Supply Chain Coordination $(10$ points$)$:

Dan McClure is tying to decide on how many copies of a book to purchase at the start of the upcoming selling season for his bookstore. The book retails at $$28\mathrm{.}00.$ The publisher sells the book to Dan for $$20\mathrm{.}00.$ Dan will dispose of all of the unsold copies of the book at $75$ percent off the retail price, at the end of the season. Dan estimates that demand for this book during the season is nomal with a mean $01100$ and a standard deviation of $42.$

a$.$ How many books should Dan order to maximize his expected profit?

b$.$ Given the order quantity in part a$,$ what is Dan's expected profit?

c$.$ The publisher's variable cost per book is $$7\mathrm{.}50.$ Given the order quantity in pate a what is the publisher's expected profit?

The publisher is thinking of of ereing the following deal to Dan. At the end of the season, the publisher will buy back unso f copies at a predetermined price of $$15\mathrm{.}00.$ However, Dan would have to bear the costs of shipping unsold copies back to the publisher at $$1\mathrm{.}00$ per copy.

d$.$ How many books should Dan order to maximize his expected profits given the buy back ofim?

$1$

c$.$ Given the order quantity in part d $,$ what is Dan's expected profit?

Assume the publisher is able on average to cam $$6$ on cach returned book net the publisher's handling costs $($some books are destroyed while others are sold at a discount and othess are sold at full price$).$ Given the order quantity in part $d,$ what is the publisher's expected profit?