APPOINTMENT SCHEDULING WITH OVERBOOKING TO MITIGATE PRODUCTIVITY LOSS FROM NO-SHOWS

Linda R. LaGanga & Stephen R. Lawrence

Mental Health Center of Denver Leeds School of Business, UCB 419 4141 East Dickenson Place University of Colorado at Boulder Denver, CO 80222 Boulder, CO 80309-0419 (303) 504-6665 (303) 492-4351 Linda.Laganga@mhcd.org Stephen.Lawrence@colorado.edu

ABSTRACT

The challenge of balancing the interests of patients with those of healthcare providers is

increased when patients fail to show up for scheduled appointments. Overbooking appointments

mitigates the lost productivity caused by no-shows but increases patient wait time and provider

overtime. In this paper, simulation analysis is used to develop and test the performance of

scheduling rules that are designed specifically to accommodate excess overbooked appointments.

Our analysis provides new insights into rules that perform well to increase provider productivity

while balancing the increased waiting time and overtime costs of overbooked schedules.

Keywords: Appointment Scheduling, No-shows, Overbooking, Service Operations, Simulation

INTRODUCTION

When patients fail to show up for their scheduled appointments, provider productivity and

effective clinic capacity are reduced (Cayirli & Veral, 2003). To mitigate this loss, health care

clinicians have experimented with a number of alternative appointment scheduling policies.

Some clinics overbook appointments by double-booking patients into common appointment

times and relying on no-shows to allow the schedule to catch up (Chung, 2002). Others have

experimented with “wave scheduling” policies that build extra appointments into a schedule to

boost provider productivity and that leave other appointment slots empty (Silver, 1975; Schroer

1

& Smith, 1977; Barron, 1980). This combination allows a schedule to catch up after a backlog

occurs, thus reducing patient waiting and reducing the need for clinic overtime. Practitioners

have reported success in managing appointment schedules with these and other similar

approaches, but their accounts have been anecdotal and do not analyze or describe how schedule

performance relates to no-show rates or other system characteristics (Chesanow, 1996; Baum,

2001; Chung, 2002).

In this paper, we build upon and extend the double-booking, block scheduling, and wave-

scheduling policies devised by practicing clinicians to develop and measure the performance of a

number of scheduling rules based on these policies. We adjust traditional appointment

scheduling performance measures to capture the operating dynamics of overbooked appointment

scheduling systems, determine their effectiveness when overbooking is used to compensate for

the lost productivity of no-shows, and provide recommendations for improving performance in

overbooked appointment scheduling systems. Our analysis is useful for schedulers and health

care providers to identify and evaluate operational and policy changes that will boost clinic

productivity and improve patient service.

OVERBOOKING AND PROVIDER PRODUCTIVITY

The practice of booking multiple appointments at the same appointment start time is intended to

reduce the time that providers wait for patients to show up, thereby increasing productivity

(Bailey, 1952). However, to increase daily productivity, an increased number of patients must be

booked and served in each clinic session. Overbooking is often interpreted as “double-

booking,” the practice of scheduling two patients to arrive at the same time (Rohleder & Klassen,

2002). Double-booking is a specific case of block-booking, which schedules a multiple number

of patients to show up at the same time, and is not the only option for overbooking.

2

Overbooking a clinic session can often be accomplished without assigning more than one patient

to the same appointment time. Depending on how schedule performance is measured, other

scheduling rules could be more effective.

Chung (2002), for example, “loads up” the schedule by double-booking the first

appointment of the hour to ensure that the provider doesn’t lose productivity if one of the two

patients doesn’t show up. He reports that this method increased his bottom line profit by almost

15 percent without increasing his overhead costs, and he describes “modified-wave” scheduling

as the practice of “loading up the front end of each hour and leaving open slots in the schedule

later on to catch up.” This is intended to prevent long patient wait times because if the physician

begins to run late, the effect isn’t cumulative. The unscheduled time at the end of the hour can

be “borrowed” if needed to serve the patients who show up and may require extra service time.

The scheduling rules that we develop and test in our simulation experiments are designed

specifically to analyze the effects of the placement of the extra appointments in an overbooked

appointment schedule. We compare traditional double-booking and other multiple-booking

scheduling patterns suggested by providers with alternative scheduling patterns that use

compressed inter-appointment arrival times instead of, or in combination with, multiple booking.

RESEARCH METHODOLOGY

We model overbooked appointment schedules with deterministic service times D and clinic

capacity N fixed as the total number of patients that can be served within the normal operating

time of a clinic session without overtime. Each patient is assigned to a specific provider, and

providers do not service each other’s patients. We assume patients who show up are punctual.

Appointment scheduling researchers have considered an alternative approach of compressing

appointment intervals to accommodate excess appointments (Vissers & Wijngaard, 1979;

3

LaGanga & Lawrence, 2007). For a given show rate S, we schedule appointments for a total of K

patients during a clinic session, where K = N/S rounded to the nearest integer (Shonick & Klein,

1977). The expected number of patients served in a clinic session is therefore NSKXE ==)(

(or is very close to N) and the total number of overbooked patients is equal to K-N.

