Deck 18: Introduction to Optimization

Question 1

What does LP stand for?

A) Linear Partition
B) Linear Program
C) Large-Scale Program
D) Linear Parameter
E) Linear Parallelism

Answer 1

B

Question 2

Consider the following three functions:
F(x) = 6x²
G(x, y) =4x - 3y + 19
H(x, y) = 3xy
Which of the following is true regarding the linearity of the functions?

A) none of the functions is linear
B) all three functions are linear
C) g(x, y) is linear, but f(x) and h(x, y) are not linear
D) g(x, y) and h(x, y) are linear, but f(x) is not linear
E) h(x, y) is linear, but f(x) and g(x, y) are not linear

Answer 2

C

Question 3

The algebraic formulation of an optimization problem must state what three things?

A) decision variables, random variables, and parameters
B) decision variables, random variables, and constraints
C) decision variables, random variables, and objective function
D) random variables, parameters, and objective function
E) decision variables, objective function, and constraints

Answer 3

E

Question 4

Which type of constraint does not allow the solution for a decision variable of an optimization problem to be less than zero?

A) negativity
B) positivity
C) anti-negativity
D) feasibility
E) non-negativity

Question 5

If a firm is using a linear program to determine production amounts of its chairs and tables, which of the following constraints must be in the model?

A) non-negativity
B) labor hours
C) machine hours
D) profit maximization
E) wood availability

Question 6

If C represents the number of chairs produced, which of the following is a proper non-negativity constraint?

A) C ≠ 0
B) C > 0
C) C ≥ 0
D) C < 0
E) C ≤ 0

Question 7

A mathematical model in which one is trying to maximize or minimize some quantity while satisfying a set of constraints is a(n):

A) objective model
B) constrained model
C) constrained optimization problem
D) decision model
E) excel model

Question 8

In Linear Programming models, what do you want to do with the objective?

A) upsize or downsize
B) rightsize or wrongsize
C) maximize or minimize
D) memorize
E) popularize

Question 9

In Linear Programming models, over what quantities do you have control?

A) decision variables
B) parameters
C) constraints
D) objective variables
E) coefficients

Question 10

A ____________________ contains explicit definitions of the decision variables, an algebraic expression of the objective function, and algebraic statements of the constraints.

A) composition
B) derivation
C) manipulation
D) formulation
E) regurgitation

Question 11

Suppose that a firm makes gears (G) and axles (A). Each gear requires 15 minutes of labor and each axle requires 30 minutes of labor. One thousand hours of labor are available. How should the labor constraint(s) be written?

A) G + A ≤ 1000
B) )25G + .5A ≤ 1000
C) G ≤ 15 and A ≤ 30
D) )25G + .5A ≥ 1000
E) 15G + 30A ≤ 1000

Question 12

When formulating optimization problems, which of the following represent the typical sequence?

A) (1) text-based formulation, (2) diagram, (3) algebraic formulation
B) (1) text-based formulation, (2) algebraic formulation, (3) diagram
C) (1) diagram, (2) algebraic formulation, (3) text-based formulation
D) (1) diagram, (2) text-based formulation, (3) algebraic formulation
E) (1) algebraic formulation, (2) diagram, (3) text-based formulation

Question 13

Large-scale optimization software can handle how many decision variables and constraints?

A) dozens
B) hundreds
C) thousands
D) millions
E) all of these answer choices are correct

Question 14

Testing of the LP model should include __________ and _____________.

A) extreme case, base case
B) extreme case, test values
C) base case, minimum value
D) base case, test values
E) test value, maximum value

Question 15

Once you've written the algebraic formulation of the problem, the next setup involved in Solving Optimization Problems is:

A) develop test cases
B) examine the results and make corrections
C) analyze and interpret the results
D) set up the Solver settings
E) write the problem formulation

Question 16

The text-based formulation of an optimization problem should state what three things?

