Economic efficiency versus accessibility: Planning of the hospital landscape in rural regions using a linear model on the example of paediatric and obstetric wards in the northeast of Germany
Abstract
Background
Costs for the provision of regional hospital care depend, among other things, on the population density and the maximum reasonable distance to the nearest hospital. In regions with a low population density, it is a challenge to plan the number and location of hospitals with respect both to economic efficiency and to the availability of hospital care close to residential areas.
We examined whether the hospital landscape in rural regions can be planned on the basis of a regional economic model using the example which number of paediatric and obstetric wards in a region in the Northeast of Germany is economically efficient and what would be the consequences for the accessibility when one or more of the three current locations would be closed.
Methods
A model of linear programming was developed to estimate the costs and revenues under different scenarios with up to three hospitals with both a paediatric and an obstetric ward in the investigation region. To calculate accessibility of the wards, geographic analyses were conducted.
Results
With three hospitals in the study region, there is a financial gap of €3.6 million. To get a positive contribution margin for all three hospitals, more cases have to be treated than the region can deliver. Closing hospitals in the parts of the region with the smallest population density would lead to reduced accessibility for about 8% of the population under risk.
Conclusions
Quantitative modelling of the costs of regional hospital care provides a basis for planning. A qualitative discussion to the locations of the remaining departments and the implementation of alternative healthcare concepts should follow.
Keywords
Regional hospital planning Economic efficiency Care close to residence Linear model, paediatric wards, obstetric wardsAbbreviations
 DEA
Data Envelopment Analysis
 DRG
Diagnosis Related Groups
 GDRG
German Diagnosis Related Groups
 InEK
Institut für das Entgeltsystem im Krankenhaus (in English: Institute for the hospital remuneration system)
 LP
Linear programming
 SFA
Stochastic Frontier Analysis
Background
In Germany, the provision of adequate inpatient healthcare is generally determined by federal law (§ 1 of the Hospital Financing Act) whereas hospital planning (including geographic location, number of beds, departments, wards, and specializations of the hospitals) is the responsibility of the 16 federal states of Germany. Hospital plans serve as an instrument to reach an adequate level of healthcare of inpatient care in determined regions [1].
In the catchment area of many rural hospitals, the population is decreasing and aging simultaneously. In consequence, hospitals in such regions have to face decreasing capacity utilization especially in paediatric and obstetric wards, which can cause both economic problems and a lower quality of medical care [2].
This situation may lead to a conflict between the necessity to concentrate capacities for quality and economic reasons and the need to provide for sufficiently available healthcare in rural regions. To find a transparent and fair balance between quality of care, cost effectiveness and accessibility, evidencebased planning of the hospital landscape is necessary.
Theoretical background
Important parameters for hospital planning are the number of inhabitants and the population density in a region, as well as the age distribution of the regional population. The relationship between population density and number of beds has been the foundation of hospital planning as early as 1946 when the United States Congress passed the “Hospital Survey and Construction Act” which became well known as the Hill–Burton Act. The formula, which was used to determine the number of beds for a hospital or region, has currently become the standard of regional hospital planning and is the foundation of hospital plans in Germany. It calculates the demand for hospital beds of a specific department i in a given region [3]:
B_{i} = Number of beds of department i required
P = Population
h_{i} = Hospital admission rate of department i [per 1000 inhabitants]
V_{i} = Average length of stay in department i in days
a_{i} = Occupancy rate in department i
r = Maximum distance [km]
AT = Total area of the region [km^{2}]
P = Population in the region
h = Admission rate
K = Total hospital cost [€]
F = Fixed cost per hospital [€]
v = variable cost per case [€]
Research question
In this analysis, we consider different scenarios in a databased planning model of paediatric and obstetric wards in the rural region Western Pomerania in the Northeast of Germany. In 2016, this region had three hospitals with wards for paediatrics and obstetrics. One of the hospitals is a university hospital (paediatric and obstetrics wards each have 24 beds); the other two hospitals are small hospitals of primary care (paediatric wards: 18 and 16 beds, respectively; obstetrics wards: 11 and 6 beds, respectively). A longrunning political discussion concerns the needed number and locations of the wards.
