Uncapacitated Multi-Item Lot Sizing with Shipment Minimization

Companies often order multiple items from suppliers. There can be environmental and economic benefits from combining these orders. Based on a company case, we formulate the problem of determining in which periods orders for multiple items should be placed to minimize the sum of the order and inventory costs. As a second objective, the goal is to limit the number of shipment periods and  we measure the impact of varying this number. We call this problem the Bi-Objective Lot Sizing Problem with Shipment Minimization (BLSPSM). This problem combines an overarching problem, namely the determination of shipment periods, with subproblems for each item, namely the determination of order periods and order quantities. We develop so-called bi-level dynamic programming heuristics for the BLSPSM, where the bi-level nature of the problem is exploited. We show that these heuristics outperform existing methods for a special case of our problem (the so-called Dynamic Joint Replenishment Problem) and find efficient frontiers very close to the optimal ones for all instances in our test set.