Problem 10 - Algebra 2

Question

Teams chosen from 30 forest rangers and 16 trainees are planting trees. An experienced team consisting of two rangers can plant 500 trees per week. A training team consisting of one ranger and two trainees can plant 200 trees per week. $$\begin{array}{|c|c|c|c|}\hline \text { Number of Teams } & {x} & {y} & {x+y} \\\ \hline \text { Number of Rangers } & {2 x} & {y} & {30} \\ \hline \text { Number of Trainees } & {0} & {2 y} & {16} \\ \hline \text { Number of Trees Planted } & {500 x} & {200 y} & {500 x+200 y} \\ \hline\end{array}$$ a. Write an objective function and constraints for a linear program that models the problem. b. How many of each type of team should be formed to maximize the number of trees planted? How many trainees are used in this solution? How many trees are planted? c. Find a solution that uses all the trainees. How many trees will be planted in this case?

Step-by-Step Solution

Answer

The maximum number of trees (7100) is planted with 11 experienced teams and 8 training teams, using all 16 trainees. The solution that uses all the trainees gives the same number of trees planted (7100).

Step 1 - Define Variables

Let the variable x represent the number of experienced teams (two rangers per team) and the variable y represent the number of training teams (one ranger and two trainees per team).

Step 2 - Write the Objective Function

The objective function represents the total number of trees planted by both types of teams. It is formulated as follows: Maximize: Z = 500x + 200y.

Step 3 - Write the Constraints from Rangers

The number of rangers used in all teams must not exceed 30. This leads to the constraint from the number of experienced teams and training teams: 2x + y ≤ 30.

Step 4 - Write the Constraints from Trainees

The number of trainees used in the training teams must not exceed 16, leading to the constraint: 2y ≤ 16 or y ≤ 8.

Step 5 - Write the Non-Negativity Constraints

Since the number of teams cannot be negative, the non-negativity constraints are: x ≥ 0 and y ≥ 0.

Step 6 - Solve the Linear Program to Maximize the Objective Function

Since both constraints are linear, and the objective function is to be maximized, use graphical methods or linear programming techniques such as the simplex method to find the values of x and y that maximize Z subject to the constraints.

Step 7 - Checking the Corners of the Feasible Region

Evaluate the objective function at each corner point of the feasible region defined by the constraints. Possible corner points include (0, 8), (15, 0), and the point where the lines 2x + y = 30 and y = 8 intersect.

Step 8 - Calculate the Intersection Point

To find the intersection of 2x + y = 30 and y = 8, substitute y into the first equation: 2x + 8 = 30, which gives x = 11. The intersection point is (11, 8).

Step 9 - Evaluate the Objective Function at the Intersection Point

Plugging the intersection point (11, 8) into the objective function Z = 500x + 200y, we get Z = 500(11) + 200(8) = 5500 + 1600 = 7100.

Step 10 - Choose the Maximal Value

Comparing the values of Z for each corner point, the maximal value is at the intersection point (11, 8), which is Z = 7100 trees.

Step 11 - Find Solution that Uses all Trainees

If all trainees must be used, then y = 8. Substitute y = 8 into the ranger constraint 2x + y = 30 to find x, leading to 2x = 22, so x = 11.

Step 12 - Calculate Number of Trees Planted Using All Trainees

Using x = 11 and y = 8 in the objective function Z = 500x + 200y gives the total number of trees planted as Z = 500(11) + 200(8) = 7100.

Key Concepts

Objective Function

In linear programming, the objective function is a mathematical expression that represents the goal of the problem you are trying to solve. It could be to maximize profit, minimize cost, or, as in our example, maximize the number of trees planted by teams of rangers and trainees.

The objective function for this scenario is given by:
\[ Z = 500x + 200y \]
where:

• \( x \) represents the number of experienced ranger teams,
• \( y \) corresponds to the number of training teams.

The coefficients 500 and 200 denote the number of trees that each team type can plant per week, respectively. Setting up the objective function correctly is essential, as it guides the entire decision-making process in linear programming.

Constraints

Every linear programming problem has limitations or constraints that must be considered. They represent restrictions on the resources or conditions that must be satisfied. In our example, the constraints are framed by the limited number of available rangers and trainees. They can be represented as follows:

• For rangers:
  \[ 2x + y \leq 30 \]
• For trainees:
  \[ 2y \leq 16 \]
• Non-negativity:
  \[ x \geq 0, y \geq 0 \]

These constraints ensure no more than 30 rangers and 16 trainees are utilized, and they prevent the solution from suggesting a negative number of teams, which is impractical.

Maximization Problem

Our exercise showcases a maximization problem, where the objective is to find the highest possible value of the objective function within the bounds of the constraints. Here, our goal is to plant the maximum number of trees. To solve such problems, one needs to identify the set of values for the decision variables—in our case, the number of experienced and training teams (\( x \) and \( y \))— that result in the highest output of the objective function without breaching any constraints.

Feasible Region

The set of all possible points (values for \( x \) and \( y \)) that satisfy the constraints of a linear programming problem is known as the feasible region. This region is crucial as it contains the potential solutions to our problem. It is often visualized graphically as a polygon on a coordinate system, where each vertex (corner) corresponds to a potential solution. The solutions at the vertices are of particular interest because, in linear programming, the optimal solution to a maximization or minimization problem will always be found at one of these vertices.

Simplex Method

The simplex method is a popular algorithm used to solve linear programming problems like our maximization example. Developed by George Dantzig in 1947, this method systematically tests the vertices of the feasible region to identify the optimal solution. It traverses edges of the polygon (feasible region) to reach the highest value of the objective function without testing all possible points. While we can use graphical methods for linear programs with two variables, the simplex method scales well for problems with many variables and constraints, making it a powerful tool in various optimization scenarios.