Chapter 8 – LP Sensitivity Analysis
True / False
1. Classical sensitivity analysis provides no information about changes resulting from a change in the coefficient of a
variable in a constraint.
a. True
b. False
2. The reduced cost for a positive decision variable is 0.
a. True
b. False
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. True
b. False
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. True
b. False
5. The 100% Rule does not imply that the optimal solution will necessarily change if the percentage exceeds 100%.
a. True
b. False
6. For any constraint, either its slack/surplus value must be zero or its dual price must be zero.
a. True
b. False
Chapter 8 – LP Sensitivity Analysis
7. A negative dual price indicates that increasing the right-hand side of the associated constraint would be detrimental to
the objective.
a. True
b. False
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. True
b. False
9. Decreasing the objective function coefficient of a variable to its lower limit will create a revised problem that is
unbounded.
a. True
b. False
10. The dual price for a percentage constraint provides a direct answer to questions about the effect of increases or
decreases in that percentage.
a. True
b. False
11. 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. True
b. False
12. For a minimization problem, a positive dual price indicates the value of the objective function will increase.
a. True
b. False
13. There is a dual price for every decision variable in a model.
a. True
b. False
Chapter 8 – LP Sensitivity Analysis
14. The amount of a sunk cost will vary depending on the values of the decision variables.
a. True
b. False
15. If the optimal value of a decision variable is zero and its reduced cost is zero, this indicates that alternative optimal
solutions exist.
a. True
b. False
16. Any change to the objective function coefficient of a variable that is positive in the optimal solution will change the
optimal solution.
a. True
b. False
17. Relevant costs should be reflected in the objective function, but sunk costs should not.
a. True
b. False
18. 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. True
b. False
19. The 100 percent rule can be applied to changes in both objective function coefficients and right-hand sides at the same
time.
a. True
b. False
Chapter 8 – LP Sensitivity Analysis
20. 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. True
Multiple Choice
21. 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.
22. 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.
23. 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.
24. 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.
Chapter 8 – LP Sensitivity Analysis
25. 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.
26. 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. each of these choices are true.
27. The 100% Rule compares
a. proposed changes to allowed changes.
b. new values to original values.
c. objective function changes to right-hand side changes.
d. dual prices to reduced costs.
28. 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
a. the maximum premium (say for overtime) over the normal price that the company would be willing to pay.
b. the upper limit on the total hourly wage the company would pay.
c. the reduction in hours that could be sustained before the solution would change.
d. the number of hours by which the right-hand side can change before there is a change in the solution point.
29. A section of output from The Management Scientist is shown here.
Variable Lower Limit Current Value Upper Limit
1 60 100 120
What will happen to the solution if the objective function coefficient for variable 1 decreases by 20?
a. Nothing. The values of the decision variables, the dual prices, and the objective function will all remain the same.
b. The value of the objective function will change, but the values of the decision variables and the dual prices will
remain the same.
c. The same decision variables will be positive, but their values, the objective function value, and the dual prices will
change.
d. The problem will need to be resolved to find the new optimal solution and dual price.
Chapter 8 – LP Sensitivity Analysis
30. A section of output from The Management Scientist is shown here.
Constraint Lower Limit Current Value Upper Limit
2 240 300 420
What will happen if the right-hand-side for constraint 2 increases by 200?
a. Nothing. The values of the decision variables, the dual prices, and the objective function will all remain the same.
b. The value of the objective function will change, but the values of the decision variables and the dual prices will
remain the same.
c. The same decision variables will be positive, but their values, the objective function value, and the dual prices
will change.
d. The problem will need to be resolved to find the new optimal solution and dual price.
31. 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.
32. The dual price measures, per unit increase in the right hand side of the constraint,
a. the increase in the value of the optimal solution.
b. the decrease in the value of the optimal solution.
c. the improvement in the value of the optimal solution.
d. the change in the value of the optimal solution.
33. Sensitivity analysis information in computer output is based on the assumption of
a. no coefficient changes.
b. one coefficient changes.
c. two coefficients change.
d. all coefficients change.
34. 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.
Chapter 8 – LP Sensitivity Analysis
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.
