Exam 8: Nonlinear Optimization Models

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.

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

In the Bass model for forecasting the adoption of a new product, the objective function

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

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.

Nonlinear programming algorithms are more complex than linear programming algorithms.

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

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

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

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

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- 10P₁ and M = 450 -15P₂ , where S = demand for sandals (in pairs), M = demand for moccasins (in pairs), P₁ = price for a pair of sandals, and P₂ = 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.

Which of the following is not true regarding a concave function?

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.

When the number of blending components exceeds the number of storage facilities, the number of feasible solutions to the blending problem

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

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

A convex function is

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.

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

Functions that are convex have a single local maximum that is also the global maximum.