Stochastic Network Interdiction

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