A) decision variables, random variables, and parameters
B) decision variables, objective function, and constraints
C) decision variables, random variables, and constraints
D) decision variables, random variables, and objective function
E) random variables, parameters, and objective function

Question 17

A diagram of the situation can help _______ the problem as well as be a(n) ______ _______ tool.

A) structure, valuable communication
B) define, algebraic development
C) bound, formulation development
D) unbound, formulation development
E) structure, unbounded formulation

Question 18

Human intelligence is not needed in which of the following steps of solving optimization problems?

A) writing the problem formulation in words
B) writing the algebraic formulation of the problem
C) developing a spreadsheet model
D) solving the problem
E) interpreting the results

Question 19

What are the allowable constraint relationship types in optimization problems?

A) <, >, =
B) <, >, =, ≤, ≥
C) =, ≤, ≥
D) <, >, ≤, ≥
E) ≠, =, ≤, ≥

Question 20

How can the following Linear Program be characterized?
Min X + Y
Subject to
X ≤ 20
Y ≤ 5
X + Y ≥ 40
X, Y ≥ 0

A) bounded and feasible
B) unbounded and feasible
C) bounded and infeasible
D) unbounded and infeasible
E) this is not a Linear Program

Question 21

What is the optimal solution to the following linear program?
Max 2X + Y
Subject to
2X + 2Y ≥ 40
X + Y ≤ 10
X ≥ 0
Y ≥ 0

A) the solution is unbounded
B) (0, 20)
C) (10, 0)
D) (20, 0)
E) the program is infeasible

Question 22

Which of the following is not one of the steps in setting up the Solver optimization problem?

A) specify the target cell
B) specify the objective cell
C) specify the changing cell(s)
D) specify the constraints
E) specify the Solver options

Question 23

Capital Co. is considering which of five different projects to undertake in order to maximize its net present value (NPV). Define X_i as a binary (0-1) variable that equals 1 if project i is undertaken and 0 otherwise, for i = 1,2,3,4,5. The NPV and required capital (in thousands) for each project are listed below.
Project Net Present Value Capital Required
1 60 7
2 50 10
3 40 6
4 20 3
5 30 12
Which of the following represents the constraint stating that project 1 can be undertaken only if project 5 is undertaken?

A) X₁ + X₅ ≤ 1
B) X₁ - X₅ ≤ 0
C) X₅ - X₁ ≤ 0
D) X₁ - X₅ ≤ 1
E) X₅ - X₁ ≤ 1

Question 24

Sausage and Cheese Ltd. prepares three gift packages containing sausages and cheeses. The "Tasters," "Succulent," and "Gourmet" gift packages contain 3 sausages and 6 cheeses, 5 sausages and 4 cheeses, and 6 sausages and 5 cheeses, respectively. There are 2500 sausages and 3000 cheeses available for packing, and demand is unlimited. Profits are $2.50, $3.50, and $4.00 for the "Tasters," "Succulent," and "Gourmet" gift packages, respectively. The goal is to maximize profits. Let T, S, and G represent the number of gift packages produced of type "Tasters," "Succulent," and "Gourmet," respectively. What is the objective function for this linear program?

A) Max T + S + G
B) Max 2.5T + 3.5S + 4G
C) Min T + S + G
D) Max 10(T + S + G)
E) Max 9T + 9S + 11G

Question 25

How can the following Linear Program be characterized?
Max X + Y
Subject to
X ≤ 34
X, Y ≥ 0

A) bounded and feasible
B) unbounded and feasible
C) bounded and infeasible
D) unbounded and infeasible
E) this is not a Linear Program

Question 26

Which of the following statements is not correct?

A) A function with multiple variables can be linear.
B) If a constraint is violated, then the problem is infeasible.
C) A Linear Program as described in Supplement B can have more than one objective function if it has only one constraint.
D) Non-negativity constraints should be added to Linear Programs when the decision variable represents units of production.
E) An unbounded solution to a mathematical program may occur if there is not a constraint stopping the objective function value from continuing towards -∞.

