Multi-criteria Group Decision Making Using A Modified Fuzzy TOPSIS Procedure

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1 Mult-crter roup Decson Mkng Usng A Modfed Fuzzy TOPSIS Procedure Soroush Sghfn Deprtment of Industrl Engneerng Shrf Unversty of Technology Tehrn Irn sghfn@mehr.shrf.edu S.Rez Hez Deprtment of Industrl Engneerng Isfhn Unversty of Technology Isfhn Irn rehez@cc.ut.c.r Abstrct-In ths pper we propose modfed Fuzzy Technque for Order Performnce by Smlrty to Idel Soluton (modfed Fuzzy TOPSIS) for the Mult-crter Decson Mkng (MCDM) problem when there s group of decson mkers. Regrdng the vlue of the truth tht fuzzy number s greter thn or equl to nother fuzzy number new dstnce mesure s proposed n ths pper. Ths dstnce mesure clcultes the dstnce of ech fuzzy number from both Fuzzy Postve Idel Soluton (FPIS) nd Fuzzy negtve Idel Soluton (FNIS). Then the lterntve whch s smultneously closer to FPIS nd frther from FNIS wll be selected s the best choce. To clrfy our proposed procedure numercl emple s dscussed. I. INTRODUCTION Decson mkng problem s the process of fndng the best opton from ll of the fesble lterntves. In lmost ll such problems the multplcty of crter for udgng the lterntves s pervsve. Tht s for mny such problems the decson mker wnts to solve multple crter decson mkng (MCDM) problem. A MCDM problem cn be concsely epressed n mtr formt s: where A 1 A... Am re possble lterntves mong whch decson mkers hve to choose C 1 C... Cm re crter wth whch lterntve performnce re mesured s the (1) () rtng of lterntve A wth respect to crteron C nd w s the weght of crteron C. A survey of the MCDM methods hs been presented by Hwng nd Yoon [1]. Technque for Order Performnce by Smlrty to Idel Soluton (TOPSIS) one of the known clsscl MCDM methods lso ws frst developed by Hwng nd Yoon [1]. It bses upon the concept tht the chosen lterntve should hve the shortest dstnce from the Postve Idel Soluton (PIS).e. the soluton tht mmzes the beneft crter nd mnmzes the cost crter; nd the frthest from the Negtve Idel Soluton (NIS).e. the soluton tht mmzes the cost crter nd mnmzes the beneft crter. In clsscl MCMD methods ncludng clsscl TOPSIS the rtngs nd the weghts of the crter re known precsely. However under mny condtons crsp dt re ndequte to model rel-lfe stutons snce humn udgments ncludng preferences re often vgue nd cnnot estmte hs preference wth n ect numercl vlue. A more relstc pproch my be to use lngustc ssessments nsted of numercl vlues tht s to suppose tht the rtngs nd weghts of the crter n the problem re ssessed by mens of lngustc vrbles. Lngul epressons for emple low medum hgh etc. re regrded s the nturl representton of the udgment. These chrcterstcs ndcte the pplcblty of fuzzy set theory n cpturng the decson mkers preference structure. Fuzzy set theory ds n mesurng the mbguty of concepts tht re ssocted wth humn beng s subectve udgment. Moreover snce n the group decson mkng evluton s resulted from dfferent evlutor s vew of lngustc vrbles ts evluton must be conducted n n uncertn fuzzy envronment. There re mny emples of pplctons of fuzzy TOPSIS n lterture (For nstnce: The evluton of servce qulty []; Inter compny comprson [3]; The pplctons n ggregte producton plnnng [4] Fclty locton selecton [5] nd lrge scle nonlner progrmmng [6]). The modfctons proposed n ths pper cn be mplemented n ll rel world pplctons of Fuzzy TOPSIS. Proceedngs of the 005 Interntonl Conference on Computtonl Intellgence for Modellng Control nd Automton nd Interntonl Conference on Intellgent Agents Web Technologes nd Internet Commerce (CIMCA-IAWTIC 05)

