MODELS FOR MULTIPLE ATTRIBUTE GROUP DECISION MAKING WITH 2-TUPLE LINGUISTIC ASSESSMENT INFORMATION

Transcription

1 Iterto Jour of Coputto Itegece Systes, Vo., No. (Septeber, 00, 5-4 MODELS FOR MULTIPLE TTRIBUTE GROUP DECISION MKING WITH -TUPLE LINGUISTIC SSESSMENT INFORMTION GUIWU WEI*, RUI LIN, XIOFEI ZHO, HONGJUN WNG Deprtet of Ecoocs d Mgeet, Chogqg Uversty of rts d Sceces Yogchu 4060, Ch *Correspodg uthor,e-:weguwu@6.co bstrct The of ths pper s to vestgte the utpe ttrbute group decso g(mgdm probes wth - tupe gustc ssesset forto, whch the forto bout ttrbute weghts s copetey ow, d the ttrbute vues te the for of gustc ssesset forto. I order to get the weght vector of the ttrbute, we estbsh two optzto odes bsed o the bsc de of trdto TOPSIS, by whch the ttrbute weghts c be detered. For the spec stutos where the forto bout ttrbute weghts s copetey uow, we estbsh soe other optzto odes. By sovg these odes, we get two spe d exct forus, whch c be used to detere the ttrbute weghts. The, bsed o the TOPSIS ethod, ccuto steps for sovg MGDM probes wth -tupe gustc ssesset forto re gve. The weghted dstces betwee every tertve d -tupe gustc postve de souto (TLPIS d -tupe gustc egtve de souto (TLNIS re ccuted. The, ccordg to the weghted dstces, the retve coseess degree to the TLPIS s ccuted to r tertves. These ethods hve exct chrcterstc gustc forto processg. They voded forto dstorto d osg whch occur forery the gustc forto processg. Fy, soe prctc expes re used to ustrte the deveoped procedures. Keywords: Group decso g; Lgustc ssesset forto; -tupe; TOPSIS. Itroducto Mg decsos wth gustc forto s usu ts fced by y decso ers [], d thus, the use of gustc pproch s ecessry []. My pproches hve bee proposed for ggregtg forto up to ow [-7, 4-]. Prtcury for the gustc utpe ttrbute group decso g probes, whch the ttrbute weghts d expert weghts te the for of re ubers, d the preferece vues te the for of gustc vrbes, pproch bsed o the LOW d LH opertors s proposed [4]. For the se decso probe, pproch bsed o the LOWG d LHG opertors s proposed [5]; pproch bsed o the EIOWG opertor s proposed [6]. The bove ethods copute wth words drecty. I 000, Herrer F. [7] deveoped -tupe gustc ode bsed o fuzzy gustc represetto ode, whch represets the gustc forto wth pr of vues ced -tupe, coposed by gustc ter d uber. -tupe gustc ode hs exct chrcterstc gustc forto processg. It voded forto dstorto d osg whch occur forery the gustc forto processg. I recet yers, ths ethod hs bee wdey used group decso g probes [9-7]. Techque for order perforce by srty to de souto (TOPSIS [8] oe of ow cssc MDM ethod, ws frst deveoped by Hwg d Yoo [9] for sovg MDM probe. TOPSIS, ow s oe of the ost cssc MDM ethods, s bsed o the de, tht the chose tertve shoud hve the shortest dstce fro the postve de souto d o the other sde the frthest dstce of the Pubshed by tts Press Copyrght: the uthors 5

