A Method for Ranking of Fuzzy Numbers Using New Weighted Distance

In this paper, the researchers proposed a modified new weighted distance method to rank fuzzy numbers. The modified method can effectively rank various fuzzy numbers, their images and overcome the shortcomings of the previous techniques. The proposed model is studied for a broad class for fuzzy numbers and class of functions the membership of which is formed on the basis of the template) 1. 0 (max) (s x x   . This article also used some comparative examples to illustrate the advantage of the proposed method. 1. INTRODUCTION In many applications, ranking of fuzzy numbers is an important component of the decision process. In addition to a fuzzy environment, ranking is a very important decision making procedure. Since Jain [4, 5] employed the concept of maximizing set to order the fuzzy numbers in 1976, many authors have investigated various ranking methods. Some of these ranking methods have been compared and reviewed by Bortolan and Degani [6], and more recently by Chen and Hwang [7]. Other contributions in this field include: an index for ordering fuzzy numbers defined by Choobineh and Li [8], ranking alternatives using fuzzy numbers studied by Dias [9], automatic ranking of fuzzy numbers using artificial neural networks proposed by Requena e.t al [10], ranking fuzzy values with satisfaction function investigated by Lee e.t al [11], ranking and defuzzification methods based on area compensation presented by Fortemps and Roubens [12], and ranking alternatives with fuzzy weights using maximizing set and minimizing set given by Raj and Kumar [13]. However, some of these methods are computationally complex and difficult to implement, and others are counterintuitive and not discriminating. Furthermore, many of them produce different ranking outcomes for the same problem. In 1988, Lee and Li [14] proposed a comparison of fuzzy numbers by considering the mean and dispersion (standard deviation) based on the uniform and the proportional probability distributions. Cheng [15] proposed the coefficient of variance (CV index), i.e.   CV (standard error)


INTRODUCTION
In many applications, ranking of fuzzy numbers is an important component of the decision process.In addition to a fuzzy environment, ranking is a very important decision making procedure.Since Jain [4,5] employed the concept of maximizing set to order the fuzzy numbers in 1976, many authors have investigated various ranking methods.Some of these ranking methods have been compared and reviewed by Bortolan and Degani [6], and more recently by Chen and Hwang [7].Other contributions in this field include: an index for ordering fuzzy numbers defined by Choobineh and Li [8], ranking alternatives using fuzzy numbers studied by Dias [9], automatic ranking of fuzzy numbers using artificial neural networks proposed by Requena e.t al [10], ranking fuzzy values with satisfaction function investigated by Lee e.t al [11], ranking and defuzzification methods based on area compensation presented by Fortemps and Roubens [12], and ranking alternatives with fuzzy weights using maximizing set and minimizing set given by Raj and Kumar [13].However, some of these methods are computationally complex and difficult to implement, and others are counterintuitive and not discriminating.Furthermore, many of them produce different ranking outcomes for the same problem.In 1988, Lee and Li [14] proposed a comparison of fuzzy numbers by considering the mean and dispersion (standard deviation) based on the uniform and the proportional probability distributions.Cheng [15] proposed the coefficient of variance (CV index), i.e.
. In this approach, the fuzzy number with smaller CV index is ranked higher, therefore Cheng's CV index also contains shortcomings.To improve Murakami e.t al.'s method, Cheng [15] proposed the distance method for ranking fuzzy numbers; i.e., . For any two fuzzy numbers i A and from [15] .In Cheng's distance method, R(A)=0.590,R(B)=0.604, and R(C)=0.662,produce the ranking order . From this result, the researchers can logically infer the ranking order of the images of these fuzzy numbers as . However, in the distance method, the ranking order remains . Obviously, the distance method also has shortcomings.Moreover, in [1] a method based on "Sign Distance" was introduced and a new method based on "Distance Minimization" was introduced by Asady e.t al.'s [3].This method has some drawbacks, i.e., for all triangular fuzzy numbers and also trapezoidal fuzzy numbers ) , , , ( , gives the same results.However it is clear that these fuzzy numbers do not place in an equivalence class.Recently, a new method based on "the left and the right spreads at some  -levels of trapezoidal fuzzy numbers" was introduced [2].This method has some shortcoming too, because, for any two symmetric trapezoidal fuzzy numbers, gives equal ordering.Having reviewed the previous methods, this article proposes here a method to use the concept of fuzzy distance, so as to find the order of fuzzy numbers.This method can distinguish the alternatives clearly.The main purpose of this article is to present a new method for ranking of fuzzy numbers.In addition to its ranking features, this method removes the ambiguities resulted from the comparison of previous ranking.The paper is organized as follows: In Section 2, we recall some fundamental results on fuzzy numbers.Proposed method for ranking of fuzzy numbers is in the Section 3. In this Section some theorems and remarks are proposed and illustrated.Discussion and comparison of this work and other methods are carried out in Section 4. The paper ends with conclusions in section 5.

