A new approach for ranking of interval type-2 trapezoidal fuzzy numbers

Abstract

Interval type-2 fuzzy sets (IT2 FSs) provide us with additional degrees of freedom to represent the uncertainty and the fuzziness of the real word than traditional type-1 fuzzy sets (T1 FSs). In many applications, ranking of interval type-2 fuzzy numbers is an important component in the decision process. In this paper, the concepts of magnitude mean value and variance of interval type-2 trapezoidal fuzzy numbers (IT2 TrFNs) are introduced and some of their properties are studied.. Then, we propose a new magnitude mean-variance possibility degree method to solve ranking of interval type-2 trapezoidal fuzzy numbers. Finally, some comparative examples are used to illustrate the advantage of the proposed method.

Keywords

Interval type-2 fuzzy numbers magnitude mean value magnitude variance possibility degree

1 Introduction

More than 35 fuzzy numbers ranking methods have been proposed since 1976. Chen and Hwang [1] thoroughly reviewed the existing approaches, and pointed out some illogical conditions that arose among them. Chen [2], Choobineh and Li [3], Cheng [4] have presented some methods, and also more recently numerous ranking techniques have been proposed and investigated by Chu, Tsao [5] and Ma, Kandel and Friedman [6]. Dubois and Prade [7] defined an interval-valued expectation of fuzzy numbers, viewing them as consonant random set. They also showed that this expectation remained additive in the sense of addition of fuzzy numbers. Carlsson and Fullér [8] proposed the concepts of possibilistic mean value and variance of fuzzy numbers and application in fuzzy numbers ranking. Abbasbandy and Hajjari [9] presented a magnitude method for ranking of trapezoidal fuzzy numbers based on the left and the right spreads at some α-levels of trapezoidal fuzzy numbers. However, the above ranking methods are based on traditional type-1 fuzzy sets (T1 FSs).

The concept of type-2 fuzzy sets (T2 FSs), initially introduced by Zadeh [10] can be regarded as an extension of the concept of T1 FSs. The main difference between the two kinds of fuzzy sets is that the memberships of T1 FSs are crisp numbers whereas the memberships of T2 FSs are T1 FSs [11]; hence, T2 FSs involve more uncertainties than T1 FSs. Since its introduction, type-2 fuzzy sets are gaining more and more attention. Because the computational complexity of using general T2 FSs is very high, to date, interval type-2 fuzzy sets (IT2 FSs) [12] are the most widely used type-2 fuzzy sets and have been successfully applied to many practical fields [13 –17]. Thus, in many applications, ranking of interval type-2 fuzzy numbers (IT2 FNs) is an important component in decision process. However, few studies have focused on the ranking IT2 FNs. Mitchell [18] proposed a ranking method for general type-2 fuzzy sets based on random embedded type-1 fuzzy numbers. Chen and Lee [19] presented a likelihood method for calculating the ranking values of trapezoidal interval type-2 fuzzy sets. Hu and Zhang [20] proposed a possibility degree method to compare two trapezoidal interval type-2 fuzzy numbers. Wu and Mendel [21], Ghorabaee and Amiri [22] proposed a centroid method for ranking interval type-2 fuzzy numbers.

In this paper, we will introduce the concepts of magnitude mean value and variance of interval type-2 fuzzy number. The new possibility degree method is proposed to rank interval type-2 fuzzy numbers based on magnitude mean value and variance. The rest of this paper is organized as follows: Section 2 contains the basic definitions of interval type-2 fuzzy sets which are used in the remaining parts of the paper. In Section 3, we will introduce the concept of magnitude mean value and variance of interval type-2 trapezoidal fuzzy number (IT2 TrFN) and study some of their properties. In Section 4, we will also introduce a new possibility degree method based on the magnitude mean value and variance to ranking IT2 TrFNs. In Section5, we present some numerical examples to illustrate the advantages of the proposed approach to ranking interval type-2 fuzzy numbers. The conclusions are discussed in Section 6.

2 Some concepts of interval type-2 fuzzy sets

Definition 1. [23] Let $\tilde{A}$ be an IT2 FSs on X, thus $\tilde{A}$ can be fully characterized by its footprint of uncertainty (FOU), which is defined as the union of all primary memberships as follows: $FOU (\tilde{A}) = ⋃_{x \in X} [μ_{\tilde{A}}^{L} (x), μ_{\tilde{A}}^{U} (x)]$ (1)

Where FOU( $\tilde{A}$ ) is a bounded region that represents the uncertainty associated with the membership grades of $\tilde{A}$ .

Definition 2. [13, 23] Assume that $\tilde{A}$ is an IT2 FSs in the universal set X, the lower α-cut of $\tilde{A}$ , denoted as ${\tilde{A}}^{L} (α)$ , is the set of the elements in X whose lower grades are greater than or equal to a specific value α, i.e. ${\tilde{A}}^{L} (α) = {x \in X | μ_{\tilde{A}}^{L} (x) \geq α}$ . The upper α-cut of $\tilde{A}$ , denoted as ${\tilde{A}}^{U} (α)$ , is the set of the elements in X whose upper grades are greater than or equal to a specific value α, i.e. ${\tilde{A}}^{U} (α) = {x \in X | μ_{\tilde{A}}^{U} (x) \geq α}$ .

Definition 3. [24, 25] Assume that $\tilde{A}$ lower memb- ership function (LMF) and an upper membership function (UMF) are two type-1 membership functions that are bounds for the FOU ( $\tilde{A}$ ) of an IT2 FNs $\tilde{A}$ . Let two T1 FSs A^L : X → [0, 1] and A^U : X → [0, 1] be the lower and upper fuzzy sets, respectively, with respect to $\tilde{A}$ . The LMF μ_{A
^L} (x) and the UMF μ_{A
^U} (x) are associated with the lower bound ${FOU}^{L} (\tilde{A})$ and the upper bound ${FOU}^{U} (\tilde{A})$ , respectively, of $FOU (\tilde{A})$ and are defined as follows:

$\begin{matrix} A^{L} = {FOU}^{L} (A) = {(x, μ_{A^{L}} (x)) | x \in X} \\ A^{U} = {FOU}^{U} (A) = {(x, μ_{A^{U}} (x)) | x \in X} \end{matrix}$ (2) where 0 ≤ μ_{A
^L} (x) ≤ μ_{A
^U} (x) ≤1 for all x.

Chen [26] introduced the concept of generalized fuzzy numbers. The difference between traditional fuzzy numbers and generalized fuzzy numbers is that the height of a traditional fuzzy number is equal to unity, whereas the height of a generalized fuzzy number is between zero and one. Accordingly, the concept of a generalized trapezoidal fuzzy number is a trapezoid-shaped fuzzy number whose height is between zero and one [26, 27].

