Now we're ready to start Robust Parameter Design and Process Robustness Studies. The goal of robust design is to make products and then processes robust, or let's say less sensitive to variability that is transmitted into those products and processes into the responses for those products that factors that really cannot be easily controlled. In practice, there are a number of those with products. There are environmental factors that cause variability. There are factors associated with how the people use those products that cause variability. In manufacturing, there are variables that can't be necessarily controlled very well. Or there's lot to lot variability and properties of material that cannot be easily controlled. Finding methodology for making products and processes robust to this problem, RPD, robust parameter design, or robust process design. Some of the ideas for this were developed by Japanese Engineer, Genichi Taguchi starting in the 1950s and they were first introduced to Western industry in the 1980s, as a lot of our industrial base went through trying to incorporate using statistical methods effectively to improve quality. Taguchi's methods generated a lot of controversy, to say the least. Some of the ideas were quite novel and in fact, it turns out that many of his ideas from a statistical design viewpoint were inefficient and some of his data analysis methods weren't always effective. So Taguchi got a lot of criticism. But I don't think that we should lose sight of the fact that he got the problem right. This is an important problem and he absolutely got the problem right. I remember a good friend of mine who was a medical researcher, he was an MD that did medical research for Center for Disease Control. He asked me one time, he said, "How do you become famous in academic medicine?" I said, "Well, you create a cure for a disease." He said, "Wrong." He said, "Name a physician that got famous for a cure in our lifetime." So I thought a little bit, and the I said, "Well, Jonas Salk and the salt polio vaccine", He said "That's right, think of another one." He was right, I'd have to go back to Fleming and penicillin, but I was really clutching at straws. He said, "The way you'd get famous in academic medicine is not by discovering the cure, by discover a new disease. As soon as you discover a new disease and immediately becomes famous and you become famous." Montgomery's syndrome, you are instantly famous. You are in all the medical books. Your are famous made. Getting the problem right is a big deal, and Taguchi got the problem right. Now, subsequent research since the 90s has produced improved approaches to solving the RPD problem, that we can incorporate within the response surface framework and we're going to talk about that. In robust design, our focus is usually on design systems that are insensitive to environmental factors. For example, the development of a paint product that goes on the exterior of a house or a building and you want that paint to be insensitive to variations in weather conditions which are never perfectly predictable and which vary all across the country. Designing products that are insensitive to variability transmitted by components in the system. Suppose you design an electronic amplifier, so the output voltages is as close to the desired target as possible. Well, but there are electrical parameters of this amplifier transistors, resistors, power supplies and they have variability. So variability in those components is transmitted in the output, designing processes so that the manufactured product is as close as possible to the desired target. Even though some of your process variables like temperature or some of your material properties are hard to control. Then, determining the operating conditions for a process so that your critical process characteristics are as close to the desired target as possible. For example, in semiconductor manufacturing, we want the oxide thickness on a wafer to be as close as possible to the desired target because that reduces the variability in thickness across the wafer. It reduces the deffectivity. So there are a lot of places where this problem comes up. Taguchi called these difficult to control factors, noise variables. Noise variables are variables that can't be controlled in the end application of either the product or the process, but they can be control for purposes of an experiment. So the objective is determine the levels of the control of variables that minimize the variability transmitted from the noise variables. By the way, if you do this, you try to do this you're going to find situations where this approach won't work. The noise factors dominate. When the noise factors dominate, about the only thing you can do is to change the design. For example, making your integrated circuits in a clean room and that would eliminate defects due to microscopic contamination. Sometimes technology can be employed to do this. Substituting an electronic components from mechanical ones can sometimes improve robustness. Taguchi had an original approach to doing this involving crossed array designs. Basically, the approach consisted of putting all of the controllable factors in one experiment, which he called an inner array, and then all of the noise factors in another experimental design, which he called an outer array and then we crossed those arrays. Here's an example. This is a leaf spring experiment, which is discussed back earlier in the book. It's one of the problems in Chapter 8. There are five factors that were studied to determine their affect on the free height of a leaf spring that's used in an automobile. Here are factors. A is temperature, B is heating time, C is transfer time, D is hold down time, and E is quench oil temperature. The quench oil temperature was the noise factor. So here's the data from this experiment. This is Table 12.1. So the four factors that are the controllable factors are A, B, C and D, and if you look carefully at this design, you would see that it's a 1.5 fraction of a two to the four. Then E appears at two levels, so it's a two to the one, if you will. There are three replicates at each of these sets of conditions. So this is your inner array and this is your outer array. Then, we summarize the data in terms of an average for each run in the inner array. Now, that's averaged across the outer array and then a standard deviation or variance, this is actually a variance measurement across all the runs. For all the runs in the inner array, across the runs of the outer array. It turns out that control by noise factor interactions play a really big role in transmitted noise. Here's an illustration of this. Here is a noise factor z, that varies naturally in application. Here are two levels of a controllable variable X. X at the low level, X at the high level. This is showing the variability transmitted into y as this noise factor z varies. You'll notice because there's no interaction between x and z. The amount of transmitted variability from z does not depend which level of x you use. But on the other hand, if you have a situation like you see here where there is strong interaction between x and z. One level of x clearly results in much reduced variability in y as z varies. So here we'll have a significant role control by noise interaction. If we can choose that low level of x, we can get much less output variability in y. So this is how this works. This control by noise interaction is key in solving robust design problems. Here is a very famous example in the Taguchi literature. This is a Connector Pull-Off Force Experiment. The connector is used in automobile engine application. Here are the design parameters for the connector. They are things like wall thicknesses and insertion depths and amount of adhesive and so on and so forth. Three levels of each factor were chosen. This is a three to the four minus two design. So this is a fractional three to the k design. Here is the outer array and these are variables that have more to do with conditioning time and temperature and humidity. Things that you can't very easily control. So this experiment has nine times eight or 72 runs. Here are the 72 runs. Those are the pull-off forces associated with those tests. Now, you notice that because every run in the outer array is crossed with every variable in the inner array that we can estimate all the control by noise interactions. But you cannot estimate any of the control by control factor interactions because this is a resolution three design. This is the three to the four minus two resolution three design. So three to the four minus two and you can guesstimate no interactions between any of the controllable factors. In fact, this presents one of the major problems with Taguchi's design strategy. The crossed array approach can produce a lot of tests, a lot of experiments, a big experimental design. There's only seven factors here, but the design has 72 runs. Even though it has 72 runs, you can't estimate any of the interactions between the control factors. You can only really estimate the control by noise interactions. So information about interaction between your controllable factors is missing. You only have seven factors. Wouldn't you think about maybe a 16 run design that's resolution four, or maybe even a 32 run design. I think there are other options that would produce fewer runs that could produce results that are at least as good.