Question 1

The 0-1 variables in the fixed cost models correspond to

Accepted Answer

A) a process for which a fixed cost occurs. 
B) the number of products produced. 
C) the number of units produced. 
D) the actual value of the fixed cost. 
A) a process for which a fixed cost occurs. 
B) the number of products produced. 
C) the number of units produced. 
D) the actual value of the fixed cost.

Question 2

To perform sensitivity analysis involving an integer linear program,it is recommended to

Accepted Answer

A) use the dual prices very cautiously. 
B) make multiple computer runs. 
C) use the same approach as you would for a linear program. 
D) use LP relaxation. 
A) use the dual prices very cautiously. 
B) make multiple computer runs. 
C) use the same approach as you would for a linear program. 
D) use LP relaxation.

Question 3

The LP Relaxation contains the objective function and constraints of the IP problem,but drops all integer restrictions.

Accepted Answer

A) True 
 B)False

Question 4

​The use of integer variables creates additional restrictions but provides additional flexibility.Explain.

Accepted Answer

The use of integer variables creates add

Question 5

List and explain four types of constraints involving 0-1 integer variables only.

Accepted Answer

1. Binary Constraints: These constraints

Question 6

Simplon Manufacturing must decide on the processes to use to produce 1650 units.If machine 1 is used,its production will be between 300 and 1500 units.Machine 2 and/or machine 3 can be used only if machine 1's production is at least 1000 units.Machine 4 can be used with no restrictions.
​
 $$\begin{array} { c c c c c } 
\text { Machine } & \begin{array} { c } 
\text { Fixed } \
\text { cost }
\end{array} & \begin{array} { c } 
\text { Variable } \
\text { cost }
\end{array} & \begin{array} { c } 
\text { Minimum } \
\text { Production }
\end{array} & \begin{array} { c } 
\text { Maximum } \
\text { Production }
\end{array} \
\hline 1 & 500 & 2.00 & 300 & 1500 \
2 & 800 & 0.50 & 500 & 1200 \
3 & 200 & 3.00 & 100 & 800 \
4 & 50 & 5.00 & \text { any } & \text { any }
\end{array}$$ (HINT: Use an additional 0 - 1 variable to indicate when machines 2 and 3 can be used. )

Accepted Answer

​
 Let @#LAT-DLM& the number of units ma

Question 7

Solve the following problem graphically.
​
 $$\begin{array} { l l } 
\text { Max } & X + 2 Y \
\text { s.t. } & 6 X + 8 Y \leq 48 \
& 7 X + 5 Y \geq 35 \
& X , Y \geq 0 \
& Y \text { is integer }
\end{array}$$ 
a.Graph the constraints for this problem.Indicate all feasible solutions.
b.Find the optimal solution to the LP Relaxation.Round down to find a feasible integer solution.Is this solution optimal?
c.Find the optimal solution.

Accepted Answer

a. The feasible region consists of the p

Question 8

Rounding the solution of an LP Relaxation to the nearest integer values provides

Accepted Answer

A) a feasible but not necessarily optimal integer solution. 
B) an integer solution that is optimal. 
C) an integer solution that might be neither feasible nor optimal. 
D) an infeasible solution. 
A) a feasible but not necessarily optimal integer solution. 
B) an integer solution that is optimal. 
C) an integer solution that might be neither feasible nor optimal. 
D) an infeasible solution.

Question 9

​Assuming W1,W2 and W3 are 0 -1 integer variables,the constraint W1 + W2 + W3 < 1 is often called a

Accepted Answer

A) ​multiple-choice constraint. 
B) ​mutually exclusive constraint. 
C) ​k out of n alternatives constraint. 
D) ​corequisite constraint. 
A) ​multiple-choice constraint. 
B) ​mutually exclusive constraint. 
C) ​k out of n alternatives constraint. 
D) ​corequisite constraint.

