Chapter 3 – Linear Programming: Sensitivity Analysis and Interpretation of Solution
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
True
Changes in constraint coefficients
2. The reduced cost for a positive decision variable is 0.
a.
True
b.
False
True
Reduced cost
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
True
Range of feasibility
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
True
Dual price
Chapter 3 – Linear Programming: Sensitivity Analysis and Interpretation of Solution
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
True
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
True
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
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
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
False
12. For a minimization problem, a positive dual price indicates the value of the objective function will increase.
a.
True
b.
False
False
Chapter 3 – Linear Programming: Sensitivity Analysis and Interpretation of Solution
13. There is a dual price for every decision variable in a model.
a.
True
b.
False
False
14. The amount of a sunk cost will vary depending on the values of the decision variables.
a.
True
b.
False
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
True
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
False
17. Relevant costs should be reflected in the objective function, but sunk costs should not.
a.
True
b.
False
True
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
False
Chapter 3 – Linear Programming: Sensitivity Analysis and Interpretation of Solution
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
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
b.
False
True
Right-hand sides
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.
a
Computer solution
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.
a
Dual price
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
Chapter 3 – Linear Programming: Sensitivity Analysis and Interpretation of Solution
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.
Slack and dual price
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.
Range of optimality
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.
Range of feasibility
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.
Dual price
Chapter 3 – Linear Programming: Sensitivity Analysis and Interpretation of Solution
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.
b
1
Range of optimality
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.
d
1
Range of feasibility
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.
c
1
Interpretation of computer output
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.
Chapter 3 – Linear Programming: Sensitivity Analysis and Interpretation of Solution
c.
the improvement in the value of the optimal solution.
d.
the change in the value of the optimal solution.
Interpretation of computer output
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.
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.
Dual price
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?
Interpretation of computer output
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.
Sunk and relevant costs
37. A cost that is incurred no matter what values the decision variables assume is
Chapter 3 – Linear Programming: Sensitivity Analysis and Interpretation of Solution
a.
a reduced cost.
b.
an optimal cost.
c.
a sunk cost.
d.
a dual cost.
1
Sunk and relevant costs
38. Sensitivity analysis is often referred to as
a.
feasibility testing.
b.
duality analysis.
c.
alternative analysis.
d.
postoptimality analysis.
d
1
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.
1
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.
d.
will always equal 0.
b
1
Dual price
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
Chapter 3 – Linear Programming: Sensitivity Analysis and Interpretation of Solution
a.
Objective functions 2), 3), and 4)
Graphical sensitivity analysis
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.
b.
c.
Dual prices are .25, .25, 0
Graphical sensitivity analysis
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.
Constraints 1 and 2 will be binding.
Chapter 3 – Linear Programming: Sensitivity Analysis and Interpretation of Solution
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
#2
48
-2.69E-11
#3
17.333333
2.6666667
a.
Give the original linear programming problem.
b.
Give the complete optimal solution.
a.
Max
4X + 6Y
s.t.
3X + 5Y ≤ 60
3X + 2Y ≤ 48
1X + 1Y ≤ 20
1
Spreadsheet solution of LPs
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
e.
Dual prices are .33, 0, .33 (The first and third values are negative.)
1
Graphical sensitivity analysis
Chapter 3 – Linear Programming: Sensitivity Analysis and Interpretation of Solution
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.
a.
Min
5X + 4Y
s.t.
4X + 3Y ≥ 60
2X + 5Y ≥ 50
9X + 8Y ≥ 144
b.
The complete optimal solution is X = 9.6, Y = 7.2, Z = 76.8, S1 = 0, S2 = 5.2, S3 = 0
1
Spreadsheet solution of LPs
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
Chapter 3 – Linear Programming: Sensitivity Analysis and Interpretation of Solution
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
=B8*B11
=C8*C11
=B12+C12
=B4*B11+C4*C11
=B5*B11+C5*C11
=B6*B11+C6*C11
8.46
4.61
yes
Spreadsheet solution of LPs
47. Use the following Management Scientist output to answer the questions.
LINEAR PROGRAMMING PROBLEM
MAX 31X1+35X2+32X3
Chapter 3 – Linear Programming: Sensitivity Analysis and Interpretation of Solution
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.
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?
b.
Constraints 1 and 2 are binding.
c.
The value of the objective function would increase by 40.
d.
The value of the objective function would decrease by 7.78.
1
Interpretation of Management Scientist output
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
Chapter 3 – Linear Programming: Sensitivity Analysis and Interpretation of Solution
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.
b.
Constraint 3 is binding.
c.
Dual prices are 0, 0, and −1.25.
They measure the improvement in Z per unit increase in each right-hand side.
values will remain the same.
1
Interpretation of Management Scientist output
49. The following linear programming problem has been solved by The Management Scientist. Use the output to answer
the questions.
Chapter 3 – Linear Programming: Sensitivity Analysis and Interpretation of Solution
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
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?
b.
Constraints 1 and 2 are binding.
Dual price 2 = 2.33. A unit increase in the right-hand side of constraint 2 will increase the
e.
1050
change.
1