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(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, be the 0-1 which is 1 if a camera is installed at opening /, and 0 otherwise, for /= 1, 2, ..., 13.) Min s.t. Room 1 Room 2 Room 3 Room 4 Room 5 Room 6 Room 7 Room 8 x₁ = 0, 1, for /= 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, X, X, X, X10*11*12*13) (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? O Room 1 O Room 2 O Room 3 O Room 4 O Room 5 O Room 6 O Room 7 O Room 8 What should the new constraint be?

(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, be the 0-1 which is 1 if a camera is installed at opening /, and 0 otherwise, for /= 1, 2, ..., 13.) Min s.t. Room 1 Room 2 Room 3 Room 4 Room 5 Room 6 Room 7 Room 8 x₁ = 0, 1, for /= 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, X, X, X, X10*11*12*13) (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? O Room 1 O Room 2 O Room 3 O Room 4 O Room 5 O Room 6 O Room 7 O Room 8 What should the new constraint be?

Practical Management Science
Practical Management Science
6th Edition
ISBN:9781337406659
Author:WINSTON, Wayne L.
Publisher:WINSTON, Wayne L.
Chapter2: Introduction To Spreadsheet Modeling
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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 through 13. **Diagram Description:** The diagram illustrates a layout of eight display rooms arranged in a grid format. There are openings between rooms, each numbered from 1 to 13, indicating possible locations for camera placement. - **Room Arrangement:** - Room 1 is connected to Room 2, Room 4, and the Entrance. - Room 2 is connected to Room 3 and Room 5. - Room 3 is connected to Room 6. - Room 4 is connected to Room 5 and Room 7. - Room 5 is connected to Room 6, Room 8, and Room 9. - Room 7 is connected to Room 8. - Room 9 is connected to Room 10. - Room 10 is connected to Room 11. - Room 11 is connected to Room 12. - Room 12 is connected to Room 13. **Security System Proposal:** 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.
Transcribed Image Text: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. **Diagram Description:** The diagram illustrates a layout of eight display rooms arranged in a grid format. There are openings between rooms, each numbered from 1 to 13, indicating possible locations for camera placement. - **Room Arrangement:** - Room 1 is connected to Room 2, Room 4, and the Entrance. - Room 2 is connected to Room 3 and Room 5. - Room 3 is connected to Room 6. - Room 4 is connected to Room 5 and Room 7. - Room 5 is connected to Room 6, Room 8, and Room 9. - Room 7 is connected to Room 8. - Room 9 is connected to Room 10. - Room 10 is connected to Room 11. - Room 11 is connected to Room 12. - Room 12 is connected to Room 13. **Security System Proposal:** 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.
### Integer Linear Programming for Camera Placement This exercise involves formulating a 0-1 integer linear programming model to help Bayside's management decide the optimal locations for installing camera systems. The problem is broken into several steps with specific constraints and objectives. #### (a) Formulating the Model - **Objective:** Minimize the total number of cameras. - **Variables:** - \( x_i \): Binary variable, where \( x_i = 1 \) if a camera is installed at opening \( i \); otherwise, \( x_i = 0 \). - \( i = 1, 2, \ldots, 13 \) - **Subject to:** - Each room needs coverage by at least one camera installed in the openings allocated to them. - Openings are associated with rooms as follows: - Room 1 - Room 2 - Room 3 - Room 4 - Room 5 - Room 6 - Room 7 - Room 8 _(Provide separate mathematical constraints for each room as needed.)_ #### (b) Solving the Model - **Goal:** Determine how many two-way cameras to purchase and their locations. - **Solution:** - The gallery should install a specific number of cameras. - Specify which openings \( (x_1, x_2, x_3, x_4, x_5, x_6, x_7, x_8, x_9, x_{10}, x_{11}, x_{12}, x_{13}) \) are covered: - \( \left( \boxed{} \right) \). #### (c) Adjusting for Additional Coverage - **Scenario:** - Management requires additional security for Room 7, aiming to have two cameras cover it. - **Required Change:** - Identify which constraint needs modification: - Options for rooms: - Room 1 - Room 2 - Room 3 - Room 4 - Room 5 - Room 6 - Room 7 - Room 8 - **New Constraint:** - Specify what the new constraint should be to ensure Room 7 has dual coverage. This problem demonstrates the application of linear programming in optimizing resource allocation, ensuring efficient security coverage in a
Transcribed Image Text:### Integer Linear Programming for Camera Placement This exercise involves formulating a 0-1 integer linear programming model to help Bayside's management decide the optimal locations for installing camera systems. The problem is broken into several steps with specific constraints and objectives. #### (a) Formulating the Model - **Objective:** Minimize the total number of cameras. - **Variables:** - \( x_i \): Binary variable, where \( x_i = 1 \) if a camera is installed at opening \( i \); otherwise, \( x_i = 0 \). - \( i = 1, 2, \ldots, 13 \) - **Subject to:** - Each room needs coverage by at least one camera installed in the openings allocated to them. - Openings are associated with rooms as follows: - Room 1 - Room 2 - Room 3 - Room 4 - Room 5 - Room 6 - Room 7 - Room 8 _(Provide separate mathematical constraints for each room as needed.)_ #### (b) Solving the Model - **Goal:** Determine how many two-way cameras to purchase and their locations. - **Solution:** - The gallery should install a specific number of cameras. - Specify which openings \( (x_1, x_2, x_3, x_4, x_5, x_6, x_7, x_8, x_9, x_{10}, x_{11}, x_{12}, x_{13}) \) are covered: - \( \left( \boxed{} \right) \). #### (c) Adjusting for Additional Coverage - **Scenario:** - Management requires additional security for Room 7, aiming to have two cameras cover it. - **Required Change:** - Identify which constraint needs modification: - Options for rooms: - Room 1 - Room 2 - Room 3 - Room 4 - Room 5 - Room 6 - Room 7 - Room 8 - **New Constraint:** - Specify what the new constraint should be to ensure Room 7 has dual coverage. This problem demonstrates the application of linear programming in optimizing resource allocation, ensuring efficient security coverage in a
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**(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 1 - [ ] Room 2 - [ ] Room 3 - [ ] Room 4 - [ ] Room 5 - [ ] Room 6 - [ ] Room 7 - [ ] Room 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 \((x_1, x_2, x_3, x_4, x_5, x_6, x_7, x_8, x_9, x_{10}, x_{11}, x_{12}, x_{13}) =\) \([_____________]\).
Transcribed Image Text:**(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 1 - [ ] Room 2 - [ ] Room 3 - [ ] Room 4 - [ ] Room 5 - [ ] Room 6 - [ ] Room 7 - [ ] Room 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 \((x_1, x_2, x_3, x_4, x_5, x_6, x_7, x_8, x_9, x_{10}, x_{11}, x_{12}, x_{13}) =\) \([_____________]\).
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