Consider a new apparel item which retails for $$85,$ costs $$45$ to procure, and can be sold

at a clearance price of $$35$ after it is no longer fashionable.

$($a$)$ Find the optimal stocking quantity and the resulting expected profits if demand D $$ N $(1200,2002).$

$($b$)$ Find the optimal stocking quantity and the resulting expected profits if demand D $$ Uniform$(400,1600).$

Hint: it may be helpful to make use of LU$($a$,$b$)($x$)$ from Problem $2.$

$($c$)$ Now instead consider $100$ retail locations owned by the same company, each with its own $($inde$-$

pendent$)$ demand Di $$ Uniform$(4,16)$; demands across locations are independent. It should be

easy to see that the optimal stocking quantity at each location is $1\%$ of that found in part $($b$)$

and the total expected profits across all locations is the same as that found in part $($b$).$ Briefly

justify why this is the case$\mathrm{.}4$

$($d$)$ Given the same scenario in part $($c$),$ what would change if we could instantaneously transship

the apparel items between any of the $100$ locations? Assuming this is the case, find the optimal

stocking quantity for the shared inventory across all locations.

$($e$)$ Assuming we had to pay a cost for the scenario in part $($d$)$ to be possible $($i$.$e$.,$ to have the

ability, technology, and$/$or right to instantaneously transship the items during this selling season

as opposed to managing the inventory at each location separately$),$ what is the most we should

be willing to pay for this possibility?

$($f$)$ Now consider two locations which can instantaneously pull their demand from the same cen$-$

tral warehouse, one with demand DA $$ N $(1200,2002)$ and the other with demand DB $$

N $(500,1502).$ Find the optimal stocking quantity for the central warehouse assuming a cor$-$

relation of $\backslash$rho $=1.$ Do the same for correlations of $\backslash$rho $=0\mathrm{.}75,\backslash$rho $=0,\backslash$rho $=0\mathrm{.}75,$ and $\backslash$rho $=1.$

$4$Something extra to think about: is the standard deviation of profits also the same as in part $($b$)?$ If you are curious about

this but not quite sure of the answer, feel free to ask about this in class or in office hours.

$3$