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प्रश्न
Maximize Z = x + y
Subject to
\[- 2x + y \leq 1\]
\[ x \leq 2\]
\[ x + y \leq 3\]
\[ x, y \geq 0\]
उत्तर
We need to maximize Z = x + y
First, we will convert the given inequations into equations, we obtain the following equations:
−2x + y = 1, x = 2, x + y = 3, x = 0 and y = 0.
The line −2x + y = 1 meets the coordinate axis at \[A\left( \frac{- 1}{2}, 0 \right)\] and B(0, 1). Join these points to obtain the line −2x + y = 1 .
Clearly, (0, 0) satisfies the inequation −2x + y ≤ 1. So, the region in xy-plane that contains the origin represents the solution set of the given equation.
x = 2 is the line passing through (2, 0) and parallel to the Y axis.The region below the line x = 2 will satisfy the given inequation.
The line x + y = 3 meets the coordinate axis at C(3, 0) and D(0, 3). Join these points to obtain the line x + y = 3.
Clearly, (0, 0) satisfies the inequation x + y ≤ 3. So, the region in xy-plane that contains the origin represents the solution set of the given equation.
Region represented by x ≥ 0 and y ≥ 0:
Since, every point in the first quadrant satisfies these inequations. So, the first quadrant is the region represented by the inequations.
These lines are drawn using a suitable scale.
The corner points of the feasible region are O(0, 0), \[G\left( 2, 0 \right)\] ,\[E\left( 2, 1 \right)\] and \[F\left( \frac{2}{3}, \frac{7}{3} \right)\]
The values of Z at these corner points are as follows.
| Corner point | Z = x + y |
| O(0, 0) | 0 + 0 = 0 |
|
\[C\left( 2, 0 \right)\]
|
2 + 0 = 2 |
|
\[E\left( 2, 1 \right)\]
|
2 +1 = 3 |
|
\[F\left( \frac{2}{3}, \frac{7}{3} \right)\]
|
\[\frac{2}{3} + \frac{7}{3} = \frac{9}{3} = 3\]
|
We see that the maximum value of the objective function Z is 3 which is at \[E\left( 2, 1 \right)\] and \[F\left( \frac{2}{3}, \frac{7}{3} \right)\]
Thus, the optimal value of Z is 3.
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