Suppose you are an inventory analyst for Red-mart and are examining the
inventory policy used to replenish one of their items: paper towels. The company
orders paper towels every 30 days from an overseas supplier with a shipping lead
time of 14 days. Average daily demand is 10 rolls with a standard deviation of 3
rolls. The company has 150 rolls in inventory              Red-mart would like to minimize its inventory holding cost while
maintaining a 98% service level.                                   

In this scenario, you're dealing with an inventory management problem, and the goal is to determine how much inventory Red-mart should hold while minimizing holding costs and ensuring a 98% service level. The key aspects of the problem are the ordering policy, lead time, demand variability, and the desired service level. Here's how we approach it:

### 1. **Key Variables**
- **Order cycle (T)**: 30 days (time between placing orders)
- **Lead time (L)**: 14 days (time for the order to arrive after it's placed)
- **Average daily demand (d)**: 10 rolls per day
- **Standard deviation of demand (σd)**: 3 rolls per day
- **Inventory level (I)**: 150 rolls
- **Service level (SL)**: 98% (probability of not running out of stock)

### 2. **Calculate Reorder Point (ROP)**  
The **Reorder Point (ROP)** determines the level of inventory at which a new order should be placed to avoid stockouts during the lead time. It accounts for both average demand and demand variability during the lead time.

First, compute the **mean demand during lead time**:

\[
\text{Mean demand during lead time} = d \times L = 10 \times 14 = 140 \text{ rolls}
\]

Next, calculate the **standard deviation of demand during lead time**. Since demand varies daily, the total demand during the lead time also has variability:

\[
\sigma_L = \sigma_d \times \sqrt{L} = 3 \times \sqrt{14} \approx 11.22 \text{ rolls}
\]

The **safety stock (SS)** ensures that there is enough inventory to meet demand fluctuations with the desired service level. For a 98% service level, we use the z-score corresponding to 98%, which is approximately **2.05**.

\[
\text{Safety Stock (SS)} = z \times \sigma_L = 2.05 \times 11.22 \approx 22.98 \text{ rolls}
\]

Finally, the **Reorder Point (ROP)** is:

\[
ROP = \text{Mean demand during lead time} + \text{Safety stock}
\]
\[
ROP = 140 + 22.98 \approx 162.98 \text{ rolls}
\]

### 3. **Decision on Inventory**
Currently, the company has 150 rolls in inventory, which is **less than** the reorder point of 162.98 rolls. This means the company needs to place an order **immediately** to avoid a stockout.

### 4. **Order Quantity**
The amount ordered will depend on the Economic Order Quantity (EOQ), which is not provided in the problem but can be calculated if we have the relevant cost data (ordering costs and holding costs). For now, we focus on the fact that an order needs to be placed to meet the ROP.

### Summary:
- The company should reorder when inventory falls below 163 rolls.
- The current inventory (150 rolls) is below the ROP, so an order should be placed immediately.
- Holding excess stock beyond this would increase inventory holding costs unnecessarily.

Would you like more details on the EOQ calculation or further clarification? Here are some follow-up questions:

---

### 5 Related Questions:
1. How do you calculate the Economic Order Quantity (EOQ) and what data is needed?
2. What are the effects of varying lead times on inventory levels?
3. How would the service level change if Red-mart wanted to increase it from 98% to 99%?
4. How does demand variability impact safety stock levels?
5. What are other ways to minimize inventory holding costs?

---

**Tip**: Regularly reviewing lead times and demand variability can help refine your inventory policy and reduce stockouts or excess stock.

Suppose you are an inventory analyst for Red-mart and are examining the inventory policy used to replenish one of their items: paper towels. The company orders paper towels every 30 days from an overseas supplier with a shipping lead time of 14 days. Average daily demand is 10 rolls with a standard deviation of 3 rolls. The company has 150 rolls in inventory. Red-mart would like to minimize its inventory holding cost while maintaining a 98% service level.

Explore how to calculate the reorder point (ROP) and safety stock for Red-mart's paper towels with a 98% service level. This example demonstrates key inventory management formulas, including demand variability and service level calculations. Suitable for business students or professionals in operations management.

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