The arithmetic of \(Z\)-numbers. Theory and applications
type: Item
instance of: scholarly article
zbMATH Open document ID: 1426.03003
DOI: 10.1142/9575
author: Akif A. Alizadeh
author: Rafik Aliev
author: O. H. Huseynov
author: R. R. Aliev
publication date: 2016-03-17T00:00:00Z
Mathematics Subject Classification ID: 03-02
Mathematics Subject Classification ID: 03E72
Mathematics Subject Classification ID: 90B50
Mathematics Subject Classification ID: 90C70
Mathematics Subject Classification ID: 91B06
Mathematics Subject Classification ID: 94D05
zbMATH DE Number: 6557423
zbMATH Keywords: $Z$-numbers
zbMATH Keywords: fuzzy sets
zbMATH Keywords: fuzzy numbers and their arithmetic
zbMATH Keywords: operations research
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1142/9575
OpenAlex ID: W4254628720
title: The arithmetic of \(Z\)-numbers. Theory and applications
review text: $Z$-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 readabi…
reviewed by: Witold Pedrycz
description: scientific article; zbMATH DE number 6557423