Some novel uncertainty measures of hesitant fuzzy sets and their applications

Abstract

The main purpose of the paper is to develop a general analytical framework of uncertainty measures, which provides a fresh new look into the uncertain information of hesitant fuzzy sets (HFSs). We first refine the axiomatic principles of hesitant fuzzy entropy based on the fuzzy factor and hesitant factor, and propose some distance-based entropy formulas of HFSs. Then, a novel hesitant fuzzy cross-entropy is defined to measure the discrimination of uncertain information of different HFSs. Meanwhile, the relationship between cross-entropy and entropy of HFSs is also discussed, and the hesitant fuzzy entropy may be divided into arithmetic average of fuzzy entropy and hesitant entropy. Moreover, some parameterized uncertainty measures are investigated, and two comparative examples are presented to demonstrate their reasonability. Finally, we apply these proposed measures in multiple attribute decision-making problem to illustrate their efficiency and applicability.

Keywords

Hesitant fuzzy entropy fuzzy factor hesitant factor cross-entropy parameterized uncertainty measures

1 Introduction

Since fuzzy theory was firstly introduced by Zadeh [33] in 1965, the theory and its applications have achieved a great success in a variety of scientific fields. In the past decades, the extensions of fuzzy set have been developed by many scholars, such as the interval-valued fuzzy set (IVFS) [26], intuitionistic fuzzy set (IFS) [1], etc. In 1975, a greater extension, called the type-2 fuzzy set (T2FS), has been given by Zadeh [35]. However, it’s very difficult to deal with the practical problems by use of T2FSs, thus hesitant fuzzy set (HFS), proposed by Torra [23] in 2009, has been viewed as an intermediate kind of fuzzy sets. The membership degree of a HFS could be extended from the single value to a set of possible values, which can reflect the human’s hesitancy more objective and manageable than other extensions of fuzzy set. As a natural generalization of HFSs, Zhu et al. [37] proposed a dual hesitant fuzzy set (DHFS) and investigated its basic operations and properties.

Recently, Rodriguez et al. [20] presented an overview on HFSs and provided a clear perspective on future directions. In fact, the main topics of HFSs have been presented in two categories: one is aggregation operators, the other is information measures. For the first case, Xu and Xia [25, 27] developed a series of basic aggregation operators of hesitant fuzzy information. Zhang [36] proposed several hesitant fuzzy power aggregation operators. Bedregal et al. [2] introduced the fuzzy negations into HFSs. Sevastjanov and Dymova [21] presented some generalized operations of HFSs based on Dempster-Shafer theory. Liao and Xu [8, 9] proposed some hybrid weighted aggregation operators, such as (quasi) HFHAA operator, (quasi) HFHAG operator and their generalized versions. In this paper, we summarize some basic operations of HFSs, such as inclusion relation, equivalent relation, complement set, intersection set and union set, etc., and prove several important properties. For the second issue. information measures, such as distance, similarity, correlation coefficients and entropy, etc., have been viewed as a fundamentally important content in fuzzy mathematics. These measures in hesitant fuzzy information environment have drawn the attention of many researchers. For instance, Xu and Xia [28, 29] introduced distance, similarity and correlation measures into HFSs and showed how they are related to each other. Chen et al. [3] and Liao et al. [14] proposed the correlation coefficients of HFSs and applied them in clustering analysis and decision-making. Li et al. [16] constructed some new distance and similarity measures in HFSs and discussed their applications in multiple criteria decision-making problems. Farhadinia [4] investigated the relationship between entropy, similarity and distance for HFS and interval-valued hesitant fuzzy set (IVHFS). Inspired by these literatures, we may actually construct some distance-based entropies for HFSs under our analytical framework. Meanwhile, some other information measures of HFSs have also been investigated, for example, Farhadinia [5] presented a series of score functions for hesitant fuzzy sets which provides us with a variety of new methods for ranking HFSs. Liao and Xu [7, 11] introduced satisfaction degree, subtraction and division operations in HFSs and applied them to multi-criteria decision-making problem. Actually, decision-making methods are of great importance in practical applications. Liao and Xu [10] extended the classical XIKOR method to accommodate hesitant fuzzy circumstances. Liao et al. [12] and Liu et al. [15] provided a new method to determine the value of consistency index of hesitant fuzzy preference relation and applied this method to group decision-making problems. Liao et al. [13] developed an approach to hesitant fuzzy multi-stage multi-criterion decision-making. Although these approaches are quite enlightening, most of them are based on the aggregation operators. In this paper, we try to avoid using these operators due to different operators may lead to entirely conflicting results. Thus we introduced a new score function to compare different HFSs and develop a novel multiple attribute decision-making approach in hesitant fuzzy information environment.

The study of entropy measures in fuzzy theory also becomes a most interesting issue. Zadeh [34] initiated the concept of fuzzy entropy to quantify the uncertainty associated with a fuzzy set. This concept has been quickly adapted to other extensions of fuzzy set, and has a wide applications in some areas such as decision-making [8, 21 , 30], pattern recognition [8, 13 , 15], clustering analysis [4], image processing [15] and personnel evaluation [23], etc. In order to measure the uncertain information of all types of fuzzy sets, entropy has been constructed in different views. For example, Huang and Yang [6] refined the constructive principles of entropy in IFSs, but they still ignored effects inducing by changes of hesitancy degree when membership equals to non-membership. Pal et al. [18] pointed out that the previous uncertain measures cannot capture all fuzziness and lack of knowledge in IFSs and proposed a generating family of measures. Additionally, cross-entropy, as another kind of uncertainty measure, is often used to describe the discrimination information between two fuzzy sets. Shang and Jiang [22] defined fuzzy cross-entropy in fuzzy sets. Vlachos and Serigiadis [24] introduced the intuitionistic fuzzy cross-entropy into IFSs, and applied it to pattern recognition, medical diagnosis and image segmentation. Xia and Xu [26] proposed some new entropy and cross-entropy measures in IFSs and discussed the relationship between entropy and cross-entropy. Mao et al. [17] provided a new framework to describe uncertainty in IFSs and proposed some novel entropy and cross-entropy measures based on the fuzzy factor and intuitionistic factor. Xu and Xia [30] first introduced a hesitant fuzzy entropy and cross-entropy and applied them in multiattribute decision-making. Zhao et al. [38] adapted a two-tuple entropy model to represent the two types of uncertainty associated with HFSs. Quirós et al. [19] gave an entropy measure definition for finite interval-valued hesitant fuzzy sets. However, the existing definitions of entropy and cross-entropy in HFSs overemphasize fuzziness of a HFS while neglecting its hesitancy. Meanwhile, it is shown that the hesitant fuzzy entropies, proposed by Farhadinia [4], Xu and Xia [30], cannot discriminate some HFSs, even though they are apparently different.

