Linear Programming Problem: Finding Condition for Maximum Z

This problem involves a Linear Programming Problem (LPP) where we are given some corner points of the feasible region and an objective function $Z = \alpha x + \beta y$, with $\alpha > 0$ and $\beta > 0$. We are told that the maximum value of this objective function Z occurs at two specific points, (5, 5) and (0, 20). We need to find the relationship between $\alpha$ and $\beta$ based on this condition.

The corner points of the feasible region are given as (0, 10), (5, 5), (5, 15), and (0, 30). However, the question states that the maximum of Z occurs at corner points (5, 5) and (0, 20). In LPP, if the maximum occurs at two distinct points, it means the objective function takes the same maximum value at both points.

Understanding Multiple Optima in LPP

In a Linear Programming Problem, if the optimal solution (maximum or minimum) occurs at two distinct points on the boundary of the feasible region, then it occurs at every point on the line segment joining these two points. This happens when the line representing the objective function for the optimal value is parallel to a boundary edge of the feasible region, and this edge contains the two points.

A fundamental property in this case is that the value of the objective function is identical at these two distinct optimal points.

Calculating Objective Function Value at Specified Points

The objective function is $Z = \alpha x + \beta y$. We are given that the maximum value of Z occurs at points (5, 5) and (0, 20).

Let's evaluate the objective function Z at each of these points:

• At the point (5, 5):
  $Z(5, 5) = \alpha(5) + \beta(5) = 5\alpha + 5\beta$
• At the point (0, 20):
  $Z(0, 20) = \alpha(0) + \beta(20) = 0 + 20\beta = 20\beta$

Equating Objective Function Values at Optimal Points

Since the maximum value of Z occurs at both (5, 5) and (0, 20), the value of the objective function must be the same at these two points. We set the expressions for $Z(5, 5)$ and $Z(0, 20)$ equal:

$\qquad 5\alpha + 5\beta = 20\beta$

Solving for the Condition on $\alpha$ and $\beta$

Now we solve this equation to find the required relationship between $\alpha$ and $\beta$.

Subtract $5\beta$ from both sides of the equation:

$\qquad 5\alpha = 20\beta - 5\beta$

$\qquad 5\alpha = 15\beta$

To isolate $\alpha$, divide both sides of the equation by 5:

$\qquad \alpha = \frac{15\beta}{5}$

$\qquad \alpha = 3\beta$

This equation $\alpha = 3\beta$ represents the condition on $\alpha$ and $\beta$ for the maximum of the objective function $Z = \alpha x + \beta y$ to occur at both points (5, 5) and (0, 20).

Matching with Given Options

Let's compare the derived condition $\alpha = 3\beta$ with the provided options:

• Option 1: $\alpha = 5\beta$
• Option 2: $5\alpha = \beta$
• Option 3: $\alpha = 3\beta$
• Option 4: $4\alpha = 5\beta$

The condition we found, $\alpha = 3\beta$, matches Option 3.

This condition implies that the ratio $\alpha/\beta = 3$. Since $\alpha, \beta > 0$, the slope of the objective function line $Z = \alpha x + \beta y$ (which is $y = -\frac{\alpha}{\beta}x + \frac{Z}{\beta}$, with slope $-\frac{\alpha}{\beta}$) is $-\frac{3\beta}{\beta} = -3$. The slope of the line segment connecting (5, 5) and (0, 20) is $\frac{20-5}{0-5} = \frac{15}{-5} = -3$. The equality of slopes confirms that the objective function line is parallel to the segment connecting these two points, consistent with the maximum occurring along this segment.

Revision Table: LPP Optimization Basics

Additional Information: Further Insights into LPP

Linear Programming is a powerful mathematical technique used across various fields, including business, economics, and engineering, for decision-making involving optimization.

• Graphical Method: For LPPs with two variables, the graphical method is intuitive. Plotting the feasible region and objective function lines helps visualize the optimal solution.
• The Role of Corner Points: The Corner Point Theorem is crucial because it limits the search for the optimum to a finite number of points. Evaluating Z only at corner points is sufficient to find the maximum or minimum.
• Understanding Multiple Solutions: When multiple optima exist, any point on the segment connecting two optimal corner points is also optimal. This provides flexibility in choosing a solution, which might be useful if there are other non-linear considerations not included in the LPP model.
• Impact of $\alpha, \beta > 0$: For $Z = \alpha x + \beta y$ with $\alpha, \beta > 0$, the objective function value generally increases as $x$ and $y$ increase. This means the maximum will tend to be further away from the origin within the feasible region. The condition $\alpha = 3\beta$ shows how the relative weights of $x$ and $y$ in the objective function determine the direction of the gradient and thus the boundary segment where the maximum lies.

The problem demonstrates a key characteristic of LPP optima occurring along an edge when the objective function's slope aligns with that edge.