Dan$$s Independent Book Store is trying to decide on how many copies of a book to purchase at the start of the upcoming selling season. The book retails at $$28,$ and the publisher$$s production cost is $$15$ per book. At the end of the season, Dan can dispose of all of the unsold copies of the book at $50\%$ off the retail price $($i$.$e$.,$ $$14$ each$),$ whereas the publisher can dispose of all of the unsold copies they buy back from Dan $($if any$)$ at $$10$ each. Dan estimates that demand for this book during the season is Normal with a mean of $1000$ and a standard deviation of $250.$

a$.$ What is the order quantity if $($hypothetically$)$ a single firm owned both the publisher and Dan$$s Book Store? What is the profit for this combined firm? $($You may assume that the combined firm will dispose of unsold copies at the larger salvage value, i$.$e$.,$ $$14$ per unit.$)$

The publisher has offered the following two schemes to Dan.

$($i$)$ The publisher sells the book to Dan at $$22.$ At the end of the season, they will not buy back the unsold copies.

$($ii$)$ The publisher sells the book to Dan at $$20.$ At the end of the season, they will buy back unsold copies at a pre$-$determined price of $$16.$

b$.$ What is the quantity that Dan should order under each scheme, to maximize his expected profits?

c$.$ What is Dan$$s profit under the two schemes? Which scheme would Dan prefer?

d$.$ What is the publisher$$s profit under Dan$$s preferred scheme in part c$?$

e$.$ What is the overall supply chain$$s$,$ i$.$e$.,$ Dan$$s and the publisher$$s$,$ profit under Dan$$s preferred scheme in part c$?$

Please focus on the parts B to E