CH 03 - Linear Programming: Sensitivity Analysis and Interpretation of Solution. Questions and Answers

/ 1. Classical sensitivity analysis provides no information about changes resulting from a change in the coefficient of a variable in a constraint. a. b. : POINTS: 1 DIFFICULTY: Easy LEARNING OBJ ...

/ 1. Classical sensitivity analysis provides no information about changes resulting from a change in the coefficient of a variable in a constraint. a. b. : POINTS: 1 DIFFICULTY: Easy LEARNING OBJECTIVES: IMS.ASWC.19.03.04 - 3.4 NATIONAL STANDARDS: United States - BUSPROG: Reflective Thinking TOPICS: 3.4 Limitations of Classical Sensitivity Analysis KEYWORDS: Bloom's: Understand 2. The reduced cost of a variable is the dual value of the corresponding nonnegativity constraint. a. b. : POINTS: 1 DIFFICULTY: Easy LEARNING OBJECTIVES: IMS.ASWC.19.03.03 - 3.3 NATIONAL STANDARDS: United States - BUSPROG: Reflective Thinking TOPICS: 3.3 Sensitivity Analysis: Computer Solution KEYWORDS: Bloom's: Understand 3. When the right-hand sides of two constraints are each increased by one unit, the objective function value will be adjusted by the sum of the constraints' dual prices. a. b. : POINTS: 1 DIFFICULTY: Moderate LEARNING OBJECTIVES: IMS.ASWC.19.03.02 - 3.2 NATIONAL STANDARDS: United States - BUSPROG: Reflective Thinking TOPICS: 3.2 Graphical Sensitivity Analysis KEYWORDS: Bloom's: Understand 4. If the range of feasibility indicates that the original amount of a resource, which was 20, can increase by 5, then the amount of the resource can increase to 25. a. b. : POINTS: 1 DIFFICULTY: Moderate LEARNING OBJECTIVES: IMS.ASWC.19.03.03 - 3.3 NATIONAL STANDARDS: United States - BUSPROG: Reflective Thinking TOPICS: 3.3 Sensitivity Analysis: Computer Solution KEYWORDS: Bloom's: Apply 5. When two or more objective function coefficients are changed simultaneously, further analysis is necessary to determine whether the optimal solution will change. a. b. : POINTS: 1 DIFFICULTY: Moderate LEARNING OBJECTIVES: IMS.ASWC.19.03.02 - 3.2 NATIONAL STANDARDS: United States - BUSPROG: Analytic TOPICS: 3.2 Graphical Sensitivity Analysis KEYWORDS: Bloom's: Understand 6. The dual value and dual price are identical for a minimization problem. a. b. : POINTS: 1 DIFFICULTY: Easy LEARNING OBJECTIVES: IMS.ASWC.19.A3.02 - A3.2 NATIONAL STANDARDS: United States - BUSPROG: Reflective Thinking TOPICS: Appendix 3.2 Sensitivity Analysis with Lingo KEYWORDS: Bloom's: Understand 7. A negative dual price indicates that increasing the right-hand side of the associated constraint would be detrimental to the objective. a. b. : POINTS: 1 DIFFICULTY: Moderate LEARNING OBJECTIVES: IMS.ASWC.19.03.02 - 3.2 NATIONAL STANDARDS: United States - BUSPROG: Reflective Thinking TOPICS: 3.2 Graphical Sensitivity Analysis KEYWORDS: Bloom's: Understand 8. In order to tell the impact of a change in a constraint coefficient, the change must be made and then the model resolved. a. b. : POINTS: 1 DIFFICULTY: Easy LEARNING OBJECTIVES: IMS.ASWC.19.03.04 - 3.4 NATIONAL STANDARDS: United States - BUSPROG: Reflective Thinking TOPICS: 3.4 Limitations of Classical Sensitivity Analysis KEYWORDS: Bloom's: Understand 9. A small change in the objective function coefficient can necessitate modifying the optimal solution. a. b. : POINTS: 1 DIFFICULTY: Easy LEARNING OBJECTIVES: IMS.ASWC.19.03.02 - 3.2 NATIONAL STANDARDS: United States - BUSPROG: Reflective Thinking TOPICS: 3.2 Graphical Sensitivity Analysis KEYWORDS: Bloom's: Understand 10. The dual price associated with a constraint is the change in the value of the solution per unit decrease in the right-hand side of the constraint. a. b. : POINTS: 1 DIFFICULTY: Moderate LEARNING OBJECTIVES: IMS.ASWC.19.03.02 - 3.2 NATIONAL STANDARDS: United States - BUSPROG: Analytic TOPICS: 3.2 Graphical Sensitivity Analysis KEYWORDS: Bloom's: Remember 11. For a minimization problem, a positive dual price indicates the value of the objective function will increase. a. b. : POINTS: 1 DIFFICULTY: Moderate LEARNING OBJECTIVES: IMS.ASWC.19.03.02 - 3.2 NATIONAL STANDARDS: United States - BUSPROG: Reflective Thinking TOPICS: 3.2 Graphical Sensitivity Analysis KEYWORDS: Bloom's: Understand 12. There is a dual price for every decision variable in a model. a. b. : POINTS: 1 DIFFICULTY: Easy LEARNING OBJECTIVES: IMS.ASWC.19.03.03 - 3.3 NATIONAL STANDARDS: United States - BUSPROG: Reflective Thinking TOPICS: 3.3 Sensitivity Analysis: Computer Solution 3.2 Graphical Sensitivity Analysis KEYWORDS: Bloom's: Understand 13. The amount of a sunk cost will vary depending on the values of the decision variables. a. b. : POINTS: 1 DIFFICULTY: Easy LEARNING OBJECTIVES: IMS.ASWC.19.03.03 - 3.3 NATIONAL STANDARDS: United States - BUSPROG: Reflective Thinking TOPICS: 3.3 Sensitivity Analysis: Computer Solution KEYWORDS: Bloom's: Understand 14. If the optimal value of a decision variable is zero and its reduced cost is zero, this indicates that alternative optimal solutions exist. a. b. : POINTS: 1 DIFFICULTY: Moderate LEARNING OBJECTIVES: IMS.ASWC.19.A3.01 - A3.1 NATIONAL STANDARDS: United States - BUSPROG: Reflective Thinking TOPICS: Appendix 3.1 Sensitivity Analysis with Excel Solver KEYWORDS: Bloom's: Understand 15. Relevant costs should be reflected in the objective function, but sunk costs should not. a. b. : POINTS: 1 DIFFICULTY: Easy LEARNING OBJECTIVES: IMS.ASWC.19.03.03 - 3.3 NATIONAL STANDARDS: United States - BUSPROG: Reflective Thinking TOPICS: 3.3 Sensitivity Analysis: Computer Solution KEYWORDS: Bloom's: Understand 16. If the range of feasibility for b1 is between 16 and 37, then if b1 = 22, the optimal solution will not change from the original optimal solution. a. b. : POINTS: 1 DIFFICULTY: Easy LEARNING OBJECTIVES: IMS.ASWC.19.03.03 - 3.3 NATIONAL STANDARDS: United States - BUSPROG: Reflective Thinking TOPICS: 3.3 Sensitivity Analysis: Computer Solution KEYWORDS: Bloom's: Apply 17. If the dual price for the right-hand side of a ≤ constraint is zero, there is no upper limit on its range of feasibility. a. b. : POINTS: 1 DIFFICULTY: Moderate LEARNING OBJECTIVES: IMS.ASWC.19.03.02 - 3.2 NATIONAL STANDARDS: United States - BUSPROG: Reflective Thinking TOPICS: 3.2 Graphical Sensitivity Analysis KEYWORDS: Bloom's: Understand 18. Increasing the right-hand side of a nonbinding constraint will not cause a change in the optimal solution. a. b. : POINTS: 1 DIFFICULTY: Easy LEARNING OBJECTIVES: IMS.ASWC.19.03.02 - 3.2 NATIONAL STANDARDS: United States - BUSPROG: Reflective Thinking TOPICS: 3.2 Graphical Sensitivity Analysis KEYWORDS: Bloom's: Understand 19. If two or more objective function coefficients are changed simultaneously, further analysis is necessary to determine whether the optimal solution will change. a. b. : POINTS: 1 DIFFICULTY: Easy LEARNING OBJECTIVES: IMS.ASWC.19.03.02 - 3.2 NATIONAL STANDARDS: United States - BUSPROG: Reflective Thinking TOPICS: 3.2 Graphical Sensitivity Analysis KEYWORDS: Bloom's: Understand Multiple Choice 20. To solve a linear programming problem with thousands of variables and constraints, a. a personal computer can be used. b. a mainframe computer is required. c. the problem must be partitioned into subparts. d. unique software would need to be developed. : a POINTS: 1 DIFFICULTY: Easy LEARNING OBJECTIVES: IMS.ASWC.19.03.03 - 3.3 NATIONAL STANDARDS: United States - BUSPROG: Reflective Thinking TOPICS: 3.3 Sensitivity Analysis: Computer Solution 3.1 Introduction to Sensitivity Analysis KEYWORDS: Bloom's: Remember 21. A negative dual price for a constraint in a minimization problem means a. as the right-hand side increases, the objective function value will increase. b. as the right-hand side decreases, the objective function value will increase. c. as the right-hand side increases, the objective function value will decrease. d. as the right-hand side decreases, the objective function value will decrease. : a POINTS: 1 DIFFICULTY: Moderate LEARNING OBJECTIVES: IMS.ASWC.19.A3.02 - A3.2 NATIONAL STANDARDS: United States - BUSPROG: Reflective Thinking TOPICS: Appendix 3.2 Sensitivity Analysis with Lingo KEYWORDS: Bloom's: Understand 22. If a decision variable is not positive in the optimal solution, its reduced cost is a. what its objective function value would need to be before it could become positive. b. the amount its objective function value would need to improve before it could become positive. c. zero. d. its dual price. POINTS: 1 DIFFICULTY: Moderate LEARNING OBJECTIVES: IMS.ASWC.19.03.03 - 3.3 NATIONAL STANDARDS: United States - BUSPROG: Reflective Thinking TOPICS: 3.3 Sensitivity Analysis: Computer Solution KEYWORDS: Bloom's: Understand 23. A constraint with a positive slack value a. will have a positive dual price. b. will have a negative dual price. c. will have a dual price of zero. d. has no restrictions for its dual price. POINTS: 1 DIFFICULTY: Moderate LEARNING OBJECTIVES: IMS.ASWC.19.03.03 - 3.3 NATIONAL STANDARDS: United States - BUSPROG: Reflective Thinking TOPICS: 3.3 Sensitivity Analysis: Computer Solution KEYWORDS: Bloom's: Understand 24. The amount by which an objective function coefficient can change before a different set of values for the decision variables becomes optimal is the a. optimal solution. b. dual solution. c. range of optimality. d. range of feasibility. 