Evaluation of Urban Rail Transit System Planning Based on Integrated Empowerment Method and Matter-Element Model

Abstract

Urban rail transit system planning is significant for alleviating traffic congestion and optimizing spatial resource allocation in cities with scarce land resources. However, the long period of rail transit construction, large-scale investment, and its planning involve a variety of factors, which require scientific and reasonable evaluation methods to ensure that its construction can realize the expected economic and social benefits. To solve this problem, this study first establishes an appropriate evaluation system by selecting suitable evaluation indicators. Then, the comprehensive assignment method combining the ordinal relationship method (G1 method) and the improved entropy weight method is applied to assign weights to the indicators in the evaluation system, and the correlation degree is calculated by combining with the matter-element model for evaluating the planning scheme of the urban rail transit system. Finally, the urban rail transit system planning scheme of Zhengzhou City is verified by example. The results show that the proposed method can balance the practical significance and dynamics of the evaluation indices, evaluate the importance of each index more objectively, and provide methodological support for dynamic decision-making in rail transportation planning in the context of a smart city, which is of guiding significance for the sustainable development of the city.

1. Introduction

The earliest definition of urban rail transit appeared in the American Academy of Engineering. It defines urban rail transit as using surface, underground, or elevated facilities, independent of other modes of transportation on the surface, using specific electric means of transportation, and operating on designated routes. Moreover, it is a public transportation system that transports passengers from the city and its immediate vicinity in large numbers and quickly using high-density frequency [1]. In the context of accelerated urbanization and rigid constraints on land resources, the rail transit system can achieve the dual sustainable development goals of traffic congestion management and intensive land use.

Efficient and reasonable urban rail transit cannot be separated from a scientific and appropriate urban rail transit planning scheme. Through the integrated evaluation of urban rail transit planning schemes, it can systematically optimize the resource allocation and enhance the scientificity of planning through the dynamic analysis of a multi-dimensional index system and the scientific integration of subjective and objective weights. While reducing construction and operation costs and avoiding network reliability risks, it promotes green, low-carbon development and social equity. Traditional evaluation methods for urban rail transit planning usually rely on a single assignment model and static index system, which have limitations such as subjective—objective split, insufficient dynamic adaptability, and difficulty in quantifying complex contradictions. Therefore, this study aims to propose a comprehensive evaluation method combining the integrated empowerment method and matter-element model, which takes into account the importance of expert experience and data objectivity and solves the problems of the irrational weight distribution of evaluation indices and insufficient dynamic adaptability.

2. Research Status

With the growth of urban rail transit systems, people are paying more and more attention to the planning and evaluation of urban rail transit. In 1882, in the urban renewal program of Madrid, Soria [2] proposed three methods of building rail transit according to the characteristics of Madrid City. The three methods were building underground, leveling at ground level, and installing viaducts. This changed the prevailing idea of building rail transit and had a positive impact on the world’s urban rail transit development. In 1977, Florin [3] and others established a model for traffic allocation, distribution, and mode choice. It provided new considerations for urban rail transit planning and made urban rail transit more complete. In 1982, Corrente built the “Covered Path Model” and the “Sub-Short Circuit Model” on the unit-importance-based rail transit network planning model proposed by Church [4]. Meanwhile, Severo [5] conducted a systematic study on the factors affecting the planning of urban rail transit systems. After the study, it was found that the exploitation and utilization of land have a significant impact on the urban rail transit system. For example, the location, exploitation, and utilization of urban residential and commercial areas have an important impact on the attendance rate of urban rail transit. According to the Severo study, an increase of ten to twenty units per acre increases Metro’s attendance by about 4%. At this point, if the quality of service is also enhanced, attendance will grow even further. The above research has pointed out the direction for the planning of the urban rail transit system step by step, and, at the same time, it has also made the urban rail transit system more complete. However, all the above studies have considered a certain aspect of the factors and studied these factors in a static manner, which cannot be analyzed in a dynamic study. Thus, in 1984, the U.S. Urban Rail Transit Administration [6] proposed the cost–effect analysis method. The analysis of cost and effect is carried out at the same time, and the two correspond to each other, dynamically controlled and analyzed in the actual situation. In 1998, Miao Yanying [7] categorized China’s urban rail transit into six forms: single line shape, switching line shape, cross shape, triangle, radial, and grid shape. The study of urban rail transit in China through these six specific forms was in line with the current situation in China at the time. In 1999, because there was no specific and systematic theoretical research program for urban rail transit at that time, Wang Zhongqiang and Gao Shilian [8] made a systematic elaboration of the network and scale of rail transit according to the situation at that time. It is of guiding significance for the future construction of rail transit in China. In 2000, Gu Baonan and Fang Jingjing analyzed the urban rail transit of Guangzhou and Nanjing and the evaluation index system established by the predecessors at that time. It also carried out a comparative study, analyzed the advantages and disadvantages of various evaluation methods, and put forward an improvement plan.

