In this research we analyze the effectiveness of the current and optimal locations of a set of existing regional assets maintained by the Department of Defense to respond to large-scale emergencies. These assets have been incrementally resourced, established, sited over the past 20 years without regard to the entire enterprise and, due to fiscal and political costs, modifications to the current structure must yield significant gains to garner approval. We formulate a multiobjective hierarchical extension of the maximal covering location problem that seeks to maximize coverage of the population within a rapid response window while minimizing modifications to the existing structure. Additionally, we prevent facilities from covering nodes located within close proximity using a modified conditional covering problem (CCP) constraint; this constraint accounts for the large impact radius that can occur in a worst-case scenario. To solve our multiobjective problem, we develop a set of non-inferior solutions using the ε-constraint method. These non-inferior solutions explicitly represent the trade-off between maximizing coverage and minimizing cost, and they offer a decision maker a set of Pareto optimal decisions to consider for implementation.
We examine the optimal location of Integrated Air Defense System (IADS) missile batteries to protect a country’s assets, formulated as a Defender-Attacker-Defender three-stage sequential, perfect information, zero-sum game between two opponents. We formulate a trilevel nonlinear integer program for this Defender-Attacker-Defender model and seek a subgame perfect Nash equilibrium (i.e., a set of attacker and defender strategies from which neither player has an incentive to deviate). Such a trilevel formulation is not solvable via conventional optimization software, and an exhaustive enumeration of the game tree based on the discrete set of strategies is only tractable for small instances. We develop and test a customized heuristic over a set of small instances having deliberate parametric variations in a designed experiment, comparing its performance to an exhaustive enumeration algorithm. Testing results indicate the enumeration approach to be severely limited for realistically sized instances, so we demonstrate the heuristic on a larger instance from the literature for which it maintains computational efficiency.
Given the ubiquitous nature of both offensive and defensive missile systems, the catastrophe-causing potential they represent, and the limited resources available to countries for missile defense, optimizing the defensive response to a missile attack is a necessary national security endeavor. For a single salvo of offensive missiles launched at a set of targets, a missile defense system protecting those targets must determine how many interceptors to fire at each incoming missile. Since such missile engagements often involve the firing of more than one attack salvo, we develop a Markov decision process model to examine the optimal fire control policy for the defender. Due to the computational intractability of using exact methods for all but the smallest problem instances, we utilize an approximate dynamic programming (ADP) approach to explore the efficacy of applying approximate methods to the problem. We obtain policy insights by analyzing subsets of the state space that reflect a range of possible defender interceptor inventories. Testing of four instances derived from a representative planning scenario demonstrates that the ADP policy provides high-quality decisions for a majority of the state space, achieving a 7.74% mean optimality gap over all states for the most realistic instance, modeling a longer-term engagement by an attacker who assesses the success of each salvo before launching a subsequent one. Moreover, the ADP algorithm requires only a few minutes of computational effort versus hours for the exact dynamic programming algorithm, providing a method to address more complex and realistically-sized instances.
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