Definition 4. [28 –30] An interval type-2 fuzzy number is called interval type-2 trapezoidal fuzzy number (IT2 TrFN), where the UMF and LMF are both trapezoidal fuzzy numbers (see Fig. 1), i.e.

$\begin{matrix} \tilde{A} & = & [{\tilde{A}}^{U}, {\tilde{A}}^{L}] \\ = & [(a_{1}^{U}, a_{2}^{U}, a_{3}^{U}, a_{4}^{U}; h^{U}), (a_{1}^{L}, a_{2}^{L}, a_{3}^{L}, a_{4}^{L}; h^{L})] \end{matrix}$ (3) where h^L and h^U denote the heights of A^L and A^U, respectively.

Let ς ∈ {L, U}, the membership function of IT2 TrFN in A^ς is expressed as follows: $μ_{A^{ς}} (x) = {\begin{matrix} \frac{h^{ς} (x - a_{1}^{ς})}{a_{2}^{ς} - a_{1}^{ς}} & if a_{1}^{ς} \leq x \leq a_{2}^{ς} \\ h^{ς} & if a_{2}^{ς} \leq x \leq a_{3}^{ς} \\ \frac{h^{ς} (a_{4}^{ς} - x)}{a_{4}^{ς} - a_{3}^{ς}} & if a_{3}^{ς} \leq x \leq a_{4}^{ς} \\ 0 & otherwise \end{matrix}$ (4)

Definition 5. [31, 32] Suppose that ${\tilde{A}}_{1}$ and ${\tilde{A}}_{2}$ are two non-negative IT2 TrFNs, where ${\tilde{A}}_{1} = ({\tilde{A}}_{1}^{U}, {\tilde{A}}_{1}^{L}) = ((a_{11}^{U}, a_{12}^{U}, a_{13}^{U}, a_{14}^{U}; h_{1}^{U}), (a_{11}^{L}, a_{12}^{L}, a_{13}^{L}, a_{14}^{L}; h_{1}^{L})$ and ${\tilde{A}}_{2} = ({\tilde{A}}_{2}^{U}, {\tilde{A}}_{2}^{L}) = ((a_{21}^{U}, a_{22}^{U}, a_{23}^{U}, a_{24}^{U}; h_{2}^{U}), (a_{21}^{L}, a_{22}^{L}, a_{23}^{L}, a_{24}^{L}; h_{2}^{L})$ . The arithmetic operations between ${\tilde{A}}_{1}$ and ${\tilde{A}}_{2}$ are defined as follows:

Addition operation $\begin{matrix} {\tilde{A}}_{1} \oplus {\tilde{A}}_{2} & = & ({\tilde{A}}_{1}^{U}, {\tilde{A}}_{1}^{L}) \oplus ({\tilde{A}}_{2}^{U}, {\tilde{A}}_{2}^{L}) \\ = & ((a_{11}^{U} + a_{21}^{U}, a_{12}^{U} + a_{22}^{U}, a_{13}^{U} + a_{23}^{U}, \\ a_{14}^{U} + a_{24}^{U}; \min (h_{1}^{U}, h_{2}^{U})), \\ (a_{11}^{L} + a_{21}^{L}, a_{12}^{L} + a_{22}^{L}, a_{13}^{L} + a_{23}^{L}, \\ a_{14}^{L} + a_{24}^{L}; \min (h_{1}^{L}, h_{2}^{L}))) \end{matrix}$

Multiplication operation $\begin{matrix} {\tilde{A}}_{1} \otimes {\tilde{A}}_{2} & = & ({\tilde{A}}_{1}^{U}, {\tilde{A}}_{1}^{L}) \otimes ({\tilde{A}}_{2}^{U}, {\tilde{A}}_{2}^{L}) \\ = & ((a_{11}^{U} \times a_{21}^{U}, a_{12}^{U} \times a_{22}^{U}, a_{13}^{U} \times a_{23}^{U}, \\ a_{14}^{U} \times a_{24}^{U}; \min (h_{1}^{U}, h_{2}^{U})), \\ (a_{11}^{L} \times a_{21}^{L}, a_{12}^{L} \times a_{22}^{L}, a_{13}^{L} \times a_{23}^{L}, \\ a_{14}^{L} \times a_{24}^{L}; \min (h_{1}^{L}, h_{2}^{L}))) \end{matrix}$

Multiplication by real number operation $\begin{matrix} k {\tilde{A}}_{1} & = & k ({\tilde{A}}_{1}^{U}, {\tilde{A}}_{1}^{L}) \\ = & (({ka}_{11}^{U}, {ka}_{12}^{U}, {ka}_{13}^{U}, {ka}_{14}^{U}; h_{1}^{U}), \\ ({ka}_{11}^{L}, {ka}_{12}^{L}, {ka}_{13}^{L}, {ka}_{14}^{L}; h_{1}^{L})) \end{matrix}$ where k > 0.

Power operation $\begin{matrix} ({\tilde{A}}_{1})^{k} & = & ({\tilde{A}}_{1}^{U}, {\tilde{A}}_{1}^{L})^{k} \\ = & ((a_{11}^{U})^{k}, (a_{12}^{U})^{k}, (a_{13}^{U})^{k}, (a_{14}^{U})^{k}; h_{1}^{U}), \\ ((a_{11}^{L})^{k}, (a_{11}^{L})^{k}, (a_{12}^{L})^{k}, (a_{13}^{L})^{k}, (a_{14}^{L})^{k}; h_{1}^{L})) \end{matrix}$

3 The magnitude mean value and variance of IT2 TrFN

Definition 6. [9] A type-1 fuzzy number $\tilde{A}$ in parametric form is pair ( $\underline{μ}, \bar{μ}$ ) of functions $\underline{μ} (x),$ $\bar{μ} (x)$ , 0 ≤ x ≤ 1, which satisfies the following requirements:

$\underline{μ} (x)$ is a bounded monotonic increasing left continuous function.

$\bar{μ} (x)$ is a bounded monotonic decreasing left continuous function.

$\underline{μ} (x) \leq \bar{μ} (x), 0 \leq x \leq 1$ .

Obviously function $\underline{μ} (x), \bar{μ} (x)$ can be considered as α-cut set of A. The trapezoidal fuzzy number A = (x₀, y₀, ρ, β) be a trapezoidal fuzzy number with interval center [x₀, y₀], left-width ρ > 0 and right-width β > 0, its parametric form is $\begin{matrix} \underline{μ} (x) & = & x_{0} - (1 - x) ρ, \\ \bar{μ} (x) & = & y_{0} + (1 - x) β \end{matrix}$

Definition 7. [9] For an arbitrary trapezoidal type-1 fuzzy number u = (x₀, y₀, ρ, β), with parametric form $u = (\underline{u} (r), \bar{u} (r))$ , the magnitude of the trapezoidal fuzzy number u is defined as

$\begin{matrix} Mag (u) \\ = \frac{1}{2} (\int_{0}^{1} (\underline{u} (r) + \bar{u} (r) + x_{0} + y_{0}) f (r) dr) \end{matrix}$ (5) where the function f (r) is a non-negative and increasing function on [0,1] with f (0) =0, f (1) =1 and $\int_{0}^{1} f (r) dr = \frac{1}{2}$ . Obviously function f (r) can be considered as a weighting function. In actual applications, function f (r) can be chosen according to the actual situation.

