Deck 13: Nonlinear Optimization Models

Question 1

A ____________ is the shadow price of a binding simple lower or upper bound on the decision variable.

A)reduced gradient
B)binding constraint
C)binary variable
D)local optimum

Answer 1

reduced gradient

Question 2

Which of the following functions yields the shape shown below? ​

A)f(X, Y) = X² + Y²
B)f(X, Y) = Xsin(2ðY) + Ysin(2ðX)
C)f(X, Y) = -X² - Y²
D)f(X, Y) = Xsin(5ðX) + Ysin(5ðY)

Answer 2

f(X, Y) = X² + Y²

Question 3

A function that is bowl-shaped down is called a ___________ function.

A)concave
B)convex
C)conic
D)linear

Answer 3

concave

Question 4

In a nonlinear optimization problem

A)the objective function is a nonlinear function of the constraints.
B)all the constraints are nonlinear only when the objective is to maximize the function of the decision variables.
C)at least one term in the objective function or a constraint is nonlinear.
D)both the objective function and the constraints must have all nonlinear terms.

Question 5

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

A)infeasible
B)unbounded
C)nonlinear
D)a global optimum

Question 6

The Lagrangian multiplier is the __________ for a constraint in a nonlinear problem.

A)shadow price
B)payoff value
C)reducing gradient
D)reduced cost

Question 7

A feasible solution is ____________ if there are no other feasible points with a smaller objective function value in the entire feasible region.

A)a global minimum
B)not a local maximum
C)not a local minimum
D)bowl-shaped

Question 8

In a nonlinear problem, the rate of change of the objective function with respect to the right-hand side of a constraint is given by the

A)slope of the contour line.
B)local optimum.
C)Reducing gradient.
D)Lagrangian multiplier.

Question 9

A function that is bowl-shaped up is called a(n) _________ function.

A)concave
B)optimal
C)convex
D)elliptical

Question 10

In reviewing the image below, which of the following functions is most likely to yield the above shape? ​ ​

A)f(X, Y) = X² + Y²
B)f(X, Y) = -X - Y
C)f(X, Y) = -X² - Y²
D)f(X, Y) = Xsin(5ðX) + Ysin(5ðY)

Question 11

The reduced gradient is analogous to the ___________ for linear models.

A)binary variable
B)binding constraint
C)reduced cost
D)objective coefficient

Question 12

If there are no other feasible points with a larger objective function value in the entire feasible region, a feasible solution is

A)an efficient frontier.
B)a global maximum.
C)not a local maximum.
D)a global minimum.

Question 13

A nonlinear function with term to the power of two is known as a

A)hyperbolic function.
B)quadratic function.
C)logarithmic function.
D)cubic function.

Question 14

A global minimum

A)is also a local maximum.
B)need not be a local maximum, but vice versa is true.
C)is also a local minimum.
D)need not be local minimum, but vice versa is true.

Question 15

If all the squared terms in a quadratic function have a negative coefficient and there are no cross-product terms, then the function is a _____ function.

A)convex quadratic
B)nonlinear objective
C)concave quadratic
D)negative elliptical

Question 16

The ___________ of a solution is a mathematical concept that refers to the set of points within a relatively close proximity of the solution.

A)objective function contour
B)neighborhood
C)regression equation
D)Lagrangian multiplier

Question 17

A feasible solution is a(n) ___________ if there are no other feasible solutions with a better objective function value in the immediate neighborhood.

A)efficient frontier
B)local optimum
C)global maximum
D)diverging function

Question 18

In reviewing the image below, the point (0, 0, 0) is a(n) __________ for the given concave function. ​

A)local maximum
B)local minimum
C)convergence point
D)endpoint

Question 19

If there are no other feasible solutions with a larger objective function value in the immediate neighborhood, then the feasible solution is known as

A)a global maximum.
B)infeasible.
C)a nonlinear solution.
D)a local maximum.

Question 20

A feasible solution is a local minimum if there are no other feasible solutions with a

A)smaller objective function value in the immediate neighborhood.
B)same objective function value in the immediate neighborhood.
C)set of points defining the minimum possible risk in the entire feasible region.
D)same objective function value in the entire feasible region.

