Problem 5 (Probabilistic Inventory Model. Total 15pts.) The SmartAuto Inc. is a car dealer that sells new cars. Customer Demands and Order Delivery Time from the car manufacturer may vary randomly from time to time.Assume that historical data shows that the Demand During Delivery Lead-time can be summarized in the following table.

Demand During Leadtime (# of Cars)		0	5	10	15	20	25
Probability		0.03	0.10	0.15	0.40	0.20	0.12

Assume that SmartAuto can sell an average 1000 cars a year, and the manager currently orders 250 cars each time from the manufacturer whenever its inventory drops to 20.

a. Calculate

i. The Safety Stock

ii. the Chance of Stockout, and the Service level,

iii. the Expected # of Demand Shortages in Each Ordering Cycle

iv. the Fill Rate

b. Assume that it will cost the Inc. $200 to hold a car in inventory for a year, and $400 for each car demand that has to be backordered due to shortage. Find the optimal Reorder point, or Safety Stock level,such that the Total Annual Costs in Cycle Stock + Safety Stock + Shortage is minimized.