Research Article Interval 2-Tuple Linguistic Distance Operators and Their Applications to Supplier Evaluation and Selection

Transcription

1 Hidawi Publishig Corporatio Mathematical Problems i Egieerig Volume 2016, Article ID , 12 pages Research Article Iterval 2-Tuple Liguistic Distace Operators ad Their Applicatios to Supplier Evaluatio ad Selectio Meg-Meg Sha, 1 Pig Li, 2 ad Hu-Che Liu 1,3 1 School of Maagemet, Shaghai Uiversity, Shaghai , Chia 2 Zhoupu Hospital Affiliated to Shaghai Uiversity of Medicie & Health Scieces, Shaghai , Chia 3 School of Ecoomics ad Maagemet, Togji Uiversity, Shaghai , Chia Correspodece should be addressed to Hu-Che Liu; hucheliu@foxmail.com Received 11 August 2016; Revised 28 October 2016; Accepted 1 November 2016 Academic Editor: Aa M. Gil-Lafuete Copyright 2016 Meg-Meg Sha et al. This is a ope access article distributed uder the Creative Commos Attributio Licese, which permits urestricted use, distributio, ad reproductio i ay medium, provided the origial work is properly cited. With respect to multicriteria supplier selectio problems with iterval 2-tuple liguistic iformatio, a ew decisio makig approach that uses distace measures is proposed. Motivated by the ordered weighted distace (OWD) measures, i this paper, we develop some iterval 2-tuple liguistic distace operators such as the iterval 2-tuple weighted distace (ITWD), the iterval 2-tuple ordered weighted distace (ITOWD), ad the iterval 2-tuple hybrid weighted distace (ITHWD) operators. These aggregatio operators are very useful for the treatmet of iput data i the form of iterval 2-tuple liguistic variables. We study some desirable properties of the ITOWD operator ad further geeralize it by usig the geeralized ad the quasi-arithmetic meas. Fially, the ew approach is utilized to complete a supplier selectio study for a actual hospital from the healthcare idustry. 1. Itroductio The ordered weighted averagig (OWA) operator [1] is a very well-kow aggregatio operator, providig a parameterized family of aggregatio operators which icludes the maximum, the miimum, ad the average. The promiet characteristic of the OWA operator is the reorderig step. A iterestig extesio of the OWA is the use of distace measures i the OWA operator. I this respect, Xu ad Che [2] developed the ordered weighted distace (OWD) measure, which is the geeralizatio of a variety of wellkow distace measures, such as the ormalized Hammig distace, the ormalized Euclidea distace, ad the ormalized geometric distace. The promiet characteristic of the OWDmeasureisthatitcarelieve(oritesify)theifluece of uduly large or uduly small deviatios o the aggregatio results by assigig them low (or high) weights. Merigó ad Gil-Lafuete [3] proposed a techique for decisio makig usig the OWA operator to calculate Hammig distace ad itroduced the ordered weighted averagig distace (OWAD) operator. The mai advatage of this operator is that it ca take ito accout the attitudial character of a decisio maker i the aggregatio process; thus the decisio maker is able to cosider the decisio problem more clearly accordig to his or her iterests. Merigó et al.[4] itroduced the probabilistic ordered weighted averagig distace (POWAD) operator, which uses a uified model betwee the probability ad the OWA operator cosiderig the degree of importace that each cocept has i the aggregatio. Zeg et al. [5] further exteded the POWAD operator to deal with ucertai eviromets represeted i the form of iterval umbers ad proposed the ucertai probabilistic ordered weighted averagig distace (UPOWAD) operator. Merigó et al. [6] studied the use of distace measures ad heavy aggregatios i the OWA operator ad preseted the heavy ordered weighted averagig distace (HOWAD) operator. It is a ew aggregatio operator that provides a parameterized family of aggregatio operators betwee the miimum distace ad the total distace operator. I additio, Merigó ad Casaovas [7] itroduced the liguistic ordered weighted averagig distace (LOWAD) operator for liguistic decisio makig. Zeg ad Su [8] ad

2 2 Mathematical Problems i Egieerig Zeg [9] cosidered the situatios with ituitioistic fuzzy ad iterval-valued ituitioistic iformatio ad developed some ituitioistic fuzzy weighted distace measures. Xu [10] developed some fuzzy ordered distace measures for group decisio makig with liguistic, iterval, triagular, or trapezoidal fuzzy preferece iformatio. Xia ad Su [11] developed the fuzzy liguistic iduced Euclidea ordered weighted averagig distace (FLIEOWAD) operator for group liguistic decisio makig, i which the criteria values take the form of fuzzy liguistic iformatio. More recetly, other differet typesofdistacemeasureshavebeeproposeditheliterature, like the ucertai iduced heavy ordered weighted averagig distace (UIHOWAD) operator [12], the cotiuous ituitioistic fuzzy ordered weighted distace (C-IFOWD) measure [13], the Pythagorea fuzzy ordered weighted averagig weighted average distace (PFOWAWAD) operator [14], the fuzzy liguistic iduced ordered weighted averagig Mikowski distace (FLIOWAMD) operator [15], the geeralized iterval-valued 2-tuple liguistic weighted distace measures [16], the geeralized hesitat fuzzy liguistic weighted distace measures [17], ad so o [18 21]. I may situatios, however, the iput argumets may take the form of iterval 2-tuple liguistic variables [25 27] because of time pressure, lack of kowledge or data, addecisiomakers limitedattetioadiformatioprocessig capabilities. Furthermore, decisio makers may use differet liguistic term sets to express their evaluatios o the established selectio criteria cosiderig their persoal backgrouds, prefereces, ad differet uderstadig levels to the alteratives. Therefore, it is ecessary to exted the ordered weighted distace measures to accommodate the iterval 2-tuple liguistic eviromet [28 30], which is also the focus of this paper. For doig so, we will develop some iterval 2-tuple liguistic distace operators such as iterval 2-tuple weighted distace (ITWD), iterval 2-tuple ordered weighted distace (ITOWD), ad iterval 2-tuple hybrid weighted distace (ITHWD) operators. These aggregatio operators are very effective to deal with situatios where the iput data are expressed i iterval 2-tuple liguistic variables. We study some desirable properties of the ITOWD operator ad further geeralize it by usig the geeralized ad the quasi-arithmetic meas obtaiig the geeralized iterval 2-tuple ordered weighted distace (GITOWD) ad the quasi-arithmetic iterval 2-tuple ordered weighted distace (Quasi-ITOWD) operators. Fially, based o the GITOWD operator, we develop a approach to group supplier evaluatio ad selectio with iterval 2-tuple liguistic iformatio ad illustrate it with a umerical example. The remaider of this paper is set out as follows. I Sectio 2, we itroduce some basic cocepts ad operatio laws of iterval 2-tuple liguistic variables. I Sectio 3, we develop the ITWD, the ITOWD, ad the ITHWD operators