Readings in Fuzzy Sets for Intelligent Systems (eBook)

Front Cover 1
Readings in Fuzzy Sets for Intelligent Systems 4
Copyright Page 5
Table of Contents 6
Acknowledgments 11
Chapter 1. 
14 
The Emergence of Fuzzy Sets and 
14 
What Fuzzy Sets Are About 16
Fuzzy Sets Versus Possibility 
17 
Degrees of Truth and Incomplete 
17 
Possibility and Probability 18
Possibility Theory and Interval Analysis 18
Fuzzy Relations 18
Where Are Fuzzy Sets Useful? 19
Organization of the Readings 21
Acknowledgments 22
References 22
Appendix A: 
24 
Appendix B: General English Books and 
28 
Appendix C: Books on Applications of 
31 
Appendix D: References on Fuzzy 
31 
Chapter 2. 
34 
Introduction 34
Further Readings 35
FUZZY SETS 40
NOTATION, TERMINOLOGY, AND BASIC OPERATIONS 42
THE CONCEPT OF A FUZZY RESTRICTION AND TRANSLATION RULES 
57 
REFERENCES 65
BIBLIOGRAPHY 65
REPRESENTATION THEOREMS FOR FUZZY CONCEPTS 78
1 INTRODUCTION 78
2 PRELIMINARIES 78
3 THE GENERAL REPRESENTATION THEOREM 79
4 REPRESENTATION THEOREM FOR L-TOPO-LOGICAL 
80 
5 L-ALGEBRAIC 
81 
6 CONCLUSIONS 82
REFERENCES 82
On Some Logical Connectives for Fuzzy Sets Theory 84
INTRODUCTION 84
1. PRELIMINARIES ON FUZZY CONNECTIVES 84
2. ON STRONG NEGATIONS AND DEMORGAN'S LAWS 86
3. SOME LOGICAL PROPERTIES OF 
87 
4. FURTHER CHARACTERIZATION OF THE 
88 
REFERENCES 89
Symmetric Summation: A Class of Operations 
90 
I. COMPLEMENTARY SETS 90
II. SET COMBINATION 90
III. SYMMETRIC SUMS 90
IV. EXAMPLES 91
V. SUMMARY 92
ACKNOWLEDGMENT 92
REFERENCES 92
On Ordered Weighted Averaging Aggregation 
93 
INTRODUCTION 93
FORMULATING OF THE AGGREGATION PROBLEM 93
GENERAL "ANDING" AND "ORING" OPERATORS 93
OWA OPERATORS 94
PROPERTIES OF OWA OPERATORS 95
QUANTIFIERS AND OWA OPERATORS 96
MEASURE OF "ANDNESS" AND "ORNESS" 97
IN A GENERAL SETTING 98
BUILDING CONSISTENT OWA OPERATORS 99
INCLUDING UNEQUAL IMPORTANCES 99
CONCLUSION 100
REFERENCES 100
FUZZY POWER SETS AND FUZZY IMPLICATION 
101 
1. Towards a theory of fuzzy power sets 101
2. Comparative semantics of fuzzy implication operators 102
3. Height, plinth and crispness of fuzzy sets 105
4. Fuzzy set-inclusions and equalities 106
5. Dis joint ness of fuzzy sets 107
6. A fuzzy set and its complement 108
7. Conservation of crispness, versus the 'bootstrap effect' 108
8. Conclusion 109
References 109
ON IMPLICATION AND INDISTINGUISHABELTTY IN THE 
110 
1. INTRODUCTION 110
2. SOME PRELIMINARIES 111
3. 
111 
4. 
113 
REFERENCES 116
A THEOREM ON IMPLICATION FUNCTIONS 
118 
0· Introduction 118
1. Background 118
2. 
