Pet Emporium (PE) at Lawrence is a local franchise of Aquatic America (AA), which sells fresh water aquariums on a national basis. AA offers PE choice of three different varieties of aquariums. PE has to order its assortment of aquariums from AA well in advance of the upcoming selling season. Aquariums are custom built and hence once the orders are placed, they cannot be modified during the selling season. Demand for each type of aquarium is normally distributed with mean 400 and a standard deviation of 100. Further, you may assume that demands for each aquarium is independent of the others. PE buys these aquariums from AA at a whole sale price of $100 per aquarium and plans to sell them at a retail price of $150 per aquarium. AA delivers the orders placed by PE in truckloads at a transportation cost of $2,000 per truckload. The transportation cost is borne by AA and other costs like unpacking and handling are negligible. Assume all orders that are placed by PE will fit into one truckload. AA does not take back any unsold stock of aquariums. However, PE can sell any unsold inventory at a discounted price of $75 per aquarium at the end of the season.

1. How many units of each aquarium type should PE order to maximize profit?

2. If PE wishes to ensure 95% in-stock probability what should its order quantity be for each type of aquarium?
3. If PE wishes to ensure 95% fill rate, what should its order quantity be for each type of aquarium?

For parts d through f, assume PE orders 500 of each type of aquariums:

4. What is PEs expected profit?

5. What is PEs expected fill rate for each type of aquarium?

6. What is in stock probability for each type of aquarium?

7. Now suppose that AA announces that the unit of truckload capacity is 1200 units of aquariums. If AA orders more than 1200 units (anything between 1201 to 2400 units), it will have to pay for two truckloads. What is AAs optimal order quantity for each type now?