Divergence measures on hesitant fuzzy sets

Abstract

Measures of difference are studied for hesitant fuzzy sets, i.e. mappings where the values are multisets in the unit interval. Locality (the change of the value of the difference is dependent only on changes in singletons) of such mappings is discussed and the class of all local divergences is characterized. Entropies for hesitant fuzzy sets are also studied, namely fuzziness entropy measure and hesitance entropy measure.

Keywords

Hesitant fuzzy set divergence measure hesitant divergence measure locality entropy

1 Introduction

The subjective substance of a natural language leads to impreciseness in describing collections of objects. Fuzzy sets are a convenient tool to represent such cases [25, 26]. During over 50 years of their research not only a number of applications have been created, but also other related theoretical models, mostly based on real life situations. One of such models are IF sets [3], combining the information on the membership and non-membership of an object to a collection.

Formally, an IF-set could be understood as a pair of ordinary fuzzy sets, but as the membership and nonmembership functions are closely related, they are studied as a single object. A similar argument justifies also other kinds of type 2 fuzzy sets, including hesitant fuzzy sets, which are objects of our research.

When a group of evaluators decides about the membership of an object to a collection, the result of their decision is an n-tuple of their opinions. In case they work together and exchange information on their decision process, these decisions cannot be considered as independent and thus they should be considered as a single object. Technically, if we do not care about the order of the evaluators, we do not need to work with n-tuples (where the order is important. On the other hand, it would be limiting to work with subsets (of all possible decisions), as the number of evaluators with an identical decisions is usually important. Hence, a multisets (see [2]) are proper objects for the values of such decision processes.

This idea leads directly to the notion of a hesitant fuzzy set [6 , 19]. We focus on a question of divergences between these objects. The motivation is obvious, as for instance in decision processes it is important to know the grade of similarity for studied objects. Our research is based on similar ideas and notions which are used in some previous concepts [11 , 22]. Also we study the related question of entropy for hesitant fuzzy sets.

2 Basic concepts

Necessary concepts to understand the remaining parts of this work are given in this section. In particular, we will focus on the definitions and notations for general concepts about fuzzy sets. They are well-known and can be found in a wide range of sources (see, for instance, the classical book Dubois and Prade [4]). The basic notations about hesitant fuzzy sets will be described in the separate section following further. The universal set or referential we will denote by X. In this framework, we can consider the following notations (the exact definitions follow later):

$℘ (X)$ will denote the set of all (crisp) subsets of the universal set X,

$℘ (X)$ will denote the set of all fuzzy subsets of X,

$ℱ_{H} (X)$ will denote the set of all hesitant fuzzy subsets of X.

A fuzzy set is characterized by a membership function which measures the degree to which a point of the referential belongs to the fuzzy set. Here, this membership function is denoted with the same symbol as the fuzzy set. Thus, for the universal set, we have that X (x) =1 for all x ∈ X and for the empty set we have that ∅ (x) =0 for all x ∈ X.

Another important concepts will be the inclusion relation and the complement set. In particular we will consider the standard Zadeh’s negation for defining the complement (see [25]).

Definition 1. Let $\tilde{A}$ be a fuzzy set in $ℱ (X)$ .

The complement of $\tilde{A}$ , which will be denoted by ${\tilde{A}}^{c}$ is a fuzzy set in $ℱ (X)$ defined by ${\tilde{A}}^{c} (x) = 1 - \tilde{A} (x)$ , for all x ∈ X.

For $\tilde{B} \in ℱ (X)$ , $\tilde{A}$ is contained in $\tilde{B}$ , which is denoted by $\tilde{A} \subseteq \tilde{B}$ if and only if $\tilde{A} (x) \leq \tilde{B} (x)$ , for all x ∈ X.

Apart from the previous operation of containment, we need to consider the concepts of intersection and union of fuzzy sets. The initial definitions were also given by Zadeh (see [25] or [26]) as follows:

The intersection of $\tilde{A}$ and $\tilde{B}$ :

$\tilde{A} \cap \tilde{B} (x) = min {\tilde{A} (x), \tilde{B} (x)}$ , for all x ∈ X.

The union of $\tilde{A}$ and $\tilde{B}$ :

$\tilde{A} \cup \tilde{B} (x) = max {\tilde{A} (x), \tilde{B} (x)}$ , for all x ∈ X.

These are the standard operations, since they were considered in the initial definition. We can notice that they coincide with the usual operations for crisp sets when we restrict the values to the set {0, 1}. However, there is a broad class of functions representing the intersection, namely the t-norms and for the union the t-conorms (see [8]). The remaining part of this section is devoted to these concepts.

The triangular norm (t-norm) is a function T : [0, 1] × [0, 1] → [0, 1] satisfying the following conditions:

T (a, b) = T (b, a), for all a, b ∈ [0, 1],

T (T (a, b) , c) = T (a, T (b, c)), for all a, b, c ∈ [0, 1],

b ≤ c ⇒ T (a, b) ≤ T (a, c), for all a, b, c ∈ [0, 1],

T (a, 1) = a, for all a ∈ [0, 1].

Therefore, the function T is a monotone, associative and commutative operation defined on [0, 1] × [0, 1] with the neutral element 1.

Some important examples of t-norms are:

the minimum t-norm: T_M (a, b) = min {a, b}, for all a, b ∈ [0, 1],

the product t-norm: T_P (a, b) = a · b, for all a, b ∈ [0, 1],

the Łukasiewicz t-norm: T_L (a, b) = max {a + b - 1, 0}, for all a, b ∈ [0, 1],

the drastic t-norm: $T_{D} (a, b) = {\begin{matrix} min {a, b}, & if max {a, b} = 1 . \\ 0, & otherwise . \end{matrix}$

For these basic t-norms, it holds that T_D ≤ T_L ≤ T_P ≤ T_M. In fact, for any t-norm T it is fulfilled that T_D ≤ T ≤ T_M.

By changing the neutral element from 1 to 0, we obtain the triangular conorm (t-conorm). A t-norm T and a t-conorm S are dual if for each a, b ∈ [0, 1] the equation T (a, b) =1 - S (1 - a, 1 - b) is fulfilled. Using t-norms and t-conorms, we can define in general the intersection and union of two fuzzy sets as follows.

Definition 2. Let $\tilde{A}, \tilde{B} \in ℱ (X)$ . Given a t-norm T and a t-conorm S,

the intersection of $\tilde{A}$ and $\tilde{B}$ is defined as the fuzzy set whose membership function is $\tilde{A} \cap_{T} \tilde{B} (x) = T (\tilde{A} (x), \tilde{B} (x))$ , for all x ∈ X;

the union of $\tilde{A}$ and $\tilde{B}$ is defined as the fuzzy set whose membership function is $\tilde{A} \cup_{S} \tilde{B} (x) = S (\tilde{A} (x), B (x))$ , for all x ∈ X.

Thus, we can denote by (X, T, S) the triple formed by the universe with the t-norm and the t-conorm used to define the intersection and the union, respectively.

3 Hesitant fuzzy sets

Hesitant fuzzy sets are one of possible generalizations of fuzzy sets [15]. Hesitant fuzzy logic was recently defined by Torra (see [18] or [19]), although the first introduction of this concept was made by Grattan-Guinnes under the name of set-valued fuzzy set (see [6]). Several papers related to this logic have been published, where the basic concepts can be found (see, for instance, [5, 22] or [23]). Hesitant fuzzy sets have already found interesting applications, for instance in decision making [7 , 21, 29].

Definition 3. Let X be a non-empty set and $ℍ \subseteq ℘ ([0, 1])$ the set of all finite non-empty subsets of the interval [0, 1]. Then the set $\tilde{A} = {(x, μ_{A} (x)) | x \in X}$ where $μ_{A} : X \to ℍ$ is a typical hesitant fuzzy set in X.

To each x ∈ X we assign the finite set μ_A (x) of values in [0, 1], named as hesitant fuzzy element and denoted by $\tilde{A} (x)$ . In general case $\tilde{A} (x) = {h_{1}, \dots, h_{n}}$ . This approach is motivated from real situations if more (in our case n) experts are evaluating and assigning the membership degrees for some element x ∈ X to the hesitant fuzzy set $\tilde{A}$ . For this reason we find it more suitable to use multisets (see [2]) as hesitant elements. We allow the particular values in the hesitant fuzzy element can be repeated, i.e. for $h_{i}, h_{j} \in \tilde{A} (x)$ also the case h_i = h_j for i ≠ j will be accepted.

The general hesitant fuzzy element $\tilde{A} (x)$ we will represent as a multiset $\tilde{A} (x) = {(h_{1}, k_{1}), (h_{2}, k_{2}), \dots (h_{n}, k_{n})},$ what means that an element h_i occurs k_i-times in the hesitant fuzzy element $\tilde{A} (x)$ for any i∈ { 1, 2, …, n }. For example, the multiset written as {a, a, a, b, c, c} is defined as { (a, 3) , (b, 1) , (c, 2)}.