For each of five show rate levels S ={0.9, 0.7, 0.5, 0.3, 0.1}, we modeled thirteen

scheduling rules. To attempt to improve schedule performance, we modeled an additional rule

for S ={0.9, 0.7} for a total of fourteen scheduling rules, as shown in Table 1. The extra rule

could not be implemented for S = {0.5, 0.3, 0.1} because the number of appointments K

scheduled at these show rates became too large to be accommodated by the additional scheduling

rule.

------------------------------- Table 1 about here.

-------------------------------

Modeling was done in two stages. First, we developed spreadsheet-based planning

models of all experimental schedules to calculate patient wait time at every appointment time

and for the entire schedule for the case in which every scheduled patient shows up. This analysis

was useful in identifying where large wait times are likely to accumulate and to support

experimentation and development of alternative scheduling rules. Then, to test the performance

of various schedules with stochastic no-shows, we simulated the operation of the planning

models for various levels of show rate S.

The Clinic Model and General Scheduling Rules

We model a realistic clinic session using the parameters of an actual outpatient psychiatric clinic

that we studied. In this clinic, a normal morning clinic session runs for four hours from 8:00

4

a.m. – 12 p.m. The service time for each patient is 20 minutes without variation so that the clinic

size is N = 12.

If every patient showed up with certainty, then there would be no need to overbook,

there would be no patient wait time, the provider would be utilized 100% of the time, and there

would be no overtime required to serve all patients. However, the clinic experiences a

significant no-show rate (S < 1), so if the target number of patients to be served remains at

N = 12, then additional overbooked appointments must be added to the clinic schedule. One way

to schedule extra appointments into the clinic session is to compress the inter-arrival time

between appointments T proportionally to the show-rate S, so that T =DS, or for the clinic under

consideration, T = 20S. Table 2 summarizes the inter-arrival times between appointments for ten

show-rate levels and appointment durations of D = 20 minutes.

------------------------------- Table 2 about here.

-------------------------------

Table 2 shows that, in most cases, the calculated compressed appointment inter-arrival

times T do not correspond to practical appointment times such as 10, 15, 20, 30, 40, 45, or 50

minutes after the hour. For the scheduling rules tested in this paper, appointment start times were

set to practical clock times, multiples of 10 or 15 minutes. Thus, for every show rate tested, we

included an adjusted compressed rule to move the calculated appointment start times forward or

backward to the nearest practical time. We also tested schedules that compressed all

appointment times uniformly by the same 5-minute multiple less than the calculated compressed

time T. In an abridged representation of our planning models, Table 3 illustrates these

scheduling rules in clock time for show rate S = 0.9. The number of appointments to be

scheduled is 12/0.9 = 13.33, rounded down to 13.

5

------------------------------- Table 3 about here.

------------------------------- The average patient wait time, maximum patient wait time, and overtime shown in Table

3 are for the case in which every one of the 13 scheduled patients shows up. The probability of

this scenario occurring, for show rate S = 0.9, is 0.2542. If every patient shows up, then, for all

of these scheduling rules in this scenario, the amount of overtime is equal to the service duration

of one filled appointment, 20 minutes, because providing service to all of the patients who are

scheduled requires one extra occurrence of service time duration beyond normal clinic capacity.

Wait time, however, varies among rules because of varying amounts of patient backlog caused

by the time intervals between appointments.

As shown in Table 3, compressing all inter-appointment times to 15 minutes leads to

average patient wait time of 30 minutes, with maximum wait time of one hour, when every

patient shows up. The advantage of such tight schedule compression is that expected provider

overtime is reduced because appointments are scheduled to begin earlier, but at the expense of

extended patient wait times. If patients and clinic administrators cannot accept these wait times

occurring in at least 25% of the clinic sessions on average, then such a compression rule should

be eliminated from consideration. For our evaluation of schedule performance, we include a

similar compressed rule for all of the show rates tested.

Analysis of Overbooking Dynamics

Overbooking a scheduling system introduces new challenges in constructing schedules and

comparing performance. The first consideration is the process of scheduling the extra K-N

appointments. Overbooking is achieved by adjusting the time intervals between appointments,

using block scheduling of multiple patients at one or more schedule times, or combinations of

these approaches to fit the extra appointments into the schedule. For high show rates, developing

6

potential schedules can be handled by fitting a small number of extra appointments into the

schedule through adding individual appointments to the baseline schedule of N appointments

spaced D time units apart. This becomes more challenging as show rates decrease because the

number of appointments that must be added to the schedule becomes large, as shown in Figure 1.

------------------------------- Figure 1 about here.

------------------------------- When K-N exceeds the number of appointment slots available, then block booking of multiple

patients at the same appointment time becomes unavoidable. For example, when S = 0.1, then K

= 120, which is almost 4 times as large as 33, the number of available scheduling times in our

clinic model, because we allow scheduling at 0, 10, 15, 20, 30, 40, 45, and 50 minutes after the

start of each of the four hours in the clinic session, and at the end of session at 12:00 p.m.

