Some new signed distances and similarity measures of interval type-2 trapezoidal fuzzy numbers and comparative study

Abstract

In many applications, we need to numerically express the difference of two objects (notions) by means of distance and similarity between the corresponding interval type-2 fuzzy sets (IT2 FSs). In this paper, we propose two new signed distances between interval type-2 trapezoidal fuzzy numbers (IT2 TrFNs) and some properties of their have been introduced. We also give an approach to construct similarity measures using the signed distance for IT2 TrFNs. Several illustrative examples are given to demonstrate the practicality and effectiveness of the proposed measures.

Keywords

Interval type-2 fuzzy numbers signed distance similarity measure

1 Introduction

The concept of type-2 fuzzy sets (T2 FSs), initially introduced by Zadeh [1] can be regarded as an extension of the concept of type-1 fuzzy sets (T1 FSs). The main difference between type-1 fuzzy sets and type-2 fuzzy sets is that the memberships of T1 FSs are crisp numbers whereas the memberships of T2 FSs are type-1 fuzzy numbers [2]; hence, T2 FSs involve more uncertainties than T1 FSs. Since its introduction, type-2 fuzzy sets are receiving 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) [3] are the most widely used type-2 fuzzy sets and have been successfully applied to many practical fields [4 –8]. Thus, in many applications, we need to numerically express the difference of two objects by means of distance and similarity between the corresponding interval type-2 fuzzy numbers (IT2 FNs). Distance measures and similarity measures have attracted attention due to their diverse applications in various areas such as remote sensing, data mining, pattern recognition and multivariate data analysis. Many researchers have conducted extensive studies on distance measures for IT2 FSs. A number of distance measures for precise numbers have been established in the literature. However, a logical problem arises when the distance is computed in an imprecise framework due to the inherent vagueness [9, 10]. Burillo et al. [11] defined the normalized Hamming and the normalized Euclidean distances, only involving two parameters, the membership degree and the non-membership degree in describing intuitionistic fuzzy sets. Atanassov [12] suggested a direct generalization of the Hamming and Euclidean distances used in classical set theory for intuitionistic fuzzy sets and interval valued fuzzy sets. Hung et al. [13] proposed a method to compute the distance between intuitionistic fuzzy sets on the basis of the Hausdorff distance. Grzegorzewski [14] gave a definition of the interval-valued fuzzy set distance based on the Hausdorff metric. Chen [15] presented a signed distance-based method to handle interval type-2 fuzzy multi-criteria group decision-making problems. Wu and Mendel [16] compared the existing some similarity measures [17 –19] based on real survey data for interval type-2 fuzzy numbers. Wei et al. [20] also given an approach to construct similarity measures using entropy measures for interval-valued intuitionistic fuzzy sets. Zhang et al. [21] presented some new entropy measures for interval-valued intuitionistic fuzzy sets and discuss their relationships with similarity measures and inclusion measures. Zhang et al. [22] proposed a new axiomatic definition of entropy of interval-valued fuzzy sets and discusses its relation with similarity measure.

The main purpose of our work here is to present some suitable distances formula for interval type-2 trapezoidal fuzzy sets (IT2 TrFSs), i.e. the continuous weighted Hamming distances, the continuous weighted Hamming distances and the continuous weighted Minkowski distances. We also study some properties of these distances, and then give an approach to construct signed distance and signed-based similarity measures for IT2 TrFNs. The rest of this paper is organized as follows: Section 2 contains the basic definitions of interval type-2 fuzzy sets are used in the remaining parts of the paper. In Section 3, we will introduce some definitions of distances and similarity measures for IT2 TrFNs and some of their properties are studied. In Section 4, we demonstrate the practicality and effectiveness of the proposed measures. The conclusions are discussed in Section 5.

2 Some concepts of interval type-2 fuzzy sets

Definition 1. [23] let $\tilde{A}$ 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. [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 membership 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, and are defined as follows:

$\begin{matrix} A^{L} & = & FO U^{L} (A) = {(x, μ_{A^{L}} (x)) | x \in X} \\ A^{U} & = & FO U^{U} (A) = {(x, μ_{A^{U}} (x)) | x \in X} \end{matrix}$ (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.

Definition 4. [26, 27] 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.

Fig. 1.

A geometrical interpretation of an IT2 TrFN $\tilde{A}$ .

$\begin{matrix} \tilde{A} & = & [{\tilde{A}}^{U}, {\tilde{A}}^{L}] \\ = & [(a_{1}^{U}, a_{2}^{U}, a_{3}^{U}, a_{4}^{U}; h^{U}), \\ (a_{1}^{L}, a_{2}^{L}, a_{3}^{L}, a_{4}^{L}; h^{L})] \end{matrix}$ (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)

3 New signed distances and similarity measures between IT2 TrFNs

Definition 5. A real function d is called a metric distance, if d has the following properties [21]

\bar{D_{3} (\tilde{A}, \tilde{B})}

$\begin{matrix} = \frac{1}{2} \sqrt{\int_{0}^{h^{U}} α [{({\tilde{A}}_{1}^{U} (α) - {\tilde{B}}_{1}^{U} (α))}^{2} + {({\tilde{A}}_{2}^{U} (α) - {\tilde{B}}_{2}^{U} (α))}^{2}] d α + \int_{0}^{h^{L}} α [{({\tilde{A}}_{1}^{L} (α) - {\tilde{B}}_{1}^{L} (α))}^{2} + {({\tilde{A}}_{2}^{L} (α) - {\tilde{B}}_{2}^{L} (α))}^{2}] d α} \end{matrix}$ (7) $\begin{matrix} D_{4} (\tilde{A}, \tilde{B}) \\ = \frac{1}{2} \sqrt{\int_{0}^{h^{U}} α {({\tilde{A}}_{1}^{U} (α) + {\tilde{A}}_{2}^{U} (α) - {\tilde{B}}_{1}^{U} (α) - {\tilde{B}}_{2}^{U} (α))}^{2} d α + \int_{0}^{h^{L}} α {({\tilde{A}}_{1}^{L} (α) + {\tilde{A}}_{2}^{L} (α) - {\tilde{B}}_{1}^{L} (α) - {\tilde{B}}_{2}^{L} (α))}^{2} d α} \end{matrix}$ (8)

$d (\tilde{A}, \tilde{B}) \geq 0$ , and $d (\tilde{A}, \tilde{B}) = 0 \Leftrightarrow \tilde{A} = \tilde{B}$ , for any two IT2 FSs and $\tilde{B}$ .

