Dynamic allocation of same-day requests in multi-physician primary care practices in the presence of prescheduled appointments

Transcription

1 Health Care Manag Sci DOI /s Dynamic allocation of same-day requests in multi-physician primary care practices in the presence of 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 of a given workday; and 2) same-day appointments, which are booked as calls come during the workday. The challenge for practices is to provide preferred time slots for prescheduled appointments and yet see as many same-day patients as possible during regular work hours. It is also important, to the extent possible, to match same-day patients with their own providers (so as to maximize continuity of care). In this paper, we present a mathematical framework (a stochastic dynamic program) for same-day patient allocation in multiphysician practices in which calls for same-day appointments come in dynamically over a workday. Allocation decisions have to be made in the presence of prescheduled appointments and without complete demand information. The objective is to maximize a weighted measure that includes the number of same-day patients seen during regular work hours as well as the continuity provided to these patients. Our experimental design is motivated by empirical data we collected at a 3-provider family medicine practice in Massachusetts. Our results show that the location of prescheduled appointments i.e. where in the day these appointments are booked has a significant impact on the number of same-day patients a practice can see during regular work hours, as well as the continuity the practice is able to provide. We find that a 2-Blocks policy which books prescheduled appointments in two clusters early morning and early afternoon works very well. We also provide a simple, easily implementable policy for schedulers to assign H. Balasubramanian (*) : S. Biehl : L. Dai : A. Muriel Department of Mechanical and Industrial Engineering, University of Massachusetts, 160 Governors Drive, Amherst 01003, USA hbalasubraman@ecs.umass.edu incoming same-day requests to appointment slots. Our results show that this policy provides near-optimal same-day assignments in a variety of settings. Keywords Appointment scheduling. primary care. Sameday access. Continuity of care. Stochastic dynamic programming. Heuristics 1 Introduction The two operational measures important for primary care practices are timely access and continuity of care. Timely access refers to the ability of patients to secure an appointment as quickly as possible, while continuity refers to the ability of patients to see their own personal physicians (PCPs) as often as possible. Timely access enables patients to obtain adequate access for acute conditions that might have resulted in costly and unnecessary emergency department visits. Continuity of care allows the patient to develop a long-term relationship with the provider and ensures that the care provided is person centered and holistic. In the ideal case, each patient exclusively sees her own provider; but a dedicated PCP model is rarely possible in practice. A patient s PCP may not be available on the day the appointment is requested. Furthermore, an acute condition requiring a same-day appointment, such as sore throat or a sprained ankle, does not always necessarily require continuity (though it may still be desired), while a patient with long-persisting chronic conditions (such as hypertension, asthma, depression) requires regular prescheduled follow-ups and will highly benefit from seeing her own PCP. Appointments, thus, can be classified into two broad types: same-day appointments (generally for acute conditions) and

2 H. Balasubramanian et al. prescheduled appointments (for annual physical exams, and monitoring and follow-up of chronic conditions). A typical primary care physician s calendar at the beginning of a workday will consist of appointments booked in advance of the day. Slots that are still unfilled can be used to satisfy calls for same-day appointments that arise as the workday progresses. Generally, prescheduled appointments are always seen by the patient s PCP while same day appointments are flexibly shared across physicians. We address two important operational questions relevant for multi-physician primary care practices in this paper. 1) How does the location of prescheduled appointment slots impact the ability of the practice to satisfy same-day demand? By location we mean where in the day these appointments are booked. 2) How should dynamically arriving requests for same-day appointments be assigned to the multiple physicians over the day? We call these two questions the location problem and the dynamic assignment problem. To investigate both questions, we propose a stochastic dynamic program to allocate incoming same-day appointment requests to physicians so as to optimize a weighted measure of timely access and continuity in a multi-physician practice. The ultimate objective is to provide guidance on locating prescheduled appointments and simple allocation rules that will allow practices to improve access and continuity. The dynamic program provides optimal same-day assignments, but may not be considered by users to be practical for daily use. It will allow us to identify simple heuristic rules that can perform well in practice. 2 Key model features and contributions We model a single 8-h workday in a multi-physician practice. Each physician s schedule consists of deterministic 20- min appointment slots. We assume deterministic slot durations because patient wait times in the clinic are not the focus 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 on August 9, 2011 at the family medicine practice we worked with. Prescheduled slots are marked with a dark shade. The unmarked slots are available to schedule same-day patients. Same-day requests specific to each physician arise at different times during the day and calls are likely to be more frequent in certain hours of the day than others. We use empirical data from a time study on call rates conducted over 10 workdays at the family medicine practice to calibrate same-day request rates in our model. Timely access in our model is the number of same-day patients seen during regular hours of a practice (an 8-h workday). Note that our objective does not consider the wait from the time of call to when the appointment is scheduled for each individual same-day request. We consider access to be appropriate so long as patients are seen during the 8 h day. This is reasonable, since urgent requests require same-day access, but are not emergencies, in which case immediate access to an ER would be more appropriate. However, we do assume that urgency of request is sufficient enough for each same-day patient to accept the earliest available same-day appointment once the physician has been decided, irrespective of how close it is to the time the request is made. Patients in our model can get assigned either to their own personal physician, or to a non-pcp physician with a slight penalty. The penalty reflects the cost of not maintaining continuity. Our model also tracks the number of same-day patients who did not get access during regular hours of a workday. These patients are either seen at the cost of overtime or decide to use an emergency room. Given these model features, the location problem implicitly considers time-of-day preferences for prescheduled patients and its impact on same-day access and continuity. For example, prescheduled appointments could be allowed to be scattered throughout the day to allow for patient time-of-day preferences. But our results reveal that such a location policy results in the inability to provide care for a significant number of same-day patients during regular work hours compared to the case where prescheduled requests are Fig. 1 Example of physician calendars at 8 am for a 3- provider practice on August 9, Prescheduled slots are shaded gray; all other slots are available for same-day patients. The treatment of certain conditions and physical exams require two slots; all other patients, including all same-day patients, are assigned to a single slot. Slots are not booked between noon and 1:00, when the entire practice closes for lunch Time Slots Provider 1 Provider 2 Provider 3 8:00 8:20 8:40 9:00 9:20 9:40 10:00 10:20 10:40 11:00 11:20 11:40 Morning Session Time Slots Provider 1 Provider 2 Provider 3 1:00 1:20 1:40 2:00 2:20 2:40 3:00 3:20 3:40 4:00 4:20 4:40 Post-lunch session

