A manufacturing company faces the aggregate planning problem shown in the table below. The

cost of regular production is $$5$ per unit, the cost of producing the same unit overtime is $$7\mathrm{.}50,$

the cost of subcontracting is $$9$ per unit, and the cost of carrying a unit in inventory from one

month to the next is $$2.$

The labor contract at the plant prohibits both overtime and subcontracting output to exceed $300$

units in any five$-$month window. The plant capacity is $600$ units per month produced using two

shifts, regardless of the number of days in a month. By policy, management wants to avoid

stockouts.

$($a$)$ Define and introduce decision variables. How many are there?

$($b$)$ Define and introduce constraints. How many are there?

$($c$)$ Define and introduce the objective function.

$($d$)$ Which of these statements is evident without even developing an aggregate plan?

i$.$ Some overtime will be needed.

ii$.$ Ending inventory must be negative.

iii. Overtime must exceed subcontracting.

iv$.$ Subcontracting must exceed regular time output.

$($e$)$ Formulate and solve the problem using the Excel solver.