# Ordered p-median problems with neighborhoods

## TL;DR

This paper introduces a new variant of the p-median problem where facility locations are uncertain within neighborhoods, proposing formulations and heuristics to optimize facility placement and assignment at minimal cost.

## Contribution

It presents a novel p-median variant with neighborhood uncertainty, along with mixed-integer nonlinear formulations and heuristic methods for solution.

## Key findings

- Developed multiple formulations for the problem.
- Designed two heuristic algorithms for large instances.
- Conducted extensive computational experiments demonstrating effectiveness.

## Abstract

In this paper, we introduce a new variant of the $p$-median facility location problem in which it is assumed that the exact location of the potential facilities is unknown. Instead, each of the facilities must be located in a region around their initially assigned location (the neighborhood). In this problem, two main decisions have to be made simultaneously: the determination of the potential facilities that must be open to serve the demands of the customers and the location of the open facilities in their neighborhoods, at global minimum cost. We present several mixed integer non-linear programming formulations for a wide family of objective functions which are common in Location Analysis: ordered median functions. We also develop two math-heuristic approaches for solving the problem. We report the results of extensive computational experiments.