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 given linear programming problem involves finding the condition on α and β such that the maximum value of the objective function \( Z = \alpha x + \beta y \) occurs at the corner points (5, 5) and (0, 20) of the feasible region. To solve this, we analyze the given corner points of the feasible region: (0,10), (5,5), (5,15), and (0,30). 

First, compare the Z values at the corner points (5,5) and (0,20):

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

Now, equate \( Z_1 \) and \( Z_2 \) since Z maximum occurs at both these points:

5⁣α+5⁣β=20⁣β

Rearranging gives:

5⁣α=15⁣β

Simplifying:

α=3⁣β

Therefore, the condition on α and β is α = 3β.

Answer 2

The slope of the objective function \( Z = \alpha x + \beta y \) is given by the formula:

\[ \text{Slope of the objective function} = -\frac{\alpha}{\beta}. \] In order to maximize \( Z \), the slope of the objective function must match the slope of the line passing through the given points (5, 5) and (0, 20).

Step 1: Calculate the slope of the line passing through the points (5, 5) and (0, 20):

The slope of a line passing through two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by the formula: \[ \text{Slope} = \frac{y_2 - y_1}{x_2 - x_1} \] Substituting the given points (5, 5) and (0, 20): \[ \text{Slope} = \frac{20 - 5}{0 - 5} = \frac{15}{-5} = -3. \]

Step 2: Equate the slope of the objective function with the slope of the line:

The slope of the objective function is \( -\frac{\alpha}{\beta} \), and we want it to be equal to the slope of the line, which is \( -3 \). Therefore, we have the equation: \[ -\frac{\alpha}{\beta} = -3. \]

Step 3: Solve for \( \alpha \):

Simplifying the equation: \[ \frac{\alpha}{\beta} = 3 \implies \alpha = 3\beta. \]

Conclusion: Thus, the correct answer is \( \alpha = 3\beta \).