## 2 Multiple-Criteria Decision-Making
### The TOPSIS Method
This composite score allows for combining the preference rates \(R_{i}^{+}\) and \(\phi(a_{i})\) of an alternative (prediction model) \(i\) in a fair manner as follows: \begin{table}
\begin{tabular}{l c c c c c c} \hline \hline  & **Acc** & **RMSE** & **TP-Rate** & **TN-Rate** & **AUC** & **G-Mean** & **F-Measure** \\ \hline Bnet & 0.81 & 0.38 & 0.86 & 0.53 & 0.79 & 0.59 & 0.80 \\ NBC & 0.66 & 0.52 & 0.67 & **0.71** & 0.76 & 0.65 & 0.67 \\ logR & **0.83** & 0.35 & **0.89** & 0.52 & 0.80 & 0.60 & 0.81 \\ MLP & 0.80 & 0.39 & 0.85 & 0.56 & 0.79 & 0.66 & 0.79 \\ SVM & 0.80 & 0.44 & 0.88 & 0.47 & 0.67 & 0.50 & 0.78 \\ RBF & 0.79 & 0.38 & 0.82 & 0.50 & 0.75 & 0.51 & 0.76 \\
1 NN & 0.77 & 0.48 & 0.80 & 0.58 & 0.69 & 0.66 & 0.77 \\ RIPPER & 0.82 & 0.37 & 0.87 & 0.53 & 0.71 & 0.61 & 0.80 \\ C4.5 & 0.80 & 0.39 & 0.85 & 0.53 & 0.71 & 0.62 & 0.79 \\ randF & **0.83** & **0.34** & **0.89** & 0.56 & **0.83** & **0.67** & **0.82** \\ \hline Weight & 0.02762 & 0.20048 & 0.08524 & 0.04286 & 0.21954 & 0.21571 & 0.20855 \\ \hline \hline \end{tabular}
\end{table} 
Table 11: Performance results averaged across the six experimental databases. \begin{table}
\begin{tabular}{l c c c c} \hline \hline  & **TOPSIS** & **PROMETHEE** \\ \hline
**Alternative** & **Rank** & \(R_{i}^{+}\) & **Rank** & \(\phi(a_{i})\) \\ \hline Bnet & (4) & 0.76538 & (5) & 0.05905 \\ NBC & (10) & 0.17492 & (8) & \(-\)0.38273 \\ logR & (2) & 0.87375 & (2) & 0.48095 \\ MLP & (6) & 0.70375 & (3) & 0.18736 \\ SVM & (8) & 0.44787 & (10) & \(-\)0.60858 \\ RBF & (5) & 0.71332 & (9) & \(-\)0.38358 \\ 1 NN & (9) & 0.31014 & (7) & \(-\)0.32921 \\ RIPPER & (3) & 0.80342 & (4) & 0.13517 \\ C4.5 & (7) & 0.70363 & (6) & \(-\)0.13937 \\ randF & (1) & 0.96282 & (1) & 0.98095 \\ \hline \hline \end{tabular}
\end{table} 
Table 12: Preference rankings given by TOPSIS and PROMETHEE. \[score(i)=\frac{R_{i}^{+}+\phi(a_{i})}{2}. \tag{6}\]

Furthermore, this score can be easily generalized to \(L\) different MCDM methods as:

\[Generalized\ score(i)=\frac{1}{L}\sum_{j\in L}value_{j}, \tag{7}\]

where \(value_{j}\) denotes the preference value given by the method \(j\). Figure 2 displays a graphical representation of the composite scores, which is a simple way of visualizing the rationale of the decisions made. It clearly shows that both random forest and logistic regression are superior to all the other classifiers and, on the other hand, the poor performance achieved by the naive Bayes, SVM and 1 NN algorithms is also apparent. ## 5 Conclusions

The present analysis supports the synergetic application of MCDM techniques for the performance assessment of credit granting decision systems. Through a series of experiments, it has been shown that the employment of an individual metric may give rise to inconsistent conclusions about what is the best prediction model for a given problem, which would lead to selecting an inappropriate method with not the most reliable results. TOPSIS and PROMETHEE, which are two well-known MCDM techniques, have been tested in the experiments applying ten prediction models (alternatives) to six real-world bankruptcy and credit data sets and using seven performance evaluation criteria. The use of single performance metrics have designated different classifiers as the most suitable alternatives. These results suggest that credit granting decision corresponds to a real-world application where the MCDM techniques are especially useful to consistently assess a pool of classifiers and help decision-makers to choose the most beneficial model. In our experiments, both TOPSIS and PROMETHEE have determined that random forest and logistic regression are the best performing prediction methods on most of the performance evaluation measures. Figure 2: Composite ranking scores. Furthermore, we have also introduced a plain score that can be easily expressed as a linear combination of the preference values given by a number of MCDM methods.