Metaheuristic algorithms for the induced P-median problem with upgrades

TL;DR

This paper introduces a metaheuristic approach using GRASP for the complex Induced p-Median Problem with Upgrades, achieving significant improvements in solution time and quality over existing mathematical programming methods.

Contribution

It presents a novel two-phase metaheuristic algorithm tailored for the IpMU, outperforming state-of-the-art exact methods in efficiency and solution quality.

Findings

Average execution time reduced by two orders of magnitude.

Achieved best known solutions in over 99% of instances.

Analyzed instance complexity to inform algorithm design.

Abstract

Facility location problems (FLPs) are a family of optimisation problems with significant social impact. This class of problems has been the subject of study since the 1960s, with classical approaches including the Weber problem and the p-Median problem. Currently, more complex variations of these problems are being investigated. In particular, the Induced p-Median Problem with Upgrades (IpMU) represents a variation of the classical p-Median problem, where the concepts of transport cost and time are separated as distinct metrics in the input graph of the problem. Furthermore, the problem includes a budget which allows one to relax the graph costs, reducing the cost of the edges, thus improving the associated routes between the designated medians and the customers. In this study, a metaheuristic algorithm, based on the Greedy Randomized Adaptive Search Procedure (GRASP), is proposed. A two-phase resolution scheme is defined, studying the median problem and the upgrading problem independently. In this approach, a larger set of state-of-the-art instances was analysed to ensure a fair comparison with previous proposals. In addition, the characteristics of the instances were studied to assess their complexity. The results obtained are promising when compared to the state-of-the-art, which is based entirely on mathematical programming models. The execution time was improved on average by two orders of magnitude for the harder instances, and the best known result was obtained in more than 99% of the tested instances.