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

$\backslash$sigma $1=8$; demand D$2$ is Normally distributed with mean $2=19$ and standard deviation $\backslash$sigma $2=4$;

demand D$3$ is Normally distributed with mean $3=17$ and standard deviation $\backslash$sigma $3=3.$ 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 $\backslash$rho $12=\backslash$rho $21=0\mathrm{.}75$ and $\backslash$rho $13=\backslash$rho $31=\backslash$rho $23=\backslash$rho $32=0\mathrm{.}75.$

Calculate expected cost of the centralized system.