The main research question of this analysis is which number of paediatric and an obstetric ward in this region is economically efficient and what would be the consequences for the accessibility of paediatric and obstetric inpatient care when one or more locations would be closed. To examine this question, three different scenarios with up to three hospital wards of the respective medical discipline were analysed.
Methods
To answer the research question, healtheconomic and geographical methods were combined. To address the economic parts of the research question, a model of linear programming (LP) was calculated to estimate the costs and revenues under different scenarios with up to three hospitals with both a paediatric and an obstetric ward in the investigation region [5, 6].
To calculate catchment areas and accessibility of the wards for the regional population, geographic analyses were conducted based on a geographical information system.
Concept of efficiency
The model described in the following section maximizes the efficiency of a set of hospitals in a region. Our analysis takes the perspective of the provider (i.e. hospitals). Other perspectives, such as the society, financer (i.e. health insurance schemes) or patients are not considered in the model. Consequently, the model presented here cannot contemplate all aspects of economic efficiency and will in particular ignore some costs, such as transport costs for patients. In the subsection “geographical analyses and population data” we will reflected on the distances which are partly reflecting the travel costs.
Generally, efficiency (E) can be defined as quotient of results (R) and resources; this quotient has to be maximized. If the results are defined as constant and the value of resources is expressed in costs (C), the efficiency quotient can be reduced to a cost minimization [7]. Our model assumes that the service units of the hospital(s) are given, i.e., all patients will be treated but the location of treatment might change, i.e.,
The provider perspective allows reducing the efficiency to a returnoninvestment (RoI) formula. The numerator are the revenues (price p times quantity q) of the hospital, the denominator are the fixed (K_{f}) and variable costs (variable unit cost v times quantity q). Efficiency is maximized by maximizing this quotient. However, as both numerator and denominator are currency units, we can also reduce this problem to the maximization of the difference between revenues and costs. Under the assumption that a hospital does not challenge its complete existence but merely optimizes its service portfolio, fixed costs are not decisionrelevant and the difference expresses the marginal contribution (m) defined as the difference between price and variable costs [8].
Consequently, we can reduce the problem of maximizing efficiency in a hospital system to a linear program under the assumption that we concentrate on the provider perspective and safeguard that results are constant [9]. At the same time, we have to assume that factor costs and variable costs are constant, i.e., models of production planning disregard economies or diseconomies of scale. Most models of production planning, furthermore, assume that fixed costs are given, but this is – as we will see in the following subsection – no prerequisite.
Linear program
About 50 years ago, a first model of linear programming (LP) was developed to allocate resources in hospitals efficiently. This development of a rational allocation model was almost in parallel with the development of the DRG (Diagnosis Related Groups) system by Robert B. Fetter at the Department of Operations Research of Yale University. DRGs describe a classification system for a lump sum billing procedure, with which hospital cases are assigned to case groups on the basis of medical data [10].
Further models followed [11, 12, 13].These models were scientifically interesting but of very little practical use as the computer capacity of that time prevented realistic LPs to be developed. In particular, the number of binary variables was limited. Consequently, several decades passed before the idea of calculating an optimum allocation in a hospital was takenup again by Meyer in 1996 [14]. Later, Meyer & Harfner showed the potential of these models for horizontal integration but they never applied it to a regional system in rural areas. The model was more relevant for urban areas where accessibility was of no interest [6].
In principle, the model maximizes for each hospital the marginal contribution. The German hospital financing system is dual, i.e., buildings, vehicles and equipment are paid by the government while running expenditures are paid by the health insurance funds based on the GDRGsystem (German Diagnosis Related Groups). Thus, only running expenditure are relevant for this analysis. Depreciation for buildings, vehicles and equipment can be ignored. However, even among the costs recovered by the DRG, the majority is fixed or stepfixed, i.e., they remain stable for some variation of outputs and then jump to a higher level where they will be stable again until another threshold of outputs is reached so that they jump again. Our model distinguishes them accordingly (e.g. each department has fixed costs while personnel are stepfixed). The DRG itself is a price per service unit and does not distinguish fixed and variable costs [15, 16].
Each hospital can treat patients of n different DRGs and receives the respective rebate d_{j} (j = 1..n).
Modell of marginal contribution analysis in a hospital
DRG 1  DRG 2  DRG 3  DRG..  DRG n2  DRG n1  DRG n  