In fact, we need to have an adequate understanding of uncertain information in HFSs before we construct uncertainty measures. In my opinion, the uncertainty of HFSs may be consist of two facets: one is fuzziness, the other is hesitancy. The former is caused by closeness between membership degree and 1/2, the latter is determined by dispersion of HFEs. Every coin has two sides, it not only brings us a convenience but also may cause some new uncertainty when membership degrees are extended to a set of some possible values. It is a pity that few scholars seem to actually realize these facts. Therefore, we need to reformulate the axioms of hesitant fuzzy entropy by introducing the fuzzy factor and hesitant factor, and construct some new distance-based entropy formulas according to the relationship between entropy and distance measure. Inspired by Mao et al. [17], we could propose a novel hesitant fuzzy cross-entropy to describe the discrimination of uncertain information of different HFSs. In general, we do not need to consider the order relations and the numbers of HFEs (hesitant fuzzy elements) while using our cross-entropy formulas, which may be the biggest advantage while comparing to ones proposed by Xu and Xia [30]. Meanwhile, we exploit the relationship between cross-entropy and entropy of HFSs under the new framework, and point out that the hesitant fuzzy entropy induced by cross-entropy could be divided into arithmetic average of fuzzy entropy and hesitant entropy. In order to reflect different effects of fuzziness and hesitancy, we may introduce a parameterized entropy and cross-entropy model. All of these seem to provide us some fresh thoughts to analyze the uncertainty of HFSs.

The remainder of the paper is structured as follows: the following section introduces some basic concepts and their operations. Section 3 proposes the new axiomatic principles of hesitant fuzzy entropy and gives some distance-based entropies. Section 4 introduces a novel hesitant fuzzy cross-entropy and investigates the relationship between cross-entropy and entropy of HFSs. A parameterized cross-entropy model and two comparative examples are presented in Section 5. We apply the proposed models in multiple attribute decision-making problem in Section 6. Section 7 concludes the whole paper.

2 Preliminaries

In this section, we review some necessary concepts, such as fuzzy set (FS) and hesitant fuzzy set (HFS), which will be needed in the following analysis. Throughout the paper, U = {x₁, x₂, ⋯ , x_n} is used frequently to denote the discourse set.

Definition 2.1. [33] A fuzzy set (FS) A on U is given by A = {< x, μ_A (x) > : x ∈ U}, where the mapping μ_A : U → [0, 1] is called membership degree of an object x to A, and satisfies 0 ≤ μ_A (x) ≤1, ∀ x ∈ U.

In order to improve the reasonability and effectiveness of group decision making methods, we often invite some experts to estimate the membership degree of a given object at the same time. In that case, this situation may be found: one expert wants to assign 0.2, the other 0.4, and so on. That is to say, they may fail to agree with each other due to their different knowledge structure and personal experience. Thus, Torra [23] extended the membership degree from the single value to a set of possible values, and firstly introduced the concept of hesitant fuzzy set (HFS) as an special extension of FS.

Definition 2.2. [23] Let P ([0, 1]) denote the family of subsets of some different values in [0,1], a hesitant fuzzy set (HFS) A on U is given by A = {< x, h_A (x) > : x ∈ U}, where the mapping $\begin{matrix} h_{A} : U \to P ([0, 1]) \\ x \to {μ_{A}^{1} (x), μ_{A}^{2} (x), \dots, μ_{A}^{n_{A} (x)} (x)} \end{matrix}$ represents the possible membership degrees of the object x to A, n_A (x) is the number of values in h_A (x), and satisfies $0 \leq μ_{A}^{i} (x) \leq 1, 1 \leq i \leq n_{A} (x), \forall x \in U$ .

Especially, a HFS A can be seen as a classical fuzzy set if the set h_A (x) contains only one element for all x ∈ U. That is, the HFSs include FSs as a special case, and the theory for HFSs may also apply to FSs. For convenience, Xia and Xu [25] called the set $α = {μ_{α}^{i} : 1 \leq i \leq n_{α}}$ the HFE (hesitant fuzzy element).

Notes. The number of different HFSs may be different, which may lead to a wrong result when compare them. Therefore, we assume that two HFSs h_A (x) and h_B (x) have different length n_A (x) and n_B (x), respectively, namely, n_A (x) ≠ n_B (x), and let n (x) = max {n_A (x) , n_B (x)}. To operate correctly, we should extend the short one until both of them have the same length. The specific methods could be seen in Chen et al. [3], Li et al. [16], Xu and Xia [28], etc.

For the sake of discussion, we assume that all HFEs have the same length in the follow-up sections. We can denote all HFSs on U by HFS (U), and some basic operations of HFSs, such as inclusion relation, the complement set, the intersection set and union set, could been seen in Xu [27]. We can easily obtain the following theorems.

Theorem 2.1. Let A, B ∈ HFS (U), we may have the following dual laws:

(A ∪ B) ^C = A^C ∩ B^C;

(A ∩ B) ^C = A^C ∪ B^C.

Theorem 2. 2. Let A, B, C ∈ HFS (U), we have

(A ∪ B) ∪ C = A ∪ (B ∪ C);

(A ∩ B) ∩ C = A ∩ (B ∩ C);

(A ∪ B) ∩ C = (A ∩ C) ∪ (B ∩ C);

(A ∩ B) ∪ C = (A ∪ C) ∩ (B ∪ C).

