Modeling single machine preemptive scheduling problems for computational efficiency

We propose two modeling approaches to solve single machine preemptive scheduling problems with tardiness related objectives. Employing the conventional time-indexed formulation, we first build a model that explicitly identifies completion times of jobs with varying release times, due dates, and processing times. The second model adopts the aggregate planning view and eliminates binary constraints. Under this approach, each job is seen as a unit demand while its due date is mapped to a period where this unit is demanded. With this mapping, the periodic job allocation decisions are transformed into periodic production decisions that are measured by fraction of demand. Consequently, instead of explicit representation of the job completion times, this model tracks the amounts of production completed and backlogged via inventory and shortage variables and conservation of units constraints. We establish that the latter model provides tighter bounds and demonstrate that it provides a more efficient platform for optimization via computational analysis employing four commonly used tardiness related criteria and a case study from a real life application. Numerical computations reveal that aggregate planning view becomes more dominant in terms of computational performance as the problem size grows.