Some properties of entropy of vague soft sets and its applications

Abstract

A vague soft set is a combination of a vague set and a soft set. In this paper, we first discuss the properties of the entropy on vague soft sets. We also propose the cross-entropy of vague soft sets, and investigate the relationship between entropy and cross-entropy. Furthermore, some numerical examples are presented to illustrate the applications of the entropy and cross-entropy of vague soft sets to decision making and pattern recognition.

Keywords

Vague set soft set vague soft set entropy cross-entropy

1 Introduction

Uncertain data are treated in many fields such as economics, engineering, environment, social science, medical science and business management with the complexities of modeling. Classical methods are not always successful, because the uncertainties in these data are of various types. To treat these uncertainties probability theory, fuzzy set theory [1], intuitionistic fuzzy set theory [2], vague set theory [3], interval mathematics [4], and other mathematical tools are useful. However, all of these theories have their own difficulties which was pointed out in [5]. Molodtsov suggested that one of the reasons for these difficulties is due to the inadequacy of the parametrization tool of the theory. To overcome these difficulties, Molodtsov [5] introduced the concept of soft sets as a new mathematical tool for dealing with uncertainties.

By definition, a soft set is a parameterized family of subsets of a universal set. Concretely, a soft set is a collection of approximate descriptions of an object. Maji et al. [6] initiated the study on hybrid structuresinvolving both fuzzy sets and soft sets. Afterwards, various kinds of extended fuzzy soft sets such as generalized fuzzy soft sets [7], intuitionistic fuzzy soft sets [8, 9], vague soft sets [10], and interval-valued fuzzy soft sets [11] were presented. A vague set is defined by a truth-membership function t _A and a false-membership function f _A, where t _A (x) is a lower bound on the grade of membership of x derived from the evidence for x, and f _A (x) is a lower bound on the negation of x derived from the evidence against x. The values of t _A (x) and f _A (x) are both defined on [0, 1] with each point in a basic set X, where t _A (x) + f _A (x) ≤1. An intuitionistic fuzzy sets in X is an object having the form A = {〈x, μ _A (x) , γ _A (x) 〉}, where the functions μ _A : X → [0, 1] and γ _A : X → [0, 1] define the degree of membership and the degree of non-membership respectively of the element x to the set A. For any x, 0 ≤ μ _A (x) + γ _A (x) ≤1. A vague soft set is a combination of a vague set and a soft set. An intuitionistic fuzzy soft set is a combination of an intuitionistic fuzzy set and a soft set. Since vague sets are equivalent to intuitionistic fuzzy sets [12], so vague soft sets are equivalent to intuitionistic fuzzy soft sets.

Entropy is an important notion to measure uncertain information. It describes the fuzziness of a set and was first introduced by Zadeh in 1965. Several authors have studied it from different points of view [13 –17]. Some authors have investigated entropy of intuitionistic fuzzy sets and interval-valued fuzzy sets. Bustince and Burillo [18] introduced the concept of entropy of intuitionistic fuzzy set and the interval-valued fuzzy set. Szmidt and Kacprzyk [19] extended De Luca and Termini’s axioms and proposed an entropy measure for intuitionistic fuzzy set. Zeng and Li [20] expressed the axioms of Szmidt and Kacprzyk using the notation of interval-valued fuzzy set. Farhadinia [21] generalized some results on the entropy of interval-valued fuzzy sets based on the intuitionistic distance. Zhang and Jiang [22] proposed two kinds of entropy measures for vague sets which are entropy and cross-entropy. They [23] also discussed some information measures for interval-valued intuitionistic fuzzy sets such as entropy, similarity measure and inclusionmeasure.

Vague soft set theory and intuitionistic fuzzy soft set theory makes description of the object world more realistic, practical and accurate in certain cases. Some authors have studied intuitionistic fuzzy soft sets from different view points [24 –29]. For example, Jiang et al. [26] presented an adjustable approach to intuitionistic fuzzy soft sets based decision making by using level soft sets of intuitionistic fuzzy soft sets and gave some illustrative examples. Jiang et al. [28] gave an axiom definition of intuitionistic entropy for an intuitionistic fuzzy soft set and a theorem which characterizes it, and transformed the structure of entropy for intuitionistic fuzzy soft sets to the interval-valued fuzzy soft sets. Wang and Qu [29] introduced the definitions of entropy, similarity measure and distance measure of vague soft sets, the relations between these measures are discussed in detail. However the results are incomplete, since very little work was discussed on the properties of entropy and its applications of intuitionistic fuzzy soft sets or vague soft sets. In this paper, we present two kinds of effective information entropy measures for vague soft sets and give some applications of these measures in decision making and pattern recognition.

The rest of this paper is organized as follows. In Section 2, we recall some basic concepts of vague sets, soft sets and vague soft sets. In Section 3, we discuss the properties of entropy on vague soft sets. In Section 4, we propose the cross-entropy between two vague soft sets. In Section 5, the applications of the entropy and the cross-entropy of vague soft sets to decision making and pattern recognition are illustrated through numerical examples. In the final Section, some concluding comments are presented.

2 Preliminaries

In this section, we will recall several definitions and results which are necessary for our paper. They are stated as follows:

Definition 2.1. [3] A vague set A in the universe U = {x ₁, x ₂, …, x _n} can be expressed by the following notion, A = {(x _i, [t _A (x _i) , 1 - f _A (x _i)]) |x _i ∈ U}, i.e A (x _i) = [t _A (x _i) , 1 - f _A (x _i)] and the condition 0 ≤ t _A (x _i) ≤1 - f _A (x _i) should hold for any x _i ∈ U, where t _A (x _i) is called the membership degree (true membership) of element x _i to the vague set A, while f _A (x _i) is the degree of nonmembership (false membership) of the element x _i to the vague set A.

Definition 2.2. [3] Let A, B be two vague sets in the universe U = {x ₁, x ₂, …, x _n}, then the union, intersection and complement of vague sets are defined as follows:

A ∪ B = {(x _i, [t _A (x _i) ∨ t _B (x _i) , (1 - f _A (x _i)) ∨ (1 - f _B (x _i))]) |x _i ∈ U},

A ∩ B = {(x _i, [t _A (x _i) ∧ t _B (x _i) , (1 - f _A (x _i)) ∧ (1 - f _B (x _i))]) |x _i ∈ U},

A ^c = {(x _i, [f _A (x _i) , 1 - t _A (x _i)]) |x _i ∈ U}.

