A company selling hot rolled steel serves 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 at each customer location is replenished only once each month, just before the month begins. The company’s goal is to cover all customer demand in every month at least 95% of the time. (a) Assuming that any shortage of steel at one of the customer locations can be made up by transporting excess steel from another of the customer locations, compute the minimum amount of steel that should be stocked in total across all of the locations at the beginning of each month so that the company will be able to accomplish this goal. (Hint: =NORMSINV(0.5) function in Excel or scipy. norm. ppf(0.5) function of the scipy package in Python returns z such that P{N(0, 1) ≤ z} = 0.5. =NORMSDIST(1.5) function in Excel or scipy. norm. cdf(1.5) function of the scipy package in Python returns the probability P{N(0, 1) ≤ 1.5}.)
(b) Suppose that to minimize its transportation costs, the company contemplates not allowing trans-shipment of steel between customer locations. Instead, the company operates a warehouse next to each customer and serves 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 steel needs to be placed nationwide (in total across all locations) at the start of each month?
(c) Explain intuitively in at most 2 sentences why your answer in Part (a) yields a smaller required inventory.