Solve the following LPP graphically:
Maximize Z $=$ 20 $x +$ 10 $y$
Subject to the following constraints
$x +$ 2 $y \leq$ 28
3 $x +$ $y \leq$ 24
$x \geq$ 2
$x$ , $y \geq$ 0

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Solution

The given constraints are

x + 2y ≤ 28

3x + y ≤ 24

x ≥ 2

x, y ≥ 0

Converting the given inequations into equations, we get

x + 2y = 28, 3x + y = 24, x = 2, x = 0 and y = 0

These lines are drawn on the graph and the shaded region ABCD represents the feasible region of the given LPP.

It can be observed that the feasible region is bounded. The coordinates of the corner points of the feasible region are A(2, 13), B(2, 0), C(4, 12) and D(8, 0).

The values of the objective function, Z at these corner points are given in the following table:

Corner Point Value of the Objective Function Z = 20x + 10y

A(2, 13) Z = 20 × 2 + 10 × 13 = 170

B(2, 0) Z = 20 × 2 + 10 × 0 = 40

C(8, 0) Z = 20 × 8 + 10 × 0 = 160

D(4, 12) Z = 20 × 4 + 10 × 12 = 200

From the table, Z is maximum at x = 4 and y = 12 and the maximum value of Z is 200.

Thus, the maximum value of Z is 200.

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Corner Point	Value of the Objective Function Z = 20x + 10y
A(2, 13)	Z = 20 × 2 + 10 × 13 = 170
B(2, 0)	Z = 20 × 2 + 10 × 0 = 40
C(8, 0)	Z = 20 × 8 + 10 × 0 = 160
D(4, 12)	Z = 20 × 4 + 10 × 12 = 200