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A simple method to optimize hospital beds capacity
International Journal of Medical Informatics (2005) 74, 39—49
A simple method to optimize hospital beds capacity J.M. Nguyena,b,∗, P. Sixc, D. Antoniolib, P. Glemaind, G. Potele, P. Lombrailb, P. Le Beuxf a
Laboratory of Medical Statistics and Informatics, 1 rue Gaston Veil, 44035 Nantes Cedex 01, France PIMESP, St. Jacques Hospital, CHU Nantes, 44093 Nantes Cedex 01, France c Department of Medical Information, CHU Angers, 49033 Angers Cedex 01, France d Department of Urology, CHU Nantes, 44093 Nantes Cedex 01, France e Department of Internal Medicine, CHU Nantes, 44093 Nantes Cedex 01, France f Laboratory of Medical Informatics, CHU Rennes, 35000 Rennes Cedex 01, France b
Received 15 April 2004 ; received in revised form 13 September 2004; accepted 14 September 2004 KEYWORDS Hospital bed capacity; Multicriteria decision; Optimization; Software
Summary Objective: The number of acute hospital beds is determined by health authorities using methods based on ratios and/or target bed occupancy rates. These methods fail to consider the variability in hospitalization demands over time. On the other hand, the implementation of sophisticated models requires the decision concerning the number of beds to be made by an expert. Our aim is to develop a new method that is as simple to use as the ratio method while minimizing the roundabout approaches of these methods. Method: A score was constructed with three parameters: number of transfers due to lack of space, number of days with no possibility for S unscheduled admissions and number of days with at least a threshold of U unoccupied beds. The optimal number of beds is the number for which both the mean and the standard deviation of the score reach their minimum. We applied this method to two internal medicine departments and one urological surgery department and we compared the solutions proposed by this method with those put forward by the ratio method. Results: The solutions proposed by this method were intermediate to those calculated by the local and national length of Stays ratio methods. Simulating an unusual increase in admission requests had no consequence on the bed number selected, indicating that the method was robust. Conclusion: Our tool represents a real alternative to the ratio methods. A software has been developed and is now available for use. © 2004 Elsevier Ireland Ltd. All rights reserved.
* Corresponding author. Tel.: +33 2 40 84 69 35; fax: +33 2 40 84 69 40.
E-mail address: [email protected] (J.M. Nguyen). URL: http://www.sante.univ-nantes.fr/med/stat/.
1386-5056/$ — see front matter © 2004 Elsevier Ireland Ltd. All rights reserved. doi:10.1016/j.ijmedinf.2004.09.001
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1. Introduction Determining the number of beds ‘objectively’ needed for a ward or a hospital is a complex problem because there are many kinds of hospital beds within each length-of-stay category (long-term, medium-term, short-term) and several specialties each with its own patient management profile. Furthermore, within each specialty, differences in operation occur across hospital departments because of practice variations. Additional differences result from the constraints imposed by site-specific characteristics of patient recruitment and the structural, organizational, geographical, historical and political environment in which the hospital operates. Determining the number of beds needed for a hospital department raises three types of problems. First, a good compromise must be found in the setting up of an operational hospital department, between the number of patients transferred because of a full department and the number of unoccupied beds. Second, changes in needs must be anticipated. Third, evidence that admissions are appropriate must be obtained or at least an evaluation of the rate of inappropriate stays must be performed. Several methods have been suggested to solve this issue. The first approach is based on a census quantifying the number of patients [1—6]. The number of beds is then determined using the ratio method. This method does not take into account the fluctuation of requested admissions over time. Stochastic simulations have also been used to quantify the number of patients [7—14]. Here, the number of beds is determined by a critical probability. The second approach uses models describing the utilization of hospital resources [15,16]. These published methods primarily aim to provide models for the patient cover process and the operation of hospital departments. Their strategy is therefore to reproduce the activity of a department using simulation and to assess the immediate number of beds required for the department. The major problem with these methods is not the number of beds needed per given day, but the number of beds to be allocated to a department over a period corresponding to the time necessary to implement a health plan. This is the issue the medical authorities need to address. The implementation of these models thus requires substantial data and logistics. Another approach is to model patient flow through the hospital using multi compartmental or multi state models [17—36]. In the latter two approaches, the methodology used to determine the optimal number of beds requires an expert opinion in all cases. Such a judg-
J.M. Nguyen et al. ment depends on the author and not on an objective and reproducible decision criterion. For these reasons, and for the sake of simplicity, the medical authorities always use ratio methods and not the models described above. Indeed, most hospitals are supposed to use ratio-based methods to determine the number of beds they require [13]. Ratio-based methods are frequently criticized because they disregard local specificities and are often considered arbitrary. In fact, the ratio method, which uses the average length of stays, overestimates the number of beds required by departments with a high average length of stay. Contrary to this, the national ratio method (case-mix) overestimates the needs of departments, which select patients with the least costly cover. Our aim is to recommend a method [37,38] that is as simple to use as ratio methods while minimizing their biased approaches. Such a tool would be a real alternative to the ratio methods. Part of this work was presented at the HPSS ICHPR 2003 conference [39].
2. Methods The optimal number of beds is defined as the number for which the following three criteria are met: the number of unoccupied beds is not excessive (productivity), the number of patients transferred because of full bed occupancy is not excessive (security), and one or perhaps several beds are available for unscheduled admissions (accessibility). A score is constructed based on these three parameters. The optimal number of beds is the number for which both the means and the standard deviation of the score reach a minimum.
2.1. Construction of the score The parameter formulation is based on the following observations. The statistical unit common to all departments is the number of operational days. Overhead and personnel charges depend only on the number of beds and on the specialty of the department. These charges do not fluctuate, except if beds are temporarily closed (fixed ‘costs’) and can therefore be expressed using a logical function with an acceptability threshold: “Beyond the threshold, the situation is no longer acceptable and each day counts”. The parameter assessing accessibility can also be expressed using a threshold: “to reach the objective of a public service, the department must have at least one or more possibility for unscheduled admission per day".
A simple method to optimize hospital beds capacity The statistical unit for ‘fixed cost’ could be the number of days beyond a threshold. On the contrary, patients who are transferred because the department is full generate costs (health risks, additional economic costs, etc.) that depend on the number of transfers: ‘Each transfer counts’. Thus, this parameter is best expressed as a function whose measurement unit is the number of transfers and no threshold is needed. In conclusion, there are two parameters to assess department capacity (the number of patients transferred and the number of days with empty beds) and one parameter to assess the possibility of unscheduled admission. 1. The number of transfers due to full bed occupancy (security). The number of transfers on a given day is the total number of patients transferred to another department for the sole reason that no beds are available. 2. The number of days with less than S beds for unscheduled admission (accessibility). Department saturation is expressed as the number of days on which no new patients can be admitted. The number of beds that must be available for new patients depends on the size and structure of the department. For instance, a medical—surgical ICU needs at least two unoccupied beds, one for a medical patient and one for a surgical patient. Similarly, a large department may need two or more unoccupied beds to ensure continuity of care. In smaller departments and in those that admit only one type of patient, a single unoccupied bed is sufficient for unscheduled admissions. 3. The number of days with more than U unoccupied beds (productivity). This parameter is the number of days on which the number of unoccupied beds is above a predefined threshold associated with an unacceptable cost.
2.2. Data For each day and for each department, the number of patients in the department, the number of patients transferred due to lack of space and the number of unsatisfied admission requests are collected. Conversely, appropriateness of each stay is defined by the physician. Patients waiting for placement or transfer to an adequate department are defined as inappropriate stay. These patients are excluded from the analysis. The study period must be representative of the activity of the ward. The total number of requested admissions is the sole data needed for the model.
