# 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.

## Keywords

Fuzzy numbers Ranking Vague value## 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**

*L*and

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

*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**

*Remark 2*

**Definition 3**

**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

*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*

**Theorem 1**

*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*:

**Theorem 2**

*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*:

**Definition 5**

### 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*

**Proposition 6**

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

**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**

- 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.

## 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.

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