2 G.W. We, R. L, X.F. Zho d H.J. Wg egtve de souto. I [], Wg d F exteded the TOPSIS to sove the group decso g probes wth -tupe gustc ssesset forto whch both the ttrbute vues d ttrbute weght te the for of gustc forto. I the process of MGDM wth gustc ssesset forto, soetes, the ttrbute vues te the for of gustc ssesset forto, d the forto bout ttrbute weghts s copetey ow or copetey uow becuse of te pressure, c of owedge or dt, d the expert s ted expertse bout the probe do. of the bove ethods, however, w be usutbe for deg wth such stutos. Therefore, t s ecessry to py tteto to ths ssue. The of ths pper s to deveop ew ethod for gustc MGDM probes wth copete weght forto bsed the trdto des of TOPSIS. I order to do so, the reder of ths pper s set out s foows. I the ext secto, we troduce soe bsc cocepts d operto ws of -tupe gustc vrbes. I Secto we deveop soe prctc ethods bsed o the trdto des of TOPSIS for gustc group decso g probe wth copete weght forto, whch s strghtforwrd d hs o oss of forto. I Secto 4, we gve soe ustrtve expes to verfy the deveoped pproch d to deostrte ts fesbty d prctcty. I Secto 5 we cocude the pper d gve soe rers.. Preres Let S { s 0,,, t} = = L be gustc ter set wth odd crdty. y be, s represets possbe vue for gustc vrbe, d t shoud stsfy the foowg chrcterstcs [7-8]: ( The set s ordered: s > s, f > ; ( Mx opertor: x ( s, s opertor: ( s, s = s, f s s ; ( M = s, f s s. For expe, S c be defed s S = { s = extreey poor( EP, s = very poor( VP, 0 s = poor( P, s = edu( M, s = good( G, 4 s5 = very good( VG, s6 = extreey good( EG} The -tupe fuzzy gustc represetto ode represets the gustc forto by es of - s,, where s s gustc be d s tupe, ( uerc vue tht represets the vue of the syboc trsto [7-8]. Defto. Let β be the resut of ggregto of the dces of set of bes ssessed gustc ter set S,.e., the resut of syboc ggregto β 0,t, beg t the crdty of S. operto. [ ] Let roud ( β such tht, [ 0, t] = d α = β be two vues, d α 0.5,0.5 the α s ced Syboc Trsto [7-8]. Fro ths cocept, Herrer F. [7-8] deveoped gustc represetto ode whch represets the s, α, gustc forto by es of -tupe ( s S d α 0.5,0.5 : s represets the gustc be of the forto; α s uerc vue expressg the vue of the trsto fro the org resut β to the cosest dex be, the gustc ter set ( s S,.e., the syboc trsto. Ths gustc represetto ode defes set of fuctos to e trsfortos betwee gustc - tupe d uerc vues: Defto. Let S = { s0, s, L, s t } be gustc ter set d [ 0,t] β vue supportg the resut of syboc ggregto operto, the, the -tupe tht expresses the equvet forto to s obted wth the foowg fucto: :, [ 0 t] S 0.5, 0.5 ( ( β ( s, = roud β = ( α = β, α 0.5, 0.5 where roud s the usu roudg operto, s hs the cosest dex be to β d α s the vue of the syboc trsto [7-8]. Defto. Let S = { s0, s, L, s t } be gustc ter set d (, fucto s α be -tupe. There s wys, such tht, fro -tupe, t returs ts β 0,t R[7-8] equvet uerc vue [ ] S [ 0 t] ( s, α α β :,, ( = = (4 Fro Deftos d, t s obvous tht the coverso of gustc ter to gustc -tupe Pubshed by tts Press Copyrght: the uthors 6

