Asymptotic optimal location of facilities in a competition between population and industries

TL;DR

This paper analyzes the asymptotic behavior of optimally placing multiple facilities in a space considering competing interests of existing industries and residents, providing density results as the number of facilities grows large.

Contribution

It introduces an asymptotic analysis framework for large-scale facility location problems involving competing measures, extending classical models.

Findings

Derived the asymptotic density of optimal facility locations as the number of facilities increases.

Established a mathematical model balancing industrial proximity and residential distance.

Provided insights into large-scale facility placement strategies.

Abstract

We consider the problem of optimally locating a given number $k$ of points in ${\mathbb R}^n$ for an integral cost function which takes into account two measures $\varphi^+$ and $\varphi^-$. The points represent for example new industrial facilities that have to be located, the measure $\varphi^+$ representing in this case already existing industries that want to be close to the new ones, and $\varphi^-$ representing private citizens who want to stay far away. The asymptotic analysis as $k\to\infty$ is performed, providing the asymptotic density of optimal locations.