A generalized similarity measure for fuzzy numbers

Abstract

Similarity measure is one of essential concepts in fuzzy systems. This study presents a generalized similarity measure for fuzzy numbers. The generalized similarity measure for fuzzy numbers takes account of the membership degree of fuzzy number. Finally, the generalized similarity measure for fuzzy numbers is compared with nine existing methods for similarity measure of fuzzy numbers by using some sets of fuzzy numbers. The comparison result shows that the generalized similarity measure proposed in this paper seems to be generalized and promising.

Keywords

Fuzzymathematics membership degree generalized similaritymeasure fuzzy decision-making similarity relationship

1 Introduction

The similarity of fuzzy numbers is one of the essential issues in human decision-making science. In the past, some studies on similarity measure for fuzzy numbers have been presented. These presented similarity measures for fuzzy numbers also have been applied to many fields such as image thresholding [4 –6], dynamic classifier aggregation [40], basic probability assignment [28], multiple-criteria decision-making [20, 30], clustering [38 , 46], leisure and tourism [17], group decision-making [1 , 45], ordering and ranking of fuzzy numbers [49], risk evaluation [9 , 36], prediction [3], fuzzy pattern recognition [11 , 43], medical diagnoses [47] and other fuzzy fields [7 , 42].

The similarity relationship between fuzzy numbers for representing and describing the fuzzy situation or the fuzzy environment is shown in Fig. 1. In Fig. 1, $\tilde{A} = (a_{1}, a_{2}, a_{3}, a_{4})$ and $\tilde{B} = (b_{1}, b_{2}, b_{3}, b_{4})$ are two trapezoidal fuzzy numbers. In the past, although a lot of studies presented many methods for similarity measure of fuzzy numbers, few made an attempt to introduce similarity measure methods which consider the membership degree of fuzzy number. Chou [16] presented a specific similarity measure with membership degree consideration. Thus, this paper makes an attempt to fill the gap in the literature by proposing a generalized similarity measure which is not only to take account of the membership degree of fuzzy number but also more generalized.

2 Concept and representation of fuzzy number

Fuzzy sets theory which has been used widely to many fields, e.g. management, engineering, decision-making, industry, education, was initially proposed by Zadeh [48]. The concept of fuzzy number is introduced briefly in this section. Let $\tilde{A} = (a_{1}, a_{2}, a_{3}, a_{4})$ be a trapezoidal fuzzy number as Fig. 2. Suppose the membership function of $\tilde{A}$ is $f_{\tilde{A}} (x)$ . $f_{\tilde{A}} (x) = {\begin{matrix} \frac{(x - a_{1})}{(a_{2} - a_{1})}, a_{1} \leq x \leq a_{2}, \\ 1, a_{2} \leq x \leq a_{3}, \\ \frac{(x - a_{4})}{(a_{3} - a_{4})}, a_{3} \leq x \leq a_{4}, \\ 0, otherwise . \end{matrix}$ $L_{\tilde{A}} (x) = \frac{(x - a_{1})}{(a_{2} - a_{1})}, a_{1} \leq x \leq a_{2},$ $R_{\tilde{A}} (x) = \frac{(x - a_{4})}{(a_{3} - a_{4})}, a_{3} \leq x \leq a_{4},$ $L_{\tilde{A}}^{- 1} (h) = a_{1} + (a_{2} - a_{1}) h, 0 \leq h \leq 1 .$ $R_{\tilde{A}}^{- 1} (h) = a_{4} + (a_{3} - a_{4}) h, 0 \leq h \leq 1 .$

$L_{\tilde{A}} (x)$ and $R_{\tilde{A}} (x)$ are the function L and the function R of the trapezoidal fuzzy number $\tilde{A}$ , respectively. $L_{\tilde{A}}^{- 1} (h)$ and $R_{\tilde{A}}^{- 1} (h)$ are the inverse functions of the function $L_{\tilde{A}} (x)$ and the function $R_{\tilde{A}} (x)$ at h-level, respectively.

Chou [14] proposed the graded multiple integration representation method for presenting the representation of fuzzy number, based on the graded mean integration representation method [8]. Here we describe the meaning as follows. Let the graded mean h-level value of fuzzy number $\tilde{A}$ is $h (L_{\tilde{A}}^{- 1} (h) + R_{\tilde{A}}^{- 1} (h)) / 2$ as Fig. 2. Then the representation of fuzzy number $\tilde{A}$ is $P (\tilde{A})$ .

$\begin{matrix} P (\tilde{A}) \\ = \int_{0}^{1} \frac{h (L_{\tilde{A}}^{- 1} (h) + R_{\tilde{A}}^{- 1} (h))}{2} dh / \int_{0}^{1} hdh \\ = \int_{0}^{1} \frac{h (a_{1} + (a_{2} - a_{1}) h + a_{4} + (a_{3} - a_{4}) h)}{2} dh / \int_{0}^{1} hdh \\ = \frac{1}{6} (a_{1} + 2 a_{2} + 2 a_{3} + a_{4}) \end{matrix}$ (1)

3 Literature review

In the past, some similarity measures for fuzzy numbers have been proposed. In this section, the existing similarity measures proposed previously are introduced as follows.

