SOME DISTANCE MEASURES FOR INTUITIONISTIC UNCERTAIN LINGUISTIC SETS AND THEIR APPLICATION TO GROUP DECISION MAKING

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1 Lecturer Bi Jiaxi, PhD Uiversity of Shaghai for Sciece ad Techology Zheiag Wali Uiversity Professor Lei Liaghai, PhD Uiversity of Shaghai for Sciece ad Techology Lecturer Peg Bo, PhD ( Correspodig author) Fuda Uiversity Zheiag Wali Uiversity pb_00@63.com SOME DISTANCE MEASURES FOR INTUITIONISTIC UNCERTAIN LINGUISTIC SETS AND THEIR APPLICATION TO GROUP DECISION MAKING Abstract. I this paper, we preset the ituitioistic ucertai liguistic weighted distace measure. It is a ew decisio maig techique that geeralized the OWD measure, havig bee proved suitable to deal with the situatio where the give iformatio is represeted i exact umerical values. We ivestigate the ew distace measure i multi-attribute decisio maig with ituitioistic ucertai liguistic iformatio. Firstly, we develop some distace measures for ituitioistic ucertai liguistic sets, icludig the ituitioistic ucertai liguistic weighted distace (IULWD) measure, the ituitioistic ucertai liguistic ordered weighted distace (IULOWD) measure, the ituitioistic ucertai liguistic ordered weighted Hammig distace (IULOWHD) measure, the ituitioistic ucertai liguistic ordered weighted Euclidea distace (IULOWED) measure, the ituitioistic ucertai liguistic hybrid weighted distace (IULHWD) measure. These developed distace measures are very suitable to deal with the situatio where the iput argumets are represeted i ituitioistic ucertai liguistic sets. The we study several desirable properties of the ew distace measures ad preset a cosesus reachig process based o the developed distace measures with ituitioistic ucertai liguistic preferece iformatio for group decisio maig. Fially, we apply the developed approach with a umerical example to group decisio maig uder ituitioistic ucertai eviromet. Keywords: Ituitioistic ucertai liguistic sets; Distace measure; Group decisio maig; Cosesus. JEL Classificatio: D8, M, M5

2 Bi Jiaxi, Lei Liaghai, Peg Bo. Itroductio I the real-life world, distace measure is a commo used tool for measurig the deviatios of differet argumets. Its theory ad methods have bee widely applied i may fields, such as decisio maig, medical diagosis,iformatio fusio, supply chai maagemet ad so o. Over the last decades, the study o distace measure has attracted great attetios, refer to [Szmidt ad Kacprzy 000; Merigó ad Gil-Lafuete 00; Zeg ad Su 0]. May authors proposed some distace measures icludig the weighted Hammig distace (WHD) measure ad the weighted Euclidea distace (WED) measure, etc. i most of the existig literature; however, these distace measures oly cosider the importace of each deviatio value. To solve the drawbac, Xu ad Che (008) itroduced the ordered weighted distace (OWD) measure based o the ordered weighted averagig (OWA) operator (Yager 988). The promiet characteristic of the OWD measure is that it ca relieve (or itesify) the ifluece of uduly large or small deviatios o the aggregatio results by assigig low (or high) weights of them ad emphasize the importace of the ordered positio of the give idividual distaces istead of weightig argumets themselves. For further research o the other distace measures usig the OWA operator ad their applicatios, please see, for example [Merigó ad Casaovas 0; Merigó ad Gil-Lafuete 00; Xu 007]. The above distace measures ust discuss the decisio iformatio is expressed i exact umerical umbers. However, i practical applicatios the available iformatio may be represeted by ucertai or fuzzy argumets, icludig ituitioistic fuzzy sets (IFSs) (Ataassov 986), iterval-value ituitioistic fuzzy sets (IVIFSs) (Ataassov ad Gargov 989), ad liguistic labels (Herrera ad Martiez 000) because of time pressure, people s limited expertise related to the problem domai ad so o. As a result, Xu (007) proposed some similarity measures of ituitioistic fuzzy sets based o the geometric distace ad the matchig fuctio model. Zeg (03) itroduced some ituitioistic fuzzy weighted distace measures, lie ituitioistic fuzzy ordered weighted distace (IFOWD) measure ad ituitioistic fuzzy hybrid weighted distace (IFHWD) measure. Recetly, motivated by the idea of the ituitioistic liguistic set (IUS) (Wag ad Li 009), Liu ad Ji (0) developed the otio of ituitioistic ucertai liguistic set (IULS), which ca be viewed as a collectio of the ituitioistic ucertai liguistic variables. Ad the, Liu (03) proposed a approach to group decisio maig based o the iterval ituitioistic ucertai liguistic sets. However, i the literature it seems that there is o ivestigatio o distace measure for aggregatig a collectio of ituitioistic ucertai liguistic sets, except of some similarity measures of ituitioistic fuzzy sets (Xu 007) ad some ituitioistic fuzzy weighted distace measures (Zeg 03). The research o the cosesus reachig process based o distace measures with ituitioistic liguistic preferece iformatio for group decisio maig is i its ifacy.