Question 27

Riker Co. is considering which of four different projects to undertake in order to maximize its net present value (NPV). Define X_i as a binary (0-1) variable that equals 1 if project i is undertaken and 0 otherwise, for i = 1,2,3,4. The NPV and required capital (in millions) for each project are listed below.
Project Net Present Value Capital Required
1 60 7
2 50 10
3 40 6
4 20 3
Which of the following represents the constraint(s) stating that project 1 must be undertaken and at least one of the other projects must be undertaken?

A) X₁ + X₂ ≤ 1, and X₁ + X₃ ≤ 1, and X₁ + X₄ ≤ 1
B) X₁ ≥ 1, and X₂ + X₃ + X₄ ≥ 1
C) X₁ + X₂ ≥ 1, and X₁ + X₃ ≥ 1, and X₁ + X₄ ≥ 1
D) X₁ ≥ 1, and X₂ + X₃ + X₄ ≤ 3
E) X₁ + X₂ + X₃ + X₄ ≥ 2

Question 28

Sausage and Cheese Ltd. prepares three gift packages containing sausages and cheeses. The "Tasters," "Succulent," and "Gourmet" gift packages contain 3 sausages and 6 cheeses, 5 sausages and 4 cheeses, and 6 sausages and 5 cheeses, respectively. There are 2500 sausages and 3000 cheeses available for packing, and demand is unlimited. Profits are $2.50, $3.50, and $4.00 for the "Tasters," "Succulent," and "Gourmet" gift packages, respectively. The goal is to maximize profits. Let T, S, and G represent the number of gift packages produced of type "Tasters," "Succulent," and "Gourmet," respectively. What is the constraint describing the sausage capacity?

A) 2.5T + 3.5S + 4G ≤ 2500
B) 3T + 5S + 6G ≥ 2500
C) 9T + 9S + 11G ≤ 2500
D) 0.8333T + 1.9444S + 2.1818G ≤ 2500
E) 3T + 5S + 6G ≤ 2500

Question 29

Capital Co. is considering which of five different projects to undertake in order to maximize its net present value (NPV). Define X_i as a binary (0-1) variable that equals 1 if project i is undertaken and 0 otherwise, for i = 1,2,3,4,5. The NPV and required capital (in thousands) for each project are listed below.
Project Net Present Value Capital Required
1 60 7
2 50 10
3 40 6
4 20 3
5 30 12
What is the proper objective function?

A) Max X₁ + X₂ + X₃ + X₄ + X₅
B) Min 60X₁ + 50X₂ + 40X₃ + 20X₄ + 30X₅
C) Max 7X₁ + 10X₂ + 6X₃ + 3X₄ + 12X₅
D) Max 60X₁ + 50X₂ + 40X₃ + 20X₄ + 30X₅
E) Min 7X₁ + 10X₂ + 6X₃ + 3X₄ + 12X₅

Question 30

Dane's aircraft muffler manufacturers have 1500 linear feet of steel on hand to manufacture the three top selling muffler sets. Super mufflers (S) provide $285 profit and common (C) mufflers' profit margin is $310, while the deluxe (D) muffler set provides a $400 profit margin. It costs Dane $310, $295, and $400 to build each muffler set, respectively. What is the objective function of Dane's aircraft muffler manufacturing?

A) Min S, C and D
B) MAX S, C, and D
C) Max 285S + 310C + 400D
D) Min 285S + 310C + 400D
E) Max 310S + 295C + 400D

Question 31

Consider the following constraints from a two-variable Linear Program.
(1) X ≥ 0
(2) Y ≥ 0
(3) X + Y ≤ 50
If constraints (2) and (3) are binding, what is the optimal solution (X, Y)?

A) (0, 0)
B) (0, 50)
C) (50, 0)
D) (50, 50)
E) not enough information is provided to answer the problem

Question 32

Solver provides many options for the solution process. For LPs, the two most commonly used are:

A) assume linear model, assume negative
B) assume non-linear model, assume negative
C) assume non-linear model, assume non-negative
D) assume linear model, assume non-negative
E) assume non-linear model, assume non-negative

Question 33

Consider the following three functions:
G(x, y) = 4x - 3y + 21
H(x, y, z) = 13x² + y + 3z
I(z) = z
Which of the following is true regarding the linearity of the functions?

