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Integer Linear Programming: True/False and Multiple Choice Questions, Exams of Mathematics

Walden UniversityMathematics

A series of true/false and multiple-choice questions related to integer linear programming. It covers topics such as lp relaxation, rounding of decision variables, the role of slack and surplus variables, and applications involving 0-1 variables. The questions assess understanding of modeling flexibility, fixed costs, and constraints in integer linear programs. It also touches on the sensitivity of solutions and the characteristics of assignment problems, providing a comprehensive review of key concepts in integer linear programming. This material is useful for students studying optimization techniques and operations research, offering a practical way to test their knowledge and comprehension of the subject matter. Questions that test understanding of the theoretical aspects and practical applications of integer linear programming, making it a valuable resource for exam preparation and self-assessment.

Typology: Exams

2024/2025

Available from 08/28/2025

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CH 07 - Integer Linear Programming
Page 1
True / False
1. The LP Relaxation contains the objective function and constraints of the IP
problem, but drops all integer restrictions.
a.
True
b.
False
ANSWER:
POINTS:
DIFFICULTY:
LEARNING OBJECTIVES:
NATIONAL STANDARDS:
TOPICS:
KEYWORDS:
2. In general, rounding large values of decision variables to the nearest integer value
causes fewer problems than rounding small values.
a.
True
b.
False
ANSWER:
POINTS:
DIFFICULTY:
LEARNING OBJECTIVES:
NATIONAL STANDARDS:
TOPICS:
KEYWORDS:
3. If the optimal solution to the LP Relaxation problem is an integer, it is the
optimal solution to the integer linear program.
a.
True
b.
False
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True / False

  1. The LP Relaxation contains the objective function and constraints of the IP problem, but drops all integer restrictions. a. True b. False ANSWER: True POINTS: 1 DIFFICULTY: Easy LEARNING OBJECTIVES: IMS.ASWC.19.07.02 - 7. NATIONAL STANDARDS: United States - BUSPROG: Reflective Thinking TOPICS: 7.2 Graphical and Computer Solutions for an All- Integer Linear Program KEYWORDS: Bloom's: Understand
  2. In general, rounding large values of decision variables to the nearest integer value causes fewer problems than rounding small values. a. True b. False ANSWER: True POINTS: 1 DIFFICULTY: Easy LEARNING OBJECTIVES: IMS.ASWC.19.07.02 - 7. NATIONAL STANDARDS: United States - BUSPROG: Reflective Thinking TOPICS: 7.2 Graphical and Computer Solutions for an All- Integer Linear Program KEYWORDS: Bloom's: Understand
  3. If the optimal solution to the LP Relaxation problem is an integer, it is the optimal solution to the integer linear program. a. True b. False

ANSWER: True POINTS: 1 DIFFICULTY: Easy LEARNING OBJECTIVES: IMS.ASWC.19.07.02 - 7. NATIONAL STANDARDS: United States - BUSPROG: Reflective Thinking TOPICS: 7.2 Graphical and Computer Solutions for an All- Integer Linear Program KEYWORDS: Bloom's: Understand

  1. Slack and surplus variables are not useful in integer linear programs. a. True b. False ANSWER: False POINTS: 1 DIFFICULTY: Easy LEARNING OBJECTIVES: IMS.ASWC.19.07.03 - 7. NATIONAL STANDARDS: United States - BUSPROG: Reflective Thinking TOPICS: 7.3 Applications Involving 0 - 1 Variables KEYWORDS: Bloom's: Understand
  2. A multiple-choice constraint involves selecting k out of n alternatives, where k ≥
  3. a. True b. False ANSWER: False POINTS: 1 DIFFICULTY: Moderate LEARNING OBJECTIVES: IMS.ASWC.19.07.04 - 7. NATIONAL STANDARDS: United States - BUSPROG: Reflective Thinking TOPICS: 7.4 Modeling Flexibility Provided by 0 - 1 Integer Variables KEYWORDS: Bloom's: Understand

NATIONAL STANDARDS: United States - BUSPROG: Analytic TOPICS: 7.4 Modeling Flexibility Provided by 0 - 1 Integer Variables KEYWORDS: Bloom's: Apply

