The arithmetic of \(Z\)-numbers. Theory and applications
$Z$-numbers developed by L. A. Zadeh offer an interesting conceptual and algorithmic framework to manage granular information. $Z$ numbers are commonly represented as a pair, $Z= (A,B)$ of fuzzy numbers where $A$ is an imprecise constraint (fuzzy number) on values of some variable of interest whereas $B$ is a descriptor of reliability (relevance) of $A$ and expressed in some probabilistic fashion. The essence of this construct is to convey a comprehensive characterization of a piece of knowledge by accommodating not only its semantics (the restriction $A$) but also the associated relevance $(B)$. The authors deliver a comprehensive, compelling and well-motivated treatise of the subject. While maintaining the required level of rigor and a formal exposure of the subject matter, the book covers the essentials of the theory and ensuing algorithms.\par The book is structured into seven chapters and delivers a systematic exposure to the concepts and detailed algorithms. Chapter 1 delivers an introduction to the concept by elaborating on a way in which semantics of restrictions is formalized and quantified. Chapter 2 includes definitions and elaborates on the main properties of $Z$-numbers. A core part of the book is devoted to operations on $Z$-numbers (algebraic operations, square, square root, maximum and minimum) presenting the calculus of continuous (Chapter 3) and discrete (Chapter 4) $Z$-numbers. The algebraic system of $Z$-numbers is presented in Chapter 5; the topics there concern distances, approximations, derivatives, and equations with $Z$-numbers. The application facet of the book is presented in Chapters 6 and 7. Chapter 6 discusses $Z$-number-based classic problems studied in operations research (linear programming, regression analysis, decision making). Selected application areas dealing with business, marketing, and planning problems are outlined in Chapter 7.\par The book would appeal to the readership by exhibiting several appealing features. The readability and lucidity of exposure is definitely one of them. The reader can enjoy the main ideas being presented in a convincing way. The material is systematically structured with a step-by-step exposure of the generic ideas and their advancements. A wealth of detailed examples is another advantage of the book. The probability of calculus, being an integral part of $Z$-numbers, is clearly inbuilt into the text and augmented by a series of illustrative and detailed numeric examples.\par Undoubtedly, the book can be strongly recommended as a reference material written by the experts in the field and offering an authoritative and well rounded exposure to the area of $Z$-numbers.
scientific article; zbMATH DE number 6557423