Stability and Robustness: Reliability in the World of Uncertainty

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ExampleProblemFormulationStability andAccuracyFunctionsLet m = 3, s = 2, J = ({1}, {2, 3}) and⎛1 2 1C 0 ⎜= ⎝ 2 1 21 3 2⎞⎟⎠.Assume also that B = (∅, {(0, 0) T , (1, 1) T }), i.e.X = {x 1 , x 2 , x 3 , x 4 }, x 1 = (0, 0, 1) T , x 2 = (0, 1, 0) T ,x 3 = (1, 0, 1) T , and x 4 = (1, 1, 0) T .Then C 0 x 1 = (1, 2, 2) T , C 0 x 2 = (2, 1, 3) T , C 0 x 3 = (2, 4, 3) T ,C 0 x 4 = (3, 3, 4) T , and JE 3 (C 0 , X) = {x 1 , x 2 }. Using formula(3), we calculate R 3 (C 0 , J, x 1 ) = 1 2 and R3 (C 0 , J, x 2 ) = 1 2 . Ithappens that for both solutions x 1 and x 2 , the stability radius isequal to 1 2 . To get more information about x1 and x 2 , we maycheck the behavior of these solutions when they becomenon-equilibria by means of calculating stability functions oninterval [0, q(C 0 , X)), q(C 0 , X) = 1 using formula (1):Sensitivity Analysis in Game Theory Yury Nikulin - p. 26/29
ExampleProblemFormulationStability andAccuracyFunctions{S(C 0 , J, x 1 , ρ) = max{S(C 0 , J, x 2 , ρ) = maxmax { 0, −1 + ρ2 − 2ρmax{0, −1 + ρ3 − 2ρ}, max{0, min{1 + 2ρ1 − ρ}, max{0,−1 + 2ρ2 − ρ } },,−1 + 2ρ3 − ρwhere ρ ∈ [0, 1).Sensitivity Analysis in Game Theory Yury Nikulin - p. 27/29
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OutlineThe World of UncertaintyRobu
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Introduction: Sensitivity analysis
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Dealing with uncertainty16/04/2009
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Classical approachKouvelis P. and Y
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Telecommunication and MST16/04/2009
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Multicommodity reformulation formin
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Dual problem16/04/2009 Yury Nikulin
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Interval data and scenariosG = ( E,
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RST problem reformulation as mixedi
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Robust Spanning Tree Problem within
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Example: weak edgesweaknon-weak12-5
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Example: Strong edgesstrong12-553-4
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CPLEX results on complete graphs16/
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Branching strategyNode d with the s
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Lower bound16/04/2009 Yury Nikulin,
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SA: preprocessing, initial solution
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SA: neighborhood search moveCase2.1
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Running time16/04/2009 Yury Nikulin
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SA summary• In spite of SA is gen
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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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Stability and Robustness: Reliability in the World of Uncertainty

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