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प्रश्न
Minimize: Z = 2x + y
Subject to: x + y ≤ 5
x + 2y ≤ 8
4x + 3y ≥ 12
x ≥ o, y ≥ o
Solve graphically.
आलेख
योग
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उत्तर
| Constraints |
Corresponding equations |
Points on | |
| X-axis | Y-axis | ||
| 1. x + y ≤ 5 | x + y = 5 | A (5 , 0) | B (0 , 5) |
| 2. x + 2y ≤ 8 | x + 2y = 8 | C (8 , 0) | D (0 ,4) |
| 3. 4x + 3y ≥ 12 | 4x + 3y = 12 | E (3 , 0) | F (0 , 4) |
From the graph, the shaded region seg EAGDF is the feasible region.
The objective function Z = 2x + y will be minimum at the vertices of the feasible region EAGDF.
G is the point of ~ntersection of the lines
x + y = 5 and x + 2y = 8
By solving this simultaneous equation, we will get
x = 2 and y = 3
∴ G(2, 3) .
Now calculate Z = 2x + y at each vertex of the feasible region EAGDF at E, Z(3,0) = 2 × 3 + 0 = 6
At A , Z(5,0) = 2 × 5 + 0 = 10
At G , Z(2,3) = 2 × 2 + 3 = 7
At D , Z(0,4) = 2 × 0 + 4 = 4
From the above calculation Z is minimum at D(0,4).
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