A Type-2 Fuzzy Rule-Based Expert System Model for Portfolio Selection

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1 Type- Fuzzy Rule-Based Epe Syse Model fo Pofolo Seleco M.H. Fazel Zaad, E. Hagol Yazd Eal: Depae of Idusal Egeeg, kab Uvesy of Techology, P.O. Bo , Teha, Ia bsac: Ths pape peses a ype- fuzzy ule based epe syse o hadle uceay cople pobles such as pofolo seleco. I a ype- fuzzy epe syse boh aecede ad coseque have ype- ebeshp fuco. Ths eseach uses dec appoach fuzzy odelg, whee he ules ae eaced auoacally by pleeg a cluseg appoach. Fo hs pupose, a ew cluse aalyss appoach based o Fuzzy C-Meas (FCM s developed o geeae pay ebeshp of ype- ebeshp fucos. ew cluse valdy de based o Xe-Be valdy de s peseed. The poposed ype- fuzzy odel s appled sock ake facos (such as sk, eu, dvded, as he pu vaables. Ths odel s esed o Teha Sock Echage (TSE. Though he esve epeeal ess, he odel has successfully seleced he os effce pofolo based o dvdual veso. The esuls ae vey ecouagg ad ca be pleeed a eal-e adg syse fo sock. Keywods: eval ype- fuzzy se, fuzzy c-eas cluseg, valdy de. Ioduco The sock ake s a cople ad dyac syse wh osy, o-saoay ad chaoc daa sees (Pees, 994. Due o coplcao ad uceay of sock ake, pofolo seleco s oe of he os challegg pobles. log wh he develope of afcal ellgece, especally ache leag ad daa g, oe ad oe eseaches y o buld auoac decso-akg syses o pedc sock ake (Kovalechuk & Vyaev, 000. og hese appoaches, sof copug echques such as fuzzy logc, eual ewoks, ad pobablsc easog daw os aeo because of he ables o hadle uceay ad ose sock ake (Vasoe & Ta, 003, 005. Though sof copug ca soewha educe he pac of ado facos, lowlevel daa ae so ucea ha hey eve behave puely adoly a soe e (Pees, 994. Hua udesadg of sock ake egaes o he hgh-level epeseave odel whch educes hdde abguy of low level daa. The elees of facal ake (such as sk, eu, flucuae because of he coues vaey of segh bewee buyg ad sellg sde. Pofolo aagee s a pacula aspec of vese heoy. Thee ae a lage ube of ways of vesg he sock echage. Thaks o he dffee echques, s possble o oba he esaed pof ad sk of a cea aco. The goal of vese decso-ake s o selec a opal pofolo ha sasfes he veso s obecve. The dffculy of hs poble s decdg abou selecg whch asse ad s sze pofolo because of he uceay of he eus. Quaave appoaches o pofolo opzao ely o how he eu ad sk of asses pofably ae defed ad easue. Each specfed odel povdes he decso ake wh a specfc se of paaees. Sce, Epes do o o whch eu odel s os appopae heefoe hey ecoed dffee opal pofolo choces. The Mode pofolo aalyss saed fo poeeg eseach wok of Makowz (95. I ode pofolo seleco heoy, he well-kow Makowz s ea vaace appoach eques zg he sk of he seleced (sock asse pofolo (easued by he vaace of s eu, whle guaaeeg a pe-esablshed eu ae ad he oal use of he avalable sag capal. fe Makowz wok, schola assues eus of dvdual secues as sochasc vaables ad ay eseaches wee focused o eedg Makowz s ea-vaace odels [3, 5, 36]. Moeove, he facal ake s affeced by seveal opobablsc facos such as vagueess ad abguy foao whch s chaacezed by lgusc descpos such as