Multiple Attribute Group Decision Making using Interval-Valued Intuitionistic Fuzzy Soft Matrix

Transcription

1 Mutpe ttbute Goup Deco Mkg ug Itev-Vued Itutotc Fuzzy Soft Mtx Sut D, Mohuy B K, Td P, St K btct- otcebe poge h bee foud deco kg pobe ce the toducto of oft et theoy by Moodtov 999 It foud tht cc oft et e ot utbe to de wth pece pete whee fuzzy oft et (FSS) e poved to be uefu Ue of tutotc fuzzy oft et (IFSS) oe effectve evoet whee guet e peeted ug ebehp d oebehp vue I th ppe we popoe gothc ppoch fo utpe ttbute goup deco kg pobe ug tev-vued tutotc fuzzy oft tx (IVIFSM) IVIFSM the tx epeetto of tevvued tutotc fuzzy oft et (IVIFSS), whee IVIFSS tu cobto of tev-vued tutotc fuzzy et d oft et theoy Fty, we popoe the cocept of IVIFSM The goth deveoped to fd out the deed tetve() bed o poduct tev-vued tutotc fuzzy oft tx, cobed choce tx, d coe vue of the et of tetve Fy, pctc expe h bee deotted to how the effectvee of the popoed goth Keywod- Mutpe ttbute goup deco kg, tevvued tutotc fuzzy oft tx, tev-vued tutotc fuzzy oft et, choce tx I INTRODUCTION Soft et theoy [] w toduced by Moodtov 999 geec thetc too fo deg wth ucet pobe whch cot be hded ug tdto thetc too oft et c be ued fo ppoxte decpto of obect wthout y etcto Due to th bece of etcto o the ppoxte decpto, oft et theoy h bee eegg coveet d ey ppcbe too pctce [] Sce t toducto, th theoy h bee uccefuy pped y dffeet fed uch deco kg [3]-[9], dt y [0], foectg [], uto [], optzto [3], textue cfcto [4], etc Sut D wth Dept of Copute Scece d Egeeg, D B C Roy Egeeg Coege, Dugpu 7306, Id (phoe , e-: ut_ce@yhooco) Mohuy B K wth Dept of Copute Scece d Egeeg, Hetge Ittute of Techoogy, Kokt 70007, Id (e-: ohuy_k@yhooco) Td P wth Dept of Copute Scece d Egeeg, Nto Ittute of Techoogy, Dugpu 7309, Id (e-: tdp@gco) St K wth Dept of Mthetc, Nto Ittute of Techoogy, Dugpu 7309, Id (e-: k k@yhooco) M et [5] peeted oe opeto fo oft et d o dcued the popete They peeted the cocept of fuzzy oft et (FSS) [6] whch bed o cobto of the fuzzy et d oft et ode Yg et [7] toduced the cocept of the tev-vued fuzzy oft et (IVFSS) by cobg the tev-vued fuzzy et d oft et d the popoed deco kg goth bed o IVFSS Deco kg pobe wee oved ft by M d Roy [8] ug oft et Roy d M, [9], popoed oft et theoetc ppoch to de wth deco kg d toduced the cocept of choce vue Cg d Egogu [3], [4] popoed the cocept of oft tx to epeet oft et Xu et [8] dcued the popete of vgue oft et Jg et [9] toduced tev-vued tutotc fuzzy oft et by cobg tev-vued tutotc fuzzy et d oft et theoy d dcued the popete Jg et [6] peeted dutbe ppoch to tutotc fuzzy oft et bed deco kg by ug eve oft et of tutotc fuzzy oft et Feg et [0] exteded the eve oft et ethod to tev-vued fuzzy oft et Q et [] geezed the ppoche toduced by Feg et [0] d Jg et [6] Q et [] o defed the oto of educt tutotc fuzzy oft et d peeted dutbe ppoch fo deco kg bed o tevvued tutotc fuzzy oft et Zhg et [] vetgted the deco kg pobe bed o tev-vued tutotc fuzzy oft et They deveoped dutbe ppoch to tev-vued tutotc fuzzy