A) none of the functions is linear
B) all three functions are linear
C) h is linear, but g and i are not linear
D) i is linear, but g and h are not linear
E) g and i are linear, but h is not linear

Question 34

Consider the mathematical program below. Which of the following choices represents an upper bound to the problem?
Max 10 - X²
Subject to
X ≥ 3

A) X = 10
B) X = 5
C) X = 3.5
D) X = -6
E) X = 0

Question 35

How can the following Linear Program be characterized?
Min X + Y
Subject to
X ≥ 20
Y ≥ -5
X + Y ≤ 23

A) bounded and feasible
B) unbounded and feasible
C) bounded and infeasible
D) unbounded and infeasible
E) this is not a Linear Program

Question 36

Consider the following two functions:
G(x, y) =4x - 3y + 21
H(x, y) = 13xy
Which of the following is true regarding the linearity of the functions?

A) neither of the functions is linear
B) both functions are linear
C) g(x, y) is linear, but h(x, y) is not linear
D) h(x, y) is linear, but g(x, y) is not linear
E) h(x, y) is not linear, and g(x, y) is only linear when y ≥ 0

Question 37

Consider the Linear Program below. Which of the choices represents the best (tightest) lower bound?
Max 2X + Y
Subject to
X + Y ≤ 10
X, Y ≥ 0

A) (∞, -∞)
B) (-∞, -∞)
C) (0, 0)
D) (5, 0)
E) (25, 16)

Question 38

What is the optimal solution (X, Y, Z) to the following linear program?
Max 2X + 4Y + 6Z
Subject to
Z ≤ 0
X + Y + Z ≤ 20
X, Y, Z ≥ 0

A) (0, 0, 20)
B) (20, 0, 0)
C) (0, 20, 0)
D) (20, 20, 20)
E) the solution is unbounded

Question 39

Consider a mathematical program where X_i represents the amount produced of item i (i = 1,2,3,4), and you want the total amount produced over all four items to equal either 100, 120, 140, or 200. If you define q_i as binary (0-1) variables (i = 1,2,3,4) and add the constraint q₁ + q₂ + q₃ + q₄ = 1, what other constraint do you need to add to the program?

A) 100q₁ + 120q₂ + 140q₃ + 200q₄ =560
B) X₁ + X₂ + X₃ + X₄ =560
C) X₁ + X₂ + X₃ + X₄ = 100q₁ + 120q₂ + 140q₃ + 200q₄
D) 100X₁ + 120X₂ + 140X₃ + 200X₄ = q₁ + q₂ + q₃ + q₄
E) X₁ + X₂ + X₃ + X₄ ≤ 200

Question 40

Capital Co. is considering five different projects. Define X_i as a binary (0-1) variable that equals 1 if project i is undertaken and 0 otherwise, for i = 1,2,3,4,5. Which of the following represents the constraint(s) stating that projects 2, 3, and 4 cannot all be undertaken simultaneously?

A) X₂ + X₃ + X₄ ≤ 3
B) X₁ + X₂ + X₃ + X₄ + X₅ ≤ 3
C) X₂ + X₃ + X₄ ≤ 1
D) X₂ + X₃ + X₄ ≤ 2
E) X₂ + X₃ ≤ 1 and X₃ + X₄ ≤ 1

Question 41

How can the following Linear Program be characterized?
Min X + 2Y
Subject to
X ≤ 20
X, Y ≥ -40

A) bounded and feasible
B) unbounded and feasible
C) bounded and infeasible
D) unbounded and infeasible
E) this is not a Linear Program

Question 42

George Dantzig developed the ___________________ in 1947 to solve Linear Programs.

