Interval type-2 fuzzy information measures and their applications to attribute decision-making approach

Abstract

Interval type-2 fuzzy set (IT2FS) offers great ability to depict high order information in reality while dealing with both extrinsic and intrinsic facets of uncertainty. In this paper, we try to develop a general framework of information measures and a flexible multiple attribute decision-making (MADM) approach in the interval type-2 fuzzy information environment. First, we propose the fuzzy factor, hesitant factor and interval factor to quantify the fuzziness, hesitancy and interval information of one IT2FS, respectively. An interval type-2 fuzzy cross-entropy has been initiated based on these three factors to measure the discrimination degree of uncertain information between two IT2FSs. Meanwhile, we exploit the axiomatic principles of interval type-2 fuzzy entropy and study the inherent relationship between cross-entropy and entropy measures. Then, some parameterized information measures are naturally investigated, and the decomposition formula suggests that the interval type-2 fuzzy entropy could be expressed as the weighted average of the fuzzy entropy, hesitant entropy and interval entropy. Finally, we construct two programming models based on the maximizing cross-entropy principle to determine attribute weights, and a novel MADM procedure is proposed and applied to a case study on the banks’ liquidity risk evaluation.

Keywords

IT2FS interval type-2 fuzzy cross-entropy interval type-2 fuzzy entropy MADM the maximizing cross-entropy principle

1 Introduction

Fuzzy sciences originated almost 50 years ago since Zadeh [33] introduced the concept of fuzzy set (FS) in 1965, and Merigó et al. [21] presented a general overview of research in the fuzzy sciences using bibliometric analysis method. In fuzzy sciences, type-2 fuzzy set (T2FS) theory, initiated by Zadeh [35] in 1975, has been viewed as an effective extension of FS due to its great ability to cope with uncertain nature of real world. In order to clearly characterize the complex structure of a T2FS, Mendel & John [19] proposed a new Representation Theorem, which provides us a convenient way to analysis the uncertain information of T2FSs. Mendel [20] further developed the fundamental theory and extended related operational laws of T2FSs. Nevertheless, we usually confront with high computational overhead when T2FSs are used to solve some practical problems. To resolve these difficulties, an interval type-2 fuzzy set (IT2FS) [18] is expressed as a special case of a general type-2 fuzzy set, which can be just considered the primary membership grade, as the secondary membership degree equals to 1. Because of their easiness and reduced computational effort in comparison with general T2FSs, IT2FSs have received increasing attention both in theory and practice fields.

How to quantify the uncertain information of a T2FS and its extensions is of great importance. Wu & Mendel [30] defined some uncertainty measures, such as centroid, cardinality, fuzziness, variance and skewness, for IT2FSs. Zhai & Mendel [36] extended these measures to general type-2 fuzzy sets. Greenfield and John [12, 13] used vertical slice and invariance under vertical translation to measure uncertainty of T2FSs and IT2FSs, respectively. Some other information measures are also studied in T2FSs, such as similarity [15], inclusion and subsethood measure [27]. Fuzzy entropy, initiated by Zadeh [34] in 1968, has been viewed as one of important tools to provide the foundation for analogical reasoning and could directly describe uncertainty of FSs. Then, fuzzy entropy has been extended to other extensions of FSs. Such as Szmidt & Kacprzyk [26] redefined the constructive axioms of intuitionistic fuzzy entropy, but they cannot express the equilibrium state of supportability and opposability if neutral evidences are indicated in the hesitancy degree. Pal et al. [22] pointed out that existing uncertainty measures cannot capture all fuzziness and lack of knowledge and proposed a generating family of measures. Xu & Xia [31], Zhao et al. [37] introduced some hesitant fuzzy entropies in hesitant fuzzy sets (HFSs). Hwang et al. [15] presented similarity, inclusion and entropy of T2FSs by use of Sugeno integral. However, they only consider the fuzziness of secondary membership degree while neglecting the fuzziness of primary membership grade, and they also have neglected the dispersity of primary membership degree. Up to now, few scholars have extended the entropy concept to IT2FSs and proposed the axiomatic principle of interval type-2 fuzzy entropy. By analyzing the structural characteristics of IT2FSs and on the basis of previous works, we need to point out that the uncertainty of an IT2FS may be divided into three parts: fuzziness, hesitancy and interval information. Based on this thought, we propose a constructive principle of interval type-2 fuzzy entropy.

The classical cross-entropy is used to depict the discrimination information between FSs or their extensions. For instance, Shang & Jiang [25], Vlachos & Sergiadis [28] discussed fuzzy cross-entropy and intuitionistic fuzzy cross-entropy and studied the relationship between these two measures. Xu & Xia [31] introduced a hesitant fuzzy cross-entropy in HFSs. It is worth noting that Mao et al. [17] and Yao et al. [32] proposed two novel kinds of cross-entropies to measure the discrimination degree of uncertain information between two IFSs and T2FSs, respectively. In reality, we need to discriminate and quantify different amount of uncertain information since they usually represent some cost-type facets, such as risk, cost, etc. Inspired by Mao et al. [17] and Yao et al. [32], we try to introduce an interval type-2 fuzzy cross-entropy in IT2FSs and study the inherent relation between cross-entropy and entropy of IT2FSs.

An increased number of multiple attribute or criteria decision-making methods (MADM or MCDM) have been developed in interval type-2 fuzzy environment. Wang et al. [29] studied the multiple attribute group decision-making (MAGDM) models by use of weighted arithmetic averaging operator and ranking-value measure. Some other scholars further extended ranking value methods and applied them in MADM problems, such as Chen & Wang [5], Chen et al. [6], Chen [7], Qin & Liu [23]. Meanwhile, Hu et al. [14] proposed a new approach based on possibility degree to solve multi-criteria decision-making problems in IT2FSs. Qin et al. [24] extended the VIKOR method based on prospect theory to accommodate interval type-2 fuzzy circumstances. Celik et al. [1] gave a comprehensive review of MCDM approaches based on IT2FSs. In addition, some researches have focus on other interval type-2 fuzzy MADM (or MAGDM or MCDM) approaches based on a variety of “traditional” decision-making techniques, such as AHP [4, 16], TOPSIS [3], PROMETHEE [8], ELECTRE [9], Preference relationship [2], and some other methods [10, 11]. However, the decision process in most of these works depends on aggregating operators, and different operators may lead to a contradict results. Meanwhile, the attribute information of these studies is often down to completely known or partly known, few scholars consider the case that the attribute weights are completely unknown. In this paper, we divide the attribute information into two cases: completely unknown and partly known, and construct programming models based on the maximizing cross-entropy principle to determine attribute weights under different cases. We also propose a new expected score function, which considers information comes from both lower membership function (LMF) and upper membership function (UMF), to avoid using aggregating operators. The step-by-step MADM procedure is developed and applied to a case study on banks’ liquidity risk evaluation. Some comparisons with other existing methods are made to verify the validity of our approach.

