Fuzzy Sets and Fuzzy Logic: Theory and Applications

Fuzzy Sets and Fuzzy Logic: Theory and Applications

by George J. Klir and Bo Yuan
Published by Prentice Hall, Upper Saddle River, NJ, 1995
574 pages, $63
ISBN 0-13-101171-5

reviewed by
Dan Simon
Innovatia Software
dansimon@innovatia.com

Fuzzy Sets and Fuzzy Logic is an impressive book. At this point, it is probably the most authoritative book dealing with fuzzy systems. The book is broadly divided into two parts. The first part, which is primarily theoretical, presents a mathematically rigorous exposition of fuzzy systems. The second part presents various applications of fuzzy logic. The text assumes a knowledge of probability and set theory, and is suitable for a one- or two-semester course at the graduate level. Each chapter ends with a list of references and a set of problems. A solutions manual is available from the publisher.

The theoretical half of the book consists of nine chapters. Chapter 1 presents an interesting history of fuzzy systems theory. It also presents an overview of both crisp sets and fuzzy sets. Chapter 2 discusses the relationship between crisp sets and fuzzy sets, including how fuzzy sets can be represented by combinations of crisp sets. It also shows how basic mathematical functions can be modified for applications to fuzzy sets. Chapter 3 covers the extension of the crisp complement, intersection, and union operations to fuzzy sets. It is shown that these operations as applied to fuzzy sets) unlike their crisp counterparts, are not unique. There are, however, "standard" fuzzy operations which have special properties. Chapter 3 also has a section on aggregation operations, which are combinations of several fuzzy sets to produce another fuzzy set. Chapter 4 deals with fuzzy numbers (which are a type of a fuzzy set) and fuzzy arithmetic. Chapters 5 and 6 deal with fuzzy relations (membership in fuzzy sets). Two fuzzy relations can be combined in a way which is similar to matrix multiplication. Inverse relations can be computed in a way which is analogous to matrix inversion. Chapter 7 introduces fuzzy measure theory, which deals with the degree of certainty of an element's membership in a crisp set. This chapter discusses three branches of fuzzy measure theory: evidence theory, possibility theory, and probability theory. Chapter 8 gives an overview of multivalued logic and fuzzy logic, including fuzzy propositions and fuzzy inference. Chapter 9 is a fascinating discussion of information theory from the five different perspectives of classical set theory, fuzzy set theory, possibility theory, evidence theory, and probability theory.

The second half of the text considers a broad range of applications of fuzzy systems theory. The contrast between the theoretical orientation of the first half of the book and the applied orientation of the second half is immediately apparent. Chapter 10 is devoted to the construction of fuzzy membership functions. Membership functions can be constructed by humans which are experts in the given field, or they can be constructed on the basis of sample data. Chapter 11 shows how to build an expert system which uses fuzzy logic. Chapter 12 discusses the application of fuzzy set theory and fuzzy logic to systems theory. The fuzzy systems which are discussed include fuzzy controllers, fuzzy neural networks, fuzzy state machines, and fuzzy dynamic systems. Chapter 13 gives an overview of fuzzy techniques for pattern recognition, and Chapter 14 discusses fuzzy databases and fuzzy information retrieval. Chapter 15 covers various aspects of fuzzy decision making and fuzzy linear programming, and Chapter 16 introduces some applications of fuzzy theory to various engineering fields. Chapter 17 is titled "Miscellaneous Applications", and discusses applications in such fields as medicine and economics. Following the main body of the text is a set of appendices which include overviews of neural networks, genetic algorithms, and rough sets. The book contains an impressive bibliography of 1731 references.

A mastery of the first (theoretical) half of this book would require a fairly high level of mathematical sophistication, and a significant expenditure of time and effort. A reader looking for a simple, straightforward overview of fuzzy systems would be better served elsewhere. On the other hand, the second (application) half of the book is fairly independent of the first half, so an engineer could get a good feeling for fuzzy applications with only a basic understanding of the theory.

In summary, this book is a complete and thorough exposition of fuzzy systems theory and application. Anyone serious about becoming an expert in fuzzy systems or contributing to the fuzzy systems literature will find ample resources and direction in this book. I highly recommend it as both a textbook and as a reference.

Home Credentials Publications White Papers

Last Revised: March 13, 2001