Background
Theoretical background
Research question
Methods
Concept of efficiency
Linear program
DRG 1 | DRG 2 | DRG 3 | DRG.. | DRG n-2 | DRG n-1 | DRG n | ||
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Revenues | x1⋅d1 | x2⋅d2 | x3⋅d3 | … | xn-2⋅dn-2 | xn-1⋅dn-1 | xn⋅dn | |
– | Direct Cost | x1⋅a1 | x2⋅a2 | x3⋅a3 | … | xn-2⋅an-2 | xn-1⋅an-1 | xn⋅an |
= | Contribution I | x1⋅(d1-a1) | x2⋅(d2-a2) | x3⋅(d3-a3) | … | xn-2⋅(dn-2-an-2) | xn-1⋅(dn-1-an-1) | xn⋅(dn-an) |
– | DRG-fixed cost | FD1 | FD2 | FD3 | … | FDn-2 | FDn-1 | FDn |
= | Contribution II | x1⋅(d1-a1)- FD1 | x2⋅(d2-a2)- FD2 | x3⋅(d3-a3)- FD3 | … | xn-2⋅(dn-2-an-2)- FDn-2 | xn-1⋅(dn-1-an-1)- FDn-1 | xn⋅(dn-an)- FDn |
– | department cost | FA1 | … | FAb | ||||
= | Contribution III | x1⋅(d1-a1)- FD1 + x2⋅(d2-a2)- FD2 - FA1 | … | xn-2⋅(dn-2-an-2)-FDn-2 + xn-1⋅(dn-1-an-1)- FDn-1 + xn⋅(dn-an)-FDn – Fab | ||||
– | hospital-fixed cost | FK | ||||||
= | profit/loss |
\( \sum \limits_{j=1}^n\left({d}_j-{a}_j\right)\cdot {x}_j-\sum \limits_{j=1}^n{FB}_j-\sum \limits_{p=1}^b{FA}_p- FK \)
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Health economic data
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Nursing care has to be available 365 days a year, 24 h a day. During the core time (two shifts a day) two nurses, otherwise one. With a weekly working time of 40 h and 8 weeks of absence from work (holiday, training, illness) this results in a minimum of nine nurses.
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An average nursing care of 5.2 h per day and patient and an average hospital stay of 3.1 days was assumed (data from the hospital controlling department).
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One paediatrician specialist and one paediatrician in training have to be available at any time. This results in a minimum staffing of five physicians plus a senior physician.
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45 min physicians’ time per day per patient for medical history, diagnostics, therapy decisions, monitoring, and documentation are assumed.
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Midwives have to be available 365 days a year, 24 h a day. With a weekly working time of 40 h and 8 weeks of absence from work (holiday, training, illness) this results in a minimum staffing of five midwifes. Since normally the head of the ward (senior midwife, included in the fixed costs) also cares for births, the minimum staffing can be reduced to four midwifes.
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The duration of a birth takes 14.5 h on average, considering all modes of delivery (spontaneous or assisted vaginal delivery, caesarean section).
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It is assumed that the number of delivery rooms is sufficiently large (no capacity limitation).
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As in the paediatric ward, a minimum staffing of nine nurses is necessary.
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For each patient, nursing care of 2.8 h per day plus 1.8 h per day for a new-born is assumed.
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The caesarean section rate is 35.6%. This result in an average length of stay of 4.9 days (normal birth: 3.6 days, caesarean section: 7.3 days).
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The minimum staffing is five doctors and one senior physician, analogue to the paediatric ward.
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The assumed physician-working time per vaginal delivery is 60 min.
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The assumed physician-working time for a caesarean section is 300 min (including 120 min anaesthesiologist). Additionally, 300 min nursing care are needed.
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Paediatrics: 152.59€/case
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Obstetrics: 244.64€/case
Number of hospitals |
3
| ||
Number of DRGs |
2
| ||
j = 1 |
delivery
| ||
j = 2 |
paediatrics
| ||
Capacity of personal category i in hospital k, i = 1..5, k = 1..3 |
k
ik
| ||
i = 1 | 1760 h | ||
i = 2 | 1760 h | ||
i = 3 | 1760 h | ||
i = 4 | 1760 h | ||
i = 5 | 1760 h | ||
Time consumption of personal category i for production of one unit of DRG j in hospital k, j = 1..2; i = 1..5; k = 1..3 | j = 1 | j = 2 | |
1 | 16.2 | ||
2 | 2.33 | ||
3 | 14.50 | ||
4 | 22.62 | ||
5 | 1.00 | ||
Rebate of DRG j, j = 1..2 | d1 = 1505.68 d2 = 3135.10 | ||
Direct cost for one case of DRG j in hostpial k, j = 1..2; k = 1..3 | a1,k = 152.59 a2,k = 244.64 | ||
Department fixed costs of department j in hospital k, j = 1..2; k = 1..3 | FA1k = 576,500 FA2k = 679,000 | ||
Fixed cost per bed for DRG j in hospital k, j = 1..2; k = 1..3 | bA1k = 15,253.47 bA2k = 20,212.24 | ||
Cost per staff of category i in hospital k, i = 1..5; k = 1..3 |
w
ik
| ||
i = 1 | 56,700 | ||
i = 2 | 101,250 | ||
i = 3 | 56,700 | ||
i = 4 | 56,700 | ||
i = 5 | 101,250 | ||
Average lengths of stay in DRG j, j = 1..2 | v1 = 3.10 v2 = 4.92 |
Geographical analyses and population data
Results
Pediatrics | Obstetrics | |||
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Number of cases in the hospital | Cases in the catchment area | Number of births in the hospital | Births in the catchment area | |
Greifswald | 1820 | 1192 | 800 | 518 |
Wolgast | 1057 | 926 | 357 | 203 |
Anklam | 496 | 898 | 280 | 293 |
Total | 3373 | 3016 | 1437 | 1014 |
Basic model including all three hospitals
Pediatric wards | Obstetric wards | Total [€] | |||||||
---|---|---|---|---|---|---|---|---|---|
Beds | Cases | Capacity utilization | Contribution margin [€] | Beds | Cases | Capacity utilization | Contribution margin [€] | ||
Wolgast | 18 | 1057 | 50% | − 494,091 | 11 | 357 | 44% | −1,169,487 | −1,663,578 |
Anklam | 16 | 496 | 26% | −1,165,970 | 6 | 280 | 63% | −1,290,992 | −2,456,962 |
Greifswald | 24 | 1820 | 64% | 49,899 | 24 | 800 | 45% | − 378,569 | − 328,670 |
Total | 58 | 3373 | 47% | −1,610,162 | 41 | 1437 | 51% | −2,839,048 | −4,449,210 |
Pediatric | Obstetric | Total [€] | |||||||
