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Exam 6: Integer Linear Programming

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​One way of solving an ILP problem is to solve its LP relaxation and round the solution up or down to the nearest integer.

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Question 21

A research director must pick a subset of research projects to fund over the next five years. He has five candidate projects, not all of which cover the entire five-year period. Although the director has limited funds in each of the next five years, he can carry over unspent research funds into the next year. Additionally, up to $30K can be carried out of the five-year planning period. The following table summarizes the projects and budget available to the research director. A research director must pick a subset of research projects to fund over the next five years. He has five candidate projects, not all of which cover the entire five-year period. Although the director has limited funds in each of the next five years, he can carry over unspent research funds into the next year. Additionally, up to $30K can be carried out of the five-year planning period. The following table summarizes the projects and budget available to the research director. ​ Define the ILP formulation for this capital budgeting problem. ​ Define the ILP formulation for this capital budgeting problem.

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Question 22

A vendor offers 5 different prices per unit depending on the quantity purchased. How many binary variables are needed to model this discounting scheme?

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Question 23

A company wants to build a new factory in either Atlanta or Columbia. It is also considering building a warehouse in whichever city is selected for the new factory. The following table shows the net present value (NPV) and cost of each facility. The company wants to maximize the net present value of its facilities, but it only has $15 million to invest. Formulate the ILP for this problem. A company wants to build a new factory in either Atlanta or Columbia. It is also considering building a warehouse in whichever city is selected for the new factory. The following table shows the net present value (NPV) and cost of each facility. The company wants to maximize the net present value of its facilities, but it only has $15 million to invest. Formulate the ILP for this problem.

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Question 24

Exhibit 6.1 The following questions pertain to the problem, formulation, and spreadsheet implementation below. A research director must pick a subset of research projects to fund over the next five years. He has five candidate projects, not all of which cover the entire five-year period. Although the director has limited funds in each of the next five years, he can carry over unspent research funds into the next year. Additionally, up to $30K can be carried out of the five-year planning period. The following table summarizes the projects and budget available to the research director. Exhibit 6.1 The following questions pertain to the problem, formulation, and spreadsheet implementation below. A research director must pick a subset of research projects to fund over the next five years. He has five candidate projects, not all of which cover the entire five-year period. Although the director has limited funds in each of the next five years, he can carry over unspent research funds into the next year. Additionally, up to $30K can be carried out of the five-year planning period. The following table summarizes the projects and budget available to the research director. The following is the ILP formulation and a spreadsheet model for the problem. -Refer to Exhibit 6.1. What formulas should go in cells D8:H8 and D11:H11 of the above Excel spreadsheet? The following is the ILP formulation and a spreadsheet model for the problem. Exhibit 6.1 The following questions pertain to the problem, formulation, and spreadsheet implementation below. A research director must pick a subset of research projects to fund over the next five years. He has five candidate projects, not all of which cover the entire five-year period. Although the director has limited funds in each of the next five years, he can carry over unspent research funds into the next year. Additionally, up to $30K can be carried out of the five-year planning period. The following table summarizes the projects and budget available to the research director. The following is the ILP formulation and a spreadsheet model for the problem. -Refer to Exhibit 6.1. What formulas should go in cells D8:H8 and D11:H11 of the above Excel spreadsheet? Exhibit 6.1 The following questions pertain to the problem, formulation, and spreadsheet implementation below. A research director must pick a subset of research projects to fund over the next five years. He has five candidate projects, not all of which cover the entire five-year period. Although the director has limited funds in each of the next five years, he can carry over unspent research funds into the next year. Additionally, up to $30K can be carried out of the five-year planning period. The following table summarizes the projects and budget available to the research director. The following is the ILP formulation and a spreadsheet model for the problem. -Refer to Exhibit 6.1. What formulas should go in cells D8:H8 and D11:H11 of the above Excel spreadsheet? -Refer to Exhibit 6.1. What formulas should go in cells D8:H8 and D11:H11 of the above Excel spreadsheet?

