Occupancy and Scheduling Models for Inpatient Obstetrics

Transcription

1 DRAFT Submission Under Review for Publication Occupancy and Scheduling Models for Inpatient Obstetrics Mark W. Isken Oakland University School of Business Administration Rochester, MI USA Timothy J. Ward Health Services Engineering, Inc. Cabin John, MD USA Abstract - Increases in the rate of births via scheduled cesarean section and induced labor have led to some challenging scheduling and capacity planning problems for hospital inpatient obstetrical units. We present a patient flow model that allows assessment of the impact of scheduling policies on occupancy variability as well as patient scheduling models for minimizing this variability. Keywords: patient flow, scheduling, obstetrics 1. Introduction Historically, the vast majority of patient arrivals to hospital obstetrical units were initiated when the mother-to-be experienced spontaneous labor. Patient arrivals were essentially random with respect to day of week and time of day and obstetrical bed occupancy could be accurately described by a Poisson process. By estimating the average number of daily patient arrivals and the average patient length of stay postpartum, occupancy of the postpartum unit could be predicted reasonably well. Recently, the Poisson process has become much less accurate at predicting peak postpartum bed requirements. It is not unusual for hospitals to experience weekly census peaks that the Poisson process predicts will occur only two or three times each year. We have worked in several hospital obstetrical units that are experiencing capacity constraints. The most pronounced capacity constraints are accommodating postpartum patients usually on Thursday and Friday. Figure 1 shows several percentiles of post-partum occupancy by time of week for a representative hospital. We call this phenomenon the occupancy whale. Figure 1. The Occupancy Whale

2 What is driving this change and how can operations research models help? In the next section we describe the changes in clinical practice that are one driver of postpartum capacity issues. An analytical patient flow model is presented in Section 3 and shown, in Section 4, to be amenable to embedding in an optimization model for patient scheduling. In Section 5, validation of the patient flow model is done via discrete event simulation. The utility of the models is demonstrated through initial computational experiments and the results presented in Section 6. Finally, conclusions and a road map for future work in this area are discussed in the last section. 2. Changes in Practice The clinical practice of obstetrics has changed significantly in the past few years. What was a process that could be accurately described by random patient arrivals has increasingly become a scheduled event. Recent increases in two clinical practices are particularly noteworthy - scheduled (non-urgent) cesarean deliveries and scheduled (non-urgent) inductions. 2.1 Cesarean Deliveries The cesarean delivery rate has increased from approximately 20% in 1996 to over 30% nationally in 2006 (Hamilton, Marting and Ventura, 2007). In some hospitals, a cesarean delivery rate of 40 percent has become the norm. Cesarean deliveries can be roughly categorized as: Emergent - resulting from complications experienced during trial-of-labor and must be performed immediately (within 20 or 30 minutes of the time the attending physician declares the patient emergent) Urgent - resulting from a troubling or worsening patient condition (usually identified during a lateterm prenatal visit) and must be performed within 24 hours (sometime referred to as add-on cases) Scheduled - resulting from non-urgent condition or event, commonly a repeat cesarean delivery patients. Sometimes performed electively, these cases are scheduled three or more days in advance. 2.2 Induction Significant practice changes have also been introduced regarding trial-of-labor patients. In the past most patients arrived at the hospital sometime after labor spontaneously started. Now, however, many patients arrive at the hospital at a scheduled time. Labor is started or induced after the patient arrives and, in most cases, the patient proceeds thru labor to a normal vaginal birth. However, a significant portion of induced labor patients require emergent cesarean delivery. Thus, there is a correlation between a high induction rate leading to a high emergent cesarean delivery rate leading to a high scheduled repeat cesarean delivery rate for subsequent pregnancies. The number of scheduled inductions in most obstetrical units has increased from near zero in 1980 to more than 20 percent of the overall birth volume in By definition, induction is not performed emergently. Inductions can be categorized as: Urgent - resulting from a troubling or worsening patient condition (usually identified during a lateterm prenatal visit) and must be performed within 24 hours Scheduled - resulting from non-urgent condition or event, often performed for post date pregnancies after 39 to 41 weeks gestation - frequently performed electively for the convenience of the patient or physician. These cases can be scheduled three or more days in advance. Unfortunately, many hospital obstetrical programs do not distinguish between urgent and scheduled inductions. Typically, the attending physician will inform the hospital of the need/desire to perform an induction the day before the requested procedure date. Essentially, every induction is treated as an Urgent 2

