Dynamic allocation of same-day requests in multi-physicianprimary care practices in the presenceof prescheduled appointments

Hari Balasubramanian & Sebastian Biehl & Longjie Dai &Ana Muriel

Received: 15 August 2012 /Accepted: 8 May 2013# Springer Science+Business Media New York 2013

Abstract Appointments in primary care are of two types: 1)prescheduled appointments, which are booked in advance ofa given workday; and 2) same-day appointments, which arebooked as calls come during the workday. The challenge forpractices is to provide preferred time slots for prescheduledappointments and yet see as many same-day patients aspossible during regular work hours. It is also important, tothe extent possible, to match same-day patients with theirown providers (so as to maximize continuity of care). In thispaper, we present a mathematical framework (a stochasticdynamic program) for same-day patient allocation in multi-physician practices in which calls for same-day appoint-ments come in dynamically over a workday. Allocationdecisions have to be made in the presence of prescheduledappointments and without complete demand information.The objective is to maximize a weighted measure that in-cludes the number of same-day patients seen during regularwork hours as well as the continuity provided to thesepatients. Our experimental design is motivated by empiricaldata we collected at a 3-provider family medicine practice inMassachusetts. Our results show that the location ofprescheduled appointments – i.e. where in the day theseappointments are booked – has a significant impact on thenumber of same-day patients a practice can see duringregular work hours, as well as the continuity the practice isable to provide. We find that a 2-Blocks policy which booksprescheduled appointments in two clusters – early morningand early afternoon – works very well. We also provide asimple, easily implementable policy for schedulers to assign

incoming same-day requests to appointment slots. Our re-sults show that this policy provides near-optimal same-dayassignments in a variety of settings.

Keywords Appointment scheduling . primary care . Same-day access . Continuity of care . Stochastic dynamicprogramming . Heuristics

1 Introduction

The two operational measures important for primary carepractices are timely access and continuity of care. Timelyaccess refers to the ability of patients to secure an appoint-ment as quickly as possible, while continuity refers to theability of patients to see their own personal physicians(PCPs) as often as possible. Timely access enables patientsto obtain adequate access for acute conditions that mighthave resulted in costly and unnecessary emergency depart-ment visits. Continuity of care allows the patient to developa long-term relationship with the provider and ensures thatthe care provided is person centered and holistic.

In the ideal case, each patient exclusively sees her ownprovider; but a dedicated PCP model is rarely possible inpractice. A patient’s PCP may not be available on the daythe appointment is requested. Furthermore, an acute condi-tion requiring a same-day appointment, such as sore throator a sprained ankle, does not always necessarily requirecontinuity (though it may still be desired), while a patientwith long-persisting chronic conditions (such as hyperten-sion, asthma, depression) requires regular prescheduledfollow-ups and will highly benefit from seeing her ownPCP.

Appointments, thus, can be classified into two broad types:same-day appointments (generally for acute conditions) and

H. Balasubramanian (*) : S. Biehl : L. Dai :A. MurielDepartment of Mechanical and Industrial Engineering, Universityof Massachusetts, 160 Governors Drive,Amherst 01003, USAe-mail: hbalasubraman@ecs.umass.edu

Health Care Manag SciDOI 10.1007/s10729-013-9242-2

prescheduled appointments (for annual physical exams, andmonitoring and follow-up of chronic conditions). A typicalprimary care physician’s calendar at the beginning of a work-day will consist of appointments booked in advance of theday. Slots that are still unfilled can be used to satisfy calls forsame-day appointments that arise as the workday progresses.Generally, prescheduled appointments are always seen by thepatient’s PCP while same day appointments are flexiblyshared across physicians.

We address two important operational questions relevantfor multi-physician primary care practices in this paper. 1)How does the location of prescheduled appointment slotsimpact the ability of the practice to satisfy same-day de-mand? By location we mean where in the day these appoint-ments are booked. 2) How should dynamically arrivingrequests for same-day appointments be assigned to the mul-tiple physicians over the day? We call these two questionsthe location problem and the dynamic assignment problem.To investigate both questions, we propose a stochastic dy-namic program to allocate incoming same-day appointmentrequests to physicians so as to optimize a weighted measureof timely access and continuity in a multi-physician prac-tice. The ultimate objective is to provide guidance on locat-ing prescheduled appointments and simple allocation rulesthat will allow practices to improve access and continuity.The dynamic program provides optimal same-day assign-ments, but may not be considered by users to be practical fordaily use. It will allow us to identify simple heuristic rulesthat can perform well in practice.

2 Key model features and contributions

We model a single 8-h workday in a multi-physician prac-tice. Each physician’s schedule consists of deterministic 20-min appointment slots. We assume deterministic slot dura-tions because patient wait times in the clinic are not thefocus of this study. At the beginning of a workday some 20-min slots are already prescheduled. Figure 1 shows the

calendars of the 3 providers at 8:00 am in the morning onAugust 9, 2011 at the family medicine practice we workedwith. Prescheduled slots are marked with a dark shade. Theunmarked slots are available to schedule same-day patients.

Same-day requests specific to each physician arise atdifferent times during the day and calls are likely to be morefrequent in certain hours of the day than others. We useempirical data from a time study on call rates conductedover 10 workdays at the family medicine practice to cali-brate same-day request rates in our model.

Timely access in our model is the number of same-daypatients seen during regular hours of a practice (an 8-hworkday). Note that our objective does not consider the waitfrom the time of call to when the appointment is scheduledfor each individual same-day request. We consider access tobe appropriate so long as patients are seen during the8 h day. This is reasonable, since urgent requests requiresame-day access, but are not emergencies, in which caseimmediate access to an ER would be more appropriate.However, we do assume that urgency of request is sufficientenough for each same-day patient to accept the earliestavailable same-day appointment once the physician hasbeen decided, irrespective of how close it is to the time therequest is made. Patients in our model can get assignedeither to their own personal physician, or to a non-PCPphysician with a slight penalty. The penalty reflects the costof not maintaining continuity. Our model also tracks thenumber of same-day patients who did not get access duringregular hours of a workday. These patients are either seen atthe cost of overtime or decide to use an emergency room.

Given these model features, the location problem implic-itly considers time-of-day preferences for prescheduled pa-tients and its impact on same-day access and continuity. Forexample, prescheduled appointments could be allowed to bescattered throughout the day to allow for patient time-of-daypreferences. But our results reveal that such a locationpolicy results in the inability to provide care for a significantnumber of same-day patients during regular work hourscompared to the case where prescheduled requests are

Time Slots Provider 1 Provider 2 Provider 3

8:008:208:409:009:209:4010:0010:2010:4011:0011:2011:40

Time Slots Provider 1 Provider 2 Provider 3

1:001:201:402:002:202:403:003:203:404:004:204:40

Morning Session Post-lunch session

Fig. 1 Example of physiciancalendars at 8 am for a 3-provider practice on August 9,2011. Prescheduled slots areshaded gray; all other slots areavailable for same-day patients.The treatment of certainconditions and physical examsrequire two slots; all otherpatients, including all same-daypatients, are assigned to a singleslot. Slots are not bookedbetween noon and 1:00, whenthe entire practice closes forlunch

H. Balasubramanian et al.

located in the morning (All-Morning Policy). The All-Morning case works very well because it allows consider-able space later in the day to schedule same-day requestsreceived throughout the day. We emphasize, however, thatAll-Morning is not realistic in practice. We only use it as anoptimal benchmark to compare other more practical locationpolicies and to quantify the tradeoffs inherent in accommo-dating patient time-of-day preferences for prescheduledslots on the one hand, and same-day access and continuityon the other. A key contribution of this paper is a newlocation policy called 2-Blocks, which does almost as wellas All-Morning with regard to both same-day access andcontinuity, but is less restrictive than All-Morning in that itoffers patients two clusters in the day to book prescheduledappointments and accommodate their preferences. Our rec-ommendation for practices is to guide prescheduled patientstowards choosing appointment times in those clusters, whilestill allowing bookings beyond them when needed.