In the 21st century, research on transportation systems has become more and more diversified, mainly focusing on the planning and design of urban rail transit, operation and management, and passenger flow prediction. Planning and design: In 2018, Cai et al. [9] studied the mechanistic model of the road network to solve the problems in the road network structure in the urban rail transit hub area. Then, based on the TOD (Transit-Oriented Development) model and the land utilization rate of the hub area, the influence range of the road network in the hub area of Tianjin Roundabout was determined, and the corresponding planning and design were proposed. A suitable road network model for Tianjin’s urban rail transit is finally summarized, which can alleviate the traffic pressure in the central city and provide a reference for the promotion of an urban comprehensive transportation network. In 2022, Ma et al. [10], based on deep learning, constructed an algorithm for the optimization of urban rail transit line network planning. Firstly, according to the urban layout and urban planning, a suitable form of rail transit line network is required. Then, we divide the types of urban rail transit stations according to the function to realize the optimization of the urban rail transit line network routes. Finally, the effectiveness of the proposed path planning optimization algorithm is verified through experiments. In 2023, Yin et al. [11] proposed a cost-oriented station spacing optimization model for factors affecting the station spacing and arrangement at Shanghai Pudong International Airport. It combines urban rail transit planning with the HDBSCAN algorithm, which provides a scientific method for local governments to plan rail transit lines. In 2024, Huang [12] et al. designed a hybrid evolutionary algorithm with chromosomes having a two-layer coding structure using the Niched Pareto genetic algorithm as the main algorithmic framework to solve the constructed two-stage robust optimization-based urban rail transit line network planning model. The validity of the model and algorithm is confirmed by example validation, and the constructed model improves the overall efficiency and resilience of the rail transit network. In 2025, Du [13] et al. proposed a two-tier algorithm based on agent-assisted fish migration optimization (SA-FMO) to solve the urban rail transit train path optimization problem. The algorithm reduces the computational cost of high-dimensional complex problems through global and local agent models, and significantly outperforms algorithms such as SHPSO and RBFPSO in 50-dimensional benchmark tests. In the real case, SA-FMO optimizes the departure frequency and turnaround stations of long- and short-distance trains on Suzhou Metro Line 2, and successfully reduces the passenger waiting time to 5.6 × 10⁴ s, a significant improvement in efficiency compared with the traditional methods. Operation and management: In 2019, Chen Jiaxu [14] established a safety evaluation index system and a safety evaluation based on the TOPSIS method for urban rail transit surrounding urban rail transit safety issues. This provides the relevant departments and urban rail transit safety management personnel with useful decision-making support and has a certain reference significance for improving the level of urban rail transit safety management. In 2019, Annunziata Esposito Amideo et al. [15] defined a weighted node assessment index for node importance and introduced a new two-tier multi-criteria optimization model. Finally, two kinds of methods, a sequence method based on a single asset indicator and an integrated method based on an RFP (Railway Fortification Problem), are used to assess the safety and protection strategy for the City of London’s rail transit system. In 2021, Wu et al. [16] analyzed the reliability performance of the rail transit network and the traffic-weighted network from the perspective of a complex network. Then, a dynamic model of the Shanghai urban rail transit line network based on the coupled tug model was established. The cascading failure process of network nodes caused by different damage schemes was simulated in conjunction with Shanghai’s urban rail transit system to measure the change in passenger-flow-weighted network reliability during the process. In 2023, Ma et al. [17] constructed a quality risk evaluation index system for urban subway construction by selecting 20 index factors from the five dimensions of “man–machine–material–method–environment”. Based on the AHP-FCE method, the quality risk of metro construction is comprehensively evaluated, and the evaluation model is applied to engineering practice for verification. This model provides a new idea for metro project construction quality risk assessment. In 2024, Yin [18] et al. constructed a two-level fuzzy evaluation model containing indicators of immunity, redundancy, and resource deployment capability. Combining the G1 weighting method and Anylogic simulation technology, the model quantitatively analyzes the resilience performance of the interchange station under daily operation, heavy passenger flow, and emergencies. The feasibility of the model is verified by taking the Xi’an North Street Interchange as an example. The model provides theoretical support for improving the anti-interference ability of the interchange station and guaranteeing the sustainability of the rail transportation network. Passenger flow prediction: In 2022, Dung David Chu Wang and Weiya Chen [19] constructed an urban rail transit model for daily and weekly passenger demand forecasting based on historical data. The optimal prediction results of each model were output with MAE, RMSE, and MSE as the performance evaluation indices of the model. In 2024, Dong [20] et al. proposed a deep-learning model incorporating spatio-temporal network features, namely, the spatio-temporal network long- and short-term memory model (TSN-LSTM), for short-term passenger flow prediction in urban rail transit. The model integrates station passenger flow trends through k-means clustering, combines temporal features, network features, and spatial features, and constructs a spatio-temporal network matrix. In 2024, Li [21] et al. proposed a long-term passenger flow prediction method that fuses complex network and Informer models, quantifying spatial dependence by analyzing station connectivity and centrality, and reducing computational complexity by using Informer’s sparse self-attention mechanism. Taking Shanghai metro swipe card data as an example, the study finds that the established model is superior to the traditional ARIMA, LSTM, and Transformer models. In 2025, Zhang [22] et al. proposed a deep-learning-based multi-step passenger flow prediction model, the Event Flow Converter Network (EF-former). The model utilizes normal and extra outflows from target stations to predict actual outflows, and also combines PIAM and MSC-MSA to extract the global and local temporal dependencies of the passenger flow data to accurately capture the patterns and trends of passenger flows during large-scale events for multi-step prediction. In 2025, Xing [23] et al. proposed a spatio-temporal fusion network for predicting passenger flow from origin to destination (OD flow) in urban rail transit trips. By constructing topological maps, static inflow–outflow similarity maps, dynamic passenger flow distribution similarity maps, and dynamic spatial correlation maps, combined with a graph convolution network (GCN) and long short-term memory network (LSTM), the complex spatio-temporal dependencies between stations are captured. The experiments are based on real data from Shanghai and Hangzhou, and the results show that the model improves by more than 4% in MAE and other metrics compared with the existing methods, and introduces train stop control strategies based on the prediction results to optimize the operation (See

Table 1. Summary of urban rail transit research directions.

Research Dimension	Author (Year)	Research Content
Planning and design	Cai et al. (2018) [9]	Planning and designing the road network of Tianjin Roundabout Hub Area based on TOD mode and land utilization rate of the hub area
	Ma et al. (2022) [10]	Establishment of a deep-learning-based optimization algorithm for urban rail transit line network planning
	Yin et al. (2023) [11]	Cost-oriented station spacing optimization model based on HDBSCAN algorithm
	Huang et al. (2024) [12]	A hybrid evolutionary algorithm with chromosomes having a two-layer coding structure is designed, and a line network planning model based on two-stage robust optimization is established
	Du et al. (2025) [13]	A two-layer algorithm based on SA-FMO is proposed to solve the train path optimization problem for urban rail transit
Operation and management	Chen (2019) [14]	Evaluation of safety of urban rail transit using TOPSIS methodology
	Annunziata Esposito Amideo et al. (2019) [15]	A sequential approach using single asset metrics and a combination of RFP were used to assess the City of London’s rail transit system safety and protection strategy
	Wu et al. (2021) [16]	Establishment of a dynamic model of Shanghai urban rail transit network based on a coupled tug model
	Ma et al. (2023) [17]	Comprehensive evaluation of metro project construction quality risk based on AHP-FCE method
	Yin et al. (2025) [18]	A two-level fuzzy evaluation model was established by combining the G1 weighting method with Anylogic simulation technique
Passenger flow prediction	Dung David Chu Wang et al. (2025) [19]	Constructed a model for forecasting daily and weekly passenger demand for urban rail transit
	Dong et al. (2024) [20]	A TSN-LSTM model was developed to predict short-duration passenger flow in urban rail transit
	Li et al. (2024) [21]	A proposed method for long-term passenger flow prediction by fusing complex networks and Informer models
	Zhang et al. (2025) [22]	A multi-step passenger flow prediction model based on deep learning is proposed
	Xing et al. (2025) [23]	A spatio-temporal fusion network for predicting OD flows in urban rail transit trips established

).

Traditional methods of evaluating urban rail transit are mostly single, static evaluation indicators, which cannot adapt to the dynamic changes in urban development. Moreover, evaluation methods mostly rely on a single assignment method, which easily leads to difficulty integrating subjective and objective information. In addition, most of the current evaluations only focus on a single specific aspect, such as only focusing on efficiency, safety, etc., and lack a systematic evaluation framework. This paper combines the matter-element model with the integrated empowerment method to overcome the problems of subjective–objective weight fragmentation and insufficient dynamic adaptability in the traditional evaluation model and proposes an evaluation framework compatible with dynamic demand and multi-objective conflict, which makes up for the limitations of the existing studies that mostly focus on a single dimension. The specific research content is as follows: Firstly, analyze the influencing factors of the urban rail transit system, select suitable evaluation indices, and establish a proper evaluation index system. Then, the subjective and objective weights are calculated by the G1 method and the improved entropy weight method, respectively, and then the weighted linear combination is used to calculate the comprehensive weights and evaluate the planning scheme of the urban rail transit system by combining with the matter-element model. Finally, the proposed method is applied to evaluate the planning scheme of the rail transit system with examples to verify the scientificity and feasibility of the proposed method, and improvement measures and suggestions are proposed for the examples according to the evaluation results.

3. Methodology

The evaluation of urban rail transit system planning involves a number of indicators, and it is susceptible to uncertainties. The integrated empowerment method–matter-element model combines the G1 method, the improved entropy method, and the matter-element model. It can effectively deal with the correlation between the indicators and the weight distribution problem, and can more comprehensively assess the comprehensive benefits of the system, so it has a certain advantage in the planning and evaluation of the urban rail transit system. The object-element model is a combination of qualitative and quantitative evaluation model [24,25], which can be used to quantitatively evaluate the urban rail transit network line planning. The G1 method is a subjective empowerment method for multi-level evaluation. Compared with the analytic hierarchy process, there is no need for a consistency test, and it can avoid the problem of inconsistency in the judgment matrix caused by the excessive number of evaluation indicators. The entropy weight method is an objective assignment method, which can better reflect the distribution of data and can more accurately assess the weights of indicators [26,27]. The evaluation method used in this paper is based on the matter-element model and correlation function theory. Using entropy theory to calculate the weight of evaluation indices, the G1 method and improved object-element entropy weight model for the evaluation of urban rail transportation system planning is established.