oft et bed deco kg ug eve oft et Cocept of fuzzy petezed tev-vued fuzzy oft et theoy h bee toduced by khzeh et [3] They tuded the opeto, the deveoped ggegto opeto to de wth deco kg pobe Mo et [4] peeted the cocept of tutotc fuzzy oft tx (IFSM) d pped t goup deco kg pobe Recety, D d K [5], [6] hve toduced the cocept of tutotc ut fuzzy oft et d hett fuzzy oft et, d o pped the deco kg pobe The go of utpe ttbute deco kg (MDM) to eect bette tetve() fo et of tetve bed o et of ttbute vue The ovety of th ppe to toduce the cocept of tev-vued tutotc fuzzy oft tx d peet gothc ppoch fo utpe ttbute goup deco kg ug IVIFSM I ou goth, goup of deco ke ugget the opo egdg et of tetve d the ttbute ug tev-vued tutotc fuzzy oft et whch e epeeted by IVIFSM Itev-vued tutotc fuzzy

2 et e foud to be uefu whee deco ke e ot ue bout ptcu vue the they woud ke to ey o tev to epeet gutc foto To fctte ou ppoch we ue cobed choce tx fo dvdu deco ke by copotg the choce pete of the et of expet Next, fo ech deco ke, we utpy the cobed choce tx wth the dvdu IVIFSM to poduce the poduct IVIFSM of tht expet Suto of poduct IVIFSM of deco ke copute the eutt IVIFSM, whee the ebehp d o-ebehp vue of ech tetve e dded to geete the coepodg weght Fy, coe d ccucy vue e ccuted to yed the deed tetve() Ret of the ppe oged foow I ecto we befy evew oe bc oto d bckgoud of oft et, fuzzy oft et, tev-vued fuzzy oft et d tev-vued tutotc fuzzy oft et Secto 3 peet tev-vued tutotc fuzzy oft tx foowed by the popoed gothc ppoch ecto 4 ce tudy h bee utted ecto 5 to vefy the pctcbty d effectvee of the popoed ethod Fy, cocuo e dw ecto 6 II PRELIMINRIES Th ecto befy evew oe bc cocept eted wth th tce Defto [] Let U be t uvee, E be et of pete, P(U) be the powe et of U, d E p ( F, ) ced oft et ove U, whee F ppg F : P( U ) I othe wod, oft et ove U ppg fo pete to P(U), d t ot et, but petezed fy of ubet of U Fo y pete e, F( e) y be codeed the et of e-ppoxte eeet of the oft et ( F, ) Defto [6] Let U be t uvee d E be et of pete (whch e fuzzy vbe) Let P ( U ) deote the et of fuzzy et of U d E p ( F, E ) ced Fuzzy Soft Set ( FSS ) ove U, whee F ppg gve by, F : E P( U ) uch tht F ( e ) = f e, whee ~ u fuzzy et Rek fuzzy oft et petezed fy of fuzzy ubet of U It uvee the et of fuzzy et of U, e, P(U) fuzzy oft et c be codeed pec ce of oft et becue t t ppg fo pete to uvee The dffeece betwee fuzzy oft et d oft et tht fuzzy oft et, the uvee to be codeed the et of fuzzy ubet of U Defto 3 [7] Let U be t uvee d E be et of pete IVF (U) deote the et of tevvued fuzzy et of U Let E p ( F, ) tev-vued fuzzy oft et ove U, whee F ppg, gve by F : IVF (U) tev-vued fuzzy oft et petezed fy of tev-vued fuzzy ubet of U, thu, t uvee the et of tevvued fuzzy et of U, e, IVF (U) Rek The cobed eut of fuzzy et d oft et theoy defed fuzzy oft et Howeve, t foud tht y e ppcto, the ebehp degee fuzzy et cot be boutey cofed It oe eobe to gve tev-vued dt to decbe ebehp degee Fo uch pot of vew, Zdeh [30] popoed the cocept of tev-vued fuzzy et whch