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Exam 11: Integer Linear Programming

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Question 1

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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.

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Question 2

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

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

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.No more than two of projects 1,2,and 3 can be chosen. d.Projects cost 100,200,150,75,and 300 respectively.The budget is 450.

A multiple choice constraint involves selecting k out of n alternatives,where k > 2.

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

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

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

Multiple choice constraints involve binary variables.

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.

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

Solve the following problem graphically. Max X + 2Y s.t.6X + 8Y < 48 7X + 5Y > 35 X,Y > 0 Y 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.

In an all-integer linear program,

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.

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

If the acceptance of project A is conditional on the acceptance of project B,and vice versa,the appropriate constraint to use is a

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.

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.

Rounded solutions to linear programs must be evaluated for

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.