Comprehensive cross-entropy and entropy measures and their applications under interval type-2 fuzzy environment

Abstract

In recent years, interval type-2 fuzzy set has played an increasingly important role in decision-making. The purpose of this paper is firstly to propose fuzzy factor and interval factor of interval type-2 fuzzy sets (IT2FSs) based on α-cuts. Then, entropy and cross-entropy formulas of IT2FSs are put forward to measure the fuzzy information and mutual information between IT2FSs based on the two factors. Furthermore, IT2FS comprehensive cross-entropy formula is brought forward to measure the relative uncertain relationships among IT2FSs based on IT2FS cross-entropy formula and total probability formula. In the next moment, a utility function of IT2FS is proposed to compare the ordering relationship among IT2FSs. On the other hands, an optimal linear programming model is constructed based on IT2FS entropy, comprehensive cross-entropy and utility function to solve the optimal alternative. Finally, a simple example is utilized to testify the practicability and effectiveness of the proposed approach.

Keywords

IT2FS fuzzy factor IT2FS interval factor cross-entropy comprehensive relative cross-entropy utility function

1. Introduction

Since the fuzzy set theory was put forward by Zadeh [3] in 1965, it has been playing an increasingly important role in many aspects. Until now, fuzzy set theory has been widely used in biomedicine, aeronautics, astronautics, statistics, decision making and other fields. In 1975, Zadeh [1] proposed type-2 fuzzy set theory, it is a type of fuzzy set theory which has a broader application and development. For example, Mendel & John [2] presented a representation theorem and provided convenience for analysis of T2FS. And, several works were invented by Tanaka [8] and Yager [13].

The concept of fuzzy entropy was put forward by Zadeh [4] in 1968, which was a kind of uncertain information measure of IT2FS. It has a very important impact on measuring the uncertainty between IT2FSs. After him, more and more experts have borrowed experience from previous many fuzzy entropy formulas of fuzzy sets were proposed and improved constantly after him. They have been making a great contribution to the uncertainty measurement of fuzzy sets. Such as N. Zhao & Z.S. Xu & F.J. Liu [5] proposed the concept of hesitant fuzzy entropy and cross-entropy; The concept of uncertainty measurement for the interval type-2 fuzzy set was proposed by Greenfield [6]; and Z.S. Xu & M.M. Xia [7] put forward hesitant fuzzy entropy and cross-entropy in multiple attribute decision making. Torra [10] proposed hesitant fuzzy sets to as an intermediate kind of fuzzy set between FSs and T2FSs. Wu and Mendel [12], Zhai and Mendel [14] discussed some uncertainty measures, such as centroid, cardinality, fuzziness, variance and skewness of IT2FSs, respectively, through studying and proposing the new method to measure the uncertainty measures for general type-2 fuzzy sets.

Cross-entropy is a kind of entropy measure that measure the relative uncertain relationship between two T2FSs or IT2FSs. The classical cross-entropy is used to depict the discrimination information between two non-crisp sets. For examples, Shang & Jiang [9], Vlachos & Sergiadis [11] discussed fuzzy cross-entropy and intuitionistic fuzzy cross-entropy and the relationship between cross-entropy and entropy measures. Based on the previous research of fuzziness, hesitancy, interval of T2FS, D.B. Yao proposed a kind of novel fuzzy factors in T2FS to measure the cross-entropy between two T2FSs or IT2FSs [15], who makes contributions to the study of entropy and cross-entropy in T2FS and IT2FS.

In this paper, it is organized as follows: In Section 1, some basic concepts and definitions about type-2 fuzzy sets are introduced. In Section 2, α-cuts is introduced, and the fuzzy factor and interval factor of T2FSs are proposed based on α-cuts. In Section 3, IT2FS cross-entropy and entropy formulas are put forward based on IT2FS fuzzy factor and interval factor, and IT2FS comprehensive cross-entropy formula is proposed based on cross-entropy and total probability formula. In Section 4, utility function of IT2FS is brought forward to rank them. In Section 5, an optimal linear programming model is constructed based on IT2FS entropy, comprehensive cross-entropy and utility function to solve the optimal alternative. In Section 6, a simple example is utilized to testify the feasibility and availability of the proposed approach in this paper. In Section 7, the conclusions arepresented.

2. Preliminaries

In this section, some basic concepts and definitions about type-2 fuzzy sets are discussed.

Definition 2.1. [1] Let X be a universe of discourse, and a type-2 membership function $0 \leq μ_{\tilde{A}} (x, u) \leq 1$ , then a type-2 fuzzy set $\tilde{A}$ can be defined as follows:

$\tilde{A} = {(x, u), μ_{\tilde{A}} (x, u) | \forall x \in X, \forall u \in J_{x} \subseteq [0, 1]} .$ (1)

For any x ∈ X, u ∈ J_x, that J_x presents the main membership of x, and has J_x ∈ [0, 1]; Meanwhile, that $0 \leq μ_{\tilde{A}} (x, u) \leq 1$ represents the secondary membership of x. When the elements of the fuzzy sets are continuous, then the type-2 fuzzy set(T2FS) $\tilde{A}$ can be represented as follows: $\tilde{A} = \int_{x \in X} \int_{u \in J_{x}} \frac{μ_{\tilde{A}} (x, u)}{(x, u)} = \int_{x \in X} (\int_{u \in J_{x}} \frac{μ_{\tilde{A}} (x, u)}{u}) / x$

The figure is shown in Fig. 1.:

Fig. 1.

3-dimensional figure of a T2FS in continuous form.

Definition 2.2. For any type-2 fuzzy set $\tilde{A}$ in a universe of discourse X, if its secondary membership $μ_{\tilde{A}} (x, u) \equiv 1$ , then $\tilde{A}$ is called an interval type-2 fuzzy set (IT2FS), and the expression is shown as follows: $\tilde{A} = \int_{x \in X} \int_{u \in J_{x}} 1 / (x, u) = \int_{x \in X} (\int_{u \in J_{x}} 1 / u) / x$ (2) For discrete spaces, ∫ is replaced by ∑.

Let $\tilde{A} = (A^{U}, A^{L})$ be an IT2FS in the universe of discourse X, then, we have $\tilde{A} = (\begin{matrix} (a_{1}^{U}, a_{2}^{U}, a_{3}^{U}, a_{4}^{U}; H (A^{U})), \\ (a_{1}^{L}, a_{2}^{L}, a_{3}^{L}, a_{4}^{L}; H (A^{L})) \end{matrix})$

And it satisfies $a_{1}^{*} \leq a_{2}^{*} \leq a_{3}^{*} \leq a_{4}^{*}$ , 0 ≤ H (A^k) ≤1, ∗, k present U or L. A^U and A^L all are represented by interval type-1 fuzzy sets, and the 2-dimensional figure is shown in Fig. 2., and meets the relationship: $a_{1}^{U} \leq a_{1}^{L} \leq a_{4}^{L} \leq a_{4}^{U}$ .

Fig. 2.

2-dimensional figure of interval type-2 fuzzy set $\tilde{A}$ .

Definition 2.3. [19] Let X be a universe of discourse, an IT2FS $\tilde{A}$ is completely determined by the union of all the primary memberships which is be called the footprint of uncertainty $(FOU (\tilde{A}))$ , and it can be expressed as follows: $\begin{matrix} F O U (\tilde{A}) = \underset{\forall x \in X}{\cup} J_{x} = {(x, u) : u \in J_{x} \subseteq [0, 1]} \end{matrix}$

Let ${\bar{μ}}_{\tilde{A}} (x)$ and ${\underline{μ}}_{\tilde{A}} (x)$ be the upper membership function (UMF) and the lower membership function (LMF) of $\tilde{A}$ , respectively. Based on the definition of FOU, for any x ∈ X, then we have: $\begin{matrix} {\bar{μ}}_{\tilde{A}} (x) = sup {FOU (\tilde{A})} = \bar{FOU (\tilde{A})} \\ {\underline{μ}}_{\tilde{A}} (x) = inf {FOU (\tilde{A})} = \underline{FOU (\tilde{A})} \end{matrix}$

Therefore, the FOU of IT2FS $\tilde{A}$ can be expressed in the following form: $\begin{matrix} F O U (\tilde{A}) = \underset{\forall x \in X}{\cup} [{\underline{μ}}_{\tilde{A}} (x), {\bar{μ}}_{\tilde{A}} (x)] \end{matrix}$

For an IT2FS $\tilde{A}$ , whose upper membership function (UMF) ${\bar{μ}}_{\tilde{A}} (x)$ and the lower membership function (LMF) ${\underline{μ}}_{\tilde{A}} (x)$ are defined as follows:

${\bar{μ}}_{\tilde{A}} = {\begin{matrix} \frac{(x - a_{1}^{U}) H (A^{U})}{a_{2}^{U} - a_{1}^{U}} a_{1}^{U} \leq x \leq a_{2}^{U} \\ H (A^{U}) a_{2}^{U} \leq x \leq a_{3}^{U} \\ \frac{(x - a_{4}^{U}) H (A^{U})}{a_{3}^{U} - a_{4}^{U}} a_{3}^{U} \leq x \leq a_{4}^{U} \end{matrix}$ (3)