We define the length $l_{\tilde{A}} (x)$ of a hesitant fuzzy element as a number of all elements in $\tilde{A} (x)$ accounting number of their occurance. It is obvious that the length need not be the same for each element x ∈ X. The number of experts may be different for any element x ∈ X. So the lenght of the general fuzzy element $\tilde{A} (x)$ we can compute as $l_{\tilde{A}} (x) = \sum_{i = 1}^{n} k_{i}$ . The length of the multiset {a, a, a, b, c, c} equals 3 + 1 +2 = 6.

Each fuzzy set can be considered as a special case of a hesitant fuzzy set in which the length of each element is one.

In case we do not obtain the information on the multiplicity of elements we denote a multiset just by { (h₁) , (h₂) , … , (h_n) }.

For example, a multiset consists only of elements 0 we will denote shortly as { (0)}. This formula can be equivalently rewritten as: $[\tilde{A} (x) = {(0)}] \Leftrightarrow [⋃ {h : h \in \tilde{A} (x)} \subseteq {0}] .$

In the following next we will use the shorter formula.

Torra introduced in [19] four cases related to a hesitant fuzzy element $\tilde{A} (x)$ . We state the following cases by considering that the some elements of $\tilde{A} (x)$ can be the same. The first and the second cases are trivial extensions of an empty set and universal set, respectively, from fuzzy set theory.

$\tilde{A}$ be the empty set iff $\tilde{A} (x) = {(0)}$ for allx ∈ X,

$\tilde{A}$ be the full set iff $\tilde{A} (x) = {(1)}$ for all x ∈ X,

The remaining two cases are not corresponding with our purposes.

complete ignorance for an element x ∈ X (i.e. all is possible): $\tilde{A} (x) = [0, 1]$ ,

set for a nonsense of x: $\tilde{A} (x) = \emptyset$ .

Complete ignorance assigns the closed interval [0, 1] to x ∈ X, which contradicts to the definition of a hesitant fuzzy element as a finite subset of [0, 1]. Also a nonsense of x contradicts with the definition of hesitant fuzzy element as a non-empty set. Therefore, we will not admit these cases in the following.

We will provide some notions related to a hesitant fuzzy element $\tilde{A} (x)$ (see also [23]).

A lower bound: ${\tilde{A}}^{-} (x) = min {h | h \in \tilde{A} (x)}$ ,

an upper bound: ${\tilde{A}}^{+} (x) = max {h | h \in \tilde{A} (x)}$ ,

an α-lower bound: ${\tilde{A}}_{α}^{-} (x) = {h \in \tilde{A} (x) | h \leq α}$ ,

an α-upper bound: ${\tilde{A}}_{α}^{+} (x) = {h \in \tilde{A} (x) | h \geq α}$ ,

a complement: ${\tilde{A}}^{c} (x) = ⋃_{h \in \tilde{A} (x)} {1 - h} .$

There are several ways to define operations between hesitant fuzzy elements. Xu [23] and Zheng [28] defined them in the following way: Let x ∈ X and $\tilde{A}, \tilde{B}$ be two hesitant fuzzy sets. Then for the hesitant fuzzy elements $\tilde{A} (x), \tilde{B} (x)$ the union (denoted by ⊍) and intersection (denoted by ⩀) are defined as follows: $(\tilde{A} ⊍ \tilde{B}) (x) = {(\tilde{A} (x))}_{α}^{+} \cup {(\tilde{B} (x))}_{α}^{+},$ where $α = max {{\tilde{A}}^{-} (x), {\tilde{B}}^{-} (x)},$ $(\tilde{A} ⩀ \tilde{B}) (x) = {(\tilde{A} (x))}_{α}^{-} \cup {(\tilde{B} (x))}_{α}^{-},$ where $α = min {{\tilde{A}}^{+} (x), {\tilde{B}}^{+} (x)} .$

Example 1. Let us consider the following hesitant fuzzy sets defined on the three-point referential X ={ x₁, x₂, x₃ } and given by their membership function scheduled in the table: $\begin{matrix} x_{1} & x_{2} & x_{3} \\ \tilde{A} & {0.3} & {0.2} & {0.7} \\ \tilde{B} & {0.3, 0.1} & {0.2, 0.1} & {0.7, 0.8, 0.9} \\ \tilde{C} & {0} & {0.1} & {1} \\ \tilde{D} & {0.5, 0.4, 0.6} & {0.5, 0.55} & {0.5, 0.9, 0.6, 0.85, 0.8} \\ \tilde{A} \cup \cdot \tilde{C} & {0.3} & {0.2} & {1} \\ \tilde{A} ⩀ \tilde{C} & {0} & {0.1} & {0.7} \\ \tilde{A} \cup \cdot \tilde{D} & {0.4, 0.5, 0.6} & {0.5, 0.55} & {0.7, 0.8, 0.85, 0.9} \\ \tilde{A} ⩀ \tilde{D} & {0.3} & {0.2} & {0.5, 0.6, 0.7} \\ \tilde{B} \cup \cdot \tilde{C} & {0.1, 0.3} & {0.1, 0.1, 0.2} & {1} \\ \tilde{B} ⩀ \tilde{C} & {0} & {0.1, 0.1} & {0.7, 0.8, 0.9} \\ \tilde{B} \cup \cdot \tilde{D} & {0.4, 0.5, 0.6} & {0.5, 0.55} & {0.7, 0.8, 0.8, 0.85, 0.9, 0.9} \\ \tilde{B} ⩀ \tilde{D} & {0.1, 0.3} & {0.1, 0.2} & {0.5, 0.6, 0.7, 0.8, 0.8, 0.85, 0.9, 0.9} \end{matrix}$

4 Divergences on hesitant fuzzy sets

The measure of the difference of two hesitant fuzzy sets is defined axiomatically based on the following properties similar to the case of fuzzy sets(see [12 –14]).

It is a nonnegative and symmetric function of the two compared fuzzy sets.

It becomes zero when the two sets coincide.

It decreases when the two subsets become “more similar” in some sense.

Definition 4. Let $ℱ_{H} (X)$ be the family of the hesitant fuzzy sets on X. A map $D_{H} : ℱ_{H} (X) \times ℱ_{H} (X) ⟶ ℝ^{+}$ is a hesitant divergence measure if for all $\tilde{A}, \tilde{B}, \tilde{C} \in ℱ_{H} (X)$ it satisfies the following conditions:

$D_{H} (\tilde{A}, \tilde{B}) = D_{H} (\tilde{B}, \tilde{A})$ ;

$D_{H} (\tilde{A}, \tilde{A}) = 0$ ;

$max {D_{H} (\tilde{A} ⊍ \tilde{C}, \tilde{B} ⊍ \tilde{C}), D_{H} (\tilde{A} ⩀ \tilde{C}, \tilde{B} ⩀ \tilde{C})} \leq D_{H} (\tilde{A}, \tilde{B})$ .

The basic problem is how can the difference between two hesitant fuzzy sets be measured. For fixed x ∈ X the corresponding hesitant fuzzy elements may have different lengths, i.e. $l_{\tilde{A}} (x) \neq l_{\tilde{B}} (x)$ in general.

Many authors (see, for instance, [5] or [22]) prefer to extend the shorter one until both of them to have the same length when we compare them. This extension can be done by adding some values. It mainly depends on the decision making risk’s preferences by anticipating of desirable outcomes. The optimists may add the maximum value, while the pessimists may add the minimum value. The result may be also different by extension the shorter one by adding different values. Let us assume that the decision makings are all pessimistic.

The distances between hesitant fuzzy sets and their modifications has been proposed in [1 , 27]. We introduce this concept in a general form.

For fixed $\tilde{A}, \tilde{B}$ let us denote $l_{x} = max {l_{\tilde{A}} (x), l_{\tilde{B}} (x)}$ , for each x ∈ X.

Proposition 1. Let X be finite, $\tilde{A}, \tilde{B} \in ℱ_{H} (X), D_{H}$ be defined in the following way: $\begin{matrix} D_{H} (\tilde{A}, \tilde{B}) \\ = {[\frac{1}{| X |} \sum_{x \in X} S_{j = 1}^{l_{x}} {| {\tilde{A}}^{σ (j)} (x) - {\tilde{B}}^{σ (j)} (x) |}^{λ}]}^{\frac{1}{λ}}, \end{matrix}$ where S is a t-conorm, $l_{x} = max {l_{\tilde{A}} (x), l_{\tilde{B}} (x)}$ for each x ∈ X, σ is a permutation of the elements of $\tilde{A} (x), \tilde{B} (x)$ into a non-increasing order and ${\tilde{A}}^{σ (j)} (x), {\tilde{B}}^{σ (j)} (x)$ are the j-th largest values in $\tilde{A} (x)$ and $\tilde{B} (x)$ , respectively. Then D_H is a hesitant divergence measure.

