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

A hybrid version of a compact genetic algorithm (cGA) is presented as approach to solve the Multi-Level Capacitated Lot Sizing Problem. The present paper extends results reported in [18]. The hybrid method combines a fix and optimize heuristic with cGA aiming to improve solutions generated by cGA. Also a linear mathematical programming model is solved to first evaluated solution provided by cGA. The performance of the hybrid compact genetic algorithm (HcGA) is evaluated over two sets of benchmark instances. The results are compared against methods from literature recently proposed for the same problem: two time-oriented decomposition heuristics and a hybrid multi-population genetic algorithm. A superior performance of HcGA is reported mainly for instances dealing with setup times and against time-oriented decomposition heuristics.

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