Group decision-making based on heterogeneous preference. relations with self-confidence

Transcription

1 Group decso-mag based o heterogeeous preferece relatos wth self-cofdece Yucheg Dog,Weq Lu, Busess School, Schua Uversty, Chegdu 60065, Cha E-mal: ycdog@scu.edu.c; wqlu@stu.scu.edu.c Fracsco Chclaa, Faculty of Techology, De Motfort Uversty Lecester, Uted Kgdom E-mal: chclaa@dmu.ac.u Erque Herrera-Vedma DECSAI, Uversty of Graada, Graada, Spa vedma@deecsa.ugr.es Fracsco Javer Cabrerzo Dept. Software Egeerg ad Computer Systems, UNED Madrd, Spa cabrerzo@ss.ued.es Abstract: Preferece relatos are very useful to express decso maers prefereces over alteratves the process of group decso-mag. However, the multple self-cofdece levels are ot cosdered exstg preferece relatos. I ths study, we defe the preferece relato wth self-cofdece by tag multple self-cofdece levels to cosderato, ad we call t the preferece relato wth self-cofdece. Furthermore, we preset a two-stage lear programmg model for estmatg the collectve preferece vector for the group decso-mag based o heterogeeous preferece relatos wth self-cofdece. Fally, umercal examples are used to llustrate the two-stage lear programmg model, ad a comparatve aalyss s carred out to show how self-cofdece levels fluece o the group decso-mag results. Keywords: Preferece relatos, self-cofdece levels, collectve preferece vector, lear programmg model.. Itroducto Preferece relatos are wdely used group decso-mag (GDM). A complete preferece relato cotas preferece elemets, ad each elemet dcates

2 the degree up to whch a alteratve s preferred to aother oe [30, 3, 33-35]. Sometmes, decso maers have o self-cofdece o the preferece formato because of tme pressure ad lmted expertse regardg the problem doma. I these stuatos, decso maers may provde ther preferece formato the form of complete preferece relatos,.e. a preferece relato wth some of ts elemets mssg [, 3, 4, 38-40]. I a complete preferece relato, the decso maer provdes all preferece formato, ad t s geerally assumed that all preferece values are provded wth the same self-cofdece level. I a complete preferece relato, two self-cofdece levels are used: () The decso maer s of self-cofdece for those preferece elemets for whch a value s provded ad () the decso maer s wthout self-cofdece for the preferece elemets for whch a value s ot gve. However, multple self-cofdece levels dfferet to the two levels case metoed above are ot cosdered exstg preferece relatos. Therefore, t would be of great mportace to provde decso maers wth tools to allow them to express multple self-cofdece levels whe provdg ther prefereces. I ths study, we propose the preferece relato wth self-cofdece by tag multple self-cofdece levels to cosderato, ad we call t the preferece relato wth self-cofdece. I the preferece relato wth self-cofdece, each elemet cossts of two compoets, the frst oe s the preferece value betwee pars of alteratves, ad the secod part, whch s defed o a lgustc terms set, represets the decso maer s self-cofdece level of ts correspodg frst part or preferece value. I practcal GDM problems, each decso maer has dfferet owledge, experece, culture ad educatoal bacgrouds. As a result, the decso maers use dfferet preferece relatos to express ther dvdual preferece formato. Three ds of preferece relatos have bee wdely vestgated: multplcatve preferece relatos [5, 6, 33, 4], addtve preferece relatos [6, 9, 3, 34-36, 4] ad lgustc preferece relatos [8, 9,, 6, 0, 8, 37]. Chclaa et al. [4, 5], Dog et al. [, 4], Herrera et al. [6], ad Herrera-Vedma et al. [] tated ad developed the GDM models wth heterogeeous preferece relatos represeted by preferece ordergs, utlty fuctos, addtve preferece relatos, multplcatve preferece relatos. Moreover, Fa et al. [8] ad Ma et al. [9] tated several optmzato-based models to tegrate heterogeeous preferece relatos. The GDM problem wth heterogeeous preferece relatos has become oe of the maor areas of GDM researches [3].

3 A mportat challege to bear md whe decso maers provde dfferet preferece relatos wth self-cofdece s how to obta the collectve soluto. I ths paper, a two-stage lear programmg model to deal wth GDM problems based o heterogeeous preferece relatos wth self-cofdece s developed. Ths two-stage lear programmg model s based o a dstace-based framewor that mmzes the formato devato betwee decso maers preferece relatos ad collectve preferece vector, ad that s preseted frst.. The rest of ths study s orgazed as follows. Secto troduces the basc owledge regardg ordal lgustc -tuples model ad the three ds of preferece relatos metoed above. Secto 3 defes the preferece relatos wth self-cofdece ad descrbes the GDM problem based o heterogeeous preferece relatos wth self-cofdece. Secto 4 proposes a dstace-based framewor that s used to develop a two-stage lear programmg model for estmatg the collectve preferece vector the GDM. Secto 5 provdes umercal examples ad a comparatve aalyss to show how self-cofdece levels fluece o the GDM results. Fally, Secto 6 cocludes the study.. Prelmares I ths secto, ad wth the am of mag ths study self-cotaed, prelmary cocepts regardg the ordal lgustc -tuple model ad the three ma type of preferece relatos used ths framewor are covered.. The ordal -tuple lgustc model The ordal -tuple lgustc model s used ths study to carry out ordal computg wth words whe dealg wth the lgustc self-cofdece levels formato. The basc otatos ad operatoal laws of ordal lgustc varables are troduced [3, 5, 5, 7], a summary of whch s provded below. Let S = { s = 0,,..., g} be a lgustc term set wth odd cardalty. The term s represets a possble value of a lgustc varable. The followg ordal orderg o set S s assumed: s > s f ad oly f >. Herrera ad Martíez preseted the ordal -tuple lgustc model [5], ad t was based the followg adapted defto: 3

