Introduction to fuzzy arithmetic : theory and applications

We were rather pleased to read the review of our book, Introduction to Fuzzy Arithmetic: Theory and Applications. This review was done quite carefully by Caroline M. Eastman of the University of South Carolina, and we are grateful to her for pointing out many interesting, positive aspects as well as some shortcomings of our book. As members of the fuzzy community, we are concerned with studies and developments of concepts and techniques basic to the analysis of uncertainty arising from human perception, thinking, and reasoning processes. In this book we present such concepts and some novel tools for dealing with uncertainties. We start our introduction with the definition for the interval of confidence [al, a2], where al and a2 represent, respectively, the lower and upper bounds of our (subjective) confidence. Next, we introduce some arithmetic operations on these numbers. We then introduce the level of presumption ue [13, 1] and, using it, introduce the uncertain or fuzzy number that is so pervasive in our reasoning process. The reviewer has rightly pointed out that in certain situations, interval arithmetic can be considered a subset of fuzzy arithmetic, the main topic of our book. However, we intentionally did not want to confuse the issue by introducing interval arithmetic and then giving a generalization. We liked our approach, as have many other researchers and students who have used the book. In our approach, we have been guided throughout by a desire to lay a firm foundation for the definition of fuzzy numbers using the basic concept of level of confidence. We have introduced many novel mathematical operations based on this concept and have presented many generalizations. In addition, we have presented several operations and functions of fuzzy numbers, such as integer modulo operations, trigonometric functions, and hyperbolic functions. These studies have been included for students as well as researchers who wish to have an extended view of the theory. We have attempted to give a thorough exposition of fuzzy numbers; this exposition is illustrated by about 115 worked-out examples, 150 diagrams, and 90 tables. We did not include problems or exercises, which would have put this book in the category of a textbook. The subtitle of the book is "Theory and Applications," but as is rightly