Cao Fuzzy Information and Engineering

Proceedings of the Second International Conference of Fuzzy Information and Engineering (ICFIE)
1. Auflage 2007
ISBN: 978-3-540-71441-5
Verlag: Springer Berlin Heidelberg
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band 40, 1032 Seiten

Reihe: Advances in Intelligent and Soft Computing

The Second International Conference on Fuzzy Information and Engineering (ICFIE2007), built on the success of previous conferences, the ICIKE2002 (Dalian China), is a major symposium for scientists, engineers and practitioners in China as well as the world to present their latest results, ideas, developments and applications in all areas of fuzzy information and knowledge engineering. It aims to strengthen relations between industry research laboratories and universities, and to create a primary symposium for world scientists in fuzzy fields such as Fuzzy Information, Fuzzy Sets and Systems, Soft Computing, Fuzzy Engineering, Fuzzy Operation Research and Management, Artificial Intelligence, Rough Sets and Its Application, Application in Fuzzy Mathematics and Systems, etc.

Autoren/Hrsg.

Cao, Bing-Yuan

Inhaltsverzeichnis

1;Title Page;1
2;Preface;5
3;Organization;7
4;Contents;10
5;Part I Fuzzy Information;19
6;Contrast Enhancement for Image by WNN and GA Combining PSNR with Information Entropy;20
6.1;Introduction;20
6.2;IBT;21
6.3;Contrast Classification for Image Based on Histogram;22
6.4;Transform Parameters Optimization by GA;22
6.5;IBT Calculation with WNN;25
6.6;Local Contrast Enhancement by Non-linear Operator;26
6.7;Algorithm Steps;27
6.8;Experimental Results;27
6.9;Conclusion;31
6.10;References;31
7;A Generalized Decision Logic Language for Information Tables;33
7.1;Introduction;33
7.2;Decision Logic Language;34
7.3;Generalized Decision Logic Language;36
7.4;Conclusions;38
7.5;References;38
8;New Similarity Measures on Intuitionistic Fuzzy Sets;39
8.1;Introduction;39
8.2;Basic Notions and Definitions of Intuitionistic Fuzzy Sets;40
8.3;New Approaches to Calculating Similarity Measures;41
8.4;Application to Pattern Recognition Problem;44
8.5;Conclusions;46
8.6;References;46
9;Information Source Entropy Based Fuzzy Integral Decision Model on Radiodiagnosis of Congenital Heart Disease;48
9.1;Introduction;48
9.2;Medical Expert Diagnosis System Model;49
9.2.1;Conformation of Attaching Function of Each Symptom;50
9.2.2;Information Analysis of Radiodiagnosis Value and Weight of Each Symptom;52
9.2.3;Fuzzy Integral Decision-Making;53
9.3;Example Analysis and Optimization Processing;54
9.3.1;Example Analysis;54
9.3.2;Optimization Processing;56
9.4;References;57
10;The Design and Research of Controller in Fuzzy PETRI NET;58
10.1;Introduction;58
10.2;The Definition of Fuzzy PETRI NET;58
10.2.1;The Definition of Fuzzy PETRI NET;58
10.2.2;The Rules of Activation;59
10.2.3;Place Invariant;59
10.3;Design of Controller;59
10.4;A Synthetical Algorithm Aiming at the Dead Lock Phenomenon;62
10.5;Dealing with Dead Lock Phenomenon;63
10.6;Summary;65
10.7;References;65
11;Fuzzy Tracing the Source of Net-Virus;67
11.1;Introduction;67
11.2;Source Tracing Modeling;68
11.3;Solutions to the Equations;71
11.4;Simulation Tests;72
11.5;Conclusion and Acknowledgements;74
11.6;References;75
12;Uncertain Temporal Knowledge Reasoning of Train Group Operation Based on Extended Fuzzy-Timing Petri Nets;76
12.1;Introduction;76
12.2; EFTN and Relative Computations;77
12.2.1;Updating Fuzzy Timestamps;77
12.2.2;Possibility Computation;78
12.3;Analysis of Temporal Uncertainty ;78
12.4;Comparing Analysis;79
12.5;Conclusion;80
12.6;References;81
13;An Improved FMM Neural Network for Classification of Gene Expression Data;82
13.1;Introduction;82
13.2;Improving FMM Neural Network Classifier;84
13.2.1;Original FMM Neural Network;84
13.2.2;Drawbacks of FMM Neural Network;85
13.2.3;Improvement of FMM Neural Network;86
13.3;Gene Selection;87
13.4;Experiment Evaluation;89
13.5;Conclusion;90
13.6;References;90
14;Using Assignment Matrix on Incomplete Information Systems Reduction of Attributes;92
14.1;Introduction;92
14.2;Incomplete Information System and Tolerance Relation;93
14.3;Assignment Matrix and Measurement of Conditional Attributes’ Significance to Decision;94
14.4;Assignment Matrix Based Attribute Reduction Algorithm;96
14.5;Example and Analysis;97
14.6;Summary;99
14.7;References;99
15;An Image Compression Algorithm with Controllable Compression Rate;100
15.1;Introduction;100
15.2;The Fuzzy Neural Network;101
15.3;The Technical Details;101
15.4;Performance;103
15.5;Conclusions;104
15.6;References;104
16;A Novel Approach for Fuzzy Connected Image Segmentation;106
16.1;Introduction;106
16.2;Preliminaries;107
16.2.1;A Framework for Fuzzy Connectedness, Relative Fuzzy Connectedness;107
16.2.2;Fuzzy k -Object;109
16.3;Algorithm;109
16.3.1;Original Algorithm;109
16.3.2;Proposed Algorithm;109
16.3.3;Algorithm Complexity Analysis;110
16.4;Experimental and Evaluation ;111
16.4.1;Experimental Result;111
16.4.2;Evaluation;111
16.5;Conculsion;113
16.6;References;114
17;Knowledge Mass and Automatic Reasoning System in Similarity Logic $C_Q$;115
17.1;Introduction and Preliminary;115
17.2;Type V Knowledge Mass, Type V Knowledge Universe, Type V Knowledge Base and Type V $Q$-Automatic Reasoning System;119
17.3;Type V True Level k Knowledge Circle, Extended Type V Knowledge Base and Extended Type V $Q$-Automatic Reasoning System;122