Proof. The map D_H is a distance by [23]. Now define the function $D_{H}^{*} (\tilde{A}, \tilde{B})$ on the finite referential X by using the Hamming distance in the followingway: $D_{H}^{*} (\tilde{A} (x), \tilde{B} (x)) = S_{j = 1}^{l_{x}} | {\tilde{A}}^{σ (j)} (x) - {\tilde{B}}^{σ (j)} (x) |,$ where S is an arbitrary t-conorm. We will show that $D_{H}^{*}$ is a divergence measure. The first and the second conditions are trivial. Let us check the thirdone.

We will consider the operations between hesitant fuzzy elements defined in the previous section. For technical reasons we extend all hesitant fuzzy elements so that $l_{x} = max {l_{\tilde{A}} (x), l_{\tilde{B}} (x), l_{\tilde{A} ⩀ \tilde{C}} (x), l_{\tilde{B} ⩀ \tilde{C}} (x)}$ for each x ∈ X. Moreover, we are considering that the permutation σ is ordering all the elements non-increasingly.

Hence, for each index j the following inequality holds: $\begin{matrix} | {(\tilde{A} ⩀ \tilde{C})}^{σ (j)} (x) - {(\tilde{B} ⩀ \tilde{C})}^{σ (j)} (x) | \\ \leq | {\tilde{A}}^{σ (j)} (x) - {\tilde{B}}^{σ (j)} (x) | . \end{matrix}$

Since the t-conorm S is monotone, we have $D_{H}^{*} (\tilde{A} ⩀ \tilde{C}, \tilde{B} ⩀ \tilde{C}) \leq D_{H}^{*} (\tilde{A}, \tilde{B})$ . Similarly $| {(\tilde{A} ⊍ \tilde{C})}^{σ (j)} (x) - {(\tilde{B} ⊍ \tilde{C})}^{σ (j)} (x) | \leq | {\tilde{A}}^{σ (j)} (x) - {\tilde{B}}^{σ (j)} (x) |$ for each j. Thus $D_{H}^{*} (\tilde{A} ⊍ \tilde{C}, \tilde{B} ⊍ \tilde{C}) \leq D_{H}^{*} (\tilde{A}, \tilde{B})$ and therefore $D_{H}^{*}$ is a hesitant divergence measure.

Let us check that D_H is also a hesitant divergence measure. The first and the second conditions hold trivially.

If a, b ∈ [0, 1] , a ≤ b and $k, λ \in ℝ^{+}$ then $a^{λ} \leq b^{λ}, k \cdot a \leq k \cdot b, a^{\frac{1}{λ}} \leq b^{\frac{1}{λ}}$ .

Applying of the previous relations recursively for each index j we concludethat $D_{H} (\tilde{A} ⩀ \tilde{C}, \tilde{B} ⩀ \tilde{C}) \leq D_{H} (\tilde{A}, \tilde{B})$ . Similarly $D_{H} (\tilde{A} ⊍ \tilde{C}, \tilde{B} ⊍ \tilde{C}) \leq D_{H} (\tilde{A}, \tilde{B})$ . Therefore the distance D_H is also a hesitant divergence measure.

By substituting basic t-conorms S_M and S_L from the previous general proposition two well-known examples of distances as well as hesitant divergence measures between hesitant fuzzy sets in a finite universe can be derived.

Example 2. The generalized normalized Euclidean distance for S = S_L is $\begin{matrix} D_{H} (\tilde{A}, \tilde{B}) \\ = {[\frac{1}{| X |} \sum_{x \in X} (\frac{1}{l_{x}} \sum_{j = 1}^{l_{x}} {| {\tilde{A}}^{σ (j)} (x) - {\tilde{B}}^{σ (j)} (x) |}^{λ})]}^{\frac{1}{λ}}, \end{matrix}$ and the generalized normalized Hausdorff distance for S = S_M is $\begin{matrix} D_{H} (\tilde{A}, \tilde{B}) \\ = {[\frac{1}{| X |} \sum_{x \in X} max_{1 \leq j \leq l_{x}} {| {\tilde{A}}^{σ (j)} (x) - {\tilde{B}}^{σ (j)} (x) |}^{λ}]}^{\frac{1}{λ}}, \end{matrix}$ where $l_{x} = max {l_{\tilde{A}} (x), l_{\tilde{B}} (x)}$ for each x ∈ X, σ is a permutation of the elements of $\tilde{A} (x), \tilde{B} (x)$ and ${\tilde{A}}^{σ (j)} (x), {\tilde{B}}^{σ (j)} (x)$ are the j-th largest values in $\tilde{A} (x)$ and $\tilde{B} (x)$ , respectively.

Thus we have two examples of hesitant divergence measures between hesitant fuzzy sets. Next we will not follow these ideas and define a local divergences between hesitant fuzzy sets.

5 Local divergences on hesitant fuzzy sets

Before the definition of local divergences we introduce some important notions. First, for a hesitant fuzzy element $\tilde{A} (x)$ we will define a score function in the following way (see [23]).

Definition 5. Let $\tilde{A} (x) = {h_{1}, \dots, h_{n}}$ be a general hesitant fuzzy element, $l_{\tilde{A}} (x)$ be its length and $ℍ \subseteq ℘ ([0, 1])$ be the set of all finite non-empty subsets of the closed interval [0, 1]. Then we define a score function $s : ℍ \to [0, 1]$ as follows: $s : \tilde{A} (x) \mapsto \frac{1}{l_{\tilde{A}} (x)} [\sum_{h \in \tilde{A} (x)} h] .$

Definition 6. Let $\tilde{A} (x), \tilde{B} (x)$ be two hesitant fuzzy elements. Then

$\tilde{A} (x)$ is superior to $\tilde{B} (x)$ (denoted by $\tilde{A} (x) > \tilde{B} (x)$ ) iff $s (\tilde{A} (x)) > s (\tilde{B} (x))$ ,

$\tilde{A} (x)$ is indifferent to $\tilde{B} (x)$ (denoted by $\tilde{A} (x) \approx \tilde{B} (x)$ ) iff $s (\tilde{A} (x)) = s (\tilde{B} (x))$ .

The score function s aggregates the elements h of the hesitant fuzzy element $\tilde{A} (x)$ into one value $s (\tilde{A} (x))$ . However it is easy to see that in general case it need not represent all values in the hesitant fuzzy element appropriately, especially in case when the differences among particular values are relatively large. If we consider the example from the Introduction, this represents a situation when the opinions of evaluators differ a lot.

Therefore we introduce another difference function, denoted by Δ that fulfills some natural properties. If the difference between ${\tilde{A}}^{+} (x)$ and ${\tilde{A}}^{-} (x)$ is great (at most 1) then the score function for $\tilde{A} (x)$ is not a good representation of $\tilde{A} (x)$ and therefore we will assign smaller importance to it. On the contrary, if the difference between ${\tilde{A}}^{+} (x)$ and ${\tilde{A}}^{-} (x)$ is small (at least 0) then we will assign more importance for the result of score function for $\tilde{A} (x)$ . Based on these ideas we introduce the following definition of weight vector and local hesitant divergence.

Definition 7. Let $\tilde{A} (x) = {h_{1}, \dots, h_{n}}$ be a general hesitant fuzzy element and $ℍ \subseteq ℘ ([0, 1])$ be the set of all finite non-empty subsets of the closed interval [0, 1]. Then we define a difference function $Δ : ℍ \to [0, 1]$ as follows: $Δ : \tilde{A} (x) \mapsto 1 - {\tilde{A}}^{+} (x) + {\tilde{A}}^{-} (x) .$

Contrary to Definition 6 the difference function describes the degree of ambiguity for the respective element. For instance, if we consider two hesitant fuzzy elements A (x) ={ 0.2, 0.8 } , B (x) = { 0.4, 0.6 }, then the score function assigns 0.5 in both cases, however for the difference function we have the results $Δ (\tilde{A} (x)) = 0.4, Δ (\tilde{B} (x)) = 0.8$ , so therefore score value for hesitant fuzzy element $\tilde{B} (x)$ is a better representation compared with $\tilde{A} (x)$ .

In case if ${\tilde{A}}^{+} (x) = {\tilde{A}}^{-} (x)$ we have $Δ (\tilde{A} (x)) = 1$ , so it is a good representation of this particular hesitant fuzzy element, as no ambiguity takes place. By using a difference function we will continue with notion of a weight vector.

Definition 8. Let X be a finite universe, $\tilde{A}$ be a hesitant fuzzy set in X. We define its weight vector α^A as follows: $α^{A} = (α_{1}^{A}, \dots, α_{n}^{A}) | α_{x}^{A} = \frac{Δ_{x}^{A}}{Δ^{A}}$ , where $Δ_{x}^{A} = Δ (\tilde{A} (x)) = 1 - {\tilde{A}}^{+} (x) + {\tilde{A}}^{-} (x),$ and $Δ^{A} = \sum_{x \in X} Δ_{x}^{A} .$

We see that α^A is a weight vector since $α_{x}^{A} \geq 0$ and $\sum_{x \in X} α_{x}^{A} = 1$ . Obviously α^A is a normalized vector of $Δ^{A} = (Δ_{1}^{A}, \dots, Δ_{n}^{A})$ . If Δ^A = 0 then we will define $α_{x} = \frac{1}{| X |}$ for all x ∈ X.

