Many decision making problems in real life are affected by uncertainty. The area of optimization
under uncertainty has been studied widely and deeply for over sixty years, and it continues to
be an active area of research. The overall aim of this thesis is to contribute to the literature by
developing (i) theoretical models that reflect problem settings closer to real life than previously considered in literature, as well as (ii) solution techniques that are scalable. The thesis focuses on two particular applications to this end, the vehicle routing problem and the problem of patient scheduling in a healthcare system.
The first part of this thesis studies the vehicle routing problem, which asks for a cost-optimal
delivery of goods to geographically dispersed customers. The probability distribution governing
the customer demands is assumed to be unknown throughout this study. This assumption
positions the study into the domain of distributionally robust optimization that has a well developed literature, but had so far not been extensively studied in the context of the capacitated vehicle routing problem. The study develops theoretical frameworks that allow for a tractable solution of such problems in the context of rise-averse optimization. The overall aim is to create a model that can be used by practitioners to solve problems specific to their requirements with minimal adaptations.
The second part of this thesis focuses on the problem of scheduling elective patients within the available resources of a healthcare system so as to minimize overall years of lives lost. This
problem has been well studied for a long time. The large scale of a healthcare system coupled
with the inherent uncertainty affecting the evolution of a patient make this a particularly
difficult problem. The aim of this study is to develop a scalable optimization model that
allows for an efficient solution while at the same time enabling a flexible modelling of each
patient in the system. This is achieved through a fluid approximation of the weakly-coupled
counting dynamic program that arises out of modeling each patient in the healthcare system
as a dynamic program with states, actions, transition probabilities and rewards reflecting the condition, treatment options and evolution of a given patient. A case-study for the National
Health Service in England highlights the usefulness of the prioritization scheme obtained as a result of applying the methodology developed in this study.