IM
.
6
2
Athletic Apparel is trying to determine how to manage their stock of t
-
shirts. For one particular size and color they have an average daily demand of
1
8
shirts, but it varies by about

Question

IM
.
6
2
 ﻿Athletic Apparel is trying to determine how to manage their stock of t
-
shirts. For one particular size and color they have an average daily demand of 
1
8
 ﻿shirts, but it varies by about 
6
 ﻿shirts per day. Their lead time for receiving new t
-
shirts is 
5
 ﻿days. They are striving for a 
8
8
%
 ﻿service level.
What should be their safety stock for this particular t
-
shirt? 
(
Display your answer to two decimal places.
)
﻿
Number
﻿
﻿
What should be their reorder point for this particular shirt? 
(
Display your answer to two decimal places.
)
﻿
Number

give full answer and give final summary

VIPSolutions · Accepted Answer

# Inventory Management Calculation

Athletic Apparel seeks to manage t-shirt inventory with the following information:

- **Average daily demand ($D$)**: 18 shirts  
- **Standard deviation of daily demand ($\sigma_D$)**: 6 shirts  
- **Lead time ($L$)**: 5 days  
- **Service level**: 88%

Let's calculate:

---

## 1. Safety Stock

### a. Find the Z-Score for 88% Service Level

Look up the Z-score corresponding to an 88% service level:

- **Service level (probability)**: .88
- **Z-score**: ≈ 1.175

### b. Calculate Standard Deviation of Demand During Lead Time

$$
\sigma_{DL} = \sigma_D \times \sqrt{L}
$$

$$
\sigma_{DL} = 6 \times \sqrt{5} \approx 6 \times 2.2361 = 13.4166
$$

### c. Safety Stock Formula

$$
\text{Safety Stock} = Z \times \sigma_{DL}
$$

$$
\text{Safety Stock} = 1.175 \times 13.4166 \approx 15.77
$$

---

## 2. Reorder Point

$$
\text{Reorder Point} = \text{Average demand during lead time} + \text{Safety Stock}
$$

- **Average demand during lead time**: $D \times L = 18 \times 5 = 90$

$$
\text{Reorder Point} = 90 + 15.77 = 105.77
$$

---

## Final Answers

- **Safety Stock:** **15.77 shirts**
- **Reorder Point:** **105.77 shirts**

---

## Summary

| Calculation      | Value (to two decimals) |
|------------------|------------------------|
| Safety Stock     | 15.77 shirts           |
| Reorder Point    | 105.77 shirts          |

Athletic Apparel should keep a safety stock of **15.77 shirts** and reorder when inventory falls to **105.77 shirts** to maintain an 88% service level for this t-shirt size and color.

VIPSolutions · Answer

# Summary of Inventory Management Steps

## 1. Safety Stock Calculation
To determine the safety stock, we first find the Z-score for an 88% service level, which is approximately 1.175. We then calculate the standard deviation of demand during the lead time and use it to find the safety stock, resulting in **15.77 shirts**.

## 2. Reorder Point Calculation
The reorder point is calculated by determining the average demand during the lead time, which amounts to **90 shirts**. By adding the safety stock of **15.77 shirts**, we find the reorder point to be **105.77 shirts**.

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