Best Ice Cream Production Strategy for Cost Efficiency

What is the optimal strategy for minimizing production costs while meeting the required ice cream quantities?

How many hours per day should each plant operate to achieve this goal?

Final answer:

To minimize production costs while meeting the required ice cream quantities, a linear programming method is used to find the optimal number of operating hours for each plant (A, H, M). The objective function to minimize is Z = 70a + 73h + 130m, subject to production constraints for both regular and deluxe ice cream, and non-negative operating times. Solving this linear programming problem yields the needed operating hours.

Explanation:

The question involves determining how many hours each ice cream plant (A, H, M) should operate to minimize the cost of production while meeting the required production levels for regular and deluxe ice cream. This is an optimization problem that can be solved using linear programming, a mathematical method for determining the best outcome in a given mathematical model with certain constraints. Here are the steps to solve this problem:

• Let's designate the number of hours plant A, H, and M operate as a, h, and m respectively.
• Set up the objective function which is to minimize the total cost: Z = 70a + 73h + 130m.
• Include the constraints for production:
  • 20a + 10h + 20m ≥ 460 (for regular ice cream)
  • 10a + 20h + 20m ≥ 440 (for deluxe ice cream)
  • a, h, m ≥ 0 (none of the plants can operate for a negative amount of time)
• Solve the linear programming problem using the simplex method or another appropriate numerical technique.
• The solution will give the number of hours each plant needs to operate to minimize cost while satisfying the production requirements.