$d (\tilde{A}, \tilde{B}) = d (\tilde{B}, \tilde{A})$ , for any two IT2 FSs $\tilde{A}$ and $\tilde{B}$

$d (\tilde{A}, \tilde{B}) + d (\tilde{B}, \tilde{C}) \geq d (\tilde{A}, \tilde{C})$ , for any three IT2 FSs $\tilde{A}$ , $\tilde{B}$ and $\tilde{C}$ .

Definition 6. Suppose that two IT2 TrFNs $\tilde{A}$ and $\tilde{B}$ have the same heights, where $\tilde{A} = ({\tilde{A}}^{U}, {\tilde{A}}^{L}) = ((a_{1}^{U}, a_{2}^{U}, a_{3}^{U}, a_{4}^{U}; h^{U}), (a_{1}^{L}, a_{2}^{L}, a_{3}^{L}, a_{4}^{L}; h^{L})$ and $\tilde{B} = ({\tilde{B}}^{U}, {\tilde{B}}^{L}) = ((b_{1}^{U}, b_{2}^{U}, b_{3}^{U}, b_{4}^{U}; h^{U}), (b_{1}^{L}, b_{2}^{L}, b_{3}^{L}, b_{4}^{L}; h^{L})$ . Then the continuous weighted Hamming distances are defined as follows: $\begin{matrix} D_{1} (\tilde{A}, \tilde{B}) & = & \frac{1}{2} (\int_{0}^{h^{U}} α (| {\tilde{A}}_{1}^{U} (α) - {\tilde{B}}_{1}^{U} (α) | + | {\tilde{A}}_{2}^{U} (α) - {\tilde{B}}_{2}^{U} (α) |) d α \\ + \int_{0}^{h^{L}} α (| ({\tilde{A}}_{1}^{L} (α) - {\tilde{B}}_{1}^{L} (α)) | + | {\tilde{A}}_{2}^{L} (α) - {\tilde{B}}_{2}^{L} (α) |) d α) \end{matrix}$ (5) $\begin{matrix} D_{2} (\tilde{A}, \tilde{B}) & = & \frac{1}{2} (\int_{0}^{h^{U}} α | {\tilde{A}}_{1}^{U} (α) + {\tilde{A}}_{2}^{U} (α) - {\tilde{B}}_{1}^{U} (α) - {\tilde{B}}_{2}^{U} (α) | d α \\ + \int_{0}^{h^{L}} α | {\tilde{A}}_{1}^{L} (α) + {\tilde{A}}_{2}^{L} (α) - {\tilde{B}}_{1}^{L} (α) - {\tilde{B}}_{2}^{L} (α) | d α) \end{matrix}$ (6)

It follows that the continuous weighted Hamming distances are nothing else but the level-weighted average of the arithmetic means of all α-cut sets. That is the weight of the arithmetic mean of α-cut sets absolute difference of two IT2 TrFNs $\tilde{A}$ and $\tilde{B}$ .

Definition 7. Suppose that two IT2 TrFNs $\tilde{A}$ and $\tilde{B}$ have the same heights, where $\tilde{A} = ({\tilde{A}}^{U}, {\tilde{A}}^{L}) = ((a_{1}^{U}, a_{2}^{U}, a_{3}^{U}, a_{4}^{U}; h^{U}), (a_{1}^{L}, a_{2}^{L}, a_{3}^{L}, a_{4}^{L}; h^{L})$ and $\tilde{B} = ({\tilde{B}}^{U}, {\tilde{B}}^{L}) = ((b_{1}^{U}, b_{2}^{U}, b_{3}^{U}, b_{4}^{U}; h^{U}), (b_{1}^{L}, b_{2}^{L}, b_{3}^{L}, b_{4}^{L}; h^{L})$ . Then the continuous weighted Euclidean distances are defined as follows:

Definition 8. Suppose that two IT2 TrFNs $\tilde{A}$ and $\tilde{B}$ have the same heights, where $\tilde{A} = ({\tilde{A}}^{U}, {\tilde{A}}^{L}) = ((a_{1}^{U}, a_{2}^{U}, a_{3}^{U}, a_{4}^{U}; h^{U}), (a_{1}^{L}, a_{2}^{L}, a_{3}^{L}, a_{4}^{L}; h^{L})$ and $\tilde{B} = ({\tilde{B}}^{U}, {\tilde{B}}^{L}) = ((b_{1}^{U}, b_{2}^{U}, b_{3}^{U}, b_{4}^{U}; h^{U}), (b_{1}^{L}, b_{2}^{L}, b_{3}^{L}, b_{4}^{L}; h^{L})$ . Let r ∈ N⁺ be a real number. Then the continuous weighted Minkowski distances are defined as follows:

$\begin{matrix} D_{r 1} (\tilde{A}, \tilde{B}) & = & \frac{1}{2} (\frac{1}{2} \int_{0}^{h^{U}} f (α) ({| {\tilde{A}}_{1}^{U} (α) - {\tilde{B}}_{1}^{U} (α) |}^{r} + {| {\tilde{A}}_{2}^{U} (α) - {\tilde{B}}_{2}^{U} (α) |}^{r}) d α \\ {+ \frac{1}{2} \int_{0}^{h^{L}} f (α) ({| ({\tilde{A}}_{1}^{L} (α) - {\tilde{B}}_{1}^{L} (α)) |}^{r} + {| {\tilde{A}}_{2}^{L} (α) - {\tilde{B}}_{2}^{L} (α) |}^{r}) d α)}^{1 / r} \end{matrix}$ (9) $\begin{matrix} D_{r 2} (\tilde{A}, \tilde{B}) & = & \frac{1}{2} (\frac{1}{2} \int_{0}^{h^{U}} f (α) {| {\tilde{A}}_{1}^{U} (α) + {\tilde{A}}_{2}^{U} (α) - {\tilde{B}}_{1}^{U} (α) - {\tilde{B}}_{2}^{U} (α) |}^{r} d α \\ {+ \frac{1}{2} \int_{0}^{h^{L}} f (α) {| {\tilde{A}}_{1}^{L} (α) + {\tilde{A}}_{2}^{L} (α) - {\tilde{B}}_{1}^{L} (α) - {\tilde{B}}_{2}^{L} (α) |}^{r} d α)}^{1 / r} \end{matrix}$ (10) where f is non-negative, monotone increasing and satisfies the normalization condition $\int_{0}^{1} f (α) d α = 1$ . If f (α) =2α and r = 1, 2 then $D_{1} (\tilde{A}, \tilde{B})$ , $D_{3} (\tilde{A}, \tilde{B})$ and $D_{2} (\tilde{A}, \tilde{B}), D_{4} (\tilde{A}, \tilde{B})$ are the special cases of the continuous weighted Minkowski distance $D_{r 1} (\tilde{A}, \tilde{B})$ and $D_{r 2} (\tilde{A}, \tilde{B})$ , respectively.

