On the Nearest Weighted Point Theorem for Ranking Fuzzy Numbers

In this study the author discusses the problem of defuzzification by minimizing the weighted distance between two fuzzy quantities. Also, this study obtains the nearest point with respect to a fuzzy number and shows that this point is unique relative to the weighted distance. By utilizing this point, a method is presented for effectively ranking fuzzy numbers and their images to overcome the deficiencies of the previous techniques. Finally, several numerical examples following the procedure indicate the ranking results to be valid.


INTRODUCTION
Representing fuzzy numbers by proper intervals is an interesting and important problem.An interval representation of a fuzzy number may have many useful applications.By using such a representation, it is possible to apply some results, in fuzzy number approaches, derived in the field of interval number analysis.For example, it may be applied to a comparison of fuzzy numbers by using the order relations defined on the set of interval numbers.Various authors in (Grzegorzewski, 2002;Saneifard, 2009a) have studied the crisp approximation of fuzzy sets.They proposed a rough theoretic definition of that crisp approximation, called the nearest interval approximation of a fuzzy set.Moreover, quite different approach to crisp approximation of fuzzy sets was applied in Chakrabarty et al. (1998).They proposed a rough theoretic definition of that crisp approximation, called the nearest ordinary set of a fuzzy set.And they suggested a construction of such a set.They discussed rather discrete fuzzy sets.Their approximation of the given fuzzy set is not unique.Thus this study will not discuss this method.
Having reviewed the previous interval approximations, this study proposes here a method to find the weighted interval approximation of a fuzzy number that it is fulfills two conditions.In the first, this interval is a continuous interval approximation operator.In the second, the parametric distance between this interval and the approximated number is minimal.

PRELIMINARIES
The basic definitions of fuzzy number are given in Chu et al. (1994), Dubois and Prade (1987), Heilpern (1992) and Kauffman and Gupta, 1991) as follows: Definition 1: A fuzzy number A in parametric form is a pair (L A , R A ) of functions L A (α) and R A () that 0 ≤ α ≤ 1 , which satisfy the following requirements: Definition 2: The trapezoidal fuzzy number A= (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 A= (x 0 , σ, β ) is called trapezoidal fuzzy number.The parametric from of triangular fuzzy number is L A (α) = x 0 -σ+ σr, R A (α) = y 0 + ββr.
If A = (x 0 , y 0 , σ, β) is a trapezoidal fuzzy number, the parametric form of it is A δ = (L Aσ , R Aσ ) that is as follows: Was f: [0, 1] → [0, 1] is a bi -symmetrical (regular) weighted, is called the bi -symmetrical (regular) weighted distance between A and B based on f.Definition 6: Grzegorzewski (2002): An operator I: F→ (set of c losed intervals in R) is called an interval approximation operator if for any A ∈ F: where, d : F→ [0, + ∞] denotes a metric defined in the family of all fuzzy numbers.Definition 7: Grzegorzewski (2002): An interval approximation operator satisfying in condition (c') for any A, B F is called the continuous interval approximation operator.

NEAREST WEIGHTED INTERVAL OF A FUZZY NUMBER
Various authors in Grzegorzewski (2002) and Saneifard (2009b) 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 intervalvalue approximation.Let A= (L A , R A ) be an arbitrary fuzzy number.This study will try to find a closed interval C dp (A) = [L C , R C ], which is the weighted interval to A with respect to metric d p .So, this study has to minimize: (4) with respect to I L and I R .In order to minimize d p it suffices to minimize: It is clear that, the parameters L C and R C which minimize Eq. ( 4) must satisfy in Therefore; this study has the following equations: (5) The parameters I L associated with the left bound and I R associated with the right bound of the nearest weighted interval can be found by using Eq. ( 5) as follows: (6) therefore L C and R C given by ( 6), minimize (d p (A, C dp (A)) Therefore, the interval: is the nearest weighted interval approximation of fuzzy number A with respect d p .Now, suppose that this study wants to approximate a fuzzy number by a crisp interval.Thus the researchers have to use an operator C dp which transforms fuzzy numbers into family of closed intervals on the real line.

NEAREST WEIGHTED POINT OF A FUZZY NUMBER
Let A = (L A , R A ) be an arbitrary fuzzy number and , is the nearest weighted point approximation to fuzzy number A and its unique.The value of N P (A) is as follows: The above equation introduces in the following Theorem.
Theorem 1: Let A = (L A , R A ) be a fuzzy number and f(α) be a bi symmetrical weighted function.Then N p (A) is nearest weighted point to fuzzy number A.
Proof: For the proof of Theorem it suffices that we replace L C = R C = N P (A) in ( 4) and then minimize function with respect N P (A).Thus this study has to minimize with respect to N P (A).It is clear that, the parameter N P (A) which minimizes Eq. ( 8) must satisfy in.
The solution is:

≤
Holds.We can write  Meanwhile, using the proposed CV index, the ranking order is A≻B≻C.From Fig. 1, it is obvious that the ranking results obtained by the existing approaches (Cheng, 1999;Chu and Tsao, 2002) are unreasonable and inconsistent.On the other hand, in Abbasbandy and Asady (2006), the ranking result is C≻B≻A hich the same as the one is obtained by the proposed method.However the authors' approach proves to be simpler in the computation procedure.Based on the analysis results from Abbasbandy and Asady (2006), the ranking results of this effort and other approaches are listed in Table 1.

CONCLUSION
In this study, the authors proposed a defuzzification using minimize of the weighted distance between two fuzzy numbers and by using this defuzzification we proposed a method for ranking of fuzzy numbers.Roughly, there not much difference in our method and theirs.The method can effectively rank various fuzzy numbers and their images.

Definition 3 :
For arbitrary fuzzy number A F (F denotes the space of fuzzy numbers) and 0≤ σ ≤1, function f: F × [σ, 1] → F such that f (A, σ) = (L Aσ , R Aσ ) is called delta -vicinity of the fuzzy number A The following values constitute the weighted averaged representative and weighted width, respectively, of the fuzzy number A: Saneifard (2010): For two arbitrary fuzzy numbers A = (L A , R A ) and B = (L B , R B ) the quantity: , therefore N P (A) actually minimize  �  (A, N P (A)) and simultaneously minimize d p (A, N P (A)).Theorem 2: The nearest weighted point approximation to a given fuzzy number A is unique.Proof: To prove the uniqueness of the operator N P (A), we show that for any C ∈ R The last sentence of the above diction is zero, hence  �  (, ) =  �  �,   ()� + 2(  () − 2 Consequently  , = , +2−2≥0 then we have  �  (, ) ≥  �  (,   ()) which completes the proof of theorem.Remark 1: Let A and B be two fuzzy numbers and λ and μ be positive numbers.Then we have   ( ±=()±()Proof: Let us suppose for all 0 ≤ α ≤ 1, A = (L A (α), R A (α)) and B = (L C , (α), R B (α)) and f(α) is a bisymetrical weighted function.Then,

Table 1 :
Comparative results of example 1