This paper considers a single machine scheduling problem with a so-called lot processing mode. In this mode, the machine is divided into some continuous lots in time, and each lot is characterized by its capacity and processing time. Multiple jobs (or parts of jobs) with different sizes can be processed in one lot and are all completed at the end of the slot. The grouping constraint is that the total size of jobs in the lot is not allowed to exceed the capacity of the lot. For this problem, the authors show that the weighted shortest processing time first (WSPT) rule is optimal for both minimizing total weighted completion time and minimizing total weighted discounted completion time.