Stability and Robustness: Reliability in the World of Uncertainty

More documents

Recommendations

Info

The Central Tree ProblemThe Central TreeProblemHeuristics for theCTPRecall that the cospanning tree ¯T of a spanning tree T is theedge complement of T in G, i.e. ¯T = (V, E ¯T), E ¯T = E\E T .Also, if a graph G has n vertices and c connected components,then the rank ρ(G) of graph G is n − c.Final RemarksHeuristics for the Central Tree Problem Yury Nikulin - p. 4/33
The Central Tree ProblemThe Central TreeProblemHeuristics for theCTPFinal RemarksRecall that the cospanning tree ¯T of a spanning tree T is theedge complement of T in G, i.e. ¯T = (V, E ¯T), E ¯T = E\E T .Also, if a graph G has n vertices and c connected components,then the rank ρ(G) of graph G is n − c.Definition. [Deo, 1966]The Central Tree Problem (shortly CTP) consists in finding centraltree T ∗ such that the rank ρ( ¯T ∗ ) of cospanning tree ¯T ∗ isminimal, i.e.ρ( ¯T ∗ ) ≤ ρ( ¯T) ∀ T ∈ T G .Heuristics for the Central Tree Problem Yury Nikulin - p. 4/33
Page 1 and 2:
OutlineThe World of UncertaintyRobu
Page 3 and 4:
OutlineOutlineThe World of Uncertai
Page 5 and 6:
Page 7 and 8:
Page 9 and 10:
Page 11 and 12:
Page 13 and 14:
Interval UncertaintyOutlineThe Worl
Page 15 and 16:
OutlineOutlineThe World of Uncertai
Page 17 and 18:
Page 19 and 20:
Page 21 and 22:
Page 23 and 24:
Page 25 and 26:
Page 27 and 28:
Page 29 and 30:
Page 31 and 32:
Page 33 and 34:
Page 35 and 36:
Page 37 and 38:
Page 39 and 40:
Page 41 and 42:
Page 43 and 44:
Page 45 and 46:
Page 47 and 48:
Page 49 and 50:
Page 51 and 52: ExampleOutlineThe World of Uncertai
Page 53 and 54: ExampleOutlineThe World of Uncertai
Page 55 and 56: OutlineOutlineThe World of Uncertai
Page 57 and 58: OutlineThe World of UncertaintyRobu
Page 59 and 60: Introduction: Sensitivity analysis
Page 61 and 62: Dealing with uncertainty16/04/2009
Page 63 and 64: Classical approachKouvelis P. and Y
Page 65 and 66: Telecommunication and MST16/04/2009
Page 67 and 68: Multicommodity reformulation formin
Page 69 and 70: Dual problem16/04/2009 Yury Nikulin
Page 71 and 72: Interval data and scenariosG = ( E,
Page 73 and 74: RST problem reformulation as mixedi
Page 75 and 76: Robust Spanning Tree Problem within
Page 77 and 78: Example: weak edgesweaknon-weak12-5
Page 79 and 80: Example: Strong edgesstrong12-553-4
Page 81 and 82: CPLEX results on complete graphs16/
Page 83 and 84: Branching strategyNode d with the s
Page 85 and 86: Lower bound16/04/2009 Yury Nikulin,
Page 87 and 88: SA: preprocessing, initial solution
Page 89 and 90: SA: neighborhood search moveCase2.1
Page 91 and 92: Running time16/04/2009 Yury Nikulin
Page 93 and 94: SA summary• In spite of SA is gen
Page 95 and 96: 16/04/2009 Yury Nikulin, Sensitivit
Page 97 and 98: Heuristics for Central Tree Problem
Page 99 and 100: Distance MeasureThe Central TreePro
Page 101: Distance MeasureThe Central TreePro
Page 105 and 106: The CTP vs. MDTPThe Central TreePro
Page 107 and 108: The CTP vs. MDTPThe Central TreePro
Page 109 and 110: Short ExampleThe Central TreeProble
Page 111 and 112: Complexity of the CTPThe Central Tr
Page 113 and 114: Complexity of the CTPThe Central Tr
Page 115 and 116: Equivalence between 0 − 1 RSTP an
Page 117 and 118: One more evidence of NP-hardnessThe
Page 119 and 120: MILP FormulationCorollary from the
Page 121 and 122: Approximation for the CTPThe Centra
Page 123 and 124: The Central TreeProblemHeuristics f
Page 125 and 126: Greedy Construction HeuristicThe Ce
Page 127 and 128: ExampleThe Central TreeProblemHeuri
Page 135 and 136: Computational ExperimentsThe Centra
Page 145 and 146: ConclusionThe Central TreeProblemHe
Page 147 and 148: Sensitivity Analysis in Game Theory
Page 149 and 150: Game descriptionProblemFormulationS
Page 151 and 152: Forming a coalition, bansProblemFor
Page 153 and 154:
EquilibriaProblemFormulationStabili
Page 155 and 156:
Original problemProblemFormulationS
Page 157 and 158:
ExampleProblemFormulationStability
Page 159 and 160:
Stability FunctionProblemFormulatio
Page 161 and 162:
SufficiencyProblemFormulationStabil
Page 163 and 164:
Formula for Stability FunctionProbl
Page 165 and 166:
Sketch of the proofProblemFormulati
Page 167 and 168:
Formula for Stability RadiusProblem
Page 169 and 170:
Independent players, no bansProblem
Page 171 and 172:
Accuracy Function and RadiusProblem
Page 173 and 174:
ExampleProblemFormulationStability
Page 175 and 176:
Concluding remarksProblemFormulatio
Page 177 and 178:
Fuzziness: one more way of dealingw
Page 179 and 180:
Crisp and Fuzzy Numbers28/04/2009 Y
Page 181 and 182:
Geometrical interpretation: Union28
Page 183 and 184:
Geometrical interpretation:Intersec
Page 185 and 186:
Fuzzy sets28/04/2009 Yury Nikulin,
Page 187 and 188:
Comparison of fuzzy numbers:strong
Page 189 and 190:
Maximal and Minimalfuzzy numbersμ(
Page 191 and 192:
Fuzzy maximum and minimum28/04/2009
Page 193 and 194:
Statistical data about flight delay
Page 195 and 196:
Possibility distribution28/04/2009
Page 197 and 198:
Fuzzy Number representing delays28/
Page 199 and 200:
28/04/2009 Yury Nikulin, Sensitivit
Page 201:
28/04/2009 Yury Nikulin, Sensitivit
show all

Stability and Robustness: Reliability in the World of Uncertainty

Create successful ePaper yourself

Delete template?

Save as template?