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Beds | Cases | Capacity utilization | Contribution margin [€] | Beds | Cases | Capacity utilization | Contribution margin [€] | ||
Wolgast | 9 | 1057 | 100% | − 356,810 | 5 | 357 | 100% | −1,048,214 | −1,405,024 |
Anklam | 5 | 496 | 100% | − 998,182 | 5 | 280 | 100% | −1,270,780 | −2,268,962 |
Greifswald | 16 | 1820 | 100% | 171,927 | 11 | 800 | 100% | −115,809 | 56,118 |
Total | 30 | 3373 | 100% | −1,186,065 | 21 | 1437 | 100% | −2,434,803 | −3,617,868 |
Scenarios with 1 and 2 hospitals
Pediatric ward | Obstetric ward | Total [€] | |||||||
---|---|---|---|---|---|---|---|---|---|
Beds (N) | Cases (N) | Capacity Utilization | Contribution Margin [€] | Beds (N) | Cases (N) | Capacity Utilization | Contribution Margin [€] | ||
Hospital | 29 | 3373 | 100% | 1,281,188 | 20 | 1437 | 100% | 806,408 | 2,087,597 |
Accessibility of the hospitals
Discussion
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Perspective: The LP models takes the perspective of the provider and does not consider societal, financer or patient perspectives. Thus, it can only optimize the system from the perspective of the providers (and partly of the financers), while other costs (e.g. transport of patients) are not include in the model.
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Constant service units: the model assumes that the number of service units in the catchment area of the three hospitals is constant. In reality, the demand will also depend on the travel distance. For a model with only three hospitals and a catchment area where all hospitals are accessible in reasonable distances, this is acceptable. Extending the model to bigger regions and more hospitals would require the definition of a distance decay curve or a maximum travel distance.
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Linearity: Linear programming assumes that all functions are linear. Consequently, economies or diseconomies of scale cannot be considered. At the same time, the models consider efficiency gains only through the digression of the fixed costs. Efficiency gains through learning effects (e.g. more routine because of larger numbers of cases) could not be included.
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Decision-Model: The linear model optimizes the efficiency under certain constraints. However, it does not allow comparing the relative efficiency of hospitals based on empirical data. Other methods, such as Data Envelopment Analysis (DEA) and Stochastic Frontier Analysis (SFA) are designed to find the bench-marks. Consequently, DEA and SFA can give interesting insights into relative efficiency based on empirical data. They might be in particular helpful to compare the efficiency of the hospitals before and after the recommendations are implemented. This calls for further research.
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Quality assumption: The linear model assumes that the quality of services which can be provided in all three hospitals is equal and does not depend on volume. This is an assumption, but our experience with “normal deliveries” and “general pediatrics” underlines that this assumption is correct.
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Data: For the calculation of the models, average values for Germany were used to calculate costs because real data was only partly available. The salaries of nurses, midwifes, and physicians are based on collective agreements, these data are quite valid. Although the salaries between the hospitals might be comparable, there are differences in the structure of the staff between a university hospital and small regional hospitals. Other fixed and variable costs are likely to be different among the hospitals. Therefore, real comparability between the hospitals is limited.
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Catchment area: The assumption for the calculation of the catchment areas, that all patients visit the nearest hospital, is certainly not completely valid. Patients may be willing to travel longer distances to be treated in the university hospital or to give birth in a hospital with special offers.
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Accessibility: which location has a good accessibility for the inhabitants of the region both by car and public transport;
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Availability of paediatric and obstetric wards in neighbouring regions;
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Medical equipment of the hospital where the paediatric and obstetric wards are located: a better medical equipment of the hospital could allow the treatment of more severe or complex patients;
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Other wards and departments in the hospital: it should be assessed, in which hospital the paediatric and obstetric wards fit best in the entire portfolio of health services of the hospital;
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Social and economic factors in the region.
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A close cooperation between small hospitals in rural regions and a compensatory alignment of the services and wards;
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A close cooperation between inpatient and outpatient providers with mutual support and compensation of services, For example is it possible to support the cooperation between outpatient midwifes and obstetric wards to ensure obstetric care in rural regions [38];
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Implementation of tele-medical connections between small hospitals and hospitals with maximum care to ensure medical standards in small hospitals maintaining only few medical specialties [39].
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Improvement of location in the public road system as well as public transport to and from hospitals [2].
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Improvement of emergency systems in order to safeguard rapid transport from the homes of patients to the hospital [40].