Innovative Applications of O.R.
An integrated approach to demand and capacity planning in outpatient clinics

https://doi.org/10.1016/j.ejor.2019.06.001Get rights and content

Highlights

  • An integrated approach for demand and capacity planning in clinics is formulated.

  • It includes a static publication policy and a state-dependent expediting policy.

  • A heuristic search is proposed for finding the joint optimal values of the policies.

  • The savings from using the optimal policies are illustrated using real data.

  • Carve-out clinics benefit more from expediting policy than advanced access clinics.

Abstract

An outpatient clinic serving two independent demand streams, one representing advance booking requests and the other same-day requests, is considered. Advance requests book their appointments through an electronic booking system for a future day, and same-day requests are served on the day they arise. A compact policy formulation is proposed that incorporates major operational levers suggested in the literature. It combines a slot publication policy, which specifies the pattern under which slots are released to the booking system, with an expediting policy that adjusts the daily workload of advance patients. Relying on a wide range of numerical experiments, a heuristic search method is developed for finding the joint publication and expediting policies, minimizing the cost of overtime slots whilst ensuring a waiting and an access constraint is met. Several managerial insights are derived using a combination of illustrative and real data, highlighting the importance of taking an integrated approach towards the operational levers captured by our policy formulation.

Introduction

Outpatient medical facilities must typically serve patients who require a same-day visit as well as those who book an appointment in advance. This applies not only to clinics that operate a “carve-out” mode of delivery, but also to those that have implemented “advanced access”. In the former, a few slots in each day are reserved for patients with urgent medical needs and the rest is available to routine patients for advance booking (Murray & Berwick, 2003). In the latter, although the primary objective is to offer every patient a same-day appointment regardless of urgency, certain patient groups such as commuters highly value the flexibility to book appointments for future days (Pope, Banks, Salisbury, & Lattimer, 2008). With both delivery modes, clinics must therefore decide how much of their daily capacity should be allocated to “advance booking” patients, and how much be left open in anticipation of “same-day” demand. Clinics may also decide on certain days to serve more pre-booked patients than originally scheduled for those days by “expediting” some patients, i.e. bringing their appointments forward, aiming to alleviate the excessive backlogs caused by a temporary surge (decline) in demand (supply). In addition to these capacity planning decisions, clinics may exercise some control over the demand streams. Adjusting the appointment scheduling window, i.e. how far in advance a patient can schedule an appointment with a provider, is an important way of achieving this especially when there is less flexibility in panel size selection (Liu, 2015). In this paper, we develop a methodology that guides clinics in making such strategic capacity and demand planning decisions in an integrated manner, and investigate the efficiency gains of such integration.

We assume advance booking patients schedule their appointments through an electronic booking system (EBS), and same-day patients must be served on the day they arise. There is a wide range of EBS’s used in primary and specialty clinics in different countries, e.g. ZocDoc in the US (Zocdoc, 2015), and ZorgDomein in the Netherlands (Dixon, Robertson, & Bal, 2010). A prime example of an EBS implemented on a large scale is that of the e-Referral Service (e-RS) (NHS Digital, 2017) deployed in the UK National Health Service (NHS), which has motivated this study. It enables routine patients referred to specialty clinics to book their first outpatient appointment online. The e-RS is linked to providers’ digital records containing information about their appointment slots, including the timings, the clinicians providing them, and whether they are publishable on the e-RS on not. The free and publishable time slots on each day are released to the e-RS a given number of days in advance as specified by the appointment scheduling window (or the “polling range” as referred to in the e-RS context) selected by the provider. These slots will then be available to routine patients seeking appointments: once a routine referral is deemed necessary for a patient, first the referring clinician (jointly with the patient) chooses where the patient must be referred to, and second the patient books the most convenient slot from a menu of available slots displayed by the e-RS for her chosen provider. The slots not released to the e-RS will be offered to urgent referrals and walk-in patients on the day they arise. Most other EBS’s enjoy the same basic functionalities as explained above for the e-RS.