hgh sk, low pof, hgh ees ae ec. so, Soe auhos have used possbly dsbuos o odel he uceay o eus, whle ohe auhos have suded he pofolo seleco poble usg fuzzy foulaos. Taaka ad Guo eplaced pobably dsbuos of eus of secues wh possbly dsbuos he odels. Calsso oduced a possblsc appoach fo selecg pofolos wh he hghes uly value ude he assupo ha he eus of asses ae apezodal fuzzy ubes. I hs pape, we develop a epe syse fo pofolo seleco based o veso s popesy fo sk. The goal of hs eseach s o develop a fuzzy odelg echas whch s capable of pleeg fve obecves: geeag a ule base auoacally fo uec daa, fdg he opal ube of ules ad fuzzy ses, Poceedgs of he h Jo Cofeece o Ifoao Sceces (008 Publshed by las Pess

2 opzg he paaees of fuzzy ebeshp fucos, Reducg he pac of vagueess ad abguy hdde sock ake suao Ipleeg he veso's pesoaly fo ceasg he obusess of he syse To acheve hese obecves, hs pape poposes a fuzzy odelg paadg by copoag a odfed FCM cluseg assocaed wh a poposed cluse valdy easue, wh cosdeg eval ype- fuzzy ses doa. The es of pape s ogazed as follows: Seco evews he ype- fuzzy ses ad syses ad he assocaed eologes. I Seco 3 he desg appoach of eval ype- fuzzy logc syse s peseed. Seco 4 peses he poposed eval ype- fuzzy syse fo pedco of sgle sock pce of a auoove aufacoy sa. Fally, coclusos ad coes o fuhe eseach ae appeaed Seco 5.. Backgoud:.. Ieval Type- Fuzzy Ses To-dae, because of educg he copuaoal copley of usg a geeal Type- fuzzy se a Type- fuzzy logc syse, os eseaches plee a IT FS. The esul s a IT FLS. I Ths pape, vey sple oduco o IT- fuzzy odelg gve fo soeoe who s ew o he feld of a IT FLS o ge o vey quckly as follow (Medel & oh 00: Defo. ype- fuzzy se, deoed, s chaacezed by a ype- ebeshp fuco, (, u X as (, u ( Whee X ad u J [0, ] u u,.e.,, u,, u X, u J 0, ( {( ( [ ]} whch 0 (, u, ca also be epessed as:, u, u 0, ( ( ( [ ] J X u J Defo. Whe all secoday gades as (, u he s a eval T FS (IT FS. Defo 3. Uceay he pay ebeshps of a IT FS,, cosss of a bouded ego ha we call he foop of uceay (FOU. I s he uo of all pay ebeshps,.e., FOU ( U X (3 The uppe ebeshp fuco (UMF ad lowe ebeshp fuco (LMF of ae wo T MFs ha boud he FOU defe as: ( FOU (, X (4 ( FOU (, X (5 Noe ha fo a IT FS, [ (, ( ], X Fg. Foo P of Uceay fo a eval ype- fuzzy se (Medel, 00.. Ieval Type- Fuzzy Logc Syses geeal T-FLS as show fg. s vey sla o he T-FLS. The ao sucual dffeece beg ha he defuzzfe block of a T-FLS s eplaced by he oupu pocessg block a T-FLS. Tha block cosss of ype-educo followed by defuzzfcao. Type-educo aps a T-FS o a T-FS, ad he defuzzfcao, as usual, aps ha T-FS o a csp ube. The sucue of he ules eas eacly he sae he T case, bu ow soe o all of he FSs volved ae T. Whe all of he aecede ad coseque ae IT FSs, he we call he esulg FLS, a eval T FLS (IT FLS. Fg.. The sucue of a ype- fuzzy logc syse (Medel, Desgg eval ype- fuzzy logc syse Thee ae dffee kds of epe syse based o fucoaly ad sucue of he bu ay ealy eseaches pay uch oe aeo o ule-base easog. Epe (Kowledge Based Syse s a poble solvg ad decso akg syse based o kowledge of s ask ad logcal ules o pocedues fo usg