Another way to interpret the impact of overbooking is as the increased work added to the system,

expressed as the load factor or multiplier of the number of appointments scheduled (Fetter &

Thompson, 1966).

Another challenge in developing overbooked schedules is that the number of possible

schedules is extremely large. Each of the 33 possible clock times in our schedule represents a

scheduling block in which the number of appointments that could be scheduled

is . The total number of possible schedules Π that can be

constructed for K appointments and J = 33 schedule times is

{ } { 33,...1,,....1,0 =∀= jKy j }

1Π

K JK

+ −⎛ ⎞= ⎜

⎝ ⎠⎟ (Fries & Marathe,

1981), as shown in Table 4 for varying show rate S, from which the number of appointments, K,

is obtained as 12/S, rounded.

------------------------------- Table 4 about here.

-------------------------------

7

Thus, for even the smallest possible number of overbooked appointments, fitting that one

extra appointment into the schedule can be done in over 73 billion different ways. Therefore, we

need some general guidelines to identify a focused set of schedules for which we can evaluate

and compare performance and choose among several alternatives that perform well. The

calculation of binomial probabilities, shown in Table 5, is useful for determining how often all

scheduled patients are likely to show up for an overbooked clinic session and for determining the

risk that more than one patient shows up at the same time when a scheduling block has more than

one appointment scheduled simultaneously. This information is useful when developing

potential schedules.

------------------------------- Table 5 about here.

------------------------------- For example, if the probability of having more than one scheduled patient show up at the

same time should be no greater than 0.25, then double-booked blocks should not be used unless

, and block sizes of 5 should not be used unless S < 0.20. For S = 0.8 and K = 15, we

can expect all scheduled patients to show up in less than 4% of the clinic sessions on average.

Unfortunately, if we assume all patients are equally likely to show up or not show up, the no-

shows could occur anywhere in the schedule. Since we cannot count on the no-shows to occur at

particular points in the schedule, we need to design and evaluate the performance of potential

schedules using the worst-case scenario in which all scheduled patients show up. We use

maximum, rather than average wait time, as a performance measure to more accurately represent

the system congestion that can occur when more patients show up during a time interval than a

provider can serve, which causes one or more patients to experience very long wait time.

LaGanga (2006) provides further analysis and examples.

50.0≤S

8

Wait time is minimized when all appointments are scheduled as late as possible and are

spread as far apart as possible. For example, to minimize wait time in our clinic model, 13

appointments would be scheduled by placing each of them 20 minutes apart. Because the

interval is equal to the service time duration, there would be no wait time, but overtime would be

maximized even if no-shows were possible because the last appointment starts at the latest

possible time. This analysis considers only reasonable schedules – if all appointments were

scheduled at the latest time, then overtime would be at its true maximum, but it would not be

reasonable to keep all of the earlier appointment times unscheduled. Next, overtime is

minimized when all appointments occur as early as possible in the schedule, which means that all

appointments are scheduled at the same time at the start of the clinic, but this schedule

maximizes patient wait time. Thus, in our planning models, the contribution to overtime is

considered as a function of the lateness of a schedule time and the number of patients scheduled

at that time, based on the observation that schedules with heavier appointment weighting toward

the end of the schedule tend to incur more overtime.

We construct and evaluate potential wave schedules, identify schedule block sizes that

contribute to congestion and overtime, and modify the schedule to attempt to alleviate these

conditions. Although we can calculate the probability of every possible number of patient shows

or no-shows occurring, the analytical calculation of expected wait time and overtime is

intractable because overtime occurs not only when capacity is exceeded, but also as the result of

the time-based appointment positions in which patients show up. Thus we turn to simulation to

model the performance of our schedules at varying patient show rates. Results and analysis are

presented in the next section.

9

SIMULATION RESULTS AND ANALYSIS

We developed a simulation model with deterministic patient no-shows for each of the 67

schedules that we developed and analyzed in our planning models. For each schedule simulated,

we completed 10,000 replications for a total of 670,000 observations of the schedules’

performance. The number of replications was determined by conducting pilot studies preceding

the main experiment that indicated that the half-widths of the 95% confidence intervals were less

than 1% of the point estimates for the performance measures of interest.

In this section, we graphically present our results to display and analyze the performance

trade-offs between wait time and overtime. For each show rate S, we consider the objective of

choosing the schedule with the best performance, measured as the minimum of the weighted sum

of maximum patient wait time, W, and provider overtime, O. Then, for π = the cost per minute of

maximum patient wait time and ω = cost per minute of provider overtime, the expected cost of

using a particular schedule is

ωπ += WC O. (1)

The results are useful in separating the schedules that should be considered for

implementation from those that should not because they are dominated by one or more others

that have smaller values for both patient wait time and provider overtime (Fries & Marathe,

1981). Following the work of Ho and Lau (1992), we can identify the best schedule that

minimizes cost among those tested by finding the point where π/ω has a value between the

slopes of two adjacent line segments on the efficient frontier.

