## 6 Conclusions  
In numerous practical transportation problems, two conflicting objectives are required to be considered simultaneously. In general, an improvement in one objective is accompanied by a compromise in the other. Therefore, the goal of bi-objective optimization is to find a set of Pareto optimal solutions that balance both objectives. Since it is inefficient and unnecessary to obtain the full set of Pareto optimal solutions for practical purposes, this study proposes that a near-optimal subset of Pareto optimal solutions is sufficient to tackle bi-objective optimization problems. In addition, we elaborate the revised \(\epsilon\)-constraint method, the improved revised \(\epsilon\)-constraint method, and the augmented \(\epsilon\)-constraint method for solving bi-objective optimization problems. A key challenge in bi-objective optimization is evaluating the quality of the obtained solutions. Therefore, we rigorously analyze the near-optimality of the subset of Pareto optimal solutions obtained by the revised \(\epsilon\)-constraint method, the improved revised \(\epsilon\)-constraint method, and the augmented \(\epsilon\)-constraint method from a mathematical perspective. To this end, we present a series of lemmas, propositions, and corollaries, which are supported by rigorous proofs. Moreover, in order to validate the effectiveness and efficiency of our methods, computational experiments are conducted based on a real-world bi-objective optimization problem. This study establishes a solid foundation for the development of bi-objective optimization by mathematically proving the near-optimality of the Pareto optimal solution subsets and offering feasible methods for finding such near-optimal subsets of Pareto optimal solutions, contributing to balancing trade-off objectives in practical transportation problems. However, this study has some limitations, which can be further investigated from the following aspects. First, future work can extend our methods to multi-objective optimization problems in the field of transportation. Second, the computational efficiency of our methods could be enhanced. Deeper exploration of advanced optimization algorithms [33] and machine learning [34; 35] could be considered in future studies. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Scale Type**} & \multirow{2}{*}{**Number of Nodes**} & \multirow{2}{*}{**ID**} & \multicolumn{4}{c}{**Weighting Method**} \\ \cline{3-6}  & & & **NOSO** & **HR (\%)** & **CPU Time (s)** \\ \hline \multirow{4}{*}{Small} & & 1 & 5 & 98.43 & 3.95 \\  & & 2 & 5 & 99.37 & 3.39 \\  & & 3 & 3 & 97.47 & 3.92 \\  & & 4 & 5 & 98.59 & 4.99 \\  & & 5 & 6 & 99.58 & 3.67 \\ \hline \multirow{4}{*}{Medium} & & Avg. & 4.80 & 98.69 & 3.98 \\ \cline{2-6}  & & 1 & 5 & 98.17 & 46.59 \\  & & 2 & 7 & 97.44 & 50.06 \\  & & 3 & 5 & 99.92 & 67.34 \\  & & 4 & 7 & 99.25 & 46.19 \\  & & 5 & 6 & 97.77 & 47.36 \\ \hline \multirow{4}{*}{Large} & & Avg. & 6.00 & 98.51 & 51.51 \\ \cline{2-6}  & & 1 & 6 & 99.97 & 322.49 \\ \cline{1-1}  & & 2 & 4 & 94.58 & 300.11 \\ \cline{1-1}  & & 3 & 4 & 98.94 & 297.81 \\ \cline{1-1}  & & 4 & 3 & 92.70 & 298.98 \\ \cline{1-1}  & & 5 & 6 & 99.33 & 303.88 \\ \hline \hline \end{tabular} Abbreviations: NOSO, number of solutions obtained; HR, hypervolume-based ratio.  
\end{table} 
Table 4: Experimental results of the weighting method. **Author Contributions:** Conceptualization, H.Z. and S.W. ; methodology, H.Z. and S.W. ; software, H.Z. ; validation, Q.R., Y.J. and S.W. ; formal analysis, H.Z. and S.W. ; investigation, Q.R. and Y.J. ; resources, Q.R. and Y.J. ; data curation, H.Z. ; writing--original draft preparation, H.Z. and S.W. ; writing--review and editing, Q.R., Y.J. and S.W. ; visualization, H.Z. ; supervision, Q.R., Y.J. and S.W. ; project administration, S.W. ; funding acquisition, Q.R., Y.J. and S.W. All authors have read and agreed to the published version of the manuscript.