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2.3. Algorithm The algorithm of the model is based on the upgrading of one virtual bed at each step and the calculation of the score for each saturation threshold and each unoccupied bed threshold. For each virtual number of beds and for each day, the algorithm counts the number of patients transferred because of full bed occupancy, the number of days on which the department is considered to be saturated (less than S beds available), and the number of days on which the number of unoccupied beds is too high (more than U beds unemployed). An example of such a calculation is given in Table 1.
2.4. Decision rule An intuitive decision-making method based on three criteria as different from one another as those defined above is to select the criterion that maximizes (or minimizes) all three criteria, if these are quantitative. This approach is equivalent to maximizing (or minimizing) the mean of the values for the three criteria. However, a decision based on minimizing the mean favors the extreme values. One way to solve this problem is to penalize the extreme values. Another solution is to minimize the standard deviation of the data. The optimization of our model is based on the minimization of both the standard deviation and the mean score for each number of beds in the model. The optimal number of beds is the number for which the minimization of both the mean and the standard deviation is reached with the lowest threshold of U unoccupied beds. For S = 0, the model with a number of beds equal to the maximum number of admission requests is the lowest productive model. For this model, the mean and the standard deviation scores are equal to 0 and the threshold U of unoccupied beds is equal to the range of the data [40]. The solution is thus only based on the statistical properties of the model. No hypothesis concerning the distribution of stays or the admission or target bed occupancy is needed.
2.5. Comparison with ratio methods 2.5.1. Local length of stay ratio The number of beds required is defined as N= =
LoS × number of patients number of days number of patients-days number of days
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Table 1
Example of a score calculation for 10—12-beds and for the thresholds S = 1, U = 1 and 2
Day
Monday Tuesday Wednesday Thursday Friday Saturday Sunday
Number of requested admissions 11 7 10 10 11 13 9
Total
71
T(x)
S1 (x)
U2 (x)
10a
11a
12a
10a
11a
12a
10a
11a
12a
10a
11a
12a
1 0 0 0 1 3 0
0 0 0 0 0 2 0
0 0 0 0 0 1 0
1 0 1 1 1 1 0
1 0 0 0 1 1 0
0 0 0 0 0 1 0
0 1 0 0 0 0 0
0 1 0 0 0 0 1
0 1 1 1 0 0 1
0 1 0 0 0 0 0
0 1 0 0 0 0 0
0 1 0 0 0 0 1
5
2
1
5
3
1
1
2
4
1
1
2
10-beds
11-beds
S = 1, U = 1 Mean S.D.
U1 (x)
S = 1, U = 2
3.67 1.89
12-beds
S = 1, U = 1
S = 1, U = 2
2.33 0.47
S = 1, U = 1
S = 1, U = 2
2.0 1.41
1.33 0.47
For 10—12-beds and with the thresholds S = 1 and U = 1(U = 2), the mean score is equal to (5 + 5 + 1)/3 = 3.67, 2.33, 2.0 (1.33) and the S.D. is equal to 1.89, 0.47, 1.41 (0.47) with S.D. = (1/n) x2 − ( x)2 /n .Both the mean and the S.D. of the score reached a minimum at 12-beds for S = 1 and U = 2. a Number of beds.
2.5.2. National length of stay ratio (case-mix method) The number of patient-days was calculated according to the national length of stay from the DRGs database. For each DRGs of the hospital department, we applied the national DRGs length of stays. patients-days expected =
n 1
national DRGi LoS
The number of beds expected according to the national database is: N=
i
1 national DRGi LoS
number of days
3. Applications The data reported here come from a study of a urological surgery department and two internal medical departments and from a study published in the literature [41] (Fig. 1 and Table 2).
3.1. Surgery department (Fig. 1, Table 2) The survey was conducted over a period of 33 days. Eight hundred and fourteen patientdays were recorded, including 741 (91%) appropriate and 73 (9%) inappropriate patient-days
Table 2 Comparison between the ratio methods and the model Number of beds required
LoS ratio
DRGs ratio
Model
Surgery department Internal medicine departments
23 43
30 49
28 48
The number of beds proposed by the model is intermediate to those calculated by the two ratio methods.
(mean = 22.45/days; min = 15; max = 29). For this surgical department, which had 29-beds, the model suggested 28-beds and predicted 1 transfer due to lack of space per month and 1 day with 12 unoccupied beds. 3.1.1. Comparison with the LoS ratio According to the 741 patients-days, it would be necessary to have at least 23-beds (741/33). 3.1.2. Comparison using the national DRGs reference During these 33 days, 98 different types of DRGs were treated by this department. According to the national reference DRGs database, the LoS expected was 7.74, compared to the 5.88 days calculated for this department. According to this ratio, 30-beds would be required {[(7.74/5.88) × 741]/33}.
A simple method to optimize hospital beds capacity
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Fig. 1. Modelling the urological surgery department. T = number of patients transferred due to lack of space during the study period. S = number of days with S beds available for unscheduled admission. U = number of days with at least U unoccupied beds. Changes in mean and standard deviation of the three parameters (T, S, U) with a threshold of more than 12 unoccupied beds: the mean and the standard deviation both reach a minimum at 28-beds.
3.2. Internal medicine departments (Table 2) Two internal medicine departments with similar patient populations recorded the number of appropriate and inappropriate patient admissions over a 65-day period, as well as the number of patients who would have been appropriate for the departments but were admitted elsewhere in the hospital during the same period. In all, 3638 patient-days were recorded, including 2782 (76.5%) appropriate patient-days in the departments or elsewhere in the hospital and 856 (23.5%) inappropriate patientdays in the departments. The number of appropriate hospitalized patients on a given day in the two departments ranged from 36 to 50. For these two departments, having 54-beds in all, our method indicated that the optimal number of beds was 48 if all inappropriate patients were immediately directed to an appropriate de-
partment. Under this hypothesis, the method predicted that the two departments would see 1.87 transfers per month because of bed shortage, 0.94 days per month with full bed occupancy and 2.81 days per month with at least 9 unoccupied beds in the two departments considered together. 3.2.1. Comparison with the LoS ratio According to the 2782 patients-days, it would be necessary to have at least 43-beds (2782/65). 3.2.2. Comparison using the national DRGs as a reference During these 65 days, 85 different types of DRGs were treated by these two departments. According to the national reference DRGs database, the LoS expected was 9.1, compared to the 8.06 days calculated for these two departments. According to this ratio, 49-beds would be required {[(9.1/8.06) × 2782]/65}.
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Table 3a Data from St George’s paper [41]: number of cots occupied per day in special care baby unit vs. number of days (permission has been given to cite these data) Number of cots occupied
No. of days
5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23
3 6 14 29 38 23 31 43 33 31 27 20 19 15 14 8 7 2 2
Table 3b Data from St George’s paper [41]: number of beds occupied per day by number of days in accident hospital (permission has been given to cite these data) Number of beds occupied
No. of days
1 2 3 4 5 6 7 8 9 10 11 12 13 14
5 12 50 72 65 41 27 28 19 21 11 5 2 7
3.3. Comparison with methods based on occupancy target A full review of the literature revealed only one study giving adequate data for our comparison. In the paper by St George [41], two studies were carried out, one for the number of cots for a baby care unit and the second for the number of beds for an emergency department (Tables 3a and 3b).
Table 4 Comparison of the number of unoccupied beds between our method and the target beds activity used by St George Number of beds required
8 9 (our method) 10 (St. George) 11 (St. George) 12 (St. George)
Number of daily unoccupied beds (total in the year) 2.6 (949) 3.4 (1249) 4.3 (1568) 5.2 (1908) 6.2 (2259)
Number of days with more than 25% of unoccupied beds (%) 204 (55.9%) 245 (67.1%) 272 (74.5%) 300 (82.2%) 300 (82.2%)
In the first case, a mean of 13-beds were occupied daily. Our model suggests the same solution as that proposed by the study, i.e. 19 cots (Fig. 2a). In the second case, a mean of six-beds were occupied daily. The author recommended a 10—12beds model whereas our method selects the nine-beds model. (Fig. 3). The daily bed occupancy rate showed three peaks of activity (August, October and December) [41], which explain the 10—12-beds model choice. The consequence of the choice made by the author is that more than 25% of beds would be unoccupied more than 75% of the year, which is no more acceptable than the solution proposed by our method (Table 4).
