Helotes Creek Art Gallery is considering installing a video camera security system to reduce its insurance premiums. A diagram of the eight display rooms that Helotes Creek uses for exhibitions is shown in the figure below $($i$.$e$.,$ Rooms A$,$ B$,$ C $...,$ H$)$; the openings between the rooms are numbered $1$ through $11.$

A security firm proposed that two$-$way cameras be installed at some room openings. Each camera has the ability to monitor the two rooms between which the camera is located. For example, if a camera were located at opening number $3,$ rooms A and D would be covered; if a camera were located at opening $10,$ rooms G and H would be covered; and so on$.$ Management decided not to locate a camera system at the entrance to the display rooms. The objective is to provide security coverage for all eight rooms using the minimum number of cameras.

Formulate a $0-1$ integer linear programming model that will enable Helotes Creek $\text{'}$s management to determine the optimal locations for the camera systems. $(32$ Points$)$

Make sure to fully specify the objective function and constraint$($s$).$

$[$Hint: the objective function to minimize the total number of cameras used is specified below.$]$

$[$Hint: In the objective function, x$\_1$ is a $0-1$ decision variable which takes the value of $1$ if a camera is installed at opening $1,$ and $0$ if no camera is installed; x$\_2$ is a $0-1$ decision variable which takes the value of $1$ if a camera is installed at opening $2,$ and $0$ if no camera is installed; x$\_3$ is a $0-1$ decision variable $...]$

$[$Hint: To provide security coverage for Room A$,$ a camera needs to be installed in at least one of the following openings: opening $3,$ opening $5.$ To provide security coverage for Room B$,$ a camera needs to be installed in at least one of the following openings: opening $5,$ opening $7,$ opening $11.$ A similar logic can be applied for Rooms C$,$ D$,$ E$,$ F$,$ G$,$ and H$.$ Hence, you need to specify a constraint for each room.$]$

Minimize x$\_1+$x$\_2+$x$\_3+$x$\_4+$x$\_5+$x$\_6+$x$\_7+$x$\_8+$x$\_9+$x$\_10+$x$\_11$

Subject to:

Use Excel Solver to find a solution for the model formulated in part $($a$).$ What is the total number of cameras that need to be purchased? Where should each of these cameras be located $($i$.$e$.,$ which openings$)?(20$ Points$)$

Suppose that management wants to provide additional security coverage for Room D$.$ Specifically, management wants Room D to be covered by two cameras. Which constraint would have to change? What should the new constraint be$?(8$ Points$)$

With the policy restriction specified in part $($c$),$ what is the total number of cameras that need to be purchased? Where should each of these cameras be located $($i$.$e$.,$ which openings$)?(8$ Points$)$