AUTHOR COPY. Some new hybrid weighted aggregation operators under hesitant fuzzy multi-criteria decision making environment

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1 Joural of Itelliget & Fuzzy Systems 26 (2014) DOI: /IFS IOS Press 1601 Some ew hybrid weighted aggregatio operators uder hesitat fuzzy multi-criteria decisio makig eviromet Huchag Liao a ad Zeshui Xu a,b, a Atai College of Ecoomics ad Maagemet, Shaghai Jiao Tog Uiversity, Shaghai, Chia b College of Scieces, PLA Uiversity of Sciece ad Techology, Najig, Chia Abstract. Hesitat fuzzy set, as a ew geeralized type of fuzzy set, is a efficiet ad powerful structure i expressig ucertaity ad vagueess ad has attracted more ad more scholars attetio. The aim of this paper is to develop some ew aggregatio operators to fuse hesitat fuzzy iformatio. The hesitat fuzzy hybrid arithmetical averagig (HFHAA) operator, the hesitat fuzzy hybrid arithmetical geometric (HFHAG) operator, the quasi HFHAA operator ad the quasi HFHAG operator are proposed ad their properties are ivestigated. O the basis of these proposed operators, some algorithms are itroduced to aid multi-criteria sigle perso decisio makig ad multi-criteria group decisio makig respectively. Some examples are provided to illustrate the practicality ad validity of our proposed procedures. Keywords: Group decisio makig, hesitat fuzzy set, hybrid weighted aggregatio operator, multi-criteria decisio makig 1. Itroductio Sice it was origially itroduced by Zadeh [1], the fuzzy set has tured out to be oe of the most efficiet decisio aid techiques providig the ability to deal with ucertaity ad vagueess. I realistic decisio makig, imprecisio may arise due to the uquatifiable iformatio, icomplete iformatio, uobtaiable iformatio, partial igorace, ad so forth [2]. To cope with imperfect ad imprecise iformatio whereby two or more sources of vagueess appear simultaeously, Zadeh s traditioal fuzzy set shows some limitatios [3]. The traditioal fuzzy set uses a crisp umber i uit iterval [0, 1] as a membership degree of a elemet to a set; however, very ofte, such a crisp umber is difficult to be determied for the Correspodig author. Zeshui Xu, Tel./Fax: ; s: xuzeshui@263.et; liaohuchag@163.com (H. Liao). decisio maker. O the other had, if a group of decisio makers are asked to evaluate the cadidate alteratives, they ofte fid some disagreemets amog themselves. Sice the decisio makers may have differet opiios over the alteratives ad they ca t persuade each other easily, a cosesus result is hard to be obtaied but a set of possible values. I such case, the traditioal fuzzy set also ca ot be used to depict the group s opiios. Hece, the classical fuzzy set has bee exteded ito several differet forms, such as the ituitioistic fuzzy set [4], the iterval-valued ituitioistic fuzzy set [5], the type 2 fuzzy set [6], the type fuzzy set [7], the fuzzy multisets (also amed the fuzzy bags) [8], ad so o [9]. All these extesios are based o the same ratioale that it is ot clear to assig the membership degree of a elemet to a fixed set [10]. Recetly, o the basis of the above extesioal forms of the fuzzy set, Torra ad Narukawa [10, 11] proposed a ew geeralized type of fuzzy set called hesitat fuzzy set (HFS), /14/$ IOS Press ad the authors. All rights reserved

2 1602 H. Liao ad Z. Xu / Some ew hybrid weighted aggregatio operators which opes ew perspectives for further research o decisio makig uder hesitat eviromets. Up to ow, the HFS has attracted more ad more scholars attetio [3, 10 18], which i tur has show the defiite advatages of the HFS over all the other exteded forms of fuzzy set. Torra [11] firstly gave the cocept of HFS ad defied some of its basic operatios. After that, Torra ad Narukawa [10] preseted a extesio priciple permittig to geeralize the existig operatios o fuzzy sets to HFSs, ad described the applicatio of this ew type of sets i the framework of group decisio makig. Xia et al. [12, 13] defied some operators ad gave a itesive study o hesitat fuzzy iformatio aggregatio techiques ad their applicatio i decisio makig. Zhu ad Xu [14, 15] proposed the hesitat fuzzy Boferroi meas ad hesitat fuzzy geometric Boferroi meas for multi-criteria decisio makig with hesitat fuzzy iformatio. Subsequetly, Xu ad Xia [16, 17] ivestigated the distace, similarity, ad correlatio measures for HFSs. Wei ad Zhao [18] ivestigated multiple attribute decisio makig with hesitat iterval-valued fuzzy iformatio ad proposed some ew Eistei aggregatio operators with hesitat iterval-valued fuzzy iformatio. Combiig ituitioistic fuzzy set ad HFS, Zhu et al. [19] proposed dual hesitat fuzzy set (DHFS) as a extesio of HFS, which cosists of two parts, i.e., the membership hesitacy fuctio ad the o-membership hesitacy fuctio. Wei et al. [20] ivestigated the multiple attribute decisio makig (MADM) problems i which attribute values take the form of hesitat triagular fuzzy iformatio, ad developed some hesitat triagular fuzzy aggregatio operators. I order to apply HFS to multi-criteria decisio makig, Liao et al. [3] ivestigated the VIKOR method withi the cotext of hesitat fuzzy circumstaces ad they [21] also studied the multiplicative cosistecy of a hesitat fuzzy preferece relatio ad the group cosesus amog differet decisio makers. Later o, to maker a more reasoable decisio, Liao ad Xu [22] proposed a satisfactio degree based iteractive decisio makig method to derive the weights of the hesitat fuzzy preferece relatio. Zhag ad Wei [23] itroduced a Shapley valued-based VIKOR method for multi-criteria decisio makig with hesitat fuzzy decisio makig. Wei et al. [24] ivestigated the hesitat fuzzy multiple attribute decisio makig with icomplete weight iformatio ad developed some optimizatio models to derive the weights. I order to hadle qualitative settigs occurred i decisio makig, Rodriguez et al. [25] itroduced the cocept of hesitat fuzzy liguistic term set (HFLTS) whose evelope is a ucertai liguistic variable [26] ad preseted a multi-criteria liguistic decisio makig model i which the decisio makers provide their assessmets by elicitig liguistic expressios. Zhu ad Xu [27] proposed some cosistecy measures for hesitat fuzzy liguistic preferece relatios. The aim of this paper is to ivestigate the aggregatio operators for HFSs. I [12], Xia ad Xu developed a family of operators to fuse hesitat fuzzy iformatio, such as the hesitat fuzzy weighted averagig (HFWA) operator, the hesitat fuzzy weighted geometric (HFWG) operator, the hesitat fuzzy ordered weighted averagig (HFOWA) operator, ad the hesitat fuzzy ordered weighted geometric (HFOWG) operator. The HFWA ad HFWG operators ca be used to weight the hesitat fuzzy argumets, but igore the importace of the ordered positio of the argumets, while the HFOWA ad HFOWG operators oly weight the ordered positio of each give argumet, but igore the importace of the argumets. To solve this drawback, the hesitat fuzzy hybrid averagig (HFHA) operator ad the hesitat fuzzy hybrid geometric (HFHG) operator were proposed to aggregate hesitat fuzzy argumets, which weight all the give argumets ad their ordered positios simultaeously. Hece, these two operators have may advatages tha the above metioed operators i aggregatig hesitat fuzzy iformatio. However, these two operators do ot satisfy the basic property amed idempotecy, which is desirable for aggregatig a fiite collectio of HFSs. Therefore, i this paper, we ited to develop some ew hesitat fuzzy hybrid weighted aggregatio operators. I additio, ispired by the quasi hesitat fuzzy ordered weighted averagig (QHFOWA) operator proposed i [13], we exted our proposed operator to more geeral forms. Cosider the powerfuless of HFS i multi-criteria decisio makig, we also give some procedures with our proposed operators for multi-criteria sigle perso decisio makig ad multi-criteria group decisio makig. To do so, the remaider of this paper is set out as follows. Sectio 2 gives some basic kowledge of HFS ad the aggregatio operators. Sectio 3 develops two ew hesitat fuzzy hybrid weighted aggregatio operators amed hesitat fuzzy hybrid arithmetical averagig (HFHAA) operator ad hesitat fuzzy hybrid arithmetical geometric (HFHAG) operator. Sectio 4 exteds the HFHAA ad HFHAG operators to the quasi HFHAA ad the quasi HFHAG operators, respectively. I Sectio 5, we apply our proposed operators to sigle