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Question 25

The setup cost incurred in preparing a machine to produce a batch of product is an example of a

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Question 26

​One useful feature of Analytic Solver Platform (ASP) is its ability to generate graphs of the optimization results.

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Question 27

How is an LP problem changed into an ILP problem?

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Question 28

Exhibit 6.2 The following questions pertain to the problem, formulation, and spreadsheet implementation below. A certain military deployment requires supplies delivered to four locations. These deliveries come from one of three ports. Logistics planners wish to deliver the supplies in an efficient manner, in this case by minimizing total ton-miles. The port-destination data, along with destination demand is provided in the following table. Exhibit 6.2 The following questions pertain to the problem, formulation, and spreadsheet implementation below. A certain military deployment requires supplies delivered to four locations. These deliveries come from one of three ports. Logistics planners wish to deliver the supplies in an efficient manner, in this case by minimizing total ton-miles. The port-destination data, along with destination demand is provided in the following table. The ports are supplied by one of two supply ships. These ships travel to a particular port where their supplies are off-loaded and shipped to the requesting destinations. Ship S1 carries 1200 tones of supplies while Ship S2 carries 1120 tons of supplies. These ships can only go to a single port and each port can only accommodate one ship. Assume the costs for a ship to travel to a port are not part of the objective function. The following is the ILP formulation and a spreadsheet model for the problem. -Refer to Exhibit 6.2. What formula would go into cell E14? The ports are supplied by one of two supply ships. These ships travel to a particular port where their supplies are off-loaded and shipped to the requesting destinations. Ship S1 carries 1200 tones of supplies while Ship S2 carries 1120 tons of supplies. These ships can only go to a single port and each port can only accommodate one ship. Assume the costs for a ship to travel to a port are not part of the objective function. The following is the ILP formulation and a spreadsheet model for the problem. Exhibit 6.2 The following questions pertain to the problem, formulation, and spreadsheet implementation below. A certain military deployment requires supplies delivered to four locations. These deliveries come from one of three ports. Logistics planners wish to deliver the supplies in an efficient manner, in this case by minimizing total ton-miles. The port-destination data, along with destination demand is provided in the following table. The ports are supplied by one of two supply ships. These ships travel to a particular port where their supplies are off-loaded and shipped to the requesting destinations. Ship S1 carries 1200 tones of supplies while Ship S2 carries 1120 tons of supplies. These ships can only go to a single port and each port can only accommodate one ship. Assume the costs for a ship to travel to a port are not part of the objective function. The following is the ILP formulation and a spreadsheet model for the problem. -Refer to Exhibit 6.2. What formula would go into cell E14? Exhibit 6.2 The following questions pertain to the problem, formulation, and spreadsheet implementation below. A certain military deployment requires supplies delivered to four locations. These deliveries come from one of three ports. Logistics planners wish to deliver the supplies in an efficient manner, in this case by minimizing total ton-miles. The port-destination data, along with destination demand is provided in the following table. The ports are supplied by one of two supply ships. These ships travel to a particular port where their supplies are off-loaded and shipped to the requesting destinations. Ship S1 carries 1200 tones of supplies while Ship S2 carries 1120 tons of supplies. These ships can only go to a single port and each port can only accommodate one ship. Assume the costs for a ship to travel to a port are not part of the objective function. The following is the ILP formulation and a spreadsheet model for the problem. -Refer to Exhibit 6.2. What formula would go into cell E14? -Refer to Exhibit 6.2. What formula would go into cell E14?

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Question 29

A company wants to select no more than 2 projects from a set of 4 possible projects. Which of the following constraints ensures that no more than 2 will be selected?

