## 3 Preventive Maintenance Modeling for Most Critical Machines Based on Delay-Time Theory  
As mentioned above, the most critical machines as evaluated in Section 2 have great influence on the operating rate of the automatic production line, and they will bring enormous economic losses to production enterprises, if the downtime of the most critical machines is too long. If the preventive maintenance is carried out separately for each of the most critical machines, which causes the frequent shutdown to the production line. Therefore, based on the theory of synchronous maintenance [23], all of the most critical machines are maintained in a group in order to reduce the frequency of the preventive maintenance. In the present study, the shortest shutdown time was regarded as the decision goal of synchronization maintenance for the most critical machines, and then the balance between the shutdown time of failure and preventive maintenance period was found, so as to minimize the total shutdown time. The relationship between the total expected value of shutdown time \(ED(T_{m})\) per unit time and the preventive maintenance period \(T_{m}\) for the most critical machines can be expressed:  
\[ED(T_{m})=\frac{t_{f}EN_{f}(T_{m})+t_{p}}{T_{m}}, \tag{4}\]  
where \(t_{f}\) is the average failure shutdown time, \(t_{p}\) is the average preventive maintenance shutdown time, and \(EN_{f}(T_{m})\) is the expected failure number for all most critical machines in the preventive maintenance period. The \(t_{f}\) and \(t_{p}\) can be gained by the failure data and maintenance data of the most critical machines. Then, the \(T_{m}\) is gained when \(ED(T_{m})\) is the shortest. According to the delay-time theory, the machine in the production line may gradually generate the potential failure, which can't cause machine shutdown temporarily. However, if the potential failure is not maintained in time, it will develop into failure and causes the machine to be shutdown. The time while a potential failure develops into a failure is delay-time \(t_{h}\). On the basis of the Akaike information criterion, the delay-time \(t_{h}\) is exponential distribution [24]. Therefore, the \(EN_{f}(T_{m})\) of the most critical machines can be calculated according to the delay-time model. The key to establishing the delay-time model is to determine the probability of potential failure \(\theta\), delay-time function \(f(h)\) and the probability of potential failure that can be detected by preventive maintenance \(r\). The maximum-likelihood method can be used to calculate the parameters in the delay-time model. Firstly, the preventive maintenance period is divided into \(z\) number of \(\Delta t\) intervals:  
\[I_{l}^{k}=[t_{k-1}+(l-1)\Delta t,\ t_{k-1}+l\Delta t](l=1,\ 2,\cdots,\ z),\ t_{k-1}+z \Delta t=t_{k}, \tag{5}\]  
where \(t_{k-1}\) is the time for the \((k-1)\)th maintenance, and \(t_{k-1}+z\Delta t=t_{k}\). Because the potential failure and its mature failure couldn't generate at the same time, it was assumed that the potential failure and failure were independent events and each potential failure generated independently. Thus, the maximum-likelihood function is expressed:  
\[L=\prod\limits_{k=1}^{n}\{\textit{Probability of potential failure }n_{k}\textit{ generating in }T_{k}\}\prod\limits_{l=1}^{n}\Big{\{}\textit{Probability of failure }m_{kl}\textit{ generating in }I_{l}^{k}\Big{\}}, \tag{6}\]  
where \(n\) is the number of preventive maintenance, \(T_{k}\) is the duration time of \(k\)-th maintenance, \(n_{k}\) is the number of potential failures detected in the \(k\)-th maintenance, and \(m_{kl}\) is the number of failures in the \(l\)-th interval of the \(k\)-th maintenance period. While a potential failure of machine generates at time \(u\), the probability of the potential failure degrades to be a failure in \((t,t+\Delta t)\) interval is:  
\[p(t,\ t+\Delta t|u)=\left\{