Implementation of a Model for Census Prediction and Control By Ralph W. Swain, Kerry E. Kilpatrick, and John J. Marsh III A model is described that predicts hospital census and computes, for each day, the number of elective admissions that will maximze the census over the short run, subject to constraints on the probability of overflow. Where a computer is available the model provides detailed predictions of census in units as small as 10 beds; used with manual computation the model allows production of tables of the recommended numbers of elective admissions to the hospital as a whole. The model has been tested in five hospitals and is part of the admissions system in two of them; implementation is described, and the results obtained are discussed.
Hospital census has a major effect on revenues, costs, and operating efficiency. Low occupancy results in high costs per patient day and can lead to reduction of staff and other resources; extremely high occupancy can strain hospital resources and lead to reduced quality of care. Even variability in census, if it is not anticipated, can cause problems in matching patient needs with hospital resources. Ancillary services such as laboratory, radiology, and pharmacy are especially vulnerable to fluctuations in census because work load in such services is a function not only of total census but also of changes in census; hence a small change in census can cause a large change in work load. The importance of census control is affirmed by the large volume of literature on the problem. Milsum, Turban, and Vertinsky [1] have provided a partial review of this literature; Shonick [2] and Shonick and Jackson [3] are also excellent sources. The basic issue addressed by census-control models is the trade-off of average occupancy against excess occupancy, patient rejection, or excessive delays before admission. In nearly all of the models proposed for this problem, a length-of-stay pattern is implicitly assumed that, together with the pattern of requests for admission, allows determination of the distribution of occupancy or of its expected value and variance. A number of models [2-6] assume a Poisson arrival process and a negative exponential length-of-stay distribution to facilitate queueing theory analysis. Other models, e.g., refs. 7-14, make somewhat less restrictive assumptions. From the viewpoint of use, these models can be separated into two HEALTH
SERVICES RESEARCH
380
Address communications and requests for reprints to Ralph W. Swain, Associate
Health Research Division, Box J-177, J. Hillis Miller Health Director, Center, University Systems of Florida, Gainesville, FL 32610. Kerry E. Kilpatrick is Director of the Health Systems Research Division, University of Florida. John J. Marsh III is Hospital Management Engineer at the Ohio State University Hospitals, Columbus.
basic types, aimed at long-term planning of the bed complement or at CENSUS PREDICTION short-term planning of elective admissions within the constraints of a AND CONTROL fixed bed supply (sometimes with fixed allocation of beds to surgical and medical services). In this article we describe a model of the second type, intended to predict census in the short run on the basis of historical data, data on the current hospital census, and elective admissions already scheduled. The simplicity of the model permits it to be adapted to the hospital's needs, data availability, and computing capability. The basic objectives of our model are similar to those suggested by Hancock et al. [15]. Their model, however, uses computer simulation to determine an acceptable fixed schedule of daily quotas for various types of admissions, without provision for modifying these quotas to allow for census variation or changes in the statistical properties of the system. Our model, based on the work of Swain [16], explicitly accounts for actual variations in census and elective reservations and allows adaptive modification of the parameters of the underlying length-of-stay distribution and rate of unscheduled arrivals. Although our model is similar to those proposed by Jewell and Fox [11] and Rubenstein [17], the results reported here provide the first detailed information relative to the implementation and evaluation of this type of model for census prediction and control.
Model Development
Starting with a given census on day t, one can predict the census on a future day t + j if one can predict elective and emergency admissions and discharges that will occur over the intervening days plus the number of patients present on day t who will still be in the hospital on day t + j. In making such a prediction, one must consider both the value being predicted and the accuracy associated with the prediction. Given the many potential sources of variation in the census, it is clear that the actual census will not be equal to the predicted census in most cases. Instead, the actual value will tend to vary around the predicted value. To account for this inherent variation, our model predicts both an expected census value and a range about the value such that there is a high probability that the actual census will fall within that range. Given the census at midnight (i.e., 2400 hr) on day t, the expected census on day t + j may be expressed as
C1=N,+Ej+ Uj
(1)
where C1 = expected census on day t + j, predicted on day t N1 = current patients, present on day t, who are predicted to be present on day t + j = elective admissions (that is, minus discharges of future net Ei elective patients) during the next j days U1 = net expected unscheduled (emergency plus urgent) admissions (that is, minus discharges of future unscheduled patients) during the next j days
WINTER 1977
381
SWAIN, Prediction of C1 and its probable range requires estimation of the KILPATRICK, means and variances of Nj, Ej, and U1. For the moment, we shall sup& MARSH pose that census is being predicted for some well-defined set of patients (e.g., in a particular service or nursing unit or in the total hospital) and that all of the model parameters refer to that set. Current Patients Current patients are classified by number of complete days they have already stayed in the hospital; the number of patients who have stayed T complete days at the time of the current census is denoted by n2. If the stays of patients in the unit are independent, then for any distribution of length of stay (LOS) with finite variance, the number of current patients with LOS equal to T who will be present after j more days will have a binomial distribution with a mean of nTPI.j, where Pr, is the probability that a patient with a current stay of T days will still be present on day t + j. Induding all values of the currently completed stay T, the expected Nj of current patients who will still be present after j more days is max T
= EnPpj N1 T=O
and the variance of Nj is max T
V2(Nf) = £ fTP1j( -PTI) Elective Admissions To avoid a second term and an additional symbol in the variance expression, we assume that all those scheduled for elective admission actually enter the hospital. Patients to be admitted in the future are considered to have a stay to date of zero; hence the probability that such a patient, admitted on day k in the future, will still be present on day j is Po,.-, for j a k. We also assume that admissions for day k occur before the census on day k is taken. Of the ek elective patients admitted on day k, the number expected to still be present in the census for day j is ekPo,j-1 for j a k. Thus, considering all elective admissions between the current time and the census at the end of day j, the mean net contribution to the census from elective admissions is
E=i ekPowj-k and the variance of E1 is
V2(EI) =
ekPo,1-k (
-
Po,-k)
HEALTH Unscheduled Admisions
RESEARCH
382
The census contribution from unscheduled admissions is handled
in the same manner as elective admissions, except that an additional variance term is used. This term expresses the variance due to un-
scheduled admissions on day j and is in addition to the variance due CENSUS to unscheduled patients admitted earlier who are still present on PREDICTION day j. The expected contribution to the census on day j from emergency AND CONTROL admissions on day k only is utPo,jk for j > k, and the total contribution over all days up to j is
Uj =
UkPo,j-
The variance of us is then
P(U1) =
[UkPo,-k (
-
Po,lk) +
A J-4]
where Qk2 is the variance of the number of unscheduled admissions on day k only. Total Census With the foregoing definitions for N1, E1, and U1 in Eq. 1, the expected value of the total census on day t + j can be expressed as c
F T=O
L uk(Po,-k) nTPTI + k= ek(Po,-k) + k=l
=m
(2)
The variance of C1 is then max
-
T
£ hnrPTi(I V2(C)= T=O
-
PT;) +
ekPo,-k(
-Po,-k)
+ [ukPo,,k(l(I PoP-t)+ (Po,rk)2] lc=1
(3)
The Confidence Interval The most natural approach to a confidence interval for the predicted census is to assume a distribution for the actual census, given information about the current census and expected admissions. For patient units of a reasonable size one can invoke the central limit theorem and assume that the actual census will be normally distributed about Cj with a variance of V2(C1). This assumption of normality implies that the prediction error is normally distributed about the predicted census value, which is not inconsistent with a Poisson distribution of long-term census or some of the other forms reported [2,3,9,18]. In this case, the e-level confidence interval for the actual census is the range from C1 - Z.V(C1) to C1 + Z.V(C1), where Za is the standard normal random variable such that 'I(ZQ) = 1 - a/2 (4 is the standard normal cumulative distribution function and a is the desired level of confidence) and V(C,) is the square root of V2(C,). In large patient units the assumption of normality for the distribution of census is acceptable, but in units with 10 or fewer beds it can lead to significant errors. A more conservative interval can be obtained by employing the Chebyshev WINTER inequality, which leads to a confidence interval ranging from C1 1977 -[V2(Cj)la]r to C1 + [V2(C,)/a]%. This interval will be at least twice 383 the other in most cases.