defed by cobg the tev-vued fuzzy et d oft et ode Defto 4 [7] tov d Ggov (989) toduced tev-vued tutotc fuzzy et (IVIFS) Let X be uve et tev-vued tutotc fuzzy et X c be expeed { x,[ ( x), ( x)],[ ( x), ( x)] x X, whee [ ( x), ( x)] [0,] d [ ( x), ( x)] [0,] e epectvey the tev-vued degee of ebehp d o-ebehp of eeet x X to, whch e oe o e depedet o ech othe The oy equeet tht the u of uppe bud of thee two tev-vued degee ot gete th, e, 0 ( x) ( x) If ( x) ( x) d ( x) ( x), x X, the the IVIFS { x,[ ( x), ( x)],[ ( x), ( x)] x X educed to tov IFS, deoted by whee { x, ( x), ( x) x X, ( x) ( x) ( x) d ( x) ( x) ( x) So, we c y tov IFS pec ce of IVIFS Fo fxed x X, obect [ ( ), ( )],[ ( ), x x x ( x)] ced tev-vued tutotc fuzzy ube (IVIFN) Let [, b],[ c, d] be IVIFN The coe fucto [8] S of c be defed ( c) ( b d ) S ( ), whee S( ) [0,] The ccucy fucto [9] H of c be defed ( c) ( b d ) H ( ), whee H ( ) [0,] Defto 5 [9] Let U be t uvee d E be et of pete IVIF (U) deote the et of tevvued tutotc fuzzy et of U Let E p ( F, ) tev-vued tutotc fuzzy oft et ove U, whee F ppg, gve by F : IVIF (U) tev-vued tutotc fuzzy oft et petezed fy of tev-vued tutotc fuzzy

3 ubet of U, thu, t uvee the et of tevvued tutotc fuzzy et of U, e, IVIF U, F( ) tev-vued tutotc fuzzy et of U F( ) c be expeed : F( ) { x,[ ( x), ( x)],[ ( x), ( x)] x X F ( ) F ( ) F ( ) F ( ) Defto 6 Choce Mtx que tx whoe ow d cou both dcte pete If choce tx, the t eeet (, ) defed foow: th th (, ),(,) whe d pete e both choce pete of the deco ke P (, ) th (0, 0),(0,0) othewe, e, whe t et oe of the th o pete ot ude choceof the deco ke I cobed choce tx, ow dcte choce pete of ge deco ke, whee cou dcte cobed choce pete (obted by the teecto of pete et) of deco ke III INTERVL-VLUED INTUITIONISTIC FUZZY SOFT MTRIX Let ( F, E) be tev-vued tutotc fuzzy oft et ove the t uvee U Let E be et of pete d E The ubet of U E uquey defed by R = {( u, e) : e, u F ( e), whch ced eto of ( F, E) The ebehp fucto of R wtte R : U E It([0,]) d defed by {[ ( ), ( )],[ ( ), ( )] F u ( e) u F ( e) u F( e) u f e F( e) R ( u, e) = [0,0], f e, whee It ([0,]) td fo the et of coed ubtev of [0,] d F ( e) F ( e) [ ( u), ( u)] F ( e) d F ( e) [ ( u), ( u)] epectvey deote the tevvued tutotc fuzzy ebehp d o-ebehp degee of the obect u octed wth the pete e If U = { x, x,, x d E = { e, e,, e, the R c be peeted foow: e e e x ( x ), ( x ), ( x ), ( x ), ( x ), ( x ), F ( e ) F ( e ) F ( e ) F ( e ) F ( e ) F ( e ) ( x ), ( x ) ( x ), ( x ) ( x ), ( x ) F ( e ) F ( e ) F ( e ) F ( e ) F ( e ) F ( e ) x ( x ), ( x ), ( x ), ( x ), ( x ), ( x ) F ( e ) ( ) ( ) ( ) ( ) ( ) F F e F e F e F e F e ( x ), ( x ) ( x ), ( x ) ( x ), ( x ) F ( e ) F ( e ) F ( e ) F ( e ) F ( e ) F ( e ) x ( x ), ( x ), ( x ), ( x ) ( x ), ( x ) F ( e ) F ( e ) F ( e ) F ( e ) F ( e ) F ( e ) ( x ), ( x ) ( x ), ( x ) ( x ), ( x ) F ( e ) F ( e ) F ( e ) F ( e ) F ( e ) F ( e ) Fo pcty, f we tke the [ ] th ety of the eto F {[ ( x ), ( x )],[ ( x ), ( x )], F ( e ) F ( e ) F ( e ) F ( e ) the the tx c be defed [ ] = The bove tx ced tev-vued tutotc fuzzy oft tx of ode coepodg to the tev-vued tutotc fuzzy oft et ( F, E) ove U Expe Let U be the et of fve dee, gve by, U = { D, D, D, D, D d E the et of fve ypto, gve by, E = { S, S, S3, S4, S5 Let ={ S, S, S3, S5 E Now uppoe tht, F : E P( U) decbe the pobe dee coepodg to et of ypto d the tev-vued tutotc fuzzy oft et ( F, E) gve by ( F, E) = { dee voved wth S = { D /[4,6],[3,4], D /[7,8],[,], D /[4,5],[3,4], D /[3,6],[,4], D /[,3],[4,6], dee voved wth S { D /[0,],[5,7], D /[7,9],[0,], D /[3,4],[3,6], D /[4,6],[,3], D /[,5],[,4], =

4 dee voved wth S { D /[3,5],[,5], D /[4,5],[,4], 3 = D3/[6,8],[0,], D4/[,],[5,8], D5/[3,5],[,4], dee voved wth S = { D /[5,7],[,3], D /[3,7],[,], D /[,],[6,7], D /[4,6][,4], D /[7,8],[,] Hece, the IVFSM [ ] c be wtte, [4,6],[3,4] [0,],[5,7] [3,5],[,5] [0,0],[0,0] [5,7],[,3] [7,8],[,] [7,9],[0,] [4,5],[,4] [0,0],[0,0] [3,7],[,] [ ] = [4,5],[3,4] [3,4],[3,6] [6,8],[0,] [0,0],[0,0 ] [,],[6,7] [3,6],[,4] [4,6],[,3] [,],[5,8] [0,0],[0,0] [4,6],[,4] [,3],[4,6] [,5],[,4] [3,5],[,4] [0,0],[0,0] [7,8],[,] Poduct of Itev-Vued Itutotc Fuzzy Soft Mtx d Choce Mtx c be peeted f the ube of cou of tev-vued tutotc fuzzy oft tx be equ to the ube of ow of the choce tx The d e d to be cofobe fo the poduct ( ) d the poduct ( ) becoe tev-vued tutotc fuzzy oft tx The poduct ( ) o deoted py by If [ ] d whee c k [ k ] p, the [ ck ] p, [x {,, x {, ], k k [ {,, {, ] k k Two tev-vued tutotc fuzzy oft tce d B e d to be cofobe fo ddto, f they e of the e ode d fte ddto the obted u o IVIFSM of the e ode Now f both = ( ) d B = ( b ) be the e ode, the the ddto of d B deoted by B d defed by [ c ] [ ] [ ], b whee c [x{, b,x{, ], b =, [{, b,{, b ] Copeet of IVIFSM ( ) deoted by ( c ), whee ( ) the tx epeetto of the tev-vued tutotc fuzzy oft et ( F, E) ( ) c the tx epeetto of the tev-vued c tutotc fuzzy oft et ( F, E) d c be defed ( ) c, c, c, c c [, ],[, ] IV LGORITHMIC PPROCH Step : Opo of et of expet / deco ke P { P, P,, P k D d d d S {,,, fo gve et of tetve {,,, d et of ttbute tutotc fuzzy oft tce e epeeted ug tev-vued Step : Choce tx (, ) P d cobed choce P (, ) C P { P, P,, P k tx of ech of the deco ke e coputed the cotext of tev-vued tutotc fuzzy et bed o the choce pete / ttbute Step 3: Poduct IVIFSM ( P IVIFSM ) fo ech deco ke ccuted by utpyg ech IVIFSM wth t cobed choce tx Step 4: Suto of thee poduct IVIFSM the eutt IVIFSM ( R ) IVIFSM Step 5: Weght W( d ) of ech tetve d {,,, etted by ddg the ebehp d o-ebehp vue of the ete of the epectve ow ( th ow) Step 6: d D, copute the coe S( d ) of d, uch tht, S( d ) = {( ) ( )/, d D Step 7: If S( d ) S( d ) d D, the tetve d eected If, uch tht, S( d ) S( d ), whee, fo hghet coe vue, the deco de ccodg to the ccucy vue decbed tep 8 Step 8: ccucy vue H( d ){,,,, defed H ( d ) = {( ) ( )/, d D If H ( d ) H ( d ) fo whch S( d ) S( d ), defed tep 7, tetve d eected If H ( d) H( d ), fo y, the d d d e eected

