Introduction to Management Science A Modeling And Case Studies Approach with Spreadsheets 6th Edition By Frederick Hillier - Test Bank

Introduction to Management Science A Modeling And Case Studies Approach with Spreadsheets 6th Edition By Frederick Hillier – Test Bank

 

Instant Download – Complete Test Bank With Answers

 

 

Sample Questions Are Posted Below

 

Intro to Management Science: Modeling and Case Studies, 6e (Hillier)

Chapter 5   What-If Analysis for Linear Programming

 

1) An optimal solution is only optimal with respect to a particular mathematical model that provides only a representation of the actual problem.

 

Answer:  TRUE

Difficulty: 1 Easy

Topic:  The Importance of What-if Analysis to Managers

Learning Objective:  Explain what is meant by what-if analysis.

Bloom’s:  Remember

AACSB:  Knowledge Application

Accessibility:  Keyboard Navigation

 

2) The purpose of a linear programming study is to help guide management’s final decision by providing insights.

 

Answer:  TRUE

Difficulty: 1 Easy

Topic:  The Importance of What-if Analysis to Managers

Learning Objective:  Summarize the benefits of what-if analysis.

Bloom’s:  Remember

AACSB:  Knowledge Application

Accessibility:  Keyboard Navigation

 

3) It is usually quite easy to find the needed data for a linear programming study.

 

Answer:  FALSE

Difficulty: 1 Easy

Topic:  The Importance of What-if Analysis to Managers

Learning Objective:  Summarize the benefits of what-if analysis.

Bloom’s:  Remember

AACSB:  Knowledge Application

Accessibility:  Keyboard Navigation

 

4) If the optimal solution will remain the same over a wide range of values for a particular coefficient in the objective function, then management will want to take special care to narrow this estimate down.

 

Answer:  FALSE

Difficulty: 2 Medium

Topic:  The Importance of What-if Analysis to Managers

Learning Objective:  Summarize the benefits of what-if analysis.

Bloom’s:  Understand

AACSB:  Knowledge Application

Accessibility:  Keyboard Navigation

 

 

5) Shadow price analysis is widely used to help management find the best trade-off between costs and benefits for a problem.

 

Answer:  TRUE

Difficulty: 1 Easy

Topic:  The Effect of Single Changes in a Constraint

Learning Objective:  Predict how the value in the objective cell would change if a small change were to be made in the right-hand side of one or more of the functional constraints.

Bloom’s:  Remember

AACSB:  Knowledge Application

Accessibility:  Keyboard Navigation

 

6) When certain parameters of a model represent managerial policy decisions, what-if analysis provides information about what the impact would be of altering these policy decisions.

 

Answer:  TRUE

Difficulty: 1 Easy

Topic:  The Importance of What-if Analysis to Managers

Learning Objective:  Summarize the benefits of what-if analysis.

Bloom’s:  Remember

AACSB:  Knowledge Application

Accessibility:  Keyboard Navigation

 

7) The term “allowable range for an objective function coefficient” refers to a constraint’s right-hand side quantity.

 

Answer:  FALSE

Difficulty: 1 Easy

Topic:  The Effect of Changes in One Objective Function Coefficient

Learning Objective:  Find how much any single coefficient in the objective function can change without changing the optimal solution.

Bloom’s:  Remember

AACSB:  Knowledge Application

Accessibility:  Keyboard Navigation

 

 

 

8) The allowable range for an objective function coefficient assumes that the original estimates for all the other coefficients are completely accurate so that this is the only one whose true value may differ from its original estimate.

 

Answer:  TRUE

Explanation:  value may differ from its original estimate.

Difficulty: 2 Medium

Topic:  The Effect of Simultaneous Changes in Objective Function Coefficients

Learning Objective:  Find how much any single coefficient in the objective function can change without changing the optimal solution.

Bloom’s:  Understand

AACSB:  Knowledge Application

Accessibility:  Keyboard Navigation

9) A shadow price indicates how much the optimal value of the objective function will increase per unit increase in the right-hand side of a constraint.

 

Answer:  TRUE

Difficulty: 1 Easy

Topic:  The Effect of Single Changes in a Constraint

Learning Objective:  Predict how the value in the objective cell would change if a small change were to be made in the right-hand side of one or more of the functional constraints.

Bloom’s:  Remember

AACSB:  Knowledge Application

Accessibility:  Keyboard Navigation

 

10) When maximizing profit in a linear programming problem, the allowable increase and allowable decrease columns in the sensitivity report make it possible to find the range over which the profitability does not change.

 

Answer:  FALSE

Difficulty: 2 Medium

Topic:  The Effect of Simultaneous Changes in Objective Function Coefficients

Learning Objective:  Find how much any single coefficient in the objective function can change without changing the optimal solution.

Bloom’s:  Understand

AACSB:  Knowledge Application

Accessibility:  Keyboard Navigation

 

 

 

11) Changing the objective function coefficients may or may not change the optimal solution, but it will always change the value of the objective function.

 

Answer:  FALSE

Difficulty: 1 Easy

Topic:  The Effect of Simultaneous Changes in Objective Function Coefficients

Learning Objective:  Find how much any single coefficient in the objective function can change without changing the optimal solution.

Bloom’s:  Remember

AACSB:  Knowledge Application

Accessibility:  Keyboard Navigation

 

12) Every change in the value of an objective function coefficient will lead to a changed optimal solution.

 

Answer:  FALSE

Difficulty: 2 Medium

Topic:  The Effect of Simultaneous Changes in Objective Function Coefficients

Learning Objective:  Find how much any single coefficient in the objective function can change without changing the optimal solution.

Bloom’s:  Understand

AACSB:  Knowledge Application

Accessibility:  Keyboard Navigation

13) When a change in the value of an objective function coefficient remains within the allowable range, the optimal solution will also remain the same.

 

Answer:  TRUE

Difficulty: 1 Easy

Topic:  The Effect of Changes in One Objective Function Coefficient

Learning Objective:  Find how much any single coefficient in the objective function can change without changing the optimal solution.

Bloom’s:  Remember

AACSB:  Knowledge Application

Accessibility:  Keyboard Navigation

 

14) According to the 100% rule for simultaneous changes in objective function coefficients, if the sum of the percentage changes exceeds 100%, the optimal solution definitely will change.

 

Answer:  FALSE

Difficulty: 2 Medium

Topic:  The Effect of Simultaneous Changes in Objective Function Coefficients

Learning Objective:  Evaluate simultaneous changes in objective function coefficients to determine whether the changes are small enough that the original optimal solution must still be optimal.

Bloom’s:  Understand

AACSB:  Knowledge Application

Accessibility:  Keyboard Navigation

 

15) Whenever proportional changes are made to all the unit profits in a problem, the optimal solution will remain the same.

 

Answer:  TRUE

Difficulty: 1 Easy

Topic:  The Effect of Simultaneous Changes in Objective Function Coefficients

Learning Objective:  Evaluate simultaneous changes in objective function coefficients to determine whether the changes are small enough that the original optimal solution must still be optimal.

Bloom’s:  Remember

AACSB:  Knowledge Application

Accessibility:  Keyboard Navigation

 

16) The term “allowable range for the right-hand-side” refers to coefficients of the objective function.

 

Answer:  FALSE

Difficulty: 1 Easy

Topic:  The Effect of Single Changes in a Constraint

Learning Objective:  Find how much the right-hand side of a single functional constraint can change before this prediction becomes no longer valid.

Bloom’s:  Remember

AACSB:  Knowledge Application

Accessibility:  Keyboard Navigation

17) If the change to a right-hand side is within the allowable range, the value of the shadow price remains valid.

 

Answer:  TRUE

Difficulty: 1 Easy

Topic:  The Effect of Single Changes in a Constraint

Learning Objective:  Find how much the right-hand side of a single functional constraint can change before this prediction becomes no longer valid.

Bloom’s:  Remember

AACSB:  Knowledge Application

Accessibility:  Keyboard Navigation

 

 

 

18) If the change to a right-hand side is within the allowable range, the solution will remain the same.

 

Answer:  FALSE

Difficulty: 1 Easy

Topic:  The Effect of Single Changes in a Constraint

Learning Objective:  Predict how the value in the objective cell would change if a small change were to be made in the right-hand side of one or more of the functional constraints.

Bloom’s:  Remember

AACSB:  Knowledge Application

Accessibility:  Keyboard Navigation

 

19) A shadow price tells how much a decision variable can be increased or decreased without changing the value of the solution.