SWAIN, KILPATRICK, & MARSH
HEALTH SERVICES RESEARCH
384
Data Requirements and Computation Alternatives The Patient Unit The choice of the smallest patient unit for which predictions are to be made influences the method used to calculate the values of Prj. As the unit becomes smaller, more detailed data are needed to obtain reasonable accuracy in the predicted occupancy levels. Some trade-off must usually be made between the definition of the patient unit and the information available in the hospital. For example, admission diagnosis offers an appealing method of categorizing patients, but available data are often related to admissions to nursing units or particular medical or surgical services or to discharge diagnoses rather than admission diagnoses. In some cases data are in such inconvenient forms that only total hospital occupancy can be predicted; in other cases data are available to support virtually any definition of patient units. An extremely detailed definition of patient units increases the computational requirements in terms of both computation time and storage space for data. In systems with detailed, easily accessible data we have attempted to deal only with units of 10 or more beds. In hospitals that maintain data only on average LOS, it is normally necessary to use the entire hospital as the basic unit.
Calculation of P2j The calculation of PTj depends on the data available, the ease with which the data can be used, and the set of factors that are chosen to derive information about the future stay of a patient. The minimal data situation usually occurs in a hospital with no internal computing system and no commercial computing service for its billing process. If, in addition, the hospital does not participate in the Professional Activity Study (PAS), there may be no practical method of obtaining the probability distribution of LOS in that hospital. If it is assumed, however, that future stay is independent of T, average LOS can be used to obtain values for PTJ* Under this assumption LOS has a geometric distribution and the probability of staying at least j more days is (1 - py, where p is the probability of discharge. The value of p can be found by equating the expected LOS, L, to the inverse of the discharge probability: L = 1/p. Thus the discharge probability is the reciprocal of L, and PTj = (1 - I/L)i. The assumption of a geometric distribution of LOS implies that PTj is not a function of T; therefore under this assumption the population does not have to be categorized by stay to date. Although this procedure is extremely simple, it yields impressive results when used to predict total hospital occupancy. A closely related approach is to assume that additional stay is independent of stay to date but is related to the particular day of the week that follows. Under this assumption the discharge probability for each day of the week is calculated by taking the reciprocal of average LOS times the relative discharge rate associated with each day of the week. These latter values can be determined by calculating the average fraction of the previous day's midnight census that remains at the next
midnight census: (ct+l - at+i)/ct, where at+, is admissions on day t + 1 CENSUS and c denotes observed census. If these values are organized by day of PREDICTION the week and normalized so that their average is one, they can be multi- AND CONTROL plied by the average discharge probability to obtain discharge probability by day of the week. If data on LOS distribution are available one can use knowledge of T. When such information is available it is usually also possible to divide the total hospital population into units based on medical or surgical services. Either PAS data summaries or tapes from the hospital billing system can provide data on the LOS distribution. Let F(T) be the cumulative distribution function for T; if F(T) is known for a given service,
PT,=P[L>(T+1)IL> T] = P[L > (T + 1)]/P(L > T) [1 -F(T + 1)]/[1-F(T)] =
where L is expected LOS, as previously.
If the data indicate the day of the week when each patient was admitted, then the cumulative distribution of LOS can be based on the day of the week as well. It has been our experience that in some settings the day on which a patient was admitted provides considerable information that can be used in census prediction. It is also possible to make use of adjusted physician estimates, but such estimates have not always proven reliable [19,20]. Other factors that may be useful are patient age, sex, and financial status, transfers from other facilities, attending physician, and the like, but as a general rule added explanatory factors cause large increases in the space required to store the resulting PTj values and in the calculation required to handle the more detailed data. Unscheduled Admissions
Unscheduled admissions include several different categories of admissions. Patients admitted after receiving care in the emergency room normally require immediate admission. Patients in the urgent category are usually admitted within 24 to 48 hours of the admission request. In addition, some elective patients are admitted on the day the request is made. In our system the first two groups are induded in the unscheduled category, but the latter group is not, because those admissions could normally be delayed if necessary. One of the objectives of a census prediction system is to reduce uncertainty in the future census, and scheduling as many patients as possible electively contributes to this goal. The final definition of unscheduled admissions that is used must be consistent with the admission policy of the particular institution. No matter how the unscheduled admissions are defined, weekday variations in such admissions must be accounted for. Both the mean rate and the variance of emergency admissions vary significantly by day of the week, and the variance in emergency admissions usually represents a large portion of the total variance in the census.
WINTER 1977
385
SWAIN, KILPATRICK, & MARSH
Prediction and Control
The framework of the basic census-prediction model allows a number of possibilities for either census prediction or census control or both. The prediction process is based on Eqs. 2 and 3 with a suitable confidence interval and the particular set of values for PTJ associated with the hospital. The control process determines fk, the pattern of additional elective admissions for day k that will maximize the predicted patient days in the hospital over the next m days, subject to a constraint on the probability that occupancy will be greater than the number of available beds. This problem may be expressed as: Maximize S
lc1 us=k
fk PO,n-k
subject to
I~~ C, + L fkPO,J-k + Zc.
386
(4)
where B is the number of beds available on the unit being controlled and Za is the standard normal random variable defined previously. This problem can be solved heuristically by finding the maximum value of fi that satisfies all m constraints (i.e., for j = 1, . . . , m) when f2, . . , fm are all zero, then taking that value of fi as a constant and finding the maximum f2 that satisfies the constraints for j = 2, ...., m when f, . . . fi are all zero, and then repeating the process for f8, and so on. This process can be further simplified by recognizing that the standard deviation is virtually independent of the values of f in most problems, and that if constraint j limits the value of fk, then all fA such that i = k + 1, k + 2, . . .j will have a value of zero. The formulation given in Eq. 4 implies that the primary objectives are a high occupancy level and avoidance of crowding to the point that patients with reservations must be turned away. These are only two of the objectives of admissions control [1]; others include the choice of a convenient day for the patient or the physician, avoidance of an unusual case mix, and the like. Because of these other objectives, the actual pattern of admissions granted during a 24-hour period does not usually match the pattern generated by the solution of Eq. 4. The computed pattern does, however, provide one feasible admissions plan that meets two of the major objectives. Admissions personnel can make good use of this information in the process of granting reservations. In a recent attempt to make the computed pattern of admissions correspond more closely to physician desires, we adapted Eq. 4 by adding absolute limits on the total number of elective admissions permitted on some days (particularly weekends) and by making the effective bed capacity, B, a function of the day of the week. This forces a lower occupancy on weekends, when nursing staff is normally smallest. ,
HEALTH SERVICES RESEARCH
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Equation 4 will automatically honor any preexisting reservations unless a given admission violates the constraint on occupancy. The concerns of the patient can be given more weight by using Eq. 4 to determine the set of dates for which admissions would be feasible in terms of both bed capacity and admission level desired for each day of the week. The actual output of the prediction system is shown in the figure on p. 387. Predicted mean values, together with the upper and lower limits of the confidence interval, are ordinarily provided for a period of 7 to 14 days, depending on the needs of the institution. With the assumption that the actual census will be normally distributed about the predicted value, it is also possible to calculate the probability that the census on each day will exceed the capacity of the unit, given current commitments. The printout includes the number of additional elective admissions that can be accepted by the unit with a reasonably low probability of overflow together with the number of currently scheduled elective admissions and the current probability of census overload. If Eq. 4 is used to determine the permissible number of additional elective admissions, the result can be used as a set of suggested levels to be achieved during the next one or two weeks. As reservations for additional admissions are granted they are added to the proper ek values; when the process is repeated for the next day these added admissions are included in computation of the next set of permissible elective admissions. If the system uses stay to date and the day of the week when admission occurred, a computer is needed both to gather data about current census and elective admission requirements and to calculate the mean and variance of the future census. In hospitals with computerized master patient files, the process of surveying the census, predicting the future census, and suggesting additional elective admissions can be incorporated into the midnight census process. If the file is manual, a tabular system can be used to summarize the current census and timesharing can be used to compute the predictions. If no computing facilities are available only the simplest model, not using stays to date, is feasible.