3.1. Establishment of the Evaluation System

The transportation situation of a city determines the level of development of that city. The problem of traffic congestion needs to be dealt with to realize the rapid economic development of the city. Therefore, the first and foremost solution for solving the traffic problem of a city is to build rail transportation. Generally speaking, macro-level factors influencing the planning of urban rail transit systems include national development orientation, the degree of economic and social development, the level of development of rail transit science and technology, and environmental protection. The micro-level factors include the city’s size and development plan, the city’s transportation demand, the characteristics of the residents’ travel, and the land utilization situation.

The evaluation indicators for existing cities in Table 2 are mostly designed based on the layout and economic conditions of specific cities (e.g., Beijing focuses on direct access to transportation hubs, environmental protection, and historical and cultural preservation, while other cities do not adequately cover similar elements), which results in a limited scope of application and makes it difficult to extend them to cities at different levels of development. Some cities (e.g., Wuhan and Nanjing) have not included the implementability of the project, the degree of difficulty of the project, and the social impacts (e.g., residents’ opinions and safety) in the evaluation framework, which may pose hidden dangers for subsequent construction. In addition, the assessment of travel efficiency is still weak. Although individual cities (e.g., Beijing and Nanjing) are involved, there is a lack of systematic quantitative indicators, making it difficult to comprehensively measure the effect of rail transit on the improvement of urban transportation problems. Early indicator systems generally ignored environmental protection requirements (e.g., carbon emissions and ecological impact) and fell short of advanced international practices (e.g., Japan’s explicit requirements for environmental protection) [28]. These shortcomings reflect the limitations of the current evaluation system in terms of universality, comprehensiveness, and foresight, which need to be integrated with international experience and dynamically optimized to enhance its scientificity and adaptability.

It is committed to optimizing the urban rail transit system by focusing on travel efficiency, construction feasibility, and engineering evaluation. It adheres to the concepts of being comprehensive, harmonious, systematic, and efficient, as well as the concept of sustainable green transportation. Based on the “Bus City Assessment and Evaluation Indicator System [29]” formulated by China’s Ministry of Transportation and Communications, the internationally renowned “Transportation Capacity and Service Quality Manual [30]”, and the “2022 Analysis Report on Evaluation Indicators for the Effectiveness of Urban Rail Transit [31]” issued by the China Railway Transportation Association, the deficiencies of the existing evaluation system are considered. And, combined with the influencing factors affecting the planning of urban rail transit systems, the evaluation index system shown in Figure 1 was established following the principles of scientificity, comparability, independence, and perfection.

Table 2. Evaluation indicators selected for urban rail transit system planning domestically and internationally [32].

City	Evaluation Indicators	Category
Guangzhou	Total length of line network Density of line network in the central area Non-linear coefficient Number of transfer nodes Rate of area covered Connection with large passenger distribution points	Wire network structure
	Transfer coefficient Line network load intensity Daily passenger volume	Operational analysis
	Difficulty of project implementation Implementability of the project Engineering construction cycle	Construction analysis
	Development value of land along the route Savings in average travel time Adaptation of line network development	Social benefit
Beijing	Fringe group Accessibility to external transportation hubs and major passenger distribution points	Strategic development
	Area coverage area Line density by region Directness of travel and ease of transfer	Wire network structure
	Passenger transportation effectiveness Total time of travel Ease of travel	Service level
	Difficulty of the project Interface with existing wired networks Construction cost	Project construction
Nanjing	Passenger flow Saving running time Interchange volume	Traffic function
	Construction cost Operating income Operating cost	Economic benefit
	Line efficiency Equalization of lines Adaptation of lines	Operational efficiency
Wuhan	Harmonization with the urban layout Coordination with urban transportation facilities Regularization of the construction of field station facilities	Infrastructure
	Wireline coverage Savings in travel time Convenience of traveling	Quality of service
	Implementability of the program Difficulty of construction Investment budget	Concrete construction
Japan	People’s lives Regional economy Regional security	Impact evaluation
	Implementability of the project Feasibility of the project Technical level	Evaluation of implementation
	Relative efficiency Project benefit	Earnings evaluation

Table 2. Evaluation indicators selected for urban rail transit system planning domestically and internationally [32].

City	Evaluation Indicators	Category
Guangzhou	Total length of line network Density of line network in the central area Non-linear coefficient Number of transfer nodes Rate of area covered Connection with large passenger distribution points	Wire network structure
	Transfer coefficient Line network load intensity Daily passenger volume	Operational analysis
	Difficulty of project implementation Implementability of the project Engineering construction cycle	Construction analysis
	Development value of land along the route Savings in average travel time Adaptation of line network development	Social benefit
Beijing	Fringe group Accessibility to external transportation hubs and major passenger distribution points	Strategic development
	Area coverage area Line density by region Directness of travel and ease of transfer	Wire network structure
	Passenger transportation effectiveness Total time of travel Ease of travel	Service level
	Difficulty of the project Interface with existing wired networks Construction cost	Project construction
Nanjing	Passenger flow Saving running time Interchange volume	Traffic function
	Construction cost Operating income Operating cost	Economic benefit
	Line efficiency Equalization of lines Adaptation of lines	Operational efficiency
Wuhan	Harmonization with the urban layout Coordination with urban transportation facilities Regularization of the construction of field station facilities	Infrastructure
	Wireline coverage Savings in travel time Convenience of traveling	Quality of service
	Implementability of the program Difficulty of construction Investment budget	Concrete construction
Japan	People’s lives Regional economy Regional security	Impact evaluation
	Implementability of the project Feasibility of the project Technical level	Evaluation of implementation
	Relative efficiency Project benefit	Earnings evaluation

3.2. Determination of Evaluation Indicator Weights

The methods of calculating weights can be categorized into the subjective empowerment method, the objective empowerment method, and the integrated empowerment method. Integrated empowerment method is a method that combines the subjective empowerment method and the objective empowerment method. The methods commonly used in urban rail transportation planning are shown in

Table 3. Commonly used methods of empowerment [33].

Typology	Method	Characteristic
Subjective empowerment method	Analytic hierarchy process (AHP)	Structured decision-making; highly interpretable; highly subjective; requires multiple consistency tests and is inefficient when there are too many indicators
	Ordinal relationship method (G1)	No consistency test required; highly adaptable; rely on expert ranking; not utilizing objective data to dynamically adjust weights
	Expert scoring method	Direct prioritization of indicators through two-by-two comparisons without complex matrix calculations; relative evaluation only; suitable for evaluation systems with a small number of indicators
	Logogram method	Direct aggregation of expert opinion, applicable to urgent or ad hoc evaluations; high degree of subjectivity and arbitrariness; lack of data support
Objective empowerment method	Entropy weight method	Strong mathematical theoretical basis; relies on data distribution, high objectivity; sensitive to outliers
	Principal component analysis (PCA)	Eliminate indicator redundancy; suitable for high-dimensional data; physical significance of principal components unclear; suitable for scenarios where evaluation indicators are highly correlated
	Coefficient of variation method	Reflects the discriminatory power of the indicator; ignores independence of the indicator; does not reflect the experience of experts

below.

3.2.1. Subjective Weights

According to Figure 1, there are five first-level indicators in the evaluation index system, and each first-level indicator corresponds to multiple second-level indicators. Combined with Table 3, the ordinal relationship method (G1 method) is used to calculate the subjective weights, which does not require a consistency test compared with the analytic hierarchy process. The calculation process is as follows:

(1)

Determine the ranking relationship between indicators and rank the importance of each indicator in the evaluation indicator system according to the importance of each evaluation indicator, taking into account the recommendations of experts:

$$D_1>D_2>\cdots D_n$$

(1)

where n is the total number of indicators.

(2)

The relative importance between neighboring indicators is judged by experts:

$$R_i={\displaystyle \frac{D_{k-1}}{D_k}} k=2,3\cdots ,n$$

(2)

(3)

Calculate the weighting factors:

$${\omega }_n^1=(1+{\displaystyle \sum _{k=2}^n{\displaystyle \sum _{j=k}^n{R}_j}}{)}^{-1}$$

(3)

${\omega }_n^1$ is the weight coefficient of the nth indicator, and the weights of the other indicators can be derived from ${\omega }_j^1=R_{j+1}{\omega }_{j+1}^1$ sequentially.