5 V CSE STUDY Let D d, d, d, d, d be the et of fve tge of het dee (Stge I, Stge II, Stge III, Stge IV, d Stge V ) Ptet beogg to Stge I e ued ot to be ffected by het dee Ptet beogg to Stge II e t tge, ptet beogg to Stge III e oe ufe tge th tge II d o o Ptet beogg to Stge V e the t tge of het dee whch uecovebe Let E be the et of fve ypto (Chet p, Pptto, Dzze, Ftg, Ftgue) gve by E =,,,, Suppoe tht goup of thee expet P = {P, P, P 3 e otog the ypto pe the kowedgebe to ech coeu bout whch dee oe key to ppe fo the ptet, whee expet P we of ypto (,, 3, 4 ), P we of ypto (,, 3, 5 ), d P 3 we of (,, 4, 5 ) ccodg to the ypto o pete obeved by the thee expet, we ue to hve the tev-vued tutotc fuzzy oft et ( F { P, E ), ( F { P, E ), d ( F { P, E ) fo expet P, P, d P 3 epectvey 3 [Step ] Let the tev-vued tutotc fuzzy oft tce of the tev-vued tutotc fuzzy oft et ( F { P, E),( F { P, E),( F { P, E ) e epectvey, 3 (0307), (03,04), (05,06), (05,06),, (0,0) (04,05) (0,04) (0,04) (04,05), (03,07), (03,06), (03,07),, (03,05) (0,03) (0,03) (0,0) (03,06), (04,06), (04,06), (04,06),, { p(, ), (03,04) (0,03) (0,04) (0,04) (04,08), (0,05), (0,05), (0,05),, (0,0) (03,04) (03,04) (03,04) (03,05), (0,03), (0,03), (03,04),, (0,04) (04,07) (05,06) (04,05) (03,05), (05,06), (0,05),, (05,07), (03,04) (03,04) (03,04) (0,03) (04,05), (04,07), (03,05),, (04,05), (03,05) (0,03) (0,03) (03,04) (03,06), (07,08), (0,04),, (04,06), { p(, k), (03,04) (0,0) (03,05) (0,04) (0,0 6), (05,07), (03,06),, (05,06), (0,0) (0,0) (0,03) (03,04) (04,05), (0,03), (04,07), (0 0,00), (04,05), (0,04) (04,05) (0,03) (03,04) (03,05), (0,07),, (03,06), (0,04), (03,04) (0,03) (03,04) (05,06) (03,07), (03,05),, (04,08), (07,08), (0,03) (03,05) (0,0) (0,0) (05,07), (05,06),, (05,06), (04,06), { p3 (, ) (0,03) (03,04) (03,04) (03,04) (04,0 6), (04,05),, (03,07), (03,04), (0,03) (03,04) (0,03) (04,05) (0,0), (04,06),, (0,03), (0,04), (05,07) (03,04) (05,06) (04,05) Hee (,,,5) be the ube of tetve (tge of het dee) d, k, (,,,5) be the ube of ttbute (ypto) [Step ] The cobed choce tce fo P, P, d P 3 e epectvey { P { P { P3 { P P3,,,,,,,,,,,,,,,,,,,,,,,,, { P P3,,,,,,,,,,,,,,,,,,,,,,,,, { P P,,,,,,,,,,,,,,,,,,,,,,,,, [Step 3] The poduct ( defed Secto 3) of IVFSM d cobed choce tce e gve beow

6 3 (0307), (03,04), (05,06), (05,06),, (,), (,), (0,0), (0,0), (,), (0,0) (04,05) (0,04) (0,04) (,) (,) (0,0) (0,0) (,) (04,05), (03,07), (03,06), (03,07),, (,), (,), (0,0), (0,0), (,), (03,05) (0,03) (0,03) (0,0) (,) (,) (0,0) (0,0) (,) (03,06), (04,06), (04,06), (04,06),, { p (, ) (,), (,), (0,0), (0,0), (,), { P (03,04) (0,03) (0,04) (0,04) (00, 00) (,) (,) (0,0) (0,0) (,) (04,08), (0,05), (0,05), (0,05),, (,), (,), (0,0), (0,0), (,), (0,0) (03,04) (03,04) (03,04) (,) (,) (0,0) (0,0) (,) (03,05), (0,03), (0,03), (03,04),, (0,0), (0,0), (0,0), (0,0), (0,0), (0,04) (04,07) (05,06) (04,05) (0,0) (0,0) (0,0) (0,0) (0,0) (0507), (0507),,, (0507), (0,0) (0,0) (0,0) (04,07), (04,07),,, (04,07), (0,0) (0,0) (0,0) (04,06), (04,06),,, (04,06), (0,03) (0,03) (0,03) (04,08), (04,08),,, (04,08), (0,0) (0,0) (0,0) (03,05), (03,05),,, (03,05), (0,04) (0,04) (0,04) { P