Whether private or public, healthcare service providers have an interest in planning facilities that are well-dimensioned to the demand for their services. A growing trend in systems with private healthcare providers is to encourage or induce the formation of hospital networks. Collaboration in networks could have various advantages. Some of those are the facilitation of collaborations that consolidate particular activities in order to achieve economies of scale and improve quality through specialisation, the dissemination of best practices, or increased capital investment strength Reames et al. (2019). Merging facilities is a commonly discussed topic in networks, raising the relevance of hospital location planning methodologies.

Planning capacity and evaluating the current use of capacity is challenging. Often, specific facilities are dimensioned based on the area that they are designed to service. In most systems with free choice of healthcare service providers, however, hospitals can compete for patients in areas without clearly circumscribed borders. Through the development of a reputation for operational efficiency, higher quality, better patient experience, superior pricing, or other competitive advantages, hospitals can attract patients away from competing facilities, even beyond their local operating area Noether (1988); Beukers et al. (2014). Thus, an approach where dimensioning is done by matching local expected demand with facilities does not sufficiently recognize the agency of hospitals, and their potential to grow.

In order to account for hospital agency, planning and evaluation models should integrate expected patient decision-making as driven by facility characteristics. Depending on the type of planning decision, models should aim to integrate those factors that are relevant within the timeframe in which the consequences of the decision play out. Facility location decisions will in most cases have an impact for decades. It is therefore meaningful to understand the transience of the factors used to evaluate particular locations. We can distinguish between describing patient choices for facilities as fully as possible, and describing patient choices in the context of those factors that are sufficiently stable over time to assume they are predominantly static over the economic life of the decision. At one extreme end, a model could be fitted that only includes distance decay, presuming that new hospital facilities start from an entirely level playing field. Accordingly, each hospital is equally competitive regardless of current size, accredited services, reputation, pricing, or other factors. Such an entirely geographical approach assumes full flexibility in other dimensions, aligning with a view of assumed regression to the norm of other relevant factors within the project lifetime. From a hospital director’s point of view, this type of model would inform on an optimal location assuming that all current competitive advantages and disadvantages are fleeting. Presumably though, such a bare-bones model will yield hospitals of various expected sizes. Since size itself is expected to affect attraction of a facility, either directly or as a proxy, an iterative process could be imagined in which resulting size differences are included in subsequent analysis rounds. On the other extreme end, a highly descriptive model could include many different factors that influence patients’ hospital choices. This model type would suffer from something like an inertia bias. Factors taken up in the model, such as the accredited services or number of affiliated specialists, might be volatile and subject to evolution over the economic life of location decisions. It could thus be argued that, in planning exercises, regression to the mean of these factors should be assumed, or that they should be left out of applied models, provided that their absence does not significantly bias the remaining model. Different project lifetimes or expectations concerning the change in input variable values can thus affect the variable set that should be included, or studying outcomes with different variables sets can align with different temporal perspectives on the project.

In this manner, though an important overlap exists between explaining behaviour and planning according to expected behaviour, the objectives could affect the applied model itself. In descriptive cases, most gravity-type models do not include many variables, often due to data availability constraints (De Beule et al. 2014). In general, size is the only variable aside from location-derived factors that is included, yielding a simple model such as described below (Bucklin 1971). The market share \(MS_\) of hospital j in block i is equal to the fraction of the utility of j perceived in block i out of the utility perceived of all alternative j in block i. The utility is related to a transformation of size \(S_\), and decays over distance \(D_\). \(A_\) represents this decayed utility for a hospital facility j.

$$\begin MS_ = \dfrac D_^}^}} \end$$

If additional variables are added, they could take several forms. First, factors could have a symmetrical effect, in the sense that a value is related to a particular facility, and does not affect utility differently at diverse angles around the facility. Suitable examples are size, general reputation, or whether the facility was recently renovated. Asymmetrical factors could also be introduced. These are factors that can take different values for each combination of facility and region, for instance, boundary friction due to language borders, or reference rates by local general practitioners.

Several model types have been used to estimate hospital admission rates in healthcare contexts, often depending on the type of objective or perspective taken. Broadly speaking, two types of perspectives are common. First, approaches that consider equal access from the patient’s point of view a primary objective. Second, approaches that consider dimensioning from the hospital’s viewpoint. In order to evaluate spatial accessibility and equity of accessibility, model types such as Floating Catchment Area (FCA) models (Zhang et al. 2021) and Kernel Density models (Spencer and Angeles 2007), cumulative opportunity models, nearest distance methods, such as Thiessen Polygons, and Huff-type models (Zhang et al. 2015) are common. Other models focus on patient choice rather than access. Fabbri and Robone (2010), for instance, analyze in and outflows from areas in the context of hospital and region characteristics. Others, such as Congdon (2000, 2001), describe the entirety of patient flows.

Fabbri and Robone (2010) focus on evaluating scale effects in the healthcare landscape, as opposed to other spatial factors in the distribution of healthcare resources. They use a Poisson Pseudo Maximum Likelihood approach to estimate the parameters, and find that technology availability and completeness, measured as the Theil index of the spread of technology within an area, yield the largest effects on patient inflows. The study reviews several diagnostic groups, and rather than modelling all admissions, it models the flows of patients that do not visit a facility in their Local Health Authority or area. They find that small and large area sizes are favourable determinants of the ability to attract inflows. Since their approach focuses on cross-border effects, it yields insight into drivers of hospital attraction, though it does not provide a model for general admissions estimation.

FCA methods generally calculate supply-to-demand metrics for particular areas. In a first step, a quantification of a supply node’s resources, such as the number of beds, is divided by the demand nodes’ need for resources, quantified with population numbers or pathology prevalence within the catchment area of the supply node. In a second step, the ratios per facility are summed up for all facilities within range for a demand node, which represents the resource accessibility of the demand node. Thus, it captures a measure of supply node utilization in the first step, and captures the supply options for a demand node in the second step. Some variants define the catchment area in more nuanced ways, with step functions over distance, or continuous decay functions reducing the weight of a supply node over a proximity metric. Some variants of these models add a third step in order to address demand overestimation problems. Demand overestimation problems for supply nodes occur since demand for a supply node in the first step is not affected by the presence of alternatives for the relevant demand nodes. In other words, it does not take into account the competitive interactions between, or realized choices for, facilities. Recently, adaptations of Floating Catchment Area models have been used to predict hospital admission patterns (Delamater et al. 2019; Bauer et al. 2020; Wang 2018).

Gravity-type models appear under various names in the literature, such as gravity models, Huff models, Multiplicative Competitive Interaction models, with minor differences in meaning. Huff showed the applicability of gravity models to trade areas, pioneering the use of such models in competitive location planning challenges (Huff 1964). Subsequently, many authors contributed to the further improvement of gravity-based methodologies and estimation methods. Nakanishi and Cooper linearized versions of gravity models, thereby increasing potential use of estimation procedures and estimator properties (Nakanishi and Cooper 1982). They refer to this type of model as a Multiplicative Competitive Interaction model. The essence of these types of models is that supply nodes exert an attraction over demand commensurate with their utility value, but that attraction decays over distance. The market share of a supply node is then the proportion of utility out of all the supply alternatives for a demand area, each supply alternative’s utility decayed according to its distance to the demand node. Various parameters can be used to quantify utility, though in practice, size is often used as the sole attraction variable due to data unavailability. Perception of utility is usually assumed to be homogeneous in a particular area, though some work has distinguished subgroups within areas and included their idiosyncrasies in one holistic model, such as Mao and Nekorchuk (2013). Advantages of gravity models are that they are intuitive, that they adopt widely use discrete choice utility theory, and they do not suffer from demand overestimation as two-step FCA-models do when applied to admissions estimation. An important disadvantage is that a lot of data is required, including data on competing nodes that is often not available.

The manner in which decay of utility occurs has been widely researched. In recent decades, exponential functions, log-logistic functions, or log-normal functions are used most commonly, as they are often shown to outcompete the power function (De Beule et al. 2014; de Mello-Sampayo 2014), which was the initial function type used by Huff and other early authors. Due to different reporting standards, it is hard to distill a prediction accuracy benchmark from the available literature for hospital facilities. In a study by Teow et al. (2018), the MAE (Mean Absolute Error) was found to be 24% of the mean of hospital admissions. Delamater et al. reported achieving a hospital visit prediction accuracy of 74%, meaning that the the choice of hospital for 74% of patients was predicted correctly. Bauer et al., using an FCA-model, reported that for about 30% of hospitals, the error of the predicted hospital admissions rate was lower than 15%. In retail context, a MAPE aggregated on destination level of 22.34% are reported by De Beule et al. (2014). In De Beule et al.’s replication of Orpana and Lampinen (2003), a MAPE of 26.78% is found.

Mao and Nekorchuk (2013) have integrated multiple travel modes into their FCA-model for hospital facility accessibility measurement. They divided the population into regions in proportion to those who use particular transport modes according to regional surveys. Similarly, Zhou et al. (2020), measured accessibility of healthcare facilities with different transportation modes. No admission rate prediction models in hospital facility context have made use of multiple transportation mode data as far as the authors could ascertain.

Only a subset of the literature in hospital facility planning is interested in estimating the number of admissions in a competitive context. At the root of this are diverging research objectives. First, measuring accessibility does not require it, nor do related models unambiguously suggest expected patient admissions. Second, hospitals in some healthcare systems do not compete for patients. Rather, policymakers plan which area a hospital serves. It therefore makes sense that not all of the literature and models in the domain report admissions estimation accuracy. For the limited number that does, the reported accuracy is measured in various ways, and hard to directly compare with each other. Our work focuses on providing models that improve on the accuracy of currently reported models in the context of competitive hospital systems. Accordingly, the accuracy of our models is analysed and reported varying three different components of the model. Concretely, distance decay specifications, the impact of different transportation modes, and different patient populations, i.e. daycare and inpatient admissions in different age groups, are compared.

The central objective of this research is to improve the methods at the disposal of policymakers and hospital administrators to optimize hospital facility location decisions. Improving accuracy of models is a primary component of that. This research reviews improvements by examining three aspects of location planning with gravity models. First, this paper intends to improve the accuracy of admission estimates of hospital facilities by identifying the best modeling methods to capture geographical impedance. Second, this paper looks into the effects of accessibility by car and public transport on the choice of patients for particular hospitals. Third, the differences in geographical impedance for inpatient and daycare hospitalizations are compared, as well as differences between age groups.