An optimization method for railway alignment schemes based on game-theoretic combination weighting and an interval number model

Abstract

Optimal selection of railway alignment schemes is a critical phase in railway line design. However, existing studies primarily adopt an engineering perspective, often overlooking economic and social factors. Current decision-making models also face limitations in quantifying qualitative indicators and balancing subjective and objective weights, thereby undermining the scientific rigor and effectiveness of the evaluation process. To address these issues, this study proposes a multi-attribute optimization model for railway alignment selection based on combined weighting and the interval number model. First, the key factors influencing railway alignment are systematically identified through expert consultations and an extensive literature review. On this basis, a multi-dimensional evaluation index system comprising 20 indicators is established, encompassing four dimensions: technical feasibility, economic rationality, environmental impact, and regional coordinated development. Subjective weights are derived using the Analytic Hierarchy Process (AHP), while objective weights are calculated through an improved CRITIC method. Game theory is then employed to integrate them. To further enhance the objectivity and robustness of the evaluation, the study develops a decision-making model based on interval number distance. This model quantifies qualitative indicators using interval numbers and ranks alternative schemes by computing their distances to an ideal solution. A case study on the Jieshipu-Pingliang section of the Dingxi-Pingliang railway confirms the method’s validity. The results show that the comprehensive weighting method reduces the influence of weight factor sensitivity on route selection. Furthermore, the scheme ranking derived from the interval number model closely aligns with the scheme recommended by the design institute and demonstrates stability. The research findings can provide a scientific reference for optimizing railway alignment in economically underdeveloped regions.

Introduction

Route selection is a critical stage in railway engineering design that determines the project’s safety, cost-effectiveness, and overall performance¹. Railway alignment directly affects engineering investment, transportation efficiency, and economic returns, while also exerting long-term and wide-ranging impacts on regional politics, economic development, culture, national defense, and the ecological environment along the corridor. Therefore, clearly identifying and defining the key indicators of railway alignment schemes is essential to support multi-scenario optimization and scientific decision-making in railway planning².

China Railway Network Planning (2016–2030) explicitly calls for optimizing the national railway network structure by expanding coverage in less developed regions, thereby supporting national strategies such as poverty alleviation and rural revitalization. The Development Plan for the Shaanxi, Gansu, and Ningxia Regions further emphasizes accelerating transportation infrastructure development and enhancing the role of railways in regional development and the preservation of revolutionary culture. In this context, integrating regional economic development needs into railway alignment schemes not only aligns with the fundamental role of railways as transportation infrastructure but also constitutes a core requirement for advancing high-quality development in economically underdeveloped regions³.

Multi-attribute decision-making problems are widespread in engineering and technology⁴. Extensive research has addressed such problems^5–8. With the continuous advancement of mathematical modeling, methods such as the Analytic Hierarchy Process (AHP), Grey Relational Analysis, the TOPSIS method, and the VIKOR method have been increasingly applied to railway route decision-making, yielding promising results. For example, Volkan Yildirim employed tomographic analysis combined with geographic information systems to determine the design scheme for the Erzincan-Trabzon section of the Turkish high-speed railway project⁹. Nedevska Ivona Z proposed a multi-attribute decision-making method for railway line planning based on WSM, AHP, and VIKOR methods and validated it through engineering case studies¹⁰. Luo Yuan et al. applied variable weight theory to determine indicator weights and used the TOPSIS method to evaluate and compare railway alignment schemes¹¹. Zhao Xiaoran et al. developed a comprehensive evaluation model that integrates the cloud model with cumulative prospect theory, enabling the ranking of alternative schemes based on their relative advantages and disadvantages¹². Han Feng et al. established a decision-making model using trapezoidal fuzzy numbers as qualitative indicators and combining them with multi-attribute utility theory. This model was applied to the comparison and selection of implementation schemes for high-speed railways traversing protected areas¹³.

Railway route selection is a complex system decision-making process that involves multiple objectives and constraints¹⁴. Current decision-making models still exhibit several limitations. First, the theoretical evaluation framework remains incomplete. Existing studies focus on engineering conditions, overlooking the complex geology and cultural value of economically undeveloped regions, leading to evaluation systems that inadequately reflect their broader development needs. Second, the calculation of indicator weights is not sufficiently reasonable. Many models overlook the interactions among evaluation factors, leading to potentially biased weight assignments and deviations in final decision outcomes. Third, the assignment of indicator values lacks objectivity. Qualitative indicators are typically quantified using either expert scoring or triangular fuzzy numbers¹⁵. The former method is easy to use, but it fails to adequately reflect the fuzzy nature of language evaluation. Although the latter method considers fuzzy characteristics, when it is used in conjunction with quantitative indicators for comprehensive distance calculation, different distance metrics need to be used separately, which complicates the evaluation process and introduces additional uncertainty.

