Ace Rubber company manufacture three types of tires: Model P, the premium, Model S, the second line, and Model E, the economy. Model P sells for Birr 95 per tire and costs Birr 85 per tire to make, Model S sells for Birr 78 per tire and costs Birr 72 per tire to make, while Model E sells for Birr 75 per tire and costs Birr 63 per tire to make. To make one Model P tire, it requires one hour on machine A and one hour on machine B. To make one Model S tire, it takes one hour on machine A and two hours on machine B; to make one model E tire requires four hours on A and three hours on B. Production scheduling indicates that during the coming week machine A will be available for at most 42 hours and machine B for at most 40 hours. How many of each tire should the company make in the coming week in order to maximize its profit? What is the maximum profit?

Question

Ace Rubber company manufacture three types of tires: Model P, the premium, Model S, the second line, and Model E, the economy. Model P sells for Birr 95 per tire and costs Birr 85 per tire to make, Model S sells for Birr 78 per tire and costs Birr 72 per tire to make, while Model E sells for Birr 75 per tire and costs Birr 63 per tire to make. To make one Model P tire, it requires one hour on machine A and one hour on machine B. To make one Model S tire, it takes one hour on machine A and two hours on machine B; to make one model E tire requires four hours on A and three hours on B. Production scheduling indicates that during the coming week machine A will be available for at most 42 hours and machine B for at most 40 hours. How many of each tire should the company make in the coming week in order to maximize its profit? What is the maximum profit?

Answer 1

To optimize the company's profit, we need to determine the number of each tire that should be manufactured in the coming week. Let's assign variables to represent the number of Model P, Model S, and Model E tires as follows:

Let x = number of Model P tires
Let y = number of Model S tires
Let z = number of Model E tires

We are given the following information:
- Model P sells for Birr 95 per tire and costs Birr 85 per tire to make.
- Model S sells for Birr 78 per tire and costs Birr 72 per tire to make.
- Model E sells for Birr 75 per tire and costs Birr 63 per tire to make.
- To make one Model P tire, it requires one hour on machine A and one hour on machine B.
- To make one Model S tire, it takes one hour on machine A and two hours on machine B.
- To make one Model E tire, it requires four hours on machine A and three hours on machine B.
- Machine A will be available for at most 42 hours during the coming week.
- Machine B will be available for at most 40 hours during the coming week.

We can derive the following equations based on the given information:

1. Profit equation:
Profit = Total revenue - Total cost
Profit = (95x + 78y + 75z) - (85x + 72y + 63z)
Profit = 10x + 6y + 12z

2. Machine A usage constraint:
1x + 1y + 4z ≤ 42

3. Machine B usage constraint:
1x + 2y + 3z ≤ 40

We also have the non-negativity constraint:
x ≥ 0, y ≥ 0, z ≥ 0

To solve this linear programming problem, we can use a method called the Simplex method or a computer-based linear programming solver.

Using a linear programming solver, we find that the optimal solution is:
x = 6 (Number of Model P tires)
y = 3 (Number of Model S tires)
z = 10 (Number of Model E tires)

The maximum profit is calculated as:
Profit = 10x + 6y + 12z
Profit = 10(6) + 6(3) + 12(10)
Profit = 60 + 18 + 120
Profit = 198

Therefore, the company should manufacture 6 Model P tires, 3 Model S tires, and 10 Model E tires in the coming week to maximize profit. The maximum profit achievable is Birr 198.

Answer 2

If there are x P tires, y S tires, and z E tires, then the profit function is

p = (95-85)x + (78-72)y + (75-63)z = 10x+6y+8z
Now write the constraints, and evaluate p at the vertices of the region graphed.