3 Dynamic scheduling & same-day requests located in the morning (All-Morning Policy). The All- Morning case works very well because it allows considerable space later in the day to schedule same-day requests received throughout the day. We emphasize, however, that All-Morning is not realistic in practice. We only use it as an optimal benchmark to compare other more practical location policies and to quantify the tradeoffs inherent in accommodating patient time-of-day preferences for prescheduled slots on the one hand, and same-day access and continuity on the other. A key contribution of this paper is a new location policy called 2-Blocks, which does almost as well as All-Morning with regard to both same-day access and continuity, but is less restrictive than All-Morning in that it offers patients two clusters in the day to book prescheduled appointments and accommodate their preferences. Our recommendation for practices is to guide prescheduled patients towards choosing appointment times in those clusters, while still allowing bookings beyond them when needed. Irrespective of what location policy a practice uses, the dynamic assignment problem deals with the allocation of each incoming same-day request to a particular physician and time slot without complete demand information. Here the competing considerations are not allowing an upcoming physician slot to go idle, yet also ensuring that patients see their own physician as much as possible. An idle slot is a lost revenue opportunity for the practice because it reduces the number of same-day patients seen during regular work hours. This cost has to be weighed against the loss of continuity for same day patients that see an unfamiliar physician. Here, our contribution is the Adaptive Threshold (AT) heuristic, which blends both access and continuity in its decision process, and provides near optimal same-day assignments. The AT heuristic works as follows. When a same-day request arrives, the scheduler looks at a predefined time window (say the next hour) to check if any of the physicians has an available slot. If a slot is indeed available, the patient is scheduled in that slot. This minimizes the probability of near term slots going idle and increases the number of patients who obtain access during regular work hours. If no slot is available within the time window, the patient is scheduled with the earliest available slot of her PCP, irrespective of how far this slot is in the day, thus keeping continuity in mind. If the PCP slot is also not available, then the patient is assigned the 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 to primary care. In Section 3 we present the details of the stochastic dynamic program. We then propose an experimental design in Section 4 to explore the impact of prescheduled appointment slot locations and investigate the allocation of same-day appointments to physicians, and present the computational results in Section 5. We discuss our conclusions and the implications for primary care practices in Section 6, and point to opportunities for future research. 3 Literature review Appointment scheduling in healthcare is an active and growing area of research. The adoption of advanced access [11], which promises patients same-day appointments, has prompted a number of questions. How many patients should a primary care physician care for? What impact do no-shows have? How does a practice deal with patient preferences and different appointment types? These questions have necessitated the use of queuing and stochastic optimization approaches that provide guidelines to practices. We restrict this review to appointment scheduling papers as they apply to access and continuity in primary care. Green et al. [6] investigate the link between panel sizes and the probability of overflow (overtime) for a physician under advanced access. They propose a simple probability model that estimates the percentage of workdays a physician will work overtime, as a function of her panel size. The principal message of their work is that for advanced access to work, the supply of physician appointment slots needs to be sufficiently higher than the average patient demand for appointments. This allows a practice to have adequate buffer capacity on days where the patient demand realizations are well above the average. Green and Savin [5] use a queuing model to determine the effect of no-shows on a physician's panel size. They develop analytical queuing expressions that estimate the size of the waiting list (the list of patient appointments scheduled in the future) as a function of panel size and no-show rates. A longer wait list indicates longer times between the date of the appointment request and the date the appointment is scheduled. In their model, no show rates increase as the waiting list grows, since patients who have to wait longer until their appointment date are more likely to not show up. This results in a paradoxical situation where physicians have low 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 considerations are important when it comes to designing physician panels. Case-mix refers to the type of patients (older versus younger; healthy patients versus patients with chronic conditions) in a physician s panel. They propose that in the long term, panels can be redesigned to improve timely access and continuity. Gupta et al. [7] conduct an empirical study of clinics in the Minneapolis metropolitan area that adopted advanced access. They provide statistics on call volumes, backlogs and number of visits with own physician (which measures continuity) and discuss options for increasing capacity at the level of the physician and clinic. Kopach et al. [17] use discrete event simulation to study the effects of clinical characteristics in an advanced

4 H. Balasubramanian et al. access scheduling environment on various performance measures such as continuity and overbooking. One of their primary conclusions is that continuity in care is affected adversely as the fraction of patients on advanced access increases. The authors mention physicians and support staff working in teams as one solution to the problem. Numerous studies have investigated the impact of no-shows and proposed overbooking strategies for single physician clinic sessions. Examples include LaGanga and Lawrence [9], Muthuraman and Lawley [12], and Chakraborty et al. [4]. Robinson and Chen [15] compare the performance of advanced access with a traditional appointment scheduling system. In the advanced access system, a practice has to deal with day to day variability but very few no shows, while in the traditional appointment system, patients book their appointments well in advance with the result that day to day variability is smoothed but patients have a higher probability of no-show. Their numerical analysis reveals that advanced access generally outperforms the traditional appointment system when the objective function is a weighted average of patients' waiting time (lead time to appointment), the doctor's idle time, and the doctor's overtime. Only when the patient waiting time is held in little regard or when the probability of no-show is small does the traditional system work better than the advanced access system. Liu et al. [10] propose new heuristic policies for dynamic scheduling of patient appointments under noshows and cancellations. They find that advanced access works 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 Wang and Gupta [16]. Qu et al [14] consider an essential question for primary care practices: how many prescheduled appointments should a physician plan for in a workday given that the physician also has to see same-day patients? Their model considers different show rates for the two appointment types. They derive conditions under which a solution for the number of prescheduled appointments to reserve is locally optimal. Balasubramanian et al [2] show a stronger result for the single physician problem, guaranteeing global optimality, by first showing that the revenue maximization function has diminishing returns under mild assumptions. They extend this result for a multi-physician practice in which prescheduled demand is always seen by a patient s PCP whereas same-day patients can be seen by any physician in the practice. Thus such a practice is fully-flexible when it comes to same-day patients. For a multi-physician practice, Balasubramanian et al [2] also present a stochastic optimization model to determine the number of prescheduled appointments each physician should plan for and how this number changes depending on how flexible physicians are in seeing same-day patients of other physicians. An important conclusion of their study is that partial flexibility where a same-day patient can be seen by either its PCP or a small subset of the physicians, thus maintaining an acceptable level of continuity comes very close to matching full flexibility with regard to the number of patients a practice is able to see per day. In Qu et al. [14] and Balasubramanian et al. [2], the total capacity of each physician (or physicians) needs to be split between prescheduled and same-day appointments. Since capacity allocation is considered at an aggregate level, where in the day prescheduled appointments are located is not modeled. Furthermore, all same-day demand is realized at once. In this paper, we consider both the location of prescheduled appointments as well as a dynamic call in process for same-day appointments. Gupta and Wang [8] and Wang and Gupta [16] model many of the key elements of a primary care practice. They consider scheduling the workday of a clinic in the presence of 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 day and also for physicians. The objective is to maximize the clinic's revenue, PCP matches, and the number of patients seen by the practice. They use a Markov Decision Process (MDP) model to obtain booking policies that provide limits on when to accept or deny requests for appointments from patients. Our approach resembles the framework of [8] and [16]in a number of ways. Both papers consider a multi-physician practice and model a sequential call-in process, as does our stochastic dynamic program. PCP matches are an important criterion in these papers and this applies to our work as well. The difference is that in [8] and [16], the call in process is for patients requesting for specific time slots on a future workday. In other words, their call in process is for prescheduled appointments. Furthermore, same-day demand in their models is realized instantly at the beginning of the day. In contrast, our call in process is for same-day appointments and occurs over the course of a workday. Allocations have to be made dynamically, without complete demand information, and in a situation where specific time slots in the day are already booked before the workday begins (prescheduled appointments) and the earlier slots turn to be unavailable and may go idle as the day progresses. We are interested in both the number of same-day patients a practice can see as well as the continuity a practice can provide to these patients. We do not model patient preferences explicitly as [8] and [16] do. But by considering the location of prescheduled appointments we are implicitly including patient time-of-day preferences. Our results capture the tradeoff between accommodating prescheduled patient choices for specific time slots and the number of same-

