Question 1184583
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.