Some Hybrid Geometric Aggregation Operators with 2-tuple Linguistic Information and Their Applications to Multi-attribute Group Decision Making

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1 Iteratoal Joural of Computatoal Itellgece Systems Vol 6 No (July Some Hybrd Geometrc Aggregato Operators wth -tuple Lgustc Iformato ad her Applcatos to Mult-attrbute Group Decso Mag Shu-Pg Wa ab a College of Iformato echology Jagx Uversty of Face ad Ecoomcs Nachag Jagx 00 Cha b Jagx Key Laboratory of Data ad Kowledge Egeerg Jagx Uversty of Face ad Ecoomcs Nachag 00 E-mal: shupgwa@6com Abstract A ew method s developed to solve mult-attrbute group decso mag (MAGDM problem whch the attrbute values attrbute weghts ad expert weghts are all the form of -tuple lgustc formato Frst the operato laws for -tuple lgustc formato are defed ad the related propertes of the operato laws are studed he some ew hybrd geometrc aggregato operators wth -tuple lgustc formato are developed volvg the -tuple hybrd weghted geometrc average (HWAG operator the -tuple hybrd lgustc weghted geometrc average (-HLWG operator ad the exteded -tuple hybrd lgustc weghted geometrc average (E-HLWG operator hese hybrd geometrc aggregato operators geeralze the exstg -tuple lgustc geometrc aggregato operators ad reflect the mportat degrees of both the gve -tuples ad the ordered postos of the -tuples I the proposed decso method usg the E-HLWG operators the dvdual overall preferece values of the alteratves are tegrated to the collectve oes of the alteratves whch are used to ra the alteratves he method ca suffcetly cosder the mportace degrees of dfferet experts ad thus releve the fluece of those ufar argumets o the decso results A real example of evaluatg uversty faculty s gve to llustrate the proposed method ad the comparso aalyss demostrates the uversalty ad flexblty of the proposed method ths paper Keywords: Mult-attrbute group decso mag Lgustc preferece -tuple lgustc formato hybrd aggregato operator Co-publshed by Atlats Press ad aylor & Fracs Copyrght: the authors 750

2 S P Wa Itroducto Mult-attrbute group decso mag (MAGDM problems wth lgustc formato arse from a wde rage of real-world stuatos (Jag et al Herrera ad Herrera-Vedma Parreras et al I lgustc MAGDM aalyss frstly experts provde ther assessmet formato from the pre-establshed lgustc term sets he the lgustc formato provded by experts s aggregated to form a collectve opo o the alteratves ad the most desrable alteratve(s ca be selected accordg to the derved collectve opo (Herrera et al Jag et al Xu 56 We 78 Mergo et al 9 Herrera et al proposed -tuple lgustc represetato model whch composed by a lgustc term ad a real umber (Herrera ad Martíez 0- he -tuple lgustc model has exact characterstc lgustc formato processg It avoded formato dstorto ad losg whch occur formerly the lgustc formato processg I recet years -tuple lgustc model has bee wdely used decso mag problems (Jag et al Herrera ad Martíez Herrera-Vedma et al Lu ad J Jag ad Fa 5 Yag ad Che 6 García et al 7 We ad Zhao 8 We 9 We ad L 0 We et al We Chag ad We Zhag ad Fa 5 Wag ad Fa 6 Wag 7 Herrera et al 8 Esplla et al 9 Martíez ad Herrera 0 Rodríguez ad Martíez We 8 developed some ew geometrc aggregato operators: the exteded -tuple weghted geometrc (E-WG ad the exteded -tuple ordered weghted geometrc (E- OWG operator he a MAGDM method s preseted based o the E-WG ad E-OWG operators Herrera ad Martíez 0 developed -tuple arthmetc averagg (AA operator -tuple weghted averagg (WA operator -tuple ordered weghted averagg (OWA operator ad exteded -tuple weghted averagg (E- WA operator Herrera ad Martez proposed aother method to solve the group decso mag problem wth mult-graularty lgustc formato hey costructed lgustc herarchy term sets ad geeralzed trasformato fuctos to ufy the multgraularty lgustc formato to the lgustc - tuples Jag ad Fa 5 proposed the -tuple weghted geometrc (WG operator ad -tuple ordered weghted geometrc (OWG operator We 9 utlzed the maxmzg devato method to solve the -tuple lgustc MAGDM wth complete attrbute weght formato We ad L 0 ad We developed grey relatoal aalyss (GRA MAGDM methods based o -tuple lgustc formato Xu et al adopted the vrtual lgustc label to replace -tuple lgustc varable ad proposed the lgustc power average operators ad the ucerta lgustc power average operators We developed three ew aggregato operators: geeralzed -tuple weghted average (G- WA operator geeralzed -tuple ordered weghted average (G-OWA operator ad duced geeralzed -tuple ordered weghted average (IG-OWA operator Chag ad We proposed a ovel techque combg -tuple ad the Ordered Weghted Averagg (OWA operator for prortzato of falures a product desg falure mode ad effect aalyss Zhag ad Fa 5 