Question 1

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

Accepted Answer

A) True 
 B)False  False

Question 2

Rounded solutions to linear programs must be evaluated for

Accepted Answer

A) feasibility and optimality. 
B) sensitivity and duality. 
C) relaxation and boundedness. 
D) each of the above is true. 
A) feasibility and optimality. 
B) sensitivity and duality. 
C) relaxation and boundedness. 
D) each of the above is true. A

Question 3

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.

Accepted Answer

a.Let P_i = the number of hoses produced on machine i
U_i = 1 if machine i is used,= 0 otherwiseb.Min 750U₁ + 500U₂ + 1000U₃ + 300U₄ + 1.25P₁ + 1.5P₂ + P₃ + 2P₄c.P₁ < 6000U₁
P₂ < 7500U₂
P₃ < 4000U₃
P₄ < 5000U₄
P₁+ P₂ + P₃ + P₄ > 10000c.U₁ + U₄ < 1

Question 4

In an all-integer linear program,

Accepted Answer

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. 
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.

Question 5

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

Accepted Answer

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. 
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.

Question 6

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

Accepted Answer

A) True 
 B)False

Question 7

Solve the following problem graphically.
Max X + 2Y
s.t.6X + 8Y  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.

Accepted Answer

a.The feasible region consists of the po

Question 8

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.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.

Accepted Answer

a.x₁ + x₂ + x₃ + x₄ + x₅

Question 9

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

Accepted Answer

A) True 
 B)False

Question 10

Sensitivity analysis for integer linear programming

Accepted Answer

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. 
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.

Question 11

Multiple choice constraints involve binary variables.

Accepted Answer

A) True 
 B)False

Question 12

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

Accepted Answer

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. 
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.

Question 13

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

Accepted Answer

A) True 
 B)False

Question 14

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.

Accepted Answer

MAX 5.2X₁ + 3.6X₂ + 3.2X₃ + 2.8X₄ S.T..35X₁ +

Question 15

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

Accepted Answer

A) True 
 B)False

Question 16

Given the following all-integer linear program:
MAX 3x1 + 2x2
s.t.3x1 + x2 1 + 3x2 1 + x2 1,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.

Accepted Answer

a.From the graph on the next page,the op

Question 17

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

Accepted Answer

A) multiple-choice constraint. 
B) k out of n alternatives constraint. 
C) mutually exclusive constraint. 
D) corequisite constraint. 
A) multiple-choice constraint. 
B) k out of n alternatives constraint. 
C) mutually exclusive constraint. 
D) corequisite constraint.

Question 18

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.

Accepted Answer

Max P₁ + 1.8P₂ + 2P₃ + 1.5P₄ + 3.6

Question 19

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

Accepted Answer

A) True 
 B)False

Question 20

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.

Accepted Answer

A) True 
 B)False

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

Rounded solutions to linear programs must be evaluated for

In an all-integer linear program,

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

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

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

Sensitivity analysis for integer linear programming

Multiple choice constraints involve binary variables.

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

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

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

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

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

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.

Introduction

Introduction to Probability

Probability Distributions

Decision Analysis

Utility and Game Theory

Forecasting

Introduction to Linear Programming

Lp Sensitivity Analysis and Interpretation of Solution

Linear Programming Applications

Distribution and Network Problems

Advanced Optimization Application

Project Scheduling: Pertcpm

Inventory Models

Waiting Line Models

Simulation

Markov Processes

Filters

Exam 11: Integer Linear Programming

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

Rounded solutions to linear programs must be evaluated for

In an all-integer linear program,

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

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

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

Sensitivity analysis for integer linear programming

Multiple choice constraints involve binary variables.

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

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

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

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

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

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.

Introduction

Introduction to Probability

Probability Distributions

Decision Analysis

Utility and Game Theory

Forecasting

Introduction to Linear Programming

Lp Sensitivity Analysis and Interpretation of Solution

Linear Programming Applications

Distribution and Network Problems

Advanced Optimization Application

Project Scheduling: Pertcpm

Inventory Models

Waiting Line Models

Simulation

Markov Processes

Filters