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 1-13. 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.

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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 1-13. 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.

 

Room
1
Entrance
Room
3
Room
Room
4
5
Room
9
5
Room
10
2
12
Room
13
Room
8
Transcribed Image Text:Room 1 Entrance Room 3 Room Room 4 5 Room 9 5 Room 10 2 12 Room 13 Room 8
a. Formulate a 0-1 integer linear programming model that will enable Bayside's management to determine the locations for the camera systems.
1if a camera is located at op ening i
0if not
Let X; =
X1 +
X12 +
Min
X2 +
X3 +
X4 +
X5 +
X6 +
Xg +
X9 +
X10 +
X11 +
X13
X1 +
X4 +
X6
Room 1
X6 +
Xg +
X12
Room 2
X1 +
X2 +
Room 3
X3 +
X4 +
x5 +
Room 4
Xs +
X12 +
X7 +
X9 +
X10
Room 5
X10 +
X13
Room 6
x2 +
Xs +
X9 +
X11
Room 7
X11 +
X13
Room 8
b. Solve the model formulated in part (a) to determine how many two-way cameras to purchase.
Solve the model formulated in part (a) to determine where the two-way cameras should be located.
Transcribed Image Text:a. Formulate a 0-1 integer linear programming model that will enable Bayside's management to determine the locations for the camera systems. 1if a camera is located at op ening i 0if not Let X; = X1 + X12 + Min X2 + X3 + X4 + X5 + X6 + Xg + X9 + X10 + X11 + X13 X1 + X4 + X6 Room 1 X6 + Xg + X12 Room 2 X1 + X2 + Room 3 X3 + X4 + x5 + Room 4 Xs + X12 + X7 + X9 + X10 Room 5 X10 + X13 Room 6 x2 + Xs + X9 + X11 Room 7 X11 + X13 Room 8 b. Solve the model formulated in part (a) to determine how many two-way cameras to purchase. Solve the model formulated in part (a) to determine where the two-way cameras should be located.
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