Irrespective of what location policy a practice uses, thedynamic assignment problem deals with the allocation of eachincoming same-day request to a particular physician and timeslot without complete demand information. Here the competingconsiderations are not allowing an upcoming physician slot togo idle, yet also ensuring that patients see their own physician asmuch as possible. An idle slot is a lost revenue opportunity forthe practice because it reduces the number of same-day patientsseen during regular work hours. This cost has to be weighedagainst the loss of continuity for same day patients that see anunfamiliar physician. Here, our contribution is the AdaptiveThreshold (AT) heuristic, which blends both access andcontinuity in its decision process, and provides near optimalsame-day assignments. The AT heuristic works as follows.When a same-day request arrives, the scheduler looks at a pre-defined time window (say the next hour) to check if any of thephysicians has an available slot. If a slot is indeed available,the patient is scheduled in that slot. This minimizes the prob-ability of near term slots going idle and increases the numberof patients who obtain access during regular work hours. If noslot is available within the time window, the patient is sched-uled with the earliest available slot of her PCP, irrespective ofhow far this slot is in the day, thus keeping continuity in mind.If the PCP slot is also not available, then the patient is assignedthe physician with the earliest available slot.

The rest of the paper is organized as follows. In Section 2,we review the appointment scheduling literature related toprimary care. In Section 3 we present the details of thestochastic dynamic program. We then propose an experimen-tal design in Section 4 to explore the impact of prescheduledappointment slot locations and investigate the allocation ofsame-day appointments to physicians, and present the com-putational results in Section 5.We discuss our conclusions andthe implications for primary care practices in Section 6, andpoint to opportunities for future research.

3 Literature review

Appointment scheduling in healthcare is an active andgrowing area of research. The adoption of advanced ac-cess [11], which promises patients same-day appoint-ments, has prompted a number of questions. How manypatients should a primary care physician care for? Whatimpact do no-shows have? How does a practice deal withpatient preferences and different appointment types?These questions have necessitated the use of queuing andstochastic optimization approaches that provide guidelinesto practices.

We restrict this review to appointment scheduling papersas they apply to access and continuity in primary care.Green et al. [6] investigate the link between panel sizesand the probability of “overflow” (overtime) for a physicianunder advanced access. They propose a simple probabilitymodel that estimates the percentage of workdays a physicianwill work overtime, as a function of her panel size. Theprincipal message of their work is that for advanced accessto work, the supply of physician appointment slots needs tobe sufficiently higher than the average patient demand forappointments. This allows a practice to have adequate buffercapacity on days where the patient demand realizations arewell above the average.

Green and Savin [5] use a queuing model to determinethe effect of no-shows on a physician's panel size. Theydevelop analytical queuing expressions that estimate the sizeof the waiting list (the list of patient appointments scheduledin the future) as a function of panel size and no-show rates.A longer wait list indicates longer times between the date ofthe appointment request and the date the appointment isscheduled. In their model, no show rates increase as thewaiting list grows, since patients who have to wait longeruntil their appointment date are more likely to not show up.This results in a paradoxical situation where physicians havelow utilization even though size of the wait list is high.Balasubramanian et al [1] and Ozen and Balasubramanian[13] show that in addition to panel size, case-mix consider-ations are important when it comes to designing physicianpanels. Case-mix refers to the type of patients (older versusyounger; healthy patients versus patients with chronic con-ditions) in a physician’s panel. They propose that in the longterm, panels can be redesigned to improve timely access andcontinuity.

Gupta et al. [7] conduct an empirical study of clinicsin the Minneapolis metropolitan area that adopted ad-vanced access. They provide statistics on call volumes,backlogs and number of visits with own physician(which measures continuity) and discuss options for in-creasing capacity at the level of the physician and clinic.Kopach et al. [17] use discrete event simulation to studythe effects of clinical characteristics in an advanced

Dynamic scheduling & same-day requests

access scheduling environment on various performancemeasures such as continuity and overbooking. One oftheir primary conclusions is that continuity in care isaffected adversely as the fraction of patients on advancedaccess increases. The authors mention physicians andsupport staff working in teams as one solution to theproblem. Numerous studies have investigated the impactof no-shows and proposed overbooking strategies forsingle physician clinic sessions. Examples includeLaGanga and Lawrence [9], Muthuraman and Lawley[12], and Chakraborty et al. [4].

Robinson and Chen [15] compare the performance ofadvanced access with a traditional appointment schedulingsystem. In the advanced access system, a practice has to dealwith day to day variability but very few no shows, while inthe traditional appointment system, patients book their ap-pointments well in advance with the result that day to dayvariability is smoothed but patients have a higher probabil-ity of no-show. Their numerical analysis reveals that ad-vanced access generally outperforms the traditionalappointment system when the objective function is aweighted average of patients' waiting time (lead time toappointment), the doctor's idle time, and the doctor's over-time. Only when the patient waiting time is held in littleregard or when the probability of no-show is small does thetraditional system work better than the advanced accesssystem. Liu et al. [10] propose new heuristic policies fordynamic scheduling of patient appointments under no-shows and cancellations. They find that advanced accessworks best when patient load is relatively low.

The papers most related to our study are Qu et al [14],Balasubramanian et al [2], Gupta and Wang [8] and Wangand Gupta [16]. Qu et al [14] consider an essential questionfor primary care practices: how many prescheduled appoint-ments should a physician plan for in a workday given thatthe physician also has to see same-day patients? Their modelconsiders different show rates for the two appointmenttypes. They derive conditions under which a solution forthe number of prescheduled appointments to reserve islocally optimal. Balasubramanian et al [2] show a strongerresult for the single physician problem, guaranteeing globaloptimality, by first showing that the revenue maximizationfunction has diminishing returns under mild assumptions.They extend this result for a multi-physician practice inwhich prescheduled demand is always seen by a patient’sPCP whereas same-day patients can be seen by any physi-cian in the practice. Thus such a practice is fully-flexiblewhen it comes to same-day patients.

For a multi-physician practice, Balasubramanian et al [2]also present a stochastic optimization model to determinethe number of prescheduled appointments each physicianshould plan for and how this number changes depending onhow flexible physicians are in seeing same-day patients of

other physicians. An important conclusion of their study isthat partial flexibility – where a same-day patient can beseen by either its PCP or a small subset of the physicians,thus maintaining an acceptable level of continuity – comesvery close to matching full flexibility with regard to thenumber of patients a practice is able to see per day.

In Qu et al. [14] and Balasubramanian et al. [2], the totalcapacity of each physician (or physicians) needs to be splitbetween prescheduled and same-day appointments. Sincecapacity allocation is considered at an aggregate level,where in the day prescheduled appointments are located isnot modeled. Furthermore, all same-day demand is realizedat once. In this paper, we consider both the location ofprescheduled appointments as well as a dynamic call inprocess for same-day appointments.

Gupta and Wang [8] and Wang and Gupta [16] modelmany of the key elements of a primary care practice. Theyconsider scheduling the workday of a clinic in the presenceof 1) multiple physicians; 2) two types of appointments(prescheduled as well as same-day appointments); and 3)prescheduled patient preferences for specific slots in a dayand also for physicians. The objective is to maximize theclinic's revenue, PCP matches, and the number of patientsseen by the practice. They use a Markov Decision Process(MDP) model to obtain booking policies that provide limitson when to accept or deny requests for appointments frompatients.

