Fuzzy Risk Analysis based on A New Approach of Ranking Fuzzy Numbers using Orthocenter of Centroids

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1 nternatonal Journal of omputer pplcatons ( ) Volume No. March Fuzzy sk nalyss based on New pproach of ankng Fuzzy Numbers usng Orthocenter of entrods N.av Shankar Mohd Lazm bdullah Y.L.P. Thoran Dept. of ppled Mathematcs Dept. of Mathematcs Dept. of ppled Mathematcs P.Phan Bushan ao Dept. of Mathematcs GS GTM Unversty Faculty of Scence Technology GS GTM Unversty GT GTM Unversty Vsakhapatnam nda Unverst Malaysa Terengganu Malaysa Vsakhapatnam nda Vsakhapatnam nda BSTT n ths paper a new approach of rankng fuzzy numbers usng orthocenter of centrods of fuzzy numbers to ts dstance from orgnal pont s proposed. The proposed method can rank all types of fuzzy numbers ncludng crsp numbers wth dfferent membershp functons. e apply the proposed rankng method to develop a new method to deal wth fuzzy rsk analyss problems. The proposed method s more flexble than the exstng methods. Keywords ankng fuzzy numbers; centro orthocenter; fuzzy rankng; fuzzy rsk analyss.. NTODUTON ankng fuzzy numbers s an mportant tool n decson makng. n fuzzy decson analyss fuzzy quanttes are used to descrbe the performance of alternatves n modelng a real-world problem. Most of the rankng procedures proposed so far n lterature [8] cannot dscrmnate fuzzy quanttes some are counterntutve. s fuzzy numbers are represented by possblty dstrbutons they may overlap wth each other hence t s not possble to order them. t s true that fuzzy numbers are frequently partal order cannot be compared lke real numbers whch can be lnearly ordered. n order to rank fuzzy quanttes each fuzzy quantty s converted nto a real number compared by defnng a rankng functon from the set of fuzzy numbers to a set of real numbers whch assgns a real number to each fuzzy number where a natural order exsts. Usually by reducng the whole of any analyss to a sngle number much of the nformaton s lost hence an attempt s to be made to mnmze ths loss. Varous rankng procedures have been developed snce 976 when the theory of fuzzy sets was frst ntroduced by Zadeh []. ankng fuzzy numbers was frst proposed by Jan [] for decson makng n fuzzy stuatons by representng the ll-defned quantty as a fuzzy set. Snce then varous procedures to rank fuzzy quanttes are proposed by varous researchers. Bortolan Degan [] revewed some of these rankng methods [-] for rankng fuzzy subsets. hen [] presented rankng fuzzy numbers wth maxmzng set mnmzng set. Dubos Prade [6] presented the mean value of a fuzzy number. Lee L [8] presented a comparson of fuzzy numbers based on the probablty measure of fuzzy events. Delgado Verdegay Vla [9] presented a procedure for rankng fuzzy numbers. ampos Munoz [] presented a subectve approach for rankng fuzzy numbers. Km Park [] presented a method of rankng fuzzy numbers wth ndex of optmsm. Yuan [] presented a crteron for evaluatng fuzzy rankng methods. Helpern [] presented the expected value of a fuzzy number. Saade Schwarzler [] presented orderng fuzzy sets over the real lne. Lou ang [5] presented rankng fuzzy numbers wth ntegral value. hoobneh L [6] presented an ndex for orderng fuzzy numbers. Snce then several methods have been proposed by varous researchers whch nclude rankng fuzzy numbers usng area compensaton dstance method decomposton prncple sgned dstance [7 8 9]. ang Kerre [ ] classfed all the above rankng procedures nto three classes. The frst class conssts of rankng procedures based on fuzzy mean spread [ ] second class conssts rankng procedures based on fuzzy scorng [ 7 ] whereas the thrd class conssts of methods based on preference relatons [ ] concluded that the orderng procedures assocated wth frst class are relatvely reasonable for the orderng of fuzzy numbers specally the rankng procedure presented by damo [7] whch satsfes all the reasonable propertes for the orderng of fuzzy quanttes. The methods presented n the second class are not dong well the methods [ 5 ] whch belong to class three are reasonable. Later on rankng

