Improved algorithms for proportionate flow shop scheduling with due-window assignment

In a recent study, Sun et al. (AOR 292:113–131, 2020) studied due-window proportionate flow shop scheduling problems with position-dependent weights. For common due-window (denoted by CONW) and slack due-window (denoted by SLKW) assignment methods, they proved that these two problems can be solved in O(n2logn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(n^2\log n)$$\end{document} time respectively, where n is the number of jobs. In this paper, we consider the same problems, and our contribution is that the CONW problem can be optimally solved by a lower-order algorithm, which runs 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, implying an improvement of a factor of n.