4. Properties of the model 4.1. Decision method robustness Robustness is defined as the capacity of the model to select the same solution when outliers are introduced into the data. Truncated normal distributions were simulated. The size of the sample represents the number of days. Each value represents the number of patients per day. First, the optimal number of beds is calculated. Next, the maximum value of the sample is incremented of one unit and the number of virtual beds is recalculated using the same range of the threshold of unoccupied beds. The results in Table 5 showed that for a period of 1 year, the number of requested admissions modifying the previous solution must be at least twice the maximum of the sample and this value increases with the size of the sample. To illustrate these simulations, the data in the paper by St George [41] were analyzed. We changed 9 days with 13 cots occupied to 9 days with 40 admission requests. To cover 95% of the period study, it will be necessary to have 21 cots (Fig. 2b). Our method does not change the number of cots required, i.e. 19.
A simple method to optimize hospital beds capacity
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Fig. 2. (a) St George’s paper [41]: 19 cots were required using the target beds occupancy method. Our method selected the same solution as that proposed by the author. (b) St George’s paper [41]: 21 cots will be required to cover 95% of the 1-year study period, if 40 requests were added over a 9-day period.
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Fig. 3. St George’s paper [41]: 10—12-beds were required for a mean of six-beds occupied. Our method selected a nine-beds model.
Outlier data did not modify the solution selected.
4.2. Reality of the decision method We evaluated the reality of the model by simulating absurd situations. Table 5
We studied the hypothetical scenario of a department with exactly 12 hospitalized patients per day for 1 month and no patients at all for 5 months. With these data, contrary to common sense, there was no fixed optimal bed number unless a threshold of more than nine unoccupied beds
Truncated normal distributions
Normal distribution truncated, mean = 20 N; S.D.; [Min.—Max.]
Optimal number of beds selected (S, U)
Number of requested admissions changing the number of beds initially selected
N = 10; S.D. = 2; [18—23] N = 20; S.D. = 2; [17—25] N = 50; S.D. = 2; [16—25] N = 100; S.D. = 2; [12—24] N = 100; S.D. = 4; [11—29] N = 500; S.D. = 2; [14—27] N = 500; S.D. = 4; [9—36] N = 1000; S.D. = 2; [4—28] N = 1000; S.D. = 4; [8—34]
23 (1, 4) 24(1, 7) 22 (1, 2) 22 (1, 2) 27 (1, 12) 23 (1, 5) 27 (1, 12) 23 (1, 5) 28 (1, 15)
26 26 51 82 51 158 196 281 432
The size of the sample represents the number of days. Each value represents the number of patients per day. For each sample the maximum value is incremented of one until the number of beds selected is modified. This procedure simulates an unusual admissions request.
A simple method to optimize hospital beds capacity was accepted. Under this constraint, the nine-beds model represents the ‘optimal solution’.
5. Discussion The model described here was designed to determine the optimal number of beds for an existing situation. It is not intended as a tool to predict future needs based on anticipated changes or to evaluate the appropriateness of resource utilization. It is also not intended to be used to model the process of a care unit. This method has been successfully applied to an Intensive Care Department [38]. The objective of this paper was to extend this method to internal medicine and surgical departments. The number of beds needed for a medical or surgical department depends on many parameters. To solve this issue, complex operational frameworks have been developed whose strategy is to reproduce the activity of a department using simulation and to assess the immediate number of beds required for this department. These methods require many model input parameters, making them dependent on hypotheses concerning their parameter distribution. Moreover, these models do not provide a reproducible rule of decision to select the best solution. The implementation of these methods requires specific software and human resources that cannot be obtained without an institutional project. These limitations make these models inaccessible to non teaching hospitals. For all of these reasons, the medical authorities continue to use ratio methods for the sake of simplicity. The ratio method, which uses the average length of stay, overestimates the number of beds required by departments with a high average length of stay. Contrary to this, the case-mix ratio method overestimates the needs of departments, which select patients with the least costly cover. None of these methods take into account the fluctuation of recruitment over time. We used a different approach in which the hospital department is viewed as a container that must neither overflow (avoid transfers due to a full department) nor remain empty and we developed a method of decision. In order not to hypothesize about length of stay, we used the number of patients present on a given day as a statistical unit. We constructed a multicriteria score exploring the main criteria involved when choosing the number of beds for a hospital department (security, accessibility, productivity). In using this score, we accepted the hypothesis that each criterion could be translated into a single unit of measurement. Among the three parameters, two
47 deal with the department’s under-capacity (number of transfers and number of saturated days) and one deals with its excess capacity (number of days with too high a threshold of vacant beds). In the absence of weighting, this score gives priority to covering needs at the expense of efficiency. In the examples we studied, we did not use any weighting. This assumes that the three parameters are numerically equivalent. It will now be necessary to perform subsequent studies to determine the sensitivity of the decision when using parameter weighting. If one maximizes (or minimizes) a score, one accepts that the disparity between each criterion is of no importance. Thus, a score taking into account three criteria (3, 0, 0) would be of equal significance to a score (1 1 1) taking into account the same three criteria. Intuitively, such an optimization could result in incoherent decisions. To avoid this problem, it is necessary to minimize the disparity between the criteria. This naturally led us to establish our decision rule by minimizing the mean and standard deviation of our score. As a decision rule, we chose to impose a joint minimization of two functions, since this always leads to a single solution [40]. On the contrary, if the decision rule was not based on a joint minimization, we would have needed to define the score’s critical mean and standard deviation values, above which the solution would be rejected. This would mean setting confidence intervals for each minimum of each function. This could constitute a main line of development for the method. The optimal bed numbers provided by our model are highly realistic, with both real-life and simulated data. Application of the method to two internal medicine departments and one surgical department led to solutions intermediate to the number of beds calculated by the local and national length of stay ratio method. The comparability of our method’s findings with those of the sophisticated methods published in the literature raise the problem of the availability of tools (BOMPS© for Windows© [21] is not available; PROMPT© and TOCHSIM© cost more than £4500 [15]). The clinicians were in agreement with our recommendations. With the simulated data, the model refused to build a department with an unacceptable occupancy rate, for instance, beds unoccupied 80% of the time (5 out of 6 months). On the contrary, it suggested a logical 19-beds solution for a stay profile having a truncated normal distribution with a mean of 18 and a standard deviation of 1.5. The robustness of the model was evaluated by simulating a massive arrival of patients on a given day. This unlikely event did not change the solution given by the model. This robustness stems from the
48 model’s mathematical properties. However, if such an event was to occur, the solutions must be found using a network health organization. Our model provides a coherent, objective, reproducible, robust and evidence-based solution to a problem that depends on several variables. The parameters used in the model represent quantitative data only that are in no way specific to a given hospital department. Seasonal demand can be taken into account by stratifying the model. Another factor that contributes to the universality of our method is that the three parameters can be weighted and no hypothesis is made about LoS. Clearly, the ‘costs’ of patient transfers (risk of life-threatening events, resources used, etc.) differ widely between an ICU and an extended-stay department. The same is true of fixed ‘costs’ related to unoccupied beds or to the inability of the department to accommodate unscheduled admissions. A weighting procedure can be used to make the model suitable for all calculations of bed requirements in a healthcare system. Therefore revaluations are needed to take into account any modifications in recruitment or practices. In conclusion, we propose a decision method to select the most suitable number of beds, which is as simple as the ratio method and represents a real alternative. A software has been developed for this method [42], which is available free of charge and is extremely easy to use. A project is now being developed to use this method to optimize a multilevel perinatal care network.
