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Simplex-Based SensitivityAnalysis and Duality: Chapter 18 Comprehensive Exam Study Guide L, Exams of Physics

Tennessee Technological University (TTU)Physics

Simplex-Based SensitivityAnalysis and Duality: Chapter 18 Comprehensive Exam Study Guide Latest Updated 2024/2025

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2023/2024

Available from 11/02/2024

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Simplex-Based Sensitivity Analysis and
Duality: Chapter 18
Comprehensive Exam Study Guide
Latest Updated 2024/2025
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Download Simplex-Based SensitivityAnalysis and Duality: Chapter 18 Comprehensive Exam Study Guide L and more Exams Physics in PDF only on Docsity!

Duality: Chapter 18

Comprehensive Exam Study Guide

Latest Updated 2024/

Duality: Chapter 18

Comprehensive Exam Study Guide

Latest Updated 2024/

CH 18 - Simplex-Based Sensitivity Analysis and Duality

True / False

  1. The range of optimality is useful only for basic variables. a. True b. False ANSWER: False POINTS: 1 DIFFICULTY: Easy LEARNING OBJECTIVES: IMS.ASWC.19.18.01 - 18. NATIONAL STANDARDS: United States - BUSPROG: Reflective Thinking TOPICS: 18.1 Sensitivity Analysis with the Simplex Tableau KEYWORDS: Bloom's: Remember
  2. The range of optimality is calculated by considering changes in the cj − zj value of the variable in question. a. True b. False ANSWER: False POINTS: 1 DIFFICULTY: Easy LEARNING OBJECTIVES: IMS.ASWC.19.18.01 - 18. NATIONAL STANDARDS: United States - BUSPROG: Reflective Thinking TOPICS: 18.1 Sensitivity Analysis with the Simplex Tableau KEYWORDS: Bloom's: Understand
  3. As long as the objective function coefficient remains within the range of optimality, the variable values will not change although the value of the objective function could. a. True b. False ANSWER: True POINTS: 1 DIFFICULTY: Easy LEARNING OBJECTIVES: IMS.ASWC.19.18.01 - 18. NATIONAL STANDARDS: United States - BUSPROG: Reflective Thinking TOPICS: 18.1 Sensitivity Analysis with the Simplex Tableau KEYWORDS: Bloom's: Understand
  4. If the simplex tableau is from a maximization converted from a minimization, the signs and directions of the inequalities that give the objective function ranges will need to be adjusted to apply to the original coefficients. a. True

Duality: Chapter 18

Comprehensive Exam Study Guide

Latest Updated 2024/

CH 18 - Simplex-Based Sensitivity Analysis and Duality

NATIONAL STANDARDS: United States - BUSPROG: Reflective Thinking TOPICS: 18.1 Sensitivity Analysis with the Simplex Tableau KEYWORDS: Bloom's: Understand

  1. The ranges for which the right-hand-side values are valid are the same as the ranges over which the dual prices are valid. a. True b. False ANSWER: True POINTS: 1 DIFFICULTY: Easy LEARNING OBJECTIVES: IMS.ASWC.19.18.01 - 18. NATIONAL STANDARDS: United States - BUSPROG: Reflective Thinking TOPICS: 18.1 Sensitivity Analysis with the Simplex Tableau KEYWORDS: Bloom's: Understand
  2. A dual price is associated with each decision variable. a. True b. False ANSWER: False POINTS: 1 DIFFICULTY: Moderate LEARNING OBJECTIVES: IMS.ASWC.19.18.01 - 18. NATIONAL STANDARDS: United States - BUSPROG: Reflective Thinking TOPICS: 18.1 Sensitivity Analysis with the Simplex Tableau KEYWORDS: Bloom's: Understand
  3. The dual price for an equality constraint is the zj value for its artificial variable. a. True b. False ANSWER: True POINTS: 1 DIFFICULTY: Easy LEARNING OBJECTIVES: IMS.ASWC.19.18.01 - 18. NATIONAL STANDARDS: United States - BUSPROG: Reflective Thinking TOPICS: 18.1 Sensitivity Analysis with the Simplex Tableau KEYWORDS: Bloom's: Understand
  4. The entries in the associated slack column of the final tableau can also be interpreted as the changes in the values of the

Duality: Chapter 18

Comprehensive Exam Study Guide

Latest Updated 2024/

current basic variables corresponding to a one-unit increase in the right-hand side.

a. True b. False ANSWER: True POINTS: 1

© Cengage. Testing Powered by Cognero. Page 2

Duality: Chapter 18

Comprehensive Exam Study Guide

Latest Updated 2024/

TOPICS: 18.1 Sensitivity Analysis with the Simplex Tableau KEYWORDS: Bloom's: Understand

  1. We can often avoid the process of formulating and solving a modified linear programming problem by using the range of optimality to determine whether a change in an objective function coefficient is large enough to cause a change in the optimal solution.