3 Hesitant fuzzy entropy for hesitant fuzzy sets

The aim of entropy measure is to quantify the uncertainty associated with either a fuzzy set or its extension. As we known, entropy measures have been widely applied in many fields, such as group decision making, pattern recognition, clustering analysis, approximate reasoning and image processing, it is significant to construct a reasonable entropy. Some scholars have proposed different hesitant fuzzy entropy measures in HFSs, such as Farhadinia [4], Xu and Xia [30], and Zhao et al. [38], etc. However, few of them notice the difference in uncertain information between FSs and HFSs. In fact, as the numbers of membership degree increase, the uncertainty in a HFS is not only determined by its fuzziness but also influenced by the hesitancy of HFEs. That is to say, we should take fully into account two facets while constructing uncertainty measures of HFSs, one is its fuzziness, and the other is the hesitancy of HFEs. The former may be reflected by the proximity of membership degree to 1/2 just as fuzziness in FSs, the latter would be displayed with the dispersive degree of all HFEs. In order to characterize the two facets of uncertain information in HFSs more convenient, along the thoughts of Mao et al. [17], we propose two new concepts: the fuzzy factor and the hesitant factor. For a given HFS A ∈ HFS (U), we denote the fuzzy factor and the hesitant factor by Δ_A (x) and σ_A (x), respectively, where $\begin{matrix} Δ_{A} (x) = \frac{1}{n (x)} \sum_{i = 1}^{n (x)} | μ_{A}^{i} (x) - 1 / 2 |, \\ σ_{A} (x) = \frac{1}{n (x)} \sum_{i = 1}^{n (x)} | μ_{A}^{i} (x) - {\bar{μ}}_{A} (x) | . \end{matrix}$ here ${\bar{μ}}_{A} (x) = \frac{1}{n (x)} \sum_{i = 1}^{n (x)} μ_{A}^{i} (x)$ is the mean value of h_A (x).

It’s easy to verify that 0 ≤ Δ_A (x) , σ_A (x) ≤ 1/ 2, and then 0 ≤ Δ_A (x) + σ_A (x) ≤1. In general, the fuzziness of a HFS A become stronger as Δ_A (x) decreases, and its hesitancy increases with σ_A (x) becomes bigger. The physical meanings of these two factors can be shown in the following example.

Example 3.1. Suppose U = {x}, and there are four experts will assess three events in terms of hesitant fuzzy information, the results are as follows: $\begin{matrix} A = {< x, {0.4, 0.5, 0.9} >}, \\ B = {< x, {0.1, 0.4, 0.5} >}, \\ C = {< x, {0.2, 0.5, 0.8} >} . \end{matrix}$

According to above formulas, we have $\begin{matrix} Δ_{A} (x) = Δ_{B} (x) = 0.1667 < Δ_{C} (x) = 0.2, \\ σ_{A} (x) = σ_{C} (x) = 0.2 > σ_{B} (x) = 0.1556 . \end{matrix}$

As above calculations shown, the fuzziness of A and B are bigger than C, while the hesitancy of A and C are bigger than B. In fact, if we consider Δ_* (x) and σ_* (x) at the same time, we could find the uncertainty of A is the strongest, but we can not make a judgment between B and C. That is why we need to construct a certain principle to determine which is more important to uncertainty of HFSs between Δ_* (x) and σ_* (x). Meanwhile, if the principle has been established, the uncertain information hidden in HFSs could be clearly measured by some combinations of the fuzzy factor and the hesitant factor, and the complex HFEs distribution h_* (x) can be transformed into a two-tuple (Δ_* (x) , σ_* (x)) without considering the HFEs’ order relation.

Inspired by above views, we only need to pay attention to the fuzzy factor and hesitant factor while defining the uncertainty measures of HFSs. Thus, the follow-up hesitant fuzzy entropy and cross-entropy measures will be constructed based on this two-tuple.

3.1 The new axiomatic principles of hesitant fuzzy entropy

In order to measure the uncertain information of HFSs, Xu and Xia [30] firstly proposed a hesitant fuzzy entropy, and then some scholars, such as Farhadinia [4] and Zhao et al. [38], have presented several hesitant fuzzy entropies in their own views. However, most of them only focus on the fuzziness of HFSs, but neglect the uncertainty caused by hesitancy of HFEs. In addition, previous entropy formulas often need to rank the HFEs or turn to distance (or similarity) measures. In fact, if we grasp the nature and source of uncertain information in HFSs, a reasonable entropy formula could be constructed easily. Based on the discussion above, we may propose a novel axiomatic principle of hesitant fuzzy entropy based on the fuzzy factor and hesitant factor. For simplicity, we suppose the universe U only has one object, i.e., U = {x}.

Definition 3.1. The hesitant fuzzy entropy is a real-valued function E with Δ_* and σ_*, i.e., E = g (Δ_*, σ_*) : HFS (U) → [0, 1]. For any A ∈ HFS (U), E (A) satisfies the following axiomatic principles:

E (A) =0 iff (if and only if) A is a crisp set, i.e., h_A (x) = {0} or {1} , ∀ x ∈ U;

E (A) ≤1 for all A ∈ HFS (U);

E (A) = E (A^C);

g (Δ_A, σ_A) is a real-valued continuous function, which is decreasing with Δ_A and increasing with second variable σ_A, where Δ_A and σ_A are the fuzzy factor and the hesitant factor, respectively.

According to the above axiomatic definition, four axioms could be demonstrated, respectively, that:

The crisp sets don’t have any uncertain information;

Uncertainty of any HFS is bounded;

One HFS and its complement set have same uncertainty;

Hesitant fuzzy entropy changes continuously, and decreases with its fuzziness and increases with its hesitancy.

It is worth noting that many scholars, such as Farhadinia [4], Xu and Xia [30], etc., have pointed out that E (A) =1 iff h_A (x) = {1/2}, they overemphasize the fuzziness while neglecting hesitancy in HFSs. However, we do not present the regular conditions when the entropy reaches its maximum in Definition 3.1, the reasons mainly lie in two aspects: one is that it’s hard to find a consistent maximum point by considering Δ_A and σ_A due to their different changing trends, the other is that the maximum point may change with the position which is more important between Δ_A and σ_A, this directly reflects the decision-makers’ understanding of uncertainty in HFSs. Therefore, it is good enough for us to point out that the hesitant fuzzy entropy is bounded and how to change with Δ_A and σ_A. It also depends on the decision-makers’ views as to how to understand the balance between fuzziness and hesitancy of HFSs. All of these flexible and easy conditions will make the entropy more suitable for applying to the real world.

