A Branch-and-Cut-and-Price Algorithm for the Capacitated Location-Routing Problem

In this paper we present an exact algorithm for the Capacitated LocationRouting Problem (CLRP) based on column and cut generation. The CLRP is formulated as a set-partitioning problem which also inherits all of the known valid inequalities for the flow formulations of the CLRP. We introduce five new families of inequalities that are shown to dominate some of the cuts from the two-index formulation. The problem is then solved by column generation, where the sub-problem consists in finding a shortest path of minimum reduced cost under capacity constraints. We first use the two-index formulation for enumerating all of the possible subsets of depot locations that could lead to an optimal solution of cost smaller than or equal to a given upper bound. For each of these subsets, the corresponding Multiple Depot Vehicle Routing Problem is solved by means of column generation. The results show that we can improve the bounds found in the literature, solve to optimality some previously open instances, and improve the upper bounds on some other.