## 4 Discussions
The dependence of reactive BRed dye removal (\(Y_{1}\)) vs. \(X_{3}\) (biosorption time) (\(X_{1}\) and \(X_{2}\) coded values are 0) indicates no _distinct maximum_ dye removal degree, thus only an _extreme local maximum_ can be considered (\(Y\) = 47.291%) for \(X_{3}\)*= +2.00 (i.e., _a local maximum_ point after a S/W contact time in biosorption step of 12 h, a dye concentration of 50 mg/L and a biosorbent concentration of 12 g/L (i.e., around 0.3 g per 25 mL) (Figure 5c). The \(X_{3}\) variable (biosorption time) has the most significant influence on the maximum value \(Y_{1}\). * for \(Y_{2}\) function (BRed dye biosorption onto biosorbent 2): the influence of biosorbent 2 concentration (\(X_{1}\)) vs. \(Y_{2}\) is evidenced in Figure 6a. It seems that there exists _a distinct maximum_ of the BRed dye removal (\(Y_{2}\)* = 67.189%) at \(X_{1}\)* = +1.4048 corresponding to a biosorbent 2 concentration of 20.4288 g/L (0.5107 g per 25 mL), for a BRed dye concentration of 50 mg/L and t\({}_{\text{biosorption}}\) of 6 h (\(X_{2}\) and \(X_{3}\) kept constant at their basic values). The dependence of BRed dye removal (\(Y_{2}\)) vs. \(X_{2}\) (dye concentration) (\(X_{1}\) and \(X_{3}\) coded values are 0) indicates a _distinct maximum_ dye removal degree (\(Y\) = 61.852%) at \(X_{2}\)* = +0.4402, corresponding to a biosorbent concentration of 14.6412 g/L (0.366 g per 25 mL), a dye concentration of 50.00 mg/L and t\({}_{\text{sorption}}\) of 6 h (Figure 6b). The dependence of reactive BRed dye removal (\(Y_{2}\)) vs. \(X_{3}\) (biosorption time) (\(X_{1}\) and \(X_{2}\) coded values are 0) indicates a _distinct maximum_ dye removal degree (\(Y\) = 66.207%) for \(X_{3}\)*= +1.1133 (i.e., _a maximum_ point after a S/W contact time in biosorption step of 8.226 h, a dye concentration of 50 mg/L and a biosorbent concentration of 12 g/L (i.e., around 0.30 g per 25 mL) (Figure 6c). Considering two independent variables at their basic values, we can conclude that the optimum values for the dye removal performance (\(Y^{*}\)) by biosorption onto residual micro-encapsulated biosorbent 1) or immobilized yeasts (biosorbent 2) biomass corresponded to a relatively high quantity of biosorbent (more than 14.64-24.00 (g/L)) and thus high dye removal efficiency can be performed working with a relatively high biosorbent concentration (i.e., \(>\)20 g/L, or \(>\)0.50 g per 25 mL of sample), a dye concentration around of 45-50.00 mg/L or more, and a biosorption contact time of at least 8 h (or ca. 12 h). The application of other _direct empirical optimization_ methodologies (i.e., univariant search or gradient methods) [13; 14; 15; 16; 17; 18; 19; 20] for the maximum findings in the case of the proposed mathematical model leads to optimum values close to that found by the application of the classical optimization method for the proposed mathematical model, i.e., \(Y_{1}\)* = 52.878% or \(Y_{2}\)* = 75.338%. The graphical representations (Figures 3-6) allowed us to visualize the 3D and 2D variation of the selected response/decision function and its maximal points for the selected experimental variation field on isolines (2D), and also when some of the independent variables were constantly kept at their basic values (3D and/or 2D). The WinSurf and Excel data processing programs were used.