2 nternatonal Journal of omputer pplcatons ( ) Volume No. March fuzzy numbers by preference rato [] left rght domnance [] area between the centrod pont orgnal pont [] ] sgn dstance [6] dstance mnmzaton [7] came nto exstence. Later n 7 Garca Lamata [8] modfed the ndex of Lou ang [5] for rankng fuzzy numbers by statng that the ndex of optmsm s not alone suffcent to dscrmnate fuzzy numbers proposed an ndex of modalty to rank fuzzy numbers. Most of the methods [8] presented above cannot dscrmnate fuzzy numbers some methods do not agree wth human ntuton whereas some methods cannot rank crsp numbers whch are a specal case of fuzzy numbers. One of the possble methods to overcome these problems s to ntroduce rankng based on Orthocentre of centrods to rank fuzzy quanttes. n a trapezodal fuzzy number frst the trapezod s splt nto three parts where the frst second thrd parts are a trangle a rectangle a trangle respectvely. Then the centrods of these three parts are calculated followed by the calculaton of the orthocentre of these centrods. Orthocentre s the pont where the three alttudes of a trangle ntersect. The alttude of a trangle s a lne whch passes through a vertex of a trangle s perpendcular to the opposte sde. Fnally a rankng functon s defned whch s the Eucldean dstance between the centrod pont the orgnal pont to rank fuzzy numbers. Most of the rankng procedures proposed n lterature use entrod of trapezod as reference pont as the entrod s a balancng pont of the trapezod. But the Orthocenter of centrods can be consdered a much more balancng pont than the centrod. Further ths method uses an ndex of optmsm to reflect the decson maker s optmstc atttude also uses an ndex of modalty that represents the neutralty of the decson maker. The work s organzed as follows: Secton brefly ntroduces the basc concepts defntons of fuzzy numbers. Secton presents the proposed new rankng method. n Secton the proposed method has been explaned wth examples whch descrbe the advantages the effcency of the method whch ranks generalzed fuzzy numbers mages of fuzzy numbers even crsp numbers. n Secton 5 the method demonstrates ts robustness by comparng wth other methods lke Lou ang[5] Yager[8] others where the methods cannot dscrmnate fuzzy quanttes do not agree wth human ntuton. n Secton 6 the proposed rankng method s appled to propose a fuzzy rsk algorthm to deal wth fuzzy rsk analyss problems. Fnally the conclusons of the work are presented n Secton 7.. FUZZY ONEPTS ND NKNG OF FUZZY NUMBES. Fuzzy concepts n ths secton some fuzzy basc defntons are presented []. Defnton. Let U be a Unverse set. fuzzy set of U s defned by a membershp functon f : U where f (x) s the degree of x n x U. Defnton. fuzzy set of Unverse set U s normal f only f sup x U f (x) Defnton. fuzzy set of Unverse set U s convex f only f f ( x ( ) y) mn f ( x) f ( y) x y U. Defnton. fuzzy set of Unverse set U s a fuzzy number ff s normal convex on U. Defnton 5. real fuzzy number s descrbed as any fuzzy subset of the real lne wth membershp functon f (x) possessng the followng propertes: () f (x) s a contnuous mappng from to the closed nterval w. w () f (x) = for all x a f s strctly ncreasng on a b f = for all x c f s strctly decreasng on c d f = for all x d () (x) () (x) (5) (x) (6) (x) real numbers. where a c d are 5

