tut1 - Algebraic Formulations Usually in class we describe...

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1 Algebraic Formulations Cleaver, an MIT Beaver Algebraic formulations sound hard. But they are not so hard. However, they do take a while to get used to. Usually in class, we describe linear programs by writing them out fully. This is fine for small linear programs, but it doesn’t work when the linear programs are very large. In that case, it helps to use algebraic formulations.
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2 On creating algebraic formulations When we create algebraic formulations, we rely on substituting notation for some of the coefficients. Let’s start with an example of a linear program. Minimize 500 x 1 + 200 x 2 + 250 x 3 + 125 x 4 subject to 50,000 x 1 + 25,000 x 2 + 20,000 x 3 + 15,000 x 4 1,500,000 0 x 1 20 0 x 2 15 0 x 3 10 0 x 4 15 Cleaver This is the MSR example from lecture 1.
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3 More on the MSR Problem In the MSR problem, we wanted to determine the number of ads of four different types of advertising media. We let x 1 , x 2 , x 3 , and x 4 denote the number of ads on TV, Radio, Mail, and Newspaper. Need to choose ads to reach at least 1.5 million people Minimize Cost Upper bound on number of ads of each type TV Radio Mail Newspaper Audience Size 50,000 25,000 20,000 15,000 Cost/Impression $500 $200 $250 $125 Max # of ads 20 15 10 15
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4 The LP Formulation again Minimize 500 x 1 + 200 x 2 + 250 x 3 + 125 x 4 subject to 50,000 x 1 + 25,000 x 2 + 20,000 x 3 + 15,000 x 4 1,500,000 0 x 1 20 0 x 2 15 0 x 3 10 0 x 4 15 Cleaver The objective is the cost of advertising. The first constraint says that the number of people who see the ads is at least 1.5 million. The remaining four constraints give upper and lower bounds on the number of showings of each of the four ads.
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5 Transforming into an algebraic problem We’ll transform this problem into an algebraic version in a couple of stages. Then we’ll show how to do it all at once. So, let’s start with the four upper bound constraints. Suppose that we let d = (d 1 , d 2 , d 3 , d 4 ) = (20, 15, 10, 15). We can then write the linear program as follows: Minimize 500 x 1 + 200 x 2 + 250 x 3 + 125 x 4 subject to 50,000 x 1 + 25,000 x
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This note was uploaded on 01/18/2012 for the course MGMT 15.053 taught by Professor Jamesorli during the Spring '07 term at MIT.

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tut1 - Algebraic Formulations Usually in class we describe...

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