# Joint Maintenance Strategy Optimization for Railway
## 3 Optimization Model of the Joint Maintenance Strategy
### Joint Maintenance Probability Model
#### 3.2.1 Maintenance Probability Model
The probability of corrective maintenance subjected to the last preventive maintenance event \(B(i^{m-1}\Delta t,\ t^{m})\) represents that the last preventive maintenance is performed at time \(i^{m-1}\Delta t\), and time \(i^{m}\in((k-1)\Delta t,\ k\Delta t)\). The damage on the tread surface is beyond the limit, and reprofiling is adopted. Similarly, based on the relationship between \(i^{m-1}\Delta t\) and \(k\Delta t\), two cases are considered in the following. 1 If \(i^{m-1}=k\ -\ 1\), it denotes that, in the next maintenance cycle after preventive maintenance, corrective maintenance is required, and event \(B(i^{m-1}\Delta t,\ t^{m})\) meets  
\[ \begin{array}{ll}B(i^{m-1}\Delta t,i^{m})&=\left\{\overline{Q}((k-1)\Delta t,i^{m}),i^{m-1}=k-1\right.\\ &=\left\{\overline{Q}(T_{m-1},T_{m-1}+t^{m}-i^{m-1}\Delta t)\right\}\end{array} \]  
where \(\overline{Q}(t_{1},t_{2})\) represents that, from time \(t_{1}\) to \(t_{2}\), tread surface damage is found to be over the limit. 2 If \(i^{m-1}<\ k-1\), it means that the cycle in which the tread surface failure occurs is not adjacent to the \(i^{m-1}\)-th cycle. Event \(B(i^{m-1}\Delta t,\ t^{m})\) satisfies  
\[ \begin{array}{ll}B(i^{m-1}\Delta t,i^{m})&=\left\{\begin{array}{ll}\frac {X((i^{m-1}+1)\Delta t)<Lp,\ldots,X((k-1)\Delta t)<Lp,i^{m}\Delta t<(k-1) \Delta t<t^{m}\\ \overline{Q}(i^{m-1}\Delta t,i^{m})}\\ \end{array}\right.\\ &=\left\{X_{T_{m-1}},T_{m-1}+M<Lp,\ldots,X_{T_{m-1}},T_{m-1}+(k-1-i^{m-1}) \Delta t<Lp,\overline{Q}(T_{m-1},T_{m-1}+t^{m}-i^{m}\Delta t)\right\}\end{array} \]  
To summarize, the probability \(P(B(i^{m-1}\Delta t,\ t^{m}))\) of the event \(B(i^{m-1}\Delta t,\ t^{m})\) is obtained as  
\[ \begin{array}{ll}P(B(i^{m-1}\Delta t,i^{m}))=\left\{\begin{array}{ll} \int^{T_{m-1}+(i^{m}-i^{m-1}\Delta t)}_{T_{m-1}}f(dt)dt,i^{m-1}=k-1\\ \int^{k-1}_{P=1}\Phi(\frac{1-p-f^{m}_{T_{m-1}},T_{m-1}+pM}{\sigma^{m}_{T_{m-1}}, T_{m-1}+pM})f^{T_{m-1}+(i^{m}-i^{m}-i^{m}\Delta t)}_{T_{m-1}}f(t)dt,i^{m-1}<k-1 \end{array}\right.\end{array}  \tag{8}\]  
3. The probability of preventive maintenance subjected to the last corrective maintenance Event \(A(t^{m-1},\ i^{m}\Delta t)\) represents the \((m-1)\)-th corrective maintenance at time \(i^{m-1}\), where \(t^{m-1}\in((k-1)\Delta t,\ k\Delta t)\), and the actual age is returned to \(T_{m-1}\). In the time interval \([k\Delta t,\ (i^{m}-\ 1)\ \Delta t]\), the flange thickness deterioration is less than \(Lp\). Preventive maintenance is required when the amount of flange thickness deterioration is greater than or equal to \(Lp\) at time \(i^{m}\Delta t\). Considering the relationship between \(k\Delta t\) and \(i^{m}\Delta t\), it can be divided into the following two cases. 1 If \(k=i^{m}\), it means that corrective maintenance is performed in the same cycle as preventive maintenance, and the event \(A(t^{m-1},\ i^{m}\Delta t)\) is denoted as  
\[ \begin{array}{ll}A(t^{m-1},i^{m}\Delta t)&=\left\{Lp\leq X(i^{m}\Delta t),Q(t^ {m-1},i^{m}\Delta t),t^{m-1}<k\Delta t=i^{m}\Delta t\right.\\ &=\left\{Lp\leq X_{T_{m-1},T_{m-1}+(i^{m}\Delta t-t^{m-1})},Q(T_{m-1},T_{m-1}+( i^{m}\Delta t-t^{m-1}))\right.\\ \end{array} \]  
2 If \(k<i^{m}\), it indicates that corrective maintenance is in a different cycle from preventive maintenance. In this case, event \(A(t^{m-1},\ i^{m}\Delta t)\) satisfies  
\[ \begin{array}{ll}A(t^{m-1},i^{m}\Delta t)&=\left\{Lp\leq X(i^{m}\Delta t),X( k\Delta t)<Lp,\ldots,X((i^{m}-1)\Delta t)<Lp,Q(t^{m-1},i^{m}\Delta t),t^{m-1}<k \Delta t<i^{m}\Delta t\right.\\ &=\left\{Lp\leq X_{T_{m-1},T_{m-1}+(i^{m}\Delta t-t^{m-1})},X_{T_{m-1},T_{m-1} +(k\Delta t-t^{m-1})}<Lp,\ldots,X_{T_{m-1},T_{m-1}+(i^{m}-1)\Delta t-t^{m-1})} <Lp,Q(T_{m-1},T_{m-1}+i^{m}\Delta t-t^{m-1})\right.\\ \end{array} \]  
Therefore, the probability \(P(A(t^{m-1},\ i^{m}\Delta t))\) of event \(A(t^{m-1},\ i^{m}\Delta t)\) is expressed as  
\[P(A(t^{m-1},i^{m}\Delta t))=\left\{