120 
3. Proof 120
4. Conclusion 123
References 123
FUZZY NUMBERS: AN OVERVIEW 125
ABSTRACT 125
I. INTRODUCTION 126
II. DEFINITIONS AND FUNDAMENTAL PRINCIPLES 126
III. THE CALCULUS OF FUZZY QUANTITIES WITH NONINTERACTIVE 
134 
IV. CALCULATIONS WITH FUZZY INTERVALS IN PRACTICE 141
V. ALTERNATIVE FUZZY INTERVAL CALCULI 145
VI. COMPARISON OF FUZZY QUANTITIES 150
VII. SOME APPLICATIONS OF FUZZY NUMBERS AND INTERVALS 154
VIII. CONCLUSION 155
LIST OF SYMBOLS 156
REFERENCES 157
A REVIEW OF SOME METHODS FOR RANKING 
162 
1. Introduction 162
2. Ranking methods 163
3. Comparative examples 166
4. Conclusions 170
References 170
SOLUTIONS IN COMPOSITE FUZZY RELATION EQUATIONS: APPLICATION TO MEDICAL DIAGNOSIS IN 
172 
PRELIMINARY DEFINITIONS AND RESULTS 
172 
FUZZY RELATION INEQUATIONS 174
FUZZY RELATIONS 174
MEDICAL DIAGNOSIS AND BROUWERIAN LOGIC 177
CONCLUSION 178
REFERENCES 178
FUZZY RELATION EQUATION UNDER A CLASS 
179 
1. Introduction 179
2 . t- nores, conorms and definitions 181
3. Resolution of sup-t fuzzy 
184 
4. LOWER SOLUTIONS OF EQUATION (4) 186
5 . LOWER SOLUTIONS OF EQUATION 
189 
6. Fuzziness measures based on t-norms and t-conorms 191
8. Approximate solutions 196
9. Some applicationa 1 aspect 197
References 199
Chapter 3. Uncertainty and Informativeness 204
Introduction 204
Further Readings 205
A Definition of a Nonprobabilistic Entropy in the 
210 
1. INTRODUCTION 210
2. ENTROPY OF A FUZZY SET 210
3. INTERPRETATION OF d(f) 212
ACKNOWLEDGMENTS 215
REFERENCES 215
On the specificity of a possibility 
216 
1. Introduction 216
2. Specificity and negative entropy 217
3. A unifying view of specificity measures 218
4. On the relationship between specificity of 
222 
5. Specificity in the continuous domain 225
6. Specificity measures modified by a similarity 
227 
7. Conclusion 228
References 228
MEASURES OF UNCERTAINTY AND INFORMATION 
230 
1 INTRODUCTION 230
2 HARTLEY'S MEASURE OF 
232 
3 POSSIBILITY DISTRIBUTIONS AND 
233 
4 CHARACTERIZATION OF 
236 
5 PROPOSED MEASURE OF 
238 
6 CONDITIONAL POSSIBILISTIC 
242 
7 CONCLUSIONS 243
ACKNOWLEDGEMENT 244
REFERENCES 244
CONDITIONAL POSSIBILITY MEASURES 246
INTRODUCTION 246
POSSIBILITY DISTRIBUTIONS AND UNCERTAINTY FUNCTIONS 247
DEFINITION OF CONDITIONAL POSSIBILITIES 249
CONDITIONAL AND MARGINAL POSSIBILITIES 250
CONDITIONAL POSSIBILITIES AND INFORMATION DISTANCE 251
COMPARISONS WITH OTHER PROPOSALS 252
REFERENCES 252
On Modeling of Linguistic Information Using Random Sets 255
I. INTRODUCTION 255
II. GENERALITIES ON RANDOM SETS 256
III. SOME MEASURES OF UNCERTAINTY 257
IV. PROBABILISTIC INTERPRETATION OF POSSIBILITY MEASURES 257
V. CONCLUDING REMARKS 259
REFERENCES 259
ON THE CONCEPT OF POSSIBILITY-PROBABILITY 
260 
1. Introduction 260
2. The concept of consistency 260
3. Consistency axioms 262
4. Conclusion 263
References 263
FUZZY MEASURES AND FUZZY 
264 
FUZZY MEASURES, FUZZY INTEGRALS AND THEIR SEMANTICS 264
REFERENCES 270
PROPERTIES OF THE FUZZY EXPECTED VALUE AND THE FUZZY EXPECTED INTERVAL IN FUZZY 
271 
1. Introduction 271
2. Fuzzy expected value 271
3. The fuzzy expected interval (FEI) 272
4. Addition of fuzzy intervals 274
5. Building a mapping table 275
6. Application to fuzzy expert systems 276
7. Conclusion 277
References 277
Fuzzy Random Variables 278
1. INTRODUCTION 278
2. INTEGRAL CALCULUS FOR SET-VALUED FUNCTIONS 278
3. FUZZY VARIABLES AND THEIR EXPECTATIONS 280
4. PROPERTIES OF THE EXPECTATION 281
5. COMPUTATION OF E(X) 282
6. CONCLUDING REMARKS 283
7. APPENDIX 283
REFERENCES 284
The Strong Law of Large Numbers for Fuzzy Random Variables 285
1. INTRODUCTION 285
2. FUZZY NUMBERS 285
3. FUZZY RANDOM VARIABLES 286
4. THE LAW OF LARGE NUMBERS 286
REFERENCES 288
Chapter 4. 
