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Survivable Network Design 229preference), we can add the following constraints to model (1):∑u k i − ∑ u k i ∗ ≤ M(x i − w i ) + ε(p i − p i ∗) ∀i ≠ i ∗ , i ∈ A (4a)k∈K k∈K∑u k i ∗ − ∑ u k i ≤ M(1 − (x i − w i )) + ε(p i ∗ − p i ) ∀i ≠ i ∗ , i ∈ A (4b)k∈K k∈K∑b i (x i − w i ) = B(4c)i∈A0 ≤ w i ∗ ≤ 1 ∀i ∗ ∈ A (4d)w i ∈ {0, 1} ∀i ∈ A − {i ∗ }, (4e)where ε is a very small constant. The addition of the constraints in (4) captures the flow-basedgreedy enemy interdiction model. If there exists more initial flow on arc i than on i ∗ , thensince x i = 1, we have by (4a) that arc i must be interdicted (by setting w i = 0). If there is a tie,but i ∗ is preferred to i, then (4a) once again implies that w i = 0. Similarly, (4b) forces w i = x iif there is less flow on i than i ∗ , or if there is a tie when i has a lower priority than i ∗ .2.3 Optimal Interdiction CaseFinally, we consider the third case in which the enemy optimally disrupts arcs so as to minimizeour profit. We assume that the enemy has complete information of our network designincluding arc capacities, flow revenues, and demands. Given a network topology x, therefore,the enemy solves a continuous multicommodity flow network interdiction problem in amin-max structure. Taking the linear dual of the inner maximization problem, this problemcan be reformulated as a disjointly constrained bilinear program (BLP), in which the enemy’sdecision variables and our dual variables associated with arc capacity constraints constitutebilinear terms in the objective function.One important property of this BLP formulation is that a global optimum can be foundamong pairs of extreme points from respective feasible regions. In particular, the enemy hasa single knapsack constraint besides bounds on variables, and hence, each extreme point hasonly one basic variable while other nonbasic variables are set at one of their bounds 0 or 1.Exploiting this fact, Lim and Smith [4] recently proposed an optimal algorithm that solveslinearized mixed integer subproblems obtained by designating one variable as basic. Thus, anexact solution can be identified after solving |A| subproblems. (See [4] for details.)Note that there exists a finite number of pairs of extreme points for disjoint polyhedral setsof the bilinear programming problem. Let P denote the set of such pairs. Furthermore, letθ x (p) denote the objective value of the bilinear program at p ∈ P given x. Then, our networkdesign problem can be formulated as follows.Maximizeũ ∑ ∑fi k u k i − ∑ c i x i + ṽzk∈K i∈A i∈Asubject to (1b), (1e), (1j), and u k i ≥ 0 ∀i ∈ A ∀k ∈ K (5b)z ≤ θ x (p) ∀p ∈ P (5c)(5a)z unrestricted. (5d)As a solution method, we prescribe the following cutting plane algorithm, BCPA that exploitsBenders cuts in an iterative fashion.