proposed the exteded -tuple ordered weghted averagg (E-OWA operator Wag ad Fa 6 proposed a OPSIS method for solvg MAGDM problems wth -tuple lgustc assessmet formato Wag 7 preseted a -tuple fuzzy lgustc evaluato model for selectg approprate agle maufacturg system relato to MC producto Herrera et al 8 developed a fuzzy lgustc methodology to deal wth ubalaced lgustc term sets Most of the proposals for solvg MAGDM problems wth -tuple lgustc formato foud lterature dd ot cosder the mportace degrees of dfferet experts However the experts have ther dfferet cultural educatoal bacgrouds experece ad owledge ad expertse related wth the problem doma Geerally speag dfferet experts act as dfferet roles the decso process Some experts may assg uduly hgh or uduly low ucerta preferece values to ther preferred or repugat obects I order to releve the fluece of these ufar argumets o the decso results ad reflect the mportace degrees of all the experts t s ecessary to pay atteto to the dfferet mportace degrees of dfferet experts the real-lfe MAGDM problems herefore ths paper develops some ew hybrd geometrc aggregato operators wth -tuple lgustc formato ad proposes a ew method for MAGDM problems wth -tuple lgustc assessmets he motvato of ths paper s based o the followg facts: ( he exstg aggregato operators wth - tuple lgustc formato are maly focused o the weghted arthmetc (geometrc average ad the ordered weghted arthmetc (geometrc average Co-publshed by Atlats Press ad aylor & Fracs Copyrght: the authors 75

3 Some hybrd geometrc aggregato operators wth -tuple lgustc formato operators here has o vestgato about the hybrd aggregato operators wth -tuple lgustc formato ( he hybrd aggregato operators ca reflect the mportat degrees of both the gve -tuples ad the ordered postos of the -tuples hey are usually used to tegrate the dvdual overall preferece values of alteratves to the collectve oes of alteratves o do so each dvdual overall preferece value should frst be weghted by usg the correspodg expert weght whch ca suffcetly reflect the mportace degrees of dfferet experts ( We 8 oly cosdered that the weght formato of attrbutes ad experts s the form of the lgustc varables he MAGDM method of We 8 ca ot deal wth the case that the weght formato of attrbutes ad experts taes the form of the -tuples However ths case may appear some real-lfe decso problems (see Secto 5 hese ew hybrd geometrc aggregato operators wth -tuple lgustc formato proposed ths paper ca effectvely overcome ths drawbac (v he proposed method ths paper s more reasoable ad flexble tha the exstg oes ad ca be applcable to real-lfe decso problems may areas such as rs vestmet performace evaluato of mltary system egeerg maagemet supply cha ad so o he rest of the paper s arraged as follows Secto troduces the otos for -tuple lgustc formato gves the operato laws ad aalyzes the propertes of the operato laws Secto presets the exstg -tuple lgustc geometrc aggregato operators ad proposes some ew hybrd geometrc aggregato operators wth -tuple lgustc formato Secto costructs the MAGDM model wth -tuple lgustc assessmets ad proposes the correspodg decso method A real applcato to evaluatg uversty faculty for teure ad promoto example s gve Secto 5 he comparso aalyss wth other method s coducted Secto 6 Cocludg remar s made Secto 7 -tuple lgustc formato I ths secto some related otos for -tuple lgustc formato are lsted the the operato laws ad propertes for -tuple lgustc formato are vestgated -tuple lgustc formato Defto Let S { s0 s s st } be a fte ad totally ordered dscrete lgustc term set wth odd cardalty where s represets a possble value for a lgustc varable [0 t ] s a umber value represetg the aggregato result of lgustc symbolc he the fucto used to obta the - tuple lgustc formato equvalet to s defed as: :[0 t] S[ 0505 ( ( s ( where roud( [ 0505 roud( s the usual roud operato s has the closest dex label to ad s the value of the symbolc traslato (Herrera ad Martíez 0- Herrera et al Defto Let S { s0 s s st } be a lgustc term set ad ( s be a lgustc -tuple here s always a fucto such that from a -tuple t returs ts equvalet umercal value [0 t] R whch s (Herrera ad Martíez 0- Herrera et al : S [ 0505 [0 t ] ( s ( From Deftos ad we ca coclude that the coverso of a lgustc term to a lgustc -tuple cossts of addg a value 0 as symbolc traslato: ( s ( s0 ( Defto Let ( s ad ( sl l be two -tuples they should have the followg propertes (Herrera ad Martíez 0- Herrera et al : If l the ( s s smaller tha ( sl l deoted by ( s ( sl l If l the a f l the ( s ad ( sl l represet the same formato deoted by ( s ( sl l b f l the ( s ( sl l c If l the ( s s bgger tha ( s deoted by ( s ( s l l l l Operato laws ad propertes for -tuple lgustc formato Defto Let ( s ad ( sl l be two -tuples ad 0 he the operato laws for -tuples are defed as follows: ( ( s ( s l l l l l l t l l ( ( s ( s f ( s ( s t ( s 0 f ( s ( s t Co-publshed by Atlats Press ad aylor & Fracs Copyrght: the authors 75

