Regulation and Control Strategy of Highway Transportation Volume in Urban Agglomerations Based on Complex Network

Abstract

Urban development within an urban agglomeration is unbalanced; the coordinated development of urban agglomerations is the core task of urban development. There are now many mechanisms and methods to promote the coordinated development of urban agglomerations; however, there is a lack of research on promoting the coordinated development of urban agglomerations from the perspective of highway transportation volume regulation. According to the physical characteristics of highway transportation networks, the logical characteristics of urban regional connectivity, and the connection characteristics of complex networks, a two-layer complex network model was designed. The objective function and constraint conditions for urban agglomeration transportation volume regulation were proposed, and the optimal solution of the highway transportation volume regulation was solved. Due to the many variables and constraints, a hierarchical solution method was adopted. A probability search iteration algorithm was proposed innovatively to solve multivariable, many-to-many allocation problems. The algorithm is universal and can be applied to solving similar problems. Taking provincial urban agglomerations as an example, the process of solving the regulation model and realizing the method was explained. The transportation volume regulation methods and strategies proposed in this study realize the best combination of macro control and micro control, static and dynamic control, coordinated development, and collaborative transportation. It is an innovative exploration and study of highway transportation volume allocation and collaborative transportation in urban agglomerations and opens up a new direction for research on the coordinated development of urban agglomerations. The coordinated development of urban agglomerations provides a guarantee for the sustainable development of urban agglomerations. Therefore, this study is also of great significance for promoting the sustainable development of urban agglomerations.

1. Introduction

Urban agglomeration is generally a regional economic unit, with one or two large cities as core cities. The concept of “agglomeration” is mutual cooperation and coordinated development among cities within an urban agglomeration; it promotes urban agglomeration to obtain greater benefits and faster development than an individual urban center. However, reference [1] demonstrated that current urban development within an urban agglomeration is unbalanced; regulation and control strategies for the coordinated development of urban agglomerations should be studied.

There are now many factors and methods to promote the coordinated development of urban agglomerations, including functional coordination, spatial coordination, industrial coordination, urban–rural coordination, and institutional coordination [2,3,4,5,6]. Most scholars study the development of urban agglomerations from the perspective of city aggregation, the flow of urban development factors, and the identification of urban characteristics. There are currently no relevant studies in the literature examining the coordinated development of urban agglomerations from the perspective of highway collaborative transportation and that consider highway transportation as the main participating factor in the development of urban agglomerations. This is one of the most important and challenging research topics in the field of urban planning and transportation management; it extends and enriches the theory of the coordinated development of urban agglomerations and opens up new directions for research on the coordinated development of urban agglomerations. The scientific basis for proposing the research direction [1] is as follows.

(1) A transportation system is the spatial structural skeleton of urban agglomeration economic development and an important bridge for connection and development between cities. Highway transportation is an important link in urban transportation. It is the main mode of medium- and short-distance transportation. It is also a bridge and link between railway, water transport, aviation, and other transportation methods [7]. Highway transportation includes features, such as high flexibility, fast speed, better convenience, and better accessibility, and plays an important role in the economic development of urban agglomerations [8];

(2) Highway transportation can be divided into two categories: expressway transportation and non-expressway transportation. Figure 1 shows the proportions of highway mileage of various transportation methods in China in 2022 [9]. Although the proportion of expressway mileage is only 3.3%, its traffic volume accounts for the highest proportion of passenger and freight volume. Expressway passenger transport turnover accounts for 54.86% of the total commercial passenger transport turnover of the whole society; freight transport turnover accounts for 43.35% of the total commercial freight transport turnover—both are approximately 50%. The research [9] shows that expressway traffic volume maintains linear growth every year;

(3) Expressway traffic volume has a positive correlation with economic development (GDP). Relevant research in the literature [1,7] shows that when expressway freight transportation volume declines, the GDP also declines. The correlation coefficient between active freight transportation capacity and the GDP was 0.54 in 2020, 0.59 in 2021, and 0.62 in 2022. This indicates that the correlation between freight transportation capacity and GDP is significant, and that freight transportation activities have a close interactive relationship with industrial activities and are closely related to the trend of economic development;

(4) Due to the relatively closed feature of expressways, traffic volume can be regulated by entering and exiting toll stations, making it easier to achieve expressway collaborative transportation, thereby promoting the coordinated development of urban agglomerations. If other highways have regulation conditions, traffic volume regulation can also be achieved.

Figure 2 shows an example of two cities sharing an expressway segment. There is a maximum traffic volume in the segment. If the sum of the outbound traffic volumes from two cities is greater than the maximum traffic volume, it will inevitably cause traffic congestion, increase the probability of traffic accidents, and have lower transportation efficiency, thus affecting urban expressway traffic volume and urban development. Therefore, it is necessary to regulate the traffic volume of the segment to achieve collaborative transportation and improve transportation efficiency. One of the regulating measures is to reduce the traffic volume of the developed city and relatively increase the traffic volume of the slow-developing city, so it can achieve collaborative transportation, and promote the coordinated development of the two cities. For an urban agglomeration composed of multiple cities, it requires studying strategies and methods that how to reasonably allocate highway transportation resources, how to regulate highway traffic volume, and how to promote the coordinated development of urban agglomerations through highway collaborative transportation in urban agglomerations.

For highway transportation, most studies are related to traffic flow and urban transportation planning. (1) The micro traffic flow theory analyzes and models the traffic characteristics of individual vehicles. The theory is based on the analysis of micro driving behavior and travel behavior characteristics, combining the complex environment with individual physiological characteristics, and forming a simulation model [10,11]. (2) The meso traffic flow theory describes the randomness and uncertainty of traffic flow, studies the real highway network traffic operation status, and describes traffic flow laws, such as the effectiveness of transportation vehicles, signal control, weather, and environmental impact [12]. (3) Macroscopic traffic flow theory introduces concepts, such as vehicle density, flow rate, and average speed to study the dynamic behavior of traffic flow at various locations and moments on a road segment, and describes the changes in traffic flow along the entire road. Transportation planning mainly focuses on the interaction between transportation and land use [13], coordinated development of transportation and the environment [14], analysis of transportation behavior characteristics, traffic surveys and data analysis [15], transportation demand forecast [16], planning decision support tools and transportation policies [17], etc.

Based on current research, there is a lack of research on the relationship between highway collaborative transportation and the coordinated development of urban agglomerations. As discussed above, we propose a new research field on the coordinated development of urban agglomerations through highway collaborative transportation. The priority regulation strategies and grouping circular game regulation strategies for highway transportation volume regulation have been proposed by the authors in reference [1,7], which belong to macro-level regulation and static regulation strategies. Based on the complex network principle, this study proposes a new regulation strategy that combines static regulation with dynamic regulation.

A highway transportation network in an urban agglomeration has the properties of self-organization, self-similarity, and a small world, and it satisfies the characteristics of a complex network. In the field of urban research, multi-layer complex networks are mainly used in transport supply chain path optimization [18,19], urban disaster simulation and vulnerability analysis [20], land use change and land use transformation direction [21], and transportation network structure change [22]. Related research on complex networks mainly focuses on the empirical research of network characteristics [23], detection of network center nodes [24], process of dynamic change and propagation [25], robustness and vulnerability [26,27], and multi-layer network theory and application [28]. There are no related studies on the applications of complex multi-layer networks for the regulation and control of transportation volume in urban agglomerations. Overall, there is no relevant reference that applies complex network theory to the research on highway collaborative transportation and coordinated development of urban agglomerations.