Definition 2.3. [3] Let A, B be two vague sets in the universe U = {x ₁, x ₂, …, x _n}. If ∀x _i ∈ U, t _A (x _i) ≤ t _B (x _i) , 1 - f _A (x _i) ≤1 - f _B (x _i), then A is called a vague subset of B, denoted by A ⊆ B, where 1 ≤ i ≤ n.

Definition 2.4. [5] Let U be an initial universe set, P (U) the power set of U, E be a set of parameters and A ⊆ E. A pair (F, A) is called a soft set over U, where F is a mapping given by F : A → P (U).

Definition 2.5. [10] Let U be an initial universe set, V (U) the set of all vague sets on U, E be a set of parameters and A ⊆ E. A pair (F, A) is called a vague soft set over U, where F is a mapping given by F : A → V (U).

Definition 2.6. [10] Let (F, A) and (G, B) be two vague soft sets over a universe U. If A ⊆ B and ∀e ∈ A, F (e) is a vague subset of G (e), then (F, A) is called a vague soft subset of (G, B). This relation is denoted by (F, A) ⊆ (G, B).

Definition 2.7. [10] Two vague soft sets (F, A) and (G, B) over a universe U are said to be vague soft equal if (F, A) is a vague soft subset of (G, B) and (G, B) is a vague soft subset of (F, A). This relation is denoted by (F, A) = (G, B).

Definition 2.8. [10] Let E = {e ₁, e ₂, …, e _n} be a parameter set. The not set of E denoted by ¬E is defined by ¬E = {¬ e ₁, ¬ e ₂, …, ¬ e _n}, where ¬e _i = not e _i.

Definition 2.9. [10] The complement of vague soft set (F, A) is denoted by (F, A) ^c and is defined by (F, A) ^c = (F ^c, ¬ A), where F ^c : ¬ A → V (U) is a mapping given by t _F^c(¬α) (x) = f _F(α) (x) , 1 - f _F^c(¬α) (x) =1 - t _F(α) (x) , ∀ ¬ α ∈ ¬ A, x ∈ U.

Clearly (F ^c) ^c is the same as F and ((F, A) ^c) ^c = (F, A).

Definition 2.10. [10] The union of two vague soft sets (F, A) and (G, B) over a universe U is a vague soft set (H, C), where C = A ∪ B and ∀e ∈ C, $t_{H (e)} (x) = {\begin{matrix} t_{F (e)} (x) e \in A - B, x \in U, \\ t_{G (e)} (x) e \in B - A, x \in U, \\ t_{F (e)} (x) \lor t_{G (e)} (x) e \in A \cap B, x \in U . \end{matrix}$

$\begin{matrix} 1 - f_{H (e)} (x) \\ = {\begin{matrix} 1 - f_{F (e)} (x) e \in A - B, x \in U, \\ 1 - f_{G (e)} (x) e \in B - A, x \in U, \\ (1 - f_{F (e)} (x)) \lor (1 - f_{G (e)} (x)) e \in A \cap B, x \in U . \end{matrix} \end{matrix}$ We denote it by (F, A) ∪ (G, B) = (H, C).

Definition 2.11. [10] The intersection of two vague soft sets (F, A) and (G, B) over a universe U is a vague soft set (H, C), where C = A ∪ B and ∀e ∈ C, $t_{H (e)} (x) = {\begin{matrix} t_{F (e)} (x) e \in A - B, x \in U, \\ t_{G (e)} (x) e \in B - A, x \in U, \\ t_{F (e)} (x) \land t_{G (e)} (x) e \in A \cap B, x \in U . \end{matrix}$

$\begin{matrix} 1 - f_{H (e)} (x) \\ = {\begin{matrix} 1 - f_{F (e)} (x) e \in A - B, x \in U, \\ 1 - f_{G (e)} (x) e \in B - A, x \in U, \\ (1 - f_{F (e)} (x)) \land (1 - f_{G (e)} (x)) e \in A \cap B, x \in U . \end{matrix} \end{matrix}$ We denote it by (F, A) ∩ (G, B) = (H, C).

3 Some properties of entropy of vague soft sets

Definition 3.1. Let E be a set of parameters. Suppose that (F, E) and (G, E) are two vague soft sets over U, we say that (F, E) ≤ (G, E) if ∀e _i ∈ E, x _j ∈ U, t _{F(e_i)} (x _j) ≤ t _{G(e_i)} (x _j) ≤ f _{G(e_i)} (x _j) ≤ f _{F(e_i)} (x _j), or t _{F(e_i)} (x _j) ≥ t _{G(e_i)} (x _j) ≥ f _{G(e_i)} (x _j) ≥ f _{F(e_i)} (x _j).

Definition 3.2. [29] Let H : VSS (U) → [0, 1] be a mapping, where VSS (U) denotes the set of all vague soft sets in U = {x ₁, x ₂, …, x _n}. For (F, E) ∈ VSS (U), H (F, E) is called the entropy of (F, E) if it satisfies the following conditions:

(H1) H (F, E) =0 ⇔ ∀ e _i ∈ E, x _j ∈ U, t _{F(e_i)} (x _j) =0, f _{F(e_i)} (x _j) =1, or t _{F(e_i)} (x _j) =1, f _{F(e_i)} (x _j) =0;

(H2) H (F, E) =1 ⇔ ∀ e _i ∈ E, x _j ∈ U, t _{F(e_i)} (x _j) = f _{F(e_i)} (x _j);

(H3) H (F, E) = H ((F, E) ^c);

(H4) If (F, E) ≤ (G, E), then H (F, E) ≤ H (G, E).

Theorem 3.1. Let (F, E) ∈ VSS (U), H (F, E) is an entropy on VSS (U). If the function f : [0, 1] → [0, 1] be strictly monotone increasing real function, and f (0) =0, f (1) =1, then H′ (F, E) = f (H (F, E)) is also an entropy on VSS (U).

Proof. It is sufficient to prove that H′ (F, E) holds the four conditions of Definition 3.2.

