Understanding the Problem: Finding Condition for Maximum Value

The problem asks us to find a relationship between the coefficients \( a \) and \( b \) of an objective function \( z = ax + by \), where \( a > 0 \) and \( b > 0 \). We are given the corner points of a feasible region in a Linear Programming problem. We need to find the condition on \( a \) and \( b \) such that the maximum value of the objective function \( z \) occurs at two specific corner points: (120,0) and (60,30).

Evaluating the Objective Function at Corner Points

In Linear Programming, the maximum or minimum value of an objective function, if it exists, always occurs at one of the corner points of the feasible region. If the maximum value occurs at two distinct corner points, it means that the objective function has the same maximum value at both of these points, and this value is greater than or equal to the value at all other corner points.

Let's evaluate the objective function \( z = ax + by \) at each of the given corner points:

• At (60,0): \( z_1 = a(60) + b(0) = 60a \)
• At (120,0): \( z_2 = a(120) + b(0) = 120a \)
• At (40,20): \( z_3 = a(40) + b(20) = 40a + 20b \)
• At (60,30): \( z_4 = a(60) + b(30) = 60a + 30b \)

Condition for Maximum at Two Points

We are told that the maximum value of \( z \) occurs at (120,0) and (60,30). This implies two conditions:

1. The value of \( z \) at (120,0) must be equal to the value of \( z \) at (60,30).
2. This common value must be greater than or equal to the values of \( z \) at the other corner points (60,0) and (40,20).

Let's use the first condition to find the relationship between \( a \) and \( b \):

\( z_2 = z_4 \)

\( 120a = 60a + 30b \)

Solving for the Relationship between a and b

Now, we solve the equation \( 120a = 60a + 30b \) for \( a \) and \( b \):

Subtract \( 60a \) from both sides:

\( 120a - 60a = 30b \)

\( 60a = 30b \)

Divide both sides by 30:

\( \frac{60a}{30} = \frac{30b}{30} \)

\( 2a = b \)

Verifying the Condition

Let's check if the condition \( 2a = b \) ensures that \( z_2 = z_4 \) is the maximum value, given that \( a > 0 \) and \( b > 0 \). If \( 2a = b \), then since \( a > 0 \), it must be that \( b > 0 \).

Under the condition \( b = 2a \):

• \( z_1 = 60a \)
• \( z_2 = 120a \)
• \( z_3 = 40a + 20b = 40a + 20(2a) = 40a + 40a = 80a \)
• \( z_4 = 60a + 30b = 60a + 30(2a) = 60a + 60a = 120a \)

We see that \( z_2 = z_4 = 120a \). We need to check if \( 120a \) is greater than or equal to \( z_1 \) and \( z_3 \).

• Compare \( z_2 \) with \( z_1 \): \( 120a \) vs \( 60a \). Since \( a > 0 \), \( 120a > 60a \). So \( z_2 > z_1 \).
• Compare \( z_2 \) with \( z_3 \): \( 120a \) vs \( 80a \). Since \( a > 0 \), \( 120a > 80a \). So \( z_2 > z_3 \).

Both conditions are satisfied. The value \( 120a \) is indeed the maximum value among all corner points when \( 2a = b \) and \( a > 0, b > 0 \).

Conclusion

The condition on \( a \) and \( b \) such that the maximum of \( z = ax + by \) occurs at (120,0) and (60,30) is \( 2a = b \).

Revision Table: Key Concepts in Linear Programming

Additional Information: Linear Programming and Optimal Solutions

Linear programming is a mathematical method for determining a way to achieve the best outcome (such as maximum profit or lowest cost) in a given mathematical model whose requirements are represented by linear relationships. These relationships are called constraints.

The set of points satisfying all constraints is called the feasible region. If the feasible region is non-empty and bounded (like a polygon), the objective function will have both a maximum and a minimum value within this region.

The fundamental theorem of linear programming states that if an optimal solution exists, it must occur at a corner point of the feasible region. If the objective function is optimized at two adjacent corner points, it is also optimized at all points on the line segment connecting these two corner points. In this problem, the maximum occurs at (120,0) and (60,30), which implies any point on the line segment connecting (120,0) and (60,30) would also yield the maximum value for \( z \) under the derived condition.

The condition \( 2a = b \) represents the slope of the objective function's level lines being equal to the slope of the line segment connecting the two optimal corner points, and this level line is the farthest from the origin in the direction of increasing \( z \) (since \( a, b > 0 \)).