3 Modes for Mutpe ttrbute Group Decso Mg cossts of ddg vue 0 s syboc trsto [7-8]: s S s,0 (5 ( Defto 4. Let ( s, d (, tupes, the [7-8] If s be two - < the ( s, s ser th ( s, If = the f =, the ( s,, ( s, represets the se forto b f < the ( s, s ser th ( s, c f > the (, ( s, Defto 5. -tupe egto opertor: s s bgger th ( ( (, α (, α eg s = t s (6 where t s the crdty of S, S = { s0, s, L, s t } [7-8]. Defto 6. Let x = {( r,,( r,, K,( r, } be set of -tupes, the -tupe rthetc e s coputed s foows [7-8] r, = r,, r S, , = (7 x = r,, r,, K, r, ( ( Defto 7. Let {( ( ( } T be set of -tupes d = (,, L, be the ω ω ω ω weghtg vector of -tupes ( r, ( =,, L, d [ 0,] = ω, =,, L,, ω =. The -tupe weghted verge s [7-8] ( r %, % = ϕ (( r,,( r,, K,( r, = ( r, ω, r% S, % 0.5, 0.5 = Defto 8. Let (, the we c [6] r d (, (8 r be two -tupes, ((,, (, (, (, = d r r r r the dstce betwee (, r d(, r.. Modes for utpe ttrbute group decso g (MGDM probes wth -tupe gustc ssesset forto The foowg ssuptos or ottos re used to represet the group decso g probes wth copete weght forto gustc settg: Let = {,, L, } be dscrete set of tertves, G { G, G,, G} ttrbutes, D { D D D} (9 = L be the set of =,, L, t be the set of decso ( ers. Suppose tht R = ( r s the group ( decso g trx, where r S s preferece vues, whch te the for of gustc vrbe, gve by the decso er D D, for the tertve wth respect to the ttrbute G ( ttrbutes G (,,, G, w= w, w, L, w s the weghtg vector of the w [ 0,], w =. = = L, where For coveece of coputto, we trsfor ( gustc decso trx R = r to -tupe gustc decso trx R ( ( ( r,0 =, the utze the decso forto gve trx R to derve the coectve over -tupe gustc decso R= r, trx ( t = =,, L,, =,, L,. (0 t ( r, = ( r, Defto 9. Let ( r, x {( r, } =,, L,, the =, ( r, ( r,,( r,,,( r, ( = L ( s ced the -tupe gustc postve de souto (TLPIS. Pubshed by tts Press Copyrght: the uthors 7

4 G.W. We, R. L, X.F. Zho d H.J. Wg Defto 0. Let ( r, {( r, } =,, L,, the =, ( r, ( r,,( r,,,( r, ( = L ( s ced the -tupe gustc egtve de souto (TLNIS. For the coveece of depcto, bsed o the - tupe gustc decso trx, we deote the tertve (,,, = L s: ((,,(,,,(, = r r L r, where (, =,, L,. ( r dcte the ttrbute vues of correspodg to the ttrbute G (,,, = L. Defto. The weghted dstces betwee d s defed s foows: (, = ( ξ, η d (, = r ( r, w = Defto. The weghted dstces betwee d s defed s foows: (, = ( ξ, η d (, = r ( r, w = (4 (5 Defto. The retve coseess of the tertve wth respect to s defed s (, = ( ξ, η ( ξ, η ( ξ, η ( ξ, η c = (6 The retve coseess (6 c be used to r tertves. The rger the retve coseess (, c s, the better the tertve s. If the forto bout the ttrbute weghts s copetey ow, the we c detere the rg of tertves d seect the best oe(s ccordce wth the retve coseess c(, (,,, = L. I the foowg, we ppy TOPSIS ethod to sove the - tupe gustc MGDM wth copetey ow weght forto. Expe. Let us suppose there s vestet copy, whch wts to vest su of oey the best opto (dpted fro []. There s pe wth fve possbe tertves to vest the oey: s cr copy; s food copy; s coputer copy; 4 4 s rs copy; 5 5 s TV copy. The vestet copy ust te decso ccordg to the foowg four ttrbutes: G s the rs yss; G s the growth yss; G s the soc-potc pct yss; 4G 4 s the evroet pct yss. The fve possbe =,, L,5 re to be evuted tertves ( usg the gustc ter set S by the three decso ers uder the bove four ttrbutes, d costruct ( the decso trces R = r =,, s foows: ( ( 5 4 G G G G 4 G P VP VG VP G P G R = VG VP G P 4 G VG EG VP 5 M VP M VP Pubshed by tts Press Copyrght: the uthors 8