The main contributions of this work are listed as follows:

(1) According to the requirements of highway transportation volume regulation in urban agglomerations and the characteristics of highway transportation networks, a two-layer complex network regulation model for highway transportation volume is proposed. The model solves the problem of combining macro-regulation with micro regulation, and static regulation with dynamic regulation. The upper network realizes static regulation of traffic volume between cities at the macro level, and the lower network realizes dynamic regulation of traffic volume between highway segments at the micro level.

(2) For solving the model, a probability search iteration algorithm is proposed innovatively. The algorithm is suitable to solve multi-variable and many-to-many allocation problems, and it is universal and can be applied to solving other similar problems.

(3) The model and algorithm provide important reference value for solving other similar problems.

(4) The study opens up a new direction for research on the coordinated development of urban agglomerations through highway collaborative transportation.

The remainder of this paper is organized as follows. Section 2 introduces definitions and theorem. Section 3 introduces a two-layer complex network model. Section 4 introduces the objective function, constraint conditions, and solution method. Section 5 introduces a case study and simulation. Section 6 provides a summary of this study.

2. Definitions and Theorem

Because the following research content involves matrix operations and objective function optimization, relevant concepts are defined in advance.

Definition 1. 

Matrix comparison.

If A and B have the same number of rows and columns as the following condition is satisfied, this is called A > B. Conversely, A < B.

$${{\sum }_{i=1}^n{\sum }_{j=1}^{m}a}_{ij}>{{\sum }_{i=1}^n{\sum }_{j=1}^{m}b}_{ij}$$

Definition 2. 

Matrix dot multiplication.

If A and B have the same number of rows and columns, A is multiplied by B, as the following formula shows; this is called matrix dot multiplication:

$$\mathrm{C}=\mathrm{A}\ast \ast \mathrm{B},c_{ij}=a_{ij}\ast b_{ij}$$

Definition 3. 

The static shortest transport route.

Among all transport routes from the city i to the city j, the route with the shortest length is called the static shortest route.

Theorem 1. 

$f\left(x,y\right)={\displaystyle \frac{1}{n}}\ast x^2-{\displaystyle \frac{1}{n^2}}{y}^2,n>0,x>0,y>0,x<y$, when the increase in y is much greater than the increase in x, the value of x is regarded as approximately constant and the value of the function is decreasing.

Proof. 

The partial derivatives of function f(x,y) with respect to x and y are as follows:

$${\displaystyle \frac{\partial f}{\partial x}}={\displaystyle \frac{2}{n}}x,{\displaystyle \frac{\partial f}{\partial y}}=-{\displaystyle \frac{2}{n^2}}y$$

(1) Because n > 0, x > 0 and ${\displaystyle \frac{\partial f}{\partial x}}>0$, the value of the function is increasing in the x direction;

(2) Because n > 0, y > 0 and ${\displaystyle \frac{\partial f}{\partial y}}<0$, the value of the function is decreasing in the y direction.

The change in the value of the function depends on the relative rate and magnitude of the change in x and y, because x < y, as follows:

(1) If x is kept constant and y increases, and x < y, the value of f(x,y) will decrease;

(2) If x increases and y increases, x < y, the function value may increase first. Then, due to the increase in y, the function value may decrease again. The functional curve in Figure 3 shows the trend. □

3. Two-Layer Complex Network Model

As discussed above, the expressway network is the main highway transportation network. An expressway has entrance and exit toll stations, is relatively closed, and is the most suitable for transportation volume regulation. Therefore, we mainly studied expressway networks and built a complex network model. The model also adapts to other highway networks.

According to the characteristics of an urban agglomeration and an expressway transportation network, a single-layer urban agglomeration transportation network model G^U and a single-layer expressway transportation network model G^L were established. The urban agglomeration transportation network model represents the correlation relationships of the transportation volume among multiple cities; the expressway transport network model represents the topological structure of the highway transportation networks in the urban agglomeration. Subsequently, a two-layer complex network of urban agglomeration transportation model was constructed. Whether it is urban transportation volume regulation or expressway transport behavior in the network structure, the relationship between any two points is two-way, and the constructed two-layer transportation network of the urban agglomeration is an undirected weighted network.

According to the definitions and characteristics of small-world networks and scale-free networks, the upper-layer network is a regular and non-scale-free network; the lower-layer network is a small-world and scale-free network. Because the upper network is a network established based on the transportation connections between major cities in the urban agglomeration, not a network established based on physical space transportation paths. Any node in the network is connected to other nodes (including its own node), and the degree of connectivity is the same, which is an n-n connection relationship. The lower-layer network is a transportation network established based on physical space traffic paths, with hub nodes, which conforms to the characteristics of a small-world network and a scale-free network.

Because the characteristics of the upper and lower networks are different, the connection relationship between the two-layer networks is also relatively complex. However, it can reflect the complex traffic connections between various cities in the urban agglomeration, between cities and highway segments, and between highway segments from the macro level to the micro level. Therefore, it is suitable for analyzing the mechanism of collaborative highway transportation in urban agglomerations and coordinated development of urban agglomerations.

3.1. Single-Layer Transportation Network Model

3.1.1. Urban Agglomeration Transportation Network Model (G^U)

The nodes of urban agglomeration transportation network model G^U are major cities and the model is defined as follows:

$$G^U=\left(V^U,E^U,W^{U1},W^{U2},A^U\right)$$

where $V^U$ is the node set, $E^U$ is the edge set, $W^{U1}$ is the basic node weight, $W^{U2}$ is the node edge weight, and $A^U$ is the adjacency matrix.

(1)

Adjacency matrix $A^U$

Assuming that there are n major cities, therefore:

$$A^U={\left\{a_{ij}^U\right\}}_{n\ast n}\text{, i = 1, 2, … n, j = 1, 2, … n.}$$

$$a_{ij}^U=\left\{\begin{array}{l}1,\text{there is the transporttion relationship between city i and city j.}\\ 0,\text{there is not the transportion relationship between city i and city j.}\end{array}\right.$$

According to the transportation relationship between cities in urban agglomerations, $A^U$ is n*n matrix with element 1.

(2)

Basic node weight matrix $W^{U1}$

The basic node weight is $W^{U1}={\left\{w_{ij}^{U1}\right\}}_{n\ast n}$ and it is defined as the node transportation intensity ${TI}_i$:

$$w_{ij}^{U1}={TI}_i={\displaystyle \frac{{HV}_i}{{2\ast TV}^U}}(0<{TI}_i<1,\sum {TI}_i=1,i=j)$$

(1)

$$w_{ij}^{U1}=0(i\ne j)$$

where ${HV}_i$ is the highway transportation volume of city i, including the outbound and inbound volumes and ${TV}^U$ is the total outbound transportation volume of an urban agglomeration. Since the sum of the outbound volume of the urban agglomeration is equal to the sum of the inbound volume of the urban agglomeration, the denominator of Equation (1) needs to be divided by 2 to ensure that the value of ${TI}_i$ is less than 1.

