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Exam 10: Nonlinear Optimization Models

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Question 1

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The profit function for two types of iPod is : Profit $=-5 x_{1}^{2}+38 x_{1}-5 x_{2}^{2}+44 x_{2}+520$ where x₁ and x₂ represent number of units of production of basic and advanced iPods, respectively. Production time required for the basic iPod is 6 hours per unit, and production time required for the advanced iPod is 8 hours per unit. Currently, 50 hours are available. The cost of hours is already factored into the profit function. a. Formulate an optimization problem that can be used to find the optimal production quantity of basic and advanced iPods. b. Solve the optimization model you formulated in part a. How much should be produced?

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a. The optimization model is, Max \( -5 x_{1}^{2}+38 x_{1}-5 x_{2}^{2}+44 x_{2}+520 \) s.t. \( 6 x_{1}+8 x_{2} \leq 50 \) \( x_{1}, x_{2} \geq 0 \) b. The optimal solution is: = 3.32 and = 3.76 with a profit of $685.8. The spreadsheet model is: 11ea4021_0f12_a5e2_b5a8_f9bf6c01eca3_TB1880_00 11ea4021_0f12_ccf3_b5a8_89b6b34b5795_TB1880_00 11ea4021_0f12_f404_b5a8_2b6ee5c6e78f_TB1880_00

Question 2

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Consider the stock return data given below. 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.

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a.
Let A = proportion of portfolio inves...

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Question 3

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

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

The exponential smoothing model is given by, $\hat{y}_{t+1}=\alpha y_{t}+(1-\alpha) \hat{y}_{t}$ where $\hat{y}$ _(t+1)=forecast of sales for period t+1 y_t=actual sales for period t $\hat{y}_{t}$ =forecast of sales for period t α=smoothing constant,0 ≤ α≤1 This model is used to predict the future based on the past data values. a. The observed values with the smoothing constant α = 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 y_t- (y_t ) ̂ 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 α is often chosen by minimizing the sum of squared forecast errors. Use Excel Solver to find the value of α that minimizes the sum of squared forecast errors.

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

Consider the return scenario for 3 types of mutual funds, shown in the following table: a. Construct the Markowitz model that maximizes expected return subject to a maximum variance of 35. b. Solve the model developed in part a. Round all your answers to three decimal places.

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: $R=-2 S^{2}-9 B^{2}-4 S B+20 S+30 B$ 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.

Reference - 10.1: Use the graph given below to answer questions -Reference - 10.1. Which of the following functions is most likely to yield the above shape?

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

The _____ option is helpful when the solution to a problem appears to depend on the starting values for the decision variables.

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

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

In a nonlinear optimization problem:

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.

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

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 ____.

Reference - 10.2: Use the graph given below to answer questions -Reference - 10.2. Which of the following functions is most likely to yield the above shape?

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

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.