Chapter 13, Problem 10P

Phillips Inc. produces two distinct products, A and B. The products do not compete with each other in the marketplace; that is, neither cost, price, nor demand for one product will impact the demand for the other. Phillips’ analysts have collected data on the effects of advertising on profits. These data suggest that, although higher advertising correlates with higher profits, the marginal increase in profits diminishes at higher advertising levels, particularly for product B. Analysts have estimated the following functions:

$\begin{array}{l}\text{Annual}\text{profit}\text{for}\text{product}\text{A}=1.2712LN\left(X_{\text{A}}\right)+17.414\\ \text{Annual}\text{profit}\text{for}\text{product}\text{B}=0.3970LN\left(X_{\text{B}}\right)+16.109\end{array}$

where X_A and X_B are the advertising amount allocated to products A and B, respectively, in thousands of dollars, profit is in millions of dollars, and LN is the natural logarithm function. The advertising budget is $500,000, and management has dictated that at least $50,000 must be allocated to each of the two products.

(Hint: To compute a natural logarithm for the value X in Excel, use the formula = LN(X). For Solver to find an answer, you also need to start with decision variable values greater than 0 in this problem.)

1. a. Build an optimization model that will prescribe how Phillips should allocate its marketing budget to maximize profit.
2. b. Solve the model you constructed in part (a) using Excel Solver.