kowledge. Boh he kowledge ad he logc ca oba wo aes so wo kds of epe syse es: dec ad dec epe syses. I dec appoach, ules desged by cosdeg epeece of specalss he puposed aea. O he ohe had, dec appoach desged syse whou pleeed epe opo saghly, bu cosuced wh kowledge esece abou poble ad based o kowledge ules desg. Because of esg hgh degee of uceay he foao used fo buldg ules such as dffee epes vew pos abou puposed poble, we plee fuzzy epe syse fo solvg ou pofolo seleco poble. So, we desg a FLS oo. Ou basc wokspace es o Sock Make Echage whee vese paaees akes place a pecse ad/o ucea ecooc evoe ha ealy ake codo chages cao be pedced accuaely. The fo os obusess we cosde eval ype- fuzzy epe syse ha coves uch ay uceaes he sudyg aea. We have o develop a eval ype- Poceedgs of he h Jo Cofeece o Ifoao Sceces (008 Publshed by las Pess

3 fuzzy syse fo pofolo seleco. Fuzzy odelg ca be defed as syse descpo wh fuzzy quaes. Sce hee s o geeal ehod fo fuzzy logc se up ad eave pocedue, o oba he bes pefoace of a FLS, he dffee pas of he FLS have o be defed. Ths sage of odelg s kow as defcao. ccodg o Zadeh s defo, gve a class of odels, syse defcao volves fdg a odel whch ay be egaded as equvale o he obecve syse wh espec o pu-oupu daa. Idefcao of a fuzzy odel s dvded o wo secos (Sugeo & Yasukawa, 993: Sucue Idefcao, ad Paaee Idefcao. Sucue Idefcao ca be dvded o wo ypes, called ype I ad ype II. Each ype s dvded o wo goups. The sucue defcao of ype dvded o subsyses: Ipu caddae: as we kow hee s fe ube of possble pus caddae fo he pus o a syse, whch should esced o cea ube. Thee s geeal soluo fo selecg fe ube of pus. So, we have o use heusc ehods o solve. Fo ou case.e. pofolo seleco vaable ca be gouped as follows:. Lgusc vaable ha be apply fo asweg quesos ha defy veso's sk popesy.. csp Vaables as sock ake paaees; Ipu vaable: I hs seco we fd a se of pu vaables fo he gve pu caddae whch affecs he pus. Thee ae soe syseac odels o solve hs poble. These odels ae based o peassged pu-oupu vaables. Sugeo ad Yasukawa ehod used fo selecg pu vaables aog caddae vaables. Ipu- oupu elaos: I Ipu-oupu elao defcao, ube of fuzzy ules ad how paog pu space ecogze. Ths poble s a cobaoal oe. We eed a heusc ehod o fd a opal pao ogehe wh a ceo. These pobles ae solved by applyg fuzzy cluseg ehods. Fo ou pupose we use odfed FCM ha ca be used by eval ype- daa. 3.. Cluseg he oupu space ad deeao of he ube of ules Cluseg [,, 5], a ao bach of daa g, s used fo dscoveg goups ad defyg paes ad dsbuos of a gve se of daa. The goal of evey cluseg algoh as a usupevsed classfcao s goupg daa elees accodg o soe (dsslay easues so ha uobvous elaos ad sucues he daa ca be evealed. Sce ou daa ad cees of cluses ae eval ype- fuzzy ses, we use a dsace easue base o dces, whch s applcable fo IT- ses. Fo achevg hs a, we oduce a deeao de ha ca appoae each IT- fuzzy dau wh ype- oe. We desg a de ha appoae each Ieval ype- fuzzy se by a ype- oe. The puposed dsace easue s based o copag each pa of fuzzy appoaed vaable by cosdeg dsaces bewee he eas ( M, M ad sadad devaos ( Σ, Σ, sepaaely. The poposed