In this paper, we propose a simple but efficient heuristic that combines construction and improvement heuristic ideas to solve multi-level lot-sizing problems. A relax-and-fix heuristic is firstly used to build an initial solution, and this is further improved by applying a fix-and-optimize heuristic. We also introduce a novel way to define the mixed-integer subproblems solved by both heuristics. The efficiency of the approach is evaluated solving two different classes of multi-level lot-sizing problems: the multi-level capacitated lot-sizing problem with backlogging and the two-stage glass container production scheduling problem (TGCPSP). We present extensive computational results including four test sets of the Multi-item Lot-Sizing with Backlogging library, and real-world test problems defined for the TGCPSP, where we benchmark against state-of-the-art methods from the recent literature. The computational results show that our combined heuristic approach is very efficient and competitive, outperforming benchmark methods for most of the test problems
