Stochastic flow-shop scheduling with minimizing the expected number of tardy jobs

In this research, minimizing the expected number of tardy jobs in a dynamic m machine flow-shop scheduling problem, i.e., \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {F_m}\left| {{r_j}\left| {{\text{E}}\left[ {\sum {{U_j}} } \right]} \right.} \right. $$\end{document} is investigated. It is assumed that the jobs with deterministic processing times and stochastic due dates arrive randomly to the flow-shop cell. The due date of each job is assumed to be normally distributed with known mean and variance. A dynamic method is proposed for this problem by which the m machine stochastic flow-shop problem is decomposed into m stochastic single-machine sub-problems. Then, each sub-problem is solved as an independent stochastic single-machine scheduling problem by a mathematical programming model. Comparison of the proposed method with the most effective rule of thumb for the proposed problem, i.e., shortest processing time first rule shows that the proposed method performs 23.9 % better than the SPT rule on average for industry-size scheduling problems.