## 2 Problem Definition  
### Problem Description  
In this section, the distribution network of MCVs while considering traffic congestion under the mixed carbon policy is defined. Here, \(G=(N,E)\), \(N=\{0\}\cup N^{\prime}=\{0,1,2...,n\}\) and \(E=\{(i,j)|i,j\in N\}\). \(G\) is the multi-compartment vehicle distribution system. \(N\) is the set of logistics nodes. \(\{0\}\) is the distribution center of multiple types of products with subdivision loading requirements. \(N^{\prime}\) is the set of customer nodes. \(E\) is the set of undirected edges. \(i,j\) is the logistics node, and \(i,j\in N\). \(d_{ij}\) is the distance from logistics node \(i\) to \(j\), and \(d_{ij}=d_{ji}\). Thus, the distance from node \(i\) to \(j\) is the same as the distance from node \(j\) to \(i\). \(V=\{1,2,3...,k\}\), where \(V\) is the set of MCVs, \(k\) is the MCV and \(k\in V\). \(M=\{1,2,3,...,m\}\). \(M\) is the set of compartments of the MCV. \(m\) is one compartment of the MCV. \(P=\{1,2,3,...,p\}\), where \(P\) is the set of product categories, \(p\) represents the product category and \(m=p\). Thus, the number of product types is the same as the number of compartments, each product type corresponds to an independent compartment and different types of products require different loading environments. \(Q_{p}\) represents the maximum capacity of compartment \(m\) corresponding to product \(p\). \(Q_{max}\) is the maximum capacity of the MCV. The maximum capacity of a MCV is equal to the sum of the maximum capacity of all its individual compartments. \(q_{jp}\) is the demand of customer \(j\) for the current type of product \(p\). The decision variable \(x_{ij}^{k}=1\) represents the MCV \(k\) traveling from node \(i\) to node \(j\), and \(x_{ij}^{k}=0\) represents other situations. The decision variable \(y_{jp}^{k}=1\) represents the demand of customer \(j\) for product \(p\) served by MCV \(k\), and \(y_{jp}^{k}=0\) represents other cases. In addition, the problem is subject to some assumptions. 1. The assumptions related to the distribution center are as follows. The quantity of different types of products is sufficient to meet the needs of different types of products at all customer nodes in the distribution area. The number of MCVs is sufficient to meet the logistics service needs of different types of products at all customer nodes. The number of compartments in the MCV corresponds to the number of product types required by the customers. In other words, the MCV can meet the needs of separate loading and joint distribution of all types of products. Different product types must be loaded strictly separately to ensure that the products will not affect each other and there will be no cross-infection. 2. The assumptions related to the client node are as follows. The demand of each customer for each product is known and cannot exceed the capacity ceiling of the corresponding compartment. Each customer's demand for all kinds of products cannot be distributed separately; that is, a customer node can only be served by one MCV. A MCV can serve multiple customer nodes. Moreover, since the maximum capacity of each compartment and entire vehicle is fixed, the MCV is constrained by the capacity ceiling in the distribution process. 3. The assumptions related to traffic congestion are as follows. There is traffic congestion in the urban distribution area. The congestion degree of the same road section is constant, whereas the congestion degree of different road sections is different. The travel speed and travel time of MCVs depend on the congestion degree of the road section, which is obtained through the BPR function. The MCV drives at a constant speed within each road section. The carbon emissions of MCVs in the distribution process are affected by the load and travel speed. In the distribution process of MCVs, the loading and unloading of products will lead to a change in the vehicle load and then affect the carbon emissions.