(a) Chen [10] presents a fuzzy similarity measure which the experts’ estimates do not necessarily have a common intersection at α-level. In addition, it can process fuzzy opinion aggregation in a more efficient manner and it does not need to use Delphi method to adjust fuzzy numbers given by the decision makers. Let $\tilde{A} = (a_{1}, a_{2}, a_{3}, a_{4})$ and $\tilde{B} = (b_{1}, b_{2}, b_{3}, b_{4})$ be two trapezoidal fuzzy numbers. Then the degree of similarity $S (\tilde{A}, \tilde{B})$ between fuzzy numbers $\tilde{A}$ and $\tilde{B}$ can be expressed as Equation (2). $S (\tilde{A}, \tilde{B}) = 1 - \frac{\sum_{i = 1}^{4} | a_{i} - b_{i} |}{4}$ (2)

If $\tilde{A}$ and $\tilde{B}$ are two triangular fuzzy numbers, $\tilde{A} = (a_{1}, a_{2}, a_{3})$ and $\tilde{B} = (b_{1}, b_{2}, b_{3})$ , then the degree of similarity $S (\tilde{A}, \tilde{B})$ between fuzzy numbers $\tilde{A}$ and $\tilde{B}$ can be calculated as Equation (3). $S (\tilde{A}, \tilde{B}) = 1 - \frac{\sum_{i = 1}^{3} | a_{i} - b_{i} |}{3}$ (3)

The larger the value of $S (\tilde{A}, \tilde{B})$ , the more the similarity measure between fuzzy numbers $\tilde{A}$ and $\tilde{B}$ .

(b) Hsieh and Chen [24] introduced a fuzzy similarity measure based on the graded mean integration representation distance. One of the advantages of the approach is that the membership degrees of fuzzy numbers are considered. Another is the easy computation procedure. Assume that there are two trapezoidal fuzzy numbers, $\tilde{A} = (a_{1}, a_{2}, a_{3}, a_{4})$ and $\tilde{b} = (b_{1}, b_{2}, b_{3}, b_{4})$ , then the degree of similarity $S (\tilde{A}, \tilde{B})$ between fuzzy numbers $\tilde{A}$ and $\tilde{B}$ can be expressed as Equation (4). $S (\tilde{A}, \tilde{B}) = \frac{1}{1 + d (\tilde{A}, \tilde{B})}$ (4)

where $d (\tilde{A}, \tilde{B}) = | P (\tilde{A}) - P (\tilde{B}) |$ . $P (\tilde{A})$ and $P (\tilde{B})$ are the graded mean integration representations of fuzzy numbers $\tilde{A}$ and $\tilde{B}$ , respectively. The larger the value of $S (\tilde{A}, \tilde{B})$ , the more the similarity measure between fuzzy numbers $\tilde{A}$ and $\tilde{B}$ .

If $\tilde{A}$ and $\tilde{B}$ are trapezoidal fuzzy numbers, $\tilde{A} = (a_{1}, a_{2}, a_{3}, a_{4})$ and $\tilde{b} = (b_{1}, b_{2}, b_{3}, b_{4})$ , then the graded mean integration representations of $\tilde{A}$ and $\tilde{B}$ , respectively, are defined as follows. $P (\tilde{A}) = \frac{a_{1} + 2 a_{2} + 2 a_{3} + a_{4}}{6}$ (5) $P (\tilde{B}) = \frac{b_{1} + 2 b_{2} + 2 b_{3} + b_{4}}{6}$ (6)

(c) Hsu and Chen [25] introduced the similarity measure of fuzzy numbers as Equation (7). The proposed approach is applied to aggregate individual fuzzy opinions into a group fuzzy consensus opinion. Assume that there are two trapezoidal fuzzy numbers, $\tilde{A} = (a_{1}, a_{2}, a_{3}, a_{4})$ and $\tilde{B} = (b_{1}, b_{2}, b_{3}, b_{4})$ , then the degree of similarity $S (\tilde{A}, \tilde{B})$ between fuzzy numbers $\tilde{A}$ and $\tilde{B}$ can be expressed as Equation (7). $S (\tilde{A}, \tilde{B}) = \frac{\int_{x} (min {u_{\tilde{A}} (x), u_{\tilde{B}} (x)}) dx}{\int_{x} max {u_{\tilde{A}} (x), u_{\tilde{B}} (x)}) dx}$ (7)

where $u_{\tilde{A}}$ , $u_{\tilde{B}}$ are the membership functions of fuzzy numbers $\tilde{A}$ and $\tilde{B}$ ; where $S (\tilde{A}, \tilde{B})$ is called as similarity measure function by Zwick et al. [50].

The larger the value of $S (\tilde{A}, \tilde{B})$ , the more the similarity measure between fuzzy numbers $\tilde{A}$ and $\tilde{B}$ .

(d) To minimize the sum of weighted dissimilarity among aggregated consensus and individual opinions. Lee [29] presents a fuzzy similarity measure which takes account of the importance of each expert in the process of aggregation. Assume there are two trapezoidal fuzzy numbers, $\tilde{A} = (a_{1}, a_{2}, a_{3}, a_{4})$ and $\tilde{B} = (b_{1}, b_{2}, b_{3}, b_{4})$ , then the degree of similarity $S (\tilde{A}, \tilde{B})$ between fuzzy numbers $\tilde{A}$ and $\tilde{B}$ can be expressed as Equation (8). $S (\tilde{A}, \tilde{B}) = 1 - \frac{{∥ \tilde{A} - \tilde{B} ∥}_{lp}}{∥ U ∥} \times 4^{- 1 / p}$ (8)

where U is the universe of discourse ${∥ \tilde{A} - \tilde{B} ∥}_{lp} = {[\sum_{i = 1}^{4} {(| a_{i} - b_{i} |)}^{p}]}^{1 / p}$ (9)and $∥ U ∥ = max (U) - min (U)$ (10)

The larger the value of $S (\tilde{A}, \tilde{B})$ , the more the similarity measure between fuzzy numbers $\tilde{A}$ and $\tilde{B}$ .

