Hesitant Fuzzy Linguistic VIKOR Method and Its Application in Qualitative MCDM

Hesitant Fuzzy Linguistic VIKOR Method and Its Application in Qualitative MultipleCriteria Decision MakingHuchang Liao,Member,IEEE,Zeshui Xu,Senior Member,IEEE,and Xiao-Jun Zeng,Member,IEEEAbstract—The hesitant fuzzy linguistic term set(HFLTS)has turned out to be a powerful andflexible technique in represent-ing decision makers’qualitative assessments in the processes of decision making.The aim of this paper is to develop a method to solve the multicriteria decision making(MCDM)problem within the context of HFLTS in which the criteria conflict with each other. To do so,the concepts of ideal solutions for a HFL-MCDM prob-lem have been introduced.In addition,in order to represent the closeness of one solution to the ideal one,we propose a sort of hesitant fuzzy linguistic measures,such as the hesitant fuzzy lin-guistic group utility measure,the hesitant fuzzy linguistic individ-ual regret measure,and the hesitant fuzzy linguistic compromise measure.Based on these measures,we develop a hesitant fuzzy linguistic VIKOR(HFL-VIKOR)method,which is motivated by the traditional VIKOR method.The general procedures for the HFL-VIKOR method are given.Some numerical examples are provided to demonstrate the advantages and practicality of our method.Finally,we make some discussions on the advantages of the HFL-VIKOR method,as well as future work.Index Terms—Hesitant fuzzy linguistic term set(HFLTS),hes-itant fuzzy linguistic VIKOR method,multiple criteria decision making(MCDM),qualitative decision making.I.I NTRODUCTIONI N MANY real-life decision making problems,experts arerequested to provide their evaluation values or preferences on a set of alternatives over some predefined criteria in order to make a decision by means of some reasoning methodologies [1].In the process of deriving thefinal decision,how to ex-press the judgments of experts naturally lays the foundation of any decision-making approaches and,thus,has been discussed by many scholars over the half past century[2].Modeling a decision maker’s cognition is a quite challengeable issue be-cause there is much vague and imprecise information.In1965,Manuscript received January3,2014;revised April15,2014;accepted August 27,2014.Date of publication September26,2014;date of current version October2,2015.This work was supported in part by the National Natural Science Foundation of China under Grant61273209,the Excellent Ph.D.Thesis Foundation of Shanghai Jiao Tong University under Grant20131216,and the Scholarship from China Scholarship Council under Grant201306230047. H.C.Liao is with the Antai College of Economics and Management,Shanghai Jiao Tong University,Shanghai200052,China.He is now with the Business School,Sichuan University,Chengdu610064,China(e-mail:liaohuchang@ ).Z.S.Xu is with the Business School,Sichuan University,Chengdu610064, China(e-mail:xuzeshui@).X.-J.Zeng is with the School of Computer Science,University of Manchester, Manchester M139PL,U.K.(e-mail:x.zeng@).Color versions of one or more of thefigures in this paper are available online at .Digital Object Identifier10.1109/TFUZZ.2014.2360556fuzzy set[3]was proposed to deal with imprecise and vague information.Nonetheless,to cope with imperfect and imprecise information,where two or more sources of vagueness appear simultaneously the traditional fuzzy sets show some limitations [4]–[6].Hence,it has been extended into several different forms, such as intuitionistic fuzzy sets,interval-valued intuitionistic fuzzy sets,type-2fuzzy sets,type-n fuzzy sets,fuzzy multisets (also named the fuzzy bags),and hesitant fuzzy sets(HFSs).All these extensions are based on the same rationale that it is not clear to assign the membership degree of an element to afixed set[7].These different forms of fuzzy sets suit problems that are defined as quantitative situations.However,in real-world decision making problems,many aspects of different activities cannot be assessed in a quantitative form,but rather in a quali-tative one[8].Using linguistic information to express experts’opinions is suitable and straightforward because it is very close to human’s cognitive processes.A common approach to model linguistic information is the fuzzy linguistic approach proposed by Zadeh[9]in1975,which represents qualitative information as linguistic variables.Although it is less precise than a number, the linguistic variable,defined as“a variable whose values are not numbers but words or sentences in a natural or artificial language,”enhances theflexibility and reliability of decision models and provides