$(20$ points$)$ Consider the distribution network of a durable goods producer that produces washing machines and dishwashers. The two products are produced in separate plants and shipped to a central distribution center $($CDC$).$ The CDC then ships products to two regional distribution centers, RDC$-$West and RDC$-$East.

The distance between the CDC and RDC$-$West is $50km$ whereas that with RDC$-$East is $250km.$

The demand for the two products in each region is quite symmetrical and the size, volume etc. of the products are also quite similar so that all planning is made for their combined demand: weekly demand at RDC$-$West is normally distributed with a mean $100$ combined units and standard deviation $10.$ Weekly demand at RDC$-$East is normally distributed with a mean $20$ combined units and standard deviation $10.$

$ss=z\sqrt[\phantom{2}]{L}{}_D,orss=z\sqrt[\phantom{2}]{L}{}_D^C,{}_D^C=\sqrt[\phantom{2}]{{\displaystyle \underset{i=1}{\overset{k}{}}}{}_{D_i}^2+2{}_{ij}{}_i{}_j}$

a$.(4$ points$)$ First focus on safety inventory. Assuming that the target cycle service level is $0\mathrm{.}95($i$.$e$.z=1\mathrm{.}96),$ and replenishment lead time is $1$ week for RDC$-$West and $2$ weeks for RDC$-$East, what is the safety inventory that would be kept at the each RDC$?$ Show your work.

b$.(4$ points$)$ If you could consolidate $($aggregate$)$ the safety stock of the regions at the CDC$,$ what would be the safety stock needed at the CDC$,$ assuming demands in the regions are independent and replenishment lead time is $2$ weeks $($show work$)?$ How does this compare to the total safety stock you calculated in part a$,$ explain.

c$.(4$ points$)$ Recalculate your answer in b$.($if you don't know how to calculate provide a qualitative answer for partial credit$)$ Assuming the correlation coefficient between East and West region demands is ${}_{\text{E}\text{a}\text{s}\text{t},\text{W}\text{e}\text{s}\text{t}}=-0\mathrm{.}5.$ Explain what you observe.