hges mthemts d lssl log [7,8]. The osslty theoy s oe of the most mott ssues of fuzzy set theoy whh s used to del wth most heome of the el wold whh thee s uetty. Ths theoy s somewht sml to lssl olty theoy s sets [5]. I ode to elmte the weesses of the osslty theoy, Lu d Lu oosed the edlty theoy s vl to the osslty theoy. Aodg to ths theoy, the uose of the edlty of fuzzy evet s the he of fuzzy evet [5]. Suose tht s fuzzy vle wth memesh futo µ, d d u e el umes. The the osslty d the eessty of fuzzy evet e oted esetvely fom the equtos d. os{ } Su µ u, u Nes{ } os{ < } Su u< µ u The edlty of fuzzy evet s equl to the thmet me vlue of the osslty d the eessty of fuzzy evet d ts vlue s lulted fom equto. C { } os{ } + Nes{ } 9

3 Bullet de l Soété Royle des Sees de Lège, Vol. 85, 6,. 9-9 As esult of the exeted vlue of fuzzy vle of s oted fom the equto. { } { } > d C d C Wth the exeted vlue, ve d sewess of fuzzy vle e lulted y the equtos 5 d 6. ] [ V 5 ] [ S 6 If,, s tgul fuzzy vle Fg.., memesh futo d ts edlty wll e s the equtos 7 d 8 [5]. Fg. : A tgul fuzzy vle othewse µ 7 } C{ 8 Theefoe, the exeted vlue, ve, d sewess of eh tgul fuzzy vle e oted fom the equtos 9, d [6]

4 Bullet de l Soété Royle des Sees de Lège, Vol. 85, 6,. 9 - α α V 6 β Whee α d β. + αβ + α 8α + α β + αβ 8β β β α α > β α β α < β S [ ] Tht shows f, the S d f the S. I tul f S s symmet, the d S. If the S heves the mxmum vlue / d, f heves the mmum vlue / [6].. Lqudty s Lqudty s oe of the most mott te fo vestos whle the fomto of the Potfolo. The lty of qu tsto of huge mout of sto t low ost d the e effet s lled "lqudty". The low e effet mes tht sset es hve ot muh hged the tevl of ode to uy [7]. I ft, lqudty d lqudty s lude the m oetes d mostutue ftos of the met, whh hve theteed eole's tl d ly ul ole the vestmet deso to uy d sell the stos. Illqudty ous whe the sto e esose to low tdg volumes hs hged muh, ft, the l of lqudty my hve egtve mt o the sto's vlue [8]. Lqudty s mult-dmesol mesue, d thee s o uque teo tht would ove ll sets of lqudty. Theefoe, sevel dffeet te e used tht eh of whh eesets dmeso of lqudty. So f, my mesues hve ee todued fo lqudty, ludg the vlue of the tsto, the ume of tstos, tdg volume, d the dffeee etwee offeed d d uy es [,7]. Theefoe lqudty of sset ot e utely lulted. Oe of the most effetve d effet solutos s to use fuzzy umes to exess the lqudty of ssets. Asset tuove te s fto tht eflet the lqudty of ssets. Hee, we ssume -th sset tuove te s show y tgul fuzzy ume l l, l, l. As esult, the otfolo tuove te x x, x,..., x s lulted though the equto. l x l x Aodg to equto 9, the lss osslty of the me vlue of the sset otfolo x x, x,..., x te e lulted. tuove l + l + l { l x} x Also ths e, osslty ve of fuzzy tuove te s used s mesue of lqudty s lulted sed o the equto.. Fuzzy mult-ojetve otmzto model fo otfolo seleto Cosde the otml otfolo seleto olem of sy ssets ude odtos of fuzzy uetty. Rtes of etu d tuove tes of ssets, the mxmum eted s of the otfolo d the mmum eted tuove te of the otfolo y vesto d mxmum ume of tyes of ssets the otfolo hve ee med y tgul fuzzy umes. Mxmzg exeted etu o otfolo, mxmzg sewess o he to ot hghe exeted etus d mmzg lqudty s smulteously, e vesto s ojetves. He eeds fo less th o equl otfolo etu s d lqudty otfolo gete th o equl to edetemed vlue. O the othe hd, vestos ted to hold q tyes of ssets the otfolos. The vles d metes e s follows. : The te of etu o sy ssets,,...,, so tht,, 9