In response to these issues, this study establishes a multidimensional evaluation index system for railway route schemes in economically underdeveloped regions, encompassing four dimensions: technical feasibility, economic rationality, environmental impact, and regional coordinated development. On the one hand, the AHP method and the CRITIC method based on entropy improvement are introduced to determine subjective and objective weights respectively. A game-theoretic optimal coordination model is used to achieve a balance between the two, thereby reducing the uncertainty and bias caused by a single weighting method. On the other hand, interval numbers are used instead of single membership degrees or scores to achieve a quantitative expression of qualitative indicators in the route scheme. Furthermore, based on the p-norm, the interval distance between the scheme and the ideal solution is constructed, achieving quantitative ranking of alternative schemes while preserving the fuzziness of expert evaluation. The proposed method is applied to the optimization of the Dingxi-Pingliang railway route, improving the practicality and scientific validity of railway route decision-making. The results provide a valuable reference for optimizing railway alignment in economically underdeveloped regions.

Research framework

The research framework is illustrated in Fig. 1. First, a multidimensional evaluation index system is developed to compare alternative railway routes in economically underdeveloped regions across four dimensions: technical feasibility, economic rationality, environmental impact, and regional coordinated development. Second, each scheme’s impact on the regional road network is assessed through accessibility, a key indicator of integration and transport efficiency. Third, subjective weights are determined using the Analytic Hierarchy Process (AHP), while objective weights are calculated using an improved CRITIC method, these are then integrated via game theory to derive balanced comprehensive weights. Fourth, an evaluation model based on interval number distances is constructed to quantitatively compare the advantages and disadvantages of each route. This model is applied in an empirical analysis of the Dingxi–Pingliang railway alignment schemes. The specific methodologies and implementation details for each phase are presented in the subsequent chapters.

Fig. 1.

Research framework.

Construction of the railway scheme evaluation index system

Analysis of evaluation factors

The optimal selection of railway alignment schemes is vital not only for ensuring engineering feasibility but also for enhancing investment efficiency, promoting ecological integrity, and supporting regional development¹⁶. To systematically assess the overall merits of different routing schemes, it is essential to adopt decision-making indicators that are scientific, effective, and practically applicable. The evaluation should be guided by the following key principles: scientific validity, systematic approach, comparability, and feasibility¹⁷.

Technical feasibility refers to the practicality of implementing a proposed railway route under complex geological and construction conditions¹⁸. Optimizing the total route length and balancing the proportions of bridges and tunnels can help reduce environmental disturbance, minimize elevation changes, and avoid unstable terrain. These design choices also help reduce gradients and shorten travel distances, thereby improving transport efficiency and operational safety. The bridge setting avoids environmental impacts on both sides of the line and facilitates the migration and foraging of wild animals¹⁹. To ensure a rational and safe project design, it is essential to systematically compare alternative routing schemes and associated engineering measures.

Economic efficiency is a key criterion in the evaluation of railway alignment schemes. The total static investment of a proposed route directly affects funding requirements and construction feasibility. Therefore, it is essential to control the investment scale within reasonable limits while ensuring that the route meets necessary transportation functionality and safety standards²⁰. In addition, from the perspective of land acquisition and demolition, the route layout should minimize land consumption and avoid disrupting existing urban planning frameworks²¹.

Railway construction and operation can exert significant impacts on the natural and ecological environment along the proposed alignment. Railway construction must be planned comprehensively under the principle of prioritizing ecological protection²². Route design should strive to minimize air and soil pollution, protect biodiversity, and avoid ecologically sensitive areas whenever possible. In cases where avoidance is not feasible, mitigation measures—such as ecological corridors, culverts, or elevated bridge structures—should be integrated with local topographical features to preserve ecosystem integrity and connectivity²³.

The selection of a railway route plays a pivotal role in promoting the coordinated development of economically underdeveloped regions, primarily through its capacity to stimulate local economies and enhance regional integration²⁴. This impact is evaluated using a combination of indicators, including the number of economic nodes intersected by the route, the transportation efficiency of scenic areas, and the degree of alignment with national spatial planning and surrounding road networks²⁵. The construction of railway infrastructure can substantially reduce travel time between key scenic destinations, thereby improving accessibility and increasing the tourist carrying capacity of these areas.

Construction of the evaluation index system

The evaluation index system serves as the basis for railway alignment scheme optimization. This study systematically reviews and categorizes the key factors influencing current alignment schemes. By synthesizing findings from existing literature, it analyzes railway alignment principles and integrates the concepts of red culture and red economic development²⁶. This study adopts a four-dimensional evaluation framework. The framework encompasses technical feasibility, economic rationality, environmental impact, and regional coordinated development. Based on these dimensions, 20 representative indicators are selected to construct a four-tier, multi-dimensional evaluation system for optimizing railway alignment schemes. The structure and interrelationships of this evaluation system are illustrated in Fig. 2. It should be noted that this indicator system is geared towards the preliminary planning stage of railway route selection. Given the limited accuracy of the data, it is necessary to use comprehensive indicators to reflect the engineering, environment and regional development in a coordinated manner. To avoid cross-redundancy among indicators, this paper explicitly defines the functional positioning and semantic boundaries of each metric.

Fig. 2.

Evaluation index system.