3 Some Distace Measures for Ituitioistic Ucertai Liguistic Sets ad their Applicatio to Group Decisio Maig Therefore, based o the ituitioistic ucertai liguistic sets (Liu ad Ji 0), i this paper, we shall develop some distace measures for ituitioistic ucertai liguistic sets, such as the ituitioistic ucertai liguistic weighted distace (IULWD) measure, the ituitioistic ucertai liguistic ordered weighted distace (IULOWD) measure ad the ituitioistic ucertai liguistic hybrid weighted distace (IULHWD) measure. These developed distace measures are very suitable to deal with the situatios where the available iformatio is represeted i ituitioistic ucertai liguistic sets. Also, they ca alleviate (or itesify) the ifluece of uduly large (or small) deviatios o the aggregatio results by assigig low (or high) weights of them. To do so, this paper is structured as follows. I Sectio, we review some commo distace measures ad the ituitioistic ucertai liguistic sets. I Sectio 3, we develop the IULWD measure, the IULOWD measure ad the IULHWD measure, ad study the various properties of them. I Sectio 4, we propose a approach to establish a cosesus reachig process of ituitioistic ucertai liguistic group decisio maig. I Sectio 5, we give a practical applicatio of the developed approach, ad the mai coclusios of the paper are summarized i Sectio 6.. Prelimiaries I this sectio, we first review some basic distaces measures, ad the itroduce the otio of the ituitioistic ucertai liguistic sets... Some distace measures The weighted Hammig distace (WHD) ad the weighted Euclidea distace (WED) are the most wide used distace measures, which are based o the ormalized Hammig distace (NHD) ad the ormalized Euclidea distace (NED) (Kacprzy 997). For two collectios of argumets A a, a,, a ad B b, b,, b, they ca be defied as follows: Defiitio. A weighted Hammig distace (WHD) of dimesio is a mappig WHD: R R that has a associated weightig (,,, ) T such that [0,], ad WHD( A, B) i ai b () i where a i ad b i is the ith argumets of the A ad B, respectively. i

4 Bi Jiaxi, Lei Liaghai, Peg Bo Defiitio. A weighted Euclidea distace (WED) of dimesio is a mappig WED: R R that has a associated weightig (,,, ) T such that [0,], ad WED( A, B) a b i i () i where a i ad b i is the ith argumets of the A ad B, respectively. Cosider a geeralizatio of both the distaces measures () ad (), a weighted distace (WD) is defied as follows: i i i i, 0 i WD a b (3) However, the above weighted distace measures tae oly the give idividual distaces ito cosideratio rather tha the ordered positios of the give idividual. Yager (988) developed the wide useful OWA operator, the promiet advatage of the OWA operator is that the iput argumets are rearraged i descedig order, ad the weights associated with the operator are the weights of the ordered positios of the iput argumets rather tha the weights of the iput argumets. Sice its appearace, the OWA operator has bee widely studied ad used i a rage of applicatios, see, for example [Herrera et al. 003; Peg et al. 0; Peg ad Ye 03; Xu 005a]. Motivated by the idea of the OWA operator, Xu ad Che (008) developed a ordered weighted distace (OWD) measure. Defiitio 3. Let A a, a,, a ad B b, b,, b be two collectios of real umbers, ad, d a b a b be the distace betwee a ad b, the OWD A, B w d a( ), b( ) (4) is called a ordered weighted distace (OWD) betwee A ad B, i which 0, w ( w, w,, w ) T is the weighted vector of the ordered positio of the d a ( ), b, where [0,] ( ) permutatio of,,,, such that w, w, ad (), (),, ( ) is ay ( ), ( ) ( ), ( ) d a b d a b (5) I the case of ad, the OWD measure is called the ordered weighted Hammig distace (OWHD) measure ad the ordered weighted Euclidea distace (OWED) measure:

