Minimizing the total completion time in a two-machine flowshop with release times

In this paper, we study a two-machine flowshop scheduling problem where a discrete processing machine is followed by a single batching machine, and each job is associated by the processing time and a release time on the first machine. The discrete machine processes one job at a time and the batching machine processes several jobs simultaneously. The objective is to find an optimal schedule which minimizes the total completion time of all jobs. We prove that the problem is strongly NP-hard, and further show that the problem with two distinct release times is NP-hard. When the job sequence is predetermined, this problem can be optimally solved in polynomial time.