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Chapter I – Fuzzy Relational Equations – Basic Concepts W. B. VASANTHA KANDASAMY FLORENTIN SMARANDACHE FUZZY RELATIONAL MAPS AND NEUTROSOPHIC RELATIONAL MAPS 2004 FUZZY RELATIONAL MAPS AND NEUTROSOPHIC RELATIONAL MAPS W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of Technology, Madras Chennai – 600036, India e-mail: [email protected] web: http://mat.iitm.ac.in/~wbv Florentin Smarandache Department of Mathematics University of New Mexico Gallup, NM 87301, USA e-mail: [email protected] 2004 2 CONTENTS Dedication Preface Chapter One FUZZY RELATIONAL EQUATIONS: BASIC CONCEPTS AND PROPERTIES 1.1 Fuzzy Relational Equations and their properties 10 1.2 Properties of Fuzzy Relations 16 1.3 Fuzzy compatibility relations and composition of fuzzy relations 22 1.4 Optimization Of FRE with Max-Product composition 32 1.5 Composite fuzzy relation equation resolution based on Archimedean triangular norms 40 1.6 Solving non-linear optimization problem with FRE constraints 48 1.7 Method of solution to fuzzy relation equations in a complete Brouwerian lattice 55 1.8 Multi-objective optimization problems with fuzzy relation equation constraints 57 1.9 Neural fuzzy relational system with a new learning algorithm 63 1.10 Unattainable Solution of FRE 67 1.11 Specificity shift in solving fuzzy relational equations 70 1.12 FRE with defuzzification algorithm for the largest solution 75 1.13 Solvability and Unique solvability of max-min fuzzy equations 81 1.14 New algorithms for solving fuzzy relation equations 85 1.15 Novel neural algorithms based on fuzzy S-rules for FRE 87 1.16 Novel neural network part I 94 1.17 Novel neural network part II 101 1.18 Simple Fuzzy control and fuzzy control based on FRE 108 3 1.19 A FRE in dynamic fuzzy systems 113 1.20 Solving FRE with a linear objective function 117 1.21 Some properties of minimal solution for a FRE 123 1.22 FRE and causal reasoning 126 1.23 Identification of FRE by fuzzy neural networks 134 1.24 Equations in classes of fuzzy relations 140 1.25 Approximate solutions and FRE and a characterization of t-norms for fuzzy sets 146 1.26 Solvability criteria for systems of FRE 156 1.27 Infinite FRE to a complete browerian lattices 162 1.28 Semantics of implication operators and fuzzy relational products 164 Chapter Two SOME APPLICATIONS OF FRE 2.1 Use of FRE in Chemical Engineering 167 2.2 New FRE to estimate the peak hours of the day for transport system 175 2.3 Study of the proper proportion of raw material mix in cement plants using FRE 188 2.4 The effect of globalization on silk weavers who are bonded labourers using FRE 189 2.5 Study of bonded labour problem using FRE 196 2.6 Data compression with FRE 196 2.7 Applying FRE to Threat Analysis 196 2.8 FRE application to medical diagnosis 197 2.9 A fuzzy relational identification algorithm and its application to predict the behaviour to a motor-drive system 197 2.10 Application of genetic algorithm to problems in chemical industries 199 2.11 Semantics of implication operators and fuzzy relational products applied to HIV/AIDS patients 211 Chapter Three SOME NEW AND BASIC DEFINITIONS ON NEUTROSOPHIC THEORY 3.1 Neutrosophic sets and neutrosophic logic 222 3.2 Fuzzy neutrosophic sets 230 4 3.3 On neutrosophic lattices 235 3.4 Neutrosophic notions: Basic concepts 239 3.5 Neutrosophic matrices and fuzzy neutrosophic matrices 243 3.6 Characteristics and significance of newer paradigm shift using indeterminacy 246 Chapter Four NEUTROSOPHIC RELATIONAL EQUATIONS AND THEIR PROPERTIES 4.1 Neutrosophic relational equations: Basic Definitions 249 4.2 Optimization of NRE with max product Composition 259 4.3 Method of solution to NRE in a complete Browerian lattice 260 4.4 Multi-objective optimisation problem with NRE constraints 260 4.5 Neutral neutrosophic relational system with a new learning algorithm 264 4.6 Unattainable solution of NRE 265 4.7 Specificity shift in solving NRE 266 4.8 NRE with deneutrofication algorithm for largest solution 268 4.9 Solving NRE with a linear objective function 269 4.10 Some properties of minimal solution for NRE 270 4.11 Application of NRE to Real-world problems 272 Chapter Five SUGGESTED PROBLEMS 279 Bibliography 283 Index 295 About the Authors 301 5 06 Dedicated to those few, young, not-so-influential, revolutionary scientists and mathematicians who support the newer paradigm shift 12 6 Preface The aim of this book is two fold. At the outset the book gives most of the available literature about Fuzzy Relational Equations (FREs) and its properties for there is no book that solely caters to FREs and its applications. Though we have a comprehensive bibliography, we do not promise to give all the possible available literature about FRE and its applications. We have given only those papers which we could access and which interested us specially. We