 

Answer:  FALSE

Difficulty: 1 Easy

Topic:  The Effect of Single Changes in a Constraint

Learning Objective:  Predict how the value in the objective cell would change if a small change were to be made in the right-hand side of one or more of the functional constraints.

Bloom’s:  Remember

AACSB:  Knowledge Application

Accessibility:  Keyboard Navigation

 

20) The allowable range gives ranges of values for the objective function coefficients within which the values of the decision variables are optimal.

 

Answer:  TRUE

Difficulty: 1 Easy

Topic:  The Effect of Changes in One Objective Function Coefficient

Learning Objective:  Find how much any single coefficient in the objective function can change without changing the optimal solution.

Bloom’s:  Remember

AACSB:  Knowledge Application

Accessibility:  Keyboard Navigation

21) When a change occurs in the right-hand side values of one of the constraints, a proportional change will occur in one of the coefficients of the objective function.

 

Answer:  FALSE

Difficulty: 2 Medium

Topic:  The Effect of Single Changes in a Constraint

Learning Objective:  Find how much the right-hand side of a single functional constraint can change before this prediction becomes no longer valid.

Bloom’s:  Understand

AACSB:  Knowledge Application

Accessibility:  Keyboard Navigation

 

22) Managerial decisions regarding right-hand sides are often interrelated and so frequently are considered simultaneously.

 

Answer:  TRUE

Difficulty: 2 Medium

Topic:  The Effect of Simultaneous Changes in the Constraints

Learning Objective:  Evaluate simultaneous changes in right-hand sides to determine whether the changes are small enough that this prediction must still be valid.

Bloom’s:  Understand

AACSB:  Knowledge Application

Accessibility:  Keyboard Navigation

 

23) If the sum of the percentage changes of the right-hand sides does not exceed 100%, then the solution will definitely remain optimal.

 

Answer:  FALSE

Difficulty: 2 Medium

Topic:  The Effect of Simultaneous Changes in the Constraints

Learning Objective:  Evaluate simultaneous changes in right-hand sides to determine whether the changes are small enough that this prediction must still be valid.

Bloom’s:  Understand

AACSB:  Knowledge Application

Accessibility:  Keyboard Navigation

 

24) A parameter analysis report re-solves the problem for a range of values of a data cell.

 

Answer:  TRUE

Difficulty: 1 Easy

Topic:  The Effect of Simultaneous Changes in the Constraints

Learning Objective:  Use Parameters with Analytic Solver to systematically investigate the effect of changing either one or two data cells to various other trial values.

Bloom’s:  Remember

AACSB:  Knowledge Application

Accessibility:  Keyboard Navigation

25) A parameter analysis report can only be used to investigate changes in a single data cell at a time.

 

Answer:  FALSE

Difficulty: 1 Easy

Topic:  The Effect of Simultaneous Changes in Objective Function Coefficients

Learning Objective:  Use Parameters with Analytic Solver to systematically investigate the effect of changing either one or two data cells to various other trial values.

Bloom’s:  Remember

AACSB:  Knowledge Application

Accessibility:  Keyboard Navigation

 

 

 

26) A parameter analysis report can be used to easily investigate the changes in any number of data cells.

 

Answer:  FALSE

Difficulty: 1 Easy

Topic:  The Effect of Simultaneous Changes in Objective Function Coefficients

Learning Objective:  Use Parameters with Analytic Solver to systematically investigate the effect of changing either one or two data cells to various other trial values.

Bloom’s:  Remember

AACSB:  Knowledge Application

Accessibility:  Keyboard Navigation

 

27) A shadow price reflects which of the following in a maximization problem?

1. A) The marginal cost of adding additional resources.
2. B) The marginal gain in the objective value realized by adding one unit of a resource.
3. C) The marginal loss in the objective value realized by adding one unit of a resource.
4. D) The marginal gain in the objective value realized by subtracting one unit of a resource.
5. E) None of the choices is correct.

 

Answer:  B

Difficulty: 1 Easy

Topic:  The Effect of Single Changes in a Constraint

Learning Objective:  Predict how the value in the objective cell would change if a small change were to be made in the right-hand side of one or more of the functional constraints.

Bloom’s:  Remember

AACSB:  Knowledge Application

Accessibility:  Keyboard Navigation

28) In linear programming, what-if analysis is associated with determining the effect of changing:

 

1. objective function coefficients.
2. right-hand side values of constraints.

III. decision variable values.

1. A) objective function coefficients and right-hand side values of constraints.
2. B) right-hand side values of constraints and decision variable values.
3. C) objective function coefficients, right-hand side values of constraints, and decision variable values.
4. D) objective function coefficients and decision variable values.
5. E) None of the choices is correct.

 

Answer:  A

Difficulty: 1 Easy

Topic:  The Importance of What-if Analysis to Managers

Learning Objective:  Explain what is meant by what-if analysis.

Bloom’s:  Remember

AACSB:  Knowledge Application

Accessibility:  Keyboard Navigation

 

29) What-if analysis can:

 

1. be done graphically for problems with two decision variables.
2. reduce a manager’s confidence in the model that has been formulated.

III. increase a manager’s confidence in the model that has been formulated.

1. A) I only.
2. B) II only.
3. C) III only.
4. D) All of the these.
5. E) I and III only.

 

Answer:  D

Difficulty: 1 Easy

Topic:  The Importance of What-if Analysis to Managers

Learning Objective:  Explain what is meant by what-if analysis.

Bloom’s:  Remember

AACSB:  Knowledge Application

Accessibility:  Keyboard Navigation

30) What-if analysis:

1. A) may involve changes in the objective function coefficients.
2. B) requires that only one parameter change while the rest are held fixed.
3. C) may involve changes in the right-hand side values.
4. D) All of the choices are correct.
5. E) None of the choices is correct.

 

Answer:  D

Difficulty: 2 Medium

Topic:  The Importance of What-if Analysis to Managers

Learning Objective:  Explain what is meant by what-if analysis.

Bloom’s:  Understand

AACSB:  Knowledge Application

Accessibility:  Keyboard Navigation

 

 

 

31) If a change is made in only one of the objective function coefficients:

1. A) the slope of the objective function line always will change.
2. B) the optimal solution always will change.
3. C) one or more of the decision variables always will change.
4. D) All of the choices are correct.
5. E) None of the choices is correct.

 

Answer:  A

Difficulty: 2 Medium

Topic:  The Importance of What-if Analysis to Managers

Learning Objective:  Enumerate the different kinds of changes in the model that can be considered by what-if analysis.

Bloom’s:  Understand

AACSB:  Knowledge Application

Accessibility:  Keyboard Navigation

 

32) If the right-hand side value of a constraint in a two variable linear programming problems is changed, then:

1. A) the optimal measure of performance may change.
2. B) a parallel shift must be made in the graph of that constraint.
3. C) the optimal values for one or more of the decision variables may change.
4. D) All of the choices are correct.
5. E) None of the choices is correct.

 

Answer:  D

Difficulty: 2 Medium

Topic:  The Effect of Single Changes in a Constraint

Learning Objective:  Find how much the right-hand side of a single functional constraint can change before this prediction becomes no longer valid.

Bloom’s:  Understand

AACSB:  Knowledge Application

Accessibility:  Keyboard Navigation

33) Which of the following are benefits of what-if analysis?

1. A) It pinpoints the sensitive parameters of the model.
2. B) It gives the new optimal solution if conditions change.
3. C) It tells management what policy decisions to make.
4. D) All of the choices are correct.
5. E) None of the choices is correct.

 

Answer:  A

Difficulty: 1 Easy

Topic:  The Importance of What-if Analysis to Managers

Learning Objective:  Summarize the benefits of what-if analysis.

Bloom’s:  Remember

AACSB:  Knowledge Application

Accessibility:  Keyboard Navigation

 

34) When even a small change in the value of a coefficient in the objective function can change the optimal solution, the coefficient is called:

1. A) optimal.
2. B) sensitive.
3. C) out of the range.
4. D) within the range.
5. E) None of the choices is correct.

 

Answer:  B

Difficulty: 1 Easy

Topic:  The Importance of What-if Analysis to Managers

Learning Objective:  Summarize the benefits of what-if analysis.