Obviously function f (r) can be considered as a weighting function. Obviously, the magnitude of a trapezoidal fuzzy number u which is defined by (5), synthetically reflects the information on every membership degree, and meaning of this magnitude is visual and natural. The resulting scalar value is used to rank the type-1 fuzzy numbers. The larger Mag(u), the larger fuzzy number. Therefore, for any two trapezoidal type-1 fuzzy numbers u and v, the ranking of u and v by the Mag(.) are asfollows:

Mag (u)> Mag (v) if and only if u ≻ v

Mag (u)< Mag (v) if and only if u ≺ v

Mag (u)= Mag (v) if and only if u ∼ v

In this section, we extended the concept about the possibilistic mean value of type-1 fuzzy numbers of Carlsson and Fullér [8]. First, let $\tilde{A}$ be an interval type-2 trapezoidal fuzzy number with α-cut $\tilde{A} (α) = [{\tilde{A}}^{U} (α), {\tilde{A}}^{L} (α)]$ , where ${\tilde{A}}^{U} (α) = [{\tilde{A}}_{1}^{U} (α), {\tilde{A}}_{2}^{U} (α)]$ , ${\tilde{A}}^{L} (α) = [{\tilde{A}}_{1}^{L} (α), {\tilde{A}}_{2}^{L} (α)]$ , we introduce $M^{*} (\tilde{A})$ , the upper magnitude mean value of $\tilde{A}$ , as

$\begin{matrix} M^{*} (\tilde{A}) & = & \frac{1}{2} \int_{0}^{h^{U}} α ({\tilde{A}}_{1}^{U} (α) + a_{2}^{U} + {\tilde{A}}_{2}^{U} (α) + a_{3}^{U}) d α \\ = & \frac{1}{4} (\frac{\int_{0}^{h^{U}} α ({\tilde{A}}_{1}^{U} (α) + a_{2}^{U}) d α}{\int_{0}^{1} α d α} + \frac{\int_{0}^{h^{U}} α ({\tilde{A}}_{2}^{U} (α) + a_{3}^{U}) d α}{\int_{0}^{1} α d α}) \\ = & \frac{1}{4} (\frac{\int_{0}^{h^{U}} Pos [{\tilde{A}}^{U} \leq {\tilde{A}}_{1}^{U} (α)] ({\tilde{A}}_{1}^{U} (α) + a_{2}^{U}) d α}{\int_{0}^{1} Pos [{\tilde{A}}^{U} \leq {\tilde{A}}_{1}^{U} (α)] d α} + \frac{\int_{0}^{h^{U}} Pos [{\tilde{A}}^{U} \geq {\tilde{A}}_{2}^{U} (α)] ({\tilde{A}}_{2}^{U} (α) + a_{3}^{U}) d α}{\int_{0}^{1} Pos [{\tilde{A}}^{U} \geq {\tilde{A}}_{2}^{U} (α)] d α}) ∥ \end{matrix}$ (6)

Where Pos denotes possibility, i.e. $Pos [{\tilde{A}}^{U} \leq {\tilde{A}}_{1}^{U} (α) = \sup_{u \leq {\tilde{A}}_{1}^{U} (α)} {\tilde{A}}^{U} (α) = α$ and $Pos [{\tilde{A}}^{U} \geq {\tilde{A}}_{2}^{U} (α) = \sup_{u \geq {\tilde{A}}_{2}^{U} (α)} {\tilde{A}}^{U} (α) = α$ .

In a similar manner, we introduce $M_{*} (\tilde{A})$ , the lower magnitude mean value of $\tilde{A}$ , as

$\begin{matrix} M_{*} (\tilde{A}) & = & \frac{1}{2} \int_{0}^{h^{L}} α ({\tilde{A}}_{1}^{L} (α) + a_{2}^{L} + {\tilde{A}}_{2}^{L} (α) + a_{3}^{L}) d α \\ = & \frac{1}{4} (\frac{\int_{0}^{h^{L}} α ({\tilde{A}}_{1}^{L} (α) + a_{2}^{L}) d α}{\int_{0}^{1} α d α} + \frac{\int_{0}^{h^{L}} α ({\tilde{A}}_{2}^{L} (α) + a_{3}^{L}) d α}{\int_{0}^{1} α d α}) \\ = & \frac{1}{4} (\frac{\int_{0}^{h^{L}} Pos [{\tilde{A}}^{L} \leq {\tilde{A}}_{1}^{L} (α)] ({\tilde{A}}_{1}^{L} (α) + a_{2}^{L}) d α}{\int_{0}^{1} Pos [{\tilde{A}}^{L} \leq {\tilde{A}}_{1}^{L} (α)] d α} + \frac{\int_{0}^{h^{L}} Pos [{\tilde{A}}^{L} \geq {\tilde{A}}_{2}^{L} (α)] ({\tilde{A}}_{2}^{L} (α) + a_{3}^{L}) d α}{\int_{0}^{1} Pos [{\tilde{A}}^{L} \geq {\tilde{A}}_{2}^{L} (α)] d α}) \end{matrix}$ (7)

Where Pos denotes possibility, i.e. $Pos [{\tilde{A}}^{L} \leq {\tilde{A}}_{1}^{L} (α) = \sup_{u \leq {\tilde{A}}_{1}^{L} (α)} {\tilde{A}}^{L} (α) = α$ and $Pos [{\tilde{A}}^{L} \geq {\tilde{A}}_{2}^{L} (α) = \sup_{u \geq {\tilde{A}}_{2}^{L} (α)} {\tilde{A}}^{L} (α) = α$ .

