Assessment of Patient Classification in Appointment System Design

This paper investigates two approaches to patient classification: using patient classification only for sequencing patient appointments at the time of booking and using patient classification for both sequencing and appointment interval adjustment. In the latter approach, appointment intervals are adjusted to match the consultation time characteristics of different patient classes. Our simulation results indicate that new appointment systems that utilize interval adjustment for patient class are successful in improving doctors' idle time, doctors' overtime and patients' waiting times without any trade‐offs. Best performing appointment systems are identified for different clinic environments characterized by walk‐ins, no‐shows, the percentage of new patients, and the ratio of the mean consultation time of new patients to the mean consultation time of return patients. As a result, practical guidelines are developed for managers who are responsible for designing appointment systems.     

The Impact of Variability and Patient Information on Health Care System Performance

In the delivery of health care services, variability in the patient arrival and service processes can cause excessive patient waiting times and poor utilization of facility resources. Based on data collected at a large primary care facility, this paper investigates how several sources of variability affect facility performance. These sources include ancillary tasks performed by the physician, patient punctuality, unscheduled visits to the facility's laboratory or X‐ray services, momentary interruptions of a patient's examination, and examination time variation by patient class. Our results indicate that unscheduled visits to the facility's laboratory or X‐ray services have the largest impact on a physician's idle time. The average patient wait is most affected by how the physician prioritizes completing ancillary tasks, such as telephone calls, relative to examining patients. We also investigate the improvement in system performance offered by using increasing levels of patient information when creating the appointment schedule. We find that the use of policies that sequence patients based on their classification improves system performance by up to 25.5%.     

Appointment Overbooking in Health Care Clinics to Improve Patient Service and Clinic Performance

The problem of no‐shows (patients who do not arrive for scheduled appointments) is particularly significant for health care clinics, with reported no‐show rates varying widely from 3% to 80%. No‐shows reduce revenues and provider productivity, increase costs, and limit patient access by reducing effective clinic capacity. In this article, we construct a flexible appointment scheduling model to mitigate the detrimental effects of patient no‐shows, and develop a fast and effective solution procedure that constructs near‐optimal overbooked appointment schedules that balance the benefits of serving additional patients with the potential costs of patient waiting and clinic overtime. Computational results demonstrate the efficacy of our model and solution procedure, and connect our work to prior research in health care appointment scheduling.     

A Universal Appointment Rule in the Presence of No‐Shows and Walk‐Ins

This study introduces a universal “Dome” appointment rule that can be parameterized through a planning constant for different clinics characterized by the environmental factors—no‐shows, walk‐ins, number of appointments per session, variability of service times, and cost of doctor's time to patients’ time. Simulation and nonlinear regression are used to derive an equation to predict the planning constant as a function of the environmental factors. We also introduce an adjustment procedure for appointment systems to explicitly minimize the disruptive effects of no‐shows and walk‐ins. The procedure adjusts the mean and standard deviation of service times based on the expected probabilities of no‐shows and walk‐ins for a given target number of patients to be served, and it is thus relevant for any appointment rule that uses the mean and standard deviation of service times to construct an appointment schedule. The results show that our Dome rule with the adjustment procedure performs better than the traditional rules in the literature, with a lower total system cost calculated as a weighted sum of patients’ waiting time, doctor's idle time, and doctor's overtime. An open‐source decision‐support tool is also provided so that healthcare managers can easily develop appointment schedules for their clinical environment.     

Optimal Sequencing of Unpunctual Patients in High‐Service‐Level Clinics

Even though patients often arrive early and out of turn for scheduled appointments in outpatient clinics, no research has been undertaken to establish whether an available provider should see an early patient right away (preempt) or wait for the patient scheduled next. This problem, which we call the “Wait‐Preempt Dilemma,” is particularly relevant for “high‐service‐level” clinics (such as psychotherapy, chiropractic, acupuncture), where preempting may cause the missing patient to wait for an excessively long time, should she show up soon. Typically, the dilemma is resolved by preemption, where the provider starts serving the patient who has already arrived to avoid staying idle. By contrast, we analytically determine the time intervals where it is optimal to preempt and those where it is optimal to wait, and find that in some cases the provider should in fact stay idle, even in the presence of waiting patients. Our results suggest that the proposed analytical method outperforms the always‐preempt policy in clinics that do not overbook and have service times longer than 30 minutes. In these cases, the analytical method dramatically reduces patient waiting times at the cost of a modest increase in overtime. By contrast, in clinics that overbook or have short service times, the two policies perform similarly, and hence the always‐preempt policy is preferable due to its simplicity. A software application is provided that clinics can readily use to solve the wait‐preempt dilemma.     