3 nternatonal Journal of omputer pplcatons ( ) Volume No. March Defnton 6. The membershp functon of the real fuzzy number s gven by f L f a x w b x c ( x) () f c x d otherwse w s a constant a c d are real f L : a b where numbers w f : c d w are two strctly monotonc contnuous functons from to the closed nterval w. t s customary to wrte a fuzzy number as ( a c f then ( a c ) w s a normalzed fuzzy number otherwse s sad to be a generalzed or non-normal fuzzy number. f the membershp functon f ( x) s pecewse lnear then s sad to be a trapezodal fuzzy number. The membershp functon of a trapezodal fuzzy number s gven by: f ( x) w( x a) b a w w( x d c d ) a x b x c c x d otherwse f w = then ( a c ) s a normalzed trapezodal fuzzy number s a generalzed or non normal trapezodal fuzzy number f w. The mage of ( a c w) s gven by ( d c a; s a partcular case fb c the trapezodal fuzzy number reduces to a trangular fuzzy number gven by ( a The value of b corresponds wth the mode or core [a d] wth the support. f w = then ( a d) s a normalzed trangular fuzzy number. s a generalzed or non normal trangular fuzzy number f w. s f L : a b w () c d w f : are strctly monotonc contnuous functons ther nverse functons g L : w a b g : w c d are L also contnuous strctly monotonc. Hence g g are ntegrable on w.n ths paper we use fuzzy arthmetcal operators shown n ()-(v) to deal wth the fuzzy arthmetcal operatons between generalzed fuzzy numbers. ssume that there are two generalzed trapezodal fuzzy numbers where ( a b c w ) ( a b c w ). The arthmetc operatons between the generalzed trapezodal fuzzy numbers are revewed from [][] [] as follows : () Fuzzy Numbers ddton = a b c w ) ( a b c w ) a a b b c c d mn w w b c d b c d ( = where a a are any real numbers. ()Fuzzy Numbers subtracton = a b c d ; ) a b c d ; ) ( w ( w = a d b c c b d a ;mn w w where a b c d a b c d are any real numbers. () Fuzzy Numbers Multplcaton = ( a b c d ;mn( w w )) where a =Mna a a d d a d d b = Mnb b b c c b c c c=max b b b c c b c c d =Mna a a d d a d d. t s obvous that f a b c d a b c d are all postve-real numbers then = a a b b c c d d ;mn( w )). ( w (v) Fuzzy Numbers Dvson : 6

4 nternatonal Journal of omputer pplcatons ( ) Volume No. March The nverse of the fuzzy number s ; w d c b a where b c d b c d a b c d a are all real numbers except zero.let a be non-zero postve real numbers. Then the dvson of s as follows: a b = ;mn( w ) w d c b a. evew of some rankng fuzzy numbers.. heng s ankng method heng [8] ranked fuzzy numbers wth the dstance method usng the Eucldean dstance between the entrod pont orgnal pont. For a generalzed Trapezodal fuzzy number ( a c the centrod s gven by: c x y wd c b a dc ab c b w d c b a 6 c b w b c a d w b c a d a d w the rankng functon x as Let y d assocated wth be two fuzzy numbers then. then then () f ()f () f. He further mproved Lee L s method by proposng the ndex of coeffcent of varaton (V) as V where s stard error s mean the fuzzy number wth smaller V s ranked hgher... ang et al. ankng method ang et al. [5] found that the centrod formulae proposed by heng [8] are ncorrect have led to some msapplcatons such as by hu Tsao []. They presented the correct centrod formulae for a generalzed fuzzy number ( a c w) as: O x y w a b c d c b d c a b dc ab d c a b the rankng functon assocated wth as x y. Let two fuzzy numbers then then then () f () f () f.. Lou ang s ankng method Lou ang [5] ranked fuzzy numbers wth total ntegral value. For a fuzzy number defned by defnton 6 the total ntegral value s defned as where T L L g ydy g ydy the rght left ntegral values of respectvely s the ndex of optmsm whch represents the degree of optmsm of a decson maker. f the total ntegral value represents a pessmstc decson maker s vew pont whch s equal to left ntegral value. f the total ntegral value represents an optmstc decson maker s vew pont s equal to the rght ntegral value when. 5 the total ntegral value represents an moderate decson maker s vew pont s equal to the mean of rght left ntegral values. For a