Acknowledgements We thank Jan L. Talmon and the Editorial Board for constructive comments. We thank Nathalie Surer for conducting the ratio method analysis. This work was conducted at the PIMESP, Hospital St Jacques, CHU Nantes, 44093 Nantes Cedex 1, France.
References [1] R.A. Lyons, K. Wareham, H.A. Hutchings, E. Major, B. Ferguson, Population requirement for adult critical-care beds: a prospective quantitative and qualitative study, Lancet 355 (2000) 595—598. [2] A.L. Jung, N.S. Streeter, Total population estimate of newborn special-care bed needs, Pediatrics 75 (1985) 993—996.
J.M. Nguyen et al. [3] C. Plati, C. Lemonidou, M. Priami, G. Baltopoulos, J. Mantas, The intensive care units in greater Athens: needs and resources, Intensive Crit. Care Nurs. 12 (1996) 340—345. [4] J. Parmanum, D. Field, J. Rennie, P. Steer, National census of availability of neonatal intensive care, BMJ 321 (2000) 727—729. [5] C. Lampl, D. Klingler, E. Deisenhammer, E. Hagenbichler, L. Neuner, B. Pesec, Hospitalization of patients with neurological disorders and estimation of the need of beds and of the related costs in Austria’s non-profit hospitals, Eur. J. Neurol. 8 (2001) 701—706. [6] S.V. Williams, How many intensive care beds are enough? Crit. Care Med. 11 (1983) 412—416. [7] X.M. Huang, A planning model for requirement of emergency beds, IMA J. Math. Appl. Med. Biol. 12 (1995) 345—353. [8] A. Bagust, M. Place, J.W. Posnett, Dynamics of bed use in accommodating emergency admissions: stochastic simulation model, BMJ 319 (1999) 155—158. [9] J.M. Lamiell, Modeling intensive care unit census, Mil. Med. 160 (1995) 227—232. [10] E.P.C. Kao, G.G. Tung, Bed allocation in a public health care delivery service, Manage. Sci. (1981). [11] E. Milne, P. Whitty, Calculation of the need for paediatric intensive care beds, Arch. Dis. Child 73 (1995) 505—507. [12] G. Romanin-Jacur, P. Facchin, Optimal planning of a pediatric semi-intensive care unit via simulation, Eur. J. Op. Res. 29 (1987) 192—198. [13] E. Toussaint, G. Herengt, P. Gillois, F. Kohler, Method to determine the bed capacity, different approaches used for the establishment planning project in the University Hospital of Nancy, Med. Info. 10 (2001) 1404—1408. [14] G. Vassilacopoulos, A simulation model for bed allocation to hospital inpatient departments, Simulation (1985) 233—241. [15] A.X. Costa, S.A. Ridley, A.K. Shahani, P.R. Haper, V. De Senna, Mathematical modelling an simulation for planning critical care capacity, Anaesthesia 58 (2003) 320—327. [16] P.R. Harper, A framework for operational modelling of hospital resources, Health Care Manage. Sci. (2002) 165—173. [17] S. McClean, P. Millard, Modelling in-patient bed usage behavior in a department of geriatric medicine, Methods Inf. Med. (1993) 79—81. [18] S. McClean, P.H. Millard, A decision support system for bed-occupancy management and planning hospitals, IMA J. Math. Appl. Med. Biol. 12 (1995) 249—257. [19] E. El-Darzi, C. Vasilakis, T. Chaussalet, P.H. Millard, A simulation modelling approach to evaluating length of stay, occupancy, emptiness and bed blocking in a hospital geriatric department, Health Care Manage. Sci. 1 (1998) 143—149. [20] J.A. Garcia-Navarro, W. Thompson, Analysis of bed usage and occupancy following the introduction of geriatric rehabilitative care in a hospital in Huesca, Spain, Health Care Manage. Sci. (2001) 63—66. [21] P.H. Millard, G. Christodoulou, C. Jagger, G.W. Harrison, S. McClean, Modelling hospital and social care bed occupancy and use by elderly people in an English health district, Health Care Manage. Sci. (2001) 57—62. [22] P. Millard, Throuput in a department of geriatric medicine: a problem of time, space and behaviour, Health Trends (1991) 20—24. [23] M. Mackay, Practical experience with bed occupancy management and planning systems: an Australian view, Health Care Manage. Sci. (2001) 47—56. [24] G.W. Harrison, Implications of mixed exponetial occupancy distributions and patient flow models for health care planning, Health Care Manage. Sci. (2001) 37—45.
A simple method to optimize hospital beds capacity [25] F. Gorusnescu, S. McClean, P. Millard, A queueing model for bed-occupancy management and planning of hospitals, JORS 53 (2002) 19—24. [26] C. Vasilakis, E. El-Darzi, A simulation study of the winter bed crisis, Health Care Manage. Sci. (2001) 31—36. [27] J.F. Pendergast, W.B. Vogel, A multistage model of hospital bed requirements, Health Serv. Res. 23 (1988) 381—399. [28] J. Sorensen, Multi-phased bed modelling, Health Serv. Manage. Res. (1996) 61—67. [29] J.C. Lowery, J.B. Martin, Design and validation of a critical care simulation model, J. Soc. Health Syst. 3 (1992) 15— 36. [30] S. Groothuis, A. Hasman, P. van Pol, N. Lencer, J. Janssen, J. Jans, J. Stappers, W. Danssen, P. Doevensans, G. van Merode, Predicting capacities required in cardiology units for heart failure patients via simulation, Comput. Methods Programs Biomed. (2003) 1—17. [31] J.C. Ridge, S.K. Jones, M.S. Nielsen, A.K. Shahani, Capacity planning for intensive care units, Eur. J. Op. Res. (1998) 346—355. [32] M.B. Dumas, Simulation modelling for hospital bed planning, Simulation (1984) 233—241. [33] E. Marcon, S. Kharraja, N. Smolski, B. Luquet, J.P. Viale, Determining the number of beds in the postanesthesia care unit: a computer simulation flow approach, Anesth. Analg. 96 (2003) 1415—1423 (Table of contents). [34] M. Utley, S. Gallivan, K. Davis, P. Daniel, P. Reeves, J. Worrall, Estimating bed requirements for an intermediate care facility, Eur. J. Op. Res. 150 (2003) 92—100.
49 [35] P.S. Pulat, B.L. Foote, S. Kasap, G.L. Splinter, M.K. Lucas, Data and simulation modelling to determine the estimated number of primary care beds in Oklahoma, Top Health Inf. Manage. 19 (1999) 36—46. [36] D. Gove, D. Hewett, A. Shahani, Towards a model for hospital case-load decision support, IMA J. Math. Appl. Med. Biol. 12 (1995) 329—338. [37] J.M. Nguyen, P. Six, D. Antonioli, P. Lombrail, P. Le Beux Beds, Simulator 1.0: a software for the modelisation of the number of beds required for a hospital department, Stud. Health Technol. Inform. 95 (2003) 310—315. [38] J.M. Nguyen, P. Six, R. Parisot, D. Antonioli, F. Nicolas, P. Lombrail, A universal method for determining intensive care unit bed requirements, Intensive Care Med. 27 (2003) 27, doi:10.1007/s00134-003-1725-z. [39] J.M. Nguyen, P. Six, D. Antonioli, P. Glemain, G. Potel, P. Le Beux, P. Lombrail, A simple stochastic method for optimizing hospital beds capacity, 2003 International Conference on Health Policy Research: Methodological Issues in Health Services and Outcomes Research, Chicago, Illinois, USA, 2003. [40] J.M. Nguyen, P. Six, D. Antonioli, P. Le Beux, A non parametric method for the detection and classification of extreme values, XXIst International Biometric Conference, Freiburg (Germany), 2002, p. 145. [41] D. St George, How many beds? Helping consultants to estimate their requirements, BMJ 297 (1988) 729—731. [42] BedsSimulator, available at, http://www.sante.univnantes.fr/, 2003.