For example, simulation results shown in Figure 2 illustrate that the efficient frontier is

formed by the three points corresponding to plotted wait time and overtime results for schedules

10

MAF, MAS, and RDI. Because we seek to minimize cost, the efficient frontier must form the

lower bound of a convex region and therefore, it consists of only these three points.

------------------------------- Figure 2 about here.

------------------------------- For example, the slope between the plotted costs for scheduling rules MAF and MAS is

89.0054.160

61.394.17=

−− (with the negative sign omitted because all the slopes are negative) and

between schedules MAS and RDI is 18.0757.42054.16

08.161.3=

−+ . Moving from left to right along

the efficient frontier, the first schedule, MAF, schedules the one overbooked appointment at the

end of the clinic session, which minimizes wait time for patients and maximizes overtime for the

provider, and according to our efficient frontier analysis, this rule results in the lowest cost for

89.0>ωπ . The next schedule, MAS, results from the rule established by Bailey (1952) and

Welch and Bailey (1952) that schedules two appointments at the same time at the start of the

clinic session and individual successive appointments at intervals equal to the average service

duration. This scheduling rule was designed to balance provider idle time with patient wait time

and would be the cost-minimizing schedule if 89.018.0 <<ωπ . The next schedule, RDI,

compresses all of the time intervals between appointments to 15 minutes, which is shorter than

service duration D = 20 and results in the schedule that minimizes provider overtime and

maximizes patient wait time. It would be the cost-minimizing schedule if 18.00 <<ωπ .

Suppose, however, that there are some practical constraints on patient wait time and provider

overtime. If providers refused to accept more than 15 minutes of overtime on average and the

11

threshold for maximum patient wait time, averaged across clinic sessions, was 20 minutes, then

schedules MAF and RDI would be eliminated from consideration. Figure 3 shows the four

schedules that perform within the practical constraints and are not dominated by any other

schedules. Tables 6 and 7 show, across all experiments, maximum waiting time and overtime,

respectively.

------------------------------- Figure 3 about here.

------------------------------- -------------------------------

Table 6 about here. ------------------------------- -------------------------------

Table 7 about here. -------------------------------

By the strict definition of the efficient frontier as a convex region formed by the line

segments between schedules, the only two points on the efficient frontier are WAV1 and MAS,

because considering from left to right the full set of four points, the absolute value of the slope of

the line segment connecting each pair is monotonically increasing rather than decreasing. The

slope of the line between the two points is 1.83, which means that if 83.1=ωπ , then the cost is

the same for these two rules, but schedule 10 performs better when 83.1>ωπ , and schedule 4

performs better when 83.1<ωπ .

------------------------------- Figure 4 about here.

-------------------------------

As shown in Figure 4, the set of schedules on the efficient frontier varies with show rate.

The points for S = 0.70 differ from those of S =0.90. At S =0.70, MAS, which is analogous to

the Bailey-Welch rule with 5, instead of 2, appointments scheduled in the initial block, is far off

12

the efficient frontier, but another block-scheduling variation is MAM, which schedules several

blocks of size 2 and is on the efficient frontier. For this show rate, WAV5 was developed to

attempt to improve performance of schedule WAV1 by reducing overtime, which did, indeed,

occur in the simulation results. The trade-off is that wait time increased with the schedule

change, but the improved rule made it onto the efficient frontier, unlike WAV1, which was

dominated by the first two rules.

At show rates S = {0.70, 0.50}, WAV4 was designed to represent the wave scheduling

pattern used by some providers to schedule repeated patterns of appointments in block sizes of

{3, 2, 1} (Baum, 2001). The schedule performs better for S = 0.70, with 13.23 minutes of

overtime and 49.078 minutes of maximum wait time, than for S = 0.50, with 24.44 minutes of

overtime and 47.638 maximum wait time (unless the cost of patient wait time is much higher

than provider overtime). But the schedule would not be considered for implementation for S =

0.70 because it is dominated by other schedules. In contrast, for S = 0.50, the schedule results

are positioned on the efficient frontier and would be considered. This illustrates that the

selection of the best schedule for a given show rate depends not only on the scheduling rule and

the show rate itself but also on the relative performance of the alternative rules. At each of the

levels of S analyzed in this study, several rules emerged for consideration, allowing for choice

according to weighting of cost factors and practical considerations.

SENSITIVITY ANALYSIS

An initial assumption in our analysis was that service times were constant, which is

consistent with psychiatric clinics that we studied. To determine the effects of variable service

times, we conducted additional simulation experiments for S = 0.70 and the five rules on or near

the efficient frontier. We modeled service time as a Gamma distribution with four parameter

13

pairs (α, β) of (16, 1.25), (4, 5), (1.78, 11.25), (1, 20) for four levels of provider service time

variability, cs = σ/μ, where σ is the standard deviation of service time and μ is its mean, for five

levels of service time variability, }0.1,75.0,5.0,25.0,0.0{=sc , including the original

experiments where cs= 0. We used the Gamma distribution because it is bounded from below by

0 (no negative service times) and is skewed to the right with a long right tail allowing the

possibility of extended service times, which may occur in handling emergency service.

