A process has $3$ stages as shown. P$1,$ P$2,$ R$1,$ M$1,$M$2,$M$3$ are $6$ machines. The number in the box $($besides the machine label$)$ is the machine capacity in $[$units$/$hour$].$

$($a$)$ What is the designed cycle time for stage $1($in $[$min$/$unit$])?$

$($b$)$ What is the designed cycle time for stage $2($in $[mi\frac{n}{u}$nit $)?$

$($c$)$ What is the designed cycle time for stage $3($in $[$min$/$unit$])?$

$($d$)$ Which stage is the bottleneck? What is the process capacity in $[$units$/$hour$]?$

$($e$)$ If you can add one machine in parallel at the bottleneck stage, what should be the largest cycle time for the new machine so that you increase the processing capacity as much as possible $([$min$/$unit$])?$