A Fuzzy Optimization Method for Multi-criteria Decisionmaking Problem Based on the Inclusion Degrees of Intuitionistic Fuzzy Sets

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1 SSN England UK Journal of nformaton and omutng Scence Vol. No Fuzzy Otmzaton ethod for ult-crtera Deconmang Problem aed on the ncluon Degree of ntutontc Fuzzy Set Yan Luo and hangru Yu + nttute of Sytem Engneerng Shangha Jao Tong Unverty Shangha 0005 hna School of nformaton anagement and Engneerng Shangha Unverty of Fnance and Economc Shangha 004 hna (Receved January acceted rl 007 btract. Th aer reent a fuzzy otmzaton method baed on the ncluon degree of ntutontc fuzzy et to olve mult-crtera decon mang roblem under fuzzy envronment. Frt the ncluon degree of ntutontc fuzzy et defned and a ere of ecfc formula of ncluon degree are reented by mean of the normal mlcaton oerator. Some formula of ncluon degree of ntutontc fuzzy et are generalzed by defnng the cardnal number of ntutontc fuzzy et. Then we gve multcrtera fuzzy decon-mang method baed on ncluon degree of ntutontc fuzzy et. Fnally we llutrate the effectvene of the method rooed n th aer by an examle. Keyword: ult-crtera Decon-mang Fuzzy Otmzaton ncluon Degree FS.. ntroducton Fuzzy et were ntroduced by Zadeh n 965 []. n the followng everal decade fuzzy et theory ha been ued for handlng fuzzy mult-crtera decon-mang roblem [-5]. The man charactertc of fuzzy et that: the memberh functon agn to each element u n a unvere of dcoure a memberh degree rangng between 0 and and the non-memberh degree equal one mnu the memberh degree.e. th memberh degree combne the evdence for u and the evdence agant u. The ngle number tell u nothng about the lac of nowledge. n real alcaton however the nformaton of an obect correondng to a fuzzy concet may be ncomlete.e. the um of the memberh degree and the nonmemberh degree of an element n a unvere correondng to a fuzzy concet may be le than one. n fuzzy et theory there no mean to ncororate the lac of nowledge wth the memberh degree. oble oluton to ue ntutontc fuzzy et (FS for hort ntroduced by tanaov n [6]. FS a an extenon of Zadeh fuzzy et wa alo aled to the decon-mang roblem [7-]. n the reent aer the ncluon degree of ntutontc fuzzy et aled to mult-crtera decon mang n fuzzy envronment. n Secton we defne the ncluon degree of ntutontc fuzzy et and reent a ere of ecfc formula of ncluon degree by mean of the normal mlcaton oerator. Then ome formula of ncluon degree of fuzzy et are generalzed to ntutontc fuzzy et by defnng the cardnal number of ntutontc fuzzy et. n Secton we gve a fuzzy method for mult-crtera deconmang method baed on ncluon degree of ntutontc fuzzy et. Fnally we llutrate the effectvene of the method rooed n th aer by an examle.. ncluon degree of FS The concet of FS an extenon of Zadeh fuzzy et. t gve u the oblty to model unnown nformaton by ung an addtonal degree. n [6] [4] FS are defned a follow: Defnton. n ntutontc fuzzy et on a unvere U defned a an obect of the followng form: + orreondng author. Tel.: ; fax: E-mal addre: yucr@tu.edu.cn. Publhed by World cademc Pre World cademc Unon

