Correlation Coefficients of Extended Hesitant Fuzzy Sets and Their Applications to Decision Making

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1 S S symmetry rticle Correlatio Coefficiets of Exteded Hesitat Fuzzy Sets ad Their pplicatios to Decisio Makig Na Lu * ad Lipi Liag School of Ecoomics ad dmiistratio, Taiyua Uiversity of Techology, Taiyua , Chia; liagtyut@26.com * Correspodece: lua23456@26.com cademic Editor: gel Garrido Received: 9 December 206; ccepted: 22 March 207; Published: 29 March 207 bstract: Exteded hesitat fuzzy sets EHFSs), which allow the membership degree of a elemet to a set represeted by several possible value-groups, ca be cosidered as a powerful tool to express ucertai iformatio i the process of group decisio makig. Therefore, we derive some correlatio coefficiets betwee EHFSs, which cotai two cases, the correlatio coefficiets takig ito accout the legth of exteded hesitat fuzzy elemets EHFEs) ad the correlatio coefficiets without takig ito accout the legth of EHFEs, as a ew extesio of existig correlatio coefficiets for hesitat fuzzy sets HFSs) ad apply them to decisio makig uder exteded hesitat fuzzy eviromets. real-world example based o the eergy policy problem is employed to illustrate the actual eed for dealig with the differece of evaluatio iformatio provided by differet experts without iformatio loss i decisio makig processes. Keywords: EHFSs; correlatio; correlatio coefficiets; decisio makig MSC: 03E72; 03E75. Itroductio Whe people make a decisio, they are usually hesitat ad irresolute for oe thig or aother, which makes it difficult to reach a fial agreemet, that is there usually exists a hesitatio or ucertaity about the degree of sureess about the fial decisio. Torra et al. [,2] proposed the hesitat fuzzy set, which permits the membership to have a set of possible values, ad discussed the relatioship betwee hesitat fuzzy sets ad taassov s ituitioistic fuzzy sets [3]. The hesitat fuzzy set is a very useful tool to deal with ucertaity; more ad more decisio makig theories ad methods uder the hesitat fuzzy eviromet have bee developed sice its appearace. Yi [4] gave some properties of operatios ad algebraic structures of hesitat fuzzy sets. Xia ad Xu [5] proposed hesitat fuzzy iformatio aggregatio techiques ad their applicatio i decisio makig. The, Xu ad Xia [6] itroduced a variety of distace measures for hesitat fuzzy sets ad their correspodig similarity measures. Meawhile, Xu ad Xia [7] defied the distace ad correlatio measures for hesitat fuzzy iformatio ad the discussed their properties i detail. Xu et al. [8] developed some hesitat fuzzy aggregatio operators with the aid of quasi-arithmetic meas ad applied them to group decisio makig problems. Gu et al. [9] ivestigated a evaluatio model for risk ivestmet with hesitat fuzzy iformatio; they utilized the hesitat fuzzy weighted averagig operator to aggregate the hesitat fuzzy iformatio correspodig to each alterative ad the raked the alteratives ad selected the most desirable oes) accordig to the score fuctio for hesitat fuzzy sets. Wei [0] developed some prioritized aggregatio operators to aggregate hesitat fuzzy iformatio ad the applied them to hesitat fuzzy multiple attribute decisio makig problems, i which the attributes are at differet priority levels. lcatud et al. [] itroduced a ovel methodology for rakig hesitat Symmetry 207, 9, 47; doi:0.3390/sym

2 Symmetry 207, 9, 47 2 of 2 fuzzy sets ad built o a recet, theoretically-soud cotributio i social choice. Che et al. [2] proposed some correlatio coefficiet formulas for hesitat fuzzy sets ad applied them to clusterig aalysis uder hesitat fuzzy eviromets. dditioally, a positio ad perspective aalysis of hesitat fuzzy sets [3] is give to show the importat role of hesitat fuzzy sets o iformatio fusio i decisio makig. However, hesitat fuzzy sets have some drawbacks. if the two decisio makers DMs) both assig the same value, we ca oly save oe value by the hesitat fuzzy elemet ad lose the other oe, which appears to be a iformatio loss problem of HFSs. Further, sice geerally the DMs have differet importace i group decisio makig [4,5] due to their differet social importace, positio i the group, previous merits, etc., for example, the loss of iformatio provided by the leadig DM may lead to ieffective results. To resolve the iformatio loss problem, Zhu ad Xu [6] itroduced the defiitio of EHFS, which is a extesio of the hesitat fuzzy set [,2]. EHFSs ca better deal with the situatios that permit the membership of a elemet to a give set havig value-groups, which ca avoid givig DMs prefereces aoymously that cause iformatio loss. EHFSs icrease the richess of umerical represetatio based o the value-groups, ehace the modelig abilities of HFSs ad ca idetify differet DMs i decisio makig, which expad the applicatios of HFSs i practice. Correlatio is oe of the most broadly applied idices i may fields ad also a importat measure i data aalysis ad classificatio, patter recogitio, decisio makig, ad so o [7 23]. s may real-world data may be fuzzy, the cocept of correlatio has bee exteded to fuzzy eviromets [2,24 26] ad ituitioistic fuzzy eviromets [27 33]. For istace, Gerstekor ad Mako [27] itroduced the correlatio coefficiets of ituitioistic fuzzy sets. Hog ad Hwag [26] also defied them i probability spaces. Mitchell [3] derived the correlatio coefficiet of ituitioistic fuzzy sets by iterpretig a ituitioistic fuzzy set as a esemble of ordiary fuzzy sets. Hug proposed a method to calculate the correlatio coefficiets of ituitioistic fuzzy sets by meas of the cetroid. ecause of the potetial applicatios of correlatio coefficiets, they have bee further exteded by ustice ad urillo [32] ad Hog [33] for iterval-valued ituitioistic fuzzy sets. Several ew methods of derivig the correlatio coefficiets for both ituitioistic fuzzy sets ad iterval-valued ituitioistic fuzzy sets have also bee proposed i [8]. I 203, Che et al. [2] proposed correlatio coefficiets of hesitat fuzzy sets ad their applicatios to clusterig aalysis. Thus, we urgetly eed to put forward the correlatio coefficiets of EHFSs to deal with these problems. I this paper, we further itroduce the correlatio of EHFSs, which is a ew extesio of the correlatio of hesitat fuzzy sets ad ituitioistic fuzzy sets. The, we utilize the weighted correlatio coefficiet to solve exteded hesitat fuzzy group decisio makig problems i which attribute values take the form of exteded hesitat fuzzy elemets. The remaider of the paper is orgaized as follows: I Sectio 2, we review some basic otios of hesitat fuzzy sets ad EHFSs; the correlatio coefficiets betwee hesitat fuzzy sets are give as a basis of the mai body of the paper i the ext sectio. I Sectio 3, we propose some correlatio coefficiets betwee EHFSs, which cotai two cases: the correlatio coefficiets takig ito accout the legth of EHFEs ad the correlatio coefficiets without takig ito accout the legth of EHFEs. I Sectio 4, we preset methods to deal with group decisio makig based o exteded hesitat fuzzy iformatio, ad a example is give to show the actual eed for dealig with the differece of evaluatio iformatio provided by differet experts without iformatio loss i decisio makig processes. Fially, i Sectio 5, some coclusios are give.