(e) Chen and Chen [12] proposed a similarity measure for fuzzy numbers by using the center-of- gravity (COG) points. Firstly, they introduce a method called the simple center of gravity method (SCGM) to calculate the center-of-gravity (COG) points of generalized fuzzy numbers, and then, use the SCGM to propose a new method to measure the degree of similarity between generalized fuzzy numbers. Finally, the proposed similarity measure is applied to develop a new method to deal with fuzzy risk analysis problems. The results show that the proposed fuzzy risk analysis method is more flexible and more intelligent than the existing methods. The method is described as follows.

Assume there are two generalized trapezoidal fuzzy numbers, $\tilde{A} = (a_{1}, a_{2}, a_{3}, a_{4}; w_{A})$ and $\tilde{B} = (b_{1}, b_{2}, b_{3}, b_{4}; w_{B})$ , where 0 ≤ a₁ ≤ a₂ ≤ a₃ ≤ a₄, and 0 ≤ b₁ ≤ b₂ ≤ b₃ ≤ b₄. The COG points $COG (\tilde{A})$ and $COG (\tilde{B})$ of fuzzy numbers $\tilde{A}$ and $\tilde{B}$ can be denoted as ( $x_{A}^{*}$ , $y_{A}^{*}$ ) and ( $x_{B}^{*}$ , $y_{B}^{*}$ ), respectively. Then the degree of similarity $S (\tilde{A}, \tilde{B})$ between fuzzy numbers $\tilde{A}$ and $\tilde{B}$ can be expressed as Equation (11).

$\begin{matrix} S (\tilde{A}, \tilde{B}) & = & [1 - \frac{\sum_{i = 1}^{4} | a_{i} - b_{i} |}{4}] \\ \times {(1 - | x_{\tilde{A}}^{*} - x_{\tilde{B}}^{*} |)}^{B (S_{\tilde{A}}, S_{\tilde{B}})} \times \frac{min (y_{\tilde{A}}^{*}, y_{\tilde{B}}^{*})}{max (y_{\tilde{A}}^{*}, y_{\tilde{B}}^{*})} \end{matrix}$ (11)

where $B (S_{\tilde{A}}, S_{\tilde{B}})$ is defined as Equation (12). $B (S_{\tilde{A}}, S_{\tilde{B}}) = {\begin{matrix} 1 \begin{matrix} \end{matrix} S_{\tilde{A}} + S_{\tilde{B}} > 0 \\ 0 \begin{matrix} \end{matrix} S_{\tilde{A}} + S_{\tilde{B}} = 0 \end{matrix}$ (12)

where $S_{\tilde{A}}$ and $S_{\tilde{B}}$ are the lengths of the base of the generalized fuzzy numbers $\tilde{A}$ and $\tilde{B}$ , respectively, defined as Equation (13). $\begin{matrix} S_{\tilde{A}} = a_{4} - a_{1} \\ S_{\tilde{B}} = b_{4} - b_{1} \end{matrix}$ (13)

(f) Deng et al. [19] presented a similarity measure based on the radius-of-gyration (ROG) points. Some sets of generalized fuzzy numbers are used to compare the proposed method with the existing similarity measures. The results show that the similarity measure can overcome the drawbacks of the existing methods. Finally, the proposed similarity measure is applied to pattern recognition.

Assume there are two generalized trapezoidal fuzzy numbers, $\tilde{A} = (a_{1}, a_{2}, a_{3}, a_{4}; w_{A})$ and $\tilde{B} = (b_{1}, b_{2}, b_{3}, b_{4}; w_{B})$ , where 0 ≤ a₁ ≤ a₂ ≤ a₃ ≤ a₄, and 0 ≤ b₁ ≤ b₂ ≤ b₃ ≤ b₄. The ROG points $ROG (\tilde{A})$ and $ROG (\tilde{B})$ of fuzzy numbers $\tilde{A}$ and $\tilde{B}$ can be denoted as ( $r_{x}^{A}$ , $r_{y}^{A}$ ) and ( $r_{x}^{B}$ , $r_{y}^{B}$ ), respectively. Then the degree of similarity $S (\tilde{A}, \tilde{B})$ between fuzzy numbers $\tilde{A}$ and $\tilde{B}$ can be expressed as Equation (14).