As shown in Figure 5, when service time changes from constant (cs = 0) to high variation

(cs = 1), the pattern of plotted points for each scheduling rule in relationship to the other

scheduling rules remains very similar, and the same scheduling rules {CAI, RNI, RDI, MAM,

WAV1, WAV4, WAV5} remain on or near the efficient frontier. For every scheduling rule,

maximum patient wait time and provider overtime increase with increased service time variation.

------------------------------- Figure 5 about here.

------------------------------- -------------------------------

Figure 6 about here. -------------------------------

Focusing on the scheduling rules on or near the efficient frontier in Figure 6 for the three

additional levels of service time variation tested, for cs = {0.25, 0.50, 0.75}, reveals that the

pattern of points on the efficient frontier remains unchanged with changes in service time

variation, and the magnitude of maximum patient wait time and provider overtime increase with

increased service time variability. Therefore, we conclude that service time variability has little

impact on the efficient frontier and is, therefore, immaterial in comparing the performance of

various scheduling rules; however, it does impact the magnitude of the performance components

─ maximum patient wait time and provider overtime ─ of the scheduling rules.

14

SUMMARY AND RECOMMENDATIONS

The simplest practical overbooked schedule, the RNI rule, compresses all inter-appointment

times by the same factor S = show rate, and then places each resulting appointment time on the

nearest practical clock time. Our simulation results show that for the full range of show rates

tested, this schedule is always among the schedules that perform well and should be considered

for implementation because it is not dominated by other policies.

In contrast, we would not recommend scheduling policies with very tight uniform

compression at any show rate because they allow no catch-up time, and large accumulations of

patient wait time occur, even with stochastic no-shows. In particular, the RDI scheduling rule

results in high patient wait times for all show rates S; this would never be acceptable to patients

or their payers and advocates, especially with the existence of alternative schedules that result in

much less wait time. Similarly, schedules that use block scheduling of multiple patients at the

same time, especially in large block sizes, are not recommended unless 50.0≤S . Otherwise,

they result in large patient wait time. Breaking large blocks into smaller ones improves

performance.

From our experiments with S = 0.90, we found that patient wait time can be avoided

entirely by scheduling one extra appointment at the end of the clinic session, resulting in an

average of only 18 minutes of overtime. If less overtime is desired, this can be accomplished

with the wave schedule that compresses selected inter-appointment times from D = 20 minutes to

15 minutes to avoid a large accumulation of patient wait time anywhere in the schedule and

results in average maximum patient wait time of about 12 minutes. Hence, we recommend that

these schedules be considered if patients and providers can accept these relatively small impacts

for the benefit of adding capacity so that one extra patient per provider per clinic session (two

15

per day if sessions run both morning and afternoon) can be offered access to services and

providers can be more productive.

For schedules under consideration, formal analysis of the benefits and costs can be

evaluated using a net utility U measure as proposed by LaGanga and Lawrence (2007):

( )U MS K N Wπ ω= − − − O (2)

where M is the marginal benefit of servicing an additional patient (other terms defined

previously).

CONCLUSIONS AND FUTURE RESEARCH

Health care providers need to manage their schedules to operate efficiently and to balance patient

needs with productivity concerns. Wave scheduling and block scheduling have been used and

recommended by providers to improve productivity and mitigate operating variation that is

detrimental to schedule performance. In this paper, we have examined the possibility of using

wave scheduling, double booking, block scheduling, and other scheduling rules to reduce lost

productivity caused by the prevalent problem of patient no-shows. We analyzed the operating

characteristics of overbooked schedules and demonstrated the challenges of developing

schedules for overbooked appointments. Scheduling complexity increases exponentially with

the number of overbooked appointments due to the new appointments that must be absorbed into

the schedule and to the increasing number of possible schedules that result. General discrete

scheduling rules were represented as practical clock-time schedules in planning models to

efficiently construct, evaluate, and improve potential schedules prior to conducting long

simulation runs that consume more resources.

Overbooking allows providers to increase their productivity and create additional

capacity to improve patients’ access to services. Our results demonstrate that, in overbooking to

16

recover capacity that would be lost to no-shows, both overtime and patient wait time increase

with increased no-show rates and with increased service time variation. Although service time

variability does impact the magnitude of patient wait time and provider overtime, it has little

impact on the set of schedules that perform best. We showed that the selection of the best

schedule for a given show rate depends not only on the scheduling rule and the show rate itself

but also on the relative performance of alternative rules.