Our approach resembles the framework of [8] and [16] ina number of ways. Both papers consider a multi-physicianpractice and model a sequential call-in process, as does ourstochastic dynamic program. PCP matches are an importantcriterion in these papers and this applies to our work as well.The difference is that in [8] and [16], the call in process isfor patients requesting for specific time slots on a futureworkday. In other words, their call in process is forprescheduled appointments. Furthermore, same-day demandin their models is realized instantly at the beginning of theday.

In contrast, our call in process is for same-day appoint-ments and occurs over the course of a workday. Allocationshave to be made dynamically, without complete demandinformation, and in a situation where specific time slots inthe day are already booked before the workday begins(prescheduled appointments) and the earlier slots turn tobe unavailable and may go idle as the day progresses. Weare interested in both the number of same-day patients apractice can see as well as the continuity a practice canprovide to these patients. We do not model patient prefer-ences explicitly as [8] and [16] do. But by considering thelocation of prescheduled appointments we are implicitlyincluding patient time-of-day preferences. Our results cap-ture the tradeoff between accommodating prescheduled pa-tient choices for specific time slots and the number of same-

H. Balasubramanian et al.

day patients that can be seen during regular working hoursof a practice.

4 Model formulation and notation

We formulate the same-day appointment allocation problemas a probabilistic dynamic program with a time-dependentarrival probability distribution for same-day requests. Atany decision epoch, if a same-day request arrives, it isallocated to a physician in the practice or rejected (i.e., thatpatient will not get a regular appointment for that day, butmay be scheduled after hours or be offered to book anappointment for the following day). We assume that thesame-day patient accepts being assigned to the earliestavailable free slot for that physician. To reflect the lossof continuity, we impose a slight penalty on non-PCP as-signments. The full details of the model – states, actions,transitions and the objective function, along with the nota-tion – are described below.

4.1 Input parameters

N Number of stages of decision making, i.e.decision epochs. The current stage of thesystem is denoted by the indexn∈[0,1,…,N–1]. We make the number ofstages large enough to ensure that at mostone patient request arrives on any singlestage.

A Duration of one single appointmentslot (measured in number of stages).

S Number of appointment slots, where S=N/A. Slots are indexed by s∈[0,1,…S–1]

M Number of physicians and thus patientpanels as well. Both – physicians andpanels – are indexed with m∈[1…M].According to this notation physician m isthe PCP of panel m.

pnm Probability of a patient requestfrom panel m arriving in stage n.

pn0 ¼ 1− ∑M

m¼1pnm Probability of no arrivals in stage n.

Qsm

1 if slot s of physicianm0s calendar is booked witha prescheduled appointment;

0 otherwise:

8>><>>:

Rmj Revenue associated with assigning apatient from panel m to physician j. Rmmis the revenue earned when a PCP sees oneof his own patients, and a lower revenue

will be applied when continuity is broken,for any Rmj where m≠j.

4.2 State variables

For n=0,1,2,…,N-1 and m=1,2,…,M, we define, amn: avail-

able (open) slots for same-day appointments for physician mfrom stage n to the end of the horizon. These variablescharacterize the state of the system at stage n and will beupdated every decision epoch. The knowledge of the exactplacement of the available slots is not necessary since apatient assigned to physician m in stage n will take theearliest of these slots. The initial status of available same-day appointments for each physician is easily calculatedfrom the given matrix of prescheduled appointments:

a0m ¼ ∑S−1

s¼01−Qsmð Þ:

4.3 Decision variables

For n=0,1,…,N-1, m=1,2,…,M and j=0,1,.., M, we let,

xnmj ¼1 if a request from panel m in stage n is

assigned to physician j;0 otherwise:

(

Here, j=0, corresponds to the refusal of the patient’srequest. The decision space is restricted by the necessary

constraint ∑Mj¼0 x

nmj ¼ 1, that is, an incoming request can be

assigned to at most one physician or refused. We reiter-ate that our model is based on an important assumption.Once the physician is chosen, the model assigns anincoming same-day request to the earliest available slotof the physician, irrespective of how near or far that slotis in the future. To illustrate consider Fig. 2. A same-dayrequest arrives for Physician 1 at time n. This requestcan be assigned to any of the M physicians but only tothe earliest available slot of each physician. These choices areindicated in the figure.

At any decision epoch n, using Qsm (the prescheduledmatrix, which is known at the beginning of the day and doesnot change) and the number of same-day slots am

n physicianm has available, we can determine the earliest availablesame-day slot for physician m. As an example, considerFig. 3 below. For a particular physician m, at somedecision epoch n which falls in slot s=n/A, supposethere are 5 total slots still to come before the end ofthe day, two of which are prescheduled, but not adjacent toeach other.

Then if amn =3 for physician m, knowing Qs+2, m=1 and

Qs+4, m=1 (these variables correspond to Slot 2 and Slot 4 inFig. 3), we can automatically come to the conclusion that

Dynamic scheduling & same-day requests

Slot 1 is the earliest available slot for assigning a same-daypatient. However, if am

n =2 then it must be true that Slot 1 hasalready been booked to accommodate a same-day request thathappened in the past; therefore Slot 3 is Physician m’s earliestavailable same-day slot. Note that the prescheduled slots canbe distributed in any fashion through the day.

This logic holds true for all physicians. Thus, the state ofthe system at stage n is completely described by the values ofamn and Qsm s=0,1,…,S-1, for all physicians m=1,2,…,M.

4.3.1 Transition function for physician m

Given a current decision in stage n to assign an incomingrequest to the jth physician, j=0,1,…,M (where j=0 would

mean that no assignment was made either because no patientarrived or because the patient was refused), the state ofphysician m would transition to am

n+1=T(amn ,j) as follows.

The transition needs to take into account whether the nextdecision epoch marks the beginning of the next slot in theschedule. If this next slot is not booked (either by aprescheduled appointment or a previous same-day request),then it will go idle. To capture how this will influence thetransition function, we first define a new indicator variableems such that

esm ¼1 if slot s has not been booked is emptyð Þ

by the time it starts; in stage n ¼ sA;

0 otherwise:

(

….N

0

.

.

.

.

.

.

.

.

.

….

n

One 20-minute Slot; A=20

Physician M

Currentdecision epoch

Physician 1

Physician 2

Same-day slots available to be assigned correspond to earliest available for each physician

Phy. 1Call arrives for Physician 1 at time n

Phy. M

Phy. 2

Fig. 2 Visual illustration ofphysician-slot allocationchoices at time n when a same-day request arrives forPhysician 1. The dark slotscorrespond to prescheduledappointments; the gray slots toalready booked same-dayappointments. Slots that are notshaded are available to beassigned

Earliest Available slot

0 n

Currentdecision epoch

N

Slot 1 Slot 2 Slot 3 Slot 4 Slot 5

0 n

Currentdecision epoch

N

Slot 1 Slot 2 Slot 3 Slot 4 Slot 5

mn = 3

mn = 2

Same-dayslot booked

Earliest Available slot

(a)

(b)

a

a

Fig. 3 Parts (a) and (b) showpossible system states andearliest available slot for asingle physician, m. The darkslots correspond toprescheduled appointments; thegray slots to already bookedsame-day appointments. Slotsthat are not shaded are availableto be assigned

H. Balasubramanian et al.

These variables are easy to calculate. The slot is notbooked if there are no initial prescheduled patientsassigned to it and, since same day appointments areassigned to the earliest available slots, no same-dayappointments have been booked on either slot s or any

later slot for physician m. Thus, esm ¼ 1 iff Qsm ¼ 0

and asAm ¼ S−sð Þ− ∑S−1

k¼sQkm (see Fig. 3). Using these new

variables, we are now ready to write the transitionfunction. There are two possible state transitions forphysician m’s available slots:

1. No change occurs if no patient is assigned to physicianm (i.e., j≠m) and either:

& the current system epoch n is within a slot (i.e., (n+1)mod A≠0), OR

& the system is transitioning to the next slot ((n+1)mod A=0), which was already booked (em

s =0) andthus not part of the count of available slots.