Combined weighting model

Subjective weight calculation-AHP method

The Analytic Hierarchy Process (AHP), proposed by Thomas L. Saaty²⁷, is a multi-level, multi-objective decision-making method widely used for the analysis of complex problems. The AHP methodology consists of three main steps: hierarchical decomposition, comparative judgment, and weight integration²⁸. In the decomposition step, complex problems are modeled as multi-level analytical hierarchy models. Comparative judgment is based on the 1 ~ 9 proportional scale theory²⁹. The relative importance of each element within the hierarchy is determined using pairwise comparisons, and a judgment matrixis constructed. Through normalization, the weight vectoris obtained.

To ensure the rationality of the judgment matrix, a consistency test must be performed. The verification formula is:

1

In the formula, denotes the average random consistency index; represents the largest eigenvalue of the judgment matrix; is the random consistency ratio. If, the matrix is considered to have acceptable consistency, and the derived weight distribution can be regarded as valid. Otherwise, the judgment matrix requires adjustment and correction.

Objective weight calculation-an improved CRITIC method

CRITIC (Criteria Importance Through Intercriteria Correlation) is an objective weighting method³⁰. This study employs an improved CRITIC method that uses entropy redundancy to characterize the contribution of indicator dispersion. Together with the conflict term from CRITIC, this forms an information measure to calculate objective weights. Given n schemes and m indicators, the original data matrix is. The specific steps are as follows:

Indicator normalization and dimensionless transformation. This study employs vector standardization to perform a dimensionless transformation of each indicator’s original data. For a benefit-type indicator, the normalization formula is:

2

For a cost-type indicator, the normalization formula is:

3

Calculate the indicator variance, which measures the dispersion of the indicator using the standard deviation. The formula is:

4

Calculate the indicator conflict. Conflict is denoted by, with the calculation formula as follows:

5

In the formula,.

Calculate the entropy values for each indicator using the entropy weighting method. This method determines the weights of various indicators by computing their information entropy. Information entropy reflects the degree of dispersion, with lower entropy values indicating more effective information³¹. The formula for information entropy is:

6

Finally, by integrating the standard deviation, indicator conflict, and entropy value, the objective weights of each indicator are obtained:

7

Game theory combination weighting

Relying exclusively on the Analytic Hierarchy Process (AHP) for subjective weighting can introduce bias, as the method depends heavily on expert judgment and lacks inherent objectivity. In contrast, the CRITIC method derives weights solely from data variability and correlation, without incorporating expert insight or contextual knowledge. Both approaches, when used in isolation, have notable limitations. To overcome these shortcomings, this study adopts a game-theoretic combination method that integrates subjective weights from AHP with objective weights calculated using an improved CRITIC method. The final comprehensive weights capture both the objectivity of statistical measures and the nuanced understanding of domain experts³². The specific implementation steps are outlined as follows:

The combined weights are determined using a game-theoretic aggregation model that minimizes the deviation between subjective and objective weights. By optimizing two linear combination coefficients, this method seeks to achieve the most satisfactory weights between expert judgment and data results. The objective function is formulated as follows:

8

By constructing a system of first-order linear differential equations, solve for the vector whose coefficients are unknown to obtain the optimal linear combination. This yields the linear equation system that optimizes the first-order derivative conditions:

9

Calculate the original optimization combination coefficients and normalize them:

10

Based on the normalized linear combination coefficients, solve for the final combination weights:

11

Railway alignment scheme optimization based on the interval number model

Quantification of qualitative indicators

The concept of interval numbers was proposed by Moore in 1965³³, primarily to address the problem of error amplification or propagation during calculations. Using interval numbers to address uncertainty can reduce the influence of human factors.

Let R denote the set of real numbers. An interval number, denoted as, is defined as a closed subset of real numbers bounded by two endpoints³⁴. Here, andare the left and right boundaries of the interval number, respectively.

Let a=[a^L,a^U] and b=[b^L,b^U] for any two interval numbers. Then a and b follow the following interval number operation rules: a + b=[a^L+b^L, a^U+b^U], a-b=[a^L-b^L, a^U-b^U], ab=[a^Lb^L,a^Ub^U], λa=[λa^L,λa^U](λ = 0).

Let Q denote the set of all interval numbers. For any interval number, the midpoint of the interval, denoted by, is defined as: . Similarly, the width of the interval, denoted by, is defined as: .

In the evaluation of railway schemes, certain indicators are inherently qualitative and difficult to quantify precisely, often relying on linguistic descriptions instead³⁵. To address this issue, this study establishes a mapping relationship between linguistic scales and interval numbers, enabling the rational quantification of qualitative indicators. First, the index value range is uniformly set to [0,10], consistent with the 10-point expert scoring system commonly used in the evaluation of existing railway lines. Second, a seven-level linguistic scale is established. [0,10] is initially divided into seven sub-intervals, establishing an initial correspondence between linguistic levels and numerical ranges. Based on this, by setting equally spaced center points within [0,10] and assigning approximately fixed half-widths, the intervals are adjusted to form a partially overlapping interval structure, more realistically reflecting the ambiguity during critical evaluation while maintaining the distinction between different levels. The correspondence between the linguistic scales of qualitative indicators and the number of intervals is shown in Table 1.

Table 1.

Linguistic scale and interval number conversion rules.