The node transportation intensity reflects the proportion of the urban transportation volume in the total transportation volume of the urban agglomeration, which is dynamic. The greater the value is, the stronger the transportation activity of the city is; this means that there are more transportation connections with other cities. It shows that the greater the basic node weight is, the larger the urban transportation volume is, and the better the urban development is. In contrast, urban development is poor.

(3)

Node edge weight matrix $W^{U2}$

The edge weight between two nodes is $W^{U2}={\left\{w_{ij}^{U2}\right\}}_{n\ast n}$. It is defined by the proportion of the transportation volume between the two cities to the total transportation volume of the urban agglomeration, as follows:

$$w_{ij}^{U2}={\displaystyle \frac{{TV}_{ij}^U}{{TV}^U}}$$

(2)

where ${TV}_{ij}^U$ is the outbound transportation volume from city i to city j and ${TV}^U$ is the total outbound transportation volume of the urban agglomeration.

The edge weight reflects the degree of transportation between two cities. The greater the weight is, the stronger the transportation links; otherwise, the links are weaker. If ${TV}^U$ is known, the following calculation can be obtained:

$$\mathrm{The}\mathrm{outbound}\mathrm{transportation}\mathrm{volume}\mathrm{from}\mathrm{city}\mathrm{i}\mathrm{to}\mathrm{city}\mathrm{j}\mathrm{is:}{TV}_{ij}^U={TV}^U\ast w_{ij}^{U2}$$

(3)

$$\mathrm{The}\mathrm{outbound}\mathrm{transportation}\mathrm{volume}\mathrm{of}\mathrm{city}\mathrm{i:}\mathrm{is}\mathrm{T}{\mathrm{S}\mathrm{V}}_i^U={\sum }_{j=1}^n{TV}_{ij}^U$$

(4)

$$\mathrm{The}\mathrm{inbound}\mathrm{transportation}\mathrm{volume}\mathrm{of}\mathrm{city}\mathrm{I}\mathrm{is:}\mathrm{T}{\mathrm{R}\mathrm{V}}_i^U={\sum }_{j=1}^n{TV}_{ji}^U$$

(5)

The total outbound transportation volume of urban agglomeration is as follows:

$${TV}^U={\sum }_{i=1}^n{TSV}_i^U={\sum }_{i=1}^n{\sum }_{j=1}^n{TV}_{ij}^U$$

(6)

According to the above relationships, the following matrix operation relationships can be derived:

$$\mathrm{Set:}{\mathrm{T}\mathrm{S}\mathrm{V}}^U={\left[\begin{array}{cccc}T{SV}_1^U,& T{SV}_2^U,& \dots & T{SV}_n^U\end{array}\right]}^T$$

$${\mathrm{T}\mathrm{R}\mathrm{V}}^U={\left[\begin{array}{cccc}T{RV}_1^U,& T{RV}_2^U,& \dots & T{RV}_n^U\end{array}\right]}^T$$

$${\mathrm{T}\mathrm{I}}^U={\left[\begin{array}{cccc}T{I}_1^U,& T{I}_2^U,& \dots & T{I}_n^U\end{array}\right]}^T$$

$$\mathrm{A}={{\left[\begin{array}{cccc}1,& 1,& \dots & 1\end{array}\right]}^T}_{n\ast 1}$$

$$\mathrm{Then:}{\mathrm{T}\mathrm{S}\mathrm{V}}^U={TV}^U\ast W^{U2}\ast A$$

(7)

$${\mathrm{T}\mathrm{R}\mathrm{V}}^U={TV}^U\ast W^{{U2}^T}\ast A$$

(8)

$${\mathrm{T}\mathrm{I}}^U={\displaystyle \frac{1}{{2\ast TV}^U}}\ast ({\mathrm{T}\mathrm{S}\mathrm{V}}^U+{\mathrm{T}\mathrm{R}\mathrm{V}}^U)$$

(9)

3.1.2. Expressway Transportation Network Model (G^U)

An expressway transport network is defined as follows:

$$G^L=\left(V^L,E^L,W^{L1},W^{L2},A^L\right)$$

where $V^L$ is the node set, $E^L$ is the edge set (road segment set), $W^{L1}$ is the basic node weight, $W^{L2}$ is the node edge weight, and $A^L$ is the adjacency matrix.

The node set of the model $G^L$ includes the following nodes: (1) urban expressway toll station, where one city has a toll station; and (2) the intersection between one expressway and another expressway, which is the crossing point from one expressway to another.

(1)

Adjacency matrix $A^L$

Assuming that there are m nodes, therefore:

$$A^L={\left\{a_{ij}^L\right\}}_{m\ast m},\text{i = 1, 2, … m, j = 1, 2, … m.}$$

$$a_{ij}^L=\left\{\begin{array}{l}1,\text{node i and j are adjacent nodes or i = j}.\\ 0,\text{node i and j are not adjacent nodes}.\end{array}\right.$$

If node 1 is the starting point and node m is the endpoint, the adjacency matrix $A^L$ is an m*m symmetric matrix. It can be proved that the upper triangular matrix reflects the forward driving relationship from node1 to node m, whereas the lower triangular matrix reflects the reverse driving relationship from node m to node 1.

(2)

Basic node weight matrix $W^{L1}$

The basic node weight is defined as the node connection degree $d_i$, which refers to the number of nodes directly connected to the node. $d_i$ reflects the direct connection degree between nodes in an expressway network. The larger $d_i$ is, the greater the basic node weight, which means that many other nodes are connected to it. Therefore, the node is more important. $d_i$ is calculated by adjacency matrix $A^L$:

$$d_{ii}=\sum _{j=1}^m{a}_{ij}^L$$

(10)

$$d_{ij}=0(i\ne j)$$

$$\mathrm{D}=W^{L1}={\left\{d_{ij}\right\}}_{m\ast m}$$

where $W^{L1}$ or D is also called the degree matrix.

(3)

Node edge weight matrix $W^{L2}$

The node edge weight between the two nodes (road segment weight), $W^{L2}={\left\{w_{ij}^{L2}\right\}}_{m\ast m}$, is defined as the ratio of the transportation volume of the road segment per unit time to the total outbound transportation volume of the upper urban agglomeration transportation network. The greater the weight is, the greater the current transportation volume, and the more important the road segment is. However, the road segment is also easily congested:

$$w_{ij}^{L2}={\displaystyle \frac{{tv}_{ij}^{L2}}{{TV}^U}}$$

(11)

where ${\mathit{t}\mathit{v}}_{\mathit{i}\mathit{j}}^{\mathit{L}2}$ is the transportation volume of the road segment per unit time and ${TV}^U$ is the total outbound transported transportation volume of the urban agglomeration.