Our experiments and analysis show that a simple overbooking policy of compressing

inter-appointment times in proportion to show rates and rounding each calculated time to the

nearest practical clock time generally works at least as well as other scheduling policies,

including double booking, block booking, and wave scheduling. These other scheduling policies

provide overbooking alternatives that might be attractive to clinics that must minimize overtime

operation, but they result in increased wait time for patients. Also, they are more challenging to

design and implement than the simple compressed scheduling rule because the number and

pattern of the extra appointments that must be fit into the schedule varies with the no-show rate

of each clinic. This makes it more difficult for large organizations with multiple clinics to

consistently and effectively manage their scheduling operations across clinics.

There are many opportunities to extend this research by further developing the planning

models, performance measures, and their predictive capabilities under more generalized

conditions. For example, further experiments can be structured to focus on improving schedules

that performed well in this study. On the other hand, when schedule performance is poor, as is

likely for very low show rates because of high overbooking levels, insights about the

performance dynamics of scheduling systems could lead to the development of alternative health

care access systems. Another area of exploration is to vary the level of overbooking. Continued

17

public interest in improving health care access and service delivery is likely to lead to further

exploration of the approaches developed in this paper.

18

REFERENCES

Bailey, N. T. (1952). A study of queues and appointment systems in hospital out-patient departments, with special reference to waiting-times. Journal of the Royal Statistical Society, Series B, 14(2), 185-199.

Baum, N .H. (2001). Control your scheduling to ensure patient satisfaction. Urology Times, 29(3), 38-43. Barron, W. M. (1980). Failed appointments: Who misses them, why they are missed, and what can be done. Primary Care, 7(4), 563-574. Cayirli, T., & Veral, E. (2003). Outpatient scheduling in health care: A review of literature. Production and Operations Management, 12(4), 519-549. Chesanow, N. (1996). Can’t stay on schedule? Here’s a solution. Medical Economics, 73(21), 174-180. Chung, M. K. (2002). Tuning up your patient schedule. Family Practice Management, 9(1), 41-48. Fetter, R. B., & Thompson, J. D .(1966). Patients’ waiting time and doctors’ idle time in the outpatient setting. Health Services Research, 1(1), 66-90. Fries, B. E., & Marathe, V. P. (1981). Determination of optimal variable-sized multiple-block appointment systems. Operations Research, 29(2), 324-345. Ho, C., & Lau, H. (1992). Minimizing total cost in scheduling outpatient appointments. Management Science, 38(12), 1750-1763. LaGanga, L. R. (2006). An examination of clinical appointment scheduling with no-shows and overbooking. Doctoral dissertation, University of Colorado, Boulder, CO. LaGanga, L. R. & Lawrence, S. R. (2007). Clinic overbooking to improve patient access and increase provider productivity. Decision Sciences, 38(2). Rohleder, T. R., & Klassen, K .J. (2002). Rolling horizon appointment scheduling: A simulation study. Health Care Management Science, 5(3), 201-209. Schroer, B .J., & Smith, H .T. (1977). Effective patient scheduling. The Journal of Family Practice, 5(3), 407-411. Shonick, W., & Klein, B. W. (1977). An approach to reducing the adverse effects of broken appointments in primary care systems. Medical Care, 15(5), 419-429.

19

Silver, M. (1975). Scheduling: Least developed art. Family Practice News, 5(23), 34. Vissers, J., & Wijngaard, J. (1979). The outpatient appointment system: Design of a simulation study. European Journal of Operational Research, 3(6), 459-463. Welch, J. D., & Bailey, N. T. (1952). Appointment systems in hospital outpatient departments. The Lancet, May 31, 1105-1108.

20

TABLES

Table 1: Scheduling Rules. Scheduling Rule Name Abb. Description

1 Compressed Appointment Interval CAI The time interval between appointments is compressed from D to

T=DS.

2 Round to Nearest Interval RNI Compressed appointment times are rounded up or down to the nearest practical clock time.

3 Round Down Interval RDI Time intervals are compressed by rounding T down to the nearest multiple of 5 minutes that is less than D.

4 Multiple Appointments in First-block MAS Schedule all K–N overbooked appointments in the first (starting)

schedule block.

5 Multiple Appointments in Multiple-blocks MAM Schedule all K–N multiple appointments in multiple blocks

6 Multiple Appointments in Last-block MAL Schedule all K–N extra appointments in last normal schedule

block.

7 Multiple Appointments in Early-blocks MAE Spread the K–N extra appointments into slots earlier than the last

slot.

8 Multiple Appointments at Finish MAF Schedule all K–N extra appointments after the last block in clinic

session.

9 Multiple Appointments Distributed-evenly MAD Spread the K–N extra appointments evenly across the clinic

session.

10 Wave Schedule 1 WAV1 Schedule blocks of 3 in the first and last appointment slot, individual appointments in other slots.

11 Wave Schedule 2 WAV2 Schedule individual appointments tightly compressed.

12 Wave Schedule 3 WAV3 Reduce patient wait time by moving some compressed appointments to later slots.

13 Wave Schedule 4 WAV4 Schedule in widely-spread blocks of 3, 2, 1 patients.

14 Wave Schedule 5 WAV5 Rearrange blocks in WAV4 to reduce patient wait time.

Table 2: Clinic size N = 12; E(X) = expected number of patients.