algoh s as follow: Sep: soe ulabeled eval ype- fuzzy daa: X,,,...,, s a IT- fuzzy ube fo,...,. Sep:choose c : he ube of cluse such ha <c< : wegheg epoeal as fuzzess easue, > ε : eao ceo, 0<ε < al fuzzy veco v 0 V0,,,..., c sep 3 : appoae each X, V wh suggesed dces as: X α. + ( α. Σ X X (6 ad V α. + ( α. Σ V V (7 Sep 4: eae : fo o To calculae U wh V ad as: (8 c [ w d + ] w d ( Σ k [ w dk + ( w dk Σ] calculae M, Σ based o U as: M M. (9 Σ Σ. (0 IF E U ( U ε sop, Else Ne (. ( To deee a appopae ube of cluses N c fo a gve daa se, a cluse valdy fuco eeds o be seleced hee. Cluse valdao efes o he poble whehe a gve fuzzy pao fs o he daa all. So s used fo evaluag he cluseg esuls by eas of a obecve fuco. Defo 4. ssue X,,,..., s ulabeled eval ype- fuzzy daa ad v 0 V0,,,..., c s al IT- fuzzy cluse cees wh he coespodg ea (M ad vaace ( Σ ad < c <, >, Is a paaee whch deees he fuzzess of he esulg cluses he sae above we defe dsace easue as: dm d( M, M M M ( d d( Σ, Σ Σ Σ ( d Σ M M + Σ (3 Σ Poceedgs of he h Jo Cofeece o Ifoao Sceces (008 Publshed by las Pess 3

4 Defo 5. The vaace of a Gaussa eval ype- se wh coespodg,, σ defe as follow: ( σ σ ( e (4 Σl + σ + ( π Z( σ 3 3 π ( σ ( + σ(σ (5 Σ ( + πσ Σl + Σ Σ od fy (6 Defo 6. Based o dsace defo, odfed Xe-Be de fo IT- fuzzy daa ca be defed as: [ v ] Ve ( clus ( [ M M + Σ Σ ] (7 c Vclus U Ve ( clus [ vl ( clus, v ( clus] e (8 v l ( clus M( vclus (9 v ( clus Ma( vclus (0 vl ( clus + v ( clus Vod fy ( σ clus V odfy ( Fo sepaao we should aze he u dsace bewee cluse cees. So, we use: Sepaao { (, fuzzyakg d v v } (3 χ σ clus ( { d ( v, v (4 } fuzzyakg 4. Ipleeao of he poposed odel sock pofolo seleco I hs seco, we pese a eval ype- fuzzy odel fo pofolo seleco Teha echage sock ake. We sudy o he 50 bes copaes of sock ake. Ipu vaables of he syse ae epeced eu, peodcal ae of eu, vaace of ae of eu, dvded of each shae, lqudy coeffce of copaes ad vaables ha show decso sk pefeece (0 lgusc vaables. The oupu of syse s he peceage of each kd of shaes o be bough by veso cosdeg he above vaables ha ca acheve he bes pefoace pofolo. The daa s odeled o a Mul-Ipu- Sgle-Oupu (MISO syse. The Oupu of he ulebase of syse s sk veso popesy. Hee, 'sk popesy' eas he aou of sk a dvdual would feel cofoable wh whe hey have vesed he oey. Two vesos wh he sae pofle chaacescs ay ake vey dffee facal choces. Wha's dffee abou he s he level of sk-aveso, o he ae chaacescs. By cosdeg how people ae uque, wh dffee backgouds, epeeces, ad belefs, a veso who s Cauous by aue ay feel cofoable wh ehe a low level, a edu level, o a hgh level of sk. ll hese facos feed o he pecepo of sk. sple way o assess veso s sk level wll be appled. Soe quesos based o veso pesoal aude used o hadlg sk ad he oal scoe s used fo deeed veso sk-aveso. Use afess hs/he e wh he ceo by usg lgusc vaables. The seps of developg of he MISO odel ae as follows:. Fo vaable seleco, he Sugeo ehod (Sugeo vaable seleco algoh s used. Fo ou eval ype- syse, we acheve 5 vaables (epeced eu, peodcal ae of eu, vaace of ae of eu, dvded of each shae, lqudy coeffce of copaes as pus