P$2(40$ points$)$: Nike has signed a quantity flexibility contract with a retailer for a seasonal product. If the retailer orders $\backslash ($ O $\backslash )$ units, Nike is willing to provide up to an additional $35$ percent if necessary. Nike's production cost is $\backslash (\backslash$$ $20\backslash ),$ and it charges the retailer a wholesale price of $\backslash (\backslash$$ $36\backslash ).$ The retailer sets the price for customers at $\backslash (\backslash$$ $55\backslash )$ per unit. Any unsold units can be sold by the retail at a salvage value of $\backslash (\backslash$$ $25\backslash ).$ Nike can only recover $\backslash (\backslash$$ $10\backslash )$ per unit for its leftover inventory. The retailer forecasts demand to be normally distributed with a mean of $4,000$ and a standard deviation $1,600.$

a$.$ How many units $\backslash ($ O $\backslash )$ should the retailer order?

b$.$ What is the expected quantity purchased by the retailer $($considering the retailer can increase the order by up to $\backslash (35\backslash \%\backslash )$ after observing demand$)?$

c$.$ What is the expected quantity sold by the retailer?

d$.$ What is the retailer's expected excess inventory?

e$.$ What is the retailer's expected profit?

f$.$ What is Nike's expected profit?