Deck 9: Integer Linear Optimization Models

Question 1

The objective function for an optimization problem is: Max 5x - 3y, with one of the constraints being x, y ≥ 0 and y integer. x and y are the only decisions variables. This is an example of a(n) _____.

A) all-integer linear program
B) mixed-integer linear program
C) LP relaxation of the integer linear program
D) binary integer linear program

Answer 1

mixed-integer linear program

Question 2

In cases where Excel Solver experiences excessive run times when solving integer linear problems, the Integer Optimality is set to _____.

A) 5%
B) 0%
C) infinity
D) a value equal to the number of integer constraints

Answer 2

5%

Question 3

The _____ of a set of points is the smallest intersection of linear inequalities that contain the set of points.

A) concave hull
B) slope
C) convex hull
D) geometry

Answer 3

convex hull

Question 4

In binary integer linear program, the integer variables take only the values:

A) 0 or 1.
B) 0 or ∞.
C) 1 or ∞.
D) 1 or -1.

Question 5

_____ is a binary integer programming problem that involves choosing which possible projects or activities provide the best investment return.

A) Capital budgeting problem
B) Fixed-cost problem
C) Market share optimization problem
D) Location problem

Question 6

The linear program that results from dropping the integer requirements for the variables in an integer linear program is known as _____.

A) convex hull
B) a mixed-integer linear program
C) LP relaxation
D) a binary integer linear program

Question 7

In order to choose the best solution for implementation, practitioners usually recommend re-solving the integer linear program several times with variations in the_____.

A) objective function
B) decision variables
C) constraint coefficients
D) integer constraints

Question 8

The optimal solution to the integer program will be an extreme point of the _____.

A) convex hull
B) objective contour
C) cutting plane
D) slope

Question 9

Binary variables are identified with the _____designation in the Solver Parameters dialog box.

A) bin
B) 0 and 1
C) int
D) dif

Question 10

Which of the following is true of the relationship between the value of the optimal integer solution and the value of the optimal solution to the LP Relaxation?

A) For integer linear programs involving minimization, the value of the optimal solution to the LP Relaxation provides an upper bound on the value of the optimal integer solution.
B) For integer linear programs involving maximization, the value of the optimal solution to the LP Relaxation provides a lower bound on the value of the optimal integer solution.
C) For integer linear programs involving minimization, the value of the optimal solution to the LP Relaxation provides a lower bound on the value of the optimal integer solution.
D) For any linear program involving either minimization or maximization, the value of the optimal solution to the LP Relaxation provides an infeasible value for the optimal integer solution.

Question 11

Which of the following approaches to solving integer linear optimization problems tries to identify the convex hull by adding a series of new constraints that do not exclude any feasible integer points?

A) Branch-and bound approach
B) Cutting plane approach
C) Trial-and-error approach
D) Convex hull approach

Question 12

Which of the following is true about the sensitivity analysis for integer optimization problems?

A) Sensitivity reports are readily available for integer optimization problems similar to the linear programming problems.
B) Because of the discrete nature of the integer optimization, Excel Solver takes much more time to calculate objective function coefficient ranges, shadow prices, and right-hand-side ranges.
C) The sensitivity analysis is not important for integer problems.
D) To determine the sensitivity of the solution to changes in model inputs for integer optimization problems, the data must be changed and the problem must be re-solveD)

Question 13

The objective function for a linear optimization problem is: Max 3x + 5y, with one of the constraints being x, y ≥ 0 and integer. x and y are the only decisions variables. This is an example of a(n) _____.

A) all-integer linear program
B) mixed-integer linear program
C) nonlinear program
D) binary integer linear program

Question 14

The imposition of integer restriction is necessary for models where:

A) nonnegativity constraints are needed.
B) variables can take negative values.
C) the decision variables cannot take fractional values.
D) possible values of variables are restricted to particular intervals.

Question 15

The _____ approach to solving integer linear optimization problems breaks the feasible region of the LP Relaxation into subregions until the subregions have integer solutions or it is determined that the solution cannot be in the subregion.

A) cutting plane
B) trial-and-error
C) breaking region
D) branch-and-bound

Question 16

The importance of _____ for integer linear programming problems is often intensified by the fact that a small change in one of the coefficients in the constraints can cause a relatively large change in the value of the optimal solution.

