# Quiz 8: Nonlinear Optimization Models

Business

Q 1Q 1

Which of the following is incorrect?
A) A global optimum is a local optimum in a nonlinear optimization problem.
B) A local maximum is a global maximum in a concave nonlinear optimization problem.
C) A global minimum is a local minimum in a convex nonlinear optimization problem.
D) A local optimum is a global optimum in a nonlinear optimization problem.

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Multiple Choice

D

Q 2Q 2

The measure of risk most often associated with the Markowitz portfolio model is the
A) portfolio average return.
B) portfolio minimum return.
C) portfolio variance.
D) portfolio standard deviation.

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Multiple Choice

C

Q 3Q 3

An investor can pick the mean-variance tradeoff that he or she is most comfortable with by looking at a graph of the
A) feasible region.
B) pooled components.
C) rolling horizon.
D) efficient frontier.

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Multiple Choice

D

Q 4Q 4

Which of the following is not a parameter of the Bass model for forecasting adoption of a new product?
A) the coefficient of innovation
B) the coefficient of interaction
C) the coefficient of imitation
D) the estimated number of people to eventually adopt the new product

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Multiple Choice

Q 5Q 5

When the number of blending components exceeds the number of storage facilities, the number of feasible solutions to the blending problem
A) is reduced.
B) is increased.
C) is unchanged.
D) is zero.

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Multiple Choice

Q 6Q 6

In the Bass model for forecasting the adoption of a new product, the objective function
A) minimizes the sum of forecast errors.
B) minimizes the sum of squared forecast errors.
C) maximizes the number of adoptions.
D) maximizes the number of adoptions and imitations.

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Multiple Choice

Q 7Q 7

Which of the following is not true regarding a concave function?
A) It is bowl-shaped down.
B) It is relatively easy to maximize.
C) It has multiple local maxima.
D) It has a single global maximum.

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Multiple Choice

Q 8Q 8

A convex function is
A) bowl-shaped up.
B) bowl-shaped down.
C) elliptical in shape.
D) sinusoidal in shape.

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Multiple Choice

Q 9Q 9

If the coefficient of each squared term in a quadratic function is positive, the function is
A) concave.
B) convex.
C) elliptical.
D) sinusoidal.

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Multiple Choice

Q 10Q 10

Components that share a storage facility are called
A) constrained components.
B) indexed components.
C) blended components.
D) pooled components.

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Multiple Choice

Q 11Q 11

The key idea behind constructing an index fund is to choose a portfolio of securities that
A) is a mix of growth-oriented and income-oriented stocks.
B) minimizes risk without sacrificing liquidity.
C) mimics the performance of a broad market index.
D) balances short-term and long-term investments.

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Multiple Choice

Q 12Q 12

Components are referred to as pooled if they
A) are shared by two or more customers
C) share a storage facility
B) have common ingredients
D) are interchangeable

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Multiple Choice

Q 13Q 13

A nonlinear optimization problem is any optimization problem in which at least one term in the objective function or a constraint is nonlinear.

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True False

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True False

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True False

Q 16Q 16

Many linear programming algorithms such as the simplex method optimize by examining only the extreme points of the feasible region.

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True False

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True False

Q 18Q 18

A feasible solution is a global optimum if there are no other feasible solutions with a better objective function value in the immediate neighborhood.

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True False

Q 19Q 19

A feasible solution is a global optimum if there are no other feasible points with a better objective function value in the feasible region.

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True False

Q 20Q 20

For a typical nonlinear problem, duals price are relatively insensitive to small changes in right-hand side values.

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True False

Q 21Q 21

The interpretation of the dual price for nonlinear models is different than the interpretation of the dual price for linear models.

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True False

Q 22Q 22

In the case of functions with multiple local optima, most nonlinear optimization software methods can get stuck and terminate at a local optimum.

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True False

Q 23Q 23

For a minimization problem, a point is a global minimum if there are no other feasible points with a smaller objective function value.

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True False

Q 24Q 24

There are nonlinear applications in which there is a single local optimal solution that is also the global optimal solution.

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True False

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True False

Q 26Q 26

The function f (X, Y) = X

^{2}+ Y^{2}has a single global minimum and is relatively easy to minimize.Free

True False

Q 27Q 27

The problem of maximizing a concave quadratic function over a linear constraint set is relatively difficult to solve.

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True False

Q 28Q 28

Each point on the efficient frontier is the maximum possible risk, measured by portfolio variance, for the given return.

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True False

Q 29Q 29

Any feasible solution to a blending problem with pooled components is feasible to the problem with no pooling.

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True False

Q 30Q 30

Any feasible solution to a blending problem without pooled components is feasible to the problem with pooled components.

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True False

Q 31Q 31

When components (or ingredients) in a blending problem must be pooled, the number of feasible solutions is reduced.

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True False

Q 32Q 32

The value of the coefficient of imitation, q, in the Bass model for forecasting adoption of a new product cannot be negative.

