Extended TOPSISs for Belief Group Decision Making

Transcription

1 J. Sev. Sc. & Maageme. 2008, : -20 Publshed Ole Jue 2008 ScRes ( Exeded TOPSISs fo Belef Goup Decso Makg hao Fu School of Maageme, Hefe Uvesy of Techology, Hefe , ha ABSTRAT Mulple abue decso aalyss (MADA) poblems he suao of belef goup decso makg (BGDM) ae a specal class of decso poblems, whee he abue evaluaos of each decso make (DM) ae epeseed by belef fucos. I ode o solve hese specal poblems, hs pape, TOPSIS (echque fo ode pefeece by smlay o deal soluo) model s exeded by hee appoaches, by whch goup pefeeces ae aggegaed dffee maes. oespodg o he hee appoaches, hee exeded TOPSIS models, he pe-model, pos-model, ad e-model, ae developed ad he pocedues ae elaboaed sep by sep. Aggegag goup pefeeces he hee exeded models especvely depeds o Dempse s ule o s modfcaos, some socal choce fucos, ad some mea appoaches. Fuhemoe, a umecal example clealy llusaes he pocedues of he hee exeded models fo BGDM. Keywods: basc belef assgme, belef goup decso makg, belef pefeeces aggegao, TOPSIS. Ioduco Recely, he ucea mulple abue decso aalyss (MADA) poblems wh a goup of decso makes (DMs) have bee wdely suded he leaue, whch he abue evaluaos ae ukow, vague, paal kow, o mpecse. The epeseave soluo s o cosuc a fuzzy TOPSIS (echque fo ode pefeece by smlay o deal soluo), a classcal modfed appoach fo ucea MADA poblems, o choose he bes oe fom a se of aleaves [2-4, 8, 20, 30]. Howeve, compaed wh he Dempse-Shafe heoy (DST) [5,23], he opeaos of fuzzy se heoy (FST) o aggegae goup pefeeces, whch ae usually he ahmecal mea, he geomec mea, o he modfcaos, ae less adapable ad avalable. Hece, hs pape uses he DST o descbe ucea MADA poblems; ha s o say, uses basc belef assgmes (bbas) o epese ucea abue evaluaos. I pacce, due o he oe-o-oe coespodece bewee he bba ad he belef fuco [23], he bba s usually ehe elced fom expes, o cosuced fom obsevao daa. To asfom qualave expes opos o bbas, some mehods have bee poposed by Wog ad Lgas [3], Byso ad Mobolu [], ad Yaghlae e al. [34]. Usg he bba o epese ucea goup abue evaluaos, oe coespodgly coves he goup decso makg (GDM) o he belef goup decso makg (BGDM). To solve MADA poblems he suao of BGDM, he ogal TOPSIS [5] s exeded by hee appoaches descbed [25]. The opeaos o aggegae goup pefeeces ae especvely he pe-opeao, pos-opea- o, ad e-opeao. Based o Yag s ule ad uly based equvale asfomao of he assessmes o dffee fames of dsceme [35], he evaluaos o dffee abues elaed o dffee fames ca be ufed o become he oes o a commo fame. Fuhemoe, he posve ad egave pefeece vecos of DM, he posve deal soluo of belef (PISB), ad he egave deal soluo of belef (NISB) ae cosuced. The pefeece vecos avod he possble paadoxes bewee he calculag aks of aleaves ad he fac of DM s pefeece, ad he PISB ad NISB ae used o deeme he aks of aleaves. The dealed exeded models ae explaed sep by sep Seco 3. The es of hs pape s ogazed as follows. I Seco 2, he elaed foudaos ae evewed. Seco 3 dscusses hee exeded models accod wh hee appoaches o aggegag goup pefeeces, he pe-opeao, pos-opeao, ad e-opeao, ode o make soluos o BGDM. A umecal example s gve Seco 4 o llusae he pocedues of hee exeded models ad he dffeeces. A las, Seco 5 cocludes hs pape. 2. Revew of Relaed Foudaos 2.. Bascs of bba I a specfc applcao doma, he DST fs defes Ω, called he fame of dsceme, coag N exhausve ad exclusve hypoheses. Le 2 Ω deoe he powe se composed of 2 N poposos of A such ha A Ω. Defo. Le Ω deoe a fame of dsceme, ad S be a pece of abay evdece souce (ES) o Ω. Thus, he bba of ES s defed by m: 2 Ω [0, ]. Ths fuco opygh 2008 ScRes