5 Some Distace Measures for Ituitioistic Ucertai Liguistic Sets ad their Applicatio to Group Decisio Maig ad, ( ), ( ) OWHD A B w d a b (6), ( ), ( ) OWED A B w d a b (7) Later, Xu (00) defied a distace measure called ituitioistic fuzzy distace (IFD) based o the cocept of ituitioistic fuzzy set (IFS), itroduced by Ataassov (986). The promiet characteristic of IFS is that it assigs to each elemet a membership degree ad a o-membership degree, while the fuzzy set (Zadeh 965) oly assigs to each elemet a membership degree. Over the last decades, may authors have paid attetio to the applicatio to group decisio maig based o IFS [Ataassov ad Gargov 989; Szmidt ad Kacprzy 000; Xu 007; Xu ad Yager 006]. Xu ad Yager (006) itroduced the ituitioistic fuzzy umbers (IFNs), which simplify the otio of the IFSs. For ay two IFNs, the ituitioistic fuzzy distace (IFD) is defied as follows: Defiitio 4. Let ad be two IFNs, the d, (8) is called the ituitioistic fuzzy distace (IFD) betwee ad. Based o the ituitioistic fuzzy distace (IFD) ad the weighted distace (WD), Zeg (03) proposed some ituitioistic fuzzy weighted distace measures, lie ituitioistic fuzzy ordered weighted distace (IFOWD) measure, ituitioistic fuzzy hybrid weighted distace (IFHWD) measure ad so o. These weighted distace measures ca deal with the situatio where the iput argumets are represeted i ituitioistic fuzzy umbers (IFNs). S s t,,,0,,, t be a fiite ad totally ordered discrete Let term set, where s represets a possible value for a liguistic variable. For example, i the case oft 3, S ca be defied as: S { s very poor, s poor, s slightly poor, s medium, 3 0 s slightly good, s good, s3 very good} Note that i the process of give iformatio aggregatig, some decisio results may do ot match ay liguistic labels exactly. To preserve all the give iformatio, Xu (004) exteded the discrete label set S to a cotiuous label S s [ q, q], where q( q t) is a sufficietly large positive umber. set If s S, we call s origial liguistic label, otherwise, we call s the virtual liguistic label.

6 Bi Jiaxi, Lei Liaghai, Peg Bo With respect to measure the deviatio betwee two liguistic variables s, s S, Xu (005b) defied the liguistic distace as follows: Defiitio 5. Let s, s S, the d( s, s ) s s (9) t is called the distace measure betwee s ad s... The ituitioistic ucertai liguistic sets Motivated by the idea of the ituitioistic liguistic set (IUS) (Wag ad Li 009) ad the ucertai liguistic variables (Xu 004), Liu ad Ji (0) itroduced the otio of ituitioistic ucertai liguistic set (IULS), which ca be defied as follows: Defiitio 6. A ituitioistic ucertai liguistic set i X is give as: A { x[[ s( x), s( x) ],( A( x), A( x))] x X} (0) where [ s( x), s( x) ] S, ( x A ) : X [0,] ad ( x A ) : X [0,] with the coditio of 0 ( x) ( x), x X. Also, the umbers ( x ) ad A A ( x ) represet the membership degree ad o-membership degree of the A elemet x to the ucertai liguistic variable [ s( x), s ( x) ], respectively. Ad if ( x) ( x) ( x), x X, the ( x ) is called the degree of A A A idetermiacy of the elemet x to the ucertai liguistic variable [ s( x), s ( x) ]. For a ituitioistic ucertai liguistic set A, Liu ad Ji (0) defied the ituitioistic ucertai liguistic variable, which ca be expressed as the quaterio [ s( x), s( x) ],( A( x), A( x)). The ituitioistic ucertai liguistic set A ca also be viewed as a collectio of the ituitioistic ucertai liguistic variables. Therefore, the ituitioistic ucertai liguistic set A ca also be deoted by A { [ s( x), s( x) ],( A( x), A( x)) x X}. For ay two ituitioistic ucertai liguistic variables ad, the operatioal laws are defied as follows: () [ s ( x ) ( x ), s ( x ) ( x )],( ( A( x ))( A( x )), A( x ) A( x )) A A

