Deck 3: Linear Programming: Formulation and Applications

Question 1

Financial planning is one of the most important areas of application for cost-benefit-tradeoff problems.

Answer 1

False

Question 2

In the algebraic form of a resource constraint, the coefficient of each decision variable is the resource usage per unit of the corresponding activity.

Answer 2

True

Question 3

When formulating a linear programming model on a spreadsheet, the decisions to be made are located in the data cells.

Answer 3

False

Question 4

Strict inequalities (i.e., < or >) are not permitted in linear programming formulations.

Question 5

It is usually quite simple to obtain estimates of parameters in a linear programming problem.

Question 6

Approximations and simplifying assumptions generally are required to have a workable model.

Question 7

When formulating a linear programming model on a spreadsheet, the constraints are located (in part) in the output cells.

Question 8

Linear programming does not permit fractional solutions.

Question 9

The objective cell is a special kind of output cell.

Question 10

Resource-allocation problems are linear programming problems involving the allocation of limited resources to activities.

Question 11

A resource constraint refers to any functional constraint with a ≥ sign in a linear programming model.

Question 12

A mathematical model will be an exact representation of the real problem.

Question 13

A key assumption of linear programming is that the equation for each of the output cells, including the objective cell, can be expressed as a SUMPRODUCT (or SUM) function.

Question 14

For cost-benefit-tradeoff problems, minimum acceptable levels for each kind of benefit are prescribed and the objective is to achieve all these benefits with minimum cost.

Question 15

In most cases, the minimum acceptable level for a cost-benefit-tradeoff problem is set by how much money is available.

Question 16

A benefit constraint refers to a functional constraint with a ≥ sign in a linear programming model.

Question 17

When formulating a linear programming problem on a spreadsheet, data cells will show the levels of activities for the decisions being made.

Question 18

Cost-benefit-tradeoff problems are linear programming problems involving the allocation of limited resources to activities.

Question 19

When formulating a linear programming model on a spreadsheet, the measure of performance is located in the objective cell.

Question 20

When studying a resource-allocation problem, it is necessary to determine the contribution per unit of each activity to the overall measure of performance.

Question 21

It is the nature of the application that determines the classification of the resulting linear programming formulation.

Question 22

Once a linear programming problem has been formulated, it is rare to make major adjustments to it.

Question 23

When dealing with huge real problems, there is no such thing as the perfectly correct linear programming model for the problem.

Question 24

In an assignment problem, it is necessary to add an integer constraint to the decision variables to ensure that they will take on a value of either 0 or 1.

Question 25

Generally, assignment problems match people to an equal number of tasks at a minimum cost.

Question 26

Blending problems are a special type of mixed linear programming problems.

Question 27

A transportation problem will always return integer values for all decision variables.

Question 28

The requirements assumption states that each source has a fixed supply of units, where the entire supply must be distributed to the destinations and that each destination has a fixed demand for units, where the entire demand must be received from the sources.

Question 29

A transportation problem requires a unit cost for every source-destination combination.

Question 30

It is fairly common to have both resource constraints and benefit constraints in the same formulation.

Question 31

The capacity row in a distribution-network formulation shows the maximum number of units than can be shipped through the network.

Question 32

Transportation problems are concerned with distributing commodities from sources to destinations in such a way as to minimize the total distribution cost.

Question 33

Having one requirement for each location is a characteristic common to all transportation problems.

Question 34

It is the nature of the restrictions imposed on the decisions regarding the mix of activity levels that determines the classification of the resulting linear programming formulation.

Question 35

A mixed linear programming problem will always contain some of each of the three types of constraints in it.

Question 36

Model formulation should precede problem formulation.

Question 37

Fixed-requirement constraints in a linear programming model are functional constraints that use an equal sign.

Question 38

Choosing the best tradeoff between cost and benefits is a managerial judgment decision.

Question 39

An assignment problem is a special type of transportation problem.