Question 10

Which of the following applications modeled in the textbook is an example of a fixed cost problem?​

Accepted Answer

A) ​supply chain design 
B) ​bank location 
C) ​capital budgeting 
D) ​product design and market share optimization 
A) ​supply chain design 
B) ​bank location 
C) ​capital budgeting 
D) ​product design and market share optimization

Question 11

Given the following all-integer linear program:
​
 Max $3 x _ { 1 } + 2 x _ { 2 }$
s.t.
$\begin{array} { l l } \text { M.t. } & 3 x _ { 1 } + 2 x _ { 2 } \ & 3 x _ { 1 } + x _ { 2 } \leq 9 \ & x _ { 1 } + 3 x _ { 2 } \leq 7 \ & - x _ { 1 } + x _ { 2 } \leq 1 \ & x _ { 1 } , x _ { 2 } \geq 0 \text { and integer } \end{array}$
 
a.​Solve the problem as a linear program ignoring the integer constraints.Show that the optimal solution to the linear program gives fractional values for both x1 and x2.
b.​What is the solution obtained by rounding fractions greater than of equal to 1/2 to the next larger number? Show that this solution is not a feasible solution.
c.What is the solution obtained by rounding down all fractions? Is it feasible?
d.​
​
Enumerate all points in the linear programming feasible region in which both x1 and x2 are integers,and show that the feasible solution obtained in (c)is not optimal and that in fact the optimal integer is not obtained by any form of rounding.

Accepted Answer

​
 a. From the graph on the next page, t

Question 12

Slack and surplus variables are not useful in integer linear programs.

Accepted Answer

A) True 
 B)False

Question 13

Grush Consulting has five projects to consider.Each will require time in the next two quarters according to the table below.
​
 $$\begin{array} { c c c c } 
\text { Project } & \text { Time in first quarter } & \text { Time in second quarter } & \text { Revenue } \
\hline \text { A } & 5 & 8 & 12000 \
\text { B } & 3 & 12 & 10000 \
\text { C } & 7 & 5 & 15000 \
\text { D } & 2 & 3 & 5000 \
\text { E } & 15 & 1 & 20000
\end{array}$$ Revenue from each project is also shown.Develop a model whose solution would maximize revenue,meet the time budget of 25 in the first quarter and 20 in the second quarter,and not do both projects C and D.

Accepted Answer

Let A = 1 if project A is selected,0 oth

Question 14

If a problem has only less-than-or-equal-to constraints with positive coefficients for the variables,rounding down will always provide a feasible integer solution.

Accepted Answer

A) True 
 B)False

Question 15

Let x1 and x2 be 0 - 1 variables whose values indicate whether projects 1 and 2 are not done or are done.Which answer below indicates that project 2 can be done only if project 1 is done?

Accepted Answer

A) x1 + x2 = 1 
B) x1 + x2 = 2 
C) x1 − x2 ≤ 0 
D) x1 − x2 ≥ 0 
A) x1 + x2 = 1 
B) x1 + x2 = 2 
C) x1 − x2 ≤ 0 
D) x1 − x2 ≥ 0

Question 16

In a model involving fixed costs,the 0 - 1 variable guarantees that the capacity is not available unless the cost has been incurred.

Accepted Answer

A) True 
 B)False

Question 17

If the LP relaxation of an integer program has a feasible solution,then the integer program has a feasible solution.

Accepted Answer

A) True 
 B)False

Question 18

Rounded solutions to linear programs must be evaluated for

Accepted Answer

A) feasibility and optimality. 
B) sensitivity and duality. 
C) relaxation and boundedness. 
D) each of these choices are true. 
A) feasibility and optimality. 
B) sensitivity and duality. 
C) relaxation and boundedness. 
D) each of these choices are true.

Question 19

Why are 0 - 1 variables sometimes called logical variables?

Accepted Answer

0 - 1 variables are sometimes called log

Question 20

The constraint x1 − x2 = 0 implies that if project 1 is selected,project 2 cannot be.

Accepted Answer

A) True 
 B)False

The 0-1 variables in the fixed cost models correspond to

To perform sensitivity analysis involving an integer linear program,it is recommended to

The LP Relaxation contains the objective function and constraints of the IP problem,but drops all integer restrictions.