4 Defto (Herrera ad Martíez [5]): Let β [0, g] be a umber the graularty terval of the lgustc term set S = { s0,..., s g } ad let = roud( β) ad α = β be two values such that [0, g] ad α [ 0.5, 0.5). The, α s called a symbolc traslato, wth roud beg the usual roudg operato. Herrera ad Martíez s model represets the lgustc formato by meas of ordal -tuples ( s, α ), where s s a smple term S ad α [ 0.5,0.5) mappg betwee ordal lgustc -tuples ad umercal values [0, g] s possble.. A oe-to-oe Defto (Herrera ad Martíez [5]): Let S = { s0,..., s g } be a lgustc term set ad β [0, g] a value represetg the result of a symbolc aggregato operato, the the ordal -tuple that expresses the equvalet formato to β s obtaed wth the followg fucto: Δ :[0, g] S [ 0.5, 0.5), where Δ ( β) = (, α), wth s s, = roud( β) α = β, α [ 0.5, 0.5). For coveece, deotg by S = S [ 0.5, 0.5) the verse fucto of Δ s Δ : S [0, g] wth Δ (( s, α)) = + α. For otato smplcty, ths paper sets Δ (( s,0)) =Δ ( s). Clearly, a orderg o the set of ordal -tuples ad a egato operator are possble to defe as follows: ) Let ( s, α ) ad ( sl, γ ) be two ordal -tuples. The: () f < l, the ( s, α ) s smaller tha ( s, γ ). () f = l, the (a) f α = γ, the ( s, α ) ad ( s, γ ) represets the same formato. (b) f α < γ, the ( s, α ) s smaller tha ( s, γ ). ) Ordal -tuple egato operator: ((, α)) ( ( (, α))). l l l Neg s =Δ g Δ s (). Preferece relatos I ths subsecto, we troduce multplcatve preferece relatos, addtve preferece relatos ad ordal -tuple lgustc preferece relatos. () Multplcatve preferece relatos Saaty troduced multplcatve preferece relatos [33]. Defto 3 [33]: Let X = { x, x,..., x } be a fte set of alteratves. A multplcatve preferece relato A= ( a ) o X s descrbed by a postve preferece relato 4

5 A X X, wth elemet a measurg o a rato scale [/9, 9] the testy of preferece of alteratve x over alteratve x. The followg terpretato s assumed: a = dcates dfferece betwee x ad x ; a = 9 dcates that x s absolutely preferred to x, ad a {,3,...,8} dcates termedate evaluatos. It s assumed that aa = ad a =. () Addtve preferece relatos Addtve preferece relatos are also called fuzzy preferece relatos [4,, 3]. Defto 4 [3]: A addtve preferece relato P o a fte set of alteratves X s a relato X X that s charactersed by a membershp fucto µ P : X X [0,], where µ ( x, x ) = p deotes the preferece degree or testy of the P alteratve x over x. The followg terpretato s assumed: p = 0.5 dcates dfferece betwee x ad x, p > 0.5 dcates a defte preferece for x over x, p = dcates the maxmum degree of preferece for x over x. It s assumed that p + p = ad p = 0.5. (3) Ordal -tuple lgustc preferece relatos Let S = { s = 0,,..., g} be a ordal lgustc term set wth odd cardalty as troduced Secto.. Defto 5 [7]: A ordal -tuple lgustc preferece relato T o a fte set of alteratves X s defed as T = ( t ), where t S deotes the degree of lgustc preferece of the alteratve x over x. The followg terpretato s assumed: t = s g dcates dfferece betwee x ad x, t > s dcates a defte preferece for x g over x, ad t < s dcates a defte preferece for x over g x. It s assumed that t = s ad t = Neg( t ). g 5

6 Itally, the decso maer expresses her/hs prefereces usg the smple ordal terms of S, ad the ordal -tuple lgustc values oly appear after operatos o smple ordal terms are carred out..3 Trastvty of prefereces Trastvty s a mportat cocept to apply to preferece relatos to assess ther ratoalty. Here, we lst there dfferet trastve propertes of preferece relatos, wth the thrd oe beg a stroger codto tha the secod oe, whch tur s stroger tha the frst oe. Let A= ( a ) be a multplcatve preferece relato. Some trastve propertes of multplcatve preferece relatos ca be descrbed as follows: (a) Wea stochastc trastvty: a, a a,,. (b) Strog stochastc trastvty: a, a a max( a, a ),,. (c) Multplcatve trastvty: aa = a,,. The equvalet propertes for addtve preferece relatos ad -tuple lgustc preferece relatos are also descrbed as follows. Let P= ( p ) be a addtve preferece relato. (a) Wea stochastc trastvty: p 0.5, p 0.5 p 0.5,,. (b) Strog stochastc trastvty: p 0.5, p 0.5 p max( p, p ),,. (c) Addtve trastvty: p = p p + 0.5,,. Let T = ( t ) be a ordal -tuple lgustc preferece relato. (a) Wea stochastc trastvty: t sg, t sg t sg,,. (b) Strog stochastc trastvty: t s, t s t max( t, t ),,. g g g (c) Addtve trastvty: Δ ( t ) =Δ ( t ) Δ ( t ) +,,. A preferece relato that verfes the stroger of the above three trastvty propertes s usually referred to as a cosstet preferece relato followg Saaty s defto of cosstecy of multplcatve preferece relatos. 6

