CONTENTS
Contents = ⅴ
List of Figures = xv
List of Code Listings = xxv
Foreword = xxvii
Acknowledgements = xxix
Preface = xxxi
1. Introduction = 1
Fuzzy System Models = 1
Logic, Complexity, and Comprehension = 1
The Idea of Fuzzy Sets = 2
Linguistic Variables = 3
Approximate Reasoning = 4
Benefits of Fuzzy System Modeling = 5
The Ability to Model Highly Complex Business Problems = 5
Improved Cognitive Modeling of Expert Systems = 6
The Ability to Model System Involving Multiple Experts = 7
Reduced Model Complexity = 7
Improved Handling of Uncertainty and Possibilities = 8
Notes = 8
2. Fuzziness and Certainty = 9
The Different Faces of Imprecision = 9
Inexactness = 10
Precision and Accuracy = 11
Accuracy and Imprecision = 12
Measurement Imprecision and Intrinsic Imprecision = 12
Ambiguity = 13
Semantic Ambiguity = 13
Visual Ambiguity = 14
Structural Ambiguity = 14
Undecidability = 15
Vagueness = 17
Probability, Possibility, and Fuzzy Logic = 18
What is Probabil = 18
Context Terms = 19
Fuzzy Logic = 19
Notes = 20
3. Fuzzy Sets = 21
Imprecision in the Everyday World = 21
Imprecise Concepts = 21
The Nature of Fuzziness = 23
Fuzziness and Imprecision = 27
Representing mprecision with Fuzzy Sets = 30
Fuzzy Sets Versus Crisp Sets = 30
Fuzzy Sets = 31
Representing Fuzzy Sets in Software = 33
Basic Properties and Characteristics of Fuzzy Sets = 35
Fuzzy Set Height and Normalization = 36
Domains, Alpha - level Sets, and Support Sets = 38
The Fuzzy Set Domain = 38
The Universe of Discourse = 40
The Support Set = 41
Fuzzy Alpha - Cut Thresholds = 41
Encoding Information with Fuzzy Sets = 44
Approximating a Fuzzy Concept = 45
Generating Fuzzy Membership Functions = 46
Linear Representations = 47
S - Curve (Sigmoid / Logistic) Representations = 51
S - Curves and Cumulative Distributions = 53
Proportional and Frequency Representations = 55
Fuzzy Numbers and "Around" Representations = 61
Fuzzy Numbers = 62
Fuzzy Quantities and Counts = 64
PI Curves = 65
Beta Curves = 69
Gaussian Curves = 75
Irregularly Shaped and Arbitrary Fuzzy Sets = 78
Truth Series Descriptions = 83
Domain - based Coordinate Memberships = 88
Triangular, Trapezoidal, and Shouldered Fuzzy Sets = 94
Triangular Fuzzy Sets = 95
Shouldered Fuzzy Sets = 97
Notes = 105
4. Fuzzy Set Operators = 107
Conventional (Crisp) Set Operations = 107
Basic Zadeh - Type Operations on Fuzzy Sets = 110
Fuzzy Set Membership and Elements = 111
The Intersection of Fuzzy Sets = 111
The Union of Fuzzy Sets = 118
The Complement (Negation) of Fuzzy Sets = 121
Counterlntuitives and the Law of Noncontradiction = 125
Non-Zadeh and Compensatory Fuzzy Set Operations = 130
General Algebraic Operations = 133
The Mean and Weighted Mean Operators = 134
The Product Operator = 138
The Heap Metaphor = 139
The Bounded Difference and Sum Operators = 141
Functional Compensatory Classes = 143
The Yager Compensatory Classes = 144
The Yager AND Operator = 144
The Yager OR Operator = 146
The Yager NOT Operator = 151
The Sugeno Class and Other Alternative NOT Operators = 153
Threshold NOT Operator = 154
The Cosine NOT Function = 155
Notes = 158
5. Fuzzy Set Hedges = 161
Hedges and Fuzzy Surface Transformers = 161
The Meaning and Interpretation of Hedges = 162
Applying Hedges = 163
Fuzzy Region Approximation = 164
Restricting a Fuzzy Region = 167
Intensifying and Diluting Fuzzy Regions = 170
The Very Hedge = 171
The Somewhat Hedge = 179
Reciprocal Nature of Very and Somewhat = 186
Contrast Intensification and Diffusion = 186
The Positively Hedge = 187
The Generally Hedge = 189
Approximating a Scalar = 200
Examples of Typical Hedge Operations = 203
Notes = 209
6. Fuzzy Reasoning = 211
The Role of Linguistic Variables = 213
Fuzzy Propositions = 215
Conditional Fuzzy Propositions = 215
Unconditional Fuzzy Propositions = 216
The Order of Proposition Execution = 216
Monotonic (Proportional) Reasoning = 217
