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In the previous video,

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we encountered many of the important
aspects related to optimizing a system.

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We covered the case with
a single variable, one factor.

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The reason why we started so simply
was because we also introduced other

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important topics such as the idea of
linear verses nonlinear behaviour.

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And how to move between coded units and
real world units; and

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the idea of using noise to judge
a model's prediction quality.

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Join me now and allow me to show
you this topic of optimization and

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why it's so important.

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To start off, I'm going to show you
why the idea of changing one factor at

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a time is not efficient.

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And let me use this example to show why.

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A grocery store is considering
varying the price of their product and

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the height of the shelf where
the product is placed on.

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Currently, the product
is priced at $3.46 and

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is displayed at one and a half meters, or
150 centimetres, above the ground.

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They make a profit of
$665 at these conditions,

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but obviously would like to increase it.

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The standard approach, and

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we've all done this before,
is to vary one factor at a time.

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It's also called Changing
One Single Thing or COST.

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So let's try that approach and
see what happens.

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Raise the price first.

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That should increase the profit.

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This is our first experiment and
at $3.54 now, we show a profit of $620.

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That's a little unexpected.

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Looks like we've gone
in the wrong direction.

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Our profit has dropped.

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So let's lower the price.

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We try $3.38 and
we'll make a profit there of $688.

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So it looks like we're going
in the right direction now.

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Let's lower that price again to $3.30 and

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then we record a profit of
$690 at those conditions.

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Like we saw in the previous video,
we're probably starting to level off here.

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Let's double check.

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Let's reduce that price a little bit more,
to $3.22 now.

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And we get a profit of $668.

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So we're almost back to
where we started off with.

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We have increased our profit and
then dropped back off again.

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So let's go back to that previous
point where we had the best profit,

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the sales price of $3.30,
and we were making $690.

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Now let's vary the shelf position.

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This is a continuous variable, but we have
discrete levels for it, 40, 45, 50, 55,

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60, up to a maximum of 200 centimetres.

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So let's try to raise the product
from 150 to 170 centimetres.

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The average height of a male
person is 170 centimetres and

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since this product is for male customers,
that should be a good height.

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But unfortunately,
the profit seems to have dropped off.

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We're down to $675, so
let's go the other direction.

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Our sixth experiment has a height
of 125 centimetres now and

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the profit recorded over there is $697.

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That's a little better than
our previous best value.

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So let's decide to go a little bit
further down to 100 centimetres and

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we get the same profit, $697.

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Seems like we've levelled out again.

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Let's try one more.

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We'll use a lower shelf
height at 80 centimetres and

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we record a profit now of $685 per hour.

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So we figured out here that
we can sell the product for

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$3.30 at a height of either 100 or
125 centimetres.

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At this point, our hourly profit is $697.

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This is still a lot better than
our starting point of $665.

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But let me show you the true surface.

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This is called a "contour plot" and

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if the term contour plot is unfamiliar,
I'll explain it in just a minute.

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Now, we never really know
what this contour plot or

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surface looks like in practice, but this
example quickly demonstrates the problem

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with the COST approach,
or the OFAT approach.

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We have not actually
achieved the optimum here.

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The company has the false belief that they
have achieved an optimum and that's really

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why when people use the COST approach,
they think they're doing a great job.

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The two variables,

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when considered independently, make you
think that you've reached the optimum, but

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we can see jointly,
there's still room for improvement.

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The COST approach does work
in some limited cases, but

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the chances are quite small.

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Now imagine doing the cost approach
with 3 variables or 4 variables.

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The chances become lower and
lower that you will succeed and

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hit that optimum, especially if there
are interactions in the system.

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We've had some good discussions about
COST and OFAT on the course forums.

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It works well in scientific labs when
you want to conclusively prove cause and

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effect between the factor and the outcome.

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But that's not what optimization is about.

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In optimization,
we already know a cause and effect exists.

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Now we want to find the best
combination of all our factors.

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In fact, a key point about
the material in this module is

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that we've already used a screening design
to eliminate the unimportant factors.

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Now our focus is only on the important
ones that actually affect the outcome.