Answer to Question #224637 in Operations Research for mafi

Question

Hint:Solve the below formulated LPM using Graphics method.

Z-Min = 1500x+2400y

Subjected to:

4x+Y>24

2x+3y>42

X+4y>36

X<14

Y<14

X, y>0 Required: Perform the below requirements by Using the Graphics method through considering

appropriate steps and procedures.

i. Solve using Graphics method?

Answer 1

Question:

Hint: Solve the below formulated LPM using Graphics method.

Z-Min = 1500x+2400y

Subjected to:

4x+Y>24

2x+3y>42

X+4y>36

X<14

Y<14

X, y>0 Required: Perform the below requirements by Using the Graphics method through considering

appropriate steps and procedures.

i. Solve using Graphics method?

Solution:

Step 1:

I will consider x as the x-axis on the graph and the y as the y-axis on the graph. Every constrain will treated as a linear equation and plotted in the graph. Then I will use the appropriate inequality conditions to mark the feasibility area solutions.

The first constraint 4x + Y > 24, is considered as the equation.

4x + Y = 24

 This indicates that on plotting the equation on a graph, the x will cut the x-axis at 6 and the y cuts y-axis at 24, forming a straight line.

Similarly, the other constraints:

2x + 3y > 42 considered equation 2x + 3y = 42

 The x-axis = 21 and y-axis = 14

X + 4y > 36 considered equation X + 4y = 36

 The x-axis = 36 and y-axis = 9

X < 14 considered equation X = 14

 The x-axis = 14 and y-axis = 0

Y < 14 considered equation Y = 14

The x-axis = 0 and y-axis = 14

Step 2:

Determining the coordinates of the extreme points.

  Extreme Point Coordinate

O (0, 0)

A (9. 0)

B (8, 4)

C (0, 6)

Step 3:

In this step, we evaluate the value of the objective at every extreme point. 

Extreme Point Coordinate (X, Y) Objective Functional Value

Z = 1500x + 2400y

O (0, 0) 1500(0) + 2400(0) = 0

A (9. 0) 1500(9) + 2400(0) = 13500

B (8, 4) 1500(8) + 2400(4) = 21600

C (0, 6) 1500(0) + 2400(6) = 14400

Step 4:

Determine the optimal value of the function.

From Step 3 above, the maximum value of the objective function. The maximum value of Z = 21600and is achieved at vertex B (8, 4). Thus, the optimal solution is given by

Answer:

X = 8

Y = 4

Objective function value = 21600