Application of nearest interval approximation of a fuzzy number to the ranking

This paper proposes a new ranking method for fuzzy numbers, which uses a defuzzification of fuzzy numbers and a weighting function. First, we define a weighted distance measure on fuzzy numbers, and then, by minimizing this distance, the weighted interval a approximations of fuzzy numbers are obtained.


Introduction
Ranking fuzzy numbers plays an important role in fuzzy decision making problems; therefore, deriving the final efficiency and powerful ranking are helpful to decision makers when solving fuzzy problems.Selecting a good ranking method can apply to choosing a desired criterion in a fuzzy environment.In recent years, many ranking methods have been introduced by researchers; some of these ranking methods have been compared and reviewed by Bortolan and Degani [1].Som other methods use statistical techniques such as simulation and hypothesis and quadratic fuzzy regression.Yager and Filve [2] proposed a ranking method with parameterized valuation functions.Tran and Duckstein [3] poposed a weighting function that represents the decision maker , s attitude.Abbasbandy and Asady [4] proposed the ranking of fuzzy numbers by sign distance.Therefore, the essential subject of paper is the weighted ranking of fuzzy numbers.

Definition 1.1 A fuzzy number A=(a,b,c,d) is called a trapezoidal fuzzy number if its membership function A(x) has the following form:
In set 1, the ranking result by our method and nine other methods is A<B<C (see Fig set 1).In set 2, the ranking result for the four methods A<B<C.our method has the same result as in other eleven papers (A<C<B).We conclude that A<C<B is better than A<B<C (see Fig set 2).
In set 3 (Fig set 3), our method has the same result as in four papers of Choobineh and Li, Yager, Chen, and Goetschel and Voxman.The ranking result by Baldwin and Guild, Yao and Wu, Abbasbandy and Asady, and Asady and Zendehnam is A<B<C.By Cheng , s distance method and that of Chu and Taso, the ranking order is A<C<B, B<C<A.  1 gives the results.In set 1, the ranking result by our method and nine other methods is A<B<C (see Fig set 1).
′ () = 0.5667   ′ () = 0.6667   ′ () = 0.7 In set 2, the ranking result for the four methods A<B<C.our method has the same result as in other eleven papers (A<C<B).We conclude that A<C<B is better than A<B<C (see Fig set 2).   ′ () = 0.5, In set 3 (Fig set 3), our method has the same result as in four papers of Choobineh and Li, Yager, Chen, and Goetschel and Voxman.The ranking result by Baldwin and Guild, Yao and Wu, Abbasbandy and Asady, and Asady and Zendehnam is A<B~C.By Cheng , s distance method and that of Chu and Taso, the ranking order is A<C<B, B<C<A.For the CV index, the ranking order is C<A<B.
be three triangular fuzzy numbers (Fig. 2).Therefore, the fuzzy numbers are ranked as B<C< .

Discussion and conclusion
In general, the aim of this paper is threefold.The first aim is to find out the nearest weighted interval approximation of a fuzzy number.The second one is to obtain the nearest weighted point approximation of fuzzy number.The last one is a new flexibility ranking method of the fuzzy numbers by their the f-weighted mean.

Figure 2 :
Figure 2: Fuzzy number A, B, C For two arbitrary fuzzy numbers A and B with []  and []  respectively, the quantity 1) Definition 1.2 The above corollary shows that, for large values m and n the interval [b,c] and the pointAre the nearest weighted interval and point to the trapezoida fuzzy number, respectively (see Fig 1).This section proposes a new ranking method by the weighting mean of a fuzzy number.