Question

GreenLawns provides a lawn fertilizer 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 $2,500 is available for this promotional campaign. Based on past experience in promoting its other services, GreenLawns obtained the following estimate of the relationship between sales and the amount spent on promotion in these two media. S = -2R2 – 10m2 - BRM + 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 (in dollars) if $2,000 is spent on radio advertising and $500 is spent on direct-mail advertising? S = $ (b) Formulate an optimization problem that can be solved to maximize sales subject to the media budget. Max s.t. < 2.5 (c) Determine the optimal amount to spend (in dollars) on radio and direct-mail advertising. (Round your answers to the nearest dollar.) radio R = $ direct-mail M = $ How much money in dollars) will be generated in sales? Total sales S = $

Answer 1

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Dale Page Sales --26_10MŠ SRM +18R+34M where R = Thousands of dollar spent endio advertising M= Thousands of dollar spent mail adres tising (a) Given amount spent on radio advertising => $ 2000 Amount spent on disect mail advertising on direct $500 Therefore, R=2, M = 0.5 Sales = -28(2)- 10 x6.50-8X2x0.5+ 18x2 + 34x0.5 > 34.5 (in thousands) Sales » 8 $ 34,500 Max. -2R2 - 10M² - 8RM + 18 R +34M (b) S. t P R + M £2.5 Scanned by TapScanner

Pa co We can solve this using lagrange multipliers. FCRM).-.-22-10MP8RM +)$R +34M CR, M) = R+M-2.5 According to je multipliers dje,m) lag range - лаа, М). LR and del(RM) da (RM) = 0 Therefore df (B,M) -4R-8M+18 LUM 2013 AR da (RM) dk deckM)-1 AM Scanned by TapScanner

Sog - 4 R-8M +18 - 1x) =D and - 20M -8 R +34- IX)=0 - I and -4R - 8 M +18 - 1 -20 - 8 R +34= 1 From 6 and -4R • - '8M+18 = -20M-8R + 34 4R +2 M-16=0 RED R+ 3M-4=0 But we have, R+M=2.5 Pe Subtract 1 and (1) R+ 3M-4- (REM). - 0-2.5 3M-4 M = -2.5 2M = 1.5 M=0,75 Scanned by TapScanner

Page Put value % Min (TV) R+ 0.75-2.5 R=1.75 * & put value of R & M in Sales function : Salles =>> -2x (1.75)?-10X (0.753-8X). 75X0.75 +18X1.75+ 34% 0.75 34.75 Cin thone ande) Therefore, Radio R = $ 1750 Direct Mail M=$ 750 Total Sales $ 34, 750 Scanned by TapScanner