## Abstract

In this paper we extend the Rectilinear 1-center problem as follows: given a set S of n demand points in the plane, simultaneously locate a facility point f and a rapid transit line (i.e. highway) h that together minimize the expression max _{p}_{∈}_{S}T_{h}(p, f) , where T_{h}(p, f) denotes the travel time between p and f. A point of S uses h to reach f if h saves time: every point p∈ S moves outside h at unit speed under the L_{1} metric, and moves along h at a given speed v> 1. We consider two types of highways: (1) a turnpike in which the demand points can enter/exit the highway only at the endpoints; and (2) a freeway problem in which the demand points can enter/exit the highway at any point. We solve the location problem for the turnpike case in O(n^{2}) or O(nlog n) time, depending on whether or not the highway’s length is fixed. In the freeway case, independently of whether the highway’s length is fixed or not, the location problem can be solved in O(nlog n) time.

Original language | English |
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Pages (from-to) | 167-179 |

Number of pages | 13 |

Journal | Annals of Operations Research |

Volume | 246 |

Issue number | 1-2 |

DOIs | |

Publication status | Published - 2016 Nov 1 |

Externally published | Yes |

## Keywords

- Facility location
- Geometric optimization
- Rectilinear 1-center problem
- Time metric

## ASJC Scopus subject areas

- Decision Sciences(all)
- Management Science and Operations Research