WEBVTT
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So let's begin
modeling this problem
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as a linear
optimization problem.
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In terms of the framework that
we have seen in the lectures,
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our objective is to maximize
the revenue to Google.
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Our decision is for each
advertiser and each query
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to decide the number of times
that advertiser's ad will
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be displayed for that query.
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Our constraints
are of two types.
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Our first constraint is that
the average amount paid by each
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advertiser, based on the
number of times Google displays
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their ad for each query, cannot
exceed the budget of that
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advertiser.
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Our second constraint is that
the total number of ads we
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display for a query
cannot exceed our estimate
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of the number of requests
that we expect to see for that
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query.
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Let's quickly review
our problem data.
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We have the average
price per display
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for each advertiser
and each query.
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We have the budget
of each advertiser.
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And we have estimates
of the number
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of requests for each query.
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So let's now think
about how to model
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all the pieces of our problem.
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First, how should we define
our decision variables?
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Well, we should define them for
each advertiser and each query.
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So for example, one of
our decision variables
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will be x_A1.
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x_A1 is the number
of times we will
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display AT&T's ad for query 1.
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Similarly, we can
define x_A2 and x_A3.
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These are the numbers of times
that we will display AT&T's ad
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for queries 2 and
3, respectively.
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Now, of course, we have
three different advertisers,
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so we will have variables
x_T1, x_T2, and x_T3.
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These variables represent
the number of times
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that we display T-Mobile's
ad for queries 1, 2,
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and 3, respectively.
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And, of course, our last
advertiser's Verizon.
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So we will have variables
x_V1, x_V2, and x_V3, which
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are the numbers of
times that we display
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Verizon's ad for queries
1, 2, and 3, respectively.
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Now, how do we compute
the revenue to Google,
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which is our objective?
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Well, we know the average
price per display,
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and our decision variables
tell us exactly how many times
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we'll display each
ad for each query.
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So all we need to do is we
need to multiply each decision
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variable with its corresponding
average price per display
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and add them all up.
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If we did this, we would have
0.5*x_A1 + 0.5*x_A2 + 1.6*x_A3
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+ 1*x_T1.
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And we can continue this.
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And the last term in
our sum will be 5*x_V3.
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To get the constraints,
we use a similar process.
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For instance, to get
how much AT&T pays,
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we multiply the AT&T variables
by their average prices
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per display and add them up.
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So we'd get 0.5*x_A1
+ 0.5*x_A2 + 1.6*x_A3.
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Now, one of our constraints,
as you will recall,
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is that this cannot exceed
AT&T's budget, which is $170.
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So our model would have the
constraint 0.5*x_A1 + 0.5*x_A2
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+ 1.6*x_A3, is less than 170.
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We can repeat this to get the
same kind of budget constraint
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for T-Mobile and Verizon.
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Now, to get the number
of times query 2 is used,
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we add up the decision variables
corresponding to query 2.
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So we have x_A2,
the number of times
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AT&T is paired with query 2.
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We have x_T2, which
is the number of times
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T-Mobile is paired
with query 2, and x_V2,
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which is the number of times
Verizon is paired with query 2.
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Now, one of our constraints
is that this cannot exceed
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the estimated number of requests
for query 2, which is 80.
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So our model would have the
constraint x_A2 + x_T2 + x_V2
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all less than 80.
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We can do this for
the other queries.
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So we can do this for
query 1 and query 3.
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And this is all
that we really need.
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So now, let's take this
problem to LibreOffice
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and actually solve it.