2 Ths study ncludes modfctons n Fuzzy Multple Crter Decson-Mkng (MCDM) theory to strengthen the comprehensveness nd resonbleness of the decson mkng process usng Fuzzy TOPSIS. Consderng the fuzzness n the decson dt nd group decson mkng process lngustc vrbles re used to ssess the weghts of ll crter nd the rtngs of ech lterntve wth respect to ech crteron. It s possble to convert the decson mtr nto fuzzy decson one nd construct weghted normlzed fuzzy decson mtr once the decson mkers' fuzzy rtngs hve been pooled. Accordng to the concept of TOPSIS we defne the Fuzzy Postve Idel Soluton (FPIS) nd the Fuzzy Negtve Idel Soluton (FNIS). Then we use new method to clculte the dstnce between two trngulr fuzzy rtngs. Usng the de of comprson between two fuzzy numbers we clculte the dstnce of ech lterntve from FPIS nd FNIS respectvely. In other words new dstnce mesure for Fuzzy TOPSIS s proposed n ths pper. Fnlly closeness coeffcent of ech lterntve s used to determne the rnkng order of ll lterntves. The hgher vlue of closeness coeffcent ndctes tht n lterntve s closer to FPIS nd frther from FNIS smultneously. The remnder of ths pper s orgnzed s follows. Secton II presents some necessry defntons nd formultons. Secton III descrbes our modfed procedure nd n secton IV of ths pper to hghlght nd clrfy our proposed procedure numercl emple s dscussed. Fnlly secton V brefly concludes. II. DEFINITIONS AND FORMULATIONS In ths secton we wll cover some bsc defntons nd formuls tht re used n our pper. Defnton 1. A fuzzy set A n unverse of dscourse X s chrcterzed by membershp functon A() whch ssoctes wth ech element n X rel number n the ntervl [01]. The functon vlue s termed the grde of membershp of n A (defned by Zdeh [7]). Defnton. A fuzzy set A of the unverse of dscourse X s conve f nd only f for ll 1 nd n X : A( 1 (1 ) ) Mn( A( 1) A( )) (3) where [01]. Defnton.3. A fuzzy set A of the unverse of dscourse X s clled norml fuzzy set mplyng tht X : A() =1. Defnton 4. A fuzzy number ñ s fuzzy subset n the unverse of dscourse X tht s both conve nd norml. Fg. 1 shows fuzzy number of the unverse of dscourse X whch s both conve nd norml. Defnton 5. The -cut of fuzzy number ñ s defned: n { : n ( ) X } (4) where [01]. n defned n Eq. (4) s non-empty bounded closed ntervl contned n X nd t cn be denoted by n =[ n l n u ] where n l nd n u re the lower nd upper bounds of the closed ntervl respectvely (defned by Zmmermnn [8]). Fg. shows fuzzy number ñ wth cuts where: [ ] n nl nu n [ nl nu ]. Defnton 6. A trngulr fuzzy number ñ cn be defned by trplet ( n1 n n3) shown n Fg. 3. The membershp functon n ( ) s defned s: n1 n1 n n n1 n3 n ( ) n n3 (5) n n3 0 otherwse. In ths pper wthout loosng ntegrty nd ust to smplfy the clcultons we ssume the fuzzy trngulr numbers to be symmetrc.e. for every -cut n s equl to ( nl n u ) / or smply n Eq. (5) ssume n n ) /. ( 1 n 3 Fg. 1. A fuzzy number ñ. Fg.. Fuzzy number ñ wth -cuts. Fg. 3. A trngulr fuzzy number ñ. Proceedngs of the 005 Interntonl Conference on Computtonl Intellgence for Modellng Control nd Automton nd Interntonl Conference on Intellgent Agents Web Technologes nd Internet Commerce (CIMCA-IAWTIC 05)

3 Defnton 7. Let m ( m m ) nd ñ= n n ) be two 1 m3 ( 1 n3 trngulr fuzzy numbers. If m = ñ then m n 1 3. Defnton 8. If ñ s trngulr fuzzy number nd n l >0 nd n u 1 for [01] then ñ s clled normlzed postve trngulr fuzzy number (see for emple [9]). Defnton 9. D s clled fuzzy mtr f t lest n entry n D s fuzzy number [10]. Defnton 10. A lngustc vrble s vrble whose vlues re lngustc terms (defned by Zdeh [7]). The concept of lngustc vrble s very useful n delng wth stutons whch re too comple or too ll-defned to be resonbly descrbed n conventonl quntttve epressons [7]. For emple weght s lngustc vrble ts vlues cn be very low low medum hgh very hgh etc. These lngustc vlues cn lso be represented by fuzzy