Optimal schedule for periodic jobs with discretely controllable processing times on two machines

In many real-world situations, the processing time of computational jobs can be shortened by lowering the processing quality. This is referred to as discretely controllable processing time, where the original processing time can be shortened to a number of levels with lower processing qualities. In this paper, we study the scheduling problem of periodic jobs with discretely controllable processing times on two machines. The problem is NP-hard, as directly solving it through dynamic programming leads to exponential computational complexity. This is because we need to memorise a set of processed jobs to avoid reprocessing. In order to address this issue, we prove the Ordered Scheduling Structure (OSS) Property and the Consecutive Decision Making (CDM) Property. The OSS Property allows us to search for an optimal solution in which jobs on the same machine are orderly started. The CDM Property allows us to memorise only two jobs to completely avoid the job reprocessing. These two properties greatly decrease the searching space, and the resultant dynamic programming solution to find an optimal solution is with pseudo-polynomial computational complexity.