Production Optimization in a Food Processing Company
How can we schedule the operations of three plants in a food processing company to produce the required amount of ice cream and minimize production costs?
Given the production capacities and operating costs of each plant, how many hours per day should each plant be scheduled to operate?
Optimizing Production Operations
To minimize the cost of production and meet the minimum requirement of 300 gallons of regular ice cream and 200 gallons of deluxe ice cream, we need to determine how many hours each plant should be scheduled to operate.
Let's break down the information given:
1. Cedarburg plant: - Produces 20 gallons of regular ice cream per hour - Produces 10 gallons of deluxe ice cream per hour - Costs $70 per hour to operate
2. Grafton plant: - Produces 10 gallons of regular ice cream per hour - Produces 20 gallons of deluxe ice cream per hour - Costs $75 per hour to operate
3. West Bend plant: - Produces 20 gallons of regular ice cream per hour - Produces 20 gallons of deluxe ice cream per hour - Costs $90 per hour to operate
We need to produce at least 300 gallons of regular ice cream and 200 gallons of deluxe ice cream. Let's assume the Cedarburg plant operates for x hours, the Grafton plant operates for y hours, and the West Bend plant operates for z hours.
Based on the given information, we can set up the following equations:
For regular ice cream: 20x + 10y + 20z ≥ 300
For deluxe ice cream: 10x + 20y + 20z ≥ 200
To minimize the cost of production, we need to consider the cost per hour of each plant and find the combination that yields the lowest total cost.
The total cost can be calculated as: Total Cost = (Cost per hour at Cedarburg) * x + (Cost per hour at Grafton) * y + (Cost per hour at West Bend) * z
Now, we need to solve this system of inequalities and find the values of x, y, and z that satisfy the minimum production requirement while minimizing the cost. This can be done using linear programming techniques or graphical methods.
Once the optimal values of x, y, and z are determined, the corresponding hours per day for each plant can be scheduled accordingly. The minimum production cost will be the total cost calculated using the optimal values.