# Improved GA-LNS Algorithm for Solving Vehicle Path Problems Considering Carbon Emissions
## 3 Model Construction  
The model presented in this paper seeks to reduce the overall distribution costs while complying with constraints related to time windows, vehicle loads, and carbon emissions. Taking these factors into account, a low-carbon logistics distribution path model that integrates time window constraints has been developed, as outlined below in accordance with the symbol definitions provided in the Abbreviations:  
\[\min T=\sum_{i=1}^{N}\sum_{j=1}^{N}\sum_{k=1}^{K}x_{ijk}d_{ij}c_{k}+\sum_{j=1} ^{N}\sum_{k=1}^{K}y_{qijk}f_{k}+c_{m}F_{e}\delta\sum_{i=1}^{N}\sum_{j=1}^{N} \sum_{k=1}^{K}x_{ijk}d_{ij} \tag{1}\]  
In Equation (2), the objective function is defined, incorporating the vehicle's fixed costs, the expenses incurred while driving, and the costs related to carbon emissions. Carbon emissions associated with the vehicle routing problem are primarily generated by the energy consumption of delivery vehicles during transit. The fuel consumption of vehicles during transportation directly contributes to the emission of carbon dioxide and other pollutants (e.g., particulate matter and nitrogen oxides), which significantly impact the environment and climate change. This study focuses on the quantitative analysis of carbon emissions, with the analysis limited to carbon dioxide (CO\({}_{2}\)) emissions, while other pollutants, such as particulate matter and nitrogen oxides, are not considered in this context. This decision was primarily based on the focus of the study and the availability of data. Additionally, a vehicle's fuel consumption per distance unit is significantly affected by its load weight, along with various other elements such as driving speed, road conditions (including terrain and traffic), and driving behaviors (such as sudden acceleration and braking patterns). The logistics and distribution model in this study primarily considers the distance and load weight. To simplify the analysis, this study focuses on the impacts of distance and load weight on fuel consumption. This paper employs the load estimation technique proposed in [35] to determine fuel consumption. The fuel consumption for each unit of distance \(F_{e}\) is treated as a linear function dependent on the vehicle load \(q\), i.e.,  
\[F_{e}=\frac{P_{1}}{P_{0}}q_{ij}+p_{0} \tag{2}\]  
The connection between carbon emissions and fuel consumption is as follows:  
\[E_{c}=\delta\times F_{e} \tag{3}\]  
The carbon emissions cost generated by vehicles during transportation is as follows:  
\[E=C_{m}\times\delta\times F_{e} \tag{4}\]Constraint conditions:  
\[\sum_{k=1}^{N}\sum_{j=1}^{K}x_{ijk}=1\;\forall i\in N\backslash\{0\} \tag{5}\]  
\[\sum_{k=1}^{K}\sum_{j=1}^{N}x_{ijk}=1\;\forall j\in N\backslash\{0\} \tag{6}\]  
\[\sum_{i=1}^{N}\sum_{k=1}^{K}x_{\alpha k}=\sum_{j=1}^{N}\sum_{k=1}^{K}x_{\alpha jk }\leq k \tag{7}\]  
\[\sum_{i=1}^{N}\sum_{j=1}^{N}x_{ijk}\leq Q\;\forall i,j\in N\backslash\{0\} \tag{8}\]  
\[\sum_{j}^{N}x_{\alpha jk}\leq 1\;k\in K \tag{9}\]  
\[h_{ik}\leq b_{i}\;\forall i\in N \tag{10}\]  
\[F_{e}=\frac{P_{1}}{P_{0}}q_{ij}+p_{0} \tag{2}\]0  
\[F_{e}=\frac{P_{1}}{P_{0}}q_{ij}+p_{0} \tag{2}\]1  
\[x_{ijk}=\left\{ \begin{array}{ll}1\;\text{\it vehicle}\;\;k\;\;\text{\it travelling }\;\;from\;\text{\it node}\;\;i\;\;\text{\it to}\;\;\text{\it node}\;\;j\\ 0\;\;\;\;\text{\it if}\;\;\text{\it not}\end{array} \right. \tag{13}\]  
\[y_{ik}=\left\{ \begin{array}{ll}1\;\text{\it customer}\;\;\;i\;\;\text{\it has }\;\;\text{\it vehicle}\;\;k\;\;\text{\it delivery}\\ 0\;\;\;\;\text{\it if}\;\;\text{\it not}\end{array} \right. \tag{14}\]  
Equations (5) and (6) stipulate that each customer location can be serviced by only one vehicle, and this can occur just once. Equation (7) describes the route taken by the transport vehicle, starting from the distribution center, completing its delivery assignments, and finally returning to the center. Equation (8) sets forth the limitations on the vehicle's capacity. Equation (9) establishes that each vehicle operates on a single circuit without any sub-circuits.