ad ivestigate some desirable properties of the ITOWD operator. I Sectio 4, we preset a approach based o the developed iterval 2-tuple liguistic distace operators to multicriteria group supplier selectio. A supplier selectio example is give i Sectio 5 to verify the proposed approach ad to demostrate its feasibility ad practicality. Fially, coclusios ad future directios are provided i Sectio Prelimiaries Tuple Liguistic Variables. The 2-tuple liguistic represetatio model was firstly preseted i [31] based o the cocept of symbolic traslatio. It is used to represet liguistic iformatio by meas of a liguistic 2-tuple, (s, α),where s is a liguistic term from the predefied liguistic term set S ad α is a umerical value represetig the symbolic traslatio. Itheclassical2-tupleliguisticapproach,therageofβ is betwee 0 ad g, whichisrelevattothegraularityof a liguistic term set. Here, β istheresultofaaggregatio of the idices of a set of labels assessed i the liguistic term set S. To overcome this restrictio, Tai ad Che [32] proposed a geeralized 2-tuple liguistic model ad traslatio fuctios. Defiitio 1. Let S{s 0,s 1,...,s g } be a liguistic term set ad let β [0,1]beavalue represetig the result of a symbolic aggregatio operatio. The the geeralized traslatio fuctioδ used to obtai the 2-tuple liguistic variable equivalet to β is defied as follows [32]: Δ:[0, 1] S [ 1 2g, 1 2g ), Δ(β)(s i,α), with s { i, i roud (β g) { αβ i, { g 1 α [ 2g, 1 2g ), where roud( ) is the usual roudig operatio, s i has the closest idex label to β, adα is the value of the symbolic traslatio. Defiitio 2. Let S{s 0,s 1,...,s g } be a liguistic term set ad let (s i,α)bea2-tuple. There exists a fuctio Δ 1 which is able to covert a 2-tuple liguistic variable ito its equivalet umerical value β [0,1].ThereversefuctioΔ 1 is defied as follows [32]: Δ 1 :S [ 1 2g, 1 ) [0, 1], 2g Δ 1 (s i,α) i (2) g +αβ. Particularly, it is ecessary to poit out that the coversio of a liguistic term ito a liguistic 2-tuple cosists of addig a value 0 as symbolic traslatio [31]: (1) s i S (s i,0). (3) The compariso of liguistic iformatio represeted by 2-tuples is carried out accordig to a ordiary lexicographic order. Defiitio 3. Let (s k,α 1 ) ad (s l,α 2 ) be two 2-tuples; the [31, 33] (1) if k<l,the(s k,α 1 ) is smaller tha (s l,α 2 );

3 Mathematical Problems i Egieerig 3 (2) if kl, the oe has the followig: (a) if α 1 α 2,the(s k,α 1 ) is equal to (s l,α 2 ); (b) if α 1 <α 2,the(s k,α 1 ) is smaller tha (s l,α 2 ); (c) if α 1 >α 2,the(s k,α 1 ) is bigger tha (s l,α 2 ) Iterval 2-Tuple Liguistic Variables. Motivated by Xu s ucertai liguistic variables [22] ad based o the defiitios of [32], Zhag [34] proposed the iterval 2-tuple liguistic represetatio model as geeralizatio of the 2- tuple liguistic variables. Due to its characteristics ad advatages, the iterval 2-tuple liguistic represetatio model has bee widely applied for dealig with ucertaity i various multicriteria decisio makig problems [35 38]. It ca be defied as follows. Defiitio 4. Let S {s 0,s 1,...,s g } be a liguistic term set. A iterval 2-tuple liguistic variable is composed of two 2-tuples, deoted by [(s i,α 1 ), (s j,α 2 )], where(s i,α 1 ) (s j,α 2 ), s i (s j ),adα 1 (α 2 ) represet the liguistic label of S ad symbolic traslatio, respectively. The iterval 2-tuple that expresses the equivalet iformatio to a iterval value [β 1,β 2 ](β 1,β 2 [0,1], β 1 β 2 ) is derived by the followig fuctio [25, 34]: Δ[β 1,β 2 ][(s i,α 1 ),(s j,α 2 )] with s i, i roud (β 1 g) s j, j roud (β 2 g) { α 1 β 1 i g, α 1 [ 1 2g, 1 2g ) { α 2 β 2 j { g, α 2 [ 1 2g, 1 2g ). O the cotrary, there is always a fuctio Δ 1 such that a iterval 2-tuple ca be coverted ito a iterval value [β 1,β 2 ](β 1,β 2 [0, 1], β 1 β 2 ) as follows: (4) Δ 1 [(s i,α 1 ),(s j,α 2 )] [ i g +α 1, j g +α 2][β 1,β 2 ]. (5) I particular, if s i s j ad α 1 α 2, the the iterval 2-tuple liguistic variable reduces to a 2-tuple liguistic variable. Note that the ucertai liguistic variable [32] is simpler tha the iterval 2-tuple liguistic variable, which ca also be used for dealig with group decisio makig with ucertai liguistic iformatio. However, compared with the ucertai liguistic variables, Zhag s iterval 2-tuple liguistic variableshavethefollowigadvatages[31,34,39].(1) The iterval 2-tuple liguistic variable has exact characteristic i liguistic iformatio processig, which ca effectively avoid iformatio distortio ad loss i the liguistic iformatio processig. I cotrast, the ucertai liguistic variable performs the retraslatio step as a approximatio process to express the results i the origial expressio domai (iitial discrete liguistic term set) provokig a lack of accuracy. (2) I the process of aggregatig ucertai liguistic iformatio, the operatioal laws of ucertai liguistic variables are ot closed. Let S {s 0,s 1,...,s 6 } be a liguistic term set; we have [s 2,s 3 ] [s 4,s 5 ] [s 6,s 8 ] ad [s 4,s 5 ] [s 2,s 3 ] [s 8,s 15 ]. Clearly, the results of operatio exceed the rage of the liguistic term set S. This problem is solved by employig the iterval 2-tuple liguistic variables. (3) Decisio makers ca express their prefereces by the use of liguistic term sets with differet graularity of ucertaity ad their judgmets ca be better expressed with iterval 2-tuples from the preestablished liguistic term sets. But the ucertai liguistic variables may lead to iflexibility for maagig the group decisio makig problem with multigraularity liguistic iformatio. Based o the operatios of ucertai liguistic variables [22],Zhag[34]furthergavesomebasicoperatioallawsof iterval 2-tuples ad proposed the iterval 2-tuple weighted average (ITWA) operator. Defiitio 5. Cosider ay three iterval 2-tuples, a [(r, α), (t, ε)], a 1 [(r 1,α 1 ), (t 1,ε 1 )], ad a 2 [(r 2,α 2 ), (t 2, ε 2 )], ad let λ [0,1]; the their operatios are defied as follows [34]: (1) a 1 a 2 [(r 1,α 1 ), (t 1,ε 1 )] [(r 2,α 2 ), (t 2,ε 2 )] Δ[Δ 1 (r 1,α 1 )+Δ 1 (r 2,α 2 ), Δ 1 (t 1,ε 1 )+Δ 1 (t 2,ε 2 )]; (2) λ a λ[(r, α), (t, ε)] Δ[λΔ 1 (r, α), λδ 1 (t, ε)]. Defiitio 6. Let a i [(r i,α i ), (t i,ε i )] (i 1,2,...,) be a set of iterval 2-tuples ad let w(w 1,w 2,...,w ) T be their associated weights, with w i [0, 1] ad i1 w i 1.The ITWA operator is defied as [34] ITWA w ( a 1, a 2,..., a ) Δ[ i1 w i Δ 1 (r i,α i ), i1 i1 (w i a i ) w i Δ 1 (t i,ε i )]. Ispired by the distace measure i ucertai liguistic eviromet [40], the distace betwee iterval 2-tuples ca be defied below. Defiitio 7. Let a 1 [(r 1,α 1 ), (t 1,ε 1 )] ad a 2 [(r 2,α 2 ), (t 2, ε 2 )] be ay two iterval 2-tuples; the (6) d ITD ( a, b) Δ [ 1 2 ( Δ 1 (r 1,α 1 ) Δ 1 (r 2,α 2 ) + Δ 1 (t 1,ε 1 ) Δ 1 (t 2,ε 2 ) )] (7) is called the iterval 2-tuple distace betwee a ad b.particularly, if the iterval 2-tuples a 1 [(r 1,α 1 ), (t 1,ε 1 )] ad a 2 [(r 2,α 2 ), (t 2,ε 2 )] are degeerated to 2-tuples a (r 1,α 1 ) ad b (r 2,α 2 ), the the iterval 2-tuple distace will become the 2-tuple distace [41].