Revenues  x_{1}⋅d_{1}  x_{2}⋅d_{2}  x_{3}⋅d_{3}  …  x_{n2}⋅d_{n2}  x_{n1}⋅d_{n1}  x_{n}⋅d_{n}  
–  Direct Cost  x_{1}⋅a_{1}  x_{2}⋅a_{2}  x_{3}⋅a_{3}  …  x_{n2}⋅a_{n2}  x_{n1}⋅a_{n1}  x_{n}⋅a_{n} 
=  Contribution I  x_{1}⋅(d_{1}a_{1})  x_{2}⋅(d_{2}a_{2})  x_{3}⋅(d_{3}a_{3})  …  x_{n2}⋅(d_{n2}a_{n2})  x_{n1}⋅(d_{n1}a_{n1})  x_{n}⋅(d_{n}a_{n}) 
–  DRGfixed cost  FD_{1}  FD_{2}  FD_{3}  …  FD_{n2}  FD_{n1}  FD_{n} 
=  Contribution II  x_{1}⋅(d_{1}a_{1}) FD_{1}  x_{2}⋅(d_{2}a_{2}) FD_{2}  x_{3}⋅(d_{3}a_{3}) FD_{3}  …  x_{n2}⋅(d_{n2}a_{n2}) FD_{n2}  x_{n1}⋅(d_{n1}a_{n1}) FD_{n1}  x_{n}⋅(d_{n}a_{n}) FD_{n} 
–  department cost  FA_{1}  …  FA_{b}  
=  Contribution III  x_{1}⋅(d_{1}a_{1}) FD_{1} + x_{2}⋅(d_{2}a_{2}) FD_{2}  FA_{1}  …  x_{n2}⋅(d_{n2}a_{n2})FD_{n2} + x_{n1}⋅(d_{n1}a_{n1}) FD_{n1} + x_{n}⋅(d_{n}a_{n})FD_{n} – Fa_{b}  
–  hospitalfixed cost  FK  
=  profit/loss  \( \sum \limits_{j=1}^n\left({d}_j{a}_j\right)\cdot {x}_j\sum \limits_{j=1}^n{FB}_j\sum \limits_{p=1}^b{FA}_p FK \) 
The model used for this paper assumes that the objective function given in the last row of Table 1 is maximized by each hospital while assuming that that total demand of patients in the area is met. If we maximize the objective function for each institution independently, it is likely that hospital specialize in a way that some DRGs are not covered [17, 18]. However, if all hospitals in a region behave in this pattern the needs of the population will not be met. Consequently, the model assumes that a region is covered by s hospitals in cooperation and that all cases must be treated. At the same time, we assume that the same DRGs are allocated to the same departments in all hospitals. The model defines the following variables:
x_{jk} = Number of treated patients in DRG j in hospital k, j = 1..n; k = 1..s; integer
K_{ik} = Units of resource i in hospital k, i = 1..m; k = 1..s
ß_{jk} \( =\left\{\begin{array}{cc}1& if\ DRG\ j\ is\ in\ the\ service\ portfolio\ of\ hospital\ k\\ {}0& else\end{array}\right. \), j = 1..n; k = 1..s
D_{pk} \( =\left\{\begin{array}{cc}1& if\ department\ p\ is\ opened\ in\ hospital\ k\\ {}0& else\end{array}\right. \), p = 1..b; k = 1..s
DTotal_{k} \( =\left\{\begin{array}{cc}1& if\ hospital\ k\ is\ opened\\ {}0& else\end{array}\right. \), k = 1..s
and constants:
k_{ik} = Capacity per service unit of resource i in hospital k, i = 1..m; k = 1..s
c_{ijk} = Consumptoin of resource i for one unit of DRG j in hospital k, j = 1..n; i = 1..m; k = 1..s
d_{j} = Rebate of DRG j, j = 1..n
a_{jk} = direct cost of one case of DRG j in hospital k, j = 1..n; k = 1..s
nvNumber of DRGs
M = \( M\in N, with\ M>\sum \limits_{j=1}^n\sum \limits_{k=1}^s{x}_{jk} \)
b = Number of departments
R_{p} = set of DRGs treated in department p, p = 1..b
FD_{jk} = DRGspecific fixed cost in hospital k, j = 1..n; k = 1..s
FA_{pk} = departmentspecific fixed cost of p in hospital k, p = 1..b; k = 1..s
FK_{k} = hospital fixed cost in hospital k; k = 1..s
B_{j} = number of patients in DRG j, j = 1..n
w_{ik} = Cost per unit of resource i in hospital k, i = 1..m; k = 1..s
The LP maximizes the total marginal contribution in the entire region:
subject to the constraints:
The first constraints safeguards that the capacity limitations are respected in each hospital. The second determines that the binary variable ß_{jk} is one if at least one patient is treated with DRG j in hospital k so that the DRGspecific fixed costs are reflected in the objective function. The third and fourth constraints do the same to the department and hospital fixed costs. Finally, the last equation safeguards that all patients are treated in the region.
Health economic data