A) objective function
B) decision variables
C) sensitivity analysis
D) optimization analysis

Question 17

The worksheet formulation for integer linear programs and linear programming problems is exactly the same except that the _____ for integer linear programs.

A) objective function using Set Objective in the Solver Parameters dialog box is set to Value Of option
B) decision variables need not be added in By Changing Variable Cells in the Solver Parameters dialog box
C) decision variables must be added in By Changing Variable Cells in the Solver Parameters dialog box along with selecting the Ignore Integer Constraints in the Integer Options dialog box
D) constraints must be added in the Solver Parameters dialog box to identify the integer variables and the value for Tolerance in the Integer Options dialog box may need to be adjusted

Question 18

Which of the following is true of rounding the solution to an integer?

A) It always produces the most optimal integer solution.
B) It never produces a feasible solution.
C) It does not affect the objective function.
D) It may or may not be feasible.

Question 19

The objective function for a linear optimization problem is: Max 3x + 2y, with one of the constraints being x, y = 0, 1. x and y are the only decision variables. This is an example of a _____.

A) nonlinear program
B) mixed-integer linear program
C) LP relaxation of the integer linear program
D) binary integer linear program

Question 20

The objective function for an optimization problem is: Min 3x - 2y, with one of the constraints being x, y ≥ 0 and integer. If the integer restriction on the variables is removed, this would be a familiar two-variable linear program; however, it would be an example of _____.

A) convex hull of the linear program
B) a mixed-integer linear program
C) LP relaxation of the integer linear program
D) a binary integer linear program

Question 21

In a production application involving a fixed cost and a variable cost, the use of _____ makes including the fixed cost possible in a production model.

A) location variables
B) noninteger constraints
C) objective function coefficients
D) binary variables

Question 22

A constraint involving binary variables that does not allow certain variables to equal one unless certain other variables are equal to one is known as a _____.

A) conditional constraint
B) corequisite constraint
C) k out of n alternatives constraint
D) mutually exclusive constraint

Question 23

Reference - 9.1. The part-worths for each of the attribute levels obtained from an initial customer survey and the subsequent regression analysis can be used to determine the:

A) customer utility value.
B) optimal solution for the regression analysis.
C) overall profit for the company.
D) overall sales achieved by the company.

Question 24

The part-worth for each of the attribute levels in a conjoint analysis is determined by _____.

A) regression analysis
B) sensitivity analysis
C) online surveys
D) word-of-mouth

Question 25

Reference - 9.1. Pink, green, and black will be _____ of the color attribute.

A) levels
B) constraints
C) regression constants
D) utility values

Question 26

According to the _____ constraint, the sum of two or more binary variables must be equal to one.

A) conditional
B) corequisite
C) multiple-choice
D) mutually exclusive

Question 27

_____ is a constraint requiring that two binary variables be equal and that thus are both either in or out of the solution together.

A) Conditional constraint
B) Corequisite constraint
C) k out of n alternatives constraint
D) Mutually exclusive constraint

Question 28

Which of the following is true about generating alternatives in binary optimization?

A) If the second-best solution is very close to optimal, it is always preferred over the true optimal solution because of factors outside the model.
B) If alternative solutions exist, it would not help management because some factors that make one alternative are not preferred over the factors that make another alternative.
C) If the solution is a unique optimal solution, it would be good for management to know how much worse the second-best solution is than the unique optimal solution.
D) If any alternative solution exists, it would only be a second-best next to the optimal solution because there is no third-best or an alternative second-best solution to any binary integer programming problem.

Question 29

Reference - 9.1. The levels - small, medium, and large of the size attribute are modeled using:

A) objective function coefficients.
B) slack variables.
C) binary variables.
D) nonlinear coefficients.

Question 30

Coming up with a product design that will have the highest utility for a sufficient number of people to ensure sufficient sales to justify making the product is known as the _____ in marketing literature.

A) capital budgeting problem
B) share of choice problem
C) fixed-cost problem
D) traveling-salesman problem

Question 31

_____ is a market research technique that can be used to learn how prospective buyers of a product value the product's attributes.

A) Part-worth analysis
B) Conjoint analysis
C) Regression analysis
D) Sensitivity analysis

Question 32

Reference - 9.1. What category does the above objective fall under?