3 H. Liao ad Z. Xu / Some ew hybrid weighted aggregatio operators 1603 perso decisio makig ad multi-criteria group decisio makig uder hesitat fuzzy eviromets. Some practical examples are provided to illustrate the use of our proposed procedures. The paper eds with some coclusios i Sectio Some basic cocepts ad hesitat fuzzy aggregatio operators Hesitat fuzzy set [10, 11], as a geeralizatio of fuzzy set, permits the membership degree of a elemet to a set preseted as several possible values betwee 0 ad 1, which ca better describe the situatios where people have hesitacy i providig their prefereces over objects i the process of decisio makig. To facilitate our presetatio, i what follows, let s review some basic cocepts ad hesitat fuzzy aggregatio operators. Defiitio 1. [10, 11] Let X be a fixed set, a hesitat fuzzy set (HFS) o X is i terms of a fuctio that whe applied to X returs a subset of [0, 1], which ca be represeted as the followig mathematical symbol: E {<x, h E (x) > x X (1) where h E (x) is a set of some values i [0, 1], deotig the possible membership degrees of the elemet x X to the set E. For coveiece, Xia ad Xu [12] called h E (x) a hesitat fuzzy elemet (HFE) ad H the set of all the HFEs. Torra ad Narukawa [10] defied the complemet, uio ad itersectio about HFEs. Based o that, Xia ad Xu [12] gave some operatioal laws o the HFEs h, h 1 ad h 2 : Defiitio 2. [12] Let h, h 1 ad h 2 be three HFEs, ad λ be a positive real umber, the (1) h λ { γ λ ; γ h { (2) λh 1 (1 γ) λ ; γ h (3) h 1 h 2 {γ 1 + γ 2 γ 1 γ 2 ; γ 1 h 1,γ 2 h 2 (4) h 1 h 2 {γ 1 γ 2 ; γ 1 h 1,γ 2 h { 2 (5) h j 1 (1 γ j ) ; (6) h j γj h j γj h j { γ j. Note that the umber of values i differet HFEs may be differet. Let l he (x) be the umber of values i h E (x). For two HFEs h 1 ad h 2, let l max{l h1,l h2. To operate correctly, Xu ad Xia [12] gave the followig regulatio, which is based o the assumptio that all the decisio makers are pessimistic: If l h1 <l h2, the h 1 should be exteded by addig the miimum value i it util it has the same legth with h 2 ;if l h1 >l h2, the h 2 should be exteded by addig the miimum value i it util it has the same legth with h 1. I this paper, we shall exted the shorter oe by addig the value of 0.5 i it, that is to say, we assume that all the decisio makers are compromise. Defiitio 3. [12] For a HFE h, s(h) 1 l h γ h γ is called the score fuctio of h, where l h is the umber of values i h. For two HFEs h 1 ad h 2,ifs(h 1 ) >s(h 2 ), the h 1 >h 2 ;ifs(h 1 ) s(h 2 ), the h 1 h 2. Defiitio 4. [21] For a HFE h, v(h) 1 l h γ i,γ j h (γ i γ j ) 2 is called the variace fuctio of h, where l h is the umber of values i h, ad v(h)is called the variace degree of h. For two HFEs h 1 ad h 2,ifv(h 1 ) >v(h 2 ), the h 1 <h 2 ;ifv(h 1 ) v(h 2 ), the h 1 h 2. Based o the score fuctio s(h) ad the variace fuctio v(h), the compariso scheme ca be developed to rak ay HFEs: If s(h 1 ) <s(h 2 ), the h 1 <h 2 ; If s(h 1 ) s(h 2 ), the 1) If v(h 1 ) <v(h 2 ), the h 1 >h 2 ; 2) If v(h 1 ) v(h 2 ), the h 1 h 2. I order to export the operatios o fuzzy sets to HFSs, Torra ad Narukawa [10] proposed a aggregatio priciple for HFEs: Defiitio 5. [10] Let E {h 1,h 2,...,h be a set of HFEs, be a fuctio o E, : [0, 1] [0, 1], the E γ {h1 h 2 h { (γ) (2) Based o the above extesio priciple, Xia ad Xu [12] developed a series of specific aggregatio operators for HFEs: Defiitio 6. [12] Let h j (j 1, 2,...,) be a collectio of HFEs, ω (ω 1,ω 2,...,ω ) T be the aggregatio-associated vector such that ω j [0, 1] ad ω j 1, the

4 1604 H. Liao ad Z. Xu / Some ew hybrid weighted aggregatio operators 1) A hesitat fuzzy weighted averagig (HFWA) operator is a mappig HFWA : H H, such that HFWA (h 1,h 2,...,h ) ( ) wj h j { γ1 h 1,γ 2 h 2,...,γ h 1 (1 γ j) w j (3) 2) A hesitat fuzzy weighted geometric (HFWG) operator is a mappig HFWG : H H, where HFWG (h 1,h 2,...,h ) h w j j γ1 h 1,γ 2 h 2,...,γ h { γw j j (4) I the case where w (1/,1/,...,1/) T, the HFWA operator reduces to the hesitat fuzzy averagig (HFA) operator ad the HFWG operator reduces to the hesitat fuzzy geometric (HFG) operator. Based o the idea of the ordered weighted averagig (OWA) operator [28], the followig operators ca be defied: Defiitio 7. [12] Let h j (j 1, 2,...,) be a collectio of HFEs, h σ(j) be the jth largest of them, ω (ω 1,ω 2,...,ω ) T be the aggregatio-associated vector such that ω j [0, 1] ad ω j 1, the 1) A hesitat fuzzy ordered weighted averagig (HFOWA) operator is a mappig HFOWA: H H, where HFOWA (h 1,h 2,...,h ) ( ) ωj h σ(j) γσ(1) h σ(1),γ σ(2) h σ(2),...,γ σ() h σ() {1 (1 γ σ(j)) ω j (5) 2) A hesitat fuzzy ordered weighted geometric (HFOWG) operator is a mappig HFOWG: H H, where of the argumets, while the HFOWA ad HFOWG operators oly weight the ordered positio of each give argumets, but igore the importace of the argumets. To solve this drawback, Xia ad Xu [12] the itroduced some hybrid aggregatio operators for hesitat fuzzy argumets, which weight all the give argumets ad their ordered positios. Defiitio 8. [12] For a collectio of HFEs h j (j 1, 2,...,), λ (λ 1,λ 2,...,λ ) T is the weight vector of them with λ j [0, 1] ad λ j 1, is the balacig coefficiet which plays a role of balace, the we defie the followig aggregatio operators, which are all based o the mappig H H with a aggregatio-associated vector ω (ω 1,ω 2,...,ω ) T such that ω j [0, 1] ad ω j 1: 1) The hesitat fuzzy hybrid averagig (HFHA) operator: HFHA (h 1,h 2,...,h ) ( ) ωj ḣ σ(j) γσ(1) ḣ σ(1), γ σ(2) ḣ σ(2),..., γ σ() ḣ σ() {1 (1 γ σ(j)) ω j (7) where ḣ σ(j) is the jth largest of ḣ λ k h k, (k 1, 2,...,). 2) The hesitat fuzzy hybrid geometric (HFHG) operator: HFHG (h 1,h 2,...,h ) ḧ ω j σ(j) { γσ(1) ḧ σ(1), γ σ(2) ḧ σ(2),..., γ σ() ḧ σ() γω j σ(j) (8) where ḧ σ(j) is the jth largest of ḧ k h λ k k, (k 1, 2,...,). Especially, if w (1/,1/,...,1/) T, the the HFHA operator reduces to the HFOWA operator, the HFHG operator reduces to the HFOWG operator. HFOWG (h 1,h 2,...,h ) h ω j σ(j) { γσ(1) h σ(1),γ σ(2) h σ(2),...,γ σ() h σ() γω j σ(j) (6) I the case where ω (1/,1/,...,1/) T, the HFOWA operator reduces to the HFA operator, ad the HFOWG operator becomes the HFG operator. It is oted that the HFWA ad HFWG operators oly weight the hesitat fuzzy argumets themselves, but igore the importace of the ordered positio 3. Some ew hesitat fuzzy hybrid weighted aggregatio operators Although the HFHA (HFHG) operator geeralizes both the HFWA (HFWG) ad HFOWA (HFOWG) operators by weightig the give importace ad the ordered positio of the argumets, there is a flaw that the operator does ot satisfy the desirable property, i.e., idempotecy. A example ca be used to illustrate this drawback.