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.

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).

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.

45 min physicians’ time per day per patient for medical history, diagnostics, therapy decisions, monitoring, and documentation are assumed.

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.

The duration of a birth takes 14.5 h on average, considering all modes of delivery (spontaneous or assisted vaginal delivery, caesarean section).

It is assumed that the number of delivery rooms is sufficiently large (no capacity limitation).

As in the paediatric ward, a minimum staffing of nine nurses is necessary.

For each patient, nursing care of 2.8 h per day plus 1.8 h per day for a newborn is assumed.

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).

The minimum staffing is five doctors and one senior physician, analogue to the paediatric ward.

The assumed physicianworking time per vaginal delivery is 60 min.

The assumed physicianworking time for a caesarean section is 300 min (including 120 min anaesthesiologist). Additionally, 300 min nursing care are needed.
 3.
Salaries: For nurses and midwifes, a gross annual salary of 42,000€ was assumed, for senior nurses and midwifes (management of the ward) 50,000. Physician specialists in training cost 75,000€/year, senior physician specialists 120,000€. 35% employer’s share has to be added.
 4.
Department fixed costs: Fixed costs per department are added for administration, cleaning, heating etc. We calculate €50,000/year per ward and 35,000€/year per delivery room. Including the salaries (which are fixed as well) these assumptions result in fixed costs for the wards of 576,500€/year for paediatric wards and 679,000€/year for obstetric wards.
 5.
Fixed cost per bed: The second kind of costs is fixed costs per bed. Here, an exact calculation was not possible. Therefore, we used average values for paediatrics and obstetrics that are available in the context of the calculation of DRGvalues for a normal birth, a caesarean section, a healthy newborn, and a newborn, born by a caesarean section. The fixed costs per bed are 15,253.47€/year for beds on the paediatrics wards and 20,212.24€/year for beds on the obstetrics wards.
 6.
Variable costs: The third kind of costs are variable costs (e.g. drugs, food). These include according to the average DRGvalues

Paediatrics: 152.59€/case

Obstetrics: 244.64€/case
 7.
Revenues: The revenues for the cases are calculated based on the base case value of the Federal State of MecklenburgWestern Pomerania (3117.36€) and the case mix indices for paediatrics (0.483) and births (1.006). This results in revenues of 1505.68€ per case in paediatrics and 3135.10€ per birth.
Basic parameters of the model
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  d_{1} = 1505.68 d_{2} = 3135.10  
Direct cost for one case of DRG j in hostpial k, j = 1..2; k = 1..3  a_{1,k} = 152.59 a_{2,k} = 244.64  
Department fixed costs of department j in hospital k, j = 1..2; k = 1..3  FA_{1k} = 576,500 FA_{2k} = 679,000  
Fixed cost per bed for DRG j in hospital k, j = 1..2; k = 1..3  bA_{1k} = 15,253.47 bA_{2k} = 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  v_{1} = 3.10 v_{2} = 4.92 
Geographical analyses and population data
To identify the potential number of patients in the study region, we calculated catchment areas of the hospitals using a Geographic information System (ArcGIS 10.0 (ESRI, Redlands, USA)). To calculate the catchment areas, it was assumed that patients visit the paediatric or obstetric ward in the nearest hospital. The travel time to the nearest hospital was calculated using the centre points of the municipalities and municipal districts as origins of the patients. The travel time to the hospital was determined alongside the road network. We included other hospitals with paediatric and obstetric wards in neighbouring regions to get realistic catchment areas.
The number of cases in outpatient paediatrics and obstetrics in the postal code areas of the study region were retrieved from the InEKdatabase of the Federal State of MecklenburgWestern Pomerania (InEK: Institut für das Entgeltsystem im Krankenhaus, in English: Institute for the hospital remuneration system). In this database, the numbers of all different DRGs are available on the level of postal code areas. These data give information on the number of cases in the population of a postal code region, not on the hospital where the DRGs were remunerated. The cases were assigned to the respective catchment areas of the hospitals using the Geographic information System. The numbers of cases in the catchment areas are the potential number of cases under the prerequisite that all patients consult the nearest hospital. The potential number of cases in the catchment area of the hospital is an indication for the limitations of the models: it is not realistic that the wards can acquire far more cases than the potential number of cases in the catchment area.
Results
Case numbers of the hospitals and in in the catchment areas based on hospital data and on regional data; Data sources: Controlling departments of the hospitals, InEK (Institute for hospital remuneration) 2014
Pediatrics  Obstetrics  

