The

Fuzzy SystemsH a n d b o o k

Second Edition

Te^hnische Universitat toinstmJNik AutomatisiaMngstechnlk

Fachgebi^KQegelup^stheorie und

D-S4283 Darrftstadt

lnvfentar-NgxC? V2^s

TU DarmstadtFB ETiT

05C

ContentsFigures xix

Code Listings xxxi

Foreword xxxiii

Acknowledgments xxxv

Preface xxxvii

Fuzzy Decision Systems: The Early Days xxxixNature of this Book xliAdapting and Using the C++ Code Library xliiThe Graphical Representation of Fuzzy Sets xlivContacting the Author xlvIcons and Topic Symbols xlvi

Notes xlviii

1. Introduction 1

Fuzzy System Models 2Logic, Complexity, and Comprehension 2The Idea of Fuzzy Sets 3Linguistic Variables 4Approximate Reasoning 6

Benefits of Fuzzy System Modeling 7The Ability to Model Highly Complex Business Problems 8Improved Cognitive Modeling of Expert Systems 8The Ability to Model Systems Involving Multiple Experts 9Reduced Model Complexity 10Improved Handling of Uncertainty and Possibilities 10

Common Objections to Fuzzy Logic 11What Can Fuzzy Logic Do? 12

Reasons to Reject Fuzzy System Solutions 13The Precise Organization 14Fuzzy Logic Is a Control Engineering Tool 14Complex Time-Series Modeling 17The Power of Conventional Expert Systems 17

vii

Contents

The Precision of Mathematical Models 18Fuzzy Model Stability 18Fuzzy Model Execution Speed 19Fuzzy Set Discovery and Correctness 22Tuning and Validating Fuzzy Systems 27The Somewhat Ad-Hoc Nature of Defuzzification 30The Problem of Combinatorial Explosion 32

Some Actual Fuzzy System Models 36Company Acquisition and Credit Analysis 36Credit Authorization 37Criminal Identification System 37Mainframe DASD Planning 38Expense Auditing 38Financial Statement Advisor 38Container Management System 38Intelligent Project Management ~ 39Integrated MRP and Production Scheduler 39Managed Health Care—Provider Fraud Detection 40Organizational Dynamics 40Loan Evaluation Advisor 41Portfolio Safety and Suitability Model 41Product Pricing Model 43Risk Underwriting 43Systems Complexity Analysis 43Notes 43

2. Fuzziness and Certainty 45

The Different Faces of Imprecision 45Inexactness 46Precision and Accuracy 48

Accuracy and Imprecision 48Measurement Imprecision and Intrinsic Imprecision 49

Ambiguity 49Semantic Ambiguity 49Visual Ambiguity 50Structural Ambiguity 51

Undecidability 52Vagueness 54

Fuzzy Logic and Interval Arithmetic 55

Contents ix

Fuzzy Logic and Probability 57What Is Probability? 57

Frequentist Probabilities 57Subjective Probabilities 58Mathematical Foundations (Briefly) 59

Confusion of Aims 59Confusion of Methods 60Likelihood and Ambiguity 60Fuzzy Probabilities 62

Bayes Theorem and Fuzzy Probability 63Fuzzy Logic 64Notes • 65

3. Fuzzy Sets 67

The Age of Science 68Imprecision in the Everyday World 70

Imprecise Concepts 70The Nature of Fuzziness 71Fuzziness and Imprecision 75Representing Imprecision wi th Fuzzy Sets 78

Fuzzy Sets 79Representing Fuzzy Sets in Software 81

Basic Properties and Characteristics of Fuzzy Sets 84Fuzzy Set Height and Normalization 84

Domains, Alpha-level Sets, and Support Sets 87The Fuzzy Set Domain 87The Universe of Discourse 89The Support Set 90Use of Psychometric Domains 90Fuzzy Alpha-Cut Thresholds 94Alpha Cuts, Transition Walls, and Control Voids 95

