Question

Corner points of the feasible region for an LPP are $(0, 2), (3, 0), (6, 0), (6, 8)$ and $(0, 5)$. Let $z = 4x + 6y$ be the objective function. The minimum value of $z$ occurs at:

• A. Only $(0, 2)$
• B. Only $(3, 0)$
• C. The mid-point of the line segment joining the points $(0, 2)$ and $(3, 0)$
• D. Any point on the line segment joining the points $(0, 2)$ and $(3, 0)$

Answer 1

The Correct Option is D

1. List the corner points:

Given points: \( (0, 2) \), \( (3, 0) \), \( (6, 0) \), \( (6, 8) \), \( (0, 5) \).

2. Evaluate the objective function \( z = 4x + 6y \) at each point:

\[ \begin{align*} (0, 2) & : z = 4(0) + 6(2) = 12 \\ (3, 0) & : z = 4(3) + 6(0) = 12 \\ (6, 0) & : z = 4(6) + 6(0) = 24 \\ (6, 8) & : z = 4(6) + 6(8) = 72 \\ (0, 5) & : z = 4(0) + 6(5) = 30 \\ \end{align*} \]

3. Determine the minimum value:

The minimum value of \( z \) is 12, achieved at both \( (0, 2) \) and \( (3, 0) \).

Since the feasible region is convex, the minimum also occurs at all points on the line segment joining \( (0, 2) \) and \( (3, 0) \).

Correct Answer: (D) Any point on the line segment joining the points (0, 2) and (3, 0)

Answer 2

The feasible region is bounded, and $z = 4x + 6y$ is the objective function. Compute $z$ at the corner points: \[ z(0, 2) = 12, \quad z(3, 0) = 12. \] Since $z$ has the same value at $(0, 2)$ and $(3, 0)$, any point on the line segment joining these points also gives the minimum value of $z$.