${\underline{μ}}_{\tilde{A}} = {\begin{matrix} \frac{(x - a_{1}^{L}) H (A^{L})}{a_{2}^{L} - a_{1}^{L}} a_{1}^{L} \leq x \leq a_{2}^{L} \\ H (A^{L}) a_{2}^{L} \leq x \leq a_{3}^{L} \\ \frac{(x - a_{4}^{L}) H (A^{L})}{a_{3}^{L} - a_{4}^{L}} a_{3}^{L} \leq x \leq a_{4}^{L} \end{matrix}$ (4)

In general, for all IT2FSs on X, its complement ${\tilde{A}}^{C}$ is defined as follows: $\begin{matrix} \begin{matrix} {\tilde{A}}^{C} & = ((A^{U})^{C}, (A^{L})^{C}) \\ = (\begin{matrix} ((1 - a_{4}^{U}), (1 - a_{3}^{U}), (1 - a_{2}^{U}), (1 - a_{1}^{U}); H (A^{U})), \\ ((1 - a_{4}^{L}), (1 - a_{3}^{L}), (1 - a_{2}^{L}), (1 - a_{1}^{L}); H (A^{L})) \end{matrix}) \end{matrix} \end{matrix}$

3. A definition of α-cuts and its applications

Definition 3.1. For an IT2FS $\tilde{A}$ , the definition of α-cuts can be expressed as follows:

$\begin{matrix} {\tilde{A}}_{α} & = & ({\bar{μ}}_{{\tilde{A}}_{α}} (x), {\underline{μ}}_{{\tilde{A}}_{α}} (x)) \\ = & {x \in X | {\bar{μ}}_{{\tilde{A}}_{α}} (x) \geq α, {\underline{μ}}_{{\tilde{A}}_{α}} (x) \geq α} \end{matrix}$ (5) The figure is shown below in Fig. 3.:

Fig. 3.

A α-cut of interval type-2 fuzzy set $\tilde{A}$ .

For any IT2FS $\tilde{A}$ , $\tilde{A}$ can be expressed by α-cuts as follows: $\tilde{A} = \underset{α \in [0, 1]}{\cup} (α μ_{{\tilde{A}}_{α}})$

Proof. Since for ∀ x ∈ X, $\begin{matrix} \begin{matrix} (⋃_{α \in [0, 1]} α μ_{{\tilde{A}}_{α}}) (x) \\ = \underset{α \in [0, 1]}{⋁} (α \land μ_{{\tilde{A}}_{α}} (x)) \\ = (\underset{α \in [0, μ_{\tilde{A}} (x)]}{⋁} α \land μ_{{\tilde{A}}_{α}} (x)) \land \\ (\underset{α \in [μ_{\tilde{A}} (x), 1]}{⋁} α \land μ_{{\tilde{A}}_{α}} (x)) \\ = \underset{α \leq μ_{\tilde{A}} (x)}{⋁} α = μ_{\tilde{A}} (x) \end{matrix} \end{matrix}$ $μ_{\tilde{A}} (x)$ respects ${\bar{μ}}_{\tilde{A}} (x)$ and ${\underline{μ}}_{\tilde{A}} (x)$ .

The eigenfunction of α-cuts in IT2FS $\tilde{A}$ is defined as follows:

$\begin{matrix} \tilde{A} & = & \int_{x \in X} \int_{u \in J_{x} \geq α} 1 / (x, u) \\ = & \int_{x \in X} (\int_{u \in J_{x} \geq α} 1 / u) / x \end{matrix}$ (6)

Definition 3.2. For any α ∈ [0, 1], and α is continuous, an IT2FS $\tilde{A}$ , we let its lower membership altitude H (A^L) = h^-, upper membership altitude H (A^U) = h⁺, and 0 ≤ h^- ≤ h⁺ ≤ 1, utilize α to cut an IT2FS, when α ∈ [0, h^-], two closed subset interval can be obtained, and be expressed as follows:

$\begin{matrix} {\tilde{A}}_{α}^{l} & = & [inf_{α \in [0, h^{-}]} {\bar{μ}}_{{\tilde{A}}_{α}} (x), inf_{α \in [0, h^{-}]} {\underline{μ}}_{{\tilde{A}}_{α}} (x)] \\ = & [{\tilde{A}}_{(α) +}^{l}, {\tilde{A}}_{(α) -}^{l}] \end{matrix}$ when the closed subset interval is on the left;

$\begin{matrix} {\tilde{A}}_{α}^{r} & = & [sup_{α \in [0, h^{-}]} {\underline{μ}}_{{\tilde{A}}_{α}} (x), sup_{α \in [0, h^{-}]} {\bar{μ}}_{{\tilde{A}}_{α}} (x)] \\ = & [{\tilde{A}}_{(α) +}^{r}, {\tilde{A}}_{(α) -}^{r}] \end{matrix}$ When α ∈ [h^-, h⁺], there is a closed subset interval ${\tilde{A}}_{α} = [{\tilde{A}}_{(α) +}^{l}, {\tilde{A}}_{(α) +}^{r}]$ . ${\tilde{A}}_{(α) +}^{l}, {\tilde{A}}_{(α) -}^{l}, {\tilde{A}}_{(α) -}^{r}, {\tilde{A}}_{(α) +}^{r}$ are obtained points, respectively. ${\begin{matrix} ⋃_{α \in [0, h^{-}]} [[{\tilde{A}}_{(α) +}^{l}, {\tilde{A}}_{(α) -}^{l}], [{\tilde{A}}_{(α) -}^{r}, {\tilde{A}}_{(α) +}^{r}]] \\ ⋃_{α \in [h^{-}, h^{+}]} [[{\tilde{A}}_{(α) +}^{l}, {\tilde{A}}_{(α) +}^{r}]] \\ 0 α \in [h^{-}, h^{+}] \end{matrix}$ (7)

So, for any α ∈ [0, 1], utilize α to cut an IT2FS $\tilde{A}$ , closed subset interval can be obtained as follows: $\begin{matrix} \tilde{A} = \underset{α \in [0, 1]}{\cup} [[{\tilde{A}}_{(α) +}^{l}, {\tilde{A}}_{(α) +}^{r}], [{\tilde{A}}_{(α) -}^{l}, {\tilde{A}}_{(α) -}^{r}]] \end{matrix}$

And the figure is shown in Fig. 4.:

Fig. 4.

Left and right closed interval based on α-cut of anIT2FS $\tilde{A}$ .

Remark. “+” represents the upper bound target of IT2FS, “-” represents the lower bound target of IT2FS; “l” represents the intercepting point on the left, “r” represents the intercepting point on the right.

4. Interval type-2 fuzzy cross-entropy

4.1. The introduction of measure of fuzziness

In this section, IT2FSs fuzzy factor and interval factor are put forward based on the concepts and properties of α-cuts. The value of the membership of IT2FSs is within [0,1], for a classical set $\tilde{A}$ , for any element x, there must have $x \in \tilde{A}$ , or $x \notin \tilde{A}$ , and we can express it by using eigenfunction:

$χ_{\tilde{A}} (x) = {\begin{matrix} 1, x \in \tilde{A} \\ 0, x \notin \tilde{A} \end{matrix}$ (8) $χ_{\tilde{A}} (x)$ is the eigenfunction of $\tilde{A}$ . Take the eigenfunction extend to fuzzy sets, the eigenfunction extend to fuzzy set is interval [0,1]. Before proposing IT2FSs fuzzy factor, we firstly introduce the definition of measure of fuzziness.

Definition 4.1.1. If the map f : F (X) → [0, 1] meets the conditions as follows: (a): Iff(if and only if) $\tilde{A} \in F (x), f (\tilde{A}) = 0$ ; (b): Iff $\tilde{A} \equiv 1 / 2, f (\tilde{A}) = 1$ ; (c): ∀x ∈ X, when $\tilde{B} \leq \tilde{A} \leq 1 / 2, f (\tilde{B}) \leq f (\tilde{A})$ , and when $1 / 2 \leq \tilde{B} \leq \tilde{A} \leq 1, f (\tilde{A}) \leq f (\tilde{B})$ ; (d): $\forall \tilde{A} \in F (X), f (\tilde{A}) = f ({\tilde{A}}^{C})$ .

The map f is said a measure of fuzziness, and $f (\tilde{A})$ is said the measure of fuzziness of fuzzy set $\tilde{A}$ . And the condition (b) and (c) indicate that closer to 0.5, more obscure, and when $\tilde{A} (x) \equiv 0.5$ , it is the most obscure, and ${\tilde{A}}^{C} (x) = 1 - \tilde{A} (x) = 0.5$ .

