# 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}C(i^{m-1}\Delta t,i^{m}\Delta t)&=\left\{\begin{array}{ ll}Ls\leq X(i^{m}\Delta t),X((i^{m-1}+1)\Delta t)<Lp,\ldots,X((i^{m}-1)\Delta t)<Lp, \\ Q(i^{m-1}\Delta t,i^{m}\Delta t),i^{m-1}+1<i^{m}\\ &=\left\{\begin{array}{ll}Ls\leq X_{T_{m-1},T_{m-1}+(i^{m}-i^{m-1})\Delta t },X_{T_{m-1},T_{m-1}+\Delta t}<Lp,\ldots,X_{T_{m-1},T_{m-1}+(i^{m}-i^{m-1}-1) \Delta t}<Lp,\\ Q(T_{m-1},T_{m-1}+(i^{m}-i^{m-1})\Delta t)\end{array}\right.\end{array}\]  
Combining the above two situations, the probability of \(P(C(i^{m-1}\Delta t,i^{m}\Delta t))\) satisfies  
\[P\left(C\left(i^{m-1}\Delta t,i^{m}\Delta t\right)\right)=\left\{\begin{array} []{ll}\left[1-\Phi\left(\frac{Ls-i^{\prime}I^{\prime}_{m-1},T_{m-1}+\Delta t }{\sigma^{\prime}_{T_{m-1},T_{m-1}+\Delta t}}\right)\right],\left[1-f\right]T ^{m-1+\Delta t}_{m-1}f(t)dt\right],i^{m-1}+1\\ \left[1-\Phi\left(\frac{Ls-i^{\prime}I^{\prime}_{m-1},T_{m-1}+(i^{m}-i^{m-1 })\Delta t}{\sigma^{\prime}_{T_{m-1},T_{m-1}+(i^{m}-i^{m-1})\Delta t}}\right) \right]\\ \prod\limits_{p=1}^{i^{m}-\Delta t-1}\Phi\left(\frac{Lp-i^{\prime}I^{\prime} _{T_{m-1},T_{m-1}+(i^{m}-i+i)\Delta t}}{\sigma^{\prime}_{T_{m-1},T_{m-1}+(i^{ m}-1+i)\Delta t}}\right)\cdot\left[1-\int_{T_{m-1}}^{T_{m-1}+(i^{m}-i^{m}-1) \Delta t}f(t)dt\right],i^{m}>i^{m-1}+1\end{array}\right. \tag{11}\]  
The probability of the wheel flange thickness exceeding the limit under the condition of the last corrective maintenance  
Event \(C(t^{m-1},i^{m}\Delta t)\) represents the \((m-1)\)-th maintenance at time \(t^{m-1}\), and \(t^{m-1}\in((k~{}-~{}1)\Delta t,~{}k\Delta t]\). In the time interval \([k\Delta t,(i^{m}-1)\Delta t]\), the wheel flange thickness deterioration is less than \(Lp\), reaching or exceeding \(Ls\) in the \(i^{m}\)-th maintenance cycle.  
According to the relationship between \(k\Delta t\) and \(i^{m}\Delta t\), the following two cases are discussed.  
1. If \(k=i^{m}\), corrective maintenance is in the same cycle as wheel flange thickness and is over the limit, and event \(C(t^{m-1},i^{m}\Delta t)\) meets \[\begin{array}{ll}C(t^{m-1},i^{m}\Delta t)&=\left\{Ls\leq X(i^{m}\Delta t ),Q(t^{m-1},i^{m}\Delta t),t^{m-1}<k\Delta t=i^{m}\Delta t\right.\\ &=\left\{Ls\leq X_{T_{m-1},T_{m-1}+(i^{m}\Delta t-i^{m-1})},Q(T_{m-1},T_{m-1}+( i^{m}\Delta t-t^{m-1}))\right.\end{array}\]
2. If \(k~{}<~{}i^{m}\), it denotes that the cycle in which the corrective maintenance is performed is not the same as the wheel flange thickness exceeding the limit. Event \(C(t^{m-1},~{}i^{m}\Delta t)\) satisfies \[\begin{array}{ll}C(t^{m-1},i^{m}\Delta t)&=\left\{Ls\leq X(i^{m}\Delta t ),X(k\Delta t)<Lp,\ldots,X((i^{m}-1)\Delta t)<Lp,Q(t^{m-1},i^{m}\Delta t),t^{m-1 }<k\Delta t<i^{m}\Delta t\\ &=\left\{Ls\leq X_{T_{m-1},T_{m-1}+(i^{m}\Delta t-i^{m-1})},X_{T_{m-1},T_{m-1}+( k\Delta t-i^{m-1})}<Lp,\ldots,X_{T_{m-1},T_{m-1}+((i^{m}-1)\Delta t-i^{m-1})}<Lp,Q(T_{m-1},T_{m-1} +i^{m}\Delta t-t^{m-1})\right.\end{array}\]  
