Semi-online scheduling with known maximum job size on two uniform machines

In this paper, we investigate the semi-online scheduling problem with known maximum job size on two uniform machines with the speed ratio s≥1. The objective is to minimize the makespan. Two algorithms are presented, where the first is optimal for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$1\leq s\leq\sqrt{2}$\end{document} , and the second is optimal for 1.559≤s≤2 and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$s\ge \frac{3+\sqrt{17}}{2}$\end{document} . In addition, the improvement on lower bounds is made for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$2<s<\frac{3+\sqrt{17}}{2}$\end{document} .