A) Capital budgeting problem
B) Covering problem
C) Fixed-cost problem
D) Product design and market share optimization problem

Question 33

The _____ is the utility value that a consumer attaches to each level of each attribute in a conjoint analysis model.

A) weightage
B) share of choice
C) part-worth
D) share of market

Question 34

Which of the following is a likely constraint on the production quantity x associated with a maximum value and a setup variable y in a fixed-cost problem?

A) x ≥ My
B) x ≤ My
C) Mx ≤ y
D) xy ≥ M

Question 35

Reference - 9.1. The available size of the trousers will be a(n) _____ in an integer programming model for this problem.

A) binary variable
B) constraint
C) attribute
D) regression constant

Question 36

In a fixed-cost problem, choosing excessively large values for the maximum production quantity will result in:

A) all reasonable levels of production.
B) no production.
C) no solution at all.
D) possibly a slow solution procedure.

Question 37

In a fixed-cost model, each fixed cost is associated with a binary variable and a specification of the:

A) upper bound for the corresponding production variable.
B) upper bound for each of the binary variable.
C) integer constraints involving the corresponding production variables.
D) objective function involving these binary variables only.

Question 38

A binary mixed-integer programming problem in which the binary variables represent whether an activity, such as a production run, is undertaken or not is known as the _____.

A) capital budgeting problem
B) share of choice problem
C) fixed-cost problem
D) covering problem

Question 39

For a location problem, if the variables are defined as xi = 1 if an outlet store is established in region i and 0 otherwise, the objective function is best defined by _____ for i = 1, 2, …, n number of outlet stores included in the problem.

A) Min(∑xi)
B) Max(∑xi)
C) Min(πxi)
D) Max(πxi)

Question 40

The sum of two or more binary variables must be less than or equal to one in _____ constraint.

A) corequisite
B) conditional
C) multiple-choice
D) mutually exclusive

Question 41

Sansuit Investments is deciding on future investment for the coming two years. The company considered four bonds to choose from and their investment details for the next two years are given in the table below:

The net worth of these four bonds is $75,000, $40,000, $25,500, and $18,000, respectively. The funds available with the company for Year 1 and Year 2 are $35,000 and $62,000, respectively. a. Develop and solve a binary integer programming model for maximizing the net worth assuming that only one of the bonds can be considered.
b. Suppose the investment has to be made on Bond B, and only two of the four bonds can be considered for investment. Modify your formulation from part a. to reflect this new situation. Of these two options, which is better?

Question 42

Greenbell Software Inc. conducted a study on its smartphone products in the market to determine which phone has the best features in terms of three prominent attributes: operating system of the phone (A or B), RAM (512MB or 1GB), and the rear camera specifications (3MP, 5MP, or 7MP) provided with the phone. Eight sample customers have participated in the study and provided the following part-worths for each of the above attributes.

Suppose the overall utility (sum of part-worths) of the current favorite Greenbell smartphone is 100 for each consumer. What new product design will maximize the share of choice for the eight consumers in the sample?

Question 43

A coffee maker vendor has set up two coffee machines, Machine 1 and Machine 2, inside an organization. The service cost incurred on Machine 1 and Machine 2 is $100 and $80, respectively. The production cost of coffee is $2 per mug for Machine 1 and $3 per mug for Machine 2. There is no service provided by the vendor on Sundays. The weekly production capacity is 1000 mugs for Machine 1 and 1200 mugs for Machine 2, and thereafter the machine needs to be serviced before any extra mug of coffee is to be served. Due to Christmas, the employee attendance at the organization is going to be low and only one machine has to be used to serve at least 800 mugs of coffee in that week in order to minimize the total cost.
Formulate and solve the integer programming model that could be used to determine the coffee machine which minimizes the cost.

Question 44

The following questions refer to an advertisement budgeting problem involving printing of five magazines represented by binary variables M1, M2, M3, M4, and M5.a. Write a constraint modeling a situation in which two of the magazines M1, M4, and M5 must be printed.
b. Write a constraint modeling a situation in which, if M2 or M3 is printed, they must both be printed.
c. Write a constraint modeling a situation in which magazine M1 or M3 must be printed, but not both.
d. Write constraints modeling a situation where M2 cannot be printed unless magazine M3 and M5 also are printed.
e. Write a constraint in which not more than 4 of all the five magazines have to be printed.
f. Write a constraint in which exactly five of the magazines are printed.

