On scheduling multiple parallel two-stage flowshops with Johnson's Rule

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(nlogn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(n \log n)$$\end{document} 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.