Definition 8. The crisp magnitude mean value of IT2 TrFN $\tilde{A}$ is the arithmetic mean of its lower magnitude and upper magnitude mean values, i.e. $\bar{M} (\tilde{A}) = \frac{1}{2} (M^{*} (\tilde{A}) + M_{*} (\tilde{A}))$ (8)

We can get the α-cut of IT2 TrFN $\tilde{A} = [{\tilde{A}}^{U}, {\tilde{A}}^{L}] = [(a_{1}^{U}, a_{2}^{U}, a_{3}^{U}, a_{4}^{U}; h^{U}), (a_{1}^{L}, a_{2}^{L}, a_{3}^{L}, a_{4}^{L}; h^{L})]$ is

$\begin{matrix} {\tilde{A}}^{U} (α) & = & [{\tilde{A}}_{1}^{U} (α), {\tilde{A}}_{2}^{U} (α)] \\ = & [a_{1}^{U} + \frac{(a_{2}^{U} - a_{1}^{U}) α}{h^{U}}, a_{4}^{U} - \frac{(a_{4}^{U} - a_{3}^{U}) α}{h^{U}}] \\ {\tilde{A}}^{L} (α) & = & [{\tilde{A}}_{1}^{L} (α), {\tilde{A}}_{2}^{L} (α)] \\ = & [a_{1}^{L} + \frac{(a_{2}^{L} - a_{1}^{L}) α}{h^{L}}, a_{4}^{L} - \frac{(a_{4}^{L} - a_{3}^{L}) α}{h^{L}}] \end{matrix}$ (9)

hrule

Then we have

$\begin{matrix} \bar{M} (\tilde{A}) & = & \frac{1}{2} (M^{*} (\tilde{A}) + M_{*} (\tilde{A})) \\ = & \frac{(h^{U})^{2}}{24} (a_{1}^{U} + 5 a_{2}^{U} + 5 a_{3}^{U} + a_{4}^{U}) \\ + \frac{(h^{L})^{2}}{24} (a_{1}^{L} + 5 a_{2}^{L} + 5 a_{3}^{L} + a_{4}^{L}) \end{matrix}$ (10)

hrule

When $a_{i}^{U} = a_{i}^{L} (i = 1, 2, 3, 4)$ and h^U = h^L = 1, the interval type-2 trapezoidal fuzzy number reduces to trapezoidal fuzzy number, the crisp magnitude mean value is $\bar{M} (\tilde{A}) = \frac{1}{12} (a_{1}^{U} + 5 a_{2}^{U} + 5 a_{3}^{U} + a_{4}^{U})$ , which is the same as the definition proposed by Abbasbandy and Hajjari [9].

The following theorem shows two very important properties of $\bar{M} : Ψ \to R$ .

Theorem 1. Suppose that two IT2 TrFNs $\tilde{A}$ and $\tilde{B}$ have the same heights, where $\tilde{A} = ({\tilde{A}}^{U}, {\tilde{A}}^{L}) = ((a_{1}^{U}, a_{2}^{U}, a_{3}^{U}, a_{4}^{U}; h_{\tilde{A}}^{U}), (a_{1}^{L}, a_{2}^{L}, a_{3}^{L}, a_{4}^{L}; h_{\tilde{A}}^{L}))$ and $\tilde{B} = ({\tilde{B}}^{U}, {\tilde{B}}^{L}) = ((b_{1}^{U}, b_{2}^{U}, b_{3}^{U}, b_{4}^{U}; h_{\tilde{B}}^{U}), (b_{1}^{L}, b_{2}^{L}, b_{3}^{L}, b_{4}^{L}; h_{\tilde{B}}^{L}))$ . Let $λ \in R$ be a real number. Then $\bar{M} (\tilde{A} \oplus \tilde{B}) \leq \bar{M} (\tilde{A}) + \bar{M} (\tilde{B})$ (11) And $\bar{M} (λ \cdot \tilde{A}) = λ \bar{M} (\tilde{A})$ (12)

If and only if $h_{\tilde{A}}^{U} = h_{\tilde{B}}^{U}$ and $h_{\tilde{A}}^{L} = h_{\tilde{B}}^{L}$ , the Equation (11) is satisfied. Thus, the crisp magnitude mean value does not meet the additive property.

Proof. According to the addition operation of Definition 5, we have $\begin{matrix} \tilde{A} \oplus \tilde{B} & = & ({\tilde{A}}^{U}, {\tilde{A}}^{L}) \oplus ({\tilde{B}}^{U}, {\tilde{B}}^{L}) \\ = & ((a_{1}^{U} + b_{1}^{U}, a_{2}^{U} + b_{2}^{U}, a_{3}^{U} + b_{3}^{U}, a_{4}^{U} + b_{4}^{U}; \min (h_{\tilde{A}}^{U}, h_{\tilde{B}}^{U})), \\ (a_{1}^{L} + b_{1}^{L}, a_{2}^{L} + b_{2}^{L}, a_{3}^{L} + b_{3}^{L}, a_{4}^{L} + b_{4}^{L}; \min (h_{\tilde{A}}^{L}, h_{\tilde{B}}^{L}))) \end{matrix}$ then we can get the α-cut of IT2 TrFN $\tilde{A} \oplus \tilde{B}$ is $\begin{matrix} (\tilde{A} \oplus \tilde{B})^{U} (α) \\ = [(\tilde{A} \oplus \tilde{B})_{1}^{U} (α), (\tilde{A} \oplus \tilde{B})_{2}^{U} (α)] \\ = [a_{1}^{U} + b_{1}^{U} + \frac{(a_{2}^{U} + b_{2}^{U} - a_{1}^{U} - b_{1}^{U}) α}{min {h_{\tilde{A}}^{U}, h_{\tilde{B}}^{U}}}, a_{4}^{U} + b_{4}^{U} - \frac{(a_{4}^{U} + b_{4}^{U} - a_{3}^{U} - b_{3}^{U}) α}{min {h_{\tilde{A}}^{U}, h_{\tilde{B}}^{U}}}] \\ (\tilde{A} \oplus \tilde{B})^{L} (α) \\ = [(\tilde{A} \oplus \tilde{B})_{1}^{L} (α), (\tilde{A} \oplus \tilde{B})_{2}^{L} (α)] \\ = [a_{1}^{L} + b_{1}^{L} + \frac{(a_{2}^{L} + b_{2}^{L} - a_{1}^{L} - b_{1}^{L}) α}{min {h_{\tilde{A}}^{L}, h_{\tilde{B}}^{L}}}, a_{4}^{L} + b_{4}^{L} - \frac{(a_{4}^{L} + b_{4}^{L} - a_{3}^{L} - b_{3}^{L}) α}{min {h_{\tilde{A}}^{L}, h_{\tilde{B}}^{L}}}] \end{matrix}$

According to the Equation (10), we have

$\begin{matrix} \bar{M} (\tilde{A} \oplus \tilde{B}) & = & \frac{(min {h_{\tilde{A}}^{U}, h_{\tilde{B}}^{U}})^{2}}{24} [a_{1}^{U} + b_{1}^{U} + 5 (a_{2}^{U} + b_{2}^{U}) + 5 (a_{3}^{U} + b_{3}^{U}) + a_{4}^{U} + b_{4}^{U}] \\ + \frac{(min {h_{\tilde{A}}^{L}, h_{\tilde{B}}^{L}})^{2}}{24} [a_{1}^{L} + b_{1}^{L} + 5 (a_{2}^{L} + b_{2}^{L}) + 5 (a_{3}^{L} + b_{3}^{L}) + a_{4}^{L} + b_{4}^{L}] \end{matrix}$ (13)