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Question 30

A company wants to build a new factory in either Atlanta or Columbia. It is also considering building a warehouse in whichever city is selected for the new factory. The following table shows the net present value (NPV) and cost of each facility. The company wants to maximize the net present value of its facilities, but it only has $16 million to invest. A company wants to build a new factory in either Atlanta or Columbia. It is also considering building a warehouse in whichever city is selected for the new factory. The following table shows the net present value (NPV) and cost of each facility. The company wants to maximize the net present value of its facilities, but it only has $16 million to invest. Based on this ILP formulation of the problem and the indicated optimal solution what formulas should go in cells F6:F14 of the following Excel spreadsheet? ​ Based on this ILP formulation of the problem and the indicated optimal solution what formulas should go in cells F6:F14 of the following Excel spreadsheet? A company wants to build a new factory in either Atlanta or Columbia. It is also considering building a warehouse in whichever city is selected for the new factory. The following table shows the net present value (NPV) and cost of each facility. The company wants to maximize the net present value of its facilities, but it only has $16 million to invest. Based on this ILP formulation of the problem and the indicated optimal solution what formulas should go in cells F6:F14 of the following Excel spreadsheet? ​ ​ A company wants to build a new factory in either Atlanta or Columbia. It is also considering building a warehouse in whichever city is selected for the new factory. The following table shows the net present value (NPV) and cost of each facility. The company wants to maximize the net present value of its facilities, but it only has $16 million to invest. Based on this ILP formulation of the problem and the indicated optimal solution what formulas should go in cells F6:F14 of the following Excel spreadsheet? ​

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Question 31

The branch & bound algorithm stops when:

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Question 32

The feasible region for the pure ILP problem

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Question 33

The feasible region for the pure ILP problem

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Question 34

A small town wants to build some new recreational facilities. The proposed facilities include a swimming pool, recreation center, basketball court and baseball field. The town council wants to provide the facilities which will be used by the most people, but faces budget and land limitations. The town has $400,000 and 14 acres of land. The pool requires locker facilities which would be in the recreation center, so if the swimming pool is built the recreation center must also be built. Also the council has only enough flat land to build the basketball court or the baseball field. The daily usage and cost of the facilities (in $1,000) are shown below. A small town wants to build some new recreational facilities. The proposed facilities include a swimming pool, recreation center, basketball court and baseball field. The town council wants to provide the facilities which will be used by the most people, but faces budget and land limitations. The town has $400,000 and 14 acres of land. The pool requires locker facilities which would be in the recreation center, so if the swimming pool is built the recreation center must also be built. Also the council has only enough flat land to build the basketball court or the baseball field. The daily usage and cost of the facilities (in $1,000) are shown below. Based on this ILP formulation of the problem and the indicated optimal solution what values should go in cells B5:G12 of the following Excel spreadsheet? ​ Based on this ILP formulation of the problem and the indicated optimal solution what values should go in cells B5:G12 of the following Excel spreadsheet? A small town wants to build some new recreational facilities. The proposed facilities include a swimming pool, recreation center, basketball court and baseball field. The town council wants to provide the facilities which will be used by the most people, but faces budget and land limitations. The town has $400,000 and 14 acres of land. The pool requires locker facilities which would be in the recreation center, so if the swimming pool is built the recreation center must also be built. Also the council has only enough flat land to build the basketball court or the baseball field. The daily usage and cost of the facilities (in $1,000) are shown below. Based on this ILP formulation of the problem and the indicated optimal solution what values should go in cells B5:G12 of the following Excel spreadsheet? ​ ​ A small town wants to build some new recreational facilities. The proposed facilities include a swimming pool, recreation center, basketball court and baseball field. The town council wants to provide the facilities which will be used by the most people, but faces budget and land limitations. The town has $400,000 and 14 acres of land. The pool requires locker facilities which would be in the recreation center, so if the swimming pool is built the recreation center must also be built. Also the council has only enough flat land to build the basketball court or the baseball field. The daily usage and cost of the facilities (in $1,000) are shown below. Based on this ILP formulation of the problem and the indicated optimal solution what values should go in cells B5:G12 of the following Excel spreadsheet? ​