(H1) H′ (F, E) =0⇔ f (H (F, E)) =0 ⇔ H (F, E) =0 ⇔ ∀e _i ∈ E, x _j ∈ U, t _{F(e_i)} (x _j) =0, f _{F(e_i)} (x _j) =1, or t _{F(e_i)} (x _j) =1, f _{F(e_i)} (x _j) =0;

(H2)H′ (F, E) =1⇔ f (H (F, E)) =1 ⇔ H (F, E) =1 ⇔ ∀e _i ∈ E, x _j ∈ U, t _{F(e_i)} (x _j) = f _{F(e_i)} (x _j);

(H3) H′ ((F, E) ^c) = f (H ((F, E) ^c)) = f (H (F, E)) = H′ (F, E);

(H4) If (F, E) ≤ (G, E), then H (F, E) ≤ H (G, E). Since f be strictly monotone increasing real function, so H′ (F, E) = f (H (F, E)) ≤ f (H (G, E)) = H′ (G, E). □

Some functions f : [0, 1] → [0, 1] which are strictly monotone increasing and f (0) =0, f (1) =1 can be given by:

$f (x) = \frac{x}{2 - x}$ ;

$f (x) = \frac{2 x}{1 + x}$ ;

f (x) = xe ^x-1;

f (x) = x ²;

Example 3.1. Let the function f : [0, 1] → [0, 1] be given by $f (x) = \frac{x}{2 - x}$ . Then for any vague soft set (F, E), $H^{'} (F, E) = \frac{H (F, E)}{2 - H (F, E)}$ is an entropy on vague soft sets.

Example 3.2. Let the function f : [0, 1] → [0, 1] be given by f (x) = xe ^x-1. Then for any vague soft set (F, E), $H^{'} (F, E) = H (F, E) e^{H (F, E) - 1}$ is an entropy on vague soft sets.

Similar to the work of Farhadinia [21], we will propose a method which allows us to construct a variety of entropies for vague soft sets using a family of entropies on VSS (U).

Theorem 3.2. Let $H = {H | H$ is the entropy on vague soft sets }. If the function g : [0, 1] ⁿ → [0, 1], a = (a ₁, a ₂, …, a _n) → g (a) satisfies the following conditions:

$g (a) = 0 \Leftrightarrow \frac{1}{n} \sum_{i = 1}^{n} a_{i} = 0$ ;

$g (a) = 1 \Leftrightarrow \frac{1}{n} \sum_{i = 1}^{n} a_{i} = 1$ ;

If $a \leq \tilde{a}$ , that is, $a_{i} \leq \tilde{a_{i}}, 1 \leq i \leq n$ , then $g (a) \leq g (\tilde{a})$ .

Then H _g (F, E) = g (H ₁ (F, E) , H ₂ (F, E) , …, H _n (F, E)),

H_{i} \in H, 1 \leq i \leq n

, is an entropy on vague soft sets.

Proof. It is sufficient to prove that H _g (F, E) holds the four conditions of Definition 3.2.

(H1) From the definition of $H$ and H _g (F, E) together with the condition (1) in Theorem 3.2, we can have

$H_{g} (F, E) = g (H_{1} (F, E), H_{2} (F, E), \dots, H_{n} (F, E)) = 0 \Leftrightarrow \frac{1}{n} \sum_{i = 1}^{n} H_{i} (F, E) = 0 \Leftrightarrow H_{i} (F, E) = 0, 1 \leq i \leq n \Leftrightarrow$ ∀e _i ∈ E, x _j ∈ U, t _{F(e_i)} (x _j) =0, f _{F(e_i)} (x _j) =1, or t _{F(e_i)} (x _j) =1, f _{F(e_i)} (x _j) =0;

(H2) From the definition of $H$ and H _g (F, E) together with the condition (2), we can get

$H_{g} (F, E) = g (H_{1} (F, E), H_{2} (F, E), \dots, H_{n} (F, E)) = 1 \Leftrightarrow \frac{1}{n} \sum_{i = 1}^{n} H_{i} (F, E) = 1 \Leftrightarrow H_{i} (F, E) = 1, 1 \leq i \leq n \Leftrightarrow \forall e_{i} \in E, x_{j} \in U, t_{F (e_{i})} (x_{j}) = f_{F (e_{i})} (x_{j})$ ;

(H3) H _g ((F, E) ^c) = g (H ₁ ((F, E) ^c) , H ₂ ((F, E) ^c) , …, H _n ((F, E) ^c)) = g (H ₁ (F, E) , H ₂ (F, E) , …, H _n (F, E)) = H _g (F, E);

(H4) If (F, E) ≤ (G, E), then H _i (F, E) ≤ H _i $(G, E), \forall H_{i} \in H$ . From the definition of H _g (F, E) together with the condition (3), we can obtain that (F, E) ≤ (G, E) leads to

H _g (F, E) = g (H ₁ (F, E) , H ₂ (F, E) , …, H _n (F, E)) ≤ g (H ₁ (G, E) , H ₂ (G, E) , …, H _n (G, E)) = H _g (G, E).

Some functions g : [0, 1] ⁿ → [0, 1] satisfying the conditions (1)–(3) listed in Theorem 3.2 can be given by:

$g (a) = \frac{1}{n} \sum_{i = 1}^{n} a_{i}$ ;

$g (a) = (\frac{1}{n} \sum_{i = 1}^{n} a_{i}) e^{1 - \frac{1}{n} \sum_{i = 1}^{n} a_{i}}$ ;

$g (a) = (\frac{1}{n} \sum_{i = 1}^{n} a_{i}) \sin (\frac{π}{2 n} \sum_{i = 1}^{n} a_{i})$ ;

$g (a) = \frac{2 \frac{1}{n} \sum_{i = 1}^{n} a_{i}}{1 + \frac{1}{n} \sum_{i = 1}^{n} a_{i}}$ ;

$g (a) = \frac{\frac{1}{n} \sum_{i = 1}^{n} a_{i}}{2 - \frac{1}{n} \sum_{i = 1}^{n} a_{i}}$ .

Example 3.3. Let $H = {H | H$ is the entropy on vague soft sets}, and the function g : [0, 1] ⁿ → [0, 1] be given by $g (a) = \frac{1}{n} \sum_{i = 1}^{n} a_{i} \sin (\frac{π}{2 n} \sum_{i = 1}^{n} a_{i})$ . Then for any vague soft set (F, E), $\begin{matrix} H_{g} (F, E) = \frac{1}{n} \sum_{i = 1}^{n} \\ H_{i} (F, E) \sin (\frac{π}{2 n} \sum_{i = 1}^{n} H_{i} (F, E)), H_{i} \in H \end{matrix}$ is an entropy on vague soft sets.

