I am a retailer. Each week I see random demand, Dn$,$ which I fill if I can. Past data indicatesthatP$($Dn $=0)=0\mathrm{.}2,$P$($Dn $=1)=0\mathrm{.}3,$P$($Dn $=2)=0\mathrm{.}3,$andP$($Dn $=3)=0\mathrm{.}2.$ 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?