2. The number of available slots is reduced by 1 if either

& a patient is assigned to physician m in stage n (i.e., j=m), OR

& a patient is not assigned to physician m in stagen (i.e., j≠m) AND the system is transitioning tothe next slot ((n+1) mod A=0) which is emptyand goes idle (em

(n+1)/A=1).

In summary, we have, amn+1=T(am

n , j) where

T anm; j� � ¼

anm if m≠ jð Þ AND ð nþ 1ð Þ mod A≠0ð ÞOR ð nþ 1ð Þ mod A ¼ 0ð Þ AND ðe nþ1ð Þ=A

m ¼ 0ÞÞÞ;

anm � 1 if m ¼ jð Þ OR ð m≠ jð Þ AND nþ 1ð Þ mod A ¼ 0ð ÞAND ðe nþ1ð Þ=A

m ¼ 1ÞÞ:

8>>>>>><>>>>>>:

Observe that if a same-day patient is assigned to physi-cian m at time n, the next slot s≥(n+1)/A of that physiciancannot go idle. The slot will be assigned to that patient, ifnot previously booked, since same-day patients are assignedto the next available slot of the physician. As a result, thenumber of available slots can never decrease by more thanone in any transition.

4.3.2 Objective function and solution approach

The objective is to maximize the revenue generated byall assignment decisions, where refused patients generateno revenue and patients seeing an unfamiliar physicianyield a lower revenue than those assigned to their PCPs.In other words, the objective is to maximize a weightedmeasure that includes the number of same day patients

seen during regular work hours as well as the continuityprovided to these patients.

The dynamic optimization problem is solved by themethod of backwards recursion. We will use the followingvector/matrix and function notation for n=0,1,2,…,N-1:

an ¼ anm� �

m¼1;2;…;M; xn ¼ xnmj

� �m ¼ 1; 2;…;Mj ¼ 0; 1;…;M ;T an; jð Þ

¼ T anm; j� �� �

m¼1;2;…;Mfor j ¼ 0; 1;…;M :

fn(an,xn) Expected revenue-to-go from stage n until the

end of the horizon when the system is in state an

and the assignment decision xn=(xmjn ) is made.

This includes the immediately realized revenue instage n and the revenue gained in the followingstages (n+1,…,N) given that the optimal decisionpolicy is followed from n+1 onwards.

fn*(an) Optimal expected revenue-to-go from stage n until

the end of the horizon when the system is in stagen and the optimal decision policy (xn)* is chosen.That is, f �n anð Þ ¼ maxXn f n an; xnð Þf g.

If the system is in a certain state an, a request from panelm is received, and the decision vector is xn=(xmj

n ), the

system will transition to a state anþ1 ¼ ∑M

j¼0xnmjT an; jð Þ. If

no request is received at stage n, then the system willtransition to an+1=T(an,0). The recursive equation can bewritten as follows.

f n an; xnð Þ ¼XMm¼1

pnmXMj¼1

xnmjRmj

!

þXMm¼1

pnm f�nþ1

XMj¼0

xnmjT an; jð Þ !

þ pn0 f�nþ1 T an; 0ð Þð Þ

with f �n anð Þ ¼ maxXn f n an; xnð Þf g as defined above. Giventhe current state and decision vector, the first term in therecursive equation captures the expected revenue earned atstage n, while the second and third terms determine theexpected revenues to go considering whether or not a patientrequest arrives at stage n, respectively.

According to the assumption that no overtime is possibleand that the working day ends at stage N we set fN(a

N,xN)=0 as the “starting point” of the backward recursion. Allfeasible state vectors from stage N backwards to stage0 are evaluated via backward recursion, leading to the initialproblem f0

*(a0).

Dynamic scheduling & same-day requests

4.4 Optimality of the All-morning location scenario

Given the above stochastic dynamic program, we nowshow that the optimal solution to the location problemis to have all prescheduled slots at the start of the day.We call this the All-Morning (AM) location scenario.See Fig. 4 below for examples of location scenarios,including All Morning.

Theorem The All-Morning location scenario, whereprescheduled appointments are placed in the very first slotsof the day, always results in maximum system revenue.

Proof Consider an All-Morning location scenario(schedule AM), with prescheduled slot locations speci-fied by the matrix Qsm

AM, and any other location scenario(schedule G, for general) with the same number ofprescheduled slots, but locations specified by the matrixQsm

G , for s=0,1,…, S-1 and m=1,2,…,M. We use thesuperscript AM and G to distinguish between the vari-ables in the two location scenarios.

We will show by induction that any assignment xnG ofincoming same-day requests to doctors in schedule G isalso feasible in schedule AM; that is, we can alwaysconstruct an assignment xnAM in the All-Morning sched-ule such that if xjm

nG=1 then xjmnAM=1, for j=1,2,…,M, and

m=0,1,…,M. In what follows, we focus on a singlephysician m.

First observe that, before any same-day demand isassigned, the number of free, future slots available tosame-day patients arriving at any time epoch n, with

A(s-1)≤n<As, is always no lower in schedule AM thanin schedule G:

anAMm ¼ S−sð Þ−XS−1k¼s

QAMkm ≥ S−sð Þ−

XS−1k¼s

QGkm ¼ anGm :

This is because more of the booked slots with Qkm=1 willoccur before time slot s in the AM schedule than in anyother schedule G. Consider the first time epoch n1 such thatthere exists a demand from a certain panel j1 that is assigned

to physician m, that is, xn1Gj1m¼ 1. Since no other same-day

appointments have been previously allocated, the number offree future slots in the All-Morning scenario can be nosmaller, that is, an1AMm ≥an1Gm as shown above. Thus physicianm has slots available to satisfy the demand for that patient inthe All-Morning schedule as well, and the incoming patientat time n1 can be assigned to physician m, i.e. xjm

nAM=1. Thepatient is booked in the first open slot of physician m aftertime epoch n1 in each of the schedules. In this way, bothschedules continue to have the same total number of bookedappointments and the open slots in the All-Morning sched-ule still occur no earlier than those in the general schedule.As a result, it continues to hold that the number of free slotsafter any time epoch n>n1 is always no lower in the All-Morning schedule, that is, am

nAM≥amnG.

Assume now that the first k same-day patients, say frompanels j1, j2…, jk, assigned to physician m under schedule G,occurred at epochs n1, n2…, nk. By the induction assumption,they were also accommodated in the All-Morning setting, so

that xnrGjrm ¼ 1 and xnrAMjrm¼ 1 for r=1,2,…,k, and resulted in

schedules satisfying amnAM≥amnG for all n>nk. Consider now the

next same-day assignment in schedule G, say from panel jk+1,

occurring at time epoch nk+1 such that xnkþ1Gjkþ1m

¼ 1. Since

ankþ1AMm ≥ankþ1G

m , we can readily schedule that patient withphysician m in the next free slot available in the All-Morning

scenario, so that xnkþ1AMjkþ1m

¼ 1 and amnAM≥amnG for all n>nk+1.

By induction, we have shown that for any demand sce-nario, the All-Morning schedule can accommodate the sameassignments, and thus provide at least the same revenue, asthose given by any other schedule. Therefore, the expectedrevenue of the All-Morning prescheduled location scenariocan be no lower than that of any other scenario.