BASIC DEFINITIONS AND NOTATIONS
The basic definitions of a fuzzy number are given in [18,19,20] as follows: with the following properties: Let  be the set of all real numbers.The researchers assume a fuzzy number A that can be expressed for all are real numbers such as and g is a real valued function that is increasing and right continuous and h is a real valued function that is decreasing and left continuous.The trapezoidal fuzzy number . In this paper, the researchers will always refer to fuzzy number A described by (1).

NEW APPROACH FOR RANKING OF FUZZY NUMBERS
In this section, the researchers will propose the ranking of fuzzy numbers associated with the metric D in F , that F denotes the space of fuzzy numbers.We will assume that the fuzzy number F A is represented by means of the following LRrepresentation: is a monotonically nonincreasing left-continuous functions.The functions denotes an "optimism/pessimism" coefficient in conducting operations on fuzzy numbers.The function is also called weighting function.In actual applications, function ) ( f can be chosen according to the actual situation.In this article, in practical case, we assume that be a function that is defined as follows: . Definition 3.3.For arbitrary fuzzy numbers A and B the quantity , is called the TRD distance between the fuzzy numbers A and B .It is easily proved that the TRD distance satisfies the following properties:

This article considers 0
A as a fuzzy origin and since This article considers the following reasonable axioms that Wang and Kerre [21] proposed for fuzzy quantities ranking.Let TR be an ordering method, S the set of fuzzy quantities for which the method TR can be applied, and  a finite subset of S .The statement "two elements A and B in  satisfy that A has a higher ranking than B when TR is applied to the fuzzy quantities in " will be written as " B A  by TR on " , " A ~B by TR on ", and " A  B by TR on " are similarly interpreted.[21], the axioms as the reasonable properties of ordering fuzzy quantities for an ordering approach TR are as follows: A-1 For an arbitrary finite subset  of S and The function TR has the properties (A-1), (A-2), ..., (A-5).Remark 3.4.
Hence, this approach can infer ranking order of the images of the fuzzy numbers.

NUMERICAL EXAMPLES
Now, the authors compare proposed method with the others in [8,22,24,25].Throughout this section, we assum that , and "optimism/pessimism" coefficient is 0.5.

B A 
In Table 3, A ~B is the results of Sign Distance method with p=1, Magnitude method, Distance Minimization and Chen method, which is unreasonable.The results of this method is the same as Sign Distance method with p=2, i.e., B A  .All the above examples show that the results of this method are reasonable results.The proposed method can be overcome the shortcoming of "Magnitude" method and "Distance Minimization" method.

CONCLUSION
In this article, the researchers proposed a new weighted distance between the two fuzzy numbers and a ranking method for the fuzzy numbers.The new method can effectively rank various fuzzy numbers, their images and overcome the shortcomings of the previous techniques.The calculations of the proposed method are simpler than the other approaches.This article also used comparative examples to illustrate the advantages of the proposed method.
set where the membership function is as

Figs. 1
Figs.1 Moreover, the distance method contradicts the CV index in ranking some The following values constitute the weighted averaged representative and weighted width, respectively, of the fuzzy number A For each arbitrary fuzzy numbers A , F B , define the ranking of A and B by TR on F , i.e.This article formulates the order  and  as A  B if and only if B A  or A ~B , A  B if and only if B A  or A ~B .

2
For an arbitrary finite subset  of S and For an arbitrary finite subset  of S and A  B and B  A by TR on , this method should have A ~B .A-3 A  B and B  C by TR on , this method should have A  C .A-4 For an arbitrary finite subset  of S and A ) > sup supp( B ), this method should have A  B. A'-4 For an arbitrary finite subset  of S and

Table 1 .
Comparative results of Example 1.

Table 3 .
Comparative results of Example 4