Hi, welcome back. In this tutorial we're going to discuss Robust Parameter Design. An experiment was run in a semiconductor manufacturing facility, involving two controllable variables and three noise variables. The experimenter decided to perform a combined array design as opposed to across array design. In this design, at the end, they'll be able to fit a response model and the controllable variables and then, main effects of the noise variables and then also, interactions between the control and the noise variables. Their goal, is to try and find operating conditions where the process mean is below 30, while also keeping the process standard deviation from all the noise variables below five. Here's the experiment, if you look closely, we have our two controllable variables, X1 and X2, and then our three noise variables, Z1, Z2, and Z3. Then we also have our response. The first 16 runs of this design are actually a two to the five minus one fractional factorial, and that has a resolution of five. So all the main effects and all the two-factor interactions are estimatable. So we'll start there and then rows run, 17 and 18, 19 and 20, are all axial values for our two controllable variables, X1 and X2. Then the last three are some center points. This design will help us be able to estimate all those effects that we're interested in. So in JMP, I'm going to show you how we can analyze this design, and then also, use a contour profile to try and find that mean response less than 30 and also, standard deviation less than five. So I'll open up JMP and I have this table all set and ready to go for us, and you'll see we have our two controllable factors, X1 and X2. I have included their coding because we want to make sure that all negative one is actually coded as a negative one and not a negative two for the axial points. So make sure that those look correct. Then we have our three noise variables, Z1, Z2, Z3, and then our response Y. So the first thing I'm going to do is just fit this model. So I'm going to go to analyze fit model. I'm going to select Y as my response, and then I'm going to add all those model terms that we're interested in estimating. So we want X1 and X2, and we want those as a response surface. So I'm going to use my response surface macro. So I got the main effects X1 and X2, their two quadratic terms and their two-factor interaction term, so all five of those. Then I'm going to add in Z1, and Z2, and Z3 because we want those main effects. So just add those. Now, we want all the interactions between the noise variables and the controllable variables. I'm not interested in interactions between noise variables. So just an easy way to do this is that I'm going to select all and use the factorial to degree macro, which will add all two-factor interactions into my model effects. But then I'm just going to go in and remove these within noise variable interactions. So we'll just remove those. So now, I just have X1 crossed with Z1, Z2, and Z3, and X2 crossed with Z1, Z2, and Z3. So those are all the terms that I'm interested in estimating. So I'm going to click run, and we see, if we fit this full model, we can come down to these parameter estimates and look at this full model. The next thing I'm going to do is I'm going to create two new columns in this data table. One is going to be for the mean response and one is going to be for the propagation of error model. So we have to do this manually. So you can create a new column, which I've already done, and you can add a column property called the Formula. Then in this, we can edit the formula, and create our means model. So our means model is really just the intercept from this full fitted model. Then our beta one times X1, beta two times X2, and then our quadratic effects in our interactions. You see that these are the same parameter estimates for this mean model, but it doesn't include any Zs because those are noise variables. We just want to include the controllable variables in this model. So we have the means model ready. Then I can also create another column for the propagation of error model or our standard deviation model. So we can figure out what this model will look like by following the formulas along in Chapter 12, and then manually adding this formula through the column features. You'll see that I have the square root of two, and after we have combined all the like-terms and distributed as we need to for this variance model, we can then take the square root and get the standard deviation or propagation of the error model. So I've already done this and edit in here. So I'm going to click okay. So now, I have these two model columns two responses that I'm interested in. So if we want to, we can also add the response limits, but remember that for the mean, we're trying to keep that below 30. Then we're also trying to keep the standard deviation below five. So a nice way for us to figure out what are good process settings for X1 and X2 to maintain our requirements, we can go to graph and we can select the contour profiler. Then once I'm in the contour profiler dialogue, I can add in my two models, which are prediction formulas using X1 and X2, and I can add them here into this prediction formula area. So I have both of those in there. I'm going to click okay. Now, I have a contour profiler, and we can say, "Well, we really want the mean to be less than 30 and we want the standard deviation to be less than five, " so doing so, we can find there is a small process window right here, where we can keep our process mean less than 30 while also keeping the process standard deviation from those three noise variables less than five. So this is our Robust Process Setting to maintain the mean and keep our standard deviation or variance within a limit. So that is how we can analyze a Robust Parameter Design by adding a mean model column and a propagation of error model column, and then using a profiler to find our process window. Thanks.