4.2. The fuzzy factor of IT2FS

Based on the introduction of measure of fuzziness, the IT2FS $\tilde{A}$ fuzzy factor is defined based on α-cuts as follows:

Definition 4.2.1. Let $\tilde{A}$ be an IT2FS, and $\tilde{A} \in IT 2 FSs (X)$ , when α is discrete, and α_i ∈ [0, 1], fuzzy measures of upper bound and lower bound of IT2FS $\tilde{A}$ are defined as follows: $f_{{\tilde{A}}^{U}} = \sqrt{\frac{2}{n}} {(\begin{matrix} \sum_{i = 1}^{n} [\begin{matrix} {| {\bar{μ}}_{{\tilde{A}}_{α_{i}}}^{l} (x) - {\bar{μ}}_{{\tilde{A}}_{(α_{i}) 0.5}}^{l} (x) |}^{2} \\ + {| {\bar{μ}}_{{\tilde{A}}_{α_{i}}}^{r} (x) - {\bar{μ}}_{{\tilde{A}}_{(α_{i}) 0.5}}^{r} (x) |}^{2} \end{matrix}] \end{matrix})}^{\frac{1}{2}}$ (9) $f_{{\tilde{A}}^{L}} = \sqrt{\frac{2}{n}} {(\begin{matrix} \sum_{i = 1}^{n} [\begin{matrix} {| {\underline{μ}}_{{\tilde{A}}_{α_{i}}}^{l} (x) - {\underline{μ}}_{{\tilde{A}}_{(α_{i}) 0.5}}^{l} (x) |}^{2} \\ + {| {\underline{μ}}_{{\tilde{A}}_{α_{i}}}^{r} (x) - {\underline{μ}}_{{\tilde{A}}_{(α_{i}) 0.5}}^{r} (x) |}^{2} \end{matrix}] \end{matrix})}^{\frac{1}{2}}$ (10)

When α ∈ [0, 1], and α is continuous, fuzzy measures of upper bound and lower bound of IT2FS $\tilde{A}$ are defined as follows: $f_{{\tilde{A}}^{U}} = \sqrt{\frac{2}{h^{+}}} {(\begin{matrix} \int_{0}^{h^{+}} [\begin{matrix} {| {\bar{μ}}_{{\tilde{A}}_{α_{i}}}^{l} (x) - {\bar{μ}}_{{\tilde{A}}_{(α_{i}) 0.5}}^{l} (x) |}^{2} \\ + {| {\bar{μ}}_{{\tilde{A}}_{α_{i}}}^{r} (x) - {\bar{μ}}_{{\tilde{A}}_{(α_{i}) 0.5}}^{r} (x) |}^{2} \end{matrix}] \end{matrix} d α)}^{\frac{1}{2}}$ (11) $f_{{\tilde{A}}^{L}} = \sqrt{\frac{2}{h^{-}}} {(\begin{matrix} \int_{0}^{h^{-}} [\begin{matrix} {| {\underline{μ}}_{{\tilde{A}}_{α_{i}}}^{l} (x) - {\underline{μ}}_{{\tilde{A}}_{(α_{i}) 0.5}}^{l} (x) |}^{2} \\ + {| {\underline{μ}}_{{\tilde{A}}_{α_{i}}}^{r} (x) - {\underline{μ}}_{{\tilde{A}}_{(α_{i}) 0.5}}^{r} (x) |}^{2} \end{matrix}] \end{matrix} d α)}^{\frac{1}{2}}$ (12)

Based on the definition of fuzzy measures of upper bound and lower bound of IT2FS $\tilde{A}$ , the fuzzy factor of IT2FS $\tilde{A}$ can be defineed as follows:

$f (\tilde{A}) = \sqrt{\frac{(f_{{\tilde{A}}^{U}})^{2} + (f_{{\tilde{A}}^{L}})^{2}}{2}}$ (13) $f (\tilde{A})$ is the fuzzy factor of IT2FS $\tilde{A}$ , $f_{{\tilde{A}}^{U}}$ and $f_{{\tilde{A}}^{L}}$ are fuzzy measures of upper bound and lower bound of IT2FS $\tilde{A}$ , respectively. When α_i ∈ [0, 1] is discrete, we have: $\begin{matrix} μ_{{\tilde{A}}_{(α_{i}) 0.5}} (x) = {\begin{matrix} 1, 0.5 \leq α_{i} \leq 1 \\ 0, 0 \leq α_{i} < 0.5 \end{matrix} \end{matrix}$

When α ∈ [0, 1], and α is continuous, we have: $\begin{matrix} μ_{{\tilde{A}}_{(α) 0.5}} (x) = {\begin{matrix} 1, 0.5 \leq α \leq 1 \\ 0, 0 \leq α < 0.5 \end{matrix} \end{matrix}$

The figure of the relationships between α-cuts and membership of upper and lower bounds in an IT2FS $\tilde{A}$ is shown in Fig. 5.:

Fig. 5.

The relationships between α-cuts and membership of upper and lower bounds in an IT2FS $\tilde{A}$ .

4.3. The interval factor of IT2FS

Definition 4.3.1. Let $\tilde{A}$ be an IT2FS, and $\tilde{A} \in IT 2 FSs (X)$ , when α is discrete, and α_i ∈ [0, 1], the interval factor of IT2FS $\tilde{A}$ can be defined as follows:

$φ_{\tilde{A}} = \frac{1}{\sqrt{2 n}} {(\begin{matrix} \sum_{i = 1}^{n} [\begin{matrix} {| {\tilde{A}}_{(α_{i}) +}^{l} - {\tilde{A}}_{(α_{i}) -}^{l} |}^{2} \\ + {| {\tilde{A}}_{(α_{i}) +}^{r} - {\tilde{A}}_{(α_{i}) -}^{r} |}^{2} \end{matrix}] \end{matrix})}^{\frac{1}{2}}$ (14)

When α_i ∈ [0, 1], and α is continuous:

$φ_{\tilde{A}} = \frac{1}{\sqrt{2 h^{+}}} {(\begin{matrix} \int_{0}^{h^{+}} [\begin{matrix} {| {\tilde{A}}_{(α) +}^{l} - {\tilde{A}}_{(α) -}^{l} |}^{2} \\ + {| {\tilde{A}}_{(α) +}^{r} - {\tilde{A}}_{(α) -}^{r} |}^{2} \end{matrix}] \end{matrix})}^{\frac{1}{2}}$ (15) $φ_{\tilde{A}}$ is the interval factor of IT2FS $\tilde{A}$ .

And the figure can be shown in Fig. 6.:

Fig. 6.

The interval difference between the upper and lower bounds in an IT2FS $\tilde{A}$ .

4.4. Cross-entropy formula of IT2FS

Now, we utilize the fuzzy factor and interval factor of IT2FS $\tilde{A}$ to develop an analytical framework to measure the discrimination degree of uncertain information between two IT2FSs based on two-tuple (f, φ).

Definition 4.4.1. Let $\tilde{A}, \tilde{B} \in IT 2 FSs (X)$ , then the IT2FSs relative cross-entropy of $\tilde{A}$ against $\tilde{B}$ is defined as follows: $\begin{matrix} RE (\tilde{A}, \tilde{B}) = \frac{- 1}{2 ln 2} (f_{\tilde{A}} ln \frac{f_{\tilde{A}}}{(f_{\tilde{A}} + f_{\tilde{B}})} + φ_{\tilde{A}} ln \frac{φ_{\tilde{A}}}{(φ_{\tilde{A}} + φ_{\tilde{B}})}) \end{matrix}$ (16)

Proposition 4.4.1. $RE (\tilde{A}, \tilde{B}) = RE ({\tilde{A}}^{C}, \tilde{B}) = RE (\tilde{A}, {\tilde{B}}^{C}) = RE ({\tilde{A}}^{C}, {\tilde{B}}^{C}) .$

Proof. According to the above formula, we can easily prove that $f_{\tilde{A}}, φ_{\tilde{A}} \in [0, 1]$ , so, we can obtain that $f_{{\tilde{A}}^{C}}, φ_{{\tilde{A}}^{C}} \in [0, 1]$ . Furthermore, we can also get $f_{\tilde{A}} = f_{{\tilde{A}}^{C}}$ , $φ_{\tilde{A}} = φ_{{\tilde{A}}^{C}}$ . Similarly, $f_{\tilde{B}} = f_{{\tilde{B}}^{C}}$ , $φ_{\tilde{B}} = φ_{{\tilde{B}}^{C}}$ . So, we can obtain that $RE (\tilde{A}, \tilde{B}) = RE ({\tilde{A}}^{C}, \tilde{B})$ , $RE ({\tilde{A}}^{C}, \tilde{B}) = RE (\tilde{A}, {\tilde{B}}^{C})$ and $RE (\tilde{A}, {\tilde{B}}^{C}) = RE ({\tilde{A}}^{C}, {\tilde{B}}^{C})$ , therefore, an equation relation is obtained: $RE (\tilde{A}, \tilde{B}) = RE ({\tilde{A}}^{C}, \tilde{B}) = RE (\tilde{A}, {\tilde{B}}^{C}) = RE ({\tilde{A}}^{C}, {\tilde{B}}^{C}) .$ Proposition 4.4.2. F (f, φ) is a real valued continuous function, and F will increase with the increase of $f_{\tilde{A}}$ , $φ_{\tilde{A}}$ , $f_{\tilde{B}}$ and $φ_{\tilde{B}}$ .