3 induction. The hospital has very little control over the number of inductions performed on a given day and a great deal of nursing management time is spent juggling true Urgent from non-urgent induction patients on a given day as beds and staff become available. 2.3 Postpartum Length of Stay Vaginal birth patients have a relatively short postpartum LOS averaging about two days. There are no significant differences in postpartum LOS as a result of labor type (induction or spontaneous labor). So, if the patient gives birth vaginally, the increase in the number of induction patients does not significantly increase postpartum LOS. However, as noted previously, induced patients are more likely to have an emergent cesarean delivery and, thus, a longer postpartum LOS. In addition, the increased induction rate influences the day-of-week distribution of postpartum length of stay. Inductions are scheduled on all weekdays but, physicians generally prefer to schedule induction patients on Mon-Tue-Wed so these patients are discharged from the hospital before the weekend. The early to mid-week scheduled induction patients therefore increase the postpartum census on Thursday and Friday of most weeks. Cesarean delivery patients typically have a long postpartum LOS of nearly four days or about twice the postpartum LOS of vaginal birth patients. There are no significant differences in postpartum LOS as a result of cesarean delivery type (emergent, urgent or scheduled). Physicians often prefer to perform cesarean deliveries on Monday or Tuesday so their patients are discharged from the hospital on Thursday or Friday, before the weekend. Urgent and Scheduled cesarean deliveries are rarely performed on the weekend. Therefore, the increasing number of Scheduled non-emergent cesarean deliveries at the beginning of each week builds the postpartum census from a low on Monday to the peak on Thursday or Friday. The postpartum census decreases dramatically on Saturday and Sunday. Therefore, the increasing cesarean delivery rate has increased postpartum census and the need for postpartum beds. Post-partum unit occupancy is affected by the interaction of multiple patient arrival streams, some random and some scheduled, along with length of stay processes that may have some day of week dependency. Since models for projecting occupancy are quite useful for effective management of obstetrical services, the objective of this work is to build occupancy projection and scheduling models for this new inpatient obstetrical environment. 3. Modeling Gallivan and Utley (2005) developed an infinite capacity, discrete time, stochastic patient flow model for predicting occupancy in a hospital unit subject to demand by a superposition of a random stream of arrivals and one or more scheduled streams of arrivals. Length of stay is modeled with discrete random variables and relatively simple analytical expressions are derived for the mean and variance of occupancy in the unit by patient type. The model assumes cyclically (weekly) repeating schedules and cyclically repeating random arrival processes in order to obtain cyclic steady state occupancy probabilities. The equations are linear in the number of scheduled patients. The linearity results partially from the fact that the model assumes infinite bed capacity. A nice feature of the Gallivan and Utley model (GUM for short) is that length of stay can be dependent on both patient type and arrival time epoch. This allows capturing phenomena such as length of stay inflation due to the stay spanning a weekend when activities in many hospital ancillaries are reduced. With this model it is very easy to explore occupancy impacts of changes in random arrival rates or patient scheduling. Furthermore, a simple normal approximation can be used to estimate occupancy percentiles 3

4 from means and variances produced by the base model. The linear nature of the mean and variance equations also facilitates creation of simple scheduling optimization models. 3.1 Extensions for Modeling Inpatient Obstetric Patient Flow Planning cycle and time bin granularity In order to capture important time of day phenomena in inpatient obstetrics, we consider a planning cycle of one week, where each day is divided into time bins of hours in duration. Typically we use a value of. A planning horizon of weeks is used in the occupancy calculations. The value of should be chosen such that the probability of any unit length of stay exceeding this value is essentially zero. A value of should suffice for obstetrical services. Within a week, there are a total of time bins. Each time bin of each day can be represented by the pair where is the time bin of day and is the day of week, for. As a computational and notational convenience, define the function This function just converts a time bin and day of week pair. to a weekly time bin that ranges from Patient Flow Patterns In our model, patients are classified into one of patient types stemming from three arrival streams of patients random arrival due to onset of labor, scheduled induction, or scheduled cesarean section. The seven patient types and their flow patterns through the various patient care units are summarized in Figure 2 and Figure 3, respectively. In practice, there are alternate configurations of physical capacity for inpatient obstetrics services the use of combined labor/delivery/recovery/postpartum (LDRP) units is one such alternative. The model presented here can easily be adapted for such alternate configurations or alternate patient type classifications. Figure 2. Patient types Figure 3. Flow pattern by patient type Let be the set of patient types who arrive randomly via spontaneous onset of labor, be the set of patient types having scheduled induction of labor, and be those patient 4

