Question 1

Let x1 ,x2 ,and x3 be 0 - 1 variables whose values indicate whether the projects are not done (0) or are done (1) .Which answer below indicates that at least two of the projects must be done?

Accepted Answer

A)  x1 + x2 + x3  2
B)  x1 + x2 + x3  2
C)  x1 + x2 + x3 = 2
D)  x1  x2 = 0

Question 2

In a model,x1  0 and integer,x2  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

Question 3

Dual prices cannot be used for integer programming sensitivity analysis because they are designed for linear programs.

Accepted Answer

A) True 
 B)False

Question 4

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. B

Question 5

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.

Question 6

Given the following all-integer linear programming problem:

$$\begin{array} { l } 
\operatorname { Max } \quad 3 x _ { 1 } + 10 x _ { 2 } \\
\text { s.t. } 2 x _ { 1 } + x _ { 2 } \leq 5 \
x _ { 1 } + 6 x _ { 2 } \leq 9 \
\begin{aligned}
x _ { 1 } - x _ { 2 } & \geq 2 \
x _ { 1 } , x _ { 2 } & \geq 0 \text { and integer }
\end{aligned} \
\end{array}$$ 
a. Solve the problem graphically as a linear program.
b. Show that there is only one integer point and it is optimal.
c. Suppose the third constraint was changed to x1 - x2 > 2.1. What is the new optimal solution to the LP? ILP?

Accepted Answer

a.x₁ = 7/3,x₂ = 1/3,obj.func.= 3...

Question 7

Market Pulse Research has conducted a study for Lucas Furniture on some designs for a new commercial office desk.Three attributes were found to be most influential in determining which desk is most desirable: number of file drawers,the presence or absence of pullout writing boards,and simulated wood or solid color finish.Listed below are the part-worths for each level of each attribute provided by a sample of 7 potential Lucas customers.  Suppose the overall utility (sum of part-worths)of the current favorite commercial office desk is 50 for each customer.What is the product design that will maximize the share of choices for the seven sample participants? Formulate and solve,using Lindo or Excel,this 0 - 1 integer programming problem.

Accepted Answer

Define the decision variables: There are $7 l _ { i j }$ decision variables, one for each level of attribute. $l _ { i j } = 1$ if Lucas chooses level $i$ for attribute $j ; 0$ otherwise. There are $7 Y _ { k }$ decision variables, one for each consumer in the sample. $Y _ { k } = 1$ if consumer $k$ chooses the Lucas brand, 0 otherwise. Define the objective function: Maximize the number of consumers preferring the Lucas brand desk. $\operatorname { Max } \quad Y _ { 1 } + Y _ { 2 } + Y _ { 3 } + Y _ { 4 } + Y _ { 5 } + Y _ { 6 } + Y _ { 7 }$ Define the constraints: There is one constraint for each consumer in the sample. $$\begin{array} { c } 5 l _ { 11 } + 26 l _ { 21 } + 20 l _ { 31 } + 18 l _ { 12 } + 11 l _ { 22 } + 17 l _ { 13 } + 10 l _ { 23 } - 50 Y _ { 1 } \geq 1 \ 18 l _ { 11 } + 11 l _ { 21 } + 5 l _ { 31 } + 12 l _ { 12 } + 16 l _ { 22 } + 15 l _ { 13 } + 26 l _ { 23 } - 50 Y _ { 2 } \geq 1 \ 4 l _ { 11 } + 16 l _ { 21 } + 22 l _ { 31 } + 7 l _ { 12 } + 13 l _ { 22 } + 11 l _ { 13 } + 19 l _ { 23 } - 50 Y _ { 3 } \geq 1 \ 12 l _ { 11 } + 8 l _ { 21 } + 4 l _ { 31 } + 18 l _ { 12 } + 9 l _ { 22 } + 22 l _ { 13 } + 14 l _ { 23 } - 50 Y _ { 4 } \geq 1 \ 19 l _ { 11 } + 9 l _ { 21 } + 3 l _ { 31 } + 4 l _ { 12 } + 14 l _ { 22 } + 30 l _ { 13 } + 19 l _ { 23 } - 50 Y _ { 5 } \geq 1 \ 6 l _ { 11 } + 15 l _ { 21 } + 21 l _ { 31 } + 8 l _ { 12 } + 17 l _ { 22 } + 20 l _ { 13 } + 11 l _ { 23 } - 50 Y _ { 6 } \geq 1 \ 9 l _ { 11 } + 6 l _ { 21 } + 3 l _ { 31 } + 13 l _ { 12 } + 5 l _ { 22 } + 16 l _ { 13 } + 28 l _ { 23 } - 50 Y _ { 7 } \geq 1 \end{array}$$ There is one constraint for each attribute. $$\begin{array} { l } l _ { 11 } + l _ { 21 } + l _ { 31 } = 1 \ l _ { 12 } + l _ { 22 } = 1 \ l _ { 13 } + l _ { 23 } = 1 \end{array}$$ Optimal Solution: Lucas should choose these product features: 1 file drawer $\left( I _ { 21 } = 1 \right)$ No pullout writing boards $\left( I _ { 22 } = 1 \right)$ Simulated wood finish $\left( I _ { 13 } = 1 \right)$ Three sample participants would choose the Lucas design: Participant $1 \left( Y _ { 1 } = 1 \right)$ Participant $5 \left( Y _ { 5 } = 1 \right)$ Participant $6 \left( Y _ { 6 } = 1 \right)$

