$2.$ Eric Brown $(35$ points$)$

Eric Brown is responsible for upgrading the wireless network for his employer. He must make sure that each region must be covered by a Node. He has identified seven possible locations to install new nodes for his network. Each node can provide service to different regions within his employer's corporate campus. The cost of installing each node and the regions that can be served by each node are given below:

Node $1$: Regions $1,2,6$; Cost $\backslash$$$750$

Node $2$: Regions $3,4,7,10$; Cost $\backslash$$$900$

Node $3$: Regions $2,3,7,9$; Cost $\backslash$$$1100$

Node $4$: Regions $1,5,6,10$; Cost $\backslash$$$1300$

Node $5$: Regions $2,4,6,8$; Cost $\backslash$$$700$

Node $6$: Regions $4,5,9,10$; Cost $\backslash$$$900$

Node $7$: Regions $1,5,7,8,9$; Cost $\backslash$$$1400$

Hint: See Unit $5$ Practice Problem $\backslash$#$3\backslash$& $4$

$($a$)$ State the decisions to be made, the objective and the constraints in words using or $=$ in the constraints. $($Handwrite answer, $5$ points$)$

$($b$)$ Letting each decision variable $\backslash (=1\backslash ),$ what is the values of the objective and the constraints? $($Handwrite answer, $5$ points$)$

$($c$)$ Define your inputs algebraically as decision variables. Then, using your objective and constraints in part $($a$),$ formulate the problem as an algebraic linear optimization model. $($Handwrite answer. $5$ points$)$

$($d$)$ Using the values in $($b$)$ for your variables, does your algebraic model in $($c$)$ give the same answers as in $($b$)?($Don$\text{'}$t hand in $-$ this is to help you check your model.$)$

$($e$)$ Construct an EXCEL model corresponding to your algebraic model in part $($b$).($Submit via Canvas, $5$ points$)$

$($f$)$ Using the values in $($b$)$ for your variables, does your Excel model in $($e$)$ give the same answers as in $($b$)?($Don$\text{'}$t hand in $-$ this is to help you check your model.$)$

$($g$)$ Optimize your EXCEL model using Solver and give the Answer Report. From the Answer Report, what is the optimal solution and the optimal objective value? $($Submit via Canvas, $5$ pts$)$

$($h$)$ How would your algebraic model change if each Region needed to be covered by at least $2$ Nodes? $($Handwrite answer, $5$ points$)$

$($i$)$ What is the optimal value of the objective in this case? $($Submit via Canvas, $5$ points$)$