Fuzzy Set Theory — and Its Applications

Most of our traditional tools for formal modelling, reasoning, and computing are crisp, deterministic and precise in character. By crisp we mean dichotomous, that is, yes-or-no-type rather than more-or-less type. In conventional dual logic, for instance, a statement can be true or false—and nothing in between. In set theory an element can either belong to a set or not, and in optimization a solution is either feasible or not. Precision assumes that the parameters of a model represent exactly either our perception of the phenomenon modelled or the features of the real system that has been modelled. Generally precision also implies that the model is unequivocal, that is, that it contains no ambiguities.

A classical (crisp) set is normally defined as a collection of elements or objects xεX which can be finite, countable, or overcountable. Each single element can either belong to or not belong to a set A, A⊆X. In the former case, the statement “x belongs to A” is true, whereas in the latter case this statement is false.

In chapter 2 the basic definition of a fuzzy set was given and the original set theoretic operations were discussed. The membership space was assumed to be the space of real numbers, membership functions were crisp functions, and the operations corresponded essentially to the operations of dual logic or Boolean algebra.

In order to prevent confusion about fuzzy measures and measures of fuzziness, we shall first briefly describe the meaning and features of fuzzy measures. In the early 1970s, Sugeno defined a fuzzy measure as follows [Sugeno 1977]: 𝓑 is a Borel field of the arbitrary set (universe) X.

One of the most basic concepts of fuzzy set theory which can be used to generalize crisp mathematical concepts to fuzzy sets is the extension principle. In its elementary form it was already implied in Zadeh’s first contribution [1965]. In the meantime, modifications have been suggested [Zadeh 1973, 1975; Jain 1976].

Fuzzy relations are fuzzy subsets of X × Y, that is, mappings from X → Y. They have been studied by a number of authors, in particular by Zadeh [1965, 1971], Kaufmann [1975], and Rosenfeld [1975]. Applications of fuzzy relations are widespread and important. We shall consider some of them and point to more possible uses at the end of this chapter. We shall exemplarily consider only binary relations. A generalization to n-ary relations is straight forward.

A fuzzy function is a generalization of the concept of a classical function. A classical function f is a mapping (correspondence) from the domain D of definition of the function into a space S; f(D)⊆S is called the range of f. Different features of the classical concept of a function can be considered to be fuzzy rather than crisp. Therefore different “degrees” of fuzzification of the classical notion of a function are conceivable.

Since L. Zadeh proposed the concept of a fuzzy set in 1965 the relationships between probability theory and possibility theory have been discussed. Both theories seem to be similar in the sense that they both are concerned with some type of uncertainty and both use the [0,1] interval for their measures as the range of their respective functions. (At least as long as one considers normalized fuzzy sets only!) Other uncertainty measures, which were already mentioned in chapter 4, also focus on uncertainty and could therefore be included in such a discussion. The comparison between probability theory and fuzzy set theory is difficult primarily for two reasons:

During the last two centuries the potential of electronic data processing (EDP) has been used to an increasing degree to support human decision making in different ways. In the sixties the management informations systems (MIS) created probably exaggerated hopes of managers. Since the late 1970s and early 1980s decision support systems (DSS) found their way into management and engineering: The youngest child of these developments are the so-called Knowledge-based expert systems or short expert systems. If one interpretes decisions rather general, that is, including evaluation, diagnosis, prediction, et cetera, then all three types could be classified as decision support systems which differ gradually with respect to the following properties:

The contents and scope of operations research has been described and defined in many different ways. Most of the people working in operations research will agree, however, that the modelling of problem situations and the search for optimal solutions to these models are undoubtedly important parts of it. The latter activity is more algorithmic, mathematical, or formal in character. The former comprises many more disciplines than mathematics, has been more neglected than mathematical research in operations research and, therefore, would probably need more new advances in theory and practice.

The terms model, theory, and law have been used with a variety of meanings, for a number of purposes, and in many different areas of our life. It is therefore necessary to defme more accurately what we mean by models, theories, and laws to describe their interrelationships and to indicate their use before we can specify the requirements they have to satisfy and the purposes for which they can be used. To facilitate our task we shall distinguish between defmitions which are given and used in the scientific area and defmitions and interpretations as they can be found in more application-oriented areas, which we will call “technologies” by contrast to “scientific disciplines.” By technologies we mean areas such as operations research, decision analysis, and information processing, even though these areas call themselves sometimes theories (i.e., decision theory) or science (i.e., computer science, management science, etc.). This is by no means a value statement. We only want to indicate that the main goals of these areas are different. While the main purpose of a scientific discipline is to generate knowledge and to come closer to truth without making any value statements, technologies normally try to generate tools for solving problems better and very often by either accepting or basing on given value schemes.

In the first 9 chapters of this book we have covered the basic foundations of the theory of fuzzy sets, as they can be considered undisputed as of today. Many more concepts and theories could not be discussed either because of space limitations or because they can not yet be considered ready for a textbook. In a recent book by Kandel [1980] 3064 references are listed which supposedly are “Key references in fuzzy pattern recognition” [Kandel 1980, p. 209]. Even though this might overestimate somewhat the total of all knowledge available in the area of fuzzy set theory today, it is an indication of rather vivid research and particular publication activities.