“
Behind the Glass
”
is a small boutique shop that sells fashionable women clothes. Every week the shop operates from Tuesday to Sunday and closes on Monday. The shop is using a fixed
-
period inventory control model. The desired service level is
9
5
%
.
Currently, a staff performs the inventory count for a particular women winter dress every Sunday after the shop closes for the day. The dress is priced at $
1
2
0
per unit. The annual holding cost per unit is
1
0
%
of product value. The daily demand for the dress follows a normal distribution with a mean of
1
0
units and standard deviation of
2
units. When the shop places a new order for the dress, it takes
3
days for the supplier to fulfill and ship the order items to
“
Behind the Glass
”
.
The supplier operates
7
days a week. The owner of this boutique store is thinking about changing the day of counting on
-
hand inventory for the dress from Sunday to Wednesday. That is
,
in the future, the owner wants to perform the inventory count for the dress every Wednesday.
1
.
In terms of order
-
up
-
to level
(
OUL
)
,
does it make any difference choosing one option over the other? Please explain through calculation. My answer to this question is
(
Yes or No
)
Question Blank
1
of
6
,
here is why Question Blank
2
of
6
.
2
.
What is the safety stock annual holding cost for the winter dress if the shop counts inventory every Sunday? The safety annual holding cost if counting on Sundays is $ Question Blank
3
of
6
.
3
.
What is the safety stock annual holding cost for the winter dress if the shop counts inventory every Wednesday? The safety annual holding cost if counting on Wednesdays is $ Question Blank
4
of
6
.
4
.
If we only focus on safety stock annual holding cost, what day in a week to perform inventory counting is more worthwhile?
Please briefly explain your conclusion. Question Blank
5
of
6
would be a better option. Here is why: Question Blank
6
of
6
.
Give step-by-step explanation
Question:
“
Behind the Glass
”
is a small boutique shop that sells fashionable women clothes. Every week the shop operates from Tuesday to Sunday and closes on Monday. The shop is using a fixed
-
period inventory control model. The desired service level is
9
5
%
.
Currently, a staff performs the inventory count for a particular women winter dress every Sunday after the shop closes for the day. The dress is priced at $
1
2
0
per unit. The annual holding cost per unit is
1
0
%
of product value. The daily demand for the dress follows a normal distribution with a mean of
1
0
units and standard deviation of
2
units. When the shop places a new order for the dress, it takes
3
days for the supplier to fulfill and ship the order items to
“
Behind the Glass
”
.
The supplier operates
7
days a week. The owner of this boutique store is thinking about changing the day of counting on
-
hand inventory for the dress from Sunday to Wednesday. That is
,
in the future, the owner wants to perform the inventory count for the dress every Wednesday.
1
.
In terms of order
-
up
-
to level
(
OUL
)
,
does it make any difference choosing one option over the other? Please explain through calculation. My answer to this question is
(
Yes or No
)
Question Blank
1
of
6
,
here is why Question Blank
2
of
6
.
2
.
What is the safety stock annual holding cost for the winter dress if the shop counts inventory every Sunday? The safety annual holding cost if counting on Sundays is $ Question Blank
3
of
6
.
3
.
What is the safety stock annual holding cost for the winter dress if the shop counts inventory every Wednesday? The safety annual holding cost if counting on Wednesdays is $ Question Blank
4
of
6
.
4
.
If we only focus on safety stock annual holding cost, what day in a week to perform inventory counting is more worthwhile?
Please briefly explain your conclusion. Question Blank
5
of
6
would be a better option. Here is why: Question Blank
6
of
6
.
Give step-by-step explanation
This is AI generated Response.
Asked by: Test User 100135
Created at: 2025-05-07 22:28:04
Final Answer:
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Let's solve each part step by step.
