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.