On Approximation of Fuzzy Sets by Crisp Sets: From Continuous Control-Oriented Defuzzification To Discrete Decision Making

Fuzzy logic and fuzzy set theory enable us to use experts’ uncertain (“fuzzy”) knowledge in decision making. The corresponding methods usually generalize known methods of decision making which are based on the crisp (non-fuzzy) knowledge about the environment. These crisp methods enable us to come up with a crisp decision: e.g., whether we should build a plant or not. When we generalize these methods to fuzzy knowledge, as a result, we usually get not a crisp decision, but rather a fuzzy decision: e.g., a decision may be “most probably it is better to build a plant”. Formally, in the simplest (“yes”-“no”) decision situation, a crisp decision procedure means that for every input x, we decide whether to choose a positive alternative A+ or a negative alternative A . Such a crisp decision can be described by a set S of all the inputs x for which the decision is positive, or, alternatively, by the characteristic function (x) of this set, i.e., by a function for which (x) = 1 if x 2 S and (x) = 0 if x 62 S. Similarly, as a result of a fuzzy decision making procedure, for every input x, we generate a degree (x) 2 [0; 1] to which choosing a positive alternative A+ is reasonable. Thus, a fuzzy decision making procedure produces a fuzzy subset of the set of all inputs, a subset which is characterized by the membership function (x).