A certain part is routed through six machines in a batch production plant. The setup and operation times for each machine are given in the table below. The batch size is $100$ and the average non$-$operation time per machine is $12$ hours. Determine $($a$)$ manufacturing lead time and $($b$)$ production rate for operation $3.$

Suppose the part is made in very large quantities on a production line in which an automated work handling system is used to transfer parts between machines. Transfer time between stations $=15$ s$.$ The total time required to set up the entire line is $150$ hours. Assume that the operation times at the individual machines remain the same.

Determine $($a$)$ manufacturing lead time for a part coming off the line, $($b$)$ production rate for operation $3,($c$)$ theoretical production rate for the entire production line?