Fuzzy Set Theory—and Its Applications

Most of our traditional tools for formal modeling, 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 modeled or the features of the real system that has been modeled. 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 x ∈ X that 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 settheoretic 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 late 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 that 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 1973a; Zadeh et al. 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 straightforward.

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 fuzzy set theory have been further discussed. Both theories seem to be similar in the sense that 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 three decades, the potential of electronic data processing (EDP) has been used to an increasing degree to support human decision making in different ways. In the 1960s, the management information systems (MISs) created probably exaggerated hopes for managers. Since the late 1970s and early 1980s, decision support systems (DSSs) found their way into management and engineering. The youngest offspring of these developments are the so-called knowledgebased expert systems or short expert systems, which have been applied since the mid-1980s to solve management problems [Zimmermann 1987, p. 3101]. It is generally assumed that expert systems will increasingly influence decisionmaking processes in business in the future.

The objective of fuzzy logic control (FLC) systems is to control complex processes by means of human experience. Thus fuzzy control systems and expert systems both stem from the same origins. However, their important differences should not be neglected. Whereas expert systems try to exploit uncertain knowledge acquired from an expert to support users in a certain domain, FLC systems as we consider them here are designed for the control of technical processes. The complexity of these processes range from cameras [Wakami and Terai 1993] and vacuum cleaners [Wakami and Terai 1993] to cement kilns [Larsen 1981], model cars [Sugeno and Nishida 1985], and trains [Yasunobu and Miamoto 1985]. Furthermore, fuzzy control methods have shifted from the original trans­lation of human experience into control rules to a more engineering-oriented approach, where the goal is to tune the controller until the behavior is sufficient, regardless of any human-like behavior.

The terms data analysis, pattern recognition, and data mining are often used synonymously, and we shall do the same here. On the one hand, this area is one of the oldest and most obvious application areas for fuzzy set theory. On the other hand, pattern recognition existed long before fuzzy sets became known.

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 modeling 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 lives. It is therefore necessary to define more accurately what we mean by models, theories, and laws in order 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 definitions given and used in the scientific area and definitions and interpretations as they can be found in more applicationoriented areas, which we will call “technologies” in 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) and sometimes science (i.e., computer science, management science, etc.). This statement is by no means a value judgment; 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 judgments, technologies normally try to generate tools for solving problems better, very often by either accepting or being based on given value schemes. Let us first turn to the area of scientific inquiry and consider the following quotation concerning the definition of the term model: “A possible realization in which all valid sentences of a theory T are satisfied is called a model of T.”