Even though several clinics serve patients in more than one stage (e.g., visit nurse and then visit doctor) and employ multiple providers in each stage, most of the previous work on appointment system design considers a simplified single-stage single-server clinic. Motivated by a real-life clinic setting, this paper aims to determine the schedule configuration of a hybrid appointment system (i.e., the number of pre-booking and same-day time slots reserved for a physician along with their positions in the schedule) for a two-stage multi-server clinic. A stochastic optimization model is developed to obtain a schedule configuration that minimizes the expected total cost - a weighted sum of excessive patient waiting time, resource idle time, resource overtime, and denied appointment requests. Owing to its computational complexity, we estimate the expected total cost using the sample average approximation method. The proposed model is verified and validated using small test instances and subject matter experts. A case study of a family medicine clinic in Pennsylvania is used to illustrate the proposed approach. The schedule generated by the proposed model results in a significantly lower expected cost compared to the approximated single-stage system's best schedule configuration and clinic's existing configuration. Further, sensitivity analysis is conducted to assess the impacts of no-show rate, service time variation, and cost ratios on the schedule configuration. Our findings demonstrate that the schedule configuration is sensitive to changes in the average no-show rate and cost ratios but is not significantly impacted by service time variation. Several managerial insights are also drawn from our analysis. Finally, we provide directions for future research that also highlights the potential to use the revenue management approach to address the problem under study.
Keywords: Appointment system design; OR in health services; Open access; Outpatient clinic; Sample average approximation; Scheduling; Stochastic optimization; Two-stage multi-server clinic.