## 2 Materials and Methods
This table is also known as the design matrix of the experiment if using the \(+\) and \(-\) notation (orthogonal coding). In the table, it is also possible to see two other common notations which are also used in terms of DOE. Similarly, the following Figure 1 shows the geometric view of a general \(2^{3}\) design. \begin{table}
\begin{tabular}{c c c c c c c} Run & A & B & C & Labelling & A & B & C \\ \hline
1 & - & - & - & (1) & 0 & 0 & 0 \\
2 & + & - & - & \(a\) & 1 & 0 & 0 \\
3 & - & + & - & \(b\) & 0 & 1 & 0 \\
4 & + & + & - & _ab_ & 1 & 1 & 0 \\
5 & - & - & + & \(c\) & 0 & 0 & 1 \\
6 & + & - & + & _ac_ & 1 & 0 & 1 \\
7 & - & + & + & _bc_ & 0 & 1 & 1 \\
8 & + & + & + & _abc_ & 1 & 1 & 1 \\ \hline \end{tabular}
\end{table} 
Table 2: Experiment table of \(2^{3}\) design with the most commonly used notations. Figure 1: Geometric view of a general \(2^{3}\) design. Source: [20]. On the other hand, if the experiment is replicated \(n\) times, the observations could also be described by a linear model, as shown in Equation (1). \[ \begin{array}{l}y_{ijkl}=\mu+\tau_{i}+\beta_{j}+\gamma_{k}+(\tau\beta)_{ij}+\\ (\tau\gamma)_{ik}+(\beta\gamma)_{jk}+(\tau\beta\gamma)_{ijk}+\epsilon_{ijkl} \end{array} \left\{ \begin{array}{l}i=1,\ldots a\\ j=1,\ldots b\\ k=1,\ldots c\\ l=1,\ldots n\end{array} \right.\end{array} \tag{1}\]  
where \(y_{ijkl}\) represents the observed variable, for example, the resulting roundness or the final outer diameter of a tube; \(\tau_{i}\) is the effect of the \(i\)-th level of factor \(A\); \(\beta_{j}\) is the effect of the \(j\)-th level of factor \(B\); \(\gamma_{k}\) is the \(k\)-th level effect of factor \(C\); and \((\tau\beta)_{ij}\), \((\tau\gamma)_{jk}\), \((\beta\gamma)_{jk}\) represent the effects of the interaction between factors. Similarly, the letters \(a\), \(b\), and \(c\) refer to the number of levels for each factor, \(r\) denotes the number of replicates, and \(\epsilon_{ijkl}\) is a random error component. Despite the fact that all of the parameters of the model are independent of the error \(\epsilon_{ijkl}\), they are all normally distributed, with a mean and value of variance of zero, expressed by _NID_ (0, \(\sigma_{T}^{2}\)). In consequence, the variance of any observation is given by Equation (2). \[\sigma_{y}^{2}=\sigma_{\tau}^{2}+\sigma_{\beta}^{2}+\sigma_{\gamma}^{2}+\sigma _{\tau\beta}^{2}+\sigma_{\tau\gamma}^{2}+\sigma_{\beta\gamma}^{2}+\sigma_{ \tau\beta\gamma}^{2}+\sigma^{2} \tag{2}\]  
Moreover, if assuming all factors of a given experiment are fixed, as is the case of the case study in this paper, then the ANOVA can be defined as presented in the following Table 3. This table has been taken and modified from [25], who cited it, modified it, and took it from [20] at the same time. For a more detailed take on the calculation of the sum of squares, the readers of this document are kindly advised to refer to the content starting from page 201 in the work of the last of the mentioned authors. Since a key part of a factorial design of this type relies on the initial calculation of the effects and their interactions, and since these are sometimes to be manually calculated in order to possibly corroborate ambiguous computer results, the authors present a set of equations for the calculation of such effects. As it can be seen, the parts of the equations in square brackets refer to the contrasts. The calculation of the effects is also key for the calculation of the sums of squares. \[A=\frac{1}{4n}\left[a+ab+ac+abc-(1)-b-c-bc\right], \tag{3}\]  
\[B=\frac{1}{4n}\left[b+ab+bc+abc-(1)-a-c-ac\right], \tag{4}\]  
\[C=\frac{1}{4n}\left[c+ac+bc+abc-(1)-a-b-ab\right], \tag{5}\]  
\[AB=\frac{abc+ab+c+(1)}{4n}-\frac{bc+b+ac+a}{4n}, \tag{6}\]