The ELECTRE Multicriteria Analysis Approach Based on Intuitionistic Fuzzy Sets

Transcription

1 The ELETRE Multicriteria alysis roach Based o Ituitioistic Fuzzy Sets Mig-he Wu ad Tig-Yu he bstract Over the last decades, ituitioistic fuzzy sets have bee alied to ay differet fields, such as logic rograig, edical diagosis, decisio akig, etc The urose of this aer is to develo a ew ethodology for solvig ulti-attribute decisio-akig robles with ituitioistic fuzzy iforatio by usig the cocet of ELETRE ethod ELETRE uses the cocet of a outrakig relatioshi We also use TOPSIS ethod to rak all of the alteratives ad to deterie the best alterative Fially, a illustrative exale is give to verify the develoed aroach ad to deostrate its racticality ad effectiveess I INTROUTION HE ituitioistic fuzzy set (IFS) was first itroduced by Ttaassov i 98 [], which is characterized by a ebershi fuctio ad a o-ebershi fuctio The IFS geeralizes the fuzzy set ad itroduced by Zadeh i 95[], ad has bee foud to be highly useful to deal with vagueess Over the last decades, IFS has bee alied to ay differet fields, such as logic rograig [3], edical diagosis [4-5], decisio akig [-0,0--0-4], etc taassov & Georgiev [3] reseted a logic rograig syste which uses a theory of IFS to odel various fors of ucertaity e et al [4] has alied i edical diagosis usig the otio of IFS theory taassov et al [] reseted IF iterretatios of the rocesses of ulti-erso ad of ulti-easureet tool with ulti-criteria decisio akig Hog ad hoi [7] rovided ew fuctios to easure the degree of accuracy i the grades of ebershi of each alterative with resect to a set of criteria rereseted by vague values Xu ad Roald [8] reseted a alicatio of the ituitioistic fuzzy hybrid geoetric (IFHG) oerator to ultile attribute decisio akig based o IFS Z Xu [9] develoed soe siilarity easures of IFS He defied the otios of ositive ideal IFS ad egative ideal IFS ad alied the siilarity easures to ultile attribute decisio akig uder ituitioistic fuzzy eviroet Szidt ad Kacrzyk [0] alied a ew easure of siilarity to aalyze the extet of agreeet i a grou of exerts The roosed easure takes ito accout ot oly a ure distace betwee ituitioistic fuzzy refereces but also exaies if the coared refereces are ore siilar or ore dissiilar to each other Li et al [] roosed a ew ethod for M Wu is with the Graduate Istitute of Busiess Maageet, ollege of Maageet, hag Gug Uiversity, Kwei-Sha, Taoyua 333, Taiwa (e-ail: richwu50@gailco) T Y he is with the eartet of Busiess diistratio, ollege of Maageet, hag Gug Uiversity, Kwei-Sha, Taoyua 333, Taiwa (corresodig author to rovide hoe: x578; fax: ; e-ail: tyche@ailcguedutw) hadlig ulti-criteria fuzzy decisio-akig robles based o IFS It allows the degrees of satisfiability ad o-satisfiability of each alterative with resect to a set of criteria to be rereseted by ituitioistic fuzzy sets, resectively he ad Wag []reseted iterval-valued fuzzy erutatio (IVFP) ethods for solvig ulti-attribute decisio akig robles based o iterval-valued fuzzy sets Li et al [3] develoed a ew ethodology to solve the ulti-attribute grou decisio-akig robles with ultile attributes beig cosidered exlicitly ad both ratigs of alteratives o attributes ad weight of attributes beig exressed usig IFS Bora et al [4] roosed the techique for order referece by siilarity to ideal solutio (TOPSIS) ethod cobied with IFS to select aroriate sulier i a grou decisio akig I this study, also ituitioistic fuzzy weighted averagig (IFW) oerator is used to aggregate all idividual decisio akers oiios for ratig iortace of criteria ad alteratives Nevertheless, the literatures show that oe of curret studies use the cocet of ELETRE ethod to solve ulti-attribute decisio-akig robles with ituitioistic fuzzy iforatio The ELETRE ethod is oe of the ethods of ultile criteria decisio akig [5]The ultile criteria decisio akig (MM) odels have two classificatios: ultile obective decisio akig (MOM) ad ultile attribute decisio akig (MM) MOM have decisio variable values which are deteried i a cotiuous or iteger doai with either a ifiitive or a large uber of choices, the best of which should satisfy the decisio aker s costraits ad referece riorities MM o the other had are geerally discrete, have a liited uber of alteratives They require both itra ad iter attributes coarisos ad ivolve exlicit tradeoff which are aroriate for the roble exlaied [] ELETRE ethod was first itroduced by Beayou et al [5]The origis of ELETRE ethods go back to 95 at the Euroea cosultacy coay SEM, which is still active today t that tie, a research tea fro SEM worked o a cocrete, ultile criteria, real-world roble regardig decisios dealig with the develoet of ew activities i firs [7] The ethod uses the cocet of a outrakig relatios Its first idea cocerig cocordace, discordace ad outrakig cocets origiated fro real-world alicatios [8] The ethod uses cocordace ad discordace idexes to aalyze the outrakig relatios aog the alteratives [9] I this aer, we develo a ew ethodology for solvig

2 ulti-attribute decisio-akig robles with ituitioistic fuzzy iforatio by usig the cocet of ELETRE ethod We also use the TOPSIS idex to rak all of the alteratives ad to deterie the best alterative II INTUITIONISTI FUZZY SETS I this sectio, the cocet ad oeratios of IFSs ad soe of the related distace easures are described ocet Let X = { x, x,, x} be a fiite uiversal set IFS i X is defied as a obect of the followig for: = { < x, μ ( x ), ν ( x ) > x Χ} ( ) where the fuctios x X μ ( x ) [ 0,], x X v ( x ) [ 0,] () efie the degree of ebershi ad the degree of o-ebershi of the eleet x X to the set X, resectively, ad for every x X, 0 μ ( x ) + ν ( x ) (3) We call π ( x ) = μ ( x ) ν ( x ) (4) as the ituitioistic idex of the eleet x i the set It is the degree of ideteriacy ebershi of the eleet x to the set It is obvious that for every x X, 0 π ( ) (5) x B Oeratio Referece [] [0] ad [] shows that soe oeratios for IFS For every two IFSs ad B the followig oeratios ad relatios are valid: () B iff x X,( μ ( x) μb ( x) & v( x) vb ( x)) ; () = B iff B & B ; (3) = {( x, v ( x), μ ( x))} Soe related distace easures I ay ractical ad theoretical robles, i order to fid the differece betwee two obects, the kowledge of distace betwee two fuzzy sets is ecessary Soe oular distace easure forula were itroduced betwee two IFSs ad B that take ito accout the ebershi degree μ, the o-ebershi degree ν, ad the hesitatio degree (or ituitioistic fuzzy idex) π i X = { x, x,, x} Soe of the ituitioistic fuzzy distace easures are as follows [-3]: --Ituitioistic Haig distace: dis (, B) = ( μ ( x ) μb ( x ) + v ( x ) vb ( x ) = + π ( x ) π B( x ) ) () --Ituitioistic Euclidea distace: dis (, B) = ( [( μ ( x ) μb ( x )) + ( ν ( x ) ν B ( x )) i= + ( π ( x ) π ( x )) ]) (7) B III INTUITIONISTI FUZZY ELETRE METHO Solutio Process ll solutio rocess i this aer are divided ito two arts They are IF ELETRE Method (Ste to 8) ad Rakig Method (Ste 5b to 8b), ad all stes show i Fig B IF ELETRE Method The IF ELETRE ethod is icluded i eight stes The stes are as follows Ste eterie the decisio atrix: Let X i IF ELETRE Method M = μ, ν, π ), ( i i i eterie the decisio atrix eterie cocordace & discordace set 3 alculate the cocordace atrix 4 alculate the discordace atrix 5eterie the cocordace doiace atrix eterie the discordace doiace atrix 7eterie the aggregate doiace atrix 8Eliiate the less favorable alteratives μ i is the degree of ebershi of the ith alterative with resect to the th attribute, ν i is the degree of o-ebershi of the ith alterative with resect to the th attribute, π is the ituitioistic idex of the ith alterative i with resect to the th attribute M is a ituitioistic fuzzy decisio atrix, where 0 μ i + νi, i =,,, =,, (8) π = μ ν (9) i i 5b eterie the cocordace doiace atrix b eterie the discordace doiace atrix 7b eterie the aggregate doiace atrix 8b eterie the best alterative Fig Ituitioistic fuzzy ELETRE ethod solutio rocess i Rakig Method

