Stochastic Dual Dynamic Programming for Multiechelon Lot Sizing with Component Substitution

This work investigates lot sizing with component substitution under demand uncertainty. The integration of component substitution with lot sizing in an uncertain demand context is important because the consolidation of the demand for components naturally allows risk-pooling and reduces operating costs. The considered problem is relevant not only in a production context, but also in the context of distribution planning. We propose a stochastic programming formulation for the static–dynamic type of uncertainty, in which the setup decisions are frozen but the production and consumption quantities are decided dynamically. To tackle the scalability issues commonly encountered in multistage stochastic optimization, this paper investigates the use of stochastic dual dynamic programming (SDDP). In addition, we consider various improvements of SDDP, including the use of strong cuts, the fast generation of cuts by solving the linear relaxation of the problem, and retaining the average demand scenarios. Finally, we propose two heuristics, namely, a hybrid of progressive hedging with SDDP and a heuristic version of SDDP. Computational experiments conducted on well-known instances from the literature show that the heuristic version of SDDP outperforms other methods. The proposed method can plan with up to 10 decision stages and 20 scenarios per stage, which results in 20 10 scenario paths in total. Moreover, as the heuristic version of SDDP can replan to account for new information in less than a second, it is convenient in a dynamic context. Summary of Contribution: We believe our paper is suitable for the mission and scope of IJOC because we design efficient algorithms to solve an operations research problem. More precisely, we investigate the use of stochastic dual dynamic programming (SDDP) for lot sizing with component substitution under demand uncertainty. In this work, we consider the static–dynamic decision framework, and a good approximation of the expected costs in this context requires us to solve the problem with a large number of scenarios of future demand. As solving the considered problem is computationally intensive, we investigate the use of SDDP, which decomposes the problem per decision stage. We study several enhancements of SDDP, such as the use of strong cuts, the incorporation of a lower bound computed with the average demand scenario, the multicut version of SDDP, and scenario sampling with randomized quasi–Monte Carlo. Despite these improvements, the convergence of SDDP remains slow. Consequently, we propose a heuristic version of SDDP and a hybrid of progressive hedging and SDDP. We present the results of an extensive computational study performed on well-known instances from the literature. The results show that the heuristic SDDP outperforms the hybrid of progressive hedging with SDDP and state-of-the-art methods from the literature. Besides, our analysis shows that component substitution can pool the risk, and it allows maintaining the same service level with less inventory. The presented methodology can be used by practitioners to size their production lots, and subsequent researchers can build upon our results to consider uncertainty in other parameters, such as lead times, yields, and production capacities.