## 5 Numerical Example
### Result Analysis
#### 5.2.3 Sensitivity Analysis  
The regret coefficient, functioning as a pivotal risk metric, effectively balances economic efficiency and operational reliability in transportation planning. Comparative analysis of regret coefficient variations (Table 13) reveals three operational regimes: (1) No feasible solutions emerge below 0.2, indicating excessively stringent constraints; (2) between 0.3 and 0.5, robust optimal strategies evolve with decreasing total costs (12.7% reduction from 0.3 to 0.5), demonstrating that lower coefficients correspond to conservative strategies with cost premiums (18.3% higher than baseline), while higher coefficients enable risk-tolerant decision-making with cost optimization; and (3) beyond 0.5, strategic configurations stabilize despite coefficient increases. This parametric sensitivity analysis establishes a flexible decision-making mechanism allowing systematic navigation from complete risk aversion to managed risk acceptance through coefficient calibration. Such adaptive optimization proves particularly advantageous for container multimodal transportation systems exposed to stochastic disruptions like port congestion and scheduled  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Policy Type** & **Cost** & **Emissions** & **Stability** \\ \hline Carbon Tax Policy & High & Medium & Medium \\ Carbon Trading Policy & Low & Medium & High \\ Mandatory Carbon Emission Policy & Medium & Low & Low \\ Carbon Offset Policy & Medium & Medium & High \\ \hline \hline \end{tabular}
\end{table}
Table 12: Comparison of the results of the regret coefficient change.  
Figure 16: Comparison of the results under different types of carbon policies.  
service delays, enabling balanced tradeoffs between cost containment and service reliability under uncertainty.  
The time confidence level \(\alpha\) represents the decision-maker's preference attitude towards the transportation time risk, and is used to measure the degree of trust in the \"reliable part\" of the triangular fuzzy number. In the above-mentioned example, with other conditions remaining unchanged, the robust total cost and the maximum regret value obtained by changing \(\alpha\) are shown in Figure 17. When \(\alpha\) increases, the total cost rises, indicating that the decision-maker tends to be conservative. In this case, it corresponds to a longer transportation time and lower risk, which is suitable for scenarios with high requirements for transportation stability. When \(\alpha\) decreases, the total cost drops, and the decision-maker tends to take risks. At this time, it corresponds to a shorter transportation time, which is suitable for scenarios with high requirements for timeliness. When \(\alpha=0.5\), the decision-maker has a relatively balanced consideration of the transportation time risk and the transportation cost economy. At this time, the maximum regret value in all scenarios is the smallest. As \(\alpha\) increases or decreases, the maximum regret value keeps increasing, indicating that the volatility of the transportation plan is greater. Decision-makers need to flexibly set \(\alpha\) according to the specific needs of multimodal transportation to balance the risk of time fluctuations and the cost.  
\begin{table}
\begin{tabular}{c c c c} \hline
**Regret** & **Robust Optimal** & **Expected Total** & **Maximum Regret** \\
**Coefficient** & **Transportation Scheme** & **Cost** & **Value** \\ \hline
0.2 & — & — & \\
0.3 & 1-R-2-R-7-R-9-R-13-H-14 & 154,297 & 0.21 \\
0.4 & 1-R-2-R-7-R-9-R-13-R-14 & 154,138 & 0.34 \\
0.5 & 1-H-2-R-7-R-9-R-13-H-14 & 154,025 & 0.46 \\
0.6 & 1-H-2-R-7-R-9-R-13-H-14 & 154,025 & 0.46 \\ \hline \end{tabular}
\end{table}
Table 13: Comparison of the results of the regret coefficient change.  
Figure 17: Effect of the period time confidence on total cost and maximum regret value.