A decision support system for bed-occupancy management and planning hospitals

IMA Journal of Mathematics Applied in Medicine & Biology (1995) 12, 249-257 A decision support system for bed-occupancy management and planning hospitals SALLY MCCLEAN Division of Mathematics, School of Information and Software Engineering, University of Ulster, Belfast, UK PETER H. MILLARD Division of Geriatric Medicine, St George's Hospital Medical School, London, UK [Received 31 August 1994 and in revised form 1 August 1995] The planning of services within a hospital is a complex task which relies on the availability of accurate data. Such data on patterns of bed occupancy enable us to develop tools which assess performance measures based on activity within a hospital and its beds, and hence they improve the efficiency of bed management and they facilitate the more effective use of resources. We report the development of a bedoccupancy-modelling package that uses a mathematical model to separate the pattern of bed occupancy in hospitals into acute, rehabilitative, and long-stay components. The bed-occupancy management and planning system analyses data downloaded from the patient administrative system. A query-by-example function is then used to separate the data into meaningful subgroups, such as age groups, or the various specialties. The underlying model is a mixed exponential where the number of terms in the mixture corresponds to the number of stages in the hospital spell, typically, acute, rehabilitative, and long stay. The software has been used to analyse data from a number of different specialties and services. A 'what-if capacility allows the planner to assess the effect of changes prior to their introduction. Keywords: bed occupancy; hospital-bed usage; decision support system. 1. Introduction With the split of health-care management in the UK into purchasers and providers, with an accompanying focus on encouraging a business approach to the provision of health care, it has become increasingly important to provide tools which assist in the development of an efficient and cost-effective service. In order to plan and provide health services there is a need for a credible methodology which is supported by appropriate data. The purchaser needs such a methodology to manage resources in a flexible manner (Dillon, 1994) while the provider wants a means of facilitating patient management and discharge planning (Millard, 1994). In each case, it is essential that the professionals are not overwhelmed by a multitude of irrelevant data. Data collection should be restricted to a necessary minimum and supported by a decision support system (DSS) which enables health-service managers to make efficient decisions based on up-to-date and relevant information. A modelling approach underpins such a methodology by using appropriately collected data to develop a mathematical model which enables us to understand 249 ( Oxford University Press 1995

250 SALLY McCLEAN AND PETER. H. MILLARD the health-care system, to predict its future behaviour, and to investigate possible scenarios. Our model explains the empirical finding that the time spent in hospital since admission by geriatric in-patients may be described by a two-term-mixedexponential distribution. The original model assumed two stages of patient behaviour, where patients are initially admitted to a short-stay or rehabilitative state, from which they either die or are discharged, or they are admitted to a second long-stay state (Harrison & Millard, 1991). Further exploration of the boundaries of the empirical observations led to the expansion of the model to include a three-term state (Harrison, 1994). The model may be used to estimate the average numbers and lengths of stay for short-, medium-, and long-stay patients and to assess the affect of changes in policy such as improving rehabilitation or converting long-stay beds to short-stay beds. The model has been implemented in a stand-alone DSS software package for bed-occupancy, management, and planning software (BOMPS) which uses manually entered or electronically downloaded data to create bed-occupancy censuses. The model has been applied to a number of specialties such as obstetrics, general surgery, orthopaedics, general and geriatric medicine, psychiatry, residential homes, and the home-help service (see Millard & McClean, 1994). The DSS has the advantage of using data which is easy to collect, to verify, and to use. The underlying model is explanatory: it explains the interaction between staff, patients, and resources, thus facilitating discussion between managers and healthservice professionals. A number of publications in the statistical literature (Harrison & Millard, 1991; McClean & Millard, 1993a, b; Irvine et al., 1994) have established the mathematical aspects of the methodology. Our model can therefore be seen as going a long way towards explaining patient behaviour, thereby providing a useful tool for planning hospital services. 2. The model The theoretical model which underpins our DSS explains why the empirical pattern of length of stay in the occupied beds fits a sum of two exponentials; conversely, the empirical distribution obtained from occupancy data, can be used to study the effect of various policy decisions on both immediate and future admission rates for the department, and it shows the benefits of policies which reduce long-stay patient numbers by improving rehabilitation (Harrison & Millard, 1991). The model was developed following an empirical observation that the pattern of duration of bed occupancy is best fitted by a two-term-mixed-exponential distribution of the form y= Ae- Bx + Ce- Dx, where x is the patient's length of stay in hospital, up to the present, and y is the number of patients who have been in the department for at least x days. This pattern is illustrated for all of the departments of geriatric medicine in the South-West Thames region in Fig. 1 where the squares represent the actual data, and the continuous curve is the fitted two-term mixed exponential. The presence of a two-term-mixed-exponential pattern is clinically explainable because the in-patient work consists of two distinct forms of clinical activity, acute/rehabilitative and long-stay care, which are organizationally distinct and have very different resource

