Brian J. Lunday

On generating maximal nondominated Benders cuts

In this paper, we explore certain algorithmic strategies for accelerating the convergence of Benders decomposition method via the generation of maximal nondominated cuts. Based on interpreting the seminal work of Magnanti and Wong (Operations Research, 29(3), 464–484, 1981) for generating nondominated cuts within a multiobjective framework, we propose an algorithmic strategy that utilizes a preemptively small perturbation of the right-hand-side of the Benders subproblem to generate maximal nondominated Benders cuts, as well as a complimentary strategy that generates an additional cut in each iteration via an alternative emphasis on decision variable weights. We also examine the computational effectiveness of solving a secondary subproblem using an objective cut as proposed by Magnanti and Wong versus identifying the Pareto-optimality region for cut generation by utilizing complementary slackness conditions. In addition, we exhibit how a standard feasibility cut can be extracted from the solution of subproblems that generate only optimality cuts through the use of artificial variables. With Magnanti and Wong’s baseline procedure approximated during implementation via the use of a core point estimation technique (Papadakos in Computers and Operations Research, 36(1), 176–195, 2009), these algorithmic strategies are tested on instances from the literature concerning the fixed charge network flow program.

A bilevel formulation of the pediatric vaccine pricing problem

We consider the characterization of optimal pricing strategies for a pediatric vaccine manufacturing firm operating in an oligopolistic market. The pediatric vaccine pricing problem (PVPP) is formulated as a bilevel mathematical program wherein the upper level models a firm that selects profit-maximizing vaccine prices while the lower level models a representative customer’s vaccine purchasing decision to satisfy a given, recommended childhood immunization schedule (RCIS) at overall minimum cost. Complicating features of the bilevel program include the bilinear nature of the upper-level objective function and the binary nature of the lower-level decision variables. We develop and test variants of three heuristics to identify the pricing scheme that will maximize a manufacturer’s profit: a Latin Hypercube Sampling (LHS) of the upper-level feasible region, an LHS enhanced by a Nelder–Meade search from each price point, and an LHS enhanced by a custom implementation of the Cyclic Coordinate Method from each price point. The practicality of the PVPP is demonstrated via application to the analysis of the 2014 United States pediatric vaccine private sector market. Testing results indicate that a robust sampling method combined with local search is the superlative solution method among those examined and, in the current market, that a manufacturer acting unilaterally has the potential to increase profit per child completing the RCIS by 35 percent (from 231.84 to 312.55 dollars) for GlaxoSmithKline, 47 percent (from 63.96 to 93.70 dollars) for Merck, and 866 percent (from 25.99 to 251.04 dollars) for Sanofi Pasteur over that obtained via current pricing mechanisms.

Approximate dynamic programming for missile defense interceptor fire control

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 (MDP) 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.

A Game Theoretic Model for the Optimal Location of Integrated Air Defense System Missile Batteries

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.

A markov decision process model for the optimal dispatch of military medical evacuation assets.

We develop a Markov decision process (MDP) model to examine aerial military medical evacuation (MEDEVAC) dispatch policies in a combat environment. The problem of deciding which aeromedical asset to dispatch to each service request is complicated by the threat conditions at the service locations and the priority class of each casualty event. We assume requests for MEDEVAC support arrive sequentially, with the location and the priority of each casualty known upon initiation of the request. The United States military uses a 9-line MEDEVAC request system to classify casualties as being one of three priority levels: urgent, priority, and routine. Multiple casualties can be present at a single casualty event, with the highest priority casualty determining the priority level for the casualty event. Moreover, an armed escort may be required depending on the threat level indicated by the 9-line MEDEVAC request. The proposed MDP model indicates how to optimally dispatch MEDEVAC helicopters to casualty events in order to maximize steady-state system utility. The utility gained from servicing a specific request depends on the number of casualties, the priority class for each of the casualties, and the locations of both the servicing ambulatory helicopter and casualty event. Instances of the dispatching problem are solved using a relative value iteration dynamic programming algorithm. Computational examples are used to investigate optimal dispatch policies under different threat situations and armed escort delays; the examples are based on combat scenarios in which United States Army MEDEVAC units support ground operations in Afghanistan.

Network interdiction to minimize the maximum probability of evasion with synergy between applied resources

In this paper, we model and solve the network interdiction problem of minimizing the maximum probability of evasion by an entity traversing a network from a given source to a designated terminus, while incorporating novel forms of superadditive synergy between resources applied to arcs in the network. Inspired primarily by operations to coordinate Iraqi and U.S. security forces seeking to interdict an evader attempting to avoid detection while transiting part of the nearly rectilinear street network in East Baghdad, this study motivates and examines either linear or concave (nonlinear) synergy relationships between the applied resources within our formulations. We also propose an alternative model for sequential overt and covert deployment of subsets of interdiction resources, and conduct theoretical as well as empirical comparative analyses between models for purely overt (with or without synergy) and composite overt-covert strategies to provide insights into absolute and relative threshold criteria for recommended resource utilization. Our empirical results confirm the value of tactical patience regarding decisions on the covert utilization of resources for network interdiction. Furthermore, considering non-integral and integral resource allocations, we identify (theoretically and empirically) parametric characteristics of instances that exhibit the relative worth of employing partially covert operations. Under the relatively more practical scenario involving integral resource allocations, we demonstrate that the composite overt-covert strategy of deploying resources has a greater potential to improve over a purely overt resource deployment strategy, both with and without synergy, particularly when costs are positively correlated, resources are plentiful, and a sufficiently high ratio of covert to overt resources exists. Moreover, should an interdictor be able to ascertain an optimal evader path, the potential and magnitude of this relative improvement for the overt-covert resource allocation strategy is significantly greater.

