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(Advertising problem) Show & Sell can advertise its products on local radio and television (TV). The advertising budget is limited to $£10,000$ a month. Each minute of radio advertising costs $£15$ and each minute of TV commercials $£300$. Show & Sell likes to advertise on radio at least twice as much as on TV. In the meantime, it is not practical to use more than $400$ minutes of radio advertising a month.

From past experience, advertising on TV is estimated to be $25$ times as effective as on radio. Determine the optimum allocation of the budget to radio and TV advertising. Model this problem as a linear programming problem

I know that this a minimisation linear programing, and the variables for the problem are TV and Radio, but how do I set up the inequalities? 

<p>EDIT: it seems that what I assumed to be minimising cost is actually a maximisation problem </p>
UniStuffz
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2 Answers2

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Let $T$ be the number of TV advertising minutes per month, and $R$ the number of radio advertising minutes per month.

The constraints are (1) $300T+15R\le10000$ (limited budget), (2) $R\ge 2T$, (3) $R\le 400$, (4) $T\ge 0$.

Note that we do not need $R\ge 0$ because it follows from (2) and (4).

We want to maximise $R+25T$ which is a measure of effectiveness.

almagest
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  • You have a typo at the first constraint. – callculus42 May 06 '16 at 17:26
  • @callculus Many thanks. Fixed. – almagest May 06 '16 at 17:27
  • @callculus What "Key Words" do you look for to know how to set this problem up. E.g. how do you know its a maximisation problem and the objective is R+25T and not say 2R+25T... – UniStuffz May 06 '16 at 17:27
  • @callculus I.e what is your methodology tp build an LP? – UniStuffz May 06 '16 at 17:29
  • From a general reading of the problem you identify $R,T$. It is then a question of writing each constraint as an inequality. Why $R+25T$. Well you are told that TV is 25 times as effective as radio. So the effectiveness of a mix of R,T is given by some multiple of $R+25T$. Then we have to optimise something. What could it be? It is hard to think of anything else except effectiveness. Note that minimising cost would not make much sense, because you could trivially do that by spending nothing! – almagest May 06 '16 at 17:31
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Start by defining your variables. "Show & Sell can advertise its products on local radio and television (TV)" so let "x" be the amount, in dollars, spent on radio advertising each month and let "y" be the amount, in dollars, spent on tv advertising each month. ( the linear programing variables are the TV and Radio" means nothing to me. "The TV and Radio" are not numbers.) (You could also have let x and y be the number of minutes of advertising for radio and tv- but you need to say that.)

"The advertising budget is limited to £10,000 a month" so $x+ y\le 10000$.

"Each minute of radio advertising costs £15 and each minute of TV commercials £300. So spending £x on radio you have x/15 minutes of advertising on radio and spending £y on tv you have y/300 minute of advertising on tv.

"Show & Sell likes to advertise on radio at least twice as much as on TV" so $x/15\ge y/300$.

"it is not practical to use more than 400 minutes of radio advertising a month" so $x/15\le 400$.

user247327
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