The effect of cancelled appointments on outpatient clinic operations

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The Effect of Cancelled Appointments on Outpatient Clinic Operations Shannon L. Harris , Jerrold H. May , Luis G. Vargas , Krista M. Foster PII: DOI: Reference:

S0377-2217(20)30087-4 https://doi.org/10.1016/j.ejor.2020.01.050 EOR 16304

To appear in:

European Journal of Operational Research

Received date: Accepted date:

31 October 2019 22 January 2020

Please cite this article as: Shannon L. Harris , Jerrold H. May , Luis G. Vargas , Krista M. Foster , The Effect of Cancelled Appointments on Outpatient Clinic Operations, European Journal of Operational Research (2020), doi: https://doi.org/10.1016/j.ejor.2020.01.050

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Highlights

    

Develop a multi-day scheduling model that includes inter- and intra-day waiting Our solution approach optimizes the model without resorting to heuristics Introduce the concept of the switch point of total demand for a scheduling horizon The switch point marks the boundary between when cancellations hurt or help a clinic Switch point is calculated without explicitly obtaining the optimal clinic schedule

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The Effect of Cancelled Appointments on Outpatient Clinic Operations Shannon L. Harrisa* [email protected]

Jerrold H. Mayb [email protected]

Luis G. Vargasb [email protected]

Krista M. Fosterc [email protected]

a

School of Business, Snead Hall, Virginia Commonwealth University, Richmond, VA USA 23284 Katz Graduate School of Business, Mervis Hall, University of Pittsburgh, Pittsburgh, PA USA 15260 c Mendoza College of Business, University of Notre Dame, Notre Dame, IN USA 46556 b

*Corresponding Author

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ABSTRACT This paper studies how appointment cancellations affect scheduling strategies in outpatient healthcare clinics. While cancellation rates in outpatient clinics have been reported to be as high as 27%, cancelled appointments are often ignored, or grouped with no-shows in healthcare scheduling models. We find that there exists a value of total demand that, when calculated over a scheduling horizon, marks the boundary between where cancellations hurt or help a clinic. We refer to that value as the switch point. Up to the switch point, clinics can achieve a greater reward when patients do not cancel. However, for values of expected total demand greater than the switch point, the clinic reward is reduced if more patients retain (do not cancel) appointments. To assist us in evaluating the switch point, we construct a mixed-integer nonlinear programming model to solve a multi-day outpatient scheduling problem. The model accounts for both inter-day (appointment day) and intra-day (appointment time slot) scheduling decisions, while balancing service benefits against service costs. We include probabilities of no-show and cancellation, which allows us to discuss how cancellations affect scheduling decisions through the switch point. The knowledge of the switch point allows a clinic to understand when appointment no-shows and cancellations negatively affect clinic service, and can assist the clinic in determining the number of patients for which it is committed to provide service, i.e., its panel size. In this paper, we discuss methodologies for calculating the switch point, and discuss its sensitivity to model parameters. Keywords: OR in health services, appointment scheduling, analytics, appointment no-shows, appointment cancellations

1 Introduction Patient behavior, such as no-shows and cancellations, lead to schedule inefficiencies and contribute to underutilization of clinic resources and/or overtime. That behavior curtails timely access to healthcare systems (IOM 2015), and may lead to lengthy patient scheduling queues and wait times at a clinic. In addition, schedule inefficiencies may reduce patient satisfaction and contribute to “poorer health outcomes” (IOM 2015, p. 11). Clinics that suffer from a greater supply of appointment requests than appointment availability must make strategic scheduling decisions in order to better serve their patients (Berg et al. 2013). No-shows have been found to be an integral part of scheduling decisions (Dantas et al. 2018, Zacharias and Pinedo 2014, LaGanga and Lawrence 2012, Robinson and Chen 2010), while, in the literature, cancellations are either not mentioned or are grouped with no-shows. Advance cancellations – cancellations that occur at least one day before the appointment is to occur – differ from no-shows and should also be considered. No-shows primarily differ from cancellations because advance cancellations may be rebooked by another patient, while no-shows can only be utilized by walk-ins. Additionally, the probabilities of cancellation and no-show may not follow the same distribution over time. Based upon analyses at representative Veterans Health Administration (VHA) clinics, advance appointment cancellation rates vary by the age of the patient and range from 15% to 27% (Goffman et al. 2017, Whittle et al. 2008), and no-show rates have also been found to exceed 20% (Whittle et al. 2008). Consequently, clinic schedulers are faced with the challenge of booking patients

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with a wide range of patient attendance behavior, and in this paper, we focus on how advance cancellations affect those scheduling decisions. The primary contribution of this paper is the concept of the total demand switch point. The switch point is associated with how the reward functions for two types of scheduling models – with and without cancellations – relate to each other. It is the value of total demand that lies at the intersection of the reward curves of the two models, and marks the boundary between when cancellations hurt or help a clinic. In the remainder of the paper, total demand refers to the expected total demand over a specific scheduling horizon. Consider a clinic with fixed capacity over a scheduling horizon. If the number of patients a clinic services is less than its capacity, then the clinic benefits from servicing additional patients. If the number of patients a clinic services is greater than its capacity, then the reward from servicing additional patients will, at some point, decrease, due to the increased cost of potential overtime and clinic waiting time. Consequently, cancellations will hurt a clinic if, because of appointment cancellations, a clinic services fewer of its scheduled appointments and, thus, operates at a lower demand level. Conversely, cancellations will help a clinic if, due to the number of patients scheduled in the appointment slots, the clinic is over capacity, and would prefer that fewer of the scheduled appointments are serviced, in order to reduce overtime and clinic waiting time costs. At the intersection of the reward curves of the models with and without cancellations lies a point at which the clinic is indifferent to cancellations, and will not profit or lose if appointments are cancelled. We refer to the value of demand at which the two curves intersect as the switch point. In this paper, we discuss the relationship of the location of that intersection (the switch point) to the clinic parameters, and provide insight as to how the concept of the switch point may assist a clinic in operational and tactical planning. Preliminary results of our analysis are as follows: a. If the total demand that a clinic schedules over a scheduling horizon is less than the switch point, a clinic can anticipate a greater expected reward if more of the scheduled patients complete their appointments. That will occur if patients retain (do not cancel) appointments during the scheduling horizon, and show for the appointments that are retained. In this scenario, cancellations require a clinic to make scheduling decisions such as reducing its capacity, or increasing the number of patients for which it is committed to provide service, i.e., its panel size (Ahmadi-Javid et al. 2017), in order to better match supply and demand for appointments. The clinic may also take tactical steps to reduce patient non-attendance, such as live-reminder calls or text messages, parking assistance, or providing transportation. b. If a clinic consistently schedules at a level of demand greater than the switch point, the clinic’s expected reward increases with increasing cancellations. In that case, the clinic benefits if fewer scheduled patients complete their appointments, and the clinic may make strategic decisions to increase its capacity or to decrease its panel size.

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Thus, determining the switch point allows a clinic to make decisions concerning daily demand booked, and to determine the appropriate panel size for the clinic. To the best of our knowledge, the prior literature has neither identified nor discussed the implications of the switch point. In this paper, we characterize when the switch point occurs, as a function of the clinic service benefits and costs, and, using computational analysis, we show how the total demand at which the switch point occurs changes with the probabilities of cancellation and show for small to moderate sized clinics. The rate is a non-linear function of the no-show and the cancellation rates. In addition, we develop a model that may be used to approximate the switch point for larger clinics, based upon model parameters. To assist in studying the switch point, we construct a mixed-integer nonlinear programming model to solve a multi-day scheduling model that balances the clinic benefits of seeing patients against the costs. We model a specialty clinic that schedules up to six patients per session; extensions to larger clinics are discussed in the conclusions. The model accounts for two types of patient nonattendance – appointment no-shows and cancellations. The probability of completion – the probability that a patient shows for an appointment given that they have not cancelled – is set to vary, based upon the appointment lead time. The lead time of an appointment is defined as the time interval between when the appointment is made and when it is scheduled to occur. We tackle both the inter-day (appointment day) and intra-day (appointment time slot) scheduling problems, which Feldman et al. (2014) identify as a gap in the literature. Furthermore, because scheduling decisions may vary widely, we outline general overbooking and scheduling strategies for a range of clinic parameters, emphasizing the influence of cancellations on the scheduling strategies. Intuitively, substantial cancellation rates could lead to more aggressive overbooking strategies, because patients may remove themselves from the schedule. Our modeling structure is directly motivated by our observations of scheduling in a hospitalbased specialty clinic. Although the impetus for this paper is the study of a healthcare clinic, our conclusions may be applicable to other service industries that operate over a scheduling horizon, under availability constraints, with users that interact with the system before service is provided, and who may no-show or cancel. The inclusion of inter-day and intra-day scheduling solutions is pertinent for appointment scheduling systems in which both day and slot placements must be considered. For example, in the healthcare industry, the clinic may schedule patients in need of infusion services in advance for a specific day, and assign a time slot after all requests for appointments on that day have been recorded (Mandelbaum et. al. 2019). Additionally, our conclusions will apply in any service industry where customers are required to submit requests for appointments online before service is provided, such as hair salons, auto repair, and legal consultations. This paper provides several contributions to the appointment scheduling literature. First, we develop a new concept, which we term the switch point, and discuss its effect on scheduling decisions. We believe that we are among the first to discuss day and slot scheduling assignments over a multi-

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day horizon in which patients may no-show or cancel, and, thus, the first to discuss the switch point. While optimal day assignment has been discussed (Feldman et al. 2014, Liu et al. 2010), we are able to tackle slot assignment as well. Our computational results are mined from a dataset created by solving our model with a variety of specifications across multiple scheduling horizons. Our novel approach to complete enumeration, as a solution technique, allows us to determine an exact solution for small to moderate sized clinic schedules, without resorting to heuristic methods, and to create an extensive dataset of model solutions. Based upon the solution results, we develop a model which may be used to calculate the switch point for larger clinics. Lastly, because scheduling strategies are a function of both the no-show and the cancellation probabilities, we show that eliminating cancellations from overbooking analysis leads to less preferred outcomes. The rest of the paper is organized as follows. Section 2 includes a review of the related literature. In Section 3, we present our model and assumptions, and Section 4 outlines our model’s properties. Section 5 details the solution technique for the model, Section 6 discusses the numerical analysis of the switch point, and Section 7 discusses some extensions of our model. Finally, Section 8 summarizes our findings and presents conclusions.

