$(30$ 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 $1.$