Problem 14 - Finite Mathematics and Applied Calculus

Question

$$ \begin{array}{ll} \text { Minimize } & c=0.4 s+0.1 t \\ \text { subject to } & 30 s+20 t \geq 600 \\ & 0.1 s+0.4 t \geq 4 \\ & 0.2 s+0.3 t \geq 4.5 \\ & s \geq 0, t \geq 0 \end{array} $$

Step-by-Step Solution

Answer

The minimum value of the objective function \(c = 0.4s + 0.1t\) occurs at one of the vertices of the feasible region. To find this, we first find the coordinates of the vertices by solving the intersection points of the boundary lines formed by the given inequalities. Then, calculate the value of 'c' at each vertex and compare them to determine the minimum value. The corresponding values of s and t will be the solution to the given linear programming problem.

Step 1: Identify the given system of inequalities and objective function

We have the following inequalities and the objective function: Objective function: \(c = 0.4s + 0.1t\) Subject to: \[ \begin{cases} 30s + 20t \geq 600 \\ 0.1s + 0.4t \geq 4 \\ 0.2s + 0.3t \geq 4.5 \\ s \geq 0 \\ t \geq 0 \end{cases} \]

Step 2: Solve each inequality for 't'

To plot the inequality on a 2D graph, we'll solve each inequality for 't': 1. \(t \geq \frac{600-30s}{20}\) 2. \(t \geq \frac{4-0.1s}{0.4}\) 3. \(t \geq \frac{4.5-0.2s}{0.3}\)

Step 3: Find the feasible region

Plot the boundary lines of each inequality on a graph along with the non-negativity constraints, and identify the area that satisfies all the inequalities. This area is called the feasible region.

Step 4: Find the coordinates of the vertices of the feasible region

For a linear programming problem, the optimal solution always occurs at one of the vertices of the feasible region. We will find these vertices by finding the intersection points of the boundary lines. Vertices are formed by the intersections of: 1. The boundary lines of inequalities 1 and 2 2. The boundary lines of inequalities 1 and 3 3. The boundary lines of inequalities 2 and 3 Solve these systems of linear equations to find the coordinates of the vertices.

Step 5: Calculate the value of the objective function 'c' at each vertex

We will compute the value of the objective function 'c' at each of the vertices found in the previous step: 1. \(c = 0.4s + 0.1t\) at vertex 1 2. \(c = 0.4s + 0.1t\) at vertex 2 3. \(c = 0.4s + 0.1t\) at vertex 3

Step 6: Determine the minimum value of the objective function

Compare the values of the objective function 'c' at each vertex and choose the lowest value. This will be the minimum value of the objective function, and the corresponding values of s and t will be the solution to the linear programming problem.

Key Concepts

Objective Function

In linear programming, the objective function is a mathematical expression that you aim to maximize or minimize. It's like a goal you want to achieve in your problem. In our problem, the objective function is given by:\[ c = 0.4s + 0.1t \]
Here, \(c\) represents the total cost we want to minimize. The variables \(s\) and \(t\) can be thought of as quantities that affect the cost. The coefficients 0.4 and 0.1 are the weights or impact each unit of \(s\) and \(t\) has on the total cost.

• The aim is to find values for \(s\) and \(t\) that make \(c\) as small as possible.
• The challenge is that \(s\) and \(t\) must satisfy certain conditions or constraints.

By minimizing this function, we solve the given problem under specific conditions.

Feasible Region

In the context of linear programming, the feasible region is the set of all possible points that satisfy all the given inequalities. These points are plotted on a graph based on the constraints. Think of it as the area where you can choose your solution from.

• To find it, you first turn each inequality into an equation by replacing the inequality symbol with \(=\).
• These equations are then plotted as lines on a graph.

The region that satisfies all inequalities simultaneously forms a polygonal shape, known as the feasible region. For our problem, after plotting all the lines implied by the inequalities, the intersection of these areas on the graph gives the feasible region.

• It is within this area that all potential solutions lie.
• The boundaries are defined by the inequality lines.

Inequalities

Inequalities set the constraints or conditions that must be met for a solution to be valid. In linear programming, they ensure the solutions lie within a certain range and comply with limitations, such as budget, resources, or time.
For our problem, the inequalities are:\[\begin{cases}30s + 20t \geq 600 \0.1s + 0.4t \geq 4 \0.2s + 0.3t \geq 4.5 \s \geq 0 \t \geq 0\end{cases}\]

• Boundary Lines: The equations of the form \(ax + by = c\) represent the boundaries of the feasible region.
• Non-negativity Constraints: \(s \geq 0\) and \(t \geq 0\) ensure the variables are not negative, aligning with real-world contexts where negative values don't make sense.

Optimization

Optimization in linear programming involves finding the best solution within the feasible region. It means selecting the set of variable values that maximize or minimize the objective function. In our case, we want to minimize the objective function \(c = 0.4s + 0.1t\).
The process involves:

• Vertices Evaluation: The vertices or corner points of the feasible region polygon are critical because they are where the optimal solutions generally exist.
• Calculating Values: The value of the objective function is calculated at each vertex. These calculations reveal which vertex provides the optimal result.

After determining the objective function's value at each vertex, the smallest value indicates the best solution for our minimization problem. This optimization technique ensures efficiency and accuracy in decision-making for resource allocation.