Question 40

Transportation problems always involve shipping goods from one location to another.

Question 41

Transportation and assignment problems are examples of fixed-requirement problems.

Question 42

When formulating a linear programming problem on a spreadsheet, which of the following is true?

A) Parameters are called data cells.
B) Decision variables are called changing cells.
C) Right hand sides are part of the constraints.
D) The objective function is called the objective cell.
E) All of the answer choices are correct.

Question 43

In linear programming formulations, it is possible to have the following types of constraints:

A) ≤.
B) >.
C) =.
D) ≤ and > only.
E) All of the answer choices are correct.

Question 44

A linear programming problem where the objective is to find the best mix of ingredients for a product to meet certain specifications is called:

A) a resource-allocation problem.
B) a blending problem.
C) a cost-benefit tradeoff problem.
D) a mixture problem.
E) None of the answer choices are correct.

Question 45

Mixed problems may have the following type of constraints:

A) ≥.
B) ≤.
C) =.
D) All of the answer choices are correct.
E) None of the answer choices are correct.

Question 46

Where is the objective cell located?

A) B2:D2
B) B2:D2, B4:D7, and G5:G7
C) B10:D10
D) E5:E7
E) G10

Question 47

Cost-benefit tradeoff problems have the following type of constraints:

A) ≥
B) ≤
C) =
D) <
E) None of the answer choices are correct.

Question 48

The transportation model method for evaluating location alternatives minimizes:

A) the number of sources.
B) the number of destinations.
C) total supply.
D) total demand.
E) total shipping cost.

Question 49

Resource-allocation problems have the following type of constraints:

A) ≥.
B) ≤.
C) =.
D) <.
E) None of the answer choices are correct.

Question 50

A linear programming model contains which of the following components?

A) Data.
B) Decisions.
C) Constraints.
D) Measure of performance.
E) All of the answer choices are correct.

Question 51

Where are the output cells located?

A) B2:D2
B) B2:D2, B4:D7, and G5:G7
C) B10:D10
D) E5:E7
E) G10

Question 52

A transportation problem with 3 factories and 4 customers will have 12 fixed-requirement constraints.

Question 53

Where are data cells located?

A) B2:D2
B) B2:D2, B4:D7, and G5:G7
C) B10:D10
D) E5:E7
E) G10

Question 54

Starting with a simple version of a model and adding to it until it reflects the real problem is called:

A) model validation.
B) model enrichment.
C) model enhancement.
D) model elaboration.
E) None of the answer choices are correct.

Question 55

A transportation problem with 3 factories and 4 customers will have 12 shipping lanes.

Question 56

In a cost-benefit-trade-off problem, management defines the maximum amount that can be spent and the objective is to maximize benefits within this cost target.

Question 57

Which of the following are categories of linear programming problems?

A) Resource-allocation problems.
B) Cost-benefit-tradeoff problems.
C) Distribution-network problems.
D) All of the above.
E) None of the above.

Question 58

A linear programming problem may return fractional solutions for a resource allocation problem.

Question 59

Using techniques to test the initial versions of a model to identify errors and omissions is called:

A) model validation.
B) model enrichment.
C) model enhancement.
D) model debugging.
E) None of the answer choices are correct.

Question 60

Where are the changing cells located?

A) B2:D2
B) B2:D2, B4:D7, and G5:G7
C) B10:D10
D) E5:E7
E) G10

Question 61

Which of the following is not information needed to use the transportation model?

A) Capacity of the sources.
B) Demand of the destinations.
C) Unit shipping costs.
D) Unit shipping distances.
E) All of the answer choices are correct.

Question 62

When formulating a transportation problem on a spreadsheet, which of the following are necessary?

A) A table of data.
B) A network representation.
C) A table for the solution.
D) A table of data and a table for the solution only.
E) All of the answer choices are correct.