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True False

Q 33Q 33

The Markowitz mean-variance portfolio model presented in the text is a convex optimization problem.

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True False

Q 34Q 34

Because most nonlinear optimization codes will terminate with a local optimum, the solution returned by the codes will be the best solution.

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True False

Q 35Q 35

It is possible for the optimal solution to a nonlinear optimization problem to lie in the interior of the feasible region.

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True False

Q 36Q 36

Investment manager Max Gaines has several clients who wish to own a mutual fund portfolio that matches, as a whole, the performance of the S&P 500 stock index. His task is to determine what proportion of the portfolio should be invested in each of the five mutual funds listed below so that the portfolio most closely mimics the performance of the S&P 500 index. Formulate the appropriate nonlinear program.

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Essay

Q 37Q 37

Financial planner Minnie Margin has a substantial number of clients who wish to own a mutual fund portfolio that matches, as a whole, the performance of the Russell 2000 index. Her task is to determine what proportion of the portfolio should be invested in each of the five mutual funds listed below so that the portfolio most closely mimics the performance of the Russell 2000 index. Formulate the appropriate nonlinear program.

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Essay

Q 38Q 38

Investment manager Max Gaines wishes to develop a mutual fund portfolio based on the Markowitz portfolio model. He needs to determine the proportion of the portfolio to invest in each of the five mutual funds listed below so that the variance of the portfolio is minimized subject to the constraint that the expected return of the portfolio be at least 4%. Formulate the appropriate nonlinear program.

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Essay

Q 39Q 39

Financial planner Minnie Margin wishes to develop a mutual fund portfolio based on the Markowitz portfolio model. She needs to determine the proportion of the portfolio to invest in each of the five mutual funds listed below so that the variance of the portfolio is minimized subject to the constraint that the expected return of the portfolio be at least 5%. Formulate the appropriate nonlinear program.

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Essay

Q 40Q 40

Shampooch is a mobile dog grooming service firm that has been quite successful developing a client base in the Dallas area. The firm plans to expand to other cities in Texas during the next few years. Shampooch would like to use its Dallas subscription data shown below to develop a model for forecasting service subscriptions in cities where it might expand. The first step is to estimate values for p (coefficient of innovation) and q (coefficient of imitation). Formulate the appropriate nonlinear program.

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Essay

Q 41Q 41

Cutting Edge Yard Care is a residential and commercial lawn service company that has been in business in the Atlanta metropolitan area for almost one year. Cutting Edge would like to use its Atlanta service subscription data below to develop a model for forecasting service subscriptions in other metropolitan areas where it might expand. The first step is to estimate values for p (coefficient of innovation) and q (coefficient of imitation). Formulate the appropriate nonlinear program.

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Essay

Q 42Q 42

MegaSports, Inc. produces two high-priced metal baseball bats, the Slugger and the Launcher, that are made from special aluminum and steel alloys. The cost to produce a Slugger bat is $100, and the cost to produce a Launcher bat is $120. We can not assume that MegaSports will sell all the bats it can produce. As the selling price of each bat model -- Slugger and Launcher -- increases, the quantity demanded for each model goes down.
Assume that the demand, S, for Slugger bats is given by S = 640 4P

_{S}and the demand, L, for Launcher bats is given by L = 450 3P_{L}_{ }where P_{S}is the price of a Slugger bat and P_{L}is the price of a Launcher bat. The profit contributions are P_{S }S 100S for Slugger bats and P_{L }L 120L for Launcher bats. Develop the total profit contribution function for this problem.Free

Essay

Q 43Q 43

Skooter's Skateboards produces two models of skateboards, the FX and the ZX. Skateboard revenue (in $l,000s) for the firm is nonlinear and is stated as (number of FXs)(5 0.2 number of FXs) + (number of ZXs)(7 0.3 number of ZXs). Skooter's has 80 labor-hours available per week in its paint shop. Each FX requires 2 labor-hours to paint and each ZX requires 3 labor-hours. Formulate this nonlinear production planning problem to determine how many FX and ZX skateboards should be produced per week at Scooter's.

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Essay

Q 44Q 44

Native Customs sells two popular styles of hand-sewn footwear: a sandal and a moccasin. The cost to make a pair of sandals is $18, and the cost to make a pair of moccasins is $24. The demand for these two items is sensitive to the price, and historical data indicate that the monthly demands are given by S = 400 10P

_{1}and M = 450 15P_{2 }, where S = demand for sandals (in pairs), M = demand for moccasins (in pairs), P_{1}= price for a pair of sandals, and P_{2}= price for a pair of moccasins. To remain competitive, Native Customs must limit the price (per pair) to no more than $60 and $75 for its sandals and moccasins, respectively. Formulate this nonlinear programming problem to find the optimal production quantities and prices for sandals and moccasins that maximize total monthly profit.Free

Essay