A DECISION SUPPORT SYSTEM FOR BED OCCUPANCY 251 2500 i Midnight bed occupancy 2335 500 1000 Length of stay of in-patients FIG. 1. (- -) Double-exponential fit to in-patient bed occupancy in a geriatric department in the South-West Thames Region on 5 December 1988, and ( ) cumulative sum of in-patients. (Reprinted with the. permission of the Royal Society of Medicine from Modelling Hospital Resources Use; a different approach to the planning of health care systems (1994), P. H. Millard & S. I. McClean, eds., p. 54.) needs. Hospital planning models based on the average length of stay and the turnover per allocated bed assume that patients all move through the system at the same rate, and ignoring the inherent heterogeneity in patient behaviour leads to planning mistakes (Vere, 1983). It is hypothesized that geriatric patients pass through two distinct stages, and patient behaviour is therefore described by a two-component model (Millard, 1992). The first state represents the clinical care of the short-stay patients, that is, those who are undergoing assessment and rehabilitation, while the second stage represents bed usage by the long-stay patients where bed availability usually comes about by death (Hodkinson & Hodkinson, 1980). In the model B and D correspond to the respective flow rates. The midnight bed-state facility generated by the main frame patient administration systems (PASs), maintained in all National Health Service hospitals provides data on admission x and occupancy y, thus permitting us to calculate the parameters of the model using routinely available data. The model may then be used to predict future movements of patients through the hospital system. Essential aspects of the model are: (i) the concept that fairly homogeneous groups of patients all progress at much the same rate; (ii) the idea that discharge rates vary depending on the destination; (iii) the belief that discharge practices vary from hospital to hospital; (iv) the observation that the length of stay in hospital beds can be described by exponential distributions.

252 SALLY MCCLEAN PETER. H. MIL LA (a) A 0 N rn (b) A o N N 2 r 2 N 2 (c) A o N \ f 1 N 2 r 2 N 2 \ v 2N 2 «. r 3 N 3 FIG 2. Compartmental-flow models. (Reprinted with the permission of the Royal Society of Medicine from Modelling Hospital Resources Use; a different approach to the planning of health care systems (1994), P. H. Millard & S. I. McClean, eds., p. 55.). The flow of patients through one-, two-, and three-compartment models is illustrated in Fig. 2, where all patients are initially admitted to acute care, for diagnosis and assessment. From here, they are either discharged or they die, at a rate r, or they are converted to long-stay patients at a rate v. Accordingly, to maintain a stable state, the rate of conversion from medium-stay to long-stay patients must equal the discharge (death) rate in long-stay patients. By making a number of plausible assumptions about discharge rates it is possible to find equations giving the numbers of patients and their length of stay in the different groups. From the model we may also calculate simple measures of performance, such as the expected length of stay on admission, which is \/v + r for acute patients or \/d for long-stay patients. The model is fitted to occupancy data using a least-squares method. A visual and statistical analysis is used to determine the best fit and to estimate the model parameters, thus enabling us to use the model to assess the performance of the department and to evaluate the effect of possible changes, planned or otherwise, in

A DECISION SUPPORT SYSTEM FOR BED OCCUPANCY 253 the system as a whole. When a change is made, the admission rate should approach the new equilibrium value, but it may take several years to reach it. Thus the long-term result of a modified practice may be quite different from the immediate short-term result. Harrison & Millard (1991) present a scenario in which the patients are treated faster but no change is made in the proportion entering long stay. First, the admission rate increases sharply, but then admissions gradually decline as the beds fill up with long-stay patients. 3. Development of the DSS The DSS was developed to simplify the stages of data analysis and to provide a framework to help hospital staff to evaluate the relative advantages of different policy decisions, such as whether to increase treatment rates or to decrease the fraction of patients who are retained as long-stay patients. A personal-computer-based DSS, known as BOMPs has been developed as St George's Hospital Medical School. This DSS has a Paradox 3.5 run-time database with a C+ + mathematics core. Using the model, managers may assess the resource and planning implications of long-stay patients and better grasp the underlying assumptions about acute, rehabilitative, and long-stay care. To increase the applicability of the software and to facilitate its use across disciplines, the BOMPS package allows for the possibility of fitting one-, two-, or three-stage models and deciding which is most appropriate. A specialty or ward group of patients may therefore be identified as: (i) single stream, that is, acute, rehabilitative, or long-stay patients only; (ii) double stream, that is, acute and rehabilitation, or rehabilitation and long stay (the default situation usually encountered in geriatric medicine); (iii) triple stream, where we may have acute, medium-stay, and long-stay or short, long, and very long stay. In addition, using the query-by-example function, we may break data down and analyse it by diagnosis, specialty, age, sex, or consultant, thus providing a facility for looking at the behaviour of different groups of patients and making comparisons. The system allows these comparisons to be made using a graphical, tabular, or report format. Using the model described in the previous section, the package produces statistics for each group giving the actual bed usage and the estimated acute and long-stay bed usage thus allowing us to evaluate and compare the performance of different groups of patients. An essential feature of BOMPS modelling is the provision of a 'what-if facility which enables managers to assess the impact of a proposed change. Typical questions which can be answered by using the 'what-if facility are: How many less patients might we treat next year if we close ten beds? If we stop admitting patients to a ward, how long will it be before the ward is empty?