On the other hand

$\begin{matrix} \bar{M} (\tilde{A}) & = & \frac{(h_{\tilde{A}}^{U})^{2}}{24} (a_{1}^{U} + 5 a_{2}^{U} + 5 a_{3}^{U} + a_{4}^{U}) + \frac{(h_{\tilde{A}}^{L})^{2}}{24} (a_{1}^{L} + 5 a_{2}^{L} + 5 a_{3}^{L} + a_{4}^{L}) \\ \bar{M} (\tilde{B}) & = & \frac{(h_{\tilde{B}}^{U})^{2}}{24} (b_{1}^{U} + 5 b_{2}^{U} + 5 b_{3}^{U} + b_{4}^{U}) + \frac{(h_{\tilde{B}}^{L})^{2}}{24} (b_{1}^{L} + 5 b_{2}^{L} + 5 b_{3}^{L} + b_{4}^{L}) \end{matrix}$ (14)

From the Equation (13) and (14), we have $\bar{M} (\tilde{A} \oplus \tilde{B}) \leq \bar{M} (\tilde{A}) + \bar{M} (\tilde{B})$

In the same way, we have $\begin{matrix} \bar{M} (λ \cdot \tilde{A}) = \frac{λ (h^{U})^{2}}{24} (a_{1}^{U} + 5 a_{2}^{U} + 5 a_{3}^{U} + a_{4}^{U}) \\ + \frac{λ (h^{L})^{2}}{24} (a_{1}^{L} + 5 a_{2}^{L} + 5 a_{3}^{L} + a_{4}^{L}) \\ = λ \bar{M} (\tilde{A}) \end{matrix}$

This ends the proof. □

Definition 9. [20] An IT2 TrFN $\tilde{A}$ is expressed $\tilde{A} = ({\tilde{A}}^{U}, {\tilde{A}}^{L}) = ((a_{1}^{U}, a_{2}^{U}, a_{3}^{U}, a_{4}^{U}; h^{U}), (a_{1}^{L}, a_{2}^{L}, a_{3}^{L}, a_{4}^{L}; h^{L})$ , then the expected value of $\tilde{A}$ is defined as follows:

hrule

$E (\tilde{A}) = \frac{(h^{U} + h^{L})}{16} \sum_{i = 1}^{4} (a_{i}^{U} + a_{i}^{L})$ (15)

When $a_{i}^{U} = a_{i}^{L} (i = 1, 2, 3, 4)$ and h^U = h^L = 1, the interval type-2 trapezoidal fuzzy number reduces to trapezoidal fuzzy number, the expected value is $E (\tilde{A}) = \frac{1}{4} (a_{1}^{U} + a_{2}^{U} + a_{3}^{U} + a_{4}^{U})$ , which is the same as the definition proposed by Liu [33].

We show now an important relationship between the expected value $E (\tilde{A})$ introduced in [20] and the crisp magnitude mean value $\bar{M} (\tilde{A})$ for an IT2 TrFN $\tilde{A}$ .

Theorem 2. Let $\tilde{A} = ({\tilde{A}}^{U}, {\tilde{A}}^{L}) = ((a_{1}^{U}, a_{2}^{U}, a_{3}^{U}, a_{4}^{U}; h^{U}), (a_{1}^{L}, a_{2}^{L}, a_{3}^{L}, a_{4}^{L}; h^{L})$ be a non-negative IT2 TrFN and h^U = h^L = h, (0 ≤ h ≤ 1), then the expected value $E (\tilde{A})$ and the crisp magnitude mean value $\bar{M} (\tilde{A})$ have

If $0 \leq h \leq \frac{3}{5}$ , then $E (\tilde{A}) \geq \bar{M} (\tilde{A})$ .

If $\frac{3}{5} \leq h \leq 1$ then $E (\tilde{A}) \geq \bar{M} (\tilde{A})$ if and only if $\frac{3 - h}{5 h - 3} \geq \frac{a_{2}^{U} + a_{3}^{U} + a_{2}^{L} + a_{3}^{L}}{a_{1}^{U} + a_{1}^{L} + a_{4}^{U} + a_{4}^{L}}$ .

if $\tilde{A}$ is a symmetric IT2 TrFN, i.e. $a_{1}^{U} + a_{4}^{U} = a_{2}^{U} + a_{3}^{U}$ and $a_{1}^{L} + a_{4}^{L} = a_{2}^{L} + a_{3}^{L}$ , then for any 0 ≤ h ≤ 1, then $E (\tilde{A}) \geq \bar{M} (\tilde{A})$ .

Proof. Firstly, we find $\begin{matrix} \bar{M} (\tilde{A}) & = & \frac{h^{2}}{24} (a_{1}^{U} + 5 a_{2}^{U} + 5 a_{3}^{U} + a_{4}^{U} \\ + a_{1}^{L} + 5 a_{2}^{L} + 5 a_{3}^{L} + a_{4}^{L}) \\ E (\tilde{A}) & = & \frac{h}{8} (a_{1}^{U} + a_{2}^{U} + a_{3}^{U} + a_{4}^{U} \\ + a_{1}^{L} + a_{2}^{L} + a_{3}^{L} + a_{4}^{L}) \end{matrix}$

Therefore, we get $\begin{matrix} E (\tilde{A}) - \bar{M} (\tilde{A}) \\ = \frac{1}{24} [(3 h - h^{2}) (a_{1}^{U} + a_{1}^{L} + a_{4}^{U} + a_{4}^{L}) \\ + (3 h - 5 h^{2}) (a_{2}^{U} + a_{3}^{U} + a_{2}^{L} + a_{3}^{L})] \end{matrix}$

Obviously, for ∀h ∈ [0, 1], 3h - h² ≥ 0, if $0 \leq h \leq \frac{3}{5}$ , then 3h - 5h² ≥ 0, therefore $E (\tilde{A}) \geq \bar{M} (\tilde{A})$ .

hrule

If $\frac{3}{5} \leq h \leq 1$ then 3h - 5h² ≤ 0, we have $E (\tilde{A}) \geq \bar{M} (\tilde{A})$ if and only if $\frac{3 - h}{5 h - 3} \geq \frac{a_{2}^{U} + a_{3}^{U} + a_{2}^{L} + a_{3}^{L}}{a_{1}^{U} + a_{1}^{L} + a_{4}^{U} + a_{4}^{L}}$

If $\tilde{A}$ is a symmetric IT2 TrFN, i.e. $a_{1}^{U} + a_{4}^{U} = a_{2}^{U} + a_{3}^{U}$ and $a_{1}^{L} + a_{4}^{L} = a_{2}^{L} + a_{3}^{L}$ , we get $\begin{matrix} E (\tilde{A}) - \bar{M} (\tilde{A}) \\ = \frac{1}{4} h (1 - h) (a_{1}^{U} + a_{1}^{L} + a_{4}^{U} + a_{4}^{L}) \geq 0 \end{matrix}$

