U$$ur Restaurant in K$$z$$lay is trying to decide how many Turkish bagels $($simit$)$ to buy for Saturday morning to serve for the breakfast. The bagels are purchased fresh in the morning from the supplier, Simit Factory $($SF$).$ U$$ur buys the bagels from Simit Factory for $2\mathrm{.}5$ TL$/$bagel and sells them for $5$ TL$/$bagel between $8$AM to $12$:$00$ Noon, during the breakfast time. Unsold bagels are salvaged for $0\mathrm{.}30$ TL$/$bagel after $12$:$00$ Noon. The unit production cost of Simit Factory is $1$TL$/$bagel$.$ U$$ur estimates that the demand for bagels on Saturday mornings is normally distributed with mean $400$ and standard deviation $100.$

a$.$ How many bagels should U$$ur order to maximize his expected profit? What is the expected profit of U$$ur$?$

b$.$ How many bagels should U$$ur order to maximize the total supply chain profit?

Simit Factory is thinking of offering the following deal to U$$ur$.$ At the end of every day, the supplier will buy back the unsold bagels at a pre$-$determined price of $1\mathrm{.}5$TL$/$bagel$.$ Shipment cost is negligible.

c$.$ How many bagels should U$$ur order to maximize his expected profits given the buyback offer? What is the expected profit of U$$ur$?$

d$.$ Assume the supplier is able on average to earn $0\mathrm{.}50$TL on each returned bagel. Given the order quantity in part c$,$ what is the supplier$$s expected profit?