  1. The constraint x 1 − x 2 = 0 implies that if project 1 is selected, project 2 cannot be. a. True b. False ANSWER: False POINTS: 1 DIFFICULTY: Easy LEARNING OBJECTIVES: IMS.ASWC.19.07.04 - 7. NATIONAL STANDARDS: United States - BUSPROG: Analytic TOPICS: 7.4 Modeling Flexibility Provided by 0 - 1 Integer Variables KEYWORDS: Bloom's: Understand
  2. A product design and market share optimization problem involves choosing a product design that maximizes the number of consumers preferring it. a. True b. False ANSWER: True POINTS: 1 DIFFICULTY: Easy LEARNING OBJECTIVES: IMS.ASWC.19.07.03 - 7. NATIONAL STANDARDS: United States - BUSPROG: Reflective Thinking TOPICS: 7.3 Applications Involving 0 - 1 Variables KEYWORDS: Bloom's: Remember
  3. 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.

a. True b. False ANSWER: True POINTS: 1 DIFFICULTY: Easy LEARNING OBJECTIVES: IMS.ASWC.19.07.02 - 7. NATIONAL STANDARDS: United States - BUSPROG: Reflective Thinking TOPICS: 7.2 Graphical and Computer Solutions for an All- Integer Linear Program KEYWORDS: Bloom's: Understand

  1. The computer output for integer programs in the textbook does not include reduced costs, dual values, or sensitivity ranges because these variables are not meaningful for integer programs. a. True b. False ANSWER: True POINTS: 1 DIFFICULTY: Easy LEARNING OBJECTIVES: IMS.ASWC.19.07.02 - 7. NATIONAL STANDARDS: United States - BUSPROG: Reflective Thinking TOPICS: 7.2 Graphical and Computer Solutions for an All- Integer Linear Program KEYWORDS: Bloom's: Understand
  2. Some linear programming problems have a special structure that guarantees the variables will have integer values. a. True b. False ANSWER: True POINTS: 1

a. True b. False ANSWER: False POINTS: 1 DIFFICULTY: Moderate LEARNING OBJECTIVES: IMS.ASWC.19.07.04 - 7. NATIONAL STANDARDS: United States - BUSPROG: Reflective Thinking TOPICS: 7.4 Modeling Flexibility Provided by 0 - 1 Integer Variables KEYWORDS: Bloom's: Understand

  1. If the LP Relaxation of an integer program has a feasible solution, then the integer program has a feasible solution. a. True b. False ANSWER: False POINTS: 1 DIFFICULTY: Moderate LEARNING OBJECTIVES: IMS.ASWC.19.07.02 - 7. NATIONAL STANDARDS: United States - BUSPROG: Reflective Thinking TOPICS: 7.2 Graphical and Computer Solutions for an All- Integer Linear Program KEYWORDS: Bloom's: Understand
  2. Binary variables can be used to model multiple-choice and mutually exclusive constraints. a. True b. False ANSWER: True POINTS: 1 DIFFICULTY: Easy

LEARNING OBJECTIVES: IMS.ASWC.19.07.04 - 7.

NATIONAL STANDARDS: United States - BUSPROG: Reflective Thinking TOPICS: 7.4 Modeling Flexibility Provided by 0 - 1 Integer Variables KEYWORDS: Bloom's: Remember

  1. Most practical applications of integer linear programming involve only 0 - 1 integer variables. a. True b. False ANSWER: True POINTS: 1 DIFFICULTY: Easy LEARNING OBJECTIVES: IMS.ASWC.19.07.03 - 7. NATIONAL STANDARDS: United States - BUSPROG: Reflective Thinking TOPICS: 7.3 Applications Involving 0 - 1 Variables KEYWORDS: Bloom's: Remember
  2. Integer linear programs provide substantial modeling flexibility and thus are harder to solve than linear programs. a. True b. False ANSWER: True POINTS: 1 DIFFICULTY: Easy LEARNING OBJECTIVES: IMS.ASWC.19.07.01 - 7. NATIONAL STANDARDS: United States - BUSPROG: Reflective Thinking TOPICS: 7.1 Types of Integer Linear Programming Models KEYWORDS: Bloom's: Understand
  3. In general, whenever rounding has a minimal impact on the objective function

b. x 1 = 4, x 2 = 0.389, x 3 = 1 c. x 1 = 2, x 2 = 3, x 3 = 0. d. x 1 = 0, x 2 = 8, x 3 = 0 ANSWER: c POINTS: 1 DIFFICULTY: Moderate LEARNING OBJECTIVES: IMS.ASWC.19.07.01 - 7. NATIONAL STANDARDS: United States - BUSPROG: Analytic TOPICS: 7.1 Types of Integer Linear Programming Models KEYWORDS: Bloom's: Analyze