2 2 hao Fu vefes he followg popees [5, 23]: A ΩmA ( ) =. () I Shafe s ogal defo, m s called basc pobably assgme (bpa) [23] wh codo m (Ø) =0. Howeve, sce asfeable belef model (TBM) was poposed as a model of uceay [28], codo m (Ø) =0 has bee omed. Subses A of Ω such ha m (A)>0 ae called focal elemes of m. Defo 2. Le a powe se o Ω be defed as 2 Ω = (B, B 2,, B ), whee = 2 Ω, he cadaly of 2 Ω. Suppose bba ( ) epeses he dsbuo o 2 Ω, hus bba = (x,x 2,,x ) sasfes: x 0, 0 -, (2) x =, =, 2,,. (3) = 0 Gve A Ω, he mass m(a) epeses he belef ha suppos A, ad ha, due o lack of he fomao ad kowledge, does o suppo ay sc subse of A. Le m ad m 2 be wo bbas defed o Ω. Sasfyg he closed wold assumpo, he omalzed Dempse s ule of combao s defed as [5,23] ( m 2 B, Ω, B = A 2 whee K m )( A) = k I m ( B) m ( ), (4) = I m ( B) m ( ), (5) B, Ω, B = 2 (m m 2 )(Ø)=0. (6) Hee, B, Ω, BI = m( B) m2( ) s he mass of he combed belef allocaed o he empy-se befoe omalzao. Dempse s ule s meagful ad ca be appled oly whe B, Ω, BI= m( B) m2( ) Bascs of TOPSIS MADM. MADM poblems ae a class of decso poblems smply deoed by 2 L A v v L v A v v L v M M M M M A v v L v m m m2 m, (7) whee A ( m) deoes he h aleave, ( ) deoes he h abue, ad v ( m, ) deoes he assessme of DM o he abue of aleave A. Suppose W=(w, w 2,, w ) such ha w = s a = wegh veco, whee w deoes he wegh of. MADM poblem solvg cludes: (a) osuc he abue se of sysem assessme ad coelae sysem pefomace ad obecve; (b) ofm he avalable aleave se fo mplemeg he obecve; (c) Evaluae all aleaves accodg o he abue se ad gve v ( m, ). (d) Apply omalzed aalyss mehodologes o MADM poblems; (e) Make choce of he bes aleave; (f) ollec ew fomao ad sa wh a ew decso pocedue fo MADM poblems f he esulg aleave ca o be acceped. Seps (a) ad (e) oe o DM, bu ohes o applcaos. I Sep (d), DM expesses hs/he pefeece accodg o he elave mpoace of evey abue, fo example, seg w TOPSIS The TOPSIS s a mpoa paccal echque o solve MADA poblems ogag fom he cocep of a dsplaced deal po fom whch he compomse soluo has he shoes dsace [36]. I he vew of Hwag ad Yoo [5], he ag of aleave depeds o he shoes dsace fom he posve deal soluo (PIS) ad he fahes dsace fom he egave deal soluo (NIS) o ad. ompaed wh he Aalyc Heachy Pocess (AHP) [22], he TOPSIS fs he cases wh a lage umbe of abues ad aleaves. I [5], Hwag ad Yoo pao abues o hee classes: beef oes, cos oes ad o-moooc oes. The dffee classes of abues coespod o dffee omalzao mehods ode o f dffee eal-wold suaos,.e. he veco omalzao, he lea omalzao, ad he o-moooc omalzao. Paccally, he TOPSIS ad s exesos ae used o solve may heoecal ad eal-wold poblems, such as decso makg wh fuzzy daa [6] o eval daa [7], decso suppo aalyss fo maeal seleco of meallc bpola plaes [24], evaluag al ag acaf ude a fuzzy evome [29], o e-compay compaso [6]. A geeal flow of TOPSIS volves: ) Nomalze decso max V= (v ) m. The decso max V s asfomed o a omalzed v max R by = ( m, ), whee m 2 v k k s he omalzed oe of v. 2) alculae weghed decso max Z=(z ) m. opygh 2008 ScRes