7 3. Preferece relatos wth self-cofdece GDM I ths secto, we defe three ds of preferece relatos wth self-cofdece ad descrbe the GDM problem based o heterogeeous preferece relatos wth self-cofdece. To eable decso maers to characterze self-cofdece levels a lgustc way, a lgustc terms set S ={ l0, l,..., l g} s used, wth the followg oe beg a possble example: S = { l = extremely poor, l = very poor, l = poor, l = slghtly poor, l = far, l = slghtly good, l = good, l = very good, l = extremely good} The decso maer uses the smple term l S to characterze hs/her self-cofdece level over the preferece value. 3. Preferece relatos wth self-cofdece Deftos of multplcatve preferece relato wth self-cofdece, addtve preferece relato wth self-cofdece ad ordal -tuple lgustc preferece relato wth self-cofdece o a fte set of alteratves X = { x, x,..., x } are gve below: Defto 6: A multplcatve preferece relato wth self-cofdece o a fte set of alteratves X, =, s relato o X X whose elemets have two * A (( a, s )) compoets, the frst oe a [ / 9,9] represetg the preferece degree or testy of the alteratve x over x, ad the secod compoet s S represetg the self-cofdece level assocated to the frst compoet. The followg codtos are assumed: aa =, a =, s = s ad s = l. g Defto 7: A addtve preferece relato wth self-cofdece o a fte set of alteratves X, =, s relato o X X whose elemets have two * P (( p, s )) compoets, the frst oe p [0,] represetg the preferece degree or testy of the alteratve x over x, ad the secod compoet s S represetg the self-cofdece level assocated to the frst compoet. The followg codtos are assumed: p + p =, p = 0.5, s = s ad s = l. g 7

8 Defto 8: A ordal -tuple lgustc preferece relato wth self-cofdece o a fte set of alteratves X, T * = ((t,s )), s relato o X X whose elemets have two compoets, the frst oe t S represetg the ordal -tuple lgustc preferece of the alteratve x over x, ad the secod compoet s S represetg the self-cofdece level assocated to the frst compoet. The followg codtos are assumed: t = Neg( t ), t = s, s = s ad s = l. g g Remar: Zadeh [4] developed the cocept of a Z-umber relates to the ssue of relablty of formato. A Z-umber s a ordered par of fuzzy umbers, the frst compoet s a restrcto (costrat) o the values whch a real-valued ucerta varable, ad the secod compoet s a measure of relablty (certaty) of the frst compoet. I our preferece relato wth self-cofdece, each elemet ca be cosdered to be a Z-umber ( some sese). I the followg, we descrbe some trastve propertes of preferece relatos wth self-cofdece. Let * A (( a, s )) = be a multplcatve preferece relato wth self-cofdece. Some trastve propertes ca be descrbed as follows: (a) Wea stochastc trastvty at the self-cofdece level a, a a,,, ad s l,,. (b) Strog stochastc trastvty at the self-cofdece level l S. l S. a, a a max( a, a ),,, ad s l,,. (c) Multplcatve trastvty at the self-cofdece level aa = a,,, ad s l,,. l S. Let * P (( p, s )) = be a addtve preferece relato wth self-cofdece. (a) Wea stochastc trastvty at the self-cofdece level l S. p 0.5, p 0.5 p 0.5,,, ad s l,,. (b) Strog stochastc trastvty at the self-cofdece level l S. p 0.5, p 0.5 p max( p, p ),,, ad s l,,. (c) Addtve trastvty at the self-cofdece level l S. 8

9 p = p p + 0.5,,, ad s l,,. Let * T (( t, s )) = be a ordal -tuple lgustc preferece relato wth self-cofdece. (a) Wea stochastc trastvty at the self-cofdece level t s, t s t s,,, ad s l,,. g g g (b) Strog stochastc trastvty at the self-cofdece level g g l S. l S. t s, t s t max( t, t ),,, ad s l,,. (c) Addtve trastvty at the self-cofdece level l S. g Δ ( t ) =Δ ( t ) Δ ( t ) +,,, ad s l,,. The tradtoal defto to characterze cosstecy of preferece relatos s usg a set of pre-establshed trastve propertes [, 6, ]. I ths paper, a preferece relato wth self-cofdece s cosdered to be acceptable cosstet f t satsfes the wea stochastc trastvty at the self-cofdece level l0 S. 3. Group decso-mag problem wth self-cofdece Let X = { x, x,..., x } be a fte set of alteratves. These alteratves have to be classfed from best to worst, usg the formato gve by a fte set of decso maers E e e e m = {,,..., }. Let C = { c, c,..., c m } be a set of ormalzed weght/mportace values assocated to the set of experts: c s the weght/mportace value of decso maer e ad c 0, m c =. As each decso maer e = E has ther ow deas, atttudes, motvatos, ad persoalty, t s qute atural to cosder that dfferet decso maers wll gve ther prefereces a dfferet way. Thus, decso maers prefereces over the set of alteratves X may be represeted oe of the followg three ways: multplcatve preferece relatos wth self-cofdece, addtve preferece relatos wth self-cofdece ad ordal -tuple lgustc preferece relatos wth self-cofdece. Wthout loss of geeralty, let * A E e e e m = {,,..., }, * P E = { e, e,..., e }, m+ m+ m * T E = { e, e,..., e } m+ m+ m be three subsets of E, represetg the set of decso maers whose preferece formato o X are expressed as multplcatve preferece relatos wth self-cofdece, addtve preferece relatos wth self-cofdece ad ordal -tuple lgustc preferece relatos 9

