SPECIAL FUZZY MATRICES FOR SOCIAL SCIENTISTS W. B. Vasantha Kandasamy e-mail: [email protected] web: http://mat.iitm.ac.in/~wbv www.vasantha.net Florentin Smarandache e-mail: [email protected] K. Ilanthenral e-mail: [email protected] INFOLEARNQUEST Ann Arbor 2007 This book can be ordered in a paper bound reprint from: Books on Demand ProQuest Information & Learning (University of Microfilm International) 300 N. Zeeb Road P.O. Box 1346, Ann Arbor MI 48106-1346, USA Tel.: 1-800-521-0600 (Customer Service) http://wwwlib.umi.com/bod/ Peer reviewers: Prof. Mihaly Bencze, University of Brasov, Romania Prof. Valentin Boju, Ph.D., Officer of the Order “Cultural Merit” Category — “Scientific Research” MontrealTech — Institut de Techologie de Montreal Director, MontrealTech Press, P.O. Box 78574 Station Wilderton, Montreal, Quebec, H3S2W9, Canada Prof. Mircea Eugen Selariu, Polytech University of Timisoara, Romania. Copyright 2007 by InfoLearnQuest and authors Cover Design and Layout by Kama Kandasamy Many books can be downloaded from the following Digital Library of Science: http://www.gallup.unm.edu/~smarandache/eBooks-otherformats.htm ISBN-10: 1-59973-030-8 ISBN-13: 978-1-59973-030-1 EAN: 9781599730301 Standard Address Number: 297-5092 Printed in the United States of America 2 CONTENTS Preface 5 Chapter One A NEW CLASS OF SPECIAL FUZZY MATRICES AND SPECIAL NEUTROSOPHIC MATRICES 7 1.1 Introduction to Fuzzy Matrices 7 1.2 Special Classes of Fuzzy Matrices 26 1.3 Special Neutrosophic matrices and fuzzy neutrosophic matrices and some essential operators using them 98 Chapter Two SPECIAL FUZZY MODELS AND SPECIAL NEUTROSOPHIC MODELS AND THEIR GENERALIZATIONS 171 2.1 Basic Description of Fuzzy Models 172 2.1.1 Definition of Fuzzy Cognitive Maps 172 2.1.2 Definition and Illustration of Fuzzy Relational Maps (FRMs) 176 2.1.3 Properties of Fuzzy Relations and FREs 180 2.2 Neutrosophy and Neutrosophic models 187 2.2.1 An Introduction to Neutrosophy 187 3 2.2.2 Some Basic Neutrosophic Structures 190 2.2.3 On Neutrosophic Cognitive Maps 196 2.2.4 Neutrosophic Relational Maps 201 2.2.5 Binary Neutrosophic Relation and their Properties 205 2.3 Special Fuzzy Cognitive Models and their Neutrosophic Analogue 216 2.4 Special FRMs and NRMs and their generalizations 237 2.5 Special FRE models and NRE models 258 2.6 Some Programming Problems for Computer Experts 263 FURTHER READING 265 INDEX 293 ABOUT THE AUTHORS 301 4 PREFACE This book is a continuation of the book, "Elementary fuzzy matrix and fuzzy models for socio-scientists" by the same authors. This book is a little advanced because we introduce a multi-expert fuzzy and neutrosophic models. It mainly tries to help social scientists to analyze any problem in which they need multi-expert systems with multi-models. To cater to this need, we have introduced new classes of fuzzy and neutrosophic special matrices. The first chapter is essentially spent on introducing the new notion of different types of special fuzzy and neutrosophic matrices, and the simple operations on them which are needed in the working of these multi expert models. In the second chapter, new set of multi expert models are introduced; these special fuzzy models and special fuzzy neutrosophic models that can cater to adopt any number of experts. The working of the model is also explained by illustrative examples. However, these special fuzzy models can also be used by applied mathematicians to study social and psychological problems. These models can also be used by doctors, engineers, scientists and statisticians. The SFCM, SMFCM, SNCM, SMNCM, SFRM, SNRM, SMFRM, SMNRM, SFNCMs, SFNRMs, etc. can give the special hidden pattern for any given special input vector. 