Multi-Compartment Vehicle Routing Problem Considering Traffic Congestion under the Mixed Carbon Policy

Abstract

The use of multi-compartment vehicles (MCVs) in urban logistics distribution is increasing. However, urban traffic congestion causes high carbon emissions in the logistics distribution, resulting in unsustainable development in urban transportation. In addition, the application of the mixed carbon policy has gradually become the first choice for energy conservation and emission reduction in some countries and regions. The transportation industry is a major carbon-emitting industry, which needs to be constrained by carbon emission reduction policies. In this context, the research on the multi-compartment vehicle routing problem (MCVRP) considering traffic congestion under the mixed carbon policy is carried out. Firstly, a mathematical model of MCVRP considering traffic congestion under the mixed carbon policy is constructed. Secondly, a two-stage variable neighborhood threshold acceptance algorithm (VNS-TA) is proposed to solve the above mathematical model. Thirdly, 14 adapted standard examples of the MCVRP are used to verify the effectiveness and optimization ability of the two-stage VNS-TA algorithm. A simulation example of the MCVRP considering traffic congestion under the mixed carbon policy is used to conduct sensitivity analyses for different scenarios. Finally, the following conclusions are drawn: (1) the two-stage VNS-TA algorithm is effective and has strong optimization ability in solving the basic MCVRP, and (2) the two-stage VNS-TA algorithm can solve and optimize the MCVRP considering traffic congestion under the mixed carbon policy, which has the effects of cost saving and energy conservation and emission reduction.

1. Introduction

Logistics distribution in cities has the characteristics of multi-variety [1], small-batch and multi-frequency [2]. The transportation and distribution of related products require more costs and vehicles, consume more energy and emit more carbon dioxide. Take waste management as an example. Traditional waste management generally leads to unnecessary waste collection or collection delay, thus increasing the annual collection cost by about 70%. Inefficient route planning also leads to an increase in the number of collection vehicles sent, road traffic volume and fuel volume of vehicles [3], which directly lead to an increase in carbon emissions. The use of multi-compartment vehicles (MCVs) is increasing in actual logistics [4], which can meet the logistics service needs of separate loading and joint distribution of different kinds of products. The application of these MCVs facilitates the joint distribution of multi-variety, small-batch and multi-frequency products in the city, such as the collection of solid waste [3], the distribution of perishable products [5] and the transportation of medical waste [6]. Compared with ordinary single-compartment vehicles (SCVs), MCVs have obvious advantages in cost reduction [4] and energy conservation and emission reduction [3]. Continuing to take waste management as an example, the application of ordinary SCVs will increase the number of vehicles and increase the fuel burden. But the application of MCVs can reduce cross-infection between wastes and carbon emissions [3]. In an experiment of source-separated waste collection in Shanghai, China, the joint collection of MCVs (multiple wastes collected by one type of MCV) reduced carbon dioxide emissions by 24.54 kg and costs by 5720.73 CNY compared with the separate collection of SCVs (multiple wastes collected by multiple types of SCVs) [7]. The broad application prospect of MCVs in many fields of urban traffic lays a realistic foundation for the multi-compartment vehicle routing problem (MCVRP) [4]. The corresponding multi-compartment vehicle routing planning has become a research hotspot in the field [8]. Therefore, compared with the capacitated vehicle routing problem (CVRP) model (based on SCVs), it is more appropriate to use the MCVRP model (based on MCVs) to study the problem.

However, with the development of cities, environmental pollution and traffic congestion problems have appeared in many cities [9], which make urban traffic unsustainable. The application of MCVs in urban logistics has to face traffic congestion problems and be constrained by carbon emission reduction policies. Correspondingly, the study of MCVRP also needs to consider the impact of traffic congestion and the constraint of emission reduction policies. On the one hand, the problem of urban traffic congestion is increasingly serious [10], which prolongs the travel time of vehicles, increases the energy consumption of vehicles and ultimately increases the carbon emissions of urban traffic [11,12,13]. For example, the operation of urban traffic has a great impact on the carbon emissions of motor vehicles, and the carbon emissions under mild traffic congestion are 4.51 times that under smooth traffic [14]. On the other hand, the transportation industry, as one of the three major carbon emission industries, is the main implementation and constraint field of energy conservation and emission reduction policies. Carbon emissions from the transport sector (from energy consumption) account for 16.2–25% of global carbon emissions. Among them, path planning is the main content of transportation, and inefficient path planning will increase energy consumption and carbon emissions [3]. In China, green and low-carbon action in transportation has been listed as one of the ten actions in the Carbon Peak Action Plan by 2030 [15]. Internationally, the combined application of carbon tax policy and carbon trading policy has gradually become the choice of energy conservation and emission reduction for relevant industries with high carbon emissions [16]. “Mixed carbon policy” refers to the parallel and comprehensive application of a carbon tax policy and a carbon trading policy. Research and practice show that the use of a mixed carbon policy has improved the efficiency and effect of emissions reduction [17,18] and improved social benefits and environment benefits [19]. At present, many countries have implemented the mixed carbon policy in related industries. For example, the European Union and Japan added carbon taxes after establishing carbon emission trading systems.

It can be seen that the application of MCVs in urban logistics to distribute or collect multi-variety and small-batch products can help save costs and energy and reduce emissions. However, at the same time, the implementation of multi-compartment vehicle path planning must also take into account the urban traffic congestion problem and the constraints of the mixed carbon policy. These further increase the challenge of multi-compartment vehicle path planning and the difficulty in solving the MCVRP. Therefore, this paper studies the model and algorithm of MCVRP considering road traffic congestion in the context of the mixed carbon policy, which has certain practical significance.

Several related studies have established a theoretical foundation for the research developed in this paper.

On the one hand, numerous scholars have discussed the vehicle routing problem (VRP) that considers traffic congestion from the perspective of a low-carbon economy. Based on the realistic impact of road congestion on multimodal transport, Cheng and Jin [20] established and solved a multimodal transport route selection model under the emissions reduction policies of carbon tax, carbon trading, mandatory emissions reduction and carbon compensation. Zhao et al. [21] constructed a mathematical model of green vehicle paths under a time-varying network (an urban distribution problem of cold chain logistics considering traffic congestion from the perspective of a low-carbon economy) and designed an improved ant colony algorithm to solve and optimize the model. Other scholars have also presented useful discussions and enriched the relevant theoretical knowledge [22,23].

However, shortcomings remain in the research literature. Most scholars construct the VRP model considering traffic congestion based on a single emissions reduction policy. Few examine path planning and algorithm design considering traffic congestion under the background of the mixed carbon policy. The parallel application of carbon tax policy and carbon trading policy has gradually become the preferred method of emissions reduction in several countries and regions. This mixed carbon policy will have an impact on route planning and a corresponding algorithm design in the context of urban traffic congestion. Therefore, based on actual circumstances, it is necessary to study the routing problem that considers traffic congestion under the mixed carbon policy.

On the other hand, scholars have conducted model and algorithm research based on the basic MCVRP and corresponding problem variants [24,25,26,27,28,29,30,31]. The MCVRP and corresponding variants are NP-hard problems [7]. The exact optimal solution for this kind of problem cannot be obtained by an exact algorithm. It can approach the optimal solution gradually through the design and solution of a heuristic algorithm. Chen et al. [32] improved the particle swarm optimization algorithm (IPSO) to solve the basic MCVRP. Wang et al. [33] designed a heuristic algorithm including internal and external cycles for a multi-warehouse distribution problem involving refined oil with multiple trip and replenishment time windows. Hübner and Ostermeier [34] considered the easily neglected loading and unloading costs in the MCVRP and used a large neighborhood search algorithm (LNS) to solve the MCVRP with loading and unloading costs. Guo et al. [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 distribution 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. Specifically, the improvement design includes: (1) five improved neighborhood structures are designed to strengthen the global exploration capability, (2) a neighbor path selection mechanism is designed to strengthen the local search capability and (3) the threshold accepting algorithm (TA) is effectively embedded in the VNS process to facilitate jumping out of the local optimum solution.

