Some generalized aggregating operators with linguistic information and their application

Some generalized aggregating operators with linguistic information and their application to multiple attribute group decision making qGui-Wu Wei ⇑Department of Economics and Management,Chongqing University of Arts and Sciences,Yongchuan,Chongqing 402160,PR Chinaa r t i c l e i n f o Article history:Received 26June 2010Received in revised form 28November 2010Accepted 14February 2011Available online 16February 2011Keywords:Multiple attribute group decision making Linguistic informationGeneralized 2-tuple weighted average (G-2TWA)operatorGeneralized 2-tuple ordered weighted average (G-2TOWA)operatorInduced generalized 2-tuple ordered weighted average (IG-2TOWA)operatora b s t r a c tWith respect to multiple attribute group decision making problems with linguistic information,some new decision analysis methods are proposed.Firstly,we develop three new aggregation operators:generalized 2-tuple weighted average (G-2TWA)operator,generalized 2-tuple ordered weighted average (G-2TOWA)operator and induced generalized 2-tuple ordered weighted average (IG-2TOWA)operator.Then,a method based on the IG-2TOWA and G-2TWA operators for multiple attribute group decision making is presented.In this approach,alternative appraisal values are calculated by the aggregation of 2-tuple linguistic information.Thus,the ranking of alternative or selection of the most desirable alterna-tive(s)is obtained by the comparison of 2-tuple linguistic information.Finally,a numerical example is used to illustrate the applicability and effectiveness of the proposed method.Ó2011Elsevier Ltd.All rights reserved.1.IntroductionMultiple attribute decision making (MADM)is a usual task in human activities (Merigó,2010;Merigó&Casanovas,2009,2010;Merigó,Casanovas,&Martínez,2010;Merigó&Gil-Lafuente,2010;Wei,2008,2009a,b;Wei,2010a–d;Wei,Lin,Zhao,&Wang,2010;Wei,Zhao,&Lin,2010;Wei,Zhao,Wang,&Lin,2010;Xu,2009;Wang,Luo,and Liu,2007).It consists of finding the most desirable alternative(s)from a given alternative set.However,un-der many conditions,for the real multiple attribute decision mak-ing problems,the decision information about alternatives is usually uncertain or fuzzy due to the increasing complexity of the socio-economic environment and the vagueness of inherent subjec-tive nature of human thinking,thus,numerical values are inade-quate or insufficient to model real-life decision problems.Indeed,human judgments including preference information may be stated in linguistic terms.Several methods have been proposed for dealing with linguistic information.The approximative computational model based on the Exten-sion Principle (Degani &Bortolan,1988).This model transforms linguistic assessment information into fuzzy numbers and usesfuzzy arithmetic to make computations over these fuzzy num-bers.The use of fuzzy arithmetic increases the vagueness.The results obtained by the fuzzy arithmetic are fuzzy numbers that usually do not match any linguistic term in the initial term set. The ordinal linguistic computational model (Delgado,Verdegay,&Vila,1993).This model is also called symbolic model which makes direct computations on labels using the ordinal structure of the linguistic term sets.But symbolic method