Vague ranking of fuzzy numbers

Abstract

In a lot of scientific models in the real world, we confront with comparing fuzzy numbers as decision-making procedures and etc. It will be interest, if we know that, comparison discuss is sometimes ambiguous. Hence, this article focus on ranking fuzzy numbers with protection ambiguity. Our idea for this work is based on this claim that ranking of two fuzzy numbers should be a vague value. However, we utilize the notion of max and min fuzzy simultaneously.

Introduction

In variety of application domains, such as decision making [26], risk assessment [13], linear programming [20], linear systems [12], and artificial intelligence [6], ranking fuzzy numbers are used. This topic has been studied by many researchers. Some researchers employed a distance for ordering of fuzzy numbers such as Abbasbandy and Asady [1], Yao and Wu [25], Allahviranloo and Adabitabar Firozja [4], Deng [21] and Janizade-Haji et al. [14]. Some researchers as [2, 15, 16] presented a defuzzification method for ranking fuzzy numbers. Vincent and Luu in [22] proposed improve their ranking method for fuzzy numbers with integral values. In [7], Deng by using ideal solutions showed a ranking approach. Fortemps and Roubens [11] introduced a ranking method based on area compensation. Some of the other researchers such as Adabitabar firozja et al. [3], Ezzati et al. [9, 10] and Modarres and Sadi-Nezhad [18] proposed a function for ranking. Wang et al. [23] defined the maximal and minimal reference sets and then proposed the ranking method based on deviation degree and relative variation of fuzzy numbers and subsequent Asady in [5] proposed a revised method of ranking LR fuzzy number based on deviation degree with Wang$^{,}$s method. Wang and Luo in [24] presented a ranking approach with positive and negative ideal points. Mahmodi Nejad and Mashinchi [17], introduced ranking fuzzy numbers based on the areas on the left and right sides of fuzzy number. In this paper, we provide a method for calculating the amount of vague value ranking fuzzy numbers.

The paper is organized as follows: The background on fuzzy concepts is presented in Sect. 2. A vague ranking of two fuzzy numbers with its properties is introduced in Sect. 3. Subsequently, in Sect. 4 some examples are presented. Finally, conclusion are drawn in Sect. 5.

Background

There are several definitions of a fuzzy number. In this paper we use the following definition.

Definition 1

[8] A set $\tilde{A}$ is a generalized left right fuzzy numbers (GLRFN) and denoted as $\tilde{A}=(a_{1},a_{2},a_{3},a_{4})_{LR}$, if it$^{,}$s membership function satisfies the following:

$$\begin{aligned} \mu _{\tilde{A}}(x)=\left\{ \begin{array}{ll} L(\frac{a_{2}-x}{a_{2}-a_{1}}), &\quad a_{1}\le x\le a_{2}, \\ 1, &\quad a_{2}\le x\le a_{3}, \\ R(\frac{x-a_{3}}{a_{4}-a_{3}}), &\quad a_{3}\le x\le a_{4},\\ 0, &\quad \mathrm{otherwise} \end{array}\right. \end{aligned}$$

(1)

where L and R are strictly decreasing functions defined on [0, 1] and satisfying the conditions:

$$\begin{aligned} \begin{array}{ll} L(t)=R(t)=1 & \quad \mathrm{if}\;t\le 0 \\ L(t)=R(t)=0 & \quad \mathrm{if}\;t\ge 1 \end{array}. \end{aligned}$$

(2)

Remark 1

Trapeziodal fuzzy numbers (TrFN) are special cases of GLRFN with $L(t)=R(t)=1-t$ and we show it as $\tilde{A}=(a_{1},a_{2},a_{3},a_{4})$.

