Deck 3: Linear Programming: Sensitivity Analysis and Interpretation of Solution

Question 1

Any change to the objective function coefficient of a variable that is positive in the optimal solution will change the optimal solution.

Answer 1

False

Question 2

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.

Answer 2

False

Question 3

If the optimal value of a decision variable is zero and its reduced cost is zero, this indicates that alternative optimal solutions exist.

Answer 3

True

Question 4

In order to tell the impact of a change in a constraint coefficient, the change must be made and then the model resolved.

Question 5

Relevant costs should be reflected in the objective function, but sunk costs should not.

Question 6

The amount of a sunk cost will vary depending on the values of the decision variables.

Question 7

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.

Question 8

Decreasing the objective function coefficient of a variable to its lower limit will create a revised problem that is unbounded.

Question 9

Classical sensitivity analysis provides no information about changes resulting from a change in the coefficient of a variable in a constraint.

Question 10

The 100 percent rule can be applied to changes in both objective function coefficients and right-hand sides at the same time.

Question 11

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.

Question 12

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.

Question 13

For a minimization problem, a positive dual price indicates the value of the objective function will increase.

Question 14

If the dual price for the right-hand side of a ≤ constraint is zero, there is no upper limit on its range of feasibility.

Question 15

There is a dual price for every decision variable in a model.

Question 16

The 100% Rule does not imply that the optimal solution will necessarily change if the percentage exceeds 100%.

Question 17

A negative dual price indicates that increasing the right-hand side of the associated constraint would be detrimental to the objective.

Question 18

The dual price for a percentage constraint provides a direct answer to questions about the effect of increases or decreases in that percentage.

Question 19

For any constraint, either its slack/surplus value must be zero or its dual price must be zero.

Question 20

The reduced cost for a positive decision variable is 0.

Question 21

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.

Question 22

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.

Question 23

A section of output from The Management Scientist is shown here. 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.

Question 24

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.

Question 25

​Sensitivity analysis is often referred to as

A) ​feasibility testing.
B) ​duality analysis.
C) ​alternative analysis.
D) ​postoptimality analysis.

Question 26

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?

Question 27

A section of output from The Management Scientist is shown here. 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.

Question 28

​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.

Question 29

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.

Question 30

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.

Question 31

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.

Question 32

When the cost of a resource is sunk, then the dual price can be interpreted as the

A) minimum amount the firm should be willing to pay for one additional unit of the resource.
B) maximum amount the firm should be willing to pay for one additional unit of the resource.
C) minimum amount the firm should be willing to pay for multiple additional units of the resource.
D) maximum amount the firm should be willing to pay for multiple additional units of the resource.

Question 33

​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.

Question 34

​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.

Question 35

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.

Question 36

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.

Question 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.

Question 38

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.

Question 39

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.

Question 40

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.

Question 41

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 OBJECTIVE COEFFICIENT RANGES RIGHT HAND SIDE RANGES
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?

Question 42

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. ​
a.Give the original linear programming problem.
b.Give the complete optimal solution.

Question 43

​How can the interpretation of dual prices help provide an economic justification for new technology?

Question 44

Eight of the entries have been deleted from the LINDO output that follows. Use what you know about linear programming to find values for the blanks.
MIN 6 X1 + 7.5 X2 + 10 X3
SUBJECT TO
2) 25 X1 + 35 X2 + 30 X3 >= 2400
3) 2 X1 + 4 X2 + 8 X3 >= 400
END
LP OPTIMUM FOUND AT STEP 2
OBJECTIVE FUNCTION VALUE
1) 612.50000 NO. ITERATIONS= 2
RANGES IN WHICH THE BASIS IS UNCHANGED:

Question 45

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

Question 46

​Describe each of the sections of output that come from The Management Scientist and how you would use each.

Question 47

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 NO. ITERATIONS= 1
RANGES IN WHICH THE BASIS IS UNCHANGED:
a.What is the solution to the problem?
b.Which constraints are binding?
c.Interpret the reduced cost for x₁.
d.Interpret the dual price for constraint 2.
e.What would happen if the cost of x₁ dropped to 10 and the cost of x₂ increased to 12?

Question 48

Portions of a Management Scientist output are shown below. Use what you know about the solution of linear programs to fill in the ten blanks.
LINEAR PROGRAMMING PROBLEM
MAX 12X1+9X2+7X3
S.T.
1) 3X1+5X2+4X3<150
2) 2X1+1X2+1X3<64
3) 1X1+2X2+1X3<80
4) 2X1+4X2+3X3>116
OPTIMAL SOLUTION
Objective Function Value = 336.000 OBJECTIVE COEFFICIENT RANGES RIGHT HAND SIDE RANGES

Question 49

​How would sensitivity analysis of a linear program be undertaken if one wishes to consider simultaneous changes for
both the right-hand-side values and objective function.

Question 50

Excel's Solver tool has been used in the spreadsheet below to solve a linear programming problem with a maximization objective function and all ≤ constraints. ​
a.Give the original linear programming problem.
b.Give the complete optimal solution.

Question 51

​Explain the two interpretations of dual prices based on the accounting assumptions made in calculating the objective
function coefficients.

Question 52

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.

Question 53

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.

Question 54

​How is sensitivity analysis used in linear programming? Given an example of what type of questions that can be
answered.

Question 55

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 OBJECTIVE COEFFICIENT RANGES RIGHT HAND SIDE RANGES
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.

Question 56

​Explain the connection between reduced costs and the range of optimality, and between dual prices and the range of
feasibility.