1. Stochastic Network Interdiction
Udom Janjarassuk
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2. 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
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3. Introduction
Network Interdiction Problem
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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
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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
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6. Formulation (cont.)
where fs(x) is the maximum flow from r to t in scenario s
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7. Formulation (cont.) uij = Capacity of arc (i,j)  A A’ = A  {r,t}
ijs =
yijs = flow on arc (i,j) in scenario s
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8. Formulation (cont.)
Maximum flow problem for scenario s
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9. Formulation (cont.)
The dual of the maximum flow problem for scenario s is
Strong Duality, we have
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10. Formulation (cont.)
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11. Formulation (cont.)
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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
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13. Formulation (cont.)
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14. Formulation (cont.)
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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
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16. Sample Average Approximation(cont.)
Lower bound on f(x)=v*
Confidence Interval
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17. Sample Average Approximation(cont.)
The (1-)-confidence interval for lower bound
Where P(N(0,1)  z)=1- 
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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
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19. Sample Average Approximation(cont.)
Confidence Interval
The (1-)-confidence interval for upper bound
Where P(N(0,1)  z)=1- 
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20. Decomposition Approach
Recall our problem in two-stages stochastic form
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21. Decomposition Approach (cont.)
and
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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)
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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
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24. Computational results (cont.)
Note: Optimal objective value in [Cormican,Morton,Wood]=10.9 with error 1%
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25. Computational results (cont.)
SNIP 7x5 example:
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26. Computational results (cont.)
Note: Optimal objective value in [Cormican,Morton,Wood]=80.4 with error 1%
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27. Further work… Solving bigger instance on computer grid
Using Decomposition Approach
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28. Thank you
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