5 Dynamic scheduling & same-day requests day patients that can be seen during regular working hours of a practice. 4 Model formulation and notation We formulate the same-day appointment allocation problem as a probabilistic dynamic program with a time-dependent arrival probability distribution for same-day requests. At any decision epoch, if a same-day request arrives, it is allocated to a physician in the practice or rejected (i.e., that patient will not get a regular appointment for that day, but may be scheduled after hours or be offered to book an appointment for the following day). We assume that the same-day patient accepts being assigned to the earliest available free slot for that physician. To reflect the loss of continuity, we impose a slight penalty on non-pcp assignments. The full details of the model states, actions, transitions and the objective function, along with the notation are described below. 4.1 Input parameters N Number of stages of decision making, i.e. decision epochs. The current stage of the system is denoted by the index n [0,1,,N 1]. We make the number of stages large enough to ensure that at most one patient request arrives on any single stage. A Duration of one single appointment slot (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 patient panels as well. Both physicians and panels are indexed with m [1 M]. According to this notation physician m is the PCP of panel m. p nm Probability of a patient request p n0 ¼ 1 M p nm from panel m arriving in stage n. Probability of no arrivals in stage n. m¼1 8 1 if slot s of physician >< m 0 s calendar is booked with Q sm a prescheduled appointment; >: 0 otherwise: R mj Revenue associated with assigning a patient from panel m to physician j. R mm is the revenue earned when a PCP sees one of his own patients, and a lower revenue 4.2 State variables will be applied when continuity is broken, for any R mj where m j. For n=0,1,2,,n-1 and m=1,2,,m, we define, a n m : available (open) slots for same-day appointments for physician m from stage n to the end of the horizon. These variables characterize the state of the system at stage n and will be updated every decision epoch. The knowledge of the exact placement of the available slots is not necessary since a patient assigned to physician m in stage n will take the earliest of these slots. The initial status of available sameday appointments for each physician is easily calculated from the given matrix of prescheduled appointments: S 1 a 0 m ¼ ð1 Q sm Þ: s¼0 4.3 Decision variables For n=0,1,,n-1, m=1,2,,m and j=0,1,.., M, we let, ( x n mj ¼ assigned to physician j; 1 if a request from panel m in stage n is 0 otherwise: Here, j=0, corresponds to the refusal of the patient s request. The decision space is restricted by the necessary constraint M j¼0 xn mj ¼ 1, that is, an incoming request can be assigned to at most one physician or refused. We reiterate that our model is based on an important assumption. Once the physician is chosen, the model assigns an incoming same-day request to the earliest available slot of the physician, irrespective of how near or far that slot is in the future. To illustrate consider Fig. 2. Asame-day request arrives for Physician 1 at time n. This request can be assigned to any of the M physicians but only to the earliest available slot of each physician. These choices are indicated in the figure. At any decision epoch n, using Q sm (the prescheduled matrix, which is known at the beginning of the day and does not change) and the number of same-day slots a n m physician m has available, we can determine the earliest available same-day slot for physician m. As an example, consider Fig. 3 below. For a particular physician m, at some decision epoch n which falls in slot s=n/a, suppose there are 5 total slots still to come before the end of the day, two of which are prescheduled, but not adjacent to each other. Then if a n m =3 for physician m, knowing Q s+2, m =1 and Q s+4, m =1 (these variables correspond to Slot 2 and Slot 4 in Fig. 3), we can automatically come to the conclusion that

6 H. Balasubramanian et al. Fig. 2 Visual illustration of physician-slot allocation choices at time n when a sameday request arrives for Physician 1. The dark slots correspond to prescheduled appointments; the gray slots to already booked same-day appointments. Slots that are not shaded are available to be assigned Call arrives for Physician 1 at time n Physician 1 Physician 2... Physician M Same-day slots available to be assigned correspond to earliest available for each physician Phy. 2 Phy. 1 Phy. M n. N Current decision epoch One 20-minute Slot; A=20 Slot 1 is the earliest available slot for assigning a same-day patient. However, if a m n =2 then it must be true that Slot 1 has already been booked to accommodate a same-day request that happened in the past; therefore Slot 3 is Physician m searliest available same-day slot. Note that the prescheduled slots can be distributed in any fashion through the day. This logic holds true for all physicians. Thus, the state of the system at stage n is completely described by the values of a m n and Q sm s=0,1,,s-1, for all physicians m=1,2,,m Transition function for physician m Given a current decision in stage n to assign an incoming request to the j th physician, j=0,1,,m (where j=0 would mean that no assignment was made either because no patient arrived or because the patient was refused), the state of physician m would transition to a m n+1 =T(a m n,j) as follows. The transition needs to take into account whether the next decision epoch marks the beginning of the next slot in the schedule. If this next slot is not booked (either by a prescheduled appointment or a previous same-day request), then it will go idle. To capture how this will influence the transition function, we first define a new indicator variable e m s such that ( 1 if slot s has not been booked ðis emptyþ e s m ¼ by the time it starts; in stage n ¼ sa; 0 otherwise: Fig. 3 Parts (a) and (b) show possible system states and earliest available slot for a single physician, m. The dark slots correspond to prescheduled appointments; the gray slots to already booked same-day appointments. Slots that are not shaded are available to be assigned 0 n a m n = 3 Current decision epoch Earliest Available slot Slot 1 Slot 2 Slot 3 Slot 4 Slot 5 N (a) am n = 2 Same-day slot booked Earliest Available slot N 0 n Current decision epoch Slot 1 Slot 2 Slot 3 Slot 4 Slot 5 (b)