Example 3.4. Let $H = {H | H$ is the entropy on vague soft sets }, and the function g : [0, 1] ⁿ → [0, 1] be given by $g (a) = \frac{2 \frac{1}{n} \sum_{i = 1}^{n} a_{i}}{1 + \frac{1}{n} \sum_{i = 1}^{n} a_{i}}$ . Then for any vague soft set (F, E), $H_{g} (F, E) = \frac{2 \frac{1}{n} \sum_{i = 1}^{n} H_{i} (F, E)}{1 + \frac{1}{n} \sum_{i = 1}^{n} H_{i} (F, E)}, H_{i} \in H$ is an entropy on vague soft sets.

Theorem 3.3. [29] Let U = {x ₁, x _2, … , x _n} be the universal set of elements and E = {e ₁, e _2, … , e _m} be the universal set of parameters. Hence (F, E) = {F (e _i) |i = 1, 2, …, m} is a family of vague soft sets. Define H (F, E) as follows: $H (F, E) = \frac{1}{m} \sum_{i = 1}^{m} H_{i} (F, E),$ where $H_{i} (F, E) = \frac{1}{n} \sum_{j = 1}^{n} \frac{1 - | t_{F (e_{i})}^{2} (x_{j}) - f_{F (e_{i})}^{2} (x_{j}) |}{1 + | t_{F (e_{i})}^{2} (x_{j}) - f_{F (e_{i})}^{2} (x_{j}) |},$ then H (F, E) is an entropy of vague soft sets.

Example 3.5. Consider the following two vague soft sets (F, E) and (G, E) over U, where U is an initial universe set denoted by U = {x ₁, x ₂} and E is a set of parameters denoted by E = {e ₁}.Let $\begin{matrix} F (e_{1}) = {[0.0, 0.5] / x_{1}, [0.0, 0.0] / x_{2}}, \\ G (e_{1}) = {[0.2, 0.6] / x_{1}, [0.4, 0.4] / x_{2}} . \end{matrix}$ Then we have $\begin{matrix} H (F, E) = 0.3, H (G, E) \approx 0.726 . \end{matrix}$ For a vague soft set(F, E), $π_{F (e)} = \frac{1}{m} \sum_{i = 1}^{m} π_{F (e_{i})}$ is called the hesitancy index of the vague soft set, in which $π_{F (e_{i})} = \frac{1}{n} \sum_{j = 1}^{n} (1 - t_{F (e_{i})} (x_{j}) - f_{F (e_{i})} (x_{j})) .$ Clearly, 0 ≤ π _F(e) ≤ 1 . Then we have $\begin{matrix} π_{F (e)} = 0.25, π_{G (e)} = 0.2 . \end{matrix}$ Hence the entropy of (F, E) is less than the entropy of (G, E), but the hesitancy index of the vague soft set (F, E) is greater than that for (G, E).

Just like fuzzy entropy measure, the entropy of vague soft sets proposed in Theorem 3.3 has the following important properties.

Theorem 3.4. Let (F, E), (G, E) be two vague soft sets in universe U = {x₁, x₂, …, x_n} and if they satisfy that for ∀e_i ∈ E, x_j ∈ U, either t_{F(e_i)} (x_j) ≤ t_{G(e_i)} (x_j), f_{F(e_i)} (x_j) ≥ f_{G(e_i)} (x_j) or t_{F(e_i)} (x_j) ≥ t_{G(e_i)} (x_j), f_{F(e_i)} (x_j) ≤ f_{G(e_i)} (x_j), then we can get $\begin{matrix} H (F, E) + H (G, E) & = & H ((F, E) \cap (G, E)) \\ + H ((F, E) \cup (G, E)) . \end{matrix}$

Proof. First, we can separate U into two parts U ₁ and U ₂, where $\begin{matrix} U_{1} = {x_{j} | t_{F (e_{i})} (x_{j}) \leq t_{G (e_{i})} (x_{j}), \\ f_{F (e_{i})} (x_{j}) \geq f_{G (e_{i})} (x_{j}), \forall e_{i} \in E}, \end{matrix}$ $\begin{matrix} U_{2} & = & {x_{j} | t_{F (e_{i})} (x_{j}) \geq t_{G (e_{i})} (x_{j}), \\ f_{F (e_{i})} (x_{j}) \leq f_{G (e_{i})} (x_{j}), \forall e_{i} \in E} . \end{matrix}$

Let (F, E) ∩ (G, E) = (P, E) and (F, E) ∪ (G, E) = (K, E). So, for all x _j ∈ U ₁, then for ∀e _i ∈ E, t _{F(e_i)} (x _j) ≤ t _{G(e_i)} (x _j) , f _{F(e_i)} (x _j) ≥ f _{G(e_i)} (x _j), i.e. t _{P(e_i)} (x _j) = t _{F(e_i)} (x _j) , f _{P(e_i)} (x _j) = f _{F(e_i)} (x _j), t _{K(e_i)} (x _j) = t _{G(e_i)} (x _j) , f _{K(e_i)} (x _j) = f _{G(e_i)} (x _j).

And for all x _j ∈ U ₂, then for ∀e _i ∈ E, t _{F(e_i)} (x _j) ≥ t _{G(e_i)} (x _j) , f _{F(e_i)} (x _j) ≤ f _{G(e_i)} (x _j), i.e. t _{P(e_i)} (x _j) = t _{G(e_i)} (x _j) , f _{P(e_i)} (x _j) = f _{G(e_i)} (x _j), t _{K(e_i)} (x _j) = t _{F(e_i)} (x _j) , f _{K(e_i)} (x _j) = f _{F(e_i)} (x _j).

