A book retailer LuvBooks is trying to decide on how many copies of a book to purchase at the start of the

upcoming selling season for its bookstore. The book retails at $$28\mathrm{.}00.$ The publisher sells the book to the

retailer for $$20\mathrm{.}00.$ LuvBooks estimates that demand for this book during the season is normal with a mean of

$100$ and a standard deviation of $42.$ The publisher$$s variable cost per book is $$7\mathrm{.}50.$

Part I $[20$pt$]$: LuvBooks offers a $75\%$ discount on the retail price of all unsold books at the end of the season,

and the discounted books are always selling quickly.

a$.[2$pt$]$ How many copies should LuvBooks order to maximize its expected profit?

b$.[5$pt$]$ Given the order quantity in part a$,$ what is LuvBooks$$ expected profit?

c$.[5$pt$]$ Given the order quantity in part a$,$ what is the publisher$$s expected profit?

d$.[3$pt$]$ If the publisher owns LuvBooks, what is the optimal order quantity?

e$.[5$pt$]$ By comparing the supply chain profit in part d with that in parts b and c$,$ what is the supply

chain efficiency level achieved by ordering the quantity in part a$?[$Hint: the supply chain efficiency

level is a $($decentralized$)$ supply chain$$s profit divided by its centralized supply chain$$s profit.