A New Distance Measure and Ranking Method for Generalized Trapezoidal Fuzzy Numbers

which permits


Introduction
The fuzzy set theory pioneered by Zadeh [1] has been extensively used.Fuzzy numbers or fuzzy subsets of the real line  are applied to represent the imprecise numerical measurements of different alternatives.Therefore, comparing the different alternatives is actually comparing the resulting fuzzy numbers.
Also in many applications of fuzzy set theory to decision making, we need to know the best select from a collection of possible solutions.This selection process may require that we rank or order fuzzy numbers.
Several researchers presented ranking methods [2][3][4][5][6][7].Various techniques are applied to compare the fuzzy numbers.Some of the exiting approaches are difficult to understand and have suffered from different plights, for example, the lack of discrimination, producing counterintuitive orderings, and ultimately resulting in inconsistent ordering if a new fuzzy number is added.Also nearly all approaches should acquire membership functions of fuzzy numbers before the ranking is performed; however, this may be infeasible in real applications.Furthermore, accuracy and efficiency should be of priority concern in the ranking process if ranking a large amount of fuzzy numbers.In light of the above discussion, specially in [6], Chen and Lu have proposed an approximate approach for ranking fuzzy numbers, in that they worked with dominance.
Also we studied here the approach is determined by convex combining the centroid point ( 0 ,  0 ) of  0 and area  of a fuzzy number that performs simple arithmetic operations for the ranking purpose, and it can be applied to rank the combination case of some fuzzy numbers and crisp numbers and the case of discrete fuzzy numbers and it is useful in ranking a large quantity of fuzzy numbers.Comparing the proposed approximate approach with the existing methods using both Bortolan and Degani's examples [3].
The methods of measuring the distance between fuzzy numbers have became important due to the significant applications in diverse fields like remote sensing, data mining, pattern recognition, multivariate data analysis and so on.Several distance measures for precise numbers are well established in the literature.Several researchers focused on computing the distance between fuzzy numbers [2,3,5,[7][8][9][10].Here we introduce a new distance between two trapezoidal fuzzy numbers by the new approach proposed for them.
This paper is organized as follows.
In Section 2, the basic concept of fuzzy number operation is brought.Section 3 introduces the ranking approach and presents some comparative examples which demonstrate the accuracy of the proposed approach over the exiting methods.A new measure between fuzzy numbers is defined in Section 4. Concluding remarks are finally made in Section 5.

Preliminaries
A fuzzy number is a convex fuzzy subset of real line  and is completely defined by its membership function.Let  be a fuzzy number, whose membership function () can generally be defined as where 0 <  ≤  Definition 1 (see [11]).Consider a general trapezoidal fuzzy number  = [, , , ; ], whose membership function is defined as  1.

The Ranking Method
Let  be a trapezoidal generalized fuzzy number.We define the particular convex combination of the fuzzy number's relative locations on the -axis,  0 , and the fuzzy number's area, , in Definitions 1 and 2, respectively, as follows where the index  ∈ [0, 1].
The following choices are related to find approximate approach for rankings fuzzy numbers based on decision maker's opinions.
A decision maker is able to denote his preference for the fuzzy number's relative locations on the -axis, , or the fuzzy number's area, , with ordering suitable index  in obtaining approximate approach for ranking fuzzy numbers.If the decision maker has a preference for  0 then  must be chosen in (1/2, 1].Also if the area  is a choice then  must be chosen in [0, 1/2).
It is clear that for  = 0, the choice is just  and for  = 1 is  0 .If  = 1/2 then both  0 and  have the same effects.Now a decision maker can rank a pair of fuzzy numbers,  and , using   () and   () based on the following rules: We can see calculation results of the proposed method in Table 1 for the twelve-pair sets in Figure 1 that were used in [4].
Comparisons of the proposed method to the exiting methods for the twelve-pair sets in Figure 1 are given in Tables 2 and 3.

A New Measure between Two Trapezoidal Generalized Fuzzy Numbers
Let  and  be two arbitrary trapezoidal fuzzy numbers with  0 and  defined by ( 1) and ( 2), respectively.The distance between  and  is defined as A decision maker can order a pair of trapezoidal fuzzy numbers,  and , too using (, ) based on the following rules: (i) (, ) > 0, then  ≻ , (ii) (, ) = 0, then  ≈ , (iii) (, ) < 0, then  ≺ .

Some Properties
Some valuable properties are described in the following, which are useful in ranking a large quantity of generalized  1.
trapezoidal fuzzy numbers simultaneously.Assume that there are three different generalized trapezoidal fuzzy numbers, , , and , to be ranked: This equation suggests that the dominance difference between one fuzzy number and the other two fuzzy numbers is insignificant, if the two fuzzy numbers are close to each other.These properties hold since (, )−(, ) = (, ).

Conclusion
Ranking fuzzy numbers is a critical task in a fuzzy decision making process.Particularly, when ranking a large quantity of fuzzy numbers and only limited information about them can be obtained, an effective, efficient, and accurate ranking method becomes necessary.The proposed ranking approach only considers convex combining of the centroid point ( 0 ,  0 ) of  0 and area  of a fuzzy number with a decision maker's optimistic perspectives, that is, actually the measurement of the degree of difference between two fuzzy numbers.This ranking is compared with some exiting ranking methods.
Here too, a new distance measure has been introduced for computing crisp distances for fuzzy numbers.

Table 2 :
Comparison of other methods for the examples in Figure

Table 3 :
Comparison of other methods for the examples in Figure