Scheduling Arrivals to a Stochastic Service Delivery System Using Copositive Cones

In this paper we investigate a stochastic appointment-scheduling problem in an outpatient clinic with a single doctor. The number of patients and their sequence of arrivals are fixed, and the scheduling problem is to determine an appointment time for each patient. The service durations of the patients are stochastic, and only the mean and covariance estimates are known. We do not assume any exact distributional form of the service durations, and we solve for distributionally robust schedules that minimize the expectation of the weighted sum of patients' waiting time and the doctor's overtime. We formulate this scheduling problem as a convex conic optimization problem with a tractable semidefinite relaxation. Our model can be extended to handle additional support constraints of the service durations. Using the primal dual optimality conditions, we prove several interesting structural properties of the optimal schedules. We develop an efficient semidefinite relaxation of the conic program and show that we can still obtain near-optimal solutions on benchmark instances. in the existing literature. We apply our approach to develop a practical appointment schedule at an eye clinic that can significantly improve the efficiency of the appointment system in the clinic, compared to an existing schedule.