This ends the proof. □

We also introduce the (magnitude) variance of $\tilde{A}$ as

Definition 10. An IT2 TrFN $\tilde{A}$ is expressed $\tilde{A} = ({\tilde{A}}^{U}, {\tilde{A}}^{L}) = ((a_{1}^{U}, a_{2}^{U}, a_{3}^{U}, a_{4}^{U}; h^{U}), (a_{1}^{L}, a_{2}^{L}, a_{3}^{L}, a_{4}^{L}; h^{L})$ , then the variance of $\tilde{A}$ is defined asfollows: $Var (\tilde{A}) = \frac{1}{2} ({Var}^{*} (\tilde{A}) + {Var}_{*} (\tilde{A}))$ (16)

Where ${Var}^{*} (\tilde{A})$ and ${Var}_{*} (\tilde{A})$ are defined asfollows $\begin{matrix} {Var}^{*} (\tilde{A}) \\ = \int_{0}^{h^{U}} {α (\frac{{\tilde{A}}_{1}^{U} (α) + a_{2}^{U} + {\tilde{A}}_{2}^{U} (α) + a_{3}^{U}}{2} - {\tilde{A}}_{1}^{U} (α) - a_{2}^{U})}^{2} d α \\ + \int_{0}^{h^{U}} {α (\frac{{\tilde{A}}_{1}^{U} (α) + a_{2}^{U} + {\tilde{A}}_{2}^{U} (α) + a_{3}^{U}}{2} - {\tilde{A}}_{2}^{U} (α) - a_{3}^{U})}^{2} d α \\ = \frac{1}{2} \int_{0}^{h^{U}} {α ({\tilde{A}}_{2}^{U} (α) + a_{3}^{U} - {\tilde{A}}_{1}^{U} (α) - a_{2}^{U})}^{2} d α \\ {Var}_{*} (\tilde{A}) \\ = \int_{0}^{h^{L}} {α (\frac{{\tilde{A}}_{1}^{L} (α) + a_{2}^{L} + {\tilde{A}}_{2}^{L} (α) + a_{3}^{L}}{2} - {\tilde{A}}_{1}^{L} (α) - a_{2}^{L})}^{2} d α \\ + \int_{0}^{h^{L}} {α (\frac{{\tilde{A}}_{1}^{L} (α) + a_{2}^{L} + {\tilde{A}}_{2}^{L} (α) + a_{3}^{L}}{2} - {\tilde{A}}_{2}^{L} (α) - a_{3}^{L})}^{2} d α \\ = \frac{1}{2} \int_{0}^{h^{L}} {α ({\tilde{A}}_{2}^{L} (α) + a_{3}^{L} - {\tilde{A}}_{1}^{L} (α) - a_{2}^{L})}^{2} d α \end{matrix}$

From the Equation (9), then we have

$\begin{matrix} Var (\tilde{A}) & = & \frac{1}{2} ({Var}^{*} (\tilde{A}) + {Var}_{*} (\tilde{A})) \\ = & \frac{(h^{U})^{2}}{2} [\begin{matrix} \frac{(a_{4}^{U} - a_{1}^{U} + a_{3}^{U} - a_{2}^{U})^{2}}{4} - \frac{(a_{4}^{U} - a_{1}^{U} + a_{3}^{U} - a_{2}^{U}) (a_{4}^{U} - a_{1}^{U} + a_{2}^{U} - a_{3}^{U})}{3} \\ + \frac{(a_{4}^{U} - a_{1}^{U} + a_{2}^{U} - a_{3}^{U})^{2}}{8} \end{matrix}] \\ + \frac{(h^{L})^{2}}{2} [\begin{matrix} \frac{(a_{4}^{L} - a_{1}^{L} + a_{3}^{L} - a_{2}^{L})^{2}}{4} - \frac{(a_{4}^{L} - a_{1}^{L} + a_{3}^{L} - a_{2}^{L}) (a_{4}^{L} - a_{1}^{L} + a_{2}^{L} - a_{3}^{L})}{3} \\ + \frac{(a_{4}^{L} - a_{1}^{L} + a_{2}^{L} - a_{3}^{L})^{2}}{8} \end{matrix}] \\ = & \frac{(m^{U})^{2} + 6 m^{U} n^{U} + 17 (n^{U})^{2}}{48} (h^{U})^{2} + \frac{(m^{L})^{2} + 6 m^{L} n^{L} + 17 (n^{L})^{2}}{48} (h^{L})^{2} \end{matrix}$ (17)

Where $m^{U} = a_{4}^{U} - a_{1}^{U}$ , $n^{U} = a_{3}^{U} - a_{2}^{U}$ and $m^{L} = a_{4}^{L} - a_{1}^{L}$ , $n^{L} = a_{3}^{L} - a_{2}^{L}$ . If $a_{i}^{U} = a_{i}^{L}$ (i = 1, 2, 3, 4) and h^U = h^L = 1, the interval type-2 trapezoidal fuzzy number reduces to trapezoidal fuzzy number, the crisp magnitude mean value is $Var (\tilde{A}) = \frac{(m^{U})^{2} + 6 m^{U} n^{U} + 17 (n^{U})^{2}}{24}$ .

The variance of $\tilde{A}$ is defined as the expected value of the squared deviations between the arithmetic mean and the endpoints of its level sets, i.e. the lower magnitude-weighted average of the squared distance between the left-hand endpoint and the arithmetic mean of the endpoints of its level sets plus the upper magnitude-weighted average of the squared distance between the right-hand endpoint and the arithmetic mean of the endpoints of its level sets. We show now that the variance of a fuzzy number is invariant to shifting.

The standard deviation of $\tilde{A}$ is defined by $σ_{\tilde{A}} = \sqrt{Var (\tilde{A})}$ (18)

Theorem 3. Let $θ \in R$ , if an IT2 TrFN $\tilde{A}$ is shifted by value θ then we get a new IT2 TrFN $\tilde{B}$ , satisfying the property $\tilde{B} (x) = \tilde{A} (x + θ)$ for all x. Then the variance of $\tilde{B}$ satisfies the following relations $Var (\tilde{B}) = Var (\tilde{A})$ (19)

