The following linear program model is used to optimize the production of four products $($X$1,$

X$2,$ X$3$ and X$4),$ with two different manufacturing processes and two different material

requirements.

Max Z $=$ $$48$X$1+$ $$52$X$2+$ $$57$X$3+$ $$60$X$4($Total profit$)$

s$.$t$.4$X$1+2\mathrm{.}1$X$2+2\mathrm{.}6$X$3+3\mathrm{.}9$X$4500($Process $1)$

$3\mathrm{.}5$X$1+4\mathrm{.}6$X$2+3\mathrm{.}5$X$3+1\mathrm{.}9$X$4600($Process $2)$

$15$X$1+21$X$2+19$X$3+22$X$43500($Material A$)$

$7\mathrm{.}5$X$1+11\mathrm{.}5$X$2+9\mathrm{.}5$X$3+10\mathrm{.}6$X$41600($Material B$)$

X$1,$ X$2,$ X$3,$ X$40$

The linear program model is solved using Excel Solver and sensitivity reports are given as

below:

Adjustable Cells

Final Reduced Objective Allowable Allowable

Cell Name Value Cost Coefficient Increase Decrease

$B$$15$ Product $129\mathrm{.}1004825\mathrm{.}351\mathrm{.}68$

$C$$15$ Product $20-17\mathrm{.}025217\mathrm{.}021$E$+30$

$D$$15$ Product $3139\mathrm{.}370571\mathrm{.}407\mathrm{.}23$

$E$$15$ Product $45\mathrm{.}440605\mathrm{.}144\mathrm{.}56$

Constraints

Final Shadow Constraint Allowable Allowable

Cell Name Value Price R$.$H$.$ Side Increase Decrease

$F$$18$ Process $15000\mathrm{.}72500268\mathrm{.}9334\mathrm{.}29$

$F$$19$ Process $26002\mathrm{.}1660012\mathrm{.}97135\mathrm{.}19$

$F$$20$ Material A $3204\mathrm{.}35035001$E$+30295\mathrm{.}65$

$F$$21$ Material B $16005\mathrm{.}011600141\mathrm{.}5348\mathrm{.}98$

Based on the sensitivity reports, answer the following questions:

$($a$)$ What is the optimal production mix and total profit? $(5$ marks$)$

$($b$)$ Holding all other factors remain unchanged, what is the profit contribution of Product $1$ so

that the optimal production mix remains the same. $(2$ marks$)$

$($c$)$ Which is the most valuable resource to the firm? Why? $(2$ marks$)$

$($d$)$ One of the four products is not produced in the optimal mix. What would the profit become

if one unit of such a product was produced while other conditions remain the same?

$(2$ marks$)$

$($e$)$ If the manufacturer can obtain an extra amount of $100$ units of Material B at the price of

$$4\mathrm{.}0$ each, should he do so or not? Explain. What is the new total profit? $(4$ marks$)$