H-Mart in the lobby of your school sells fresh Pizzas of four types, Bacon, Ham, Cheese and Vege. The mart has to order its sandwiches from a dinning service everyday morning before 5:30 AM. Once orders sent to the dinning service, they cannot be modified during the same day. Demand for each type of sandwich is normally distributed with mean 60 and a standard deviation of 16. Further, you may assume that demands for each type of sandwiches are independent of the others. The mart orders these sandwiches from the dinning service at a whole sale price of $3 each and plans to sell them at a retail price of $5 per sandwich. The dinning service delivers the orders placed in truckloads at a transportation cost of $20 per truckload. The transportation cost is borne the dinning service and other costs like unpacking and handling are negligible. Assume all orders that are placed by the mart will fit into one truckload. The dinning service does not take back any unsold sandwiches. However, the mart can sell any unsold inventory at a discounted price of $1.5 after 5:00 PM every day.
(a) How many sandwiches of each type should the mart order to maximize profit?
(b) If the mart wishes to ensure a 98% in-stock level, how many of each type of sandwiches should the mart order?
(c) If the mart wishes to ensure a 98% fill rate, how many of each type of sandwiches should the mart order?