10 wth self-cofdece, respectvely. The questo s how to obta a collectve soluto to the GDM problem based o heterogeeous preferece relatos wth self-cofdece level. 4. A two-stage lear programmg model I ths secto, we frst propose a dstace-based framewor that ams to mmze the formato devato betwee decso maers preferece relatos ad the collectve preferece vector, whch later s used to develop a two-stage lear programmg model to solve the GDM problem at had. 4.. A dstace-based framewor c c c c T Let w = ( w, w,..., w ) be the collectve prorty preferece vector of the decso maers, where = ad w c 0 for. I geeral, there are dffereces betwee c w = the dvdual preferece formato ad the collectve soluto, whch ca be measured as follows: () Let * A = (( a, s )) be a multplcatve preferece relato wth self-cofdece. The error betwee the preferece value a ad the collectve prorty preferece vector c w s [8, 33] c c ε = w wa, =,,..., m,, =,,..., (5) () Let * P = (( p, s )) be a addtve preferece relato wth self-cofdece. The error betwee the preferece value p ad the collectve prorty preferece vector c w s [8, 3] ( c c ) 0.5 ε = w w + p, = m+, m+,..., m,, =,,..., (6) (3) Let * T = (( t, s )) be a ordal -tuple lgustc preferece relato wth self-cofdece. The error betwee the preferece value t ad the collectve prorty preferece vector ( c c g ) ( w w t ) c w ca be smlarly defed as ε = + Δ, = m +, m +,..., m,, =,,..., (7) If the dvdual preferece relatos are cosstet, the t s ε = 0. 0

11 Whe the error ε s at a self-cofdece level of s ( s S ), the followg formato devato ca be troduced z =Δ s ε, =,,..., m,, =,,..., (8) ( ) The level of self-cofdece s Eq. (8) determes the magfcato of error ε : the larger ts value, the larger magfcato wll be the error ε assged to the correspodg preferece value. I the followg, a dstace-based framewor that mmzes the formato devato betwee decso maers preferece relatos ad the collectve preferece vector s troduced. The followg obectve fucto are troduced for metrc p, m p / p cα z = = = (9) m z= ( ( ) ) where α s a ormalzato coeffcet. Due to the varyg domas adopted for the varety of preferece formats, α s used to ormalze the measure of the formato devato of each decso maer to elmate the fluece of heterogeeous preferece relatos GDM. The ormalzato coeffcets α s determed based o the sze of the coeffcet matrx G. Geerally, accordg to matrx theory ad related research [9], the value of α ca be calculated as follows, α =, =,,..., m (0) sp where sp s the Frobeus orm of matrx G. Whe G has real umber elemets, the T Frobeus orm s sp = λ, where λ m s the greatest egevalue of ( G ) G. m The obectve fucto (9) s affected by parameter p : the -orm dstace ( p = ) s Mahatta dstace; the -orm dstace ( p = ) s the Eucldea dstace; the fty orm dstace ( p = ) s the Chebyshev dstace. I ths study, we study the -orm ad the fty orm dstaces. For p = ad p =, the above obectve fuctos are expressed as Eq. () ad (), respectvely. m z m cα z = = = = ()

12 m z = c α ( max z ) () m =, =,,..., 4. A two-stage lear programmg model based o the dstace-based framewor I ths subsecto, we develop a two-stage lear programmg model for estmatg the collectve preferece vector for the GDM, whch s based o the prevously troduced dstace-based framewor. I the frst stage, we set p = to mmze the sum of all formato devato of all decso maers to obta a set of collectve preferece vectors. I the secod stage, we set p = ad mmze the maxmal formato devato of decso maers to select the optmal collectve preferece vector from the soluto set of the frst stage. () Frst stage: p = We use three trasformed varables model (): y = ε, d =Δ ( t ) ad b =Δ ( s ). The frst stage lear programmg model s expressed as follows: m m z = c α z (a) = = = s.t. w a w c y = 0, =,,,m ;, =,,..., ; < (b) (w w ) p y = 0, = m +,m +,,m ;, =,,...,; < (c) (w w c )+ g d y = 0, = m +,m +,,m;, =,,...,; < (d) z b y 0, =,,,m;, =,,..., ; < (e) z + b y 0, =,,,m;, =,,..., ; < ( f ) w c +w c w c =, (g) w c 0, =,,..., (h) z 0, =,,,m;, =,,..., ; < () I model (3), costrats () b () d express the errors betwee the dvdual preferece formato ad the collectve preferece vector; costrats () e ad ( f ) (3) guaratee that z Δ s ε ; costrat ( g ) guaratees that the prorty vector s ( ) ormalzed to sum to oe; ad fally, costrats ( h ) ad () guaratee that varables c w ad c z are oegatve. () Secod stage: p = It s possble that there are multple optmal solutos to the frst stage model. The secod stage model s model (), ad further selects the optmal collectve preferece vector

