Palin’s Muffler Shop has one standard muffler that fits a large variety of cars. The shop wishes to establish a periodic review system to manage inventory of this standard muffler. Use the information in the following table to determine the optimal inventory target level (or order-up-to level).

 

 	 	 	 	 	 	 	 
Annual demand	 	2,850	mufflers	Ordering cost	$	55	per order
Standard deviation of daily demand	 	6	mufflers per working day	Service probability	 	91	%
Item cost	$	30.00	per muffler	Lead time	 	4	working days
Annual holding cost	 	40	% of item value	Working days	 	285	per year
Review period	 	16	working days	 	 	 	 
 

a. What is the optimal target level (order-up-to level)? (Use Excel's NORMSINV() function to find the correct critical value for the given α-level. Do not round intermediate calculations. Round "z" value to 2 decimal places and final answer to the nearest whole number.)

 



b. If the service probability requirement is 97 percent, the optimal target level will:

 

multiple choice
Increase
Decrease
Stay the same



### Part A: Optimal Target Level Calculation

To determine the **optimal target level (or order-up-to level)**, we will use the formula for the periodic review inventory system:

\[
T = D \cdot (L + R) + z \cdot \sigma_d \cdot \sqrt{L + R}
\]

Where:
- $T$ = optimal target level (order-up-to level)
- $D$ = average daily demand
- $L$ = lead time (in days)
- $R$ = review period (in days)
- $z$ = safety factor (determined from the service probability)
- $\sigma_d$ = standard deviation of daily demand

**Step 1: Calculate average daily demand ($D$):**

\[
D = \frac{\text{Annual demand}}{\text{Working days per year}} = \frac{2850}{285} = 10 \text{ mufflers per day}
\]

**Step 2: Identify the lead time and review period:**

- $L = 4$ working days (lead time)
- $R = 16$ working days (review period)

**Step 3: Calculate the safety factor ($z$) for a 91% service probability:**

To determine the safety factor ($z$) for a 91% service level, we use Excel's `NORMSINV()` function. For a service probability of 91%:

\[
z = \text{NORMSINV}(0.91) \approx 1.34
\]

**Step 4: Calculate the standard deviation during the lead time plus review period:**

The total period during which variability matters is $L + R = 4 + 16 = 20$ days.

\[
\sigma_{L+R} = \sigma_d \cdot \sqrt{L + R}
\]

Given that $\sigma_d = 6$ mufflers/day:

\[
\sigma_{L+R} = 6 \cdot \sqrt{20} = 6 \cdot 4.47 \approx 26.82 \text{ mufflers}
\]

**Step 5: Calculate the optimal target level:**

Now, using the formula for the target level:

\[
T = D \cdot (L + R) + z \cdot \sigma_{L+R}
\]

Substituting the values:

\[
T = 10 \cdot 20 + 1.34 \cdot 26.82
\]
\[
T = 200 + 35.93 \approx 235.93
\]

Rounding to the nearest whole number:

\[
T \approx 236 \text{ mufflers}
\]

So, the optimal order-up-to level is **236 mufflers**.

---

### Part B: Impact of a 97% Service Probability

If the service probability requirement increases to **97%**, the corresponding $z$-value will change. For a service level of 97%, using `NORMSINV(0.97)`:

\[
z = \text{NORMSINV}(0.97) \approx 1.88
\]

A higher $z$-value leads to a higher safety stock. Therefore, the **optimal target level will increase**.

---

### Final Answer:
1. The optimal order-up-to level for a 91% service probability is **236 mufflers**.
2. If the service probability increases to 97%, the optimal target level will **increase**.

---

Would you like more details or have any questions? Here are some related questions to expand on:

1. How does the standard deviation of demand affect the order-up-to level?
2. What happens if the review period is shortened to 10 days?
3. How would a higher ordering cost impact the optimal target level?
4. What role does lead time variability play in setting the target level?
5. How can Palin’s Muffler Shop reduce holding costs in this scenario?

**Tip:** When managing inventory, regularly updating demand forecasts and reviewing lead times can significantly improve inventory efficiency!

Palin’s Muffler Shop has one standard muffler that fits a large variety of cars. The shop wishes to establish a periodic review system to manage inventory of this standard muffler. Use the information in the following table to determine the optimal inventory target level (or order-up-to level). What is the optimal target level (order-up-to level)? If the service probability requirement is 97 percent, will the optimal target level increase, decrease, or stay the same?

Learn how to calculate the optimal inventory target level for Palin's Muffler Shop using a periodic review system. This example walks through demand forecasts, lead time, review period, service probability, and statistical safety stock calculations. Includes key formulas like T = D * (L + R) + z * σ_d * √(L + R). Ideal for business or operations management students.

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