# Lesson 12: Robust Parameter Designs

Lesson 12: Robust Parameter Designs## Overview

In what we have discussed so far in the context of optimization only the average location of the response variable has been taken into account. However, from another perspective the variation of the response variable could be of major importance as well. This variation could be due to either usual noise of the process or randomness in the nature of one or more controllable factors of the process.

The Robust Parameter Design (RPD) approach initially proposed by Japanese engineer, Genichi Taguchi, seeks a combination of controllable factors such that two main objectives are achieved:

- The mean or average location of the response is at the desired level, and
- The variation or dispersion of the response is as small as possible.

Taguchi proposed that only some of the variables cause the variability of the process, which he named noise variables or uncontrollable variables. Please note that noise variables may be controllable in the laboratory, while in general they are a noise factor, and uncontrollable. An important contribution of RPD efforts is to identify both the controllable variables and the noise variables and find settings for the controllable variable such that the variation of response due to noise factors is minimized.

The general ideas of Taguchi widely spread throughout the world; however, his philosophy and methodology to handle RPD problems caused lots of controversy among statisticians. With the emergence of Response Surface Methodology (RSM), many efficient approaches were proposed which could nicely handle RPD problems. In what follows, RSM approaches for Robust Parameter Design will be discussed.

## Objectives

- Understanding the general idea of Robust Parameter Design approaches
- Getting familiar with Taguchi’s crossed array design and its relative weaknesses
- Understanding combined array design and response model approach to RPD

# 12.1 - Crossed Array Design

12.1 - Crossed Array DesignCrossed array design was originally propose by Taguchi. These designs consist of an **inner array** and an **outer array**. The inner array consists of the controllable factors while the outer array consists of the noise factors. The main feature of this design is that these two arrays are “crossed”; that is, every treatment combination in the inner array is run in combination with every treatment combination in the outer array. Table 12.2 is an example of crossed array design, where the inner array consists of four controllable factors and outer array consists of three noise factors. Note the typo in the levels of the 6th column of data. It should be {+,-,+}.

Crossed array designs provide sufficient information about the interaction between controllable factors and noise factors existing in the model which is an integral part of RPD problems. However, it can be seen that crossed array design may result in a large number of runs even for a fairly small number of controllable and noise factors. An alternative for these designs are **combined array designs** which is discussed in the next section.

The dominant method used to analyze crossed array designs is to model the mean and variance of the response variable separately, where the sample mean and variance can be calculated for each treatment combination in the inner array across all combinations of outer array factors. Consequently, these two new response variables can be considered as a dual response problem where the response variance needs to be minimized while response mean could be maximized, minimized or set close to a specified target. The text book has an example about the leaf spring experiment in which the resulting dual response problem has been solved by the overlaid contour plots method (See Figure 12.6) for multiple response problems, discussed in section 11.3.4.

# 12.2 - Combined Array Design

12.2 - Combined Array DesignThe combined array design approach treats all the variables the same, no matter they are controllable or noise. These models are capable of modeling the main effects of controllable and noise factors and also their interactions. To illustrate, consider a case with two controllable and one noise factor. Equation 12.1 in the textbook gives a first-order model:

\(y = \beta _{0}+\beta _{1}x _{1}+\beta _{2}x _{2}+\beta _{12}x _{1}x _{2}+\gamma _{1}z_{1}+\delta _{11}x _{1}z_{1}+\delta _{21}x _{2}z_{1}+\varepsilon \)

where the \(\beta_i\) are coefficients of controllable factors, \(\beta_{12}\) is the coefficient of the interaction of controllable factors, \(\gamma_1\) is the coefficient of the noise factor and \(\delta_{ij}\) are the coefficients of interaction between controllable and noise factors. As can be seen, the response model approach puts all of the variables, no matter they are controllable or noise, in a single experimental design. There exist some assumptions which are mentioned as follows:

- \(\epsilon\) is a random variable with mean zero and variance \(\sigma^2\)
- Noise factors are random variables (although controllable in the experiment) with mean zero and variance \(\sigma ^{2}_{z} \)
- If there exist several noise factors their covariance is zero

Under these general assumptions, we will find the mean and variance for the given example, as following:

\(E(y) = \beta _{0}+\beta _{1}x _{1}+\beta _{2}x _{2}+\beta _{12}x _{1}x _{2} \)

and

\(Var(y) = \sigma ^{2}_{z}\left (\gamma _{1}+\delta _{11}x_{1}+\delta _{21}x_{2} \right )^{2}+\sigma ^{2}\)

Notice that although the variance model involves only controllable variables it also considers the interaction regression coefficients between the controllable and noise factors.

Finally, as before, we perform the optimization using any dual response approach like overlaid contours, desirability functions or etc. (Example 12.1 from the textbook is a good example of overlaid contour plots approach).

From the design point of view, using any resolution V (or higher) design for the two-level factor designs is efficient. Because these designs allow any main effect or two-factor interaction to be estimated separately, assuming that three and higher factor interactions are negligible.