The single machine total weighted completion time scheduling problem with the sum-of-processing time based models: Strongly NP-hard

- We analyse the single machine total weighted completion time scheduling with learning and aging effects.
- The job processing times depend on the sum of the normal job processing times.
- The considered problem with the learning effect is proved to be strongly NP-hard.
- The considered problem with the aging effect is proved to be strongly NP-hard.
- A parallel branch and bound algorithm is constructed for the general version of the problem.

Although the single machine scheduling problem to minimize the total weighted completion times with the sum-of-processing time based learning or aging effects have been known for a decade, it is still an open question whether these problems are strongly NP-hard. We resolve this issue and prove them to be strongly NP-hard with the learning effect as well as with the aging effect. Furthermore, we construct an exact parallel branch and bound algorithm for the problem with general sum-of-processing time based models, which can solve optimally moderate problem instances in reasonable time.