# Comparison of Profit-Based Multi-Objective Approaches for Feature Selection in Credit Scoring
## 3 Methods
### Non-Dominated Binary Grasshopper Optimization Algorithm (NSBGOA)
Four potential classification outcomes exist, as per Table 1. When a good borrower is rejected, the lender loses the return on investment (ROI). Additionally, accepting a bad borrower results in loss of the benefit, \(b\) \(\in\) [0, 1], expressed by the Equation (7), with exposure at default (EAD), loss given default (LGD), and principal amount (A) of the loan [33]. \[b=\frac{LGD*EAD}{A} \tag{7}\]

The benefit, \(b\), depends on how much of the loan is repaid in full. * \(b\) = 0 with probability \(\mathrm{p_{0}}\) that the loan is repaid in full,
* \(b\) = 1 with probability \(\mathrm{p_{1}}\) that no portion of the loan is repaid,
* \(b\) is uniformly distributed within (0,1) with \(\mathrm{g}(b)=1-\mathrm{p_{0}}-\mathrm{p_{1}}\). Finally, EMP is given by equation 8 below with prior probabilities of default (\(\mathrm{\tau_{0}}\)), prior probabilities non-default (\(\mathrm{\tau_{1}}\)), predicted cumulative density functions (\(\mathrm{F_{0}}\) and \(\mathrm{F_{1}}\)), and constant ROI. \[EMP=\int_{0}^{1}(b*\ \pi_{0}F_{0}-ROI*\ \pi_{1}F_{1})*g(b)\ \mathrm{d}b \tag{8}\]

### Performance Metrics

To evaluate the output of many- and multi-objective optimizations, the hypervolume indicator (HV) may be used. HV, which can be used to evaluate convergence and distribution, is denoted by Equation (9), where \(\lambda_{m}\) is the m-dimensional Lebesgue measure and \(m\) is the number of objectives [34]. It calculates the volume of objective space dominated by the Pareto Front approximation, \(P\), and delimited from above by the reference point \(r\) such that \(z\in P\), \(z\) dominates \(r\). \[\mathrm{HV}(P,r)=\lambda_{m}(\bigcup_{z}\frac{\omega}{\epsilon}P[z;r]) \tag{9}\]

## 4 Empirical Evaluation

### Problem Formulation

With credit-scoring data identified, objectives may be listed. To achieve this, the following definitions are given:

1. _Available features_\(X\), (Equation (10)) a set of \(j\) variables that could be used to predict loan repayment. 2. _Cardinality (number of features)_, \(N\), (Equation (11)) is the number of selected features per solution. 3. _Expected Maximum Profit, EMP_, (Equation (8)) a profit-based measure for credit scoring. 4. _Ease of explanation_, \(C\), (Equation (12)) a vector representing the ease of explaining each variable to stakeholders. 5. _Default status, \(D\), (Equation (13)) is a vector with loan repayment information for each borrower. 6. _Borrower information, \(B\),_ (Equation (14)) a matrix with feature values for each borrower. \[A\text{\emph{Available}}\ \text{\emph{Features}},\ \ X=\left\{x_{1},x_{2},x_{3},\ \ldots x_{j}\right\} \tag{10}\]

\[Cardinality,\ \ N=\left\{n_{1},n_{2},n_{3},\ \ldots\right\} \tag{11}\]

\[\text{\emph{Ease}}\ \text{\emph{of}}\ \text{\emph{explanation}},\ \ =\left\{c_{1},c_{2},c_{3},\ \ldots\right\} \tag{12}\]

\[\text{\emph{Default}}\ \text{\emph{status}},\ \ D=\left\{d_{1},d_{2},d_{3},\ \ldots d_{i}\right\} \tag{13}\]

\[\text{\emph{Borrower}}\ \text{\emph{Information}},\ B=\left[ \begin{array}{cccc}b_{11} &b_{12}&\cdots&b_{1j}\\ b_{21}&b_{22}&\cdots&\cdots\\ \vdots&\vdots&\ddots&\vdots\\ b_{m1}&\cdots&\cdots&\ddots\end{array} \right] \tag{14}\]

\[\text{\emph{possible}}\ \text{\emph{solution}},\ P=\left\{x_{1},x_{2},x_{3},\ \ldots x_{n}\right\} \tag{15}\]

Each solution, \(P\) (Equation (15)) is comprised of \(\mathbf{n}\) features such that \(P\subset X\). The goal is to select a set, \(S=\left\{s_{1},s_{2},s_{3},\ \ldots s_{p}\right\}\) so that each element is a distinct subset of \(X\) resulting in non-dominated objective values.