Chapter 11 – Integer Linear Programming
True / False
1. The LP Relaxation contains the objective function and constraints of the IP problem, but drops all integer restrictions.
a. True
b. False
2. In general, rounding large values of decision variables to the nearest integer value causes fewer problems than rounding
small values.
a. True
b. False
3. 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.
a. True
b. False
4. If the optimal solution to the LP relaxation problem is integer, it is the optimal solution to the integer linear program.
a. True
b. False
5. Slack and surplus variables are not useful in integer linear programs.
a. True
b. False
6. A multiple choice constraint involves selecting k out of n alternatives, where k ≥ 2.
a. True
b. False
7. In a model involving fixed costs, the 0 – 1 variable guarantees that the capacity is not available unless the cost has been
incurred.
Chapter 11 – Integer Linear Programming
a. True
8. If x1 + x2 ≤ 500y1 and y1 is 0 – 1, then if y1 is 0, x1 and x2 will be 0.
a. True
b. False
9. The constraint x1 + x2 + x3 + x4 ≤ 2 means that two out of the first four projects must be selected.
a. True
b. False
10. The constraint x1 − x2 = 0 implies that if project 1 is selected, project 2 cannot be.
a. True
b. False
11. The product design and market share optimization problem presented in the textbook is formulated as a 0-1 integer
linear programming model.
a. True
b. False
12. 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.
a. True
b. False
13. 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.
a. True
b. False
Chapter 11 – Integer Linear Programming
14. Dual prices cannot be used for integer programming sensitivity analysis because they are designed for linear
programs.
a. True
b. False
15. Some linear programming problems have a special structure that guarantees the variables will have integer values.
a. True
b. False
16. Generally, the optimal solution to an integer linear program is less sensitive to the constraint coefficients than is a
linear program.
a. True
b. False
17. The classic assignment problem can be modeled as a 0-1 integer program.
a. True
b. False
18. If Project 5 must be completed before Project 6, the constraint would be x5 − x6 ≤ 0.
a. True
b. False
19. If the LP relaxation of an integer program has a feasible solution, then the integer program has a feasible solution.
a. True
b. False
20. Multiple choice constraints involve binary variables.
Chapter 11 – Integer Linear Programming
a. True
b. False
21. Most practical applications of integer linear programming involve only 0 -1 integer variables.
a. True
b. False
22. Integer linear programs are harder to solve than linear programs.
a. True
b. False
Multiple Choice
23. 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
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
25. 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.
Chapter 11 – Integer Linear Programming
26. 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.
27. 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
28. 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.
29. 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.
30. 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.
31. 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?
Chapter 11 – Integer Linear Programming
a. x1 + x2 = 1
b. x1 + x2 = 2
c. x1 − x2 ≤ 0
d. x1 − x2 ≥ 0
32. 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
33. 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.
34. 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.
35. 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.
36. Modeling a fixed cost problem as an integer linear program requires
Chapter 11 – Integer Linear Programming
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.
37. 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.
38. 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.
39. 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
40. Which of the following applications modeled in the textbook is an example of a fixed cost problem?
a. supply chain design
b. bank location
c. capital budgeting
d. product design and market share optimization
41. LP relaxation refers to
a. eliminating nonbinding constraints
b. rounding down non-integer valued decision variables
c. dropping the integer requirements for the decision variables
d. accepting a non-optimal, but feasible, solution
Chapter 11 – Integer Linear Programming
42. A requirement that two 0-1 variables both be either in or out of a solution together is a
a. k out of n alternatives constraint
b. conditional constraint
c. mutually exclusive constraint
d. corequisite constraint
Subjective Short Answer
43. 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.
Chapter 11 – Integer Linear Programming
44. 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.
45. 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.
Chapter 11 – Integer Linear Programming
b. The optimal relaxed solution is at X = 4.5, Y = 1.75, and Z = 46.25.
The rounded solution is X = 5, Y = 2.
c. The optimal solution is at X = 6, Y = 1, and Z = 47.
POINTS: 1
TOPICS: Graphical solution
46. Consider a capital budgeting example with five projects from which to select. Let xi = 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.
47. Grush Consulting has five projects to consider. Each will require time in the next two quarters according to the table
below.
Project Time in first quarter Time in second quarter Revenue
A 5 8 12000
B 3 12 10000
C 7 5 15000
D 2 3 5000
E 15 1 20000
Chapter 11 – Integer Linear Programming
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.
48. 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.
Machine Fixed Cost to Set
Up Production Run Variable Cost
Per Hose Capacity
1 750 1.25 6000
2 500 1.50 7500
3 1000 1.00 4000
4 300 2.00 5000
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.
49. 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.
Shipping Cost to Destination
Sources Construction Cost Seattle Denver Kansas City Capacity
San Diego –— 5 7 8 2,500
Houston —- 10 8 6 2,500