17.4;The Type V Level (k,j) Perfection of Extended Type V Knowledge Base $K^V$;126
17.5;Conclusion;128
17.6;References;128
18;Image Segmentation by Multi-level Thresholding Based on C-Means Clustering Algorithms and Fuzzy Entropy;130
18.1;Introduction;130
18.2;Image Segmentation Based on C-Means Clustering Algorithms;131
18.2.1;C-Means Clustering Algorithms;131
18.2.2;C-Means Clustering Algorithms for Image Segmentation;132
18.3;Perform Post Process Using Fuzzy Entropy;133
18.3.1;Fuzzy Entropy and Membership Function;133
18.3.2;Perform Post Process Using Fuzzy Entropy;133
18.4;Experimental Results;134
18.5;Conclusion;138
18.6;References;138
19;Part II Fuzzy Sets and Systems;139
20;Extension Principle of Interval-Valued Fuzzy Set;140
20.1; Introduction;140
20.2;Preliminaries;141
20.3;Maximal Extension Principle of Interval-Valued Fuzzy Set;144
20.4;Minimal Extension Principle of Interval-Valued Fuzzy Set;146
20.5;Generalized Extension Principle of Interval-Valued Fuzzy Set;150
20.6;Conclusion;151
20.7;References;152
21;On Vague Subring and Its Structure;153
21.1;Introduction;153
21.2;Preliminaries;153
21.3;Vague Ring and Its Substructures;154
21.4;The Homomorphism of Vague Ring;157
21.5;References;158
22;A Common Generalization of Smooth Group and Vague Group;159
22.1;Introduction;159
22.2;Common Generalization of Smooth Group and Vague Group;159
22.3;The Relationships Among Fuzzy Groups Based on Fuzzy Equalities;161
22.4;Conclusion;163
22.5;References;163
23;Sequences of Fuzzy-Valued Choquet Integrable Functions;164
23.1;Introduction;164
23.2;Preliminaries and Propositions;164
23.3;Uniform Integrability, Uniform Absolute Continuity and Uniform Boundedness;166
23.4;Some Convergence Theorems;170
23.5;Conclusions;172
23.6;References;173
24;Generalized Root of Theories in Propositional Fuzzy Logical Systems;174
24.1;Introduction;174
24.2;Preliminaries;175
24.2.1;Logic Systems: ${\L}uk$, $G\ddot{o}d$, $\Pi$ and$L^{\ast}$} \label;175
24.2.2;Generalized Deduction Theorems in ${\L}uk$, $G\ddot{o}d$, $\Pi$ and;177
24.3;Basic Definitions and Properties;178
24.4;Results in the n-Valued ukasiewicz Logic System;180
24.5;Results in the $G\ddot{o}del$ Fuzzy Logic System;181
24.6;Results in the R_o -- Fuzzy Logic System;182
24.7;Conclusion Remarks;183
24.8;References;184
25;Countable Dense Subsets and Countable Nested Sets;185
25.1;Introduction;185
25.2;Countable Decomposition Theorems;186
25.3;Countable Representation Theorems;190
25.4;Conclusion;194
25.5;References;195
26;The Construction of Power Ring;196
26.1;Introduction;196
26.2;Concepts of HX Ring and Power Ring;196
26.3;Construction of HX Ring and Power Ring;198
26.4;References;202
27;On Fuzzy Ideals in BCH-Algebras;203
27.1;Introduction;203
27.2;Preliminaries;203
27.3;Fuzzy Quasi-associate Ideals of BCH-Algebras;204
27.4;References;207
28;Minimization of Lattice Automata;209
28.1;Introduction;209
28.2;Lattice Automata;209
28.3;Refining Congruence and Quotient Automata;210
28.4;Equivalent and Minimal Automata;213
28.5;Minimization Algorithm;215
28.6;An Example;217
28.7;Conclusions;220
28.8;References;220
29;Fixed Points in $M$-Fuzzy Metric Spaces;221
29.1;Introduction;221
29.2;Compatible Mappings of Type (*);224
29.3;Common Fixed Point Theorems;226
29.4;References;229
30;Common Fixed Points of Maps on Intuitionistic Fuzzy Metric Spaces;231
30.1;Introduction;231
30.2;Preliminaries;232
30.3;Result;233
30.4;References;240
31;Generalized Fuzzy $B$-Algebras;241
31.1;Introduction;241
31.2;Preliminary ;241
31.3;($\alpha,\beta$,)-Fuzzy $B$-Algebras;243
31.4;References;248
32;Fuzzy Set Theory Applied to $QS$-Algebras;249
32.1;Introduction;249
32.2;Preliminary;249
32.3;Fuzzy $QS$-Subalgebra;251
32.4;Fuzzy Topological $QS$-Algebra;256
32.5;References;257
33;$L$-Topological Dynamical System and Its Asymptotic Properties;258
33.1;Introduction;258
33.2;Dynamical Systems on F-Lattice;259
33.2.1;Systems;260
33.2.2;Subsystems;262
33.3;Asymptotic Properties;263
33.3.1;Recurrence;263
33.3.2;$ \omega $ -Limit Sets;264
33.3.3;Topological Conjugacy;265
33.4;Conclusion;266
33.5;References;266
34;Integrating Probability and Quotient Space Theory: Quotient Probability;267
34.1;Introduction;267
34.2;Quotient Probability and Some Results;268
34.2.1;Probability Update and Spaces Fusion;269
34.2.2;Distance Between Two Different Quotient Probabilities;271
34.3;Conclusion;273
34.4;References;273
35;.-Convergence Theory of Filters in L.-Spaces;275
35.1;Introduction;275
35.2;Preliminaries;275
35.3;.-Convergence of Filters;277
35.4;Relationships Among .-Convergence of Filters, .-Convergence of Nets and .-Convergence of Ideals;279
35.5;Some Applications of .-Convergence of Filters;281
35.6;References;282
36;On the Perturbation of Fuzzy Matrix Equations with $.$-T Composition;284
36.1;Preliminaries;284
36.2;Fuzzy Solution-Invariant Matrix;286
36.3;Fuzzy Perturbation Issues;289
36.4;References;294
37;Normal Distribution Fuzzy Sets;295
37.1;Introduction;295
37.2;Normal Distribution Fuzzy Sets;296
37.3;Some Properties of Union, Complementation and Intersection;299
37.4;The Relationship Among Fuzzy Sets, Intuitionistic Fuzzy Sets and Normal Distribution Fuzzy Sets;300
37.5;Conclusion;303
37.6;References;303
38;Probabilistic Fuzzy Hypernear-Rings;305
38.1;Introduction ;305
38.2;Preliminaries;306
38.3;Probabilistic Fuzzy Hypernear-Rings;307
