  In this paper we study a proportionate flow shop of batching machines with
release dates and a fixed number $m \geq 2$ of machines. The scheduling problem
has so far barely received any attention in the literature, but recently its
importance has increased significantly, due to applications in the industrial
scaling of modern bio-medicine production processes. We show that for any fixed
number of machines, the makespan and the sum of completion times can be
minimized in polynomial time. Furthermore, we show that the obtained algorithm
can also be used to minimize the weighted total completion time, maximum
lateness, total tardiness and (weighted) number of late jobs in polynomial time
if all release dates are $0$. Previously, polynomial time algorithms have only
been known for two machines.
