Model and exact solution for a two-echelon inventory routing problem

The classic version of the Inventory Routing Problem considers a system with one supplier that manages the inventory level of a set of customers. The supplier defines when and how much products to supply and how to combine customers in routes while minimising storage and transportation costs. We present a new version of this problem that considers a two-echelon system with indirect deliveries and routing decisions at both levels. In this variant, the products are delivered to customers through distribution centres to meet demands with a minimum total cost. We propose a mathematical formulation and a branch-and-cut algorithm combined with a two-step matheuristic to solve the proposed problem for different inventory policies and routing configurations. Intrinsic new valid inequalities to the two-echelon system are introduced. We analyse the efficiency of the new valid inequalities as well as the already known valid inequalities from the literature. Computational experiments are presented for a new set of benchmark instances. The results show that, for the simplest inventory policy, the proposed method is able to solve small and some medium-scale instances to the proven optimality and find feasible solutions for all instances.