2 Literature Scheduling in outpatient healthcare clinics has been studied for some time, beginning with Bailey (1952). Since then, models have been extended to consider various patient attendance behavior, patient preference, and patient classifications. Cayirli and Veral (2003), Gupta and Denton (2008), and Ahmadi-Javid et al. (2017) present reviews of developments and general methodologies used in scheduling models, and outline possible open challenges in healthcare scheduling. To our knowledge, a concept such as the switch point has not been discussed in the scheduling literature, and, thus, we position our paper based upon the scheduling model we developed to discuss the switch point. Table 1: Positioning of This Paper Within the Existing Literature

Paper Rohleder and Klassen (2002) Kaandorp and Koole (2007) Hassin and Mendel (2008) Glowacka et al. (2009) Klassen and Yoogalingam (2009) Liu et al. (2010) Robinson and Chen (2010) Cayirli et al. (2012) LaGanga and Lawrence (2012) Feldman et al. (2014) Zacharias and Pinedo (2014) Samorani and LaGanga (2015) Parizi and Ghate (2016) Lee et al. (2018) This Paper

X

Day Assignment X

X

X

Multi-Day

Slot Assignment X X X X X X X X

X

X X

X X

X X

X

X

6

X X

No-Shows X X X X X X X X X X X X X X X

Cancellations

X

X

X X

An open challenge in Gupta and Denton (2008) suggests that more models need to consider both direct and indirect waiting. Typically, direct waiting time, the time that a patient spends waiting in a clinic, is considered in single-day scheduling models, and it is known to be affected by patient noshows. Multi-day scheduling models are more likely to consider indirect waiting time, the time that a patient waits outside of the clinic (e.g., the time that a patient’s appointment request is deferred). With a multi-day scheduling horizon, a patient may cancel her appointment or no-show for her appointment if it has not been cancelled. To frame our work in the current literature, we review multi-day scheduling models that consider both types of waiting, and the single-day models that address patient no-show behavior. Table 1 shows how our paper adds to the existing literature. The majority of work in scheduling is focused on the sequencing or blocking of patients within a single day, while balancing a combination of direct waiting, doctors’ idle time, and clinic overtime. Samorani and LaGanga (2015) develop a multi-day scheduling model that accounts for indirect waiting and day-dependent no-show predictions. Patients are assigned to a clinic day, and are sequenced based upon their assignment to a show or no-show class. Overbooking is done using slot compression, as opposed to multi-booking a slot. They find that scheduling over a longer horizon is beneficial to a clinic that uses heterogeneous no-show probabilities, due to the increased flexibility. Rohleder and Klassen (2002) develop a multi-day scheduling model in which patients may no-show. Their model includes several demand patterns, overbooking rules, and delay rules; and they find that the correct choice of slot overbooking versus indirect waiting is dependent on patient needs. While both of those papers consider scheduling, and, thus overbooking, for multi-day horizons, they do not include patient cancellation behavior in their models. To our knowledge, Liu et al. (2010), Feldman et al. (2014), and Parizi and Ghate (2016) are the only papers that account for no-shows and cancellations in a multi-day scheduling model. Those papers tackle the multi-day problem with Markov Decision Process (MDP) techniques, but the models do not include slot assignment. Due to the complexity of the models, they design heuristics or relaxation techniques to solve them. Liu et al. (2010) calculate delay-dependent no-show and cancellation probabilities, and use them as inputs to an MDP. They compare the performance of their proposed heuristic policies – which include patient behavior – with benchmark heuristics, and provide insight for a “best” scheduling policy based upon clinic parameters. Feldman et al. (2014) build a similar dynamic scheduling model, but include patient preference. They find that clinics can achieve better performance when the current schedule and patient preference set are taken into account simultaneously. Parizi and Ghate (2016) calculate no-show and cancellation probabilities that are delay and job-type dependent. They manipulate the structure of their MDP model to develop a Lagrangian relaxation technique. All three papers tackle the inter-day scheduling problem in the presence of cancellations. However, there is no discussion of how including cancellations may affect scheduling decisions or outcomes. Furthermore, the addition of slot assignment is listed as an important, open research question in all three papers. They agree that the increased complexity – of an

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already complex problem – makes the inclusion of slot assignment a problem that a researcher should be creative in addressing. We consider that problem in the paper. Because the papers that account for no-shows and cancellations in a multi-day scheduling model do not consider slot assignment, we now review the optimization techniques used in single-day models, which do consider slot assignment, and seek to extend that work in order to tackle the day and slot assignment problem. In particular, we review papers that develop models with a single-day and a single server, with patients who may no-show for their appointments. Kaandorp and Koole (2007), Hassin and Mendel (2008), Klassen and Yoogalingam (2009), and Robinson and Chen (2010) develop models that consider patients with homogenous no-show probabilities. In Kaandorp and Koole (2007), the objective function includes direct waiting time, overtime, and doctors’ idle time. Service times are exponential, and inter-arrival times are found to have a dome shape. Hassin and Mendel (2008) investigate how schedules should be formulated with patient no-shows. They find a similar dome result, as did Kaandorp and Koole (2007), and note that, as the probability of show decreases, the inter-arrival times at the beginning and end of the day decrease. Klassen and Yoogalingam (2009) use a simulation optimization approach. They find that introducing no-shows into their model leads to slot double-booking. Robinson and Chen (2010) compare open access scheduling – in which patients can only make same-day appointments – with traditional systems, in which patients can make appointments an arbitrary number of days in advance. Patients may no-show for appointments in the traditional system, and service times are deterministic. They find that the ability to defer patients to a day later than the one requested may improve clinic performance when the length of the clinic day is close to the expected workload. We extend the work in those papers by including a multi-day horizon and patient cancellations in our model, and by comparing models with and without cancellations. Glowacka et al. (2009), Cayirli et al. (2012), LaGanga and Lawrence (2012), Zacharias and Pinedo (2014), and Lee et al. (2018) also build single-day models, but consider heterogeneous noshow probabilities. Glowacka et al. (2009) use an association rule mining (ARM) model to predict noshow probabilities, and find that such an approach for including no-show behavior can significantly improve the performance of appointment scheduling. Cayirli et al. (2012) introduce walk-ins into their model, and use simulation and non-linear regression to determine a universal “dome” appointment rule, given clinic parameters. LaGanga and Lawrence (2012) vary no-show probabilities, based upon the appointment slot, in an analytical optimization model. Zacharias and Pinedo (2014) develop a similar model, but allow for high and low no-show probabilities. Lee et al. (2018) consider patient heterogeneity in service times and no-shows, and develop a load-balancing schedule that surpasses the benchmark dome methods. We add to this stream of research by developing an analytical optimization problem that is a generalization of the models in LaGanga and Lawrence (2012) and in Zacharias and Pinedo (2014). Our model extends to multiple days, includes cancellations, and provides both day and slot assignments.

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In this paper, we develop and discuss the concept of the switch point of total demand. A

concept somewhat analogous to the switch point, threshold curves, has been discussed in capacity management. Chapman and Carmel (1992) discussed how threshold curves may be used in sensitive healthcare applications to manage demand and to determine when dynamic pricing should be offered. In our context, we assume that the healthcare services are not price sensitive, and, therefore, capacity management should be managed by the panel size and clinic size/length of day. While models in the literature have included cancellations, there has been no discussion on how scheduling decisions vary based upon cancellation rates. We begin that discussion by investigating a multi-day scheduling horizon, and study the effect of cancellations on clinic service.

