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

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

A classical problem in appointment scheduling, with applications in health care, 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. Part 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. While it was known that SVF is not always optimal, many papers have found that it performs well in practice and simulation. We give theoretical justification for these observations by proving quantitative worst-case bounds on the ratio between the cost incurred by the SVF rule and the minimum attainable cost, in a number of settings. We also show that under quite general conditions, this ratio approaches 1 as the number of patients grows large, showing that the SVF rule is asymptotically optimal. While this viewpoint in terms of 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

Dynamic Surgery Assignment of Multiple Operating Rooms With Planned Surgeon Arrival Times

International audienceThis paper addresses the dynamic assignment of a given set of surgeries to multiple identical operating rooms (ORs). Surgeries have random durations and planned surgeon arrival times. Surgeries are assigned dynamically to ORs at surgery completion events. The goal is to minimize the total expected cost incurred by surgeon waiting, OR idling, and OR overtime. We first formulate the problem as a multi-stage stochastic programming model. An efficient algorithm is then proposed by combining a two-stage stochastic programming approximation and some look-ahead strategies. A perfect information-based lower bound of the optimal expected cost is given to evaluate the optimality gap of the dynamic assignment strategy. Numerical results show that the dynamic scheduling and optimization with the proposed approach significantly improve the performance of static scheduling and First Come First Serve (FCFS) strategy

Performance analysis and scheduling strategies for ambulatory surgical facilities

Ambulatory surgery is a procedure that does not require an overnight hospital stay and is cost effective and efficient. The goal of this research is to develop an ASF operational model which allows management to make key decisions. This research develops and utilizes the simulation software ARENA based model to accommodate: (a) Time related uncertainties – Three system uncertainties characterize the problem (ii) Surgery time variance (ii) Physician arrival delay and (iii) Patient arrival delay; (b) Resource Capture Complexities – Patient flows vary significantly and capture/utilize both staffing and/or physical resources at different points and varying levels; and (c) Processing Time Differences – Patient care activities and surgical operation times vary by type and have a high level of variance between patient acuity within the same surgery type. A multi-dimensional ASF non-clinical performance objective is formulated and includes: (i) Fixed Labor Costs – regular time staffing costs for two nurse groups and medical/tech assistants, (i i) Overtime Labor Costs – staffing costs beyond the regular schedule, (i i i) Patient Delay Penalty – Imputed costs of waiting time experienced patients, and (iv) Physician Delay Penalty – Imputed costs of physicians having to delay surgical procedures due to ASF causes (limited staffing, patient delays, blocked OR, etc.). Three ASF decision problems are studied: (i) Optimize Staffing Resources Levels - Variations in staffing levels though are inversely related to patient waiting times and physician delays. The decision variable is the number of staff for three resource groups, for a given physician assignment and surgery profile. The results show that the decision space is convex, but decision robustness varies by problem type. For the problems studied the optimal levels provided 9% to 28% improvements relative to the baseline staffing level. The convergence rate is highest for less than optimal levels of Nurse-A. The problem is thus amenable to a gradient based search. (ii) Physician Block Assignment - The decision variables are the block assignments and the patient arrivals by type in each block. Five block assignment heuristics are developed and evaluated. Heuristic #4 which utilizes robust activity estimates (75% likelihood) and generates an asymmetrical resource utilization schedule, is found to be statistically better or equivalent to all other heuristics for 9 out of the 10 problems and (iii) Patient Arrival Schedule – Three decision variables in the patient arrival control (a) Arrival time of first patient in a block (b) The distribution and sequence of patients for each surgery type within the assigned windows and (c) The inter arrival time between patients, which could be constant or varying. Seven scheduling heuristics were developed and tested. Two heuristics one based on Palmers Rule and the other based on the SPT (Shortest Processing Time) Rule gave very strong results

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

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