The nearest trapezoidal fuzzy number to a fuzzy quantity

doi:10.1016/j.amc.2003.07.025

Applied Mathematics and Computation

Volume 156, Issue 2, 6 September 2004, Pages 381-386

https://doi.org/10.1016/j.amc.2003.07.025 Get rights and content

Abstract

In this paper, we introduce a fuzzy trapezoidal approximation using the metric (distance) between two fuzzy numbers. An application of this approximation is also investigated.

Introduction

Defuzzification methods have been widely studied for some years and were applied to fuzzy expert systems [1], [2], [3]. The major idea behind these methods was to obtain a typical value from a given fuzzy set according to some specified characters, such as central gravity, median, etc. In other words, each defuzzification method provides a correspondence from the set of all fuzzy sets into the set of real numbers. Obviously, in defuzzification methods which replace a fuzzy set by a single number, we generally loose too many important information. Also, an interval approximation for fuzzy numbers is considered in [4], which in, a problem in fuzzy area is converted into interval arithmetic area.

In this paper, we use the concept of the trapezoidal fuzzy number, and introduce approach to defuzzify a general fuzzy quantity. The basic idea of the new method is to obtain the “nearest” trapezoidal fuzzy number which is related to a fuzzy quantity.

In Section 2, we recall some fundamental results on fuzzy numbers. In Section 3, to obtain “nearest” trapezoidal fuzzy number method is proposed and illustrated with an example.

Section snippets

Preliminaries

Definition 1

A fuzzy number is a fuzzy set like $u: R →I=[0,1]$ which satisfies:

1.
u is upper semicontinuous,
2.
u(x)=0 outside some interval [c,d],
3.
There are real numbers a,b such that c⩽a⩽b⩽d and
- 3.1
  u(x) is monotonic increasing on [c,a],
- 3.2
  u(x) is monotonic decreasing on [b,d],
- 3.3
  u(x)=1, a⩽x⩽b.

The set of all these fuzzy numbers is denoted by E. An equivalent parametric is also given in [5] as follows.

Definition 2

A fuzzy number u in parametric form is a pair $(u, u ̄)$ of function $u (r), u ̄ (r),0⩽r⩽1$ , which satisfies the following requirements:

1.
$u$

The nearest trapezoidal fuzzy number to a given fuzzy number

In this section, we will propose the nearest trapezoidal approximation. Suppose v is a fuzzy number with parametric form $(v (r), v ̄ (r))$ . Given v we will try to find a trapezoidal fuzzy number $u(v (1)−h_{1}^{2}, v ̄ (1)+h_{2}^{2},σ,β)$ which is the nearest to v with respect to metric D. Hence, let parametric form of u is $(u (r), u ̄ (r))$ .

Now we have to minimize $D(v,u[v (1)−h_{1}^{2}, v ̄ (1)+h_{2}^{2},σ,β])= ∫_{0}^{1} u (r)− v (r)^{2} d r+∫_{0}^{1} (u ̄ (r)− v ̄ (r))^{2} d r^{1/2},$ with respect to $u$ and $u ̄$ . In order to minimize D(v,u) it suffices to minimize function d(h₁

Conclusions

In the paper, we have suggested a interest approach to trapezoidal approximation of general fuzzy numbers. The proposed method leads to the trapezoidal fuzzy number which is the best one with respect to a certain measure of distance between fuzzy numbers, D(u,v). Our method is simple and natural and can be applied in anywhere, for example in fuzzy linear programming and so on. Moreover, it is unique in (E,D).

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There are more references available in the full text version of this article.

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The nearest trapezoidal fuzzy number to a fuzzy quantity

Abstract

Introduction

Section snippets

Preliminaries

The nearest trapezoidal fuzzy number to a given fuzzy number

Conclusions

Fuzzy Sets Systems

Fuzzy Sets Systems

Fuzzy Sets Systems

Fuzzy Sets Systems