GreenLawns provides a lawn fertilizing and weed control service. The company is
ID: 3227327 • Letter: G
Question
GreenLawns provides a lawn fertilizing and weed control service. The company is adding a special aeration treatment as a low-cost extra service option, which 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 = -2R2 10M2 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.
GreenLawns provides a lawn fertilizing and weed control service. The company is adding a special aeration treatment as a low-cost extra service option, which 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 = -2R2 10M2 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.
(a) What is the value of sales if $2,000 is spent on radio advertising and $1,000 is spent on direct-mail advertising? $ (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. Max R2 + M2 + RM + R + M s.t R + M - Select your answer -<>=Item 9 R, M - Select your answer -<>=Item 11 (c) Determine the optimal amount to spend on radio and direct-mail advertising. How much in sales will be generated? Amount spent on radio advertising = $ Amount spent on direct-mail advertising = $ Total Sales = $Explanation / Answer
S = -2R2 10M2 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.
(a) What is the value of sales if $2,000 is spent on radio advertising and $1,000 is spent on direct-mail advertising?
total sales where R = $ 2000 and M = $ 1000
S = -2(2)2 10(1)2 8(2)(1) + 18(2) + 34(1)
S = -8 - 10 - 16 + 36 + 34 = $36000
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.
Max Total sales S =-2R2 10M2 8RM + 18R + 34M,
s.t :
R+M < = 3
R, M >= 0
I have solved it in solver in excel or one can solve it by using differentiation method
(c) The results are
Amount spent on radio advertising = $ 2500
Amount spent on direct-mail advertising = $ 500
Total Sales = -2R2 10M2 8RM + 18R + 34M = -2(2.5)2 10(0.5)2 8(2.5)(0.5) + 18(2.5) + 34(0.5)
= $ 37000
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