Scheduling Arrivals to a Stochastic Service Delivery System Using Copositive Cones

Abstract

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 solve for distributionally robust schedules that minimize the expectation of the weighted sum of patients' waiting time and 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.

Appointment Scheduling with Limited Distributional Information

“…Taking the dual of this linear program leads to the following proposition. This result can also be found in Kong et al (2013).…”

“…The paper most relevant to ours is Kong et al (2013). Under the assumption of given mean, covariance matrix, and nonnegative support of job durations, they derive a copositive programming formulation for the appointment scheduling problem.…”

“…The authors propose a semidefinite programming relaxation as a solution approach. It is notable that, by specifying the complete covariance matrix, Kong et al (2013) consider a cross-moment model (CMM) of uncertainty. In contrast, by assuming knowledge of only marginal moments, we consider a marginal moment model (MMM) of uncertainty.…”

Patient Appointments in Ambulatory Care

“…They show that the total expected overage and underage cost is convex under a sufficient condition on cost parameters, and drive bounds on the number of samples required to obtain near-optimal solutions with a high probability. Kong et al (2010) model the appointment scheduling problem as a robust minmax problem in which they assume the mean and covariance estimates of the service durations are known, and use the worst-case distribution to obtain the schedule. The authors show that in a congested system with two types of patients, there is an optimal schedule such that the probability of waiting for each patient is identical for most appointments, except the first and last few slots.…”

Mitigating Delays and Unfairness in Appointment Systems

“…Currently, the majority of studies take a weighted average of the combinations among patients' waiting time and the physician's idle time and overtime as an optimization criterion, and they exploit different methods to solve. Three main streams are based on queueing theory (Wang 1993(Wang , 1999Green and Savin 2008;Hassin and Mendel 2008), stochastic programming (Robinson andChen 2003, Denton andGupta 2003), and robust optimization (Mittal and Stiller 2011;Kong et al 2013;Mak et al 2014Mak et al , 2015 frameworks. However, the second concern is that because the decisions are very sensitive to the prescribed weight for each participant, how to provide an accurate interpretation and estimation of these weights is a crucial issue (Mondschein and Weintraub 2003).…”