3 Symmetry 207, 9, 47 3 of 2 2. Prelimiaries I this sectio, we carry out a brief itroductio to EHFSs ad correlatio coefficiets of HFSs as a basis of the mai body of the paper. 2.. Several asic Cocepts about HFSs ad EHFSs Torra et al. [,2] firstly proposed the cocept of a hesitat fuzzy set, which is defied as follows: Defiitio. Let X be a fixed set; a hesitat fuzzy set o X is defied i terms of a fuctio h that whe applied to X returs to a fiite subset of [0,], which ca be represeted as the followig mathematical symbol [,2]: = {< x, h x) > x X }, ) where h x) is a set of some differet values of [0,], deotig the possible membership degrees of the elemet x X to. For coveiece, we call h x) a hesitat fuzzy elemet deoted by h. Zhu et al. [6] defied a EHFS, which is a extesio of the hesitat fuzzy set, i terms of a fuctio that returs a fiite set of membership value-groups. Defiitio 2. Let X be a fixed set, h D x) = γ D h D x){γ D } D =,..., m) be HFSs o X. The, a EHFS,that is H hd, is defied as [6]: H hd x) = h x)... h m x) = For coveiece, we call: H = h,..., h m = γ h x),γ 2 h 2 x),...,γ m h m x) γ h x),γ 2 h 2 x),...,γ m h m x) {< x, γ x),..., γ m x)) > x X}. 2) {γ,..., γ m )} 3) a exteded hesitat fuzzy elemet EHFE) ad let u = γ,..., γ m ); the, we call u a membership uit MU), ased o u, a EHFE H, ca also be idicated by: H = u h m x) {u} = γ h x),γ 2 h 2 x),...,γ m h m x) {γ,..., γ m )}. 4) From Defiitio 2, we ca see that EHFS icreases the richess of umerical represetatio based o the value-groups, ehaces the modelig abilities of hesitat fuzzy sets ad ca idetify differet decisio makers i decisio makig processes, which expad the applicatios of hesitat fuzzy sets i practice. HFSs ca be used to costruct EHFSs. O the cotrary, EHFSs ca reduce to HFSs. The existig sets, icludig fuzzy sets, ituitioistic fuzzy sets, fuzzy multisets, type-2 fuzzy sets, dual hesitat fuzzy sets ad especially hesitat fuzzy sets, ca hadle a more exemplary ad flexible access to assig values for each elemet i the domai. Example. Let X = {x, x 2 } be the referece set, Hx ) = {0.2, 0.4), 0.2, 0.5), 0.3, 0.4), 0.3, 0.5)} ad Hx 2 ) = {0., 0.4), 0., 0.5)} be the EHFEs of x i i =, 2) to a set, respectively. The H ca be cosidered as a EHFS, i.e., = {< x, {0.2, 0.4), 0.2, 0.5), 0.3, 0.4), 0.3, 0.5)} >, < x 2, {0., 0.4), 0., 0.5)} >}. To compare the EHFEs, Zhu et al. [6] gave the cocepts of score fuctio ad deviatio fuctio:

4 Symmetry 207, 9, 47 4 of 2 Defiitio 3. For a MU, u = γ,..., γ m ), the we call su) = /) γ u γ the score fuctio of u, where is the umber of memberships i u. For ay two MUs, u ad u 2, if su ) > su 2 ), the u u 2 ; if su ) = su 2 ), the u u 2, where deotes be superior to ad meas be idifferet to [6]. Defiitio 4. For a MU, u = γ,..., γ m ), let su) be the score fuctio of u, the we call pu) = [/) γ u γ su)) 2 ] /2 the deviatio fuctio of HFSs, where is the umber of memberships i u [6]. ased o the score fuctio ad the deviatio fuctio, we develop the followig compariso law. Defiitio 5. Let u ad u 2 be two MUs, su ) ad su 2 ) the scores of u ad u 2, respectively, ad pu ) ad pu 2 ) the deviatio degrees of u ad u 2, respectively, the [6]: ) if su ) < su 2 ), the u u 2 ; 2) if su ) = su 2 ), the ) if pu ) = pu 2 ), the u is equivalet to u 2, deoted by u u 2 ; 2) if pu ) < pu 2 ), the u is superior to u 2, deoted by u u 2 ; 3) if pu ) > pu 2 ), the u is superior to u 2, deoted by u u 2. The compariso laws of fuzzy set theory [,4,6,34] play a importat role i decisio makig problems, ad the score fuctio ad accuracy fuctio of EHFEs are the basis of the mai body of the ext part Correlatio Coefficiet of Hesitat Fuzzy Sets Correlatio coefficiets are a effective tool for addressig the relatioship betwee elemets with ucertai iformatio that have bee deeply studied [2,24 27]. Che et al. [2] itroduced the iformatioal eergy, correlatio ad correlatio coefficiets of hesitat fuzzy sets. For a hesitat fuzzy elemet h, let σ :, 2,..., ), 2,..., ) be a permutatio satisfyig h σj) h σj+) for j =, 2,..., ad h σj) be the j-th largest value i h; the iformatioal eergy of hesitat fuzzy sets is give as follows: Defiitio 6. Let be a hesitat fuzzy set o a uiverse of discourse X = {x, x 2,..., x } deoted as = {< x i, h x i ) > x i X}. The, the iformatioal eergy of is defied as [2]: E HFS ) = i= l i l i where l i = lh x i )) represets the umber of values i h x i ), x i X. j= h 2 σj) x i)), 5) Defiitio 7. Let ad be two hesitat fuzzy sets o a uiverse of discourse X = {x, x 2,..., x } deoted as = {< x i, h x i ) > x i X} ad = {< x i, h x i ) > x i X}, respectively. The, the correlatio betwee ad is defied as [2]: C HFS, ) = i= l i l i j= h σj) x i )h σj) x i )), 6) here, l i = max{lh x i )), lh x))} for each x i i X, where lh x i )) ad lh x i )) represet the umber of values i h x i ) ad h x i ), respectively. Whe lh x i )) = lh x i )), oe ca make them have the same umber of elemets through addig some values to the hesitat fuzzy elemet, which has less values. ccordig to the pessimistic priciple, the smallest elemet will be added. Therefore, if lh x i )) < lh x i )), h x i ) should be exteded by addig the miimum value i it util it has the same legth as h x i ).

