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 the figure below; 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. (Let x_i be the 0-1 which is 1 if a camera is installed at opening i, and 0 otherwise, for i = 1, 2, ..., 13.)

Min

s.t.Room 1

Room 2

Room 3

Room 4

Room 5

Room 6

Room 7

Room 8

x_i = 0, 1, for i = 1, 2, , 13

(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. Which constraint would have to change? What should the new constraint be?

(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.

The gallery should install cameras with (X1, X2, X3, X4, X5, X6, X7, Xg, X9, X10, X11, X12, X13) = The gallery should install cameras with (X1, X2, X3, X4, X5, X6, X7, X3, X5, X10, X11, X12, X13) =