## 5 Optimization Results and Discussion  
The results obtained by solving the NLP optimization problem, and by deriving the Pareto-optimal fronts, are presented in the following forms:  
* The Pareto-optimal fronts for several opposite objectives (Figures 3-5);
* The dynamics of the accounted species predicted by the HSMDM, for all the considered FBR operating alternatives (that is, the NLP-optimal, or the Pareto-optimal), with constant or variable feeding, are comparatively displayed in (Figures 6-10);
* Predictions of the fitted HSMDM constructed by [PERSON] [11] (adopted in this paper) compared to the experimental data of [PERSON] [72] are presented in (Figures 6 and 8-10);
* A comparison of all FBR operating alternatives in terms of TRP production and raw material consumption is given in Table 5, even if the batch time of the [PERSON] et al. [55] experiment is much shorter. It is worth mentioning that the FBR operation in Table 5 is considered in several alternative forms: (i) a time-step-wise variable feeding (Equations (1)-(6), and Figures 6-10), (ii) a constant non-optimal feeding (literature results); or (iii) a constant but optimal feeding (this paper, the NLP-optimal, or the Pareto-optimal) (Figures 6-10). In Table 5, the substrate (GLC) consumption for the FBR case was evaluated with the following relationship:  
\[m_{GLC}\ =\ \sum\
olimits_{j=1}^{N_{div}}c_{glcj}^{feed}F_{L,j}\Delta t_{j} \ \ ;\ \ \ \Delta t_{j}\ =\ t_{f}/N_{div} \tag{16}\]  
By analyzing the resulting FBR optimal operating modes (plots 2-4 in Figures 6-10, and Table 5) compared to those of the sub-optimal (nominal) operation of [PERSON] [72] (plot 1 in the same figures), several observations can be derived, as follows:  
* The HSMDM adopted in this paper fairly fits the experimental data of [PERSON] [72], being adequate enough to derive engineering evaluations of the FBR optimal operation. 2. Even if the FBR is operated under the nominal, non-optimal conditions of (Table 1), the modified _E. coli_ T5 strain of [PERSON] [72] reported a much higher (45\(\times\)) TRP production compared to the wild strain of [PERSON] et al [55]. Such a result is explained by a much higher GLC uptake rate in the modified _E. coli_ T5 strain, where the PTS import system of GLC was replaced by a more effective one based on the galactose permease/glucokinase (GalP/Glk), by modulating the gene expression of GalP/Glk [47; 72]. Figure 4: The Pareto-optimal operating mode of the FBR in terms of required constant feed flow rate, for various amounts of maximum TRP produced. The blue curve is the “rough” Pareto-optimal front. The black curve illustrates the approximated Pareto-optimal front by using the cubic spline procedure of Matlab(tm) (function “CSAPS”), with a smoothing constant of \(10^{-20}\). The marked points correspond to those of the Pareto-optimal curve of Figure 4.  
Figure 3: The Pareto-optimal front for the analyzed FBR of Table 1 in terms of two opposite objectives, namely maximum TRP production vs. minimum substrate (GLC) consumption. The blue curve is the “rough” Pareto-optimal front. The black curve illustrates the approximated Pareto-optimal front by using the cubic spline procedure of Matlab(tm) (function “CSAPS”), with a smoothing constant of \(10^{-20}\). For this problem, detailed in Equation (11), a solution was obtained by imposing the control variable limits given in Equation (12). The end point was chosen as being the most favourable solution of this optimization problem. However, the break point can be chosen as well, according to the suggestions of [PERSON] [94]. 3. The efficiency in the TRP production of the optimally operated FBR with a variable feeding is significantly higher (237 mM) compared to the same FBR, but with a constant NLP optimal feeding (219 mM), even if the same modified _E. coli_ T5 strain was employed in both cases. 4.