3 Model Description As a tool to investigate the effect of cancellations on outpatient clinics, we construct a model that represents a clinic that services patients on an appointment basis (see Table 2 for a summary of our notation). Patients request appointments for a day in a scheduling horizon of length H. There are N appointment slots each day the clinic is open, and the length of each appointment slot is fixed and constant. The assumption of fixed service times is a common assumption in the scheduling literature (e.g., Robinson and Chen 2010, LaGanga and Lawrence 2012, Lee et al. 2018). That assumption makes the model more tractable, and is appropriate for clinics with low variability of service times, such as routine elective surgery clinics, mental health clinics, and routine care clinics (LaGanga and Lawrence 2012). Additionally, assuming fixed service times allows us to lay the groundwork for a schedule that focuses on the sources of variability of concern in this paper – appointment no-shows and cancellations. The clinic expects daily patient demand of qi, i = {1, …, H}, that is received by the clinic at H

the beginning of each day in the scheduling horizon. We assume that qi ≥ N and

q i 1

i

 M , so that M

≥ H×N and the clinic may need to overbook (Zacharias and Pinedo 2014, Lee et al. 2018). When qi > N, the clinic overbooks the additional patients on day i, or on a future day in the scheduling horizon. The goal of the clinic scheduler is to schedule all expected demand during the scheduling horizon, so that all patients may be seen in a timely fashion. The decision variables for our model detail how each day’s demand should be scheduled, assuming there is at least one person in each slot of the day. Thus, our model provides a scheduling template for a future scheduling horizon, given the clinic parameters. Further assumptions concerning clinic interaction with patients are as follows. We assume that the clinic has a single provider, or does not permit providers to share patients. When multiple patients are present at the beginning of a time slot, all but one patient must wait, and the others are serviced using a FCFS discipline (Zacharias & Pinedo 2014). No patients are booked, a priori, to

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overtime slots. The clinic is permitted to use overtime, and the daily overtime is not bounded, so that all patients scheduled on a day are seen that day. The clinic cannot change or remove appointments from the schedule unless requested by the patient. The current model does not account for a clinic that accepts walk-ins. The following assumptions detail how the patient interacts with the clinic. Patients request an appointment for a day in the scheduling horizon. Patients are scheduled to arrive at the beginning of their assigned time slot, and they are assumed to arrive punctually, if at all. A patient may cancel her appointment before it occurs, or no-show, given she has not cancelled. In the remainder of this paper, we refer to the probability of show as the conditional probability of a patient appearing for an appointment, given that she has not cancelled. We refer to an appointment for which the patient shows, given that she has not cancelled, as a completed appointment. Table 2: Notation Parameter H N M S i d d–i qi ηd-i θd-i pd-i = ηd-i×θd-i sidj

Ldj Z(d) π w(k) y(k) A(i) δ Π(S) W(S) O(S) I(S) R(S)

Definition length of the scheduling horizon number of regular time slots per day number of patients scheduled for the clinic horizon minimum number of slots needed to service all patients in a clinic day clinic schedule for the scheduling horizon day the appointment is requested day the appointment is scheduled indirect waiting time expected demand on day i = {1,…,H} average show rate for patients with d-i days of indirect waiting time average retention rate for patients with d-i days of indirect waiting time average appointment completion rate for patients with d-i days of indirect waiting time probability of k patients in backlog at the end of slot j on day d for schedule S number of patients from day i scheduled to arrive at the beginning of slot j on day d probability that k patients arrive at the beginning of slot j from the sidj scheduled patients maximum backlog at the end of slot j on day d for schedule probability of g patients showing in slot j on day d from day i appointment requests number of patients from prior day’s requests assigned to slot j on day d expected number of completed appointments on day d marginal net benefit of seeing an additional patient marginal cost of k units of direct waiting time marginal cost of k units of over time total indirect waiting time cost for day i penalty incurred for each unit of indirect waiting time clinic service benefit function for schedule S direct waiting time function for schedule S overtime function for schedule S indirect waiting time function for schedule S expected clinic service reward for schedule S (R(S) = Π(S) – W(S) – O(S) – I(S)

Let i denote the day for which a patient would like to schedule an appointment, and d the day the appointment is assigned. The indirect waiting, or lead time of the appointment, is given by (d – i). We denote by pd-i the probability that an appointment with lead time (d – i) is completed on the day of the appointment. The probability of completion, pd-i, is a function of two elements: (1) the probability that the patient has not cancelled during the lead time and (2) the probability that the patient shows for the appointment. Let θd-i, (0 ≤ θd-i ≤ 1) denote the probability that a patient does not cancel (or retains)

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an appointment with lead time (d – i), and let ηd-i, (0 ≤ ηd-i ≤ 1) denote the probability that a patient shows, given that she has not cancelled an appointment with lead time of (d – i). Then pd-i is defined as pd-i = ηd-i×θd-i. That functional form combines both cancellations and no-shows into a single function. The core analysis of this paper focuses on how changes in θd-i affect pd-i, and thus a clinic’s scheduling decisions. The probabilities in our model are assumed to represent the average behavior across the patient base, so all patients with the same lead time are assigned the same probability of completion, pd-i. Patients who are scheduled on their request day are permitted to cancel and to noshow; patients who cancel are assumed to not rebook within the scheduling horizon. Due to our model design, a slot for which an appointment was cancelled is not refilled. Despite the fact that appointments that are cancelled are not refilled, the separation of pd-i into show and retention components allows a clinic to estimate the probability of completion for appointments based upon both types of patient behavior, and does not require that the evolution of the two probabilities over the appointment lead time be identical. Our solution model may be considered as myopic, in that all appointment requests must be scheduled within the H days of the scheduling horizon, regardless of the request day. For example, if H = 5 and a patient requests an appointment for day i = 4, the patient may only be scheduled on day 4 or 5. Our approach is similar to the solution method in Samorani and LaGanga (2015), where the authors build a myopic model to solve a scheduling problem that includes day-dependent no-shows. As noted in Samorani and LaGanga (2015, pp 248-249), a myopic multi-day model is still more general than the models in the current literature that consider patient no-show behavior over a single day (Robinson and Chen 2010, LaGanga and Lawrence 2012, Glowacka et al. 2009, Muthuraman and Lawley 2008, Zeng et al. 2010). In order to include advance cancellations, a multiday model must be considered. Recent papers have tackled the multi-day scheduling problem, which considers both patient no-shows and cancellations, by modeling the problem as a Markov decision process (MDP) (Liu et al. 2010, Feldman et al. 2014, Parizi and Ghate 2016). However, those papers only take into account the inter-day problem, and determine the clinic day for a scheduled appointment. Tackling both the inter-day and the intra-day problems with an MDP is computationally complex (Feldman et al. 2014), and hence, it has not been pursued with that methodology. Because of the complexity of alternative methods, and the desire to provide a solution for an inter-day and an intra-day scheduling problem while considering no-shows and cancellations, we proceed with a myopic model. Our model may apply to a clinic that books appointments for a scheduling window after all requests are received and provides a next step in answering how cancellations will influence clinic decisions. The assumption that scheduling decisions are delayed until after observing all demand in the scheduling horizon is common when scheduling elective surgeries (Gupta and Denton 2008) and infusion services (Mandelbaum et. al. 2019). In practice, clinics may define a cut-off time

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for which they stop accepting new patient requests, and schedule all appointments simultaneously (Dexter et al. 1999). The goal of the clinic scheduler is to book patients so that the clinic’s expected service reward, R(S), is maximized. As in LaGanga and Lawrence (2012) and in Zacharias and Pinedo (2014), our reward function assigns a benefit to the clinic for patients who complete appointments, and a penalty for service costs, such as patient waiting and the use of overtime. Because we tackle the inter-day and the intra-day problems, the service cost of our model accounts for indirect waiting (the lead time a patient experiences before an appointment), and direct waiting (patient waiting time while in the clinic) costs. The service cost components are affected by the schedule, the number of people who are expected to retain or to show, and the patient backlog. Some of the prior work in scheduling includes an idle time cost term in the objective function of the scheduling problem. When maximizing the benefit of seeing patients, the objective function increases for each slot in which a patient is treated, and is penalized for the time that a patient waits for an appointment (both direct and indirect waiting), as well as for overtime. If minimizing the doctor’s idle time, the objective function increases for each slot without a patient, but it does not necessarily take into account the cost to patients for direct and for indirect waiting time. Because the clinic’s mission is to serve patients, and demand is exogenous, it is desirable to consider the problem from the patients’ standpoint, and to find the schedule which most benefits the patients. Thus, as in LaGanga and Lawrence (2012) and in Zacharias and Pinedo (2014), we choose to maximize the clinic’s benefit from seeing patients. 3.1 Patient Backlog The patient backlog at the end of a slot represents the number of patients who have shown up for a clinic appointment, but who have not been seen by a provider by the end of their assigned timeslot. The backlog level is fundamental to the calculation of direct waiting time and overtime, and drives the model complexity and the optimal overbooking strategy of the model solution. Let sidj denote the number of patients scheduled to arrive at the beginning of slot j on day d, d

from day i demand (1≤ i ≤ d), and s dj   sidj . Then the clinic schedule, S, is a 1 × (HN) row vector i 1

of the s dj , and the number of patients scheduled over the scheduling horizon is, M = SeT, where e = (1,…,1).1 Let

denote the probability that there are k patients in backlog at the end of slot j, (1≤

), on day d, (1 ≤ d ≤ H) for schedule , where N d ( S )  N  K dN represents the latest

j ≤ possible

appointment

slot

at

the

clinic

on

day

d,

including

overtime

and

d

K dN ( S )  K d , N 1 ( S )   sidN  1 is the maximum backlog at end of slot N on day d. For each slot, i 1

is computed recursively, based upon the number of patients who arrive during slot j and the 1

Because the paper tackles both inter-day and intra-day schedules, the row vector 𝐻 rows and 𝑁 columns.

