Question 1

Why are 0-1 variables sometimes called logical variables?

Accepted Answer

0 - 1 variables are sometimes called log

Question 2

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 3

Which of the following is the most useful contribution of integer programming?

Accepted Answer

A) finding whole number solutions where fractional solutions would not be appropriate 
B) using 0-1 variables for modeling flexibility 
C) increased ease of solution 
D) provision for solution procedures for transportation and assignment problems 
A) finding whole number solutions where fractional solutions would not be appropriate 
B) using 0-1 variables for modeling flexibility 
C) increased ease of solution 
D) provision for solution procedures for transportation and assignment problems

Question 4

Solve the following problem graphically. $\operatorname { Max } \quad\quad\quad 5 \mathrm { X } + 6 \mathrm { Y }$
s.t.  $\quad\quad\quad17 \mathrm { X } + 8 \mathrm { Y } \leq 136$
$\quad\quad\quad\quad3 \mathrm { X } + 4 \mathrm { Y } \leq 36$
$\quad\quad\quad\mathrm { X } , \mathrm { Y } \geq 0$ and integer 
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 is those integer v

Question 5

Some linear programming problems have a special structure which guarantees that the variables will have integer values.

Accepted Answer

A) True 
 B)False

Question 6

The objective of the product design and market share optimization problem presented in the textbook is to choose the levels of each product attribute that will maximize the number of sampled customers preferring the brand in question.

Accepted Answer

A) True 
 B)False

Question 7

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

The answer of Simplon Manufacturing must decide on the processes...

Question 8

In a model, x1 $\ge$  0 and integer, x2 $\ge$  0, and x3 = 0, 1. Which solution would not be feasible?

Accepted Answer

A) x1 = 5, x2 = 3, x3 = 0 
B) x1 = 4, x2 = .389, x3 = 1 
C) x1 = 2, x2 = 3, x3 = .578 
D) x1 = 0, x2 = 8, x3 = 0 
A) x1 = 5, x2 = 3, x3 = 0 
B) x1 = 4, x2 = .389, x3 = 1 
C) x1 = 2, x2 = 3, x3 = .578 
D) x1 = 0, x2 = 8, x3 = 0

Question 9

The graph of a problem that requires x1 and x2 to be integer has a feasible region

Accepted Answer

A) the same as its LP relaxation. 
B) of dots. 
C) of horizontal stripes. 
D) of vertical stripes. 
A) the same as its LP relaxation. 
B) of dots. 
C) of horizontal stripes. 
D) of vertical stripes.

Question 10

Explain how integer and 0-1 variables can be used in an objective function to minimize the sum of fixed and variable costs for production on two machines.

Accepted Answer

Integer and 0-1 variables can be used in

Question 11

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 12

The product design and market share optimization problem presented in the textbook is formulated as a 0-1 integer linear programming model.

Accepted Answer

A) True 
 B)False

Question 13

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

Accepted Answer

A) True 
 B)False

Question 14

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 the above is true. 
A) feasibility and optimality. 
B) sensitivity and duality. 
C) relaxation and boundedness. 
D) each of the above is true.

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 $\le$ 0 
D) x1 -x2 $\ge$  0 
A) x1 + x2 = 1 
B) x1 + x2 = 2 
C) x1 -x2 $\le$ 0 
D) x1 -x2 $\ge$  0

Question 16

Sensitivity analysis for integer linear programming

Accepted Answer

A) can be provided only by computer. 
B) has precisely the same interpretation as that from linear programming. 
C) does not have the same interpretation and should be disregarded. 
D) is most useful for 0-1 models. 
A) can be provided only by computer. 
B) has precisely the same interpretation as that from linear programming. 
C) does not have the same interpretation and should be disregarded. 
D) is most useful for 0-1 models.

Question 17

The Westfall Company has a contract to produce 10,000 garden hoses for a large discount chain. Westfall has four different machines that can produce this kind of hose. Because these machines are from different manufacturers and use differing technologies, their specifications are not the same. $\begin{array}{cccc}
& \text { Fixed Cost to Set } & \text { Variable Cost } & \
\text { Machine } & \text { Up Production Run } & \text { Per Hose } & \text { Capacity }\\hline
1 & 750 & 1.25 & 6000 \
2 & 500 & 1.50 & 7500 \
3 & 1000 & 1.00 & 4000 \
4 & 300 & 2.00 & 5000
\end{array}$ 
a.This problem requires two different kinds of decision variables.Clearly define each kind.
b.The company wants to minimize total cost.Give the objective function.
c.Give the constraints for the problem.
d.Write a constraint to ensure that if machine 4 is used, machine 1 cannot be.