Bloom’s:  Remember

AACSB:  Knowledge Application

Accessibility:  Keyboard Navigation

35) In a problem with 4 decision variables, the 100% rule indicates that each objective coefficient can be safely increased by what amount without invalidating the current optimal solution?

1. A) 25%.
2. B) 25% of the allowable increase of that coefficient.
3. C) 100%.
4. D) 25% of the range of optimality.
5. E) It can’t be determined from the information given.

 

Answer:  B

Explanation:  With 4 decision variables an increase of 25% of the allowable increase would result in a total change of 100% (4 × 25%), which does not violate the 100% rule.

Difficulty: 2 Medium

Topic:  The Effect of Simultaneous Changes in Objective Function Coefficients

Learning Objective:  Evaluate simultaneous changes in objective function coefficients to determine whether the changes are small enough that the original optimal solution must still be optimal.

Bloom’s:  Analyze

AACSB:  Knowledge Application

Accessibility:  Keyboard Navigation

 

 

36) variable cells

Cell	Name	Final Value	Reduced Cost	Objective Coefficient	Allowable Increase	Allowable Decrease
$B$6	Activity 1	3	0	30	23	17
$C$6	Activity 2	6	0	40	50	10
$D$6	Activity 3	0	–7	20	7	1E+30

 

Constraints

Cell	Name	Final Value	Shadow Price	Constraint R.H. Side	Allowable Increase	Allowable Decrease
$E$2	Resource A	20	7.78	20	10	12.5
$E$3	Resource B	30	6	30	50	10
$E$4	Resource C	18	0	40	1E+30	22

 

What is the optimal objective function value for this problem?

1. A) It cannot be determined from the given information.
2. B) $7.78
3. C) $240
4. D) $90
5. E) $330

 

Answer:  E

Difficulty: 2 Medium

Topic:  The Importance of What-if Analysis to Managers

Learning Objective:  Describe how the spreadsheet formulation of the problem can be used to perform any of these kinds of what-if analysis.

Bloom’s:  Evaluate

AACSB:  Analytical Thinking

Accessibility:  Keyboard Navigation

 

 

37) Variable cells

Cell	Name	Final Value	Reduced Cost	Objective Coefficient	Allowable Increase	Allowable Decrease
$B$6	Activity 1	3	0	30	23	17
$C$6	Activity 2	6	0	40	50	10
$D$6	Activity 3	0	–7	20	7	1E+30

 

Constraints

Cell	Name	Final Value	Shadow Price	Constraint R.H. Side	Allowable Increase	Allowable Decrease
$E$2	Resource A	20	7.78	20	10	12.5
$E$3	Resource B	30	6	30	50	10
$E$4	Resource C	18	0	40	1E+30	22

 

What is the allowable range for the objective coefficient for Activity 2?

1. A) −10 ≤ A2 ≤ 50
2. B) −44 ≤ A2 ≤ 16
3. C) −4 ≤ A2 ≤ 56
4. D) 30 ≤ A2 ≤ 90
5. E) 20 ≤ A2 ≤ 80

 

Answer:  D

Explanation:  The objective coefficient for Activity 2 is 40. The allowable decrease is 10 and the allowable increase is 50. Therefore, the allowable range for this objective coefficient is (40 − 10) ≤ A2 ≤ (40 + 50) → 30 ≤ A2 ≤ 90 .(40-10)≤A2≤(40+50)→30≤A2≤90 (40-10)≤A2≤(40+50)→30≤A2≤90

Difficulty: 2 Medium

Topic:  The Effect of Changes in One Objective Function Coefficient

Learning Objective:  Find how much any single coefficient in the objective function can change without changing the optimal solution.

Bloom’s:  Apply

AACSB:  Knowledge Application

Accessibility:  Keyboard Navigation

 

 

38) Variable cells

Cell	Name	Final Value	Reduced Cost	Objective Coefficient	Allowable Increase	Allowable Decrease
$B$6	Activity 1	3	0	30	23	17
$C$6	Activity 2	6	0	40	50	10
$D$6	Activity 3	0	–7	20	7	1E+30

 

Constraints

Cell	Name	Final Value	Shadow Price	Constraint R.H. Side	Allowable Increase	Allowable Decrease
$E$2	Resource A	20	7.78	20	10	12.5
$E$3	Resource B	30	6	30	50	10
$E$4	Resource C	18	0	40	1E+30	22

 

What is the allowable range for the right-hand-side for Resource C?

1. A) 18 ≤ RHSc ≤ ∞
2. B) ∞ ≤ RHSc≤ 62
3. C) −2 ≤RHSc ≤ ∞
4. D) − ∞ ≤ RHSc ≤ 40
5. E) 0 ≤ RHSc≤ 22

 

Answer:  A

Explanation:  The right-hand side for Resource C is 40. The allowable decrease is 22 and the allowable increase is ∞ (infinity). Therefore, the allowable range for this right-hand side is (40 − 22) ≤ RHSc ≤ (40 + ∞ ) → 18 ≤ RHSc ≤ ∞.

Difficulty: 2 Medium

Topic:  The Effect of Single Changes in a Constraint

Learning Objective:  Find how much the right-hand side of a single functional constraint can change before this prediction becomes no longer valid.

Bloom’s:  Understand

AACSB:  Knowledge Application

Accessibility:  Keyboard Navigation

 

 

 

39) Variable cells

Cell	Name	Final Value	Reduced Cost	Objective Coefficient	Allowable Increase	Allowable Decrease
$B$6	Activity 1	3	0	30	23	17
$C$6	Activity 2	6	0	40	50	10
$D$6	Activity 3	0	–7	20	7	1E+30

 

Constraints

Cell	Name	Final Value	Shadow Price	Constraint R.H. Side	Allowable Increase	Allowable Decrease
$E$2	Resource A	20	7.78	20	10	12.5
$E$3	Resource B	30	6	30	50	10
$E$4	Resource C	18	0	40	1E+30	22

 

If the coefficient for Activity 1 in the objective function changes to $40, then the objective function value:

 

77. A) will increase by $77.80.
78. B) will increase by $23.
79. C) will increase by $30.
80. D) will remain the same.
81. E) can only be discovered by resolving the problem.

 

Answer:  C

Explanation:  Increasing the objective function coefficient for Activity 1 to 40 is an increase of 10 (40 − 30 = 10). Since the allowable increase for this objective function coefficient is 23, this is within the allowable range and the optimal solution will not change. The final value for Activity 1 is 3, so increasing the objective function coefficient by 10 leads to an increase in the objective function of 30 {(40 − 30) × 3 = 30}.

Difficulty: 3 Hard

Topic:  The Effect of Changes in One Objective Function Coefficient

Learning Objective:  Find how much any single coefficient in the objective function can change without changing the optimal solution.

Bloom’s:  Evaluate

AACSB:  Analytical Thinking

Accessibility:  Keyboard Navigation

 

 

 

40) Variable cells

Cell	Name	Final Value	Reduced Cost	Objective Coefficient	Allowable Increase	Allowable Decrease
$B$6	Activity 1	3	0	30	23	17
$C$6	Activity 2	6	0	40	50	10
$D$6	Activity 3	0	–7	20	7	1E+30

 

Constraints

Cell	Name	Final Value	Shadow Price	Constraint R.H. Side	Allowable Increase	Allowable Decrease
$E$2	Resource A	20	7.78	20	10	12.5
$E$3	Resource B	30	6	30	50	10
$E$4	Resource C	18	0	40	1E+30	22

 

If the coefficient for Activity 3 in the objective function changes to $30, then the objective function value:

70. A) will increase by $70.
71. B) is $0.
72. C) will increase by $30.
73. D) will remain the same.
74. E) will increase by an unknown amount.

 

Answer:  E

Explanation:  Increasing the objective function coefficient for Activity 3 to 30 is an increase of 10 (30 − 20 = 10). Since the allowable increase for Activity 3 is 7, this change exceeds the allowable increase and the impact on the optimal solution cannot be determined without re-running the optimization.

Difficulty: 3 Hard

Topic:  The Effect of Changes in One Objective Function Coefficient

Learning Objective:  Find how much any single coefficient in the objective function can change without changing the optimal solution.