4 S P Wa ( ( s ( s l l ( st 0 f ( s ( sl l t ( ( s ( ( s ( sl l f ( s ( sl l t ( st 0 f ( s t (v ( (v s ( ( s f ( s t ( st 0 f ( ( s t (( ( s f ( ( s t ( l l ( s s ( sl l ( sl l (( ( s f ( ( s t ( sl l ( st 0 f ( ( s t It should be oted that f the -tuple lgustc formato comes from dfferet lgustc term sets (e mult-graularty lgustc formato they have to be coverted to the fuzzy sets defed the basc lgustc term set by meas of a trasformato fucto (Herrera et al the they ca be operated usg the above operato laws If ( s ( sl l t the addto for ( s ad ( sl l may be cosdered as the maxmum -tuple ( st 0 o the lgustc term set S he other operatos ca be terpreted aalogously Obvously Defto ca assure that the operato results regardg -tuples ad lgustc terms must be [0 t ] whch s accordace wth the CWW scheme (Rodríguez ad Martíez As far as we ow however there s less vestgato o the operato laws of -tuples Defto gves the operato laws of -tuples whch ca be used to drectly compute for -tuple lgustc formato Defto s a terestg ad valuable wor for -tuples although there maybe lose lttle formato uder some stuatos How to defe more reasoable operato laws of -tuples wll be further researched the future I the followg suppose that a gve lgustc term set s S { s0 s s s8} we gve some examples to llustrate the above Defto Example ( s0 ( s0 ( ( s0 ( s0 ( ( s0 ( s0 ( s70 ( s80 Example ( s0 ( s0 ( ( s0 ( s0 (5 ( s 08 ( s0 ( s0 ( s80 Example 06( s 0 (06 ( s 0 (8 ( s08 ( s0 ( s Example ( s0 (( ( s0 ( (566 ( s 08 ( s 0 ( s 0 8 ( s0 ( 0 Example 5 ( s s0 (( ( s0 ( (566 ( s0566 ( s ( s 0 0 ( s80 From Defto the followg propertes are prove: ( s ( sl l ( sl l ( s ( s ( sl l ( sl l ( s (( s ( sl l ( s ( sl l (( s ( s ( s ( s ( s 0 5 For ay ( s there have [( s ( sl l] ( s [( s ( s ] [( sl l ( s ] ad [( s ( sl l] ( s ( s [( s ( s ] l l Some geometrc aggregato operators wth - tuple lgustc formato I ths secto we frst preset the exstg -tuple lgustc geometrc aggregato operators ad the propose some ew hybrd geometrc aggregato operators wth -tuple lgustc formato he exstg -tuple lgustc geometrc aggregato operators Based o Deftos ad the exstg -tuple lgustc geometrc aggregato operators are preseted ths subsecto For coveece let be the set composed of all -tuples Defto 5 Let x {( r a( r a ( r a} be a set of -tuples ad w ( w w w be the weghtg vector of -tuples ( r a ( satsfyg that 0 w ( ad w he -tuple weghted geometrc (WG average operator s defed as (Jag ad Fa 5 w WG w (( r a( r a ( r a ( ( r a ( Co-publshed by Atlats Press ad aylor & Fracs Copyrght: the authors 75

5 Some hybrd geometrc aggregato operators wth -tuple lgustc formato Lemma Let a 0 w 0 ( ad w the w wa ( a wth equalty f ad oly f a a a (orra ad Naruawa heorem I Defto 5 the argumet of delta Eq w ( ( r a s defed [0 t ] Proof By Defto we ow that : S [ 0505 [0 t ] hus 0 ( r a t ( Sce the weghted vector satsfes that 0w ( ad yelds by Lemma that w ( w w w w t w 0 ( r a w ( r a w( t ( t herefore w r a t ( [0 ] ad the proof of heorem s completed Defto 6 Let x {( r a( r a ( r a} be a set of -tuples he -tuple ordered weghted geometrc (OWG average operator of dmeso s a mappg OWG : so that OWG (( r a ( r a ( r a w w ( ( r( a( (5 where w ( w w w s the weghted vector correlatg wth OWG satsfyg that 0 ( ad w w ( ( ( ( s a permutato of ( such that ( r( a( ( r( a( for ay (Jag ad Fa 5 Remar I Defto 6 sce the weghted vector w ( w w w satsfes that 0 w ( ad w the argumet of delta w Eq (5 ( r ( ( a s also defed [0 t ] whch ca be prove by the smlar way to heorem Defto 7 Let x {( r a( r a ( r a} be a set of -tuples ad C (( c b( c b ( c b be the lgustc weghtg vector of -tuples ( r a ( he exteded -tuple weghted geometrc (E-WG average operator s defed as (We 8 E-WG (( r a ( r a ( r a C ( c b ( c b r a ( ( ( (6 Remar I Defto 7 sce the power dex ( c b ( c b satsfes that 0 ( ( c b ( c b ad ( c b ( c b the argumet of delta Eq (6 ( c b ( c b ( ( r a s defed [0 t ] whch ca be easly prove by the smlar way to heorem Defto 8 Let x {( r a( r a ( r a} be a set of -tuples he exteded -tuple ordered weghted geometrc (E-OWG average operator of dmeso s a mappg E-OWG : so that E-OWG (( r a ( r a ( r a L ( l ( l ( ( ( r( a( l l l (7 where L (( ( ( s the lgustc weghted vector correlatg wth E OWG ( ( ( ( s a permutato of ( such that ( r( a( ( r( a( for ay (We 8 Remar I Defto 8 sce the power dex ( l ( l satsfes that 0 ( ( l ( l ( l ad the argumet of delta Eq (7 ( l ( l ( l ( ( r( a( s also defed [0 t ] whch ca be easly prove by the smlar way to heorem Co-publshed by Atlats Press ad aylor & Fracs Copyrght: the authors 75