7 Dynamic scheduling & same-day requests These variables are easy to calculate. The slot is not booked if there are no initial prescheduled patients assigned to it and, since same day appointments are assigned to the earliest available slots, no same-day appointments have been booked on either slot s or any later slot for physician m. Thus, e s m ¼ 1 iff Q sm ¼ 0 and a sa m ¼ ðs s S 1 Q km k¼s Þ (see Fig. 3). Using these new variables, we are now ready to write the transition function. There are two possible state transitions for physician m s availableslots: 1. No change occurs if no patient is assigned to physician m (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 (e s m =0) and thus 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 stage n (i.e., j m) AND the system is transitioning to thenextslot((n+1) mod A=0) which is empty and goes idle (e (n+1)/a m =1). In summary, we have, a m n+1 =T(a m n, j) where T a n m ; j 8 a n m if ðm jþ AND ðððn þ 1 Þ mod A 0 Þ OR ðððn þ 1Þ mod A ¼ 0Þ AND ðe ð m nþ1þ=a ¼ 0ÞÞÞ; >< ¼ a n m 1ifðm ¼ j Þ OR ð ð m j >: ÞAND ððn þ 1Þ mod A ¼ 0Þ AND ðe ðnþ1þ=a ¼ 1ÞÞ: m Observe that if a same-day patient is assigned to physician m at time n, the next slot s (n+1)/a of that physician cannot go idle. The slot will be assigned to that patient, if not previously booked, since same-day patients are assigned to the next available slot of the physician. As a result, the number of available slots can never decrease by more than one in any transition Objective function and solution approach The objective is to maximize the revenue generated by all assignment decisions, where refused patients generate no revenue and patients seeing an unfamiliar physician yield a lower revenue than those assigned to their PCPs. In other words, the objective is to maximize a weighted measure that includes the number of same day patients seen during regular work hours as well as the continuity provided to these patients. The dynamic optimization problem is solved by the method of backwards recursion. We will use the following vector/matrix and function notation for n=0,1,2,,n-1: a n ¼ a n m m¼1;2; ;M ; xn ¼ x n mj m ¼ 1; 2; ; M j ¼ 0; 1; ; M ; Ta ð n ; jþ ¼ T a n m ; j f n (a n,x n ) f n * (a n ) m¼1;2; ;M for j ¼ 0; 1; ; M: Expected revenue-to-go from stage n until the end of the horizon when the system is in state a n and the assignment decision x n =(x n mj )ismade. This includes the immediately realized revenue in stage n and the revenue gained in the following stages (n+1,,n) given that the optimal decision policy is followed from n+1 onwards. Optimal expected revenue-to-go from stage n until the end of the horizon when the system is in stage n and the optimal decision policy (x n )* is chosen. That is, f nð an Þ ¼ max X nff n ða n ; x n Þg. If the system is in a certain state a n, a request from panel m is received, and the decision vector is x n =(x n mj ), the system will transition to a state a nþ1 ¼ M x n mj Tan ð ; jþ.if no request is received at stage n, then the system will transition to a n+1 =T(a n,0). The recursive equation can be written as follows. f n ða n ; x n Þ ¼ XM p nm m¼1 þ XM m¼1 X M j¼1 p nm f nþ1 x n mj R mj X M j¼0! þ p n0 f nþ1ð Tan ð ; 0ÞÞ j¼0 x n mj Tan ð ; jþ with f nð an Þ ¼ max X nff n ða n ; x n Þg as defined above. Given the current state and decision vector, the first term in the recursive equation captures the expected revenue earned at stage n, while the second and third terms determine the expected revenues to go considering whether or not a patient request arrives at stage n, respectively. According to the assumption that no overtime is possible and that the working day ends at stage N we set f N (a N,x N )= 0 as the starting point of the backward recursion. All feasible state vectors from stage N backwards to stage 0 are evaluated via backward recursion, leading to the initial problem f * 0 (a 0 ).!

8 H. Balasubramanian et al. 4.4 Optimality of the All-morning location scenario Given the above stochastic dynamic program, we now show that the optimal solution to the location problem is 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, where prescheduled appointments are placed in the very first slots of the day, always results in maximum system revenue. Proof Consider an All-Morning location scenario (schedule AM), with prescheduled slot locations specified by the matrix Q sm AM, and any other location scenario (schedule G, for general) with the same number of prescheduled slots, but locations specified by the matrix Q sm G, for s=0,1,, S-1 and m=1,2,,m. We use the superscript AM and G to distinguish between the variables in the two location scenarios. We will show by induction that any assignment x ng of incoming same-day requests to doctors in schedule G is also feasible in schedule AM; that is, we can always construct an assignment x nam in the All-Morning schedule such that if x jm ng =1 then x jm nam =1, for j=1,2,,m, and m=0,1,,m. In what follows, we focus on a single physician m. First observe that, before any same-day demand is assigned, the number of free, future slots available to same-day patients arriving at any time epoch n, with Morning 2Blocks Afternoon Patient centered Slot P1 P2 P3 Slot P1 P2 P3 Slot P1 P2 P3 Slot P1 P2 P Fig. 4 The four location scenarios for prescheduled appointments A(s-1) n<as, is always no lower in schedule AM than in schedule G: a nam m ¼ ðs sþ XS 1 k¼s Q AM km ðs s Þ XS 1 k¼s Q G km ¼ ang m : This is because more of the booked slots with Q km =1 will occur before time slot s in the AM schedule than in any other schedule G. Consider the first time epoch n 1 such that there exists a demand from a certain panel j 1 that is assigned to physician m, that is, x n 1G j 1 m ¼ 1. Since no other same-day appointments have been previously allocated, the number of free future slots in the All-Morning scenario can be no smaller, that is, a n 1AM m a n 1G m as shown above. Thus physician m has slots available to satisfy the demand for that patient in the All-Morning schedule as well, and the incoming patient at time n 1 can be assigned to physician m, i.e. x nam jm =1. The patient is booked in the first open slot of physician m after time epoch n 1 in each of the schedules. In this way, both schedules continue to have the same total number of booked appointments and the open slots in the All-Morning schedule still occur no earlier than those in the general schedule. As a result, it continues to hold that the number of free slots after any time epoch n>n 1 is always no lower in the All- Morning schedule, that is, a nam m a ng m. Assume now that the first k same-day patients, say from panels j 1, j 2, j k, assigned to physician m under schedule G, occurred at epochs n 1, n 2, n k. By the induction assumption, they were also accommodated in the All-Morning setting, so that x n rg jr m ¼ 1 and xn ram j r m ¼ 1 for r=1,2,,k, andresultedin schedules satisfying a nam m a ng m for all n>n k. Consider now the next same-day assignment in schedule G, say from panel j k+1, occurring at time epoch n k+1 such that x n kþ1g j kþ1 m ¼ 1. Since a n kþ1am m a n kþ1g m, we can readily schedule that patient with physician m in the next free slot available in the All-Morning scenario, so that x n kþ1am j kþ1 m ¼ 1 and a m nam a ng m for all n>n k+1. By induction, we have shown that for any demand scenario, the All-Morning schedule can accommodate the same assignments, and thus provide at least the same revenue, as those given by any other schedule. Therefore, the expected revenue of the All-Morning prescheduled location scenario can 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 m to her earliest available slot is optimal, regardless of the prescheduled location scenario: The proof would proceed in a very similar manner as above. Assigning a later slot increases the chance that earlier slots of the physician would go idle; it also reduces available slots