2 Journal of nformaton and omutng Scence ( {( u v u U} μ where the functon u : U [0] and v : U [0] defne the degree of memberh and the degree of non-memberh of the element u U n reectvely and for every u U : + v 0 μ Obvouly each ordnary fuzzy et may be wrtten a {( u μ μ u U} of ntutontc fuzzy et on a unvere U wll be denoted by FS(U.. For mlcty the cla.. ncluon degree baed on mlcatonal oerator Defnton f the mang : FS ( U FS( U [ 0] atfe ( ( ; ( ( U φ 0; ( ( mn( ( ( then we call ( the ncluon degree of n and call a ncluon degree functon on FS(U. Defnton f mang R : [ 0 ] [ 0 ] atfe ( R( 0 0 ( R( 0 0 R( 0 R( then R called fuzzy mlcatonal oerator (brefly mlcaton. Defnton 4 f mang T : [ 0 ] [ 0 ] atfe ( T ( a b T ( b a ( a c b d T ( a b T ( c d ( T ( T ( a b c T ( a T ( b c (v T ( a a( a [0] then t called trangular norm (brefly t-norm. f T atfe (-( and (v' T ( 0 a a( a [0] then we call t trangular conorm (brefly t-conorm. Theorem Let FS ( U and R be an mlcaton. f R atfe ( a b [0] and a b R( a b ( R(ab non-decreang wth reect to b non-ncreang wth reect to a then the followng are the ncluon degree functon of FS: [ ] [ ] ( nf[ λr( μ μ + ( λ R( v v ] λ [ 0 ] u U ( nf T ( R( μ R( v v μ where T : 0 0 a t-norm Proof. We only rove. The roof of mlar. ( μ μ v v u U R( μ μ R( v v ( nf T ( ( ( U φ nf T ( R( 0 R( 0 nf T ( 00 0 u U ( μ μ μ v v v R ( a R ( b Snce non-decreang and non-ncreang we have J emal for ubcrton: ublhng@wu.org.u

3 48 T Y. Luou et al: Fuzzy Otmzaton ethod for ult-crtera Decon-mang Problem ( μ μ R( μ μ R( v v R( v v ( R( μ μ v v T ( R( μ μ R( v v T ( R( μ μ v v nt T ( R( μ μ R( v v U ( ( ( ( R nt u n a mlar we can get. Theorem ume that U a fnte unvere and R an mlcaton. f R atfe ( a b [0] and a b R( a b ( R(ab non-decreang wth reect to b non-ncreang wth reect to a then the followng are the ncluon degree functon of FS: 4 λ [ 0 ] ( [ R( μ μ + ( λ R( v v ] λ U ( T ( R( μ R( v v 4 μ U where U denote the cardnalty of U and : [ 0 ] [ 0 ] T a t-norm. The roof of theorem mlar to the roof of theorem. The mlcaton R atfy the condton of above theorem a b [0] : ( Luaewcz mlcaton: R L ( a b mn( a + b ( Goguen mlcaton: ( a b R π ( Gödel mlcaton: ( a b R G b mn a b (v Gane-Recher mlcaton: ( a b (v R 0 -mlcaton: R ( a b π R GR max f f ( a b f f a b 0 a > b. f f f f a b a > b a 0 a > 0. a b a > b... ncluon degree baed on the cardnalte of et Defnton 5 Let U be a fnte et FS ( U. The cardnalty of defned a + μ v t eay to rove that the followng ncluon functon of fuzzy et - [5] tll hold to FS. Theorem Let U be a fnte et FS ( U of FS: 7 5 ( ( φ φ 5 0. Then the followng - are ncluon degree functon ; ( U ; ( U φ ; otherwe U ; otherwe J emal for contrbuton: edtor@c.org.u

4 Journal of nformaton and omutng Scence ( ( ; 0 ( φ or U. otherwe.. Generaton of ncluon degree Theorem 4 Let be an ncluon degree functon on FS(U and mang : [ 0 ] [ 0 ( h( h( ; ( h ( a b non-decreang wth reect to a and b then G( h( ( ( ( FS( U h ] atfe ncluon degree of n and G an ncluon degree on FS(U. roof. ( ( ( G ( h( ( G( U φ h( ( U φ ( φ U h( 00 0 ( h ( ( ( ( ( ( ( h( ( ( G ( G ( n the mlar way we can get G ( G ( Theorem 5 Let be ncluon degree functon on FS(U and ( h( h( ; : [ 0 ] [ 0 ] h atfe ( h ( a b non-decreang wth reect to a and b then G ( h( ( ( ( FS( U ncluon degree of n and G an ncluon degree on FS(U. The roof of theorem 5 mlar to the roof of theorem 4.. ult-crtera fuzzy decon-mang baed on ncluon degree Defnton 6 (ult-crtera fuzzy decon-mang roblem [7]. Let be a et of alternatve and let be a et of crtera where ume that the charactertc of the alternatve { } { } m n are reented by the FS hown a follow: {( μ v ( μ v ( v } n μn where μ ndcate the degree to whch the alternatve atfe crteron ndcate the degree to v * whch the alternatve doe not atfy crteron μ v L n; m. ( ( ume that there a decon-maer who want to chooe an alternatve whch atfe the crtera and or whch atfe the crteron. Th decon-maer requrement rereented by the followng exreon: and and and or t noted that we ay an alternatve atfe a crteron f t meet ome derable level of an evaluaton crteron. The atfacton gradual and characterzed by a dual nformaton: a degree of atfacton and a degree of non-atfacton. We now ue the ncluon degree of FS to olve the muft-crtera fuzzy decon-mang roblem (defnton n [7].The bac dea mlar to the TOPSS (Technque for Order Preference by Smlarty to n J emal for ubcrton: ublhng@wu.org.u

