# Payment Maintenance Decision Making Based on Optimization Models
## 4 Model Application
### Solving the Models  
Prior to solving the models, the premaintenance PCI, maintenance area, maintenance cost per unit area, and maintenance threshold for each road were first fed into the model as input parameters. Subsequently, the optimization models were implemented, and the values of the decision-making variables and the objective functions were calculated by  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**FID** & **Name** & **Grade** & **Maintenance Area (m\({}^{2}\))** & **PCIi** \\ \hline
1 & AnQing Rd. & Main road & 1984 & 68.63 \\
2 & BaiYanWan Rd. & Branch road & 87 & 98.75 \\
3 & BanQao Rd. & Branch road & 364 & 80.36 \\
4 & YanHe N. Rd. & Branch road & 298 & 86.08 \\
5 & North 1 st Ring Exp. & Expressway & 1002 & 75.63 \\
......... &...... &...... &...... \\
70 & YanHe S. Rd. & Branch road & 236 & 88.98 \\
71 & South 1 st Ring Exp. & Expressway & 320 & 78.69 \\
72 & NingXi Rd. & Branch road & 0 & 100.00 \\
73 & NongXin Rd. & Branch road & 125 & 92.52 \\
74 & PiHe Rd. & Main road & 695 & 71.31 \\
75 & QianShan Rd. & Expressway & 789 & 78.95 \\......... &...... &...... &...... \\
145 & Changjiang Middle & Main road & 302 & 86.36 \\
146 & ZhenXin Rd. & Branch road & 0 & 100.00 \\
147 & ZhiWuYuan E. Rd. & Branch road & 123 & 90.20 \\
148 & ZhiWuYuan S. Rd. & Branch road & 231 & 89.54 \\
149 & HaiTang Rd. & Main road & 33 & 99.15 \\ \hline \hline \end{tabular}
\end{table}
Table 3: PCI and other attributes of roads in Shushan District.  
using sequential quadratic programming and the NSGA-II algorithm. Finally, the optimal pavement maintenance scheme was determined based on the results obtained from the model. The first model was solved using the sequential quadratic programming method.  
Because the multiobjective model could be solved by a GA, the second model was solved by using the NSGA-II algorithm in our research. The calculation steps involved in the NSGA-II algorithm are as follows. First, multiple sets of optimization solutions for pavement maintenance are randomly generated and are referred to as the solution population. Subsequently, these solution sets are ranked based on the maximum pavement maintenance quality and the minimum pavement maintenance cost calculated for each set of solutions. This process is also known as rapid nondominated sorting of the solution population. Next, a new generation of solutions is obtained through selection, crossover, and mutation of the solution population. The new solutions are the offspring of the first-generation solutions. Starting from the second generation, the parent and offspring populations are combined to perform rapid nondominated sorting. Simultaneously, the crowding degrees are calculated for each individual in the nondominated layer. Suitable individuals are then selected according to their crowdedness and the nondominated relationship to form a new parent population. Finally, a new offspring population is generated through selection, crossover, and mutation of the new parent population.  
The final solution set of the objective function can be obtained by repeating the aforementioned process continuously until the maximum number of iterations is reached. NSGA-II can select multiple nondominated individuals that form an optimal set of trade-off solutions called the Pareto set. The pavement maintenance scheme can then be formulated according to the solution results. Some researchers have proposed simpler coding methods to represent complex evolutionary phenomena. These methods can realize heuristic searches in the complex search space and can determine the global optimal solution of the objective function with high probability through a simplified genetic process.