# MILP-Based Approach for High-Altitude Region Pavement Maintenance Decision Optimization
## 6 Extension of the Maintenance Decision Model
### Robust Solution and Analysis  
Obtaining a solution that is minimally affected by uncertainties is the focus of robust optimization. The pessimistic solution obtained under the pessimistic condition is definitely a robust solution, but the corresponding result is likely to be overly conservative. In response to this, [PERSON] et al. [34] proposed a robust optimization method based on the acceptable objective variation range (AOVR). This method requires the decision variables to be independent of the uncertain parameters and aims to find a robust solution such that, under the maximum fluctuation caused by the uncertain parameters, the fluctuation range of the objective function value does not exceed the acceptable range. Based on this, a robust optimization model as shown in Equation (28) is further proposed. \[ \begin{split}\text{min}& F^{\prime}=\mathbf{A}_{*}^{ \text{T}}\mathbf{X}+\mathbf{c}_{*}\\ s.t.&\left\{\begin{array}{l}\sum\limits_{j\in \mathbf{J}}\sum\limits_{i\in\left(X_{i,j}-x_{i,j,*}\right)^{2}\leq\varepsilon \\ \widetilde{\mathbf{e}}_{p}^{\text{T}}\mathbf{X}\leq\widetilde{h}_{p},\forall p \in\left\{1,2,\cdots,N_{c}\right\}\\ \mathbf{X}\in\left\{0,1\right\}\end{array}\right.\end{split}  \tag{28}\]  
where \(\mathbf{A}_{*}\) and \(\mathbf{c}_{*}\), respectively, represent the coefficient vector and constant term vector formed by the nominal values of each uncertain parameter in the optimistic model. \(\mathbf{X}\) represents the decision variable vector. \(\widetilde{\mathbf{e}}_{p}\) and \(\widetilde{h}_{p}\) represent the coefficient vector and constant term of the constraint equation, respectively, taking the parameters corresponding to the pessimistic model. \(N_{c}\) represents the number of constraints. \(x_{i,j,*}\) represents the value of \(x_{i,j}\) in the optimistic model solution. \(\varepsilon\) is the acceptable variation range, which limits the fluctuation range of the solution and the objective function value. This range should be as small as possible (but too small may result in no feasible solution) to obtain a robust solution as close as possible to the optimistic solution. Since \(x_{i,j}\) is a binary variable, the robust optimization model shown in Equation (28) also belongs to the MILP problem, which can be directly solved using the Gurobi solver. For this, based on the parameters in Section 5.1 and Strategy 5, we set \(\widetilde{C}_{j}\)= \([0.95\times C_{j},1.05\times C_{j}]\) (i.e., the cost variation coefficient is between 0.95 and 1.05), and set \(\varepsilon\) to 50 to obtain a robust solution. Table 10 displays the optimization results for the optimistic solution. Figure 6 shows some results corresponding to the pessimistic solution and the robust solution. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Segment** & **Approach** & **Segment** & **Approach** & **Segment** & **Approach** \\ \hline
1 & 1 & 11 & 3 & 21 & 3 \\
2 & 1 & 12 & 1 & 22 & 3 \\
3 & 1 & 13 & none & 23 & 1 \\
4 & 1 & 14 & 3 & 24 & 3 \\
5 & 1 & 15 & 1 & 25 & 1 \\
6 & 1 & 16 & 1 & 26 & 1 \\
7 & 3 & 17 & 3 & 27 & 1 \\
8 & 3 & 18 & 3 & 28 & 3 \\
9 & 3 & 19 & 2 & 29 & 3 \\
10 & 1 & 20 & none & 30 & 1 \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Results of the optimistic solution. At this point, the total cost is CNY 1,363,383, which is below the CNY 1,500,000 threshold. The annual costs are CNY 594,700, CNY 567,340, and CNY 201,343, all of which are under CNY 600,000, meeting the relevant constraints. However, when the unit area maintenance cost is in the worst-case condition (\(1.05\times C_{j}\)), the total cost increases to CNY 1,506,897, exceeding the limit (CNY 1,500,000). The annual costs are CNY 657,300, CNY 627,060, and CNY 222,537, with the first two years exceeding CNY 600,000, thereby failing to meet the relevant constraints. Consequently, the optimistic solution becomes infeasible. This indicates that as the unit area maintenance cost gradually increases, the optimistic solution may transition from feasible to infeasible.