## 1 Introduction
[35] included the cost of carbon emissions in the total transportation cost of the MCVRP and proposed a three-dimensional ant colony optimization algorithm (TDACO) to solve it. In addition, in studies on the MCVRP solution algorithm, the variable neighborhood search algorithm (VNS) also shows a strong optimization ability [36; 37; 38; 39]. The cited studies demonstrate that the existing MCVRP model and algorithm research results are fruitful. They have established a foundation for the construction of the model and the design and innovative improvement of the algorithm presented in this study. However, there is a lack of extended research based on a dual background of urban traffic congestion and the implementation of the mixed carbon policy. The impact of road traffic congestion and the mixed carbon policy on MCVRP model construction and algorithm design has not been considered. In the current real-world context, MCVs are increasingly widely used in urban logistics distribution. MCVRP research in the urban context needs to consider traffic congestion, as well as emission reduction policy constraints. In addition, the mixed carbon policy has become the emissions reduction choice in numerous countries. Therefore, it is necessary to study the MCVRP considering traffic congestion under the mixed carbon policy. It is also necessary to further ameliorate the process and design of the VNS algorithm to improve its ability and effectiveness in solving the MCVRP in a practical context. Therefore, this paper considers the constraint of the mixed carbon policy, as well as the impact of traffic congestion, and constructs the mathematical model of MCVRP considering traffic congestion under the mixed carbon policy. Among them, the objective of the mathematical model is to minimize the total cost--that is, the sum of the vehicle management and use cost, transportation cost and mixed carbon emission cost. Then, the paper proposes a two-stage VNS-TA to solve the mathematical model. Finally, the study uses adapted standard examples and a simulation example to perform numerical experiments and results analyses and summarizes the research conclusion. A MCVRP that considers traffic congestion under the mixed carbon policy is described as follows. A distribution center adopts MCVs to perform joint transportation and dis tribution for customers with multiple mutually exclusive product demands. And the distribution process is affected by urban traffic congestion and constrained by the mixed carbon policy. Moreover, a variety of mutually exclusive products must be correctly loaded into their corresponding compartments. Each compartment of the vehicle, as well as the entire vehicle, may not be overloaded. The demand for the multi-variety of products of all customers is quantitative. Each MCV departing from the distribution center must return to the distribution center after completing the distribution tasks of all customer nodes along the current route. Finally, the distribution center needs to achieve the goal of minimizing the sum of the mixed carbon emission cost, vehicle management and usage cost and transportation cost. This study on the solution and optimization of the described problem can extend the basic MCVRP mathematical model and improve the corresponding solution algorithm, thereby providing theoretical value. This study is innovative in the following manner. First, in terms of the model, the impacts of the mixed carbon policy and traffic congestion are simultaneously introduced into the MCVRP mathematical model construction. Specifically, the objective function is constructed considering (1) the impact of traffic congestion on travel speed and travel time and (2) the impact of travel speed and load on the carbon emission rate and fuel rate. Secondly, in terms of the algorithm, a two-stage VNS-TA algorithm is designed based on the basic VNS algorithm.