## 4 Case Study and Results Discussion
### Analysis 1: Covariate Weight Sensitivity Analysis  
The first analysis aimed to assess the performance of the covariate weight and band estimation process relative to the preliminary solution shown in Table 3, exploring the impact of these covariate weights on the decision policy. Additionally, it investigated whether the optimization parameters identified as optimal in Ref. [3] could enhance the accuracy of the decision rule. Furthermore, a sensitivity analysis was conducted on the latter solution to examine whether minor modifications to the optimization parameters could lead to variations in overall reliability metrics. A preliminary sensitivity analysis was conducted to determine the optimal GA hyperparameters. By changing the values of the gene population (from 50 to 500 in steps of 50), crossover probability (from 0 to 1 in steps of 0.1), and mutation probability (from 0 to 1 in steps of 0.1), approximately 4000 cases were created. An extract of these cases is shown in \begin{table}
\begin{tabular}{c c c} \hline \hline
**Covariates** & **Measurement** & **Description** \\ \hline IFS & A & Failure or working current. \\ C\({}_{2}\)H\({}_{4}\) & ppm & Ethylene gas in the transformer oil. \\ C\({}_{2}\)H\({}_{6}\) & ppm & Ethane gas in the transformer oil. \\ CH\({}_{4}\) & ppm & Methane gas in the transformer oil. \\ GTF & ppm & Typical failure gases in the transformer oil. \\ CO\({}_{2}\) & ppm & Carbon dioxide gas in the transformer oil. \\ Co & ppm & Carbon monoxide gas in the transformer oil. \\ TCC & ppm & Total combustible gases in the transformer oil. \\ TGC-CO & ppm & Total combustible gases excluding Co in the transformer oil. \\ R.Die & KV & Dielectric strength of the transformer oil in kilovolt. \\ \% Hum & \% & Humidity percentage in the interior of the transformer. \\ \hline \hline \end{tabular}
\end{table} 
Table 1: _Cont._ \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Date** & **FT** & **Interv** & **ID** &
\begin{tabular}{c} **Dda** \\ **(MVA)** \\ \end{tabular} & **Tint (\({}^{\circ}\)C)** & \(\mathbf{C_{2}H_{4}\) **(\%)** & **R.Die (kV)** \\ \hline
259,180 & 162,770 & 0 & 1 & 18.1 & 64 & 2.1392 & 64.1948 \\
259,610 & 163,210 & 0 & 1 & 16.4 & 54 & 2.1082 & 64.0284 \\
283,300 & 186,900 & 0 & 1 & 16.6 & 59 & 3.3276 & 63.7482 \\
270,580 & 156,630 & 0 & 2 & 8.9 & 53 & 1.6896 & 66.2913 \\
271,010 & 157,060 & 0 & 2 & 12.8 & 50 & 1.7452 & 65.9824 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Representation of the dataset. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(\beta\) & \(\eta\) & \(\gamma_{Dda}\) & \(\gamma_{Tint}\) & \(\gamma_{dmd}\) & \(\gamma_{R,Die}\) \\ \hline
1.4913 & 117,531 & 0.0038 & 0.4012 & 0.0404 & 2.7561 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Covariate weights and Weibull parameters obtained by the _fmincon_ method. Table 4 with the top best and worst cases. The optimal results indicated the selection of a population of 500 genes, the crossover probability was set to one, the gene distribution of each offspring was set to 0.1, the mutation probability was equal to 0.7, and each value in the offspring underwent mutation by applying a Gaussian additive mutation with a mean of zero, a standard deviation of one, and an independent probability of 0.3 for mutation occurrence. Finally, 1000 generations were taken into account. A total of eight cases were defined for the analysis. Case 1 corresponded to the original solution provided by the Chilean company, as detailed in Table 3. Case 2 represented the optimal solution identified based on the partial log-likelihood score from [3], utilizing the optimization parameters IPOPT, NS transformation, and IPCRidge. The remaining cases involved sensitivity analyses applied to the optimization parameters of case 2. Table 5 outlines the optimization parameters, log-likelihood scores, Weibull parameters, log-likelihood scores, and resulting covariate weights used for each solution.