A small city toy store manager forecasts that the demand for the new Lite$-$Brite Super Brite HD Barbie Editions will be increased during holiday season this year. She did some research and received following possibilities for demand. She realizes the Barbie edition is perishable and may not sell after the season! As a consultant you should advise her how many Lite$-$Brite Super Brite HD Barbie Editions she should order to maximize her expected profit?

$$ D $<=500$ with probability $0\mathrm{.}5$

$$ D $<=550$ with probability $0\mathrm{.}75$

$$ The shop buys toys from a distributor at a unit cost of $$15.$

$$ The shop sells toys at a unit price of $$25$

$$ Any unsold toys has a salvage value of $$7.$

$($Note: D is the demand, not the demand rate in EOQ model; and we assumed this specific toy is perishable.$)$

a$)$ What is the value of $501$st toy?

b$)$ What is the value of $551$st toy

c$)$ What is the optimal order number to maximize the profit?

d$)$ Give the exact value of optimal order number to maximize the profit.