o ou syse fuhe oe veso lguscally aswes o 0 quesos ae pus o ou syse oo.. Modfed FCM algoh (seco 3- s pleeed o cluse he quesos whch ae base of easug veso s sk popesy. The, a ew cluse valdy de ( a odfcao veso of Xe-Be de s used o deeae he os suable ube of ules (c; sce we cluse he aveage of aswes gve o each queso he selecg eval ype- fuzzy daa look a good choce. To Fo educe copley, we esae each IT- fuzzy se wh a de. The puposed de s ype- fuzzy se. The paaees of Modfed FCM algoh ae assued as follows: eao ceo ε 0 -, weghg epoe.3, c a 4. The bes ube of cluses based o hs cluse valdy de s obaed by he u value of he cluse valdy de ha s happeed ube 3, so ou syse coas 3 ules. 3. The oupu ebeshp values ae poeced oo he pu spaces o geeae he ebeshp values of pus. We assue ha pus ad oupu ebeshp fucos ae Gaussa MF. 4. Fo geeag eval ype- fuzzy ule bases ha he aecede ad coseque ses ae eval ype- ses, we use Gaussa Pay MF wh Ucea Mea ad fed sadad devao. We have obaed 5 ohs sock foao ha we use oh foao daa fo geeag ou ules ad use 3 las peod daa fo esg ou odel. fe cluseg, we use he opu ube of cluse 3. So, he ube of ules s 3, oo. 4.. Poposed ype- fuzzy odel We ceae a eval ype- FLS based o ype- FLS. Sla o he ype- FLS he eval ype- FLS uses sgleo fuzzfcao; we have wo opos fo selecg -o: ad poduc -o. Mada feece ad ceod ype-educo ae pleeed. I also uses he sae ube of fuzzy ses ad he sae ules as he eval ype- FLS. The oly dffeece ow s ha he aecede ad coseque ses ae eval ype- whch has a fed Poceedgs of he h Jo Cofeece o Ifoao Sceces (008 Publshed by las Pess 4

5 sadad devao ad ucea eas ha akes o values a eval,.e., (Medel, 000.I fg 3 Gaussa Ieval ype- ule base fo easug veso sk popesy s show. s show fg.3, hee ae hee pus. Hee, he feece syse s ada ype. The oupu of ou FLS s sk popesy of veso. The advaage of syse s deeao sk aveso by cosdeg he pesoaly ad aco ha be show by veso dffee sock ake suaos. So, by pleeg hs ulebase epe syse each veso has a uque pofolo. The oupu of ou ulebase s a eval ype- fuzzy se ha us be ype- educed ad he defuzzfy. We use ceod ype educo ad ceod defuzzfe. The esul s a csp ube ha s pu o he pofolo seleco odel. We use copose pogag odel ode o solve pofolo seleco odel. 4.. Fuzzy pofolo seleco odel I hs seco, pofolo seleco poble foulaed as a opzao poble wh ulple obecves. We assue ha a veso allocaes hs/he wealh aog asses offeg ado aes of eu. We oduce soe oaos as follows: : The ae of epeced eu o he -h asse, : The popoo of oal fuds vesed he -h asse, d : The aual dvded o he -h asse, : The aveage -oh pefoace of he -h asse, : Rsk aveso of veso as a sue fo easug veso sk popesy level I he poposed ul obecve asse pofolo seleco poble, we cosde he followg obecves ad cosas Obecves Sho e eu Fg. 3. Ieval ype- ule base of easug veso sk popesy Fo asses pofolo (,,..., he epeced sho e eu of he pofolo s epessed as: R ( (5 ca be deeed by hsocal daa. ual dvded The aual dvded of a pofolo s epessed as D ( d (6 Rsk The se-absolue devao of eu of he pofolo (,...,, below he epeced eu ove he pas peod,,,..., T ca be epessed as: V ( X {0, ( } ( + ( (7 