New cuts and a branch-cut-and-price model for the multi-vehicle covering tour problem

The multi-vehicle covering tour problem (m-CTP) involves a graph in which the set of vertices is partitioned into a depot and three distinct subsets representing customers, mandatory facilities, and optional facilities. Each customer is linked to a specific subset of optional facilities that define its coverage set. The goal is to determine a set of routes with minimal cost that satisfy the following constraints: each route begins and ends at the depot; every mandatory facility is visited exactly once on a single route; each route visits not more than p facilities and have a maximum cost of q; for each customer, at least one optional facility from its coverage set must be visited by one of the routes. In this paper, we present the following contributions for the m-CTP: an exact branch-cut-and-price algorithm; a new family of capacity-like cuts; and a new set of benchmark instances. We report several experiments that prove the effectiveness of the proposed algorithm and cuts. The results show that the proposed algorithm outperforms the best exact method from the literature and that the proposed cuts further improve its performance by one order of magnitude. The proposed algorithm and cuts allow us to effectively solve 287 out of 288 lature instances.