Design experiments that combine mixture components with process factors

In two previous articles, we have covered the typical strategy for design of experiments

1. Fine-tune the formulation via mixture design (1); and

2. Optimize the process with factorial design and response surface methods (2).

To keep things simple, these two steps usually are handled separately by the chemist and chemical engineer, respectively. Interactions between compositional variables and process factors cannot be revealed, however, by this simplistic approach. So, in this article, we show you how to do a comprehensive experiment that combines mixture components with process factors in one "crossed" design.

A relatively simple case study

To illustrate how to do a crossed mixture/ process design, we present a relatively simple case study from "Experiments with Mixtures" by Cornell (3). (This textbook provides a wealth of statistical detail on DOE for mixtures, including the crossing of process factors.) The case study involves various blends of three vinyl plasticizers (X^sub 1^, X^sub 2^, X^sub 3^) processed at two levels each for extrusion rate (Z^sub 1^) and drying temperature (Z^sub 2^). Many other components go into a vinyl formulation (stabilizers, lubricants, drying agents, and resins), but the percentages of all nonplasticizer ingredients were held fixed throughout the experiment; so, they are nonfactors.

Figure 1 illustrates the crossed design. The triangles represent the mixtures, which must be repeated at the four combinations of the process factors (Z^sub 1^ and Z^sub 2^).

The scales on the mixture diagrams go from zero to one based on the relative proportions of the three ingredients. The vertices represent pure component blends (X^sub 1^, X^sub 2^, and X^sub 3^). Binary blends (50/50 combinations of any two plasticizers) occur at the midpoints of the sides on the triangle. The interior space, empty in this case, represents three-part blends. The points on the mixture plots come from a standard experimental design called a "simplex lattice." These designs can be tailored for the degree of polynomial you want to fit: linear (lst degree), quadratic (2nd degree), or cubic (3rd degree). The experiment on the vinyl formulation was done with a 2nd-degree simplex lattice, thus revealing any two-component interactions between plasticizers. If you choose this design for your experimental work, we recommend that you augment it with a three-part blend called a "centroid." Results from this blend will reveal potential problems with lack of fit in the quadratic mode.

Table 1 shows the experimental design in terms of coded factor levels:

* for mixture components from 0 to 1 for least to most, respectively; and

* for process factors from -1 to +1 representing lowest to highest levels, respectively.

For proprietary reasons, the experimenters did not reveal the actual units of measure for the variables, but this does not matter for our purposes. All calculations are done in coded form in any case. The experimenters measured the effects of these variables on the thickness of an automotive seat cover. Again, for proprietary reasons, the results have been rescaled; so, no units of measure are reported, but the relative results remain relevant. The entire run (design) was replicated to gain statistical power - all told, there's a total of 48 runs (six blends at four process combinations done two times). To simplify tabulation, each mixture/process combination is shown as a unique row listed in a standard order, with the results of replicate runs appearing together. The actual experiment, however, was completely randomized to insure against lurking factors such as material degradation, machine wear, ambient changes, and the like. Randomization is an essential element of good statistical design.

Creating

a mathematical model

The experimenters wanted better control of the thickness response. The desired results depended upon the model of automobile. For example, thicker vinyl seats might be needed for a pickup truck aimed at the rugged outdoors type. On the other hand, a thinner cover might be needed to cut costs on the low-end "econocar." The outcome of a statistically significant DOE is a polynomial model that can be used to predict the response at any combination of tested variables. As you can see from the derivation below, the models for crossed mixture/process designs can be very cumbersome, even for a relatively simple study like the one done on the vinyl seat covers. In this case, crossing the six-term mixture model IMAGE FORMULA 17

The letter Y symbolizes the response. The first equation (with the X variables) represents the mixture part of the design. The Greek letter beta indicates the unknown coefficients. Another article in this series (1) describes how these polynomials are constructed to account for the overall constraint that all mixture components must sum to one. Mixture models can be recognized by their lack of an intercept. The one shown above is second order. The secondorder terms, such as (beta)^sub 12^X^sub 1^X^sub 2^, reveal interactions.

The second equation (comprised of Z variables) represents the process side of the design. It's sometimes called a "factorial" model. The Greek letter cE indicates the unknown coefficients. For more details on this model and more complex polynomials used for response surface methods, see Ref. 2.

The fitted equation for the vinyl study is given in the box below.

This predictive model will be used to generate response surface graphs, which make interpretation much easier than looking at all the coefficients. For those of you, however, who want to dissect the equation, notice that the first line contains only mixture components (X variables). It represents the blending properties averaged over the various process conditions. The second line reveals the linear effect of the first process factor (Z^sub 1^), which shifts the mean response at any given combination of mixture components. The third line shows the linear effect of the second process factor (Z^sub 2^). The last line represents interactions between process factors and the mixture. When these complex interactions are present, you will see the shape of the response surface change as process conditions are varied.