good results in differentfields: Nevertheless,similar to fuzzy sets,the fuzzy linguistic ap-proach has some limitations as follows:1)Regarding the information modeling,the fuzzy linguis-tic approach defines a priori via linguistic terms,which prevents other choices to express preferences in a richer way.2)Meanwhile,decision makers are restricted to use only oneterm to express their assessments,which may not always reflect experts’cognition completely and perfectly[10].3)As for the computational process,since the informationrepresentation model of fuzzy linguistic approach is dis-crete in a continuous domain,the traditional models of computing with words(CWW)may lose information es-pecially in the approximation processes.To circumvent theseflaws and to improve the accuracy and understandability of the processes of CWW,different linguistic representation models have been introduced,such as the two-tuple fuzzy linguistic representation model[11],the linguistic model based on type-2fuzzy set[12],the virtual linguistic model [13],the proportional two-tuple model[14],etc.However,all these extended models are still very limited due to the fact that they are based on the elicitation of single and simple terms that1063-6706©2014IEEE.Personal use is permitted,but republication/redistribution requires IEEE permission.See /publications standards/publications/rights/index.html for more information.should encompass and describe the information provided by ex-perts regarding a linguistic variable[15].When experts hesitate among different linguistic terms and feel the need to use a more complex linguistic term that is not usually defined in the linguis-tic term set to depict their assessments,the aforementioned fuzzy linguistic approaches are out of use.Thus,motivated by the HFS [7],Rodr´ıguez et al.[15]proposed the concept of hesitant fuzzy linguistic term set(HFLTS),which provides a different and great flexible form to represent the assessments of experts.Since the HFLTS provides a new and more powerful tech-nique to represent the qualitative judgments of experts,it has attracted more and more scholars’attentions in recent years and has shown its efficiency and validation in aiding practi-cal decision making processes.After introducing the definition of HFLTS and some operations over HFLTSs,Rodr´ıguez et al.[15]applied the HFLTS to linguistic multicriteria decision mak-ing(L-MCDM)in which experts provide their evaluations by linguistic expressions based on comparative terms,such as“be-tween very low and medium,”or by simple linguistic terms, such as“very low;low;medium;high;very high.”Based on the HFLTS and context-free grammars,Rodr´ıguez et al.[8] then proposed a new linguistic group decision-making model that facilitates the elicitation offlexible and riches linguistic ex-pressions close to the cognitive models of experts for express-ing linguistic preferences.For further application of HFLTS to decision making,Zhu and Xu[16]defined the hesitant fuzzy linguistic preference relation(HFLPR)as a tool to collect and represent experts preferences and investigated the consistency measure of HFLPR.In order to apply the HFLTSs into the MCDM problems more deeply,Liao et al.[17]paid their at-tention to the basic characteristics of HFLTS and proposed a family of distance and similarity measures for HFLTSs and then implemented them into MCDM problems within the context of hesitant fuzzy linguistic circumstance.Wei et al.[18]developed some comparison methods and studied the aggregation theory for HFLTSs.Beg and Rashid[19]proposed a TOPSIS-based method for MCDM in which the opinions of experts are rep-resented by ter,Liu and Rodr´ıguez[20]proposed another TOPSIS-based method in which all the HFLTSs are transformed into fuzzy envelopes.As for a MCDM problem with hesitant fuzzy linguistic infor-mation,it is common that the criteria involved in the problem conflict with each other.Thus,it is highly probable that there is no solution optimizing all criteria simultaneously.This leads the decision making processes to be very tough and intractable.In such a situation,how tofind the best possible solution is an im-portant and urgent question needed to be answered.To the best of our knowledge,up to now,there is no work having addressed this issue.Hence,in this paper,we are going tofill this gap by de-veloping an approach for MCDM within the context of HFLTSs. The aim of MCDM is tofind sufficiently good alternative(s) from a collection of alternatives,which attains the goal of each criteria.In general cases of MCDM with crisp numerical evalua-tions,there are two different solution concepts:one