Gang Aft Agley, a manufacturing company, faces the aggregate planning problem shown in the table below. Cost of regular production is $$5$ per unit, the cost of producing the same unit on overtime is $$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.$

Month Forecast Beginning Inventory Regular Time Overtime Subcontracting Ending Inventory

Jan Feb Mar Apr May $5007501200650300100$

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 they?

$1$

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

$($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. Turn in your answers as well as the Excel file.