3 Exeded TOPSISs fo Belef Goup Decso Makg 3 The omalzed max R s asfomed o a weghed decso max Z such ha z =w ( m, ), whee w deoes he wegh of such ha w =. 3) Deeme PIS ad NIS. The PIS ad NIS ae especvely = A + = { z +, z + 2,, z + }={( max z Ω b ), ( m z Ω c )}, A - = { z, z 2,, z }={( m z Ω b ), ( max z Ω c )}, whee Ω b ad Ω c ae beef abue se ad cos abue se, especvely. 4) ompue he sepaao measues of each aleave fom he PIS ad NIS. The sepaao measues of each aleave fom he PIS ad NIS ae especvely + ( ) 2 = + =, =, 2,, m, D z z ( ) 2 = =, =, 2,, m. D z z 5) alculae he closeess coeffce of each aleave. The closeess of each aleave ca be defed as D R =,=, 2,, m. + D + D 6) Rak he pefeece ode. The aleave se deoed by A ( m) s aked by meas of R, whch dcaes wha he bes aleave s Dscusso The ogal TOPSIS has he ably o effecvely solve geeal MADM poblems fo oe DM, whch ca easly exeded o deal wh he suao of GDM. I he wok of Shh e al. [25], hey cosuced a eal exeded model of TOPSIS fo GDM, whch he seps wee updaed volvg he decso max omalzao, dsace measues, ad aggegao opeaos. Oe ca obvously ealze ha he eal model eve fs exeal exesos of TOPSIS assocaed wh he pe-opeao ad pos-opeao. Fuhemoe, s o suable fo he eal exeso of TOPSIS hs sudy, whee ucea goup evaluaos ae epeseed by bbas. I Seco 3, hee exeded models fo BGDM, ecely eseached by Fu ec. [0-2], ae elaboaed sep by sep, coespodg o he pe-opeao, pos-opeao, ad e-opeao. 3. Soluos o Belef Goup Decso Makg Accodg o he classes of goup pefeece aggegao poposed by Shh e al. [25], we exed he ogal TOPSIS o be avalable fo BGDM suao by hee appoaches, coespodg o he pe-opeao, pos-opeao, ad e-opeao. Thee exeded TOPSIS models ae especvely amed as pe-model, pos-model, ad e-model. The dealed pocedues of he hee models ae epeed as follows. 3.. Pe-model The pe-model s composed of he followg seps. Sep : osuc al goup belef decso maces (BDMs). The al BDM of each DM ca be defed as follows: L A y y y A y y y M M M M M A y y L y 2 2 L L 2 m m m2 m whee A ( m) deoes he h aleave, ( ) deoes he h abue, ad y ( m,, T) deoes he belef assessme of DM o he abue of aleave A. Le Ω ( ) be he fame of dsceme used o geeae he assessmes o he abue. I ems of Defo 2, we have y = B Ω = ( b, b 2, K, b ), whee = 2. Ω (8) ovee o decde he PISB ad NISB, he dsbuo of powe se o Ω s specfed Defo 3. Defo3. Le Ω be he fame of dsceme used o geeae he assessmes o he abue ( ), ad 2 Ω = ( B, B2, K, B ) be he dsbuo of a abay Ω powe se o Ω, whee = 2. Suppose he cadaly of B k s ceasg alog he cease of k. Fuhemoe, we assume B = Ø (empy-se), B 2 ad B 3 especvely coespod o he sgle posve deal eleme (SPIE) ad he sgle egave deal eleme (SNIE) of Ω. The ogal TOPSIS eques a ufom dmeso fo he assessmes o evey quaave abue. The hee exesos of TOPSIS fo BGDM suao ae also cosaed by hs equeme. Tha s o say, he vaous fames, Ω ( ), have o be asfomed o a ufed fame Ω so ha evey abue ca be assessed a ufom, cosse ad compable mae. The asfomao fom Ω ( ) o Ω s spulaed opygh 2008 ScRes