3.2.2. Objective Weights

Combined with Table 3, the entropy weight method was chosen to calculate the objective weights. The entropy weight method can effectively reduce the influence of subjective judgment on weight allocation, thus making the comprehensive evaluation results more objective and reasonable. In information theory, the degree of disorganization of a system’s information is reflected by the value of entropy. The smaller the value of entropy, the less disordered the system is. Thus, information entropy can be used to assess the degree of orderliness of the acquired system information and its utility [34,35]. However, the traditional entropy method has certain shortcomings. That is, when the entropy value of an indicator tends toward 1, a small change between each entropy value causes a multiplicative change between the entropy weights of different indicators. Therefore, the entropy weight method model improved by Lu Dongyao [36] is applied to determine the weights. The improved entropy weight method is modeled as follows:

$${\omega }_j^{\prime }={\displaystyle \frac{{\displaystyle \sum _{k=1}^m{E}_k+2-3E_j}}{{\displaystyle \sum _{l=1}^m({\displaystyle \sum _{k=1}^m{E}_k+2-3E_l})}}} j=1, 2, \cdots , m$$

(4)

Assuming that a and b are any two indicators in the indicator system, the weight coefficients are ${\omega }_a^{\prime }$ and ${\omega }_b^{\prime }$, and the entropy values are E_a and E_b, respectively, which can be obtained from Equation (4):

$$\left|{\omega }_a^{\prime }-{\omega }_b^{\prime }\right|=\left|{\displaystyle \frac{{\displaystyle \sum _{k=1}^m{E}_k+2-3E_a}}{{\displaystyle \sum _{l=1}^m\left({\displaystyle \sum _{k=1}^m{E}_k+2-3E_l}\right)}}}-{\displaystyle \frac{{\displaystyle \sum _{k=1}^m{E}_k+2-3E_b}}{{\displaystyle \sum _{l=1}^m\left({\displaystyle \sum _{k=1}^m{E}_k+2-3E_l}\right)}}}\right|=\left|{\displaystyle \frac{3\left(E_a-E_b\right)}{{\displaystyle \sum _{l=1}^m\left({\displaystyle \sum _{k=1}^m{E}_k+2-3E_l}\right)}}}\right|\left(a,b=1,2,\dots ,m.a\ne b\right)$$

(5)

When the entropy values of a and b are infinitely close to each other, i.e., when $\left|E_a-E_b\right|=\epsilon \left(\epsilon \to 0\right)$, $\left|{\omega }_a^{\prime }-{\omega }_b^{\prime }\right|=\left|{\displaystyle \frac{3\epsilon }{{\displaystyle \sum _{l=1}^m\left({\displaystyle \sum _{k=1}^m{E}_k+2-3E_l}\right)}}}\right|$. Since ε is a tiny variable and ${\displaystyle \sum _{l=1}^m\left({\displaystyle \sum _{k=1}^m{E}_k+2-3E_l}\right)}$ is determined, the magnitude of change of $\left|{\omega }_a^{\prime }-{\omega }_b^{\prime }\right|$ is consistent with ε. Therefore, when the improved entropy weight method is applied to calculate the weights, even when the entropy value of the evaluation indices is close to 1 and a tiny change occurs, the corresponding weights will not change by multiples.

3.2.3. Integrated Weights

To make the assessment model reflect both the expert’s judgment of the importance of the indicator and the importance of the data information objectively, many scholars often use a weighted linear combination to determine the comprehensive weights as follows:

$${\omega }_j=\alpha {\omega }_j^1+(1-\alpha ){\omega }_j^2$$

(6)

${\omega }_j^1,{\omega }_j^2$ are the subjective and objective weights of the jth indicator, respectively, and α is the preference coefficient for subjective weights. In this study, α is taken as 0.5, which means that subjective and objective weights are considered equally important.

3.3. Constructing Matter-Element Models

(1)

Evaluation of object element and object-element matrix

Setting the thing as N, its features as C, and the quantities corresponding to each feature as X, the object-element model uses these three elements to describe the thing and forms them into the basic elements of an ordered triad R = (N, C, X), referring to R as the object element [37]. The planning level of the studied urban rail transit system is N. According to the established evaluation index system, there are 20 features and corresponding eigenvalues in the established evaluation model for urban rail transit system planning. Then, the object-element matrix for urban rail transit planning evaluation is as follows:

$$R=(N,C,X)=\left[\begin{array}{ccc}N& C_1& X_1\\ & C_2& X_2\\ & ⋮& ⋮\\ & C_n& X_n\end{array}\right]$$

(7)

If there are m things and these m objects have common features, suppose these m things have n common features, ${\mathrm{C}}_1,{\mathrm{C}}_2,{\mathrm{C}}_3,{\mathrm{C}}_4,\dots ,{\mathrm{C}}_{\mathrm{n}}$, respectively. The quantity value corresponding to the characteristic ${\mathrm{C}}_{\mathrm{i}}$ of thing ${\mathrm{N}}_{\mathrm{j}}$ is ${\mathrm{X}}_{\mathrm{i}\mathrm{j}}$; then, the complex object-element matrix of these m objects is as follows:

$$R_{nm}=\left[\begin{array}{cccc}C& N_1& ⋮& N_m\\ C_1& X_{11}& ⋮& X_{1m}\\ ⋮& ⋮& ⋮& ⋮\\ C_n& X_{n1}& ⋮& X_{nm}\end{array}\right]$$

(8)

(2)

Determine the classical domain of the object element, the section domain, and the object element to be evaluated

(1)

Classical domain

The urban rail transit system planning scheme is classified into three levels, excellent, good, and poor, where excellent is level 1, good is level 2, and poor is level 3. The three levels represent three objects, and their object-element matrix, ${\mathrm{R}}_{\mathrm{j}}(\mathrm{j}=\mathrm{1,2},3)$, are as follows:

$$R_j=(N_j,C,X_j)=\left[\begin{array}{ccc}{N}_j& C_1& X_{j1}\\ & C_2& X_{j2}\\ & ⋮& ⋮\\ & C_n& X_{jn}\end{array}\right]=\left[\begin{array}{ccc}{N}_j& C_1& (a_{j1},b_{j1})\\ & C_2& (a_{j2},b_{j2})\\ & ⋮& ⋮\\ & C_n& (a_{jn},b_{jn})\end{array}\right]$$

(9)

In the equation, ${\mathrm{X}}_{\mathrm{j}\mathrm{i}}$ is the range of values of ${\mathrm{N}}_{\mathrm{j}}$ with respect to ${\mathrm{C}}_{\mathrm{j}}$, ${\mathrm{X}}_{\mathrm{j}\mathrm{i}}=({\mathrm{a}}_{\mathrm{j}\mathrm{i}},{\mathrm{b}}_{\mathrm{j}\mathrm{i}})$, and $({\mathrm{a}}_{\mathrm{j}\mathrm{i}},{\mathrm{b}}_{\mathrm{j}\mathrm{i}})$ is the range of values of each indicator of the given level of the planning scheme of the urban rail transit system, which is called the classical domain.

