Exam 6: Integer Linear Programming

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

A company is planning next month's production. It has to pay a setup cost to produce a batch of X₄'s so if it does produce a batch it wants to produce at least 100 units. Which of the following pairs of constraints show the relationship(s) between the setup variable Y₄ and the production quantity variable X₄?

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. Formulate the ILP for this problem.

An ILP problem has 5 binary decision variables. How many possible integer solutions are there to this problem?

Which of the following are potential pitfalls of using a non-zero integer tolerance factor in the Risk Solver Platform (RSP)?

The objective function value for the ILP problem can never

How is an LP problem changed into an ILP problem?

How are general integrality requirements indicated in the Excel Risk Solver Platform (RSP)?

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

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 Risk Solver Platform (RSP) task pane for the above Excel spreadsheet? Objective Cell: Variables Cells: Constraints Cells:

A company is developing its weekly production plan. The company produces two products, A and B, which are processed in two departments. Setting up each batch of A requires $60 of labor while setting up a batch of B costs $80. Each unit of A generates a profit of $17 while a unit of B earns a profit of $21. The company can sell all the units it produces. The data for the problem are summarized below. What is the appropriate formula to use in cell E8 of the following Excel implementation of the ILP model for this problem?

A cellular phone company wants to locate two new communications towers to cover 4 regions. The company wants to minimize the cost of installing the two towers. The regions that can be covered by each tower site are indicated by a 1 in the following table: Based on this ILP formulation of the problem and the solution (X₁, X₂, X₃, X₄) = (1, 1, 0, 0) what values should go in cells B6:G14 of the following Excel spreadsheet?

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 cells B11:E11 and cells F8:F10?

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 formula should go in cell D15 of the above Excel spreadsheet?

A company is developing its weekly production plan. The company produces two products, A and B, which are processed in two departments. Setting up each batch of A requires $60 of labor while setting up a batch of B costs $80. Each unit of A generates a profit of $17 while a unit of B earns a profit of $21. The company can sell all the units it produces. The data for the problem are summarized below. The decision variables are defined as X_i = the amount of product i produced Y_i = 1 if X_i > 0 and 0 if X_i = 0 What is the objective function for this problem?

A company will be able to obtain a quantity discount on component parts for its three products, X₁, X₂ and X₃ if it produces beyond certain limits. To get the X₁ discount it must produce more than 50 X₁'s. It must produce more than 60 X₂'s for the X₂ discount and 70 X₃'s for the X₃ discount. How many binary variables are required in the formulation of this problem?

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 cells G8:G10?

A cellular phone company wants to locate two new communications towers to cover 4 regions. The company wants to minimize the cost of installing the two towers. The regions that can be covered by each tower site are indicated by a 1 in the following table: Based on this ILP formulation of the problem what formulas should go in cells F6:F14 of the following Excel spreadsheet?

The following ILP is being solved by the branch and bound method. You have been given the initial relaxed IP solution. Complete the entries for the 3 nodes and label the arcs when you branch on X2. Initial solution X₁ = 5.0 X₂ = 7.5 Obj = 550