2.1.1 Selection of Construction Stage Investment Engineering Feature Indicators

Based on research on relevant literature, "Railway Line Design Specification" (TB10098-2017), Civil Engineering Standard Measurement Method (CESMM3), and various standards of the International Union of Railways (UIC), international environment, construction conditions, and technical standards are the main factors affecting overseas HSR construction investment considering external influencing factors. Given the complexity and diversity of these factors, this paper intends to use a fishbone diagram to analyze and sort them out, as shown in Fig. 1.

From the perspective of the HSR construction project, based on the project breakdown structure (PBS) decomposition of an overseas HSR, the construction cost of overseas projects includes expenses in subgrade engineering, bridge engineering, tunnel engineering, track engineering, station engineering, ``four-power engineering'', and environmental protection engineering. Through an analysis of relevant literature and combined with expert opinions, this article initially selected the following engineering characteristics as indicators: subgrade proportion, subgrade width, embankment proportion, and cutting proportion in subgrade engineering; proportion of extra-large bridges and proportion of large-to-medium-sized bridges in bridge engineering, proportion of large tunnels and proportion of extra-large tunnels in tunnel engineering; track type and number of main tracks in track engineering <sup>[10]</sup>; station building area in transmission station engineering <sup>[11]</sup>. This article does not consider them because the impact of ``four power'' engineering and environmental protection engineering is relatively small.

Fig. 1. Fishbone diagram of overseas HSR construction investment amount based on external factor influence.

2.1.2 Selection of Engineering Characteristic Index System during Operational Phase

Similar to the selection method of the engineering characteristic index system during the construction investment phase, this article combined literature and data analysis and considered external influencing factors, believing that the international environment, construction conditions, operational characteristics ^[12], and technical standards have significant effects on the investment during the operational phase ^[13]. Fig. 2 shows the fishbone diagram analysis results.

Based on the PBS decomposition of overseas HSR and from the perspective of the projects themselves, the preliminarily selected investment characteristics for the operation phase include the width and protection type of the subgrade, the proportion of large and medium-sized bridges, the proportion of extra-large bridges, the proportion of medium and long tunnels, the proportion of extra-long tunnels, the type of railway track, and the number of main tracks.

Fig. 2. Fishbone diagram of the investment amount in the operation stage of overseas HSR based on external factors.

2.1.3 Determination of Investment Characteristics in the Construction and Operation Phases of Overseas HSR

The questionnaire designed based on the Likert's five-point scale was distributed, and SPSS software was used to analyze and process the questionnaire data. The characteristics met the reliability and validity requirements. Each characteristic with a score of four or above, rated as important by more than 90% of the experts, was determined as a principle for determining engineering characteristics. Finally, the investment characteristics for the construction and operation phases of overseas HSR were determined, as listed in Tables 1 and 2.

2.3.1 Adaptability of PSO-BPNN Investment Decision Model

Chinese international contractors often use the traditional norm unit price method for investment estimation in overseas HSR construction. Although improvements have been made to adapt to local conditions, weaknesses, such as lagging, linearity, and simple fitting, still exist, and the estimation calculation has a heavy burden and a low accuracy. Therefore, this paper proposed the PSO-BPNN model considering the randomness, complexity, and non-linearity of the investment system of overseas HSR construction projects. The PSO algorithm overcomes the defects of traditional clustering methods, such as dependence on the selection of initial values and easy fall into local extreme values. It helps improve the reliability of selecting similar cases before prediction, improving the accuracy of the prediction results. The cooperation with the BPNN algorithm greatly reduces the workload of investment prediction and improves the calculation efficiency and accuracy.

2.3.2 Select Similar Cases using PSO Clustering Analysis

The relationship between the cases needs to be analyzed first when performing cluster analysis on the samples. The K-means clustering method was chosen to establish the model, and the Euclidean distance was used to operate between the samples and the cluster centers, as shown in Eq. (1). Similarity $s=1-d\left(xy\right)$.

where $n$ represents the number of samples; $x_{i}$ represents the $\mathrm{i}$$^{\mathrm{th}}$ sample; $m$ represents the number of data feature dimensions included in each sample.