Proof. Since the two parts of $RE (\tilde{A}, \tilde{B})$ have same form, so, they have same monotonicity. Hence, just prove one of them. Its monotonicity is proved to be equivalent to the monotonicity of the follow functional equation: F (x₁, x₂) = - x₁ ln(x₁/(x₁ + x₂)). Where x₁, x₂ ∈ [0, 1], therefore, partial derivatives are shown as follows: $\begin{matrix} {\begin{matrix} \frac{\partial F}{\partial x_{1}} = & ln (\frac{x_{1} + x_{2}}{x_{1}}) - \frac{x_{2}}{x_{1} + x_{2}} \\ \frac{\partial F}{\partial x_{2}} = & \frac{x_{1}}{x_{1} + x_{2}} \end{matrix} \end{matrix}$

Since x₁, x₂ ∈ [0, 1], so, we can obtain that $\frac{\partial F}{\partial x_{2}} > 0$ , F will increase with the increase of x₂. Since the monotonicity of partial derivative of the function F with respect to x₁ cannot be proved, so, we have to take the second partial derivative about x₁:

$\frac{\partial^{2} \begin{matrix} F \end{matrix}}{\partial x_{1}^{2}} = \frac{1}{x_{1} + x_{2}} - \frac{1}{x_{1}} + \frac{x_{2}}{{(x_{1} + x_{2})}^{2}} = - \frac{x_{2}^{2}}{x_{1} {(x_{1} + x_{2})}^{2}} \leq 0$ Since F′ (x₁) _min = f′ (1) ≥ 0 . So, F with respect to x₁ is monotonically increasing. Therefore, we can come to the conclusion that: F will increase with the increases of $f_{\tilde{A}}$ , similarly, F will increase with the increase of $f_{\tilde{A}}$ and $φ_{\tilde{A}}$ , $f_{\tilde{B}}$ and $φ_{\tilde{B}}$ . So, F increases with the increases of $f_{\tilde{A}}$ , $φ_{\tilde{A}}$ , $f_{\tilde{B}}$ and $φ_{\tilde{B}}$ .

Proposition 4.4.3. For any $\tilde{A}, \tilde{B} \in IT 2 FSs (X)$ , $0 \leq RE (\tilde{A}, \tilde{B}) \leq 1$ .

Proof. Through the Proposition 4.4.2, we can obtain that for any two IT2FSs $\tilde{A}, \tilde{B} \in IT 2 FSs (X)$ , the IT2FSs relative cross-entropy of $\tilde{A}$ against $\tilde{B}$ $RE (\tilde{A}, \tilde{B})$ increases with the increases of $f_{\tilde{A}}$ , $φ_{\tilde{A}}$ , $f_{\tilde{B}}$ and $φ_{\tilde{B}}$ . So, $RE (\tilde{A}, \tilde{B})_{\min} = 0$ , $RE (\tilde{A}, \tilde{B})_{\max} = RE (1, 1) = - \frac{1}{2 ln 2} (ln (1 / 2) + ln (1 / 2)) = \frac{2 ln 2}{2 ln 2} = 1$ . So, for any $\tilde{A}, \tilde{B} \in IT 2 FSs (X)$ , $0 \leq RE (\tilde{A}, \tilde{B}) \leq 1$ .

5. Comprehensive cross-entropy based on total probability formula

In Multi-Attribute Decision Making (MADM) problems under interval type-2 fuzzy environment, suppose that their have n alternatives A_i (1 ≤ i ≤ n), m attributes C_j (1 ≤ j ≤ m), the attribute weight vector ω = (ω₁, ω₂, ·· · , ω_m), and $\sum_{j = 1}^{m} ω_{j} = 1, ω_{j} \geq 0$ . In multi-attribute decision making, we should consider not only the merits and demerits of the fuzzy set, but also the relative uncertainty measures between the fuzzy sets. Based on this theory, in this section, a kind of comprehensive cross-entropy theory under a certain attribute in IT2FSs is put forward. An attribute C_j in multi-attribute decision making, the cross-entropy that a_ij corresponds to a_kj (i ≠ k) is RE (a_ij, a_kj), an axiomatic criterion based on the smaller the better of relative cross-entropy, under an attribute C_j. If the cross-entropy that a_ij corresponds to a_kj RE (a_ij, a_kj) (i ≠ k) is smaller, it should has greater weight. And we all know that in Multiple Attribute Decision Making (MADM) problems, the influence is mutual, so that is in an attribute C_j, two evaluates a_ij has influence on a_kj, likewise, a_kj also influence on a_ij (i ≠ k). We suppose that there are four evaluation values of four alternatives under attribute C_j {a_ij, a_kj, a_lj, a_nj}, so the figure of the mutual influence of them can be shown as Fig. 7.:

Fig. 7.

The relative cross-entropy relationship of {a_ij, a_kj, a_lj, a_nj} under attribute C_j.

Next, introduce an algorithm that use the total probability formula to calculate the corresponding weights. Firstly, the decision matrix is unified dimension, and then the cost attributes are transformed into the benefit attributes, and the normalized matrix is obtained. Then assume under anattribute C_j.

Let q_(i,k) (i ≠ k) on behalf of probability of relative cross-entropy RE (a_ij, a_kj) respects to $\sum_{k = 1, k \neq i}^{n} R E (a_{i j}, a_{k j})$ under an attribute C_j, so, one equation is obtained:

$q_{(i, k)} = \frac{\sqrt{1 / R E (a_{i j}, a_{k j})}}{\sum_{k = 1, k \neq i}^{n} (R E (a_{i j}, a_{k j}))}$ (17)

Similarly,

$q_{(k, i)} = \frac{\sqrt{1 / R E (a_{k j}, a_{i j})}}{\sum_{i = 1, i \neq k}^{n} (R E (a_{k j}, a_{i j}))}$ (18)

In summary, we get the expression of comprehensive cross-entropy CE (a_ij) of IT2FS a_ij under attribute C_j:

$C E (a_{i j}) = \frac{1}{2} [\begin{matrix} \sum_{k = 1, k \neq i}^{n} R E (a_{i j}, a_{k j}) \times q_{(i, k)} \\ + \sum_{i = 1, i \neq k}^{n} R E (a_{k j}, a_{i j}) \times q_{(k, i)} \end{matrix}]$ (19)

6. IT2FS entropy based on fuzzy factor and interval factor

In Section 4, we have proposed fuzzy factor and interval factor of IT2FSs, and according to the Definition 4.2 and 4.3. For any IT2FS, when the fuzzy factor and interval factor are greater, the IT2FS will be more obscure. In this section, based on IT2FSs fuzzy factor and interval factor, IT2FSs entropy formula is proposed as follows: $CE (\tilde{A}) = \frac{1}{2 ln 2} (f_{\tilde{A}} ln (1 + f_{\tilde{A}}) + φ_{\tilde{A}} ln (1 + φ_{\tilde{A}}))$ (20)

Proposition 6.1. For any IT2FS $\tilde{A}$ , $\tilde{A} \in IT 2 FSs (X)$ , $E (\tilde{A}) = E ({\tilde{A}}^{C})$ .

Proof. We have proofed that for any $\tilde{A} \in IT 2 FSs (X)$ , $f_{\tilde{A}}$ is the fuzzy factor of IT2FS $\tilde{A}$ , $φ_{\tilde{A}}$ is the interval factor of IT2FS $\tilde{A}$ ,and $f_{\tilde{A}} = f_{\tilde{A^{C}}} \in [0, 1]$ , $φ_{\tilde{A}} = φ_{\tilde{A^{C}}} \in [0, 1]$ , so, $\begin{matrix} E (\tilde{A}) & = & \frac{1}{2 ln 2} (f_{\tilde{A}} ln (1 + f_{\tilde{A}}) + φ_{\tilde{A}} ln (1 + φ_{\tilde{A}})) \\ = & \frac{1}{2 ln 2} (f_{\tilde{A^{C}}} ln (1 + f_{\tilde{A^{C}}}) + φ_{\tilde{A^{C}}} ln (1 + φ_{\tilde{A^{C}}})) \\ = & E (\tilde{A^{C}}) . \end{matrix}$ Proposition 6.2. For any IT2FS $\tilde{A}$ , $\tilde{A} \in IT 2 FSs (X)$ , $E (\tilde{A})$ increases with the increase of $f_{\tilde{A}}$ and $φ_{\tilde{A}}$ .

Proof. Since that $f_{\tilde{A}} \in [0, 1]$ , $φ_{\tilde{A}} \in [0, 1]$ , we assume that $x = f_{\tilde{A}} \in [0, 1]$ , $y = φ_{\tilde{A}} \in [0, 1]$ , $E (\tilde{A}) = z (x, y)$ . So, $E (\tilde{A}) = z (x, y) = \frac{1}{2 ln 2} (x ln (1 + x) + y ln (1 + y))$ . Therefore, partial derivatives are shown as follows: $\begin{matrix} {\begin{matrix} \frac{\partial z}{\partial x} = \frac{1}{2 ln 2} (ln (1 + x) + \frac{x}{1 + x}) \geq 0; \\ \frac{\partial z}{\partial y} = \frac{1}{2 ln 2} (ln (1 + y) + \frac{y}{1 + y}) \geq 0 . \end{matrix} \end{matrix}$ So, $E (\tilde{A})$ increases with the increase of $f_{\tilde{A}}$ and $φ_{\tilde{A}}$ .

