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

Question

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Respuesta :

MrRoyal MrRoyal · Answer 1 · 2021-06-12T00:17:41+02:00

Answer:

2 units of x and 2 units of y

Explanation:

The model can be represented as:

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

So, we have:

Max [tex]z = 300x + 250y[/tex] --- the objective function

Subject to:

[tex]9x + 5y \le 30[/tex]

[tex]12x + 15y \le 60[/tex]

[tex]x,y > 0[/tex]

Multiply the first equation by 3

[tex]9x + 5y \le 30[/tex] becomes

[tex]27x + 15y \le 90[/tex]

Subtract [tex]12x + 15y \le 60[/tex] from [tex]27x + 15y \le 90[/tex]

[tex]27x - 12x + 15y - 15y \le 90 - 60[/tex]

[tex]15x \le 30[/tex]

Divide by 15

[tex]x \le 2[/tex]

Substitute 2 for x in [tex]9x + 5y \le 30[/tex]

[tex]9 * 2 + 5y \le 30[/tex]

[tex]18 + 5y \le 30[/tex]

Collect like terms

[tex]5y \le 30 - 18[/tex]

[tex]5y \le 12[/tex]

Divide by 5

[tex]y \le 2.4[/tex]

y must be an integer;

So:

[tex]y \le 2[/tex]

So, we have:

[tex](x,y) \le (2,2)[/tex]

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