This paper is organized as follows. In Section 2, we briefly introduce some basic concepts. In Section 3, the interval type-2 fuzzy cross-entropy and entropy are proposed, and the relationship between these two measures is also investigated. We exploit some parameterized information measures for IT2FSs in Section 4. A novel MADM approach and expected score function are presented and applied to a case study concerning banks’ liquidity risk evaluation in Section 5. Section 6 concludes the whole paper.

2 Preliminaries

In this section, we first give a brief review of some concepts related to T2FSs and IT2FSs, and their basic operations, which will be used in the following sections.

Definition 2.1. [35] A type-2 fuzzy set (T2FS) is a fuzzy set whose membership values are type-1 fuzzy sets on [0,1].

Definition 2.2. [19] A type-2 fuzzy set, denoted by A, is characterized by a type-2 membership function μ_A (x, u), where x ∈ X and u ∈ J_x ⊆ [0, 1], that is, $\begin{matrix} A & = & {((x, u), μ_{A} (x, u)) : \\ \forall x \in X, u \in J_{x} \subseteq [0, 1]} \end{matrix}$ in which 0 ≤ μ_A (x, u) ≤1, A is also expressed as $A = \int_{x \in X} \int_{u \in J_{x}} μ_{A} (x, u) / - (x, u), J_{x} \subseteq [0, 1],$ where ∬ denotes union over all admissible x and u. For a universe of discrete discourse, ∫ is replaced by ∑.

Definition 2.3. [23] Let A be a type-2 fuzzy set in the universe X represented by a type-2 fuzzy membership function μ_A (x, u). If all μ_A (x, u) =1, then A is called an interval type-2 fuzzy set (IT2FS). An IT2FS could be viewed as a special case of the corresponding T2FS, which is defined as follows: $\begin{matrix} A & = & \int_{x \in X} \int_{u \in J_{x}} 1 / - (x, u) \\ = & \int_{x \in X} (\int_{u \in J_{x}} 1 / - u) / - x, J_{x} \subseteq [0, 1] . \end{matrix}$

In addition, an IT2FS is completely determined by the union of all primary memberships which is called the footprint of uncertainty (FOU), the FOU of A denoted by FOU (A) may be expressed as follows: $FOU (A) = \underset{\forall x \in X}{\cup} J_{x} = {(u, x) : u \in J_{x} \subseteq [0, 1]} .$

For an IT2FS A, let A^U (x) and A^L (x) be the upper membership function (UMF) and the lower membership function (LMF) of A, respectively. That is to say, for any x ∈ X, we have $\begin{matrix} A^{U} (x) & = & sup {FOU (A)} = \bar{FOU (A)}, \\ A^{L} (x) & = & inf {FOU (A)} = \underline{FOU (A)} . \end{matrix}$

Therefore, the FOU of an IT2FS A can be also expressed in the following form: $FOU (A) = \underset{\forall x \in X}{\cup} [A^{L} (x), A^{U} (x)] .$

For the sake of simplicity, we denote by A = (A^L, A^U) a trapezoidal interval type-2 fuzzy set (TIT2FS) in X, as shown in Fig. 1, where $\begin{matrix} A & = & ((a_{1}^{U}, a_{2}^{U}, a_{3}^{U}, a_{4}^{U}; m (A^{U}), n (A^{U})), \\ (a_{1}^{L}, a_{2}^{L}, a_{3}^{L}, a_{4}^{L}; m (A^{L}), n (A^{L}))) \end{matrix}$ and satisfy $0 \leq a_{1}^{*} \leq a_{2}^{*} \leq a_{3}^{*} \leq a_{4}^{*}$ , 0 ≤ m (A^*) ≤ n (A^*) ≤1, and * represses L or U. Both of A^U and A^L are type-1 fuzzy sets and such that $a_{1}^{U} \leq a_{1}^{L} \leq a_{4}^{L} \leq a_{4}^{U}$ . Especially, if $a_{2}^{U} = a_{3}^{U}$ and $a_{2}^{L} = a_{3}^{L}$ , then the trapezoidal interval type-2 fuzzy set A reduces to a triangular interval type-2 fuzzy set.

Fig.1

A trapezoidal interval type-2 fuzzy set A.

Since the trapezoidal interval type-2 fuzzy set is the typical case of a general IT2FS, many scholars have used the trapezoidal IT2FSs rather than the general IT2FSs to analysis the practical problems, such as Qin and Liu [23], Qin et al. [24], Wang et al. [29], Wu and Mendel [30]. Therefore, the following sections are also discussed based on the trapezoidal IT2FSs. For an IT2FS A, whose UMF A^U (x) and LMF A^L (x) are defined by $\begin{matrix} A^{U} (x) & = & {\begin{matrix} \frac{(x - a_{1}^{U}) m (A^{U})}{a_{2}^{U} - a_{1}^{U}} & a_{1}^{U} \leq x < a_{2}^{U} \\ \frac{(n (A^{U}) - m (A^{U}))}{a_{3}^{U} - a_{2}^{U}} (x - a_{2}^{U}) + m (A^{U}) & a_{2}^{U} \leq x < a_{3}^{U} \\ \frac{(x - a_{4}^{U}) n (A^{U})}{a_{3}^{U} - a_{4}^{U}} & a_{3}^{U} \leq x < a_{4}^{U} \\ 0 & otherwise \end{matrix} \\ A^{L} (x) & = & {\begin{matrix} \frac{(x - a_{1}^{L}) n (A^{L})}{a_{2}^{L} - a_{1}^{L}} & a_{1}^{L} \leq x < a_{2}^{L} \\ \frac{(m (A^{L}) - n (A^{L}))}{a_{3}^{L} - a_{2}^{L}} (x - a_{2}^{L}) + n (A^{L}) & a_{2}^{L} \leq x < a_{3}^{L} \\ \frac{(x - a_{4}^{L}) m (A^{L})}{a_{3}^{L} - a_{4}^{L}} & a_{3}^{L} \leq x < a_{4}^{L} \\ 0 & otherwise \end{matrix} \end{matrix}$

