Quality Bike Products $($QBP$)$ is a national distributor of bicycle products $($bikes$,$ service parts, and other accessories, some of which are company$-$designed$)$ for small, independent bike shops across the US$.$ The final assembly suppliers for several of their company$-$designed line of bicycles are mainly located in Taiwan. These bicycle suppliers have an average lead$-$time of about $4$ months between the time an order is placed by QBP and the associated shipment is received at QBP$$s distribution center in Bloomington, MN$.$ Because of this long lead$-$time, QBP positions most of its annual bicycle inventory in its distribution center in anticipation of the main selling season, which runs from April to September. As a result, most orders to these bicycle suppliers are placed in early January for delivery in April.

One of QBP$$s bicycle models for the upcoming season is known as the Surley$-007.$ This will be the first offering of this new model. The wholesale price for the Surley$-007($i$.$e$.,$ the price QBP charges bike shops$)$ will be $$750$ per bicycle. Based on an analysis of prior sales history for $$like$-$skus$,$ seasonal demand at this price point for the Surley$-007$ is estimated to be normally distributed with a mean of $10,000$ and a standard deviation of $3,500.$ In terms of expenses, QBP must pay the supplier $$200$ per bicycle ordered as well as an additional $$10$ per bicycle for shipping $($including customs duties, etc.$).$ QBP also incurs a seasonal holding cost, which is accounted as $10\%$ of QBP$$s purchased cost. For the Surley$-007,$ this results in a holding cost of $$200$ x $10\%=$ $$20$ per bicycle. QBP also pays its sales staff a $15\%$ commission, computed again as a $\%$ of QBP$$s purchased cost, for each bicycle sold to a bike shop.

How many Surley$-007$ bicycles should QBP order to maximize its expected profit?$$ Explain your reasoning

What expected profit level does this order quantity imply for QBP$?$

What order quantity would you suggest if QBP wants to support a $90\%$ in$-$stock probability target?