Question 63

A grocery store manager must decide how to best present a limited supply of milk and cookies to its customers. Milk can be sold by itself for a profit of $1.50 per gallon. Cookies can likewise be sold at a profit of $2.50 per dozen. To increase appeal to customers, one gallon of milk and a dozen cookies can be packaged together and are then sold for a profit of $3.00 per bundle. The manager has 100 gallons of milk and 150 dozen cookies available each day. The manager has decided to stock at least 75 gallons of milk per day and demand for cookies is always 140 dozen per day. To maximize profits, how much of each product should the manager stock.
Which of the following is the objective function for the grocer's problem?

A) Max P = 1.5M + 2.5C + 3B
B) Min P = 1.5M + 2.5C + 3B
C) Max P = 2.5M + 1.5C + 3B
D) Max P = 2.5M + 3C + 1.5B
E) Min P = 1.5M + 1.5C + 3B

Question 64

An assignment problem:

A) is a special transportation problem.
B) will always have an integer solution.
C) has all supplies and demands equal to 1.
D) None of the answer choices are correct.
E) All of the answer choices are correct.

Question 65

A freelance writer must choose how to spend her time working on several different types of projects.
Newspaper stories take 3 hours to write and pay a flat rate of $45 per story. Magazine articles take much longer to write (25 hours) but pay significantly better ($400 per article). Proofreading is often tedious, but the writer can always find proofreading jobs that pay $20 per hour. The writer wants to maximize her income, but doesn't want to work more than 45 hours per week. Additionally, she dislikes proofreading so she would like to spend no more than 7 hours per week on that task. Both newspaper stories and magazine articles must be completed in the week they are started (HINT: use an integer constraint to be sure that all newspaper and magazine jobs are finished within a week).
Which of the following is the constraint that limits the amount of time the writer will work each week?

A) 3N + 25M + P ≥ 45
B) 3N + 25M + P ≤ 45
C) 3N + 25M + P ≤ 7
D) 45N + 400M + 20P ≤ 45
E) 45N + 400M + 20P ≥ 45

Question 66

Applications of assignment problems may include:

A) matching personnel to jobs.
B) assigning machines to tasks.
C) designing bussing routes.
D) matching personnel to jobs and assigning machines to tasks only.
E) matching personnel to jobs, assigning machines to tasks, and designing bussing routes.

Question 67

A grocery store manager must decide how to best present a limited supply of milk and cookies to its customers. Milk can be sold by itself for a profit of $1.50 per gallon. Cookies can likewise be sold at a profit of $2.50 per dozen. To increase appeal to customers, one gallon of milk and a dozen cookies can be packaged together and are then sold for a profit of $3.00 per bundle. The manager has 100 gallons of milk and 150 dozen cookies available each day. The manager has decided to stock at least 75 gallons of milk per day and demand for cookies is always 140 dozen per day. To maximize profits, how much of each product should the manager stock.
The manager's problem falls within which classification?

A) Resource-allocation
B) Cost-benefit-trade-off
C) Mixed problems
D) Transportation problems
E) Assignment problems

Question 68

A grocery store manager must decide how to best present a limited supply of milk and cookies to its customers. Milk can be sold by itself for a profit of $1.50 per gallon. Cookies can likewise be sold at a profit of $2.50 per dozen. To increase appeal to customers, one gallon of milk and a dozen cookies can be packaged together and are then sold for a profit of $3.00 per bundle. The manager has 100 gallons of milk and 150 dozen cookies available each day. The manager has decided to stock at least 75 gallons of milk per day and demand for cookies is always 140 dozen per day. To maximize profits, how much of each product should the manager stock.
Which of the following is the constraint that limits the amount of milk the store will use (both in bundles and sold separately) each day?