Proposition 6.3. For any IT2FS $\tilde{A}$ , $E (\tilde{A}) \in [0, 1] .$

Proof. Since that $f_{\tilde{A}} \in [0, 1]$ , $φ_{\tilde{A}} \in [0, 1]$ , and $E (\tilde{A})$ increases with the increase of $f_{\tilde{A}}$ and $φ_{\tilde{A}}$ . So, when $f_{\tilde{A}} = 0, φ_{\tilde{A}} = 0$ , the entropy of IT2FS $\tilde{A}$ $E (\tilde{A})$ is minimum. $E (\tilde{A})_{\min} = E (0, 0) = 0$ . Similarly, when $f_{\tilde{A}} = φ_{\tilde{A}} = 1$ , $E (\tilde{A})$ is maximum, $E (\tilde{A})_{\max} = \frac{1}{2 ln 2} (ln 2 + ln 2) = \frac{2 ln 2}{2 ln 2} = 1 .$ So, for any IT2FS $\tilde{A}$ , $E (\tilde{A}) \in [0, 1] .$

7. IT2FS utility function

In this section, a utility function of IT2FS is put forward to measure the superiority-inferiority among IT2FSs based on fuzzy-cut.

Definition 7.1. Let $\tilde{A}$ be an IT2FS, utility function of IT2FS $\tilde{A}$ can be defined based on λ-cuts and σ-cuts as follows: $\begin{matrix} U (\tilde{A}) & = & \int_{0}^{h^{+}} \int_{a_{1}^{U}}^{a_{4}^{U}} (\begin{matrix} \frac{{\tilde{A}}_{(λ) l}^{U} + {\tilde{A}}_{(λ) r}^{U}}{2} \\ + x ({\tilde{A}}_{(λ) r}^{U} - {\tilde{A}}_{(λ) l}^{U}) \end{matrix}) {\bar{μ}}_{\tilde{A}} (x) dxd λ \\ + & \int_{0}^{h^{-}} \int_{a_{1}^{L}}^{a_{4}^{L}} (\begin{matrix} \frac{{\tilde{A}}_{(σ) l}^{L} + {\tilde{A}}_{(σ) r}^{L}}{2} \\ + y ({\tilde{A}}_{(σ) r}^{L} - {\tilde{A}}_{(σ) l}^{L}) \end{matrix}) {\underline{μ}}_{\tilde{A}} (y) dyd σ \end{matrix}$ (21)

“l” represents the intercepting point on the left, “r” represents the intercepting point on the right. ${\tilde{A}}_{(λ) l}^{U} = \frac{λ (a_{2}^{U} - a_{1}^{U})}{h^{+}} + a_{1}^{U}$ , ${\tilde{A}}_{(λ) r}^{U} = a_{4}^{U} - \frac{λ (a_{4}^{U} - a_{3}^{U})}{h^{+}}$ , ${\tilde{A}}_{(σ) l}^{L} = \frac{σ (a_{2}^{L} - a_{1}^{L})}{h^{-}} + a_{1}^{L}$ , ${\tilde{A}}_{(σ) r}^{L} = a_{4}^{L} - \frac{σ (a_{4}^{L} - a_{3}^{L})}{h^{-}}$ , $\frac{{\tilde{A}}_{(λ) l}^{U} + {\tilde{A}}_{(λ) r}^{U}}{2} = \frac{λ (a_{2}^{U} + a_{3}^{U} - a_{1}^{U} - a_{4}^{U})}{2 h^{+}} + \frac{a_{1}^{U} + a_{4}^{U}}{2},$ $\frac{{\tilde{A}}_{(σ) l}^{U} + {\tilde{A}}_{(σ) r}^{U}}{2} = \frac{σ (a_{2}^{L} + a_{3}^{L} - a_{1}^{L} - a_{4}^{L})}{2 h^{-}} + \frac{a_{1}^{L} + a_{4}^{L}}{2},$ ${\tilde{A}}_{(λ) r}^{U} - {\tilde{A}}_{(λ) l}^{U} = \frac{λ (a_{2}^{U} + a_{3}^{U} - a_{1}^{U} - a_{4}^{U})}{h^{+}} + (a_{1}^{U} + a_{3}^{U} - a_{2}^{U} - a_{4}^{U}),$ ${\tilde{A}}_{(σ) r}^{L} - {\tilde{A}}_{(σ) l}^{L} = \frac{σ (a_{2}^{L} + a_{3}^{L} - a_{1}^{L} - a_{4}^{L})}{h^{-}} + (a_{1}^{L} + a_{3}^{L} - a_{2}^{L} - a_{4}^{L}) .$

So, the utility function of the IT2FS $\tilde{A}$ can be calculated as follows: $\begin{matrix} U (\tilde{A}) & = & \int_{0}^{h^{+}} \int_{a_{1}^{U}}^{a_{4}^{U}} (\begin{matrix} \frac{{\tilde{A}}_{(λ) l}^{U} + {\tilde{A}}_{(λ) r}^{U}}{2} \\ + x ({\tilde{A}}_{(λ) r}^{U} - {\tilde{A}}_{(λ) l}^{U}) \end{matrix}) {\bar{μ}}_{\tilde{A}} (x) dxd λ \\ + & \int_{0}^{h^{-}} \int_{a_{1}^{L}}^{a_{4}^{L}} (\begin{matrix} \frac{{\tilde{A}}_{(σ) l}^{L} + {\tilde{A}}_{(σ) r}^{L}}{2} \\ + y ({\tilde{A}}_{(σ) r}^{L} - {\tilde{A}}_{(σ) l}^{L}) \end{matrix}) {\underline{μ}}_{\tilde{A}} (y) dyd σ \\ = & \frac{h^{+}}{2} \int_{a_{1}^{U}}^{a_{4}^{U}} (\begin{matrix} \frac{(a_{1}^{U} + a_{2}^{U} + a_{3}^{U} + a_{4}^{U})}{2} \\ + x (a_{3}^{U} + a_{4}^{U} - a_{1}^{U} - a_{2}^{U}) \end{matrix}) {\bar{μ}}_{\tilde{A}} (x) dx \\ + & \frac{h^{-}}{2} \int_{a_{1}^{L}}^{a_{4}^{L}} (\begin{matrix} \frac{(a_{1}^{L} + a_{2}^{L} + a_{3}^{L} + a_{4}^{L})}{2} \\ + y (a_{3}^{L} + a_{4}^{L} - a_{1}^{L} - a_{2}^{L}) \end{matrix}) {\underline{μ}}_{\tilde{A}} (y) dy . \end{matrix}$ We assume that $K = \frac{a_{1}^{U} + a_{2}^{U} + a_{3}^{U} + a_{4}^{U}}{4},$ $P = \frac{a_{3}^{U} + a_{4}^{U} - a_{1}^{U} - a_{2}^{U}}{2},$ $M = \frac{a_{1}^{L} + a_{2}^{L} + a_{3}^{L} + a_{4}^{L}}{4},$ $N = \frac{a_{3}^{L} + a_{4}^{L} - a_{1}^{L} - a_{2}^{L}}{2}$ .

So, the formula can be reduced as follows: $\begin{matrix} U (\tilde{A}) & = & \int_{0}^{h^{+}} \int_{a_{1}^{U}}^{a_{4}^{U}} (\begin{matrix} \frac{{\tilde{A}}_{(λ) l}^{U} + {\tilde{A}}_{(λ) r}^{U}}{2} \\ + x ({\tilde{A}}_{(λ) r}^{U} - {\tilde{A}}_{(λ) l}^{U}) \end{matrix}) {\bar{μ}}_{\tilde{A}} (x) dxd λ \\ + & \int_{0}^{h^{-}} \int_{a_{1}^{L}}^{a_{4}^{L}} (\begin{matrix} \frac{{\tilde{A}}_{(σ) l}^{L} + {\tilde{A}}_{(σ) r}^{L}}{2} \\ + y ({\tilde{A}}_{(σ) r}^{L} - {\tilde{A}}_{(σ) l}^{L}) \end{matrix}) {\underline{μ}}_{\tilde{A}} (y) dyd σ \\ = & \frac{h^{+}}{2} \int_{a_{1}^{U}}^{a_{4}^{U}} (\begin{matrix} \frac{(a_{1}^{U} + a_{2}^{U} + a_{3}^{U} + a_{4}^{U})}{2} \\ + x (a_{3}^{U} + a_{4}^{U} - a_{1}^{U} - a_{2}^{U}) \end{matrix}) {\bar{μ}}_{\tilde{A}} (x) dx \\ + & \frac{h^{-}}{2} \int_{a_{1}^{L}}^{a_{4}^{L}} (\begin{matrix} \frac{(a_{1}^{L} + a_{2}^{L} + a_{3}^{L} + a_{4}^{L})}{2} \\ + y (a_{3}^{L} + a_{4}^{L} - a_{1}^{L} - a_{2}^{L}) \end{matrix}) {\underline{μ}}_{\tilde{A}} (y) dy \\ = & \frac{(a_{1}^{U} + a_{2}^{U} + a_{3}^{U} + a_{4}^{U}) h^{+}}{4} \int_{a_{1}^{U}}^{a_{4}^{U}} {\bar{μ}}_{\tilde{A}} (x) dx \\ + & \frac{(a_{3}^{U} + a_{4}^{U} - a_{1}^{U} - a_{2}^{U}) h^{+}}{2} \int_{a_{1}^{U}}^{a_{4}^{U}} x {\bar{μ}}_{\tilde{A}} (x) dx \\ + & \frac{(a_{1}^{L} + a_{2}^{L} + a_{3}^{L} + a_{4}^{L}) h^{-}}{4} \int_{a_{1}^{L}}^{a_{4}^{L}} {\underline{μ}}_{\tilde{A}} (y) dy \\ + & \frac{(a_{3}^{L} + a_{4}^{L} - a_{1}^{L} - a_{2}^{L}) h^{-}}{2} \int_{a_{1}^{L}}^{a_{4}^{L}} y {\underline{μ}}_{\tilde{A}} (y) dy \\ = & {Kh}^{+} \int_{a_{1}^{U}}^{a_{4}^{U}} {\bar{μ}}_{\tilde{A}} (x) dx + {Ph}^{+} \int_{a_{1}^{U}}^{a_{4}^{U}} x {\bar{μ}}_{\tilde{A}} (x) dx \\ + & {Mh}^{-} \int_{a_{1}^{L}}^{a_{4}^{L}} {\underline{μ}}_{\tilde{A}} (y) dy + {Nh}^{-} \int_{a_{1}^{L}}^{a_{4}^{L}} y {\underline{μ}}_{\tilde{A}} (y) dy . \end{matrix}$ (22)

The figure of upper and lower bounds fuzzy-cuts of IT2FS $\tilde{A}$ are shown in Fig. 8. and Fig. 9.:

Fig. 8.