In general, all IT2FSs on X could be denoted by IT2FS (X), and the complement set (denoted by A^U) of an IT2FS A could be defined as [20]: $A^{C} = ((A^{C})^{U}, (A^{C})^{L}) = (\begin{matrix} (1 - a_{4}^{U}, 1 - a_{3}^{U}, 1 - a_{2}^{U}, 1 - a_{1}^{U}; m (A^{U}), n (A^{U})), \\ (1 - a_{4}^{L}, 1 - a_{3}^{L}, 1 - a_{2}^{L}, 1 - a_{1}^{L}; m (A^{L}), n (A^{L})) \end{matrix})$

3 The general framework of information measures in IT2FSs

Up to now, many kinds of information measures have been constructed to describe uncertainty of fuzzy set and its extensions, such as centroid, cardinality, variance, skewness, similarity, distance and correlation degree, etc. However, among these measures, entropy is considered as one of direct and effective tools to quantify uncertainty. For examples, Shang & Jiang [25] for FSs, Vlachos & Sergiadis [28], Mao et al. [17] for IFSs, Xu & Xia [31], Zhao et al. [37] for HFSs, Takáč [27], Yao et al. [32] for T2FSs. Unfortunately, few scholars have developed these entropy measures in IT2FSs. Inspired by Mao et al. [17] and Yao et al. [32], we try to construct an interval type-2 fuzzy cross-entropy to measure the discrimination degree of uncertainty between two IT2FSs and induce a new interval type-2 fuzzy entropy to quantify uncertainty of one IT2FS.

3.1 Some preparations

As we know, different extensions of FS have different types of uncertainty, for example, a FS’s uncertainty is its fuzziness, an IFS’s uncertainty includes its fuzziness and intuitionism, and a T2FS’s uncertainty is consist of fuzziness and hesitancy.

By analyzing characteristic of an IT2FS, we may point out that the uncertain information of an IT2FS could be divided into three facets: fuzziness, hesitancy and interval information. Specifically, fuzziness is reflected by the average closeness between primary membership degree and 1/2 in UMF and LMF, the hesitancy is displayed with dispersity degree of all primary membership degrees in UMF and LMF, and the interval information is determined by area of region between UMF and LMF. In order to quantify these three types of uncertainty in an IT2FS, we propose three new concepts (namely, the fuzzy factor, hesitant factor and interval factor) as follows:

For a given IT2FS A ∈ IT2FS (X), we denote the fuzzy factor, hesitant factor and interval factor by Δ_A, σ_A and S_A, respectively, where $Δ_{A} = \sqrt{\frac{Δ_{A^{U}}^{2} + Δ_{A^{L}}^{2}}{2}},$ (1) here $Δ_{A^{U}} = \frac{1}{| X |} \int_{a_{1}^{U}}^{a_{4}^{U}} | A^{U} (x) - 1 / - 2 | dx$ , $Δ_{A^{L}} = \frac{1}{| X |} \times \int_{a_{1}^{L}}^{a_{4}^{L}} | A^{L} (x) - 1 / - 2 | dx$ are fuzziness of UMF and LMF respectively. $σ_{A} = \sqrt{\frac{σ_{A^{U}}^{2} + σ_{A^{L}}^{2}}{2}},$ (2) here $σ_{A^{U}} = \frac{1}{| X |} \int_{a_{1}^{U}}^{a_{4}^{U}} | A^{U} (x) - {\bar{u}}^{U} | dx$ , $σ_{A^{L}} = \frac{1}{| X |} \times \int_{a_{1}^{L}}^{a_{4}^{L}} | A^{L} (x) - {\bar{u}}^{L} | dx$ are hesitancy of UMF and LMF, and ${\bar{u}}^{U} = \frac{1}{| X |} \int_{a_{1}^{U}}^{a_{4}^{U}} A^{U} (x) dx$ , ${\bar{u}}^{L} = \frac{1}{| X |} \int_{a_{1}^{L}}^{a_{4}^{L}} A^{L} (x) dx$ are the means of UMF and LMF, respectively.

$\begin{matrix} S_{A} & = & \frac{1}{| X |} (\int_{a_{1}^{U}}^{a_{4}^{U}} A^{U} (x) dx - \int_{a_{1}^{L}}^{a_{4}^{L}} A^{L} (x) dx) \\ = & {\bar{u}}^{U} - {\bar{u}}^{L} . \end{matrix}$ (3)

It’s easy to verify that Δ_{A
^U}, Δ_{A
^L}, σ_{A
^U}, σ_{A
^L} ∈ [0, 1/ - 2] ⇒ Δ_A, σ_A ∈ [0, 1/ - 2] and S_A ∈ [0, 1]. Moreover, we also have Δ_A = Δ_{A
^C}, σ_A = σ_{A
^C}, S_A = S_{A
^C}. It’s worth noting that the fuzziness of A becomes stronger while Δ_A decreases, its hesitancy increases with σ_A and interval information increases with S_A. So the whole uncertainty of an IT2FS could be adequately characterized by a three-tuple (Δ_*, σ_*, S_*).

3.2 Interval type-2 fuzzy cross-entropy

In reality, it’s of great importance to accurately measure uncertainty of IT2FSs since it often represents cost, risk, etc. However, the traditional cross-entropy is used to measure the discrimination information, which includes determination information and uncertain information. Mao et al. [17] and Yao et al. [32] proposed a novel cross-entropy of IFSs and T2FSs to describe their discrimination of uncertain information, respectively. Inspired by above references, we try to develop an analytical framework to measure the discrimination degree of uncertain information between two IT2FSs based on three-tuple (Δ_*, σ_*, S_*).

Definition 3.1. Let A, B ∈ IT2FS (X), then the interval type-2 fuzzy cross-entropy of A against B is defined as:

$\begin{matrix} CE (A, B) : & = & Δ_{A} ln \frac{Δ_{A}}{\frac{1}{2} (Δ_{A} + Δ_{B})} \\ + σ_{A} ln \frac{σ_{A}}{\frac{1}{2} (σ_{A} + σ_{B})} \\ + S_{A} ln \frac{S_{A}}{\frac{1}{2} (S_{A} + S_{B})} . \end{matrix}$ (4)

However, one can observe that CE (A, B) is not symmetric with respect to its arguments. Then a symmetric form is given as $DE (A, B) = CE (A, B) + CE (B, A) .$ (5)

Theorem 3.1. Let A, B ∈ IT2FS (X), DE (A, B) is the symmetric type-2 fuzzy cross-entropy between A and B, then we have

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

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

0 ≤ DE (A, B) ≤3 ln 2.

