We explored the effect of stirring, pre-heat time, and brand of popcorn on the key characteristics of the popcorn in a 23 factorial with center points (click here for Part I).

From that first Design of Experiment (DOE), we came up with some interesting conclusions and, as to be expected from any good experiment, many more intriguing questions.

In this follow up, we’ll try to answer some of those questions as we augment the first DOE to learn more about making the best popcorn possible to ensure “movie night” will be thoroughly enjoyed for years to come.

In the first DOE, we achieved the best taste, texture, and volume of popcorn with constant stirring (a level of 1.0) and no pre-heating (a similar conclusion can be drawn from the graph of this design in Figure 1). The last factor, brand of popcorn wasn’t statistically significant.

A key finding in the first DOE was the detection of significant curvature which we made possible by running center points. The curvature hinted that we may achieve better taste by only stirring a fraction of the time. This question begged more study: Who wouldn’t want better tasting popcorn while doing less work? To answer this question, we augmented the original factorial design to fit the curvature by adding points to fit a quadratic model.

Augmenting the Design
Augmenting the design in Design-Expert® software was easy. From the Design Tools menu, we chose Augment Design. The default augment was an RSM Optimal design, which is a response surface method (RSM). RSM designs find the optimum process settings by fitting higher order models (in this case quadratic) and looking for peaks or valleys. Normally, one would augment from a factorial design to a standard central composite design (CCD), but in this case we had a categoric factor (popcorn type) which precluded that option.

Therefore, using the computer-generated optimal algorithm (via RSM optimal design) was the only way to go. The default augment for a quadratic model was four model points. That means, by adding just four more points to the existing data, we should be able to resolve which quadratic terms will properly model the curvature through the center points. However, after evaluating our design using the Fraction of Design Space* (FDS) plot, we chose to add 6 points to give us higher precision. Evaluation is always an important step after building a design, especially with an augment like this.

The Results
With the new data in hand, it was apparent that the significant curvature detected in the first DOE was due to a significant A2 term in the model for both taste and unpopped kernels (UPKs). In Figure 1, you can see the curvature in the new model, which wasn’t present after the first DOE (see Figure 1, Part 1 of the series). This is a key difference between the factorial and RSM designs.

In the RSM design, we are fitting curves, so a maximum can be found in the middle of the multilevel space. This isn’t possible with the two-level factorial design, which fits straight lines. For instance, look at the curve in Figure 2 for high preheat (B+). It shows that a level of about 0.6 stirring will give the best taste, with taste degrading with more or less stirring. Also, notice that the LSD bars (I-beams) are now shorter.

The shorter bars indicate higher confidence, which is a consequence of having more runs in the DOE. In the first DOE, we had said there was not a statistical difference between B- and B+ at constant stirring, because the LSD bars overlapped. However, now that we have more data and smaller LSD bars, it’s clear that there is a statistically significant difference. No pre-heat (B-) is the better choice. From the no pre-heat (B-) curve in the figure, it looks like we will need constant stirring (1.0) to get the best taste.

Optimization usually comes down to a series of trade-offs. In our case, we started by looking at the model graphs and then setting goals on each individual response in numerical optimization. This analysis showed that a stirring fraction of 0.76 would minimize UPKs, but a better taste, texture, and volume of popcorn were still found at constant stirring (1.0 fraction).

Using the prediction node in Design-Expert, we compared the predictions for these two possible solutions. The comparison showed that the solution with shorter stirring would decrease the number of UPKs by only two kernels, but it degraded the taste and texture by almost 1 full unit on the 1 to 5 rating scale. Therefore, when we used the numerical optimization to simultaneously search for the optimum with goals set for all four responses, we found that the solution of constant stirring was best (with no pre-heat). In other words, to get the best taste, we must accept a few more UPKs as a trade-off.

Since it was confirmed that the brand of popcorn doesn’t matter, we’ll go with the cheap popcorn to save some money. And now, for the crown jewel of RSM designs…look back at the 3D surface plot in Figure 1. This curvy graph clearly supports the solution we found in optimization namely, the best tasting popcorn is achieved with a high level of stirring and no pre-heat.

Augmenting to a response surface design allowed us to model the curvature and get a better picture of what was going on in the center of the design space. We found that the A2 term was needed to fit the center points. In other words, the stirring factor was causing the curvature, not the pre-heat.

We also found that we could minimize UPKs by stirring about three-fourths of the time, but this would degrade the taste and texture too much to be worth it. With RSM, we were able to better characterize the system and increase our process knowledge. In this case, we confirmed our original conclusion. Constant stirring and no pre-heat seems to be the best way to go.

After these experiments, I’m sure my wife and I will never look upon our Whirley Pop popper quite the same way again. More importantly, we can be confident that we’re getting the most out of our favorite night of the week, “movie night.”

For Part I of Whirley Pop DOE, click here

Brooks Henderson is DOE Consultant at Stat-Ease.
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