Minimizing the number of tardy job units under release time constraints

We study two single-machine scheduling problems: Minimizing the weighted and unweighted number of tardy units, when release times are present. Fast strongly polynomial algorithms are given for both problems: For problems with n jobs, we give algorithms which require O(n log n) and O(n2) steps, for the unweighted and weighted problems respectively. Our results also imply an extension of the family of very efficiently solvable transportation problems, as well as these which are greedily solvable using the “Monge sequence” idea.