Stability and Robustness: Reliability in the World of Uncertainty

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OutlineThe World of UncertaintyRobust OptimizationStability AnalysisConclusionGeneric Combinatorial Optimization ProblemProblem FormulationIf the sets E and T are fixed, then an instance of m-criteriacombinatorial optimization problem is uniquely determined by thematrix C ∈ R m×n+ .Denote P m (C) the set of all Pareto optimal solutions.Yury NikulinStability and Robustness: Reliability in the World of Uncertainty
OutlineThe World of UncertaintyRobust OptimizationStability AnalysisConclusionMeasuring deviation from the optimumRelative ErrorFor t 0 ∈ P m (C 0 ) and C ∈ R m×n+ , we introduced a so-called relativeerror of t 0 :ε P (C, t 0 ) = maxt∈TTransformation in scalar casemin f i (C, t 0 ) − f i (C, t).i∈N m f i (C, t)In the scalar case, i.e. for m=1, the Pareto set transforms into the setof optimal solutions. Therefore the relative error ε P (C, t 0 ) convertsinto:f 1 (C, t 0 ) − min f 1(C, t)t∈Tε P (C, t 0 ) =min f 1(C, t)t∈T.Yury NikulinStability and Robustness: Reliability in the World of Uncertainty
Page 1 and 2: OutlineThe World of UncertaintyRobu
Page 3 and 4: OutlineOutlineThe World of Uncertai
Page 13 and 14: Interval UncertaintyOutlineThe Worl
Page 41: OutlineThe World of UncertaintyRobu
Page 51 and 52: ExampleOutlineThe World of Uncertai
Page 53 and 54: ExampleOutlineThe 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 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
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SA summary• In spite of SA is gen
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16/04/2009 Yury Nikulin, Sensitivit
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Heuristics for Central Tree Problem
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Distance MeasureThe Central TreePro
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Distance MeasureThe Central TreePro
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The Central Tree ProblemThe Central
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The CTP vs. MDTPThe Central TreePro
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The CTP vs. MDTPThe Central TreePro
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Short ExampleThe Central TreeProble
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Complexity of the CTPThe Central Tr
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Complexity of the CTPThe Central Tr
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Equivalence between 0 − 1 RSTP an
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One more evidence of NP-hardnessThe
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MILP FormulationCorollary from the
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Approximation for the CTPThe Centra
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The Central TreeProblemHeuristics f
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Greedy Construction HeuristicThe Ce
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ExampleThe Central TreeProblemHeuri
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Computational ExperimentsThe Centra
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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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Stability and Robustness: Reliability in the World of Uncertainty

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