Application Results
HEALTH SERVICES RESEARCH
388
The complete census-prediction system has been in use since April 1975 at the Ohio State University Hospitals and for a shorter time (on an experimental basis) in four hospitals in Gainesville, Florida. At Ohio State a total of 679 beds are monitored by the system and predictions are made for 22 separate medical and surgical services as well as for the occupancy of all 679 beds. The system is integrated with the hospital's admission, discharge, and transfer system. All data required for the system are maintained in on-line files, so the actual operation of the system is fully automatic and no manual data collection is required. Census predictions and suggested additional admissions are supplied as part of the midnight census reports, and a
graphic presentation of the predicted occupancy is provided to the CENSUS PREDICTION director of admitting. AND CONTROL of a variety different system a such of utility the In evaluating factors must be considered, including the cost of implementing and operating the prediction system, the accuracy of the predictions, the effect on hospital occupancy and overflow conditions, and the effect on convenience for both patients and physicians. Developing the programs for system operation required about six man-weeks for the Ohio State installation and also for the University of Florida installation. In future installations development time could be reduced by use of one of these program packages. The daily calculations require less than 10 seconds in most situations and cost less than $10 per day. Table 1 indicates the accuracy of the Ohio State system for a 60-day period in the first year of operation; it shows the percentage of days on which the actual census was outside the confidence intervals calculated from one to seven days in advance. Several important points are implicit in these data. It is dear that the accuracy of the model varies with the service being predicted and with the number of days in the prediction horizon. In general, accuracy decreased as the prediction Table 1. Frequency of Prediction Error: Percentage of Days in 60-day Trial on Whic Actual Census was Outside Predicted 90-percent-Confidence Interval, for Predictions Made One to Seven Days in Advance Prediction lead time, days Service census predicted
H
1
L
H
2
L
H
3
L
H
4
L
H
5
L
H
6
L
H
7
L
0 0 0 0 0 0 0 1.6 1.6 0 1.6 5.0 0 3.3 16.6 1.6 13.3 1.6 10.0 0 8.3 15.0 11.6 11.6 8.3 8.3 3.3 1.6 8.3 0 8.3 5.0 8.3 6.6 5.0 5.0 10.0 8.3 11.6 6.6 10.0 1.6 6.6 3.3 10.0 1.6 11.6 0 10.0 0 10.0 0 18.3 0 23.3 0 21.6 0 28.3 0 1.6 8.3 5.0 13.3 3.3 15.0 1.6 10.0 0 10.0 11.6 10.0 11.6 8.3 11.6 5.0 11.6 3.3 10.0 0 16.6 16.6 18.3 20.0 16.6 15.0 13.3 5.0 10.0 5.0 0 18.3 0 20.0 0 20.0 0 16.6 0 15.0 5.0 8.3 10.0 5.0 8.3 1.6 10.0 3.3 8.3 0 23.3 5.0 25.0 3.3 26.6 0 21.6 0 28.3 0 6.6 8.3 3.3 8.3 6.6 8.3 3.3 10.0 1.6 11.6 0 23.3 23.3 23.3 20.0 21.6 15.0 13.3 13.3 8.3 8.3 11.6 16.6 28.3 21.6 23.3 21.6 28.3 13.3 21.6 6.6 6.6 10.0 10.0 66.0 11.6 53.3 8.3 50.0 8.3 43.3 0 36.6 8.3 EYEt* ........0 10.0 20.0 13.3 18.3 5.0 15.0 1.6 5.0 1.6 0 10.0 D/S .......... 6.6 5.0 30.0 5.0 31.6 6.6 23.3 5.0 13.3 6.6 8.3 3.3 SURGt ....... 5.0 3.3 5.0 3.3 8.3 6.6 5.0 21.6 3.3 6.6 0 8.3 T/St* ........ S. 0 6.6 18.3 6.6 20.0 8.3 18.3 3.3 13.3 1.6 8.3 5.0 GU ......... 5.0 3.3 43.3 5.0 43.3 0 43.3 0 43.3 0 51.6 0 GYNt* ....... 1.6 * H indicates actual census above upper confidence interval limit; L indicates actual census below
S. 0 1.6 3.3 5.0 8.3 11.6 3.3 1.6 5.0 1.6 8.3 11.6 5.0 P/S* ......... 6.6 ENTt ........ 6.6
INF ......... CRC ......... ENDOt ...... MEDt ....... HEMt ...... CARt ........ REN ......... GE ......... DERM* ...... A/P* ......... NEU* ........ N/St* .. ORTI .......
5.0 3.3 6.6 8.3 5.0 3.3 8.3 0.5 8.3 10.0 11.6 6.6 8.3 13.3 25.0 53.3 11.6 10.0 8.3 10.0 31.6
3.3 3.3 8.3 5.0 6.6 8.3 5.0 10.0 11.6 16.0 8.3 20.0
1.6 16.6 11.6 11.6 3.3 0 8.3 10.0 15.0 16.6 6.6 1.6 6.6 16.6 33.0 68.3 16.6 28.3 15.0 16.6 40.0
lower limit. t Error at 5-percent significance level in input data on expected emergency admissions. * Error at 10-percent significance level in input data on LOS distribution.
SWAIN,
MILPATRICK,
Table 2. Number of Days of Excess or Near-excess Occupancy During Three-month Periods in 1974 and 1975 (Overflow defined as fewer than 20 beds empty) empty
35-39
30-34
25 29
...................
................ ................
20-24 ................ 15-19 10-14
................ ................
1974 9
1975 3
16 10 5 4 4
12 12 13 3
0
was made further in advance. Since all of the confidence intervals were for 90-percent confidence, approximately 10 percent of the predictions should be outside the confidence interval. It can be seen in Table 1 that a higher degree of accuracy was achieved for some services whereas for others accuracy was worse than would be expected; in such cases the primary source of error was most often in the data base used. For many of the services, errors occurred either in the number of unscheduled admissions or in the distribution of the LOS. In the Ohio State system both of these types of errors are spotted by a separate program that reads a history of the inputs and outputs of the predictor program and then calculates data such as those in Table 1, a history of the number of unscheduled admissions for each day of the week, and the LOS distribution for those patients discharged during the prediction period. Errors are most severe when both unscheduled admissions and LOS are incorrect in the same direction. For example, for the ophthalmology service (EYE) the number of unscheduled admissions in the data base was significantly higher than the actual number and the actual LOS of patients discharged was significantly lower than that used for prediction. In general, predictions are quite sensitive to the accuracy of the supporting data. Table 1 also indicates the effect of not accounting for variance due to day of the week in the predictions; this is shown by the fluctuations of the error across the number of days in the prediction horizon. For most services the error rate was lower for predictions seven days in advance than for predictions four or five days ahead, because a sevenday horizon tends to cancel the day-of-the-week variations in LOS. The benefits that result from the use of a prediction system are difficult to quantify. For the period July through September in 1975, average occupancy was 90.26 percent, 0.5 percent higher than for the same period in 1974; this increase represents $125,000 in revenue. It is obviously difficult to assocate this small increase with any change HEALTH such as the use of the prediction system. However, the primary problem ESRVIC'ES RESEARCH in this institution before adoption of the system was periodic overflows (considered, at Ohio State, to be occasions when fewer than 20 of the 390 679 beds are available). Table 2 shows the number of days when
excess or near-excess occupancy occurred during July through Sep- CENSUS tember in 1974 and 1975. The system reduced the occurrence of excess PREDICTION occupancy in spite of the slight increase in average occupancy it AND CONTROL achieved. This change indicates increased control over the census. An additional benefit, more difficult to quantify, resulted from replacing the intuition of the admitting director with estimates made on a consistent basis: this reduced the variation in performance that had been observed in the director's absence. The Ohio State system is used in an advisory rather than a mandatory way, so it is possible to grant elective reservations that may not be consistent with the recommended pattern. This flexibility allows patient and physician convenience to be considered in the admitting office; patients and physicians are usually not aware of the system's existence. In consequence, the system has not complicated the process of obtaining admission reservations. In the four hospitals in Gainesville, where the system has been implemented experimentally, the primary interest has been in evaluating the accuracy of predictions in widely varying administrative settings. The results of these experiments have been analyzed by Nguyen [21]. The system has been operating for nearly a year at Shands Teaching Hospital in Gainesville, where both the LOS and the admission day of the week are used in the prediction process. The absolute errors in one-day predictions were recorded for the pediatric units over a 50-day experimental period; the average error was 4 percent, with a range from zero to 14 percent. At the Gainesville Veterans Administration Hospital the census for all 480 beds was predicted. One-day-ahead and seven-day-ahead predictions over a 45-day trial showed a maximum error of less than 5 percent; for each prediction horizon the average error was less than 3 percent, or 15 patients. In the VA hospital also, LOS and admission day of the week were used to predict census. As would be expected, the relative error rate tends to decrease as the census increases. Experiments were also conducted at North Florida Regional Hospital and Alachua General Hospital to study the accuracy and the utility of census prediction. In both settings it was dear that regular prediction of census was useful, despite the fact that one of these hospitals is privately owned and maintains high occupancy and the other is a public hospital with relatively low occupancy.
Other Implementation Possibilities
In the hospitals described, computers were available. Many smaller hospitals do not have computers available to them but they still need census control. It is possible to adapt the prediction model and the optimization model for use without an inhouse computer. Information about LOS would not be used; rather, the total patient count of each unit would be used for prediction together with discharge probabilities WINTER such as the day-of-the-week probabilities described in the section on 1977 calculation of PTj. When these values are combined with information on current census and unscheduled admissions, it is possible to make
391
SWAIN, KILPATRICK, & MARSH
Table 3. Number of Elective Admissions Recommended on Tuesday, Given a Monday Midnight Census in a 259-bed Hospital Monday census
Tue
242 . 243 . 244 . 245 . 246 . 247 . 248 . 249 . 250 . 251 . 252 . 253 . 254 .
