ELEMENTARY FUZZY MATRIX THEORY AND FUZZY MODELS 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] AUTOMATON Los Angeles 2007 1 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/ This book has been peer reviewed and recommended for publication by: Dr. Liu Huaning, Department of Mathematics, Northwest University, Xi’an, Shaanxi, P.R.China. Dr. L. Bojilov, Bulgarian Academy of Sciences, Sofia, Bulgaria. Prof. Ion Goian, Department of Algebra, Number Theory and Logic, State University of Kishinev, R. Moldova. Copyright 2007 by Automaton, W. B. Vasantha Kandasamy, Florentin Smarandache and Ilanthenral 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): 1-59973-005-7 ISBN(13): 978-1-59973-005-9 (EAN) 9781599730059 Standard Address Number: 297-5092 Printed in the United States of America 2 CONTENTS Preface 4 Dedication 6 Chapter One BASIC MATRIX THEORY AND FUZZY MATRIX THEORY 7 1.1 Basic Matrix Theory 7 1.2 Basic Concepts on Fuzzy Matrices 33 1.3 Basic Concepts on Graphs 47 Chapter Two DESCRIPTION OF SIMPLE FUZZY MODELS AND THEIR APPLICATIONS TO REAL WORLD PROBLEMS 71 2.1 Description of Simple Fuzzy Matrix Model 72 2.2 Definition of Fuzzy Cognitive Maps with real world model representation 146 2.3 Definition and Illustration of Fuzzy Relational Maps 205 2.4 Introduction to Bidirectional Associative Memories (BAM) Model and their Application 238 2.5 Description of Fuzzy Associative Memories (FAM) model and their illustrations 264 2.6 Fuzzy Relational Equations (FRE) and their application 277 REFERENCE 317 INDEX 345 ABOUT THE AUTHORS 352 3 PREFACE This book aims to assist social scientists to analyze their problems using fuzzy models. The basic and essential fuzzy matrix theory is given. The book does not promise to give the complete properties of basic fuzzy theory or basic fuzzy matrices. Instead, the authors have only tried to give those essential basically needed to develop the fuzzy model. The authors do not present elaborate mathematical theories to work with fuzzy matrices; instead they have given only the needed properties by way of examples. The authors feel that the book should mainly help social scientists who are interested in finding out ways to emancipate the society. Everything is kept at the simplest level and even difficult definitions have been omitted. Another main feature of this book is the description of each fuzzy model using examples from real-world problems. Further, this book gives lots of references so that the interested reader can make use of them. This book has two chapters. In Chapter One, basic concepts about fuzzy matrices are introduced. Basic notions of matrices are given in section one in order to make the book self- contained. Section two gives the properties of fuzzy matrices. Since the data need to be transformed into fuzzy models, some elementary properties of graphs are given. Further, this section provides details of how to prepare a linguistic questionnaire to make use of in these fuzzy models when the data related with the problem is unsupervised. Chapter Two has six sections. Section one deals with basic fuzzy matrix theory and can be used in a simple and effective way for analyzing supervised or unsupervised data. The simple elegant graphs related with this model can be understood even by a layman. The notion of Fuzzy Cognitive Maps (FCMs) model is introduced in the second section. This model is illustrated by a few examples. It can give the hidden pattern of the problem under analysis. The generalization of the FCM models, which are known as Fuzzy Relational Maps (FRMs), 4 come handy when the attributes related with the problem can be divided into two disjoint sets. This model comes handy when the number of attributes under study is large. This is described in section three. This also gives a pair of fixed points or limit cycle which happens to be the hidden pattern of the dynamical system. Bidirectional Associative Memories (BAM) model is described in the fourth section of this chapter. They are time or period dependent and are defined in real intervals. One can make use of them when the change or solution is time- dependent. This is also illustrated using real-world problems. The fifth section deals with Fuzzy Associative Memories (FAM) model and the model comes handy when one wants the gradations of each and every attribute under study. This model is also described and its working is shown through examples. The last section of this chapter deals with the Fuzzy Relational Equations (FRE) model. This model is useful when there are a set of predicted results and the best solution can be constructed very close, or at times, even equal to the predicted results. The working of this model is also given. Thus the book describes simple but powerful and accurate models that can be used by social scientists. 