38.4;Conclusions;309
38.5;References;310
39;Isomorphic Fuzzy Sets and Fuzzy Approximation Space;311
39.1;Introduction;311
39.2;Isomorphism and Homomorphism of Fuzzy Sets;312
39.3;Fuzzy Sets and Approximation Spaces;314
39.4;Fuzzy Relations and Fuzzy Approximation Spaces;316
39.5;Conclusion;320
39.6;References;321
40;Part III Soft Computing;322
41;Particle Swarm Optimization Algorithm Design for Fuzzy Neural Network;323
41.1;Introduction;323
41.2;Fuzzy Neural Network Architecture;323
41.3;Particle Swarm Optimization with Division of Work;324
41.4;Pruning Algorithm;326
41.5;Numerical Simulations;327
41.6;Conclusion;327
41.7;References;328
42;A Method of Intuitionistic Fuzzy Reasoning Based on Inclusion Degree and Similarity Measure;329
42.1;Introduction;329
42.2;Inclusion Degree Based on Fuzzy Implication Operators;330
42.3;Similarity Measure Between IFSs Based on Inclusion Degree;331
42.4;Intuitionistic Fuzzy Reasoning Based on Similarity Measure;333
42.4.1;Single Rule Case of IFMP;333
42.4.2;Multi-rules Case of IFMP;334
42.4.3;Numerical Example;334
42.5;Conclusion;335
42.6;References;336
43;A Discrete Particle Swarm Optimization Algorithm for the Multiobjective Permutation Flowshop Sequencing Problem;337
43.1;Introduction;337
43.2;The Discrete Particle Swarm Optimization Algorithm for the MPFSP;338
43.2.1;Discrete PSO for the Single Objective PFSP;339
43.2.2;The Proposed DPSO for the MPFSP;339
43.3;Experimental Results;341
43.3.1;The 5 Small Problems with the Objectives of Minimizing the Makespan and the Total Tardiness;341
43.3.2;The 20 Problems of Taillard with the Objectives of Minimizing the Makespan and the Total Flowtime;343
43.4;Conclusions;344
43.5;References;345
44;An Optimization Method for Fuzzy c-Means Algorithm Based on Grid and Density;346
44.1;Introduction;346
44.2;Grid and Density;346
44.3;Initialization Method for Fuzzy c-Means Algorithm Based on Grid and Density;347
44.3.1;Extract Approximate Clustering Center Algorithm Based on Grid and Density;347
44.3.2;Initialization;348
44.4;Experiments;348
44.5;Conclusions;349
44.6;References;350
45;Obstacle Recognition and Collision Avoidance of a Fish Robot Based on Fuzzy Neural Networks;351
45.1;Distance Scanning System for Fish Robots;351
45.2;Distance Scanning System for Fish Robots;352
45.3;Obstacle Recognition System;355
45.3.1;Scanning Obstacles;355
45.3.2;Recognition of Shapes and Estimation of Approaching Angles;356
45.3.3;Experiments;357
45.4;Conclusions;358
45.5;References;358
46;Generalization of Soft Set Theory: From Crisp to Fuzzy Case;359
46.1;Introduction;359
46.2;Theory of Fuzzy Soft Set;360
46.2.1;Soft Set Theory;360
46.2.2;Fuzzy Soft Set Theory;361
46.3;Operators on Fuzzy Soft Set;363
46.3.1;Fuzzy Logic Operators;363
46.3.2;Operators on Fuzzy Soft Set;363
46.4;Application of Fuzzy Soft Set;365
46.5;Conclusion;367
46.6;References;367
47;A New QPSO Based BP Neural Network for Face Detection;369
47.1;Introduction;369
47.2;Quantum Particle Swarm Optimization;370
47.2.1;PSO Algorithm;370
47.2.2;QPSO Algorithm;370
47.3;Improved BPNN Learning Algorithm Based on QPSO;371
47.4;Face Detection Based on the Improved Learning Algorithm;373
47.4.1;Preprocessing;373
47.4.2;Designing the Network;373
47.4.3;Training;373
47.4.4;Detection;374
47.5;Experiment Results;374
47.6;Conclusion;376
47.7;References;377
48;ECC-Based Fuzzy Clustering Algorithm;378
48.1;Introduction;378
48.2;The Hierarchical Model of ECC;379
48.3;ECC-Based Vector Space Model;380
48.4;ECC-Based Fuzzy Clustering Algorithm;382
48.5;Validation;383
48.6; Experimental Results and Analysis;384
48.7;Conclusion;386
48.8;References;386
49;A New Algorithm for Attribute Reduction Based on Discernibility Matrix;387
49.1;Introduction;387
49.2;Basic Concepts;388
49.2.1;Decision Tables and Pawlak Reduction;388
49.2.2;Discernibility Matrices;389
49.3;Algorithms;391
49.3.1;Algorithm Principle;391
49.3.2;Algorithm Describing;392
49.3.3;Complexity Analysis of the Algorithm;393
49.4;Experimental Analysis;393
49.5;Conclusions;394
49.6;References;394
50;Optimal Computing Budget Allocation Based Compound Genetic Algorithm for Large Scale Job Shop Scheduling;396
50.1;Introduction;396
50.2;Job Shop Scheduling Problem Formulation;397
50.3;Heuristic and Search Techniques for Job Shop Scheduling Problem;398
50.4;Genetic Algorithm for Job Shop Scheduling;399
50.4.1;The Encoding Representation and Fitness Value;399
50.4.2;The Selection, Crossover and Mutation Operator;400
50.4.3;Genetic Algorithm Scheme Designing;401
50.5;Optimal Computing Budget Allocation Algorithm for GA Parameters Selection;402
50.5.1;Problem Statement;402
50.5.2;The Method for Algorithm Comparison and Selection;403
50.5.3;Optimal Computing Budget Allocation in Algorithm Comparison;404
50.6;Optimal-Computing-Budget-Allocation Based Compound Genetic Algorithm;406
50.7;Computational Experiments and Analysis;407
50.8;Conclusions;409
50.9;References;409
51;Part IV Fuzzy Engineering;411
52;Multi-Immune-Agent Based Power Quality Monitoring Method for Electrified Railway;412
52.1;Introduction;412
52.2;Immune Agent Model;413
52.3;Immune Supervision Networks Model;413
52.4;Architecture and Algorithm of Immune Agent;414
52.5;Cooperative Multi-Immune-Agent Based Power Quality Monitoring Systems;415
52.5.1;Immune-Terminal Parameter;416
52.5.2;Control Algorithm;418
52.5.3;Control Decision Algorithm;419
52.5.4;Resolving Conflicts;420