have taken those papers which in our opinion could be transformed for neutrosophic study. The second importance of this book is that for the first time we introduce the notion of Neutrosophic Relational Equations (NRE) which are analogous structure of FREs. Neutrosophic Relational Equations have a role to play for we see that in most of the real-world problems, the concept of indeterminacy certainly has its say; but the FRE has no power to deal with indeterminacy, but this new tool NRE has the capacity to include the notion of indeterminacy. So we feel the NREs are better tools than FREs to use when the problem under investigation has indeterminates. Thus we have defined in this book NREs and just sketched its probable applications. This book has five chapters. The first chapter is a bulky one with 28 sections. These sections deal solely with FREs and their properties. By no means do we venture to give any proof for the results for this would make our book unwieldy and enormous in size. For proofs, one can refer the papers that have been cited in the bibliography. The second chapter deals with the applications of FRE. This has 10 sections: we elaborately give the applications of FRE in flow rates in chemical industry problems, preference and determination of peak hour in the transportation problems, the social problems faced by bonded laborers etc. Chapter three for the first time defines several new neutrosophic concepts starting from the notion of neutrosophic fuzzy set, neutrosophic fuzzy matrix, neutrosophic lattices, neutrosophic norms etc. and just indicate some of its important analogous properties. This chapter has six sections which are solely devoted to the introduction of several neutrosophic concepts which are essential for the further study of NRE. 7 Chapter four has eleven sections. This chapter gives all basic notions and definitions about the NREs and introduces NREs. Section 4.11 is completely devoted to suggest how one can apply NREs in the study of real world problems. We suggest many problems in chapter five for the reader to solve. This is the third book in the Neutrosophics Series. The earlier two books are Fuzzy Cognitive Maps and Neutrosophic Cognitive Maps (http://gallup.unm.edu/~smarandache/NCMs.pdf) and Analysis of Social Aspects of Migrant Labourers Living With HIV/AIDS Using Fuzzy Theory and Neutrosophic Cognitive Maps: With Specific Reference to Rural Tamil Nadu in India (http://gallup.unm.edu/~smarandache/NeutrosophyAIDS.pdf). Finally, we thank Meena Kandasamy for the cover design. We thank Kama Kandasamy for the layout of the book and for drawing all the figures used in this book. She displayed an enormous patience that is worthy of praise. We owe deep thanks to Dr.K.Kandasamy for his patient proof-reading of the book. Without his help this book would not have been possible. W.B.VASANTHA KANDASAMY FLORENTIN SMARANDACHE 8 Chapter One FUZZY RELATIONAL EQUATIONS: BASIC CONCEPTS AND PROPERTIES The notion of fuzzy relational equations based upon the max-min composition was first investigated by Sanchez [84]. He studied conditions and theoretical methods to resolve fuzzy relations on fuzzy sets defined as mappings from sets to [0,1]. Some theorems for existence and determination of solutions of certain basic fuzzy relation equations were given by him. However the solution obtained by him is only the greatest element (or the maximum solution) derived from the max-min (or min-max) composition of fuzzy relations. [84]’s work has shed some light on this important subject. Since then many researchers have been trying to explore the problem and develop solution procedures [1, 4, 10-12, 18, 34, 52, 75-80, 82, 108, 111]. The max-min composition is commonly used when a system requires conservative solutions in the sense that the goodness of one value cannot compensate the badness of another value [117]. In reality there are situations that allow compensatability among the values of a solution vector. In such cases the min operator is not the best choice for the intersection of fuzzy sets, but max- product composition, is preferred since it can yield better or at least equivalent result. Before we go into the discussion of these Fuzzy Relational Equations (FRE) and its properties it uses and applications we just describe them. This chapter has 28 sections that deal with the properties of FRE, methods of solving FRE using algorithms given by several researchers and in some cases methods of neural networks and genetic algorithm is used in solving problems. A complete set of references is given in the end of the book citing the names of all researchers whose research papers have been used. 9 1.1 Binary Fuzzy Relation and their properties It is well known fact that binary relations are generalized mathematical functions.
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