Some heSitant fuzzy geometric operatorS and their application to multiple attribute group deciSion making

. Hesitant fuzzy set (HFS), a generalization of fuzzy set (FS), permits the membership degree of an element of a set to be represented as several possible values between 0 and 1. In this paper, motivated by the extension principle of HFs, we export Einstein operations on FSs to HFs, and develop some new aggregation operators, such as the hesitant fuzzy Einstein weighted geometric operator, hesitant fuzzy Einstein ordered weighted geometric operator, and hesitant fuzzy Einstein hybrid weighted geometric operator, for aggregating hesitant fuzzy elements. In addition, we discuss the correlations between the proposed aggregation operators and the existing ones respectively. Finally, we apply the hesitant fuzzy Einstein weighted geometric operator to multiple attribute group decision making with hesitant fuzzy information. Some numerical examples are given to illustrate the proposed aggregation operators.

introduction Aggregation operators, usually taking the forms of mathematical functions, are common techniques to fuse all the input individual data into a single one. Due to their great importance in the information processing of extensive areas, such as decision making, pattern recognition, information retrieval, medical diagnosis, data mining, machine learning, etc., the investigation on aggregation operators has been receiving much attention from both researchers and practitioners over the last decades (Yager, Kacprzyk 1997;Calvo et al. 2002;Xu, Da 2003;Torra, Narukawa 2007;Beliakov et al. 2007;Grabisch et al. 2009). Three of the most common geometric operators for aggregating arguments are the weighted geometric (WG) operator (Saaty 1980;Aczl, Saaty 1983;Willett, Sharda 1991;Benjamin et al. 1992;Xu 2000;Xu, Da 2003), the ordered weighted geometric (OWG) operator Chiclana et al. 2001;Xu, Da 2003), which based on the ordered weighted geometric (OWG) operator (Yager 1988) and the geometric mean, and the hybrid weighted geometric (HWG) operator (Xu, Da 2003). The WG operator first weights all the given arguments and then aggregates all these weighted arguments into a collective one. The fundamental aspect of the OWG operator is the reordering step before aggregating all the ordered weighted arguments into a collective one. The HWG operator generalizes both the OWG and WG operators and reflects the importance degrees of both the given argument and the ordered position of the argument.
In the real-life world, due to the increasing complexity of the socioeconomic environment and the lack of knowledge or data about the problem domain, crisp data are sometimes unavailable. Thus, the input arguments may be vague or fuzzy in nature. Besides fuzzy sets (FSs) by Zadeh (1965), several extensions of this concept have been introduced in the literature, for example, intuitionistic fuzzy sets (Atanassov 1986), interval-valued fuzzy sets (Zadeh 1973), type 2 fuzzy sets (Mizumoto, Tanaka 1976;Dubois, Prade 1980), fuzzy multisets (Yager 1986;Chakrabarty, Despi 2007) and hesitant fuzzy sets (HFSs) (Torra, Narukawa 2009;Torra 2010). IFSs are equivalent to interval-valued fuzzy sets (Atanassov, Gargov 1989;Cornelis et al. 2004). The membership of an element to a type 2 fuzzy set is defined in terms of a FS on the domain of memberships. IFSs can be seen, from a mathematical point of view, as a particular case of type 2 fuzzy sets (Dubois et al. 2005). Fuzzy multisets, or fuzzy bags, permit us to have multiple occurrences of the elements. The basic elements of a HFS are hesitant fuzzy elements (HFEs) , each of which is characterized by a membership degree consisting of a set of possible values. Although all HFSs can be represented as fuzzy multisets, the operations on fuzzy multisets do not apply properly on HFSs. Torra and Narukawa (2009) and Torra (2010) showed that the envelope of HFS is an IFS, and proved that the operations applied to the envelope of HFS are consistent with the ones of IFS. It can be proved that HFSs can also be represented as type 2 fuzzy sets and IFS is a particular case of HFS.
In many practical situations, particularly in the process of group decision making under uncertainty and anonymity, the experts may come from different research areas and thus have different backgrounds and levels of knowledge, skills, experience, and personality, the experts may not have enough expertise or possess a sufficient level of knowledge to precisely express their preferences over the objects, and then, they usually have some uncertainty in providing their preferences. Moreover, the experts have only assigned a small and finite set in providing their preferences, where the difficulty may be caused by a doubt between a few different values. In such cases, the data or preferences given by the experts may be appropriately expressed in HFEs. For example, in multiple attribute group decision-making (MAGDM) problems, anonymity is required in order to protect the decision makers' privacy or avoid influencing each other, such as presidential election, blind peer review of thesis, etc., in which