Problem 34 - Algebra and Trigonometry

Question

Consider the objective function \(z=A x+B y(A>0\) and \(B>0\) ) subject to the following constraints: \(2 x+3 y \leq 9\) \(x-y \leq 2, x \geq 0,\) and \(y \geq 0 .\) Prove that the objective function will have the same maximum value at the vertices \((3,1)\) and \((0,3)\) if \(A=\frac{2}{3} B\).

Step-by-Step Solution

Answer

The coefficients of \(x\) and \(y\) in the objective function must be in the ratio \(A = \frac{2}{3} B\) in order for the objective function to have the same maximum value at the vertices (3,1) and (0,3).

Step 1: Calculate the Value of the Objective Function at the Vertices

We substitute the coordinates of the vertices into the objective function to find its values. For the vertex at (3,1), the value of \(z\) is \(z = A*3 + B*1\), and for the vertex at (0,3), the value of \(z\) is \(z = A*0 + B*3\).

Step 2: Equate the Values of the Objective Function from Step 1

For \(z\) to have the same maximum value at both vertices, the values obtained from Step 1 should be equal i.e. \(A*3 + B*1 = A*0 + B*3\). Simplifying this equation leads to \(3A + B = 3B\).

Step 3: Simplify the Equation from Step 2 to Find the Relationship between A and B

Rewriting the equation from Step 2, we get \(3A = 2B\), or \(A = \frac{2}{3} B\). Therefore, the objective function will have the same maximum value at the vertices (3,1) and (0,3) if \(A = \frac{2}{3} B\).

Key Concepts

Objective Function

In linear programming, the objective function is a mathematical expression that you aim to either maximize or minimize. It is based on variables and coefficients which represent certain real-world quantities. In our problem, the objective function is represented by the formula \( z = Ax + By \), where \( A > 0 \) and \( B > 0 \). Here, \( x \) and \( y \) are the decision variables, and \( A \) and \( B \) are their respective coefficients, indicating how much each variable contributes to the total value of \( z \). The goal in this problem is to find the maximum value of \( z \). Let's delve deeper into how this objective function interacts with other components of the linear programming model.

Vertex Analysis

Vertex analysis is an essential step in solving linear programming problems. The feasible region, defined by the constraints of the problem, typically forms a polygon on the graph. The special characteristic of linear programming is that the maximum or minimum value of the objective function must occur at one of the vertices (corners) of this polygon.

• For our problem, the vertices of the feasible region are (3,1) and (0,3).
• By calculating the value of the objective function at each vertex, we can identify which gives the maximum or minimum value.
• This method simplifies the computational task by focusing only on a limited set of potential solutions.

This makes vertex analysis a powerful tool whenever you are dealing with linear constraints.

Constraints

Constraints are conditions that must be met within a linear programming problem. They define the feasible region within which solutions must lie. In our scenario, the constraints are:

• \(2x + 3y \leq 9\)
• \(x - y \leq 2\)
• \(x \geq 0\)
• \(y \geq 0\)

These constraints are represented as linear inequalities that limit the possible values \( x \) and \( y \) can take. They ensure that any chosen solution meets all necessary conditions. It's crucial to correctly identify and graph these constraints to accurately determine the potential feasible region for vertex analysis.

Optimization

Optimization in linear programming is about finding the best possible solution from a set of feasible solutions as defined by constraints. The aim is to maximize or minimize the objective function. In the given problem, optimization is about maximizing the objective function \( z = Ax + By \) at certain points in the feasible region. By substituting the coordinates of the vertices (3,1) and (0,3) into the objective function, it becomes evident that the function achieves the same maximum value at both of these points if \( A = \frac{2}{3} B \).

• First, calculate \( z \) at vertex (3,1): \( z = 3A + B \)
• Next, calculate \( z \) at vertex (0,3): \( z = 3B \)
• Set these equations equal to one another to find the relationship \( 3A + B = 3B \) which simplifies to \( A = \frac{2}{3} B \).

This example illustrates how changing coefficients in an objective function can alter the optimization process and outcome.