4 4 Mathematical Problems i Egieerig 3. Iterval 2-Tuple Liguistic Distace Operators The ordered weighted averagig distace (OWAD) operator [3] is a extesio of the traditioal Hammig distace by usig the OWA operator, which provides a parameterized family of aggregatio operators ragig from the miimum to the maximum distace. For two real umbers sets A {a 1,a 2,...,a } ad B{b 1,b 2,...,b },theowadoperator is defied as follows. Defiitio 8. A OWAD operator of dimesio is a mappig OWAD: R R Rwhich has a associated weightig vector ω(ω 1,ω 2,...,ω ) T,withω j [0, 1] ad ω j 1, such that OWAD ( a 1,b 1, a 2,b 2,..., a,b ) ω j d j, (8) where d j represets the jth largest of the idividual distace a i b i. Further,XuadChe[2]developedaorderedweighted distace (OWD) measure, which geeralizes a variety of wellkow distace measures ad aggregatio operators. Defiitio 9. A OWD measure of dimesio is a mappig OWD: R R Rwhich has a associated weightig vector ω(ω 1,ω 2,...,ω ) T,withω j [0, 1] ad ω j 1,such that OWD ( a 1,b 1, a 2,b 2,..., a,b ) ( 1/λ ω j d λ j ), where d j is the jth largest of the idividual distace a i b i.ifλ 1,thetheOWDmeasureisreducedtothe OWAD operator; if λ1ad ω (1/,1/,...,1/) T,the theowdmeasureisreducedtotheormalizedhammig distace Iterval 2-Tuple Liguistic Distace Operators. The OWAD ad the OWD operators have oly bee used i the situatios i which the iput argumets are exact values. However, judgmets of people deped o persoal psychological aspects such as experiece, learig, situatio, ad state of mid. It is more suitable for decisio makers to provide their prefereces by meas of liguistic variables rather tha umerical oes. For coveiece, let S be the set of all iterval 2-tuples, let S be the set of all 2-tuples, ad let A { a 1, a 2,..., a } ad B { b 1, b 2,..., b } be two sets of iterval 2-tuples. Based o (7), we defie a iterval 2-tuple weighted distace (ITWD) operator as follows. Defiitio 10. A ITWD operator of dimesio is a mappig ITWD: S S S, whichhasaassociatedweight (9) vector w(w 1,w 2,...,w ) T with w i [0, 1] ad i1 w i 1, such that ITWD ( a 1, b 1, a 2, b 2,..., a, b ) i1 w i d ITD ( a i, b i ), (10) where d ITD ( a i, b i ) is the iterval 2-tuple distace betwee a i ad b i. I particular, if w (1/,1/,...,1/) T, the the ITWD becomes the iterval 2-tuple ormalized distace (ITND) operator: ITND ( a 1, b 1, a 2, b 2,..., a, b ) 1 i1 d ITD ( a i, b i ). (11) If the sets of iterval 2-tuples A { a 1, a 2,..., a } ad B { b 1, b 2,..., b } are degeerated to the sets of 2-tuples A { a 1, a 2,..., a } ad B { b 1, b 2,..., b }, the the ITWD isreducedtothe2-tupleweighteddistace(twd)operator: TWD ( a 1, b 1, a 2, b 2,..., a, b ) i1 w i d TD ( a i, b i ), (12) where d TD ( a i, b i ) is the 2-tuple distace betwee a i ad b i. BasedotheOWAadtheITWDoperators,wedefie a iterval 2-tuple ordered weighted distace (ITOWD) operator as follows. Defiitio 11. A ITOWD operator of dimesio is a mappig ITOWD: S S S, whichhasaassociated weight vector ω (ω 1,ω 2,...,ω ) T,withω j [0, 1] ad ω j 1,suchthat ITOWD ( a 1, b 1, a 2, b 2,..., a, b ) ω j d ITD ( a σ(j), b σ(j) ), (13) where d ITD ( a σ(j), b σ(j) ) is the jth largest of the iterval 2-tuple distace d ITD ( a i, b i ). Particularly, if there is a tie betwee d ITD ( a i, b i ) ad d ITD ( a j, b j ), the we replace each of d ITD ( a i, b i ) ad d ITD ( a j, b j ) by their average (d ITD ( a i, b i )+d ITD ( a j, b j ))/2 i the process of aggregatio. If k itemsaretied,thewereplace these by k replicas of their average. If ω(1/,1/,...,1/) T, the the ITOWD becomes the ITND; if the positio of d ITD ( a i, b i ) is the same as the ordered positio of d ITD ( a σ(j), b σ(j) ), the the ITWD is obtaied. Moreover, if A ad B are degeerated to A ad B, the the ITOWD is

5 Mathematical Problems i Egieerig 5 reduced to the 2-tuple ordered weighted distace (TOWD) operator: Proof. Let max{d ITD ( a i, b i )} t,adletmi{d ITD ( a i, b i )} r; the TOWD ( a 1, b 1, a 2, b 2,..., a, b ) ω j d TD ( a σ(j), b σ(j) ), (14) where d TD ( a σ(j), b σ(j) ) is the jth largest of the 2-tuple distace d TD ( a i, b i ). Similar to the OWAD operator, the ITOWD operator is commutative, mootoic, idempotet, ad bouded. These properties ca be show with the followig theorems. Theorem 12 (commutativity-owa aggregatio). Assume that f is the ITOWD operator. If ( a 1, b 1, a 2, b 2,..., a, b ) is ay permutatio of the argumets ( a 1, b 1, a 2, b2,..., a, b ),the f( a 1, b 1, a 2, b 2,..., a, b ) ω j d ITD ( a σ(j), b σ(j) ) f( a 1, b 1, a 2, b 2,..., a, b ) Therefore, ω j d ITD ( a σ(j), b σ(j) ) mi {d ITD ( a i, b i )} ω j tt ω j rr ω j t, ω j r. f( a 1, b 1, a 2, b 2,..., a, b ) max {d ITD ( a i, b i )}. (20) (21) f( a 1, b 1, a 2, b 2,..., a, b ) f( a 1, b 1, a 2, b 2,..., a, b (15) ). Theorem 13 (commutativity-distace measure). Assume that f is the ITOWD operator; the f( a 1, b 1, a 2, b 2,..., a, b ) f( b 1, a 1, b 2, a 2,..., b, a ). (16) Theorem 14 (mootoicity). Assume that f is the ITOWD operator. If d ITD ( a i, b i ) d ITD ( a i, b i ),foralli,the f( a 1, b 1, a 2, b 2,..., a, b ) f( a 1, b 1, a 2, b 2,..., a, b (17) ). Theorem 15 (idempotecy). Assume that f is the ITOWD operator. If d ITD ( a i, b i )d,foralli,the f( a 1, b 1, a 2, b 2,..., a, b ) d. (18) The proofs of the above theorems are straightforward ad thus are omitted. Theorem 16 (bouded). Assume that f is the ITOWD operator; the mi {d ITD ( a i, b i )} f( a 1, b 1, a 2, b 2,..., a, b ) max {d ITD ( a i, b i )}. (19) Aother importat issue is the determiatio of the weightig vector associated with the ITOWD operator. I the literature, various methods have bee suggested for the OWA weights geeratio, which ca also be implemeted for the ITOWD operator, such as the ormal distributio based method [42], the maximum Bayesia etropy method [43], adtheleastsquaresbasedmethod[44].ispiredby[8,45], ithefollowig,wegivethreewaystodetermietheitowd weights. (1) Let d ITD ( a σ(j), b σ(j) ) ω j 1 d ITD ( a σ(j), b,,2,...,; (22) σ(j) ) the ω j+1 ω j 0,,2,..., 1,ad 1 ω j 1. (2) Let e d ITD( a σ(j), b σ(j) ) ω j,,2,...,; 1 (23) e d ITD( a σ(j), b σ(j) ) the 0 ω j+1 ω j,,2,..., 1,ad 1 ω j 1. (3) Let d ITD ( a σ(j), b σ(j) ) 1 1 d ITD ( a σ(j), b σ(j) ), d(d ITD ( a σ(j), b σ(j) ), d ITD ( a σ(j), b σ(j) )) d ITD ( a σ(j), b σ(j) ) d ITD ( a σ(j), b σ(j) ) ; (24)

6 6 Mathematical Problems i Egieerig ω j the we defie 1 d(d ITD ( a σ(j), b σ(j) ), d ITD ( a σ(j), b σ(j) )) 1 (1 d(d ITD ( a σ(j), b σ(j) ), d ITD ( a σ(j), b σ(j) ))), from which we get ω j 1 ω j 1.,2,...,, (25) 0, j 1,2,...,,ad Note that the weight vector derived from (22) is a mootoic decreasig sequece, the weight vector derived from (23) is a mootoic icreasig sequece, ad the weight vector derived from (25) combies the above two cases; that is, the closer the value d ITD ( a σ(j), b σ(j) ) to the mea d ITD ( a σ(j), b σ(j) ), the larger the weight ω j. Clearly, the fudametal characteristic of the ITWD operator is that it cosiders the importace of each give iterval 2-tuple distace, whereas the fudametal characteristic of the ITOWD operator is the reorderig step, ad it weights all the ordered positios of the iterval 2-tuple distaces istead of weightig the give iterval 2-tuple distaces themselves. Motived by the idea of the liguistic hybrid geometric averagig (LHGA) operator [42], i the followig, we develop a iterval 2-tuple hybrid weighted distace (ITHWD) operator that weights both the give iterval 2-tuple distaces ad their ordered positios. Defiitio 17. A ITHWD operator of dimesio is a mappig ITHWD: S S S, whichhasaassociated weight vector ω (ω 