Crossed 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.

The 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.