Managing appointments with waiting time targets and random walk-ins

The random arrivals of walk-in patients significantly affect the daily operations of healthcare facilities. To improve the performance of outpatient departments, this paper attempts to make an appointment schedule by considering walk-ins and the waiting time target (WTT) for appointment patients. A stochastic programming model is proposed to solve this problem with the objective of minimizing the weighted patient waiting and makespan cost. A non-decreasing waiting cost function is used to capture the WTT fulfillment of appointment patients, whereas walk-ins incur a linear waiting cost. A finite-horizon Markov Decision Process model is formulated to establish the optimal real-time scheduling policy under a given appointment schedule. The appointment schedule is determined by a two-stage stochastic programming approximation and a local search improvement. Structural properties of the optimal appointment scheduling and real-time scheduling policies are established. In particular, it is shown that appointment overbooking is allowed only at the end of the regular session, and the optimal real-time scheduling policy is an easy-to-implement threshold policy with bounded sensitivity. Numerical experiments based on real data are performed to investigate the influence of different parameters and to compare different schedules. The optimal schedule demonstrates superior performance by allowing reasonable waiting times for appointment patients depending on their WTTs. Managerial insights are also provided to hospital managers. Finally, the basic model is extended by incorporating random service times and random arrivals of appointment patients. The latter includes the random number of patients that show up for service or call for appointments, and the random arrival time (unpunctuality). Appointment overbooking strategies are shown to have different structures under some stochastic factors.