Single machine preemptive scheduling to minimize the weighted number of late jobs with deadlines and nested release/due date intervals

This paper is devoted to the following version of the single machine preemptive scheduling problem of minimizing the weighted number of late jobs. A processing time, a release date, a due date and a weight of each job are given. Certain jobs are specified to be completed in time, i.e., their due dates are assigned to be deadlines, while the other jobs are allowed to be completed after their due dates. The release/due date intervals are nested, i.e., no two of them overlap (either they have at most one common point or one covers the other). Necessary and sufficient conditions for the completion of all jobs in time are considered, and an O(n log n) algorithm (where n is the number of jobs) is proposed for solving the problem of minimizing the weighted number of late jobs in case of oppositely ordered processing times and weights.