7 Some Distace Measures for Ituitioistic Ucertai Liguistic Sets ad their Applicatio to Group Decisio Maig () [ s ( x ) ( x ), s ( x ) ( x )],( A( x ) A( x), A( x ) A( x) A( x ) A( x)) (3) [ s, s ],( ( ( x )),( ( x )) ), 0 ( x) ( x) A A Ad the above operatioal results are still ituitioistic ucertai liguistic variables. To compare ay two ituitioistic ucertai liguistic variables ad, Liu ad Ji (0) proposed a simple method as below: If the expected value E( ) E( ), the is smaller tha, deoted ; by If E( )= E( ), the ) If the accuracy fuctio H( ) H( ), the ; ) If H( ) H( ), the ad represet the same iformatio, deoted. by 3. Some ituitioistic ucertai liguistic distace measures I this sectio, we first defie a distace measure for each pair of ituitioistic ucertai liguistic variable, ad the develop some ituitioistic ucertai liguistic distace measures. At last, we study the various properties of them. Defiitio 7. For ay two ituitioistic ucertai liguistic variables [ s, s ],( ( x ), ( x )) ad [ s, s ],( ( x ), ( x )), ( x) ( x) the d, ( x) ( x) ( d ([ s ( x ), s ( x ) ],[ s ( x ), s ( x ) ]) d (( ( x ), ( x )),( ( x ), ( x )))) (( ( x ) ( x ) ( ) ( ) ) / ( ) ( ) ( ) ( ) ) x x t x x x x 4 () is called the ituitioistic ucertai liguistic distace (IULD) betwee ad. Cosider two IULSs A { [ s ( x), s ( x) ],( A( x), A( x)) x X} ad A B { [ s ( ), ( )],( ( ), ( )) } B x sb x B x B x x X X x, x,, x, we let Ax ( ) ad Bx ( ), the the ituitioistic ucertai liguistic sets A ad B o ca be deoted by A ad B,,, A,,,. Based o the

8 Bi Jiaxi, Lei Liaghai, Peg Bo iformatio above, we ca calculate the distace betwee the ituitioistic ucertai liguistic sets A ad B utilizig the IULD betwee i ad i, i,,,. Defiitio 8. Let A ad B,,, ituitioistic ucertai liguistic variables, the IULWHD IULD,,, be two sets of d A, B d (, ) () is called a ituitioistic ucertai liguistic weighted Hammig distace (IULWHD) betwee A ad B. Defiitio 9. Let A ad B,,, ituitioistic ucertai liguistic variables, the, ( (, )) IULWED IULD,,, be two sets of d A B d (3) is called a ituitioistic ucertai liguistic weighted Euclidea distace (IULWED) betwee A ad B, where (,,, ) T is the weightig vector of the diuld (, ) such that [0,],. Combie Eqs. () ad (3) to the followig form: diulwd A, B ( diuld (, )) (4) which is called a ituitioistic ucertai liguistic weighted distace (IULWD) betwee A ad B. I the case of ad, the IULWD measure is reduced to the IULWHD measure () ad the IULWED measure (3), respectively. Based o the OWD measure (4) ad the IULWD measure (4), we ca defie a ituitioistic ucertai liguistic ordered weighted distace (IULOWD) measure as follows: Defiitio 0. Let A,,, ad B,,, be two sets of ituitioistic ucertai liguistic variables, the diulowd A, B w ( diuld ( ( ), ( ) )) (5) is called a ituitioistic ucertai liguistic ordered weighted distace (IULOWD) betwee A ad B, where w ( w, w,, w ) T is the weighted vector of the,, with the coditio w [0,], ordered positio of the diuld ( ) ( )