6 S P Wa he ew hybrd geometrc aggregato operators wth -tuple lgustc formato It ca be see from Deftos 7 ad 8 that the E-WG operator weghts the -tuple lgustc argumets whle the E-OWG operator weghts the ordered postos of the -tuple lgustc argumets stead of weghtg the argumets themselves herefore weghts represet dfferet aspects both the E-WG ad E-OWG operators However both the operators cosder oly oe of them o solve ths drawbac based o Deftos ad some ew hybrd geometrc aggregato operators wth -tuple lgustc formato are developed the followg Defto 9 Let x {( r a( r a ( r a} be a set of -tuples If HWG : so that HWG (( r a ( r a ( r a w ω r( a ( w ( ( ( (8 where w ( w w w s the weghted vector correlatg wth HWG satsfyg that 0 ( ad w w ( r( a ( s the th largest -tuple of -tuples ( r a ( wth ( r a ( r a ω ( s the weght vector of -tuples ( r a ( satsfyg that 0 ( ad s the balacg coeffcet ( ths case f ω ( goes to ( the ( r a goes to ( r a ( he the fucto HWG s called the -tuple hybrd weghted geometrc average operator of dmeso heorem I Defto 9 the argumet of delta Eq w (8 ( ( r( ( a s defed [0 t ] Proof Accordg to (v of Defto we have ( r a ( r a t (( ( r a f ( ( r a t ( s 0 f ( ( r a t Hece Defto 9 ( r a [0 t] ad thus ( r a [0 t ] ( ( I addto sce the weghted vector w ( w w w satsfes that 0 w ( ad w t follows from Lemma that w r( a ( w r ( a ( 0 ( ( ( w ( t t w Namely ( ( r( ( a s defed [0 t ] Example 6 Assume that ( r a ( s0 ( r a ( s0 ( r a ( s0 ( r a ( s0 w =( ad 0 ω =( the ( r a ( s 0 ( s 0089 ( r a ( s 0 ( s ( r a ( s0 ( s 0 ad 0 ( r a ( s0 ( s0 herefore ( r a ( s 05 ad hus ( ( 7 ( r a ( s 0 ( ( ( r a ( s 0 ( ( ( r a ( s 0089 ( ( HWG (( r a ( r a ( r a ( r a w ω ( ( ( r( ( w a (9980 ( s 000 heorem he OWG operator s a specal case of the HWG operator Proof Let ( the ( r a ( r a ( r a ( hs completes the proof of heorem heorem he WG operator s a specal case of the HWG operator Proof Let w ( the HWG (( r a ( r a ( r a w ω w r( a ( r( a ( r a ( ( ( ( ( ( ( ( (( Co-publshed by Atlats Press ad aylor & Fracs Copyrght: the authors 755

7 Some hybrd geometrc aggregato operators wth -tuple lgustc formato ( ( ( r a ( ( ( r a ω r a r a r a WG (( ( ( whch completes the proof of heorem From heorems ad we ow that the HWG operator frst weghts the gve argumets ad the reorders the weghted argumets descedg order ad weghts these ordered argumets ad fally aggregates all the weghted argumets to a collectve oe he HWG operator geeralzes both the WG ad OWG operators he HWG operator reflects the mportat degrees of both the gve -tuples ad the ordered postos of the -tuples Defto 0 Let x {( r a( r a ( r a} be a set of -tuples If -HLWG : so that -HLWG (( r a ( r a ( r a L ω ( l ( l r ( a ( ( ( ( (9 where L (( l ( l ( l s the lgustc weghted vector correlatg wth -HLWG ( r( a ( s the th largest -tuple of -tuples ( r a ( wth ( r a ( r a ω ( s the weght vector of -tuples ( r a ( satsfyg that 0 ( ad s the balacg coeffcet ( ths case f ω ( goes to ( the ( r a goes to ( r a ( he the fucto - HLWG s called the -tuple hybrd lgustc weghted geometrc average operator of dmeso Remar Accordg to (v of Defto ( r a ( r a t (( ( r a f ( ( r a t ( s 0 f ( ( r a t Hece Defto 0 ( r a [0 t ] Meawhle sce the power dex that ( l 0 ( l ( ( ( l ( l satsfes ( l ( ad ( l the argumet of delta Eq (9 ( l ( l ( ( r ( a ( s defed [0 t ] whch ca be easly prove by the smlar way to heorem Example 7 Assume that ( l ( s0 ( l ( s0 ( l ( s0 ( l ( s50 ( r a ( s0 ( r a ( s0 ( r a ( s0 ( r a ( s0 ad ω =( the 0 ( r a ( s0 ( s ( r a ( s0 ( s ( r a ( s0 ( s 0 0 ( r a ( s0 ( s0 herefore ( r( a ( ( s7 05 ( r( a ( ( s0 ( r( a ( ( s 0 ad ( r( a ( ( s0089 hus -HLWG (( r a ( r a ( r a ( r a L ω ( l ( l r ( a ( ( ( ( (765 ( s0765 heorem 5 he E-OWG operator s a specal case of the -HLWG operator Proof Let ( the ( r a ( r a ( r a ( hs completes the proof of heorem 5 Defto Let x {( r a( r a ( r a} be a set of -tuples If E-HLWG : so that E-HLWG (( r a ( r a ( r a LC ( l ( l r ( a ( ( ( ( (0 where L (( l ( l ( l s the lgustc weghted vector correlatg wth E-HLWG ( r a s the th largest -tuple of -tuples ( ( Co-publshed by Atlats Press ad aylor & Fracs Copyrght: the authors 756

