Partially Concurrent Open Shop Scheduling with Integral Preemptions

Partially-concurrent open shop scheduling (PCOSS) was recently introduced as a common generalization of the well-known open shop scheduling model and the concurrent open shop scheduling model. PCOSS was shown to be NP-hard even when there is only one machine and all operations have unit processing time. In the present paper we study PCOSS problems with integral processing times that allow preemptions at integral time points. A special and simple subclass of the problems at focus is that of unit processing times, which is considered separately. For these two cases a schedule is related to the colouring of a graph called the conflict graph, which represents the operations that cannot be performed concurrently. This enables us to extract insights and solutions from the well-studied field of graph colouring and apply them to the recently introduced PCOSS model. We then focus on two special cases of the problem—the case where the conflict graph is perfect, and the case of uniform PCOSS, in which all the jobs, including their conflicts, are identical. The development of the PCOSS model was motivated from a real-life timetabling project of assigning technicians to a fleet of airplanes. The latter case of uniform PCOSS correlates to instances in which the fleet of airplanes is homogeneous.