(2)

Section domain

All classes of urban rail transit systems are planned with physical elements:

$$R_m=(N,C,X_m)=\left[\begin{array}{ccc}N& C_1& X_{1m}\\ & C_2& X_{2m}\\ & ⋮& ⋮\\ & C_n& X_{nm}\end{array}\right]=\left[\begin{array}{ccc}N& C_1& (a_{1m},b_{1m})\\ & C_2& (a_{2m},b_{2m})\\ & ⋮& ⋮\\ & C_n& (a_{nm},b_{nm})\end{array}\right]$$

(10)

${\mathrm{R}}_{\mathrm{m}}$ is the maximum range of values of the n characteristic indicators for each level of urban rail transit system planning and evaluation. N is the level of all the urban rail transit system planning schemes. ${\mathrm{X}}_{\mathrm{i}\mathrm{m}}$ is the range of values of N with respect to ${\mathrm{C}}_{\mathrm{i}}$. ${\mathrm{X}}_{\mathrm{i}\mathrm{m}}=\left({\mathrm{a}}_{\mathrm{i}\mathrm{m},}{\mathrm{b}}_{\mathrm{i}\mathrm{m}}\right)$, where $\left({\mathrm{a}}_{\mathrm{i}\mathrm{m},}{\mathrm{b}}_{\mathrm{i}\mathrm{m}}\right)$ is called the section domain of the planning object element of the urban rail transit planning system.

(3)

The object element to be evaluated

Define the element to be evaluated at the level of each planning scheme for the planning of the urban rail transit system as follows:

$$R_0=(N_0,C,X)=\left[\begin{array}{ccc}{N}_0& C_1& X_{j1}\\ & C_2& X_{j2}\\ & ⋮& ⋮\\ & C_n& X_{jn}\end{array}\right]$$

(11)

where ${\mathrm{N}}_0$ is the level of the urban rail transit system planning scheme to be evaluated. ${\mathrm{C}}_1,{\mathrm{C}}_2,{\mathrm{C}}_3,{\mathrm{C}}_4,\dots ,{\mathrm{C}}_{\mathrm{n}}$ is the characteristic parameter of ${\mathrm{N}}_0$, and ${\mathrm{X}}_1,{\mathrm{X}}_2,{\mathrm{X}}_3,\dots ,{\mathrm{X}}_{\mathrm{n}}$ is the specific real value of ${\mathrm{C}}_1,{\mathrm{C}}_2,{\mathrm{C}}_3,{\mathrm{C}}_4,\dots ,{\mathrm{C}}_{\mathrm{n}}$.

3.4. Evaluation of Urban Rail Transit System Planning Schemes Based on Matter-Element Model

Correlation measures and correlation functions are important tools for quantifying and analyzing the interrelationships between variables, and they provide a framework for analyzing the extent to which variables interact with each other in different domains [38,39]. While both tools have their advantages and disadvantages, when used in combination, they can provide deeper, more comprehensive insights for data analysis. The meaning of the correlation function representation of urban rail transit system planning is the degree to which the actual data of the object element R₀ to be evaluated about its characteristic parameters belong to the urban rail transit system planning scheme at levels N₁, N₂, and N₃, respectively. Therefore, the correlation function for urban rail transit system planning is defined as follows:

$$K_j(X_i)=\left\{\begin{array}{c}{\displaystyle \frac{-\rho (X_i,X_{ji})}{\left|X_{ji}\right|}},X_i\in X_{ji}\\ {\displaystyle \frac{\rho (X_i,X_{ji})}{\rho (X_i,X_{im})-\rho (X_i,X_{ji})}},X_i\notin X_{ji}\end{array}\right.$$

(12)

$$\rho (X_i,X_{ji})=\left|X_i-{\displaystyle \frac{1}{2}}\left(a_{ij}+b_{ij}\right)\right|-{\displaystyle \frac{1}{2}}(b_{ji}-a_{ji})$$

(13)

where $({\mathrm{X}}_{\mathrm{i}},{\mathrm{X}}_{\mathrm{i}\mathrm{m}})=|{\mathrm{X}}_{\mathrm{i}}-{\displaystyle \frac{1}{2}}({\mathrm{a}}_{\mathrm{i}\mathrm{m}}+{\mathrm{b}}_{\mathrm{i}\mathrm{m}})|-{\displaystyle \frac{1}{2}}({\mathrm{b}}_{\mathrm{i}\mathrm{m}}-{\mathrm{a}}_{\mathrm{i}\mathrm{m}})$, ${\mathrm{X}}_{\mathrm{j}\mathrm{i}}=({\mathrm{a}}_{\mathrm{i}\mathrm{j}},{\mathrm{b}}_{\mathrm{i}\mathrm{j}})$, ${\mathrm{X}}_{\mathrm{i}\mathrm{m}}=({\mathrm{a}}_{\mathrm{i}\mathrm{m}},{\mathrm{b}}_{\mathrm{i}\mathrm{m}})$. ${\mathrm{K}}_{\mathrm{j}}({\mathrm{X}}_{\mathrm{i}})$ denotes the extent to which the actual data ${\mathrm{X}}_{\mathrm{i}}$ of the characteristic parameters of the object element ${\mathrm{R}}_0$ to be evaluated belong to the level j of the urban rail transit system plan ${\mathrm{N}}_{\mathrm{j}}$. Therefore, the correlation of the object element to be evaluated with respect to the urban rail transit system planning at level j is as follows:

$$K_j(N_0)={\displaystyle \sum _{i=1}^n\omega K_j(X_i)}$$

(14)

The $\omega (\mathrm{i}=\mathrm{1,2},3\dots ,\mathrm{n})$ in equation is the weight coefficient of each characteristic parameter:

$$N_j=\max \left\{K_j(N_0),j=1,2,3\right\}$$

(15)

where x denotes the evaluation level of the object element y to be evaluated with respect to level j of the urban rail transit system plan. Finding ${\mathrm{K}}_{\mathrm{j}}\left({\mathrm{N}}_0\right)$ is a maximum, the rail transit system plan for the evaluation object element ${\mathrm{R}}_0$ corresponding to ${\mathrm{N}}_{\mathrm{j}}$ is level j.

In summary, the overall idea of the evaluation methodology proposed in this study is shown in Figure 2 below.

4. Case Analysis

4.1. Case Selection

The feasibility of the evaluation model developed is studied using Zhengzhou as an example. Zhengzhou City has a complex topography and landscape (Figure 3), dominated by plains and hills. The area of Zhengzhou City is about 7567 km², of which the number of permanent residents in the Erqi and Jinshui districts has exceeded one million people. A rail transit planning Scheme 1 in Zhengzhou City has six lines, namely, Line 1, Line 2, Line 3, Line 4, Line 5, and Line 6. The total length is about 206.3 km. A planning Scheme 2 is designed with five operational lines, totaling 187.8 km in length, with a total of 15 interchange stations. The basic information for Scheme 1 and Scheme 2 is shown in

Table 4. Basic information on planning programs.

	Scheme 1	Scheme 2
Operating lines	6 lines	5 lines
Total length	206.3 km	187.8 km
Number of Interchange Stations	17	15
Line density	0.66 km/km2	0.51 km/km2
Average station spacing	2.10 km	2.14 km
Estimated costs	79,848 million	76,532 million
Planning map

below. The quantitative data related to basic operation and planning in this study are from the real-time data of the Zhengzhou Metro Group statistics, and other quantitative data refer to the “Urban Rail Transportation 2023 Annual Statistics and Analysis Report” published by the China Urban Rail Transportation Association. Qualitative data are scored and summarized by experts.

4.2. Specific Values of Evaluation Indicators and Criteria for Their Delineation

4.2.1. Specific Values

When evaluating and analyzing the rail transit planning scheme in Zhengzhou City, use the evaluation index system established in Section 3, as shown in Table 2. The concrete values of the evaluation indices are mainly measured according to the official data given by Zhengzhou Metro Group and the field research. Concrete values for each indicator are shown in

Table 5. Specific figures for evaluation indicators.

