Problem $1$: $(20$ points$)$ Suppose that a manager needs to order a component

to satisfy next week's demand. The probability that there will be $x($integer$)$

units of demand is $\frac{1}{20}$ where $$1$$$1\mathrm{.}8$$$0\mathrm{.}310.$

The manager can pay $$1$ per unit $to$ order some components from a regular

channel $in$ advance. $Ifit$ turns out that the amount $is$ not sufficient $to$ satisfy

all demands, the manager needs $to$ place $an$ expediting orders $at$ $$1\mathrm{.}8$ per unit $to$

cover the shortage. Unused component can $be$ sold $in$ the second$-$hand market

$at$ $$0\mathrm{.}3$ per unit. Determine

$(8$ points$)$ The optimal number $of$ units that the manager should order

from the regular channel;

$(4$ points$)$ The expected number $of$ units $of$ expediting orders;

$(4$ points$)$ The expected total purchase cost $(include$ both regular and

expediting orders$)$;

$(4$ points$)$ The expected salvage value $of$ unused units sold $in$ the second$-$

hand market;