9 Dynamic scheduling & same-day requests (capacity) in the system and therefore may result in lower expected revenue. This is why in the stochastic dynamic program, we always assign to the earliest available slot of the chosen physician m. 2) The optimality of the All-Morning location scenario holds even if the revenues associated with PCP and non-pcp assignments were patient dependent. The theorem above shows that any feasible assignment in a particular location scenario is also feasible in the All-Morning location scenario, and thus would lead to the All-Morning location scenario achieving at least the same total revenue even in the presence of patient-dependent revenues. 5 Experimental design 5.1 Prescheduled location scenarios As discussed earlier, one of the principal goals is to study the impact of the location of prescheduled appointments on the number of same-day appointments a practice can accept during regular working hours, as well as the continuity it can provide to same-day requests. Towards this end, we propose four location scenarios, shown in Fig. 4. In the first, which we call the All-Morning scenario, a practice books all its prescheduled slots in a single cluster first thing in the morning. In All-Afternoon, the opposite is true: the practice schedules all its prescheduled appointments in a single cluster at the end of the day. In the third scenario, 2- Blocks, half the prescheduled slots are scheduled first thing in the morning the other half booked early in the afternoon (right after lunch). The motivation behind 2-Blocks is that it provides greater choice to prescheduled patients (early morning and lunch time are favored slots) while allowing time for sameday patient requests to arrive before the same-day slots. This scenario was suggested by a practice we work closely with. Some patients may still prefer other times, but guiding most patients to these two blocks should provide similar results. Finally, we have the Patient-Centered scenario, in which the prescheduled slots are scattered throughout the day, but nevertheless have greater density in the mornings and late afternoons which we have observed to be the most sought after prescheduled appointment times (working patients, school kids, etc.). We call this scenario patient-centered since the practice allows slots to be booked at any time that the patient prefers. The four scenarios are shown in Fig. 4. We test these four prescheduled scenarios in a detailed experimental setup. Our objective, as discussed earlier, is to maximize a weighted measure of timely access and continuity. Timely access is simply the number of same-day patients seen in a day. Continuity is included by assigning a slight penalty if a same-day request is seen by a physician other than the patient s PCP. If a same-day patient sees her own PCP, the objective function gets the full revenue of 1.0, but if a patient sees a non-pcp physician the revenue is 0.9. The impact of the lack of continuity is difficult to quantify. We have kept the penalty low, at 10 %, since the loss of continuity for a same-day appointment (typically for an acute non-recurring condition) is not clinically significant. This was anecdotally confirmed by the physicians we interacted with. Furthermore, many patients might still prefer to see their PCP, but they are generally also willing to forgo this requirement in return for quick access to an unfamiliar physician. In our model, patients can be refused an appointment. In practice, this may mean that these patients have to be scheduled after the regular working hours of the practice, resulting in overtime. Or, they may be asked to come the next day, and block a slot in the next day s schedule, just as a prescheduled patient. Either way, refusals can be thought of as a cost to the practice and reflect the lack of timely access. 5.2 Practice details All our experiments involve 3 physicians. This practice size is realistic, since 78 % of the group practices in the United States employ 5 physicians or less [3]. Even in large group practices (academic practices for example) physicians are often divided into subgroups of 3 5 physicians. Further, the family medicine practice we have worked closely with for this paper employs 3 providers. On the computational side, the state space of the dynamic program grows significantly with the increase in the number of physicians. The 3- physician case, however, is tractable and we believe the insights we gain from it can be generalized for larger physician cases. Weassumethelengthoftheworkdaytobe8h,or 480 min and will consider each minute as a decision epoch; that is, N=480. Each appointment slot in primary care is typically 20 min, and thus we set A=20. Thus during an 8 h workday a physician will have a total of S=24 appointment slots. Practices also book 40 min prescheduled appointments (for example physical exams and follow up for diabetes patients), but these can be thought of as two 20-min prescheduled appointments scheduled in succession. All same-day appointments are typically 20 min slots, although it may vary significantly from practice to practice; we know of practices that 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, to allow for the first slot in the day to be filled and for the requests on the last slot of the day to be fulfilled.

10 H. Balasubramanian et al. 5.3 Call probabilities and workload settings We assume that no more than one call arrives in any minute of the 8-h call horizon. The probability of a call arriving in any minute for a physician can either be assumed to be constant (time homogeneous) or to vary from hour to hour (time dependent). The time-dependent case is clearly more realistic, as the frequency of calls that a practice gets can vary from hour to hour. To calculate call probabilities by the hour, we conducted a detailed 9-day time study of incoming calls at a 3-provider family medicine practice. A graph of how call frequencies vary from hour to hour for 5 of the 9 days is shown in Fig. 5. The dark line shows the average 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 of which same day appointment requests are only a subset. But in the absence of more detailed data, we can assume that patient calls for same-day appointments also follow similar trends. Let C k be the number of calls observed in hour k in our time-study. For an 8-h workday, the total number of calls is C ¼ 8 k¼1 Ck. Thus C k /C gives the proportion of calls received in hour k of the day. We now explain how we use these call frequencies to determine the per minute call probabilities for each physician. Suppose physician m has P m booked prescheduled slots and S m available same-day slots (this is the P/S setting for the physician, which we explain shortly). Let D m be the average daily same-day demand for physician m. Thesame day workload for physician m is defined as the ratio of average same-day demand to the available same-day capacity, expressed as a percentage: D m =S m 100%. By setting D m we can obtain the desired workload for each physician. To achieve a certain D m, we need to set the per minute same-day call probabilities used in the dynamic program. Recall that p n,m denotes the probability that a same-day call arrives for physician m in minute n of the time-horizon. The mean number of same-day requests for physician m in hour k, denoted by D k,m, is equal to the mean of the sum of 60 Bernoulli trials, one for each minute n in hour k. Therefore for a given hour k, D k,m =60 p n,m for all n such that n/60 = k. For an 8-h workday, we expect to see D m same-day requests. Therefore, D m ¼ 8 k¼1 D k;m. To reflect the variations from hour to hour (the time dependent arrivals case), we set p n,m values to ensure that for each hour k, the ratio D k;m =D m equals the ratio C k /C observed in our time-study. The overall practice workload is M m¼1 D m 100= M m¼1 S m. 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 demand observed in US primary care practices. We consider symmetric practices and asymmetric practices. In a symmetric practice, all physicians have identical workloads. They also have the same number of booked prescheduled appointments and available sameday slots. We use P/S to indicate the number of booked prescheduled slots (P) and available same-day slots (S) for each physician. For example, we test the 3-physician, 8/16 symmetric case in which each physician has the same workload (either 80 % or 100 % or 120 %). The 8/16 case implies that each physician in the practice expects a larger number of same-day appointments compared to prescheduled slots. This may happen at urgent care centers, where familiarity or continuity is not as important as the need to address acute needs quickly. We also test the 16/8 symmetric case, which represents a family medicine practice that books much of its calendar in advance so Fig. 5 Call frequencies for the different hours of the day for 5 workdays in July and August, 2011

