Security System Design. The Bayside Art Gallery is considering installing a video

camera security system to reduce its insurance premiums. A diagram of Bayside's

eight exhibition rooms is shown in the figure in the next page; the openings between

the rooms are numbered $1$ to $13.$ 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.LO $2,LO4,LO5$

a$.$ Formulate a binary integer linear programming model that will enable

Bayside's management to determine the locations for the camera systems.

b$.$ Solve the model formulated in part $($a$)$ to determine how many two$-$way

cameras to purchase and where they should be located.

c$.$ Suppose that management wants to provide additional security coverage for

room $7.$ Specifically, management wants room $7$ to be covered by two cameras.

How would the model you formulated in part $($a$)$ have to change to

accommodate this policy restriction?

d$.$ With the policy restriction specified in part $($c$),$ determine how many two$-$way

camera systems will need to be purchased and where they should be located.

$($A schematic shows a rectangular floor plan. An entrance on the left lead to $8$ rooms. Rooms $1$ and $2$ are of equal size. Room $1$ leads to rooms $3$ and $4.$ Room $2$ leads to rooms $5$ and $6.$ Rooms $3,4,$ and $5$ lead to room $7$ which is the largest room. Room $6$ leads to room $8.)$