we do not know which attributes that the decision makers are respectively familiar with, and thus, leading us to consider all the situations in order to get more reasonable decision results. But the existing methods only consider the minor situations that each decision maker is good at evaluating all the attributes, which hardly happen. HFS is very useful in avoiding such issues in which each attribute can be described as a HFE defined in terms of the opinions of decision makers (Torra, Narukawa 2009). Up to now, some authors (Torra, Narukawa 2009;Torra 2010;Xu, Xia 2011a, b;Rodriguez et al. 2012;Zhu et al. 2012a, b;Wei 2012;Yu et al. 2012) have paid attention to the HFS theory. Torra and Narukawa (2009) and Torra (2010) proposed the concept of HFS, which is deferent from other extensions exist for FSs, and also introduced some basic operations on HFSs. Torra and Narukawa (2009) presented an extension principle of HFSs, which permits to generalize existing operations on FSs to HFSs, and also discussed their use in decision making. Xu and Xia (2011a, b) proposed a variety of distance measures for HFSs, and particularly developed a number of hesitant ordered weighted distance measures and hesitant ordered weighted similarity measures, which can alleviate the influence of unduly large (or small) deviations on the aggregation results by assigning them low (or high) weights.  developed a series of aggregation operators for hesitant fuzzy information, and applied them to solve decision making problems.  developed several series of aggregation operators for hesitant fuzzy information with the aid of quasi-arithmetic means, and gave a group decision making method under hesitant fuzzy environment based on the developed aggregation operators and the weight-determined technics. Rodriguez et al. (2012) introduced the concept of hesitant fuzzy linguistic term set (HFLTS) to increase the flexibility and richness of linguistic elicitation, and presented a multi-criteria linguistic decision-making model in which experts provide their assessments by using linguistic expressions based on comparative terms. Zhu et al. (2012a) introduced the dual hesitant fuzzy set (DHFS), and investigated some basic operations, properties and an extension principle for DHFSs. The results were illustrated by a practical example of group forecasting. Motivated by the ideal of prioritized aggregation operators (Yager 2008), Wei (2012) developed some prioritized aggregation operators for aggregating hesitant fuzzy information, and utilized these operators to develop some approaches to solve the hesitant fuzzy multiple attribute decision making problems in which the attributes are in different priority level. Zhu et al. (2012b) extended the geometric Bonferroni mean (GBM) to hesitant fuzzy environment, and defined a hesitant fuzzy geometric Bonferroni mean, and applied it to multi-criteria decision making. Yu et al. (2012) extended the generalized Bonferroni mean (GBM) to hesitant fuzzy environment and proposed the generalized hesitant fuzzy Bonferroni mean (GHFBM). Then they proposed an approach based on proposed operator for multiple criteria group decision making under hesitant fuzzy environment.
The aforementioned hesitant aggregation operators Xu, Xia 2011b;Wei 2012;Zhu et al. 2012b;Yu et al. 2012) are mainly based on product triangular norm (t-norm) and its dual triangular conorm (t-conorm) (probabilistic sum), which are the most commonly used ones in decision making applications (Schweizer, Sklar 1983;Hájek 1998). The product and Einstein t-norms are two prototypical examples of the class of strict t-norms. For an intersection on FS, a good alternative to the product t-norm is the Einstein t-norm, which typically gives the same smooth approximations as the product t-norm. Equivalently, for an intersection on FS, a good alternative to probabilistic sum is the Einstein sum. Liu (2011, 2012) introduced Einstein operations on IFSs, and studied some intuitionistic fuzzy aggregation operators with the help of Einstein operations. However, it seems that in the literature there is little investigation on aggregation techniques using the Einstein operations for aggregating a collection of HFEs. In this paper, we shall develop some geometric aggregation operators based on Einstein t-norm and its dual t-conorm, and give an application of these operators to MAGDM. To do so, this paper is structured as follows. In Section 1, we briefly review some basic concepts related to HFSs and the existing geometric operators for aggregating HFEs. In Section 2, we introduce some Einstein operations on HFSs, and develop some novel geometric aggregation operators, such as the hesitant fuzzy Einstein weighted geometric (HFWG ε ) operator, hesitant fuzzy ordered Einstein weighted geometric (HFOWG ε ) operator, and hesitant fuzzy Einstein hybrid weighted geometric (HFHWG ε ) operator, for aggregating a collection of HFEs. In addition, we make some comparisons between the proposed operator and ones proposed by . In Section 3, we apply the HFWG ε operator to MAGDM with hesitant fuzzy information. In the last section, we have a conclusion.

