## 3 Methodology
### Stochastic Structural Relationship Programming (SSRP) Model  
This paper proposes a novel SSRP model based on the neural network architecture to unveil the existing endogenous relationships among performance relative to the ideal solutions and rail section normalized attributes, while identifying the relevant structural cause-effect relationships that may exist. Precisely, neural networks are employed on unveiling endogeneity among rail section performance and its normalized attributes in terms of the residuals produced by the following models.  
Model 1: Performance-f(PG; MCR; NODY; LC; I; B; LC.RSL; IRT; AP; PI).  
Model 2: PG-f(Performance; MCR; NODY; IC; I; B; LC.RSL; IRT; AP; PI).  
Model 3: MCR-f(PG; Performance; NODY; IC; IC; I; B; LC.RSL; IRT; AP; PI).  
Model 4: NODY-f(PG; MCR; Performance; IC; I; B; LC.RSL; IRT; AP; PI).  
Model 5: IC-f(PG; MCR; NODY; Performance; LC; I; B; LC.RSL; IRT; AP; PI).  
Model 6: LC-f(PG; MCR; NODY; IC; Performance; I; B; LC.RSL; IRT; AP; PI).  
Model 7: I-f(PG; MCR; NODY; IC; LC; Performance; B; LC.RSL; IRT; AP; PI).  
Model 8: B-f(PG; MCR; NODY; IC; IC; I; Performance; LC.RSL; IRT; AP; PI).  
Model 9: LC.RSL-f(PG; MCR; NODY; IC; LC; I; B; Performance; IRT; AP; PI).  
Model 10: IRT-f(PG; MCR; NODY; IC; LC; I; B; LC.RSL; Performance; AP; PI).  
Model 11: AP-f(PG; MCR; NODY; IC; LC; I; B; LC.RSL; IRT; Performance; PI).  
Model 12: PI-f(PG; MCR; NODY; IC; LC; I; B; LC.RSL; IRT; AP; Performance).  
Rail section length (RSL); predominant gauge (PG); minimum curve radius (MCR);  
operational days per year (NODY); installed capacity (IC); linked capacity (LC); idleness (I); bottleneck (B); linked capacity.rail section length (LC.RSL); increasing ramp tax (IRT); auxiliary power (AP); percentage of idleness (PI).  
The optimal neural network architecture for models (1)-(12) is formed by two layers with ten neurons each. A 10-fold cross-validation procedure was adopted, yielding a minimal Mean Average Error (MAE) of 0.0018. This architecture suggests that there may be non-linear hidden relationships between predictors. In fact, neural networks were employed here to the detrimment of other approaches, such as linear regressions, due to their ability to unveil non-linear relationships between dependent variables and their predictors. This feature is particularly useful as long as information entropy embeds in itself partial measurements of the non-linear correlation between variables. As further discussed, information entropy is the cornerstone for analyzing the distributional profile of the residuals of each one of these 12 regressions, in the search for evidence that could support endogeneity claims.  
Therefore, residuals for these 12 neural networks are subsequently used to generate a full set of conditional residual distributions between the respective dependent variable pairs identified in each model. These conditional residual distributions allow exploration of the directional relationships that may exist among variables. The novel SSRP model is structured in two consecutive steps that allow the unveiling of endogeneity while identifying relevant cause-effect relationships among the remainder variables. These steps are described next.  
Step 1: Minimal Endogenous Relationship Variance  
The relative importance of models (1)-(12) in explaining the feedback process between normalized attributes and performance levels in Brazilian rail sections, besides the endogenous nature of these variables, were explored, respectively, by the variances of each model and the covariances between models. Variances and covariances of the residuals (\(R_{i}\)) of these 12 models are simultaneously minimized by a non-linear stochastic optimization problem, as presented in model (13), where \(w_{i}\) stands for the weights--which range from 0