Theorem 1. The proposed distances formula (5) and (10) satisfies all three conditions in Definition 5.

Proof. The first and the second properties being trivially true for (5) and (10), we only establish the third property. Without loss of generality, we take $D_{1} (\tilde{A}, \tilde{B})$ as an example, we have: $\begin{matrix} D_{1} (\tilde{A}, \tilde{B}) + D_{1} (\tilde{B}, \tilde{C}) \\ = \frac{1}{2} (\int_{0}^{h^{U}} α (| {\tilde{A}}_{1}^{U} (α) - {\tilde{B}}_{1}^{U} (α) | + | {\tilde{A}}_{2}^{U} (α) - {\tilde{B}}_{2}^{U} (α) |) d α \\ + \int_{0}^{h^{L}} α (| ({\tilde{A}}_{1}^{L} (α) - {\tilde{B}}_{1}^{L} (α)) | + | {\tilde{A}}_{2}^{L} (α) - {\tilde{B}}_{2}^{L} (α) |) d α) \\ + \frac{1}{2} (\int_{0}^{h^{U}} α (| {\tilde{B}}_{1}^{U} (α) - {\tilde{C}}_{1}^{U} (α) | + | {\tilde{B}}_{2}^{U} (α) - {\tilde{C}}_{2}^{U} (α) |) d α \\ + \int_{0}^{h^{L}} α (| ({\tilde{B}}_{1}^{L} (α) - {\tilde{C}}_{1}^{L} (α)) | + | {\tilde{B}}_{2}^{L} (α) - {\tilde{C}}_{2}^{L} (α) |) d α) \end{matrix}$

We easily find $\begin{matrix} | {\tilde{A}}_{1}^{U} (α) - {\tilde{B}}_{1}^{U} (α) | + | {\tilde{B}}_{1}^{U} (α) - {\tilde{C}}_{1}^{U} (α) | \\ \geq | {\tilde{A}}_{1}^{U} (α) - {\tilde{B}}_{1}^{U} (α) + {\tilde{B}}_{1}^{U} (α) - {\tilde{C}}_{1}^{U} (α) | \\ = | {\tilde{A}}_{1}^{U} (α) - {\tilde{C}}_{1}^{U} (α) | \\ | {\tilde{A}}_{2}^{U} (α) - {\tilde{B}}_{2}^{U} (α) | + | {\tilde{B}}_{2}^{U} (α) - {\tilde{C}}_{2}^{U} (α) | \\ \geq | {\tilde{A}}_{2}^{U} (α) - {\tilde{B}}_{2}^{U} (α) + {\tilde{B}}_{2}^{U} (α) - {\tilde{C}}_{2}^{U} (α) | \\ = | {\tilde{A}}_{2}^{U} (α) - {\tilde{C}}_{2}^{U} (α) | \end{matrix}$

and $\begin{matrix} | {\tilde{A}}_{1}^{L} (α) - {\tilde{B}}_{1}^{L} (α) | + | {\tilde{B}}_{1}^{L} (α) - {\tilde{C}}_{1}^{L} (α) | \\ \geq | {\tilde{A}}_{1}^{L} (α) - {\tilde{B}}_{1}^{L} (α) + {\tilde{B}}_{1}^{L} (α) - {\tilde{C}}_{1}^{L} (α) | \\ = | {\tilde{A}}_{1}^{L} (α) - {\tilde{C}}_{1}^{L} (α) | \\ | {\tilde{A}}_{2}^{L} (α) - {\tilde{B}}_{2}^{L} (α) | + | {\tilde{B}}_{2}^{L} (α) - {\tilde{C}}_{2}^{L} (α) | \\ \geq | {\tilde{A}}_{2}^{L} (α) - {\tilde{B}}_{2}^{L} (α) + {\tilde{B}}_{2}^{L} (α) - {\tilde{C}}_{2}^{L} (α) | \\ = | {\tilde{A}}_{2}^{L} (α) - {\tilde{C}}_{2}^{L} (α) | \end{matrix}$

Then we get $\begin{matrix} D_{1} (\tilde{A}, \tilde{B}) + D_{1} (\tilde{B}, \tilde{C}) \\ \geq \frac{1}{2} (\int_{0}^{h^{U}} α (| {\tilde{A}}_{1}^{U} (α) - {\tilde{C}}_{1}^{U} (α) | + | {\tilde{A}}_{2}^{U} (α) - {\tilde{C}}_{2}^{U} (α) |) d α \\ + \int_{0}^{h^{L}} α (| ({\tilde{A}}_{1}^{L} (α) - {\tilde{C}}_{1}^{L} (α)) | + | {\tilde{A}}_{2}^{L} (α) - {\tilde{C}}_{2}^{L} (α) |) d α) \\ = D_{1} (\tilde{A}, \tilde{C}) \end{matrix}$

which ends the proof. □

We can get the α-cut of IT2 TrFN $\tilde{A} = [{\tilde{A}}^{U}, {\tilde{A}}^{L}] [(a_{1}^{U}, a_{2}^{U}, a_{3}^{U}, a_{4}^{U}; h^{U}), (a_{1}^{L}, a_{2}^{L}, a_{3}^{L}, a_{4}^{L}; h^{L})]$ and $\tilde{B} = ({\tilde{B}}^{U}, {\tilde{B}}^{L}) = ((b_{1}^{U}, b_{2}^{U}, b_{3}^{U}, b_{4}^{U}; h^{U}), (b_{1}^{L}, b_{2}^{L}, b_{3}^{L}, b_{4}^{L}; h^{L})$ is $\begin{matrix} {\tilde{A}}^{U} (α) = [{\tilde{A}}_{1}^{U} (α), {\tilde{A}}_{2}^{U} (α)] \\ = [a_{1}^{U} + \frac{(a_{2}^{U} - a_{1}^{U}) α}{h^{U}}, a_{4}^{U} - \frac{(a_{4}^{U} - a_{3}^{U}) α}{h^{U}}] \end{matrix}$ (11) $\begin{matrix} {\tilde{A}}^{L} (α) = [{\tilde{A}}_{1}^{L} (α), {\tilde{A}}_{2}^{L} (α)] \\ = [a_{1}^{L} + \frac{(a_{2}^{L} - a_{1}^{L}) α}{h^{L}}, a_{4}^{L} - \frac{(a_{4}^{L} - a_{3}^{L}) α}{h^{L}}] \end{matrix}$ and $\begin{matrix} {\tilde{B}}^{U} (α) = [{\tilde{B}}_{1}^{U} (α), {\tilde{B}}_{2}^{U} (α)] \\ = [b_{1}^{U} + \frac{(b_{2}^{U} - b_{1}^{U}) α}{h^{U}}, b_{4}^{U} - \frac{(b_{4}^{U} - b_{3}^{U}) α}{h^{U}}] \end{matrix}$ (12) $\begin{matrix} {\tilde{B}}^{L} (α) = [{\tilde{B}}_{1}^{L} (α), {\tilde{B}}_{2}^{L} (α)] \\ = [b_{1}^{L} + \frac{(b_{2}^{L} - b_{1}^{L}) α}{h^{L}}, b_{4}^{L} - \frac{(b_{4}^{L} - b_{3}^{L}) α}{h^{L}}] \end{matrix}$