numbers. Defnton 11. Let m =( m 1 m m3 ) nd n =( n 1 n n3 ) be two trngulr fuzzy numbers n vlble methods n lterture of Fuzzy TOPSIS method wth trngulr fuzzy numbers (see for emple [11]) the verte method s usully defned to clculte the dstnce between fuzzy numbers s: d( m n ) [ (( m1 n1 ) ( m n ) ( m3 n3) ))]. (6) 3 Here we develop new dstnce mesure usng the fuzzy comprson functon tht cn be substtute wth (6). Suppose fuzzycmp( m n ) represents the fuzzy comprson functon of two gven dscrete fuzzy numbers ( m nd n ). We defne the fuzzycmp( m n ) be the truth of tht the fuzzy number m be greter thn or equl to the fuzzy number n. Usng fuzzy logc t cn be logclly defned s: fuzzycmp( m n )= m[mn{ m ( m ) n ( n ) }] : m n (7) where m nd n re the unverse elements of dscrete fuzzy numbers m nd n respectvely. Usng (7) we defne the dstnce of two fuzzy numbers m nd n : d( m n )= fuzzycmp( m n) - fuzzycmp( n m ). (8) Eq. (8) descrbes the de tht more closer two fuzzy numbers be more nerer re the truths of one beng greter thn or equl to the other. Usng (8) new Fuzzy TOPSIS procedure for mult-crter group decson mkng wll be descrbed n net sesson. III. MODIFIED FUZZY TOPSIS METHOD The wll be descrbed method s very sutble for solvng the group decson mkng problem under fuzzy envronment. In ths pper the mportnce weghts of vrous crter nd the rtngs of qulttve crter re consdered s lngustc vrbles. These lngustc vrbles cn be epressed n postve trngulr fuzzy numbers s Tbles I nd II. A. Clcultons The mportnce weght of ech crteron cn be obtned by ether drectly ssgn or ndrectly usng prwse comprsons [1]. Here t s suggested tht the decson mkers use the lngustc vrbles (shown n Tbles I nd II) to evlute the mportnce of the crter nd the rtngs of lterntves wth respect to vrous crter. Assume tht decson group hs K persons then the mportnce of the crter nd the rtng of lterntves wth respect to ech crteron cn be clculted s: 1 1 [ ( ) ( )...( ) K ] (9) K 1 w [ w ( ) w ( )...( ) w ] (10) k K k w 1 K where nd re the rtng nd the mportnce weght of the kth decson mker nd (+) ndctes the fuzzy rthmetc summton functon. As stted prevously fuzzy mult-crter group decsonmkng problem cn be concsely epressed n mtr formt s: n D... (11) m 1 m... mn W w... 1 w wn. (1) where k k nd w re lngustc vrbles tht cn be shown by trngulr fuzzy numbers : ( b c ) nd w ( w w w ). 1 3 To vod the complcted normlzton formul used n clsscl TOPSIS n some ppers (see for emple [11]) the lner scle trnsformton s used to trnsform the vrous crter scles nto comprble scle. Therefore t s possble to obtn the normlzed fuzzy decson mtr denoted by R : R r (13) mn where B nd C re the set of beneft crter nd cost crter respectvely nd: b c r ( ) B; * * * (14) c c c r ( c c * b ) C; (15) m c f B; (16) mn f C. (17) The normlzton method mentoned bove s to preserve the property tht the rnges of normlzed trngulr fuzzy numbers belong to [01]. In ths pper to vod these Proceedngs of the 005 Interntonl Conference on Computtonl Intellgence for Modellng Control nd Automton nd Interntonl Conference on Intellgent Agents Web Technologes nd Internet Commerce (CIMCA-IAWTIC 05)

4 computtons nd mke more esy nd prctcl procedure we smply defne ll of fuzzy numbers n ths ntervl to omt the need of normlzton method. Constructng the fuzzy numbers sclble nd n [01] we vod clcultons (14) through (17) nd therefore we hve: r nd R D. Consderng the dfferent mportnce of ech crteron one cn now construct the weghted normlzed fuzzy decson mtr s: V v 1... m 1... n (18) m n where v ( ) r w. Accordng to the weghted normlzed fuzzy decson mtr we know tht the elements v re normlzed postve trngulr fuzzy numbers nd ther rnges belong to the closed ntervl [0 1]. Then we cn defne the Fuzzy Postve- Idel