# Improved GA-LNS Algorithm for Solving Vehicle Path Problems Considering Carbon Emissions
## 1 Introduction  
Following the introduction of the vehicle path optimization problem by [PERSON] et al. [1], numerous studies have been conducted in this area. Many researchers have conducted systematic scientific studies from the perspective of reducing delivery distances and overall logistics costs, resulting in the derivation of various research variants [2, 3, 4]. As customer demands for timely deliveries increase, ensuring on-time performance has emerged as a core competency within the logistics industry. Consequently, vehicle path optimization concerning time windows has become a significant area of research, contributing to enhanced customer satisfaction and optimized resource allocation [5, 6, 7]. Vehicle capacity constraints are critical in the distribution process, as they directly influence logistics efficiency and the optimal utilization of resources. To address these challenges, an increasing number of studies are focusing on vehicle path optimization under capacity constraints, with the resulting findings offering valuable insights for companies seeking to enhance their operational efficiency [8, 9, 10]. As e-commerce continues to develop, logistics and distribution are playing an increasingly vital role in urban life. However, significant carbon dioxide emissions have a detrimental impact on the environment. The implementation of green transportation has become an essential trend for energy conservation and emissionreduction in contemporary society, with a focus on strategies to reduce energy consumption and carbon emissions emerging as a central theme in academic research. [PERSON] et al. [11] were the first to incorporate carbon dioxide emissions into the vehicle path problem model, successfully balancing the conflict between cost and carbon emission reduction targets, thereby effectively controlling the rise in economic costs. [PERSON] et al. [12] proposed a novel G-VRP model aimed at further reducing CO\({}_{2}\) emissions by decreasing vehicle fuel consumption. This series of scientific studies demonstrates that incorporating environmental factors into vehicle path optimization can yield more sustainable solutions, particularly when time windows and carbon emission constraints are integrated into the vehicle routing problem. These models exhibit significant potential for enabling companies to achieve a win-win scenario regarding cost reduction and environmental goals [13; 14; 15]. At the same time, in the context of a competitive market, customer satisfaction is increasingly recognized as a key aspect of corporate competitiveness, and some scholars have further integrated customer satisfaction and carbon emissions into multi-objective optimization models, which not only enhances logistics efficiency but also offers a novel approach for companies to improve market competitiveness while minimizing environmental impact [16; 17; 18]. Furthermore, studies have demonstrated that vehicle load management plays a critical role in optimizing transport routes, and effective load management can significantly reduce fuel consumption and carbon emissions, thereby enhancing the economic efficiency and environmental responsibility of companies [19; 20; 21]. In addressing the Vehicle Routing Problem (VRP) and its variants, heuristic algorithms are extensively employed due to their efficiency and adaptability in dealing with large-scale issues [22; 23]. Heuristic algorithms that are widely used include genetic algorithms, ant colony optimization methods, taboo search techniques, and simulated annealing approaches. Among these, genetic algorithms are particularly prevalent in vehicle routing applications owing to their robust global search capabilities and adaptability [24; 25; 26].