Problem 3 (Correlation). Many companies such as Walmart, Costco, and Amazon take advantage of "risk pooling" to reduce their inventories. To illustrate how risk pooling works, consider a company with warehouses in Pittsburgh and Philadelphia to serve its customers in these two cities. Many different products are held in inventory. The idea of risk pooling will become clear by focusing on one such product. Weekly demand for one of the company's products is 40 units on average in Pittsburgh with a standard deviation of 20 units, and 80 units on average in Philadelphia with a standard deviation of 30 units. The correlation between these two demands is = 0.1. To protect against stockouts, the company maintains an inventory at each warehouse equal to the expected weekly demand locally plus two standard deviations of this weekly demand. The company is considering replacing its warehouses in Pittsburgh and Philadelphia by a more centrally located warehouse in Harrisburg. The inventory level at the Harrisburg warehouse will be based on the total demand in Pittsburgh and Philadelphia. Let us call D this total weekly (random) demand. Inventory at the Harrisburg warehouse will be equal to the expected value of D plus two standard deviations of D.

(3.1) What are the current inventory levels at the Pittsburgh and Philadelphia warehouses? (3.2) What would be the inventory needed at the Harrisburg warehouse? What would be the decrease in inventory achieved by switching to a single warehouse in Harrisburg as compared to (3.1)?