# Low-Carbon Logistics Distribution Vehicle Routing Optimization Based on INNC-GA
## 3 Model Construction
### Construction of the Model  
The model developed in this study aims to minimize vehicle fixed costs, transportation costs, and carbon emission costs, taking into account time window constraints, vehicle load constraints, and speed constraints. Based on the notation provided, the low-carbon logistics and distribution path model considering time windows is formulated as follows:  
\[\min T=\sum_{i=1}^{N}\sum_{j=1}^{N}\sum_{k=1}^{K}x_{ij\
eq}d_{ij}c_{k}+\sum_{j =1}^{N}\sum_{k=1}^{K}y_{\varphi\beta}f_{k}+c_{m}F_{e}\delta\sum_{i=1}^{N}\sum_ {j=1}^{N}\sum_{k=1}^{K}x_{ijk}d_{ij} \tag{2}\]  
Equation (2) defines the objective function, which includes the vehicle fixed cost, vehicle travel cost, and carbon emission cost. The vehicle fixed cost is composed of vehicle maintenance, transfer, and workers' wages, with each vehicle being accounted for with a driver. The vehicle travel cost accounts for expenses incurred during the vehicle's journey.  
Carbon emissions result from fuel consumption during vehicle travel. The logistics distribution model in this study focuses on path length and cargo weight. To simplify fuel consumption analysis, the study utilizes the load estimation method from literature [22]. This method considers fuel consumption per unit distance \(F_{e}\) as a linear function of the vehicle load \(q\), that is  
\[F_{e}=\frac{P_{0}}{P_{1}}q_{ij}+P_{0} \tag{3}\]  
And the relationship between fuel consumption and carbon emissions is  
\[E_{c}=\delta\times F_{e} \tag{4}\]  
Then the cost of the carbon emissions produced by the vehicle during transport is  
\[E=C_{m}\times\delta\times F_{e} \tag{5}\]  
Constraints are  
\[\sum_{k=1}^{N}\sum_{j=1}^{K}x_{ijk}=1\ \ \forall i\in N\backslash\{0\} \tag{6}\]  
\[\sum_{k=1}^{K}\sum_{j=1}^{N}x_{ijk}=1\ \ \forall j\in N\backslash\{0\} \tag{7}\]  
Equations (6) and (7) indicate that only one vehicle can perform the delivery service at a customer point and only once.  
\[\sum_{i=1}^{N}\sum_{k=1}^{K}x_{\alpha ik}=\sum_{j=1}^{N}\sum_{k=1}^{K}x_{ \alpha jk}\leq k \tag{8}\]  
Equation (8) illustrates that the transport vehicle initiates its journey from the distribution center, fulfills the distribution task, and ultimately returns to the same distribution center.  
\[\sum_{i=1}^{N}\sum_{j=1}^{N}x_{ijk}\leq Q\ \ \ \forall i,j\in N\backslash\{0\} \tag{9}\]Equation (9) indicates that the vehicle capacity limits.  
\[\sum_{j}^{N}x_{\mathit{ok}}\leq 1\quad k\in K \tag{10}\]  
Equation (10) indicates that there is only one circuit per vehicle and no sub-circuits.  
\[h_{ik}\leq b_{i}\quad\forall i\in N \tag{11}\]  
Equation (11) indicates that the customer point must be reached before the latest time the customer can receive service.  
\[a_{i}\leq w_{ik}+h_{ik}\leq b_{i}\quad\forall i\in N,k\in K \tag{12}\]  
Equation (12) specifies that the service time must fall within the customer's designated time window.  
\[t_{ij}+w_{ik}+h_{ik}+s_{ik}+H\big{(}1-x_{\bar{i}\bar{j}k}\big{)}\leq h_{ik}\quad \forall i,j\in N,k\in K \tag{13}\]  
Equation (13) indicates that the service time of both the front and back customers must satisfy their respective time window constraints.  
\[x_{ijk}=\left\{\begin{array}{l}1\;\mathit{vehicle}\;k\;\mathit{travelling}\; \mathit{from\;node}\;i\;\mathit{to\;node}\;j\\ 0\;\mathit{if}\;\mathit{not}\;\mathit{not}\;\mathit{not}\;\mathit{not}\; \mathit{not}\;\mathit{not}\;\mathit{not}\;\mathit{not}\;\mathit{not}\;\mathit{ (15)}\end{array}\right. \tag{14}\]  
\[y_{ik}=\left\{\begin{array}{l}1\;\mathit{customer}\;i\;\mathit{has\; vehicle}\;k\;\mathit{delivery}\\ 0\;\mathit{if}\;\mathit{not}\;\mathit{not}\;\mathit{not}\;\mathit{not}\; \mathit{not}\;\mathit{not}\;\mathit{not}\;\mathit{not}\;\mathit{not}\;\mathit{ (15)}\end{array}\right. \tag{15}\]  
Equations (14) and (15) are decision variables, where \(i\) and \(j\) are arbitrary customer points, and k is an arbitrary vehicle.