4 4 hao Fu as Poposo. Poposo. Le Ω be he fame of dsceme used o geeae he assessmes o he abue ( ). The assessmes o Ω ca be equvalely ad aoally asfomed o he oes o a commo fame of dsceme Ω. I fac, Poposo s clealy coec sce wo echques, a ule based oe ad a uly based oe, ae vesgaed o accomplsh he asfomao Poposo [35]. Fom Poposo, y Eq (8) ca be asfomed o a dsbuo o Ω. Theefoe, he belef abue evaluaos of each DM o each aleave ae ufed he se of dsbuos o Ω. I he followg, we suppose y deoes a dsbuo o Ω. Sep 2: Aggegae goup BDMs o fom a oal BDM. Fom Sep, we kow he BDM of each DM as defed Eq (8). Wh he omalzed Dempse s ule of combao [5, 23], goup BDMs ae combed o fom a oal BDM. Le he oal BDM be defed he followg: 2 L A x x L x A x x L x M M M M M A x x L x m m m2 m Ω whee x = B Ω = ( b, b2, K, b ), = 2, m,. Gve ay eleme x he oal BDM, we T have x = y, whee he opeao deoes he = omalzed Dempse s ule of combao as specfed Eqs (4) o (6). Hee, we suppose all expes have he same mpoace. Sep 3: Nomalze he oal BDM. Dffee fom he ogal TOPSIS, x s o a eal umbe bu a omalzed dsbuo o Ω, he Sep ca be omed. Sep 4: Assg a oal wegh veco W o he abue se. Le W deoe he wegh veco of each DM assged o he abue se. We have W = (, 2,, w w K w ), T, w = =. The oal wegh veco W ca be defed as he ahmecal mea of all W ( T), whch s W= (w, w 2,, w ) such ha (9) w T w = T =,. (0) Sep 5: Deeme he oal PISB ad NISB. Befoe deemg he oal PISB ad NISB, fs of all we defe he PISB ad NISB Defo 4, owg o he dsbuo specfcao Defo 3. Defo4. Based o he specfcao Defo 3, gve he abue ( ), o mae whehe s he beef abue o he cos abue, s PISB ad NISB ae especvely (0,,0, K,0) ad (0,0,, 0, K,0). Accodg o Defo 4, by combg he PISB ad NISB of each abue, we acheve he oal PISB ad NISB of oal BDM. Sep 6: alculae he sepaao measues of each aleave fom he oal PISB ad NISB. Fom Sep 5, he oal PISB ad NISB ca be especvely deoed by S ( ) = (0,,0, K,0, L, 0,,0, K,0) ad S ( ) = (0,0,,0, K,0, L, 0,0,,0, K,0). Fuhemoe, ode o pecsely eflec he pefeece of each DM ad he physcal mplcao of each subse of Ω he dsbuo o 2 whe calculag he sepaao measues of each aleave fom he PISB ad NISB, we defe he posve pefeece veco (PPV) + (,, +,, + β K βk K β ) ad he egave pefeece veco (NPV) (,,,, β K βk K β ) of each DM fo he dsbuo o 2 whee β + Ω k =, β Ω k =, = 2. k = k = Though odeed compaso of ay wo dffee subses Ω of he dsbuo o 2 he PPV ad NPV of DM ca be acheved. We posulae β + k > 0, β k > 0, f k>, ad + βk = βk = 0, f k=, so as o keep all avalable fomao. Le he posve goup pefeece veco (PGPV) ad egave goup pefeece veco (NGPV) especvely be (,, +,, + + β K βk K β ) ad (,,,, β K βk K β ) such ha β + k =, β k =, we hus have k = k = + T + k βk T = β =, () opygh 2008 ScRes

5 Exeded TOPSISs fo Belef Goup Decso Makg 5 β k T βk T = =. (2) The PPV ad NPV ca effecvely avod he possble paadoxes bewee calculag esuls ad he fac of DM s pefeece as well as physcal mplcaos of wolds Ω. Hece, he sepaao measues of each aleave fom he oal PISB ad NISB ae expessed as ad D = w β k ( bk S(( ) + k ) (3) = = k= 2 D = w β k ( bk S(( ) + k ) (4) k= Ω whee m, = 2, wh he appoach of Euclda dsace [9]. Sep 7: ompue he closeess coeffce each aleave fo goup. E * of The closeess coeffce of each aleave ca be defed as + E = D /( D + D ) ( m). (5) * The lage he value of Sep 8: Rak he pefeece ode. * E, he bee he aleave. * I ems of E, a se of aleaves wll be aked a cemeal ode epeseg goup pefeeces Pos-model The pos-model s paally he same as he pe-model. Afe he pocedue of ogal TOPSIS, he ak of each aleave epeseg goup pefeeces s deemed, aded by oe of socal choce fucos [4], such as