Linguistic scale (cost-type)	Linguistic scale (benefit-type)	Interval range
Very small	Very bad	[0.0, 1.5]
Smaller	Worst	[1.3, 3.0]
Small	Bad	[2.8, 4.5]
Average	Average	[4.3, 5.5]
Large	Good	[5.3, 7.0]
Larger	Better	[6.8, 8.5]
Very large	Very good	[8.3, 10.0]

Construction of the interval number decision matrix

Letdenote the set of railway alignment schemes, anddenote the set of evaluation indicators. Depending on its nature, takes the valuefor quantitative indicators andfor qualitative indicators. Then, the interval number decision matrix A, containing n schemes and m indicators, can be represented as:

12

Where, andrepresent the lower and upper bounds of the interval number, respectively.

Calculation of the distance between interval numbers

Due to differences in the attributes and dimensions of the evaluation indicators, direct comparison and analysis are not possible. To eliminate the influence of data dimensions on the calculation results, the interval number decision matrix A is normalized using vector normalization formulas (2) and (3), respectively, to obtain the normalized matrix.

13

Based on the comprehensive weights and the normalized decision matrix, a weighted normalized decision matrixis constructed. The positive ideal solutionand the negative ideal solutionof the decision schemes are determined according to formulas (14) and (15). When the indicator is a benefit-type indicator, the positive ideal solutionand the negative ideal solutionare:

14

When the indicator is a cost-type indicator, the positive ideal solutionand the negative ideal solutionare:

15

Using formulas (16) and (17), calculate the distancefrom each alternative solution to the positive ideal solution and the distanceto the negative ideal solution, respectively.

16

17

The TOPSIS method can be effectively applied to the evaluation of railway route schemes³⁶. Drawing on the relative proximity calculation method of TOPSIS, the difference between the proposed scheme and the ideal scheme is calculated based on the relative distance of the interval numbers. The smaller the relative distance, the smaller the difference between the alternative solution and the ideal solution, indicating that the solution is better. This is calculated using formula (18).

18

Case study

Study area

To verify the accuracy and rationality of the proposed evaluation model, this study selected the alignment scheme for the section from Jieshipu to Pingliang on the Dingxi-Pingliang Railway as a case study for analysis. This section was chosen for its strategic geographic location and its importance to national economic development. The study area encompasses the administrative regions of Gansu Province and the Ningxia Hui Autonomous Region, which are traversed by the proposed alignment. Specifically, the key cities along the corridor include Dingxi, Baiyin, Guyuan, and Pingliang. These cities are not only significant nodes in regional transportation networks but also play important roles in supporting economic growth, promoting regional integration, and advancing the coordinated development of western China.

This study comprehensively considers a range of factors in the evaluation of route schemes, including the functional zoning of the Liupanshan National Nature Reserve, the current urban layout and overall regional planning, the distribution of economic centers along the corridor, the impact on environmentally protected zones, and the existing transportation infrastructure. Based on these considerations, three initial alignment schemes—Scheme A, Scheme B, and Scheme C—are proposed. In addition, given the limited road network coverage in Zhuanglang County and the strategic role of Huating City as a key energy supply and freight distribution hub in eastern Gansu, a fourth alternative, Scheme D, is proposed. As a result, a total of four alignment schemes are analyzed for the Jieshipu–Pingliang section. The schematic diagram of the route schemes is shown in Fig. 3. Relevant engineering information for the route schemes is presented in Table 2. Engineering data are derived from actual design documentation.

Fig. 3.

Route schemes. (ArcGIS 10.8 software was used to create this map: https://www.esri.com).

Table 2.

Engineering information for the four route schemes.

Indicators	Scheme A	Scheme B	Scheme C	Scheme D
Railway length	109.66	112.76	113.800	144.150
Roadbed length	29.92	23.43	21.64	24.33
Total bridge length	25.44	23.83	23.92	22.31
Total tunnel length	54.30	65.50	68.23	97.51
Proportion of bridges and tunnels	72.72	79.22	80.98	83.12
Construction difficulty	Large	Average	Average	Average
Total static investment	150.73	159.35	161.77	212.75
Total regional population	106.5	106.5	95.5	139
Aggregate GDP	286.14	286.14	250.66	362.61
Land acquisition and demolition	44,917	56,029	36,818	113,825
Impact on nature reserves	Average	Larger	Smaller	Small
Length within the protection zone of water source areas	3.1	0	2.3	0
Length of sections crossing forest parks	1.5	1.5	1.5	9.8
Total length of sections crossing ecologically sensitive areas	11.0	13.4	15.7	9.8
Ability to promote local economic development	Better	Average	Better	Average
Contribution to the development of areas	Better	Good	Better	Average
Accessibility of tourist attractions	39.667	35.333	40.500	28.500
Total number of economic nodes passed thought	4	4	3	4
Coordination with urban planning	Average	Good	Good	Bad
Compatibility with adjacent transport networks	Good	Average	Good	Bad

Scheme A: The route extends westward from Pingliang Station, establishing Guyuan Liupanshan Station to the southeast of Liupanshan Town. It then curves southwest to cross the Liupanshan Mountain, with Longde Station located approximately 6 km west of Longde County. After departing Longde, the alignment continues westward along the Yuhe River valley, reaching the northern side of the Qinglan Expressway, where Jingning Station is situated. Finally, the route turns west again, crossing both the Qinglan Expressway and National Highway G312 before reaching its terminal destination.

