In many real-world problems fuzzy sets allow us to represent vague concepts expressed in natural language. The membership function of a fuzzy set A can be denoted by µA: X → [0,1]. Each fuzzy set should be uniquely defined by one particular membership function. Consider a fuzzy set where membership function is defined in Equation (1). This is one of the general formulae of a parameterized family of membership functions described in Klir and Yuan (1995):
This fuzzy set expresses, in a particular form, the general concept of a class of real numbers that are close to r. When the non-negative parameter ρ increases, the graph of µA(x) becomes narrower. The function has the following properties: µA(r) = 1 and µA(x) < 1 for all x ≠ r. For a complete discussion of fuzzy sets we refer to (Klir and Yuan, 1995; Zimmermann, 2001; Wang & Klir, 2013).