25. The range of feasibility measures a. the right-hand-side values for which the objective function value will not change. b. the right-hand-side values for which the values of the decision variables will not change. c. the right-hand-side values for which the dual prices will not change. d. All of these are correct. 26. An objective function reflects the relevant cost of labor hours used in production rather than treating them as a sunk cost. The correct interpretation of the dual price associated with the labor hours constraint is the a. maximum premium (say for overtime) over the normal price that the company would be willing to pay. b. upper limit on the total hourly wage the company would pay. c. reduction in hours that could be sustained before the solution would change. d. number of hours by which the right-hand side can change before there is a change in the solution point. 27. The graphical solution procedure is useful only for linear programs involving a. two decision variables. b. more than two decision variables. c. a single constraint. d. None of these are correct. 28. An improvement in the value of the objective function per unit increase in a right-hand side is the a. sensitivity value. b. constraint coefficient. c. slack value. d. None of these are correct. 29. The amount the objective function coefficient of a decision variable would have to improve before that variable would have a positive value in the solution is the a. dual price. b. surplus variable. c. reduced cost. d. upper limit. 30. Based on the per-unit increase in the right-hand side of the constraint, the dual price measures the a. increase in the value of the optimal solution. b. decrease in the value of the optimal solution. c. improvement in the value of the optimal solution. d. change in the value of the optimal solution. 31. Sensitivity analysis information in computer output is based on the assumption that a. no coefficient changes. b. one coefficient changes. c. two coefficients change. d. all coefficients change. 32. When the cost of a resource is sunk, then the dual price can be interpreted as the a. minimum amount the firm should be willing to pay for one additional unit of the resource. b. maximum amount the firm should be willing to pay for one additional unit of the resource. c. minimum amount the firm should be willing to pay for multiple additional units of the resource. d. maximum amount the firm should be willing to pay for multiple additional units of the resource. 33. Which of the following is NOT a question ed by standard sensitivity analysis information? a. If the right-hand-side value of a constraint changes, will the objective function value change? b. Over what range can a constraint's right-hand-side value change without the constraint's dual price possibly changing? c. By how much will the objective function value change if the right-hand-side value of a constraint changes beyond the range of feasibility? d. By how much will the objective function value change if a decision variable's coefficient in the objective function changes within the range of optimality? 34. The cost that varies depending on the values of the decision variables is a a. reduced cost. b. relevant cost. c. sunk cost. d. dual cost. 35. A cost that is incurred no matter what values the decision variables assume is a(n) a. reduced cost. b. optimal cost. c. sunk cost. d. dual cost. 36. Sensitivity analysis is sometimes referred to as a. feasibility testing. b. duality analysis. c. alternative analysis. d. postoptimality analysis. 37. Sensitivity analysis is concerned with how certain changes affect the a. feasible solution. b. unconstrained solution. c. optimal solution. d. degenerative solution. 