Deck 8: Nonlinear Programming

Question 1

A quadratic programming problem is a special type of linear programming problem.

Answer 1

False

Question 2

A local maximum is always a global maximum in a nonlinear programming problem.

Answer 2

False

Question 3

Applying separable programming requires having profit graphs that are smooth curves.

Answer 3

False

Question 4

Sometimes the Solver can return different solutions when optimizing a nonlinear programming problem.

Question 5

Nonlinear programming problems with decreasing marginal returns are generally easier to solve then nonlinear programming problems with increasing marginal returns.

Question 6

Excel's curve fitting method is used to find the values of the parameters for an equation that best fit data.

Question 7

A nonlinear function may contain a product of two variables.

Question 8

In separable programming, if an activity violates the proportionality assumption it must have increasing marginal returns.

Question 9

If C1:C6 are all changing cells, then SUMPRODUCT(C1:C3, C4:C6) is a linear function.

Question 10

When the marginal return from an activity decreases on a continuous basis, the profit graphs will consist of a series of line segments.

Question 11

If the slope of a graph never increases but sometimes decreases as the level of the activity increases, then it is said to have decreasing marginal returns.

Question 12

If D1 is a data cell, and C1 and C2 are changing cells, then IF(D1 >= 2, C1, C2) is a linear function.

Question 13

In separable programming, each activity that violates the proportionality assumption is separated into parts with a new variable for each part.

Question 14

In problems where the objective is to minimize the total cost of the activities, an activity is said to have decreasing marginal returns if the slope of its cost graph never increases but sometimes decreases as the level of the activity increases.

Question 15

Excel's curve fitting method is used to graph a nonlinear equation.

Question 16

Linear programming assumes that the profit from each activity is proportional to the level of that activity.

Question 17

Having activities with decreasing marginal returns is the only way that the proportionality assumption can be violated.

Question 18

If C1 is a changing cell, then ROUND(C1) is a linear function.

Question 19

In some cases of separable programming, the profit graphs will be curves rather than a series of line segments.

Question 20

Separable programming is applicable when there are increasing or decreasing marginal returns.

Question 21

Evolutionary Solver uses an algorithm based on genetics, evolution, and survival of the fittest.

Question 22

Two runs of the Evolutionary Solver on the same problem will typically yield the same solution.

Question 23

Sometimes Evolutionary Solver will make a random change in a member of the population.

Question 24

If the RSPE Model Analysis indicates that the model is a NSP, then the GRG Nonlinear search method is the best one to use.

Question 25

If the RSPE Model Analysis indicates that the model is NLP Convex, then only the Evolutionary Solver can be counted on to yield near optimal solutions.

Question 26

It now is common practice for professional managers of large stock portfolios to use computer models based partially on separable programming.

Question 27

Trying different starting points and picking the best solution will always yield the optimal solution to a nonlinear programming problem.

Question 28

The multistart feature in Solver can be used for nonlinear programming problems to systematically try a number of different starting points.

Question 29

The additivity assumption of linear programming states that each term in the objective function is the sum of two or more variables.

Question 30

If the data cells are in column D and the changing cells are in column C, which of the following are linear formulas in a spreadsheet?
I) =SUMPRODUCT(D1:D6, C1:C6)
II) =SUMPRODUCT(C1:C3, C4:C6)
III) =SUM(C1:C6)

A) I only
B) II only
C) III only
D) I, II, and III
E) I and III only

Question 31

The members of the population used to create the next generation are picked randomly by the Evolutionary Solver.

Question 32

If the data cells are in column D and the changing cells are in column C, which of the following are not linear formulas in a spreadsheet?
I) =IF(D1 >= 6, C1, C2)
II) =ROUND(C3)
III) =ABS(C3)

A) I only
B) II only
C) III only
D) I, II, and III
E) II and III only

Question 33

Profit = 3x1 + 2x2 + 9x1x2 is an example of a nonlinear function.

Question 34

The Nonlinear Solver keeps track of a large set of candidate solutions, called the population.

Question 35

The additivity assumption can be violated by nonlinear programming because of cross-product terms involving the product of two variables.

Question 36

When applying nonlinear programming to portfolio selection, a trade-off is being made between the expected return and the risk associated with the investment.

Question 37

The risk for a portfolio is decreased when the particular stocks tend to move up and down together.

Question 38

When there are decreasing marginal returns:

A) the slope of the graph never increases but sometimes decreases.
B) the slope of the graph never decreases but sometimes increases.
C) the graph always consists of a smooth curve.
D) the graph always consists of a series of line segments.
E) separable programming should not be used.

Question 39

Evolutionary Solver is often faster than the standard Solver at solving linear programming problems.