11 Dynamic scheduling & same-day requests that patients can see their own provider, but also sees the occasional same-day appointment should an urgent need arise. The family medicine practice we have worked with lies very close to the 16/8 profile. We also consider asymmetric practices. Asymmetric practices reflect the reality that some physicians are overworked or under-worked in relation to others. Senior physicians may be more popular and therefore may have larger panels than physicians who are still in the early years of their practice. In the 3-physician asymmetric cases we consider, one physician is set such that she is overworked in relation to the overall practice workload; the second will be balanced, or in other words, her workload will match that of the practice; and the third physician will be underworked. Note these non-identical workloads are achieved by adjusting the per minute call probabilities for each physician, which we have discussed earlier. We test both symmetric and asymmetric 16/8 as well as 8/16 cases under the three workload settings, 80 %, 100 % and 120 %. 5.4 Non-identical P/S settings In all the cases discussed above, we have assumed that the ratio of prescheduled to same-day slots for each physician is the same. But in practice, certain physicians, because they take care of more patients with chronic conditions, may have more prescheduled follow-ups and annual exams than others. Moreover, many practices use a nurse practitioner who over time develops a small panel of her own, but whose calendar is mostly free so that same-day patients of other physicians panels can be seen. Thus providers can differ in their P/S settings. To address this reality, we consider cases in which Physicians 1, 2 and 3 have P/S settings of 8/16, 12/12 and 16/8, respectively. Each physician may be overworked (O: 120 % workload), underworked (U: 80 % workload) or balanced (B: 100 % workload). We consider practices whose configurations are: BOU, BUO, OBU and UBO. To illustrate, BOU refers to the case where Physician 1 has a workload of 100 % (B) and hence an expected same-day demand 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 an expected same-day demand of 8*0.8=6.4. Table 1 summarizes all parameters and their possible settings. 6 Results We now present the results of our computational experiments. We follow the settings and notation introduced in the previous section. The results are divided into two parts. We first demonstrate the impact of the location of prescheduled appointments. Next we propose simple heuristics that practices can use and test the performance of these heuristics against the optimal assignment of same-day requests provided by the stochastic dynamic program. 6.1 Location of prescheduled appointments For both the P/S settings of 8/16 for each physician and 16/8 for each physician, we compare the 4 prescheduled location scenarios under symmetric and asymmetric workloads. Figure 6 shows the relative percentage loss of each location scenario with regard to the objective function of the dynamic program. For convenience, we call this objective function revenue, though it also reflects the level of timely access and continuity provided to same-day patients. All-Morning is used as the revenue benchmark. It is the best of the four scenarios since it allows the practice to accommodate the most sameday 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 % 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

12 H. Balasubramanian et al. Comparison of revenue Symm. time dependent (8/16) Morning 2 Blocks Afternoon Patient centered Comparison of Revenue Symm. time dependent (16/8) Morning 2 Blocks Afternoon Patient centered Comparison of revenue Asymm. time dependent (8/16) Morning 2 Blocks Afternoon Patient centered Comparison of Revenue Asymm. time dependent (16/8) Morning 2 Blocks Afternoon Patient centered Fig. 6 Percent losses in revenue for the 4 prescheduled location scenarios under symmetric and asymmetric 8/16 and 16/8 cases. All-Morning is the benchmark scenario, with the highest revenue (0 % loss) As expected, the worst of the 4 scenarios is All-Afternoon, since a practice will have to refuse same-day calls that come in the afternoon (or see them after regular working hours). In the 8/16 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 and 16/8 cases respectively compared to All-Morning. Thus, in terms of percent losses in revenue, Patient-Centered and All- Afternoon perform worse when the number of prescheduled appointments is higher than the number of available same-day appointments. This is because in the 16/8 case, with a larger number of prescheduled appointments, a higher portion of the end of the day is blocked off, causing all of the same-day patientrequestsreceivedduringthatperiodtogounfulfilled (or seen after regular hours). The 2-Blocks scenario, which divides the prescheduled appointments between early morning and early afternoon, is within 2.5 % of the All-Morning scenario in all cases. This is an important result, since it implies that practices can strike a middle ground between the extremes of All- Morning and All-Afternoon. While early morning and lateafternoon slots may be preferred by patients, a practice can guide, to the extent possible, patients towards slots early in the afternoon (immediately after lunch) rather than late in the day. In the 2-Block design, same-day (urgent) requests received during the early hours, can be scheduled more promptly than in the All-Morning case, which would result in increased patient satisfaction. While Fig. 6 presents revenue results benchmarked in relation to All-Morning, Table 2 lists the expected values of revenue, PCP assignments, referrals to unfamiliar physicians, and refusals under various settings. These absolute values give a more comprehensive picture of access and continuity under the four location policies. Notice, as an example, that in the 100 % Asymmetric, time-dependent, 8/16, All-Morning case, the practice sees about 45 same day patients in a day (i.e. PCP Assignments+Referrals); in the 2- Blocks case, the practice will see 44 patients. In the Patient- Centered and All-Afternoon cases, the practice will see 41 and 31 same-day patients respectively. While All-Afternoon implies significantly reduced same-day access, the Patient Centered case remains a feasible policy. This is especially true if the practice is willing to incur some overtime. In the 100 % Asymmetric, time-dependent case in Table 2, notice that the expected refusals in the Patient-Centered scenario is around 7 patients. Refusals can be considered as overtime work for physicians in the practice. If so, then the 7 patients can be seen after regular work hours by the 3 physicians, resulting in an expected overtime of just over two 20 min slots on average per physician. Using these results, a practice can decide whether such overtime is reasonable enough

13 Dynamic scheduling & same-day requests to accommodate prescheduled patient preferences for time of day. Table 2 also reveals another important finding: the All- Morning and 2-Blocks scenarios provide high levels of continuity to same-day patients (see PCP assignments and Referrals columns). Continuity levels are considerably lower for the All- Afternoon case. The Patient-Centered scenario does better on PCP Assignments than the All-Afternoon case, but not as well as All-Morning and 2-Blocks. Here too, a practice can evaluate whether allowing time of day preferences for prescheduled appointments is worth the loss of continuity for same-day patients. When the P/S setting is different for each of the three physicians (see explanation for BOU, BUO, OBU and UBO cases in the previous section), we find that our findings remain robust, as seen in Fig Blocks still remains close in performance (within 2.5 %) to All-Morning. All- Afternoon is significantly worse than the benchmark (around 40 % worse), while Patient-Centered performs around % worse. 6.2 Dynamic assignment: evaluation of heuristics In primary care practices a decision has to be made within the time period of a phone call or a personal conversation (in the case of walk-ins). While the stochastic dynamic program can be used to provide guidance on allocation decisions, practices will also rely on heuristic rules that are easy to understand and implement. In this section, we first propose three heuristics and then test their performance against the optimal values obtained from the stochastic dynamic program. To motivate heuristics for the dynamic assignment problem, it is helpful to recall the competing considerations a scheduler faces. When a call for a same-day appointment comes in, the scheduler has to use information from physician calendars: how many same-day slots each physician has available and which slots in the day are already booked with prescheduled appointments. She must also consider whether 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

