Mathematical modeling for the development of tra ﬃ c based on the theory of system dynamics

: This paper is concerned with mathematical modeling for the development of Shandong tra ﬃ c. The system dynamics model of the development of tra ﬃ c in Shandong is established. In terms of this model, it is shown that highway operation as well as rail transit promotes the development of tra ﬃ c, while tra ﬃ c accidents inhibit tra ﬃ c development. Moreover, the maximum error between the output data and the statistics bureau, based on which some forecasts for the development of tra ﬃ c in the future are given, is obtained, some suggestions and optimization schemes for tra ﬃ c development are given. Finally, a neural network model of the development of Shandong tra ﬃ c is also derived.


Introduction
Recently, the traffic situation has been improved, which is reflected by the amount of passenger on a highway of 15 billion people in 2018, and the total length of rail lines in China has reached 5761.4 kilometers, therefore, the study of the traffic system by different theories is crucial.For example, based on mathematical modeling, reference [11] shows that the emergency vehicle management solution can reduce travel times of EVs without causing any performance degradation of normal vehicles.Zhu [30] proposed a bayesian updating approach based on the dirichlet model to describe the traffic system performance.Via machine learning, Saleem [21] proposed a fusion-based intelligent traffic congestion control system to alleviate traffic congestion in smart cities.
The traffic system is a complex dynamic system whose influencing factors are complex and diverse, thus, the mathematical modeling of traffic system is difficult.In order to understand the dynamic behavior of traffic systems comprehensively, the system dynamic (SD) method is one of the most powerful tools [10].The theory of system dynamics founded by Forrester [7] can be used to simplify the multi-variable system, and it is widely used to deal with social and economic problems.As for the application of the theory of system dynamics, one can refer to [4,18,25,26], to name but a few.The basic concepts of system dynamics are as follows: (1) Level variable, represented by a rectangle in VENSIM software, are affected by rate variables, which can be expressed by integrating the rate variables.(2) Rate variables are time functions, and they determine the level variable.
(3) For ease of communication and clarity, it is often to define auxiliary variables, which are neither stock nor flow.(4) The constant variable is a constant, and it can be characterized by table functions.(5) Table functions conveniently represent nonlinear relationships.(6) Flow graph is a characteristic diagram in system dynamics.It simplifies the equations of models.
In the sequel, the flow graph can be transformed into VENSIM equations.
The modeling steps for system dynamics in VENSIM software are shown by Figure 1.

Building the SD model
Analyzing the result Recently, research based on system dynamic theory mainly focuses on urban traffic, low-carbon traffic as well as traffic policies.According to the restriction policy, Wen [27] calculated that the implementation of the tail licensing restriction policy could effectively reduce carbon dioxide emissions by 3.86%.Moreover, they found that the vehicle license limit policy could effectively curb the growth of car ownership, and alleviate traffic congestion.Jia [14] established an SD model of traffic congestion charge and subsidy.Furthermore, it can be found that zero-subsidy and low charge reduce carbon dioxide emissions.He [10] concluded that the railway occupation rate was directly proportional to the railway length.Additionally, in order to assess the relation between numbers of available public traffic and traffic congestion in Jakarta, Sardjono [24] developed an SD model for traffic conditions, which can reduce traffic congestion.Based on road accident statistics, Victor [16] proposed the methodology of evaluated long-term trends in the dynamics of traffic safety improvement.Rajput [20] present a system dynamics simulation model to reduce traffic congestion by implementing an Intelligent Transportation System in metropolitan cities of India.Ye [29] concluded that highway investment had a certain pulling effect on economic growth based on a concrete analysis on the relationship between highway construction and economic growth.
In China, the traffic system mainly includes taxi, road vehicles, private cars and rail transit [8].The interactive communication between them forms a complex dynamic system of traffic.To the knowledge of the author, there still lacks proper mathematical models to study traffic systems, including the six subsystem by the SD method.Based on the actual situation of the traffic system in Shandong Province, this paper proposes an SD model to describe the relationship among the six subsystem which contains the highway vehicle subsystem, the private car subsystem, the highway mileage subsystem, the rail transit subsystem and the traffic accident subsystem.Based on the SD model of the traffic system, some forecasts for the development of traffic in the future are given and some suggestions and optimization schemes for traffic development are given.This model is helpful for improving the convenience of traffic service and to promote stable traffic.