P (03,05), (05,06), (0,05),, (05,07), (,), (,), (0,0), (,), (0,0), (03,04) (03,04) (03,04) (0,03) (,) (,) (0,0) (,) (0,0) (04,05), (04,07), ( 03,05),, (04,05), (,), (,), (0,0), (,), (0,0), (03,05) (0,03) (0,03) (03,04) (,) (,) (0,0) (,) (0,0) (03,06), (07,08), (0,04),, (04,06), { p(, k) (,), (,), (0,0), (,), (0,0), { P (03,04) (0,0) (03,05) (0,04) (,) (,) (0,0) (,) (0,0) (0,06), (05,07), (03,06),, (05,06), (0,0), (0,0), (0,0), (0,0), (0,0), (0,0) (0,0) (0,03) (03,04) (0,0) ( 0,0) (0,0) (0,0) (0,0) (04,05), (0,03), (04,07),, (04,05), (,), (,), (0,0), (,), (0,0), (0,04) (04,05) (0,03) (03,04) (,) (,) (0,0) (,) (0,0) (05,0 7), (05,07),, (05,07),, (0,03) (0,03) (0,03) (04,07), (04,07),, ( 04,07),, (0,03) (0,03) (0,03) (07,08), (07,08),, (07,08),, (0,0) (0,0) (0,0) (05,07), (05,07),, (05,07),, (0,0) (0,0) (0,0) (04,07), (04,07),, (04,07),, (0,03) (0,03) (0,03) { P P3 (03,05), (0,07),, (03,06), (0,04), (,), (,), (,), (0,0), (0,0), (03,04) (0,03) (03,04) (05,06) (,) (,) (,) (0,0) (0,0) (03,07), (03,05), ( 00,00), (04,08), (07,08), (,), (,), (,), (0,0), (0,0), (0,03) (03,05) (0,0) (0,0) (,) (,) (,) (0,0) (0,0) (05,07), (05,06),, (05,06), (04,06), { p3(, ) (0,0), (0,0), (0,0), (0,0), (0,0), { P3 (0,03) (03,04) (03,04) (03,04) (0,0) (0,0) (0,0) (0,0) (0,0) (04,06), (04,05),, (03,07), (03,04), (,), (,), (,), (0,0), (0,0), (0,03) (03,04) (0,03) (04,05) (,) (,) (,) (0,0) (0,0) (0,0), (04,06),, (0,03), (0,04), (,), (,), (,), (0,0), (0,0), (05,07) (03,04) (05,06) (04,05) (,) (,) (,) (0,0) (0,0) (03, 07), (03,07), (03,07),,, (0,03) (0,03) (0,03) (07,08), (07,08), (07,08),,, (0,0) (0,0) (0,0) (05,07), (05,07), (05,07),,, (0,03) (0,03) (0,03) (04,07), (04,07), (04,07),,, (0,03) (0,03) (0,03) (04,06), (04,06), (04,06),,, (03,04) (03,04) (03,04) { P P

7 [Step 4] The u of thee poduct tev-vued tutotc fuzzy oft tce ( ), ( ), ( 0 0,0 0 ), (0 0,0 0 ), ( ), (0 5,0 7 ), (0 5,0 7 ), ( 0 0,0 0 ), (0 5,0 7 ), (0 0,0 0 ), (0,0 ) (0,0 ) ( 0 0,0 0 ) (0 0,0 0 ) (0,0 ) (0,0 3 ) (0,0 3 ) ( 0 0,0 0 ) (0,0 3 ) (0 0,0 0 ) (0 4,0 7 ), (0 4,0 7 ), ( 0 0,0 0 ), (0 0,0 0 ), (0 4,0 7 ), (0 4,0 7 ), (0 4,0 7 ), ( 0 0,0 0 ), (0 4,0 7 ), (0 0,0 0 ), (0,0 ) (0,0 ) ( 0 0,0 0 ) (0 0,0 0 ) (0,0 ) (0,0 3 ) (0,0 3 ) ( 0 0,0 0 ) (0,0 3 ) (0 0,0 0 ) (0 4,0 6 ), (0 4,0 6 ), ( 0 0,0 0 ), (0 0,0 0 ), (0 4,0 6 ), (0 7,0 8 ), (0 7,0 8 ), ( 0 0, 0 0 ), (0 7,0 8 ), (0 0,0 0 ), (0,0 3 ) (0,0 3 ) ( 0 0,0 0 ) (0 0,0 0 ) (0,0 3 ) (0,0 ) (0,0 ) ( 0 0,0 0 ) (0,0 ) (0 0,0 0 ) (0 4,0 8 ), (0 4,0 8 ), ( 0 0,0 0 ), (0 0,0 0 ), (0 4,0 8 ), (0 5,0 7 ), (0 5,0 7 ), ( 0 0,0 0 ), (0 5,0 7 ), ( 0 0,0 0 ), (0,0 ) (0,0 ) ( 0 0,0 0 ) (0 0,0 0 ) (0,0 ) (0,0 ) (0,0 ) ( 0 0,0 0 ) (0,0 ) ( 0 0,0 0 ) (0 3,0 5 ), (0 3,0 5 ), ( 0 0,0 0 ), ( 0 0,0 0 ), (0 3,0 5 ), (0 4,0 7 ), (0 4,0 7 ), ( 0 0,0 0 ), (0 4,0 7 ), (0 0,0 0 ), (0,0 4 ) (0,0 4 ) ( 0 0,0 0 ) (0 0,0 0 ) (0,0 4 ) (0,0 3 ) (0,0 3 ) ( 0 0,0 0 ) (0,0 3 ) (0 0,0 0 ) (0 3,0 7 ), (0 3,0 7 ), ( 0 3,0 7 ), (0 0,0 0 ), (0 0,0 0 ), (0 5,0 7 ), ( 0 5,0 7 ), (0 3,0 7 ), (0 5,0 7 ), (0 5,0 7 ), (0,0 3 ) (0,0 3 ) ( 0,0 3 ) ( 0 0,0 0 ) (0 0,0 0 ) (0,0 ) ( 0,0 ) (0,0 3 ) (0,0 3 ) (0,0 ) (0 7,0 8 ), (0 7,0 8 ), ( 0 7,0 8 ), (0 0,0 0 ), ( 0 0,0 0 ), (0 7,0 8 ), ( 0 7,0 8 ), (0 7,0 8 ), (0 4,0 7 ), (0 4,0 7 ), (0,0 ) (0,0 ) ( 0,0 ) (0 0,0 0 ) ( 0 0,0 0 ) (0,0 ) ( 0, 0 ) (0,0 ) (0,0 3 ) (0,0 ) (0 5,0 7 ), (0 5,0 7 ), ( 0 5,0 7 ), (0 0,0 0 ), ( 0 0,0 0 ), (0 7,0 8 ), ( 0 7,0 8 ), (0 5,0 7 ), (0 7,0 8 ), (0 4,0 6 ), (0,0 3 ) (0,0 3 ) ( 0,0 3 ) (0 0,0 0 ) ( 0 0,0 0 ) (0,0 ) ( 0,0 ) (0,0 3 ) (0,0 ) ( 0,0 3 ) (0 4,0 7 ), (0 4,0 7 ), ( 0 4,0 7 ), (0 0,0 0 ), (0 0,0 0 ), (0 5,0 8 ), ( 0 5,0 8 ), (0 4,0 7 ), (0 5,0 7 ), ( 0 4,0 8 ), (0,0 3 ) (0,0 3 ) ( 0,0 3 ) (0 0,0 0 ) (0 0,0 0 ) (0,0 ) ( 0,0 ) (0,0 3 ) (0,0 ) ( 0,0 ) (0 4,0 6 ), (0 4,0 6 ), ( 0 4,0 6 ), (0 0,0 0 ), (0 0,0 0 ), (0 4,0 7 ), ( 0 4,0 7 ), (0 4,0 6 ), (0 4,0 7 ), ( 0 3,0 5 ), (0 3,0 4 ) (0 3,0 4 ) ( 0 3,0 4 ) (0 0,0 0 ) (0 0,0 0 ) (0,0 3 ) ( 0,0 3 ) (0 3,0 4 ) (0,0 3 ) ( 0,0 4 ) [Step 5] Now the weght of vou tge of het dee W ( d ),,,, 5 e ccuted foow: [ , ], [ 3, 35], W ( d ) [ , [06,] ] [9, 45], [30, 37], Sy, W ( d ), ( 3 ), [06,] W d [06,] [3, 38], [9, 3], W ( d 4 ), W ( d 5 ) [06,] [08,7] [Step 6] Scoe of vou tge of het dee c be coputed foow: S( d ) {(3 06) (35 )/ S( d ) {(9 06) (45 )/ 85 S( d ) {(30 06) (37 )/ 45 3 S( d ) {(3 06) (38 )/ 4 S( d ) {(9 08) (3 7)/ 3 5 [Step 7] Sce coe of d xu, the ptet ude codeto beog to Stge II, e, t tge of het dee pe the coectve opo of the goup of expet VI CONCLUSIONS Th ppe peet gothc ppoch fo utpe ttbute goup deco kg ug tev-vued tutotc fuzzy oft tx Fty we popoe tevvued tutotc fuzzy oft tx d defe oe of t eevt opeto Next we popoe goth ug cobed choce tx, coe fucto, d ccucy fucto The popoed ethod yed opt tetve() whch efect the coectve opo of goup of deco ke We hve peeted ce tudy eted wth edc dgo I th tudy, we ue opo of goup of expet bout coo et of ypto whee opo of oe expet bout ubet of ypto ght be g due to ck of kowedge o expeece Expet povde the opo ug tev-vued tutotc fuzzy oft et whch e epeeted te of tevvued tutotc fuzzy oft tce Idvdu tevvued tutotc fuzzy oft tce e utped wth epectve cobed choce tce The eutt dvdu poduct tce e ued up to fd out the f tx whee ebehp vue of ech tetve e dded to geete the weght of tetve Next, coe vue e ccuted Moe coe w ed to bette tetve Whe thee te coe vue, the ccucy vue e ued fo f deco tetve wth oe ccucy vue w be the opt choce Futue cope of th eech wok ght be to vetgte the vou popete of tev-vued tutotc fuzzy oft tx d ppy the to utbe ucet deco kg pobe REFERENCES [] D Moodtov, Soft et theoy ft eut, Coput Mth pp vo 37, o (4 5), pp 9 3, 999 [] Kh, Dtce d ty eue fo oft et, New Mthetc d Ntu Coputto, vo 6, o 3, pp 3 334, 00 [3] N Cg d S Egogu, Soft tx theoy d t deco kg, Copute d Mthetc wth ppcto, vo 59, o 0, pp , 00 [4] N Cg d S Egogu, Soft et theoy d u t deco kg, Euope Jou of Opeto Reech, vo 07, o, pp , 00 [5] F Feg, Y B Ju, X Lu, d L L, dutbe ppoch to fuzzy oft et bed deco kg, Jou of Coputto d pped Mthetc, vo 34, o, pp 0 0, 00 [6] Y Jg, Y Tg, d Q Che, dutbe ppoch to tutotc fuzzy oft et bed deco kg, pped Mthetc Modeg, vo 35, o, pp , 0 [7] Z Kog, L Go, d L Wg, Coet o fuzzy oft et theoetc ppoch to deco kg pobe, Jou of Coputto d pped Mthetc, vo 3, o, pp , 009