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can also be thought of as a matrix with

backlog from the prior slot. We assume that the number of patients who complete their appointment during a slot is a binomially distributed random variable with parameters sidj and pd-i. Let

denote

the probability that k patients arrive at the beginning of slot j from the sidj scheduled patients, given they have not cancelled. The probability

is dependent on pd-i, and thus on θd-i and pd-i. Then,

can be expressed as follows:





 B 0 ( S )  0  1  B1 ( S ) 0 1 j 1 11 j 11 j 1 j 1 11 j   k Ki , j 1 ( S ) B1 j ( S )     B1l j 1 ( S )11k jl 1  l 0 

for k  0 for 1  k  K dj ( S )

0 B10 (S)  1 a B10 (S)

 0, a 

(1) 

d

where K dj ( S )  K d , j 1 ( S )   sidj  1 is the maximum backlog at end of slot j on day d. i 1

The first line of the expression in Equation (1) is the probability of zero patients in backlog at the end of slot j. A zero patient state can occur in two ways. First, when there are no patients in backlog at the end of slot j-1, and zero or one person arrives at the beginning of slot j and is serviced during that slot. Second, when one patient is in backlog at the end of slot j-1, zero patients show at the beginning of slot j, and the patient waiting at the end of slot j-1 is serviced in slot j. The second line of the expression in Equation (1) is the joint probability of l patients in backlog at the end of slot j-1 and k-l+1 patients from slot j demand showing, and thus summing to k patients in backlog at the end of slot j, because one patient is serviced during slot j. The day 1 backlog equations are similar to the equations in the current single-day scheduling literature (Zacharias and Pinedo 2014, LaGanga and Lawrence 2012). For our multi-day model, we extend Equation (1) so that it applies to all subsequent days in the scheduling horizon. To accomplish that, we add an additional term to account for patients who are scheduled on day d from prior days’ demand, and who may have cancelled during their lead time. The inclusion of the additional term is a novel update to the analytical model, which, to our knowledge, has not been considered in the scheduling literature. Figure 1 shows the possible inflow of patients into slot j, for day d, d ≥ 1, that would result in k patients in backlog at the end of slot j. In general, the backlog at the end of slot j is affected by l (the number of patients in backlog at the end of slot j-1), g (the number of patients who are assigned from previous days’ requests to the current day), the number of patients assigned from the current day’s requests, and the person serviced in slot j.

13

Figure 1. Possible Inflow of Patients into Slot j so as to Result in k Patients in Backlog at the End of Slot j

Let

denote the probability of g patients showing at the beginning of slot j on day d from

day i appointment requests. Then the recursive expression for g

g g a  idj    ia1,d , j idj

is given by:

i  1,.., d  1

a 0

 00dj  1  0adj  0, a  As in Equation (1) for

(2) 

, the outcome of g patients showing at the beginning of slot j results

from accumulating assignments from prior days. Patients present in a slot may be assigned to the slot from that day’s demand, assigned to the slot from previous days’ demands, or trickled down as patients who were not serviced in prior slots that day. Our formulation allows us to account for all the possible ways in which g patients could show from all prior days in the scheduling horizon. The decrease in the probability of completion as a function of indirect waiting is captured in the λ term. d 1

Let Ldj ( S )   sidj denote the number of people from prior days’ requests assigned to slot j on day i 1

d. Then, the backlog equation for day d, d > 1 is given by





0 1 0 1 0 1 0 0  B 0 ( S ) 0 d 1, d , j ddj  ddj  Bd , j 1 ( S ) d 1, d , j ddj  Bd , j 1 ( S ) d 1, d , j ddj  d , j 1  k K d , j 1 ( S ) Ldj ( S ) Bdj (S)   k l  g 1 l  B ( S ) d 1  d , j 1   dg1,d , j ddj  l 0 g 0 

for k  0 for 1  k  K dj ( S )

(3) Equation (3) for day d, d > 1 follows logic similar to that of Equation (1) for day 1, but also accounts for cancellations and for previous days’ assignments. Demand arrives from three avenues: prior slots, prior days, and the current slot’s demand. The first line of the expression, for k = 0, is an expression for the number of ways demand can arrive from the three avenues so as to result in a zero backlog at the end of slot j. The second line is an expression for achieving k > 0 patients in backlog at the end of slot j, with l patients from prior slots, g from prior days, and k–l–g+1 from the current slot’s demand. 3.2 Clinic Service Benefit

14

The clinic is assumed to receive a benefit for every patient serviced in the scheduling horizon. The benefit corresponds to the financial profit or goodwill received from attending to a patient, and thus is applicable to both not-for-profit and for-profit organizations. The clinic receives a benefit, π, for seeing a patient. Assuming that the total benefit to the clinic is linear in the number of patients who show, the expected service benefit function for schedule S is given by: H

  S     Z d  ,

(4)

d 1

d

N

where Z  d    sidj  pd i denotes the expected number of completed appointments on day d. i 1 j 1

3.3 Service Costs In our model, we consider three types of service costs: direct waiting time, overtime, and indirect waiting time. Direct waiting time occurs if a clinic chooses to overbook an appointment slot, and multiple patients show for that slot. Overtime occurs when surplus demand is not cleared by the end of the regular clinic day. Thus, the calculation of direct waiting time and overtime is dependent on the backlog equation. Indirect waiting time cost is accrued for all patients who are not given an appointment on the day they requested. Direct and indirect waiting costs are primarily a function of the value of patients’ time, and may be difficult to quantify. The clinic’s part of overtime costs may be primarily due to additional personnel costs, and may be more easily measured. We incorporate the three costs in a way similar to other papers in the literature. 3.1.1 Direct Waiting Time Cost Direct waiting time is the time spent by patients waiting in a clinic for service. The cost of patient waiting includes patient dissatisfaction, loss of patient goodwill, and potential loss of future business. Let w(k) represent the waiting cost function, where k denotes the number of appointment slots a patient must wait for service, which is equal to the number of patients in backlog who arrived before that patient. We assume that w(k) is a non-decreasing convex function on the non-negative integers, and that w(0) = 0. Our formulation for direct waiting time is an extension of the equation in LaGanga and Lawrence (2012). We modify their equations to calculate the waiting time at the end of a slot, and we calculate direct waiting time over a scheduling horizon. The expected waiting time cost across all appointment slots and possible levels of backlog is given by: H N d ( S ) K dj ( S )

W S  

 

d 1 j 1

k 0

Bdjk ( S )  w  k 

Because our formulation of direct waiting time incorporates the backlog functions, it represents the sum of the expected individual patient waiting costs incurred across the scheduling horizon. 3.1.2 Overtime Cost

15

(5)

Overtime costs are incurred when a clinic overbooks appointment slots, and surplus demand is not cleared before the end of the regular clinic day. In this paper, we do not allow for a priori booking of patients to slots outside of the regular clinic day. The overtime costs measure the cost of provider time, wages paid to clinic staff, and loss of goodwill with the affected patients. Let y(k) represent the overtime cost incurred per slot of overtime used, given the schedule S. The function y(k) is assumed to be convex and non-decreasing on the non-negative integers, and to behave similarly to the direct waiting cost function. The expected clinic overtime costs over the scheduling horizon are given by: H K dN ( S )

O S   



d 1 k 0

k BdN (S)  y k  ,

(6)

where each term of the summation represents the expected overtime penalty for k patients waiting for service at the end of the last scheduled appointment slot. Similar to Equation (5), Equation (6) also accounts for patient no-shows and cancellations within the backlog equation. 3.1.3 Indirect Waiting Time Cost Indirect waiting time is accrued for each day a patient is delayed service. We assume that patients prefer to be seen as close to their request day as possible. A penalty, δ, is incurred by the clinic for each day a patient’s appointment is scheduled later than their request day, even if the patient later cancels or no-shows the appointment. Our formulation is similar to the indirect waiting calculation in Samorani and LaGanga (2015). The total indirect waiting cost for schedule S is given by: H

I  S    A  i  ,

(7)

i 1

H N

where A i    sidj   d  i  denotes the total patient delay for patients who request an d i j 1

appointment for day i. 3.4 Clinic Service Reward The clinic service reward (CSR) links the expected service benefit and costs for a clinic. The goal of a clinic is to schedule patients in a manner that maximizes the expected CSR, where the CSR is equal to the service benefit minus the waiting time costs and the overtime cost. Given the prior definitions, the CSR, R(S), is given by: N d ( S ) K dj ( S ) K dN ( S ) H   k k R  S      Z  d    A  d     Bdj ( S )  w  k    BdN (S)  y k     d 1  j 1 k 0 k 0 

Expression (8) allows for the probabilities of show and retention to be any functions that are decreasing in increasing indirect wait time. Additionally, any increasing convex functions may be used to represent the waiting time and overtime cost functions.

