# Joint Maintenance Strategy Optimization for Railway
## 3 Optimization Model of the Joint Maintenance Strategy
### Joint Maintenance Probability Model
#### 3.2.1 Maintenance Probability Model  
In the proposed maintenance strategy, there are two main reasons for reprofiling the wheel. One is that the wheel tread's surface damage exceeds the limit and is found at daily inspection. The other is that the flange thickness is below \(Sd_{L}\) at periodic inspection. In the following, the probabilities of these two cases are investigated in detail. Let \(Sd^{m}\) and \(Sd_{b}\) donate the flange thickness after the \(m\)-th reprofiling and the flange thickness measured before reprofiling. Generally, \(Sd^{m}=Sd_{H}\), and the value of \(Sd^{m}\) may be different if \(Sd_{b}\) is greater than \(Sd_{H}\) at corrective maintenance. (1) The probability of preventive maintenance subjected to the last preventive maintenance event \(A(i^{m-1}\Delta t,\,i^{m}\Delta t)\) indicates that the amount of flange thickness degradation detected at the periodic inspection time \(i^{m}\Delta t\) is greater than or equal to \(Lp\), and the \(m\)-th preventive maintenance must be performed, where \(Lp\) is the maximum allowable flange thickness degradation amount for preventive maintenance, i.e., \(Lp=Sd^{m-1}-\ Sd_{L}\). In particular, \(Lp=Sd\ -\ Sd_{L}\) at the first maintenance. After the \((m-1)\)-th preventive maintenance, the actual age of the wheel is returned to \(T_{m-1}\). The deterioration of the wheel flange thickness in the interval \((i^{m-1}\Delta t,\ (i^{m}-\ 1)\Delta t]\) is all less than \(Lp\), and there is no failure on the tread surface. With respect to the relationship between the two maintenance times \(i^{m-1}\Delta t\) and \((i^{m}-\ 1)\Delta t\), the discussion is divided into two cases. 1 If \(i^{m}=i^{m-1}+1\), it means that the two maintenance cycles are adjacent. In this case, event \(A(i^{m-1}\Delta t,\ i^{m}\Delta t)\) satisfies  
\[A(i^{m-1}\Delta t,\ i^{m}\Delta t) = \begin{cases}Lp\leq X(i^{m}\Delta t),Q(i^{m-1}\Delta t,i^{m} \Delta t),i^{m}\Delta t=(i^{m-1}+1)\Delta t\\ Lp\leq X_{T_{m-1},T_{m-1}+\Delta t},Q(T_{m-1},T_{m-1}+\Delta t)\end{cases} \]  
where \(X(i\Delta t)\) represents the wear amount of the flange thickness detected at time \(i\Delta t\), and \(Q(t_{1},\ t_{2})\) represents that the tread surface damage does not exceed the limit during the time interval \((t_{1},\ t_{2}]\). 2 If \(i^{m}>\ i^{m-1}+1\), it means that the two preventive maintenance periods are not adjacent to each other, and the deterioration of the wheel flange is less than \(Lp\) in the time interval \((i^{m-1}\Delta t,\ (i^{m}-\ 1)\ \Delta t]\). Event \(A(i^{m-1}\Delta t,\ i^{m}\ \Delta t)\) satisfies  
\[ \begin{array}{ll}A(i^{m-1}\Delta t,i^{m}\Delta t)&=\left\{\begin{array}{ll}Lp \leq X(i^{m}\Delta t),X((i^{m-1}+1)\Delta t)<Lp,\ldots,X((i^{m}-1)\Delta t)<Lp, \\ Q(i^{m}-1\Delta t,i^{m}\Delta t),i^{m-1}+1<i^{m}\\ \end{array}\right.\\ &=\left\{\begin{array}{ll}Lp\leq X_{T_{m-1}},T_{m-1}+(i^{m}-i^{m-1})\Delta t,X_{T_{m-1}},T_{m-1}+M<Lp,\ldots,X_{T_{m-1}},T_{m-1}+(i^{m}-i^{m-1})\Delta t<Lp, \\ Q(T_{m-1},T_{m-1}+(i^{m}\Delta t-i^{m-1}\Delta t))\\ \end{array}\right.\end{array} \]  
To summarize, by adapting the deterioration rule and the age-reduction factor, the probability \(P(A(i^{m-1}\Delta t,\ i^{m}\Delta t))\) of the event \(A(i^{m-1}\Delta t,\ i^{m}\Delta t)\) is obtained as  
\[P(A(i^{m-1}\Delta t,i^{m}\Delta t))=\left\{ \begin{array}{ll}\left[1-\Phi \left(\frac{1-p-f^{m}_{T_{m-1}},T_{m-1}+M}{\sigma^{m}_{T_{m-1}},T_{m-1}+M} \right)\right]\cdot\left[1-f^{m-1}_{T_{m-1}}+M}_{T_{m-1}}(t)dt\right]\cdot i^{ m-1}=i^{m-1}+1\\ \left[1-\Phi\left(\frac{1-p-f^{m}_{T_{m-1}},T_{m-1}+(i^{m}-i^{m-1})\Delta t}{ \sigma^{m}_{T_{m-1}},T_{m-1}+(i^{m}-i^{m-1})\Delta t}\right)\right]\\ \left[1-\Phi\left(\frac{1-p-f^{m}_{T_{m-1}},T_{m-1}+(i^{m}-i^{m-1})\Delta t}{ \sigma^{m}_{T_{m-1}},T_{m-1}+(i^{m}-i^{m-1})\Delta t}\right)\cdot\left[1-f^{m -1}_{T_{m-1}}+(i^{m}-i^{m-1})\Delta t\right.f(t)dt\right]\cdot i^{m}_{j}>i^{m- 1}+1\\ \end{array} \right. \tag{7}\]  
2.