decson maker the larger the value of s the hgher s the degree of optmsm... Garca Lamata s ankng method Garca Lamata [8] modfed the ndex of Lou ang [5] for rankng fuzzy numbers. Ths method use an ndex of optmsm to reflect the decson maker s optmstc atttude whch s not enough to dscrmnate fuzzy numbers but rather t also uses an ndex of modalty that represents the neutralty of the decson maker. For a fuzzy number defned by defnton 6 Garca Lamata [8] proposed an ndex assocated wth the rankng as the convex combnaton: L are S where S M M T s the area of the core of the fuzzy number whch s equal to b for a trangular fuzzy number defned by ( a the average value of the plateau n case of a trapezodal fuzzy number gven 7

5 nternatonal Journal of omputer pplcatons ( ) Volume No. March by ( a c w) s the ndex of modalty that represents the mportance of central value s the degree of aganst the extreme values optmsm of the decson maker T meanng as defned n secton.... POPOSED NKNG METHOD has ts own The entrod of a trapezod s consdered as the balancng pont of the trapezod (Fg.). Dvde the trapezod nto three plane fgures. These three plane fgures are a trangle (PB) a rectangle (BPQ) a trangle (QD) respectvely. Let the entrods of the three plane fgures be G G & G respectvely. The orthocenter of these entrods G G & G s taken as the pont of reference to defne the rankng of generalzed trapezodal fuzzy numbers. The reason for selectng ths pont as a pont of reference s that each entrod pont are balancng ponts of each ndvdual plane fgure the orthocentre of these entrod ponts s a much more balancng pont for a generalzed trapezodal fuzzy number. Orthocentre s the pont where the three alttudes of a trangle ntersect. The alttude of a trangle s a lne whch passes through a vertex of a trangle s perpendcular to the opposte sde.. Therefore ths pont would be a better reference pont than the entrod pont of the trapezod. w P ( w) Q (c w) G G G O (a ) B ( ) (c ) D (d ) Fg.. Orthocenter of entrods onsder a generalzed trapezodal fuzzy number ( a c w) (Fg.). The entrods of the a b w three plane fgures are G b c w c d w G G respectvely. Equaton of the lne G G w s y G does not le on the lne G G. Therefore G G are non-collnear they form a trangle. e defne the Orthocentre O x y of the trangle wth G vertces G G G of the generalzed trapezodal fuzzy number ( a c w) O as x y c b c a bc b d 6w w s a specal case for trangular fuzzy number ( a.e. c =b the Orthocentre of centrods s gven by O ( x y ) ( b a)( d b) w w For a generalzed trapezodal fuzzy number ( a c wth orthocentre of centrods x y O () () defned by Eq. () we defne the ndex assocated wth the rankng as y x where s the ndex of optmsm whch represents the degree of optmsm of a decson maker. f we have a pessmstc decson maker s vew pont whch s equal to the dstance of the orthocentre from y-axs. f we have an optmstc decson maker s vew pont s equal to the dstance of the orthocentre from x-axs when.5 we have the moderate decson maker s vew pont s equal to the mean of the dstances of orthocentre from y x axes. The larger the value of s the hgher the degree of optmsm of the decson maker. The ndex of optmsm s not alone suffcent to dscrmnate fuzzy numbers as ths uses only the extreme values of the orthocentre of centrods. Hence we upgrade ths by usng an ndex of modalty whch represents the mportance of central value along wth ndex of optmsm. For a generalzed trapezodal fuzzy number ( a c w) wth orthocentre of centrods x y O defned by Eq.() we defne the ndex assocated wth the rankng as S M where s the ndex of modalty whch represents the mportance of central value aganst the extreme values x y S M s the mode assocated wth the fuzzy number whch s equal to b for a trangular fuzzy number ( a the average value of the plateau for a trapezodal fuzzy number s the one whch s defned n defnton. Here represents the weght of the central value s the weght assocated wth the extreme values x y. 8

