A LINGUISTIC-VALUED WEIGHTED AGGREGATION OPERATOR TO MULTIPLE ATTRIBUTE GROUP DECISION MAKING WITH QUANTITATIVE AND QUALITATIVE INFORMATION

Transcription

1 A LINGUISTIC-VALUED WEIGHTED AGGREGATION OPERATOR TO MULTIPLE ATTRIBUTE GROUP DECISION MAKING WITH QUANTITATIVE AND QUALITATIVE INFORMATION XIAOBING LI Itellget Cotrol Developmet Ceter Southwest Jaotog Uversty Chegdu 6003 P.R. Cha DA RUAN Belga Nuclear Research Cetre (SCK CEN 400 Mol & Ghet Uversty 9000 Get Belgum JUN LIU School of Computg ad Mathematcs Faculty of Egeerg Uversty of Ulster Jordastow Newtowabbey BT37 0QB Norther Irelad UK YANG XU Itellget Cotrol Developmet Ceter Southwest Jaotog Uversty Chegdu 6003 P.R. Cha I selectg a optoal alteratve a evromet of multple attrbute group decso makg dfferet attrbutes of the alteratve are ofte cosdered as wth quattatve ad qualtatve formato. Cosequetly decso makg problems may clude preferece formato dfferet formats. I ths paper a lattce-based lgustc-valued weghted aggregato (LVWA operator s proposed for multple attrbute group decso makg wth o-totally ordered lgustc-valued formato. The some trasformato fuctos for ufyg dfferet formats of preferece formato are revewed ad summarzed. Fally a example s llustrated how to use the LVWA operator ad trasformato fuctos for multple attrbute group decso makg. Keywords: Trasformato fuctos multple attrbute group decso makg lgustc-valued weghted aggregato (LVWA operator lgustc-valued lattce mplcato algebra lattce theory. Itroducto Multple attrbute group decso makg (MAGDM addresses the problems of choosg a optmal choce that has the hghest degree of satsfacto by multple experts assessmets from a set of alteratves that are characterzed terms of ther attrbutes. Geerally multple attrbute group decso makg problems follow a commo scheme composed by the three phases: ( Evaluato phase: Experts are asked to gve preferece values to each attrbute of each alteratve. ( Aggregato phase: It combes dvdual preferece values to obta a collectve preferece value for each alteratve. (3 Explotato phase: It orders the collectve preferece values to obta the best alteratves. I the frst phase experts are asked to provde ther prefereces o each attrbute of each alteratve. Usually the formato s expressed by meas of umercal values such as exact values terval values fuzzy umbers etc. However real world huma begs are costatly makg decsos uder a lgustc evromet. For example whe evaluatg the comfort or desg of a car lgustc labels lke good far poor are usually used; evaluatg a 74

2 Xaobg L Da Rua Ju Lu Yag Xu the speed of a car lgustc labels lke very fast fast slow ca be used ad evaluatg studets performaces ther courses lgustc labels lke bad medum good ca be used. As a result t s ecessary to cosder aggregatos of lgustc formato. To date several methods have bee proposed for dealg wth lgustc formato for stace: ( The exteso prcple based method for operatos o fuzzy umbers that support the sematcs of the lgustc labels 4 5. ( The symbolc method for computatos o the dexes of the lgustc terms 6 ; both the methods ( ad ( process the results the tal expresso domas whch produce the cosequet loss of formato ad hece the lack of precso 7. (3 A fuzzy lgustc model based method for the lgustc formato wth a par of values called - tuple composed by a lgustc term ad a umber 7-. Alog wth the model ths method deals wth the - tuple wthout loss of formato. (4 The drect computg wth words method -3. I ths paper we follow the 4th method to aggregate lgustc-valued formato for group decso makg. At preset a umber of researches have focused o group decso makg wth lgustc preferece. Herrera et al. developed a cosesus model for group decso makg uder lgustc assessmets 7 ad combed the lgustc ordered weghted averagg (LOWA operator wth lgustc preferece relatos ad the cocept of domace ad o-domace to show ts use the feld of group decso makg based