3.2 Some distance-based hesitant fuzzy entropy measures

Just as the classical fuzzy theory shown, there are some inherent relationships between entropy and distance. Recently, some scholars, such as Farhadinia [4], Quirs et al. [19], Xu and Xia [30] and Zhao et al. [38], have studied these relations, and presented some entropy formulas based on distance measures. Under the above new framework of hesitant fuzzy entropy, we could also propose some novel entropy formulas induced by distance measures. In general, if we suppose U = {x}, the uncertain information of any A ∈ HFS (U) may be only determined by the two-tuple (Δ_* (x) , σ_* (x)). We denote an ideal set M^* by (0,1/2), namely, the uncertainty of M^* is the most strongest. Thus, a HFS A has a bigger entropy value if it is close to M^* while a smaller entropy value if it is far from M^*. Based on this thought, we may establish the following relationship between hesitant fuzzy entropy and distance measure. $E (A) : = H (D^{normal} (A, M *)),$ (1)where D^normal (A, M *) is the normalized distance between A and M^*, and the mapping H : [0, 1] → [0, 1] is a strictly decreasing real-valued continuous function.

In fact, there are many H-functions in Math, such as H (y) =1 - y, 1 - ye^y-1, cos(πy)/2, etc. For simplicity, we only take H (y) =1 - y for example in the follow-up discussions. According to Equation (1), we may construct some distance-based hesitant fuzzy entropy formulas as follows.

(1) The Euclidean distance-based hesitant fuzzy entropy: $\begin{matrix} E (A) = 1 - D_{E}^{normal} (A, M *) \\ = 1 - \sqrt{\frac{Δ_{A}^{2} (x) + {(σ_{A} (x) - 1 / 2)}^{2}}{2}} . \end{matrix}$ (2)

(2) The Hamming distance-based hesitant fuzzy entropy: $\begin{matrix} E (A) = 1 - D_{H}^{normal} (A, M *) \\ = 1 - \frac{| Δ_{A} (x) - 0 | + | σ_{A} (x) - 1 / 2 |}{2} \\ = \frac{3}{4} + \frac{1}{2} (σ_{A} (x) - Δ_{A} (x)) . \end{matrix}$ (3)

(3) The general distance-based hesitant fuzzy entropy: $\begin{matrix} E (A) = 1 - D_{G}^{normal} (A, M *) \\ = 1 - {\frac{Δ_{A}^{λ} (x) + | σ_{A} (x) - {1 / 2 |}^{λ}}{2}}^{1 / λ}, (λ > 0) . \end{matrix}$ (4)

It is easy to verify the above formulas satisfy the axiomatic principles in Definition 3.1. We can also extend the above formulas to U = {x₁, x₂, ⋯ , x_m}. $E (A) = 1 - \frac{1}{m} \sum_{j = 1}^{m} \sqrt{\frac{Δ_{A}^{2} (x_{j}) + {(σ_{A} (x_{j}) - 1 / 2)}^{2}}{2}} .$ (5) $E (A) = \frac{3}{4} + \frac{1}{2 m} \sum_{j = 1}^{m} (σ_{A} (x_{j}) - Δ_{A} (x_{j})) .$ (6) $E (A) = 1 - \frac{1}{m} \sum_{j = 1}^{m} {\frac{Δ_{A}^{λ} (x_{j}) + | σ_{A} (x_{j}) - 1 / 2 |^{λ}}{2}}^{1 / λ} .$ (7)

4 Discrimination measures of uncertain information of HFSs

Actually, information of one HFS is consist of two parts: one is determinate information, the other is uncertain information. The classical cross-entropy is mainly used to measure the whole discrimination information between two non-crisp sets, such as Huang and Yang [6], Shang and Jiang [22], Vlachos and Sergiadis [24], etc., for intuitionistic fuzzy sets Xu and Xia [30] for HFSs. In reality, uncertain information often performs as the risk or cost of one event, it is of great importance to accurately measure the discrimination of uncertain information among the different HFSs. Meanwhile, the hesitant fuzzy cross-entropy, proposed by Xu and Xia [30], has not captured the nature of uncertainty in HFSs due to they only focus on the fuzziness while neglecting the hesitancy of HFEs. Motivated by Mao et al. [17], we now construct a novel analytical framework to reveal the discrimination of uncertain information based on the two-tuple (Δ_* (x) , σ_* (x)), and study the relationship between cross-entropy and entropy for HFSs.

4.1 Hesitant fuzzy cross-entropy

For simplicity, we suppose U = {x}, and a new hesitant fuzzy cross-entropy between two different HFSs based on the fuzzy factor and the hesitant factor could be defined as follows.

Definition 4.1. Let A, B ∈ HFS (U), then the hesitant fuzzy cross-entropy of A against B based on two-tuple (Δ_* (x) , σ_* (x)) is defined as $\begin{matrix} CE (A, B) = Δ_{A} (x) ln \frac{Δ_{A} (x)}{\frac{1}{2} (Δ_{A} (x) + Δ_{B} (x))} \\ + σ_{A} (x) ln \frac{σ_{A} (x)}{\frac{1}{2} (σ_{A} (x) + σ_{B} (x))} \end{matrix}$ (8)

However, one can observe that CE (A, B) is not symmetric with respect to its arguments. Then, a symmetric hesitant fuzzy cross-entropy is given as $DE (A, B) = CE (A, B) + CE (B, A) .$ (9)

Theorem 4.1. Let A, B ∈ HFS (U), DE (A, B) is the symmetric hesitant fuzzy cross-entropy between A and B, then it satisfies following properties.

DE (A, B) = DE (B, A);

DE (A, B) = DE (A, B^C) = DE (A^C, B) = DE (A^C, B^C);

0 ≤ DE (A, B) ≤2 ln 2.

Proof. From Definition 4.1 and the properties of Δ_* (x) and σ_* (x), (1) and (2) are straightforward. We only prove the property (3).

Since 0 ≤ Δ_A (x) + σ_A (x) ≤1, by use of Shannon’s inequality, we have CE (A, B) ≥0 ⇒ DE (A, B) ≥0.

Meanwhile, 0 ≤ Δ_A (x) , σ_A (x) ≤ 1/2, then we have CE (A, B) ≤ (Δ_A (x) + σ_B (x)) ln 2 ≤ ln 2, Therefore, we yield DE (A, B) ≤2 ln 2. □

Now, we extend the above symmetric hesitant fuzzy cross-entropy to the finite universe situation U = {x₁, x₂, ⋯ , x_m}.