The upper bound fuzzy-cut of IT2FS $\tilde{A}$ .

Fig. 9.

The lower bound fuzzy-cut of IT2FS $\tilde{A}$ .

8. A novel multi-attribute decision-making approach

8.1. Analysis of decision-making

Since many MADM approaches are put forward in IT2FSs, a decision-making approach is developed by using of the utility function, comprehensive cross-entropy and entropy measures in interval type-2 fuzzy information environment, which proposed in this paper. And we only consider the case that the attribute weight is complete unknown. We suppose that there are n alternatives A_i (1 ≤ i ≤ n), m attributes C_j (1 ≤ j ≤ m), and the attribute weight vector ω = {ω₁, ω₂, ⋯ , ω_m} and for any ω_j that $ω_{j} \in [0, 1], \sum_{j = 1}^{m} ω_{j} = 1$ . All evaluation information are presented by IT2FSs, and they constitute a decision-making matrix A = (a_ij) _n×m, where a_ij is described by IT2FS, which is the evaluation of estimate by the decision maker for the alternative x_i with respect to attribute C_j. The attribute type is generally divided into two types: benefit and cost, so, when the attribute type is not same, then normalize original decision matrix A = (a_ij) _n×m, obtain normalized decision matrix D = (d_ij) _n×m. Where d_ij is the normalized form of a_ij:

$d_{ij} = {\begin{matrix} a_{ij} if C_{j} is a benefit attribute \\ (a_{ij})^{C} if C_{j} is a cost attribute \end{matrix}$ (23)

8.2. Construction of the optimal model

At the front of this paper, IT2FSs entropy and cross-entropy formulas are proposed to measure the uncertain information of IT2FSs. Among, the IT2FSs entropy formula is used to measure the uncertainty information of IT2FSs, and the cross-entropy formula is used to measure the uncertain information and mutual influence relationship between IT2FSs. And the entropy is smaller, that the uncertain information of IT2FS is less, and it will be more stable. Similarly, the cross-entropy is smaller, that the mutual influence relationship between IT2FSs is less, which means that the mutual influence relationship between alternatives are less, decision system will be more stable. So, based on above analysis, we construct an optimal decision model based on utility function, comprehensive cross-entropy and entropy.

${\begin{matrix} {m a x R}_{i} (ω) = \sum_{j = 1}^{m} (\frac{U (d_{i j})}{C E (d_{i j}) + E (d_{i j})}) ω_{j} \\ ω_{j} \geq 0, \sum_{j = 1}^{m} ω_{j} = 1, j = 1, 2, \dots, m \end{matrix}$ (24)

Obviously, the higher the R_i (ω), the better the alternative x_i. In order to solve the optimal decision model, the optimal solution can be obtained ω_i = (ω_1i, ω_2i, ⋯ , ω_mi) (i = 1, 2, ⋯ , n). According to the equation, the optimal attribute weight vector can be established: $ω = \sum_{i = 1}^{n} θ_{i} ω_{(i)}$ , where θ = (θ₁, θ₂, ⋯ , θ_n) ^T, $\sum_{i = 1}^{n} θ_{i}^{T} θ_{i} = 1$ , θ_i ≥ 0. Then the matrix form can be shown as follows:

$\begin{matrix} ω = (\begin{matrix} ω_{(11)} & ω_{(12)} \dots & ω_{(1 n)} \\ ω_{(21)} & ω_{(22)} \dots & ω_{(2 n)} \\ ⋮ & ⋱ & ⋮ \\ ω_{(m 1)} & ω_{(m 2)} \dots & ω_{(mn)} \end{matrix}) (\begin{matrix} θ_{1} \\ θ_{2} \\ ⋮ \\ θ_{n} \end{matrix}) = Φ θ \end{matrix}$ (25)

Where $\begin{matrix} \begin{matrix} Φ = (\begin{matrix} ω_{(11)} & ω_{(12)} \dots & ω_{(1 n)} \\ ω_{(21)} & ω_{(22)} \dots & ω_{(2 n)} \\ ⋮ & ⋱ & ⋮ \\ ω_{(m 1)} & ω_{(m 2)} \dots & ω_{(mn)} \end{matrix}) \end{matrix} \end{matrix}$

The optimal decision-making model is expressed as follows:

$\begin{matrix} R_{i} (ω) & = & \sum_{j = 1}^{m} (\frac{U (d_{i j})}{C E (d_{i j}) + E (d_{i j})}) ω_{j} \\ = & \sum_{j = 1}^{m} r_{i j} ω_{j} = ω^{T} r_{i} = {(Φ θ)}^{T} r_{i} \end{matrix}$ (26)

If we want to determine the weight vector θ = (θ₁, θ₂, ⋯ , θ_n) ^T, the values of R_i (ω) (i = 1, 2, ⋯ , n) should be maximum. That is maxR (ω) = (R₁ (ω) , R₂ (ω) , ⋯ , R_n (ω)) ^T, which is a multi-objective optimal linear programming model. Under the equality constraint $\sum_{i = 1}^{n} θ_{i}^{T} θ_{i} = 1$ , θ_i ≥ 0. So, based on the above analysis, an optimal linear programming model is established as follows:

${\begin{matrix} \max {(R (ω))}^{T} R (ω) = \sum_{i = 1}^{n} {(R_{i} (ω))}^{2} \\ \sum_{i = 1}^{n} θ_{i}^{T} θ_{i} = 1, θ_{i} \geq 0, (i = 1, 2, \dots, n) \end{matrix}$ (27)

We assume that $\begin{array}{l} y (θ) = {(R (ω))}^{T} R (ω) \\ = (R_{1} (ω), …, R_{n} (ω)) (R_{1} (ω), …, {(R_{n} (ω))}^{T} \\ = ({(Φ θ)}_{r 1}^{T}, …, ({(Φ θ)}_{r n}^{T}) ({(Φ θ)}_{r 1}^{T}, …, {(Φ θ)}_{r n}^{T})^{T} \\ = (Φ^{T} θ_{r 1}^{T}, …, Φ^{T} θ_{r n}^{T}) (Φ^{T} θ_{r 1}^{T}, …, Φ^{T} θ_{r n}^{T})^{T} \\ = θ^{T} (R^{T} Φ) T (R^{T} Φ) θ \end{array}$ We let W = (R^TΦ) ^T (R^TΦ), so, P^T = ((R^TΦ) ^T (R^TΦ)) = (R^TΦ) ^T (R^TΦ) = P. So, there is no deny that P is a nonnegative real symmetrical matrix. And it is easy to verify that max ((ω^TPω)/(ω^Tω)) = γ_max. Where γ_max is the largest eigenvalue of P and ω is a nonzero vector. Based on the analysis of the matrix, weight vector θ = (θ₁, θ₂, ⋯ , θ_n) ^T can be solved, which is an eigenvector of γ_max. After normalizing the weight vector θ = (θ₁, θ₂, ⋯ , θ_n) ^T, the attribute weight vector ω = {ω₁, ω₂, ⋯ , ω_m} can be obtained.

8.3. The steps of decision-making approach

The novel multi-attribute decision-making steps which based on utility function, comprehensive cross-entropy and entropy in IT2FSs is shown as follows: Step 1. Construct the interval type-2 fuzzy original matrix A (as shown in Table 2);

Step 2. Normalize original decision matrix A = (a_ij) _n×m, obtain normalized decision matrix D = (d_ij) _n×m;

Step 3. Calculate utility function, the comprehensive cross-entropy and entropy of the normalized matrix D = (d_ij) _n×m, and an optimal decision-making model based on utility function divides comprehensive cross-entropy and entropy is constructed and the optimal attribute weight is solved:

$R = (r_{ij})_{n \times m} {(\frac{U (d_{ij})}{CE (d_{ij}) + E (d_{ij})})}_{n \times m}$ (28)

${\begin{matrix} \underset{i}{m a x R} (ω) = \sum_{j = 1}^{m} (\frac{U (d_{i j})}{C E (d_{i j}) + E (d_{i j})}) ω_{j} \\ ω_{j} \geq 0, \sum_{j = 1}^{m} ω_{j} = 1, j = 1, 2, \dots, m \end{matrix}$ (29)

Step 4. Utilize interval type-2 fuzzy set weighted arithmetic average (WAA) operator to aggregate decision information and obtain utility function U (x_i) of the alternatives, the greater, the better.