o the LOWA operator 8. Later Herrera et al. preseted a cosesus model complete lgustc framework for group decso makg guded by cosstecy ad cosesus measures 9. Z.S. Xu proposed a ucerta lgustc ordered weghted averagg (ULOWA aggregato operator ad ucerta lgustc hybrd aggregato (ULHA operator ad developed a approach to multple attrbute group decso makg wth ucerta lgustc formato based o the ULOWA ad ULHA operators 0. Although there are may aggregato operators to aggregate lgustc formato they ca oly aggregate learly ordered lgustc formato. Note that there exst comparable lgustc terms such as slghtly false ad very true. So t s ecessary to fd a algebra for modelg the orderg relato of the atural laguage terms. Lattce theory s a well-developed brach of a abstract algebra for modelg the orderg relato the real world. Lattce-valued algebra for modelg lgustc values would be a possble choce. To establsh theores ad methods to smultaeously deal wth fuzzess ad comparablty of processed obect tself ad ucertaty the course of formato processg Xu combed a lattce wth mplcato algebra ad establshed the lattce mplcato algebra 4 whch provdes a ecessary foudato for the processg of comparable formato. I addto there are some research works o comparable formato processg. A evaluato method wth comparable formato s preseted Ref. 3. Lattce-valued lgustc-based decso makg method s dscussed Ref.. A model for hadlg lgustc terms the framework of lattcevalued logc s preseted Ref. 4. I Ref. 30 the LVWA operator based o lgustc-valued lattce mplcato algebra s preseted. I ths paper based o the LVWA operator a approach to solve multple attrbute group decso makg wth comparable lgustc-valued formato s establshed. I Ref. a ew method for sesory evaluato of dustral products wth ucerta formato s preseted. I ths approach sesory data provded by dfferet evaluators are trasformed to measures of cosstecy o fuzzy satsfacto degrees. Based o these measures of cosstecy the aggregated formato for all evaluators ad all attrbutes ad measure the dssmlarty betwee evaluators ad betwee used evaluatos attrbutes s obtaed. The effectveess of ths method has bee valdated the fabrc had evaluato for a umber of samples of ktted cotto. O the other had multple attrbute group decso makg dfferet types of attrbutes ether quattatve or qualtatve eed to be cosdered. Therefore the decso makg problems may clude may dfferet types of preferece formato such as umber terval ad lgustc values. I order to deal wth these preferece formato dfferet formats some researches have bee doe 6-9. Ths paper also ams at developg a ew method for ufyg 75

3 A lgustc-valued weghted aggregato operator preferece formato dfferet formats to the format for lgustc-valued formato. The paper s orgazed as follows: Secto brefly gves basc deftos of lattce mplcato algebra ad lgustc-valued lattce mplcato algebra. Secto 3 troduces the LVWA operator ad dscusses ts propertes. Secto 4 studes trasformato fuctos for ufyg preferece formato dfferet formats. Secto 5 proposes a approach for multple attrbute group decso makg based o the LVWA operator wth a lgustc-valued lattce mplcato algebra preferece set. Secto 6 llustrates how to use the proposed approach. The paper s cocluded Secto 7.. Prelmares I ths secto we recall some basc cocepts about lattce mplcato algebra 4 ad lgustc truth-valued lattce mplcato algebra 5. For some detals of lattce mplcato algebra we refer to Ref Lattce Implcato Algebra Defto. Let ( L ' be a bouded lattce wth a order-reversg voluto ad the uversal bouds O I : L L L be a mappg. ( L ' OI s called a lattce mplcato algebra f the followg axoms hold for all x yz L: (I x ( y z = y ( x z ; (I x x = I ; (I 3 x y = y x ; (I 4 x y = y x = I mples x = y ; (I 5 ( x y y = ( y x x. (I 6 ( x y z = ( x z ( y z ; (I 7 ( x y z = ( x z ( y z ; Theorem. Let L be a lattce mplcato algebra. The for ay x yz L: ( If y z the x y x z ; ( If x y the x z y z ; (3 O x = I ; (4 I x = x; (5 x y f ad oly f x y = I ; (6 x y x y. Example. (Boolea algebra Let ( L be a Boolea lattce. For ay x y L defe x y = x y. The ( L ' s a lattce mplcato algebra. Example. (Łukasewcz mplcato algebra o fte chas Cosder a set L = { a = L }. For ay k defe a ak = amax{ k} a ak = am{ k} = a + a ak = am{ + k }. The ( L s a lattce mplcato algebra. I the followg sectos the lattce mplcato algebra ( L s deoted by L smply uless emphaszed... Lgustc-valued lattce mplcato algebra Defto. Let ( L I O ( = L be a famly of lattce mplcato algebras. The L= L = { a L a a L} = s called a drect product of lattce mplcato algebras. Theorem. Let L ( = L be a lattce mplcato algebra. If the operators o L= L = are defed as follows respectvely: for ay a L a ( b b L b L = a L a ( b b L b = b a b L a b a L a ( b b L b = b a b L a b a L a ( b b L b = b a b L a b a L a = a L a the ( L s also a lattce mplcato algebra. Remark. Let L = L L where L ( = be a fte-cha-type lattce mplcato algebra. The L s a lattce mplcato algebra. Defto.3 Let ML={ b b } be a lgustc term set where b be a atoym of b ad b b term of ther meags atural laguage such as poor ad good false ad true etc. Defe the same 76

4 Xaobg L Da Rua Ju Lu Yag Xu operators o ML as the oes Example.. The ML s a lattce mplcato algebra called a meta-lgustc lattce mplcato algebra. Example.3 Let ML={good poor}. The operators o ML are defed as the same Example.. The ML s a meta-lgustc lattce mplcato algebra. Defto.4 Let MW= { a = L } ad a ( = L be lgustc modfers used to modfy the meta-lgustc terms. The set MW s ordered the sese that a a f ad oly f. The operators o MW are defed as the same Example. the (MW a a s a lattce mplcato algebra called a lattce mplcato algebra wth modfers. Example.4 Let MW= {absolutely (Abbr. to Ab hghly (Abbr. to H very (Abbr. to Ve qute (Abbr. to Qu exactly (Abbr. to Ex almost (Abbr. to Al rather (Abbr. to Ra somewhat (Abbr. to So slghtly (Abbr. to Sl} be a set of lgustc modfers. The the cha Ab H Ve Qu Ex Al Ra So Sl s a lgustc-modfer lattce mplcato algebra wth operatos defed as Example.. Defto.5 Let MW= { a a L a } be a lattce mplcato algebra wth modfers ML={ b b } be a meta lattce mplcato algebra deote L = { b L b b L b } whch Hasse dagram s show as Fg.. The operatos o L s defed as follows: b = b b b = a b b. Defe a mappg f : f : MW ML L b b = b f( b =. b b = b The f s a somorphc mappg. k l k l a b a b Fgure. Hasse Dagram of L. Defe the operatos o MW ML as follows: b b = f ( f b f b k l k l b b = f ( f b f b k l k l b b = f ( f b f b b = f (( f b. a b a b M 3 a b a b k l k l The MW ML s called lgustc-valued lattce mplcato algebra whch Hasse dagram s show Fg.. a b a b a b M a b 3 a b a b a b a b a b a b 3 a b a b a b a b a b 3 M a b a b a b Fgure. Hasse Dagram of MW ML. Example.5 Let MW= {absolutely hghly very qute exactly almost rather somewhat slghtly} be a set of lgustc modfers ad ML={good poor}. So MW ML={absolutely good hghly good very good qute good exactly good almost good rather good somewhat good slghtly good absolutely poor hghly poor very poor qute poor exactly poor almost poor rather poor somewhat poor slghtly poor}. The M 77

5 A lgustc-valued weghted aggregato operator (MW ML O (.e. slghtly poor I (.e. absolutely good s a lgustc-valued lattce mplcato algebra. I the followg secto we wll use ths lgustc-valued lattce mplcato algebra as a lgustc assessmet set to represet the preferece or the mportat weght deoted shortly as S. 3. A lgustc-valued aggregato operator for multple attrbute group decso makg Yager troduced a ordered weghted averagg (OWA operator defed as follows 5. Defto 3. A OWA operator of dmeso s a mappg OWA: R R that has assocated a vector w= ( w w L w such that w [0] = L ad w = =. Furthermore OWA w a L a = wb = where b s the th largest of the a. However the OWA operator ca oly be used the stuatos