16

(8)

The decision variables for R(S) are the sidj variables, which state on which day and slot the daily demand, qi, should be scheduled. R(S) is defined on a lattice, i.e., on the positive integer values of its components. To generate solutions, we solve the following problem (P1): Maximize R(S) s.t. H N

 sidj  qi

(P1)

i

(9)

d i j 1

d

 sidj  1

d , j

(10)

i 1

Constraint (9) ensures that all demand scheduled equals the expected demand. Constraint (10) ensures that there is at least one patient scheduled in each slot. Proposition 1 in Section 4, explains why that will be true, as long as the demand for each day is at least N (qi ≥ N) and the indirect waiting cost is greater than 0 (δ > 0). For a given demand vector q = (q1, …, qH), the solution to P1 gives the optimal day and slot allocation of the demand, denoted by Sq* . For a given value of total demand, M, let QM = H

{q = (q1, …, qH) | qi ≥ N and

q i 1

i

 M }. Thus, there are as many Sq* solutions for a given M as the

cardinality of QM, |QM|.

4 Model Properties In this section, we define the properties of the scheduling model that motivate our solution algorithm and our discussion of the switch point. In the propositions and in the remainder of this paper, we refer to the schedule derived from solving problem P1 as the optimal schedule for a given set of parameters; we do not posit that it is the optimal schedule achieved by exhausting all possible parameter values. Proofs of the propositions are included in the Appendix. A common proposition in single day appointment scheduling states that overbooking is not optimal if an empty slot is available. Proposition 1 provides the conditions for which that concept extends to a multi-day schedule. First, if the demand for each day is at least N (qi ≥ N), then, for each individual day, the single day proposition holds. Thus, an empty slot will only occur in a schedule with overbooking if it is optimal to make a patient incur indirect waiting, as opposed to being assigned to their requested day. That will never be optimal when the indirect waiting time cost is greater than zero. PROPOSITION 1. A clinic schedule for one or more days in the scheduling horizon, with at least one patient assigned to each appointment slot on every day, has a greater expected clinic service reward than does a clinic schedule with overbooked slots, when open slots are available, under the following conditions: (i) the demand for each day is at least N (qi ≥ N) and, (ii) there is a non-zero cost of patient indirect waiting (δ > 0).

17

Proposition 1 allows us to only consider schedules in which at least one patient is scheduled in each clinic slot on each day. Proposition 2 states the directional concavity of the reward function R(S), which stems from the fact that the direct waiting time and the overtime costs are both convex, and the expected benefit is linear and thus directionally concave. PROPOSITION 2. The reward function R(S) is directionally concave. A directionally concave function is both submodular and component-wise concave (Zacharias and Pinedo 2017). Those two properties imply that, after at least one patient is scheduled in each slot in the scheduling horizon, the clinic can expect diminishing returns for each additional patient that is scheduled (see also Zacharias and Pinedo 2017), and thus influence the existence of the switch point. In the next proposition, we show that the switch point exists. Let the expected clinic reward for a clinic without cancellations be denoted by R(S), and the expected clinic reward for a clinic with cancellations be denoted by Rc(S) = Πc(S) – Wc(S) – Oc(S) – Ic(S) = Πc(S) – Cc(S) – Ic(S), where Cc(S) = Wc(S) + Oc(S). PROPOSITION 3. C(S) – Cc(S) is directionally convex, therefore there exists a switch point between R(S) and Rc(S). Let Rc(S) and R(S) denote, respectively, the expected clinic service reward functions for a schedule, with and without cancellations. Note that the indirect waiting time cost, I(S), does not depend on cancellations, so I(S) is equal to Ic(S) for all schedules. Thus, for the purposes of discussing the switch point, Rc(S) and R(S) may be written as Rc(S) = Πc(S) – Cc(S) and R(S) = Π(S) – C(S), respectively, or as benefit minus costs. The switch point occurs when Rc(S) = R(S), or when Π(S) – Πc(S) = C(S) – Cc(S). In the proof of the proposition, we show that the intersection is guaranteed to occur when C(S) – Cc(S) is directionally convex. A generic example of the behavior of Π(S), Πc(S), C(S), and Cc(S) as functions of M is depicted in Figure 2a. For simplicity, we depict the functions in Figure 2a and 2b as continuous, although their support is a lattice. In Figure 2a, the benefits and costs for Rc(S) and R(S) are depicted as individual curves. The shaded areas indicate the difference in benefits, Π(S) – Πc(S), and difference in costs, C(S) – Cc(S), between the two models. As total demand increases, the differences both increase due to the decrease in benefit and costs when appointments are cancelled. M* denotes the location of the switch point, which occurs when C(S) – Cc(S) equals Π (S) – Πc(S). In Figure 2b, the differences are depicted as individual curves. For values of total demand less than M*, for the same number of patients scheduled, C(S) – Cc(S) < Π(S) – Πc(S) (i.e., Rc(S) < R(S)), and a clinic expects less reward if appointments are cancelled. For values of total demand greater than M*, C(S) – Cc(S) > Π(S) – Πc(S) (i.e., Rc(S) > R(S)), and a clinic expects a greater reward if appointments are cancelled. The relationship between R(Sc) and R(S) is dependent upon the retention rate of the clinic, θdi,

and the clinic parameter values. Because of the complexity of the cost function, it is beyond the

scope of the current paper to provide a closed form function that calculates the size of the schedule at

18

which the switch point occurs. In Section 6.5, we develop an estimation model that may be used to calculate a switch point based upon model parameters.

(a)

(b) Figure 2. Existence of the Switch Point. 2a. Relationship between the cost and benefit functions at the switch point, M* 2b. Relationship between the differences in the cost and benefit functions at the switch point, M*

5 Solution Techniques Due to the complexity of optimizing a non-linear integer program, we initially solved P1 using complete enumeration. The number of candidate schedules to be enumerated increases quickly as H and N increase. Therefore, we developed a partial enumeration technique that allows us to find an optimal solution. As the size of the problem increases, the time necessary, even for a partial enumeration technique, may grow to be too large to be practical for real-time solution. In such situations, there are two alternatives. One, the scheduling problem could be solved not in real-time, or with an efficient heuristic, such as the one suggested by Federgruen and Groenevelt (1986). Two, the approach we take in this paper, is to approximate the switch point for larger problems based upon problem characteristics without having to calculate the schedule. Our approximation function is detailed in Section 6.5. P1 can be thought of as two separate problems: (1) how many people to schedule on each day (the inter-day or scheduling problem) and (2) how to sequence the patients on each day (the intra-day or sequencing problem). The sequencing problem for day i is a function of the number of patients scheduled on day i from prior days; the number of patients who request appointments on day i and who are scheduled for appointments on day i; and the number of patients from day i deferred to future

19

days. The scheduling problem determines the optimal number of patients to transfer between days in the scheduling horizon. By construction, an optimal solution to the scheduling problem is the aggregation of all optimal single day sequences. Thus, for our solution technique, we enumerate sequences by day, and then combine schedules, as opposed to enumerating schedules for all days at the same time. We outline the partial enumeration process and provide a detailed write-up in the Appendix.

6 Numerical Experiments In this section, we discuss the behavior of the switch point across a series of parameter values. The parameter values used to generate the results cover an array of clinic types, and align with the parameters used by comparable models in the literature (e.g., Robinson and Chen 2010, LaGanga and Lawrence 2012, Salzarulo et al. 2015). We illustrate that the expected CSR functions are directionally concave and that a switch point exists, as shown in Proposition 3. We also explore the sensitivity of the switch point to changes in the scheduling horizon, H; the clinic capacity, N; and the probability of completion, ηd-i. Because the focus of this paper is on the effect of cancellations (or non-retained appointments), we specifically discuss the sensitivity of the switch point to θd-i, a component of pd-i. In order to perform numerical experiments, we must first define functional forms for the probability of retention, the probability of show, direct waiting time, and overtime. In the no-show prediction literature, it has been widely suggested that the lead time of an appointment and the probability of show are inversely related (e.g., Galluci et al. 2005, Whittle et al. 2008, Goffman et al. 2017). While Wang and Gupta (2011) found no significant correlation in their data between lead time and the probability of show, in this paper we assume patients are more likely to complete appointments with a shorter lead time. For each day of indirect waiting, (d-i), we assume that the probability of show decreases by a factor of α, and that the probability of retention decreases by a factor of β. Thus, we assume that ηd-i = η0×αd-i and θd-i = θ0 ×βd-i, where η0 and θ0 are the average probabilities of show and of retention, respectively. Thus, the probability of completion as a function of the lead time is as follows: pd-i = ηd-iθd-i ×αd-iβd-i. We assume linear functions for the direct waiting time and overtime functions (LaGanga and Lawrence 2012), which are given by w(k) = ω×k and y(k) = σ×k, respectively. Therefore, the clinic incurs a penalty of ω for each unit of direct waiting, and a penalty of σ for each unit of overtime incurred. We first illustrate the switch point over scheduling horizons of H = {2, 3, 4, 5} days for N = 4. Those possibilities might correspond to a four-hour session of four 60-minute appointment time slots. We assume that patients show with probability η0 = 0.90 and that the probability of show decreases by 5% for each day the patient incurs indirect waiting, so that α = 0.95. Appointment slots are retained with probability θ0 = {0.85, 0.9, 0.95, 1}; which, together with the probability of show, correspond to a probability of completion for a same day appointment of p0 = {0.765, 0.81, 0.855, 0.9}. For all analyses when θ0 = 1, β = 1. That represents a model in which appointment slots are not