Bi‐Criteria Scheduling of Surgical Services for an Outpatient Procedure Center

Uncertainty in the duration of surgical procedures can cause long patient wait times, poor utilization of resources, and high overtime costs. We compare several heuristics for scheduling an Outpatient Procedure Center. First, a discrete event simulation model is used to evaluate how 12 different sequencing and patient appointment time‐setting heuristics perform with respect to the competing criteria of expected patient waiting time and expected surgical suite overtime for a single day compared with current practice. Second, a bi‐criteria genetic algorithm (GA) is used to determine if better solutions can be obtained for this single day scheduling problem. Third, we investigate the efficacy of the bi‐criteria GA when surgeries are allowed to be moved to other days. We present numerical experiments based on real data from a large health care provider. Our analysis provides insight into the best scheduling heuristics, and the trade‐off between patient and health care provider‐based criteria. Finally, we summarize several important managerial insights based on our findings.     

Are Patients Patient? The Role of Time to Appointment in Patient Flow

The current state of outpatient healthcare delivery is characterized by capacity shortages and long waits for appointments, yet a substantial fraction of valuable doctors’ capacity is wasted due to no‐shows. In this study, we examine the effect of wait to appointment on patient flow, specifically on a patient's decision to schedule an appointment and to subsequently arrive to it. These two decisions may be dependent, as appointments are more likely to be scheduled by patients who are more patient and are thereby more likely to show up. To estimate the effect of wait on these two decisions, we introduce the willingness to wait (WTW), an unobservable variable that affects both bookings and arrivals for appointments. Using data from a large healthcare system, we estimate WTW with a state‐of‐the‐art non‐parametric method. The WTW, in turn, allows us to estimate the effect of wait on no‐shows. We observe that the effect of increased wait on the likelihood of no‐shows is disproportionately greater among patients with low WTW. Thus, although reducing the wait to an appointment will enable a provider to capture more patient bookings, the effects of wait time on capacity utilization can be non‐monotone. Our counterfactual analysis suggests that increasing wait times can sometimes be beneficial for reducing no‐shows.     

Clinic Overbooking to Improve Patient Access and Increase Provider Productivity*

The problem of patient no‐shows (patients who do not arrive for scheduled appointments) is significant in many health care settings, where no‐show rates can vary widely. No‐shows reduce provider productivity and clinic efficiency, increase health care costs, and limit the ability of a clinic to serve its client population by reducing its effective capacity. In this article, we examine the problem of no‐shows and propose appointment overbooking as one means of reducing the negative impact of no‐shows. We find that patient access and provider productivity are significantly improved with overbooking, but that overbooking causes increases in both patient wait times and provider overtime. We develop a new clinic utility function to capture the trade‐offs between these benefits and costs, and we show that the relative values that a clinic assigns to serving additional patients, minimizing patient waiting times, and minimizing clinic overtime will determine whether overbooking is warranted. From the results of a series of simulation experiments, we determine that overbooking provides greater utility when clinics serve larger numbers of patients, no‐show rates are higher, and service variability is lower. Even with highly variable service times, many clinics will achieve positive net results with overbooking. Our analysis provides valuable guidance to clinic administrators about the use of appointment overbooking to improve patient access, provider productivity, and overall clinic performance.     

