Problem $1$: ABC store sells chocolate gift boxes for Valentine's Day every year. The order needs to be

placed two months in advance.

Suppose that ABC store pays $$25$ for each box that they order, and ABC store sells the chocolate gift boxes

at $$40$ each. After Valentine's Day, ABC store can sell all unsold boxes of chocolate at $50\%$ off. In case of a

stock$-$out, $50\%$ of the customers buy a "hello kitty" doll instead, and the rest of the customers buy nothing.

The ABC store pays $$20$ for each hello kitty, and sells for $$25$ each. Since the hello kitty dolls are non$-$

perishable, the campus bookstore holds a very large inventory of hello kitty to satisfy all demand before or on

Valentine's day. Assume that the total demand for the chocolate gift boxes before or on Valentine's day is

Normally distributed with mean $200$ and standard deviation $50.$

a$)$ With the goal of profit maximization, how many boxes should be ordered by ABC store two months

in advance?

b$)$ In this case, what is the expected profit of the chocolate boxes?

c$)$ In this case, what is the expected profit of the hello kitty dolls that are from the stockout of the

chocolate boxes?

d$)$ This year, the supplier of the chocolate boxes offers an additional delivery option on Valentine's Day

for an extra per$-$unit premium p$.$ By then, ABC store will know for certain what the demand will be$.$

Assume that inventory from the additional delivery can satisfy demand at any point in time. What is

the maximum premium $p$ that ABC store would be willing to pay to exercise this option?