9 Some Distace Measures for Ituitioistic Ucertai Liguistic Sets ad their Applicatio to Group Decisio Maig w. (), (),, ( ) is ay permutatio of d ( ), ( ) d ( ), ( ),,,, such that (6) Especially, if, the the IULOWD measure is called a ituitioistic ucertai liguistic ordered weighted Hammig distace (IULOWHD) measure: ( ) ( ) d A, B w d (, ) (7) IULOWHD IULD Ad if, the IULOWD measure is called a ituitioistic ucertai liguistic ordered weighted Euclidea distace (IULOWED) measure:, ( (, )) d A B w d (8) IULOWED IULD ( ) ( ) From the distace measure Eqs. (4) ad (5), we ow that the IULWD measure weights the give idividual distaces while the IULOWD measure weights the ordered positios of the give idividual distaces istead of weightig the idividual distaces themselves. Therefore, weights represet differet aspects i both the two measures. To overcome the drawbac, we develop a ituitioistic ucertai liguistic hybrid weighted distace (IULHWD) measure, which is defied as follows: Defiitio. Let A,,, ad B,,, be two sets of ituitioistic ucertai liguistic variables, the diulhwd A, B wdiuld ( ( ), ( ) ) (9) is called a ituitioistic ucertai liguistic hybrid weighted distace (IULHWD) measure betwee A ad B, where w ( w, w,, w ) T is the weightig vector associated with the IULHWD measure, DIULD ( ( ), ( ) ) is the th largest of the weighted distace (, ) IULD D ( DIULD (, ) diuld (, ),,,, ), ad (,,, ) T is the weightig vector of the diuld (, ) such that [0,],, is the balacig coefficiet. A,,, 5 ( [ s, s ],(0.6,0.3), [ s 4, s ], Example. Let

10 Bi Jiaxi, Lei Liaghai, Peg Bo (0.5,0.4), [ s, s ],(0.7,0.), [ s, s ],(0.4,0.5), [ s, s ],(0.,0.6) ) ad B,,, 5 ( [ s 3, s ],(0.5,0.4), [ s3, s4],(0.7,0.3), [ s0, s],(0.,0.5), [ s3, s4],(0.4,0.), [ s, s],(0.8,0.) be two sets of ituitioistic ucertai liguistic variables, the by utilizig Eq. (), we ca get, d, 0.485, diuld, IULD diuld 3, , IULD 4 4 d, IULD 5 5 d, 0.875, Suppose that (0.5, 0.3, 0., 0.5, 0.), ad without loss of geerality, let, the we ca get DIULD (, ) Similarly, we have D ( IULD, ) , D ( IULD, ) , D ( IULD, ) , D ( IULD, ) The weight vector associated with the ituitioistic ucertai liguistic hybrid weighted distace (IULHWD) measure w (0.,0.4,0.3,0.4,0.) T, which is derived by usig the Gaussia distributio based method, for more details, refer to Xu (005a). The we ca get the IULHWD betwee A ad B by utilizig Eq. (9): 5 diulhwd A, B wdiuld ( ( ), ( ) ) ( / ) Theorem The IULWD measure is a special case of the IULHWD measure. Proof. Let w (,,, ) T, the diulhwd A, B wdiuld ( ( ), ( ) ) d IULWD ( d (, )) IULD A, B which completes the proof of Theorem. D (, ) IULD ( d (, )) IULD

11 Some Distace Measures for Ituitioistic Ucertai Liguistic Sets ad their Applicatio to Group Decisio Maig Theorem. The IULOWD measure is a special case of the IULHWD measure. Proof. Let (,,, ) T, the D (, ) ( d (, )) ( d (, )) IULD ( ) ( ) IULD ( ) ( ) IULD ( ) ( ) which completes the proof of Theorem. From the IULHWD measure ad above aalysis, we ca get: () By computatioal aalysis, we ow that the IULHWD measure ca relieve (or itesify) the ifluece of uduly large or small differece idividual o the aggregatio results by assigig them low (or high) weights. () The IULHWD measure geeralizes both the IULWD ad IULOWD measure ad reflects the importace degrees of both the give idividual distaces ad their ordered positios. (3) I fact, firstly, the IULHWD measure weights the give idividual distaces, ad the reorders the weighted idividual distaces i descedig order ad weights these ordered idividual distaces by the IULHWD weights. At last, we process these idividual distaces ito a collective oe uder the parameter. 4. A approach to group decisio maig based o ituitioistic ucertai liguistic variables Cosider a group decisio maig problem with ituitioistic ucertai liguistic iformatio. Let X x, x,, x be a discrete set of alteratives, d D(,,, m) be the set of decisio maers (DMs), ad u u u u m (,,, ) T be the weight vector of DMs, with the coditio m u 0, u. The DMs d (,,, m ) provide their prefereces with ituitioistic ucertai liguistic variable (,,, ) over all the alteratives X respect to a criterio. For computatioal coveiece, the x preferece vectors of all the DMs d are deoted by: (,,, ),,,, m (0) Based o the above decisio iformatio, we shall develop a approach to reachig cosesus of group opiios utilize the IULHWD measure as follows: Step :Calculate the collective preferece vector (,,, ) by usig the ituitioistic ucertai liguistic weighted average operator, ad we have