	X11 (km)	X12 (%)	X13 (Number)	X14 (km)	X21 (Score)
Scheme 1	206.3	31	17	2.10	7.10
Scheme 2	187.8	29	15	2.14	7.21
	X22 (100 million)	X23 (year)	X24 (score)	X31 (km/h)	X32 (%)
Scheme 1	798.48	9	8.21	80	2.38
Scheme 2	765.32	7	7.12	75	2.59
	X33 (time)	X34 (thousands of people/day·km)	X41 (thousands of people)	X42 (%)	X43 (h)
Scheme 1	1.36	3.89	125.23	28	10.28
Scheme 2	1.29	3.49	121.31	24	9.38
	X51 (score)	X52 (score)	X53 (score)	X54 (score)	X55 (score)
Scheme 1	8.56	7.94	8.74	8.34	8.01
Scheme 2	8.16	8.19	7.69	8.01	7.23

.

4.2.2. Delineation Criteria

The evaluation indices of urban rail transit are classified into three grades, excellent, good, and poor, and are denoted by N₁, N₂, and N₃ respectively. The criteria for dividing the evaluation indicators are shown in

Table 6. Criteria for the division of evaluation indices for urban rail transit system planning.

Name of the Index (Unit)	$\mathbf{Level} {\mathbf{N}}_1$	$\mathbf{Level} {\mathbf{N}}_2$	$\mathbf{Level} {\mathbf{N}}_3$
Total length of line network ${\mathrm{X}}_{11}$ (km)	(190, 200)	(200, 210)	(210, 220)
Density of line network in the central area ${\mathrm{X}}_{12}$ (%)	(30, 35)	(25, 30)	(20, 25)
Number of interchange stations ${\mathrm{X}}_{13}$ (number)	(15, 20)	(10, 15)	(5, 10)
Average station spacing ${\mathrm{X}}_{14}$ (km)	(1.0, 1.5)	(1.5, 2.0)	(2.0, 2.5)
Degree of difficulty in construction ${\mathrm{X}}_{21}$ (score)	(8, 10)	(7, 8)	(6, 7)
Investment estimation ${\mathrm{X}}_{22}$ (100 million)	(600, 700)	(700, 800)	(800, 900)
Length of construction cycle ${\mathrm{X}}_{23}$ (year)	(3, 6)	(6, 10)	(10, 13)
Implementability of the program ${\mathrm{X}}_{24}$ (score)	(8, 10)	(7, 8)	(6, 7)
Average speed ${\mathrm{X}}_{31}$ (km/h)	(80, 100)	(60, 80)	(40, 60)
Unevenness coefficient of passenger flow section ${\mathrm{X}}_{32}$ (%)	(1.5, 2.0)	(2.0, 2.5)	(2.5, 3.0)
Average number of transfers ${\mathrm{X}}_{33}$ (time)	(1.2, 1.3)	(1.3, 1.4)	(1.4, 1.5)
Line load strength ${\mathrm{X}}_{34}$ (thousands of people/day·km)	(4, 5)	(3, 4)	(1, 2)
Average daily passenger capacity ${\mathrm{X}}_{41}$ (thousands of people)	(130, 140)	(120, 130)	(110, 120)
Percentage of trips using rail transit ${\mathrm{X}}_{42}$ (%)	(30, 35)	(25, 30)	(20, 25)
Travel time savings ${\mathrm{X}}_{43}$ (h)	(15, 20)	(10, 15)	(5, 10)
Urban land utilization ${\mathrm{X}}_{51}$ (score)	(8, 10)	(7, 8)	(6, 7)
Harmonization with urban layout ${\mathrm{X}}_{52}$ (score)	(8, 10)	(7, 8)	(6, 7)
Adaptation to urban development ${\mathrm{X}}_{53}$ (score)	(8, 10)	(7, 8)	(6, 7)
Satisfaction with urban needs ${\mathrm{X}}_{54}$ (score)	(8, 10)	(7, 8)	(6, 7)
Protection of the environment ${\mathrm{X}}_{55}$ (score)	(8, 10)	(7, 8)	(6, 7)

.

4.3. Calculation of the Integrated Weights of Evaluation Indicators

The calculation of the integrated weights of evaluation indicators is carried out according to the calculation steps of the G1 method in 3.2 to determine the subjective weights, according to the principle of the improved entropy weight method to determine the objective weights, and, finally, according to Formula (6) to determine the integrated weights, as shown in

Table 7. Integrated weights of evaluation indices.

Evaluation Indicators	Subjective Weights	Objective Weights	Integrated Weights	Evaluation Indicators	Subjective Weights	Objective Weights	Integrated Weights
X11	0.04675	0.04997	0.04836	X33	0.04535	0.04996	0.04766
X12	0.06670	0.04996	0.05833	X34	0.04172	0.04996	0.04584
X13	0.05951	0.04996	0.05474	X41	0.06327	0.05068	0.05698
X14	0.04754	0.04996	0.04875	X42	0.05814	0.04997	0.05406
X21	0.06159	0.04996	0.05578	X43	0.04959	0.04996	0.04978
X22	0.05294	0.04996	0.05145	X51	0.04642	0.04996	0.04819
X23	0.05078	0.04996	0.05037	X52	0.05275	0.04996	0.05136
X24	0.05078	0.04996	0.05037	X53	0.04431	0.04996	0.04714
X31	0.05079	0.04997	0.05038	X54	0.03798	0.04996	0.04397
X32	0.04353	0.04996	0.04675	X55	0.02954	0.04996	0.03975

.

4.4. Establishment of Matter-Element Model and Evaluation Results

(1)

Establishment of matter-element model

According to the established evaluation model of urban rail transit system planning, Scheme 1 and Scheme 2 are taken as the object elements to be evaluated to evaluate and analyze the planning scheme of Zhengzhou City’s rail transit system.

The classical domain for the three levels of urban rail transit system planning are as follows:

$$R_1=\left[\begin{array}{ccc}excellent& X_{11}& (190,200)\\ & X_{12}& (30,35)\\ & X_{13}& (15,20)\\ & X_{14}& (1.0,1.5)\\ & X_{21}& (8,10)\\ & X_{22}& (600,700)\\ & X_{23}& (3,6)\\ & X_{24}& (8,10)\\ & X_{31}& (80,100)\\ & X_{32}& (1.5,2.0)\\ & X_{33}& (1.2,1.3)\\ & X_{34}& (4,5)\\ & X_{41}& (130,140)\\ & X_{42}& (30,35)\\ & X_{43}& (15,20)\\ & X_{51}& (8,10)\\ & X_{52}& (8,10)\\ & X_{53}& (8,10)\\ & X_{54}& (8,10)\\ & X_{55}& (8,10)\end{array}\right]R_2=\left[\begin{array}{ccc}\mathrm{good}& X_{11}& (200,210)\\ & X_{12}& (25,30)\\ & X_{13}& (10,15)\\ & X_{14}& (1.5,2.0)\\ & X_{21}& (7,8)\\ & X_{22}& (700,800)\\ & X_{23}& (6,10)\\ & X_{24}& (7,8)\\ & X_{31}& (60,80)\\ & X_{32}& (2.0,2.5)\\ & X_{33}& (1.3,1.4)\\ & X_{34}& (3,4)\\ & X_{41}& (120,130)\\ & X_{42}& (25,30)\\ & X_{43}& (10,15)\\ & X_{51}& (7,8)\\ & X_{52}& (7,8)\\ & X_{53}& (7,8)\\ & X_{54}& (7,8)\\ & X_{55}& (7,8)\end{array}\right]R_3=\left[\begin{array}{ccc}poor& X_{11}& (210,220)\\ & X_{12}& (20,25)\\ & X_{13}& (5,10)\\ & X_{14}& (2.0,2.5)\\ & X_{21}& (6,7)\\ & X_{22}& (800,900)\\ & X_{23}& (10,13)\\ & X_{24}& (6,7)\\ & X_{31}& (40,60)\\ & X_{32}& (2.5,3.0)\\ & X_{33}& (1.4,1.5)\\ & X_{34}& (1,2)\\ & X_{41}& (110,120)\\ & X_{42}& (20,25)\\ & X_{43}& (5,10)\\ & X_{51}& (6,7)\\ & X_{52}& (6,7)\\ & X_{53}& (6,7)\\ & X_{54}& (6,7)\\ & X_{55}& (6,7)\end{array}\right]$$