easily results in a loss of information caused by the use of the round operator. The 2-tuple linguistic computational model (Herrera &Martí-nez,2000a,b,2001;Herrera,Martínez,&Sánchez,2005).This model uses the 2-tuple linguistic representation and computa-tional model to make linguistic computations.Thus,multiple attribute decision making problems under 2-linguistic environment is an interesting research topic having received more and more attention from researchers during the last several years.Herrera and Martínez (1991)show 2-tuple linguistic information processing manner can effectively avoid the loss and distortion of information.It has a distinct advantage over other lin-guistic processing methods in accuracy and reliability.Herrera and Martínez (2000a)developed 2-tuple arithmetic average (TAA)oper-ator,2-tuple weighted average (TWA)operator,2-tuple ordered weighted average (TOWA)operator and extended 2-tuple weighted average (ET-WA)operator.Jiang and Fan (2003)proposed the 2-tuple ordered weighted geometric (TOWG)operator.Wang and0360-8352/$-see front matter Ó2011Elsevier Ltd.All rights reserved.doi:10.1016/j.cie.2011.02.007qThis manuscript was processed by area editor Imed Kacem.⇑Tel.:+862349891870.E-mail address:weiguiwu@Fan(2003)proposed a Technique for Order Preference by Similarity to Ideal Solution(TOPSIS)method for solving multiple attribute group decision making problems with linguistic assessment infor-mation.Herrera-Viedma,Martinez,Mata,and Chiclana(2005)pre-sented a model of consensus support system to assist the experts in all phases of the consensus reaching process of group decision mak-ing problems with multi-granular linguistic preference relations. Herrera et al.(2005)presented a group decision making process for managing non-homogeneous information.The non-homoge-neous information can be represented as values belonging to do-mains with different nature as linguistic,numerical and interval valued or can be values assessed in label sets with different granu-larity,multi-granular linguistic information.Liao,Li,and Lu(2007) presented a model for selecting an ERP system based on linguistic information processing.Herrera,Herrera-Viedma,and Martínez (2008)proposed a fuzzy linguistic methodology to deal with unbal-anced linguistic term sets.Jiang,Fan,and Ma(2008)developed a method for group decision making with multi-granularity linguistic assessment information.Tai and Chen(2009)developed a evalua-tion model for intellectual capital based on computing with linguis-tic variable.Wang(2009)presented a2-tuple fuzzy linguistic computing approach to deal with heterogeneous information and information loss problems during the processes of subjective eval-uation integration which is based on the group decision-making scenario to assist business managers to measure the performance of New Product Development(NPD)manipulates the heteroge-neous integration processes and avoids the information loss effec-tively.Zhang and Chu(2009)developed fuzzy group decision making for multi-format and multi-granularity linguistic judg-ments in quality function deployment.Fan,Feng,Sun,and Ou (2009)evaluated knowledge management capability of organiza-tions by using a fuzzy linguistic method.Fan and Liu(2010)devel-oped a method for group decision making based