Metaheuristic algorithms for the induced P-median problem with upgrades

TL;DR

A metaheuristic algorithm, based on the Greedy Randomized Adaptive Search Procedure (GRASP), is proposed, which shows promising results when compared to the state-of-the-art, which is based entirely on mathematical programming models.

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.

Figures (7)

• Figure 1: Example instance for the IpMU. It has a budget of $B=2$, a capacity to improve the edges of $u_a=c^2_a$, a uniform demand of $\omega_i=1$ and $p=2$ medians have to be established.
• Figure 2: Solution for example instance depicted in Figure \ref{['fig:ejemplo']}
• Figure 3: Average execution time comparison between GRASP and FL1 in the set of instances of the state-of-the-art espejo2023p.
• Figure 4: Bayesian Inference Statistical Test.
• Figure 5: Representation of the Search Space Graph (SSG) for an instance of type $P$