A simple method to optimize hospital beds capacity J.M. Nguyena,b,∗, P. Sixc, D. Antoniolib, P. Glemaind, G. Potele, P. Lombrailb, P. Le Beuxf a
Laboratory of Medical Statistics and Informatics, 1 rue Gaston Veil, 44035 Nantes Cedex 01, France PIMESP, St. Jacques Hospital, CHU Nantes, 44093 Nantes Cedex 01, France c Department of Medical Information, CHU Angers, 49033 Angers Cedex 01, France d Department of Urology, CHU Nantes, 44093 Nantes Cedex 01, France e Department of Internal Medicine, CHU Nantes, 44093 Nantes Cedex 01, France f Laboratory of Medical Informatics, CHU Rennes, 35000 Rennes Cedex 01, France b
Received 15 April 2004 ; received in revised form 13 September 2004; accepted 14 September 2004 KEYWORDS Hospital bed capacity; Multicriteria decision; Optimization; Software
Summary Objective: The number of acute hospital beds is determined by health authorities using methods based on ratios and/or target bed occupancy rates. These methods fail to consider the variability in hospitalization demands over time. On the other hand, the implementation of sophisticated models requires the decision concerning the number of beds to be made by an expert. Our aim is to develop a new method that is as simple to use as the ratio method while minimizing the roundabout approaches of these methods. Method: A score was constructed with three parameters: number of transfers due to lack of space, number of days with no possibility for S unscheduled admissions and number of days with at least a threshold of U unoccupied beds. The optimal number of beds is the number for which both the mean and the standard deviation of the score reach their minimum. We applied this method to two internal medicine departments and one urological surgery department and we compared the solutions proposed by this method with those put forward by the ratio method. Results: The solutions proposed by this method were intermediate to those calculated by the local and national length of Stays ratio methods. Simulating an unusual increase in admission requests had no consequence on the bed number selected, indicating that the method was robust. Conclusion: Our tool represents a real alternative to the ratio methods. A software has been developed and is now available for use. © 2004 Elsevier Ireland Ltd. All rights reserved.
* Corresponding author. Tel.: +33 2 40 84 69 35; fax: +33 2 40 84 69 40.
E-mail address: [email protected] (J.M. Nguyen). URL: http://www.sante.univ-nantes.fr/med/stat/.
1386-5056/$ — see front matter © 2004 Elsevier Ireland Ltd. All rights reserved. doi:10.1016/j.ijmedinf.2004.09.001
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1. Introduction Determining the number of beds ‘objectively’ needed for a ward or a hospital is a complex problem because there are many kinds of hospital beds within each length-of-stay category (long-term, medium-term, short-term) and several specialties each with its own patient management profile. Furthermore, within each specialty, differences in operation occur across hospital departments because of practice variations. Additional differences result from the constraints imposed by site-specific characteristics of patient recruitment and the structural, organizational, geographical, historical and political environment in which the hospital operates. Determining the number of beds needed for a hospital department raises three types of problems. First, a good compromise must be found in the setting up of an operational hospital department, between the number of patients transferred because of a full department and the number of unoccupied beds. Second, changes in needs must be anticipated. Third, evidence that admissions are appropriate must be obtained or at least an evaluation of the rate of inappropriate stays must be performed. Several methods have been suggested to solve this issue. The first approach is based on a census quantifying the number of patients [1—6]. The number of beds is then determined using the ratio method. This method does not take into account the fluctuation of requested admissions over time. Stochastic simulations have also been used to quantify the number of patients [7—14]. Here, the number of beds is determined by a critical probability. The second approach uses models describing the utilization of hospital resources [15,16]. These published methods primarily aim to provide models for the patient cover process and the operation of hospital departments. Their strategy is therefore to reproduce the activity of a department using simulation and to assess the immediate number of beds required for the department. The major problem with these methods is not the number of beds needed per given day, but the number of beds to be allocated to a department over a period corresponding to the time necessary to implement a health plan. This is the issue the medical authorities need to address. The implementation of these models thus requires substantial data and logistics. Another approach is to model patient flow through the hospital using multi compartmental or multi state models [17—36]. In the latter two approaches, the methodology used to determine the optimal number of beds requires an expert opinion in all cases. Such a judg-
J.M. Nguyen et al. ment depends on the author and not on an objective and reproducible decision criterion. For these reasons, and for the sake of simplicity, the medical authorities always use ratio methods and not the models described above. Indeed, most hospitals are supposed to use ratio-based methods to determine the number of beds they require [13]. Ratio-based methods are frequently criticized because they disregard local specificities and are often considered arbitrary. In fact, the ratio method, which uses the average length of stays, overestimates the number of beds required by departments with a high average length of stay. Contrary to this, the national ratio method (case-mix) overestimates the needs of departments, which select patients with the least costly cover. Our aim is to recommend a method [37,38] that is as simple to use as ratio methods while minimizing their biased approaches. Such a tool would be a real alternative to the ratio methods. Part of this work was presented at the HPSS ICHPR 2003 conference [39].
2. Methods The optimal number of beds is defined as the number for which the following three criteria are met: the number of unoccupied beds is not excessive (productivity), the number of patients transferred because of full bed occupancy is not excessive (security), and one or perhaps several beds are available for unscheduled admissions (accessibility). A score is constructed based on these three parameters. The optimal number of beds is the number for which both the means and the standard deviation of the score reach a minimum.
2.1. Construction of the score The parameter formulation is based on the following observations. The statistical unit common to all departments is the number of operational days. Overhead and personnel charges depend only on the number of beds and on the specialty of the department. These charges do not fluctuate, except if beds are temporarily closed (fixed ‘costs’) and can therefore be expressed using a logical function with an acceptability threshold: “Beyond the threshold, the situation is no longer acceptable and each day counts”. The parameter assessing accessibility can also be expressed using a threshold: “to reach the objective of a public service, the department must have at least one or more possibility for unscheduled admission per day".
A simple method to optimize hospital beds capacity The statistical unit for ‘fixed cost’ could be the number of days beyond a threshold. On the contrary, patients who are transferred because the department is full generate costs (health risks, additional economic costs, etc.) that depend on the number of transfers: ‘Each transfer counts’. Thus, this parameter is best expressed as a function whose measurement unit is the number of transfers and no threshold is needed. In conclusion, there are two parameters to assess department capacity (the number of patients transferred and the number of days with empty beds) and one parameter to assess the possibility of unscheduled admission. 1. The number of transfers due to full bed occupancy (security). The number of transfers on a given day is the total number of patients transferred to another department for the sole reason that no beds are available. 2. The number of days with less than S beds for unscheduled admission (accessibility). Department saturation is expressed as the number of days on which no new patients can be admitted. The number of beds that must be available for new patients depends on the size and structure of the department. For instance, a medical—surgical ICU needs at least two unoccupied beds, one for a medical patient and one for a surgical patient. Similarly, a large department may need two or more unoccupied beds to ensure continuity of care. In smaller departments and in those that admit only one type of patient, a single unoccupied bed is sufficient for unscheduled admissions. 3. The number of days with more than U unoccupied beds (productivity). This parameter is the number of days on which the number of unoccupied beds is above a predefined threshold associated with an unacceptable cost.
2.2. Data For each day and for each department, the number of patients in the department, the number of patients transferred due to lack of space and the number of unsatisfied admission requests are collected. Conversely, appropriateness of each stay is defined by the physician. Patients waiting for placement or transfer to an adequate department are defined as inappropriate stay. These patients are excluded from the analysis. The study period must be representative of the activity of the ward. The total number of requested admissions is the sole data needed for the model.
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2.3. Algorithm The algorithm of the model is based on the upgrading of one virtual bed at each step and the calculation of the score for each saturation threshold and each unoccupied bed threshold. For each virtual number of beds and for each day, the algorithm counts the number of patients transferred because of full bed occupancy, the number of days on which the department is considered to be saturated (less than S beds available), and the number of days on which the number of unoccupied beds is too high (more than U beds unemployed). An example of such a calculation is given in Table 1.