Encoding Information wi th Fuzzy Sets 99Expressing a Fuzzy Concept 100

Fuzzy Numbers 100Fuzzy Qualifiers 102

Generating Fuzzy Membership Functions 103Linear Representations 104

Contents

S-Curve (Sigmoid/Logistic) Representations 109S-Curves and Cumulative Distributions 111Proportional and Frequency Representations 113

Fuzzy Numbers and "Around" Representations 119Fuzzy Numbers 119Fuzzy Quantities and Counts 121PI Curves 123Beta Curves 127Gaussian Curves 133

Triangular, Trapezoidal, and Shouldered Fuzzy Sets 136Triangular Fuzzy Sets 138Shouldered Fuzzy Sets 140

Irregularly Shaped and Arbitrary Fuzzy Sets 149Truth Series Descriptions 154Domain-Based Coordinate Memberships 159Notes 165

4. Fuzzy Set Operators 167

Conventional (Crisp) Set Operations 167Basic Zadeh-Type Operations on Fuzzy Sets 168

Fuzzy Set Membership and Elements 168The Intersection of Fuzzy Sets 172The Union of Fuzzy Sets 178The Complement (Negation) of Fuzzy Sets 182

Counterintuitives and the Law of Noncontradiction 186Non-Zadeh and Compensatory Fuzzy Set Operations 191General Algebraic Operations 194

The Mean and Weighted Mean Operators 194The Product Operator 198

The Heap Metaphor 199The Bounded Difference and Sum Operators 201

Functional Compensatory Classes 202The Yager Compensatory Operators 203

The Yager AND Operator 204The Yager OR Operator 205The Yager NOT Operator 209

Contents xi

The Sugeno Class and Other Alternative NOT Operators 211Threshold NOT Operator 212The Cosine NOT Function 212

Notes 216

5. Fuzzy Set Hedges 217

Hedges and Fuzzy Surface Transformers 217The Meaning and Interpretation of Hedges 218Importance of Hedges in Fuzzy Modeling 219

Dynamically Created Fuzzy Sets 219Reducing Rule Complexity 221

Applying Hedges 222Predicate and Consequent Hedges 223

Fuzzy Region Approximation ? 223Restricting a Fuzzy Region 227Intensifying and Diluting Fuzzy Regions 230

The Very Hedge 231The Somewhat Hedge 239Reciprocal Nature of Very and Somewhat 245

Contrast Intensification and Diffusion 246The Positively Hedge 246The Generally Hedge 249Approximating a Scalar 260Examples of Typical Hedge Operations 263Notes , 268

6. Fuzzy Reasoning 269

The Role of Linguistic Variables 271Fuzzy Propositions 273

Conditional Fuzzy Propositions 274Unconditional Fuzzy Propositions 275The Order of Proposition Execution 275

Monotonic (Proportional) Reasoning 275Monotonic Reasoning with Complex Predicates 282

xii Contents

The Fuzzy Compositional Rules of Inference 284The Min-Max Rules of Implication 284The Fuzzy Additive Rules of Implication 285

Accumulating Evidence with the Fuzzy Additive Method 286Fuzzy Implication Example 289Correlation Methods 293

Correlation Minimum 293Correlation Product 295

The Minimum Law of Fuzzy Assertions 297Methods of Decomposition and Defuzzification 303

Composite Moments (Centroid) 307Composite Maximum (Maximum Height) 309Hyperspace Decomposition Comparisons 310Preponderance of Evidence Technique 310Other Defuzzification Techniques 314

The Average of Maximum Values ^ 315The Average of the Support Set 315The Far and Near Edges of the Support Set 316The Center of Maximums 317

Singleton Geometry Representations 324Notes , 328

7. Fuzzy Models 329

The Basic Fuzzy System 329The Fuzzy Model Overview 330The Model Code View 332Code Representation of Fuzzy Variables 333Incorporating Hedges in the Fuzzy Model 336Representing and Executing Rules in Code 337Setting Alpha-Cut Thresholds 339Including a Model Explanatory Facility 340