Soluton (FPIS A*) nd Fuzzy Negtve Idel Soluton (FNIS A ) s: * * * * A ( v1 v... v ) n (... (19) A v1 v vn ) * where v =( ) nd v =( ). The dstnce of ech lterntve A (=1 m) from A* nd A cn be clculted s: d d * n 1 n 1 * d( v v ) 1... m d( v v ) 1... m (0) where d( ) s the new dstnce mesurements between two fuzzy numbers s s proposed by Eq. (8). Moreover closeness coeffcent s usully defned to determne the * rnkng order of ll lterntves once the d nd d of ech lterntve A (=1 m) hs been clculted. The closeness coeffcent of ech lterntve s clculted s [11]: d CC 1... m. (1) * d d Obvously ccordng to Eq. (1) n lterntve A would be closer to FPIS (.e. A* defned n Eq. (19)) nd frther from FNIS (.e. A defned n Eq. (19)) s CC pproches 1. In other words the closeness coeffcent clculted by Eq. (1) cn determne the rnkng order of ll lterntves nd ndcte the best one mong set of gven fesble lterntves. B. Implementng procedure The dscussed modfed clcultons for mult-crter group decson mkng fuzzy TOPSIS procedure cn now be mplemented n followng generl mplementng steps proposed by Chen [11]: Step 1. Form commttee of decson-mkers then dentfy the evluton crter. Step. Choose the pproprte lngustc vrbles for the mportnce weght of the crter nd the lngustc rtngs for lterntves wth respect to crter. Step 3. Aggregte the weght of crter to get the ggregted fuzzy weght w of crteron C nd pool the decson mkers' opnons to get the ggregted fuzzy rtng of lterntve A under crteron C. Step 4. Construct the (normlzed) fuzzy decson mtr (Eq. (13) or n our procedure Eq.(11)). Step 5: Construct the weghted (normlzed) fuzzy decson mtr (Eq.(18)). Step 6: Determne FPIS nd FNIS (Eq. (19)). Step 7: Clculte the dstnce of ech lterntve from FPIS nd FNIS respectvely usng Eq. (0) wth modfed dstnce mesure defned by Eq (8). Step 8: Clculte the closeness coeffcent of ech lterntve (Eq. (1)). Step 9: Accordng to the closeness coeffcent determne the rnkng order of ll lterntves. IV. NUMERICAL EXAMPLE The proposed modfed Fuzzy TOPSIS procedure nd requred clcultons hve been coded usng MATLAB 6.5 on Pentum III pltform runnng wndows XP nd mny rel world pplctons hve been tested mplementng the wrtten procedure. Hereby to llustrte our proposed pproch of ths pper we wll dscuss numercl emple. Assume tht unversty X desres to hre professor for techng fuzzy theory course. A commttee of three epert decson mkers D1 D nd D3 hs been formed to conduct the ntervew wth three elgble cnddtes nmely nd nd to select the most sutble cnddte. Fve beneft crter re consdered: (1) Publctons nd reserches (C1) () Techng sklls (C) (3) Prctcl eperences n ndustres nd corportons (C3) (4) Pst eperences n techng (C4) (5) Techng dscplne (C5). The proposed method s ppled to solve ths problem nd the computtonl procedure s summrzed s follows: Step 1. The decson mkers use the lngustc weghtng vrbles (shown n Tble I) to ssess the mportnce of the crter (shown n Tble III). Step. The decson mkers use the lngustc rtng vrbles (shown n Tble II) to evlute the rtng of lterntves wth respect to ech crteron. Fnl ggregted results re clculted nd presented n Tble IV s the lngustc fuzzy decson mtr. Step 3. The lngustc evlutons (shown n Tbles III nd IV) re converted nto symmetrc trngulr fuzzy numbers n order to construct the fuzzy decson mtr. Proceedngs of the 005 Interntonl Conference on Computtonl Intellgence for Modellng Control nd Automton nd Interntonl Conference on Intellgent Agents Web Technologes nd Internet Commerce (CIMCA-IAWTIC 05)