5 H. Liao ad Z. Xu / Some ew hybrid weighted aggregatio operators 1605 Example 1. Assume h 1 {0.3, 0.3, 0.3, h 2 {0.3, 0.3, 0.3 ad h 3 {0.3, 0.3, 0.3 are three HFEs, whose weight vector is λ (1, 0, 0) T, ad the aggregatio-associated vector is also ω (1, 0, 0) T. The, ḣ h 1 3h 1 (1 (1 0.3) 3, 1 (1 0.3) 3, 1 (1 0.3) 3) (0.657, 0.657, 0.657) ; ḣ h 2 0 h 2 (1 (1 0.3) 0, 1 (1 0.3) 0, 1 (1 0.3) 0) (0, 0, 0) ; ḣ h 3 0 h 3 (1 (1 0.3) 0, 1 (1 0.3) 0, 1 (1 0.3) 0) (0, 0, 0). Obviously, s ( ḣ 1 ) >s (ḣ2 ) s (ḣ3 ). By usig Equatio (7), we have HFHA (h 1,h 2,h 3 ) 3 ( ) ωj ḣ σ(j) γσ(1) ḣ σ(1), γ σ(2) ḣ σ(2), γ σ(3) ḣ σ(3) {1 (1 γ σ(1) ) 1 (1 γ σ(2) ) 0 (1 γ σ(3) ) 0 (0.657, 0.657, 0.657) / {0.3, 0.3, 0.3 Aalogously, ḧ 1 h 31 1 h 3 1 ( 0.3 3, 0.3 3, 0.3 3) (0.027, 0.027, 0.027) ; ḧ 2 h 30 2 h 0 2 (0.3 0, 0.3 0, 0.3 0) (0, 0, 0) ; ḧ 3 h 30 3 h 0 3 (0.3 0, 0.3 0, 0.3 0) (0, 0, 0). HFHG (h 1,h 2,h 3 ) 3 ḧ ω j σ(j) γσ(1) ḧ σ(1), γ σ(2) ḧ σ(2), γ σ(3) ḧ σ(3) { γ 1 σ(1) γ0 σ(2) γ0 σ(3) (0, 0, 0) / {0.3, 0.3, 0.3 Sice idempotecy is the most importat property for every aggregatio operators [29] but the HFHA ad HFWG operators do t meet this basic property, we eed to develop some ew hybrid aggregatio operators which also weight the importace of each argumet ad its ordered positio simultaeously. I this sectio below, we focus o solvig this problem ad try to develop some ew hybrid operators for HFSs. Cosider the HFOWA operator give as Equatio (5), it is equivalet to the followig form: HFOWA (h 1,h 2,...,h ) ( ) ωε(j) h j (9) where h j be the ε(j)th largest elemet of h j (j 1, 2,...,). Ispired by this, supposig the weightig vector of the elemets is λ (λ 1,λ 2,...,λ ) T, i order to weight the positio ad the elemet simultaeously, we ca use such a form as ε(j) h j, which weights both the positio ad the elemet. After ormalizatio, a ew hesitat fuzzy hybrid arithmetical averagig operator ca be geerated. Defiitio 9. For a collectio of HFEs h j (j 1, 2,..., ), a hesitat fuzzy hybrid arithmetical averagig (HFHAA) operator is a mappig HFHAA: H H, defied by a associated weightig vector ω (ω 1,ω 2,...,ω ) T with ω j [0, 1] ad ω j 1, such that ε(j) h j HFHAA (h 1,h 2,...,h ), (10) ε(j) where ε : {1, 2,..., {1, 2,..., is the permutatio such that h j is the ε(j)th largest elemet of the collectio of HFEs h j (j 1, 2,...,), ad λ (λ 1,λ 2,...,λ ) T is the weightig vector of the HFEs h j (j 1, 2,...,), with λ j [0, 1] ad λ j 1. Theorem 1. For a collectio of HFEs h j (j 1, 2,...,), the aggregated value by usig the HFHAA operator is also a HFE, ad HFHAA (h 1,h 2,...,h ) γ 1 h 1,γ 2 h 2,...,γ h 1 (1 γ j ) ε(j) (11)

6 1606 H. Liao ad Z. Xu / Some ew hybrid weighted aggregatio operators where ω (ω 1,ω 2,...,ω ) T is a associated weightig vector with ω j [0, 1] ad ω j 1, ε : {1, 2,..., {1, 2,..., is the permutatio such that h j is the ε(j)th largest elemet of the collectio of HFEs h j (j 1, 2,...,), ad λ (λ 1,λ 2,...,λ ) T is the weightig vector of the HFEs h j (j 1, 2,...,), with λ j [0, 1] ad λ j 1. Proof. From the defiitio of HFS, it is obvious that the aggregated value by usig the HFHAA operator is also a HFE. By usig the operatioal law (2) give i Defiitio 2, we have ε(j) h j λ j ω 1 (1 γ) ε(j) ε(j) γ h j j 1, 2,...,. ε(j) Summatio all these weighted HFEs h j ε(j) (j 1, 2,...,) by usig the operatioal law (5) give i Defiitio 2, we ca derive HFHAA (h 1,h 2,...,h ) ε(j) h j ε(j) 1 (1 γ) γ h j ε(j) ζ 1 h 1,ζ 2 h 2,...,ζ h 1 (1 ζ j ) where h j ε(j) ε(j) h j ad (12) (1 γ j ) γ 1 h 1,γ 2 h 2,...,γ h 1 (1 γ j ) ε(j) ε(j) which completes the proof of Theorem 1. Example 2. Let h 1 {0.2, 0.4, 0.5, h 2 {0.2, 0.6 ad h 3 {0.1, 0.3, 0.4 be three HFEs, whose weight vector is λ (0.15, 0.3, 0.55) T, ad the aggregatioassociated vector is ω (0.3, 0.4, 0.3) T. At first, comparig h 1, h 2 ad h 3 by usig the score fuctio give as Defiitio 3, we have s(h 1 ) , s(h 2 ) , s(h 3 ) Sice s(h 2 ) >s(h 1 ) >s(h 3 ), we obtai h 2 >h 1 > h 3 ad hece ε(1) 2,ε(2) 1,ε(3) 3. The λ 1 ω ε(1) 3 ε(j) , λ 2 ω ε(2) λ 3 ω ε(3) , ε(j) ε(j) The, by usig Equatio (10), we ca calculate that HFHAA (h 1,h 2,h 3 ) 3 ε(j) h j ε(j) ζ j 1 (1 γ) ε(j), γ h j,j 1, 2,..., Combiig Equatios (12) ad (13), we obtai HFHAA (h 1,h 2,...,h ) γ 1 h 1,γ 2 h 2,...,γ h (13) γ 1 h 1,γ 2 h 2,γ 3 h (1 γ j ) ε(j) 3 ε(j) γ 1 h 1,γ 2 h 2,γ 3 h 3 {1 (1 γ 1 ) 0.19 (1 γ 2 ) (1 γ 3 ) 0.524