Bloom’s:  Analyze

AACSB:  Analytical Thinking

Accessibility:  Keyboard Navigation

 

 

41) Variable cells

Cell	Name	Final Value	Reduced Cost	Objective Coefficient	Allowable Increase	Allowable Decrease
$B$6	Activity 1	3	0	30	23	17
$C$6	Activity 2	6	0	40	50	10
$D$6	Activity 3	0	–7	20	7	1E+30

 

Constraints

Cell	Name	Final Value	Shadow Price	Constraint R.H. Side	Allowable Increase	Allowable Decrease
$E$2	Resource A	20	7.78	20	10	12.5
$E$3	Resource B	30	6	30	50	10
$E$4	Resource C	18	0	40	1E+30	22

 

If the coefficient of Activity 1 in the objective function changes to $10, then:

1. A) the original solution remains optimal.
2. B) the problem must be resolved to find the optimal solution.
3. C) the shadow price is valid.
4. D) the shadow price is not valid.
5. E) None of the above.

 

Answer:  B

Explanation:  Decreasing the objective function coefficient for Activity 1 to 10 is a decrease of 20 (30 − 10 = 20). Since the allowable decrease for Activity 1 is 17, this change exceeds the allowable decrease and the impact on the optimal solution cannot be determined without re-running the optimization.

Difficulty: 3 Hard

Topic:  The Effect of Changes in One Objective Function Coefficient

Learning Objective:  Find how much any single coefficient in the objective function can change without changing the optimal solution.

Bloom’s:  Analyze

AACSB:  Analytical Thinking

Accessibility:  Keyboard Navigation

 

 

 

42) Variable cells

Cell	Name	Final Value	Reduced Cost	Objective Coefficient	Allowable Increase	Allowable Decrease
$B$6	Activity 1	3	0	30	23	17
$C$6	Activity 2	6	0	40	50	10
$D$6	Activity 3	0	–7	20	7	1E+30

 

Constraints

Cell	Name	Final Value	Shadow Price	Constraint R.H. Side	Allowable Increase	Allowable Decrease
$E$2	Resource A	20	7.78	20	10	12.5
$E$3	Resource B	30	6	30	50	10
$E$4	Resource C	18	0	40	1E+30	22

 

If the right-hand side of Resource A changes to 10, then the objective function value:

 

12. A) will decrease by $12.50.
13. B) will decrease by $125.
14. C) will decrease by $77.80.
15. D) will remain the same.
16. E) can only be discovered by resolving the problem.

 

Answer:  C

Explanation:  Decreasing the constraint right-hand side for Resource A to 10 is a decrease of 10 (20 − 10 = 10). Since the allowable decrease for this constraint right-hand side is 12.5, this is within the allowable range and the shadow price still applies. A decrease of 10 in the constraint right-hand side leads to a decrease of 77.80 in the objective function—multiply the change in the constraint right-hand side by the shadow price {(20 − 10) × 7.78 = 77.80}.

Difficulty: 3 Hard

Topic:  The Effect of Single Changes in a Constraint

Learning Objective:  Predict how the value in the objective cell would change if a small change were to be made in the right-hand side of one or more of the functional constraints.

Bloom’s:  Evaluate

AACSB:  Analytical Thinking

Accessibility:  Keyboard Navigation

 

 

43) Variable cells

Cell	Name	Final Value	Reduced Cost	Objective Coefficient	Allowable Increase	Allowable Decrease
$B$6	Activity 1	3	0	30	23	17
$C$6	Activity 2	6	0	40	50	10
$D$6	Activity 3	0	–7	20	7	1E+30

 

Constraints

Cell	Name	Final Value	Shadow Price	Constraint R.H. Side	Allowable Increase	Allowable Decrease
$E$2	Resource A	20	7.78	20	10	12.5
$E$3	Resource B	30	6	30	50	10
$E$4	Resource C	18	0	40	1E+30	22

 

If the right-hand side of Resource B changes to 10, then the objective function value:

120. A) will decrease by $120.
121. B) will decrease by $60.
122. C) will decrease by $20.
123. D) will remain the same.
124. E) can only be discovered by resolving the problem.

 

Answer:  E

Explanation:  Decreasing the constraint right-hand side for Resource B to 10 is a decrease of 20 (30 − 10 = 20). Since the allowable decrease for Resource B is 10, this change exceeds the allowable decrease and the impact on the optimal solution cannot be determined without re-running the optimization.

Difficulty: 2 Medium

Topic:  The Effect of Single Changes in a Constraint

Learning Objective:  Find how much the right-hand side of a single functional constraint can change before this prediction becomes no longer valid.

Bloom’s:  Understand

AACSB:  Knowledge Application

Accessibility:  Keyboard Navigation

 

 

44) Variable cells

Cell	Name	Final Value	Reduced Cost	Objective Coefficient	Allowable Increase	Allowable Decrease
$B$6	Activity 1	3	0	30	23	17
$C$6	Activity 2	6	0	40	50	10
$D$6	Activity 3	0	−7	20	7	1E+30

 

Constraints

Cell	Name	Final Value	Shadow Price	Constraint R.H. Side	Allowable Increase	Allowable Decrease
$E$2	Resource A	20	7.78	20	10	12.5
$E$3	Resource B	30	6	30	50	10
$E$4	Resource C	18	0	40	1E+30	22

 

If the right-hand side of Resource B changes to 10, then:

1. A) the original solution remains optimal.
2. B) the problem must be resolved to find the optimal solution.
3. C) the shadow price is valid.
4. D) the shadow price is not valid.
5. E) None of the choices is correct.

 

Answer:  D

Explanation:  Decreasing the constraint right-hand side for Resource B to 10 is a decrease of 20 (30 − 10 = 20). Since the allowable decrease for Resource B is 10, this change exceeds the allowable decrease and the impact on the optimal solution cannot be determined without re-running the optimization.

Difficulty: 2 Medium

Topic:  The Effect of Single Changes in a Constraint

Learning Objective:  Find how much the right-hand side of a single functional constraint can change before this prediction becomes no longer valid.

Bloom’s:  Understand

AACSB:  Knowledge Application

Accessibility:  Keyboard Navigation

 

 

 

45) Variable cells

Cell	Name	Final Value	Reduced Cost	Objective Coefficient	Allowable Increase	Allowable Decrease
$B$6	Activity 1	3	0	30	23	17
$C$6	Activity 2	6	0	40	50	10
$D$6	Activity 3	0	–7	20	7	1E+30

 

Constraints

Cell	Name	Final Value	Shadow Price	Constraint R.H. Side	Allowable Increase	Allowable Decrease
$E$2	Resource A	20	7.78	20	10	12.5
$E$3	Resource B	30	6	30	50	10
$E$4	Resource C	18	0	40	1E+30	22

 

If the coefficients of Activity 1 and Activity 2 in the objective function are both increased by $10, then:

 

1. A) the optimal solution remains the same.
2. B) the optimal solution may or may not remain the same.
3. C) the optimal solution will change.
4. D) the shadow prices are valid.
5. E) None of the choices is correct.

 

Answer:  A

Explanation:  Applying the 100% rule, the change in Activity 1 is 43.48% of the allowable increase . The change in Activity 2 is 20% of the allowable increase . The total change of 6.3.48% (43.48% + 20%) does not exceed 100%, so the optimal solution will not change.

Difficulty: 3 Hard

Topic:  The Effect of Simultaneous Changes in Objective Function Coefficients

Learning Objective:  Evaluate simultaneous changes in objective function coefficients to determine whether the changes are small enough that the original optimal solution must still be optimal.

Bloom’s:  Evaluate

AACSB:  Analytical Thinking

Accessibility:  Keyboard Navigation

 

 

 

46) Variable cells

Cell	Name	Final Value	Reduced Cost	Objective Coefficient	Allowable Increase	Allowable Decrease
$B$6	Activity 1	3	0	30	23	17
$C$6	Activity 2	6	0	40	50	10
$D$6	Activity 3	0	–7	20	7	1E+30

 

Constraints

Cell	Name	Final Value	Shadow Price	Constraint R.H. Side	Allowable Increase	Allowable Decrease
$E$2	Resource A	20	7.78	20	10	12.5
$E$3	Resource B	30	6	30	50	10
$E$4	Resource C	18	0	40	1E+30	22

 

If the right-hand side of Resource B is increased by 30, and the right-hand side of Resource C is decreased by 10, then:

1. A) the optimal solution remains the same.
2. B) the optimal solution will change.
3. C) the shadow prices are valid.
4. D) the shadow prices may or may not be valid.
5. E) None of the choices is correct.

