## 2 Multiple-Criteria Decision-Making
### The TOPSIS Method
Finally, the global net preference flow, which is calculated as \(\phi(a_{i})=\phi^{+}(a_{i})-\phi^{-}(a_{i})\), indicates how an alternative \(a_{i}\) is outranking (\(\phi(a_{i})>0\)) or outranked (\(\phi(a_{i})<0\)) by all the other alternatives on all the evaluation criteria. As a result, the alternative \(a_{i}\) with the maximum global net preference flow will be deemed to be the best. The general PROMETHEE methodology can be easily implemented in the form of a stepwise procedure as defined in Algorithm 2. ```
1: For each pair \((a_{i},a_{j})\) of a finite set of alternatives \(A=\{a_{1},a_{2},\ldots,a_{M}\}\), compute aggregated preference indices \(\pi(a_{i},a_{j})=\sum\limits_{k=1}^{N}P_{k}(a_{i},a_{j})w_{k}\) and \(\pi(a_{j},a_{i})=\sum\limits_{k=1}^{N}P_{k}(a_{j},a_{i})w_{k}\)
2: Compute the positive and negative preference flows \(\phi^{+}(a_{i})=\frac{1}{M-1}\sum\limits_{a\in A}\pi(a_{i},A)\) and \(\phi^{-}(a_{i})=\frac{1}{M-1}\sum\limits_{a\in A}\pi(A,a_{i})\)
3: Compute the net preference flow for each alternative as \(\phi(a_{i})=\phi^{+}(a_{i})-\phi^{-}(a_{i})\)
```

**Algorithm 2** PROMETHEE

The global net preference flow \(\phi(a_{i})\) indicates how an alternative \(a_{i}\) is outranking (\(\phi(a_{i})>0\)) or outranked (\(\phi(a_{i})<0\)) by all the remaining alternatives on all the evaluation criteria. As a result, the alternative \(a_{i}\) with the maximum global net preference flow will be identified as the best one. ## 3 Experiments

A series of experiments were conducted to assess the performance of a pool of classifiers through the TOPSIS and PROMETHEE tools for some credit granting decision problems, with the purpose of demonstrating that the synergetic application of MCDM models makes better decisions than using a single measure to determine the best performing prediction algorithm. The TOPSIS and PROMETHEE techniques were run with the Sanna open source software [34], whereas the classifiers were tested in the WEKA environment [35] using their default parameters (see Table 2):* Artificial neural networks: Bayesian belief network (Bnet), multilayer perceptron (MLP), and radial basis function (RBF);
* Statistical models: naive Bayes classifier (NBC), logistic regression (logR), support vector machine (SVM) and nearest neighbor classifier (1 NN);
* Rule-based classifier: RIPPER;
* Decision trees: C4.5 and random forest (randF). ### Data Sets

Table 3 reports some characteristics of the six real-life credit data sets used for the experiments, including the number of input or explanatory variables, the total number of instances and the number of instances in each class, and the imbalance ratio (IR) calculated as the ratio of the number of instances in the minority class to the number of instances in the majority class. The Australian and German databases were obtained from the UCI Machine Learning Database Repository ([[http://archive.ics.uci.edu/ml/](http://archive.ics.uci.edu/ml/)]([http://archive.ics.uci.edu/ml/](http://archive.ics.uci.edu/ml/))). The Australian database contains 690 samples of credit card applicants, 307 of which were labeled as solvent and 383 as unable to pay their debts; each sample is described by 14 input variables. The German credit database represents a credit screening application, comprising cases on 24 explanatory variables for a total of 1000 applicants: 700 were considered as creditworthy and 300 were labeled as non-creditworthy. The Iranian database is an adaptation of a customers' data set of a small private bank [36]. It contains 950 observations tagged as non-defaulters and 50 as defaulters, where each sample is formed by 27 explanatory variables. The Polish database consists of financial information regarding 120 firms registered over a 2-year period [37], with a total of 112 bankrupt and 128 non-bankrupt accounts. The [PERSON] database [38] comprises the data of 1225 applicants for a credit product, each one being shaped by 12 input attributes.