The Fuzzy Systems Handbook Second Edition Te^hnische Universitat to instmJNik AutomatisiaMngstechnlk Fachgebi^KQegelup^stheorie und D-S4283 Darrftstadt lnvfentar-NgxC? V2^s TU Darmstadt FB ETiT 05C Contents Figures xix Code Listings xxxi Foreword xxxiii Acknowledgments xxxv Preface xxxvii Fuzzy Decision Systems: The Early Days xxxix Nature of this Book xli Adapting and Using the C++ Code Library xlii The Graphical Representation of Fuzzy Sets xliv Contacting the Author xlv Icons and Topic Symbols xlvi Notes xlviii 1. Introduction 1 Fuzzy System Models 2 Logic, Complexity, and Comprehension 2 The Idea of Fuzzy Sets 3 Linguistic Variables 4 Approximate Reasoning 6 Benefits of Fuzzy System Modeling 7 The Ability to Model Highly Complex Business Problems 8 Improved Cognitive Modeling of Expert Systems 8 The Ability to Model Systems Involving Multiple Experts 9 Reduced Model Complexity 10 Improved Handling of Uncertainty and Possibilities 10 Common Objections to Fuzzy Logic 11 What Can Fuzzy Logic Do? 12 Reasons to Reject Fuzzy System Solutions 13 The Precise Organization 14 Fuzzy Logic Is a Control Engineering Tool 14 Complex Time-Series Modeling 17 The Power of Conventional Expert Systems 17 vii Contents The Precision of Mathematical Models 18 Fuzzy Model Stability 18 Fuzzy Model Execution Speed 19 Fuzzy Set Discovery and Correctness 22 Tuning and Validating Fuzzy Systems 27 The Somewhat Ad-Hoc Nature of Defuzzification 30 The Problem of Combinatorial Explosion 32 Some Actual Fuzzy System Models 36 Company Acquisition and Credit Analysis 36 Credit Authorization 37 Criminal Identification System 37 Mainframe DASD Planning 38 Expense Auditing 38 Financial Statement Advisor 38 Container Management System 38 Intelligent Project Management ~ 39 Integrated MRP and Production Scheduler 39 Managed Health Care—Provider Fraud Detection 40 Organizational Dynamics 40 Loan Evaluation Advisor 41 Portfolio Safety and Suitability Model 41 Product Pricing Model 43 Risk Underwriting 43 Systems Complexity Analysis 43 Notes 43 2. Fuzziness and Certainty 45 The Different Faces of Imprecision 45 Inexactness 46 Precision and Accuracy 48 Accuracy and Imprecision 48 Measurement Imprecision and Intrinsic Imprecision 49 Ambiguity 49 Semantic Ambiguity 49 Visual Ambiguity 50 Structural Ambiguity 51 Undecidability 52 Vagueness 54 Fuzzy Logic and Interval Arithmetic 55 Contents ix Fuzzy Logic and Probability 57 What Is Probability? 57 Frequentist Probabilities 57 Subjective Probabilities 58 Mathematical Foundations (Briefly) 59 Confusion of Aims 59 Confusion of Methods 60 Likelihood and Ambiguity 60 Fuzzy Probabilities 62 Bayes Theorem and Fuzzy Probability 63 Fuzzy Logic 64 Notes • 65 3. Fuzzy Sets 67 The Age of Science 68 Imprecision in the Everyday World 70 Imprecise Concepts 70 The Nature of Fuzziness 71 Fuzziness and Imprecision 75 Representing Imprecision with Fuzzy Sets 78 Fuzzy Sets 79 Representing Fuzzy Sets in Software 81 Basic Properties and Characteristics of Fuzzy Sets 84 Fuzzy Set Height and Normalization 84 Domains, Alpha-level Sets, and Support Sets 87 The Fuzzy Set Domain 87 The Universe of Discourse 89 The Support Set 90 Use of Psychometric Domains 90 Fuzzy Alpha-Cut Thresholds 94 Alpha Cuts, Transition Walls, and Control Voids 95 Encoding Information with Fuzzy Sets 99 Expressing a Fuzzy Concept 100 Fuzzy Numbers 100 Fuzzy Qualifiers 102 Generating Fuzzy Membership Functions 103 Linear Representations 104 Contents S-Curve (Sigmoid/Logistic) Representations 109 S-Curves and Cumulative Distributions 111 Proportional and Frequency Representations 113 Fuzzy Numbers and "Around" Representations 119 Fuzzy Numbers 119 Fuzzy Quantities and Counts 121 PI Curves 123 Beta Curves 127 Gaussian Curves 133 Triangular, Trapezoidal, and Shouldered Fuzzy Sets 136 Triangular Fuzzy Sets 138 Shouldered Fuzzy Sets 140 Irregularly Shaped and Arbitrary Fuzzy Sets 149 Truth Series Descriptions 154 Domain-Based Coordinate Memberships 159 Notes 165 4. Fuzzy Set Operators 167 Conventional (Crisp) Set Operations 167 Basic Zadeh-Type Operations on Fuzzy Sets 168 Fuzzy Set Membership and Elements 168 The Intersection of Fuzzy Sets 172 The Union of Fuzzy Sets 178 The Complement (Negation) of Fuzzy Sets 182 Counterintuitives and the Law of Noncontradiction 186 Non-Zadeh and Compensatory Fuzzy Set Operations 191 General Algebraic Operations 194 The Mean and Weighted Mean Operators 194 The Product Operator 198 The Heap Metaphor 199 The Bounded Difference and Sum Operators 201 Functional Compensatory Classes 202 The Yager Compensatory Operators 203 The Yager AND Operator 204 The Yager OR Operator 205 The Yager NOT Operator 209 Contents xi The Sugeno Class and Other Alternative NOT Operators 211 Threshold NOT Operator 212 The Cosine NOT Function 212 Notes 216 5. Fuzzy Set Hedges 217 Hedges and Fuzzy Surface Transformers 217 The Meaning and Interpretation of Hedges 218 Importance of Hedges in Fuzzy Modeling 219 Dynamically Created Fuzzy Sets 219 Reducing Rule Complexity 221 Applying Hedges 222 Predicate and Consequent Hedges 223 Fuzzy Region Approximation ? 