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 5 DEDICATION We dedicate this book to the great Tamil political leader Jeeva (1907-1963) on the occasion of his centennial celebration. He was an eminent orator, poet, litterateur and an uncompromising politician. He started his public life as a freedom fighter, and then entered the Self Respect Movement under Thanthai Periyar to become one of his powerful propagandists campaigning against superstitions and untouchability. He faced imprisonment several times and in the brunt of severe state repression built the CPI in Tamil Nadu. Moreover, his contribution to Tamil literary criticism was exemplary. It is significant to note that the CPI leader Jeeva was a mathematical genius in his early years; but he sacrificed his thirst for higher education in order to work for the exploited people. His radical ideology changed the Tamil political landscape, but perhaps if he had worked in the field of mathematics he would have revolutionized it too. 6 Chapter One BASIC MATRIX THEORY AND FUZZY MATRIX THEORY This chapter has three sections. In section one; we give some basic matrix theory. Section two recalls some fundamentals of fuzzy matrix theory. Section three gives the use of mean and standard deviation in matrices. 1.1 Basic Matrix Theory In this section we give some basic matrix theory essential to make the book a self contained one. However the book of Paul Horst on Matrix Algebra for social scientists [92] would be a boon to social scientists who wish to make use of matrix theory in their analysis. We give some very basic matrix algebra which is need for the development of fuzzy matrix theory and the related fuzzy model used for the analysis of socio-economic and psychological problems. However these fuzzy models have been used by applied mathematicians, to study social and psychological problems. These models are very much used by doctors, engineers, scientists, industrialists and statisticians. Here we proceed on to give some basic properties of matrix theory. 7 A matrix is a table of numbers with a finite number of rows and finite number of columns. The following 3 0 1 1 0 5 8 9 8 1 3 7 is an example of a matrix with three rows and four columns. Since data is always put in the table form it is very easy to consider the very table as matrix. This is very much seen when we use fuzzy matrix as the fuzzy models or fuzzy dynamical systems. So we are mainly going to deal in fuzzy model data matrices which are got from feelings, not always concrete numbers. When we speak of the table we have the rows and columns clearly marked out so the table by removing the lines can become a matrix with rows and columns. The horizontal entries of the table form the rows and the vertical entries forms the columns of a matrix. Since a table by any technical person (an analyst or a statistician) can have only real entries from the set of reals, thus we have any matrix to be a rectangular array of numbers. We can interchange the rows or columns i.e., the concepts or the entities or the attributes are interchanged. Let ⎡ 2 4 3 1 0 ⎤ ⎢ ⎥ 5 7 8 −9 4 ⎢ ⎥ ⎢ 7 −6 5 4 −3⎥ A = ⎢ ⎥ 8 2 1 0 1 ⎢ ⎥ ⎢10 1 0 4 −7⎥ ⎢ ⎥ ⎢⎣−2 2 5 9 8 ⎥⎦ be a 6 × 5 matrix i.e., a matrix with six rows and five columns, suppose we wish to interchange the sixth row with the second row we get the interchanged matrix as A' 8 ⎡ 2 4 3 1 0 ⎤ ⎢ ⎥ −2 2 5 9 8 ⎢ ⎥ ⎢ 7 −6 5 4 −3⎥ A' = ⎢ ⎥. 8 2 1 0 1 ⎢ ⎥ ⎢10 1 0 4 −7⎥ ⎢ ⎥ ⎢⎣ 5 7 8 −9 4 ⎥⎦ Now if one wants to interchange the first and fourth column one gets ⎡ 1 4 3 2 0 ⎤ ⎢ ⎥ 9 2 5 −2 8 ⎢ ⎥ ⎢ 4 −6 5 7 −3⎥ A'' = ⎢ ⎥. 0 2 1 8 1 ⎢ ⎥ ⎢ 4 1 0 10 −7⎥ ⎢ ⎥ ⎢⎣−9 7 8 5 4 ⎥⎦ Some times the interchange of row or column in a matrix may not be allowed in certain fuzzy models. If a matrix has only one column but any number of rows then the matrix is called as a column matrix. ⎡ 9 ⎤ ⎢ ⎥ 12 ⎢ ⎥ ⎢10⎥ C = ⎢ ⎥ 3 ⎢ ⎥ ⎢ 7 ⎥ ⎢ ⎥ ⎢⎣−2⎥⎦ is a column matrix. This is a special case of a matrix and sometimes known as the column vector. If a matrix has only one row then we call such a matrix to be a row matrix or a row vector it is also a special case of a matrix. R = [8 9 12 14 –17 10 1 –2 5], 9 R is a row matrix or a row vector. If a matrix has more number of rows than columns then we call it as a vertical matrix. ⎡ 3 1 2 ⎤ ⎢ ⎥ 0 5 9 ⎢ ⎥ ⎢−10 2 3 ⎥ X = ⎢ ⎥ 11 7 −8 ⎢ ⎥ ⎢ 6 9 10⎥ ⎢ ⎥ ⎢⎣ 7 −1 16⎥⎦ is a vertical matrix. A matrix which has more number of rows than columns will be known as the horizontal matrix. ⎡3 0 12 7 −9 8 −10 9⎤ Y = ⎢ ⎥ ⎣1 9 10 −15 7 1 7 3⎦ is a horizontal matrix. The number of rows and columns in a matrix is called the order of a matrix. ⎡ 3 0 −7 8 9 10 13⎤ ⎢ ⎥ 9 8 9 −11 6 5 −9 ⎢ ⎥ ⎢11 0 11 2 1 0 8 ⎥ A = ⎢ ⎥ 12 3 −7 6 5 −4 3 ⎢ ⎥ ⎢ 7 −5 2 1 −2 3 4 ⎥ ⎢ ⎥ ⎢⎣−9 6 −5 6 7 8 −9⎥⎦ is 6 × 7 matrix. In any matrix ⎡3 −1 6 1 8⎤ ⎢ ⎥ 8 9 3 2 1 A = ⎢ ⎥ ⎢4 2 1 4 6⎥ ⎢ ⎥ ⎣5 6 5 6 5⎦ which is a 4 × 5 matrix has 20 elements in total, is a rectangular matrix. 10