A) M + B ≥ 100
B) M + B ≤ 100
C) M + B ≥ 75
D) M + B ≤ 75
E) M ≤ 100

Question 69

A firm has 4 plants that produce widgets. Plants A, B, and C can each produce 100 widgets per day. Plant D can produce 50 widgets per day. Each day, the widgets produced in the plants must be shipped to satisfy the demand of 3 customers. Customer 1 requires 75 units per day, customer 2 requires 100 units per day, and customer 3 requires 175 units per day. The shipping costs for each possible route are shown in the table below:
$\begin{array}{c}Shipping~ Costs&&&Customer\\per~unit~Plant&&1&2&3\\\mathrm{A} && \$ 25 & \$ 35 & \$ 15 \\\mathrm{~B} && \$ 20 & \$ 30 & \$ 40 \\\mathrm{C} && \$ 40 & \$ 35 & \$ 20 \\\mathrm{D} && \$ 15 & \$ 20 & \$ 25\end{array}$
The firm needs to satisfy all demand each day, but would like to minimize the total costs.
The objective function for the firm's problem will have how many terms?

A) 5
B) 7
C) 10
D) 12
E) 14

Question 70

A grocery store manager must decide how to best present a limited supply of milk and cookies to its customers. Milk can be sold by itself for a profit of $1.50 per gallon. Cookies can likewise be sold at a profit of $2.50 per dozen. To increase appeal to customers, one gallon of milk and a dozen cookies can be packaged together and are then sold for a profit of $3.00 per bundle. The manager has 100 gallons of milk and 150 dozen cookies available each day. The manager has decided to stock at least 75 gallons of milk per day and demand for cookies is always 140 dozen per day. To maximize profits, how much of each product should the manager stock.
What is the maximum daily profit that the grocery store can achieve?

A) $515
B) $485
C) $455
D) $425
E) $395

Question 71

A firm has 4 plants that produce widgets. Plants A, B, and C can each produce 100 widgets per day. Plant D can produce 50 widgets per day. Each day, the widgets produced in the plants must be shipped to satisfy the demand of 3 customers. Customer 1 requires 75 units per day, customer 2 requires 100 units per day, and customer 3 requires 175 units per day. The shipping costs for each possible route are shown in the table below:
$\begin{array}{c}Shipping~ Costs&&&Customer\\per~unit~Plant&&1&2&3\\\mathrm{A} && \$ 25 & \$ 35 & \$ 15 \\\mathrm{~B} && \$ 20 & \$ 30 & \$ 40 \\\mathrm{C} && \$ 40 & \$ 35 & \$ 20 \\\mathrm{D} && \$ 15 & \$ 20 & \$ 25\end{array}$
The firm needs to satisfy all demand each day, but would like to minimize the total costs.
Which of the following constraints is unnecessary for this problem (xi,j is the number of widgets shipped from factory i to customer j)?

A) xA,1 + xA,2 + xA,3 ? 100
B) xB,1 + xB,2 + xB,3 ? 100
C) xC,1 + xxC,2 + xxC,3 ? 100
D) xA,1 + xB,1 + xxC,1 ? 75
E) xA,1, xA,2, xA,3, xB,1, xB,2, xB,3, xC,1, xC,2, xC,3 integer

Question 72

A freelance writer must choose how to spend her time working on several different types of projects.
Newspaper stories take 3 hours to write and pay a flat rate of $45 per story. Magazine articles take much longer to write (25 hours) but pay significantly better ($400 per article). Proofreading is often tedious, but the writer can always find proofreading jobs that pay $20 per hour. The writer wants to maximize her income, but doesn't want to work more than 45 hours per week. Additionally, she dislikes proofreading so she would like to spend no more than 7 hours per week on that task. Both newspaper stories and magazine articles must be completed in the week they are started (HINT: use an integer constraint to be sure that all newspaper and magazine jobs are finished within a week).
The writer's problem falls within which classification?

A) Resource-allocation.
B) Cost-benefit-trade-off.
C) Mixed problems.
D) Transportation problems.
E) Assignment problems.

