## 4 Fed-Batch Bioreactor Optimization Problem  
### Preliminary Considerations  
The experimentally validated FBR HSMDM (Tables 2-4) adopted in this paper achieves a reasonable compromise between its level of detail (the number of accounted intra-cellular species and reaction pathways, included in the lumped key modules [a-c], Figure 2), and its predictive power for the bulk-phase species of interest (that is, X growth, GLC depletion, excreted TRP, and PYR, depicted in Tables 2-4). The adequacy of this model vs. the experimental data (Figures 6, 8, 9 and 10) [11] gives it enough credentials to make predictions of a satisfactory quality within the investigated experimental domain. The optimal FBR operation alternatives derived in this paper are more complex than the simple operation presented by [PERSON] [72], characterized by a constant non-optimal feeding over the FBR batch (Table 1), and the simulated species blue curves in Figures 6-10). In the present paper, other FBR feeding alternatives will be explored. Thus, (i) the batch time is divided in N\({}_{\rm div}\) (equal lengths time-arcs), and (ii) the control variables are kept constant over every time-arc only at the optimal values determined from solving an optimization problem (i.e., the maximization of the TRP production). The time intervals of equal lengths \(\Delta\)t = t\({}_{\rm f}\) /N\({}_{\rm div}\) are obtained by dividing the batch time t\({}_{\rm f}\) into N\({}_{\rm div}\) parts t\({}_{\rm-1}\leq\) t \(\leq\) t\({}_{\rm j}\), where t\({}_{\rm j}\) = j\(\Delta\)t are switching points (where the reactor input is continuous and differentiable). The time intervals for the present case study with an adopted N\({}_{\rm div}\) = 5 are shown in Footnote (a) of (Table 2). Additionally, even if a FBR with a constant feeding is chosen, due to its simple operation, the constant values of the control variables will be determined by applying a NLP optimizer, or a Pareto-optimal front technique. So as not to complicate the engineering calculations, the main assumption in Equations (5) and (6) is the following: On each time-step-wise arc, index j = 1, \(\ldots\), N\({}_{\rm div}\), the control variables \(F_{L,j}\) and \(c^{feed}_{glc,j}\) are kept constant. Of course, the values on each time-arc do not have to be necessarily equal to each other. To summarize, in the FBR case with a variable feeding, according to Equations (1)-(6), for every j = 1, \(\ldots\), N\({}_{\rm div}\) time-'arcs', the following control variables are to be determined:  
* The time-step-wise continuously added liquid feed flow rate F\({}_{\rm L,j}\) (\(j\) = 0, \(\ldots\), N\({}_{\rm div}\)\(-\) 1); F\({}_{\rm L,0}\) = F\({}_{\rm L}\) (t = 0) of the substrate solution;
* The inlet concentrations [GLC]\({}_{\rm inlet,j}\) (\(j\) = 0, \(\ldots\), N\({}_{\rm div}\)\(-\) 1) for each of the added solutions of F\({}_{\rm L,j}\). Usually, the fed F\({}_{\rm L,j}\) present different concentrations, to be determined together with the initial concentration; that is, [GLC]\({}_{0}\) = [GLC](t = 0) = [GLC]\({}_{\rm inlet,0}\). The FBR time-'arcs' are defined before Equation (5). For an adopted N\({}_{\rm div}\) = 5 time-'arcs', the FBR optimal operation is determined in two alternatives:  
FBR operation alternative (A): A constant feeding along the entire batch, as follows:  
\[F_{L,0}\ =\ F_{L,1}\ =\ F_{L,2}\ =\ F_{L,3}\ =\ F_{L,4}\ =\ \mbox{constant},\]  
\[c^{feed}_{glc,0}\ =\ c^{feed}_{glc,1}\ =\ c^{feed}_{glc,2}\ =\ c^{feed}_{glc,3}\ =\ c^{feed}_{glc,A}\ =\ \mbox{constant}.\]  
This case corresponds to the nominal FBR non-optimal operating conditions of Chen [72]; the control variables \(F_{L,j}\) and \(c^{feed}_{glc,j}\) being kept constant on each time-arc at the non-optimal values given in (Table 1).