2.4. Decision rule An intuitive decision-making method based on three criteria as different from one another as those defined above is to select the criterion that maximizes (or minimizes) all three criteria, if these are quantitative. This approach is equivalent to maximizing (or minimizing) the mean of the values for the three criteria. However, a decision based on minimizing the mean favors the extreme values. One way to solve this problem is to penalize the extreme values. Another solution is to minimize the standard deviation of the data. The optimization of our model is based on the minimization of both the standard deviation and the mean score for each number of beds in the model. The optimal number of beds is the number for which the minimization of both the mean and the standard deviation is reached with the lowest threshold of U unoccupied beds. For S = 0, the model with a number of beds equal to the maximum number of admission requests is the lowest productive model. For this model, the mean and the standard deviation scores are equal to 0 and the threshold U of unoccupied beds is equal to the range of the data [40]. The solution is thus only based on the statistical properties of the model. No hypothesis concerning the distribution of stays or the admission or target bed occupancy is needed.
2.5. Comparison with ratio methods 2.5.1. Local length of stay ratio The number of beds required is defined as N= =
LoS × number of patients number of days number of patients-days number of days
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Table 1
Example of a score calculation for 10—12-beds and for the thresholds S = 1, U = 1 and 2
Day
Monday Tuesday Wednesday Thursday Friday Saturday Sunday
Number of requested admissions 11 7 10 10 11 13 9
Total
71
T(x)
S1 (x)
U2 (x)
10a
11a
12a
10a
11a
12a
10a
11a
12a
10a
11a
12a
1 0 0 0 1 3 0
0 0 0 0 0 2 0
0 0 0 0 0 1 0
1 0 1 1 1 1 0
1 0 0 0 1 1 0
0 0 0 0 0 1 0
0 1 0 0 0 0 0
0 1 0 0 0 0 1
0 1 1 1 0 0 1
0 1 0 0 0 0 0
0 1 0 0 0 0 0
0 1 0 0 0 0 1
5
2
1
5
3
1
1
2
4
1
1
2
10-beds
11-beds
S = 1, U = 1 Mean S.D.
U1 (x)
S = 1, U = 2
3.67 1.89
12-beds
S = 1, U = 1
S = 1, U = 2
2.33 0.47
S = 1, U = 1
S = 1, U = 2
2.0 1.41
1.33 0.47
For 10—12-beds and with the thresholds S = 1 and U = 1(U = 2), the mean score is equal to (5 + 5 + 1)/3 = 3.67, 2.33, 2.0 (1.33) and the S.D. is equal to 1.89, 0.47, 1.41 (0.47) with S.D. = (1/n) x2 − ( x)2 /n .Both the mean and the S.D. of the score reached a minimum at 12-beds for S = 1 and U = 2. a Number of beds.
2.5.2. National length of stay ratio (case-mix method) The number of patient-days was calculated according to the national length of stay from the DRGs database. For each DRGs of the hospital department, we applied the national DRGs length of stays. patients-days expected =
n 1
national DRGi LoS
The number of beds expected according to the national database is: N=
i
1 national DRGi LoS
number of days
3. Applications The data reported here come from a study of a urological surgery department and two internal medical departments and from a study published in the literature [41] (Fig. 1 and Table 2).
3.1. Surgery department (Fig. 1, Table 2) The survey was conducted over a period of 33 days. Eight hundred and fourteen patientdays were recorded, including 741 (91%) appropriate and 73 (9%) inappropriate patient-days
Table 2 Comparison between the ratio methods and the model Number of beds required
LoS ratio
DRGs ratio
Model
Surgery department Internal medicine departments
23 43
30 49
28 48
The number of beds proposed by the model is intermediate to those calculated by the two ratio methods.
(mean = 22.45/days; min = 15; max = 29). For this surgical department, which had 29-beds, the model suggested 28-beds and predicted 1 transfer due to lack of space per month and 1 day with 12 unoccupied beds. 3.1.1. Comparison with the LoS ratio According to the 741 patients-days, it would be necessary to have at least 23-beds (741/33). 3.1.2. Comparison using the national DRGs reference During these 33 days, 98 different types of DRGs were treated by this department. According to the national reference DRGs database, the LoS expected was 7.74, compared to the 5.88 days calculated for this department. According to this ratio, 30-beds would be required {[(7.74/5.88) × 741]/33}.
A simple method to optimize hospital beds capacity
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Fig. 1. Modelling the urological surgery department. T = number of patients transferred due to lack of space during the study period. S = number of days with S beds available for unscheduled admission. U = number of days with at least U unoccupied beds. Changes in mean and standard deviation of the three parameters (T, S, U) with a threshold of more than 12 unoccupied beds: the mean and the standard deviation both reach a minimum at 28-beds.
3.2. Internal medicine departments (Table 2) Two internal medicine departments with similar patient populations recorded the number of appropriate and inappropriate patient admissions over a 65-day period, as well as the number of patients who would have been appropriate for the departments but were admitted elsewhere in the hospital during the same period. In all, 3638 patient-days were recorded, including 2782 (76.5%) appropriate patient-days in the departments or elsewhere in the hospital and 856 (23.5%) inappropriate patientdays in the departments. The number of appropriate hospitalized patients on a given day in the two departments ranged from 36 to 50. For these two departments, having 54-beds in all, our method indicated that the optimal number of beds was 48 if all inappropriate patients were immediately directed to an appropriate de-
partment. Under this hypothesis, the method predicted that the two departments would see 1.87 transfers per month because of bed shortage, 0.94 days per month with full bed occupancy and 2.81 days per month with at least 9 unoccupied beds in the two departments considered together. 3.2.1. Comparison with the LoS ratio According to the 2782 patients-days, it would be necessary to have at least 43-beds (2782/65). 3.2.2. Comparison using the national DRGs as a reference During these 65 days, 85 different types of DRGs were treated by these two departments. According to the national reference DRGs database, the LoS expected was 9.1, compared to the 8.06 days calculated for these two departments. According to this ratio, 49-beds would be required {[(9.1/8.06) × 2782]/65}.
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Table 3a Data from St George’s paper [41]: number of cots occupied per day in special care baby unit vs. number of days (permission has been given to cite these data) Number of cots occupied
No. of days
5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23
3 6 14 29 38 23 31 43 33 31 27 20 19 15 14 8 7 2 2
Table 3b Data from St George’s paper [41]: number of beds occupied per day by number of days in accident hospital (permission has been given to cite these data) Number of beds occupied
No. of days
1 2 3 4 5 6 7 8 9 10 11 12 13 14
5 12 50 72 65 41 27 28 19 21 11 5 2 7
3.3. Comparison with methods based on occupancy target A full review of the literature revealed only one study giving adequate data for our comparison. In the paper by St George [41], two studies were carried out, one for the number of cots for a baby care unit and the second for the number of beds for an emergency department (Tables 3a and 3b).
Table 4 Comparison of the number of unoccupied beds between our method and the target beds activity used by St George Number of beds required
8 9 (our method) 10 (St. George) 11 (St. George) 12 (St. George)
Number of daily unoccupied beds (total in the year) 2.6 (949) 3.4 (1249) 4.3 (1568) 5.2 (1908) 6.2 (2259)
Number of days with more than 25% of unoccupied beds (%) 204 (55.9%) 245 (67.1%) 272 (74.5%) 300 (82.2%) 300 (82.2%)
In the first case, a mean of 13-beds were occupied daily. Our model suggests the same solution as that proposed by the study, i.e. 19 cots (Fig. 2a). In the second case, a mean of six-beds were occupied daily. The author recommended a 10—12beds model whereas our method selects the nine-beds model. (Fig. 3). The daily bed occupancy rate showed three peaks of activity (August, October and December) [41], which explain the 10—12-beds model choice. The consequence of the choice made by the author is that more than 25% of beds would be unoccupied more than 75% of the year, which is no more acceptable than the solution proposed by our method (Table 4).