In order to facilitate the calculation, this paper mainly study the continuous weighted Euclidean distances, then we have

$\begin{matrix} D_{3} (\tilde{A}, \tilde{B}) & = & \frac{1}{2} ({(h^{U})}^{2} [\frac{{(m_{1}^{U})}^{2} + {(m_{2}^{U})}^{2}}{2} + \frac{2 (m_{1}^{U} n_{1}^{U} + m_{2}^{U} n_{2}^{U})}{3} + \frac{{(n_{1}^{U})}^{2} + {(n_{2}^{U})}^{2}}{4}] \\ {+ {(h^{L})}^{2} [\frac{{(m_{1}^{L})}^{2} + {(m_{2}^{L})}^{2}}{2} + \frac{2 (m_{1}^{L} n_{1}^{L} + m_{2}^{L} n_{2}^{L})}{3} + \frac{{(n_{1}^{L})}^{2} + {(n_{2}^{L})}^{2}}{4}])}^{1 / 2} \end{matrix}$ (13) where $m_{1}^{U} = a_{1}^{U} - b_{1}^{U}, n_{1}^{U} = a_{2}^{U} + b_{1}^{U} - a_{1}^{U} - b_{2}^{U}$ and $m_{2}^{U} = a_{4}^{U} - b_{4}^{U}, n_{2}^{U} = a_{3}^{U} + b_{4}^{U} - a_{4}^{U} - b_{3}^{U}$ and $m_{1}^{L} = a_{1}^{L} - b_{1}^{L}, n_{1}^{L} = a_{2}^{L} + b_{1}^{L} - a_{1}^{L} - b_{2}^{L}$ and $m_{2}^{L} = a_{4}^{L} - b_{4}^{L}, n_{2}^{L} = a_{3}^{L} + b_{4}^{L} - a_{4}^{L} - b_{3}^{L}$ .

and

$\begin{matrix} D_{4} (\tilde{A}, \tilde{B}) \\ = \frac{1}{2} ({(h^{U})}^{2} [\frac{{(p^{U})}^{2}}{2} + \frac{2 p^{U} q^{U}}{3} + \frac{{(q^{U})}^{2}}{4}] \\ + {{(h^{L})}^{2} [\frac{{(p^{L})}^{2}}{2} + \frac{2 p^{L} q^{L}}{3} + \frac{{(q^{L})}^{2}}{4}])}^{1 / 2} \end{matrix}$ (14) where $p^{U} = a_{1}^{U} + a_{4}^{U} - b_{1}^{U} - b_{4}^{U}, q^{U} = a_{2}^{U} + a_{3}^{U} + b_{1}^{U} + b_{4}^{U} - a_{1}^{U} - a_{4}^{U} - b_{2}^{U} - b_{3}^{U}$ and $p^{L} = a_{1}^{L} + a_{4}^{L} - b_{1}^{L} - b_{4}^{L}, q^{L} = a_{2}^{L} + a_{3}^{L} + b_{1}^{L} + b_{4}^{L} - a_{1}^{L} - a_{4}^{L} - b_{2}^{L} - b_{3}^{L}$ .

From the above the continuous weighted Euclidean distances, we also can define the signed distance as follows:

Definition 9. Let level 1 fuzzy number ${\tilde{0}}_{1}$ map onto the y-axis at x = 0, and $\tilde{A} = [(a_{1}^{U}, a_{2}^{U}, a_{3}^{U}, a_{4}^{U}; h^{U}), (a_{1}^{L}, a_{2}^{L}, a_{3}^{L}, a_{4}^{L}; h^{L})]$ be a non-negative IT2 TrFN. The signed distance from $\tilde{A}$ to ${\tilde{0}}_{1}$ is expressed as follows:

$\begin{matrix} D_{3} (\tilde{A}, {\tilde{0}}_{1}) & = & \frac{1}{2} (\frac{{(h^{U})}^{2}}{12} [{(a_{1}^{U} + a_{2}^{U})}^{2} + 2 {(a_{2}^{U})}^{2} + 2 {(a_{3}^{U})}^{2} + {(a_{3}^{U} + a_{4}^{U})}^{2}] \\ {+ \frac{{(h^{L})}^{2}}{12} [{(a_{1}^{L} + a_{2}^{L})}^{2} + 2 {(a_{2}^{L})}^{2} + 2 {(a_{3}^{L})}^{2} + {(a_{3}^{L} + a_{4}^{L})}^{2}])}^{1 / 2} \end{matrix}$ (15) and

$\begin{matrix} D_{4} (\tilde{A}, {\tilde{0}}_{1}) & = & \frac{1}{2} (\frac{{(h^{U})}^{2}}{12} [{(a_{1}^{U} + a_{2}^{U} + a_{3}^{U} + a_{4}^{U})}^{2} + 2 {(a_{2}^{U} + a_{3}^{U})}^{2}] \\ + {\frac{{(h^{L})}^{2}}{12} [{(a_{1}^{L} + a_{2}^{L} + a_{3}^{L} + a_{4}^{L})}^{2} + 2 {(a_{2}^{L} + a_{3}^{L})}^{2}])}^{1 / 2} \end{matrix}$ (16)

Property 1. Let $\tilde{A} = [(a_{1}^{U}, a_{2}^{U}, a_{3}^{U}, a_{4}^{U}; h^{U}), (a_{1}^{L}, a_{2}^{L}, a_{3}^{L}, a_{4}^{L}; h^{L})]$ be a non-negative IT2 TrFN. $\tilde{A}$ is located at ${\tilde{0}}_{1}$ if and only if $D_{3} (\tilde{A}, {\tilde{0}}_{1}) = 0, or D_{4} (\tilde{A}, {\tilde{0}}_{1}) = 0 .$

Property 2. Let $\tilde{A}, \tilde{B}$ and ${\tilde{0}}_{1}$ be three non-negative IT2 TrFNs, where ${\tilde{0}}_{1} = [(0, 0, 0, 0; 1), (0, 0, 0, 0; 1)]$ . $\tilde{A}$ is farther from ${\tilde{0}}_{1}$ than $\tilde{B}$ if and only if $D_{3} (\tilde{A}, {\tilde{0}}_{1}) > D_{3} (\tilde{B}, {\tilde{0}}_{1})$ or $D_{4} (\tilde{A}, {\tilde{0}}_{1}) > D_{4} (\tilde{B}, {\tilde{0}}_{1})$ .