5 Modes for Mutpe ttrbute Group Decso Mg G G G G 4 M G P P P VP M P R = G M G EP 4 VG P P G 5 EG EP VP M G G G G 4 P M VP VP VP EP G G R = M G P EG 4 EG VP VP M 5 P VP M VP Frsty, we trsfor gustc decso trx ( R = r to -tupe gustc decso trx R ( ( ( r,0 = s foows ( G,0 ( P,0 ( VP,0 ( VG,0 ( VP,0 ( G,0 ( P,0 ( G,0 (,0 (,0 (,0 (,0 ( G,0 ( VG,0 ( EG,0 ( VP,0 ( M,0 ( VP,0 ( M,0 ( VP,0 ( M,0 ( G,0 ( P,0 ( P,0 ( P,0 ( VP,0 ( M,0 ( P,0 (,0 (,0 (,0 (,0 ( VG,0 ( P,0 ( P,0 ( G,0 ( EG,0 ( EP,0 ( VP,0 ( M,0 ( P,0 ( M,0 ( VP,0 ( VP,0 ( VP,0 ( EP,0 ( G,0 ( G,0 (,0 (,0 (,0 (,0 ( EG,0 ( VP,0 ( VP,0 ( M,0 ( P,0 ( VP,0 ( M,0 ( VP,0 R = VG VP G P R = G M G EP R = M G P EG The, we utze Eq. (0 to derve the coectve over R= r, s -tupe gustc decso trx ( foows: ( M,0 ( M,0 ( VP,0. ( M, 0. ( VP,0. ( P, 0. ( M,0 ( M,0. (,0 (, 0. (,0. (, 0. ( VG,0 ( M, 0. ( M,0 ( M, 0. ( G, 0. ( VP, 0. ( P,0. ( P,0. R= G M M M If the forto bout the ttrbute weghts s copetey ow s foows: w = ( 0.000,0.000,0.00,0.800 The, we utze the pproch deveoped to get the ost desrbe tertve(s. Step. Defg the TLPIS d TLNIS s ( r, = (( VG,0, ( M,0 (, M,0. (, M,0. T ( r, = (( VP,0., ( VP,0., ( VP,0., ( P,0. T Step. Ccutg the dstces of ech tertve fro TLPIS d TLNIS by Eq. (4-5 ξ, η = VP,0.09, ξ, η = VP, 0.60 ( ( ( ( ( ξ, η = ( EP,0.40,( ξ4, η4 = ( EP,0.47 ( ξ5, η5 = ( P, 0.447,( ξ, η = ( P,0.0 ( ξ, η = ( VP,0.67,( ξ, η = ( P, 0. ( ξ4, η4 = ( P, 0.0,( ξ5, η5 = ( VP, Step. Ccutg the retve coseess degree of ech tertve fro TLPIS by Eq. (6 ξ, η = EP,0.48, ξ, η = VP, 0.5 ( ( ( ( ( ξ, η = ( VP, 0.99 (, ξ4, η4 = ( VP, 0.0 ( ξ5, η5 = ( EP,0.6 Step 4. Rg the tertves ( =,, L,5 ccordce wth the retve coseess degree ( ξ, η : f 4 f f f 5, d thus the ost desrbe tertve s. I the foowg, we sh ppy TOPSIS ethod to sove the -tupe gustc MGDM wth copetey ow weght forto. H s the set of the ow weght forto, whch c be costructed by the foowg fors[0-], for : For. we Pubshed by tts Press Copyrght: the uthors 9

6 G.W. We, R. L, X.F. Zho d H.J. Wg rg: w w w w α, α > 0 ; For. strct rg: w w ; For. rg of dffereces: w w, for ; For 4. rg wth utpes: w βw, 0 ; For 5. terv for: α w α ε, 0 α < α ε. The bsc prcpe of the TOPSIS ethod s tht the chose tertve shoud hve the shortest dstce fro the postve de souto d the frthest dstce fro the egtve de souto. Obvousy, for the weght vector gve, the ser d(, the rger (, β d d s, the better tertve s. But the forto bout ttrbute weghts s copetey ow. So, order to get the d(, d d(,, frsty, we ust ccute the weght forto. So, we c estbsh the foowg utpe obectve optzto odes (M. d (M. to ccute the weght forto: (M. ( ξ, η = ( r, ( r, w = subectto: w H, =,, L,. (M.x ( ξ, η = ( r, ( r, w = subectto: w H, =,, L,. Sce ech tertve s o-feror, so there exsts o preferece reto o the the tertves. The, we y ggregte the bove utpe obectve optzto odes wth equ weghts to the foowg utpe obectve optzto odes (M. d (M.4: ( ξη = ( ( ξ η (M.x,, = = ( = = subect to: ( r, ( r, w ( ξη = ( ( ξ η (M.4x,, = = ( = = subect to: ( r, ( r, w ccordg to fucto, utpe obectve optzto odes (M. d (M.4 c be trsfored to the sge obectve optzto odes (M.5 d (M.6: ( ξη ( ( ξ η (M.5, =, = = ( ( r, ( r, w = = subectto: ( ξη ( ( ξ η (M.6x, =, = = ( ( r, ( r, w = = subectto: By sovg the odes (M.5 d (M.6, we get the opt souto w ( w, w,, w (,,, = L d w = w w L w, whch c be used s the weght vector of ttrbutes. The, we c get ( ξ, η Pubshed by tts Press Copyrght: the uthors 0

7 Modes for Mutpe ttrbute Group Decso Mg d ( ξ, η by Equtos (4-5, respectvey. The we utze (6 to derve the retve coseess c(, (,,, = L, by whch we c r the tertves (,,, seect the best oe(s. = L d If the forto bout ttrbute weghts s copetey uow, we c costruct the foowg sge obectve optzto odes: ( ξη ( ( ξ η (M.7, =, = = ( (, (, = = subectto: w =, w 0, =,, L,. = (M.8x ( ξη, = ( ( ξ, η = = r, r, w r r w ( ( ( = = subectto: w =, w 0, =,, L,. = To sove the odes (M.7 d (M.8, we get two spe d exct foru for deterg the ttrbute weghts s foows: w = = = = ( r, ( r, ( r, ( r, =,, L,. (7, w = = = = ( r, ( r, ( r, ( r, =,, L,. (8 whch c be used s the weght vector of ttrbutes. Obvousy, w 0 (, (,,, for. The, we c get d = L d (, (,, d = L by Equtos (4-5 respectvey. The we utze (6 to derve the retve coseess c(, (,,, = L, by whch we c r the tertves (,,, seect the best oe(s. = L d Expe. For the MGDM probe cosdered Expe, suppose tht the forto bout the ttrbute weghts s prty ow s foows: { H = 0.05 w 0.0,0.8 w 0., 0.5 w 0.,0.5 w w [0,], =,,,4, w = 4 = The by odes (M.5 d (M.6, we c estbsh the foowg two sge-obectve progrg odes: ( ξη, = 8.00w 4.w.67w.67w4 Subectto: x ( ξη, = 0.w 7.w 6.w 4.67w4 Subectto: To sove these odes, we get the weght vector of ttrbutes: w = ( ,0.800,0.5, w = ( 0.076,0.800,0.784, }, Pubshed by tts Press Copyrght: the uthors

8 G.W. We, R. L, X.F. Zho d H.J. Wg by Eq.(4,5, we get ξ, η = VP,0.00, ξ, η = VP, 0.47 ( ( ( ( ( ξ, η = ( EP,0.45,( ξ4, η4 = ( EP,0.469 ( ξ5, η5 = ( P,0. 48,( ξ, η = ( VP,0.009 ( ξ, η = ( VP,0.47,( ξ, η = ( P, 0.4 ( ξ4, η4 = ( P, 0.44,( ξ5, η5 = ( EP,0.445 By Eq.(6, we hve ( ξ, η = ( EP,0.495,( ξ, η = ( VP, 0.70 ( ξ, η = ( VP, 0.08 (, ξ4, η4 = ( VP, 0. ( ξ, η = ( EP, Sce c, fc, fc, fc, f c, ( ( ( 4 ( ( 5 the f 4 f f f 5, Hece, the ost desrbe tertve s. If the forto bout ttrbute weghts s copetey uow, the by (7-8, we hve w = ,0.7,0.096,0.9 ( w = ( 0.095,0.86,0.548, by Eq.(4,5, we get ξ, η = VP,0.04, ξ, η = VP, 0.4 ( ( ( ( ( ξ, η = ( EP,0.45,( ξ4, η4 = ( EP,0.48 ( ξ5, η5 = ( P, 0.4,( ξ, η = ( VP,0.056 ( ξ, η = ( VP,0.87,( ξ, η = ( P, 0.40 ( ξ4, η4 = ( P, 0.95,( ξ5, η5 = ( EP,0.47 By Eq.