In the following each pair of hesitant fuzzy sets will be compared as a pair of fuzzy sets using previous notions about score function with appropriate weight vector.

Therefore we will define $D_{H} (\tilde{A}, \tilde{B}) = D (s (\tilde{A}), s (\tilde{B}))$ , where $s (\tilde{A}) (x) = s (\tilde{A} (x))$ for each element x ∈ X. So the set operations ⊍, ⩀ can be replaced by ∪_S, ∩ _T, where S, T are triangular conorm or triangular norm, respectively.

Definition 9. A hesitant divergence measure D_H has the local property (is local), if for all $\tilde{A}, \tilde{B} \in ℱ_{H} (X)$ and for all x ∈ X, there is $\begin{matrix} D_{H} (\tilde{A}, \tilde{B}) - D_{H} (\tilde{A} \cup_{S} {x}, \tilde{B} \cup_{S} {x}) \\ = h_{x} (a_{x}, b_{x}), \end{matrix}$ where $a_{x} = s (\tilde{A} (x))$ and $b_{x} = s (\tilde{B} (x))$ .

Theorem 1. (Representation Theorem) D_H is local if and only if $D_{H} (\tilde{A}, \tilde{B}) = \sum_{x \in X} h_{x} (a_{x}, b_{x}),$ with h_x (a_x, b_x) = α_x · h (a_x, b_x), where

$α_{x} = \frac{1}{2} α_{x}^{A} + \frac{1}{2} α_{x}^{B}$ , where $α_{x}^{A}$ , $α_{x}^{B}$ are x-th coordinates of the weight vectors α^A, α^B of the hesitant fuzzy sets $\tilde{A}, \tilde{B}$ , respectively.

h (x, y) is a function with the following properties:

h (x, x) =0, for all x∈ [0, 1] ;

h (x, y) = h (y, x), for all x, y∈ [0, 1] ;

h (x, y) ≥ max {h (S (x, z) , S (y, z)) , h (T (x, z) , T (y, z))} , for all x, y, z ∈ [0, 1] .

Proof. Let the map D_H be a local divergence. We will show that it can be expressed as a sum: $D_{H} (\tilde{A}, \tilde{B}) = \sum_{x \in X} h_{x} (a_{x}, b_{x}) .$ Since D_H is local, from the definition we obtain $D_{H} (\tilde{A}, \tilde{B}) - D_{H} (\tilde{A} \cup_{S} {x}, \tilde{B} \cup_{S} {x}) = h_{x} (a_{x}, b_{x}) = α_{x} \cdot h (a_{x}, b_{x}) .$

We apply recursively the previous equation for other elements from X. Therefore: $\begin{matrix} D_{H} (\tilde{A}, \tilde{B}) \\ = D_{H} (\tilde{A} \cup_{S} {x}, \tilde{B} \cup_{S} {x}) + h_{x} (a_{x}, b_{x}) \\ = D_{H} (\tilde{A} \cup_{S} {x} \cup_{S} {y}, \tilde{B} \cup_{S} {x} \cup_{S} {y}) \\ + h_{x} (a_{x}, b_{x}) + h_{y} (a_{y}, b_{y}) \\ = \dots = D_{H} (\tilde{A} \cup_{S} X, \tilde{B} \cup_{S} X) + \sum_{x \in X} h_{x} (a_{x}, b_{x}) \\ = \underset{0}{\underset{︸}{D_{H} (X, X)}} + \sum_{x \in X} h_{x} (a_{x}, b_{x}) . \end{matrix}$

Now we must verify that the function $h : [0, 1] \times [0, 1] \to ℝ$ fulfills the properties (i)-(iii) from the previous definition. We define the hesitant fuzzy sets $\tilde{A}, \tilde{B}, \tilde{C} \in ℱ_{H} (X)$ for which $s (\tilde{A} (x)) = a_{x}, s (\tilde{B} (x)) = b_{x}, s (\tilde{C} (x)) = c_{x}$ for all x ∈ X. Since ∑_x∈Xα_x = 1 we have the following results:

$0 = D_{H} (\tilde{A}, \tilde{A}) = \sum_{x \in X} h_{x} (a_{x}, a_{x}) = \sum_{x \in X} α_{x} \cdot h (a_{x}, a_{x}) = h (a_{x}, a_{x}) \cdot \sum_{x \in X} α_{x} = h (a_{x}, a_{x}) .$ Hence h (a_x, a_x) =0.

$h (a_{x}, b_{x}) = h (a_{x}, b_{x}) \cdot \sum_{x \in X} α_{x} = \sum_{x \in X} α_{x} \cdot h (a_{x}, b_{x}) = \sum_{x \in X} h_{x} (a_{x}, b_{x}) = D_{H} (\tilde{A}, \tilde{B}) = D_{H} (\tilde{B}, \tilde{A}) = \sum_{x \in X} h_{x} (b_{x}, a_{x}) = \sum_{x \in X} α_{x} \cdot h (b_{x}, a_{x}) = h (b_{x}, a_{x}) \cdot \sum_{x \in X} α_{x} = h (b_{x}, a_{x})$ . Hence h (a_x, b_x) = h (b_x, a_x).

Since D_H is a hesitant divergence measure we have: $h (a_{x}, b_{x}) = h (a_{x}, b_{x}) \cdot \sum_{x \in X} α_{x} = \sum_{x \in X} α_{x} \cdot h (a_{x}, b_{x}) = \sum_{x \in X} h_{x} (a_{x}, b_{x}) = D_{H} (\tilde{A}, \tilde{B}) ⩾ D_{H} (\tilde{A} \cap_{T} \tilde{C}, \tilde{B} \cap_{T} \tilde{C}) = \sum_{x \in X} h_{x} (T (a_{x}, c_{x}), T (b_{x}, c_{x})) = \sum_{x \in X} α_{x} \cdot h (T (a_{x}, c_{x}), T (b_{x}, c_{x})) = h (T (a_{x}, c_{x}), T (b_{x}, c_{x})) \cdot \sum_{x \in X} α_{x} = h (T (a_{x}, c_{x}), T (b_{x}, c_{x}))$ .

Hence h (T (a_x, c_x) , T (b_x, c_x)) ≤ h (a_x, b_x).

Similarly, since $D (\tilde{A}, \tilde{B}) \geq D (\tilde{A} \cup_{S} \tilde{C}, \tilde{B} \cup_{S} \tilde{C})$ we can show that h (S (a_x, c_x) , S (b_x, c_x)) ≤ h (a_x, b_x).

So we have h (a_x, b_x) ≥ max {h (T (a_x, c_x) , T (b_x, c_x)) , h (S (a_x, c_x) , S (b_x, c_x))}.

Let us show the converse implication. Let $D_{H} : ℱ_{H} (X) \times ℱ_{H} (X) \to ℝ^{+}$ be a map, which can be expressed as a sum $D_{H} (\tilde{A}, \tilde{B}) = \sum_{x \in X} h_{x} (a_{x}, b_{x}),$ where the function $h : = \frac{h_{x}}{α_{x}}$ fulfills the conditions (i)-(iii) from the previous definition. Now we will show that D_H is a hesitant divergence measure and D_H is local.

D_H is a hesitant divergence measure:

$D_{H} (\tilde{A}, \tilde{A}) = \sum_{x \in X} h_{x} (a_{x}, a_{x}) = \sum_{x \in X} α_{x} \cdot h (a_{x}, a_{x}) = h (a_{x}, a_{x}) = 0$ ,

$D_{H} (\tilde{A}, \tilde{B}) = \sum_{x \in X} h_{x} (a_{x}, b_{x}) = \sum_{x \in X} α_{x} \cdot h (a_{x}, b_{x}) = h (a_{x}, b_{x}) = h (b_{x}, a_{x}) = \sum_{x \in X} α_{x} \cdot h (b_{x}, a_{x}) = \sum_{x \in X} h_{x} (b_{x}, a_{x}) = D_{H} (\tilde{B}, \tilde{A})$ ,

$D_{H} (\tilde{A} \cap_{T} \tilde{C}, \tilde{B} \cap_{T} \tilde{C}) = \sum_{x \in X} h_{x} (T (a_{x}, c_{x}), T (b_{x}, c_{x})) = \sum_{x \in X} α_{x} \cdot h (T (a_{x}, c_{x}), T (b_{x}, c_{x})) = h (T (a_{x}, c_{x}), T (b_{x}, c_{x})) \leq h (a_{x}, b_{x}) = \sum_{x \in X} α_{x} \cdot h (a_{x}, b_{x}) = \sum_{x \in X} h_{x} (a_{x}, b_{x}) = D_{H} (\tilde{A}, \tilde{B})$ . $D_{H} (\tilde{A} \cup_{S} \tilde{C}, \tilde{B} \cup_{S} \tilde{C}) = \sum_{x \in X} h_{x} (S (a_{x}, c_{x}), S (b_{x}, c_{x})) = \sum_{x \in X} α_{x} \cdot h (S (a_{x}, c_{x}), S (b_{x}, c_{x})) = h (S (a_{x}, c_{x}), S (b_{x}, c_{x})) \leq h (a_{x}, b_{x}) = \sum_{x \in X} α_{x} \cdot h (a_{x}, b_{x}) = \sum_{x \in X} h_{x} (a_{x}, b_{x}) = D_{H} (\tilde{A}, \tilde{B})$ .

