91Ó°ÊÓ

In linear programming, the concept of a feasible region is a fundamental aspect. It represents the set of all possible points that satisfy a given set of constraints. These constraints are typically inequalities involving decision variables. For our exercise, the constraints are:

• \(x + y \leq 6\)
• \(-x + 3y \leq 4\)
• \(x \geq 0\)
• \(y \geq 0\)

To find the feasible region, we first convert these inequalities into equations and then graph them. By plotting the lines represented by the equations and noting on which side of the line the inequality holds true, we determine the area that satisfies all constraints simultaneously.
This shaded area, where all conditions meet, is the feasible region. It's important to ensure that the feasible region is indeed bounded, meaning not extending infinitely, otherwise we cannot proceed to find an optimal solution.

An objective function in linear programming tells us what we want to optimize, which can be either maximizing or minimizing a particular quantity. In this exercise, the objective function we are tasked to maximize is given by:\[ p = x + 5y \]This function represents our goal, based on decision variables \(x\) and \(y\). The coefficients in front of these variables indicate how much each one contributes to the function's overall value.
In our exercise, the challenge is to find the values of \(x\) and \(y\) within the feasible region that make \(p\) as large as possible. To do this, we evaluate the objective function at each vertex of the feasible region.

Graphing constraints involves plotting each of the individual equations derived from inequalities on a coordinate plane. For example, from our exercise:

• Convert \(x + y \leq 6\) into the line equation \(x + y = 6\).
• Convert \(-x + 3y \leq 4\) into the line equation \(-x + 3y = 4\).

Each equation becomes a line on the graph, creating boundaries for the feasible region. We shade the side of each line where the inequality holds true, and where all shaded regions overlap corresponds to the feasible region.

The intersections of these lines, especially those lying within the non-negative quadrant (since \(x \geq 0\) and \(y \geq 0\)), form the vertices of the feasible region. Be careful to graph accurately as errors here could lead to incorrect vertices, impacting later steps.

Once the feasible region has been determined and its vertices identified, the next step is vertices evaluation. This step involves checking each corner point of the feasible area to find the value of the objective function at these points.From our exercise, we identified vertices at:

• \((2, 4)\)
• \((6, 0)\)
• \((0, \frac{4}{3})\)

For each vertex, substitute values into the objective function \(p = x + 5y\) to determine which gives the highest value. This comparison allows us to identify the optimal vertex, which is essential for solving the problem.
Only by evaluating the objective function at each vertex can we be sure of finding the best possible solution within the defined boundaries.