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

Question 1

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.

Answer 1

False

Question 2

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

Answer 2

False

Question 3

If two or more objective function coefficients are changed simultaneously,further analysis is necessary to determine whether the optimal solution will change.

Answer 3

True

Question 4

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 5

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 6

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

Question 7

When two or more objective function coefficients are changed simultaneously,further analysis is necessary to determine whether the optimal solution will change.

Question 8

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 9

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

Question 10

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

Question 11

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 12

A small change in the objective function coefficient can necessitate modifying the optimal solution.

Question 13

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

Question 14

The dual value and dual price are identical for a minimization problem.

Question 15

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

Question 16

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

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

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 19

Increasing the right-hand side of a nonbinding constraint will not cause a change in the optimal solution.

Question 20

The reduced cost of a variable is the dual value of the corresponding nonnegativity constraint.

Question 21

Based on the per-unit increase in the right-hand side of the constraint,the dual price measures the

A)increase in the value of the optimal solution.
B)decrease in the value of the optimal solution.
C)improvement in the value of the optimal solution.
D)change in the value of the optimal solution.

Question 22

The graphical solution procedure is useful only for linear programs involving

A)two decision variables.
B)more than two decision variables.
C)a single constraint.
D)None of these are correct.

Question 23

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 24

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 25

A cost that is incurred no matter what values the decision variables assume is a(n)

A)reduced cost.
B)optimal cost.
C)sunk cost.
D)dual cost.

Question 26

An improvement in the value of the objective function per unit increase in a right-hand side is the

A)sensitivity value.
B)constraint coefficient.
C)slack value.
D)None of these are correct.

Question 27

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)All of these are correct.

Question 28

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 29

Sensitivity analysis information in computer output is based on the assumption that

A)no coefficient changes.
B)one coefficient changes.
C)two coefficients change.
D)all coefficients change.

Question 30

​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 31

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 32

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 change 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 33

Sensitivity analysis is sometimes referred to as

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

Question 34

Sensitivity analysis is concerned with how certain changes affect the

A)feasible solution.
B)unconstrained solution.
C)optimal solution.
D)degenerative solution.

Question 35

The dual price for a < constraint will

A)always be < 0.
B)always be > 0.
C)be < 0 in a minimization problem and > 0 in a maximization problem.
D)always equal 0.

Question 36

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 37

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 38

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 the

A)maximum premium (say for overtime)over the normal price that the company would be willing to pay.
B)upper limit on the total hourly wage the company would pay.
C)reduction in hours that could be sustained before the solution would change.
D)number of hours by which the right-hand side can change before there is a change in the solution point.

Question 39

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 40

The optimal solution of this linear programming problem is at the intersection of constraints 1 and 2. ​
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 41

The binding constraints for this problem are the first and second. ​
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 42

Consider the following linear program: The graphical solution to the problem is shown below.From the graph,we see that the optimal solution occurs at x₁ = 5,x₂ = 3,and z = 46.
a.Calculate the range of optimality for each objective function coefficient.
b.Calculate the dual price for each resource.

Question 43

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 44

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.