This paper tackles the two-machine chain-reentrant flow shop scheduling problem with the no-wait constraint; we assume that each job passes from the first machine to the second and returns back to the first machine in order to execute its last operation. The objective is to minimize the makespan. In this work, we prove that the symmetric case of this problem, which is proven to be \(\mathcal NP\)-hard in the strong sense, remains \(\mathcal NP\)-hard. Then we provide two polynomial subproblems. For the main problem’s resolution, we propose two new efficient heuristics as well as two improved lower bounds that consistently outperform the existing methods. Additionally, we provide an effective Branch & Bound algorithm that can solve up to 100 jobs for some types of instances. These contributions not only enhance the theoretical comprehension of the problem but also offer efficient solutions supported by extensive statistical analysis over randomly generated instances.

本文解决了带无等待约束的两机链式可重入流水车间调度问题；我们假设每个作业从第一台机器传递到第二台机器，然后返回到第一台机器以执行其最后一个操作。目标是最大限度地缩短完工时间。在这项工作中，我们证明了这个问题的对称情况，在强意义上被证明是 \(\mathcal NP\)-hard，仍然是 \(\mathcal NP\)-hard。然后我们提供两个多项式子问题。对于主要问题的解决，我们提出了两种新的有效启发式方法以及两个改进的下界，其性能始终优于现有方法。此外，我们还提供了有效的分支限界算法，可以为某些类型的实例解决最多 100 个作业。这些贡献不仅增强了对问题的理论理解，而且还通过对随机生成的实例进行广泛的统计分析来提供有效的解决方案。