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The goal when solving scheduling problems is to generate a high-quality schedule that satisfies every scheduling requirement. When scheduling healthcare providers, the quality of a schedule is often measured through provider satisfaction, a crucial issue that affects provider morale and patient safety. Manually generating a schedule for healthcare providers, as is often done in practice, can require a significant amount of time and effort. Additionally, since identifying a schedule that satisfies every scheduling requirement is challenging, it may not be practical to also consider all of the additional scheduling preferences that lead to improved provider satisfaction. Using computer-based mathematical programming to solve scheduling problems can dramatically decrease the time required to generate a schedule while also greatly improving the quality of the schedule. However, there are additional challenges associated with solving scheduling problems with computer-aided scheduling methods. This dissertation addresses some of these scheduling challenges in relation to scheduling healthcare providers. Specifically, we study three healthcare provider scheduling problems in this dissertation and propose methods for overcoming challenges associated with solving them. In the first problem, surgeons must be assigned to both operating and clinical rooms while satisfying many scheduling requirements. For this problem, we elaborate on the challenges we experienced while developing a mathematical scheduling model and show how the use of alternative variable definitions allowed us to overcome those challenges. In doing so, we explore the art of modeling and its impacts on solving a real-world scheduling problem. In the second scheduling problem we address, medical residents must be scheduled for their training rotations. For this problem, we expand on the previously discussed concept of using alternative decision variables by showing how different decision variable definitions can be used to simplify complex scheduling rules and improve computational performance. In both of the first two problems, it is desirable to maximize the number of individual scheduling requests that can be satisfied. Satisfying every scheduling request, however, is typically not possible. For solving the third scheduling problem we address, a resident shift scheduling problem, we develop a novel approach for resolving conflicting scheduling requests. Our approach identifies the exhaustive collection of maximally-feasible and minimally-infeasible request sets which can then be used by the decision maker to determine their preferred schedule.