Consider the following linear program:

Max

$1$

$$

$+$

$2$

$$

s$.$t$.$

$1$

$$

$<=$

$5$

$1$

$$

$<=$

$4$

$2$

$$

$+$

$2$

$$

$=$

$12$

$$

$,$

$$

$>=$

$0$

Show the feasible region.

What are the extreme points of the feasible region?

Find the optimal solution using the graphical procedure.

RMC$,$ Inc., is a small firm that produces a variety of chemical products. In a particular production process, three raw materials are blended $($mixed together$)$ to produce two products: a fuel additive and a solvent base. Each ton of fuel additive is a mixture of

$2$

$5$

ton of material $1$ and

$3$

$5$

of material $3.$ A ton of solvent base is a mixture of

$1$

$2$

ton of material $1,$

$1$

$5$

ton of material $2,$ and

$3$

$10$

ton of material $3.$ After deducting relevant costs, the profit contribution is $$40$ for every ton of fuel additive produced and $$30$ for every ton of solvent base produced.

RMC$$s production is constrained by a limited availability of the three raw materials. For the current production period, RMC has available the following quantities of each raw material:

Raw Material

Amount Available for Production

Material $1$

$20$ tons

Material $2$

$5$ tons

Material $3$

$21$ tons

Assuming that RMC is interested in maximizing the total profit contribution, answer the following:

What is the linear programming model for this problem?

Find the optimal solution using the graphical solution procedure. How many tons of each product should be produced, and what is the projected total profit contribution?

Is there any unused material? If so$,$ how much?

Are any of the constraints redundant? If so$,$ which ones?