The transportation volume matrix of road segments is as follows:

$$FM={\left\{{tv}_{ij}^{L2}\right\}}_{m\ast m}$$

and then

$$FM={TV}^U\ast W^{L2}$$

(12)

Assuming that ${VI}_i^L$ represents the inbound transportation volume of node i and ${VO}_i^L$ represents the outbound transportation volume of node i, as follows:

$${VI}_i^L={\sum }_{j=1}^k{tv}_{ji}^{L2}$$

(13)

$${VO}_i^L={\sum }_{j=1}^k{tv}_{ij}^{L2}$$

(14)

If there is only one adjacent node from i to j, then ${VO}_i^L$ is the transportation volume from node i to node j. According to the above relationships, the following matrix operation relations can be derived:

$$\mathrm{Set:}{\mathrm{V}\mathrm{I}}^L={\left[\begin{array}{cccc}{VI}_1^L,& {VI}_2^L,& \dots & {VI}_m^L\end{array}\right]}^T$$

$${\mathrm{V}\mathrm{O}}^L={\left[\begin{array}{cccc}{VO}_1^L,& {VO}_2^L,& \dots & {VO}_m^L\end{array}\right]}^T$$

$$\mathrm{B}={{\left[\begin{array}{cccc}1,& 1,& \dots & 1\end{array}\right]}^T}_{m\ast 1}$$

$$\mathrm{Then:}{\mathrm{V}\mathrm{O}}^L=\mathrm{F}\mathrm{M}\ast \ast A^L\ast B$$

(15)

$${\mathrm{V}\mathrm{I}}^L={\mathrm{F}\mathrm{M}}^T\ast \ast {A^L}^T\ast B$$

(16)

3.2. Two-Layer Complex Network

A multi-layer complex network is composed of many single-layer networks and the single-layer networks are connected with each other. Compared to the single-layer network, the multi-layer complex network is closer to the spatial structure of the multi-mode transportation network and it reflects the heterogeneity of different layer networks. According to the characteristics of the transportation network of urban agglomeration, the single-layer model ${\mathit{G}}^{\mathit{U}}$ and ${\mathit{G}}^{\mathit{L}}$ are connected to form a two-layer complex network, model G, of urban agglomeration transportation. As shown in Figure 4, the mathematical model is described as follows:

$$G^{double}=G^U+G^L$$

$$V^{double}=V^U+V^L$$

$$E^{double}=E^U+E^L+E^{UL}$$

$$A^{double}=A^U+A^L+A^{UL}$$

G is an undirected double-layer complex network, $A^{UL}$ is the upper- and lower-node connection matrix, $A^{UL}={\left(a_{i_U{j}_L}^{UL}\right)}_{n\ast m}$, and $a_{i_U{j}_L}^{UL}$ is the interlayer connection relationship, as follows:

$$a_{i_U{j}_L}^{UL}=\left\{\begin{array}{l}1,\text{there is the direct path relationship between upper node }{\mathrm{i}}_{\mathrm{U}}\text{and lower node }{\mathrm{j}}_{\mathrm{L}.}\\ 0,\text{there is the indirect path relationship between upper node }{\mathrm{i}}_{\mathrm{U}}\text{and lower node }{\mathrm{j}}_{\mathrm{L}.}\end{array}\right.$$

$$W^{double}=\left[\begin{array}{cc}{W}^U& W^{UL}\\ W^{LU}& W^L\end{array}\right]$$

$$W^{UL}=W^{LU}=\left[\begin{array}{ccc}{w}_{1_U{1}_L}& \dots & w_{1_U{m}_L}\\ ⋮& \ddots & ⋮\\ w_{n_U{1}_L}& \dots & w_{n_U{m}_L}\end{array}\right]$$

$W^{UL}$ and $W^{LU}$ are the interlayer edge weight matrix, and its element $w_{i_U{j}_L}^{UL}$ is the edge weight between the upper node $i_U$ and lower node $j_L$, which is the static shortest transport route from the upper node $i_U$ to the lower node $j_L$ (see Definition 3).

For the convenience of analyzing the problem, the node numbering of the lower toll station must be consistent with the node numbering of the upper corresponding city; the numbers of adjacent nodes must be assigned in sequential order according to the connection order of the nodes. Therefore, when the node numbering of the lower layer is equal to that of the same city in the upper layer, the edge weight between the layers expresses the length of the static shortest transport route between the two cities ($j_L=j_U$).

The inter-layer edge weight can be assigned to multiple types of values. For example, it can be defined as the ratio of the outbound transportation volume from the upper city node $i_U$ to the lower node $j_L$ (${CT}_{i_U{j}_L}$) to the total outbound transportation volume of the urban agglomeration (${TV}^U$), as follows:

$$w_{i_U{j}_L}^{UL}={\displaystyle \frac{{CT}_{i_U{j}_L}}{{TV}^U}}$$

(17)

When the node numbering of the lower layer is equal to that of the same city in the upper layer, the result as as follows:

$${CT}_{i_U{j}_L}={TV}_{ij}^U,w_{i_U{j}_L}^{UL}=w_{ij}^{U2},\mathrm{i}=i_U,\mathrm{j}=j_L.$$

(18)

From the relationship between the upper and lower layers of the network, we can see that: (1) the node transportation volume allocation scheme of the upper-layer network affects the road segment transportation volume of the lower layer; (2) the capacity of the road segment of the lower layer (maximum transportation volume) and the transport time cost (transport route) constrain the transportation volume allocation scheme between the upper cities; (3) the upper-layer network mainly solves the problem of coordinated development among cities in the urban agglomeration and cooperates with the lower-layer network to solve the problem of the collaborative transportation of expressway networks.

4. Transportation Volume Regulation Model

4.1. Objective Function and Constraint Conditions

As discussed above, the purpose of transportation volume regulation and control is to balance the urban outbound transportation volume and to promote the coordinated development between cities. On the other hand, the total outbound transportation volume of the urban agglomeration does not exceed the maximum transportation volume of the transport network and ensures that each road segment does not cause congestion. The basic weight of the upper urban node is defined as the node highway transportation intensity ${\mathit{T}\mathit{I}}_{\mathit{i}}$, which reflects the transportation volume of the city. The average transportation intensity of the nodes in the urban agglomeration is calculated as follows:

$$\overline{TI}={\displaystyle \frac{1}{n}}{\sum }_{i=1}^n{TI}_i$$

(19)

If the transportation intensity of each city is close to the average value, this is a satisfactory allocation scheme that can realize the requirement of coordinated development. The satisfaction variance is set as the objective function, and the minimum value is considered as follows:

$$\min f={\displaystyle \frac{1}{n}}{\sum }_{i=1}^n{\left({TI}_i-\overline{TI}\right)}^2$$

(20)

4.1.1. Constraint Conditions of the Upper-Layer Network

The total outbound transportation volume of the urban agglomeration:

$${\sum }_{i=1}^n{\sum }_{j=1}^n{TV}_{ij}^U\le {TV}^U$$

(21)

The outbound transportation volume of city i:

$$\mathrm{T}{\mathrm{S}\mathrm{V}}_i^U={\sum }_{j=1}^n{TV}_{ij}^U\ge \mathrm{T}{\mathrm{S}\mathrm{V}}_{i-min}^U$$

(22)

The inbound transportation volume of city i:

$$\mathrm{T}{\mathrm{R}\mathrm{V}}_i^U={\sum }_{j=1}^n{TV}_{ji}^U\ge \mathrm{T}{\mathrm{R}\mathrm{V}}_{i-min}^U$$

(23)

The outbound transportation volume between cities:

$${TV}_{ij}^U\ge {TV}_{ij-min}^U$$

(24)

where $\mathrm{T}{\mathrm{S}\mathrm{V}}_{i-min}^U$ is the minimum outbound transportation volume of city i, $\mathrm{T}{\mathrm{R}\mathrm{V}}_{i-min}^U$ is the minimum inbound transportation volume of city i, ${TV}_{ij-min}^U$ is the minimum outbound transportation volume between cities, and ${TV}^U$ is the total outbound transportation volume of the urban agglomeration.

