Fuzzy Sets

Introduction

In this chapter we have collected together the basic ideas from fuzzy sets and fuzzy functions needed for the book. Any reader familiar with fuzzy sets, fuzzy numbers, the extension principle, α-cuts, interval arithmetic, and fuzzy functions may go on and have a look at Sections 2.5-2.7. In Section 2.5 we present a method that we have used in the past of maximizing/minimizing a fuzzy number $\overline{Z}$ which represents the value of some objective function in a fuzzy optimization problem. In Section 2.6 we are concerned with ordering a finite set of fuzzy numbers from smallest to largest to be used in our fuzzy Monte Carlo studies. Basically, given two fuzzy numbers $\overline{M}$ and $\overline{N}$, we need a method of deciding which of the following three possibilities is true: $\overline{M} < \overline{N}$, $\overline{M} \approx \overline{N}$, $\overline{M} > \overline{N}$. Three methods are discussed in Section 2.6. Section 2.7 discusses dominated and undominated fuzzy vectors needed in Chapter 9. Fuzzy vectors are vectors made up of fuzzy numbers. A good general reference for fuzzy sets and fuzzy logic is [4] and [19].

Our notation specifying a fuzzy set is to place a “bar” over a letter. So $\overline{A}$, $\overline{B}, \ldots$, $\overline{X}$, $\overline{Y},\ldots$, $\overline{\alpha}$, $\overline{\beta},\ldots,$ will all denote fuzzy sets.