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

Since the maximum value of $Z$ is obtained at $(0, 2)$ and $(3, 0)$, we have the following system of equations for the objective function at these points: - At $(0, 2)$: $Z = 0 \cdot a + 2b = 2b$ - At $(3, 0)$: $Z = 3a + 0 \cdot b = 3a$ For the maximum value of $Z$ to be the same at both points, we set $2b = 3a$, which gives the relation: \[ a = \frac{2}{3}b \] Thus, the correct relation is $a = 3b$.