An Evaluation of Mathematical Programming and Lower-Bound Methods for Hybrid Flow Shop Problems With a Makespan Criterion

This paper considers the hybrid flow shop scheduling problem, where jobs are processed in
$m$
stages with the same route of the stage. Each stage has identical parallel machines for processing jobs. Some mathematical programming formulations and lower bound calculations have been proposed in the literature for such cases. Nevertheless, there is a lack of complete comparisons of these mathematical programming formulations and lower bounds in the hybrid flow shop literature. This paper proposes a new mixed integer programming model and two new lower bounds based on the bin-packing concept for the considered problem. To evaluate the proposed model, two sets of small and small-to-medium problems are used to compare our model with the existing models. Moreover, two propositions are discussed for lower bounds. The experimental results show that the proposed mixed integer programming model efficiently found optimal solutions because it needs a smaller number of binary variables and a smaller number of constraints, and the proposed lower bound can also serve as a strong indicator to evaluate the distances between the solutions obtained by heuristic algorithms and the optimal solution.