7 H. Liao ad Z. Xu / Some ew hybrid weighted aggregatio operators 1607 {0.1490, , , , , , , , , , , , , , , , , HFHAA operator. This characteristic is differet from the HFHA operator. Aalogously, we also ca develop the HFHAG operator for HFSs: Theorem 2. (Idempotecy) If h j h (j 1, 2,...,), the HFHAA (h 1,h 2,...,h ) h. Proof. Accordig to Defiitio 9, we have HFHAA (h 1,h 2,...,h ) ε(j) h j ε(j) ε(j) h ε(j) h h ε(j) ε(j) Thus, HFHAA (h 1,h 2,...,h ) h, which completes the proof of Theorem 1. Example 3. Let s use our developed HFHAA operator to calculate Example 1. We have HFHAA (h 1,h 2,h 3 ) 3 ε(j) h j ε(j) γ 1 h 1,γ 2 h 2,γ 3 h 3 {1 (1 γ 1 ) 1 (1 γ 2 ) 0 (1 γ 3 ) 0 {0.3, 0.3, 0.3 h 1 h 2 h 3, which satisfies the property of idempotecy. This is also cosistet with our ituitio. From this example, we ca see that our proposed HFHAA operator is more reasoable tha the HFHA operator developed by Xia ad Xu [12]. By usig the differet maifestatio of weightig vector, the HFHAA operator ca be reduced ito some special cases. For example, if the associated weightig ( T vector ω 1, ) 1,..., 1, the the HFHAA operator reduces to the HFWA operator; if λ ( 1, 1,..., 1 ) T, the the HFHAA operator reduces to the HFOWA operator. It must be poited out that the weightig operatio of the ordered positio ca be sychroized with the weightig operatio of the give importace by the Defiitio 10. For a collectio of HFEs h j (j 1, 2,..., ), a hesitat fuzzy hybrid arithmetical geometric (HFHAG) operator is a mappig HFHAG: H H, defied by a associated weightig vector ω (ω 1,ω 2,...,ω ) T with ω j [0, 1] ad ω j 1, such that HFHAG (h 1,h 2,...,h ) ( ) ε(j) hj, (14) where ε : {1, 2,..., {1, 2,..., beig the permutatio such that h j is the ε(j)th largest elemet of the collectio of HFEs h j (j 1, 2,...,), ad λ (λ 1,λ 2,...,λ ) T is the weightig vector of the HFEs h j (j 1, 2,...,), with λ j [0, 1] ad λ j 1. Theorem 3. For a collectio of HFEs h j (j 1, 2,...,), the aggregated value by usig the HFHAG operator is also a HFE, ad HFHAG (h 1,h 2,...,h ) γ 1 h 1,γ 2 h 2,...,γ h γ ε(j) j (15) where ω (ω 1,ω 2,...,ω ) T is a associated weightig vector with ω j [0, 1] ad ω j 1, ε : {1, 2,..., {1, 2,..., is the permutatio such that h j is the ε(j)th largest elemet of the collectio of HFEs h j (j 1, 2,...,), ad λ (λ 1,λ 2,...,λ ) T is the weightig vector of the HFEs h j (j 1, 2,...,), with λ j [0, 1] ad λ j 1. Proof. Similar to Theorem 1, the aggregated value by usig the HFHAG operator is also a HFE. By usig the operatioal law (1) give i Defiitio 2, we have ( hj ) ε(j) j 1, 2,...,. γ h j γ ε(j),

8 1608 H. Liao ad Z. Xu / Some ew hybrid weighted aggregatio operators Accordig to the operatioal law (6) give i Defiitio 2, we ca derive Proof. Sice h j h, the γ j γ. Hece, accordig to Equatio (14), we have HFHAG (h 1,h 2,...,h ) ( ) hj where ξ 1 h 1,ξ 2 h 2,...,ξ h ξ j ε(j) (16) h j ( ε(j) λ ) j ω λ h j ω ε(j) j ad ξj γ ε(j), (17) γ h j j 1, 2,..., Combiig Equatios (16) ad (17), we ca obtai HFHAG (h 1,h 2,...,h ) γ 1 h 1,γ 2 h 2,...,γ h γ which completes the proof of Theorem 3. ε(j) j Example 4. Let s use the HFHAG operator to fuse the HFEs h 1, h 2 ad h 3 i Example 2. Accordig to Theorem 3, we have HFHAG (h 1,h 2,h 3 ) λ j ω γ ε(j) γ 1 h 1,γ 2 h 2,...,γ h j { γ 0.19 γ 1 h 1,γ 2 h 2,γ 3 h 1 γ γ {0.1391, , , , , , , , , , , , , , , , , HFHAG (h 1,h 2,...,h ) γ 1 h 1,γ 2 h 2,...,γ h γ γ 1 h 1,γ 2 h 2,...,γ h γ ε(j) j ε(j) ε(j) {γ h. γ h Thus, HFHAG (h 1,h 2,...,h ) h, which completes the proof of Theorem 4. Example 5. Let s use our proposed HFHAG operator to calculate Example 1. We have HFHAG (h 1,h 2,h 3 ) ( ) hj ε(j) γ 1 h 1,γ 2 h 2,γ 3 h 3 {γ 1 1 γ0 2 γ0 3 {0.3, 0.3, 0.3 h 1 h 2 h 3, which meas the HFHAG operator satisfies idempotecy, which i other words is more reasoable tha Xia ad Xu [12] s HFHG operator. Especially, if the associated weightig vector ω ( ) 1 T,, 1,..., 1 the the HFHAG operator reduces ( to the HFWG operator; if λ 1 T,, ) 1,..., 1 the the HFHAG operator reduces to the HFOWG operator. With the HFHAG operator, the weightig operatio of the ordered positio also ca be sychroized with the weightig operatio of the give importace, while the HFHG operator does ot have this characteristic. 4. Quasi hesitat fuzzy hybrid arithmetical aggregatio operators Theorem 4. (Idempotecy) If h j h (j 1, 2,...,), the HFHAG (h 1,h 2,...,h ) h. Combiig the HFOWA operator give as Defiitio 7 with the quasi-arithmetical average [30], Xia et al. [13] developed the QHFOWA operator:

9 H. Liao ad Z. Xu / Some ew hybrid weighted aggregatio operators 1609 Defiitio 11. [13] Let h j (j 1, 2,...,) be a collectio of HFEs ad h ρ(j) the jth largest of them. Let QHFOWA : H H,if QHFOWA ( h j j 1, 2,..., ) γρ(j) h ρ(j),,2,..., { ( g 1 ) ω jg(γ ρ(j) ), (18) the QHFOWA is called a quasi hesitat fuzzy ordered weighted averagig (QHFOWA) operator, where g(γ) is a strictly cotiuous mootoic fuctio, ω (ω 1,ω 2,...,ω ) T is the associated weight vector with ω j 1, ad ω j 0, j 1, 2,...,. Similarly, we ca propose the QHFOWG operator as follows: Defiitio 12. Let h j (j 1, 2,...,) be a collectio of HFEs ad h ρ(j) the jth largest of them. Let QHFOWG : H H,if QHFOWG ( h j j 1, 2,..., ) γρ(j) h ρ(j),,2,..., { ( ) g 1 g ω j (γ ρ(j) ), (19) the QHFOWG is called a quasi hesitat fuzzy ordered weighted geometric (QHFOWG) operator, where g(γ) is a strictly cotiuous mootoic fuctio, ω (ω 1,ω 2,...,ω ) T is the associated weight vector with ω j 1, ad ω j 0, j 1, 2,...,. Motivated by Defiitios 11 ad 12, if we replace the arithmetical average ad the arithmetical geometric average i Defiitios 9 ad 10 with the quasi arithmetical average, respectively, the the QHFHAA ad QHFHAG operators will be obtaied, which are i mathematical forms as below: Defiitio 13. For a collectio of HFEs h j (j 1, 2,...,), λ (λ 1,λ 2,...,λ ) T is the weight vector of them with λ j [0, 1] ad λ j 1, the we defie the followig aggregatio operators, which are all based o the mappig H H with a aggregatioassociated vector ω (ω 1,ω 2,...,ω ) T such that ω j [0, 1] ad ω j 1, ad a cotiuous strictly mootoic fuctio g(γ): 1) The quasi hesitat fuzzy hybrid arithmetical averagig (QHFHAA) operator: QHFHAA (h 1,h 2,...,h ) λ g 1 j ω ε(j) g ( ) h j ε(j) γ 1 h 1,γ 2 h 2,...,γ h 1 (1 g(γ j )) g 1 ε(j) (20) 2) The quasi hesitat fuzzy hybrid arithmetical geometric (QHFHAG) operator: QHFHAG (h 1,h 2,...,h ) g 1 ( g(hj ) ) ε(j) γ 1 h 1,γ 2 h 2,...,γ h (g(γ j )) g 1 ε(j) (21) where ε : {1, 2,..., {1, 2,..., is the permutatio such that h j is the ε(j)th largest elemet of the collectio of HFEs h j (j 1, 2,...,), ad λ (λ 1,λ 2,...,λ ) T is the weightig vector of the HFEs h j (j 1, 2,...,), with λ j [0, 1] ad λ j 1. Note that whe assigig differet weightig vector of ω or λ or choosig differet types of fuctio g(γ), the QHFHAA ad QHFHAG operators will reduce to may special cases, which ca be set out as follows: ( If the associated weightig vector ω 1, 1,..., 1 ) T, the the QHFHAA operator reduces to the QHFWA operator show as: QHFHAA (h 1,h 2,...,h ) ( g 1 λ j g ( ) ) h j γ 1 h 1,γ 2 h 2,...,γ h 1 (1 g(γ j )) λ j g 1