The results of traditional K-means clustering are influenced strongly by the initial values and are prone to fall into local extrema, which may not yield the optimal clustering ^[14]. In contrast, the PSO algorithm uses the population as the base information and finds the optimal solution through constant iterations. During each iteration, $x_{i}$ and $v_{i}$ are updated according to Eqs. (2) and (3), and the fitness function is used to calculate the fitness value and update individual and global best values. The algorithm terminates when the maximum number of iterations is reached, or the fitness value falls below the preset threshold.

where $v_{i}$ represents the velocity of the $\mathrm{i}$$^{\mathrm{th}}$ particle, $p_{i}$ and $g_{i}$ represents the $\mathrm{i}$$^{\mathrm{th}}$ individual and global extrema, respectively; $\mathrm{w}$ is the inertia weight factor; $\mathrm{q}$ represents the current iteration number; $c_{1}$ and $c_{2}$ are the acceleration coefficients; $r_{1}$ and $r_{2}$ are random numbers between 0 and 1.

2.3.3 Prediction of Construction Stage Investment based on BPNN Model

The BPNN is a multi-layer feedforward neural network with the characteristics of simple structure, multiple adjustable parameters, and good operability. The network learns through the forward propagation of signals and backward propagation of errors and takes quantized engineering feature data as the model input and investment objectives as the output. When the actual output value does not match the expected output value, the process of error backpropagation is initiated, and the output error value is reduced continuously through repeated training, resulting in an accurate prediction target.

2.3.4 Prediction of Operational Stage Investment based on FIS

This study focuses on the full-life-cycle investment of overseas HSR. One hundred years was selected as the analysis and calculation period of the full life cycle of overseas HSR based on the "Railway Engineering Construction Standards" released by the National Railway Administration because most overseas HSR systems are still in operation. The BPNN was used to predict the first 20 years, while the FIS was used to determine the trend of operation investment for the next 80 years based on expert experience in the case of no historical data available. The FIS is a system based on the theory of fuzzy sets and fuzzy inference methods that can process fuzzy information. Its establishment involves five steps ^[15]: (1) fuzzification of the exact quantity; (2) generation of "if-then" conditional rules; (3) inference; (4) compositional operation; (5) defuzzification.

3.2.1 Selection of Similar Cases

The engineering characteristics of the construction phase investment of the three alternative routes were quantified using the quantification standards; Table 8 lists the results.

This article took the L1 line of the LY HSR as an example to demonstrate owing to the similarity in the prediction approach. The PSO clustering method was used to extract cases with high similarity to the L1 line from the database of construction stage investments of 32 overseas HSR. MATLAB was used for PSO clustering analysis, and the optimal clustering was obtained when the number of clusters was two, and the maximum iteration number was 500. Fig. 5 shows the clustering results and centers. Based on PSO clustering, 18 similar projects were extracted from the database, of which 16 projects were used as training samples; the other two projects were used as validation samples.

Similarly, through the Matlab software, 20 engineering cases similar to the investment plan of the L1 line of the LY HSR were obtained.

Fig. 5. Clustering results.

Fig. 6. BPNN training process.

Fig. 7. Convergence process of BPNN.

3.2.2 Investment Prediction of L1 Line Construction Phase based on BPNN

(1) Building BPNN

The BPNN model was used to predict the engineering investment objective of the L1 line. The Sigmoid function was selected. There were 19 input nodes and one output node, representing 19 engineering feature indicators and the investment amount in the construction phase. The number of hidden layer nodes was 2${\times}$19+1=39.

(2) Running BPNN

Based on the data from the 18 similar projects, 16 of them were used as training samples, and the remaining two were used for testing. The allowable error was 10$^{-10}$, and the maximum iteration number was 500. Figs. 6 and 7 show the training process of the data and the convergence process of the data, respectively.

(3) Validation of BPNN

The output results of the model were not fixed because of the randomness of the initial weights and thresholds. Therefore, this paper reduced the error by taking the average value of 15 sets of model training, and the results are presented in Table 9. A comparison of the verification samples with the actual values showed that the relative errors were ${-}$1.85% and 1.45%, respectively, within the allowable range of ${\pm}$3%. Similarly, this paper predicted the 20-year operating investment of the L1 line scheme based on 20 similar engineering cases in the database, and the predicted operating investment for the first 20 years of the scheme was 7.45 million pounds per kilometer, with a reasonable error.