 

Answer:  D

Explanation:  Both changes are within the allowable range for the resources. Applying the 100% rule, the change in Resource B is 60% of the allowable increase . The change in Resource C is 45.5% of the allowable decrease . The total change of 105.5% (60% + 45.5%) is greater than 100%, so the shadow prices may or may not remain valid.

Difficulty: 2 Medium

Topic:  The Effect of Simultaneous Changes in the Constraints

Learning Objective:  Evaluate simultaneous changes in right-hand sides to determine whether the changes are small enough that this prediction must still be valid.

Bloom’s:  Understand

AACSB:  Knowledge Application

Accessibility:  Keyboard Navigation

 

 

47) Variable cells

Cell	Name	Final Value	Reduced Cost	Objective Coefficient	Allowable Increase	Allowable Decrease
$B$6	Activity 1	3	0	30	23	17
$C$6	Activity 2	6	0	40	50	10
$D$6	Activity 3	0	–7	20	7	1E+30

 

Constraints

Cell	Name	Final Value	Shadow Price	Constraint R.H. Side	Allowable Increase	Allowable Decrease
$E$2	Resource A	20	7.78	20	10	12.5
$E$3	Resource B	30	6	30	50	10
$E$4	Resource C	18	0	40	1E+30	22

 

Which parameter is most sensitive to an increase in its value?

1. A) The objective coefficient of Activity 1.
2. B) The objective coefficient of Activity 2.
3. C) The objective coefficient of Activity 3.
4. D) All of the choices are correct.
5. E) None of the choices is correct.

 

Answer:  C

Explanation:  The allowable increase for Activity 3 (7 according to the sensitivity report) is smaller than the allowable increases for both Activity 1 (allowable increase of 23) and Activity 2 (allowable increase of 50).

Difficulty: 2 Medium

Topic:  The Effect of Changes in One Objective Function Coefficient

Learning Objective:  Find how much any single coefficient in the objective function can change without changing the optimal solution.

Bloom’s:  Understand

AACSB:  Knowledge Application

Accessibility:  Keyboard Navigation

 

 

48) Variable cells

Cell	Name	Final Value	Reduced Cost	Objective Coefficient	Allowable Increase	Allowable Decrease
$B$6	Activity 1	0	425	500	1E+30	425
$C$6	Activity 2	27.5	0.0	300	500	300
$D$6	Activity 3	0	250	400	1E+30	250

 

Constraints

Cell	Name	Final Value	Shadow Price	Constraint R.H. Side	Allowable Increase	Allowable Decrease
$E$2	Benefit A	110	0	60	50	1E+30
$E$3	Benefit B	110	75	110	1E+30	46
$E$4	Benefit C	137.5	0	80	57.5	1E+30

 

What is the optimal objective function value for this problem?

1. A) It cannot be determined from the given information.
2. B) 1,200
3. C) 975
4. D) 8,250
5. E) 500

 

Answer:  D

Explanation:  The objective function value is calculated by multiplying the final value of each variable by the appropriate objective function coefficient. (425 × 0) + (300 × 27.5) + (400 × 0) = 8,250.

Difficulty: 3 Hard

Topic:  The Importance of What-if Analysis to Managers

Learning Objective:  Enumerate the different kinds of changes in the model that can be considered by what-if analysis.

Bloom’s:  Analyze

AACSB:  Analytical Thinking

Accessibility:  Keyboard Navigation

 

 

49) Variable cells

Cell	Name	Final Value	Reduced Cost	Objective Coefficient	Allowable Increase	Allowable Decrease
$B$6	Activity 1	0	425	500	1E+30	425
$C$6	Activity 2	27.5	0.0	300	500	300
$D$6	Activity 3	0	250	400	1E+30	250

 

Constraints

Cell	Name	Final Value	Shadow Price	Constraint R.H. Side	Allowable Increase	Allowable Decrease
$E$2	Benefit A	110	0	60	50	1E+30
$E$3	Benefit B	110	75	110	1E+30	46
$E$4	Benefit C	137.5	0	80	57.5	1E+30

 

What is the allowable range for the objective function coefficient for Activity 3?

1. A) 150 ≤ A3 ≤ ∞
2. B) 0 ≤ A3 ≤ 650
3. C) 0 ≤ A3 ≤ 250
4. D) 400 ≤ A3 ≤ ∞
5. E) 300 ≤ A3 ≤ 500

 

Answer:  A

Explanation:  The objective coefficient for Activity 3 is 400. The allowable decrease is 250 and the allowable increase is ∞ (infinity). Therefore, the allowable range for this objective coefficient is (400 − 250) ≤ A3 ≤ (40 + ∞) → 150 ≤ A3 ∞.

Difficulty: 3 Hard

Topic:  The Effect of Changes in One Objective Function Coefficient

Learning Objective:  Find how much any single coefficient in the objective function can change without changing the optimal solution.

Bloom’s:  Analyze

AACSB:  Analytical Thinking

Accessibility:  Keyboard Navigation

 

 

50) Variable cells

Cell	Name	Final Value	Reduced Cost	Objective Coefficient	Allowable Increase	Allowable Decrease
$B$6	Activity 1	0	425	500	1E+30	425
$C$6	Activity 2	27.5	0.0	300	500	300
$D$6	Activity 3	0	250	400	1E+30	250

 

Constraints

Cell	Name	Final Value	Shadow Price	Constraint R.H. Side	Allowable Increase	Allowable Decrease
$E$2	Benefit A	110	0	60	50	1E+30
$E$3	Benefit B	110	75	110	1E+30	46
$E$4	Benefit C	137.5	0	80	57.5	1E+30

 

What is the allowable range of the right-hand-side for Resource A?

1. A) –∞ ≤ RHSA≤ 60
2. B) 0 ≤ RHSA≤ 110
3. C) –∞ ≤ RHSA≤ 110
4. D) 110 ≤ RHSA≤ 1600
5. E) 0 ≤ RHSA≤ 160

 

Answer:  C

Explanation:  The right-hand side for Resource A is 60. The allowable decrease is ∞ (infinity) and the allowable increase is 50. Therefore, the allowable range for this right-hand side is (60 – ∞ ) ≤ RHSA ≤ (60 + 50 ) → −∞ ≤ RHSA ≤ 110.

Difficulty: 3 Hard

Topic:  The Effect of Single Changes in a Constraint

Learning Objective:  Find how much the right-hand side of a single functional constraint can change before this prediction becomes no longer valid.

Bloom’s:  Analyze

AACSB:  Analytical Thinking

Accessibility:  Keyboard Navigation

 

 

 

51) Variable cells

Cell	Name	Final Value	Reduced Cost	Objective Coefficient	Allowable Increase	Allowable Decrease
$B$6	Activity 1	0	425	500	1E+30	425
$C$6	Activity 2	27.5	0.0	300	500	300
$D$6	Activity 3	0	250	400	1E+30	250

 

Constraints

Cell	Name	Final Value	Shadow Price	Constraint R.H. Side	Allowable Increase	Allowable Decrease
$E$2	Benefit A	110	0	60	50	1E+30
$E$3	Benefit B	110	75	110	1E+30	46
$E$4	Benefit C	137.5	0	80	57.5	1E+30

 

If the coefficient for Activity 2 in the objective function changes to $400, then the objective function value:

 

500. A) will increase by $7,500.
501. B) will increase by $2,750.
502. C) will increase by $100.
503. D) will remain the same.
504. E) can only be discovered by resolving the problem.

 

Answer:  B

Explanation:  Increasing the objective function coefficient for Activity 2 to 400 is an increase of 100 (400 − 300 = 100). Since the allowable increase for Activity 2 is 500, this change is within the range of optimality and the final values for the variables will not change. The new objective function value is calculated by multiplying the final value of each variable by the appropriate objective function coefficient. (425 × 0) + (400 × 27.5) + (400 × 0) = 11,000, which is an increase of 2,750 over the original objective function value of 8,250.

Difficulty: 3 Hard

Topic:  The Effect of Changes in One Objective Function Coefficient

Learning Objective:  Find how much any single coefficient in the objective function can change without changing the optimal solution.

