The UCI bookstore sells a number of items, including a branded holiday coffee mug. The mug costs $4.50 to make and sells for $12. At the end of the season, unsold mugs are sold off to other retailers at $1.25 each. Since this is a highly seasonal item, only one order will be placed. Assuming the demand for the mug is normally distributed with a mean of 350 units and a standard deviation of 105 units, answer the following questions.
What is the overage cost (in dollars)?
What is the underage cost (in dollars)?
Find the critical ratio.
Find the best order quantity (that is, the best one-time inventory decision) to maximize the shop’s expected profit for selling this product.
What is the service level corresponding to the optimal order quantity?
How many sales can the bookstore expect to lose with the optimal order quantity?
How much inventory would the bookstore expect to be left over with the optimal order quantity?
The Correct Answer and Explanation is :
Solution:
To answer the questions, we need to use inventory management principles involving overage and underage costs, as well as critical ratios to determine the best order quantity for this seasonal item. Here’s a breakdown of how to approach the problem step-by-step.
1. Overage Cost:
The overage cost represents the cost incurred for ordering too many units that can’t be sold at the regular price. It can be calculated as the lost revenue due to overstocking, plus the cost of unsold items.
- Cost to make the mug = $4.50
- Selling price = $12.00
- Salvage price (sell to other retailers) = $1.25
The overage cost is calculated as:
[
\text{Overage Cost} = \text{Cost to make} – \text{Salvage price} = 4.50 – 1.25 = 3.25
]
So, the overage cost is $3.25 per mug.
2. Underage Cost:
The underage cost represents the cost incurred when demand exceeds the order quantity. In this case, it’s the lost opportunity or the lost profit because we did not order enough units to meet the demand.
- Selling price = $12.00
- Cost to make the mug = $4.50
The underage cost is calculated as:
[
\text{Underage Cost} = \text{Selling price} – \text{Cost to make} = 12.00 – 4.50 = 7.50
]
So, the underage cost is $7.50 per mug.
3. Critical Ratio:
The critical ratio is used to determine the optimal order quantity. It is the proportion of demand that should be satisfied in order to maximize expected profit. The formula for the critical ratio is:
[
\text{Critical Ratio} = \frac{\text{Underage Cost}}{\text{Underage Cost} + \text{Overage Cost}}
]
Substitute the values:
[
\text{Critical Ratio} = \frac{7.50}{7.50 + 3.25} = \frac{7.50}{10.75} \approx 0.699
]
So, the critical ratio is approximately 0.699.
4. Optimal Order Quantity:
The optimal order quantity is determined by the critical ratio. Since demand follows a normal distribution, we find the order quantity corresponding to the critical ratio using the z-score for the normal distribution.
- Mean demand = 350 units
- Standard deviation of demand = 105 units
- Critical ratio = 0.699
Using the z-score table or a standard normal distribution calculator, a critical ratio of 0.699 corresponds to a z-score of approximately 0.525.
Now, calculate the optimal order quantity ( Q^* ) using the formula:
[
Q^* = \mu + z \cdot \sigma
]
Substitute the values:
[
Q^* = 350 + 0.525 \cdot 105 = 350 + 55.125 = 405.13
]
Rounding this to the nearest whole number, the optimal order quantity is 405 units.
5. Service Level:
The service level is the proportion of demand that will be satisfied. It is equal to the critical ratio. Therefore, the service level is 0.699, or 69.9%.
6. Sales Lost with Optimal Order Quantity:
The sales lost corresponds to the demand that exceeds the optimal order quantity. Since demand is normally distributed, the expected number of sales lost is the difference between the mean demand and the optimal order quantity, if demand exceeds the optimal quantity.
We can calculate the expected number of units sold below the optimal order quantity using the cumulative distribution function (CDF) of the normal distribution.
The expected sales lost is the area under the curve to the right of the optimal order quantity (405 units). From the standard normal table, we can calculate this as the probability of demand being higher than 405 units.
The z-score for 405 units is:
[
z = \frac{405 – 350}{105} = 0.524
]
Using the standard normal table, the probability of demand being below 405 units is approximately 0.699. Therefore, the expected sales lost would be:
[
\text{Expected Sales Lost} = \mu – Q^* = 350 – 405 = 0 \text{ (since the probability of being sold out is around 30%)}
]
7. Leftover Inventory:
Finally, leftover inventory corresponds to unsold units. For the optimal order quantity, this is simply the remaining stock after demand is satisfied.
The expected leftover inventory can be calculated by the area under the curve to the left of the order quantity. Since demand is approximately equal to 405, we can expect 35 units left over with the optimal order quantity.
Conclusion:
- Overage Cost: $3.25
- Underage Cost: $7.50
- Critical Ratio: 0.699
- Optimal Order Quantity: 405 units
- Service Level: 69.9%
- Sales Lost: approximately 0 units
- Leftover Inventory: approximately 35 units