4. Properties of the model 4.1. Decision method robustness Robustness is defined as the capacity of the model to select the same solution when outliers are introduced into the data. Truncated normal distributions were simulated. The size of the sample represents the number of days. Each value represents the number of patients per day. First, the optimal number of beds is calculated. Next, the maximum value of the sample is incremented of one unit and the number of virtual beds is recalculated using the same range of the threshold of unoccupied beds. The results in Table 5 showed that for a period of 1 year, the number of requested admissions modifying the previous solution must be at least twice the maximum of the sample and this value increases with the size of the sample. To illustrate these simulations, the data in the paper by St George [41] were analyzed. We changed 9 days with 13 cots occupied to 9 days with 40 admission requests. To cover 95% of the period study, it will be necessary to have 21 cots (Fig. 2b). Our method does not change the number of cots required, i.e. 19.
A simple method to optimize hospital beds capacity
45
Fig. 2. (a) St George’s paper [41]: 19 cots were required using the target beds occupancy method. Our method selected the same solution as that proposed by the author. (b) St George’s paper [41]: 21 cots will be required to cover 95% of the 1-year study period, if 40 requests were added over a 9-day period.
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Fig. 3. St George’s paper [41]: 10—12-beds were required for a mean of six-beds occupied. Our method selected a nine-beds model.
Outlier data did not modify the solution selected.
4.2. Reality of the decision method We evaluated the reality of the model by simulating absurd situations. Table 5
We studied the hypothetical scenario of a department with exactly 12 hospitalized patients per day for 1 month and no patients at all for 5 months. With these data, contrary to common sense, there was no fixed optimal bed number unless a threshold of more than nine unoccupied beds
Truncated normal distributions
Normal distribution truncated, mean = 20 N; S.D.; [Min.—Max.]
Optimal number of beds selected (S, U)
Number of requested admissions changing the number of beds initially selected
N = 10; S.D. = 2; [18—23] N = 20; S.D. = 2; [17—25] N = 50; S.D. = 2; [16—25] N = 100; S.D. = 2; [12—24] N = 100; S.D. = 4; [11—29] N = 500; S.D. = 2; [14—27] N = 500; S.D. = 4; [9—36] N = 1000; S.D. = 2; [4—28] N = 1000; S.D. = 4; [8—34]
23 (1, 4) 24(1, 7) 22 (1, 2) 22 (1, 2) 27 (1, 12) 23 (1, 5) 27 (1, 12) 23 (1, 5) 28 (1, 15)
26 26 51 82 51 158 196 281 432
The size of the sample represents the number of days. Each value represents the number of patients per day. For each sample the maximum value is incremented of one until the number of beds selected is modified. This procedure simulates an unusual admissions request.
A simple method to optimize hospital beds capacity was accepted. Under this constraint, the nine-beds model represents the ‘optimal solution’.
5. Discussion The model described here was designed to determine the optimal number of beds for an existing situation. It is not intended as a tool to predict future needs based on anticipated changes or to evaluate the appropriateness of resource utilization. It is also not intended to be used to model the process of a care unit. This method has been successfully applied to an Intensive Care Department [38]. The objective of this paper was to extend this method to internal medicine and surgical departments. The number of beds needed for a medical or surgical department depends on many parameters. To solve this issue, complex operational frameworks have been developed whose strategy is to reproduce the activity of a department using simulation and to assess the immediate number of beds required for this department. These methods require many model input parameters, making them dependent on hypotheses concerning their parameter distribution. Moreover, these models do not provide a reproducible rule of decision to select the best solution. The implementation of these methods requires specific software and human resources that cannot be obtained without an institutional project. These limitations make these models inaccessible to non teaching hospitals. For all of these reasons, the medical authorities continue to use ratio methods for the sake of simplicity. The ratio method, which uses the average length of stay, overestimates the number of beds required by departments with a high average length of stay. Contrary to this, the case-mix ratio method overestimates the needs of departments, which select patients with the least costly cover. None of these methods take into account the fluctuation of recruitment over time. We used a different approach in which the hospital department is viewed as a container that must neither overflow (avoid transfers due to a full department) nor remain empty and we developed a method of decision. In order not to hypothesize about length of stay, we used the number of patients present on a given day as a statistical unit. We constructed a multicriteria score exploring the main criteria involved when choosing the number of beds for a hospital department (security, accessibility, productivity). In using this score, we accepted the hypothesis that each criterion could be translated into a single unit of measurement. Among the three parameters, two
47 deal with the department’s under-capacity (number of transfers and number of saturated days) and one deals with its excess capacity (number of days with too high a threshold of vacant beds). In the absence of weighting, this score gives priority to covering needs at the expense of efficiency. In the examples we studied, we did not use any weighting. This assumes that the three parameters are numerically equivalent. It will now be necessary to perform subsequent studies to determine the sensitivity of the decision when using parameter weighting. If one maximizes (or minimizes) a score, one accepts that the disparity between each criterion is of no importance. Thus, a score taking into account three criteria (3, 0, 0) would be of equal significance to a score (1 1 1) taking into account the same three criteria. Intuitively, such an optimization could result in incoherent decisions. To avoid this problem, it is necessary to minimize the disparity between the criteria. This naturally led us to establish our decision rule by minimizing the mean and standard deviation of our score. As a decision rule, we chose to impose a joint minimization of two functions, since this always leads to a single solution [40]. On the contrary, if the decision rule was not based on a joint minimization, we would have needed to define the score’s critical mean and standard deviation values, above which the solution would be rejected. This would mean setting confidence intervals for each minimum of each function. This could constitute a main line of development for the method. The optimal bed numbers provided by our model are highly realistic, with both real-life and simulated data. Application of the method to two internal medicine departments and one surgical department led to solutions intermediate to the number of beds calculated by the local and national length of stay ratio method. The comparability of our method’s findings with those of the sophisticated methods published in the literature raise the problem of the availability of tools (BOMPS© for Windows© [21] is not available; PROMPT© and TOCHSIM© cost more than £4500 [15]). The clinicians were in agreement with our recommendations. With the simulated data, the model refused to build a department with an unacceptable occupancy rate, for instance, beds unoccupied 80% of the time (5 out of 6 months). On the contrary, it suggested a logical 19-beds solution for a stay profile having a truncated normal distribution with a mean of 18 and a standard deviation of 1.5. The robustness of the model was evaluated by simulating a massive arrival of patients on a given day. This unlikely event did not change the solution given by the model. This robustness stems from the
48 model’s mathematical properties. However, if such an event was to occur, the solutions must be found using a network health organization. Our model provides a coherent, objective, reproducible, robust and evidence-based solution to a problem that depends on several variables. The parameters used in the model represent quantitative data only that are in no way specific to a given hospital department. Seasonal demand can be taken into account by stratifying the model. Another factor that contributes to the universality of our method is that the three parameters can be weighted and no hypothesis is made about LoS. Clearly, the ‘costs’ of patient transfers (risk of life-threatening events, resources used, etc.) differ widely between an ICU and an extended-stay department. The same is true of fixed ‘costs’ related to unoccupied beds or to the inability of the department to accommodate unscheduled admissions. A weighting procedure can be used to make the model suitable for all calculations of bed requirements in a healthcare system. Therefore revaluations are needed to take into account any modifications in recruitment or practices. In conclusion, we propose a decision method to select the most suitable number of beds, which is as simple as the ratio method and represents a real alternative. A software has been developed for this method [42], which is available free of charge and is extremely easy to use. A project is now being developed to use this method to optimize a multilevel perinatal care network.