5 Symmetry 207, 9, 47 5 of 2 Defiitio 8. Let ad be two hesitat fuzzy sets o a uiverse of discourse X = {x, x 2,..., x }, deoted as = {< x i, h x i ) > x i X} ad = {< x i, h x i ) > x i X}, respectively. The, the correlatio coefficiet betwee ad is defied as [2]: ρ HFS, ) = C HFS,) CHFS,) C HFS,) = i= l l i i j= h σj) x i )h σj) x i )) i= l i l i j= h 2 σj) x i)) i= l i l i j= h 2 σj) x i)). 7) Theorem. The correlatio coefficiet betwee two hesitat fuzzy sets ad satisfies the followig properties [2]: ) ρ HFS, ) = ρ HFS, ); 2) 0 ρ HFS, ) ; 3) ρ HFS, ) =, if =. 3. Correlatio ad Correlatio Coefficiets of EHFSs The correlatio ad correlatio coefficiets of hesitat fuzzy sets were itroduced by Che et al. [2] to solve practical decisio makig problems. I this sectio, we itroduce the iformatioal eergy, correlatio ad correlatio coefficiets of EHFSs as a ew extesio. Let X = {x, x 2,..., x } be a discrete uiverse of discourse, be a EHFS o X deoted as = {< x i, u H {u x i )} > x i X}. The MUs of a EHFE are usually give i disorder, ad for coveiece, we arrage them i a decreasig order. ased o Defiitios 3 5, for a EHFE H, let σ :, 2,.., ), 2,..., ) be a permutatio satisfyig u σi) u σi+), i =, 2..., ad H σi) be the j-th largest value i H. s is show i Example, = {< x, {0.2, 0.4), 0.2, 0.5), 0.3, 0.4), 0.3, 0.5)} >, < x 2, {0., 0.4), 0., 0.5)} >}, so we obtai that EHFEs Hx ) = {0.2, 0.4), 0.2, 0.5), 0.3, 0.4), 0.3, 0.5)} ad Hx 2 ) = {0., 0.4), 0., 0.5)}, accordig to Defiitios 3 5, as s0.2, 0.4)) = 0.3 s0.2, 0.5)) = 0.35 s0.3, 0.4)) = 0.35 s0.3, 0.5)) = 0.4 ad p0.2, 0.5)) = 0.22 p0.3, 0.4)) = 0.07, the H σ x ) = {0.3, 0.5), 0.3, 0.4), 0.2, 0.5), 0.2, 0.4)}; similarly, H σ x 2 ) = {0., 0.5), 0., 0.4)}. It is oted that the umber of values i differet EHFEs may be differet. To compute the correlatio coefficiets betwee two EHFSs, let = max{lh x i )), lh x i ))} for each x i i X, where lh x i )) ad lh x i )) represet the umber of MUs i H x i ) ad H x i ), respectively. Whe lh x i )) = lh x i )), oe ca make them have the same umber of MUs through addig some elemets to the EHFE, which has less MUs. Similarly, = max{lux i )), ux i ))} for each x i i X, where lu x i )) ad lu x i )) represet the umber of values i u x i ) ad u x i ), respectively. Motivated by the optimized parameter, Zhu et al. [6] gave the followig defiitios. Defiitio 9. For a MU, u = γ,..., γ m }, let u = mi{γ γ u} ad u + = max{γ γ u} be the miimum ad maximum memberships i u, respectively, ad ς0 ς ) be the optimized parameter, the we call γ = ςu + + ς)u a added membership [6]. For two EFHFEs with differet umbers of MUs, we further utilize the optimized parameter to obtai a MU. Defiitio 0. Give a EHFE, H hd = γ h,...,γ m h m {γ,..., γ m )} D =,..., m), let h D ad h+ D be the miimum ad maximum memberships i h D, respectively, ad ς0 ς ) be the optimized parameter, the a added MU is defied as ũ = γ,..., γ m ), where γ = ςu + + ς)u D =,..., m) [6]. Similar to the existig works [2], we defie the iformatioal eergy for EHFSs ad the correspodig correlatio.

6 Symmetry 207, 9, 47 6 of 2 Defiitio. Let be a EHFS o a uiverse of discourse X = {x, x 2,..., x }, deoted as = {< x i, u H {ux i )} > x i X}. The, the iformatioal eergy of is defied as: E EHFS ) = i= S s j= k= x i ) {γ x i )) 2 u σj) x i ) Hx i )})), 8) where ad are the umber of MUs i H ad values i MU u, respectively, S s is a fuctio that idicates a summatio of all values i the set of u σj) x i ) i Hx i ), γ x i ) is the k-th largest membership i u to x i X ad u σj) x i ) is the j-th largest MUs i H. Defiitio 2. Let ad be two EHFSs o a uiverse of discourse X = {x, x 2,..., x }, deoted as = {< x i, u H {u x i )} > x i X} ad = {< x i, u H {u x i )} > x i X}, respectively. The, the correlatio betwee ad is defied as: C EHFS, ) = i= S s j= k=,γ x i ) {γ x i)γ x i ) u σj) x i) H x i ), u σj) x i ) H x i )})), 9) here, = =, = =, S s is a fuctio that idicates a summatio of all values i the set of u σj) x i ) i Hx i ), γ ad u σj) x i) ad u σj) x i) ad γ x i ) are the k-th largest memberships i u ad u, respectively, x i ) are the j-th largest MUs i H ad H, respectively. It is obvious that the correlatio of two EHFSs satisfies the followig properties: ) C EHFS, ) = E EHFS ); 2) C EHFS, ) = C EHFS, ). Defiitio 3. Let ad be two EHFSs o a uiverse of discourse X = {x, x 2,..., x }, deoted as = {< x i, u H {u x i )} > x i X} ad = {< x i, u H {u x i )} > x i X}, respectively. The, the correlatio coefficiet betwee ad is defied as: ρ EHFS, ) = C EHFS, ), 0) C EHFS, ) C EHFS, ) where: C EHFS, ) = i= C EHFS, ) = C EHFS, ) = S s i= i= j= k= S s S s j= k= j= k=,γ x i ) {γ x i)γ x i ) u σj) x i) H x i ), u σj) x i ) H x i )})), x i) {γ x i)) 2 u σj) x i) H x i )})), x i ) {γ x i )) 2 u σj) x i ) H x i )})).

7 Symmetry 207, 9, 47 7 of 2 Theorem 2. The correlatio coefficiet betwee two EHFSs ad satisfies the followig properties: ) ρ EHFS, ) = ρ EHFS, ); 2) 0 ρ EHFS, ) ; 3) ρ EHFS, ) =, if =. Proof. ) It is straightforward. 2) The iequality ρ EHFS, ) 0 is obvious. elow, let us prove ρ EHFS, ) : C EHFS, ) = S s i= j= k= = S s j= k= S s j= k= S s j= k= γ γ γ γ = j= x ) u σj) x 2) u σj) x) uσj) S s j=,γ,γ S s,γ k= S s S s j=,γ x ) u σj) x 2 ) u σj) x ) u σj) k= k= k= S s S s x ) u σj) k= k= x i ) {γ x ) {γ x 2 ) {γ x {γ ) x ) u σj) x i)γ x )γ x 2)γ x )γ {γ x ) {γ x 2) u σj) x 2 ) u σj) x i ) u σj) x i) H x i ), u σj) x i ) H x i )})) x ) u σj) x ) H x ), u σj) x ) H x )}))+ x 2 ) u σj) x 2) H x 2 ), u σj) x 2 ) H x 2 )})) x ) u σj) x ) H x ), u σj) x ) H x )})) x ) u σj) x ) H x )})) x ) u σj) x ) H x )})) + {γ x 2 ) {γ x ) u σj) x ) u σj) x 2) u σj) x 2) H x 2 )})) x 2 ) u σj) x 2 ) H x 2 )})) {γ σj) x ) {γ x ) u σj) x ) H x )})) x ) u σj) x ) H x )})), usig the Cauchy Schwarz iequality: x y + x 2 y 2,..., x y ) 2 x 2 + x2 2 +,..., x2 )y 2 + y2 2 +,..., y2 ), where x, x 2,..., x ) R, y, y 2,..., y ) R ; we obtai:

8 Symmetry 207, 9, 47 8 of 2 C EHFS, )) 2 [ S s S s [ [ = [ i= S s S s i= S s j= k= j= k= j= k= S s j= k= j= k= j= k= S s S s x ) u σj) x 2) u σj) x ) u σj) j= k= j= k= x ) u σj) x 2 ) u σj) x ) u σj) x ) {γ x )) 2 u σj) x ) H x )}))+ x 2) {γ x 2)) 2 u σj) x 2) H x 2 )})) +..., + x ) {γ x )) 2 u σj) x i) H x )}))] x ) {γ x )) 2 u σj) x ) H x )}))+ x 2 ) {γ x 2 )) 2 u σj) x 2 ) H x 2 )})) +..., + x ) {γ x )) 2 u σj) x i ) H x )}))] = C EHFS, ) C EHFS, ). x i) {γ x i)) 2 u σj) x i) H x i )}))] x i ) {γ x i )) 2 u σj) x i ) H x i )}))] Therefore, C EHFS, ) C EHFS, ) C EHFS, ). Therefore, 0 ρ EHFS, ). 3) = γ σj) x i) = γ σj) x i ), x i X ρ EHFS, ) =. ased o the cocepts of HFSs, EHFSs ad their iformatioal eergies, the correlatios ad the correlatio coefficiets, we ca easily obtai the followig remark: Remark. If EHFSs reduce to HFSs, the iformatioal eergy, the correlatio ad the correlatio coefficiet about EHFSs will reduce to the iformatioal eergy, the correlatio ad the correlatio coefficiet about HFSs, respectively. I what follows, we give a ew formula of calculatig the correlatio coefficiet, which is similar to that used i HFSs [2]: Defiitio 4. Let ad be two EHFSs o a uiverse of discourse X = {x, x 2,..., x }, deoted as = {< x i, u H {u x i )} > x i X} ad = {< x i, u H {u x i )} > x i X}, respectively. The, the correlatio coefficiet betwee ad is defied as: ρ EHFS2, ) = C EHFS, ) max{c EHFS, ), C EHFS, )}. ) Theorem 3. The correlatio coefficiet of two EHFSs ad, ρ EHFS2, ), follows the same properties listed i Theorem 2.

9 Symmetry 207, 9, 47 9 of 2 Proof. The process to prove Properties ) ad 3) is aalogous to that i Theorem 2; we do ot repeat it here. 2) ρ EHFS2, ) 0 is obvious. We ow oly prove ρ EHFS2, ). ased o the proof process of Theorem 2, we have C EHFS, ) C EHFS, ) C EHFS, ), ad the C EHFS, ) max{c EHFS, ), C EHFS, )}; thus, ρ EHFS2, ). Example 2. Let ad be two EHFSs i X = {x, x 2 }, ad = {< x, {0.3, 0.4, 0.5), 0.3, 0.4, 0.6)}, < x 2, {0.4, 0.3, 0.2), 0.4, 0.3, 0.), 0.4, 0.3, 0.5)} >}, = {< x, {0., 0.2, 0.5)}, < x 2, {0.4, 0.4, 0.2), 0.4, 0.4, 0.)} >}. y applyig Equatio 9), we calculate: C EHFS, ) = E EHFS ) = S s γ i= j= k= ad similarly: x i) {γ x i)) 2 u σj) x i) H x i )})) = 2 [ ) )] + 3 [ ) ) )] = 0.307, C EHFS, ) = E EHFS ) = S s γ i= j= k= x i ) {γ x i )) 2 u σj) x i ) H x i )})) = 2 [ ) 2] + 3 [ ) ) )] = With ς = 0.5, we obtai: C EHFS, ) = S s i= j= k= γ,γ x i ) {γ x i)γ = 2 [ ) )] [ ) ) )] 3 = x i ) u σj) x i) H x i ), u σj) x i ) H x i )})) Fially, we ca calculate the correlatio coefficiet ρ EHFS, ) as: C ρ EHFS, ) = EHFS,) = = , CEHFS,) C EHFS,) ad similarly, we ca calculate the correlatio coefficiet ρ EHFS2, ) as: C ρ EHFS2, ) = EHFS,) max{c EHFS,),C EHFS,)} = = From Example 2, we ca fid that differet results are obtaied by extedig differet values i the short EHFE, so we preset several ew correlatio coefficiets of EHFSs, ot takig ito accout the legth of EHFEs ad the arragemet of their possible value-groups.

10 Symmetry 207, 9, 47 0 of 2 Defiitio 5. Let H ad H be ay two MUs with u H, the: du, H ) = mi u H γ σi) u,γ σi) σi) γ u γσi) 2) is called the distace betwee the value u i H ad the EHFE H ; by u, we deote the value i H such that du, H ). If there is more that oe value i H such that du, H ), the u = mi{u u H, γ σi) u,γ σi) γ σi) u γσi) } = du, H ). For coveiece, u = {γ } ad u = {γ }. It is obvious that the above distace du, H ) satisfies the followig properties: ) du, H ) = dh, u ); 2) 0 du, H ) ; 3) du, H ) = 0 if ad oly if u = u for ay u H, where u = u meas γ σi) γ σi) u, γ σi) u. = γ σi), Defiitio 6. Let ad be two EHFSs o a uiverse of discourse X = {x, x 2,..., x }, deoted as = {< x i, u H {u x i )} > x i X} ad = {< x i, u H {u x i )} > x i X}, respectively. The, the correlatio betwee ad is defied as: C EHFS2, ) = i= x i ) x i ) S s H x i )})) + x i ) x i ) S s x i ) x i ) j= k= x i ) x i ) j= k= γ x i ) x i ) {γ x i) {γ x i)γ x i ) u σj) x i) x i )γ x i ) u σj) x i ) H x i )}))), where ad are the umbers of exteded hesitat fuzzy elemets H ad H, respectively. dditioally, γ x i), γ x i ), γ x i ) ad γ x i ) are show i Defiitio 5. It is easy to prove that the above correlatio C EHFS2, ) satisfies the followig theorem: Theorem 4. Let ad be ay two EHFSs i X; the correlatio C EHFS2, ) satisfies: ) C EHFS2, ) = C EHFS2, ); 2) C EHFS2, ) = 2E EHFS2 ) with E EHFS2 ) = i= x i ) j= x i ) k= x i ) x i ) S s x i) {γ x i)) 2 u σj) x i) H x i )})). ccordig to the correlatio of EHFSs, the correlatio coefficiet of EHFSs is give as follows: Defiitio 7. Let ad be ay two EHFSs i X; the correlatio coefficiet betwee ad is defied as: ρ EHFS3, ) = C EHFS2, ) E EHFS2 )E EHFS2 ) + E EHFS2 )E EHFS2 ), 3) where: E EHFS2 ) = i= E EHFS2 ) = i= E EHFS2 ) = i= x i ) x i ) x i ) x i ) S s x i ) j= x i ) x i ) S s x i ) j= x i ) k= k= x i ) S s x i ) j= x i ) k= x i) {γ x i)) 2 u σj) x i) H x i )})), x i ) {γ x i )) 2 u σj) x i ) H x i )})), x i) {γ x i )) 2 u σj) x i) H x i )})),

