View - Universidad de Almería

226 J. Cole Smith, Fransisca Sudargho, and Churlzu Limmight not only be to generate initial revenues for us, but might also be used as decoys to trickthe opponent into making a poor interdiction decision, if their algorithm relies on these initialflows.In the second stage, the enemy acts to inflict damage to our network arcs by reducing thecapacity of certain arcs (perhaps setting some of their capacities to zero). This action is oftenreferred to as interdiction. The enemy’s objective function can vary depending on its goal. Inparticular, we consider three different cases of the enemy’s objective function. The first casehandles the situation in which the enemy destroys the largest-capacity arcs in our networkdesign. The second case deals with the problem in which the enemy destroys arcs having thelargest initial commodity flows. In the third case, the enemy makes an effort to minimize thetotal profit obtained from transporting commodities to the destination nodes in the network.The first two cases might represent the case in which our enemy intends to maximally disruptour network, but is acting according to a heuristic strategy due to real-time considerationsor due to the complexity of the system. For all of these cases, it is important to note thatthe enemy can destroy portions of arcs by removing some of their capacity, rather than beingrestricted to integer interdiction actions.In the third stage, after the enemy interdiction, some of the arcs in the network will beinaccessible and we will modify the flow in the surviving proportion of the arcs in order tomeet the demand. If some arcs in the network that consist of some demand flows should failand become completely or partially unavailable, another flow pattern must be built on theexisting (previously constructed) arcs. There might not be any possible flows at all if thereexists a small cut-set of network arcs that are inexpensive to destroy. Under this condition,redundant arcs and nodes in the network design are needed to ensure the survivability of thenetwork, given the importance of meeting demands under inopportune situations.The network interdiction problem has received much attention in the literature due to itsapplications in military and homeland security operations. Such papers form the basis forsolving the last two echelons of the problem considered in this paper. Wollmer [6] proposesan algorithm that interdicts a prescribed number of arcs in a network in order to minimizethe follower’s maximal flow. Wood [7] provides an integer programming formulation for adiscrete interdiction problem, and provided an extension of the model to allow for continuousinterdiction, multiple sources and sinks, undirected networks, multiple interdiction resources,and multiple commodities. This work is continued by Cormican et al. [1], in which the leaderminimizes the expected maximum flow, given uncertainties regarding the success of interdictionand arc capacities. A different interdiction problem is examined by Fulkerson andHarding [2], who examine the problem of maximizing the shortest source-sink path in thepresence of arc-extension costs, which serve as interdiction costs. A recent study by Israeliand Wood [3] develops two decomposition algorithms using super-valid inequalities and setcovering master problems. Finally, Lim and Smith [4] examine the multicommodity flow networkinterdiction problem under assumptions of discrete and continuous enemy action.2. Models and Solution MethodsIn this section, we describe models and solution methods under three different interdictionscenarios. To facilitate our discussion, consider a graph G(N, A), with node set N and arcset A. Associated with each arc i ∈ A are a nonnegative construction cost c i , a nonnegativedisruption cost b i , and a mutual arc capacity q i . Define RS(j) and F S(j), ∀j ∈ N, to be theset of arcs going to and going from node j, respectively. Furthermore, define K to be theset of commodities, and let d j kdenote the demand of commodity k ∈ K at node j ∈ N. Ifd j k > 0, then j is a supply node of commodity k, while dj k< 0 implies that j is a destinationof commodity k. Without loss of generality, we assume that ∑ j∈N dj k= 0, ∀k ∈ K. We are