Appointment Scheduling with No‐Shows and Overbooking

We study an overbooking model for scheduling arrivals at a medical facility under no‐show behavior, with patients having different no‐show probabilities and different weights. The scheduler has to assign the patients to time slots in such a way that she minimizes the expected weighted sum of the patients' waiting times and the doctor's idle time and overtime. We first consider the static problem, where the set of patients to be scheduled and their characteristics are known in advance. We partially characterize the optimal schedule and introduce a new sequencing rule that schedules patients according to a single index that is a function of their characteristics. Then we apply our theoretical results and conclusions from numerical experiments to sequential scheduling procedures. We propose a heuristic solution to the sequential scheduling problem, where requests for appointments come in gradually over time and the scheduler has to assign each patient to one of the remaining slots that are available in the schedule for a given day. We find that the no‐show rate and patients' heterogeneity have a significant impact on the optimal schedule and should be taken under consideration.     

Panel Size and Overbooking Decisions for Appointment‐Based Services under Patient No‐Shows

Many service systems that work with appointments, particularly those in healthcare, suffer from high no‐show rates. While there are many reasons why patients become no‐shows, empirical studies found that the probability of a patient being a no‐show typically increases with the patient's appointment delay, i.e., the time between the call for the appointment and the appointment date. This paper investigates how demand and capacity control decisions should be made while taking this relationship into account. We use stylized single server queueing models to model the appointments scheduled for a provider, and consider two different problems. In the first problem, the service capacity is fixed and the decision variable is the panel size; in the second problem, both the panel size and the service capacity (i.e., overbooking level) are decision variables. The objective in both cases is to maximize some net reward function, which reduces to system throughput for the first problem. We give partial or complete characterizations for the optimal decisions, and use these characterizations to provide insights into how optimal decisions depend on patient's no‐show behavior in regards to their appointment delay. These insights especially provide guidance to service providers who are already engaged in or considering interventions such as sending reminders in order to decrease no‐show probabilities. We find that in addition to the magnitudes of patient show‐up probabilities, patients' sensitivity to incremental delays is an important determinant of how demand and capacity decisions should be adjusted in response to anticipated changes in patients' no‐show behavior.     

The Effect of Integrated Scheduling and Capacity Policies on Clinical Efficiency

In outpatient healthcare clinics, capacity, patient flow, and scheduling are rarely managed in an integrated fashion, so a question of interest is whether clinic performance can be improved if the policies that guide these decisions are set jointly. Despite the potential importance of this issue, we find surprisingly few studies that look at how the allocation of capacity, paired with various appointment scheduling policies and different patient flow configurations, affects patient flow and clinical efficiency. In this paper, we develop an empirically based discrete‐event simulation to examine the interactions between patient appointment policies and capacity allocation policies (i.e., the number of available examination rooms) and how they jointly affect various performance measures, such as resource utilization and patient waiting time. Findings suggest that scheduling lower‐variance, shorter appointments earlier in the clinic (and, conversely, higher‐variance, longer appointments later) results in less overall patient waiting without reducing physician utilization or increasing clinic duration. Additionally, exam rooms exhibited classic bottleneck behavior: there was no effect on physician utilization by adding exam rooms beyond a certain threshold, but too few exam rooms were devastating to clinic throughput. Some significant interactions between these variables were observed, but were not influential to the level of managerial concern. Clinicians' intuition about managing capacity in healthcare settings may differ substantially from best policies.     

Optimal Choice for Appointment Scheduling Window under Patient No‐Show Behavior

Observing that patients with longer appointment delays tend to have higher no‐show rates, many providers place a limit on how far into the future that an appointment can be scheduled. This article studies how the choice of appointment scheduling window affects a provider's operational efficiency. We use a single server queue to model the registered appointments in a provider's work schedule, and the capacity of the queue serves as a proxy of the size of the appointment window. The provider chooses a common appointment window for all patients to maximize her long‐run average net reward, which depends on the rewards collected from patients served and the “penalty” paid for those who cannot be scheduled. Using a stylized M/M/1/K queueing model, we provide an analytical characterization for the optimal appointment queue capacity K, and study how it should be adjusted in response to changes in other model parameters. In particular, we find that simply increasing appointment window could be counterproductive when patients become more likely to show up. Patient sensitivity to incremental delays, rather than the magnitudes of no‐show probabilities, plays a more important role in determining the optimal appointment window. Via extensive numerical experiments, we confirm that our analytical results obtained under the M/M/1/K model continue to hold in more realistic settings. Our numerical study also reveals substantial efficiency gains resulted from adopting an optimal appointment scheduling window when the provider has no other operational levers available to deal with patient no‐shows. However, when the provider can adjust panel size and overbooking level, limiting the appointment window serves more as a substitute strategy, rather than a complement.     