Question 21

Excel Solver's __________ is based on a method that searches for an optimal solution by iteratively adjusting a population of candidate solutions.

A)Evolutionary Solver
B)Goal Seeker
C)Simplex LP
D)GRG Nonlinear

Question 22

The __________forecasting model uses nonlinear optimization to forecast the adoption of innovative and new technologies in the marketplace.

A)Hauck
B)LMS
C)Markowitz
D)Bass

Question 23

Using the graph below, the feasible region for the function represented in the graph is

A)-1 £ X £ 1, -1 £ Y £ 1.
B)-1.5 £ X £ 1, 0 £ Y £ 8.
C)-1.5 £ X £ 2.0, -1.5 £ Y £ 2.0.
D)0 £ X £ 1, 0 £ Y £ 1.

Question 24

The measure of risk most often associated with the Markowitz portfolio model is the

A)expected return of the portfolio.
B)annual interest on the portfolio.
C)variance of the portfolio's return.
D)number of investments listed in the portfolio.

Question 25

Using the graph below, which of the following is true of the above function? ​

A)It has single local minimum.
B)It has multiple local optima.
C)It has single local maximum.
D)It has no maxima and minima.

Question 26

In the Bass forecasting model, the ___________measures the likelihood of adoption, assuming no influence from someone who has already purchased (adopted) the product.

A)coefficient of correlation
B)coefficient of imitation
C)coefficient of independence
D)coefficient of innovation

Question 27

In the Bass forecasting model, the __________ measures the likelihood of adoption due to a potential adopter being influenced by someone who has already adopted the product.

A)coefficient of innovation
B)coefficient of imitation
C)coefficient of regression
D)coefficient of the objective function

Question 28

Which of the following is a second way of formulating the Markowitz model?

A)Maximizing the expected return of the portfolio subject to a constraint on variance
B)Minimizing the expected return of the portfolio subject to a constraint on variance.
C)Maximizing the variance of the portfolio subject to a constraint on the expected return of the portfolio
D)Maximizing the variance of the portfolio with no constraint needed for the expected return of the portfolio

Question 29

If the portfolio variance were equal to zero, the amount of risk would be

A)unity.
B)a positive number greater than 1.
C)negative always.
D)zero.

Question 30

The __________ option in Excel Solver is helpful when the solution to a problem appears to depend on the starting values for the decision variables.

A)Restart
B)Convergence
C)Derivatives
D)Multistart

Question 31

Which of the following conclusions can be drawn from the below figure using the Bass forecasting model? (Note: Bass forecasting model is given by: F_t = (p + q[Ct _{- 1} /m]) (m - Ct _{- 1}),
Where m = the number of people estimated to eventually adopt the new product,
Ct _{- 1} = the number of people who have adopted the product through time t - 1,
Q = the coefficient of imitation, and
P = the coefficient of innovation.)

A)q < p
B)q > p
C)m < q
D)p > m

Question 32

A(n) __________ is a set of points defining the minimum possible risk for a set of return values.

A)contour
B)efficient frontier
C)unity constraint
D)reduced gradient

Question 33

The portfolio variance is the

A)sum of the squares of the deviations from the mean value under each scenario.
B)average of the sum of the squares of the deviations from the mean value under each investment scenario.
C)average of the product of the squares of the deviations from the mean value under each scenario.
D)average of the sum of the deviations from the mean value under each investment scenario.

Question 34

A portfolio optimization model used to construct a portfolio that minimizes risk subject to a constraint requiring a minimum level of return is known as

A)capital budgeting pricing model.
B)market share optimization model.
C)Hauck maximum variance portfolio model.
D)Markowitz mean-variance portfolio model.

Question 35

One of the ways to use the Bass forecasting model is to wait until several periods of data for the problem under consideration are available. This is known as the ___________ approach.

A)branch-and-bound
B)cutting plane
C)rolling-horizon
D)sensible-period

Question 36

Solving nonlinear problems with local optimal solutions is performed using _____________, in Excel Solver, which is based on more classical optimization techniques.

