Problem $2.$ Three distribution centers face Normally distributed demands $D_1,D_2,$ and $D_3,$

respectively. Demand $D_1$ is Normally distributed with mean ${}_1=22$ and standard deviation

${}_1=8$; demand $D_2$ is Normally distributed with mean ${}_2=19$ and standard deviation ${}_2=4$;

demand $D_3$ is Normally distributed with mean ${}_3=17$ and standard deviation ${}_3=3.$ All

three distribution centers face a Newsvendor problem with overage cost $h=1$$ and underage cost

$p=15$$$.$ Calculate expected cost of the decentralized system.

$($A$)$ Suppose demands across the three distribution centers are independent. Calculate expected

cost of the centralized system.

$($B$)$ Suppose correlation coefficients between all demand pairs are $0\mathrm{.}75.$ Calculate expected cost

of the centralized system.

$($C$)$ Suppose correlation coefficients are ${}_{12}={}_{21}=0\mathrm{.}75$ and ${}_{13}={}_{31}={}_{23}={}_{32}=-0\mathrm{.}75.$

Calculate expected cost of the centralized system.