Scheme B: The route begins at Pingliang Station and proceeds through the experimental zone of the Liupanshan Provincial Nature Reserve, entering Jingyuan County. The proposed Jingyuan Station is situated in Yanzhi Village, approximately 2 km northeast of the county seat. From Jingyuan, the route continues across the Liupan Mountain, following the experimental zone of the Liupanshan National Nature Reserve, and reaches Longde Station, which is planned to be 10 km southwest of Longde County. The route then turns southwest into Jingning County, with Jingning Station located north of the county seat. After leaving the station, the line follows the Gaojie River valley to the final destination.

Scheme C: The route diverges from Pingliang Station and runs parallel to the Baoji-Zhongwei Railway. The alignment first passes through the experimental zone of the Liupan Mountain Provincial Nature Reserve, then enters Jingyuan County, where Jingyuan Station is planned at Yanzhi Village, located approximately 2 km northeast of the county seat. Exiting the station, the line crosses the experimental zone of the Liupan Mountain National Nature Reserve, crosses Liupan Mountain, and establishes Caowu Station south of Caowu Town. After leaving the station, the lline turns northwest, establishing Jingning Station at Chengchuan Town south of Jingning County as the final terminus.

Scheme D: The route departs from Pingliang Station, turns southward, and consecutively crosses Kongtong Avenue and the Fuyin Expressway. It then establishes stations at Cedi Town in Huating City and Nanping Town in Zhuanglang County. Subsequently, it follows the provincial highway along the Jing-Tian Expressway corridor to establish a station in Jingning County before reaching its final terminus.

The extent to which each route scheme improves the regional road network is evaluated through the lens of accessibility, which serves as a key indicator of regional integration and transportation efficiency. One of the primary goals of the Dingxi-Pingliang Railway is to alleviate the underdeveloped transport conditions in economically underdeveloped regions by enhancing connectivity. This is achieved by reducing both temporal and spatial distances between these historically significant but remote regions and the broader national transportation network. To quantify this effect, the study employs GIS-based spatial analysis techniques, specifically using the Raster Calculator and the Multi-Value Extraction to Point tool. The accessibility calculation results for the four route schemes are shown in Fig. 4.

Fig. 4.

Accessibility distribution map of different solutions. (ArcGIS 10.8 software was used to create this map: https://www.esri.com).

Data sources

The design materials used for this study were provided by the China Railway First Survey and Design Institute Group Co., Ltd. The data involved in the study includes geographical and administrative divisions, natural conditions such as topography and geology, socio-economic information including population and economic indicators, as well as regional transportation networks and the distribution of tourism attractions. To facilitate unified data management and analysis, this study uses ArcGIS 10.8 software for statistical quantification management and the creation of relevant maps.

The administrative division data primarily includes: provincial, prefecture-level city, and county-level administrative boundaries; the spatial areas of provincial, prefecture-level city, and county-level administrative boundaries; and county-level regional points. This data is mainly sourced from publicly available data provided by the China National Basic Geographic Information Center (http://ngcc.sbsm.gov.cn/). The terrain data comes from the Geospatial Data Cloud Platform (https://www.gscloud.cn). Regional population and socioeconomic data primarily originate from the Statistical Yearbooks of the Gansu Provincial Bureau of Statistics (http://tjj.gansu.gov.cn/) and the Ningxia Hui Autonomous Region Bureau of Statistics (http://tj.nx.gov.cn/).

The data used in this study also includes survey questionnaire data. Ten experts with relevant field expertise were also invited to participate in the evaluation and consultation. The expert panel comprises five scholars from universities, whose research areas cover transportation engineering, regional planning, geographic information science, and ecological and environmental protection; and five engineering and technical personnel from railway design units, who possess extensive practical experience in railway route selection and design, engineering cost analysis, and environmental impact assessment. The selection of experts follows the principle of combining professional diversity and practical experience to ensure the comprehensiveness and authority of the evaluation results.

Determination of combined weighting

This study employs the Analytic Hierarchy Process (AHP), which has good applicability for subjective weighting. All judgment matrices pass consistency testing (CR < 0.1). Results of subjective weighting calculations are summarized in Table 3.

Table 3.

Subjective weights of each indicator.

Indicator number	Criterion layer weights	Indicator layer	Indicator layer weights	Subjective weight
B1	0.3525	C1	0.3236	0.1141
		C2	0.1390	0.0490
		C3	0.1390	0.0490
		C4	0.1390	0.0490
		C5	0.2020	0.0712
		C6	0.0574	0.0202
B2	0.2043	C7	0.2212	0.0452
		C8	0.4566	0.0933
		C9	0.1202	0.0246
		C10	0.2020	0.0413
B3	0.1302	C11	0.3863	0.0503
		C12	0.2487	0.0324
		C13	0.2487	0.0324
		C14	0.1163	0.0151
B4	0.3130	C15	0.2987	0.0935
		C16	0.1394	0.0436
		C17	0.0933	0.0292
		C18	0.0627	0.0196
		C19	0.2279	0.0713
		C20	0.1780	0.0557

To ensure the scientific validity and reliability of the expert ratings, this study subjected the expert opinions to a consistency analysis. The coefficient of variation (CV) for the qualitative indicators ranged from 0.12 to 0.23, all falling below the threshold of 0.25; furthermore, Kendall’s Coefficient of Concordance (W) was 0.82, and the Chi-square test yielded a p-value of < 0.01, indicating a high degree of overall consistency among the expert ratings. For indicators exhibiting significant dispersion, adjustments can be made through feedback-based revisions or by excluding extreme values.

