Fashionables is a franchisee of The Limited, the well-known retailer of fashionable clothing. Prior to the winter

Question:

Fashionables is a franchisee of The Limited, the well-known retailer of fashionable clothing. Prior to the winter season, The Limited offers Fashionables the choice of five different colors of a particular sweater design. The sweaters are knit overseas by hand, and because of the lead times involved, Fashionables will need to order its assortment in advance of the selling season. As per the contracting terms offered by The Limited, Fashionables also will not be able to cancel, modify, or reorder sweaters during the selling season. Demand for each color during the season is normally distributed with a mean of 500 and a standard deviation of 200. Further, you may assume that the demands for each sweater are independent of those for a different color. The Limited offers the sweaters to Fashionables at the wholesale price of $40 per sweater and Fashionables plans to sell each sweater at the retail price of $70 per unit.
The Limited delivers orders placed by Fashionables in truckloads at a cost of $2,000 per truckload. The transportation cost of $2,000 is borne by Fashionables. Assume unless otherwise specified that all the sweaters ordered by Fashionables will fit into one truck- load. Also assume that all other associated costs, such as unpacking and handling, are negligible. The Limited does not accept any returns of unsold inventory. However, Fashionables can sell all of the unsold sweaters at the end of the season at the fire-sale price of $20 each.

a. How many units of each sweater type should Fashionables order to maximize its expected profit?
b. If Fashionables wishes to ensure a 97.5 percent in-stock probability, what should its order quantity be for each type of sweater? For parts c and d, assume Fashionables orders 725 of each sweater.
c. What is Fashionables' expected profit?
d. What is the stockout probability for each sweater?
e. Now suppose that The Limited announces that the unit of truckload capacity is 2,500 total units of sweaters. If Fashionables orders more than 2,500 units in total (actually, from 2,501 to 5,000 units in total), it will have to pay for two truckloads. What now is Fashionables' optimal order quantity for each sweater?