where the put argumets are the exact umercal values. I the real world huma begs are costatly makg decsos uder a lgustc evromet. Hece t s ecessary to vestgate lgustc-valued formato aggregato. Remark 3.: There have bee some exstg works o lgustc-valued formato aggregato cludg Yager s work such as Refs. 4 5 but they are all based o the totally ordered lgustc term set. I the followg we shall vestgate a lgustcvalued weghted aggregato operator whch ca be used stuatos where the aggregated argumets are gve the form of lgustc values whch may be comparable. Defto 3. A mappg LVWA: S S s called a lgustc-valued weghted aggregato (LVWA operator f LVWA w a L a = ( w a = where S s a evaluato set whch s a lgustcvalued lattce mplcato algebra ad cludes both comparable ad comparable lgustc terms commoly used atural laguage where w= ( w w L w s a weght vector ad w s the weght of lgustc-valued a wth w S a S ad = L. Remark 3.: Yager s aggregato method Ref. 3 s a specal case of the proposed method whch s lmted to the totally ordered lgustc term set. The LVWA operator has the followg propertes: Theorem 3. (Mootocty Let A= a L a ad C = ( c c L c be argumet vectors. If for each ( = L a c the LVWA w( A LVWA w ( C. Proof. Sce LVWA w ( A = ( w a ad = LVWA w ( C = ( w c = the result follows drectly from the property a c. Theorem 3. (Commutatvty Let A = a L a be a ordered argumet vector A = a L a s ay permutato of the elemets A the LVOWA w ( A = LVOWA w ( A. Proof. Suppose that w= ( w w L w s the weghtg vector of lgustc-valued a ( = L. The ( w a = ( w a. = LVWA w a L a = ; LVWA w a L a = Hece LVOWA w ( A = LVOWA w ( A. Theorem 3.3 (Idempotece If w = I ad a = a = ( = L the LVWA w a L a = a. Proof. Sce a = a t follows that LVWA w a L a = ( w a = = ( w a = ( w a= I a = a. = = 78

6 Xaobg L Da Rua Ju Lu Yag Xu Theorem 3.4 Let w= ( I I L I. The LVWA w a L a = f [ a ]. 4. Trasformato schemes for ufyg dfferet formats of preferece formato To obta evaluato results of all the alteratves multple attrbute group decso makg dfferet formats of preferece formato eed to be ufed to a commo format. The lgustc-valued preferece formato set S s chose as the commo format. The ufyg steps are gve as follows: Step : ormalzato of quattatve preferece formato Step : ormalzato of qualtatve preferece formato Step 3: trasformato for umercal preferece formato to lgustc-valued preferece formato. The cocrete trasformato methods wll be gve the followg subsectos. 4.. Normalzato of quattatve preferece formato Geerally there exst sx kds of attrbutes: proft cost fxato terval devato ad devatg terval. I Refs. 7-9 the methods of ormalzg the above sx kds of attrbutes are gve ad expressed as follows: a Proft attrbute r = max{ a } m{ a } Cost attrbute r = a a α Fxato attrbute r = max{ a α } Iterval attrbute r = a β max a m a β m a β β Devato attrbute max( q a a q a [ q q] r max[ m( max( ] = q a a q a [ q q] Devatg terval attrbute max( q a a q a q q max[ m( max( ] = q a a q a q q r [ ] 0 [ ] where a deotes the orgal value of attrbute A for alteratve X r represets the ormalzed value of a α ad β are fxed values [ q q ] s a terval. The larger the proft attrbute value the better the attrbute whle the larger the cost attrbute value the worse the attrbute. The fxato attrbute meas that the closer to a fxed value α attrbute value the better the attrbute. Further we ca kow that the closer to or cluded a terval [ q q ] values the better the attrbute. The larger of the dstace of devato attrbute values to a fxed value are the better of the attrbute s. Devato terval attrbute meas that the larger the dstace of devato attrbute values to a fxed terval the better the attrbute. Remark 4. Accordg to the above formula the orgal values ca be ormalzed wth the terval [0 ]. 