3 X M = X x x X X I the decisio atrix M, have of alteratives (fro to ) ad of attributes(fro x to x ) The subective iortace of attributes, W, are give by the decisio aker(s) For exale, attribute x has attribute weight w, x has attribute weight w ad the su of weight of all attributes fro x to x are equal to Ste eterie the cocordace ad discordace sets: It use the cocet of IFS relatio to idetify (deterie) cocordace ad discordace set For exale, we ca classify differet tyes of the cocordace sets as strog cocordace set or oderate cocordace set or weak cocordace set It ca also be classify the discordace sets by the sae cocet Let X i = ( μi, ν i, π i ), where μ i, ν i, π are defied at st ste i The strog cocordace set of k ad l is coosed of all criteria for which k is referred to l I other words, The strog cocordace set ca forulate as = μ μ, ν < ν ad π < π } (0) w w, { k l k l k l The oderate cocordace set is defied as = { μk μl, ν k < ν l ad π k π l} () The weak cocordace set is defied as = { μk μladν k ν l} () The strog discordace set is coosed of all criteria for which k is ot referred to l The strog discordace set ca forulate as = { μk < μl, ν k ν l ad π k π l} (3) The oderate discordace set is defied as = { μ k < μl, ν k ν l ad π k < π l} (4) The weak discordace set is defied as = { μ k < μladν k < ν l} (5) The decisio aker(s) give the weight i differet sets For exale, the strog cocordace set have it ow weight Weight of, ad are resectively, w, w w,, w ad w,, Ste 3 alculate the cocordace atrix: The relative value of the cocordace sets are easured by eas of the cocordace idex The cocordace idex is equal to the su of the weights associated with those criteria ad relatio which are cotaied i the cocordace sets Therefore, the cocordace idex c betwee k ad l is defied as: c = w w + w w + w w () where w, w, w are weight i differet sets ad defied i ste ad w are weight of attributes that are also defied i ste Ste 4 alculate the discordace atrix: The discordace idex d is defied as follows: ax w dis( X k, Xl) d = (7) axdis( X, X ) J k ( μ μ ) + ( ν ν ) + ( π π ) ) k, X l ) = (8) k l k l k l where w is equal to w or w or w that deed o the differet tyes of discordace sets ad defied i ste Ste 5 eterie the cocordace doiace atrix: This atrix ca be calculated with the aid of a threshold value for the cocordace idex k will oly have a chace of doiatig l, if its corresodig cocordace idex c exceeds at least a certai threshold value c ie, c c, ad c k =, k l l=, l k c = (9) ( ) O the basis of the threshold value, a Boolea atrix F ca be costructed, the eleets of which are defied as f =, if c c ; f = 0, if c < c The each eleet of o the atrix F reresets a doiace of oe alterative with resect to aother oe Ste eterie the discordace doiace atrix: This atrix is costructed i a way aalogous to the F atrix o the basis of a threshold value d to the discordace idices The eleets of g of the discordace doiace atrix G are calculated as d _ k =, k l l=, l k d = ( ) g =, if d d ; g = 0, if d > d l (0) lso the uit eleets i the G atrix rereset the doiace relatioshis betwee ay two alteratives Ste 7 eterie the aggregate doiace atrix: This ste is to calculate the itersectio of the cocordace doiace atrix F ad discordace doiace atrix G The resultig atrix, called the aggregate doiace atrix E, is defied by eas of its tyical eleets e as: e = f g () Ste 8 Eliiate the less favorable alteratives: The aggregate doiace atrix E gives the artial-referece orderig of the alteratives If e =, the k is referred to l for both the cocordace ad discordace criteria, but k still has the chace of beig doiated by the other alteratives Hece the coditio that k is ot doiated by ELETRE rocedure is, e =, for at least oe l, l =,,,, k l ;