11 Symmetry 207, 9, 47 of 2 E EHFS2 ) = i= x i ) x i ) S s x i ) j= x i ) k= x i ) {γ x i )) 2 u σj) x i ) H x i )})). Theorem 5. The correlatio coefficiet ρ EHFS3, ) for ay two EHFSs ad i X satisfies: ) ρ EHFS3, ) = ρ EHFS3, ); 2) 0 ρ EHFS3, ) ; 3) ρ EHFS3, ) =, if =. Proof. ) It is straightforward. 2) From Defiitio 7, it is apparet that ρ EHFS3, ) 0. For ρ EHFS3, ), usig the Cauchy Schwarz iequality: x y + x 2 y 2,..., x y ) 2 x 2 + x2 2 +,..., x2 )y 2 + y2 2 +,..., y2 ), where x, x 2,..., x ) R, y, y 2,..., y ) R, we drive: i= x i ) x i ) S s = x ) x ) S s x 2 ) x 2 ) S s x ) x ) S s = x ) S s x 2 ) S s x ) S s i= i= x i ) x i ) j= k= x ) x ) j= k= x 2 ) x 2 ) j= k= x ) x ) j= k= x ) x ) j= k= x 2 ) x 2 ) j= k= x ) x ) j= k= x i ) x i ) S s x i ) x i ) S s x ) u σj) x 2) u σj) x ) u σj) x i ) x i ) j= k= x i ) x i ) j= k= = C EHFS, ) C EHFS, ). Namely, i= x i ) H x i )}))) Similarly, oe ca have i= x i ) H x i )}))) Thus, x i ) S s x i ) j= x i ) x ) u σj) x 2) u σj) x ) u σj) x i) {γ x ) {γ x 2) {γ x ) {γ { γ x ) x ) x i)γ x i ) u σj) x i) H x i )}))) 2 x )γ x ) u σj) x ) H x )}))+ x 2)γ x 2 ) u σj) x 2) H x 2 )})) x )γ x ) u σj) x ) H x )}))) 2 γ x ) x ) x ) uσj) x ) H x )}))+ { γ x 2) x 2) γ x 2 ) x 2 ) x 2 ) uσj) x 2) H x 2 )})) { γ x ) x ) γ x ) x ) x ) uσj) x ) H x )}))) 2 k= E EHFS2 )E EHFS2 ). x i ) S s x i ) j= x i ) k= E EHFS2 )E EHFS2 ). x i) {γ x i)) 2 u σj) x i) H x i )}))) x i) {γ x i )) 2 u σj) x i) H x i )})) x i) {γ x i ) {γ x i)γ x i ) u σj) x i) x i )γ x i ) u σj) x i )

12 Symmetry 207, 9, 47 2 of 2 C EHFS2, ) E EHFS2 )E EHFS2 ) + E EHFS2 )E EHFS2 ) The result is obtaied. 3) = C EHFS2, ) = E EHFS2 ) = E EHFS2 ) = E EHFS2 ) = E EHFS2 ) ρ WEHFS3, ) =. Similar to the correlatio coefficiet of Defiitio 4, a modified form of the correlatio coefficiet of EHFSs is defied by: where: C EHFS3, ) = i= C EHFS3, ) = i= ρ EHFS4, ) = 2 C EHFS3, ) + E EHFS2 )E EHFS2 ) x i ) x i ) S s x i ) j= x i ) x i ) x i ) S s x i ) j= x i ) k= k= C EHFS3, ) E EHFS2 )E EHFS2 ) x i) {γ x i ) {γ ), 4) x i)γ x i ) u σj) x i) H x i )})), x i )γ x i ) u σj) x i ) H x i )}))). Theorem 6. The correlatio coefficiet ρ EHFS4, ) for ay two EHFSs ad i X satisfies: ) ρ EHFS4, ) = ρ EHFS4, ); 2) 0 ρ EHFS4, ) ; 3) ρ EHFS4, ) =, if =. Proof. Similar to the proof of Theorem 2, we ca easily obtai the coclusios. Ispired by Defiitio 4, the correlatio coefficiets of EHFSs, for ay two EHFSs ad i X, are defied as follows: ρ EHFS5, ) = C EHFS2, ) max{e EHFS2 ), E EHFS2 )} + max{e EHFS2 ), E EHFS2 )}, 5) ρ EHFS6, ) = 2 C EHFS3, ) max{e EHFS2 ), E EHFS2 )} + C EHFS3, ) max{e EHFS2 ), E EHFS2 ), 6) )} Theorem 7. The correlatio coefficiets ρ EHFSi, ) i = 5, 6) for ay two EHFSs ad i X satisfies: ) ρ EHFSi, ) = ρ EHFSi, ); 2) 0 ρ EHFSi, ) ; 3) ρ EHFSi, ) =, if =. Proof. Similar to the proofs of Theorems 2 ad 5, the coclusios obviously hold. Example 3. Now, we calculate Example 2 by the ew correlatio coefficiets ρ EHFSi, ) i = 3, 4, 5, 6) without takig ito accout the legth of EHFEs. The calculatio process is give as follows: E EHFS2 ) = i= x i ) x i ) S s x i ) x i ) j= k= x i) {γ x i)) 2 u σj) x i) H x i )})) = 2 [ ) )] + 3 [ ) ) )] = 0.307,

13 Symmetry 207, 9, 47 3 of 2 E EHFS2 ) = E EHFS2 ) = E EHFS2 ) = i= x i ) x i ) S s x i ) x i ) j= k= x i ) {γ x i )) 2 u σj) x i ) H x i )})) = ) ) )] = 0.250, i= x i ) x i ) S s x i ) x i ) j= k= x i) {γ x i )) 2 u σj) x i) H x i )})), = ) ) )] = , i= x i ) x i ) S s x i ) x i ) j= k= x i ) {γ x i )) 2 u σj) x i ) H x i )})) = 2 [ ) )] + 3 [ ) ) )] = 0.267, C EHFS3, ) = C EHFS3, ) = i= x i ) x i ) S s x i ) x i ) j= k= x i) {γ x i)γ x i ) u σj) x i) H x i )})) = 2 [ ) )] + 3 [ ) ) )] 3 = , i= x i ) x i ) S s x i ) x i ) j= k= x i ) {γ = ) ) )] 3 = 0.227, x i )γ x i ) u σj) x i ) H x i )}))) C EHFS2, ) = C EHFS3, ) + C EHFS3, ) = Fially, we ca calculate the correlatio coefficiets: ρ EHFS3, ) = ρ EHFS4, ) = 2 ρ EHFS5, ) = ρ EHFS6, ) = 2 C EHFS2,) E EHFS2 )E EHFS2 )+ C EHFS3,) + E EHFS2 )E EHFS2 ) E EHFS2 )E EHFS2 ) = , C EHFS 3,) E EHFS2 )E EHFS2 ) ) = , C EHFS2,) max{e EHFS2 ),E EHFS2 )}+max{e EHFS2 ),E EHFS2 )} = , C EHFS3,) max{e EHFS2 ),E EHFS2 )} + C EHFS3,) max{e EHFS2 ),E EHFS2 )} ) = To save all of the iformatio provided by the DMs, distiguish them from each other ad cosider their differet importace i decisio makig, we ow propose the weighted exteded hesitat correlatio coefficiets cosiderig DMs as follows. ssume a decisio makig problem with m DMs. For ay MU, u = {γ, γ 2,..., γ m }, the weights of DMs are ω D D =, 2,..., m) with ω D [0, ] ad m D= =. Let γ ω D = ω D γ D be memberships associated with the DMs weights. O the other had, i practical applicatios, the elemets x i i =, 2,..., ) i the uiverse X have differet weights.