223 Restricting a Fuzzy Region 227 Intensifying and Diluting Fuzzy Regions 230 The Very Hedge 231 The Somewhat Hedge 239 Reciprocal Nature of Very and Somewhat 245 Contrast Intensification and Diffusion 246 The Positively Hedge 246 The Generally Hedge 249 Approximating a Scalar 260 Examples of Typical Hedge Operations 263 Notes , 268 6. Fuzzy Reasoning 269 The Role of Linguistic Variables 271 Fuzzy Propositions 273 Conditional Fuzzy Propositions 274 Unconditional Fuzzy Propositions 275 The Order of Proposition Execution 275 Monotonic (Proportional) Reasoning 275 Monotonic Reasoning with Complex Predicates 282 xii Contents The Fuzzy Compositional Rules of Inference 284 The Min-Max Rules of Implication 284 The Fuzzy Additive Rules of Implication 285 Accumulating Evidence with the Fuzzy Additive Method 286 Fuzzy Implication Example 289 Correlation Methods 293 Correlation Minimum 293 Correlation Product 295 The Minimum Law of Fuzzy Assertions 297 Methods of Decomposition and Defuzzification 303 Composite Moments (Centroid) 307 Composite Maximum (Maximum Height) 309 Hyperspace Decomposition Comparisons 310 Preponderance of Evidence Technique 310 Other Defuzzification Techniques 314 The Average of Maximum Values ^ 315 The Average of the Support Set 315 The Far and Near Edges of the Support Set 316 The Center of Maximums 317 Singleton Geometry Representations 324 Notes , 328 7. Fuzzy Models 329 The Basic Fuzzy System 329 The Fuzzy Model Overview 330 The Model Code View 332 Code Representation of Fuzzy Variables 333 Incorporating Hedges in the Fuzzy Model 336 Representing and Executing Rules in Code 337 Setting Alpha-Cut Thresholds 339 Including a Model Explanatory Facility 340 The Advanced Fuzzy Modeling Environment 345 The Policy Concept 345 Understanding Hash Tables and Dictionaries 346 Creating a Model and Associated Policies 353 Managing Policy Dictionaries 357 Loading Default Hedges 358 Contents xiii Fundamental Model Design Issues 360 Integrating Application Code with the Modeling System 361 Tasks at the Module Main Program Level 362 Connecting the Model to the System Control Blocks 362 Allocating and Installing the Policy Structure 363 Defining Solution (Output) Variables 363 Creating and Storing Fuzzy Sets in Application Code 364 Creating and Storing Fuzzy Sets in a Policy's Dictionary 365 Loading and Creating Hedges 366 Segmenting Application Code into Modules 369 Maintaining Addressability to the Model 369 Establishing the Policy Environment 369 Initializing the Fuzzy Logic Work Area for the Policy 370 Locating the Necessary Fuzzy Sets and Hedges 371 Exploring a Simple Fuzzy System Model 372 Exploring a More Extensive Pricing Policy 384 The Interpretation of Model Results 395 Undecidable Models 396 Compatibility Index Metrics 399 The Idea of a Compatibility Index 399 The Unit Compatibility Index 400 Scaling Expected Values by the Compatibility Index 409 The Statistical Compatibility Index • 410 Selecting Height Measurements 414 Measuring Variability in the Model 414 Notes 415 8. Fuzzy Systems: Case Studies 417 A Fuzzy Steam Turbine Controller 418 The Fuzzy Control Model 418 The Fuzzy Logic Controller 418 The Conventional PID Controller 419 The Steam Turbine Plant Process 420 Designing the Fuzzy Logic Controller 422 Running the Steam Turbine FLC Logic 424 xiv Contents The New Product Pricing Model (Version 1) 428 Model Design and Objectives 429 The Model Execution Logic 430 Create the Basic Price Fuzzy Sets 431 Create the Run-Time Model Fuzzy Sets 431 Execute the Price Estimation Rules - 432 Defuzzify to Find Expected Value for Price 438 Evaluating Defuzzification Strategies 438 The New Product Pricing Model (Version 2) 452 Model Design Strategies 452 The Model Execution Logic 453 Create the Basic Fuzzy Sets 453 Create the Run-Time Model Fuzzy Sets 454 Execute the Price Estimation Rules ^ 456 The New Product Pricing Model (Version 3) 462 The Model Execution Logic 462 Execute the Price Estimation Rules 462 Defuzzify to Find Expected Value for Price 468 The New Product Pricing Model (P&L Version) 468 Design for the P&L Model 469 Model Execution and Logic 470 Using Policies to Calculate Price and Sales Volume 473 A Project Risk Assessment Model 475 The Model Design 475 Model Application Issues 476 Model Execution Logic 479 Executing the Risk Assessment Rules 481 Notes 486 9. Building Fuzzy Systems: A System Evaluation and Design Methodology 489 Evaluating Fuzzy System Projects 489 The Ideal Fuzzy System Problem 490 Fuzzy