Fuzzy Numbers

A fuzzy set A is called a fuzzy number if it satisfies the following conditions:

  • 1. A is normal, that is h (A) = 1.
  • 2. A is convex.
  • 3. The membership function (x) is at least piecewise continuous.

Triangular Fuzzy Number

A fuzzy number A = ^aL, aN, aR J (Figure 3.1) is said to be a triangular fuzzy number (TFN) when the membership function is given by

The TFN A = [aL, aN, aR J may be expressed into an ordered pair function by using а-cut as follows:

Trapezoidal Fuzzy Number

A fuzzy number A = |^aL, aNL, aNR, aR J (Figure 3.2) is said to be trapezoidal fuzzy number (TRFN) when the membership function is given by

Again, the TRFN may be expressed into an ordered pair function through а-cut in the following manner:

FIGURE 3.1

Triangular fuzzy number (TFN).

FIGURE 3.2

Trapezoidal fuzzy number (TRFN).

Fuzzy Arithmetic

Let us consider [x(a), x(a)] and [y (a), у (a) to be two fuzzy numbers, then the fuzzy arithmetic (Zimmermann 1991; Hanss 2005) may be written as

From this discussion, we may say that the intervals are the special cases of fuzzy numbers. For each membership value (a), we get an interval. In this regard, if we consider a TFN A = [aL, aN, aR ], then the same may be represented by using a-cut as follows:

If we take a=0, then we have the interval [aL, aR] and at a = 1, the interval [aN, aN] becomes crisp.

As such, a TFN A = |^aL, aN, aR ] may be represented into an ordered pair (interval) form by using a-cut as follows:

Here, f (a) and f (a) are monotonic increasing and decreasing functions, respectively. Using these functions, we have modified the interval arithmetic (а-cut of fuzzy numbers) as follows:

where

 
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