preliminary
The FS, an extension of the classical notion of set, was introduced by Zadeh (1965). definition 1. Let a set X be fixed, a FS F on X is defined as: where F µ is a mapping from X to the closed interval [0,1], and for each x X ∈ , ( ) F x µ is called the degree of membership of x in X.
The set theoretical operations have had an important role since in the beginning of FS theory. Starting from Zadeh's operations min and max many other operators were introduced in the fuzzy set literature (Zadeh 1965). All types of the particular operators were included in the general concepts of t-norms and t-conorms (Schweizer, Sklar 1983;Hájek 1998), which satisfy the requirements of the conjunction and disjunction operators, respectively. They are the most general families of binary functions that map the unit square into the unit interval, i.e. Here, we introduce some examples of the t-norms and t-conorms (Schweizer, Sklar 1983;Hájek 1998): -Zadeh-intersection min is a t-norm, Zadeh-union max is a t-conorm; -Algebraic product ⋅ is a t-norm and Algebraic sum + is a t-conorm, where: ( , ) x y xy ⋅ = , ˆ( , ) x y x y xy + = + − ; -Einstein product ε  is a t-norm and Einstein sum + ε is a t-conorm, where: , 1 x y xy The most accepted one extension of the FS is the notion of IFS (Atanassov 1986), which is characterized by a membership function and a non-member function.
definition 2. Let a set X be fixed, an IFS A in X is defined as: for all x X ∈ , and they denote the degrees of membership and non-membership of element x X ∈ to the set A, respectively. Let ≤ for x X ∈ . However, when giving the membership degree of an element on FS, the difficulty of establishing the membership degree is not because we have a margin of error, or some possibility distribution on the possibility values, but because we have several possible values. For such cases, Torra and Narukawa (2009) and Torra (2010) proposed another generation of FS as follows.
definition 3. Let X be a reference set, then hesitant fuzzy set on X is defined in terms of a function h that when applied to X returns a subset of [0, 1].
To be easily understood,  express the HFS as follows: definition 4. Let X be a fixed set, a HFS E on X is defined as: x is a set-valued function from X to the power set of the unit interval (i.e. [0,1] 2 ) and denotes the possible membership degrees of the element x X ∈ to the set E. For convenience, let Ω be the set of all HFSs on X. Given To compare the HFEs, Xia and Xu (2011) Based on the above algebraic operational laws of HFEs and Definition 5,  proposed some geometric aggregation operators for aggregating HFEs as listed below: For a collection of n HFEs ( 1,2, )  . The hesitant fuzzy weighted geometric (HFWG) operator: The hesitant fuzzy ordered weighted geometric (HFOWG) operator: HFOWG ( , , , ) , , The hesitant fuzzy hybrid weighted geometric (HFHWG ε ) operator: . Note that the HFWG, HFOWG and HFHWG operators extend the WG, OWG, and HWG operators to aggregate HFEs, respectively.

hesitant fuzzy einstein geometric averaging aggregation operators
In this section, we first introduce the extension principle for extending functions to HFEs proposed by Torra and Narukawa (2009 where Θ  the extension of an operator Θ on a set of HFEs , considers all the values in such sets and the application of Θ on them.
Naturally, this extension principle permits us to consider alternative operations for sum and product on HFEs, e.g. Einstein sum and Einstein product. Let If n is any a positive integer and h is a HFE of H, then the power multiplication operation n h ε ⋅ is a mapping from Z H + × to H: where: Mathematical induction can be used to prove that Eq. (8) holds for all positive integers n. Eq. (8) is called ( ) P n . Basis: Show that the statement ( ) P n holds for 1 n = . The statement ( ) P n amounts to the statement (1) P : . In the right-hand side of the equation, The two sides are equal, so the statement ( ) P n is true for 1 n = . Thus it has been shown the statement (1) P holds. Inductive step: Show that if ( ) P n holds, then also ( 1) P n + holds. Assume ( ) P n holds (for some unspecified value of n). It must then be shown that ( 1) P n + holds, that is: (  Technological and Economic Development of Economy, 2014, 20(3): 371-390 Thereby showing that indeed ( 1) P n + holds. Since both the basis and the inductive step have been proved, it has now been proved by mathematical induction that ( ) P n holds for any positive integer n.