$\begin{matrix} S (\tilde{A}, \tilde{B}) = [1 - \frac{\sum_{i = 1}^{4} | a_{i} - b_{i} |}{4}] \\ \times {(1 - | r_{y}^{A} - r_{y}^{B} |)}^{B (S_{\tilde{A}}, S_{\tilde{B}})} \times \frac{min (r_{x}^{A}, r_{x}^{B})}{max (r_{x}^{A}, r_{x}^{B})} \end{matrix}$ (14)

where $B (S_{\tilde{A}}, S_{\tilde{B}})$ is defined as Equation (15). $B (S_{\tilde{A}}, S_{\tilde{B}}) = {\begin{matrix} 1 \begin{matrix} \end{matrix} S_{\tilde{A}} + S_{\tilde{B}} > 0 \\ 0 \begin{matrix} \end{matrix} S_{\tilde{A}} + S_{\tilde{B}} = 0 \end{matrix}$ (15)

where $S_{\tilde{A}}$ and $S_{\tilde{B}}$ are the lengths of the base of the generalized fuzzy numbers $\tilde{A}$ and $\tilde{B}$ , respectively, defined as Equation (16). $\begin{matrix} S_{\tilde{A}} = a_{4} - a_{1} \\ S_{\tilde{B}} = b_{4} - b_{1} \end{matrix}$ (16)

(g) Chou [15] introduced a similarity measure for fuzzy numbers. Finally, the proposed method is applied to the graphic analysis. The disadvantage of this approach for measuring the similarity of fuzzy numbers is regardless the membership degree of fuzzy number. Assume that there are two trapezoidal fuzzy numbers, $\tilde{A} = (a_{1}, a_{2}, a_{3}, a_{4})$ and $\tilde{B} = (b_{1}, b_{2}, b_{3}, b_{4})$ , then the degree of similarity $S (\tilde{A}, \tilde{B})$ between fuzzy numbers $\tilde{A}$ and $\tilde{B}$ can be expressed as Equation (17). $S (\tilde{A}, \tilde{B}) = \frac{\int (\tilde{A} \cap \tilde{B}) dh}{\int (\tilde{A} \cup \tilde{B}) dh}$ (17)

The larger the value of $S (\tilde{A}, \tilde{B})$ , the more the similarity measure between fuzzy numbers $\tilde{A}$ and $\tilde{B}$ .

(h) Allahviranloo et al. [2] defined the distance measure entitled generalized Hausdorff distance between two trapezoidal generalized fuzzy numbers. Then the similarity measure is defined by using another distance and combining with generalized Hausdorff distance. Assume there are two generalized trapezoidal fuzzy numbers, $\tilde{A} = (a_{1}, a_{2}, a_{3}, a_{4}; w_{A})$ and $\tilde{B} = (b_{1}, b_{2}, b_{3}, b_{4}; w_{B})$ , where 0 ≤ a₁ ≤ a₂ ≤ a₃ ≤ a₄, and 0 ≤ b₁ ≤ b₂ ≤ b₃ ≤ b₄. Then the degree of similarity $S (\tilde{A}, \tilde{B})$ between fuzzy numbers $\tilde{A}$ and $\tilde{B}$ can be expressed as Equation (18). $S (\tilde{A}, \tilde{B}) = {\begin{matrix} 1 - \frac{d (\tilde{A}, \tilde{B})}{d_{\infty}^{g} (\tilde{A}, \tilde{B})}, d (\tilde{A}, \tilde{B}) < d_{\infty}^{g} (\tilde{A}, \tilde{B}) \\ 1 - \frac{d (\tilde{A}, \tilde{B})}{m}, d (\tilde{A}, \tilde{B}) = d_{\infty}^{g} (\tilde{A}, \tilde{B}) \end{matrix}$ (18)

where m = max {a₂ + a₄, b₂ + b₄}. It is clear m ≠ 0 because a₂, a₄, b₂, and b₄ are non-negative real numbers and m = 0 is only for zero singleton fuzzy numbers. $\begin{matrix} If d_{\infty}^{g} (\tilde{A}, \tilde{B}) = 0 then d (\tilde{A}, \tilde{B}) = 0 \\ so d (\tilde{A}, \tilde{B}) = d_{\infty}^{g} (\tilde{A}, \tilde{B}) \\ and S (\tilde{A}, \tilde{B}) = {1 - \frac{d (\tilde{A}, \tilde{B})}{m} . \end{matrix}$

(i) Adabitabar Firozja et al. [1] presented a similarity measure based on the interval distance measurement. Assume there are two fuzzy numbers $\tilde{A}$ and $\tilde{B}$ . Then the degree of similarity $S (\tilde{A}, \tilde{B})$ between fuzzy numbers $\tilde{A}$ and $\tilde{B}$ can be calculated as Equation (19). $S (\tilde{A}, \tilde{B}) = [\frac{1}{1 + \bar{d}}, 1 - \frac{\underline{d}}{1 + \bar{d}}]$ (19)where $[\underline{d}, \bar{d}]$ is the interval distance measure between $\tilde{A}$ and $\tilde{B}$ . The similarity measure has the following properties: $\begin{matrix} [0, 0] < S (\tilde{A}, \tilde{B}) \leq [1, 1] . \\ S (\tilde{A}, \tilde{B}) = S (\tilde{B}, \tilde{A}) . \\ If \tilde{A} = \tilde{B} then S (\tilde{A}, \tilde{B}) = [1, 1] . \\ If \tilde{A} \leq \tilde{B} \leq \tilde{C}, then \\ S (\tilde{A}, \tilde{C}) \leq \min {S (\tilde{A}, \tilde{B}), S (\tilde{B}, \tilde{C})} . \end{matrix}$

