$.$

Teddy Bower is an outdoor clothing and accessories chain that purchases a line or

parkas at $$10$ each from its Asian supplier, TeddySports. Unfortunately, at the time of order placement, demand is still uncertain. Teddy Bower forecasts that its demand is

normally distributed with mean of$2100$ and standard deviation of $1200.$ Teddy Bower sells this parkas at $$22$ each. Unsold parkas have little salvage value; Teddy Bower

virtually gives them away to charity.

$(1)$ How many parkas should Teddy Bower buy from TeddySports to maximize its

expected profit?

$($ii$)$ If Teddy Bower wishes to ensure $98\mathrm{.}5\%$ fill rate, how many parkas should it order?

$($iii$)$ IfTeddy Bower wishes to ensure $98\mathrm{.}5\%$ in$-$stock probability, how many parkas should it order?

Assume Teddy Bower orders $3000$ parkas.

$($iv$)$Evaluate Teddy Bower's expected profit.

$($v$)$ Evaluate Teddy Bower's fill rate and stockout probability.

$[$D$(0\mathrm{.}11)=0\mathrm{.}5438$; D$0\mathrm{.}12)=0\mathrm{.}5478$; D $(2.17)=0.9850$ ; D$(0\mathrm{.}75)=0\mathrm{.}7734$; L$(1\mathrm{.}54)=0\mathrm{.}0267$; L$(1\mathrm{.}55)=0\mathrm{.}0261$; L$(0\mathrm{.}75)=0\mathrm{.}1312]$