46 45 44 43 42 42 41 40 39 38 37 36 35 34 33 33 32 31 30 29 28 27 26 25 24 24 23 22 21 20 19 18 17 16 15
255 . 256 . 257 . 258 . 259 .
14 14 13 12 11
220 . 221 . 222 .
223 . 224 . 225 . 226 . 227 . 228 . 229 . 230 . 231 .
232 . 233 . 234 . 235 . 236 . 237 . 238 . 239 . 240 . 241
Day of admission Fri Thu Wed
Sat
19 19 19 19 19 19 19 20 19 19
21 21 21 21 21
20 20 20 20 20 20 20 19 20 20 20 20 20 20 20
19 19 19 19 19
25 25 24 24 25 25 25 25 25 25
20 21 21 21 21
20 19 20 20 20
19 20 19 19 19
25 24 25 25 25
21 21 21 21 21
20 20 20 20
19 19 19 19 19
25 25 25 25 25
21 21 21 21 21
20 19 19 19 19
25 25 25 25
21 21 21 22 22
19 20 20 20 20 20 20 20 19 19
22 21 21 21 21
19 20 20 20 20
21 21 21 21 21 20 20 21 21 21
20
19 19 19 20 20 20 19 19 19 19
24 24 25 25 25
25 25 25 25 25 25 25 25 25 25 25
census predictions and to calculate the total elective admission pattern that will maximize the expected census, subject to a probabilistic constraint on excess occupancy. When this approach is taken an HEALTH elective admissions table such as that shown in Table 3 is used to SERVICES control census. Table 3 shows elective admissions recommended on RESEARCH Tuesday for a hospital with 259 controllable beds (no obstetric, nursery, or intensive care beds); since the day of the week is used in determining
392
discharge probabilities, a separate table must be provided for each CENSUS day of the week. Table 3 indicates that if 240 patients are present on PREDICTION AND CONTROL Tuesday, then the pattern of admissions that will maximize occupancy with a 0.05 chance of overflow is to admit 28 patients on Tuesday, 19 on Friday, and 25 on Saturday, if possible. The higher value on Saturday is due to the high discharge rate on Saturday. Such tables are generated by using daily discharge rates for P,1 in Eq. 4 and then solving the equation for each possible census level. Data for such tables can be collected in one week from data normally maintained at most hospitals. The resulting tables are similar to those that Kolesar's approach [13] would produce, but the data and computing effort required are considerably less.
Discussion
The model that has been presented captures the major aspects of the hospital census process and provides a high degree of flexibility of implementation. Given minimal data, it is possible to establish a simple tabular system for guidance in planning additional elective admissions. If both computing power and additional data are available, there is virtually no limit to the degree of detail that can be used to predict census. Experience with the applied system indicates that both stay to date and admission day of the week provide important predictive information that should be used if a computer is available. It also appears that relatively small data errors can have a significant effect on accuracy of prediction. In computer-based systems indusion of a mechanism to evaluate predictor performance and to identify potential sources of error permits prediction quality to be assessed. Results indicate that good census prediction enhances control of census so that high occupancy is maintained while periods of unacceptably high occupancy are reduced. Census prediction can have various uses, depending on the setting. In hospitals with high occupancy, avoiding census overflows and admission cancellations is of particular importance. Hospitals with lower occupancy are more interested in predictions for short-term staffing decisions. In installing a prediction system the hospital's specific objectives must be considered and the system specifically designed to meet those needs. Hospitals that have had the greatest success with this system are those in which it was tailored to meet specific needs and in which the admitting director understood the concepts underlying the model. Several modifications of the model may be useful in future implementations. One such modification would allow other resources in addition to beds to determine recommended additional admissions: for example, available nursing staff could be a second constraint on census [22]. WINTER A more important area for future additions to the model is the 1977 problem of dealing with information about the probable number of additional elective admissions when a particular admission goal is set.
393
SWAIN, KILPATRICK, & MARSH
Barber [8] suggests that a set of conditional probabilities relating actual admissions to admission goals might be used to represent this information. However, the set of conditional probabilities required is immense, and it is not clear that data on these probabilities can be easily obtained. It is more likely that a mechanism for capturing the intuition or informal knowledge of the admitting director will be useful in this problem. If information concerning probable admissions could be effectively represented, then several interesting census-control problems could be attacked. In particular, controlling census variance as well as total census would be feasible. It would then be possible to establish admission levels for overbedded hospitals, in which nursing units might be opened or closed depending on occupancy levels. The costs associated with such openings and dosings of units could then be weighed against the benefits of maximum occupancy in choosing the desired admission pattern. In summary, the process of predicting hospital census and identifying desirable pattems of future admissions appears to be both practical and worthwhile in a variety of hospital environments. It seems likely that quantitative models of the hospital census process will play an increasing role in the admissions policies of many hospitals. Acknowledgments The authors wish to thank Lee Daniel, Khanh-Luu Thi Nguyen, Paul T. Grebe, Julie Kennedy, and William Walker of the Health Systems Research Division of the University of Florida for their technical support in implementing the model. We also thank the administration and staff of the participating hospitals for their support and cooperation.
REFERENCES 1. Milsum, J.H., .. Turban, and I. Vertinsky. Hospital admispion systems: Their evaluation and management. Manage Sci 19:646 Feb. 1973. 2. Shonick, W. A stochastic model for occupancy-related random variables in generalacute hospitals. J Am Stat Assoc 65:1474 Dec. 1970. 3. Shonick, W. and J.R. Jackson. An improved model for occupancy-related random variables in general-acute hospitals. Oper Res 21:952 July-Aug. 1973. 4. Bailey, N.T.J. Queuing for medical care. Ajpp Stat 3:137 Nov. 1954. 5. Young, J.P. Stabilization of inpatient bed occupancy through control of admissions. Hospitals 39:41 Oct. 1, 1965. 6. Young, J.P. Administrative control of multiple-channel queueing systems with parallel input streams. Oper Res 14:145 Jan. 1966. 7. Balintfy, J.L. A S astic Model for the Analysis and Prediction of Admissions and Discharges in Hospitals. In C.W. Churchman and M. Verhulst (eds.), Management Sciences: Models and Techniques, Vol. 2, pp. 288-299. New York:
Pergamon, 1960. 8. Barber, R.W. A unified model for scheduling elective admissions. Health Sem Res
HEALTH SE3RVICES
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394
12:407 Winter 1977. 9. Bithel, J.F. A class of discrete-time models for the study of hospital admission systems. Oper Res 17:48 Jan. 1969. 10. Connors, M.M. A stochastic elective admissions scheduling algorithm. Health Serv Res 5:308 Winter 1970. 11. Jewell, W. and B. Fox. Decision systems for scheduling elective patients into hospitals. Working paper, Department of Operations Research, University of Cailfomia at Berkeley, 1964. 12. Kao, E.P.C. A semi-Markovian model with application to hospital planning. IEEE Trans Syst Man Cybern SMC3:327 July 1973.
13. Kolesar, P. A Markovian model for hospital admission scheduling. Manage Sci 16:384 Feb. 1970. 14. Robinson, G., P. Wing, and L. Davis. Computer simulation of hospital patient scheduling systems. Health ServeRs 3:130 Siummer 1968. 15. Hancock, W.M., D.M. Warner, S. Heda, and P. Fuhs. Admission Scheduling and Control Systems. In J.R. Griffith, W.M. Hancock, and F.C. Munson (eds.), Cost Control in Hospitals, pp. 150-185. Ann Arbor, MI: Health Administration Press, 1976. 16. Swain, R. A general model for hospital census prediction and control. Paper presented at ORSA/TIMS Joint National Meeting, Boston, MA, Apr. 22-24, 1974. 17. Rubenstein, L.S. Computerized hospital inpatient a ions sheduling systema model. Working paper, Kaiser Foundation Health Plan, Los Angeles, CA, 1975. 18. DuFour, R.G. Predicting hospital bed needs. Health Sewv Res 9:62 Spring 1974. 19. Robinson, G., L. Davis, and G. Johnson. The physician as an estimator of hospital stay. Hum Factors 8:201 June 1966. 20. Robinson, G., L. Davis, and R. Liefer. Prediction of hospital length of stay. Health Sev Res 1:287 Winter 1966. 21. Nguyen, K.-L.T. Models for Hospital Census Prediction and Allocation. Doctoral dissertation, Department of Industrial and Systems Engineering, University of Florida, 1977. 22. Swain, R. and J. Marsh. A general model for prediction and control of hospital census. Working paper, Health Systems Research Division, University of Florida, 1976.