5 types having a scheduled cesarean delivery. As in GUM, the component arrival streams for patient types are modeled as a discrete time, non-stationary Poisson arrival process. Denote by the total combined Poisson arrival rate for these patient types in time bin of day, for. For the scheduled patient types, let be the number of elective inductions scheduled for bin, and the number of scheduled cesarean sections. In order to decompose these three arrival streams into seven patient types, define four branching probabilities: probability that a patient arriving in spontaneous labor will receive labor augmentation, probability that a patient arriving in spontaneous labor, and not receiving labor augmentation, will require a cesarean section, probability that a patient arriving in spontaneous labor, and receiving labor augmentation, will require a cesarean section, probability that a patient undergoing induced labor will require a cesarean section. The set of patient care units, U, is. While the original GUM considered a single treatment centre, the problem we faced necessitated extending the model to capture patient flow through a network of units. Patient types 1, 3, and 5 use routing pattern, types 2, 4, and 6 use route and type 7 uses. For each patient type, let be the number of units visited in their route through the system and be the unit visited on the stop in the route for patient type Also, let be the source unit for the stop for patient type, for. For example, Each patient type, unit pair,, can be classified into one of four sets according to whether h corresponds to spontaneous labor (random arrival) or a scheduled arrival and whether the visit to unit u is the initial unit upon admission or is a result of a transfer in from a previously visited unit see TABLE 1. TABLE 1 Classification of patient visits to units Set Description Patient type, unit pairs Random arrival, admit to initial unit (1,1), (2,1), (3,1), (4,1) Random arrival, transfer in (1,2) (1,4) (2,3) (2,2) (2,4) (3,2) (3,4) (4,3) (4,2) (4,4) Scheduled arrival, admit to initial (5,1), (6,1), (7,3) unit Scheduled arrival, transfer in (5,2), (5,4) (6,3), (6,2), (6,4) (7,2), (7,4) 5

6 For each patient type and associated unit to which they arrive to the system, let be the arrival rate to unit for patient type in bin of day, for. For, these are Poisson rates while for, they are deterministic rates stemming from scheduled patients. Using the probabilities defined above, we have (1) (2) (3) (4) (5) (6) (7) Discharge processes Length of stay is modeled with arrival epoch specific discrete empirical persistence distributions (Gallivan and Utley, 2005). Let denote the probability that a patient of type h, arriving on unit u in bin i of day j is still present k periods later. Define the set to be the set of all such that patient type visits unit at some point during their stay. For patient type, LD is the first unit visited and the arrival process of these patients to the Recovery unit is their discharge process from LD. Since these patients have time varying Poisson arrival processes, their discharges processes are also time-varying Poisson processes (Whitt and Massey, 1993). Computation of the discharge rates for such patients follows from the infinite capacity assumption and discrete length of stay distributions. Let be the discharge rate of patient type h from unit u in bin i of day j - understood to be equal to zero for. The cyclical nature of the weekly planning cycle causes a slight notational and computational complication. For example, consider computation of the discharge rate in time bin 3 of day 5 for some patient type. The patients discharged in this period include all patients admitted in this same week prior to this time period with a length of stay resulting in discharge in the time bin 3 of day 5. For patients admitted in the same week in some time bin, we must have and the length of stay equal to. However, discharges in time bin 3 of day 5 can also be the result of patients admitted in previous weeks. For those admitted in the previous week in time bin, we have to consider both the case of in determining which length of stay probabilities to use in the computation of discharge rates. We introduce the following convenience function to handle this complication. Let, 6

7 Then, the discharge rates can be computed by (8) Given the initial unit arrival rates (1)-(7), the discharge rates (8) and the patient routes, the arrival rates can be computed for all,. For example, (arrival rate of patient type 1 to Recovery = discharge rate of patient type 1 from LD) and patient type 1 to Postpartum = discharge rate of patient type 1 from Recovery). (arrival rate of Notice that the arrival process into postpartum consists of a superposition of multiple discharge processes corresponding to different patient types. While the discharge process from labor and delivery for scheduled arrivals is neither deterministic (nor Poisson), as an approximation we assume that the arrival processes for all inter-unit transfers are deterministic for scheduled patient types,. Of course, the model sensitivity to this simplifying approximation requires testing. This approximation along with the model extensions described above allows decomposition of the network of patient care units and analysis of each in isolation using GUM Mean and variance of occupancy Let and be the mean and variance of the number of type patients in unit in time bin i of day j, respectively. A straight forward generalization of Equation 2 in (Gallivan and Utley, 2005) leads to the following expression for for all (9) Note that for negative values of the third subscript in the last term in the summation, the term takes a value of zero. Likewise, generalizing Equation 4 from (Gallivan and Utley, 2005), leads to the following expression for for, scheduled patient types at all unit visited. (10) As mentioned above, all transfer processes for random (spontaneous labor) arrivals are approximated by Poisson processes and thus we have for. The infinite capacity nature of the model allows computation of the overall mean and variance of each unit s occupancy by summing over the patient type specific occupancies given by (9)-(11). (11) 7