12 Bi Jiaxi, Lei Liaghai, Peg Bo u u u,,,, () m m Step :Calculate the distace diuld (, ) of each preferece value give by the decisio maer d ad the correspodig collective preferece with ituitioistic ucertai liguistic variable by usig Eq. (). Step 3:By usig Eq. (9), we calculate the IULHWD measure betwee the preferece vectors ad : diulhwd, wdiuld ( ( ), ( ) ) () where w (,,, ) T w w w is the weightig vector associated with the IULHWD measure, ca be derived by usig the Gaussia distributio based method (Xu 005a), DIULD ( ( ), ( ) ) is the th largest of the weighted distace DIULD (, ) ( DIULD (, ) diuld (, ),,,, ), ad (,,, ) T is the weightig vector of the d (, ) such that [0,],. IULD Step 4 : A discussio o the cosesus reachig process for group decisio maig: () Let be the threshold value of acceptable cosesus, if all, d (,,, m) IULHWD, the the group is of acceptable cosesus. Therefore, it ca be determied by the group i practical applicatios. () Otherwise, if there exists some 0 d,, such that, IULHWD 0 the we shall retur 0 (together with as a referece) to the decisio maer d for revaluatio, ad repeat this cosesus reachig process util, IULHWD 0 d or the umber of rouds reach the maximum which is predefied by the group so as to avoid stagatio. 5. Illustrative example Let us cosider a decisio maig problem of evaluatig port logistics system for vulerability ad promotio discussed i (Zhag et al. 0). Amog may criterios of the system evaluatio, cargo throughput is the mai criterio used.

13 d 3 4 Some Distace Measures for Ituitioistic Ucertai Liguistic Sets ad their Applicatio to Group Decisio Maig There are four port cadidates x X (,,3, 4) ad three decisio maers (DMs) d D(,,3) (whose weightig vector is u (0.4,0.3,0.3) T ). Ad each decisio maer d provides his/her prefereces with ituitioistic ucertai liguistic variable (,,3;,,3, 4) over all the port cadidates x, show i Table. Table. Decisio matrix with IULSs x x x 3 x 4 [ s, s ],(0.3,0.6) [ s0, s],(0.7,0.) [ s, s],(0.4,0.6) [ s0, s],(0.4,0.3) [ s, s ],(0.4,0.3) [ s, s0],(0.8,0.) [ s 4, s ],(0.6,0.) [ s, s3],(0.,0.8) d [ s, s ],(0.,0.3) [ s, s],(0.7,0.) [ s, s],(0.4,0.5) [ s 3, s ],(0.3,0.4) d3 3 For computatioal coveiece, the prefereces of all the DMs d D(,,3) are deoted by the vector forms as follows: (,,, ) ( [ s, s ],(0.3,0.6), [ s, s ],(0.7,0.), [ s, s ],(0.4,0.6), [ s, s ],(0.4,0.3) ) 0 (,, 3, 4) ( [ s, s ],(0.4,0.3), [ s, s0],(0.8,0.), [ s, s ],(0.6,0.), [ s, s ],(0.,0.8) ) ( 3, 3, 33, 34) ( [ s, s3],(0.,0.3), [ s, s],(0.7,0.), [ s, s],(0.4,0.5), [ s 3, s ],(0.3,0.4) ) Calculate the collective preferece vector by usig u u u,,,3,4 3 3 We ca have (,,, ) 3 4 ( [ s, s ],(0.3,0.4), [ s, s ],(0.7,0.), [ s, s ],(0.5,0.3), [ s, s ],(0.4,0.5) ) Ad the, by usig Eq. (), we ca calculate the distace diuld (, ) of each preferece value give by the DM d ad the correspodig collective preferece with ituitioistic ucertai liguistic variable :