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
Contribution margin including the wards in 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 
Contribution margin including the wards in all three hospitals while optimizing the number of beds to obtain full capacity utilization
Pediatric  Obstetric  Total [€]  

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
A linear model was calculated concentrating all paediatric cases and births in one hospital by adding the following constraints:
Cases form baseline at 2014 treated at one hospital, minimum number of beds
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 
Since we had the assumption that the services offered in the three hospitals are equivalent, the location of the hospital does not matter if we only consider financial aspects. A concentration on two hospitals generates a negative result of − 237,000 € (data not shown).
Accessibility of the hospitals
With three hospitals in the region, 15% of the children < 18 years and 14% of the women between 15 and 50 years have more than 20 min travel time to the nearest hospital. However, all patients reach the nearest hospital within 30 min travel time.
If the obstetric and paediatric wards have their locations in Greifswald and Anklam (map a) or in Greifswald and Wolgast (map b) it is possible to reach the nearest hospital in at most 40 min for all patients in the region. If only the location in Greifswald would remain (Fig. 3 (c)) a part of the patients would have a longer drive by car. This concerns mainly people from the island Usedom, in the most eastern part of the study region. About 8% of the inhabitants under 18 years (n = 2060) of the study region and 8% of the women between 15 and 50 years (n = 2800) would be affected by travel times of more than 40 min by car.
Discussion
The findings resulting from the LP model stress the conflict between accessibility (expressed in distances to the health care provider) and efficiency (expressed in cost per patient or inhabitant) which has been discussed frequently. For instance, Berwick et al. discuss the “triple” aim of health care, i.e. “improving the individual experience of care; improving the health of populations; and reducing the per capita costs of care for populations” while physical accessibility is a major factor of “improving the health of populations” [21]. This conflict was also a drive of the Declaration of Alma Ata on Primary Health Care in 1978 as an innovative way to reconcile this conflict inspiring the discussion on distributional ethics in health care [22, 23, 24]. Other authors stress that this conflict is the healthrelated expression of the general tradeoff between equality and efficiency which is an underlying principle of economics [25, 26, 27]. However, many of these papers and discussions remain on a theoretical and in particular nonempirical level. The results presented in this paper demonstrate – based on the example of a remote German region – the conflict between costs of providing services and accessibility with real data.
The current discussions in Germany focus on the question whether the high number of small and unprofitable hospitals is still needed and should be subsidized by the government. The number of hospitals in Germany has been declining from 2411 hospitals in 1991 to 1956 in 2015 [19]. The closures mostly affect unspecialized hospitals in particular (but not only) in rural areas as they are seen as less efficient [28]. The linear models in this analysis showed a similar effect: too many hospitals with identical departments in a region with a low population density and, consequently, a low number of potential patients endanger the economic efficiency of the hospitals.
The basic model demonstrates a critical economic situation. Even if all three hospitals in the study region operate at the maximum capacity utilization, there is still a coverage gap of € 3.6 million. The optimization model points out that the case number is too small to allow positive financial results for more than one hospital. To get a positive contribution margin, every hospital would have to treat at least 894 births (13 beds) und 1587 paediatric cases (14 beds), which the region cannot deliver. Consequently, an analysis based only on the provider perspective would call for a consequent closure of the paediatric and obstetric departments in the two smaller hospitals.
However, the Government of the German states hesitate to base their decisions on mathematical models. For instance, Kuntz et al. developed a DEAbased model allowing efficiencybased resource allocation for one westGerman state but the results were never applied [29, 30].
A consequence of closing down hospitals is that a part of the patients have a longer travel time to the next hospital, which could have an influence on access and utilization. In an analysis of reimbursement data of statutory health insurances, it was stated, that younger patients (< 30 years) on average travel longer distances than older patients. Patients in rural regions have twice as long travel times compared to patients in urban areas [31]. Stentzel at al examined in the same study region as our analysis (Western Pomerania), whether a longer travel time to an inpatient gynaecologist or GP practice leads to a lower utilization of those providers. However, no significant association was found here [32]. In a systematic review on minimum standards for spatial accessibility of primary care, different results on reasonable travel times from the USA, Germany and Austria were included. It was found that a travel time of 30 min for primary care for at least 90% of the population is acceptable from the perspective of the patients. The accepted travel time tends to be lower in urban regions [33]. Comparing these results to the geographic results of our analyses, the travel times to the nearest hospital might be ok for most of the population also in case of closing down one of the smaller hospitals with accordingly longer travel times and travel costs.
The results of the model presented here were considered by the government of the Federal State of MecklenburgWestern Pomerania but were not highly influential. The attempt to close down the paediatric and obstetric departments in the two smaller hospitals inspired some “civil unrest” including demonstration walks and high election results for a rightwing partly. Consequently, the government had to consider much more aspects than only the costs and efficiency of hospitals.

