The average part produced in a certain batch manufacturing plant must be processed sequentially through an average of eight operations. Twenty $(20)$ new batches of parts are launched each week. Average operation time is $6$ min, average setup time is $5$ hours, average batch size is $25$ parts, and average nonoperation time per batch is $10$ hours per machine. There are $18$ machines in the plant. Any machine can be set up for any type of bateh processed in the plant. The plant operates $75$ production hours per week. Determine $($a$)$ manufacturing lead time for an average part, $($b$)$ plant capacity if all machines could be operated at $100\%$ utilization, $($c$)$ plant utilization, and $($d$)$ work$-$in$-$process $($number of parts$-$in$-$process$).($e$)$ How would you expect the nonoperation time to be affected by plant utilization?