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 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 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) = ______ (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? Room 1Room 2 Room 3Room 4Room 5Room 6Room 7Room 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, x12, x13) = _________

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 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 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) = (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? Room 1Room 2 Room 3Room 4Room 5Room 6Room 7Room 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, x12, x13) = _____

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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Question

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 (x1, x2, x3, x4, x5, x6, x7, x8, x9, 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?

Room 1Room 2 Room 3Room 4Room 5Room 6Room 7Room 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, x₁₂, x₁₃) = _________

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