Question 8

Your express package courier company is drawing up new zones for the location of drop boxes for customers.The city has been divided into the seven zones shown below.You have targeted six possible locations for drop boxes.The list of which drop boxes could be reached easily from each zone is listed below. $\begin{array}{c}
\begin{array}{lll}
\text {Zone}\
\hline\text {Downtown Financial}\
\text {Downtown Legal}\
\text {Retail South}\
\text {Retail East}\
\text {Manufacturing North}\
\text {Manufacturing East}\
\text {Corporate West}\
\end{array}
\begin{array}{c}
\text {Can Be Served By Locations:}\
\hline1,2,5,6 \
2,4,5 \
1,2,4,6 \
3,4,5 \
1,2,5 \
3,4 \
1,2,6
\end{array}
\end{array}$
 Let xi = 1 if drop box location i is used,0 otherwise.Develop a model to provide the smallest number of locations yet make sure that each zone is covered by at least two boxes.

Accepted Answer

Question 9

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 10

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.

Question 11

Tom's Tailoring has five idle tailors and four custom garments to make.The estimated time (in hours)it would take each tailor to make each garment is listed below.(An 'X' in the table indicates an unacceptable tailor-garment assignment.) $\quad $$\quad $$\quad $$\quad $$\quad $$\quad $$\quad $$\quad $$\quad $$\quad $$\quad $$\quad $$\quad $$\quad $$\quad $$\quad $$\text { Tail or }$
$$\begin{array} { l c c c c c } 
\text { Garment } & 1 & 2 & 3 & 4 & 5 \
\hline \text { Wedding gown } & 19 & 23 & 20 & 21 & 18 \
\text { Clown costume } & 11 & 14 & \mathrm { X } & 12 & 10 \
\text { Admiral's uni form } & 12 & 8 & 11 & \mathrm { X } & 9 \
\text { Bullfighter's outfit } & \mathrm { X } & 20 & 20 & 18 & 21
\end{array}$$ Formulate and solve an integer program for determining the tailor-garment assignments that minimize the total estimated time spent making the four garments.No tailor is to be assigned more than one garment and each garment is to be worked on by only one tailor.

Accepted Answer

The answer of Tom's Tailoring has five idle tailors and...

Question 12

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.

Question 13

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 14

In an all-integer linear program,

Accepted Answer

A)  all objective function coefficients must be integer.
B)  all right-hand side values must be integer.
C)  all variables must be integer.
D)  all objective function coefficients and right-hand side values must be integer.

Question 15

Which of the following applications modeled in the textbook does not involve only 0 - 1 integer variables?

Accepted Answer

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

Question 16

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

Question 17

The constraint x1 + x2 + x3 + x4 F1F1F1S1F1F1F10 2 means that two out of the first four projects must be selected.

Accepted Answer

A) True 
 B)False

Question 18

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 19

If Project 5 must be completed before Project 6,the constraint would be x5 F1F1F1S1F1F1F10 x6 F1F1F1S1F1F1F10 0.

