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Annals of Operations Research

, Volume 226, Issue 1, pp 89–99 | Cite as

Distance measure and the \(p\)-median problem in rural areas

  • Kenneth Carling
  • Mengjie Han
  • Johan HåkanssonEmail author
  • Pascal Rebreyend
Article

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.

Keywords

Dense network Location model Optimal location  Simulated annealing Travel-time Urban areas 

Notes

Acknowledgments

Financial support from the Swedish Retail and Wholesale Development Council is gratefully acknowledged.

References

  1. Al-khedhairi, A. (2008). Simulated annealing metaheuristic for solving p-median problem. International Journal of Contemporary Mathematical Sciences, 3(28), 1357–1365.Google Scholar
  2. Bach, L. (1981). The problem of aggregation and distance for analyses of accessibility and access opportunity in location-allocation models. Environment & Planning A, 13, 955–978.CrossRefGoogle Scholar
  3. Berman, O., & Krass, D. (1998). Flow intercepting spatial interaction model: A new approach to optimal location of competitive facilities. Location Science, 6, 41–65.CrossRefGoogle Scholar
  4. Brimberg, J., & Love, R. F. (1993). General considerations on the use of the weighted l-p norm as an empirical distance measure. Transportation Science, 27(4), 341–349.CrossRefGoogle Scholar
  5. Brimberg, J., & Love, R. F. (1995). Estimating distances. In Z. Drezner (Ed.), Facility location: A survey of applications and methods (pp. 9–32). Berlin: Springer.CrossRefGoogle Scholar
  6. Carling, K., Han, M., & Håkansson, J. (2012). Does Euclidean distance work well when the \(p\)-median model is applied in rural areas? Annals of Operations Research, 201(1), 83–97.CrossRefGoogle Scholar
  7. Daskin, M. S. (1995). Network and discrete location: models, algorithms, and applications. New York: Wiley.CrossRefGoogle Scholar
  8. Dijkstra, E. W. (1959). A note on two problems in connexion with graphs. Numerische Mathematik, 1, 269–271.CrossRefGoogle Scholar
  9. Drezner, T., & Drezner, Z. (2007). The gravity \(p\)-median model. European Journal of Operational Research, 179, 1239–1251.CrossRefGoogle Scholar
  10. Francis, R. L., Lowe, T. J., Rayco, M. B., & Tamir, A. (2009). Aggregation error for location models: Survey and analysis. Annals of Operations Research, 167, 171–208.CrossRefGoogle Scholar
  11. Hakimi, S. L. (1964). Optimum locations of switching centers and the absolute centers and medians of a graph. Operations Research, 12(3), 450–459.CrossRefGoogle Scholar
  12. Hale, T. S., & Moberg, C. R. (2003). Location science research: A review. Annals of Operations Research, 32, 21–35.CrossRefGoogle Scholar
  13. Han, M., Håkansson, J., & Rebreyend, P., (2012). How does the use of different road networks effect the optimal location of facilities in rural areas? Working papers in transport, tourism, information technology and microdata analysis, Dalarna university, 2012:02.Google Scholar
  14. Han, M., Håkansson, J., Rebreyend, P., (2013). How do different densities in a network affect the optimal location of service centers? Working papers in transport, tourism, information technology and microdata analysis, Dalarna university, 2013:15.Google Scholar
  15. Handler, G. Y., & Mirchandani, P. B. (1979). Location on networks: Theorem and algorithms. Cambridge, MA: MIT Press.Google Scholar
  16. Hillsman, E. L., & Rhoda, R. (1978). Errors in measuring distances from population to service centers. Annals of Regional Science, 12, 74–88.CrossRefGoogle Scholar
  17. Kariv, O., & Hakimi, S. L. (1979). An algorithmic approach to network location problems. Part 2: The p-median. SIAM Journal of Applied Mathematics, 37, 539–560.CrossRefGoogle Scholar
  18. Kirkpatrick, S., Gelatt, C., & Vecchi, M. (1983). Optimization by simulated annealing. Science, 220(4598), 671–680.CrossRefGoogle Scholar
  19. Love, R. F., & Morris, J. G. (1972). Modeling inter-city road distances by mathematical functions. Operational Research Quarterly, 23, 61–71.CrossRefGoogle Scholar
  20. Murray, A. T., & Church, R. L. (1996). Applying simulated annealing to location-planning models. Journal of Heuristics, 2, 31–53.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Kenneth Carling
    • 1
  • Mengjie Han
    • 2
  • Johan Håkansson
    • 3
    Email author
  • Pascal Rebreyend
    • 4
  1. 1.Department of Statistics, School of Technology and Business StudiesDalarna universityFalunSweden
  2. 2.School of Technology and Business StudiesDalarna universityFalunSweden
  3. 3.Department of Human Geography, School of Technology and Business StudiesDalarna universityFalunSweden
  4. 4.Department of Computer Science, School of Technology and Business StudiesDalarna university FalunSweden

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