Robust Appointment Scheduling with Heterogeneous Costs

Designing simple appointment systems that under uncertainty in service times, try to achieve both high utilization of expensive medical equipment and personnel as well as short waiting time for patients, has long been an interesting and challenging problem in health care. We consider a robust version of the appointment scheduling problem, introduced by Mittal et al. (2014), with the goal of finding simple and easy-to-use algorithms. Previous work focused on the special case where per-unit costs due to under-utilization of equipment/personnel are homogeneous i.e., costs are linear and identical. We consider the heterogeneous case and devise an LP that has a simple closed-form solution. This solution yields the first constant-factor approximation for the problem. We also find special cases beyond homogeneous costs where the LP leads to closed form optimal schedules. Our approach and results extend more generally to convex piece-wise linear costs. For the case where the order of patients is changeable, we focus on linear costs and show that the problem is strongly NP-hard when the under-utilization costs are heterogeneous. For changeable order with homogeneous under-utilization costs, it was previously shown that an EPTAS exists. We instead find an extremely simple, ratio-based ordering that is 1.0604 approximate

Robust Appointment Scheduling

Health care providers are under tremendous pressure to reduce costs and increase quality of their services. It has long been recognized that well-designed appointment systems have the potential to improve utilization of expensive personnel and medical equipment and to reduce waiting times for patients. In a widely influential survey on outpatient scheduling, Cayirli and Veral (2003) concluded that the "biggest challenge for future research will be to develop easy-to-use heuristics." We analyze the appointment scheduling problem from a robust-optimization perspective, and we establish the existence of a closed-form optimal solution--arguably the simplest and best `heuristic\u27 possible. In case the order of patients is changeable, the robust optimization approach yields a novel formulation of the appointment scheduling problem as that of minimizing a concave function over a supermodular polyhedron. We devise the first constant-factor approximation algorithm for this case

Designing cyclic appointment schedules for outpatient clinics with scheduled and unscheduled patient arrivals

We present a methodology to design appointment systems for outpatient clinics and diagnostic facilities that offer both walk-in and scheduled service. The developed blueprint for the appointment schedule prescribes the number of appointments to plan per day and the moment on the day to schedule the appointments. The method consists of two models that are linked by an algorithm; one for the day process that governs scheduled and unscheduled arrivals on the day and one for the access process of scheduled arrivals. Appointment schedules that balance the waiting time at the facility for unscheduled patients and the access time for scheduled patients, are calculated iteratively using the outcomes of the two models. The method is of general nature and can therefore also be applied to scheduling problems in other sectors than health care

Adaptive Appointment Systems with Patient Preferences

Patients\u27 satisfaction with an appointment system when they attempt to book a nonurgent appointment is affected by their ability to book with a doctor of choice and to book an appointment at a convenient time of day. For medical conditions requiring urgent attention, patients want quick access to a familiar physician. For such instances, it is important for clinics to have open slots that allow same-day (urgent) access. A major challenge when designing outpatient appointment systems is the difficulty of matching randomly arriving patients\u27 booking requests with physicians\u27 available slots in a manner that maximizes patients\u27 satisfaction as well as clinics\u27 revenues. What makes this problem difficult is that booking preferences are not tracked, may differ from one patient to another, and may change over time. This paper describes a framework for the design of the next generation of appointment systems that dynamically learn and update patients\u27 preferences and use this information to improve booking decisions. Analytical results leading to a partial characterization of an optimal booking policy are presented. Examples show that heuristic decision rules, based on this characterization, perform well and reveal insights about trade-offs among a variety of performance metrics important to clinic managers

Performance of the smallest-variance-first rule in appointment sequencing

A classic problem in appointment scheduling with applications in healthcare concerns the determination of the patients' arrival times that minimize a cost function that is a weighted sum of mean waiting times and mean idle times. One aspect of this problem is the sequencing problem, which focuses on ordering the patients. We assess the performance of the smallest-variance-first (SVF) rule, which sequences patients in order of increasing variance of their service durations. Although it is known that SVF is not always optimal, it has been widely observed that it performs well in practice and simulation. We provide a theoretical justification for this observation by proving, in various settings, quantitative worstcase bounds on the ratio between the cost incurred by the SVF rule and the minimum attainable cost. We also show that, in great generality, SVF is asymptotically optimal, that is, the ratio approaches one as the number of patients grows large. Although evaluating policies by considering an approximation ratio is a standard approach in many algorithmic settings, our results appear to be the first of this type in the appointment-scheduling literature

Appointment sequencing: Why the Smallest-Variance-First rule may not be optimal

We study the design of a healthcare appointment system with a single physician and a group of patients whose service durations are stochastic. The challenge is to find the optimal arrival sequence for a group of mixed patients such that the expected total cost of patient waiting time and physician overtime is minimized. While numerous simulation studies report that sequencing patients by increasing order of variance of service duration (Smallest-Variance-First or SVF rule) performs extremely well in many environments, analytical results on optimal sequencing are known only for two patients. In this paper, we shed light on why it is so difficult to prove the optimality of the SVF rule in general. We first assume that the appointment intervals are fixed according to a given template and analytically investigate the optimality of the SVF rule. In particular, we show that the optimality of the SVF rule depends on two important factors: the number of patients in the system and the shape of service time distributions. The SVF rule is more likely to be optimal if the service time distributions are more positively skewed, but this advantage gradually disappears as the number of patients increases. These results partly explain why the optimality of the SVF rule can only be proved for a small number of patients, and why in practice, the SVF rule is usually observed to be superior, since most empirical distributions of the service durations are positively skewed, like log-normal distributions. The insights obtained from our analytical model apply to more general settings, including the cases where the service durations follow log-normal distributions and the appointment intervals are optimized

Scheduling arrivals to a stochastic service delivery system using copositive cones

10.1287/opre.2013.1158Operations Research613711-726OPRE