InterCriteria Analysis for selection of specific growth rate models of batch cultivation by Saccharomyces cerevisiae yeast for ethanol production

: In this study we have developed an application of a new method for multicriteria decision analysis namely InterCriteria Analysis (ICA). The method is based on the apparatus of the index matrices and the intuitionistic fuzzy sets. The ICA has been used to evaluate and select specific growth rate models for cultivation by the Saccharomyces cerevisiae yeast. Different unstructured models Monod, Mink, Tessier, Moser, Aiba, Andrews, Haldane, Luong, Edward, and Han-Levenspiel have been considered in order to explain the cell growth kinetics. The application of the ICA for the specific growth rate of glucose has shown that there are many correlation connections between the investigated models. The models have been reduced only to Monod and Mink . Each of the two models can be used for modelling.


Introduction
The ethanol is the most important organic compound which has a wide application in different industry fields: food, perfumery-cosmetic, chemical, millwright, etc. In recent years enormous attention is paid to the ethanol production as a fuel. Ethanol production from

Kinetic model of the fed-batch and batch processes
The mathematical model of the process is based on the mass balance equations by perfect mixing in bioreactor. The batch model for Saccharomyces cerevisiae yeast is obtained at feed flow rate F = 0 [19]: where: X -cell concentration, g/l; S -glucose concentration, g/l; E -ethanol concentration, g/l; t -time, h; ρ(S) -specific growth rate from glucose, h -1 ; YS/X and YS/E -yield coefficients, g/g.

Specific growth rate models
The models for the growth rate of glucose ρ(S) rate is unknown, so the work investigates ten  [16,17,23,29,30]. The models are shown in Table 1. Table 1. Investigated specific growth rate models dependent on glucose

Model Equation Model Equation
The denotations in Table 1 are as follows: ρm -maximum growth rate, h -1 ; KS -Monod saturation constants for cell growth on glucose, g/l; α -Moser constant; KSI -inhibition constants for cell growth on glucose, g/l; K -constant in Edward model, g/l; Sm -critical inhibitor concentrations, above which the reactions stops, g/g; m, n -constants in the Luong and the Han-Levenspiel models.
All denominators in the models M1-M10 are different and larger than zero, for example (KS + S) > 0, etc.

Criteria of evaluation of the model parameters
The mathematical estimation of the model parameters is based on the minimization of some quantities that can be calculated and the estimation of a function of parameters. If the model under consideration is linear, the estimation is generally an easy task. However, there is no general theory for nonlinear parameter estimations. The least-squares error is commonly employed as a criterion to inspect how close the computed profiles of the state variables come to the experimental observations [32]: where J -criteria for minimization; x -vector of estimated parameters in specific growth rate models, -maximal values of biomass, glucose and ethanol; tj -time partitions, h.

Criteria for model validation
The best dependences are defined by criteria of minimization: Experimental correlation coefficient R 2 for kinetics variables: For number of the experiments N = 12, and number of the kinetics variables Q = 3. Full formulas of statistical criteria are presented in [26].
We have developed an algorithm and a program on Compaq Visual FORTRAN 90 to determine the parameters in the models and the computing criteria. For solving the nonlinear problem (4) we have used BCPOL with double precision from IMSL Library of COMPAQ Visual FORTAN 90 [15].
The experimental and simulated results of the ten models for the specific grown rate of Saccharomyces cerevisiae are shown from Figure 1 to Figure 3.

Applications of ICA for selection of the specific growth rate models
For the sake of clarity, we will show the rules for determining the positive consonance, the negative consonance and the dissonance between the criteria. Atanassova et al. [14] have discussed an important aspect of the ICA approach related to the possibilities for defining the intuitionistic fuzzy threshold values that help discriminate between the positive consonance, the negative consonance and the dissonance between the criteria (Figure 4). The triangular zone for the negative consonance (NC) from Figure 4 corresponds to where the pairs of the criteria which exhibit NC will be located. Formally, this area can be expressed as: The triangular zone for positive consonance (PC) from Figure 4 corresponds to where the pairs of criteria which exhibit PC will be located. Formally, this area can be expressed as: The pentagonal zone for dissonance (D) from Figure 4 corresponds to the place where the pairs of criteria which are in D will be located. Formally, this area can be expressed as: For selecting growth rate models we have to have many high values of the membership function (µ) because we have accepted α=0.95 and β=0.05 in this study. The method is used for selection of growth rate models from lactose and oxygen.
The values of membership function (µ) and non-membership function (ν) have been calculated with the help of the software developed by our colleagues for the realization of the method [18].
For determination of the positive consonance, negative consonance and dissonance the following threshold values have been assumed ] The index matrix for membership function (µ) of different specific growth rate models is shown in Table 3.  Table 3. Index matrix for membership function of different specific growth rate models The interpretations of the intuitionistic fuzzy triangle for the different models in the real boundaries of µ and ν is shown in Figure 5a. For the sake of clarity, Figure 5b  When selecting a suitable model, we basically start using the model with less complexity (fewer parameters to be identified). Now let us look at Table 3 and The M8 has a model connection (M7 -M8 -M9). However, M7 and M9 models are dropped out of the first group. Hence, model M8 is also dropping from the research.
Models M4 and M10 have dissonance with all others and therefore fall out. This is also shown in Figure 1 to Figure 3.
All other combinations of models where we have dissonance are not included in the study.
The application of the ICA has shown that we have a very high degree of agreement between the tested models. Of all ten models there are only two Monod and Minik.

Conclusions
The application of the InterCriteria Analysis for determination of the correlation connections between the specific growth rates has shown that eight models can be eliminated from all ten initial models. Only the models of Monod and Mink have remained. In the end, we choose the Monod model for a specific growth rate of batch process for cultivation of Saccharomyces cerevisiae yeast.
The application of ICA for establishment of correlation connection between different models for the specific growth rate proved to be very useful. In this way using of ICA for a negative consonance was not correctly chosen (they do not reflect the real situation) and for this aim we have to search another description of the process.