1) Risk Pooling (20 points) A mail-order firm has four regional warehouses. Weekly demand at each warehouse is normally distributed with a mean of 10,000 units and a standard deviation of 2,000 units. The company purchases each unit of product at $10. The annual holding cost of one unit of product is 25% of its value. Each order incurs an ordering cost of $1,000 (primarily from fixed transportation costs), and the lead time is 1 week. The company wants the probability of stocking out during the lead time at each warehouse to be no more than 5%. Assume 50 working weeks in a year.

(a) (2 points) What is the optimal order quantity for one single warehouse?

(b) (2 points) Assuming that each warehouse operates independently, how much safety stock does each warehouse hold?

(c) (2 points) How much average inventory is held at each warehouse?

(d) (2 points) What are the combined annual holding costs and ordering costs for all four warehouses?

(e) (2 points) On average, how long - in weeks - does a unit of product spend in the warehouse before being sold?

Assume that the firm has centralized all inventories in a single warehouse and that the probability of stocking out during the lead time is still no more than 5%.

(f) (2 points) What is this optimal order quantity for the central warehouse?

(g) (2 points) How much safety stock does the central warehouse hold?

(h) (2 points) How much average inventory does the company hold now?

(i) (2 points) What are the annual holding cost and ordering cost for this central warehouse?

(j) (2 points) On average, how long - in weeks - does a unit of product spend in the central warehouse before being sold?

2. Forecasting (14 points) (a) (2 points) Look at the following time series. Would a 3-week or a 6-week moving average forecast provide a better forecast? Why or why not?

(b) (2 points) For the same time series, would an exponential smoothing forecast with α = 0.2 provide a better forecast than a forecast with α = 0.6? Why or why not?

(c) (3 points) When looking at the formula for exponential smoothing, one could get the impression that older actual data are simply ignored, because only the actual demand of the previous period, At−1, will be considered. In comparison, simple or weighted moving averages seem to use more previous periods. Is this entirely true? Why or why not?

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