52.6;Experiment Evaluation;420
52.6.1;Evaluation;421
52.6.2;Solution to Conflicts;422
52.7;Conclusions;422
52.8;References;422
53;BF Hot Metal Silicon Content Prediction Using Unsupervised Fuzzy Clustering;424
53.1;Introduction;424
53.2;General Scheme and Principle of the Methods;425
53.2.1;Established Temporal Patterns;426
53.2.2;Fuzzy Clustering by State Recognition;426
53.2.3;Fit a Prediction Model to Each Fuzzy Cluster;427
53.2.4;Predicting by a Combination of the Models;427
53.3;Simulation of Predictive Algorithm;427
53.4;Conclusions and Discussion;430
53.5;Refences;430
54;Speech Emotion Pattern Recognition Agent in Mobile Communication Environment Using Fuzzy-SVM;432
54.1;Introduction;432
54.2;Proposed System;433
54.3;Emotional Feature Extractions and Optimization;435
54.3.1;Pre-processing;435
54.3.2;Emotional Feature Extraction;436
54.3.3;SFS Feature Optimization;436
54.3.4;MA (Moving Average) Filter to Minimize Noise Effect;437
54.4;Experimental Results;438
54.4.1;Speech Database and Experimental Setup;438
54.4.2;SFS Feature Optimization Experiment;439
54.4.3;Classification Results for Five Emotional States;439
54.4.4;Classification Results with Two Emotional States;441
54.5;Conclusion;442
54.6;References;442
55;Efficient Wavelet Based Blind Source Separation Algorithm for Dependent Sources;444
55.1;Introduction;444
55.2;BSS Model and Indeterminacies;445
55.3;CWT Based BSS Algorithm for Dependent Sources;446
55.3.1;Problem Statement and Why Use WT;446
55.3.2;How to Use CWT and the Basic Algorithm;448
55.4;CWT Based General BSS Algorithm;449
55.4.1;$m$=2 Mixtures and $n$ Source Signals;449
55.4.2;General Case: $n$ Mixtures and $n$ Source Signals;450
55.5;Procedure of the BSS Algorithm;451
55.6;Simulation Results;452
55.7;Conclusion;453
55.8;References;454
56;An Incentive Mechanism for Peer-to-Peer File Sharing;455
56.1;Introduction;455
56.2;Relation Work;456
56.3;Incentive Mechanism;456
56.4;Experiments;459
56.5;Conclusions and Future Work;460
56.6;References;460
57;Application of the BP Neural Network in the Checking and Controlling Emission of Vehicle Engines;461
57.1;Introduction;461
57.2;Model of the BP Neural Network;461
57.2.1;Construction of the BP Neural Network;462
57.2.2;Equation of the BP Neural Network;462
57.3;Process of the Neural Network Leaning;463
57.4;Application of the BP Neural Network in Checking and Controlling Emission;463
57.5;Result;466
57.6;References;466
58;Oil and Gas Pipeline Limited Charge Optimum Maintenance Decision-Making Analysis Based on Fuzzy-Gray-Element Theory and Fuzzy Analytical Hierarchy Process;468
58.1;Foreword;468
58.2;Theoretical Basis of Fuzzy-Gray-Element;469
58.2.1;Concept of Fuzzy-Gray-Element;469
58.2.2;Correlation Analysis;469
58.3;Fuzzy Hierarchy Analysis Method Assurance the Maintenance Measure Weight;470
58.3.1;The Analytic Hierarchy Process Brief Introduction;470
58.3.2;Maintenance Measure Support Layer Model Establishment;470
58.3.3;Establishment Triangle Fuzzy Number Complementary Judgment Matrix;471
58.3.4;Calculation Triangle Fuzzy Number Weight Vector of Maintenance Factor;471
58.3.5;Calculation Weight of Maintenance Factor;472
58.4;The Fuzzy-Grey-Element Correlative Decision-Making Model;472
58.4.1;The Establishment of the Maintenance Project Decision Model;472
58.4.2;Determination of Optimum Project Based on Fuzzy-Gray-Element Theory;474
58.5;Conclusion;475
58.6;References;476
59;The Fuzzy Seepage Theory Based on Fuzzy Structuring Element Method;477
59.1;Introduction;477
59.2;Fuzzy Structuring Element and Fuzzy Value Function;478
59.2.1;Fuzzy Structuring Element;478
59.2.2;Analysis Expression of Fuzzy Value Function;479
59.3;Establish of Fuzzy Seepage Model;480
59.3.1;Classical Landfill Gas Seepage Model;480
59.3.2;Fuzzy Qualified Differential Equation;481
59.3.3;Fuzzy Seepage Equation;481
59.4;Fuzzy Solution of Fuzzy Seepage Model;482
59.4.1;Analysis Solution of One-Dimension Ideal Seepage Model;482
59.4.2;Expressible Problem of Fuzzy Differential Equation Solution;483
59.4.3;Analysis Expression of Fuzzy Seepage Model;483
59.5;Conclusions;484
59.6;References;484
60;Ecological Safety Comprehensive Evaluation on Mineral-Resource Enterprises Based on AHP;485
60.1;Introduction;485
60.2;Method of Ecological Safety Evaluation for MRE;486
60.2.1;Meaning of Ecological Safety Evaluation for MRE;486
60.2.2;Index System of Ecological Safety Evaluation;486
60.2.3;Method of Evaluating the Ecological Safety of MRE;487
60.3;Example for the Applying of Comprehensive Evaluation on Ecological Safety of MRE;490
60.4;Conclusion;493
60.5;References;493
61;A Note on the Optimal Makespan of a Parallel Machine Scheduling Problem;494
61.1;Introduction;494
61.2;Level Algorithm;495
61.3;The Errors in Proofs for the Optimal Makespan;496
61.3.1;The Error in the Proof of $C_{\textmd{max}}\leq l_{\textmd{max}}+c$ in \cite{Tang};497
61.3.2;The Errors in the Proof of Cmax = lmax + c in [7];498
61.4;A New Algorithm and the Proof for the OptimalMakespan of $P|p_j=1,intree|C_{\textmd{max}}$;501
61.5;Conclusion;506
61.6;References;506
62;Part V Fuzzy Operation Research and Management;507
63;Advances in Fuzzy Geometric Programming;508
63.1;Introduction;508
63.2;Fuzzy Geometric Programming;509
63.3;Present Situation of Fuzzy Geometric Programming;510
63.4;Future Development of Fuzzy Geometric Programming;511
63.5;References;512