14 Symmetry 207, 9, 47 4 of 2 Let w = w i, w 2,..., w ) T be the weight vector of x i i =, 2,..., ) with w i 0, i =, 2,..., ad i= w i =, we further exted the correlatio coefficiet formulas give i Table. ρ of EHFSs ρ WEHFS ω, ω ) ρ WEHFS2 ω, ω ) ρ WEHFS3 ω, ω ) ρ WEHFS4 ω, ω ) 2 ρ WEHFS5 ω, ω ) ρ WEHFS6 ω, ω ) 2 Table. The correlatio coefficiets of EHFSs. Correlatio Coefficiet Formulas C WEHFS ω, ω ) CWEHFS ω, ω ) C WEHFS ω, ω ) C WEHFS ω, ω ) max{c WEHFS ω, ω ),C WEHFS ω, ω ) C WEHFS2 ω, ω ) EWEHFS2 ω )E WEHFS2 ω )+ E WEHFS2 ω )E WEHFS2 ω) C WEHFS3 ω, ω ) EWEHFS2 ω )E WEHFS2 ω ) + C WEHFS3 ω, ω ) EWEHFS2 ω )E WEHFS2 ω) ) C WEHFS2 ω, ω ) max{e WEHFS2 ω ),E WEHFS2 ω )}+max{e WEHFS2 ω ),E WEHFS2 ω)} C WEHFS3 ω, ω ) max{e WEHFS2 ω ),E WEHFS2 ω )} + C WEHFS3 ω, ω ) max{e WEHFS2 ω ),E WEHFS2 ω)} ) where: C WEHFS ω, ω ) = i= C WEHFS ω, ω ) = C WEHFS ω, ω ) = E WEHFS2 ω ) = E WEHFS2 ω ) = E WEHFS2 ω) = E WEHFS2 ω) = i= i= i= i= w i S s i= i= w i S s w i S s w i S s w i S s j= k= w i S s w i S s j= k= j= k= x i ) x i ) j= k= x i ) x i ) j= k= x i ) x i ) j= k= x i ) x i ) j= k= C WEHFS3 ω, ω ) = i= w is s x i ) j= x i ),γ k= C WEHFS3 ω, ω ) = i= w is s x i ) j= x i ) k= C WEHFS2, ) = C WEHFS3, ) + C WEHFS3, ). x i ) {γ ωd x i )γ ωd x i ) u σj) x i) H x i ),,γ,γ u σj) x i ) H x i )}))), x i ) {γ ωd x i )) 2 u σj) x i) H x i ), u σj) x i ) H x i )}))), x i ) {γ ωd x i )) 2 u σj) x i) H x i ), u σj) x i ) H x i )}))), x i) {γ ωd x i )) 2 u σj) x i) H x i )}))), x i ) {γ ωd x i )) 2 u σj) x i ) H x i )}))), x i) {γ x ω i )) 2 u σj) x i) H x i )}))), D x i ) {γ ω x i )) 2 u σj) x i ) H x i )}))). D x i) {γ ω D x i )γ ω x i ) u σj) x i) H x i )}))), D x i ) {γ ω D x i )γ x ω i ) u σj) x i ) H x i )}))). D It ca be see that if w i = /, /,..., /) T ad ω i = /m, /m,..., /m), the ρ WEHFSi ω, ω ) i =, 2, 3,..., 6) reduce to ρ EHFSi, ) i =, 2, 3,..., 6). dditioally, it is easy to prove that ρ WEHFSi ω, ω ) i =, 2, 3,..., 6) also have the followig properties:

15 Symmetry 207, 9, 47 5 of 2 Theorem 8. Let w = w i, w 2,..., w ) T be the weight vector of x i i =, 2,..., ) with w i 0, i =, 2,..., ad i= w i = ad ω = ω i, ω 2,..., ω ) T be the weight vector of DMs with ω i 0, i =, 2,..., ad i= ω i = ; the correlatio coefficiets ρ WEHFSi ω, ω ) i =, 2, 3,..., 6) betwee two EHFSs ad satisfy: ) ρ WEHFSi ω, ω ) = ρ WEHFSi ω, ω ); 2) 0 ρ WEHFSi ω, ω ) ; 3) ρ WEHFSi ω, ω ) =, if =. However, sometimes, the exact weights w i of elemets x i are ukow, we preset the weighted exteded hesitat correlatio coefficiet of EHFEs as Table 2. Table 2. The correlatio coefficiets of EHFEs. ρ of EHFEs Correlatio Coefficiet Formulas ρ EHFE ω, ω ) ρ EHFE2 ω, ω ) ρ EHFE3 ω, ω ) ρ EHFE4 ω, ω ) 2 ρ EHFE5 ω, ω ) ρ EHFS6 ω, ω ) 2 C EHFE ω, ω ) CEHFE ω, ω ) C EHFE ω, ω ) C EHFE ω, ω ) max{c EHFE ω, ω ),C EHFE ω, ω ) C EHFE2 ω, ω ) EEHFE2 ω )E EHFE2 ω )+ E EHFE2 ω )E EHFE2 ω) C EHFS3 ω, ω ) EEHFE2 ω )E EHFE2 ω ) + C EHFE3 ω, ω ) EEHFE2 ω )E EHFE2 ω) ) C EHFE2 ω, ω ) max{e EHFE2 ω ),E EHFE2 ω )}+max{e EHFE2 ω ),E EHFE2 ω)} C EHFE3 ω, ω ) max{e EHFE2 ω ),E EHFE2 ω )} + C EHFE3 ω, ω ) max{e EHFE2 ω ),E EHFE2 ω)} ) where: C EHFE ω, ω ) = S s j= k= C EHFE ω, ω ) = S s C EHFE ω, ω ) = S s E EHFE2 ω ) = E EHFE2 ω ) = E EHFE2 ω) = E EHFE2 ω) = x i ) S s x i ) S s x i ) S s x i ) S s j= k= j= k= x i ) x i ) j= k= x i ) x i ) j= k= x i ) x i ) j= k= x i ) x i ) j= k= C EHFE3 ω, ω ) = x i ) S s x i ) j= x i ),γ k= x i ) {γ ωd x i )γ ωd x i ) u σj) x i) H x i ),,γ,γ u σj) x i ) H x i )})), x i ) {γ ωd x i )) 2 u σj) x i) H x i ), u σj) x i ) H x i )})), x i ) {γ ωd x i )) 2 u σj) x i) H x i ), u σj) x i ) H x i )})), x i) {γ ωd x i )) 2 u σj) x i) H x i )})), x i ) {γ ωd x i )) 2 u σj) x i ) H x i )})), x i) {γ x ω i )) 2 u σj) x i) H x i )})), D x i ) {γ ω x i )) 2 u σj) x i ) H x i )})). D x i) {γ ω D x i )γ ω x i ) u σj) x i) H x i )}), D