on multi-granularity uncertain linguistic information.Wei(2010a)extended Technique for Order Preference by Similarity to Ideal Solution(TOP-SIS)method to2-tuple linguistic multiple attribute group decision making with incomplete weight information.Wei(2010b)pro-posed some extended geometric aggregating operators with2-tu-ple linguistic information.Chang and Wen(2010)developed a approach for Design Failure Mode and Effect Analysis(DFMEA) combining2-tuple and the OWA operator.The traditional generalized aggregating operators and induced generalized aggregating operators are generally suitable for aggre-gating the information taking the form of numerical values,and yet they will fail in dealing with linguistic information.The aim of this paper is to extend the traditional generalized aggregating opera-tors to linguistic environment and develop three new generalized aggregation operators:generalized2-tuple weighted average(G-2TWA)operator,generalized2-tuple ordered weighted average (G-2TOWA)operator and induced generalized2-tuple ordered weighted average(IG-2TOWA)operator.Furthermore,a method based on the IG-2TOWA and G-2TWA operators for multiple attri-bute group decision making is presented.In this approach,alterna-tive appraisal values are calculated by the aggregation of2-tuple linguistic information.Thus,the ranking of alternative or selection of the most desirable alternative(s)is obtained by the comparison of2-tuple linguistic information.In doing so,the remainder of this paper is set out as follows.In the next section,we introduce some basic concepts and operational laws of2-tuple linguistic variables. In Section3we develop some generalized weighted average oper-ator with linguistic assessment information.In Section4we devel-op some generalized ordered weighted average operator with linguistic assessment information.In Section5we develop some induced generalized ordered weighted average operator with lin-guistic assessment information.In Section6we apply the IG-2TOWA and G-2TWA operator to multiple attribute group decision making with linguistic assessment information.In Section7,we give an illustrative example about risk analysis to verify the devel-oped approach and to demonstrate its feasibility and practicality. In Section8we conclude the paper and give some remarks.2.PreliminariesLet S¼s if i¼1;2;...;tj g be a linguistic term set with odd cardi-nality.Any label,s i represents a possible value for a linguistic var-iable,and it should satisfy the following characteristics(Herrera& Martínez,2000a;Herrera&Martínez,2000b;Herrera&Martínez, 2001):(1)The set is ordered:s i>s j,if i>j;(2)Max operator:max(-s i,s j)=s i,if s i P s j;(3)Min operator:min(s i,s j)=s i,if s i6s j.For example,S can be defined asS¼f s1¼extremely poor;s2¼very poor;s3¼poor;s4¼medium;s5¼good;s6¼very good;s7¼extremely good gHerrera and Martinez(2000a,2000b)developed the2-tuple fuzzy linguistic representation model based on the concept of sym-bolic translation.It is used for representing the linguistic assess-ment information by means of a2-tuple(s i,a i),where s i is a linguistic label from predefined linguistic term set S and a i is the value of symbolic translation,and a i2½À0:5;0:5Þ.Definition1.Let b be the result of an aggregation of the indices of a set of labels