Problem $3$: The Bayside Art Gallery is considering installing a video camera security system to reduce its

insurance premiums. A diagram of the eight display rooms that Bayside uses for exhibitions is shown in Figure

$7\mathrm{.}13$ ; the openings between the rooms are numbered $1$ through $13.$ 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 $4,$ rooms $1$ and $4$ would be covered;

if a camera were located at opening $11,$ rooms $7$ and $8$ 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 two$-$way cameras.

Formulate a binary $(0-1)$ integer linear programming model that will enable Bayside's management to determine

the locations for the camera systems.