The two-echelon location-routing problem: A comparative analysis of novel and existing compact formulations

• Two novel compact formulations for the two-echelon location routing problem. • New valid inequalities for the proposed and existing formulations. • A theoretical comparison of the linear programming relaxations of the formulations. • Computational experiments to assess how the formulations perform in a MIP solver. • 125 new best known lower bounds for benchmark instances and 55 new optimal solutions. The two-echelon location-routing problem (2E-LRP) is a well-known problem in the literature that is commonly used to address applications in which deliveries occur at two levels. It concerns the location of facilities and the routing of vehicle fleets. Most studies addressing this problem and its variants rely on mixed-integer programming (MIP) formulations that are compact (i.e., have a polynomial number of variables and constraints). Although the formulations with two-index arc variables tend to perform better than those with vehicle index variables in vehicle routing problems, most of the literature on the 2E-LRP is based on the latter. In this paper, we present a comparative analysis of three compact formulations for the 2E-LRP: a literature-based formulation with vehicle index variables, and two novel formulations with two-index arc variables. Additionally, we propose enhancements for the literature-based formulation and polynomial valid inequalities for all of them. The linear programming relaxations of these formulations are thoroughly compared, showing that those of the two-index formulations are stronger. Extensive computational experiments evaluate the performance of the formulations on a general-purpose MIP solver. The results show that the formulations with vehicle index variables, despite being the standard approach in the literature, lead to poor solver performance, failing to find feasible solutions even for instances with only 50 customers. In fact, the best performance comes from the novel formulations, one of which leads to feasible solutions for all benchmark instances evaluated. Valid inequalities can be used to improve this performance even further. These experiments resulted in the discovery of 125 new best known lower bounds and 55 new optimal solutions (out of 131 benchmark instances evaluated).