1. Robust Optimization: Some Tools Based in JMP® to Enhance Traditional Taguchi Methods
2015 JMP Discovery Summit, San Diego
Author: Steve Olsen, Sr. Principal Reliability Engineer, Ogden Tech Center

2. Why robust optimization?
The philosophy behind Taguchi DOE’s is very powerful: There are some factors which are impossible, difficult or expensive to control. Taguchi calls these ‘noise factors’ and places them in an ‘outer array’ which create the replicates in each row of the control factor DOE (inner array). The signal-to-noise (S/N) ratio analysis method is used to find the optimum spot in the design where performance is minimally sensitive to variation of the noise factors.
SN and Mean calculated from spread in results across noise factors
Inner array
Outer array

3. What are some limitations of the traditional Taguchi approach?
Taguchi developed a menu of orthogonal arrays for the inner array. They are designed to break up interaction confounding patterns and are efficient screening designs. However:
The Taguchi arrays do not allow for estimating interactions in control factors.*
The control factors have to be fitted to one of the Taguchi ‘cookbook’ arrays, which often results in empty columns and wasted resources
Traditional Taguchi examples in the literature don’t lend themselves to more complex outer arrays for the noise factors
The traditional graphical S/N method often requires manual Excel-based evaluation
The S/N ratio graphical analysis method lacks a statistically robust method for separating signal from noise in the experiment (no ‘p-values’ are created)
Can JMP help overcome some of these limitations?
*See Wu, Wu, “Taguchi Methods for Robust Design” chapter 6 for Taguchi philosophy on interactions between control factors.

4. Alternative to ‘cookbook’ Taguchi orthogonal arrays
Suppose we want to run the following experiment (all factor levels are categorical):
1 control factor with 2 levels and 7 control factors with 3 levels, 8 control factors total.
This would fully utilize a Taguchi L18{21 x 37} orthogonal array, one of the more popular and useful of the Taguchi menu arrays.
What would happen if we created this matrix using the JMP Custom Design feature?
Answer: JMP default creates essentially the same matrix as the L18, (18 runs, and the ‘Color Map On Correlations” shows confounding patterns are fragmented)
We don’t need to use the ‘cookbook’ arrays; we can use the Custom Design feature to create efficient inner arrays (and also the outer array if necessary)
Creating the inner array with the Custom Design feature also allows for investigation of interactions between control factors, if that is desired

5. Using the JMP Fit Model tool to create the S/N and mean analysis graphs
Assume we create a Robust Optimization DOE by crossing a custom inner array with a custom outer array. How do we get the traditional Taguchi S/N ratio and mean analysis graphs without a manual effort in Excel?
Assume the complete DOE array is in an Excel worksheet, and the mean and S/N ratio columns have been calculated.
The S/N ratio for each row in the matrix is calculated with the following formula:
Demonstrate example of S/N analysis in JMP

6. An alternate way to look at data from a Taguchi experiment
To consider a method to gain some additional insight, we start by restating the goal of robust optimization in traditional DOE language:
In Taguchi language: We want to find settings of the control factors that minimize the effect of the noise factors on the output.
In traditional DOE language: We want to discover where there are interactions between the control factors and noise factors, and set the control factors to minimize the effect of those interactions on the output.
Given the way we have re-stated the problem, how might JMP assist in enhancing the Taguchi approach? The following example will demonstrate.
Start by creating a new data table in JMP containing the same data as in the above example, but transpose the outer array data into a single column and create replicates for each inner array row to match up with the transposed data

7. What about the JMP Profiler noise factor function?
Experienced JMP users will recall that there is an algorithm within the JMP ‘Profiler’ method in the ‘Graph’ menu that can automatically find the flattest response for noise factors. However, this only works if your noise factors are numeric continuous. The Autoliv study discussed below required categorical noise factors.

8. Real world example
Pyrotechnic airbag inflators utilize proprietary pyrotechnic chemicals to create rapid gas output for deploying automobile airbags. Repeatable performance (low variation) is essential to provide robust occupant restraint. Airbag inflator output is measured by deploying inside a sealed tank and recording the tank pressure output vs. time. Metrics of interest are:
Onset (when the gas starts flowing after the electric signal is given)
Rate of gas output during the deployment (referred to as slope)
Maximum pressure when the reaction is complete
Inflator function is dependent on the chemical reactions of the various pyrotechnic materials, and also on the physical properties of mechanical components.

9. Noise strategy
An important step in Robust Optimization is determining the noise strategy.
In the case of optimization of an airbag inflator, Autoliv was interested in understanding how they could reduce the sensitivity of inflator output to lot-to-lot variation of certain influential components and pyrotechnic materials.
We had a choice whether to use one of two different technologies for one of the components.
Which choice for the component in question would yield the most robust performance (least sensitive to lot-to-lot variation)?

10. Noise strategy (cont)
Four components were chosen to be included as noise factors in the outer array. These selections were plausible suspects, but it is important to note that we had to be exploratory; we didn’t know for sure what the effect of lot-to-lot variation would be. One purpose of the DOE was to gain more specific understanding of these noise factors.
The team decided to pull samples from the production line once per week for six weeks, for each of the four selected components. This yielded an outer array of 4 noise factors, six levels per factor (46).
Given the complexity of the noise strategy, this experiment would not even be feasible without using a relatively sparse array. The JMP Custom Design tool makes this a trivial problem. We chose an outer array with 92 runs for this experiment. (It could have been done with as few as 21.)
Note this outer array is significantly more complex than most examples of Taguchi DOE’s in the literature.

11. Control factor (inner array)
For the control factors, the inner array was very simple: There was one factor with two levels. (Taguchi experiments with only one control factor are called “Robust Assessment”, but the evaluation method is identical.)
Goal of the experiment:
Which of two technologies for Component B would yield an inflator that was least sensitive to lot-to-lot variation of the four influential components in the outer array.
92 inflators were built and tested for each technology of Component B, or 184 total.

12. Taguch S/N analysis
For onset and slope (most interested in optimizing), Technology 2 for Component B is clearly better in reducing sensitivity to lot-to-lot variation of the factors with a dB gain of greater than 2.
Does this analysis yield any kind of detailed information about the four noise factors?
Demonstrate alternate Profiler method in JMP

13. Results
(Component B Level 1)
Component A has an effect on onset, but that effect is dampened when Component B uses Technology 2 (note curve is the same shape but flatter)
Component B Level 1 lot-to-lot variation has an effect on slope, but Component B Level 2 does not
(Component B Level 2)
The Profiler analysis agrees with the Taguchi S/N analysis (Technology 2 of Comp B is better for onset and slope), but with significantly more insight about the what’s going on with the noise factors.

14. Summary
JMP can contribute several enhancements to the traditional Taguchi Robust Optimization methods:
Use of the Custom Design tool for the inner and outer arrays allows more flexibility in choice of factors and less waste of resources, especially when the noise strategy is more complex
JMP provides a method to automate the traditional Taguchi graphing analysis
JMP provides statistical output to help separate signals from common cause variation
Especially in the case of more complex outer arrays, the Profiler analysis method yields more insight for the noise factors compared to the traditional S/N ratio method (see also Case Study 2 in paper)
Note to JMP developers: Added functionality to optimize categorical noise factors (in addition to numeric continuous) in the ‘Graph/Profiler’ would be a nice addition!