5 Bullet de l Soété Royle des Sees de Lège, Vol. 85, 6,. 9 - R : Potfolo etu te : The mxmum etle etu s of the otfolo fo the vesto, so tht,, l,,...,, so tht l l, l, l l : Potfolo tuove te l : The mmum etle tuove s of the otfolo fo the vesto, so tht l l, l, l x : Ivestmet to sy ssets,,..., q : The mxmum ume of tyes of ssets the otfolo, so tht q q, q, q Fo modelg d fomultg ths olem s mthemtl exesso, we suggest the followg otfolo otmzto model P whh s sed o me-ve-sewess model. mx mx R SR x S x 5 P m s.t. Vl VR l l x sg x q x,,,..., V l x The ojetve futo shows the mxmzto of exeted otfolo etus. The ojetve futo 5 sttes sewess mxmzto o the he to ot hghe exeted etus d the ojetve futo 6 dtes the mmzto of lqudty s. Cost 7 esues tht the otfolo etu s s smlle th edetemed fuzzy ume. Cost 8, shows tht the vege vlue of the otfolo tuove te s ot less th the oosed mmum level of l. Cost 9, shows tht the sum of ssets the otfolo s equl to oe. Cost, dtes tht the ume of ssets the otfolo must e less th o equl to the gve fuzzy umeq. Whe you wt to defuzzy the fuzzy ume q, y el ume my e oted, se the ume of tyes of stos luded the otfolo s eessly tege, theefoe ode to oveome ths olem, y usg the floo futo ovetg ths ume to tege oe d the t s used modelg d solvg. Cost shows esole lmts. Usg the equtos 9- fuzzy model P hs ee de-fuzzy d lssl model P s oted s follows. P mx mx R SR + + x [ ] x 9

6 Bullet de l Soété Royle des Sees de Lège, Vol. 85, 6,. 9 - m + s.t. 8 9 Vl 9 l l x l l x + l l x + l l l x l l x l l x + l l l l l x l l + 6 l x l x 8 x + x x + + l x x x + + l + l x + x l + x sg x q + q + 6 x,,,..., x + l 6 q l x + l l x x Soluto Methodology To solve the otfolo seleto otmzto olem the my studes, the ext methods suh s le ogmmg, ole ogmmg, d gol ogmmg hve ee used. But se t hs ee ove tht otfolo seleto otmzto olem suh P model hs hgh omlexty, hevg otml soluto fo the olem wth lge dmesos d eve medum esole tme s ot ossle [,5,6]. Thus my ttemts hve ee efomed to ot e-otml d hgh qulty solutos y emloyg heust d met-heust methods sted of ext methods [6,9]. Oe of met-heust methods tht oely hs oed wth solvg ths d of olem s geet lgothm [,]. Met-heust NSGA-II lgothm s oe of the most vestle d oweful geet lgothms fo solvg mult-ojetve otmzto olems d ts effey solvg vous olems hs ee ove. Ths lgothm ws todued y De et l. []. I ths e, ode to solve the mult-ojetve otfolo seleto olem otmzto P tht ts model ws eseted the evous seto, mult-ojetve NSGA-II lgothm hve develoed s follows.. Reesetto of homosome d geeto of tl oulto O the olem of otml otfolo seleto, soluto x x, x,..., x e eeseted y homosome,,...,, so tht gees j e estted the ge of[,]. Fo exmle -th homosome wll e s follows. C