10 1610 H. Liao ad Z. Xu / Some ew hybrid weighted aggregatio operators while the QHFHAG operator reduces to the QHFWG operator show as: (l(h j )) ε(j) e ε(j) h j ε(j) ε(j) QHFHAG (h 1,h 2,...,h ) ( g 1 ( g(hj ) ) ) λ j γ 1 h 1,γ 2 h 2,...,γ h (g(γ j )) λ j. g 1 If the argumets weight vector λ ( 1, 1,..., 1 ) T, the the QHFHAA operator reduces to the QHFOWA operator give as Defiitio 11, while the QHFHAG operator reduces to the QHFOWG operator give as Defiitio 12; If g(γ) γ, the the QHFHAA operator reduces to the HFHAA operator give as Defiitio 9, while the QHFHAG operator reduces to the HFHAG operator give as Defiitio 10. It is obvious ad herei we do t show some proofs. If g(γ) l γ, the the QHFHAA operator reduces to the HFHAG operator give as Defiitio 10, while the QHFHAG operator reduces to the HFHAA operator give as Defiitio 9. The derivatio ca be show as below: QHFHAA (h 1,h 2,...,h ) e e ε(j) l(h j) ε(j) ε(j) l(h j) ( ) hj while, / 1 ε(j) ε(j) HFHAG (h1,h 2,...,h ) QHFHAG (h 1,h 2,...,h ) e λ (l(h j )) j ω ε(j) HFHAA (h 1,h 2,...,h ) Some other special cases ca also be costructed by choosig differet types of the fuctio g(γ) for the QHFHAA ad QHFHAG operators, such as g(γ) γ λ, g(γ) 1 (1 γ) λ, g(γ) si((π / 2)γ), g(γ) 1 si((π / 2)(1 γ)), g(γ) cos((π / 2)γ), g(γ) 1 cos((π / 2)(1 γ)), g(γ) ta((π / 2)γ), g(γ) 1 ta((π / 2)(1 γ)), g(γ) λ γ, g(γ) 1 b 1 γ, ad so o (see Xia et al. [13] for more details). I the followig, we try to ivestigate the properties of the QHFHAA ad QHFHAG operators: Theorem 5. (Idempotecy) If h j h (j 1, 2,...,), the QHFHAA (h 1,h 2,...,h ) h, QHFHAG (h 1,h 2,...,h ) h. Proof. Accordig to the Defiitio 13, we ca obtai QHFHAA (h 1,h 2,...,h ) λ g 1 j ω ε(j) g ( ) h j ε(j) λ g 1 j ω ε(j) g (h) ε(j) g 1 ( g (h) ε(j) ε(j) ) g 1 (g (h)) h, QHFHAG (h 1,h 2,...,h ) g 1 ( g(hj ) ) ε(j) g 1 λ (g(h)) j ω ε(j) ε(j) g 1 (g(h)) ε(j) g 1 (g (h)) h, which completes the proof of Theorem 5.

11 H. Liao ad Z. Xu / Some ew hybrid weighted aggregatio operators A approach to multi-criteria decisio makig uder hesitat fuzzy eviromet based o the proposed operators Sice it was firstly proposed by Torra ad Narukawa [10, 11], the HFS has bee ivestigated by may scholars from differet poits of view. It also has bee applied to multi-criteria decisio makig ad shows may advatages tha other exteded forms of fuzzy set due to its particular powerfuless ad efficiecy i represetig ucertaity ad vagueess. The HFE [12, 16] is very close to the huma s cogitive process whe evaluatig the cadidate alteratives, because it ca assig a set of possible values of membership degree of a elemet to a give set, while other exteded forms of fuzzy set, like ituitioistic fuzzy set, iterval valued ituitioistic fuzzy set, liguistic fuzzy set, ca ot be used to represet this situatio i case people has several possible evaluatio values at the same time. I additio, due to the icreasig complexity of socio-ecoomic eviromets, it is less ad less possible for a sigle decisio maker to take all relevat aspects of the problem ito cosideratio. I order to get a more reasoable decisio result, formig a group, such as the board of directors of a compay, whose members come from differet fields is a commo way used i realistic applicatio. However, disagreemets may arise from the differece i the decisio makers subjective evaluatios of the decisio problem [31]. Oce they caot persuade each other, the HFE ca be applied to maitai all of those origial assessmets provided by the decisio makers, while all the other exteded forms of fuzzy set, eve icludig fuzzy multisets [8], caot be used to depict this situatio exactly. Moreover, i practical settig of group decisio makig, aoymity is eeded i order to protect the privacy of the decisio makers or esure oiterferece opiios accumulated. Thus, it is atural to cosider all the assessmets so as to get more reasoable decisio results. This also ca oly be represeted by HFEs. Sice the HFE ca be used ot oly i multi-criteria sigle perso decisio makig, but also i multicriteria group decisio makig, i this sectio, we try to apply the proposed HFHAA, HFHAG, QHFHAA ad QHFHAG operators to multi-criteria sigle perso decisio makig ad multi-criteria group decisio makig, respectively Multi-criteria decisio makig with hesitat fuzzy iformatio Whe a decisio maker iteds to evaluate a collectio of alteratives A {A 1,A 2,...,A m with respect to the predetermied criteria C {C 1,C 2,...,C, he/she may fid it is hard to give a sigle value or a sigle iterval for the membership degree of a elemet to a give set but a set of possible values due to the complexity of the problem ad the icomplete iformatio. For example, suppose that a perso wats to buy a laptop, there are several brads for him/her to choose, such as ThikPad, Apple, Acer, HP ad so forth. The perso fids it s hard for him to decide which is the best eve oly over a sigle criterio amed, for istace, appearace, because if he/she prefers the color of the alterative a to the alterative b with the membership degree 0.7, the desig of the alterative b may be better tha the alterative a with the membership degree 0.8. I this case, the decisio maker caot represet the judgmet of the alterative a to the alterative b over the criterio appearace, which cosists of color ad desig as sub-criteria, i traditioal fuzzy set but oly i HFE as {0.7, 0.8. Such case is commo i our daily life; hece, we eed to develop some decisio models withi the cotext of HFEs to aid the decisio maker. Based o the developed aggregatio operators, we ca propose a procedure for the decisio maker to select the best choice with hesitat fuzzy iformatio, which ivolves the followig steps: Algorithm 1: Step 1. Costruct the hesitat fuzzy decisio matrix. The decisio maker determies the relevat criteria of the potetial alteratives ad gives the evaluatio iformatio i the form of HFEs of the alteratives with respect to the criteria. Whe the decisio maker is asked to compare the alteratives over criteria, he/she may have several possible values accordig to the subcriteria. Thus, i this situatio, it is atural to set out all the possible evaluatios for a alterative uder certai criteria give by the decisio maker, which is represeted as HFE. He/She also determies the importace degrees λ j (j 1, 2,...,) for the relevat criteria accordig to his/her prefereces. Meawhile, sice differet alteratives may have differet focuses ad advatages, to reflect this issue, the decisio make