The procedure of the novel MADM approach can be shown in Fig. 10.:

Fig. 10.

Procedure of the new MADM approach.

8.4. A practical example

We suppose there are five banks (x₁, x₂, x₃, x₄, x₅) will be evaluated and four attributes will be considered in reality: the ability to ensure at all times an adequate corresponding balance between cash inflows and cash outflows (a₁), the ability to coordinate the issuing by the bank of short, medium and long term financing instruments (a₂), the ability to optimize the costs of refinancing, striking a trade-off balance between liquidity and profitability (a₃), and the ability to optimize, for banks structured as banking groups, the intra-group management of cash flows, with the aim of reducing dependence on external financial requirements, by means of cash pooling techniques or other optimization instruments (a₄). The evaluation information on the five banks with respect to each attribute is characterized by IT2FSs, which are contained in the interval type-2 fuzzy original matrix, as shown in Table 2. The corresponding evaluation information attribute values of five banks are given in the form of interval two fuzzy. We suppose the decision maker expect to form linguistic terms (Table 1) to give the linguistic value to evaluate liquidity risk with interval type-2 fuzzy information. Table 1 shows the linguistic terms “Very Low” (VL), “Low” (L), “Medium Low” (ML), “Medium” (M), “Medium High” (MH), “High” (H), “Very High” (VH) and their corresponding interval type-2 fuzzy sets, respectively.

Table 1
Linguistic terms and their corresponding IT2FSs

Linguistic terms Interval type-2 fuzzy sets

Very Low(VL) {(0,0,0,0.1;1,1),(0,0,0,0.05;0.9,0.9)}

Low(L) {(0,0.1,0.1,0.3;1,1),(0.05,0.1,0.1,0.2;0.9,0.9)}

Medium Low(ML) {(0.1,0.3,0.3,0.5;1,1),(0.2,0.3,0.3,0.4;0.9,0.9)}

Medium(M) {(0.3,0.5,0.5,0.7;1,1),(0.4,0.5,0.5,0.6;0.9,0.9)}

Medium High(MH) {(0.5,0.7,0.7,0.9;1,1),(0.6,0.7,0.7,0.8;0.9,0.9)}

High(H) {(0.7,0.9,0.9,1;1,1),(0.8,0.9,0.9,0.95;0.9,0.9)}

Very High(VH) {(0.9,1,1,1;1,1),(0.95,1,1,1;0.9,0.9)}

Linguistic terms	Interval type-2 fuzzy sets
Very Low(VL)	{(0,0,0,0.1;1,1),(0,0,0,0.05;0.9,0.9)}
Low(L)	{(0,0.1,0.1,0.3;1,1),(0.05,0.1,0.1,0.2;0.9,0.9)}
Medium Low(ML)	{(0.1,0.3,0.3,0.5;1,1),(0.2,0.3,0.3,0.4;0.9,0.9)}
Medium(M)	{(0.3,0.5,0.5,0.7;1,1),(0.4,0.5,0.5,0.6;0.9,0.9)}
Medium High(MH)	{(0.5,0.7,0.7,0.9;1,1),(0.6,0.7,0.7,0.8;0.9,0.9)}
High(H)	{(0.7,0.9,0.9,1;1,1),(0.8,0.9,0.9,0.95;0.9,0.9)}
Very High(VH)	{(0.9,1,1,1;1,1),(0.95,1,1,1;0.9,0.9)}

Table 2

The evaluation information on five banks

Banks	a ₁	a ₂	a ₃	a ₄
Bank 1 (x₁)	H	MH	H	ML
Bank 2 (x₂)	MH	M	H	H
Bank 3 (x₃)	VH	ML	H	M
Bank 4 (x₄)	H	L	VH	M
Bank 5 (x₅)	H	MH	M	MH

The realization of decision-making steps is shown as follows:

Step 1. Construct the interval type-2 fuzzy original matrix A = (a_ij) _n×m (as shown in Table 2).

Step 2. Normalize original decision matrix A = (a_ij) _n×m, obtain normalized decision matrix D = (d_ij) _n×m: Since that the four attributes are all benefit attributes, don’t need to normalize the original matrix A. So, D = (d_ij) _n×m = (a_ij) _n×m = A.

Step 3. Calculate the comprehensive cross-entropy and entropy of the normalized D = (d_ij) _n×m, and an optimal decision-making model based on utility function divides comprehensive cross-entropy and entropy is constructed and the optimal attribute weight is solved: $\begin{matrix} U = (\begin{matrix} 0.2101 & 0.2295 & 0.2101 & 0.1022 \\ 0.2295 & 0.1704 & 0.2101 & 0.2101 \\ 0.0688 & 0.1022 & 0.2101 & 0.1704 \\ 0.2101 & 0.0591 & 0.0688 & 0.1704 \\ 0.2101 & 0.2295 & 0.1704 & 0.2295 \end{matrix}) \end{matrix}$ Let P = (p_ij) _n×m = (CE (d_ij) + E (d_ij)) _n×m, so, we can obtain that $\begin{matrix} P = (\begin{matrix} 0.1470 & 0.1598 & 0.1481 & 0.1544 \\ 0.1467 & 0.1549 & 0.1481 & 0.1519 \\ 0.1391 & 0.1498 & 0.1481 & 0.1570 \\ 0.1470 & 0.1501 & 0.1396 & 0.1570 \\ 0.1470 & 0.1498 & 0.1524 & 0.1515 \end{matrix}) \end{matrix}$ $\begin{matrix} R = (\begin{matrix} 1.4293 & 1.5230 & 1.4186 & 0.6619 \\ 1.5644 & 1.1001 & 1.4186 & 1.3831 \\ 0.4946 & 0.6822 & 1.4186 & 1.0854 \\ 1.4293 & 0.3937 & 0.4928 & 1.0854 \\ 1.4293 & 1.5320 & 1.1181 & 1.5149 \end{matrix}) \end{matrix}$

And based on the equations (24), (25), (26), (27), an optimal attribute weight is obtained: $\begin{matrix} ω = (0.2738, 0.2080, 0.2930, 0.2252) \end{matrix}$ Step 4. Utilize interval type-2 fuzzy set weighted arithmetic average (WAA) operator to aggregate decision information and utility function U (x_i) of the alternatives are obtained: U (x₁) =0.1898, U (x₂) =0.2072, U (x₃) =0.1400, U (x₄) =0.1284, U (x₅) =0.2069.

Because, U (x₂) > U (x₅) > U (x₁) > U (x₃) > U (x₄), so, the ranking order of the alternatives x₁, x₂, x₃, x₄, x₅ is x₂ ≻ x₅ ≻ x₁ ≻ x₃ ≻ x₄. This result coincides with the ones shown in [16] and [17].

8.5. Comparison with other existing methods

In order to testify the feasibility and effective- ness of the approach which proposed in this paper, some previous methods are utilized to compare the ranking results. In order to keep the consistency of weights information, we let that the attribute weight ω = (0.2738, 0.2080, 0.2930, 0.2252). (1) Based on the cross-entropy method put forward by Yao [16], the collective scores of alternatives are obtained as follows: R (x₁) =0.0934, R (x₂) =0.1016, R (x₃) =0.0590, R (x₄) =0.0502, R (x₅) =0.1006. Because, R (x₂) > R (x₅) > R (x₁) > R (x₃) > R (x₄), so, the ranking order of the alternatives x₁, x₂, x₃, x₄, x₅ is x₂ ≻ x₅ ≻ x₁ ≻ x₃ ≻ x₄. (2) Based on the ranking value formula proposed by Chen et al. [17], a decision matrix is obtained as follows: $\begin{matrix} R = (\begin{matrix} 1.5863 & 1.1550 & 1.5863 & 0.3750 \\ 1.1550 & 0.7250 & 1.5863 & 1.5863 \\ 1.8644 & 0.3750 & 1.5863 & 0.7250 \\ 1.5863 & 0.1306 & 1.8644 & 0.7250 \\ 1.5863 & 1.1550 & 0.7250 & 1.1550 \end{matrix}) \end{matrix}$ $\begin{matrix} AD = (1.5557, 0.7099, 1.4697, 0.9133) \end{matrix}$

So, we can obtain that: R (x₁) =4.1366, R (x₂) =4.5800, R (x₃) =3.5999, R (x₄) =3.2788, R (x₅) =4.4047.

Because R (x₂) > R (x₅) > R (x₁) > R (x₃) > R (x₄), so, the ranking order of the alternatives x₁, x₂, x₃, x₄, x₅ is x₂ ≻ x₅ ≻ x₁ ≻ x₃ ≻ x₄. (3) Based on the possibility degree proposed by Hu et al. [18], the possibility degree matrix P is obtained as follows: $\begin{matrix} P = (\begin{matrix} 0.5000 & 0.2654 & 0.5829 & 0.7253 & 0.6348 \\ 0.7346 & 0.5000 & 0.7876 & 0.8303 & 0.6834 \\ 0.4171 & 0.2124 & 0.5000 & 0.5992 & 0.5036 \\ 0.2747 & 0.1697 & 0.4008 & 0.5000 & 0.4085 \\ 0.3652 & 0.3166 & 0.4964 & 0.5915 & 50000 \end{matrix}) \end{matrix}$ R (x₁) =0.1104, R (x₂) =0.1518, R (x₃) =0.0866, R (x₄) =0.0627, R (x₅) =0.0885.

