Revisiting Continuous p-Hub Location Problems with the L1 Metric

TL;DR

This paper analyzes continuous p-hub location problems using the L1 metric, deriving solutions for simple cases and proposing a simulation-based method for complex scenarios, with a case study on defibrillator deployment.

Contribution

It provides closed-form solutions for low-dimensional cases and introduces a simulation-based approximation for complex multi-hub problems with real-world applications.

Findings

Closed-form solutions for 1D and 2D with up to two hubs.

A simulation-based method for larger 2D problems.

Application to optimize AED deployment in Virginia Beach.

Abstract

Motivated by emerging urban applications in commercial, public sector, and humanitarian logistics, we revisit continuous $p$-hub location problems in which several facilities must be located in a continuous space such that the expected minimum Manhattan travel distance from a random service provider to a random customer through exactly one hub facility is minimized. In this paper, we begin by deriving closed-form results for a one-dimensional case and two-dimensional cases with up to two hubs. Subsequently, a simulation-based approximation method is proposed for more complex two-dimensional scenarios with more than two hubs. Moreover, an extended problem with multiple service providers is analyzed to reflect real-life service settings. Finally, we apply our model and approximation method using publicly available data as a case study to optimize the deployment of public-access automated external defibrillators in Virginia Beach.