6 nternatonal Journal of omputer pplcatons ( ) Volume No. March For any decson maker whether pessmstc ( ) optmstc ( ) or neutral(. 5 ) the rankng functon of the trapezodal fuzzy number ( a c whch maps the set of all fuzzy numbers to a set of real numbers s defned as x y whch s the Eucldean dstance from the orthocentre of the centrods as defned from the orgnal pont. e defne rankng between fuzzy numbers as: Let be two fuzzy numbers then then then then n ths case the case ( ) f case ( ) f case ( ) f dscrmnaton of fuzzy numbers s not possble. n ths case we use S M the rankng has been done as follows : (a) f (b) f. then then.numel EXMPLES Example. 7 Let ; ; 5 where B be two normalzed postve trangular fuzzy numbers. Then O x y.6666 x y.5 O B.. Therefore 8 B B. t s observed that the above rankng order s unaltered even by usng the ndex of modalty proposed n secton rrespectve of the decson maker. Example....5;.5...; Let B be two normalzed trapezodal fuzzy numbers wth opposte sgn. Then x y..6 x y..6. O O B. Therefore 686 B 686. dscrmnaton of fuzzy numbers s not possble. Now by usng case () as defned n secton we have () for a pessmstc decson maker.. B.. as.... B. () for a optmstc decson maker..6 B..6 as B () for a neutral decson maker.5.. B...5 as.... B Thus we see that the rankng order s same n all the three cases. Example. Let...5;.5...; B be two trapezodal fuzzy numbers n example.. Then.5...; B...5; x y..6 x y..6. O O B.. Therefore 686 B 686 dscrmnaton of fuzzy numbers s not possble.e have ()for an pessmstc decson maker.. B.. as.... B ()for an optmstc decson maker 9

7 nternatonal Journal of omputer pplcatons ( ) Volume No. March..6 B..6 as B () for a neutral decson maker.5.. B...5 as.... B Thus we see that the rankng order s same n all the three cases. From examples.. we see that B B. Example. be a normalzed trapezodal fuzzy number B ; be a crsp number. Then O x y.. Let...5; x y.. O B Therefore. 55 B. B t s observed that the above rankng order s unaltered even by usng the ndex of modalty proposed rrespectve of the decson maker. Example.5 Let...5;.8 B be two generalzed trapezodal fuzzy numbers....5; Then O x y..8 x y..66 O B.. Therefore 6 B 58 B t s observed that the above rankng order s unaltered even by usng the ndex of modalty proposed n secton rrespectve of decson maker. 5. OMPTVE STUDY Example 5. onsder four normalzed trapezodal fuzzy numbers...;..5.8;...9;.6.7.8; whch were ranked earler by Yager[8] Fortemps oubens[7] Lou ang[5] hen Lu [] as shown n Table. t can be seen from Table that none of the methods dscrmnates fuzzy numbers. Yager[8] Fortemps oubens [7] methods faled to dscrmnate the fuzzy numbers whereas the methods of Lou ang[5] hen Lu [] cannot dscrmnate the fuzzy numbers. By usng our method we have x y.. x y.5.6 O O x y..5 x y.7.. O O Therefore > > > t s observed that the above rankng order s unaltered even by usng the ndex of modalty proposed n secton rrespectve of decson maker. Example 5. Let...5; be a normalzed trapezodal fuzzy number B ; be a crsp number. heng [8] ranked fuzzy numbers wth the dstance method usng the Eucldean dstance between the entrod pont orgnal pont. whereas hu Tsao [] proposed a rankng functon whch s the area between the centrod pont orgnal pont.ther centrod formulae are gven by x y