Proof. The α-cut of IT2 TrFN $\tilde{B} = [(a_{1}^{U} + θ, a_{2}^{U} + θ, a_{3}^{U} + θ, a_{4}^{U} + θ; h^{U}), (a_{1}^{L} + θ, a_{2}^{L} + θ, a_{3}^{L} + θ, a_{4}^{L} + θ; h^{L})]$ is $\begin{matrix} {\tilde{B}}^{U} (α) & = & [{\tilde{A}}_{1}^{U} (α) + θ, {\tilde{A}}_{2}^{U} (α) + θ] \\ {\tilde{B}}^{L} (α) & = & [{\tilde{A}}_{1}^{L} (α) + θ, {\tilde{A}}_{2}^{L} (α) + θ] \end{matrix}$

hrule

Then, we have $\begin{matrix} Var (\tilde{B}) \\ = \frac{1}{2} ({Var}^{*} (\tilde{B}) + {Var}_{*} (\tilde{B})) \\ = \frac{1}{4} (\int_{0}^{h^{U}} {α (({\tilde{A}}_{2}^{U} (α) + a_{3}^{U} + θ) - ({\tilde{A}}_{1}^{U} (α) + a_{2}^{U} + θ))}^{2} d α \\ + \int_{0}^{h^{L}} {α (({\tilde{A}}_{2}^{L} (α) + a_{3}^{L} + θ) - ({\tilde{A}}_{1}^{L} (α) + a_{2}^{L} + θ))}^{2} d α) \\ = \frac{1}{2} ({Var}^{*} (\tilde{A}) + {Var}_{*} (\tilde{A})) = Var (\tilde{A}) \end{matrix}$

This ends the proof. □

Definition 11. Suppose that two IT2 TrFNs $\tilde{A}$ and $\tilde{B}$ have the same heights, where $\tilde{A} = ({\tilde{A}}^{U}, {\tilde{A}}^{L}) = ((a_{1}^{U}, a_{2}^{U}, a_{3}^{U}, a_{4}^{U}; h_{\tilde{A}}^{U}), (a_{1}^{L}, a_{2}^{L}, a_{3}^{L}, a_{4}^{L}; h_{\tilde{A}}^{L})$ and $\tilde{B} = ({\tilde{B}}^{U}, {\tilde{B}}^{L}) = ((b_{1}^{U}, b_{2}^{U}, b_{3}^{U}, b_{4}^{U}; h_{\tilde{B}}^{U}), (b_{1}^{L}, b_{2}^{L}, b_{3}^{L}, b_{4}^{L}; h_{\tilde{B}}^{L})$ . The covariance between IT2 TrFNs $\tilde{A}$ and $\tilde{B}$ is defined as

$\begin{matrix} Cov (\tilde{A}, \tilde{B}) \\ = \frac{1}{4} (\int_{0}^{h^{U}} α ({\tilde{A}}_{2}^{U} (α) + a_{3}^{U} - {\tilde{A}}_{1}^{U} (α) - a_{2}^{U}) \\ ({\tilde{B}}_{2}^{U} (α) + b_{3}^{U} - {\tilde{B}}_{1}^{U} (α) - b_{2}^{U}) d α \\ + \int_{0}^{h^{L}} α ({\tilde{A}}_{2}^{L} (α) + a_{3}^{L} - {\tilde{A}}_{1}^{L} (α) - a_{2}^{L}) \\ ({\tilde{B}}_{2}^{L} (α) + b_{3}^{L} - {\tilde{B}}_{1}^{L} (α) - b_{2}^{L}) d α) \end{matrix}$ (20)

Taking into consideration that $supp (\tilde{A}) = [{\tilde{A}}^{U} (0), {\tilde{A}}^{L} (0)]$ , where ${\tilde{A}}^{U} (0) = [{\tilde{A}}_{1}^{U} (0), {\tilde{A}}_{2}^{U} (0)]$ , ${\tilde{A}}^{L} (0) = [{\tilde{A}}_{1}^{L} (0), {\tilde{A}}_{2}^{L} (0)]$ , and $supp (\tilde{B}) = [{\tilde{B}}^{U} (0), {\tilde{B}}^{L} (0)]$ , where ${\tilde{B}}^{U} (0) = [{\tilde{B}}_{1}^{U} (0), {\tilde{B}}_{2}^{U} (0)]$ , ${\tilde{B}}^{L} (0) = [{\tilde{B}}_{1}^{L} (0), {\tilde{B}}_{2}^{L} (0)]$ , we find that the covariance measures how much the products of α-levels sets of two fuzzy numbers A and B are close to the product of α-levels sets of the universal fuzzy sets in supp( $\tilde{A}$ ) and in supp( $\tilde{B}$ ).

From $\begin{matrix} (a_{3}^{U} + {\tilde{A}}_{2}^{U} (0) - a_{2}^{U} + {\tilde{A}}_{1}^{U} (0)) (b_{3}^{U} + {\tilde{B}}_{2}^{U} (0) - b_{2}^{U} + {\tilde{B}}_{1}^{U} (0)) \\ \geq (a_{3}^{U} + {\tilde{A}}_{2}^{U} (α) - a_{2}^{U} + {\tilde{A}}_{1}^{U} (α)) (b_{3}^{U} + {\tilde{B}}_{2}^{U} (α) - b_{2}^{U} + {\tilde{B}}_{1}^{U} (α)) > 0 \end{matrix}$ and $\begin{matrix} (a_{3}^{L} + {\tilde{A}}_{2}^{L} (0) - a_{2}^{L} + {\tilde{A}}_{1}^{L} (0)) (b_{3}^{L} + {\tilde{B}}_{2}^{L} (0) - b_{2}^{L} + {\tilde{B}}_{1}^{L} (0)) \\ \geq (a_{3}^{L} + {\tilde{A}}_{2}^{L} (α) - a_{2}^{L} + {\tilde{A}}_{1}^{L} (α)) (b_{3}^{L} + {\tilde{B}}_{2}^{L} (α) - b_{2}^{L} + {\tilde{B}}_{1}^{L} (α)) > 0 \end{matrix}$

For any α ∈ [0, 1], it follows that $\begin{matrix} \frac{(h^{U})^{2}}{8} ({\tilde{A}}_{2}^{U} (0) + a_{3}^{U} - {\tilde{A}}_{1}^{U} (0) - a_{2}^{U}) ({\tilde{B}}_{2}^{U} (0) + b_{3}^{U} - {\tilde{B}}_{1}^{U} (0) - b_{2}^{U}) \\ + \frac{(h^{L})^{2}}{8} ({\tilde{A}}_{2}^{L} (0) + a_{3}^{L} - {\tilde{A}}_{1}^{L} (0) - a_{2}^{L}) ({\tilde{B}}_{2}^{L} (0) + b_{3}^{L} - {\tilde{B}}_{1}^{L} (0) - b_{2}^{L}) \\ > Cov (\tilde{A}, \tilde{B}) > 0 \end{matrix}$ which holds for any pair of IT2 TrFNs $\tilde{A}$ and $\tilde{B}$ in supp( $\tilde{A}$ ) and in supp( $\tilde{B}$ ), respectively.

For any two IT2 TrFNs $\tilde{A}$ and $\tilde{B}$ , then we have following situations.

If the crisp magnitude mean values and variance are $\bar{M} (\tilde{A}) = \bar{M} (\tilde{B})$ and $Var (\tilde{A}) = Var (\tilde{B})$ , then the ranking order is $\tilde{A} \sim \tilde{B}$ .

