Matthew J. Robbins

The Weighted Set Covering Game: A Vaccine Pricing Model for Pediatric Immunization

The United States pediatric vaccine manufacturing market is analyzed using a static Bertrand oligopoly pricing model that characterizes oligopolistic interactions between asymmetric firms in a homogeneous multiple product market. Firms satisfy demand by appropriately pricing and selling its given set of bundles, where each bundle contains one or more products. In analyzing the pediatric vaccine market, a bundle is a vaccine, where each vaccine contains one or more immunogenic antigens. Consumers seek to purchase at least one of each antigen at an overall minimum cost. Demand is captured by defining a weighted set covering optimization problem, with the weights (prices) controlled by firms engaged in Bertrand competition. A repeated game version of the model enables multiple interactions between firms, allowing examination of tacit collusion. An iterative improvement algorithm is defined that constructs a pure strategy Nash equilibrium (some in the limiting sense) for the static game. Sufficient conditions...

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.

Analytics for vaccine economics and pricing: insights and observations

Pediatric immunization programs in the USA are a successful and cost–effective public health endeavor, profoundly reducing mortalities caused by infectious diseases. Two important issues relate to the success of the immunization programs, the selection of cost–effective vaccines and the appropriate pricing of vaccines. The recommended childhood immunization schedule, published annually by the CDC, continues to expand with respect to the number of injections required and the number of vaccines available for selection. The advent of new vaccines to meet the growing requirements of the schedule results: in a large, combinatorial number of possible vaccine formularies. The expansion of the schedule and the increase in the number of available vaccines constitutes a challenge for state health departments, large city immunization programs, private practices and other vaccine purchasers, as a cost–effective vaccine formulary must be selected from an increasingly large set of possible vaccine combinations to satisfy the schedule. The pediatric vaccine industry consists of a relatively small number of pharmaceutical firms engaged in the research, development, manufacture and distribution of pediatric vaccines. The number of vaccine manufacturers has dramatically decreased in the past few decades for a myriad of reasons, most notably due to low profitability. The contraction of the industry negatively impacts the reliable provision of pediatric vaccines. The determination of appropriate vaccine prices is an important issue and influences a vaccine manufacturer’s decision to remain in the market. Operations research is a discipline that applies advanced analytical methods to improve decision making; analytics is the application of operations research to a particular problem using pertinent data to provide a practical result. Analytics provides a mechanism to resolve the challenges facing stakeholders in the vaccine development and delivery system, in particular, the selection of cost–effective vaccines and the appropriate pricing of vaccines. A review of applicable analytics papers is provided.

An Analytical Comparison of Social Network Measures

Network science spans many different fields of study, ranging from psychology to biology to the social sciences. A number of descriptive network measures have been identified for use within these fields; however, little research examines the relationships of these measures for possible statistical dependence. The research presented in this paper uses Spearmans rank correlation coefficient to examine the statistical dependence between pairs of 24 widely accepted social network measures. Confidence intervals are compared to determine whether computation times between measures in the same correlation group are significantly different. We use a three-factor, four-level, full-factorial experimental design to construct a test set of 64 unique network topologies. The three factors of interest are the network structural properties of size, cluster ability, and the scale-free parameter. A set of 320 networks are generated from a power law degree distribution using a random graph generation algorithm. Results indicate that there exists high correlation among 14 of the 24 tested network measures, many of which also exhibit statistically significant differences with respect to computation time. These findings are of interest to analysts seeking to identify measures that provide similar ranked outcomes and where computational efficiency is an important consideration.

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.

A symmetric capacity-constrained differentiated oligopoly model for the United States pediatric vaccine market with linear demand

The United States pediatric vaccine market is examined using Bertrand–Edgeworth–Chamberlin price competition. The proposed game captures interactions between symmetric, capacity-constrained manufacturers in a differentiated, single-product market with linear demand. Results indicate that a unique pure strategy equilibrium exists in the case where the capacities of the manufacturers are at their extreme. For the capacity region where no pure strategy equilibrium exists, there exists a mixed strategy equilibrium where the distribution function, its support, and the expected profit of the manufacturers are characterized. Three game instances are introduced to model the United States pediatric vaccine market. In each instance, the manufacturers are assumed to have equal capacity in producing vaccines. Vaccines are differentiated based upon the number of reported adverse medical events for that vaccine. Using a game-theoretic model, equilibrium prices are computed for each monovalent vaccine. Results indicate that the equilibrium prices for monovalent vaccines are lower than the federal contract prices. The numerical results provide both a lower and upper bound for the vaccine equilibrium prices in the public sector, based on the capacity of the vaccine manufacturers. Results illustrate the importance of several model parameters such as market demand and vaccine adverse events on the equilibrium prices. Supplementary materials are available for this article. Go to the publisher’s online edition of IIE Transactions for datasets, additional tables, detailed proofs, etc.

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.

Heterogeneous surface-to-air missile defense battery location: a game theoretic approach

In the context of an air defense missile-and-interceptor engagement, a challenge for the defender is that surface-to-air missile batteries often must be located to protect high-value targets dispersed over a vast area, subject to which an attacker may observe the disposition of batteries and subsequently develop and implement an attack plan. To model this scenario, we formulate a two-player, extensive form, three-stage, perfect information, zero-sum game that accounts for, respectively, a defender’s location of batteries, an attacker’s launch of missiles against targets, and a defender’s assignment of interceptor missiles from batteries to incoming attacker missiles. The resulting trilevel math programming formulation cannot be solved via direct optimization, and it is not suitable to solve via full enumeration for realistically-sized instances. We instead adapt the game tree search technique Double Oracle, within which we embed either of two alternative heuristics to solve an important subproblem for the attacker. We test and compare these solution methods to solve a designed set of 52 instances having parametric variations, from which we derive insights regarding the nature of the underlying problem. Enhancing the solution methods with alternative initialization strategies, our superlative methodology attains the optimal solution for over 75% of the instances tested and solutions within 3% of optimal, on average, for the remaining 25% of the instances, and it is promising for realistically-sized instances, scaling well with regard to computational effort.