Fuzzy Logic With Engineering Applications PDF

This page intentionally left blank FUZZY LOGIC WITH ENGINEERING APPLICATIONS Second Edition This page intentionally left blank FUZZY LOGIC WITH ENGINEERING APPLICATIONS Second Edition Timothy J. Ross University of New Mexico, USA Copyright  2004 John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex PO19 8SQ, England Telephone (+44) 1243 779777 Email (for orders and customer service enquiries): [email protected] Visit our Home Page on www.wileyeurope.com or www.wiley.com All Rights Reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except under the terms of the Copyright, Designs and Patents Act 1988 or under the terms of a licence issued by the Copyright Licensing Agency Ltd, 90 Tottenham Court Road, London W1T 4LP, UK, without the permission in writing of the Publisher. 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British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library ISBN 0-470-86074-X (Cloth) 0-470-86075-8 (Paper) Typeset in 10/12pt Times NewRoman by Laserwords Private Limited, Chennai, India Printed and bound in Great Britain by TJ International, Padstow, Cornwall This book is printed on acid-free paper responsibly manufactured from sustainable forestry in which at least two trees are planted for each one used for paper production. This book is dedicated to the memories of my father, Jack, and my sister, Tina – the two behavioral bookends of my life. This page intentionally left blank CONTENTS About the Author xiii Preface to the Second Edition xv 1 Introduction 1 The Case for Imprecision 2 An Historical Perspective 3 The Utility of Fuzzy Systems 6 Limitations of Fuzzy Systems 8 The Allusion: Statistics and Random Processes 10 Uncertainty and Information 12 Fuzzy Sets and Membership 13 Chance versus Fuzziness 15 Sets as Points in Hypercubes 17 Summary 19 References 19 Problems 20 2 Classical Sets and Fuzzy Sets 24 Classical Sets 25 Operations on Classical Sets 27 Properties of Classical (Crisp) Sets 28 Mapping of Classical Sets to Functions 32 Fuzzy Sets 34 Fuzzy Set Operations 35 Properties of Fuzzy Sets 36 Noninteractive Fuzzy Sets 41 Alternative Fuzzy Set Operations 42 Summary 43 References 43 Problems 44 viii CONTENTS 3 Classical Relations and Fuzzy Relations 52 Cartesian Product 53 Crisp Relations 53 Cardinality of Crisp Relations 55 Operations on Crisp Relations 56 Properties of Crisp Relations 56 Composition 57 Fuzzy Relations 58 Cardinality of Fuzzy Relations 59 Operations on Fuzzy Relations 59 Properties of Fuzzy Relations 59 Fuzzy Cartesian Product and Composition 59 Tolerance and Equivalence Relations 66 Crisp Equivalence Relation 66 Crisp Tolerance Relation 67 Fuzzy Tolerance and Equivalence Relations 68 Value Assignments 71 Cosine Amplitude 72 Max–Min Method 74 Other Similarity Methods 74 Other Forms of the Composition Operation 74 Summary 75 References 75 Problems 76 General Relations 76 Value Assignments and Similarity 85 Equivalence Relations 88 Other Composition Operations 88 4 Properties of Membership Functions, Fuzzification, and Defuzzification 90 Features of the Membership Function 91 Various Forms 93 Fuzzification 94 Defuzzification to Crisp Sets 96 λ-cuts for Fuzzy Relations 98 Defuzzification to Scalars 99 Summary 112 References 113 Problems 114 5 Logic and Fuzzy Systems 120 Part I Logic 120 Classical Logic 121 Tautologies 126 Contradictions 128 Equivalence 128 CONTENTS ix Exclusive Or and Exclusive Nor 129 Logical Proofs 130 Deductive Inferences 132 Fuzzy Logic 134 Approximate Reasoning 137 Other Forms of the Implication Operation 141 Part II Fuzzy Systems 142 Natural Language 143 Linguistic Hedges 145 Fuzzy (Rule-Based) Systems 148 Graphical Techniques of Inference 151 Summary 162 References 163 Problems 165 6 Development of Membership Functions 178 Membership Value Assignments 179 Intuition 179 Inference 180 Rank Ordering 181 Neural Networks 182 Genetic Algorithms 193 Inductive Reasoning 200 Summary 206 References 208 Problems 209 7 Automated Methods for Fuzzy Systems 212 Definitions 213 Batch Least Squares Algorithm 216 Recursive Least Squares Algorithm 220 Gradient Method 223 Clustering Method 228 Learning From Example 231 Modified Learning From Example 234 Summary 242 References 243 Problems 243 8 Fuzzy Systems Simulation 245 Fuzzy Relational Equations 250 Nonlinear Simulation Using Fuzzy Systems 251 Fuzzy Associative Memories (FAMs) 254 Summary 264 References 265 Problems 265 x CONTENTS 9 Rule-base Reduction Methods 274 Fuzzy Systems Theory and Rule Reduction 275 New Methods 275 Singular Value Decomposition 276 Combs Method 282 SVD and Combs Method Examples 284 Summary 303 References 304 Problems 304 Singular Value Decomposition 304 Combs Method for Rapid Inference 306 10 Decision Making with Fuzzy Information 308 Fuzzy Synthetic Evaluation 310 Fuzzy Ordering 312 Nontransitive Ranking 315 Preference and Consensus 317 Multiobjective Decision Making 320 Fuzzy Bayesian Decision Method 326 Decision Making under Fuzzy States and Fuzzy Actions 335 Summary 349 References 350 Problems 350 Ordering and Synthetic Evaluation 350 Nontransitive Ranking 352 Fuzzy Preference and Consensus 353 Multiobjective Decision Making 355 Bayesian Decision Making 357 11 Fuzzy Classification and Pattern Recognition 362 Part I Classification 362 Classification by Equivalence Relations 363 Crisp Relations 363 Fuzzy Relations 365 Cluster Analysis 369 Cluster Validity 370 c-Means Clustering 370 Hard c-Means (HCM) 371 Fuzzy c-Means (FCM) 379 Fuzzy c-Means Algorithm 382 Classification Metric 387 Hardening the Fuzzy c-Partition 389 Similarity Relations from Clustering 391 Part II Pattern Recognition 392 Feature Analysis 393 Partitions of the Feature Space 393 CONTENTS xi Single-Sample Identification 394 Multifeature Pattern Recognition 400 Image Processing 412 Syntactic Recognition 420 Formal Grammar 422 Fuzzy Grammar and Syntactic Recognition 424 Summary 429 References 429 Problems 430 Exercises for Equivalence Classification 430 Exercises for Fuzzy c-Means 431 Exercises for Classification Metric and Similarity 434 Exercises for Fuzzy Vectors 435 Exercises for Multifeature Pattern Recognition 436 Exercises for Syntactic Pattern Recognition 444 Exercises for Image Processing 444 12 Fuzzy Arithmetic and the Extension Principle 445 Extension Principle 445 Crisp Functions, Mapping, and Relations 446 Functions of Fuzzy Sets – Extension Principle 447 Fuzzy Transform (Mapping) 448 Practical Considerations 450 Fuzzy Arithmetic 455 Interval Analysis in Arithmetic 457 Approximate Methods of Extension 459 Vertex Method 459 DSW Algorithm 462 Restricted DSW Algorithm 465 Comparisons 466 Summary 469 References 469 Problems 470 13 Fuzzy Control Systems 476 Control System Design Problem 478 Control (Decision) Surface 479 Assumptions in a Fuzzy Control System Design 480 Simple Fuzzy Logic Controllers 480 Examples of Fuzzy Control System Design 481 Aircraft Landing Control Problem 485 Fuzzy Engineering Process Control [Parkinson, 2001] 492 Classical Feedback Control 492 Classical PID Control 494 Fuzzy Control 496 Multi-input, Multi-output (MIMO) Control Systems 500 Fuzzy Statistical Process Control 504 xii CONTENTS Measurement Data – Traditional SPC 505 Attribute Data – Traditional SPC 510 Industrial Applications 517 Summary 518 References 519 Problems 521 14 Miscellaneous Topics 537 Fuzzy Optimization 537 One-dimensional Optimization 538 Fuzzy Cognitive Mapping 544 Fuzzy Cognitive Maps 545 System Identification 550 Fuzzy Linear Regression 555 The Case of Nonfuzzy Data 557 The Case of Fuzzy Data 558 Summary 567 References 567 Problems 568 Fuzzy Optimization 568 System Identification 569 Regression 570 Cognitive Mapping 571 15 Monotone Measures: Belief, Plausibility, Probability, and Possibility 572 Monotone Measures 573 Belief and Plausibility 574 Evidence Theory 578 Probability Measures 582 Possibility and Necessity Measures 583 Possibility Distributions as Fuzzy Sets 590 Possibility Distributions Derived from Empirical Intervals 592 Deriving Possibility Distributions from Overlapping Intervals 593 Redistributing Weight from Nonconsonant to Consonant Intervals 595 Comparison of Possibility Theory and Probability Theory 600 Summary 601 References 603 Problems 603 Appendix A Axiomatic Differences between Fuzzy Set Theory and Probability Theory 610 Appendix B Answers to Selected Problems 614 Index of Examples and Problems by Discipline 621 Index 623 ABOUT THE AUTHOR Timothy J. Ross is Professor and Regents’ Lecturer of Civil Engineering at the University of New Mexico. He received his PhD degree in Civil Engineering from Stanford University, his MS from Rice University, and his BS from Washington State University. Professor Ross has held previous positions as Senior Research Structural Engineer, Air Force Weapons Laboratory, from 1978 to 1986; and Vulnerability Engineer, Defense Intelligence Agency, from 1973 to 1978. Professor Ross has authored more than 120 publications and has been active in the research and teaching of fuzzy logic since 1983. He is the founding Co-Editor- in-Chief of the International Journal of Intelligent and Fuzzy Systems and the co-editor of Fuzzy Logic and Control: Software and Hardware Applications, and most recently co-editor of Fuzzy Logic and Probability Applications: Bridging the Gap. Professor Ross is a Fellow of the American Society of Civil Engineers. He consults for industry and such institutions as Sandia National Laboratory and the National Technological University, and is a current Faculty Affiliate with the Los Alamos National Laboratory. He was recently honored with a Senior Fulbright Fellowship for his sabbatical study at the University of Calgary, Alberta, Canada. This page intentionally left blank PREFACE TO THE SECOND EDITION The second edition of this text has been ‘‘on the drawing board’’ for quite some time. Since the first edition was published, in 1995, the technology of fuzzy set theory and its application to systems, using fuzzy logic, has moved rapidly. Developments in other theories such as possibility theory and evidence theory (both being elements of a larger collection of methods under the rubric ‘‘generalized information theories’’) have shed more light on the real virtues of fuzzy logic applications, and some developments in machine computation have made certain features of fuzzy logic much more useful than in the past. In fact, it would be fair to state that some developments in fuzzy systems are quite competitive with other, linear algebra-based methods in terms of computational speed and