## 4 Combination of Objective Functions  
Many optimization problems make use of Single Objective (SO) functions which are well suited to the global optimal search. Usually, these functions are associated with the predicted variance/errors such as the Root Mean Square Error (_RMSE_) and coefficient of determination (\(R^{2}\)). Most of the engineering practices reveal that parameters optimization is intrinsically high dimensional, nonlinear and combinatorial [30]. Furthermore, each SO function refers to specific behavioral errors in the model. Therefore, the optimal model search needs to be performed by combining different objective functions where each of them places emphasis on a particular error's feature. The combination of objective functions leads to an improvement in the best local search of the model and provides the highest accuracy.  
In this paper, the analytical and experimental nature of the model's uncertainties are analyzed. The analytical uncertainty is expressed in Equation (7) by the term \(\sigma_{Y_{lin}}^{2}\). It reflects the variability in the predicted model due to the first-order derivatives of the observed quantity with respect to each explanatory variable. The experimental uncertainty refers to the variance in the model's error, calculated as shown in Equation (8):  
\[\sigma^{2}=\frac{\sum_{i=1}^{n}\left(\tilde{Y}_{i}-Y_{i}\right)^{2}}{(n-1)} \tag{8}\]  
where \(Y_{i}\) refers to the \(i\)-th observed value and \(\tilde{Y}_{i}\) is the fitted \(i\)-th value obtained through the EPR procedure. The term \(\sigma^{2}\) is evaluated based on the observed data points; therefore, its experimental nature is clear. The experimental variance is adopted in the definition of common objective functions, such as the \(R^{2}\) or _RMSE_.  
A combination between the two described error-based functions is proposed as a novel OF suitable for EPR-based technique. The basic idea lies in modifying the common mathematical formulation of \(R^{2}\) by replacing the model's variance with a combination between experimental and analytical variance, according to Equation (9):  
\[R_{N}^{2}=1-\left[\frac{(n-1)\cdot\sigma^{2}}{\sum\limits_{i=1}^{n}\left(Y_{ i}-\sum\limits_{i=1}^{n}Y_{i}/n\right)^{2}}\right]^{(1-f(\sigma_{Y_{lin}}))} \tag{9}\]  
where \(f(\sigma_{Ylin})\) is a function of the analytical standard deviation, while the denominator represents the mean square errors of the data points. The combination is expressed as:  
\[f(\sigma_{Ylin})=\delta\cdot\left(\frac{\sigma_{Ylin}}{\mu}\right)^{\beta} \tag{10}\]  
\[\delta=(0;1) \tag{11}\]  
The function \(f(\sigma_{Ylin})\) is defined as a polynomial expression of the ratio between the analytical standard deviation (\(\sigma_{Ylin}\)) and the experimental mean value (\(\mu\)) of the observed variable. This ratio represents the coefficient of variation of the analytical model, while \(\beta\) is the exponential coefficient of Equation (10) and \(\delta\) is a dummy variable assuming 0-1 values. The influence of the analytical dispersion on the optimal search procedure is given by \(\beta\) in the case that \(\delta=1\). Moreover, when the dummy variable is equal to zero, Equation (9) coincides with the standard formulation of the \(R^{2}\). The model that shows the best trade-off between the new proposed formulation of accuracy (\(R_{N}^{2}\)) and complexity (number of model parameters) will be selected as the optimal one.