A new three-machine shop scheduling: complexity and approximation algorithm

The paper investigates a new three-machine shop scheduling problem that arises from many production systems, such as the garment assembly line, etc. In such scenarios, each job consists of three operations, each of which has to be non-preemptively processed by one specific machine. In contrast with the classical three-machine shop scheduling, the processing order of the operations of each job is partially restricted. In particular, the first two operations are ordered and all the same for all jobs, while the third operation is not restricted. The objective is to minimize the makespan. We show the problem is NP-hard in the ordinary sense and present a polynomial time approximation algorithm with a worst case performance ratio of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\frac{3}{2}$\end{document}.