1. Irwin's sells a particular model of fan, with most sales being made in the summer months. Irwin's makes a one-time purchase of the fans prior to each summer season at $40 each. It sells each fan for $70. Any unsold fan at the end of the summer season are marked down to $29 and sold in the special fall sale. Virtually all marked down fans are sold. The following is number of sales of fans during the past 10 summers: 30,50,30,60,10,40,30,30,20,40.

a) Estimate the mean and the standard deviation of demand for the fans each summer.

b) Assuming that demand each summer follows a normal distribution, determine the optimal number of fans to buy prior to each summer season.

c) What will be the number of fans to purchase if Irwin's desires to have a 90% chance of not running out of stock during a summer season?

2. James Long, the owner of Longs Inc., is a distributor of one size fit all T-shirts. T-shirts are purchased from the manufacturer in Asia before the beginning of the season and are stocked and sold to resellers. At the end of the season, unsold T-shirts are all sold to the local discount stores at a deep discount. Consider the following situation regarding a T-shirt, which comes in two popular colors, red and green.

Assuming that demand for T-shirts are Normal,

a) Find the optimal quantities to be ordered for each T-shirt color and the overall average total cost for the season associated with this operation.

Instead of ordering the T-shirts with colors to the Asian Manufacturer and then stocking them at the warehouse, James is considering ordering white T-shirts to the manufacturer and then complete the dying of the T-shirts locally, as needed. This way, only white T-shirts will be stocked as dying can be completed expeditiously in no time. Each white T-shirt will cost $6. Unsold white T-shirts will be salvaged at the end of the season at $4.5. In addition, the unit holding cost of stocking white a T-shirt for the season is $0.3. Finally, the dying cost will be $1 per T-shirt.

b) Which supply chain concept(s) discussed in class is (are) being considered here?

c) Assuming that the demand for Red and Green T-shirts are independent, calculate the optimal order quantity for the white T-shirts, the average total cost for the season associated with this operation. Based on your analysis, should James take this action?

3. Z&G Ltd. is an Italian upscale manufacturer of eyewear. DV Inc. is one of Z&G's retailers in the United States. To match UV's stylish assortment, DV owns only small boutique stores located in trendy locations. Let's focus on one of DV's stores in Beverly Hills, CA. Z&G manufactures the sunglasses in Europe and the replenishment lead-time to U.S. is long. Furthermore, the selling season for sunglasses is short as styles change significantly from year to year. As a result, DV receives only one delivery of Z&G glasses before each season. As with any fashion product, some styles sell out quickly while others are left over at the end of the season.

Consider Z&G's entry-level sunglasses for the coming season, the Bassano. DV purchases each one of those pairs for $75 and sells them for $115. Z&G's production and shipping costs per pair are $35. At the end of the season, DV offers a deep discount to sell the leftover glasses. Specifically, the Beverly Hills store will be able to fetch $25 per leftover of Bassano. DV's Beverly Hills store has estimated that the demand during the season for Bassano follows a Normal distribution with a mean 250 and standard deviation of 125 glasses.

a) Find the optimal stocking level at DV-Beverly Hills, his expected profit as well as Z&G's expected profit. Then find the expected total profit for supply chain.

b) Suppose Z&G sells each pair of Bassanos to DV at $35. Find DV's optimal order quantity, his expected profit, DV's expected profit and the total expected profit for the supply chain (this is the solution to the centralized case).

c) Buy Back Contract: Suppose Z&G sells Bassanos at $75 a pair and offers to buy back the leftovers from DV at the end of the season. Under this arrangement, DV will ship the leftover sunglasses to Z&G. The shipping cost is $1.50 per sunglass and is paid by Z&G. Furthermore, Z&G is able to sell the leftover sunglasses to one of his European outlets for $26.50 per pair. What should be Z&G's optimal Buy Back price that achieves the highest total SC expected profit (coordinates the channels)? What are the expected profits for Z&G and DV under this arrangement?

d) Revenue Sharing: Find the parameters of a revenue sharing contract that results in exactly the same expected profits for the parties as in (c)