Exam 13: Nonlinear Optimization Models

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

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

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

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

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: Horizontal Coordinate Vertical Coordinate B 9 11 C 14 18 D 18 22 E 13 16 F 17 21 Formulate a model to find the optimal location of the point A.

In a nonlinear optimization problem

A global minimum

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

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

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: $\text { Minimize } \sum _ { k = 1 } ^ { N } \left[ C _ { k } Q _ { k } + \frac { S _ { k } D _ { k } } { Q _ { k } } + i C _ { k } \frac { Q _ { k } } { 2 } \right]$ s.t. $\sum _ { k = 1 } ^ { N } C _ { k } Q _ { k } \leq B$ $Q _ { k } \geq 0 \text { for } k = 1,2 , \ldots N$ a. Set up a spreadsheet model and for the following data: Product 1 Product 2 Product 3 Annual Demand 1250 1550 1450 Product Cost \ 120 \ 90 \ 105 Order Cost \ 110 \ 175 \ 140 B \ 30,000 i 0.3 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.)

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.

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

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 exponential smoothing model is given by $\hat { y } _ { t + 1 } = \alpha y _ { t } + ( 1 - \alpha ) \hat { y } _ { t }$ where $\hat { y } _ { t + 1 } = \text { forecast of sales for period } t + 1$ $y _ { t } = \text { actual sales for period } t$ $\hat { y } _ { t } = \text { forecast of sales for period } t$ $\alpha = \text { smoothing constant, } 0 \leq \alpha \leq 1$ 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 $y _ { t } - \hat { y _ { t } }$ 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 Squared Forecast Day Observed Value Forecast Forecast Error Error 1 15 15.00 0.0000 0.0000 2 12 15.00 -3.0000 9.0000 3 14 13.65 0.3500 0.1225 4 18 13.81 4.1925 17.5771 5 15 15.69 -0.6941 0.4818 6 16 15.38 0.6182 0.3822 7 15.66 5.3400 28.5159 8 21 18.06 -0.0630 0.0040 9 18 18.03 -6.0346 36.4169 10 12 15.32 3.6809 13.5494 11 19 16.98 5.0245 25.2458 12 22 19.24 -5.2365 27.4211 13 14 16.88 4.1199 16.9737 14 21 18.73 1.2660 1.6026 15 20 19.30 -0.3037 0.0922 Sum of the squared forecast 177.39 error 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.

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

Consider the stock return data given below. Stock Month 1 Month 2 Month 3 Month 4 Month 5 A 12.07 10.12 14.54 46.58 -19.34 B 15.95 4.16 6.31 -2.74 6.54 C 30.52 16.51 34.25 45.62 -27.21 D 32.42 21.36 13.84 8.12 -6.84 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.

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.

The portfolio variance is the

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

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