1 General Remarks on Robust Solutions .......................... 1
Y. Ermoliev, M. Makowski, and K. Marti
References ...................................................... 7
Part I
Modeling of Uncertainty and Probabilistic Issues
2 On Joint Modelling of Random Uncertainty and Fuzzy
Imprecision ................................................. 11
Olgierd Hryniewicz
2.1 Introduction ........................................... 11
2.2 Generalizations of Classical Probability and Their
Applications in Decision Making ........................ 13
2.2.1 Measures of Uncertainty and Criteria of Their
Evaluation ...................................... 13
2.2.2 Probability ..................................... 15
2.2.3 Dempster-Shafer Theory of Evidence and
Possibility Theory .............................. 17
2.2.4 Imprecise Probabilities and Their
Generalizations ................................. 20
2.3 Fuzzy Random Variables and Fuzzy Statistics ............ 22
2.4 Applications of Fuzzy Statistics in Systems Analysis ... 29
2.4.1 Example 1: Verification of the Kyoto Protocol ... 29
2.4.2 Example 2: Sequential Testing of a Hypothesis
About the Mean Value in the Normal
Distribution .................................... 31
2.5 Conclusions ............................................ 33
References .................................................. 35
3 On the Approximation of a Discrete Multivariate
Probability Distribution Using the New Concept of t-Cherry
Junction Tree ............................................... 39
Edith Kovács and Tamas Szántai
3.1 Introduction ........................................... 39
3.2 Preliminaries .......................................... 40
3.2.1 Notations ....................................... 40
3.2.2 Cherry Tree and t-Cherry Tree ................... 41
3.2.3 Junction Tree ................................... 42
3.3 f-Cherry-Junction Tree ................................. 43
3.3.1 Construction of a t-Cherry-Junction Tree ........ 43
3.3.2 The Approximation of the Joint Distribution
Over X by the Distribution Associated to
a t-Cherry-Junction Tree ........................ 44
3.3.3 The Relation Between the Approximations
Associated to the First-Order Dependence Tree
and t-Cherry-Junction Tree ...................... 47
3.4 Some Practical Results of Our Approximation and
Discussions ............................................ 50
References .................................................. 56
Part II
Robust Solutions Under Uncertainty
4 Induced Discounting and Risk Management ..................... 59
T. Ermolieva, Y. Ermoliev, G. Fischer, and M. Makowski
4.1 Introduction ........................................... 59
4.2 Standard and Stopping Time Induced Discounting ......... 62
4.3 Time Declining Discount Rates .......................... 65
4.4 Endogenous Discounting ................................. 68
4.5 Dynamic Risk Profiles and CVaR Risk Measure ............ 71
4.6 Intertemporal Inconsistency ............................ 73
4.7 Concluding Remarks ..................................... 75
References .................................................. 76
5 Cost Effective and Environmentally Safe Emission Trading
Under Uncertainty ........................................... 79
T. Ermolieva, Y. Ermoliev, G. Fischer, M. Jonas, and
M. Makowski
5.1 Introduction ........................................... 79
5.2 Uncertainties and Trends of Carbon Fluxes .............. 82
5.3 Detectability of Emission Changes ...................... 84
5.4 Trade Equilibrium Under Uncertainty .................... 86
5.5 Dynamic Bilateral Trading Processes .................... 90
5.6 Computerized Multi-agent Decentralized Trading
System ................................................. 92
5.7 Myopic Market Processes ................................ 93
5.8 Concluding Remarks ..................................... 96
References .................................................. 97
6 Robust Design of Networks Under Risks ...................... 101
Y. Ermoliev, A. Gaivoronski, and M. Makowski
6.1 Introduction .......................................... 101
6.2 Cooperative Provision of Advanced Mobile Data
Services .............................................. 104
6.3 Simplified Model of the Service Portfolio ............. 106
6.3.1 Description of Services ........................ 106
6.3.2 Profit Model of an Actor ....................... 108
6.3.3 Service Portfolio: Financial Perspective ....... 110
6.4 Modeling of Collaborative Service Provision ........... 113
6.4.1 Service Provision Capacities ................... 114
6.4.2 Risk/Return Industrial Expectations ............ 115
6.4.3 Pricing ........................................ 116
6.4.4 Revenue Sharing Schemes ........................ 116
6.5 Properties of the Models and Implementation Issues .... 118
6.6 Case Study ............................................ 119
6.7 Dynamics of Attitudes ................................. 122
6.7.1 Simplified Model: Direct and Indirect
Interdependencies .............................. 123
6.7.2 Model Formulation .............................. 125
6.7.3 Bayesian Networks and Markov Fields ............ 130
6.7.4 Sensitivity Analysis ........................... 131
6.7.5 General Interdependencies ...................... 133
6.8 Conclusion ............................................ 136
References ................................................. 136
Part III
Analysis and Optimization of Technical Systems and
Structures Under Uncertainty
7 Optimal Ellipsoidal Estimates of Uncertain Systems:
An Overview and New Results ................................ 141
F.L. Chernousko
7.1 Introduction .......................................... 141
7.2 Reachable Sets ........................................ 142
7.3 Ellipsoidal Bounds .................................... 145
7.4 Optimality ............................................ 146
7.5 Equations of Ellipsoids ............................... 148
7.6 Transformation of the Equations ....................... 150
7.7 Properties of Optimal Ellipsoids ...................... 152
7.8 Generalizations ....................................... 153
7.9 Applications .......................................... 154