(6, we hve ( ξ, η = ( VP, 0.497,( ξ, η = ( VP, 0. ( ξ, η = ( VP, 0.06 (, ξ4, η4 = ( VP, 0.4 ( ξ, η = ( EP, Sce (, f ( 4, f (, f (, f ( 5, c c c c c the f 4 f f f 5, Hece, the ost desrbe tertve s. Besdes, the dvtge of the pproch preseted ths pper s cer usg coputg wth word represetto ode, -tupe gustc represetto tht ows us to ggregte gustc forto wthout osg t. 4. Cocuso I ths pper, we hve vestgted the probe of - tupe gustc utpe ttrbute group decso-g wth copetey ow ttrbute weght forto. odfed TOPSIS yss ethod s proposed. I order to get the ttrbute weght, we estbsh the utpe obectve optzto odes bsed o the bsc de of the trdto TOPSIS. The, by er equ weghted ethod, the utpe obectve optzto odes c be trsfored to two sgeobectve progrg ode. By sovg the sgeobectve progrg odes, we c get the ttrbute weght forto. For the spec stutos where the forto bout ttrbute weghts s copetey uow, we estbsh soe other optzto odes. By sovg these odes, we get two spe d exct foru, whch c be used to detere the ttrbute weghts. The, the weghted dstces betwee every tertve d TLPIS d TLNIS re ccuted. The, ccordg to the weghted dstces, the retve coseess degree to the TLPIS s ccuted to r tertves. They voded forto dstorto d osg whch occur forery the gustc forto processg. Fy, ustrtve expe s gve. These ethods hve exct chrcterstc gustc forto processg. By coprg wth the TOPSIS ethod proposed terture [], the pproch preseted ths pper proves to be effectve to sove the MGDM probes wth -tupe gustc ssesset forto, whch the forto bout ttrbute weghts s copetey ow, d the ttrbute vues te the for of gustc ssesset forto. I the future, we sh exted TOPSIS ethod to sove the -tupe gustc utpe ttrbute group decso-g wth ubced gustc ter sets. cowedget The uthor s very grtefu to the edtor d the oyous referees for ther sghtfu d costructve coets d suggestos, whch hve bee very hepfu provg the pper. The wor ws supported by the Hutes d Soc Sceces Foudto of Pubshed by tts Press Copyrght: the uthors

9 Modes for Mutpe ttrbute Group Decso Mg Mstry of Educto of the Peope s Repubc of Ch (No.09XJ6000 Refereces [] Z. S. Xu, Ucert Mutpe ttrbute Decso Mg: Methods d ppctos, Tsghu Uversty Press, Beg, 004. [] M., Degdo F. Herrer, E. Herrer-Ved d L. Mrtez, Cobg Nuerc d Lgustc Iforto Group Decso Mg, Iforto Sceces 07( [] R. Deg d G. Borto, The probe of gustc pproxto cc decso g, Iterto Jour of pproxte Resog ( ( [4] Z. S. Xu, ote o gustc hybrd rthetc vergg opertor utpe ttrbute group decso g wth gustc forto, Group Decso d Negotto 5(6 ( [5] Z. S. Xu, ethod bsed o gustc ggregto opertors for group decso g wth gustc preferece retos, Iforto Sceces 66( ( [6] Z. S. Xu, Exteded IOWG opertor d ts use group decso g bsed o utpctve gustc preferece retos, erc Jour of pped Sceces ( ( [7] F.Herrer d L. Mrtíez, -tupe fuzzy gustc represetto ode for coputg wth words, IEEE Trsctos o Fuzzy Systes 8 ( [8] F. Herrer d L. Mrtíez, ode bsed o gustc -tupes for deg wth utgrur herrchc gustc cotexts ut-expert decso-g, IEEE Trsctos o Systes, M, d Cyberetcs ( [9] X. R. Wg, Z. P. F, ethod for group decso g probes wth dfferet