8 w d w y x = w d c b a dc ab c b wd c b a 6c b b c a d w b c a d a d w c b a dc ab c b w b a b c d c b a 6c b c w d Example 5. Let 57; B be two normalzed trangular fuzzy numbers. Then O x y x y O B 5 5 ; 8 nternatonal Journal of omputer pplcatons ( ) Volume No. March Both these centrod formulae cannot rank crsp numbers whch are a specal case of fuzzy numbers as t can be seen from the above formulae that the denomnator n the frst coordnate of ther centrod formulae s zero hence centrod of crsp numbers are undefned for ther formulae. By usng our method we have O x y.. x y.. O B Therefore. 55 B. B t s observed that the above rankng order s unaltered even by usng the ndex of modalty proposed n secton rrespectve of the decson maker. y x = w d c b a dc ab c b w d w b a b c d c b a 6c b c Therefore B 5. 6 B t s observed that the above rankng order s unaltered even by usng the ndex of modalty proposed n secton rrespectve of decson maker....5; be a normalzed Example 5. Let trangular fuzzy number ; B be a crsp number. heng [8] ranked fuzzy numbers wth the dstance method usng the Eucldean dstance between the entrod pont orgnal pont. hereas hu Tsao [] proposed a rankng functon whch s the area between the centrod pont orgnal pont.ther centrod formulae are gven by y x = w d c b a dc ab c b w d c b a 6c b c ( a d)( w) w b ( b c a d) ( a d) w Both these centrod formulae cannot rank crsp numbers whch are a specal case of fuzzy numbers as t can be seen from the above formulae that the denomnator n the frst coordnate of ther centrod formulae s zero hence centrod of crsp numbers are undefned for ther formulae. By usng our method we have x y..66 x y. O O B Therefore. 58 B. 5 B t s observed that the above rankng order s unaltered even by usng the ndex of modalty proposed n secton rrespectve of the decson maker. 6. PPLTON OF THE POPOSED FUZZY NKNG METHOD TO FUZZY SK NLYSS n ths secton the proposed fuzzy method s appled to fuzzy rsk analyss problems. Let... n be n manufactures producng the products respectvely. Suppose the product produced by the manufacturer s composed of p sub-components

9 ... p n. falure of component the manufacture k To know the probablty of of component produced by we use the evaluatng terms k respectvely called the probablty of falure severty of loss of the sub-component k k p. The structure of the fuzzy rsk s shown n Fg. The nne-member lngustc term set s shown n Table s used for representng the lngustc terms ther correspondng generalzed fuzzy numbers respectvely. Each lngustc term shown n Table s defned n the unverse of dscourse []. The algorthm to deal wth fuzzy rsk analyss s presented as follows: Step : Fnd the probablty of falure of each component made by manufacturer where n by usng the fuzzy weghted mean method the generalzed fuzzy number arthmetc operatons to ntegrate the factors of each sub-component where k p k p k where k by r r r r w. ; p.e. k s a generalzed fuzzy number gven Step : Fnd the orthocenter of centrods rankng ndces... n of each... fuzzy number n O x y c b k by c a bc b d x y. 6w k usng w The larger the value of the hgher s the probablty of falure of component manufacturer. made by the Step : n case when the rankng ndces n of the fuzzy numbers n are equal then use the cases () to () n secton to fnd the rankng ndex of the fuzzy numbers whch nvolves the decson maker s optmstc atttude along wth the mportance of central value spreads. The nternatonal Journal of omputer pplcatons ( ) Volume No. March larger the value of the hgher s the probablty of falure of component made by the manufacturer. To llustrate the fuzzy rsk analyss of the proposed method let be three manufacturers producng the products respectvely. Suppose the product produced by the manufacturer s composed of three sub-components. To know the probablty of falure of the component use the evaluatng terms produced by the manufacturer k k. e respectvely called the probablty of falure severty of loss of the subcomponent k k. The nne-member lngustc term set shown n Table s used to represent the lngustc terms ther correspondng generalzed fuzzy numbers respectvely. The lngustc values of the evaluatng terms k k of the sub-components k made by the manufacturer are shown n Table [].n the last column of Table denotes the degree of k confdence of the decson maker s opnon wth respect to the sub-component k made by the manufacturer where k. For the above problem the fuzzy rsk analyss s evaluated by usng the proposed method as: Step: ombne the evaluatng tems of each component k k of sub-components k made by the manufacturer by usng the fuzzy arthmetc operatons where n are gven by k p ;.7 ; ; ;.8. ; Step: Usng O x y c b x y c a bc b d 6w the orthocenter of centrods of each fuzzy number. w

