Question

Solve the following linear programming problems by graphical method.

Minimize Z = 20x₁ + 40x₂ subject to the constraints 36x₁ + 6x₂ ≥ 108; 3x₁ + 12x₂ ≥ 36; 20x₁ + 10x₂ ≥ 100 and x₁, x₂ ≥ 0.

Answer 1

Given that 36x₁ + 6x₂ ≥ 108

Let 36x₁ + 6x₂ = 108

6x₁ + x₂ = 18

x1	0	3	2
x2	18	0	6

Also given that 3x₁ + 12x₂ ≥ 36

Let 3x₁ + 12x₂ = 36

x₁ + 4x₂ = 12

x1	0	12	4
x2	3	0	2

Also given that 20x₁ + 10x₂ ≥ 100

Let 20x₁ + 10x₂ = 100

2x₁ + x₂ = 10

x1	0	5	4
x2	10	0	2

The feasible region satisfying all the conditions is ABCD.

The coordinates of the comer points are A(12, 0), B(4, 2), C(2, 6) and D(0, 18).

Corner points	Z = 20x1 + 40x2
A(12, 0)	240
B(4, 2)	80 + 80 = 160
C(2, 6)	40 + 240 = 280
D(0, 18)	720

The minimum value of Z occurs at B(4, 2)

∴ The optimal solution is x₁ = 4, x₂ = 2 and Z_min = 160