AA Manufacturing produces component X that is supplied to the assembly of product A and product B. Two units of X are needed for assembly of one unit of A, and three units of X are needed for assembly of one unit of B. The weekly demand for Product A is normally distributed with a mean of 15 and a standard deviation of 4, and the weekly demand for Product B is normally distributed with a mean of 10 and a standard deviation of 2. Assume that the demands for Product A and Product B are independent. The lead time to produce component X is 4 weeks. AA Manufacturing decides to use a (Q, R)-policy to manage the inventory level of X. If on-hand inventory is not enough to meet demand, the excess demand is backordered. AA's service goal is to limit the demand met through backorders to at most 5% of the total demand. The fixed cost of ordering Component X is $800. The holding cost of Component X is $5 per unit per week.
What is the probability that AA will run out of stock of X in a given cycle?
