## 4 Model Solving
### Solution Algorithm Design  
WOA is characterized by its advantages of few algorithmic parameters and robust global search capabilities [39]; however, conventional WOA implementations are fundamentally incompatible with multimodal transportation route optimization problems due to their exclusive applicability to continuous optimization domains, coupled with inherent limitations in local search efficacy that predispose the algorithm to premature convergence in combinatorial search spaces. This study proposes a multi-strategy enhanced whale optimization algorithm (IWOA) employing discrete encoding mechanisms, featuring three principal innovations: First, Sobol sequences are implemented during population initialization to ensure low-discrepancy solution distribution, significantly reducing computational overhead in the preliminary optimization phase. Second, the canonical position update mechanism is fundamentally reconceptualized through discrete operator transformations, enabling effective adaptation to combinatorial optimization problems. Third, strategic enhancements, including variable neighborhood search (VNS) operators for local intensification and elitist preservation mechanisms, are integrated to systematically expand the solution space while mitigating local optima entrapment risks. This hybrid design paradigm synergistically balances exploration-exploitation dynamics, thereby improving global convergence properties and solution quality for complex multimodal transportation optimization challenges. The algorithm was implemented using MATLAB R2021a, and its core parameters are explicitly defined as follows: initial population size, maximum iterations, VNS adoption rate, and elitism preservation rate. (1) Coding method  
The algorithm employs a positive integer-based hybrid encoding strategy, where the core representation integrates both feasible multimodal path \(x_{ij}^{m}\) and corresponding transportation mode \(y_{i}^{mk}\) information into a unified one-dimensional vector. Specifically, the first n positions of the encoding vector store the sequential node indices along the transportation path, while the subsequent n \(-\) 1 positions encode the transportation modes utilized between consecutive node pairs. The mathematical expression and schematic diagram of this encoding method are shown in Equation (46) and Figure 3, respectively. Numbers denote nodes in the path, and R/H/W represent road transportation, railway transportation, and waterway transportation, respectively. \[X=[\{x_{1},x_{2},\ldots x_{n}\}|\{y_{1},y_{2},\ldots y_{n-1}\}\,] \tag{46}\]  
Figure 3: Schematic diagram of coding structure. (2) Sobol sequence initialization population  
The Sobol sequence, a low-discrepancy sequence demonstrating superior uniformity characteristics in high-dimensional space, systematically addresses the issue of solution clustering through its quasi-random sampling mechanism. By generating solution populations that uniformly cover the search space with minimized dispersion, it fundamentally enhances initialization quality and optimization efficiency. This approach not only substantially mitigates population aggregation during the initialization phase but also achieves comprehensive spatial coverage of the solution domain. Consequently, it establishes favorable initial conditions for global optimization while significantly enhancing the probability of algorithmic convergence to globally optimal solutions through improved exploration-exploitation balance. (3) Whale individual location update  
Given the distinctive combinatorial optimization characteristics inherent in multi-modal transportation route selection problems, the canonical position update mechanism of the whale optimization algorithm has been fundamentally reconceptualized through discrete state transition operators, as formally presented in Equation (47).