D_H is a local hesitant divergence: $D_{H} (\tilde{A}, \tilde{B}) - D_{H} (\tilde{A} \cup_{S} {x}, \tilde{B} \cup_{S} {x}) = \sum_{t \in X} h_{t} (a_{t}, b_{t}) - \sum_{t \in X} h_{t} [(\tilde{A} \cup_{S} {x}) (t), (\tilde{B} \cup_{S} {x}) (t)] = \sum_{t \in X} α_{t} \cdot h (a_{t}, b_{t}) - \sum_{t \in X} α_{t} \cdot h [(\tilde{A} \cup_{S} {x}) (t), (\tilde{B} \cup_{S} {x}) (t)] = \sum_{t \in X \ {x}} α_{t} \cdot h (a_{t}, b_{t}) + α_{x} \cdot h (a_{x}, b_{x}) - [\sum_{t \in X \ {x}} α_{t} \cdot h (a_{t}, b_{t}) - α_{x} \cdot h (1, 1)] = α_{x} \cdot h (a_{x}, b_{x}) - α_{x} \cdot h (1, 1) = α_{x} \cdot h (a_{x}, b_{x}) - 0 = h_{x} (a_{x}, b_{x}) .$

We see that D_H is local by definition.

However, we must add and clarify one remark to understand the last step in the previous proof. We have defined the following: $(\tilde{A} \cup_{S} {x}) (t) = {\begin{matrix} a, & if t \neq x . \\ 1, & if t = x . \end{matrix} (\tilde{B} \cup_{S} {x}) (t) = {\begin{matrix} b, & if t \neq x . \\ 1, & if t = x . \end{matrix} h [(\tilde{A} \cup_{S} {x}) (t), (\tilde{B} \cup_{S} {x}) (t)] = = {\begin{matrix} h (a, b), & if t \neq x . \\ h (1, 1) = 0, & if t = x . \end{matrix}$

This completes the proof.

Example 3. The generalized normalized Euclidean distance is an example of a hesitant divergence, which is local. $\begin{matrix} D_{H} (\tilde{A}, \tilde{B}) \\ = {[\frac{1}{| X |} \sum_{x \in X} (\frac{1}{l_{x}} \sum_{j = 1}^{l_{x}} {| {\tilde{A}}^{σ (j)} (x) - {\tilde{B}}^{σ (j)} (x) |}^{λ})]}^{\frac{1}{λ}}, \end{matrix}$ where $l_{x} = max {l_{\tilde{A}} (x), l_{\tilde{B}} (x)}$ for each x ∈ X, σ is a permutation of the elements of $\tilde{A} (x), \tilde{B} (x)$ and ${\tilde{A}}^{σ (j)} (x), {\tilde{B}}^{σ (j)} (x)$ are the j-th largest values in $\tilde{A} (x)$ and $\tilde{B} (x)$ , respectively.

Recall that the divergence $D_{H}^{*}$ is defined as follows: $D_{H}^{*} (\tilde{A} (x), \tilde{B} (x)) = \sum_{j = 1}^{l_{x}} | {\tilde{A}}^{σ (j)} (x) - {\tilde{B}}^{σ (j)} (x) | .$

From defining the functions h_j by following $h_{j} (x) : = α_{j} \cdot | {\tilde{A}}^{σ (j)} (x) - {\tilde{B}}^{σ (j)} (x) |$ , where $\sum_{j = 1}^{l_{x}} α_{j} = 1$ we have that $D_{H}^{*}$ is a local divergence and also D_H is local.

Example 4. The generalized normalized Hausdorff distance is an example of a hesitant divergence, which is not local. $\begin{matrix} D_{H} (\tilde{A}, \tilde{B}) \\ = {[\frac{1}{| X |} \sum_{x \in X} max_{1 \leq j \leq l_{x}} {| {\tilde{A}}^{σ (j)} (x) - {\tilde{B}}^{σ (j)} (x) |}^{λ}]}^{\frac{1}{λ}}, \end{matrix}$ where $l_{x} = max {l_{\tilde{A}} (x), l_{\tilde{B}} (x)}$ for each x ∈ X, σ is a permutation of the elements of $\tilde{A} (x), \tilde{B} (x)$ and ${\tilde{A}}^{σ (j)} (x), {\tilde{B}}^{σ (j)} (x)$ are the j-th largest values in $\tilde{A} (x)$ and $\tilde{B} (x)$ , respectively.

In this case we will define the divergence $D_{H}^{*}$ as follows: $D_{H}^{*} (\tilde{A} (x), \tilde{B} (x)) = max_{1 \leq j \leq l_{x}} | {\tilde{A}}^{σ (j)} (x) - {\tilde{B}}^{σ (j)} (x) | .$

It is not possible to express $D_{H}^{*}$ as a sum of the functions h_j and therefore the divergence measure $D_{H}^{*}$ is not local. Also the divergence D_H cannot be local.

6 Comparison of divergences D and D_H

In this section we will study relationship between divergence measures D and hesitant divergences D_H. First we recall the definition of a divergence measure and a local divergence measure between fuzzy sets (see [12]).

Definition 10. Let X be a finite nonempty set, let T, S be a triangular norm and a conorm. A map $D : ℱ (X) \times ℱ (X) ⟶ ℝ$ is a divergence measure if for all $\tilde{A}, \tilde{B} \in ℱ (X)$ it satisfies the followingconditions:

$D (\tilde{A}, \tilde{B}) = D (\tilde{B}, \tilde{A})$ ;

$D (\tilde{A}, \tilde{A}) = 0$ ;

$max {D (\tilde{A} \cup_{S} \tilde{C}, \tilde{B} \cup_{S} \tilde{C}), D (\tilde{A} \cap_{T} \tilde{C}, \tilde{B} \cap_{T} \tilde{C})} \leq D (\tilde{A}, \tilde{B})$ for all $\tilde{C} \in ℱ (X)$ .

Definition 11. A divergence measure D has the local property (is local), if for all $\tilde{A}, \tilde{B} \in ℱ (X)$ and for all x ∈ X, we have $\begin{matrix} D (\tilde{A}, \tilde{B}) - D (\tilde{A} \cup_{S} {x}, \tilde{B} \cup_{S} {x}) \\ = h (\tilde{A} (x), \tilde{B} (x)) . \end{matrix}$

Theorem 2. D is local if and only if $D (\tilde{A}, \tilde{B}) = \sum_{x \in X} h (\tilde{A} (x), \tilde{B} (x)),$ where h (x, y) is a function from [0, 1] × [0, 1] into $ℝ$ with the following properties:

h (x, x) =0, for all x∈ [0, 1] ;

h (x, y) = h (y, x), for all x, y∈ [0, 1] ;

h (x, y) ≥ max {h (S (x, z) , S (y, z)) , h (T (x, z) , T (y, z))} , for all x, y, z ∈ [0, 1] .

From hesitant divergences D_H to fuzzy divergences D

Consider now a divergence measure D_H defined on a finite universal set X. Recall that every fuzzy set $\tilde{A}$ can be represented as a hesitant fuzzy set considering one-point hesitant fuzzy element for each x ∈ X, i.e. $s (\tilde{A}) = \tilde{A}$ . It allows us to state the following proposition.

Proposition 2. Let D_H be a hesitant divergence measure and $\tilde{A}, \tilde{B}$ be the fuzzy sets. Then the map $D : ℱ (X) \times ℱ (X) \to ℝ$ given by $D (\tilde{A}, \tilde{B}) = D_{H} (\tilde{A}, \tilde{B})$ is a divergence for fuzzy sets.