8 S P Wa c ( b ( r a ( wth ( r a ( r a C (( c b( c b ( c b s the lgustc weght vector of -tuples ( r a ( s the balacg coeffcet he the fucto E-HLWG s called the exteded -tuple hybrd lgustc weghted geometrc average operator of dmeso heorem 6 I Defto the argumet of delta Eq (0 ( l ( l ( ( r ( a ( s defed [0 t ] Proof Accordg to ( of Defto we get that ( ( c b f ( c b t c ( b ( st 0 f ( c b t Accordg to (v of Defto c ( b ( r a ( r a ( c b ( c b (( ( r a f ( ( r a t ( c b ( st 0 f ( ( r a t Hece Defto ( r a [0 t ] ad thus ( r a [0 t ] ( ( satsfes Moreover sce the power dex that ( l ( l ( l 0 ( l t yelds by Lemma that ( l ( l 0 ( ( r ( a ( ( l ( l ( l ( l ( ad ( r a ( ( ( l ( t t ( l herefore the argumet of delta Eq (0 s also defed [0 t ] Example 8 Assume that ( l ( s0 ( l ( s0 ( l ( s0 ( l ( s50 ( r a ( s0 ( r a ( s0 ( r a ( s000 ( r a ( s00 ( c b ( s0 ( c b ( s0 ( c b ( s0 ( c b ( s0 the ( s ( r 0 a ( s0 ( s00 ( s ( r 0 a ( s0 ( s07 ( s ( r 0 a ( s000 ( s00080 ad ( s ( r 0 a ( s00 ( s0057 herefore ( r( a ( ( s 07 ( r( a ( ( s00 ( r( a ( ( s0057 ad ( r( a ( ( s00080 hus E-HLWG (( r a ( r a ( r a ( r a LC ( l ( l r ( a ( ( ( ( (66 ( s 056 MAGDM model ad method wth -tuple Lgustc assessmets I the followg we apply the -tuple hybrd geometrc aggregato operators to solve the MAGDM problems wth -tuple lgustc assessmets MAGDM model descrpto wth -tuple lgustc assessmets hs subsecto descrbes the MAGDM problem wth - tuple lgustc assessmets Let A { A A A m } be a dscrete set of m possble alteratves ad F { a a a } be a fte set of attrbutes where A deotes the th alteratve ad a deotes the th attrbute Let D{ D D D t } be a fte set of t experts where D deotes the th expert he expert D provdes hs/her assessmet formato of a alteratve A o a attrbute a as a -tuple r ( s accordg to a predefed lgustc term set S where s S ad [ 0505 ( m t hus the experts assessmet formato ca be represeted by Co-publshed by Atlats Press ad aylor & Fracs Copyrght: the authors 757

9 Some hybrd geometrc aggregato operators wth -tuple lgustc formato the -tuple lgustc decso matrxes R ( r m ( t Suppose that the weght formato of attrbutes ad experts also ca be represeted by the -tuple lgustc formato Let W (( w ( w ( w be the -tuple lgustc weght vector of the attrbutes a ( ad C (( c b( c b ( ct bt be the -tuple lgustc weght vector of the experts D ( t where w S c S [ 0505 ad b [ 0505 ( t he problem cocered ths paper s how to ra alteratves or select the most desrable alteratve(s amog the fte set A based o the - tuple lgustc assessmet formato gve by the experts ad the -tuple lgustc weght formato of attrbutes ad experts he MAGDM method wth -tuple Lgustc assessmet Iformato I ths subsecto we propose a ew method based o the E-WG ad E-HLWG operators to solve the MAGDM problems wth -tuple lgustc assessmets he process ad algorthm may be summarzed as follows Step Utlzed the E-WG operator to tegrate the th le elemets of the decso matrx R the dvdual overall preferece value of the alteratve A gve by the expert D s derved as follows: z ( s E-WG (( s ( s ( s W ( w ( w ( ( ( s (( ( ( where ( W w w w be the - tuple lgustc weght vector of the attrbutes s S ad [ 0505 Step Used the E-HLWG operator to tegrate all the dvdual overall preferece values z ( s ( t of alteratve A the collectve overall preferece value of alteratve A s obtaed as follows: t t z ( s E-HLWG (( s ( s ( s LC ( l ( t t ( l ( ( ( ( s ( where L (( l ( l ( l t t s the lgustc weghted vector correlatg wth E-HLWG ( ( ( s s the th largest -tuple of -tuples t( c b ( s ( t wth ( s ( s C (( c b( c b ( ct bt s the -tuple lgustc weght vector of experts Step Ra all the alteratves ad select the best oe(s accordace wth z ( s ( m If ay alteratve has the hghest z value the t s the best alteratve 5 A real applcato to evaluatg uversty faculty for teure ad promoto I ths subsecto a real case study of evaluatg uversty faculty for teure ad promoto s examed to llustrate the proposed method ths paper Nachag Uversty of Cha teds to evaluate fve facultes for teure ad promoto he fve faculty caddates (alteratves are Iformato techology faculty A Software faculty A Humates faculty A Mathematcs faculty A ad Chemstry faculty A 5 respectvely he uversty commttee vtes four experts D ( from the other famous uverstes to evaluate these facultes Sce the expert D has egaged uversty evaluato for may years ad accumulated rch experece the uversty commttee ames the expert D as the group leader whch s resposble for the whole evaluatg wor Geerally may attrbutes should be used to evaluate these facultes o mprove the effcecy ad rapdly mae decso three attrbutes are chose by the four experts after prelmary screeg hese attrbutes are teachg a research a ad servce a respectvely hese attrbutes are all qualtatve attrbutes t s reasoable for the experts to use lgustc varables or -tuples to represet the evaluato formato of the facultes wth respectve to the attrbutes Cosequetly the fve faculty caddates are to be evaluated usg the -tuple lgustc formato accordg to the lgustc term set: S ={ s 0 = extremely poor (bad s = very poor (bad s = poor (bad s = slghtly poor (bad s = far (mportat s 5 = slghtly good (mportat s 6 = good (mportat s 7 = very good (mportat s 8 = extremely good (mportat} Co-publshed by Atlats Press ad aylor & Fracs Copyrght: the authors 758