(j) Chou [16] introduces a similarity measure based on the membership degree consideration. Assume there are two fuzzy numbers, $\tilde{A}$ and $\tilde{B}$ . Then the degree of similarity $S (\tilde{A}, \tilde{B})$ between fuzzy numbers $\tilde{A}$ and $\tilde{B}$ can be calculated as Equation (20). $S (\tilde{A}, \tilde{B}) = \frac{\int h (A \cap B) dh}{\int h (A \cup B) dh} = \frac{\sum_{i} S_{i}^{h}}{\sum_{I} S_{I}^{h}}$ (20)

where $S_{i}^{h}$ is a part of $\tilde{A} \cap \tilde{B}$ . And $\tilde{A} \cap \tilde{B}$ can be separated into i parts, including $S_{1}^{h}$ , $S_{2}^{h}$ , $S_{3}^{h}$ , … , $S_{i}^{h}$ . In other words, $\sum_{i} S_{i}^{h} = \tilde{A} \cap \tilde{B}$ . Where $S_{I}^{h}$ is a part of $\tilde{A} \cup \tilde{B}$ . And $\tilde{A} \cup \tilde{B}$ can be separated into I parts, including $S_{1}^{h}, S_{2}^{h}, S_{3}^{h}, \dots, S_{I}^{h}$ . In other words, $\sum_{I} S_{I}^{h} = \tilde{A} \cup \tilde{B}$ .

4 A generalized similarity measure method

4.1 Generalized similarity measure for fuzzy numbers

Although the above-mentioned similarity measure for fuzzy numbers proposed by Chou [16] took account of the membership degree of fuzzy number, it is only a specific similarity measure for fuzzy numbers. Therefore, based on the fuzzy similarity measure proposed by Chou [16], this study further proposes a generalized similarity measure for fuzzy numbers. The generalized similarity measure is not only with membership degree consideration, but also is appropriate to measure the similarity of most fuzzy numbers. Assume there are two trapezoidal fuzzy numbers in Fig. 1, $\tilde{A} = (a_{1}, a_{2}, a_{3}, a_{4})$ and $\tilde{B} = (b_{1}, b_{2}, b_{3}, b_{4})$ . Then the generalized similarity measure is defined as Equation (21). $S (\tilde{A}, \tilde{B}) = {\begin{matrix} 1, \tilde{A} \cap \tilde{B} = \tilde{A} \cup \tilde{B}, \\ \frac{\sum_{i} S_{i}^{\tilde{A} \cap \tilde{B}}}{\sum_{I} S_{I}^{\tilde{A} \cup \tilde{B}}}, \tilde{A} \cap \tilde{B} < \tilde{A} \cup \tilde{B}, \\ 0, \tilde{A} \cap \tilde{B} = φ \end{matrix}$ (21)

$\begin{matrix} S_{i}^{\tilde{A} \cap \tilde{B}} \subseteq S_{I}^{\tilde{A} \cup \tilde{B}} \\ = {S_{R_{\tilde{A}} - L_{\tilde{A}}, L_{\tilde{A}} R_{\tilde{A}} L_{\tilde{B}} R_{\tilde{B}}}^{h}, S_{R_{\tilde{B}} - L_{\tilde{B}}, L_{\tilde{A}} R_{\tilde{A}} L_{\tilde{B}} R_{\tilde{B}}}^{h}, \\ S_{L_{\tilde{B}} - L_{\tilde{A}}, L_{\tilde{A}} L_{\tilde{B}} R_{\tilde{A}} R_{\tilde{B}}}^{h}, S_{R_{\tilde{A}} - L_{\tilde{B}}, L_{\tilde{A}} L_{\tilde{B}} R_{\tilde{A}} R_{\tilde{B}}}^{h}, \\ S_{R_{\tilde{B}} - R_{\tilde{A}}, L_{\tilde{A}} L_{\tilde{B}} R_{\tilde{A}} R_{\tilde{B}}}^{h}, S_{L_{\tilde{B}} - L_{\tilde{A}}, L_{\tilde{A}} L_{\tilde{B}} R_{\tilde{B}} R_{\tilde{A}}}^{h}, \\ S_{R_{\tilde{B}} - L_{\tilde{B}}, L_{\tilde{A}} L_{\tilde{B}} R_{\tilde{B}} R_{\tilde{A}}}^{h}, S_{R_{\tilde{A}} - R_{\tilde{B}}, L_{\tilde{A}} L_{\tilde{B}} R_{\tilde{B}} R_{\tilde{A}}}^{h}, \\ S_{L_{\tilde{A}} - L_{\tilde{B}}, L_{\tilde{B}} L_{\tilde{A}} R_{\tilde{A}} R_{\tilde{B}}}^{h}, S_{R_{\tilde{A}} - L_{\tilde{A}}, L_{\tilde{B}} L_{\tilde{A}} R_{\tilde{A}} R_{\tilde{B}}}^{h}, \\ S_{R_{\tilde{B}} - R_{\tilde{A}}, L_{\tilde{B}} L_{\tilde{A}} R_{\tilde{A}} R_{\tilde{B}}}^{h}, S_{L_{\tilde{A}} - L_{\tilde{B}}, L_{\tilde{B}} L_{\tilde{A}} R_{\tilde{B}} R_{\tilde{A}}}^{h}, \\ S_{R_{\tilde{B}} - L_{\tilde{A}}, L_{\tilde{B}} L_{\tilde{A}} R_{\tilde{B}} R_{\tilde{A}}}^{h}, S_{R_{\tilde{A}} - R_{\tilde{B}}, L_{\tilde{B}} L_{\tilde{A}} R_{\tilde{B}} R_{\tilde{A}}}^{h}, \\ S_{R_{\tilde{B}} - L_{\tilde{B}}, L_{\tilde{B}} R_{\tilde{B}} L_{\tilde{A}} R_{\tilde{A}}}^{h}, S_{R_{\tilde{A}} - L_{\tilde{A}}, L_{\tilde{B}} R_{\tilde{B}} L_{\tilde{A}} R_{\tilde{A}}}^{h}} \end{matrix}$ where