CENSUS PREDICTION AND CONTROL
WINTER 1977
395
Hospital census has a major effect on revenues, costs, and operating efficiency. Low occupancy results in high costs per patient day and can lead to reduction of staff and other resources; extremely high occupancy can strain hospital resources and lead to reduced quality of care. Even variability in census, if it is not anticipated, can cause problems in matching patient needs with hospital resources. Ancillary services such as laboratory, radiology, and pharmacy are especially vulnerable to fluctuations in census because work load in such services is a function not only of total census but also of changes in census; hence a small change in census can cause a large change in work load. The importance of census control is affirmed by the large volume of literature on the problem. Milsum, Turban, and Vertinsky [1] have provided a partial review of this literature; Shonick [2] and Shonick and Jackson [3] are also excellent sources. The basic issue addressed by census-control models is the trade-off of average occupancy against excess occupancy, patient rejection, or excessive delays before admission. In nearly all of the models proposed for this problem, a length-of-stay pattern is implicitly assumed that, together with the pattern of requests for admission, allows determination of the distribution of occupancy or of its expected value and variance. A number of models [2-6] assume a Poisson arrival process and a negative exponential length-of-stay distribution to facilitate queueing theory analysis. Other models, e.g., refs. 7-14, make somewhat less restrictive assumptions. From the viewpoint of use, these models can be separated into two HEALTH
SERVICES RESEARCH
380
Address communications and requests for reprints to Ralph W. Swain, Associate
Health Research Division, Box J-177, J. Hillis Miller Health Director, Center, University Systems of Florida, Gainesville, FL 32610. Kerry E. Kilpatrick is Director of the Health Systems Research Division, University of Florida. John J. Marsh III is Hospital Management Engineer at the Ohio State University Hospitals, Columbus.
basic types, aimed at long-term planning of the bed complement or at CENSUS PREDICTION short-term planning of elective admissions within the constraints of a AND CONTROL fixed bed supply (sometimes with fixed allocation of beds to surgical and medical services). In this article we describe a model of the second type, intended to predict census in the short run on the basis of historical data, data on the current hospital census, and elective admissions already scheduled. The simplicity of the model permits it to be adapted to the hospital's needs, data availability, and computing capability. The basic objectives of our model are similar to those suggested by Hancock et al. [15]. Their model, however, uses computer simulation to determine an acceptable fixed schedule of daily quotas for various types of admissions, without provision for modifying these quotas to allow for census variation or changes in the statistical properties of the system. Our model, based on the work of Swain [16], explicitly accounts for actual variations in census and elective reservations and allows adaptive modification of the parameters of the underlying length-of-stay distribution and rate of unscheduled arrivals. Although our model is similar to those proposed by Jewell and Fox [11] and Rubenstein [17], the results reported here provide the first detailed information relative to the implementation and evaluation of this type of model for census prediction and control.
Model Development
Starting with a given census on day t, one can predict the census on a future day t + j if one can predict elective and emergency admissions and discharges that will occur over the intervening days plus the number of patients present on day t who will still be in the hospital on day t + j. In making such a prediction, one must consider both the value being predicted and the accuracy associated with the prediction. Given the many potential sources of variation in the census, it is clear that the actual census will not be equal to the predicted census in most cases. Instead, the actual value will tend to vary around the predicted value. To account for this inherent variation, our model predicts both an expected census value and a range about the value such that there is a high probability that the actual census will fall within that range. Given the census at midnight (i.e., 2400 hr) on day t, the expected census on day t + j may be expressed as
C1=N,+Ej+ Uj
(1)
where C1 = expected census on day t + j, predicted on day t N1 = current patients, present on day t, who are predicted to be present on day t + j = elective admissions (that is, minus discharges of future net Ei elective patients) during the next j days U1 = net expected unscheduled (emergency plus urgent) admissions (that is, minus discharges of future unscheduled patients) during the next j days
WINTER 1977
381
SWAIN, Prediction of C1 and its probable range requires estimation of the KILPATRICK, means and variances of Nj, Ej, and U1. For the moment, we shall sup& MARSH pose that census is being predicted for some well-defined set of patients (e.g., in a particular service or nursing unit or in the total hospital) and that all of the model parameters refer to that set. Current Patients Current patients are classified by number of complete days they have already stayed in the hospital; the number of patients who have stayed T complete days at the time of the current census is denoted by n2. If the stays of patients in the unit are independent, then for any distribution of length of stay (LOS) with finite variance, the number of current patients with LOS equal to T who will be present after j more days will have a binomial distribution with a mean of nTPI.j, where Pr, is the probability that a patient with a current stay of T days will still be present on day t + j. Induding all values of the currently completed stay T, the expected Nj of current patients who will still be present after j more days is max T
= EnPpj N1 T=O
and the variance of Nj is max T
V2(Nf) = £ fTP1j( -PTI) Elective Admissions To avoid a second term and an additional symbol in the variance expression, we assume that all those scheduled for elective admission actually enter the hospital. Patients to be admitted in the future are considered to have a stay to date of zero; hence the probability that such a patient, admitted on day k in the future, will still be present on day j is Po,.-, for j a k. We also assume that admissions for day k occur before the census on day k is taken. Of the ek elective patients admitted on day k, the number expected to still be present in the census for day j is ekPo,j-1 for j a k. Thus, considering all elective admissions between the current time and the census at the end of day j, the mean net contribution to the census from elective admissions is
E=i ekPowj-k and the variance of E1 is
V2(EI) =
ekPo,1-k (
-
Po,-k)
HEALTH Unscheduled Admisions
RESEARCH
382
The census contribution from unscheduled admissions is handled
in the same manner as elective admissions, except that an additional variance term is used. This term expresses the variance due to un-
scheduled admissions on day j and is in addition to the variance due CENSUS to unscheduled patients admitted earlier who are still present on PREDICTION day j. The expected contribution to the census on day j from emergency AND CONTROL admissions on day k only is utPo,jk for j > k, and the total contribution over all days up to j is
Uj =
UkPo,j-
The variance of us is then
P(U1) =
[UkPo,-k (
-
Po,lk) +
A J-4]
where Qk2 is the variance of the number of unscheduled admissions on day k only. Total Census With the foregoing definitions for N1, E1, and U1 in Eq. 1, the expected value of the total census on day t + j can be expressed as c
F T=O
L uk(Po,-k) nTPTI + k= ek(Po,-k) + k=l
=m
(2)
The variance of C1 is then max
-
T
£ hnrPTi(I V2(C)= T=O
-
PT;) +
ekPo,-k(
-Po,-k)
+ [ukPo,,k(l(I PoP-t)+ (Po,rk)2] lc=1
(3)
The Confidence Interval The most natural approach to a confidence interval for the predicted census is to assume a distribution for the actual census, given information about the current census and expected admissions. For patient units of a reasonable size one can invoke the central limit theorem and assume that the actual census will be normally distributed about Cj with a variance of V2(C1). This assumption of normality implies that the prediction error is normally distributed about the predicted census value, which is not inconsistent with a Poisson distribution of long-term census or some of the other forms reported [2,3,9,18]. In this case, the e-level confidence interval for the actual census is the range from C1 - Z.V(C1) to C1 + Z.V(C1), where Za is the standard normal random variable such that 'I(ZQ) = 1 - a/2 (4 is the standard normal cumulative distribution function and a is the desired level of confidence) and V(C,) is the square root of V2(C,). In large patient units the assumption of normality for the distribution of census is acceptable, but in units with 10 or fewer beds it can lead to significant errors. A more conservative interval can be obtained by employing the Chebyshev WINTER inequality, which leads to a confidence interval ranging from C1 1977 -[V2(Cj)la]r to C1 + [V2(C,)/a]%. This interval will be at least twice 383 the other in most cases.
SWAIN, KILPATRICK, & MARSH
HEALTH SERVICES RESEARCH
384
Data Requirements and Computation Alternatives The Patient Unit The choice of the smallest patient unit for which predictions are to be made influences the method used to calculate the values of Prj. As the unit becomes smaller, more detailed data are needed to obtain reasonable accuracy in the predicted occupancy levels. Some trade-off must usually be made between the definition of the patient unit and the information available in the hospital. For example, admission diagnosis offers an appealing method of categorizing patients, but available data are often related to admissions to nursing units or particular medical or surgical services or to discharge diagnoses rather than admission diagnoses. In some cases data are in such inconvenient forms that only total hospital occupancy can be predicted; in other cases data are available to support virtually any definition of patient units. An extremely detailed definition of patient units increases the computational requirements in terms of both computation time and storage space for data. In systems with detailed, easily accessible data we have attempted to deal only with units of 10 or more beds. In hospitals that maintain data only on average LOS, it is normally necessary to use the entire hospital as the basic unit.