(4) Comparison with actual investment

The prediction results of the model showed that the construction phase investment of the L1 line was 53 million pounds per kilometer, the operating investment for the first 20 years was 7.45 million pounds per kilometer, and the annual operating investment was 373,500 pounds per kilometer. Among the actual data collected, the actual investment for the construction phase of the L1 line was 54 million pounds per kilometer, with an error of 1.85%. Because its operation started on November 14, 2007, the total actual investment during the operation phase was approximately 4.9 million pounds per kilometer, and the annual operating investment was 376,900 pounds per kilometer, with an error of 0.64% compared to the model calculation. The errors were within the allowable range of ${\pm}$3%. The comparison revealed reasonable prediction results of the model.

This study analyzed the data from both the BPNN method and actual engineering aspects. First, the prediction accuracy of the BPNN was tested again using two sets of verification samples based on training 16 samples to determine the feasibility of the investment decision-making model. This demonstrated the accuracy of the prediction from the method level. Second, the predicted values of the construction and operating investments of the LY HSR in the first 20 years were compared with the actual values. The errors were within the reasonable range of ${\pm}$3%, which verified the accuracy of the prediction from the actual engineering perspective.

3.2.3 Prediction of the L1 Line Operating Investment based on Fuzzy Inference

Owing to the lack of historical data on HSR operation, this paper predicted the future operating trend of the L1 line in the future 20-100 years, i.e., the ratio of the annual operating investment amount of the latter 80 years to that of the first 20 years, based on expert experience using the fuzzy inference method.

First, the fuzzy indicators affecting the operating investment for the latter 80 years were inferred to be environmental factors, operational characteristic factors, and scale factors, and the membership functions of the trend of operating investment changes were obtained through expert experience. Figs. 8-12 present the FIS

structure and membership function graphs. Fig. 8 shows the set fuzzy system. The input is the environmental factors, operational characteristics factors, and scale factors. The output is the change ratio of operational investment. Figs. 9-12 present the fuzzy setting of the system. The triangular membership function (trimf) was selected as the membership function.

Fuzzy inference relationships were established between various engineering characteristics based on expert experience, and Fig. 13 shows the trend of operating investment changes.

According to expert experience, an interval of (0, 10) was used as a standard to evaluate the environmental factors, operating characteristic factors, and scale factors. The environmental factors were rated as two owing to the stable political environment and geological conditions and relatively low investment costs of the L1 line. As the number of years of opening to traffic and the average daily passenger flow was also low, the operating characteristic factors were rated as two. The route length and proportion of bridges and tunnels were relatively high, so it was rated as eight. The input variable for environmental factors, operating characteristic factors, and scale factors were (2, 2, 8), and Fig. 14 shows the results.

According to Fig. 14, the ratio of annual operating investment in the latter 80 years to that in the first 20 years was five, so the annual operating investment in the latter 80 years was 47.36 million pounds per kilometer. Based on the future operating characteristics of the L1 line and research on the overseas HSR operating investment in academia and industry, academic and industry studies on overseas HSR operating investments showed that it was feasible to forecast the operating investments using this method.

Similarly, the full-life-cycle investments of the L2 and L3 lines were predicted, as shown in Table 10. According to the principle of optimal total investment, alternative schemes were compared, and the results showed that the L1 line scheme had the lowest full-life-cycle investment, so the L1 line scheme was finally determined as the optimal investment plan.

In practical cases, the traditional quota calculation method was used to predict and verify the construction-period investment of different investment schemes multiple times. The operating-period investment was predicted and demonstrated continuously in combination with the actual operation of the LY HSR and the opinions of the operating party. Eventually, the L1 line scheme was selected, which was consistent with the prediction result of this paper. This result demonstrated the rationality of the model established in this paper.

Fig. 8. FIS structure diagram.

Fig. 9. Membership function graph of environmental factors.

Fig. 10. Membership function graph of operating characteristic factors.

Fig. 11. Membership function graph of scale factors.

Fig. 12. Membership function graph of operation investment amount variation trend.

Fig. 13. Display of fuzzy inference rules in FIS.

Fig. 14. Investment trend during the operation phase of the L1 line.