Accepted Answer

A) True 
 B)False

Question 20

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 an objective function to minimize the sum of fixed and variable costs for production on two machines by representing the decision variables for the production process.

The integer variables can represent the number of units produced on each machine, as production quantities are typically whole numbers. These integer variables can be used to determine the production levels on each machine in order to minimize the total cost.

The 0-1 variables can be used to represent whether a particular machine is used or not. For example, if a 0-1 variable is set to 1, it indicates that the machine is being used for production, while a value of 0 indicates that the machine is not being used. By including these 0-1 variables in the objective function, the model can determine the most cost-effective allocation of production across the two machines.

By including both integer and 0-1 variables in the objective function, the model can optimize the production process to minimize the sum of fixed and variable costs while meeting production requirements. This approach allows for a more accurate representation of the production process and can lead to significant cost savings for the company.

Exam 11: Integer Linear Programming

Let x₁ ,x₂ ,and x₃ be 0 - 1 variables whose values indicate whether the projects are not done (0) or are done (1) .Which answer below indicates that at least two of the projects must be done?

Correct Answer
verified

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

Correct Answer
verified

Dual prices cannot be used for integer programming sensitivity analysis because they are designed for linear programs.

Correct Answer
verified

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

Correct Answer
verified
B

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

Correct Answer
verified

Correct Answer
verified
a.x₁ = 7/3,x₂ = 1/3,obj.func.= 3...
View Answer

Correct Answer
verified

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

Correct Answer
verified

Sensitivity analysis for integer linear programming

Correct Answer
verified

Correct Answer
verified

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

Correct Answer
verified

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.

Correct Answer
verified

In an all-integer linear program,

Correct Answer
verified

Which of the following applications modeled in the textbook does not involve only 0 - 1 integer variables?

Correct Answer
verified

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

Correct Answer
verified

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

Correct Answer
verified

Correct Answer
verified

If Project 5 must be completed before Project 6,the constraint would be x₅  x₆  0.

Correct Answer
verified

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.

Exam 11: Integer Linear Programming

Let x1 ,x2 ,and x3 be 0 - 1 variables whose values indicate whether the projects are not done (0) or are done (1) .Which answer below indicates that at least two of the projects must be done?

Correct AnswerverifiedShow Answer

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

Correct AnswerverifiedShow Answer

Dual prices cannot be used for integer programming sensitivity analysis because they are designed for linear programs.

Correct AnswerverifiedShow Answer

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

Correct AnswerverifiedB

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

Correct AnswerverifiedShow Answer

Correct Answerverifieda.x1 = 7/3,x2 = 1/3,obj.func.= 3...View AnswerShow Answer

Correct AnswerverifiedShow Answer

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

Correct AnswerverifiedShow Answer

Sensitivity analysis for integer linear programming

Correct AnswerverifiedShow Answer

Correct AnswerverifiedShow Answer

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

Correct AnswerverifiedShow Answer

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.

Correct AnswerverifiedShow Answer

In an all-integer linear program,

Correct AnswerverifiedShow Answer

Which of the following applications modeled in the textbook does not involve only 0 - 1 integer variables?

Correct AnswerverifiedShow Answer

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

Correct AnswerverifiedShow Answer

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

Correct AnswerverifiedShow Answer

Correct AnswerverifiedShow Answer

If Project 5 must be completed before Project 6,the constraint would be x5  x6  0.

Correct AnswerverifiedShow Answer

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.

Let x₁ ,x₂ ,and x₃ be 0 - 1 variables whose values indicate whether the projects are not done (0) or are done (1) .Which answer below indicates that at least two of the projects must be done?

Correct Answer
verified

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

Correct Answer
verified

Correct Answer
verified

Correct Answer
verified
B

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

Correct Answer
verified

Correct Answer
verified
a.x₁ = 7/3,x₂ = 1/3,obj.func.= 3...
View Answer

Correct Answer
verified

Correct Answer
verified

Correct Answer
verified

Correct Answer
verified

Correct Answer
verified

Correct Answer
verified

Correct Answer
verified

Correct Answer
verified

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

Correct Answer
verified

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

Correct Answer
verified

Correct Answer
verified

If Project 5 must be completed before Project 6,the constraint would be x₅  x₆  0.

Correct Answer
verified