Answer Happy

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. Entrance Room 1 Room 2 1 12 Room 3 Room 4 Room 5 10 Room 6 2 9 13 Room 7 11 Room ® 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 cight rooms using the minimum number of two-way cameras.
(a) Formulate a D-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 0 otherwise, for i=1,2,..., 13.) Min s.t. Room 1 Room 2 Room 3 Roar 4 Room 5 Room 6 Room 7 Roar 8 x = 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. The gallery should install cameras with (x₁, x₂, X3, X4, X5, X6, X7, X8i X₁ X10 X11 X₁2, X13) = ( (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? O Room 1 O Room 2 O Room 3 O Room 4 Ⓒ Room 5 Ⓒ Room 6 O Room 7 O Room 8 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, X8, X9, X10, X11′ ×12, ×13) =