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Stochastic Network Interdiction

Stochastic Network Interdiction. Udom Janjarassuk. Outline. Introduction Model Formulation Dual of the maximum flow problem Linearize the nonlinear expression Sample Average Approximation Decomposition Approach Computational Results Further Work. Introduction.

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Stochastic Network Interdiction

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  1. Stochastic Network Interdiction Udom Janjarassuk

  2. Outline • Introduction • Model Formulation • Dual of the maximum flow problem • Linearize the nonlinear expression • Sample Average Approximation • Decomposition Approach • Computational Results • Further Work

  3. Introduction • Network Interdiction Problem

  4. Introduction (cont.) • Stochastic Network Interdiction Problem (SNIP) • Uncertain successful interdiction • Uncertain arc capacities • Goal: minimize the expected maximum flow • This is a two-stages stochastic integer program • Stage 1: decide which arcs to be interdicted • Stage 2: maximize the expected network flow • Applications • Interdiction of terrorist network • Illegal drugs • Military

  5. Formulation • Directed graph G=(N,A) • Source node rN, Sink node tN • S = Set of finite number of scenarios • ps = Probability of each scenario • K = budget • hij = cost of interdicting arc (i,j)A

  6. Formulation (cont.) where fs(x) is the maximum flow from r to t in scenario s

  7. Formulation (cont.) • uij = Capacity of arc (i,j) A • A’ = A {r,t} • ijs = • yijs = flow on arc (i,j) in scenario s

  8. Formulation (cont.) Maximum flow problem for scenario s

  9. Formulation (cont.) The dual of the maximum flow problem for scenario s is Strong Duality, we have

  10. Formulation (cont.)

  11. Formulation (cont.)

  12. Linearize the nonlinear expression • Linearize xijijs • Let zijs = xijijs xij = 0  zijs = 0 xij = 1  zijs = ijs • Then we have zijs – Mxij <= 0 – ijs+ zijs <= 0 ijs– zijs + Mxij<= M where M is an upper bound for ijs , here M = 1

  13. Formulation (cont.)

  14. Formulation (cont.)

  15. Sample Average Approximation • Why? • Impossible to formulate as deterministic equivalent with all scenarios • Total number of scenarios = 2m, m = # of interdictable arcs • Sample Average Approximations • Generate N samples • Approximate f(x) by

  16. Sample Average Approximation(cont.) • Lower bound on f(x)=v* • Confidence Interval

  17. Sample Average Approximation(cont.) • The (1-)-confidence interval for lower bound Where P(N(0,1)  z)=1- 

  18. Sample Average Approximation(cont.) • Upper bound on f(x) • Estimate of an upper bound (For a fixed x) • Generate T independent batches of samples of size N • Approximate by

  19. Sample Average Approximation(cont.) • Confidence Interval • The (1-)-confidence interval for upper bound Where P(N(0,1)  z)=1- 

  20. Decomposition Approach • Recall our problem in two-stages stochastic form

  21. Decomposition Approach (cont.) and

  22. Decomposition Approach (cont.) • E[Q(x, s)] is piecewise linear, and convex • The problem has complete recourse – feasible set of the second-stage problem is nonempty • The solution set is nonempty • Integer variables only in first stage • Therefore, the problem can be solve by decomposition approach (L-Shaped method)

  23. Computational results SNIP 4x9 example: Note: 1. Only arcs with capacity in ( ) are interdictable 2. The successful of interdiction = 75% 3. Total budget K = 6

  24. Computational results (cont.) Note: Optimal objective value in [Cormican,Morton,Wood]=10.9 with error 1%

  25. Computational results (cont.) SNIP 7x5 example:

  26. Computational results (cont.) Note: Optimal objective value in [Cormican,Morton,Wood]=80.4 with error 1%

  27. Further work… • Solving bigger instance on computer grid • Using Decomposition Approach

  28. Thank you

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