Main Line Auto Distributor is an auto parts supplier to local garage shops. None of its customers have the space or capital to store all of the possible parts they might need so they order parts from Main Line several times a day. To provide fast service, Main Line uses three pickup trucks to make its own deliveries. Each Friday evening, Main Line orders additional inventory from its supplier. The supplier delivers early Monday morning. Delivery costs are significant, so Main Line only orders on Fridays. Consider part A153QR, or part A for short. Part A costs Main Line $175 and Main Line sells it to garages for $200. If a garage orders part A and Main Line is out of stock, then the garage finds the part from some other distributor. Main Line has its own capital and space constraints and estimates that each unit of part A costs $0.50 to hold in inventory per week. (Assume you incur the $0.50 cost for units left in inventory at the end of the week, not $0.50 for your average inventory during the week or $0.50 for your inventory at the start of the week.) Average weekly demand for this part follows a Poisson distribution with mean 1.5 units. Suppose it is Friday evening and Main Line currently doesn't have any part A's in stock. The distribution and loss functions for a Poisson distribution with mean 1.5 can be found in Appendix B.
a. How many part A's should Main Line order from the supplier?
b. Suppose Main Line orders three units. What is the probability Main Line is able to satisfy all demand during the week?
c. Suppose Main Line orders four units. What is the probability Main Line is not able to satisfy all demand during the week?
d. If Main Line seeks to hit a target in-stock probability of 99.5 percent, then how many units should Main Line order?
e. Suppose Main Line orders five units. What is Main Line's expected holding cost for the upcoming week?