Single machine lot scheduling to minimize the total weighted (discounted) completion time

In this work, we investigate the problem of single machine lot scheduling where each lot contains one or more jobs and is of identical processing time. Jobs with different sizes are splittable and shall be processed in consecutive lots. Any completed (part of a) job is delivered to the customer on its completion. We prove that the WSPT (Weighted Shortest Processing Time first) rule is optimal for both models of minimizing total weighted completion time and minimizing total weighted discounted completion time.