A batch production operation has a machine setup time of $3\mathrm{.}0$ hours and a processing time of $1\mathrm{.}60$min per cycle. Two parts are produced each cycle. No tool handling time is included in the cycle. Part handling time each cycle is $45sec.$ It consists of the worker obtaining two starting work units from a parts tray, loading them into the machine, and then after processing, unloading the completed units and placing them into the same tray. Each tray holds $24$ work units. When all of the starting work units have been replaced with completed units, the tray of completed parts is moved aside and a new tray of starting parts is moved into position at the machine. This irregular work element takes $3\mathrm{.}0$min. Batch quantity is $2400$ units the plant work $3$ shift $8$ hr each for $6$ days a week. Determine

$($a$)$ Average cycle time,

$($b$)$ Time to complete the batch, and

$($c$)$ Average production rate,

$($d$)$ Plant capacity