Two observations follow as a result of the above theorem:

1) Assigning incoming requests allocated to physician mto her earliest available slot is optimal, regardless ofthe prescheduled location scenario: The proof wouldproceed in a very similar manner as above. Assigning alater slot increases the chance that earlier slots of thephysician would go idle; it also reduces available slots

Slot P1 P2 P3 Slot P1 P2 P3 Slot P1 P2 P3 Slot P1 P2 P31 1 1 12 2 2 23 3 3 34 4 4 45 5 5 56 6 6 67 7 7 78 8 8 89 9 9 910 10 10 1011 11 11 1112 12 12 1213 13 13 1314 14 14 1415 15 15 1516 16 16 1617 17 17 1718 18 18 1819 19 19 1920 20 20 2021 21 21 2122 22 22 2223 23 23 2324 24 24 24

Patient centeredAfternoon2BlocksMorning

Fig. 4 The four location scenarios for prescheduled appointments

H. Balasubramanian et al.

(capacity) in the system and therefore may result inlower expected revenue. This is why in the stochasticdynamic program, we always assign to the earliestavailable slot of the chosen physician m.

2) The optimality of the All-Morning location scenario holdseven if the revenues associated with PCP and non-PCPassignments were patient dependent. The theorem aboveshows that any feasible assignment in a particular locationscenario is also feasible in the All-Morning location sce-nario, and thus would lead to the All-Morning locationscenario achieving at least the same total revenue even inthe presence of patient-dependent revenues.

5 Experimental design

5.1 Prescheduled location scenarios

As discussed earlier, one of the principal goals is to studythe impact of the location of prescheduled appointments onthe number of same-day appointments a practice can acceptduring regular working hours, as well as the continuity it canprovide to same-day requests.

Towards this end, we propose four location scenarios, shownin Fig. 4. In the first, which we call the “All-Morning” scenario,a practice books all its prescheduled slots in a single cluster firstthing in the morning. In “All-Afternoon”, the opposite is true:the practice schedules all its prescheduled appointments in asingle cluster at the end of the day. In the third scenario, “2-Blocks”, half the prescheduled slots are scheduled first thing inthe morning the other half booked early in the afternoon (rightafter lunch). The motivation behind 2-Blocks is that it providesgreater choice to prescheduled patients (early morning andlunch time are favored slots) while allowing time for same-day patient requests to arrive before the same-day slots. Thisscenario was suggested by a practice we work closely with.Some patients may still prefer other times, but guiding mostpatients to these two blocks should provide similar results.

Finally, we have the “Patient-Centered” scenario, inwhich the prescheduled slots are scattered throughoutthe day, but nevertheless have greater density in themornings and late afternoons which we have observedto be the most sought after prescheduled appointmenttimes (working patients, school kids, etc.). We call thisscenario patient-centered since the practice allows slotsto be booked at any time that the patient prefers. Thefour scenarios are shown in Fig. 4.

We test these four prescheduled scenarios in a detailedexperimental setup. Our objective, as discussed earlier, is tomaximize a weighted measure of timely access and conti-nuity. Timely access is simply the number of same-daypatients seen in a day. Continuity is included by assigning

a slight penalty if a same-day request is seen by a physicianother than the patient’s PCP. If a same-day patient sees herown PCP, the objective function gets the full “revenue” of1.0, but if a patient sees a non-PCP physician the revenue is0.9. The impact of the lack of continuity is difficult toquantify. We have kept the penalty low, at 10 %, since theloss of continuity for a same-day appointment (typically foran acute non-recurring condition) is not clinically signifi-cant. This was anecdotally confirmed by the physicians weinteracted with. Furthermore, many patients might still pre-fer to see their PCP, but they are generally also willing toforgo this requirement in return for quick access to anunfamiliar physician.

In our model, patients can be refused an appointment. Inpractice, this may mean that these patients have to be sched-uled after the regular working hours of the practice, resultingin overtime. Or, they may be asked to come the next day, andblock a slot in the next day’s schedule, just as a prescheduledpatient. Either way, refusals can be thought of as a cost to thepractice and reflect the lack of timely access.

5.2 Practice details

All our experiments involve 3 physicians. This practice sizeis realistic, since 78 % of the group practices in the UnitedStates employ 5 physicians or less [3]. Even in large grouppractices (academic practices for example) physicians areoften divided into subgroups of 3–5 physicians. Further, thefamily medicine practice we have worked closely with forthis paper employs 3 providers. On the computational side,the state space of the dynamic program grows significantlywith the increase in the number of physicians. The 3-physician case, however, is tractable and we believe theinsights we gain from it can be generalized for larger phy-sician cases.

We assume the length of the workday to be 8 h, or480 min and will consider each minute as a decisionepoch; that is, N=480. Each appointment slot in primarycare is typically 20 min, and thus we set A=20. Thusduring an 8 h workday a physician will have a total ofS=24 appointment slots. Practices also book 40 minprescheduled appointments (for example physical examsand follow up for diabetes patients), but these can bethought of as two 20-min prescheduled appointmentsscheduled in succession. All same-day appointmentsare typically 20 min slots, although it may vary signif-icantly from practice to practice; we know of practicesthat schedule patients in as little as 10 min increments.The appointment-taking hours of operation start 20 mins(one slot) ahead of the visits and ends 20mins earlier, toallow for the first slot in the day to be filled and for therequests on the last slot of the day to be fulfilled.

Dynamic scheduling & same-day requests

5.3 Call probabilities and workload settings

We assume that no more than one call arrives in any minuteof the 8-h call horizon. The probability of a call arriving inany minute for a physician can either be assumed to beconstant (time homogeneous) or to vary from hour to hour(time dependent). The time-dependent case is clearly morerealistic, as the frequency of calls that a practice gets canvary from hour to hour. To calculate call probabilities by thehour, we conducted a detailed 9-day time study of incomingcalls at a 3-provider family medicine practice. A graphof how call frequencies vary from hour to hour for 5 ofthe 9 days is shown in Fig. 5. The dark line shows theaverage call frequency. There are no calls from 12:30–1:30 pm since the entire practice is closed for lunch.Note that we counted all phone calls to the practice ofwhich same day appointment requests are only a subset.But in the absence of more detailed data, we canassume that patient calls for same-day appointments alsofollow similar trends.

Let Ck be the number of calls observed in hour k in ourtime-study. For an 8-h workday, the total number of calls is

C ¼ ∑8k¼1 C

k . Thus Ck/C gives the proportion of calls re-ceived in hour k of the day. We now explain how we usethese call frequencies to determine the per minute call prob-abilities for each physician.

Suppose physician m has Pm booked prescheduled slotsand Sm available same-day slots (this is the P/S setting for

the physician, which we explain shortly). Let Dm be theaverage daily same-day demand for physician m. The sameday workload for physician m is defined as the ratio ofaverage same-day demand to the available same-day

capacity, expressed as a percentage: Dm=Sm� � � 100%. By

setting Dm we can obtain the desired workload for eachphysician.

To achieve a certain Dm, we need to set the per minutesame-day call probabilities used in the dynamic program.Recall that pn,m denotes the probability that a same-day callarrives for physician m in minute n of the time-horizon. Themean number of same-day requests for physician m in hourk, denoted by Dk,m, is equal to the mean of the sum of 60Bernoulli trials, one for each minute n in hour k. Thereforefor a given hour k, Dk,m=60 pn,m for all n such that ⌈n/60⌉=k. For an 8-h workday, we expect to see Dm same-day

requests. Therefore, Dm ¼ ∑8k¼1 Dk;m. To reflect the varia-

tions from hour to hour (the time dependent arrivals case),we set pn,m values to ensure that for each hour k, the ratio

Dk;m=Dm equals the ratio Ck/C observed in our time-study.The overall practice workload is ∑M

m¼1 Dm

� �� 100=

∑Mm¼1 Sm

� �. To keep our analysis meaningful for practices,

we restrict ourselves to practice workloads of 80 %, 100 %and 120 %. The high workload cases reflect the high demandobserved in US primary care practices.