1,ω 2,...,ω ) T,withω j [0, 1] ad ω j 1,suchthat ITHWD ( a 1, b 1, a 2, b 2,..., a, b ) ω j d ITD ( a σ(j), bσ(j) ), (26) where d ITD ( a σ(j), bσ(j) ) is the jth largest of the weighted iterval 2-tuple distace d ITD ( a i, bi )(d ITD ( a i, bi ) w i d ITD ( a i, bi ), i 1,2,...,), w (w 1,w 2,...,w ) T is the weight vector of d ITD ( a i, b i ) (i 1,2,...,),withw i [0,1] ad i1 w i 1,ad is the balacig coefficiet. I particular, if w (1/,1/,...,1/) T, the the ITHWDisreducedtotheITOWDoperator;ifω (1/, 1/,...,1/) T, the the ITHWD is reduced to the ITWD operator. Moreover, if A ad B are degeerated to A ad B, the the ITHWDisreducedtothe2-tuplehybridweighteddistace (THWD) operator: where d TD ( a σ(j), bσ(j) ) is the jth largest of the weighted 2- tuple distace d TD ( a j, bj )(d TD ( a j, bj )w i d TD ( a i, b i ), i 1,2,...,), w (w 1,w 2,...,w ) T is the weight vector of d TD ( a j, bj ) (i 1,2,...,),withw i [0, 1] ad i1 w i 1, ad is the balacig coefficiet Geeralizatios of the ITOWD Operator. I what follows, geeralizatios of the ITOWD operator are preseted by usig the geeralized ad the quasi-arithmetic meas. Defiitio 18. A geeralized iterval 2-tuple ordered weighted distace (GITOWD) operator of dimesio is a mappig GITOWD: S S S, which has a associated weight vector ω(ω 1,ω 2,...,ω ) T,withω j [0,1]ad ω j 1,such that GITOWD ( a 1, b 1, a 2, b 2,..., a, b ) ( ω j d λ ITD ( a σ(j), b σ(j) )) 1/λ, (28) where d ITD ( a σ(j), b σ(j) ) is the jth largest of the iterval 2- tuple distace d ITD ( a i, b i ) ad λ is a parameter such that λ (, + ) {0}. Similar to the OWA ad the GOWA operators [1, 46], the GITOWD operator has may desirable properties, such as commutativity, mootoicity, boudedess, ad idempotecy. Particularly, if there are ties betwee iterval 2-tuple distaces, as i the case of the ITOWD operator, we replace each of the tied argumets by their geeralized mea i the process of aggregatio. If A ad B are degeerated to A ad B, the we ca get the geeralized 2-tuple ordered weighted distace (GTOWD) operator. The GITOWD operator provides a parameterized family of aggregatio operators. I order to study this family, we ca aalyze the weightig vector ω or the parameter λ. By choosig a differet maifestatio of the weightig vector i the GITOWD operator, we are able to obtai differet types of distace operators: (i) The iterval 2-tuple maximum distace is foud if ω 1 1ad ω j 0,forallj 1. (ii) The iterval 2-tuple miimum distace is foud if ω 1ad ω j 0,forallj. (iii) The geeralized iterval 2-tuple ormalized distace (GITND) operator is formed whe ω j 1/,forallj. (iv) The geeralized iterval 2-tuple weighted distace (GITWD) operator is obtaied whe the positio of d ITD ( a i, b i ) is the same as the ordered positio of d ITD ( a σ(j), b σ(j) ). THWD ( a 1, b 1, a 2, b 2,..., a, b ) ω j d TD ( a σ(j), bσ(j) ), (27) Some special cases ca also be obtaied with the chage of the parameter λ: (i) If λ1, the the GITOWD is reduced to the ITOWD operator.

7 Mathematical Problems i Egieerig 7 (ii) If λ 0, the the GITOWD is reduced to the iterval 2-tuple ordered weighted geometric distace (ITOWGD) operator. (iii) If λ 1, the the GITOWD is reduced to the iterval 2-tuple ordered weighted harmoic distace (ITOWHD) operator. (iv) If λ 2, the the GITOWD is reduced to the iterval 2-tuple ordered weighted Euclidea distace (ITOWED) operator. (v) If λ3, the the GITOWD is reduced to the iterval 2-tuple ordered weighted cubic distace (ITOWCD) operator. Defiitio 19. A quasi-arithmetic iterval 2-tuple ordered weighted distace (Quasi-ITOWD) operator of dimesio is a mappig Quasi-ITOWD: S S S, whichhasa associated weight vector ω (ω 1,ω 2,...,ω ) T,withω j [0, 1] ad ω j 1,suchthat Quasi-ITOWD ( a 1, b 1, a 2, b 2,..., a, b ) g 1 ( ω j g(d ITD ( a σ(j), b σ(j) ))), (29) where d ITD ( a σ(j), b σ(j) ) is the jth largest of the iterval 2-tuple distace d ITD ( a i, b i ) ad g is a geeral cotiuous strictly mootoe fuctio. As we ca see, the GITOWD operator is a particular case of the Quasi-ITOWD operator whe g(x) x λ.if A ad B are degeerated to A ad B, the we ca get the quasi-arithmetic 2-tuple ordered weighted distace (Quasi- TOWD) operator. Note that all properties ad particular cases commeted i the GITOWD operator ca also be discussed i this geeralizatio. 4. The Proposed Multicriteria Group Supplier Selectio Method I this sectio, we develop a approach based o the proposed iterval 2-tuple liguistic distace operators for solvig multicriteria group supplier selectio problems. Supposethatagroupsupplierselectioproblemhasl decisio makers DM k (k1,2,...,l), m alteratives A i (i 1,2,...,m),ad decisio criteria C j (j 1,2,...,). Each decisio maker DM k is give a weight V k >0 (k1,2,...,l) satisfyig l k1 V k 1toreflect his/her relative importace i the supplier selectio process. Let D k (d k ij ) m be the liguistic decisio matrix of the kth decisio maker, where d k ij is the liguistic iformatio provided by DM k o the assessmet of A i with respect to C j. I additio, decisio makers may use differet liguistic term sets to express their assessmet values. Next, we apply the ITWA ad the GITOWD operators for multicriteria group supplier selectio uder iterval 2-tuple liguistic eviromet. Step 1. Covert the liguistic decisio matrix D k (d k ij ) m ito the iterval 2-tuple liguistic decisio matrix R k ( r k ij ) m ([(r k ij, 0), (tk ij, 0)]) m, wherer k ij,tk ij S,S{s 0,s 1,...,s g } ad r k ij tk ij. Suppose that DM k provides his assessmets i a set of five liguistic terms S: S {s 0 very poor,s 1 poor,s 2 medium,s 3 good,s 4 very good}. The liguistic iformatio provided i the decisio matrix D k ca be coverted ito correspodig iterval 2-tuple liguistic assessmets accordig to the followig ways: (i) A certai grade such as poor, which ca be writte as [(s 1, 0), (s 1,0)] (ii) A iterval such as poor-medium, which meas that the assessmet of a alterative cocerig the criterio uder cosideratio is betwee poor ad medium. This ca be expressed as [(s 1, 0), (s 2,0)]. Remark 20. I particular supplier selectio problems, there exist may situatios where iformatio may be uquatifiable due to its ature, or the precise quatitative iformatio may be uavailable or the cost for its computatio is too high. Thus, it is more reasoable ad atural for decisio makers to make their judgmets by usig liguistic expressios. Geerally, three mai methods have bee itroduced for dealig with qualitative assessmets [31, 39, 47]. The first method is based o membership fuctios [48], which coverts liguistic iformatio ito fuzzy umbers by meas of a membership fuctio. However, this method led to a certai degree of iformatio loss i the trasformatio process. The secod method is based o liguistic symbols [49] that made computatios o the subscripts of liguistic terms ad was easy to operate. However, this approach may lead to iflexibility for differet sematics. The third method is based o liguistic 2-tuples [29], which ca avoid the iformatio distortio ad loss i liguistic iformatio processig ad has bee widely utilized for maagig liguistic MCDM problems. Therefore, to deal with liguistic iformatio more reasoably ad accurately, the decisio maskers assessmets o alterative suppliers are first trasformed ito iterval 2-tuples i the proposed approach. Step 2. Utilize the ITWA operator r ij [(r ij,α ij ),(t ij,ε ij )] ITWA ( r 1 ij, r2 ij,..., rl ij ) Δ[ l k1 V k Δ 1 (r k ij,0), l k1 V k Δ 1 (t k ij,0)], i1,2,...,m,,2,...,, (30) to aggregate all the iterval 2-tuple liguistic decisio matrices R k (k 1,2,...,l) ito a collective iterval 2-tuple liguistic decisio matrix R ( r ij ) m.

