Welcome back to our course on experimental design. The text reference is chapter 11 for this material and what we're seeing on the slide here is the chapter outline. And we're going to cover all of the material on basic response surface methods which takes us through Section 11.4. And then we're going to talk about a fairly new topic in experimental design, but I consider part of response surface methods. And that's experiments with computer models, that's section 11.5. And then we're going to go back and talk about another special case of a response surface experiment, and that's experiments with mixtures. And as we'll see when we get into that material, mixture experiments are very very common. And they require a special type of approach both to design and analysis, which we'll spend some time on. Now the primary focus of the previous modules in this course, and most of the chapters, in fact, in the book. Up through chapter 9 and 10 has been on factor screening. And in factor screening, we generally use factorial experiments, we often use fractional factorial experiments. There's a lot of emphasis on two-level designs. And the primary thing we're trying to do in factor screening is to to discover which factors are important. What are the active effects in the system? Well, once you've answered that question, there's another question that comes up in many situations. And that is what levels of the active factors do we use to get the best results? That's an optimization question and RSM or response surface methods is all about optimization. So the types of experiments were talking about here generally have their focus on optimization. And they typically assume that we've done some factor screening. And we know what the important or active factors are. RSM is relatively new in the history of experimental design. The notion of statistical design really began, as you know, in the early 1900's but RSM dates from the late 40s and early 1950s. With early applications primarily in the chemical industry. Although it's spread after some years to other parts of the process industries, including Pharmaceuticals. And modern applications of RSM, which began probably in the 80s, span many many industrial settings, many business settings. Lots of people use response surface methods today. Basically response surface methods is a collection of mathematical optimization. And statistical experimental design and modeling techniques that we basically use for modeling and analysis of problems. Where your response is influenced by several factors and we've identified what those factors are typically through variable screening. And our objective is to optimize the response, RSM basically has a three step approach. First of all, run an experiment that enables you to find a suitable approximation for y, the response. As a function of your controllable process variables using least squares. And typically the models that we use here for these approximating functions are low order polynomials. Then if we're not really close to the optimum in this initial experiment. We want something to move as efficiently and rapidly as we can toward the region of the optimum. And then when we get close to the optimum. And one of the ways we identify getting close to the optimum is that we start to see curvature in the response function. When we when we find that condition, generally will find a new approximation for y = f(x), a new model. And generally that is a higher order polynomial model, something that accounts for the curvature. And then we go through the RSM optimization procedure. The types of models that we typically use in response surface work are for screening. We typically use the classic main effects plus two-factor interaction type model that you see at the top of the slide. For that moving in the direction of the optimum part of RSM. We typically use a method called steepest ascent and steepest ascent relies almost exclusively on first order models. You notice that this first order model does not include an interaction term. And then when we get in the vicinity of the optimum, our optimization work is typically done with a second-order model. And that's the model you see down at the bottom of the slide. This is a pictorial representation of the sequential nature of RSM. Beginning with, of course, factor screening and factor screening is often done somewhere in the vicinity of our current operating conditions. And sometimes from a factor screening experiment we are able to determine that we're probably not very close to the optimum. For example, if we run a screening experiment around the current conditions here. The largest responses that you would see are probably in the the middle 70s. And if this is some measure of yield, you would like to think that the yield should be higher. So maybe we need to move toward a new region that is more likely to contain the optimum. Now the way we do that is, we look for a path of improvement that moves us toward the direction of the optimum. That is called the method of steepest ascent and we'll see an example of how steepest descent works very shortly. And we follow this path of steepest descent until we get to some place where we have an indication that we're getting near the optimum. And here you notice that we would be climbing and climbing and climbing, the response would continue to increase. And then somewhere in this vicinity, the response would start to go down again. That's an indication that we're probably near the optimum. At that point, then we would want to get a new model for the system. And that model should probably be one that accounts for the curvature in the response function. And then we would find the optimum operating conditions using that model.