A)Goal Seeker
B)Linear Regression
C)GRG Nonlinear
D)Simplex LP

Question 37

In the Bass forecasting model, parameter m

A)measures the likelihood of adoption due to a potential adopter being influenced by someone who has already adopted the product.
B)measures the likelihood of adoption, assuming no influence from someone who has already adopted the product.
C)refers to the number of people estimated to eventually adopt the new product.
D)refers to the number of people who have already adopted the new product.

Question 38

In reviewing the image below, what is the minimum value for this function?

A)-8
B)0
C)-1
D)1

Question 39

One of the ways to formulate the Markowitz model is to

A)maximize the variance of the portfolio subject to a constraint on the expected return of the portfolio.
B)minimize the expected return of the portfolio subject to a constraint on variance.
C)minimize the variance of the portfolio subject to a constraint on the expected return of the portfolio.
D)minimize the expected return of the portfolio with no constraint on variance.

Question 40

Using the graph given below, which of the following equations is most likely to yield the above curve?

A)f(X, Y) = Xlog(2ðY) + Ylog(2ðX)
B)f(X, Y) = X - Y
C)f(X, Y) = -X² - Y²
D)f(X, Y) = Xsin(5ðX) + Ysin(5ðY)

Question 41

Consider the following data on the returns from bonds. Develop and solve the Markowitz portfolio model using a required expected return of at least 15 percent. Assume that the 8 scenarios are equally likely to occur. Use this model to construct an efficient frontier by varying the expected return from 2 to 18 percent in increment of 2 percent and solving for the variance. Round all your answers to three decimal places.

Question 42

A Steel Manufacturing company has two production facilities that manufacture Dishwashers. Production costs at the two facilities differ because of varying labor costs, local property taxes, type of material used, volume, and so on. For Plant A, the weekly costs for producing a number of units of Dishwashers is expressed as a function
TC_A(X) = X² - 2X + 12000
where X is the weekly production volume and TC_A(X) is the weekly cost for Plant A. Plant B's weekly production costs are given by
TC_B(Y) = Y² + 8Y + 10000
where Y is the weekly production volume and TC_B(Y) is the weekly cost for Plant B. The manufacturer would like to produce 50 dishwashers per week at the lowest possible cost.
a. Formulate a mathematical model that can be used to determine the optimal number of dishwashers to produce each week at each facility.
b. Solve the optimization model to determine the optimal number of dishwashers to produce at each facility.

Question 43

Roger is willing to promote and sell two types of smart watches, X and Y, at his outlet. The demand for these two watches are as follows.
D_X = -0.45P_X + 0.34P_Y + 242
D_Y = 0.2P_X - 0.58P_Y+ 282
where, D_X is the demand for watch X, P_X is the selling price of watch X, D_Y is the demand for watch Y, and P_Y is the selling price of watch Y.
Rogers wishes to determine the selling price that maximizes revenue for these two products. Develop the revenue function for these two models, and find the revenue maximizing prices.

Question 44

The exponential smoothing model is given by where This model is used to predict the future based on the past data values.
a. The observed values with the smoothing constant a = 0.45 are given in the below table. The third column of the table displays the forecast values obtained using the above model. The forecasted error is calculated in the fourth column, and the square of the forecast error and the sum of squared forecast errors are given in fifth column. Construct this table in your spreadsheet model using the formula above. (Hint: The first forecast value is same as the observed value.)
Alpha = 0.45
b. The value of á is often chosen by minimizing the sum of squared forecast errors. Use Excel Solver to find the value of á that minimizes the sum of squared forecast errors.

Question 45

Consider the stock return data given below. Develop and solve the Markowitz model that maximizes expected return subject to a maximum variance of 35. Use this model to construct an efficient frontier by varying the maximum allowable variance from 25 to 55 in increments of 5 and solving for the maximum return for each.

Question 46

Consider the following data on the returns from bonds.
a. Construct the Markowitz portfolio model using a required expected return of at least 15 percent. Assume that the 8 scenarios are equally likely to occur.
b. Solve the model using Excel Solver.

Question 47

Consider the economic order quantity (EOQ) model for multiple products that are independent except for a budget restriction. The following model describes this situation
Let D_k = annual demand for product k
C_k = unit cost of product k
S_k = cost per order placed for product k
i = inventory carrying charge as a percentage of the cost per unit
B = the maximum amount of investment in goods
N = number of products
The decision variables are Q_k, the amount of product k to order. The model is: s.t.
a. Set up a spreadsheet model and for the following data:
b. Solve the problem using Excel Solver. (Hint: For Solver to find a solution, you need to start with decision variable values that are greater than 0.)

