Stochastic Network Interdiction Udom Janjarassuk 12302021 1 Outline

Stochastic Network Interdiction Udom Janjarassuk 12/30/2021 1

Outline Introduction n Model Formulation n Dual of the maximum flow problem n Linearize the nonlinear expression n Sample Average Approximation n Decomposition Approach n Computational Results n Further Work n 12/30/2021 2

Introduction n Network Interdiction Problem 12/30/2021 3

Introduction (cont. ) n Stochastic Network Interdiction Problem (SNIP) n n Goal: minimize the expected maximum flow This is a two-stages stochastic integer program n n n Uncertain successful interdiction Uncertain arc capacities Stage 1: decide which arcs to be interdicted Stage 2: maximize the expected network flow Applications n n n Interdiction of terrorist network Illegal drugs Military 12/30/2021 4

Formulation n n n 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 12/30/2021 5

Formulation (cont. ) where fs(x) is the maximum flow from r to t in scenario s 12/30/2021 6

Formulation (cont. ) n uij = Capacity of arc (i, j) A A’ = A {r, t} n ijs = n yijs = flow on arc (i, j) in scenario s n 12/30/2021 7

Formulation (cont. ) Maximum flow problem for scenario s 12/30/2021 8

Formulation (cont. ) The dual of the maximum flow problem for scenario s is Strong Duality, we have 12/30/2021 9

Formulation (cont. ) 12/30/2021 10

Formulation (cont. ) 12/30/2021 11

Linearize the nonlinear expression n Linearize xij ijs zijs = xij ijs xij = 0 xij = 1 n Then we have n Let zijs = 0 zijs = ijs zijs – Mxij <= 0 – ijs + zijs <= 0 ijs – zijs + Mxij <= M where M is an upper bound for ijs , here M = 1 12/30/2021 12

Formulation (cont. ) 12/30/2021 13

Formulation (cont. ) 12/30/2021 14

Sample Average Approximation n Why? n n n Impossible to formulate as deterministic equivalent with all scenarios Total number of scenarios = 2 m, m = # of interdictable arcs Sample Average Approximations n n 12/30/2021 Generate N samples Approximate f(x) by 15

Sample Average Approximation(cont. ) n Lower bound on f(x)=v* n Confidence Interval 12/30/2021 16

Sample Average Approximation(cont. ) n The (1 - )-confidence interval for lower bound Where P(N(0, 1) z )=1 - 12/30/2021 17

Sample Average Approximation(cont. ) n Upper bound on f(x) n Estimate of an upper bound (For a fixed x) n n Generate T independent batches of samples of size N Approximate by 12/30/2021 18

Sample Average Approximation(cont. ) n Confidence Interval Where P(N(0, 1) z )=1 - n The (1 - )-confidence interval for upper bound 12/30/2021 19

Decomposition Approach 12/30/2021 20

Decomposition Approach (cont. ) and 12/30/2021 21

Decomposition Approach (cont. ) E [Q(x, s)] is piecewise linear, and convex n The problem has complete recourse – feasible set of the second-stage problem is nonempty n The solution set is nonempty n Integer variables only in first stage n Therefore, the problem can be solve by decomposition approach (L-Shaped method) n 12/30/2021 22

Computational results SNIP 4 x 9 example: Note: 1. Only arcs with capacity in ( ) are interdictable 2. The successful of interdiction = 75% 3. Total budget K = 6 12/30/2021 23

Computational results (cont. ) Note: Optimal objective value in [Cormican, Morton, Wood]=10. 9 with error 1% 12/30/2021 24

Computational results (cont. ) SNIP 7 x 5 example: 12/30/2021 25

Computational results (cont. ) Note: Optimal objective value in [Cormican, Morton, Wood]=80. 4 with error 1% 12/30/2021 26

Further work… n Solving bigger instance on computer grid n Using Decomposition Approach 12/30/2021 27

Thank you 12/30/2021 28