Proof. Let us verify the properties of a divergence measure for D:

$D (\tilde{A}, \tilde{B}) = D (s (\tilde{A}), s (\tilde{B})) = D_{H} (\tilde{A}, \tilde{B}) = D_{H} (\tilde{B}, \tilde{A}) = D (s (\tilde{B}), s (\tilde{A})) = D (\tilde{B}, \tilde{A})$ ,

$D (\tilde{A}, \tilde{A}) = D (s (\tilde{A}), s (\tilde{A})) = D_{H} (\tilde{A}, \tilde{A}) = 0$ ,

$D (\tilde{A} \cap_{T} \tilde{C}, \tilde{B} \cap_{T} \tilde{C}) = D (s (\tilde{A}) \cap_{T} s (\tilde{C}), s (\tilde{B}) \cap_{T} s (\tilde{C})) = D_{H} (\tilde{A} \cap_{T} \tilde{C}, \tilde{B} \cap_{T} \tilde{C}) \leq D_{H} (\tilde{A}, \tilde{B}) = D (s (\tilde{A}), s (\tilde{B})) = D (\tilde{A}, \tilde{B})$ and $D (\tilde{A} \cup_{S} \tilde{C}, \tilde{B} \cup_{S} \tilde{C}) = D (s (\tilde{A}) \cup_{S} s (\tilde{C}), s (\tilde{B}) \cup_{S} s (\tilde{C})) = D_{H} (\tilde{A} \cup_{S} \tilde{C}, \tilde{B} \cup_{S} \tilde{C}) \leq D_{H} (\tilde{A}, \tilde{B}) = D (s (\tilde{A}), s (\tilde{B})) = D (\tilde{A}, \tilde{B})$ .

Therefore D is a divergence.

From fuzzy divergences D to hesitant divergences D_H

Let us consider the hesitant fuzzy set $\tilde{A}$ . Since the length of the hesitant fuzzy elements is not constant in general, we cannot apply division of the hesitant fuzzy set into a finite number of fuzzy sets (similar to IF-sets, see [11]). However it is possible to follow a procedure assigning a score function to each hesitant fuzzy element, but $\tilde{A} \neq s (\tilde{A})$ in general. The result is represented as a fuzzy set.

Proposition 3. Let D be a divergence measure and $\tilde{A}, \tilde{B}$ be the hesitant fuzzy sets. Then the map $D_{H} : ℱ_{H} (X) \times ℱ_{H} (X) \to ℝ$ given by $D_{H} (\tilde{A}, \tilde{B}) = D (s (\tilde{A}), s (\tilde{B})),$ where s is a score function defined by $s (\tilde{A}) (x) = s (\tilde{A} (x))$ for each element x ∈ X, is a hesitant divergence measure.

Proof. Let us verify the properties of a hesitant divergence for D_H:

$D_{H} (\tilde{A}, \tilde{B}) = D (s (\tilde{A}), s (\tilde{B})) = D (s (\tilde{B}), s (\tilde{A})) = D_{H} (\tilde{B}, \tilde{A})$ ,

$D_{H} (\tilde{A}, \tilde{A}) = D (s (\tilde{A}), s (\tilde{A})) = 0$ ,

$D_{H} (\tilde{A} \cap_{T} \tilde{C}, \tilde{B} \cap_{T} \tilde{C}) = D (s (\tilde{A} \cap_{T} \tilde{C}), s (\tilde{B} \cap_{T} \tilde{C})) \leq D (s (\tilde{A}), s (\tilde{B})) = D_{H} (\tilde{A}, \tilde{B})$ , and $D_{H} (\tilde{A} \cup_{S} \tilde{C}, \tilde{B} \cup_{S} \tilde{C}) = D (s (\tilde{A} \cup_{S} \tilde{C}), s (\tilde{B} \cup_{S} \tilde{C})) \leq D (s (\tilde{A}), s (\tilde{B})) = D_{H} (\tilde{A}, \tilde{B})$ .

We see that D_H is a hesitant divergence measure.

Let us study a relationship between locality of divergence D and locality of hesitant divergence D_H. These results are contained in the following proposition:

Proposition 4. Let X be finite.

Let D_H be a hesitant divergence measure defined on $ℱ_{H} (X) \times ℱ_{H} (X)$ and D be the fuzzy divergence measure restricted on $ℱ (X) \times ℱ (X)$ by $D (\tilde{A}, \tilde{B}) = D_{H} (\tilde{A}, \tilde{B}) .$

If D_H is local then |X| · D is local.

Let D be the fuzzy divergence measure defined on $ℱ (X) \times ℱ (X)$ and D_H be the hesitant divergence measure extended on $ℱ_{H} (X) \times ℱ_{H} (X)$ by $D_{H} (\tilde{A}, \tilde{B}) = D (s (\tilde{A}), s (\tilde{B})) .$

If D is local then α · D_H is local,

where $α = \frac{1}{2} \cdot α^{A} + \frac{1}{2} \cdot α^{B}$ and α^A, α^B are weight vectors for $\tilde{A}, \tilde{B}$ , respectively.

Proof. Using the fact that the $\tilde{A}, \tilde{B}$ are fuzzy sets we can write $α_{x} = \frac{1}{| X |}$ for each x ∈ X. Considering that D_H is local we have: $\begin{matrix} D (\tilde{A}, \tilde{B}) & = & D_{H} (\tilde{A}, \tilde{B}) = \sum_{x \in X} h_{x} (a_{x}, b_{x}) \\ = & \sum_{x \in X} α_{x} \cdot h (a_{x}, b_{x}) = \frac{1}{| X |} \cdot \sum_{x \in X} h (a_{x}, b_{x}) . \end{matrix}$

Hence $| X | \cdot D (\tilde{A}, \tilde{B}) = \sum_{x \in X} h (a_{x}, b_{x})$ .

We conclude that |X| · D is a local divergence.

To show the second part we will denote $s (\tilde{A}) (x) = a_{x}$ and $s (\tilde{B}) (x) = b_{x}$ . Considering that D is local we have: $D_{H} (\tilde{A}, \tilde{B}) = D (s (\tilde{A}), s (\tilde{B})) = \sum_{x \in X} h (a_{x}, b_{x}) = \frac{1}{| α |} \cdot [α \cdot \sum_{x \in X} h (a_{x}, b_{x})] = \frac{1}{| α |} \cdot \sum_{x \in X} α_{x} \cdot h (a_{x}, b_{x}) = \frac{1}{| α |} \cdot \sum_{x \in X} h_{x} (a_{x}, b_{x}) .$

Hence $α \cdot D_{H} (\tilde{A}, \tilde{B}) = \sum_{x \in X} h_{x} (a_{x}, b_{x})$ and α · D_H is a local divergence.

7 Entropy measures for hesitant fuzzy sets

The study of entropy measures in the fuzzy set theory can be traced to De Luca and Termini in 1972. The main aim was to quantify the uncertainty associated to a fuzzy set. This concept has been adapted to other types and extensions of fuzzy sets, such as Atanassov’s IF sets (see [3]), interval-valued fuzzy sets (see [5]) or even interval-valued hesitant fuzzy sets (see [17]). In our paper we will propose the entropy measure adapted to the hesitant fuzzy sets. We will focus on two cases of entropies such as fuzziness entropy measure and hesitance entropy measure.

Definition 12. Let $f_{F} : ℱ_{H} \to [0, 1]$ be a mapping, $\tilde{A}, \tilde{B} \in ℱ_{H}$ . The map f_F is said to be a fuzziness entropy measure associated to a hesitant divergence measure D_H if it satisfies the following properties:

$f_{F} (\tilde{A}) = 0$ iff $\tilde{A} (x) = {(0), (1)}$ for all x ∈ X,

$f_{F} (\tilde{A}) = 1$ iff $\tilde{A} = \tilde{E}$ , where $\tilde{E}$ is an equilibrium set, i.e. $\tilde{E} (x) = {(0.5)}$ for all x ∈ X,

$f_{F} (\tilde{A}) = f_{F} ({\tilde{A}}^{c})$ ,

$f_{F} (\tilde{A}) \leq f_{F} (\tilde{B})$ for which $D_{H} (\tilde{A}, \tilde{E}) \geq D_{H} (\tilde{B}, \tilde{E})$ .

We state two other ways to construct a fuzziness entropy measure associated to a local hesitant divergence measure (see also [5]).

Proposition 5. Let Z : [0, 1] → [0, 1] be a strictly decreasing real function and D_H be a local hesitant divergence measure from Definition 5oc with the following properties:

h (a, 0.5) =0 ⇔ a = 0.5,

h (a, 0.5) =0.5⇔ a ∈ { 0, 1 },

h (a, 0.5) = h (1 - a, 0.5) for all a ∈ [0, 1].

Then, for a hesitant fuzzy set $\tilde{A}$ the function $f_{F} (\tilde{A}) = \frac{Z (2 \cdot D_{H} (\tilde{A}, \tilde{E})) - Z (1)}{Z (0) - Z (1)}$ is a fuzziness entropy measure based on the corresponding divergence D_H.

Proof. If $\tilde{A} (x) = {(0)}$ for all x ∈ X₁ or $\tilde{A} (x) = {(1)}$ for all x ∈ X₂ then $D_{H} (\tilde{A}, \tilde{E}) = \sum_{x \in X_{1}} h_{x} (0, 0.5) + \sum_{x \in X_{2}} h_{x} (1, 0.5)$ , where X = X₁ ∪ X₂.