The research paper is organized as follows. Section 2 describes the MCVRP considering traffic congestion under the mixed carbon policy, analyzes the cost composition and constructs the mathematical model. In Section 3, the two-stage VNS-TA algorithm is proposed, and the initial solution generation, as well as pre-optimization and re-optimization stages, are designed. In Section 4, based on the example setting and parameter setting, numerical experiments and results analyses of the adapted standard examples, as well as a simulation example, are performed. Section 5 presents the study’s conclusions and considers directions for future research.

2. Problem Definition

2.1. Problem Description

In this section, the distribution network of MCVs while considering traffic congestion under the mixed carbon policy is defined. Here, G $=(N,E)$, $N=\left\{0\right\}\cup N^{\prime }=\{\mathrm{0,1},2...,n\}$ and $E=\left\{\right(i,j\left)\right|i,j\in N\}$. $G$ is the multi-compartment vehicle distribution system. $N$ is the set of logistics nodes. $\left\{0\right\}$ 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 $i$. $V=\{\mathrm{1,2},3...,k\}$, where $V$ is the set of MCVs, $k$ is the MCV and $k\in V$. $M=\{\mathrm{1,2},3,...,m\}$. $M$ is the set of compartments of the MCV. $m$ is one compartment of the MCV. $P=\{\mathrm{1,2},3,...,p\}$, 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.

• 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.
• 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.
• 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. Similarly, a change in traffic congestion will lead to a change in vehicle travel speed, which will also affect carbon emissions. These effects are obtained through the MEET formula.
• The assumptions relevant for the mixed carbon policy are as follows. In order to control the carbon emissions of urban traffic, the government implements the mixed carbon policy in the distribution area. The “mixed carbon policy” refers to the simultaneous implementation of a carbon tax policy and a carbon trading policy. Considering the superposition of carbon emission reduction policies, the implementation intensity of the two emission reduction policies is set by the government. The total implementation intensity is 1, the intensity of the two carbon policies is greater than or equal to 0 and less than or equal to 1. The carbon tax price and carbon emissions control coefficient are set by the government. The carbon trading price depends on the carbon trading market. The initial carbon quota is provided by the government, and the distribution center does not need to bid for and buy the initial carbon quota.

2.2. Cost Analysis

2.2.1. Travel Time and Travel Speed Considering the Congestion Level

The travel speed and travel time of a MCV change with the change in the traffic congestion degree, which further leads to changes in vehicle management and use cost, transportation cost and mixed carbon emission cost. The following two formulas are obtained in reference [20,40]. The travel time considering traffic congestion can be calculated according to Equation (1). The travel speed considering traffic congestion can be calculated according to Equation (2). Here, $s$ indicates that traffic congestion is not considered. $t_{ij}^{ks}$ represents the free-flow travel time—that is, the travel time of MCV $k$ on road $d_{ij}$ without consideration for traffic congestion. $t_{ij}^k$ represents the travel time of MCV $k$ on road $d_{ij}$ while considering traffic congestion. $v_{ij}^{ks}$ represents the free-flow travel speed—that is, the travel speed of MCV $k$ on road $d_{ij}$ without consideration for traffic congestion. $v_{ij}^k$ represents the travel speed of MCV $k$ on road $d_{ij}$ while considering traffic congestion.$T{R}_{ij}^a$ represents the actual traffic flow of $d_{ij}$. $T{R}_{ij}^d$ represents the initial design traffic flow of $d_{ij}$. $T{R}_{ij}^a/T{R}_{ij}^d$ represents the congestion degree coefficient of $d_{ij}$. $\alpha$ and $\beta$ represent the parameters of the BPR function. This function reflects the relationship between the travel time and the traffic flow of the road section, and the parameter values are recommended by the Bureau of Public Road (renamed to Federal Highway Administration) [40].

$$t_{ij}^k=t_{ij}^{ks}·\left[1+\alpha {\left(\frac{T{R}_{ij}^a}{T{R}_{ij}^d}\right)}^{\beta }\right]$$

(1)

$$v_{ij}^k=\frac{d_{ij}}{t_{ij}^k}=\frac{d_{ij}}{t_{ij}^{ks}·\left[1+\alpha {\left(\frac{T{R}_{ij}^a}{T{R}_{ij}^d}\right)}^{\beta }\right]}=\frac{\frac{d_{ij}}{t_{ij}^{ks}}}{\left[1+\alpha {\left(\frac{T{R}_{ij}^a}{T{R}_{ij}^d}\right)}^{\beta }\right]}=\frac{v_{ij}^{ks}}{\left[1+\alpha {\left(\frac{T{R}_{ij}^a}{T{R}_{ij}^d}\right)}^{\beta }\right]}$$

(2)

2.2.2. Vehicle Management and Use Cost

It is more practical to adopt a calculation method for vehicle management and use costs that change with the travel time [21]. There are three types of expenses for each MCV dispatched by the distribution center: (1) vehicle dispatch cost, (2) vehicle usage cost and (3) labor cost. Among them, a change in travel time will lead to changes in the vehicle usage cost and labor cost. The travel time of the vehicle on the current route is the sum of the travel time of the vehicle on all the road segments along the current route. The vehicle management and use cost $C_1$ of the MCV can be calculated according to Equation (3), where $\phi$ represents the unit vehicle dispatch cost (CNY/vehicle), $\mu$ represents the unit time vehicle usage cost (CNY/hour) and $\psi$ represents the unit time labor cost (CNY/hour).

$$C_1=\phi \cdot \sum _{j\in N^{\prime }}\sum _{k\in V}{x}_{0j}^k+\left(\mu +\psi \right)\cdot \sum _{i\in N}\sum _{j\in N}\sum _{k\in V}\left(x_{ij}^k\cdot t_{ij}^k\right)$$

(3)

2.2.3. Vehicle Transportation Cost

The travel speed and load changes caused by loading and unloading during distribution affect both the carbon emissions rate and the fuel rate. Therefore, it is more realistic to use the methodology for calculating transport emissions and energy consumption (MEET) to calculate the vehicle transportation cost (fuel cost) [20,21,22,23,41]. First, the carbon emission rate (g/km) of the MCV under the current road congestion degree can be calculated according to Equation (4). ${\tau }_{0~6}$ are predefined parameters, and the values of different types of trucks differ. $\nu$ represents the travel speed of the MCV on the road with the current congestion degree. Second, the load correction factor corresponding to the carbon emission rate can be calculated according to Equation (5). ${\varpi }_{0~7}$ are predefined parameters, and the values of different types of trucks differ. $r$ represents the ratio of the sum of the actual load of all compartments of the MCV on the current road to the maximum capacity of the entire vehicle. Third, the carbon emission rate (kg/km) of MCV $k$ on the current road section $d_{ij}$ under traffic congestion situation can be calculated according to Equation (6). Then, the fuel consumption rate (kg/km) of MCV $k$ on the current road section $d_{ij}$ under traffic congestion situation can be calculated according to Equation (7). For referene, 2.32 kg/L is the carbon emission coefficient of gasoline. Finally, the transportation fuel consumption TF and vehicle transportation cost $C_2$ generated during the driving process can be calculated according to Equations (8) and (9). $\theta$ represents the unit fuel cost.

$${\xi }_v={\tau }_0+{\tau }_1\cdot v+{\tau }_2\cdot v^2+{\tau }_3\cdot v^3+\frac{{\tau }_4}{v}+\frac{{\tau }_5}{v^2}+\frac{{\tau }_6}{v^3}$$

(4)

$$\kappa ={\varpi }_0+{\varpi }_1\cdot r+{\varpi }_2\cdot r^2+{\varpi }_3\cdot r^3+{\varpi }_4\cdot v+{\varpi }_5\cdot v^2+{\varpi }_6\cdot v^3+\frac{{\varpi }_7}{v}$$

(5)

$$C_{ij}^k=\frac{{\xi }_v\cdot \kappa }{1000}$$

(6)

$$F_{ij}^k=\frac{C_{ij}^k}{2.32}$$

(7)

$$TF=\sum _{i\in N}\sum _{j\in N}\sum _{k\in V}{x}_{ij}^k\cdot t_{ij}^k\cdot v_{ij}^k\cdot F_{ij}^k$$

