- 1 - Processing An Opinion Poll Using Fuzzy Techniques

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1 - - Pocessg A Oo Poll Usg Fuzzy Techues by Da Peu Vaslu ABSTRACT: I hs ae we deal wh a mul cea akg oblem, based o fuzzy u daa : he uose s o comae he effec of dffee mecs defed o he sace of fuzzy umbes o he dyamcs of he uue akg. As he ma sume, we use fuzzy syhess o medae dffee akgs ; hs esec, he covoluo oduc of fuzzy umbes occus.a sysem of fuzzy weghs s used, o eable he odeg cea. KEYWORDS: agula fuzzy umbe ; smlay coeffce ; fuzzy mul-cea decso makg ; dscee omed fuzzy se INTRODUCTION : I yg o make a avalable decso subec o mecse ad mulcea suaos, a decso make s eued o use a fuzzy mulcea decso makg mehod. Fuzzy Mul-Cea Decso Makg (MCDM) s based o fuzzy mul- abue ad mulobecve decso-makg mehodologes. Thee ae may MCDA / MCDM mehods use oday, such as fuzzy AHP, fuzzy TOPSIS, eacve fuzzy mulobecve sochasc lea ogammg, fuzzy mulobecve dyamc ogammg, gey fuzzy mulobecve omzao, fuzzy mulobecve geomec ogammg, ad moe. Ufouaely, dffee mehods may yeld dffee esuls fo a gve oblem. Choosg he bes MCDA / MCDM mehod fom a ls of such mehods s u a dffcul mul-cea decso makg oblem.ths choce deeds o he oblem a had ad may deede o whch model he decso make s mos comfoable wh. I he case we ll use a oo oll o deeme fo eamle a sgle fal akg fo a se of obecs, dffee vewos should be eaed hough a mulcea ocedue. I such a case, he daa ca be eaed as beg fuzzy daa : he fuzzy chaace aeas o be geeaed by he dvesy of fucos of he obec o be suded ( see fo eamle Rule Based Fuzzy Classfcao usg Suashg Fucos, by Zsol Gea e al., Szeged Uv. Hugay ). Amog he mos used mehods hs aea, we meo he Aalyc Heachy Pocess (AHP), whch wll bw used as a model develog ou ow veso. I a evous ae () we ed o classfy some vesos A, A,, A wh esec o a se of gve cea C, C,, C m.

2 - - Pa of hese cea wee uales, he se beg deoed by I + = { C,, C k } ;he ohes wee falues,deoed by I - = { C k+,, C m }. I s ossble ha oe of hs wo ses I + o I - be emy. We ll sa wh a daa ma D, coag he ags of he vesos accodg o each ceo seaaely ( see Fg. ) C C C m A m Fg. A m A m The meag of he uaes Fg. s he bgge, he bee. The se of cea we deal ae weghed cea: he wegh veco s deoed by W = { w } =,m, w > 0 ; w ++ w m =.(see Fg.). C C C m A m Fg. A m A m W w w w m By followg he mehod eosed (), o deeme he uue akg of he vesos A, A,, A, hee efom he e ses: : he omed ma C= s comued,whee ( see ()) () m : hee deemes he omed weghed ma ^ D v,wh ( see ()),;,m -3:The coecvy vecos D v w () { } D { } ae comued by(see (3)), m,, m

3 - 3 - ma m m ma,,,, },daca C I },daca C I },daca C I },daca C I (3.a) (3.b) 4:The Eucldea dsaces bewee he vecos D +, D - ad he ows of ae cosdeed: hs dsace vecos ae deoed by Q + = ( Q - = ( ) =, ) =,, ^ D v,;,m 5: Fally, he ag veco S = { s } =, fo he vesos A, A,, A s gve by ( see (4)) s ( 4 ) Coseuely, a veso A s eve bee, he moe he value of s s lage. To llusae hese, le s cosde he e eamle : le he ma D be as below ( fg. 3) C C C 3 C 4 oal A 3 7 Fg.3 A A W 0, 0, 0,3 0,4 The emedae ma C, whose elemes ae gve by (see Fg. 4 ), becomes m C C C 3 C 4 A,34 0,756 0,378 0,378

4 - 4 - FIG. 4.. A 0,333 0,667 A 3 0,707 0,354,06 0,707 W 0, 0, 0,3 0,4 The coesodg ma ~ D s eseed below (Fg.5 ) C C C 3 C 4 A 0,3 0,5 0,3 0,5 Fg. 5.. A 0,033 0,00 0,00 0,400 A 3 0,07 0,07 0,38 0,83 Suose fs ha all he cea ae uales : hs suao, he comoes of D +, D - ae gve by (5) ma, },. ; m },. (5), The able Fg. 6 s hus obaed : D + 0,3 0,00 0,38 0,400 Fg. 6 D - 0,033 0,07 0,3 0,5 The ag veco S, whose elemes ae gve by s,becomes 0,58 S = 0,674 0,580 Sce s > s 3 > s, we ll fally oba he uue classfcao C C3 C. I he case all he cea ae falues, he vecos D + ad D - swa,so, deog by S he ew ag veco, we ll obvously have (6) S + S = ( 6 )