Theorem 2. Let $\tilde{A} = [(a_{1}^{U}, a_{2}^{U}, a_{3}^{U}, a_{4}^{U}; h^{U}), (a_{1}^{L}, a_{2}^{L}, a_{3}^{L}, a_{4}^{L}; h^{L})]$ be a non-negative IT2 TrFN, then $D_{4} (\tilde{A}, {\tilde{0}}_{1}) \geq D_{3} (\tilde{A}, {\tilde{0}}_{1})$ .

Proof. For $\tilde{A}$ be a non-negative IT2 TrFN, then we have $\begin{matrix} (a_{1}^{U} + a_{2}^{U} + a_{3}^{U} + a_{4}^{U})^{2} \\ \geq (a_{1}^{U} + a_{2}^{U})^{2} + (a_{3}^{U} + a_{4}^{U})^{2} \\ (a_{1}^{L} + a_{2}^{L} + a_{3}^{L} + a_{4}^{L})^{2} \\ \geq (a_{1}^{L} + a_{2}^{L})^{2} + (a_{3}^{L} + a_{4}^{L})^{2} \end{matrix}$ and $\begin{matrix} (a_{2}^{U} + a_{3}^{U})^{2} \geq (a_{2}^{U})^{2} + (a_{3}^{U})^{2} \\ (a_{2}^{L} + a_{3}^{L})^{2} \geq (a_{2}^{L})^{2} + (a_{3}^{L})^{2} \end{matrix}$ thus $\begin{matrix} (a_{1}^{U} + a_{2}^{U} + a_{3}^{U} + a_{4}^{U})^{2} + 2 (a_{2}^{U} + a_{3}^{U})^{2} \\ \geq (a_{1}^{U} + a_{2}^{U})^{2} + (a_{3}^{U} + a_{4}^{U})^{2} + 2 (a_{2}^{U})^{2} + 2 (a_{3}^{U})^{2} \\ (a_{1}^{L} + a_{2}^{L} + a_{3}^{L} + a_{4}^{L})^{2} + 2 (a_{2}^{L} + a_{3}^{L})^{2} \\ \geq (a_{1}^{L} + a_{2}^{L})^{2} + (a_{3}^{L} + a_{4}^{L})^{2} + 2 (a_{2}^{L})^{2} + 2 (a_{3}^{L})^{2} \end{matrix}$ then we get $D_{4} (\tilde{A}, {\tilde{0}}_{1}) \geq D_{3} (\tilde{A}, {\tilde{0}}_{1}) . □$

Considering the above mentioned signed distance, the distance between two interval type-2 trapezoidal fuzzy numbers and $\tilde{B}$ is calculated the signed-based distance as follows: $d_{1} (\tilde{A}, \tilde{B}) = | D_{3} (\tilde{A}, {\tilde{0}}_{1}) - D_{3} (\tilde{B}, {\tilde{0}}_{1}) |$ (17) $d_{2} (\tilde{A}, \tilde{B}) = | D_{4} (\tilde{A}, {\tilde{0}}_{1}) - D_{4} (\tilde{B}, {\tilde{0}}_{1}) |$ (18)

Theorem 3. The proposed signed-based distance formulas (17) and (18) satisfy all three metric distance conditions in Definition 5.

Proof. The first and the second properties being trivially true for (17) and (18), we only establish the third property. Without loss of generality, we take $d_{1} (\tilde{A}, \tilde{B})$ as an example, we have: $\begin{matrix} d_{1} (\tilde{A}, \tilde{B}) + d_{1} (\tilde{B}, \overset{⌢}{C}) \\ = | D_{3} (\tilde{A}, {\tilde{0}}_{1}) - D_{3} (\tilde{B}, {\tilde{0}}_{1}) | \\ + | D_{3} (\tilde{B}, {\tilde{0}}_{1}) - D_{3} (\tilde{C}, {\tilde{0}}_{1}) | \\ \geq | D_{3} (\tilde{A}, {\tilde{0}}_{1}) - D_{3} (\tilde{B}, {\tilde{0}}_{1}) \\ + D_{3} (\tilde{B}, {\tilde{0}}_{1}) - D_{3} (\tilde{C}, {\tilde{0}}_{1}) | \\ = d_{1} (\tilde{A}, \overset{⌢}{C}) \end{matrix}$ which ends the proof. □

Definition 10. A real function S is called the similarity measure between two IT2 TrFSs, if S has the following properties [20, 22]

$S (\tilde{A}, \tilde{B}) = 1 \Leftrightarrow \tilde{A} = \tilde{B}$ , for any two IT2 TrFSs $\tilde{A}$ and $\tilde{B}$ .

$S (\tilde{A}, \tilde{B}) = S (\tilde{B}, \tilde{A}),$ for any two IT2 TrFSs $\tilde{A}$ and $\tilde{B}$ .

For all three IT2 Tr FSs $\tilde{A}$ , $\tilde{B}$ and $\tilde{C}$ if $\tilde{A} \subseteq \tilde{B} \subseteq \tilde{C}$ , then $S (\tilde{A}, \tilde{C}) \leq S (\tilde{A}, \tilde{B})$ and $S (\tilde{A}, \tilde{C}) \leq S (\tilde{B}, \tilde{C})$ .

Theorem 4. For any two IT2 TrFSs $\tilde{A}$ and $\tilde{B}$ , then let $S_{1} (\tilde{A}, \tilde{B}) = \frac{1}{d_{1} (\tilde{A}, \tilde{B}) + 1}$ (19) $S_{2} (\tilde{A}, \tilde{B}) = \frac{1}{d_{2} (\tilde{A}, \tilde{B}) + 1}$ (20) then $S_{1} (\tilde{A}, \tilde{B})$ and $S_{2} (\tilde{A}, \tilde{B})$ are the similarity measures between two IT2 TrFSs.