---
## **Given Data**
- **Shop open:** Tuesday to Sunday (closed Monday)
- **Review period:** Either Sunday or Wednesday each week (fixed-period system)
- **Order lead time:** 3 days (supplier operates 7 days/week)
- **Service level:** 95% (\(Z = 1.645\))
- **Demand per day (\(\mu_D\)):** 10 units
- **Std deviation per day (\(\sigma_D\)):** 2 units
- **Unit cost:** $120
- **Annual holding cost rate:** 10%
- **Days per year:** 365
---
### **1. Does changing the counting day affect the Order-up-to Level (OUL)?**
**The OUL formula (periodic review):**
\[
OUL = \text{Expected demand during protection period} + \text{Safety stock}
\]
- **Protection period** = Review period (\(T\)) + Lead time (\(L\))
#### **Option A: Counting every Sunday**
- Review period (\(T\)): 7 days (from Sunday to next Sunday)
- Lead time (\(L\)): 3 days
- **Protection period:** \(7 + 3 = 10\) days
#### **Option B: Counting every Wednesday**
- Review period (\(T\)): 7 days (from Wednesday to next Wednesday)
- Lead time (\(L\)): 3 days
- **Protection period:** \(7 + 3 = 10\) days
**But wait:**
We must check **when orders arrive** relative to opening days.
- Supplier operates 7 days, so lead time is always 3 days (no difference).
- The protection period is always 10 days under both options.
#### **So, does OUL differ?**
**Calculation:**
\[
OUL = \mu_D \times (T+L) + Z \times \sigma_D \times \sqrt{T+L}
\]
For both options: \(T+L = 10\) days
So,
\[
OUL = 10 \times 10 + 1.645 \times 2 \times \sqrt{10}
\]
\[
OUL = 100 + 1.645 \times 2 \times 3.162
\]
\[
OUL = 100 + 1.645 \times 6.324
\]
\[
OUL = 100 + 10.4
\]
\[
OUL \approx 110.4 \text{ units}
\]
**Conclusion:**
**My answer is:**
**No** (Question Blank 1)
**Here is why:**
The order-up-to level does **not** change because the protection period (review interval + lead time) is the same regardless of which day the inventory is counted. Thus, the OUL will be identical for both counting on Sunday and counting on Wednesday. (Question Blank 2)
---
### **2. Safety stock annual holding cost (counting on Sundays)**
**Safety stock formula:**
\[
SS = Z \cdot \sigma_D \cdot \sqrt{T + L}
\]
\[
SS = 1.645 \cdot 2 \cdot \sqrt{10}
\]
\[
SS = 1.645 \cdot 2 \cdot 3.162
\]
\[
SS = 1.645 \cdot 6.324
\]
\[
SS \approx 10.4 \text{ units}
\]
**Annual holding cost per unit:**
\[
\text{Annual holding cost per unit} = \$120 \times 10\% = \$12
\]
**Total annual holding cost for safety stock:**
\[
\text{Total} = SS \times \text{Annual holding cost per unit}
\]
\[
\text{Total} = 10.4 \times 12 = \$124.8
\]
**The safety annual holding cost if counting on Sundays is $**
**124.8** (Question Blank 3)
---
### **3. Safety stock annual holding cost (counting on Wednesdays)**
As established above, the protection period is **still 10 days**, so the calculation is identical:
\[
SS = 1.645 \cdot 2 \cdot \sqrt{10} \approx 10.4 \text{ units}
\]
\[
\text{Total annual holding cost} = 10.4 \times 12 = \$124.8
\]
**The safety annual holding cost if counting on Wednesdays is $**
**124.8** (Question Blank 4)
---
### **4. Which day is more worthwhile for inventory counting (focusing only on safety stock holding cost)?**
**Conclusion:**
**Either day** (Sunday or Wednesday) is equally worthwhile **(Question Blank 5)**
**Here is why:**
The safety stock (and thus safety stock holding cost) depends only on the protection period (review interval + lead time), which is 10 days in both cases. Therefore, changing the inventory counting day **does not affect** the safety stock holding cost. (Question Blank 6)
---
## **Summary Table for Your Blanks**
| Question No. | Answer |
|--------------|---------------|
| 1 | No |
| 2 | The order-up-to level does not change because the protection period (review interval + lead time) is the same regardless of which day the inventory is counted. Thus, the OUL will be identical for both counting on Sunday and counting on Wednesday. |
| 3 | 124.8 |
| 4 | 124.8 |
| 5 | Either day |
| 6 | The safety stock (and thus its annual holding cost) is the same for both options, since the protection period is unchanged. |
---
### **Step-by-step summary**
1. The OUL is the same for either day because the protection period (review period + lead time) is unchanged.
2. Safety stock is the same for either day (10.4 units), so annual holding cost is $124.8 in both cases.
3. Thus, either day is equally worthwhile from a safety stock holding cost perspective, because the calculation is based only on the protection period, which is the same for both cases.
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