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 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 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 solve this problem, we can use the concept of linear programming, specifically the technique called the simplex method. First, we need to define decision variables and the objective function.

Let's define:
P = the number of Model P tires to produce
S = the number of Model S tires to produce
E = the number of Model E tires to produce

The objective function is to maximize the profit, which is given by:
Profit (P) = (selling price - cost to make) * quantity
Therefore, the total profit is given by:
Total Profit (TP) = Profit (P) + Profit (S) + Profit (E)

Now, let's calculate the profit for each type of tire:
Profit (P) = (95 - 85) * P
Profit (S) = (78 - 72) * S
Profit (E) = (75 - 63) * E

The next step is to define the constraints. We have two constraints in this problem:
1. Machine A usage: 1P + 1S + 4E ≤ 42
2. Machine B usage: 1P + 2S + 3E ≤ 40

Now, we can set up the linear programming problem:
Maximize: TP = (95 - 85) * P + (78 - 72) * S + (75 - 63) * E
Subject to:
1P + 1S + 4E ≤ 42
1P + 2S + 3E ≤ 40
P, S, E ≥ 0 (non-negativity constraint)

Now, we can solve this linear programming problem using techniques like the simplex method or software like Microsoft Excel's Solver add-in.

By solving this linear programming problem, we will find the values of P, S, and E that maximize the profit.