64;A Method for Estimating Criteria Weights from Intuitionistic Preference Relations;514
64.1;Introduction;514
64.2;Preliminaries;515
64.3;Consistent Intuitionistic Preference Relation;516
64.4;A Method for Estimating Criteria Weights;517
64.5;Conclusions;521
64.6;References;521
65;The Area Compensation Method About Fuzzy Order and Its Application;524
65.1;Introduction;524
65.2;Basic Concept;525
65.3;Ranking Fuzzy Numbers;526
65.4;Approach to the Fuzzy Linear Programming with Fuzzy Variables ;528
65.5;Extensions for the Area Compensation Ranking Method ;531
65.6; Conclusion ;532
65.7;References;533
66;A Method for the Priority Vector of Fuzzy Reciprocal Matrix;534
66.1;Introduction;534
66.2;Some Properties of Fuzzy Consistent Matrix;535
66.3;Algorithm;539
66.4;Numerical Examples;542
66.5;Conclusion;543
66.6;References;543
67;The Theory of Fuzzy Logic Programming;545
67.1;Introduction;545
67.2;Syntax and Semantics of Fuzzy Horn Clause Logic;545
67.3;Fuzzy Proof Theory;550
67.4;Fuzzy Procedural Interpretation;552
67.5;Conclusions;553
67.6;References;553
68;Multiobjective Matrix Game with Vague Payoffs;554
68.1;Introduction;554
68.2;Definition and Order Function of Vague Set;555
68.3;Model of Multiobjective Two-Person Zero-Sum Matrix Game Based on Vague Set;556
68.4;Solutions of Multiobjective Two-Person Zero-Sum Matrix Game Based on Vague Set;557
68.5;Conclusion;561
68.6;References;561
69;Fuzzy Geometric Object Modelling;562
69.1;Introduction;562
69.2;Modelling Fuzzy Geometric Objects Using Smooth Unit Step Function and Implicit Functions;564
69.2.1;Method 1: Fuzzification of a Solid Geometric Object;565
69.2.2;Method 2: Fuzzification of the Boundary of a Geometric Object;566
69.2.3;Method 3: Adding a Fuzzy Boundary to an Ordinary Solid Geometric Object;567
69.3;Modelling Fuzzy Geometric Objects Using Geometric Meshes and Parametrically Defined Geometric Shapes;568
69.4;Fuzzy Geometric Objects Blending;568
69.5;Shape Preserving Fuzzy Geometric Blending;569
69.6;References;572
70;A Fuzzy Portfolio Selection Methodology Under Investing Constraints;575
70.1;Introduction;575
70.2;Possibilistic Mean and Variance;576
70.3;A Fuzzy Portfolio Selection Model Under Investing Constraints ;578
70.4;Numerical Example;581
70.5;Conclusions;582
70.6;References;583
71;The Interaction Among Players in Fuzzy Games;584
71.1;Introduction;584
71.2;Interaction for Crisp Cooperative Games;585
71.3;Interaction Among Players in Games with Fuzzy Coalitions;586
71.3.1;Fuzzy Coalitions and $LP$-Derivative;586
71.3.2;Mutual Independence Among Levels Based on $s$;588
71.3.3;Interaction Among Levels of $LP(|LP|\geq 2)$ with Respect to Players of $M$;589
71.3.4;The Interaction Among Players of $P(|P|\geq 2)$;590
71.4;Conclusion;593
71.5;References;593
72;Novel Method for Fuzzy Hybrid Multiple Attribute Decision Making;594
72.1;Introduction;594
72.2;The Description for the Problem of Fuzzy Hybrid Multiple Attribute Decision Making;595
72.3;The Principle and Method of the Decision Making;595
72.3.1;Processing Data for Indexes;595
72.3.2;The New Method Based on the Grey Relational Degree;597
72.3.3;The Basic Steps of the New Decision Making Method;599
72.4;An Application Example ;600
72.5;Conclusions;601
72.6;References;602
73;Auto-weighted Horizontal Collaboration Fuzzy Clustering;603
73.1;Introduction;603
73.2;Prelimimary;604
73.2.1;Fuzzy C-Means (FCM);604
73.2.2;Horizontal Collaboration Fuzzy C-Means;604
73.3;The Determining of Weights in HC-FCM;606
73.3.1;Measure of Partition Similarity;606
73.3.2;Encouragement Approach;607
73.3.3;Penalty Approach;609
73.4;Conclusions;610
73.5;References;611
74;A Web-Based Fuzzy Decision Support System for Spare Parts Inventory Control;612
74.1;Introduction;612
74.2;The Framework for SPICDSS;613
74.2.1;The General Integrated Framework;613
74.2.2;The Criticality Class Evaluation Methodology;613
74.2.3;The Web-based Replenishment DSS (WRDSS);616
74.3;The Development of SPICDSS and Application;618
74.3.1;Prototype System;618
74.3.2;Application Analysis;619
74.4;Conclusions;619
74.5;References;619
75;Fuzzy Ranking for Influence Factor of Injury Surveillance Quality in Hospital;621
75.1;Introduction[5][6];621
75.1.1;Factors from the Filling Staff Include;622
75.1.2;Factors from the Injury Patients Include;622
75.1.3;Factors from the Staff of Collecting and Checking Include;622
75.1.4;Factors from the Supervising and Guidance Include;623
75.1.5; Factors from the Process of Inputting Include;623
75.1.6;Factors from the Group of Surveillance Work Include;623
75.1.7;Factors from the Quality Control and Managing Include;623
75.2;Some Basic Concepts[1][2];623
75.3;0.50-0.90 Scale[3];624
75.4;Ranking Based on the Fuzzy Complementary Judgement Matrix[4];625
75.5; Conclusion;628
75.6;References;629
76;Decision-Making Rules Based on Belief Interval with D-S Evidence Theory;630
76.1;Introduction;630
76.2;Decision Rule Based on the Belief Interval;631
76.2.1;The Decision Rule on $Bel$;632
76.2.2;The Decision Rule on $Pl$;632
76.2.3;The Colligation Rule;632
76.2.4;Unitary Operator;633
76.3;Example[5];635
76.4;Conclusion;638
76.5;References;638
77;An Optimization Model for Stimulation of Oilfield at the Stage of High Water Content;639
77.1;Introduction;639
77.2;Source of Data for Professional System ;640
77.3;Method of Optimization for the Adjustment of Oilfield Development;640