Monotonic Reasoning with Complex Predicates = 223
The Fuzzy Compositional Rules of Inference = 226
The Min=Max Rules of Implication = 226
The Fuzzy Additive Rules of Implication = 227
Accumulating Evidence with the Fuzzy Additive Method = 227
Fuzzy Implication Example = 231
Correlation Methods = 235
Correlation Minimum = 235
Correlation Product = 236
The Minimum Law of Fuzzy Assertions = 239
Methods of Decomposition and Defuzzification = 245
Composite Moments (Centroid) = 249
Composite Maximum (Maximum Height) = 250
Hyperspace Decomposition Comparisons = 251
Preponderance of Evidence Technique = 252
Other Defuzzification Techniques = 256
The Average of Maximum Values = 257
The Average of the Support Set = 257
The Far and Near Edge of the Support Set = 258
The Center of Maximums = 259
Singleton Geometry Representations = 266
Notes = 269
7. Fuzzy Models = 271
The Basic Fuzzy System = 271
The Fuzzy Model Overview = 271
The Model Code View = 273
Code Representation of Fuzzy Variables = 274
Incorporating Hedges in the Fuzzy Model = 277
Representing and Executing Rules in Code = 278
Setting Alpha-Cut Thresholds = 280
Including a Model Explanatory Facility = 281
The Advanced Fuzzy Modeling Environment = 285
The Policy Concept = 286
Understanding Hash Tables and Dictionaries = 287
Creating a Model and Associated Policies = 294
Managing Policy Dictionaries = 298
Loading Default Hedges = 299
Fundamental Model Design Issues = 301
Integrating Application Code with the Modeling System = 302
Tasks at the Module Main Program Level = 302
Connecting the Model to the System Control Blocks = 303
Allocating and Installing the Policy Structure = 304
Defining Solution (Output) Variables = 304
Creating and Storing Fuzzy Sets in Application Code = 304
Creating and Storing Fuzzy Sets in a Policy's Dictionary = 306
Loading and Creating Hedges = 307
Segmenting Application Code into Modules = 310
Maintaining Addressability to the Model = 310
Establish the Policy Environment = 311
Initialize the Fuzzy Logic Work Area for the Policy = 311
Locate the Necessary Fuzzy Sets and Hedges = 312
Exploring a Simple Fuzzy System Model = 313
Exploring a More extensive Pricing Policy = 325
Fuzzy Set and Data Representational Issues = 335
Fuzzy Set and Model Variables = 336
Semantic Decomposition of PROFIT = 337
Fuzzy Set Naming Conventions = 340
The Meaning and Degree of Fuzzy Set Overlap = 341
Control Engineering Perspectives on Overlap and Composition = 346
Highly Overlapping Fuzzy Regions = 349
Boolean and Semi - Fuzzy Variables = 350
Using oolean Filters = 350
Applying Explicit Degrees of Membership = 351
Uncertain and Noisy Data = 353
Fuzzy and Uncertain Numbers = 353
Handling Uncertain and Noise Data = 357
Inferencing with Fuzzy Data = 358
The Interpretation of Model Results = 360
Undecidable Models = 361
Compatibility Index Metrics = 365
The Idea of a Compatibility Index = 365
The Unit Compatibility Index = 366
Scaling Expected Values by the Compatibility Index = 375
The Statistical Compatibility Index = 376
Selecting Height Measurements = 377
Notes = 377
8. Fuzzy Systems : Case Studies = 379
A Fuzzy Steam Turbine Controller = 379
The Fuzzy Control Model = 379
The Fuzzy Logic Controller = 380
The Conventional PID Controller = 381
The Stream Turbine Plant Process = 382
Designing the Fuzzy Logic Controller = 382
Running the Steam Turbine FLC Logic = 386
The New Product Pricing Model (Version 1) = 389
Model Design and Objectives = 389
The Model Execution Logic = 390
Create the Basic Price Fuzzy Sets = 391
Create the Run - Time Model Fuzzy Sets = 392
Execute the Price Estimation Rules = 392
Defuzzify to Find Expected Value for Price = 398
Evaluation Defuzzification Strategies = 399
The New Product Pricing Model (Version 2) = 413
Model Design Strategies = 413
The Model Execution Logic = 414
Create the Basic Fuzzy Sets = 414