Calculation of P2j The calculation of PTj depends on the data available, the ease with which the data can be used, and the set of factors that are chosen to derive information about the future stay of a patient. The minimal data situation usually occurs in a hospital with no internal computing system and no commercial computing service for its billing process. If, in addition, the hospital does not participate in the Professional Activity Study (PAS), there may be no practical method of obtaining the probability distribution of LOS in that hospital. If it is assumed, however, that future stay is independent of T, average LOS can be used to obtain values for PTJ* Under this assumption LOS has a geometric distribution and the probability of staying at least j more days is (1 - py, where p is the probability of discharge. The value of p can be found by equating the expected LOS, L, to the inverse of the discharge probability: L = 1/p. Thus the discharge probability is the reciprocal of L, and PTj = (1 - I/L)i. The assumption of a geometric distribution of LOS implies that PTj is not a function of T; therefore under this assumption the population does not have to be categorized by stay to date. Although this procedure is extremely simple, it yields impressive results when used to predict total hospital occupancy. A closely related approach is to assume that additional stay is independent of stay to date but is related to the particular day of the week that follows. Under this assumption the discharge probability for each day of the week is calculated by taking the reciprocal of average LOS times the relative discharge rate associated with each day of the week. These latter values can be determined by calculating the average fraction of the previous day's midnight census that remains at the next
midnight census: (ct+l - at+i)/ct, where at+, is admissions on day t + 1 CENSUS and c denotes observed census. If these values are organized by day of PREDICTION the week and normalized so that their average is one, they can be multi- AND CONTROL plied by the average discharge probability to obtain discharge probability by day of the week. If data on LOS distribution are available one can use knowledge of T. When such information is available it is usually also possible to divide the total hospital population into units based on medical or surgical services. Either PAS data summaries or tapes from the hospital billing system can provide data on the LOS distribution. Let F(T) be the cumulative distribution function for T; if F(T) is known for a given service,
PT,=P[L>(T+1)IL> T] = P[L > (T + 1)]/P(L > T) [1 -F(T + 1)]/[1-F(T)] =
where L is expected LOS, as previously.
If the data indicate the day of the week when each patient was admitted, then the cumulative distribution of LOS can be based on the day of the week as well. It has been our experience that in some settings the day on which a patient was admitted provides considerable information that can be used in census prediction. It is also possible to make use of adjusted physician estimates, but such estimates have not always proven reliable [19,20]. Other factors that may be useful are patient age, sex, and financial status, transfers from other facilities, attending physician, and the like, but as a general rule added explanatory factors cause large increases in the space required to store the resulting PTj values and in the calculation required to handle the more detailed data. Unscheduled Admissions
Unscheduled admissions include several different categories of admissions. Patients admitted after receiving care in the emergency room normally require immediate admission. Patients in the urgent category are usually admitted within 24 to 48 hours of the admission request. In addition, some elective patients are admitted on the day the request is made. In our system the first two groups are induded in the unscheduled category, but the latter group is not, because those admissions could normally be delayed if necessary. One of the objectives of a census prediction system is to reduce uncertainty in the future census, and scheduling as many patients as possible electively contributes to this goal. The final definition of unscheduled admissions that is used must be consistent with the admission policy of the particular institution. No matter how the unscheduled admissions are defined, weekday variations in such admissions must be accounted for. Both the mean rate and the variance of emergency admissions vary significantly by day of the week, and the variance in emergency admissions usually represents a large portion of the total variance in the census.
WINTER 1977
385
SWAIN, KILPATRICK, & MARSH
Prediction and Control
The framework of the basic census-prediction model allows a number of possibilities for either census prediction or census control or both. The prediction process is based on Eqs. 2 and 3 with a suitable confidence interval and the particular set of values for PTJ associated with the hospital. The control process determines fk, the pattern of additional elective admissions for day k that will maximize the predicted patient days in the hospital over the next m days, subject to a constraint on the probability that occupancy will be greater than the number of available beds. This problem may be expressed as: Maximize S
lc1 us=k
fk PO,n-k
subject to
I~~ C, + L fkPO,J-k + Zc.
386
(4)
where B is the number of beds available on the unit being controlled and Za is the standard normal random variable defined previously. This problem can be solved heuristically by finding the maximum value of fi that satisfies all m constraints (i.e., for j = 1, . . . , m) when f2, . . , fm are all zero, then taking that value of fi as a constant and finding the maximum f2 that satisfies the constraints for j = 2, ...., m when f, . . . fi are all zero, and then repeating the process for f8, and so on. This process can be further simplified by recognizing that the standard deviation is virtually independent of the values of f in most problems, and that if constraint j limits the value of fk, then all fA such that i = k + 1, k + 2, . . .j will have a value of zero. The formulation given in Eq. 4 implies that the primary objectives are a high occupancy level and avoidance of crowding to the point that patients with reservations must be turned away. These are only two of the objectives of admissions control [1]; others include the choice of a convenient day for the patient or the physician, avoidance of an unusual case mix, and the like. Because of these other objectives, the actual pattern of admissions granted during a 24-hour period does not usually match the pattern generated by the solution of Eq. 4. The computed pattern does, however, provide one feasible admissions plan that meets two of the major objectives. Admissions personnel can make good use of this information in the process of granting reservations. In a recent attempt to make the computed pattern of admissions correspond more closely to physician desires, we adapted Eq. 4 by adding absolute limits on the total number of elective admissions permitted on some days (particularly weekends) and by making the effective bed capacity, B, a function of the day of the week. This forces a lower occupancy on weekends, when nursing staff is normally smallest. ,
HEALTH SERVICES RESEARCH
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Equation 4 will automatically honor any preexisting reservations unless a given admission violates the constraint on occupancy. The concerns of the patient can be given more weight by using Eq. 4 to determine the set of dates for which admissions would be feasible in terms of both bed capacity and admission level desired for each day of the week. The actual output of the prediction system is shown in the figure on p. 387. Predicted mean values, together with the upper and lower limits of the confidence interval, are ordinarily provided for a period of 7 to 14 days, depending on the needs of the institution. With the assumption that the actual census will be normally distributed about the predicted value, it is also possible to calculate the probability that the census on each day will exceed the capacity of the unit, given current commitments. The printout includes the number of additional elective admissions that can be accepted by the unit with a reasonably low probability of overflow together with the number of currently scheduled elective admissions and the current probability of census overload. If Eq. 4 is used to determine the permissible number of additional elective admissions, the result can be used as a set of suggested levels to be achieved during the next one or two weeks. As reservations for additional admissions are granted they are added to the proper ek values; when the process is repeated for the next day these added admissions are included in computation of the next set of permissible elective admissions. If the system uses stay to date and the day of the week when admission occurred, a computer is needed both to gather data about current census and elective admission requirements and to calculate the mean and variance of the future census. In hospitals with computerized master patient files, the process of surveying the census, predicting the future census, and suggesting additional elective admissions can be incorporated into the midnight census process. If the file is manual, a tabular system can be used to summarize the current census and timesharing can be used to compute the predictions. If no computing facilities are available only the simplest model, not using stays to date, is feasible.
