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

Question 1

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

Answer 1

True

Question 2

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.

Answer 2

B

Question 3

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.

Answer 3

D

Question 4

Which of the following is not a question answered by sensitivity analysis?

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 5

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 6

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 7

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 8

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 9

The reduced cost for a positive decision variable is 0.

Question 10

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 11

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

Question 12

The dual value on the nonnegativitiy constraint for a variable is that variable's

A) sunk cost.
B) surplus value.
C) reduced cost.
D) relevant cost.

Question 13

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 14

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 15

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.

Question 16

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 17

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 18

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 19

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 20

To solve a linear programming problem with thousands of variables and constraints

A) a personal computer can be used.
B) a mainframe computer is required.
C) the problem must be partitioned into subparts.
D) unique software would need to be developed.

Question 21

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

Question 22

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

Question 23

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

Question 24

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

Question 25

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

Question 26

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

Question 27

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

Question 28

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

Question 29

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

Question 30

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
END
LP OPTIMUM FOUND AT STEP 2
OBJECTIVE FUNCTION VALUE
1) 612.50000
NO. ITERATIONS= 2
RANGES IN WHICH THE BASIS IS UNCHANGED:

Question 31

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

Question 32

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

Question 33

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

Question 34

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

Question 35

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 36

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.