Bloom’s:  Analyze

AACSB:  Analytical Thinking

Accessibility:  Keyboard Navigation

 

 

 

52) Variable cells

Cell	Name	Final Value	Reduced Cost	Objective Coefficient	Allowable Increase	Allowable Decrease
$B$6	Activity 1	0	425	500	1E+30	425
$C$6	Activity 2	27.5	0.0	300	500	300
$D$6	Activity 3	0	250	400	1E+30	250

 

Constraints

Cell	Name	Final Value	Shadow Price	Constraint R.H. Side	Allowable Increase	Allowable Decrease
$E$2	Benefit A	110	0	60	50	1E+30
$E$3	Benefit B	110	75	110	1E+30	46
$E$4	Benefit C	137.5	0	80	57.5	1E+30

 

If the coefficient for Activity 1 in the objective function changes to $50, then the objective function value:

450. A) will decrease by $450.
451. B) is $0.
452. C) will decrease by $2750.
453. D) will remain the same.
454. E) can only be discovered by resolving the problem.

 

Answer:  E

Explanation:  Decreasing the objective function coefficient for Activity 1 to 50 is a decrease of 450 (500 − 50 = 450). Since the allowable decrease for Activity 1 is 425, this change exceeds the allowable decrease and the impact on the optimal solution cannot be determined without re-running the optimization.

Difficulty: 2 Medium

Topic:  The Effect of Changes in One Objective Function Coefficient

Learning Objective:  Find how much any single coefficient in the objective function can change without changing the optimal solution.

Bloom’s:  Understand

AACSB:  Knowledge Application

Accessibility:  Keyboard Navigation

 

 

53) Variable cells

Cell	Name	Final Value	Reduced Cost	Objective Coefficient	Allowable Increase	Allowable Decrease
$B$6	Activity 1	0	425	500	1E+30	425
$C$6	Activity 2	27.5	0.0	300	500	300
$D$6	Activity 3	0	250	400	1E+30	250

 

Constraints

Cell	Name	Final Value	Shadow Price	Constraint R.H. Side	Allowable Increase	Allowable Decrease
$E$2	Benefit A	110	0	60	50	1E+30
$E$3	Benefit B	110	75	110	1E+30	46
$E$4	Benefit C	137.5	0	80	57.5	1E+30

 

If the coefficient of Activity 2 in the objective function changes to $100, then:

1. A) the original solution remains optimal.
2. B) the problem must be resolved to find the optimal solution.
3. C) the shadow price is valid.
4. D) the shadow price is not valid.
5. E) None of the choices is correct.

 

Answer:  A

Explanation:  Decreasing the objective function coefficient for Activity 2 to 100 is a decrease of 200 (300 − 100 = 200). Since the allowable decrease for Activity 2 is 300, this change is within the range of optimality and the final values for the variables will not change.

Difficulty: 2 Medium

Topic:  The Effect of Changes in One Objective Function Coefficient

Learning Objective:  Find how much any single coefficient in the objective function can change without changing the optimal solution.

Bloom’s:  Understand

AACSB:  Knowledge Application

Accessibility:  Keyboard Navigation

 

 

 

54) Variable cells

Cell	Name	Final Value	Reduced Cost	Objective Coefficient	Allowable Increase	Allowable Decrease
$B$6	Activity 1	0	425	500	1E+30	425
$C$6	Activity 2	27.5	0.0	300	500	300
$D$6	Activity 3	0	250	400	1E+30	250

 

Constraints

Cell	Name	Final Value	Shadow Price	Constraint R.H. Side	Allowable Increase	Allowable Decrease
$E$2	Benefit A	110	0	60	50	1E+30
$E$3	Benefit B	110	75	110	1E+30	46
$E$4	Benefit C	137.5	0	80	57.5	1E+30

 

If the right-hand side of Resource B changes to 80, then the objective function value:

 

750. A) will decrease by $750.
751. B) will decrease by $1,500.
752. C) will decrease by $2,250.
753. D) will remain the same.
754. E) can only be discovered by resolving the problem.

 

Answer:  C

Explanation:  Decreasing the constraint right-hand side for Resource B to 80 is a decrease of 30 (110 − 80 = 30). Since the allowable decrease for this constraint right-hand side is 46, this is within the allowable range and the optimal solution remains unchanged. A decrease of 20 in the constraint right-hand side leads to a decrease of 2,250 in the objective function—multiply the change in the constraint right-hand side by the shadow price {(110 − 80) × 75 = 2,250}.

Difficulty: 3 Hard

Topic:  The Effect of Single Changes in a Constraint

Learning Objective:  Predict how the value in the objective cell would change if a small change were to be made in the right-hand side of one or more of the functional constraints.

Bloom’s:  Analyze

AACSB:  Analytical Thinking

Accessibility:  Keyboard Navigation

 

 

 

55) Variable cells

Cell	Name	Final Value	Reduced Cost	Objective Coefficient	Allowable Increase	Allowable Decrease
$B$6	Activity 1	0	425	500	1E+30	425
$C$6	Activity 2	27.5	0.0	300	500	300
$D$6	Activity 3	0	250	400	1E+30	250

 

Constraints

Cell	Name	Final Value	Shadow Price	Constraint R.H. Side	Allowable Increase	Allowable Decrease
$E$2	Benefit A	110	0	60	50	1E+30
$E$3	Benefit B	110	75	110	1E+30	46
$E$4	Benefit C	137.5	0	80	57.5	1E+30

 

If the right-hand side of Resource C changes to 140, then the objective function value:

137. A) will increase by $137.50.
138. B) will increase by $57.50.
139. C) will increase by $80.
140. D) will remain the same.
141. E) can only be discovered by resolving the problem.

 

Answer:  E

Explanation:  Increasing the constraint right-hand side for Resource C to 140 is an increase of 60 (140 − 80 = 60). Since the allowable increase for Resource C is 57.5, this change exceeds the allowable decrease and the impact on the optimal solution cannot be determined without re-running the optimization.

Difficulty: 2 Medium

Topic:  The Effect of Single Changes in a Constraint

Learning Objective:  Find how much the right-hand side of a single functional constraint can change before this prediction becomes no longer valid.

Bloom’s:  Understand

AACSB:  Knowledge Application

Accessibility:  Keyboard Navigation

 

 

56) Variable cells

Cell	Name	Final Value	Reduced Cost	Objective Coefficient	Allowable Increase	Allowable Decrease
$B$6	Activity 1	0	425	500	1E+30	425
$C$6	Activity 2	27.5	0.0	300	500	300
$D$6	Activity 3	0	250	400	1E+30	250

 

Constraints

Cell	Name	Final Value	Shadow Price	Constraint R.H. Side	Allowable Increase	Allowable Decrease
$E$2	Benefit A	110	0	60	50	1E+30
$E$3	Benefit B	110	75	110	1E+30	46
$E$4	Benefit C	137.5	0	80	57.5	1E+30

 

If the right-hand side of Resource C changes to 130, then:

1. A) the original solution remains optimal.
2. B) the problem must be resolved to find the optimal solution.
3. C) the shadow price is valid.
4. D) the shadow price is not valid.
5. E) None of the choices is correct.

 

Answer:  C

Explanation:  Increasing the constraint right-hand side for Resource C to 130 is an increase of 50 (130 − 80 = 50). Since the allowable increase for Resource C is 57.5, this change is within the allowable limits and the shadow prices remain valid.

Difficulty: 2 Medium

Topic:  The Effect of Single Changes in a Constraint

Learning Objective:  Find how much the right-hand side of a single functional constraint can change before this prediction becomes no longer valid.

Bloom’s:  Understand

AACSB:  Knowledge Application

Accessibility:  Keyboard Navigation

 

 

 

57) Variable cells

Cell	Name	Final Value	Reduced Cost	Objective Coefficient	Allowable Increase	Allowable Decrease
$B$6	Activity 1	0	425	500	1E+30	425
$C$6	Activity 2	27.5	0.0	300	500	300
$D$6	Activity 3	0	250	400	1E+30	250

 

Constraints

Cell	Name	Final Value	Shadow Price	Constraint R.H. Side	Allowable Increase	Allowable Decrease
$E$2	Benefit A	110	0	60	50	1E+30
$E$3	Benefit B	110	75	110	1E+30	46
$E$4	Benefit C	137.5	0	80	57.5	1E+30

 

If the objective coefficients of Activity 2 and Activity 3 are both decreased by $100, then:

 

1. A) the optimal solution remains the same.
2. B) the optimal solution may or may not remain the same.
3. C) the optimal solution will change.
4. D) the shadow prices are valid.
5. E) None of the choices is correct.

 

Answer:  A

Explanation:  Applying the 100% rule, the change in Activity 2 is 23.53% of the allowable decrease . The change in Activity 3 is 40% of the allowable decrease .  The total change of 63.53% (23.53% + 40%) does not exceed 100%, so the optimal solution will not change.