8 [8] P K M, R Roy, d R Bw, ppcto of oft et deco kg pobe, Copute d Mthetc wth ppcto, vo 44, o (8-9), pp , 00 [9] R Roy d P K M, fuzzy oft et theoetc ppoch to deco kg pobe, Jou of Coputto d pped Mthetc, vo 03, o, pp 4 48, 007 [0] Y Zou d Z Xo, Dt y ppoche of oft et ude copete foto, Kowedge-Bed Syte, vo, o 8, pp , 008 [] Z Xo, K Gog, d Y Zou, cobed foectg ppoch bed o fuzzy oft et, Jou of Coputto d pped Mthetc, vo 8, o, pp , 009 [] S J Kythk d G S Sgh, fuzzy oft food ode, Mthetc d Copute Suto, vo 80, o 5, pp , 00 [3] D V Kovkov, V M Kobov, d D Moodtov, Soft et theoy-bed optzto, Jou of Copute d Syte Scece Iteto, vo 46, o 6, pp , 007 [4] M M Muhf, S Segupt, d K Ry, Textue cfcto ug ove, oft et theoy bed cfcto goth, : PJ Ny, SK Ny, HY Shu (Ed), Poceedg of the 7th Cofeece o Copute Vo, 006 Lectue Note Copute Scece, vo 385, pp [5] P K M, R Bw, d R Roy, Soft et theoy, Coput Mth pp, vo 45, pp , 003 [6] P K M, R Bw, d R Roy, Fuzzy oft et, J Fuzzy Mth, vo 9, o 3, pp , 00 [7] X B Yg, T Y L, J Y Yg, Y L, d D Y Yu, Cobto of tev-vued fuzzy et d oft et, Coput Mth pp, vo 58, pp 5 57, 009 [8] W Xu d J M, Vgue oft et d the popete, Coput Mth pp, vo 59, pp , 00 [9] Y Jg d Y Tg, Itev-vued tutotc fuzzy oft et d the popete, Coput Mth pp, vo 60, pp , 00 [0] F Feg, Y L, d V Leoeu-Fote, ppcto of eve oft et deco kg bed o tev-vued fuzzy oft et, Coput Mth pp, vo 60, pp , 00 [] H Q, X M, T Hew, d J M Z, dutbe ppoch to tev-vued tutotc fuzzy oft et bed deco kg, : N T Nguye, C G K, d Jk (Ed), Poceedg of CIIDS, 0, LNI, vo 659, pp 80 89, Spge-Veg Be Hedebeg [] Z Zhg, C Wg, D T, d K L, ove ppoch to tevvued tutotc fuzzy oft et bed deco kg, pped Mthetc Modeg, 03, ( pe) [3] S khzeh, R Seh, d N H, Fuzzy petezed tev-vued fuzzy oft et, pp Mth Sc, vo 67, pp , 0 [4] J Mo, D Yo, d C Wg, Goup deco kg ethod bed o tutotc fuzzy oft tce, pped Mthetc Modeg, vo 37, pp , 03 [5] S D d S K, Itutotc Mut Fuzzy Soft Set d t ppcto Deco Mkg, : P M et (Ed), Poceedg of Ffth Iteto Cofeece o Ptte Recogto d Mche Itegece (PReMI), Kokt, Decebe 0-4, 03, Lectue Note Copute Scece, vo 85, pp [6] S D d S K, The Hett Fuzzy Soft Set d t ppcto Deco Mkg, Poceedg of Iteto Cofeece o Fcet of Ucette d ppcto (ICFU), Kokt, Decebe 5-7, 03 ( pe) [7] K tov d G Ggov, Itev-vued tutotc fuzzy et Fuzzy Set d Syte, vo 3, o 3, pp , 989 [8] S Che d J T, Hdg utcte fuzzy deco-kg pobe bed o vgue et theoy, Fuzzy Set d Syte, vo 67, o, pp 63-7, 994 [9] Z Xu, Method fo ggegtg tev-vued tutotc fuzzy foto d the ppcto to deco kg, Coto d Deco, vo, pp 5-9, 007 [30] L Zdeh, The cocept of gutc vbe d t ppcto to ppoxte eog I, Ifoto Scece, vo 8, o 3, pp 99 49, 975