New valid inequalities for the Two-Echelon Capacitated Vehicle Routing Problem

In a multi-echelon distribution system, goods are delivered to customers by processing and consolidating items through intermediate depots called satellites. In two-level distribution systems, the first level involves urban trucks that ship items from city distribution centers, located on the outskirts of cities, to satellites. The second level involves city freighters that collect items from satellites and deliver them to customers. The objective of the Two-Echelon Capacitated Vehicle Routing Problem (2ECVRP) is to plan delivery routes of vehicles in such a system at a minimum cost. This problem has been treated using a matheuristic [16], a Branch-and-Cut algorithm [12], a Branch-and-Cut-and-Price one [20], and an ad-hoc exact method [4]. The latter obtains the best results so far and solves instances up to five satellites and 100 customers. The problem has also been tackled using heuristics [7, 11, 6, 22, 1]. The last three achieve the best performance. In this paper, we suggest a new class of valid inequalities, inspired from depot capacity constraints proposed in [5] for the location-routing problem. The new family of cuts has a significant impact on the strength of the linear relaxation of the problem formulation. We embedded separation of new inequalities in a Branch-and-Cut-and-Price algorithm and solved to optimality 24 open instances of the problem with up to 200 customers and 10 satellites.