Question $2.$ The daily demand for a spare engine part is a random variable whose distribution,based on past experience, is given below:

The parts cost $$1500$ each and $$5000$ each when purchased on an emergency basis. Unused parts can be scrapped for $10\%$ of their purchase price.

Part A$.$ How many parts should be acquired to maximize expected profit?

Note: Even though the setting described here is not exactly the standard newsvendor problem with revenue and salvage value, the same concept of underage $($shortage$)$ and overage $($excess$)$ costs apply; you just need to derive these costs with the parameters described in this problem.

Part B$.$ Now suppose the planning horizon is over $400$ days with only one purchase opportunity in advance of the $400$ day period and the daily demand following the same distribution as before. Assume demands from one day to the next are independent. The part is expected to be obsolete after $400$ days with the same salvage value as in part A$.$ Other cost parameters are the same as in Part A as well. Furthermore, holding costs for unused parts are based on a daily interest rate of $0\mathrm{.}08\%.$

Using critical ratio, how many parts should be acquired in advance of the $400-$day period?

Note: You may use the Central Limit Theorem to approximate the demand over $400$ days with a normal distribution.