1 Introduction

1.1 Background significance

The sustained economic development puts forward the requirements of diversification, safety, convenience and high quality in transportation, and the development of the transportation industry has entered a new stage of development. Container multimodal transportation has the advantages of unified documents, short transportation links, short time and low cost, and has become the main transportation method at this stage. However, the container multimodal transportation system is actually more complicated and requires a lot of organization and coordination. When planning and coordinating the transportation process, some characteristics of the system must be considered to achieve the optimization of synergy. The BP neural network algorithm has advantages in solving complex problems of internal mechanisms, with non-mapping capabilities and generalization capabilities, a high degree of self-learning capabilities, adaptive capabilities, and fault tolerance [1]. Therefore, it is of great significance to study the synergy evaluation model of container multimodal transportation based on BP neural network.

1.2 Related work

Container multimodal transportation has gradually become the focus of everyone's research, and there are more and more research results on it. Brcic D [2] discussed how container multimodal transportation can save external costs, and analyzed relevant data by reviewing previous research and published works. His research analyzes the data of previous years. The current multimodal transport model has undergone new changes, and the data are of little reference. In a given multimodal freight network, Kim Ns [3] clearly established a scale economy model based on the number, distance and vehicle size, and used genetic algorithms to calculate acceptable routes, modes and vehicle sizes. Although his model has high accuracy, it is more complicated in calculation and takes time and effort. Once the idea of synergy was put forward, it was widely used. Chai J [4] assumes that human motion learning uses the concept of motion collaboration and studies whether this concept can also be observed in the deep reinforcement learning of robots. In a simulated environment, a joint spatial collaboration analysis of multiple jointly operating agents is performed, and two most advanced deep reinforcement learning algorithms are used for training. His research did not choose a more appropriate evaluation model when evaluating the synergy effect, which led to the deviation of the final result.

1.3 Innovative points in this paper

In order to further analyze the synergy effect of container multi-modal transportation, this paper combines the concept of synergy effect and BP neural network algorithm. The innovations of this paper are as follows: (1) Based on the BP neural network algorithm, the comprehensive evaluation of multi-attribute decision-making method is used to establish a complete synergy evaluation model of container multimodal transportation. (2) Solve the model, convert the problem, and find the shortest route from the start point to the end point without exceeding the cost and time constraints. (3) In the simulation experiment, Matlab is used to solve the transportation time and total cost between cities, and it is found that the time and cost of multimodal transportation are less than those of a single transportation method.

2 Evaluation of synergy effect of container multimodal transport based on BP neural network

2.1 BP neural network algorithm

  • (1) Artificial neural network classification.

Artificial neural networks process information by simulating the network processing and memory information of the brain, and have the characteristics of nonlinearity, non-limitation, very qualitative and non-convexity. The connection between two nodes represents the weighted value, called weight [5]. At present, common neural network models include BP neural network, RBF neural network, wavelet neural network and so on.

BP neural network adopts the training method with time training to carry out error back propagation in the system. The steepest descent method is mainly used. By repeatedly correcting the weights and thresholds through the reverse propagation of the error, the sum of squares of the network error can reach the preset value. The topological structure of the BP neural network model is divided into input layer, hidden layer and output layer, which are often used in the fields of pattern recognition and adaptive control [6].

RBF neural network is also a kind of feedforward neural network. The radial basis function is used as the basis function, and different basis functions correspond to different training data. It has the local characteristics of only producing effective nonzero response in a very small local range, and the convergence speed of learning is faster.

The wavelet neural network is formed based on the wavelet analysis theory, and the discrete wavelet transform is introduced into the neural network. The activation function is the wavelet function, and then the radiation transform is used to construct the connection between the wavelet transform and the neural network model [7].

  • (2) Structure and calculation of BP neural network.

BP neural network is mainly divided into input layer, hidden layer and output layer. The input data are passed to the output layer through processing, and finally the equations related to input and weights are calculated. Neurons will only affect the neurons in the next layer and will not affect other neurons. The output layer compares the output function with the expected value. If the expected output value cannot be reached, the weight will be backpropagated. Adjust the weights and thresholds of the network by comparing the actual output with the expected output, and perform the same calculation as the previous step again. Through continuous repetitive calculations, the actual output will continue to approach the expected output.

