Question

The corner points of the feasible region of a Linear Programming Problem are $(0, 2)$, $(3, 0)$, $(6, 0)$, $(6, 8)$, and $(0, 5)$. If $Z = ax + by; \, (a, b>0)$ be the objective function, and maximum value of $Z$ is obtained at $(0, 2)$ and $(3, 0)$, then the relation between $a$ and $b$ is :

• A. $a = b$
• B. $a = 3b$
• C. $b = 6a$
• D. $a = 3b$

Hint:

In Linear Programming, the maximum and minimum values of the objective function often occur at the corner points of the feasible region. Use the corner point method to determine the value of the objective function at these points.

Answer 1

The Correct Option is B

The maximum value of $Z$ occurs at points $(0, 2)$ and $(3, 0)$. Evaluating the objective function at these points yields:
- At $(0, 2)$: $Z = 0 \cdot a + 2b = 2b$
- At $(3, 0)$: $Z = 3a + 0 \cdot b = 3a$
To ensure the maximum value of $Z$ is identical at both points, we equate these expressions: $2b = 3a$. This establishes the relationship:
\[a = \frac{2}{3}b\]
Therefore, the correct relationship is $a = 3b$.