If the crisp magnitude mean values is $\bar{M} (\tilde{A}) = \bar{M} (\tilde{B})$ and $Var (\tilde{A}) < Var (\tilde{B})$ , then the ranking order is $\tilde{A} ≻ \tilde{B}$ .

If the crisp magnitude mean values is $\bar{M} (\tilde{A}) = \bar{M} (\tilde{B})$ and $Var (\tilde{A}) > Var (\tilde{B})$ , then the ranking order is $\tilde{A} ≺ \tilde{B}$ .

If the crisp magnitude mean values is $\bar{M} (\tilde{A}) < \bar{M} (\tilde{B})$ , then the ranking order is $\tilde{A} ≺ \tilde{B}$ .

4 The new mean-variance possibility degree of IT2 TrFN

In this section, we present a new possibility degree method for calculating the ranking values of interval type-2 trapezoidal fuzzy sets.

Definition 12. [34] Two interval numbers a = [a^L, a^U] and a = [a^L, a^U], then the possibility degree is defined as follows:

$\begin{matrix} p (a \geq b) \\ = min {max (\frac{a^{U} - b^{L}}{a^{U} - a^{L} + b^{U} - b^{L}}, 0), 1} \end{matrix}$ (21)

hrule

We use the “3σ” principle in statistics, the confidence intervals of the two IT2 TrFNs $\tilde{A}$ and $\tilde{B}$ are ${\tilde{A}}_{CI} = [\bar{M} (\tilde{A}) - 3 \sqrt{Var (\tilde{A})}, \bar{M} (\tilde{A}) + 3 \sqrt{Var (\tilde{A})}]$ and ${\tilde{B}}_{CI} = [\bar{M} (\tilde{B}) - 3 \sqrt{Var (\tilde{B})}, \bar{M} (\tilde{B}) + 3 \sqrt{Var (\tilde{B})}]$ , respectively. Therefore, we use confidence intervals to define the magnitude mean-variance possibility degree as follows:

$\begin{matrix} p (\tilde{A} \geq \tilde{B}) \\ = min {max (\frac{\bar{M} (\tilde{A}) + 3 \sqrt{Var (\tilde{A})} - \bar{M} (\tilde{B}) + 3 \sqrt{Var (\tilde{B})}}{6 \sqrt{Var (\tilde{A})} + 6 \sqrt{Var (\tilde{B})}}, 0), 1} \end{matrix}$ (22)

If n interval type-2 trapezoidal fuzzy numbers need to be compared, then two corresponding possibility degree matrixes can be obtained as follows: $P = (p_{ij})_{n \times n} = (p ({\tilde{A}}_{i} \geq {\tilde{A}}_{j}))_{n \times n}$ (23)

Then, the ranking value of interval trapezoidal type-2 fuzzy set $Rank ({\tilde{A}}_{i})$ is calculated as follows [35] $Rank ({\tilde{A}}_{i}) = \frac{1}{n (n - 1)} (\sum_{k = 1}^{n} p ({\tilde{A}}_{i} ≻ {\tilde{A}}_{k}) + \frac{n}{2} - 1)$ (24)

Where 1 ≤ i ≤ n and $\sum_{i = 1}^{n} Rank ({\tilde{A}}_{i}) = 1$ . The larger ranking value $Rank ({\tilde{A}}_{i})$ , the greater the IT2 FS ${\tilde{A}}_{i}$ .

5 Numerical example

To illustrate the rationality and efficiency of the crisp magnitude mean value and variance in ranking IT2 TrFNs, some examples proposed by them are employed to compare the new method with the existing methods.

Example 1. In this section, we consider the 32 word FOUs of IT2 TrFNs from Wu and Mendel [21], the magnitude mean values and variances are shown in Table 1.

From Table 1, visual examination shows that the magnitude mean-variance ranking is reasonable; it also coincides with the meanings of the words. We can see the range of magnitude mean ranking method ( $\bar{M} ({\tilde{A}}_{i})$ ) from the smallest term to largest term is [0.15, 9.9], while the range of centroid-based ranking method ( $c ({\tilde{A}}_{i})$ ) from the smallest term to largest term is [0.47, 9.69]. Obviously, the distribution of the former is more reasonable. The four smallest terms and the four largest terms are more compact, thus variance is relatively less. The magnitude variance (MVar) is less than centroid variance (CVar) in the four smallest terms and the four largest terms. The biggest advantage of our method is that it does not require complex calculations; yet centroid-based method should be computed by KM algorithms.

Example 2. In the following, we use thirteen fuzzy sets provided by Bortolan and Degani [36] which are used for computing and comparing the ranking values. We compute the mean-variance possible degrees to compare those fuzzy numbers based on different definitions proposed by Chang et al. [37], Chen and Lee [19], Hu et al. [20] and Equation (22) developed by us respectively. The calculated results are shown in Table 2.

According to Table 2, some drawbacks of other methods can be observed. Comparisons between these existing methods and the proposed one can be described as follows:

From Set 1, the same results can be obtained based on methods proposed by Chang et al., Hu et al., and us, respectively, while Chen’s method fails to produce a correct order. The reason is that Chen’s method, overall, has enhanced the influence of the value of right branch.

From Set 10, the result from Chen’s method is inconsistent with other approaches, because the membership effect of the interval type-2 fuzzy number has been enlarged.

From Set 9 and Set 11, although the result from Hu et al.’s method is consistent with other approaches, but the value $Rank ({\tilde{A}}_{1}) = 0$ and $Rank ({\tilde{A}}_{2}) = 1$ is unreasonable, because all elements of UMF and LMF are of an equal status.

According to the above examples, we know that the new method is simpler than some existing methods for ranking interval type-2 fuzzy numbers. Besides, the new method ranks fuzzy numbers based on two different criteria, namely, the magnitude mean value and variance of the interval type-2 fuzzy numbers. So, the new method can overcome the shortcomings of some existing methods for ranking fuzzy numbers. The simplicity of the proposed method is its major advantage. The computational steps of the proposed method are less than those of other methods, while it gets the same results in most cases.

6 Conclusions

We introduce the notations of crisp magnitude mean value, variance of interval type-2 fuzzy numbers, and some of their properties are studied in this paper. The relationship of crisp magnitude mean and the expected value given by Hu et al. is investigated. These concepts are consistent with the extension principle and with the well-known definition in probability theory. Then, we propose a new mean-variance possibility degree method to solve ranking problems of interval type-2 fuzzy numbers. The proposed method can effectively rank interval type-2 fuzzy numbers and their images. We also used comparative examples to illustrate the advantages of the proposed method.

Footnotes

Acknowledgments

The work was partially supported by the Natural Science Foundation of Jiangsu Province of China (No. BK20130242), the Fundamental Research Fund for the Central University of China (Nos. 2015B28014, 2015B23914).

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