Minimizing total completion time of jobs on a flexible two machine flowshop

This paper considers the following scheduling problem on a flowshop composed of two flexible machines: N identical jobs must be processed in the same sequence through a two-machine flowshop. Each job must be processed for a total of three times on both machines. The job is first processed on machine 1 with a processing time p1. It is then processed on machine 1 or 2 for the second time. If it is processed on machine 1, the processing time is p12, while the processing time is p21 if processed on machine 2. Finally, the job is processed for the third time on machine 2 with a processing time p2. A proper sequence of processing is sought such that the total processing time of n jobs is minimized. For such a scheduling problem, different optimal algorithms are constructed corresponding to different cases.