Proof. It’s easy to prove that DE (A, B) verifies the properties (1) and (2) by its definition and above discussion in subsection 3.1, we only prove (3).

(3) Since Δ_A, σ_A ∈ [0, 1/ - 2] and S_A ∈ [0, 1], by use of Shannon’s inequality, we have CE (A, B) ≥0 ⇒ DE (A, B) ≥0. Meanwhile, the relation CE (A, B) ≤ (Δ_A + σ_A + S_A) ln 2 ≤ 3 ln 2/ - 2 is true, so we have DE (A, B) ≤3 ln 2. □

3.3 Relationship between cross-entropy and entropy of IT2FSs

Since Zadeh [34] proposed fuzzy entropy in 1968 to describe FS’s fuzziness, this concept has gradually become one of powerful tools to measure uncertainty while its has been extended to other extensions of FSs both in its form and connotation, such as Szmidt & Kacprzk [26], Pal et al. [22], Mao et al. [17] developed intuitionistic fuzzy entropy, Xu & Xia [31], Zhao et al. [37] defined hesitant fuzzy entropy, and Hwang et al. [15], Takáč [27] have given some basic axioms of entropy for T2FSs. Based on these previous works, we now propose some axiomatic principles of an interval type-2 fuzzy entropy as follows.

Definition 3.2. The interval type-2 fuzzy entropy is a real-valued function E with Δ_*, σ_* and S_*, i.e., E = f (Δ_*, σ_*, S_*) : IT2FS (X) → [0, 1]. For any A ∈ IT2FS (X), E (A) satisfies the following axiomatic principles:

E (A) =0 iff (if and only if) A is a crisp set, i.e., Δ_A = 1/ - 2, σ_A = S_A = 0;

E (A) ≤1 for any A ∈ IT2FS (X);

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

f (Δ_A, σ_A, S_A) is a real-valued continuous function, which decreases with the first variable Δ_A and increases with second variable σ_A and third variable S_A.

The classical relationship between entropy and cross-entropy have been studied by Vlaschos & Sergiadis [28], Xu & Xia [31], that is, E (A) = θ × DE (A, A^C), where θ is a normalized parameter. However, this relation may be false in our models due to DE (A, A^C) =0 for any A ∈ IT2FS (X). Motivated by Mao et al. [17], we denote a crisp set by C^*, i.e., Δ_{C
^*} = 1/ - 2, σ_{C
^*} = 0, S_{C
^*} = 0 then DE (A, C^*) could be viewed as a method to measure the amount of uncertainty of an IT2FS A, which agrees with the intuitive concept of interval type-2 fuzzy entropy.

Theorem 3.2. Let A ∈ IT2FS (X), C^* is a crisp set, then

$\begin{matrix} E (A) \\ = \frac{1}{3 ln 2} DE (A, C^{*}) \\ = \frac{1}{3 ln 2} (\begin{matrix} (Δ_{A} + \frac{1}{2}) ln \frac{1}{Δ_{A} + 1 / - 2} \\ + Δ_{A} ln (2 Δ_{A}) + σ_{A} ln 2 + S_{A} ln 2 \end{matrix}) \end{matrix}$ (6) is an interval type-2 fuzzy entropy of A.

Proof. (1) If A is a crisp set, namely, A = C^* or (C^*) ^C, then DE (A, C^*) =0 ⇒ E (A) =0. Conversely, let E (A) =0, i.e., DE (A, C^*) =0. By Definition 3.2, we have Δ_A = Δ_{C
^*} = 1/ - 2, σ_A = σ_{C
^*} = 0, S_A = S_{C
^*} = 0, these imply that A = C^* or (C^*) ^C, namely, A is a crisp set.

(2) According to Theorem 3.1, we know E (A) ≤1 and E (A) = E (A^C).

(3) In order to prove E (A) satisfies the principle (4) in Definition 3.2, it suffices to prove that the following function: $\begin{matrix} g (x, y, z) & = & x ln (2 x) - (x + 1 / - 2) \\ ln (x + 1 / - 2) + y ln 2 + z ln 2 \end{matrix}$ where x, y ∈ [0, 1/ - 2] , z ∈ [0, 1], decreases with its first argument x and increases as its second argument y and third argument z. Taking the partial derivative of g (x, y, z) with respect to x, y, z, respectively, we yield ${\begin{matrix} \frac{\partial g (x, y, z)}{\partial x} = ln (\frac{2 x}{x + 1 / - 2}) \leq 0 \\ \frac{\partial g (x, y, z)}{\partial y} = y ln 2 \geq 0 \\ \frac{\partial g (x, y, z)}{\partial z} = z ln 2 \geq 0 \end{matrix}$

Let x = Δ_A, y = σ_A and z = S_A, then E (A) = g (Δ_A, σ_A, S_A)/ - 3 ln 2 decreases with Δ_A and increases as σ_A and S_A. □

3.4 Decomposition formula of uncertain information for IT2FSs

Vlachos & Sergiadis [28] had proposed an unified formulation to analysis the relation of entropy, fuzziness and intuitionism for IFSs. Mao et al. [17] also gave a decomposition formula of intuitionistic fuzzy entropy. Based on these works, we try to construct a similar decomposition formula of interval type-2 fuzzy entropy to study inherent connections of fuzziness, hesitancy and interval information. In order to clearly illustrate this relationship, we introduce three kinds of entropies to discuss different effects of each factor (the fuzzy factor, hesitant factor and interval factor).

Definition 3.3. The fuzzy entropy is a real-valued function E_F with Δ_*, i.e., E_F = f (Δ_*) : IT2FS (X) → [0, 1]. For any A ∈ IT2FS (X), E_F (A) satisfies the following axiomatic principles:

E_F (A) =0 iff Δ_A = 1/ - 2;

E_F (A) =1 iff Δ_A = 0, namely, A^U (x) = A^L (x) = 1/ - 2, ∀ x ∈ X;

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

f (Δ_A) decreases with the fuzzy factor Δ_A.

Definition 3.4. The hesitant entropy is a real-valued function E_H with σ_*, i.e., E_H = f (σ_*) : IT2FS (X) → [0, 1]. For any A ∈ IT2FS (X), E_H (A) satisfies the following axiomatic principles:

E_H (A) =0 iff σ_A = 0, namely A^U (x) and A^L (x) are constant;

E_H (A) =1 iff σ_A = 1/ - 2;

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

f (σ_A) increases with the hesitant factor σ_A.

