Hesitant Fuzzy Variable and Distribution

: In recent decades, the hesitant fuzzy set theory has been used as a main tool to describe the hesitant fuzzy phenomenon, which usually exists in multiple attributes of decision making. However, in the general case concerning numerous decision-making problems, values of attributes are real numbers, and some decision makers are hesitant about these values. Consequently, the possibility of taking a number contains several possible values in the real number interval [0, 1]. As a result, the hesitant possibility of hesitant fuzzy events cannot be inferred from the given hesitant fuzzy set which only presents the hesitant membership degree with respect to an element belonging to this one. To address this problem, this paper explores the axiomatic system of the hesitant possibility measure from which the hesitant fuzzy theory is constructed. Firstly, a hesitant possibility measure from the pattern space to the power set of [0, 1] is deﬁned, and some properties of this measure are discussed. Secondly, a hesitant fuzzy variable, which is a symmetric set-valued function on the hesitant possibility measure space, is proposed, and the distribution of this variable and one of its functions are studied. Finally, two examples are shown in order to explain the practical applications of the hesitant fuzzy variable in the hesitant fuzzy graph model and decision-making considering hesitant fuzzy attributes. The relevant research results of this paper provide an important mathematical tool for hesitant fuzzy information processing from another new angle different from the theory of hesitant fuzzy sets, and can be utilized to solve decision problems in light of the hesitant fuzzy value of multiple attributes.


Introduction
The hesitant fuzzy set, which is an extended version of the fuzzy set [1], is characterized by hesitant membership grades that contain several possible real numbers in an interval [0, 1] [2], and is illustrated by decomposition theorems and extension principles concerning this concept [3]. In the 2010s, the theory and application in respect to the hesitant fuzzy set were investigated for making a group decision. For one thing, Xia et al. introduced some operators with respect to hesitant fuzzy sets and utilized these operators to deal with hesitant fuzzy information [4]. For another thing, on the basis of the assumption that elements of a hesitant fuzzy set are increasingly arranged and the length of two elements is the same, Xia et al. proposed some distance measures of hesitant fuzzy sets [5]. Consequently, they considered connections of these measures and presented several kinds of similarity measures for hesitant fuzzy sets [6]. After that, Xia et al. extended the preference relation to the hesitant fuzzy case and introduced some relevant properties [7]. Zhang et al. developed the normalization extension and group extension concerning the hesitant fuzzy preference relation for describing the consensus process in terms of hesitant fuzzy information [8,9]. Zhu et al. defined a consensus index for measuring the agreement between the group and arbitrary individual hesitant fuzzy preference relation considering hesitant fuzzy attributes [10]. Xu et al. introduced a consensus model with respect to the hesitant fuzzy preference relation and presented feedback mechanisms [11]. Zhang et al. established some consistency models in order to obtain missing elements for incomplete hesitant fuzzy preference relations. In addition, some aggregation operators have been introduced to fuse hesitant fuzzy sets in decision making [12][13][14][15][16]. These operators are able to provide powerful guarantees for decision makers to handle complex situations. In order to determine better group decisions, some methods based on hesitant fuzzy sets or the extended hesitant fuzzy sets have been introduced, for instance, the hesitant fuzzy TOP-SIS [17], m-polar hesitant fuzzy TOPSIS approach [18], hesitant fuzzy PROMETHEE [19], hesitant fuzzy ELECTRE [20], hesitant fuzzy VIKOR [21], hesitant fuzzy TODIM [22], hesitant fuzzy QUALIFLEX [23], hesitant N-soft sets decision method [24], necessary and possible hesitant fuzzy sets method [25], dual extended hesitant fuzzy sets method [26], hesitant fuzzy N-soft sets method [27] and hesitant fuzzy LINMAP [28].
In recent years, the hesitant fuzzy set theory has been used as a main mathematical tool to describe the hesitant fuzzy phenomenon in the research of multiple attributes of decision making. However, a hesitant fuzzy set only presents several possible values on the unit interval [0, 1], which defines the hesitant membership of an element belonging to this hesitant fuzzy set, and the hesitant possibility of hesitant fuzzy events cannot be inferred from the given hesitant fuzzy sets [2]. Because it is the general case that the value of an attribute is a real number in multiple attributes of group decision making problems, decision makers are hesitant about this real number corresponding to the attribute, that is, the possibility of taking this real number contains several possible values in the interval [0,1]. In this situation, the hesitant fuzzy model is more appropriate to describe hesitant fuzzy events. On the one hand, probability measures can explain the probability of the occurrence of random events and a random variable is essentially a measurable function from the probability measure space to the real value line [29]. On the other hand, as a pair of dual measures, the possibility measure and necessity measure can show the possibility and necessity of fuzzy events, and the basic concept of fuzzy theory is a fuzzy variable, which is a measurable function from measure space to the real set. Similarly, a credibility measure is the self-dual fuzzy measure, which can measure the credibility of fuzzy events [30][31][32]. Therefore, the axiomatic system with respect to the hesitant possibility of hesitant fuzzy events has important theoretical significance in the study of the hesitant fuzzy phenomenon; this axiomatic framework is based on the hesitant fuzzy variable, which is a symmetric set-valued function with respect to the union operation of the hesitant fuzzy variable, and can provide theoretical support for hesitant fuzzy information processing.
In this paper, the main contributions include several points: (1) Proposing a hesitant possibility measure, a hesitant necessary measure and a hesitant credibility measure and studying some properties with respect to these three set-valued measures. (2) Defining a hesitant fuzzy variable on the hesitant possibility measure space, and discussing the distribution of this variable and one of its functions and listing several common hesitant fuzzy variables, including triangle, trapezoid, normal and exponential types. (3) Introducing the concept of the hesitant fuzzy graph based on the proposed hesitant fuzzy variable. The rest of this paper is structured as follows: In Section 2, preliminaries are illustrated. Section 3 presents the axiomatic system with respect to the hesitant possibility of hesitant fuzzy events. Two applications are showed in Section 4. Some conclusions and future works are stated in the final section.

