Question 1

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 exp ansi on
(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

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

Question 2

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

The answer of Tower Engineering Corporation is considering undertaking several...

Question 3

Simplon Manufacturing must decide on the processes to use to produce 1650 units.If machine 1 is used,its production will be between 300 and 1500 units.Machine 2 and/or machine 3 can be used only if machine 1's production is at least 1000 units.Machine 4 can be used with no restrictions.
​
 $$\begin{array} { c c c c c } 
\text { Machine } & \begin{array} { c } 
\text { Fixed } \
\text { cost }
\end{array} & \begin{array} { c } 
\text { Variable } \
\text { cost }
\end{array} & \begin{array} { c } 
\text { Minimum } \
\text { Production }
\end{array} & \begin{array} { c } 
\text { Maximum } \
\text { Production }
\end{array} \
\hline 1 & 500 & 2.00 & 300 & 1500 \
2 & 800 & 0.50 & 500 & 1200 \
3 & 200 & 3.00 & 100 & 800 \
4 & 50 & 5.00 & \text { any } & \text { any }
\end{array}$$ (HINT: Use an additional 0 - 1 variable to indicate when machines 2 and 3 can be used. )

Accepted Answer

The answer of Simplon Manufacturing must decide on the processes...

Question 4

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.
​
 $$\begin{array} { c c c c } 
\text { Machine } & \begin{array} { c } 
\text { Fixed Cost to Set } \
\text { Up Production Run }
\end{array} & \begin{array} { c } 
\text { Variable Cost } \
\text { Per Hose }
\end{array} & \text { Capacity } \
\hline 1 & 750 & 1.25 & 6000 \
2 & 500 & 1.50 & 7500 \
3 & 1000 & 1.00 & 4000 \
4 & 300 & 2.00 & 5000
\end{array}$$ 
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

The answer of The Westfall Company has a contract to...

Question 5

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.
​
​
  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...

Question 6

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

Accepted Answer

The answer of Market Pulse Research has conducted a study...

Question 7

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.

Accepted Answer

The answer of Explain how integer and 0-1 variables can...

Question 8

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

Accepted Answer

The answer of Consider a capital budgeting example with five...

Question 9

Explain how integer and 0-1 variables can be used in a constraint to enable production.

Accepted Answer

The answer of Explain how integer and 0-1 variables can...

Question 10

A requirement that two 0-1 variables both be either in or out of a solution together is a​
A)​k out of n alternatives constraint
B)​conditional constraint
C)​mutually exclusive constraint
D)​corequisite constraint

Accepted Answer

This requirement is a corequisite constraint because it specifies that both variables must either both be included or both be excluded, implying they are co-required in any solution.

Question 11

Why are 0 - 1 variables sometimes called logical variables?

Accepted Answer

The answer of Why are 0 - 1 variables sometimes...

Question 12

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

The answer of Solve the following problem graphically.
​
 \[\begin{array} {...

Question 13

LP relaxation refers to​
A)​eliminating nonbinding constraints
B)​rounding down non-integer valued decision variables
C)​dropping the integer requirements for the decision variables
D)​accepting a non-optimal,but feasible,solution

Accepted Answer

LP relaxation refers to dropping the requirement that decision variables must be integers, allowing for non-integer values. This allows the problem to be more easily solved using linear programming techniques, but may result in non-optimal, non-integer solutions.

Question 14

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. $$\begin{array} { l l } \text { Zone } & \text { Can Be Served By Locations: } \ \hline \text { Downtown Financial } & 1,2,5,6 \ \text { Downtown Legal } & 2,4,5 \ \text { Retail South } & 1,2,4,6 \ \text { Retail East } & 3,4,5 \ \text { Manufacturing North } & 1,2,5 \ \text { Manufacturing East } & 3,4 \ \text { Corporate West } & 1,2,6 \end{array}$$

Accepted Answer

The answer of Your express package courier company is drawing...

Question 15

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.

Accepted Answer

The answer of Solve the following problem graphically.
​
a.Graph the constraints...

Question 16

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

Accepted Answer

The answer of ​The use of integer variables creates additional...

Question 17

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

The answer of Grush Consulting has five projects to consider.Each...

Question 18

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

The answer of Solve the following problem graphically.
​
 \[\begin{array} {...

Question 19

Tom's Tailoring has five idle tailors and four custom garments to make.The estimated time (in hours)it would take each tailor to make each garment is listed below.(An 'X' in the table indicates an unacceptable tailor-garment assignment. )
​​
​
  Formulate and solve an integer program for determining the tailor-garment assignments that minimize the total estimated time spent making the four garments.No tailor is to be assigned more than one garment and each garment is to be worked on by only one tailor.

Accepted Answer

The answer of Tom's Tailoring has five idle tailors and...

Question 20

Given the following all-integer linear program: Max $3 x _ { 1 } + 2 x _ { 2 }$ s.t. $\begin{array} { l l } \text { M.t. } & 3 x _ { 1 } + 2 x _ { 2 } \ & 3 x _ { 1 } + x _ { 2 } \leq 9 \ & x _ { 1 } + 3 x _ { 2 } \leq 7 \ & - x _ { 1 } + x _ { 2 } \leq 1 \ & x _ { 1 } , x _ { 2 } \geq 0 \text { and integer } \end{array}$ 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

The answer of Given the following all-integer linear program:
​
 Max...

Quiz 11: Integer Linear Programming

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.

Explain how integer and 0-1 variables can be used in a constraint to enable production.

A requirement that two 0-1 variables both be either in or out of a solution together is a

Why are 0 - 1 variables sometimes called logical variables?

LP relaxation refers to

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.

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

Quiz 11: Integer Linear Programming

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.

Explain how integer and 0-1 variables can be used in a constraint to enable production.

A requirement that two 0-1 variables both be either in or out of a solution together is a​

Why are 0 - 1 variables sometimes called logical variables?

LP relaxation refers to​

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.

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

A requirement that two 0-1 variables both be either in or out of a solution together is a

LP relaxation refers to

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.

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