35. Which of the following is not a question answered 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 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?
36. 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.
37. A cost that is incurred no matter what values the decision variables assume is
a. a reduced cost.
b. an optimal cost.
c. a sunk cost.
d. a dual cost.
38. Sensitivity analysis is often referred to as
a. feasibility testing.
b. duality analysis.
c. alternative analysis.
d. postoptimality analysis.
39. Sensitivity analysis is concerned with how certain changes affect
a. the feasible solution.
b. the unconstrained solution.
c. the optimal solution.
d. the degenerative solution.
Chapter 8 – LP Sensitivity Analysis
40. The dual price for a < constraint
a. will always be < 0.
b. will always be > 0.
c. will be < 0 in a minimization problem and > 0 in a maximization problem.
Subjective Short Answer
41. 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
42. The optimal solution of the linear programming problem is at the intersection of constraints 1 and 2.
Max 2x1 + x2
s.t. 4x1 + 1x2 ≤ 400
4x1 + 3x2 ≤ 600
1x1 + 2x2 ≤ 300
x1 , x2 ≥ 0
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.
43. The binding constraints for this problem are the first and second.
Min x1 + 2x2
s.t. x1 + x2 ≥ 300
2x1 + x2 ≥ 400
2x1 + 5x2 ≤ 750
x1 , x2 ≥ 0
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.
44. 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.
Input Section
Objective Function Coefficients
X Y
4 6
Constraints Avail.
#1 3 5 60
#2 3 2 48
#3 1 1 20
Output Section
Variables 13.333333 4
Profit 53.333333 24 77.333333
Constraint Usage Slack
#1 60 1.789E-11
Chapter 8 – LP Sensitivity Analysis
#2 48 -2.69E-11
#3 17.333333 2.6666667
a. Give the original linear programming problem.
b. Give the complete optimal solution.
45. 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.
Input Section
Objective Function Coefficients
X Y
5 4
Constraints Req’d
#1 4 3 60
#2 2 5 50
#3 9 8 144
Output Section
Variables 9.6 7.2
Profit 48 28.8 76.8
Constraint Usage Slack
#1 60 1.35E-11
#2 55.2 -5.2
#3 144 -2.62E-11
a. Give the original linear programming problem.
b. Give the complete optimal solution.
Chapter 8 – LP Sensitivity Analysis
46. Use the spreadsheet and Solver sensitivity report to answer 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 $.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?
A B C D E
1
2 Input Information
3 Var. 1 Var. 2 (type) Avail.
4 Constraint 1 2 5 < 40
5 Constraint 2 3 1 < 30
6 Constraint 3 1 1 > 12
7
8 Profit 5 4
9
10 Output Information
11 Variables
12 Profit = Total
13
14 Resources Used Slack/Surplus
15 Constraint 1
16 Constraint 2
17 Constraint 3
18
19
Sensitivity Report
Changing Cells
Final Reduced Objective Allowable Allowable
Cell Name Value Cost Coefficient Increase Decrease
$B$12 Variable 1 8.461538462 0 5 7 3.4
$C$12 Variable 2 4.615384615 0 4 8.5 2.333333333
Constraints
Final Shadow Constraint Allowable Allowable
Cell Name Value Price R.H. Side Increase Decrease
$B$15 constraint 1 Used 40 0.538461538 40 110 7
$B$16 constraint 2 Used 30 1.307692308 30 30 4.666666667
$B$17 constraint 3 Used 13.07692308 0 12 1.076923077 1E+30
Chapter 8 – LP Sensitivity Analysis
47. Use the following Management Scientist output to answer the questions.
LINEAR PROGRAMMING PROBLEM
MAX 31X1+35X2+32X3
S.T.
