Island Supermarket operates a fixed-time period inventory system for its dairy products. The
review period is every 30 days, and the lead time is 7 days. The demand during the lead time
and review period is normally distributed with a mean of 1500 units and a standard deviation
of 200 units. The target service level is 95%, and the current inventory level is 400 units.
a) Calculate the target inventory level.
b) Calculate the order quantity. 
c) What is the stockout risk?


To solve this problem, we will use the fixed-time period inventory system formulae. Here's how we can approach it:

### a) Calculate the Target Inventory Level
The **target inventory level** (TIL) is determined using the formula:

\[
TIL = d \cdot (R + L) + z \cdot \sigma_{d(R+L)}
\]

Where:
- \( d \) is the **mean daily demand**.
- \( R \) is the **review period** in days.
- \( L \) is the **lead time** in days.
- \( z \) is the **z-value** corresponding to the desired service level.
- \( \sigma_{d(R+L)} \) is the **standard deviation of demand** during the review period and lead time.

#### Step 1: Determine the mean and standard deviation of demand during the lead time and review period

- The mean demand during the review period and lead time is \( d(R + L) \), where \( d \) is the mean demand and \( R + L = 30 + 7 = 37 \) days.
  
\[
d(R + L) = 1500 \, \text{units}
\]

- The standard deviation of demand during the lead time and review period is given by:
\[
\sigma_{d(R+L)} = \sigma_d \sqrt{R + L}
\]
Where \( \sigma_d = 200 \, \text{units} \) and \( R + L = 37 \) days:

\[
\sigma_{d(R+L)} = 200 \times \sqrt{37} \approx 200 \times 6.08 = 1216 \, \text{units}
\]

#### Step 2: Determine the z-value for a 95% service level
From standard normal distribution tables, the z-value for a 95% service level is approximately:

\[
z = 1.645
\]

#### Step 3: Calculate the Target Inventory Level (TIL)

\[
TIL = 1500 + (1.645 \times 1216)
\]

\[
TIL \approx 1500 + 1998 = 3498 \, \text{units}
\]

So, the target inventory level is approximately **3498 units**.

### b) Calculate the Order Quantity
The order quantity (Q) is the difference between the target inventory level (TIL) and the current inventory level (I):

\[
Q = TIL - I
\]

Where \( I = 400 \, \text{units} \):

\[
Q = 3498 - 400 = 3098 \, \text{units}
\]

Thus, the order quantity is **3098 units**.

### c) What is the Stockout Risk?
The **stockout risk** is the complement of the service level:

\[
\text{Stockout risk} = 1 - \text{Service level}
\]

Given the service level is 95%:

\[
\text{Stockout risk} = 1 - 0.95 = 0.05 \, \text{or} \, 5\%
\]

Thus, the stockout risk is **5%**.

### Summary:
a) Target Inventory Level: **3498 units**  
b) Order Quantity: **3098 units**  
c) Stockout Risk: **5%**

Would you like further details or explanations on any of these calculations?

#### Follow-up Questions:
1. How does increasing the service level affect the target inventory level and stockout risk?
2. What would happen to the order quantity if the current inventory level was different?
3. How would changing the review period (e.g., to 40 days) impact the target inventory level?
4. What is the relationship between demand variability (standard deviation) and stockout risk?
5. How can the supermarket reduce stockout risk without holding excessive inventory?

#### Tip:
Balancing service level and stockout risk is crucial in inventory management. Higher service levels require more safety stock, which increases holding costs.

Island Supermarket operates a fixed-time period inventory system for its dairy products. The review period is every 30 days, and the lead time is 7 days. The demand during the lead time and review period is normally distributed with a mean of 1500 units and a standard deviation of 200 units. The target service level is 95%, and the current inventory level is 400 units. a) Calculate the target inventory level. b) Calculate the order quantity. c) What is the stockout risk?

Learn how to calculate the target inventory level, order quantity, and stockout risk for a fixed-time period inventory system with a 95% service level. This problem includes step-by-step solutions using formulas for demand forecasting, standard deviation, and normal distribution. Suitable for college students in inventory management.

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