Therefore, we can obtain that $\begin{matrix} H_{i} ((F, E) \cap (G, E)) \\ = \frac{1}{n} \sum_{j = 1}^{n} \frac{1 - | t_{P (e_{i})}^{2} (x_{j}) - f_{P (e_{i})}^{2} (x_{j}) |}{1 + | t_{P (e_{i})}^{2} (x_{j}) - f_{P (e_{i})}^{2} (x_{j}) |} \\ = \frac{1}{n} {\sum_{x_{j} \in U_{1}} \frac{1 - | t_{F (e_{i})}^{2} (x_{j}) - f_{F (e_{i})}^{2} (x_{j}) |}{1 + | t_{F (e_{i})}^{2} (x_{j}) - f_{F (e_{i})}^{2} (x_{j}) |} \\ + \sum_{x_{j} \in U_{2}} \frac{1 - | t_{G (e_{i})}^{2} (x_{j}) - f_{G (e_{i})}^{2} (x_{j}) |}{1 + | t_{G (e_{i})}^{2} (x_{j}) - f_{G (e_{i})}^{2} (x_{j}) |}} \end{matrix}$

$\begin{matrix} H_{i} ((F, E) \cup (G, E)) \\ = \frac{1}{n} \sum_{j = 1}^{n} \frac{1 - | t_{K (e_{i})}^{2} (x_{j}) - f_{K (e_{i})}^{2} (x_{j}) |}{1 + | t_{K (e_{i})}^{2} (x_{j}) - f_{K (e_{i})}^{2} (x_{j}) |} \\ = \frac{1}{n} {\sum_{x_{j} \in U_{1}} \frac{1 - | t_{G (e_{i})}^{2} (x_{j}) - f_{G (e_{i})}^{2} (x_{j}) |}{1 + | t_{G (e_{i})}^{2} (x_{j}) - f_{G (e_{i})}^{2} (x_{j}) |} \\ + \sum_{x_{j} \in U_{2}} \frac{1 - | t_{F (e_{i})}^{2} (x_{j}) - f_{F (e_{i})}^{2} (x_{j}) |}{1 + | t_{F (e_{i})}^{2} (x_{j}) - f_{F (e_{i})}^{2} (x_{j}) |}} \end{matrix}$ So, H _i ((F, E) ∩ (G, E)) + H _i ((F, E) ∪ (G, E)) $\begin{matrix} = \frac{1}{n} {\sum_{x_{j} \in U_{1} \cup U_{2} = U} \frac{1 - | t_{F (e_{i})}^{2} (x_{j}) - f_{F (e_{i})}^{2} (x_{j}) |}{1 + | t_{F (e_{i})}^{2} (x_{j}) - f_{F (e_{i})}^{2} (x_{j}) |} \\ + \sum_{x_{j} \in U_{1} \cup U_{2} = U} \frac{1 - | t_{G (e_{i})}^{2} (x_{j}) - f_{G (e_{i})}^{2} (x_{j}) |}{1 + | t_{G (e_{i})}^{2} (x_{j}) - f_{G (e_{i})}^{2} (x_{j}) |}} \\ = H_{i} (F, E) + H_{i} (G, E) \end{matrix}$ Hence, H (F, E) + H (G, E) = H ((F, E) ∩ (G, E)) + H ((F, E) ∪ (G, E)). □

We introduce two new notions for vague soft sets in order to get other properties of the entropy.

Definition 3.3. Let (F, E) be vague soft set, ∀e ∈ E, x ∈ U, t _F(e) (x) ≥ f _F(e) (x), then (F, E) is called a positive definite vague soft set.

Definition 3.4. Let (F, E) be vague soft set, ∀e ∈ E, x ∈ U, t _F(e) (x) ≤ f _F(e) (x), then (F, E) is called a negative definite vague soft set.

Theorem 3.5. Let (F, E) be a positive definite vague soft set, then H ((F, E) ∩ (F, E) ^c) = H ((F, E) ^c), H ((F, E) ∪ (F, E) ^c) = H (F, E).

Proof. If (F, E) be a positive definite vague soft set, then ∀e ∈ E, x ∈ U, t _F(e) (x) ≥ f _F(e) (x). Since t _F^c(¬e) (x) = f _F(e) (x), 1 - f _F^c(¬e) (x) =1 - t _F(e) (x), so t _F(e) (x) ≥ t _F^c(¬e) (x), 1 - f _F(e) (x) ≥1 - f _F^c(¬e) (x). Therefore, we have (F, E) ⊇ (F, E) ^c. Hence H ((F, E) ∩ (F, E) ^c) = H ((F, E) ^c), H ((F, E) ∪ (F, E) ^c) = H (F, E). □

Theorem 3.6. Let (F, E) be a negative definite vague soft set, then H ((F, E) ∩ (F, E) ^c) = H (F, E), H ((F, E) ∪ (F, E) ^c) = H ((F, E) ^c).

Proof. It is similar to the proof of Theorem 3.5. □

Theorem 3.7. Let (F, E) and (G, E) be vague soft sets over U, and (F, E) ⊆ (G, E), then

∀e _i ∈ E, x _j ∈ U, if t _{F(e_i)} (x _j), $t_{G (e_{i})} (x_{j}) \in [\frac{1}{2}, 1]$ , then H (F, E) ≥ H (G, E);

If (F, E) is a positive definite vague soft set, then ∀e ∈ E, x ∈ U, H (F, E) ≥ H (G, E);

If (F, E) is a negative definite vague soft set, then ∀e ∈ E, x ∈ U, H (F, E) ≤ H (G, E);

Proof. Since (F, E) ⊆ (G, E), then ∀e _i ∈ E, x _j ∈ U, t _{F(e_i)} (x _j) ≤ t _{G(e_i)} (x _j) , f _{F(e_i)} (x _j) ≥ f _{G(e_i)} (x _j) , thus $t^{2} F_{(e i)} (x_{j}) - f^{2} F_{(e i)} (x_{j}) \leq t^{2} G_{(e i)} (x_{j}) - f^{2} G_{(e i)} (x_{j}) .$ (1) ∀e _i ∈ E, x _j ∈ U, if t _{F(e_i)} (x _j), $t_{G (e_{i})} (x_{j}) \in [\frac{1}{2}, 1]$ , then , thus $\begin{array}{l} 1 - | t^{2} F_{(e i)} (x_{j}) - f^{2} F_{(e i)} (x_{j}) | \\ \geq 1 - | t^{2} G_{(e i)} (x_{j}) - f^{2} G_{(e i)} (x_{j}) | \\ 1 + | t^{2} F_{(e i)} (x_{j}) - f^{2} F_{(e i)} (x_{j}) | \\ \leq 1 + | t^{2} G_{(e i)} (x_{j}) - f^{2} G_{(e i)} (x_{j}) | . \end{array}$ Therefore $\begin{matrix} \frac{1}{m} \sum_{i = 1}^{m} \frac{1}{n} \sum_{j = 1}^{n} \frac{1 - | t_{F (e_{i})}^{2} (x_{j}) - f_{F (e_{i})}^{2} (x_{j}) |}{1 + | t_{F (e_{i})}^{2} (x_{j}) - f_{F (e_{i})}^{2} (x_{j}) |} \\ \geq \frac{1}{m} \sum_{i = 1}^{m} \frac{1}{n} \sum_{j = 1}^{n} \frac{1 - | t_{G (e_{i})}^{2} (x_{j}) - f_{G (e_{i})}^{2} (x_{j}) |}{1 + | t_{G (e_{i})}^{2} (x_{j}) - f_{G (e_{i})}^{2} (x_{j}) |}, \end{matrix}$ i.e. H (F, E) ≥ H (G, E).