1) 3X1+5X2+2X3>90
2) 6X1+7X2+8X3<150
3) 5X1+3X2+3X3<120
OPTIMAL SOLUTION
Objective Function Value = 763.333
Variable Value Reduced Cost
X1 13.333 0.000
X2 10.000 0.000
X3 0.000 10.889
Constraint Slack/Surplus Dual Price
1 0.000 −0.778
2 0.000 5.556
3 23.333 0.000
OBJECTIVE COEFFICIENT RANGES
Variable Lower Limit Current Value Upper Limit
X1 30.000 31.000 No Upper Limit
X2 No Lower Limit 35.000 36.167
X3 No Lower Limit 32.000 42.889
RIGHT HAND SIDE RANGES
Constraint Lower Limit Current Value Upper Limit
1 77.647 90.000 107.143
2 126.000 150.000 163.125
3 96.667 120.000 No Upper Limit
a. Give the solution to the problem.
Chapter 8 – LP Sensitivity Analysis
b. Which constraints are binding?
c. What would happen if the coefficient of x1 increased by 3?
d. What would happen if the right-hand side of constraint 1 increased by 10?
48. Use the following Management Scientist output to answer the questions.
MIN 4X1+5X2+6X3
S.T.
1) X1+X2+X3<85
2) 3X1+4X2+2X3>280
3) 2X1+4X2+4X3>320
Objective Function Value = 400.000
Variable Value Reduced Cost
X1 0.000 1.500
X2 80.000 0.000
X3 0.000 1.000
Constraint Slack/Surplus Dual Price
1 5.000 0.000
2 40.000 0.000
3 0.000 −1.250
OBJECTIVE COEFFICIENT RANGES
Variable Lower Limit Current Value Upper Limit
X1 2.500 4.000 No Upper Limit
X2 0.000 5.000 6.000
X3 5.000 6.000 No Upper Limit
RIGHT HAND SIDE RANGES
Constraint Lower Limit Current Value Upper Limit
1 80.000 85.000 No Upper Limit
2 No Lower Limit 280.000 320.000
3 280.000 320.000 340.000
a. What is the optimal solution, and what is the value of the profit contribution?
b. Which constraints are binding?
c. What are the dual prices for each resource? Interpret.
d. Compute and interpret the ranges of optimality.
e. Compute and interpret the ranges of feasibility.
Chapter 8 – LP Sensitivity Analysis
49. The following linear programming problem has been solved by The Management Scientist. Use the output to answer
the questions.
LINEAR PROGRAMMING PROBLEM
MAX 25X1+30X2+15X3
S.T.
1) 4X1+5X2+8X3<1200
2) 9X1+15X2+3X3<1500
OPTIMAL SOLUTION
Objective Function Value = 4700.000
Variable Value Reduced Cost
X1 140.000 0.000
X2 0.000 10.000
X3 80.000 0.000
Constraint Slack/Surplus Dual Price
1 0.000 1.000
2 0.000 2.333
OBJECTIVE COEFFICIENT RANGES
Variable Lower Limit Current Value Upper Limit
X1 19.286 25.000 45.000
X2 No Lower Limit 30.000 40.000
X3 8.333 15.000 50.000
RIGHT HAND SIDE RANGES
Constraint Lower Limit Current Value Upper Limit
Chapter 8 – LP Sensitivity Analysis
1 666.667 1200.000 4000.000
2 450.000 1500.000 2700.000
a. Give the complete optimal solution.
b. Which constraints are binding?
c. What is the dual price for the second constraint? What interpretation does this have?
d. Over what range can the objective function coefficient of x2 vary before a new solution point becomes optimal?
e. By how much can the amount of resource 2 decrease before the dual price will change?
f. What would happen if the first constraint’s right-hand side increased by 700 and the second’s decreased by 350?
50. LINDO output is given for the following linear programming problem.
MIN 12 X1 + 10 X2 + 9 X3
SUBJECT TO
2) 5 X1 + 8 X2 + 5 X3 >= 60
3) 8 X1 + 10 X2 + 5 X3 >= 80
END
LP OPTIMUM FOUND AT STEP 1
OBJECTIVE FUNCTION VALUE
1) 80.000000
VARIABLE VALUE REDUCED COST
X1 .000000 4.000000
X2 8.000000 .000000
X3 .000000 4.000000
ROW SLACK OR SURPLUS DUAL PRICE
2) 4.000000 .000000
3) .000000 −1.000000
NO. ITERATIONS= 1
RANGES IN WHICH THE BASIS IS UNCHANGED:
OBJ. COEFFICIENT RANGES