20

cancelled on any day in the scheduling horizon; thus, the probability of completion is equal to the probability of show. The model when θ0 = 1 and β = 1 is used to calculate the switch point relative to models that assume that θ0 < 1. When θ0 ≠ 1, we assume that β = 0.95, so that the probability of retention decreases by 5% for each day the patient incurs indirect waiting. The values of α and β allow us to generate probabilities of completion that decrease in a way similar to that of the probabilities found in Galluci et al. (2005) and Liu et al. (2010). For all analyses, the benefit per patient, π, is normalized to 1, and, in a way similar to that of other scheduling papers (e.g., Robinson and Chen 2010, LaGanga and Lawrence 2012, Salzarulo et al. 2015) we set the overtime parameter to be 50% greater than the benefit of seeing a patient. Direct and indirect waiting costs are set at 5% for all analyses. We refer to a unique combination of parameters as a profile. Thus, for each H, we generate solutions for four profiles. For each profile, problem P1 is solved for a series of demand sets. A demand set is the vector H

of expected daily demands, q = (q1, …, qH), for the scheduling horizon, and

q i 1

i

is the total demand

for the demand set, or M. We evaluate demand sets for which qi ranges from N to 2×N, i = {1,…, H}. For example, when N = 4, H = 2, and M = 10, we solve the scheduling problem for the following demand sets: (q1, q2)

{(4,6), (5,5), (6,4)}. Evaluating our model over a range of demand sets allows

us to analyze the reward function across several values of total demand, and find the value of M at which the switch point exists. Because R(S) is directionally concave, and any linear combination of directionally concave functions is directionally concave, we depict the expected CSR for a value of H

total demand, M, as the average of the expected CSRs for all q for which

q i 1

i

M.

6.1 Characterization of the Switch Point The expected CSR curves for H = {2, 3, 4, 5}, N = 4, and θ0 = {0.85, 0.9, 0.95, 1} are shown in Figure 3. As anticipated, the expected CSR for the model where patients are assumed to not cancel, θ0 = 1, has the greatest expected CSR when M = H×N. As the number of patients in the schedule increases, the curve for which θ0 = 1 decreases and crosses the curves that represent models in which appointment slots are retained at rates of 5% to 15%. Each of those crossing points represents a switch point between the model for which θ0 = 1 and a model with a particular value of θ0 < 1.

21

Figure 3. Depiction of the Switch Points for N = 4 and H = {2, 3, 4, 5} (η0 = 0.9, α = β = 0.95, ω = δ = 0.05, σ = 1.5)

Table 3 lists the switch point values for the curves in Figure 3. For example, the curve for which θ0 = 1 crosses the curve for which θ0 = 0.85 near M = 11. When θ0 = 0.9 or 0.95, the switch point values are similar. When the retention rate decreases to 85% for longer scheduling horizons, the switch point increases by one. The switch point decreases as the probability of retention increases, because more people who are scheduled are expected to complete appointments, and, therefore, the capacity constraint occurs at a lower value of total demand. The switch point marks the boundary between when cancellations hurt or help a clinic. Thus, a clinic that operates with the clinic parameters as outlined above, schedules over a five-day horizon, and observes that 85% of appointment slots are retained, should target scheduling 28 patients in the available 20 slots. If the clinic consistently schedules less than 28 patients, and appointment slots are retained at the 85% rate, the clinic can achieve a better expected reward when more patients attend, and the clinic should incentivize patients to not cancel. If the clinic consistently schedules more than 28 patients, then the clinic is expected to generate more reward when patients cancel. Because the goal of the clinic is to service patients in need of care, operating while expected reward is greater when patients cancel is not preferable. Operating beyond the switch point also indicates that if a patient calls to cancel, the clinic should reschedule the patient outside of the current scheduling window. The switch point can also be calculated as the average number of patients to be scheduled in each slot of the scheduling horizon, or the slot switch point. That value is shown in parentheses next to the switch point in Table 3. In practice, clinics that schedule based upon a patient’s probability of show, η0, would like to book a minimum of ⁄

patients per slot, in order to achieve, on average, one

patient per slot to show for an appointment. In this paper, we consider a patient’s probability of completion, which includes the probability that a patient retains an appointment. When η0 = 0.9 and θ0 = {0.85, 0.9, 0.95}, p0 = {0.765, 0.81, 0.855}, and a minimum of 1.31, 1.23, and 1.17 patients per slot should be scheduled, respectively, to achieve, on average, one completed appointment per slot. The

22

values in Table 3 are slightly more conservative due to the fact that our model seeks to find the maximum number of patients a clinic should schedule before it would be over capacity if all patients showed. Additionally, in our experiments we assume that the minimum demand for a clinic is at least the number of slots available, so our results will typically suggest at least one overbooked patient. If a clinic finds that it is consistently operating at a level greater than or equal to the switch point, the clinic can decide to make strategic decisions, such as decreasing its panel size or increasing the capacity of the clinic, in order to reduce the use of overtime. The values in the table indicate that the slot switch point is relatively stable across values of θ0 and H. While the clinic parameters influence the switch point, the slot switch point for a parameter profile – which can be calculated when H = 2 – can be used as an indication of the switch point across the horizon. Next, we investigate the switch point for larger values of N when H = 2. Table 3: Switch Point Values when N = 4. Slot Switch Point Shown in Parentheses (η0 = 0.9, α = β = 0.95, ω = δ = 0.05, σ = 1.5) H 2 3 4 5

θ0 = 0.85 11 (1.38) 17 (1.42) 22 (1.38) 28 (1.4)

θ0 = 0.9 11 (1.38) 16 (1.33) 22 (1.38) 27 (1.35)

θ0 = 0.95 11 (1.38) 16 (1.33) 21 (1.31) 27 (1.35)

6.2 Effect of Clinic Capacity on the Switch Point In this section, we continue to set H = 2, and vary N = {4, 5, 6}. Those possibilities might correspond to a four-hour session with 60, 48, or 40-minute appointments, respectively. The additional parameters of the model are η0 = 0.9, α = β = 0.95, ω = δ = 0.05, σ = 1.5, and θ0 = {0.85, 0.95, 1}. We solve problem P1 for demand sets for which q1 ranges from N to 3×N, and q2 ranges from N to (4×N) – q1. That allows for the demand in the scheduling horizon to double across both days in the scheduling horizon. The results of the analyses are given in Figure 4. The daily switch point values are listed in Table 4. We also list the slot switch point values in parentheses, which indicate how many patients per slot the clinic should book in order to prevent disruptions from cancellations. Note that the daily switch point values in Table 4 are relatively stable across different values of N, and are similar to the slot switch point values displayed in Table 3.

23

Figure 4. Depiction of the Switch Points for H = 2, and N = {4, 5, 6} (η0 = 0.9, α = β = 0.95, ω = δ = 0.05, σ = 1.5) Table 4: Daily Switch Point Values when H = 2. Slot Switch Point Values Shown in Parentheses (η0 = 0.9, α = β = 0.95, ω = δ = 0.05, σ = 1.5) θ0 = 0.85 5.5 (1.38) 7 (1.40) 8 (1.33)

N 4 5 6

θ0 = 0.95 5.5 (1.38) 6.5 (1.30) 7.6 (1.25)

6.3 Marginal Analysis In prior analyses, we varied the scheduling horizon and the clinic capacity to illustrate the switch point. In this section, we illustrate the sensitivity of the daily switch point to changes in β, the change in θ0 over the scheduling horizon, and to changes in the probability of show, η0. The slot switch point is more sensitive to changes in η0 than it is to changes in β. Thus, it is more important for a clinic to accurately estimate the show probability of a patient than it is to accurately estimate how the cancellation rate changes over a scheduling horizon. 6.3.1 Effect of the Changes in Cancellation Probability on the Slot Switch Point In the previous tables, we observed that the slot switch point appears to be relatively insensitive to changes in H and N. Table 5 illustrates that the slot switch point also appears to be relatively insensitive to changes in β. Table 5: Slot Switch Point Values when H = 2. (η0 = 0.9, α = 0.95, ω = δ = 0.05, σ = 1.5) Switch point values N 4 4 4 5 5 5 6

β 0.75 0.85 0.95 0.75 0.85 0.95 0.75

θ0 = 0.85 1.38 1.38 1.38 1.40 1.40 1.40 1.33

24

θ0 = 0.95 1.38 1.38 1.38 1.30 1.30 1.30 1.25

Switch point values β 0.85 0.95

N 6 6

θ0 = 0.85 1.33 1.33

θ0 = 0.95 1.25 1.25

6.3.2 Effect of Show Probability on the Slot Switch Point Table 6 illustrates that the slot switch point is more sensitive to the change in the probability of show than it is to changes in the probability of retention. As η0 increases, the slot switch point decreases, and a clinic may schedule fewer patients per slot in order to mitigate the negative effect of cancellations. Table 6: Slot Switch Point Values when H = 2. (α = β = 0.95, ω = δ = 0.05, σ = 1.5) Switch point values θ0 = 0.85 θ0 = 0.95 1.88 1.75 1.63 1.50 1.38 1.38 1.90 1.90 1.60 1.50 1.40 1.30 1.92 1.83 1.58 1.50 1.33 1.25

