Galaxy Cloud Services operates several data centers across the United States containing servers that store and process the data on the Internet. Suppose that Galaxy Cloud Services currently has five outdated data centers: one each in Michigan, Ohio, and California and two in New York. Management is considering increasing the capacity of these data centers to keep up with increasing demand. Each data center contains servers that are dedicated to Secure data and to Super Secure data. The cost to update each data center and the resulting increase in server capacity for each type of server are as follows.

Data Center

NumberData CenterCost $($$ millions$)$Secure ServersSuper Secure

Servers$1$Michigan$4\mathrm{.}090602$New York $13\mathrm{.}580403$New York $22\mathrm{.}050304$Ohio$2\mathrm{.}020305$California$3\mathrm{.}54080$

The projected needs are for a total increase in capacity of $90$ Secure servers and $90$ Super Secure servers. Management wants to determine which data centers to update to meet projected needs and, at the same time, minimize the total cost of the added capacity.

$($a$)$

Formulate a binary integer programming model that could be used to determine the optimal solution to the capacity increase question facing management.

Let xi $=$

$1$ if data center i is updated$0$ otherwise

for

i $=1,2,3,4,5.$

Give your objective function in millions of

dollars.

Min

$4\mathrm{.}0$x$1+3\mathrm{.}5$x$2+2\mathrm{.}0$x$3+2\mathrm{.}0$x$4+3\mathrm{.}5$x$5$

s$.$t$.$Secure

$90$x$1+80$x$2+50$x$3+20$x$4+40$x$590$

Super Secure

$60$x$1+40$x$2+30$x$3+30$x$4+80$x$590$

$($b$)$

Solve the model formulated in part $($a$)$ to provide a recommendation for management. Which data center$($s$)$ do you recommend Galaxy Cloud Services update? $($Select all that apply.$)$

MichiganNew York $1$New York $2$OhioCalifornia