## 4 Application Proposed Method for Attribute Weighting on a Case Study
### Implementation of the Case Study
weighted aggregation model was implemented, as detailed in Equation (10). For the classification of alternatives, based on the results obtained in the alternative evaluation, the Jenks method, a known interval classification method, was employed. \[Y=\sum_{i=1}^{n}x_{i}e^{x_{i}}, \tag{10}\]  
where \(x_{i}\) represents variable \(i\), and \(w_{i}\) is the weight associated with variable \(i\). To carry out a comprehensive evaluation of the proposed method's performance, measurements were conducted using specific metrics to enable a detailed understanding of its effectiveness. In this context, the Mean Absolute Error (MAE) and Mean Absolute Percentage Error (MAPE) were employed. In Equations (11) and (12), the mathematical expression for the Mean Absolute Error and Mean Absolute Percentage Error is presented. These metrics were chosen to accurately determine the proximity of the estimated results to those presented using the marginalization index. The primary objective of this evaluation is to comprehend the extent to which the obtained results align with the actual values of the marginalization index. Mean Absolute Error (MAE):  
\[MAE=\frac{1}{n}|y_{i}-\hat{y_{i}}| \tag{11}\]  
where  
\(n\) is the total number of observations;  
\(y_{i}\) is the real value of observation \(i\);  
\(\hat{y_{i}}\) is the estimated value of observation \(i\). Mean Absolute Percentage Error (MAPE):  
\[MAPE=\frac{1}{n}{\sum_{i=1}^{n}\frac{|y_{i}-\hat{y_{i}}|}{|y_{i}|}}100 \tag{12}\]  
where  
\(n\) is the total number of observations;  
\(y_{i}\) is the real value of observation \(i\);  
\(\hat{y_{i}}\) is the estimated value of observation \(i\). Based on the calculations performed, an MAE of 0.244 and an MAPE of 11.33% are obtained. These results indicate that the estimates are approximately 0.24 units away from the actual values, with a deviation of 11.3%. In a dataset comprising 1814 observations, it is noteworthy that, for 1381 of them, the absolute distance between the estimate and the reference value is zero. For another 422 observations, the distance is one, and finally, for the remaining 11 observations, the distance is two. These findings demonstrate that the estimates closely align with the actual data, supporting the effectiveness and precision of the proposed method. Additionally, it is observed that the estimated values follow a similar trend to the behavior of the marginalization index. Figure 3 provides a graphical representation of the comparison between the estimated and actual values of the marginalization level for 100 locations. Phase 4. Sensitivity analysis  
In this phase, a sensitivity analysis was conducted to assess the robustness and stability of the obtained model. The sensitivity analysis aims to understand how the model results vary when adjusting sensitive parameters, such as attribute weights. Random numbers were generated in the interval [0, 0.5] for each attribute and added to the weights of the obtained model. This process was repeated in 10 experiments. The best result exhibited a Mean Absolute Error of 0.15 and a Mean Absolute Percentage Error of 10.03%, surpassing the values in the previous stage. Specifically, it was observed that approximately 77% of the data showed an absolute distance equal to zero between the estimated value and the reference value. The absolute distance is equal to 1 for 23% of the data. Finally, the absolute distance between the estimated value and the reference value is 2 for 0.3% of the data. This analysis provides valuable information about the model's performance in different scenarios, contributing to a more detailed understanding of its behavior. Based on the obtained results and with the help of the Geographic Information System QGIS version 3.34.3, a cartographic representation of locations with their respective levels of marginalization in Mexico City was performed, using a scale of five colors.