Scheduling a real-world hybrid flow shop with variable processing times using Lagrangian relaxation

This paper studies a real-world hybrid flow shop problem arising from the steelmaking continuous casting process, which is the bottleneck of the iron and steel production. There are a variety of features to be taken into account, in particular the batch constraints and the variable processing times in the last stages. Based on a time-index formulation and machine capacity relaxation, three Lagrangian relaxation (LR) approaches are presented for addressing this scheduling problem. The three LR approaches decompose the relaxed problem into job-level problems, batch-level problems and machine-level problems, respectively. These subproblems are solved based on polynomial dynamic programming algorithms. The corresponding Lagrangian dual (LD) problems are solved by an efficient subgradient algorithm with global convergence. Computational results and comparisons demonstrate that the approach adopting job-level decomposition is most efficient among three approaches, whereas the approach adopting batch-level decomposition is most effective.