η0 0.6 0.75 0.9 0.6 0.75 0.9 0.6 0.75 0.9

N 4 4 4 5 5 5 6 6 6

6.4 Characterization of the Schedule at the Switch Point We now discuss the day and slot scheduling decisions before, at, and beyond the switch point. As in Section 6.1, we let H = 2, N = 4, η0 = 0.9, α = 0.95, and β = 0.95. When θ0 = 0.85, the switch point occurs at M = 11. As a point of comparison, we show schedules for θ0 = 1 and θ0 = 0.85 for demand sets (5,5), (6,5), and (6,6). Those demand sets represent the day 1 and day 2 demands that result in the greatest expected CSR when demand is equal to the values of M before, at, and beyond the switch point. Table 7: Daily Schedules when N=4 and H = 2 (η0 = 0.9, α = β = 0.95, ω = δ = 0.05, σ = 1.5) Day 1

Day 2

Demand Set

θ0

Slot 1

Slot 2

Slot 3

Slot 4

Slot 1

Slot 2

Slot 3

Slot 4

(5, 5)

0.85

2

1

1

1

2

1

1

1

(5, 5)

1

2

1

1

1

2

1

1

1

(6, 5)

0.85

2

2

1

1

2

1

1

1

(6, 5)

1

2

1

1

2

2

1

1

1

(6, 6) (6, 6)

0.85 1

2 2

2 1

1 1

1 2

2 2

2 1

1 1

1 2

The day and slot schedules for the demand sets are shown in Table 7. As in prior research, the schedules have a front-loaded pattern, and the first slot of the day is always overbooked (Bailey 1952, Robinson and Chen 2010, LaGanga and Lawrence 2012, Zacharias and Pinedo 2014). That result serves to “prime the pump,” and to help ensure that the doctor is not idle at the beginning of the day. The slot assignments for the patients in the same demand set vary, based upon the value of θ0. When

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θ0 = 1, the probability of completion for the patients is greater, and the overbooked patients are booked closer to the end of the day. That is also a common scheduling result, where, as the probability of completion increases, overbooking is spread out more throughout the day (LaGanga and Lawrence 2012, Zacharias and Pinedo 2014). 6.5 The Switch Point as a Function of Model Parameters Exact switch points can be calculated from the solutions of the scheduling model. However, it is not necessary to solve the scheduling model to calculate a switch point. Because the direct waiting time and overtime functions are recursive, finding a closed form solution for the switch point based on the optimization model is computationally intensive. In this section, we outline how the switch point can be estimated for larger scheduling problems, based on model parameters and the solutions to smaller scheduling problems. The switch point estimation model allows for the calculation of a switch point for larger clinic sizes, where the solution of problem P1 may be computationally expensive or lengthy. The switch point is the smallest value of the total demand such that the expected CSR when all patients retain appointments (θ0 = 1) is less than the expected CSR for any other value of θ0. In order to provide a general expression for the switch point as a function of the parameters modeled in this paper, we tune a quadratic regression that models the expected CSR curve – as depicted in Figures 3 and 4 – as a function of total demand, N, η0, and θ0. We then find the intersection of the curves when cancellations are not considered, θ0 = 1, and when patients are assumed to cancel, θ0 = {0.75, 0.85, 0.95}. The switch point lies at the intersection of those curves, and the estimated CSR curves may be used to estimate a switch point for other parameters without solving the full scheduling problem. The parameters H and β are not included in the regression, as the calculation of the switch point was found to be robust to changes in those variables. We limit this numerical study to the functional dependence of the switch point on N, η0, and θ0, as those are the core parameters discussed in this paper. A topic of future work is to complete a larger study that would also include the additional parameters, H, α, β, ω, δ, and, σ. The approach of using a quadratic regression to model scheduling output is similar to the one used in Zacharias and Pinedo (2017) to predict optimal overbooking levels. The quadratic regression equation we tune to estimate an expected CSR curve is shown in Equation (11), where total demand is denoted by M.

E  CSR | M , N ,0 ,0   a0  a1N  a20  a30  a4 M  a5 N 2  a6 N0  a7 N0  a8 NM  a902  a1000  a110 M  a1202  a130 M  a14 M 2

,

(11)

Table 8 lists the parameters of the model that are used for training and testing the quadratic regression. Those values for the training and test sets allow us to determine how well the switch point may be estimated when extrapolating to greater clinic sizes.

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Table 8: Model Parameters Used in Quadratic Regression when H = 2, α = β = 0.95, ω = δ = 0.05, and σ = 1.5 Parameter N η0 θ0

Training Set {2, 3, 4, 5, 6} {0.6, 0.75, 0.9} {0.75, 0.85. 0.95}

Test Set {7, 8} {0.6, 0.75, 0.9} {0.75, 0.85. 0.95}

The results of the quadratic regression are summarized in Table 9. All variables are significant at the 0.05 level. Table 9: Result of Quadratic Regression Model Intercept Coefficients

Coefficients

a1

a2

a3

a4

a5

a6

a7 4.066

-23.880

-5.664

30.294

26.909

2.284

-0.306

4.839

a8

a9

a10

a11

a12

a13

a14

0.217

-10.429

-15.376

-1.559

-8.362

-1.304

-0.037

The fit of the quadratic model is measured by how well it can be used to estimate the switch point for combinations of N, θ0, and η0. To calculate the switch point, we find the value of total demand where the regressions for θ0 = 1 and θ0 = {0.75, 0.85, 0.95} intersect. Many of the terms cancel, and the equation for the switch point as a function of N, θ*, and η0 is given by Equation (12), where SP denotes the switch point, θ* is the retention probability of the model that assumes patients may cancel, and  x  denotes the integer ceiling of x.

   a3  a7 N  a100  a12 1  *       30.3  4.1N  15.40  8.4 1  *    E  SP | N ,0 ,*      , (12) a13 1.304     The results of estimating the switch point for N = 7 and N = 8 while varying θ* and η0 are shown in Table 10. The switch point values in the table are the total demand across a two-day scheduling horizon that mark the boundary of when cancellations hurt or help a clinic. Table 10: Estimated and Actual Switch Point Values N 7 7 7 8 8 8

η0 0.6 0.75 0.9 0.6 0.75 0.9

θ* = 0.75 Estimated Actual 25 28 23 23 21 20 28 32 26 26 24 22

θ* = 0.85 Estimated Actual 24 27 22 22 20 19 27 30 25 25 24 21

θ* = 0.95 Estimated Actual 23 25 22 21 20 18 27 29 25 23 23 20

The estimated switch points follow the same trend as the actual values, with the values increasing with N and decreasing as θ* and η0 increase. Those results are intuitive. As θ* and η0 increase, more patients are expected to complete appointments, and the value of total demand for which the clinic may benefit from cancellations occurs at a lower value. The estimates tend to be very accurate for η0 = 0.75. When η0 = 0.6, the estimates tend to underestimate the switch point and be less

27

accurate, and when η0 = 0.9, the switch point is overestimated and more accurate. The level of accuracy when η0 = 0.6 for greater values of N may be due to the fact that too limited a range of total demand was considered in the optimization runs to provide enough data for the estimation function. For example, when N = 4 and η0 = 0.6, the switch point occurs at a value of total demand greater than 16; a number that represents double the number of appointment slots we permitted to be scheduled across the two-day horizon. An intuitive lower bound for the switch point, across any value of the parameters, is the total demand value at which the model that does not consider cancellations, i.e., when θ0 = 1, reaches its maximum, or the maximum preferred demand. As shown in this paper, when the show rate is not low, as in η0 = 0.6, a loose upper bound for the switch point is the value of total demand associated with scheduling 𝑁

𝐻 patients in the horizon, or double the number of appointment slots across the

horizon. The intuition for the lower bound is as follows. The maximum preferred demand for a scheduling model that does not consider cancellations should be less than the maximum preferred demand for a scheduling model in which patients are assumed to cancel. That is because when cancellations are considered, fewer patients are expected to complete appointments, and more patients must be scheduled in order to achieve the same expected reward. Additionally, both expected reward functions are equal to zero if no patients are scheduled. Thus, at the maximum preferred demand of the no-cancellation model, and at every value of total demand less than that, the cancellation model should have a lesser expected reward, and the intersection of the two curves has not occurred. From Proposition 2, the two curves are guaranteed to cross, and thus, the intersection, i.e. the switch point, occurs at a value greater than the maximum preferred demand for the no-cancellation model. In this section, we introduced a modeling method that a clinic may use to estimate switch point values for larger clinics, based upon the results from exact calculations for smaller clinics. The results in this section apply to a two-day scheduling horizon but may be extended for larger horizons. In Section 6.1, we discussed how the switch point is reasonably robust to changes in the length of the scheduling horizon, thus, an estimated switch point value when H = 2 may be converted to a slot switch point and multiplied by the number of days in a scheduling horizon, in order to obtain an estimate for any value of H.

7 Model Extensions 7.1 Effect of the Probability of Cancellation on the Daily Schedule We further consider the effect of θ0 on the number of patients booked on each day. As in Section 6.4, we let H = 2, N = 4, η0 = 0.9, α = 0.95, and β = 0.95. We consider three values of θ0 to reflect different retention rates for all possible demand sets for which q1 ranges from N to 3×N, and q2 ranges from N to (4×N) – q1. For demand sets with q1 ≤ q2, the number of patients scheduled on each day appears to be insensitive to changes in θ0.

