Question $3[10$ points$]$ A small appliance manufacturer must meet $($on time$)$ the following

demands: $3000$ units in quarter $1$;$2000$ units in quarter $2,4000$ units in quarter $3.$ Each quarter, up

to $27000$ 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 demand.

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

ending inventor is incurred.

$($a$)$ Develop an LP model to minimize the total cost of meeting the demands of the next three

quarters. Assume that $1000$ usable units are available at the beginning of quarter $1.$

$($Hint: Assume in quarter $1,$ the manufacturer produces X$1$ units with regular$-$time labor, Y$1$ units

with overtime labor, the production costs are $40x1+60Y1.$ Then the manufacturer has $1000+$

$0\mathrm{.}8(x1+Y1)$ units that can be used to meet demand. After meeting demand $($i$.$e$.,3000)$ in quarter

$1,1000+0\mathrm{.}8(x1+Y1)-3000$ units will be left, but $10\%$ of them will spoil. In other words, the

final leftover after quarter $1$ is $0\mathrm{.}9(1000+0\mathrm{.}8(x1+Y1)-3000)--$ which is also the usable units

that can meet the demand in quarter $2--$ and each unit of this leftover will incur $$15$ inventory

cost$.)$

$($b$)$ Solve the problem using Excel. $($Fractional solution is accepted$)$

$($c$)$ What are the binding constraints and the non$-$binding constraints?

$($d$)$ For those non$-$binding constraints, what are the corresponding slacks?