Halifax Cigar stocks Cuban cigars that have variable lead times because of the difficulty in importing the​ product: Lead time is normally distributed with an average of 7 weeks and a standard deviation of 1 week. Demand is also a variable and normally distributed with a mean of 250 cigars per week and a standard deviation of 27 cigars. Refer to the standard normal tableLOADING... for​ z-values.
This exercise contains only parts a and b.
Part 2
​a) For a 98​% service​ level, what is the​ ROP?
The reorder point is __ cigars ​
Part 3
​b) What is the ROP for a 99​% service​ level?
The reorder point is ___ cigars ​

To solve this problem, we need to calculate the Reorder Point (ROP) for two different service levels. The ROP is determined by the following formula:

\[
\text{ROP} = (\text{Demand per week} \times \text{Lead time}) + Z \times \sigma_d
\]

Where:
- \(\text{Demand per week}\) is the average demand per week.
- \(\text{Lead time}\) is the average lead time in weeks.
- \(Z\) is the Z-value corresponding to the desired service level.
- \(\sigma_d\) is the standard deviation of demand during the lead time.

First, let's calculate the standard deviation of demand during the lead time (\(\sigma_d\)):

\[
\sigma_d = \sigma_{\text{demand}} \times \sqrt{\text{Lead time}}
\]
Where:
- \(\sigma_{\text{demand}} = 27\) cigars per week (given).
- \(\text{Lead time} = 7\) weeks (given).

\[
\sigma_d = 27 \times \sqrt{7} \approx 27 \times 2.646 = 71.442
\]

Now, we'll calculate the ROP for both service levels.

### (a) 98% Service Level

For a 98% service level, the Z-value is approximately 2.05.

\[
\text{ROP} = (250 \times 7) + 2.05 \times 71.442
\]
\[
\text{ROP} = 1750 + 146.4561 \approx 1896 cigars
\]

### (b) 99% Service Level

For a 99% service level, the Z-value is approximately 2.33.

\[
\text{ROP} = (250 \times 7) + 2.33 \times 71.442
\]
\[
\text{ROP} = 1750 + 166.4589 \approx 1916 cigars
\]

### Final Answers:
- **(a) For a 98% service level, the ROP is approximately 1896 cigars.**
- **(b) For a 99% service level, the ROP is approximately 1916 cigars.**

Would you like more details or have any further questions?

---

1. How do you find Z-values for other service levels?
2. What if the standard deviation of demand changes?
3. How would the ROP change if lead time increases?
4. Can the service level be less than 98%?
5. How would you calculate the Z-value if not provided?
6. What factors influence the standard deviation of demand?
7. How does an increase in demand affect the ROP?
8. Can the lead time standard deviation be calculated similarly?

**Tip:** Understanding Z-values is crucial for determining service levels, as they directly influence inventory management decisions.

Learn how to calculate the Reorder Point (ROP) for inventory management using statistical methods. This example covers determining ROP for a 98% and 99% service level based on variable lead times and demand. Understand the impact of demand variability and lead time on inventory decisions.

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