associated accuracy. To wait eight years to publish this second edition has been, perhaps, too long. On the other hand, the technology continues to move so fast that one is often caught in that uncomfortable middle-ground not wanting to miss another important development that could be included in the book. The pressures of academia and the realities of life seem to intervene at the most unexpected times, but now seems the best time for this second edition. There are sections of the first text that have been eliminated in the second edition; I shall have more to say on this below. And there are many new sections – which are included in the second edition – to try to capture some of the newer developments; the key word here is ‘‘some’’ as it would be completely impossible to summarize or illustrate even a small fraction of the new developments of the last eight years. As with any book containing technical material, the first edition contained errata that have been corrected in this second edition. A new aid to students, appearing in this edition, is a section at the end of the book which contains solutions to selected end-of-chapter problems. As with the first edition, a solutions manual for all problems in the second edition can be obtained by qualified instructors by visiting http://www.wileyeurope.com/go/fuzzylogic. One of the most important explanations I shall describe in this preface has to do with what I call the misuse of definitional terms in the past literature on uncertainty representational theories; in this edition I use these terms very cautiously. Principal among these terms is the word ‘‘coherence’’ and the ubiquitous use of the word ‘‘law.’’ To begin xvi PREFACE TO THE SECOND EDITION with the latter, the axioms of a probability theory referred to as the excluded middle will hereinafter only be referred to as axioms – never as laws. The operations due to De Morgan also will not be referred to as a law, but as a principle... since this principle does apply to some (not all) uncertainty theories (e.g., probability and fuzzy). The excluded middle axiom (and its dual, the axiom of contradiction) are not laws; Newton produced laws, Kepler produced laws, Darcy, Boyle, Ohm, Kirchhoff, Bernoulli, and many others too numerous to list here all developed laws. Laws are mathematical expressions describing the immutable realizations of nature. It is perhaps a cunning, but now exposed, illusion first coined by probabilists in the last two centuries to give their established theory more legitimacy by labeling their axioms as laws. Definitions, theorems, and axioms collectively can describe a certain axiomatic foundation describing a particular kind of theory, and nothing more; in this case the excluded middle and other axioms (see Appendix A) can be used to describe a probability theory. Hence, if a fuzzy set theory does not happen to be constrained by an excluded middle axiom, it is not a violation of some immutable law of nature like Newton’s laws; fuzzy set theory simply does not happen to have an axiom of the excluded middle – it does not need, nor is constrained by, such an axiom. In fact, as early as 1905 the famous mathematician L. E. J. Brouwer defined this excluded middle axiom as a principle in his writings; he showed that the principle of the excluded middle was inappropriate in some logics, including his own which he termed intuitionism. Brouwer observed that Aristotelian logic is only a part of mathematics, the special kind of mathematical thought obtained if one restricts oneself to relations of the whole and part. Brouwer had to specify in which sense the principles of logic could be considered ‘‘laws’’ because within his intuitionistic framework thought did not follow any rules, and, hence, ‘‘law’’ could no longer mean ‘‘rule’’ (see the detailed discussion on this in the summary of Chapter 5). In this regard, I shall take on the cause advocated by Brouwer almost a century ago. In addition, the term coherence does not connote a law. It may have been a clever term used by the probabilists to describe another of their axioms (in this case a permutation of the additivity axiom) but such cleverness is now an exposed prestidigitation of the English language. Such arguments of the past like ‘‘no uncertainty theory that is non-coherent can ever be considered a serious theory for describing uncertainty’’ now carry literally no weight when one considers that the term coherence is a label and not an adjective describing the value of an axiomatic structure. I suppose that fuzzy advocates could relabel their axiom of strong-truth functionality to the ‘‘law of practicability’’ and then claim that any other axiomatic structure that does not use such an axiom is inadequate, to wit ‘‘a theory that violates the practicability axiom is a violation of the law of utility,’’ but we shall not resort to this hyperbole. With this edition, we will speak without the need for linguistic slight-of-hand. The combination of a fuzzy set theory and a probability theory is a very powerful modeling paradigm. This book is dedicated to users who are more interested in solving problems than in dealing with debates using misleading jargon. To end my discussion on misleading definitional terms in the literature, I have made two subtle changes in the material in Chapter 15. First, following prof. Klir’s lead of a couple years ago, we no longer refer to ‘‘fuzzy measure theory’’ but instead describe it now as ‘‘monotone measure theory’’. The former phrase still causes confusion when referring to fuzzy set theory; hopefully this will end that confusion. And, in Chapter 15 in describing the monotone measure, m, I have changed the phrase describing this measure from a ‘‘basic probability assignment (bpa)’’ to a ‘‘basic evidence assignment (bea)’’. Here we attempt to avoid confusion with any of the terms typically used in probability theory. PREFACE TO THE SECOND EDITION xvii As with the first edition, this second edition is designed for the professional and academic audience interested primarily in applications of fuzzy logic in engineering and technology. Always I have found that the majority of students and practicing professionals are interested in the applications of fuzzy logic to their particular fields. Hence, the book is written for an audience primarily at the senior undergraduate and first-year graduate levels. With numerous examples throughout the text, this book is written to assist the learning process of a broad cross section of technical disciplines. The book is primarily focused on applications, but each of the book’s chapters begins with the rudimentary structure of the underlying mathematics required for a fundamental understanding of the methods illustrated. Chapter 1∗ introduces the basic concept of fuzziness and distinguishes fuzzy uncer- tainty from other forms of uncertainty. It also introduces the fundamental idea of set membership, thereby laying the foundation for all material that follows, and presents membership functions as the format used for expressing set membership. The chapter sum- marizes an historical review of uncertainty theories. The chapter reviews the idea of ‘‘sets as points’’ in an n-dimensional Euclidean space as a graphical analog in understanding the relationship between classical (crisp) and fuzzy sets. Chapter 2 reviews classical set theory and develops the basic ideas of fuzzy sets. Operations, axioms, and properties of fuzzy sets are introduced by way of comparisons with the same entities for classical sets. Various normative measures to model fuzzy intersections (t-norms) and fuzzy unions (t-conorms) are summarized. Chapter 3 develops the ideas of fuzzy relations as a means of mapping fuzziness from one universe to another. Various forms of the composition operation for relations are presented. Again, the epistemological approach in Chapter 3 uses comparisons with classical relations in developing and illustrating fuzzy relations. This chapter also illustrates methods to determine the numerical values contained within a specific class of fuzzy relations, called similarity relations. Chapter 4 discusses the fuzzification of scalar variables and the defuzzification of membership functions. The chapter introduces the basic features of a membership function and it discusses, very briefly, the notion of interval-valued fuzzy sets. Defuzzification is necessary in dealing with the ubiquitous crisp (binary) world around us. The chapter details defuzzification of fuzzy sets and fuzzy relations into crisp sets and crisp relations, respec- tively, using lambda-cuts, and it describes a variety of methods to defuzzify membership functions into scalar values. Examples of all methods are given in the chapter. Chapter 5 introduces the precepts of fuzzy logic, again through a review of the relevant features of classical, or a propositional, logic. Various logical connectives and operations are illustrated. There is a thorough discussion of the various forms of the implication operation and the composition operation provided in this chapter. Three different inference methods, popular in the literature, are illustrated. Approximate reasoning, or reasoning under imprecise (fuzzy) information, is also introduced in this chapter. Basic IF–THEN rule structures are introduced and three graphical methods for inferencing are presented. Chapter 6 provides several classical methods of developing membership functions, including methods that make use of the technologies of neural networks, genetic algorithms, and inductive reasoning. Chapter 7 is a new chapter which presents six new automated methods which can be used to generate rules and membership functions from observed or measured input–output ∗ Includes sections taken from Ross, T., Booker, J., and Parkinson, W., 2002, Fuzzy Logic and Probability Applications: Bridging the Gap, reproduced by the permission of Society for Industrial and Applied Mathematics, Philadelphia, PA. xviii PREFACE TO THE SECOND EDITION data. The procedures are essentially computational methods of learning. Examples are pro- vided to illustrate each method. Many of the problems at the end of the chapter will require software; this software can be downloaded from: www.wileyeurope.com/go/fuzzylogic. Beginning the second category of chapters in the book highlighting applications, Chapter 8 continues with the rule-based format to introduce fuzzy nonlinear simulation and complex system modeling. In this context, nonlinear functions are seen as mappings of information ‘‘patches’’ from the input