The calculation of the neural network first needs to initialize the weights and thresholds of each node randomly. After the initialization is completed, the activation function of the hidden layer can complete the function conversion and realize the system control. Currently commonly used activation functions include linear function, Sigmoid function, Gaussian function, etc. [8, 9]. The expressions of the functions are shown in Formula 1 to Formula 3:

$$ f\left( n \right) = {\text{an}} $$
(1)
$$ f\left( n \right) = \frac{1}{{1 + {\text{e}}^{ - n} }} $$
(2)
$$ f\left( n \right) = {\text{e}}^{{ - \frac{n}{\mu }}} $$
(3)

Then set the input signal and the corresponding output signal, input the signal from the input layer until the final actual output is obtained, and calculate the output of each node in the hidden layer and the output layer [10]. Through error back propagation, the node weights of the output layer and hidden layer are adjusted until the error is reduced to the desired range.

  • (3) Advantages and disadvantages of BP neural network.

BP neural network has the general advantages of neural network. First of all, it has a strong non-mapping ability. BP neural network is actually a mapping relationship between input and output, which has advantages in solving complex problems of internal mechanisms. Secondly, it has generalization ability, which can deal with unknown similar situations through learning, and the network model can apply learning results to the learning of new knowledge. Third, it has a high degree of self-learning and self-adaptation capabilities. During the training, the weights and thresholds of the network can be modified through sample comparison, and finally adjustment and learning are completed [11]. Finally, it has better fault tolerance. Errors in local or partial learning links will not cause very serious consequences to the overall training results.

Although BP neural network has unique data processing capabilities, it also has certain defects. The algorithm has a slow convergence speed, especially when the processing object is more complex, its efficiency will be greatly reduced. In addition, the algorithm is too dependent on samples, so you need to be very careful when selecting samples, because the reliability of the samples determines the accuracy of the algorithm.

When choosing a BP neural network to write an algorithm, it is necessary to maximize its strengths and avoid weaknesses, make full use of its advantages, and try to avoid or even solve its shortcomings.

2.2 Container multimodal transport

  • (1)Decision-making principles for container multimodal transport.

The container multimodal transport uses standardized containers as the transport unit. After at least 2 transport methods, the intermodal transport operator organizes the transfer from the receiving location to the delivery location. The basis of container multimodal transportation decision-making is the analysis of the technical and economic characteristics of various transportation modes, which is also a prerequisite for choosing the optimal combination of transportation modes and transportation routes [12]. Container multimodal transportation decision-making includes planning routes, selecting transportation methods, scheduling transportation tools and empty containers, etc., to achieve maximum efficiency and benefit on the premise of achieving transportation goals. Therefore, some principles need to be followed when making decisions.

Principles of safety and integrity. Safety is the most basic and most important condition of transportation. During transportation, the integrity of goods and the safety of transportation tools must be ensured [13]. Secondly, the transportation link must be placed in the entire logistics, seeking the overall optimization rather than the optimal transportation cost.

The principle of combining the best service level and the lowest total cost. By saving freight and management costs, the profits of the company can be increased. However, while reducing the total cost, the container multimodal transport company must provide customers with better services, further enhance the company's reputation and market share, and establish a corporate image.

The principle of combining economic and social benefits. Under the market economy system, the primary criterion for enterprise decision-making is to measure the feasibility of decision-making based on profitability, because enterprises are independent commodity producers and operators [14]. However, the survival and development of an enterprise is closely linked to the development of society, and it is impossible for an enterprise to exist without the social environment. Therefore, the decision-making on container multimodal transportation should take into account economic and social benefits, and harmoniously combine corporate profits and social benefits.

  • (2)Influencing factors of container multimodal transport.

The decision-making of container multimodal transportation can be divided into strategic decision-making and tactical decision-making. Different decision types have different influencing factors.

Strategic decisions appear when the shipper company chooses a carrier. Consignor companies need to consider many factors when choosing a carrier. The first is the characteristics of various modes of transportation and related costs, because cost is an important factor. In a fiercely competitive environment, transportation time is the primary factor. Therefore, it is necessary to make a choice based on the product's requirements for cost and time [15]. The second point is the distribution of factories and warehouses, including the production and storage of goods at various distribution points, and the location of the warehouse will directly affect the design of the transportation network and the choice of routes. The third point is the type of product cargo. Low-value cargo has a weak freight capacity and low requirements for speed, while high-value cargo has a strong freight capacity and high requirements for speed. The fourth point needs to consider transportation resources and transportation networks. If the cost of the transportation method is low, but the accessibility is poor, other transportation methods are generally considered, and some transportation methods are expensive but have strong accessibility, so a comprehensive comparison is required [16]. The fifth point is the strength and confidence level of the carrier, which requires a certain amount of registered capital, a sound business network and a good reputation. The last point is the tax and fee situation. Different carriers have different transportation organizations and different taxes and fees. This will directly affect the transportation cost.

