Chapter 10, Problem 1P

GreenLawns provides a lawn fertilizing and weed control service. The company is adding a special aeration treatment as a low-cost extra service option that it hopes will help attract new customers. Management is planning to promote this new service in two media: radio and direct-mail advertising. A media budget of $3,000 is available for this promotional campaign. Based on past experience in promoting its other services, GreenLawns has obtained the following estimate of the relationship between sales and the amount spent on promotion in these two media:

$S=-2R^2-10M^2-8RM+18R+34M,$

where

S = total sales in thousands of dollars

R = thousands of dollars spent on radio advertising

M = thousands of dollars spent on direct-mail advertising

GreenLawns would like to develop a promotional strategy that will lead to maximum sales subject to the restriction provided by the media budget.

1. a. What is the value of sales if $2,000 is spent on radio advertising and $1,000 is spent on direct-mail advertising?
2. b. Formulate an optimization problem that can be solved to maximize sales subject to the media budget of spending no more than $3,000 on total advertising.
3. c. Determine the optimal amount to spend on radio and direct-mail advertising. How much in sales will be generated?

To determine

To find the amount of sales generated when $2,000 and $1,000 is spend on radio and direct-mail advertising.

Explanation of Solution

To find the amount of sales generated, substitute $2,000 and $1,000 is spend on radio and direct-mail advertising in the below relationship equation.

$\begin{array}{l}S=-2R^2-10M^2-8RM+18R+34M\\ =-2(2^2)-10(1^2)-8(2)(1)+18(2)+34(1)\\ =36\end{array}$

Therefore, sales of $36,000 will be generated with the budget allocated for promotion.

To determine

To formulate optimization problem to maximize sales when total advertising cost is not more than $3,000.

Explanation of Solution

The formulation is for the problem is shown below,

$\begin{array}{c}Max-2R^2-10M^2-8RM+18R+34M\\ s.t.\end{array}$

$\begin{array}{c}R+M\le 3000\\ R,M\ge 0\end{array}$

To determine

To solve and find the optimal amount to be spend on advertising to determine the amount of sales generated.

Explanation of Solution

Formula:

Solver:

Output:

The solution is R = $2,500 and M = $500 with Sales of $37,000