space to information ‘‘patches’’ of the output space, instead of the ‘‘point-to-point’’ idea taught in classical engineering courses. Fidelity of the simulation is illustrated with standard functions, but the power of the idea can be seen in systems too complex for an algorithmic description. This chapter formalizes fuzzy associative memories (FAMs) as generalized mappings. Chapter 9 is a new chapter covering the area of rule-base reduction. Fuzzy systems are becoming popular, but they can also present computational challenges as the rule-bases, especially those derived from automated methods, can become large in an exponential sense as the number of inputs and their dimensionality grows. This chapter summarizes two relatively new reduction techniques and provides examples of each. Chapter 10 develops fuzzy decision making by introducing some simple concepts in ordering, preference and consensus, and multiobjective decisions. It introduces the powerful concept of Bayesian decision methods by fuzzifying this classic probabilistic approach. This chapter illustrates the power of combining fuzzy set theory with probability to handle random and nonrandom uncertainty in the decision-making process. Chapter 11 discusses a few fuzzy classification methods by contrasting them with classical methods of classification, and develops a simple metric to assess the goodness of the classification, or misclassification. This chapter also summarizes classification using equivalence relations. The algebra of fuzzy vectors is summarized here. Classification is used as a springboard to introduce fuzzy pattern recognition. A single-feature and a multiple-feature procedure are summarized. Some simple ideas in image processing and syntactic pattern recognition are also illustrated. Chapter 12 summarizes some typical operations in fuzzy arithmetic and fuzzy num- bers. The extension of fuzziness to nonfuzzy mathematical forms using Zadeh’s extension principle and several approximate methods to implement this principle are illustrated. Chapter 13 introduces the field of fuzzy control systems. A brief review of control system design and control surfaces is provided. Some example problems in control are provided. Two new sections have been added to this book: fuzzy engineering process control, and fuzzy statistical process control. Examples of these are provided in the chapter. Chapter 14 briefly addresses some important ideas embodied in fuzzy optimization, fuzzy cognitive mapping, fuzzy system identification, and fuzzy regression. Finally, Chapter 15 enlarges the reader’s understanding of the relationship between fuzzy uncertainty and random uncertainty (and other general forms of uncertainty, for that matter) by illustrating the foundations of monotone measures. The chapter discusses monotone measures in the context of evidence theory and probability theory. Because this chapter is an expansion of ideas relating to other disciplines (Dempster–Shafer evidence theory and probability theory), it can be omitted without impact on the material preceding it. Appendix A of the book shows the axiomatic similarity of fuzzy set theory and probability theory and Appendix B provides answers to selected problems from each chapter. Most of the text can be covered in a one-semester course at the senior undergraduate level. In fact, most science disciplines and virtually all math and engineering disciplines PREFACE TO THE SECOND EDITION xix contain the basic ideas of set theory, mathematics, and deductive logic, which form the only knowledge necessary for a complete understanding of the text. For an introductory class, instructors may want to exclude some or all of the material covered in the last section of Chapter 6 (neural networks, genetic algorithms, and inductive reasoning), Chapter 7 (automated methods of generation), Chapter 9 on rule-base reduction methods, and any of the final three chapters: Chapter 13 (fuzzy control), Chapter 14 (miscellaneous fuzzy applications), and Chapter 15 on alternative measures of uncertainty. I consider the applications in Chapter 8 on simulations, Chapter 10 on decision making, Chapter 11 on classification, and Chapter 12 on fuzzy arithmetic to be important in the first course on this subject. The other topics could be used either as introductory material for a graduate-level course or for additional coverage for graduate students taking the undergraduate course for graduate credit. The book is organized a bit differently from the first edition. I have moved most of the information for rule-based deductive systems closer to the front of the book, and have moved fuzzy arithmetic toward the end of the book; the latter does not disturb the flow of the book to get quickly into fuzzy systems development. A significant amount of new material has been added in the area of automated methods of generating fuzzy systems (Chapter 7); a new section has been added on additional methods of inference in Chapter 5; and a new chapter has been added on the growing importance of rule-base reduction methods (Chapter 9). Two new sections in fuzzy control have been added in Chapter 13. I have also deleted materials that either did not prove useful in the pedagogy of fuzzy systems, or were subjects of considerable depth which are introduced in other, more focused texts. Many of the rather lengthy example problems from the first edition have been reduced for brevity. In terms of organization, the first eight chapters of the book develop the foundational material necessary to get students to a position where they can generate their own fuzzy systems. The last seven chapters use the foundation material from the first eight chapters to present specific applications. The problems in this text are typically based on current and potential applications, case studies, and education in intelligent and fuzzy systems in engineering and related technical fields. The problems address the disciplines of computer science, electrical engineering, manufacturing engineering, industrial engineering, chemical engineering, petroleum engi- neering, mechanical engineering, civil engineering, environmental engineering, engineering management, and a few related fields such as mathematics, medicine, operations research, technology management, the hard and soft sciences, and some technical business issues. The references cited in the chapters are listed toward the end of each chapter. These refer- ences provide sufficient detail for those readers interested in learning more about particular applications using fuzzy sets or fuzzy logic. The large number of problems provided in the text at the end of each chapter allows instructors a sizable problem base to afford instruction using this text on a multisemester or multiyear basis, without having to assign the same problems term after term. I was most fortunate this past year to have co-edited a text with Drs. Jane Booker and Jerry Parkinson, entitled Fuzzy Logic and Probability Applications: Bridging the Gap, published by the Society for Industrial and Applied Mathematics (SIAM), in which many of my current thoughts on the matter of the differences between fuzzy logic and probability theory were noted; some of this appears in Chapters 1 and 15 of this edition. Moreover, I am also grateful to Prof. Kevin Passino whose text, Fuzzy Control, published by Prentice Hall, illustrated some very recent developments in the automated generation of membership xx PREFACE TO THE SECOND EDITION functions and rules in fuzzy systems. The algorithms discussed in his book, while being developed by others earlier, are collected in one chapter in his book; some of these are illustrated here in Chapter 7, on automated methods. The added value to Dr. Passino’s material and methods is that I have expanded their explanation and have added some simple numerical examples of these methods to aid first-time students in this field. Again I wish to give credit either to some of the individuals who have shaped my thinking about this subject since the first edition of 1995, or to others who by their simple association with me have caused me to be more circumspect about the use of the material contained in the book. In addition to the previously mentioned colleagues Jane Booker and Jerry Parkinson, who both overwhelm me with their knowledge and enthusiasm, my other colleagues at Los Alamos National Laboratory have shaped or altered my thinking critically and positively: Scott Doebling, Ed Rodriquez, and John Birely for their steadfast support over the years to investigate alternative uncertainty paradigms, Jason Pepin for his useful statistical work in mechanics, Cliff Joslyn for his attention to detail in the axiomatic structure of random sets, Brian Reardon for his critical questions of relevance, François Hemez and Mark Anderson for their expertise in applying uncertainty theory to validation methods, Kari Sentz for her humor and her perspective in linguistic uncertainty, Ron Smith and Karen Hench for their collaborations in process control, and Steve Eisenhawer and Terry Bott for their early and continuing applications of fuzzy logic in risk assessment. Some of the newer sections of the second edition were first taught to a group of faculty and students at the University of Calgary, Alberta, during my most recent sabbatical leave. My host, Prof. Gopal Achari, was instrumental in giving me this exposure and outreach to these individuals and I shall remain indebted to him. Among this group, faculty members Drs. Brent Young, William Svrcek, and Tom Brown, and students Jeff Macisaac, Rachel Mintz, and Rodolfo Tellez, all showed leadership and critical inquiry in adopting many fuzzy skills into their own research programs. Discussions with Prof. Mihaela Ulieru, already a fuzzy advocate, and her students proved useful. Finally, paper collaborations with Ms. Sumita Fons, Messrs. Glen Hay and James Vanderlee all gave me a feeling of accomplishment on my ‘‘mission to Canada.’’ Collaborations, discussions, or readings from Drs. Lotfi Zadeh, George Klir, and Vladik Kreinovich over the past few years have enriched my understanding in this field immeasurably. In particular, Dr. Klir’s book of 1995 (Fuzzy Sets and Fuzzy Logic) and his writings in various journals collectively have helped me deepen my understanding of some of the nuances in the mathematics of fuzzy logic; his book is referenced in many places in this second edition. I wish to thank some of my recent graduate students who have undertaken projects, MS theses, or PhD dissertations related to this field and whose hard work for me and alongside me has given me a sense of pride in their own remarkable tenacity and productive efforts: Drs. Sunil Donald and Jonathan Lucero and Mr. Greg Chavez, and Mss. Terese Gabocy Anderson and Rhonda Young. There have been numerous students over the past eight years who have contributed many example problems for updating the text; unfortunately too numerous to mention in this brief preface. I want to thank them all again for their contributions. Four individuals need specific mention because they have contributed some sections to this text. I would like to thank specifically Dr. Jerry Parkinson for his contributions to Chapter 13 in the areas of chemical process control and fuzzy statistical process control, Dr. Jonathan Lucero for his contributions in developing the material in Chapter 9 for rule-reduction methods (which form the core of his PhD dissertation), Greg Chavez for his PREFACE TO THE SECOND EDITION xxi text preparation of many of the new, contributed problems in this text and of the material in Chapter 7, and Dr. Sunil Donald for one new section in Chapter 15 on empirical methods to generate possibility distributions. I am most grateful for financial support over the past three years while I have generated most of the background material in my own research for some of the newer material in the book. I would like to thank the Los Alamos National Laboratory, Engineering and Science Applications Division, the University of New Mexico, and the US–Canadian Fulbright Foundation for their generous support during this period of time. With so many texts covering specific niches of fuzzy logic it is not possible to summarize all these important facets of fuzzy set theory and fuzzy logic in a single textbook. The hundreds of edited works and tens of thousands of archival papers show clearly that this is a rapidly growing technology, where new discoveries are being published every month. It remains my fervent hope that this introductory textbook will assist students and practising professionals to learn, to apply, and to be comfortable with fuzzy set theory and fuzzy logic. I welcome comments from all readers to improve this textbook as a useful guide for the community of engineers and technologists who will become knowledgeable about the potential of fuzzy system tools for their use in solving the problems that challenge us each day. Timothy J. Ross Santa Fe, New Mexico This page intentionally left blank CHAPTER 1 INTRODUCTION It is the mark of an instructed mind to rest satisfied with that degree of precision which the nature of the subject admits, and not to seek exactness where only an approximation of the truth is possible. Aristotle, 384–322 BC Ancient Greek philosopher Precision is not truth. Henri E. B. Matisse, 1869–1954 Impressionist painter All traditional logic habitually assumes that precise symbols are being employed. It is therefore not applicable to this terrestrial life but only to an imagined celestial existence. Bertrand Russell, 1923 British philosopher and Nobel Laureate We must exploit our tolerance for imprecision. Lotfi Zadeh Professor, Systems Engineering, UC Berkeley, 1973 The quotes above, all of them legendary, have a common thread. That thread represents the relationship between precision and uncertainty. The more uncertainty in a problem, the less precise we can be in our understanding of that problem. It is ironic that the oldest quote, above, is due to the philosopher who is credited with the establishment of Western logic – a binary logic that only admits the opposites of true and false, a logic which does not admit degrees of truth in between these two extremes. In other words, Aristotelian logic does not admit imprecision in truth. However, Aristotle’s quote is so appropriate today; it is a quote that admits uncertainty. It is an admonishment that we should heed; we should balance the precision we seek with the uncertainty that exists. Most engineering texts do not address the uncertainty in the information, models, and solutions that are conveyed Fuzzy Logic with Engineering Applications, Second Edition T. J. Ross  2004 John Wiley & Sons, Ltd ISBNs: 0-470-86074-X (HB); 0-470-86075-8 (PB) 2 INTRODUCTION within the problems addressed therein. This text is dedicated to the characterization and quantification of uncertainty within engineering problems such that an appropriate level of precision can be expressed. When we ask ourselves why we should engage in this pursuit, one reason should be obvious: achieving high levels of precision costs significantly in time or money or both. Are we solving problems that require precision? The more complex a system is, the more imprecise or inexact is the information that we have to characterize that system. It seems, then, that precision and information and complexity are inextricably related in the problems we pose for eventual solution. However, for most of the problems that we face, the quote above due to Professor Zadeh suggests that we can do a better job in accepting some level of imprecision. It seems intuitive that we should balance the degree of precision in a problem with the associated uncertainty in that problem. Hence, this book recognizes that uncertainty of various forms permeates all scientific endeavors and it exists as an integral feature of all abstractions, models, and solutions. It is the intent of this book to introduce methods to handle one of these forms of uncertainty in our technical problems, the form we have come to call fuzziness. THE CASE FOR IMPRECISION Our understanding of most physical processes is based largely on imprecise human reasoning. This imprecision (when compared to the precise quantities required by computers) is nonetheless a form of information that can be quite useful to humans. The ability to embed such reasoning in hitherto intractable and complex problems is the criterion by which the efficacy of fuzzy logic is judged. Undoubtedly this ability cannot solve problems that require precision – problems such as shooting precision laser beams over tens of kilometers in space; milling machine components to accuracies of parts per billion; or focusing a microscopic electron beam on a specimen the size of a nanometer. The impact of fuzzy logic in these areas might be years away, if ever. But not many human problems require such precision – problems such as parking a car, backing up a trailer, navigating a car among others on a freeway, washing clothes, controlling traffic at intersections, judging beauty contestants, and a preliminary understanding of a complex system. Requiring precision in engineering models and products translates to requiring high cost and long lead times in production and development. For other than simple systems, expense is proportional to precision: more precision entails higher cost. When considering the use of fuzzy logic for a given problem, an engineer or scientist should ponder the need for exploiting the tolerance for imprecision. Not only does high precision dictate high costs but also it entails low tractability in a problem. Articles in the popular media illustrate the need to exploit imprecision. Take the ‘‘traveling salesrep’’ problem, for example. In this classic optimization problem a sales representative wants to minimize total distance traveled by considering various itineraries and schedules between a series of cities on a particular trip. For a small number of cities, the problem is a trivial exercise in enumerating all the possibilities and choosing the shortest route. As the number of cities continues to grow, the problem quickly approaches a combinatorial explosion impossible to solve through an exhaustive search, even with a computer. For example, for 100 cities there are 100 × 99 × 98 × 97 × · · · × 2 × 1, or about 10200 , possible routes to consider! No computers exist today that can solve this problem through a brute-force enumeration AN HISTORICAL PERSPECTIVE 3 of all the possible routes. There are real, practical problems analogous to the traveling salesrep problem. For example, such problems arise in the fabrication of circuit boards, where precise lasers drill hundreds of thousands of holes in the board. Deciding in which order to drill the holes (where the board moves under a stationary laser) so as to minimize drilling time is a traveling salesrep problem [Kolata, 1991]. Thus, algorithms have been developed to solve the traveling salesrep problem in an optimal sense; that is, the exact answer is not guaranteed but an optimum answer is achievable – the optimality is measured as a percent accuracy, with 0% representing the exact answer and accuracies larger than zero representing answers of lesser accuracy. Suppose we consider a signal routing problem analogous to the traveling salesrep problem where we want to find the optimum path (i.e., minimum travel time) between 100,000 nodes in a network to an accuracy within 1% of the exact solution; this requires significant CPU time on a supercomputer. If we take the same problem and increase the precision requirement a modest amount to an accuracy of 0.75%, the computing time approaches a few months! Now suppose we can live with an accuracy of 3.5% (quite a bit more accurate than most problems we deal with), and we want to consider an order-of-magnitude more nodes in the network, say 1,000,000; the computing time for this problem is on the order of several minutes [Kolata, 1991]. This remarkable reduction in cost (translating time to dollars) is due solely to the acceptance of a lesser degree of precision in the optimum solution. Can humans live with a little less precision? The answer to this question depends on the situation, but for the vast majority of problems we deal with every day the answer is a resounding yes. AN HISTORICAL PERSPECTIVE From an historical point of view the issue of uncertainty has not always been embraced within the scientific community [Klir and Yuan, 1995]. In the traditional view of science, uncertainty represents an undesirable state, a state that must be avoided at all costs. This was the state of science until the late nineteenth century when physicists realized that Newtonian mechanics did not address problems at the molecular level. Newer methods, associated with statistical mechanics, were developed which recognized that statistical averages could replace the specific manifestations of microscopic entities. These statistical quantities, which summarized the activity of large numbers of microscopic entities, could then be