In the previous video, we encountered many of the important aspects related to optimizing a system. We covered the case with a single variable, one factor. The reason why we started so simply was because we also introduced other important topics such as the idea of linear verses nonlinear behaviour. And how to move between coded units and real world units; and the idea of using noise to judge a model's prediction quality. Join me now and allow me to show you this topic of optimization and why it's so important. To start off, I'm going to show you why the idea of changing one factor at a time is not efficient. And let me use this example to show why. A grocery store is considering varying the price of their product and the height of the shelf where the product is placed on. Currently, the product is priced at $3.46 and is displayed at one and a half meters, or 150 centimetres, above the ground. They make a profit of $665 at these conditions, but obviously would like to increase it. The standard approach, and we've all done this before, is to vary one factor at a time. It's also called Changing One Single Thing or COST. So let's try that approach and see what happens. Raise the price first. That should increase the profit. This is our first experiment and at $3.54 now, we show a profit of $620. That's a little unexpected. Looks like we've gone in the wrong direction. Our profit has dropped. So let's lower the price. We try $3.38 and we'll make a profit there of $688. So it looks like we're going in the right direction now. Let's lower that price again to $3.30 and then we record a profit of $690 at those conditions. Like we saw in the previous video, we're probably starting to level off here. Let's double check. Let's reduce that price a little bit more, to $3.22 now. And we get a profit of $668. So we're almost back to where we started off with. We have increased our profit and then dropped back off again. So let's go back to that previous point where we had the best profit, the sales price of $3.30, and we were making $690. Now let's vary the shelf position. This is a continuous variable, but we have discrete levels for it, 40, 45, 50, 55, 60, up to a maximum of 200 centimetres. So let's try to raise the product from 150 to 170 centimetres. The average height of a male person is 170 centimetres and since this product is for male customers, that should be a good height. But unfortunately, the profit seems to have dropped off. We're down to $675, so let's go the other direction. Our sixth experiment has a height of 125 centimetres now and the profit recorded over there is $697. That's a little better than our previous best value. So let's decide to go a little bit further down to 100 centimetres and we get the same profit, $697. Seems like we've levelled out again. Let's try one more. We'll use a lower shelf height at 80 centimetres and we record a profit now of $685 per hour. So we figured out here that we can sell the product for $3.30 at a height of either 100 or 125 centimetres. At this point, our hourly profit is $697. This is still a lot better than our starting point of $665. But let me show you the true surface. This is called a "contour plot" and if the term contour plot is unfamiliar, I'll explain it in just a minute. Now, we never really know what this contour plot or surface looks like in practice, but this example quickly demonstrates the problem with the COST approach, or the OFAT approach. We have not actually achieved the optimum here. The company has the false belief that they have achieved an optimum and that's really why when people use the COST approach, they think they're doing a great job. The two variables, when considered independently, make you think that you've reached the optimum, but we can see jointly, there's still room for improvement. The COST approach does work in some limited cases, but the chances are quite small. Now imagine doing the cost approach with 3 variables or 4 variables. The chances become lower and lower that you will succeed and hit that optimum, especially if there are interactions in the system. We've had some good discussions about COST and OFAT on the course forums. It works well in scientific labs when you want to conclusively prove cause and effect between the factor and the outcome. But that's not what optimization is about. In optimization, we already know a cause and effect exists. Now we want to find the best combination of all our factors. In fact, a key point about the material in this module is that we've already used a screening design to eliminate the unimportant factors. Now our focus is only on the important ones that actually affect the outcome.