To summarize, the occurrence probability \(P(C(t^{m-1},~{}i^{m}\Delta t))\) is obtained as \[P\big{(}C\big{(}t^{m-1},i^{m}\Delta t\big{)}\big{)}=\left\{\begin{array}{l} \left[1-\Phi\left(\frac{Ls-t_{m-1}^{f}t_{m-1,1}\tau_{m-1+(i^{m}\Delta t-i^{m-1})}}{ \sigma_{T_{m-1},T_{m-1}+(i^{m}\Delta t-i^{m-1})}^{f}}\right)\right]\cdot\left[ 1-f_{m-1}^{T_{m-1}+(i^{m}\Delta t-i^{m-1})}f(t)dt\right]\right.k\Delta t=i^{m} \Delta t\\ \left[1-\Phi\left(\frac{Ls-t_{m-1}^{f}t_{m-1,1}t^{m}(\Delta t-i^{m-1})}{\sigma _{T_{m-1},T_{m-1}+(i^{m}\Delta t-i^{m-1})}^{f}}\right)\right]\\ \frac{i^{m}-k}{\prod\limits_{p=1}^{m}\Phi\left(\frac{Lp-t_{m-1}^{f}t_{m-1,1}t^ {m}+(i^{m}\Delta t-i^{m-1})}{\sigma_{T_{m-1},T_{m-1}+(i^{m}\Delta t-i^{m-1})}^ {f}}\right)\cdot\left(1-f_{T_{m-1}+(i^{m}\Delta t-i^{m-1})}^{f}f(t)dt\right) \right.}t^{m-1}<k\Delta t<i^{m}\Delta t\end{array}\right. \tag{12}\]  
In addition, the probability that the diameter of the wheel exceeds the limit also needs to be calculated.  
(7) The probability of the wheel diameter exceeding the limit  
When carrying out correction maintenance, if \(Sd_{b}>\ Std_{H}\), the wheel diameter reduction by reproriling is about 1.5-2 mm to eliminate the tread surface damage. Otherwise, the reduction in the wheel diameter is \((Sd_{H}-Sd_{b})\ \times\ B\), where \(B\) is the proportional coefficient, representing the cost of the wheel diameter to restore 1 mm of flange thickness. Therefore, the reduction in the diameter at the \(m\)-th reproriling is  
\[Cut(m)=\left\{\begin{array}{l}1.5\sim 2,Sd_{b}\geq Sd_{H}\\ (Sd_{H}-Sd_{b})\times B,Sd_{b}<Sd_{H}\end{array}\right.\]  
Event \(L(t^{m-1},\ i\Delta t)\) means that the \((m-1)\)-th reproriling occurs at time \(t^{m-1}\), where \(t\in((k\ -\ 1)\ \Delta t,\ k\Delta t]\), and the actual age is returned to \(T_{m-1}\). The amount of wheel diameter deterioration exceeds the limit \(Lsw^{l}\) at time \(i\Delta t\). The wheel diameter limit threshold \(Lsw^{l}\) is updated after the wheel is reprorfiled at \(t^{m-1}\). The initial limit is \(Lsw\), and \(Lsw^{l}\)= \(Lsw-\sum\limits_{j=1}^{m-1}\text{Cut}(j)\).  
The relationship between the two maintenance cycles is discussed in the following, which has three cases.  
1 If \(i=k\), the reproriling and periodic inspection occur in the same cycle, and then  
\[\begin{array}{ll}L(t^{m-1},i\Delta t)&=\left\{Xw(t,k\Delta t)\geq Lsw^{l},k =i\\ &=\left\{X_{t^{m-1},i\Delta t-i^{m-1}}^{f}\geq Lsw^{l}\right.\end{array}\right.\]  
where \(Xw(t_{1},t_{2})\) represents the wear amount of the wheel diameter in the time interval \((t_{1},t_{2})\).  
2 If \(i=k+1\), the cycle in which the reproriling is carried out is adjacent to the periodic inspection cycle.  
\[\begin{array}{ll}L(t^{m-1},i\Delta t)&=\left\{Xw(t^{m-1},i\Delta t)\geq Lsw ^{l},Xw(t^{m-1},k\Delta t)<Lsw^{l},i=k+1\right.\\ &=\left\{X_{T_{m-1},T_{m-1}+(i\Delta t-i^{m-1})}^{d}\geq Lsw^{l},X_{T_{m-1},T_ {m-1}+(i\Delta t-i^{m-1})}^{d}<Lsw^{l},\ldots,X_{T_{m-1},T_{m-1}+((i-1)\Delta t -i^{m-1})}^{d}<Lsw^{l}\right.\\ \left.\end{array}\right.\end{array}