## 3 Preventive Maintenance Modeling for Most Critical Machines Based on Delay-Time Theory
\begin{array}{l}(1-r)^{n-k}(F(t+\Delta t-u)-F(t-u )),\ T_{k-1}<u<T_{k},\ k=1,\ 2,\ \cdots,\ n-1\\ F(t+\Delta t-u)-F(t-u),\ T_{n-1}<u<t\\ F(t+\Delta t-u),\ t<u<t+\Delta t\\ 0,\ u>t+\Delta t\end{array} \right., \tag{7}\]  
where \(F(t_{h})=1-\exp(-\eta t_{h})\) is the distribution function of the delay time \(t_{h}\). While the occurrence of potential failure obeys the Homogeneous Poisson process (HPPP) and the occurrence rate is \(\theta\), the failure occurrence rate \(v(t)\) at time \(t\) is expressed:  
\[v(t)=\theta(\sum\limits_{k=1}^{n-1}{(1-r)^{n-k}(F(t-T_{k-1})-F(t-T_{k}))+F(t- T_{k-1})}),\ (T_{n-1}<t\leq T_{n}) \tag{8}\]  
From Equation (8), it can be found that the occurrence of failure obeys the Non-homogeneous Poisson Process (NHPP). Thus, the average number of failures during the time interval \(I_{l}^{k}\) is:  
\[EN_{f}(I_{l}^{k})=\int\limits_{t}^{t+\Delta t}v(t)dt=\theta\int\limits_{t}^{t +\Delta t_{h}-1}(1-r)^{n-k}[F(x-T_{k-1})-F(x-T_{k})]dx+\theta\int\limits_{t}^{ t+\Delta t}F(x-T_{n-1})dx. \tag{9}\]  
The probability of \(m_{kl}\) failures generated during the \(I_{l}^{k}\) is:  
\[P_{kl}=\frac{\left[EN_{f}(I_{l}^{k})\right]^{m_{kl}}e^{-EN_{f}(t,\ t+\Delta t) }}{m_{kl}!}. \tag{10}\]  
Meanwhile, the probability that the potential failure that generates at time \(u\) can be detected in the \(T_{k}\) is expressed:  
\[p(T_{k}|u)=\left\{ \begin{array}{l}(1-r)^{k-g}\times r\times(1-F(T_{i}-u)),\ T_{g-1}<u<T_{g},\ g=1,\ 2,\ \cdots,\ k-1\\ r\times(1-F(T_{k}-u)),\ T_{k-1}<u<T_{k}\\ 0,\ u>T_{k}\end{array} \right.. \tag{11}\]According to Equation (11), the expected number of potential failures that are detected in the \(T_{k}\) can be expressed:  
\[EN_{p}(T_{k})=\theta\int\limits_{0}^{\infty}p(T_{k}|u)\,du=\theta\sum\limits_{l= 1}^{k-1}{(1-r)^{k-l}}{\int_{T_{l-1}}^{T_{l}}{[1-F(T_{k}-u)]du}}+\theta r{\int_{T _{k-1}}^{T_{k}}{[1-F(T_{k}-u)]du}}. \tag{12}\]  
As mentioned above, the occurrence of potential failure is assumed to obey the Homogeneous Poisson process (HPP), and, on the basis of Equation (12), the probability of the \(n_{k}\)-th potential failure detected in the \(T_{k}\) is:  
\[P_{k}=\frac{(EN_{p}(T_{k}))^{n_{k}}e^{-EN_{p}(T_{k})}}{n_{k}!}. \tag{13}\]  
Equations (10) and (13) are taken into Equation (6) and take its natural logarithm, and then the logarithm Likelihood function is gained as follows:  
\[\ln L=\sum\limits_{k=1}^{n}{[n_{k}\ln EN_{p}(T_{k})-EN_{p}(T_{k})]}+\sum \limits_{k=1}^{n}{\sum\limits_{l=1}^{z}{[m_{kl}\ln EN_{f}(l_{l}^{k})]}}-\sum \limits_{k=1}^{n}{\sum\limits_{l=1}^{z}{[EN_{f}(l_{l}^{k})]}}. \tag{14}\]  
The \(m_{kl}\) and \(n_{k}\) obtained based on the failure and maintenance data of the production line are taken into Equation (14), and the value of \(\theta\),\(f(t_{h})\) and \(r\) can be gained by solving the maximum value of \(\ln L\). The \(EN_{f}(T_{m})\) is gained accordingly to take the \(\theta\),\(f(t_{h})\) and \(r\) into Equation (9). Finally, the preventive maintenance period of the most critical machines can be determined by Equation (4). According to the above model, the optimal preventive maintenance period for the most critical machines can be calculated. Meanwhile, it is necessary to ensure the reliability of the most critical machine in the preventive maintenance period that can bring the shortest shutdown time. The time between failure (TBF) of the production line is subject to Weibull distribution [17; 23]:  
\[F_{w}(t)=1-\exp[-(t/\alpha)^{\beta}],t\geq 0. \tag{15}\]  
The maximum likelihood estimation method can be used to solve the scale parameter \(\alpha\) and shape parameter \(\beta\) of each machine. The cumulative failure probability of each machine can be calculated by taking the preventive maintenance period \(T_{m}\) into Equation (15), and then the cumulative failure probabilities are compared with experience value from the literature or specified value from enterprise to confirm.