4.. Normalzato of qualtatve preferece formato The lgustc values are desged to express preferece formato of qualtatve attrbutes by decso makers. I ths paper all lgustc values are selected from lgustc-valued lattce mplcato algebra S defed Example.5. Two kds of attrbutes proft ad cost are cosdered. The methods of qualtatve attrbutes are gve as follows: Proft attrbute: ths case as the preferece formato s expressed by lgustc values we keep the orgal values as the ormalzed values. Cost attrbute: r = a where a deotes the orgal value of attrbute A for alteratve X s a egato operator r represets the ormalzed value of a a ad r are all lgustc values S Trasformato fucto for umercal attrbute values to lgustc-valued attrbute values After the trasformato of the orgal attrbute values ormalzed attrbute values are expressed by umber 79

7 A lgustc-valued weghted aggregato operator terval [0 ] or lgustc values S. We eed to ufy these two kds of preferece formato. Sce the attrbute values belogg to the terval [0 ] are comparable we select the subset S = { s0 s L s } of S such that S oly cotas comparable lgustc values ad these lgustc values satsfy the followg codtos: A egato operator : s = s such that = ( + s the cardalty A m ad a max operator the lgustc term set: s s To aggregate the preferece formato a trasformato fucto for umercal attrbute values to lgustc-valued attrbute values s gve as follows: τ : [0] S τ = s[ a] where [ ] s the roudg operato + s the cardalty of S. Utlzg the trasformato fucto the umercal attrbute values ca be trasferred to lgustc-valued attrbute values. 5. A approach based o the LVWA operator to multple attrbute decso makg wth lgustc-valued formato Cosder a multple attrbute group decso makg problem wth dfferet formats of preferece formato. Assume that S s a evaluato set that s a lgustc-valued lattce mplcato algebra ad cludes both comparable ad comparable atural lgustc terms used to dcate preferece formato. Let X = { x x L x } be a dscrete set of alteratves ad U = { u u L u m } be a set of attrbutes. Let D = { d d L d l } be a set of decso makers ad w= ( w w L w l be the weght vector of decso makers where wk S k = L l. Suppose that ( k ( ( k A% = a m s the decso matrx where a ( k s a preferece value whch takes the forms of umber terval or lgustc value gve by the decso maker d k D for alteratve x X wth respect to attrbutes u U. Group decso makg problems are composed by the followg four phases: ( Evaluato phase: The experts are asked to gve the preferece values to each attrbute of each alteratve. ( Trasformato phase: All the preferece values are expressed a uque lgustc-valued doma. (3 Aggregato phase: It combes the dvdual prefereces to obta a collectve preferece value for each alteratve. (4 Explotato phase: It orders the collectve preferece values to obta the best alteratves. I the followg a approach to multple attrbute group decso makg wth lgustc-valued formato s gve based o the LVWA operator. Step : Experts gve preferece formato a% ( k = L m = L k = L l. Step : Utlze the decso formato gve matrx ( k A % ad the methods of trasformato Secto 4 to derve all the ormalzed lgustc values. Step 3: Utlze the LVWA operator: a % = LVWA ( k a L a ( k ( k ( k w m k = L l = L ( to derve the dvdual overall preferece value a% k of alteratve x where w= ( w w L w m s a weght vector ad w s the weght of lgustc-valued a wth w S = L l. Step 4: Utlze the LVWA operator: a%=lvwa a a L a = L ( ( ( l w( to derve the collectve overall preferece value a%of alteratve x where w= ( w w L w l s the weght vector of decso makers wth w S = L. Step 5: Rak all the alteratves x ad select the optmal oe(s accordg to a%. The optmal alteratve s x X that a% s maxmal. Step 6: Ed. 6. A llustratve example To llustrate how the proposed method works we wll gve a smple example 3 to evaluate the set of cars A = { x =Chevrolet x =Buck x 3 = Toyota}. Let U = { u u u3 u4} where u = comfort u = repar frequecy = cost = maxmum speed (whose vector weghts be w= (79086 ad values of attrbutes u ad u are lgustc values whle values of attrbutes ad are umercal values. Three kds of cars (alteratves x ( = 3 are to be evaluated usg the term set 80