7 Bullet de l Soété Royle des Sees de Lège, Vol. 85, 6,. 9 - The eltosh etwee fesle soluto d homosome e eseted s the exesso, esug tht j x j s lwys tue. j x j, j,,..., Aodg to equto, the soluto ssoted wth homosome C s s follows. x The tl oulto sze s osdeed equl to tege d the ove-metoed oess wll e eeted wth the ume utl the tl oulto wll e odued d they wll e dted y the otto ofc, C,..., C o sze.. Hdlg the ost of mxmum ume of tyes of stos the otfolo To stsfy ost of mxmum ume of tyes of stos the otfolo ost 8, dom ume less th o equl s geeted d s lled t. The, f the ume of gees hghe th zeo the homosome s lge th t, we hold t umes of the lgest vlues of x d ut the othe x s zeo. I the ext ste, to modfy homosomes j d meet the ost 7, g the equto s used d omlzto of gees wll e oduted. Fo exmle, f the ume of gees of the homosome s, the mxmum ume of tyes of stos the otfolo s 6, the t 5 wll e heved. Chomosome C eseted the seto ove s modfed s follows, whh s lled C. The soluto ssoted wth the homosomec, hs ee omlzed odg to equto, d t s lled x'. j C x' Cossove d mutto oetos My fmous ossove oetos fo homosomes the otml otfolo seleto olem e vlle []. We use sgle-ot ossove oeto. Fst, two homosomes e seleted s ets. A dom ume etwee d - hs ee geeted to deteme the ossg ot of two homosomes. The two hld homosomes e geeted y movg gees of two et homosomes fom the ossg ot. Cossg ot Pet Pet Afte lyg sgle-ot ossove oeto Chld

8 Bullet de l Soété Royle des Sees de Lège, Vol. 85, 6,. 9 - Chld Afte the geeto of hld homosomes, due to ost of the mxmum ume of tyes of stos the otfolo d the equto, they wll e omlzed d modfed f eeded to hve homosomes tht e vld. If, fo exmle, the mxmum ume of tyes of stos luded the otfolo e 6 d t 5, we hve: Afte modfto d vldto Chld Chld To y muttos homosomes, fst two gees fom homosomes e domly seleted d the vlues e exhged wth eh othe, suh s the followg exmle: Rdom seleto of two gees Itl homosome Mutted homosome vluto of the solutos sed o NSGA-II lgothm I the NSGA-II lgothm, fst, the ume of hlde Q t e ult, usg the etl oulto P t, esetvely usg ossove d mutto oetos. Two ets d hlde oultos hve ee omed wth eh othe d ete the oulto R t wth the sze of N. The the fst o-domted sotg oh wll e used to tegoze the ete oulto R t. Aodg to ths oh, the solutos wll e ut dffeet tegoes wth dffeet o-domted fots. The totl ume of tegoes hs ee show wth. All solutos tegoes o the fst fot F e the est o-domted solutos of the oulto d the wost solutos of the oulto wll e the lst fot F. It s le tht tems of the vlue of ftess, the most ftess wll e ssged to the solutos of the fst fot oulto d ths ftess wll e deesed the ext fots so tht, the lowest ftess wll e ssged to the solutos of the lst fot. I Fg.., the mge of tegozto of the soluto sed o the fst o-domted sotg oh fo the olem wth two-ojetve mmzto hs ee show [9]. 97

9 Bullet de l Soété Royle des Sees de Lège, Vol. 85, 6,. 9 - Fg. : Fst o-domted sotg oh I the ext ste, sed o sotg solutos, the ext geeto wll e flled wth these fots. Fllg the ext geeto P t+ hs ee stted wth the est o-domted fot d the esetvely wth the seod o-domted fot d the thd d so o, t otues utl P t+ would e flled. Se the sze s equl to N ll memes ot e led P t+ d the emg solutos wll e esly deleted. Fg.. shows the efome of the lgothm NSGA- II [9]. Fg. : Shem of efome of the lgothm NSGA-II I egd of the solutos elmted the lst fot y the eltsm oeto moe slls should e used d the solutos the e wth less owded should e mted. I ft, to oseve the le of the desty of the solutos, the solutos whh e smlle owdg e hve oty to fll P t+.. Comuttol Results I ode to evlute the efome of the model d the oosed soluto, dom otfolos wee seleted fom mog the to 5 omes Teh Sto xhge the fouth qute of 5. The tgul dstuto futo of tuove d etu tes hve ee oted fo 5 omes hose though hstol dt fom fl softwe Rhvd Nov dug to 5. Mxmum of tyes of stos the otfolo seleted ws osdeed s dom fuzzy ume q tht, the mout of whh s detemed y vestos. Polems wee lssfed to two gous wth smll d lge sze. Polems wth the mxmum ume of tyes of stos wee osdeed s smll-sle olems d wth mmum of 5 tyes of stos wee osdeed s lge-sle olems. I ode to ome d evlute the efome of the oosed soluto method, two VGA d NSGA lgothms wee lso develoed. The ove-metoed thee lgothms hve ee oded wth MATLAB R softwe. To u the ogm oded, esol omute wth. GHz seve oe oesso d 8 GB m memoy hs ee used. I ode to esue f omso mog the thee lgothms develoed, the metes wee set detlly. The tl oulto sze s d ossove d mutto tes e esetvely. d.8. Nume of geetos odued s odto fo stog the lgothm ws led d wee set s,. Fo eh of the olems odued, eh lgothm ws u fve tmes d the esults e exessed s the vege of the vlue oted. Thee e umeous d dvese dtos to ome d evlute the efome of mult-ojetve metheust lgothms. I ths e two useful des of the qulty d dseso hve ee used fo omso. Qulty dex omes the qulty of the Peto solutos oted y eh method. To lulte the qulty dex, ll solutos oted y ll thee methods hs ee levelled togethe d t s sefed tht how my eet s ssged to eh method [9]. The hghe the eetge, the hghe qulty lgothm hs. Dseso dex s used fo detemg the o-domted solutos foud o the otml fote. Dseso dex s defed s follows: D N mx x t y t 98

