A new method for ranking of fuzzy numbers based on sign distance

In this paper, a new approach to rank fuzzy numbers based on sign distance is proposed. Our proposed method is extension of sign distance method and to decrease the set of equivalence callas of fuzzy numbers. Illustrative examples are included to show the advantage of the proposed method.


Introduction
In many applications, ranking of fuzzy numbers is an important and prerequisite procedure for decision makers.Firstly, in 1976, Jain [16,17] proposed a method for ranking of fuzzy numbers, then a large of variety of methods have been developed to rank fuzzy numbers.Wang and kerre [22,23] classified the ordering method into three categories and proposed seven reasonable properties to evaluate the ordering method.In 2007, Asady and Zendehnam [4] proposed a new method based on "Distance Minimizing", and then in 2009, Abbasbandy and Hajjari [2] proposed a new method for ranking of trapezoidal fuzzy numbers and showed that their new method overcomes to some drawbacks of distance minimizing.But, by their new method, all trapezoidal nested symmetric fuzzy numbers with different spreads and also all triangular nested symmetric fuzzy numbers with different spreads are in the same order.Then Ezzati et al. in [12] proposed a new method that solved such problems, but this method also has drawback.Chen and Tang [5] presented the ranking method for the nonnormal p-norm trapezoidal fuzzy numbers based on integral value approach, presented by Liou and Wang [18].Amit Kumar et al. in [24] modified the Chen and Tang [5] approach for ranking of nonnormal p-norm trapezoidal fuzzy numbers, and also in [25] they pointed out the shortcommings of the Liou and Wang [18] approach for the ranking of L-R type generalize fuzzy numbers.In [26] author's compared different methods by means of numerical simulations.Abbasbandy et al. are pointed out the shortcomings of "sign distance method" and in order to solve the problems, means not effectively rank image of fuzzy numbers and inconsistent with human intuition, they are presented a revised method.Abbasbandy et al. are pointed out the shortcomings of "sign distance method" and in order to solve the problems presented a revised method.They stated that these problems were not effectively in rank image of fuzzy numbers and were inconsistent with human intuition.This paper is organized as follows: In Section 2, we review some basic definitions and results on fuzzy numbers.In Section 3, we proposed new method for ranking of fizzy numbers.Comparing the proposed ranking method with some other approaches, some numerical examples are given in Section 4. Finally, Section 5 gives our concluding remarks.

Preliminaries and notations
There are various definitions for the concept of fuzzy numbers ( [10,14,15]) Definition 2.1.[15] A fuzzy number is a fuzzy set like u : R → [0, 1] satisfying the following properties: (iii) There are real numbers a, b, c and d such that a ≤ b ≤ c ≤ d and and the membership function u can be express as are left and right membership function of fuzzy number u, respectively.
Definition 2.2.[20] An arbitrary fuzzy number in the parametric form is represented by an ordered pair of functions (u(r), u(r)), 0 ≤ r ≤ 1, which satisfies the following requirements: 1. u(r) is a bounded left-continuous non-decreasing function over [0, 1].

u(r) is a bounded left-continuous non-increasing function over
is a fuzzy set where the membership function is as Then it is well-known that for any r ∈ [0, 1], [u] r is a bounded closed interval.
For arbitrary u = (u(r), u(r)), v = (v(r), v(r)) addition and scaler multiplication are defined by extension principle and equivalently represented as The collection of all fuzzy numbers with addition and multiplication as defined by equations (2.1) is denoted by E, which is a convex cone.

Proposed Method
In this section, we will propose the ranking of fuzzy numbers associated with the metric D in E in what follows.
The membership function of 0 ∈ R can be defined as follows: We consider u 0 as a fuzzy origin and since u 0 ∈ E, left fuzziness σ and right fuzziness β are 0, so for each Definition 3.2.Let γ : E → {−1, 1} be a function that is defined for arbitrary fuzzy number u as follows: Remark 3.2.If sup(supp u) < 0 or sup u(r) < 0 then γ(u) = −1.
For any two fuzzy numbers u, v ∈ E, define the ranking of u, v by d on E, i.e.
Then, this article formulates the order ≽ and ≼ as u Reasonable axioms that Wang and Kerre [22] had proposed for fuzzy numbers ranking are studied in the following theorem.
Let R be an ordering method, S the set of fuzzy numbers for which the method R can be applied, and A a finite subset of S. The statement "two elements u and v in A satisfy that u has a higher ranking than v when R is applied to the fuzzy numbers in A " will be written as "u ≻ v by R on A ". "u ∼ v by R on A ", and "u ≽ v by R on A " are similarly interpreted.The following proposition shows the reasonable properties (see [22])) of the ordering approach, R.
Theorem 3.1.Let E be the set of fuzzy numbers for which the proposed method can be applied, and A and A ′ are two arbitrary finite subsets of E. The following statements hold.

We obtain the ranking order u ≼ v by R on A ′ if and only if u ≼ v by R on
Proof.The proofs of A 1 , A 2 , A 3 , A 5 , A 7 are clear.We will proof A 4 , A ′ 4 and A 6 .

Numerical examples
In this section, we apply many examples to illustrate the proposed approach to rank trapezoidal fuzzy numbers.Example 4.2.Consider the four fuzzy numbers A = (0, 0.4, 0.7, 0.8), B = (0.2, 0.5, 0.9) and C = (0.1, 0.6, 0.8) which are taken from [29] (see Figure 4.2).then ranking order is A ≺ B ≺ C, which is logically reasonable.However, many ranking approaches prove that two symmetric fuzzy numbers A and B have the same order.To compare with other ranking methods.Clearly the proposed ranking approach can overcome the shortcoming of the inconsistency of many approaches in ranking fuzzy numbers such as those are shown in Table 1.This example also shows the strong discrimination power of the proposed ranking approach and its advantages.
Moreover, based on Wang's computations [28] the ranking result is Result of the Example 14 by new proposed method for p = 2, the Wang's relative preference relation [28], and Sign Distance method [3] for p = 2 are shown in Table 2.

Conclusion
In this paper, we presented a new approach to rank of all trapezoidal fuzzy numbers based on sign distance method.Some properties studied in details and finally using some numerical examples, we presented the advantage of the proposed method.
otherwise and if b = c then u is a triangular fuzzy number, denoted by u = (a, b, d).Therefore, triangular fuzzy numbers are special cases of trapezoidal fuzzy numbers.For 0

Figure 1 :
Figure 1: Fuzzy numbers A, B, C in Example 4.2.

Table 1 :
Comparative results of Example 4.2.