Question:
The feasible region of a linear programming problem with objective function \( Z = 5x + 7y \) is shown below :
The feasible region of a linear programming problem with objective function \( Z = 5x + 7y \) is shown below :
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Tip 1: Always check the origin first for minimization problems if it's part of the feasible region.
Tip 2: Linear objective functions change most rapidly in the direction of their coefficients.
Tip 2: Linear objective functions change most rapidly in the direction of their coefficients.
Updated On: Sep 10, 2026
- \( 8 \)
- \( 29 \)
- \( 35 \)
- \( 43 \)
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The Correct Option is D
Solution and Explanation
Concept:
• Corner Point Theorem: The optimal (maximum or minimum) value of an objective function in an LPP occurs at the vertices (corner points) of the feasible region.
Step 1: Identify the corner points of the shaded region
From the provided graph, the vertices are:
• \( (0, 0) \)
• \( (0, 2) \)
• \( (3, 4) \)
• \( (7, 0) \)
Step 2: Evaluate \( Z = 5x + 7y \) at each corner point
At \( (0, 0) \): \( Z = 5(0) + 7(0) = 0 \)
At \( (0, 2) \): \( Z = 5(0) + 7(2) = 14 \)
At \( (3, 4) \): \( Z = 5(3) + 7(4) = 15 + 28 = 43 \)
At \( (7, 0) \): \( Z = 5(7) + 7(0) = 35 \)
Step 3: Identify the maximum and minimum values
From the calculated values: \[ Z_{\text{max}} = 43 \quad \text{at } (3, 4) \] \[ Z_{\text{min}} = 0 \quad \text{at } (0, 0) \]
Step 4: Calculate the difference
\[ \text{Difference} = Z_{\text{max}} - Z_{\text{min}} \] \[ \text{Difference} = 43 - 0 = 43 \]
• Corner Point Theorem: The optimal (maximum or minimum) value of an objective function in an LPP occurs at the vertices (corner points) of the feasible region.
Step 1: Identify the corner points of the shaded region
From the provided graph, the vertices are:
• \( (0, 0) \)
• \( (0, 2) \)
• \( (3, 4) \)
• \( (7, 0) \)
Step 2: Evaluate \( Z = 5x + 7y \) at each corner point
At \( (0, 0) \): \( Z = 5(0) + 7(0) = 0 \)
At \( (0, 2) \): \( Z = 5(0) + 7(2) = 14 \)
At \( (3, 4) \): \( Z = 5(3) + 7(4) = 15 + 28 = 43 \)
At \( (7, 0) \): \( Z = 5(7) + 7(0) = 35 \)
Step 3: Identify the maximum and minimum values
From the calculated values: \[ Z_{\text{max}} = 43 \quad \text{at } (3, 4) \] \[ Z_{\text{min}} = 0 \quad \text{at } (0, 0) \]
Step 4: Calculate the difference
\[ \text{Difference} = Z_{\text{max}} - Z_{\text{min}} \] \[ \text{Difference} = 43 - 0 = 43 \]
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