## 2 Optimization Using Response Surface Methodology  
Before the 1950s, the methods employed for process optimization analyzed only one independent variable at a time, disregarding the influence of other variables. This approach resulted in imprecise data and an elevated number of experiments [6]. The response surface methodology (RSM), developed by [PERSON] and his colleagues in the 1950s, introduced a significant advancement by requiring fewer experiments to evaluate multiple parameters and their interactions [7; 8]. The term Response Surface Methodology originated from the graphical representation obtained after fitting the mathematical model. This tool was initially developed as a modeling technique widely applied in industries such as chemical and pharmaceutical sectors [9], as well as in the food industry and fields like biological and medical sciences [10]. The versatility of RSM lies in its ability to analyze multiple independent variables, including their interactions, allowing for the identification of additive, synergistic, or antagonistic effects on one or more responses. This makes it particularly useful for making predictions, optimizing processes, and enhancing their interpretation [11]. Additionally, its implementation requires a reduced number of experiments without compromising result accuracy [12]. RSM is composed of a set of mathematical and statistical techniques that can be used to define the relationships between the response variable and the independent variables. It  
are determined to be statistically insignificant, those coefficients are excluded, resulting in a linear equation [16]. When selecting the model, it must undergo statistical validation to ensure it accurately reflects the relationship between the variables. This validation is achieved through an analysis of variance (ANOVA), which assesses the model's predictive accuracy by examining the coefficient of determination (R\({}^{2}\)). In addition, other methods, such as the lack-of-fit test, mean absolute deviation and residual analysis are utilized [17]. The visualization of the predicted model equation can be illustrated through contour and response surface plots. These three-dimensional representations, generated by response surface methodology (RSM), highlight the relationship between dependent and independent variables. Both contour and surface plots demonstrate how the response varies as independent variables change. Contour plots enhance our understanding of the shape of the response surface. When the target point--whether minimum, maximum, or any point within the operational range--resides at the center of the system (experimental area), the contour plots typically depict a circular or elliptical shape [1,16]. In the food industry, it is common to optimize multiple responses simultaneously, a task that is inherently more complex than optimizing processes focused on a single response. To tackle this challenge, a desirability function is employed, serving as a multi-criteria methodology. This approach assigns a desirability score to each individual response, operating under the premise that the overall \"quality\" of a product or process is deemed unacceptable if any one of the quality characteristics falls outside the \"desired\" range. The method identifies the operating conditions (X) that yield the \"most desirable\" response values. For each response Yi(x), a desirability function di(Yi) assigns values between 0 and 1 [1,14]:  
\[\mathrm{D}=\left((\mathrm{d}_{1}*\mathrm{Y}_{1})(\mathrm{d}_{2}*\mathrm{Y}_{2 })\ldots(\mathrm{d}_{n}*\mathrm{Y}_{n})\right)^{\frac{1}{h}} \tag{4}\]  
where di(Yi) = 0 represents a completely undesirable value, and di(Yi) = 1 indicates an ideal or completely desirable value. Individual desirabilities are then combined using the geometric mean, resulting in overall desirability (D) (Equation (4)). To confirm the validity of the generated equation, the difference between predicted and experimental responses should be less than 5% [1].