An ad campaign for a new snack chip will be conducted in a limited geographical

Question

An ad campaign for a new snack chip will be conducted in a limited geographical area and can use TV time, radio time, and newspaper ads. Information about each medium is shown below.

Medium Cost Per Ad # Reached Exposure Quality
TV 500 10000 30
Radio 200 3000 40
Newspaper 400 5000 25

The number of TV ads cannot exceed the number of radio ads by more than 10% (for example, if there are 100 radio ads, then there cannot be more 110 TV ads). The number of radio ads cannot exceed the number of newspaper ads by more than 13.

If in addition the advertising budget is $10000, develop the model that will maximize the number reached and achieve an exposure quality of at least 1000.

(assume that fractional numbers of ads are ok).

Find the optimal solution using Management Scientist, Lindo, Excel Solver, or online interactive LP software. List the values of the objective function and the decision variables in the optimal solution you

Explanation / Answer

First let's define decision variables. Basically our decision is to decide how many of each type of ad we want. So call them t, r, and n

t = # of TV ads to buy
r = # of Radio ads to buy
n = # of news ads to buy

Now we decide the objective function. The objective function is simply to maximize the number reached. So we set up a linear function that defines how many people are reached. A TV ad reaches 10000 people per ad, a radio ad reaches 3000 people per ad, and a news ad reaches 5000 per ad. So our objective function is simply

max z = 10,000t + 3,000r + 5,000n

Now we decide on our constraints. There are three constraints.

First the number of tv ads can't exceed radio ads by more than 4. So the constraint looks like this
t ? r + 4 .. and if we bring all the variables to the left side, we get
t - r ? 4

Second, we only have $10,000 to spend. So set up a linear function for the cost.
500t + 200r + 400n ? 10,000

Last, we have to receive exposure quality of at least 1000. So
t + r + n ? 1000
....
So if I put all these together my model looks like the one below
--------------------------------------
max z = 10,000t + 3,000r + 5,000n
s.t.
t - r ? 4
500t + 200r + 400n ? 10,000
t + r + n ? 1000
t, r, n ? 0
---------------
Except I get no feasible solution, are you sure those exposures are correct, and we need exposur of 1000.

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