Liver transplantation is the only available therapy for any acute or chronic condition resulting in irreversible liver dysfunction. The liver allocation system in the U.S. is administered by the United Network for Organ Sharing (UNOS), a scientific and educational nonprofit organization. The main components of the organ procurement and transplant network are Organ Procurement Organizations (OPOs), which are collections of transplant centers responsible for maintaining local waiting lists, harvesting donated organs and carrying out transplants. Currently in the U.S., OPOs are grouped into 11 regions to facilitate organ allocation, and a three-tier mechanism is utilized that aims to reduce organ preservation time and transport distance to maintain organ quality, while giving sicker patients higher priority. Livers are scarce and perishable resources that rapidly lose viability, which makes their transport distance a crucial factor in transplant outcomes. When a liver becomes available, it is matched with patients on the waiting list according to a complex mechanism that gives priority to patients within the harvesting OPO and region. Transplants at the regional level accounted for more than 50% of all transplants since 2000.

This dissertation focuses on the design of regions for liver allocation hierarchy, and includes optimization models that incorporate geographic equity as well as uncertainty throughout the analysis. We employ multi-objective optimization algorithms that involve solving parametric integer programs to balance two possibly conflicting objectives in the system: maximizing efficiency, as measured by the number of viability adjusted transplants, and maximizing geographic equity, as measured by the minimum rate of organ flow into individual OPOs from outside of their own local area. Our results show that efficiency improvements of up to 6% or equity gains of about 70% can be achieved when compared to the current performance of the system by redesigning the regional configuration for the national liver allocation hierarchy.

We also introduce a stochastic programming framework to capture the uncertainty of the system by considering scenarios that correspond to different snapshots of the national waiting list and maximize the expected benefit from liver transplants under this stochastic view of the system. We explore many algorithmic and computational strategies including sampling methods, column generation strategies, branching and integer-solution generation procedures, to aid the solution process of the resulting large-scale integer programs. We also explore an OPO-based extension to our two-stage stochastic programming framework that lends itself to more extensive computational testing. The regional configurations obtained using these models are estimated to increase expected life-time gained per transplant operation by up to 7% when compared to the current system.

This dissertation also focuses on the general question of designing efficient algorithms that combine column and cut generation to solve large-scale two-stage stochastic linear programs. We introduce a flexible method to combine column generation and the L-shaped method for two-stage stochastic linear programming. We explore the performance of various algorithm designs that employ stabilization subroutines for strengthening both column and cut generation to effectively avoid degeneracy. We study two-stage stochastic versions of the cutting stock and multi-commodity network flow problems to analyze the performances of algorithms in this context.

Keywords. Organ allocation, large-scale medical optimization, column generation, branch-and-price, multi-objective decision making, two-stage stochastic programming.