To determine the objective weights of the indicators, this study employs an improved CRITIC method. In accordance with the principles of the entropy weight method, an entropy value of zero indicates either the presence of extreme conditions or an inappropriate indicator selection. As shown in Table 4, all indicators yield entropy values greater than zero, confirming the rationality of the selected indicators.

Table 4.

Standard deviation, entropy value, correlation coefficient, and objective weight of each indicator.

Indicator	Standard deviation	Entropy value	Correlation coefficient	Objective weight
C1	0.4604	0.7907	5.4045	0.0413
C2	0.4219	0.7762	12.0942	0.0885
C3	0.4088	0.7411	5.7461	0.0404
C4	0.4118	0.7651	5.8574	0.0421
C5	0.4354	0.6263	8.2134	0.0533
C6	0.5000	0.7925	9.8508	0.0778
C7	0.4420	0.7877	5.3351	0.0401
C8	0.4325	0.6282	5.2725	0.0342
C9	0.4217	0.6757	5.3140	0.0356
C10	0.4244	0.7625	4.9874	0.0362
C11	0.4318	0.6637	12.8022	0.0857
C12	0.5277	0.5346	5.9447	0.0386
C13	0.5000	0.7925	5.6862	0.0449
C14	0.4435	0.7155	7.9119	0.0560
C15	0.5774	0.5000	6.8881	0.0453
C16	0.4827	0.7539	4.7686	0.0360
C17	0.4601	0.5086	4.6438	0.0275
C18	0.5000	0.7925	10.0878	0.0796
C19	0.4800	0.7586	8.1185	0.0614
C20	0.4800	0.7586	4.7101	0.0356

The final combined weights are determined using a game-theoretic approach. Figure 5 presents a visual comparison of the subjective, objective, and combined weights.

Fig. 5.

Comparison of weights of primary and secondary indicators. (a) Comparison of weights of primary indicators, (b) Comparison of weights of secondary indicators.

As illustrated in Fig. 5, certain deviations exist between the weights derived from the Analytic Hierarchy Process (AHP) and those obtained using the improved CRITIC method. The most influential factors are railway length, total regional population, and the ability to promote local economic development. Among these, railway length has the greatest impact on route selection. This is because the railway divides the area’s existing surface landscape, and the longer the line, the more significant the disruption. The geographical environment is highly sensitive to engineering construction, and the cost of restoration after disruption is prohibitively high. The total regional population and the ability to promote local economic development reflect the scheme’s influence on regional growth, the extent of the population served, and its consistency with national spatial planning objectives. These factors align with the socio-economic functions and planning requirements of transportation infrastructure projects. Game-theoretic weighting effectively balances subjective and objective weights. It preserves the justified emphasis on key indicators through expert judgment while effectively correcting inherent data patterns through objective weighting. This integration enhances the scientific rigor and rationality of the overall weighting system.

Evaluation and optimization of railway route schemes

It should be noted that the four alternative schemes (Schemes A, B, C, and D) addressed in this study have all successfully passed compliance reviews during the preliminary design phase. None of the schemes traverse the core zones of nature reserves; furthermore, any crossings of water source protection zones or forest parks are situated within their respective experimental or buffer zones. Additionally, the proportion of occupied prime farmland for each scheme remains within statutory limits. Consequently, in the context of the current case, it is unnecessary to conduct a preliminary screening based on rigid constraints; instead, all four schemes proceed directly to the comprehensive evaluation stage.

According to the linguistic scale and interval number conversion rules defined in Table 3, qualitative evaluations for each scheme are translated into corresponding interval numbers. The quantitative results of the qualitative indicators are shown in Fig. 6.

Fig. 6.

Qualitative indicators quantification.

Based on the quantitative description results obtained from Fig. 6 and the known indicator data in Table 3, the original interval number decision matrix A. Using Eqs. (2) and (3), the original decision matrix is normalized to obtain the normalized decision matrix. Next, based on the comprehensive indicator weights and the normalized decision matrix, the weighted normalized decision matrixis obtained.

Based on the weighted normalized decision matrix, the positive ideal solutionand negative ideal solutionof the decision scheme are determined using formulas (14) and (15), respectively.

According to the interval distance formulas (16) and (17), taking p = 1, the distances between schemes A, B, C, and D and the positive and negative ideal schemes are calculated. The relative distances of the interval numbers are calculated by formula (18).

To ensure the accuracy and scientific validity of the optimization method, the interval distance between the candidate scheme and the ideal scheme is calculated for p = 2, 3, 4, and ∞ using consistent steps. The detailed results are shown in Table 5. By examining distances across different dimensions, it can be observed that the interval number model exhibits excellent robustness.