We consider symmetric practices and asymmetricpractices. In a symmetric practice, all physicians haveidentical workloads. They also have the same number ofbooked prescheduled appointments and available same-day slots. We use P/S to indicate the number of bookedprescheduled slots (P) and available same-day slots (S)for each physician.

For example, we test the 3-physician, 8/16 symmetriccase in which each physician has the same workload(either 80 % or 100 % or 120 %). The 8/16 case impliesthat each physician in the practice expects a larger num-ber of same-day appointments compared to prescheduledslots. This may happen at urgent care centers, wherefamiliarity or continuity is not as important as the needto address acute needs quickly. We also test the 16/8symmetric case, which represents a family medicinepractice that books much of its calendar in advance so

Fig. 5 Call frequencies for thedifferent hours of the day for5 workdays in July and August,2011

H. Balasubramanian et al.

that patients can see their own provider, but also sees theoccasional same-day appointment should an urgent needarise. The family medicine practice we have worked withlies very close to the 16/8 profile.

We also consider asymmetric practices. Asymmetricpractices reflect the reality that some physicians areoverworked or under-worked in relation to others. Seniorphysicians may be more popular and therefore may havelarger panels than physicians who are still in the early yearsof their practice. In the 3-physician asymmetric cases weconsider, one physician is set such that she is overworked inrelation to the overall practice workload; the second will bebalanced, or in other words, her workload will match that ofthe practice; and the third physician will be underworked.Note these non-identical workloads are achieved byadjusting the per minute call probabilities for each physi-cian, which we have discussed earlier. We test both sym-metric and asymmetric 16/8 as well as 8/16 cases under thethree workload settings, 80 %, 100 % and 120 %.

5.4 Non-identical P/S settings

In all the cases discussed above, we have assumed that theratio of prescheduled to same-day slots for each physician isthe same. But in practice, certain physicians, because theytake care of more patients with chronic conditions, mayhave more prescheduled follow-ups and annual exams thanothers. Moreover, many practices use a nurse practitionerwho over time develops a small panel of her own, but whosecalendar is mostly free so that same-day patients of otherphysicians’ panels can be seen. Thus providers can differ intheir P/S settings.

To address this reality, we consider cases in whichPhysicians 1, 2 and 3 have P/S settings of 8/16, 12/12 and16/8, respectively. Each physician may be overworked (O:120 % workload), underworked (U: 80 % workload) or

balanced (B: 100 % workload). We consider practiceswhose configurations are: BOU, BUO, OBU and UBO. Toillustrate, BOU refers to the case where Physician 1 has aworkload of 100 % (B) and hence an expected same-daydemand of 16*1=16; Physician 2’s workload is 120 % (O)and hence has an expected same-day demand of 12*1.2=14.4; and Physician 3’s workload is 80 % and hence has anexpected same-day demand of 8*0.8=6.4. Table 1 summa-rizes all parameters and their possible settings.

6 Results

We now present the results of our computational experi-ments. We follow the settings and notation introduced inthe previous section. The results are divided into two parts.We first demonstrate the impact of the location ofprescheduled appointments. Next we propose simple heu-ristics that practices can use and test the performance ofthese heuristics against the optimal assignment of same-dayrequests provided by the stochastic dynamic program.

6.1 Location of prescheduled appointments

For both the P/S settings of 8/16 for each physician and16/8 for each physician, we compare the 4 prescheduledlocation scenarios under symmetric and asymmetricworkloads. Figure 6 shows the relative percentage lossof each location scenario with regard to the objectivefunction of the dynamic program. For convenience, wecall this objective function revenue, though it also re-flects the level of timely access and continuity providedto same-day patients. All-Morning is used as the reve-nue benchmark. It is the best of the four scenarios sinceit allows the practice to accommodate the most same-day appointments during the 8-h working day.

Table 1 Summary of parameters and their possible settings

S: Capacity of each physician 24 slots per day

M: Number of physicians in practice 3

N: Length of the day 480 min (8 h)

A: Length of an appointment slot 20 min (deterministic)

Prescheduled location scenarios All Morning, all afternoon, 2-Blocks, patient-centered

Per-minute call probabilities for same-day appointments Homogenous; time dependent (from empirical data)

Practice workload settings 80 %, 100 %, 120 %

Workload among physicians Symmetric, asymmetric

P/S settings [16/8, 16/8, 16/8]

[8/16, 8/16, 8/16]

[8/16, 12/12, 16/8]

“Revenue” if a same-day patient is seen by patient’s PCP 1.0

“Revenue” if a same-day patient is seen by unfamiliar physician 0.90

Dynamic scheduling & same-day requests

As expected, the worst of the 4 scenarios is All-Afternoon,since a practice will have to refuse same-day calls that come inthe afternoon (or see them after regular working hours). In the8/16 case, All-Afternoon is around 30 % worse than the All-Morning case, while in the 16/8 case, it is around 60 % worse.Patient-Centered performs relatively better compared to All-Afternoon, yet is around 10% and 20% worse in the 8/16 and16/8 cases respectively compared to All-Morning. Thus, interms of percent losses in revenue, Patient-Centered and All-Afternoon perform worse when the number of prescheduledappointments is higher than the number of available same-dayappointments. This is because in the 16/8 case, with a largernumber of prescheduled appointments, a higher portion of theend of the day is blocked off, causing all of the same-daypatient requests received during that period to go unfulfilled(or seen after regular hours).

The 2-Blocks scenario, which divides the prescheduledappointments between early morning and early afternoon, iswithin 2.5 % of the All-Morning scenario in all cases. Thisis an important result, since it implies that practices canstrike a middle ground between the extremes of All-Morning and All-Afternoon. While early morning and late-afternoon slots may be preferred by patients, a practice canguide, to the extent possible, patients towards slots early inthe afternoon (immediately after lunch) rather than late inthe day. In the 2-Block design, same-day (urgent) requests

received during the early hours, can be scheduled morepromptly than in the All-Morning case, which would resultin increased patient satisfaction.

While Fig. 6 presents revenue results benchmarked inrelation to All-Morning, Table 2 lists the expected valuesof revenue, PCP assignments, referrals to unfamiliar physi-cians, and refusals under various settings. These absolutevalues give a more comprehensive picture of access andcontinuity under the four location policies. Notice, as anexample, that in the 100 % Asymmetric, time-dependent,8/16, All-Morning case, the practice sees about 45 same daypatients in a day (i.e. PCPAssignments+Referrals); in the 2-Blocks case, the practice will see 44 patients. In the Patient-Centered and All-Afternoon cases, the practice will see 41and 31 same-day patients respectively. While All-Afternoonimplies significantly reduced same-day access, the PatientCentered case remains a feasible policy. This is especiallytrue if the practice is willing to incur some overtime. In the100 % Asymmetric, time-dependent case in Table 2, noticethat the expected refusals in the Patient-Centered scenario isaround 7 patients. Refusals can be considered as overtimework for physicians in the practice. If so, then the 7 patientscan be seen after regular work hours by the 3 physicians,resulting in an expected overtime of just over two 20 minslots on average per physician. Using these results, a prac-tice can decide whether such overtime is reasonable enough

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Fig. 6 Percent losses in revenue for the 4 prescheduled location scenarios under symmetric and asymmetric 8/16 and 16/8 cases. All-Morning isthe benchmark scenario, with the highest revenue (0 % loss)

H. Balasubramanian et al.

to accommodate prescheduled patient preferences for timeof day.

