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

Question 1

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.

Answer 1

B

Question 2

Output from a computer package is precise and answers should never be rounded.

Answer 2

False

Question 3

A section of output from The Management Scientist is shown here. $\frac { \text { Constraint } } { 2 }$$\frac { \text { Lower Limit } } { 240 }$$\frac { \text { Current Value } } { 300 }$$\frac { \text { Upper Limit } } { 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.

Answer 3

The problem will need to be resolved to find the new optimal solution and dual price.

Question 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.

Question 5

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 6

The amount by which an objective function coefficient would have to improve before it would be possible for the corresponding variable to assume a positive value in the optimal solution is called the

A)reduced cost.
B)relevant cost.
C)sunk cost.
D)dual price.

Question 7

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 8

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

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

Question 9

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 10

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 11

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 the above is true.

Question 12

The amount that 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 13

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 14

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 15

The reduced cost for a positive decision variable is 0.

Question 16

The dual price measures, per unit increase in the right hand side,

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 17

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 18

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 19

A section of output from The Management Scientist is shown here. $$\begin{array} { c c c c } \frac { \text { Variable } } { 1 } & \frac { \text { Lower Limit } } { 60 } & \frac { \text { Current Value } } { 100 } & \frac { \text { Upper Limit } } { 120 }\end{array}$$ 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 20

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 21

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

Question 22

The dual price associated with a constraint is the improvement in the value of the solution per unit decrease in the right-hand side of the constraint.

Question 23

The binding constraints for this problem are the first and second.
Min
x₁ + 2x₂
s.t.
x₁ + x₂ $\ge$ 300
2x₁ + x₂ $\ge$ 400
2x₁ + 5x₂ $\le$ 750
x₁ , x₂ $\ge$ 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 24

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

Question 25

In a linear programming problem, the binding constraints for the optimal solution are
5X + 3Y $\le$ 30
2X + 5Y $\le$ 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 26

Decision variables must be clearly defined before constraints can be written.

Question 27

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

Question 28

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

Question 29

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 30

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

Question 31

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

Question 32

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

Question 33

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

Question 34

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

Question 35

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

Question 36

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

Question 37

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

Question 38

The optimal solution of the linear programming problem is at the intersection of constraints 1 and 2.
Max
2x₁ + x₂
s.t.
4x₁ + 1x₂ $\le$ 400
4x₁ + 3x₂ $\le$ 600
1x₁ + 2x₂ $\le$ 300
x₁ , x₂ $\ge$ 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 39

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

Question 40

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

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
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

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 $$\begin{array} { c c c } \text { Variable } & \text { Value } & \text { Reduced Cost } \\\text { X1 } &- & 0.000 \\\text { X2 } &24.000&- \\\text { X3 } &- &3.500 \\\hline\end{array}$$ $$\begin{array} { c c c } \text { Constraint } & \text { Slack/Surplus } & \text { Dual Price } \\\hline 1 & 0.000 & 15.000 \\2 & --& 0.000 \\3 & --& 0.000 \\4 & 0.000 &-- \\\hline\end{array}$$ $\text { OBJECTIVE COEFFICIENT RANGES }$
$\begin{array}{cccc}\text { Variable } & \text { Lower Limit } & \text { Current Value } & \text { Upper Limit }\\\hline\mathrm{X} 1 & 5.400 & 12.000 & \text { No Upper Limit } \\\mathrm{X} 2 & 2.000 & 9.000 & 20.000 \\\mathrm{X} 3 & \text { No Lower Limit } & 7.000 & 10.500\end{array}$
$\text { RIGHT HAND SIDE RANGES }$
$\begin{array}{lllr}\text { Constraint } & \text { Lower Limit }& \text { Current Value } & \text { Upper Limit }\\\hline1 & 145.000 & 150.000 & 156.667 \\2 &- &- & 64.000 \\3 & -&-& 80.000 \\4 &110.286 &116.000& 120.000\end{array}$

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 $\le$ constraints. Input Section
$\begin{array}{l}\text { Objective Function Coefficients }\\ X & Y \\ 4 & 6\end{array}$

$\begin{array}{|c|c|c|c|}\hline \text { Constraints } & & & \text { Avail. } \\\hline \# 1 & 3 & 5 & 60 \\\hline \# 2 & 3 & 2 & 48 \\\hline \# 3 & 1 & 1 & 20 \\\hline\end{array}$
$\text { Output Section }$
$\begin{array}{|l|c|c|c|}\hline \text { Variables } & 13.333333 & 4 & \\\hline \text { Profit } & 53.333333 & 24 & 77.333333 \\\hline\end{array}$

