A note on special optimal batching structures to minimize total weighted completion time

A batch is a subset of jobs which must be processed jointly in either serial or parallel form. The algorithmic aspects of the batching scheduling problems have been extensively studied in the literature. This paper presents necessary and sufficient conditions of the existence of optimal batch sequences for the single machine, batching, total weighted completion time scheduling problems on two batching ways: (1) all jobs form one batch; (2) each batch contains a single job. This kind of conditions can help us to recognize some special optimal schedules quickly.