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.

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.

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.

DecisionModel: 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 benchmarks. 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.

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.

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.

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.

Accessibility: which location has a good accessibility for the inhabitants of the region both by car and public transport;

Availability of paediatric and obstetric wards in neighbouring regions;

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;

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;

Social and economic factors in the region.

A close cooperation between small hospitals in rural regions and a compensatory alignment of the services and wards;

Conversion of hospitals into regional ambulatory healthcare centers to grossly reduce fixed costs [36, 37];

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];

Implementation of telemedical connections between small hospitals and hospitals with maximum care to ensure medical standards in small hospitals maintaining only few medical specialties [39].

Improvement of location in the public road system as well as public transport to and from hospitals [2].

Improvement of emergency systems in order to safeguard rapid transport from the homes of patients to the hospital [40].
Conclusion
Summarizing we can conclude that the conflict between accessibility and hospital cost per patient is obvious. Rural areas require a higher number of smaller hospitals in order to safeguard acceptable access times, but this will lead to costs and losses in hospitals challenging their existence. This conflict must be expressed and discussed in a transparent way. Mathematical modelling and geographic information systems are an excellent way to base these discussions on transparency and facts. However, even in a transparent process the conflict will not be solved unless innovative forms of health care delivery are developed and applied. This is a call for all health policy makers to invest creativity into regional hospital planning going beyond the hospital.
Notes
Acknowledgements
Not applicable.
Funding
This research did not receive any funding.
Availability of data and materials
The datasets used and/or analysed during the current study are available from the corresponding author on reasonable request.
Authors’ contributions
SF, NvdB, and WH designed the study. SF conceptualized and calculated the linear model. FR and US conceptualized and calculated the geographic analyses. NvdB, SF, and WH interpreted the DRG data. NvdB and SF were major contributors in writing the manuscript. All authors read and approved the final manuscript.
Ethics approval and consent to participate
Not applicable.
Consent for publication
Not applicable.
Competing interests
The authors declare that they have no competing interests.
Publisher’s Note
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References
 1.Krankenhausplan 2012 des Landes MecklenburgVorpommern. 2018. Available at: https://www.regierungmv.de/Landesregierung/wm/gesundheit/Gesundheitsversorgung/Krankenhauswesen/. Accessed 9 Jul 2018.
 2.Kozhimannil KB, Hung P, Prasad S, Casey M, McClellan M, Moscovice IS. Birth volume and the quality of obstetric care in rural hospitals. J Rural Health. 2014;30(4):335–43.CrossRefPubMedGoogle Scholar
