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228 J. Cole Smith, Fransisca Sudargho, and Churlzu LimIn order to force the w-variables to match the enemy’s true solution, we can enact the followingcutting plane method. Suppose that we solve (1) and obtain variables ˆx and ŵ. Letw i be the true action of the enemy for i ∈ A. For each i ∈ A such that w i < ŵ i , we add thefollowing constraint:∑i−1w i ≤ w i x i + (min{1 − w i , b g /b i }) (1 − ˆx g )x g . (2)g=1Each such constraint is then passed back to (1), and the model is resolved until w = ŵ. Essentially,(2) imposes an upper bound on w i of w i , unless some other higher-preference arcsthat are not currently being built will be built in the next iteration, thus forcing the enemy toexpend resources on other arcs. There exist several other necessary conditions for w-feasibilitythat can be stated a priori in (1), but we omit them here for the sake of brevity.A model that can be solved statically makes use of the fact that there will exist only one w-variable value that can be fractional, as in the case in any knapsack decision problem. Definebinary decision variables z i ∀i ∈ A equal to one if and only if x i = 1, all constructed arcswith an index smaller than i are completely destroyed, and all constructed arcs with an indexgreater than i are not affected by the enemy. Arc i itself may be interdicted in any manner.The total enemy interdiction must exhaust their entire budget. (This forces us to build atleast enough capacity so that the enemy can destroy B units; we can remove this restrictiveassumption by use of some simple modeling tricks.) These restrictions are enforced as follows:∑z i = 1g∈Aw i ≤(3a)i∑z g ∀i ∈ A (3b)g=1|A|∑w i ≥ x i − z g ∀i ∈ A (3c)g=iw i ≤ x i ∀i ∈ A (3d)∑b i (x i − w i ) = B.(3e)i∈AOur preliminary computational experience has shown that incorporating (3) into (1), and solvingas a static integer program, is a much more effective method of solving this problem thanby using the initial cutting-plane method developed above.2.2 Flow-Based Greedy Interdiction CaseIn this case, the enemy wishes to interdict the arcs having the most initial flows on them. Sincethe enemy decision does not depend on a simple set of binary decision variables, the foregoingalgorithm must be modified for this case.Once again, we determine an index i ∗ such that arc i ∗ may be partially destroyed, implyingthat every arc with a greater flow than i ∗ must be completely destroyed, while all arcs witha smaller flow than i ∗ cannot be interdicted. Unlike the previous case, we do not define adecision variable to determine the identity of i ∗ , but must solve one integer program for eachpossible value that i ∗ can take. If there exists an arc that has the same amount of flow as i ∗ , wewill use the enemy’s secondary priority to determine whether or not it is destroyed. Lettingp i ∀i ∈ A be the enemy’s secondary priority for arc i (a smaller number denotes a higher