# The Six Sigma Guide to Robust Design

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## The Taguchi Approach to Robust Design

Genichi Taguchi deserves credit for emphasizing the importance of considering variation in performance as well as average performance. His approach to robust design involves experiments that include both controllable inputs and noise variables. The noise variables, which are expected to affect variation in performance, are actually controlled in Taguchi's experiments, which may be possible in a laboratory setting.

A Taguchi robust design consists of an inner array involving only controllable inputs and an outer array involving only noise variables. All of the runs in the outer array are conducted at each combination of settings of the controllable inputs in the inner array, thereby creating an overall crossed array design that potentially can involve a large number of runs. A simple example will illustrate this arrangement.

### The Inner and Outer Arrays

Consider a situation in which five controllable inputs, X1, X2, X3, X4, and X5, and two noise variables, Z1 and Z2, are to be investigated. The inner array, involving only X1, X2, X3, X4, and X5, could be what Taguchi calls an L8 array, shown in Table 31-1 (Taguchi uses 1 and 2, instead of -1 and 1, to denote the low and high levels of each factor). This array is a 252 fractional factorial design in which each main effect is aliased with one or more 2-factor interaction effects.

#### Table 31-1. Inner Array

 X1 X2 X3 X4 X5 -1 -1 -1 -1 -1 -1 -1 -1 1 1 -1 1 1 -1 -1 -1 1 1 1 1 1 -1 1 -1 1 1 -1 1 1 -1 1 1 -1 -1 1 1 1 -1 1 -1

The outer array, involving only Z1 and Z2, could be a 2-level full factorial design, as shown in Table 31-2.

#### Table 31-2. Outer Array

 Z1 Z2 -1 -1 -1 1 1 -1 1 1

The complete design is shown in Table 31-3.

#### Table 31-3. Combined Inner and Outer Arrays

 X1 X2 X3 X4 X5 Z1 Z2 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 1 -1 -1 -1 -1 -1 1 -1 -1 -1 -1 -1 -1 1 1 -1 -1 -1 1 1 -1 -1 -1 -1 -1 1 1 -1 1 -1 -1 -1 1 1 1 -1 -1 -1 -1 1 1 1 1 -1 1 1 -1 -1 -1 -1 -1 1 1 -1 -1 -1 1 -1 1 1 -1 -1 1 -1 -1 1 1 -1 -1 1 1 -1 1 1 1 1 -1 -1 -1 1 1 1 1 -1 1 -1 1 1 1 1 1 -1 -1 1 1 1 1 1 1 1 -1 1 -1 1 -1 -1 1 -1 1 -1 1 -1 1 1 -1 1 -1 1 1 -1 1 -1 1 -1 1 1 1 1 -1 1 1 -1 -1 -1 1 -1 1 1 -1 -1 1 1 -1 1 1 -1 1 -1 1 -1 1 1 -1 1 1 1 1 -1 -1 1 -1 -1 1 1 -1 -1 1 -1 1 1 1 -1 -1 1 1 -1 1 1 -1 -1 1 1 1 1 1 -1 1 -1 -1 -1 1 1 -1 1 -1 -1 1 1 1 -1 1 -1 1 -1 1 1 -1 1 -1 1 1

The mean and the standard deviation of performance at each combination of settings for the five controllable inputs can be calculated from the four runs at that combination of settings. It is then possible to identify controllable inputs that impact:

• Average performance but not variation in performance
• Variation in performance but not average performance
• Both average performance and variation in performance
• Neither average performance nor variation in performance.

Equations can be developed that relate:

• Average performance to controllable inputs
• Variation in performance to controllable inputs.

To draw valid inferences about the significance of individual terms in the latter equation, the logarithm (base e or base 10) of the standard deviation should be used as the response variable instead of the standard deviation itself.

### Some Limitations

There are some limitations in using this approach to robust design. One obvious challenge is the large number of runs required for even modest numbers of controllable inputs and noise variables. Another is the aliasing among individual and joint effects of the controllable inputs. And finally, if the experiment is carried out by conducting all of the runs at each combination of settings of the controllable inputs before changing to another combination of settings of the controllable inputs, this design becomes a split plot design and the analysis of results must be carried out in an appropriate manner. An alternative and more efficient approach to robust design, using response surface designs that include both controllable inputs and noise variables as factors, is described later in this chapter.

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