## 3 New Criteria for Model Comparison
### Multiple Criteria Decision Making Using Maximum  
In \(MCDM\), the most important issue is determining the weights based on fitness. Various methods exist for determining the weights, such as the Monte Carlo simulation, entropy, and the technique for performance by similarity with an ideal solution (TOPSIS). In particular, \(MCDM\) through ranking (\(MCDM\)), which considers both quantitative and qualitative methods simultaneously, has been proposed [46]. However, because the weights were differentiated through ranking, even if the difference between the criteria of the two models was large, the difference in the reflected weights was the same. For instance, suppose that three random models have mean-squared error (\(MSE\)) values, as listed in Table 2. If the first model had an \(MSE\) of 100, the second had an \(MSE\) of 20, and the third had an \(MSE\) of 10, the difference in rank between the first and second models was 1, and the difference in \(MSE\) values was 80. The difference in rank between the second and third models was also one, but the difference in the \(MSE\) values was 10. The problem with simply reflecting the rankings is that if multiple models are equally spaced and perform well, a considerably small difference can result in a large penalty.  
In this study, the previous problem was addressed using a multi-criteria decision method using maximum (\(MCDM\)), which reflects the size of the criteria, instead of simply applying a ranking by dividing the values corresponding to the column criteria into ranges while computing the \(MCDM\). Matrix \(M\) was constructed to utilize this size. The components of the matrix \(M\) are calculated by Equation (8) through \(w_{ij}=\frac{C_{ij}}{\max C_{j}}(i=1,2,\cdots,m,j=1,2,\cdots,n)\), where the maximum value of each criterion according to each model is distributed, utilizing the size.  
\[M=\begin{pmatrix}w_{11}&w_{12}&\dots&w_{1k}\\ w_{21}&w_{22}&&w_{2k}\\ &\vdots&&\ddots&\vdots\\ w_{s1}&w_{s2}&\dots&w_{sk}\end{pmatrix} \tag{8}\]  
The matrix \(Y\odot M\) is defined by the product of the respective components of matrix \(M\) and matrix \(Y\) after the transformation process explained in 3.2, and follows from Equation (9).  
\[Y\odot M=\begin{pmatrix}w_{11}y_{11}&w_{12}y_{12}&\dots&w_{1k}y_{1k}\\ w_{21}y_{21}&w_{22}y_{22}&&w_{2k}y_{2k}\\ &\vdots&&\ddots&\vdots\\ w_{s1}y_{s1}&w_{s2}y_{s2}&\dots&w_{sk}y_{sk}\end{pmatrix} \tag{9}\]  
We define \(MCDM\) by Equation (10) using \(YM_{1}\) to \(YM_{s}\) as the components that are summed in each row of the combined matrix \(Y\odot M\). The component sizes were compared to determine the overall excellence of the model based on smaller values.  
\[MCDM=\begin{pmatrix}w_{11}y_{11}+w_{12}y_{12}+\cdots+w_{1k}y_{1k}\\ w_{21}y_{21}+w_{22}y_{22}+\cdots+w_{2k}y_{2k}\\ &\vdots\\ w_{s1}y_{s1}+w_{s2}y_{s2}+\cdots+w_{sk}y_{sk}\end{pmatrix}=\begin{pmatrix}YM_{ 1}\\ YM_{2}\\ \vdots\\ YM_{s}\end{pmatrix} \tag{10}\]  
This study utilizes \(MCDM\), which improves the existing method of judging the goodness of models, and proposes a new \(MCDM\) that reflects the continuous value of criteria. We propose and utilize a new \(MCDM\) to assess the goodness of fit of models with complex evaluations and multifaceted interpretations.  
\begin{table}
\begin{tabular}{c c c c c} \hline
**Model** & **MSE** & **Rank** & **Weight-Rank** & **Weight-Max** \\ \hline Model 1 & 100 & 3 & 0.50 & 1.0 \\ \hline Model 2 & 20 & 2 & 0.33 & 0.2 \\ \hline Model 3 & 10 & 1 & 0.17 & 0.1 \\ \hline \end{tabular}
\end{table}
Table 2: Examples of \(MCDM\).