AN APPROXIMATION APPROACH FOR RANKING FUZZY NUMBERS BASED ON WEIGHTED INTERVAL - VALUE

-In the present paper, the researchers discuss the problem of weighted interval approximation of fuzzy numbers. This interval can be used as a crisp set approximation with respect to a fuzzy quantity. Then, by using this, the researchers propose a novel approach to ranking fuzzy numbers. This method can effectively rank various fuzzy numbers, their images and overcome the shortcomings of the previous techniques.

In addition to its ranking features, this method removes the ambiguities resulted and overcome the shortcomings from the comparison of previous ranking.
The paper is organized as follows: In Section 2, this article recall some fundamental results on fuzzy numbers.In Section 3, a crisp set approximation of a fuzzy number is obtained.In this Section some remarks are proposed and illustrated.Proposed method for ranking fuzzy numbers is in the Section 4. Discussion and comparison of this work and other methods are carried out in Section 5.

BASIC DEFINITIONS AND NOTATIONS
The basic definitions of a fuzzy number are given in [9,10,11,12,13,19,20] as follows.Definition 1.Let be a universe set.A fuzzy set of is defined by a membership function , where , indicates the degree of in .
Definition 2. A fuzzy subset of universe set is normal iff , where is the universe set.Definition 3. A fuzzy subset of universe set is convex iff .In this article symbols and denotes the minimum and maximum operators, respectively.Definition 4. A fuzzy set is a fuzzy number iff is normal and convex on .Definition 5. A trapezoidal fuzzy number is a fuzzy number with a membership function defined by : (1) which can be denoted as a quartet .Definition 6.The -cut of a fuzzy number , where is a set defined as According to the definition of a fuzzy number it is seen at once that every -cut of fuzzy number is a closed interval.Hence, this article has , where and .A space of all fuzzy numbers will be denoted by .Definition 7. [4].For two arbitrary fuzzy numbers and with -cut sets and respectively, the quantity , (2) is the weighted distance between and , where And the is nonnegative and increasing on with and .Definition 8. [25].An operator is called an interval approximation operator if for any , core , where denotes a metric defined in the family of all fuzzy numbers.Definition 9. [25].An interval approximation operator satisfying in condition for any is called the continuous interval approximation operator.

WEIGHTED INTERVAL -VALUE APPROXIMATION
Various authors in [18] and [25] have studied the crisp approximation of fuzzy sets.They proposed a rough theoretic definition of that crisp approximation, called the nearest ordinary set and nearest interval approximation of a fuzzy set.In this section, the researchers will propose another approximation called the weighted interval-value approximation.Let be an arbitrary fuzzy number and be its -cut set.
This article will try to find a closed interval , which is the weighted interval to with respect to metric .Since each interval with constant -cuts for all is a fuzzy number, hence, suppose , i.e. , .So, this article has to minimize (3) with respect to and , where In order to minimize it suffices to minimize It is clear that, the parameters and which minimize Eq.(3) must satisfy in Therefore, this article has the following equations: (4) The parameter associated with the left bound and associated with the right bound of the weighted interval-value can be found by using Eq. ( 4) as follows: (5) Remark 1.Since and , therefore and given by ( 5), minimize Therefore, the interval , ( 6) is theweighted interval-value approximation of fuzzy number with respect to .
Remark 2. In special case, if this article assume that , therefore is weighted interval-value possibilistic mean [26].Then, let this article wants to approximate a fuzzy number by a crisp interval.Thus, the researchers have to use an operator which transforms fuzzy numbers into family of closed intervals on the real line.Lemma 1. [4].Theorem 1.The operator is an interval approximation operator, i.e. is a continuous interval approximation operator.
Proof.It is easy to verify that the conditions and are hold.For the proof of , let and be two fuzzy numbers, then .Via to Lemma (1), there is .

It means that when , then this article has
It shows that our weighted interval-value approximation is continuous interval approximation.

COMPARISON FUZZY NUMBERS BY WEIGHTED INTERVAL
In this Section, the researchers will propose the ranking of fuzzy numbers associated with the weighted interval-value.Let be an arbitrary fuzzy number that is characterized by ( 1) and be its -cut and be its the weighted interval-value.Since ever weighted interval-value can be used as a crisp set approximation of a fuzzy number, therefore, the resulting interval is used to rank the fuzzy numbers.Thus, is used to rank fuzzy numbers.This article considers the following reasonable axioms that Wang and Kerre [21] proposed for fuzzy quantities ranking.Let be an ordering method, the set of fuzzy quantities for which the method can be applied, and a finite subset of .The statement "two elements and in satisfy that has a higher ranking than when is applied to the fuzzy quantities in" will be written as " by on " , " by on ", and " by on " are similarly interpreted.The axioms as reasonable properties of ordering fuzzy quantities for an ordering approach are as follows: A- Hence, this article can infer ranking order of the images of the fuzzy numbers.
By using this new approach =[1.66,3.00],=[2.00,3.33]and =[2.33,2.66].Hence, the ranking order is too.Obviously, the results obtained by "Sign distance" and "Distance Minimization" methods are unreasinable.To compare with some of the other methods in [23], the reader can refer to Table 3.Furthermore, to aforesaid example =[-3.00,-1.66],=[-3.33,-2.00]and =[-2.66,-2.33],consequently the ranking order of the images of three fuzzy number is .Clearly, this proposed method has consistency in ranking fuzzy numbers and their images, which could not be guaranteed by CV-index method.All the above examples show that the results of this method are reasonable results.This method can overcome the shortcomings of "Magnitude method" and "Distance Minization" method.

CONCLUSION
In this paper, the researchers proposed a method to rank fuzzy numbers.This method used a crisp set approximation of a fuzzy number that this operator leads to the interval which is the best one with respect to a certain measure of distances between fuzzy numbers.The method can effectively rank various fuzzy numbers and their images (normal/ nonnormal/trapezoidal/general), and overcome the shortcomings which are found in the other methods.

Table 1 .
Comparative results of Example 1.

Table 2 .
Comparative results of Example 2.

Table 3 .
Comparative results of Example 3.