Question $3(30$ points$)$ A DHL e$-$commerce fulfillment center $($hereafter refers to as FC$)$ in Ontario, California is supplied by a national distribution center $($hereafter refers to as DC$)$ in Memphis, Tennessee. One of the women$$s running shoes is Saucony Endorphin Elite and Saucony sells this model of shoes exclusively through FC$.$ Here is what we know regarding the Endorphin Elite $($hereafter refers to as EE$)$: The sales of EE follow a normal distribution, the average demand is $120$ units per day and the standard deviation of demand is $20$ units per day. Cost of goods sold at the FC: $$100$ per unit. Inventory carrying cost percentage: $25\%$ for all inventory $($DC$,$ FC$,$ and in$-$transit$)$ per year. Order cost at the store: $$50$ per order. Lead time from DC to FC: $2$ days and FC requires an in$-$stock probability of $95\%$ during the lead time. Assuming both FC and DC are operating $365$ days per year. If the FC uses re$-$order point inventory policy and orders by the economic order quantity, what is the average inventory level at FC$?$ Please provide details of your calculations. How much is the average in$-$transit inventory between the DC and the FC$?$ Please provide details of your calculations, including the results of each step, such as E$($Z$).$ If out$-$of$-$stock costs FC $$10$ per unit, what is the expected annual out$-$of$-$stock cost? Please provide details of your calculations. If Memphis DC also uses re$-$order point policy and orders $1,000$ units of EE from the factory each time. The lead time is $5$ days for DC and the service level required by DC is $90\%.$ How much is the average inventory level at the Memphis DC$?$ If the FC decides to change the inventory policy to order on a weekly basis $($order every $7$ days on every Monday$)$ while maintaining the same service level of $95\%,$ and if on a particular Monday FC still has $200$ units, how many units of EE shoes does FC need to order from DC on that Monday? $$