Final answer:
Dan should use the newsvendor model to calculate the critical ratio from costs and revenues, and then determine the Z-score that corresponds to that ratio using a normal distribution with a mean demand of 1000 and a standard deviation of 250 to find the optimal order quantity.
Step-by-step explanation:
To maximize expected profits, Dan's Independent Book Store should determine the optimal number of books to order using the newsvendor model, which focuses on demand forecast and cost-profit analysis. Given the information that the book retails for $28.00, costs Dan $20.00, and unsold copies will be sold at 50% off retail price, we need to calculate the critical ratio (CR) which is the understock cost divided by the sum of understock and overstock costs. The cost of understock is the profit per book not sold when demand exists, which is $28-$20 = $8. The cost of overstock is the loss per unsold book, determined by the retail price minus discount price minus wholesale price ($28-(0.5*$28)-$20) which results in $2.
The CR is $8 / ($8 + $2) = 0.8. Using the cumulative distribution function (CDF) of the normal distribution with a mean of 1000 and standard deviation of 250, we identify the Z-score that corresponds to a CR of 0.8. This score indicates the demand level Dan should aim for to maximize expected profit. If the Z-score is not readily accessible, Dan might use an approximation or software to find the closest value.
Finally, Dan must order the number of books that equates to that critical fractile (or percentile) of the demand distribution. Without the actual Z-score value or access to a standard normal table within this context, the specific order quantity cannot be calculated. However, the actual number should be slightly higher than the average demand to aim for the 80th percentile of demand considering the provided mean and standard deviation.