The use of integer variables creates additional restrictions but provides additional flexibility.Explain.

List and explain four types of constraints involving 0-1 integer variables only.

Rounding the solution of an LP Relaxation to the nearest integer values provides

Assuming W₁,W₂ and W₃ are 0 -1 integer variables,the constraint W₁ + W₂ + W₃ < 1 is often called a

Which of the following applications modeled in the textbook is an example of a fixed cost problem?

Slack and surplus variables are not useful in integer linear programs.

If a problem has only less-than-or-equal-to constraints with positive coefficients for the variables,rounding down will always provide a feasible integer solution.

Let x₁ and x₂ be 0 - 1 variables whose values indicate whether projects 1 and 2 are not done or are done.Which answer below indicates that project 2 can be done only if project 1 is done?

In a model involving fixed costs,the 0 - 1 variable guarantees that the capacity is not available unless the cost has been incurred.

If the LP relaxation of an integer program has a feasible solution,then the integer program has a feasible solution.

Rounded solutions to linear programs must be evaluated for

Why are 0 - 1 variables sometimes called logical variables?

The constraint x₁ − x₂ = 0 implies that if project 1 is selected,project 2 cannot be.

Introduction

Introduction to Probability

Probability Distributions

Decision Analysis

Utility Game Theory

Time Series Analysis Forecasting

Intro to Linear Programming

LP Sensitivity Analysis

LP Applications

Distribution Network Models

Advanced Optimization Applications

Project Scheduling

Inventory Models

Waiting Line Models

Simulation

Markov Processes

Filters

Exam 11: Integer Linear Programming

The 0-1 variables in the fixed cost models correspond to

To perform sensitivity analysis involving an integer linear program,it is recommended to

The LP Relaxation contains the objective function and constraints of the IP problem,but drops all integer restrictions.

​The use of integer variables creates additional restrictions but provides additional flexibility.Explain.

List and explain four types of constraints involving 0-1 integer variables only.

Rounding the solution of an LP Relaxation to the nearest integer values provides

​Assuming W1,W2 and W3 are 0 -1 integer variables,the constraint W1 + W2 + W3 < 1 is often called a

Which of the following applications modeled in the textbook is an example of a fixed cost problem?​

Slack and surplus variables are not useful in integer linear programs.

If a problem has only less-than-or-equal-to constraints with positive coefficients for the variables,rounding down will always provide a feasible integer solution.

Let x1 and x2 be 0 - 1 variables whose values indicate whether projects 1 and 2 are not done or are done.Which answer below indicates that project 2 can be done only if project 1 is done?

In a model involving fixed costs,the 0 - 1 variable guarantees that the capacity is not available unless the cost has been incurred.

If the LP relaxation of an integer program has a feasible solution,then the integer program has a feasible solution.

Rounded solutions to linear programs must be evaluated for

Why are 0 - 1 variables sometimes called logical variables?

The constraint x1 − x2 = 0 implies that if project 1 is selected,project 2 cannot be.

Introduction

Introduction to Probability

Probability Distributions

Decision Analysis

Utility Game Theory

Time Series Analysis Forecasting

Intro to Linear Programming

LP Sensitivity Analysis

LP Applications

Distribution Network Models

Advanced Optimization Applications

Project Scheduling

Inventory Models

Waiting Line Models

Simulation

Markov Processes

Filters

The use of integer variables creates additional restrictions but provides additional flexibility.Explain.

Assuming W₁,W₂ and W₃ are 0 -1 integer variables,the constraint W₁ + W₂ + W₃ < 1 is often called a

Which of the following applications modeled in the textbook is an example of a fixed cost problem?

Let x₁ and x₂ be 0 - 1 variables whose values indicate whether projects 1 and 2 are not done or are done.Which answer below indicates that project 2 can be done only if project 1 is done?

The constraint x₁ − x₂ = 0 implies that if project 1 is selected,project 2 cannot be.