14 Bi Jiaxi, Lei Liaghai, Peg Bo d ( IULD, ) 0.56, d ( IULD, ) 0.8, d ( IULD, ) , d ( IULD, ) ; d ( IULD, ) 0.4, d ( IULD, ) 0.344, d ( IULD, ) , d ( IULD, ) ; d ( IULD, ) , d ( IULD, ) 0.9 3, d ( IULD, ) , d ( IULD, ) Without loss of geerality, let ad (0.,0.35,0.5,0.3) T, the weight vector associated with the ituitioistic ucertai liguistic hybrid weighted distace measure w (0.55,0.345,0.345,0.55) T, which is derived by the Gaussia distributio based method (Xu (005a)). The we calculate the IULHWD measure betwee the preferece vectors ad by usig Eq. (): d, d,, 0., 0.78 IULHWD IULHWD d, 0.0 IULHWD 3. Suppose the threshold value of acceptable cosesus is 0.50, the we ca get d, 0.50 IULHWD Now, we eed to retur (together with as a referece) to the DM d for revaluatio. Suppose the revaluated is (,, 3, 4) ( [ s0, s],(0.4,0.), [ s, s0],(0.7,0.), [ s 3, s ],(0.5,0.), [ s, s3],(0.,0.6) ) Ad we ca calculate the collective preferece vector oce more (,,, ) 3 4 ( [ s, s ],(0.3,0.3), [ s, s ],(0.6,0.), [ s 0., s0.8 ],(0.3,0.4), [ s 0.3, s0.7 ],(0.4,0.4) ) Respectively, we ca have the IULHWD measure betwee the preferece vectors ad by usig Eqs. () ad () (let ): d, d,, 0.088, 0.46 IULHWD IULHWD d, 0.03 IULHWD 3. As we ca see, the recalculated umerical results are less tha 0.50, that is d, 0.50 (,,3). Thus, all the distaces are less tha the IULHWD predefied threshold value of acceptable cosesus, which idicates that the group reaches cosesus. Moreover, the process of group reaches cosesus i the cases of ca be discussed similar to the case of. 6. Coclusios I this paper, we have suggested several extesios of the commo used distace measures whe dealig with ucertai liguistic situatios. We first itroduce the otio of ituitioistic ucertai liguistic sets ad ituitioisic ucertai liguistic

15 Some Distace Measures for Ituitioistic Ucertai Liguistic Sets ad their Applicatio to Group Decisio Maig variables, ad the develop some ew distace measures to accommodate the situatios where the give argumets are ituitioistic ucertai liguistic iformatio, icludig the IULWD measure, the IULOWD measure, ad the IULHWD measure ad so o. Also, we have studied several desirable properties of the IULHWD measure, which ca alleviate the ifluece of uduly large (or small) deviatios o the aggregatio results by assigig them low (or high) weights. We have preseted a ew group decisio maig process based o the IULHWD measure ad fially we have focused o a applicatio i a group decisio maig problem of evaluatig port logistics system for vulerability ad promotio. I the future, we shall cotiue worig i extedig the distace measures to deal with the situatios where the iput argumets are expressed i other ucertai iformatio icludig iterval ituitioistic ucertai vaviables, triagular ituitioistic fuzzy values or pure liguistic labels. We will also ivestigate differet types of applicatios i other domais based o the distace measures. Acowledgemets The authors are very grateful to the editor ad the aoymous referees for their isightful ad costructive commets ad suggestios. This wor was supported by the Philosophy ad Social Sciece Plaig Proect of Shaghai (No. 04EZZ00),Shaghai Muicipal Govermet Decisio-maig Cosultatio Proect (No.05-Z-A0-C), the Chia Postdoctoral Sciece Foudatio (No. 05M57494), Zheiag Provice Natural Sciece Foudatio (No.LQ5G00003) ad Nigbo Natural Sciece Foudatio (No.05A607). REFERENCES [] K.T. Ataassov (986), Ituitioistic Fuzzy Sets. Fuzzy Sets ad Systems, 0, 87-96; [] K.T. Ataassov ad G. Gargov (989), Iterval Valued Ituitioistic Fuzzy Sets. Fuzzy Sets ad Systems, 3, ; [3] F. Herrera, E. Herrera-Viedma ad F. Chiclaa (003), A study of the Origi ad Uses of the Ordered Weighted Geometric Operator i Multicriteria Decisio Maig. Iteratioal Joural of Itelliget Systems, 8, ; [4] F. Herrera ad L. Martiez (000), A Tupple Fuzzy Liguistic Represetatio Model for Computig with Words. IEEE Trasactios o Fuzzy Systems, 8, ; [5] J. Kacprzy (997), Multistage Fuzzy Cotrol: A Model-based Approach to Cotrol ad Decisio-maig; Wiley, Chichester;