Question 40

Mutation is the technique used to create the next generation of solutions in the Evolutionary Solver.

Question 41

Decreasing marginal returns violates which assumption of linear programming?

A) The proportionality assumption
B) The divisibility assumption
C) The additivity assumption
D) All of the choices are correct.
E) None of the choices is correct.

Question 42

The following chart shows a relationship between advertising expenditures and sales. Which of the following describes the chart in terms of a linear relationship?

A) The chart shows a linear relationship.
B) The chart shows decreasing marginal returns.
C) The chart shows a proportional relationship.
D) The chart shows increasing marginal returns.
E) The chart is piecewise linear.

Question 43

Separable programming will always find the optimal solution when the following is true:

A) The profit or cost graph is piecewise linear.
B) There are decreasing marginal returns.
C) The profit or cost graph is piecewise linear and there are decreasing marginal returns must both be true.
D) Separable programming only finds an approximate solution.
E) None of the choices is correct.

Question 44

If a model uses IF or ROUND functions that incorporate the changing cells, then running the RSPE Analyze Model without Solving feature will typically say the model is of what type?

A) Linear
B) NLP Convex
C) QP Convex
D) NSP
E) None of the choices is correct.

Question 45

One reason that a manager may choose to use a nonlinear model to analyze a problem is

A) Nonlinear models are easier to solve than linear models.
B) Nonlinear models may provide greater precision than linear models.
C) Nonlinear techniques such as Evolutionary Solver provide optimal results.
D) Nonlinear models are easier to understand than linear models.
E) Nonlinear models take less time to solve than linear models.

Question 46

The measure of risk for pairs of stocks in a portfolio selection problem is called:

A) the covariance of the return.
B) the variance of the return.
C) the expected return.
D) decreasing marginal return.
E) None of the choices is correct.

Question 47

A nonlinear function may contain which of the following?
I) A term that contains a single variable with an exponent of 1.
II) A term that contains a single variable with an exponent of 2.
III) A term that is a constant times the product of two variables.

A) I only
B) II only
C) III only
D) I, II, and III
E) II and III only

Question 48

Which of the following techniques is appropriate when a nonlinear programming problem has multiple local optima?
I) Running Solver many times with different starting points.
II) Using the multistart feature to try different starting points.
III) Using Evolutionary Solver.

A) I only
B) II only
C) III only
D) I, II, and III
E) II and III only

Question 49

The following chart shows a relationship between advertising expenditures and sales. Which of the following describes the chart in terms of a linear relationship?

A) The chart shows a linear relationship.
B) The chart shows decreasing marginal returns.
C) The chart shows a proportional relationship.
D) The chart shows discontinuities.
E) The chart is piecewise linear.

Question 50

The measure of risk in a portfolio selection problem is called:

A) the covariance of the return.
B) the variance of the return.
C) the expected return.
D) decreasing marginal return.
E) None of the choices is correct.

Question 51

Which of the following can be part of a nonlinear profit graph?
I) Decreasing marginal returns.
II) Increasing marginal returns.
III) Discontinuities.

A) I only
B) II only
C) III only
D) I, II, and III
E) I and II only

Question 52

A nonlinear programming problem may have:
I) Activities with increasing marginal returns.
II) Activities with decreasing marginal returns.
III) Nonlinear functional constraints.

A) I only
B) II only
C) III only
D) I, II, and III
E) II and III only

Question 53

Evolutionary Solver is based on which of the following concepts?
I) Genetics.
II) Evolution.
III) Survival of the Fittest.

A) I only
B) II only
C) III only
D) I, II, and III
E) II and III only

Question 54

The following chart shows a relationship between advertising expenditures and sales. Which of the following describes the chart in terms of a linear relationship?

A) The chart shows a linear relationship.
B) The chart shows increasing marginal returns.
C) The chart shows a proportional relationship.
D) The chart shows discontinuities.
E) The chart is piecewise linear.

Question 55

Which of the following is an example of a nonlinear function?

A) Profit = 5x­1 + 7x2­ − 2x22
B) Profit = 8x1x2 − x12 − 4x22
C) Profit = x1 + 6x2 + 3x1x2
D) All of the choices are correct.
E) None of the choices is correct.

Question 56

A linear function may contain which of the following?
I) A term that contains a single variable with an exponent of 1.
II) A term that contains a single variable with an exponent of 2.
III) A term that is a constant times the product of two variables.

A) I only
B) II only
C) III only
D) I, II, and III
E) II and III only

Question 57

Which of the following are disadvantages of the Evolutionary Solver?
I) It does not deal well with complicated objective functions.
II) It does perform well on models with many constraints.
III) It is easily trapped at local optima.

