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प्रश्न
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.
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उत्तर
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 Zmax = 230
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