Question

The MFG Corporation is planning to produce and market three different products. Let x₁, x₂ and x₃ denote the number of units of the three respective products to be produced. The preliminary estimates of their potential profitability are as follows. For the first 15 units produced of Product 1, the unit profit would be approximately $360. The unit profit would be only $30 for any additional units of Product 1. For the first 20 units produced of Product 2, the unit profit is estimated at $240. The unit profit would be $120 for each of the next 20 units and $90 for any additional units. For the first 10 units of Product 3, the unit profit would be $450. The unit profit would be $300 for each of the next 5 units and $180 for any additional units. Certain limitations on the use of needed resources impose the following constraints on the production of the three products: x₁ + x₂ + x₃ ≤ 60 3x₁ + 2x₂ ≤ 200 x₁ + 2x₃ ≤ 70 Management wants to know what values of x₁, x₂ and x₃ should be chosen to maximize the total profit. a. Plot the profit graph for each of the three products. b. Use separable programming to formulate a linear programming model on a spreadsheet for this problem. Then solve the model. What is the resulting recommendation to management about the values of x₁, x₂ and x₃ to use?

Answer 1

The values of x₁, x₂, and x₃ that maximize the total profit are the solution to the linear programming model that helps to achieve maximum profitability while considering resource constraints.

The objective is to maximize the total profit, which can be represented as total Profit (P) = Profit from Product 1 + Profit from Product 2 + Profit from Product 3

Profit from Product 1:

For the first 15 units of Product 1, the profit per unit is $360, and for any additional units, it's $30. So, the profit from Product 1 can be calculated as:

Profit₁ = 360x₁ (for x₁ ≤ 15)

Profit₁ = 360 * 15 + 30(x₁ - 15) (for x₁ > 15)

Profit from Product 2:

For the first 20 units of Product 2, the profit per unit is $240, for the next 20 units, it's $120, and for any additional units, it's $90. So, the profit from Product 2 can be calculated as:

Profit₂ = 240x₂ (for x₂ ≤ 20)

Profit₂ = 240 * 20 + 120(x₂ - 20) (for x₂ > 20)

Profit from Product 3:

For the first 10 units of Product 3, the profit per unit is $450, for the next 5 units, it's $300, and for any additional units, it's $180. So, the profit from Product 3 can be calculated as:

Profit₃ = 450x₃ (for x₃ ≤ 10)

Profit₃ = 450 * 10 + 300(x₃ - 10) (for x₃ > 10)

Constraints:

• Resource constraint: x₁ + x₂ + x₃ ≤ 60
• Resource constraint: 3x₁ + 2x₂ ≤ 200
• Resource constraint: x₁ + 2x₃ ≤ 70.