A) I only
B) II only
C) III only
D) I and II
E) None of these

Question 58

Evolutionary Solver is best suited to which kinds of problems?

A) Linear programs
B) Nonlinear programs with difficult objective functions
C) Nonlinear programs with decreasing marginal returns and no discontinuities
D) Nonlinear programs with many constraints
E) None of the choices is correct.

Question 59

The requirement that each term in the objective function only contains a single variable is in a linear program is referred to as:

A) the proportionality assumption.
B) the divisibility assumption.
C) the additivity assumption.
D) a nonlinear function.
E) None of the choices is correct.

Question 60

Which of the following are advantages of the Evolutionary Solver?

A) The complexity of the objective function does not matter.
B) It will always find the optimal solution.
C) It is faster than the standard Solver.
D) It always finds the same solution.
E) None of the choices is correct.

Question 61

After some experimentation, you have observed the following data concerning the relationship between marketing and sales. Use Excel's curve fitting method to find the best quadratic equation to model this relationship.
$\begin{array}{|l|l|l|l|l|l|}\hline \begin{array}{l}\text { Marketing } \\(\$)\end{array} & 100 & 110 & 120 & 130 & 140 \\\hline \begin{array}{l}\text { Sales } \\(\$ 100)\end{array} & 310 & 500 & 600 & 690 & 780 \\\hline\end{array}\begin{array}{r|r|r|r|r|r|}\hline \\150 & 160 & 170 & 180 & 190 & 200 \\\hline \\850 & 925 & 1,000 & 1,050 & 1,09 & 1,120 \\\hline\end{array}$

A) ?0.055x2 + 24x ? 1516
B) 0.055x2 + 24x ? 1516
C) ?0.055x2 ? 24x ? 1516
D) ?0.055x2 + 24x + 1516
E) ?0.11x2 + 24x ? 1516

Question 62

Note: This problem requires Excel.
You have noticed that paying higher wages attracts more productive employees. However, you are concerned that there may be a limit to this relationship. Some experimentation has convinced you that the relationship between wages paid (x) and profits can be modelled by the equation Profit = 25x − 0.1x2 + 200. Use a Nonlinear Solver tool to determine the level of wages that will maximize profits.

A) 25
B) 50
C) 75
D) 100
E) 125

Question 63

Which of the following statements about solving maximization problems with Excel's Nonlinear Solver?
I) Nonlinear Solver will always find the global maximum.
II) Nonlinear Solver will always find a local maximum but not necessarily the global maximum.
III) With diminishing returns, Nonlinear Solver will always find the global maximum.

A) I only
B) II only
C) III only
D) Only II and III
E) I, II, and III

Question 64

Note: This problem requires Excel.
A firm offers three different prices on its products, depending upon the quantity purchased. Since available resources are limited, the firm would like to prepare an optimal production plan to maximize profits. Product 1 has the following profitability: $10 each for the first 50 units, $9 each for units 51-100, and $8 for each unit over 100. Product 2's profitability is $20 each for the first 25 units, $19 each for units 26-50, and $18 each for each unit over 50. The products each require 3 raw materials to produce (see table below for usages and available quantities).
$\begin{array}{|c|r|r|r|}\hline\begin{array}{c}\text { Raw } \\\text { Material }\end{array} & \begin{array}{c}\text { Product 1 usage } \\\text { (pounds per unit) }\end{array} & \begin{array}{c}\text { Product 2 usage } \\\text { (pounds per unit) }\end{array} & \begin{array}{c}\text { Available Quantity } \\\text { (pounds) }\end{array} \\\hline \text { A } & 5 & 12 & 1,000 \\\hline \text { B } & 4 & 10 & 2,000 \\\hline \text { C } & 7 & 6 & 1,500 \\\hline\end{array}$
Use separable programming to find the optimal production plan.

A) 100 units Product 1, 100 units Product 2
B) 100 units Product 1, 50 units Product 2
C) 140 units Product 1, 25 units Product 2
D) 25 units Product 1, 140 units Product 2
E) 100 units Product 1, 140 units Product 2

Question 65

Which of the following profit functions has a logarithmic form?
I) x2 + 3x − 4
II) 4 ln x + 7
III) 3x − 4

A) I only
B) II only
C) III only
D) I and II only
E) I, II, and III

Question 66

If the RSPE Model Analysis indicates that the model is quadratic, which of the following is TRUE?

A) The model can be solved using linear programming tools.
B) The Evolutionary Solver will always find the optimal solution.
C) Using the Nonlinear Solver will never find the optimal solution.
D) The model contains functions such as "IF" or "ROUND."
E) None of the choices is true.