14 H. Balasubramanian et al. Comparison of Revenue UBO -5.00% % % % % Morning 2 Blocks Afternoon Patient centered Comparison of Revenue -5.00% % % % % OBU Morning 2 Blocks Afternoon Patient centered BUO BOU Comparison of Revenue -5.00% % % % % Morning 2 Blocks Afternoon Patient centered Comparison of Revenue -5.00% % % % % Morning 2 Blocks Afternoon Patient centered Fig. 7 Percent differences in revenue for the 4 prescheduled location scenarios when the P/S settings for the three physicians are 8/16, 12/12 and 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 request rates are time-dependent. All-Morning is the benchmark scenario, with the highest revenue (0 % loss) physician, who might have a slot available at a future point in the day, or an unfamiliar physician who might have a slot available very soon and which might go idle. Note that idle slots also mean that fewer same-day patients will be seen by the practice during regular work hours. The recurring dilemma the scheduler faces is whether to provide continuity or prevent idle slots. The following dynamic assignment heuristics are based on enforcing continuity, minimizing idle slots and striking a balance between the two. Primary first (PF) In this heuristic, the scheduler assigns an incoming same-day request to the PCP first, so long as the PCP has same-day slots available. While Primary First maximizes continuity, the fact that other physicians may have slots that go idle is not considered. If the PCP has no same-day slots, the scheduler then assigns the request to the physician with the most slots available. Most slots (MS) Here the scheduler books appointments with the physician who has the most slots available. This reduces the number of slots that will go idle but does not consider 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 acute conditions typically does not outweigh the need to see a physician quickly. Adaptive threshold In Adaptive Threshold, we define a predetermined window (or threshold) that the scheduler looks ahead. Within this window, the scheduler searches for the earliest available slot and assigns the patient to that slot. The scheduler does not consider whether the slot belongs to the patient s PCP or a physician unfamiliar to the patient. If there is no slot available in the time window, then the scheduler uses a PCP-first strategy. If the PCP has no slots available, the scheduler assigns the patient to a physician with the earliest available slot. The motivation behind Adaptive Threshold is to prevent short term available slots from going idle and hence allowing the practice to see more same-day patients in a day but also to maximize continuity whenever possible. The appropriate value of the threshold will differ depending on the particular setting, workload, asymmetry, etc. Notice that when the time window is as large as the length of the workday (8 h), Adaptive Threshold reduces to a simple earliest slot strategy. That is the scheduler looks for the earliest available slot and

15 Dynamic scheduling & same-day requests assigns it to the same-day patient irrespective of whether the slot belongs to the patient s PCP or otherwise. Consequently, the Earliest Slot strategy minimizes the number of slots that go idle. Setting the window to 0, on the other hand, corresponds to the Primary First Strategy, which maximizes continuity. In all the heuristics described above, we still use the assumption that was made in the stochastic dynamic program: we always assign an incoming same-day request to the earliest slot of the chosen physician. Figure 8 shows the percentage losses in revenue for the three heuristics in the symmetric, time dependent 8/16 case under 80 %, 100 % and 120 % workloads. Here the benchmark (0 % loss) is the optimal value of the stochastic dynamic program. First, we see that all heuristics are within 4 6 % of the optimal value. This suggests that a practice s performance is much more sensitive to the location of prescheduled appointments than it is to the allocation policies. The heuristics performance declines the most in the 120 % All Afternoon case. However, the allocation policies do differ from each other in their performance. We find that Adaptive Threshold performs 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 % of the optimal. We also see that Primary First performs well in under-utilized scenarios but its performance declines as the workload increases. Interestingly, in all except for the low workload all morning scenarios, the optimal threshold (or look-ahead window) for the Adaptive Threshold heuristic is at least 1 slot. That is, there is always a benefit in filling up that next slot that is at risk of going idle and causing shortages later. Note that in the all-morning case the patient requests accumulate through the morning making this risk almost negligible in an underutilized setting. Asymmetry in the arrival rates is correlated with higher threshold values, as there is a higher probability of later slots going idle for the underutilized physician, The optimal threshold is 1 slot in 54 % of the cases run and never beyond 8 slots (2 h 20mins). In the large majority of the cases with a high threshold, the gains as more slots are added to the look-ahead window is negligible. Even in the case when the optimal look-ahead window is of 8 slots, around 70 % of the gains are obtained with a look-ahead window of just one slot. 6.3 Modeling patient same-day slot preference using simulation Recall that in stochastic Dynamic Program (DP), Primary First (PF), Most Slots (MS) and Adaptive Threshold (AT), we always pick the earliest available slot of the chosen physician to schedule a same-day request. In reality, same- All Morning 2-Blocks Revenue Loss in % -1.00% -2.00% -3.00% -4.00% -5.00% Revenue Loss in % -1.00% -2.00% -3.00% -4.00% -5.00% -6.00% PrimFirst MostSlots AdThresh -6.00% PrimFirst MostSlots AdThresh All Afternoon Patient Centered Revenue Loss in % -1.00% -2.00% -3.00% -4.00% -5.00% Revenue Loss in % -1.00% -2.00% -3.00% -4.00% -5.00% -6.00% PrimFirst MostSlots AdThresh -6.00% PrimFirst MostSlots AdThresh 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 of the stochastic dynamic program

16 H. Balasubramanian et al. day patients may also have time of day preferences. They may prefer a later slot instead of the earliest available. Such slot preferences and associated state transitions are too complicated to be captured mathematically by the stochastic dynamic program. We therefore used discrete event simulation to model the possibility that a patient calling in may randomly choose any of the available slots in the day. The practice 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 appointments later in the day, they are in effect like short notice prescheduled appointments. The effect of booking a sameday patient later in the day rather than earlier is the same as the effect of booking prescheduled appointments later in the day (see All-Afternoon results above). As our results will demonstrate, there is a tradeoff between allowing same-day patient preferences and the amount of overtime work needed to satisfy requests not seen during regular work hours. The discrete event simulation that models RA works as follows: 1. In each minute of the workday, starting from n=0, the practice can receive a same-day request, based on the perminute call probabilities calculated based on our empirical study. The setting is thus identical to that considered in the stochastic dynamic programs and heuristics. 2. If no call arrives, the simulation moves to the next minute, updating slot availabilities (state of the system) as appropriate. 3. If a call does come, we assume the patient chooses any of the available same-day slots in the day, with equal probability, irrespective of whether they belong to her PCP or not. In other words, the patient will not be assigned always to the earliest-available slot of a chosen physician as in the stochastic dynamic program, but may pick any available future slot with equal probability. The practice agrees to this and assigns the patient to the chosen slot. In the absence of actual data on time-ofday and PCP preferences, such a probability structure models the greatest possible variation in patient preferences for same-day slots. 4. After the slot assignment is made, the system performance measures (revenue, PCP assignments, referrals, and refusals) are updated. The simulation moves to the next minute and the process is repeated. We use 50,000 replications of the 8-h workday to estimate practice performance measures under Random Assignment. We test the simulation using the same prescheduled location scenarios and data inputs that we used for the stochastic dynamic program and the heuristics. Table 3 provides the expected values of revenue, PCP assignments, referrals and refusals for the stochastic dynamic program (DP), the three heuristics (PF, MS, and AT) and the simulation of random preferences (RA), for the case in which the 3 physicians have P/S settings of 8/16, 12/12 and 16/8, and have identical workloads (each physician has a workload of 100 %). We cluster all methods/heuristics under the 4 prescheduled location scenarios. These results are representative of what we found in other experimental settings. Key insights are summarized below: 1. Not unsurprisingly, allowing patients to book same-day slots later (as in RA) rather than the earliest available (DP and the 3 heuristics) results in significantly total lower revenue, under each of the 4 prescheduled location scenarios. Lower revenue implies fewer same-day patients will be seen during regular work hours. To compensate, the practice will have to allow for greater overtime work. See Refusals column in above table/figure: RA has the highest number of refusals. RA thus provides us with a benchmark on how much a practice will lose by not guiding same-day requests to earliest available slots. In reality, a practice s performance is likely to lie between the DP optimal value and the RA objective value. The more a practice is able to guide patients to earlier available slots, the more its performance will be closer to the DP optimal value. 2. Even under the random slot assignment rule, the prescheduled scenarios All-Morning and 2-Blocks (revenues of and 23.64) still perform better than Patient- Centered and All-Afternoon (revenues of and 21.60). It is worthwhile to note that All-Morning s performance gain becomes significantly greater in this case. 3. Table 3 also provides perspective on how the heuristics, Primary First, Most Slots and Adaptive Threshold, differ from each other. In terms of revenue, Adaptive Threshold is the best. But the differences in revenue between the heuristics are smallwhencomparedtothedifferencesinpcpassignments and referrals. When it comes to maximizing PCP assignments and minimizing referrals, Primary First is clearly superior. Most Slots, unsurprisingly, has the lowest PCP assignments and the largest number of referrals. In contrast, Adaptive Threshold does much better than Most Slots in PCP assignments and referrals and thus provides an adequate balance between timely access and continuity. 7 Conclusions, implications for practice, and future research The implications of our computational experiments are clear. They demonstrate that the location of prescheduled