he Boda fuco hs pape. Sep : osuc al goup BDMs. The Sep s he same as Sep of pe-model. Sep 2: Nomalze he BDM of each DM. Same as Sep 3 of pe-model, he Sep ca be omed. Sep 3: Assg he wegh veco W o he abue se fo each DM. We suppose W deoes he wegh veco of DM assged o he abue se, whee W = ( w, w2, K, w ), T, w =. = Sep 4: Deeme he PISB ad NISB of each DM. As specfed Defo 3, he PISB ad NISB of each DM ae especvely deoed by S ( ) = (0,,0, K,0, L, 0,,0, K,0) ad S ( ) = (0,0,,0, K,0, L, 0,0,,0, K,0), whee T. Sep 5: alculae he sepaao measues of each aleave fom he PISB ad NISB of each DM. Smla o Sep 6 of pe-model, he sepaao measues of each aleave fom he PISB ad NISB fo each DM ae expessed as ad D = w β k ( bk S(( ) + k ) (6) = = k= 2 D = w β k ( bk S(( ) + k ) (7) k= whee ( b, b 2, K, b ) = Ω = 2. Sep 6: ompue he closeess coeffce aleave fo each DM. y, m, T, * E of each The closeess coeffce of each aleave fo each DM ca be defed as * E = /( + D D + D ), (8) whee m, T. Sep 7: Rak he pefeece ode of each DM. I ems of E *, a se of aleaves wll be aked a cemeal ode epeseg he pefeece of each DM, whee T. Sep 8: Gve he Boda scoe of each aleave accodg o he pefeece ode of each DM. Suppose he pefeece ode of DM s B fkf B fkf Bm, whee B ( m) s he same as A ( m). The Boda scoe of B s m-, he oes of B 2 ad B m ae especvely m- ad 0, ad he es may be deduced by aalogy. Sep 9: Aggegae he Boda scoe of each aleave gve by each DM. opygh 2008 ScRes

6 6 hao Fu Le he Boda scoe vecos of each aleave epeseg he pefeece of DM ad goup pefeeces be especvely (,,,, S K S K Sm) ad ( S,,,, K S K Sm). We have S T = S, m. (9) = Sep 0: Rak he pefeece ode fo goup. Accodg o ( S,,,, K S K Sm), we ak he pefeece ode of a se of aleaves fo goup Ie-model The e-model s smla o he eal TOPSIS model of Shh e al. [25]. I combes he dvdual sepaao measues of each aleave fom he PISB ad NISB o fom goup measues wh he TOPSIS pocedue. The fs fve Seps of e-model ae he same as Seps o 5 of pos-model. Sep 6: ombe he dvdual measues of each aleave fom he PISB ad NISB o fom goup measues. Fom Sep 5 of pos-model, we acheve he dvdual measues of each aleave fom he PISB ad NISB, D + D whch ae especvely ad ( m, T). Thus, he goup measues of each aleave ae especvely T + + D = D = ad (20) T D = D. (2) = The opeao ca be he ahmecal mea, he geomec mea, o he modfcaos. I hs pape, he ahmecal mea s ou choce. Seps 7 ad 8 ae he same as Seps 7 ad 8 of pe-model. As meoed above, hee exeded models ae smla o each ohe may Seps. The ma dffeeces le he aggegao of goup pefeeces. I he pe-model, haks o wo saeges of Dempse s ule modfcao (e.g. [8, 9, 26-27, 32-33]) ad souce modfcao (e.g. [7, 3, 2]) amg a combg coflcg belefs, he pefeece coflcs bewee dffee DMs ca be effecvely deal wh. I he pos-model, some socal choce fucos [4] ca be seleced o guaaee goup pefeeces aggegao s aoal ad avalable dffee applcaos. I he e-model, he ahmecal mea, he geomec mea, o he modfcaos ae used o aggegae he dvdual sepaao measues of each aleave fom he PISB ad NISB. I pacce, how o selec he appopae exeded model depeds o how o selec he appopae appoach o aggegag goup pefeeces, whch s he mos suable oe fo eal-wold poblems. 