12 1612 H. Liao ad Z. Xu / Some ew hybrid weighted aggregatio operators also gives the orderig weights ω j (j 1, 2,...,) for differet criteria. Step 2. Utilize the developed aggregatio operators to obtai the HFEs h i, (i 1, 2,...,m) for the alteratives A i, (i 1, 2,...,m). Take the HFHAA operator as a example: Table 1 Hesitat fuzzy decisio matrix C 1 C 2 C 3 A 1 {0.6, 0.7, 0.9 {0.6, 0.8 {0.3, 0.6, 0.9 A 2 {0.7, 0.9 {0.4, 0.5, 0.8, 0.9 {0.4, 0.8 A 3 {0.6, 0.8 {0.6, 0.7, 0.9 {0.3, 0.5, 0.7 A 4 {0.6, 0.8, 0.9 {0.7, 0.9 {0.2, 0.4, 0.7 h i HFHAA (h i1,h i2,...,h i ) γ i1 h i1,γ i2 h i2,...,γ i h i 1 (1 γ ij ) i 1, 2,...,m. ε(ij) ε(ij) (22) Step 3. Compute the score values s(h i ) (i 1, 2,...,m)ofh i (i 1, 2,...,m) by Defiitio 3 ad the variace values v(h i ) (i 1, 2,...,m) of h i (i 1, 2,...,m) by Defiitio 4. Step 4. Get the priority of the alteratives A i (i 1, 2,...,m) by rakig s(h i ) ad v(h i )(i 1, 2,...,m). We ow use a decisio makig problem to illustrate our method: Example 6. Let s cosider a customer who iteds to buy a car. There are four alteratives A 1,A 2,A 3,A 4 uder cosideratio ad the customer takes three criteria ito accout to determie which car to buy: (1) C 1 : Quality of the car, which cosists of three sub-criteria: S 1 : Safety, S 2 : Aerod. degree, ad S 3 : Remedy for quality problems; (2) C 2 : Overall cost of the product, which cosists of four sub-criteria: S 4 : Product price, S 5 : Fuel ecoomy, S 6 : Tax, ad S 7 : Maiteace costs; (3) C 3 : Appearace of the car, which cosists of three sub-criteria: S 8 : Desig; S 9 : Color, ad S 10 : Comfort. The weight iformatio of these three criteria is also determied by the customer as λ (0.5, 0.3, 0.2) T. I additio, cosider the fact that differet cars may focus o differet properties; for example, some cars are promiet i security with high price, while some cars are cheap but with low appearace. To reflect this cocer, the customer gives aother weight vector ω (0.6, 0.2, 0.2) T for each criterio, which deotes that the most promiet feature of the car assigs more weight while the remaiders assig less weight. To select the most desirable car, our proposed operators ca be used to aggregate the decisio iformatio adequately. Suppose that we utilize the HFHAA operator to obtai the HFEs h i for the cars A 1,A 2,A 3,A 4. Take A 2 as a example, we have h 2 HFHAA (h 21,h 22,h 23 ) HFHAA({0.7, 0.9, {0.4, 0.5, 0.8, 0.9, {0.4, 0.8). ( ) Sice s (h 21 ) 2 0.8, s (h 22 ) ( ) ( ) , s (h 23 ) 2 0.6, the h 21 >h 22 >h 23. Thus, ε(21) 1,ε(22) 2,ε(23) 3. The, λ 1 ω ε(21) 3 ε(2j) , As metioed above, it is appropriate for the customer to represet his/her preferece assessmets i hesitat fuzzy sets to maitai the origial evaluatio iformatio adequately, which are show i the hesitat fuzzy decisio matrix H ( h ij (see Table 1). Note )43 that the criteria have two differet types such as beefit type ad cost type. The customer should take this ito accout i the process of determiig the preferece values. λ 2 ω ε(22) λ 3 ω ε(23) , ε(2j) ε(2j) Thus, by usig Equatio (11), we ca calculate that h 2 HFHAA (h 21,h 22,h 23 ) HFHAA({0.7, 0.9, {0.4, 0.5, 0.8, 0.9, {0.4, 0.8)

13 H. Liao ad Z. Xu / Some ew hybrid weighted aggregatio operators 1613 γ 21 h 21,γ 22 h 22,γ 23 h 23 {1 (1 γ 21 ) 0.75 (1 γ 22 ) 0.15 (1 γ 23 ) 0.1 {0.6423, , , , , , , , , , , , , , , Similarly, we ca calculate differet results by usig the HFHAA operator for other alteratives, A 1,A 3, ad A 4. Here we will ot list them for vast amouts of data. Fially, we ca compute the score values s(h i ) (i 1, 2, 3, 4) ad the variace values v(h i ) (i 1, 2, 3, 4) of h i (i 1, 2, 3, 4). By rakig s(h i )(i 1, 2, 3, 4), we ca get the priorities of the alteratives A i (i 1, 2, 3, 4). Sice s(h 1 ) , s(h 2 ) , s(h 3 ) 0.716, ad s(h 4 ) , we get s(h 4 ) > s(h 1 ) >s(h 3 ) >s(h 2 ), the h 4 h 1 h 3 h 2, i.e., the car A 4 is the most desirable choice for the customer. If we use Xia ad Xu s HFHA operator to solve this problem, the we have h 1 HFHA (h 11,h 12,h 13 ) HFHA({0.6, 0.7, 0.9,{0.6, 0.8,{0.3, 0.6, 0.9) {0.6438,0.6670, ,0.6856, , , , , , , , , , , , , , ; h 2 HFHA (h 21,h 22,h 23 ) HFHA ({0.7, 0.9,{0.4, 0.5, 0.8, 0.9,{0.4, 0.8) {0.7097, ,0.7191, ,0.7618, , , ,0.8920, , , , , , , ; h 3 HFHA (h 31,h 32,h 33 ) HFHA ({0.6, 0.8,{0.6, 0.7, 0.9,{0.3, 0.5, 0.7) {0.6438,0.6579,0.6783, , , , , , ,0.8091, , , , , , , , ; h 4 HFHA (h 41,h 42,h 43 ) HFHA ({0.6, 0.8, 0.9,{0.7, 0.9,{0.2, 0.4, 0.7) {0.6564,0.6680, , , ,0.7493, , , ,0.8489, , , , , , , , Sice s(h 1 ) , s(h 2 ) , s(h 3 ) , ad s(h 4 ) , we get s(h 2 ) >s(h 4 ) > s(h 1 ) >s(h 3 ), the h 2 h 4 h 1 h 3. With Xia ad Xu s HFHA operator, the car A 2 turs out to be the most desirable choice for the customer, ad all the other cars are i the same rak. Meawhile, whe usig Xia ad Xu s HFHA operator, we eed to calculate ḣ λ k h k first ad compare them, ad the calculate ω j ḣ σ(j), after which, we shall compute the aggregatio values with ( ) ωj ḣ σ(j). Sice the computatio with hesitat fuzzy sets is very complex, the results derived via Xia ad Xu s HFHA operator is hard to be obtaied. As for our proposed HFHAA operator, the weightig operatio of the ordered positio is sychroized with the weightig operatio of the give importace, which is i the mathematical form as ε(j). Sice both λ j ad ω ε(j) are crisp umbers, we oly eed to calculate / ε(j) h j ε(j), which makes our proposed HFHAA operator is easier to calculate tha Xia ad Xu s HFHA operator Multi-criteria group decisio makig with hesitat fuzzy iformatio As preseted at the begiig of this sectio, the HFE shows may defiite advatages i represetig impreciseess ad vagueess i group decisio makig. I this subsectio, we try to develop a procedure for multicriteria group decisio makig withi the cotext of HFEs. Cosider a group decisio makig problem uder ucertaity. Let A {A 1,A 2,...,A m be the set of alteratives, C {C 1,C 2,...,C be the set of criteria ad E {e 1,e 2,...,e p be the set of decisio makers. Suppose that the decisio maker e k provides all the possible evaluated values uder the criterio C j for the alterative A i deoted by a HFE h (k) ij ad costructs the decisio matrix H k (h (k) ij ) m. He/She also determies the importace degrees λ (k) (j 1, 2,...,) for j the relevat criteria accordig to his/her prefereces. Meawhile, sice differet alteratives may have differet focuses ad advatages, to reflect this issue, the decisio maker also gives the orderig weights ω (k) j (j 1, 2,...,) for differet criteria. Suppose