S K

Calculated K

Rounded T = 20S Extra

Appointments E(X) 1.0 12.00 12 20 0 12.00 0.9 13.33 13 18 1 11.70 0.8 15.00 15 16 3 12.00 0.7 17.14 17 14 5 11.90 0.6 20.00 20 12 8 12.00 0.5 24.00 24 10 12 12.00 0.4 30.00 30 8 18 12.00 0.3 40.00 40 6 28 12.00 0.2 60.00 60 4 48 12.00 0.1 120.00 120 2 108 12.00

21

Table 3: Appointment times adjusted for overbooking. Wait time and overtime are calculated for the case in which every scheduled patient shows up; S = 0.90; D = 20 minutes.

Baseline: Calculated Calculated CompressNo Overbooking T = S D Adjusted to 15 min

8:00 AM 8:00 AM 8:00 AM 8:00 AM8:20 AM 8:18 AM 8:20 AM 8:15 AM8:40 AM 8:36 AM 8:40 AM 8:30 AM9:00 AM 8:54 AM 8:50 AM 8:45 AM9:20 AM 9:12 AM 9:10 AM 9:00 AM9:40 AM 9:30 AM 9:30 AM 9:15 AM

10:00 AM 9:48 AM 9:50 AM 9:30 AM10:20 AM 10:06 AM 10:10 AM 9:45 AM10:40 AM 10:24 AM 10:20 AM 10:00 AM11:00 AM 10:42 AM 10:40 AM 10:15 AM11:20 AM 11:00 AM 11:00 AM 10:30 AM11:40 AM 11:18 AM 11:20 AM 10:45 AM

11:36 AM 11:40 AM 11:00 AMTotal Appts 12 13 13 13

Average Wait 0.00 12.00 11.54 30.00Maximum Wait 0 24 20 60

Overtime 0 20 20 20

Table 4: Exponential Growth of Feasible Schedules. Show Rate

SAppointments

KPossible Schedules

1 12 21,090,682,6130.9 13 73,006,209,0450.8 15 751,616,304,5490.7 17 6,499,270,398,1590.6 20 125,994,627,894,1350.5 24 4,355,031,703,297,2800.4 30 450,883,717,216,035,0000.3 40 285,219,402,396,401,000,0000.2 60 5,680,916,595,331,740,000,000,0000.1 120 744,986,412,434,507,000,000,000,000,000,000

22

Table 5: Probabilities that patients show up for varying show rate S, K total appointments scheduled, and number of patients scheduled at the same appointment time (block size).

Probabilitythat all

K scheduledpatients Probability that more than one patient shows up when block size is:

S K show up 2 3 4 5 6 7 8 91 12 1 1 1 1 1 1 1 1 1

0.9 13 0.2542 0.81 0.972 0.9963 0.9995 0.9999 1 1 1 10.8 15 0.0352 0.64 0.896 0.9728 0.9933 0.998 0.9996 0.9999 1 10.7 17 0.0023 0.49 0.784 0.9163 0.9692 0.989 0.996 0.9987 0.9996 0.999860.6 20 3.6562E-05 0.36 0.648 0.8208 0.913 0.959 0.981 0.9915 0.996 0.998320.5 24 5.9605E-08 0.25 0.500 0.6875 0.8125 0.891 0.938 0.9648 0.980 0.989260.4 30 1.1529E-12 0.16 0.352 0.5248 0.663 0.767 0.841 0.8936 0.929 0.953640.3 40 1.2158E-21 0.09 0.216 0.3483 0.4718 0.580 0.671 0.7447 0.804 0.850690.2 60 1.1529E-42 0.04 0.104 0.1808 0.2627 0.345 0.423 0.4967 0.564 0.624190.1 120 1.0000E-120 0.01 0.028 0.0523 0.0815 0.114 0.150 0.1869 0.225 0.2639

101

Table 6: Maximum waiting time across all experiments. Show Rate S Scheduling

Rule 90% 70% 50% 30% 10% CAI 16.375 32.641 42.218 51.739 58.928RNI 14.508 33.093 42.218 52.394 58.915RDI 42.757 71.933 116.85 68.555 219.94MAS 16.054 64.088 109.78 154.36 198.15MAM 19.884 46.658 43.81 53.493 62.149MAL 16.188 64.23 110.22 153.78 197.63MAE 16.266 53.607 62.229 87.028 91.457MAF 0 50.212 100.19 147.75 195.56MAD 12.141 42.568 57.96 84.001 88.389

WAV1 12.32 34.942 56.906 53.43 65.138WAV2 12.223 42.416 53.298 70.828 60.318WAV3 15.949 38.394 42.676 66.316 60.556WAV4 19.543 49.078 47.638 61.564 60.958WAV5 15.83 42.888