4. Numecal Example To clealy llusae he pocedues of hee exeded models, a umecal example s show as follows. Fom Tables o 3, oe ca kow al goup BDMs, ad he pefeece vecos ad wegh veco of each DM. Thee ae wo abues, hee aleaves, ad hee DMs hs example. Two abues ad 2 ae he beef oe ad he cos oe, especvely. Suppose Ω = {good, commo}, Ω 2 = {small, bg, commo}, Ω = {fs, secod, hd}, accodg o Poposo, he assessmes o Ω ad Ω 2 ca be equvalely asfomed o he oes o Ω. I ems of Defo 3, he powe se o Ω s {{Ø}, {fs}, {hd}, {secod}, {fs, hd}, {fs, secod}, {secod, hd}, {fs, secod, hd}}. As specfed Defo 4, he PISB ad NISB ae especvely (0,,0,0,0,0,0,0,0,,0,0,0,0,0,0) ad (0,0,,0, 0,0,0,0,0,0,,0,0,0,0,0). The decso pocedues of hee exeded models wll be peseed as follows. I he pe-model, goup belef evaluaos ae fsly combed o fom he oal BDM dsplayed Table 4, wh he omalzed Dempse s ule of combao. Afewads, accodg o Eq (0), he oal wegh veco W= (0.6, 0.4) s geeaed fom he wegh vecos Table 3. Based o he daa Table 2, he PGPV ad NGPV ae compued especvely as (0,0.03,0.207,0.092, 0.207,0.05,0.207,0.207) ad (0,0.384,0.055,0.63,0.055, 0.233,0.055,0.055), ems of Eqs () ad (2). Wh he above esuls, he oal sepaao measues ad he closeess coeffce of each aleave ae obaed Table 5, accodg o Eqs (3) o (5). Fom Table 5, he pefeece ode of hee aleaves s kow o be A f A 3 f A 2, whee he oao f meas po. I he pos-model, fs of all he dvdual sepaao measues ad he closeess coeffce of each aleave ae compued Table 6. The Boda scoe ad ak of each aleave fo goup ae geeaed fom he daa Table 6 ad show Table 7. Accodg o Table 7, hee aleaves ae aked by he pefeece ode A f A 2 =A 3. I he e-model, he sepaao measues ad closeess coeffce of each aleave fo goup ae acheved Table 8, o he bass of he daa Table 6. Thee aleaves ae aked wh he pefeece ode A f A 2 f A 3 accodg o Table 8. opygh 2008 ScRes

7 Exeded TOPSISs fo Belef Goup Decso Makg 7 Table. Ial goup BDMs The hee pefeece odes coespodg o hee exeded models ae pa-wse dffee. The medao ad he equemes of a eal applcao decde whch ode s he bes oe ad whch exeded model should be appled. Especally, f he medao oly was o kow he bes aleave, s uecessay o dffeeae he hee odes. A 2 DM (0,0.6,0,0,0,0.4,0,0) (0,0.3,0.2,0,0,0.5,0,0) DM2 (0,0.5,0,0.2,0,0.3,0,0) (0,0.5,0.2,0,0,0,0.3,0) DM3 (0,0.4,0,0.2,0,0.4,0,0) (0,0.4,0,0.4,0,0.2,0,0) DM (0,0.2,0,0.5,0,0,0.3,0) (0,0.6,0.2,0,0,0.2,0,0) A2 DM2 (0,0.3,0,0.5,0,0.2,0,0) (0,0.4,0.,0,0,0,0.5,0) DM3 (0,0.4,0,0.3,0,0.3,0,0) (0,0.5,0.3,0,0,0.2,0,0) DM (0,0.2,0,0.8,0,0,0,0) (0,0.2,0.4,0,0,0,0.4,0) A3 DM2 (0,0.7,0,0,0,0.3,0,0) (0,0.4,0.2,0.4,0,0,0,0) DM3 (0,0.6,0,0.,0,0.3,0,0) (0,0.2,0.6,0,0,0.2,0,0) ( β,, ) Table 2. The pefeece vecos of each DM (,, ) + + K β8 β K β8 DM (0,0.04,0.2,0.,0.2,0.06,0.2,0.2) (0,0.4,0.05,0.5,0.05,0.25,0.05,0.05) DM2 (0,0.03,0.2,0.2,0.2,0.05,0.2,0.2) (0,0.3,0.09,0.4,0.09,0.2,0.09,0.09) DM3 (0,0.02,0.22,0.06,0.22,0.04,0.22,0.22) (0,0.45,0.025,0.2,0.025,0.25,0.025,0.025) Table 3. The wegh veco of each DM w w 2 DM DM DM Table 4. The oal goup BDM 2 A (0,0.83,0,0.,0,0.07,0,0) (0,0.73,0,0.27,0,0,0,0) A2 (0,0.7,0,0.83,0,0,0,0) (0,0.8,0.3,0.07,0,0,0,0) A3 (0,0.65,0,0.35,0,0,0,0) (0,0.2,0,0.8,0,0,0,0) Table 5. The sepaao measues ad closeess coeffce of each aleave he pe-model D+ D- E*= D-/( D-+ D+) ak A A A opygh 2008 ScRes