© Cengage. Testing Powered by Cognero. Page 3

Duality: Chapter 18

Comprehensive Exam Study Guide

Latest Updated 2024/

CH 18 - Simplex-Based Sensitivity Analysis and Duality

a. True b. False ANSWER: True POINTS: 1 DIFFICULTY: Easy LEARNING OBJECTIVES: IMS.ASWC.19.18.01 - 18. NATIONAL STANDARDS: United States - BUSPROG: Reflective Thinking TOPICS: 18.1 Sensitivity Analysis with the Simplex Tableau KEYWORDS: Bloom's: Understand

  1. Within the concept of duality is the original formulation of a linear programming problem known as the primal problem.

a. True b. False ANSWER: True POINTS: 1 DIFFICULTY: Easy LEARNING OBJECTIVES: IMS.ASWC.19.18.02 - 18. NATIONAL STANDARDS: United States - BUSPROG: Reflective Thinking TOPICS: 18.2 Duality KEYWORDS: Bloom's: Remember

Multiple Choice

  1. Dual prices and ranges for objective function coefficients and right-hand-side values are found by considering a. dual analysis. b. optimality analysis. c. ranging analysis. d. sensitivity analysis. ANSWER: d POINTS: 1 DIFFICULTY: Moderate LEARNING OBJECTIVES: IMS.ASWC.19.18.01 - 18. NATIONAL STANDARDS: United States - BUSPROG: Reflective Thinking TOPICS: 18.1 Sensitivity Analysis with the Simplex Tableau KEYWORDS: Bloom's: Understand
  2. For the basic feasible solution to remain optimal, a. all cj − zj values must remain ≤ 0.

Duality: Chapter 18

Comprehensive Exam Study Guide

Latest Updated 2024/

CH 18 - Simplex-Based Sensitivity Analysis and Duality

DIFFICULTY: Easy LEARNING OBJECTIVES: IMS.ASWC.19.18.01 - 18. NATIONAL STANDARDS: United States - BUSPROG: Reflective Thinking TOPICS: 18.1 Sensitivity Analysis with the Simplex Tableau KEYWORDS: Bloom's: Understand

  1. A one-sided range of optimality a. always occurs for nonbasic variables. b. always occurs for basic variables. c. indicates changes in more than one coefficient. d. indicates changes in a slack variable's coefficient. ANSWER: a POINTS: 1 DIFFICULTY: Moderate LEARNING OBJECTIVES: IMS.ASWC.19.18.01 - 18. NATIONAL STANDARDS: United States - BUSPROG: Reflective Thinking TOPICS: 18.1 Sensitivity Analysis with the Simplex Tableau KEYWORDS: Bloom's: Understand
  2. A linear programming problem with the objective function 3x 1 + 8x 2 has the optimal solution x 1 = 5, x 2 = 6. If c 2 decreases by 2 and the range of optimality shows 5 ≤ c 2 ≤ 12, the value of Z a. will decrease by 12. b. will decrease by 2. c. will not change. d. cannot be determined from this information. ANSWER: a POINTS: 1 DIFFICULTY: Challenging LEARNING OBJECTIVES: IMS.ASWC.19.18.01 - 18. NATIONAL STANDARDS: United States - BUSPROG: Analytic TOPICS: 18.1 Sensitivity Analysis with the Simplex Tableau KEYWORDS: Bloom's: Apply
  3. The improvement in the value of the optimal solution per unit increase in a constraint's right-hand side is a. the slack value. b. the dual price. c. never negative. d. the 100% rule. ANSWER: b

Duality: Chapter 18

Comprehensive Exam Study Guide

Latest Updated 2024/

POINTS: 1

DIFFICULTY: Moderate LEARNING OBJECTIVES: IMS.ASWC.19.18.01 - 18. NATIONAL STANDARDS: United States - BUSPROG: Reflective Thinking TOPICS: 18.1 Sensitivity Analysis with the Simplex Tableau © Cengage. Testing Powered by Cognero. Page 5

Duality: Chapter 18

Comprehensive Exam Study Guide

Latest Updated 2024/

TOPICS: 18.1 Sensitivity Analysis with the Simplex Tableau KEYWORDS: Bloom's: Analyze

  1. The number of constraints to the dual of the following problem is

Max Z (^) = 3x 1 + 2x 2 + 6x 3 © Cengage. Testing Powered by Cognero. Page 6

Duality: Chapter 18

Comprehensive Exam Study Guide

Latest Updated 2024/

CH 18 - Simplex-Based Sensitivity Analysis and Duality

s.t. (^) 4x 1 + 2x 2 + 3x 3 ≥ 100 2x 1 + x 2 − 2x 3 ≤ 200 4x 2 + x 3 ≥ 200 a. 1. b. 2. c. 3. d. 4. ANSWER: c POINTS: 1 DIFFICULTY: Moderate LEARNING OBJECTIVES: IMS.ASWC.19.18.02 - 18. NATIONAL STANDARDS: United States - BUSPROG: Analytic TOPICS: 18.2 Duality KEYWORDS: Bloom's: Apply

