A note on study on proportionate flowshop scheduling with due-date assignment and position-dependent weights

In a recent paper (Lv and Wang, Study on proportionate flowshop scheduling with due-date assignment and position-dependent weights. Optim. Lett., 2021. https://doi.org/10.1007/s11590-020-01670-4), Lv and Wang studied due date proportionate flowshop scheduling problems with position-dependent weights. For common due date and slack due date assignments, they proved that these two problems can be solved in 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} time respectively, where n is the number of jobs. For slack due date assignment, there is an error. We will show the error by a counter example and explain why it is incorrect.