## 2 Experimental Setup
### ANOVA, RSM, and Multiple Response Optimization  
ANOVA analysis was employed in this research to evaluate the design parameters and to identify which factors significantly affected the output response. ANOVA enumerates a list of the regression coefficients, namely, the interactive effects, and the \(p\)-value to determine the statistically significant effects. If the \(p\)-value is less than 0.05, that indicates a 95% confidence level of the model's significance.  
RSM, a response surface methodology, is used to predict the COF and wear rate that are present. RSM [25] is a compilation of statistical and mathematical techniques that are employed to create the model and to optimize the independent variables of a physical system. The outputs (responses) and the input factors have their relationship as an empirical model function, noted as:  
\[f=y(A,B,C)+\varepsilon \tag{3}\]  
where \(f\) represents the dependent outputs (i.e., COF, wear rate); \(y\) is the function of response; \(\varepsilon\) represents the statistical error; and, \(A,B,\) and \(C\) are the parameters of aluminum nanoparticles (wt %), SDS (wt %), and the water content of glycerol (wt %), respectively.  
Figure 2: (**a**) Thrust collar friction and wear test platform; (**b**) Schematic drawing of the working principle for the tribotest used to evaluate the friction force and wear.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Factors**} & \multirow{2}{*}{**Parameters**} & \multicolumn{6}{c}{**Coded Variables Level**} \\ \cline{3-6}  & & **1** & **2** & **3** & **4** & **5** \\ \hline A & Concentration of nanoparticles (wt \%) & 0 & 0.25 & 0.5 & 0.75 & 1 \\ B & Concentration of SDS (wt \%) & 0 & 0.5 & 1 & 1.5 & 2 \\ C & Water content of glycerol (wt \%) & 10 & 20 & 30 & 40 & 50 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Control factors and their levels for experiments. SDS, sodium dodecyl sulfate.  
Second-degree polynomial formulae are used to describe the effect of one output in dependence of the variables input, noted as:  
\[f=a_{0}+\sum_{j=1}^{k}a_{j}x_{j}+\sum_{j=1}^{k}a_{jj}x_{j}^{2}+\sum_{i<j=2}^{k} a_{ij}x_{i}x_{j}+\epsilon, \tag{4}\]  
where \(x_{j}\) is the variables factor; \(a_{0},a_{j},a_{jj}\), and \(a_{ij}\) are constants; and \(\epsilon\) is the random error of observation response.  
In this research, the optimization of multiple responses (i.e., both COF and wear rate) are the desirability functions. Each response of \(y_{j}\) (target or object) is converted into an individual desirability function \(d_{j}\) in the range:  
\[0\leq d_{j}\leq 1 \tag{5}\]  
The overall desirability functions are associated in the form:  
\[D=\left(d_{1}d_{2}d_{3}\ldots\ldots d_{m}\right)^{\frac{1}{m}}, \tag{6}\]  
where \(D,\ m\) are desirability and the number of responses, respectively.  
If the target (or objective) for response \(y\) reaches the minimum value:  
\[d=\left\{\begin{array}{cc}1&y<T\\ \left(\frac{U-y}{U-T}\right)^{r}&T\leq y\leq U\\ 0&y>U\end{array}\right. \tag{7}\]  
where \(T,\ U,\) and \(r\) are the target, upper boundary, and weight of the desirability function, respectively. The \(r\) value depends on the target, \(r>1\) means close to the target, \(0<r<1\) signifies the less important target.