Determination of Membership Functions

Abstract

In our natural world and daily lives, we experience all kinds of phenomena; broadly speaking, we can divide them into two types: phenomena of certainty and phenomena of uncertainty. The class of uncertain phenomena can further be subdivided into random (stochastic) phenomena and fuzzy phenomena. Therefore, we have three categories of phenomena and their associated mathematical models: 1. Deterministic mathematical models-This is a class of models where the relationships between objects are fixed or known with certainty. 2. Random (stochastic) mathematical models-This is a class of models where the relationships between objects are uncertain or random in nature. 3. Fuzzy mathematical models-This is a class of models where objects and relationships between objects are fuzzy. The main distinction between random phenomena and fuzzy phenomena is that random events themselves have clear and well-defined meaning, whereas a fuzzy concept does not have a precise extension because it is hard to judge if an object belongs to the concept. We may say that randomness is a deficiency of the law of causality and that fuzziness is a deficiency of the law of the excluded middlc. Probability theory applies the random concept to generalized laws of causality-laws of probability. Fuzzy set theory applies the fuzzy property to the generalized law of the excluded middle-the law of membership from fuzziness. Probability reflects the internal relations and interactions of events under certain conditions. It could be very objective if a stable frequency is available from re-