Question 1

Integer linear programs are harder to solve than linear programs.

Accepted Answer

A) True 
 B)False

Question 2

A multiple choice constraint involves selecting k out of n alternatives,where k F1F1F1S1F1F1F10 2.

Accepted Answer

A) True 
 B)False

Question 3

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

Accepted Answer

A) True 
 B)False

Question 4

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

Accepted Answer

A) True 
 B)False

Question 5

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

Accepted Answer

A) True 
 B)False

Question 6

If the LP relaxation of an integer program has a feasible solution,then the integer program has a feasible solution.

Accepted Answer

A) True 
 B)False

Question 7

Solve the following problem graphically.
 $$\begin{array} { l l } 
\operatorname { Max } & x + 2 y \
\text { s.t. } & 6 x + 8 y \leq 48 \
& 7 x + 5 y \geq 35 \
& x , y \geq 0 \text { and } y \text { is integer }
\end{array}$$ 
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

Solve the following problem graphically.
 $$\begin{array} { l l } 
\text { Max } & 5 x + 6 y \
\text { s.t. } & 17 x + 8 y \leq 136 \
& 3 x + 4 y \leq 36 \
& x , y \geq 0 \text { and integer }
\end{array}$$ 
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 is those integer v

Question 9

Given the following all-integer linear program:
 Max $15 x _ { 1 } + 2 x _ { 2 }$

s. t. $7 x _ { 1 } + x _ { 2 } \leq 23$
$3 x _ { 1 } - x _ { 2 } \leq 5$
$x _ { 1 } , x _ { 2 } \geq 0$ 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.

Accepted Answer

a.x₁ = 2.8,x₂ = 3.4,Obj.func.= 48.8 b.x₁ =

Question 10

Solve the following problem graphically.
 $$\begin{array} { l l } 
\text { Min } & 6 x + 11 y \
\text { s.t. } & 9 x + 3 y \geq 27 \
& 7 x + 6 y \geq 42 \
& 4 x + 8 y \geq 32 \
& x , y \geq 0 \text { and integer }
\end{array}$$ 
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.

Accepted Answer

a.The feasible region is the set of inte

Question 11

Why are 0 - 1 variables sometimes called logical variables?

Accepted Answer

0 - 1 variables are sometimes called log

Question 12

Give a verbal interpretation of each of these constraints in the context of a capital budgeting problem.
a.x1 F1F1F1S1F1F1F10 x2 F1F1F1S1F1F1F10 0
b.x1 F1F1F1S1F1F1F10 x2 = 0
c.x1 + x2 + x3 F1F1F1S1F1F1F10 2

Accepted Answer

a. The first constraint states that the

Question 13

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 these choices are true. 
A) feasibility and optimality. 
B) sensitivity and duality. 
C) relaxation and boundedness. 
D) each of these choices are true.

Question 14

Grush Consulting has five projects to consider.Each will require time in the next two quarters according to the table below. $$\begin{array} { c c c c } 
\text { Project } & \text { Time in first quarter } & \text { Time in second quarter } & \text { Revenue } \
\hline \text { A } & 5 & 8 & 12000 \
\text { B } & 3 & 12 & 10000 \
\text { C } & 7 & 5 & 15000 \
\text { D } & 2 & 3 & 5000 \
\text { E } & 15 & 1 & 20000
\end{array}$$ 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.

Accepted Answer

Let A = 1 if project

Question 15

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

Accepted Answer

A) True 
 B)False

Question 16

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

Question 17

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

Accepted Answer

A) True 
 B)False

Question 18

Modeling a fixed cost problem as an integer linear program requires

Accepted Answer

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. 
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.

Question 19

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. $$\begin{array} { l c c c c } 
& { \text { Capital Outlays } } &&& \text { Projected } \
\text { Options } & \text { Year 1 } & \text { Year 2 } & \text { Year 3 } & \text { B enefits } \
\hline \text { New Product R\&D } & .35 & .55 & .75 & 5.2 \
\text { Plant Expansion } & .50 & .50 & 0 & 3.6 \
\text { Modernization } & .35 & .40 & .45 & 3.2 \
\text { Real Estate } & .50 & 0 & 0 & 2.8
\end{array}$$ Formulate and solve this problem as a binary programming problem.

Accepted Answer

The answer of Kloos Industries has projected the availability of...

Question 20

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. $$\begin{array} { l | c | c c c | c } 
& &{ \text { Shipping Cost to Destination: } } & \
{ \text { Sources } } & \begin{array} { c } 
\text { Construction } \
\text { Cost }
\end{array} & \text { Seattle } & \text { Denver } & \begin{array} { c } 
\text { Kansas } \
\text { City }
\end{array} & \text { Capacity } \
\hline \text { San Diego } & \cdots & 5 & 7 & 8 & 2,500 \
\text { Houston } & \cdots & 10 & 8 & 6 & 2,500 \
\text { Tulsa } & 350,000 & 9 & 4 & 3 & 10,000 \
\text { St. Louis } & 200,000 & 12 & 6 & 2 & 10,000 \
\text { Portland } & 480,000 & 4 & 10 & 11 & 10,000 \
\hline & \text { Demand } & 3,000 & 8,000 & 9,000 &
\end{array}$$ Develop a model whose solution would reveal which plants to build and the optimal shipping schedule.

Accepted Answer

The answer of Hansen Controls has been awarded a contract...

Integer linear programs are harder to solve than linear programs.

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.

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

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

If the LP relaxation of an integer program has a feasible solution,then the integer program has a feasible solution.

Why are 0 - 1 variables sometimes called logical variables?

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

Rounded solutions to linear programs must be evaluated for

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

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

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

Modeling a fixed cost problem as an integer linear program requires

Introduction

Introduction to Probability

Probability Distributions

Decision Analysis

Utility and Game Theory

Times Series Analysis and Forecasting

Introduction to Linear Programming

Linear Programming: Sensitivity Analysis and Interpretation of Solution

Linear Programming Applications in Marketing, Finance, and Operations Management

Distribution and Network Models

Advanced Optimization Applications

Project Scheduling: Pertcpm

Inventory Models

Waiting Line Models

Simulation

Markov Processes

Filters

Exam 11: Integer Linear Programming

Integer linear programs are harder to solve than linear programs.

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.

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

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

If the LP relaxation of an integer program has a feasible solution,then the integer program has a feasible solution.

Why are 0 - 1 variables sometimes called logical variables?

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

Rounded solutions to linear programs must be evaluated for

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

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

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

Modeling a fixed cost problem as an integer linear program requires

Introduction

Introduction to Probability

Probability Distributions

Decision Analysis

Utility and Game Theory

Times Series Analysis and Forecasting

Introduction to Linear Programming

Linear Programming: Sensitivity Analysis and Interpretation of Solution

Linear Programming Applications in Marketing, Finance, and Operations Management

Distribution and Network Models

Advanced Optimization Applications

Project Scheduling: Pertcpm

Inventory Models

Waiting Line Models

Simulation

Markov Processes

Filters

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