Fuzzy Sets and Logic
The fuzzy sets theory was created in response of expressing mathematically real world situations in which definitions have not clear boundaries. For example, “the high mountains of a country”, “the young people of a city”, “the good players of a team”, etc. The notion of a fuzzy set (FS) was introduced by Zadeh (1965) as follows:
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Definition: A FS on the universal set U of the discourse (or a fuzzy subset of U) is a set of ordered pairs of the form Α = {(x, mΑ(x)): xU}, defined in terms of a membership function mΑ: U [0,1] that assigns to each element of U a real value from the interval [0,1].
The value mΑ(x) us called the membership degree of x in A. The greater is mΑ(x), the better x satisfies the characteristic property of A. The definition of the membership function is not unique depending on the user’s subjective data, which is usually based on statistical or empirical observations. However, a necessary condition for a FS to give a reliable description of the corresponding real situation is that its membership function’s definition satisfies the common sense. Note that, for reasons of simplicity, many authors identify a FS with its membership function.
A crisp subset A of U can be considered as a FS in U with mΑ(x) = 1, if xA and mΑ(x) = 0, if x A. In this way most properties and operations of crisp sets can be extended to corresponding properties and operations of Fs. For general facts on FS we refer to the book of Klir & Folger (1988).
Fuzzy Logic (FL), based on FS theory, constitutes a generalization and complement of the classical bi-valued logic that finds nowadays many applications to almost all sectors of human activities (e.g. Chapter 6 of Klir & Folger, 1988, Voskoglou, 2011, 2012, 2015 etc.). Due to its nature of characterizing the ambiguous real life situations with multiple values, FL offers, among others, rich resources for assessment purposes, which are more realistic than those of the classical logic (Voskoglou, 2011, 2012, 2015, etc).
The FL approach for a problem’s solution involves the following steps:
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Fuzzification of the problem’s data by representing them with properly defined FSs.
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Evaluation of the fuzzy data by applying principles and methods of FL in order to express the problem’s solution in the form of a unique FS.
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Defuzzification of the problem’s solution in order to “translate” it in our natural language for use with the original real-life problem.
One of the most popular defuzzification methods is the Center of Gravity (COG) technique. When using it, the fuzzy outcomes of the problem’s solution are represented by the coordinates of the COG of the membership’s function graph of the FS involved in the solution (van Broekhoven & De Baets, 2006).