5 TABLE I Lngustc Vrbles For The Importnce Weght Of Ech Crteron Very low (VL) ( ) Low (L) (0.0; ) Medum low (ML) ( ) Medum (M) ( ) Medum hgh (MH) ( ) Hgh (H) ( ) Very hgh (VH) ( ) TABLE III The Importnce Weght Of Ech Crteron ven By Decson Mkers For The Numercl Emple Of Ths Pper Crteron Decson Mkers D1 D D3 C1 H VH MH C VH VH VH C3 VH H H C4 VH VH VH C5 M MH MH TABLE II Lngustc Vrbles For The Rtngs Very poor (VP) ( ) Poor (P) ( ) TABLE IV The Fnl Aggregted Results Obtned From Rtngs ven By DecsonMkers For The Numercl Emple Of Ths Pper Crteron Alterntve Lngustc Vrble Medum poor (MP) ( ) M Fr (F) ( ) C1 Medum good (M) ( ) ood () ( ) Very good () ( ) C M Step 4. The (normlzed) fuzzy decson mtr s constructed usng Eq. (13) or smply n ths pper Eq. (11). Step 5. The weghted normlzed fuzzy decson mtr s constructed usng Eq. (18). Step 6. FPIS nd FNIS re defned s : A* = [(1 1 1); (1 1 1); (1 1 1); (1 1 1); (1 1 1)] A = [(0 0 0); (0 0 0); (0 0 0); (0 0 0); (0 0 0)]. Step 7. The dstnce of ech cnddte from FPIS nd FNIS re clculted respectvely usng Eq. (0) nd the new dstnce mesure proposed by Eq. (8). Step 8. The closeness coeffcent s clculted for ech cnddte. The results re: CC 1 = 0.56; CC = 0.8; CC 3 = 0.69 Step 9. Accordng to the these closeness coeffcents the rnkng order of the three cnddtes wll be nd respectvely. Obvously the best selecton s cnddte hvng greter closeness coeffcent. C3 C4 C5 F F Proceedngs of the 005 Interntonl Conference on Computtonl Intellgence for Modellng Control nd Automton nd Interntonl Conference on Intellgent Agents Web Technologes nd Internet Commerce (CIMCA-IAWTIC 05)

6 V. CONCLUSION In ths pper we consdered the mult-crter decson mkng problem when there s group of decson mkers. Whle crsp dt re ndequte to model the rel lfe stutons n MCDM we modfed vlble procedures n the TOPSIS technque when decson mkers use lngustc vrbles. Regrdng the vlue of the truth tht fuzzy number s greter thn (or equl to) nother fuzzy number new dstnce mesure ws proposed n ths pper to clculte the dstnce of ech fuzzy number from both Fuzzy Postve Idel Soluton (FPIS) nd Fuzzy negtve Idel Soluton (FNIS). In secton II of ths pper defntons ws gven nd n secton III we completely descrbed our procedure. To clrfy our proposed procedure numercl emple ws gven nd dscussed n secton IV. Usng our wrtten functons n MATLAB 6.5 we solved the numercl emple nd presented the computtonl steps of our modfed proposed fuzzy TOPSIS procedure. REFERENCES [1] Hwng C.L. Yoon K. Multple Attrbutes Decson Mkng Methods nd Applctons Sprnger Berln Hedelberg [] Tsur S.H. Chng T.Y. Yen C.H. The evluton of rlne servce qulty by fuzzy MCDM Toursm Mngement 00 3 pp [3] Deng H. Yeh C.H. Wlls R.J. Inter-compny comprson usng modfed TOPSIS wth obectve weghts Computers & Opertons Reserch pp [4] Wng R.C. Lng T.F. Applcton of fuzzy mult-obectve lner progrmmng to ggregte producton plnnng Computers & Industrl Engneerng pp [5] Chu T.C. Fclty locton selecton usng fuzzy TOPSIS under group decsons Interntonl Journl of Uncertnty Fuzzness nd Knowledge-Bsed Systems 00 10(6) pp [6] Abo-Sn M.A. Amer A.H. Etensons of TOPSIS for multobectve lrge-scle nonlner progrmmng problems Appled Mthemtcs nd Computton 004 Artcle n press. [7] L.A. Zdeh Fuzzy sets Inform. nd Control pp [8] H.J. Zmmermnn Fuzzy Set Theory nd ts Applctons nd edn. Kluwer Acdemc Publshers Boston/Dordrecht/London [9] D.S. Neg Fuzzy nlyss nd optmzton Ph.D. Thess Deprtment of Industrl Engneerng Knss Stte Unversty [10] Buckley J.J. Fuzzy herrchcl nlyss Fuzzy Sets nd Systems pp [11] Chen C.T. Etensons of the TOPSIS for group decsonmkng under fuzzy envronment Fuzzy Sets nd Systems pp [1] Hsu H.M. Chen C.T. Fuzzy herrchcl weght nlyss model for multcrter decson problem J. Chnese Inst. Industrl Eng (3) pp Proceedngs of the 005 Interntonl Conference on Computtonl Intellgence for Modellng Control nd Automton nd Interntonl Conference on Intellgent Agents Web Technologes nd Internet Commerce (CIMCA-IAWTIC 05)