Split demand and deliveries in an integrated three-level lot sizing and replenishment problem

We address a three-level lot sizing and replenishment problem (3LSRP), an extension of the production routing problem. We consider one production plant, with limited capacity, that produces items over a discrete and finite planning horizon. The items are sent from the plant to the warehouses using direct capacitated shipments, and routes are designed to deliver the goods from the warehouses to the retailers. The objective is to minimize the sum of all operational costs. We contribute to the literature on integrated problems by introducing the concept of demand splitting which allows the demand from a specific period to be satisfied by deliveries over multiple periods. Our second contribution lies in the development of the two heuristics that we propose: a top-down approach and a bottom-up approach. The production decisions and the transportation decisions between the warehouses and the retailers are the leading decisions in the top-down and the bottom-up approach, respectively. We compare them to a branch-and-cut algorithm that we also developed. We run computational experiments to assess the performance of each heuristic. We analyze the impact of allowing splitting possibilities. The results show that the top-down approach obtains better performance in terms of cost, except when we allow demand splitting only. The bottom-up approach leads to a greater use of delivery splitting. Results also show that we obtain large gains thanks to the splitting possibilities.