Deck 7: Integer Linear Programming

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?

A)x₁ + x₂ + x₃ $\ge$ 2
B)x₁ + x₂ + x₃ $\le$ 2
C)x₁ + x₂ + x₃ = 2
D)x₁ -x₂ = 0

Answer 1

x₁ + x₂ + x₃ $\ge$ 2

Question 2

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.

Answer 2

A

Question 3

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.

Answer 3

C

Question 4

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 5

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

Question 6

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 7

In a model, x₁ $\ge$ 0 and integer, x₂ $\ge$ 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 8

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

Question 9

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 10

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 $\le$ 0
D)x₁ -x₂ $\ge$ 0

Question 11

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 12

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

Question 13

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.

Question 14

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 15

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

Question 16

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 17

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

Question 18

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.

Question 19

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 20

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 21

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.

Question 22

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

Question 23

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

Question 24

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

Question 25

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. Develop a model whose solution would reveal which plants to build and the optimal shipping schedule.

Question 26

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.

Question 27

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. $\begin{array}{cccc}& \text { Fixed Cost to Set } & \text { Variable Cost } & \\\text { Machine } & \text { Up Production Run } & \text { Per Hose } & \text { Capacity }\\\hline1 & 750 & 1.25 & 6000 \\2 & 500 & 1.50 & 7500 \\3 & 1000 & 1.00 & 4000 \\4 & 300 & 2.00 & 5000\end{array}$
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.

Question 28

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 29

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

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.

Question 31

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

Question 32

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.

Question 33

Why are 0-1 variables sometimes called logical variables?

Question 34

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

Question 35

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

Question 36

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

Question 37

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

Question 38

Grush Consulting has five projects to consider. Each will require time in the next two quarters according to the table below. 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.

Question 39

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

Question 40

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.

Question 41

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. Let x_i = 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.

Question 42

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: 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.

Question 43

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. (HINT: Use an additional 0-1 variable to indicate when machines 2 and 3 can be used.)