is the pareto optimal solution and the other is the compromise solution.The pareto optimal solution is defined as“if one criterion is to be improved,at least one other criterion has to be made worse”[21].The Pareto optimal solution is always a set of noninfe-rior alternatives,thus it cannot satisfy our initial purpose which is to select one alternative best satisfying the overall criteria from a set of alternatives.On the other hand,the compromise solution is defined by using a compromise programming[22], which is based on the distance measure determining the close-ness of a particular solution to the ideal/infeasible solution. Compromise programming provides an effective way to handle conflicting criteria.Based on the compromise programming, many multicriteria ranking methods have been introduced,in which the most popular one is the VIKOR(vlsekriterijumska optimizacija i kompromisno resenje in serbian,meaning mul-ticriteria optimization and compromise solution)method[23]. The main idea of VIKOR is to determine a compromise solution, which provides a maximum“group utility”for the“majority”and a minimum“individual regret”for the“opponent,”for a MCDM problem with noncommensurable and conflicting cri-teria by mutual concessions[23]–[25].Up to now,the VIKOR method has been applied widely in many differentfields,such as the postearthquake sustainable reconstruction[24],mountain destination choosing[25],alternative bus fuel modes selection [26],alternative hydropower systems assessment[27],financial performance evaluation[28],and so forth[29]–[44].The classi-cal VIKOR method has also been extended into many different forms,such as the fuzzy VIKOR[34]–[37],the interval val-ued VIKOR[38],[39],the intuitionistic fuzzy VIKOR[40],the interval-valued intuitionistic fuzzy VIKOR[41],and the hesitant fuzzy VIKOR[42]–[44].Noticing that the VIKOR method is so powerful in handling MCDM problems with conflicting criteria, we are motivated to investigate the extended form of VIKOR within the hesitant fuzzy linguistic circumstances,which con-sists of the basic idea in this paper.The remainder of this paper is organized as follows:Section II gives some basic knowledge on fuzzy linguistic approach and HFLTSs.The basic operations and distance measure over HFLTSs are also reviewed in this section.Section III describes the procedures of standard VIKOR method and also the ad-vances of different forms of VIKOR methods.In Section IV, we propose the concepts of hesitant fuzzy linguistic ideal so-lutions and some different types of hesitant fuzzy linguistic measures for the HFL-MCDM problem.Based on these con-cepts and measures,the HFL-VIKOR method is developed to derive the compromise solution(s)for a HFL-MCDM problem. For the simplicity of application,the procedure for the HFL-VIKOR method is also presented.Further,some practical ex-amples are given to illustrate the validity and applicability of the proposed method.In Section V,the analysis and discussion on the advantages of the HFL-VIKOR method is given,as well as the future work about this method.The paper ends in Section VI with some concluding remarks.II.F UZZY L INGUISTIC A PPROACH AND H ESITANT F UZZYL INGUISTIC T ERM S ETA.Fuzzy Linguistic ApproachIn practical MCDM problem,many aspects of the prob-lem cannot be assessed in a quantitative form,but rather in a qualitative one.Meanwhile,experts sometimes may feel moreLIAO et al.:HESITANT FUZZY LINGUISTIC VIKOR METHOD AND ITS APPLICATION IN QUALITATIVE MULTIPLE CRITERIA1345Fig.1.Semantics of virtual linguistic terms.comfortable and straightforward to express their opinions bylinguistic information directly because it is close to the humancognitive processes.Thus,the fuzzy linguistic approach[9],asa common way to model linguistic information,has attracted alot of attention.In such an approach,experts’opinions are takenas the values of a linguistic variable,which is established by alinguistic descriptors and its semantics[45].Once the expertsprovide the linguistic evaluation information,the following stepis to translate these linguistic inputs into a machine manipula-tive format in which the computation can be carried out.Suchtranslation is conducted by some fuzzy tools.Meanwhile,theoutputs of the CWW model should also be easy to convertinto the linguistic information.To do so,Xu[46]proposed thesubscript-symmetric additive linguistic term set,which is shownasS={s t|t=−τ,...,−1,0,1,...