4.1.2. Constraint Conditions of the Lower-Layer Network

According to transportation direction, the lower expressway network is divided into two directional driving networks: the forward driving network and the reverse driving network (because the lanes on both sides of an expressway are isolated). The expressway network can be reconstituted according to the dynamic fastest transport route. Based on the dynamic fastest transport route algorithm [1], the fastest driving route is found from the starting point ${\mathit{i}}_{\mathit{L}}$ (corresponding to the upper layer ${\mathit{i}}_{\mathit{U}}$) to the endpoint ${\mathit{j}}_{\mathit{L}}$ (corresponding to the upper ${\mathit{j}}_{\mathit{U}}$), and the fastest forward driving route network is thus formed. By setting ${\mathit{j}}_{\mathit{L}}$ as the starting point and ${\mathit{i}}_{\mathit{L}}$ as the endpoint, the fastest reverse driving route network can be obtained. In theory, the structures of the two networks should be the same; they have the same segments but different directions. However, because it is the dynamic fastest route network, not the static shortest route network, the structure of the two expressway networks may be different. We assume that the structures of the two expressway networks are exactly the same but with different directions.

The dynamic fastest route network is composed of the multiple fastest paths and there are the same road segments between these paths; therefore, the structure of the fastest dynamic route network is simpler than that of the original expressway network.

The congestion rate OD matrix is set as RC [2]; $c_{ij}$ represents the congestion rate of the road segment (between 0–1) and the maximum congestion rate is $c_{ij-max}$. The current transportation volume OD matrix is RF and ${rf}_{ij}$ represents the current transportation volume on the road segment. The newly added transportation volume OD matrix is NF, where ${nf}_{ij}$ represents the newly added transportation volume on the road segment. The maximum transportation volume on the road segment in the driving expressway network is $f_{ij-max}$.

The constraint conditions are set as follows:

$$c_{ij}\le c_{ij-max}$$

(25)

$${rf}_{ij}+{nf}_{ij}\le f_{ij-max}$$

(26)

$${{nf}_{ij}=tv}_{ij}^{L2}$$

(27)

Equations (21)–(27) are the constraint conditions of the objective function.

4.2. Solving Method for the Objective Function

(1)

Objective function analysis

The objective function (20) is transformed into the matrix operation, which is convenient for programming, as follows:

$${\overline{TI}={\displaystyle \frac{1}{n}}(TI}_1+{TI}_2+\dots {TI}_n)$$

$$\begin{array}{l}\min f={\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n}{\left({TI}_i-\overline{TI}\right)}^2\\ ={\displaystyle \frac{1}{n}}\left[\left({TI}_1^2-2{TI}_1\ast \overline{TI}+{\overline{TI}}^2\right)+\left({TI}_2^2-2{TI}_2\ast \overline{TI}+{\overline{TI}}^2\right)+\dots +\left({TI}_n^2-2{TI}_n\ast \overline{TI}+{\overline{TI}}^2\right)\right]\\ ={\displaystyle \frac{1}{n}}\ast \left({TI}_1^2+{TI}_2^2+\dots {TI}_n^2\right)-{\displaystyle \frac{1}{n^2}}{({TI}_1+{TI}_2+\dots {TI}_n)}^2\end{array}$$

$$\mathrm{Set:}\mathrm{B}={(1,1,\dots 1)}_{1\ast n}$$

$$TI={({TI}_1,{TI}_2,\dots {TI}_n)}^T$$

$$\mathrm{Then}\text{:}\mathit{min}f={\displaystyle \frac{1}{n}}\ast {TI}^T\ast TI-{\displaystyle \frac{1}{n^2}}{(B\ast TI)}^2$$

(28)

Based on Equations (7)–(9), and (28), the objective function can be calculated by the matrix operation.

When n is large (n ≥ 4), the objective function is as follows:

$$\left({TI}_1^2+{TI}_2^2+\dots +{TI}_n^2\right)<<{({TI}_1+{TI}_2+\dots +{TI}_n)}^2$$

(29)

Based Theorem 1, the value of the objective function is decreasing.

Theoretically, there is a global minimum value (0) for the objective function; that is, the transportation intensity of each city node is equal to the average value. However, it is impossible in actual transportation allocation. Therefore, the only relative minimum optimal solution can be obtained: $X_k,f\left(X_k\right)\to 0$. The objective Function (20) has upper- and lower-layer constraints; the lower-layer constraints are related to the road segment conditions and there are many constraint conditions and relations. The objective function optimization problem is a multivariable and complex nonlinear optimization problem. If general nonlinear objective function optimization methods are used, such as the Lagrange multiplier method, it involves n*n + k variables (k is the number of constraints). The solution becomes complex when n is relatively large. The simulated annealing algorithm, Gauss–Newton algorithm, and gradient descent algorithm involve solving partial derivatives. The partial derivative of the objective function is not easy to solve and is also very complex.

Based on the regulation and control characteristics of transportation volume, we propose a hierarchical optimization strategy. The basic idea is to optimize the objective function to obtain the transportation volume between cities according to the upper constraints and then use the lower constraints to regulate the transportation volume and obtain the final transportation volume between cities. A probability iteration algorithm is proposed to solve the objective function optimization. The probability iteration algorithm is an original, innovative algorithm and is suitable for the optimization of dozens of variables similar to the above objective function.

Among the upper constraint conditions, as long as (24) is satisfied, (22) and (23) must be satisfied; therefore, only two constraint conditions are required:

$${\sum }_{i=1}^n{\sum }_{j=1}^n{TV}_{ij}^U\le {TV}^U$$

(30)

$${TV}_{ij}^U\ge {TV}_{ij-min}^U$$

(31)

$$\mathrm{Because}{TV}_{ij}^U={TV}^U\ast w_{ij}^{U2},{TV}^U \text{is a constant.}$$

(32)

$$\mathrm{Then},\mathrm{the}\mathrm{variable}\mathrm{iteration}\mathrm{is}{TV}_{\left(ij\right)K}^U={TV}_{\left(ij\right)K-1}^U+∆{TV}_{\left(ij\right)K}^U$$

(33)

$$\mathrm{Then},\mathrm{the}\mathrm{converted}\mathrm{weight}\mathrm{iteration}\mathrm{is}\mathrm{as}\mathrm{follows:}{W}_k^{U2}=W_{k-1}^{U2}+{∆W}_k^{U2}$$

(34)

The initial value of transportation volume is set as the minimum transportation volume between cities as: ${TV}_0={\left\{{TV}_{ij-min}^U\right\}}_{n\ast n}$

$$\mathrm{The}\mathrm{initial}\mathrm{value}\mathrm{of}\mathrm{the}\mathrm{weights:}{W}_0^{U2}={\displaystyle \frac{1}{{TV}^U}}\ast {TV}_0$$

(35)

$$\mathrm{The}\mathrm{range}\mathrm{of}\mathrm{weight}\mathrm{variation:}{w_{ij-0}^{U2}\le w}_{ij}^{U2}\le w_{ij-max}^{U2}$$

(36)

