Minimize Z = 3x1 + 2x2 subject to the constraints 5x1 + x2 ≥ 10; x1 + x2 > 6; x1+ 4x2 ≥ 12 and x1, x2 ≥ 0.

Question

Solve the linear programming problem by graphical method.

Minimize Z = 3x₁ + 2x₂ subject to the constraints 5x₁ + x₂ ≥ 10; x₁ + x₂ > 6; x₁+ 4x₂ ≥ 12 and x₁, x₂ ≥ 0.

Answer 1

Given that 5x₁ + x₂ ≥ 10

Let 5x₁ + x₂ = 10

x1	0	2
x2	10	0

Also given that x₁ + x₂ ≥ 6

Let x₁ + x₂ = 6

x1	0	6
x2	6	0

Also given that x₁ + 4x₂ ≥ 12

Let x₁ + 4x₂ = 12

x1	0	12
x2	3	0

To get C

5x₁ + x₂ = 10 … (1)

x₁ + x₂ = 6 … (2)

(1) – (2) ⇒ 4x₁ = 4

⇒ x₁ = 1

x = 1 substitute in (2)

⇒ x₁ + x₂ = 6

⇒ 1 + x₂ = 6

⇒ x₂ = 5

∴ C is (1, 5)

To get B

x₁ + x₂ = 6

x₁ + 4x₂ = 12

(1) – (2) ⇒ -3x₂ = -6

x₂ = 2

x₂ = 2 substitute in (1), x₁ = 4

∴ B is (4, 2)

The feasible region satisfying all the conditions is ABCD.

The co-ordinates of the comer points are A(12, 0), B(4, 2), C(1, 5) and D(0, 10).

The minimum value of Z occours at C(1, 5).

∴ The optimal solution is x₁ = 1, x₂ = 5 and Z_min = 13