## 2 Problem Definition
### Cost Analysis
#### 2.2.4 Mixed Carbon Emission Cost  
\"Mixed carbon emission cost\" refers to the sum of the carbon tax cost and the carbon trading cost incurred by the distribution center when carbon tax and carbon trading policies are implemented simultaneously. \"Carbon tax cost\" refers to the amount of the tax on carbon emissions generated during the distribution process. \"Carbon trading cost\" has two meanings. When the carbon emissions generated in the distribution process exceed the carbon quota, it is the \"cost\" paid by the distribution center to purchase the equivalent of the excess on the carbon trading market. When the carbon emissions generated are less than the carbon quota, it is the \"income\" obtained by the distribution center by selling this remaining part of the quota on the carbon trading market. Furthermore, the carbon emissions are calculated based on the carbon emission rate. Then, the mixed carbon emission cost \(C_{3}\) is calculated based on the carbon tax cost and the carbon trading cost, as shown in Equation (10). \(w_{1}\) and \(w_{2}\) are the weights of the carbon tax cost and carbon trading cost, respectively. \(\omega\) represents the carbon emission coefficient (calculated by the MEET formula). \(\varepsilon\) represents the carbon tax price. \(\lambda\) represents the carbon trading price. \  
\[\sum_{i\in S}\sum_{j\in S}x_{ij}^{k}\leq|S|-1\ \ \ \forall k\in V,\forall S \subseteq N^{\prime} \tag{16}\]  
\[y_{jp}^{k}\cdot q_{jp}\leq Q_{p}\ \ \forall k\in V,\forall p\in P,\forall j\in N^{\prime} \tag{17}\]  
\[\sum_{j\in N^{\prime}}y_{jp}^{k}\cdot q_{jp}\leq Q_{p}\ \ \forall k\in V,\forall p\in P \tag{18}\]  
\[\sum_{j\in N^{\prime}}\sum_{p\in P}y_{jp}^{k}\cdot q_{jp}\leq Q_{max}\ \ \forall k\in V \tag{19}\]  
\[t_{ij}^{k}=t_{ij}^{ks}\cdot\left[1+\alpha\left(\frac{TR_{ij}^{q}}{TR_{ij}^{d} }\right)^{\beta}\right]\ \ \ \ \ \forall i\in N,\forall j\in N^{\prime},\forall k\in V \tag{20}\]  
\[v_{ij}^{k}=\frac{v_{ij}^{ks}}{\left[1+\alpha\left(\frac{TR_{ij}^{s}}{TR_{ij}^ {d}}\right)^{\beta}\right]}\ \ \ \ \ \forall i\in N,\forall j\in N^{\prime},\forall k\in V \tag{21}\]  
The objective function (11) represents minimizing the total cost--that is, minimizing the sum of the vehicle management and use cost, transportation cost and mixed carbon emission cost. Among them, the travel time that changes with the degree of congestion affects the vehicle management and use cost, and the travel speed that changes with the degree of congestion affects the fuel consumption rate, which affects the transportation cost and the mixed carbon emission cost. Constraint (12) indicates that the path has continuity: a customer only accepts the service of one MCV, a customer can only be visited once and each customer can obtain the service. Constraint (13) indicates that all product demands of one customer are loaded and distributed by one MCV. Constraint (14) indicates that the MCV departs from the distribution center and must return to the distribution center after completing its distribution task. Constraint (15) indicates that the empty path is eliminated and that the MCV is allowed to serve one customer. \(S\) represents the set of all customers of MCV \(k\) on the distribution path. Constraint (16) represents the elimination of subloops. Constraint (17) indicates that the demand for any type of product from any customer shall not exceed the maximum capacity of the corresponding loading compartment of the service vehicle. Constraint (18) indicates that the sum of the demand of a certain product from all customers on the distribution path of each MCV shall not exceed the maximum capacity \(Q_{p}\) of the corresponding loading compartment. Constraint (19) indicates that the sum of the demands of all product types of all customer nodes on the distribution path of each MCV shall not exceed the maximum capacity \(Q_{max}\) of the entire vehicle. Constraint (20) indicates that the travel time of any vehicle on any road segment is affected by traffic congestion.