Question

A Medla Selectlon Problem 3.8.6 Another type of allocation problem arises in the selection of various media to be employed in an advertising campaign. Thể usual objective in such an advertising campaign is to maximize its total elffective advertising expusures, subject to mune- tary constraints and constraints upun the number of exposures possible tor tne individual communication media. To illustrate the use of linear programming in media selection, consider the following problem situation. The Edward Belk Advertising Agency is preparing an advertising campaign for a group of travel agencies. These travel agencies have decided that their "target customer" should have the following characteristics, with relative importance indicated in parenthesis. 1. Female, married (0.2). 2. Age: 25-40 (0.3). 3. Household Income: above $20,000 annually (0.5). Furthermore, they do not want to spend more than S50,000 for their advertising campaign. In formulating an advertising campaign for these travel agencies, the Belk Agency has made a careful analysis of three available media and has compiled the following set of relevant data. Media No. 1-Wanen's Maganinr Dae Item Na. 2-Radie Na. J-Television Reader charecteristics I. Female, married 2. Age: 2540 1, Incume: $20,000, ur mure annually C'and per adlvertisement Minimum number uf ads allowed Maximum numier of ads allkrwed Audience size/media N 70 60% 45% 40% 25% $4,000 SI.500 32,000 10 20 750,000 1,000,000 1.500,000 In this problem situation the objective is to maximize the total effective exposures for all the advertising employed. Now the coefficients of the obiective function are the products of the audience sizes for the various media multiplied by the "effectiveness coefficients" for the various media. The effectiveness coeffi- cients for the three possible media are computed as follows:" Efectiveness coefTicient-media no. I 0.800.2) 0.60(0.3) 0.40(0.5) - 0.54 Efectiveness coefficient-media no. 2 = 0.70(0.2 ) + 0.50(0.3) +0.35(0.5) - 0.465 Effectiveness cuefficient-media no. 3 0.600.2)+ 0.45(0.3) + 0.25(0.5) - 0.38 (3-601 Then, the coefficients of the objective function can he computed as the products of the ellectiveness coefficients and the audience sizes. For example,, the obiec- tive function coefficient r, for media nuo. 1, is cumputed as: (3-61) -0.54(750,000) = 405.000

Answer 1

Short Answer: We shall spend the advertising budget of $50,000 on 20 womens' magazine ads, 10 radio ads and 0 TV ads.

Long Answer:

Continuing as explained in the textbooks' solution,

Coefficient of Objective function for Media no.1 = 0.54 x 750,000 = 405,000

Coefficient of Objective function for Media no.2 = 0.465 x 1,000,000 = ‭465,000‬

Coefficient of Objective function for Media no.3 = 0.38 x 1,500,000 = 570,000‬

Let number of ads of media no. 1, 2 and 3 be x, y and z respectively.

Therefore the objective function would become:

Maximize K = 405,000x + 465,000‬y + 570,000‬z

Which would be subject to the constrains,

Cost: 1500x + 2000y + 4000z <= 50,000

No. of ads of Media No.1: 10 <= x <= 20

No. of ads of Media No.1: 5 <= y <= 10

No. of ads of Media No.1: 5 <= z <= 10

On solving the above linear programming problem, we find:

The optimal solution value is K = 10575000
x = 13.33
y = 5
z = 5

But now consider that we are not under the compulsion to use every media (megazine, radio and TV), therefore we may also choose to use one or two and not use the remaining, if that helps us in maximum exposure. In such a case, we may choose to set the lower limit of x, y and z to be zero in our constrains. Solving the problem for this case yields:

The optimal solution value is K = 12750000
x = 20
y = 10
z = 0

As the overall exposure (K) of this case is higher than previous, we shall go with these values.

Therefore, we shall spend the advertising budget of $50,000 on 20 womens' magazine ads, 10 radio ads and 0 TV ads.

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