Lower and upper bounds for the two-echelon capacitated location-routing problem

In this paper, we introduce two algorithms to address the two-echelon capacitated location-routing problem (2E-CLRP). We introduce a branch-and-cut algorithm based on the solution of a new two-index vehicle-flow formulation, which is strengthened with several families of valid inequalities. We also propose an adaptive large-neighbourhood search (ALNS) meta-heuristic with the objective of finding good-quality solutions quickly. The computational results on a large set of instances from the literature show that the ALNS outperforms existing heuristics. Furthermore, the branch-and-cut method provides tight lower bounds and is able to solve small- and medium-size instances to optimality within reasonable computing times. ► We present a new modeling framework for the 2E-CLRP that decomposes the problem into two CLRPs. ► We introduce a compact two-index formulation for the 2E-CLRP inspired from a compact formulation of the CLRP. ► We introduce the first exact algorithm for the 2E-CLRP, namely a branch-and-cut algorithm based on the new formulation with additional cuts. ► We introduce a ALNS metaheuristic that outperforms previous methods.