5 The working of these SFREs, SMFREs and their neutrosophic analogues depends heavily upon the problems and the experts' expectation. The authors have given a long list for further reading which may help the socio scientists to know more about SFRE and SMFREs. We thank Dr. K. Kandasamy and Meena, without their unflinching support, this book would have never been possible. W.B.VASANTHA KANDASAMY FLORENTIN SMARANDACHE ILANTHENRAL. K 6 Chapter One A NEW CLASS OF SPECIAL FUZZY MATRICES AND SPECIAL NEUTROSOPHIC MATRICES In this chapter for the first time we introduce some new classes of special fuzzy matrices and illustrate them with examples. Also we give the main type of operations carried out on them. All these special fuzzy matrices will be used in the special fuzzy models which will be introduced in chapter two of this book. This chapter has three sections. In sections one we introduce the notion of fuzzy matrices and give the operations used on them like min max operations or max min operations. In section two we introduce the new classes of special fuzzy matrices define special operations on them and illustrate them with examples. In section three neutrosophic matrices, special operations on them are introduced and described. Several illustrative examples are given to make the operations explicit. 1.1 Introduction to Fuzzy Matrices Here we just recall the definition of fuzzy matrices for more about these concepts please refer [106]. Throughout this book the unit interval [0, 1] denotes the fuzzy interval. However in certain fuzzy matrices we also include the interval [–1, 1] to be 7 the fuzzy interval. So any element a ∈ [–1, 1] can be positive or ij negative. If a is positive then 0 < a ≤ 1, if a is negative then ij ij ij –1 ≤ a ≤ 0; a = 0 can also occur. So [0, 1] or [–1, 1] will be ij ij known as fuzzy interval. Thus if A = (a ) is a matrix and if in particular a ∈ [0, 1] ij ij (or [–1, 1]) we call A to be a fuzzy matrix. So all fuzzy matrices are matrices but every matrix in general need not be a fuzzy matrix. Hence fuzzy matrices forms a subclass of matrices. Now we give some examples of fuzzy matrices. Example 1.1.1: Let ⎡0.3 0.1 0.4 1 ⎤ ⎢ ⎥ A= 0.2 1 0.7 0 ⎢ ⎥ ⎢⎣0.9 0.8 0.5 0.8⎥⎦ be a matrix. Every element in A is in the unit interval [0, 1]. Thus A is a fuzzy matrix. We can say A is a 3 × 4 rectangular fuzzy matrix. Example 1.1.2: Let ⎡0.1 1 0 0.3 0.6 0.2⎤ ⎢ ⎥ 1 0.5 1 0.8 0.9 1 ⎢ ⎥ B=⎢ 0 1 0.3 0.9 0.7 0.5⎥ ⎢ ⎥ 0.4 0 1 0.6 0.3 1 ⎢ ⎥ ⎢⎣ 1 0.8 0.9 0.3 0.8 0 ⎥⎦ be a fuzzy matrix B is a 5 × 5 square fuzzy matrix. It is clear all the entries of B are from the unit interval [0, 1]. Example 1.1.3: Consider the matrix ⎡1 0 1 ⎤ ⎢ ⎥ C= −1 1 −1 ⎢ ⎥ ⎢⎣0 −1 1 ⎥⎦ 8 C is a matrix, its entries are from the set {–1, 0,1}. C is a 3 × 3 square fuzzy matrix. Example 1.1.4: Let A = [0 0.3 0.1 0.5 1 0.8 0.9 1 0]. A is a 1 × 9 fuzzy matrix will also be known as the fuzzy row vector or row fuzzy matrix. Example 1.1.5: Let ⎡ 1 ⎤ ⎢ ⎥ 0.3 ⎢ ⎥ ⎢0.7⎥ T= ⎢ ⎥ 0.5 ⎢ ⎥ ⎢0.1⎥ ⎢ ⎥ ⎢⎣0.9⎥⎦ be 6 × 1 fuzzy matrix. T is also know as the fuzzy column vector or fuzzy column matrix. Thus if A = [a a … a ] where a 1 2 n i ∈ [0, 1]; 1 ≤ i ≤ n, A will be known as the fuzzy row matrix or the fuzzy row vector. Let ⎡b ⎤ 1 ⎢ ⎥ b ⎢ 2⎥ B=⎢b ⎥ 3 ⎢ ⎥ (cid:35) ⎢ ⎥ ⎢⎣b ⎥⎦ m where b ∈ [0, 1], 1 ≤ j ≤ m, B will be known as the fuzzy j column vector or the fuzzy column matrix. Let A = (a ) with a ∈ [0, 1], 1 ≤ i ≤ n and 1 ≤ j ≤ n. A will ij ij be known as the n × n fuzzy square matrix. Suppose C = (c ) ij with c ∈ [0, 1]; 1 ≤ i ≤ n and 1 ≤ j ≤ m then C is known as the n ij × m rectangular fuzzy matrix. We have seen examples of these types of fuzzy matrices. A = [0 0 … 0] will be know as the zero 9