Minimal Total Weighted Tardiness in Tight-Tardy Single Machine Preemptive Idling-Free Scheduling

Two possibilities of obtaining the minimal total weighted tardiness in tight-tardy single machine preemptive idling-free scheduling are studied. The Boolean linear programming model, which allows obtaining the exactly minimal tardiness, becomes too time-consuming as either the number of jobs or numbers of job parts increase. Therefore, a heuristic based on remaining available and processing periods is used instead. The heuristic schedules 2 jobs always with the minimal tardiness. In scheduling 3 to 7 jobs, the risk of missing the minimal tardiness is just 1.5 % to 3.2 %. It is expected that scheduling 12 and more jobs has at the most the same risk or even lower. In scheduling 10 jobs without a timeout, the heuristic is almost 1 million times faster than the exact model. The exact model is still applicable for scheduling 3 to 5 jobs, where the averaged computation time varies from 0.1 s to 1.02 s. However, the maximal computation time for 6 jobs is close to 1 minute. Further increment of jobs may delay obtaining the minimal tardiness at least for a few minutes, but 7 jobs still can be scheduled at worst for 7 minutes. When scheduling 8 jobs and more, the exact model should be substituted with the heuristic.