assessed in a linguistic term set S,i.e.,the result of a symbolic aggregation operation,b e[1,t],and t the cardinality of S. Let i=round(b)and a=bÀi be2values,such that i e[1,t]and a2À0:5;0:5Þ½;then a is called a symbolic translation(Herrera& Martínez,2000a;Herrera&Martínez,2000b).Definition 2.Let S={s1,s2,...,s t}be a linguistic term set and b e[1,t]be a value representing the result of a symbolic aggrega-tion operation;then2-tuple that expresses the equivalent informa-tion to b is obtained with the following function:D:½1;t !S½À0:5;0:5Þð1ÞDðbÞ¼s i;i¼roundðbÞalpha¼bÀi;a2½À0:5;0:5Þ&ð2Þwhere round(Á)is the usual round operation,s i has the closest index label to b and a is the value of the symbolic translation(Herrera& Martínez,2000a,2000b).Definition 3.Let S={s1,s2,...,s t}be a linguistic term set and (s i,a i)be a2-tuple;a function DÀ1can be defined,such that,from a 2-tuple(s i,a i)it return its equivalent numerical value b e[1,t]&R, which is obtained with the following function:(Herrera&Martínez, 2000a;Herrera&Martínez,2000b)DÀ1:SÂÀ0:5;0:5Þ½!½1;t ð3ÞDÀ1ðs i;aÞ¼iþa¼bð4ÞDefinition4.(Herrera&Martínez,2000a).Let(s k,a k)and(s l,a l)be two2-tuples,with each one representing a linguistic assessment:(1)If k<l then(s k,a k)is smaller than(s l,a l).(2)If k=l then(i)if a k=a l,then(s k,a k)=s l,a l);(ii)if a k<a l then(s k,a k)<(s l,a l);(iii)if a k>a l then(s k,a k)>(s l,a l).G.-W.Wei/Computers&Industrial Engineering61(2011)32–38333.Generalized weighted average operator with linguistic informationZhao,Xu,Ni,and Liu (2010)proposed the generalized weighted average (GWA)operator which is the generalization of the WA operator.Definition 5.A GWA operator of dimension n is a mapping GWA:R n ?R ,such thatGWA ða 1;a 2;...;a n Þ¼Xn j ¼1x j a k j!1=kð5Þwhere x =(x 1,x 2,...,x n )T be the weight vector ofa j (j =1,2,...,n ),and x j >0,Pn j ¼1x j ¼1and k is a parameter such that k 2ðÀ1;þ1Þ.It’s obvious that GWA operator include a wide range of aggrega-tion operators such as the WA,the weighted geometric (WG)oper-ator,the weighted harmonic average (WHA)operator (Chen,Liu,&Sheng,2004),the weighted quadratic average (WQA),and a lot of other cases.In the following,we extend the GWA operator to linguistic envi-ronment and develop the generalized 2-tuple weighted average (G-2TWA)operator.Definition 6.Let {(r 1,a 1),(r 2,a 2),...,(r n ,a n )}be a set of 2-tuple,An G-2TWA operator of dimension n is a mapping G-2TWA:S n ?S ,furthermore,G-2TWA ððr 1;a 1Þ;ðr 2;a 2Þ;...;ðr n ;a n ÞÞ¼D X n j ¼1x j D À1ðr j ;a j Þk!1=k 0@1A ð6Þwhere x =(x 1,x 2,...,x n )T be the weight vector of (r j ,a j )(j =1,2,...,n ),and x j >0,P nj ¼1x j ¼1,and k is a parameter such that k 2ðÀ1;þ1Þ.Now we consider four special cases of the G-2TWA operator:(1)When k ¼1,the G-2TWA operator become the 2TWA oper-ator(Herrera &Martínez,2000a).2TWA ððr 1;a 1Þ;ðr 2;a 2Þ;...;ðr n ;a n ÞÞ¼DX n j ¼1x j D À1ðr j ;a j Þ!ð7ÞIn particular,if x j ¼1;for all (r j ,a j ),we get the 2TA operator.(2)When k !0,the G-2TWA operator become the 2TWGoperator.2TWG ððr 1;a 1Þ;ðr 2;a 2Þ;...;ðr n ;a n ÞÞ¼DY n j ¼1ðD À1ðr j ;a j ÞÞx j!ð8ÞIn particular,if x j ¼1n;for all (r j ,a j ),we get the 2TG operator.(3)When k ¼À1,the G-2TWA operator become the 2TWHAoperator.2TWHA ððr 1;a 1Þ;ðr 2;a 2Þ;...;ðr n ;a n ÞÞ¼D 1P nj ¼1x jD À1ðr j ;a j ÞB BB @1C C C Að9ÞIn particular,if x j ¼1n;for all (r j ,a j ),we get the 2THA operator.