hesitant fuzzy einstein weighted geometric (hfWg ε ) operator
Similar to the HFWG operator (i.e. (4) With the Einstein operational laws of HFEs, the HFWG ε operator, i.e. Eq. (9), can be transformed into the following form by induction on n.
Theorem 3. Let ( 1,2, , ) j h j n =  be a collection of HFEs, then their aggregated value by using the HFWG ε operator is also a HFE and proof. The first result follows quickly from Theorem 2. Below we prove Eq. (11) by using mathematical induction on n. We first prove that Eq. (11) holds for 2 n = .
Based on Definition 5 and the Einstein operational laws of HFEs, the HFOWG ε operator (15) can be transformed into the following form.
Theorem 5. Let ( 1,2, , ) j h j n =  be a collection of HFEs, then their aggregated value by using the HFOWG ε operator is also an HFE, and where ( (1), (2), , ( )) n σ σ σ  is a permutation of (1,2, , ) n  such that ( 1) Note that the HFOWG and HFOWG ε operators are developed based on the idea of the OWA operator (Yager 1988). The main characterization of the OWA operator is its reordering step. Several methods have been developed to obtain the OWA weights. Yager (1988) used linguistic quantifiers to compute the OWA weights. O'Hagan (1988) generated the OWA weights with a predefined degree of orness by maximizing the entropy of the OWA weights. Filev and Yager (1998) obtained the OWA weights based on the exponential smoothing. Yager and Filev (1999) got the OWA weights from a collection of samples with the relevant aggregated data. Xu and Da (2002) obtained the OWA weights under partial weight information by establishing a linear objective-programming model. Especially, based on the normal distribution (Gaussian distribution), Xu (2005) developed a method to obtain the OWA weights, whose prominent characteristic is that it can relieve the influence of unfair arguments on the decision result by assigning low weights to those "false" or "biased" ones. corollary 2. The HFOWG and HFOWG ε operators have the following relation: 1 2 1 2 HFOWG ( , , , ) HFOWG ( , , , ), where ( Thus by Eq. (16), it follows that: If we use the HFOWG operator, developed by Xia and Xu (2011) (i.e. (5) described in Section 1), to aggregate the HFEs ( 1,2,3) j h j = , then we have:

hesitant fuzzy einstein hybrid weighted geometric (hfhWg ε ) operator
The HFWG ε operator weights the hesitant fuzzy argument itself. The HFOWG ε operator weights the values instead of weighting the arguments. This is so because each j ω is attached to the j th value in decreasing order without considering from which argument the value comes. Therefore, weights represent different aspects in both aggregation operators. However, although both points of view are meaningful in a single problem, both aggregation operators present the drawback of considering only one of them. To solve this drawback, it   which completes the proof of Corollary 4. Technological and Economic Development of Economy, 2014, 20( {0. 1455,0.1610,0.1668,0.2262,0.2491,0.2499,0.2577,0.2749,0.2842, 0.2901,0.3183 3289,0.3768,0.4115,0.4243,0.4324,0.4707,0.4848}.