(8)

$$C_2=\theta \cdot TF=\theta ·\sum _{i\in N}\sum _{j\in N}\sum _{k\in V}{x}_{ij}^k\cdot t_{ij}^k\cdot v_{ij}^k\cdot F_{ij}^k$$

(9)

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). $\epsilon$ represents the carbon tax price. $\lambda$ represents the carbon trading price. $CQ$ represents the carbon quota. Additionally, when $w_1=1$ and $w_2=0$, the mixed carbon emission cost is the carbon tax cost only. When $w_1=0$ and $w_2=1$, the mixed carbon emission cost is the carbon trading cost only. Notably, the pressure of emissions reduction generated by the implementation of the mixed carbon policy is not doubled—that is, $w_1+w_2=1$.

$$\begin{gathered} C_3=w_1\cdot \epsilon \cdot \omega \cdot TF+w_2\cdot \lambda \cdot \left(\omega \cdot TF-CQ\right)=\omega \cdot \left(w_1\cdot \epsilon +w_2\cdot \lambda \right)\cdot TF-w_2\cdot \lambda \cdot CQ \\ =\omega \cdot \left(w_1\cdot \epsilon +w_2\cdot \lambda \right)\cdot \sum _{i\in N}\sum _{j\in N}\sum _{k\in V}{x}_{ij}^k\cdot t_{ij}^k\cdot v_{ij}^k\cdot F_{ij}^k-w_2\cdot \lambda \cdot CQ \end{gathered}$$

(10)

2.3. Mathematical Formulation

$$\begin{gathered} min \phi \cdot \sum _{j\in N^{\prime }}\sum _{k\in V}{x}_{0j}^k+\left(\mu +\psi \right)\cdot \sum _{i\in N}\sum _{j\in N}\sum _{k\in V}\left(x_{ij}^k\cdot t_{ij}^k\right)+\theta \cdot \sum _{i\in N}\sum _{j\in N}\sum _{k\in V}{x}_{ij}^k\cdot t_{ij}^k\cdot v_{ij}^k\cdot F_{ij}^k +\omega \cdot \left(w_1\cdot \epsilon +w_2\cdot \lambda \right) \\ \cdot \sum _{i\in N}\sum _{j\in N}\sum _{k\in V}{x}_{ij}^k\cdot t_{ij}^k\cdot v_{ij}^k\cdot F_{ij}^k-w_2\cdot \lambda \cdot CQ \end{gathered}$$

(11)

$$\sum _{i\in N}{x}_{ij}^k=\sum _{i\in N}{x}_{ji}^k=1 \forall j\in N^{\prime },\forall k\in V$$

(12)

$$\sum _{p\in P}{y}_{jp}^k=p \forall j\in N^{\prime },\forall k\in V$$

(13)

$$\sum _{j\in N^{\prime }}{x}_{0j}^k=\sum _{j\in N^{\prime }}{x}_{j0}^k \forall k\in V$$

(14)

$$\left|S\right|\ge 1 \forall S\subseteq N^{\prime }$$

(15)

$$\sum _{i\in S}\sum _{j\in S}{x}_{ij}^k\le \left|S\right|-1 \forall k\in V,\forall S\subseteq N^{\prime }$$

(16)

$$y_{jp}^k\cdot q_{jp}\le Q_p \forall k\in V,\forall p\in P,\forall j\in N^{\prime }$$

(17)

$$\sum _{j\in N^{\prime }}{y}_{jp}^k\cdot q_{jp}\le Q_p \forall k\in V,\forall p\in P$$

(18)

$$\sum _{j\in N^{\prime }}\sum _{p\in P}{y}_{jp}^k\cdot q_{jp}\le Q_{max} \forall k\in V$$

(19)

$$t_{ij}^k=t_{ij}^{ks}·\left[1+\alpha {\left(\frac{T{R}_{ij}^a}{T{R}_{ij}^d}\right)}^{\beta }\right] \forall i\in N,\forall j\in N^{\prime },\forall k\in V$$

(20)

$$v_{ij}^k=\frac{v_{ij}^{ks}}{\left[1+\alpha {\left(\frac{T{R}_{ij}^a}{T{R}_{ij}^d}\right)}^{\beta }\right]} \forall i\in N,\forall j\in N^{\prime },\forall k\in V$$

(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. Constraint (21) indicates that the travel speed of any vehicle on any road segment is affected by traffic congestion.

3. Solution Approach

In the studies on the MCVRP solution algorithm, the VNS algorithm has shown a strong optimization ability [36,37,38,39]. In addition, through a large number of experiments and the comparison of various algorithms in the early stage, it was found that the VNS algorithm is superior in the solution process and the TA algorithm is superior in the solution acceptance process. Finally, the VNS algorithm and TA algorithm are selected to design the solution algorithm of this study. The results mainly present the effectiveness analysis of the improvement before and after the algorithm design. The large number of experiments in the early stage will not be carried out here.

In this section, based on the respective advantages of the VNS algorithm and the TA algorithm, a two-stage VNS-TA algorithm is designed to solve the mathematical model of MCVRP considering traffic congestion under the mixed carbon policy. The advantages of the VNS algorithm are fewer parameters, fewer links and easy operation and improvement [42]. The advantages of the TA algorithm are high efficiency and adaptive change [43]. The specific algorithm is designed as follows.

In the stage of initial solution generation and pre-optimization, the scanning method is first used to generate the initial solution. Then, the local search process is performed to pre-optimize the initial solution. Finally, the stopping criterion of the maximum number of iterations is applied to generate a better pre-optimized initial scheme. In the re-optimization stage, the VNS-TA algorithm is designed to improve the pre-optimized initial scheme. First, five improved neighborhood structures are designed based on traditional neighborhood search operators, which promote the disturbance and neighborhood variability of the algorithm. Second, the neighbor path selection mechanism is designed, and the local search operators are combined to perform an efficient local optimization of the neighborhood solution. The local optimal solution can be obtained through this procedure. Then, the TA algorithm is embedded into the VNS algorithm, and the tolerance coefficient of the phased decline is designed to realize the gradual and slow reduction of the threshold acceptance range. This design appropriately accepts poor-quality solutions, expands the search range of the solution space and balances the search depth and search breadth of the VNS algorithm. Finally, the re-optimized scheme is output after a certain number of iterations.

For ease of understanding, the word “scheme” is used in the algorithm output, such as in the phrases “initial scheme” and “re-optimized scheme”. The term “solution” is used during the operation of the algorithm, such as in the phrases “initial solution”, “current solution”, “current optimal solution”, “neighborhood solution”, “local optimal solution” and “global optimal solution”.

3.1. Stage of Initial Solution Generation and Pre-Optimization

3.1.1. Process of the Algorithm

The algorithm process of initial solution generation and pre-optimization is as Algorithm 1.

Algorithm 1: Initial solution generation and pre-optimization.

The maximum number of iterations $i_{max}=round\left(n/5\right)$  Initial solution $x_{initial}=\left[\right]$  //Loop:   for $i=1$ to $i_{max}$ do     Use the scanning method to generate the initial solution $x_{initial}$.     Use the local optimization process to obtain the pre-optimized initial solution $x_{initial}\prime$.     if $f\left({x_{initial}}^{\prime }\right)<f\left(x_{initial}\right)$      $x_{initial}={x_{initial}}^{\prime }$     end if   end for  //Output:  Pre-optimized initial scheme.

3.1.2. Initial Solution Generation Based on the Scanning Method

From the perspective of mathematical model solving, the process of obtaining the initial solution by the scanning method is subject to all the constraints of the mathematical model of the problem in this study. Therefore, these solutions can be identified as feasible solutions. Next, this paper will explain how to obtain this feasible initial solution from the perspective of algorithm design.

