Find graphically, the maximum value of z = 2*x* + 5*y*, subject to constraints given below :

2x + 4y ≤ 83

x + y ≤ 6

x + y ≤ 4

x ≥ 0, y≥ 0

#### Solution

The given constraints are

2*x* + 4*y* ≤ 8 .....(1)

3*x* + *y* ≤ 6 .....(2)*x* + *y* ≤ 4 .....(3)*x* ≥ 0, *y* ≥ 0

We need to maximise the objective function z = 2*x* + 5*y*.

Converting the inequations into equations, we obtain the lines 2*x* + 4*y* = 8, 3*x* + *y* = 6, *x* + *y* = 4, *x* = 0 and *y* = 0.

These lines are drawn and the feasible region of the LPP is shaded.

The coordinates of the corner points of the feasible region are O(0, 0), A(0, 2), B(1.6, 1.2) and C(2, 0).

The value of the objective function at these points are given in the following table.

Points | Value of the objective function z = 2x + 5y |

O(0, 0) A(0, 2) B(1.6, 1.2) C(2, 0) |
2 × 0 + 5 × 0 = 0 2 × 0 + 5 × 2 = 10 2 × 1.6 + 5 × 1.2 = 9.2 2 × 2 + 5 × 0 = 4 |

Out of these values of *z*, the maximum value of *z *is 10 which is attained at the point (0, 2). Thus, the maximum value of *z *is 10.