Table 5.

Calculation results.

Dimension	d+				d-				D				Optimal scheme
	A	B	C	D	A	B	C	D	A	B	C	D
p = 1	0.0585	0.1017	0.0917	0.1171	0.1306	0.0874	0.0972	0.0718	0.3094	0.5378	0.4854	0.6200	A
p = 2	0.0571	0.0998	0.0874	0.1123	0.1248	0.0820	0.0944	0.0694	0.3139	0.5489	0.4806	0.6180	A
p = 3	0.0569	0.0996	0.0864	0.1117	0.1238	0.0874	0.0942	0.0689	0.3148	0.5328	0.4783	0.6186	A
p = 4	0.0545	0.0976	0.0841	0.1115	0.1234	0.0806	0.0838	0.0687	0.3065	0.5476	0.5009	0.6189	A
p=∞	0.0568	0.0996	0.0858	0.1114	0.1231	0.0803	0.0942	0.0685	0.3157	0.5534	0.4767	0.6193	A

As shown in Fig. 7, the ranking of the schemes under different p-values is Scheme A > Scheme C > Scheme B > Scheme D, proving that the optimal result is not accidental but has parameter robustness. Therefore, Scheme A is the recommended route.

Fig. 7.

Processing results of Interval number method.

Considering the impact of weight changes on the decision-making results, this paper further conducts a sensitivity analysis of the game theory-based combined weighting results. A combination of single-index perturbation and Monte Carlo random simulation is used. Each time, one index is selected, and its weight is adjusted by ± 10% from its original value. The weights of other indices are normalized, and the relative proximity and ranking of each scheme are recalculated. Simultaneously, assuming that the weights of each index follow a uniform distribution within a given interval near their original values, a large number of weight samples are randomly generated. Each weight group is evaluated and ranked, and the frequency of each scheme obtaining the optimal ranking is statistically analyzed. The analysis results show that, within a reasonable weight perturbation range, although the proximity values of each scheme fluctuate to some extent, the overall ranking remains A > C > B > D. The probability of the original optimal scheme obtaining first place is much higher than that of other schemes, and no change in the optimal scheme occurs. This indicates that the combined weighting-interval distance evaluation model constructed in this paper has good robustness to the weight parameters and can maintain stable and reliable decision results even when there is some uncertainty in the weights.

Analysis of the reasonableness of the results

From a technical feasibility perspective, Scheme D has the longest mountain-crossing tunnel and encounters the most challenging geological conditions, making it the most difficult to implement. Scheme A has the smallest overall mountain-crossing tunnel and better geological conditions, making it easier to implement than Scheme D. The remaining schemes have comparable levels of difficulty in engineering implementation.

From an economic feasibility perspective: Scheme A, which follows the valleys of the Xiehe and Yuhe rivers, has the shortest route length and a lower proportion of bridges and tunnels. Compared to schemes B, C, and D, it saves 862 million yuan, 1.104 billion yuan, and 6.202 billion yuan in construction costs, respectively. This significantly reduces project costs and subsequent operation and maintenance costs.

From the perspective of the impact on environmentally sensitive areas, Scheme D, Scheme C and Scheme A have relatively small overall disturbances to environmentally sensitive areas. Among them, Scheme A performs better in avoiding key sensitive areas such as the core area of ​​nature reserves, wetland parks, centralized drinking water source protection areas and cultural relics and historical sites, while Scheme B has a relatively large impact on ecologically sensitive areas such as nature reserves.

From the perspective of regional coordinated development, this route connects tourism cities and scenic spots, forming a tourism corridor that integrates cultural history and natural landscapes. Scheme A and Scheme B pass through more tourist attractions, which is more conducive to driving economic development in the economically underdeveloped regions of Shaanxi, Gansu, and Ningxia, promoting the prosperity of local tourism, and driving regional coordinated development.

Based on multi-dimensional analysis and model optimization ranking rules. The results from the proposed model are consistent with those obtained by design experts through quantitative calculations and qualitative analysis, verifying the model’s applicability and reliability.

Discussion

Optimizing railway route schemes presents a complex multi-criteria decision-making problem. The core lies in developing an evaluation index system that is scientifically rigorous, representative, and practically applicable. However, existing unified standards for railway alignment in economically underdeveloped regions remain poor, and related research is relatively sparse. Route selection extends beyond considerations of construction costs, operational efficiency, and technical feasibility. It also influences the surrounding natural environment, the preservation of tourism resources, and the broader goals of regional coordinated development. Therefore, when constructing an evaluation index system for railway alignment schemes, it is crucial to move beyond traditional engineering and economic metrics. The system also incorporates the specific development needs of economically underdeveloped regions, the protection of red heritage resources, and the region’s ecological sensitivity. To address this need, this study proposes an evaluation index system comprising 20 carefully selected indicators across four dimensions: technical feasibility, economic rationality, environmental impact, and regional coordinated development.