Based on (30), the sum of the weights must satisfy the following condition:

$${\sum }_{i=1}^n{\sum }_{j=1}^n{w}_{ij}^{U2}\le 1$$

(37)

Theoretically, if any set of weights $W_k^{U2}$ satisfies (36) and (37), the objective function satisfies the following conditions:

$$f\left(W_k^{U2}\right)<f\left(W_{k-1}^{U2}\right)$$

(38)

$$f\left(W_k^{U2}\right)\to 0$$

(39)

Thus, $W_k^{U2}$ can be considered an effective solution, that is, a Pareto solution. To obtain the optimal efficient solution as far as possible, if the iterative algorithm is adopted, the iterative end conditions should be as follows:

$$\left|{\sum }_{i=1}^n{\sum }_{j=1}^n{w}_{ij}^{U2}-1\right|\le \beta ,\mathrm{empirical}\mathrm{value:}\beta =0.01.$$

(40)

$$f\left(W_k^{U2}\right)<\delta ,\mathrm{empirical}\mathrm{value:}\delta =0.1.$$

(41)

According to the analysis, the total outbound transportation volume of a city accounts for approximately 0.4 of the total transportation volume of an urban agglomeration, which is generally a central city. The maximum outbound transportation volume between cities accounts for approximately 0.4 of the transportation volume of the city. Therefore,$w_{ij-max}^{U2}=0.4\ast 0.4=0.16$. For the actual calculation, the value is reduced or increased according to the transportation volume requirements of each city.

(2)

Probability iteration algorithm

The random search is involved in intelligent optimization algorithms such as the particle swarm optimization (PSO) algorithm [29], genetic algorithm [30], and evolutionary algorithm [31]. Based on the random search principle and characteristics of transportation volume allocation and regulation, we propose a probability iteration algorithm.

The process of transportation volume allocation can be regarded as a Markov process and the iterative process of the weights is set as follows:

$${∆W}_k^{U2}=P\ast {∆W}_{k-1}^{U2}$$

(42)

$$W_k^{U2}=W_{k-1}^{U2}+{∆W}_k^{U2}$$

(43)

where ${∆W}_0^{U2}={\displaystyle \frac{W_{max}^{U2}-W_0^{U2}}{m}}$, m is related to the initial weight value $W_0^{U2}$, which is generally 100 or 1000, and $W_{max}^{U2}$ is the maximum weight. P is a probability transfer matrix or Markov matrix. As long as the probability transfer matrix P satisfies certain conditions, when the number of iterations k reaches a certain value, Equation (42) eventually reaches a stable state:

$${∆W}_k^{U2}={∆W}_{k-1}^{U2}$$

(44)

P must satisfy the following conditions: (1) P is a random matrix and the sum of all column elements is 1. In other words, $0\le p_{ij}<1$, $\sum _{j=1}^m{p}_{ij}=1$; (2) P is irreducible and the corresponding digraph is strongly connected, that is, there is a path between each pair of nodes. Here, the transportation volume can be allocated between any two urban individuals. Thus, the P matrix meets the irreducible requirement; and (3) P is aperiodic. Because P is a prime matrix, it is aperiodic.

Based on (42), (43) and Definiton 1, because $0<{∆w}_{ij}^{U2}<<1$ (much less than 1) and $0{\le p}_{ij}<1$, each iteration can be considered as approximating the efficient solution by satisfying the conditions with a small random step based on the initial weight value:

$$W_k^{U2}>W_{k-1}^{U2}$$

The sum of the weights after each iteration is ${\sum }_{i=1}^n\sum _{j=1}^n{w}_{ij}^{U2}\to 1$. Therefore, an efficient solution exists that satisfies condition (40). According to Theorem 1, $f\left(W_k^{U2}\right)\to 0$, and the efficient solution also satisfies condition (41) in theory. It has been proven by a large number of programming experiments that there is an efficient solution that satisfies the conditions of (40) and (41) simultaneously.

The following conclusions are drawn. For the probability iteration algorithm, there is an efficient solution in the process in which the random search tends to be stable and satisfies the end iterative conditions (40) and (41). Even if a solution satisfying the end iteration is not found, the iterative algorithm is convergent and the final stable solution is also an efficient solution.

The implementation process of the probability iteration algorithm is as follows.

Step 1. Initial parameter settings, including the total transportation volume of urban agglomeration, minimum transportation volume between cities, maximum transportation volume between cities, iterative parameters (m, δ, β), and probability matrix P;

Step 2. Calculate initial weights ($W_0^{U2}$), the max weights ($W_{max}^{U2}$), and weight change ($∆W_0^{U2}$);

Step 3. Calculate the updated weight ($W_k^{U2}$) and value of the objective function ($f\left(W_k^{U2}\right))$;

Step 4. Determine whether the iteration end conditions ((40) and (41)) are satisfied.

Yes, go to next step;

No, go to Step 3;

Step 5. End the iteration and output the result.

Taking an urban agglomeration with four cities as an example, the initial conditions are as follows:

The total sent transportation volume of the urban agglomeration is 10,000

The initial weights are as follows:

$$W_0^{U2}=\begin{array}{ccc}0.145& 0.031& \begin{array}{cc}0.033& 0.022\end{array}\\ 0.011& 0.012& \begin{array}{cc}0.013& 0.014\end{array}\\ \begin{array}{c}0.021\\ 0.012\end{array}& \begin{array}{c}0.025\\ 0.015\end{array}& \begin{array}{cc}\begin{array}{c}0.026\\ 0.017\end{array}& \begin{array}{c}0.028\\ 0.018\end{array}\end{array}\end{array}$$

The maximum weights are:

$$W_{max}^{U2}=\begin{array}{ccc}0.18& 0.16& \begin{array}{cc}0.16& 0.16\end{array}\\ 0.16& 0.16& \begin{array}{cc}0.16& 0.16\end{array}\\ \begin{array}{c}0.16\\ 0.16\end{array}& \begin{array}{c}0.16\\ 0.16\end{array}& \begin{array}{cc}\begin{array}{c}0.16\\ 0.16\end{array}& \begin{array}{c}0.16\\ 0.16\end{array}\end{array}\end{array}$$

The iterative parameters are δ = 0.1, β = 0.01, m = 100.

P is set as the following matrix and the sum of the elements in each column is 1:

$$\mathrm{P}=\begin{array}{ccc}0.4& 0.1& \begin{array}{cc}0.4& 0.1\end{array}\\ 0.3& 0.2& \begin{array}{cc}0.3& 0.2\end{array}\\ \begin{array}{c}0.2\\ 0.1\end{array}& \begin{array}{c}0.3\\ 0.4\end{array}& \begin{array}{cc}\begin{array}{c}0.2\\ 0.1\end{array}& \begin{array}{c}0.3\\ 0.4\end{array}\end{array}\end{array}$$

The calculation results are as follows. The final weights are:

$$W^{U2}=\begin{array}{ccc}0.1740& 0.0671& \begin{array}{cc}0.0687& 0.0582\end{array}\\ 0.0402& 0.0482& \begin{array}{cc}0.0488& 0.0503\end{array}\\ \begin{array}{c}0.0504\\ 0.0415\end{array}& \begin{array}{c}0.0612\\ 0.0513\end{array}& \begin{array}{cc}\begin{array}{c}0.0618\\ 0.0528\end{array}& \begin{array}{c}0.0643\\ 0.0543\end{array}\end{array}\end{array}$$

The final transportation volume between cities (${TV}_{ij}^U$) is as follows:

$${TV}^U=\begin{array}{ccc}1740& 671& \begin{array}{cc}687& 582\end{array}\\ 402& 482& \begin{array}{cc}488& 503\end{array}\\ \begin{array}{c}504\\ 415\end{array}& \begin{array}{c}612\\ 513\end{array}& \begin{array}{cc}\begin{array}{c}618\\ 528\end{array}& \begin{array}{c}643\\ 543\end{array}\end{array}\end{array}$$

The total weight is 0.9931 and the total calculated transportation volume is 9931.

