The Arithmetic of ---NNNuuummmbbbeeerrrsss Theory and Applications 9575_9789814675284_tp.indd 1 5/3/15 6:36 pm May2,2013 14:6 BC:8831-ProbabilityandStatisticalTheory PST˙ws TThhiiss ppaaggee iinntteennttiioonnaallllyy lleefftt bbllaannkk The Arithmetic of ---NNNuuummmbbbeeerrrsss Theory and Applications Rafi k A Aliev Azerbaijan State Oil Academy, Azerbaijan Oleg H Huseynov Azerbaijan State Oil Academy, Azerbaijan Rashad R Aliyev Eastern Mediterranean University, Turkey Akif A Alizadeh Azerbaijan University, Azerbaijan World Scientifi c NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI 9575_9789814675284_tp.indd 2 5/3/15 6:36 pm Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. THE ARITHMETIC OF Z-NUMBERS Theory and Applications Copyright © 2015 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. ISBN 978-981-4675-28-4 Printed in Singapore Steven - The Arithmetic of Z-Numbers.indd 1 4/3/2015 9:29:02 AM 9575-The Arithmetic of Z-numbers Dedication Dedicated to the memory of my wife Aida Alieva Rafik Aliev To my parents and grandparents Oleg Huseynov To my parents and family Rashad Aliev To the memory of my father Vali Alizadeh Akif Alizadeh May2,2013 14:6 BC:8831-ProbabilityandStatisticalTheory PST˙ws TThhiiss ppaaggee iinntteennttiioonnaallllyy lleefftt bbllaannkk 9575-The Arithmetic of Z-numbers Preface Real-world information is imperfect and we often use natural language (NL) in order to represent this feature of the former. On the one hand, such information is often characterized by fuzziness. This implies that we often impose soft constraints on values of variables of interest. On the other hand, what is very important is that it is not sufficient to take into account only fuzziness when dealing with real-world imperfect information. The other essential property of information is its partial reliability. Indeed, any estimation of values of interest, be it precise or soft, are subject to the confidence in sources of information we deal with – knowledge, assumptions, intuition, envision, experience – which, in general, cannot completely cover the whole complexity of real-world phenomena. Thus, fuzziness from the one side and partial reliability form the other side are strongly associated to each other. In order to take into account this fact, L.A. Zadeh suggested the concept of a Z-number as a more adequate formal construct for description of real-world information. A Z-number is an ordered pair Z =(A,B) of fuzzy numbers used to describe a value of a variable X , where A is an imprecise constraint on values of X and B is an imprecise estimation of reliability of A and is considered as a value of probability measure of A. The concept of a Z-number has a potential for many applications, especially in the realms of computation with probabilities and events described in NL. Of particular importance are applications in economics, decision analysis, risk assessment, prediction, anticipation, planning, biomedicine and rule-based manipulation of imprecise functions and relations. Thus, real-world information is often represented in a framework of Z-number based evaluations. Such information is referred to as Z- information. The main critical problems that naturally arises in processing Z-information are computation and reasoning with Z- information. The existing literature devoted to computation with Z- numbers is quite scarce. Unfortunately, there is no general and vii 9575-The Arithmetic of Z-numbers viii The Arithmetic of Z-numbers. Theory and Applications computationally effective approach to computations with Z-numbers. There is a need in development of a universal approach to computations with Z-numbers which can be relatively easily applied for solving a wide spectrum of real-world problems in control, decision analysis, optimization and other areas. Computation and reasoning with Z- information are characterized by propagation of restrictions, that is, they are restriction-based computation and reasoning. As it is mentioned by L.A. Zadeh, the principal types of restrictions are probabilistic restrictions, possibilistic restrictions and combinations of probabilistic and possibilistic restrictions. Indeed, Z-information falls within the category of possibilistic-probabilistic restrictions. Nowadays, the existing literature devoted to computation and reasoning with restrictions includes well-developed approaches and theories to deal with pure probabilistic or pure possibilistic restrictions. For computation with probabilistic restrictions as probability distributions, the well-known probabilistic arithmetic is used. Fuzzy arithmetic deals with possibilistic constraints, which describe objects as classes with “unsharp” boundaries. Unfortunately, up to day there is no approach to computation and reasoning with objects described by combination of probabilistic and possibilistic restrictions, such as Z-numbers. Arithmetic of Z-numbers is a basis of a future mathematical formalism to process Z-information. Arithmetic of Z-numbers is greater than just “mechanical sum” of probabilistic arithmetic and fuzzy arithmetic, it is a synergy of these two counterparts. Consequently, development of this arithmetic requires generalization of the extension principle to deal with a fusion of probabilistic and possibilistic restrictions. In turn, computation of restrictions is computation of functions and functionals that involves optimization problems, particularly, mathematical programming and variational problems. Nowadays there is no arithmetic of Z-numbers suggested in the existing literature. The suggested book is the first to present a comprehensive and self-contained theory of Z-arithmetic and its applications. Many of the concepts and techniques described in the book are original and appear in the literature for the first time. 9575-The Arithmetic of Z-numbers Preface ix This book provides a detailed method in arithmetic of continuous and discrete Z-numbers. We also provide the necessary knowledge in its connections to other types of theories of uncertain computations. In addition, we discuss widely application of Z-numbers in variety of methods of operations research, economics, business and medicine. Let us emphasize that many numbers, especially, in fields such as economics and decision analysis, are in reality Z-numbers, but they are not treated as such, because it is much simpler to compute with numbers than with Z-numbers. Basically, the concept of a Z-number is a step toward formalization of the remarkable human capability to make rational decisions in an environment of imprecision and uncertainty. The book is organized into 7 chapters. The first chapter includes papers of L.A. Zadeh: L.A. Zadeh. Toward a restriction-centered theory of truth and meaning (RCT). Information Sciences, 248, 2013, 1–14; and L.A. Zadeh, A note on Z-numbers, Information Sciences, 181, 2011, 2923-2932. In the first section, the restriction centered theory, RCT, is considered which may be viewed as a step toward formalization of everyday reasoning and everyday discourse. Unlike traditional theories— theories which are based on bivalent logic—RCT is based on fuzzy logic. In the second section, the general concepts of a Z-number and Z+-number are suggested which have a potential for many applications. Also, sound theoretical foundation of computation of different functions of Z- numbers are suggested. For the present book to be self-containing, foundations of fuzzy sets theory, fuzzy logic and fuzzy mathematics which are used as the formal basis of the suggested theory of computation with Z-numbers are given in Chapter 2. We would like to mention that this chapter contains a material on a spectrum of computations with uncertain and imprecise information including the basics of interval arithmetic, probabilistic arithmetic, and fuzzy arithmetic. In this chapter we also give properties of continuous and discrete Z-numbers. Operations on continuous Z-numbers are explained in Chapter 3. Arithmetic operations such as addition, standard subtraction, multiplication and standard division are considered. Also, square and square root of a continuous Z-number are given. Chapter 4 provides a