# Joint Maintenance Strategy Optimization for Railway
## 3 Optimization Model of the Joint Maintenance Strategy
### Deterioration Model  
Considering the wear of the wheel caused by stochastic shock, the non-homogeneous Poisson process describes the general stochastic process well and is often used to express the rule of external action [26]. The number of shocks \(N(t)\) occurring within the time interval \((t_{k-1},\ t]\) satisfies  
\[P(N(t){=}\ n)=\frac{[at^{b}-at_{k-1}^{b}]^{n}}{n!}{\exp}\Big{\{}-[at^{b}-at_{k- 1}^{b}]\Big{\}},\ n= 0,1,2,\ldots \tag{1}\]  
where \(a>0\) represents the number of external shocks per unit of time, and \(a=a^{f}\) and \(a=a^{d}\) represent the wheel flange and diameter. \(b>0\) represents the relationship between the number of shocks and the working time. Similarly, \(b=b^{f}\) and \(b=b^{d}\) indicate the wheel flange and diameter, respectively.  
The damage of each shock obeys the same and independent normal distribution, i.e., the amount of deterioration caused by the \(i\)-th shock is \(Y_{i}\ \sim\ (\mu,\sigma^{2})\), \(\mu=\mu^{f}\) and \(\mu=\mu^{d}\) indicate the mean value of the amount of deterioration of each shock on wheel flange and diameter and \(\sigma^{2}{=(\sigma^{f})^{2}}\) and \(\sigma^{2}{=(\sigma^{d})^{2}}\) indicate the variance of the performance degradation of each shock of the wheel flange and diameter, respectively. Generally, the number of shocks to the wheelset and the amount of deterioration affected by the shock is related to the operating environment and the wheel manufacturing level, so they can be considered independent of each other.  
Therefore, the deterioration \(X_{t_{k-1},t}\) in time interval \((t_{k-1},\ t]\) is the accumulation of \(Y_{i}\) caused by \(N_{t_{k-1},t}\) shocks, shown as  
\[X_{t_{k-1},t}=\sum_{i=1}^{N_{t_{k-1},t}}Y_{i} \tag{2}\]  
The stochastic process \(X_{t_{k-1},t}\) is a compound Poisson process. Using the total probability formula, the distribution function of the deterioration \(X_{t_{k-1},t}\) is expressed as  
\[P(X_{t_{k-1},t}<x)=P(\sum_{i=1}^{N_{t_{k-1},t}}Y_{i}<x)=\sum_{n=1}^{\infty}P(N _{t_{k-1},t}=n)P(\sum_{i=1}^{n}Y<x) \tag{3}\]  
In order to facilitate the calculation, \(X_{t_{k-1},t}\) is considered a linear combination of the infinite number of independent normal distributions. Considering the central limit theorem, it can approximately be expressed as a normal distribution, i.e.,  
\[X_{t_{k-1},t}\sim N(\mu_{t_{k-1},t},\sigma^{2}_{t_{k-1},t}) \tag{4}\]  
where \(\mu_{t_{k-1},t}\) and \(\sigma^{2}_{t_{k-1},t}\) are the time-varying parameters. Based on the characteristic of the compound Poisson process, \(\mu_{t_{k-1},t}=a\mu(t^{b}-t^{b}_{k-1})\), \(\sigma^{2}_{t_{k-1},t}=a(\mu^{2}+\sigma^{2})(t^{b}-t^{b}_{k-1})\).  
For the wheel flange and wheel diameter, the amount of deterioration within the \(i-\mathrm{th}\) maintenance cycle is \(X^{f}_{(i-1)\Delta t,i\Delta t}\) and \(X^{d}_{(i-1)\Delta t,i\Delta t}\), respectively. If the last reprofiling of the wheel is in the \(k\)-th (\(1\leq k<i\)) maintenance cycle, the daily inspection and the periodic inspection cannot eliminate the deterioration. The wear of the wheel flange at time \(i\Delta t\) is the accumulation within \((k\Delta t,\ i\Delta t]\), i.e., \(\sum_{j=k}^{i-1}X^{f}_{j\Delta t,(j+1)\Delta t}\). For the wheel diameter, the wear is accumulated continuously, and the amount of deterioration at time \(i\Delta t\) is \(\sum_{j=1}^{i}X^{d}_{(j-1)\Delta t,j\Delta t}\).  
The tread receives damage such as scratches due to sliding or other reasons during wheel running, and the wheel surface is inspected after the vehicle returns back to the depot. Assuming that the surface failure of the wheelset follows the Weibull distribution, the probability density function is \(f(t)\), as follows:  
\[f(t)=\frac{m}{\eta}(\frac{t}{\eta})^{m-1}\exp[-(\frac{t}{\eta})^{m}],t\geq 0 \tag{5}\]  
where \(m\) is the shape parameter, and \(\eta\) is the scale parameter.