13 from the optmal solutos of the frst stage model. The secod stage lear programmg model s gve as follows: m z = m c α (z max ) (a) = m c α z =z (b) = = = w c a w c y = 0, =,,,m ;, =,,..., ; < (c) (w c w c ) p y = 0, = m +,m +,,m ;, =,,...,; < (d) (w c w c )+ g d y = 0, = m +,m +,,m;, =,,...,; < (e) s.t. z b y 0, =,,,m;, =,,..., ; < ( f ) z + b y 0, =,,,m;, =,,..., ; < (g) z max z 0, =,,,m;, =,,..., ; < (h) w c +w c +...+w c =, () w c 0, =,,..., ( ) z 0, =,,,m;, =,,..., ; < () I model (4), costrat ( b ) esures that oly those optmal soluto(s) to the frst (4) stage model are feasble the secod stage model, ad costrat ( h ) fds z. The rest max of costrats are detcal to the costrats of the frst stage model. The two-stage lear programmg model s straghtforward ad easy to uderstad ad formulate, ad t ca be solved very lttle computatoal tme usg readly avalable software such as LINGO. 5. Numercal aalyss I ths secto, we use three umercal examples to llustrate our two-stage lear programmg model, ad the we mae a comparatve aalyss to show the fluece of self-cofdece levels o the group decso mag results. 5. Example We cosder the followg example, whch cludes three decso maers e ( =,,3) ad four alteratves x ( =,,3,4). Suppose that the mportace degree of each decso maer s equal, c = /3, =,,3. The decso maer e provdes hs/her preferece formato by the multplcatve preferece relato wth self-cofdece * A, the decso maer e provdes hs/her preferece formato by the addtve preferece relato wth self-cofdece * P, the decso maer e 3 provdes hs/her preferece formato by the ordal -tuple lgustc preferece relato wth self-cofdece These preferece relatos wth self-cofdece satsfy the wea stochastc trastvty at the self-cofdece level l. 0 3 T.

14 A P T (, l8) (, l5) (, l6) (4, l3) 3 (, l ) (, l ) (, l ) (, l ) = 3 4 (3, l6) (3, l4) (, l8) (7, l6) (, l3) (4, l) (, l6) (, l8) 4 7 * (0.5, l8) (0.5, l5) (0.6, l) (0.9, l3) (0.5, l ) (0.5, l ) (0.8, l ) (0.6, l ) (0., l3) (0.4, l4) (0., l7) (0.5, l8) * = (0.4, l ) (0., l6 ) (0.5, l8 ) (0.8, l7 ) ( s4, l8) ( s, l5) ( s, l) ( s4, l4) ( s, l ) ( s, l ) ( s, l ) ( s, l ) ( s4, l4) ( s4, l7) ( s5, l0) ( s4, l8) = ( s7, l ) ( s, l3 ) ( s4, l8 ) ( s3, l0 ) Based o the frst stage model (3), we ca obta the value of obectve fucto z =.7. Based o the secod stage model (4), we ca obta z = 0.7 ad the collectve c T preferece vector w = (0.6,0.6,0.4,0.06). 5. Example We cosder the secod example, whch cludes four decso maers e ( =,,3,4) ad three alteratves x ( =,,3). Suppose that the mportace degree of each decso maer s equal, c = /4, =,,3, 4. The decso maer e provdes hs/her preferece formato by the multplcatve preferece relato wth self-cofdece * A, the decso maers e ad e 3 provde ther preferece formato by the addtve preferece relatos wth self-cofdece * P ad P, the decso maer e 4 provdes hs/her preferece formato by the ordal -tuple lgustc preferece relato wth self-cofdece The correspodg matrces are gve as follows, *4 T. 4

15 (, l8) (, l6) (3, l7) = (, ) (, ) (, ) 4 (, l7) (4, l5) (, l8) 3 * A l6 l8 l5 (0.5, l ) (0.3, l ) (0.4, l ) = (0.7, ) (0.5, ) (0., ) (0.6, l4) (0.8, l) (0.5, l8) * P l5 l8 l (0.5, l ) (0.4, l ) (0.9, l ) = (0.6, ) (0.5, ) (0.5, ) (0., l5) (0.5, l0) (0.5, l8) P l6 l8 l0 ( s, l ) ( s, l ) ( s, l ) = (, ) (, ) (, ) ( s0, l4) ( s4, l4) ( s4, l8) *4 T s5 l6 s4 l8 s4 l4 Based o the frst stage model (3), we ca obta the value of obectve fucto z =.87. Based o the secod stage model (4), we ca obta z =.3 ad the c T collectve preferece vector w = (0.34,0.54,0.). 5.3 Example 3 The thrd example cludes sx decso maers e ( =,,..., 6) ad fve alteratves x ( =,,..., 5). Suppose that c = /6, =,,..., 6. The decso maers e ad e provde the multplcatve preferece relatos wth self-cofdece * A ad * A, the decso maers e ad e 3 provde the addtve preferece relatos wth self-cofdece P ad *4 P, the decso maers e 5 ad e 6 provde the ordal -tuple lgustc preferece relatos wth self-cofdece *5 T ad *6 T. 5