When the EBS shows no available slot for the first choice provider of an advance booking patient, the patient might switch to a different provider, or insist on being served by that particular provider (due to, e.g., geographical proximity or reputation of the provider). Similar to Jiang, Pang, and Savin (2012), we call these two categories of advance booking patients “flexible” and “dedicated”, respectively. Some EBS’s have built-in features enabling dedicated patients to enforce an appointment with their chosen provider when the EBS shows no available slot. In the e-RS, for instance, the patient can click on the “Defer to Provider” option to inform the provider, who would subsequently contact the patient to book her an appointment on a day typically beyond the polling range. Alternatively, in some EBS’s patients are advised to phone the clinic directly when they cannot find an appointment online. Flexible patients on the other hand forgo the difficulties and potentially longer delays associated with securing an appointment through these alternative routes and seek care elsewhere.

To enable the clinics to effectively manage the demand for and supply of appointments through an EBS, we propose a novel and compact policy formulation that combines a slot publication policy with a backlog-dependent expediting policy. We define the slot publication policy as a two-dimensional policy where the first dimension specifies the number of slots in each period of time that are made publishable to the EBS by the clinic, and the second dimension determines the number of periods in advance that such slots are released to the EBS. The slot publication policy may be viewed as a combination of same-day reservation level and appointment scheduling window studied separately in the literature. The expediting policy, on the other hand, specifies the number of additional pre-booked patients the clinic must serve in each time period (by bringing some appointments forward), depending on the size of backlog at the start of the period. It is motivated by the “advance scheduling with expediting” paradigm proposed by Truong and Ruzal-Shapiro (2019).

Implementation of the slot publication policy is straightforward as EBS’s typically have the flexibility of adjusting both dimensions. Implementing the expediting policy is more of a challenge as not all patients may agree to be expedited. Those patients, as argued in Truong and Ruzal-Shapiro (2019), can simply opt out. Others however may see it as an opportunity to be seen earlier, possibly on a day that would not have been available when they originally booked their appointments. Therefore, it is unlikely to be difficult to find the required number, which as we show later in our case studies rarely exceeds one patient per time period, when there is a large backlog (which is when the policy is activated). There is also some administrative cost for the clinic to reschedule patients. Our models enable the clinics to quantify the benefits of expediting policy, hence implement it only if the benefits outweigh the costs. Besides, appointment rescheduling is already a common practice in many clinics, e.g. for filling the slots vacated by patients’ cancellations (Truong & Ruzal-Shapiro, 2019) or backfilling the slots remaining empty on each day (NHS Digital, 2016).

The combination of slot publication and expediting policies gives clinics a framework to plan their capacity and demand optimally depending on the optimality criteria they choose. Here we define the optimal joint policy as the one that minimizes the average daily number of required overtime slots whilst ensuring that advance patients’ access and waiting time requirements are met. Overtime slots are necessary when the total number of advance and same-day patients on a day exceeds the number of regular slots. The access requirement restricts the average number of advance flexible patients forced to switch provider as a result of no slots being available online. The waiting time constraint ensures the average time between the request for an appointment and the earliest available slot does not exceed a given threshold.

To search for the optimal expediting and publication policies, first we develop a queueing model that represents the evolution of appointment backlog given specific policies. A major feature of our queueing model is that it does not require patients to take the earliest available slot. In fact, in clinics supported by EBS’s patients may take any of the slots displayed on the system based on their preferences. This along with patient expediting violates the first-come first-serve (FCFS) discipline, and also creates “holes” in the backlog. We show that our model is still accurate in these circumstances as long as a mild condition is met. As such our model is able to capture the dynamics of backlog under EBS’s without detailed information on patients’ preferences. Next, relying on extensive numerical experiments, we partially characterize the structure of optimal policies, leading to the development of a fast heuristic search for systems with a bi-level expediting policy. The accuracy and reliability of policies obtained from the heuristic are confirmed through comparison with complete enumeration and simulation.

