$(30$ points$)$ Consider a closed$-$loop supply chain subject to both finite product life cycle and limited

component durability. The sales rate follows an isosceles trapezoid with $T=6$ years, $Q=24000,$

and $z=2.$ The maximum number of lives $n$ is $2$ and the collection rate $c$ is $0\mathrm{.}5$ for the product.

Remanufactured and new items are perfect substitutes. Assume that $C_{\text{m}\text{a}\text{n}}=4,C_{\text{r}\text{e}\text{m}\text{a}\text{n}}=1,$

$C_{\text{c}\text{o}\text{l}\text{l}}=0\mathrm{.}5,$ and $C_{\text{d}\text{i}\text{s}\text{p}}=0\mathrm{.}5.$ Let $t_{}$ denote the time interval between the sale of a product and

its possible resale after being collected and remanufactured. Calculate the average production

cost difference $C($a$)$ when $t_{}=3$ years and $($b$)$ when $t_{}=2$ years.

$(40$ points$)$ 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.

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.

$(30$ points$)$ Consider the two$-$period demand$-$based diffusion model in Lecture $10.$ Now suppose

that the market is partitioned into three $($rather than two$)$ segments according to consumers'

willingness to purchase remanufactured items:

A fraction ${}_1$ of the consumers in period $2$ wants to buy remanufactured items if available

and new items otherwise $($the functionality$-$oriented segment$).$

A fraction ${}_2$ of the consumers in period $2$ wants to buy only remanufactured items $($the

remanufactured$-$item$-$oriented segment$).$

The remaining fraction $1-{}_1-{}_2$ of the consumers in period $2$ wants to buy only new

items $($the newness$-$conscious segment$).$

In period $2,$ the remanufactured$-$item$-$oriented customers execute their product purchases earlier

than the functionality$-$oriented customers.

$($a$)(15$ points$)$ Formulate the total profit in periods $1$ and $2$ as a function of the sales volume

in period $1.$ Assume $m=1$ and use the notation $p,q,,,{}_1,{}_2,r_r,c_r,r_n,$ and $c_n.$

$($b$)(15$ points$)$ Assume $m=1,p=0\mathrm{.}2,q=0\mathrm{.}5,=0\mathrm{.}9,=0\mathrm{.}8,{}_1=0\mathrm{.}25,{}_2=0\mathrm{.}25,$

$r_r-c_r=10,$ and $r_n-c_n=5.$ Plot the total profit in periods $1$ and $2$ versus the sales

volume in period $1($that can take values from the set $\{0\mathrm{.}001,0\mathrm{.}002,$dots,$0\mathrm{.}200\}.$ Find the

optimal sales volume in period $1.$