10 Bullet de l Soété Royle des Sees de Lège, Vol. 85, 6,. 9 - I equto, t x y t eesets the ulde dste etwee the two solutos of xt d yt o the otml fote. Test esults fo smll-sle olems e ovded Tle d fo lge-sle olems Tle [9]. Tle : Comso of esults oted fo smll-sle olems Polem fomto Qulty dex Dseso dex Polem No Nume of tyes of stos The mxmum ume of tyes of stos the otfolo,,5,5,7,5,6,,6,5,7,5,6,5,7,,8 5,5,7,6,6 VGA NSGA NSGA-II VGA NSGA NSGA-II Tle. Comso of the esults oted fo lge-sle olems Polem fomto Qulty dex Dseso dex Polem No Nume of tyes of stos The mxmum ume of tyes of stos the otfolo 6,9,,5,7 5,7,8,, 8,8,9,,5,, 5,6, VGA NSGA NSGA-II VGA NSGA NSGA-II The esults of yg out tests ove tht the oosed model d develoed NSGA-II lgothm hve hgh efome fo modelg d solvg the mult-ojetve otmzto otfolo olem ude fuzzy uetty odtos. The oosed soluto method sed o NSGA-II lgothm hs oty to odue Peto solutos, odg to oth the qulty d dseso des the smll d lge-sle olems omso wth VGA d NSGA lgothms. 5. Comuttol Results I ths e, fuzzy mult-ojetve ogmmg model sed o me-ve-sewess model ws develoed fo otml otfolo seleto tht ludes less smlfyg ssumtos th the othe vlle models. The oosed model ludes thee ojetve futos of mxmzg exeted etu, mxmzg sewess o the he to g hghe exeted etus d mmzg lqudty s. I ths model, tes of etu, tes of tuove, d the mxmum ume of tyes of stos luded the otfolo wee osdeed tgul fuzzy umes. To solve ths olem eltst o-domted sotg geet lgothm NSGA-II ws develoed. To ome d evlute the efome of the model d the oosed soluto, the to 5 omes Teh Sto xhge the fouth qute of 5 wee osdeed d sed o hstol dt, the tgul dstuto futo of exeted tes of etu d fl tuove tes e oted. Poosed soluto method wth two veto evluto geet lgothm VGA d odomted sotg geet lgothm NSGA ws omed d lyzed sed o two qulty d dseso dtos smll d lge-sle olems. The esults show tht the oosed soluto sed o NSGA-II lgothm hs the vey etle efome fo the oduto of Peto solutos, wth hgh qulty d vety d hghe dseso th the VGA d NSGA lgothms. I othe wods, the oosed method gves the moe otml otfolo to the vestos. 99