First, the distribution center is set as the pole $o,$ and the customer node $p$ is randomly selected. The connection between the node $p$ and distribution center $o$ is taken as the polar axis, and all other customer nodes are converted into polar coordinates $\left(r,t\right)$. Second, customer nodes are assigned to the MCV one by one according to the order of the angles $t$ in the polar coordinates from small to large. At this point, whether the current customer node is assigned to the current MCV needs to be limited by two constraints. The first is the compatibility constraint between different types of products and compartments. The second is the maximum capacity limit of different compartments of the MCV. If the load capacity of at least one compartment exceeds its maximum capacity constraint after allocation, the customer node allocation of the current MCV is terminated. At the same time, a new MCV is sent to reopen a path. The current customer node is taken as the initial node of the reopened path, and the remaining customer nodes are allocated one by one. This process is repeated to complete the assignment of all client nodes and generate the initial solution. Note that the number of nodes in a path and the number of paths in a solution do not affect the operation of the program, such as the path formed by a single client node and the solution formed by a single path.

3.1.3. Pre-Optimization of the Initial Solution Based on the Local Optimization Process

Notably, the allocation of all client nodes is determined by the angle $t$ of the polar coordinates, and the distance $r$ is not considered. Therefore, the solution generation process has some limitations, and the initial solution can still be further optimized. In this regard, Section 3.2.3 is used to pre-optimize the generated initial solution. That is, the program module of Section 3.2.3 is applied to pre-optimize the initial solution. Because this program module is the key design part of the VNS-TA algorithm in the re-optimization stage, the principle and process of the module are presented in detail in Section 3.2.3. This part will not be introduced here.

3.1.4. Initial Solution Acceptance of the Stopping Criterion Based on the Maximum Number of Iterations

After the initial solution generation and pre-optimization, the stopping criterion with the maximum number of iterations is used to retain and output the better initial scheme. The maximum number of iterations affects the diversity and quality of the initial scheme. When the maximum number of iterations is small, the diversity of the initial scheme is better, but the quality is worse. When the maximum number of iterations is large, the quality of the initial scheme is better, but the diversity is worse. Therefore, the appropriate maximum number of iterations is designed to output a better pre-optimized initial scheme with a certain degree of diversity.

3.2. Re-Optimization Stage and VNS-TA Algorithm

3.2.1. Idea and Process

To more fully develop and improve the pre-optimized initial scheme, the VNS-TA is designed. The VNS algorithm is realized through the shaking process and the local search process [44,45]. First, the neighborhood structures are sequentially used to perform neighborhood transformation operations on the current solution to obtain the neighborhood solution. Second, a local search process is performed on the neighborhood solution to obtain the local optimal solution. Then, the two processes are repeated to realize the exploration of the current solution space. Finally, the current solution continues to approach the global optimal solution after several iterations. The process of the TA algorithm functions to gradually narrow the threshold acceptance range and to determine whether to accept the new solution. There are two cases for accepting a new solution. One is when the value of the objective function of the new solution is lower than that of the current optimal solution. The other is that the objective function value of the new solution is higher than that of the current optimal solution, but it does not exceed the threshold acceptance range. Otherwise, reject the new solution.

The VNS-TA algorithm proceeds as follows. First, five improved neighborhood structures are designed based on the traditional neighborhood operators. The five improved neighborhood structures are successively used to perform neighborhood transformation operations on the pre-optimized initial scheme to obtain the neighborhood solution. This design increases the diversity of the neighborhood solutions and improves the global exploration ability. Second, a neighbor path selection mechanism is designed, which combines local search operators to optimize the neighborhood solution locally. The local search operators consist of 2-opt*, swap (exchange), insert and 2-opt. This design reduces the number of invalid searches, speeds up the convergence and improves the local optimization effect. Then, the TA algorithm is effectively embedded in the VNS algorithm. The tolerance coefficient of the phased decline is designed to realize the gradual and slow reduction of the threshold acceptance range. The solution acceptance process is oriented by the current optimal solution and benchmarked by the current threshold acceptance range. This design can perform a breadth search for solutions in the early stage of iteration and depth search for solutions in the late stage of iteration. Finally, the stopping criterion of the maximum number of iterations is used to stop the iteration loop, and the re-optimized scheme is obtained and output.

A “breadth search for solutions” means that, in the early stage of iteration, the threshold acceptance range is large, and the tolerance for poor-quality solutions is large. The poor-quality solutions are appropriately retained to increase the diversity of solutions and expand the search space of solutions. A “depth search for solutions” means that, in the later stage of iteration, the threshold acceptance range is small, and the tolerance for poor-quality solutions is small. Only solutions with a small gap from the current optimal solution are retained, and these solutions are constantly optimized to improve the quality of the solutions. The algorithm process is as Algorithm 2.

Algorithm 2: The VNS-TA algorithm in the re-optimization stage.

Tolerance coefficient $rho$, decline rate of tolerance coefficient $phi$ and neighborhood structure set $N_k\left(k=\mathrm{1,2},\dots ,k_{max}\right)$, where $k_{max}=5$, and the maximum number of iterations $p_{max}=500$.  The current solution $x_{current}=x_{initial}$, and the current optimal solution $x_{current\_optimal}=x_{initial}$.  //Loop:    for $p=1$ to $p_{max}$ do                   $k=1$     while $k\le k_{max}$ do      Shaking process: select the neighborhood structure $N_k$ to perform the neighborhood transformation operation on $x_{current}$ sequentially and obtain the neighborhood solution $x_{neighborhood}$.      Local optimization process: based on the neighbor path selection mechanism, select two paths in $x_{neighborhood}$ for the local search and optimization successively and obtain the local optimal solution $x_{local\_optimal}$.      if $f\left(x_{local\_optimal}\right)\le f\left(x_{current}\right)$            $x_{current}=x_{local\_optimal}, k=1$        if $f\left(x_{current}\right)<f\left(x_{current\_optimal}\right)$              $x_{current\_optimal}=x_{current}$        end if      else        Change in the threshold acceptance range: $T=f\left(x_{current\_optimal}\right)·rho$        if $f\left(x_{local\_optimal}\right)-f\left(x_{current\_optimal}\right)<T$            $x_{current}=x_{local\_optimal},k=1$        else                $k=k+1$        end if      end if     end while     Change of the tolerance coefficient: $rho=rho·phi$    end for  //Output:  Re-optimized scheme.

3.2.2. Shaking Process

The improved neighborhood structures are used sequentially to perform neighborhood transformation operation on the better initial scheme of the input. This study designs five improved neighborhood structures. The design principle is to extensively explore the neighborhood solution space within a limited number of neighborhood structures and a limited running time. Therefore, on the one hand, transformation randomness is added to the neighborhood structures. On the other hand, the shaking ideas of several traditional neighborhood search operators are learned to increase the transformation types. In addition, to further improve the global search ability of the algorithm, the infeasible neighborhood solution is retained after the shaking process [42]. At the same time, the node sequence of the infeasible neighborhood solution is adjusted and recovered according to the capacity limit constraints, etc. In other words, the infeasible neighborhood solution is transformed into a feasible neighborhood solution on the basis of preserving the original path sequence. Due to space limitations, this process will not be elaborated here, and additional details were provided in a previous study [42]. The specific design ideas and steps for the five improved neighborhood structures are as follows.

• Integrated-k-opt neighborhood. The integrated-k-opt neighborhood is designed based on the idea of the k-opt operator. The k-opt operator process is as follows: arbitrarily remove $k$ different arcs on the current path, add $k$ new arcs to reconnect to form new paths and, finally, generate $\left(k-1\right)!2^{k-1}$ new solutions. The steps for the integrated-k-opt neighborhood are as follows.
  • Step 1
    Randomly generate an integer from $2$ to $6$ and assign it to $k$; that is, randomly select a transformation.
  • Step 2
    Determine the value of $k$, randomly select $k$ different client nodes on $q$ to form sequence $i$ and sort the client nodes in ascending order to obtain a new node sequence $t$, where $q$ is the original path sequence.
  • Step 3
    Assign the original path sequence $q$ to $newq$.
  • Step 4
    Flip all nodes between $t\left(1\right)$ and $t\left(2\right)$ on $q$ and assign them to all nodes between $t\left(1\right)$ and $t\left(2\right)$ on $newq$.
  • Step 5
    When $k>2$, $j$ loops from $2$ to $k-1$ and executes the following statements.

