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. Room 3 2 Entrance Room 1 3 Room 4 5 Room 7 7 8 Room 5 9 A blueprint of a gallery with eight rooms, one entrance and thirteen doorways is shown. The gallery is laid out in a grid of three columns and four rows. Some rooms take up more than one row. The left-hand side contains rooms 1 and 2, the middle column rooms 3, 4, 5, and 6, and the right-hand side rooms 7 and 8. The following list contains each doorway and the rooms it connects. Entrance: Room 1 Opening 5: Rooms 4 Opening 10: Rooms 5 Opening 1: Rooms 1 and and 6 and 3 Opening 6: Rooms 1 Opening 11: Rooms 7 Opening 2: Rooms 3 and 2 and 8 and 7 Opening 7: Rooms 4 Opening 12: Rooms 2 Opening 3: Rooms 3 and 5 and 6 and 4 Opening 8: Rooms 2 Opening 13: Rooms 6 Opening 4: Rooms 1 and and 4 Opening 9: Rooms 5 and 7 and 8 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 be the 0-1 which is 1 if a camera is installed at opening and o otherwise, for i = 1, 2, ..., 13.) Min s.t. Room 1 s.t. Room 1 Room 2 Room 3 Room 4 Room 5 Room 6 Room 7 Room 8 X; = 0, 1, for i = 1, 2, ..., 13 (b) ***This Question is optional***** ******No grade Points*********** Solve the model formulated in part (a) to determine how many two-way cameras to purchase and where they should be located. The gallery should install cameras with (X2, X2, X3, X4, X5, X6, X7, X3, X5, X101*111*121*13) = (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? Room 1 Room 2 Room 3 Room 4 OOOOOO Room 5 Room 6 Room 7 Room 8 What should the new constraint be