  1. Given the simplex tableau for the optimal primal solution, a. the values of the dual variables can be found from the cj − zj values of the slack/surplus variable columns. b. the values of the dual surplus variables can be found from the cj − zj values of the primal decision variable columns. c. the value of the dual objective function will be the same as the objective function value for the primal problem. d. All of these are correct. ANSWER: d POINTS: 1 DIFFICULTY: Moderate LEARNING OBJECTIVES: IMS.ASWC.19.18.02 - 18. NATIONAL STANDARDS: United States - BUSPROG: Reflective Thinking TOPICS: 18.2 Duality KEYWORDS: Bloom's: Understand
  2. An LP maximization problem with all less-than-or-equal-to constraints and nonnegativity requirements for the decision variables is known as a. a canonical form for a minimization problem. b. a canonical form for a maximization problem. c. always unbounded. d. None of these are correct. ANSWER: b POINTS: 1 DIFFICULTY: Moderate

Duality: Chapter 18

Comprehensive Exam Study Guide

Latest Updated 2024/

CH 18 - Simplex-Based Sensitivity Analysis and Duality

Subjective Short Answer

  1. For the following linear programming problem

Max Z (^) 0.5x 1 + 6x 2 + 5x 3 s.t. (^) 4x 1 + 6x 2 + 3x 3 ≤ 24 1x 1 + 1.5x 2 + 3x 3 ≤ 12 3x 1 + x 2 ≤ 12

the final tableau is

Basis (^) cB

x 1

x 2 6

x 3 5

s 1 0

s 2 0

s 3 0 x 2 6 1 1 0 0.22^ −0.22^0 2. x 3 5 0 0 1 0.11^ −0.44^0 2. s 3 0 2.33^0 0 −0.22^ 0.22^1 9. zj 4 6 5 0.77^ 0.88^0 29. cj − zj 0.5^0 0 −0.77^ −0.88^0

a. (^) Find the range of optimality for c1, c2, c3, c4, c5, and c 6. b. (^) Find the range of feasibility for b 1 , b 2 , and b 3.

ANSWER: a. (^) c 1 ≤ 4 2.5 < c 2 ≤ 10 3 < c 3 ≤ 12 c 4 ≤ 0. c 5 ≤ 0. −1.5 ≤ c 6 ≤ 3.

b. (^) 12 ≤ b 1 ≤ 48 6 ≤ b 2 ≤ 24 b 3 ≥ 2.

POINTS: 1 DIFFICULTY: Challenging LEARNING OBJECTIVES: IMS.ASWC.19.18.01 - 18. NATIONAL STANDARDS: United States - BUSPROG: Analytic TOPICS: 18.1 Sensitivity Analysis with the Simplex Tableau KEYWORDS: Bloom's: Analyze

  1. For the following linear programming problem

Comprehensive Exam Study Guide

 - Duality: Chapter - Latest Updated 2024/
  • Max Z −2x 1 + x 2 − x
  • s.t. 2x 1 + x 2 ≤ - 1x 1 + x 2 + x 3 ≥
    • © Cengage. Testing Powered by Cognero. Page

Duality: Chapter 18

Comprehensive Exam Study Guide

Latest Updated 2024/

y

1 −^ y 2 + 3y 3 ≤^5

POINTS: 1 DIFFICULTY: Challenging LEARNING OBJECTIVES: IMS.ASWC.19.18.02 - 18.

© Cengage. Testing Powered by Cognero. Page 9

Duality: Chapter 18

Comprehensive Exam Study Guide

Latest Updated 2024/

CH 18 - Simplex-Based Sensitivity Analysis and Duality

NATIONAL STANDARDS: United States - BUSPROG: Analytic TOPICS: 18.2 Duality KEYWORDS: Bloom's: Create

  1. For the following linear programming problem

Max (^) 10x 1 + 12x 2 s.t. (^) 1x 1 + 2x 2 ≥ 40

5x 1 + 8x 2 ≤ 160 1x 1 + 1x 2 ≤ 40 x 1 , x 2 ≥ 0

the final tableau is

Basis (^) cB

x 1 10

x 2 12

s 1 0

s 2 0

s 3 0 x 2 12 0 1 −2.5^ −0.5^0 x 1 10 1 0 4 1 0 0 s 3 0 0 0 −1.5^ −0.5^1 zj 10 12 10 4 0 240 cj − zj 0 0 −10^ −4^0

a. (^) Find the range of optimality for c 1 and c 2. b. (^) Find the range of feasibility for b 1 , b 2 , and b 3. c. (^) Find the dual prices. ANSWER: a. (^) 7.5 ≤ c 1 < ∞ −∞ < c 2 ≤ 16

b. (^) 32 ≤ b 1 < 0 160 ≤ b 2 ≤ 200 20 ≤ b 3 < ∞

c. (^) u 1 = 10 u 2 = 4 u 3 = 0 POINTS: 1 DIFFICULTY: Challenging LEARNING OBJECTIVES: IMS.ASWC.19.18.01 - 18. NATIONAL STANDARDS: United States - BUSPROG: Analytic