So, he epeced se-absolue devao of eu of he pofolo (,,..., below he epeced eu s gve by: ( + ( T T (8 V ( X V ( X T T Use V( o easue he pofolo sk. fe ha we defe a uly fuco based o sk ad eu of he pofolo ad veso s sk popesy easued by ulebase epe syse as follows: U R( X V ( X (9 We should aze hs uly fuco. Lqudy Lqudy s he degee of pobably of beg able o cove a vese o cash whou ay sgfca loss value. We desg a educed de Poceedgs of he h Jo Cofeece o Ifoao Sceces (008 Publshed by las Pess 5

6 by pleeed heusc foao abou copaes lqudy degee. L( L (30 Cosas Capal budge cosa o he asses: (3 Of all he asses a gve se, he veso would lke o pck up he oes ha hs/he subecve esae ae lkely o yeld he geaes pofolo effec. I s o ecessay ha all he asses he gve se ay cofgue he pofolo as well. Ivesos would dffe as egads he ube of asses hey ca effecvely hadle a pofolo. No sho sellg of asses: 0,,...,. ( The decso poble The cosaed ul-obecve pofolo seleco poble s ow foulaed as: Maze U R( X V ( X Maze D( d Maze E( L( E L Subec o 0,,...,. ( Copose pogag odel fo puposed pofolo seleco Wh he feld of ul-cea decso akg, Copose pogag (CP s as a echque based o he seleco of hose effce soluos eaes o he deal po, whch s coposed of he opu values of each dvdual obecve fuco. Zeley saes ha aleaves ha ae close o he deal ae pefeed o hose ha ae fahe fo because beg as close as possble o he peceved deal s he aoale of hua choce. I he coe of hs wok, hs leads us o he cocep of closeess bewee fuzzy ubes Calculag fuzzy deal ad fuzzy a-deal soluos I ode o calculae he deal soluo s ecessay o oba he dvdual opu of each obecve. I s ecessay o solve each fuzzy sgle obecve pobles wh cosa cosas. We calculae deal ad a-deal soluo fo each 3 poble, especvely ( U, D, L ad ( U, D, L ha a-deal soluos wll be obaed by zg he obecve fucos sepaaely. fe obag fuzzy deal ad a-deal soluos e sep s zao of y dsace bewee he obecve veco ad he deal soluo. Fo desgg he copose odel of ou odel we should use he cocep of dvdual ege. ll of he obecve have abue of he ype oe s bee so we wll defe he dvdual ege of he pofolo s uly o he deal uly as: U U P( X (34 d( au So he dvdual ege fo secod ad hd obecve fuco defes as: D DP X d ( ( (35 ad L LP ( X d3( (36 al I ode o oba copose pogag obecve fuco we wll aggegae ad wegh he dvdual eges usg a weghed obecve fuco. The he followg poble ca be foulaed: 3 M L w d ( s.. (37 0,,...,. Copose soluo coespodg wh dsace zes he weghed su of he dvdual eges fo he deal ( U, D, L. The Weghed syse s based o he poace of ceo ad specfed s pay value by epe. The os poace causes he os wegh. I he able show he aou of puposed wegh ha s used fo opzao he copose pogag odel. Obecve fuco uly fuco Obecve fuco dvded Obecve fuco 3 lqudy Table. Puposed weghs apply fo copose pogag odel Ths wegh ue by ug syse seveal es ad hs ca be vewed as a good choce. Daa of sock ake call fo a ecel shee ad by applyg lgo 8 we solve pofolo seleco poble. Fo a veso ha gve he followg aswe o he queso as saed able he oupu of ulebase syse (veso's sk popesy s 8.6(he age of sk popesy of a dvdual veso s bewee 0 ad 0. So hs veso