14 1614 H. Liao ad Z. Xu / Some ew hybrid weighted aggregatio operators that the weight vector of the decisio makers is σ {σ 1,σ 2,...,σ p. The, based o the developed aggregatio operators, we give a method for group decisio makig with hesitat fuzzy iformatio, which ivolves the followig steps: Algorithm 2: Step 1. Utilize the HFWA (or HFWG) operator show as Defiitio 6 to aggregate all the idividual hesitat fuzzy decisio matrix H k (h (k) ij ) m (k 1, 2,..., p) ito the collective hesitat fuzzy decisio matrix H (h ij ), where { h ij (k) γ ij h(k) ij,k1,2,...,p 1 p (1 γ(k) k1 ij )σ k, i 1, 2,...,m,j 1, 2,..., (23) or { p h ij (k) γ ij h(k) ij,k1,2,...,p k1 (γ(k) ij )σ k, i 1, 2,...,m,j 1, 2,..., (24) Step 2. Utilize the QHFHAA (or QHFHAG) operator to obtai the HFEs h i (i 1, 2,...,m) for the alteratives A i (i 1, 2,...,m), where or h i QHFHAA (h i1,h i2,...,h im ) γ ij h ij,,2,..., 1 (1 g(γ j )) g 1 i 1, 2,...,m h i QHFHAG (h 1,h 2,...,h ) γ ij h ij,,2,..., (g(γ j )) g 1 i 1, 2,...,m ε(j) ε(j) (25) (26) Step 3. Compute the score values s(h i )(i 1, 2,...,m) of h i (i 1, 2,...,m) by Defiitio 3 ad the variace values v(h i )ofh i (i 1, 2,...,m) by Defiitio 4. Step 4. Get the priority of the alteratives A i (i 1, 2,...,m) by rakig s(h i ) ad v(h i )(i 1, 2,...,m). We ow cosider a multi-criteria group decisio makig problem that cocers evaluatig ad rakig work systems safety (adapted from [32]) to illustrate our method: Example 7. Maitaiig the safety of work systems i workplace is oe of the most importat compoets of safety maagemet withi a effective maufacturig orgaizatio. There are may factors which affect the safety system simultaeously. Accordig to the statistical aalysis of the past work accidets i a maufacturig compay i Akara, Turkey, Dağdevire ad Yüksel [32] foud there are four sorts of factors which affect the safety system: ζ 1 : Orgaizatioal factors, which ivolve job rotatio, workig time, job completio pressure, ad isufficiet cotrol; ζ 2 : Persoal factors, which cosist of isufficiet preparatio, isufficiet resposibility, tedecy of risky behavior, ad lack of adaptatio; ζ 3 : Job related factors, which ca be divided ito job related fatigue, reduced operatio times due to dagerous behaviors, ad variety ad dimesio of job related iformatio; I additio, it is ot possible to assume that the effects of all factors of work safety are the same i all cases. Hece, by usig the fuzzy aalytic hierarchy process (FAHP) method, Dağdevire ad Yüksel [32] costructed a hierarchical structure to depict the factors ad sub-factors, ad the determied the weight vector of these three factors, which is λ (0.388, 0.3, 0.312). Three experts e 1,e 2,e 3 from differet departmets, whose weight vector is σ {0.4, 0.3, 0.3, are gathered together to evaluate three cadidate work systems A 1,A 2,A 3 accordig to the above predetermied factors ζ 1,ζ 2,ζ 3. However, sice these factors effectig work system safety have o-physical structures, it is hard for the experts to represet their preferece by usig crisp fuzzy umbers. HFEs are appropriate for them to use i expressig these prefereces realistically sice they may have a set of possible values whe evaluatig these behavioral ad qualitative factors. Thus, the hesitat fuzzy judgmet matrices H k (h (k) ij ) 34 (k 1, 2, 3) are costructed by the experts, show as Tables 2 4. Furthermore, cosiderig the fact that differet experts are familiar with differet research fields, ad meawhile, differet work systems

15 H. Liao ad Z. Xu / Some ew hybrid weighted aggregatio operators 1615 may focus o differet partitios, the experts may wat to give more weights to the criterio which is more promiet. Hece, aother weight vectors are determied by the experts accordig to their prefereces, which are ω (1) (0.4, 0.3, 0.3), ω (2) (0.5, 0.3, 0.2) ad ω (3) (0.4, 0.4, 0.2). To get the optimal work system, the followig steps are give: Step 1. Utilize the aggregatio operator to fuse all the idividual hesitat fuzzy decisio matrices H k (h (k) ij ) 34 (k 1, 2, 3) ito the collective hesitat fuzzy decisio matrix H (h ij ) 34. Here we use the HFWA operator to fuse the idividual hesitat fuzzy decisio matrix. Thus, we have h ij γ (k) ij h(k) ij,k1,2,3 i 1, 2, 3,j 1, 2, 3, 4 { 1 3 k1 (1 γ(k) Table 2 The hesitat fuzzy decisio matrix H (1) ij )σ k, (27) ζ 1 ζ 2 ζ 3 A 1 {0.6 {0.7 {0.4, 0.5 A 2 {0.6, 0.8 {0.5, 0.9 {0.7 A 3 {0.4, 0.5 {0.3 {0.6 Table 3 The hesitat fuzzy decisio matrix H (2) ζ 1 ζ 2 ζ 3 A 1 {0.2, 0.4 {0.3, 0.5 {0.4 A 2 {0.8 {0.7 {0.6, 0.7 A 3 {0.4 {0.3, 0.6 {0.5, 0.7 Table 4 The hesitat fuzzy decisio matrix H (3) ζ 1 ζ 2 ζ 3 A 1 {0.5 {0.3, 0.4 {0.6 A 2 {0.7, 0.9 {0.8 {0.5, 0.6 A 3 {0.3, 0.4 {0.4, 0.5 {0.8 Take h 23 as a example: h 23 γ (k) 23 h(k) 23,k1,2,3 { 1 3 k1 (1 γ(k) 23 )σ k { 1 (1 0.7) 0.4 (1 0.6) 0.3 (1 0.5) 0.3, 1 (1 0.7) 0.4 (1 0.6) 0.3 (1 0.6) 0.3, 1 (1 0.7) 0.4 (1 0.7) 0.3 (1 0.5) 0.3, 1 (1 0.7) 0.4 (1 0.7) 0.3 (1 0.6) 0.3 {0.619, 0.644, 0.65, Similarly, other fused values ca be obtaied, ad the the collective hesitat fuzzy matrix ca be derived as below: {0.473, {0.501, 0.524, 0.549, 0.57 {0.469, H {0.702, 0.774, 0.786, {0.674, {0.619, 0.644, 0.65, {0.372, 0.4, 0.442, {0.332, 0.367, 0.435, {0.653, Step 2. Utilize the aggregatio operator (such as the QHFHAA or QHFHAG operator) to obtai the HFEs h i (i 1, 2, 3) for the alteratives A i (i 1, 2, 3). Here we use the QHFHAA operator to fuse the collective HFEs ad let g(γ) γ, the we ca get h 1 {0.4825, , 0.493, , , , , , , , , , , , , ; h 2 {0.6811, 0.685, 0.686, , , , , 0.731, , , , , , , , , 0.769, , , , , 0.776, , , , , , , , , , ; h 3 {0.4894, , , , , 0.506, 0.509, , , , , , , , , 0.525, , , , , , , , , , , , , , , ,

