Question

Maximize the objective function subject to the constraints \(x+4 y \leq 20, x+y \leq 8\) \(3 x+2 y \leq 21, x \geq 0\), and \(y \geq 0\) $$z=12 x+5 y$$

Short answer

The maximum value of \( z \), given the constraints, is the largest of the values obtained in Step 3. The solution is the \((x,y)\) pair that corresponds to this maximum possible value for \( z \).

Step-by-step solution

Identify the feasible region

The feasible region is the region where all the constraints are satisfied. It is found by plotting the constraints on a graph. We first plot the lines \( x+4y = 20 \), \( x+y = 8 \), and \( 3x+2y = 21 \). Next, considering the inequalities, we shade the region that satisfies all these inequalities.

Identify the corner/vertices points of the feasible region

This is done by finding the points of intersection of the lines defined by the constraints. The points are found by solving the equations simultaneously.

Evaluate the objective function at the corner points

The optimal solution to the linear programming problem occurs at one (or more) of the corner points. We substitute each corner point into the objective function \( z=12x+5y \), and choose the point that yields the maximum \( z \).

Key concepts

Feasible Region

In linear programming, the feasible region represents the set of all possible solutions that satisfy a given set of constraints. In simpler terms, it's the area on a graph where all the constraints overlap and hold true at the same time. To find the feasible region, we begin by plotting each constraint as an equation on a graph. For example, if we have the constraint \( x+4y \leq 20 \), we would draw the line \( x+4y=20 \) and shade the region below the line, since \( y \) must be less to satisfy the inequality.
Next, repeat this for the rest of the constraints. For instance, plot \( x+y=8 \) and \( 3x+2y=21 \) as lines, shading the appropriate side for each inequality. The feasible region is where all shaded areas overlap. Typically, this is a polygon bounded by the intersection of these lines. The points where the lines intersect are called the vertices or corner points of the feasible region. Every vertex is crucial because any potential optimal solution to a linear programming problem will be located at one of these points.
Remember that to be part of the feasible region, solutions must also satisfy non-negativity constraints, meaning \( x \geq 0 \) and \( y \geq 0 \). Thus, the feasible region is always located in the first quadrant of the coordinate plane where both \( x \) and \( y \) are zero or positive.

Objective Function

The objective function in a linear programming problem is what we aim to optimize. This could mean either maximizing or minimizing its value. For example, if we're trying to maximize profits or minimize costs, our objective function will represent these goals. In the given problem, the objective function is defined as \( z = 12x + 5y \). Here, \( z \) is the variable we intend to maximize, representing a value of interest for our purpose, like revenue or utility.
The coefficients of \( x \) and \( y \) (12 and 5, respectively) indicate the contribution of each respective variable to the objective function. Therefore, each unit increase in \( x \) contributes 12 units to \( z \), and each unit increase in \( y \) contributes 5 units to \( z \).
To solve the problem, once the feasible region is determined, we evaluate the objective function at each vertex of the feasible region. The vertex that results in the highest value of \( z \) is considered the optimal solution, satisfying both our objective of maximizing \( z \) and all the given constraints.

Constraints

Constraints are the limitations or conditions that must be satisfied in a linear programming problem. They define the boundaries of the feasible region by setting limits on the combinations of variables that are allowed within the solution space. Constraints are typically expressed as linear inequalities involving the decision variables.
For example, the constraints in the given problem are:

• \( x+4y \leq 20 \)
• \( x+y \leq 8 \)
• \( 3x+2y \leq 21 \)
• \( x \geq 0 \)
• \( y \geq 0 \)

Each of these inequalities provides a boundary to the feasible region. The first three constraints involve linear combinations of \( x \) and \( y \) and represent conditions like resource limitations or capacity restrictions in practical scenarios. The last two constraints enforce non-negativity, ensuring that the values of \( x \) and \( y \) remain practical and applicable to real-world situations, where negative quantities would not make sense.
Solving these inequalities allows you to graphically determine the feasible region. The solution to the problem will be found within this region, satisfying all the constraints. Once the feasible region is plotted, it becomes easier to visually identify and then calculate, where the best solutions lie.