Stochastic Network Interdiction

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Stochastic Network Interdiction

• Udom Janjarassuk

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Outline

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

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Introduction

• Network Interdiction Problem

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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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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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Formulation (cont.)
where fs(x) is the maximum flow from r to t in
scenario s
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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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Formulation (cont.)
Maximum flow problem for scenario s
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Formulation (cont.)
The dual of the maximum flow problem for scenario
s is
Strong Duality, we have
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Formulation (cont.)
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Formulation (cont.)
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Linearize the nonlinear expression

• Linearize xij?ijs
• Let zijs xij?ijs
• xij 0 ? zijs 0
• xij 1 ? zijs ?ijs
• Then we have
• zijs Mxij lt 0
• ?ijs zijs lt 0
• ?ijs zijs Mxij lt M
• where M is an upper bound for ?ijs , here M 1

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Formulation (cont.)
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Formulation (cont.)
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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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Sample Average Approximation(cont.)

• Lower bound on f(x)v
• Confidence Interval

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Sample Average Approximation(cont.)

• The (1-?)-confidence interval for lower bound

Where P(N(0,1) ? z?)1- ?
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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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Sample Average Approximation(cont.)

• Confidence Interval
• The (1-?)-confidence interval for upper bound

Where P(N(0,1) ? z?)1- ?
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Decomposition Approach

• Recall our problem in two-stages stochastic form

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Decomposition Approach (cont.)
and
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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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Computational results
SNIP 4x9 example
Note 1. Only arcs with capacity in ( ) are
interdictable 2. The successful of
interdiction 75 3. Total budget K
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Computational results (cont.)
Note Optimal objective value in
Cormican,Morton,Wood10.9 with error 1
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Computational results (cont.)
SNIP 7x5 example
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Computational results (cont.)
Note Optimal objective value in
Cormican,Morton,Wood80.4 with error 1
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Further work

• Solving bigger instance on computer grid
• Using Decomposition Approach

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Thank you