Because R (x₂) > R (x₁) > R (x₅) > R (x₃) > R (x₄), so, the ranking order of the alternatives x₁, x₂, x₃, x₄, x₅ is x₂ ≻ x₁ ≻ x₅ ≻ x₃ ≻ x₄.

Table 3
Comparisons of the four methods

Methods Order of banks

cross-entropy method [16] x₂ ≻ x₅ ≻ x₁ ≻ x₃ ≻ x₄

ranking value method [17] x₂ ≻ x₅ ≻ x₁ ≻ x₃ ≻ x₄

possibility degree method [18] x₂ ≻ x₁ ≻ x₅ ≻ x₃ ≻ x₄

The method in this paper x₂ ≻ x₅ ≻ x₁ ≻ x₃ ≻ x₄

From the comparison above, it is very clear that the ranking of the four methods are essentially in agreement. The result illustrates the approach proposed in this paper is reasonable and scientific. Comparison with the three methods, the approach which proposed in this paper has four advantages as follows: (1) In this paper, we firstly introduce definition and concept of α-cuts, and fuzzy factor and interval factor of IT2FSs are proposed based on α-cuts. The fuzziness of membership degree of interval type-2 fuzzy sets can be calculated more accurately, and also the degree of interval difference between upper and lower bounds of interval type-2 fuzzy sets can be calculated more accurately. (2) The fuzzy factor and the interval factor of interval type-2 fuzzy sets are put forward by fuzziness and interval of interval type-2 fuzzy sets, which fully reflects the relationship between fuzzy numbers, membership of fuzzy numbers and fuzzy factor and interval factor of interval type-2 fuzzy set. (3) Based on the fuzzy factor and the interval factor of IT2FSs, entropy and cross-entropy of IT2FSs are brought forward to measure the uncertain information and mutual uncertain influence relationship among IT2FSs. Furthermore, comprehensive cross-entropy formula is proposed to measure the relative uncertain relationships among IT2FSs. When calculate the comprehensive cross-entropy, we consider more comprehensively. Not only the influence that one fuzzy number on other fuzzy numbers under different alternatives is considered, but also the influence of other fuzzy numbers on that fuzzy number under different alternatives. (4) A utility function of interval type-2 fuzzy set is put forward in this paper to compare the superiority-inferiority among IT2FSs. And an optimal linear programming model is constructed to solve the optimal decision alternative based on IT2FS entropy, comprehensive cross-entropy and utility function, the approach makes the decision-making more scientific and systematic.

9. Conclusions

In recent years, interval type-2 fuzzy set has received increasingly public attention in decision-making. The purpose of this paper is to develop a new approach to deal with multi-attribute decision making(MADM) problems under interval type-2 fuzzy environment. The main process is shown as follows: Firstly, analyze and introduce the concept of α-cuts, and fuzzy factor and interval factor of interval type-2 fuzzy sets(IT2FSs) are put forward based on α-cuts. Then, entropy and cross-entropy formulas of IT2FSs are proposed to measure the fuzzy information and mutual uncertain relationships between IT2FSs based on the two factors. Furthermore, IT2FS comprehensive cross-entropy formula is brought forward to measure the relative uncertain relationships among IT2FSs based on IT2FS cross-entropy formula and total probability formula. In the next moment, a utility function of IT2FS is put forward to compare the ordering relationship among IT2FSs. After that, an optimal linear programming model is constructed based on IT2FS entropy, comprehensive cross-entropy and utility function to solve the optimal alternative. Finally, a simple example is utilized to verify the reasonability and effectiveness of the proposed approach. Interval type-2 fuzzy sets are the extension of interval-valued fuzzy sets and type-1 fuzzy sets, which is more widely used in the real world. In future research, we will focus on researching and analyzing the similarities and differences between interval type-2 fuzzy sets, interval-valued fuzzy sets and type-1 fuzzy sets, and analyzing the similarities and differences between their membership degrees. In the future, we will further study and extend properties of IT2FSs, IVFSs and TIFSs in more decision-making problems, such as transportation, prospect theory analysis, supplier selection and so on.

Footnotes

Acknowledgments

The authors are highly thankful to any anonymous referee for their careful reading and insightful comments, and their constructive opinions and suggestions will generate an improved version of this article. The work is supported by the National Nature Science Foundation of China (No. 6180601), the Talent Support Key Project for Outstanding Young of Colleges and Universities in 2016 (No.gxyq ZD2016453), the Natural Science Foundation of Anhui Province, and the Project of Graduate Academic Innovation of Anhui University. *Corresponding author: Junjun Mao, Tel: +86 1365560 0410 E-mail: maojunjun@ahu.edu.cn

References

L.A.

Zadeh , The concept of a linguistic variable and its application to approximate reasoning[J], Information Science 8(3) (1975), 199–249.

J.M.

Mendel and

R.I.

John , Type-2 fuzzy sets made simple, IEEE Transactions on Fuzzy Systems 10 (2002), 117–127.

L.A.

Zadeh , Fuzzy sets [J], Information and Control 8(3) (1965), 338–353.

L.A.

Zadeh , Probability measures of fuzzy events, Journal of Mathematical Analysis and Applications 23 (1968), 421–427.

Zhao ,

Z.S.

Xu and

F.J.

Liu , Uncertainty measures for Hesitant fuzzy information, International Journal of Intelligent Systems 30 (2015), 818–836.

Greenfield , Uncertainty Measurement for the Interval Type-2 Fuzzy Set, International Conference on Artificial Intelligence and Soft Computing. Springer International Publishing, (2016), 183–194.

Z.S.

Xu and

M.M.

Xia , Hesitant fuzzy entropy and cross-entropy and their use in multi-attribute decision making, International Journal of Intelligent Systems 27 (2012), 799–822.

M, Mizumoto and

Tanaka , Some properties of fuzzy sets of type-2, Information and Control 31 (1976), 312–340.

X.G.

Shang and

W.S.

Jiang , A note on fuzzy information measures, Pattern Recognition Letters 18 (1997), 425–432.

10.

Torra , Hesitant fuzzy sets, International Journal of Intelligent Systems 25 (2010), 529–539.

11.

I.K.

Vlachos and

G.D.

Sergiadis , Intuitionistic fuzzy information-applications to pattern recognition, Pattern Recognition Letters 28 (2007), 197–206.

12.

D.R.

Wu and

J.M.

Mendel , Uncertainty measures for interval type-2 fuzzy sets, Information Sciences 177 (2007), 5378–5393.

13.

R.R.

Yager , Fuzzy subsets of type II in decisions, Journal of Cybernetics 10 (1980), 137–159.

14.

D.Y.

Zhai and

J.M.

Mendel , Uncertainty measures for general Type-2 fuzzy sets, Information Sciences 181 (2011), 503–518.

15.

D.B.

Yao ,

X.X.

Liu ,

C.C.

Wang , et al., Type-2 fuzzy crossentropy and entropy measures and their applications, Journal of Intelligent & Fuzzy Systems 30 (2016), 2169–2180.

16.

D.B.

Yao and

C.C.

Wang , Interval type-2 fuzzy information measures and their applications to attribute decision-making approach, Journal of Intelligent & Fuzzy Systems 33 (2017), 1–13.

17.

S.M.

Chen ,

M.W.

Yang ,

L.M.

Lee and

S.W.

Yang , A new method for fuzzy multiple attributes group decision making based on ranking interval type-2 fuzzy sets, International Conference on Machine Learning & Cybernetics 1 (2011), 142–147.

18.

J.H.

Hu ,

Zhang ,

X.H.

Chen and

Y.M.

Liu , Multi-criteria decision-making method based on possibility degree of interval type-2 fuzzy number, Knowledge-Based Systems 43 (2013), 21–29.

19.

J.M.

Mendel , Type-2 fuzzy sets and systems: An overview, IEEE Computational Intelligence Magazine 2(1) (2007), 20–29.

Methods	Order of banks
cross-entropy method [16]	x₂ ≻ x₅ ≻ x₁ ≻ x₃ ≻ x₄
ranking value method [17]	x₂ ≻ x₅ ≻ x₁ ≻ x₃ ≻ x₄
possibility degree method [18]	x₂ ≻ x₁ ≻ x₅ ≻ x₃ ≻ x₄
The method in this paper	x₂ ≻ x₅ ≻ x₁ ≻ x₃ ≻ x₄

Comprehensive cross-entropy and entropy measures and their applications under interval type-2 fuzzy environment

Abstract

Keywords

1. Introduction

2. Preliminaries

4.1. The introduction of measure of fuzziness

8.1. Analysis of decision-making

Table 3 Comparisons of the four methods Methods Order of banks cross-entropy method [16] x2 ≻ x5 ≻ x1 ≻ x3 ≻ x4 ranking value method [17] x2 ≻ x5 ≻ x1 ≻ x3 ≻ x4 possibility degree method [18] x2 ≻ x1 ≻ x5 ≻ x3 ≻ x4 The method in this paper x2 ≻ x5 ≻ x1 ≻ x3 ≻ x4

Footnotes

Acknowledgments

References