From the properties of h we have $D_{H} (\tilde{A}, \tilde{E}) = 0.5$ and hence $f_{F} (\tilde{A}) = \frac{Z (1) - Z (1)}{Z (0) - Z (1)} = 0$ .

If $\tilde{A} (x) = {(0), (1)}$ then Δ_x = α_x = 0. Therefore $h_{x} (s (\tilde{A} (x)), 0.5) = 0$ , where s is a score function defined in the previous section.

Conversely, for $f_{F} (\tilde{A}) = 0$ we have $D_{H} (\tilde{A}, \tilde{E}) = \sum_{x \in X} h_{x} = 0.5$ , thus $\tilde{A} (x) = {(0), (1)}$ by the second condition of f.

If $\tilde{A} (x) = {(0.5)}$ for all x ∈ X then Δ_x = 1 and $α_{x} = \frac{1}{| X |}$ . In this case $D_{H} (\tilde{A}, \tilde{E}) = \sum_{x \in X} h_{x} (0.5, 0.5) = 0$ and hence $f_{F} (\tilde{A}) = \frac{Z (0) - Z (1)}{Z (0) - Z (1)} = 1$ .

Conversely, for $f_{F} (\tilde{A}) = 1$ we have $D_{H} (\tilde{A}, \tilde{E}) = \sum_{x \in X} h_{x} = 0$ and so $\tilde{A} (x) = {(0.5)}$ by the first condition for f.

From the third condition for h we have that $D_{H} (\tilde{A}, \tilde{E}) = \sum_{x \in X} h_{x} (s (\tilde{A} (x)), 0.5) = \sum_{x \in X} h_{x} (s ({\tilde{A}}^{c} (x)), 0.5) = D_{H} ({\tilde{A}}^{c}, \tilde{E}) .$ Therefore $f_{F} (\tilde{A}) = f_{F} ({\tilde{A}}^{c})$ by the definition of f_F.

The inequality $D_{H} (\tilde{A}, \tilde{E}) \geq D_{H} (\tilde{B}, \tilde{E})$ implies $f_{F} (\tilde{A}) \leq f_{F} (\tilde{B})$ since Z is a strictly decreasing function.

Remark 1. In many practical cases we can consider the three examples of the function Z : [0, 1] → [0, 1]:

Z (t) =1 - t^λ, where λ > 0,

$Z (t) = \frac{1 - t}{1 + t}$ ,

Z (t) =1 - t · e^t-1.

Proposition 6. Let Φ : [0, 1] → [0, 1] be a strictly monotone decreasing real function with Φ (0) =1, Φ (1) =0 and D_H be a local hesitant divergence measure with the boundary condition h (1, 0) =1. Then $f_{F} (\tilde{A}) = Φ (D_{H} (A, A^{c}))$ is a fuzziness entropy measure based on the corresponding divergence D_H.

Proof. If $\tilde{A} (x) = {(0), (1)}$ for all x ∈ X then $D_{H} (\tilde{A}, {\tilde{A}}^{c}) = \sum_{x \in X} h_{x} (1, 0) = 1$ . Then $f_{F} (\tilde{A}) = Φ (1) = 0$ .

If $\tilde{A} (x) = {(0.5)}$ for all x ∈ X then $D_{H} (\tilde{A}, {\tilde{A}}^{c}) = \sum_{x \in X} h_{x} (0.5, 0.5) = 0$ . Then $f_{F} (\tilde{A}) = Φ (0) = 1$ .

Since $D_{H} (\tilde{A}, {\tilde{A}}^{c}) = D_{H} ({\tilde{A}}^{c}, \tilde{A})$ we have $f_{F} (\tilde{A}) = Φ (D_{H} (A, A^{c})) = Φ (D_{H} (A^{c}, A)) = f_{F} ({\tilde{A}}^{c})$ .

If $D_{H} (\tilde{A}, \tilde{E}) \geq D_{H} (\tilde{B}, \tilde{E})$ then $f_{F} (\tilde{A}) = Φ (D_{H} (A, A^{c})) \leq Φ (D_{H} (B, B^{c})) = f_{F} (\tilde{B})$ since Φ is a strictly decreasing function.

At the end of this proof we remark that all examples from previous Remark 1 are fulfilling required boundary conditions.

Now we are going to introduce the second type of entropy measure on HFS.

Definition 13. Let $f_{H} : ℱ_{H} \to [0, 1]$ be a mapping, $\tilde{A}, \tilde{B} \in ℱ_{H}$ . The map f_H is said to be a hesitance entropy measure, if it satisfies the following properties:

$f_{H} (\tilde{A}) = 0$ iff $l_{\tilde{A}} (x) = 1$ for all x ∈ X,

$f_{H} (\tilde{A}) = 1$ iff there exists $x \in X | l_{\tilde{A}} (x) \geq l_{\tilde{B}} (y)$ for all y ∈ X and for an arbitrary hesitant fuzzy set $\tilde{B}$ ,

$f_{H} (\tilde{A}) = f_{H} ({\tilde{A}}^{c})$ ,

$f_{H} (\tilde{A}) \leq f_{H} (\tilde{B})$ iff $l_{\tilde{A}} (x) \leq l_{\tilde{B}} (x)$ for allx ∈ X.

The second condition from the previous definition assumes that an arbitrary hesitant fuzzy set $\tilde{A}$ is not a fuzzy set in a special case. It means that for each $\tilde{A} \in ℱ_{H}$ there is an x ∈ X such that $l_{\tilde{A}} (x) \geq 2$ . The same requirement must be fulfilled also in the following example.

Example 5. Let us denote l_max (x) for the maximal length attained by some hesitant fuzzy element corresponding to the element x ∈ X for which $l_{\max} (x) \geq l_{\tilde{A}} (x)$ for an arbitrary hesitant fuzzy set $\tilde{A}$ . We define the function f_H in the following way: $f_{H} (\tilde{A}) = \frac{S_{x \in X} [l_{\tilde{A}} (x) - 1]}{S_{x \in X} [l_{\max} (x) - 1]},$ where S is a triangular conorm.

We will show that f_H is a hesitance entropy measure.

If $l_{\tilde{A}} (x) = 1$ for all x ∈ X then $f_{H} (\tilde{A}) = 0$ by definition. Also the converse is immediate.

If there is an element x ∈ X such that $l_{\tilde{A}} (x) \geq l_{\tilde{B}} (y)$ for all y ∈ X and for an arbitrary hesitant fuzzy set $\tilde{B}$ , then $l_{\tilde{A}} (x) = l_{\max} (x)$ . Therefore $f_{H} (\tilde{A}) = 1$ .

For a hesitant fuzzy set $\tilde{A}$ and its complement ${\tilde{A}}^{c}$ we have $l_{\tilde{A}} (x) = l_{{\tilde{A}}^{c}} (x)$ for all x ∈ X. Hence $f_{H} (\tilde{A}) = f_{H} ({\tilde{A}}^{c})$ by definition.

If $l_{\tilde{A}} (x) \leq l_{\tilde{B}} (x)$ for all x ∈ X then $f_{H} (\tilde{A}) \leq f_{H} (\tilde{B})$ since S is monotone.

We introduce the concept of entropy based on the both previous ones.

Definition 14. Let $f : ℱ_{H} \to [0, 1] \tilde{A}, \tilde{B} \in ℱ_{H}$ . Then f is a joint fuzziness-hesitance entropy measure associated to a divergence measure D_H if it satisfies the following:

$f (\tilde{A}) = 0$ iff $\tilde{A} (x) = {(0), (1)}$ or $l_{\tilde{A}} (x) = 1$ for all x ∈ X,

$f (\tilde{A}) = 1$ iff $\tilde{A} (x) = {(0.5)}$ and $l_{\tilde{A}} (x) \geq l_{\tilde{B}} (x)$ for all x ∈ X and for an arbitrary hesitant fuzzy set $\tilde{B}$ ,

$f (\tilde{A}) = f ({\tilde{A}}^{c})$ ,

$f (\tilde{A}) \leq f (\tilde{B})$ for which $D_{H} (\tilde{A}, \tilde{E}) \geq D_{H} (\tilde{B}, \tilde{E})$ and $l_{\tilde{A}} (x) \leq l_{\tilde{B}} (x)$ for all x ∈ X.

Proposition 7. Let f_F, f_H be a fuzziness and hesitance entropy measure, respectively, T be a triangular norm without zero divisors and $\tilde{A} \in ℱ_{H}$ . Then the map $f : ℱ_{H} \to [0, 1]$ defined as $f (\tilde{A}) = T (f_{F} (\tilde{A}), f_{H} (\tilde{A}))$ is a joint fuzziness-hesitance entropy measure on HFS.

