Problem 34 - College Algebra

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

Under the condition \(A=\frac{2}{3}B\), the objective function \(z=A x+B y\) attains equal maximum value at both vertices (3,1) and (0,3).

Step 1: Calculation of Objective Function at Vertex (3,1)

Substitute the values \(x=3\) and \(y=1\) in the objective function \(z=A x+B y\). At \(x=3, y=1\), the equation becomes \(z=3A+B.\)

Step 2: Calculation of Objective Function at Vertex (0,3)

Substitute the values \(x=0\) and \(y=3\) in the objective function \(z=A x+B y\). At \(x=0, y=3\), the equation becomes \(z=3B.\)

Step 3: Condition for Equal Maximum Values

The values of the objective function at vertices (3,1) and (0,3) will be equal if \(3A+B=3B\) or \(A=\frac{2}{3} B\). This is the given condition. Hence, if \(A= \frac{2}{3} B\), the objective function will have the same maximum value at the vertices (3,1) and (0,3).

Key Concepts

Objective Function

An objective function in linear programming is a mathematical expression that defines the goal of the optimization process. It is usually a linear equation that represents either a cost that needs to be minimized or a profit that needs to be maximized. In the given exercise, the objective function is represented as \(z=Ax+By\), with \(A>0\) and \(B>0\), indicating that both coefficients are positive and contribute to the maximization of the function.

In the context of the exercise provided, calculating the objective function's value at different points—such as the given vertices (3,1) and (0,3)—can help determine where the maximum value occurs. By substituting these points into the function, the exercise demonstrates how the relationship between coefficients \(A\) and \(B\), specifically \(A=\frac{2}{3}B\), ensures the maximum value of the objective function is the same at both vertices. Understanding the nature of the objective function is essential in linear programming, as it guides the decision-making process and determines the optimal solution.

Constraints in Linear Programming

Constraints in linear programming are the restrictions or limitations on the values that the variables in an optimization problem can take. Practical problems often have limitations, such as resource availability, which are represented as linear equations or inequalities in a linear programming model.

In the provided exercise, the constraints are \(2x+3y \leq 9\), \(x-y \leq 2\), \(x \geq 0\), and \(y \geq 0\). These boundaries form a feasible region, within which the objective function's optimal value must lie. The exercise focuses on proving a specific value relationship between \(A\) and \(B\), so that the vertices (3,1) and (0,3) of the feasible region result in the same objective function value. Grasping the concept of constraints is critical as they define the feasible region which, in turn, contains the solution to the optimization problem. Understanding how to interpret and solve these inequalities is a fundamental skill in linear programming.

Linear Programming Vertices

Vertices in linear programming are the corner points of the feasible region, which is the area defined by the intersection of all the constraints. According to the fundamental theorem of linear programming, if there is an optimal solution to a linear programming problem, it will occur at one or more of the vertices of the feasible region.

In the exercise, the vertices in question are \((3,1)\) and \((0,3)\). These points represent potential solutions where the objective function may reach its maximum value. The exercise shows that under the condition \(A=\frac{2}{3}B\), the objective function takes on the same maximum value at both vertices. This concept illustrates one of the key aspects of linear programming - often, it's unnecessary to evaluate the objective function over the entire feasible region, as checking the vertices can be sufficient to find the optimal value. Learning how to identify and evaluate the vertices of the feasible region is an important aspect of solving linear programming problems effectively.