77.4;Evaluation of Development Status of Oilfield;640
77.4.1;Method for Predicting the Pressure of Single-Phase Flow;640
77.4.2;Prediction of Residual Oil Saturation;641
77.5;Optimization of Adjustment Project by Fuzzy Theory;641
77.5.1;The First Step;642
77.5.2;The Second Step;642
77.5.3;The Third Step;643
77.6;Method of Integral Optimization;644
77.7;Diagram for the Idea of the Integral Optimization of Adjustment of Water Flooding Oilfield;644
77.8;Applications of the Optimization Model;644
77.8.1;Working out of Development Plan ;644
77.8.2;Application Effect of the Model;645
77.9;Conclusions;646
77.10;References;646
78;New Research of Coefficient-Fuzzy Linear Programming;648
78.1;Introduction;648
78.2;Membership Function of FLP;649
78.3;Solution Method;651
78.3.1;Optimal Solution of FLP;651
78.3.2;Algorithm;652
78.4;Numerical Example;652
78.5;Conclusion;653
78.6;References;653
79;Part VI Artificial Intelligence;655
80;Robust Fuzzy Control for Uncertain Nonlinear Systems with Regional Pole and Variance Constraints;656
80.1;Introduction;656
80.2;Problem Description and Preliminaries;657
80.3;Main Results;661
80.4;Solving Procedures;664
80.5;Conclusions;664
80.6;References;665
81;The Existence of Fuzzy Optimal Control for the Semilinear Fuzzy Integrodifferential Equations with Nonlocal Conditions;666
81.1;Introduction;666
81.2;Preliminaries;667
81.3;Fuzzy Optimal Control;670
81.4;References;674
82;A PSO-Based Approach to Rule Learning in Network Intrusion Detection;675
82.1;Introduction;675
82.2;Standard Particle Swarm Optimization;676
82.3;PSO for Rule Learning;677
82.3.1;Coding Scheme;677
82.3.2;Fitness Function;678
82.3.3;Inertia Weight;678
82.3.4;Algorithm Description;679
82.4;Experiment Results;679
82.5;Conclusions;681
82.6;References;682
83;Similarity Mass and Approximate Reasoning;683
83.1;Introduction;683
83.2;The Construction of $Q$-Formula Mass and $Q$-Logic;684
83.3;Type V Simple Approximate Reasoning Based on $Q$-Logic $C_Q $;684
83.4;Type V Multiple Approximate Reasoning Based on $Q$-Logic $C_Q$;686
83.5;Type V Completeness and Type V Perfection of Knowledge Base K in $Q$-Logic $C_Q$ ;688
83.6;Conclusion ;692
83.7;References;692
84;Intelligent Knowledge Query Answering System Based on Short Message;694
84.1;Introduction;694
84.2;System Framework;695
84.3;Short Message Processing Module;695
84.3.1;Hardware Condition;695
84.3.2;Correlative AT Command and PDU Data Format Analyses;695
84.3.3;Processing Flow of Short Message;697
84.4;Knowledge Query Module;697
84.4.1;Query Mode of Users;697
84.4.2;Knowledge Organization;697
84.4.3;Query Sentences Classification;698
84.4.4;Examples of Parsing Query Sentence;698
84.4.5;Drawing Answer;700
84.5;Conclusion;701
84.6;References;701
85;Comparison Study on Different Core Attributes;702
85.1;Introduction;702
85.2;Preliminary;703
85.3;Core Attributes of Simplified Discernibility Matrix Based on Hu's Discernibility Matrix;704
85.4;Core Attributes of Simplified Discernibility Matrix Based on Positive Region;705
85.5;Core Attributes of Simplified Discernibility Matrix Based on Information Entropy ;706
85.6;Comparison Study on Three Kinds of Core Attributes;710
85.7;Conclusion;711
85.8;References;712
86;Controllability for the Impulsive Semilinear Fuzzy Integrodifferential Equations;713
86.1;Introduction;713
86.2;Existence and Uniqueness of Fuzzy Solution;714
86.3;Controllability;717
86.4;Example;720
86.5;References;722
87;Remodeling for Fuzzy PID Controller Based on Neural Networks;723
87.1;Introduction;723
87.2;Fuzzy PID Controller;724
87.3;Remodeling for an Equivalent NN of Fuzzy PID Controller;728
87.4; Control Simulation;730
87.5;Conclusion;734
87.6;References;734
88;Monitoring for Healthy Sleep Based on Computational Intelligence Information Fusion;735
88.1;Introduction;735
88.2;Scheme of Monitoring Sleep Fidget;736
88.3;Information Fusion Strategy Based on NN;737
88.4;Experimental System for Input/Output Data Acquisition;739
88.5;NN Model of Monitoring Sleep Fidget;741
88.6;Precision Verification of Sleep Fidget Model;745
88.7;Conclusions;747
88.8;References;747
89;Minimization of Mizumoto Automata;748
89.1;Introduction;748
89.2;Mizumoto Automata and Their Equivalent Canonical Form;749
89.3;Minimization of NA;750
89.4;References;751
90;Transformation of Linguistic Truth Values During the Sensor Evaluation;753
90.1;Introduction;753
90.2;Basic Concepts and Hypotheses;754
90.3;Transformation Models;755
90.3.1;Point to Point Model;755
90.3.2;Fuzzy to Point Model;756
90.3.3;Point to Fuzzy Set Model;756
90.3.4;Fuzzy Set to Fuzzy Set Model;756
90.4;Example;757
90.5;Conclusions;758
90.6;References;759
91;Guaranteed Cost Control for a Class of Fuzzy Descriptor Systems with Time-Varying Delay;760
91.1;Introduction;760
91.2;Preliminaries and Problem Formulation;761
91.3;Main Results;762
91.4;Example;767
91.5;Conclusions;767
91.6;References;768
92;The Research and Simulation on the Walking Trajectory of the Hexapod Walking Bio-robot;769
92.1;Foreword;769
92.2;The Brief Introduction About the Integral Structure of the Bionic Hexapod Walking Robot;769
92.3;The Gait Theory Analyses on the Bionic Hexapod Walking Robot;770
92.4;The Trajectory Choices of Bionic Hexapod Walking Robot;771
92.5;The Moving Trajectory Simulation of the Bionic Hexapod Walking Robot;772
92.5.1;The Creation of Bionic Hexapod Walking Robot’s Feet-Tip Trajectory Curve;772
