Problem
Xavier operates a computer store that sells a single popular model. The monthly demand for this model is normally distributed with a mean of 400 and a standard deviation of 100 computers (demand is uncorrelated across months). Xavier purchases each computer for $1,000 and sells it for $1,500. All unsatisfied demand is lost to competitors. Xavier has the option of reordering computers at the beginning of each month, before demand for that month is observed. This order arrives with a one-month delay. For example, an order placed at the beginning of January arrives at the beginning of February. Each order of computers has a fixed delivery cost of $10,000 (regardless of the number of computers included in that order). Every computer remaining in inventory at the end of each month incurs a holding cost of $50. The model Xavier sells will be discontinued in two years; the manufacturer has agreed to purchase Xavier's leftover inventory when this model is discontinued (in two years), at a price of $500. Xavier's current strategy is as follows: At the beginning of each month, after receiving the previous month's order and before observing demand, check the inventory. If the inventory is below 400 units (the so-called reorder point), order 800 units (order quantity). If the inventory is greater than or equal to 400, do not order any computers this month.
I. Build a model to simulate the profit and lost sales of Xavier's computer store over the next two years. Assume that the current inventory (at the beginning of the first month) is 1000 computers.
II. What is Xavier's average profit over the next two years?
i. What are the 5th and 95th percentiles of Xavier's profit over the next two years?
ii. How many units of unsatisfied demand (lost sales) does Xavier's expect over the next two years?
iii. Holding the order quantity constant at 800 units, what is the optimal reorder point to maximize Xavier's expected profit over the next two years?