Definition 3.5. The interval entropy is a real-valued function E_I with S_*, i.e., E_I = f (S_*) : IT2FS (X) → [0, 1]. For any A ∈ IT2FS (X), E_I (A) satisfies the following axiomatic principles:

E_I (A) =0 iff S_A = 0, namely, A^U (x) = A^L (x) , ∀ x ∈ X;

E_I (A) =1 iff A^U (x) =1, A^L (x) =0;

E_I (A) = E_I (A^C);

f (S_A) increases with the hesitant factor S_A.

According to the above definitions, fuzzy entropy E_F (A) measures uncertain information caused by fuzziness of A, hesitant entropy E_H (A) describes uncertainty induced by its hesitancy, and interval entropy E_I (A) depicts the uncertainty arise from its interval information. If we rewrite Equation (6) as

$\begin{matrix} E (A) & = & \frac{1}{3 ln 2} ((Δ_{A} + \frac{1}{2}) ln \frac{1}{Δ_{A} + 1 / - 2} \\ + Δ_{A} ln (2 Δ_{A})) + \frac{1}{3} σ_{A} + \frac{1}{3} S_{A} \\ = & \frac{1}{3} E_{F} (A) + \frac{1}{3} E_{H} (A) + \frac{1}{3} E_{I} (A) \end{matrix}$ (7) where $\begin{matrix} E_{F} (A) & = & \frac{1}{ln 2} ((Δ_{A} + \frac{1}{2}) ln \frac{1}{Δ_{A} + 1 / - 2} \\ + Δ_{A} ln (2 Δ_{A})) \\ E_{H} (A) & = & σ_{A}, E_{I} (A) = S_{A} . \end{matrix}$

By Definitions 3.3, 3.4 and 3.5, it’s easy to verify that E_F (A), E_H (A) and E_I (A) are fuzzy entropy, hesitant entropy and interval entropy of A, respectively. From Equation (7), we may conclude that an interval type-2 fuzzy entropy E (A) could be viewed as the arithmetic average of fuzzy E_F (A), hesitant entropy E_H (A) and interval entropy E_I (A).

4 The parameterized information measures of IT2FSs

It’s evident to find that the coefficients of E_F (A), E_H (A) and E_I (A) are all 1/3 in Equation (7). This indicates that E_F (A), E_H (A) and E_I (A) have same effect on E (A) in the view of Mathematics, which implies that fuzziness, hesitancy and interval information have the same position to the global uncertainty of one IT2FS. Naturally, if we consider these three facets have different effects on the whole uncertainty, we need introduce some parameters to adjust and control them.

Definition 4.1. Let A, B ∈ IT2FS (X) and p, q, r > 0, then the parameterized type-2 fuzzy cross-entropy of A against B is defined as

$\begin{matrix} {CE}^{p, q, r} (A, B) : & = & Δ_{A}^{p} ln {(\frac{Δ_{A}^{p}}{\frac{1}{2} (Δ_{A}^{p} + Δ_{B}^{p})})}^{p} \\ + σ_{A}^{q} ln {(\frac{σ_{A}^{q}}{\frac{1}{2} (σ_{A}^{q} + σ_{B}^{q})})}^{q} \\ + S_{A}^{r} ln {(\frac{S_{A}^{r}}{\frac{1}{2} (S_{A}^{r} + S_{B}^{r})})}^{r} \end{matrix}$ (8)

Obviously, CE^p,q,r (A, B) is also not symmetric with its arguments, so a symmetric parameterized type-2 fuzzy cross-entropy could be obtained in similar way: ${DE}^{p, q, r} (A, B) = {CE}^{p, q, r} (A, B) + {CE}^{p, q, r} (B, A) .$ (9)

Theorem 4.1. Let A, B ∈ IT2FS (X) and p, q, r > 0, DE^p,q,r (A, B) is the symmetric parameterized interval type-2 fuzzy cross-entropy between A and B, then it satisfies:

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

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

0 ≤ DE^p,q,r (A, B) ≤ (p + q + r) ln 2.

Proof. The proof is trivial. □

In similar way, the parameterized interval type-2 fuzzy cross-entropy may induce a parameterized interval type-2 fuzzy entropy as following theorem.

Theorem 4.2. Let A ∈ IT2FS (X) and p, q, r > 0, C^* is a crisp set, then

$\begin{matrix} E^{p, q} (A) \\ = \frac{1}{(p + q) ln 2} {DE}^{p, q, r} (A, C^{*}) \\ = \frac{1}{(p + q + r) ln 2} (p ((Δ_{A}^{p} + \frac{1}{2^{p}}) \\ ln \frac{1}{Δ_{A}^{p} + 1 / - 2^{p}} + Δ_{A}^{p} ln (2 Δ_{A}^{p}) + 2^{- p} ln 2^{1 - p}) \\ + q σ_{A}^{q} ln 2 + {rS}_{A}^{r} ln 2) \end{matrix}$ (10) is a parameterized interval type-2 fuzzy entropy of A.

Proof. Similar to the process of proof in Theorem 3.2, it’s easy to verify E^p,q,r (A) satisfies the principles (1–3) in Definition 3.2. So we only prove Equation (10) satisfies the principle (4).

Obviously, E^p,q,r (A) fulfils the principle (4) if we are able to show that the function $\begin{matrix} g (x, y, z) \\ = p (x^{p} ln (2 x^{p}) - (x^{p} + 1 / - 2^{p}) ln (x^{p} + 1 / - 2^{p}) \\ + 2^{- p} ln 2^{1 - p}) + {qy}^{q} ln 2 + {rz}^{r} ln 2 \end{matrix}$ where x, y ∈ [0, 1/ - 2] , z ∈ [0, 1] , p, q, r > 0, de-creases with its first argument x and increases with its second argument y and third argument z. Taking the partial derivative of g (x, y, z) with respect tox, y and z, respectively, we yield ${\begin{matrix} \frac{\partial g (x, y, z)}{\partial x} = p^{2} x^{p - 1} ln (\frac{2 x^{p}}{x^{p} + 1 / - 2^{p}}) \leq 0 \\ \frac{\partial g (x, y, z)}{\partial y} = q^{2} y^{q - 1} ln 2 \geq 0 \\ \frac{\partial g (x, y, z)}{\partial z} = r^{2} z^{r - 1} ln 2 \geq 0 \end{matrix}$

