Rocky Mountain Tire Center sells 7 comma 000 ​go-cart tires per year. The ordering cost for each order is ​$40​, and the holding cost is 40​% of the purchase price of the tires per year. The purchase price is ​$19 per tire if fewer than 200 tires are​ ordered, ​$16 per tire if 200 or morelong dashbut fewer than 5 comma 000long dashtires are​ ordered, and ​$15 per tire if 5 comma 000 or more tires are ordered.
Part 2
​a) How many tires should Rocky Mountain order each time it places an​ order?
Rocky​ Mountain's optimal order quantity is
​b) What is the total cost of this​ policy?
Total annual cost of ordering optimal order size=

To solve this problem, we will use the Economic Order Quantity (EOQ) model to determine the optimal order quantity for Rocky Mountain Tire Center. The EOQ formula is given by:

\[
EOQ = \sqrt{\frac{2DS}{H}}
\]

where:
- $D$ is the annual demand,
- $S$ is the ordering cost per order,
- $H$ is the holding cost per unit per year.

### Step 1: Define the parameters
- **Annual demand ($D$)** = 7,000 tires per year.
- **Ordering cost ($S$)** = $40 per order.
- **Holding cost ($H$)** = 40% of the purchase price per tire per year.

Given that the purchase price depends on the order quantity, we need to calculate the EOQ for each price range and then compare the total costs.

### Step 2: Calculate EOQ for each price range

1. **For $ P = $19 $ per tire** (if fewer than 200 tires are ordered):
   - Holding cost $ H = 0.4 \times 19 = $7.6 $ per tire per year.
   \[
   EOQ_1 = \sqrt{\frac{2 \times 7000 \times 40}{7.6}} = \sqrt{\frac{560000}{7.6}} \approx 272.05 \text{ tires}
   \]
   Since this quantity exceeds 200, it is not valid for this price range.

2. **For $ P = $16 $ per tire** (if 200 or more but fewer than 5,000 tires are ordered):
   - Holding cost $ H = 0.4 \times 16 = $6.4 $ per tire per year.
   \[
   EOQ_2 = \sqrt{\frac{2 \times 7000 \times 40}{6.4}} = \sqrt{\frac{560000}{6.4}} \approx 296.98 \text{ tires}
   \]
   This quantity is valid as it is between 200 and 5,000.

3. **For $ P = $15 $ per tire** (if 5,000 or more tires are ordered):
   - Holding cost $ H = 0.4 \times 15 = $6 $ per tire per year.
   \[
   EOQ_3 = \sqrt{\frac{2 \times 7000 \times 40}{6}} = \sqrt{\frac{560000}{6}} \approx 305.51 \text{ tires}
   \]
   Since 305.51 tires is below 5,000, this price range isn't valid either.

### Step 3: Determine the optimal order quantity and total cost
Given that only the second EOQ (296.98 tires) falls within a valid range, Rocky Mountain Tire Center should order 296.98 tires (rounded to 297).

### Total Annual Cost
The total annual cost (TAC) is calculated by adding the ordering cost, holding cost, and purchase cost:

\[
TAC = \left(\frac{D}{Q}\right) \times S + \left(\frac{Q}{2}\right) \times H + D \times P
\]

For $Q = 297$ tires:
- Ordering cost $ \frac{7000}{297} \times 40 \approx 943.77 $
- Holding cost $ \frac{297}{2} \times 6.4 = 950.4 $
- Purchase cost $ 7000 \times 16 = 112000 $

Total cost:
\[
TAC \approx 943.77 + 950.4 + 112000 \approx 113894.17
\]

### Answers:
a) Rocky Mountain's optimal order quantity is **297 tires**.

b) The total annual cost of ordering the optimal order size is **$113,894.17**.

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Would you like more details or have any questions?

Here are some related questions:
1. How does the holding cost affect the EOQ?
2. What would happen if the ordering cost increased?
3. How would the EOQ change if the annual demand doubled?
4. What is the impact of bulk discounts on inventory policies?
5. Can the EOQ model be applied to perishable goods?
6. What assumptions are made in the EOQ model?
7. How would a change in interest rates affect holding costs?
8. What is the role of safety stock in inventory management?

**Tip:** Regularly reviewing and adjusting order quantities based on changing demand or cost conditions can significantly reduce total costs.

Learn how to calculate the optimal order quantity and total annual cost for Rocky Mountain Tire Center using the Economic Order Quantity (EOQ) model. This problem covers inventory management concepts, including ordering costs, holding costs, and purchase price variations based on order quantities.

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