Single machine lot scheduling with non-uniform lot capacities and processing times

In this paper, we investigate the single machine lot scheduling problem in which each lot contains one or more jobs (or part of jobs). Jobs with different sizes are splittable and should be processed in consecutive lots. Any completed (part of a) job is delivered to the customer on its completion. We generalize the problem studied in Zhang et al. (Inf Process Lett 142:46–51, 2019) such that lots may have different capacities and processing times, and the sizes of jobs are unbounded. We show that for this generalized case, the WSPT (Weighted Shortest Processing Time first) rule is still optimal for both minimizing total weighted completion time and minimizing total weighted discounted completion time.