Strategies for Appointment Policy Design with Patient Unpunctuality

Appointment policy design is complicated by patients who arrive earlier or later than their scheduled appointment time. This article considers the design of scheduling rules in the presence of patient unpunctuality and how they are impacted by various environmental factors. A simulation optimization framework is used to determine how to improve performance by adjusting the schedule of appointments. Prior studies (that did not include patient unpunctuality) have found that a scheduling policy with relatively consistent appointment interval lengths in the form of a dome or plateau dome rule to perform well in a variety of clinic environments. These rules still perform reasonably well here, but it is shown that a combination of variable‐length intervals and block scheduling are better at mitigating the effects of patient unpunctuality. In addition, performance improves if the use of this policy increases toward the end of the scheduling session. Survey and observational data collected at multiple outpatient clinics are used to add realism to the input parameters and develop practical guidelines for appointment policy decision making.     

Reserving Capacity for Urgent Patients in Primary Care

This paper examines the effect of the common practice of reserving slots for urgent patients in a primary health care practice on two service quality measures: the average number of urgent patients that are not handled during normal hours (either handled as overtime, referred to other physicians, or referred to the emergency room) and the average queue of non‐urgent or routine patients. We formulate a stochastic model of appointment scheduling in a primary care practice. We conduct numerical experiments to optimize the performance of this system accounting for revenue and these two service quality measures as a function of the number of reserved slots for urgent patients. We compare traditional methods with the advanced‐access system advocated by some physicians, in which urgent slots are not reserved, and evaluate the conditions under which alternative appointment scheduling mechanisms are optimal. Finally, we demonstrate the importance of patient arrival dynamics to their relative performance finding that encouraging routine patients to call for same‐day appointments is a key ingredient for the success of advanced‐access.     

Improving Performance in Outpatient Appointment Services with a Simulation Optimization Approach

Outpatient health care service providers face increasing pressure to improve the quality of their service through effective scheduling of appointments. In this paper, a simulation optimization approach is used to determine optimal rules for a stochastic appointment scheduling problem. This approach allows for the consideration of more variables and factors in modeling this system than in prior studies, providing more flexibility in setting policy under various problem settings and environmental factors. Results show that the dome scheduling rule proposed in prior literature is robust, but practitioners could benefit from considering a flatter, “plateau‐dome.” The plateau–dome scheduling pattern is shown to be robust over many different performance measures and scenarios. Furthermore, because this is the first application of simulation optimization to appointment scheduling, other insights are gleaned that were not possible with prior methodologies.     

Sequencing and Scheduling Appointments with Potential Call‐In Patients

This paper studies appointment scheduling for a combination of routine patients who book well in advance and last‐minute patients who call for an appointment later that same day. We determine when these same‐day patients should be scheduled throughout the day, and how the prospect of their arrivals affects the appointment times of the routine patients. By formulating the problem as a stochastic linear program, we are able to incorporate random and heterogeneous service times and no‐show rates, ancillary physician tasks, and appointment delay costs for same‐day patients who prefer to see the doctor as early as possible. We find that the optimal patient sequence is quite sensitive to the no‐show probabilities and the expected number of same‐day patients. We also develop two simple heuristic solutions to this combinatorial sequencing problem.     

