Vague ranking of fuzzy numbers

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 2 A aÀlevel interval of fuzzy numberÃ is denoted as: This definition is showed in Fig. 1.

Definition 4 [3]
Let U ¼ fu 1 ; u 2 ; u 3 ; :::; u n g, a vague set A in U is characterized by a truth-membership function t A : U ! ½0; 1 and a false-membership function f A : U ! ½0; 1, where t A ðu i Þ is a lower bound on the grade of membership of u i derived from the evidence for u i , f A ðu i Þ is a lower bound on the negation of u i derived from the evidence against u i , and For an arbitrary element a 2 ½0; 1, we assume that a is the same as [a, a], namely, a ¼ ½a; a. For any A ¼ ½a 1 ; a 2 and B ¼ ½b 1 ; b 2 , we can popularize operators such þ and -and have Vague ranking of two fuzzy numbers GivenÃ;B 2 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 suppðÃÞ \ suppðBÞ 6 ¼ / then we can not define a crisp rank forÃ;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 rateÃ B as t A"B and false rateÃ B as f A"B in rankingÃ andB as follows: where the signs n and Sf:g show subtract of Venn diagram and area, respectively. For this purpose, consider the following Fig. 2. Where geometrically, t A"B and f A"B defined above is as follows: SðA\BÞ 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 SðA ¼ BÞ is common surface of A and B.
As mentioned in above we used the notion area for ranking. Hence, we introduce two theorems in below. h Theorem 1 [19] If points Aðx 1 ; y 1 Þ, Bðx 2 ; y 2 Þ and Cðx 3 ; y 3 Þ be arbitrarily coordinates are triangular vertexes in anti-clock wise sense then the area of triangle MABC is determined as follows: Theorem 2 [19] The area of any regular polygon with P j ðx j ; y j Þ, j ¼ 1; :::; n vertex in anti-clock wise sense is as follows: S p1P2:::Pn ¼ 1 2 Definition 5 Assume thatÃ andB be two GLRFNs, validity rating of A " B is belong to interval ½t A"B ; 1 À f A"B where t A"B minimum accuracy and 1 À f A"B maximum accuracy and we show with vague value of rankÃ " B with VRðA " BÞ where

Some properties
ForÃ andB 2 E LR and k 2 R: Proposition 1 VRðA " BÞ is a vague value.
Proof With Remark 1. proof is evident.

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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