## 4 Case Study and Results Discussion
### Analysis 1: Covariate Weight Sensitivity Analysis
In Table 5, the \(Solver\) column indicates the optimization methodology: _finncon_ for the original solver, IPOPT for the traditional solver, and GA for genetic algorithms. The \(Scaler\) column indicates the type of scaler: NS using the custom solution transformation and MinMax employing the scaling with values between zero and one. The \(Bounds\) column specifies the technique for establishing covariate weight constraints: Fixed for bounds set in custom software based on expert criteria and IPCR using the proposed method for bound determination. \(LL\) represents the partial log-likelihood score from Equation (3), where a lower value technically indicates a better solution. Lastly, the Weibull parameters and covariate weights are displayed in the remaining columns. Here, case 1 stood out as having the worst LL score, while case 2 achieved the best score, although it remained to be seen whether this also translated into a good maintenance policy. Furthermore, the cases that utilized the IPOPT solver outperformed those that used the GA solver. This suggested that the solver method significantly impacted the quality of the solution, as evidenced by similar \(LL\) scores and \(\beta\) and \(\eta\) values. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Population Size** & **Crossover Probability** & **Mutation Probability** & **LL** \\ \hline
500 & 1 & 0.7 & 757.955 \\
300 & 0.6 & 0.6 & 757.957 \\
500 & 1 & 0.8 & 759.175 \\  & & \(\vdots\) & \\
100 & 0.2 & 1 & 877.462 \\
200 & 0.3 & 0.5 & 880.133 \\
50 & 0.9 & 1 & 883.064 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Extract of the sensitivity analysis to obtain the best GA hyper-parameters, sorted by log-likelihood scores. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**Case** & **Solver** & **Scaler** & **Bounds** & **LL** & \(\beta\) & \(\eta\) & \(\gamma_{Data}\) & \(\gamma_{Init}\) & \(\gamma_{\text{cid}}\) & \(\gamma_{\text{R.D.}}\) \\ \hline
1 & _fmincon_ & NS & Fixed & 760.94 & 1.49 & 117531 & 0.00 & 0.40 & 0.04 & 2.76 \\
2 & IPOPT & MinMax & IPCR & 747.14 & 2.86 & 145706 & 0.00 & 0.00 & 0.08 & 0.07 \\
3 & IPOPT & NS & IPCR & 748.35 & 2.78 & 144569 & 0.00 & 0.00 & 0.07 & 1.96 \\
4 & GA & NS & IPCR & 756.79 & 1.67 & 127421 & 0.00 & 0.19 & 0.05 & 1.05 \\
5 & IPOPT & NS & \(\infty\) & 748.35 & 2.78 & 144571 & 0.00 & 0.00 & 0.07 & 1.96 \\
6 & GA & NS & \(\infty\) & 756.79 & 1.67 & 127414 & 0.00 & 0.19 & 0.05 & 1.05 \\
7 & IPOPT & MinMax & Fixed & 748.31 & 2.78 & 144960 & 0.00 & 0.00 & 0.07 & 0.20 \\
8 & IPOPT & NS & Fixed & 748.35 & 2.78 & 144572 & 0.00 & 0.00 & 0.07 & 1.96 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Optimization parameters, log-likelihood scores, and covariate weights considered for each case. One noteworthy finding was that varying the bound determination method produced nearly identical results in terms of covariate weights and LL score. This was evident in cases 3, 5, and 8, as well as cases 4 and 6. Consequently, the total number of cases was streamlined to five: cases 1, 2, and 7 were designated as M1, M2, and M3, respectively. Cases 3, 5, and 8 were consolidated as M4, while cases 4 and 6 were represented by M5. Table 6 shows these modifications. Before proceeding with the analysis, it is essential to highlight a specific modification applied exclusively to cases using the MinMax scaler (M2 and M3). When calculating the \(f(\gamma,z)\) values for these cases, they had a magnitude of about \(1\times 10^{-2}\). This resulted in centroids with matching magnitudes. Consequently, when assessing the risk using Equation (1), the term \(exp(f(\gamma,z))\) became closer to one, which implied that the conditional reliability was not penalized by the covariates, resulting in curves closely aligned with the reliability computed solely from the Weibull parameters. An example illustrating this can be seen in Figure 2. Here, the blue curve represents the estimated reliability using only the Weibull parameters (without any covariates).