Preliminaries
Definition 1 [33]. If Γ is a domain of discourse and I is any index set, an ample field A is a collection of subsets of Γ with the following conditions: (Γ, A) is called an ample space. Evidently, (Γ, P (Γ)) is a special ample space in which P (Γ) is the power set of Γ.
Definition 2 [34]. Let (Γ, A) be an ample space. A fuzzy measure on an ample field A is defined by a function µ from A to [0, ∞] with two conditions: Particularly, µ : A → [0, ∞] is called the lower semicontinuous when: is called upper semicontinuous when: Definition 3 [35]. Let (Γ, A) be an ample space Γ and I be an arbitrary index set. A set function Pos : A → [0, 1] is called the possibility measure on A when the following conditions hold: The triple (Γ, A, Pos) is defined as a possibility measure space. Set functions Nec : A → [0, 1] and Cr : A → [0, 1] are, respectively, called the necessity measure and credibility measure on A if the following formulas hold for the arbitrary set B ∈ A.
When A, B ∈ Γ, operations A + B and A × B are presented in the following formulas.
Any B k ∈ A is a hesitant fuzzy event, and the triple (Γ, A, Pos H ) is a hesitant possibility measure space. In this paper, the finite real values of [0, 1] are discussed with respect to the hesitant possibility measure Pos H , which is a symmetric mapping with respect to the union operation of subsets in an ample A. Particularly, (1) For two sets B 1 ∈ A and B 2 ∈ A, the following formulation holds, where B 1 and B 2 are mutually independent: The triple (Γ, A, Cr H ) is a hesitant credibility measure space. By Definition 5, some properties of hesitant possibility measure were discussed on the basis of a theorem.
(2) Boundedness. According to (1), it was evident that the boundedness held. (3) Lower semicontinuity. The following topology was introduced: Therefore, the following aspects were discussed: according to the monotonicity of Pos H (B k ), the following relation was true.
By the monotonicity of Pos H and the condition we obtained y n k ∈ Pos H (B n k ) with relation y n 1 ≤ y n 2 ≤ · · · ≤ y n k · · · , and t k ≤ y n k for According to Equation (1), we needed to prove Pos H (B k ) in the following steps, that is, we chose t n ∈ Pos H (B n ) in order to obtain t = lim n→∞ t n . Supposing t n k − t < ε for k ≥ k 0 , when n < n k 0 , we selected an arbitrary t n ∈ Pos H (B n ). When n ≥ n k 0 , k ≥ k 0 ; therefore we consider the two cases. For one thing, when t n k ≤ t n k+1 , t n k , t n k +1 , t n k +2 , · · · , t n k+1 was complemented by t n k and the following method.
Based on the conditions, The above process was possible. For another thing, when t n k > t n k+1 , we could obtain y n k ∈ Pos H (B n k ) with y n k ≤ t n k+1 based on Pos H (B n k ) ≤ Pos H (B n k +1 ); therefore, we let t n k = y n k and repeated the completion of the first case such that t n k , t n k +1 , t n k +2 , · · · , t n k+1 . From the above process of constructing {t n }, we had |t n − t| < ε when n ≥ n k 0 , that is, Pos H (B k ). According to the above, the lower semicontinuity was true based on Equations (3)-(5).
, and from the definition of the hesitant possibility measure, we had x 1 , By the order relation on the operation of Formula (1) and the definition of the S norm, the following result could be obtained.
) and the strong subadditivity were proved.