$\begin{matrix} S_{R_{\tilde{A}} - L_{\tilde{A}}, L_{\tilde{A}} R_{\tilde{A}} L_{\tilde{B}} R_{\tilde{B}}}^{h} \\ = \int_{l}^{u} (R_{\tilde{A}}^{- 1} (h) - L_{\tilde{A}}^{- 1} (h)) hdh / \\ \int_{l}^{u} hdh, L_{\tilde{A}}^{- 1} \leq R_{\tilde{A}}^{- 1} \leq L_{\tilde{B}}^{- 1} \leq R_{\tilde{B}}^{- 1} \\ S_{R_{\tilde{B}} - L_{\tilde{B}}, L_{\tilde{A}} R_{\tilde{A}} L_{\tilde{B}} R_{\tilde{B}}}^{h} \\ = \int_{l}^{u} (R_{\tilde{B}}^{- 1} (h) - L_{\tilde{B}}^{- 1} (h)) hdh / \\ \int_{l}^{u} hdh, L_{\tilde{A}}^{- 1} \leq R_{\tilde{A}}^{- 1} \leq L_{\tilde{B}}^{- 1} \leq R_{\tilde{B}}^{- 1} \\ S_{L_{\tilde{B}} - L_{\tilde{A}}, L_{\tilde{A}} L_{\tilde{B}} R_{\tilde{A}} R_{\tilde{B}}}^{h} \\ = \int_{l}^{u} (L_{\tilde{B}}^{- 1} (h) - L_{\tilde{A}}^{- 1} (h)) hdh / \\ \int_{l}^{u} hdh, L_{\tilde{A}}^{- 1} \leq L_{\tilde{B}}^{- 1} \leq R_{\tilde{A}}^{- 1} \leq R_{\tilde{B}}^{- 1} \\ S_{R_{\tilde{A}} - L_{\tilde{B}}, L_{\tilde{A}} L_{\tilde{B}} R_{\tilde{A}} R_{\tilde{B}}}^{h} \\ = \int_{l}^{u} (R_{\tilde{A}}^{- 1} (h) - L_{\tilde{B}}^{- 1} (h)) hdh / \\ \int_{l}^{u} hdh, L_{\tilde{A}}^{- 1} \leq L_{\tilde{B}}^{- 1} \leq R_{\tilde{A}}^{- 1} \leq R_{\tilde{B}}^{- 1} \end{matrix}$