92.5.2;The Virtual Prototype Model Construction of Bionic Hexapod Walking Robot[7];772
92.5.3;Bionic Hexapod Walking Robot’s Kinematic Inverse Solution;774
92.6;End;776
92.7;References;776
93;Research of Group Decision Consensus Degree Based on Extended Intuitionistic Fuzzy Set;777
93.1;Introduction;777
93.2;The Definition of Extended IFS;777
93.3;The Definition of Distance of Extended IFS;778
93.4;Example Analysis;780
93.5;Conclusions;781
93.6;References;781
94;Part VII Rough Sets and Its Application;782
95;A Novel Approach to Roughness Measure in Fuzzy Rough Sets;783
95.1;Introduction;783
95.2;Preliminaries;784
95.3;The Novel Approach to Roughness Measure in Fuzzy Rough Sets;785
95.4;An Example;787
95.5;Conclusions;788
95.6;References;788
96;Rough Communication of Dynamic Concept;789
96.1;Introduction;789
96.2;Two Direction S-Rough Sets and a - Generation of Two Direction Assistant Sets;790
96.3;Rough Communication of Dynamic Concept;791
96.4;Example;792
96.5;Conclusions;793
96.6;References;794
97;Some Entropy for Rough Fuzzy Sets;796
97.1;Introduction;796
97.2;Rough Fuzzy Sets;797
97.2.1; The Expression of Rough Fuzzy Sets;797
97.2.2;The Cardinalities of RFSs;798
97.3;Entropy for RFSs;799
97.4;Conclusions;805
97.5;References;805
98;A Fuzzy Measure Based on Variable Precision Rough Sets;806
98.1;Introduction;806
98.2;Basic Notions Related to Rough Sets;807
98.2.1;Pawlak Rough Sets;807
98.2.2;Information Systems;807
98.2.3;Variable Precision Rough Sets;808
98.3;A Fuzzy Measure Based on VPRS;809
98.3.1;Basic Notions of Fuzzy Sets;809
98.3.2;A Fuzzy Measure Based on VPRS;810
98.4;Conclusion;813
98.5;References;814
99;Rough Sets of System;816
99.1;Introduction;816
99.2;Rough Sets of System;817
99.3;Discussion About the Rough Sets of System;819
99.4;Some Properties of Rough Sets of System;820
99.5;Some Examples of Rough Sets of System;821
99.5.1;Rough Sets on the Real Line;821
99.5.2;Rough Sets of the Additive Group of Integers ;822
99.6;Conclusions ;823
99.7;References;823
100;Synthesis of Concepts Based on Rough Set Theory;824
100.1;Introduction;824
100.2;Rough Sets;825
100.3;Synthesis of Concepts Based on Rough Set Theory;825
100.4;Reduction Based on the Ideas of Synthesis of Concepts;828
100.5;Conclusion and Further Research;831
100.6;References;831
101;.-Tolerance Relation-Based RS Model in IFOIS;833
101.1;Introduction;833
101.2;Basic Theories;834
101.2.1;Incomplete Information System;834
101.2.2;Fuzzy Objective Information System ;835
101.3;.-Tolerance Relation;836
101.4;Rough Set Model in Incomplete and Fuzzy Objective Information System;837
101.4.1;The Concept of Incomplete and Fuzzy Objective Information System [19];837
101.4.2;Rough Set Model;838
101.5;Precision Reduction in Incomplete and Fuzzy Objective Information System;840
101.5.1;Basic Theory of Precision Reduction ;840
101.5.2;Precision Reduction Algorithm;842
101.6;Conclusion;842
101.7;References;843
102;Granular Ranking Algorithm Based on Rough Sets;845
102.1;Introduction;845
102.2;The Granular Ranking Algorithm;846
102.2.1;The Thought of Algorithm Designing;846
102.2.2;The Framework of Algorithm;848
102.2.3;The Description of Granular Ranking Algorithm;848
102.2.4;The Complexity of Algorithm;850
102.3;Experiment;850
102.3.1;Dataset;850
102.3.2;Hit Rate;851
102.3.3;The Result of Experiment;851
102.4;Conclusion;852
102.5;References;853
103;Remote Sensing Image Classification Algorithm Based on Rough Set Theory;854
103.1;Introduction;854
103.2;Principles of Rough Sets;855
103.3;Remote Sensing Image Classification Model Based on Rough Set Theory;856
103.4;Results of Experiment and Analysis;857
103.5;Conclusion;858
103.6;References;859
104;Topological Relations Between Vague Objects in Discrete Space Based on Rough Model;860
104.1;Introduction;860
104.2;Rough Model;861
104.2.1;The Definition of Region and Boundary of Raster Space;861
104.2.2;Definition of Rough Model;863
104.3;Study of Topological Relations Between Vague Objects Based on Rough Model and RCC-D-8;865
104.3.1;The Constraints of Lower and Upper Approximate Regions;865
104.3.2;Study of Topological Relations;867
104.4;Conclusions;867
104.5;References;868
105;Part VIII Application in Fuzzy Mathematics and Systems;870
106;A Least Squares Fuzzy SVM Approach to Credit Risk Assessment;871
106.1;Introduction;871
106.2;Methodology Formulation;872
106.2.1;SVM (By Vapnik [25]);872
106.2.2;FSVM (By Lin and Wang [24]);874
106.2.3;Least Squares FSVM;875
106.3;Experiment Analysis;877
106.4;Conclusions;879
106.5;References;879
107;Similarity Measures on Interval-Valued Fuzzy Sets and Application to Pattern Recognitions;881
107.1;Introduction;881
107.2;Degree of Similarity Between IVFSs and Similarity Measures;882
107.3;Applications of the Similarity Measures to Pattern Recognitions;887
107.4;References;889
108;Updating of Attribute Reduction for the Case of Deleting;890
108.1;Introduction;890
108.2;Preliminaries;891
108.3;Improvement of Discernibility Matrix and Updating of a Core;892
108.4;Updating Principle of Attribute Reduction;894
108.5;Updating Algorithm of Attribute Reduction Based on Discernibility Matrix;895
108.6;Conclusions;898
108.7;References;898
109;The Research of the Exposition Economy Model;900
109.1;Introduction;900
109.2;Investment Multiplier of Exposition;900
109.3;The Value of Development Potential of a City’s Exposition Economy;902
109.4;The Pull Model of Exposition Economy on National Economy;903
