Two-stage open-shop scheduling with a two-machine flow shop as a stage: approximation algorithms and empirical experiments

We study a scheduling environment that finds many real-world manufacturing applications, in which there is a close connection between a hybrid multiprocessor open shop and multiple parallel identical flow shops. In this environment, there is an extended two-stage open shop, where in one stage we have a set of parallel identical machines, while in the other we have a two-machine flow shop. Our objective is to minimize the makespan, that is, the latest completion time of all jobs. We pursue approximation algorithms with provable performance, and we achieve a 2-approximation when the number of parallel identical machines is constant or is part of the input; we also design a 5/3-approximation for the special case where there is only one machine in the multiprocessor stage, which remains weakly NP-hard. Our empirical experiments show that both approximation algorithms perform much better in simulated instances; their average ratios over the proposed lower bound are around 1.5 and 1.2, respectively.