16 A A * * (, l8) (, l7) (4, l5) (5, l) (5, l0) (, l7) (, l8) (4, l) (7, l4) (3, l) (, l5) (, l) (, l8) (, l) (, l4) = 4 4 (, l) (, l4) (, l) (, l8) (, l3) 5 7 (, l0) (, l) (, l4) (, l3) (, l8) 5 3 (, l8) (, l) (, l4) (3, l) (4, l) (, l) (, l8) (3, l3) (3, l5) (3, l) (, l4) (, l3) (, l8) (, l) (, l4) = 3 (, l) (, l5) (, l) (, l8) (, l) 3 3 (, l) (, l) (, l4) (, l) (, l8) 4 3 (0.5, l8) (0.53, l3) (0.56, l3) (0.56, l) (0.6, l) (0.47, l ) (0.5, l ) (0.8, l ) (0.6, l ) (0.8, l ) (0.44, ) (0., ) (0.5, ) (0.6, ) (0.7, ) (0.44, l) (0.4, l) (0.4, l3) (0.5, l8) (0.6, l0) (0.4, l) (0., l4) (0.3, l7) (0.4, l0) (0.5, l8) P = l3 l5 l8 l3 l7 (0.5, l8) (0.7, l4) (0.75, l3) (0.95, l) (0.6, l3) (0.3, l ) (0.5, l ) (0.55, l ) (0.8, l ) (0.5, l ) (0.5, ) (0.45, ) (0.5, ) (0.7, ) (0.6, ) (0.05, l) (0., l) (0.3, l5) (0.5, l8) (0.85, l3) (0.4, l3) (0.5, l5) (0.4, l5) (0.5, l3) (0.5, l8) *4 P = l3 l l8 l5 l5 ( s4, l8) ( s6, l6) ( s8, l) ( s7, l5) ( s5, l8) ( s, l ) ( s, l ) ( s, l ) ( s, l ) ( s, l ) = (, ) (, ) (, ) (, ) (, ) ( s, l5) ( s, l3) ( s0, l7) ( s4, l8) ( s6, l) ( s3, l8) ( s, l4) ( s, l) ( s, l) ( s4, l8) *5 T s0 l s l s4 l8 s8 l7 s7 l ( s4, l8) ( s7, l) ( s7, l4) ( s5, l5) ( s6, l3) ( s, l ) ( s, l ) ( s, l ) ( s, l ) ( s, l ) = (, ) (, ) (, ) (, ) (, ) ( s3, l5) ( s, l6) ( s3, l3) ( s4, l8) ( s6, l) ( s, l3) ( s0, l) ( s3, l3) ( s, l) ( s4, l8) *6 T s l4 s3 l7 s4 l8 s5 l3 s5 l3 6

17 Based o the frst stage model (3), we ca obta the value of obectve fucto z = Based o the secod stage model (4), we ca obta z = 0.50 ad the collectve c T preferece vector w = (0.40,0.34,0.0,0.05,0.0). 5.4 Comparatve aalyss I ths subsecto, we study the fluece of dfferet self-cofdece levels o the GDM results. Cosder the followg sx matrces * A, * P, T, * A, * P ad T. The matrces * A ad * A have the same preferece values but dfferet self-cofdece levels wth matrx * A Example. The matrces * P ad * P have the same preferece values but dfferet self-cofdece levels wth matrx * P Example. The matrces T ad T have the same preferece values but dfferet self-cofdece levels wth matrx Example. The correspodg matrces are gve as follows, T A P T (, l8) (, l4) (, l6) (4, l7) 3 (, l ) (, l ) (, l ) (, l ) = 3 4 (3, l6) (3, l) (, l8) (7, l0) (, l7) (4, l3) (, l0) (, l8) 4 7 * (0.5, l8) (0.5, l4) (0.6, l) (0.9, l5) (0.5, l ) (0.5, l ) (0.8, l ) (0.6, l ) (0., l5) (0.4, l6) (0., l3) (0.5, l8) * = (0.4, l ) (0., l6 ) (0.5, l8 ) (0.8, l3 ) ( s4, l8) ( s, l4) ( s, l7) ( s4, l) ( s, l ) ( s, l ) ( s, l ) ( s, l ) ( s4, l) ( s4, l) ( s5, l6) ( s4, l8) = ( s7, l7 ) ( s, l3 ) ( s4, l8 ) ( s3, l6 ) A P (, l8) (, l8) (, l8) (4, l8) 3 (, l ) (, l ) (, l ) (, l ) = 3 4 (3, l8) (3, l8) (, l8) (7, l8) (, l8) (4, l8) (, l8) (, l8) 4 7 * (0.5, l8) (0.5, l8) (0.6, l8) (0.9, l8) (0.5, l ) (0.5, l ) (0.8, l ) (0.6, l ) (0., l8) (0.4, l8) (0., l8) (0.5, l8) * = (0.4, l8 ) (0., l8 ) (0.5, l8 ) (0.8, l8 ) T ( s4, l8) ( s, l8) ( s, l8) ( s4, l8) ( s, l ) ( s, l ) ( s, l ) ( s, l ) ( s4, l8) ( s4, l8) ( s5, l8) ( s4, l8) = ( s7, l8 ) ( s, l8 ) ( s4, l8 ) ( s3, l8 ) The sx matrces cota two groups: the oe group s the matrces * A, * P ad T, the other group s the matrces * A, * P ad T. Usg the two-stage lear programmg model obtas the GDM results for each group. The value of obectve fucto z, the value 7