Analysis of variance (ANOVA)

shows the overall equation to be highly significant (according to the F-test, which is a measure of the ratio of signal to noise, the probability that only random variability is at work - that is, the risk of a false positive outcome - is less than 0.0001 or, in shorthand, Prob>F of <0.0001). You will observe, however, that some of the coefficients in the model are at or near zero. These terms could be eliminated via manual reduction or by use of a standard computerized algorithm such as backward stepwise regression. In this case, there's no advantage to model reduction, because of the presence of statistically significant higher-- IMAGE FORMULA 24order interactions such as X^sub 1^Z^sub 1^Z^sub 2^ (Prob>F of <0.01) and X^sub 1^X^sub 2^Z^sub 1^Z^sub 2^ (Prob>F of <0.01), which must be supported by their lower-order "parents." (These results justify the application of the crossed design tool, because they involve combinations of mixture and process. Interactions such as these never would be revealed by traditional one-factor-at-a-time (OFAT) experiments, or even by more-sophisticated DOE done separately on mixture vs. process.) In any case, removal of insignificant terms causes little impact on the predictions or the resulting response surface maps that you will focus on for your reports; so, we advise that you work with the complete model.

Using graphs to tell the story

A simple way to interpret the results is to produce contour plots of the thickness vs. the composition of the three plasticizers (X^sub 1^X^sub 2^, and X^sub 3^) at all four combinations of the two process factors - extrusion rate (Z^sub 1^) and drying temperature (Z^sub 2^) - as seen in Figure 2.

Notice how the shape of the contours changes as the process conditions vary. From this, you now know that the impact of the different plasticizers depends upon how the vinyl is processed. But, which direction to go will depend upon what thickness you desire. Let's assume you want to maximize thickness. In this case, you will want to adjust the process conditions to either the high level of extrusion rate with drying temperature low (+, -), or the low rate at the high temperature (-, +). It makes more sense to go for the higher rate for production purposes - which means that you want the lower drying temperature. Figure 3 shows a three-dimensional (3-D) version of the contour plot at these conditions.

The peak thickness occurs when the X^sub 3^ plasticizer is excluded from the formulation. The maximum response comes from a binary blend of X^sub 1^ and X^sub 2^, with somewhat better results with more of the X^sub 1^, plasticizer. Numerical optimization, performed on the same DOE software used to fit the data and generate the plots (4), reveals a peak at the 60/40 blend of X^sub 1^/X^sub 2^, which produces a (scaled) thickness of 14.7.

Playing the "what-if" game

The solution noted above might work only for certain models of automobiles, such as heavy duty trucks. What if a thinner vinyl is needed for cheaper vehicles? Let's go back to Figure 2 and reconsider our options. Notice that the lower valued contours occur when you set process conditions to either the low level of extrusion rate with drying temperature low (-, -), or the high rate at the high temperature (+, +). It makes more sense to go for the higher rate for production purposes - which means that you must go to the higher drying temperature. Figure 4 depicts a 3-D version of the contour plot at these conditions.

At the processing conditions noted above, the vinyl is thinnest at two points:

* a binary blend of X^sub 1^ and X^sub 2^ plasticizers (along the X^sub 1^ - X^sub 2^ edge where X^sub 3^ is 0); and

* a binary blend of X^sub 1^ and X^sub 3^ plasticizers (along the X^sub 1^ - X^sub 3^ edge where X^sub 2^ is 0).

The first option is intriguing, because it means that you could use nearly the same blend as that needed for the thick vinyl, but process it differently to make it thin. For example, a 50/50 blend of X^sub 1^/X^sub 2^ at the high levels of both process factors produces a scaled thickness of 5. By simply lowering drying temperature, the same blend produces a thickness of 14.5. Figure 5 shows the interaction of the process factors at the 50/50 blend of X^sub 1^/X^sub 2^.

The results for high extrusion rate (going from upper left to lower right) form the operating line. To get any thickness from 14.5 down to 5, simply adjust drying temperature from high to low, respectively. When making predictions such as this, however, always remember that actual results may differ due to variations in the blending, the process, the sampling, and the testing. In addition, the model itself may be off somewhat, because it's based on sample data. For example, for the outcome of 14.5, the statistical prediction interval at 95% confidence is 10.67 to 18.33. In other words, when doing confirmation runs, don't be surprised to see individual outcomes somewhat above or below the predictions.

Looking at the wide intervals of prediction may lead to further experimentation aimed at "robust design" that will stabilize the process performance. The DOE arsenal includes a tool called "propagation of error" (or POE) that can be applied to reduce error transmitted from poorly controlled factors. This and other aspects of robust design, a relatively new arena for DOE, may become the subject of a future article for this series.

More-complicated crossed designs

What we've illustrated is a very simple mixture/process design that involves only three mixture components and two process factors. Additional mixture or process variables probably would push the number of combinations beyond the practical limits of material and time. For example, going to a third process factor on a three-component mixture pushes the total runs to 48. Other complications may arise, such as:

* constraints on individual components in the mixture; and

* added categorical factors such as who supplies what type of material.

Good DOE software can set up "optimal" designs that minimize the number of experiments regardless of constraints. The last reference provides details on how this can be accomplished (5).

A potent combination

The case study on the automotive vinyl shows how you can apply advanced tools of DOE to simultaneously optimize your mixture formulation and processing conditions, taking advantage of complex interactions in the system. Response surface graphics, which can be produced with statistical software, make it easy to find the peak performance. If you must juggle many responses to keep your product in specification, numerical optimization approaches are available to manipulate the predictive models to find the "sweet spot" for both mixture and process variables.