In this video, we're going to explore our linear optimization
model further.
We're going to use it to answer some "what if" questions
and to conduct some sensitivity analysis.
So here, we have a spreadsheet that
is formatted very similarly to the spreadsheets
that we've used in Video 5 and Video 6.
So we have the data up here, we have the price-per-click,
the click-through-rate, the average price per display,
the budgets, the query estimates.
Below those, we have the variables.
So we have the cells corresponding
to the decision variables.
We have the cell corresponding to the objective.
And to the right of these, we have
cells that contain the values of the decision variables
and a cell that contains the value of the revenue
from our original solution from Video 5.
So what we're going to do is, we're going to change our data,
and we're going to see how the solution changes
and how the objective value changes and compare it
to our original solution.
So as one of the questions that we might consider, let's
consider the following question.
What would happen if the click-through-rate of AT&T with
query one increased from 0.10 to 0.5?
So to answer this question, let's crawl up
in the spreadsheet until we hit the click-through-rate.
And let's change the click-through-rate
from 0.1 to 0.5.
Now, if we do this, you may have noticed
that the average price per display for AT&T in query one
also changed.
So of course, this makes sense, because the average price
per display is just the click-through-rate multiplied
by the price-per-click.
And here, the way we've set up the spreadsheet
is so that these cells are exactly
the product of the corresponding cells.
So the cells that correspond to the click-through-rate
and the price-per-click for that respective query and advertiser
combination.
So our average price per display has changed appropriately.
And so now, we just scroll down until we see our variables
and we see our objective.
And let's click on Tools.
Let's open up the Solver.
And we have the Solver configured the exact same way
from last time.
So we don't need to do anything here.
And so now, all we have to do is just
hit Solve and click on Keep Result, and voila.
We have a new solution.
So now, several things have changed with the solution,
if you can see.
So the first thing is that the allocations have changed.
So for instance, we allocate query one and AT&T 68 times.
So we decide to show AT&T's ad with query one 68 times,
as opposed to the original solution, where
we did it 40 times.
And we can also see that AT&T is never shown in query two
or query three in our new solution,
whereas before, it was shown 40 times for query two and 80
times for query three.
Similarly, we show T-Mobile 72 times with query one,
whereas before, we only showed it 100 times.
And we also showed T-Mobile with query three
14 times, whereas before, we didn't show it
at all with query three.
And Verizon's allocations say the same as before.
In terms of the revenue, our revenue
has gone up slightly from $428 in the original solution
to $430 in the new solution.
Now, this may seem like a small amount.
But actually, this is the most that we can hope to achieve.
And the reason for this is, if we scroll down,
if we look at our budgets, so the budget for AT&T
is 170, for T-Mobile, 100, and for Verizon, it's 160.
If we add up these values, you can
see that actually the sum of these values is 430.
Now, this isn't a coincidence.
In fact, if you think about it, this
makes sense, because what Google earns from each advertiser
is exactly how much that advertiser spends.
And if the most that each advertiser spends
is that advertiser's budget, then the most
that Google could hope to earn is in fact
the sum of these budgets.
So in fact, we are attaining the highest possible revenue
that we can hope to attain in this case.
So that was rather interesting.
And now, let's change back the click-through-rate
from 0.5 back to the original value of 0.1.
And let's answer another question.
So the question that we'd now like to answer
is, what would happen if AT&T's budget increased
from 170 to 200?
So for example, AT&T calls us and tells us that actually they
can afford more advertisements.
So how would that change our solution?
Well, in this case, let's just find AT&T's budget data.
So in this case, it is the cell here.
And let's change it from 170 to 200.
Now, let's scroll down to our variables and our objective.
And let's just set them back to zero.
And now, let's go to Tools again, let's open up
the Solver, and let's hit Solve.
We get 428, which is actually the same objective
that we got from before.
And let's just click on Keep Result
and take a look at the solution.
Now, interestingly, this new solution
is actually exactly the same as the old solution.
So what happened here?
Why didn't this change anything?
Well, actually, if you recall from the previous solution,
in the previous solution, we actually
only used $168 of AT&T's budget.
And in the previous solution, AT&T's budget was $170.
So in the previous solution, we didn't actually
use up all of AT&T's budget.
And since this constraint was not binding,
then increasing this constraint beyond 170--
so increasing the budget from 170 to 200--
won't actually have an effect on the solution.
So this is why the solution didn't change.
And in fact, in this case, we didn't really
need to change the data and to solve the problem again,
we could've deduced this from actually
looking at the budget values.
So these are examples of two questions
that we might consider in this setting.
And so this concludes our exploration
of this problem in LibreOffice.
In the next video, we'll return to the slides,
and we'll discuss some ways that we
can extend the problem beyond the formulation
that we've been thinking about here.
And we'll also summarize what we've discussed so far.
So see you in PowerPoint.