The tactical strategy is carried out by the carrier in the transportation organization in order to achieve the established transportation goals. Carriers also need to consider many factors when organizing transportation. The first is the transportation cost of each transportation mode. Compare them and choose the combination with the strongest profitability. Generally, when carrying out “door-to-door” service, the terminal chooses road transportation, and the trunk line considers railway and water transportation. Under the premise that the transportation time is required, sometimes the trunk line will also choose road transportation. The second point is the delivery time. Different types of products can bear different transportation costs. The delivery arrival time and other requirements need to be considered according to the nature of the product and market requirements. The third point needs to consider the feasibility and connectivity of various modes of transportation, shorten the conversion time, reduce the number of loading and unloading times and costs, and avoid cargo damage and difference [17]. The fourth point is to consider the required lead time of various modes of transportation. Waterway, railway and air transportation all require a certain lead time. The road has the shortest lead time and can be shipped immediately. Finally, the quality of transportation services must be considered, including the rate of damage to goods, the rate of loss and the punctuality rate.

  • (3)Features and advantages of container multimodal transport.

Container multimodal transport has the characteristics of high investment, large scale, high efficiency, excellent collaboration and standardization. In the entire intermodal transportation system, the maintenance and management of containers, transportation vehicles, loading and unloading equipment, and the entire system all require a lot of investment. Container multimodal transport is a typical large-scale industry, with large ships and capital-intensive. Container multimodal transportation can effectively avoid waste of labor and slow turnover, thereby improving the efficiency of the transportation process. Because multimodal transportation has a good organizational form, it can combine transportation networks, business operators, transfer stations and agency networks to carry out efficient and accurate information transmission and document circulation, and effectively integrate all departments involved in transportation. The international standards of container size have standardized the size and volume of goods, and the facilities layout of the means of transportation, transfer stations, relevant laws and regulations, and document procedures have gradually become standardized.

Container multimodal transportation has the advantages of various transportation methods and connects them. The road transportation is convenient and the time is short, the railway transportation is safe, and the cost of water transportation is low. In the actual transportation swelling, after a comparative analysis of various transportation methods, learn from each other's strengths to make up for their weaknesses, and give play to the advantages of different transportation methods, so that the transportation process will be more reasonable and efficient, and maximize benefits.

When container multimodal transportation is adopted, the cargo is packaged with the container as the carrier, which can avoid cargo damage to a greater extent during the loading and unloading process, and improve the overall quality of transportation. Since the size of the container is an international standard, loading and unloading is quick and convenient, shortening the time required. Intermodal transportation will make reasonable plans for the route and select the appropriate transportation method, which can also shorten the transportation time.

Multimodal transportation involves multiple modes of transportation, which seems more complicated. However, through the overall management of the intermodal operator, the cargo owner only needs to sign and handle the relevant formalities and contracts once, and the intermodal carrier is responsible for the rest of the issues, which can greatly save the owner’s time and cost. Moreover, container multimodal transportation will analyze the cost of different transportation methods, and then choose the transportation method combination with the lowest total cost, and the cost will be greatly reduced.

Container multimodal transportation will improve transportation efficiency because it is composed of multiple methods and has the advantages of different methods [18]. Due to the development of multimodal transport, operators of various transport modes can continue to learn from others, so management and decision-making strategies will continue to change. Through continuous improvement of concepts and continuous cooperation with others, reasonable and effective transportation routes and methods can be selected to better improve the overall efficiency of transportation.

2.3 Synergy evaluation

  • (1)Synergy theory

The idea of synergy emphasizes that the integrity of the system is reflected in the operation of the enterprise. The various components of the system stimulate and interact with each other to produce an overall effect and a structural effect. This effect cannot be brought about by any single component [19]. Collaboration expresses that one plus one is greater than two offline, and the company's collaboration can bring value greater than the total value of each company.

Synergy refers to the overall effect of the cooperation between various subsystems in the system, that is, the result of synergy. No matter what kind of system it is, if many substances gather under external action, the external action will make the state of internal matter more and more expand to the limit [20]. At this time, various subsystems begin to coordinate, thereby reducing external pressure, and at the same time generating a synergistic effect, making the system from disorder to order.

