Q$1(40$ points$)$: The UTD cafeteria offers scones for $$2\mathrm{.}50$ each from $8$ am to $3$ pm $.$ The

scones are ordered from their supplier at the start of each day and delivered before the store

opens. The supplier charges $75$ cents per scone. If$,$ at $3$ pm $,$ some scones are left unsold,

they are sold back to the supplier for $50$ cents each. If a customer asks for a scone before $3$

pm but the cafeteria has run out, the customer always buys a bag of chips for $$1$ instead.

$($Assume the chips are always in stock and they are purchased from the same supplier for

$40$ cents each.$)$ Demand for scones before $3$ pm at the cafeteria is variable but can only take

values between $40$ and $50,$ with probabilities given in the following table.

a$.$ What is the expected number of customers who want to buy a scone on a given day?

$(5$ points$)$

a$.$ What is the expected number of customers who want to buy a scone on a given day?

b$.$ Yesterday the cafeteria ordered $45$ scones and $49$ customers came wanting to buy a

scone between $8$ am and $3$ pm $.$ What was the profit $($including$,$ if any, the profit on

chips sold instead of scones$)?$

c$.$ How much is the understocking cost?

d$.$ How much is the overstocking cost?

e$.$ What is the optimal number of scones to order at the start of each day?