# Joint Maintenance Strategy Optimization for Railway
## 4 Numerical Analysis
### Parameter Setting  
Taking a wheel as an example, according to the maintenance regulations, the wheel flange thickness limit is in the range of 26 mm \(\leq Sd\leq\) 32 mm, and the wheel diameter should be in the range of 790 mm \(\leq D\leq\) 860 mm. In addition, damages such as scratches and spalling of the tread may cause the wheel to be emergency reprofiled and polished. The inspection and maintenance standards for the wheel tread are stated in Table 3[39]. The actual historical measurement data of the vehicle wheel were studied, and the wheel diameter and wheel flange thickness were measured over a period of time, as shown in Figure 7. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Item** & **Check Items** & **The Reprofilling Standard** \\ \hline \multirow{8}{*}{Tread} & & 1. Scratch limit: \\  & & More than one scratch length \(\leq\) 75 mm \\  & & The length of more than two scratches is between 50\(\sim\)75 mm \\  & Check the wheel tread for & The length of more than four scratches is between 25–50 mm \\  & scratches and spalling & Damage depth > 0.8 mm \\  & & 2. Spalling limit: \\  & & One spalling length \(\leq\) 30 mm \\  & & Two spalling damage lengths \(\leq\) 20 mm \\  & & Spalling depth \(\leq\) 1 mm \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Tread repair standards. Figure 7: Measurement data of wheel diameter and flange thickness. The parameters of the shock deterioration model are estimated by maximum likelihood estimation. The degradation value at time \(t_{i}\) (\(i=1,2,3,\dots\) ) is \(y_{i}\). By combining parameters, the performance degradation rule can be expressed as  
\[P(y_{i}\leq Y)=\Phi(\frac{Y-g(t^{b}-T_{j}^{b})}{\sqrt{h(t^{b}-T_{j}^{b})}}) \tag{25}\]  
where \(g\) and \(h\) are the mean and variance changes in a unit of time, respectively, i.e., \(g=a\mu\) and \(h=a(\mu^{2}+\sigma^{2})\). \(b\) indicates the relationship between the number of shocks and the working time. \(Y\) is the degradation threshold, and \(T\) is the time of the last maintenance. Therefore, if parameters \(g\), \(h\) and \(b\) are determined, the likelihood function can be constructed and simplified it to obtain  
\[ \begin{array}{l}L(g,h,w\bigg{|}(t_{1},y_{1}),(t_{2},y_{2}),\dots,(t_{j},y_{j} ))=\frac{1}{2\pi l^{2}}[\frac{1}{|h_{1}t_{1}^{b}(t_{2}^{b}-t_{1}^{b})\dots(t_{ j}^{b}-t_{j}-1^{b})}]^{1/2}\\ \exp\biggl{\{}-\frac{1}{2h}[(\frac{y_{1}^{2}}{t_{1}^{b}}+\frac{y_{2}^{2}}{(t_ {2}^{b}-t_{1}^{b})}+\dots+\frac{y_{j}^{2}}{(t_{j}^{b}-t_{j-1}^{b})})+g^{2}t_{ j}^{b}-2g\frac{g}{i=1}\frac{g}{i}y_{i}]\biggr{\}}\end{array}  \tag{26}\]  
The logarithm at both ends of the above equation, as well as the derivative of parameters \(g\), \(h\) and \(w\), are taken. The derivative formula is set to zero, and the following Equation (27) is obtained. The equations are solved to obtain the estimated value \(\hat{g}\), \(\hat{h}\) and \(\hat{b}\). \[\left\{ \begin{array}{l}gt_{j}^{b}-\sum\limits_{i=1}^{S}y_{i}=0\\ jh-2g\sum\limits_{i=1}^{S}y_{i}-g^{2}t_{j}^{b}+\biggl{(}\frac{y_{1}^{2}}{t_{1} ^{b}}+\frac{y_{2}^{2}}{t_{2}^{b}-t_{1}^{b}}+\dots+\frac{y_{j}^{2}}{t_{j}^{b}- t_{j-1}^{b}}\biggr{)}=0\\ g^{2}t_{j}^{b}\ln t_{j}+h\biggl{(}\ln t_{1}+\frac{t_{2}^{b}\ln t_{2}-t_{1}^{b} \ln t_{1}}{t_{2}^{b}-t_{1}^{b}}+\dots+\frac{t_{j}^{b}\ln t_{j}-t_{j-1}^{b}\ln t _{j-1}}{t_{2}^{b}-t_{1}^{b}}\biggr{)}\\ -\biggl{[}\frac{\ln t_{1}}{t_{1}^{b}}y_{1}^{2}+\frac{t_{2}^{b}\ln t_{2}-t_{1} ^{b}\ln t_{1}}{\bigl{(}t_{2}^{b}-t_{1}^{b}\bigr{)}^{2}}y_{2}^{2}+\dots+\frac{t _{j}^{b}\ln t_{j}-t_{j-1}^{b}\ln t_{j-1}}{\bigl{(}t_{j}^{b}-t_{j-1}^{b}\bigr{)} ^{2}}y_{1}^{2}\biggr{]}=0\end{array} \right.