The Pyrotec Company produces three electrical products-clocks, radios and toasters. These products have the following resources requirements :
Resource Requirements
Cost/Unit Labor Hours/Unit
Clock $7 2
Radio 10 3
Toaster 5 2
The manufacturer has a daily production budget of $2,000 and maximum of 660 hours of labor. Maximum daily customer demand is for 200 clocks, 300 radios, and 150 toasters. Clocks sell for $15, radios for $20, and toasters for $12.The company wants to know the optimal product mix that will maximize profit.
a) Formulate a linear programming model for this problem;
b) Solve the model by using the computer.
a) To formulate a linear programming model for this problem, we need to define decision variables, objective function, and constraints.
Decision Variables:
Let's define the decision variables as follows:
x1 = Number of clocks produced
x2 = Number of radios produced
x3 = Number of toasters produced
Objective Function:
The objective is to maximize profit, which can be represented as the total revenue minus the total production cost. The revenue for each product can be calculated by multiplying the selling price with the quantity produced:
Revenue of clocks = 15 * x1
Revenue of radios = 20 * x2
Revenue of toasters = 12 * x3
The total revenue can be expressed as:
Total Revenue = 15 * x1 + 20 * x2 + 12 * x3
The total production cost can be calculated by multiplying the cost per unit with the quantity produced:
Cost of clocks = 7 * x1
Cost of radios = 10 * x2
Cost of toasters = 5 * x3
The total production cost can be expressed as:
Total Production Cost = 7 * x1 + 10 * x2 + 5 * x3
Therefore, the objective function can be formulated as:
Maximize Z = Total Revenue - Total Production Cost
Z = (15 * x1 + 20 * x2 + 12 * x3) - (7 * x1 + 10 * x2 + 5 * x3)
Constraints:
1. Budget Constraint: The total production cost should not exceed the daily production budget of $2,000.
7 * x1 + 10 * x2 + 5 * x3 ≤ 2000
2. Labor Constraint: The total labor hours used should not exceed the maximum of 660 hours.
2 * x1 + 3 * x2 + 2 * x3 ≤ 660
3. Demand Constraints: The quantities produced for each product should not exceed the maximum daily customer demand.
x1 ≤ 200 (maximum demand for clocks)
x2 ≤ 300 (maximum demand for radios)
x3 ≤ 150 (maximum demand for toasters)
Non-Negativity Constraints:
x1, x2, x3 ≥ 0 (quantities produced cannot be negative)
b) To solve the model, we can use linear programming software or optimization libraries in programming languages like Python, R, or MATLAB. We need to input the objective function, constraints, and variable bounds into the solver and obtain the optimal solution in terms of the values of x1, x2, and x3 that maximize the objective function Z.