Table 2 also reveals another important finding: the All-Morning and 2-Blocks scenarios provide high levels of continu-ity to same-day patients (see PCP assignments and Referralscolumns). Continuity levels are considerably lower for the All-Afternoon case. The Patient-Centered scenario does better onPCPAssignments than the All-Afternoon case, but not as well asAll-Morning and 2-Blocks. Here too, a practice can evaluatewhether allowing time of day preferences for prescheduled ap-pointments is worth the loss of continuity for same-day patients.

When the P/S setting is different for each of the threephysicians (see explanation for BOU, BUO, OBU and UBOcases in the previous section), we find that our findingsremain robust, as seen in Fig. 7. 2-Blocks still remains closein performance (within 2.5 %) to All-Morning. All-Afternoon is significantly worse than the benchmark(around 40 % worse), while Patient-Centered performsaround 10–15 % worse.

6.2 Dynamic assignment: evaluation of heuristics

In primary care practices a decision has to be made withinthe time period of a phone call or a personal conversation (inthe case of walk-ins). While the stochastic dynamic programcan be used to provide guidance on allocation decisions,practices will also rely on heuristic rules that are easy tounderstand and implement. In this section, we first proposethree heuristics and then test their performance against theoptimal values obtained from the stochastic dynamicprogram.

To motivate heuristics for the dynamic assignment prob-lem, it is helpful to recall the competing considerations ascheduler faces. When a call for a same-day appointmentcomes in, the scheduler has to use information from physi-cian calendars: how many same-day slots each physicianhas available and which slots in the day are already bookedwith prescheduled appointments. She must also considerwhether the patient should be scheduled with her own

Table 2 Expected values of revenue, PCP assignments, referrals to unfamiliar physicians, and refusals for the 8/16 symmetric time dependent(STD) and asymmetric time dependent (ATD) cases

Here scenario 1 refers to All-Morning; 2 to 2-Blocks; 3 to All-Afternoon; and 4 to Patient-Centered. In all there are 48 cases

Dynamic scheduling & same-day requests

physician, who might have a slot available at a future pointin the day, or an unfamiliar physician who might have aslot available very soon and which might go idle. Notethat idle slots also mean that fewer same-day patientswill be seen by the practice during regular work hours.The recurring dilemma the scheduler faces is whether toprovide continuity or prevent idle slots. The followingdynamic assignment heuristics are based on enforcingcontinuity, minimizing idle slots and striking a balancebetween the two.

Primary first (PF) In this heuristic, the scheduler assigns anincoming same-day request to the PCP first, so long as thePCP has same-day slots available. While Primary Firstmaximizes continuity, the fact that other physicians mayhave slots that go idle is not considered. If the PCP has nosame-day slots, the scheduler then assigns the request to thephysician with the most slots available.

Most slots (MS) Here the scheduler books appointmentswith the physician who has the most slots available. Thisreduces the number of slots that will go idle but does notconsider continuity. This is a valid strategy for practices,because it allows them to balance the workload of different

physicians and the resulting loss of continuity for acuteconditions typically does not outweigh the need to see aphysician quickly.

Adaptive threshold In Adaptive Threshold, we define apredetermined window (or threshold) that the schedulerlooks ahead. Within this window, the scheduler searchesfor the earliest available slot and assigns the patient to thatslot. The scheduler does not consider whether the slot be-longs to the patient’s PCP or a physician unfamiliar to thepatient. If there is no slot available in the time window, thenthe scheduler uses a PCP-first strategy. If the PCP has noslots available, the scheduler assigns the patient to a physi-cian with the earliest available slot.

The motivation behind Adaptive Threshold is to preventshort term available slots from going idle – and henceallowing the practice to see more same-day patients in aday – but also to maximize continuity whenever possible.The appropriate value of the threshold will differ dependingon the particular setting, workload, asymmetry, etc.

Notice that when the time window is as large as thelength of the workday (8 h), Adaptive Threshold re-duces to a simple earliest slot strategy. That is thescheduler looks for the earliest available slot and

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Fig. 7 Percent differences in revenue for the 4 prescheduled locationscenarios when the P/S settings for the three physicians are 8/16, 12/12and 16/8 and the workloads of each physician in relation to that of the

practice are given by UBO, OBU, BUO and BOU. Same-day requestrates are time-dependent. All-Morning is the benchmark scenario, withthe highest revenue (0 % loss)

H. Balasubramanian et al.

assigns it to the same-day patient – irrespective ofwhether the slot belongs to the patient’s PCP or other-wise. Consequently, the Earliest Slot strategy minimizesthe number of slots that go idle. Setting the window to0, on the other hand, corresponds to the Primary FirstStrategy, which maximizes continuity.

In all the heuristics described above, we still use theassumption that was made in the stochastic dynamic pro-gram: we always assign an incoming same-day request tothe earliest slot of the chosen physician.

Figure 8 shows the percentage losses in revenue for thethree heuristics in the symmetric, time dependent 8/16 caseunder 80 %, 100 % and 120 % workloads. Here the bench-mark (0 % loss) is the optimal value of the stochasticdynamic program. First, we see that all heuristics are within4–6 % of the optimal value. This suggests that a practice’sperformance is much more sensitive to the location ofprescheduled appointments than it is to the allocation poli-cies. The heuristics’ performance declines the most in the120 % All Afternoon case.

However, the allocation policies do differ from each otherin their performance. We find that Adaptive Threshold per-forms consistently better than Primary First and Most Slots.Indeed, except for the 120 % workload case in the All-Afternoon scenario, Adaptive Threshold is within 1 % ofthe optimal. We also see that Primary First performs well inunder-utilized scenarios but its performance declines as the

workload increases. Interestingly, in all except for the lowworkload all morning scenarios, the optimal threshold (orlook-ahead window) for the Adaptive Threshold heuristic isat least 1 slot. That is, there is always a benefit in filling upthat next slot that is at risk of going idle and causingshortages later. Note that in the all-morning case the patientrequests accumulate through the morning making this riskalmost negligible in an underutilized setting. Asymmetry inthe arrival rates is correlated with higher threshold values, asthere is a higher probability of later slots going idle for theunderutilized physician,

The optimal threshold is 1 slot in 54 % of the cases runand never beyond 8 slots (2 h 20mins). In the large majorityof the cases with a high threshold, the gains as more slots areadded to the look-ahead window is negligible. Even in thecase when the optimal look-ahead window is of 8 slots,around 70 % of the gains are obtained with a look-aheadwindow of just one slot.

6.3 Modeling patient same-day slot preference usingsimulation

Recall that in stochastic Dynamic Program (DP), PrimaryFirst (PF), Most Slots (MS) and Adaptive Threshold (AT),we always pick the earliest available slot of the chosenphysician to schedule a same-day request. In reality, same-

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Fig. 8 Percent losses in revenue for the 3 heuristics in the symmetric time dependent (STD) 8/16 case. The benchmark here is the optimal value ofthe stochastic dynamic program

Dynamic scheduling & same-day requests

day patients may also have time of day preferences. Theymay prefer a later slot instead of the earliest available. Suchslot preferences and associated state transitions are too com-plicated to be captured mathematically by the stochasticdynamic program. We therefore used discrete event simula-tion to model the possibility that a patient calling in mayrandomly choose any of the available slots in the day. Thepractice agrees to assign the patient to the slot she chooses.We call this the Random Assignment (RA) heuristic.

Notice, however, that if same-day patients prefer appoint-ments later in the day, they are in effect like short noticeprescheduled appointments. The effect of booking a same-day patient later in the day rather than earlier is the same asthe effect of booking prescheduled appointments later in theday (see All-Afternoon results above). As our results willdemonstrate, there is a tradeoff between allowing same-daypatient preferences and the amount of overtime work neededto satisfy requests not seen during regular work hours.