17 Dynamic scheduling & same-day requests Table 3 Expected values of revenue, PCP assignments, referrals to unfamiliar physicians, and refusals for a practice with 3 physicians with P/S settings 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 the stochastic 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 maximize revenue); and RA is Random Assignment. The Percent column indicates fraction relative to the DP optimal value, which is the baseline. For example, 99 % indicates the revenue is 0.99 of the Baseline appointments has a significant impact on the number of same-day patients that can be seen during regular work hours and also on the continuity a practice is able to provide. While locating prescheduled appointments as early as possible during the working day is by far the best strategy, we found that also offering early afternoon slots (as in 2- Blocks) for prescheduled appointments does not compromise a practice s performance significantly. Our model allows practices to quantify the tradeoffs inherent in allowing time-of-day preferences for prescheduled appointments on the one hand, and the amount of overtime a practice incurs and the continuity it is able to provide to same-day patients on the other. The results for the Patient-Centered location policy illustrate this tradeoff well. In general, practices should be aware of the need to guide patient choices when appointments for a future workday are booked. We do not propose to convince patients to choose a slot they do not want, but suggest to use any flexibility they may have in their preferences in order to improve the clinic's performance. By guiding patients choices, it is possible to satisfy both patients' preferences for time slots as well as see more same-day patients without incurring overtime or refusing appointments. As for same-day patient slot preferences, we found that assigning an incoming request to the earliest available slot of the chosen physician not only makes our model computationally tractable, but is a policy that reduces the number of requests seen after regular work hours. Simulation of the case where same-day patients are allowed to select their appointment times randomly show that it results in ~25 % decrease in performance. The results for the dynamic assignment problem suggest that practices can use simple, easily implementable policies for allocating patients as calls arrive during the day and still achieve near-optimal results. We propose a simple lookahead strategy, Adaptive Threshold, which makes sure that slots do not go idle in the short term but also attempts to maximize PCP matches wherever possible. Adaptive

18 H. Balasubramanian et al. Threshold balances timely access and continuity and produces near-optimal same-day assignments under a variety of cases in our experimental design. Our model does have some important assumptions. While we consider the optimal assignment of incoming same-day requests, we assume that the decision on how many total same-day slots each physician should make available in a workday has already been made. We also do not consider no-show rates for the prescheduled appointments in our model. The no-show rates of the practice we worked with, while lower than the rates quoted in the literature, were still non-trivial, in the 8 10 % range. However, it is clear where in the day a practice should overbook to counter no-shows. The greater the density of prescheduled appointments in the combined calendars of the physicians, the more likely that a no-show will occur. For example, in the 2-Blocks scenario, the density of prescheduled appointments is early in the morning as well as early afternoon. These are good places to overbook appointments. Naturally, there is a price in terms of additional patient in-clinic waiting if the predicted no-shows do not occur. But capturing this tradeoff has to be the subject of a different model. Finally, appointment durations in our model are deterministic but they do vary in practice, and especially so in primary care where patient conditions are diverse. We did not consider this aspect since our goal was not to capture inclinic waiting and provider and nurse utilization. An important direction for future research, therefore, is the integration of the findings of this paper with the results of a patient sequencing model with uncertain appointment durations, where the objectives are to minimize patient waiting and provider idle time. Acknowledgments This work was funded in part by from the National Science Foundation (NSF CMMI ). Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation. References 1. Balasubramanian H, Banerjee R, Denton B, Naessens J, Wood D, Stahl J (2010) Improving clinical access and continuity using physician panel redesign. J Gen Intern Med 25(10): Balasubramanian H, Muriel A, Wang L, (2011) The impact of flexibility and capacity allocation on the performance of primary care practices, online at the Flexible Services and Manufacturing Journal. 3. Bodenheimer T, Pham H (2010) Primary care: current problems and proposed solutions. Health Aff 29(5): Chakraborty S, Muthuraman K, Lawley M (2010) Sequential clinical scheduling with patient no-shows and general service time distributions. IIE Transactions 42(5): Green LV, Savin S (2008) Reducing delays for medical appointments: a queueing approach. Oper Res 56(6): Green LV, Savin S, Murray M (2007) Providing timely access to care: what is the right patient panel size? Jt Comm J Qual Patient Saf 33: Gupta D, Potthoff S, Blowers D, Corlett J (2006) Performance metrics for advanced access. J Healthc Manag 51(4): Gupta D, Wang L (2008) Revenue management for a primary-care clinic in the presence of patient choice. Oper Res 56(3): LaGanga L, Lawrence S (2007) 2007, Clinic overbooking to improve patient access and increase provider productivity. Decision Sciences 38:2 10. Liu N, Ziya S, Kulkarni V (2010) Dynamic scheduling of outpatient appointments under patient no-shows and cancellations. Manuf Serv Oper Manag 12(2): Murray M, Berwick DM (2003) Advanced access: reducing waiting and delays in primary care. J Am Med Assoc 289(8): Muthuraman K, Lawley M (2008) Stochastic overbooking model for outpatient clinical scheduling with no shows. IIE Transactions 40(9): Ozen A, Balasubramanian H (2012) The impact of case mix on timely access to appointments in a primary care group practice. Health Care Manag Sci. doi: /s y 14. Qu X, Rardin R, Williams JAS, Willis D (2007) Matching daily healthcare provider capacity to demand in advanced access scheduling systems. Eur J Oper Res 183(2): Robinson L, Chen R (2010) A comparison of traditional and open access policies for appointment scheduling. Manuf Serv Oper Manag 12(2): Wang W, Gupta D (2011) Adaptive appointment systems with patient preferences. Manuf Serv Oper Manag 13(3): Kopach R, DeLaurentis P, Lawley M, Muthuraman K, Ozsen L, Rardin R, Wan H, Intrevado P, Qu X, Willis D (2007) Effects of clinical characteristics on successful open access scheduling. Health Care Manag Sci 10(2):