290 
Introduction 290
Further Readings 291
An Experiment in Linguistic Synthesis with a Fuzzy 
296 
Introduction 296
The Plant to be Controlled 296
The Controller 296
Results and Conclusions 298
Appendix 299
References 302
ANALYSIS OF A FUZZY LOGIC 
303 
1. Introduction 303
2 . The fuzzy logic controller 303
3. Multilevel relay analogy 304
4. Properties of a multilevel relay: Describing function 305
5. The Prediction of oscillations 306
6. Discussion 308
7. Conclusion 308
8. Appendix 308
Acknowlegdement 310
References 310
SELECTION OF PARAMETERS FOR A FUZZY LOGIC 
311 
1. Introduction 311
2. A review of the fuzzy logic controller 311
3. Parameters of the fuzzy logic controller 312
4. Conclusion 317
Acknowledgements 317
References 318
MODELLING CONTROLLERS USING 
319 
INTRODUCTION 319
FORMATION OF THE FUZZY 
319 
EXAMPLE 322
DISCUSSION 323
REFERENCES 325
HONORARY FELLOWSHIPS 326
THE USE OF FUZZY SETS FOR THE TREATMENT OF 
327 
1. Introduction 327
2. Fuzzy discretisation of a real space 327
3. Comparative properties of union and intersection of 2 fuzzy sets in X and 
328 
4. The principle of use 328
5. Conclusion 328
References 329
A Linguistic Self-Organizing 
330 
1. INTRODUCTION 330
2. THEORY OF THE SOC 331
3. EXPERIMENTS WITH THE SOC 337
4. CONCLUSION 344
REFERENCES 345
APPENDIX A: 
345 
Some Properties of Fuzzy Feedback Systems 346
I. INTRODUCTION 346
II. OPEN LOOP FUZZY SYSTEMS 346
III. THE CONCEPT OF STABILITY IN FUZZY SYSTEMS 347
IV. CLOSED LOOP FUZZY SYSTEMS 347
VI. A CONTROL PROBLEM 349
VI. CONCLUSION AND DISCUSSION 349
REFERENCES 349
CONTROL OF A CEMENT KILN BY FUZZY 
350 
1. INTRODUCTION 350
2. FUZZY CONTROL PRINCIPLES 351
3. 
353 
4. FUZZY CONTROL OF CEMENT KILN 357
5. 
359 
REFERENCES 360
APPENDIX: BASIC PCL PRINCIPLES 360
FUZZY CONTROL FOR AUTOMATIC TRAIN 
361 
INTRODUCTION 361
AUTOMATIC TRAIN OPERATION CONTROL 
361 
TRAIN OPERATION BY A HUMAN 
362 
FUZZY CONTROL 362
THE FUZZY CONTROLLED 
363 
SIMULATION 364
CONCLUSIONS 364
REFERENCES 365
FUZZY CONTROLLED ROBOT ARM PLAYING TWO 
368 
1. Introduction 368
2. The control algorithm of the two-dimensional ping-pong robot 368
3. Experimental result 372
4. Conclusions 372
References 373
A Fuzzy Logic Programming 
374 
ABSTRACT 374
INTRODUCTION 374
CONTROL RULE PROGRAMMING 376
SYSTEM DEVELOPMENT 378
SUMMARY 379
References 379
A Reinforcement 
381 
ABSTRACT 381
INTRODUCTION 381
FUZZY LOGIC CONTROL 381
REINFORCEMENT LEARNING, CREDIT ASSIGNMENT, AND 
383 
THE ARIC ARCHITECTURE 384
APPLYING ARIC TO CART-POLE BALANCING 387
RESULTS 389
RELATION TO OTHER RESEARCH 391
CONCLUSIONS 392
ACKNOWLEDGMENTS 393
References 393
CHARACTERIZATION 
394 
SPECIAL PROBLEMS USING THE PESSIMISTIC CRITERION 395
METHODS OF SOLUTION 396
PROPERTIES OF THE OPTIMAL FUZZY CONTROL 
396 
REFERENCES 398
Fuzzy Identification of Systems and Its 
400 
I. INTRODUCTION 400
II. FORMAT OF FUZZY IMPLICATION AND 
400 
III. ALGORITHM OF IDENTIFICATION 402
IV. APPLICATION TO FUZZY MODELING 410
V. CONCLUSION 415
REFERENCES 416
Fuzzy Modeling of 
417 
ABSTRACT 417
INTRODUCTION 417
THE STATIC SUGENO'S FUZZY MODEL 417
QUASILINEAR FUZZY MODELS (QLFMs) OF NONLINEAR 
417 
IDENTIFICATION OF THE QLFM 419
TRANSFER FUNCTION AND STATE-SPACE DESCRIPTION OF 
420 
CONCLUSION 421
References 421
Chapter 5. 
