# Yuma 4×4

##### Media and Communications 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
earns from each advertiser is exactly how much
that advertiser spends. And if the most that
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