s a sky oe. Opu pofolo s cosuced he ae he veso acheve he bes pof fo vesgao ude hs/he sk pefes. The esul of puposed odel bef able as follow: Sybol Rao.96%.6% 85.97% 0.47% Table 3. Resuls of desged coposed pogag odel By copag esul wh he foao of hese socks we ca cosde s a bes oe, because by eas of hs odel we selec socks wh hgh pof by cosug veso sk aveso level ad hgh lqudy ao ad appopae dvded. Poceedgs of he h Jo Cofeece o Ifoao Sceces (008 Publshed by las Pess 6

7 Slghly Makes o dffeece Makes o dffeece 5. Coclusos ad fuue woks: Hghly Hghly The focus of hs pape was o develop a eval ype- fuzzy epe syse fo sock pofolo seleco of he 50 os appopae copaes fo he Teha Sock Echage (TSE. fe vesgag he syse doa, he pus ad oupu of he syse wee deeed. The he sock pofolo seleco odel was desged a ulplepu-sgle-oupu (MISO syse ha has 3 pus ad oe oupu (veso sk popesy. We have used dec appoach o fuzzy syse odelg by pleeg he poposed cluse valdy de fo deeg he ube of ules fuzzy cluseg appoach. The Sugeo ad Yasukawa ehod was used o selecg vaables fo ule-base fuzzy logc syses. fe ha, he oupu ebeshp value was poeced oo he pu spaces o geeae he ebeshp values of pu vaables, ad he ebeshp fucos of pus ad oupu wee ued. The, he ype- Mada feece syse was pleeed ha he esul s eval ype- fuzzy ube. Fo geeag eval ype- fuzzy ule base he Gaussa Pay MF wh Ucea Mea ad fed sadad devao was used; he eval ype- feece ege, poduc o -o opoally, su s-o ad he ceod Type Reduco was used afe ha ceod defuzzfcao was doe o ge he esul. Fally, we have pleeed he poposed fuzzy odel sock pofolo seleco of he 50 os appopae copaes fo he Teha Sock Echage (TSE. Fo valdao of syse we used las hee oh daa ad copaed s esul wh he esul of he poposed ype- fuzzy odel. The poposed syse shows s supeoy wh espec o obusess, flebly. The syse ay be used by sues o pupose opu sock pofolo by cosdeg pesoaly of he cles. lso ca be used by dvdual veso fo oe pofable adg sock ake. 6. Refeece: [] M.R. debeg, Cluse alyss fo pplcao, cadec Pess, NewYok, 973. [] J.C. Bezdek, Fuzzy aheacs pae classfcao, Ph.D. Dsseao, Coell Uvesy, Ihaca, NY, 973. [3] J.C. Bezdek, Cluse valdy wh fuzzy ses, J. Cybee. 3 ( [4] J.C. Bezdek, Pae Recogo wh Fuzzy Obecve Fuco lgohs, Pleu Pess, NewYok, 98. [5] J.C. Bezdek, R.J. Hahaway, M.J. Sab, e al., Covegece heoy fo fuzzy c-eas: coue-eaples ad epas, IEEE Tas. Syses, Ma Cybee. SMC7 ( Makes o dffeece Slghly Hghly Table. case aswes of a dvdual veso gve o 0 quesos 877. Slghly Hghly [6] [3] J.C. Bezdek, N.R. Pal, Cluse valdao wh geealzed Du s dces, : N. Kasabov, G. Coghll (Eds., Poc. 995 Secod NZ Iea. [7] J.C. Bezdek, N.R. Pal, Soe ew dces of cluse valdy, IEEE Tas. Syses, Ma ad Cybee. 8 ( [8] Chug.F,Rhee.H,Hwag.C, 00, eval ype- fuzzy pecepo, IEEE [9] J.C. Du, fuzzy elave of he ISODT pocess ad s use deecg copac, wellsepaaed cluses, J. Cybee. 