τ}(1) where the mid linguistic label s0represents an assessment of“indifference,”and the rest of them are placed symmetricallyaround it.In particular,s−τand sτare the lower and upperbounds of linguistic labels used by the experts in practical ap-plications,τis a positive integer,and S satisfies the followingconditions;1)Ifα>β,then sα>sβ.2)The negation operator is defined:neg(sα)=s−α,espe-cially,neg(s0)=s0.It should be noted that the linguistic term set S is a discretelinguistic term set,thus is not convenient for calculation andanalysis.In order to preserve all given linguistic information,Xu[46]extended the discrete linguistic term set to a continuouslinguistic term set¯S={sα|α∈[−q,q]},where q(q>τ)is a sufficiently large positive integer.In general,the linguistic termsα(sα∈S)is determined by experts,and the extended linguis-tic term(also named virtual linguistic term)¯sα(¯sα∈¯S)only appears in computation.The virtual linguistic term provides a tool to compute with the linguistic terms.The mapping between virtual linguistic terms and their corresponding semantics is easy to build,as shown in Fig.1[17].B.Hesitant Fuzzy Linguistic Term SetNote that the(virtual)linguistic term set use only one term to express the uncertain information of an expert.When an expert cannot provide a single or simple term to describe his/her preference because he/she is thinking of several terms at the same time or looking for a more complex linguistic term thatis not usually defined in the linguistic term set,the(virtual)linguistic term set is out of use.In quantitative setting,whenan expert considers several values to determine the membershipof an element to a set,the concept of HFS was introduced[7].The HFS,permitting the membership of an element to a givenset represented by several possible values between0and1,iscapable to determine the membership degree especially whenthe experts have several different values on it[47].Definition1[7]:Let X be afixed set,a HFS on X is in terms of a function that when applied to X returns a subset of[0,1].Motivated by the idea of HFS,Rodr´ıguez et al.[15]intro-duced the HFLTS,whose envelope is an uncertain linguisticvariable[48].The HFLTS has been investigated by many schol-ars after being proposed(for more details,see[8]and[15]–[20]).Definition2[15]:Let S={s0,...,sτ}be a linguistic term set.A HFLTS H S is an orderedfinite subset of the consecutivelinguistic terms of S.Note:For the linguistic term set S={s0,...,sτ}given in Definition1,when its subscripts are not symmetric,some problems will arise(for more details,see[17]).To overcome these problems,Liao et al.[17]replaced the linguistic term set S={s0,...,sτ}by the subscript-symmetric linguistic term set S={s i|i=−τ,...,−1,0,1,...,τ}.Although the HFLTS can be used to elicit several linguis-tic values for a linguistic variable,it is still not similar to thehuman way of thinking and reasoning.Thus,Rodr´ıguez et al.[15]further proposed a context-free grammar to generate sim-ple but elaborated linguistic expressions that are more similar tothe human expressions and can be easily represented by meansof HFLTSs.The grammar G H is a four-tuple(V N,V T,I,P),where V N is a set of nonterminal symbols,V T is the set of termi-nals’symbols,I is the starting symbol,and P is the productionrules that are defined in an extended Backus–Naur form[49].Definition3[15]:Let S be a linguistic term set,G H be acontext-free grammar.The element of G H=(V N,V T,I,P)aredefined as follows:V N={ primary term , composite termunary relation , binary relationconjunction }V T={lower than,greater than,at least,at most between,and,s−τ,...,s−1,s0,s1...,sτ} I∈V NP={I::= primary term | composite termcomposite term ::= unary relationprimary term | binary relationconjunction primary termprimary term ::=s−τ|···|s−1|s0|s1|···|sτunary relation ::=lower than|greater than1346IEEE TRANSACTIONS ON FUZZY SYSTEMS,VOL.23,NO.5,OCTOBER 2015binary relation ::=between conjunction ::=and }.Note that in the above definition,the brackets enclose optionalelements and the symbol “|”indicates alternative elements.The expressions ll generated by the context-free grammar G H may be either single-valued linguistic terms s t ∈S or lin-guistic expressions.The transformation function E G H can be used to transform the expressions ll that are produced by G H into HFLTS.Definition 4[15]:Let E G H be a function that transforms linguistic expressions ll ∈S ll ,obtained by using G H ,into the HFLTS H S .S is the linguistic term set used by G H ,and S ll is the expression domain generated by G HE G H :S ll →H S .