Question

The corner points of the feasible region for an L.P.P. are (0, 10), (5, 5), (5, 15), and (0, 30). If the objective function is Z = αx + βy, α, β > 0, the condition on α and β so that maximum of Z occurs at corner points (5, 5) and (0, 20) is:

• A. α = 5β
• B. 5α = β
• C. α = 3β
• D. 4α = 5β

Answer 1

Answer: (C)

α = 3β

Answer 2

Solution: The slope of the objective function $Z = \alpha x + \beta y$ is given by $-\frac{\alpha}{\beta}$. To maximize $Z$, the slope of the objective function must match the slope of the line passing through the points (5, 5) and (0, 20).

The slope of the line passing through (5, 5) and (0, 20) is:

$\text{Slope} = \frac{20 - 5}{0 - 5} = -3$.

Equating this with the slope of the objective function:

$-\frac{\alpha}{\beta} = -3 \implies \alpha = 3\beta$.

Thus, the correct answer is $\alpha = 3\beta$.