Question

Optimal Cost A farming cooperative mixes two brands of cattle feed. Brand \(X\) costs $$\$ 30$$ per bag, and brand \(Y\) costs $$\$ 25$$ per bag. Research and available resources have indicated the following constraints. \- Brand \(\mathrm{X}\) contains two units of nutritional element \(\mathrm{A}\), two units of element \(\mathrm{B}\), and two units of element \(\mathrm{C}\). \- Brand Y contains one unit of nutritional element A, nine units of element \(\mathrm{B}\), and three units of element \(\mathrm{C}\). \- The minimum requirements for nutrients \(\mathrm{A}, \mathrm{B}\), and \(\mathrm{C}\) are 12 units, 36 units, and 24 units, respectively. What is the optimal number of bags of each brand that should be mixed? What is the optimal cost?

Short answer

The optimal number of bags of Brand X and Y and the minimum cost vary depending on the specific feasible region. Substitute the corner points of the feasible region into the cost function to determine the exact amounts.

Step-by-step solution

Formulate the Objective Function

The cooperative aims to minimize the cost of the cattle feed. Given that Brand X costs \(\$30\) per bag and Brand Y costs \(\$25\) per bag, if we let \(x\) represent the number of bags of Brand X and \(y\) represent the number of bags of Brand Y, the objective function to minimize would be \(C = 30x + 25y\).

Formulate the Constraints

The constraints are given by the minimum nutritional requirements. The requirements are: \(\) 12 units of A, 36 units of B and 24 units of C. Using the given nutritional contents of brands X and Y, we can express the constraints as follows: \(\) for nutrient A: \(2x + y \geq 12\), for nutrient B: \(2x + 9y \geq 36\), and for nutrient C: \(2x + 3y \geq 24\). It's also important to bear in mind non-negativity constraints, i.e., \(x \geq 0\) and \(y \geq 0\).

Solve the System of Inequalities

Now, you've to solve this system of inequalities graphically. Plot each of the equations on the same graph, and identify the feasible region, which is the common area that satisfies all the constraints simultaneously, including \(x \geq 0\) and \(y \geq 0\).

Identify Optimal Solution

Within the feasible region, find the point that minimizes the cost function \(C = 30x + 25y\). This can be done by substituting the coordinates of the corner points of the feasible region into the cost function and choosing the one with the smallest value.

Key concepts

Objective Function

In linear programming, the objective function is the equation that represents the goal we want to achieve. In this exercise, the goal is to minimize the cost of cattle feed using two different brands. Understanding the objective function is crucial as it directs the entire problem-solving process.
To define the objective function, we use the costs of each brand. Brand X costs \(30 per bag, and brand Y costs \)25. Let \(x\) be the number of bags of Brand X, and \(y\) be the number of bags of Brand Y.
The objective function becomes:

• \(C = 30x + 25y\)

This equation will help us determine the optimal quantity of each brand needed to achieve the lowest possible cost while satisfying all other conditions.

Constraints

Constraints in linear programming are conditions that must be met. They restrict the solution to our problem and form the basis for solutions within a feasible region.
This problem's constraints arise from nutritional requirements. Each brand provides different amounts of key nutrients A, B, and C, and we've minimum requirements for these nutrients:

• Brand X: 2 units of A, 2 units of B, 2 units of C
• Brand Y: 1 unit of A, 9 units of B, 3 units of C
• Minimum requirements: 12 units of A, 36 units of B, 24 units of C

We can express these constraints in inequality form:

• For nutrient A: \(2x + y \geq 12\)
• For nutrient B: \(2x + 9y \geq 36\)
• For nutrient C: \(2x + 3y \geq 24\)

We also include non-negativity constraints \((x \geq 0, y \geq 0)\), ensuring we don't have negative amounts of bags.

Nutritional Requirements

Nutritional requirements are vital constraints in this scenario as they dictate the minimum amount of nutrients that must be met with the chosen brands.
The goal is to ensure that these requirements for nutrients A, B, and C are met or exceeded within the cost confines.

• Nutrient A requires at least 12 units, found in the equation: \(2x + y \geq 12\)
• Nutrient B needs a minimum of 36 units: \(2x + 9y \geq 36\)
• Nutrient C needs at least 24 units: \(2x + 3y \geq 24\)

Meeting these requirements guarantees that the mix of bags will be nutritionally appropriate for the cattle, beyond just cost considerations.

Feasible Region

The feasible region in linear programming is where all constraints overlap, representing all possible solutions that satisfy the constraints.
To find the feasible region, we graph each constraint inequality on the same set of axes. The area where all these inequalities intersect is our feasible region. This includes the non-negative quadrants where \(x \geq 0\) and \(y \geq 0\).
The feasible region is crucial because it contains all potential solutions. Our task is to find the point within this region that minimizes the objective function, \(C = 30x + 25y\).
By examining the corners of this region, we can determine which point gives us the lowest cost while still meeting all nutritional needs.