Theorem 7.
A hesitant necessity measure Nec H on an ample space (Γ, A) has the following properties.
From the above (1), boundedness was easily proved.
Since the hesitant possibility measure Pos H satisfied the lower semicontinuity, we had , and by the definition of the hesitant necessity measure, On the basis of the monotonicity hesitant possibility measure From the order relation on P ([0, 1]), Formula (1) and the definition of S norm, hawse had the following result.
According to the above (2), boundedness could be easily proved.
According to Formula (1), t 2 ). Moreover, from the definition of the hesitant credibility measure Cr H , (4) We proved the fifth property.
From the monotonicity of the hesitant necessary measure Nec H , we obtained Therefore, the proof of property (5) was completed. It is evident that this is a hesitant possibility measure, and has the following formulas.
From the definition of independence among hesitant fuzzy events, B 4 , B 5 is mutually independent, and B 5 , B 6 is the same as B 4 , B 5 . However, B 4 , B 6 is not mutually independent. In addition, from the definition of the hesitant necessity measure and hesitant credibility measure, we could have the following formulas:

Hesitant Fuzzy Variable
Definition 9. Suppose that (Γ, A, Pos H ) is a hesitant possibility measure space, R is the real number set and P (R) is the power set of R. A real-valued function ξ : Γ → R is a hesitant fuzzy variable if, and only if, {γ|ξ(γ) ≤ t } ∈ A for any t ∈ R. Especially, for any B ∈ P (R), a set-valued mapping is defined as follows: Therefore, we had the following theorem: Theorem 10. The set-valued mapping based on Equation (6) is a hesitant possibility measure on P (R).
This result was proved if Pos Hξ (B) satisfied the conditions of the hesitant possibility measure.