8 8 Mathematical Problems i Egieerig Step 3. Determie the ideal level of each criterio i order to characterize the collective ideal alterative r ( r 1, r 2,..., r ),where r j [(r j,α j ), (t j,ε j )],,2,...,. (31) Step 4. Calculate the separatio measure S + i of each alterative from the ideal alterative by usig the GITOWD operator: his assessmets usig B;DM 3 provides her assessmets usig C. These liguistic term sets are deoted as follows: A{a 0 very poor (VP), a 1 poor (P), a 2 medium (M), a 3 good (G), a 4 very good (VG)}, B{b 0 very poor (VP), b 1 poor (P), b 2 S + i GITOWD ( r i1, r 1, r i2, r 2,..., r ij, r j ) ( 1/λ ω j d λ ITD ( r iσ(j), r σ(j) )), i1,2,...,m, (32) medium poor (MP), b 3 medium (M), b 4 medium good (MG), b 5 good (G), b 6 very good (VG)}, C{c 0 extra poor (EP), c 1 very poor (VP), c 2 (33) where d λ ITD ( r iσ(j), r σ(j) ) is the jth largest of the iterval 2-tuple distace d ITD ( r ij, r j ); ω(ω 1,ω 2,...,ω ) T is the weightig vector of the GITOWD operator such that ω j [0,1] ad ω j 1.Notethatitispossibletocosiderawiderage of GITOWD operators such as those described i Sectio 3. Step 5. Rak all the alteratives A i (i1,2,...,m)ad select the best oe(s) accordig to the icreasig order of their separatio measures. Step 6. Ed. 5. A Illustrative Example 5.1. Example Illustratio. I this sectio, we develop a illustrative example of the ew approach i a group decisiomakigproblemofsupplierselectio.supposethata tertiary care hospital desires to select the most appropriate supplier for oe of the key medical devices i the geeral aesthesia process. After prelimiary screeig, six suppliers, A i (i 1,2,...,6), have remaied as alteratives for further evaluatio. I order to evaluate the alterative suppliers ad select the best oe, a expert committee of three decisio makers, DM 1,DM 2 ad DM 3, has bee formed. The selectio decisio is made o the basis of oe objective ad five criteria C j (j 1,2,...,5). These criteria, which are critical for the supplier selectio, are defied as follows: C 1 : techical capability C 2 : delivery performace C 3 :productquality C 4 : flexibility C 5 : price/cost The three decisio makers employ differet liguistic term sets to assess the suitability of the suppliers with respect to the above selectio criteria. Specifically, DM 1 provides his assessmets by usig the liguistic term set A;DM 2 provides poor (P), c 3 medium poor (MP), c 4 medium (M), c 5 medium good (MG), c 6 good (G), c 7 very good (VG), c 8 extra good (EG)}. The liguistic assessmets of the six alteratives o each criterio provided by the three decisio makers are preseted i Table 1. With this iformatio, we ca make a aggregatio i order to make a decisio. First, we covert the liguistic decisiomatrixshowitable1itotheiterval2-tupleliguistic decisio matrix R k ([(r k ij, 0), (tk ij, 0)]) 6 5,whichis depicted i Table 2. The, we aggregate the iformatio of the three experts to obtai a collective iterval 2-tuple liguistic decisiomatrix.weusetheitwaoperatortoobtaithis matrix while assumig that V (0.3, 0.4, 0.3) T.Theresults are show i Table 3. Accordig to their objectives, the group of experts establishes the collective ideal supplier show i Table 4. Itisowpossibletodevelopdifferetmethodsbasedo the GITOWD operator for the selectio of the optimum supplier. I this example, we cosider the iterval 2-tuple maximum distace, the iterval 2-tuple miimum distace, the ITND, the ITWD, the ITHWD, the ITOWD, the ITOWGD, the ITOWHD, the ITOWED, ad the ITOWCD operators. Forcoveiece,weassumethefollowigweightigvector: ω (0.112, 0.236, 0.304, 0.236, 0.112) T, which is derived by the ormal distributio based method [42]. The aggregated results are preseted i Tables 5 ad 6 ad the rakigs of the alterative suppliers for each particular case are show i Table 7. As we ca see, depedig o the distace operator used, therakigordersofthesixsuppliersaredifferet.dueto the fact that each particular type of the GITOWD operator may lead to differet results, the decisio maker ca select for his decisio the oe that is i closest accordace with his iterests. However, i this example, it is clear that the best choice is A 2, although i some exceptioal situatios the alterative suppliers such as A 1, A 4,orA 5 could be optimal.

