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. 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 xi be the 0-1 which is 1 if a camera is installed at opening i, and 0 otherwise, for i = 1, 2, ..., 13.) Min x1+x2+x3+x4+x5+x6+x7+x8+x9+x10+x11+x12+x13 s.t.Room 1 x1+x4+x6≥1 Room 2 x6+x8+x12≥1 Room 3 x1+x2+x3≥1 Room 4 x3+x4+x5+x7≥1 Room 5 x7+x8+x9+x10≥1 Room 6 x10+x12+x13≥1 Room 7 x2+x5+x9+x11≥1 Room 8 x11+x13≥1 xi = 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, x8, x9, x10, x11, x12, x13) = x1,x7,x11,x12

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. 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 xi be the 0-1 which is 1 if a camera is installed at opening i, and 0 otherwise, for i = 1, 2, ..., 13.) Min x1+x2+x3+x4+x5+x6+x7+x8+x9+x10+x11+x12+x13 s.t.Room 1 x1+x4+x6≥1 Room 2 x6+x8+x12≥1 Room 3 x1+x2+x3≥1 Room 4 x3+x4+x5+x7≥1 Room 5 x7+x8+x9+x10≥1 Room 6 x10+x12+x13≥1 Room 7 x2+x5+x9+x11≥1 Room 8 x11+x13≥1 xi = 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, x8, x9, x10, x11, x12, x13) = x1,x7,x11,x12

Name: The Bayside Art Gallery is considering installing a video camera secur
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Description: 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

Practical Management Science

6th Edition

ISBN:9781337406659

Author:WINSTON, Wayne L.

Publisher:WINSTON, Wayne L.

Chapter12: Queueing Models

Section: Chapter Questions

Problem 59P

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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

x1+x2+x3+x4+x5+x6+x7+x8+x9+x10+x11+x12+x13

s.t.Room 1

x1+x4+x6≥1

Room 2

x6+x8+x12≥1

Room 3

x1+x2+x3≥1

Room 4

x3+x4+x5+x7≥1

Room 5

x7+x8+x9+x10≥1

Room 6

x10+x12+x13≥1

Room 7

x2+x5+x9+x11≥1

Room 8

x11+x13≥1

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

(x₁, x₂, x₃, x₄, x₅, x₆, x₇, x₈, x₉, x₁₀, x₁₁, x₁₂, x₁₃) =

x1,x7,x11,x12

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