The Bayside Art Gallery is considering installing a video camera security system to reduce its insurance premiums. A dia
Posted: Tue Jul 05, 2022 11:42 am
The Bayside Art Gallery is considering installing a video camerasecurity system to reduce its insurance premiums. A diagram of theeight display rooms that Bayside uses for exhibitions is shown inthe figure below; the openings between the rooms are numbered 1through 13.
A blueprintof a gallery with eight rooms, one entrance and thirteen doorwaysis shown. The gallery is laid out in a grid of three columns andfour rows. Some rooms take up more than one row. The left-hand sidecontains rooms 1 and 2, the middle column rooms 3, 4, 5, and 6, andthe right-hand side rooms 7 and 8. The following list contains eachdoorway and the rooms it connects.
A security firm proposed that two-way cameras be installed atsome room openings. Each camera has the ability to monitor the tworooms between which the camera is located. For example, if a camerawere 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 becovered; and so on. Management decided not to locate a camerasystem at the entrance to the display rooms. The objective is toprovide security coverage for all eight rooms using the minimumnumber of two-way cameras.
(a)
Formulate a 0-1 integer linear programming model that willenable Bayside's management to determine the locations for thecamera systems. (Let xi bethe 0-1 which is 1 if a camera is installed atopening 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
xi = 0, 1, for i = 1,2, , 13
(b)
Solve the model formulated in part (a) to determine how manytwo-way cameras to purchase and where they should be located.
The gallery should install cameraswith (x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13)=
.
(c)
Suppose that management wants to provide additional securitycoverage for room 7. Specifically, management wants room 7 to becovered by two cameras. Which constraint would have to change?
Room 1Room 2 Room 3Room 4Room 5Room 6Room7Room 8
What should the new constraint be?
(d)
With the policy restriction specified in part (c), determine howmany two-way camera systems will need to be purchased and wherethey will be located.
The gallery should install cameraswith (x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13)=
.
A blueprintof a gallery with eight rooms, one entrance and thirteen doorwaysis shown. The gallery is laid out in a grid of three columns andfour rows. Some rooms take up more than one row. The left-hand sidecontains rooms 1 and 2, the middle column rooms 3, 4, 5, and 6, andthe right-hand side rooms 7 and 8. The following list contains eachdoorway and the rooms it connects.
A security firm proposed that two-way cameras be installed atsome room openings. Each camera has the ability to monitor the tworooms between which the camera is located. For example, if a camerawere 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 becovered; and so on. Management decided not to locate a camerasystem at the entrance to the display rooms. The objective is toprovide security coverage for all eight rooms using the minimumnumber of two-way cameras.
(a)
Formulate a 0-1 integer linear programming model that willenable Bayside's management to determine the locations for thecamera systems. (Let xi bethe 0-1 which is 1 if a camera is installed atopening 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
xi = 0, 1, for i = 1,2, , 13
(b)
Solve the model formulated in part (a) to determine how manytwo-way cameras to purchase and where they should be located.
The gallery should install cameraswith (x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13)=
.
(c)
Suppose that management wants to provide additional securitycoverage for room 7. Specifically, management wants room 7 to becovered by two cameras. Which constraint would have to change?
Room 1Room 2 Room 3Room 4Room 5Room 6Room7Room 8
What should the new constraint be?
(d)
With the policy restriction specified in part (c), determine howmany two-way camera systems will need to be purchased and wherethey will be located.
The gallery should install cameraswith (x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13)=
.