16 Symmetry 207, 9, 47 6 of 2 C EHFE3 ω, ω ) = x i ) S s x i ) j= x i ) k= x i ) {γ ω D x i )γ x ω i ) u σj) x i ) H x i )})), D C EHFE2, ) = C EHFE3, ) + C EHFE3, ). dditioally, it is easy to prove that ρ EHFEi ω, ω ) i =, 2, 3,..., 6) also have the followig properties: Theorem 9. Let ω = ω i, ω 2,..., ω ) T be the weight vector of DMs with ω i 0, i =, 2,..., ad i= ω i = ; the correlatio coefficiets ρ EHFEi ω, ω ) i =, 2, 3,..., 6) betwee two EHFSs ad satisfy: ) ρ EHFEi ω, ω ) = ρ EHFEi ω, ω ); 2) 0 ρ EHFEi ω, ω ) ; 3) ρ EHFEi ω, ω ) =, if =. 4. pplicatio of the Weighted Correlatio Coefficiets of the Exted Hesitat Fuzzy Eviromet I this sectio, we shall utilize the weighted correlatio coefficiets of EHFEs ad EHFSs to decisio makig with exteded hesitat fuzzy iformatio. t first, we utilize the weighted correlatio coefficiets of EHFSs for decisio makig problems with exteded hesitat fuzzy iformatio. For a decisio makig problem with exteded hesitat fuzzy iformatio, let = {, 2,..., m } be a discrete set of alteratives ad S = {S, S 2,..., S } be a set of attributes. If the decisio makers provide several values for the alterative i i =, 2,..., m) uder the attribute C j j =, 2,..., ), these values ca be cosidered as a EHFE H ij j =, 2,.., ; i =, 2,..., m). Therefore, we ca elicit a exteded hesitat fuzzy decisio matrix H = H ij ) m, where H ij i =, 2,.., m; j =, 2,..., ) is i the form of exteded hesitat fuzzy elemets. I multiple attribute decisio makig eviromets, the cocept of ideal poit has bee used to help the idetificatio of the best alterative i the decisio set. lthough the ideal alterative does ot exist i the real world, it does provide a useful theoretical costruct to evaluate alteratives. Therefore, we defie each ideal EHFE H i the ideal alterative = {< s j, H > s j S} j =, 2,.., ). Method : Step. Use the iformatio give by decisio makers to establish exteded hesitat fuzzy model ad costruct the exteded hesitat fuzzy decisio matrix H = H ij ) m by EHFEs. Step 2. ssume that the weights of decisio makers ω D ad attributes w ad a stadard EHFE H = {,,, )} are kow; calculate the weighted correlatio coefficiets betwee a alterative i i =, 2,.., m) ad the ideal alterative by usig the formulas ρ WEHFSi ω, ω ) i =, 2,..., 6) see Table ). Step 3. Rak the alteratives i accordace with the values of ρ WEHFSi ω, ω ) i =, 2,..., 6). We may obtai differet results ad rakigs to aalyze differet weighted correlatio coefficiets. Step 4. Select the best alterative accordig to the maximum values of the weighted correlatio coefficiets ρ WEHFSi ω, ω ) i =, 2,..., 6). Step 5. Ed. However, sometimes, the exact weights w i of elemets x i are ukow; we preset the weighted exteded hesitat correlatio coefficiet of EHFEs with the Dempster Shafer belief structure [35]. Let C ij be a payoff to the alterative i, ad the state of ature is S j, C = C ij ) m a payoff matrix ad ς the optimized parameter. The DMs kowledge of the states of ature is captured i terms of a belief structure p with the focal elemets, 2,..., r, each of which is associated with a weight p k ), where r k= p k ) =. We ow develop the followig approach to deal with group decisio makig. The method is similar to Zhu et al. s see [6] i detail) as follows.

17 Symmetry 207, 9, 47 7 of 2 Method 2: Step. Use the iformatio give by decisio makers to establish the exteded hesitat fuzzy model ad costruct the exteded hesitat fuzzy decisio matrix H = H ij ) m by EHFEs. Step 2. ssume a stadard EHFE H = {,,, )} ad the optimized parameter, the calculate the correlatio coefficiets betwee H ad H ij by the exteded hesitat correlatio coefficiets of EHFEs i Table 2. Let C ij be equal to the weighted correlatio coefficiets, ad costruct the payoff matrix of ρ EHFEij ω, ω )i =, 2,..., 6, j =, 2,..., 5), deoted as C ij. Step 3. Calculate the belief fuctio p about the states of ature see [6] i detail). Step 4. Utilize the optimized parameter to calculate the collectio of weights [36,37], which are used i the OW aggregatio for each cardiality of focal elemets see [6] i detail). Step 5. Determie the payoff collectio, M ik = {C ij S j k }, which is a set of payoffs that are possible if we select the alterative i ad the focal elemet k occurs, ad calculate the aggregated payoff, V ik = OWM ik ) see [6] i detail). Step 6. Calculate C i = r k= V ik p k ), ad select the alterative that has the best geeralized expected value as the optimal alterative see [6] i detail). Step 7. Ed. Example 4. [6] Eergy is a idispesable factor for the social-ecoomic developmet of societies. Thus, the correct eergy policy affects ecoomic developmet ad the eviromet; the most appropriate eergy policy selectio is very importat. Suppose that there are five alteratives eergy projects) i i =, 2, 3, 4, 5) to be ivested ad four criteria to be cosidered: S -techological; S 2 -evirometal; S 3 -socio-political; S 4 -ecoomic. Five DMs are ivited to evaluate the performaces of the five alteratives. I order to avoid givig DMs prefereces aoymously give by [6] ad deal with this eergy policy problem without iformatio loss, the DMs D k k =, 2, 3, 4, 5) provide their prefereces over all of the alteratives i i =, 2,..., 5) with respect to the criteria S j j =, 2, 3, 4) based o hesitat fuzzy sets, the Zhu et al. [6] saved the DMs prefereces by a exteded hesitat fuzzy matrix H = H ij ) 5 4 show i Table 3 see [6] for detail). Table 3. Exteded hesitat fuzzy decisio matrix. S S 2 {0.3, 0.4, 0.3, 0.4, 0.5)} {0.7, 0.8, 0.3, 0.8, 0.6), 0.7, 0.8, 0.4, 0.8, 0.6)} 2 {0.3, 0.4, 0.5, 0.2, 0.5), 0.3, 0.4, 0.5, 0.3, 0.5)} {0.5, 0.6, 0.5, 0.6, 0.6)} 3 { )} {0.5,0.6,0.7,0.6,0.5), 0.6,0.6,0.7,0.6,0.5), 0.5,0.6,0.8,0.6,0.5), 0.6,0.6,0.8,0.6,0.5)} 4 {0.3, 0.2, 0.2, 0.3, 0.)} {0.6, 0.5, 0.7, 0.5, 0.5)} 5 {0.3, 0.4, 0.6, 0.2, 0.2), 0.3, 0.3, 0.6, 0.2, 0.2)} {0.6, 0.8, 0.5, 0.4, 0.6), 0.6, 0.8, 0.5, 0.5, 0.6)} {0.3,0.4,0.2,0.3,0.2), 0.4,0.4,0.2,0.3,0.2,), 0.3,0.4,0.3,0.3,0.2), 0.4,0.4,0.3,0.3,0.2)} S 3 S 4 {0.6,0.5,0.5,0.4,0.6)} 2 {0.6,0.4,0.5,0.3,0.5), 0.6,0.4,0.4,0.3,0.5)} {0.3,0.4,0.5,0.2,0.2),0.3,0.4,0.4,0.2,0.2)} 3 { ), 0.7,0.3,0.8,0.8,0.6)} {0.7,0.8,0.7,0.8,0.8)} 4 {0.4,0.3,0.2,0.3,0.5)} {0.3,0.2,0.7,0.2,0.)} 5 {0.7,0.5,0.6,0.8,0.6)} {0.6,0.4,0.5,0.4,0.6), 0.7,0.4,0.5,0.4,0.6)}

