Problem 2 A factory uses a type of machine that wears out quickly, and never lasts more than four months. For this type of machine, the probability that a new machine fails during its first month of operation is 10%. The probability that a one-month old machine fails during its second month of operation is 20%, and the probability that a two-month old machine fails during its third month of operation is 50%. If the machine completes its third month, then it is sure to fail by the end of the fourth month. At the beginning of each month the factory manager must decide whether or not to replace the current machine with a new machine. It costs $500 to purchase a new machine. If a machine fails during a month, the factory incurs a cost of $1,000 (due to factory downtime) and must replace the machine with a new machine. Thus, if a machine fails during a month the cost incurred for that month is $1,000 + $500 = $1,500. a) Calculate the expected cost for the month if the factory starts the month with a machine that is 1. New 2. One month old 3. Two months old b) Calculate the expected cost for the month if the factory starts the month by replacing the current machine. Three maintenance policies are under consideration: 1. Replace the machine when it fails or when it is 3 months old, whichever comes first. 2. Replace the machine when it fails or when it is 2 months old, whichever comes first. 3. Replace the machine when it fails or when it is 1 month old, whichever comes first. d) Develop a Markov chain model for each of the three policies above and determine which policy will give the lowest average monthly cost