254 SALLY MCCLEAN AND PETER. H. MILLARD What increase in the rate of discharge would counterbalance bed closures? How many more patients would we be able to admit if we increased the proportion rehabilitated by 5%? 4. Using the DSS The main menu of BOM PS is displayed as a top-bar menu which contains all the available main commands displayed as pull-down menus. The menus are organized in a top-down hierarchical manner, so that one menu leads to another submenu. The eight command menus are as follows: INPUT TOOLS QUERY BOMPS GRAPHS REPORTS HELP EXIT The eight main commands have the following functions. INPUT allows the entry of data into the DSS, manually, electronically in ASCII code or from Paradox database files. TOOLS allows the manipulation of data in the DSS; tables can be viewed, created, sorted, modified, renamed, copied, deleted, emptied, and prepared for analysis. QUERY allows simple query-by-example, for example, by age, sex, or specialty groupings. BOMPS fits the model to the data using the least-squares method and alters the derived parameters to perform the optimum displays; the fit is first of a single-exponent equation, then the calculated parameters are used to test for the presence of double or triple fits; the goodness of fit is displayed visually as well as statistically. GRAPH presents the results graphically. REPORTS produces standard reports. HELP provides a HELP facility. EXIT leaves the DSS. The DSS may therefore be seen to consist of a combination of standard database functions for inputting, processing, and reporting on data and a modelling capability which allows health-service managers to describe the health-care system and carry out 'what-if analyses. Since completing the initial work on geriatric departments, the occupancy distribution in many other types of departments has been measured. In most departments the cumulative bed-occupancy data fits a sum of two exponentials but the specific rate coefficients vary widely from one department to another, depending on the characteristic treatment times and the fraction of the patients who need long-term care. Thus, in general surgery, the calculated best fit to the bed-occupancy data for the 150 patients occupying general surgical beds at St George's Hospital showed two groups of patients. Although the expected length of stay for all patients on admission

A DECISION SUPPORT SYSTEM FOR BED OCCUPANCY 255 TABLE 1 Comparative bed-usage statistics estimated from the mixed-exponential fit between the curve and the bed-occupancy data in geriatric medicine, general medicine, and general surgery Specialty Bed-occupancy statistics Geriatric medicine General medicine General surgery Numbers Numbers of occupied beds Calculated number of beds Admissions per day Annual admissions Group I in-patients Group II in-patients Discharges per day group I patients Discharges per day group II patients Rehabilitation benefit 163 156.0 2.8 1033.0 100.2 55.9 2.8 0.05 21.0 228 222.8 21.6 7878.8 155.1 67.7 20.0 1.6 4.2 150 146.2 34.8 12692.1 95.7 50.4 50.4 1.9 6.3 Expected average stay in days Overall average Average group I patients Average group II patients 55.1 35.4 1158.0 10.3 7.2 43.6 4.2 2.8 26.4 Half-life in days Group I patients Group II patients 24.2 802.3 4.6 29.9 1.5 18.0 Percentages Group I patients Admitted patients who will be group I only Admitted patients who will become group II Group II patients 64.2 98.3 1.7 35.8 69.6 92.8 7.2 30.4 65.5 94.5 5.5 34.5 Rates Admission per day per bed Releases per in-patient per day group I Conversion per in-patient per day group I to group II Releases per in-patient per day from group II 0.18 0.028 0.005 0.048 0.097 0.129 0.010 0.023 0.238 0.343 0.020 0.038 is estimated to be 4.2 days, a two-compartment model predicts 65.5% of patients will have short stays with an expected stay of 2.8 days and 34.5% will have longer stays with an expected stay of 26.4 days. Similarly, a two-component model is found in general medicine. From data for 228 patients occupying general medical beds at St George's Hospital, the overall estimated length of stay was 10.3 days. But 69.6% of the patients stayed 7.2 days while 30.4% had an expected length of stay of 43.6 days. In contrast, the 163 geriatric medical beds had 64.2% of in-patients with a stay of 35.4 days and 35.8% of patients will stay 1158 days. The difference illustrates why overlooking bed occupancy leads to mistakes. Thus the model showed that the key to the management of hospital beds lies in the care of complex patients. Though

