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Question
Solve the following L. P. P. graphically:Linear Programming
Minimize Z = 6x + 2y
Subject to
5x + 9y ≤ 90
x + y ≥ 4
y ≤ 8
x ≥ 0, y ≥ 0
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Solution
To draw the feasible region, construct table as follows:
| Inequality | 5x + 9y ≤ 90 | x + y ≥ 4 | y ≤ 8 |
| Corresponding equation (of line) | 5x + 9y = 90 | x + y = 4 | y = 8 |
| Intersection of line with X-axis | (18, 0) | (4, 0) | − |
| Intersection of line with Y-axis | (0, 10) | (0, 4) | (0, 8) |
| Region | Origin side | Non-origin side | Origin side |
Shaded portion ABCDE is the feasible region, whose vertices are A(4, 0), B(18, 0), C,
D(0, 8) and E(0, 4).
C is the point of intersection of the lines y = 8 and 5x + 9y = 90.
Putting y = 8 in 5x + 9y = 90, we get
5x + 72 = 90
∴ x = 18/5
∴ C = `(18/5, 8)`
Here, the objective function is Z = 6x + 2y,
Z at A(4, 0) = 6(4) + 2(0) = 24
Z at B(18, 0) = 6(18) + 2(0) = 108
Z at `C(18/5,8) = 6(18/5)` + 2(8)
= 188/5 = 37.6
Z at D(0, 8) = 6(0) + 2(8) = 16
Z at E(0, 4) = 6(0) + 2(4) = 8
∴ Z has minimum value 8 at E(0, 4).
∴ Z is minimum, when x = 0 and y = 4.
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