Optimizing Stochastic Models in Health Care: Appointment Scheduling and Disease Testing

We consider two different problems: appointment scheduling and asymptomatic disease testing.

For the appointment scheduling problem, the goal is to assign appointment times to minimize a weighted sum of patient wait times, doctor idle time, and clinic overtime. We make the assumption that patients are unpunctual with respect to assigned appointment times and distributional information on unpunctuality is available. We first consider a model with heterogeneous patient distributions in both service time and unpunctuality. This is a complex system that requires heuristic approaches. We are able to show the benefits of capturing patient heterogeneity in addition to the superior performance of our heuristics. Our best methods do not scale well to large patient systems; thus, we consider a second model that allows a large number of patients. For large systems, we assume patient homogeneity; however, patient unpunctuality is permitted to be time-heterogeneous. With this model, we examine the fluid limits of the queue processes to develop a fluid control problem that seeks an asymptotically optimal appointment schedule in the form of an RCLL function. This problem is difficult to solve analytically, so we propose a numerical scheme that converts the control problem into a quadratic program. We examine asymptotically optimal appointment schedules under various unpunctuality distributions, then the superior performance of these schedules in discrete-event simulations.
For the asymptomatic disease testing problem, we consider the individual decision-maker problem of choosing when to use disease test kits from a limited supply. We assume an underlying SIR Markov model with split states for asymptomatic and symptomatic states. As only symptoms are directly observable, the decision process is modeled as a partially-observable Markov decision process for deciding when to use tests. The goal is to produce simple instructions for the average consumer to follow. We derive policies that do not require probability computations by the user. Under certain assumptions, we are able to prove that these policies are optimal. Last, we examine a community simulation where infection probabilities are dependent on community infected. Our methods are shown to outperform existing baselines.odels in Health Care: Appointment Scheduling and Disease Testing

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