A) Complex Method
B) Multiplex Method
C) Simplex Method
D) Superplex Method
E) Perplex Method

Question 43

Consider the following constraints from a two-variable Linear Program.
(1) X ≥ 0
(2) Y ≥ 0
(3) 10X + 4Y ≤ 110
(4) 5X - Y ≤ 40
If constraints (3) and (4) are binding, what is the optimal solution (X, Y)?

A) (11, 0)
B) (0, 27.5)
C) (0, -40)
D) (9, 5)
E) not enough information is provided to answer the problem

Question 44

A constraint in Excel Solver consists of what three pieces of information?

A) Target Cell, relationship operator, and Constraint
B) Target Cell, Changing Cells, and Constraint
C) Cell Reference, relationship operator, and Constraint
D) Cell Reference, Changing Cells, and Constraint
E) Target Cell, Changing Cells, and Cell Reference

Question 45

When was the Simplex Method developed?

A) 1927
B) 1937
C) 1947
D) 1957
E) 1967

Question 46

In the "Solver Options" box of Excel Solver, what should be checked to ensure that the Simplex Method is used to solve the model?

A) Assume Non-Negative
B) Use Automatic Scaling
C) Newton
D) Dantzig
E) Assume Linear Model

Question 47

Consider the following constraints from a two-variable Linear Program.
(1) X ≥ 0
(2) Y ≥ 0
(3) X + Y ≤ 20
(4) 2X + 5Y ≤ 70
If constraints (3) and (4) are binding, what is the optimal solution (X, Y)?

A) (0, 20)
B) (20, 0)
C) (10, 10)
D) (0, 14)
E) (35, 0)

Question 48

Consider the following three functions:
G(x, y) = 4x - 9y²
H(x, y, z) = 13x + y + 3z -6
I(y, z) = z - y
Which of the following is true regarding the linearity of the functions?

A) none of the functions is linear
B) all three functions are linear
C) h is linear, but g and i are not linear
D) i is linear, but g and h are not linear
E) h and i are linear, but g is not linear

Question 49

Which Excel tool provides solutions to Linear Programs?

A) Solver
B) Answer
C) Simplexer
D) Computer
E) Goal Seek

Question 50

Which of the following statements is correct?

A) Given a Linear Program with a maximization objective, the optimal objective function value may increase if a ≥ constraint is added to the program.
B) Given a Linear Program with a maximization objective, the optimal objective function value may increase if a ≤ constraint is added to the program.
C) Given a Linear Program with a minimization objective, the optimal objective function value cannot increase if a ≥ constraint is added to the program.
D) Given a Linear Program with a minimization objective, the optimal objective function value cannot increase if a ≤ constraint is added to the program.
E) Given a Linear Program with a maximization objective, the optimal objective function value may decrease if a ≥ constraint is added to the program.

Question 51

Constraints at their limits at the optimal solution of a Linear Program, that is, with the left-hand-side value equal to the right-hand-side value, are called _________________________.

A) equality constraints
B) non-binding constraints
C) binding constraints
D) limiting constraints
E) synergistic constraints

Question 52

What in Excel Solver corresponds to the decision variables in the algebraic model?

A) Optimization Cells
B) Target Cells
C) Changing Cells
D) Decision Cells
E) Parameter Cells

Question 53

In the "Solver Options" box of Excel Solver, what should be checked to ensure that all decision variables are ≥ 0?

A) Assume Linear Model
B) Use Automatic Scaling
C) Assume Positive
D) Assume Non-Negative
E) Assume Nonzero

Question 54

What in Excel Solver corresponds to the objective function in the algebraic model?

A) Optimization Cell
B) Target Cell
C) Changing Cell
D) Constraint Cell
E) Parameter Cell

Question 55

The two primary Excel tools for diagnosing problems in models are ___________________.

A) Error Auditing and Formula Checking
B) Error Auditing and Formula Auditing
C) Error Checking and Formula Auditing
D) Error Checking and Formula Checking
E) Error Diagnosing and Formula Debugging

Question 56

In the Excel Solver "Add Constraint" box, what two additional choices are available under the relationship operator list besides ≤, ≥, and =?