Question 45

A manufacturer makes two types of rubber, Butadiene and Polyisoprene. The plant has two machines, Machine-1 and Machine-2, and both of them are used to make the rubber strips. Machine-1 is available 180 hours per month and Machine-2 is available 200 hours per month. Manufacturing one strip of Butadiene requires 2.75 hours on Machine-1 and 3 hours on Machine-2. For processing one strip of Polyisoprene, it takes 3.5 hours on Machine-1 and 4 hours on Machine-2. Formulate an all-integer model that will determine how many units of each type of the rubber should be used to maximize the manufacturer's contribution to profit if he gets a profit of $20 on Butadiene and $26 on Polyisoprene?

Question 46

Four shipping containers can altogether take a maximum of 20 tons for one shipment order. The following table provides details on the weight (in tons) and value of each container:

Develop a binary integer model that will determine the two containers and the quantity that should be considered for shipment using these two containers to maximize the value of the shipment.

Question 47

BBest Ink Printing Co. received an order to print a minimum of 50,000 tickets for a concert. They have three printing machines available to meet the order they received. The set-up cost of these machines and the unit cost/ticket printed using each machine along with their maximum production are provided in the table below:
a. Formulate a binary integer linear programming model to find two machines that could be used to print the required number of tickets in order to minimize the cost.
b. Solve the problem in part a.

Question 48

Andrew is ready to invest $200,000 in stocks and he has been provided nine different alternatives by his financial consultant. The following stocks belong to three different industrial sectors and each sector has three varieties of stocks each with different expected rate of return. The average rate of return taken for the past ten years is provided with each of the nine stocks.

The decision will be based on the constraints provided below:
-Exactly 5 alternatives should be chosen.
-Any stock chosen can have a maximum investment of $55,000.
-Any stock chosen must have a minimum investment of at least $25,000.
-For the Airlines sector, the maximum number of stocks that can be chosen is two.
-The total amount invested in Banking must be at least as much as the amount invested in Agriculture.
Formulate and solve a model that will decide Andrew's investment strategy to maximize his expected annual return.

Question 49

A manufacturer wants to construct warehouses in six different locations of the city to supply dry cells to his customers on time. The manufacturer wants to construct the minimum number of warehouses, to meet the market demand on time, such that at least one warehouse is within 40 miles from each other warehouse. The following table provides the distance (in miles) between the locations.

Formulate and solve an integer linear program that can be used to determine the minimum number of warehouses needed to be constructed. What are their locations?

Question 50

Greenbell Software Inc. conducted a study on its smartphone products in the market to determine which phone has the best features in terms of three prominent attributes: operating system of the phone (A or B), RAM (512MB or 1GB), and the rear camera specifications (3MP, 5MP, or 7MP) provided with the phone. Eight sample customers have participated in the study and provided the following part-worths for each of the above attributes.

Assume the overall utility (sum of part-worths) of the current favorite Greenbell smartphone for customers 1 to 5 is 105 and customers 6 to 8 is 90. What new product design will maximize the share of choice for the eight consumers in the sample?

Question 51

Light-Twilight Inc. currently has four workshops for manufacturing light bulbs and five warehouses where the produced light bulbs are shipped from the workshops for storage. The cost, demand, and volume details are given as below:
a. Formulate a mixed-integer programming model to identify two workshops the management should retain in order to fulfil the estimated demand that minimizes the cost.
b. Solve the model you formulated in part a. What is the optimum cost?

Question 52

ROFiL Pizza Delivery is planning to open new stores in different other regions so that the excess demand for the pizza home delivery services is met. The locations under consideration and the potential areas (coded in numbers 1-9) that can be reached quickly from the considered locations for home delivery are given in the following table:

Formulate an integer programming model that could be used to find the minimum number of stores to open in order to cover customers of all areas for the home delivery service.

Question 53

A manufacturer makes two types of rubber, Butadiene and Polyisoprene. The plant has two machines, Machine-1 and Machine-2, and both of them are used to make the rubber strips. Machine-1 is available 180 hours per month and Machine-2 is available 200 hours per month. Manufacturing one strip of Butadiene requires 2.75 hours on Machine-1 and 3 hours on Machine-2. For processing one strip of Polyisoprene, it takes 3.5 hours on Machine-1 and 4 hours on Machine-2. Formulate an all-integer model that will determine how many units of each type of the rubber should be used to maximize the manufacturer's contribution to profit if he gets a profit of $20 on Butadiene and $26 on Polyisoprene?

