## 2 Optimization Using Response Surface Methodology
### Box-Behnken Design (BBD)  
The Box-Behnken matrix for three factors is a spherical and rotational design that, when visualized in a cube, consists of the central point and the midpoints of the edges [12; 26]. The Box-Behnken design (BBD) is an efficient response surface methodology tool that optimizes experimental variables with a minimal number of runs while maintaining high symmetry  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline  & & \multicolumn{6}{c}{**Central Composite Design (CCD)**} \\ \hline
**Objective** & **Raw** & **Technique** & **Independent** & **Levels** & **Optimal** & **R2** & **Desirability** & **References** \\
**Function** & **Materials** & & & & & & & \\ \hline Maximizing & & Microwave- & Power & 500–700 W & 600 W & & & \\ extract yield and & & assisted & Temperature & 30–50 \({}^{\circ}\)C & 40 \({}^{\circ}\)C & 0.9923 & - & [18] \\ content & & & Extraction time & 20–40 min & 30 min & & & \\ \hline Minimize aw, & & & & & & & & \\ maximize pH, & & & & & & & & \\ density and flow & & & & & & & & \\ index (n) and & & & & & & & & \\ keep S stable & & & & & & & & \\ \hline Minimize & & & & & & & & \\ particle size, & & & & & & & & \\ maximize & & & & & & & & \\ loading capacity & & & & & & & & \\ and & & & & & & & & \\ encapsulation, & & & & & & & & \\ and keep the zeta & & & & & & & & \\ potential in the & & & & & & & & \\ range & & & & & & & & \\ \hline Maximizing & & & & & & & & \\ extract of & & & & & & & & \\ betacyanin, & & & & & & & & \\ betaanthin and & & & & & & & & \\ total phenolic & & & & & & & & \\ content & & & & & & & \\ \hline Minimize acid & & & & & & & & \\ ascorbic content & & & & & & & & \\ and a* & & & & & & & & \\ degradation, & & & & & & & & \\ maximum b*, & & & & & & & & \\ and rehydration & & & & & & & & \\ capacity & & & & & & & \\ \hline Minimize drying & & & & & & & & \\ time, gumminess & & & & & & & & \\ and maximizing & & & & & & & & \\ ascorbic acid, L* & & & & & & & & \\ value, chewiness & & & & & & & \\ \hline Minimize & & & & & & & & \\ mineral and & & & & & & & & \\ carbohydrate & & & & & & & & \\ requirement, & & & & & & & \\ maximizing & & & & & & & & \\ protein and lipid & & & & & & & \\ requirement & & & & & & & \\ \hline Maximizing & & & & & & & & \\ overall & & & & & & & & \\ acceptability & & & & & & & & \\ \hline \hline \end{tabular}
\end{table}
Table 1: Applications of optimization using response surface methodology with central composite designs.  
and rotatability [27]. The number of experiments required to construct the model is defined by Equation (6):  
\[\mathrm{N}=2\mathrm{k}*(\mathrm{k}-1)+\mathrm{Co} \tag{6}\]  
It organizes factorial points at the midpoints of the cube's edges, along with replicated runs at the central point, significantly reducing the required runs compared to designs such as the central composite design (CCD). Although it allows the inclusion of both numerical and categorical factors, the use of categorical factors increases the number of runs. Its specialized structure and high resolution make it suitable for quadratic models, maximizing experimental information at a lower operational cost. In the food and pharmaceutical industries, BBD has proven valuable for the development and optimization of products and processes, standing out for its flexibility, precision, and ability to evaluate experimental effects and global errors in complex designs [27,28].  
Table 2 provides an overview of various studies employing RSM, specifically using Box-Behnken designs.