(3)When k ¼2,the G-2TWA operator become the 2TWQAoperator.2TWQA ððr 1;a 1Þ;ðr 2;a 2Þ;...;ðr n ;a n ÞÞ¼D X n j ¼1x j D À1ðr j ;a j Þ2!1=20@1Að10ÞIn particular,if x j ¼1n ;for all (r j ,a j ),we get the 2TQA operator.4.Generalized ordered weighted average operator with linguistic informationThe GOWA operator (Yager,2004)is a generalization of the OWA operator (Yager,1988,Yager,2007a,2007b;Chen,Liu and Sheng,2004;Liu,2006a,2006b,2007;Wu,Li,Li,&Duan,2009)by using generalized means.It can be defined as follows.Definition 7.A GOWA operator of dimension n is a mapping GOWA:R n ?R that has an associated weight vectorw =(w 1,w 2,...,w n )T such that w j >0and Pn j ¼1w j ¼1.Furthermore,GOWA ða 1;a 2;...;a n Þ¼X n j ¼1w j a k r ðj Þ!1=kð11Þwhere (r (1),r (2),ÁÁÁ,r (n ))is a permutation of (1,2,ÁÁÁ,n ),suchthat a r ðj À1ÞP a r ðj Þfor all j =2,ÁÁÁ,n and k is a parameter such that k 2ðÀ1;þ1Þ:It’s obvious that GOWA operator include a wide range of aggregation operators such as the OWA operator (Yager,1988),the ordered weighted geometric (OWG)operator (Chiclana,Herrera,&Herrera-Viedma,2000;Xu &Da,2003),the ordered weighted harmonic average (OWHA)operator (Chen,Liu and Sheng,2004),the ordered weighted quadratic average (OWQA),and a lot of other cases.In the following,we extend the GOWA operator to linguistic environment and develop the generalized 2-tuple weighted aver-age (G-2TOWA)operator.Definition 8.Let {(r 1,a 1),(r 2,a 2),...,(r n ,a n )}be a set of 2-tuple,An G-2TOWA operator of dimension n is a mapping G-2TOWA:S n ?S that has an associated weight vector w =(w 1,w 2,...,w n )Tsuch that w j >0and Pn j ¼1w j ¼1.Furthermore,G-2TOWA ððr 1;a 1Þ;ðr 2;a 2Þ;...;ðr n ;a n ÞÞ¼D Xn j ¼1w j D À1ðr r ðj Þ;a r ðj ÞÞk!1=k 0@1A ð12Þwhere (r (1),r (2),ÁÁÁ,r (n ))is a permutation of (1,2,ÁÁÁ,n ),suchthat ðr r ðj À1Þ;a r ðj À1ÞÞP ðr r ðj Þ;a r ðj ÞÞfor all j =2,ÁÁÁ,n ,and k is a param-eter such that k 2ðÀ1;þ1Þ:Now we consider four special cases of the G-2TOWA operator:(1)When k ¼1,the G-2TOWA operator reduces to the 2TOWAoperator(Herrera &Martínez,2000a).2TOWA ððr 1;a 1Þ;ðr 2;a 2Þ;...;ðr n ;a n ÞÞ¼DX n j ¼1w j D À1ðr r ðj Þ;a r ðj ÞÞ!ð13Þ(2)When k !0;the G-2TOWA operator reduces to the 2TOWGoperator (Jiang &Fan,2003).2TOWG ððr 1;a 1Þ;ðr 2;a 2Þ;...;ðr n ;a n ÞÞ¼DY n j ¼1ðD À1ðr r ðj Þ;a r ðj ÞÞÞw j!ð14Þ(3)When k ¼À1;the G-2TOWA operator reduces to the 2TOW-HA operator.2TOWHA ððr 1;a 1Þ;ðr 2;a 2Þ;...;ðr n ;a n ÞÞ¼D 1Pj ¼1w jD ðr r ðj Þ;a r ðj ÞÞBBB @1C C C Að15Þ34G.-W.Wei /Computers &Industrial Engineering 61(2011)32–38(4)When k ¼2;the G-2TOWA operator reduces to the 2TOWQAoperator.2TWQA ððr 1;a 1Þ;ðr 2;a 2Þ;...;ðr n ;a n ÞÞ¼D Xn j ¼1w j D À1ðr r ðj Þ;a r ðj ÞÞ2!1=20@1A ð16Þ5.Induced generalized ordered weighted average operator withlinguistic informationThe induced generalized OWA (IGOWA)operator was intro-duced by Merigóand Gil-Lafuente (2009)and it represents an extension of the GOWA operator,with the difference that the reor-dering step of the IGOWA operator is not defined by the values of the arguments a i ,but rather by order-inducing variables u i ,where the ordered position of the arguments a i depends upon the values of the u i .It can be defined as follows:Definition 9.An IGOWA