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) =

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.

Chapter2: Introduction To Spreadsheet Modeling

Section: Chapter Questions

Problem 20P: Julie James is opening a lemonade stand. She believes the fixed cost per week of running the stand...

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Question

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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