Question 73

A freelance writer must choose how to spend her time working on several different types of projects.
Newspaper stories take 3 hours to write and pay a flat rate of $45 per story. Magazine articles take much longer to write (25 hours) but pay significantly better ($400 per article). Proofreading is often tedious, but the writer can always find proofreading jobs that pay $20 per hour. The writer wants to maximize her income, but doesn't want to work more than 45 hours per week. Additionally, she dislikes proofreading so she would like to spend no more than 7 hours per week on that task. Both newspaper stories and magazine articles must be completed in the week they are started (HINT: use an integer constraint to be sure that all newspaper and magazine jobs are finished within a week).
Which of the following is the objective function for the writer's problem?

A) Max R = 45N + 400M − 20P
B) Min R = 3N + 25M + P
C) Max R = 45N + 400M + 20P
D) Min R = 3N + 25M + 20P
E) Max R = 3N + 400M + 20P

Question 74

A firm has 4 plants that produce widgets. Plants A, B, and C can each produce 100 widgets per day. Plant D can produce 50 widgets per day. Each day, the widgets produced in the plants must be shipped to satisfy the demand of 3 customers. Customer 1 requires 75 units per day, customer 2 requires 100 units per day, and customer 3 requires 175 units per day. The shipping costs for each possible route are shown in the table below:
$\begin{array}{c}Shipping~ Costs&&&Customer\\per~unit~Plant&&1&2&3\\\mathrm{A} && \$ 25 & \$ 35 & \$ 15 \\\mathrm{~B} && \$ 20 & \$ 30 & \$ 40 \\\mathrm{C} && \$ 40 & \$ 35 & \$ 20 \\\mathrm{D} && \$ 15 & \$ 20 & \$ 25\end{array}$
The firm needs to satisfy all demand each day, but would like to minimize the total costs.
What is the minimum daily shipping cost that the firm can achieve?

A) $6,725
B) $7,125
C) $7,525
D) $7,925
E) $8,325

Question 75

A freelance writer must choose how to spend her time working on several different types of projects.
Newspaper stories take 3 hours to write and pay a flat rate of $45 per story. Magazine articles take much longer to write (25 hours) but pay significantly better ($400 per article). Proofreading is often tedious, but the writer can always find proofreading jobs that pay $20 per hour. The writer wants to maximize her income, but doesn't want to work more than 45 hours per week. Additionally, she dislikes proofreading so she would like to spend no more than 7 hours per week on that task. Both newspaper stories and magazine articles must be completed in the week they are started (HINT: use an integer constraint to be sure that all newspaper and magazine jobs are finished within a week).
What is the optimal mix of jobs for the writer to accept each week?

A) N = 13, M = 0, P = 6
B) N = 12, M = 0, P = 7
C) N = 6, M = 1, P = 2
D) N = 5, M = 1, P = 5
E) N = 6, M = 1, P = 0

Question 76

A firm has 4 plants that produce widgets. Plants A, B, and C can each produce 100 widgets per day. Plant D can produce 50 widgets per day. Each day, the widgets produced in the plants must be shipped to satisfy the demand of 3 customers. Customer 1 requires 75 units per day, customer 2 requires 100 units per day, and customer 3 requires 175 units per day. The shipping costs for each possible route are shown in the table below:
$\begin{array}{c}Shipping~ Costs&&&Customer\\per~unit~Plant&&1&2&3\\\mathrm{A} && \$ 25 & \$ 35 & \$ 15 \\\mathrm{~B} && \$ 20 & \$ 30 & \$ 40 \\\mathrm{C} && \$ 40 & \$ 35 & \$ 20 \\\mathrm{D} && \$ 15 & \$ 20 & \$ 25\end{array}$
The firm needs to satisfy all demand each day, but would like to minimize the total costs.
The firm's problem falls within which classification?

A) Resource-allocation
B) Cost-benefit-trade-off
C) Transshipment problems
D) Transportation problems
E) Assignment problems