## 5 Computational Experiments
### Experimental Results  
The basic \(\epsilon\)-constraint method, the revised \(\epsilon\)-constraint method, the improved revised \(\epsilon\)-constraint method, the augmented \(\epsilon\)-constraint method, and the weighting method are implemented to solve the problem and find a series of solutions. We conduct five sets of small-scale instances, five sets of medium-scale instances, and five sets of large-scale instances, corresponding to 60 nodes, 200 nodes, and 500 nodes in the urban traffic network, respectively. The time limit for each instance is set to half an hour. Tables 2-4 show the computing time and value of performance metrics (i.e., number of solutions obtained (NOSO) and hypervolume-based ratio (HR)). The solution sets of the basic \(\epsilon\)-constraint method, the revised \(\epsilon\)-constraint method, the improved revised \(\epsilon\)-constraint method, the augmented \(\epsilon\)-constraint method, and the weighting method are denoted by \(S_{1}\), \(S_{2}\), \(S_{3}\), \(S_{4}\), and \(S_{5}\), respectively. Let \(S\) represent the union of solution sets of all the five methods, i.e., \(S=S_{1}\cup S_{2}\cup S_{3}\cup S_{4}\cup S_{5}\). The maximum objective values of solutions in \(S\) in the first dimension and the second dimension, represented by \(O_{1}^{\text{max}}\) and \(O_{2}^{\text{max}}\), determine the position of the reference point in the two-dimensional plane. For each obtained solution \(\mathbf{x}\), the rectangle whose vertices are \(\mathbf{x}\) and the reference point (\(O_{1}^{\text{max}}\), \(O_{2}^{\text{max}}\)), with each edge perpendicular to the coordinate axes, is denoted by \(g_{\mathbf{x}}\). The hypervolume of solution set \(S_{k}\), \(k\in\{1,2,3,4,5\}\), and \(S\), denoted by \(h_{k}\) and \(h\), are the hypervolume of the union of \(g_{\mathbf{x}}\) for \(\mathbf{x}\in S_{k}\) and the union of \(g_{\mathbf{x}}\) for \(\mathbf{x}\in S\), respectively. Then, the value of the metric HR for method \(k\), i.e., \(HR(S_{k})\), can be calculated as \(h_{k}/h\). Generally, the higher values of metrics NOSO and HR indicate better performance of the obtained solutions. To strengthen readability, the CPU time is rounded to two decimal places, and the value of HR is expressed as a percentage with two decimal places. As shown in Tables 2-4, the difficulty for all five methods to solve the safe transportation route selection problem increases as the number of nodes in the urban traffic network grows. In terms of solution quality, for the same instance, the basic \(\epsilon\)-constraint method, the improved revised \(\epsilon\)-constraint method, and the augmented \(\epsilon\)-constraint method are able to find more solutions with broader coverage than the weighting method. This is because the weighting method cannot generate nonsupported efficient solutions, even if these solutions are Pareto optimal. Additionally, it is worth noting that compared with the improved revised \(\epsilon\)-constraint method and the augmented \(\epsilon\)-constraint method developed in this study, the basic \(\epsilon\)-constraint method requires extra steps to remove weakly efficient solutions that are not Pareto optimal. In terms of the computing time, for the same instance, the revised \(\epsilon\)-constraint method is the most time-consuming. However, the improved revised \(\epsilon\)-constraint method and the augmented \(\epsilon\)-constraint method, developed by enhancing the revised \(\epsilon\)-constraint method, show relatively high computational efficiency. Specifically, for small-, medium-, and large-scale instances, the average solution times of the improved revised \(\epsilon\)-constraint method and the augmented \(\epsilon\)-constraint method are no more than 5.16% and 2.64% of that of the basic \(\epsilon\)-constraint method, and no more than 23.38% and 11.94% of that of the weighting method, respectively.