High School

A factory produces chocolate and candy. To produce 100 kilograms of chocolate, the factory uses:

- Machine A for 1 hour
- Machine B for 4 hours
- Machine C for 2 hours

To produce 100 kilograms of candy, the factory uses:

- Machine A for 2 hours
- Machine B for 1 hour
- Machine C for 1 hour

The factory earns 600 pounds for each 100 kilograms of chocolate and 400 pounds for each 100 kilograms of candy. Machines A and B belong to the factory and can be run for free 24 hours per day. However, machine C is rented from a different company and costs 10 pounds per hour to run. Machine C can be run up to 24 hours a day.

Write down a linear programming (LP) model to maximize the factory profit per day. Explain what each of the variables in the LP formulation means.

Answer :

Maximize Profit = 600C + 400D, subject to 24C + 2D ≤ 24, 4C + D ≤ 24, 2C + D ≤ 24, 10(2C + D) ≤ Budget, C ≥ 0, D ≥ 0.

To formulate the linear programming (LP) model, let's define the decision variables and objective function first.

Decision Variables:

Let's define the following decision variables:

- Let C represent the number of times the factory produces 100 kilograms of chocolate.

- Let D represent the number of times the factory produces 100 kilograms of candy.

Objective Function:

The objective is to maximize the profit per day. Since the profit depends on the quantities of chocolate and candy produced, the objective function is as follows:

Maximize: Profit = 600C + 400D

Constraints:

1. Machine A constraint: The available hours for machine A can be represented as 24C + 2D (as 1 hour is required for chocolate and 2 hours for candy for each production).

- Constraint 1: 24C + 2D ≤ 24 (as there are 24 hours available in a day).

2. Machine B constraint: The available hours for machine B can be represented as 4C + D (as 4 hours are required for chocolate and 1 hour for candy for each production).

- Constraint 2: 4C + D ≤ 24 (as there are 24 hours available in a day).

3. Machine C constraint: The available hours for machine C can be represented as 2C + D (as 2 hours are required for chocolate and 1 hour for candy for each production). Since machine C is rented and costs 10 pounds per hour, this cost needs to be considered.

- Constraint 3: 2C + D ≤ 24 (as there are 24 hours available in a day).

- Constraint 4: 10(2C + D) ≤ Budget (to ensure the cost of renting machine C is within the budget).

4. Non-negativity constraints: The number of times the factory produces chocolate and candy cannot be negative.

- Constraint 5: C ≥ 0

- Constraint 6: D ≥ 0

In summary, the LP model can be written as follows:

Maximize: Profit = 600C + 400D

Subject to:

1. 24C + 2D ≤ 24

2. 4C + D ≤ 24

3. 2C + D ≤ 24

4. 10(2C + D) ≤ Budget

5. C ≥ 0

6. D ≥ 0

The objective is to find the values of C and D that maximize the profit while satisfying the constraints. The LP solver can be used to solve this model, providing the optimal values for C and D, and consequently, the maximum profit.

Learn more about profit here: https://brainly.com/question/28856941

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