7.9.1 Two-Sided Estimates in Optimal Control ......... 154
7.9.2 Two-Sided Bounds on Time for the Time-Optimal
Problem ........................................ 155
7.9.3 Suboptimal Control ............................. 155
7.9.4 Differential Games ............................. 156
7.9.5 Control of Uncertain Systems ................... 157
7.9.6 Other Applications ............................. 157
7.9.7 State Estimation in the Presence of
Observation Errors ............................. 158
7.10 Ellipsoidal vs. Interval Analysis ..................... 159
7.11 Conclusions ........................................... 160
References ................................................. 160
8 Expected Total Cost Minimum Design of Plane Frames
by Means of Stochastic Linear Programming Methods .......... 163
Kurt Marti
8.1 Introduction .......................................... 164
8.1.1 Plastic Analysis of Structures ................. 164
8.1.2 Limit (Collapse) Load Analysis of Structures
as a Linear Programming Problem ................ 165
8.1.3 Plastic and Elastic Design of Structures ....... 167
8.2 Plane Frames .......................................... 168
8.2.1 Yield Condition in Case of M - N-
Interaction .................................... 173
8.2.2 Approximation of the Yield Condition by Using
Reference Capacities ........................... 180
8.3 Stochastic Optimization ............................... 183
8.3.1 Violation of the Yield Condition ............... 184
8.3.2 Cost Function .................................. 185
8.3.3 Choice of the Cost Factors ..................... 186
8.3.4 Total Costs .................................... 187
8.3.5 Discretization Methods ......................... 189
8.3.6 Complete Recourse .............................. 190
References ................................................. 191
Part IV
Analysis and Optimization of Economic and Engineering
Systems Under Uncertainty
9 Uncertainty in the Future Nitrogen Load to the Baltic
Sea Due to Uncertain Meteorological Conditions ............. 195
Jerzy Bartnicki
9.1 Introduction .......................................... 195
9.2 Nitrogen Emissions .................................... 198
9.2.1 National Emission Ceilings According to EU
NEC Directive .................................. 198
9.2.2 National Emission Ceilings According
to Gothenburg Protocol ......................... 199
9.2.3 Nitrogen Emission Projections Used in the
Model Runs ..................................... 200
9.3 Computed Nitrogen Depositions for 2010 ................ 201
9.3.1 Unified EMEP Model ............................. 202
9.3.2 Calculated Depositions to Sub-basins and
Catchments of the Baltic Sea ................... 203
9.4 Uncertainty Due to Meteorological Variability ......... 203
9.5 Conclusions ........................................... 207
References ................................................. 207
10 Planning Sustainable Agricultural Development Under
Risks ...................................................... 209
G. Fischer, T. Ermolieva, and L. Sun
10.1 Introduction .......................................... 209
10.2 Cooperation and Co-existence for Risk Sharing ......... 211
10.3 Agricultural Planning Under Risks ..................... 214
10.3.1 A Simulation Model ............................. 214
10.3.2 A Simplified Production Model .................. 216
10.3.3 A Rebalancing Production-Allocation
Algorithm ...................................... 217
10.4 Stochastic Production Allocation Model ................ 218
10.5 Numerical Experiments ................................. 221
10.6 Conclusions ........................................... 225
References ................................................. 226
11 Dealing with Uncertainty in GHG Inventories: How to Go
About It? .................................................. 229
Matthias Jonas, Thomas White, Gregg Marland, Daniel
Lieberman, Zbigniew Nahorski, and Sten Nilsson
11.1 Introduction .......................................... 230
11.2 Does Uncertainty Matter? .............................. 232
11.3 State of the Art of Analyzing Uncertain Emission
Changes ............................................... 233
11.4 How to Deal with Uncertainty? ......................... 238
11.5 Conclusions ........................................... 241
References ................................................. 242
12 Uncertainty Analysis of Weather Controlled Systems ......... 247
K.J. Keesman and T. Doeswijk
12.1 Introduction .......................................... 247
12.2 Preliminaries ......................................... 249
12.2.1 Bulk Storage Model ............................. 249
12.2.2 Weather Forecasts .............................. 249
12.2.3 Cost Function .................................. 250
12.2.4 Receding Horizon Optimal Control ............... 250
12.3 Weather Forecast Uncertainty and Error Analysis ...... 252
12.3.1 Open Loop Evaluation ........................... 252
12.3.2 Closed Loop Evaluation ......................... 254
12.4 Discussion ............................................ 256
12.5 Concluding Remarks .................................... 257
References ................................................. 257
13 Estimation of the Error in Carbon Dioxide Column
Abundances ................................................. 259
Mitsuhiro Tomosada, Koji Kanefuji, Yukio Matsumoto,
Hiroe Tsubaki, and Tatsuya Yokota
13.1 Introduction .......................................... 259
13.2 Trace Gas Measurement by Satellite Remote Sensing ..... 261
13.2.1 Observations of Trace Gases with Various
Sensors ........................................ 261
13.2.2 GOSAT Mission .................................. 262
13.2.3 Previous Error Analysis ........................ 264
13.3 Error Evaluation and Results .......................... 266
13.3.1 Retrieval Method ............................... 266
13.3.2 Error Evaluation ............................... 268
13.3.3 Error Evaluation Results ....................... 270
13.4 Conclusions .............................................. 276
References .................................................... 276
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