Application Results
HEALTH SERVICES RESEARCH
388
The complete census-prediction system has been in use since April 1975 at the Ohio State University Hospitals and for a shorter time (on an experimental basis) in four hospitals in Gainesville, Florida. At Ohio State a total of 679 beds are monitored by the system and predictions are made for 22 separate medical and surgical services as well as for the occupancy of all 679 beds. The system is integrated with the hospital's admission, discharge, and transfer system. All data required for the system are maintained in on-line files, so the actual operation of the system is fully automatic and no manual data collection is required. Census predictions and suggested additional admissions are supplied as part of the midnight census reports, and a
graphic presentation of the predicted occupancy is provided to the CENSUS PREDICTION director of admitting. AND CONTROL of a variety different system a such of utility the In evaluating factors must be considered, including the cost of implementing and operating the prediction system, the accuracy of the predictions, the effect on hospital occupancy and overflow conditions, and the effect on convenience for both patients and physicians. Developing the programs for system operation required about six man-weeks for the Ohio State installation and also for the University of Florida installation. In future installations development time could be reduced by use of one of these program packages. The daily calculations require less than 10 seconds in most situations and cost less than $10 per day. Table 1 indicates the accuracy of the Ohio State system for a 60-day period in the first year of operation; it shows the percentage of days on which the actual census was outside the confidence intervals calculated from one to seven days in advance. Several important points are implicit in these data. It is dear that the accuracy of the model varies with the service being predicted and with the number of days in the prediction horizon. In general, accuracy decreased as the prediction Table 1. Frequency of Prediction Error: Percentage of Days in 60-day Trial on Whic Actual Census was Outside Predicted 90-percent-Confidence Interval, for Predictions Made One to Seven Days in Advance Prediction lead time, days Service census predicted
H
1
L
H
2
L
H
3
L
H
4
L
H
5
L
H
6
L
H
7
L
0 0 0 0 0 0 0 1.6 1.6 0 1.6 5.0 0 3.3 16.6 1.6 13.3 1.6 10.0 0 8.3 15.0 11.6 11.6 8.3 8.3 3.3 1.6 8.3 0 8.3 5.0 8.3 6.6 5.0 5.0 10.0 8.3 11.6 6.6 10.0 1.6 6.6 3.3 10.0 1.6 11.6 0 10.0 0 10.0 0 18.3 0 23.3 0 21.6 0 28.3 0 1.6 8.3 5.0 13.3 3.3 15.0 1.6 10.0 0 10.0 11.6 10.0 11.6 8.3 11.6 5.0 11.6 3.3 10.0 0 16.6 16.6 18.3 20.0 16.6 15.0 13.3 5.0 10.0 5.0 0 18.3 0 20.0 0 20.0 0 16.6 0 15.0 5.0 8.3 10.0 5.0 8.3 1.6 10.0 3.3 8.3 0 23.3 5.0 25.0 3.3 26.6 0 21.6 0 28.3 0 6.6 8.3 3.3 8.3 6.6 8.3 3.3 10.0 1.6 11.6 0 23.3 23.3 23.3 20.0 21.6 15.0 13.3 13.3 8.3 8.3 11.6 16.6 28.3 21.6 23.3 21.6 28.3 13.3 21.6 6.6 6.6 10.0 10.0 66.0 11.6 53.3 8.3 50.0 8.3 43.3 0 36.6 8.3 EYEt* ........0 10.0 20.0 13.3 18.3 5.0 15.0 1.6 5.0 1.6 0 10.0 D/S .......... 6.6 5.0 30.0 5.0 31.6 6.6 23.3 5.0 13.3 6.6 8.3 3.3 SURGt ....... 5.0 3.3 5.0 3.3 8.3 6.6 5.0 21.6 3.3 6.6 0 8.3 T/St* ........ S. 0 6.6 18.3 6.6 20.0 8.3 18.3 3.3 13.3 1.6 8.3 5.0 GU ......... 5.0 3.3 43.3 5.0 43.3 0 43.3 0 43.3 0 51.6 0 GYNt* ....... 1.6 * H indicates actual census above upper confidence interval limit; L indicates actual census below
S. 0 1.6 3.3 5.0 8.3 11.6 3.3 1.6 5.0 1.6 8.3 11.6 5.0 P/S* ......... 6.6 ENTt ........ 6.6
INF ......... CRC ......... ENDOt ...... MEDt ....... HEMt ...... CARt ........ REN ......... GE ......... DERM* ...... A/P* ......... NEU* ........ N/St* .. ORTI .......
5.0 3.3 6.6 8.3 5.0 3.3 8.3 0.5 8.3 10.0 11.6 6.6 8.3 13.3 25.0 53.3 11.6 10.0 8.3 10.0 31.6
3.3 3.3 8.3 5.0 6.6 8.3 5.0 10.0 11.6 16.0 8.3 20.0
1.6 16.6 11.6 11.6 3.3 0 8.3 10.0 15.0 16.6 6.6 1.6 6.6 16.6 33.0 68.3 16.6 28.3 15.0 16.6 40.0
lower limit. t Error at 5-percent significance level in input data on expected emergency admissions. * Error at 10-percent significance level in input data on LOS distribution.
SWAIN,
MILPATRICK,
Table 2. Number of Days of Excess or Near-excess Occupancy During Three-month Periods in 1974 and 1975 (Overflow defined as fewer than 20 beds empty) empty
35-39
30-34
25 29
...................
................ ................
20-24 ................ 15-19 10-14
................ ................
1974 9
1975 3
16 10 5 4 4
12 12 13 3
0
was made further in advance. Since all of the confidence intervals were for 90-percent confidence, approximately 10 percent of the predictions should be outside the confidence interval. It can be seen in Table 1 that a higher degree of accuracy was achieved for some services whereas for others accuracy was worse than would be expected; in such cases the primary source of error was most often in the data base used. For many of the services, errors occurred either in the number of unscheduled admissions or in the distribution of the LOS. In the Ohio State system both of these types of errors are spotted by a separate program that reads a history of the inputs and outputs of the predictor program and then calculates data such as those in Table 1, a history of the number of unscheduled admissions for each day of the week, and the LOS distribution for those patients discharged during the prediction period. Errors are most severe when both unscheduled admissions and LOS are incorrect in the same direction. For example, for the ophthalmology service (EYE) the number of unscheduled admissions in the data base was significantly higher than the actual number and the actual LOS of patients discharged was significantly lower than that used for prediction. In general, predictions are quite sensitive to the accuracy of the supporting data. Table 1 also indicates the effect of not accounting for variance due to day of the week in the predictions; this is shown by the fluctuations of the error across the number of days in the prediction horizon. For most services the error rate was lower for predictions seven days in advance than for predictions four or five days ahead, because a sevenday horizon tends to cancel the day-of-the-week variations in LOS. The benefits that result from the use of a prediction system are difficult to quantify. For the period July through September in 1975, average occupancy was 90.26 percent, 0.5 percent higher than for the same period in 1974; this increase represents $125,000 in revenue. It is obviously difficult to assocate this small increase with any change HEALTH such as the use of the prediction system. However, the primary problem ESRVIC'ES RESEARCH in this institution before adoption of the system was periodic overflows (considered, at Ohio State, to be occasions when fewer than 20 of the 390 679 beds are available). Table 2 shows the number of days when
excess or near-excess occupancy occurred during July through Sep- CENSUS tember in 1974 and 1975. The system reduced the occurrence of excess PREDICTION occupancy in spite of the slight increase in average occupancy it AND CONTROL achieved. This change indicates increased control over the census. An additional benefit, more difficult to quantify, resulted from replacing the intuition of the admitting director with estimates made on a consistent basis: this reduced the variation in performance that had been observed in the director's absence. The Ohio State system is used in an advisory rather than a mandatory way, so it is possible to grant elective reservations that may not be consistent with the recommended pattern. This flexibility allows patient and physician convenience to be considered in the admitting office; patients and physicians are usually not aware of the system's existence. In consequence, the system has not complicated the process of obtaining admission reservations. In the four hospitals in Gainesville, where the system has been implemented experimentally, the primary interest has been in evaluating the accuracy of predictions in widely varying administrative settings. The results of these experiments have been analyzed by Nguyen [21]. The system has been operating for nearly a year at Shands Teaching Hospital in Gainesville, where both the LOS and the admission day of the week are used in the prediction process. The absolute errors in one-day predictions were recorded for the pediatric units over a 50-day experimental period; the average error was 4 percent, with a range from zero to 14 percent. At the Gainesville Veterans Administration Hospital the census for all 480 beds was predicted. One-day-ahead and seven-day-ahead predictions over a 45-day trial showed a maximum error of less than 5 percent; for each prediction horizon the average error was less than 3 percent, or 15 patients. In the VA hospital also, LOS and admission day of the week were used to predict census. As would be expected, the relative error rate tends to decrease as the census increases. Experiments were also conducted at North Florida Regional Hospital and Alachua General Hospital to study the accuracy and the utility of census prediction. In both settings it was dear that regular prediction of census was useful, despite the fact that one of these hospitals is privately owned and maintains high occupancy and the other is a public hospital with relatively low occupancy.
Other Implementation Possibilities
In the hospitals described, computers were available. Many smaller hospitals do not have computers available to them but they still need census control. It is possible to adapt the prediction model and the optimization model for use without an inhouse computer. Information about LOS would not be used; rather, the total patient count of each unit would be used for prediction together with discharge probabilities WINTER such as the day-of-the-week probabilities described in the section on 1977 calculation of PTj. When these values are combined with information on current census and unscheduled admissions, it is possible to make
391
SWAIN, KILPATRICK, & MARSH
Table 3. Number of Elective Admissions Recommended on Tuesday, Given a Monday Midnight Census in a 259-bed Hospital Monday census
Tue
242 . 243 . 244 . 245 . 246 . 247 . 248 . 249 . 250 . 251 . 252 . 253 . 254 .