Difficulty: 3 Hard

Topic:  The Effect of Simultaneous Changes in Objective Function Coefficients

Learning Objective:  Evaluate simultaneous changes in objective function coefficients to determine whether the changes are small enough that the original optimal solution must still be optimal.

Bloom’s:  Analyze

AACSB:  Analytical Thinking

Accessibility:  Keyboard Navigation

 

 

 

58) Variable cells

Cell	Name	Final Value	Reduced Cost	Objective Coefficient	Allowable Increase	Allowable Decrease
$B$6	Activity 1	0	425	500	1E+30	425
$C$6	Activity 2	27.5	0.0	300	500	300
$D$6	Activity 3	0	250	400	1E+30	250

 

Constraints

Cell	Name	Final Value	Shadow Price	Constraint R.H. Side	Allowable Increase	Allowable Decrease
$E$2	Benefit A	110	0	60	50	1E+30
$E$3	Benefit B	110	75	110	1E+30	46
$E$4	Benefit C	137.5	0	80	57.5	1E+30

 

If the right-hand side of Resource C is increased by 40, and the right-hand side of Resource B is decreased by 20, then:

1. A) the optimal solution remains the same.
2. B) the optimal solution will change.
3. C) the shadow price is valid.
4. D) the shadow price may or may not be not valid.
5. E) None of the choices is correct.

 

Answer:  D

Explanation:  Both changes are within the allowable increase/decrease. Applying the 100% rule, the change in Resource B is 43.48% of the allowable decrease . The change in Resource C is 69.57% of the allowable increase . The total change of 113% (43.48% + 69.57%) is greater than 100%, so the shadow prices do not necessarily remain valid.

Difficulty: 3 Hard

Topic:  The Effect of Simultaneous Changes in the Constraints

Learning Objective:  Evaluate simultaneous changes in right-hand sides to determine whether the changes are small enough that this prediction must still be valid.

Bloom’s:  Analyze

AACSB:  Analytical Thinking

Accessibility:  Keyboard Navigation

 

 

 

59) A parameter analysis report can be used to investigate the changes in how many data cells at a time?

1. A) 1
2. B) 2
3. C) 3
4. D) All of the these.
5. E) 1 or 2.

 

Answer:  E

Difficulty: 1 Easy

Topic:  The Importance of What-if Analysis to Managers

Learning Objective:  Use Parameters with Analytic Solver to systematically investigate the effect of changing either one or two data cells to various other trial values.

Bloom’s:  Remember

AACSB:  Knowledge Application

Accessibility:  Keyboard Navigation

60) The allowable range for an objective function coefficient indicates

1. A) The prices a firm is allowed to charge for its product.
2. B) The largest error in estimating objective coefficients that will not affect the optimal solution.
3. C) The amount of each resource available for use.
4. D) The shadow price of each resource.
5. E) The price a firm would be willing to obtain more of a resource.

 

Answer:  B

Difficulty: 1 Easy

Topic:  The Importance of What-if Analysis to Managers

Learning Objective:  Enumerate the different kinds of changes in the model that can be considered by what-if analysis.

Bloom’s:  Remember

AACSB:  Knowledge Application

Accessibility:  Keyboard Navigation

 

 

 

61) To determine if an increase in an objective function coefficient will lead to a change in final values for decision variables, an analyst can do which of the following?

 

1. Compare the increase in the objective function coefficient to the allowable decrease.
2. Compare the increase in the objective function coefficient to the allowable increase.

III. Rerun the optimization to see if the final values change.

1. A) I only.
2. B) II only.
3. C) III only.
4. D) I and III only.
5. E) II and III only.

 

Answer:  E

Difficulty: 2 Medium

Topic:  The Effect of Changes in One Objective Function Coefficient

Learning Objective:  Find how much any single coefficient in the objective function can change without changing the optimal solution.

Bloom’s:  Understand

AACSB:  Knowledge Application

Accessibility:  Keyboard Navigation

62) The Solver report that shows the allowable ranges for objective function coefficients, allowable ranges for constraint right-hand sides, and shadow prices is called the

1. A) Range report.
2. B) Sensitivity report.
3. C) Parameter report.
4. D) Solution report.
5. E) Answer report.

 

Answer:  B

Difficulty: 1 Easy

Topic:  The Importance of What-if Analysis to Managers

Learning Objective:  Describe how the spreadsheet formulation of the problem can be used to perform any of these kinds of what-if analysis.

Bloom’s:  Remember

AACSB:  Knowledge Application

Accessibility:  Keyboard Navigation

 

 

 

63) Activity 1 has an objective function coefficient allowable increase of 30. Activity 2 has an objective function coefficient allowable increase of 60. If both activities objective function coefficient increases by 20, what will happen to the final values in the optimal solution?

1. A) The optimal solution remains the same.
2. B) The optimal solution may or may not remain the same.
3. C) The optimal solution will change.
4. D) The shadow prices are valid.
5. E) None of the choices is correct.

 

Answer:  A

Explanation:  Applying the 100% rule, the change in Activity 1 is 66.67% of the allowable decrease .The change in Activity 2 is 33.33% of the allowable decrease . The total change of 100% (66.67% + 33.33%) does not exceed 100%, so the optimal solution will not change.

Difficulty: 3 Hard

Topic:  The Effect of Simultaneous Changes in Objective Function Coefficients

Learning Objective:  Evaluate simultaneous changes in objective function coefficients to determine whether the changes are small enough that the original optimal solution must still be optimal.

Bloom’s:  Analyze

AACSB:  Analytical Thinking

Accessibility:  Keyboard Navigation

64) Resource B has right-hand side allowable decrease of 50. Resource C has right-hand side allowable decrease of 100. If the right-hand side of Resource B decreases by 30 and the right-hand side of Resource C decreases by 40, then

1. A) the original solution remains optimal.
2. B) the problem must be resolved to find the optimal solution.
3. C) the shadow prices remain valid.
4. D) the shadow prices do not remain valid.
5. E) None of the choices is correct.

 

Answer:  C

Explanation:  Applying the 100% rule, the change in Resource B is 60% of the allowable decrease .The change in Resource C is 40% of the allowable decrease . The total change of 100% (60% + 40%) does not exceed 100%, so the shadow prices remain valid.

Difficulty: 3 Hard

Topic:  The Effect of Simultaneous Changes in Objective Function Coefficients

Learning Objective:  Evaluate simultaneous changes in objective function coefficients to determine whether the changes are small enough that the original optimal solution must still be optimal.

Bloom’s:  Analyze

AACSB:  Analytical Thinking

Accessibility:  Keyboard Navigation

 

65) Note: This question requires access to Solver.

In the following linear programming problem, what is the allowable increase for the objective function coefficient for variable x?

 

Maximize P = 3x + 15y

subject to             2x + 4y ≤ 12

5x + 2y ≤ 10

and      x ≥ 0, y ≥ 0.

1. A) 3
2. B) 4.5
3. C) 9
4. D) 15
5. E) ∞ (infinity)

 

Answer:  B

Explanation:  The sensitivity report (see below) shows that the allowable increase for variable x is 4.5.

 

Variable cells

Cell	Name	Final Value	Reduced Cost	Objective Coefficient	Allowable Increase	Allowable Decrease
$B$3	x	0	–4.5	3	4.5	1E+30
$C$4	y	3	0	15	1E+30	9

 

Constraints

Cell	Name	Final Value	Shadow Price	Constraint R.H. Side	Allowable Increase	Allowable Decrease
$F$9	*y =	12	3.75	12	8	12
$F$10	*y =	6	0	10	1E+30	4

 

Difficulty: 2 Medium

Topic:  The Effect of Changes in One Objective Function Coefficient

Learning Objective:  Describe how the spreadsheet formulation of the problem can be used to perform any of these kinds of what-if analysis.

Bloom’s:  Analyze

AACSB:  Technology

Accessibility:  Keyboard Navigation

 

 

 

66) Note: This question requires access to Solver.

In the following linear programming problem, what is the allowable increase in the right-hand side of the first constraint?

 

Maximize P = 3x + 15y

subject to             2x + 4y ≤ 12

5x + 2y ≤ 10

and         x ≥ 0, y ≥ 0.

