A firm must meet the following demands on time:

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. Given the firm$$s quality control system, $20\%$ of all units produced in any period are found unsuitable and cannot be used to meet demand. Furthermore, due to the partly perishable nature of the product, at the end of each quarter, $10\%$ of all units on hand spoil and cannot be used to meet any future demands. After each quarter$$s demand is satisfied and spoilage is accounted for, a storage cost of $$15$ per unit is assessed against the quarter$$s ending inventory. Note that the sequence of events in each period is:

production $($with $20\%$ defects$),$

sales,

spoilage,

storage.

$1,000$ usable units are available at the beginning of quarter $1,$ i$.$e$.,$ this is the beginning inventory.

Prepare a linear program to help the firm minimize the total cost of meeting demand in the next three quarters.

Note: Functions such as maximum, minimum, if$/$then$,$ etc. are not linear functions. Their usage will violate the requirements of a $$linear model$$ and consequently, they are not acceptable.

Specify all the variables needed to solve the linear program $($production$,$ inventory, regular and overtime$).$ Clearly define the decision variables for full credit.

Specify the objective.