Definition 4.2. Let A, B ∈ HFS (U) and U = {x₁, x₂, ⋯ , x_m}, then the hesitant fuzzy cross- entropy of A against B is defined as $\begin{matrix} CE (A, B) = \sum_{j = 1}^{m} (Δ_{A} (x_{j}) ln \frac{Δ_{A} (x_{j})}{\frac{1}{2} (Δ_{A} (x_{j}) + Δ_{B} (x_{j}))} \\ + σ_{A} (x_{j}) ln \frac{σ_{A} (x_{j})}{\frac{1}{2} (σ_{A} (x_{j}) + σ_{B} (x_{j}))}) \end{matrix}$ (10)

Correspondingly, DE (A, B) = CE (A, B) + CE(B, A) is its symmetric form, and we also get 0≤ DE (A, B) ≤ 2mln2.

4.2 Relationship between hesitant fuzzy entropy and its cross-entropy

As we discussed above, DE (A, B) in Definition 4.2 could be used to describe the discrimination of uncertain information of A and B when E (A) in Definition 3.1 measures uncertainty of A, thus there may be some relationships between them. The traditional references, such as Vlachos and Sergiadis [24], Xu and Xia [30], etc., suggest that E (A) = DE (A, A^C). However, this relationship may be false since DE (A, A^C) =0 under the framework of Definition 4.2. Following the idea of Mao et al. [17], if we denote the crisp set by C^*, i.e., Δ_{C
^*} (x) = 1/ 2 and σ_{C
^*} (x) =0 for all x ∈ U, then DE (A, C^*) actually displays the discrimination of uncertain information between a HFSs A and a crisp set C^*. As we known, C^* is determinate and has not uncertainty, thus DE (A, C^*) may be viewed as a tool to measure amount of uncertain information of A. In this sense, hesitant fuzzy cross-entropy proposed above could induce a new hesitant fuzzy entropy according to the following theorem.

Theorem 4.2. Let A ∈ HFS (U), |U| = m, C^* is a crisp set, then $\begin{matrix} E (A) = \frac{1}{2 m ln 2} DE (A, C^{*}) \\ = \frac{1}{2 m ln 2} \sum_{j = 1}^{m} ((Δ_{A} (x_{j}) + \frac{1}{2}) \times ln \frac{1}{Δ_{A} (x_{j}) + 1 / 2} \\ + Δ_{A} (x_{j}) ln (2 Δ_{A} (x_{j})) + σ_{A} (x_{j}) ln 2) \end{matrix}$ (11)is a hesitant fuzzy entropy of A.

Proof. The principles (1–3) in Definition 3.1 could be easily obtained according to Definition 4.2 and Theorem 4.1, we only prove the principle (4).

In order to prove E (A) satisfies the principle (4), it suffices to prove that the following function: $g (x, y) = x ln (2 x) - (x + 1 / 2) ln (x + 1 / 2) + y ln 2$ where x, y ∈ [0, 1/2], is decreasing with x and increasing for y. Taking the partial derivative of g (x, y)with x and y, respectively, we yield ${\begin{matrix} \frac{\partial g (x, y)}{\partial x} = ln (\frac{2 x}{x + 1 / 2}) \leq 0 \\ \frac{\partial g (x, y)}{\partial y} = y ln 2 \geq 0 \end{matrix}$

Let x = Δ_A (x_j) , y = σ_A (x_j), for a given object x_j ∈ U, g (Δ_A (x_j) , σ_A (x_j)) decreases with Δ_A (x_j) and increases with σ_A (x_j). Thus the summation of all objects in U also holds the same monotony, that is, the principle (4) is true. □

4.3 Decomposition formula of uncertain information for HFSs

In fact, uncertain information of HFSs is mainly consist of fuzziness and hesitancy, the former is determined by the closeness between membership degree and 1/2, the latter is controlled by the dispersive level of HFEs. That is to say, for a given A∈ HFS (U), the two-tuple (Δ_A (x) , σ_A (x)) determines the all uncertainty of A. Inspired by Mao et al. [17], we may propose two new entropy measures of HFSs based on the fuzzy factor Δ_A (x) and the hesitant factor σ_A (x), respectively.

Definition 4.3. The fuzzy entropy is a real-valued function E_F with Δ_*, i.e., E_F = f (Δ_*) : HFS (U) → [0, 1]. For any A ∈ HFS (U), E_F (A) satisfies the following axiomatic principles:

E_F (A) =0 iff h_A (x) = {0} or {1} or {0,1};

E_F (A) =1 iff h_A (x) = {1/2} , ∀ x ∈ U;

E_F (A) = E_F (A^C);

f (Δ_A) is a real-valued continuous function, which is decreasing with the fuzzy factor Δ_A.

Definition 4.4. The hesitant entropy is a real-valued function E_H with σ_*, i.e., E_H = u (σ_*) : HFS (U) → [0, 1]. For any A ∈ HFS (U), E_H (A) satisfies the following axiomatic principles:

E_H (A) =0 iff A is a fuzzy set;

E_H (A) =1 iff h_A (x) = {0, 1} , ∀ x ∈ U;

E_H (A) = E_H (A^C);

u (σ_A) is a real-valued continuous function, which is increasing with the hesitant factor σ_A.

In above definitions, fuzzy entropy E_F (A) measures uncertain information caused by fuzziness in one HFS, while hesitant entropy E_H (A) depicts uncertain information induced by its hesitancy. If we rewrite Equation (11) as $\begin{matrix} E (A) = \frac{1}{2 m ln 2} \sum_{j = 1}^{m} ((Δ_{A} (x_{j}) + \frac{1}{2}) ln \frac{1}{Δ_{A} (x_{j}) + 1 / 2} \\ + Δ_{A} (x_{j}) ln (2 Δ_{A} (x_{j}))) + \frac{1}{2 m} \sum_{j = 1}^{m} σ_{A} (x_{j}) \\ = \frac{1}{2} E_{F} (A) + \frac{1}{2} E_{H} (A) \end{matrix}$ (12)where $\begin{matrix} E_{F} (A) = \frac{1}{2 m ln 2} \sum_{j = 1}^{m} ((Δ_{A} (x_{j}) + \frac{1}{2}) ln \frac{1}{Δ_{A} (x_{j}) + 1 / 2} \\ + Δ_{A} (x_{j}) ln (2 Δ_{A} (x_{j}))), \\ E_{H} (A) = \frac{1}{m} \sum_{j = 1}^{m} σ_{A} (x_{j}) . \end{matrix}$

According to Definitions 4.3 and 4.4, it’s easy to verify that E_F (A) is a fuzzy entropy of A and E_H (A) is its hesitant entropy. That is to say hesitant fuzzy entropy E (A) Equation (12) could be viewed as the arithmetic average of fuzzy entropy E_F (A) and hesitant entropy E_H (A).