4 e ik = 0, for at all i, i =,,,, i k, i l This coditio aears difficult to aly, but the doiated alteratives ca be easily idetified i the E atrix If ay colu of the E atrix has at least oe eleet of, the this colu is ELETREcally doiated by the corresodig row(s) Hece we sily eliiate ay colu(s) which have a eleet of Rakig Method with TOPSIS idex Because of the IF ELETRE Method ca ot rak all of the alteratives we utilize TOPSIS idex to rak the Yoo ad Hwag [5] develoed the TOPSIS ethod based o the cocet that the chose alterative should have the shortest distace fro the ideal solutio ad the farthest fro the egative-ideal solutio The stes are as follows Ste 5b eterie the cocordace doiace atrix: It use the ositive-ideal solutio of TOPSIS, if c * is the biggest value i the cocordace atrix, the calculate c = c * - c () ad deterie the cocordace doiace atrix Ste b eterie the discordace doiace atrix : Let d * is the biggest value i the discordace atrix, the calculate d = d * - d (3) ad deterie the discordace doiace atrix Ste 7b eterie the aggregate doiace atrix P: P = 3 ( ) The eleet( ) of P is defie as follows: (4) d = (5) c + d where c are the eleet of cocordace doiace atrix ad d are the eleets of discordace doiace atrix Ste 8b eterie the best alterative: Usig the result of ste 7b, we ca calculate the ix evaluatio value of alteratives, the forula as below =, k =,,, () k l=, l k Thus, the best alterative * ca be geerated so that = ax{ } (7) * k ad the alteratives are raked accordig to the icreasig order of IV Nuerical exale I this sectio, we reset a uerical exale coected with a decisio akig roble Suose that the roosed ELETRE ethod with IFS evaluates the ituitioistic fuzzy decisio atrix which refers to of alteratives o 4 of attributes The ituitioistic fuzzy decisio atrix M i ste is give M = x x (035,033,03) (0,044,034) (04,034,04) (0,04,03) (0,0,073) (09,047,034) (009,038,053) (043,09,08) (037,05,007) (034,039,07) (05,034,04) (09,035,03) x3 x4 (03,059,08) (00,087,003) (03,08,04) (044,039,07) (0,034,055) (09,0,05) (044,04,03) (04,0,033) (037,035,08) (034,05,04) (030,08,04) (045,048,007) ssue that the subective iortace of attributes, W, is give by the decisio aker, W=[w,w,w 3,w 4 ]= [0, 0, 03, 04] lyig ste, deterie the cocordace ad discordace sets The decisio aker also give the relative weight (W ) W = [ w, w, w, w, w, w ] = [,,,,, ] The strog cocordace set,3 =,3,3,3 3,4 For exale, = {}, which is i the d (horizotal) row ad st (vertical) colu of strog cocordace set is ={ }, which is i the st row ad 3th colu of strog 3 cocordace set is ety, ad so forth The oderate cocordace set 3,4 4 4 = 3,4 4,3 3,4 4,3,4 The weak cocordace set, = 4,3,3,,3,4 4 4 The strog discordace set,3,3,3 3,4 = 3,3 The oderate discordace set

5 3,4 4 3,4 3,4,3, =,3 The weak discordace set,3 4,3,3 = ,,3 lyig ste 3, calculate the cocordace atrix is = For exale, = w = w w + w w + w w + w w ad = = = w w w w3 w w4 w w w = = = lyig ste 4, calculate the discordace atrix is = For exale: ax dis( X, X ) 0850 d = = = 07, ax dis( X, X ) 047 where J, X ) = ( ( )) ( ) + ( ) + ( 03 04) = 0054,, X ) = ( ( )) ( 0 0) + ( ) + ( ) = 0034, ( ) ( ) ( ) 3, X 3 ) = ( ( )) = 0787, 4, X 4 ) = ( ( )) ( 0 044) + ( ) + ( ) = 047 ad w X w w dis( X X dis( X X ( 0 0) + ( ), ) = ( ( 4 3 ( ) )) = , ( 03 03) + ( ) 3, 3) = ( ( 8 3 ( 08 04) )) = , ( 0 044) + ( ) 4, 4) = ( ( 39 + ( ) )) = lyig ste 5, deterie the cocordace doiace atrix is The average cocordace idex is _ c = k = l= c = The cocordace doiace atrix is F = lyig ste, deterie the discordace doiace atrix is The average discordace idex is _ d = k = l= d = The discordace doiace atrix is G = lyig ste 7, deterie the aggregate doiace atrix (E) is E = lyig ste 8, eliiate the less favorable alteratives is The atrix E reders the followig overrakig relatioshis:, 3, 4 3, 4 5, 5, 5, 5 3,, 3 (illustrated with Fig )