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Question 35

A company wants to build a new factory in either Atlanta or Columbia. It is also considering building a warehouse in whichever city is selected for the new factory. The following table shows the net present value (NPV) and cost of each facility. The company wants to maximize the net present value of its facilities, but it only has $16 million to invest. A company wants to build a new factory in either Atlanta or Columbia. It is also considering building a warehouse in whichever city is selected for the new factory. The following table shows the net present value (NPV) and cost of each facility. The company wants to maximize the net present value of its facilities, but it only has $16 million to invest. Based on this ILP formulation of the problem what is the optimal solution to the problem? Based on this ILP formulation of the problem what is the optimal solution to the problem? A company wants to build a new factory in either Atlanta or Columbia. It is also considering building a warehouse in whichever city is selected for the new factory. The following table shows the net present value (NPV) and cost of each facility. The company wants to maximize the net present value of its facilities, but it only has $16 million to invest. Based on this ILP formulation of the problem what is the optimal solution to the problem?

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Question 36

Exhibit 6.1 The following questions pertain to the problem, formulation, and spreadsheet implementation below. A research director must pick a subset of research projects to fund over the next five years. He has five candidate projects, not all of which cover the entire five-year period. Although the director has limited funds in each of the next five years, he can carry over unspent research funds into the next year. Additionally, up to $30K can be carried out of the five-year planning period. The following table summarizes the projects and budget available to the research director. Exhibit 6.1 The following questions pertain to the problem, formulation, and spreadsheet implementation below. A research director must pick a subset of research projects to fund over the next five years. He has five candidate projects, not all of which cover the entire five-year period. Although the director has limited funds in each of the next five years, he can carry over unspent research funds into the next year. Additionally, up to $30K can be carried out of the five-year planning period. The following table summarizes the projects and budget available to the research director. The following is the ILP formulation and a spreadsheet model for the problem. -Refer to Exhibit 6.1. What values would you enter in the Analytic Solver Platform task pane for the above Excel spreadsheet? ​ Objective Cell: ​ Variables Cells: ​ Constraints Cells: The following is the ILP formulation and a spreadsheet model for the problem. Exhibit 6.1 The following questions pertain to the problem, formulation, and spreadsheet implementation below. A research director must pick a subset of research projects to fund over the next five years. He has five candidate projects, not all of which cover the entire five-year period. Although the director has limited funds in each of the next five years, he can carry over unspent research funds into the next year. Additionally, up to $30K can be carried out of the five-year planning period. The following table summarizes the projects and budget available to the research director. The following is the ILP formulation and a spreadsheet model for the problem. -Refer to Exhibit 6.1. What values would you enter in the Analytic Solver Platform task pane for the above Excel spreadsheet? ​ Objective Cell: ​ Variables Cells: ​ Constraints Cells: Exhibit 6.1 The following questions pertain to the problem, formulation, and spreadsheet implementation below. A research director must pick a subset of research projects to fund over the next five years. He has five candidate projects, not all of which cover the entire five-year period. Although the director has limited funds in each of the next five years, he can carry over unspent research funds into the next year. Additionally, up to $30K can be carried out of the five-year planning period. The following table summarizes the projects and budget available to the research director. The following is the ILP formulation and a spreadsheet model for the problem. -Refer to Exhibit 6.1. What values would you enter in the Analytic Solver Platform task pane for the above Excel spreadsheet? ​ Objective Cell: ​ Variables Cells: ​ Constraints Cells: -Refer to Exhibit 6.1. What values would you enter in the Analytic Solver Platform task pane for the above Excel spreadsheet? ​ Objective Cell: ​ Variables Cells: ​ Constraints Cells:

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Question 37

A company has four projects, numbered 1 through 4. If any project is selected for implementation, each lower-numbered project must also be selected for implementation. Formulate the constraints to enforce these conditions.

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Question 38

A production company wants to ensure that if Product 1 is produced, production of Product 1 not exceed production of Product 2. Which of the following constraints enforce this condition?

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Question 39

For minimization problems, the optimal objective function value to the LP relaxation provides what for the optimal objective function value of the ILP problem?

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Question 40
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