## 3 Proposed Methods
### Improved Grey Wolf-Harris Hawks Hybrid Algorithm
[43] in 2019, the HHO algorithm is an emerging intelligent optimization method that has shown strong potential in logistics-related problems, such as the traveling salesman problem and vehicle routing [44; 45]. Despite these promising results, its application in multimodal transport optimization remains limited and has not been systematically investigated in the literature. During the exploration phase of the HHO algorithm, when \(\mid\!E\!\mid\!\mid\!\geq 1\), the [PERSON] hawk switches between two position update strategies depending on the value of \(q\). If \(q\geq 0.5\), the hawk selects a random location within the population range for its next move. If \(q<0.5\), the hawk adjusts its position by considering the distance between the prey and the group's central location. These two update strategies are presented in Equation (30), while Equation (31) defines the calculation of the group center. \[X(t+1)= \begin{cases}X_{rand}(t)-r_{1}|X_{rand}(t)-2r_{2}X(t)|&,\quad q\geq 0.5,\\ (X_{rabbit}(t)-X_{m}(t))-r_{3}(B_{1}+r_{4}(B_{u}-B_{1})),&\quad q<0.5,\end{cases}  \tag{30}\]  
\[X_{m}(t)=\frac{1}{N}\sum_{i=1}^{N}X_{i}(t), \tag{31}\]  
where \(X_{rand}(t)\) represents the current position of a randomly selected individual, while \(X_{rabbit}(t)\) denotes the prey's position. \(X_{m}(t)\) corresponds to the central position of the hawk flock, which is typically calculated as the average position of all the hawks in the population. This central position plays a guiding role in the HHO algorithm, allowing the hawks to coordinate their movements toward the prey during the optimization process. The set of random factors \(\{r_{1}\!\mid\!i=1,2,3,4\}\) and \(q\) are values within the range (0, 1). Additionally, \(B_{1}\) and \(B_{u}\) define the upper and lower bounds of the parameters being optimized. The escape energy \(E\) gradually decreases with the number of iterations, as formulated in Equation (32), reflecting the diminishing evasion capacity of the prey over time. \[E=2E_{0}\left(1-\frac{t}{T}\right), \tag{32}\]  
where \(E_{0}\) represents the initial random energy value within the range (\(-1,1\)), \(t\) denotes the current iteration count, and \(T\) indicates the maximum number of iterations. When \(\mid\!E\!\mid\!<1\), the HHO algorithm enters the exploitation phase and applies different hunting strategies based on the prey's escape behavior. Under the soft encirclement condition, where \(\mid\!E\!\mid\!\geq 0.5\) and \(r\geq 0.5\), the hawks update their position according to the prey's escape pattern. \[X(t+1)=\Delta X(t)-E\!\mid\!J\times X_{rabbit}(t)-X(t)\!\mid, \tag{33}\]  
where \(\Delta X(t)\) represents the difference between the prey and hawk positions, and \(J\) denotes the jump distance during the rabbit's escape process, as expressed in Equations (34) and (35). \[\Delta X(t)=X_{rabbit}(t)-X(t), \tag{34}\]\[J=2(1-r_{5}). \tag{35}\]  
Upon satisfying the conditions \(\mid\!E\!\mid\!<\!0.5\) and \(r\geq 0.5\), the algorithm transitions into the hard encirclement phase, during which the hawks fixate on the prey, gradually reduce the search radius, and update their positions, as shown in Equation (36). \[X(t+1)=X_{rabbit}(t)-E|\Delta X(t)|. \tag{36}\]  
When \(\mid\!E\!\mid\geq\!0.5\) and \(r<0.5\), the hawks either perform a high-speed dive to softly encircle the prey if it demonstrates strong escape ability or initiate Levy flight behavior when predation fails. The updated position is expressed in Equation (37). The formulation of Levy flight is provided in Equation (38). \[X(t+1)=\left\{ \begin{array}{ll}Y=X_{rabbit}(t)-E|J\times X_{rabbit}(t)-X(t)|,& \mbox{if }F(Y)<F(X(t)),\\ Z=Y+S\times LF(D)&,\mbox{if }F(Z)<F(X(t)),\end{array} \right.