Create the Run-Time Model Fuzzy Sets = 415
Execute the Price Estimation Rules = 417
The New Product Pricing Model (Version 3) = 423
The Model Execution Logic = 423
Execute the Price Estimation Rules = 423
Defuzzify to Find Expected Value for Price = 429
The New Product Pricing Model (P & L Version) = 430
Design for the P & L Model = 430
Model Execution and Logic = 432
Using Policies to Calculate Price and Sales Volume = 434
A Project Risk Assessment Model = 436
The Model Design = 437
Model Application Issues = 438
Model Execution Logic = 440
Executing the Risk Assessment Rules = 443
Notes = 448
9. Building Fuzzy Systems = 449
Evaluating Fuzzy System Projects = 449
The Ideal Fuzzy System Problem = 450
Fuzzy Model Characteristics = 450
Fuzzy Control Parameters = 450
Multiple Experts = 453
Elastic Relationships Among Continuous Variables = 454
Complex, Poorly Understood, or Nonlinear Problems = 455
Uncertainties, probabilities, and possibilities in data = 455
Building Fuzzy System Models = 457
The Fuzzy Desing Methodology = 459
Define the Model Functional and Operational Characteristics = 459
Define the System in Terms of an Input - Process - Output Model = 459
Localize the Model in the Production System = 460
Segment Model into Functional and Operational Components = 461
Isolate the Critical Performance Variables = 461
Choose the Mode of Solution Variables = 461
Resolve Basic Performance Criteria = 462
Decide on a Level of Granularity = 462
Determine Domain of the Model Variables = 462
Determine the Degree of Uncertainty in the Data = 463
Define the Limits of Operability = 463
Establish Metrics for Model Performance Requirements = 464
Define the Fuzzy Sets = 464
Determine the Type of Fuzzy Measurement = 464
Choose the Shape of the Fuzzy Set (Its Surface Morphology) = 465
Elicit a Fuzzy Set Shape = 466
Select an Appropriate Degree of Overlap = 467
Decide on the Space Correlation Metrics = 467
Ensure That the Sets Are Conformally Mapped = 467
Write the Rules = 468
Write the Ordinary Conditional Rules = 469
Enter Any Unconditional Rules = 469
Select Compensatory Operators for Special Rules = 469
Review the Rule Set and Add Any Hedges = 469
Add Any Alpha Cuts to Individual Rules = 470
Enter the Rule Execution Weights = 470
Define the Defuzzification Method for Each Solution Variable = 470
Notes = 471
10. Using the Fuzzy Code Libraries = 473
Linking Code to your Application = 473
Formal and Warning Messages = 474
System and Client Error Diagnostics = 474
Software Status Codes = 476
Modeling and Utility Software = 477
Symbolic Constants, Global Data, and Prototypes = 477
Data Structures = 477
Fuzzy Logic Functions = 478
The Fuzzy System Modeling Functions = 481
Miscellaneous Tools and Utilities = 482
Demonstration and Fuzzy Model Programs = 484
Description of Fuzzy Logic Functions = 486
FzyAboveAlfa = 486
FzyAddFZYctl = 487
FzyAND = 490
FzyApplyAlfa = 491
FzyApplyAND = 492
FzyApplyHedge = 494
FzyApplyNOT = 497
FzyApplyOR = 500
FzyAutoScale = 502
FzyBetaCurve = 503
FzyCompAND = 505
FzyCompOR = 508
FzyCondProposition = 510
FzyCoordSeries = 513
FzyCopySet = 515
FzyCopyVector = 516
FzyCorrMinimum = 517
FzyCorrProduct = 518
FzyCreateHedge = 520
FzyCreateSet = 522
FzyDefuzzify = 529
FzyDisplayFSV = 534
FzyDrawSet = 535
FzyExamineSet = 538
FzyFindFSV = 540
FzyFindPlateau = 541
FzyGetCoordinates = 543
FzyGetHeight = 545
FzyGetMembership = 546
FzyGetScalar = 547
FzyImplMatrix = 549
FzyInitCIX = 551
FzyInitFDB = 552
FzyInitFZYctl = 553
FzyInitHDB = 554
FzyInitVector = 554
FzyInterpVec = 555
FzyIsNormal = 556
FzyLinearCurve = 557
FzyMemSeries = 559
FzyMonotonicLogic = 561
FzyNormalizeSet = 563
FzyOR = 564
FzyPiCurve = 565
FzyPlotSets = 567
FzySCurve = 570
FzyStatCompIndex = 572
FzySupportSet = 574
FzyTrueSet = 575
FzyUnCondProposition = 576
FzyUnitCompIndex = 578
Glossary = 581
Bibliography = 603
Index = 607
