Question 1

If Project 5 must be completed before Project 6, the constraint would be x₅- x₆ $\le$ 0.

Accepted Answer

The answer of If Project 5 must be completed before...

Question 2

Your express package courier company is drawing up new zones for the location of drop boxes for customers. The city has been divided into the seven zones shown below. You have targeted six possible locations for drop boxes. The list of which drop boxes could be reached easily from each zone is listed below. Let x_i = 1 if drop box location i is used, 0 otherwise. Develop a model to provide the smallest number of locations yet make sure that each zone is covered by at least two boxes.

Accepted Answer

The answer of Your express package courier company is drawing...

Question 3

In a model involving fixed costs, the 0 - 1 variable guarantees that the capacity is not available unless the cost has been incurred.

Accepted Answer

In a model involving fixed costs, the 0 - 1 variable is used to represent whether or not the capacity is available. A value of 0 indicates that the capacity is not available and a value of 1 indicates that it is. The fixed cost is incurred regardless of whether or not the capacity is actually used. Therefore, the 0 - 1 variable ensures that the fixed cost has been incurred before the capacity can be considered available.

Question 4

Dual prices cannot be used for integer programming sensitivity analysis because they are designed for linear programs.

Accepted Answer

Dual prices are derived from the simplex algorithm used to solve linear programs and are not applicable to integer programming problems, which have a different solution methodology.

Question 5

Tower Engineering Corporation is considering undertaking several proposed projects for the next fiscal year. The projects, the number of engineers and the number of support personnel required for each project, and the expected profits for each project are summarized in the following table:
  Formulate an integer program that maximizes Tower's profit subject to the following management constraints:

Accepted Answer

The answer of Tower Engineering Corporation is considering undertaking several...

Question 6

If the LP relaxation of an integer program has a feasible solution, then the integer program has a feasible solution.

Accepted Answer

The answer of If the LP relaxation of an integer...

Question 7

The objective of the product design and market share optimization problem presented in the textbook is to choose the levels of each product attribute that will maximize the number of sampled customers preferring the brand in question.

Accepted Answer

This is correct as the goal of the product design and market share optimization problem is indeed to determine the optimal levels of product attributes to maximize the number of customers who prefer the brand.

Question 8

The constraint x₁ + x₂ + x₃ + x₄ F$\le$ 2 means that two out of the first four projects must be selected.

Accepted Answer

The constraint actually means that the sum of the products of xS1U1BijS1U1B0 and F1F1F1S1 for i from 1 to 4 should be less than or equal to F1F1F10. It does not specify that two out of the four projects must be selected.

Question 9

Simplon Manufacturing must decide on the processes to use to produce 1650 units. If machine 1 is used, its production will be between 300 and 1500 units. Machine 2 and/or machine 3 can be used only if machine 1's production is at least 1000 units. Machine 4 can be used with no restrictions.
  (HINT: Use an additional 0 - 1 variable to indicate when machines 2 and 3 can be used.)

Accepted Answer

The answer of Simplon Manufacturing must decide on the processes...

Question 10

Hansen Controls has been awarded a contract for a large number of control panels. To meet this demand, it will use its existing plants in San Diego and Houston, and consider new plants in Tulsa, St. Louis, and Portland. Finished control panels are to be shipped to Seattle, Denver, and Kansas City. Pertinent information is given in the table.
  Develop a model whose solution would reveal which plants to build and the optimal shipping schedule.

Accepted Answer

The answer of Hansen Controls has been awarded a contract...

Question 11

If x₁ + x₂ $\le$500y₁ and y₁ is 0 - 1, then if y₁ is 0, x₁ and x₂ will be 0.

Accepted Answer

If y₁ is 0, then the left side of the inequality becomes x₂ F1F1F1S1≤F1F1F10. Since F1F1F1S1 has a maximum value of 1, the only way for the inequality to hold is if x₂ is also 0. Therefore, both x₁ and x₂ must be 0 if y₁ is 0.

Question 12

The constraint x₁-x₂ = 0 implies that if project 1 is selected, project 2 cannot be.

Accepted Answer

The answer of The constraint x₁-x₂ = 0 implies that...

Question 13

Generally, the optimal solution to an integer linear program is less sensitive to the constraint coefficients than is a linear program.

Accepted Answer

Integer linear programs (ILPs) are generally more sensitive to changes in constraint coefficients than linear programs (LPs) because the feasible region for ILPs is discrete, and small changes can lead to different optimal solutions or make a previously feasible solution infeasible.

Question 14

The classic assignment problem can be modeled as a 0-1 integer program.

Accepted Answer

The classic assignment problem can be formulated as a 0-1 integer program, where we want to assign n tasks to n agents at minimum cost.

Question 15

A multiple choice constraint involves selecting k out of n alternatives, where k $\ge$ 2.

Accepted Answer

The answer of A multiple choice constraint involves selecting k...

Question 16

If a problem has only less-than-or-equal-to constraints with positive coefficients for the variables, rounding down will always provide a feasible integer solution.