(2) If (F, E) is a positive definite vague soft set,then ∀e _i ∈ E, x _j ∈ U, , thus $\begin{array}{l} 1 - | t^{2} F_{(e i)} (x_{j}) - f^{2} F_{(e i)} (x_{j}) | \\ \geq 1 - | t^{2} G_{(e i)} (x_{j}) - f^{2} G_{(e i)} (x_{j}) | \\ 1 + | t^{2} F_{(e i)} (x_{j}) - f^{2} F_{(e i)} (x_{j}) | \\ \leq 1 + | t^{2} G_{(e i)} (x_{j}) - f^{2} G_{(e i)} (x_{j}) | . \end{array}$ Therefore $\begin{matrix} \frac{1}{m} \sum_{i = 1}^{m} \frac{1}{n} \sum_{j = 1}^{n} \frac{1 - | t_{F (e_{i})}^{2} (x_{j}) - f_{F (e_{i})}^{2} (x_{j}) |}{1 + | t_{F (e_{i})}^{2} (x_{j}) - f_{F (e_{i})}^{2} (x_{j}) |} \\ \geq \frac{1}{m} \sum_{i = 1}^{m} \frac{1}{n} \sum_{j = 1}^{n} \frac{1 - | t_{G (e_{i})}^{2} (x_{j}) - f_{G (e_{i})}^{2} (x_{j}) |}{1 + | t_{G (e_{i})}^{2} (x_{j}) - f_{G (e_{i})}^{2} (x_{j}) |}, \end{matrix}$ i.e. H (F, E) ≥ H (G, E).

(3) It is similar to the proof of (2). □

4 Cross-entropy of vague soft sets

In this section, we focus on investigating the cross-entropy of vague soft sets. Based on the definition of the cross-entropy of vague sets [22], we introduce the concept of the cross-entropy of vague soft sets.

Definition 4.1. Let (F, E) and (G, E) be two vague soft sets in the universe U = {x ₁, x ₂, …, x _n}, E = {e ₁, e _2, … , e _m} be the universal set of parameters. The cross-entropy of vague soft sets (F, E) and (G, E) is defined by $D ((F, E), (G, E)) = \frac{1}{m} \sum_{i = 1}^{m} D_{i} ((F, E), (G, E)),$ where $\begin{matrix} D_{i} ((F, E), (G, E)) \\ = \sum_{j = 1}^{n} {\frac{t_{F (e_{i})} (x_{j}) + 1 - f_{F (e_{i})} (x_{j})}{2} ln \frac{2 [t_{F (e_{i})} (x_{j}) + 1 - f_{F (e_{i})} (x_{j})]}{[t_{F (e_{i})} (x_{j}) + 1 - f_{F (e_{i})} (x_{j})] + [t_{G (e_{i})} (x_{j}) + 1 - f_{G (e_{i})} (x_{j})]} \\ + \frac{1 - t_{F (e_{i})} (x_{j}) + f_{F (e_{i})} (x_{j})}{2} ln \frac{2 [1 - t_{F (e_{i})} (x_{j}) + f_{F (e_{i})} (x_{j})]}{[1 - t_{F (e_{i})} (x_{j}) + f_{F (e_{i})} (x_{j})] + [1 - t_{G (e_{i})} (x_{j}) + f_{G (e_{i})} (x_{j})]}} \end{matrix}$ D ((F, E) , (G, E)) can also be called the divergence measure between vague soft sets (F, E) and (G, E), which indicates the discrimination degree of vague soft set (F, E) from (G, E). It is obvious that D ((F, E) ^c, (G, E) ^c) = D ((F, E) , (G, E)).

However, D ((F, E) , (G, E)) is not symmetric. So, it should be modified to a symmetric form of cross-entropy for vague soft sets as

$\begin{matrix} D^{*} ((F, E), (G, E)) & = & D ((F, E), (G, E)) \\ + D ((G, E), (F, E)) . \end{matrix}$ Similar to the symmetric form of cross-entropy for vague sets, we can obtain that D ^∗ ((F, E) , (G, E)) ≥0, and D ^∗ ((F, E) , (G, E)) =0 when ∀e _i ∈ E, x _j ∈ U, t _{F(e_i)} (x _j) = t _{G(e_i)} (x _j) , f _{F(e_i)} (x _j) = f _{G(e_i)} (x _j).

Theorem 4.1. Let (F, E), (G, E) be two vague soft sets in universe U = {x₁, x₂, …, x_n} and if they satisfy that for ∀e_i ∈ E, x_j ∈ U, either t_{F(e_i)} (x_j) ≤ t_{G(e_i)} (x_j), f_{F(e_i)} (x_j) ≥ f_{G(e_i)} (x_j) or t_{F(e_i)} (x_j) ≥ t_{G(e_i)} (x_j), f_{F(e_i)} (x_j) ≤ f_{G(e_i)} (x_j), then we can get $\begin{matrix} D^{*} ((F, E), (G, E)) = D^{*} ((F, E) \cup (G, E), \\ (F, E) \cap (G, E)) . \end{matrix}$

Proof. It is similar to the proof of Theorem 3.4. □

According to the work of Zhang et al. [22], we can also similarly define another new entropy of vague soft set (F, E) as

$\begin{array}{l} h (F, E) = \frac{1}{m} \sum_{i = 1}^{m} h_{i} (F, E), \\ where \\ h_{i} (F, E) = - \frac{1}{n \ln 2} \sum_{j = 1}^{n} {\frac{t_{F (e_{i})} (x_{j}) + 1 - f_{F (e_{i})} (x_{j})}{2} \ln \frac{t_{F (e_{i})} (x_{j}) + 1 - f_{F (e_{i})} (x_{j})}{2} \\ + \frac{1 - t_{F (e_{i})} (x_{j}) + f_{F (e_{i})} (x_{j})}{2} \ln \frac{1 - t_{F (e_{i})} (x_{j}) + f_{F (e_{i})} (x_{j})}{2}} . \end{array}$

Moreover, the entropy h (F, E) can be deduced by the symmetric form of cross-entropy D ^∗ ((F, E) , (G, E)) of vague soft sets as follows:

Theorem 4.2. If (F, E), (G, E) be two vague soft sets in universe U = {x ₁, x ₂, …, x _n}, then $h (F, E) = 1 - \frac{1}{2 n ln 2} D^{*} ((F, E), (F, E)^{c}),$ where (F, E) ^c is the complement of vague soft set (F, E).