1. A) 8
2. B) 10
3. C) 12
4. D) 15
5. E) ∞ (infinity)

 

Answer:  A

Explanation:  The sensitivity report (see below) shows that the allowable increase for the right-hand side of constraint 1 is 8.

 

Variable cells

Cell	Name	Final Value	Reduced Cost	Objective Coefficient	Allowable Increase	Allowable Decrease
$B$3	x	0	–4.5	3	4.5	1E+30
$C$4	y	3	0	15	1E+30	9

 

Constraints

Cell	Name	Final Value	Shadow Price	Constraint R.H. Side	Allowable Increase	Allowable Decrease
$F$9	*y =	12	3.75	12	8	12
$F$10	*y =	6	0	10	1E+30	4

 

Difficulty: 2 Medium

Topic:  The Effect of Single Changes in a Constraint

Learning Objective:  Describe how the spreadsheet formulation of the problem can be used to perform any of these kinds of what-if analysis.

Bloom’s:  Analyze

AACSB:  Technology

Accessibility:  Keyboard Navigation

 

 

 

67) Note: This question requires access to Solver.

In the following linear programming problem, how much would the firm be willing to pay for an additional 5 units of Resource A?

 

Maximize P = 3x + 15y

subject to              2x + 4y ≤ 12 (Resource A)

5x + 2y ≤ 10 (Resource B)

and         x ≥ 0, y ≥ 0.

1. A) It is impossible to determine.
2. B) 7.50
3. C) 11.25
4. D) 15
5. E) 18.75

 

Answer:  E

Explanation:  The sensitivity report (see below) shows that the allowable increase for the right-hand side of the first constraint is 8. Since the change is within this allowable increase, the shadow price remains valid. Therefore, the firm would be willing to pay up to 18.75 {5×3.75} for the additional 5 units of Resource A.

 

Variable cells

Cell	Name	Final Value	Reduced Cost	Objective Coefficient	Allowable Increase	Allowable Decrease
$B$3	x	0	–4.5	3	4.5	1E+30
$C$4	y	3	0	15	1E+30	9

 

Constraints

Cell	Name	Final Value	Shadow Price	Constraint R.H. Side	Allowable Increase	Allowable Decrease
$F$9	*y =	12	3.75	12	8	12
$F$10	*y =	6	0	10	1E+30	4

 

Difficulty: 3 Hard

Topic:  The Effect of Single Changes in a Constraint

Learning Objective:  Describe how the spreadsheet formulation of the problem can be used to perform any of these kinds of what-if analysis.

Bloom’s:  Analyze

AACSB:  Technology

Accessibility:  Keyboard Navigation

 

 

 

68) Note: This question requires access to Solver.

In the following linear programming problem, how much would the firm be willing to pay for an additional 5 units of Resource B?

 

Maximize P = 3x + 15y

subject to             2x + 4y ≤ 12 (Resource A)

5x + 2y ≤ 10 (Resource B)

and        x ≥ 0, y ≥ 0.

1. A) Nothing
2. B) 11.25
3. C) 15
4. D) 18.75
5. E) It is impossible to determine.

 

Answer:  A

Explanation:  The sensitivity report (see below) shows that the allowable increase for the right-hand side of the second constraint is ∞ (infinity). Since the change is within this allowable increase, the shadow price remains valid. However, the shadow price of 0 indicates that the firm does not require any more of Resource B, so the firm will not be willing to pay anything to obtain 5 additional units.

 

Variable cells

Cell	Name	Final Value	Reduced Cost	Objective Coefficient	Allowable Increase	Allowable Decrease
$B$3	x	0	–4.5	3	4.5	1E+30
$C$4	y	3	0	15	1E+30	9

 

Constraints

Cell	Name	Final Value	Shadow Price	Constraint R.H. Side	Allowable Increase	Allowable Decrease
$F$9	*y =	12	3.75	12	8	12
$F$10	*y =	6	0	10	1E+30	4

 

Difficulty: 3 Hard

Topic:  The Effect of Single Changes in a Constraint

Learning Objective:  Describe how the spreadsheet formulation of the problem can be used to perform any of these kinds of what-if analysis.

Bloom’s:  Analyze

AACSB:  Technology

Accessibility:  Keyboard Navigation

 

 

 

69) In robust optimization, what is meant by the term “soft constraint”?

1. A) A constraint that is not violated.
2. B) A constraint that has a shadow price of zero.
3. C) A constraint that can be violated slightly without serious repercussions.
4. D) A constraint that can be violated dramatically without serious repercussions.
5. E) A constraint that cannot be violated.

 

Answer:  C

Difficulty: 1 Easy

Topic:  Robust Optimization

Learning Objective:  Describe the goal of robust optimization and how it is implemented with independent parameters.

Bloom’s:  Remember

AACSB:  Knowledge Application

Accessibility:  Keyboard Navigation

70) In robust optimization, what is meant by the term “hard constraint”?

1. A) A constraint that is not violated.
2. B) A constraint that has a shadow price of zero.
3. C) A constraint that can be violated slightly without serious repercussions.
4. D) A constraint that can be violated dramatically without serious repercussions.
5. E) A constraint that cannot be violated.

 

Answer:  E

Difficulty: 1 Easy

Topic:  Robust Optimization

Learning Objective:  Describe the goal of robust optimization and how it is implemented with independent parameters.

Bloom’s:  Remember

AACSB:  Knowledge Application

Accessibility:  Keyboard Navigation

 

71) One approach to robust optimization is to modify the original optimization problem by

1. A) Assigning average values to each uncertain parameter.
2. B) Assigning conservative values to each uncertain parameter.
3. C) Assigning optimistic values to each uncertain parameter.
4. D) Assigning random values to each uncertain parameter.
5. E) Assigning precise values to each uncertain parameter.

 

Answer:  B

Difficulty: 2 Medium

Topic:  Robust Optimization

Learning Objective:  Describe the goal of robust optimization and how it is implemented with independent parameters.

Bloom’s:  Understand

AACSB:  Knowledge Application

Accessibility:  Keyboard Navigation

72) When conducting robust optimization

 

1. The right-hand side of each ≤ constraint should be replaced with the minimum value.
2. The right-hand side of each ≤ constraint should be replaced with the maximum value.

III. The right-hand side of each ≥ constraint should be replaced with the maximum value.

1. A) I only
2. B) II only
3. C) III only
4. D) I and III only
5. E) II and III only

 

Answer:  D

Difficulty: 2 Medium

Topic:  Robust Optimization

Learning Objective:  Describe the goal of robust optimization and how it is implemented with independent parameters.

Bloom’s:  Understand

AACSB:  Knowledge Application

Accessibility:  Keyboard Navigation

 

73) When conducting robust optimization

 

1. Use the maximum value of each objective function coefficient for a maximization problem.
2. Use the minimum value of each objective function coefficient for a maximization problem.

III. Use the maximum value of each objective function coefficient for a minimization problem.

1. A) I only
2. B) II only
3. C) III only
4. D) I and III only
5. E) II and III only

 

Answer:  E

Difficulty: 2 Medium

Topic:  Robust Optimization

Learning Objective:  Describe the goal of robust optimization and how it is implemented with independent parameters.

Bloom’s:  Understand

AACSB:  Knowledge Application

Accessibility:  Keyboard Navigation

 

 

74) A chance constraint

 

1. Replaces the right-hand side with the minimum value.
2. Allows the objective function coefficients to be replaced with random numbers.

III. Ensures that the chance constraint will never be violated.

1. Can be used to model a soft constraint which can be violated at times.
2. A) I only
3. B) II only
4. C) III only
5. D) IV only
6. E) I and II only

 

Answer:  D

Difficulty: 2 Medium

Topic:  Chance Constraints With Analytic Solver

Learning Objective:  Use chance constraints to deal with constraints that actually can be violated a little bit.

Bloom’s:  Understand

AACSB:  Knowledge Application

Accessibility:  Keyboard Navigation

 

75) Chance constraints are an available option in

 

1. Graphical linear programming.
2. The Solver tool included with Excel.

III. Analytic Solver.

1. A) I only
2. B) II only
3. C) III only
4. D) II and III only
5. E) I, II, and III

 

Answer:  C

Difficulty: 2 Medium

Topic:  Chance Constraints With Analytic Solver

Learning Objective:  Use chance constraints to deal with constraints that actually can be violated a little bit.

Bloom’s:  Understand

AACSB:  Knowledge Application

Accessibility:  Keyboard Navigation