Minimizing total weighted tardiness and overtime costs for single machine preemptive scheduling

- We study the single machine preemptive scheduling problem with both regular and overtime modes.
- The objective is to minimize the total tardiness and overtime costs.
- We propose a heuristic solution methodology for the problem.
- The efficiency of the heuristic is tested with upper bounds generated by the mathematical model.

This paper studies the scheduling of a finite set of jobs on a single resource that operates under both regular and overtime capacity modes. Jobs, which can be preempted, have associated release and due dates. Limited overtime capacity can be utilized to reduce tardiness. However, since overtime is costly, justification of the overtime use depends on the trade-off between the tardiness and the overtime costs. The overall objective is to minimize the total cost of tardiness and overtime. To achieve this objective, we develop a holistic method composed of three-stages. We first provide a heuristic based on an effective priority rule for the base case where no overtime capacity is considered. This heuristic is later employed in the first-stage to produce a compact non-delay schedule built based on the assumption that overtime capacity incurs no additional cost. In the second stage, the overtime usage is reduced by shifting workload and generating a full-delay schedule without altering the tardiness of jobs produced in the first stage. The third stage improves the total costs by altering the tardiness of jobs in return for savings in overtime utilization. Using computational tests, we compare the performance of our heuristics to the upper bounds generated by the exact mixed-integer programming formulation. The results show that the proposed method is efficient in obtaining solutions that are considerably better than the generated upper bounds in significantly short times and as such, it can be quite useful as an effective solution approach especially for large size problems.