The Advanced Fuzzy Modeling Environment 345The Policy Concept 345Understanding Hash Tables and Dictionaries 346Creating a Model and Associated Policies 353Managing Policy Dictionaries 357Loading Default Hedges 358

Contents xiii

Fundamental Model Design Issues 360Integrating Application Code with the Modeling System 361Tasks at the Module Main Program Level 362

Connecting the Model to the System Control Blocks 362Allocating and Installing the Policy Structure 363Defining Solution (Output) Variables 363Creating and Storing Fuzzy Sets in Application Code 364Creating and Storing Fuzzy Sets in a Policy's Dictionary 365Loading and Creating Hedges 366

Segmenting Application Code into Modules 369Maintaining Addressability to the Model 369Establishing the Policy Environment 369Initializing the Fuzzy Logic Work Area for the Policy 370Locating the Necessary Fuzzy Sets and Hedges 371

Exploring a Simple Fuzzy System Model 372Exploring a More Extensive Pricing Policy 384The Interpretation of Model Results 395

Undecidable Models 396Compatibility Index Metrics 399

The Idea of a Compatibility Index 399The Unit Compatibility Index 400Scaling Expected Values by the Compatibility Index 409The Statistical Compatibility Index • 410Selecting Height Measurements 414Measuring Variability in the Model 414

Notes 415

8. Fuzzy Systems: Case Studies 417

A Fuzzy Steam Turbine Controller 418The Fuzzy Control Model 418

The Fuzzy Logic Controller 418The Conventional PID Controller 419

The Steam Turbine Plant Process 420Designing the Fuzzy Logic Controller 422Running the Steam Turbine FLC Logic 424

xiv Contents

The New Product Pricing Model (Version 1) 428Model Design and Objectives 429The Model Execution Logic 430

Create the Basic Price Fuzzy Sets 431Create the Run-Time Model Fuzzy Sets 431Execute the Price Estimation Rules - 432Defuzzify to Find Expected Value for Price 438

Evaluating Defuzzification Strategies 438The New Product Pricing Model (Version 2) 452

Model Design Strategies 452The Model Execution Logic 453

Create the Basic Fuzzy Sets 453Create the Run-Time Model Fuzzy Sets 454Execute the Price Estimation Rules ^ 456

The New Product Pricing Model (Version 3) 462The Model Execution Logic 462

Execute the Price Estimation Rules 462Defuzzify to Find Expected Value for Price 468

The New Product Pricing Model (P&L Version) 468Design for the P&L Model 469Model Execution and Logic 470Using Policies to Calculate Price and Sales Volume 473

A Project Risk Assessment Model 475The Model Design 475Model Application Issues 476Model Execution Logic 479Executing the Risk Assessment Rules 481Notes 486

9. Building Fuzzy Systems:

A System Evaluation and Design Methodology 489

Evaluating Fuzzy System Projects 489The Ideal Fuzzy System Problem 490Fuzzy Model Characteristics 490

Fuzzy Control Parameters 490Multiple Experts 493Elastic Relationships Among Continuous Variables 494Complex, Poorly Understood, or Nonlinear Problems 494Uncertainties, Probabilities, and Possibilities in Data 495

Contents xv

Fuzzy Set and Data Representational Issues 496Variable and Parameter Decomposition 497

Semantic Decomposition of Profit 497Fuzzy Set Naming Conventions 501

The Meaning and Degree of Fuzzy Set Overlap 503Control Engineering Perspectives on Overlap and Composition 508Highly Overlapping Fuzzy Regions 511

Designing and Eliciting Fuzzy Sets 512Knowledge Engineering 512

A Knowledge Acquisition Methodology 513Voted-For Distributions 515Statistical Properties of the Data 517Psychometrics of Fuzzy Set Evolution 521Fuzzy set implications: cross-over point ^

and the voting of populations. 524Best Estimate of the Variable's Semantics (SWAG) 525Automatic Variable Decomposition 526

Boolean and Semi-Fuzzy Variables 529Using Boolean Filters 529Applying Explicit Degrees of Membership 530

