Stability and Robustness: Reliability in the World of Uncertainty

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Exact algorithms•Kouvelis and Yu 1997 – first formulation for finite set of scenarios. Ithas been proven that the problem is NP-hard for bounded set S andstrongly NP-hard for unbounde set S.•Yaman, Karasan and Pinar (2001) – basic theoretical backgrounds forthe problem•Aron and Hentenryck (2002) – one more evidence of NP-hardness: theRST problem is related to the well-known central tree problem. Twocases when the RST problem is easily solvable:a) no edge intervals have intersectionsb) graph is complete and all edge costs take their value from the sameinterval•Montemanni and Gambardella (2003) – branch and bound procedurewhich outperforms all exact algorithms known before16/04/2009 Yury Nikulin, Sensitivity analysis 17
Robust Spanning Tree Problem withinterval data16/04/2009 Yury Nikulin, Sensitivity analysis 18
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OutlineThe World of UncertaintyRobu
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OutlineOutlineThe World of Uncertai
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Interval UncertaintyOutlineThe Worl
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OutlineOutlineThe World of Uncertai
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Page 23 and 24: OutlineThe World of UncertaintyRobu
Page 51 and 52: ExampleOutlineThe World of Uncertai
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Page 55 and 56: OutlineOutlineThe World of Uncertai
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: RST problem reformulation as mixedi
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 and 102: Distance MeasureThe Central TreePro
Page 103 and 104: The Central Tree ProblemThe Central
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
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Greedy Construction HeuristicThe Ce
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ExampleThe Central TreeProblemHeuri
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The Central TreeProblemHeuristics f
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Computational ExperimentsThe Centra
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The Central TreeProblemHeuristics f
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ConclusionThe Central TreeProblemHe
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Sensitivity Analysis in Game Theory
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Game descriptionProblemFormulationS
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Forming a coalition, bansProblemFor
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EquilibriaProblemFormulationStabili
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Original problemProblemFormulationS
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ExampleProblemFormulationStability
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Stability FunctionProblemFormulatio
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SufficiencyProblemFormulationStabil
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Formula for Stability FunctionProbl
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Sketch of the proofProblemFormulati
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Formula for Stability RadiusProblem
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Independent players, no bansProblem
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Accuracy Function and RadiusProblem
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ExampleProblemFormulationStability
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Concluding remarksProblemFormulatio
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Fuzziness: one more way of dealingw
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Crisp and Fuzzy Numbers28/04/2009 Y
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Geometrical interpretation: Union28
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Geometrical interpretation:Intersec
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Fuzzy sets28/04/2009 Yury Nikulin,
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Comparison of fuzzy numbers:strong
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Maximal and Minimalfuzzy numbersμ(
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Fuzzy maximum and minimum28/04/2009
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Statistical data about flight delay
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Possibility distribution28/04/2009
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Fuzzy Number representing delays28/
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28/04/2009 Yury Nikulin, Sensitivit
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28/04/2009 Yury Nikulin, Sensitivit
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Stability and Robustness: Reliability in the World of Uncertainty

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