Stochastic appointment scheduling with patient-and-time-dependent probability distributions

Outpatient appointment scheduling, in settings ranging from primary care practices to specialty consultation clinics, requires balancing provider idle time, overtime, and patient waiting time. Uncertainty in service times, patient punctuality, and attendance makes scheduling a challenging task, resulting in high clinic costs and low patient satisfaction. Addressing these uncertain factors is particularly difficult because they follow patient-and-time-dependent probability distributions. Existing models fail to simultaneously incorporate such distributions and often assume identical distributions for all patients in all time slots, mainly due to the increased modeling and solution complexity. In this research, we propose a novel stochastic programming model that captures an exponential number of scenarios using a polynomial number of variables and constraints without relying on sampling methods. We incorporate patient-dependent probability distributions for service times, and patient-and-time-dependent distributions for no-shows and arrival times, and we discuss how these distributions can be estimated from routinely collected electronic health record data. To enhance the model, we explore the impact of personalized reminders on no-show rates and scheduling effectiveness. Besides, we investigate how changes in attendance rates, influenced by incentives or negative clinic experiences, impact the appointment scheduling efficiency. The results demonstrate that, using our proposed model, we optimally solve several instances with up to 14 patients in a reasonable computational time. The generated schedules substantially outperform classical Bailey-type appointment rules, whose best-performing variant increases the total expected cost by 97% on average relative to our schedules. Moreover, the generated schedules reduce total costs by 34% on average by incorporating patient-dependent service times, 12% by considering patient-and-time-dependent unpunctuality, and 67% by integrating patient-and-time-dependent no-shows. Furthermore, we show that personalized reminders have the potential to reduce total costs by 23%, and sensitivity analyses confirm that these benefits are robust to moderate estimation errors in the input distributions. Our approach has a significant potential to improve the efficiency of healthcare services by reducing patient waiting time, provider idle time, and overtime costs, and it highlights the value of personalized communication strategies in improving patient attendance.