A dynamic programming approach for the two-product capacitated lot-sizing problem with concave costs - Bunn, Kevin A.; Ventura, José A. (European journal of operational research, 2023)
•A pseudo-polynomial algorithm to solve the two-product capacitated dynamic lot-sizing problem is proposed.•The structure of an optimal solution is analyzed with respect to a type of period called regeneration period, which is a period where the inventory of one or both products reach zero.•A master problem network is used to determine the location of regeneration periods.•A subproblem network finds an optimal arrangement of orders between consecutive regeneration periods.•Numerical experiments are performed to show the efficiency of the proposed algorithm.•We show how to extend this approach to any number of products. In this paper, we analyze a two-product multi-period dynamic lot-sizing problem with a fixed capacity constraint in each period. Each product has a known demand in each period that must be satisfied over a finite planning horizon. The aim of this problem is to minimize the overall cost of placing orders and carrying inventory across all periods. The structure of an optimal solution is analyzed with respect to a type of period called regeneration period, which is a period where the inventory of one or both products reach zero. We show that there is an optimal arrangement of placing orders between consecutive regeneration periods. We propose a pseudo-polynomial algorithm to solve the two-product problem. First, we show how the optimal ordering pattern between two consecutive regeneration periods can be solved using a shortest path problem. Then, we explain how the optimal locations for regeneration periods can be found by solving a shortest path problem on a different network, where each arc corresponds to the shortest path in a subproblem network. We then show how this approach can be scaled up to a three-product problem and generalize this technique to any number of products, as long as it is small.