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.
Given:
Cost/Unit Labor Hours/Unit
Clock $7 2
Radio 10 3
Toaster 5 2
Implicitly, all quantities produced daily are to be positive, so
=>
c ≥ 0
r ≥ 0
t ≥ 0
The manufacturer has a daily production budget of $2,000
=>
7c+10r+5t ≤ 2000
and maximum of 660 hours of labor.
=>
2c+3r+2t ≤ 660
Maximum daily customer demand is for 200 clocks, 300 radios, and 150 toasters.
=>
c ≤ 200
r ≤ 300
t ≤ 150
Clocks sell for $15, radios for $20, and toasters for $12.The company wants to know the optimal product mix that will maximize profit.
=>
profit for clocks = 15-7 = 8
profit for radios = 20-10 = 10
profit for toasters = 12-5 = 7
Also, labour cost is not provided, so the 600 hours will be assumed a fixed cost.
=>
Z=8c+10r+7t (objective function)
Feed in all these inequalities to your linear programming solver.
I get c=178, r=0, t=150
for Z=2474, 656 hours, and cost = $1996
Check my results.
Check my
27523
Answer
a) To formulate a linear programming model for this problem, we need to define the decision variables, constraints, and objective function.
Decision Variables:
Let's define the decision variables as follows:
C = number of clocks produced
R = number of radios produced
T = number of toasters produced
Objective Function:
The objective is to maximize profit. The profit is calculated by subtracting the cost from the revenue generated by selling the products. The revenue is the product of the number of units sold and their respective selling prices. The cost is the sum of the resources used multiplied by their respective costs per unit. Therefore, the objective function can be written as:
Maximize: 15C + 20R + 12T - 7C - 10R - 5T
Constraints:
1. Cost constraint: The total cost of production should not exceed the daily production budget of $2,000.
7C + 10R + 5T <= 2000
2. Labor hour constraint: The total labor hours used should not exceed the maximum labor hours available of 660.
2C + 3R + 2T <= 660
3. Demand constraint: The number of units produced should not exceed the maximum daily customer demand for each product type.
C <= 200 (for clocks)
R <= 300 (for radios)
T <= 150 (for toasters)
Also, the decision variables should be greater than or equal to zero:
C, R, T >= 0
b) To solve the model using a computer, you can use a linear programming solver or an optimization software. There are many options available, such as Microsoft Excel Solver, MATLAB, Python libraries (e.g., PuLP, scipy), or commercial software like Gurobi or CPLEX.
Here's an example of solving this model using PuLP, a linear programming library in Python:
```python
import pulp
# Create the Linear Programming problem
prob = pulp.LpProblem("Product Mix Optimization", pulp.LpMaximize)
# Define decision variables
C = pulp.LpVariable("Clocks", lowBound=0)
R = pulp.LpVariable("Radios", lowBound=0)
T = pulp.LpVariable("Toasters", lowBound=0)
# Define the objective function
prob += 15 * C + 20 * R + 12 * T - 7 * C - 10 * R - 5 * T
# Define the constraints
prob += 7 * C + 10 * R + 5 * T <= 2000, "Cost Constraint"
prob += 2 * C + 3 * R + 2 * T <= 660, "Labor Hour Constraint"
prob += C <= 200, "Clock Demand Constraint"
prob += R <= 300, "Radio Demand Constraint"
prob += T <= 150, "Toaster Demand Constraint"
# Solve the problem
prob.solve()
# Print the optimal solution
print("Optimal Solution:")
for v in prob.variables():
print(v.name, "=", v.varValue)
# Print the maximum profit
print("Maximum Profit = $", pulp.value(prob.objective))
```
The output will provide the optimal quantities of clocks, radios, and toasters that maximize profit, as well as the maximum profit value.