38. The dual price for a < constraint will Subjective Short 39. In a linear programming problem, the binding constraints for the optimal solution are: 5X + 3Y ≤ 30 2X + 5Y ≤ 20 a. Fill in the blanks in the following sentence: As long as the slope of the objective function stays between _______ and _______, the current optimal solution point will remain optimal. b. Which of these objective functions will lead to the same optimal solution? (1) 2X + 1Y (2) 7X + 8Y (3) 80X + 60Y (4) 25X + 35Y 40. The optimal solution of this linear programming problem is at the intersection of constraints 1 and 2. a. Over what range can the coefficient of x1 vary before the current solution is no longer optimal? b. Over what range can the coefficient of x2 vary before the current solution is no longer optimal? c. Compute the dual prices for the three constraints. 41. The binding constraints for this problem are the first and second. a. Keeping c2 fixed at 2, over what range can c1 vary before there is a change in the optimal solution point? b. Keeping c1 fixed at 1, over what range can c2 vary before there is a change in the optimal solution point? c. If the objective function becomes Min 1.5x1 + 2x2, what will be the optimal values of x1, x2, and the objective function? d. If the objective function becomes Min 7x1 + 6x2, what constraints will be binding? e. Find the dual price for each constraint in the original problem. 42. Excel's Solver tool has been used in the spreadsheet below to solve a linear programming problem with a maximization objective function and all ≤ constraints. a. Give the original linear programming problem. b. Give the complete optimal solution. 43. Excel's Solver tool has been used in the spreadsheet below to solve a linear programming problem with a minimization objective function and all ≥ constraints. 44. Use the spreadsheet and Excel Solver sensitivity report to these questions. a. What is the cell formula for B12? b. What is the cell formula for C12? c. What is the cell formula for D12? d. What is the cell formula for B15? e. What is the cell formula for B16? f. What is the cell formula for B17? g. What is the optimal value for x1? h. What is the optimal value for x2? i. Would you pay $0.50 each for up to 60 more units of resource 1? j. Is it possible to figure the new objective function value if the profit on product 1 increases by a dollar, or do you have to rerun Solver? 45. A large sporting goods store is placing an order for bicycles with its supplier. Four models can be ordered: the adult Open Trail, the adult Cityscape, the girl's Sea Sprite, and the boy's Trail Blazer. It is assumed that every bike ordered will be sold, and their profits, respectively, are 30, 25, 22, and 20. The LP model should maximize profit. The store needs to worry about several conditions. One of these is space to hold the inventory. An adult's bike needs two feet, but a child's bike needs only one foot. The store has 500 feet of space. There are 1200 hours of assembly time available. The child's bikes need 4 hours of assembly time, the Open Trail needs 5 hours, and the Cityscape needs 6 hours. The store would like to place an order for at least 275 bikes. a. Formulate a model for this problem. b. Solve your model with any computer package available to you. c. How many of each kind of bike should be ordered, and what will the profit be? d. What would the profit be if the store had 100 more feet of storage space? e. If the profit on the Cityscape increases to 35, will any of the Cityscape bikes be ordered? f. Over what range of assembly hours is the dual price applicable? g. If we require 5 more bikes in inventory, what will happen to the value of the optimal solution? h. Which resource should the company work to increase, inventory space or assembly time? 46. Consider the following linear program: The graphical solution to the problem is shown below. From the graph, we see that the optimal solution occurs at x1 = 5, x2 = 3, and z = 46. a. Calculate the range of optimality for each objective function coefficient. b. Calculate the dual price for each resource. : a. Ranges of optimality: 14/3 ≤ c1 ≤ 7 and 5 ≤ c2 ≤ 15/2 b. The dual price for the first resource is 0, for the second resource is 2, and for the third is 1. POINTS: 1 DIFFICULTY: Challenging LEARNING OBJECTIVES: IMS.ASWC.19.03.01 - 3.1 NATIONAL STANDARDS: United States - BUSPROG: Analytic TOPICS: 3.1 Introduction to Sensitivity Analysis KEYWORDS: Bloom’s: Analyze