Design of Experiments – Inner and Outer Array Analysis

October 22th, 2012

Tags: Design of Experiments, Experiment Design, Statistical Experiment

Dr. Taguchi's methods of Design of Experiments provide a means for minimizing the effect of factors that can't be controlled, by controlling the factors that are controllable.  Thus, the process or product is made robust in the face of uncontrollable factors. Dr. Taguchi calls these uncontrollable variables noise factors.  A noise factor causes definite variation, but can't be removed from a product’s design, or a manufacturing process.

To determine how to achieve process control and minimization of the noise factor impact a technique called Inner and Outer Array Analysis is employed.  The array that represents the experiment design is referred to as the inner array.  The array that represents the noise factor conditions is placed on the outside of the experiment is referred to as an outer array. 

To illustrate the concept of the inner and outer array two L4(23)  orthogonal array tables will be used.  The inner array represents three factors, A, B, and AxC, which the experiment is studying.  The outer array represents noise factors, A' and B', that normally are not controlled.  These noise factors could be raw materials that have been sorted into two different grades, where the grades are the levels in the outer array.  It is less costly to a process if raw material that falls anywhere within its distribution is useable versus having to sort it at receiving.  Although to conduct the experiment would require this sorting.  Environmental factors such as humidity, or factory temperature could also be used as noise factors.  The point is to identify the control factors in the inner array that will make the process robust against uncontrollable, or noise factors in the outer array.

Conducting the experiment with the noise factors requires more experimentation and potentially some logistics difficulty, but the rewards are great if control factors are identified in the inner array.  The control factors allow the process to be run effectively regardless of the level of the noise factors.  The first experiment run would consist of factors A and B, at level 1.  This run would be repeated four times with the noise factors A' and B', at the levels laid out in the outer array.  The outer array has 4 runs which require the noise factors to be set at different levels.  The complete inner array experiment must be run 4 times, as dictated by the outer array.

When the raw analysis is conducted the noise factors are used to account for a portion of the sum of squares in the analysis.  This information about the noise factors is kept separate from the error until it is clear whether, or not, it can be pooled.  If a noise factor accounts for a large amount of the variation in the experiment it would not be proper to lump it into the error term automatically and consider it as unexplained variation.  It is explained by the level changes of the noise factor.  It is hoped that the analysis will point out control factors that are robust against the noise factors.

In the signal-to-noise (s/n) analysis the noise factors are not separated out.  The s/n calculation for each experiment run captures the data of all the outer array runs.  The s/n analysis allows the experimenter to identify the factors from the inner array that affect both the centering and the variability of the outcome, regardless of the induced noise factor variation.  This type of analysis is one of the keys to robust process and product design.

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