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

Hint:

When working with linear objective functions in optimization problems, the slope of the objective function can provide useful insights. The key is to match the slope of the objective function with the slope of the constraint or boundary line to maximize or minimize the objective. This technique is particularly useful in linear programming problems where the goal is to optimize a linear function subject to certain constraints. Always ensure that you equate the slopes carefully when solving such problems.

Answer 1

The Correct Option is C

The linear programming problem requires determining the condition on \( \alpha \) and \( \beta \) for which the objective function \( Z = \alpha x + \beta y \) attains its maximum value at the corner points (5, 5) and (0, 20) of the feasible region. The corner points of the feasible region are given as (0,10), (5,5), (5,15), and (0,30).

We first evaluate the objective function at the specified corner points (5,5) and (0,20):

• At (5,5): \( Z_1 = 5\alpha + 5\beta \)
• At (0,20): \( Z_2 = 0\alpha + 20\beta = 20\beta \)

Since the maximum value of Z occurs at both these points, we set \( Z_1 \) equal to \( Z_2 \):

5⁣α+5⁣β=20⁣β

Rearranging the terms to isolate \( \alpha \):

5⁣α=15⁣β

Simplifying the equation:

α=3⁣β

Thus, the condition on \( \alpha \) and \( \beta \) is \( \alpha = 3\beta \).