Stochastic Network Interdiction Udom Janjarassuk 12/1/2018

Outline Introduction Model Formulation Sample Average Approximation Dual of the maximum flow problem Linearize the nonlinear expression Sample Average Approximation Decomposition Approach Computational Results Further Work 12/1/2018

Introduction Network Interdiction Problem 12/1/2018

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 12/1/2018

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 12/1/2018

Formulation (cont.) where fs(x) is the maximum flow from r to t in scenario s 12/1/2018

Formulation (cont.) uij = Capacity of arc (i,j)  A A’ = A  {r,t} ijs = yijs = flow on arc (i,j) in scenario s 12/1/2018

Formulation (cont.) Maximum flow problem for scenario s 12/1/2018

Formulation (cont.) The dual of the maximum flow problem for scenario s is Strong Duality, we have 12/1/2018

Formulation (cont.) 12/1/2018

Formulation (cont.) 12/1/2018

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 12/1/2018

Formulation (cont.) 12/1/2018

Formulation (cont.) 12/1/2018

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 12/1/2018

Sample Average Approximation(cont.) Lower bound on f(x)=v* Confidence Interval 12/1/2018

Sample Average Approximation(cont.) The (1-)-confidence interval for lower bound Where P(N(0,1)  z)=1-  12/1/2018

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 12/1/2018

Sample Average Approximation(cont.) Confidence Interval The (1-)-confidence interval for upper bound Where P(N(0,1)  z)=1-  12/1/2018

Decomposition Approach Recall our problem in two-stages stochastic form 12/1/2018

Decomposition Approach (cont.) and 12/1/2018

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) 12/1/2018

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 12/1/2018

Computational results (cont.) Note: Optimal objective value in [Cormican,Morton,Wood]=10.9 with error 1% 12/1/2018

Computational results (cont.) SNIP 7x5 example: 12/1/2018

Computational results (cont.) Note: Optimal objective value in [Cormican,Morton,Wood]=80.4 with error 1% 12/1/2018

Further work… Solving bigger instance on computer grid Using Decomposition Approach 12/1/2018

Thank you 12/1/2018