# Comparison of Profit-Based Multi-Objective Approaches for Feature Selection in Credit Scoring
## 3 Methods
### Non-Dominated Binary Grasshopper Optimization Algorithm (NSBGOA)
\begin{table}
\begin{tabular}{c c c c} \hline \hline  & & \multicolumn{2}{c}{**Predicted Class**} \\ \cline{3-4}  & & **Non-Default** & **Default** \\ \hline \multirow{3}{*}{**Actual**} & \multirow{3}{*}{**Non-default**} & Benefit: ROI & Cost: -ROI \\  & & Probability: \(\pi_{1}\)\((1-\mathrm{F_{1}})\) & Probability: \(\pi_{1}\)\(F_{1}\) \\ \cline{2-4}  & & Cost: -LGD \(\times\) EAD/A \\ \cline{2-4}  & & Probability: \(\pi_{0}\)\((1-\mathrm{F_{0}})\) & Probability: \(\pi_{0}\)\(F_{0}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Confusion matrix for Expected Maximum Profit. ### Contribution

NSGA-II was used for feature selection in credit scoring with two objectives, profit and number of features [7], and different base classifiers were applied. However, its performance on different base classifiers was not compared to that of other multi-objective methods. Feature selection was conducted by introducing data-acquisition costs as constraints to a support vector machine (SVM) classifier for credit scoring [15]. Although the performance of NSGA-II and NSGA-III on different problems was compared, these comparisons were limited to test problems, not feature selection. Interestingly, it was found that their performance varied depending on the test problem and number of objectives. This leaves open the question of which multi-objective method and base classifier is most suitable for credit-scoring problems. Furthermore, the performance of the NSBGOA algorithm with different base classifiers is still in question. This research aims to answer these open research questions. ### Data and Objectives

To test the various algorithms, the German credit dataset [35], which contains 1000 entries with 700 being non-default and 300 being default. There are 20 features in the initial dataset: 13 qualitative and 7 numerical. For the purposes of this evaluation, the variables that were judged to have ambiguous definitions were given ease of explanation values of 0, namely V2, V3, V12, V14, V17, and V18. Meanwhile, the remaining variables were assigned values of 1. The targets of the optimization are given in Table 2.

### Analysis

Numeric variables went through max-min rescaling, and class imbalance was achieved with the random over-sampling examples technique (ROSE). An assortment of packages in CRAN R was used to conduct analysis. The NSGA-III and NSGA-II algorithms were implemented with the \"_mmoo_\" package. The analysis of NSBGOA was implemented with an appropriately modified version of the GOA function from the \"_metaheuristicOpt_\" package. To calculate the objective values for each subset, three classifiers from the \"_caret_\" package were trained by using tenfold cross-validation, and the best model was selected based on the EMP (\"_EMP_\" package). The parameters for EMP evaluation were set to p\({}_{0}\) = 0.55, p\({}_{1}\) = 0.1, and ROI = 0.2644, as proposed by the authors in Reference [18]. The three base classifiers were regression (LR), support vector machine with a linear kernel (SVM LIN), and artificial neural networks (NN). Following this, the values of the remaining objectives were calculated according to Table 2. For comparison, a LASSO [20] regression model with alpha = 1 and lambda = 0 was fit on the data, using tenfold cross-validation to maximize EMP. Additionally, a single-objective genetic algorithm (GA) [36] was used for feature selection with the fitness function being a tenfold cross-validation classifier trained to maximize EMP. Lastly, the three base classifiers were also trained on all the original features to maximize EMP. Population size of 10 and maximum iteration number of 50 was used for NSGA-III, NSBGOA, NSGA-II, and GA methods.