## 2 Multiple-Criteria Decision-Making  
Over the past several years, MCDM models have acquired a great relevance because this paradigm presents a number of features that make it especially suitable for analyzing hard real-life problems. One of the fundamental features of the MCDM methodologies refers to the fact that most of them can cope with both quantitative and qualitative data, along with the subjective opinions and/or the preferences of experts [24]. From a theoretical viewpoint, MCDM is a powerful component of operations research that encompasses some analytical tools and techniques to appraise the strengths and weaknesses of a set of \(M\) competing alternatives \(A=\{a_{1},a_{2},\ldots,a_{M}\}\) evaluated on a family of \(N\) (usually conflicting) criteria of different nature \(C=\{c_{1},c_{2},\ldots,c_{N}\}\), with the objective of making an accurate decision regarding the preference judgment of the decision-maker [25; 26]. Thus, an MCDM problem can be generally represented by means of a \((M\times N)\) decision matrix as that shown in Table 1.  
Choosing the best alternative requires combining partial evaluations of each alternative into an aggregated value by using an aggregation operator \(\Psi:A\to R\) that relates a global value \(\Psi(a_{i})\) to alternative \(a_{i}\). This aggregation operator depends on the preferences of the analyst, which can be expressed regarding the relevance of criteria through weights \(w=\{w_{1},w_{2},\ldots,w_{N}\}\in[0,1]^{N}\). Thus, the aggregation operator can be defined as  
\[\Psi(a_{i})=\sum_{j=1}^{N}w_{j}a_{i}^{j}, \tag{1}\]  
where \(a_{i}^{j}\) are the partial evaluations of the alternative \(a_{i}\). MCDM methods can be categorized into two general groups [27]: the multi-objective decision-making approach assumes a theoretically infinite (or a very large) number of alternatives, whereas the multi-attribute decision-making requires the assessment of a finite number of alternatives, which corresponds to the most common situation in financial decision-making problems (e.g., credit approval applications). A rather different taxonomy identifies four categories [28]: (i) multi-objective mathematical programming, (ii) multi-attribute utility/value theory, (iii) outranking relations, and (iv) preference disaggregation analysis. As already pointed out, the present work concentrates on the outranking relations approach because it is recognized as one of the most effective ways to face the complexity of business and financial decision-making problems. In addition, unlike other MCDM techniques, the outranking relations methods are able to deal with any kind of problematics. Performance assessment of classification algorithms requires dealing with various complementary criteria of interest, typically weighting the gains of each criterion against the others. Taking this into account, \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \(c_{1}\) & \(c_{2}\) & \(\cdots\) & \(c_{N}\) \\ \hline \(a_{1}\) & \(z_{11}\) & \(z_{12}\) & \(\cdots\) & \(z_{1N}\) \\ \(a_{2}\) & \(z_{21}\) & \(z_{22}\) & \(\cdots\) & \(z_{2N}\) \\ \(\vdots\) & \(\vdots\) & \(\vdots\) & \(\ddots\) & \(\vdots\) \\ \(a_{M}\) & \(z_{M1}\) & \(z_{M2}\) & \(\cdots\) & \(z_{MN}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Decision matrix for a general MCDM problem (\(z_{ij}\) denotes the value of alternative \(a_{i}\) assessed by criterion \(c_{j}\)). choosing the best performing prediction model can be considered as a particular MCDM problem, where \(M\) represents the number of prediction models (alternatives) and \(N\) expresses the number of performance assessment measures (criteria). In the framework of credit risk analysis, the MCDM techniques ought to allow analysts and decision-makers to pick up the algorithm that yields a closely optimal compromise between the evaluation criteria.