Question

Solve the following linear programming problems by graphical method.

Maximize Z = 40x₁ + 50x₂ subject to constraints 3x₁ + x₂ ≤ 9; x₁ + 2x₂ ≤ 8 and x₁, x₂ ≥ 0.

Answer 1

Given that 3x₁ + x₂ ≤ 9

Let 3x₁ + x₂ = 9

x1	0	3
x2	9	0

Also given that x₁ + 2x₂ ≤ 8]

Let x₁ + 2x₂ = 8

x1	0	8
x2	4	0

3x₁ + x₂ = 9 ………(1)

x₁ + 2x₂ = 8 ……..(2)

6x₁ + 2x₂ = 18 ……..(3) [Multiply by 2 for eq. (1)]

− 5x₁ = − 10

x₁ = 2

x₁ = 2 substitute in (1)

3(2) + x₂ = 9

x₂ = 3

The feasible region satisfying all the conditions is OABC.

The co-ordinates of the corner points are O(0, 0), A(3, 0), B(2, 3), C(0, 4)

Corner points	Z = 40x1 + 50x2
O(0, 0)	0
A(3, 0)	120
B(2, 3)	40 × 2 + 50 × 3 = 80 + 150 = 230
C(0, 4)	200

The maximum value of Z occurs at B(2, 3).

∴ The optimal solution is x₁ = 2, x₂ = 3 and Z_max = 230