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.

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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.

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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.

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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!