Elsevier

Information Sciences

Volume 176, Issue 16, 22 August 2006, Pages 2405-2416
Information Sciences

Ranking of fuzzy numbers by sign distance

https://doi.org/10.1016/j.ins.2005.03.013Get rights and content

Abstract

Several different strategies have been proposed for ranking of fuzzy numbers. These include methods based on the coefficient of variation (CV index), distance between fuzzy sets, centroid point and original point, and weighted mean value. Each of these techniques has been shown to produce non-intuitive results in certain cases. In this paper we propose a modification of the distance based approach called the sign distance, which is both efficient to evaluate and able to overcome the shortcomings of the previous techniques. The calculation of the proposed method is far simpler than the other approaches.

Introduction

In many applications, ranking of fuzzy numbers is an important component of the decision process. Following the pioneering work of Jain [10], [11] and Dubois and Prade [8], who used maximizing sets to order fuzzy numbers, numerous ranking techniques have been proposed and investigated. Some of them have been compared and contrasted in Bortolan and Degani [3], and more recently in Chu and Tsao [7]. Cheng [5] proposed the distance method for ranking of fuzzy numbers, i.e.,R(u)=x¯2+y¯2,wherex¯=abxuLdx+bcxdx+cdxuRdxabuLdx+bcdx+cduRdx,y¯=01ru̲dr+01ru¯dr01u̲dr+01u¯dr,uL, uR are the left and right membership functions of fuzzy number u, and (u̲,u¯) is the parametric form (see Definition 1.1, Definition 1.2). The resulting scalar value R(u) is used to rank the fuzzy numbers; if R(ui) < R(uj), then ui  uj. If R(ui) > R(uj), then ui  uj; if R(ui) = R(uj), then ui  uj. Chu and Tsao [7] reviewed Cheng’s method [5] and claimed that it has shortcomings. For example, consider three triangular fuzzy numbers, u1 = (0.3, 0.1, 0.2), u2 = (0.32, 0.15, 0.26), and u3 = (0.4, 0.15, 0.3) from [5], [7]. By Cheng’s distance method, R(u1) = 0.590, R(u2) = 0.604, and R(u3) = 0.662, producing the ranking order u1  u2  u3 (see Fig. 1). Consequently, we can logically infer that the ranking order of the images of these fuzzy numbers (opposite with respect to origin, [7]) is −u1  u2  u3. However, by distance method, the ranking order remains −u1  u2  u3. Consequently, it also has shortcomings.

Cheng [5] proposed the coefficient of variance (CV index), i.e. CV = σ (standard error)/∣μ∣ (mean), μ  0, σ > 0. In this approach, the fuzzy number with smaller CV index is ranked higher. In Table 1, sets 1, 2 and 4, we illustrate that Cheng’s CV index also contains shortcomings.

Chu and Tsao [7] proposed the area between the centroid point and original point for ranking; i.e. S(u)=x¯ y¯. This method for some fuzzy numbers is unreasonable (for more details see Example 2). Yager [18] proposed weighted mean value (or centroid, -xμu(x)dx/-μu(x)dx) to define ordering.

Having reviewed the previous methods, we now turn to introduce our proposed approach, termed as sign distance method. We consider a fuzzy origin for fuzzy numbers, then according to the distance of fuzzy numbers with respect to this origin we rank them.

The basic definitions of a fuzzy number are given in [9], [20], [21] as follows.

Definition 1.1

A fuzzy number is a fuzzy set like u:RI=[0,1] which satisfies:

  • 1.

    u is upper semi-continuous,

  • 2.

    u(x) = 0 outside some interval [a, d],

  • 3.

    There are real numbers a, b such that a  b  c  d and

    • (a)

      u(x) is monotonic increasing on [a, b],

    • (b)

      u(x) is monotonic decreasing on [c, d],

    • (c)

      u(x) = 1, b  x  c.

The membership function u can be expressed asu(x)=uL(x)axb,1bxc,uR(x)cxd,0otherwise,where uL:[a,b][0,1] and uR:[c,d][0,1] are left and right membership functions of fuzzy number u. An equivalent parametric form is also given in [12] as follows.

Definition 1.2

A fuzzy number u in parametric form is a pair (u̲,u¯) of functions u̲(r),u¯(r), 0  r  1, which satisfy the following requirements:

  • 1.

    u(r) is a bounded monotonic increasing left continuous function,

  • 2.

    u¯(r) is a bounded monotonic decreasing left continuous function,

  • 3.

    u̲(r)u¯(r), 0  r  1.

The trapezoidal fuzzy number u = (x0, y0, σ, β), with two defuzzifier x0, y0, and left fuzziness σ > 0 and right fuzziness β > 0 is a fuzzy set where the membership function is asu(x)=1σ(x-x0+σ)x0-σxx0,1x[x0,y0],1β(y0-x+β)y0xy0+β,0otherwise,and its parametric form isu̲(r)=x0-σ+σr,u¯(r)=y0+β-βr.

The addition and scalar multiplication of fuzzy numbers are defined by the extension principle and can be equivalently represented in [9], [20], [21] as follows.

For arbitrary u=(u̲,u¯), v=(v̲,v¯) we define addition (u + v) and multiplication by real scalar k > 0 as(u+v̲)(r)=u̲(r)+v̲(r),(u+v¯)(r)=u¯(r)+v¯(r),(ku̲)(r)=ku̲(r),(ku¯)(r)=ku¯(r).The collection of all fuzzy numbers with addition and multiplication as defined by (1), (2) is denoted by E, which is a convex cone. The image (opposite) of u = (x0, y0, σ, β), can be defined by −u = (−y0, x0, β, σ) (see [19], [20], [21]).

Definition 1.3

For arbitrary fuzzy numbers u=(u̲,u¯) and v=(v̲,v¯), the functionDp(u,v)=01|u̲(r)-v̲(r)|pdr+01|u¯(r)-v¯(r)|pdr1/p(p1)is the distance between u and v (see [1], [13], [14]).

Section snippets

Ranking of fuzzy numbers by sign distance

In this section, we will propose the ranking of fuzzy numbers associated with the metric D in E.

The membership function of aR is ua(x) = 1, if x = a, and ua(x) = 0, if x  a. Hence if a = 0, we will haveu0(x)=1x=0,0x0.We consider u0 as a fuzzy origin and since u0  E, left fuzziness σ and right fuzziness β are 0, so for each u  EDp(u,u0)=01(|u̲(r)|p+|u¯(r)|p)dr1/p(p1).

Definition 2.1

Let γ:E{-1,1} be a function that is defined as follows:uE:γ(u)=sign01(u̲(r)+u¯(r))dr,whereγ(u)=1ifsign01(u̲+u¯)(r)dr0,-1ifsign01(u̲

Conclusions

In spite of many ranking methods, no one can rank fuzzy numbers with human intuition consistently in all cases. Shortcomings are found in ranking fuzzy numbers with the coefficient of variation (CV index), distance between fuzzy sets, centroid point and original point, and weighted mean value. To overcome shortcomings we proposed sign distance method. It can effectively rank various fuzzy numbers and their images. The calculations of the proposed method is far simpler than the other approaches.

Acknowledgments

The authors wish to express their thanks to the referees for comments which improved the paper.

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