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 info about the image 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 1 Rooms 1 and 3 Opening 2 Rooms 3 and 7 Opening 3 Rooms 3 and 4 Opening 4 Rooms 1 and 4 Opening 5 Rooms 4 and 7 Opening 6 Rooms 1 and 2 Opening 7 Rooms 4 and 5 Opening 8 Rooms 2 and 5 Opening 9 Rooms 5 and 7 Opening 10 Rooms 5 and 6 Opening 11 Rooms 7 and 8 Opening 12 Rooms 2 and 6 Opening 13 Rooms 6 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 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 The gallery should install cameras with (x 1 , x 2 , x 3 , x 4 , x 5 , x 6 , x 7 , x 8 , x 9 , x 10 , x 11 , x 12 , x 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 (Which constraint to choose from these ) Room1 Room 2 Room 3 Room 4 Room 5 Room 6 Room 7 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 (x 1 , x 2 , x 3 , x 4 , x 5 , x 6 , x 7 , x 8 , x 9 , x 10 , x 11 , x 12 , x 13 )

Question

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    info about the image  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 1  Rooms 1 and 3 Opening 2  Rooms 3 and 7 Opening 3  Rooms 3 and 4 Opening 4  Rooms 1 and 4 Opening 5  Rooms 4 and 7 Opening 6  Rooms 1 and 2 Opening 7  Rooms 4 and 5 Opening 8  Rooms 2 and 5 Opening 9  Rooms 5 and 7 Opening 10  Rooms 5 and 6 Opening 11  Rooms 7 and 8 Opening 12  Rooms 2 and 6 Opening 13  Rooms 6 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 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  The gallery should install         cameras with (x 1 , x 2 , x 3 , x 4 , x 5 , x 6 , x 7 , x 8 , x 9 , x 10 , x 11 , x 12 , x 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    (Which constraint to choose from these  ) Room1 Room 2 Room 3 Room 4 Room 5 Room 6 Room 7 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 (x 1 , x 2 , x 3 , x 4 , x 5 , x 6 , x 7 , x 8 , x 9 , x 10 , x 11 , x 12 , x 13 )

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Question: 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

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