256 SALLY MCCLEAN AND PETER. H. MILLARD TABLE 2 A 'what-if' analysis showing the changes that would compensate for a 10% reduction in allocated beds Specialty Geriatric medicine General medicine General surgery Performance changed Current Future Current Future Current Future Group I stay (days) Group I rehabilitated (%) Group II stay (days) 35.4 98.2 1158.0 29.9 98.8 833.8 7.2 92.8 43.6 6.2 94.5 29.2 2.8 94.5 26.4 2.3 96.3 18.8 Time to realization (days) 1461 207 131 extreme complexity represents only a small proportion of patients admitted 1.7% in geriatric medicine, 7.2% in general medicine, and 5.5% in general surgery complex patients occupy one-third of the beds (see Table 1). In a few departments, such as psychiatry, it is necessary to use a sum of three exponentials to fit the occupancy distribution, suggesting that there are not only patients who require long-term rehabilitation but that a few patients require even longer-term residency care. In a 600-bed psychiatric hospital, bed-occupancy analysis showed that the average stay of long-term residents in psychiatry is 100 days, two years, or 25 years. 4.1 ' What-if analysis Table 2 shows the changes that would compensate for a 10% reduction in allocated beds for the department in Table 1. Notice that three different approaches achieve the same result. Shortening the length of stay of group I patients involves speeding up the discharge of 1033 geriatric patients, 7879 general medical patients, and 12 692 surgical patients. Whereas improving in-patient management requires a 0.6% rehabilitative improvement in outcome in geriatric medicine, a 1.7% improvement in general medicine, and a 1.8% improvement in general surgery. Understanding this paradox is the key to improving health care. 5. Conclusions Health-service managers are currently using the DSS to assess the impact of hospital closures in London, and the College of Physicians is using it to assess the impact of the Community Care Act on patient throughput in hospitals. The DSS which we have developed facilitates tactical management of hospitals by allowing the short-stay and long-stay components of bed occupancy to be evaluated. However it also supports strategic management by allowing various scenarios to be assessed. We may thus gain insight into the nature of hospital planning and provide a more effective and efficient service.

A DECISION SUPPORT SYSTEM FOR BED OCCUPANCY 257 REFERENCES DILLON, A., 1994. What the general manager needs. In: Bed Occupancy Modelling and Planning (P. H. Millard & S. I. McClean, eds.), pp. 3-5. London: The Royal Society of Medicine Press. HARRISON, G. W., 1994. Compartment models of hospital patient occupancy patterns. In: Bed Occupancy Modelling and Planning (P. H. Millard & S. I. McClean, eds.), pp. 53-61. London: The Royal Society of Medicine Press. HARRISON, G. W., & MILLARD, P. H., 1991. Balancing acute and long stay care: The mathematics of throughput in departments of geriatric medicine. Methods Inform. Med. 30,211-8. HODKINSON, H. M., & HODKINSON, I.. 1980. Death and discharge from a geriatric department. Age Ageing 9, 22-229. IRVINE, V., MCCLEAN, S., & MILLARD, P. H., 1994. Stochastic models for geriatric in-patient behaviour. IMA J. Math. Appl. Med. Biol. 11, 207-16. MCCLEAN, S. I., & MILLARD, P. H., 1993a. Modelling in-patient bed usage behaviour in a department of Geriatric Medicine. Methods Inform. Med. 32, 79-81. MCCLEAN, S. I., & MILLARD, P. H., 1993b. Patterns of length of stay after admission in geriatric medicine: an event history approach. The Statistician 42, 263-74. MILLARD, P. H., 1992. Throughput in a department of geriatric medicine: A problem of time, space and behaviour. Health Trends 24, 20-24. MILLARD, P. H., 1994. What the clinician needs. In: Bed Occupancy Modelling and Planning (P. H. Millard & S. I. McClean, eds.), pp. 13-17. London: Royal Society of Medicine Press. MILLARD, P. H., & MCCLEAN, S. I. (eds.) 1994. Bed Occupancy Modelling and Planning. London: The Royal Society of Medicine Press. VERE, D. W., 1983. Assessing and allocating beds in acute medicine in East London. Br. Med. J. 287, 849-50.