8 8 hao Fu Table 6. The sepaao measues ad he closeess coeffce of each aleave he pos-model S+ S- E* A A2 A3 DM DM DM DM DM DM DM DM DM Table 7. The Boda scoe ad ak of each aleave Boda scoe A 5 A2 2 2 A3 2 2 ak Table 8. The sepaao measues ad closeess coeffce of each aleave he e-model D+ D- E*= D-/( D-+ D+) ak A A A oclusos Though epeseg he ucea abue evaluaos of a goup of DMs o aleaves by bbas, he commo GDM s exeded o he BGDM. To solve he MADA poblems he suao of BGDM, we develop hee exeded TOPSIS models, he pe-model, pos-model, ad e-model, assocaed wh hee appoaches o aggegag goup pefeeces, he pe-opeao, pos-opeao, ad e-opeao. Fo he BGDM, hee exeded models ae elaboaed sep by sep, based o he equvale asfomao of he assessmes o dffee fames of dsceme, he PISB ad NISB, ad he PPV ad NPV of each DM. Fuhemoe, a umecal example clealy llusaes he pocedues of hee exeded models. The elably of expes may be a mpoa faco o fluece ou mehod. If a goup of expes have dffee elably, he bbas may be dscoued [23] befoe used he hee models. The dscoug appoach s oduced he ogal wok of Shafe [23]. I paccal applcaos, how o decde he elably of expes may be a poblem dffcul o solve [9]. The compuaoal complexy may be a poblem fo ou mehod s o he powe se of a fame of dsceme. I fac, he umecal examples Seco 4 ae solved by he pogam made by Mcosof Vsual wh seveal secods. By esg adomly seleced daa, we fd ha whe Ω <3, he soluos ca be obaed wh seveal secods. Noe ha fo he MADA poblems he suao of BGDM, Ω <3 s geeally eough o povde he sasfacoy sevce fo expes. If Ω s oo lage, expes wll have dffcules o make decsos. Theefoe, he compuaoal complexy of ou mehod ca be effecvely solved by he compue pogam ad he eal cosas of expes decso makg. 6. Ackowledgeme Ths eseach s suppoed by he Naoal Naual Scece Foudao of ha (No ad ) ad opygh 2008 ScRes

9 Exeded TOPSISs fo Belef Goup Decso Makg 9 he Foudao of Hefe Uvesy of Techology (No. 0804F). We would lke o hak he aoymous evewes fo he cosucve commes helpg us o mpove hs pape cosdeably. REFERENES [] N. Byso, A. Mobolu, A Pocess fo Geeag Quaave Belef Fucos, Euopea Joual of Opeaoal Reseach, 5(3), 999, pp [2].T. he, Exesos of he TOPSIS fo Goup Decso-Makg ude Fuzzy Evome, Fuzzy Ses ad Sysems, 4(), 2000, pp. -9. [3].T. he,.t. L, ad S.F. Huag, A Fuzzy Appoach fo Supple Evaluao ad Seleco Supply ha Maageme, Ieaoal Joual of Poduco Ecoomcs, 02(2), 2006, pp [4] T.. hu, Facly Locao Seleco Usg Fuzzy TOPSIS ude Goup Decsos, Ieaoal Joual of Uceay, Fuzzess ad Kowledge-Based Sysems, 0(6), 2002, pp [5] A. Dempse, Uppe ad Lowe Pobables Iduced by Mulvalued Mappg, Aals of Mahemacal Sascs, 38, 967, pp [6] H. Deg,.H. Yeh, ad R. J. Wlls, Ie-ompay ompaso Usg Modfed TOPSIS wh Obecve Weghs, ompues & Opeaos Reseach, 27(0), 2000, pp [7] Y. Deg, W. Sh, Z Zhu, ad Q Lu, ombg belef fucos based o dsace of evdece, Decso Suppo Sysems, 38(3), 2004, pp [8] D. Dubos, H. Pade, Repeseao ad combao of uceay wh belef fucos ad possbly measues, ompuaoal Iellgece, 4, 998, pp [9] Z. Eloued, K. Melloul, ad P. Smes, Belef Decso Tees: Theoecal Foudaos, Ieaoal Joual of Appoxmae Reasog, 28(2-3), 200, pp [0]. Fu, S.L. Yag, X. J, A Pe-Exeso of TOPSIS fo Belef Goup Decso Makg, Ieaoal ofeece o Weless ommucaos, Newokg ad Moble ompug, WOM, 2007, pp []. Fu, S.L. Yag, Soluos o Belef Goup Decso Makg Usg Exeded TOPSIS, Ieaoal ofeece o Maageme Scece ad Egeeg, 2007, pp [2]. Fu, S.L. Yag, W.X. Lu, A Exeded TOPSIS fo Belef Goup Decso