    ○

    Step 5.1

    Flip all nodes between $t\left(j\right)+1$ and $t\left(j+1\right)$ on $newq$ and assign them to all nodes between $t\left(j\right)+1$ and $t\left(j+1\right)$ on $newq$.

  • Step 6
    Output $newq$.
• Integrated-cross-exchange neighborhood and integrated-icross-exchange neighborhood. The integrated-cross-exchange neighborhood and integrated-icross-exchange neighborhood are designed based on the ideas of swap, cross-exchange, icross-exchange and three-point operators. The swap operator process is as follows: two different nodes (or node fragments) on the path are randomly selected for exchange. The cross-exchange operator process is as follows: two node fragments are placed in one another's position without changing direction. The icross-exchange operator process is as follows: the two node fragments to be exchanged are flipped and placed in one another's position. The three-point procedure is as follows: a single node and two adjacent nodes on the path are selected for exchange. Note that the steps of the integrated-icross-exchange neighborhood are nearly identical to those of the integrated-cross-exchange neighborhood, except for the action of changing the direction of the node fragment before the exchange. The steps for the integrated-cross-exchange neighborhood are as follows.
  • Step 1
    Obtain the number $n$ of all client nodes of path $q$ and assign the original path sequence $q$ to the new path sequence $newq$.
  • Step 2
    Randomly select two different client nodes in $q$ and assign the nodes to $i_1$ and $i_2$ according to the position order, where $i_1$ and $i_2$ represent the anchor points of two nodes (or node fragments) to be exchanged.
  • Step 3
    Randomly generate $1$ or $2$.
  • Step 4
    When $1$ is generated, execute the following statements.

    ○

    Step 4.1

    Randomly select the number $b$ of nodes to be exchanged, where $1\le b\le 5$.

    ○

    Step 4.2

    Deal with extreme points. When $i_1\ge n-b+2$, assign $n-b+1$ to $i_1$. When $i_2\le b-1$, assign $b$ to $i_2$.

    ○

    Step 4.3

    For $newq$. If $b=1$, exchange nodes $i_1$ and $i_2$. If $b\ge 2$, exchange two client node fragments $i_1~i_1+b-1$ and $i_2-b+1~i_2$ without changing direction. Gain $newq$.

  • Step 5
    When $2$ is generated, execute the following statements.

    ○

    Step 5.1

    Randomly select the number $a$ of nodes of the client node fragment to be exchanged with a single client node, where $2\le a\le 5$.

    ○

    Step 5.2

    Deal with the extreme points. When $i_2\le a-1$, assign $a$ to $i_2$.

    ○

    Step 5.3

    For $newq$. Exchange the single node $i_1$ and the node fragment $i_2-a+1~i_2$ without changing direction. Gain $newq$.

  • Step 6
    Remove duplicate nodes in $newq$ without changing the sequence order of the client nodes.
  • Step 7
    Output $newq$.
• Integrated-cross-insert neighborhood and integrated-icross-insert neighborhood. The integrated-cross-insert neighborhood and integrated-icross-insert neighborhood are designed based on the ideas of the insert, cross-exchange and icross-exchange operators. The insert operator process is as follows: two points on the path are randomly selected, and one point (or node fragment) is inserted in front of or behind the other node. Note that the steps of the integrated-icross-insert neighborhood are nearly identical to those of the integrated-cross-insert neighborhood, except for the action of changing the direction of the node fragment before the insertion. The steps for the integrated-cross-insert neighborhood are as follows.
  • Step 1
    Obtain the number $n$ of all client nodes of path $q$.
  • Step 2
    Randomly select two different client nodes in $q$, and assign the nodes to $j_1$ and $j_2$ according to the position order, where $j_1$ and $j_2$ are the two anchor points for the insertion operation to be performed.
  • Step 3
    Randomly generate $1$ or $2$.
  • Step 4
    When $1$ is generated, execute the following statements.

    ○

    Step 4.1

    Randomly select the number $b$ of client nodes to be inserted, where $1\le b\le 5$.

    ○

    Step 4.2

    For $q$. If $j_1<n-b+2$, insert client nodes $j_1~j_1+b-1$ between client nodes $j_2$ and $j_2+1$ to form a new path sequence $newq$. Note that both single-node and multi-node insertions are included.

    ○

    Step 4.3

    Otherwise, assign $q$ directly to $newq$.

  • Step 5
    When $2$ is generated, execute the following statements.

    ○

    Step 5.1

    Randomly select the number $a$ of client nodes to be inserted, where $1\le a\le 5$.

    ○

    Step 5.2

    For $q$. If $j_2>a-1$, insert client nodes $j_2-a+1~j_2$ between client nodes $j_1$ and $j_1+1$ to form a new path sequence $newq$.

    ○

    Step 5.3

    Otherwise, assign $q$ directly to $newq$.

  • Step 6
    Remove duplicate nodes in $newq$ without changing the sequence order of the client nodes.
  • Step 7
    Output $newq$.

3.2.3. Local Optimization Process

From the perspective of mathematical model solving, the neighborhood solution can be retained as the local optimal solution as long as the fitness function value of the neighborhood solution becomes smaller. In addition, it should be made clear that (1) the local optimal solution changes with continuous iteration and is not static, and (2) the local optimal solutions obtained by different local optimization processes are different, which depends on the design rules of the algorithm designer. Next, this paper will explain how to obtain the local optimal solution from the perspective of algorithm design.

First, based on the designed neighbor path selection mechanism, two paths are selected successively according to the order of the search value from large to small. Then, local search operators, such as 2-opt*, swap, insert and 2-opt, are used to perform a local search and optimization on the two selected paths.

(1)

Neighbor path selection mechanism. This design is based on the neighbor optimization mechanism [42], the idea of gradually expanding the local search space of solutions [28] and the empirical basis for the construction of VNS [46]. The mechanism selects two paths for local search and optimization according to the order of the search value and optimization space from large to small. There are a few key points to note here. First, the closer the two paths are in physical space, the easier it is to generate invalid intersections. In this case, there is a larger search value and optimization space between the two paths, and eliminating invalid intersections can realize the optimization of the solution. When the distance between the two paths increases, the search value and optimization space correspondingly decrease. Second, the closer the two paths are, the closer the “distance” is between the new solution and the old solution in the solution space. In this case, there are more high-quality common features between the two solutions, and it is easier to obtain a better, new solution based on the old solution. In addition, note that the maximum number of intervals between two paths is $maxp=floor\left(numel\left(Rout\right)/2\right)$—that is, the rounding down of $1/2$ of the number of paths. The explanation is that choosing ${Subpath}_m$ and ${Subpath}_{m+p}$ is equivalent to choosing ${Subpath}_m$ and ${Subpath}_{m-numel\left(Rout\right)+p}$. For example, in the case of $numel\left(Rout\right)=8$, the choice of ${Subpath}_1$ and ${Subpath}_4$ when $p=3$ is equivalent to the choice of ${Subpath}_4$ and ${Subpath}_1$ when $p=5$. Moreover, when two paths are repeatedly selected, the search value and optimization space decrease and the invalid repeated search and running time increase.

(2)

Local search operators. Based on the two selected paths, the 2-opt*, swap and insert operators are used for the local search between paths. The 2-opt operator is used to expand the search and optimization within the path for the two paths. Specifically, the 2-opt* operator process is as follows: select one arc in each of the two paths for elimination to obtain four path fragments and generate two new paths different from the original two paths by reconnecting them. Note that the reconnection of four fragments results in two connection modes, as shown in Figure 1a. The swap operator process is as follows: select a node in each of the two paths for exchange to form two new paths, as shown in Figure 1b. The insert operator process is as follows: select a node in each of the two paths and insert one node in front of or behind the other node to form two new paths, as shown in Figure 1c. Notably, the insert operator may cause the disappearance of the path. Therefore, a mechanism of “eliminating the empty path $\to$ exiting the loop $\to$ re-entering the loop” is designed to help the local optimization process continue. The 2-opt operator process is as follows: select two different nodes on the path and flip the two nodes and all nodes in between to obtain a new path, as shown in Figure 1d. In addition, the local search will lead to changes in the load and driving distance of certain MCVs. Therefore, it is necessary to rejudge the new path that is formed. If the new solution satisfies the vehicle capacity limit and the acceptance criterion of the solution, the solution is accepted. Otherwise, the solution is not accepted.