Let x = Δ_A, y = σ_A and z = S_A, E^p,q,r (A) = g (Δ_A, σ_A, S_A)/ - [(p + q + r) ln 2] decreases with Δ_A and increases with σ_A and S_A. □

We can also divide Equation (10) into three parts:

$\begin{matrix} E^{p, q, r} (A) & = & \frac{p}{p + q + r} E_{F}^{p} (A) + \frac{q}{p + q + r} E_{H}^{q} (A) \\ + \frac{r}{p + q + r} E_{I}^{r} (A) . \end{matrix}$ (11) where $\begin{matrix} E_{F}^{p} (A) & = & \frac{1}{ln 2} ((Δ_{A}^{p} + \frac{1}{2^{p}}) ln \frac{1}{Δ_{A}^{p} + 1 / - 2^{p}} \\ + Δ_{A}^{p} ln (2 Δ_{A}^{p}) + 2^{- p} ln 2^{1 - p}), \\ E_{H}^{q} (A) & = & σ_{A}^{q}, E_{I}^{r} (A) = S_{A}^{r} . \end{matrix}$

We could verify that $E_{F}^{p} (A)$ , $E_{H}^{q} (A)$ and $E_{I}^{r} (A)$ are the fuzzy entropy, hesitant entropy and interval entropy of A, respectively. That is to say interval type-2 fuzzy entropy E^p,q,r (A) could be viewed as the weighted average of $E_{F}^{p} (A)$ , $E_{H}^{q} (A)$ and $E_{I}^{r} (A)$ . Specially, if let p = q = r = 1, Equation (8–11) may naturally degenerate to Equation (5–7). If let p + q + r = 1, Equation (11) could be expressed as $E^{p, q, r} (A) = {pE}_{F}^{p} (A) + {qE}_{H}^{q} (A) + {rE}_{I}^{r} (A) .$ (12)

In general, if p > q > r, it shows that fuzziness of A has a stronger effect on the whole uncertainty than its hesitancy and interval information, and other situations may also be explained in the similar way. However, we should choose the consistent parameters p, q and r in the same practical problem, and make comparison for different events under a fixed entropy distribution.

5 Applications

In this section, we solve multiple attribute decision-making (MADM) problems under interval type-2 fuzzy environment by use of above information measures. We suppose there are m alternatives x_i (1 ≤ i ≤ m) and n attributes e_j (1 ≤ j ≤ n) with the attribute weight vector $ω_{j} \in [0, 1], \sum_{j = 1}^{n} ω_{j} = 1$ . Let A = (a_ij) _m×n be an interval type-2 fuzzy original matrix, where a_ij is described by IT2FS, which is provided by the decision maker for the alternative x_i with respect to attribute e_j. We also suppose a normalized decision matrix D = (d_ij) _m×n, where d_ij is the normalized form of a_ij.

5.1 A new MADM approach based on information measures

Based on above notations, a new MADM method by use of the proposed information measures could be described in the following steps, and the procedure of the MADM approach can be shown in Fig. 2.

Fig.2

The procedure of the new MADM approach.

Step 1. Collect the decision-making information about alternatives and attributes, and identify the original matrix A = (a_ij) _m×n.

Step 2. Calculate the decision matrix D = (d_ij) _m×n by normalizing the matrix A according to attributes’ types. $d_{ij} = {\begin{matrix} a_{ij} & if e_{j} is a benefit attribute \\ (a_{ij})^{C} & if e_{j} is a cos t attribute \end{matrix}$ where (a_ij) ^C is the complement set of a_ij.

Step 3. Calculate the expected score s_ij of each alternative x_i under every attribute e_j, and transform the decision matrix D into a expected score matrix S = (s_ij) _m×n, where s_ij satisfies $s_{ij} = \frac{s_{ij}^{L} + s_{ij}^{U}}{2},$ (13) here $s_{ij}^{L} = \int_{d_{ij 1}^{L}}^{d_{ij 4}^{L}} {xd}_{ij}^{L} (x) d x, s_{ij}^{U} = \int_{d_{ij 1}^{U}}^{d_{ij 4}^{U}} {xd}_{ij}^{U} (x) d x$ are expected scores of LMF $d_{ij}^{L} (x)$ and UMF $d_{ij}^{U} (x)$ respectively, and $d_{ij 1}^{L}, d_{ij 4}^{L}, d_{ij 1}^{U}, d_{ij 4}^{U}$ are the value range of the interval type-2 fuzzy set d_ij.

Step 4. Determine the attribute weights according to two different cases, one is the attribute weights are completely unknown, the other is partly known. Then we try to construct two linear programming models to deal with these two cases.

Case 1. If the attribute weights are completely unknown, we determine the weights based on the maximizing cross-entropy principle, that is, a small cross-entropy under an attribute indicates that such an attribute has a small influence on the overall value of alternative. On the contrary, if cross-entropy under an attribute is big, it implies that the attribute plays an important role. The bigger the cross-entropy of alternative under a certain attribute, the larger the corresponding weight of the attribute, which is conversely assigned a smaller weight. Based on this principle, we may construct a linear programming model M(I) as follows: $M (I) : {\begin{matrix} max D (ω) = \sum_{j = 1}^{n} \sum_{i = 1}^{m} \sum_{k = 1}^{m} DE (d_{ij}, d_{kj}) ω_{j} \\ s . t . \sum_{j = 1}^{n} ω_{j}^{2} = 1, ω_{j} \geq 0, 1 \leq j \leq n \end{matrix}$

By means of Lagrange multiplier method and normalization, we have $ω_{j} = \frac{\sum_{i = 1}^{m} \sum_{k = 1}^{m} DE (d_{ij}, d_{kj})}{\sum_{j = 1}^{n} \sum_{i = 1}^{m} \sum_{k = 1}^{m} DE (d_{ij}, d_{kj})}, 1 \leq j \leq n .$ (14)

Similarly, if we use the parameterized cross-entropy do construct the model M(I), we may obtain the attribute weights as follows:

$\begin{matrix} ω_{j} & = & \frac{\sum_{i = 1}^{m} \sum_{k = 1}^{m} {DE}^{p, q, r} (d_{ij}, d_{kj})}{\sum_{j = 1}^{n} \sum_{i = 1}^{m} \sum_{k = 1}^{m} {DE}^{p, q, r} (d_{ij}, d_{kj})}, \\ 1 \leq j \leq n, p, q, r > 0 \end{matrix}$ (15)

Case 2. If the attribute weights are partly known, and suppose the weights information be given as ω ∈ Φ, where Φ is the information set related to attribute weights. According to the thoughts of M(I), we can construct another linear programming model M(II) to determine the weights as follows: $M (II) : {\begin{matrix} max D (ω) = \sum_{j = 1}^{n} \sum_{i = 1}^{m} \sum_{k = 1}^{m} DE (d_{ij}, d_{kj}) ω_{j} \\ s . t . ω \in Φ, \sum_{j = 1}^{n} ω_{j} = 1, ω_{j} \geq 0, 1 \leq j \leq n \end{matrix}$

The model M(II) is a linear programming problem, which can be easily solved by many classical algorithms such as simple method, the interior point algorithm and so on.

