Lindner Congress Bookstore sells a unique calculator to college students. The demand for this calculator has a normal distribution with an average daily demand of twelve units and a standard deviation of five units per day. The lead time for this calculator is very stable at six days. Compute the standard deviation of demand during lead time, and determine the safety stock and statistical reorder point that result in 5 percent stockouts. Do not round intermediate calculations. Round your answers to two decimal places.

To solve this problem, we'll go through the following steps:

1. **Determine the standard deviation of demand during the lead time.**
2. **Compute the Z-score for a 5% stockout.**
3. **Calculate the safety stock.**
4. **Find the statistical reorder point.**

---

### 1. **Standard Deviation of Demand During Lead Time (σ_DL)**

The standard deviation of demand during lead time is calculated using the formula:

\[
\sigma_{DL} = \sigma_D \times \sqrt{L}
\]

Where:
- \(\sigma_D = 5\) units per day (standard deviation of daily demand),
- \(L = 6\) days (lead time).

Let's calculate \(\sigma_{DL}\):

\[
\sigma_{DL} = 5 \times \sqrt{6}
\]

---

### 2. **Z-Score for 5% Stockouts**

A stockout probability of 5% corresponds to the 95th percentile of the normal distribution. Using a standard normal distribution table, we find that the Z-score for a 5% stockout probability is approximately:

\[
Z = 1.645
\]

---

### 3. **Safety Stock**

Safety stock is calculated using the formula:

\[
\text{Safety Stock} = Z \times \sigma_{DL}
\]

Where:
- \(Z = 1.645\),
- \(\sigma_{DL}\) is what we computed earlier.

---

### 4. **Statistical Reorder Point**

The statistical reorder point is the sum of the expected demand during the lead time and the safety stock:

\[
\text{Reorder Point} = \text{Expected Demand during Lead Time} + \text{Safety Stock}
\]

The expected demand during lead time is:

\[
\text{Expected Demand during Lead Time} = \mu_D \times L
\]

Where:
- \(\mu_D = 12\) units per day (average daily demand),
- \(L = 6\) days (lead time).

Let's calculate the values.### Calculations:

1. **Standard Deviation of Demand During Lead Time**:  
   \[
   \sigma_{DL} = 12.25 \, \text{units}
   \]

2. **Safety Stock**:  
   \[
   \text{Safety Stock} = 20.15 \, \text{units}
   \]

3. **Statistical Reorder Point**:  
   \[
   \text{Reorder Point} = 92.15 \, \text{units}
   \]

These results indicate that with a 5% stockout probability, the bookstore should reorder when the inventory reaches 92.15 units, maintaining a safety stock of 20.15 units to protect against variability in demand during lead time.

Would you like further details or have any questions?

---

Here are 5 related questions to explore further:
1. How would a shorter lead time affect the reorder point and safety stock?
2. What would happen to the stockout probability if the Z-score increased?
3. How is the safety stock affected by changes in the standard deviation of demand?
4. How would you calculate the reorder point if the lead time was variable?
5. How does the normal distribution assumption influence inventory management?

**Tip:** Increasing safety stock reduces the chance of stockouts but raises holding costs.

Learn how to compute the reorder point and safety stock for a calculator sold at Lindner Congress Bookstore. This problem involves calculating the standard deviation of demand during lead time, the safety stock, and the reorder point using a normal distribution with a 5% stockout probability. Suitable for undergraduate students in operations management.

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