The concept of coordination illustrates the self-rules of objective practice from disorder to order. The theory of self-organization refers to the powerful synergy effect brought to the system through self-organization, but not all systems can be self-organized. First of all, the system must be an open system and must always maintain the exchange of external materials, information and energy required by the system [21]. Second, the external effects of the system must be effective through the internal mechanism of the system. Third, the relationship between the elements in the system is non-linear. Finally, the generation and development of the self-organization and orderly structure of the system are caused by fluctuations. Only under these conditions can the system be self-organized and form the internal power of the system without external influence.

  • (2)Synergies of container multimodal transport

The collaborative ability of container multimodal transport can ensure that it maintains its advantages in the transportation process. The research on this ability was initially a tangible resource facility, and has now developed into a research based on information, optimization technology, standards and other intangible resources [22]. The container multimodal transport coordination capability is based on modern logistics technology and efficient intermodal management, logistics standards and information technology.

The foundation and important performance of the entire container multimodal transport coordination capability is the internal coordination capability of the container multimodal transport node. Only by realizing the internal coordination of the node itself, the elements such as resources and information can flow smoothly within the enterprise, laying the foundation for the coordination of the entire container multimodal transport [23]. The cooperation of container multimodal transportation is the cooperation between nodes. Each node can quickly integrate with upstream and downstream nodes to realize the overall coordination of container multimodal transportation. Only when synergy is achieved within and between nodes, the supply chain where the container multimodal transport is located will have rapid response capabilities and strong competitiveness.

The core of the container multimodal transport coordination capability is the comprehensive sharing of standards and information. Container multimodal transportation is carried out by nodes. Therefore, unified standards and high-level information sharing can help the process become smoother, accelerate the integration between nodes, and have better integration depth.

In the container multimodal transport management system, based on the intangible resources of the enterprise, it is supported by tangible resources such as modern communication technology, enterprise management software, and logistics technology. Standards and information sharing are the core to realize the collaboration between the nodes of the container multimodal transport, and the overall function of the container multimodal transport is doubled, bringing greater synergy effects [24].

  • (3)Evaluation method

At present, the commonly used comprehensive evaluation methods include operations research and other mathematical methods, technical and economic analysis methods, statistical analysis methods, etc.

Operations research and other mathematical methods can be divided into multi-attribute decision-making methods, data envelopment analysis, analytic hierarchy process and fuzzy comprehensive evaluation methods. Multi-attribute decision-making is sorted and evaluated by changing more to less, stratifying the sequence, directly seeking non-inferior solutions, and rearranging the order. It has the advantages of accurate description of the evaluation object, which can simultaneously evaluate multiple indicators, multiple decision makers, and dynamic enterprises, but it cannot involve objects with fuzzy factors. The data envelopment analysis model deals with the problem of multi-index input and output, and evaluates the relative effectiveness of different enterprises of the same nature. It has the advantage of a larger system that can evaluate multi-factor input and output. But it can only evaluate the relativity of enterprise indicators, not the actual development level [25]. The analytic hierarchy process combines qualitative and quantitative methods to make pairwise comparisons. It has high reliability and small errors, but it cannot evaluate too many objects at the same time. The fuzzy comprehensive evaluation method expresses the final evaluation result in the form of fuzzy set, which can obtain multi-level results according to different possibilities, avoiding some of the drawbacks of traditional methods, but cannot solve the problem of information duplication between indicators.

Technical economic analysis methods include economic analysis method and technology evaluation method, which are evaluated through cost value analysis, value function analysis and benefit analysis. The advantage is that the meaning is relatively clear and the comparability is strong, but the disadvantage is that it is difficult to model and is not suitable for objects with more evaluations.

Statistical analysis methods include principal component and cluster analysis. The object of principal component analysis is related economic variables dominated by common factors. It studies the internal structure of the original matrix and finds out irrelevant comprehensive indicators. Cluster analysis calculates the distance between indicators and performs systematic clustering. Statistical analysis methods have the advantages of comprehensiveness, strong comparability, and objective evaluation. There are also shortcomings that the meaning of the function is not clear, the demand for statistical data is large, and the true development level cannot be reflected.

3 Experiments on establishment of synergy evaluation model of container multimodal transport based on BP neural network

3.1 Model building

  • (1)Data preprocessing and hidden layer

The data in this article comes from part of the route data of some container multimodal transport companies, and the multi-attribute decision-making method is used to comprehensively evaluate the synergy effect of container multimodal transport. The data are normalized to convert different input components into equal positions. The final synergy output also needs to be preprocessed, which can reduce numerical errors.