The section domain for urban rail transit system planning is as follows:

$$R_m=(N,C,X_m)=\left[\begin{array}{ccc}scheme& X_{11}& (190,220)\\ & X_{12}& (20,30)\\ & X_{13}& (5,20)\\ & X_{14}& (1.0,2.5)\\ & X_{21}& (6,10)\\ & X_{22}& (600,900)\\ & X_{23}& (3,13)\\ & X_{24}& (6,10)\\ & X_{31}& (40,100)\\ & X_{32}& (1.5,3.0)\\ & X_{33}& (1.2,1.5)\\ & X_{34}& (1,5)\\ & X_{41}& (110,140)\\ & X_{42}& (20,35)\\ & X_{43}& (5,20)\\ & X_{51}& (6,10)\\ & X_{52}& (6,10)\\ & X_{53}& (6,10)\\ & X_{54}& (6,10)\\ & X_{55}& (6,10)\end{array}\right]$$

Object elements can reflect the connection between the quality and quantity of an object, and can better describe the process of changing its characteristics. For the convenience of research and calculation, the evaluation indices established according to the planning scheme of the Zhengzhou City Railway Transportation System have the same characteristic elements, so the homogeneous characteristic elements are used to represent the evaluation elements of these two schemes [40].

$$R=\left[\begin{array}{ccccc}N& N_1& N_2& \dots & N_m\\ C& V_1& V_2& \dots & V_m\end{array}\right]=\left[\begin{array}{ccccc}N& N_1& N_2& \dots & N_m\\ C_1& V_{11}& V_{12}& \dots & V_{1m}\\ C_2& V_{21}& V_{22}& \dots & V_{2m}\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ C_n& V_{n1}& V_{n2}& \dots & V_{nm}\end{array}\right]=\left[\begin{array}{ccc}& 1& 2\\ X_{11}& 206.3& 187.7\\ X_{12}& 31& 29\\ X_{13}& 17& 15\\ X_{14}& 2.1& 2.14\\ X_{21}& 7.1& 7.21\\ X_{22}& 798.48& 765.32\\ X_{23}& 9& 7\\ X_{24}& 8.21& 7.12\\ X_{31}& 80& 75\\ X_{32}& 2.38& 2.59\\ X_{33}& 1.36& 1.29\\ X_{34}& 3.89& 3.49\\ X_{41}& 125.23& 121.31\\ X_{42}& 28& 24\\ X_{43}& 10.28& 9.38\\ X_{51}& 8.56& 8.16\\ X_{52}& 7.94& 8.19\\ X_{53}& 8.74& 7.69\\ X_{54}& 8.34& 8.01\\ X_{55}& 8.01& 7.23\end{array}\right]$$

(2)

Evaluation results

The correlation of individual indicators under each level of each evaluation program and the combined correlation of each level in Program 1 and Program 2 were calculated according to Equations (11)–(14). The results are shown in

Table 8. Correlation of each evaluation index with each level of Scheme 1.

Name of the Index (Unit)	$\mathbf{Level} {\mathbf{N}}_1$	$\mathbf{Level} {\mathbf{N}}_2$	$\mathbf{Level} {\mathbf{N}}_3$
Total length of line network ${\mathrm{X}}_{11}$ (km)	−0.3150	0.3700	−0.2162
Density of line network in the central area ${\mathrm{X}}_{12}$ (%)	0.2000	−0.2000	−0.6000
Number of interchange stations ${\mathrm{X}}_{13}$ (number)	0.4000	−0.4000	−0.7000
Average station spacing ${\mathrm{X}}_{14}$ (km)	−0.6000	−0.2000	0.2000
Degree of difficulty in construction ${\mathrm{X}}_{21}$ (score)	−0.4500	0.1000	−0.0833
Investment estimation ${\mathrm{X}}_{22}$ (100 million)	−0.4924	0.0152	−0.0147
Length of construction cycle ${\mathrm{X}}_{23}$ (year)	−0.4286	0.2500	−0.2000
Implementability of the program ${\mathrm{X}}_{24}$ (score)	0.1050	−0.1050	−0.4033
Average speed ${\mathrm{X}}_{31}$ (km/h)	0.0000	0.0000	−0.5000
Unevenness coefficient of passenger flow section ${\mathrm{X}}_{32}$ (%)	−0.3800	0.2400	−0.1622
Average number of transfers ${\mathrm{X}}_{33}$ (time)	−0.3000	0.4000	−0.2000
Line load strength ${\mathrm{X}}_{34}$ (thousands of people/day·km)	−0.0902	0.1100	−0.6300
Average daily passenger capacity ${\mathrm{X}}_{41}$ (thousands of people)	−0.2441	0.4770	−0.2602
Percentage of trips using rail transit ${\mathrm{X}}_{42}$ (%)	−0.2222	0.4000	−0.3000
Travel time savings ${\mathrm{X}}_{43}$ (h)	−0.4720	0.0560	−0.0504
Urban land utilization ${\mathrm{X}}_{51}$ (score)	0.2800	−0.2240	−0.5200
Harmonization with urban layout ${\mathrm{X}}_{52}$ (score)	−0.0300	0.0600	−0.3264
Adaptation to urban development ${\mathrm{X}}_{53}$ (score)	0.3700	−0.0300	−0.5800
Satisfaction with urban needs ${\mathrm{X}}_{54}$ (score)	0.1700	−0.1700	−0.4467
Protection of the environment ${\mathrm{X}}_{55}$ (score)	0.0200	−0.0050	−0.3367
Comprehensive correlation	−0.0190	0.0065	−0.0592

and

Table 9. Correlation of each evaluation index with each level of Scheme 2.

Name of the Index (Unit)	$\mathbf{Level} {\mathbf{N}}_1$	$\mathbf{Level} {\mathbf{N}}_2$	$\mathbf{Level} {\mathbf{N}}_3$
Total length of line network ${\mathrm{X}}_{11}$ (km)	−1.3120	−1.2200	−1.1100
Density of line network in the central area ${\mathrm{X}}_{12}$ (%)	−0.1429	0.2000	−0.4000
Number of interchange stations ${\mathrm{X}}_{13}$ (number)	0.0000	0.0000	−0.5000
Average station spacing ${\mathrm{X}}_{14}$ (km)	−0.6400	−0.2800	0.2800
Degree of difficulty in construction ${\mathrm{X}}_{21}$ (score)	−0.3950	0.2100	−0.1479
Investment estimation ${\mathrm{X}}_{22}$ (100 million)	−0.3266	0.3468	−0.2084
Length of construction cycle ${\mathrm{X}}_{23}$ (year)	−0.2000	0.2500	−0.4286
Implementability of the program ${\mathrm{X}}_{24}$ (score)	−0.4400	0.1200	−0.0600
Average speed ${\mathrm{X}}_{31}$ (km/h)	−0.1667	0.2500	−0.3750
Unevenness coefficient of passenger flow section ${\mathrm{X}}_{32}$ (%)	−0.5900	−0.1800	0.1800
Average number of transfers ${\mathrm{X}}_{33}$ (time)	0.1000	−0.1000	−0.5500
Line load strength ${\mathrm{X}}_{34}$ (thousands of people/day·km)	0.2525	0.4900	−0.4967
Average daily passenger capacity ${\mathrm{X}}_{41}$ (thousands of people)	−0.4345	0.1310	−0.1038
Percentage of trips using rail transit ${\mathrm{X}}_{42}$ (%)	−0.6000	−0.2000	0.2000
Travel time savings ${\mathrm{X}}_{43}$ (h)	−0.5620	−0.1240	0.1240
Urban land utilization ${\mathrm{X}}_{51}$ (score)	0.0800	−0.0800	−0.3867
Harmonization with urban layout ${\mathrm{X}}_{52}$ (score)	0.0950	−0.0950	−0.3967
Adaptation to urban development ${\mathrm{X}}_{53}$ (score)	−0.1550	0.3100	−0.2899
Satisfaction with urban needs ${\mathrm{X}}_{54}$ (score)	0.0050	−0.0500	−0.5050
Protection of the environment ${\mathrm{X}}_{55}$ (score)	−0.3850	0.2300	−0.1575
Comprehensive correlation	−0.0501	0.0020	−0.0544

.

