Department Gallery is considering installing a video camera security system to reduce remiums. A diagram of the eight display rooms that Bayside uses for exhibiin 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.

a$.$ Formulate a $0-1$ 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 your model 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 will be located.