SD model
In highway vehicle subsystems, the number of operating vehicles is affected by highway mileage and highway density [6].Generally speaking, highway has positive effects on the number of vehicles.In addition, the scrap of private cars can force people to take public traffic, which requires the increasing of investment in public traffic [3].Thus, the above factors are defined as the influencing factors of the number of operating vehicles.
In taxi subsystems, taxi growth and taxi scrap are considered as two main factors which lead to the overall change in the number of taxis.As for the private car subsystems, it is used in traffic frequently, but it brings a lot of exhaust pollution [13].In highway mileage subsystem, with the increasing of all kinds of vehicles, highway construction has been accelerated.On the contrary, old roads reduce highway mileage [6].Highway mileage is an important embodiment of traffic development.In order to indicate the situation of highway mileage, we represent it by old roads and the number of vehicles.
In rail transit subsystems, with the increase of damage to the track, the length of rail transit has decreased.Track is influenced by the use of various vehicles and track investment [9,28].Then, the number of private cars and track damage are chosen as the influencing factors of rail transit length.
In traffic accident subsystems, the national economy can be affected by traffic accidents, and a suitable traffic condition can reduce the number of occurrences of traffic accidents effectively.Public traffic can affect the number of traffic accidents directly, which changes the choice of travel modes.Hence, we can get the influencing factors for the number of operating vehicles.
In terms of the fact indicated above, the following parameters are selected as Level variables: traffic accidents, number of road vehicles, highway mileage, number of taxis, number of private cars owned and length of rail transit line.For the sake of convenience, in the following context, all the elements are summarized in Table 1.In addition, the scenario diagram of the SD model is displayed in Figure 2.
According to the scenario diagram of traffic shown in Figure 2, each subsystem is associated with one state variable, such as T 1 , T 9 , T 15 , T 20 , T 26 , T 33 and so on.All the subsystems and their influence parameters can be found in Table 1.
In order to evaluate the situation of traffic, an SD model of the traffic system in Shandong Province was established by VENSIM software, the feedback process of flow diagram is marked by a blue arrow in Figure 3. Furthermore, it can be obtained that the influence between the parameters is multiplex and the influences between the six subsystems are mutual.

Mathematical model
In light of the flow diagram (see Figure 3), the SD model can be established by translating the SD flow diagram into VENSIM equations, and thus the level variable can be expressed as the following integral equation [25] in which, Level(t) represents level variables, Rate in and Rate in signify inflow rate variables and outflow rate variables respectively.Thus, we have where s is time at any time between the initial time t o to the current time t, the unit of time assumed here is a year.On the other hand, we find = T 36 (t) − T 37 (t). (2.3) Following Increment = Growth rate × Original data, we have (2.4) In traffic accident subsystems, the mortality rate is affected by travel modes, highway vehicles and taxis, and these are often selected as the analysis objects [15,23], rail transit can decrease the accident rate [17].In light of the data on the traffic accidents, we can find that the increase in the number of car leads to an increase in the number of traffic accidents, and the implementation of rail transit as well as positive policy can reduce the number of accidents.In addition, education on the safety awareness for people can also reduce the number of accidents.With respect to the highway subsystem, the number of cars is proportional to the number of roads and the miles of highway [12].Investment in traffic can increase the mileage of highway [29].In taxi subsystems, the expenditure on taking a taxi is higher than use of public traffic, but it is necessary to increase taxis reasonably, and the use of taxis is accompanied by the scrapping of taxis [19].As for the private car subsystem, private cars are contradictory to public traffic, and they are positively related to highway mileage [3,6].In rail transit subsystem, the increase of the length of rail transit results in increasing of the investment of rail [9].On the other hand, there exists the conflict between rail transit and private car [28].The state of GDP is an important index of national development and it can reflect the government investment on traffic infrastructure construction.As indicated above, we obtain the following expressions on the rest of variables, (2.5)

Optimization of SD model
This subsection is devoted to optimize the SD model (see Figures 4 and 5).By comparing the output data with real data, four factors were selected in the model to judge the relative error of an SD model.The data of the National Bureau of Statistics and output date are expressed by R and C respectively, O denotes the output data of optimized model, and the relative error of the optimized model is signified by Oe.The following expressions to calculate the Oe of four curves is defined as Table 2 shows that the relative error of the output data is less than the limit of error of 10% in system dynamics, which means that the precision of the model is enough.