connected in a model with appropriate macroscopic variables [Klir and Yuan, 1995]. Now, the role of Newtonian mechanics and its underlying calculus which considered no uncertainty was replaced with statistical mechanics which could be described by a probability theory – a theory which could capture a form of uncertainty, the type generally referred to as random uncertainty. After the development of statistical mechanics there has been a gradual trend in science during the past century to consider the influence of uncertainty on problems, and to do so in an attempt to make our models more robust, in the sense that we achieve credible solutions and at the same time quantify the amount of uncertainty. Of course, the leading theory in quantifying uncertainty in scientific models from the late nineteenth century until the late twentieth century had been probability theory. However, the gradual evolution of the expression of uncertainty using probability theory was challenged, first in 1937 by Max Black, with his studies in vagueness, then with the 4 INTRODUCTION introduction of fuzzy sets by Lotfi Zadeh in 1965. Zadeh’s work had a profound influence on the thinking about uncertainty because it challenged not only probability theory as the sole representation for uncertainty, but the very foundations upon which probability theory was based: classical binary (two-valued) logic [Klir and Yuan, 1995]. Probability theory dominated the mathematics of uncertainty for over five centuries. Probability concepts date back to the 1500s, to the time of Cardano when gamblers recognized the rules of probability in games of chance. The concepts were still very much in the limelight in 1685, when the Bishop of Wells wrote a paper that discussed a problem in determining the truth of statements made by two witnesses who were both known to be unreliable to the extent that they only tell the truth with probabilities p1 and p2 , respectively. The Bishop’s answer to this was based on his assumption that the two witnesses were independent sources of information [Lindley, 1987]. Probability theory was initially developed in the eighteenth century in such landmark treatises as Jacob Bernoulli’s Ars Conjectandi (1713) and Abraham DeMoiver’s Doctrine of Chances (1718, 2nd edition 1738). Later in that century a small number of articles appeared in the periodical literature that would have a profound effect on the field. Most notable of these were Thomas Bayes’s ‘‘An essay towards solving a problem in the doctrine of chances’’ (1763) and Pierre Simon Laplace’s formulation of the axioms relating to games of chance, ‘‘Memoire sur la probabilite des causes par les evenemens’’ (1774). Laplace, only 25 years old at the time he began his work in 1772, wrote the first substantial article in mathematical statistics prior to the nineteenth century. Despite the fact that Laplace, at the same time, was heavily engaged in mathematical astronomy, his memoir was an explosion of ideas that provided the roots for modern decision theory, Bayesian inference with nuisance parameters (historians claim that Laplace did not know of Bayes’s earlier work), and the asymptotic approximations of posterior distributions [Stigler, 1986]. By the time of Newton, physicists and mathematicians were formulating different theories of probability. The most popular ones remaining today are the relative frequency theory and the subjectivist or personalistic theory. The later development was initiated by Thomas Bayes (1763), who articulated his very powerful theorem for the assessment of subjective probabilities. The theorem specified that a human’s degree of belief could be subjected to an objective, coherent, and measurable mathematical framework within subjective probability theory. In the early days of the twentieth century Rescher developed a formal framework for a conditional probability theory. The twentieth century saw the first developments of alternatives to probability theory and to classical Aristotelian logic as paradigms to address more kinds of uncertainty than just the random kind. Jan Lukasiewicz developed a multivalued, discrete logic (circa 1930). In the 1960’s Arthur Dempster developed a theory of evidence which, for the first time, included an assessment of ignorance, or the absence of information. In 1965 Lotfi Zadeh introduced his seminal idea in a continuous-valued logic that he called fuzzy set theory. In the 1970s Glenn Shafer extended Dempster’s work to produce a complete theory of evidence dealing with information from more than one source, and Lotfi Zadeh illustrated a possibility theory resulting from special cases of fuzzy sets. Later in the 1980s other investigators showed a strong relationship between evidence theory, probability theory, and possibility theory with the use of what was called fuzzy measures [Klir and Wierman, 1996], and what is now being termed monotone measures. Uncertainty can be thought of in an epistemological sense as being the inverse of information. Information about a particular engineering or scientific problem may be AN HISTORICAL PERSPECTIVE 5 incomplete, imprecise, fragmentary, unreliable, vague, contradictory, or deficient in some other way [Klir and Yuan, 1995]. When we acquire more and more information about a problem, we become less and less uncertain about its formulation and solution. Problems that are characterized by very little information are said to be ill-posed, complex, or not sufficiently known. These problems are imbued with a high degree of uncertainty. Uncertainty can be manifested in many forms: it can be fuzzy (not sharp, unclear, imprecise, approximate), it can be vague (not specific, amorphous), it can be ambiguous (too many choices, contradictory), it can be of the form of ignorance (dissonant, not knowing something), or it can be a form due to natural variability (conflicting, random, chaotic, unpredictable). Many other linguistic labels have been applied to these various forms, but for now these shall suffice. Zadeh posed some simple examples of these forms in terms of a person’s statements about when they shall return to a current place in time. The statement ‘‘I shall return soon’’ is vague, whereas the statement ‘‘I shall return in a few minutes’’ is fuzzy; the former is not known to be associated with any unit of time (seconds, hours, days), and the latter is associated with an uncertainty that is at least known to be on the order of minutes. The phrase, ‘‘I shall return within 2 minutes of 6pm’’ involves an uncertainty which has a quantifiable imprecision; probability theory could address this form. Vagueness can be used to describe certain kinds of uncertainty associated with linguistic information or intuitive information. Examples of vague information are that the data quality is ‘‘good,’’ or that the transparency of an optical element is ‘‘acceptable.’’ Moreover, in terms of semantics, even the terms vague and fuzzy cannot be generally considered synonyms, as explained by Zadeh : ‘‘usually a vague proposition is fuzzy, but the converse is not generally true.’’ Discussions about vagueness started with a famous work by the philosopher Max Black. Black defined a vague proposition as a proposition where the possible states (of the proposition) are not clearly defined with regard to inclusion. For example, consider the proposition that a person is young. Since the term ‘‘young’’ has different interpretations to different individuals, we cannot decisively determine the age(s) at which an individual is young versus the age(s) at which an individual is not considered to be young. Thus, the proposition is vaguely defined. Classical (binary) logic does not hold under these circumstances, therefore we must establish a different method of interpretation. Max Black, in writing his 1937 essay ‘‘Vagueness: An exercise in logical analysis’’ first cites remarks made by the ancient philosopher Plato about uncertainty in geometry, then embellishes on the writings of Bertrand Russell (1923) who emphasized that ‘‘all traditional logic habitually assumes that precise symbols are being employed.’’ With these great thoughts as a prelude to his own arguments, he proceeded to produce his own, now-famous quote: It is a paradox, whose importance familiarity fails to diminish, that the most highly developed and useful scientific theories are ostensibly expressed in terms of objects never encountered in experience. The line traced by a draftsman, no matter how accurate, is seen beneath the microscope as a kind of corrugated trench, far removed from the ideal line of pure geometry. And the ‘‘point-planet’’ of astronomy, the ‘‘perfect gas’’ of thermodynamics, or the ‘‘pure- species’’ of genetics are equally remote from exact realization. Indeed the unintelligibility at the atomic or subatomic level of the notion of a rigidly demarcated boundary shows that such objects not merely are not but could not be encountered. While the mathematician constructs a theory in terms of ‘‘perfect’’ objects, the experimental scientist observes objects of which 6 INTRODUCTION the properties demanded by theory are and can, in the very nature of measurement, be only approximately true. More recently, in support of Black’s work, Quine states: Diminish a table, conceptually, molecule by molecule: when is a table not a table? No stipulations will avail us here, however arbitrary. If the term ‘table’ is to be reconciled with bivalence, we must posit an exact demarcation, exact to the last molecule, even though we cannot specify it. We must hold that there are physical objects, coincident except for one molecule, such that one is a table and the other is not. Bruno de Finetti , publishing in his landmark book Theory of Probability, gets his readers’ attention quickly by proclaiming, ‘‘Probability does not exist; it is a subjective description of a person’s uncertainty. We should be normative about uncertainty and not descriptive.’’ He further emphasizes that the frequentist view of probability (objectivist view) ‘‘requires individual trials to be equally probable and stochastically independent.’’ In discussing the difference between possibility and probability he states, ‘‘The logic of certainty furnishes us with the range of possibility (and the possible has no gradations); probability is an additional notion that one applies within the range of possibility, thus giving rise to graduations (‘more or less’ probable) that are meaningless in the logic of uncertainty.’’ In his book, de Finetti gives us warnings: ‘‘The calculus of probability can say absolutely nothing about reality,’’ and in referring to the dangers implicit in attempts to confuse certainty with high probability, he states We have to stress this point because these attempts assume many forms and are always dangerous. In one sentence: to make a mistake of this kind leaves one inevitably faced with all sorts of fallacious arguments and contradictions whenever an attempt is made to state, on the basis of probabilistic considerations, that something must occur, or that its occurrence confirms or disproves some probabilistic assumptions. In a discussion about the use of such vague terms as ‘‘very probable’’ or ‘‘practically certain,’’ or ‘‘almost impossible,’’ de Finetti states: The field of probability and statistics is then transformed into a Tower of Babel, in which only the most naive amateur claims to understand what he says and hears, and this because, in a language devoid of convention, the fundamental distinctions between what is certain and what is not, and between what is impossible and what is not, are abolished. Certainty and impossibility then become confused with high or low degrees of a subjective probability, which is itself denied precisely by this falsification of the language. On the contrary, the preservation of a clear, terse distinction between certainty and uncertainty, impossibility and possibility, is the unique and essential precondition for making meaningful statements (which could be either right or wrong), whereas the alternative transforms every sentence into a nonsense. THE UTILITY OF FUZZY SYSTEMS Several sources have shown and proven that fuzzy systems are universal approximators [Kosko, 1994; Ying et al., 1999]. These proofs stem from the isomorphism between two algebras: an abstract algebra (one dealing with groups, fields, and rings) and a linear algebra THE UTILITY OF FUZZY SYSTEMS 7 (one dealing with vector spaces, state vectors, and transition matrices) and the structure of a fuzzy system, which is comprised of an implication between actions and conclusions (antecedents and consequents). The reason for this isomorphism is that both entities (algebra and fuzzy systems) involve a mapping between elements of two or more domains. Just as an algebraic function maps an input variable to an output variable, a fuzzy system maps an input group to an output group; in the latter these groups can be linguistic propositions or other forms of fuzzy information. The foundation on which fuzzy systems theory rests is a fundamental theorem from real analysis in algebra known as the Stone–Weierstrass theorem, first developed in the late nineteenth century by Weierstrass , then simplified by Stone. In the coming years it will be the consequence of this isomorphism that will make fuzzy systems more and more popular as solution schemes, and it will make fuzzy systems theory a routine offering in the classroom as opposed to its previous status as a ‘‘new, but curious technology.’’ Fuzzy systems, or whatever label scientists eventually come to call it in the future, will be a standard course in any science or engineering curriculum. It contains all of what algebra has to offer, plus more, because it can handle all kinds of information not just numerical quantities. More on this similarity between abstract or linear algebras and fuzzy systems is discussed in Chapter 9 on rule-reduction methods. While fuzzy systems are shown to be universal approximators to algebraic functions, it is not this attribute that actually makes them valuable to us in understanding new or evolving problems. Rather, the primary benefit of fuzzy systems theory is to approximate system behavior where analytic functions or numerical relations do not exist. Hence, fuzzy systems have high potential to understand the very systems that are devoid of analytic formulations: complex systems. Complex systems can be new systems that have not been tested, they can be systems involved with the human condition such as biological or medical systems, or they can be social, economic, or political systems, where the vast arrays of inputs and outputs could not all possibly be captured analytically or controlled in any conventional sense. Moreover, the relationship between the causes and effects of these systems is generally not understood, but often can be observed. Alternatively, fuzzy systems theory can have utility in assessing some of our more conventional, less complex systems. For example, for some problems exact solutions are not always necessary. An approximate, but fast, solution can be useful in making preliminary design decisions, or as an initial estimate in a more accurate numerical technique to save computational costs, or in the myriad of situations where the inputs to a problem are vague, ambiguous, or not known at all. For example, suppose we need a controller to bring an aircraft out of a vertical dive. Conventional controllers cannot handle this scenario as they are restricted to linear ranges of variables; a dive situation is highly nonlinear. In this case, we could use a fuzzy controller, which is adept at handling nonlinear situations albeit in an imprecise fashion, to bring the plane out of the dive into a more linear range, then hand off the control of the aircraft to a conventional, linear, highly accurate controller. Examples of other situations where exact solutions are not warranted abound in our daily lives. For example, in the following quote from a popular science fiction movie, C-3PO: Sir, the possibility of successfully navigating an asteroid field is approximately 3,720 to 1! Han Solo: Never tell me the odds! Characters in the movie Star Wars: The Empire Strikes Back (Episode V), 1980 8 INTRODUCTION we have an illustration of where the input information (the odds of navigating through an asteroid field) is useless, so how does one make a decision in the presence of this information? Hence, fuzzy systems are very useful in two general contexts: (1) in situations involving highly complex systems whose behaviors are not well understood, and (2) in situations where an approximate, but fast, solution is warranted. As pointed out by Ben-Haim , there is a distinction between models of systems and models of uncertainty. A fuzzy system can be thought of as an aggregation of both because it attempts to understand a system for which no model exists, and it does so with information that can be uncertain in a sense of being vague, or fuzzy, or imprecise, or altogether lacking. Systems whose behaviors are both understood and controllable are of the kind which exhibit a certain robustness to spurious changes. In this sense, robust systems are ones whose output (such as a decision system) does not change significantly under the influence of changes in the inputs, because the system has been designed to operate within some window of uncertain conditions. It is maintained that fuzzy systems too are robust. They are robust because the uncertainties contained in both the inputs and outputs of the system are used in formulating the system structure itself, unlike conventional systems analysis which first poses a model, based on a collective set of assumptions needed to formulate a mathematical form, then uncertainties in each of the parameters of that mathematical abstraction are considered. The positing of a mathematical form for our system can be our first mistake, and any subsequent uncertainty analysis of this mathematical abstraction could be misleading. We call this the Optimist’s dilemma: find out how a chicken clucks, by first ‘‘assuming a spherical chicken.’’ Once the sphericity of the chicken has been assumed, there are all kinds of elegant solutions that can be found; we can predict any number of sophisticated clucking sounds with our model. Unfortunately when we monitor a real chicken it does not cluck the way we predict. The point being made here is that there are few physical and no mathematical abstractions that can be made to solve some of our complex problems, so we need new tools to deal with complexity; fuzzy systems and their associated developments can be one of these newer tools. LIMITATIONS OF FUZZY SYSTEMS However, this is not to suggest that we can now stop looking for additional tools. Realistically, even fuzzy systems, as they are posed now, can be described as shallow models in the sense that they are primarily used in deductive reasoning. This is the kind of reasoning where we infer the specific from the general. For example, in the game of tic-tac-toe there are only a few moves for the entire game; we can deduce our next move from the previous move, and our knowledge of the game. It is this kind of reasoning that we also called shallow reasoning, since our knowledge, as expressed linguistically, is of a shallow and meager kind. In contrast to this is the kind of reasoning that is inductive, where we infer the general from the particular; this method of inference is called deep, because our knowledge is of a deep and substantial kind – a game of chess would be closer to an inductive kind of model. We should understand the distinction between using mathematical models to account for observed data, and using mathematical models to describe the underlying process by LIMITATIONS OF FUZZY SYSTEMS 9 which the observed data are generated or produced by nature [Arciszewski et al., 2003]. Models of systems where the behavior can be observed, and whose predictions can only account for these observed data, are said to be shallow, as they do not account for the underlying realities. Deep models, those of the inductive kind, are alleged to capture the physical process by which nature has produced the results we have observed. In his Republic (360 BC), Plato suggests the idea that things that are perceived are only imperfect copies of the true reality that can only be comprehended by pure thought. Plato was fond of mathematics, and he saw in its very precise structure of logic idealized abstraction and separation from the material world. He thought of these things being so important, that above the doorway to his Academy was placed the inscription ‘‘Let no one ignorant of mathematics enter here.’’ In Plato’s doctrine of forms, he argued that the phenomenal world was a mere shadowy image of the eternal, immutable real world, and that matter was docile and disorderly, governed by a Mind that was the source of coherence, harmony, and orderliness. He argued that if man was occupied with the things of the senses, then he could never gain true knowledge. In his work the Phaedo he declares that as mere mortals we cannot expect to attain absolute truth about the universe, but instead must be content with developing a descriptive picture – a model [Barrow, 2000]. Centuries later, Galileo was advised by his inquisitors that he must not say that his mathematical models were describing the realities of nature, but rather that they simply were adequate models of the observations he made with his telescope [Drake, 1957]; hence, that they were solely deductive. In this regard, models that only attempt to replicate some phenomenological behavior are considered shallow models, or models of the deductive kind, and they lack the knowledge needed for true understanding of a physical process. The system that emerges under inductive reasoning will have connections with both evolution and complexity. How do humans reason in situations that are complicated or ill-defined? Modern psychology tells us that as humans we are only moderately good at deductive logic, and we make only moderate use of it. But we are superb at seeing or recognizing or matching patterns – behaviors that confer obvious evolutionary benefits. In problems of complication then, we look for patterns; and we simplify the problem by using these to construct temporary internal models or hypotheses or schemata to work with [Bower and Hilgard, 1981]. We carry out localized deductions based on our current hypotheses and we act on these deductions. Then, as feedback from the environment comes in, we may strengthen or weaken our beliefs in our current hypotheses, discarding some when they cease to perform, and replacing them as needed with new ones. In other words, where we cannot fully reason or lack full definition of the problem, we use simple models to fill the gaps in our understanding; such behavior is inductive. Some sophisticated models may, in fact, be a complex weave of deductive and inductive steps. But, even our so-called ‘‘deep models’’ may not be deep enough. An illustration of this comes from a recent popular decision problem, articulated as the El Farol problem by W. Brian Arthur. This problem involves a decision-making scenario in which inductive reasoning is assumed and modeled, and its implications are examined. El Farol is a bar in Santa Fe, New Mexico, where on one night of the week in particular there is popular Irish music offered. Suppose N bar patrons decide independently each week whether to go to El Farol on this certain night. For simplicity, we set N = 100. Space in the bar is limited, and the evening is enjoyable if things are not too crowded – specifically, if fewer than 60% of the possible 100 are present. There is no way to tell the number coming for sure in advance, therefore a bar patron goes – deems it worth going – if he expects fewer 10 INTRODUCTION than 60 to show up, or stays home if he expects more than 60 to go; there is no need that utilities differ much above and below 60. Choices are unaffected by previous visits; there is no collusion or prior communication among the bar patrons; and the only information available is the numbers who came in past weeks. Of interest is the dynamics of the number of bar patrons attending from week to week. There are two interesting features of this problem. First, if there were an obvious model that all bar patrons could use to forecast attendance and on which to base their decisions, then a deductive solution would be possible. But no such model exists in this case. Given the numbers attending in the recent past, a large number of expectational models might be reasonable and defensible. Thus, not knowing which model other patrons might choose, a reference patron cannot choose his in a well-defined way. There is no deductively rational solution – no ‘‘correct’’ expectational model. From the patrons’ viewpoint, the problem is ill-defined and they are propelled into a realm of induction. Second, any commonality of expectations gets disintegrated: if everyone believes few will go, then all will go. But this would invalidate that belief. Similarly, if all believe most will go, nobody will go, invalidating that belief. Expectations will be forced to differ, but not in a methodical, predictive way. Scientists have long been uneasy with the assumption of perfect, deductive rationality in decision contexts that are complicated and potentially ill-defined. The level at which humans can apply perfect rationality is surprisingly modest. Yet it has not been clear how to deal with imperfect or bounded rationality. From the inductive example given above (El Farol problem), it would be easy to suggest that as humans in these contexts we use inductive reasoning: we induce a variety of working hypotheses, act upon the most credible, and replace hypotheses with new ones if they cease to work. Such reasoning can be modeled in a variety of ways. Usually this leads to a rich psychological world in which peoples’ ideas or mental models compete for survival against other peoples’ ideas or mental models – a world that is both evolutionary and complex. And, while this seems the best course of action for modeling complex questions and problems, this text stops short of that longer term goal with only a presentation of simple deductive models, of the rule-based kind, that are introduced and illustrated in Chapters 5–8. THE ALLUSION: STATISTICS AND RANDOM PROCESSES The uninitiated often claim that fuzzy set theory is just another form of probability theory in disguise. This statement, of course, is simply not true (Appendix A formally rejects this claim with an axiomatic discussion of both probability theory and fuzzy logic). Basic statistical analysis is founded on probability theory or stationary random processes, whereas most experimental results contain both random (typically noise) and nonrandom processes. One class of random processes, stationary random processes, exhibits the following three characteristics: (1) The sample space on which the processes are defined cannot change from one experiment to another; that is, the outcome space cannot change. (2) The frequency of occurrence, or probability, of an event within that sample space is constant and cannot change from trial to trial or experiment to experiment. (3) The outcomes must be repeatable from experiment to experiment. The outcome of one trial does not influence the outcome of a previous or future trial. There are more general classes of random processes than the class mentioned here. However, fuzzy sets are not governed by these characteristics. THE ALLUSION: STATISTICS AND RANDOM PROCESSES 11 Stationary random processes are those that arise out of chance, where the chances represent frequencies of occurrence that can be measured. Problems like picking colored balls out of an urn, coin and dice tossing, and many card games are good examples of stationary random processes. How many of the decisions that humans must make every day could be categorized as random? How about the uncertainty in the weather – is this random? How about your uncertainty in choosing clothes for the next day, or which car to buy, or your preference in colors – are these random uncertainties? How about your ability to park a car; is this a random process? How about the risk in whether a substance consumed by an individual now will cause cancer in that individual 15 years from now; is this a form of random uncertainty? Although it is possible to model all of these forms of uncertainty with various classes of random processes, the solutions may not be reliable. Treatment of these forms of uncertainty using fuzzy set theory should also be done with caution. One needs to study the character of the uncertainty, then choose an appropriate approach to develop a model of the process. Features of a problem that vary in time and space should be considered. For example, when the weather report suggests that there is a 60% chance of rain tomorrow, does this mean that there has been rain on tomorrow’s date for 60 of the last 100 years? Does it mean that somewhere in your community 60% of the land area will receive rain? Does it mean that 60% of the time it will be raining and 40% of the time it will not be raining? Humans often deal with these forms of uncertainty linguistically, such as, ‘‘It will likely rain tomorrow.’’ And with this crude assessment of the possibility of rain, humans can still make appropriately accurate decisions about the weather. Random errors will generally average out over time, or space. Nonrandom errors, such as some unknown form of bias (often called a systematic error) in an experiment, will not generally average out and will likely grow larger with time. The systematic errors generally arise from causes about which we are ignorant, for which we lack information, or that we cannot control. Distinguishing between random and nonrandom errors is a difficult problem in many situations, and to quantify this distinction often results in the illusion that the analyst knows the extent and character of each type of error. In all likelihood nonrandom errors can increase without bounds. Moreover, variability of the random kind cannot be reduced with additional information, although it can be quantified. By contrast, nonrandom uncertainty, which too can be quantified with various theories, can be reduced with the acquisition of additional information. It is historically interesting that the word statistics is derived from the now obsolete term statist, which means an expert in statesmanship. Statistics were the numerical facts that statists used to describe the operations of states. To many people, statistics, and other recent methods to represent uncertainty like evidence theory and fuzzy set theory, are still the facts by which politicians, newspapers, insurance sellers, and other broker occupations approach us as potential customers for their services or products! The air of sophistication that these methods provide to an issue should not be the basis for making a decision; it should be made only after a good balance has been achieved between the information content in a problem and the proper representation tool to assess it. Popular lore suggests that the various uncertainty theories allow engineers to fool themselves in a highly sophisticated way when looking at relatively incoherent heaps of data (computational or experimental), as if this form of deception is any more palatable than just plain ignorance. All too often, scientists and engineers are led to use these theories as a crutch to explain vagaries in their models or in their data. For example, in