operator of dimension n is a mapping IGOWA:R n ?R defined by an associated weight vectorw =(w 1,w 2,...,w n )T such that w j >0and P nj ¼1w j ¼1,a set of order-inducing variables u i ,and a parameter k 2½À1;þ1 ,accord-ing to the following formula:IGOWA ðh u 1;a 1i ;h u 2;a 2i ;ÁÁÁ;h u n ;a n iÞ¼X n j ¼1w j a k r ðj Þ!1=kð17Þa r (j )is the a i value of the GOWA pair h u i ,a i i having the j th largestu i (u i e [0,1]),and u i in h u i ,a i i is referred to as the order-inducing var-iable and a i are the argument variables.It’s obvious that IGOWA operator include a wide range of aggre-gation operators such as the IOWA operator (Yager &Filev,1999),the induced ordered weighted geometric (IOWG)operator (Chicl-ana,Herrera,Herrera-Viedma,&Alonso,2004;Xu &Da,2003),the induced ordered weighted harmonic average (IOWHA)opera-tor (Chen,2004),the induced ordered weighted quadratic average (IOWQA),and a lot of other cases.In the following,we shall extend the IGOWA operators to accommodate the situations where the input arguments are lin-guistic variables and develop the induced generalized 2-tuple or-dered weighted averaging (IG-2TOWA)operator.Definition 10.Let (h u 1,(r 1,a 1)i ,h u 2,(r 2,a 2)i ,...,h u n ,(r n ,a n )i )be a set of 2-tuple,An IG-2TOWA operator of dimension n is a mapping IG-2TOWA:S n ?S ,furthermore,IG-2TOWA w ðh u 1;ðr 1;a 1Þi ;h u 2;ðr 2;a 2Þi ;ÁÁÁ;h u n ;ðr n ;a n ÞiÞ¼D Xn j ¼1w j D À1ðr r ðj Þ;a r ðj ÞÞk !1=k 0@1A ð18Þwhere w =(w 1,w 2,ÁÁÁ,w n )T is a weighting vector,such that w j >0,P nj ¼1w j ¼1,j =1,2,ÁÁÁ,n ,(r r (j ),a r (j ))is the (r i ,a i )value of the IG-2TOWA pair h u i ,(r i ,a i )i having the jth largest u i (u i e [0,1]),and u i in h u i ,(r i ,a i )i is referred to as the order-inducing variable and (r i ,a i )as the linguistic variables.Now we consider four special cases of the IG-2TOWA operator:(1)When k ¼1,the IG-2TOWA operator reduces to the I-2TOWA operator.I-2TOWA w ðh u 1;ðr 1;a 1Þi ;h u 2;ðr 2;a 2Þi ;ÁÁÁ;h u n ;ðr n ;a n ÞiÞ¼DPn j ¼1w j D À1ðr r ðj Þ;a r ðj Þ !ð19Þ(2)When k !0;the IG-2TOWA operator reduces to the I-2TOWG operator.I-2TOWG w ðh u 1;ðr 1;a 1Þi ;h u 2;ðr 2;a 2Þi ;ÁÁÁ;h u n ;ðr n ;a n ÞiÞ¼D Y n j ¼1D À1ðr r ðj Þ;a r ðj ÞÞÀÁw j!ð20Þ(3)When k ¼À1;the IG-2TOWA operator reduces to the I-2TOWHA operator.I-2TOWHA w ðh u 1;ðr 1;a 1Þi ;h u 2;ðr 2;a 2Þi ;ÁÁÁ;h u n ;ðr n ;a n ÞiÞ¼D 1Pnj ¼1w jD À1ðr r ðj Þ;a r ðj ÞÞ0BB B @1C C C Að21Þ(4)When k ¼2;the IG-2TOWA operator reduces to the I-2TOW-QA operator.I-2TOWQA w ðh u 1;ðr 1;a 1Þi ;h u 2;ðr 2;a 2Þi ;ÁÁÁ;h u n ;ðr n ;a n ÞiÞ¼D Xn j ¼1w j D À1ðr r ðj Þ;a r ðj ÞÞ2!1=20@1A ð22ÞEspecially,if w =(1/n ,1/n ,...,1/n )T ,then IG-2TOWA is reduced to the generalized 2-tuple averaging (G-2TA)operator;if u i =i ,forall i ,where i is the ordered position of the ~ai ,the IG-2TOWA is re-duced to the generalized 2-tuple weighted averaging (G-2TWA)operator;if u i ¼~ai ,for all i ,then IG-2TOWA is reduced to the gen-eralized 2-tuple ordered weighted averaging (G-2TOWA)operator.6.An approach to multiple attribute group decision making with linguistic assessment informationLet A ={A 1,A 2,...,A m }be a discrete set of alternatives,and G ={G 1,G 2,...,G n }be the set of attributes,x =(x 1,x 2,...,x n )is the weighting vector of the attributes G j (j =1,2,...,n ),wherex j e [0,1],Pn j ¼1x j ¼1.Let D ={D 1,D 2,...,D t }be the set of decision makers,and m =(m 1,m 2,...,m t )be the weight vector of decisionmakers,where m k e [0,1],P tk ¼1m k ¼1.Suppose that ~R k ¼ð~r ðk Þij Þm Ânis the decision matrix,where ~r ðk Þij 2~S is a preference value,which takes the form of linguistic variables,given by the decision maker D k e D ,for the alternative A i e A with respect to the attribute G j e G .To get the best alternative(s),the follows steps are involved:In the following,we apply the IG-2TOWA and G-2TWA operator to multiple attribute group decision making with linguistic assess-ment information.The method involves the following steps:Step 1.Transforming linguistic decision matrix ~R k ¼ð~r ðk Þij Þm Âninto 2-tuple linguistic decision matrix ~R k ¼ð~r ðk Þij Þm Ân ¼ðr ðk Þij ;0Þm Ân .Step 2.Utilize the decision information given in matrix ~Rk ,and the IG-2TOWA operator which has associated weighting vector w =(w 1,w 2,ÁÁÁ,w t )T~r ij ¼ðr ij ;a ij Þ¼IG-2TOWA w ðh m 1;ð~r ð1Þij ;0Þi ;h m 2;ð~r ð2Þij ;0Þi ;ÁÁÁ;h m t ;ð~r ðt Þij ;0ÞiÞ;i ¼1;2;...;m ;j ¼1;2;...;n :ð23Þto aggregate all the decision matrices ~Rk ðk ¼1;2;...;t Þinto a collec-tive decision matrix ~R ¼ð~r ij Þm Ân ,where m =(m 1,m 2,...,m t )be theweighting vector of decision makers.Step 3.Utilize the decision information given in matrix ~R,and the G-2TWA operator~r i ¼ðr i ;a i Þ¼G2TWA x ð~r i 1;~r i 2;...;~r in Þ;ði ¼1;2;...;m Þð24Þto derive the collective overall preference values ~r i ði ¼1;2;...;m Þof the alternative A i ,where x =(x 1,x 2,...,x n )Tis the weighting vector of the attributes.G.-W.Wei /Computers &Industrial Engineering 61(2011)32–3835Step4.Rank all the alternatives A i(i=1,2,...,m)and select the best one(s)in accordance with~r iði¼1;2;...;mÞ.If any alternative has the highest~r i value,then,it is the most important alternative.Step5.End.7.Illustrative exampleLet us suppose there is an investment company,which wants to invest a sum of money in the best option(adapted from Herrera and Martínez(2000b)).There is a panel withfive possible alternatives to invest the money:r A1is a car company;s A2is a food company; t A3is a computer company;u A4is an arms company;v A5is a TV company.The investment company must take a decision accord-ing to the following six attributes:r G1is thefinancial risk;s G2is the technical risk;t G3is the production risk;u G4is the market risk;v G5is the management risk;w G6is the environment risk. Thefive possible alternatives A iði¼1;2;...;5Þare to be evaluated using the linguistic term setS¼f s1¼extremely poor;s2¼very poor;s3¼poor;s4¼medium;s5¼good;s6¼very good;s7¼extremely good gby the three decision makers D kðk¼1;2;3Þ(whose weighting vec-tor m=(0.35,0.40,0.25)T)under the above six attributes,and con-struct,respectively,the linguistic decision matrices are shown in Tables1–3.In the following,we shall utilize the proposed approach in this paper getting the most desirable alternative(s):Firstly,transforming linguistic decision information given in Ta-bles1–3into2-tuple linguistic decision matrix and utilizing the IG-2TOWA operator which has associated weighting vector w=(0.30, 0.50,0.20)T,the aggregating results are shown in Tables4–7.Secondly,by utilizing the decision information given in Tables 4–7,and the G-2TWA operators,x=(0.12,0.15,0.18,0.25,0.2, 