Single machine preemptive scheduling with fixed jobs to minimize tardiness related criteria

We consider the single machine preemptive scheduling problem with some fixed jobs being previously given. The fixed jobs are already fixed in the schedule. The remaining jobs are to be assigned to the remaining time-slots of machine in such a way that they do not overlap each other and do not overlap with the fixed jobs. The objective is to minimize a tardiness related criterion. If the jobs are processed without preemption, it has been implied in the literature that this problem is strongly NP-hard. Suppose now that the jobs are processed preemptively, and that some specified free jobs must be on-time under the schedule. The considered problem will be denoted by 1|FB.pmtn.ℛ on-time|F, where F is a tardiness related criterion. We show that the considered problem is polynomially equivalent to the problem 1|FB,pmtn|F. We also show that 1|FB,pmtn|∑wjUj is polynomially equivalent to 1||∑wjUj. Consequently, the complexity of the problem 1|FB.pmtn.ℛ on-time|F is classified according to different choices of F.