5 Uncertain measures with parameters

It’s evident to find that the coefficients of E_F (A) and E_H (A) are both 1/2 by analyzing the decomposition formula of E (A). (Equation (12)). This indicates that E_F (A) and E_H (A) have same effect on E (A) in the view of Mathematics, which also means fuzziness and hesitancy have same position to the overall uncertainty of one HFS. Naturally, if we consider E_F (A) and E_H (A) have different effects on E (A), we need to introduce some parameters to adjust and control their effects. The main works of the following sections are to propose some uncertain information measures with parameters and give some comparative examples.

5.1 Hesitant fuzzy entropy and cross-entropy with parameters

Similar to the process of Section 4, firstly, we should define hesitant fuzzy cross-entropy with parameters.

Definition 5.1. Let A, B ∈ HFS (U), U = {x₁, x₂, ⋯ , x_m} and p, q > 0, then the parameterized hesitant fuzzy cross-entropy of A against B is defined as $\begin{matrix} {CE}^{p, q} (A, B) : = \sum_{j = 1}^{m} {Δ_{A}^{p} (x_{j}) ln {(\frac{Δ_{A}^{p} (x_{j})}{\frac{1}{2} (Δ_{A}^{p} (x_{j}) + Δ_{B}^{p} (x_{j}))})}^{p} \\ + σ_{A}^{q} (x_{j}) ln {(\frac{σ_{A}^{q} (x_{j})}{\frac{1}{2} (σ_{A}^{q} (x_{j}) + σ_{B}^{q} (x_{j}))})}^{q}} \end{matrix}$ (13)

Obviously, CE^p,q (A, B) is also not symmetric with its arguments, so a symmetric parameterized hesitant fuzzy cross-entropy is given in similar way: ${DE}^{p, q} (A, B) = {CE}^{p, q} (A, B) + {CE}^{p, q} (B, A) .$ (14)

Theorem 5.1. Let A, B ∈ HFS (U), U = {x₁, x₂, ⋯ , x_m} and p, q > 0, DE^p,q (A, B) is the parameterized hesitant fuzzy cross-entropy between A and B, then it satisfies following properties.

DE^p,q (A, B) = DE^p,q (B, A);

DE^p,q (A, B) = DE^p,q (A, B^C) = DE^p,q(A^C, B) = DE^p,q (A^C, B^C);

0 ≤ DE^p,q (A, B) ≤ (p + q) m ln 2.

Proof. The proof is trivial. □

Theorem 5.2. Let A ∈ HFS (U), |U| = m, C^* is a crisp set, and p, q > 0, then $\begin{matrix} E^{p, q} (A) = \frac{1}{(p + q) m ln 2} {DE}^{p, q} (A, C^{*}) \\ = \frac{1}{(p + q) m ln 2} \\ \times \sum_{j = 1}^{m} {p (\begin{matrix} Δ_{A}^{p} (x_{j}) ln (2 Δ_{A}^{p} (x_{j})) - (Δ_{A}^{p} (x_{j}) + 1 / 2^{p}) \\ \times ln (Δ_{A}^{p} (x_{j}) + 1 / 2^{p}) + 2^{- p} ln 2^{1 - p} \end{matrix}) \\ + q σ_{A}^{q} (x_{j}) ln 2} \end{matrix}$ (15)is a hesitant fuzzy entropy of A.

Proof. Similar to Theorem 4.2, we only prove Equation (15) satisfies the principle (4) in Definition 3.1.

Obviously, E^p,q (A) fulfils the principle (4) if we are able to show that the function: $\begin{matrix} g (x, y) = p (x^{p} ln (2 x^{p}) - (x^{p} + 1 / 2) ln (x^{p} + 1 / 2) \\ + 2^{- p} ln 2^{1 - p}) + qy ln 2, \end{matrix}$ where x, y ∈ [0, 1/2] , p, q > 0, is decreasing with x and increasing for y. Taking the partial derivative of g (x, y) with x and y, respectively, we yields ${\begin{matrix} \frac{\partial g (x, y)}{\partial x} = p^{2} x^{p - 1} ln (\frac{2 x^{p}}{x^{p} + 1 / 2^{p}}) \leq 0 \\ \frac{\partial g (x, y)}{\partial y} = q^{2} y^{q - 1} ln 2 \geq 0 \end{matrix}$

We can also divide Equation (15) into two parts: $E^{p, q} (A) = \frac{p}{(p + q)} E_{F}^{p} (A) + \frac{q}{(p + q)} E_{H}^{q} (A) .$ (16)where $\begin{matrix} E_{F}^{p} (A) = \frac{1}{m ln 2} \sum_{j = 1}^{m} {\begin{matrix} Δ_{A}^{p} (x_{j}) ln (2 Δ_{A}^{p} (x_{j})) - (Δ_{A}^{p} (x_{j}) + 1 / 2^{p}) \\ \times ln (Δ_{A}^{p} (x_{j}) + 1 / 2^{p}) + 2^{- p} ln 2^{1 - p} \end{matrix}}, \\ E_{H}^{q} (A) = \frac{1}{m} \sum_{j = 1}^{m} σ_{A}^{q} (x_{j}) . \end{matrix}$

We can also verify that $E_{F}^{p} (A)$ and $E_{H}^{q} (A)$ are the fuzzy entropy and hesitant entropy of A, respectively.

