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Exam 7: Integer Linear Programming

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Question 21

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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.

E) All of the above
F) B) and C)

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Question 22

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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.

E) B) and D)
F) B) and C)

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Question 23

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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.

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

The solution to the LP Relaxation of a maximization integer linear program provides

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 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?

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

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?

Rounding the solution of an LP Relaxation to the nearest integer values provides

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

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

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.

​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.

​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

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

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

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

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

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.