10 S P Wa by the four experts uder these three attrbutes he - tuple lgustc decso matrxes provded by each expert are respectvely as follows: ( s00 ( s0 ( s80 ( s0 ( s0 ( s7 0 R ( s0 ( s0 ( s60 ( s0 ( s5 0 ( s70 ( s7 0 ( s80 ( s00 ( s0 ( s0 ( s60 ( s5 0 ( s0 ( s60 R ( s0 ( s7 0 ( s60 ( s0 ( s0 ( s70 ( s60 ( s7 0 ( s80 ( s0 ( s0 ( s70 ( s0 ( s0 ( s5 0 R ( s0 ( s0 ( s60 ( s50 ( s8 0 ( s70 ( s7 0 ( s70 ( s0 ad ( s0 ( s00 ( s70 ( s0 ( s50 ( s8 0 R ( s0 ( s60 ( s80 ( s0 ( s50 ( s8 0 ( s60 ( s0 ( s0 Wth ever creasg complexty real-lfe uversty evaluato maagemet t s very dffcult to gve precsely the lgustc assessmet formato o the expert weghts ad attrbute weghts accordg to the gve lgustc term set advace For example the experts th that the attrbute a s mportat ad the weght may be s 6 but less tha s 6 thus the weght of attrbute a ca be represeted usg the lgustc - tuple ( w ( s6 0 After the egotato ad vestgato of the experts they determe the -tuple lgustc weght vector W (( w ( w ( w of the attrbutes where ( w ( s8 0 ( w ( s0 ad ( w ( s6 0 As the stated earler the expert D amed as the group leader has rch experece owledge ad specalty uversty evaluato Obvously hs mportace degree s extremely hgh ad may be s 8 but less tha s 8 therefore the weght of expert D ca be represeted usg the lgustc -tuple ( c ( s8 0 Aalogously the -tuple lgustc weght vector C (( c b( c b( c b( c b of the experts ca be obtaed where ( c b ( s 5 0 ( c b ( s0 ( c b ( s8 0 ad ( c b ( s0 Next we adopt the proposed method ths paper to solve ths faculty evaluato problem Step Combed the decso matrx R ad W (( w ( w ( w wth the E-WG operator the dvdual overall preferece value of the faculty A gve by expert D s geerated as follows: z ( s E-WG W (( s ( s ( s ( w ( w s s ( ( ( ( 05 Smlarly we have z ( s ( s5 06 z ( s ( s 066 z ( s ( s 08 z5 ( s5 5 ( s0068 z ( s ( s0007 z ( s ( s50500 z ( s ( s 0 z ( s ( s05 z5 ( s5 5 ( s7009 z ( s ( s500 z ( s ( s 0670 z ( s ( s 0065 z ( s ( s60060 z5 ( s5 5 ( s50575 z ( s ( s0887 z ( s ( s5 060 z ( s ( s 009 z ( s ( s006 ad z5 ( s5 5 ( s075 Step For the E-HLWG operator assume that the correlated -tuple weghted vector s L (( l ( l ( l ( l where ( l ( s0 ( l ( s50 ( l ( s7 0 ad ( l ( s60 he used C (( c b( c b( c b( c b ad the E-HLWG operator to tegrate all the dvdual overall preferece values z ( s ( of faculty A the collectve overall preferece value of faculty A s thus derved as follows: Co-publshed by Atlats Press ad aylor & Fracs Copyrght: the authors 759

11 Some hybrd geometrc aggregato operators wth -tuple lgustc formato ad z E-HLWG (( s ( s ( s ( s LC ( l ( l ( ( s s ( ( ( ( 056 I the same way we have z E-HLWG (( s ( s ( s ( s LC ( l ( l ( ( s s ( ( ( ( 0690 z E-HLWG (( s ( s ( s ( s LC ( l ( l ( ( s s ( ( ( ( 0099 z E-HLWG (( s ( s ( s ( s LC ( l ( l ( ( s s ( ( ( ( 005 z E-HLWG (( s ( s ( s ( s 5 LC ( l ( l ( ( s 5 5 s ( ( ( ( 005 Step Sce z z5 z z z the rag order of the facultes s A A 5 A A A ad therefore the best faculty s Software faculty A 6 Comparso aalyss wth the smlar method o llustrate the superorty of the proposed method we use the proposed method ths paper to solve the vestmet selecto problem of We 8 ad the coduct a comparso aalyss A vestmet compay wats to vest a sum of moey the best opto here s a pael wth fve possble alteratves to vest the moey: a car compay A a food compay A a computer compay A a arms compay A ad a V compay A 5 he vestmet compay must tae a decso accordg to the four attrbutes: the rs aalyss a the growth aalyss a the socal-poltcal mpact aalyss a ad the evrometal mpact aalyss a he fve possble alteratves A ( 5 are to be evaluated usg the lgustc term set S ={ s 0 =extremely poor (EP s =very poor (VP s =poor (P s =medum (M s =good (G s 5 = very good (VG s 6 =extremely good (EG} by three decso maers D (= uder the above four attrbutes hey respectvely costruct the decso matrces R ( r 5 (= as follows: M G P P P VP M P R G M G EP VG P P G EG EP VP M P M VP VP VP EP G G R M G P EG EG VP VP M P VP M VP ad G P VP VG VP G P G R VG VP G P G VG EG VP M VP M G I We 8 the lgustc weght vector of the attrbutes s H ( s s s s usg the lgustc term 0 set S ={ s 0 =extremely mportat s =very mportat s = mportat s =medum s =bad s 5 = very bad s 6 =extremely bad} For the E-OWG operator of We 8 the correlated lgustc weghted vector s tae as V ( s s s (Note that all the subscrpts the 5 lgustc term sets S ad S are mus order to be ufed wth Defto We suppose that the weght vector of decso maers s ω ( s s s accordg to the lgustc term set S I addto for the E-HLWG operator of ths paper we also tae the correlated lgustc weghted vector as V ( s 5 s s Applyg the proposed method ths paper the above lgustc decso matrces the lgustc weght vectors of the attrbutes ad experts ad the correlated lgustc weghted vector should be frstly trasformed to -tuple lgustc forms by usg Eq ( he the collectve overall preferece values of alteratves ca be obtaed able lsts the collectve overall preferece values of alteratves obtaed by the method We 8 ad method of ths paper Co-publshed by Atlats Press ad aylor & Fracs Copyrght: the authors 760

