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Exam 7: Integer Linear Programming

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Question 31

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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.

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Question 32

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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.

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Question 33

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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.

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.

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.

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

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

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.)

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

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.

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.

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

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?

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

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

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

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

constraints involve binary variables.

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

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