9 Mathematical Problems i Egieerig 9 Table 1: Liguistic assessmets of the suppliers. Decisio makers DM 1 DM 2 DM 3 Alteratives Criteria C 1 C 2 C 3 C 4 C 5 A 1 G-VG M-G G M G A 2 VG G VG M-G G A 3 M M P-M G-VG M A 4 G G-VG M G G-VG A 5 M VG G VG G A 6 G M-G VG G M-G A 1 VG M G-VG M G A 2 MG VG G MG MG A 3 M-G MG M MG G A 4 G VG G MG-G M A 5 M-G M G VG VG A 6 MG M-G G G VG A 1 M-MG G MG G M A 2 VG VG MG VG G A 3 VG M G MG VG A 4 EG VG G G VG A 5 G MG G-VG G MG A 6 M M-G G G MG Table 2: Iterval 2-tuple liguistic decisio matrix. Decisio makers DM 1 DM 2 DM 3 Alteratives Criteria C 1 C 2 C 3 C 4 C 5 A 1 [(a 3, 0), (a 4, 0)] [(a 2, 0), (a 3, 0)] [(a 3, 0), (a 3, 0)] [(a 2, 0), (a 2, 0)] [(a 3, 0), (a 3,0)] A 2 [(a 4, 0), (a 4, 0)] [(a 3, 0), (a 3, 0)] [(a 4, 0), (a 4, 0)] [(a 2, 0), (a 3, 0)] [(a 3, 0), (a 3,0)] A 3 [(a 2, 0), (a 2, 0)] [(a 2, 0), (a 2, 0)] [(a 1, 0), (a 2, 0)] [(a 3, 0), (a 4, 0)] [(a 2, 0), (a 2,0)] A 4 [(a 3, 0), (a 3, 0)] [(a 3, 0), (a 4, 0)] [(a 2, 0), (a 2, 0)] [(a 3, 0), (a 3, 0)] [(a 3, 0), (a 4,0)] A 5 [(a 2, 0), (a 2, 0)] [(a 4, 0), (a 4, 0)] [(a 3, 0), (a 3, 0)] [(a 4, 0), (a 4, 0)] [(a 3, 0), (a 3,0)] A 6 [(a 3, 0), (a 3, 0)] [(a 2, 0), (a 3, 0)] [(a 4, 0), (a 4, 0)] [(a 3, 0), (a 3, 0)] [(a 2, 0), (a 3,0)] A 1 [(b 6, 0), (b 6, 0)] [(b 3, 0), (b 3, 0)] [(b 5, 0), (b 6, 0)] [(b 3, 0), (b 3, 0)] [(b 5, 0), (b 5,0)] A 2 [(b 5, 0), (b 5, 0)] [(b 6, 0), (b 6, 0)] [(b 5, 0), (b 5, 0)] [(b 4, 0), (b 4, 0)] [(b 4, 0), (b 4,0)] A 3 [(b 3, 0), (b 5, 0)] [(b 4, 0), (b 4, 0)] [(b 3, 0), (b 3, 0)] [(b 4, 0), (b 4, 0)] [(b 5, 0), (b 5,0)] A 4 [(b 5, 0), (b 5, 0)] [(b 6, 0), (b 6, 0)] [(b 5, 0), (b 5, 0)] [(b 4, 0), (b 5, 0)] [(b 3, 0), (b 3,0)] A 5 [(b 3, 0), (b 5, 0)] [(b 3, 0), (b 3, 0)] [(b 5, 0), (b 5, 0)] [(b 6, 0), (b 6, 0)] [(b 6, 0), (b 6,0)] A 6 [(b 4, 0), (b 4, 0)] [(b 3, 0), (b 5, 0)] [(b 5, 0), (b 5, 0)] [(b 5, 0), (b 5, 0)] [(b 6, 0), (b 6,0)] A 1 [(c 4, 0), (c 5, 0)] [(c 6, 0), (c 6, 0)] [(c 5, 0), (c 5, 0)] [(c 6, 0), (c 6, 0)] [(c 4, 0), (c 4,0)] A 2 [(c 7, 0), (c 7, 0)] [(c 7, 0), (c 7, 0)] [(c 5, 0), (c 5, 0)] [(c 7, 0), (c 7, 0)] [(c 6, 0), (c 6,0)] A 3 [(c 7, 0), (c 7, 0)] [(c 4, 0), (c 4, 0)] [(c 6, 0), (c 6, 0)] [(c 5, 0), (c 5, 0)] [(c 7, 0), (c 7,0)] A 4 [(c 8, 0), (c 8, 0)] [(c 7, 0), (c 7, 0)] [(c 6, 0), (c 6, 0)] [(c 6, 0), (c 6, 0)] [(c 7, 0), (c 7,0)] A 5 [(c 6, 0), (c 6, 0)] [(c 5, 0), (c 5, 0)] [(c 6, 0), (c 7, 0)] [(c 6, 0), (c 6, 0)] [(c 5, 0), (c 5,0)] A 6 [(c 4, 0), (c 4, 0)] [(c 4, 0), (c 6, 0)] [(c 6, 0), (c 6, 0)] [(c 6, 0), (c 6, 0)] [(c 5, 0), (c 5,0)] Table 3: Collective iterval 2-tuple liguistic decisio matrix. C 1 C 2 C 3 C 4 C 5 A 1 Δ[0.775, 0.888] Δ[0.575, 0.650] Δ[0.746, 0.813] Δ[0.575, 0.575] Δ[0.708, 0.708] A 2 Δ[0.896, 0.896] Δ[0.888, 0.888] Δ[0.821, 0.821] Δ[0.679, 0.754] Δ[0.717, 0.717] A 3 Δ[0.613, 0.746] Δ[0.567, 0.567] Δ[0.500, 0.575] Δ[0.679, 0.754] Δ[0.746, 0.746] A 4 Δ[0.858, 0.858] Δ[0.888, 0.963] Δ[0.708, 0.708] Δ[0.717, 0.783] Δ[0.688, 0.763] A 5 Δ[0.575, 0.708] Δ[0.688, 0.688] Δ[0.783, 0.821] Δ[0.925, 0.925] Δ[0.813, 0.813] A 6 Δ[0.642, 0.642] Δ[0.500, 0.783] Δ[0.858, 0.858] Δ[0.783, 0.783] Δ[0.738, 0.813]

10 10 Mathematical Problems i Egieerig Table 4: Collective ideal supplier. C 1 C 2 C 3 C 4 C 5 r Δ[0.8, 0.9] Δ[0.9, 1] Δ[0.8, 0.9] Δ[0.9, 1] Δ[0.8, 0.9] Table 5: Aggregated results 1. Max Mi ITND ITWD ITHWD A 1 Δ[0.375] Δ[0.019] Δ[0.189] Δ[0.208] Δ[0.196] A 2 Δ[0.233] Δ[0.050] Δ[0.106] Δ[0.106] Δ[0.092] A 3 Δ[0.383] Δ[0.104] Δ[0.241] Δ[0.271] Δ[0.273] A 4 Δ[0.200] Δ[0.025] Δ[0.108] Δ[0.116] Δ[0.109] A 5 Δ[0.263] Δ[0.048] Δ[0.124] Δ[0.117] Δ[0.101] A 6 Δ[0.308] Δ[0.050] Δ[0.162] Δ[0.159] Δ[0.145] Table 6: Aggregated results 2. ITOWD ITOWGD ITOWHD ITOWED ITOWCD A 1 Δ[0.184] Δ[0.131] Δ[0.080] Δ[0.224] Δ[0.252] A 2 Δ[0.094] Δ[0.080] Δ[0.071] Δ[0.111] Δ[0.128] A 3 Δ[0.240] Δ[0.224] Δ[0.208] Δ[0.253] Δ[0.265] A 4 Δ[0.108] Δ[0.091] Δ[0.072] Δ[0.121] Δ[0.130] A 5 Δ[0.111] Δ[0.084] Δ[0.068] Δ[0.140] Δ[0.162] A 6 Δ[0.158] Δ[0.136] Δ[0.115] Δ[0.176] Δ[0.191] 5.2. Comparative Discussio. To further evaluate the proposed iterval 2-tuple liguistic method, we coduct a comparative aalysis with some previous liguistic decisio makig methods, which iclude the oe based o the ucertai liguistic weighted averagig (ULWA) ad the ucertai liguistic hybrid aggregatio (ULHA) operators [22], the oe based o the ucertai liguistic weighted geometric mea (ULWGM) ad the ucertai liguistic hybrid geometric mea (ULHGM) operators [23], ad the oe based o the ucertai liguistic weighted harmoic mea (ULWHM) ad the ucertai liguistic hybrid harmoic mea (ULHHM) operators [24]. The rakig results of the six alteratives derived by usig these methods are preseted i Table 8. Note that the liguistic term set S {s 0,s 1,...,s 6 } is used for evaluatig the alteratives i the compared methods. From Table 8, it ca be observed that the rakig orders of the alteratives obtaied by the methods of Xu [22] ad Wei [23] are exactly the same as those determied by proposed approach whe the ITND, the ITWD, the ITOWD, the ITOWED, ad the ITOWCD operators are applied. Further, the rakig of Park et al. s [24] approach is i lie with theproposedmethodusigtheithwdoperator.thus, the proposed supplier evaluatio ad selectio method is validated. However, compared with the listed methods, the proposed approach usig iterval 2-tuple liguistic distace operators is more reasoable ad flexible for solvig supplier selectio problems because of the followig: (i) It has exact characteristic i liguistic iformatio processig ad ca effectively avoid the loss ad distortio of iformatio i traditioal liguistic computatioal models. (ii) The liguistic term sets with differet graularity of ucertaity ca be used by decisio makers for assessig alteratives. This eables the decisio makers to express their judgmets more realistically. (iii) By usig a wide rage of distace operators, we ca take differet potetial situatios ito cosideratio ad provide a more complete picture for supplier evaluatio ad selectio. Thus, it is easier to select the alterative that better fits the iterests of the decisio maker. 