422 
Introduction 422
Further Readings 423
THE LOGIC OF INEXACT CONCEPTS 430
I. INTRODUCTION 430
II. A PARADOX 431
III. RTSOLUTION OF THE PARADOX 432
IV. REPRESENTING INEXACT CONCEPTS 435
V. THE AL.GEBRA OF INEXACT 
439 
VI. OPTIMIZATION AND THE 
442 
VII. IMPLICATION AND NEGATION 445
VIII. THE LOGIC OF INEXACT CONCEPTS 448
ACKNOWLEDGMENT 453
BIBLIOGRAPHY 453
Fuzzy Logic and the Resolution Principle 455
1. Introduction 455
2. Fuzzy Logic 456
3. Satisfiability in Fuzzy Logic 457
4. The Concept of Logical Consequence in Fuzzy Logic 461
5. Conclusions 464
REFERENCES 465
Fundamentals of Fuzzy 
466 
ABSTRACT 466
INTRODUCTION 466
CONFIDENCE, THE FUZZY RESOLUTION PRINCIPLE, AND 
467 
THE WEIGHT OF RULE 469
FUZZY RESOLUTION IN FUZZY FIRST-ORDER PREDICATE 
471 
CONCLUSION 472
Epistemic entrenchment and 
474 
1. Introduction 474
2. Necessity measures and their qualitative counterpart 475
3. Epistemic entrenchment and qualitative necessity measures 476
4. Dealing with uncertain knowledge bases 477
5. Possibilistic reasoning and belief revision 479
6. Possibility theory and Spohn's ordinal conditional functions 480
7. Conclusion 481
Acknowledgement 481
References 482
Fuzzy and Probability Uncertainty Logics 483
1. INTRODUCTION 483
2. MIN/MAX CONNECTIVES IN PROBABILITY LOGICS 483
3. A FORMAL BASIS FOR THE COMPARISON OF VARIOUS LOGICS OF 
484 
4. DERIVATION OF RESCHER'S 
487 
5. DERIVATION OF LN1, 
487 
6. SEMANTICS FOR THE LOGICS IN TERMS OF POPULATION RESPONSES 488
7. SUMMARY AND CONCLUSIONS 489
ACKNOWLEDGMENTS 490
REFERENCES 490
Possibility Theory 
491 
Abstract 491
1. Introduction 491
2. Basic Properties of Possibility Distributions 494
3. Translation Rules and Meaning Representation 499
4. Inference from Soft Data and Mathematical Programming 504
5. Examples of Inference from Soft Data 509
6. Evidence, Certainty and Possibility 514
7. Concluding Remark 519
8. References and Related Papers 519
AXIOMATIC APPROACH TO IMPLICATION FOR 
522 
1. Introduction 522
2. Some properties of implication rules found in the literature 522
3. Discussion of the properties desirable in an implication rule 528
4. Axioms for implication and derivation of classes of suitable functions 530
5. Conclusions 533
References 535
AN APPROACH TO FUZZY REASONING METHOD 536
1. INTRODUCTION 536
2 . PRELIMINARIES 536
3 . FUZZIFICATION OF 
536 
4 . RULES OF INFERENCE USING FUZZY ASSERTIONS 538
5 . COMPARISON WITH OTHER LOGICAL SYSTEMS 540
6. ILLUSTRATIVE EXAMPLES 540
7. CONCLUDING REMARKS 541
ACKNOWLEDGEMENTS 542
REFERENCES 542
SOME CONSIDERATIONS ON FUZZY CONDITIONAL INFERENCE 543
1. INTRODUCTION 543
2. Fuzzy sets—notation, terminology and basic operations 543
3. Fuzzy conditional inference 544
4. Formalization of improved methods 550
5. Concluding remarks 557
References 558
A FUZZY SYSTEM MODLE BASDE ON THE LOGICAL STRUCTEUR 559
ABSTRATC 559
KEYWORSD 559
INTRODUCTION 559
FUZZY 
559 
THE MODEL OF FUZZY SYSTEM 561
CONCLUDING REMARKS 566
APPENDIX 567
