## 4 Model Solving
### Solution Algorithm Design
This algorithmic adaptation specifically addresses the discrete nature of transportation mode selection and path sequencing decisions, ensuring operational compatibility with the problem's inherent combinatorial optimization structure. \[X^{i+1}=\left\{ \begin{aligned} Cross_{1}\big{(}X^{i},X^{*} \big{)},\,p<0.5\psi\bigg{|}\overset{\rightarrow}{A}\bigg{|}\leq 1\\ Neighborhood\big{(}X^{i},X^{*}\big{)},\,p<0.5\psi\bigg{|}\overset{ \rightarrow}{A}\bigg{|}>1\\ Cross_{2}\big{(}X^{i},X^{*}\big{)},\,p\geq 0.5\end{aligned} \right. \tag{47}\] \[\overset{\rightarrow}{A}=2\overset{\rightarrow}{a}\cdot \overrightarrow{r_{1}}-\overset{\rightarrow}{a}\] (48) \[a=2-\frac{2t}{T_{\max}} \tag{49}\]  
Within the algorithmic framework, \(t\) denotes the current iteration count, \(T_{\max}\) represents the maximum allowable iterations, while \(\overrightarrow{r_{1}}\) is a stochastically generated vector uniformly distributed within the interval \([0,1]\), serving as a randomization operator in the position update mechanism.\(X^{i+1}\) denotes the position-updated whale agent, \(X^{i}\) represents the current whale agent in the solution space, and \(X^{*}\) corresponds to the currently identified optimal whale agent within the evolutionary population. The \(Cross_{1}\) operator in the proposed evolutionary algorithm operates through the following rigorous mechanism: Initially, it identifies common intermediate transshipment node positions \(\alpha_{1}\)(excluding origin and destination nodes) shared between two parental paths. Subsequently, the algorithm performs reciprocal exchange of both nodal sequences and transportation mode assignments from position \(\alpha_{1}\) to the destination between whale agents \(X^{i}\) and \(X^{*}\), thereby generating two novel offspring individuals. Crucially, this crossover mechanism is constrained to nodes with transshipment capabilities and is triggered exclusively when the stochastic pairing of encoded whale agents satisfies the prerequisite condition of shared intermediate transshipment nodes, ensuring operational validity and solution feasibility in multimodal transportation networks. This operational mechanism is schematically represented in Figure 4.  
Diverging from the \(Cross_{1}\), \(Cross_{2}\) operator imposes a minimum requirement of two crossover points for valid path segment exchange between parental routes. The operational protocol mandates (1) identification of at least two shared nodal positions (excluding origin-destination pairs) between parental encodings and (2) reciprocal exchange of path segmentsand corresponding transportation mode assignments bounded by these crossover points. Should the cardinality of shared nodes prove insufficient for dual-crossover operations, the algorithm initiates a stochastic path regeneration procedure to ensure offspring feasibility. This constrained crossover mechanism, graphically illustrated in Figure 4, maintains solution validity through rigorous topological preservation while enhancing genetic diversity through adaptive recombination strategies. This operational mechanism is schematically represented in Figure 5.  
_Neighborhood_ operates by generating perturbed solutions through systematic application of predefined neighborhood operators to incumbent solutions, followed by rigorous validity verification procedures. Within this algorithmic architecture, four distinct neighborhood structures are implemented, with each whale agent stochastically selecting one operator per iteration under a uniform probability distribution across all available neighborhood transformation mechanisms.