18 Symmetry 207, 9, 47 8 of 2 i) We ow use Method 2 to solve the decisio makig problem first as a compariso to Zhu et al. [6]. Step. The decisio makers D k k =, 2, 3, 4, 5) provide their prefereces over all of the alteratives i i =, 2,..., 5) with respect to the criteria S j j =, 2, 3, 4) show i Table 3. Step 2. Let = {,,,, )} be the ideal values of the alterative see as a stadard EHFE H, ς = 0.75 be the optimized parameter ad ω = 0.3, 0., 0.3, 0.2, 0.) be the weightig vector of the DMs. y Table 2, we ca calculate the correlatio coefficiets betwee H ad H ij. The, costruct the payoff matrix show as Tables 4 8. Table 4. The payoff matrix of ρ EHFE ω, ω ). S S 2 S 3 S Table 5. The payoff matrix of ρ EHFE2 ω, ω ). S S 2 S 3 S Table 6. The payoff matrix of ρ EHFE3 ω, ω ). S S 2 S 3 S Table 7. The payoff matrix of ρ EHFE4 ω, ω ). S S 2 S 3 S Table 8. The payoff matrix of ρ EHFE5 ω, ω ) ad ρ EHFE6 ω, ω ). S S 2 S 3 S

19 Symmetry 207, 9, 47 9 of 2 Step 3. The DMs aalyze the eergy policy problem so as to obtai the probabilistic iformatio about the states of ature. ssume that the DMs; kowledge of the states of ature cosists of the followig belief structure, show i Table 9 see [6]). Table 9. elief structure. Focal Elemet Weights w) = {S, S 3 } = {S 2, S 4 } = {S, S 3, S 4 } 0.6 Step 4. I order to cotrast with the method s results give by [6], we still use the O Haga method [36] to obtai weightig vectors associated with the OW operators for various umbers of argumets. Sice ς = 0.75, the we ca get the weightig vectors show i Table 0 see [6]). Table 0. Weightig vectors for various umbers of argumets. Number of rgumets w w 2 w Step 5. We get V ik for all i ad ki =, 2, 3, 4; k =, 2, 3, 4, 5) as M ik = {C ij S j k } ad V ik = OWM ik ). Step 6. Calculate C i = r k= V ik p k ), the results are give i Table. Thus, 3 is the optimal alterative closest to the ideal values of alterative, which is the same with Zhu ad Xu see the Example i [6]). Step 7. Ed. Table. Weightig vectors for various umbers of argumets Rakigs Results about ρ EHFE Results about ρ EHFE Results about ρ EHFE Results about ρ EHFE Results about ρ EHFE Results about ρ EHFE s ca be see from Table, 3 is the optimal alterative closest to the ideal values of alterative, which is the same with Zhu ad Xu see the Example i [6]), but the rakigs are ot always the same: the rakigs of ρ EHFE2, ρ EHFE5, ρ EHFE6 ad Zhu ad Xu [6] are idetical, but the rakigs of ρ EHFE, ρ EHFE3 ad ρ EHFE4 are differet from [6]. Obviously, the method, which depeds o the formulas of ρ EHFEi i =, 2, 3, 4, 5, 6), is practical ad effective. ii) s the weights of the elemets x i i =, 2,..., ) i the uiverse X are easily accessible i practical applicatios, i other words, the weights of S i i =, 2, 3, 4) i this example are give by decisio makers. ssume that w i = 0.3, 0.2, 0.2, 0.3), the we ca deal with the decisio makig problem by Method as follows: Step. Costruct the exteded hesitat fuzzy decisio matrix H = H ik ) m by EHFEs, H ij i =,..., q; j =,..., ), show i Table 3. Steps 2 ad 3. Calculate the weighted correlatio coefficiet betwee a alterative i i =, 2,.., m) ad the ideal alterative by usig the formulas ρ WEHFSi, ) i =, 2,..., 6). The results are give i Table 2.

20 Symmetry 207, 9, of 2 Step 4. s ca be see from the results, we get that 3 is the optimal alterative closest to the ideal values of the alterative. Step 5. Ed. Table 2. Weightig vectors for various umbers of argumets Rakigs Results o f ρ EHFS Results o f ρ EHFS Results o f ρ EHFS Results o f ρ EHFS Results o f ρ EHFS Results o f ρ EHFS The example idicates that the proposed decisio makig methods are simple ad effective uder exteded hesitat fuzzy eviromets. 5. Coclusios I this study, we develop some correlatio coefficiets betwee EHFSs, which cotai two cases: the correlatio coefficiets takig ito accout the legth of EHFEs ad the correlatio coefficiets without takig ito accout the legth of EHFEs. We also have studied some properties of these correlatio coefficiets. t last, we give two methods to deal with decisio makig problems uder exteded hesitat fuzzy eviromets, ad a real-world example based o the eergy policy problem is employed to illustrate the actual eed for dealig with the differece of evaluatio iformatio provided by differet experts without iformatio loss i decisio makig processes. s EHFSs are a ew powerful tool to express ucertai iformatio i the process of group decisio makig, we will give more studies o the theory ad applicatios i the future. uthor Cotributios: Na Lu ad Lipi Liag cotributed to the results; Na Lu wrote the paper. Coflicts of Iterest: The authors declare o coflict of iterest. Refereces. Torra, V. Hesitat fuzzy sets. It. J. Itell. Syst. 200, 25, Torra, V.; Narukawa, Y. O hesitat fuzzy sets ad decisio. I Proceedigs of the 8th IEEE Iteratioal Coferece o Fuzzy Systems, Jeju Islad, Korea, ugust 2009; pp taassov, K. Ituitioistic fuzzy sets. Fuzzy Set Syst. 986, 25, Xia, M.; Xu, Z. Hesitat fuzzy iformatio aggregatio i decisio makig. It. J. ppox. Reaso. 20, 52, Yi, L. ote o operatios of hesitat fuzzy sets. It. J. Comput. It. Syst. 204, 8, Xu, Z.; Xia, M. Distace ad similarity measures for hesitat fuzzy sets. If. Sci. 20, 8, Xu, Z.; Xia, M. O distace ad correlatio measures of hesitat fuzzy iformatio. It. J. Itell. Syst. 20, 26, Xu, Z.; Xia, M.; Che, N. Some hesitat fuzzy aggregatio operators with their applicatio i group decisio makig. Group Decis. Negot. 203, 22, Gu, X.; Wag, Y.; Yag,. method for hesitat fuzzy multiple attribute decisio makig ad its applicatio to risk ivestmet. J. Coverg. If. Techol. 20, 6, Wei, G. Hesitat fuzzy prioritized operators ad their applicatio to multiple attribute decisio makig. Kowl. ased Syst. 202, 3, lcatud, J.; Calle, R.; Torrecillas, M. Hesitat Fuzzy Worth: iovative rakig methodology for hesitat fuzzy subsets. ppl. Soft Comput. 206, 38, Che, N.; Xu, Z.; Xia, M. Correlatio coefficiets of hesitat fuzzy sets ad their applicatios to clusterig aalysis. ppl. Math. Model. 203, 37,

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