  1. Rounded solutions to linear programs must be evaluated for a. feasibility and optimality. b. sensitivity and duality. c. relaxation and boundedness. d. All of these are correct. ANSWER: a POINTS: 1 DIFFICULTY: Moderate LEARNING OBJECTIVES: IMS.ASWC.19.07.02 - 7. NATIONAL STANDARDS: United States - BUSPROG: Reflective Thinking TOPICS: 7.2 Graphical and Computer Solutions for an All- Integer Linear Program KEYWORDS: Bloom's: Understand
  2. Rounding the solution of an LP Relaxation to the nearest integer values provides a(n) a. feasible but not necessarily optimal integer solution. b. integer solution that is optimal. c. integer solution that might be neither feasible nor optimal. d. infeasible solution. ANSWER: c

POINTS: 1

DIFFICULTY: Moderate LEARNING OBJECTIVES: IMS.ASWC.19.07.02 - 7. NATIONAL STANDARDS: United States - BUSPROG: Reflective Thinking TOPICS: 7.2 Graphical and Computer Solutions for an All- Integer Linear Program KEYWORDS: Bloom's: Understand

  1. The solution to the LP Relaxation of a maximization integer linear program provides a(n) a. upper bound for the value of the objective function. b. lower bound for the value of the objective function. c. upper bound for the value of the decision variables. d. lower bound for the value of the decision variables. ANSWER: a POINTS: 1 DIFFICULTY: Moderate LEARNING OBJECTIVES: IMS.ASWC.19.07.02 - 7. NATIONAL STANDARDS: United States - BUSPROG: Reflective Thinking TOPICS: 7.2 Graphical and Computer Solutions for an All- Integer Linear Program KEYWORDS: Bloom's: Understand
  2. The graph of a problem that requires x 1 and x 2 to be integer has a feasible region a. the same as its LP Relaxation. b. consisting of dots. c. consisting of horizontal stripes. d. consisting of vertical stripes. ANSWER: b POINTS: 1 DIFFICULTY: Moderate LEARNING OBJECTIVES: IMS.ASWC.19.07.02 - 7.
  1. Let x 1 and x 2 be 0 - 1 variables whose values indicate whether projects 1 and 2 are/are not done. Which of the following answers indicates that project 2 can be done only if project 1 is done? a. x 1 + x 2 = 1 b. x 1 + x 2 = 2 c. x 1 − x 2 ≤ 0 d. x 1 − x 2 ≥ 0 ANSWER: d POINTS: 1 DIFFICULTY: Moderate LEARNING OBJECTIVES: IMS.ASWC.19.07.04 - 7. NATIONAL STANDARDS: United States - BUSPROG: Analytic TOPICS: 7.4 Modeling Flexibility Provided by 0 - 1 Integer Variables KEYWORDS: Bloom's: Analyze
  2. Let x 1 , x 2 , and x 3 be 0 - 1 variables whose values indicate whether the projects are not done (0) or are done (1). Which of the following answers indicates that at least two of the projects must be done? a. x 1 + x 2 + x 3 ≥ 2 b. x 1 + x 2 + x 3 ≤ 2 c. x 1 + x 2 + x 3 = 2 d. x 1 − x 2 = 0 ANSWER: a POINTS: 1 DIFFICULTY: Moderate LEARNING OBJECTIVES: IMS.ASWC.19.07.04 - 7. NATIONAL STANDARDS: United States - BUSPROG: Analytic TOPICS: 7.4 Modeling Flexibility Provided by 0 - 1 Integer Variables KEYWORDS: Bloom's: Analyze
  1. If the acceptance of project A is conditional on the acceptance of project B, and vice versa, the appropriate constraint to use is a a. multiple-choice constraint. b. k out of n alternatives constraint. c. mutually exclusive constraint. d. corequisite constraint. ANSWER: d POINTS: 1 DIFFICULTY: Moderate LEARNING OBJECTIVES: IMS.ASWC.19.07.04 - 7. NATIONAL STANDARDS: United States - BUSPROG: Reflective Thinking TOPICS: 7.4 Modeling Flexibility Provided by 0 - 1 Integer Variables KEYWORDS: Bloom's: Understand
  2. In an all-integer linear program, a. all objective function coefficients must be integer. b. all right-hand-side values must be integer. c. all variables must be integer. d. all objective function coefficients and right-hand-side values must be integer. ANSWER: c POINTS: 1 DIFFICULTY: Moderate LEARNING OBJECTIVES: IMS.ASWC.19.07.01 - 7. NATIONAL STANDARDS: United States - BUSPROG: Reflective Thinking TOPICS: 7.1 Types of Integer Linear Programming Models KEYWORDS: Bloom's: Understand
  3. To perform sensitivity analysis involving an integer linear program, it is best to a. use the dual prices very cautiously.

