Sweetie, a chocolate manufacturer, produces heart shaped chocolates for Valentine's Day.

A$102$ makes a single order decision at the end of January every year for that year's Valentine's Day. The

retail price for the chocolate is $$18.$ Sweetie charges A$102$ $$14$ for the chocolate and incurs a production

cost of $$10.$ A stock$-$out situation at A$102$ results in $$9$ and at Sweetie $$12$ per unit. Both Sweetie and A$102$

can sell an unsold chocolate to a $3^{rd}$ party for $$6$ per unit.

a$)$ Write the profit function of $A102$ for a given realization of demand $($hint: $P_A(Q|x)).$

b$)$ Write the expected profit function of $A102($hint: $\{$:$E[P^A(Q)]).$

c$)$ Write the profit function of Sweetie for a given realization of demand.

d$)$ Write the expected profit function of Sweetie.

e$)$ What are the costs of overage and underage for A$102?$

f$)$ Assume that demand for chocolates is uniformly distributed between $1000$ and $2500.$ What is the

optimal order size of A$102?$

g$)$ Calculate the optimal expected profits of the parties under the uniformly distributed demand

assumption.

h$)$ Assume that demand for chocolates is normally distributed with mean $1500$ and variance $900.$

What is the optimal order size of A$102?$

i$)$ Assume that demand for chocolates is exponentially distributed with mean $1500.$ What is the

optimal order size of A$102?$

Suppose that Sweetie changed its current policy and allow A$102$ to return $20\%$ of his initial purchase

for $$8$ and order quantity of A$102$ is $1000.$

j$)$ Write the profit function of A$102$ for a given realization of demand.

k$)$ Write the expected profit function of A$102.$

l$)$ Write the profit function of Sweetie for a given realization of demand.

m$)$ Write the expected profit function of Sweetie.

n$)$ Is this return policy optimal? Why or why not? What should the optimal policy satisfy?