$\begin{matrix} S_{R_{\tilde{B}} - R_{\tilde{A}}, L_{\tilde{A}} L_{\tilde{B}} R_{\tilde{A}} R_{\tilde{B}}}^{h} \\ = \int_{l}^{u} (R_{\tilde{B}}^{- 1} (h) - R_{\tilde{A}}^{- 1} (h)) hdh / \\ \int_{l}^{u} hdh, L_{\tilde{A}}^{- 1} \leq L_{\tilde{B}}^{- 1} \leq R_{\tilde{A}}^{- 1} \leq R_{\tilde{B}}^{- 1} \\ S_{L_{\tilde{B}} - L_{\tilde{A}}, L_{\tilde{A}} L_{\tilde{B}} R_{\tilde{B}} R_{\tilde{A}}}^{h} \\ = \int_{l}^{u} (L_{\tilde{B}}^{- 1} (h) - L_{\tilde{A}}^{- 1} (h)) hdh / \\ \int_{l}^{u} hdh, L_{\tilde{A}}^{- 1} \leq L_{\tilde{B}}^{- 1} \leq R_{\tilde{B}}^{- 1} \leq R_{\tilde{A}}^{- 1} \\ S_{R_{\tilde{B}} - L_{\tilde{B}}, L_{\tilde{A}} L_{\tilde{B}} R_{\tilde{B}} R_{\tilde{A}}}^{h} \\ = \int_{l}^{u} (R_{\tilde{B}}^{- 1} (h) - L_{\tilde{B}}^{- 1} (h)) hdh / \\ \int_{l}^{u} hdh, L_{\tilde{A}}^{- 1} \leq L_{\tilde{B}}^{- 1} \leq R_{\tilde{B}}^{- 1} \leq R_{\tilde{A}}^{- 1} \\ S_{R_{\tilde{A}} - R_{\tilde{B}}, L_{\tilde{A}} L_{\tilde{B}} R_{\tilde{B}} R_{\tilde{A}}}^{h} \\ = \int_{l}^{u} (R_{\tilde{A}}^{- 1} (h) - R_{\tilde{B}}^{- 1} (h)) hdh / \\ \int_{l}^{u} hdh, L_{\tilde{A}}^{- 1} \leq L_{\tilde{B}}^{- 1} \leq R_{\tilde{B}}^{- 1} \leq R_{\tilde{A}}^{- 1} \\ S_{L_{\tilde{A}} - L_{\tilde{B}}, L_{\tilde{B}} L_{\tilde{A}} R_{\tilde{A}} R_{\tilde{B}}}^{h} \\ = \int_{l}^{u} (L_{\tilde{A}}^{- 1} (h) - L_{\tilde{B}}^{- 1} (h)) hdh / \\ \int_{l}^{u} hdh, L_{\tilde{B}}^{- 1} \leq L_{\tilde{A}}^{- 1} \leq R_{\tilde{A}}^{- 1} \leq R_{\tilde{B}}^{- 1} \\ S_{R_{\tilde{A}} - L_{\tilde{A}}, L_{\tilde{B}} L_{\tilde{A}} R_{\tilde{A}} R_{\tilde{B}}}^{h} \\ = \int_{l}^{u} (R_{\tilde{A}}^{- 1} (h) - L_{\tilde{A}}^{- 1} (h)) hdh / \\ \int_{l}^{u} hdh, L_{\tilde{B}}^{- 1} \leq L_{\tilde{A}}^{- 1} \leq R_{\tilde{A}}^{- 1} \leq R_{\tilde{B}}^{- 1} \\ S_{R_{\tilde{B}} - R_{\tilde{A}}, L_{\tilde{B}} L_{\tilde{A}} R_{\tilde{A}} R_{\tilde{B}}}^{h} \\ = \int_{l}^{u} (R_{\tilde{B}}^{- 1} (h) - R_{\tilde{A}}^{- 1} (h)) hdh / \\ \int_{l}^{u} hdh, L_{\tilde{B}}^{- 1} \leq L_{\tilde{A}}^{- 1} \leq R_{\tilde{A}}^{- 1} \leq R_{\tilde{B}}^{- 1} \\ S_{L_{\tilde{A}} - L_{\tilde{B}}, L_{\tilde{B}} L_{\tilde{A}} R_{\tilde{B}} R_{\tilde{A}}}^{h} \\ = \int_{l}^{u} (L_{\tilde{A}}^{- 1} (h) - L_{\tilde{B}}^{- 1} (h)) hdh / \end{matrix}$

$\begin{matrix} \int_{l}^{u} hdh, L_{\tilde{B}}^{- 1} \leq L_{\tilde{A}}^{- 1} \leq R_{\tilde{B}}^{- 1} \leq R_{\tilde{A}}^{- 1} \\ S_{R_{\tilde{B}} - L_{\tilde{A}}, L_{\tilde{B}} L_{\tilde{A}} R_{\tilde{B}} R_{\tilde{A}}}^{h} \\ = \int_{l}^{u} (R_{\tilde{B}}^{- 1} (h) - L_{\tilde{A}}^{- 1} (h)) hdh / \\ \int_{l}^{u} hdh, L_{\tilde{B}}^{- 1} \leq L_{\tilde{A}}^{- 1} \leq R_{\tilde{B}}^{- 1} \leq R_{\tilde{A}}^{- 1} \\ S_{R_{\tilde{A}} - R_{\tilde{B}}, L_{\tilde{B}} L_{\tilde{A}} R_{\tilde{B}} R_{\tilde{A}}}^{h} \\ = \int_{l}^{u} (R_{\tilde{A}}^{- 1} (h) - R_{\tilde{B}}^{- 1} (h)) hdh / \\ \int_{l}^{u} hdh, L_{\tilde{B}}^{- 1} \leq L_{\tilde{A}}^{- 1} \leq R_{\tilde{B}}^{- 1} \leq R_{\tilde{A}}^{- 1} \\ S_{R_{\tilde{B}} - L_{\tilde{B}}, L_{\tilde{B}} R_{\tilde{B}} L_{\tilde{A}} R_{\tilde{A}}}^{h} \\ = \int_{l}^{u} (R_{\tilde{B}}^{- 1} (h) - L_{\tilde{B}}^{- 1} (h)) hdh / \\ \int_{l}^{u} hdh, L_{\tilde{B}}^{- 1} \leq R_{\tilde{B}}^{- 1} \leq L_{\tilde{A}}^{- 1} \leq R_{\tilde{A}}^{- 1} \\ S_{R_{\tilde{A}} - L_{\tilde{A}}, L_{\tilde{B}} R_{\tilde{B}} L_{\tilde{A}} R_{\tilde{A}}}^{h} \\ = \int_{l}^{u} (R_{\tilde{A}}^{- 1} (h) - L_{\tilde{A}}^{- 1} (h)) hdh / \\ \int_{l}^{u} hdh, L_{\tilde{B}}^{- 1} \leq R_{\tilde{B}}^{- 1} \leq L_{\tilde{A}}^{- 1} \leq R_{\tilde{A}}^{- 1} \end{matrix}$