Definition 2

A $\alpha {\text {-}}level$ interval of fuzzy number $\tilde{A}$ is denoted as:

$$[\tilde{A}]^\alpha =\left[ A_{l}(\alpha ),A_{r}(\alpha )]=[a_{2}- (a_{2}-a_{1})L_{A}^{-1}(\alpha ),a_{3}+ (a_{4}-a_{3})R_{A}^{-1}(\alpha )\right] $$

(3)

Remark 2

Suppose, $\lambda \in R$ then

$$\begin{aligned} \begin{array}{c} \tilde{A}+\lambda =(a_{1}+\lambda ,a_{2}+\lambda ,a_{3}+\lambda ,a_{4}+\lambda )_{LR} \\ \lambda \tilde{A}=\left\{ \begin{array}{ll}(\lambda a_{1},\lambda a_{2},\lambda a_{3},\lambda a_{4})_{LR} &\lambda \ge 0 \\ (\lambda a_{4},\lambda a_{3},\lambda a_{2},\lambda a_{1})_{LR} &\lambda < 0 \end{array}\right. \end{array} \end{aligned}$$

(4)

Definition 3

[3] Let $[A]^{\alpha }=[A_{l}(\alpha ),A_{r}(\alpha )]$ and $[B]^{\alpha }=[B_{l}(\alpha ),B_{r}(\alpha )];~~\alpha \in [0,1]$ be two $\alpha -$cuts of fuzzy numbers. We get

$$[\max \{A,B\}]^{\alpha }=\max \{[A]^{\alpha },[B]^{\alpha }\}=[\max \{A_{l}(\alpha ),B_{l}(\alpha )\},\max \{A_{r}(\alpha ),B_{r}(\alpha )\}] $$

and

$$ [\min \{A,B\}]^{\alpha }=\min \{[A]^{\alpha },[B]^{\alpha }\}=[\min \{A_{l}(\alpha ),B_{l}(\alpha )\},\min \{A_{r}(\alpha ),B_{r}(\alpha )\}]. $$

This definition is showed in Fig. 1.

Fig. 1

Fuzzy max and fuzzy min of triangular fuzzy numbers

Definition 4

[3] Let $U=\{u_1,u_2,u_3,...,u_n\}$, a vague set A in U is characterized by a truth-membership function $t_A: U \rightarrow [0, 1]$ and a false-membership function $f_A: U \rightarrow [0, 1]$, where $t_A(u_i)$ is a lower bound on the grade of membership of $u_i$ derived from the evidence for $u_i$, $f_A(u_i)$ is a lower bound on the negation of $u_i$ derived from the evidence against $u_i$, and $t_A(u_i)+ f_A(u_i)\le 1$. The grade of membership of $u_i$ in the vague set A is vague value where bounded by a subinterval $[t_A(u_i), 1-f_A(u_i)]$ of [0, 1]. Simply expressed, $A(u_i)=[t_A(u_i),1-f_A(u_i)]$.

For an arbitrary element $a\in [0,1]$, we assume that a is the same as [a, a], namely, $a=[a,a]$. For any $A=[a_1,a_2]$ and $B=[b_1,b_2]$, we can popularize operators such $+$ and − and have $A+B=[a_1+b_1,a_2+b_2]$, $A-B=[a_1-b_2,a_2-b_1]$. Furthermore, we have $A=B \Leftrightarrow a_1=b_1,a_2=b_2$, $A\le B \Leftrightarrow a_1\le b_1,a_2\le b_2$ and $A<B \Leftrightarrow a_1<b_1,a_2<b_2.$

Vague ranking of two fuzzy numbers

Given $\tilde{A},~\tilde{B}\in E_{LR}$, are two fuzzy numbers. Regarding to many methods and shortcoming in ranking for fuzzy numbers, it is show that ranking is not deterministic. In other words, we know if $\mathrm{supp}(\tilde{A})\cap \mathrm{supp}(\tilde{B})\ne \phi$ then we can not define a crisp rank for $\tilde{A},~\tilde{B}$. Therefore, we claim that ranking of two fuzzy numbers should be a vague value. Some researcher are used the max or min notion for ranking the fuzzy number. But, we utilize the notion of max and min simultaneously. It is trivial maximum of two fuzzy numbers is greater than or equal both of them and minimum of two fuzzy numbers is less than or equal both of them. Therefore, we will present true rate $\tilde{A}\le \tilde{B}$ as $t_{A\preceq B}$ and false rate $\tilde{A}\le \tilde{B}$ as $f_{A\preceq B}$ in ranking $\tilde{A}$ and $\tilde{B}$ as follows:

