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Question
Solve the following linear programming problem graphically:
Maximize: Z = x + 2y
Subject to constraints:
x + 2y ≥ 100,
2x – y ≤ 0
2x + y ≤ 200,
x ≥ 0, y ≥ 0.
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Solution
Given, Maximize: Z = x + 2y
Subject to constraints:
x + 2y ≥ 100
2x – y ≤ 0
2x + y ≤ 200
x ≥ 0, y ≥ 0
Changing inequations to equations, we get
x + 2y = 100 ...(i)
2x – y = 0 ...(ii)
2x + y = 200 ...(iii)
For equation (i)
| x | 0 | 100 |
| y | 50 | 0 |
For equation (ii)
| x | 0 | 50 |
| y | 0 | 100 |
For equation (iii)
| x | 100 | 0 |
| y | 0 | 200 |
The feasible solution is bounded
| Maximum Z = x + 2y | |
| For A(20, 40) | Z = 20 + 2 × 40 = 100 |
| For B(100, 0) | Z = 100 + 0 = 100 |
| For C(50, 100) | Z = 50 + 2 × 100 = 250 `rightarrow` Max. |
Hence, the value of x = 50 and y = 100 and maximum Z = 250.
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