## 3 Modal Construction
### Model Construction Under Hybrid Uncertainty Conditions  
The fuzzy transit time is represented through a triangular fuzzy number \(\tilde{t}_{ij}^{m}=(t_{ij}^{m1},t_{ij}^{m2},t_{ij}^{m3})\), comprising three critical parameters: minimum transit duration \(t_{ij}^{m1}\), maximum transit duration \(t_{ij}^{m3}\), and most likely transit duration \(t_{ij}^{m2}\), satisfying the ordinal relationship \(t_{ij}^{m1}\leq t_{ij}^{m2}\leq t_{ij}^{m3}\). The minimum duration \(t_{ij}^{m1}\) corresponds to idealized operational conditions characterized by unimpeded road networks, optimal vehicle performance, and favorable meteorological circumstances. Conversely, the maximum duration \(t_{ij}^{m3}\) accounts for multiple adverse scenarios, including severe traffic congestion, recurrent mechanical failures, and extreme weather adversities. The modal value \(t_{ij}^{m2}\) is statistically derived through empirical analysis of historical transit time distributions across comparable transportation operations. The membership function governing this fuzzy temporal representation is formally articulated in Equation (29), mathematically defining the plausibility distribution across the temporal spectrum. The membership function graph is shown in Figure 2.  
\[\mu_{\tilde{t}}^{m}(x)=\left\{ \begin{array}{ll}\frac{x-t_{ij}^{m1}}{t_{ij}^{ m2}-t_{ij}^{m1}},t_{ij}^{m1}\leq x<t_{ij}^{m2}\\ 1,x=t_{ij}^{m2}\\ \frac{t_{ij}^{m3}-x}{t_{ij}^{m3}-t_{ij}^{m2}},t_{ij}^{m2}<x\leq t_{ij}^{m3}\\ 0,\text{other}\end{array} \right.,\forall m\in M,i,j\in A \tag{29}\]Consequently, the total transit duration within transportation operations is formally operationalized through Equations (30) and (31), which is methodologically integrated into the foundational optimization framework established in Section 3.2.3 to ensure temporal parameter consistency across the analytical model architecture. \[\widetilde{T}=\widetilde{t}_{D}^{arrive}-t^{go}=\sum\limits_{i\in A}\left(t_{i}^ {trans}+\widetilde{t}_{i}^{unit}+\sum\limits_{j\in A}x_{ij}^{m}\widetilde{t}_{ ij}^{m}\right) \tag{30}\]  
\[\widetilde{t}_{j}^{arrive}=\left\{ \begin{array}{cc}t^{go}&,\forall j\in O\\ \sum\limits_{i\in A}\sum\limits_{m\in M}x_{ij}^{m}.(\widetilde{t}_{i}^{ leave}+\widetilde{t}_{ij}^{m})&,\forall i\in N\cup D\end{array} \right. \tag{31}\]  
This study employs a scenario-based robust regret model to characterize freight rate volatility, where unit transportation prices are differentially parameterized across discrete operational scenarios according to realistic market conditions. Consequently, the integrated optimization framework addressing hybrid uncertainties is mathematically formalized through the system of Equations (31)-(38), which systematically incorporates stochastic pricing scenarios and fuzzy temporal parameters within a unified analytical structure. \[Z_{r}^{t}=\min\sum\limits_{h\in H}p_{h}\cdot C_{h}^{t}(x) \tag{32}\]  
\[\min C_{h}^{t}(x)=C_{1,h}+C_{2}+\widetilde{C}_{3}+C_{4}^{t} \tag{33}\]  
\[Z_{r}^{s}=\min\sum\limits_{h\in H}p_{h}\cdot C_{h}^{s}(x) \tag{34}\]  
\[\min C_{h}^{s}(x)=C_{1,h}+C_{2}+\widetilde{C}_{3}+C_{4}^{s} \tag{35}\]  
\[Z_{r}^{c}=\min\sum\limits_{h\in H}p_{h}\cdot C_{h}^{c}(x) \tag{36}\]  
\[\min C_{h}^{c}(x)=C_{1,h}+C_{2}+\widetilde{C}_{3} \tag{37}\]  
\[Z_{r}^{o}=\min\sum\limits_{h\in H}p_{h}\cdot C_{h}^{o}(x) \tag{38}\]  
\[\min C_{h}^{o}(x)=C_{1,h}+C_{2}+\widetilde{C}_{3}+C_{4}^{o} \tag{39}\]  
Figure 2: Membership function of triangular fuzzy numbers.