Question 54

A coffee manufacturing company has two different plants that roast imported coffee beans. After roasting, the plants produce three types of coffee beans, A, B, and C. The company has contracted with a chain of cafes to provide per week, 20 tons of coffee bean A, 11 tons of coffee bean B, and 18 tons of coffee bean C. The two plants have the same capacity, but diverse operational features as below:

Formulate and solve all-integer model that will determine how many tons of each type of coffee beans are produced in each plant by minimizing the total cost.

Question 55

ROFiL Pizza Delivery is planning to open new stores in different other regions so that the excess demand for the pizza home delivery services is met. The locations under consideration and the potential areas (coded in numbers 1-9) that can be reached quickly from the considered locations for home delivery are given in the following table:

Solve an integer programming model that could be used to find the minimum number of stores to open in order to cover customers of all areas for the home delivery service.

Question 56

A chocolate making company largely produces one particular type of crunchy chocolate bar. There are two machines in the plant that produces this chocolate bar. The maintenance costs per day incurred on these two machines are $100 and $120, respectively. The manufacturing cost per chocolate bar is $2.5 for Machine-1 and $2 for Machine-2. The maximum daily production capacity for Machine-1 and Machine-2 are 1100 and 1250, respectively, and the company must produce at least 1000 chocolate bars per day. Develop and solve an integer programming model for minimizing the total cost.

Question 57

Delisshious Toasty Chocolates Company largely produces two types of chocolate bars, Almond Tasty and Cashew Crunchy. The maintenance costs per day incurred to produce the two types of chocolate bars are $100 and $120, respectively. The manufacturing cost per chocolate bar is $2 for Almond Tasty and $2.5 for Cashew Crunchy. The daily production capacity of Almond Tasty and Cashew Crunchy chocolate bars are 1100 and 1250, respectively. There is only one machine to produce whichever type of chocolate bar will be produced that day. Only one type of bar can be made on a given day.
Let C₁ = the number of Almond Tasty chocolate bars produced
C₂ = the number of Cashew Crunchy chocolate bars are produced
Y₁ = 1 if the machine produces Almond Tasty; 0, otherwise
Y₂ = 1 if the machine produces Cashew Crunchy; 0, otherwise a. Write a constraint that sets the next day's maximum production of Almond Tasty to 1100.
b. Write a constraint that sets the next day's maximum production of Cashew Crunchy to 1250.
c. Write a constraint that requires that production be set up for exactly one of the two chocolates bars.
d. Write the cost function to be minimized.

Question 58

FinFone Paper Mill is a small-scale paper making company which has four making machines to produce four different types of papers. Each type of paper must go through processing on each of four machines. The manufacturing time (in minutes) per unit of paper produced is in the following table:

The maximum time allotted for each machine is 30 hours per week and at least 100 units of each type of paper should be made using these machines during the week.
Profit per unit is:

Develop and solve an all-integer model that will determine, using the available machine time, the number of units of each paper type to be produced in order to meet the weekly demand and to maximize the profit.

Question 59

Andrew is ready to invest $200,000 in stocks and he has been provided nine different alternatives by his financial consultant. The following stocks belong to three different industrial sectors and each sector has three varieties of stocks each with different expected rate of return. The average rate of return taken for the past ten years is provided with each of the nine stocks.

The decision will be based on the constraints provided below:
-Exactly 5 alternatives should be chosen.
-One stock can have a maximum invest of $55,000.
-Any stock chosen must have a minimum investment of at least $25,000.
-For the Airlines sector, the maximum number of stocks chosen should be two.
-The total amount invested in Banking must be at least as much as the amount invested in Agriculture.
Now, formulate a model that will decide Andrew's investment strategy to maximize his expected annual return.

Question 60

Sansuit Investments is deciding on future investment for the coming two years. The company considered four bonds to choose from and their investment details for the next two years are given in the table below:

The net worth of these four bonds is $75,000, $40,000, $25,500, and $18,000, respectively. The funds available with the company for Year 1 and Year 2 are $35,000 and $62,000, respectively. Develop and solve a binary integer programming model for maximizing the net worth.