probability applications the assumption of independent random variables is often assumed 12 INTRODUCTION to provide a simpler method to prescribe joint probability distribution functions. An analogous assumption, called noninteractive sets, is used in fuzzy applications to develop joint membership functions from individual membership functions for sets from different universes of discourse. Should one ignore apparently aberrant information, or consider all information in the model whether or not it conforms to the engineers’ preconceptions? Additional experiments to increase understanding cost money, and yet, they might increase the uncertainty by revealing conflicting information. It could best be said that statistics alone, or fuzzy sets alone, or evidence theory alone, are individually insufficient to explain many of the imponderables that people face every day. Collectively they could be very powerful. A poem by J. V. Cunningham titled ‘‘Meditation on Statistical Method’’ provides a good lesson in caution for any technologist pondering the thought that ignoring uncertainty (again, using statistics because of the era of the poem) in a problem will somehow make its solution seem more certain. Plato despair! We prove by norms How numbers bear Empiric forms, How random wrongs Will average right If time be long And error slight; But in our hearts Hyperbole Curves and departs To infinity. Error is boundless. Nor hope nor doubt, Though both be groundless, Will average out. UNCERTAINTY AND INFORMATION Only a small portion of the knowledge (information) for a typical problem might be regarded as certain, or deterministic. Unfortunately, the vast majority of the material taught in engineering classes is based on the presumption that the knowledge involved is deterministic. Most processes are neatly and surreptitiously reduced to closed-form algorithms – equations and formulas. When students graduate, it seems that their biggest fear upon entering the real world is ‘‘forgetting the correct formula.’’ These formulas typically describe a deterministic process, one where there is no uncertainty in the physics of the process (i.e., the right formula) and there is no uncertainty in the parameters of the process (i.e., the coefficients are known with impunity). It is only after we leave the university, it seems, that we realize we were duped in academe, and that the information FUZZY SETS AND MEMBERSHIP 13 we have for a particular problem virtually always contains uncertainty. For how many of our problems can we say that the information content is known absolutely, i.e., with no ignorance, no vagueness, no imprecision, no element of chance? Uncertain information can take on many different forms. There is uncertainty that arises because of complexity; for example, the complexity in the reliability network of a nuclear reactor. There is uncertainty that arises from ignorance, from various classes of randomness, from the inability to perform adequate measurements, from lack of knowledge, or from vagueness, like the fuzziness inherent in our natural language. The nature of uncertainty in a problem is a very important point that engineers should ponder prior to their selection of an appropriate method to express the uncertainty. Fuzzy sets provide a mathematical way to represent vagueness and fuzziness in humanistic systems. For example, suppose you are teaching your child to bake cookies and you want to give instructions about when to take the cookies out of the oven. You could say to take them out when the temperature inside the cookie dough reaches 375◦ F, or you could advise your child to take them out when the tops of the cookies turn light brown. Which instruction would you give? Most likely, you would use the second of the two instructions. The first instruction is too precise to implement practically; in this case precision is not useful. The vague term light brown is useful in this context and can be acted upon even by a child. We all use vague terms, imprecise information, and other fuzzy data just as easily as we deal with situations governed by chance, where probability techniques are warranted and very useful. Hence, our sophisticated computational methods should be able to represent and manipulate a variety of uncertainties. Other representations of uncertainties due to ambiguity, nonspecificity, beliefs, and ignorance are introduced in Chapter 15. FUZZY SETS AND MEMBERSHIP The foregoing sections discuss the various elements of uncertainty. Making decisions about processes that contain nonrandom uncertainty, such as the uncertainty in natural language, has been shown to be less than perfect. The idea proposed by Lotfi Zadeh suggested that set membership is the key to decision making when faced with uncertainty. In fact, Zadeh made the following statement in his seminal paper of 1965: The notion of a fuzzy set provides a convenient point of departure for the construction of a conceptual framework which parallels in many respects the framework used in the case of ordinary sets, but is more general than the latter and, potentially, may prove to have a much wider scope of applicability, particularly in the fields of pattern classification and information processing. Essentially, such a framework provides a natural way of dealing with problems in which the source of imprecision is the absence of sharply defined criteria of class membership rather than the presence of random variables. As an example, we can easily assess whether someone is over 6 feet tall. In a binary sense, the person either is or is not, based on the accuracy, or imprecision, of our measuring device. For example, if ‘‘tall’’ is a set defined as heights equal to or greater than 6 feet, a computer would not recognize an individual of height 5 11.999 as being a member of the set ‘‘tall.’’ But how do we assess the uncertainty in the following question: Is the person nearly 6 feet tall? The uncertainty in this case is due to the vagueness or ambiguity of the adjective nearly. A 5 11 person could clearly be a member of the set of ‘‘nearly 6 feet tall’’ 14 INTRODUCTION people. In the first situation, the uncertainty of whether a person, whose height is unknown, is 6 feet or not is binary; the person either is or is not, and we can produce a probability assessment of that prospect based on height data from many people. But the uncertainty of whether a person is nearly 6 feet is nonrandom. The degree to which the person approaches a height of 6 feet is fuzzy. In reality, ‘‘tallness’’ is a matter of degree and is relative. Among peoples of the Tutsi tribe in Rwanda and Burundi a height for a male of 6 feet is considered short. So, 6 feet can be tall in one context and short in another. In the real (fuzzy) world, the set of tall people can overlap with the set of not-tall people, an impossibility when one follows the precepts of classical binary logic (this is discussed in Chapter 5). This notion of set membership, then, is central to the representation of objects within a universe by sets defined on the universe. Classical sets contain objects that satisfy precise properties of membership; fuzzy sets contain objects that satisfy imprecise properties of membership, i.e., membership of an object in a fuzzy set can be approximate. For example, the set of heights from 5 to 7 feet is precise (crisp); the set of heights in the region around 6 feet is imprecise, or fuzzy. To elaborate, suppose we have an exhaustive collection of individual elements (singletons) x , which make up a universe of information (discourse), X. Further, various combinations of these individual elements make up sets, say A, on the universe. For crisp sets an element x in the universe X is either a member of some crisp set A or not. This binary issue of membership can be represented mathematically with the indicator function, 1, x∈A χA (x) = (1.1) 0, x ∈ A where the symbol χA (x) gives the indication of an unambiguous membership of element x in set A, and the symbols ∈ and ∈ denote contained in and not contained in, respectively. For our example of the universe of heights of people, suppose set A is the crisp set of all people with 5.0 ≤ x ≤ 7.0 feet, shown in Fig. 1.1a. A particular individual, x1 , has a height of 6.0 feet. The membership of this individual in crisp set A is equal to 1, or full membership, given symbolically as χA (x1 ) = 1. Another individual, say, x2 , has a height of 4.99 feet. The membership of this individual in set A is equal to 0, or no membership, hence χA (x2 ) = 0, also seen in Fig. 1.1a. In these cases the membership in a set is binary, either an element is a member of a set or it is not. Zadeh extended the notion of binary membership to accommodate various ‘‘degrees of membership’’ on the real continuous interval [0, 1], where the endpoints of 0 and 1 A 5 6 7 χA µχH 1 1 0 5 6 7 x 0 5 6 7 x (a) (b) FIGURE 1.1 Height membership functions for (a) a crisp set A and (b) a fuzzy set H. CHANCE VERSUS FUZZINESS 15 conform to no membership and full membership, respectively, just as the indicator function does for crisp sets, but where the infinite number of values in between the endpoints can represent various degrees of membership for an element x in some set on the universe. The sets on the universe X that can accommodate ‘‘degrees of membership’’ were termed by Zadeh as ‘‘fuzzy sets.’’ Continuing further on the example on heights, consider a set H consisting of heights near 6 feet. Since the property near 6 feet is fuzzy, there is not a unique membership function for H. Rather, the analyst must decide what the membership function, denoted µH , should look like. Plausible properties of this function might be (1) normality (µH (6) = 1), (2) monotonicity (the closer H is to 6, the closer µH is to 1), and (3) symmetry (numbers equidistant from 6 should have the same value of µH ) [Bezdek, 1993]. Such a membership function is illustrated in Fig. 1.1b. A key difference between crisp and fuzzy sets is their membership function; a crisp set has a unique membership function, whereas a fuzzy set can have an infinite number of membership functions to represent it. For fuzzy sets, the uniqueness is sacrificed, but flexibility is gained because the membership function can be adjusted to maximize the utility for a particular application. James Bezdek provided one of the most lucid comparisons between crisp and fuzzy sets [Bezdek, 1993]. It bears repeating here. Crisp sets of real objects are equivalent to, and isomorphically de

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