6. Coclusios I this paper, we have developed some iterval 2-tuple liguistic distace operators icludig the iterval 2-tuple weighted distace (ITWD), the iterval 2-tuple ordered weighted distace (ITOWD), ad the iterval 2-tuple hybrid weighted distace (ITHWD) operators. These distace operators are very suitable to deal with the decisio iformatio represeted i iterval 2-tuple argumets uder multigraular liguistic cotext. We have give three ways to determie the associated weightig vectors ad studied some desired properties of the ITOWD operator. Moreover, further geeralizatios of the ITOWD operator have bee preseted by usig the geeralized ad the quasi-arithmetic meas. The results are the geeralized iterval 2-tuple ordered weighted distace (GITOWD) ad the quasi-arithmetic iterval 2- tuple ordered weighted distace (Quasi-ITOWD) operators. The developed iterval 2-tuple liguistic distace operators ca be applied i may situatios already cosidered with the distace measures such as i statistics, ecoomics, soft computig, ad fuzzy set theory. I this paper, we have appliedthemtomulticriteriagroupsupplierselectiowith iterval 2-tuple liguistic iformatio. I additio, a case example from the healthcare idustry has bee give to verify the developed method ad to demostrate its practicality ad effectiveess. The results showed that this approach provides more complete iformatio for decisio makig because it ca cosider a wide rage of future scearios accordig to theiterestsofthedecisiomaker. I the future, we expect to preset further extesios to the proposed approach by addig ew characteristics such as the use of iducig variables or probabilistic aggregatios i thedecisioprocessadcosiderthepotetialapplicatioof thedevelopediterval2-tupleliguisticdistaceoperatorto the problems i other fields. Competig Iterests The authors declare that there are o competig iterests. Ackowledgmets This work was supported by the Natioal Natural Sciece Foudatio of Chia (os ad ), the

11 Mathematical Problems i Egieerig 11 Table 7: Rakigs of the alterative suppliers. Rakig Rakig Max A 4 A 2 A 5 A 6 A 1 A 3 ITOWD A 2 A 4 A 5 A 6 A 1 A 3 Mi A 1 A 4 A 5 A 6 A 2 A 3 ITOWGD A 2 A 5 A 4 A 1 A 6 A 3 ITND A 2 A 4 A 5 A 6 A 1 A 3 ITOWHD A 5 A 2 A 4 A 1 A 6 A 3 ITWD A 2 A 4 A 5 A 6 A 1 A 3 ITOWED A 2 A 4 A 5 A 6 A 1 A 3 ITHWD A 2 A 5 A 4 A 6 A 1 A 3 ITOWCD A 2 A 4 A 5 A 6 A 1 A 3 Table 8: Rakig comparisos. Xu s method [22] Wei s method [23] Park et al. s method [24] r j Rakig r j Rakig r j Rakig A 1 [s 4.13,s 4.29 ] 5 [s 3.98,s 4.11 ] 5 [s 3.88,s 3.99 ] 5 A 2 [s 5.00,s 5.00 ] 1 [s 4.85,s 4.85 ] 1 [s 4.82,s 4.82 ] 1 A 3 [s 3.86,s 3.98 ] 6 [s 3.67,s 3.76 ] 6 [s 3.49,s 3.56 ] 6 A 4 [s 4.83,s 4.97 ] 2 [s 4.67,s 4.81 ] 2 [s 4.54,s 4.67 ] 3 A 5 [s 4.80,s 4.93 ] 3 [s 4.61,s 4.75 ] 3 [s 4.55,s 4.67 ] 2 A 6 [s 4.53,s 4.72 ] 4 [s 4.37,s 4.59 ] 4 [s 4.28,s 4.50 ] 4 Natioal Social Sciece Foudatio (o. 15CGL003), the Program for Professor of Special Appoitmet (Youg Easter Scholar) at Shaghai Istitutios of Higher Learig (o. QD ), ad the Shaghai Sciece ad Techology Iovatio Actio Pla Soft Sciece Foudatio (o ). Refereces [1] R. R. Yager, O ordered weighted averagig aggregatio operators i multicriteria decisio makig, IEEE Trasactios o Systems, Ma, ad Cyberetics, vol. 18, o. 1, pp , [2] Z.XuadJ.Che, Orderedweighteddistacemeasure, Joural of Systems Sciece ad Systems Egieerig,vol.17,o.4,pp , [3] J. M. Merigó ad A. M. Gil-Lafuete, New decisio-makig techiques ad their applicatio i the selectio of fiacial products, Iformatio Scieces,vol.180,o.11,pp , [4] J. M. Merigó, Y. Xu, ad S. Zeg, Group decisio makig with distace measures ad probabilistic iformatio, Kowledge- Based Systems,vol.40,pp.81 87,2013. [5] S. Zeg, J. M. Merigó, ad W. Su, The ucertai probabilistic OWA distace operator ad its applicatio i group decisio makig, Applied Mathematical Modellig., vol. 37, o. 9, pp , [6] J.M.Merigó,M.Casaovas,adS.Z.Zeg, Distacemeasures with heavy aggregatio operators, Applied Mathematical Modellig,vol.38,o.13,pp ,2014. [7] J. M. Merigó ad M. Casaovas, Decisio makig with distace measures ad liguistic aggregatio operators, Iteratioal Joural of Fuzzy Systems, vol. 12, o. 3, pp , [8] S. Zeg ad W. Su, Ituitioistic fuzzy ordered weighted distace operator, Kowledge-Based Systems,vol.24,o.8,pp , [9] S. Zeg, Some Ituitioistic Fuzzy Weighted Distace Measures ad Their Applicatio to Group Decisio Makig, Group Decisio ad Negotiatio,vol.22,o.2,pp ,2013. [10] Z. S. Xu, Fuzzy ordered distace measures, Fuzzy Optimizatio ad Decisio Makig,vol.11,o.1,pp.73 97,2012. [11] S. Xia ad W. Su, Fuzzy liguistic iduced Euclidea OWA distace operator ad its applicatio i group liguistic decisio makig, Iteratioal Joural of Itelliget Systems,vol.29, o. 5,pp ,2014. [12] W. Su, C. Zhag, ad S. Zeg, Ucertai iduced heavy aggregatio distace operator ad its applicatio to decisio makig, Cyberetics ad Systems,vol.46,o.3-4,pp ,2015. [13] L. Zhou, F. Ji, H. Che, ad J. Liu, Cotiuous ituitioistic fuzzy ordered weighted distace measure ad its applicatio to group decisio makig, Techological ad Ecoomic Developmet of Ecoomy,vol.22,o.1,pp.75 99,2016. [14] S.Zeg,J.Che,adX.Li, Ahybridmethodforpythagorea fuzzy multiple-criteria decisio makig, Iteratioal Joural of Iformatio Techology & Decisio Makig,vol.15,o.02,pp , [15] S. Xia, W. Su, S. Xu, ad Y. Gao, Fuzzy liguistic iduced OWA Mikowski distace operator ad its applicatio i group decisio makig, PAA. Patter Aalysis ad Applicatios, vol. 19, o. 2, pp , [16] H. Cheg, F. Meg, ad K. Che, Several geeralized itervalvalued 2-tuple liguistic weighted distace measures ad their applicatio, Iteratioal Joural of Fuzzy Systems,2016. [17] F. Meg ad X. Che, A hesitat fuzzy liguistic multigraularity decisio makig model based o distace measures, JouralofItelligetadFuzzySystems, vol. 28, o. 4, pp ,2015. [18] J. M. Merigó ad M. Casaovas, Decisio-makig with distace measures ad iduced aggregatio operators, Computers & Idustrial Egieerig,vol.60,o.1,pp.66 76,2011. [19] H. Zhag ad L. Yu, New distace measures betwee ituitioistic fuzzy sets ad iterval-valued fuzzy sets, Iformatio Scieces,vol.245,pp ,2013. [20] M. Düğeci, A ew distace measure for iterval valued ituitioistic fuzzy sets ad its applicatio to group decisio makig problems with icomplete weights iformatio, Applied Soft Computig Joural,vol.41,pp ,2016.