fors of preferece forto, Jour of Northester Uversty (Ntur Scece 4( ( [0] F. Herrer, E. Herrer-Ved, J. Verdegy, gustc decso process group decso g, Group Decso d Negotto 5( ( [] X. R. Wg, Z. P. F, Method for group decso g bsed o -tupe gustc forto processg, Jour of Mgeet Scece Ch 6(5 (00-5. [] F. We, C.. Lu, S. Y. Lu, ethod for group decso g wth gustc forto bsed o ucert forto processg, Operto Reserch d geet Scece 5( ( [] Y. P. Jg, Z. P. F, pproch to group decso-g probes bsed o -tupe gustc sybo operto, Systes Egeerg d Eectrocs 5( ( [4] H. Y. L, Z. P. F, Mut-crter group decso g ethod bsed o -tupe gustc forto processg, Jour of Northester Uversty (Ntur Scece 4( [5] H. Y. L, Z. P. F, Coprehesve ut-ttrbute group evuto wth gustc ssesset forto, Jour of Northester Uversty (Ntur Scece 6(7 ( [6] X. W. Lo, Y. L, G. M. Dog, ut-ttrbute group decso-g pproch deg wth gustc ssesset forto, Syste Egeerg-Theory & Prctce 6(9 ( [7] Y. P. Jg, Z. P. F, Property yss of the ggregto opertors for -tupe gustc forto, Cotro d Decso 8( [8] Y.J. L, T.Y. Lu, C.L. Hwg, TOPSIS for MOD, Europe Jour of Operto Reserch 76( ( [9] C.L. Hwg, K. Yoo, Mutpe ttrbute Decso Mg Methods d ppctos, Sprger, Ber Hedeberg, 98. [0] P. S. Pr, S. H. K, W. C. Yoo, Estbshg strct doce betwee tertves wth spec type of copete forto, Europe Jour of Operto Reserch 96( [] K.S. Pr d S.H. K, Toos for terctve ut-ttrbute decso g wth copetey detfed forto, Europe Jour of Operto Reserch 98 ( [] S.H. K, S.H. Cho d J.K. K, terctve procedure for utpe ttrbute group decso g wth copete forto: rge-bsed pproch, Europe Jour of Operto Reserch 8 ( [] S.H. K d B.S. h, Iterctve group decso g procedure uder copete forto, Europe Jour of Operto Reserch 6 ( [4]F. Herrer, E. Herrer-Ved, L. Mrtíez, fuzzy gustc ethodoogy to de wth ubced gustc ter sets. IEEE Trsctos o Fuzzy Systes 6( ( [8] G.W. We d R. L, Method of grey reto yss for utpe ttrbute group decso g bsed o -tupe gustc forto, Jour of Systes Egeerg d Eectrocs 0(9 ( [6] G.W. We, -tupe gustc utpe ttrbute group decso g wth copete ttrbute weght forto, Jour of Systes Egeerg d Eectrocs 0( ( Pubshed by tts Press Copyrght: the uthors

10 G.W. We, R. L, X.F. Zho d H.J. Wg [7]G.W. We, Ucert gustc hybrd geoetrc e opertor d ts ppcto to group decso g uder ucert gustc evroet, Iterto Jour of Ucertty, Fuzzess, Kowedge-Bsed Systes 7( ( [8] X.B. L, D. Ru, J. Lu d Y. Xu, gustcvued weghted ggregto opertor to utpe ttrbute group decso g wth quttve d quttve forto, Iterto Jour of Coputto Itegece Systes ( ( [9] J. M, D. Ru, Y. Xu d G. Zhg, fuzzy-set pproch to tret detercy d cosstecy of gustc ters ut-crter decso g, Iterto Jour of pproxte Resog, 44(( [0] L. Mrtíez, D. Ru, F. Herrer, E. Herrer- Ved, P.P. Wg, Lgustc decso g: Toos d ppctos, Iforto Sceces 79(4 ( [] L. Mrtíez, J. Lu, D. Ru, J.B. Yg, Deg wth heterogeeous forto egeerg evuto processes, Iforto Sceces 77(7 ( [] X.W. Lo, Y L d B. Lu, ode for seectg ERP syste bsed o gustc forto processg, Iforto Systes (7 ( Pubshed by tts Press Copyrght: the uthors 4