This study adopts a combination of subjective and objective weighting methods to determine the comprehensive weights of evaluation indicators. The integration of subjective and objective weights into a comprehensive set of weights improves alignment between evaluation results and the practical needs of railway construction in economically underdeveloped regions. This study proposes an interval-number distance model to optimize the selection of railway alignment schemes. By constructing an interval decision matrix for each alternative and calculating the interval distance between each scheme and the ideal solution, the model enables rapid generation of evaluation results, thereby facilitating a clear comparison of the relative merits of each option. This method leverages the strengths of interval numbers in representing information uncertainty. It avoids the inaccuracies that result from forcing imprecise or fuzzy indicators into exact values, thereby reducing error propagation. As a result, the method significantly enhances the stability and credibility of the evaluation outcomes. We conduct a weight sensitivity analysis using single-index perturbation and Monte Carlo simulation. The results show that within a reasonable range of weight changes, scheme A still has the highest optimal probability, thus verifying the robustness of the evaluation results.

The evaluation results for the Dingxi-Pingliang Railway alignment schemes align with the recommended option from the design unit, thereby validating the scientific soundness and practical applicability of the decision-making model proposed in this study. This consistency highlights the model’s effectiveness in supporting rational infrastructure planning under complex, multi-criteria conditions. This railway section traverses the economically underdeveloped regions of Shaanxi and Gansu provinces, an area of significant cultural and historical importance in western China. Simultaneously, it lies within the ecologically sensitive Loess Plateau, characterized by hilly terrain, rugged topography, complex geological formations, and severe soil erosion. Despite the region’s abundance of tourism resources, much of it remains inaccessible due to insufficient transportation infrastructure. Therefore, future development strategies should prioritize closing connectivity gaps by accelerating railway construction in such areas. Improved transportation networks can serve as powerful enablers of regional economic growth, promote tourism development, and support the revitalization of historically disadvantaged areas, fostering a path toward sustainable and inclusive development.

Conclusions

The optimal railway route selection is a complex systems engineering decision-making problem involving multiple objectives and constraints. Especially in economically underdeveloped regions, the route selection not only relates to engineering investment and operational efficiency, but also has a profound impact on regional coordinated development and ecological environmental protection. To fully leverage the role of railway lines in driving regional economic and social development, this paper takes the railway line schemes from Jieshipu to Pingliang as an example and conducts a series of studies on the constraints of railway alignment options, and the construction and application of a railway alignment optimization model. The main conclusions are as follows:

1. This paper breaks through the limitations of traditional route selection that only focuses on engineering technology and economic indicators. In the context of railway route selection in economically underdeveloped region, this study identifies 20 indicators with clear definitions and well-established boundaries. These indicators span four dimensions: technical feasibility, economic rationality, environmental impact, and regional coordinated development. Based on these dimensions, a comprehensive and hierarchically structured evaluation index system is developed. The system ensures logical rigor and provides a robust framework for the systematic assessment and optimal selection of railway route schemes.

2. By integrating expert judgment with information derived from statistical data, this study proposes a combined weighting method based on game theory. This method merges subjective weights obtained through the Analytic Hierarchy Process (AHP) with objective weights derived from the entropy-based improved CRITIC method. To address qualitative indicators, interval number theory is applied to develop standardized representation rules and corresponding quantification methods. Building on this foundation, an evaluation model for railway route schemes is constructed using interval number distances. By comparing the distances between interval numbers, the model enables rapid and accurate evaluation of alternative schemes, thereby enhancing both decision-making efficiency and reliability. Compared with existing methods, the method proposed in this paper can take into account technical, economic, environmental and regional development objectives in the optimization of railway route options, and coordinate the handling of uncertainties in weights and index values, thereby obtaining more systematic and robust decision results.

3. The model is applied to a real-world engineering case of the Jieshipu to Pingliang section of the Dingxi-Pingliang Railway, and a multi-dimensional comprehensive evaluation is conducted based on the combined weight and interval distance model. The results show that under different p-norm settings, scheme A is the optimal scheme, consistent with the final recommended scheme of the design institute. This result verifies the scientific nature and engineering applicability of the method presented in this paper, and can provide some reference for the decision-making problem of railway alignment schemes in economically underdeveloped areas.

Based on this, this study still has certain limitations, and can be further deepened in the following directions: First, the “multi-dimensional indicator system-game theory combination weighting-interval distance model” proposed in this paper can be extended to other railway planning decision-making issues such as railway station site selection, hub introduction mode selection, and integrated connection between conventional and high-speed railways; Second, it is necessary to conduct comparative application studies under different types of regions and different planning and management systems to test the model’s universality and adaptability, and provide more solid decision support for the high-quality development of railways in other underdeveloped regions and even a wider area.

Author contributions

D.Y.: Conceptualization, Investigation, Resources, Writing—original draft. F.H.: Conceptualization, Methodology, Validation, Data curation, Writing—original draft. Y.G.: Methodology, Validation, Investigation, Writing—review and editing. Z.L.: Resources, Writing—review and editing.

Funding

This work is supported by the National Science Foundation of China (No. 52562051) and Special Funds for Guiding Local Scientific and Technological Development by the Central Government (No. 24ZYQA044).

Data availability

The datasets used and obtained during the current study are available from the corresponding author on reasonable request.

Declarations

Competing interests

The authors declare no competing interests.

Informed consent

Informed consent was obtained from all individual participants included in the study.