## 3 Preventive Maintenance Modeling for Most Critical Machines Based on Delay-Time Theory
Preventive Maintenance Modeling for Secondary Critical Machines Based on Reliability-Maintenance Cost  
The secondary critical machines as evaluated in Section 2 have lower influence on the operating rate of the automatic production line compared with that of the most critical machines, and the production enterprises pay more attention to save maintenance costs for the secondary critical machines. Because the failures of the secondary critical machines may also lead to the shutdown of the automatic production line, it is also necessary to ensure the reliability of the secondary machines to ensure the operating rate of production line. Firstly, the reliability model of the secondary critical machines is established to ensure the machine reliability requirement in the preventive maintenance period. In the reliability model, the failure probability of secondary critical machine needs to be lower than an upper limit. For the \(e\) secondary critical machines evaluated in Section 2, the cumulative failure probability \(F_{m}(T_{s})\) of the \(m\)-th secondary critical machine during the preventive maintenance period \(T_{s}\) for the secondary critical machines is:  
\[F_{m}(T_{s})=1-\exp[-(T_{s}/\alpha_{m})^{\beta_{m}}], \tag{16}\]  
where \(\alpha_{m}\) and \(\beta_{m}\) are the scale parameter and shape parameter in the Weibull distribution, respectively. To ensure that \(F_{m}(T_{s})\) is less than the upper limit \(F_{\max}\), \(T_{s}\) must meet the following condition:  
\[T_{s}=\min\{T_{se}\}=\min\{\alpha_{m}\bigg{[}\ln\bigg{(}\frac{1}{1-F_{\max}} \bigg{)}\bigg{]}^{1/\beta_{m}}\bigg{\}}, \tag{17}\]  
where \(T_{se}\) is the maximum preventive maintenance period for each secondary critical machine under the reliability constraints. Then, the maintenance cost model is established to optimize the total maintenance costs of the secondary critical machines during a production period. It is assumed that the TBF of the machine is Weibull distribution. In the preventive maintenance period of the secondary critical machines, the expected failure number of the \(m\)-th machine is [25; 26]:  
\[EN_{f}^{m}(T_{s})=(\frac{T_{s}}{(\alpha_{m})})^{\beta_{m}}, \tag{18}\]  
The total maintenance costs of machine in a production period \(T_{J}\) are comprised of the preventive maintenance costs and the machine failure loss, which can be expressed as [27]:  
\[TC(T_{s})=nC_{o}+\sum_{m=1}^{e}n_{m}C_{m}+\sum_{m=1}^{e}n_{m}C_{fm}(\frac{T_{ se}}{\alpha_{m}})^{\beta_{m}}, \tag{19}\]  
where \(n\) is the number of preventive maintenance in the production period, \(C_{0}\) is the fixed cost for each preventive maintenance, \(n_{m}\) is the number of preventive maintenance for the \(m\)-th secondary critical machine, \(C_{m}\) is the cost of parts replacement for the preventive maintenance of the \(m\)-th secondary critical machine, \(C_{fm}\) is the average cost of the breakdown maintenance for the \(m\)-th secondary critical machine. In order to minimize the maintenance costs based on meeting the requirement of machine reliability, the multi-objective function of reliability-maintenance cost is proposed:  
\[ \begin{array}{l}\max T_{s};\\ \min TC(T_{s})=nC_{0}+\sum\limits_{m=1}^{e}N_{m}C_{m}+\sum\limits_{m=1}^{e}N_{ m}C_{fm}(\frac{T_{PMm}}{\alpha_{m}})^{\beta_{m}};\\ T_{se}\geq T_{s}\times Y_{m},\;F_{m}(T_{s})\leq F_{\max};\\ T_{se}\leq\alpha_{m}\Big{\{}\ln\Big{[}\frac{1}{1-F_{\max}}\Big{]}\Big{\}}^{1/ \beta_{m}}\end{array} , \tag{20}\]  
where \(Y_{m}=[T_{se}/T_{s}]\), namely \(Y_{m}\) is the integer value of \(T_{se}/T_{s}\). \(T_{PMm}\) is the preventive maintenance period of the \(m\)-th secondary critical machine and \(T_{PMm}=T_{s}\times Y_{m}\), \(N_{m}\) is the number of preventive maintenance for the \(m\)-th machine in the production period and \(N_{m}=[T_{j}/T_{PMm}]\), \(n\) is the number of preventive maintenance in the production cycle and \(n=\max\{N_{m}\}\).