3.2.4. Acceptance Criteria for the Solution

In the TA algorithm, whether to accept the current solution depends on the comparison between it with the current optimal solution. First, when the fitness function value of the current solution is less than that of the current optimal solution, as shown in Equation (22), the current solution is retained. Second, when the fitness function value of the current solution is greater than that of the current optimal solution but does not exceed the current threshold tolerance range, as shown in Equation (23), the current solution is retained. It should be clarified here that (1) the current solution is the feasible solution obtained under the current number of iterations, (2) the current optimal solution is the optimal feasible solution running the iteration so far and (3) the global optimal solution is the optimal solution of the mathematical model of the problem. The current optimal solution is not the global optimal solution. NP-hard problems are almost impossible to solve by accurate algorithms unless $P=NP$ [47].

$$fitness\left(x_{current}\right)-fitness\left(x_{current\_optimal}\right)<0$$

(22)

$$0<fitness\left(x_{current}\right)-fitness\left(x_{current\_optimal}\right)<T_i$$

(23)

Next, this paper will explain how to obtain the local optimal solution from the perspective of algorithm design. The TA algorithm is effectively embedded in the VNS algorithm, and the threshold acceptance range is used as the benchmark to judge whether the solution is accepted. The TA algorithm is implemented by the threshold acceptance range change formula and the tolerance coefficient change formula, as shown in Equations (24) and (25). Among them, $T_i$ represents the threshold acceptance range of the current solution at the $i\mathrm{t}\mathrm{h}$ iteration, $fitness\left(x_{current\_optimal}\right)$ represents the objective function value of the current optimal solution at the $i\mathrm{t}\mathrm{h}$ iteration, ${rho}_i$ represents the tolerance coefficient of the $i\mathrm{t}\mathrm{h}$ iteration, ${rho}_{j+1}$ represents the tolerance coefficient of the $j+1\mathrm{t}\mathrm{h}$ iteration, ${rho}_j$ represents the tolerance coefficient of the $j\mathrm{t}\mathrm{h}$ iteration and $phi$ represents the rate of decline in the tolerance coefficient. Equation (24) shows that, with the increase in the tolerance coefficient and the objective function value of the current optimal solution, the threshold acceptance range will also increase. Equation (25) shows that, as the decline rate of the tolerance coefficient becomes larger, the decline of the tolerance coefficient slows, and the narrowing speed of the threshold acceptance range also slows.

$$T_i=fitness\left(x_{current\_optimal}\right)·{rho}_i$$

(24)

$${rho}_{j+1}={rho}_j·phi$$

(25)

In addition, a stage decline strategy for the tolerance coefficient (Strategy 1) is designed to achieve the following two points. First, in the current inner loop, the tolerance coefficient is unchanged, and the change in the threshold acceptance range only emerges from the update of the current optimal solution. Second, when entering the next outer cycle from the current outer cycle, the tolerance coefficient decreases once, and the threshold acceptance range achieves a “jump down”. To verify the effectiveness of the stage decline strategy for the tolerance coefficient, an exponential decline strategy for the tolerance coefficient (Strategy 2) is designed for comparison. An example is given to illustrate the difference between the two strategies. The VNS process is repeated $\mathrm{f}\mathrm{i}\mathrm{v}\mathrm{e}$ times, and the shaking process and local optimization process are performed $\mathrm{t}\mathrm{h}\mathrm{r}\mathrm{e}\mathrm{e}$ times in each VNS. The initial tolerance coefficient is $0.5$, and the decline rate of the tolerance coefficient is $0.8$. The changes in the tolerance coefficients of Strategy 1 and Strategy 2 are shown in Figure 2.

The framework of the algorithm based on Strategy 1 and Strategy 2 is as Algorithms 3 and 4.

Algorithm 3: Framework based on Strategy 1.

1 The outer loop begins 2  The inner loop begins 3    Shaking process 4    Local optimization process 5    Threshold accepts range change 6    Decide whether to accept the new solution 7  The inner loop ends 8  Tolerance coefficient change 9 The outer loop ends

Algorithm 4: Framework based on Strategy 2.

1 The outer loop begins 2  The inner loop begins 3    Shaking process 4    Local optimization process 5    Threshold accepts range change 6    Decide whether to accept the new solution 7    Tolerance coefficient change 8  The inner loop ends 9 The outer loop ends

The time complexity of the algorithm based on Strategy 1 tends to be the same as that based on Strategy 2. We compare the algorithmic complexity of the two strategies horizontally. In terms of details, the time complexity of Strategy 1 is slightly higher than that of Strategy 2. The threshold acceptance range of Strategy 1 converges more slowly, and the number of execution statements for inferior solutions increases. The threshold acceptance range of Strategy 2 converges faster, and the number of execution statements for inferior solutions decreases. On the whole, the time complexity of the algorithms under the two strategies tends to be consistent; that is, the difference in the decreasing trend of the tolerance coefficient is not particularly large, as shown in Figure 2.

4. Numerical Experiments and Results

4.1. Example and Parameter Setting

4.1.1. Example Setting

In this study, two types of examples are designed for the experiments and analyses. Among them, the adapted standard example of the basic MCVRP is used to analyze the solution quality and solution timeliness of the two-stage VNS-TA algorithm. The simulation example of the MCVRP considering traffic congestion under the mixed carbon policy is used to analyze the effects of cost saving and energy conservation and emission reduction of the two-stage VNS-TA algorithm.

Firstly, with reference to the literature [29], 7 international standard examples of CVRP are adapted into 14 MCVRP examples. Based on the vrpnc2-S2 example, numerical experiments are used to analyze the effectiveness of the two-stage VNS-TA algorithm in solving the basic MCVRP. The two-stage VNS-TA algorithm is used to calculate 14 adapted standard examples to obtain results. The results obtained are compared with those of the IPSO [32] and IVNS [42] to verify the optimization ability of our algorithm.

Secondly, with reference to the relevant literature and based on the vrpnc1-S2 example, a simulation example vrpnc1-S2’ of the MCVRP considering traffic congestion under the mixed carbon policy is designed. This example is used to simulate a real situation. Consider a multi-compartment vehicle distribution system in which customers demand two different categories of products with split-compartment loading and distribution requirements. The distribution center uses two-compartment vehicles. In addition, this distribution center area suffers from traffic congestion and is subject to the mixed carbon policy. Sensitivity analyses of different scenarios are used to analyze the optimization effect, cost saving effect and energy conservation and emission reduction effect. It should be noted that the design of the simulation example refers to relevant research literature and reports. Simulation examples are not real cases.

Although, in a real situation, traffic congestion and multi-variety product distribution are more complex, and the formulation of the mixed carbon policy also needs to consider various aspects, this paper simplifies these problems. Among them, in the simulation example, only the distribution of two categories of products is considered, the traffic congestion scenarios are simplified into three and the mixed carbon policy scenarios are simplified into five.

The simulation example is set as follows. The coordinates of the customer, the demand of the customer and the capacity of the vehicle are adapted by referring to the literature [29]. The unit of distance is km. The unit of product demand is kg. The unit of maximum load for each compartment and the entire vehicle is also in kg. The free-flow travel speed of the vehicle in the uncongested road section is $80$ km/h [48]. The parameters of the BPR function are $0.15$ and $4$ [20]. The cost of vehicle dispatch is $150$ CNY/vehicle [21]. The vehicle use cost and labor cost after dispatching vehicles are $5$ CNY/hour and $10$ CNY/hour, respectively [21]. The fuel cost is $7.9$ CNY/L [48]. The predefined parameters and load correction factor of the vehicles [21] are $\left\{\mathrm{0,110,0},\mathrm{0,0},\mathrm{0.000375,8702}\right\}$ and $\left\{\mathrm{1.27,0.0614,0},\mathrm{0.0011,0.00235,0},\mathrm{0,1.33}\right\}$. The carbon emission coefficient is $2.32$ kg CO₂/L [21]. Referring to China’s carbon trading website and relevant reports from several countries, we set the carbon tax price at $0.06$ CNY/kg and the carbon trading price at $0.05$ CNY/kg.