$$\begin{aligned} \begin{array}{c} t_{A\preceq B}=S\{\{\max \{\tilde{A},\tilde{B}\}{\setminus } \tilde{A}\}\bigcup \{\min \{\tilde{A},\tilde{B}\}{\setminus } \tilde{B}\}\}\\ f_{A\preceq B}=S\{\{\max \{\tilde{A},\tilde{B}\}{\setminus } \tilde{B}\}\bigcup \{\min \{\tilde{A},\tilde{B}\}{\setminus } \tilde{A}\}\} \end{array} \end{aligned}$$

(5)

where the signs ${\setminus }$ and $S\{.\}$ show subtract of Venn diagram and area, respectively. For this purpose, consider the following Fig. 2.

Fig. 2

Comparison of two fuzzy numbers $\tilde{A}, \tilde{B}$

Where geometrically, $t_{A\preceq B}$ and $f_{A\preceq B}$ defined above is as follows:

$$\begin{aligned} \begin{array}{ccc} t_{A\preceq B}=\frac{S(A<B)}{S},&f_{A\preceq B}=\frac{S(B<A)}{S},&S=S(A<B)+S(B<A)+S(A=B) \end{array} \end{aligned}$$

(6)

$S(A<B)$ is part of surface B that is located on the right side of A or part of surface A that is located on the left side of B and $S(A=B)$ is common surface of A and B.

Remark 3

For $\tilde{A},~\tilde{B}\in E_{LR}$, $t_{A\preceq B}+f_{A\preceq B}\le 1$.

Proof

$$ t_{A\preceq B}+f_{A\preceq B}=\frac{S(A<B)}{S}+\frac{S(B<A)}{S}=\frac{S(A<B)+S(B<A)}{S(A<B)+S(B<A)+S(A=B)}\le 1 $$

As mentioned in above we used the notion area for ranking. Hence, we introduce two theorems in below. $\square$

Theorem 1

[19] If points $A(x_1,y_1)$, $B(x_2,y_2)$ and $C(x_3,y_3)$ be arbitrarily coordinates are triangular vertexes in anti-clock wise sense then the area of triangle $\vartriangle ABC$ is determined as follows:

$$\begin{aligned} S_{\vartriangle ABC}=\frac{1}{2}\left\{ \left| \begin{array}{cc} x_1&y_1\\ x_2&y_2 \end{array}\right| +\left| \begin{array}{cc} x_2&y_2\\ x_3&y_3 \end{array}\right| +\left| \begin{array}{cc} x_3&y_3\\ x_1&y_1 \end{array}\right| \right\} \end{aligned}$$

(7)

Theorem 2

[19] The area of any regular polygon with $P_j(x_j,y_j)$, $j=1,...,n$ vertex in anti-clock wise sense is as follows:

$$\begin{aligned} S_{p_1P_2...P_n}=\frac{1}{2}\left\{ \left| \begin{array}{cc} x_1&y_1\\ x_2&y_2 \end{array}\right| +\left| \begin{array}{cc} x_2&y_2\\ x_3&y_3 \end{array}\right| +\cdots +\left| \begin{array}{cc} x_n&y_n\\ x_1&y_1 \end{array}\right| \right\} \end{aligned}$$

(8)

Definition 5

Assume that $\tilde{A}$ and $\tilde{B}$ be two GLRFNs, validity rating of $A\preceq B$ is belong to interval $[t_{A\preceq B},1-f_{A\preceq B}]$ where $t_{A\preceq B}$ minimum accuracy and $1-f_{A\preceq B}$ maximum accuracy and we show with vague value of rank $\tilde{A}\preceq \tilde{B}$ with $\mathrm{VR}(A\preceq B)$ where

$$ \mathrm{VR}(A\preceq B)=[t_{A\preceq B},1-f_{A\preceq B}] $$

(9)

Some properties

For $\tilde{A}$ and $\tilde{B}\in {E_{LR}}$ and ${{\lambda }\in R}$:

Proposition 1

$VR(A\preceq B)$ is a vague value.