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Furthermore, when the total demand for two days is an even number, θ0 does not affect the daily schedule totals. If q1 ≥ q2 and the total demand is odd, so that it cannot be distributed equally across two days, we observe that likelihood to schedule day 1 demand on day 2 is increasing with θ0. Table 11 displays those observations for five demand sets. Similar results were observed when the show probability was decreased to η0 = 0.75, but the schedules are less likely to defer patients to the next day. Table 11: Daily Schedules when N=4 and H = 2 (η0 = 0.9, α = β = 0.95, ω = δ = 0.05, σ = 1.5) Day 1

Day 2

Demand Set (7,4)

θ0 0.75

Schedule (6,5)

Slot 1 3

Slot 2 1

Slot 3 1

Slot 4 1

Slot 1 2

Slot 2 1

Slot 3 1

Slot 4 1

(7,4)

0.85

(6,5)

2

2

1

1

2

1

1

1

(7,4)

0.95

(5,6)

2

1

1

1

2

1

2

1

(7,5)

0.75

(6,6)

3

1

1

1

3

1

1

1

(7,5)

0.85

(6,6)

2

2

1

1

2

2

1

1

(7,5)

0.95

(6,6)

2

1

2

1

2

1

2

1

(8,5)

0.75

(7,6)

3

2

1

1

3

1

1

1

(8,5)

0.85

(6,7)

2

2

1

1

3

1

2

1

(8,5)

0.95

(6,7)

2

1

2

1

2

2

1

2

(8,6)

0.75

(7,7)

3

2

1

1

3

2

1

1

(8,6)

0.85

(7,7)

3

1

2

1

3

1

2

1

(8,6)

0.95

(7,7)

2

2

1

2

2

2

1

2

(8,7)

0.75

(8,7)

3

2

2

1

3

2

1

1

(8,7)

0.85

(8,7)

3

1

2

2

3

1

2

1

(8,7)

0.95

(7,8)

2

2

1

2

2

2

1

3

7.2 Generalization to a Single-Day Model with Multiple Classes The results of our model generalize to scheduling multiple classes of patients on a single day, where class

patients complete appointments with probability pc. When H = 4, our solution technique

generates results for up to four types of patients scheduled on the final day of the horizon. Because patients are typified based upon their lead time, and their probability of completion depends on the lead time, using a four day scheduling horizon is analogous to assuming a single clinic day with four classes of patients. Zacharias and Pinedo (2014) present a general rule for scheduling two classes of patients in a clinic day. They find that if patients are assumed to have the same direct waiting time cost, patients should be booked in alternating vertical and horizontal segments, where “all customers in a vertical segment and the immediately following horizontal segment in an optimal schedule are sequenced in decreasing order of their no-show probability” (Zacharias and Pinedo 2014, page 4). Vertical segments indicate overbooked slots, and horizontal segments indicate the last patient in a vertical segment, grouped with the patients up to the next overbooked slot. Their scheduling rule would result in a schedule in which patients with a lower probability of show, or patients booked from prior days,

29

are overbooked in a single slot, and patients with a greater probability of show, or patients from the current day’s demand, are booked in subsequent slots. The schedules generated from our model illustrate similar results. When N = 6, H = 2, η0 = 0.6, α = β = 0.95, ω = δ = 0.05, σ = 1.5, and θ0 = 1, the optimal number of patients to schedule over the scheduling horizon is 19. In Figure 5, we depict the schedule for the demand set (13, 6) when θ0 = 0.85 and 1; the patients scheduled on day 2 from day 1 demand are depicted as shaded squares. When θ0 = 0.85, we obtain the same result as in Zacharias and Pinedo (2014). The patients from day 1 have the smallest probability of show, and are all overbooked in the first slot of the day. The six patients from day 2 are scheduled in the subsequent slots. When θ0 = 1, one of the day 1 patients is overbooked in the first slot of the day. Note that the patients scheduled on day 2 from day 1 demand, when θ0 = 1, are assumed to not cancel, and complete their appointments with a 5% decrease in show probability. The schedule in Figure 5 also illustrates the importance of specifying a probability of cancellation in a scheduling model. If, when θ0 = 1, all three overbooked patients from day 1 are overbooked in the first slot, the expected direct waiting for all subsequent patients in the schedule would increase, and, while the clinic would service the same number of people, it would do so in a way that is more disruptive to the clinic and to the patients who are made to wait.

Figure 5. Daily Schedules when N=6 and H = 2 (η0 = 0.6, α = β = 0.95, ω = δ = 0.05, σ = 1.5)

8 Summary and Conclusions In this paper, we examine how appointment cancellations affect scheduling decisions, by introducing the switch point of total demand. Clinic no-shows have been found to be disruptive to a clinic schedule, and we contribute to the literature on outpatient scheduling by also considering the disruption that may be caused by appointment cancellations. While comparing models that include cancellations with those that do not, we find that there is a value of clinic demand, referred to as the switch point, which clinics can use as a boundary for total demand, to determine when cancellations help or hurt the clinic. To our knowledge, we are the first to discuss and compare those two types of models, and the concept of the switch point. The concept of the switch point may be used operationally by a clinic as follows. If a clinic consistently schedules fewer appointments than the switch point value for a given set of parameters, it

30

can achieve a greater expected reward if more patients complete their scheduled appointments. In this scenario, the clinic should take steps to try to reduce non-attendance, by providing live-reminder calls or other incentives, increasing the panel size of the clinic, or reducing the number of appointment slots available. If the clinic consistently schedules more appointments than the switch point value, then the clinic is expected to generate greater reward when patients cancel. Because the goal of the clinic is to service patients in need of care, operating while expected reward is greater when patients cancel is not preferable. Thus, a clinic may choose to reduce its panel size, or to increase the capacity of the clinic, in order to be able to meet patient demand while using its resources economically. The results of this paper were calculated for smaller clinics, of up to six patient slots. The regression model outlined in Section 6.5 provides a tool that a clinic may use to estimate the switch point for larger clinics. We found that the estimation model can predict the actual switch point to within two patients when extrapolated up to eight appointments per day. A clinic size of eight appointments per day is typical in clinics such as routine elective surgery clinics or mental health clinics, the type of clinic considered in this paper. For clinics that operate with greater than eight appointments per day, the scheduling problem might be solved with a heuristic method, or by using a computing system powerful enough to calculate a timely solution. Additionally, while we presented the number of appointment slots to represent the total number in a clinic day, our results may also apply if the number of slots represents the number of appointments in a clinic session, in situations where a provider may hold multiple, separate sessions in a single day. Our research makes several contributions. First, we introduce and provide a proof for the existence of a switch point, and illustrate when it occurs for several sample clinic parameters. While a clinic would like to operate at optimal capacity, the switch point allows a clinic to determine if it is operating at a level where, if patients were to cancel, it would be detrimental to the clinic. We examine the relationship between the switch point and the various clinic parameters, and propose a method to solve for the switch point for a given set of parameters without explicitly obtaining the optimal clinic schedule. That allows for a more general representation of the switch point, and allows a clinic to estimate switch point values for larger clinics sizes. Second, we develop a multi-day scheduling model that can be used to generate appointment schedules for outpatient clinics. Our model differs from prior models, in that we assume patients may cancel their appointments if they are made to incur indirect waiting. The inclusion of direct and indirect waiting is listed as an open literature topic by Gupta and Denton (2008). Third, we propose a solution technique that can be used to solve our model to optimality, without resorting to heuristic techniques. Finally, we show that our results follow the scheduling assumptions of prior papers in the literature on outpatient scheduling, and provide a brief discussion of how cancellations also affect the slot in which a patient is scheduled. In prior papers that consider cancellations, only the day of the appointment is specified. Our model

31

allows for a day and slot designation, which is important to an outpatient clinic servicing patients with minimal wait and overtime. We observe that there are several opportunities for future research. The first extension of our model might be to relax the assumption of fixed service times for each patient. While that assumption may be appropriate for some clinics, it does not apply to all clinics. In an attempt to bridge the gap between the appointment scheduling literature and scheduling practice, many papers have begun to incorporate stochastic services times and multiple-servers (Mandelbaum et al. 2019, Zacharias and Pinedo 2017). Those topics can be a valuable addition to the discussion of the switch point. A second extension would be to allow heterogeneous probabilities of show and probabilities of retention for each patient, and direct and indirect waiting time costs that may also be idiosyncratic. Incorporating such heterogeneity would increase the number of candidate schedules, but could better align the final solution with the clinic patient base. Additionally, an extension of our model might be to determine how the inter- and intra-day scheduling problem can be solved with a non-myopic model. While we do not pursue that in this paper, due to the complexity of such a problem (Feldman et al. 2014), we believe our results provide insight into how cancellations influence scheduling decisions over a multiday horizon. Lastly, an interesting extension might be to fully characterize the switch point for all model parameters, and to work towards a closed form solution for when it will occur. Acknowledgements We are grateful to the associate editor and the reviewers for their constructive and helpful comments on earlier versions of this paper.

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