Consider the following demand scenario:

Quantity Probability

$20003\%$

$21008\%$

$220015\%$

$230030\%$

$240017\%$

$250012\%$

$260010\%$

$27005\%$

The manufacturer produces at a cost of $$20/$unit$.$ The distributor sells to customers for

$$50/$unit during season and unsold units are sold for $$10/$unit after season. Suppose

the manufacturer is make$-$to$-$order $($i$.$e$.,$ the distributor must order before it receives

demand from end customers$).$

a$)$ What is the system optimal production quantity and the supply chain expected

profit under global optimization?

b$)$ Plot the supply chain$$s average profit for production quantities between $2000$ and

$2700$ units and confirm your answer to part a$.$

c$)$ Suppose the manufacturer sells to the distributor at $$40/$unit$,$ how much will the

distributor order? What is the expected profit for the manufacturer and distributor?

d$)$ Find a buyback contract such that it is coordinating. i$.$e$,$ the distributor$$s order

quantity is the same as the global optimum solution.

e$)$ Find a buyback contract such that both the manufacturer and distributor enjoy a

higher expected profit than $($c$).$ What is the expected profit for the manufacturer

and the distributor?