Proof. The first and the second properties being trivially true for (19) and (20), we only establish the third property. Without loss of generality, we take $S_{1} (\tilde{A}, \tilde{B})$ as an example,

If $\tilde{A} \subseteq \tilde{B} \subseteq \tilde{C}$ , then we have $\begin{matrix} d_{1} (\tilde{A}, \tilde{B}) = | D_{3} (\tilde{A}, 0) - D_{3} (\tilde{B}, 0) | \\ \leq | D_{3} (\tilde{A}, 0) - D_{3} (\tilde{C}, 0) | = d_{1} (\tilde{A}, \tilde{C}) \\ d_{1} (\tilde{B}, \tilde{C}) = | D_{3} (\tilde{B}, 0) - D_{3} (\tilde{C}, 0) | \\ \leq | D_{3} (\tilde{A}, 0) - D_{3} (\tilde{C}, 0) | = d_{1} (\tilde{A}, \tilde{C}) \end{matrix}$

Therefore $\begin{matrix} S_{1} (\tilde{A}, \tilde{B}) \frac{1}{d_{1} (\tilde{A}, \tilde{B}) + 1} \\ \geq \frac{1}{d_{1} (\tilde{A}, \tilde{C}) + 1} = S_{1} (\tilde{A}, \tilde{C}) \\ S_{1} (\tilde{B}, \tilde{C}) = \frac{1}{d_{1} (\tilde{B}, \tilde{C}) + 1} \\ \geq \frac{1}{d_{1} (\tilde{A}, \tilde{C}) + 1} = S_{1} (\tilde{A}, \tilde{C}) \end{matrix}$

Which ends the proof. □

4 Comparative example

To compare the performance of the proposed signed distance and similarity measure, we use 32 word FOUs with the UMF and LMF values as shown in Table 1, which was already presented by Wu and Mendel [16].

Table 1
Parameters of the 32 word FOUs

Rank Word UMF LMF

1 None to very (0,0, 0.14, 1.97;1) (0,0, 0.05, 0.66;1)

2 Teeny-weeny (0,0, 0.14, 1.97;1) (0,0, 0.01, 0.13;1)

3 A smidgen (0,0, 0.26, 2.63;1) (0,0, 0.05, 0.63;1)

4 Tiny (0,0, 0.36, 2.63;1) (0,0, 0.05, 0.63;1)

5 Very small (0,0, 0.64, 2.47;1) (0,0, 0.10, 1.16;1)

6 Very little (0,0, 0.64, 2.63;1) (0,0, 0.09, 0.99;1)

7 A bit (0.59, 1.50, 2.00, 3.41;1) (0.79, 1.68, 1.68, 2.21;0.74)

8 Little (0.38, 1.50, 2.50, 4.62;1) (1.09, 1.83, 1.83, 2.21;0.53)

9 Low amount (0.09, 1.25, 2.50, 4.62;1) (1.67, 1.92, 1.92, 2.21;0.30)

10 Small (0.09, 1.50, 3.00, 4.62;1) (1.79, 2.28, 2.28, 2.81;0.40)

11 Somewhat small (0.59, 2.00, 3.25, 4.41;1) (2.29, 2.70, 2.70, 3.21; 0.42)

12 Some (0.38, 2.50, 5.00, 7.83;1) (2.88, 3.61, 3.61, 4.21; 0.35)

13 Some to moderate (1.17, 3.50, 5.50, 7.83;1) (4.09, 4.65, 4.65, 5.41;0.40)

14 Moderate amount (2.59, 4.00, 5.50, 7.62;1) (4.29, 4.75, 4.75, 5.21;0.38)

15 Fair amount (2.17, 4.25, 6.00, 7.83;1) (4.79, 5.29, 5.29, 6.02;0.41)

16 Medium (3.59, 4.75, 5.50, 6.91;1) (4.86, 5.03, 5.05, 5.14;0.27)

17 Modest amount (3.59, 4.75, 6.00, 7.41;1) (4.79, 5.30, 5.30, 5.71;0.42)

18 Good amount (3.38, 5.50,7.50, 9.62;1) (5.79, 6.50, 6.50, 7.21;0.41)

19 Sizeable (4.38, 6.50, 8.00, 9.41;1) (6.79, 7.38, 7.38, 8.21;0.49)

20 Quite a bit (4.38, 6.50, 8.00, 9.41;1) (6.79, 7.38, 7.38, 8.21;0.49)

21 Considerable amount (4.38, 6.50, 8.25, 9.62;1) (7.19, 7.58, 7.58, 8.21;0.37)

22 Substantial amount (5.38, 7.50, 8.75, 9.81;1) (7.79, 8.22, 8.22, 8.81;0.45)

23 A lot (5.38, 7.50, 8.75, 9.83;1) (7.69, 8.19, 8.19, 8.81;0.47)

24 High amount (5.38, 7.50, 8.75, 9.81;1) (7.79, 8.30, 8.30, 9.21;0.53)

25 very sizable (5.38, 7.50,9.00, 9.81;1) (8.29, 8.56, 8.56, 9.21;0.38)

