Galaxy Cloud Services operates several data centers across the United States that contain servers which 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 Suppose that Galaxy Cloud Services currently has five outdated data centers: one each in Michigan, Ohio, and California and two in New York.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 is as follows:

Data Center Cost $($$ millions$)$ Secure Servers Super Secure Servers

Michigan $2\mathrm{.}55030$

New York $13\mathrm{.}58040$

New York $23\mathrm{.}54080$

Ohio $49060$

California $22030$

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. If required, round your answers to one decimal place.

Min x$1+$ x$2+$ x$3+$ x$4+$ x$5$

s$.$t$.$

x$1+$ x$2+$ x$3+$ x$4+$ x$5$ Secure Server

x$1+$ x$2+$ x$3+$ x$4+$ x$5$ Super Secure Server

x$1,$ x$2,$ x$3,$ x$4,$ x$5=0,1$

$($b$)$ Solve the model formulated in part $($a$)$ to provide a recommendation for management. If required, round your answers to one decimal place.

Optimal solution:

Total Cost: $ Millions