The remaining transportation volume is 10,000 – 9931 = 69.

The transportation intensity of each city is 0.3395, 0.2092, 0.2363, and 0.2149.

The value of the objective function is 0.0028.

(3)

Algorithm analysis

Convergence speed: After a large number of calculation experiments, it has been proven that the algorithm converges quickly, and the optimal solution that meets the conditions can be found when the number of iterations is between 20 and 30.

Computational complexity: The algorithm only needs to set parameters such as the probability matrix and initial weights. The calculation process is the matrix operation; therefore, the algorithm is concise and has very low computational complexity.

Sensitivity to initial conditions: The calculation results show that the result of the transposed P (${\mathbf{P}}^{\mathit{T}}$, the sum of row elements is also 1) is the same as that of P; the result of exchanging P between rows or columns is unchanged. When the element value of P is changed, the result remained unchanged. The algorithm is insensitive to the initial conditions.

Applicability: If the optimization problem satisfies the following conditions, the algorithm can be applied to solve it: (1) The sum of the variables is a fixed value; (2) The variables are greater than or equal to zero; and (3) The solution result is an n*n matrix and it is a many-to-many allocation problem.

Compared with the particle swarm optimization algorithm and genetic optimization algorithm, the final weight, unallocated transportation volume, and transportation intensity of each node are almost the same; however, there is a slight difference in the transportation volume between cities. Because of the iterative accuracy, the sum of the transportation volume of each city solved by the iteration algorithm may be less than the set total transportation volume, or it may be larger than the set total transportation volume. The remaining transportation volume can be allocated to the slowest-developing city, or the excess transportation volume can be deducted from the central city.

4.3. Regulation and Correction of Transportation Volume Based on Lower-Layer Constraints

Based on (27) and (28), we have:

$${rf}_{ij}+{tv}_{ij}^L\le f_{ij-max}$$

(45)

Therefore,

$${tv}_{ij}^L\le f_{ij-max}-{rf}_{ij}$$

(46)

If ${tv}_{ij}^L>f_{ij-max}-{rf}_{ij}$, it should be regulated, and the newly added transportation volume should be reduced, as follows:

$$∆{tv}_{ij}^L={tv}_{ij}^L-(f_{ij-max}-{rf}_{ij})$$

(47)

The reduced transportation volume of ${TV}_{ij}^U$ (adjusted according to the proportion) is:

$${∆TV}_{ij}^U={\displaystyle \frac{{TV}_{ij}^U}{{tv}_{ij}^L}}\ast ∆{tv}_{ij}^L$$

(48)

After regulation and control, the transportation volume between cities is as follows:

$${TV}_{ij}^{U-New}={TV}_{ij}^U-{∆TV}_{ij}^U$$

(49)

The regulated and reduced transportation volume (${∆TV}_{ij}^U)$ can be allocated to some freight that does not need to be sent urgently, which can be transported through national or provincial highways, reducing expressway transport pressure.

5. Case Analysis and Simulation

We take the regulation and control of expressway traffic volume in the urban agglomeration of Jilin Province in China as an example to illustrate the regulation process, which includes nine major cities: Changchun (CC); Jilin (JL); Baicheng (BC); Songyuan (SY); Siping (SP); Liaoyuan (LY); Tonghua (TH); Baishan (BS); and Yanbian (YB). Figure 5 shows the double-layer network. Figure 6 shows the lower-layer expressway network. Figure 7 shows the simplified expressway network, which includes nine cities, five expressways, seventeen nodes, and twenty-six road segments.

5.1. Transportation Volume Allocation Between Cities

The transportation volume allocation among the nine cities in the urban agglomeration was solved with a total of 81 variables. The probability iteration algorithm was used; the transportation volume was taken as an integer. The other places were retained four places after the decimal point and five places after the decimal point for the value of the objective function; a better optimal allocation result was obtained. The value of the probability matrix P was as follows. The sum of each column was 1.

The total weight was 0.9956 and the total calculated transportation volume was 9956.

The remaining transportation volume was 10,000 – 9956 = 44.

The transportation intensity of each city was 0.1666, 0.1127, 0.1021, 0.1131, 0.0967, 0.1045, 0.0975, 0.098, and 0.1087.

The value of the objective function was 0.00042.

$$\mathrm{P}=\begin{array}{ccccccccc}0.1& 0.1& 0.1& 0.1& 0.1& 0.1& 0.1& 0.1& 0.1\\ 0.1& 0.2& 0.1& 0.1& 0.2& 0.1& 0.1& 0.1& 0.2\\ 0.1& 0.1& 0.2& 0.1& 0.1& 0.1& 0.1& 0.1& 0.1\\ 0.1& 0.1& 0.1& 0.1& 0.1& 0.1& 0.1& 0.2& 0.1\\ 0.1& 0.1& 0.1& 0.2& 0.1& 0.1& 0.1& 0.1& 0.1\\ 0.2& 0.1& 0.1& 0.1& 0.1& 0.1& 0.1& 0.1& 0.1\\ 0.1& 0.1& 0.1& 0.1& 0.1& 0.2& 0.1& 0.1& 0.1\\ 0.1& 0.1& 0.1& 0.1& 0.1& 0.1& 0.2& 0.1& 0.1\\ 0.1& 0.1& 0.1& 0.1& 0.1& 0.1& 0.1& 0.1& 0.1\end{array}$$

The final transportation volume between cities (${\mathit{T}\mathit{V}}_{\mathit{i}\mathit{j}}^{\mathit{U}}$):

$$X=\begin{array}{ccccccccc}377& 307& 150& 148& 148& 158& 140& 139& 138\\ 289& 93& 95& 96& 104& 83& 105& 114& 94\\ 166& 112& 95& 96& 110& 119& 121& 110& 100\\ 167& 124& 127& 127& 122& 121& 113& 132& 121\\ 133& 98& 101& 102& 106& 95& 87& 106& 96\\ 126& 110& 112& 113& 112& 130& 132& 121& 111\\ 112& 99& 101& 103& 96& 96& 98& 117& 107\\ 127& 102& 106& 96& 91& 89& 91& 90& 100\\ 116& 123& 116& 217& 111& 120& 122& 131& 121\end{array}$$

5.2. Regulation of Transportation Volume Based on Lower-Layer Constraints

Figure 8 shows the dynamic shortest route network structure of the transportation between cities based on Figure 7. Taking road segment 1–2 as an example, the transportation volumes from Baicheng (BC) to Songyuan (SY), Changchun (CC), Jilin (JL), and Yanbian (YB) are first sent through road segment 1–2. Therefore, the newly added transportation volume on the segment is as follows:

$${TV}_{12}^U+{TV}_{13}^U+{TV}_{14}^U+{TV}_{15}^U={tv}_{12}^L$$

(50)

For road segment 2-1, there is only the transportation volume from Songyuan (SY) to Baicheng (BC); thus, the newly added transportation volume is ${\mathit{T}\mathit{V}}_{21}^{\mathit{U}}$.

