On the complexity of proportionate open shop and job shop problems

In this paper, we present NP\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{N}\mathcal{P}$$\end{document}-hardness proofs and exhibit linear-time algorithms for proportionate two-machine open shop and job shop problems with respect to the maximum lateness, the makespan with release dates, the total weighted completion times and the number of just-in-time jobs.