 3.Trambacz J. Grundlagen und Lehrbegriffe der Gesundheitsökonomie. Lehrbegriffe und Grundlagen der Gesundheitsökonomie. Heidelberg: Springer; 2016.CrossRefGoogle Scholar
 4.Fleßa S, Gieseler V. Die Rolle der Krankenhäuser im ländlichen Raum. In: Herbst M, Dünkel F, Stahl B, editors. Daseinsvorsorge und Gemeinwesen im ländlichen Raum. Wiesbaden: Springer; 2016. p. 43–60.CrossRefGoogle Scholar
 5.Fleßa S, Ehmke B, Herrmann R. Optimierung des Leistungsprogramms eines Akutkrankenhauses : neue Herausforderungen durch ein fallpauschaliertes Vergütungssystem. BFuP. 2006;58:585–99.Google Scholar
 6.Meyer M, Harfner A. Spezialisierung und Kooperation als Strukturoptionen für deutsche Krankenhäuser im Lichte computergestützter Modellrechnungen. Wiesbaden: Springer; 1999.CrossRefGoogle Scholar
 7.Muennig P, Bounthavong M. Costeffectiveness analysis in health: a practical approach. New York: John Wiley & Sons; 2016.Google Scholar
 8.Fleßa S. Systemisches Krankenhausmanagement. Berlin: DeGruyter; 2018.CrossRefGoogle Scholar
 9.Dantzig G. Linear programming and extensions. Princeton: Princeton University Press; 2016.Google Scholar
 10.Gurfield RM, Clayton SC. Analytical hospital planning: a pilot study of resource allocation using mathematical programming in a cardiac unit. Santa Monica FM5893RC: RAND Memorandum; 1969.Google Scholar
 11.Shuman LJ, Young JP, Naddor E. Manpower mix for health services – a prescriptive regional planning model. HSR. 1971;2:103–19.Google Scholar
 12.Shuman LJ, Wolfe H, Spears RD. The role of operations research in regional health planning. In: Kwak N, Schmitz HH, Schniederjans MJ, editors. Operations Research – Application in health Care planning:5. New York, London: Lanham; 1984.Google Scholar
 13.Dowling WL. Hospital production – a linear programming model. Toronto/London: Lexington; 1976.Google Scholar
 14.Meyer M. Das optimale FallklassenProgramm eines Krankenhauses. Führen und Wirtschaften im Krankenhaus. 1996;13(1):14–8.Google Scholar
 15.Vogl M. Hospital financing: calculating inpatient capital costs in Germany with a comparative view on operating costs and the English costing scheme. Health Policy. 2014;115(23):14151.CrossRefPubMedGoogle Scholar
 16.Fallpauschalenkatalog. Available at https://www.gdrg.de/GDRGSystem_2018/FallpauschalenKatalog/FallpauschalenKatalog_2018. Accessed 22 Nov 2018.
 17.Ellis R, McGuire T. Hospital response to prospective payment: horal Hazard, selection and practice style effects. J Health Econ. 1996;15:257–77.CrossRefPubMedGoogle Scholar
 18.Newhouse JP. Reimbursing health plans and health providers: selection vs. efficiency in production. J Econ Lit. 1996;34:1236–63.Google Scholar
 19.Meinfeld H. Personalbedarfsermittlung von Hebammen in Kliniken. Hebammenforum; 2011. p. 5.Google Scholar
 20.DRGBrowser. Available at: http://www.gdrg.de/. Accessed 29 Mar 2018.
 21.Berwick DM, Nolan TW, Whittington J. The triple aim: care, health, and cost. Health Aff. 2008;27(3):759–69.CrossRefGoogle Scholar
 22.World Health Organization (WHO). AlmaAta 1978: primary health care. Report on the International Conference on Primary Health Care, 6–12. Genf: World Health Organization (WHO); 1978.Google Scholar
 23.Bankowski Z, Bryant JH, Gallagher J. Ethics, equity and health for all. Geneva: Publisher, CIOMS, WHO; 1997.Google Scholar
 24.Lindholm L, Rosen M, Emmelin M. An epidemiological approach towards measuring the tradeoff between equity and efficiency in health policy. Health Policy. 1996;35:205–16.CrossRefPubMedGoogle Scholar
 25.Fleßa S. Gesundheitsreformen in Entwicklungsländern. Frankfurt am Main: Lembeck; 2003.Google Scholar
 26.Okun AM. Equality and efficiency  the big tradeoff. Washington DC: The Brookings Institution; 1975.Google Scholar
 27.Berman PA. Selective primary health care: is efficient sufficient? Soc Sci Med. 1982;16:1054–9.CrossRefPubMedGoogle Scholar
 28.Fareed N. Size matters: a metaanalysis on the impact of hospital size on patient mortality. Int J Evid Based Healthc. 2012;10(2):103–11.CrossRefPubMedGoogle Scholar
 29.Staat M. Efficiency of hospitals in Germany: a DEAbootstrap approach. Appl Econ. 2006;38(19):2255–63.CrossRefGoogle Scholar
 30.Kuntz L, Scholtes S. Wirtschaftlichkeitsanalyse mittels Data Envelopment Analysis zum Krankenhausbetriebsvergleich. In: Krankenhausmanagement. Wiesbaden: Gabler Verlag; 1999. p. 187–206.CrossRefGoogle Scholar
 31.Schang L, Kopetsch T, Sundmacher L. Travel times of patients to ambulatory care physicians in Germany. Bundesgesundheitsblatt Gesundheitsforschung Gesundheitsschutz. 2017;60(12):1383–92.CrossRefPubMedGoogle Scholar
 32.Stentzel U, Bahr J, Fredrich D, Piegsa J, Hoffmann W, van den Berg N. Is there an association between spatial accessibility of outpatient care and utilization? Analysis of gynecological and general care. BMC Health Serv Res. 2018;18(1):322.CrossRefPubMedPubMedCentralGoogle Scholar
 33.Voigtländer S, Deiters T. Minimum standards for the spatial accessibility of primary care: a systematic review. Gesundheitswesen. 2015;77(12):949–57.CrossRefPubMedGoogle Scholar
 34.Jun GT, Morris Z, Eldabi T, Harper P, Naseer A, Patel B, Clarkson JP. Development of modelling method selection tool for health services management: from problem structuring methods to modelling and simulation methods. BMC Health Serv Res. 2011;11:108.CrossRefPubMedPubMedCentralGoogle Scholar
 35.Pitt M, Monks T, Crowe S, Vasilakis C. Systems modelling and simulation in health service design, delivery and decision making. BMJ Qual Saf. 2016;25(1):38–45.CrossRefPubMedGoogle Scholar
 36.Landtag MecklenburgVorpommern. Älter werden in MecklenburgVorpommern. 2016. Available at: https://www.landtagmv.de/fileadmin/media/Dokumente/Ausschuesse/EnqueteKommission/EK_Aelterwerden_web.pdf. Accessed: 24 Feb 2017.Google Scholar
 37.Gesetz zur Reform der Strukturen der Krankenhausversorgung (Krankenhausstrukturgesetz – KHSG). 2015. Available at: https://www.bgbl.de/xaver/bgbl/start.xav?startbk=Bundesanzeiger_BGBl&jumpTo=bgbl115s2229.pdf#__bgbl__%2F%2F*%5B%40attr_id%3D%27bgbl115s2229.pdf%27%5D__1531164748012. Accessed at 9 Jul 2018.
 38.Hung P, Kozhimannil KB, Casey MM, Moscovice IS. Why are obstetric units in rural hospitals closing their doors? HSR. 2016;51(4):1546–60.PubMedGoogle Scholar
 39.van den Berg N, Schmidt S, Stentzel U, Mühlan H, Hoffmann W. The integration of telemedicine concepts in the regional care of rural areas: possibilities, limitations, perspectives. Bundesgesundheitsblatt Gesundheitsforschung Gesundheitsschutz. 2015;58(4):367–73.CrossRefPubMedGoogle Scholar
 40.Fleßa S, et al. Der Telenotarzt als Innovation des Rettungswesens im ländlichen Raum–eine gesundheitsökonomische Analyse für den Kreis VorpommernGreifswald. Die Unternehmung. 2016;70(3):248–62.CrossRefGoogle Scholar
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