It is well-known that the classical Johnson’s Rule leads to optimal schedules on a two-stage flowshop. However, it is still unclear how Johnson’s Rule would help in approximation algorithms for scheduling an arbitrary number of parallel two-stage flowshops with the objective of minimizing the makespan. Thus within the paper, we study the problem and propose a new efficient algorithm that incorporates Johnson’s Rule applied on each individual flowshop with a carefully designed job assignment process to flowshops. The algorithm is successfully shown to have a runtime \(O(n \log n)\) and an approximation ratio 7/3, where n is the number of jobs. Compared with the recent PTAS result for the problem (Dong et al. in Eur J Oper Res 218(1):16–24, 2020), our algorithm has a larger approximation ratio, but it is more efficient in practice from the perspective of runtime.

众所周知，经典的约翰逊规则可以在两阶段流水作业中产生最佳调度。然而，目前尚不清楚约翰逊规则如何帮助近似算法调度任意数量的并行两阶段流程，以最小化完工时间为目标。因此，在本文中，我们研究了该问题并提出了一种新的有效算法，该算法将应用于每个单独流水作业的约翰逊规则与精心设计的流水作业分配流程相结合。该算法被成功证明具有运行时间\(O(n \log n)\)和近似比率 7/3，其中n是作业数量。与该问题最近的 PTAS 结果（Dong et al. in Eur J Oper Res 218(1):16–24, 2020）相比，我们的算法具有更大的逼近率，但从以下角度来看，它在实践中更加高效：运行。