Dynamic Inter-day and Intra-day Scheduling

Abstract

There are two timescales involved in accessing appointment-based medical services: indirect delay (order of days, weeks) and (order of minutes, hours). Indirect delay is defined as the time gap between the appointment request and the offered appointment. Direct delay is the physical waiting experienced by patients at the medical facility before they see their provider. In the literature, the inter(intra)-day appointment scheduling problem is concerned with the design of optimal scheduling strategies so that healthcare providers utilize resources efficiently and patients experience short indirect(direct) delays. Both problems, separately, have received significant attention in the literature and are still active areas of research. The simultaneous consideration of appointment day (inter-day) and time of day (intra-day) in dynamic scheduling decisions is an important theoretical and practical problem that has remained open due to its complex structure and large dimensionality. One particular challenge is that the output of inter-day scheduling becomes a dynamic input for intra-day scheduling. The scope of this article is to provide the first analytical model and optimization framework to address this joint problem effectively and efficiently. We prove new theoretical results in discrete convex analysis regarding constrained multimodular function minimization. We leverage these novel results and dynamic programming tools to characterize an optimal policy. We derive theoretical upper and lower bounds for the joint problem, based on which we develop a heuristic solution with a theoretically guaranteed optimality gap. Extensive numerical experiments indicate that the optimality gap is less than 1% for practical instances of the problem.

Keywords: dynamic programming, discrete convexity, appointment scheduling, advance scheduling