Proof

With Remark 1. proof is evident.

Proposition 2

$t_{A\preceq B}=f_{B\preceq A}$, $f_{A\preceq B}=t_{B\preceq A}$.

Proof

With Eq. (6) proof is evident.

Proposition 3

$VR(A\preceq B)=1-VR(B\preceq A)$

Proof

Regarding to Eq. (9) and Proposition 2. $1-\mathrm{VR}(B\preceq A)=[1,1]-[t_{B\preceq A},1-f_{B\preceq A}]=[f_{B\preceq A},1-t_{B\preceq A}]=[t_{A\preceq B},1-f_{A\preceq B}]=\mathrm{VR}(A\preceq B)$

Proposition 4

$VR(\lambda A\preceq \lambda B)= \left\{ \begin{array}{ll} VR(A\preceq B) &\quad 0\le \lambda ,\\ VR(B\preceq A) &\quad otherwise. \end{array}\right.$

Proof

Regarding to the Eqs. (9) and (6); if $\lambda \ge 0$ $\mathrm{VR}(\lambda A\preceq \lambda B)=[t_{\lambda A\preceq \lambda B},1-f_{\lambda A\preceq \lambda B}]=[t_{A\preceq B},1-f_{A\preceq B}]=\mathrm{VR}(A\preceq B)$ And if $\lambda <0$

$\mathrm{VR}(\lambda A\preceq \lambda B)=[t_{\lambda A\preceq \lambda B},1-f_{\lambda A\preceq \lambda B}]=[t_{B\preceq A},1-f_{B\preceq A}]=\mathrm{VR}(B\preceq A)$

Proposition 5

$VR(\lambda +A\preceq \lambda + B)=VR(A\preceq B)$.

Proof

Regarding Eqs. (9) and (6)

$$ \mathrm{VR}(\lambda +A\preceq \lambda + B)=[t_{\lambda +A\preceq \lambda +B},1-f_{\lambda +A\preceq \lambda +B}]=[t_{A\preceq B},1-f_{A\preceq B}]=\mathrm{VR}(A\preceq B). $$

Proposition 6

If $a_{4}\le b_{1}$ then $VR(\tilde{A}\preceq \tilde{B})=[1,1]$.

Proof

Regarding to Eqs. (9) and (6) proof is evident.

Proposition 7

If $\tilde{A}$ and $\tilde{B}$ are two GLRFNs then only one of the following relationship is established:

$VR(A\preceq B)=VR(B\preceq A)$, $VR(A\preceq B)\le VR(B\preceq A)$ and $VR(A\preceq B)\ge VR(B\preceq A)$.

Proof

If $t_{A\preceq B}=t_{B\preceq A}$ then with Proposition 2, $f_{A\preceq B}=f_{B\preceq A}$ therefore $\mathrm{VR}(A\preceq B)=\mathrm{VR}(B\preceq A)$. If $t_{A\preceq B}< t_{B\preceq A}$ then with Proposition 2, $f_{A\preceq B}<f_{B\preceq A}$ therefore $\mathrm{VR}(A\preceq B)<\mathrm{VR}(B\preceq A)$. If $t_{A\preceq B}> t_{B\preceq A}$ then with Proposition 2, $f_{A\preceq B}>f_{B\preceq A}$ therefore $\mathrm{VR}(A\preceq B)>\mathrm{VR}(B\preceq A)$.

Definition 6

If $\tilde{A}$ and $\tilde{B}$ are two GLRFNs, validity rating of $A\preceq B$ is belong to interval $\mathrm{VR}(A\preceq B)=[t_{A\preceq B},1-f_{A\preceq B}]$ where $t_{A\preceq B}$ minimum accuracy and $1-f_{A\preceq B}$ maximum accuracy. With Proposition 9, we define ranking method as follows:

• 1. If $\mathrm{VR}(A\preceq B)=\mathrm{VR}(B\preceq A)=[0.5,0.5]$, then can be said ${\tilde{A}}\approx {\tilde{B}}$.

