Question 1

In a model,x₁ ≥ 0 and integer,x₂ ≥ 0,and x₃ = 0,1.Which of the following solutions would NOT be feasible?

Accepted Answer

x₃ must be either 0 or 1, but in option C, x₃ is 0.578, which is not allowed.

Question 2

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

Accepted Answer

The constraints x₁ and x₂ must be integers, which means the feasible solutions are discrete points on the graph, represented as dots. Continuous variables would allow for a region (such as stripes or an area), but integer constraints limit the solutions to specific points.

Question 3

Solve the following problem graphically.  ​ 
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.  ​...

Question 4

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

Accepted Answer

The correct choice is A because it ensures that the sum of the variables is at least 2, meaning at least two projects must be done.

Question 5

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

Accepted Answer

The LP relaxation provides an upper bound for the value of the objective function because it relaxes the integrality constraints of the decision variables, allowing for fractions as solutions instead of just integers. Thus, the relaxed LP solution will be at least as large as the integer solution.

Question 6

In an all-integer linear program,

Accepted Answer

In all-integer linear program, only the variables are required to be integers, while the objective function coefficients and right-hand-side values can be any real number. This means that the correct choice is C, where all variables must be integers.

Question 7

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 corequisite constraint is appropriate when the acceptance of one task or project is dependent on the acceptance of another task or project. In this scenario, the acceptance of project A is dependent on the acceptance of project B, and vice versa. Therefore, a corequisite constraint is the most appropriate choice in this case.

Question 8

Most practical applications of integer linear programming involve

Accepted Answer

Most practical applications of integer linear programming involve primarily 0-1 integer variables because they are used to model decision-making situations where choices are binary (e.g., yes/no, true/false, included/excluded).

Question 9

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

Accepted Answer

The answer of Which of the following applications modeled in...

Question 10

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

Accepted Answer

Sensitivity analysis involving an integer linear program is best done by making multiple computer runs. This is because integer programs have additional constraints that make the analysis complex, and the solution may change drastically based on even small changes in the data. Multiple runs can help in understanding the range of valid solutions and how sensitive they are to changes in the input data. Using dual prices cautiously and the same approach as for linear programs may not be sufficient for capturing the integer constraints and the combinatorial nature of the problem. LP relaxation can be used as an approximation, but it may not reflect the integer solutions accurately.

Question 11

In general,whenever rounding has a minimal impact on the objective function and constraints,most managers find it acceptable.

Accepted Answer

Rounding can be used to simplify calculations and make the problem easier to solve, as long as the rounding does not significantly affect the results. Therefore, managers generally find it acceptable to use rounding when it has minimal impact on the objective function and constraints.

Question 12

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

Accepted Answer

In fixed cost models, the 0-1 variables represent whether or not a particular process is selected, which incurs a fixed cost. Therefore, the correct choice is A. B and C are not correct as they refer to the quantity of products or units produced, which are not related to fixed costs. D is also not correct as the 0-1 variables do not represent the actual value of the fixed cost, but rather whether or not it is incurred.

Question 13

Let x1 and x2 be 0-1 variables whose values indicate whether projects 1 and 2 are/are not done.Which of the following answers indicates that project 2 can be done only if project 1 is done?

Accepted Answer

The constraint x1 - x2 ≥ 0 ensures that if project 2 is done (x2 = 1), project 1 must also be done (x1 = 1), since otherwise the left hand side of the constraint would be negative. Therefore, D is the correct choice.

Question 14

Rounded solutions to linear programs must be evaluated for

Accepted Answer

Rounded solutions to linear programs must be evaluated for feasibility to ensure they still meet all the constraints of the original problem, and for optimality to ensure they are still the best possible solutions given the rounding. Sensitivity, duality, relaxation, and boundedness are important concepts in linear programming but are not directly related to the evaluation of rounded solutions.

Question 15

Sensitivity analysis for integer linear programming

Accepted Answer

The answer of Sensitivity analysis for integer linear programming
...

Question 16

Assuming W₁,W_2,and W₃ are 0-1 integer variables,the constraint W₁ + W₂ + W₃ < 1 is often called a

Accepted Answer

A mutually exclusive constraint specifies that only one variable can take on a value of 1 at a time. In this case, the constraint ensures that at most one of the three variables W₁, W₂, and W₃ can be 1, making it a mutually exclusive constraint.

Question 17

Rounding the solution of an LP Relaxation to the nearest integer values provides a(n)

Accepted Answer

Rounding the solution of an LP Relaxation to the nearest integer values might result in a solution that does not satisfy the integer problem's constraints, making it neither feasible nor optimal.

Question 18

Solve the following problem graphically.  ​ 
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.  ​...

Question 19

Modeling a fixed cost problem as an integer linear program requires

Accepted Answer

Modeling fixed costs as an integer linear program involves using 0-1 variables that reflect whether or not each cost is incurred. By using 0-1 variables, we ensure that each fixed cost is either included (1) or excluded (0) from the optimal solution, thus making the solution integer. Including the fixed costs in the objective function is not enough to model the problem as an integer linear program, as it would lead to a continuous solution. Multiple-choice constraints and LP relaxation are not relevant to modeling fixed cost problems.

Question 20

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

Accepted Answer

The answer of Which of the following is the most...

Quiz 7: Integer Linear Programming

In a model,x₁ ≥ 0 and integer,x₂ ≥ 0,and x₃ = 0,1.Which of the following solutions would NOT be feasible?

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

Solve the following problem graphically. 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.

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

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

In an all-integer linear program,

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

Most practical applications of integer linear programming involve

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

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

In general,whenever rounding has a minimal impact on the objective function and constraints,most managers find it acceptable.

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

Let x1 and x2 be 0-1 variables whose values indicate whether projects 1 and 2 are/are not done.Which of the following answers indicates that project 2 can be done only if project 1 is done?

Rounded solutions to linear programs must be evaluated for

Sensitivity analysis for integer linear programming

Assuming W₁,W_2,and W₃ are 0-1 integer variables,the constraint W₁ + W₂ + W₃ < 1 is often called a

Rounding the solution of an LP Relaxation to the nearest integer values provides a(n)

Solve the following problem graphically. 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.

Modeling a fixed cost problem as an integer linear program requires

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

Quiz 7: Integer Linear Programming

In a model,x1 ≥ 0 and integer,x2 ≥ 0,and x3 = 0,1.Which of the following solutions would NOT be feasible?

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

Solve the following problem graphically. ​ 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.

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

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

In an all-integer linear program,

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

Most practical applications of integer linear programming involve

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

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

In general,whenever rounding has a minimal impact on the objective function and constraints,most managers find it acceptable.

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

Let x1 and x2 be 0-1 variables whose values indicate whether projects 1 and 2 are/are not done.Which of the following answers indicates that project 2 can be done only if project 1 is done?

Rounded solutions to linear programs must be evaluated for

Sensitivity analysis for integer linear programming

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

Rounding the solution of an LP Relaxation to the nearest integer values provides a(n)

Solve the following problem graphically. ​ 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. ​

Modeling a fixed cost problem as an integer linear program requires

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

In a model,x₁ ≥ 0 and integer,x₂ ≥ 0,and x₃ = 0,1.Which of the following solutions would NOT be feasible?

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

Solve the following problem graphically. 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.

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

Assuming W₁,W_2,and W₃ are 0-1 integer variables,the constraint W₁ + W₂ + W₃ < 1 is often called a

Solve the following problem graphically. 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.