8 Xaobg L Da Rua Ju Lu Yag Xu S = { 9 = absolutely good 8 =hghly good 7 =very good 6 =qute good 5 =exactly good 4 =almost good 3 =rather good =somewhat good =slghtly good 90 =slghtly poor 80 =somewhat poor 70 =rather poor 60 =almost poor 50 =exactly poor 40 =qute poor 30 =very poor 0 =hghly poor 0 =absolutely poor} by four decso makers dk ( k = 3 4 (whose weght vector ω = ( uder these four attrbutes as lsted Tables -4 respectvely. Step : Ufy the attrbute values to a lgustc values Step.: Utlze the trasformato fuctos gve Secto 4. a m{ a } r = ad r = max{ a } a to derve the ormalzed preferece formato of attrbute ad u4 respectvely. Step.: Utlze the trasformato fuctos gve Secto 4. to derve the ormalzed preferece formato of attrbute u ad u respectvely. Table. Preferece formato gve by decso maker d Step.3: Utlze the trasformato fucto gve Secto 4.3 to ufy all the preferece formato. We chose the S = { s 9 = 9 = absolutely good s 8 = 8 =hghly good s 7 = 7 =very good s 6 = 6 =qute good s 5 = 5 =exactly good s 4 = 50 =exactly poor s 3 = 40 =qute poor s = 30 =very poor s = 0 =hghly poor s 0 = 0 =absolutely poor}. It s obvous that S s a lear order. After the above three steps we ca get Tables 5-8. Step : Utlze the preferece formato gve Table 5 ad the LVWA operator (Let w= (79086 ( k ( ( ( ( a% =LVWA % k a% k a% k a% k w 3 4 k = 3 4 = 3 to derve the dvdual overall ( k preferece value a% of the alteratve x : ( a% =LVWA w( = (7 9 ((90 7 (8 7 (6 8 = 7 Smlarly we have a% = 8 a% = 6 a% = 6 a% = 8 0 ( ( 3 ( ( Table. Preferece formato gve by decso maker d u u u x x u u u x x Table 3. Preferece formato gve by decso maker d 3 Table 4. Preferece formato gve by decso maker d 4 u u u x x u u u x x

9 A lgustc-valued weghted aggregato operator Table 5. Normalzed referece formato of Table Table 6. Normalzed referece formato of Table u u u x x u u u x x Table 7. Normalzed referece formato of Table 3 Table 8. Normalzed referece formato of Table 4 u u u x x u u u x x ( 3 (3 3 (3 a% = 6 0 a% = 8 a% = 7 0 a% = 6 a% = 8 a% = 7 a% = 6 0 (4 (3 (4 (4 3 Step 3: Utlze the weght vector of decso makers ω = ( ad the LVWA operator: ( ( (3 (4 a % = LVWA ω % a % a % a % ( = 3 to aggregate the dvdual overall preferece values ( k a% ( k = 34 ad thus get the collectve overall preferece value a% of alteratve x : ( ( (3 (4 a%= LVWA ω % a % a % a % = (50 8 (7 6 (80 8 (9 8 = 8 Smlarly we have a% = 7 0 a%= Step 4: Rak all the alteratves x ad select the optmal oe(s accordg wth a%. The optmal alteratve s x X that a% s maxmal. Thus the optmal oe s x that s Chevrolet. 6. Coclusos I ths paper a lgustc-valued weghted aggregato operator was proposed whch ca be used the stuatos where the evaluato value set s a ototally ordered lgustc term set based o a lgustcvalued lattce mplcato algebra. I order to deal wth preferece formato dfferet formats trasformato methods were summarzed. Fally a method for a multple attrbute group decso makg s developed based o the LVWA operator ad trasformato methods. Advatages of ths approach are as follows: ( It does ot requre all lgustc terms to have a total order. ( It permts to compute wth preferece formato dfferet formats. Ackowledgemets We would lke to express our thaks to the support of the Cha-Fladers Blateral Proect (Grat No. OS05 the Natoal Natural Scece Foudato of Cha (Grat No ad the Specalzed Research Foudato for the Doctoral Program of Hgher Educato of Cha (Grat No Refereces. Z.S. Xu A method based o lgustc aggregato operators for group decso makg wth lgustc preferece relatos Iformato Sceces 66: Z.S. Xu EOWA ad EOWG operators for aggregatg lgustc labels based o lgustc preferece relatos Iteratoal Joural of Ucertaty Fuzzess ad Kowledge-Based Systems :