26 Large (5.98, 7.75, 8.60, 9.52;1) (8.03, 8.36, 8.36, 9.17;0.57)

27 Very large (7.37, 9.41, 10, 10;1) (8.72, 9.91, 10, 10;1)

28 Humongous amount (7.37, 9.59, 10, 10;1) (8.95, 9.93, 10, 10;1)

29 Huge amount (7.37, 9.73, 10, 10;1) (9.34, 9.95, 10, 10;1)

30 Very high amount (7.37, 9.82,10, 10;1) (9.37, 9.95, 10, 10;1)

31 Extreme amount (7.37, 9.82, 10, 10;1) (9.74, 9.98, 10, 10; 1)

32 Maximum amount (8.68, 9.91, 10, 10;1) (9.61, 9.97, 10, 10;1)

Rank	Word	UMF	LMF
1	None to very	(0,0, 0.14, 1.97;1)	(0,0, 0.05, 0.66;1)
2	Teeny-weeny	(0,0, 0.14, 1.97;1)	(0,0, 0.01, 0.13;1)
3	A smidgen	(0,0, 0.26, 2.63;1)	(0,0, 0.05, 0.63;1)
4	Tiny	(0,0, 0.36, 2.63;1)	(0,0, 0.05, 0.63;1)
5	Very small	(0,0, 0.64, 2.47;1)	(0,0, 0.10, 1.16;1)
6	Very little	(0,0, 0.64, 2.63;1)	(0,0, 0.09, 0.99;1)
7	A bit	(0.59, 1.50, 2.00, 3.41;1)	(0.79, 1.68, 1.68, 2.21;0.74)
8	Little	(0.38, 1.50, 2.50, 4.62;1)	(1.09, 1.83, 1.83, 2.21;0.53)
9	Low amount	(0.09, 1.25, 2.50, 4.62;1)	(1.67, 1.92, 1.92, 2.21;0.30)
10	Small	(0.09, 1.50, 3.00, 4.62;1)	(1.79, 2.28, 2.28, 2.81;0.40)
11	Somewhat small	(0.59, 2.00, 3.25, 4.41;1)	(2.29, 2.70, 2.70, 3.21; 0.42)
12	Some	(0.38, 2.50, 5.00, 7.83;1)	(2.88, 3.61, 3.61, 4.21; 0.35)
13	Some to moderate	(1.17, 3.50, 5.50, 7.83;1)	(4.09, 4.65, 4.65, 5.41;0.40)
14	Moderate amount	(2.59, 4.00, 5.50, 7.62;1)	(4.29, 4.75, 4.75, 5.21;0.38)
15	Fair amount	(2.17, 4.25, 6.00, 7.83;1)	(4.79, 5.29, 5.29, 6.02;0.41)
16	Medium	(3.59, 4.75, 5.50, 6.91;1)	(4.86, 5.03, 5.05, 5.14;0.27)
17	Modest amount	(3.59, 4.75, 6.00, 7.41;1)	(4.79, 5.30, 5.30, 5.71;0.42)
18	Good amount	(3.38, 5.50,7.50, 9.62;1)	(5.79, 6.50, 6.50, 7.21;0.41)
19	Sizeable	(4.38, 6.50, 8.00, 9.41;1)	(6.79, 7.38, 7.38, 8.21;0.49)
20	Quite a bit	(4.38, 6.50, 8.00, 9.41;1)	(6.79, 7.38, 7.38, 8.21;0.49)
21	Considerable amount	(4.38, 6.50, 8.25, 9.62;1)	(7.19, 7.58, 7.58, 8.21;0.37)
22	Substantial amount	(5.38, 7.50, 8.75, 9.81;1)	(7.79, 8.22, 8.22, 8.81;0.45)
23	A lot	(5.38, 7.50, 8.75, 9.83;1)	(7.69, 8.19, 8.19, 8.81;0.47)
24	High amount	(5.38, 7.50, 8.75, 9.81;1)	(7.79, 8.30, 8.30, 9.21;0.53)
25	very sizable	(5.38, 7.50,9.00, 9.81;1)	(8.29, 8.56, 8.56, 9.21;0.38)
26	Large	(5.98, 7.75, 8.60, 9.52;1)	(8.03, 8.36, 8.36, 9.17;0.57)
27	Very large	(7.37, 9.41, 10, 10;1)	(8.72, 9.91, 10, 10;1)
28	Humongous amount	(7.37, 9.59, 10, 10;1)	(8.95, 9.93, 10, 10;1)
29	Huge amount	(7.37, 9.73, 10, 10;1)	(9.34, 9.95, 10, 10;1)
30	Very high amount	(7.37, 9.82,10, 10;1)	(9.37, 9.95, 10, 10;1)
31	Extreme amount	(7.37, 9.82, 10, 10;1)	(9.74, 9.98, 10, 10; 1)
32	Maximum amount	(8.68, 9.91, 10, 10;1)	(9.61, 9.97, 10, 10;1)

As seen in Table 2 and Fig. 2, only the results corresponding to our proposed new signed distances methods and the signed distance show increasing orders. However, our approaches have a higher distinguish for these words. From the distance curves of Fig. 2, 32 words can be grouped into three classes: small-sounding words (the smallest six words), medium-sounding words (from 7th word to word 26th), and large-sounding words (the last six words). The result is consistent with centroid ranking methods about 32 word FOUs by Wu and Mendal [16].

Table 2

The distance comparison between the first element and the other 32 word FOUs

	Hamming distance [11]	Euclidean distance [12]	Hausdorff distance [13]	Signed distance [15]	New signed distance
					$d_{1} (\tilde{A}, \tilde{B})$	$d_{2} (\tilde{A}, \tilde{B})$
(1,2)	0.2159	0.3543	0.4318	0.0072	0.0162	0.0162
(1,3)	0.0855	0.1204	0.1483	0.0153	0.1092	0.1092
(1,4)	0.0950	0.1333	0.1673	0.0261	0.1260	0.1260
(1,5)	0.2197	0.2504	0.2476	0.0443	0.1793	0.1793
(1,6)	0.2037	0.2308	0.2242	0.0441	0.1915	0.1915
(1,7)	0.4955	0.5819	0.6151	0.3108	0.8314	1.2269
(1,8)	0.4525	0.5644	0.6476	0.3748	0.9840	1.3489
(1,9)	0.3978	0.5378	0.6426	0.3726	0.9031	1.1753
(1,10)	0.4220	0.5623	0.6599	0.4171	1.0708	1.4177
(1,11)	0.4259	0.5774	0.6813	0.4791	1.1925	1.6743
(1,12)	0.4141	0.5723	0.6889	0.7253	1.9796	2.5537
(1,13)	0.3914	0.5498	0.6686	0.8677	2.2863	3.1239
(1,14)	0.3344	0.5035	0.5704	0.9331	2.3668	3.3491
(1,15)	0.3545	0.5194	0.6084	0.9852	2.5568	3.5896
(1,16)	0.2622	0.4427	0.4510	0.9874	2.3825	3.4546
(1,17)	0.2903	0.4688	0.4866	1.0363	2.6552	3.8164
(1,18)	0.3139	0.4895	0.5384	1.2577	3.3011	4.6447
(1,19)	0.2725	0.4522	0.4596	1.3947	3.7415	5.3379
(1,20)	0.2725	0.4522	0.4596	1.3947	3.7415	5.3379
(1,21)	0.2697	0.4569	0.4735	1.4189	3.6401	5.1781
(1,22)	0.2318	0.4176	0.3975	1.5543	4.1057	5.8806
(1,23)	0.2338	0.4185	0.3978	1.5538	4.1390	5.9276
(1,24)	0.2383	0.4208	0.3975	1.5645	4.2694	6.1137
(1,25)	0.2338	0.4251	0.4103	1.5811	4.0627	5.8120
(1,26)	0.2079	0.3896	0.4319	1.5836	4.3828	6.2988
(1,27)	0.1854	0.3718	0.2675	1.8708	6.4506	9.2421
(1,28)	0.1747	0.3574	0.2585	1.8852	6.4860	9.2945
(1,29)	0.1609	0.3404	0.2515	1.8991	6.5266	9.3540
(1,30)	0.1579	0.3352	0.2470	1.9051	6.5384	9.3713
(1,31)	0.1479	0.3243	0.2470	1.9101	6.5639	9.4078
(1,32)	0.1164	0.2879	0.1770	1.9303	6.6357	9.5157

Fig. 2.

The distance between the first element and the other 31 words.

Assuming that the ranking order of 32 word FOUs is valid, we compare the performance of the proposed distance with the normalized Hamming distance [11], normalized Euclidean distance [12], normalized Hamming distance based on Hausdorff metric [13, 14] and signed distance [15]. Based on the Definition5, for example, the distance between “None to very little” and “Teeny-weeny” should be smaller than the distance between “None to very little” and “A smidgen”. Hence, the distances between the first word FOU (None to very little) with the other 31 word FOUs should be in increasing order.

