A capacity constrained industrial process consists of $2$ sequential steps. Step $2$ has a total setup time of $40$ minutes per batch while step $1$ has no setup time. The activity time for step $1$ is $2$ minutes per part, for step $2$ is $1$ minute per part? An entire batch must be processed at each step before the next step can begin to work on parts from that same batch.

a$)$ Assuming a batch size of $20,$ what is the capacity $($flow rate$)$ of the entire process and what is the bottleneck?

b$)$ What batch size will yield the optimal output rate for this process?