Please show your calculation and write down the formula(s) that you have used to obtain the final answer. Do not just write down the final answer.

Mango Electronics is introducing a new music device called SuperPod. The annual demand of this new product faced by Mango Electronics is given by the following demand curve: annual demand = 2,000,000 2,000p, where p. is the unit selling price. It costs Mango Electronics $100 to produce each SuperPod. (a) [6 points] If Mango Electronics sells the product directly to end consumers, what is the optimal unit selling price, p*, that Mango Electronics should set? What is Mango Electronics optimal annual profit obtained from this new product? (b) [6 points] If Mango Electronics does not sell the product to the end consumers, but it sells it to a retailer at a unit price of p*, then what is the optimal selling price that the retailer should set? What is optimal annual supply chain profit (i.e., Mango Electronics' profit plus the retailer's profit) obtained from this new product? (c) [10 points] The difference between the results of (a) and (b) indicates that double marginalization leads to a significant loss in supply chain profit. Explain how Mango Electronics can employ a volume-based discount to eliminate the loss of profit due to double marginalization. (Note: You need to show all the detailed calculations. You also need to determine the upper and lower limits of the discount price and specify the condition(s) for the retailer to receive the discount.) > The Gravity Maze is a hot toy this Christmas, and the manufacturer has decided to ration supply to all retailers. A large retail chain owns two channelsa high-service channel and a discount channel. The retailer plans to sell the toy at a unit price of $40 in the high-service channel and a unit price of $24 in the discount channel. Any units sent to the high-service channel cannot be redirected to the discount channel. At the end of the selling season, the leftover units at the high-service channel will have zero salvage value. The manufacturer sends 330,000 Gravity Mazes to the retailer. The retailer has forecast that the demand for the toy at the high-service channel is normally distributed, with a mean of 320,000 and a standard deviation of 121,000. Assume that all the units sent to the discount channel will sell out. How many Gravity Mazes should the retailer send to the discount channel? In 2009, Benson Computers ran a chain of 64 retail stores all over North America. Consider one type of computers sold by Benson. Demand for this computer at each store on any given day was independently and normally distributed with a mean of 25 units and a standard deviation of 7.5 units. At the beginning of 2010, Benson decided it was time to close their retail stores, put up a website, and begin filling customer orders from a single centralized warehouse. It takes 16 days to ship the computers from the supplier's location to the centralized warehouse. (a) [8 points] By consolidating the demand into a single warehouse, what will be the resulting standard deviation of daily demand for this computer faced by Benson (assuming Benson's demand characteristics before and after the consolidation are identical)? (b) [8 points] Benson takes physical possession of inventory when it leaves the supplier and grants possession of inventory to customers when it leaves Benson's shipping dock. In the consolidated distribution scenario, what is the expected level of Benson's in-transit inventory of this computer