Step 5. Calculate collective scores of each alternative by use of r = Sω^T, and rank all alternative according to their collective scores. So the optimal alternative x_{k
^*} is obtained by $k^{*} = arg max_{k} {r_{k} : 1 \leq k \leq m}$ .

5.2 A practical example

In order to illustrate the feasibility and reasonability of the proposed MADM method, we shall give a practical example about liquidity risk evaluation for banks. 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 {(0.1,0.3,0.3,0.5;1,1),

Low (ML) (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 {(0.5,0.7,0.7,0.9;1,1), (0.6,0.7,0.7,0.8;0.9,0.9)}

High (MH)

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	{(0.1,0.3,0.3,0.5;1,1),
Low (ML)	(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	{(0.5,0.7,0.7,0.9;1,1), (0.6,0.7,0.7,0.8;0.9,0.9)}
High (MH)
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)}

The financial crisis of 2007 that triggered the turmoil in the financial markets has demonstrated the central role that effective processes for managing liquidity risk play in maintaining both the stability of individual banks and the soundness of the entire banking system. In the narrow sense, liquidity risk is the potential inability of a bank to meet punctually and in a cost-effective way its envisaged contractual payment obligations when they fall due. Thus the liquidity for the bank takes on an essentially protective meaning that is able to continuously ensure a state of technical solvency, and liquidity risk evaluation is of great importance for banks all over the world.

Here, we suppose there are five banks (x₁, x₂, ⋯ , 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 (e₁), the ability to coordinate the issuing by the bank of short, medium and long term financing instruments (e₂), the ability to optimize the costs of refinancing, striking a trade-off balance between liquidity and profitability (e₃), 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 (e₄). 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.

Table 2

The evaluation information on five banks

Banks	e ₁	e ₂	e ₃	e 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 (as shown in Table 2).

Step 2. Since the four attributes are all benefit attributes, so we do not normalize the original matrix A, that is to say D = A.

Step 3. According to Equation (13), we calculate the expected scores of LMF and UMF and the overall expected score matrix as follows: $\begin{matrix} S^{LMF} & = & [\begin{matrix} 0.0795 & 0.0840 & 0.0795 & 0.0360 \\ 0.0840 & 0.0600 & 0.0795 & 0.0795 \\ 0.0005 & 0.0360 & 0.0795 & 0.0600 \\ 0.0795 & 0.0105 & 0.0005 & 0.0600 \\ 0.0795 & 0.0840 & 0.0600 & 0.0840 \end{matrix}], \\ S^{UMF} & = & [\begin{matrix} 0.1300 & 0.1400 & 0.1300 & 0.0600 \\ 0.1400 & 0.1000 & 0.1300 & 0.1300 \\ 0.0017 & 0.0600 & 0.1300 & 0.1000 \\ 0.1300 & 0.0200 & 0.0017 & 0.1000 \\ 0.1300 & 0.1400 & 0.1000 & 0.1400 \end{matrix}], \\ S & = & [\begin{matrix} 0.1047 & 0.1120 & 0.1047 & 0.0480 \\ 0.1120 & 0.0800 & 0.1047 & 0.1047 \\ 0.0011 & 0.0480 & 0.1047 & 0.0800 \\ 0.1047 & 0.0153 & 0.0011 & 0.0800 \\ 0.1047 & 0.1120 & 0.0800 & 0.1120 \end{matrix}] . \end{matrix}$

Step 4. If the attribute weights are completely unknown (Case 1), and we set some parameters value, such as p = 0.5, q = 0.3, r = 0.2, the weights could be calculated by use of Equations (14 and 15) based on Equations (5 and 9), respectively (see in Table 3).

Table 3

Attribute weights under different situation

	ω ₁	ω ₂	ω ₃	ω ₄
Equation (14)	0.4567	0.0432	0.4569	0.0432
Equation (15)	0.4671	0.0327	0.4675	0.0327
M(II) (Equation (5))	0.3000	0.1000	0.4000	0.2000
M(II) (Equation (9))	0.3000	0.1000	0.4000	0.2000

If the attribute weights are partly known (Case 2), and assume the information set of attribute weightsis ω ∈ Φ = {0.2 ≤ ω₁ ≤ 0.3, 0.1 ≤ ω₂ ≤ 0.2, 0.2 ≤ ω₃ ≤ 0.4 0.2 ≤ ω₄ ≤ 0.4}, we may obtain the weights by solving the model M(II), and find that the attribute weights are the same while using different cross-entropies (Equation (5 and 9)) in objective function of M(II) (as shown in Table 3).

Step 5. Calculate the collective scores of each bank by under different weights and rank all banks (as shown in Table 4).

Table 4

The collective scores under different entropies

	Equation (14)	Equation (15)	M(II)
x ₁	0.1026	0.1031	0.0941
x ₂	0.1070	0.1073	0.1044
x ₃	0.0539	0.0536	0.0630
x ₄	0.0525	0.0526	0.0494
x ₅	0.0941	0.0937	0.0970
Ranking Results	x₂ ≻ x₁ ≻ x₅	x₂ ≻ x₁ ≻ x₅	x₂ ≻ x₅ ≻ x₁
	≻x₃ ≻ x₄	≻x₃ ≻ x₄	≻x₃ ≻ x₄

It is worth noting that the bigger the score of a certain bank, the lower the corresponding liquidity risk. Therefore, we find that the ranking results are consistent while using Equations (14 and 15), and the bank 2 has the lowest liquidity risk and the bank 4 has the highest liquidity risk, that is to say bank 2 has the strongest solvency ability and stability. If we impose some information on attribute weights, we find the overall conclusion is changeless while the position between bank 1 and bank 5 is reversed, which implies that the weights information will play a role in ranking results.

