Scaling and entropy in p-median facility location along a line

TL;DR

This paper analyzes the p-median facility location problem along a line, deriving a scaling law for optimal placement related to population density, and explores solution variability using simulations.

Contribution

It introduces an analytical scaling law for optimal facility placement along a line and examines how deviations affect solution space and scaling behavior.

Findings

Optimal facility length scales inversely with the square root of population density.

Scaling law confirmed through numerical examples on US highways and the Mississippi River.

Deviations from optimal placement increase the number of solutions and alter scaling exponents.

Abstract

The p-median problem is a common model for optimal facility location. The task is to place p facilities (e.g., warehouses or schools) in a heterogeneously populated space such that the average distance from a person's home to the nearest facility is minimized. Here we study the special case where the population lives along a line (e.g., a road or a river). If facilities are optimally placed, the length of the line segment served by a facility is inversely proportional to the square root of the population density. This scaling law is derived analytically and confirmed for concrete numerical examples of three US Interstate highways and the Mississippi River. If facility locations are permitted to deviate from the optimum, the number of possible solutions increases dramatically. Using Monte Carlo simulations, we compute how scaling is affected by an increase in the average distance to the nearest facility. We find that the scaling exponents change and are most sensitive near the optimum facility distribution.