16 Bi Jiaxi, Lei Liaghai, Peg Bo [6] E. Szmidt ad J. Kacprzy. (000) Distaces betwee Ituitioistic Fuzzy Sets. Fuzzy Sets ad Systems, 4, ; [7] P. D. Liu ad F. Ji (0), Methods for Aggregatig Ituitioistic Ucertai Liguistic Variables ad their Applicatio to Group Decisio Maig. Iformatio Scieces, 05, 58-7; [8] P. D. Liu (03), Some Geometric Aggregatio Operators Based o Iterval Ituitioistic Ucertai Liguistic Variables ad their Applicatio to Group Decisio Maig. Applied Mathematical Modellig, 37, ; [9] J. M. Merigó ad A.M. Gil-Lafuete (00), New Decisio Maig Techiques ad their Applicatio i the Selectio of Fiacial Products. Iformatio Scieces, 80, ; [0] J. M. Merigó ad M. Casaovas (0), Decisio Maig with Distace Measures ad Iduced Aggregatio Operators. Computers ad Idustrial Egieerig, 60, 66-76; [] B. Peg, C. M. Ye ad S. Z. Zeg (0), Ucertai Pure Liguistic Hybrid Harmoic Averagig Operator ad Geeralized Iterval Aggregatio Operator Based Approach to Group Decisio Maig. Kowledge-Based Systems, ; [] B. Peg ad C. M. Ye (03), A Approach Based o the Iduced ucertai Pure Liguistic Hybrid Harmoic Averagig Operator to Group Decisio Maig. Ecoomic Computatio ad Ecoomic Cyberetics Studies ad Research, 47(4), 75-95; ASE Publishig, Bucharest; [3] J. Q. Wag ad J. J. Li (009), The Multi-criteria Group Decisio Maig Method Based o Multi-graularity Ituitioistic Two Sematics. Sciece & Techology Iformatio, 33, 8-9; [4] Z. S. Xu (004), Ucertai Liguistic Aggregatio Operators Based Approach to Multiple Attribute Group Decisio Maig uder ucertai Liguistic Eviromet. Iformatio Scieces, 68, 7-84; [5] Z. S. Xu (005a), A Overview of Methods for Determiig OWA Weights. Iteratioal Joural of Itelliget Systems, 0, ; [6] Z. S. Xu (005b), Deviatio Measures of Liguistic Preferece Relatios i Group Decisio Maig. Omega, 33, 49-54; [7] Z.S. Xu (007), Some Similarity Measures of Ituitioistic Fuzzy Sets ad their Applicatios to Multiple Attribute Decisio Maig. Fuzzy Optimizatio ad Decisio Maig, 6, 09-; [8] Z.S. Xu (00), A Deviatio-based Approach to Ituitioistic Fuzzy Multiple Attribute Group Decisio Maig. Group Decisio ad Negotiatio, 9, 57-76; [9] Z.S. Xu ad J. Che (008), Ordered Weighted Distace Measure. Joural of Systems Sciece ad Systems Egieerig, 7, ; [0] Z.S. Xu ad R.R. Yager (006), Some Geometric Aggregatio Operators Based o Ituitioistic Fuzzy Sets. Iteratioal Joural of Geeral Systems, 35, ;

17 Some Distace Measures for Ituitioistic Ucertai Liguistic Sets ad their Applicatio to Group Decisio Maig [] R.R. Yager (988), O Ordered Weighted Averagig Aggregatio Operators i Multi-criteria Decisio Maig. IEEE Trasactios o Systems, Ma, ad Cyberetics, 8, 83-90; [] L.A. Zadeh (965), Fuzzy Sets. Iformatio ad Cotrol, 8, ; [3] S. Z. Zeg (03), Some Ituitioistic FuzzyWeighted Distace Measures ad their Applicatio to Group Decisio Maig. Group Decisio ad Negotiatio,, 8-98; [4] S. Z. Zeg ad W. H. Su (0), Liguistic Iduced Geeralized Aggregatio Distace Operators ad their Applicatio to Decisio Maig. Ecoomic Computatio ad Ecoomic Cyberetics Studies ad Research, 46, 55-7; ASE Publishig, Bucharest; [5] W. X. Zhag, T. Y. Xi ad R. F. Zhag (0), A Case Research o Vulerability of Logistics System i the Tiai Port. Eergy Procedia, 5,