46 45 44 43 42 42 41 40 39 38 37 36 35 34 33 33 32 31 30 29 28 27 26 25 24 24 23 22 21 20 19 18 17 16 15
255 . 256 . 257 . 258 . 259 .
14 14 13 12 11
220 . 221 . 222 .
223 . 224 . 225 . 226 . 227 . 228 . 229 . 230 . 231 .
232 . 233 . 234 . 235 . 236 . 237 . 238 . 239 . 240 . 241
Day of admission Fri Thu Wed
Sat
19 19 19 19 19 19 19 20 19 19
21 21 21 21 21
20 20 20 20 20 20 20 19 20 20 20 20 20 20 20
19 19 19 19 19
25 25 24 24 25 25 25 25 25 25
20 21 21 21 21
20 19 20 20 20
19 20 19 19 19
25 24 25 25 25
21 21 21 21 21
20 20 20 20
19 19 19 19 19
25 25 25 25 25
21 21 21 21 21
20 19 19 19 19
25 25 25 25
21 21 21 22 22
19 20 20 20 20 20 20 20 19 19
22 21 21 21 21
19 20 20 20 20
21 21 21 21 21 20 20 21 21 21
20
19 19 19 20 20 20 19 19 19 19
24 24 25 25 25
25 25 25 25 25 25 25 25 25 25 25
census predictions and to calculate the total elective admission pattern that will maximize the expected census, subject to a probabilistic constraint on excess occupancy. When this approach is taken an HEALTH elective admissions table such as that shown in Table 3 is used to SERVICES control census. Table 3 shows elective admissions recommended on RESEARCH Tuesday for a hospital with 259 controllable beds (no obstetric, nursery, or intensive care beds); since the day of the week is used in determining
392
discharge probabilities, a separate table must be provided for each CENSUS day of the week. Table 3 indicates that if 240 patients are present on PREDICTION AND CONTROL Tuesday, then the pattern of admissions that will maximize occupancy with a 0.05 chance of overflow is to admit 28 patients on Tuesday, 19 on Friday, and 25 on Saturday, if possible. The higher value on Saturday is due to the high discharge rate on Saturday. Such tables are generated by using daily discharge rates for P,1 in Eq. 4 and then solving the equation for each possible census level. Data for such tables can be collected in one week from data normally maintained at most hospitals. The resulting tables are similar to those that Kolesar's approach [13] would produce, but the data and computing effort required are considerably less.
Discussion
The model that has been presented captures the major aspects of the hospital census process and provides a high degree of flexibility of implementation. Given minimal data, it is possible to establish a simple tabular system for guidance in planning additional elective admissions. If both computing power and additional data are available, there is virtually no limit to the degree of detail that can be used to predict census. Experience with the applied system indicates that both stay to date and admission day of the week provide important predictive information that should be used if a computer is available. It also appears that relatively small data errors can have a significant effect on accuracy of prediction. In computer-based systems indusion of a mechanism to evaluate predictor performance and to identify potential sources of error permits prediction quality to be assessed. Results indicate that good census prediction enhances control of census so that high occupancy is maintained while periods of unacceptably high occupancy are reduced. Census prediction can have various uses, depending on the setting. In hospitals with high occupancy, avoiding census overflows and admission cancellations is of particular importance. Hospitals with lower occupancy are more interested in predictions for short-term staffing decisions. In installing a prediction system the hospital's specific objectives must be considered and the system specifically designed to meet those needs. Hospitals that have had the greatest success with this system are those in which it was tailored to meet specific needs and in which the admitting director understood the concepts underlying the model. Several modifications of the model may be useful in future implementations. One such modification would allow other resources in addition to beds to determine recommended additional admissions: for example, available nursing staff could be a second constraint on census [22]. WINTER A more important area for future additions to the model is the 1977 problem of dealing with information about the probable number of additional elective admissions when a particular admission goal is set.
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Barber [8] suggests that a set of conditional probabilities relating actual admissions to admission goals might be used to represent this information. However, the set of conditional probabilities required is immense, and it is not clear that data on these probabilities can be easily obtained. It is more likely that a mechanism for capturing the intuition or informal knowledge of the admitting director will be useful in this problem. If information concerning probable admissions could be effectively represented, then several interesting census-control problems could be attacked. In particular, controlling census variance as well as total census would be feasible. It would then be possible to establish admission levels for overbedded hospitals, in which nursing units might be opened or closed depending on occupancy levels. The costs associated with such openings and dosings of units could then be weighed against the benefits of maximum occupancy in choosing the desired admission pattern. In summary, the process of predicting hospital census and identifying desirable pattems of future admissions appears to be both practical and worthwhile in a variety of hospital environments. It seems likely that quantitative models of the hospital census process will play an increasing role in the admissions policies of many hospitals. Acknowledgments The authors wish to thank Lee Daniel, Khanh-Luu Thi Nguyen, Paul T. Grebe, Julie Kennedy, and William Walker of the Health Systems Research Division of the University of Florida for their technical support in implementing the model. We also thank the administration and staff of the participating hospitals for their support and cooperation.
REFERENCES 1. Milsum, J.H., .. Turban, and I. Vertinsky. Hospital admispion systems: Their evaluation and management. Manage Sci 19:646 Feb. 1973. 2. Shonick, W. A stochastic model for occupancy-related random variables in generalacute hospitals. J Am Stat Assoc 65:1474 Dec. 1970. 3. Shonick, W. and J.R. Jackson. An improved model for occupancy-related random variables in general-acute hospitals. Oper Res 21:952 July-Aug. 1973. 4. Bailey, N.T.J. Queuing for medical care. Ajpp Stat 3:137 Nov. 1954. 5. Young, J.P. Stabilization of inpatient bed occupancy through control of admissions. Hospitals 39:41 Oct. 1, 1965. 6. Young, J.P. Administrative control of multiple-channel queueing systems with parallel input streams. Oper Res 14:145 Jan. 1966. 7. Balintfy, J.L. A S astic Model for the Analysis and Prediction of Admissions and Discharges in Hospitals. In C.W. Churchman and M. Verhulst (eds.), Management Sciences: Models and Techniques, Vol. 2, pp. 288-299. New York:
Pergamon, 1960. 8. Barber, R.W. A unified model for scheduling elective admissions. Health Sem Res
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12:407 Winter 1977. 9. Bithel, J.F. A class of discrete-time models for the study of hospital admission systems. Oper Res 17:48 Jan. 1969. 10. Connors, M.M. A stochastic elective admissions scheduling algorithm. Health Serv Res 5:308 Winter 1970. 11. Jewell, W. and B. Fox. Decision systems for scheduling elective patients into hospitals. Working paper, Department of Operations Research, University of Cailfomia at Berkeley, 1964. 12. Kao, E.P.C. A semi-Markovian model with application to hospital planning. IEEE Trans Syst Man Cybern SMC3:327 July 1973.
13. Kolesar, P. A Markovian model for hospital admission scheduling. Manage Sci 16:384 Feb. 1970. 14. Robinson, G., P. Wing, and L. Davis. Computer simulation of hospital patient scheduling systems. Health ServeRs 3:130 Siummer 1968. 15. Hancock, W.M., D.M. Warner, S. Heda, and P. Fuhs. Admission Scheduling and Control Systems. In J.R. Griffith, W.M. Hancock, and F.C. Munson (eds.), Cost Control in Hospitals, pp. 150-185. Ann Arbor, MI: Health Administration Press, 1976. 16. Swain, R. A general model for hospital census prediction and control. Paper presented at ORSA/TIMS Joint National Meeting, Boston, MA, Apr. 22-24, 1974. 17. Rubenstein, L.S. Computerized hospital inpatient a ions sheduling systema model. Working paper, Kaiser Foundation Health Plan, Los Angeles, CA, 1975. 18. DuFour, R.G. Predicting hospital bed needs. Health Sewv Res 9:62 Spring 1974. 19. Robinson, G., L. Davis, and G. Johnson. The physician as an estimator of hospital stay. Hum Factors 8:201 June 1966. 20. Robinson, G., L. Davis, and R. Liefer. Prediction of hospital length of stay. Health Sev Res 1:287 Winter 1966. 21. Nguyen, K.-L.T. Models for Hospital Census Prediction and Allocation. Doctoral dissertation, Department of Industrial and Systems Engineering, University of Florida, 1977. 22. Swain, R. and J. Marsh. A general model for prediction and control of hospital census. Working paper, Health Systems Research Division, University of Florida, 1976.
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