1. Sprit Halloween sells various Halloween-related seasonal merchandise during the fall season throughout the US. It often rented spaces at any location for a few months in fall to sell its merchandise. Consider the problem of a specific merchandise, the Elvis wig, at the local Spirit Halloween store. The store manager estimates the weekly demand for the wig to be normally distributed with mean 74 and standard deviation 22.

a. Suppose that the local store has 98 Elvis wigs in stock at the beginning of a week, and that no additional receipts are scheduled for the week. What then is the chance of a stock out during the week?

b. How much inventory of the Elvis wig does the store need for the week if it wants the stock-out probability to be 2% or less?

c. What is the chance that the weekly demand is within 20% of the average/man value? d. (5 points) Consider a second wig, “2099: the wig of the future” that is also sold by the local Sprit Halloween store. The weekly demand for this wig is identical to that of the Elvis wig (i.e., normally distributed with mean 74 and standard deviation 22). Furthermore, the demand for the two wigs are perfectly substitutable, implying, any customer looking for the Elvis wig will buy the wig of the future if the former is out-of-stock, and the vice versa. As a result, the store manager is thinking of managing the inventory of these two items together. How much total inventory (i.e., Elvis plus the wig of the future) wigs does the store need for the week if it wants the stock-out probability to be 2% or less?

2. Consider, once again, the problem of the Elvis wig at the local Sprit Halloween store described in question 1 above and ignore the existence of the wig of the future.

a. Suppose that the local Sprit Halloween store has 98 Elvis wigs in stock at the beginning of a week and that no additional receipts are scheduled for the week. How much is the expected fill rate? What fraction of the demand will not be satisfied from the available inventory then?

b. Suppose that the local Sprit Halloween store has 98 Elvis wigs in stock at the beginning of a week and that no additional receipts are scheduled for the week. How much is the expected leftover inventory at the end of the week?

c. If the local Sprit Halloween store finds it acceptable that the expected lost sales of the Elvis wig is 10 or lower, how many wigs should it have at the beginning of the week?