Parameter analysis
Based on the SD model, ten debugging models are established, in which T is taken as the objective function, and T 3 , T 10 , T 30 and T 40 are taken as the design variables.By using these debugging models, the optimal design of traffic can be obtained to provide a better traffic control strategy.Based on the results of debugging, the effects of different debugging models on the traffic are evaluated.
From Figure 6, which shows the current optimization curve of T , we have: (1) The variation of T associated with the traffic accident subsystem is the maximal one in scheme 1 (see Figure 6a), which means that the government should take measures to reduce the number of traffic accidents.(2) Although there exist relationships between T and all parameters of the global system, the road density is more important than others, therefore, by adjusting road density, we can control the process of development of traffic effectively.As the length of highway can increase T , the government should improve the quality and length of a high utilization factor of highways must be kept.(3) Figure 6 shows the comparison between current optimization curves and the schemes proposed.
By Figure 6c and Table 3, we assert that T 30 exerts dominant influence on the increasing of T , and the best way to promote development of traffic is to improve the quality and length of highway and rail transit [5].(4) Figure 7 indicates that the relationship between debugging two parameters and debugging one parameter is not linear.Furthermore, the effect of two parameters is better than one parameter.

Data estimation
Table 3 shows that the relative error of T 1 , T 9 , T 15 , T 20 and T 33 are under 6.83% and thus the model has high accuracy.In light of this SD model, Table 4 displays the next ten years developments of traffic in Shandong Province, and it is found that the following trends of traffic in the future.
(1) The number of traffic accidents is increasing every year, which may be related to the growth of private cars and taxis.The growth of private cars and taxis is inevitable.Therefore, the government must take active measures to reduce the number of traffic accidents.(2) Due to the needs of the citizen and environmental protection, the number of operating vehicles will always keep rising.The increase of operating vehicles means that the number of buses on remote roads and the diversification of routes are improved, which is convenient for citizens.(3) The larger number of all kinds of vehicles leads to the increase in loading of roads, which will bring more traffic accidents.Obviously, the length of highway mileage should be enlarged.( 4) Although the number of taxis will continue to rise in the next ten years, its growth is slower than other means of traffic.Taxis and buses are convenient for people to take, but the taxi results in more serious pollution than buses.Furthermore, unlike the bus, the taxi is not as safe.Therefore, the number of taxis will decrease in the future.(5) The number of private cars will grow rapidly in the future.The private cars not only bring convenience to people, but also bring great burden to traffic.The citizens should appropriately reduce the use of private cars and take public traffic.(6) The rail transit's length will maintain rapid growth in the next decades, which means that government will pay more attention to the development of rail transit, and the rail transit can greatly reduce traffic accidents and environmental pollution.where the meaning of the coefficients of (2.7) can be found in (A.1).The five neurons are represented by T 3 , T 14 , T 19 , T 25 and T 28 .The capacitance of each neuron is fixed to be 1.γ i = 1 T i (i = 3, 14, 19, 25, 28) are the resistances of per neuron [1,2,22].If T 0 is the initial value, I signifies the expected value, S i are the years to achieve I of each neuron, and T s is T 0 after s years, then the neural network model is Model 2.9 shows that the length of time to achieve the expected value of each subsystem are 1.29, 0.43, 12.42, 0.17, 0.92, and 1.86 years respectively.Obviously, the maximum is 12.42 years, which is the length of time to arrive the expected value.The SD and neural network models are advantageous to improve the speed and efficiency of the development of the traffic system.

Conclusions
This paper considers the development of the traffic system of Shandong Province, based on the system dynamics method, and the SD model of traffic system related to the six subsystem which contains the highway vehicle subsystem, the private car subsystem, the highway mileage subsystem, the rail transit subsystem and the traffic accident subsystem, in light of which prediction on the future situation of the traffic system of Shandong Province is derived.Furthermore, the neural network model of the problem considered in paper is also given.In terms of the SD model and neural network model, we can give some policies to force the traffic systems to develop to the situation we want.For example, the government need to develop rail transit and take some measures to reduce the number of traffic accidents.The model is helpful to improve the convenience of traffic services and to promote to development the stable traffic.Furthermore, based on the neural network model, we can investigate the dynamics of the traffic system, and we will pursue this line in the future.

Use of AI tools declaration
The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Figure 1 .
Figure 1.Modeling steps of system dynamics.

. 6 )T 3 ,
T 10 , T 14 , T 17 , T 19 , T 25 , T 28 , T 30 , T 34 , T 38 and T 40 are selected as the optimized parameters to examine.There are four optimization models related to the eleven objectives, see Figure 5.It is shown that the optimized curves O 1:4 are close to the real curves R 1:4.It can be seen that the O curves are closer to the R curves.

Figure 4 .
Figure 4. Optimization results of the SD model.

Figure 6 .
Figure 6.Debugging results of one parameter.

Figure 7 .
Figure 7. Debugging results of two parameters.

Table 1 .
All parameters in the SD model.

Table 2 .
The relative error of SD model.
Note: The date of this table comes from China's National Bureau of Statistics.

Table 3 .
Growth rate of Debugged models.

Table 4 .
Data estimation based on SD model.