# Preventive Maintenance Decision-Making Optimization Method for Airport Runway Composite Pavements
## 2 Methods
### Long-Term Maintenance Decision-Making Optimization Method
#### 2.3.4 State Transfer Variable  
The state transfer equation describes the transformation from a certain state to the next being affected by the decision variable. According to the principle of DP, the state variable \(s_{k+1}\) in stage \((k+1)\) is entirely determined by the state variable \(s_{k}\) and decision variable \(x_{k}\) in stage \(k\). The state transfer equation is expressed as \(s_{k+1}=T_{k}(s_{k},x_{k})\). The decision variable \(x_{k}\) is 0 when no maintenance is implemented on composite pavements or 1 otherwise. Thus, the state transfer equation of \(s_{k}^{1}\) and \(s_{k}^{2}\) is a piecewise function related to \(x_{k}\).  
When \(x_{k}\) is 0 at the beginning of a stage, no maintenance is implemented on the composite pavement region, and thus, both the improvement of the _PCI_ and the additional cost of maintenance are 0 until the next stage. As for \(s_{k+1,n1}^{1}(PCI_{k=t+1,n1})\), if no preventive maintenance is implemented on region \(n\) at the beginning of stage \((t+1)\), its variation will follow the tendency of the \(PCI_{k=t,ni}\) curve. \(PCI_{t+1>k>t,n1}\) can be calculated based on Equations (1)-(6) and the 13 regression coefficients in Table 3, which vary as \(ESAL\) during stage \((t+1)\). The state transfer equation of \(s_{k=t+1,ni}^{1}\) (\(i=0\)) is shown in Equation (10):  
\[T_{t+1>k>t,ni}^{1}=s_{t+1>k>t,ni}^{1}-s_{k=t,ni}^{1}=PCI_{t+1>k>t,ni}-PCI_{k=t,ni} \tag{10}\]  
As for \(s_{k=t+1,n1}^{2}(c_{k=t+1,n1})\), it remains the same as \(s_{k=t,n1}^{2}\) because of no additional maintenance costs in stage \((t+1)\). The state transfer equation of \(s_{k=t+1,n1}^{2}\) is shown in Equation (11):  
\[T_{t+1>k>t,ni}^{2}=s_{k=t+1,ni}^{2}-s_{k=t,ni}^{2}=c_{k=t+1,ni}-c_{k=t,ni}=0 \tag{11}\]  
When \(x_{k}\) is 1, certain maintenance is implemented on the composite pavement region, which influences both the deterioration tendency of _PCI_ and the preventive maintenance cost. As for \(s_{k=t+1,n1}^{1}(PCI_{k=t+1,n1})\), a jump in _PCI_ (_PJ_) occurs immediately after maintenance, followed by a slower deterioration (\(DRR\)) during stage \((t+1)\). The \(PCI_{k=t+1,n1}\) curve after various types of preventive maintenance can be calculated from corresponding \(PI\) regression models and \(DRR\) statistical results. The state transfer equation of \(s_{k=t+1,ni}^{1}\)(\(i=1\) to 5) is the same as Equation (10).  
As for \(s_{k=t+1,n1}^{2}(c_{k=t+1,n1})\), it equals \(s_{k=t,ni}^{2}\) plus \(\Delta c_{k=t+1,ni}\), which is the additional maintenance cost in stage \((t+1)\). \(\Delta c_{k=t+1,ni}\) is calculated based on Equation (12) considering inflation:  
\[\Delta c_{k=t+1,ni}=\Delta c_{k=t,ni}(1+R_{k=t})=\Delta c_{k=1,ni}\prod_{k=1} ^{t}\left(1+R_{k}\right) \tag{12}\]  
where \(\Delta c_{k=1,ni}\) is the cost of maintenance method \(i\) implemented in region \(n\) at the beginning of the initial stage (\(k=1\)), and \(R_{k}\) is the inflation rate in stage \(k\).  
Thus, the state transfer equation of \(s_{k=t+1,ni}^{2}\) (\(i=1\) to 5) is as shown in Equation (13):  
\[T_{t+1>k>t,ni}^{2}=s_{k=t+1,ni}^{2}-s_{k=t,ni}^{2}=\Delta c_{k=1,n1}\prod_{k=1 }^{t}\left(1+R_{k}\right) \tag{13}\]