11 Bullet de l Soété Royle des Sees de Lège, Vol. 85, 6,. 9 - Cosdeg the ost of tstos d othe tyes of s modelg s well s develog ovtve hyd lgothms to solve the olem fo futue eseh e eommeded. RFRNCS [] Nv Chshem, A., Yousef KChg, R.. Detemto of the otml otfolo usg fuzzy gol ogmmg tehque. Joul of Fl geeg d Mgemet Seutes, vol. 9,. 7-. I Pes [] Vfe, F., Letft, S., Adl, A.. Desgg otfolo otmzto model usg fuzzy mthemtl ogmmg Cse Study: B Mell I Ivestmet Comy. Joul of Idustl Mgemet Fulty of Humtes, Islm Azd Uvesty of Sdj, vol. 7, o.,. -9. I Pes [] Lu, Y.J., Zhg, W.G.. Fuzzy otfolo otmzto model ude el ostts. Isue: Mthemts d oom, vol. 5, o., [] Alezhd, A., Zohehd, M., K, M., ht, M., sfd, N.. xteso of Potfolo Seleto Polem wth Fuzzy Gol Pogmmg: A Fuzzy Alloted Potfolo Aoh. Joul of Otmzto Idustl geeg, vol., o. 9, [5] Shhmohmmd, M., mm, L., Ze Mehjed, Y.. A hyd tellget lgothm fo otfolo seleto usg fuzzy me-ve-sewess. Itetol Joul of Idustl geeg & Poduto Mgemet, vol., o., I Pes [6] L, X., Q, Z., K, S.. Me-ve-sewess model fo otfolo seleto wth fuzzy etus. uoe Joul of Oetol Reseh, vol., o., [7] Ghht, A.R., Njf, A.A.. A Fuzzy Aoh to Me-CDR Potfolo Otmzto. Itetol Joul of Aled Oetol Reseh, vol., o., [8] Khlf, H.A., Zeld, R.A.. Fuzzy ogmmg oh fo otfolo seleto olems wth fuzzy oeffeets. Itetol Joul of Setf Kowledge, vol., o. 7,. -7. [9] Zhou, R., Zh, Y., C, R., Tog, G. 5. A Me-Ve Hyd-toy Model fo Potfolo Seleto wth Fuzzy Retus. toy, vol. 7, o. 5,. 9-. [] Amm,., Khlf, H.A.. Fuzzy otfolo otmzto qudt ogmmg oh. Chos, Soltos & Ftls, vol. 8, o. 5, [] Hug, X. 7. A ew esetve fo otml otfolo seleto wth dom fuzzy etus. Ifomto Sees, vol. 77, o., [] Gut, P., Mehlwt, M.K., Sxe, A. 8. Asset otfolo otmzto usg fuzzy mthemtl ogmmg. Ifomto Sees, vol. 78, o. 6, [] Sht Al-Hmd, S.A., Hemmt, M., sfd, M.. Alto of mult-ojetve geet lgothms NSGAII the otml otfolo seleto of Sto xhge. Joul of Mgemet, vol., o.,. -. I Pes [] Delv, M.R., Bgh, A., Adolgh, A., Kzem, J. 5. Alto of geet lgothms fo mult-ojetve otmzto fltes otfolos Cse Study of Mell fltes Isfh ove. Aoutg d Audtg Reseh, vol. 7, o. 7,. -5. I Pes [5] Lu, B. d Lu, Y.K.. xeted Vlue of Fuzzy Vle d Fuzzy xeted Vlue Models. I Tstos o Fuzzy Systems, vol., o., [6] Wg, Z. d T, F.. A Note of the xeted Vlue d Ve of Fuzzy Vles. Itetol Joul of Nole See, vol. 9, o., [7] slm Bdgol, G., Sej,. 8. otfolo seleto Usg thee mesues of me etu, the stdd devto of etus d lqudty the Teh Sto xhge. Aoutg d udtg studes, vol. 5, o. 5,. -6. I Pes [8] Hyd, S.A., Fllh Shms, M.F., Hshem N.. Studyg the eltosh etwee lqudty s d e the Teh Sto xhge. Joul of Fl geeg d Mgemet Seutes, vol., o. 9, I Pes [9] Lw, K., Qu, R., Kedll, G.. A leg-guded mult-ojetve evolutoy lgothm fo osted otfolo otmzto. Aled Soft Comutg, vol.,