This paper considers the problem of scheduling n jobs on a singlemachine. A fixed processing time and an execution interval are associated with each job.Preemption is not allowed. The objective is to find a feasible job sequence that minimizesthe number of tardy jobs. On the basis of an original mathematical integer programmingformulation, this paper shows how good-quality lower and upper bounds can be computed.Numerical experiments are provided for assessing the proposed approach.

This paper considers the problem of scheduling n jobs on a single machine. A fixed processing time and an execution interval are associated with each job. Preemption is not allowed. On the basis of analytical and numerical dominance conditions, an efficient integer linear programming formulation is proposed for this problem, aiming at minimizing the maximum lateness (L max). Experiments have been performed by means of a commercial solver that show that this formulation iseffective on large problem instances. A comparison with thebranch-and-bound procedure of Carlier is provided.

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(nlogn) 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.