## 2 Materials and Methods
### Design of Experiment  
The Taguchi method provides a simple, efficient, and systematic approach to determine the optimum parameters. In addition, the Taguchi method provides a more simplified way to setup the combination of experiment parameters compared to the factorial method, instead of testing all possible combinations of parameters available. The parameters tested in this experiment are milling time, size of stainless-steel ball, and milling speed. These parameters were used to produce homogenous and high purity nanoparticles of less than 100 nm consistently. Table 1 shows the parameters selected and the levels for each parameter. In the preparation of palm oil ash nanoparticles by the ball milling technique, the Taguchi proposed to use the orthogonal array to screen the experimental conditions and means. Furthermore, by using this method, the least of the experimental process could be developed based on the number of factors and levels. In addition, the advantage of this method was to achieve the best combination of design parameters and optimal results. Corresponding to the main factors namely milling time, milling speed, size of balls, and each has three levels shown in Table 1, an orthogonal array L9 of Taguchi design was established for numerical experiments. Due to the orthogonal array, the analysis and calculation time reduce significantly, hence becomes more efficient. A combination of structured parameters in nine experiments each with three replications with the obtained results of the L9 array is shown in Table 2.  
In the Taguchi method, to determine the effect of each parameter on the production of nanoparticle of palm oil ash (POA), the signal to noise ratio based on mean value needs to be calculated for each of the parameter combinations, respectively. The S/N ratio was used to measure the quality characteristic deviating from the desired value. The value of S/N from each experiment was used to determine the optimum parameters and the most influential parameters to the results. The term'signal' represents the desirable value (mean) for the output characteristic and the term 'noise' represents the undesirable value (SD) for the output characteristic [20]. The S/N ratio formula for the static design was divided into three categories, viz. 'nominal is the best', 'larger the better', and'smaller the better'. In this experimental design, the'smaller the better' formula was chosen to analyze the production of POA  
\begin{table}
\begin{tabular}{c c c c} \hline
**Parameter** & **Level 1** & **Level 2** & **Level 3** \\ \hline Milling Time & 6 h & 12 h & 24 h \\ Milling speed & 100 rpm & 130 rpm & 170 rpm \\ Size of stainless-steel ball & 10 mm & 16 mm & 20 mm \\ \hline \end{tabular}
\end{table}
Table 1: Parameters selected and levels for each parameter.  
\begin{table}
\begin{tabular}{c c c c c} \hline
**Experiment** & **Milling Time** & **Milling Speed** & **Size of** & **Mean Particle Size** \\
**(hour)** & **(rpm)** & **Balls (mm)** & **(\(\mu\)m)** \\ \hline
1 & 6 h & 100 rpm & 10 mm & 0.48 \(\pm\) 0.06 \\
2 & 6 h & 130 rpm & 16 mm & 0.41 \(\pm\) 0.07 \\
3 & 6 h & 170 rpm & 20 mm & 0.23 \(\pm\) 0.06 \\
4 & 12 h & 100 rpm & 16 mm & 0.28 \(\pm\) 0.09 \\
5 & 12 h & 130 rpm & 20 mm & 0.19 \(\pm\) 0.04 \\
6 & 12 h & 170 rpm & 10 mm & 0.15 \(\pm\) 0.08 \\
7 & 24 h & 100 rpm & 20 mm & 0.11 \(\pm\) 0.07 \\
8 & 24 h & 130 rpm & 10 mm & 0.13 \(\pm\) 0.04 \\
9 & 24 h & 170 rpm & 16 mm & 0.09 \(\pm\) 0.07 \\ \hline \end{tabular}
\end{table}
Table 2: Experimental results for the particle size (mean \(\pm\) SD, replicated three times).  
nanoparticle. The S/N ratio was calculated for each factor level combination. The formula (Equation (1)) for the '_smaller-is-better_' S/N ratio can be expressed as:  
\[Signal-to-noise\left(S/N\right)\ =-10\log\frac{1}{n}+\sum_{i=1}^{n}Y_{i}^{2} \tag{1}\]  
where  
\(Y\) is a response to the given factor level combination  
\(n\) is a number of responses in the factor level combination.