(2)The linguistic expression generated by G H using the produc-tion rules are converted into HFLTS by means of the followingtransformations.1)E G H (s t )={s t |s t ∈S }.2)E G H (at most s m )={s t |s t ∈S and s t ≤s m }.3)E G H (lower than s m )={s t |s t ∈S and s t <s m }.4)E G H (at least s m )={s t |s t ∈S and s t ≥s m }.5)E G H (greater than s m )={s t |s t ∈S and s t >s m }.6)E G H (between s m and s n )={s t |s t ∈S and s m ≤s t ≤s n }.With the transformation function E G H defined as Definition 4,it is easy to transform the initial linguistic ex-pressions into HFLTS.C.Basic Operations and Distance Measures Over HesitantFuzzy Linguistic Term SetsRodr´ıguez et al.[15]defined the complement,union,and intersection of HFLTSs.Definition 5[15]:For three HFLTSs H S ,H 1S,and H 2S ,the following operations are defined:1)Lower bound:H −S=min (s t )=s k ,s t ∈h S and s t ≥s k ,∀t .2)Upper bound:H +S=max (s t )=s k ,s t ∈h S and s t ≤s k ,∀t .3)H cS =S −H S ={s t |s t ∈S and s t /∈H S }.4)H 1S ∪H 2S ={s t |s t ∈H 1S or s t ∈H 2S}.5)H 1S ∩H 2S ={s t |s t ∈H 1S and s t ∈H 2S}.Note that different HFLTSs have different number of lin-guistic terms in most cases.In order to operate correctly when comparing two HFLTSs,Liao et al.[17]introduced a method to add linguistic terms in a HFLTS.Assume a HFLTS b ={b l |l =1,...,#b }(#b is the number of linguistic terms in b ),let b +and b −be the maximum and minimum linguistic terms in b respectively,and ξ(0≤ξ≤1)be an optimized parameter,then we can add the linguistic term–b =ξb +⊕(1−ξ)b −(3)to the HFLTS.The max,min,and the average added linguistic terms correspond to b ∗,b ∗,and b A respectively,where b ∗=b +,b ∗=b −,and b A =(b +⊕b −)/2.Its clear that b ∗and b ∗correspond with the optimism and pessimism rules,respectively.The optimized parameter,which is used to reflect the decision makers’risk preferences,is provided by the decision makers.Without loss of any generality,in this paper,we take ξ=1/2.After giving the fundamental axioms for distance measures,Liao et al.[17]developed a family of distance measures of HFLTSs and that of different sets of HFLTSs (for more details,see [17]).In this paper,we use the Euclidean distance betweentwo HFLTSs H 1Sand H 2S as a representation,which is in the mathematical form as follows:d ed (H 1S,H 2S )= 1L L l =1|δ1l −δ2l |2τ+12 1/2(4)where H 1S=∪s δ1l ∈H 1S{s δ1l |l =1,...,#H 1S }(#H 1S is thenumber of linguistic terms in H 1S )and H 2S =∪s δ2l ∈H 2S{s δ2l |l =1,...,#H 2S }are two HFLTSs with #H 1S =#H 2S =L (other-wise,we can extend the shorter one by adding the linguisticterms given as (3)).The linguistic terms s δt l in H tS(t =1,2)are arranged in ascending order.III.A DV ANCES IN VIKOR M ETHODThe VIKOR method,originally introduced by Opricovic [23],is proposed to solve the MCDM problem with conflicting and noncommensurable (different units)criteria.The VIKOR method is an efficient tool to find a compromise solution from a set of conflicting criteria,where the compromise means an agreement established by mutual concessions.It introduces the multicriteria ranking index based on the particular measure of “closeness”to the “ideal”solution,named L p −metric ,which is used as an aggregation function in the compromise program-ming [22].For a classical MCDM problem with I alternatives a i (i =1,2,...,I ),the evaluation values of the alternatives over the criteria c j (j =1,2,...,J )are given as f ij (i =1,2,...,I ;j =1,2,...,J ).The discrete form of L p −metric over the al-ternatives a i (i =1,2,...,I )in compromise programming was determined asL p,i =⎛⎝J j =1ωj f +j −f ijf +j −f −j p ⎞⎠1p1≤p ≤∞,i =1,2,...,I(5)where ωj (j =1,2,...,J )are the corresponding weights of criteria,f +j =max i f ij and f −j =min i f ij are the best and worst values of a i over the benefit-type criterion c j ,respec-tively.Within the VIKOR method,the metric L 1,i and L ∞,i are used to formulate overall (or total)satisfactory for the major-ity of criteria and “sacrifice”of each individual criterion.TheLIAO et al.:HESITANT FUZZY LINGUISTIC VIKOR METHOD AND ITS APPLICATION IN QUALITATIVE MULTIPLE CRITERIA1347 VIKOR method provides a maximum“overall(or total)satis-factory”for the majority of criteria and a minimum“sacrifice”of each individual criterion.The compromise ranking procedureof the ordinary VIKOR method can be set out as follows.Step1:Find f+j and f−j.Step2:Compute the values of group utility and individual regret over alternatives a i(i=1,2,...,I)by the equations:S i=L1,i=Jj=1ωjf+j−f ijf+j−f−j(6)R i=L∞,i=maxjωjf+j−f ijf+j−f−j.