10 Xaobg L Da Rua Ju Lu Yag Xu 3. Z.S. Xu Devato measures of lgustc preferece relatos group decso makg Omega 33: J. Ma W.J. L Y. Xu ad Z.M. Sog A model for hadlg lgustc terms the framework of lattce-valued logc LF( X 004 IEEE Iteratoal Coferece O Systems Ma ad Cyberetcs pp R.R. Yager O ordered weghted averagg aggregato operatos multcrtera decso makg IEEE Trasactos o Systems Ma ad Cyberetcs 8: Y. Xu D. Rua K.Y. Q ad J. Lu Lattcevalued logc: a alteratve approach to treat fuzzess ad comparablty Sprger-Verlag Hedelberg F. Herrera L. Martez J.L. Verdegay A model of cosesus group decso makg uder lgustc assessmets Fuzzy Sets ad Systems 78: F. Herrera L. Martez J.L Verdegay Drect approach processes group decso makg usg lgustc OWA operators Fuzzy Sets ad Systems 79: F. Herrera L. Martez J.L. Verdegay A ratoal cosesus model group decso makg usg lgustc assessmets Fuzzy Sets ad System 88: Z.S. Xu Ucerta lgustc aggregato operators based approach to multple attrbute group decso makg uder ucerta lgustc evromet Iformato Scece 68: Y. Xu X. Y. Zeg Ludovc Koehl A Itellget Sesory Evaluato Method for Idustral Products Characterzato It. J. of Iformato Techology & Decso Makg 007 6(: N. Bryso A. Mobolur A acto learg evaluato procedure for multple crtera decso makg problems Europea Joural of Operatoal Research 96: D. Meg H.D. Ja Z.Q. Zhag ad Y. Xu Lgustc truth-value lattce-valued logc system wth mportat coeffcet ad ts applcato to evaluato system Iteratoal Joural of Computer Scece ad Network Secutty 6(6: P.P. Bossoe K.S. Decker Selectg ucertaty calcul ad graularty: a expermet tradg-off precso ad complexty : L.H. Kaal J.F. Lemmer (Eds. Ucertaty Artfcal Itellgece North-Hollad Amsterdam pp R. Dega G. Bortola The problem of lgustc approxmato clcal decso makg Iteratoal Joural of Approxmate Reasog : M. Delgado J.L. Verdegay M.A. Vla O aggregato operatos of lgustc labels Iteratoal Joural of Itellget Systems 8: F. Herrera L. Martı ez A -tuple fuzzy lgustc represetato model for computg wth words IEEE Trasactos o Fuzzy Systems 8: F. Herrera L. Martı ez A model based o lgustc -tuples for dealg wth mult-graular herarchcal lgustc cotexts mult-expert decso-makg IEEE trasactos o systems ma ad cyberetcs-part B Cyberetcs 3: F. Herrera L. Martı ez A approach for combg lgustc ad umercal formato based o -tuple fuzzy lgustc represetato model decso-makg Iteratoal Joural of Ucertaty Fuzzess Kowledge-based Systems 8: M. Delgado F. Herrera E. Herrera-Vedma M.J. Mart-Bautsta L. Martez M.A. Vla A commucato model based o the -tuple fuzzy lgustc represetato for a dstrbuted tellget aget system o teret Soft Computg 6: F. Herrera L. Martez P.J. Sachez Maagg o-homogeeous formato group decso makg Europea Joural of Operatoal Research 66: J. Lu Y. Xu D. Rua ad L. Martez lattcevalued lgustc-based decso makg method IEEE Iteratoal Coferece o Graular Computg(Grc 005 (Beg Cha July pp R.R. Yager A ew methodology for ordal multple obectve decsos based o fuzzy sets : D. Dubos H. Prade ad R.R. Yager eds. Fuzzy Sets for Itellget System Sa Maeto CA: Morga Kaufma Publshers pp Y. Xu. Lattce mplcato algebra Joural of Southwest Jaotog Uversty 89(: ( Chese 83

11 A lgustc-valued weghted aggregato operator 5. Y. Xu S.W. Che ad J. Ma Lgustc truthvalued lattce mplcato algebra ad ts propertes IMACS Mult-coferece o Computatoal Egeerg System Applcato pp Lus Martez J Lu D. Rua J. B. Yag Dealg wth heterogeeous formato egeerg evaluato processes Iformato Sceces 77(7: Hwag C L ad Yoo K. Multple attrbute decso makg- methods ad applcatos. A State-of-the Art Survey. (Sprger-Verlag Berl Herdelberg Z. T. Che Aalyss of decso makg. Scetfc press Beg S. L. Lu W. H. Qu Studes o the basc theores for MADM System Egeerg: Theory ad Practce (: ( Chese 30. X.B. L D. Rua Y. Xu. A ew lgustc-valued aggregato operator to multple attrbute group decso makg. Proc. 007 Iteratoal Coferece o Itellget Systems ad Kowledge Egeerg (ISKE007 Oct Chegdu Cha pp