Robust goal programming using different robustness echelons via norm-based and ellipsoidal uncertainty sets

The notion of robust goal programming (RGP) using cardinality-constrained robustness via interval-based uncertainty was first examined over a decade ago. Since then, the RGP methodology has not been widely researched, specifically when considering different uncertainty sets to implement. Within this context, this paper compares interval-based and norm-based uncertainty sets using cardinality-constrained robustness. Strict robustness using ellipsoidal uncertainty sets is also examined in the RGP realm. The aforementioned methods are demonstrated for a simple instance from the literature, and the results are summarized. Conclusions are made regarding the proposed RGP models when likened to a similar RGP model seen in the literature. Further, the suitability of each RGP model is offered when a decision makers risk preference or computing availability are taken into consideration. Inferences are made regarding the effectiveness of each uncertainty set in the context of solutions that are relatively unaffected by data uncertainty – that is, robust solutions.

Approximate dynamic programming for the dispatch of military medical evacuation assets

Military medical planners must consider the dispatching of aerial military medical evacuation (MEDEVAC) assets when preparing for and executing major combat operations. The launch authority seeks to dispatch MEDEVAC assets such that prioritized battlefield casualties are transported quickly and efficiently to nearby medical treatment facilities. We formulate a Markov decision process (MDP) model to examine the MEDEVAC dispatching problem. The large size of the problem instance motivating this research suggests that conventional exact dynamic programming algorithms are inappropriate. As such, we employ approximate dynamic programming (ADP) techniques to obtain high quality dispatch policies relative to current practices. An approximate policy iteration algorithmic strategy is applied that utilizes least squares temporal differencing for policy evaluation. We construct a representative planning scenario based on contingency operations in northern Syria both to demonstrate the applicability of our MDP model and to examine the efficacy of our proposed ADP solution methodology. A designed computational experiment is conducted to determine how selected problem features and algorithmic features affect the quality of solutions attained by our ADP policies. Results indicate that the ADP policy outperforms the myopic policy (i.e., the default policy in practice) by up to nearly 31% with regard to a lifesaving performance metric, as compared for a baseline scenario. Moreover, the ADP policy provides decreased MEDEVAC response times and utilization rates. These results benefit military medical planners interested in the development and implementation of cogent MEDEVAC tactics, techniques, and procedures for application in combat situations with a high operations tempo.

The coastal seaspace patrol sector design and allocation problem

In this paper, we model and solve the problem of designing and allocating coastal seaspace sectors for steady-state patrolling operations by the vessels of a maritime protection agency. The problem addressed involves optimizing a multi-criteria objective function that minimizes a weighted combination of proportional measures of the vessels’ distances between home ports and patrol sectors, the sector workload, and the sector span. We initially assure contiguity of each patrol sector in our mixed-integer programming formulation via an exponential number of subtour elimination constraints, and then propose three alternative solution methods, two of which are based on reformulations that suitably replace the original contiguity representation with a polynomial number of constraints, and a third approach that employs an iterative cut generation procedure based on identifying violated subtour elimination constraints. We further enhance these reformulations with symmetry defeating constraints, either in isolation or in combination with a suitable perturbation of the objective function using weighted functions based on such constraints. Computational comparisons are provided for solving the problem using the original formulation versus either of our three alternative solution approaches for a representative instance. Overall, a model formulation based on Steiner tree problem (STP) constructs and enhanced by the reformulation-linearization technique (RLT) yielded the best performance.

Using agent-based modeling and a designed experiment to simulate and analyze a new air-to-air missile

New weapons system analysis is a field with much interest and study due to the enduring requirement for militaries to improve their set of tactical capabilities. Moreover, as development, testing, fielding, and employment of any new weapon system can be quite costly, justifications of acquisition decisions must be deliberate and thorough to improve necessary capabilities at the least possible cost. Informing these decisions, via analyses of the weapons systems’ benefits and costs, yields better decisions. Our goal herein is to demonstrate a sound methodology to efficiently attain information about the potential benefits, known as key performance parameters (KPPs), of a particular weapon system. Utilizing a simple, unclassified scenario, we identify benefits that the small advanced capability missile (SACM) concept provides, and we demonstrate a basis for further investigation into the tactics used to leverage its capabilities. Within this study, we substitute unclassified data from Lockheed Martin’s Cuda prototype for the SACM concept. Furthermore, we discuss how each of the chosen study factors influences the air combat scenario. Ultimately, we establish the usefulness of a designed experimental approach to analysis of agent-based combat simulation models, which yields useful insights during the acquisition process about the complex interactions of different actors on the battlefield.