In Lecture $9$ we derived optimal acquisition and sorting policies in the case of used

product condition variability. Assume that used products are acquired for a unit cost of $$4.$ The

pdf f$($x$)$ and the cdf F$($x$)$ of the condition are given as follows.

f$($x$)=($

$1/32$ if $0<=$ x $<=8,$

$1/16$ if $8<=$ x $<=20,$

and F$($x$)=($

x$/32$ if $0<=$ x $<=8,$

$($x $4)/16$ if $8<=$ x $<=20.$

The above distribution implies an average cost to remanufacture of $$11\mathrm{.}5.$ Unit sales revenue is

$$20$ and unit shortage penalty is $$6.$ Demand has a continuous uniform distribution between $1000$

and $2000.$

$($a$)(25$ points$)$ Find the optimal fraction of acquired used products that must be remanufactured.

$($b$)(10$ points$)$ Find the optimal number of used products that must be remanufactured.

$($c$)(5$ points$)$ Find the optimal number of used products that must be acquired.