Accepted Answer

If all constraints have positive coefficients, rounding down the variables will ensure that each constraint is satisfied, as decreasing the value of a variable can only make the constraint easier to satisfy. Additionally, since we are rounding down, we know that the resulting values will be integers, thus providing a feasible integer solution.

Question 17

The product design and market share optimization problem presented in the textbook is formulated as a 0-1 integer linear programming model.

Accepted Answer

The product design and market share optimization problem presented in the textbook is indeed formulated as a 0-1 integer linear programming model.

Question 18

Grush Consulting has five projects to consider. Each will require time in the next two quarters according to the table below.
  Revenue from each project is also shown. Develop a model whose solution would maximize revenue, meet the time budget of 25 in the first quarter and 20 in the second quarter, and not do both projects C and D.

Accepted Answer

The answer of Grush Consulting has five projects to consider....

Question 19

Multiple choice constraints involve binary variables.

Accepted Answer

Multiple choice constraints involve binary variables, where the variables can take on only one of two possible values, typically 0 or 1.

Question 20

Some linear programming problems have a special structure that guarantees the variables will have integer values.

Accepted Answer

This special structure is called an integer programming problem, which is a type of linear programming problem but with the additional constraint that the variables must take on integer values. Not all linear programming problems have this structure, but some do.

Quiz 7: Integer Linear Programming

If Project 5 must be completed before Project 6, the constraint would be x₅- x₆ $\le$ 0.

In a model involving fixed costs, the 0 - 1 variable guarantees that the capacity is not available unless the cost has been incurred.

Dual prices cannot be used for integer programming sensitivity analysis because they are designed for linear programs.

If the LP relaxation of an integer program has a feasible solution, then the integer program has a feasible solution.

The objective of the product design and market share optimization problem presented in the textbook is to choose the levels of each product attribute that will maximize the number of sampled customers preferring the brand in question.

The constraint x₁ + x₂ + x₃ + x₄ F $\le$ 2 means that two out of the first four projects must be selected.

If x₁ + x₂ $\le$ 500y₁ and y₁ is 0 - 1, then if y₁ is 0, x₁ and x₂ will be 0.

The constraint x₁-x₂ = 0 implies that if project 1 is selected, project 2 cannot be.

Generally, the optimal solution to an integer linear program is less sensitive to the constraint coefficients than is a linear program.

The classic assignment problem can be modeled as a 0-1 integer program.

A multiple choice constraint involves selecting k out of n alternatives, where k $\ge$ 2.

If a problem has only less-than-or-equal-to constraints with positive coefficients for the variables, rounding down will always provide a feasible integer solution.

The product design and market share optimization problem presented in the textbook is formulated as a 0-1 integer linear programming model.

Multiple choice constraints involve binary variables.

Some linear programming problems have a special structure that guarantees the variables will have integer values.

Quiz 7: Integer Linear Programming

If Project 5 must be completed before Project 6, the constraint would be x5- x6 ≤\le≤ 0.

In a model involving fixed costs, the 0 - 1 variable guarantees that the capacity is not available unless the cost has been incurred.

Dual prices cannot be used for integer programming sensitivity analysis because they are designed for linear programs.

If the LP relaxation of an integer program has a feasible solution, then the integer program has a feasible solution.

The objective of the product design and market share optimization problem presented in the textbook is to choose the levels of each product attribute that will maximize the number of sampled customers preferring the brand in question.

The constraint x1 + x2 + x3 + x4 F ≤\le≤ 2 means that two out of the first four projects must be selected.

If x1 + x2 ≤\le≤ 500y1 and y1 is 0 - 1, then if y1 is 0, x1 and x2 will be 0.

The constraint x1-x2 = 0 implies that if project 1 is selected, project 2 cannot be.

Generally, the optimal solution to an integer linear program is less sensitive to the constraint coefficients than is a linear program.

The classic assignment problem can be modeled as a 0-1 integer program.

A multiple choice constraint involves selecting k out of n alternatives, where k ≥\ge≥ 2.

If a problem has only less-than-or-equal-to constraints with positive coefficients for the variables, rounding down will always provide a feasible integer solution.

The product design and market share optimization problem presented in the textbook is formulated as a 0-1 integer linear programming model.

Multiple choice constraints involve binary variables.

Some linear programming problems have a special structure that guarantees the variables will have integer values.

If Project 5 must be completed before Project 6, the constraint would be x₅- x₆ $\le$ 0.

The constraint x₁ + x₂ + x₃ + x₄ F $\le$ 2 means that two out of the first four projects must be selected.

If x₁ + x₂ $\le$ 500y₁ and y₁ is 0 - 1, then if y₁ is 0, x₁ and x₂ will be 0.

The constraint x₁-x₂ = 0 implies that if project 1 is selected, project 2 cannot be.

A multiple choice constraint involves selecting k out of n alternatives, where k $\ge$ 2.