Proof. Since (F, E) ^c is the complement of vague soft set (F, E), then ∀e _i ∈ E, x _j ∈ U, t _{F^c(¬e_i)} (x _j) = f _{F(e_i)} (x _j), f _{F^c(¬e_i)} (x _j) = t _{F(e_i)} (x _j). And according to the cross-entropy formula between two vague soft sets, we have $\begin{matrix} D_{i} ((F, E), (F, E)^{c}) \\ = \sum_{j = 1}^{n} {\frac{t_{F (e_{i})} (x_{j}) + 1 - f_{F (e_{i})} (x_{j})}{2} ln \frac{2 [t_{F (e_{i})} (x_{j}) + 1 - f_{F (e_{i})} (x_{j})]}{[t_{F (e_{i})} (x_{j}) + 1 - f_{F (e_{i})} (x_{j})] + [t_{F^{c} (\neg e_{i})} (x_{j}) + 1 - f_{F^{c} (\neg e_{i})} (x_{j})]} \\ + \frac{1 - t_{F (e_{i})} (x_{j}) + f_{F (e_{i})} (x_{j})}{2} ln \frac{2 [1 - t_{F (e_{i})} (x_{j}) + f_{F (e_{i})} (x_{j})]}{[1 - t_{F (e_{i})} (x_{j}) + f_{F (e_{i})} (x_{j})] + [1 - t_{F^{c} (\neg e_{i})} (x_{j}) + f_{F^{c} (\neg e_{i})} (x_{j})]}} \\ = \sum_{j = 1}^{n} {\frac{t_{F (e_{i})} (x_{j}) + 1 - f_{F (e_{i})} (x_{j})}{2} ln \frac{t_{F (e_{i})} (x_{j}) + 1 - f_{F (e_{i})} (x_{j})}{2} \\ + \frac{1 - t_{F (e_{i})} (x_{j}) + f_{F (e_{i})} (x_{j})}{2} ln \frac{1 - t_{F (e_{i})} (x_{j}) + f_{F (e_{i})} (x_{j})}{2}} + n ln 2 \end{matrix}$ and $\begin{matrix} D_{i} ((F, E)^{c}, (F, E)) \\ = \sum_{j = 1}^{n} {\frac{1 + t_{F (e_{i})} (x_{j}) - f_{F (e_{i})} (x_{j})}{2} \\ ln \frac{1 + t_{F (e_{i})} (x_{j}) - f_{F (e_{i})} (x_{j})}{2} \end{matrix}$

$\begin{matrix} + \frac{1 - t_{F (e_{i})} (x_{j}) + f_{F (e_{i})} (x_{j})}{2} \\ ln \frac{1 - t_{F (e_{i})} (x_{j}) + f_{F (e_{i})} (x_{j})}{2}} + n ln 2 \end{matrix}$ Therefore $\begin{matrix} D_{i}^{*} ((F, E)^{c}, (F, E)) \\ = D_{i} ((F, E), (F, E)^{c}) + D_{i} ((F, E)^{c}, (F, E)) \\ = 2 \sum_{j = 1}^{n} {\frac{1 + t_{F (e_{i})} (x_{j}) - f_{F (e_{i})} (x_{j})}{2} \\ ln \frac{1 + t_{F (e_{i})} (x_{j}) - f_{F (e_{i})} (x_{j})}{2} \\ + \frac{1 - t_{F (e_{i})} (x_{j}) + f_{F (e_{i})} (x_{j})}{2} \\ ln \frac{1 - t_{F (e_{i})} (x_{j}) + f_{F (e_{i})} (x_{j})}{2}} + 2 n ln 2 \end{matrix}$ Thus $h_{i} (F, E) = 1 - \frac{1}{2 n ln 2} D_{i}^{*} ((F, E), (F, E)^{c}) .$ Hence $h (F, E) = 1 - \frac{1}{2 n ln 2} D^{*} ((F, E), (F, E)^{c}) .$ □

5 Applications

Vague soft set is a suitable tool to better model and process imperfect information. In this section, we give some numeric examples to show the applications of the entropy and symmetric cross-entropy of vague soft sets to decision making and pattern recognition problems.

Using the entropy defined in Theorem 3.4, we introduce the following approach to solve decision making problem:

Suppose that there exist p alternatives represented by vague soft sets (F _i, E) , i = 1, 2, …, p.

Compute the entropy H (F _i, E).

Select the smallest one, denoted by H (F _k, E), from H (F _i, E) , i = 1, 2, …, p. Then (F _k, E) is the best choice.

Example 5.1. Let U = {x ₁, x ₂, x ₃, x ₄} be a set of four houses, E = {beautiful, cheap, large} = {e ₁, e ₂, e ₃} be a set of parameters. Suppose that Mr.X, Mr.Y and Mr.Z want to buy a house. The vague soft sets (F _i, E) , i = 1, 2, 3 describe attractiveness of the houses to them, respectively. $\begin{matrix} (F_{1}, E) = \\ {e_{1} = {[0.3, 0.5] / x_{1}, [0.2, 0.6] / x_{2}, [0.5, 0.7] / x_{3}, [0.8, 0.9] / x_{4}}, \\ e_{2} = {[0.4, 0.6] / x_{1}, [0.7, 0.7] / x_{2}, [0.8, 0.9] / x_{3}, [0.1, 0.2] / x_{4}}, \\ e_{3} = {[0.3, 0.4] / x_{1}, [0.2, 0.3] / x_{2}, [0.5, 0.6] / x_{3}, [0.2, 0.4] / x_{4}}}, \end{matrix}$

