Scheduling two projects with controllable processing times in a single-machine environment

We consider two single-machine scheduling problems in which two competing projects share one common resource. Each project has multiple interim assessments, and its own jobs are ordered completely. A tardy job incurs a tardiness penalty cost which can be avoided by compressing some jobs, which requires an additional cost. The performance measure of each project is the total tardiness penalty cost plus the total compression cost. The first problem minimizes the weighted sum of the performance measures of two projects, and the second problem minimizes the performance measure of one project with a constraint on that of the other. We show that the first problem is solvable in strongly polynomial time and the second is weakly NP-hard. Furthermore, we analyze how the computational complexity of each problem changes if the number of projects is more than two.