The calculation method for the newly added transportation volume of other road segments is the same as that of the above road segment, 1-2 or 2-1. The sum of the current transportation volume and newly added transportation volume cannot exceed the allowable maximum transportation volume; that is, conditions (26) or (27) should be satisfied. If it exceeds the amount, it must be regulated to meet the constraints.

Taking five cities connected by the G12 expressway as an example, the regulation process is illustrated. At the current time, the OD matrix of the transportation volume sent between cities through the G12 expressway is presented in

Table 1. Transportation volume sent between cities.

City	YB	JL	CC	SY	BC
YB	0	21	38	12	19
JL	38	0	121	69	35
CC	48	239	0	146	121
SY	16	48	21	0	32
BC	19	32	38	71	0

.

The road segment setting is shown in Figure 9. The condition settings are shown in

Table 2. Conditions and regulation results.

	Forward Road Segment (A-B)				Reverse Road Segment (B-A)
Name	${\mathit{L}}_1$	${\mathit{L}}_2$	${\mathit{L}}_3$	${\mathit{L}}_4$	${\mathit{L}}_4$	${\mathit{L}}_3$	${\mathit{L}}_2$	${\mathit{L}}_1$
${cr}_{max}$	0.7	0.7	0.7	0.7	0.7	0.7	0.7	0.7
Current speed	90 Km/h	66 Km/h	90 Km/h	90 Km/h	90 Km/h	90 Km/h	80 Km/h	90 Km/h
Estimated cars	38	130	86	45	73	68	110	61
New cars	90	225	267	32	171	112	287	64
Estimated $cr$	0.32	0.82	0.82	0	0.69	0.54	0.86	0.29
Regulation	No	Yes	Yes	No	No	No	Yes	No
Regulated volume $∆f$	0	−105	−103	0	0	0	−147	0

Note: in Table 2, ${\mathit{c}\mathit{r}}_{\mathit{m}\mathit{a}\mathit{x}}$ represents the allowable maximum congestion rate on the road segment; current speed represents the current vehicle driving speed; estimated cars represents the predicted number of vehicles on the current road segment; new cars represents the number of vehicles to be added on the current road segment; estimated cr represents the congestion rate after adding new vehicles on the current road segment.

. The regulation results are shown in Table 2. According to the calculation in reference [1], the number of vehicles on the road segment is between 0 and 250; the congestion rate is 0 ≤ cr ≤ 0.7.

For example, the newly added transportation volume on the forward road segment ${\mathit{L}}_2$ is the sum calculated as follows based on Table 1: ${\mathit{t}\mathit{v}}_{43}^{\mathit{L}}=$ 121 + 69 + 35 = 225. Current cars are ${\mathit{r}\mathit{f}}_{43}$ = 130 and total cars are 225 + 130 = 355. Therefore, the excess volume is 355 – 250 = 105, which is the reduced volume. The reduced volume between cities is calculated according to Equation (48).

The reduced volume from Jilin to Changchun: 121/225 × 105 = 56.

The final volume from Jilin to Changchun: 121 − 56 = 65.

The reduced volume from Jilin to Songyuan: 69/225 × 105 = 32.

The final volume from Jilin to Songyuan: 69 − 32 = 37.

The reduced volume from Jilin to Baicheng: 35/225 × 105 = 17.

The final volume from Jilin to Baicheng: 35 − 17 = 18.

The calculations of the forward road segment ${\mathit{L}}_3$ and reverse road segment ${\mathit{L}}_2$ are the same as that above. The regulated and reduced transportation volume can be transported through national or provincial highways, or it can be delayed until the road segments are not congested. It can also be sent during off-peak hours.

It should be noted that the above data are not accurate real-time data but are used to explain the regulation process and methods based on the actual situation.

6. Conclusions

6.1. Summary

Currently, the vehicles dispatched from cities to highway transportation networks are in a state of spontaneous organization in urban agglomerations. There is a lack of comprehensive consideration at the macro level regarding the allocation and distribution of traffic volume in highway transportation networks. This study establishes a coordination mechanism from the perspective of highway transportation volume regulation, promoting the coordinated development of urban agglomerations through highway collaborative transportation, and providing new methods for studying the coordinated development of urban agglomerations. The research results of the paper are as follows: (1) From the macro level, in order to ensure the coordinated development of cities within an urban agglomeration, when the transportation capacity of the highway transportation network is limited, the highway transportation volume of each city must be reasonably allocated, which is called a static regulation. From the micro level, in order to ensure that each road segment in the highway transportation network is not congested, according to the current traffic flow of the road segment, the road segment traffic volume between various cities must be adjusted in real time, which is called a dynamic regulation. The best way to combine the two regulation modes is to use multiple-layer complex networks. In this paper, a two-layer complex network model is established. The upper layer is a network composed of major cities in the urban agglomeration, and the lower layer is a network composed of various nodes of expressways. The two networks are connected by expressway transportation. The network model can describe the traffic volume correlation between cities in urban agglomerations, between the cities and network road segments, and between the road segments. It provides an important model for studying highway collaborative transportation in urban agglomerations and the coordinated development of urban agglomerations; (2) Taking the urban highway transportation intensity as a variable, an objective function for the coordinated development of urban agglomerations was established and constraint conditions were given. Due to too many constraints, a hierarchical solution method was adopted according to the characteristics of transportation volume regulation; (3) During the process of solving the model, a probability search iteration algorithm was explored that was suitable for solving many-to-many allocation problems and optimization problems with dozens of variables. The algorithm is concise, with few parameters, a fast convergence speed, convenient for programming, and is easily implemented.

According to the size of regional space, urban agglomerations can be divided into three types: large, medium, and small. Although our example is a provincial small urban agglomeration, the two-layer complex network model established in the paper can be applied to other types of urban agglomerations and has good scalability and adaptability for the following reasons: (1) There are no preconditions, limitations, and assumptions for the establishment of the network model; (2) The nodes in the network are the main cities in the urban agglomeration, expressway toll stations, and intersection nodes. There are no special nodes. These nodes exist in any kind of urban agglomeration; (3) The objective function we proposed is not a unique objective function for solving the coordinated development of urban agglomerations. Other types of objective functions can also be proposed based on other performance indicators, but the solution method is the same, which is to adopt a hierarchical solution strategy; (4) The probability search iteration algorithm is very suitable for solving many-to-many optimal allocation problems. It uses matrix operations to solve dozens of variables. Other optimization algorithms cannot realize the function.

6.2. Future Research Directions

This paper contributes to the research field of coordinated development of urban agglomerations by presenting innovative and practical research work. In the future, we will combine the regulation of highway transportation volume with other indicators, such as the urban population growth rate, GDP growth rate, and spatial change rate, to study the dynamic simulation of urban development and the coordinated development of urban agglomerations, providing a reference for relevant government departments to formulate regional economic development plans.