$(40$ points$)$ I am a retailer. Each week I see random demand, $D_n,$ which I fill if I can. Past data

indicates that $P(D_n=0)=0\mathrm{.}2,P(D_n=1)=0\mathrm{.}3,P(D_n=2)=0\mathrm{.}3,$ and $P(D_n=3)=0\mathrm{.}2.$

Demand that I cannot fill is backlogged. I pay $$1$ per item per period I have on hand, and $$5$

per item per period I have on backlog. I have a supplier in China with lead time $3$ weeks and

unit purchasing cost $$1,$ and a supplier in Baku with lead time $2$ weeks and unit purchasing

cost $$3.$ I will use a single$-$index policy.

$($a$)$ Find the optimal base$-$stock level if $=1.$

$($b$)$ Find the optimal base$-$stock level if $=2.$

$($c$)$ Find the safety stock levels for the single$-$index policies found in parts $($a$)$ and $($b$).$

$($d$)$ Find the average single$-$period costs for the single$-$index policies found in parts $($a$)$ and

$($b$).$ Which one is better?