4.5. Analysis of Evaluation Results

Analyzing the above tables, the maximum combined correlation of Scheme 1 and Scheme 2 is ${\mathrm{K}}_2\left(\mathrm{s}\mathrm{c}\mathrm{h}\mathrm{e}\mathrm{m}\mathrm{e}1\right)=0.0065$ and ${\mathrm{K}}_2\left(\mathrm{s}\mathrm{c}\mathrm{h}\mathrm{e}\mathrm{m}\mathrm{e}2\right)=0.0020$; hence, Scheme 1 has the maximum correlation, and Scheme 1 is the best planning scheme.

In terms of the structural characteristics of rail transit lines, the structural characteristics of the rail transit network are poor in both schemes, so the focus can be on optimizing the structural characteristics of the urban rail transit lines to make the network planning reasonable and correct. From the viewpoint of the practicability of the construction of engineering projects, the practicability of the construction of the project is good for both Scheme 1 and Scheme 2. Scheme 2 has the maximum relevance, and, thus, Scheme 2 has the best practicability. Scheme 1 is in the poor range of practicability of project construction, particularly in terms of the length of the construction period, and, therefore, for Scheme 1, emphasis should be placed on reducing its construction period. In addition, the degree of difficulty of construction and the investment estimate should be appropriately reduced. Given the operational effectiveness of rail transit, both Scheme 1 and Scheme 2 have good operating results, while Scheme 1 has the maximum correlation. Thus, Scheme 1 has the best operating results. In the operational effect of the rail transit Scheme 2, the imbalance coefficient of the passenger section is poor, so it should be improved appropriately to reduce the imbalance coefficient of the passenger section. In terms of the social and economic benefits, the social and economic benefits of Scheme 1 are good, and those of Scheme 2 are poor. In terms of the proportion of trips using rail transit, we should consider rationalizing the operation schedule to attract people to use rail transit for their trips. In terms of the harmonization with the city and sustainability, Scheme 1 is excellent, and Scheme 2 is good. Scheme 2 is worse in terms of the compatibility with urban development and protection of the environment.

As can be seen in Table 7, the results of the integrated weights reflect the effective integration of subjective and objective information. For example, the integrated weight of the line network density in the central area is 0.05833, and its subjective weight is significantly higher than the objective weight, indicating that it is subjectively more biased towards the service capacity of the core area. At the same time, the improved entropy weighting method verifies the actual importance of this indicator through data discretization. From the integrated weights, it can be seen that the proposed subjective–objective integrated weight not only avoids the weight bias caused by the expert preference of the G1 method but also makes up for the misjudgment caused by the data noise of the entropy weight method. From the evaluation results, it can be seen that the matter-element model solves the problem of insufficient adaptability caused by the traditional method due to the fixed threshold, takes into account the practical significance of the evaluation indices, and verifies the scientificity, reasonableness, and applicability of the method proposed in this study.

From Table 7, Table 8 and Table 9 as a whole, the highest weighted evaluation indicator is the central area line network density X₁₂, which belongs to the N₁ level in Scheme 1, with a correlation of 0.2000. X₁₂ in Scheme 2 belongs to the N₂ level, with a correlation of −0.1429. A high central area line network density can effectively alleviate the traffic pressure in the core area of the city and enhance the convenience of travel for residents. Scheme 1 achieves a high coverage of the core area and enhances the coordination with the urban layout through a higher line density and many interchange stations. The line load intensity X₃₄ of Scheme 1 belongs to class N₂, and its correlation is significantly higher than that of classes N₁ and N_3, which indicates that the distribution of passenger flow on the line network of Scheme 1 is more balanced and the operation efficiency is higher. Although the total length of the line network of Scheme 1 is longer, the number of interchange stations and the coverage area ratio are higher, which enhances the attractiveness of passenger flow and keeps the line load intensity in a reasonable range. A high central area line network density can effectively alleviate the traffic pressure in the core area of the city and enhance the convenience of travel for residents. Scheme 1 achieves a high coverage of the core area and enhances the coordination with the urban layout through a higher line density and many interchange stations.

5. Summary

The development and improvement of rail transportation is a symbol of a city’s strength, which will not only make the city’s transportation layout more reasonable, ease the increasingly congested traffic environment, and improve the city’s efficiency, but also promote the improvement of the urban environment, reduce carbon emissions, and promote the green and sustainable development of the city. Urban rail transit projects involve many factors and large-scale investments. To save social resources and reflect the coordinated development of the society, economy, and environment, decision-makers must pay attention to the planning of urban rail transportation networks. A scientific and reasonable urban rail transit network planning and evaluation method is not only an important tool for analysis and judgment, comparison and improvement, and optimization and decision-making, but also an important way to effectively connect the theoretical system of urban rail transit network planning with the planning method, and an important means of coordinating the green development of the city. Combined with the full text, the following conclusions can be drawn:

(1)

By comparing and analyzing the existing evaluation indices of rail transit system planning at home and abroad, a set of systematic, reasonable, and comprehensive evaluation index systems has been established based on the concept of sustainable green transportation, which lays a solid theoretical foundation for the subsequent research.

(2)

The integrated empowerment method and the improved entropy weight method are used to combine the evaluation indices, and the integrated empowerment method and the matter-element model evaluation method for urban rail transit network planning evaluation are established by applying the theory of matter-element model and correlation function in topology. This evaluation method combines subjectivity and objectivity, which significantly improves the scientific and practical adaptability of planning evaluation.

(3)

The established evaluation method is applied to evaluate the urban rail transit planning scheme of Zhengzhou City as an example. The evaluation results of the example verify the reliability and scientificity of the proposed method. According to the methodology proposed in this study, through an integrated weights and correlation analysis, planning authorities can identify the shortcomings in key indicators and prioritize the allocation of resources to optimize weak links. The dynamic threshold design of the matter-element model can adapt to the differentiated needs of different cities. Planners can adjust the classical domain scope based on local data and flexibly balance the goals of the economy, social benefit, and environmental protection. In the future, the planning department can embed this model into the decision support system and iteratively optimize it with real-time data to enhance the scientific and adaptive nature of urban rail transit planning.

As a fast, efficient, and convenient mode of transportation, the urban rail transit system brings a lot of convenience to people. But, to achieve the expected results, it is necessary that we carry out rational planning and design, and evaluate and analyze the design scheme. The evaluation analysis has many shortcomings. Only the most basic evaluation analysis needs to be improved and optimized. There are a number of evaluation methods available, but, regardless of which one is used, there are shortcomings as well as problems with the scope of application. In this study, there are still some limitations in choosing a combination of the integrated empowerment method and matter-element modeling, in which the improved entropy weighting method provides a relatively objective way of assigning weights and alleviates the problem of sudden changes in weights. However, in practical application, how to choose initial variables and how to deal with these variables still have some subjectivity. The initial variable selection still relies on expert experience, and, in the future, we can consider combining machine learning and other algorithms to achieve the automation and dynamic optimization of variable selection and reduce human intervention. In addition, the current method applies to the medium- and long-term evaluation of rail transit planning in large- and medium-sized cities, but the adaptability to megacities or emerging intelligent transportation systems needs further verification. This subjectivity may be further amplified when combining mixture metamodels. Therefore, how to improve these methods and how to establish new evaluation methods are new issues to be considered in the future.