Everett Ltd is a company who sells CNC machines to universities for educational purposes. The company observes the number of CNC machines on its hand (call it i) at the beginning of a month. If at the beginning of a month i ≤ 1, then the company orders 3 - i machines (so, the number of inventories is completed to 3). If i ≥ 2, then 0 machines are ordered. It is known that delivery of all ordered machines is immediate. The customers are equally likely to demand 0, 1, 2, or 3 machines from Everett Ltd. during the month. After fulfilling these demands, the company observes the inventory level at the beginning of the next month.
a) Construct the one step transition matrix for the above process.
b) What proportion of time no inventories exists (0 units) on hand at the beginning of a typical month?
c) Assume that there is a shortage cost of $1000 per machine for each demand that is not met by Everett Ltd. during a month. What is the expected amount of this shortage cost for a month?