Scheduling Arrivals to a Stochastic Service Delivery System Using Copositive Cones

We develop a convex programming approach to the appointment system design problem in a single server facility, using a (stochastic) network flow model to capture the waiting time performance measure of each patient in the system. We solve a robust min-max problem, using a representative “worst case” distribution matching the prescribed means and covariance estimates of the service durations of the patients, to determine the optimal schedule. Using this approach, the scheduling problem (finding the arrival time of the patients) can be determined by solving a semidefinite programming relaxation. Our analysis and numerical results yield several interesting insights on the nature of the optimal appointment system design the scheduling decisions obtained by planning against the “worst case” distribution performed exceedingly well even for several other distributions with the same mean and covariance parameters. Furthermore, for multi-class appointment system, if patients with higher variability are to be seen first before patients with lower variability, the optimal schedule generally exhibits “Bailey’s Rule + break” structure, with first few patients coming “close together” at the beginning, and a higher than usual interval for the last patient in the first class. Our analysis also reveals an interesting characteristic of the optimal appointment system except for the first and last few slots (those with zero consultation interval allocated to them), the chances of waiting for consultation service is identical for patients choosing all other slots in the system! ∗Department of Decision Sciences, NUS. Email: [email protected] †Department of Industrial Engineering and Logistics Management, HKUST. Email: [email protected] ‡Department of Decision Sciences, NUS. Email: [email protected] §Department of Decision Sciences, NUS. Email: [email protected]