Online scheduling of two-machine flowshop with lookahead and incompatible job families

This paper considers online scheduling on two unit flowshop machines, which there exists unbounded parallel-batch scheduling with incompatible job families and lookahead intervals. The unit flowshop machine means that the processing time of any job on each machine is unit processing time. The objective is to minimize the maximum completion time. The lookahead model means that an online algorithm can foresee the information of all jobs arriving in time interval (t,t+β]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(t,t+\beta ]$$\end{document} at time t. There exist incompatible job families, that is, jobs belonging to different families cannot be processed in the same batch. In this paper, we address the lower bound of the proposed problem, and provide a best possible online algorithm of competitive ratio 1+αf\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1+\alpha _f$$\end{document} for 0≤β<1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0\le \beta <1$$\end{document}, where αf\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha _f$$\end{document} is the positive root of the equation (f+1)α2+(β+2)α+β-f=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${(f+1)} \alpha ^{2}+(\beta +2) \alpha +\beta -f=0$$\end{document} and f is the number of incompatible job families which is known in advance.