The selection of hidden layer nodes is very important for the establishment of BP neural network. Too many hidden layers will lead to longer training time and increase errors. If the number of hidden layers is too small, the convergence speed will slow or even not converge, so that there is no solution. Therefore, the range of the number of neurons in the hidden layer is shown in Formula 4:

$$ K = \sqrt {a + b} + i $$
(4)

Among them, a, b are the number of input and output units, and i is a constant between 1 and 10. The number of hidden layer nodes applicable to this article is 8.

  • (2)Learning rate and training function

The learning rate is a fixed value, so Matlab is needed to determine the learning rate of the model in this article. Different learning rates can be selected for training, and the best learning rate is finally determined to be 0.06. The network transfer function selects the logsig function. The maximum number of iterations is 800, and traingdx is selected as the training function.

3.2 Model solving

According to the above objective function and constraint conditions, this research constructs a transportation network diagram, transforming the original problem into the shortest route under the premise of cost and time constraints.

As shown in Fig. 1, construct a virtual starting point and ending point, Q and P, respectively. The solution of the model can be transformed into: finding the shortest route from Q to P without exceeding the cost and time constraints.

Fig.1
figure 1

Schematic diagram of transportation network

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3.3 Simulation experiment

The above-mentioned virtual starting point and ending point are 2 hinterlands. In the simulation experiment, it is assumed that hinterland Q and hinterland P have 4 cities, respectively, so that there will be 16 routes between these 8 cities, which can reach the end point from the starting point. Use Matlab to solve the transportation time and total cost between these 8 cities.

4 Discussion on performance of synergy evaluation model of container multimodal transport based on BP neural network

4.1 Number of iterations and training error

Perform performance testing on the algorithm model, record its fitness value after every 250 iterations, and observe its convergence.

As shown in Fig. 2, when the number of iterations is 800, the algorithm starts to converge, after which it remains stable, with an adaptation value of 12. This shows that the algorithm model has a faster convergence speed.

Fig. 2
figure 2

Iteration situation

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The model is trained 8 times, the expected output and actual output value of each training are recorded, and the training error is calculated to test the accuracy of the model. The recorded results are as follows:

As shown in Fig. 3, the error values for different training times are calculated to be 0.0263, 0.0312, 0.0376, 0.0745, 0.0004, 0.0099, 0.0111 and 0.0285. The smallest error is the fifth training. The error between the actual output and the expected output is only 0.0004, and the largest error is only 0.0745. This shows that the established BP neural network-based container multimodal transport synergy evaluation model has high accuracy.

Fig. 3
figure 3

Training error

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4.2 Time and cost comparison

There are many routes between the starting point Q and the ending point P. We have selected 5 routes for comparison and named them A, B, C, D, and E, respectively. Comparing the time spent on these 5 routes using a single transportation method and a container multimodal transportation method, the results are as follows:

As shown in Table 1, no matter which route, the time consumed by a single transportation mode is much higher than the time consumed by container multimodal transportation. The most obvious one is Route B. The multimodal transport time is 38.5 h, while the single transport time is 74.3 h, which is a full 48.18% less. This shows that container multimodal transportation can greatly save transportation time and improve transportation efficiency.

Table 1 Time comparison
Full size table

In order to further test the ability of container multimodal transportation under the BP neural network algorithm to save costs. Calculate the total cost of a single transportation mode and compare it with the total cost of the container multimodal transportation mode. The results are as follows:

As shown in Fig. 4, no matter which route, the total cost of a single transportation method is higher than the total cost of container multimodal transportation. The highest cost savings is route B, which saves 50.02%, followed by the E route, a saving of 44.75%. The least C route can also save 24.43%. This shows that multimodal container transportation can save costs, enhance the competitiveness of enterprises, and maximize benefits.

Fig. 4
figure 4

Cost comparison

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5 Conclusions

Container multimodal operation is an advanced modern transportation method with the characteristics of high investment, large scale, high efficiency, excellent collaboration and standardization. Container multimodal transport has the advantages of various modes of transportation and can complement each other; it can avoid cargo damage to a greater extent during the transportation process and improve the overall quality of transportation; it can also improve transportation efficiency and reduce costs.

This paper establishes a multimodal transport synergy evaluation model for containers based on the BP neural network algorithm, and improves its objectivity and evaluation accuracy. The model building process is easy to operate and highly accurate. Through iteration and training, it was found that this algorithm has fast convergence speed and good performance.

Due to the limited time and knowledge, the same batch of sample data was used in the algorithm training and simulation experiments in this study, which may affect the actual accuracy of the algorithm. Moreover, the BP neural network itself has some defects, so it should be improved in design. These problems will be further improved in the next research work.