18 of obectve fucto z ad the collectve preferece vectors of three groups for Example are preseted the Table. Table The comparso results regardg the three groups for Example z z c w * * ( A, P, T ) (0.6,0.6,0.4,0.06) T ( A, P, T ) (0.40,0.30,0.0,0.0) T * * ( A, P, T ) (0.46,0.4,0.6,0.04) T * * Furthermore, we mae the equvalet comparatve aalyss for the GDM of Example. Cosder the followg eght matrces * A, * P, P, *4 T, * A, * P, P ad *4 T. The correspodg matrces are gve as follows, (, l8) (, l7) (3, l) * A = (, l7) (, l8) (, l6) 4 (, l) (4, l6) (, l8) 3 (0.5, l8) (0.3, l) (0.4, l5) * P = (0.7, l) (0.5, l8) (0., l4) (0.6, l5) (0.8, l4) (0.5, l8) (0.5, l8) (0., l3) (0.6, l3) P = (0.8, l3) (0.5, l8) (0.7, l6) (0.4, l3) (0.3, l6) (0.5, l8) ( s4, l8) ( s3, l5) ( s8, l) *4 T = ( s5, l5) ( s4, l8) ( s4, l8) ( s0, l) ( s4, l8) ( s4, l8) (, l8) (, l8) (3, l8) * A = (, l8) (, l8) (, l8) 4 (, l8) (4, l8) (, l8) 3 (0.5, l8) (0.3, l8) (0.4, l8) * P = (0.7, l8) (0.5, l8) (0., l8) (0.6, l8) (0.8, l8) (0.5, l8) (0.5, l8) (0., l8) (0.6, l8) P = (0.8, l8) (0.5, l8) (0.7, l8) (0.4, l8) (0.3, l8) (0.5, l8) ( s4, l8) ( s3, l8) ( s8, l8) *4 T = ( s5, l8) ( s4, l8) ( s4, l8) ( s0, l8) ( s4, l8) ( s4, l8) The eght matrces cota two groups: the oe group s the matrces * A, * P, P ad *4 T, the other group s the matrces * A, * P, P ad *4 T. The comparso results regardg three groups for Example are preseted the Table. 8

19 Table The comparso results regardg the three groups for Example z z c w * * *4 ( A, P, P, T ).87.3 (0.34,0.54,0.) T ( A, P, P, T ) (0.3,0.6,0.5) T * * *4 ( A, P, P, T ) (0.0,0.40,0.40) T * * *4 Fally, we mae the equvalet comparatve aalyss for the GDM of Example 3. Cosder the followg twelve matrces whch have the same preferece values but dfferet self-cofdece levels wth the correspodg matrces of Example 3. A * A A * * (, l8) (, l8) (4, l5) (5, l6) (5, l7) (, l8) (, l8) (4, l5) (7, l4) (3, l6) (, l5) (, l5) (, l8) (, l7) (, l) = 4 4 (, l6) (, l4) (, l7) (, l8) (, l) 5 7 (, l7) (, l6) (, l) (, l) (, l8) 5 3 (, l8) (, l8) (4, l8) (5, l8) (5, l8) (, l8) (, l8) (4, l8) (7, l8) (3, l8) (, l8) (, l8) (, l8) (, l8) (, l8) = 4 4 (, l8) (, l8) (, l8) (, l8) (, l8) 5 7 (, l8) (, l8) (, l8) (, l8) (, l8) 5 3 (, l8) (, l7) (, l7) (3, l4) (4, l6) (, l7) (, l8) (3, l5) (3, l8) (3, l6) (, l7) (, l5) (, l8) (, l7) (, l5) = 3 (, l4) (, l8) (, l7) (, l8) (, l8) 3 3 (, l6) (, l6) (, l5) (, l8) (, l8) 4 3 9

20 A * (, l8) (, l8) (, l8) (3, l8) (4, l8) (, l8) (, l8) (3, l8) (3, l8) (3, l8) (, l8) (, l8) (, l8) (, l8) (, l8) = 3 (, l8) (, l8) (, l8) (, l8) (, l8) 3 3 (, l8) (, l8) (, l8) (, l8) (, l8) 4 3 (0.5, l8) (0.53, l7) (0.56, l8) (0.56, l6) (0.6, l8) (0.47, l7) (0.5, l8) (0.8, l5) (0.6, l5) (0.8, l8) P = (0.44, l8) (0., l5) (0.5, l8) (0.6, l6) (0.7, l) (0.44, l6) (0.4, l5) (0.4, l6) (0.5, l8) (0.6, l6) (0.4, l8) (0., l8) (0.3, l) (0.4, l6) (0.5, l8) (0.5, l8) (0.53, l8) (0.56, l8) (0.56, l8) (0.6, l8) (0.47, l8) (0.5, l8) (0.8, l8) (0.6, l8) (0.8, l8) P = (0.44, l8) (0., l8) (0.5, l8) (0.6, l8) (0.7, l8) (0.44, l8) (0.4, l8) (0.4, l8) (0.5, l8) (0.6, l8) (0.4, l8) (0., l8) (0.3, l8) (0.4, l8) (0.5, l8) (0.5, l8) (0.7, l4) (0.75, l5) (0.95, l5) (0.6, l6) (0.3, l4) (0.5, l8) (0.55, l8) (0.8, l7) (0.5, l7) *4 P = (0.5, l5) (0.45, l8) (0.5, l8) (0.7, l7) (0.6, l6) (0.05, l5) (0., l7) (0.3, l7) (0.5, l8) (0.85, l5) (0.4, l6) (0.5, l7) (0.4, l6) (0.5, l5) (0.5, l8) (0.5, l8) (0.7, l8) (0.75, l8) (0.95, l8) (0.6, l8) (0.3, l8) (0.5, l8) (0.55, l8) (0.8, l8) (0.5, l8) *4 P = (0.5, l8) (0.45, l8) (0.5, l8) (0.7, l8) (0.6, l8) (0.05, l8) (0., l8) (0.3, l8) (0.5, l8) (0.85, l8) (0.4, l8) (0.5, l8) (0.4, l8) (0.5, l8) (0.5, l8) ( s4, l8) ( s6, l6) ( s8, l8) ( s7, l7) ( s5, l6) ( s, l6) ( s4, l8) ( s6, l5) ( s7, l5) ( s7, l8) *5 T = ( s0, l8) ( s, l5) ( s4, l8) ( s8, l3) ( s7, l5) ( s, l7) ( s, l5) ( s0, l3) ( s4, l8) ( s6, l6) ( s3, l6) ( s, l8) ( s, l5) ( s, l6) ( s4, l8) ( s4, l8) ( s6, l8) ( s8, l8) ( s7, l8) ( s5, l8) ( s, l8) ( s4, l8) ( s6, l8) ( s7, l8) ( s7, l8) *5 T = ( s0, l8) ( s, l8) ( s4, l8) ( s8, l8) ( s7, l8) ( s, l8) ( s, l8) ( s0, l8) ( s4, l8) ( s6, l8) ( s3, l8) ( s, l8) ( s, l8) ( s, l8) ( s4, l8) 0