Uncertain and Noisy Data 532Handling Uncertain and Noisy Data 536Inferencing with Fuzzy Data 537

Building Fuzzy System Models 539The Fuzzy Design Methodology 542

Define the Model's Functional and Operational Characteristics 542Define the System in Terms of an Input-Process-Output Model 543Localize the Model in the Production System 543Segment the Model into Functional and Operational Components 544Isolate the Critical Performance Variables 544Choose the Mode of Solution Variables 545Resolve Basic Performance Criteria 545

Decide on a Level of Granularity 545Determine Domain of the Model Variables 546Determine the Degree of Uncertainty in the Data 546Define the Limits of Operability 547Establish Metrics for Model Performance Requirements 547

xvi Contents

Define the Fuzzy Sets 547Determine the Type of Fuzzy Measurement 547Choose the Shape of the Fuzzy Set (Its Surface Morphology) 548Elicit a Fuzzy Set Shape 549Select an Appropriate Degree of Overlap 550Decide on the Space Correlation Metrics 550Ensure that the Sets are Conformally Mapped 550

Write the Rules 553Write the Ordinary Conditional Rules 554Enter Any Unconditional Rules 554Select Compensatory Operators for Special Rules 554Review the Rule Set and Add Any Hedges 555Add Any Alpha Cuts to Individual Rules - 555Enter the Rule Execution Weights 555

Define the Defuzzification Method for Each Solution Variable 556Notes 556

10. Using the Fuzzy Code Libraries 557

The Code and Interface Libraries 557General Software Issues 559

System and Client Error Diagnostics 559Software Status Codes 561Information and Warning Messages 561

Using Dynamic Link Library (DLL) Files 562The Visual Basic Module Definitions 563Using the DLL Names in Visual Basic 564

Modeling and Utility Software 566Symbolic Constants, Global Data, and Prototypes 566Data Structures 566Fuzzy Logic Functions 567

The Fuzzy System Modeling Functions 567Miscellaneous Tools and Utilities 571Demonstration and Fuzzy Model Programs 572

Description of Fuzzy Logic Functions 573FzyAboveAlfa 574FzyAddFZYctl 575FzyAND 577FzyApplyAlfa 578FzyApplyAND 580FzyApplyHedge 582FzyApplyNOT 585

Contents x v i i

FzyApplyOR 586FzyAutoScale 588FzyBetaCurve 588FzyCompAND 590FzyCompOR 591FzyCondProposition 593FzyCoordSeries 595FzyCopySet 597FzyCopyVector 598FzyCorrMinimum 599FzyCorrProduct 600FzyCreateHedge 601FzyCreateSet - 604FzyDefuzzify 611FzyDisplayFSV 616FzyDrawSet 617FzyExamineSet 619FzyFindFSV 622FzyFind Plateau 622FzyGetCoordinates 624FzyGetHeight 626FzyGetMembership 627FzyGetScalar 628FzylmplMatrix 629FzylnitCIX 632FzylnitFDB 632FzylnitFZYctl 633FzylnitHDB 634FzylnitVector 634FzylnterpVec 635FzylsNormal 636FzyLinearCurve 637FzyMemSeries 639FzyMonotonicLogic 641FzyNormalizeSet 642FzyOR 643FzyPiCurve 644FzyPlotSets 646FzySCurve 649FzyStatComplndex 651FzySupportSet 653FzyTrueSet 654FzyUnCondProposition 654FzyUnitComplndex 656

xviii Contents

A. Appendix—The Combs Method for Rapid Inference 659

The Combinatorial Problem: Fuzzy Logic's Achilles' Heel 659How Does the URC Affect the Multiplication of Rules? 665How Does the URC Work? 669Modeling Each Input's Relative Importance 672And Speaking of Tuning . . . 674Design Considerations 676An HMO Scheduling Program 677Conclusion 678Acknowledgments 678References 678Biography 680

Glossary 681

Bibliography 703

Index 707