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.