## 2 Methodology
### Reliability Modeling of Marine Oil System
Assuming that the lifetime follows a normal distribution \(\mathcal{N}(\mu,\sigma^{2})\), The cumulative distribution function is \(F(t;\mu,\sigma)=\Phi\Big{(}\frac{t-\mu}{\sigma}\Big{)}\), the survival function (reliability function) is \(S(t;\mu,\sigma)=1-\Phi\Big{(}\frac{t-\mu}{\sigma}\Big{)}\), and the log-likelihood function is:  
\[\ell_{\text{Normal}^{\prime}}(\mu,\sigma)=\sum_{i=1}^{N}\text{ln}\Bigg{[} \frac{1}{\sqrt{2\pi\sigma}}e^{-\frac{(t_{i}-\mu)^{2}}{2\sigma^{2}}}\Bigg{]}=- N\text{ln}\ (\sqrt{2\pi}\sigma)-\sum_{i=1}^{N}\frac{(t_{i}-\mu)^{2}}{2\sigma^{2}} \tag{16}\]  
Assuming that the lifetime follows an exponential distribution with parameter \(\lambda\) (i.e., mean lifetime \(1/\lambda\) ), its probability density function is \(f(t;\lambda)=\lambda e^{-\lambda t}\), the survival function is \(S(t;\lambda)=e^{-\lambda t}\), and the log-likelihood function is:  
\[\ell_{\text{Exp}}(\lambda)=\sum_{i=1}^{N}\text{ln}\Big{(}\lambda e^{-\lambda t _{i}}\Big{)}=N\text{ln}\lambda-\lambda\sum_{i=1}^{N}t_{i} \tag{17}\]  
By taking the derivative and solving for MLE, we obtain \(\hat{\lambda}=\frac{N}{\Sigma_{n}^{N}t_{i}}\). Assuming that the lifetime follows a Weibull distribution with shape parameter \(\beta\) and scale parameter \(\eta\), its probability density function is \(f(t;\beta,\eta)=\frac{\beta}{\eta}\Big{(}\frac{t}{\eta}\Big{)}^{\beta-1}e^{-(t /\eta)^{\beta}}\), the survival function is \(S(t;\beta,\eta)=e^{-(t/\eta)^{\beta}}\), and the log-likelihood function is:  
\[\ell_{\text{Weiball}}(\beta,\eta)=\sum_{i=1}^{N}\text{ln}\Bigg{[}\frac{\beta }{\eta}\Big{(}\frac{t_{i}}{\eta}\Big{)}^{\beta-1}e^{-\Big{(}\frac{t_{i}}{ \eta}\Big{)}^{\beta}}\Bigg{]}=N\text{ln}\ \beta-N\text{ln}\ \eta+(\beta-1)\sum_{i=1}^{N}\text{ln}\ t_{i}-\sum_{i=1}^{N}\left(\frac{t_{i} }{\eta}\right)^{\beta} \tag{18}\]  
The MLE estimates \(\hat{\beta}\),\(\hat{\eta}\) must be obtained numerically. AIC (Akaike Information Criterion) and BIC (Bayesian Information Criterion) are used to evaluate model performance. The formulas are \(\text{AIC}=-2\ell(\hat{\theta})+2k\) and \(\text{BIC}=-2\ell(\hat{\theta})+k\text{ln}N\), where \(\ell(\hat{\theta})\) is the maximum log-likelihood value. \(k\) is the number of model parameters: NPHM (Normal): \(k=2(\mu,\sigma)\). EPHM (Exponential): \(k=1(\lambda)\). WPHM (Weibull): \(k=2(\beta,\eta)\). \(N\) is the number of samples. The data calculated using this method are shown in \[u_{(t,p)}=\frac{1}{M_{t}}\sum_{m=1}^{M_{t}}q_{(m,t,p)} \tag{11}\]0  
In this study, the reliability modeling of key components within the lubricating oil system is based on appropriate proportional hazards models (PHMs), with failure modes identified from both aging and random failure mechanisms. Although parameters such as lubricating oil temperature, water content, and base number (BN) are not explicitly listed as independent influencing variables, their effects are implicitly considered in the component-level failure modeling. Specifically, these oil characteristics influence the degradation mechanisms of system components. For instance, elevated oil temperature accelerates bearing wear and seal aging in pumps and valves; water contamination contributes to corrosion and scaling in oil coolers; and the decline in BN indicates lubricant degradation, which is closely tied to component aging and reduced lubrication effectiveness. These physical and chemical changes manifest as observable failure modes such as leakage, blockage, and reduced heat exchange efficiency, all of which are accounted for in the failure mode definitions of the respective components. Therefore, the state of the oil at a component's location is inherently reflected in its failure behavior and corresponding hazard rate function. Modeling the oil's influence indirectly through component condition and failure history avoids redundancy while ensuring that its impact on system reliability is not neglected. This approach maintains the simplicity and focus of the model without compromising its fidelity.