8 (12) (13) 4. Scheduling Optimization Several interesting schedule optimization problems can now be posed. Consider the case of trying to schedule a fixed number of induction and cesarean deliveries per week in such a way that the postpartum occupancy levels are smoothed as much as possible across the days of the week. The decision variables are the number of scheduled cesareans, and the number of scheduled inductions, by time bin and day of week. There may be practical and physical limits to the number of such cases done in any particular time bin, day of week, or for the entire week. Similarly, there may be minimum levels for these decision variables. Such constraints can be handled by the use of appropriate upper and lower bounds on the decision variables. Let upper (lower) bound on number of scheduled deliveries in time bin i of day j,, upper (lower) bound on number of scheduled deliveries in day j, upper (lower) bound on number of scheduled deliveries per week, In order to smooth the postpartum occupancy across days of the week, we introduce two, non-negative, dummy variables,. These are used to bound the mean occupancy across the week from above and below, respectively. By minimizing the difference between these bounds, mean occupancy is smoothed across the week. Minimize (minimize the gap between bounds on postpartum occupancy) Subject to (M.1) (define gap to be positive) (M.2) (bound mean occupancy in postpartum (u=4)) (M.3) (daily bounds on scheduled patients) 8

9 , (M.4) (weekly bounds on scheduled patients) (M.5a) Equations (9), (12) (M.5b) Equations (10)-(11), (13) (calculation of mean occupancy by unit) (calculation of variance of occupancy by unit) (M.6) (M.7) Equations (1)-(7) (calculation of discharge rates) (calculation of arrival rates to initial unit) (M.8) (conservation of flow for patient transfers between units) (M.9) and (upper and lower bounds on schedule slots by time bin) This is a standard mixed integer linear program that can be solved easily with any number of solvers. A variant of this model can be used to try to maximize the number of slots for scheduled inductions and cesarean deliveries for a given volume of randomly arriving spontaneous labor patients and a fixed level of bed capacity in each unit. Let be the capacity of patient care unit u. Now, a non-linear constraint can be constructed that will limit the probability that a patient needing a bed in a unit, will find that unit full, using a normal distribution to approximate the occupancy distribution (as done in GUM). The common standardized Z-score can be computed for each unit and time bin via. Adding a constraint to force to be greater than, say, 1.645, would ensure that the overflow probability is less than or equal to The addition of this non-linear constraint complicates the problem and necessitates the use of a mixed integer non-linear solver. We have had very good experience solving this model using the freely available, open source solver, Bonmin ( 9

10 5. Validation via Simulation 5.1 Preliminary Validation Runs In order to assess the impact of various assumptions made in the patient flow model, we developed a discrete event simulation model of the same inpatient obstetrical patient flow system. The specific assumptions we were most concerned about included the discrete time nature of the model, the use of deterministic rates for the flow of scheduled patients between units, and the use of the normal distribution to estimate occupancy percentiles. The model was created using Java and the Java based discrete event library Simkit ( Simkit (Buss, 2002 and Buss and Sanchez, 2002) is freely available, open source and is based on the very general theoretical foundation of event graphs (Schruben, 1983). Our preliminary validation runs have focused the two most important units labor & delivery and the postpartum unit. In Figure 4, we see a comparison of occupancy over the week for the two main patient care units for a representative scenario. The schedule suggested by the optimization model was used as input to the simulation model. Both the mean and 95 th percentile of occupancy match extremely closely for both units Figure 4. Comparison of optimization and simulation models 6. Using the Models The GUM based optimization model has been implemented in the widely used algebraic modeling language, AMPL (Fourer, Gay and Kernighan, 1993). For mixed integer linear variants of the model, we use GLPK as the solver ( For mixed integer non-linear variants, we use Bonmin ( Both solvers are free and open source optimizers. A translator, written in AMPL, converts optimization model inputs and outputs into specially formatted text files which serve as inputs to the Java based simulation model. The pieces fit together as shown in Figure 5. 10

