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(nlog n) 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.