ANSWER: a POINTS: 1 DIFFICULTY: Easy LEARNING OBJECTIVES: IMS.ASWC.19.07.03 - 7. NATIONAL STANDARDS: United States - BUSPROG: Reflective Thinking TOPICS: 7.3 Applications Involving 0 - 1 Variables KEYWORDS: Bloom's: Understand

  1. Assuming W 1 , W 2, and W 3 are 0 - 1 integer variables, the constraint W 1 + W 2 + W 3 < 1 is often called a a. multiple-choice constraint. b. mutually exclusive constraint. c. k out of n alternatives constraint. d. corequisite constraint. ANSWER: b POINTS: 1 DIFFICULTY: Moderate LEARNING OBJECTIVES: IMS.ASWC.19.07.04 - 7. NATIONAL STANDARDS: United States - BUSPROG: Reflective Thinking TOPICS: 7.4 Modeling Flexibility Provided by 0 - 1 Integer Variables KEYWORDS: Bloom's: Understand
  2. Which of the following applications modeled in the textbook does NOT involve only 0 - 1 integer variables? a. supply chain design b. bank location c. capital budgeting d. product design and market share optimization ANSWER: a POINTS: 1 DIFFICULTY: Moderate

LEARNING OBJECTIVES: IMS.ASWC.19.07.03 - 7.

NATIONAL STANDARDS: United States - BUSPROG: Reflective Thinking TOPICS: 7.3 Applications Involving 0 - 1 Variables KEYWORDS: Bloom's: Remember Subjective Short Answer

  1. Solve the following problem graphically. Max 5X + 6Y s.t. 17X + 8Y ≤ 136 3X + 4Y ≤ 36 X, Y ≥ 0 and integer a. Graph the constraints for this problem. Indicate all feasible solutions. b. Find the optimal solution to the LP Relaxation. Round down to find a feasible integer solution. Is this solution optimal? c. Find the optimal solution. ANSWER: a. The feasible region is those integer values in the space labeled Feasible Region.

b. The optimal LP relaxed solution occurs at X = 1.538, Y = 4.846, with Z = 11.231. The rounded down solution occurs at X = 1.538, Y = 4. c. The optimal solution is at X = 2.667, and Y = 4, and Z = 10.667. POINTS: 1 DIFFICULTY: Challenging LEARNING OBJECTIVES: IMS.ASWC.19.07.02 - 7. NATIONAL STANDARDS: United States - BUSPROG: Analytic TOPICS: 7.2 Graphical and Computer Solutions for an All- Integer Linear Program KEYWORDS: Bloom's: Analyze

  1. Solve the following problem graphically. Min 6X + 11Y s.t. 9X + 3Y ≥ 27 7X + 6Y ≥ 42 4X + 8Y ≥ 32 X, Y ≥ 0 and integer

a. Graph the constraints for this problem. Indicate all feasible solutions. b. Find the optimal solution to the LP Relaxation. Round up to find a feasible integer solution. Is this solution optimal? c. Find the optimal solution. ANSWER: a. The feasible region is the set of integer points in the area labeled Feasible Region. b. The optimal relaxed solution is at X = 4.5, Y = 1.75, and Z = 46.25. The rounded solution is X = 5, Y = 2. c. The optimal solution is at X = 6, Y = 1, and Z = 47. POINTS: 1 DIFFICULTY: Challenging LEARNING OBJECTIVES: IMS.ASWC.19.07.02 - 7. NATIONAL STANDARDS: United States - BUSPROG: Analytic TOPICS: 7.2 Graphical and Computer Solutions for an All- Integer Linear Program KEYWORDS: Bloom's: Evaluate

  1. Consider a capital budgeting example with five projects from which to select. Let xi = 1 if project i is selected, 0 if not, for i = 1, … , 5. Write the appropriate