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The 1-median and 1-highway problem

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  • Díaz-Báñez, J.M.
  • Korman, M.
  • Pérez-Lantero, P.
  • Ventura, I.

Abstract

In this paper we study a facility location problem in the plane in which a single point (median) and a rapid transit line (highway) are simultaneously located in order to minimize the total travel time of the clients to the facility, using the L1 or Manhattan metric. The highway is an alternative transportation system that can be used by the clients to reduce their travel time to the facility. We represent the highway by a line segment with fixed length and arbitrary orientation. This problem was introduced in [Computers & Operations Research 38(2) (2011) 525–538]. They gave both a characterization of the optimal solutions and an algorithm running in O(n3logn) time, where n represents the number of clients. In this paper we show that the previous characterization does not work in general. Moreover, we provide a complete characterization of the solutions and give an algorithm solving the problem in O(n3) time.

Suggested Citation

  • Díaz-Báñez, J.M. & Korman, M. & Pérez-Lantero, P. & Ventura, I., 2013. "The 1-median and 1-highway problem," European Journal of Operational Research, Elsevier, vol. 225(3), pages 552-557.
  • Handle: RePEc:eee:ejores:v:225:y:2013:i:3:p:552-557 DOI: 10.1016/j.ejor.2012.09.028
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    References listed on IDEAS

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    1. Durier, Roland & Michelot, Christian, 1985. "Geometrical properties of the Fermat-Weber problem," European Journal of Operational Research, Elsevier, vol. 20(3), pages 332-343, June.
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    1. Díaz-Báñez, J.M. & Korman, M. & Pérez-Lantero, P. & Ventura, I., 2014. "Locating a single facility and a high-speed line," European Journal of Operational Research, Elsevier, vol. 236(1), pages 69-77.

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