Acknowledgements We thank Jan L. Talmon and the Editorial Board for constructive comments. We thank Nathalie Surer for conducting the ratio method analysis. This work was conducted at the PIMESP, Hospital St Jacques, CHU Nantes, 44093 Nantes Cedex 1, France.
References [1] R.A. Lyons, K. Wareham, H.A. Hutchings, E. Major, B. Ferguson, Population requirement for adult critical-care beds: a prospective quantitative and qualitative study, Lancet 355 (2000) 595—598. [2] A.L. Jung, N.S. Streeter, Total population estimate of newborn special-care bed needs, Pediatrics 75 (1985) 993—996.
J.M. Nguyen et al. [3] C. Plati, C. Lemonidou, M. Priami, G. Baltopoulos, J. Mantas, The intensive care units in greater Athens: needs and resources, Intensive Crit. Care Nurs. 12 (1996) 340—345. [4] J. Parmanum, D. Field, J. Rennie, P. Steer, National census of availability of neonatal intensive care, BMJ 321 (2000) 727—729. [5] C. Lampl, D. Klingler, E. Deisenhammer, E. Hagenbichler, L. Neuner, B. Pesec, Hospitalization of patients with neurological disorders and estimation of the need of beds and of the related costs in Austria’s non-profit hospitals, Eur. J. Neurol. 8 (2001) 701—706. [6] S.V. Williams, How many intensive care beds are enough? Crit. Care Med. 11 (1983) 412—416. [7] X.M. Huang, A planning model for requirement of emergency beds, IMA J. Math. Appl. Med. Biol. 12 (1995) 345—353. [8] A. Bagust, M. Place, J.W. Posnett, Dynamics of bed use in accommodating emergency admissions: stochastic simulation model, BMJ 319 (1999) 155—158. [9] J.M. Lamiell, Modeling intensive care unit census, Mil. Med. 160 (1995) 227—232. [10] E.P.C. Kao, G.G. Tung, Bed allocation in a public health care delivery service, Manage. Sci. (1981). [11] E. Milne, P. Whitty, Calculation of the need for paediatric intensive care beds, Arch. Dis. Child 73 (1995) 505—507. [12] G. Romanin-Jacur, P. Facchin, Optimal planning of a pediatric semi-intensive care unit via simulation, Eur. J. Op. Res. 29 (1987) 192—198. [13] E. Toussaint, G. Herengt, P. Gillois, F. Kohler, Method to determine the bed capacity, different approaches used for the establishment planning project in the University Hospital of Nancy, Med. Info. 10 (2001) 1404—1408. [14] G. Vassilacopoulos, A simulation model for bed allocation to hospital inpatient departments, Simulation (1985) 233—241. [15] A.X. Costa, S.A. Ridley, A.K. Shahani, P.R. Haper, V. De Senna, Mathematical modelling an simulation for planning critical care capacity, Anaesthesia 58 (2003) 320—327. [16] P.R. Harper, A framework for operational modelling of hospital resources, Health Care Manage. Sci. (2002) 165—173. [17] S. McClean, P. Millard, Modelling in-patient bed usage behavior in a department of geriatric medicine, Methods Inf. Med. (1993) 79—81. [18] S. McClean, P.H. Millard, A decision support system for bed-occupancy management and planning hospitals, IMA J. Math. Appl. Med. Biol. 12 (1995) 249—257. [19] E. El-Darzi, C. Vasilakis, T. Chaussalet, P.H. Millard, A simulation modelling approach to evaluating length of stay, occupancy, emptiness and bed blocking in a hospital geriatric department, Health Care Manage. Sci. 1 (1998) 143—149. [20] J.A. Garcia-Navarro, W. Thompson, Analysis of bed usage and occupancy following the introduction of geriatric rehabilitative care in a hospital in Huesca, Spain, Health Care Manage. Sci. (2001) 63—66. [21] P.H. Millard, G. Christodoulou, C. Jagger, G.W. Harrison, S. McClean, Modelling hospital and social care bed occupancy and use by elderly people in an English health district, Health Care Manage. Sci. (2001) 57—62. [22] P. Millard, Throuput in a department of geriatric medicine: a problem of time, space and behaviour, Health Trends (1991) 20—24. [23] M. Mackay, Practical experience with bed occupancy management and planning systems: an Australian view, Health Care Manage. Sci. (2001) 47—56. [24] G.W. Harrison, Implications of mixed exponetial occupancy distributions and patient flow models for health care planning, Health Care Manage. Sci. (2001) 37—45.
A simple method to optimize hospital beds capacity [25] F. Gorusnescu, S. McClean, P. Millard, A queueing model for bed-occupancy management and planning of hospitals, JORS 53 (2002) 19—24. [26] C. Vasilakis, E. El-Darzi, A simulation study of the winter bed crisis, Health Care Manage. Sci. (2001) 31—36. [27] J.F. Pendergast, W.B. Vogel, A multistage model of hospital bed requirements, Health Serv. Res. 23 (1988) 381—399. [28] J. Sorensen, Multi-phased bed modelling, Health Serv. Manage. Res. (1996) 61—67. [29] J.C. Lowery, J.B. Martin, Design and validation of a critical care simulation model, J. Soc. Health Syst. 3 (1992) 15— 36. [30] S. Groothuis, A. Hasman, P. van Pol, N. Lencer, J. Janssen, J. Jans, J. Stappers, W. Danssen, P. Doevensans, G. van Merode, Predicting capacities required in cardiology units for heart failure patients via simulation, Comput. Methods Programs Biomed. (2003) 1—17. [31] J.C. Ridge, S.K. Jones, M.S. Nielsen, A.K. Shahani, Capacity planning for intensive care units, Eur. J. Op. Res. (1998) 346—355. [32] M.B. Dumas, Simulation modelling for hospital bed planning, Simulation (1984) 233—241. [33] E. Marcon, S. Kharraja, N. Smolski, B. Luquet, J.P. Viale, Determining the number of beds in the postanesthesia care unit: a computer simulation flow approach, Anesth. Analg. 96 (2003) 1415—1423 (Table of contents). [34] M. Utley, S. Gallivan, K. Davis, P. Daniel, P. Reeves, J. Worrall, Estimating bed requirements for an intermediate care facility, Eur. J. Op. Res. 150 (2003) 92—100.
49 [35] P.S. Pulat, B.L. Foote, S. Kasap, G.L. Splinter, M.K. Lucas, Data and simulation modelling to determine the estimated number of primary care beds in Oklahoma, Top Health Inf. Manage. 19 (1999) 36—46. [36] D. Gove, D. Hewett, A. Shahani, Towards a model for hospital case-load decision support, IMA J. Math. Appl. Med. Biol. 12 (1995) 329—338. [37] J.M. Nguyen, P. Six, D. Antonioli, P. Lombrail, P. Le Beux Beds, Simulator 1.0: a software for the modelisation of the number of beds required for a hospital department, Stud. Health Technol. Inform. 95 (2003) 310—315. [38] J.M. Nguyen, P. Six, R. Parisot, D. Antonioli, F. Nicolas, P. Lombrail, A universal method for determining intensive care unit bed requirements, Intensive Care Med. 27 (2003) 27, doi:10.1007/s00134-003-1725-z. [39] J.M. Nguyen, P. Six, D. Antonioli, P. Glemain, G. Potel, P. Le Beux, P. Lombrail, A simple stochastic method for optimizing hospital beds capacity, 2003 International Conference on Health Policy Research: Methodological Issues in Health Services and Outcomes Research, Chicago, Illinois, USA, 2003. [40] J.M. Nguyen, P. Six, D. Antonioli, P. Le Beux, A non parametric method for the detection and classification of extreme values, XXIst International Biometric Conference, Freiburg (Germany), 2002, p. 145. [41] D. St George, How many beds? Helping consultants to estimate their requirements, BMJ 297 (1988) 729—731. [42] BedsSimulator, available at, http://www.sante.univnantes.fr/, 2003.