12 12 Mathematical Problems i Egieerig [21]C.Lu,J.X.You,H.C.Liu,adP.Li, Health-carewaste treatmet techology selectio usig the iterval 2-tuple iduced TOPSIS method, Iteratioal Joural of Evirometal Research ad Public Health,vol.13,o.6,p.562,2016. [22] Z. S. Xu, Ucertai liguistic aggregatio operators based approach to multiple attribute group decisio makig uder ucertai liguistic eviromet, Iformatio Scieces, vol. 168,o.1 4,pp ,2004. [23] G.-W. Wei, Ucertai liguistic hybrid geometric mea operator ad its applicatio to group decisio makig uder ucertai liguistic eviromet, Iteratioal Joural of Ucertaity, Fuzziess ad Kowledge-Based Systems,vol.17,o.2,pp , [24] J. H. Park, M. G. Gwak, ad Y. C. Kwu, Ucertai liguistic harmoic mea operators ad their applicatios to multiple attribute group decisio makig, Computig, vol. 93, o. 1, pp , [25] H. Zhag, Some iterval-valued 2-tuple liguistic aggregatio operators ad applicatio i multiattribute group decisio makig, Applied Mathematical Modellig, vol. 37, o. 6, pp , [26] J.-Q. Wag, D.-D. Wag, H.-Y. Zhag, ad X.-H. Che, Multicriteria group decisio makig method based o iterval 2- tuple liguistic iformatio ad Choquet itegral aggregatio operators, Soft Computig, vol. 19, o. 2, pp , [27] A. Sigh, A. Gupta, ad A. Mehra, Eergy plaig problems with iterval-valued 2-tuple liguistic iformatio, Operatioal Research,2016. [28] H.-C. Liu, J.-X. You, ad X.-Y. You, Evaluatig the risk of healthcare failure modes usig iterval 2-tuple hybrid weighted distace measure, Computers & Idustrial Egieerig,vol.78, pp ,2014. [29] M. Sha, J. You, ad H. Liu, Some iterval 2-tuple liguistic harmoic mea operators ad their applicatio i material selectio, Advaces i Materials Sciece ad Egieerig, vol. 2016, Article ID , 13 pages, [30]F.Meg,M.Zhu,adX.Che, Somegeeralizeditervalvalued 2-tuple liguistic correlated aggregatio operators ad their applicatio i decisio makig, Iformatica,vol.27,o.1, pp , [31] F. Herrera ad L. Martíez, A 2-tuple fuzzy liguistic represetatio model for computig with words, IEEE Trasactios o Fuzzy Systems,vol.8,o.6,pp ,2000. [32] W.-S. Tai ad C.-T. Che, A ew evaluatio model for itellectual capital based o computig with liguistic variable, Expert Systems with Applicatios, vol. 36, o. 2, pp , [33] F. Herrera ad L. Martíez, A model based o liguistic 2- tuples for dealig with multigraular hierarchical liguistic cotexts i multi-expert decisio-makig, IEEE Trasactios o Systems, Ma, ad Cyberetics Part B: Cyberetics, vol.31, o. 2, pp , [34] H. Zhag, The multiattribute group decisio makig method based o aggregatio operators with iterval-valued 2-tuple liguistic iformatio, Mathematical ad Computer Modellig, vol.56,o.1-2,pp.27 35,2012. [35] X. Liu, Z. Tao, H. Che, ad L. Zhou, A ew iterval-valued 2-tuple liguistic boferroi mea operator ad its applicatio to multiattribute group decisio makig, Iteratioal Joural of Fuzzy Systems,2016. [36] Q.Wu,P.Wu,Y.Zhou,L.Zhou,H.Che,adX.Ma, Some 2-tuple liguistic geeralized power aggregatio operators ad their applicatios to multiple attribute group decisio makig, Joural of Itelliget ad Fuzzy Systems, vol.29,o.1,pp , [37] H. C. Liu, J. X. You, P. Li, ad Q. Su, Failure mode ad effect aalysis uder ucertaity: a itegrated multiple criteria decisio makig approach, IEEE Trasactios o Reliability, vol.65,o.3,pp ,2016. [38] H. Liu, J. You, S. Che, ad Y. Che, A itegrated failure mode ad effect aalysis approach for accurate risk assessmet uder ucertaity, IIE Trasactios, vol. 48, o. 11, pp , [39] L. Martíez ad F. Herrera, A overview o the 2-tuple liguistic model for computig with words i decisio makig: extesios, applicatios ad challeges, Iformatio Scieces, vol. 207, pp. 1 18, [40] Z. Xu, Ucertai Multi-Attribute Decisio Makig: Methods ad Applicatios, Spriger, New York, NY, USA, [41] G.-W. Wei, Extesio of TOPSIS method for 2-tuple liguistic multiple attribute group decisio makig with icomplete weight iformatio, Kowledge ad Iformatio Systems, vol. 25,o.3,pp ,2010. [42] Z. Xu, A overview of methods for determiig OWA weights, Iteratioal Joural of Itelliget Systems, vol.20,o.8,pp , [43] G. Yari ad A. R. Chaji, Maximum Bayesia etropy method for determiig ordered weighted averagig operator weights, Computers & Idustrial Egieerig,vol.63,o.1,pp , [44] B. S. Ah ad H. Park, Least-squared ordered weighted averagig operator weights, Iteratioal Joural of Itelliget Systems,vol.23,o.1,pp.33 49,2008. [45] Z.S.XuadM.M.Xia, Distaceadsimilaritymeasuresfor hesitat fuzzy sets, Iformatio Scieces, vol. 181, o. 11, pp , [46] R. R. Yager, Geeralized OWA aggregatio operators, Fuzzy Optimizatio ad Decisio Makig, vol.3,o.1,pp , [47] J. Q. Wag, J. T. Wu, J. Wag, H. Y. Zhag, ad X. H. Che, Multi-criteria decisio-makig methods based o the Hausdorff distace of hesitat fuzzy liguistic umbers, Soft Computig,vol.20,o.4,pp ,2016. [48] L. A. Zadeh, The cocept of a liguistic variable ad its applicatio to approximate reasoig. I, Iformatio Scieces,vol.8, o.3,pp ,1975. [49] Z. Xu, A method based o liguistic aggregatio operators for group decisio makig with liguistic preferece relatios, Iformatio Scieces,vol.166,o.1 4,pp.19 30,2004.

13 Advaces i Operatios Research Hidawi Publishig Corporatio Advaces i Decisio Scieces Hidawi Publishig Corporatio Joural of Applied Mathematics Algebra Hidawi Publishig Corporatio Hidawi Publishig Corporatio Joural of Probability ad Statistics The Scietific World Joural Hidawi Publishig Corporatio Hidawi Publishig Corporatio Iteratioal Joural of Differetial Equatios Hidawi Publishig Corporatio Submit your mauscripts at Iteratioal Joural of Advaces i Combiatorics Hidawi Publishig Corporatio Mathematical Physics Hidawi Publishig Corporatio Joural of Complex Aalysis Hidawi Publishig Corporatio Iteratioal Joural of Mathematics ad Mathematical Scieces Mathematical Problems i Egieerig Joural of Mathematics Hidawi Publishig Corporatio Hidawi Publishig Corporatio Hidawi Publishig Corporatio Discrete Mathematics Joural of Hidawi Publishig Corporatio Discrete Dyamics i Nature ad Society Joural of Fuctio Spaces Hidawi Publishig Corporatio Abstract ad Applied Aalysis Hidawi Publishig Corporatio Hidawi Publishig Corporatio Iteratioal Joural of Joural of Stochastic Aalysis Optimizatio Hidawi Publishig Corporatio Hidawi Publishig Corporatio