10 nternatonal Journal of omputer pplcatons ( ) Volume No. March ther respectve rankng ndexes are calculated shown as follows: x y.5.9 ;. 57 O O x y ;.789 O x y ; the rankng order of the fuzzy s numbers. That s the rankng order of the rsk of the manufacturer s > >. That s the component made by the Manufacturer has the hghest probablty of falure. 7. ONLUSONS Many approaches n rankng fuzzy numbers fal to clearly dscrmnate the fuzzy numbers. Therefore n ths paper a rankng method based on orthocenter of centrods s proposed a new method to deal wth fuzzy rsk analyss s presented whch s based on rankng fuzzy numbers. Frst a new method to rank fuzzy numbers s proposed. The proposed fuzzy rankng method overcomes some of the drawbacks of the exstng methods as they faled to dscrmnate fuzzy numbers could not rank crsp numbers whch are a specal case of fuzzy numbers. nother mportant feature of the proposed fuzzy rankng method s that t takes nto consderaton the decson maker optmstc atttude as well as the ndex of modalty whch tells the mportance of central value spreads whle rankng fuzzy numbers. e also appled the proposed fuzzy rankng method to deal wth fuzzy rsk analyss where the evaluatng values are represented by fuzzy numbers. The proposed method s useful method to deal wth fuzzy rsk analyss as t consders the decson maker s vew n both the stages. Table. omparson of varous rankng methods Fuzzy rankng Yager [8] Fortemps&oubens[7] Lou & ang[5] hen [] Proposed method ankng order > > >

11 nternatonal Journal of omputer pplcatons ( ) Volume No. March omponent made by the manufacturer probablty of falure: Sub-component: Probablty of falure: Severty of loss: Sub-component: Probablty of falure: Severty of loss: Sub-component: p Probablty of falure: Severty of loss: p p Fgure: The structure of fuzzy rsk analyss Table : Lngustc terms ther correspondng generalzed fuzzy numbers Lngustc terms Generalzed fuzzy numbers bsolutely-low (...;.) Very-low (...7;.) Low (...8.;.) Farly-low (.7..6.;.) Medum ( ;.) Farly-hgh ( ;.) Hgh ( ;.) Very-hgh ( ;.) bsolutely-hgh (...;.) Table : Lngustc values of the evaluatng tems of the sub-components made by the manufacturer Manufacturer Subcomponents Lngustc values of the severty of loss Lngustc values of the probablty of falure =low =farly-low =farly-hgh =medum =very-low =farly-hgh =low =very-hgh =farly-hgh =farly-hgh.9

12 EFEENES [] L.. Zadeh Fuzzy sets nformaton control 8 (965) 8-5. []. Jan Decson makng n the presence of fuzzy varables EEE Trans. on Sys. Man ybernetcs 6 (976) [] S.M. Bass H. Kwakernaak atng rankng of multple-aspect alternatves usng Fuzzy sets utomatca (977) []. Jan procedure for mult aspect decson makng usng fuzzy sets. nt. J. Syst. Sc. 8 (978) -7. [5] J. F. Baldwn N..F Guld omparson of fuzzy sets on the same decson spacefuzzy Sets Systems (979) -. [6].. Yager On choosng between fuzzy subsets Kybernetes 9 (98) 5-5. [7] J.M. damo Fuzzy decson trees Fuzzy Sets Systems (98) 7-9 [8].. Yager procedure for orderng fuzzy subsets of the unt nterval nformaton Scences (98) -6. [9]. hang ankng of fuzzy utltes wth trangular membershp functons Proceedngs of nternatonal onference on Polcy nalyss Systems 98 pp [] E. Kerre The use of fuzzy set theory n electrocardologcal dagnostcs n: M.M. Gupta E. Sanchez (Eds.) pproxmate easonng n Decson-nalyss North- Holl msterdam 98 pp [] D. Dubos H. Prade ankng fuzzy numbers n the settng of possblty theory nformaton Scences (98) 8-. [] G. Bortolan. Degan revew of some methods for rankng fuzzy subsets Fuzzy Sets Systems 5 (985) -9. [] S.