Optimal design of spatial distribution networks

TL;DR

This paper investigates the optimal spatial distribution of public facilities to minimize average distance to residents, confirming a two-thirds power law relation with population density, and explores optimal network linking strategies.

Contribution

It introduces a novel analytic and numerical approach to determine the optimal distribution and network design of facilities based on population density.

Findings

Optimal facility density increases with population density to the two-thirds power.

Numerical methods confirm the analytic predictions for the US.

Examples of minimal-cost network configurations are provided.

Abstract

We consider the problem of constructing public facilities, such as hospitals, airports, or malls, in a country with a non-uniform population density, such that the average distance from a person's home to the nearest facility is minimized. Approximate analytic arguments suggest that the optimal distribution of facilities should have a density that increases with population density, but does so slower than linearly, as the two-thirds power. This result is confirmed numerically for the particular case of the United States with recent population data using two independent methods, one a straightforward regression analysis, the other based on density dependent map projections. We also consider strategies for linking the facilities to form a spatial network, such as a network of flights between airports, so that the combined cost of maintenance of and travel on the network is minimized. We show specific examples of such optimal networks for the case of the United States.