Proof. Since $\tilde{A} (x) = {(0), (1)}$ for all x ∈ X we have $f (\tilde{A}) = T (f_{F} (\tilde{A}), f_{H} (\tilde{A})) = T (0, f_{H} (\tilde{A})) = 0$ . Similarly, since $l_{\tilde{A}} (x) = 1$ for all x ∈ X, there is $f (\tilde{A}) = T (f_{F} (\tilde{A}), f_{H} (\tilde{A})) = T (f_{F} (\tilde{A}), 0) = 0$ .

Conversely, if $f (\tilde{A}) = 0$ then $f_{F} (\tilde{A}) = 0$ or $f_{H} (\tilde{A}) = 0$ , by properties of T. Now $\tilde{A} (x) = {(0), (1)}$ or $l_{\tilde{A}} (x) = 1$ for all x ∈ X, by definition of f_F and f_H, respectively.

If $\tilde{A}$ is an equilibrium set, i.e. $\tilde{A} (x) = {(0.5)}$ and $l_{\tilde{A}} (x) \geq l_{\tilde{B}} (x)$ for all x ∈ X and for an arbitrary hesitant fuzzy set $\tilde{B}$ then $f (\tilde{A}) = T (f_{F} (\tilde{A}), f_{H} (\tilde{A})) = T (1, 1) = 1$ .

Conversely, if $f (\tilde{A}) = 1$ then $f_{F} (\tilde{A}) = 1$ and $f_{H} (\tilde{A}) = 1$ , simultaneously, by properties of T. Hence by definition $\tilde{A} (x) = {(0.5)}$ and $l_{\tilde{A}} (x) \geq l_{\tilde{B}} (x)$ for every x ∈ X and a hesitant fuzzy set $\tilde{B}$ .

For the hesitant fuzzy set $\tilde{A}$ and its complement ${\tilde{A}}^{c}$ we have $f_{F} (\tilde{A}) = f_{F} ({\tilde{A}}^{c})$ and $f_{H} (\tilde{A}) = f_{H} ({\tilde{A}}^{c})$ . Therefore $f (\tilde{A}) = T (f_{F} (\tilde{A}), f_{H} (\tilde{A})) = T (f_{F} ({\tilde{A}}^{c}), f_{H} ({\tilde{A}}^{c})) = f ({\tilde{A}}^{c})$ .

Finally, we have $f_{F} (\tilde{A}) \leq f_{F} (\tilde{B})$ since $D_{H} (\tilde{A}, \tilde{E}) \geq D_{H} (\tilde{B}, \tilde{E})$ and $f_{H} (\tilde{A}) \leq f_{H} (\tilde{B})$ since $l_{\tilde{A}} (x) \leq l_{\tilde{B}} (x)$ for all x ∈ X. So $f (\tilde{A}) = T (f_{F} (\tilde{A}), f_{H} (\tilde{A})) \leq T (f_{F} (\tilde{B}), f_{H} (\tilde{B})) = f (\tilde{B})$ by the monotonicity of T.

Example 6. Let us consider the following hesitant fuzzy sets defined on X ={ x₁, x₂, x₃ } given by their membership function in the table: $\begin{matrix} x_{1} & x_{2} & x_{3} \\ \tilde{A} & {0.3} & {0.2} & {0.7} \\ \tilde{B} & {0.3, 0.1} & {0.2, 0.1} & {0.7, 0.8, 0.9} \\ \tilde{C} & {0} & {0.1} & {1} \\ \tilde{D} & {0.5, 0.4, 0.6} & {0.5, 0.55} & {0.5, 0.9, 0.6, 0.85, 0.8} \end{matrix}$

We will construct the function f_F as in the Proposition 6 and a function f_H from the Example 5. Next, we suppose that D_H is a local hesitant divergence measure corresponding to the proposal in the Theorem 1 (Representation Theorem). Two cases of the joint fuzziness-hesitance entropy measure aggregating by t-norm T∈ { T_M, T_P } will be shown.

Define the local hesitant divergence measure D_H in the following way: $D_{H} (\tilde{A}, \tilde{B}) = \sum_{i = 1}^{3} α_{i} \cdot | s (\tilde{A} (x_{i})) - s (\tilde{B} (x_{i})) |,$ where s is a score function and α is a weight vector. We use the function Φ (t) =1 - t from the Proposition 6 and the t-conorm S = S_M from the Example 5. The weight vectors are: $α^{A} = (\frac{1}{3}, \frac{1}{3}, \frac{1}{3}), α^{B} = (0.32, 0.36, 0.32), α^{C} = (\frac{1}{3}, \frac{1}{3}, \frac{1}{3}), α^{D} = (0.34, 0.40, 0.26)$ .

We can express the divergence between the hesitant fuzzy sets $\tilde{A}$ and $\tilde{B}$ as follows: $\begin{matrix} D_{H} (\tilde{A}, \tilde{B}) \\ = [(0.5 \cdot \frac{1}{3} + 0.5 \cdot 0.32) \cdot | 0.3 - 0.2 |] \\ + [(0.5 \cdot \frac{1}{3} + 0.5 \cdot 0.36) \cdot | 0.2 - 0.15 |] \\ + [(0.5 \cdot \frac{1}{3} + 0.5 \cdot 0.32) \cdot | 0.7 - 0.8 |] \\ = (\frac{49}{150} \cdot 0.1) + (\frac{52}{150} \cdot 0.05) + (\frac{49}{150} \cdot 0.1) \\ = \frac{31}{375} \approx 0.083 . \end{matrix}$

Now we can compute the fuzziness f_F, hesitance f_H and the joint fuzziness-hesitance entropy measure f for each of the hesitant fuzzy sets $\tilde{A}, \tilde{B}, \tilde{C}$ and $\tilde{D}$ .

A sample calculation is presented for $\tilde{A}$ .

the fuzziness entropy measure f_F:

$f_{F} (\tilde{A}) = Φ (D_{H} (A, A^{c})) = 1 - [(\frac{1}{3} \cdot | 0.3 - 0.7 | + (\frac{1}{3} \cdot | 0.2 - 0.8 |) + (\frac{1}{3} \cdot | 0.7 - 0.3 |)] = 1 - \frac{7}{15} = \frac{8}{15} \approx 0.53$ .

the hesitance entropy measure f_H:

$f_{H} (\tilde{A}) = \frac{S_{x \in X} [l_{\tilde{A}} (x) - 1]}{S_{x \in X} [l_{\max} (x) - 1]} = \frac{max {0, 0, 0}}{max {2, 1, 4}} = 0$ .

the joint fuzziness-hesitance entropy measure f:

for a t-norm T_M: $f (\tilde{A}) = T_{M} (f_{F} (\tilde{A}), f_{H} (\tilde{A})) = min {0.53, 0} = 0$ ,

for a t-norm T_P: $f (\tilde{A}) = T_{P} (f_{F} (\tilde{A}), f_{H} (\tilde{A})) = 0.53 \cdot 0 = 0$ .

All the results are figured in the following table: $\begin{matrix} f_{F} & f_{H} & f = T_{M} (f_{F}, f_{H}) & f = T_{P} (f_{F}, f_{H}) \\ \tilde{A} & 0.53 & 0 & 0 & 0 \\ \tilde{B} & 0.36 & 0.5 & 0.36 & 0.18 \\ \tilde{C} & 0.07 & 0 & 0 & 0 \\ \tilde{D} & 0.86 & 1 & 0.86 & 0.86 \end{matrix}$

We conclude that $f_{F} (\tilde{C}) \leq f_{F} (\tilde{B}) \leq f_{F} (\tilde{A}) \leq f_{F} (\tilde{D})$ and $f_{H} (A) = f_{H} (\tilde{C}) \leq f_{H} (\tilde{B}) \leq f_{H} (\tilde{D})$ . Finally we have that $f (\tilde{A}) = f (\tilde{C}) = 0$ and $f (\tilde{D}) > f (\tilde{B})$ for both t-norms T∈ { T_M, T_P }. Therefore a joint fuzziness-hesitance entropy measure attains the maximal value for the hesitant fuzzy set $\tilde{D}$ .

8 Conclusions

We have extended the results from divergencies and entropy for fuzzy sets to hesitant fuzzy sets. As an immediate further research direction we see the related questions for interval valued hesitant fuzzy sets. Some of our results are limited to a finite universe – extensions for hesitant fuzzy sets defined on infinite sets that require a suitable integration would be also interesting.

Footnotes

Acknowledgements

The research in this communication has been supported in part by MINECO-TIC2014-59543-P its financial support is gratefully acknowledged.

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Divergence measures on hesitant fuzzy sets

Abstract

Keywords

1 Introduction

2 Basic concepts

3 Hesitant fuzzy sets

4 Divergences on hesitant fuzzy sets

5 Local divergences on hesitant fuzzy sets

6 Comparison of divergences D and D H

7 Entropy measures for hesitant fuzzy sets

8 Conclusions

Footnotes

Acknowledgements

References

6 Comparison of divergences D and D_H