A computer manufacturer sells its notebook model through a web-based reseller, which it buys at a unit cost of $ 200 and sells at a unit price of $ 500. The product life cycle is so short that the Distributor has only one chance to order inventory (stock) before technology becomes obsolete and a new model is released. At the beginning of the cycle, the distributor orders a level of stock based on an uncertain demand in the retail stores (retailers). Based on its previous experience, the distributor estimates that a reasonable demand model is a uniform distribution with a minimum of 1,000 and a maximum of 8,000 laptops. Items in stock are ultimately sold, returned, or thrown away. Customers place orders on the Web, and the distributor tries to satisfy their orders with the inventory (Stock) available. If there is no inventory available, the demands are lost. The computer manufacturer offers the reseller the following return policy: $ 100 is paid for each unit returned at the end of the product life cycle, but only up to a maximum of 20% of the original number of units ordered. Excess inventory that cannot be returned to the manufacturer is collected as waste material by a recycling center, without affecting costs or revenue. The decision faced by the distributor is to order an appropriate level of inventory

a. Suppose there is no limit to the return of excess laptops. How many laptops should the dealer order to maximize their expected profits? Obtain a graph of ordered units vs. expected profits.

b. When considering the ceiling imposed by the manufacturer, how many laptops must the dealer order to maximize their expected profits? Obtain a graph of ordered units vs. expected profits. Obtain a 95% confidence interval for the level of units to sort.

c. If no return policy is considered, how many laptops should the dealer order to maximize their expected profits? Obtain a 95% confidence interval for the expected utility.

d. What would be the corresponding expected benefit to the manufacturer if the manufacturing cost is $ 125 per laptop, and the reseller uses the policy in part (b)?