The simulation scenarios are set as follows. On the one hand, it is assumed that there are three traffic congestion scenarios, namely, the no congestion scenario, normal congestion scenario and very congested scenario. The coefficient matrix of the congestion degree for the three scenarios is generated [20]. Here, all elements in the coefficient matrix of the congestion degree of the no congestion scenario are $0$. All elements in the coefficient matrix of the congestion degree of the normal congestion scenario are random numbers within $\left[\mathrm{0,3}\right]$. All elements in the coefficient matrix of the congestion degree of the very congested scenario are $3$. On the other hand, it is assumed that there are five mixed carbon policy scenarios, namely, S1, S2, S3, S4 and S5. The specific explanation is as follows. $w_1:w_2$ is the weight ratio of the carbon tax cost and carbon trading cost in the mixed carbon emission cost. The simulation of five mixed carbon policy scenarios is realized by changing the weight ratio. The weight ratios and meanings of the five scenarios are shown in

Table 1. Cost weight ratios and implications of different mixed carbon policy scenarios.

Category of Scenarios	Weight Ratio between Costs	Meaning of Scenario
S1	$w_1:w_2=1:0$	Only the carbon tax policy is adopted.
S2	$w_1:w_2=0.75:0.25$	The carbon tax policy is dominant, and the carbon trading policy is supplemented.
S3	$w_1:w_2=0.5:0.5$	The carbon tax policy and the carbon trading policy are equal.
S4	$w_1:w_2=0.25:0.75$	The carbon trading policy is dominant, and the carbon tax policy is supplemented.
S5	$w_1:w_2=0:1$	Only the carbon trading policy is adopted.

.

In addition, the historical method is used to calculate the carbon quota of a single distribution activity of the distribution center, as shown in Equation (26). $\mu$ is the carbon quota. $\tau$ represents the carbon emission intensity in the historical base year (tons CO₂/100 million ton-km freight turnover). $\chi$ represents the control coefficient (dimensionless). $\gamma$ represents the freight turnover in the approved year (100 million ton-km freight turnover), which can be calculated according to Equation (27). In the simulation example vrpnc1-S2', $\nu$ represents the average of the total transportation distance of multiple running tests, and $\delta$ represents the total demand. Here, the emissions control coefficient is $99.5\mathrm{\%}$ [49]. According to the road freight data in the Galaxy Transportation and China Statistical Yearbook (2018), the carbon emissions intensity is $0.0000793$ kgCO₂/kg-km freight turnover. For additional information regarding the concepts of carbon intensity and freight turnover, please refer to the method [49].

$$\mu =\tau ·\chi ·\gamma$$

(26)

$$\gamma =\nu ·\delta$$

(27)

Furthermore, in Section 4.1.2 and Section 4.2, other variables and procedures are maintained unchanged, and the results of $\mathrm{f}\mathrm{i}\mathrm{v}\mathrm{e}$ runs under different coefficients or strategies are obtained. The maximum number of iterations in the re-optimization stage is $500$. To facilitate comparisons with research results in the literature and to analyze the effectiveness of the designed algorithm, the total distribution distance is used as the objective function value, and five improved neighborhood structures are used to form the neighborhood structure set. In Section 4.3, the relevant impacts of different traffic congestion scenarios and different mixed carbon policy scenarios on the multi-compartment vehicle path planning are examined and compared. In each scenario, the maximum number of iterations in the re-optimization stage is $100$, and the results of $\mathrm{f}\mathrm{i}\mathrm{v}\mathrm{e}$ runs are obtained. The total cost, including the mixed carbon emission cost, vehicle management and usage cost and transportation cost, is used as the objective function value. The integrated-icross-insert neighborhood and integrated-icross-exchange neighborhood are used to form the neighborhood structure set. In this way, a comparative analysis can be performed while appropriately reducing the running time. In addition, the related indicators of the objective function value and running time of the results of multiple runs are adopted for comparison, such as the optimal value, worst value, average value, standard variance, range and range percentage. MATLAB R2016b was used for the calculations, and the operating environment was a Windows 10 Professional Edition, Intel(R)Core(TM)i5-4210H CPU @ 2.90 GHz processor, 4 G memory and 64-bit operating system.

4.1.2. Parameter Setting

The tolerance coefficients are set to 0.025, 0.05, 0.075 and 0.1. With the increase in the tolerance coefficient, the solution quality gradually decreases, the running time increases greatly and the solution timeliness worsens. When the tolerance coefficient is 0.025, the optimal value of the objective function value is 868.606, and the average running time is 1019.405 s. The reduction rates of the tolerance coefficients are set to 0.05, 0.35, 0.65 and 0.95. When the decrease rate of the tolerance coefficient is 0.95, the solution quality is the best, and the optimal value, worst value and average value of the objective function value are all optimal. The gap in the solution timeliness is not large, and the maximum differences of the optimal value, the worst value and the average value of the running time are 86.6703 s, 100.2706 s and 30.0703 s, respectively. The maximum number of iterations in the initial solution generation and pre-optimization stage are set to integer values of $1$, $n/5$, $2n/5$, $3n/5$, $4n/5$ and $n$, where $n$ is the number of customers. With the increase in the maximum number of iterations, the optimal value, worst value, average value, standard variance, range and range percentage of the objective function value gradually decrease, the quality and stability of the initial scheme improve and the diversity decreases.

Therefore, considering the solution quality, solution timeliness and diversity of the initial scheme, the tolerance coefficient, the decline rate of the tolerance coefficient and the maximum number of iterations are, finally, set as 0.025, 0.95 and the integer of $n/5$, respectively.

4.2. Effectiveness and Optimization Ability Analyses of the Two-Stage VNS-TA Algorithm for Solving the Basic MCVRP

4.2.1. Effectiveness Analysis of the Improved Neighborhood Structure Set

The classical swap, insert, 2-opt, 3-opt and three-point operators are used to form the traditional neighborhood structure set, which is compared with the improved neighborhood structure set. The optimal value, worst value and average value of the objective function value of the algorithm based on the improved neighborhood structure set are 2.4083, 0.2263 and 1.3227 lower than those of the algorithm based on the traditional neighborhood structure set, respectively. It can be observed that the solution quality of the two-stage VNS-TA based on the improved neighborhood structure set is higher. In addition, there is little difference in solution stability between the algorithm based on the improved neighborhood structure set and the algorithm based on the traditional neighborhood structure set. These results indicate that the constructed neighborhood structure performs better in obtaining high-quality solutions (Figure 3).

4.2.2. Effectiveness Analysis of the Neighbor Path Selection Mechanism

We set the upper limit $maxp$ of the maximum interval of the path traversal to 1, 2, 3, and $floor\left(numel\left(Rout\right)/2\right)$. With the increase in $maxp$, the optimal value, worst value and average value of the objective function value all decrease, and the quality of all the solutions improves. The downward trend of the objective function value becomes slower, and the optimization degree decreases, indicating that the increase in the path interval leads to a decrease in the search value and optimization space. Moreover, with the increase in $maxp$, the running time increases gradually. This result indicates that it is necessary to properly set the maximum interval of the path traversal to balance the solution quality and solution timeliness. These results reveal the effectiveness of the designed neighbor path selection mechanism (Figure 4).

4.2.3. Effectiveness Analysis of the Acceptance Criteria of the Solution

The optimal acceptance criterion (Strategy 3), the simulated annealing (SA) acceptance criterion (Strategy 4), the acceptance criterion of the stage decline tolerance coefficient strategy (Strategy 1) and the acceptance criterion of the exponential decreasing tolerance coefficient strategy (Strategy 2) are compared. The optimal value of the objective function value of Strategy 1 is better than that of the other strategies, and the solution quality of Strategy 1 is higher. There is little difference in the stability of the solutions for all the acceptance criteria. In addition, the algorithm based on Strategy 4 has the longest running time. Its average running time is 2.08 times, 1.93 times and 1.87 times that of Strategy 3, Strategy 2 and Strategy 1, respectively. This is because Strategy 4 judges and accepts solutions with random probability, which substantially increases the number of invalid searches. In particular, contrast Strategy 1 with Strategy 2. The algorithm based on Strategy 2 has a slightly shorter running time, a slightly faster convergence speed and a poor solution quality. The algorithm based on Strategy 1 has a slightly longer running time and a higher solution quality. The algorithm based on Strategy 1 performs better. This is because Strategy 1 avoids the problems of too-fast convergence and premature maturity, accounts for the solution quality and solution timeliness and jumps out of the local optimal solution. It can be observed that the design of the stage decline tolerance coefficient of Strategy 1 is more effective (Figure 5).