• 2. If $\mathrm{VR}(A\preceq B)=\mathrm{VR}(B\preceq A)=[0,1]$, then can be said ${\tilde{A}}={\tilde{B}}$.

• 3. If $\mathrm{VR}(A\preceq B)< \mathrm{VR}(B\preceq A)$ then can be said ${\tilde{B}}\preceq {\tilde{A}}$.

• 4. If $\mathrm{VR}(A\preceq B)=[1,1]$ or $\mathrm{VR}(B\preceq A)=[0,0]$ then can be said ${\tilde{A}}<{\tilde{B}}$.

Numerical examples

For description the proposed method some examples constructed as follow [4, 21].

Set 1 $A_{1}=(0.4,0.9,1)$, $A_{2}=(0.4,0.7,1)$, $A_{3}=(0.4,0.5,1)$ where show in Fig. 3.

$\mathrm{VR}(A_{2}\le A_{1})=[0.4,1]$, with Proposition 3 $\mathrm{VR}(A_{1}\le A_{2})=[0,0.6]$ therefore according to Definition 2, ${A_{1}}\succeq {A_{2}}$.

$\mathrm{VR}(A_{3}\le A_{1})=[0.82,1]$ and obviously $\mathrm{VR}(A_{1}\le A_{3})=[0,0.18]$ therefore, ${A_{1}}\succeq {A_{3}}$.

$\mathrm{VR}(A_{3}\le A_{2})=[0.4,1]$ and it is trivial that $\mathrm{VR}(A_{2}\le A_{3})=[0,0.6]$ therefore, ${A_{2}}\succeq {A_{3}}$.

Set 2 $A_{1}=(0.2,0.5,0.8)$, $A_{2}=(0.4,0.5,0.6)$, where show in Fig. 4.

$\mathrm{VR}(A_{1}\le A_{2})=[0.5,0.5]$ therefor $\mathrm{VR}(A_{2}\le A_{1})=[0.5,0.5]$ hence ${A_{1}}\approx {A_{2}}$.

Set 3 $A_{1}=(0.5,0.7,0.9)$, $A_{2}=(0.3,0.7,0.9)$, $A_{3}=(0.3,0.4,0.7,0.9)$, where show in Fig. 5.

$\mathrm{VR}(A_{3}\le A_{2})=[0.333,1]$, $\mathrm{VR}(A_{3}\le A_{1})=[0.56,1]$ and $\mathrm{VR}(A_{2}\le A_{1})=[0.333,1]$ as before, it follows that, ${A_{1}}\succeq {A_{2}}$, ${A_{2}}\succeq {A_{3}}$ and ${A_{1}}\succeq {A_{3}}$.

And we consider another example for comparing the current method with

Set 4 $A_{1}=(0.1,0.6,0.7)$, $A_{2}=(0.2,0.4,0.9)$ where show in Fig. 6.

$\mathrm{VR}(A_{2}\le A_{1})=[0.1931417,0.83182047]$, that shows ${A_{2}}\succeq {A_{1}}$.

Fig. 3

Set 1-$\{ \tilde{A_1},\tilde{A_2}, \tilde{A_3}\}$

Fig. 4

Set 2-$\{ \tilde{A_1}, \tilde{A_2}\}$

Fig. 5

Set 3-$\{ \tilde{A_1}, \tilde{A_2}, \tilde{A_3}\}$

Fig. 6

Set 4-$\{ \tilde{A_1}, \tilde{A_2}\}$

Conclusions

In this paper, we showed that ranking of two fuzzy numbers should be a vague value. For this reason, we utilize the notion of max and min simultaneously in order to determining the ambiguity rate in ranking of two fuzzy numbers. It is shown that this approach verifies some properties as stability, transition and complement.