Problem 4: Risk pooling You own a small novelty store in a resort area. (We have a friend whose son-in-law owns several such shops.) Among other things, you sell pens that have names engraved on them (AI, Albert, Alfred, Alice, Andrew, Barbara, ...). In all you have 80 different names on the pens. You reorder the pens using a QR inventory policy and you want to ensure that the probability of stocking out of a pen with a particular name is less than 0.05 during a leadtime. The mean demand for each SKU (stock keeping unit) is 12 pens during a lead time. Demand for each SKU follows a Poisson distribution. The holding cost per pen per year is 50 cents. In this problem, we will only be concerned about the safety stock costs. Feel free to use reasonable approximations in the computations below. a) What should the reorder point be for each SKU? b) What is the average safety stock for each SKU? c) Over all 80 SKUs, what is the expected cost of the safety stock on an annual basis? d) Now suppose that a friend comes by and is offering a machine that will engrave any name on the pen on the spot while a customer waits a brief period of time. He will lease the machine to you for $150 per year. You realize that the use of this machine will result in a number of cost changes for you. In particular: a. You will possibly save money since you would have only one SKU and you could save money on the safety stock. b. You could engrave any name on the pen and you would not be limited to the 80 names that you now have in inventory. This is an issue since, for example, in my immediate family we typically find only my name on such pens. My wife's name and the names of our daughters never seem to make the cut. C. You would need to devote a bit of time to operating the machine for each sale. d. You would get to know your customers better during the time you operate the machine. e. Your life would be far simpler since you would have to manage 1 SKU (generic pens) instead of 80 different SKUs. f. Any fixed ordering costs would virtually vanish since you would incur such a cost for only 1 generic pen instead of incurring the cost for each of the 80 different SKUs. It is very hard for you to estimate the benefits/costs of items (b)-(f) above, but you can handle item (a), the change in inventory. i. ii. iv. V. vi. What is the average leadtime demand now for the generic pens (without names on them)? What is the standard deviation of this lead time demand? What should the reorder point be for the generic pens? What is the safety stock? What is the annual cost of this safety stock? What is the annual savings in safety stock due to aggregating the demand? At $150 per year, does the safety stock savings pay for the lease on the machine? If your friend increases the price to $200 per year, would it be worth it to lease the machine? In your opinion, if you consider all the benefits in (b)-(f) above, would a lease price of $250 be worth it? There is no right or wrong answer to this, I just want to see your thinking vii. v