109.5;The Theoretical Analysis of the Model;904
109.6;Conclusion;906
109.7;References;907
110;A Region-Based Image Segmentation Method with Kernel FCM;908
110.1;Introduction;908
110.2;Region Segmentation Process;909
110.3;Integrated Feature Extraction;909
110.3.1;Extracting Texture Features;909
110.3.2;Extracting Other Features;911
110.4;Region-Based Color Image Segmentation;911
110.4.1;The Number of the Optimal Clusters;912
110.4.2;Label and Segment Image;913
110.5;Experiment Results;915
110.6;Conclusion and Future Work;916
110.7;References;916
111;An Efficient Threshold Multi-group-Secret Sharing Scheme;917
111.1;Introduction;917
111.2;The Proposed Scheme;918
111.2.1;System Parameters;918
111.2.2;Secret Distribution;918
111.2.3;Secret Reconstruction;920
111.3;Numerical Example;921
111.4;Analyses and Discussions;922
111.4.1;Security Analysis;922
111.4.2;Performance Analysis;923
111.5;Conclusions;924
111.6;References;924
112;A Morphological Approach for Granulometry with Application to Image Denoising;925
112.1;Introduction;925
112.2;Fuzzy Logical Operators;926
112.3;Operations of Fuzzy Sets;926
112.4;Convex Fuzzy Sets;929
112.5;Granulometry;931
112.6;Experiment Results;933
112.7;Conclusion;934
112.8;References;935
113;A Hybrid Decision Tree Model Based on Credibility Theory;936
113.1;Introduction;936
113.2;Credibility Theory;937
113.3;Hybrid Decision Trees Model;938
113.3.1;Fuzzification of Numerical Numbers;938
113.3.2;Hybrid Decision Trees Model;940
113.4;Experimental Results and Discussion;941
113.4.1;Experimental Results;941
113.4.2;Discussion;943
113.5;Conclusion;944
113.6;References;944
114;A Region-Based Image Retrieval Method with Fuzzy Feature;946
114.1;Introduction;946
114.2;Image Segmentation;947
114.3;Region Fuzzy Feature Extraction;948
114.4;Similarities Between Images Computation;950
114.5;Experiment Results and Analysis;951
114.6;Conclusions and Further Work;953
114.7;References;954
115;Association Rule Mining of Kansei Knowledge Using Rough Set;955
115.1;Introduction;955
115.2;Frameworks;956
115.3;Rough Set Theory;956
115.3.1;Review and Background;956
115.3.2;Definition;957
115.3.3;Reduction and Core Computing;958
115.4;Association Rule Mining Based on Rough Set;959
115.4.1;Association Rule Definition;959
115.4.2;Association Rule Algorithm Using Rough Set;960
115.5;Case Study;960
115.5.1;Product Knowledge Representation System Construction;960
115.5.2;Attributes Reduction;961
115.5.3;Strong Rule Extracting;962
115.6;Conclusion and Future Works;963
115.7;References;963
116;A Survey of Fuzzy Decision Tree Classifier Methodology;965
116.1;Introduction;965
116.2;Preliminaries;966
116.3;Potentials and Problems with Fuzzy Decision Tree Classifiers;967
116.4;Special Issues of a Fuzzy Decision Tree Classifier;968
116.4.1;Attribute Selection Criteria in Fuzzy Decision Trees;969
116.4.2;Inference for Decision Assignment;970
116.4.3;Stopping Criteria;971
116.5;Summary and Conclusions;972
116.6;References;973
117;The Optimization for Location for Large Commodity’s Regional Distribution Center;975
117.1;Problem Statement;975
117.2;The Model Hypothesis and Data Processing;976
117.3;The Theory and the Algorithm of Optimization Model for Location;977
117.4;The Calculation Result and Processing;983
117.5;Conclusion;984
117.6;References;985
118;Fuzzy Evaluation of Different Irrigation and Fertilization on Growth of Greenhouse Tomato;986
118.1;Introduction;986
118.2;Materials and Methods;987
118.2.1;Experimental Materials;987
118.2.2;Experimental Design;987
118.2.3;Experimental Method;987
118.2.4;Sampling Collection and Analysis;989
118.3;Result and Analysis;989
118.3.1;Confirm of Model Fuzzy Synthetic Evaluation;989
118.3.2;The Solve of Fuzzy Synthetic Evaluation Model;990
118.3.3; The Result of Fuzzy Synthesis Evaluation Model;991
118.4;Conclusion;992
118.5;References;992
119;The Solution of Linear Programming with LR-Fuzzy Numbers in Objective Function;994
119.1;Introduction;994
119.2;Preliminaries;995
119.3;Fuzzy Linear Programming Problem and Fuzzy Max Order;996
119.4;Possibility and Necessity Maximization Problems;1000
119.5;Numerical Examples;1003
119.6;References;1004
120;On Relationships of Filters in Lattice Implication Algebra;1006
120.1;Introduction;1006
120.2;Preliminaries;1007
120.3;$FL$-Filter of Lattice Implication Algebra;1008
120.4;On FL-Filter and NF-Filter of Lattice Implication Algebra;1010
120.5;Conclusions;1013
120.6;References;1013
121;Study on Adaptive Fuzzy Control System Based on Gradient Descent Learning Algorithm;1015
121.1;Introduction;1015
121.2;How to Design a General Fuzzy Control System;1016
121.3;Adaptive Fuzzy Control Algorithm;1017
121.3.1; System Structure;1017
121.3.2; Adaptive Control Algorithm;1017
121.4;Fuzzy Control Rules;1019
121.5;Simulation Results and Discussions;1021
121.5.1;Step Response;1021
121.5.2; Ramp Response;1023
121.5.3; Acceleration Response;1023
121.5.4; Sine Response;1023
121.5.5; Discussions;1024
121.6;Stability Discussion;1025
121.7;Conclusions;1025
121.8;References;1026
122;Uncertainty Measure of Fuzzy Rough Set;1027
122.1;Introduction;1027
122.2;Prelimilary;1027
122.3;Rough Entropy of Fuzzy Rough Set;1029
122.3.1;Information Entropy and Rough Entropy of Fuzzy Knowledge R;1029
122.3.2;Rough Entropy of Fuzzy Rough Set;1030
122.4;Conclusions;1032
122.5;References;1032
123;Author Index;1034