$2.(40$ points$)$ A semiconductor chip manufacturer produces three different types of processors: dual$-$core, quad$-$core, and hexa$-$core. The production rate is $600$ units per day for each processor type. Each time the manufacturing facility switches from producing one processor type to another, the production should be stopped for two days for configuration of the machinery. Demands for processors, unit production costs, and fixed configuration costs are as follows:

The inventory holding cost per unit per year is $\backslash (25\backslash \%\backslash )$ of the unit production cost for each processor type. Assume $365$ days in a year. The firm implements a rotation cycle policy: the manufacturing facility is set up exactly once for each processor type in each cycle.

$($a$)(10$ points$)$ How many units of each processor type should the firm produce each time? How many times should the firm produce a batch of dual$-$core processors each year?

$($b$)(10$ points$)$ What is the optimal total annual cost $($inventory holding costs plus production costs plus configuration costs$)$ for these three processor types?

$($c$)(10$ points$)$ What is the maximum inventory level for each processor type?

$($d$)(10$ points$)$ What percentage of the time is the manufacturing facility busy with producing each processor type? What percentage of the time is the facility being setup? What percentage of the time is the facility idle?