We're continuing our discussion of mixture experiments in this lecture. In the previous lecture, we talked really about the simplest form of mixture experiments. That's where there are no constraints on the individual component proportions. But constraints are actually pretty common. For example, a very typical kind of constraint on a mixture component is a lower bound. For example, you may have to have a certain minimum amount l_i or minimum proportion l_i of component x_i in the mixture for it to even work. So that means that l_i less than or equal to x_i less or equal to 1 is a constraint that has to be incorporated into the design. Well, it turns out that this is pretty easy to do because when you have only lower bound constraints, the feasible design region is still a simplex, but it's inscribed inside that original simplex. So you can still use the simplex class of designs, but you have to figure out how to fit them inside the region. The way we typically simplify that problem is to introduce what are called pseudo-components. The pseudo-components are defined in terms of equation, 11.29. In other words, the pseudo-component x prime sub i is equal to the original component x_i minus its lower bound divided by 1 minus the sum of all of the lower bounds. By the way, the sum of all those lower bounds has to be less than one. It turns out that with this definition, the sum of the pseudo-components will now be exactly equal to 1. Here's an example of a problem that includes constraints. What we're trying to do here is to formulate an automotive clear coat paint. This is an example of a constraints to mixture experiment where we have both lower and upper bound constraints. There are three components, a monomer x_1, a crosslinker x_2, and a resin x_3. The experimenter wants his clear coat to have a particular Knoop hardness of at least 25 and wants the percentage of solids in the clear coat to be below 30. So it's a three-component mixture. But look at the constraints, x_1 plus x_2 plus x_3 must be equal to 100. But x_1 ranges between five and 25 percent, x_2 ranges between 25 and 40 percent, and x_3 ranges between 50 and 70 percent. So they're not only just lower bounds, they are lower and upper bounds. Well, when you have both lower and upper bounds that are active, then the feasible region is no longer a simplex. In fact, this is what the feasible region now looks like. You noticed that it looks like a simplex, but this corner has been cut off over here. So a standard simplex design is not going to work. When we have this type of mixture problem with both lower and upper bound constraints, typically, what we do is use either a D or an I-optimal design. In this particular example, I used a D-optimal design. The D-optimal design is shown in the Figure 11.45. It uses the vertices, the edge centers, the overall centroid, and then the check runs halfway between the centroid and the vertices as the candidate points. This is a 14 run design. So this is a 14 run design. There are replicates here, here, here, and here, 10 distinct rooms and four replicates. So we should now be able to fit a quadratic response surface model for both hardness and for solids. Here is the data, by the way, from the experiment. It's Table 11.20. Then here is the model fitting for the hardness response. It turns out that the BC blending term is probably not significant and neither is the AB blending term. So you could possibly eliminate those from the model if you so desire. I left them in. Here is the model for the solids response. All of the components here are significant at at least the five percent level. So there's the model in terms of pseudo-components for hardness. So here is a contour plot of hardness that's on the left. Remember, we want the hardness to be at least 25, and the percent solids is shown as a contour plot in the figure on the right. So to come up with the appropriate formulation or recipe, a fairly straightforward thing to do would be to overlay the contour plots. So here's an overlay of contour plots showing you a region where the solids are below 30 and where the hardness is at least 25. So there is a fairly large feasible region in here where we can formulate this product and get a automotive clear coat that satisfies our objectives. So that would give you a lot of latitude in terms of maybe ease of manufacturing, ease of application in the car plant or cost. That would be another factor that you can consider in choosing the formulation. Mixture experiment occur a lot. They are a very, very widely used type of experimental design in the food and beverage industry, in the pharmaceuticals industry, and in manufacturing where we have different types of processes involving coding and wet chemical etching and other kinds of applications involving chemical mixtures. Of course, they also play a great role in formulating paints, and adhesives, and many, many other types of commercial products. In addition to the D and I-optimality criteria, there are other criteria that could be used to create mixture designs. Distance-based designs are an approach that spreads the design points out to maximize some measure of distance between them. This often creates a design that has more runs in the interior then either a D or an I-optimal design would use. Another approach is the extreme vertices approach, which starts by constructing a design with vertices at any of the centers of edges and then adding the centroids of all the subspaces until the desired number of points in the design has been achieved. Space-filling is another way to do this. JMP has a space filling algorithm that essentially maximizes the product of all the distances between all potential design points. Then I created one of those 12 run space-filling design in Table 11.24, and then Table 11.23 has a 12 run I-optimal design just for comparison. You notice that there are a number of runs in the interior here for the I-optimal design, but many more interior runs in the JMP design. Here are pictures that illustrate that the runs in the the I-optimal design are a little harder to see, but they're in all of the usual spaces, and some of these runs are replicated. This is the 12 run space-filling design. Notice how this does spread the points out more uniformly over the design space. Here is a comparison of the fraction of design space plots for these two designs. The I-optimal design is the upper figure. So this is the fraction of design space plot for the 12 run I-optimal design. This is the fraction of design space plot for the 12 run space-filling design. Notice that the space-filling design is uniformly better in terms of relative prediction variance over this entire design space than is the I-optimal design. So I think that in many cases where you might be considering the optimal design, you probably ought to think about the space-filling design as an option. There are other ways to create space-filling designs. There are several other algorithms for doing it, but this is just an illustration of one of those and that's the space-filling algorithm that's in JMP.