Pos Hξ
(2) For arbitrary B i ∈ P (R), i ∈ I, we could obtain Therefore, Pos Hξ : P (R) → P ([0, 1]) is a hesitant possibility measure on P (R). When t ∈ R, we defined the following function on the basis of the hesitant fuzzy variable ξ : Γ → R with respect to the hesitant possibility measure space (Γ, A, Pos H ).
Thus, a hesitant possibility measure Pos Hξ on P (R) was determined by Equation (7) with the following formula: Therefore, from an original hesitant possibility measure space (Γ, A, Pos H ), a new hesitant possibility measure space (R, P (R), Pos Hξ ) was induced by using the hesitant fuzzy variable ξ on the space (Γ, A, Pos H ). Moreover, the hesitant fuzzy variable could be studied according to the induced hesitant possibility measure space (R, P (R), Pos Hξ ). As a result, we had the definition of the hesitant possibility distribution. Definition 11. The set-valued function µ ξ (t), t ∈ R defined by Equation (7) is called the hesitant possibility distribution of the hesitant fuzzy variable ξ.
ξ is discrete if X = {t 0 , t 1 , t 2 · · · } is the set of all values of the hesitant fuzzy variable ξ on the space (Γ, A, Pos H ), and let µ i = Pos H {ξ = t i }(i = 0, 1, 2, · · · ).Thus, the hesitant possibility of hesitant fuzzy events related to a discrete hesitant fuzzy variable can directly be determined by the hesitant possibility, such as Similarly, for a continuous hesitant fuzzy variable ξ, we could obtain the same results by the hesitant possibility distribution µ ξ , for instance where Pos Hξ {ξ ≥ a}, Nec Hξ {ξ ≥ a} and Cr Hξ {ξ ≥ a} are, respectively, called the hesitant possibility, hesitant necessity and hesitant credibility of the hesitant fuzzy event ξ ≥ a.

Several Common Continuous Hesitant Fuzzy Variables and Their Distribution
Example 2. The hesitant possibility distribution µ ξ (t) of triangle hesitant fuzzy variable ξ.
where λ > 0, and is shorthand for λ. For instance, the hesitant possibility distribution of λ = 2 is showed in Figure 4.   is also a hesitant fuzzy variable, and its hesitant possibility distribution is presented as follows.

The Distribution of Functions of Hesitant Fuzzy Variable and the Distribution of Sum of Hesitant Fuzzy Variables
Theorem 12. Provided that(Γ, A, Pos H )is a hesitant possibility measure space,ξ is a hesitant fuzzy variable, and g is a real-valued function; then, g(ξ) is also a hesitant fuzzy variable, and its hesitant possibility distribution is presented as follows.
Proof. From the above conditions, we had Therefore, g(ξ) was a hesitant fuzzy variable; thus, Theorem 13. Supposing that (Γ, A, Pos H ) is a hesitant possibility measure space, ξ and η are mutually independent hesitant fuzzy variables, then ξ + η is also a hesitant fuzzy variable, and its hesitant possibility distribution is presented as follows.

Application of Hesitant Fuzzy Variables
In this section, we introduced two examples to show how hesitant fuzziness can be used to model for hesitant fuzzy graph and hesitant fuzzy group decision making.

Hesitant Fuzzy Graph Based on Hesitant Fuzzy Variable
The complexity of thinking and the cognitive differences among people have determined that it is difficult for people to reach an agreement on the same issue, they often have different views and, finally, they show a hesitant state. How can we more objectively reflect people's different preferences and hesitations, and how close can we come to the real thinking mode of human beings. References [37][38][39][40] utilize the hesitant fuzzy set to describe the degree of hesitation in the human thinking process, while this paper, from another perspective, shows people's hesitation more flexibly. This section mainly discussed the application of the hesitant fuzzy variable in simulating human subjective thinking and proposed a hesitant fuzzy graph according to the hesitant fuzzy variable.

Definition 14.
Suppose that G = (V, E) is a graph, V = {v 1 , v 2 , · · · , v n } is a vertex set of the graph G, E = e ij is an edge collection, (V × V, P (V × V), Pos H ) is the hesitant possibility measure space and ξ ij is a hesitant fuzzy variable on (V × V, P (V × V), Pos H ). If edges e ij of the graph G = (V, E) are presented by ξ ij , G = (V, E)is called the hesitant fuzzy graph, which is shorthand for G = V, E ξ , Pos H .
From the above definition, the hesitant fuzzy variables show the existence and the hesitant fuzzy degree of vertexes and, in this sense, it can more objectively simulate the hesitant fuzzy degree in people's thinking process. When ξ ij = 1, it explains that there is an edge e ij and its hesitant possibility is Pos H ξ ij = 1 ∈ P ([0, 1]); when ξ ij = 0, it explains that there is no edge e ij , and its hesitant possibility is Pos H ξ ij = 0 ∈ P ([0, 1]).
Therefore, a hesitant fuzzy graph consists of two components, the first one being the topological structure of the graph and the other one being the distribution table of the hesitant fuzzy variables corresponding to edges. For instance, Figure 5 shows the topology of a hesitant fuzzy graph, and Table 1 shows the distribution table of the related hesitation possibility and hesitation credibility.

