A company selling hot rolled steel operates a national warehouse in Tennessee to serve 10 customer locations dispersed around the country. The monthly demand at each customer location is normally distributed with mean 5 million metric tons and standard deviation 1 million metric tons. The demands at different customer locations are independent of each other. The steel inventory is replenished every month, and replenishment during the month is not desired. The company's goal is to cover all customer demand in every month at least 95% of the time.

a) Compute the minimum amount of steel that should be stocked in the warehouse at the beginning of each month, such that the company will be able to accomplish this goal.

(Hint: Recall that if A and B are independent events, then P{A and B} = P{A} P{B}. In particular if X and Y are independent random variables, then P{X ≥ x and Y ≥ y} = P{X ≥ x} P{Y ≥ y}. (0.5) function in Excel returns z such that P{N(0,1) ≤ z} = 0.5. (1.5) function in Excel returns the probability P{N (0, 1) ≤ 1.5}.)

b) Primarily aimed to minimize its transportation costs, the company now considers operating a warehouse next to each customer and servicing each customer only from the warehouse next to them. The company's goal is still to cover all customer demand in every month at least 95% of the time with minimum inventory. In other words, the company would still like to see no customers lacking steel in 95% of the months. How much extra inventory needs to be placed nationwide? The steel inventory at each warehouse is replenished every month, and replenishment during the month, as well as shipping steel rolls from other warehouses is not desired: recall the aim of minimizing transportation costs.

c) Suppose the company has n customers with the same monthly demand distribution and the company decides to serve each of the n customers from warehouses next to each of these customers with the objective to cover all customer demand in every month at least 1-α fraction of the time, with α ≤ 0.05. In other words, the company would still like to see no customers lacking steel in 100(1 - α)% of the months. The steel inventory at each warehouse is replenished every month, and replenishment during the month, as well as shipping steel rolls from other warehouses is not desired. Express, in terms of n, the percentage increase in required nationwide inventory in excess of average demand with n warehouses compared with the required inventory in excess of average demand if the company had operated a single warehouse.