# Preventive Maintenance Decision-Making Optimization Method for Airport Runway Composite Pavements
## 2 Methods
### Optimization Incremental Dynamic Programming Algorithm  
Multi-objective decision-making optimization based on DP is a multi-stage solution process. Incremental DP (IDP) is a widely used backward algorithm from the final stage to the initial stage. In other words, the maintenance decision of the final stage is made first, followed by that of the former stage successively. Finally, decisions for all stages are made, and the long-term maintenance plan is completed during the entire period. Considering both remaining pavement service performance and reducing the influence of inflation, maintenance should be implemented at the beginning of each stage. Thus, the \(PCI\) at the beginning of stage \(k\) (\(s_{kni}^{1}\)) equals the \(PCI\) at the end of stage \((k-1)\) (\(s_{(k-1)ni}^{1}\)) plus the \(PJ\) calculated from the regression models. Moreover, the \(PCI\) at the end of stage \(k\) (\(s_{kni}^{1}\)) equals \(s_{kni}^{1}\) minus the deterioration value, which is less than that with no maintenance by \(DRR\) (\(\%\)). In the first step of iterative computation based on IDP, \(s_{k-K,n,i=1}^{1}\) and \(s_{k-K,n,i=1}^{2}\) at the final stage, are calculated from the state transfer equation when \(x_{kni}\) is 0, which means no maintenance is implemented during the whole planning period. This is the most economical maintenance plan because of zero cost, but it perhaps does not meet the requirement of \(PCI_{\min}\). Thus, the implementation of some maintenance technology may be required in advance. The maintenance decision combination of all the regions in stage \(k\) is \(p_{k}=\bigl{[}i_{n=1}\quad i_{n=2}\quad\cdots\quad i_{n=N}\bigr{]}\). The set of non-inferior solutions in stage \(K\) consists of all \(p_{K}\) satisfying constraints from Equations (19)-(22). The optimal solution \(p_{K}^{*}\) is determined as shown in Figure 1, where the dots represent the \(C_{K}\) and \(\overline{PCI_{K}}\) corresponding to each \(p_{K}\); the horizontal and vertical lines represent the limits of \(C_{\max,K}\) and \(PCI_{\min}\); and the dashed oblique line is the optimal split line. The vertical distance between the dot and the optimal split line is positive if the dot is above the line otherwise the opposite is true. Furthermore, \(p_{K}^{*}\) corresponds to the dot where the maximum vertical distance exists. In the subsequent steps, the set of non-inferior solutions in stage \(k\) (\(k<K\)) consists of \(p_{k}\) and \(p_{k+1}^{*},\ldots,\)\(p_{K}^{*}\), and \(p_{k}^{*}\) is determined using the same method. Based on the IDP backward algorithm, the optimal preventive maintenance plan could be determined, which is a combination of \(p_{1}^{*},\ldots,p_{K}^{*}\). For composite pavement consisting of \(N\) regions, which can be treated using \(I\) types of maintenance technology, there are \((I+1)^{N}\) types of decision combinations per stage as well as numerous calculations. In terms of long-term preventive maintenance decision-making, calculation costs must be reduced via optimizing the IDP algorithm. IDP consists of two parts: a set of non-inferior solutions and the optimal solution. The two calculation processes were optimized as follows. In the calculation process of non-inferior solutions, some \(p_{k}\) may exceed the constraint of \(C_{\max,k}\), or different \(p_{k}\) may correspond to the same \(\overline{PCI}_{k}\) and \(C_{k}\). Taking a 3-year preventive maintenance plan as an example, the composite pavement consists of three regions and two available maintenance technologies. \(PCI_{\min}\) is 90 and \(C_{\max,k=3}\) is 13, and the costs of the three decision variables \(x_{0}\), \(x_{1}\), and \(x_{2}\) are 0, 5, and 8, respectively. Furthermore, the \(P\) is 0, 4, and 6, respectively, when \(PCI_{b}\) is 94. Figure 2a plots the deterioration of the \(PCI\). Considering the constraint \(C_{\max,k=3}=13\), 16 decision combinations are available, much fewer than that (3\({}^{3}\) = 27 combinations) without \(C_{\max,k=3}\).