12 S P Wa able he collectve overall preferece values of alteratves obtaed by both methods Alteratves A A A A A 5 Rag result We 8 ( s -05 ( s 0 ( s 05 ( s 0 ( s -0 A A A A5 A hs paper ( s ( s 0685 ( s 050 ( s 00 ( s -09 A A A A5 A It s easly see from able that the rag results obtaed by the method We 8 ad the method of ths paper are slghtly dfferet he dfferece s the rag order of A ad A e A A by the former whle A A by the latter he worst alteratve s A by both methods but the best alteratve by the former s A whle the best alteratve by the latter s A Compared wth the former the ma advatages of the latter maly le the followg: ( he latter suffcetly taes the mportace degrees of dfferet experts to cosderato Before utlzg the E-HLWG operator the dvdual overall preferece values of alteratves should be frstly weghted by the expert weghts ad the the collectve overall preferece values of alteratves ca be obtaed However the former s based o the E-WG ad E- OWG operators whch does t cosder the mportace degrees of dfferet experts at all I fact dfferet experts act as dfferet roles the decso process (such as the expert D Secto 5 Some experts may assg uduly hgh or uduly low ucerta preferece values to ther preferred or repugat obects o releve the fluece of these ufar argumets o the decso results ad reflect the mportace degrees of all the experts the latter frst weghts each dvdual overall preferece value by usg the correspodg expert weght ad the utlzes the E-HLWG operator to aggregate all the dvdual weghted overall preferece values of each alteratve to the collectve oes of alteratves herefore the E-HLWG or -HLWG operator ca mae the decso results more reasoable through assgg low weghts to those false or based argumets hese advatages ca ot be reflected the former ( he former s oly sutable for the case where the weght formato of attrbutes s the form of the lgustc varables whereas the latter ca deal wth the three cases: the lgustc varables the -tuples ad umercal values for the weght formato of attrbutes ad experts If the weght formato of experts s gve by lgustc varables or -tuples the E-HLWG operator ca be used to tegrate the dvdual overall preferece values of alteratves to the collectve oes If the weght formato of experts s gve by the umercal values we ca use the -HLWG operator to replace the E-HLWG operator to derve the collectve overall preferece values of alteratves whch demostrates that the latter s of uversalty ad flexblty 7 Coclusos he tradtoal aggregato operators are geerally sutable for aggregatg the formato tag the form of umercal values ad yet they wll fal dealg wth lgustc formato A ew decso method s proposed for the MAGDM problem wth -tuple lgustc assessmets Frstly the operato laws for - tuple lgustc formato are defed After revewg the exstg -tuple lgustc geometrc aggregato operators some ew hybrd geometrc aggregato operators wth -tuple lgustc formato are developed cludg HWG -HLWG ad E-HLWG operators he HWG operator geeralzes both the WG ad OWG operators he E-OWG operator s a specal case of the -HLWG operator he decso method proposed ths paper s based o hybrd geometrc aggregato operators whch ca suffcetly cosder the mportace degrees of dfferet experts ad thus releve the fluece of those ufar argumets o the decso results he proposed hybrd geometrc aggregato operators wth -tuple lgustc formato elarge the research cotet o - tuple lgustc formato ad erch the deas for solvg the MAGDM problems wth lgustc formato However how to reasoably determe the lgustc (or -tuple lgustc weghted vector correlatg wth these hybrd geometrc aggregato operators s a crtcal problem whch wll be vestgated the ear future Acowledgemets he author would le to tha Edtors--chef Dr L Martíez Lopez ad aoymous revewers for ther sghtful ad costructve commets hs wor was partally supported by the Natoal Natural Scece Foudato of Cha (Nos the Humates Socal Scece Programmg Proect of Mstry of Educato of Cha (No 09YGC6007 the Natural Scece Foudato of Jagx Provce of Cha (No 0BAB00 ad the Scece ad echology Proect of Jagx provce educatoal departmet of Cha (Nos GJJ65 ad GJJ70 Co-publshed by Atlats Press ad aylor & Fracs Copyrght: the authors 76

