## 3 Methodology
### Artificial Neural Network Metamodels  
In this investigation, the phenomenological dynamic model of the fermentation system depicted in Figure 1 was developed to compute the concentrations of the eight chemical species, acetone, acetic acid, butanol, butyric acid, ethanol, glucose, biomass, and water, as a function of time for each set of the five decision variables. When the concentrations in the fermenter do not change within a given tolerance, the system is assumed to have achieved steady state. Then the four objectives (butanol productivity, overall butanol concentration, sugar conversion, and number of membrane modules) are evaluated. To obtain a complete Pareto domain, this procedure needs to be repeated a large number of times. A methodology to achieve a well-defined Pareto domain was therefore developed. In order to develop the ANNs for the integrated fermentation-membrane pervaporation system, the available datasets from the UD experimental runs were divided into training and validation sets. The dataset consisted of 40 input-output pairs in which a U\({}_{30}\)(5\({}^{\text{s}}\)) and U\({}_{10}\)(5\({}^{\text{s}}\)) were assigned for training and validation of the neural networks, respectively. The architecture of the ANN with inputs and different responses is schematically depicted in Figure 3. The inputs to the ANNs are the five decision variables: dilution rate (D), inlet glucose concentration (S\({}_{\text{o}}\)), cell retention factor (\(\alpha_{8/12}\)), number of pervaporation modules in series (N\({}_{\text{mod}}\)), and number of stacked membranes in their respective membrane module (N\({}_{\text{Stack}}\)). Two strategies were used in the derivation of ANNs. The first one was aimed at replacing the entire continuous integrated ABE fermentation process, whereby each of the four objective functions (butanol productivity, overall butanol concentration, sugar conversion, and number of membrane modules in series) is modelled. For this first strategy, four ANNs were derived and could be used independent of the actual integrated fermentation model. For a given set of decision variables, the four neural networks can be used in the multiobjective optimization algorithm to calculate the four objective functions. The Pareto domain can therefore be circumscribed using the ANN surrogate models. In the second strategy, it is desired to use the actual integrated ABE fermentation model to converge to steady-state operation but use ANNs to predict the initial concentrations of the main species within the fermentation broth (acetone, acetic acid, butanol, butyric acid, ethanol, glucose, bacteria, and water). The motivation of the latter strategy, with the derivation of eight ANNs, is to converge more rapidly to a steady state, resulting in reduced computation time. For all 12 ANNs (four for the objectives and eight for the initial species concentrations) derived in this investigation, the architecture corresponded to an input layer of six neurons: the bias, one hidden layer of three neurons including the bias, and an output layer of one neuron, as shown in Figure 3. The activation function for the neurons of the hidden and output layers is a sigmoid function, and the input and output variables were scaled between 0 and 1. Given the reduced number of data pointsgenerated by the experimental uniform design and available for training the neural network, it was decided to limit the number of neurons in the hidden layer to three. Moreover, the sum of squares of the prediction errors based on the validation data set was evaluated at each iteration, and the set of weights corresponding to the minimum sum of squares of the errors was the one used for the predictions. The Pareto domain can now be obtained using three different routes. First, the genetic algorithm can be used along with the four ANNs of the four objective functions.