Question 1

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

Accepted Answer

The use of integer variables creates additional restrictions because they can only hold whole numbers and cannot store decimal values. However, they provide additional flexibility because they can be used for a wide range of purposes such as counting, indexing, and performing mathematical operations. Additionally, integer variables are more memory efficient and faster to process compared to other data types, making them a preferred choice in many programming scenarios. Overall, while integer variables may have limitations, their versatility and efficiency make them a valuable tool in programming.

Question 2

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

Accepted Answer

A) True 
 B)False  True

Question 3

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

Question 4

In general, rounding large values of decision variables to the nearest integer value causes fewer problems than rounding small values.

Accepted Answer

A) True 
 B)False

Question 5

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

Accepted Answer

A) True 
 B)False

Question 6

Solve the following problem graphically.
Max
X + 2Y
s.t.
6X + 8Y $\le$ 48
7X + 5Y $\ge$35
X, Y$\ge$ 0
Y 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

The answer of Solve the following problem graphically.
Max
X + 2Y
s.t.
6X...

Question 7

The solution to the LP Relaxation of a minimization problem will always be less than or equal to the value of the integer program minimization problem.

Accepted Answer

A) True 
 B)False

Question 8

If the acceptance of project A is conditional on the acceptance of project B, and vice versa, the appropriate constraint to use is a

Accepted Answer

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

Question 9

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 10

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 11

Explain how integer and 0-1 variables can be used in a constraint to enable production.

Accepted Answer

The answer of Explain how integer and 0-1 variables can...

Question 12

If the optimal solution to the LP relaxation problem is integer, it is the optimal solution to the integer linear program.

Accepted Answer

A) True 
 B)False

Question 13

Solve the following problem graphically.
Min
6X + 11Y
s.t.
9X + 3Y $\ge$ 27
7X + 6Y $\ge$42
4X + 8Y $\ge$ 32
X, Y $\ge$ 0 and integer
a.Graph the constraints for this problem.Indicate all feasible solutions.
b.Find the optimal solution to the LP Relaxation.Round up to find a feasible integer solution.Is this solution optimal?
c.Find the optimal solution.

Accepted Answer

The answer of Solve the following problem graphically.
Min
6X + 11Y
s.t.
9X...

Question 14

Modeling a fixed cost problem as an integer linear program requires

Accepted Answer

A)  adding the fixed costs to the corresponding variable costs in the objective function.
B)  using 0-1 variables.
C)  using multiple-choice constraints.
D)  using LP relaxation.

Question 15

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  ≥\ge≥  2
B)  x1 + x2 + x3  ≤\le≤  2
C)  x1 + x2 + x3 = 2
D)  x1 -x2 = 0

Question 16

Hansen Controls has been awarded a contract for a large number of control panels. To meet this demand, it will use its existing plants in San Diego and Houston, and consider new plants in Tulsa, St. Louis, and Portland. Finished control panels are to be shipped to Seattle, Denver, and Kansas City. Pertinent information is given in the table. $\begin{array}{l|c|rcc|c}
&&&(\text { Shipping Cost to Destination: }&\
&\text { Construction }&&&\text { Kansas }\
\text { Sources } & \text { Cost } & \text { Seattle } & \text {Denver } & \text { City } & \text { Capacity }\
\hline \text { San Diego } & \cdots & 5 & 7 & 8 & 2,500 \
\text { Houston } & \cdots & 10 & 8 & 6 & 2,500 \
\text { Tulsa } & 350,000 & 9 & 4 & 3 & 10,000 \
\text { St. Louis } & 200,000 & 12 & 6 & 2 & 10,000 \
\text { Portl and } & 480,000 & 4 & 10 & 11 & 10,000\\hline
&\text { Demand }& 3,000 & 8,000 & 9,000\
\end{array}$ Develop a model whose solution would reveal which plants to build and the optimal shipping schedule.

Accepted Answer

The answer of Hansen Controls has been awarded a contract...

Question 17

A multiple choice constraint involves selecting k out of n alternatives, where k $\ge$2.

Accepted Answer

A) True 
 B)False

Question 18

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

The answer of Grush Consulting has five projects to consider....

Question 19

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.

Question 20

Tower Engineering Corporation is considering undertaking several proposed projects for the next fiscal year. The projects, the number of engineers and the number of support personnel required for each project, and the expected profits for each project are summarized in the following table: $\begin{array}{lllllll}&&&\text { Project }\
&1&2&3&4&5&6\\hline
\text { Engineers Required } & 20 & 55 & 47 & 38 & 90 & 63 \
\text { Support Personnel Required } & 15 & 45 & 50 & 40 & 70 & 70 \
\text { Profit }(\$ 1,000,000 s) & 1.0 & 1.8 & 2.0 & 1.5 & 3.6 & 2.2
\end{array}$ Formulate an integer program that maximizes Tower's profit subject to the following management constraints:
1)
Use no more than 175 engineers
2)
Use no more than 150 support personnel
3)
If either project 6 or project 4 is done, both must be done
4)
Project 2 can be done only if project 1 is done
5)
If project 5 is done, project 3 must not be done and vice versa
6)
No more than three projects are to be done.

Accepted Answer

The answer of Tower Engineering Corporation is considering undertaking several...

Exam 7: Integer Linear Programming

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

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

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

In general, rounding large values of decision variables to the nearest integer value causes fewer problems than rounding small values.

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

The solution to the LP Relaxation of a minimization problem will always be less than or equal to the value of the integer program minimization problem.

If the acceptance of project A is conditional on the acceptance of project B, and vice versa, the appropriate constraint to use is a

In an all-integer linear program,

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

Explain how integer and 0-1 variables can be used in a constraint to enable production.

If the optimal solution to the LP relaxation problem is integer, it is the optimal solution to the integer linear program.

Modeling a fixed cost problem as an integer linear program requires

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?

A multiple choice constraint involves selecting k out of n alternatives, where k $\ge$ 2.

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

Exam 7: Integer Linear Programming

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

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

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

In general, rounding large values of decision variables to the nearest integer value causes fewer problems than rounding small values.

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

The solution to the LP Relaxation of a minimization problem will always be less than or equal to the value of the integer program minimization problem.

If the acceptance of project A is conditional on the acceptance of project B, and vice versa, the appropriate constraint to use is a

In an all-integer linear program,

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

Explain how integer and 0-1 variables can be used in a constraint to enable production.

If the optimal solution to the LP relaxation problem is integer, it is the optimal solution to the integer linear program.

Modeling a fixed cost problem as an integer linear program requires

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?

A multiple choice constraint involves selecting k out of n alternatives, where k ≥\ge≥ 2.

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

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

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?

A multiple choice constraint involves selecting k out of n alternatives, where k $\ge$ 2.