$\begin{matrix} ((F_{2}, E) = \\ ({e_{1} = {[0.8, 0.9] / x_{1}, [0.4, 0.7] / x_{2}, [0.3, 0.5] / x_{3}, [0.2, 0.6] / x_{4}}, \\ (e_{2} = {[0.1, 0.5] / x_{1}, [0.3, 0.6] / x_{2}, [0.4, 0.9] / x_{3}, [0.3, 0.4] / x_{4}}, \\ (e_{3} = {[0.0, 0.0] / x_{1}, [0.4, 0.5] / x_{2}, [0.6, 0.7] / x_{3}, [0.4, 0.6] / x_{4}}}, \end{matrix}$

$\begin{matrix} (F_{3}, E) = \\ {e_{1} = {[0.4, 0.5] / x_{1}, [0.6, 0.7] / x_{2}, [0.2, 0.5] / x_{3}, [0.1, 0.2] / x_{4}}, \\ e_{2} = {[0.4, 0.8] / x_{1}, [0.5, 0.8] / x_{2}, [0.3, 0.6] / x_{3}, [0.1, 0.3] / x_{4}}, \\ e_{3} = {[0.2, 0.4] / x_{1}, [0.6, 0.7] / x_{2}, [0.4, 0.8] / x_{3}, [0.5, 0.6] / x_{4}}}, \end{matrix}$

As we know, the less uncertainty information each attractiveness has, the larger possibility they will buy a house. Hence, we can rank all buyers according to the entropy values of the corresponding vague soft sets.

By the afore-defined entropy formula of vague soft sets, we can compute

H (F ₁, E) ≈0.584, H (F ₂, E) ≈0.568, H (F ₃, E) ≈0.638.

i.e. H (F ₂, E) < H (F ₁, E) < H (F ₃, E).

Therefore, from the computation above the real estate agency can know Mr.Y has larger possibility to buy a house than Mr.X and Mr.Z.

We apply the symmetric form of cross-entropy of vague soft sets to solve pattern recognition problem:

Step 1: Suppose that there exist p patterns represented by vague soft sets (F _i, E) , i = 1, 2, …, p. And suppose that there is a sample to be recognized, which is represented by a vague soft set (G, E).

Step 2: Compute the symmetric form of cross-entropy D ^∗ ((F _i, E) , (G, E)) between (F _i, E) and (G, E).

Step 3: Select the smallest one, denoted by D ^∗ ((F _k, E) , (G, E)), from D ^∗ ((F _i, E) , (G, E)) , i = 1, 2, …, p. Then (G, E) belongs to the pattern (F _k, E).

Example 5.2. Assume (F _i, E) , i = 1, 2, 3 are given three known patterns, which correspond to classifications C ₁, C ₂ and C ₃, respectively, and an unknown sample (G, E). Suppose we have the following data: $\begin{matrix} (F_{1}, E) = \\ {e_{1} = {[0.1, 0.2] / x_{1}, [0.5, 0.6] / x_{2}, [0.3, 0.7] / x_{3}}, \\ e_{2} = {[0.8, 0.9] / x_{1}, [0.0, 0.1] / x_{2}, [0.4, 0.5] / x_{3}}, \\ e_{3} = {[0.2, 0.5] / x_{1}, [0.3, 0.4] / x_{2}, [0.8, 0.9] / x_{3}}}, \end{matrix}$ $\begin{matrix} (F_{2}, E) = \\ {e_{1} = {[1.0, 1.0] / x_{1}, [0.7, 0.8] / x_{2}, [0.4, 0.7] / x_{3}}, \\ e_{2} = {[0.6, 0.8] / x_{1}, [0.2, 0.7] / x_{2}, [0.9, 1.0] / x_{3}}, \\ e_{3} = {[0.5, 0.6] / x_{1}, [0.2, 0.4] / x_{2}, [0.7, 0.9] / x_{3}}}, \end{matrix}$ $\begin{matrix} (F_{3}, E) = \\ {e_{1} = {[0.4, 0.6] / x_{1}, [0.2, 0.3] / x_{2}, [0.5, 0.7] / x_{3}}, \\ e_{2} = {[0.3, 0.5] / x_{1}, [0.2, 0.6] / x_{2}, [0.4, 0.6] / x_{3}}, \\ e_{3} = {[0.7, 0.9] / x_{1}, [0.0, 0.1] / x_{2}, [0.5, 0.9] / x_{3}}}, \end{matrix}$ $\begin{matrix} (G, E) = \\ {e_{1} = {[0.2, 0.3] / x_{1}, [0.4, 0.7] / x_{2}, [0.5, 0.8] / x_{3}}, \\ e_{2} = {[0.8, 0.9] / x_{1}, [0.4, 0.5] / x_{2}, [0.1, 0.3] / x_{3}}, \\ e_{3} = {[0.4, 0.6] / x_{1}, [0.1, 0.2] / x_{2}, [0.7, 0.8] / x_{3}}}, \end{matrix}$

Our aim is to distinguish which classification the unknown pattern (G, E) belongs to. We can compute the symmetric form of cross-entropy D ^∗ ((F _i, E) , (G, E)) , i = 1, 2, 3 as follows: $\begin{matrix} D^{*} ((F_{1}, E), (G, E)) \approx 0.165, D^{*} ((F_{2}, E), \\ (G, E)) \approx 0.115, D^{*} ((F_{3}, E), (G, E)) \approx 0.197 . \end{matrix}$ Clearly, the D ^∗ ((F ₂, E) , (G, E)) is the smallest one. Hence the pattern (G, E) should be classified to C ₂.

6 Conclusion

Entropy is an important notion for measuring uncertain information. In this paper, we discuss the properties of the entropy of vague soft sets, and propose a new information measure which is cross-entropy. The entropy can measure the uncertainty in a vague soft set, while the cross-entropy can measure the discrimination information between two vague soft sets. Two kinds of information entropy measures can be used in many fields such as decision making, pattern recognition, image processing, approximate reasoning, fuzzy control, and so on. Two numerical examples are also given to illustrate the applications of the entropy and cross-entropy of vague soft sets.

Acknowledgments

The author would like to thank the anonymous referees for their constructive comments as well as helpful suggestions from the associate editor which helped in improving this paper significantly. The works described in this paper are supported by the National Natural Science Foundation of China under Grant nos. 11171271, 11326048; the Postdoctoral Science Foundation of China under Grant nos. 2013M532079, 2014T70932; the Science Research Foundation of Northwest University under Grant no.12NW04.

Conflict of interests

The author declare that there is no conflict of interests regarding the publication of this paper.

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