Problem 55 - College Algebra and Calculus: An Applied Approach

Question

The given linear programming problem has an unusual characteristic. Sketch a graph of the solution region for the problem and describe the unusual characteristic. Find the maximum value of the objective function and where it occurs. Objective function. \(z=3 x+4 y\) Constraints. \(\begin{aligned} x & \geq 0 \\ y & \geq 0 \\ x+y & \leq 1 \\ 2 x+y & \leq 4 \end{aligned}\)

Step-by-Step Solution

Answer

The maximum value of the objective function \(z = 3x + 4y\) under the constraints is achieved at the vertex (0, 1) and the maximum value is 4.

Step 1: Graph the Constraints

Plot the line for each equation on a graph and shade the feasible region. The feasible region will be a polygon with vertices at the intersection of the boundary lines. Remember, we are dealing with the inequalities \(x \geq 0\), \(y \geq 0\), \(x+y \leq 1\), and \(2x+y \leq 4\), hence the feasible region will lie in the quadrant where \(x \geq 0\) and \(y \geq 0\). It will be bounded by the lines \(x+y = 1\) and \(2x+y = 4\).

Step 2: Identify the Vertices of the Feasible Region

Identify the vertices of the feasible region. These vertices will be the intersection points of the equations. For this example, the vertices are (0, 0), (0, 1) and (1, 0).

Step 3: Evaluate the Objective function at each Vertex

To find the maximum value of the function, evaluate the objective function at each of the vertices. Whichever vertex produces the highest value for the objective function will indicate the location of the maximum value. For this case, substitute the vertices into the objective function \(z = 3x + 4y\) and solve.

Step 4: Conclusion

The vertex that gives the greatest value for the objective function is the solution of the linear programming problem. Find the maximum value of z from the obtained values at each vertex.

Key Concepts

Objective Function

The objective function is a key component in a linear programming problem. It represents the formula that needs to be maximized or minimized. In our exercise, the objective function is given by the equation \[ z = 3x + 4y \]. This equation defines the target outcome, and the goal is to find the point(s) in the feasible region that maximize \( z \).
In simpler terms, think of the objective function as the equation that gives us the profit or cost we want to optimize. Each point (combination of \(x\) and \(y\)) on the graph gives a different value of \(z\). By solving the linear programming problem, we are essentially finding which point gives us the best value of \(z\).

Constraints

Constraints are conditions that the solution must satisfy in a linear programming problem. They are represented by a set of inequalities that restrict the values that variables can take.
In our example, the constraints are:

• \(x \geq 0\)
• \(y \geq 0\)
• \(x + y \leq 1\)
• \(2x + y \leq 4\)

These inequalities form the boundaries of the feasible region on a graph.
Each constraint represents a line, and the area where all conditions are met is the feasible region. Constraints help in narrowing down the possible solutions to a specific range, making it easier to find the optimal solution for the objective function.

Feasible Region

The feasible region is the area where all the constraints of a linear programming problem intersect. This region represents all possible solutions that satisfy the boundary conditions.
In our example, after graphing the constraints, the feasible region appears as a polygon on the graph. It is located in the first quadrant because of the non-negativity constraints \(x \geq 0\) and \(y \geq 0\).
The boundaries of this region are formed by the lines \(x + y = 1\) and \(2x + y = 4\). This polygon is crucial because the solution to the linear programming problem will always lie on one of the vertices of this feasible region.
Understanding the feasible region helps in visualizing where exactly the optimum solution lies.

Vertices

Vertices are the corner points of the feasible region. In linear programming problems, the optimum solution can be found at these vertices.
To identify the vertices, we examine where the constraint lines intersect each other and the axes. In our problem, the feasible region is defined by the intersection points of the inequalities:

• (0, 0) - Intersection of \(x \geq 0\) and \(y \geq 0\)
• (0, 1) - Intersection of \(x + y = 1\) and \(y \geq 0\)
• (1, 0) - Intersection of \(x + y = 1\) and \(x \geq 0\)

By calculating the objective function at each vertex, we can determine which vertex yields the highest objective function value. This is critical as, for linear functions, the solution to a linear programming problem will always be located at one of these vertices. This method makes it very efficient to find the optimal solution.