Over the past few decades, appointment scheduling has become a valuable decision tool in healthcare systems. Appointment scheduling policies are tools for managing many complex problems that continuously plague healthcare service systems. Such issues include the patient's waiting time, resource underuse, capacity allocation, panel selection, service coordination, provider's overtime, patient's no-show behavior, arrival patterns of patients, patients' service time requirements, patient's preferences, heterogeneity of patient's population, provider-team collaboration, and patient's unpunctuality. Experts have proposed many appointment scheduling schemes in the past; some of these schemes would not satisfy the system's dynamic need over time, and others are still applicable in practice. Moreover, the peculiarity and localization of system settings remain a monumental challenge to finding universal solutions to many appointment scheduling problems in healthcare clinics. For the operations management field, accurately capturing the natural system characteristics in model formulation means a herculean task -- where the derivation of appropriate mathematical models is possible, tractable solutions may be non-existent or hard to find. Thus, researchers often resort to solvable models that mimic reality and provide useful insights for guiding decision policies in practice. One such system features that pose challenges is the heterogeneity, springing from the stochastic patient's behavior and mix of providers in team collaboration practices.

Motivated by some practices' observable needs and the need to fill some existing gaps in the literature, this dissertation focuses on solving some appointment scheduling problems in healthcare settings with heterogeneous system characteristics. Here, the primary sources of heterogeneity are in two folds -- the arrival patterns of patients' requests for service, and physician and non-physician providers mix in team collaboration. Towards achieving its broad objective, this dissertation presents three (3) novel studies with their contributions listed as follows.

First is the study that determines the optimal appointment scheduling decision in the presence of a patient's no-show behavior and random walk-ins arriving in a specified time window. The study proposes a two-stage optimization model to determine the optimal time window for the arrival of walk-ins and corresponding optimal appointment schedule for regular patients to minimize the total cost of patients' waiting times, and physician's idleness and overtime. Theoretical results demonstrate that the proposed model possesses a fine structural property (multimodularity) that allows for global optimal-solution via local search algorithms. Therefore, a variable neighborhood descent (VND) algorithm is proposed to solve the optimization problem -- the algorithm performs well compared with some standard local search algorithms. Numerical analysis suggests that the policy that stipulates an optimal time window for walk-ins' arrival is more effective than the general random walk-in arrival (open walk-in) policy when walk-ins' arrival rate is moderate relative to the clinic's capacity. In particular, utilizing the optimal time window policy can lead to a fair reduction in system cost and preservation of an increased level of patients' access to care.

The optimal time window policy improves system efficiency and minimizes the effect of variability due to walk-in arrivals. However, the policy would not effectively mitigate the impact of variability when there is a high influx of walk-in arrivals. Scheduling these walk-ins as same-day patients would help improve the coordination of the system workload and increase its efficiency. Hence, another study is proposed to investigate the appointment scheduling decisions for routine and same-day patients. When scheduling routine and same-day appointments, the typical decision rule is the "carve-out scheduling" (COS) policy. The COS system, which stipulates that the decision-maker cannot assign two patients of different classes to a slot, may often experience increasing underutilization and long waiting times. Alternatively, the policy "mixed appointment scheduling" (MAS), which allows for possible assignment of the two patient classes to a slot, may be desirable. Whether the MAS policy would perform better than the COS, and under what business settings are the performance gaps worthwhile, are questions that this study seeks to address. This study answers these questions by proposing a new model to determine the optimal appointment scheduling decision for routine and same-day patients under the MAS condition. It is shown that the priority service discipline, together with uncertainty in demand, destroys the multimodularity of the objective function in the component of the same-day schedule. Thus, the joint problem of determining the optimal routine and same-day schedules becomes hard to solve. This study present various characterizations that allow for the efficient identification of the optimal decision policy. The numerical results reveal that the MAS policy helps to lower system cost, increase utilization, and reduce service delay when the same-day demand to clinic capacity ratio is within a moderate limit.

The heterogeneity from the patient side poses challenges for the appointment scheduling system but managing the heterogeneous mix of medical personnel supply is equally challenging. The aging population and increasing chronic disease load are rapidly changing the face of primary care delivery, with mid-level (e.g., nurse) practitioners providing growing proportion of patient care. Potential differences in the quality of care offered by physicians and nurse practitioners may affect patient preferences, thus leading to patient choice behavior. This dissertation proposes another study that focuses on the problem of appointment scheduling for physician--nurse teams in the presence of patient choice and no-shows. This study proposes a novel model that accounts for decision-making by patients in a system with two provider types. It proves that the presence of patient choice, in general, breaks down the multimodularity of a medical practice's cost function, creating challenges for the numerical identification of optimal appointment schedules. At the same time, the study derives the sufficient conditions under which the multimodularity is restored. To counter the computational challenges arising in the general setting, an easy-to-implement heuristic is proposed and its optimality in the absence of patient no-shows is proved. The numerical results show how the ratio of qualities of care delivered by nurses and physicians affect the profitability of the medical practice, enabling the analysis of the trade-offs involved in hiring a nurse practitioner.