In this sense, E^p,q (A) could be viewed as weighted average of $E_{F}^{p} (A)$ and $E_{H}^{q} (A)$ . Specially, if let p + q = 1, Equation (16) may be presented as: $E^{p, q} (A) = {pE}_{F}^{p} (A) + {qE}_{H}^{q} (A) .$ (17)

General speaking, if p > q, it shows that fuzziness has the stronger effect on uncertainty of A than its hesitancy, the effect of hesitancy performs more stronger than its fuzziness while p < q, and the two facets have same effect on uncertainty of A if p = q. However, we should choose the consistent parameters p and q in the same practical problem, and make comparison for different events under a fixed entropy distribution. In order to quantify the relationship between p and q and give some feasible guidance for choosing appropriate parameters in reality, we only consider the situation p + q = 1 and let q = θp for simplicity, where θ ∈ (0, + ∞). The general rules could been seen in Table 1.

5.2 Comparative examples

The classical cross-entropy is mainly to measure the discrimination information of different non-crisp sets, and hesitant fuzzy cross-entropies firstly proposed by Xu and Xia [30] are no exception. However, we try to construct a more precise measure to describe the discrimination of uncertain information of HFSs, which provides a fresh new look into the nature of uncertainty in HFSs. The new hesitant fuzzy cross-entropy defined in the paper focus on both fuzziness and hesitancy, and the main difference between the new measures and the previous cross-entropy may be presented in the following example.

Example 5. 1. Let U = {x}, A, B ∈ HFS (U), where $\begin{matrix} h_{A} (x) = {0.2, 0.5, 0.9}, \\ h_{B} (x) = {0.1, 0.5, 0.8}, \end{matrix}$ are HFEs of A and B. A hesitant fuzzy cross-entropy proposed by Xu and Xia [30] is given as $\begin{matrix} C (A, B) = \frac{1}{(1 - 2^{1 - s}) n} \\ \sum_{i = 1}^{n} {\begin{matrix} \frac{μ_{A}^{(i) s} (x) + μ_{B}^{(i) s} (x)}{2} + \frac{{(1 - μ_{A}^{(n - i + 1)} (x))}^{s} + {(1 - μ_{B}^{(n - i + 1)} (x))}^{s}}{2} - \\ {(\frac{μ_{A}^{(i)} (x) + μ_{B}^{(i)} (x)}{2})}^{s} - {(\frac{(1 - μ_{A}^{(n - i + 1)} (x)) + (1 - μ_{B}^{(n - i + 1)} (x))}{2})}^{s} \end{matrix}}, s > 1 . \end{matrix}$ (18)

If we suppose s = 2, by use of Equation (18), we have $C (A, B) = 0.0067 > 0 .$

However, it’s easy to find that Δ_A (x) = Δ_B (x) =0.3 and σ_A (x) = σ_B (x) =0.2444, thus $DE (A, B) = {DE}^{p, q} (A, B) = 0, \forall p, q > 0 .$

So there is no discrimination of uncertain information between A and B, but they have discrimination of determinate information. That is not to say C (A, B) is more precise than DE (A, B) or DE^p,q (A, B), but scopes and purposes of their application are different.

Actually, there have existed some researches about hesitant fuzzy entropy, such as Farhadinia [4], Xu and Xia [30], etc. We suppose U = {x}, A ∈ HFS (U), and list some entropies as follows:

$E_{1} (A) = \frac{1}{n (\sqrt{2} - 1)} \sum_{i = 1}^{n} [\begin{matrix} sin \frac{π (μ_{A}^{(i)} + μ_{A}^{(n - i + 1)})}{4} + \\ sin \frac{π (2 - μ_{A}^{(i)} - μ_{A}^{(n - i + 1)})}{4} - 1 \end{matrix}] .$ (19)

$E_{2} (A) = - \frac{1}{n ln 2} \sum_{i = 1}^{n} [\begin{matrix} \frac{μ_{A}^{(i)} + μ_{A}^{(n - i + 1)}}{2} ln \frac{μ_{A}^{(i)} + μ_{A}^{(n - i + 1)}}{2} + \\ \frac{2 - μ_{A}^{(i)} - μ_{A}^{(n - i + 1)}}{2} ln \frac{2 - μ_{A}^{(i)} - μ_{A}^{(n - i + 1)}}{2} \end{matrix}] .$ (20)

$E_{3} (A) = 1 - \frac{1}{(1 - 2^{1 - s}) n} \sum_{i = 1}^{n} [\begin{matrix} \frac{μ_{A}^{(i) s} + {(1 - μ_{A}^{(n - i + 1)})}^{s}}{2} - \\ {(\frac{μ_{A}^{(i)} + 1 - μ_{A}^{(n - i + 1)}}{2})}^{s} \end{matrix}], s > 1 .$ (21)

$E_{4} (A) = 1 - \frac{2}{n} \sum_{i = 1}^{n} | μ_{A}^{(i)} - 0.5 | .$ (22)

In order to show the importance of the proposed entropies, we may compare the new entropy measures to the above existing formulas in Example 5.2.

Example 5.2. Let U = {x}, h_A (x) = {0.1, 0.5, 0.9}, h_B (x) = {0.4, 0.5, 0.6}, h_C (x) = {0.2, 0.8} and h_D (x) = {0, 1, 0.2, 0.3} are four HFEs. Without loss of generality, we set some parameters: s = 2, λ = 1.5. Applying the existing entropies Equations (19–22) and the proposed formulas Equations (5–7, 11, 15) to the HFSs A, B, C and D, respectively. The comparative results could be seen in Table 2.

As shown in Table 2, we find that the previous entropies cannot distinguish some of HFSs. For example, E₁ and E₂ cannot distinguish A and B, E₃ could not distinguish A, B and C, E₄ cannot distinguish C and D. However, the entropy formulas proposed in this paper could distinguish all of them well, and most of uncertainty in these four HFSs measured by different entropies display the consistent trends expect E^p,q, namely, E (A) > E (C) > E (B) > E (D). The performances of uncertainty in these HFSs by use of E^p,q with different parameters are different because of the two facets, that is, the fuzziness and hesitancy, have different position. We have E (B) > E (A) > E (C) > E (D) if p = 0.7, q = 0.3, and E (C) > E (A) > E (B) > E (D) if p = 0.3, q = 0.7. Therefore, different values of p, q reflect different

attitude and preference of decision-makers, which make the entropy more flexible. Although two or more HFSs have the same entropy is entirely possible, all proposed entropy measures may be more precise and detailed than other entropies in presenting uncertainty of HFSs under the framework of Definition 3.1.