5 50 Y. Luou et al: Fuzzy Otmzaton ethod for ult-crtera Decon-mang Problem deal Soluton [6-0]. Frtly the deal oluton and the ant-deal oluton are contructed where the deal oluton and the ant-deal oluton are reectvely the bet and the wort oluton uoed but not extng n the et of alternatve. Then we comare the ncluon degree of the deal oluton n alternatve and the ncluon degree of alternatve n the ant-deal oluton. The alternatve contanng the deal oluton maxmally a well a beng contaned by the negatvely deal oluton mnmally the bet choce. For th uroe we ntroduce the followng defnton. m n be the et of crtera Defnton 7 Let ( be the et of alternatve and ( ( The deal oluton and the ant-deal oluton atfyng the crtera {( ( ( } g g g g g g are defned a follow: G μ v μ v μ v ( μ v μ v μ v μ v μ v μ v {( μ v ( μ v ( μ v } ( b b D( b b Defnton 8 The ncluon degree of the deal oluton n alternatve and the ncluon degree d ( of alternatve n the ant-deal oluton are reectvely defned a follow: where denote the ncluon degree functon ( max( ( G ( G b b D (5 d ( mn( ( ( (6 {( μ v ( μ v ( μ v } ( v { } m μb b Defnton 9 The ranng ndex of alternatve ( m defned a follow: D( ( + D( d The rocedure of olvng mult-crtera fuzzy decon-mang roblem (Defnton 6 a follow: ( calculate the deal oluton and the ant-deal oluton atfyng the crtera G calculate the deal oluton and the ant-deal oluton atfyng the crtera ; G ( calculate the ncluon degree ( G ; m; d( of G n and the ncluon degree ( (7 of n ( calculate the ncluon degree D ( of the deal oluton n alternatve and the ncluon degree of n the ant-deal oluton; (v calculate the ranng ndex of alternatve ( m (v f there ext { m uch that 0 } max ( m then alternatve the bet choce. 0 0 The reaon for ntroducng the deal oluton and the ant-deal oluton multaneouly n above method that when two alternatve contan the deal oluton by the ame ncluon degree we ntroduce the negatvely deal oluton for dfferentatng whch alternatve ueror. Then the alternatve contaned by the ant-deal oluton wth the le ncluon degree the better choce. 4. n examle Let be fve alternatve and let be three crtera. ume that the 4 5 charactertc of the alternatve are rereented by the FS hown a follow: J emal for contrbuton: edtor@c.org.u

6 Journal of nformaton and omutng Scence ( {( ( 0.0. ( ( 0.0. ( ( 0.0 } {( ( 0.0. ( ( 0.0. ( ( 0.0. } {( ( ( ( ( ( 0.0. } {( ( ( ( ( ( } {( ( ( ( ( ( } and aume that the decon-maer want to chooe an alternatve whch atfe the crtera or whch atfe the crteron. Frtly we contruct the deal oluton and the negatvely deal oluton atfyng the crtera and atfyng crteron. Tae G {( ( } G {( 0.60 } {( ( 0.0.4} {( } n theorem a the ncluon degree functon and tae ( U R λ.e. ( μ ( u μ ( u + R( v ( u v ( u where we chooe Luaewcz mlcaton R L a mlcaton R.e. R L ( a b mn( a + b a b [ 0]. So we obtan the ncluon degree ( G of G n and the ncluon degree of n ( ; 4 5 lted n table and ( 4 5 (G ( Table : ncluon degree of G n and n 4 5 (G ( Table. ncluon degree of G n and n Ung formulae (5 and (6 we get the ncluon degree ( ( D of the deal oluton n alternatve and the ncluon degree d of n the negatvely deal oluton. We ft them n table. 4 5 D( d( Table : ncluon degree of G n and n From (7 we get the ranng ndex of alternatve a follow: Therefore alternatve 5 the bet choce. 5. oncluon Th aer reent a fuzzy otmzaton method baed on the ncluon degree of ntutontc fuzzy et to olve mult-crtera decon mang roblem under fuzzy envronment. ncluon degree a quantty decrbng that a et contaned by another et and quanttatve decrton of contanment relaton. t hold the uncertanty of the relaton. The ncluon degree theory and FS theory are the mortant tool n tudyng the uncertan nowledge. The rooed method n the aer ha been demontrated by an examle llutratng the ower of the aroach to olve mult-crtera fuzzy decon mang roblem. Th reearch wor not only develo and enrche the fundamental theory of FS but alo rovde a new dea J emal for ubcrton: ublhng@wu.org.u

