This work studies a combinatorial optimization problem encountered in industrial production planning: the single-item multi-resource lot-sizing problem with inventory bounds and lost sales. The demand to be satisfied by the production plan is subject to uncertainty and only probabilistically known. We consider a multi-stage decision process with a Decision–Hazard–Decision information structure in which decisions are made at each stage both before and after the uncertainty is revealed. Such a setting has not yet been studied for stochastic lot-sizing problems, and the resulting problem is modeled as a multi-stage stochastic integer program. We propose a solution approach based on an approximate stochastic dynamic programming algorithm. It relies on a decomposition of the problem into single-stage sub-problems and on the estimation at each stage of the expected future costs. Due to the Decision–Hazard–Decision information structure, each nested single-stage sub-problem is itself a two-stage stochastic integer program. We therefore introduce a Benders decomposition scheme to reduce the computational effort required to solve each nested sub-problem, and present a special-purpose polynomial-time algorithm to efficiently solve the single-scenario second-stage sub-problems involved in the Benders decomposition. The results of extensive simulation experiments carried out on large-size randomly generated instances are reported. They demonstrate the practical benefit, in terms of the actual production cost, of using the proposed approach as compared to a naive deterministic optimization approach based on the expected demand.