Calculated distances between the first element, “None to very little”, and elements 7 and 14, “A bit” and “Moderate amount”, with the normalized Hamming, normalized Euclidean distances and normalized Hamming based on Hausdorff metric are: $\begin{matrix} d_{H} (w_{1}, w_{7}) = 0.4955, \\ d_{H} (w_{1}, w_{14}) = 0.3344 \\ d_{E} (w_{1}, w_{7}) = 0.5819, \\ d_{E} (w_{1}, w_{14}) = 0.5035 \\ d_{H} H (w_{1}, w_{7}) = 0.6151, \\ d_{H} H (w_{1}, w_{14}) = 0.5704 \end{matrix}$

These are in contradiction with definition5, because w₇ < w₁₄ but d_H (w₁, w₇) > d_H (w₁, w₁₄), d_E (w₁, w₇) > d_E (w₁, w₁₄) and d_HH (w₁, w₇) > d_HH (w₁, w₁₄).

As seen in Fig. 2, neglecting the above mentioned limitation of the signed distance, the results obtained by this method has the same trend as our method, but it seems that three other methods are not appropriates for interval type-2 fuzzy sets.

In this section, four existing similarity measure [16] for IT2 FSs are briefly reviewed, and then two new signed-based similarity measures, having reduced computational cost, is proposed. As seen in Table 3 and Fig. 3, only the results corresponding to our proposed new signed-based similarity measures methods, Mitchell’s method, Gorzalczany’s method and Jaccard’s similarity measure show decreasing orders. However, our approaches have a higher distinguish for these words. Zeng’s method is contradiction with the facts.

Fig. 3.

The similarity comparison between the first element and the other 32 word FOUs.

Table 3

The similarity comparison between the first element and the other 32 word FOUs

	Mitchell’s method [17]	Gorzalczany’s method [18]	Jaccard’s method [16]	Zeng’s method [19]	New similarity measure
					$S_{1} (\tilde{A}, \tilde{B})$	$S_{2} (\tilde{A}, \tilde{B})$
(1,2)	0.57	1	0.80	0.93	0.9841	0.9841
(1,3)	0.62	1	0.77	0.92	0.9016	0.9016
(1,4)	0.61	1	0.75	0.91	0.8881	0.8881
(1,5)	0.60	1	0.64	0.84	0.8480	0.8480
(1,6)	0.60	1	0.65	0.86	0.8393	0.8393
(1,7)	0.11	0.25	0.11	0.58	0.5460	0.4491
(1,8)	0.09	0.27	0.11	0.63	0.5040	0.4257
(1,9)	0.13	0.31	0.16	0.66	0.5254	0.4597
(1,10)	0.11	0.29	0.13	0.62	0.4829	0.4136
(1,11)	0.06	0.21	0.08	0.60	0.4561	0.3739
(1,12)	0.04	0.20	0.05	0.62	0.3356	0.2814
(1,13)	0.01	0.09	0.01	0.63	0.3043	0.2425
(1,14)	0	0	0	0.69	0.2970	0.2299
(1,15)	0	0	0	0.66	0.2812	0.2179
(1,16)	0	0	0	0.75	0.2956	0.2245
(1,17)	0	0	0	0.72	0.2736	0.2076
(1,18)	0	0	0	0.70	0.2325	0.1772
(1,19)	0	0	0	0.74	0.2109	0.1578
(1,20)	0	0	0	0.74	0.2109	0.1578
(1,21)	0	0	0	0.74	0.2155	0.1619
(1,22)	0	0	0	0.78	0.1959	0.1453
(1,23)	0	0	0	0.77	0.1946	0.1444
(1,24)	0	0	0	0.77	0.1898	0.1406
(1,25)	0	0	0	0.77	0.1975	0.1468
(1,26)	0	0	0	0.80	0.1858	0.1370
(1,27)	0	0	0	0.82	0.1342	0.0976
(1,28)	0	0	0	0.86	0.1336	0.0971
(1,29)	0	0	0	0.83	0.1329	0.0966
(1,30)	0	0	0	0.84	0.1327	0.0964
(1,31)	0	0	0	0.85	0.1322	0.0961
(1,32)	0	0	0	0.89	0.1310	0.0951

Specifically speaking, Mitchell’s method is not satisfied the property (2) in definition10 [16]. We see that Gorzalczany’s method indicates “very large (27)”, “humongous amount (28)”, “huge amount (29)”, “very high amount (30)”, “extreme amount (31)” and “maximum amount (32)” are equivalent, which is counter-intuitive because their FOUs are not completely the same. Examining Zeng’s method, we see that all similarities are larger than 0.50, i.e., their method gives large similarity whether or not $\tilde{A}$ and $\tilde{B}$ overlap. Examining this column, we see that the similarity generally decreases and then increases as two words get further away, whereas a monotonically decreasing trend is expected. From the beginning of 14th words, all the similarity is equal to 0, so the distinction of Jaccard’s similarity method is too low. Also, simulations show that our method is faster than the Jaccard’s method.

According to the above examples, we know that the new signed distance methods and signed-based similarity measures are practicality and effectiveness for ranking interval type-2 fuzzy numbers. The new distance methods and signed-based similarity measures methods can overcome the shortcomings of some existing methods for ranking interval type-2 fuzzy numbers. The computational cost of the above four IT2 FS similarity measure is heavy because the computation formula requires direct enumeration of all embedded T1 FSs [16]. Therefore, the biggest advantage of our methods is that it does not require complex calculations. The computational steps of the proposed method are less than those of other methods, while it gets the same results in most cases.

5 Conclusions

We present some suitable distances formula for interval type-2 trapezoidal fuzzy sets, i.e. the continuous weighted Hamming distances, the continuous weighted Hamming distances and the continuous weighted Minkowski distances, and some of their properties are studied. And then we give an approach to construct signed distance and signed-based similarity measures for IT2 TrFNs. 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. The shortcoming of this method is that we only study the distance measure and similarity measure of the interval type-2 trapezoidal fuzzy numbers, but do not consider the other types of interval type-2 fuzzy numbers.

The ordered weighted averaging (OWA) operator assigns the ith weight to the ith largest input value. The OWA operator has become a very developed area of research in the scope of both applied and theoretical information science [28 –31]. In future studies, we will further consider the OWA operator techniques to defuzzification of IT2 FSs and applied to the distances and similarity measures of IT2 FSs.

Footnotes

Acknowledgments

The authors are very grateful to the editor and the anonymous reviewers for their constructive comments and suggestions that have led to an improved version of this paper. 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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