# Joint Maintenance Strategy Optimization for Railway
## 3 Optimization Model of the Joint Maintenance Strategy
### Joint Maintenance Probability Model
#### 3.2.1 Maintenance Probability Model
\begin{array}{ll}\left[1-\Phi\left(\frac {Lp-\mu^{\prime}_{T_{m-1},T_{m-1}+(i^{m}\Delta t),t^{m-1}}}{\sigma^{\prime}_{T_ {m-1},T_{m-1}+(i^{m}\Delta t-t^{m-1})}}\right)\right]\cdot\left[1-\int_{T_{m-1} }^{T_{m-1}+(i^{m}\Delta t-t^{m-1})}f(t)dt\right],k=i^{m}\\ \left[1-\Phi\left(\frac{Lp-\mu^{\prime}_{T_{m-1},T_{m-1}+(i^{m}\Delta t,t^{m-1 })}}{\sigma^{\prime}_{T_{m-1},T_{m-1}+(i^{m}\Delta t-t^{m-1})}}\right)\right] \cdot\left[1-\int_{T_{m-1}}^{T_{m-1}+(i^{m}\Delta t-t^{m-1})}f(t)dt\right],t^{ m-1}<k\Delta t<i^{m}\Delta t\\ \end{array} \right.\]  
(4) The probability of corrective maintenance subjected to the last corrective maintenance event \(B(t^{m-1}\Delta t,\ t^{m})\) performs the \((m-1)\)-th maintenance at time \(t^{m-1}\), where \(t^{m-1}\in((k_{1}-1)\cdot t,\ k_{1}\Delta t]\), and the actual service age is returned to \(T_{m-1}\). At time \(t^{m}\in((k_{2}-1)\cdot t,\ k_{2}\Delta t]\), the damage on the wheel tread is over the limit, and the \(m\)-th reprofiling is performed. The relationship between \(k_{1}\) and \(k_{2}\) has two cases, which are discussed in the following. 1 If \(k_{1}\)= \(k_{2}\), the two corrective maintenance are in the same cycle. \[ \begin{array}{ll}B(t^{m-1},t^{m})&=\left\{\overline{Q}(t^{m-1},t^{m}),k_{1 }=k_{2}\right.\\ &=\left\{\overline{Q}(T_{m-1},T_{m-1}+t^{m}-t^{m-1})\right.\\ \end{array} \]  
2 If \(k_{2}\ \geq\ k_{1}\)+1, the corrective maintenance cycles are different. \[ \begin{array}{ll}B(t^{m-1},t^{m})&=\left\{X(k_{1}\Delta t)<Lp,\ldots,X((k_{2 }-1)\Delta t)<Lp,\overline{Q}(t^{m-1},t^{m}),t^{m-1}<k_{1}\Delta t\leq(k_{2}-1 )\Delta t<i^{m}\right.\\ &=\left\{\begin{array}{ll}X_{T_{m-1},T_{m-1}+(k_{1}\Delta t-t^{m-1})}<Lp, \ldots,X_{T_{m-1},T_{m-1}+[(k_{2}-1)\Delta t-t^{m-1}]}<Lp,\\ \overline{Q}(T_{m-1},T_{m-1}+t^{m}-t^{m-1})\end{array}\right.\\ \end{array} \]  
In the above discussion, the probability of event \(B(t^{m-1}\Delta t,\ t^{m})\) is  
\[ \begin{array}{ll}P(B(t^{m-1},t^{m}))=\left\{\begin{array}{ll}\int_{T_{m-1 }}^{T_{m-1}+(i^{m}\Delta t-t^{m-1})}f(t)dt,k_{1}=k_{2}\\ \prod\limits_{p=1}^{k_{2}-k_{1}}\Phi(\frac{Lp-\mu^{\prime}_{T_{m-1},T_{m-1}+(i _{1}-1)\Delta t-(i_{1}-1)\Delta t-(\mu-t^{m-1})}}{\sigma^{\prime}_{T_{m-1},T_{m- 1}+(i_{1}-1)\Delta t-(m-1)}})\cdot\int_{T_{m-1}}^{T_{m-1}+(i^{m}-t^{m-1})}f(t) dt,t^{m-1}<k_{1}\Delta t\leq(k_{2}-1)\Delta t<t^{m}\\ \end{array}\right.\end{array}  \tag{10}\]  
When the train operation with the wheel flange thickness is below the usage limit, it brings damage and hidden loss to the vehicle. Therefore, the probability that the wheel flange thickness exceeds the limit during a maintenance cycle needs to be calculated. (5) The probability of the wheel flange thickness exceeding the limit under the condition of the last preventive maintenance  
Event \(C(i^{m-1}\Delta t,\ i^{m}\Delta t)\) means that the \((m-1)\)-th preventive maintenance is performed at time \(i^{m-1}\Delta t\), and the actual age is returned to \(T_{m-1}\). In the \(i^{m}\)-th cycle, the wheel flange thickness deterioration exceeds \(Ls\), where \(Ls\) is the maximum wear value of the wheel flange thickness. Similarly, \(Ls=Sd^{m-1}-~{}Sd_{L}\). In particular, \(Ls=SD~{}-~{}Sd_{L}\) at the first maintenance. The following two cases are discussed, which are determined by the two times, \(i^{m-1}\Delta t\) and \(i^{m}\Delta t\). 1. If \(i^{m}=i^{m-1}+1\), it represents that the wheel flange thickness exceeds the limit in the adjacent maintenance cycle after the last reproriling, and event \(C(i^{m-1}\Delta t,~{}i^{m}\Delta t)\) satisfies \[ \begin{array}{ll}C(i^{m-1}\Delta t,i^{m}\Delta t)&=\left\{Ls\leq X(i^{m} \Delta t),Q(i^{m-1},i^{m}),i^{m}\Delta t=(i^{m-1}+1)\Delta t\right.\\ &=\left\{Ls\leq X_{T_{m-1},T_{m-1}+\Delta t},Q(T_{m-1},T_{m-1}+\Delta t)\right. \end{array} \]
2. If \(i^{m}>~{}i^{m-1}+1\), it means that the maintenance cycle of the previous reproriling is not adjacent to the \(i^{m}\)-th maintenance cycle. \[