A store uses an (s,S) inventory policy for a particular product. Every Friday evening after the store closes, the inventory level is checked. If the stock level is greater than s, then no action is taken. Otherwise, enough stock is procured over the weekend so that, when the store reopens on Monday morning, the inventory level is S = 8. Suppose the demand for the product during a given week is a Poisson random variable having a mean of 4 items per week. If the demand exceeds the inventory during a given week, i.e. if a shortage occurs, then the excess demand is lost.

Task 1: Let f(n) denote the probability that the demand is n during a given week, and let R(n) denote the probability that the demand is greater than n during a given week. Construct expressions for f(n) and R(n).

Task 2: Define X(t) so that {X(t), t = 0, 1, } can be modeled as a discrete-time Markov chain. The store pays $5 in inventory costs for every unit on the shelves at the end of the week. In addition, the store estimates it loses $35 in future sales when they a have a week with a shortage. Replenishment orders cost the store $15 plus $10 per item. The store sells each item for $22. Suppose that s = 3.

Task 3: Construct the transition probability matrix for {X(t), t = 0, 1, } using f(n) and R(n). Task 4: If the starting inventory on Monday morning is 6 units, determine the probability that a shortage occurs by Friday afternoon.

Task 5: If the starting inventory on Monday morning is 6 units, determine the expected value of the inventory level on Friday afternoon.