$\begin{array}{|c|c|c|}\hline \text { Constraint } & \text { Usage } & \text { Slack } \\\hline \# 1 & 60 & 1.789 \mathrm{E}-11 \\\hline \# 2 & 48 & -2.69 \mathrm{E}-11 \\\hline \# 3 & 17.333333 & 2.6666667 \\\hline\end{array}$
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 $\ge$ constraints. $\text { Input Section }$
$\begin{array}{l}\text { Objective Function Coefficients }\\\begin{array}{c|c}\hline X & Y \\\hline 5 & 4\end{array}\end{array}$

$\begin{array}{|c|c|c|c|}\hline \text { Constraints } & & & \text { Req'd. } \\\hline \# 1 & 4 & 3 & 60 \\\hline \# 2 & 2 & 5 & 50 \\\hline \# 3 & 9 & 8 & 144 \\\hline\end{array}$
$\text { Output Section }$
$\begin{array}{|l|c|c|c|}\hline \text { Variables } & 9.6 & 7.2 & \\\hline \text { Profit } & 48 & 28.8 & 76.8 \\\hline\end{array}$

$\begin{array}{|c|c|c|}\hline \text { Constraint } & \text { Usage } & \text { Slack } \\\hline \# 1 & 60 & 1.35 \mathrm{E}-11 \\\hline \# 2 & 55.2 & -5.2 \\\hline \# 3 & 144 & -2.62 \mathrm{E}-11 \\\hline\end{array}$
a. Give the original linear programming problem.
b. Give the complete optimal solution.

Question 45

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 $$\begin{array} { c c c } \text { VARIABLE } & \text { VALUE } & \text { REDUCED COST } \\\text { X1 } & .000000 & 4.000000 \\\text { X2 } & 8.000000 & .000000 \\\text { X3 } & .000000 & 4.000000\end{array}$$ $\begin{array}{rrr}\text { ROW } & \text { SLACK OR SURPLUS } & \text { DUAL PRICE }\\\text { 2) } & 4.000000 & .000000 \\3) & .000000 & -1.000000\end{array}$ NO. ITERATIONS= 1
RANGES IN WHICH THE BASIS IS UNCHANGED: $$\begin{array} { c c c c } && { \text { OBJ. COEFFICIENT RANGES } } \\ & \text { CURRENT } & \text { ALLOWABLE } & \text { ALLOWABLE } \\\text { VARIABLE }&\text { COEFFICIENT }& \text { INCREASE } & \text { DECREASE } \\\hline \text { X1 } & 12.000000 & \text { INFINITY } & 4.000000 \\\text { X2 } & 10.000000 & 5.000000 & 10.000000 \\\text { X3 } & 9.000000 &\text { INFINITY } & 4.000000\\\end{array}$$ $\begin{array}{cccc}&&&\text { RIGHT HAND SIDE RANGES }\\&\text { CURRENT } & \text { ALLOWABLE }& \text { ALLOWABLE }\\\text { ROW } & \text { RHS } & \text { INCREASE } & \text { DECREASE } \\\hline2 & 60.000000 & 4.000000 & \text { INFINITY } \\3 & 80.000000 & \text { INFINITY } & 5.000000\end{array}$
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 46

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 47

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 $$\begin{array} { c r c } \text { Variable } & \text { Value } & \text { Reduced Cost } \\\text { X1 } & 0.000 & 1.500 \\\text { X2 } & 80.000 & 0.000 \\\text { X3 } & 0.000 & 1.000\end{array}$$ $\begin{array} { c c c } \text { Constraint } & \text { Slack/Surplus } & \text { Dual Frice } \\ 1 & 5.000 & 0.000 \\ 2 & 40.000 & 0.000 \\ 3 & 0.000 & - 1.250 \end{array}$
OBJECTIVE COEFFICIENT RANGES
$\begin{array}{lllc}\text { Variable }&\text { Lower Limit } & \text { Current Value } & \text { Upper Limit }\\\hline\mathrm{X} 1 & 2.500 & 4.000 & \text { No Upper Limit } \\\mathrm{X} 2 & 0.000 & 5.000 & 6.000 \\\mathrm{X} 3 & 5.000 & 6.000 & \text { No Upper Limit }\end{array}$
$\text { RIGHT HAND SIDE RANGES }$
$\begin{array}{cccc}\text { Constraint }&\text { Lower Limit } & \text { Current Value } & \text { Upper Limit }\\\hline1 & 80.000 & 85.000 & \text { No Upper Limit } \\2 & \text { No Lower Limit } & 280.000 & 320.000 \\3 & 280.000 & 320.000 & 340.000\end{array}$
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.