Question 1184583: Bruno Fruscalzo decided to start a small production facility in Sydney to sell gelato to the local restaurants. His local milk supplier charges $ 0.50 per kg of milk plus a $20 delivery fee (the $ 20 fee is independent of the amount ordered; i.e., $20 is the order cost). Bruno's holding cost is $ 0.03 per kg per month. He needs 9000 kg of milk per month.
Note: (if your obtained values are NOT an integer, please round to Nearest 10th; otherwise it does not match the answer).
Suppose Bruno orders 9000 kg each time. His average inventory is
4500
kg.
Suppose Bruno orders 7000 kg each time. He will place
15.43
orders with his supplier each year.
To minimize the sum of the ordering and holding costs, Bruno should order
3464.10
kg from his supplier with each order.
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Let's break down Bruno's gelato milk ordering costs and find the optimal order quantity.
**1. Current Scenario (9000 kg orders):**
* **Annual Demand:** 9000 kg/month * 12 months/year = 108,000 kg/year
* **Order Quantity:** 9000 kg
* **Number of Orders per Year:** 108,000 kg / 9000 kg/order = 12 orders/year
* **Average Inventory:** 9000 kg / 2 = 4500 kg (as stated)
* **Annual Holding Cost:** 4500 kg * $0.03/kg/month * 12 months/year = $1620
* **Annual Ordering Cost:** 12 orders * $20/order = $240
* **Annual Purchase Cost:** 108,000 kg * $0.50/kg = $54,000
* **Total Annual Cost:** $1620 + $240 + $54,000 = $55,860
**2. Scenario with 7000 kg orders:**
* **Order Quantity:** 7000 kg
* **Number of Orders per Year:** 108,000 kg / 7000 kg/order ≈ 15.43 orders/year (as stated)
* **Average Inventory:** 7000 kg / 2 = 3500 kg
* **Annual Holding Cost:** 3500 kg * $0.03/kg/month * 12 months/year = $1260
* **Annual Ordering Cost:** 15.43 orders * $20/order ≈ $308.60
* **Annual Purchase Cost:** 108,000kg * $0.50/kg = $54,000
* **Total Annual Cost:** $1260 + $308.60 + $54,000 = $55,568.60
**3. Economic Order Quantity (EOQ):**
The EOQ formula minimizes the total ordering and holding costs. It's given by:
EOQ = √(2 * D * S / H)
Where:
* D = Annual demand (108,000 kg)
* S = Ordering cost ($20)
* H = Annual holding cost per unit ($0.03/kg/month * 12 months/year = $0.36/kg/year)
EOQ = √(2 * 108,000 kg * $20 / $0.36/kg)
EOQ = √(4,320,000 / 0.36)
EOQ = √12,000,000
EOQ ≈ 3464.10 kg (as stated)
**4. Optimal Ordering Cost and Holding Cost:**
With the EOQ, the annual holding cost and the annual ordering cost are roughly equal.
* **Annual Holding Cost:** (3464.10 kg / 2) * $0.36/kg = $623.54
* **Annual Ordering Cost:** (108,000 kg / 3464.10 kg) * $20 = $623.54
* **Annual Purchase Cost:** 108,000kg * $0.50/kg = $54,000
* **Total Annual Cost:** $623.54 + $623.54 + $54,000 = $55,247.08
**Conclusion:**
Bruno should order approximately 3464.10 kg of milk per order to minimize his total costs. This will result in approximately 31.2 orders per year.
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