5.3 Comparison with other existing methods

In order to verify the validity of the proposed MADM approach, we make some comparisons with other previous methods, such as ranking value method [6, 23], possibility degree method [14], by calculating the above example. In order to keep the consistency of weights information, we let weight vector ω = {0.3, 0.1, 0.4, 0.2} in these previous methods.

(1) According to the ranking value formula proposed by Chen et al. [6], and we may establish the ranking matrix and calculate the average agreement (AD), then the comprehensive ranking value of each bank is shown as follows: $\begin{matrix} R (x_{1}) & = & 4.1175, R (x_{2}) = 4.5136, R (x_{3}) = 3.6742, \\ R (x_{4}) & = & 2.9183, R (x_{5}) = 3.8451 \end{matrix}$

Thus the ranking order of all banks is x₂ ≻ x₁ ≻ x₅ ≻ x₃ ≻ x₄.

(2) By use of combined ranking value formula proposed by Qin & Liu [23] and IT2OWA operator, we have $\begin{matrix} R (x_{1}) & = & 1.6542, R (x_{2}) = 1.8426, R (x_{3}) = 1.6839, \\ R (x_{4}) & = & 1.4291, R (x_{5}) = 1.7366 \end{matrix}$

So all banks are ranked as x₂ ≻ x₅ ≻ x₁ ≻ x₃ ≻ x₄.

(3) Based on the possibility degree proposed by Hu et al. [14] and TIT2-WAA operator, we can obtain the possibility degree matrix P as follows: $P = [\begin{matrix} 0.5 & 0.4887 & 0.5078 & 0.5242 & 0.4962 \\ 0.5113 & 0.5 & 0.5191 & 0.5354 & 0.5075 \\ 0.4922 & 0.4809 & 0.5 & 0.5164 & 0.4809 \\ 0.4758 & 0.4646 & 0.4836 & 0.5 & 0.4720 \\ 0.5038 & 0.4925 & 0.5191 & 0.5280 & 0.5 \end{matrix}]$

Then we obtain the ranking vector r = (0.1008, 0.1037, 0.0985, 0.0948, 0.1022), so the ranking result is x₂ ≻ x₅ ≻ x₁ ≻ x₃ ≻ x₄.

All ranking results including both he proposed method and previous approaches could be shown in Table 5.

Table 5
Comparisons of four MADM methods

Methods Order of banks

Ranking value method [6] x₂ ≻ x₁ ≻ x₅ ≻ x₃ ≻ x₄

Combined value method [23] x₂ ≻ x₅ ≻ x₁ ≻ x₃ ≻ x₄

Possibility computing method [14] x₂ ≻ x₅ ≻ x₁ ≻ x₃ ≻ x₄

The above score function method x₂ ≻ x₅ ≻ x₁ ≻ x₃ ≻ x₄

(M(II))

Methods	Order of banks
Ranking value method [6]	x₂ ≻ x₁ ≻ x₅ ≻ x₃ ≻ x₄
Combined value method [23]	x₂ ≻ x₅ ≻ x₁ ≻ x₃ ≻ x₄
Possibility computing method [14]	x₂ ≻ x₅ ≻ x₁ ≻ x₃ ≻ x₄
The above score function method	x₂ ≻ x₅ ≻ x₁ ≻ x₃ ≻ x₄
(M(II))

From Table 5, it is clear that most of four methods have the same results, this verifies the method we proposed is reasonable and validity in this paper. We can also find some advantages while comparing with other previous methods as follows:

The method of determining attribute weights is more reasonable. We construct two programming models based on the maximizing cross-entropy principle, which divides the weights information into two cases: completely unknown and partly known. However, all weights in Chen et al. [6] are assumed equal and not consider the importance of attribute information. Qin & Liu [23] and Hu et al. [14] only consider the situation that weights are partly known. Meanwhile, the maximizing cross-entropy method considers simultaneously at least three aspect of information of both LMF and UMF: fuzziness, hesitancy and interval information. Comparing with these three methods, the flexible of our approach is much higher than previous methods, and our method has a solid theoretical foundation.

The proposed MADM method does not depend on the aggregating operators. In fact, different aggregating operators may loss different levels of information, which might lead to some contradict results. Such as Chen et al. [6] adopted arithmetic operations, Qin & Liu [23] used IT2OWA operator and Hu et al. [14] considered TIT2-WAA operator. However, our method transforms the decision matrix to a score matrix and avoids using aggregating operators, which changes the complex interval type-2 fuzzy information into the simple real numbers. Comparing with existing methods, the computational complexity of our approach is much lower and the decision-making process is more straightforward.

According to the above stated comparisons and analysis, our proposed MADM approach is better than other three methods.

6 Conclusions

Interval type-2 fuzzy sets (IT2FSs) have received increasing attention due to its great ability to deal with imprecise and ambiguous information in real word. The purpose of this paper is to develop a general framework of interval type-2 fuzzy information measures and propose a new MADM approach in interval type-2 fuzzy information environment. The main results could be concluded as follows:

Firstly, we point out, for the first time, that uncertainty of an IT2FS is composed of its fuzziness, hesitancy and interval information, which could be quantified by fuzzy factor, hesitant factor and interval factor. Thus, we initiate an interval type-2 fuzzy cross-entropy and entropy measures to describe the discrimination degree of uncertainty between two IT2FSs and uncertainty level of one IT2FS, respectively. We also study the inherent connection between two kinds of entropies. Meanwhile, we extend these information measures to parameterized case, and find interval type-2 fuzzy entropy could be viewed as the weighted average of fuzzy entropy, hesitant entropy and interval entropy.

Secondly, we construct two optimal models based on the maximizing cross-entropy principle to determine the attribute weights, and develop a novel MADM method in interval type-2 fuzzy information environment by using the proposed information measures. Meanwhile, a case study concerning banks’ liquidity risk evaluation is provided to illustrate the applicability and flexibility of the above approaches.

In future, we shall extend the information measures to general type-2 fuzzy environment, with expectations that could be made applicable to other similar decision-making problems, such as supplier selection, public transportation evaluation and social network analysis and so on.

Footnotes

Acknowledgments

The authors are highly grateful to any anonymous for their careful reading and insightful comments, and the views and opinions expressed are those of the authors. The work is supported by the National Nature Science Foundation of China (No. 71473036), the Talent Introduction Project of Anhui University and the Natural Science Key Project of Anhui Sanlian University (No. kjzd 2016 001).

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