Question 48

Mark and his friends are planning for a holiday party. Data on longitude, latitude, and number of friends at each of the 10 locations are given below. Mark would like to identify the location for the holiday party such that it minimizes the demand-weighted distance, where demand is the number of friends at each location. Find the optimal location for the party. The distance between two cities can be approximated by the following formula. where lat1 and long1 are the latitude and longitude of city 1, and lat2 and long2 are the latitude and longitude of city 2. (Hint: Notice that all longitude values given for this problem are negative. Make sure that you do not check the option for Make Unconstrained Variables Non-Negative in Solver.)

Question 49

Develop a model that minimizes semivariance for the data given below with a required return of 15 percent. Define a variable for each scenario and let with = 0. Then make the objective function: Min . Solve the model you developed with a required expected return of at least 15 percent.

Question 50

Jeff is willing to invest $5000 in buying shares and bonds of a company to gain maximum returns. From his past experience, he estimates the relationship between returns and investments made in this company to be:
R = -2S² - 9B² - 4SB + 20S + 30B.
where,
R = total returns in thousands of dollars
S = thousands of dollars spent on Shares
B = thousands of dollars spent on Bond
Jeff would like to develop a strategy that will lead to maximum return subject to the restriction provided on amount available for investment.
a. What is the value of return if $3,000 is invested in shares and $2,000 is invested bonds of the company?
b. Formulate an optimization problem that can be solved to maximize the returns subject to investing no more than $5,000 on both share and bonds.
c. Determine the optimal amount to invest in shares and bonds of the company. How much return will Jeff gain? Round all your answers to two decimal places.

Question 51

Consider the stock return data given below.
a. Construct the Markowitz model that maximizes expected return subject to a maximum variance of 35.
a. Round all your answers to three decimal places.
b. Solve the model developed in part

Question 52

Jim must solve a nonlinear optimization problem where point A should be within a radius of 15 centimeters from each of the points B, C, D, E and F. The decision variables are defined as below.
X = horizontal coordinate of point A
Y = vertical coordinate of point A
The data on the distances is given below: Formulate and solve a model that minimizes the maximum distance from point A to each of the points B, C, D, E, and F. Round all your answers to three decimal places.

Question 53

Jim must solve a nonlinear optimization problem where point A should be within a radius of 15 centimeters from each of the points B, C, D, E and F. The decision variables are defined as below.
X = horizontal coordinate of point A
Y = vertical coordinate of point A
The data on the distances is given below: Formulate a model to find the optimal location of the point A.

Question 54

Gatson manufacturing company is willing to promote 2 types of tires: Economy tire and Premium tire. These two tires are independent of each other in terms of demand, cost, price, etc. An analytics team of this company has estimated the profit functions for both the tires as
Monthly profit for Economy tire = 49.2415 IN(X_A) + 180.414
Monthly profit for Premium tire = 84.344 IN(X_B) - 150.112
where X_A and X_B are the advertising amount allocated to Economy tire and Premium tire, respectively, and IN is the natural logarithm function. The advertising budget is $200,000, and management has dictated that at least $20,000 must be allocated to each of the two tires.
(Hint: To compute a natural logarithm for the value X in Excel, use the formula = IN(X). For Solver to find an answer, you also need to start with decision variable values greater than 0 in this problem.)
Develop and solve an optimization model that will prescribe how the company should allocate its marketing budget to maximize profit.

Question 55

An Electrical Company has two manufacturing plants. The cost in dollars of producing an Amplifier at each of the two plants is given below. The cost of producing Q₁ Amplifiers at first plant is:
65Q₁ + 4Q₁²+ 90
and the cost of producing Q₂ Amplifiers at the second plant is
20Q₂ + 2Q₂²+ 120
The company needs to manufacture at least 60 Amplifiers to meet the received orders. How many Amplifiers should be produced at each of the plant to minimize the total production cost? Round the answers to two decimal places and the total cost to the nearest dollar value.