Makg, Ieaoal ofeece o Fuzzy Sysems ad Kowledge Dscovey, 2007, pp [3] R. Hae, Ae aleaves o Dempse s ule of combao eal aleaves? ommes o Abou he belef fuco combao ad he coflc maageme poblem, Ifomao Fuso, 3(4), 2002, pp [4].L. Hwag, M.J. L, Goup Decso Makg ude Mulple ea, Bel: Spge-Velag, Bel, 987. [5].L. Hwag, ad K. Yoo, Mulple Abue Decso Makg, Bel: Spge-Velag, Bel, 98. [6] G.R. Jahashahloo, F.H. Lof, ad M. Izadkhah, Exeso of he TOPSIS Mehod fo Decso-Makg Poblems wh Fuzzy Daa, Appled Mahemacs ad ompuao, 8(2), 2006, pp [7] G.R. Jahashahloo, F.H. Lof ad M. Izadkhah, A Algohmc Mehod o Exed TOPSIS fo Decso-Makg Poblems wh Ieval Daa, Appled Mahemacs ad ompuao, 75(2), 2006, pp [8] M.S. Kuo, G.H. Tzeg, ad W.. Huag, Goup Decso-Makg Based o oceps of Ideal ad A-deal Pos a Fuzzy Evome, Mahemacs ad ompue Modellg, vol. 45, o. 3-4, , Feb [9] E. Lefeve, O. olo, ad P. Vaooebeghe, Belef fuco combao ad coflc maageme, Ifomao Fuso, 3, 2002, pp [20] D.F. L, ompomse Rao Mehod fo Fuzzy Mul-abue Goup Decso Makg, Appled Sof ompug, 7(3), 2007, pp [2].K. Muphy, ombg belef fucos whe evdece coflcs, Decso Suppo Sysems, 29(), 2000, pp. -9. [22] T.L. Saay, The Aalyc Heachy Pocess (edo 2), RWS publcao, Psbugh, PA, 990. [23] G. Shafe, A Mahemacal Theoy of Evdece, Pceo: Pceo Uvesy Pess, Pceo, 976. [24] A. Shaa, O. Savadogo, TOPSIS Mulple-cea Decso Suppo Aalyss fo Maeal Seleco of Meallc Bpola Plaes fo Polyme Elecolye Fuel ell, Joual of Powe Souces, 59(2), 2006, pp [25] H.S. Shh, H.J. Shyu, ad E.S. Lee, A Exeso of TOPSIS fo Goup Decso Makg, Mahemacal ad ompue Modellg, 45(7-8), 2007, pp [26] P. Smes, The combao of evdece he asfeable belef model, IEEE Tasaco o Pae Aalyss ad Mache Iellgece, 2(5), 990, pp [27] P. Smes, Aalyzg he combao of coflcg belef fucos, Ifomao Fuso, 8(4), 2007, pp [28] P. Smes, K. Kees, The Tasfeable Belef Model, Afcal Iellgece, 66 (2), 994, pp opygh 2008 ScRes

10 20 hao Fu [29] T.. Wag, T.H. hag, Applcao of TOPSIS Evaluag Ial Tag Acaf ude a Fuzzy Evome, Expe Sysems wh Applcaos, 33(4), 2007, pp [30] Y.M. Wag, Y. Luo, ad Z.S. Hua, A Noe o Goup Decso-Makg Based o oceps of Ideal ad A-deal Pos a Fuzzy Evome, Mahemacal ad ompue Modellg, 46(9-0), 2007, pp [3] S.K.M. Wog, P. Lgas, Repeseao of Qualave Use Pefeece by Quaave Belef Fucos, IEEE Tasacos o Kowledge ad Daa Egeeg, 6(), 994, pp [32] R.R. Yage, O he Dempse-Shafe Famewok ad New ombao Rules, Ifomao Sceces, 4(2), 987, pp [33] R.R. Yage, Quas-assocave opeaos he combao of evdece, Kybeees, 6, 987, [34] A.Be Yaghlae, T. Deoeux, ad K. Melloul, osucg Belef Fucos fom Qualave Expe Opos, Ifomao ad ommucao Techologes, ITTA 06,, 2006, pp [35] J.B. Yag, Rule ad Uly Based Evdeal Reasog Appoach fo Mulabue Decso Aalyss ude Uceaes, Euopea Joual of Opeaoal Reseach, 3, 200, pp [36] M. Zeley, A ocep of ompomse Soluos ad he Mehod of he Dsplaced Ideal, ompues ad Opeaos Reseach, (3-4), 974, pp AUTHOR S BIOGRAPHY hao Fu eceved hs M.S. degee Mechacal ad Elecoc Egeeg fom Huazhog Uvesy of Scece ad Techology, Wuha, Hube, ha He oed Hefe Uvesy of Techology, whee he s cuely a lecue a School of Maageme. He s a eseache of Key Laboaoy of Pocess Opmzao ad Iellge Decso-makg, Msy of Educao, he same school. Hs eseach eess clude decso scece ad echology, fomao sysems ad egeeg, ad he smulao of complex decso ask. opygh 2008 ScRes