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In this video, we're going to
explore our linear optimization

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model further.

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We're going to use it to
answer some "what if" questions

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and to conduct some
sensitivity analysis.

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So here, we have
a spreadsheet that

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is formatted very similarly
to the spreadsheets

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that we've used in
Video 5 and Video 6.

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So we have the data up here,
we have the price-per-click,

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the click-through-rate, the
average price per display,

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the budgets, the
query estimates.

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Below those, we
have the variables.

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So we have the
cells corresponding

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to the decision variables.

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We have the cell corresponding
to the objective.

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And to the right
of these, we have

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cells that contain the values
of the decision variables

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and a cell that contains
the value of the revenue

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from our original
solution from Video 5.

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So what we're going to do is,
we're going to change our data,

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and we're going to see
how the solution changes

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and how the objective value
changes and compare it

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to our original solution.

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So as one of the questions
that we might consider, let's

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consider the following question.

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What would happen if the
click-through-rate of AT&T with

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query one increased
from 0.10 to 0.5?

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So to answer this
question, let's crawl up

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in the spreadsheet until we
hit the click-through-rate.

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And let's change the
click-through-rate

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from 0.1 to 0.5.

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Now, if we do this,
you may have noticed

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that the average price per
display for AT&T in query one

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also changed.

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So of course, this makes sense,
because the average price

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per display is just the
click-through-rate multiplied

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by the price-per-click.

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And here, the way we've
set up the spreadsheet

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is so that these
cells are exactly

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the product of the
corresponding cells.

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So the cells that correspond
to the click-through-rate

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and the price-per-click for that
respective query and advertiser

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combination.

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So our average price per display
has changed appropriately.

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And so now, we just scroll
down until we see our variables

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and we see our objective.

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And let's click on Tools.

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Let's open up the Solver.

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And we have the Solver
configured the exact same way

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from last time.

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So we don't need to
do anything here.

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And so now, all we
have to do is just

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hit Solve and click on
Keep Result, and voila.

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We have a new solution.

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So now, several things have
changed with the solution,

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if you can see.

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So the first thing is that
the allocations have changed.

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So for instance, we allocate
query one and AT&T 68 times.

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So we decide to show AT&T's
ad with query one 68 times,

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as opposed to the
original solution, where

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we did it 40 times.

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And we can also see that AT&T
is never shown in query two

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or query three in
our new solution,

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whereas before, it was shown
40 times for query two and 80

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times for query three.

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Similarly, we show T-Mobile
72 times with query one,

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whereas before, we only
showed it 100 times.

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And we also showed
T-Mobile with query three

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14 times, whereas
before, we didn't show it

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at all with query three.

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And Verizon's allocations
say the same as before.

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In terms of the
revenue, our revenue

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has gone up slightly from
$428 in the original solution

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to $430 in the new solution.

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Now, this may seem
like a small amount.

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But actually, this is the most
that we can hope to achieve.

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And the reason for this
is, if we scroll down,

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if we look at our budgets,
so the budget for AT&T

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is 170, for T-Mobile, 100,
and for Verizon, it's 160.

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If we add up these
values, you can

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see that actually the sum
of these values is 430.

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Now, this isn't a coincidence.

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In fact, if you
think about it, this

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makes sense, because what Google
earns from each advertiser

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is exactly how much
that advertiser spends.

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And if the most that
each advertiser spends

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is that advertiser's
budget, then the most

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that Google could hope
to earn is in fact

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the sum of these budgets.

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So in fact, we are attaining
the highest possible revenue

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that we can hope to
attain in this case.

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So that was rather interesting.

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And now, let's change back
the click-through-rate

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from 0.5 back to the
original value of 0.1.

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And let's answer
another question.

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So the question that
we'd now like to answer

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is, what would happen if
AT&T's budget increased

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from 170 to 200?

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So for example, AT&T calls us
and tells us that actually they

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can afford more advertisements.

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So how would that
change our solution?

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Well, in this case, let's
just find AT&T's budget data.

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So in this case, it
is the cell here.

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And let's change
it from 170 to 200.

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Now, let's scroll down to our
variables and our objective.

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And let's just set
them back to zero.

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And now, let's go to
Tools again, let's open up

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the Solver, and let's hit Solve.

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We get 428, which is
actually the same objective

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that we got from before.

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And let's just
click on Keep Result

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and take a look at the solution.

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Now, interestingly,
this new solution

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is actually exactly the
same as the old solution.

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So what happened here?

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Why didn't this change anything?

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Well, actually, if you recall
from the previous solution,

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in the previous
solution, we actually

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only used $168 of AT&T's budget.

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And in the previous solution,
AT&T's budget was $170.

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So in the previous
solution, we didn't actually

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use up all of AT&T's budget.

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And since this constraint
was not binding,

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then increasing this
constraint beyond 170--

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so increasing the
budget from 170 to 200--

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won't actually have an
effect on the solution.

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So this is why the
solution didn't change.

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And in fact, in this
case, we didn't really

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need to change the data and
to solve the problem again,

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we could've deduced
this from actually

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looking at the budget values.

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So these are examples
of two questions

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that we might consider
in this setting.

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And so this concludes
our exploration

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of this problem in LibreOffice.

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In the next video, we'll
return to the slides,

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and we'll discuss
some ways that we

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can extend the problem
beyond the formulation

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that we've been
thinking about here.

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And we'll also summarize
what we've discussed so far.

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So see you in PowerPoint.