Using our models, we conduct a series of numerical experiments to derive managerial insights. The first insight is that there is substantial value in adopting an integrated approach towards the three operational levers captured by slot publication and expediting policies. In particular, we observe that setting the two dimensions of slot publication policy in a joint rather than sequential manner is likely to result in large efficiency savings. Further, we show that applying the expediting policy without fully revising the publication policy is less likely to create improvement. The second insight is that deployment of a backlog-dependent expediting policy would enable the clinics to release fewer slots to the EBS per unit of time but over a longer period. This reduces the overtime cost, as compared to the publication policy alone, whilst giving patients more choice over appointment days. The scale of reductions in overtime cost is likely to be large in a carve-out clinic where the majority of patients are advance booking. The administrative burden is also small as we observe that the optimal policy typically involves only one or two expedited visits when activated, and the frequency of its activation is small when the corresponding savings are large. In a typical advanced access clinic, on the other hand, the savings from expediting policy are much smaller, especially when the ratio of advance booking patients is 25% as cited in Murray and Tantau (2000).

Section snippets

Literature review

Our work is related to the appointment scheduling literature, see Cayirli and Veral (2003) and Gupta and Denton (2008) for comprehensive reviews. This literature can be divided to “intra-day” and “multi-day” scheduling. In intra-day scheduling, the focus is on scheduling appointments in a single day. Examples include Koeleman and Koole (2012) and Cayirli, Yang, and Quek (2012). Multi-day scheduling, on the other hand, is concerned with allocating appointment requests arising in each day to

Problem formulation

We assume the clinic provides services for two independent demand streams, one representing same-day requests and the other advance booking requests. Note that in the carve-out mode the urgency of care determines the group to which a patient belongs, while in the advanced access the preference of a patient is the major identifier. We divide the time axis into equally spaced intervals (periods), e.g. days, numbered 1,2,3,, and assume a nominal capacity of r regular slots is available in each

Dynamics of appointment queue

The focus here is only on advance booking patients as same-day requests do not influence the backlog. We develop a state-dependent discrete bulk service queue with slot cancellation and customer no-show to represent the evolution of backlog. Our queueing model is bulk service as a batch of customers is served in each interval; see Izady (2015) for further details. It is also a state-dependent model in service capacity, arrival process, and no-show probability. The dependence of service capacity

Numerical analysis of appointment queue

Define n(i)=n+e(i) as the nominal service capacity available to advance requests, and let x=(x0,x1,,xm) be the stationary distribution for the discrete-time Markov chain characterized by Eq. (3), where m is a sufficiently large upper-bound for the numbers in the system. One can find the the stationary queue length probabilities by solving balance equations xϕ=x, with ϕ=[ϕij] the transition probability matrix specified below.

Proposition 2

The transition probabilities ϕij=P(Xt+1=j|Xt=i) of the Markov chain

Heuristic search

The optimization model in (1) is intractable in its general form. This is due to many different functional forms that can be considered for piecewise function e(i). Various realistic features included in our model also adds to the complexity. To make the problem tractable, we restrict e(i) to bi-level functions represented ase(i)={0,i<Δ,Υ,iΔ.This simplification enables us to derive insight into the structure of optimal policy using numerical experiments, leading to the development of a

Numerical results

In this section, we apply the models developed in the paper to two different outpatient environments. One is based on representative data for an advanced access clinic, and the other is based on real data from a specialty clinic. The objective here is first to validate the structural properties conjectured in Section 6, second to derive management insight into the behaviour of optimal policies in different circumstances, and finally to validate the reliability of the heuristic search. We note

Conclusions

EBS’s have become an integral part of modern outpatient clinics. Apart from giving patients a greater choice over the location and time of their treatments, these systems give providers more flexibility in managing their supply and demand. At a strategic level, this flexibility is achieved by enabling the clinics to set the number of slots allocated to different patient groups and determine the timing of their release to the booking system. Booking managers often rely on personal experience or

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