Locating a single facility and a high-speed line

TL;DR

This paper addresses a facility location problem involving the simultaneous placement of a single facility and a high-speed transit line in the plane, optimizing total travel time for clients under Manhattan metric.

Contribution

It introduces an $O(n^3)$-time algorithm for locating a facility and highway with flexible entry/exit points, and characterizes solutions based on highway speed.

Findings

Developed an $O(n^3)$ algorithm for the problem.

Provided a detailed solution characterization based on highway speed.

Enhanced understanding of facility and highway placement in Manhattan metric.

Abstract

In this paper we study a facility location problem in the plane in which a single point (facility) and a rapid transit line (highway) are simultaneously located in order to minimize the total travel time from the clients to the facility, using the $L_1$ or Manhattan metric. The rapid transit line is given by a segment with any length and orientation, and is an alternative transportation line that can be used by the clients to reduce their travel time to the facility. We study the variant of the problem in which clients can enter and exit the highway at any point. We provide an $O(n^3)$-time algorithm that solves this variant, where $n$ is the number of clients. We also present a detailed characterization of the solutions, which depends on the speed given in the highway.