Question 67

Which of the following statements about Solver's Multistart option are TRUE?

A) A nonlinear problem will always have the solution for any starting point.
B) Multistart always uses 100 random starting points.
C) Using Multistart guarantees that Solver will find the optimal solution.
D) Multistart may not find the optimal solution.
E) Multistart works well with functions such as "IF" and "ROUND."

Question 68

The following chart shows the relationship between marketing and sales. Which of the following statements is TRUE?
I) There is a local minimum when marketing expenditure equals $120.
II) There is a local maximum when marketing expenditure equals $120.
III) There is a global maximum when marketing expenditure equals $160.

A) I only
B) II only
C) III only
D) I and II only
E) II and III only

Question 69

Note: This problem requires Excel.
The marketing department has determined that the relationship between marketing expenditures (x) and sales can be modelled by the equation Sales = 100x − x2 + 20. Use a Nonlinear Solver tool to determine the level of marketing expenditure that will maximize sales.

A) 40
B) 50
C) 60
D) 70
E) 80

Question 70

The following chart shows the relationship between marketing and sales. Which of the following statements is TRUE?
I) There is a local maximum when marketing expenditure equals $120.
II) There is a local minimum when marketing expenditure equals $140.
III) There is a local maximum when marketing expenditure equals $160.

A) I only
B) II only
C) III only
D) Only I and II
E) I, II, and III

Question 71

Note: This problem requires Excel.
You have noticed that paying higher wages attracts more productive employees. However, you are concerned that there may be a limit to this relationship. Some experimentation has convinced you that the relationship between daily wages paid (x) and profits can be modelled by the equation Profit = 50x − 0.5x2 + .001x3 + 200. The range of wages you are willing to consider is from $0 to $500 per day. Use the Evolutionary Solver tool to determine the level of wages that will maximize profits.

A) 200
B) 300
C) 400
D) 500
E) Cannot be determined

Question 72

If a problem can be modelled as a separable program with three line segments, how many decision variables is it likely to have?

A) 1
B) 2
C) 3
D) Only 1 or 2
E) 1, 2, or 3

Question 73

Note: This problem requires Excel.
A firm offers two different prices on its products, depending upon the quantity purchased. Since available resources are limited, the firm would like to prepare an optimal production plan to maximize profits. Product 1 has the following profitability: $75 each for the first 25 units and $60 for each unit over 25. Product 2's profitability is $200 each for the first 50 units and $100 each for each unit over 50. The products each require two raw materials to produce (see table below for usages and available quantities).
$$\begin{array} { | c | r | r | r | } \hline\begin{array} { c } \text { Raw } \\\text { Material }\end{array} & \begin{array} { c } \text { Praduct l usage } \\\text { (gallons per unit) }\end{array} & \begin{array} { c } \text { Praduct 2 usage } \\\text { (gallons per unit) }\end{array} & { \begin{array} { c } \text { Available Quantity } \\\text { (gallons) }\end{array} } \\\hline \text { A } & 10 & 20 & 1,500 \\\hline \text { B } & 5 & 7 & 2,000 \\\hline\end{array}$$
Use separable programming to find the optimal production plan.

A) 100 units Product 1, 100 units Product 2
B) 50 units Product 1, 50 units Product 2
C) 100 units Product 1, 25 units Product 2
D) 25 units Product 1, 100 units Product 2
E) 100 units Product 1, 50 units Product 2

Question 74

After some experimentation, you have observed the following data concerning the relationship between marketing and sales. Use Excel's curve fitting method to find the best quadratic equation to model this relationship.
$\begin{array}{|l|c|c|c|c|c}\hline \begin{array}{l}\text { Marketing } \\(\$)\end{array} & 100 & 110 & 120 & 130 & 140 \\\hline \begin{array}{l}\text { Sales } \\(\$ 100)\end{array} & 310 & 400 & 500 & 600 & 700 \\\hline\end{array}\begin{array}{|r|r|r|r|r|r|}\hline \\150 & 160 & 170 & 180 & 190 & 200 \\\hline \\850 & 975 & 1,100 & 1,300 & 1,425 & 1,600 \\\hline\end{array}$

A) 0.11x2 + 1.4x + 40
B) 0.05x2 ? 1.1x ? 44
C) 0.05x2 + 1.1x + 44
D) 0.11x2 ? 1.4x ? 40
E) 0.05x2 ? 1.1x + 44

Question 75

Which of the following profit functions has a quadratic form?
I) x2 + 3x − 4
II) 3x2 + 4x + 6
III) 3x − 4

A) I only
B) II only
C) III only
D) I and II only
E) I, II, and III