Experiments 5D - The concept of contour plots and which objectives should we maximize
Channel: Kevin Dunn
Date: 2015-08-10

Videos used in the Coursera course: Experimentation for Improvement. Join the course for FREE at https://www.coursera.org/learn/experimentation

These videos are also part of the free online book, "Process Improvement using Data", http://yint.org/pid

Full script for the video: http://yint.org/scripts/5D
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Let's come back to the idea of contour lines, shown here in dashed blue lines. Contour lines are lines that show equal height. If you walk along a contour line, you are not going up and you are not going down. Here's a photo of Mount Fuji in Japan on the left, and a contour plot of the mountain is shown on the right. Notice that these small numbers here on the plot tell how high up the mountain we are. 

Response surface methods can be thought of, at least conceptually, using the following analogy. Imagine you're in a basement in someone else's house and the lights suddenly go out. You grab a ski pole and start to tap the ground to test the slope of the floor. You know that the door is the highest point in the basement and you want to reach that door. Now also imagine that you want to reach that point using the minimum number of taps -- for whatever reason. This is exactly how we treat response surface methods. 

The objective is to find the maximum on the contour plot using that pole. Each time you touch the surface, you're observing the system and get an outcome value. But when you do that, it is a really expensive process. 

Each time you touch the surface or run an experiment, it might cost hundreds of dollars. How do you get to that optimum efficiently? How do you know that you've reached the optimum with only the information from the taps of the pole? Remember the lights are out, so you cannot see what's around you. 

That's what the videos in the section are all about, getting to the top of that surface with as few uses of that ski pole as possible. People often ask, what should this mountain surface look like? What does it consist of? Well, it is usually your outcome variable, something like total sales that you want to maximize, or the number of unburned popcorn, or the height of plants that you are growing. 

If you're ever stuck thinking about which variable to maximize, you should consider profit. Profit is a good choice because it includes the income and accounts for all the expenses. Let me give you an example. In many engineering systems, we can make things faster by increasing the temperature, but that energy is not free. 

As we go to higher and higher temperature and make things faster, our costs are also increasing. The calculated value of profit takes all of those into account in a single number. You'll find that profit, as an outcome variable, often has the shape of a mountain. It is a great way to incorporate multiple objectives into a single quantity. More on that in the next video. 

Now, you don't always have to climb the mountain. Sometimes you want to be in the valley and get the minimum value. Consider the wastewater treatment example from earlier in the course. We wanted to minimize the amount of pollution. In this course, all our examples will be for maximizing, but minimizing is just the opposite of maximizing. 

In fact, those with a background in optimization theory will know that maximization is just the negative of minimization. For example, in the waste water problem, if we tried to maximize the negative pollution, that is the same as minimizing pollution. 

Now I'll wrap up this video and leave you with some questions to think about. 
How do we climb up that mountain? 
Which direction should we go up and how big should our steps be? 
How do we handle nonlinear surfaces as we climb up? 
When do we stop? 

Because remember, we don't have a contour map, so we need to have a way to know that we've really reached the peak.