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प्रश्न
Solve the following L.P.P. by graphical method :
Minimize : Z = 7x + y subject to 5x + y ≥ 5, x + y ≥ 3, x ≥ 0, y ≥ 0.
उत्तर
To draw the feasible region, construct table as follows:
| Inequality | 5x + y ≥ 5 | x + y ≥ 3 |
| Corresponding equation (of line) | 5x + y = 5 | x + y = 3 |
| Intersection of line with X-axis | (1, 0) | (3, 0) |
| Intersection of line with Y-axis | (0, 5) | (0, 3) |
| Region | Non-origin side | Non-origin side |
Shaded portion XABCY is the feasible region,
whose vertices are A(3, 0), B and C (0, 5).
B is the point of intersection of the lines x + y = 3 and 5x + y = 5
Solving the above equations, we get
x = `(1)/(2), y = (5)/(2)`
∴ B ≡ `(1/2, 5/2)`
Here, the objective function is Z = 7x + y
Z at A(3, 0) = 7(3) + 0 = 21
Z at B`(1/2, 5/2) = 7(1/2) + (5)/(2)`
= `(7)/(2) + (5)/(2)` = 6
Z at C (0, 5) = 7(0) + 5 = 5
∴ Z has minimum value 5 at C(0, 5).
∴ Z is minimum, when x = 0 and y = 5.
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In the figure, ABCD represents
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The coordinates of C are obtained by solving equations
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| A | (12, 0) | 4(12) + 5(0) | 48 | |
| B | ( ___, ___ ) | 4( ___) + 5(___ ) | ______ | ______ |
| C | ( ___, ___ ) | 4( ___) + 5(___ ) | ______ | |
| D | (0, 10) | 4(0) + 5(10) | 50 |
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