Stochastic Optimization in Disaster Operations Management: Medical Capacity Planning During Epidemic, Optimal Subsidy Policy For Renewal Technology Adoption and Data-driven Robust Supply Chain Optimization

Abstract

Each of the problems addressed in this thesis falls under the broader umbrella of disaster operations management, where the focus is on minimizing the impact of unforeseen disruptions on critical systems. Whether it’s managing the surge in healthcare demand during a pandemic, promoting sustainable energy adoption to mitigate long-term environmental crises, or optimizing supply chains to withstand production disruptions, the core challenge remains the same: to design resilient systems capable of navigating uncertainty and resource constraints. These problems highlight the need for well-informed, strategic decision-making to ensure that essential services continue to function even in the face of crises. The first problem deals with managing medical equipment capacity during the early spread of an infection in a region, a critical research problem during COVID pandemic. After a brief introduction to infectious disease modeling, we develop a model for a regional decision- maker to analyze the requirement of medical equipment capacity in the early stages of a spread of infections. We use the model to propose and evaluate ways to manage limited equipment capacity. Early stage infection growth is captured by a stochastic differential equation (SDE) and is part of a two-period community spread and shutdown model. We use the running-maximum process of a geometric Brownian motion to develop a performance metric, probability of breach, for a given capacity level. Decision-maker estimates costs of economy versus health and the time till the availability of a cure; we develop a heuristic rule and an optimal formulation that use these estimates to determine the required medical equipment capacity. We connect the level of capacity to a menu of actions, including the level and timing of shutdown, shutdown effectiveness, and enforcement. Our results show how these actions can compensate for the limited medical equipment capacity in a region. We next address the sharing of medical equipment capacity across regions and its impact on the breach probability. In addition to traditional risk-pooling, we identify a peak-timing effect depending on when infections peak in different regions. We show that equipment sharing may not benefit the regions when capacity is tight. A coupled SDE model captures the messaging coordination and movement across regional borders. Numerical experiments on this model show that under certain conditions, such movement and coordination can synchronize the infection trajectories and bring the peaks closer, reducing the benefit of sharing capacity. We then turn our attention to the problem of devising optimal subsidy to encourage adoption of solar technology in a region. Each household in a population characterized by income heterogeneity faces random demand for electricity and decides if and when it should adopt a solar product, rooftop solar or community solar. A central planner, aiming to meet an adoption level target within a set time, offers net metering and subsidy on solar products and minimizes its total cost. Our focus is on analyzing the interactions of three new features we add to the literature: income diversity, availability of community solar, and consideration of adoption timing. We develop a bilevel optimization formulation to derive the optimal subsidy policy. The upper level (planner’s) problem is a constrained non-linear optimization model in which the planner aims to minimize the average subsidy cost. The lower level (household’s) problem is an optimal stopping formulation, which captures the adoption decisions of the households. We derive a closed-form expression for the distribution of optimal adoption time of households for a given subsidy policy. We show that the planner’s problem is convex in the case of homogeneous subsidy for the two products. Our results underscore the importance for planners to consider three factors - adoption level target, time target, and subsidy budget - simultaneously as they work in tandem to influence the adoption outcome. The planners must also consider the inclusion of community solar in their plans because, as we show, community and rooftop solar attract households from different sides of the income spectrum. In the presence of income inequality, the availability of community makes it easier to meet solar adoption targets.The third problem we consider is a classic mixed integer programming formulation for sup- ply chain optimization of a semi-conductor supplier in presence of supply disruptions. We provided a deterministic formulation for optimal outbound network routing. We integrate the past publicly available data in the optimization model and show the shortcomings of its optimal solution using a numerical experiment. We then designed a practical robust formulation to handle uncertain production capacity. We numerically study the optimal solutions for low and high tolerance to deviation from deterministic optimal solution and generate optimal production and transportation plan which is not only more resilient to random shocks but also requires marginal change from current plan.