23

Table 7: Overtime across all experiments. Show Rate S Scheduling

Rule 90% 70% 50% 30% 10% CAI 7.19 23.7 33.77 41.53 48.4RNI 7.48 23.62 33.77 41.3 48.6RDI -1.08 5.55 6.03 24.77 -0.06MAS 3.61 10.83 14.36 16 17.47MAM 11.26 11.74 28.73 37.78 38.35MAL 16.02 64.21 110.21 153.78 197.63MAE 15.15 49.06 49.31 75.3 77.87MAF 17.94 70.16 120.18 167.74 215.55MAD 16.5 58.78 66.84 93.77 96.42

WAV1 10.46 28.31 24.83 33.38 35.16WAV2 15.23 32.59 20.64 17.87 38.78WAV3 4.66 33.73 34.91 24.08 43.19WAV4 7.16 13.23 24.44 23.2 36.64WAV5 7.16 14.72

24

FIGURES

Figure 1: Total and extra appointments scheduled due to overbooking.

Number of appointments scheduled

0

20

40

60

80

100

120

140

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

Show Rate

App

oint

men

ts

0.00

2.00

4.00

6.00

8.00

10.00

12.00

Load

Fac

tor

Total Scheduled

Extra forOverbookingLoad Factor fromOverbooking

Figure 2: The efficient frontier of simulation results for S = 0.90.

Show Rate = 0.90

MAS, 16.054, 3.61

MAF, 0, 17.94

RDI, 42.757, -1.08

-5

0

5

10

15

20

0 5 10 15 20 25 30 35 40 45

Max Patient Wait

Ove

rtim

e

CAI

RNI

RDI

MAS

MAM

MAL

MAE

MAF

MAD

WAV1

WAV2

WAV3

WAV4

WAV5

Slope=0.89

Slope=0.18

Scheduling Rule Description as implemented for S=0.90 MAF Schedules the one extra appointment at

the end of clinic session. MAS Bailey-Welch Rule, schedules the extra

appointment to form a block of 2 appointments at the start of the clinic session.

RDI Compresses the time between appointments from T = 20S =18 minutes down to 15 minutes.

25

Figure 3: Schedules that are within practical constraints and not dominated by others.

Show Rate = 0.90

MAS

WAV1

WAV3

RNI

-5

0

5

10

15

20

0 5 10 15 20 25 30 35 40 45

Max Patient Wait

Ove

rtim

e

CAI

RNI

RDI

MAS

MAM

MAL

MAE

MAF

MAD

WAV1

WAV2

WAV3

WAV4

WAV5

Scheduling Rule Patient Wait

Overtime Slope, (between schedules)

WAV1 12.32 10.46 1.36, (10,2)

RNI 14.508 7.48 1.96, (2,12)

WAV3 15.95 4.66 10, (12,4)

MAS 16.064 3.61

26

Figure 4: Simulation Results for S = {0.7, 0.5}.

Show Rate = 0.70

0

20

40

60

80

100

120

0 20 40 60 80 100 120Maximum Patient Wait

Ove

rtim

eCAI

RNI

RDI

MAS

MAM

MAL

MAE

MAF

MAD

WAV1

WAV2

WAV3

WAV4

WAV5

Show Rate = 0.50

0

20

40

60

80

100

120

0 20 40 60 80 100 120Maximum Patient Wait

Ove

rtim

e

CAI

RNI

RDI

MAS

MAM

MAL

MAEMAF

MAD

WAV1

WAV2

WAV3

WAV4

27

Figure 5: Comparison of plots for constant service time (cs = 0) and high variation cs = 1.0.

Show Rate = 0.70, Coefficient of Variation = 0

0

20

40

60

80

100

120

0 20 40 60 80 100 120Maximum Patient Wait

Ove

rtim

eCAI

RNI

RDIMAS

MAM

MALMAE

MAF

MAD

WAV1WAV2

WAV3

WAV4WAV5

Show Rate = 0.70, Coefficient of Variation = 1.0

0

20

40

60

80

100

120

0 20 40 60 80 100 120Maximum Patient Wait

Ove

rtim

e

CAI

RNI

RDI

MAS

MAM

MAL

MAE

MAF

MAD

WAV1

WAV2

WAV3

WAV4

WAV5

28

Figure 6: Schedules on the efficient frontier for }75.0,5.0,25.0{=sc .

Show Rate = 0.70, Coefficient of Variation = 0.25

020406080

100120140

0 25 50 75 100 125Maximum Patient Wait

Ove

rtim

e

CAI

RNI

MAM

WAV1

WAV4

WAV5

Show Rate = 0.70, Coefficient of Variation = 0.50

020406080

100120140

0 25 50 75 100 125Maximum Patient Wait

Ove

rtim

e

CAI

RNI

MAM

WAV1

WAV4

WAV5

Show Rate = 0.70, Coefficient of Variation = 0.75

020406080

100120140

0 25 50 75 100 125Maximum Patient Wait

Ove

rtim

e

CAI

RNI

MAM

WAV1

WAV4

WAV5

29