Question

Linear Progamming 3: A manufacturer wants to maximize the profit of two products.  Product X yields a profit of $ 2.50 per unit, and product Y yields a profit of $3.20 per unit.  Market tests and available resources have indicated the following constraints:

• The combined production level should not exceed 1200 units per month.
• The demand for product Y is no more than half the demand for product X.
• The production level of product X is less than or equal to 600 units plus three times the production level of product Y.

Show all steps. Answer the question fully!

Use and show how  systems of equations  obtain critical corners!

Answer 1

Solution:

We have the following information available.

OBJECTIVE FUNCTION

2.5X₁ + 3.2X₂

MAX = = 2.5x + 3.2x2 subject to *1 + x2s 1200 *1 - 2x220 *1 - 3X25 600 and X1, X220;

Hint to draw constraints 1. To draw constraint x1 + x2 = 1200 (1) Treat it as x: +x2 = 1200 When X1 = 0 then x2 = ? = (0) + x2 = 1200 X = 1200 When x2 = 0 then x1 = ? = x1 + (0) = 1200 = 1200 *: 0 1200 X2 1200 0 2. To draw constraint X1 - 2x2 > 0 – (2) Treat it as x1 - 2x2 = 0 When xy = 0 then xy = ? = (0) - 2x2 = 0 = -2x = 0 0 212-3 = 0 When xy = 0 then x1 = ? > X.-200) = 0 x = 0 *100 *200

3. To draw constraintx: - 3x2 < 600 - (3) Treat it as x1 - 3x2 - 600 When xy = 0 then = (0) - 3x2 = 600 = - 3x2 = 600 = x2 600 - 3 -200 When x2 = 0 then x1 = ? > *1-3(0) = 600 3x = 600 * 0 600 22 -2000

xnyo 1200 xo x-3y=100 1200 1000 500 400 200 xht +x1 -1000 500 500 1000 200 400

each of these extreme points is as follows: The value of the objective function Extreme Point Coordinates Lines through Extreme Point Objective function value := 2.5x1 -3.2x2 0(0,0) 2.500) + 3.200) = 0 4(800, 400) 2.5(800) - 3.2(400) = 3280 2 + x1 - 2x220 4-X120 1 + x3+x2 = 1200 2- -2x20 1- x: +22 $ 1200 3- X1 - 3x2 < 600 3- X1 - 3x2 s 600 5 + x220 B(1050, 150) 2.5(1050) + 3.2(150) = 3105 C(600.0) 2.5(600) + 3.2(0) = 1500 The maximum value of the objective function : = 3280 occurs at the extreme point (800, 400). Hence, the optimal solution to the given LP problem is : x1 = 300, x2 = 400 and max z = 3280.

To obtain Maximum Profit

We need to produce 800 units of product X and 400 units of Product Y. So, that we earn a profit of $ 3280