"Six sigma" is the new rallying cry for quality improvement in the chemical process industry. For example, Dow aims to generate an extra $1.5 billion per year in profits after training 50,000 of its employees on the methods of six sigma.3 Statistical tools play a key role in achieving savings of this magnitude. In fact, "sigma" is a Greek letter that statisticians use as a symbol for standard deviation -- a measure of variability. If a manufacturer achieves a six-sigma buffer from its nearest specification, they will experience only 3.4 off-grades per million lots. This translates to better than 99.99966% of product being in specification. To illustrate what this level of performance entails, imagine playing 100 rounds of golf a year: At six sigma you'd miss only 1 putt every 163 years!4

Of all the statistical tools used within six sigma, design of experiments (DOE) offers the most power for making breakthroughs. By way of a case-study example, this article demonstrates how DOE can be applied to development of a coatings system to achieve optimal performance with minimum variability, thus meeting the objectives of six sigma programs.

Figure 1: Propagation of Error Transmitted from Control Factor X1 (Non-Linear Response)

Minimizing Propagation of Error (POE) from Varying Inputs

Hopefully the operators of your coatings systems will be more exacting when adding ingredients. However, at the very least, you can expect some variation due to inherent limitations in equipment. How will these variations affect product quality and process efficiency? Can you do anything to make your system more robust to variations in component levels and process settings? The answers to these questions can be supplied by way of an advanced form of DOE called "response surface methods" (RSM). This statistical tool produces maps of product and product performance, similar to topographical displays of elevation, as a function of the input variables that you (or your clients) control. The objective of six sigma is to "find the flats" -- the high plateaus of product quality and process efficiency that do not get affected much by variations in component levels or factor settings. You can find these desirable operating regions visually, by looking over the 3-D renderings of response surfaces, or more precisely by way of a mathematical procedure called "propagation of error" (POE).

Figure 2: Response Surface Graph of POE Transmitted from Control Factor X1

Assume that this factor exhibits a constant variation shown on the graph as a difference with magnitude delta (D). This variation, or error, will be transmitted to the measured response to differing degrees depending on the shape of the curve at any particular setting. In this example, because the curve flattens out as the control factor increases, a setting at the higher level causes less POE. Therefore, you see a narrower difference (DY) on the response axis as a result of setting the factor at the higher, rather than lower, level.

With the aid of some calculus, the POE itself can be graphed as a continuous function. In this case the original response surface can be described by the following quadratic equation.

Figure 3: Propagation of Error Transmitted from Control Factor X2 (Linear Response)

Figure 4: Achieving Variability Reduction by Intelligent Manipulation of Factor Levels

Obviously, reducing response variation is a highly desirable outcome. But what if the absolute level of response gets shifted out of specification? Ideally, you will find another control factor that affects response in a linear fashion, such as that shown in Figure 3. (We added the bell-shaped curves to indicate that variation will likely be distributed according to a normal distribution.)

Figure 5: Therabath Paraffin Therapy Bath

Let's see how these concepts can be applied to a six-sigma project on a coatings system.

Figure 6a: Effect of Powder Coating Components on Adhesion with Bake Time and Temperature at Mid-Levels

A Case Study on POE:
Powder Coating a Metal Part

The tank holding the molten wax is made out of stamped steel. It's then powder coated electrostatically with an epoxy-based paint. The coating must withstand temperatures approaching 130 degF. and exposure to salt-water that collects from the sweat of the Therabath users.

The units are sold with a life-time guarantee. Very few units get returned, but in those that do, two types of off-grade predominate.

• Blisters that appear spontaneously after some time in use
  
• Scratches that become noticeably corroded.

Figure 6b: / Effect of Powder Coating Components on Hardness with Bake Time and Temperature at Mid-Levels

Designing An Experiment That Combines Mixture with Process Variables

The mixture variables (A, B and C) were added to a constant of 90 weight % of the powder. The other 10% was made up of bisphenol A, aluminum oxide and silica, all held in constant proportion. The paint chemists expected the three main powder-coating components to interact in complex ways. To properly model such behavior, we wanted to formulate a sufficient number of unique blends to fit a "special cubic" mixture model:

Figure 7a: Effect of Process Factors on Adhesion with the Powder Coating Formulated at Mid-Levels (centroid)

Figure 7b: Effect of Process Factors on Hardness with the Powder Coating Formulated at Mid-Levels (centroid)

These two models were crossed to account for possible interactions between mixture and process. For example, the ideal coating formulation may differ depending on the choice of bake time and temperature, which could be a function of the particular type oven available for processing the tanks. The crossed model contains 42 terms (7 from the special cubic mixture model multiplied by 6 from the quadratic process model). With the aid of computer software,6 a d-optimal design (described in a previous article2) was generated to provide the ideal combinations of variables for fitting response data to the combined mixture-process model. We added five extra unique points for estimating lack-of-fit for the model, plus five replicates of existing points to estimate pure error. We also incorporated one center point (combination #3 on the list), representing the standard operating conditions, for a grand total of 53 combinations.

The results (simulated) of this experiment can be seen in the Appendix. The combinations are organized for easier viewing, with replicate points lumped together. However, experiments like this should always be performed as randomly as possible to offset any lurking variables, such as ambient humidity.

Figure 8: Effect of Process Factors on POE of Hardness

Generating 3D Surfaces to See Impact of Inputs on Responses and POE

Figure 9: Sweet Spots for Processing Powder Coat Made by Optimal Formulation

Figure 10: Response Surface for Adhesion as a Function of Bake Time and Temperature with Optimal Formulation of Powder Coat

Table 1: Input Variables for Powder Coating Study

Finding a Setup that Meets All Specifications with Minimum POE

The input variables aren't listed in the table, but they do get constrained within their previously specified ranges. Never extrapolate outside of your experimental region because the predictive models probably won't work out there. Stay in bounds or possibly face some dangerous consequences!

Table 2: Goals for Powder Coating Study

Two "sweet spots" can be seen in the process space (bake temperature versus bake time) by overlaying contour plots for hardness and adhesion, with unacceptable regions shaded out (see Figure 9). (The formulation of the powder-coat is set at the optimal component levels shown in Table 3.)

Table 3: Optimal Combinations for Mixing and Baking Powder Coat

Conclusion

For more information on design of experiments, contact Stat-Ease Inc., 2021 East Hennepin Ave., Suite 191, Minneapolis, MN 55413; phone 612/378.9449; fax 612/378.2152; e-mail Mark@StatEase.com; or Circle Number 80.

References