3 ( [0] Ea, M.R.,Tu kse,i.b., Goldebeg,., 999, ufed paaeezed foulao of easog fuzzy odelg ad cool, Fuzzy Ses ad Syses 08, 59-8 [] Y. Fukuyaa, M. Sugeo, ew ehod of choosg he ube of cluses fo he fuzzy c-eas ehod, : Poc. Ffh Fuzzy Syses Syp., 989, pp [] I. Gah,.B. Geva, Usupevsed opal fuzzy cluseg, IEEE Tas. Pae al. Mach. Iell. ( [3] Gupa, P., Mehlawa, M.K., ad Saea,., 007, sse Pofolo Opzao Usg Fuzzy Maheacal Pogag, Maheacal Pogag, Ifoao Sceces [4] Hsdal E., The IF THEN ELSE saee ad eval-values fuzzy ses of hghe ype, I. J. Ma-Mache Sudes, vol. 5, pp , 98. [5] Huag.X, 007, Mea-se vaace odels fo fuzzy pofolo seleco, Joual of Copuaoal ad ppled Maheacs [6] Huag.X, 007, Rsk cuve ad fuzzy pofolo seleco, Copues ad Maheacs wh pplcaos [7] Joh, R. I., Fuzzy Ses of Type- J. of dvaced Copuaoal Iellgece, 3(6, , 999a. [8] Joh, R. I., Type Fuzzy Ses Epe Updae, Vol., No, Sue 999, ISSN , 999. [9] Kak.N.N, Medel.J.M, 999, Type- Fuzzy Logc Syses, IEEE TRNSCTIONS ON FUZZY SYSTEMS, VOL. 7, NO. 6, DECEMBER 999 [0] Kak, N. N. ad J. M. Medel, pplcaos of Type- Fuzzy Logc Syses: Hadlg he Uceay ssocaed Wh Suveys, Poc. FUZZ-IEEE'99, Seoul, Koea, ugus 999. [] Kak, N. N. ad J. M. Medel, Ceod of a ype- fuzzy se, Ifoao Sceces, vol. 3, pp. 95-0, 00. Poceedgs of he h Jo Cofeece o Ifoao Sceces (008 Publshed by las Pess 7

8 [] Kak, N. N. ad J. M. Medel, Ioduco o Type- Fuzzy Logc Syses, Poc. 998 IEEE FUZZ Cof., pp , choage, K, May 998a. [3] Kak, N. N. ad J. M. Medel, Type- Fuzzy Logc Syses: Type-Reduco, Poc. IEEE Cofeece o Syses, Ma ad Cybeecs, pp , Sa Dego C, Oc., 998. [4] Kak, N. N., J. M. Medel ad Q. Lag Type- Fuzzy Logc Syses," IEEE Tas.o Fuzzy Syses, vol. 7, pp , Dec [5] D.W. K, K.H. Lee, D. Lee, O cluse valdy de fo esao of he opal ube of fuzzy cluses, Pae Recogo 37 (004, [6] Lag, Q., Medel, J.M.,000, Ieval Type- Fuzzy Logc Syses: Theoy ad Desg, IEEE TRNSCTIONS ON FUZZY SYSTEMS, VOL. 8, NO. 5, [7] Lu, N.K. ad Lee, K.K, " Iellge Busess dvso Syse fo Sock Ivese", Epe Syses, vol.4, p.9-39, 997. [8] Medel.J.M,007, dvaces ype- fuzzy ses ad syses, Ifoao Sceces 77,84 0 [9] Medel, J. M. ad H. Wu, Popees of he Ceod of a Ieval Type- Fuzzy Se, Icludg he Ceod of a Fuzzy Gaule, Poc. IEEE FUZZ Cofeece, pp , Reo, NV, May 005. [30] Medel, J. M. ad R. I. Joh, Fudaeal Decoposo of Type- Fuzzy Ses, Poceedgs of Jo 9h IFS Wold Cogess ad 0h NFIPS I l. Cof., Vacouve, Bsh Coluba, Caada, July 5-8, 00. [3] Medel, J. M. ad R. I. Joh, Type- Fuzzy Ses Made Sple, IEEE Tas. o Fuzzy Syses, vol. 0, pp. 7-7, pl 00. [3] Medel, J. M. R. I. Joh ad F. Lu, O Usg Type- Fuzzy Se Maheacs o Deve Ieval Type- Fuzzy Logc Syses, Poc. Noh eca Fuzzy Ifo. Pocessg Socey (NFIPS, pp , bo, MI, Jue, 005. [33] Medel, J. M., Copug Wh Wods, Whe Wods Ca Mea Dffee Thgs o Dffee People, Poc. of Thd Ieaoal ICSC Syposu o Fuzzy Logc ad pplcaos, Rochese Uv., Rochese, NY, Jue 999. [34] Medel, J. M ad Joh, R. I ad Lu, F, Ieval Type- Fuzzy Logc Syses Made Sple, IEEE Tas o Fuzzy Syses, pp. - 4, 006. [35] N.R. Pal, J.C. Bezdek, O cluse valdy fo fuzzy c-eas odel, IEEE Tas. Fuzzy Syses 3 (3 ( [36] Veche.E, Beúdez.J.D, Segua.J.V, 007, Fuzzy pofolo opzao ude dowsde sk easues, Fuzzy Ses ad Syses 58, [37] X.L. Xe, G. Be, valdy easue fo fuzzy cluseg, IEEE Tas. Pae al. Mach. Iell. 3 ( Poceedgs of he h Jo Cofeece o Ifoao Sceces (008 Publshed by las Pess 8