0.1)T is the weighting vector of the attributes,we derive the collec-tive overall preference values of the alternatives.The aggregating results are shown in Table8.Finally,according to the aggregating results shown in Table8, the ordering of the alternatives are shown in Table9.Note that> means‘‘preferred to’’.As we can see,depending on the aggregation operators used,the ordering of the strategies is slightly different. Therefore,depending on the aggregation operators used,the re-sults may lead to different decisions.However,the best alternative is A3.From the above analysis,wefind an interesting phenomena that there are some common advantages in these operators:(1)From Definitions6and8,we know that the G-2TWA operator weights the linguistic arguments while the G-2TOWA operator weights the ordered positions of the linguistic arguments instead of weighting the arguments themselves;From Definitions10,we know that the main difference of the IG-2TOWA operator is that the reordering step of the IG-2TOWA is carried out with order-inducing variables,rather than depending on the values of the lin-guistic arguments.(2)Both the G-2TWA and the G-2TOWA opera-tors are special case of the IG-2TOWA operator.However,we also find difference among these operators,for the same multiple attri-bute group decision making problems with linguistic information, if we emphasize the individual influence,the methods based on the geometric aggregating operators are available;if we emphasize the group’s influence,the methods based on the arithmetic aggregat-ing operators are available;if we emphasize to aggregate central tendency data with linguistic information,the methods based on the harmonic aggregating operators are available.8.ConclusionThe traditional generalized aggregation operators and induced generalized aggregation operators are generally suitable for aggre-gating the information taking the form of numerical values,and yet they will fail in dealing with linguistic information.In this paper, with respect to multiple attribute group decision making(MAG-DM)problems in which both the attribute weights and the expert weights take the form of real numbers,attribute values take the form of linguistic variables,some new decision making analysis methods are developed.Firstly,some operational laws of linguistic variables are introduced.Then,we develop three new aggregation operators:generalized2-tuple weighted average(G-2TWA)opera-tor,generalized2-tuple ordered weighted average(G-2TOWA) operator and induced generalized2-tuple ordered weightedTable1Decision matrix~R1.G1G2G3G4G5G6A1P M VP VP EP PA2VP EP G G VP MA3VG G VG EG M VGA4EG VP VP M VG GA5P VP M VP P VGTable2Decision matrix~R2.G1G2G3G4G5G6A1M VP P P M VPA2P VP M P VG MA3G EG G EG G VGA4VG P P G M GA5EG EP VP M P EPTable3Decision matrix~R3.G1G2G3G4G5G6A1G P VP M EP EPA2VP G P G M MA3VG EG G EG G VGA4G VG EG VP EG PA5M VP M VP P EPTable4Decision matrix~R(k¼1).G1G2G3G4G5G6A1(M,À0.30)(P,0.20)(VP,0.30)(P,À0.30)(VP,À0.10)(VP,0.30) A2(VP,0.30)(VP,0.10)(M,0.30)(M,0.40)(M,À0.40)(M,0.00) A3(VG,À0.30)(VG,0.00)(VG,À0.10)(EG,0.00)(G,À0.50)(VG,0.00) A4(VG,0.30)(P,0.10)(P,0.30)(M,À0.10)(VG,À0.40)(G,À0.40) A5(M,0.40)(VP,À0.30)(P,0.40)(P,À0.40)(P,0.00)(M,À0.50) 36G.-W.Wei/Computers&Industrial Engineering61(2011)32–38。