# Route Optimization for Open Vehicle Routing Problem (OVRP): A Mathematical and Solution Approach
## 3 Materials and Methods
### Solution and Analysis of Results
#### 3.4.2 Heuristic Algorithms
In the P-n55-k7 instance, complexity increases due to the larger number of nodes and vehicles, resulting in greater challenges for route optimization. In this scenario, H4 again stands out for its ability to approach the costs obtained by LKH-3 and AMPL, highlighting its robustness and adaptability to more complex instances. However, the differences in costs are more pronounced compared to E-n30-k3, highlighting the importance of using more advanced methods for larger scale scenarios. In synthesis, it highlights the effectiveness of the H4 heuristic in providing competitive solutions, especially for instances of higher complexity, and emphasizes the need to balance between computational efficiency and solution quality when selecting the appropriate heuristic for different instance sizes in the OVRP. Figure 7: Comparative results between heuristics. ## 5 Conclusions

This research takes a comprehensive dive into the open vehicle routing problem (OVRP), a fundamental logistics challenge in today's mobility and trade landscape. By adopting a multidimensional approach, we have formulated and solved the OVRP using advanced tools such as AMPL and compared results with state of the art algorithms such as LKH-3. Detailed experimentation and comparison with the Best Known Solution (BKS) have revealed the complexity of the OVRP. Despite the implemented mathematical methods up to heuristics, such as Greedy and Nearest Neighbor, the OVRP has offered remarkable solutions. The synergy between Nearest Neighbor and 2-opt stands out especially for its efficiency. Experimentation supported by the AMPL formulation and the LKH-3 algorithm provided solutions very close to the optimum, with a difference of less than 5% with respect to the Best Known Solution (BKS). The heuristics, especially H4 (Nearest Neighbor + Local Search), were shown to be highly competitive, reducing costs by approximately 10% compared to other heuristics such as H1 (Greedy) and H2 (Nearest Neighbor) in several instances. Similarly, the diversity of instances and their impact on the efficiency of the algorithms is highlighted, where instance C1, with 51 nodes and 5 vehicles, and instance E-n23-k3, with 23 nodes and 3 vehicles, showed how complexity and geographical distribution affect the formulation of solutions. In these problems, AMPL and LKH-3 managed to approach BKS with less than 3% deviation, while the heuristics varied more significantly, with H4 again standing out for its performance. Another more detailed analysis, taking as reference the most prominent instances E-n30-k3 and P-n55-k7, highlights the effectiveness of H4, which achieved with respect to costs just 7% higher than those obtained by AMPL and LKH-3, in contrast to H1 and H2, which presented differences of up to 15%. This quantitative analysis shows that the applied heuristics provide fast and acceptable solutions, where the combination of Nearest Neighbor and Local Search is especially powerful, reducing the gap with the most advanced methods. The main innovation of this research lies in the implementation of a comprehensive and comparative approach to solving the open vehicle routing problem (OVRP) using both advanced mathematical and heuristic methods. Unlike previous studies that focus on specific routing and vehicle problems, this research covers 58 instances, including up to 200 nodes and 17 vehicles, addressing a diversity of problems from multiple depots to heterogeneous fleets and time constraints. The combination of the AMPL formulation and the LKH-3 algorithm, along with the synergy of heuristics such as Greedy and Nearest Neighbor with 2-opt, has demonstrated remarkable efficiency, whereby this integration of advanced methods outperforms previous solutions such as ant colony algorithms, tabu search methods, and constrained programming models, demonstrating superiority in both efficiency and versatility. Finally, it is recommended to explore the integration of emerging technologies, such as machine learning or artificial intelligence-based optimization, to address OVRP.