13 Some hybrd geometrc aggregato operators wth -tuple lgustc formato ad the Excellet Youg Academc alet Support Program of Jagx Uversty of Face ad Ecoomcs Refereces Jag Y P Fa Z P Ma J A method for group decso-mag wth mult-graularty lgustc assessmet formato Iformato Sceces 78( ( Herrera F Herrera-Vedma E Lgustc decso aalyss: Steps for solvg decso problems uder lgustc formato Fuzzy Sets ad Systems 5( Parreras R O Eel P Ya Mart J S C Palhares R M A flexble cosesus scheme for multcrtera group decso mag uder lgustc assessmets Iformato Sceces 80(7 ( Herrera F Martíez L Sáchez P J Maagg o-homogeeous formato group decsomag Europea Joural of Operatoal Research 66( ( Xu Z S A method based o lgustc aggregato operators for group decso mag wth lgustc preferece relatos Iformato Sceces 66 ( Xu Z S Ucerta lgustc aggregato operators based approach to multple attrbute group decso mag uder ucerta lgustc evromet Iformato Sceces 68 ( We G W Ucerta lgustc hybrd geometrc mea operator ad ts applcato to group decso mag uder ucerta lgustc evromet Iteratoal Joural of Ucertaty Fuzzess ad Kowledge-Based Systems 7( ( We G W A method for multple attrbute group decso mag based o the E-WG ad E-OWG operators wth -tuple lgustc formato Expert Systems wth Applcato 7( ( Mergo J M Casaovas M Martíez L Lgustc aggregato operator for lgustc decso mag based o the Dempster-Shafer theory of evdece Iteratoal Joural of Ucertaty Fuzzess ad Kowledge-Based Systems 8( ( Herrera F Martíez L A -tuple fuzzy lgustc represetato model for computg wth words IEEE rasactos o Fuzzy Systems 8 ( Herrera F Martíez L A approach for combg lgustc ad umercal formato based o -tuple fuzzy lgustc represetato model decsomag Iteratoal Joural of Ucertaty Fuzzess Kowledge-Based Systems 8 ( Herrera F Martíez L A model based o lgustc -tuple for dealg wth mult-graular herarchcal lgustc cotexts mult-expert decso mag IEEE rasactos o Systems Ma ad Cyberetcs (00 7- Herrera-Vedma E Martez L Mata F Chclaa F A cosesus support system model for group decso-mag problems wth mult-graular lgustc preferece relatos IEEE rasactos o Fuzzy Systems ( Lu P J F Methods for aggregatg tutostc ucerta lgustc varables ad ther applcato to group decso mag Iformato Sceces (0 do: 006/s000 5 Jag Y P Fa Z P Property aalyss of the aggregato operators for -tuple lgustc formato Cotrol ad Decso 8(6 ( Yag W Che Z P New aggregato operators based o the Choquet tegral ad -tuple lgustc formato Expert Systems wth Applcatos 9 ( García J M apa Moral M J del Martíez M A Herrera-Vedma E A cosesus model for group decso mag problems wth lgustc terval fuzzy preferece relatos Expert Systems wth Applcatos 9 ( We G W Zhao X F Some depedet aggregato operators wth -tuple lgustc formato ad ther applcato to multple attrbute group decso mag Expert Systems wth Applcatos 9 ( We G W -tuple lgustc multple attrbute group decso-mag wth complete attrbute weght formato Systems Egeerg ad Electrocs 0( ( We G W L R Method of grey relatoal aalyss for multple attrbute group decso-mag based o -tuple lgustc formato Systems Egeerg ad Electrocs 0(9 ( Xu Y J Mergó José M Wag H M Lgustc power aggregato operators ad ther applcato to multple attrbute group decso mag Appled Mathematcal Modellg 6( ( We G W Grey relatoal aalyss method for - tuple lgustc multple attrbute group decso mag wth complete weght formato Expert Systems wth Applcato 8(5 ( We G W Some geeralzed aggregatg operators wth lgustc formato ad ther applcato to Co-publshed by Atlats Press ad aylor & Fracs Copyrght: the authors 76

14 S P Wa multple attrbute group decso mag Computers & Idustral Egeerg 6( (0-8 Chag K H We C A ovel effcet approach for DFMEA combg -tuple ad the OWA operator Expert Systems wth Applcatos 7( ( Zhag Y Fa Z P A approach to lgustc multple attrbute decso mag wth lgustc formato based o ELOWA operator Systems Egeer ( ( Wag X R Fa Z P Method for group decsomag based o -tuple lgustc formato processg Joural of Maagemet Scece Cha 6(5 ( Wag W P Evaluatg ew product developmet performace by fuzzy lgustc computg Expert Systems wth Applcatos 6(6 ( Herrera F Herrera-Vedma E Martíez L A fuzzy lgustc methodology to deal wth ubalaced lgustc term sets IEEE rasactos o Fuzzy Systems 6( ( Esplla M Lu J Martíez L A exteded herarchcal lgustc model for decso-mag problems Computatoal Itellgece 7 ( ( Martíez L Herrera F A overvew o the -tuple lgustc model for computg wth words decso mag: extesos applcatos ad challeges Iformato Sceces 07( (0-8 Rodríguez R M Martíez L A aalyss of symbolc lgustc computg models decso mag Iteratoal Joural of Geeral Systems ( (0-6 Herrera F Herrera-Vedma E Martíez L A fuso approach for maagg mult-graularty lgustc term sets decso-mag Fuzzy Sets ad Systems ( orra V Naruawa Y Modelg Decsos: Iformato Fuso ad Aggregato Operators (Sprger-Verlag Berl Germay 007 Co-publshed by Atlats Press ad aylor & Fracs Copyrght: the authors 76