FUZZY LOGIC WITH ENGINEERING APPLICATIONS Third Edition Fuzzy L ogic w ith Engine e ring A pplic ations, Third Edition Timothy J. Ross © 2010 John Wiley & Sons, Ltd. ISBN: 978-0-470-74376-8 FUZZY LOGIC WITH ENGINEERING APPLICATIONS Third Edition Timothy J. Ross University of New Mexico, USA A John Wiley and Sons, Ltd., Publication Thiseditionfirstpublished2010 ©2010JohnWiley&Sons,Ltd Firsteditionpublished1995 Secondeditionpublished2004 Registeredoffice JohnWiley&SonsLtd,TheAtrium,SouthernGate,Chichester,WestSussex,PO198SQ,United Kingdom Fordetailsofourglobaleditorialoffices,forcustomerservicesandforinformationabouthowtoapply forpermissiontoreusethecopyrightmaterialinthisbookpleaseseeourwebsiteatwww.wiley.com. Therightoftheauthortobeidentifiedastheauthorofthisworkhasbeenassertedinaccordancewith theCopyright,DesignsandPatentsAct1988. 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LibraryofCongressCataloging-in-PublicationData Ross,TimothyJ. Fuzzylogicwithengineeringapplications/TimothyJ.Ross.–3rded. p.cm. Includesbibliographicalreferencesandindex. ISBN978-0-470-74376-8(cloth) 1. Engineeringmathematics. 2. Fuzzylogic. I. Title. TA331.R742010 620.001(cid:2)511313–dc22 2009033736 AcataloguerecordforthisbookisavailablefromtheBritishLibrary. ISBN:978-0-470-74376-8 Setin10/12ptTimesRomanbyLaserwordsPvtLtd,Chennai,India PrintedinSingaporebyFabulousPrintersPteLtd This book is dedicated to my brother Larry, my cousin Vicki Ehlert and my best friends Rick and Judy Brake, all of whom have given me incredible support over the past 5 years. Thank you so much for helping me deal with all my angst! CONTENTS About the Author xiii Preface to the Third Edition xv 1 Introduction 1 The Case for Imprecision 2 A Historical Perspective 3 The Utility of Fuzzy Systems 6 Limitations of Fuzzy Systems 8 The Illusion: Ignoring Uncertainty and Accuracy 10 Uncertainty and Information 13 The Unknown 14 Fuzzy Sets and Membership 14 Chance Versus Fuzziness 16 Sets as Points in Hypercubes 18 Summary 20 References 20 Problems 21 2 Classical Sets and Fuzzy Sets 25 Classical Sets 26 Operations on Classical Sets 28 Properties of Classical (Crisp) Sets 29 Mapping of Classical Sets to Functions 32 Fuzzy Sets 34 Fuzzy Set Operations 35 Properties of Fuzzy Sets 37 Alternative Fuzzy Set Operations 40 Summary 41 References 42 Problems 42 viii CONTENTS 3 Classical Relations and Fuzzy Relations 48 Cartesian Product 49 Crisp Relations 49 Cardinality of Crisp Relations 51 Operations on Crisp Relations 52 Properties of Crisp Relations 52 Composition 53 Fuzzy Relations 54 Cardinality of Fuzzy Relations 55 Operations on Fuzzy Relations 55 Properties of Fuzzy Relations 55 Fuzzy Cartesian Product and Composition 55 Tolerance and Equivalence Relations 62 Crisp Equivalence Relation 63 Crisp Tolerance Relation 64 Fuzzy Tolerance and Equivalence Relations 65 Value Assignments 68 Cosine Amplitude 69 Max–Min Method 71 Other Similarity Methods 71 Other Forms of the Composition Operation 72 Summary 72 References 73 Problems 73 4 Properties of Membership Functions, Fuzzification, and Defuzzification 89 Features of the Membership Function 90 Various Forms 92 Fuzzification 93 Defuzzification to Crisp Sets 95 λ-Cuts for Fuzzy Relations 97 Defuzzification to Scalars 98 Summary 110 References 111 Problems 112 5 Logic and Fuzzy Systems 117 Part I Logic 117 Classical Logic 118 Proof 124 Fuzzy Logic 131 Approximate Reasoning 134 Other Forms of the Implication Operation 138 Part II Fuzzy Systems 139 Natural Language 140 Linguistic Hedges 142 CONTENTS ix Fuzzy (Rule-Based) Systems 145 Graphical Techniques of Inference 148 Summary 159 References 161 Problems 162 6 Development of Membership Functions 174 Membership Value Assignments 175 Intuition 175 Inference 176 Rank Ordering 178 Neural Networks 179 Genetic Algorithms 189 Inductive Reasoning 199 Summary 206 References 206 Problems 207 7 Automated Methods for Fuzzy Systems 211 Definitions 212 Batch Least Squares Algorithm 215 Recursive Least Squares Algorithm 219 Gradient Method 222 Clustering Method 227 Learning From Examples 229 Modified Learning From Examples 233 Summary 242 References 242 Problems 243 8 Fuzzy Systems Simulation 245 Fuzzy Relational Equations 250 Nonlinear Simulation Using Fuzzy Systems 251 Fuzzy Associative Memories (FAMS) 255 Summary 264 References 265 Problems 266 9 Decision Making with Fuzzy Information 276 Fuzzy Synthetic Evaluation 278 Fuzzy Ordering 280 Nontransitive Ranking 283 Preference and Consensus 285 Multiobjective Decision Making 289 Fuzzy Bayesian Decision Method 294 Decision Making Under Fuzzy States and Fuzzy Actions 304 Summary 317