Grey Information (eBook)

Preface 6
Contents 9
Introduction 13
1.1 Scientific Background for the Appearance of Grey Systems Theory 13
1.2 Fundamental Concepts and Principles of Grey Systems 15
1.3 Comparison Between Several Nondeterministic Methods 19
1.4 Main Contents in Grey Systems Theory 20
1.5 Role of Grey Systems Theory in the Development of Science 22
1.6 Positions of Grey Systems Theory in the Spectrum of Interdisciplinary Sciences 23
1.7 Grey Systems in the Content of Uncertain Information 25
Grey Numbers and Their Operations 34
2.1 Grey Numbers 34
2.2 Whitenization of Grey Numbers and Degree of Greyness 37
2.3 Operations of Interval Grey Numbers 41
2.4 Measures of Grey Numbers 44
2.5 Information Content of Grey Numbers 49
Grey Equations and Grey Matrices 55
3.1 Grey Algebraic Equations and Grey Di erential Equations 55
3.2 Grey Matrices and Their Operations 56
3.3 Several Special Grey Matrices 60
3.4 Singularities of Grey Matrices 62
3.5 Grey Characteristic Values and Vectors 64
Generation of Grey Sequences 67
4.1 Introduction 67
4.2 Generation Based on Average 69
4.3 Operators of Sequences 71
4.4 Smooth Sequences 80
4.5 Stepwise and Smooth Ratios 83
4.6 Accumulating and Inverse Accumulating Generation Operators 85
4.7 Randomness of Sequences of Accumulating Generations 89
4.8 Grey Exponentiality of Accumulating Generations 91
Grey Incidence Analysis 95
5.1 Introduction 95
5.2 Grey Incidence Factors and Set of Grey Incidence Operators 97
5.3 Metric Spaces 100
5.4 Degrees of Grey Incidences 103
5.5 Absolute Degree of Grey Incidence 111
5.6 Relative Degree of Grey Incidence 123
5.7 Synthetic Degree of Grey Incidence 127
5.8 Order of Grey Incidences 128
5.9 Preference Analysis 130
5.10 Practical Applications 142
Grey Clusters and Grey Statistical Evaluations 149
6.1 Introduction 149
6.2 Clusters of Grey Incidences 150
6.3 Clusters with Variable Weights 154
6.4 Clusters with Fixed Weights 163
6.5 Grey Evaluation Based on Triangular Whitenization Functions 172
6.6 Grey Statistics 174
6.7 Entropy of Coe cient Vector of Grey Evaluations 179
6.8 Practical Examples 184
Grey Systems Modeling 200
7.1 The Thought of Five-Step-Modeling 200
7.2 Grey Di erential Equations 203
7.3 Model: GM(1,1) 206
7.4 Model: Remnant GM(1,1) 226
7.5 Model Group of GM(1,1) Type 231
7.6 GM(1,N) and GM(0,N) 237
7.7 GM(2,1) and Verhulst Model 244
Grey Combined Models 253
8.1 Econometric Models 254
8.2 Cobb-Douglas Model 262
8.3 Markov Model 266
8.4 Combined Time Series Model 270
8.5 Combined Predictions 273
Grey Prediction 282
9.1 Test of Grey Prediction Models 282
9.2 Predictions of Sequences 284
9.3 Interval Predictions 288
9.4 Disaster Predictions 296
9.5 Seasonal Disaster Predictions 300
9.6 Stock-Market-Like Predictions 306
9.7 Systems Predictions 312
9.8 Practical Applications 318
Grey Decisions 322
10.1 Introduction 322
10.2 Grey Target Decisions 325
10.3 Grey Incidence Decisions 332
10.4 Grey Development Decisions 343
10.5 Grey Statistical Decisions 348
10.6 Grey Cluster Decisions 354
10.7 Multiple-Target-Situation Decisions with a Synthesized Target 358
10.8 Grey Stratified Decisions 365
Grey Programming 374
11.1 Introduction 374
11.2 Linear Programming Models with Grey Parameters 376
11.3 Grey Linear Programming of Prediction Type 380
11.4 Several Theorems on Positioned Solutions of LPGP 384
11.5 Satisfactory Solutions of Grey Linear Programming 389
11.6 Quasi-Optimal Solutions of Grey Linear Programming 396
11.7 Grey 0-1 Programming 402
11.8 Grey Nonlinear Programming Without Constraints 411
11.9 Grey Nonlinear Programming with Constraints 414
Grey Input and Output 421
12.1 Basic Concepts for Grey Input and Output 421
12.2 P-F Theorems of Grey Non-Negative Matrices 424
12.3 Responsibility and Infiuence Coefficients 430
12.4 Optimal Input-Output Models 434
12.5 Dynamic Input-Output Models 440
12.6 Practical Applications 444
Grey Control 450
13.1 Introduction 450
13.2 Grey Linear Control Systems 453
13.3 Grey Transfer Functions and Special Links 457
13.4 Matrices of Grey Transfer Functions 462
13.5 Control with Abandonment 464
13.6 Control of Grey Incidences 465
13.7 Control of Grey Predictions 466
13.8 Practical Applications 468
References 476
Index 505