ON MODE AND IMPLICATION IN APPROXIMATE REASONING 568
INTRODUCTION 568
MODUS PONENS IN BOOLEAN, PROBABILISTIC 
568 
MODUS PONENS GENERATING 
569 
MODUS TOLLENS GENERATING FUNCTIONS 571
CONCLUDING REMARKS 572
ACKNOWLEDGEMENTS 572
REFERENCES 572
FUZZY INFERENCES AND CONDITIONAL 
573 
1. Introduction 573
2. Conditional possibility distributions 573
3. Joint possibility distributions 574
4. Non-acting and non-interactivity 575
5. Conclusion 576
Acknowledgements 577
References 577
Fuzzy Modus Ponens: A New Model Suitable for Applications in Knowledge-Based 
578 
I. INTRODUCTION 578
II. GENERALIZED MODUS PONENS 578
III. CLASSICAL RULES OF IMPLICATION 579
IV. INTUITIVE PROPERTIES OF THE IMPLICATION 580
V. GMP WITH A SUP-TM COMPOSITION 581
VI. THE MODUS PONENS 581
VII. CONCLUSIONS 586
References 586
Using Approximate Reasoning to 
588 
1. Introduction 588
2. A Theory of Approximate Reasoning 588
3. Default Variables and Possibility Qualification 591
4. Typical Default Reasoning Data 593
5. Conclusion 594
REFERENCES 594
FUZZY SET SIMULATION MODELS IN A SYSTEMS 
595 
THE POINT OF DEPARTURE 595
CONTENTION 595
IMPLEMENTATION 596
VALIDATION OF A SEMANTICAL MODEL 599
EPILOGUE: VERBAL MODELS IN A SYSTEM DYNAMICS ENVIRONMENT 602
REFERENCES 603
Chapter 
606 
Introduction 606
Further Readings 606
Numerical Methods for Fuzzy Clustering 612
ABSTRACT 612
1. INTRODUCTION 612
2. DEFINITIONS AND NOMENCLATURE 612
3. SOME USEFUL RESULTS 613
4. FORMULAS BASED ON MEAN CLUSTER DENSITY (CLUSTERING I AND II) 615
5. AN IMPROVED METHOD 623
6. DISCUSSION 627
ACKNOWLEDGMENT 627
REFERENCES 627
A Physical Interpretation of Fuzzy ISODATA 628
I. INTRODUCTION 628
II. AN ELECTRICAL ANALOG FOR FUZZY ISODATA 628
REFERENCES 629
Convex Decompositions of Fuzzy Partitions 630
I. INTRODUCTION AND CONCLUSIONS 630
II. PARTITION SPACES 630
III. THE DIMENSION OF FUZZY PARTITION SPACE 631
IV.MINIMAX 
632 
V. RECLASSIFICATION DECOMPOSITION 635
VI. CONVEX DECOMPOSITION AND THE PARTITION COEFFICIENT 637
VII. SUMMARY 640
ACKNOWLEDGEMENT 640
REFERENCES 641
New Results in Fuzzy Clustering Based on the 
642 
I. INTRODUCTION 642
II. ON INDISTINGUISHABILITY RELATIONS 642
IV. CONCLUDING REMARKS 645
REFERENCES 645
The fuzzy geometry of image subsets 646
1. Introduction 646
2. Fuzzy subsets [2] 646
3. Connectedness and surroundedness 
647 
4. Adjacency 648
5. Convexity [4-6] and starshapedness 648
6. Area, perimeter, and compactness [7] 649
7. Extent and diameter [8] 649
8. Shrinking and expanding, medial axes, 
650 
9. Gray-level-dependent properties splitting and
651 
10. Representation of fuzzy subsets 651
11. Concluding remarks 652
References 652
Fuzzy Confidence Measures in Midlevel Vision 653
I. INTRODUCTION 653
II. THE LINGUISTIC CONFIDENCE STRUCTURE 
653 
III. APPLICATIONS 656
IV. CONCLUSION 658
REFERENCES 659
Fuzzy sets and generalized Boolean retrieval systems 661
1. Introduction 661
2. Fuzzy retrieval 663
3. Generalized queries 664
4. Thresholds 665
5. Other approaches 666
6. Performance measures 668
