Question 1

A negative dual price for a constraint in a minimization problem means

Accepted Answer

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) 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 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.

Accepted Answer

A) True 
 B)False

Question 3

The amount by which an objective function coefficient can change before a different set of values for the decision variables becomes optimal is the

Accepted Answer

A) optimal solution. 
B) dual solution. 
C) range of optimality. 
D) range of feasibility. 
A) optimal solution. 
B) dual solution. 
C) range of optimality. 
D) range of feasibility.

Question 4

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

Accepted Answer

A) True 
 B)False

Question 5

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.

Accepted Answer

a.Min
5X + 4Y
s.t.4X + 3Y @#LAT-DLM& 60

Question 6

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 $\begin{array}{ccc}
\text { VARIABLE } & \text { VALUE } & \text { REDUCED COST } \
\mathrm{X} 1 &-&1.312500\
\mathrm{X}_{2} &-&-\
\mathrm{X} 3 &27.500000&-
\end{array}$

$\begin{array}{ccc}
\text { ROW }& \text { SLACK OR SURPLUS } & \text { DUAL PRICE } \\hline
2)&-&-.125000\3)&-&-781250
\end{array}$ NO. ITERATIONS= 2
RANGES IN WHICH THE BASIS IS UNCHANGED: $\begin{array}{lrrr}
&&&\text { OBJ. COEFFICIENT RANGES }\
& \text { CURRENT}& \text {ALLOWABLE }& \text {ALLOWABLE }\
\text { VARIABLE }&\text { COEFFICIENT } & \text { INCREASE } & \text { DECREASE }\\hline
\mathrm{X} 1 &6.000000&-&-\
\mathrm{X} 2 & 7.500000 & 1.500000 & 2.500000 \
\mathrm{X} 3 & 10.000000 & 5.000000 & 3.571429
\end{array}$ $\begin{array}{cccc}&&&\text { RIGHT HAND SIDE RANGES }\
&\text { CURRENT } & \text { ALLOWABLE }& \text { ALLOWABLE }\
\text { ROW } & \text { RHS } & \text { INCREASE } & \text { DECREASE } \\hline
2 & 2400.000000 & 1100.000000 & 900.000000 \
3 & 400.000000 & 240.000000 & 125.714300
\end{array}$

Accepted Answer

It is easiest to calculate the values in

Question 7

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

Accepted Answer

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

Question 8

A constraint with a positive slack value

Accepted Answer

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. 
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 9

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

Accepted Answer

A) True 
 B)False

Question 10

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.

Accepted Answer

a.Max
4X + 6Y
s.t.3X + 5Y @#LAT-DLM&60
3

Question 11

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

Accepted Answer

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? 
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 12

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

Accepted Answer

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. 
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 13

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

Accepted Answer

A) True 
 B)False

Question 14

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

Accepted Answer

a.-5/3 and -2/5
b.Ob

Question 15

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?

Accepted Answer

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. 
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 16

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?

Accepted Answer

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. 
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 17

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

Accepted Answer

Sensitivity analysis is used in linear p

Question 18

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

Accepted Answer

A) True 
 B)False

Question 19

The reduced cost for a positive decision variable is 0.

Accepted Answer

A) True 
 B)False

Question 20

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.

Accepted Answer

A) True 
 B)False

A negative dual price for a constraint in a minimization problem means

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.

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

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

A constraint with a positive slack value

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

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

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

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

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?

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

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

The reduced cost for a positive decision variable is 0.

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.

Introduction

An Introduction to Linear Programming

Linear Programming Applications in Marketing, Finance and Operations Management

Advanced Linear Programming Applications

Distribution and Network Problems

Integer Linear Programming

Nonlinear Optimization Models

Project Scheduling: Pertcpm

Inventory Models

Waiting Line Models

Simulation

Decision Analysis

Multicriteria Decisions

Forecasting

Markov Processes

Linear Programming: Simplex Method

Simplex-Based Sensitivity Analysis and Duality

Solution Procedures for Transportation and Assignment Problems

Minimal Spanning Tree

Dynamic Programming

Filters

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

A negative dual price for a constraint in a minimization problem means

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.

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

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

A constraint with a positive slack value

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

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

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

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

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?

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

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

The reduced cost for a positive decision variable is 0.

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.

Introduction

An Introduction to Linear Programming

Linear Programming Applications in Marketing, Finance and Operations Management

Advanced Linear Programming Applications

Distribution and Network Problems

Integer Linear Programming

Nonlinear Optimization Models

Project Scheduling: Pertcpm

Inventory Models

Waiting Line Models

Simulation

Decision Analysis

Multicriteria Decisions

Forecasting

Markov Processes

Linear Programming: Simplex Method

Simplex-Based Sensitivity Analysis and Duality

Solution Procedures for Transportation and Assignment Problems

Minimal Spanning Tree

Dynamic Programming

Filters