11 Figure 5. Prototype system architecture for model components We envision creating an open source software framework to allow others to use these models to explore their own versions of these problems. A generic representation of a possible future is show in Figure 6. Web browser interface Facility configuration and capacity Birth volume Scheduling policies Input data transformations Simulation model facility, volume and LOS inputs Scheduling optimization inputs Scheduling optimization model Patient flow simulation model Occupancy projections Scheduling parameters Enhanced occupancy projections Utilization and patient flow congestion measures Figure 6. A possible implementation path 7. Experimenting with the Model In order to illustrate how the scheduling model might be used, we constructed the following simple example. Consider a large hospital with approximately 9000 births annually. Of those, 65% are random arrivals of patients in labor, 15% are scheduled cesareans and the rest are scheduled inductions. On a weekly basis this results in 25 scheduled cesareans and 35 scheduled induction patients. 7.1 Scenarios Scenario 1 represents a smooth schedule. Each weekday, 7 inductions are scheduled and 5 cesareans are scheduled. For the inductions, 4 are scheduled in the 8a-12p time bin and 3 in the 1p-4p bin. For the inductions, all 5 are scheduled in the 8a-12p bin. No procedures are scheduled on weekends. In Scenario 2, cases are still only allowed on weekends during the same time bins. However, the optimization model is used to suggest a schedule that minimizes the variability in postpartum occupancy across the days of the week. Upper bounds on the number of daily scheduled cesareans and scheduled 11

12 inductions were each set to 12. Similarly, upper bounds of 8 were placed on the number of scheduled cesareans and scheduled inductions that could be done in any one time bin. Scenario 3 was identical to Scenario 2 except that a small number of cases (3 cesareans and 4 inductions) are allowed on Saturday. Again, the optimization model was used to find a good schedule. 7.2 Results Figure 7 shows the resulting mean occupancy in postpartum for the three scenarios and TABLE 2 shows the associated schedules. In Scenario 1, even though the schedule is uniform across the weekday, the occupancy in postpartum still exhibits high occupancy late in the week (we call this the whale effect due to the shape of the graph) due to the lack of weekend cases. Scenarios 2 and 3 both flatten postpartum occupancy by increasing the number of cases scheduled on Mondays and Fridays while reducing the number on Tuesdays and Wednesdays. Scenario 3 illustrates the potential, in this example, for a small number of Saturday cases to further reduce postpartum occupancy variability across the days of the week. TABLE 2 Schedules by scenario Sch edu led In du ctio ns Sched u led C-Sectio n s S cen ario Time S M T W T F S S M T W T F S a-12p p a-12p p a-12p p Figure 7. Average postpartum occupancy 12

13 8. Conclusions We have developed an analytical patient flow model for analyzing hospital inpatient obstetrics services. The model is an extended version of a model originally developed by Gallivan and Utley (2005). Our extensions include approximations for handling flow of patients within a network of patient care units as well as modeling time of day in addition to day of week. The form of this model makes it well suited to embedding in optimization based scheduling models. Preliminary experimental results show that the models can be solved with widely available solvers and that these scheduling models can be used to smooth daily variability in the postpartum unit. Currently we are in the process of piloting the model at several hospitals as well as working on the underlying software architecture for eventual release as an open source project. REFERENCES: Buss, A. (2002). Component-based simulation modeling using Simkit. Proceedings of the 2002 Winter Simulation Conference, ed., E. Yücesan, C.-H. Chen, J. L. Snowdon, and J. M. Charnes. Institute of Electrical and Electronics Engineers, Piscataway, New Jersey. Buss, Arnold H. and Paul J. Sánchez. (2002) Building complex models with LEGOs (listener event graph objects). Proceedings of the 2002 Winter Simulation Conference, E. Yücesan, C.-H. Chen, J. L. Snowdon,and J. M. Charnes, eds. Fourer, R., Gay, D.M., and Kernighan, B.W. (1993) AMPL: A Modeling Language for Mathematical Programming. San Francisco: The Scientific Press. Gallivan, S. and Utley, M. (2005) Modelling admissions booking of elective in-patients into a treatment centre. IMA Journal of Management Mathematics, 16: Hamilton, B.E., Martin J.A., and Ventura S.H. (2007) Births: Preliminary data for National vital statistics report; 56(7), Hyattsville, Maryland: National Center for Health Statistics. Whitt, W. and Massey, W.A. (1993), Networks of infinite-server queues with non-stationary Poisson input. Queueing Systems, 13: Schruben, L. (1983) Simulation modeling with event graphs. Communications of the ACM. 26: