Abstract
The \(p\)-median model is used to locate \(P\) facilities to serve a geographically distributed population. Conventionally, it is assumed that the population patronizes the nearest facility and that the distance between the resident and the facility may be measured by the Euclidean distance. Carling et al. (Ann Oper Res 201(1):83–97, 2012) compared two network distances with the Euclidean in a rural region with a sparse, heterogeneous network and a non-symmetric distribution of the population. For a coarse network and \(P\) small, they found, in contrast to the literature, the Euclidean distance to be problematic. In this paper we extend their work by use of a refined network and study systematically the case when \(P\) is of varying size (1–100 facilities). We find that the network distance give almost as good a solution as the travel-time network. The Euclidean distance gives solutions some 2–13 % worse than the network distances, and the solutions tend to deteriorate with increasing \(P\). Our conclusions extend to intra-urban location problems.
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Notes
The population data used in this study comes from Statistics Sweden, and is from 2002 (www.scb.se).
The road networks are provided by The National Road Data Base (NVDB). NVDB was formed in 1996 on behalf of the government and now operated by the Swedish Transport Agency. NVDB is divided into national roads, local roads and streets. The national roads are owned by the national public authorities, and the construction of them funded by a state tax. The local roads or streets are built and owned by private persons or companies or by the municipalities. Data was extracted spring 2011 and represents the network of the winter of 2011.
Arguments leading to other objective functions can be found elsewhere see e.g. Berman and Krass (1998) and Drezner and Drezner (2007). For instance, a heterogeneous population raises the issue of whether attributes such as the number of residents, average income, educational level, and so on should be considered. To maintain focus, we adhere to the objective function mentioned above.
Facilities are always located at a node in line with the result of Hakimi (1964). Residents are assumed to start the travel at their nearest node, and reaching it by a travel of the Euclidean distance. This assumption is inconsequential in this dense road network.
Due to the large number of candidate nodes and demand points we found the construction of conventional OD-matrices very time-consuming and expensive in its storage considering the gigantic number of routes of no relevance for the optimization. However, results from the Dijkstra algorithm were saved and re-used when needed.
It was not computationally feasible to use LR at the full scale of this problem. The comparison was done using 10 % of candidate nodes of the full network, the choice of candidate nodes corresponds to the ones used in Han et al. (2013) derived from road classes 0–4 in NVDB. This simplification implies that location of facilities may not occur along the smallest roads in the region.
Fig. 2 
Relative difference (%) between the SA-solution and the upper bound of LR for both Euclidian distance (\(E\), right bar) and travel-time (\(T\), left bar)
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Financial support from the Swedish Retail and Wholesale Development Council is gratefully acknowledged.
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Carling, K., Han, M., Håkansson, J. et al. Distance measure and the \(p\)-median problem in rural areas. Ann Oper Res 226, 89–99 (2015). https://doi.org/10.1007/s10479-014-1677-4
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DOI: https://doi.org/10.1007/s10479-014-1677-4