A) gin and bin
B) int and gin
C) big and int
D) inf and bin
E) int and bin

Question 57

If the solution to an optimization problem violates two constraints but satisfies three, it is a(an) ________.

A) suboptimal solution
B) infeasible solution
C) inoptimal solution
D) unbounded solution
E) simplex solution

Question 58

In the "Solver Parameters" box of Excel Solver, what is clicked to actually solve the problem?

A) "GO"
B) "OK"
C) "SOLVE"
D) "CLOSE"
E) "START"

Question 59

For an optimization problem a(an) __________________ violates at least one of the constraints.

A) infeasible solution
B) inoptimal solution
C) suboptimal solution
D) unbounded solution
E) simplex solution

Question 60

How can the following Linear Program be characterized?
Min X + 2Y
Subject to
X ≤ 20
Y ≤ 5
X, Y ≥ -40

A) bounded and feasible
B) unbounded and feasible
C) bounded and infeasible
D) unbounded and infeasible
E) this is not a Linear Program

Question 61

The ________________________ for a constraint is the amount the optimal objective value will change if the right-hand-side of the constraint is increased by one unit.

A) Changing Price
B) Opportunity Cost
C) Shadow Price
D) Opportunity Price
E) Shadow Cost

Question 62

A feasible solution to an optimization problem is a particular combination of decision variables that satisfies some of the constraints.

Question 63

Constraints that are not at their limits at the optimal solution of a Linear Program, that is, with the left-hand-side value not equal to the right-hand-side value, are called _________________________.

A) inequality constraints
B) non-binding constraints
C) binding constraints
D) limiting constraints
E) shadow constraints

Question 64

The Answer Report Target Cell, Adjustable Cell, and Constraint sections all include:

A) base value, original value
B) original value, final value
C) base value, final value
D) original value, test value
E) bounded value, un-bounded value

Question 65

The algebraic formulation of an optimization problem states the decision variables, objective function, and constraints in algebraic form.

Question 66

In Linear Programming, users have control over quantities called parameters.

Question 67

The advantage of formal optimization modeling is that it can simultaneously consider the effects of alternate decisions to produce the best overall decision according to the objective.

Question 68

To retain model flexibility while using Solver you must:

A) use appropriate color codes
B) enter specific values
C) ensure the inputs and outputs are separated
D) use only cell references
E) use Slack

Question 69

The constraints in an optimization model must all be of one type, for example, all ≤, all ≥, or all =.

Question 70

All mathematical programs should include non-negativity constraints.

Question 71

What do you need to do before using Solver?

A) take an Excel class
B) enter solver settings into Excel
C) define all constraints
D) enter Solver options
E) have a working, flexible spreadsheet model

Question 72

Strict inequalities (<, >) are not used in optimization problems.

Question 73

Given a Linear Program with a minimization objective, the optimal objective function value may decrease if a ≥ constraint is added to the program.

Question 74

Not-equals-to constraints (≠) are allowable in optimization problems.

Question 75

At the optimal solution of a Linear Program, the difference between the right-hand-side value and the left-hand-side value of a constraint is the ____________________.

A) shadow
B) difference
C) shortfall
D) slack
E) cushion

Question 76

One use of the Answer report is:

A) to develop the final answer.
B) to duplicate the output results in a new form.
C) as a debugging tool.
D) as a duplication tool.
E) as a results summary printout.

Question 77

Three reports are available when Solver has successfully found an optimal solution. These are _______.

A) Answer Report, Sensitivity Report, and Limits Report
B) Answer Report, Sensitivity Report, and Constraints Report
C) Answer Report, Limits Report, and Constraints Report
D) Solution Report, Sensitivity Report, and Limits Report
E) Solution Report, Sensitivity Report, and Constraints Report

Question 78

An optimal solution to an optimization problem is always feasible.

Question 79

The optimal solution to an optimization problem is the feasible solution with the largest (for a "min" problem) objective function value.

Question 80

In Linear Programming (LP) problems, all of the relationships among the variables are linear.