Deck 7: Integer Linear Programming

Question 1

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.

Answer 1

True

Question 2

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

Answer 2

True

Question 3

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.

Answer 3

True

Question 4

constraints involve binary variables.

Question 5

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

Question 6

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

Question 7

The classic assignment problem can be modeled as a 0-1 integer program.

Question 8

If the LP relaxation of an integer program has a feasible solution, then the integer program has a feasible solution.

Question 9

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

Question 10

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

Question 11

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

Question 12

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

Question 13

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

Question 14

Generally, the optimal solution to an integer linear program is less sensitive to the constraint coefficients than is a linear program.

Question 15

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

Question 16

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

Question 17

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 18

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

Question 19

A constraint involves selecting k out of n alternatives, where k ≥ 2.

Question 20

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

Question 21

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 22

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

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

Question 23

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

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

Question 24

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 25

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 26

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 27

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 28

​Most practical applications of integer linear programming involve only 0 -1 integer variables.

Question 29

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 30

Solve the following problem graphically.
Max
5X + 6Y
s.t.
17X + 8Y ≤ 136
3X + 4Y ≤ 36
X, Y ≥ 0 and 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 31

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 32

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 33

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 34

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 35

​Integer linear programs are harder to solve than linear programs.

Question 36

Rounded solutions to linear programs must be evaluated for

A) feasibility and optimality.
B) sensitivity and duality.
C) relaxation and boundedness.
D) each of these choices are true.

Question 37

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 38

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 39

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 40

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 41

Solve the following problem graphically.
Min
6X + 11Y
s.t.
9X + 3Y ≥ 27
7X + 6Y ≥ 42
4X + 8Y ≥ 32
X, Y ≥ 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 42

​Give a verbal interpretation of each of these constraints in the context of a capital budgeting problem.
a. x1 − x2 ≥ 0
b. x1 − x2 = 0
c. x1 + x2 + x3 ≤ 2

Question 43

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 44

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 45

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 46

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

Question 47

​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 48

​Why are 0 - 1 variables sometimes called logical variables?

Question 49

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. ​
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 50

Kloos Industries has projected the availability of capital over each of the next three years to be $850,000, $1,000,000, and $1,200,000, respectively. It is considering four options for the disposition of the capital:
(1)
Research and development of a promising new product
(2)
Plant expansion
(3)
Modernization of its current facilities
(4)
Investment in a valuable piece of nearby real estate
​
Monies not invested in these projects in a given year will NOT be available for following year's investment in the projects. The expected benefits three years hence from each of the four projects and the yearly capital outlays of the four options are summarized in the table below in $1,000,000's.
In addition, Kloos has decided to undertake exactly two of the projects, and if plant expansion is selected, it will also modernize its current facilities. ​
Formulate and solve this problem as a binary programming problem.

Question 51

Consider a capital budgeting example with five projects from which to select. Let x_i = 1 if project i is selected, 0 if not, for i = 1,...,5. Write the appropriate constraint(s) for each condition. Conditions are independent.
a.Choose no fewer than three projects.
b.If project 3 is chosen, project 4 must be chosen.
c.If project 1 is chosen, project 5 must not be chosen.
d.Projects cost 100, 200, 150, 75, and 300 respectively. The budget is 450.
e.No more than two of projects 1, 2, and 3 can be chosen.

Question 52

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.

Question 53

Given the following all-integer linear program:
Max
3x₁ + 2x₂
s.t.
3x₁ + x₂ ≤ 9
x₁ + 3x₂ ≤ 7
−x₁ + x₂ ≤ 1
x₁, x₂ ≥ 0 and integer
​
a.
​
Solve the problem as a linear program ignoring the integer constraints. Show that the optimal solution to the linear program gives fractional values for both x₁ and x₂.
b.
​
What is the solution obtained by rounding fractions greater than of equal to 1/2 to the next larger number? Show that this solution is not a feasible solution.
c.
What is the solution obtained by rounding down all fractions? Is it feasible?
d.
​
​
Enumerate all points in the linear programming feasible region in which both x₁ and x₂ are integers, and show that the feasible solution obtained in (c) is not optimal and that in fact the optimal integer is not obtained by any form of rounding.

Question 54

Given the following all-integer linear program:
​
Max 15x₁ + 2x₂
​ s. t. 7x₁ + x₂ < 23
3x₁ - x₂ < 5
x₁, x₂ > 0 and integer
​
a. Solve the problem as an LP, ignoring the integer constraints.
b. What solution is obtained by rounding up fractions greater than or equal to 1/2? Is this the optimal integer solution?
c. What solution is obtained by rounding down all fractions? Is this the optimal integer solution? Explain.
d. Show that the optimal objective function value for the ILP is lower than that for the optimal LP.
e. Why is the optimal objective function value for the ILP problem always less than or equal to the corresponding LP's optimal objective function value? When would they be equal? Comment on the MILP's optimal objective function compared to the corresponding LP & ILP.

Question 55

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

Question 56

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.)

Question 57

A business manager for a grain distributor is asked to decide how many containers of each of two grains to purchase to fill its 1,600 pound capacity warehouse. The table below summarizes the container size, availability, and expected profit per container upon distribution.
​ ​
a. Formulate as a linear program with the decision variables representing the number of containers purchased of each grain. Solve for the optimal solution.
b. What would be the optimal solution if you were not allowed to purchase fractional containers?
c. There are three possible results from rounding an LP solution to obtain an integer solution:
(1) the rounded optimal LP solution will be the optimal IP solution;
(2) the rounded optimal LP solution gives a feasible, but not optimal IP solution;
(3) the rounded optimal LP solution is an infeasible IP solution.
​
For this problem (i) round down all fractions; (ii) round up all fractions; (iii) round off (to the nearest integer) all fractions (NOTE: Two of these are equivalent.) Which result above (1, 2, or 3) occurred under each rounding method?

Question 58

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 59

Solve the following problem graphically.
Max
X + 2Y
s.t.
6X + 8Y ≤ 48
7X + 5Y ≥ 35
X, Y ≥ 0
Y is 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 60

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.)
​​ ​
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.

Question 61

​List and explain four types of constraints involving 0-1 integer variables only.