## Computation over fuzzy quantities.(English)Zbl 0859.94035

Boca Raton, FL: CRC Press. xi, 157 p. (1994).
The book presents in a unified approach the results of more than twenty years of research the author has done on a quite down-to-earth problem that many practitioners of number data processing encounter in their daily work: the difficulty of “handling of vagueness hidden in numerical data” (cf. page v), which means handling the vagueness of various practical problems. The “vagueness” which the author has in mind is that resulting from the imprecise information which one has to deal with in computer treatment of real-life problems when subjective estimates of essential parameters replace exact or even approximate evaluations based on repeated measurements or extensive computation. It became common practice during the last decade that imprecise information of this nature is formalized in terms of fuzzy sets. Representing imprecise numbers by fuzzy sets may simplify the formal representation of the problem but, in itself, it does not provide solutions. Once a formal model of a given problem is established one still has to do arithmetic operations with the data a priori known to be imprecise in order to establish relations and connections which lead to a computed solution. Nevertheless, “vague numbers or vague data cannot be processed by the methods used for exact values” (p. vi).
The present work describes simple but rigorous rules of processing vague information. It starts with introducing the notion of a fuzzy quantity (see Chapter 2). Mathematically speaking, a fuzzy quantity is a $$[0,1]$$-valued function $$\mu$$ with a real variable which vanishes outside an interval and whose supremum is exactly 1. The value $$\mu(x)$$ is the subjectively established degree of certainty that $$x$$ is exactly the quantity one has to deal with. In other words, a fuzzy quantity is a particular fuzzy subset of the real line. Thus, the well-known logical operations with fuzzy sets can be performed (they are surveyed in Chapter 1). In addition to the logical operations the author introduces some “arithmetic operations” with fuzzy quantities (Chapter 3) and analyzes them in Chapters 4-7. This leads to a quite solid mathematical basis for a series of algorithmic methods of solving practical problems involving fuzzy quantities discussed and illustrated in the last four Chapters.
What makes these algorithmic methods convincing and attractive is not their intrinsic mathematical rigor (most of the methods are essentially heuristic techniques of reaching solutions to specific decision making problems) but their simplicity and similarity with classical methods of solving problems of the same type. What differentiates the methods proposed by the author from the classical ones (for instance, the author’s approach to the critical path problem versus the well-established critical path method) is the fact that the author abolishes the fundamental assumption of the classical methods that given data are indubitable. Once this is done, the author proposes to operate with the given fuzzy quantities in such natural ways that the reader is almost convinced that the solutions reached are indeed optimal. Obviously, there is no mathematical guarantee that the methods of decision making the author proposes will generate “optimal” solutions in the standard sense of the term. As in any heuristic method “infallibility” is a matter of experience. Beyond this, as a mathematician, experimenting with the decision making methods described in this book I remained with the feeling of walking on a yet to be explored field and that the concepts and the techniques which are dealt with here are worth further and deeper mathematical investigation.
This is a well-written book and it can be a tool for solving decision making problems with fuzzy data (this is, in fact, its main purpose) as well as a starting point for mathematicians looking for interesting research queries.
Reviewer: D.Butnariu (Haifa)

### MSC:

 94D05 Fuzzy sets and logic (in connection with information, communication, or circuits theory) 94-02 Research exposition (monographs, survey articles) pertaining to information and communication theory 03E72 Theory of fuzzy sets, etc.