5 - 5 - so (7) 0,74 S = 0,36 (7) 0,40 he ew fal classfcao beg ow C C3 C Covesely, f some of he cea ae uales ad he ohes ae falues,say I + = { C,C } ; I - = { C 3, C 4 }, we ll have he e calculaos: ma m,3,3 }; }; ma m,3,3 }; }; 3 3 m ma,3,3 3 3 }; }; 4 4 m ma,3,3 4 4 }; }; (6) ad so (8) he, D + 0,3 0,00 0,3 0,5 D - 0,033 0,07 0,38 0,400. ( 8 ) 0,049 0,34 Q + = 0,76 ş Q - = 0,75 0,79 0,3 ad fally ( 9 ) 0,875 S = 0,388.( 9 ) 0,306 The fal classfcao becomes ow C C C3 == // == Fo a oo oll, dffee u daa occus. I he followg we ll secfy he caegoy of daa ha wll be used.

6 - 6 - A omed dscee fuzzy se s a obec havg he e asec X subec o he e coveos : - { }, ae he elemes of he se X - { }, ae he degees of aaeece of he se X : so, s he degee o whch he eleme belogs o he se X ; oe ha hs degees of aaeece sasfy he omaly codos : 0,( ), ;. Two fuzzy ses, ae eual f X ad Y,( ), ad dso, f 0,( ),. Fo wo omed fuzzy ses X,Y, dffee smlay coeffces ae defed, fo eamle : ( X, Y ) ; R ( X, Y ). Boh of hs coeffces have he e oees : - ae sub- uay, o- egave ( 0 ( X, Y ) ; 0 R ( X, Y ) ) - f he coeffce euals zeo, he he fuzzy ses ae dso - f he coeffce euals, he he fuzzy ses ae eual Le fo eamle X 3 3 4, Y 0, 0, 3 0, 5 0, 4 0, 5 0, be wo omed fuzzy ses : le s calculae he smlay coeffces : fs, we have o eed X, Y o ge he same se of agumes,.e.

7 - 7 - X , Y 0, 0, 3 0, , 4 0, 5 0,, ha s, 0, ; 0, 3 ;3 0, 5;4 0 ; 0; 0, 4;3 0, 5;4 0, 0, 37 Fally, ( X, Y ) 0, 96 0, 38 0, 4 ; 0, 37 R ( X, Y ) 0, 0, 38 0, 4 95 Geeally, he values of hs wo smlay coeffces ae close eough. Noe ha a commo eal umbe,, coesods o he fuzzy se wll be called a cs umbe ( esecvely, a cs se ). Le s ow eu o ou oblem. X : such a obec le s cosde he case of a oo oll : fo a cs daa se, a daa able wh he e shae wll be obaed : TABLE : ceo ceo ceo k classfcable k obec classfcable k obec classfcable obec k The values hs able ae laces he akgs, accodg o each cea : so, {,,, } s a emuao of {,,, }. Ths s he case whe oly oe ee s volved If moe ha oe ee s mlcaed, he dvege vewos ca be summazed as Table below

8 - 8 - TABLE : classfcable obec classfcable obec ceo ceo ceo k X = X = X = X = X k = X k = k classfcable obec X = X = X k = k k k k k k k k The samles Table, amely ae obaed as follows : askg all he ees abou he ak of obec wh esec o h he ceo, a eceage of hem oed o lace h. The esuls ae cosdeed o be fuzzy ses ; coseuely, wo moe oblems ae o be solved : how o deeme a uue fuzzy classfcao se fo each ceo, ad he, how o deeme a fally cs akg. Fo he fs oblem, he mmum oeao was used, ~ X m{x,x,,x },,k Ths oo was chose, fo he aks have he meag he smalle, he bee. If coay ( e he geae, he bee ), he mamum oeao wll be efeed. Fo he secod oblem : o esablsh a uue cs akg based o he uue fuzzy oe, we ll assg o he obecs aks accodg o he magude of smllay coeffce bewee ~ X ad X, fo evey, k.

9 - 9 - Refeeces. Belo, V., e al O shocomg of Say s Mehod of Aalycal Heaches. Omega, 7-30, Dye, J.S. Remaks o he Aalyc Heachy Pocess. : Maageme Scece 36, 49-58, Feek A.Loosma, Mul-Cea Decso Aalyss va Rao ad Dffeece Judgmes, Kluwe Acamedc Publshes, : D. B. Skalak, A. Nculescu-Mzl, R. Cauaa: Classfe Loss ude Mec Uceay : Poc.8 h Euoea Cof. ECML 07 5 :Lazm Abdullah & Nuadah Zam, Rakg of he facos assocaed wh oad accdes usg Coelao Aalyss ad Fuzzy TOPSIS, he Ausala Joual of Basc ad Aled Sc, 4():34-30,00