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Survivable Network Design 227given per-unit flow revenue fk i for transmitting a unit of commodity k over arc i for eachi ∈ A and k ∈ K. This flow revenue accommodates the flow cost as well as the reward asnecessary. That is, fk i includes the reward for commodity k if it enters a destination node of k,and has this reward subtracted from the flow cost if it exits a destination node for k. If neitheror both of these cases hold, fk i is simply the negative per-unit flow cost associated with arc iand commodity k. The enemy’s interdiction is subject to a budget limitation of B, given thedifferent disruption costs b i for each arc i ∈ A. We will introduce some dummy arcs in thenetwork to ensure the feasibility of our models; these dummy arcs have zero construction andflow costs, and large enough interdiction costs and capacities to ensure that they cannot beeffectively destroyed by the enemy.The first set of decision variables determines whether or not we construct an arc. Let x i ,∀i ∈ A, be a binary decision variable that equals to 1 if arc i is constructed and 0 otherwise.For the flow decision variables, we let u i k and vi k, ∀i ∈ A, ∀k ∈ K, be the decision variables thatrepresent actual flows of commodity k before and after the enemy’s interdiction, respectively.Also, let w i ∈ [0, 1], ∀i ∈ A, represent the remaining percentage of arc i after the enemydestroys his/her preferred set of arcs. Although w i is determined by the enemy, we view it asa decision variable induced by our choice of x-variables. Finally, we define ũ and ṽ to be theweights of the initial flow profit and final flow profit, respectively.2.1 Capacity-Based Greedy Interdiction CaseIn this subsection, suppose that the enemy repeatedly destroys arcs with the largest capacityuntil the budget B is exhausted. Assume that the enemy breaks a tie with a certain preferenceorderknown a priori. Order the arc indices i = 1, . . . , |A| so that the enemy prefers arc i to arci + 1. A first attempt at formulating this problem is given as follows.Maximizesubject toũ ∑ ∑fi k uk i + ṽ ∑ ∑fi k vk i − ∑ c i x ik∈K i∈A k∈K i∈A i∈A∑c i x i ≤ Ci∈A∑i∈F S(j)∑i∈F S(j)u k i −v k i −∑i∈RS(j)∑i∈RS(j)(1a)(1b)u k i = dk j ∀k ∈ K ∀j ∈ N (1c)v k i = dk j ∀k ∈ K ∀j ∈ N (1d)∑u k i ≤ q i x i ∀i ∈ A (1e)k∈K∑vi k ≤ q i w i ∀i ∈ A (1f)k∈Kw i ≤ x i ∀i ∈ A (1g)w i ≥ 0 ∀i ∈ A (1h)u k i , vk i ≥ 0 ∀i ∈ A ∀k ∈ K (1i)x i ∈ {0, 1} ∀i ∈ A. (1j)Note that (1b) is a construction budget constraint, (1c)-(1d) are flow conservations before andafter enemy’s action, respectively, and (1e)-(1f) are arc capacities before and after enemy’saction, respectively. Also, (1g) imply that the remaining portion of an arc can have a positivevalue only when its arc is constructed. However, these conditions are only necessary for thew-solution to this problem to reflect the true decision of the enemy, and are certainly notsufficient.