Scheduling to maximize the weighted number of on-time jobs on parallel machines with bounded job-rejection

We study a scheduling problem on parallel identical machines, where the objective function is maximizing the weighted number of jobs completed exactly at their due-dates. The scheduler may reject a subset of the jobs, and the total permitted rejection cost is assumed to be bounded. Thus, when a decision has to be made regarding a certain job, the following options need to be considered: either (i) identify the set of machines on which the job can be scheduled on time, and assign the job to one of these machines, or (ii) reject the job (if possible), or (iii) delay the job. A pseudo-polynomial dynamic programming algorithm is introduced for this NP-hard problem. Medium size problems are solved in reasonable running times. We then study a different version of the problem, in which job-dependent due-windows are considered, and the job processing times are assumed to be identical. This version is proved to be NP-hard, and a pseudo-polynomial dynamic programming algorithm is introduced as well. Our numerical study indicates that medium size instances can be handled for this version. In addition, for both problems, an alternative solution procedure based on integer-linear-programming formulation is introduced.