Problem: Risk Pooling

The Sweater Factory stocks specialty cashmere sweaters that carry the University of Georgia logo. The company started this line to sell to two local firms (Firm A and Firm B) who use these items as gifts for senior executives and important customers. The sweaters are available in two colors, with red sweaters made exclusively for Firm A and black sweaters made exclusively for Firm B. Both red and black sweaters cost c = $20 per unit, but red sweaters sell for p = $50, while black sweater are priced at p = $30. The sweaters have no value after the selling season, i.e., v = $0 for both colors. Because of long lead times, the Sweater Factory must order well in advance of the selling season.

Before placing an order for red and black sweaters, you call on your contacts at Firm A and Firm B. Neither of these contacts can tell you how many units their firm will ultimately order, but the contact at Firm A states that there is a 50% chance that they will want 50 units and a 50% chance they will want 100. Similarly, the contact at Firm B states that there is a 50% chance that they will want 100 units and a 50% chance they will want 200. No other quantities are considered possible.

a. How many units of red sweaters should be ordered to maximize expected profit? What is the associated expected profit?

b. How many units of black sweaters should be ordered to maximize expected profit? What is the associated expected profit?

c. The manufacturer of this line of sweater advises you that you may order sweaters uncolored. The advantage of an uncolored sweater is that it can be dyed by a local supplier during the selling season, allowing it to satisfy demand for red sweaters if necessary or black sweaters otherwise. The downside is the added cost - the cost to the Sweater Factory to deliver an uncolored sweater is $20, but it costs another $1 to dye the sweater (note that the latter cost is only incurred if the sweater is sold. Given this alternative, how many red, black, and uncolored sweaters should be ordered to maximize expected profit? What is the associated expected profit?