Deck 7: Integer Linear Programming

Question 1

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

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

Answer 1

B

Question 2

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

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.

Answer 2

A

Question 3

Modeling a fixed cost problem as an integer linear program requires

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.

Answer 3

B

Question 4

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

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 5

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

Question 6

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.

Question 7

If the optimal solutionASW8 to the LP relaxation problem is integer, it is the optimal solution to the integer linear program.

Question 8

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

Question 9

Most practical applications of integer linear programming involve

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.

Question 10

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

Question 11

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

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

Question 12

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

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 13

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?

A)x₁ + x₂ = 1
B)x₁ + x₂ = 2
C)x₁ - x₂ < 0
D)x₁ - x₂ > 0

Question 14

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

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

Question 15

Rounded solutions to linear programs must be evaluated for

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

Question 16

In an all-integer linear program,

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 17

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)x₁ + x₂ + x₃ > 2
B)x₁ + x₂ + x₃ < 2
C)x₁ + x₂ + x₃ = 2
D)x₁ - x₂ = 0

Question 18

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

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 19

Sensitivity analysis for integer linear programming

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 20

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

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 21

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

Question 22

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

Question 23

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

Question 24

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

Question 25

If x₁ + x₂ < 500y₁ and y₁ is 0 - 1, then if y₁ is 0, x₁ and x₂ will be 0.

Question 26

A multiple choice constraint involves selecting k out of n alternatives, where k > 2.

Question 27

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

Question 28

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

Question 29

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.

Question 30

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.