Exam 7: Nonlinear Optimization Models

If a convex function is multiplied by a negative constant,the result:

Exhibit 7-3 A company has the following historical data on the number of ad exposures and the corresponding number of units sold of one of its products: -[Part 1] Refer to Exhibit 7-3.Formulate a nonlinear optimization model to find the parameters of a function of the form: f(x)= ax^b to model demand for its product as a function of ad exposures (x).

Exhibit 7-2 A soda producer makes and sells two products,Classic Cola and Diet Cola.During the planning period,if the producer spends x₁ dollars on promotion of Classic Cola,it can sell 100x₁^0.5 cases of Classic Cola,and if it spends x₂ dollars on promotion of Diet Cola,it can sell 10x₂^0.75 cases of Diet Cola.Each case of Classic Cola sells for $12.00 and costs $0.95 to produce and ship to customers,while each case of Diet Cola sells for $12.50 and costs $1.00 to produce and ship to customers.A total of $7,500 is available for promotion during the planning period. -Refer to Exhibit 7-2.Formulate and solve a nonlinear optimization model to help this soda producer identify the best promotional strategies for its two products.

The sum of two concave functions is concave.

In advertising response models,a typical source of nonlinearity is a decreasing marginal effect,where each extra ad gains fewer exposures than the previous ad.

In pricing models,two products are substitutes for each other if a larger price for one product tends to induce customers to demand less of the other.

Exhibit 7-1 A company manufactures two products.If it charges price p₁ for product 1 and price p₂ for product 2,it can sell quantities q₁ = 55 - 3p₁ + 2p₂ and q₂ = 75 + 2p₁ -2p₂ for products 1 and 2,respectively.It costs the company $20 to produce a unit of product 1 and $65 to produce a unit of product 2. -Refer to Exhibit 7-1.Suppose the company is required by regulation to charge the same price for both products.How many units of each product should the company produce in that case? What prices should it charge,to maximize profit?

In some nonlinear models,Solver will find the optimal solution only if the starting solution is reasonably close to the optimal solution.

Exhibit 7-1 A company manufactures two products.If it charges price p₁ for product 1 and price p₂ for product 2,it can sell quantities q₁ = 55 - 3p₁ + 2p₂ and q₂ = 75 + 2p₁ -2p₂ for products 1 and 2,respectively.It costs the company $20 to produce a unit of product 1 and $65 to produce a unit of product 2. -Refer to Exhibit 7-1.How many units of each product should the company produce? What prices should it charge,to maximize profit?

Exhibit 7-2 A soda producer makes and sells two products,Classic Cola and Diet Cola.During the planning period,if the producer spends x₁ dollars on promotion of Classic Cola,it can sell 100x₁^0.5 cases of Classic Cola,and if it spends x₂ dollars on promotion of Diet Cola,it can sell 10x₂^0.75 cases of Diet Cola.Each case of Classic Cola sells for $12.00 and costs $0.95 to produce and ship to customers,while each case of Diet Cola sells for $12.50 and costs $1.00 to produce and ship to customers.A total of $7,500 is available for promotion during the planning period. -Refer to Exhibit 7-2.Suppose the producer can double the promotional budget.Formulate and solve a nonlinear optimization model to help this soda producer identify the best promotional strategies for its two products in that case.Does the change in profit justify the budget increase? Does the proportional amount spent promoting the two products remain the same?

Which of the following types of fit should be selected in Excel's Trendline function if a constant elasticity curve is to be used to model demand?

Which of the following is not one of the reasons an optimization model can become nonlinear?

Exhibit 7-4 You are given the following means,standard deviations,and correlations for the annual return on three stocks.The means are 0.08,0.10,and 0.15.The standard deviations are 0.15,0.20,and 0.30.The correlation between stocks 1 and 2 is 0.62,between stocks 1 and 3 is 0.32,and between stocks 2 and 3 is 0.43. -Refer to Exhibit 7-4.Suppose you set the weights in the portfolio to a maximum of 0.5 for each stock.Is it possible to achieve a 12% return? What is the portfolio standard deviation in that case?

Exhibit 7-4 You are given the following means,standard deviations,and correlations for the annual return on three stocks.The means are 0.08,0.10,and 0.15.The standard deviations are 0.15,0.20,and 0.30.The correlation between stocks 1 and 2 is 0.62,between stocks 1 and 3 is 0.32,and between stocks 2 and 3 is 0.43. -[Part 2] Refer to Exhibit 7-4.Determine the minimum variance portfolio that yields an expected annual return of at least 0.12.How has the portfolio changed from your answer in Part 1?

Exhibit 7-1 A company manufactures two products.If it charges price p₁ for product 1 and price p₂ for product 2,it can sell quantities q₁ = 55 - 3p₁ + 2p₂ and q₂ = 75 + 2p₁ -2p₂ for products 1 and 2,respectively.It costs the company $20 to produce a unit of product 1 and $65 to produce a unit of product 2. -Refer to Exhibit 7-1.Suppose the company must produce a minimum of 20 units of each product.How many units of each product should the company produce in that case? What prices should it charge,to maximize profit?

The amount invested is one of the required inputs in a portfolio optimization model.

Solver is guaranteed to solve certain types of nonlinear programming models.

Which of the following is not one of the conditions for maximization in a nonlinear problem?

Solver can get stuck at a global optimum and never find the local optimum.

Solver is guaranteed to find the global minimum (if it exists)if the objective function is concave and the constraints are linear.