16 1616 H. Liao ad Z. Xu / Some ew hybrid weighted aggregatio operators The computatioal process of h 2 ca be illustrated as a example: ( ) Sice s (h 21 ) , ( ) s (h 22 ) , ( ) s (h 23 ) , the, h 21 >h 22 >h 23. Thus, ε(21) 1, ε(22) 2, ε(23) 3, ad, λ 1 ω ε(21) 3 ε(2j) , λ 2 ω ε(22) λ 3 ω ε(23) , ε(2j) ε(2j) Therefore, by usig Equatio (20), we ca calculate that h 2 QHFHAA (h 21,h 22,h 23 ) γ 21 h 21,γ 22 h 22,γ 23 h 23 { 1 (1 γ 21 ) 0.56 (1 γ 22 ) (1 γ 23 ) {0.6811, 0.685, 0.686, , , , , 0.731, , , , , , , , , 0.769, , , , , 0.776, , , , , , , , , , ; Step 3. Compute the score values s(h i ) of h i (i 1, 2, 3) by Defiitio 3, ad the we have s(h 1 ) , s(h 2 ) , ad s(h 3 ) Step 4. Sice s(h 2 ) >s(h 3 ) >s(h 1 ), the we get h 2 h 3 h 1, which meas ζ 2 is the most desirable work system. 6. Cocludig remarks I this paper, we have poited out the drawbacks of some proposed aggregatio operators for HFEs, ad the some ew hybrid hesitat fuzzy weighted aggregatio operators, such as the hesitat fuzzy hybrid arithmetical averagig (HFHAA) operator, the hesitat fuzzy hybrid arithmetical geometric (HFHAG) operator, the quasi HFHAA operator ad the quasi HFHAG operator have bee proposed ad their properties have bee ivestigated. We have foud that all of our developed operators satisfyig the property amed idempotecy. Furthermore, we have applied our proposed operators to multi-criteria sigle perso decisio makig ad multi-criteria group decisio makig respectively. Ackowledgemets The authors are very grateful to the aoymous reviewers for their isightful ad costructive commets ad suggestios that have led to a improved versio of this paper. The work was supported i part by the Natioal Natural Sciece Foudatio of Chia (No ad ). Refereces [1] L.A. Zadeh, Fuzzy sets, Iformatio ad Cotrol 8 (1965), [2] R.V. Rao, Decisio makig i the maufacturig eviromet usig graph theory ad fuzzy multiple attribute decisio makig methods, Spriger-Verlag, Lodo, [3] H.C. Liao ad Z.S. Xu, A VIKOR-based method for hesitat fuzzy multi-criteria decisio makig, Fuzzy Optimizatio ad Decisio Makig, (2013), doi: /s [4] K.T. Ataassov, Ituitioistic fuzzy sets, Fuzzy Sets ad Systems 20 (1986), [5] K.T. Ataassov ad G. Gargov, Iterval-valued ituitioistic fuzzy sets, Fuzzy Sets ad Systems 31 (1989), [6] M. Mizumoto ad K. Taaka, Some properties of fuzzy sets of type 2, Iformatio ad Cotrol 31 (1976), [7] D. Dubois ad H. Prade, Fuzzy sets ad systems-theory ad applicatios, Academic Press, [8] R.R. Yager, O the theory of bags, Iteratioal Joural of Geeral System 13 (1986), [9] J. Motero, D. Gómez ad H. Bustice, O the relevace of some families of fuzzy sets, Fuzzy sets ad Systems 158 (2007), [10] V. Torra ad Y. Narukawa, O hesitat fuzzy sets ad decisio, The 18th IEEE Iteratioal Coferece o Fuzzy Systems, Jeju Islad, Korea, 2009, pp [11] V. Torra, Hesitat fuzzy sets, Iteratioal Joural of Itelliget Systems 25 (2010), [12] M.M. Xia ad Z.S. Xu, Hesitat fuzzy iformatio aggregatio i decisio makig, Iteratioal Joural of Approximate Reasoig 52 (2011), [13] M.M. Xia, Z.S. Xu ad N. Che, Some hesitat fuzzy aggregatio operators with their applicatio i group decisio makig, Group Decisio ad Negotiatio 8 (2011). doi: /s

17 H. Liao ad Z. Xu / Some ew hybrid weighted aggregatio operators 1617 [14] B. Zhu ad Z.S. Xu, Hesitat fuzzy Boferroi meas for multi-criteria decisio makig, Joural of the Operatioal Research Society (2013). doi: /jors [15] B. Zhu, Z.S. Xu ad M.M. Xia, Hesitat fuzzy geometric Boferroi meas, Iformatio Scieces 205 (2012), [16] Z.S. Xu ad M.M. Xia, Distace ad similarity measures for hesitat fuzzy sets, Iformatio Scieces 181 (2011), [17] Z.S. Xu ad M.M. Xia, O distace ad correlatio measures of hesitat fuzzy iformatio, Iteratioal Joural of Itelliget Systems 26 (2011), [18] G.W. Wei ad X. Zhao, Iduced hesitat iterval-valued fuzzy Eistei aggregatio operators ad their applicatio to multiple attribute decisio makig, Joural of Itelliget ad Fuzzy Systems 24 (2013), [19] B. Zhu, Z.S. Xu ad M.M. Xia, Dual hesitat fuzzy set, Joural of Applied Mathematics (2012). doi: / [20] G.W. Wei, H.J. Wag, X.F. Zhao ad R. Li, Hesitat triagular fuzzy iformatio aggregatio i multiple attribute decisio makig, Joural of Itelliget ad Fuzzy Systems. doi: /IFS [21] H.C. Liao, Z.S. Xu ad M.M. Xia, Multiplicative cosistecy o hesitat fuzzy preferece relatio ad the applicatio o group decisio makig, Iteratioal Joural of Iformatio Techology & Decisio Makig (2013), Techical report. [22] H.C. Liao ad Z.S. Xu, Satisfactio degree based iteractive decisio makig method uder hesitat fuzzy eviromet with icomplete weights, Iteratioal Joural of Ucertaity, Fuzziess ad Kowledge-Based Systems (2013), accepted for publicatio. [23] N. Zhag ad G.W. Wei, A multiple criteria hesitat fuzzy decisio makig with Shapley value-based VIKOR method, Joural of Itelliget ad Fuzzy Systems 24 (2013), [24] G.W. Wei, H.J. Wag, X.F. Zhao ad R. Li, Approaches to hesitat fuzzy multiple attribute decisio makig with icomplete weight iformatio, Joural of Itelliget ad Fuzzy Systems. doi: /IFS [25] R.M. Rodríguez, L. Martíez ad F. Herrera, Hesitat fuzzy liguistic term sets for decisio makig, IEEE Trasactios o Fuzzy Systems 1 (2012), [26] Z.S. Xu, Ucertai liguistic aggregatio operators based approach to multiple attribute group decisio makig uder ucertai liguistic eviromet, Iformatio Scieces 168 (2004), [27] B. Zhu ad Z.S. Xu, Cosistecy measures for hesitat fuzzy liguistic preferece relatios, IEEE Trasactios o Fuzzy Systems. doi: /TFUZZ [28] R.R. Yager, O ordered weighted averagig aggregatio operators i multi-criteria decisio makig, IEEE Trasactios o Systems, Ma, ad Cyberetics 18 (1988), [29] J. Li ad Y. Jiag, Some hybrid weighted averagig operators ad their applicatio to decisio makig, Iformatio Fusio (2011). doi: /j.iffus [30] J. Fodor, J.L. Marichal ad M. Roubes, Characterizatio of the ordered weighted averagig operators, IEEE Trasactio Fuzzy Systems 3 (1995), [31] P.L. Yu, A class of solutios for group decisio problems, Maagemet Sciece 19 (1973), [32] M. Dağdevire ad İ. Yüksel, Developig a fuzzy aalytic hierarchy process (AHP) model for behavior-based safety maagemet, Iformatio Scieces 178 (2008),