7. Summary and conclusions 668
References 669
Appendix A: 
671 
A FUZZY REPRESENTATION OF DATA FOR 
673 
1. Introduction 673
2. Organization of a fuzzy database 673
3. Redundancy and determinancy properties 675
4. Application 676
5. Conclusions 679
References 679
FREEDOM-O: A FUZZY DATABASE SYSTEM 680
1. INTRODUCTION 680
2. POSSIBILITY DISTRIBUTIONS 680
3. FUZZY DATABASES 682
4. DATA MANIPULATION LANGUAGE 683
5. EXAMPLES OF QUERY STATEMENTS 686
6. CONCLUSIONS 688
REFERENCES 688
WEIGHTED FUZZY PATTERN MATCHING 689
1. Introduction 689
2. Fuzzy pattern matching 689
3. Importance assignment 693
4. Conclusion 697
References 697
FUZZY QUERYING WITH SQL: EXTENSIONS AND 
699 
1. Introduction 699
2. Non-fuzzy sets based interpretations 699
3. Extending the SQL language 701
4. Some implementation aspects 704
5. Conclusions and future works 706
References 707
FILIP: A FUZZY INTELLIGENT INFORMATION 
708 
1. INTRODUCTION 708
2. INTELLIGENT INFORMATION SYSTEM 
709 
3. CONCEPT LEARNING IN FILIP 714
4. CONCLUSION 720
REFERENCES 721
Chapter 7. 
722 
Introduction 722
Further Readings 723
DECISION-MAKING WITH A 
730 
1. Introduction 730
2. Preliminary definitions 730
3. Fuzzy preference relations and nondominated alternatives 731
4 . Transitive fuzzy preference relations 732
5. Properties of unfuzzy nondominated elements 734
6. Existence of unfuzzy nondominated elements 735
7. Summary 736
Acknowledgement 736
References 736
STRUCTURE OF FUZZY BINARY RELATIONS 737
Introduction 737
1. Fuzzy sets 737
2. V-correspondences, V-relations, and V-mappings 738
3. Structure of V-equivalences 739
4. V-preferences 741
5. Relations between transitivity properties 742
6. V-quasi-orders 744
7. V-choice functions 745
8. Scalar V-criteria and quasi-transitive V-preferences 746
9. Stability 747
10. Indices of fuzziness 747
11. Stability of the transitivity property 748
References 749
Some properties of choice functions based 
751 
1. Introduction 751
2. The operators 
752 
3. Choice functions for valued pairwise comparisons 753
4. Some properties of DR(y), NR(y), SDR( y) and 
756 
5. Rationality properties of the choice functions 759
6. Some particular cases 761
7. Conclusions 762
Acknowledgment 762
References 762
A NEW METHODOLOGY FOR ORDINAL MULTIOBJECTIVE DECISIONS BASED ON FUZZY 
764 
INTRODUCTION 764
MULTIOBJECTIVE DECISION MAKING 764
FUZZY SET APPROACH 765
SET THEORY AND LOGIC 766
A MODEL USING ORDINAL INFORMATION 766
EXAMPLE 767
DISCUSSION OF PROPERTIES OF THE MODEL 768
REFERENCES 769
Decision Making Under Uncertainty 
770 
PART I: SURVEY 
770 
PART II: EXAMPLE 777
PART III: CONCLUSION 787
REFERENCES 789
A Linguistic Approach to Decisionmaking 
790 
I. INTRODUCTION 790
II. A MULTICHOICE DECISION PROBLEM 790
III. TRUTH QUALIFICATION AND LINGUISTIC APPROXIMATION 792
IV. AN INVESTMENT DECISION PROBLEM 793
V. DISCUSSION 795
VI. CONCLUSION 796
ACKNOWLEDGMENT 796
REFERENCES 797
GROUP DECISION MAKING WITH FUZZY MAJORITIES REPRESENTED 
798 
1. INTRODUCTION 798