Accepted Answer

a.Let P_i = the number of hoses produced o

Question 18

The solution to the LP Relaxation of a maximization integer linear program provides

Accepted Answer

A) an upper bound for the value of the objective function. 
B) a lower bound for the value of the objective function. 
C) an upper bound for the value of the decision variables. 
D) a lower bound for the value of the decision variables. 
A) an upper bound for the value of the objective function. 
B) a lower bound for the value of the objective function. 
C) an upper bound for the value of the decision variables. 
D) a lower bound for the value of the decision variables.

Question 19

The constraint x1 + x2 + x3 + x4 $\le$ 2 means that two out of the first four projects must be selected.

Accepted Answer

A) True 
 B)False

Question 20

Most practical applications of integer linear programming involve

Accepted Answer

A) only 0-1 integer variables and not ordinary integer variables. 
B) mostly ordinary integer variables and a small number of 0-1 integer variables. 
C) only ordinary integer variables. 
D) a near equal number of ordinary integer variables and 0-1 integer variables. 
A) only 0-1 integer variables and not ordinary integer variables. 
B) mostly ordinary integer variables and a small number of 0-1 integer variables. 
C) only ordinary integer variables. 
D) a near equal number of ordinary integer variables and 0-1 integer variables.

Why are 0-1 variables sometimes called logical variables?

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

Which of the following is the most useful contribution of integer programming?

Some linear programming problems have a special structure which guarantees that the variables will have integer values.

The objective of the product design and market share optimization problem presented in the textbook is to choose the levels of each product attribute that will maximize the number of sampled customers preferring the brand in question.

In a model, x₁ $\ge$ 0 and integer, x₂ $\ge$ 0, and x₃ = 0, 1. Which solution would not be feasible?

The graph of a problem that requires x₁ and x₂ to be integer has a feasible region

Explain how integer and 0-1 variables can be used in an objective function to minimize the sum of fixed and variable costs for production on two machines.

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.

The product design and market share optimization problem presented in the textbook is formulated as a 0-1 integer linear programming model.

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

Rounded solutions to linear programs must be evaluated for

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?

Sensitivity analysis for integer linear programming

The solution to the LP Relaxation of a maximization integer linear program provides

The constraint x₁ + x₂ + x₃ + x₄ $\le$ 2 means that two out of the first four projects must be selected.

Most practical applications of integer linear programming involve

Introduction

An Introduction to Linear Programming

Linear Programming: Sensitivity Analysis and Interpretation of Solution

Linear Programming Applications in Marketing, Finance and Operations Management

Advanced Linear Programming Applications

Distribution and Network Problems

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 7: Integer Linear Programming

Why are 0-1 variables sometimes called logical variables?

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

Which of the following is the most useful contribution of integer programming?

Some linear programming problems have a special structure which guarantees that the variables will have integer values.

The objective of the product design and market share optimization problem presented in the textbook is to choose the levels of each product attribute that will maximize the number of sampled customers preferring the brand in question.

In a model, x1 ≥\ge≥ 0 and integer, x2 ≥\ge≥ 0, and x3 = 0, 1. Which solution would not be feasible?

The graph of a problem that requires x1 and x2 to be integer has a feasible region

Explain how integer and 0-1 variables can be used in an objective function to minimize the sum of fixed and variable costs for production on two machines.

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.

The product design and market share optimization problem presented in the textbook is formulated as a 0-1 integer linear programming model.

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

Rounded solutions to linear programs must be evaluated for

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?

Sensitivity analysis for integer linear programming

The solution to the LP Relaxation of a maximization integer linear program provides

The constraint x1 + x2 + x3 + x4 ≤\le≤ 2 means that two out of the first four projects must be selected.

Most practical applications of integer linear programming involve

Introduction

An Introduction to Linear Programming

Linear Programming: Sensitivity Analysis and Interpretation of Solution

Linear Programming Applications in Marketing, Finance and Operations Management

Advanced Linear Programming Applications

Distribution and Network Problems

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

In a model, x₁ $\ge$ 0 and integer, x₂ $\ge$ 0, and x₃ = 0, 1. Which solution would not be feasible?

The graph of a problem that requires x₁ and x₂ to be integer has a feasible region

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 + x₃ + x₄ $\le$ 2 means that two out of the first four projects must be selected.