A new approach for ranking fuzzy numbers

In this paper, we propose a novel approach for constructing a preference fuzzy distance measure. For this purpose, we propose a new and effective preference ordering based on the Abbasbandy and Hajjari’s approach [2].


Introduction
In order to rank fuzzy numbers, one fuzzy number needs to be evaluated and compared with the others, but this may not be easy.Jain [4,5] proposed a method using concept of maximizing set to order the fuzzy numbers in 1976.Some of these ranking methods have been compared and reviewed by Bortolan and Degani [6].Chen and Hwang [29] proposed fuzzy multiple attribute decision making in 1992, Choobineh and Li [8] proposed an index for ordering fuzzy numbers in 1993.Also, Dias [9] ranked alternatives by ordering fuzzy numbers in 1993, Fortemps and Roubens [12] presented ranking and defuzzication methods based on area compensation in 1996.In 1998, Lee and Li [14] ranked fuzzy numbers based on two different criteria, namely, the fuzzy mean and the fuzzy spread of fuzzy numbers.So, Cheng [15] proposed the coefficient of variance (CV index) to improve Lee and Li's ranking method.Chu and Tsao [24] pointed out that the shortcomings of Cheng's method and suggested to rank fuzzy numbers with the area between the centroid point and the point of origin.Wang and co-workers [30] found that the centroid formulae provided by the paper [15,24] are incorrect and have led to some misapplications.So, they presented the correct centroid formulae for the fuzzy numbers and justify them from the viewpoint of analytical geometry.Also, Chu's method still has some drawbacks, which is expressed by Deng et.al [30] and carried out to modify by applying radius of gyration in 2006.However, Deng's method also still has some drawbacks, i.e. it can not rank negative fuzzy numbers correctly (numerical example is illustrated in section 3).Notice that Deng's method is independent from centroid point of fuzzy numbers [30].Also, Abbasbandy et.al [1] proposed a ranking method based on sign distance in 2006, Asady et.al proposed an ordering approach by distance minimization [3].Asady's method has some drawbacks, i.e. for all triangular fuzzy numbers u = (x 0 , σ , β ) where x 0 = σ −β 4 and also trapezoidal fuzzy number u = (x 0 , y 0 , σ , β ) such that x 0 + y 0 = σ −β 2 gives the same results.So, Abbasbandy and Hajjari [2] proposed magnitude of fuzzy numbers in 2009 to improve Asady's method.Also, Abbasbandy's method has some drawbacks (numerical example is illustrated in section 3).These mentioned ranking methods are also placed in first or second class of Kerre's [21]categories.In this paper, we proposed a new ranking method for fuzzy numbers which is improved Abbasbandy's approach [2].Some algebraic properties of our proposed method are given, then is introduced an interval distance between two arbitrary fuzzy number.

Preliminaries
The basic definitions of a fuzzy number are given in [18,19,20]as follows: Definition 2.1.A fuzzy number is a fuzzy set like u : R→ [0, 1] which satisfies: 3. There are real numbers a, b such as a ≤ b ≤ c ≤ d and The membership function u is presented as Definition 2.2.A fuzzy number u in parametric form is a pair (u, u) of functions u(r), u(r), 0 ≤ r ≤ 1, which satisfy the following requirements: 1. u(r) is a bounded non-decreasing left continuous function in (0, 1], and right continuous at 0, 2. u(r) is a bounded non-increasing left continuous function in (0, 1], and right continuous at 0, The trapezoidal fuzzy number u = (x 0 , y 0 ; σ , β ), with two defuzzifier x 0 , y 0 , and left fuzziness σ > 0 and right fuzziness β > 0 is a fuzzy set where the membership function is as If x 0 = y 0 , then u is called triangular fuzzy number and we write u = (x 0 ; σ , β ).The support of fuzzy number u is defined as follows: where {x|u(x) > 0} is closure of set {x|u(x) > 0}.
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New preference ordering for fuzzy numbers
For an arbitrary fuzzy number u = (x 0 , y 0 ; σ , β ), with parametric form u = (u(r), u(r)), we define the modified magnitude of the fuzzy number u as where the function f (r) is non-negative and increasing function on [0, 1] with f (0) = 0, f (1) = 1 and and λ ∈ [0, 1] is a decision maker parameter.
Obviously, modified magnitude of fuzzy number inherit most of properties of magnitude.So they have similar interpretation such that modified magnitude of fuzzy number u which is defined by ( 2), reflects the information on every membership degree and even have flexible behavior in compare of magnitude approach [2].So, for two arbitrary fuzzy number u and v ∈ E, we define preference ranking of u and v by the MMag(., λ ) on E(set of all fuzzy numbers) for all λ ∈ [0, 1] as follows: (1) Also we formulate the order ≽ and ≼ as u 3.1 Some algebraic properties of MMag(., λ ) Here, we investigate the algebraic properties of modified magnitude of fuzzy number (2).
A-2 For an arbitrary finite subset A of S and (u, v) ∈ A 2 ; u ≽ v and v ≽ u by MMag on A , this method should have u ∼ v.

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A-3 For an arbitrary finite subset A of S and (u, v, w) ∈ A 3 ; u ≽ v and v ≽ w by MMag on A , this method should have u ≽ w.
A-4 For an arbitrary finite subset A of S and (u, v) ∈ A 2 ; inf supp(u)>sup supp(v), this method should have u ≽ v.
A ′ -4 For an arbitrary finite subset A of S and (u, v) ∈ A 2 ; inf supp(u)>sup supp(v), this method should have u ≻ v.
A-5 Let S , S ′ be two arbitrary finite sets of fuzzy quantities in which MMag can be applied and u , v are in S ∩ S ′ .
This method obtain the ranking order u ≽ v on S ′ iff u ≽ v on S.
A-6 Let u, v, u + w and v + w be elements of S.

Examples
Here, we take some illustrative examples to show the ability of proposed method.
For more detail see Table1.

. 1 )
where u L : [a, b] → [0, 1] and u R : [c, d] → [0, 1] are left and right membership functions of fuzzy number u.Another definition for a fuzzy number is as follows:

Table 1 :
Comparative results of example 3.3