7 5 Y. Luou et al: Fuzzy Otmzaton ethod for ult-crtera Decon-mang Problem for the alcaton of FS theory. 6. cnowledgement Th reearch wor uorted by the Natural Scence Fund of hna (# Reference [] Zadeh L.. Fuzzy et. nformaton and ontrol 965; 8(: [] ortolan G. Degan R. revew of ome method for ranng fuzzy ubet. Fuzzy Set and Sytem 985; 5( (985 : -9. [] Lou T. S. Wang. J. Ranng fuzzy number wth ntegral value. Fuzzy Set and Sytem 99; 50(: [4] Tran L. and Ducten L. omaron of fuzzy number ung a fuzzy dtance meaure. Fuzzy Set and Sytem 00; 0(: 4. [5] Wang X. Z. and Kerre E.E. Reaonable roerte for the orderng of fuzzy quantte ( and (. Fuzzy Set and Sytem 00; 8(: [6] tanaov K. ntutontc fuzzy et V TKR Seon Sofa June 98 (Deoed n entral Scence- Techncal Lbrary of ulg. cademy of Scence 697/84 (n ulgaran. [7] hen S.. and Tan J.. Handlng mult-crtera fuzzy decon-mang roblem baed on vague et theory. Fuzzy Set and Sytem 994; 67(: 6 7. [8] Hong D.H.and ho.h. ult-crtera fuzzy decon-mang roblem baed on vague et theory. Fuzzy Set and Sytem 000; 4: 0. [9] Szmdt E. and Kacrzy J. Grou decon mang under ntutontc fuzzy reference relaton. Proc. of the Seventh nternatonal onference on PU 998; [0] tanaov K. Pa G. and Yager R. ntutontc fuzzy nterretaton of mult-meaurement tool mult-crtera decon mang. Proc. of the Sxth nternatonal onference on ntutontc Fuzzy Set Varna 4 Setember 00 Lecture Note on ntutontc Fuzzy Set 00; 8(: [] tanaov K. Pa G. Yager R.and tanaova V. ntutontc fuzzy grou nterretaton of mult-eron multcrtera decon mang. Proc. of the Thrd onference of the Euroean Socety for Fuzzy Logc and Technology EUSFLT 00. Zttau 0 Setember 00; [] Pa G. Yager Y.and tanaov K. ntutontc fuzzy grah nterretaton of mult-eron mult-crtera decon mang: Generalzed net aroach. Proc. of 004 econd nternatonal EEE onference ntellgent Sytem 004; : [] tanaov K. ntutontc fuzzy et. Fuzzy Set and Sytem 986; 0: [4] tanaov K. ntutontc fuzzy et. Phyca-Verlag Hedelberg New Yor 999. [5] Fan J. Xe W.and Pe J. Subethood meaure: new defnton. Fuzzy Set and Sytem 999; 06: [6] hen. T. Extenon of the TOPSS for grou decon-mang under fuzzy envronment. Fuzzy Set and Sytem 000; 4(: -9. [7] hu T..and Ln Y.. fuzzy TOPSS method for robot electon. The nternatonal Journal of dvanced anufacturng Technology 00; : [8] Wang Y..and Elhag T. Fuzzy TOPSS method baed on alha level et wth an alcaton to brdge r aement. Exert Sytem wth lcaton 006; (: [9] bo-snna..and mer. H. Extenon of TOPSS for mult-obectve large-cale nonlnear rogrammng roblem. led athematc and omutaton 005; 6(: [0] Lang G. S. Fuzzy D baed on deal and ant-deal concet. Euroean Journal of Oeratonal Reearch 999; (: J emal for contrbuton: edtor@c.org.u

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