-H. hen ankng fuzzy numbers wth maxmzng set mnmzng set Fuzzy Sets Systems 7 (985) -9. []. Kolodzeczyk Orlovsky oncept of decsonmakng wth fuzzy preference relaton Further results Fuzzy Sets Systems 9 (986) -. [5] K. Nakamura Preference relatons on a set of fuzzy utltes as a bass for decson makng. Fuzzy Sets Systems (986) 7-6. [6] D. Dubos H. Prade The mean value of a fuzzy number Fuzzy Sets Systems (987) 79-. nternatonal Journal of omputer pplcatons ( ) Volume No. March =very-low =medum.9 =low =farly-low.95 =farly-hgh =hgh.8 =very-low =farly-hgh. [7] X. ang class of approaches to orderng alternatves MSc Thess Tayuan Unversty Technology 987 (n hnese). [8] E. S. Lee.-J. L omparson of fuzzy numbers based on the probablty measure of fuzzy events omput. Math. pplc. 5() (988) [9] M. Delgado J.L. Verdegay M.. Vla procedure for rankng fuzzy numbers Fuzzy Sets Systems 6 (988) 9-6. [] L. ampos. Munoz subectve approach for rankng fuzzy numbers Fuzzy Sets Systems 9 (989) 5-5. [] K. Km K. S. Park ankng fuzzy numbers wth ndex of optmsm Fuzzy Sets Systems 5 (99) -5. [] Y. Yuan rtera for evaluatng fuzzy rankng methods Fuzzy Sets Systems (99) []. S. Helpern The expected value of a fuzzy number Fuzzy Sets Systems 7 (99) [] J. J. Saade H. Schwarzler Orderng fuzzy sets over the real lne: n approach based on decson makng under uncertanty Fuzzy Sets Systems 5 (99) 7-6. [5] T. S. Lou M. - J. ang ankng fuzzy numbers wth ntegral value Fuzzy Sets Systems 5 (99) [6] F. hoobneh H. L n ndex for orderng fuzzy numbers Fuzzy Sets Systems 5 (99) [7] P.Fortemps M.oubens ankng defuzzfcaton methods based on area compensaton Fuzzy Sets Systems 8 (996) 9-. [8]. H. heng new approach for rankng fuzzy numbers by dstance method Fuzzy Sets Systems 95 (998) 7-7. [9] J. S. Yao. K. u ankng fuzzy numbers based on decomposton prncple sgned dstance Fuzzy Sets Systems 6 () [] X. ang E. E. Kerre easonable propertes for the orderng of fuzzy quanttes () Fuzzy Sets Systems 8 () [] X. ang E. E. Kerre easonable propertes for the orderng of fuzzy quanttes () Fuzzy Sets Systems 8 ()

13 nternatonal Journal of omputer pplcatons ( ) Volume No. March [] M. Modarres S. S.-Nezhad ankng fuzzy numbers by preference rato Fuzzy Sets Systems 8 () 9-6. [] L.-H. hen H.-. Lu n approxmate approach for rankng fuzzy numbers based on left rght domnance omputers Mathematcs wth pplcatons () [] T.-. hu. - T. Tsao ankng fuzzy numbers wth an area between the entrod pont orgnal pont omputers Mathematcs wth pplcatons () - 7. [5] Y.- M. ang J.- B. Yang D.- L. Xu K.- S. hn On the centrods of fuzzy numbers Fuzzy Sets Systems 57 (6) [6] S. bbasby B. sady ankng of fuzzy numbers by sgn dstance nformaton Scences 76 (6) 5-6. [7] B. sady. Zendehnam ankng fuzzy numbers by dstance mnmzaton ppled Mathematcal Modellng (7) [8] M. S. Garca M. T. Lamata modfcaton of the ndex of lou wang for rankng fuzzy number nternatonal Journal of Uncertanty Fuzzness Knowledge-Based Systems 5 () (7) -. [9] Y.-J. ang H.-S. Lee The revsed method of rankng fuzzy numbers wth an area between the centrod orgnal ponts omputers Mathematcs wth pplcatons 55 (8) -. [].-. hen H.-. Tang ankng nonnormal p- norm trapezodal fuzzy numbers wth ntegral value omputers Mathematcs wth pplcatons 56 (8) -6. [] S.H. hen Operatons on fuzzy numbers wth functon prncpal Tamkang J. Manag. Sc. Vol.6 no. pp [] Kaufmann. Gupta M.M. (985) ntroducton to Fuzzy arthmetcs : Theory [] applcatons Van Nostr enhold New York. [] -H. Hseh S-H hen model algorthm of fuzzy product postonng; nformaton Scences vol. no. pp