6 Application

In reality, software evaluation is an increasingly important problem in any sector of human activity. Now, there is a company wants to evaluate five kinds of software (x₁, x₂, ⋯ , x₅) and determines which one is best for production. Four attributes will be considered in evaluation, such as functionality (a₁), portability (a₂), usability (a₃) and maturity (a₄). In order to improve the reasonability and accuracy of evaluation, the company invites four experts with different backgrounds and levels of knowledge, skills, experience, etc. to make such an evaluation will lead to a information difference. To clearly reflect the differences of the opinions of different experts, all evaluation data are represented by the HFSs and listed in Table 3.

Step 1. Calculate the score matrix S = (s_kj) _5×4 according to the following score function: $s_{kj} = {\begin{matrix} μ_{kj}^{(1)} + 0.5 (μ_{kj}^{(4)} - μ_{kj}^{(1)}) & if Δ_{kj} = 0 \\ μ_{kj}^{(1)} + \frac{μ_{kj}^{(1)}}{μ_{kj}^{(1)} + 1 - μ_{kj}^{(4)}} (μ_{kj}^{(4)} - μ_{kj}^{(1)}) & if Δ_{kj} > 0 \end{matrix}$ where $Δ_{kj} = \frac{1}{4} \sum_{i = 1}^{4} | μ_{kj}^{(i)} - 0.5 |$ , and $μ_{kj}^{(i)}$ is the ith smallest membership degree of object x_k under the attribute a_j.

Therefore, the evaluation data in Table 3 could be transformed into the following score matrix S: $S = [\begin{matrix} 0.3333 & 0.6000 & 0.2000 & 0.4000 \\ 0.6000 & 0.8000 & 0.8000 & 0.7143 \\ 0.4000 & 0.2000 & 0.6000 & 0.2000 \\ 0.8000 & 0.5000 & 0.7143 & 0.6000 \\ 0.1667 & 0.5000 & 0.3333 & 0.2000 \end{matrix}] .$

Step 2. In order to determine the attribute weights, we may construct the following criterion: since uncertain information often describes the cost and risk in reality, a rational decision-maker should determine the attribute weights on the basis of determination information. Hesitant fuzzy entropy E is used to measure the uncertainty of HFSs, thus the difference between 1 and the entropy E may reflect the deterministic degree of HFSs. Therefore, we may point out that the weight of an attribute a_j increases as the value of (1 - E (a_j)) under a_j becomes bigger and vice versa. Specially, we can obtain the attribute weights ω = {ω₁, ω₂, ω₃, ω₄} as following formulas, which are in line with Mao et al. [17]. $ω_{j} = \frac{1 - E (a_{j})}{\sum_{k = 1}^{4} (1 - E (a_{k}))} or \frac{1 - E^{p, q} (a_{j})}{\sum_{k = 1}^{4} (1 - E^{p, q} (a_{k}))}, p, q > 0 .$

Let λ = 1.5, p = 0.7, q = 0.3, by use of hesitant fuzzy entropy measures Equations (5–7, 11, 15), the attribute weights are given in Table 4. Moreover, we also calculate the weight vector with entropies proposed by Farhadinia [4], Xu and Xia [30], namely Equations (19 and 20), to make some comparative analysis (in Table 4).

The evolutionary trend of the attribute weight vector under seven entropies could be revealed in Fig. 1. It is easy to find that the general trends by use of different formulas are consistent, that is, ω₃ > ω₁ > ω₄ > ω₂. However, the entropies Equations (5–7, 11 and 15) based on new principles are more similar than two other vectors Equations (19, 20) proposed by Farhadinia [4], Xu and Xia [30]. Therefore, we may conclude that entropies based on the same principles often produce clustering effects.

Step 3. Calculate collective scores of each alternative by means of formula r = Sω^T (see in Table 5), and rank all objects according to their collective scores.

From the above table, although we use different entropy formulas, including other entropies proposed by Farhadinia [4], Xu and Xia [30], we still obtain a consistent result, namely, x₂ ≻ x₄ ≻ x₁ ≻ x₃ ≻ x₅. Thus, the optimal object is x₂ and the company may choose x₂ to product. Moreover, the difference among these weight vectors may be inadequate to influence the final decision. In other words, the proposed entropies may perform as well as or better than others.

The example clearly indicates that the proposed decision-making method is simple and effective under hesitant fuzzy information environments. Furthermore, fuzzy set is a special case of the HFSs, so the proposed approaches in this paper can be used to solve not only decision-making problems with HFSs but also relevant problems of FSs.

7 Conclusions

The HFS has been viewed as an efficient soft tool in describing uncertainty, fuzziness and vagueness in reality. Recently, how to quantify the uncertain information of HFSs has drawn great attention both in the theoretic and practical field. Therefore, we try to develop a general framework of constructing hesitant fuzzy cross-entropy and entropy measures. Owing to the fact that most of existing entropies in HFSs overemphasize their fuzziness while neglecting their hesitancy, we point out that uncertainty of HFSs comes mainly from their fuzziness and hesitancy, which could be characterized by the fuzzy factor and hesitant factor, respectively.

Based on above idea, the main contribution may be summarized in the following ways. Firstly, we refine the constructive principles of hesitant fuzzy entropy, and the systematic transformation of distance into entropy for HFSs is exploited and some corresponding distance-based entropies are presented, too. Secondly, in order to describe the discrimination of uncertain information in HFSs, we propose a new hesitant fuzzy cross-entropy, and the relationship between cross-entropy and entropy of HFSs could also be investigated, it suggests that the hesitant fuzzy entropy may be divided into arithmetic average of fuzzy entropy and hesitant entropy. Moreover, hesitant fuzzy entropy and cross-entropy with parameters have been proposed, the relation between these two measures has been also studied in the similar way. We also make comparison between the proposed measures

In the future, we shall apply the proposed cross-entropy and entropy measures into other aspects, such as pattern recognition, group decision-making, clustering analysis and image processing. In addition, as this paper just is a theoretical study about entropy measures of HFSs, we may try to develop a more efficient algorithm to deal with the information in this big data time.

Footnotes

Acknowledgments

The authors are highly grateful to any anonymous referee for their careful reading and insightful comments, and the views and opinions expressed are those of the authors. The work is supported by the National Nature Science Foundation of China (Nos. 71273048, 71473036), and the Founding of Jiangsu Innovation Program for Graduate Education (KYZZ15_0070).

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