Question

Consider a model in which two products, x and y, are produced. There are 30 pounds of material and 60 hours of labor available. It requires 9 pounds of material and 12 hours of labor to produce a unit of x, and 5 pounds of material and 15 hours of labor to produce a unit of y. The profit for x is $300 per unit, and the profit for y is $250 per unit.
Required:
How many units of x and y to produce to maximize profit, the model is

Answer 1

Answer:

2 units of x and 2 units of y

Explanation:

The model can be represented as:

$\begin{array}{cccc} & {x} & {y} & {} & {Materials} & {9} & {5} & {30} & {Labor} & {12} & {15} & {60} & {} & {300} & {250} \ \end{array}$

So, we have:

Max $z = 300x + 250y$ --- the objective function

Subject to:

$9x + 5y \le 30$

$12x + 15y \le 60$

$x,y > 0$

Multiply the first equation by 3

$9x + 5y \le 30$ becomes

$27x + 15y \le 90$

Subtract $12x + 15y \le 60$ from $27x + 15y \le 90$

$27x - 12x + 15y - 15y \le 90 - 60$

$15x \le 30$

Divide by 15

$x \le 2$

Substitute 2 for x in $9x + 5y \le 30$

$9 * 2 + 5y \le 30$

$18 + 5y \le 30$

Collect like terms

$5y \le 30 - 18$

$5y \le 12$

Divide by 5

$y \le 2.4$

y must be an integer;

So:

$y \le 2$

So, we have:

$(x,y) \le (2,2)$

Hence, the company must product 2 units of x and 2 units of y