A small appliance manufacturer must meet $($on time$)$ the following demands: quarter $1,3000$ units; quarter $2,2000$ units; quarter $3,4000$ units. Each quarter, up to

$2700$ units can be produced with regular$-$time labor, at a cost of $$40$ per unit. During each quarter, an unlimited number of units can be produced with overtime labor, at a cost of $$60$ per unit. Of all units produced, $20\%$ are unsuitable and cannot be used to meet demand.

Also, at the end of each quarter, $10\%$ of all units on hand spoil and cannot be used to meet any future demands.

After each quarter's demand is satisfied and spoilage is accounted for, a cost of $$15$ per unit in ending inventory is incurred.

a$.$ Determine how to minimize the total cost