Managing appointments with waiting time targets and random walk-ins

The random arrivals of walk-in patients significantly affect the daily operations of healthcare facilities. To improve the performance of outpatient departments, this paper attempts to make an appointment schedule by considering walk-ins and the waiting time target (WTT) for appointment patients. A stochastic programming model is proposed to solve this problem with the objective of minimizing the weighted patient waiting and makespan cost. A non-decreasing waiting cost function is used to capture the WTT fulfillment of appointment patients, whereas walk-ins incur a linear waiting cost. A finite-horizon Markov Decision Process model is formulated to establish the optimal real-time scheduling policy under a given appointment schedule. The appointment schedule is determined by a two-stage stochastic programming approximation and a local search improvement. Structural properties of the optimal appointment scheduling and real-time scheduling policies are established. In particular, it is shown that appointment overbooking is allowed only at the end of the regular session, and the optimal real-time scheduling policy is an easy-to-implement threshold policy with bounded sensitivity. Numerical experiments based on real data are performed to investigate the influence of different parameters and to compare different schedules. The optimal schedule demonstrates superior performance by allowing reasonable waiting times for appointment patients depending on their WTTs. Managerial insights are also provided to hospital managers. Finally, the basic model is extended by incorporating random service times and random arrivals of appointment patients. The latter includes the random number of patients that show up for service or call for appointments, and the random arrival time (unpunctuality). Appointment overbooking strategies are shown to have different structures under some stochastic factors.     

Emergency Department Staff Planning to Improve Patient Care and Reduce Costs

In the face of high staffing costs, uncertain patient arrivals, and patients unsatisfied with long wait times, staffing of medical emergency departments (EDs) is a vexing problem. Using empirical data collected from three active EDs, we develop an analytic model to provide an effective staffing plan for EDs. Patient demand is aggregated into discrete time buckets and used to model the stochastic distribution of patient demand within these buckets, which considerably improves model tractability. This model is capable of scheduling providers with different skill profiles who work either individually or in teams, and with patients of varying acuity levels. We show how our model helps to balance staffing costs and patient service levels, and how it facilitates examination of important ED staffing policies.     

Optimal Policy for a Stochastic Scheduling Problem with Applications to Surgical Scheduling

We consider the stochastic, single‐machine earliness/tardiness problem (SET), with the sequence of processing of the jobs and their due‐dates as decisions and the objective of minimizing the sum of the expected earliness and tardiness costs over all the jobs. In a recent paper, Baker ( 2014 ) shows the optimality of the Shortest‐Variance‐First (SVF) rule under the following two assumptions: (a) The processing duration of each job follows a normal distribution. (b) The earliness and tardiness cost parameters are the same for all the jobs. In this study, we consider problem SET under assumption (b). We generalize Baker's result by establishing the optimality of the SVF rule for more general distributions of the processing durations and a more general objective function. Specifically, we show that the SVF rule is optimal under the assumption of dilation ordering of the processing durations. Since convex ordering implies dilation ordering (under finite means), the SVF sequence is also optimal under convex ordering of the processing durations. We also study the effect of variability of the processing durations of the jobs on the optimal cost. An application of problem SET in surgical scheduling is discussed.     

A stochastic programming model for outpatient appointment scheduling considering unpunctuality

Many papers on outpatient appointment scheduling assume that patients arrive on time. However, unpunctuality is a stochastic factor that is inevitable in practice, which leads to patients arriving out of order. A schedule may not be reasonable if a clinic neglects the influence of patient unpunctuality. This paper addresses the outpatient scheduling problem considering unpunctuality (OS-U) by developing a stochastic programming model. We compare the performance of the OS-U system with the strict-punctuality (OS-P) system. We illustrate that the model has an exact and unified formula for cases of patients arriving in the appointment order and arriving out of order. The OS-U problem is solved by using Benders decomposition combined with the sample average approximation (BD-SAA) technique to determine the global optimal set of appointment intervals with the goal of minimizing the weighted sum of all patient waiting times, doctor idle times, and overtime. Numerical experiments indicate that the appointment rule changes when considering unpunctuality, although the set of optimal appointment intervals still takes the shape of dome (interval width increases at first, then remains nearly constant and eventually decreases for the last patients). The OS-P system schedules the first two patients together at the start of a session, whereas the OS-U system schedules them with different appointment times and requires a longer slot between the first two patients if patients tend to arrive early rather than late. The variance of unpunctuality has little impact. The no-show probability has a greater influence on system performances in an OS-U system than those in an OS-P system.