6 the ultile criteria decisio akig ethods 4 Usig the Rakig rocess lyig ste 5b, calculate cocordace outrakig atrix is(c*= 087) The cocordace outrakig atrix = lyig ste b, calculate discordace outrakig atrix is(d*= ) The cocordace outrakig atrix = lyig ste 7b, deterie the aggregate outrakig atrix is P = lyig ste 8b, deterie the best alterative is = 034, = 0 55, 3 = , 4 = 05989, 5 = 0 48, = 0 58 The otial rakig order of the alteratives is give by The best alterative is V ONLUING REMRKS I this study, we have rovided a ew ethodology for solvig ulti-attribute decisio-akig robles with ituitioistic fuzzy iforatio by usig the cocet of ELETRE ethod The ew aroach itegrate the cocet of outrakig relatioshi of ELETRE ethod We also used the Rakig ethod with TOPSIS idex to rak all of the alteratives ad to deterie the best alterative We also illustrated uerical exale to deostrate its racticality ad effectiveess I a future research, we shall utilize the cocet of iterval-valued ituitioistic fuzzy sets to develo 5 3 Fig The overrakig relatioshis that atrix E redered REFERENES [] K T taassov, Ituitioistic fuzzy sets, Fuzzy sets ad Systes, vol 0, 87-9, 98 [] L Zadeh, Fuzzy Sets, Iforatio ad otrol, vol 8, , 95 [3] K taassov, ad Georgiev, Ituitioistic fuzzy rolog, Fuzzy Sets ad Systes, vol 53, 8, 993 [4] S K e, R Biswas, ad R Roy, alicatio of ituitioistic fuzzy sets i edical diagosis, Fuzzy Sets ad Systes, vol7, 09 3, 00 [5] Kharal, Hoeoathic drug selectio usig Ituitioistic Fuzzy Sets, Hoeoathy, vol 98, 35 39, 009 [] K taassov, G Pasi, ad R R Yager, Ituitioistic fuzzy iterretatios of ulti-criteria ultierso ad ulti-easureet tool decisio akig, It Joural of Systes Sciece, vol 3, o4, , 005 [7] H Hog, ad H hoi, Multicriteria fuzzy decisio-akig robles based o vague set theory, Fuzzy Sets ad Systes, vol 4, 03 3, 000 [8] Z S Xu, ad R R Roald, Soe geoetric aggregatio oerators based o ituitioistic fuzzy sets, It Joural of Geeral Systes, vol 35, o 4, , 00 [9] Z Xu, Soe siilarity easures of ituitioistic fuzzy sets ad their alicatios to ultile attribute decisio akig, Fuzzy Otiizatio ad ecisio Makig, vol, 09-, 007 [0] E Szidt, ad J Kacrzyk, cocet of siilarity for ituitioistic fuzzy sets ad its use i grou decisio akig, i Proc of It Joit of o Neural Networks & IEEE It of o Fuzzy Systes, Budaest, Hugary, 004, 9 34 [] L Li, X H Yua, ad Z Q Xia, Multicriteria fuzzy decisio-akig ethods based o ituitioistic fuzzy sets, J out Syste Sci, vol 73, 84-88, 007 [] T Y he, ad J Wag, Iterval-valued fuzzy erutatio ethod ad exerietal aalysis o cardial ad ordial evaluatios, J out Syste Sci (009), doi:00/css [3] F Li, Y Wag, S Liu, ad F Sha, Fractioal rograig ethodology for ulti-attribute grou decisio-akig usig IFS, lied Soft outig, vol 9, 9-5, 009 [4] F E Bora, M Kurt, ad kay, ulti-criteria ituitioistic fuzzy grou decisio akig for selectio of sulier with TOPSIS ethod, Exert Systes with licatios (009), doi:00/eswa [5] L Hwag, ad K Yoo, Multile ttribute ecisio Makig Berli: Sriger-Verlag, 98, 5-40 [] Shaia, ad O Savadogo, o-coesatory coroised solutio for aterial selectio of biolar lates for olyer electrolyte ebrae fuel cell (PEMF) usig ELETRE IV, Electrochiica cta, vol 5, , 00 [7] J Figueria, S Greco, ad M Ehrgott, Multile riteria ecisio alysis: State of the rt Surve, New York: Sriger, 005 [8] B Roy, ad Vaderoote, overview o "The Euroea school of M: Eergece,basic features ad curret works", Euroea Joural of Oeratioal Research, vol 99, -7, 997 [9] diel Teixeira de leida, Multicriteria decisio odel for outsourcig cotracts selectio based o utility fuctio ad ELETRE ethod, outers & Oeratios Research, vol 34, , 007 [0] K T taassov, More o ituitioistic fuzzy sets, Fuzzy sets ad Systes, vol 33, o, 37-45, 989 [] K T taassov, Ituitioistic fuzzy sets: theory ad alicatios New York : Physica-Verlag, 999, ch [] E Szidt ad J Kacrzyk, istaces betwee ituitioistic fuzzy sets, Fuzzy Sets ad Systes, vol 4, , 000 [3] P Grzegorzewski, istaces betwee ituitioistic fuzzy sets ad/or iterval-valued fuzzy sets based o the Hausdorff etric, Fuzzy Sets ad Systes, vol 48, 39 38, 004