where u, l are the upper bound and the lower bound of membership functions of S_i. Assume there are two fuzzy numbers $\tilde{A} = (0.1, 0.3, 0.5, 0.7)$ and $\tilde{B} = (0.2, 0.4, 0.6, 0.8)$ in Fig. 1. By Equation (21), the simple similarity measure for fuzzy numbers $\tilde{A}$ and $\tilde{B}$ can be obtained easily as follows. $\begin{matrix} S (\tilde{A}, \tilde{B}) = \frac{\sum_{i} S_{i}^{\tilde{A} \cap \tilde{B}}}{\sum_{I} S_{I}^{\tilde{A} \cup \tilde{B}}} \\ = S_{R_{\tilde{A}} - L_{\tilde{B}}, L_{\tilde{A}} L_{\tilde{B}} R_{\tilde{A}} R_{\tilde{B}}}^{h} \div (S_{L_{\tilde{B}} - L_{\tilde{A}}, L_{\tilde{A}} L_{\tilde{B}} R_{\tilde{A}} R_{\tilde{B}}}^{h} \\ + S_{R_{\tilde{A}} - L_{\tilde{B}}, L_{\tilde{A}} L_{\tilde{B}} R_{\tilde{A}} R_{\tilde{B}}}^{h} + S_{R_{\tilde{B}} - R_{\tilde{A}}, L_{\tilde{A}} L_{\tilde{B}} R_{\tilde{A}} R_{\tilde{B}}}^{h}) \\ = {\int_{l}^{u} (R_{\tilde{A}}^{- 1} (h) - L_{\tilde{B}}^{- 1} (h)) hdh / \int_{l}^{u} hdh} \\ \div {\int_{l}^{u} (L_{\tilde{B}}^{- 1} (h) - L_{\tilde{A}}^{- 1} (h)) hdh / \int_{l}^{u} hdh \\ + \int_{l}^{u} (R_{\tilde{A}}^{- 1} (h) - L_{\tilde{B}}^{- 1} (h)) hdh / \int_{l}^{u} hdh \end{matrix}$ $\begin{matrix} + \int_{l}^{u} (R_{\tilde{B}}^{- 1} (h) - R_{\tilde{A}}^{- 1} (h)) hdh / \int_{l}^{u} hdh} \\ = {\int_{l}^{u} (0.7 + (0.5 - 0.7) h \\ - (0.2 + (0.4 - 0.2) h) hdh / \int_{l}^{u} hdh} \\ \div {\int_{l}^{u} (0.2 + (0.4 - 0.2) h \\ - (0.1 + (0.3 - 0.1) h) hdh / \int_{l}^{u} hdh \\ + \int_{l}^{u} (0.7 + (0.5 - 0.7) h \\ - (0.2 + (0.4 - 0.2) h) hdh / \int_{l}^{u} hdh \\ + \int_{l}^{u} (0.8 + (0.6 - 0.8) h \\ - (0.7 + (0.5 - 0.7) h) hdh / \int_{l}^{u} hdh} = \frac{7}{13} \end{matrix}$

4.2 Comparing the generalized method with the existing methods

This study illustrates the proposed generalized fuzzy similarity measure with six sets of fuzzy numbers presented in Chou [16] shown as follows. This study compares the results for the generalized similarity measure with those for the nine exiting methods proposed by Chen [10], Hsieh and Chen [24], Hsu and Chen [25], Lee [29], Chen and Chen [12], Deng et al. [19], Chou [15], Allahviranloo et al. [2] and Chou [16]. A comparison of the results for the generalized similarity measure and the existing nine methods is shown in Table 1.

Set 1: $\tilde{A} = (0 . 1, 0 . 2, 0 . 3, 0 . 4)$ , $\tilde{B} = (0.1, 0.25, 0.25, 0 . 4)$

Set 2: $\tilde{A} = (0 . 1, 0 . 2, 0 . 3, 0 . 4)$ , $\tilde{B} = (0 . 1, 0 . 2, 0.3, 0 . 4)$

Set 3: $\tilde{A} = (0 . 1, 0 . 2, 0 . 3, 0 . 4)$ , $\tilde{B} = (0 . 4, 0 . 55, 0.55, 0 . 7)$

Set 4: $\tilde{A} = (0 . 3, 0 . 3, 0 . 3, 0 . 3)$ , $\tilde{B} = (0 . 3, 0 . 3, 0.3, 0 . 3)$

Set 5: $\tilde{A} = (0 . 1, 0 . 4, 0 . 7)$ , $\tilde{B} = (0 . 3, 0 . 4, 0 . 5)$

Set 6: $\tilde{A} = (0 . 2, 0 . 3, 0 . 5, 0 . 6)$ , $\tilde{B} = (0 . 3, 0 . 4, 0 . 5)$

5 Conclusions

This paper fills this gap in the current literature by establishing a generalized similarity measure for fuzzy numbers. The generalized similarity measure is not only to take account of the degree of membership of fuzzy number, but also can be applied to measure the similarity of most fuzzy numbers. Finally, the proposed generalized similarity measure for fuzzy numbers is compared with the existing similarity measures by using some sets of various fuzzy numbers. The results show that the proposed generalized method for similarity measure seems to be more generalized and promising. The generalized similarity measure can be applied to solve fuzzy similarity problems in clustering, multiple-criteria decision-making, group decision-making, image thresholding, risk analysis, prediction, pattern recognition, diagnose, management and engineering in fuzzy environment.

Footnotes

Acknowledgments

The author is grateful to the Editor-in-Chief, Professor Reza Langari, and the three anonymous referees for their great comments and valuable suggestions that helped the author improve this paper.

This research work was partially supported by the Ministry of Science and Technology of Republic of China under Grant No. MOST 103-2410-H-022-002.

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