21 ( s4, l8) ( s7, l7) ( s7, l7) ( s5, l6) ( s6, l5) ( s, l7) ( s4, l8) ( s5, l8) ( s7, l4) ( s8, l3) *6 T = ( s, l7) ( s3, l8) ( s4, l8) ( s5, l5) ( s5, l8) ( s3, l6) ( s, l4) ( s3, l5) ( s4, l8) ( s6, l7) ( s, l5) ( s0, l3) ( s3, l8) ( s, l7) ( s4, l8) ( s4, l8) ( s7, l8) ( s7, l8) ( s5, l8) ( s6, l8) ( s, l8) ( s4, l8) ( s5, l8) ( s7, l8) ( s8, l8) *6 T = ( s, l8) ( s3, l8) ( s4, l8) ( s5, l8) ( s5, l8) ( s3, l8) ( s, l8) ( s3, l8) ( s4, l8) ( s6, l8) ( s, l8) ( s0, l8) ( s3, l8) ( s, l8) ( s4, l8) The twelve matrces also cota two groups: the oe group s the matrces * A, * A, P, *4 P, *5 T ad *6 T, the other group s the matrces * A, * A, P, *4 P, *5 T ad *6 T. The comparso results regardg the three groups for Example 3 are preseted the Table 3. Table 3 The comparso results regardg the three groups for Example 3 z z c w * * *4 *5 *6 ( A, A, P, P, T, T ) (0.40,0.34,0.0,0.05,0.0) T ( A, A, P, P, T, T ) (0.3,0.30,0.0,0.0,0.08) T * * *4 *5 *6 ( A, A, P, P, T, T ) (0.36,0.30,0.8,0.09,0.07) T * * *4 *5 *6 From Table, Table ad Table 3, we otce that dfferet self-cofdece levels lead to dfferet collectve preferece vectors. Thus the self-cofdece levels have certa fluece o the GDM results. 6. Coclusos I ths study, we defe a ew d of preferece relato, called preferece relato wth self-cofdece. The, we preset a two-stage lear programmg model to deal wth the GDM problem based o heterogeeous preferece relatos wth self-cofdece. The ma cotrbutos preseted are as follows: () Ths study defes the preferece relatos wth self-cofdece, whch allow decso maers to have multple self-cofdece levels to express ther prefereces regardg pars of alteratves.

22 () We propose a two-stage lear programmg model for estmatg the collectve preferece vector the GDM based o heterogeeous preferece relatos wth self-cofdece. Fally, a comparso study s coducted to demostrate the fluece of self-cofdece levels o the GDM results. It wll be a terestg future research to fd out possble relatoshps amog prefereces, self-cofdece assessmets ad results. Meawhle, the cosesus problem s a hot topc GDM [, 7, 0, 7, 3, 37], ad t wll be terestg to vestgate the cosesus reachg model GDM based o heterogeeous preferece relatos wth self-cofdece. Acowledgmets Ths wor was supported part by NSF of Cha uder Grats Nos ad 7574, FEDER fuds uder Grat TIN P, ad the Adalusa Excellece Proect Grat TIC-599. Refereces [] S. Aloso, F. Chclaa, F. Herrera, E. Herrera-Vedma, J. Alcalá-Fdez, C. Porcel. A cosstecy-based procedure to estmate mssg parwse preferece values. Iteratoal Joural of Itellget Systems 3 (008) [] S. Aloso, I. J. Pérez, F. J. Cabrerzo, E. Herrera-Vedma. A Lgustc Cosesus Model for Web.0 Commutes. Appled Soft Computg 3 () (03) [3] X. Che, H. J. Zhag, Y. C. Dog. The fuso process wth heterogeeous preferece structures group decso mag: A survey. Iformato Fuso 4 (05) [4] F. Chclaa, F. Herrera, E. Herrera-Vedma. Itegratg three represetato models fuzzy multpurpose decso mag based o fuzzy preferece relatos. Fuzzy Sets ad Systems 97 (998) [5] F. Chclaa, F. Herrera, E. Herrera-Vedma. Itegratg multplcatve preferece relatos a multpurpose decso-mag model based o fuzzy preferece relatos. Fuzzy Sets ad Systems (00) [6] F. Chclaa, E. Herrera-Vedma, S. Aloso, F. Herrera. Cardal cosstecy of recprocal preferece relatos: A Characterzato of multplcatve trastvty. IEEE Trasactos o Fuzzy Systems 7 () (009) 4-3.

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