Group Decision Making Based on Hesitant Fuzzy Variable
It could be potentially significant to describe a subjective opinion with hesitation by using hesitant fuzzy modelling. One of the most general situations is that a clear opinion of an individual is not necessarily on a practical question, but their hesitant feeling is stronger than others. In this sense, an individual's hesitant judgement about an issue may be characterized by a hesitant fuzzy variable. As an example, a company manager may want to determine a collective opinion of his subordinates in regards to the quantity of allocated money, which applies to research and development activities.

Group Decision Making Based on Hesitant Fuzzy Variable
It could be potentially significant to describe a subjective opinion with hesitation by using hesitant fuzzy modelling. One of the most general situations is that a clear opinion of an individual is not necessarily on a practical question, but their hesitant feeling is stronger than others. In this sense, an individual's hesitant judgement about an issue may be characterized by a hesitant fuzzy variable. As an example, a company manager may want to determine a collective opinion of his subordinates in regards to the quantity of allocated money, which applies to research and development activities. Let ξ i = {number of money the subordinate i should be put into research and development activities}. The hesitant possible distributions of ξ i (i = 1, 2, 3, 4, 5, 6) might be as follows. {0.0, 0.02} 880, 000 < x Now, suppose that a manager obtained the hesitant possible distribution describing the opinion of each subordinate, which is represented by a hesitant fuzzy variable. The question we deal with is the following: is there a way of making an inference for the hesitant possible distribution of a hesitant fuzzy variable associated with the group opinion from the hesitant possible distributions of hesitant fuzzy variables corresponding to the issues of each subordinate? Where the group opinion is only modeled except for an interactive group decision, we could assume that there are six constants α 1 , α 2 , α 3 , α 4 , α 5 , α 6 with ∑ 6 i=1 α i = 1, and α i represent the importance of the ith subordinate's opinion about the group opinion. For instance, when α 1 = 0.15 and α 2 = 0.45, the opinion of the second subordinate is three times more important than the first. We present that the hesitant fuzzy variable η = ∑ 6 i=1 α i ξ i is a reasonable explanation of the group opinion. As a result, the hesitant possible distribution of η would be inferred by the following steps based on the hesitant possible distributions of (ξ 1 , ξ 2 , ξ 3 , ξ 4 , ξ 5 , ξ 6 ).
Although the introduced example is only a rough description of an application of the hesitant fuzzy variable in a group decision-making framework, it showed that the hesitant fuzzy variable concept appears to be convenient for modelling hesitant fuzzy decision-making problems.

Conclusions
This paper introduced the hesitant possibility measure, hesitant necessity measure and hesitant credibility measure on an ample space (Γ, A), and discussed some properties concerning these three set-valued measures, which, respectively, corresponded to the hesitant possibility measure space, hesitant necessity measure space and hesitant credibility measure space. On the hesitant fuzzy possibility measure space, a hesitant fuzzy variable was defined. Consequently, the distribution of this variable and one of its functions were introduced, including several common hesitant fuzzy variables such as the triangle hesitant fuzzy variable, trapezoid hesitant fuzzy variable, normal hesitant fuzzy variable and exponential hesitant fuzzy variable. As a result, the axiomatic system with regard to the hesitant possibility of hesitant fuzzy events was constructed. This axiomatic framework was based on the hesitant fuzzy variable and could provide theoretical support for hesitant fuzzy information processing, which was illustrated by two practical applications in the fourth section of this paper. In the future, according to obtained results of the hesitant fuzzy variable, the hesitant fuzzy theory could be enriched in digital numerical characteristics of the hesitant fuzzy variable such as the mathematical expectation, variance, correlation coefficient, covariance and all sorts of moments. Besides these notions, the multidimensional hesitant fuzzy variable and its distribution are also directions of further research in the future.