A new approach for ranking fuzzy numbers by distance method

Abstract

Many ranking methods have been proposed so far. However, there is yet no method that can always give a satisfactory solution to every situation; some are counterintuitive, not discriminating; some use only the local information of fuzzy values; some produce different rankings for the same situation. For overcoming the above problems, we propose a new method for ranking fuzzy numbers by distance method. Our method is based on calculating the centroid point, where the distance means from original point to the centroid point ( x 0, y 0 ), and the x 0 index is the same as Murakami et al.'s x 0 . However, the y 0 index is integrated from the inverse functions of an LR-type fuzzy number. Thus, we use ranking function R( A) = √ x 2 + y 2 (distance index) as the order quantities in a vague environment. Our method can rank more than two fuzzy numbers simultaneously, and the fuzzy numbers need not be normal. Furthermore, we also propose the coefficient of variation (CV index) to improve Lee and Li's method [Comput. Math. Appl.15 (1988) 887–896]. Lee and Li rank fuzzy numbers based on two different criteria, namely, the fuzzy mean and the fuzzy spread of the fuzzy numbers, and they pointed out that human intuition would favor a fuzzy number with the following characteristics: higher mean value and at the same time lower spread. However, when higher mean value and at the same time higher spread/or lower mean value and at the same time lower spread exists, it is not easy to compare its orderings clearly. Our CV index is defined as CV = σ (standard error)/μ (mean), which can overcome Lee and Li's problem efficiently. In this way, our proposed method can also be easily calculated by the “Mathematica” package to solve problems of ranking fuzzy numbers. At last, we present three numerical examples to illustrate our proposed method, and compare with other ranking methods.