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.

中文翻译：

---

以两机流水车间为阶段的两阶段开放车间调度：逼近算法和经验实验

我们研究了一个调度环境，它发现了许多现实世界的制造应用程序，其中混合多处理器开放式车间和多个并行的相同流程车间之间存在密切联系。在这个环境中，有一个扩展的两阶段开放式车间，在一个阶段，我们有一组平行的相同机器，而在另一个阶段，我们有一个两机流水车间。我们的目标是最小化 makespan，即所有作业的最晚完成时间。我们追求性能可证明的近似算法，当并行相同机器的数量是恒定的或者是输入的一部分时，我们实现了 2-近似；我们还为多处理器阶段只有一台机器的特殊情况设计了一个 5/3 近似，它仍然是弱 NP-hard。我们的经验实验表明，这两种近似算法在模拟实例中的表现都要好得多；它们对拟议下限的平均比率分别约为 1.5 和 1.2。