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Exam 7: Integer Linear Programming

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Question 21

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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 seven potential Lucas customers.​ ​ Suppose the overall utility (sum of part-worths)of the current favorite commercial office desk is 50 for each customer.What product design will maximize the share of choices for the seven sample participants? Using Excel,formulate and solve this 0-1 integer programming problem.

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Define the decision variables:There are ...

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Question 22

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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

E) A) and B)
F) A) and C)

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Question 23

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Your express package courier company is drawing up new zones for the location of drop boxes for customers.The city has been divided into seven zones.You have targeted six possible locations for drop boxes.The list of the seven zones and 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,but make sure that each zone is covered by at least two boxes.​

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. ​

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

Modeling a fixed cost problem as an integer linear program requires

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

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

In a model,x₁ ≥ 0 and integer,x₂ ≥ 0,and x₃ = 0,1.Which of the following solutions would NOT be feasible?

Rounding the solution of an LP Relaxation to the nearest integer values provides a(n)

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

Integer linear programs provide substantial modeling flexibility and thus are harder to solve than linear programs.​

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.

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

In an all-integer linear program,

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

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 (integer linear programming)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 optimal objective function of the MILP (mixed-integer linear programming)compared to the corresponding LP and ILP.

The computer output for integer programs in the textbook does not include reduced costs,dual values,or sensitivity ranges because these variables are not meaningful for integer programs.​

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.

Let x1 and x2 be 0-1 variables whose values indicate whether projects 1 and 2 are/are not done.Which of the following answers indicates that project 2 can be done only if project 1 is done?