4.2.4. Comparison with Other Algorithms

The results of 14 examples of the two-stage VNS-TA algorithm are compared with those of the IPSO algorithm [32] and IVNS algorithm [42]. Among the 14 examples, the two-stage VNS-TA algorithm optimizes 12 solutions. Specifically, the maximum reduction value is 37.712, and the maximum optimization percentage is $2.61\mathrm{\%}$. The optimal values of the other two examples are consistent with the objective function values of the current optimal solutions of the other two algorithms. In addition, the standard variance of the objective function values of the 14 examples solved by the two-stage VNS-TA algorithm is mainly between 0.3625 and 9.8541. Only the vrpnc11-S2 example has a large standard variance of 45.5913. This result indicates that the designed algorithm has good solution stability overall. It can be observed that the two-stage VNS-TA algorithm has a strong optimization ability (

Table 2. Comparison of the objective function values of the results of the two-stage VNS-TA algorithm and other algorithms.

Example of Calculation	IPSO	IVNS	Two-Stage VNS-TA
			Optimal Value	Value of Reduction	Percentage of Optimization	Average Value	Standard Variance
vrpnc1-S2	552.19	550.6958	550.6958	0	0.00%	551.5908	0.7308
vrpnc1-S3	549.69	547.9608	547.9608	0	0.00%	548.3637	0.3625
vrpnc2-S2	933.72	874.2201	868.606	−5.6141	−0.64%	872.4336	2.3628
vrpnc2-S3	980.56	883.5585	878.328	−5.2305	−0.59%	883.9418	4.1727
vrpnc3-S2	941.71	873.992	864.8485	−9.1435	−1.05%	870.6048	5.0803
vrpnc3-S3	942.17	872.7455	872.2247	−0.5208	−0.06%	879.6754	6.7235
vrpnc4-S2	1236.16	1091.991	1084.4192	−7.5718	−0.69%	1090.7992	3.274
vrpnc4-S3	1272.13	1117.096	1102.6068	−14.4892	−1.30%	1112.9597	9.6698
vrpnc5-S2	1632.95	1393.437	1364.9778	−28.4592	−2.04%	1374.8837	6.147
vrpnc5-S3	1658.33	1446.556	1408.844	−37.712	−2.61%	1418.3487	9.4022
vrpnc11-S2	1317.87	1094.415	1092.8736	−1.5414	−0.14%	1150.4099	45.5913
vrpnc11-S3	1327.51	1214.479	1204.437	−10.042	−0.83%	1215.6311	8.1814
vrpnc12-S2	916.37	828.0884	820.5216	−7.5668	−0.91%	822.2101	1.2296
vrpnc12-S3	993.57	924.0961	923.737	−0.3591	−0.04%	931.5986	9.8541

).

In general, the two-stage VNS-TA algorithm is effective and has a strong optimization ability in solving the MCVRP.

4.3. Scenario Simulation Analysis of the Two-Stage VNS-TA Algorithm to Solve the MCVRP Considering Traffic Congestion under the Mixed Carbon Policy

4.3.1. Sensitivity Analysis of Different Traffic Congestion Scenarios

The path planning under the no congestion and very congested scenario tends to be consistent. The total cost, vehicle management and use cost, transportation cost, mixed carbon emission cost, total fuel consumption and total carbon emission in the very congested scenario are 3519.505 CNY, 1257.196 CNY, 2227.959 CNY, 34.35 CNY, 282.0201 L and 654.2866 kg higher than those in the no congestion scenario, respectively. These results show that the cost, energy consumption and carbon emissions in the case of the very congested scenario increase compared with those in the case of the no congestion scenario. Furthermore, when the congestion degree coefficient changes consistently, the change rate of the mixed carbon emission cost is the largest. This result indicates that traffic congestion has the most serious negative impact on energy conservation and emission reduction. In addition, focus on the planning path in the output results while taking customer 20 as an example. In the no congestion scenario, $20\to 35$ is the frequently selected section under multiple runs. However, $20\to 29$ becomes the frequently selected section in the normal congestion scenario. The congestion degree coefficient of $20\to 35$ is 2.3 and $20\to 29$ is 1.67. This outcome shows that the two-stage VNS-TA algorithm has the ability to avoid road congestion (Figure 6).

4.3.2. Sensitivity Analysis of Different Mixed Carbon Policy Scenarios

First, in scenario S4, both the mixed carbon emission cost and total cost are suboptimal, and the vehicle management and use cost are the highest. The carbon emission is the lowest. This indicates that the emission reduction effect is the best in scenario S4, and the emission reduction actions have less impact on the economic benefits. Second, in scenario S2, the carbon emissions, total cost, total driving distance and total fuel consumption are the highest. This finding indicates that the distribution center expends many financial and material resources, but the final economic benefits and emissions reduction effect are very low. It can be seen that scenario S2 is not conducive to energy conservation and emission reduction in distribution centers. In addition, it can be observed in scenarios S1, S2 and S3 that the cost of vehicle management and use is relatively small but that the total cost and total carbon emissions remain high. This result indicates that the reduction in vehicle management and use cost cannot reduce the final total cost and total carbon emissions. Finally, in scenario S5, the distribution center plans to travel more distance and pays more vehicle management and use cost, the total cost is the smallest, the effect of carbon emissions reduction is suboptimal and the carbon emission cost is relatively the lowest. This finding indicates that scenario S5 is suitable for the distribution center to consider both economic benefits and energy conservation and emission reduction (Figure 7).

4.3.3. Convergence Analysis Based on the Total Cost and Carbon Emissions

From the perspective of the current solutions, some current solutions that exceed the objective function value of the current optimal solution are also retained. With the increase in the number of iterations, the better solution is obtained again. This result indicates that the design of Strategy 1 can expand the search scope and help to find better solutions. From the perspective of the current optimal solutions, the total cost is gradually reduced, but the carbon emissions are slightly increased. This result indicates that the constructed model further considers the carbon emissions without sacrificing the total economic benefits. In addition, for both the current solution and the current optimal solution, the total cost and total carbon emissions decrease as the number of iterations increases. This result indicates that the designed algorithm has a certain optimization ability in solving MCVRP considering traffic congestion under the mixed carbon policy (Figure 8).

In general, the two-stage VNS-TA algorithm can solve and optimize the MCVRP considering traffic congestion under the mixed carbon policy, which has the effects of cost saving and energy conservation and emission reduction.

5. Conclusions

MCVs have broad application prospects in urban logistics distribution. However, urban traffic congestion is increasingly serious, resulting in an increase in carbon emissions, and the constraints on carbon emissions from urban traffic are becoming stricter. In addition, the application of the mixed carbon policy has gradually become the first choice for energy conservation and emission reduction in some countries and regions. In this context, it is of practical significance to carry out research on MCVRP considering traffic congestion under the mixed carbon policy. Based on the above analyses, this study constructed a mathematical model of MCVRP considering traffic congestion under the mixed carbon policy and designed a two-stage VNS-TA algorithm to solve the model. The following conclusions are drawn from the study. First, the two-stage VNS-TA algorithm designed in this study is effective and has a strong optimization ability in solving the basic MCVRP. Second, the two-stage VNS-TA algorithm can solve and optimize the MCVRP considering traffic congestion under the mixed carbon policy, which has the effects of cost saving and energy conservation and emission reduction. In subsequent research, we will consider the practical applications of the model and algorithm and provide references for multi-compartment vehicle path planning in real cities through the study of real cases.