2. ZADEH'S DISPOSITION - BASED REPRESENTATION OF COMMONSENSE KNOWLEDGE, AND ITS 
799 
3. GROUP DECISION MAKING WITH A FUZZY MAJORITY 800
4. REMARKS ON "SOFT" DEGREES OF CONSENSUS 806
5. CONCLUDING REMARKS 806
LITERATURE 806
Applications of Fuzzy Set Theory to Mathematical Programming 808
1. INTRODUCTION 808
2. SYMMETRICAL MODELS 808
3. NONSYMMETRICAL MODELS 811
4. DUALITY AND SENSITIVITY ANALYSIS 813
5. EXTENSIONS 816
6. MULTIPLE OBJECTIVE PROGRAMMING 819
7. FUTURE PERSPECTIVES 821
REFERENCES 822
Fuzzy Versus Stochastic Approaches to Multicriteria 
823 
1. INTRODUCTION 823
2. SHORT DESCRIPTION OF STRANGE AND FLIP 824
3. DIDACTIC EXAMPLE 826
4. COMPARISON OF STRANGE AND FLIP 832
5. CONCLUSIONS 833
REFERENCES 834
FUZZY SETS IN FEW CLASSICAL OPERATIONAL RESEARCH PROBLEMS 835
1. INTRODUCTION 835
2. FLOWS IN NETWORKS WITH FUZZY CAPACITY CONSTRAINTS 836
3. FUZZY APPROACH TO PROJECT ANALYSIS 839
4. FUZZY ASSIGNMENT PROBLEM 843
5. FINAL REMARKS 845
FOOTNOTES 847
REFERENCES 847
FUZZY ZERO-BASE BUDGETING 848
References 852
THE FUZZY MATHEMATICS OF FINANCE 853
1. Introduction 853
2. Future and present value 854
3 . Fuzzy annuities 857
4. Fuzzy cash flows 858
5. Summary and conclusions 861
References 861
Chapter 8. 
862 
Introduction 862
Further Readings 862
RECONCILIATION OF THE YES-NO VERSUS GRADE OF MEMBERSHIP DUALISM IN HUMAN 
867 
1 . INTRODUCTION 867
2 . THE GRADE OF MEMBERSHIP CONCEPT 867
3 . THE MAX MIN OPERATIONS 869
4 . FORMER POSTULATES AND DIFFUCULTIES 870
5 . RESOLUTION OF THE DUALISM OF THE LAW OF THE EXCLUDED MIDDLE (LEM) VERSUS GRADABLE CONCEPTS AND OF AFFIRMATION-NEGATION VERSUS 
871 
6 . CONCLUDING REMARKS 872
REFERENCES 873
A MODEL FOR THE MEASUREMENT OF MEMBERSHIP 
874 
1. Introduction 874
2. The fundamental measurement of fuzziness 874
3. The construction of membership functions 878
4. The meaningfulness of operations on membership 883
5. Conclusions 885
Acknowledgement 885
References 885
THE CONCEPT OF GRADE OF MEMBERSHIP 887
1. Nature of the problem 887
2. Two approaches to the problem 887
3. The semantic approach to fuzzy reasoning 888
4. Decision theory 889
5. Application to assertions 890
6. Mathematical foundations 892
7. Simple assertions 893
8. The concept of grade of membership 895
9. The definition of connectives 897
10. The logic of assertions 898
References 900
ADAPTIVE INFERENCE IN FUZZY KNOWLEDGE NETWORKS 901
I. Knowledge Nets vs. 
901 
II. Combining Fuzzv Knowledge Networks 902
III. ADAPTIVE INFERENCE THROUGH CONCOMITANT VARIATION 904
A Formulation of Fuzzy Automata and its Application 
905 
I. INTRODUCTION 905
II. FOR MULATION OF FUZZY 
905 
III. SPECIAL CASES OF FUZZY AUTOMATA 907
IV. FUZZY AUTOMATA AS MODELS 
908 
VI. 
911 
APPENDIX I 911
APPENDIX II: 
912 
REFERENCES 913
LEARNING OF FUZZY PRODUCTION RULES FOR MEDICAL DIAGNOSIS 914
1. INTRODUCTION 914
2. OVERVIEW OF THE SYSTEM 915
3. FUZZY PRODUCTION RULES 917
4. EXPERIMENTAL RESULTS 920
5. CONCLUDING REMARKS 922
ACKNOWLEDGEMENTS 924
REFERENCES 924
Index 926