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

Dan McClure is trying 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 normal with a mean of $100$ 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 part a$,$ what is the publisher's expected profit?

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

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

$1$

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

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