(7)The objective function min i S i implies tofind a solution with maximum group utility,which also can be seen as“majority”rule.The objective function min i R i indicates to minimize in-dividual regret of the opponent.In order tofind a compromise solution,we should minimize these two objective functions si-multaneously.Step3:Calculate the values of Q i(i=1,2,...,I)by the relationQ i=υS i−S+S−−S++(1−υ)R i−R+R−−R+(8)where S+=min i S i,S−=max i S i,R+=min i R i,R−= max i R i,andυis the weight of the strategy of the majority of criteria or the maximum group utility.Without loss of gen-erality,it takes the value0.5.The smaller the value of Q i,the better the alternative a i.Step4:Rank alternatives A i(i=1,2,...,I)according to the values of S i,R i,and Q i.The results are three ranking lists.Step5:Determine the best solution or a compromise solution.To ensure the uniqueness of thefinal alternative,the following two qualifications must be satisfied simultaneously:C1:Q(a(2))−Q(a(1))≥1J−1,where a(1)and a(2)are thealternatives with thefirst and second positions in the ranking list,respectively;C2:a(1)should also be the best ranked by S i and R i. Unfortunately,these two conditions often cannot be attained simultaneously.Thus,a set of compromise solutions are de-rived,which is of the most critical significance for the VIKOR method.If the condition C1is not satisfied,then we shall explore the maximum value of M according to the equationQ(a(M))−Q(a(1))<1J−1.(9)All the alternatives a(i)(i=1,2,...,M)are the compromise solutions.If the condition C2is not satisfied,then the alternatives a(1) and a(2)are the compromise solutions.Since it was proposed by Opricovic[23],the VIKOR method has been investigated by many scholars and has been applied to manyfields.Opricovic and Tzeng[24]applied the ordi-nary VIKOR into the postearthquake reconstruction problem in Central Taiwan.They[25]further made a comparison between the VIKOR and TOPSIS methods and applied these two methods to select the destinations for a mountain climber.The difference between the VIKOR and TOPSIS methods can be clarified as follows:1)The VIKOR method uses linear normalization and thenormalized value does not depend on the evaluation unit of a criterion;while the TOPSIS method uses the vector normalization and the normalized value could be different for different evaluation unit of a particular criterion.2)In addition,the VIKOR method introduces an aggregatingfunction which combines all criteria,the relative impor-tance of the criteria,and a balance between the total and individual satisfactions,while the TOPSIS method only considers the distances from the ideal point and from the negative-ideal point without considering their relative importance.3)As Opricovic and Tzeng[25]pointed out,the highestranked alternative by the VIKOR method is the closest to the ideal solution,but the highest ranked alternative by the TOPSIS method is the best in terms of the ranking index and is not always the closest to the ideal solution.To show the difference between the VIKOR and TOPSIS methods,Tzeng et al.[26]used these two methods respec-tively to select a moderate fuel mode for buses in urban area of Taiwan with several types of fuels such as electricity,fuel cell(hydrogen),and methanol considered as alternative-fuel ter on,Opricovic and Tzeng[27]enhanced the VIKOR method with a stability analysis determining the weight stability intervals and with tradeoffs analysis,and then com-pared it deeply with the TOPSIS method,the PROMETHEE method,and the ELECTRE method.They applied these four methods respectively to evaluate alternative hydropower sys-tems on the Drina River in the former Yugoslavia.Yalcin et al.[28]used the VIKOR method to evaluate thefinancial perfor-mance of the companies in the Turkish manufacturing sectors traded on Istanbul Stock Exchange(ISE)by considering both the accounting-basedfinancial performance measures and the value-basedfinancial performance measures together in a fuzzy environment.Besides the aforementioned applications of the classical VIKOR method,the VIKOR method has also been improved by many scholars.Based on the concept of regret theory,Huang et al.[29]proposed a revised model of VIKOR,which defines the regret as the choiceless utility.Chang[30]developed a mod-ified VIKOR method to avoid numerical difficulties in solving problems by the classical VIKOR method.Liou et al.[31]mod-ified the VIKOR method by replacing afixed common number of criteria for all alternatives with a set of criteria for each al-ternative and provided a method to rank the unimproved gaps of alternatives,and based on this,they then used the modified VIKOR method to improve the domestic airlines’service qual-ity in Taiwan.Since sometimes approaching the target values is desirable,Jahan et al.[32]modified the VIKOR method by us-ing a novel normalization technique.Kaya and Kahraman[33] proposed an integrated VIKOR-AHP methodology,in which the。