## 4 Fed-Batch Bioreactor Optimization Problem
### Optimization Problem Formulation
#### 4.2.2 Single Objective Function (\(\Omega\)) Optimization (NLP)  
By considering the mentioned control variables in Equation (7), the optimization of the FBR with a variable feeding consists of simultaneously determining its optimal initial load with its feeding strategy for every time interval ('arc') during the batch, leading to the maximization of [TRP] production, outlined as follows:  
\[\mathrm{Find\ the\ control\ variables\ values\ of\ Equations}\ (7)-(9),\ \mathrm{to\ reach} \tag{10}\] \[\mathrm{Max}\ \mathbf{\Omega},\ \mathrm{where}:\quad\mathbf{\Omega}\ =\ \mathrm{Max}\ [\mathrm{TRP}(t)],\ \mathrm{with}(t)\ \epsilon\ [0,\ tf]\]  
The [TRP](\(t\)) dynamics in Equation (10) are evaluated by solving the FBR ODE dynamic model [Equations (1)-(6)] over the whole batch time (\(t\)) \(\in\) [0, \(t\)]. The choice of the number of time-'arcs' (N\({}_{\rm div}\)): The FBR operating strategy with a variable feeding (alternative B) implies a time-step-wise variable feeding over an adopted (N\({}_{\rm div}\) = 5), with equal time-arcs that cover the whole batch time. Each time-arc j\({}^{\prime}\) (j = 1, \(\ldots\), N\({}_{\rm div}\)) is characterized by optimal levels of the feed flow rate \(F_{L,j}\), and of the \(c^{feed}_{glc,j}\). It is self-evident that, over a time interval, the control variables are kept constant. Of course, the optimal values on various intervals may differ from each other. A brief survey of the FBR optimization literature [5; 18; 26] reveals that a small number (N\({}_{\rm div}\)) \(<\) 10 is commonly used due to its multiple advantages, extensively pointed out by [PERSON] [5; 32]. Additionally, FBR operation, using a large N\({}_{\rm div}\) time-arc, can present special operating problems when including PAT (Process Analytical Technology) tools [91]. Thus, to not complicate the computational analysis, (N\({}_{\rm div}\)) = 5 was adopted, with equal time-arcs covering the batch time \(t_{f}\) = 100 h.  
2.3 Multi-Objective Optimization by Using the Pareto-Optimal Front with an Optimal but Constant FBR Feeding  
When more than one objective function is simultaneously considered, the optimization problem is more difficult to solve. For multi-objective optimization, several alternatives can be utilized [92; 93]. One elegant option is to obtain the set of Pareto-optimal solutions, also called a Pareto front, for the case of at least two adverse objectives [94]. A Pareto solution is one where any improvement in one objective can only take place at the cost of the other objective. In fact, the Pareto curve is the geometrical place of all the points (problem solutions) that realize the best compromise between the two considered objective functions. The choice of a solution (a point) from this Pareto curve is subjective, usually dependent on other criteria not included in the Pareto optimization. For the present case study of an FBR operated with a constant optimal feeding (in terms of feed flow rate \(\mathrm{F_{L}}\) and inlet [GLC]), several opposing objectives can be considered at the batch end, such as maximum TRP production, minimum substrate [GLC] consumption, minimum initial necessary viable biomass [X]o, minimum feed flow rate \(\mathrm{F_{L}}\) (to avoid overly diluting the bioreactor content), and others. Of course, the Pareto-optimal fronts can be obtained by using any pair of these opposite objective functions. In the considered case study, the following four opposite objectives are considered two-by-two, based on their adverse effect in the process model, as follows:  
Max. TRP production -vs.- min. substrate [GLC] consumption;  
Max. TRP production -vs.- min. initial necessary viable biomass [X]o;  
Max. TRP production -vs.- min. feed flow rate \(\mathrm{F_{L}}\) (i.e., minimum increase in reactor liquid volume). Due to the limited efficiency of the applied Pareto optimizer (Section 4.2.5), the rough (slightly oscillating) Pareto curves obtained are difficult to interpret.