The discrete event simulation that models RA works asfollows:

1. In each minute of the workday, starting from n=0, thepractice can receive a same-day request, based on the per-minute call probabilities calculated based on our empiricalstudy. The setting is thus identical to that considered in thestochastic dynamic programs and heuristics.

2. If no call arrives, the simulation moves to the nextminute, updating slot availabilities (state of the system)as appropriate.

3. If a call does come, we assume the patient chooses anyof the available same-day slots in the day, with equalprobability, irrespective of whether they belong to herPCP or not. In other words, the patient will not beassigned always to the earliest-available slot of a chosenphysician as in the stochastic dynamic program, butmay pick any available future slot with equal probabil-ity. The practice agrees to this and assigns the patient tothe chosen slot. In the absence of actual data on time-of-day and PCP preferences, such a probability structuremodels the greatest possible variation in patient prefer-ences for same-day slots.

4. After the slot assignment is made, the system perfor-mance measures (revenue, PCP assignments, referrals,and refusals) are updated. The simulation moves to thenext minute and the process is repeated.

We use 50,000 replications of the 8-h workday to esti-mate practice performance measures under RandomAssignment. We test the simulation using the sameprescheduled location scenarios and data inputs that we usedfor the stochastic dynamic program and the heuristics.

Table 3 provides the expected values of revenue, PCPassignments, referrals and refusals for the stochastic

dynamic program (DP), the three heuristics (PF, MS, andAT) and the simulation of random preferences (RA), for thecase in which the 3 physicians have P/S settings of 8/16,12/12 and 16/8, and have identical workloads (each physi-cian has a workload of 100 %). We cluster allmethods/heuristics under the 4 prescheduled location sce-narios. These results are representative of what we found inother experimental settings. Key insights are summarizedbelow:

1. Not unsurprisingly, allowing patients to book same-dayslots later (as in RA) rather than the earliest available(DP and the 3 heuristics) results in significantly totallower revenue, under each of the 4 prescheduled loca-tion scenarios. Lower revenue implies fewer same-daypatients will be seen during regular work hours. Tocompensate, the practice will have to allow for greaterovertime work. See “Refusals” column in abovetable/figure: RA has the highest number of refusals.RA thus provides us with a benchmark on how mucha practice will lose by not guiding same-day requests toearliest available slots. In reality, a practice’s perfor-mance is likely to lie between the DP optimal valueand the RA objective value. The more a practice is ableto guide patients to earlier available slots, the more itsperformance will be closer to the DP optimal value.

2. Even under the random slot assignment rule, theprescheduled scenarios All-Morning and 2-Blocks (reve-nues of 27.08 and 23.64) still perform better than Patient-Centered and All-Afternoon (revenues of 15.27 and21.60). It is worthwhile to note that All-Morning’s per-formance gain becomes significantly greater in this case.

3. Table 3 also provides perspective on how the heuristics,Primary First, Most Slots and Adaptive Threshold, dif-fer from each other.

In terms of revenue, Adaptive Threshold is the best.But the differences in revenue between the heuristics aresmall when compared to the differences in PCP assign-ments and referrals. When it comes to maximizing PCPassignments and minimizing referrals, Primary First isclearly superior. Most Slots, unsurprisingly, has the low-est PCP assignments and the largest number of referrals.In contrast, Adaptive Threshold does much better thanMost Slots in PCP assignments and referrals and thusprovides an adequate balance between timely access andcontinuity.

7 Conclusions, implications for practice, and futureresearch

The implications of our computational experiments areclear. They demonstrate that the location of prescheduled

H. Balasubramanian et al.

appointments has a significant impact on the number ofsame-day patients that can be seen during regular workhours and also on the continuity a practice is able to provide.While locating prescheduled appointments as early as pos-sible during the working day is by far the best strategy, wefound that also offering early afternoon slots (as in 2-Blocks) for prescheduled appointments does not compro-mise a practice’s performance significantly. Our model al-lows practices to quantify the tradeoffs inherent in allowingtime-of-day preferences for prescheduled appointments onthe one hand, and the amount of overtime a practice incursand the continuity it is able to provide to same-day patientson the other. The results for the Patient-Centered locationpolicy illustrate this tradeoff well.

In general, practices should be aware of the need toguide patient choices when appointments for a futureworkday are booked. We do not propose to convincepatients to choose a slot they do not want, but suggestto use any flexibility they may have in their preferences

in order to improve the clinic's performance. By guidingpatients’ choices, it is possible to satisfy both patients'preferences for time slots as well as see more same-daypatients without incurring overtime or refusing appoint-ments. As for same-day patient slot preferences, we found thatassigning an incoming request to the earliest available slot ofthe chosen physician not only makes our model computation-ally tractable, but is a policy that reduces the number ofrequests seen after regular work hours. Simulation of the casewhere same-day patients are allowed to select their appoint-ment times randomly show that it results in ~25 % decrease inperformance.

The results for the dynamic assignment problem suggestthat practices can use simple, easily implementable policiesfor allocating patients as calls arrive during the day and stillachieve near-optimal results. We propose a simple look-ahead strategy, Adaptive Threshold, which makes sure thatslots do not go idle in the short term but also attempts tomaximize PCP matches wherever possible. Adaptive

Table 3 Expected values of revenue, PCP assignments, referrals to unfamiliar physicians, and refusals for a practice with 3 physicians with P/Ssettings of 8/16, 12/12 and 16/8, respectively, and 100 % workload for each physician

The results are organized by the 4 prescheduled location scenarios: All-Morning; 2-Blocks; All-Afternoon; and Patient-Centered. DP indicates thestochastic dynamic program, which provides the optimal allocation of same-day requests and hence forms the baseline; PF indicates Primary First;MS Most Slots; AT Adaptive Threshold (where the threshold or look-ahead window is chosen independently for each scenario to maximizerevenue); and RA is Random Assignment. The “Percent” column indicates fraction relative to the DP optimal value, which is the baseline. Forexample, 99 % indicates the revenue is 0.99 of the Baseline

Dynamic scheduling & same-day requests

Threshold balances timely access and continuity and pro-duces near-optimal same-day assignments under a variety ofcases in our experimental design.

Our model does have some important assumptions.While we consider the optimal assignment of incomingsame-day requests, we assume that the decision on howmany total same-day slots each physician should makeavailable in a workday has already been made. We also donot consider no-show rates for the prescheduled appoint-ments in our model. The no-show rates of the practice weworked with, while lower than the rates quoted in theliterature, were still non-trivial, in the 8–10 % range.However, it is clear where in the day a practice shouldoverbook to counter no-shows. The greater the density ofprescheduled appointments in the combined calendars of thephysicians, the more likely that a no-show will occur. Forexample, in the 2-Blocks scenario, the density ofprescheduled appointments is early in the morning as wellas early afternoon. These are good places to overbookappointments. Naturally, there is a price in terms of addi-tional patient in-clinic waiting if the predicted no-shows donot occur. But capturing this tradeoff has to be the subject ofa different model.

Finally, appointment durations in our model are deter-ministic but they do vary in practice, and especially so inprimary care where patient conditions are diverse. We didnot consider this aspect since our goal was not to capture in-clinic waiting and provider and nurse utilization. An impor-tant direction for future research, therefore, is the integrationof the findings of this paper with the results of a patientsequencing model with uncertain appointment durations,where the objectives are to minimize patient waiting andprovider idle time.

Acknowledgments This work was funded in part by from theNational Science Foundation (NSF CMMI 1031550). Any opinions,findings, and conclusions or recommendations expressed in this mate-rial are those of the authors and do not necessarily reflect the views ofthe National Science Foundation.

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