decision making based on hesitant fuzzy information
The MAGDM problems are widespread in real life decision situations. In some practical problems, such as presidential election or the blind peer review of thesis, the experts propose the preferences or opinions for the alternatives with anonymous in order to protect their privacy or avoid influencing each other. In such situations, HFE permits us to represent the rating of the alternative on the attribute given by several experts, so we use a hesitant fuzzy decision matrix to describe the group decision making problems.  In what follows, we apply the HFWG ε and HFWG operators proposed by  to hesitant fuzzy MAGDM, which involves the following steps.
Step 1. Obtain the normalized hesitant fuzzy decision matrix. In general, attributes can be classified into two types: benefit attributes and cost attributes. In other words, the attribute set G can be divided into two subsets: 1 G and 2 G , which are the subset of benefit attributes and cost attributes, respectively. Furthermore, 1 2 G G G ∪ = and 1 2 G G ∩ = Φ, where Φ is empty set. The decision matrix D needs to be normalized besides all the attributes ( 1,2, , ) j g j m =  are of the same type. In this paper we choose the following normalization formula to update the hesitant fuzzy decision matrix D: Step 3. Rank the order of all alternatives. Utilize the method in Definition 5 to compute the scores of the overall rating values i h ( 1,2, , ) i n =  , and rank all the alternatives ( 1,2, , ) in descending order, finally select the most desirable alternative(s) with the largest overall rating value.
We consider a MAGDM problem involves the prioritization of a set of information technology improvement projects (adapted from (Ngwenyama, Bryson 1999)) is used to illustrate the developed HFWG ε operator. x -Quality Management Information, 2 x -Customer Order Tracking, 3 x -Inventory Control and 4 x -Budget Analysis. Technological and Economic Development of Economy, 2014, 20(3): 371-390 The committee is concerned that the projects are prioritized from highest to lowest potential contribution to the firm's strategic goal of gaining competitive advantage in the industry. In assessing the potential contribution of each project, three factors are considered as follows (it should be noted that three factors are benefit attributes): -Productivity 1 ( ) g : The productivity factor assesses the potential of a proposed project to increase effectiveness and efficiency of the firm's manufacturing and service operations; -Differentiation 2 ( ) g : The differentiation factor assesses the potential of a proposed project to fundamentally differentiate the firm's products and services from its competitors, and to make them more desirable to its customers; -Management 3 ( ) g : The management factor assesses the potential of a proposed project to assist management in improving their planning, controlling and decision-making activities. Additionally suppose that the weight vector of the attributes is: In order to avoid influencing each other, the members of the committee are required to provide their preferences in anonymity, so the committee, which includes five members, represents the characteristics of the projects i x ( 1,2,3,4) i = by the HFEs ij h ( 1,2,3,4; 1,2,3) i j = = with respect to the factors j g ( 1,2,3) j = , listed in Table 2. In the following, we use the developed HFWG ε operator to get the optimal project.
Step 1. Because all the factors are benefit attributes, the hesitant fuzzy decision matrix D need not be updated.
Step 2 Step 3. Compute the scores of the overall rating values i h ( 1,2,3,4) i = by Definition 5, and rank all the alternatives i x ( 1,2,3,4) i = in accordance with i h ( 1,2,3,4) i = in descending order. The scores of the overall rating values and the rankings of alternatives are listed in Table 3. Finally select the most desirable alternative 2 x with the largest overall rating value. x x x x   

In
Step 2, if we use the HFWG operator proposed by  (i.e. (4) described in Section 1) to aggregation the values of the alternatives, then the scores of the overall rating values ( 1,2,3,4) i h i = and the rankings of the alternatives ( 1,2,3,4) i x i = are also listed in Table 3.
From Table 3, we can see that the values obtained by the HFWG ε operator are always greater than the ones obtained by the HFWG operator for the same aggregation rating values corresponding to the alternatives, but the rankings of alternatives by both different aggregation operators are slightly different, and the best projects are the same, i.e. the project 3 x . remark 1. Using the HFWG ε operator rather than the HFWG operator, the decision-maker has more optimistic attitude in aggregation process. Using different techniques reflects the decision maker's optimistic (or pessimistic) attitude in aggregation process. Therefore, in general, the different aggregation operators do not always return the same ranking orders and the same alternatives.
conclusion Motivated by the extension principle of HFSs, we have extended the Einstein operations on FSs to HFSs, and have developed some new hesitant fuzzy aggregation operators, including the HFWG ε operator, HFOWG ε operator, and HFHWG ε operator. Then, we have applied the HFWG ε to the DMGDM problem with anonymity. It is worth point out that these aggregation operators are the same effective tools as the aggregation operators proposed by , for aggregating hesitant fuzzy information. It is worth pointing that the proposed aggregation operators not only have the same good nature as those proposed by , but also can reflect the decision-makers more optimistic attitude in aggregation process.

acknowledgment
This work was supported by the National Natural Science Foundation of China (NSFC) (71171048, 71371049), the PhD Program Foundation of Chinese Ministry of Education (20120092110038), Technological and Economic Development of Economy, 2014, 20(3): 371-390