Ranking of fuzzy numbers by sign distance
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.,whereuL, uR are the left and right membership functions of fuzzy number u, and 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. . This method for some fuzzy numbers is unreasonable (for more details see Example 2). Yager [18] proposed weighted mean value (or centroid, ) 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 which satisfies: u is upper semi-continuous, u(x) = 0 outside some interval [a, d], There are real numbers a, b such that a ⩽ b ⩽ c ⩽ d and u(x) is monotonic increasing on [a, b], u(x) is monotonic decreasing on [c, d], u(x) = 1, b ⩽ x ⩽ c.
The membership function u can be expressed aswhere and 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 of functions , 0 ⩽ r ⩽ 1, which satisfy the following requirements:
- 1.
u(r) is a bounded monotonic increasing left continuous function,
- 2.
is a bounded monotonic decreasing left continuous function,
- 3.
, 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 asand its parametric form is
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 , we define addition (u + v) and multiplication by real scalar k > 0 asThe 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 and , the functionis 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 is ua(x) = 1, if x = a, and ua(x) = 0, if x ≠ a. Hence if a = 0, we will haveWe consider u0 as a fuzzy origin and since u0 ∈ E, left fuzziness σ and right fuzziness β are 0, so for each u ∈ E Definition 2.1 Let be a function that is defined as follows:where
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.
References (21)
- et al.
A note on “A new approach for defuzzification”
Fuzzy Sets Syst.
(2002) - et al.
A review of some methods for ranking fuzzy numbers
Fuzzy Sets Syst.
(1985) Ranking fuzzy numbers with maximizing set and minimizing set
Fuzzy Sets Syst.
(1985)A new approach for ranking fuzzy numbers by distance method
Fuzzy Sets Syst.
(1998)- et al.
An index for ordering fuzzy numbers
Fuzzy Sets Syst.
(1993) - et al.
A new fuzzy arithmetic
Fuzzy Sets Syst.
(1999) - et al.
Correction to “A new approach for defuzzification”
Fuzzy Sets Syst.
(2002) - et al.
A new approach for defuzzification
Fuzzy Sets Syst.
(2000) - et al.
Reasonable properties for the ordering of fuzzy quantities I
Fuzzy Sets Syst.
(2001) - et al.
Reasonable properties for the ordering of fuzzy quantities II
Fuzzy Sets Syst.
(2001)
Cited by (260)
Evaluation and ranking of fuzzy sets under equivalence fuzzy relations as α−certainty and β−possibility[Formula presented]
2024, Expert Systems with ApplicationsA user-configurable and explainable framework for ranking fuzzy numbers
2022, Expert Systems with ApplicationsThe alpha-ordering for a wide class of fuzzy sets of the real line: the particular case of fuzzy numbers
2024, Computational and Applied MathematicsRanking of Fuzzy Numbers on the Basis of New Fuzzy Distance
2024, International Journal of Fuzzy SystemsTHE EXACT DEFUZZIFICATION METHOD UNDER POLYNOMIAL APPROXIMATION OF VARIOUS FUZZY SETS
2024, Yugoslav Journal of Operations ResearchHealth status assessment of radar systems at aerospace launch sites by fuzzy analytic hierarchy process
2023, Quality and Reliability Engineering International