Railway engineering construction period index-based sorting method and system

Abstract

This invention relates to the field of railway engineering schedule management and intelligent decision-making technology, and in particular to a method and system for ranking railway engineering schedule indicators. The method includes constructing a multi-dimensional schedule indicator system, establishing a hesitant fuzzy decision matrix, introducing a β threshold to construct a covering approximate space, calculating attribute dependencies to reduce redundant indicators, reconstructing the matrix, and calculating a comprehensive evaluation value, ultimately achieving a scientific ranking of schedule indicators. This application can effectively handle uncertainties, incompleteness, and attribute redundancy while preserving the multiple hesitancy and fuzziness of evaluation information, thereby improving the efficiency and accuracy of schedule indicator screening and ranking.

Description

Technical Field

[0001] This invention relates to the field of railway engineering schedule management technology, and more specifically, to a method and system for ranking railway engineering schedule indicators. Background Technology

[0002] In railway construction, the rational determination and dynamic optimization of schedule targets are crucial aspects of project management and decision-making. Due to the characteristics of railway projects, such as large track spans, complex structural types, significantly varying construction environments, and numerous participating units, the schedule is influenced by a variety of factors, including natural conditions, construction organization, resource allocation, technical solutions, and management capabilities. To achieve scientific control of project progress, it is typically necessary to construct a multi-dimensional schedule indicator system. This system comprehensively evaluates and ranks the schedule indicators for different plans or construction stages, thereby providing a basis for decision-making in schedule planning and adjustments.

[0003] Currently, existing railway engineering schedule indicator decisions mostly employ traditional statistical analysis, analytic hierarchy process (AHP), or classic multi-attribute decision models to weight and comprehensively evaluate influencing factors. However, in practical applications, some schedule factors are difficult to quantify precisely, and evaluation information often relies on experience-based judgment, interval estimation, or verbal description, resulting in strong subjectivity and high uncertainty. Differences in risk preferences and understanding of key processes among different decision-makers further enhance the hesitation and ambiguity of information, easily leading to indicator redundancy and decreased discriminative ability. Although fuzzy set theory has improved the expression of uncertain information to some extent, traditional and some extended fuzzy set methods still struggle to effectively screen and reduce indicators under conditions of multiple hesitations, incomplete information, and strong attribute correlations. Furthermore, existing methods generally lack dynamic screening mechanisms based on thresholds, limiting decision-making efficiency and reliability.

[0004] Therefore, how to effectively reduce, screen, and rank schedule indicators while fully preserving the uncertainties in railway engineering schedule evaluation information, and improve the efficiency and scientific nature of the decision-making process, has become an urgent technical problem to be solved in current research on railway engineering schedule indicator decision-making. Summary of the Invention

[0005] In view of this, the present invention proposes a method and system for ranking railway engineering schedule indicators, aiming to solve the problem that existing railway engineering schedule indicator decision-making methods are difficult to effectively screen, reduce and rank schedule indicators in evaluation environments with multiple uncertainties, strong hesitation and incomplete information, resulting in indicator redundancy and insufficient discrimination ability.

[0006] This invention proposes a method for ranking railway engineering schedule indicators, including: Construct a multi-dimensional schedule indicator system for railway engineering, establish an attribute set based on factors affecting the schedule such as natural conditions, construction organization, resource allocation, technical solutions and management level, and establish the decision-making object based on candidate schedule control schemes or indicators; Based on the hesitant fuzzy set, a hesitant fuzzy decision matrix is ​​constructed for each decision-maker's evaluation information for each decision object under each attribute; The hesitant fuzzy decision matrix is ​​transformed into a covering approximation space, a β-covering relation is constructed based on the parameter β, and upper and lower approximation sets are established. The dependency of each attribute is calculated based on the upper and lower approximation sets. The attribute set is then reduced according to the dependency and redundant schedule indicators are removed. The hesitant fuzzy decision matrix is ​​reconstructed based on the reduced attribute set, the comprehensive evaluation value of each decision object is obtained, and all decision objects are sorted according to the comprehensive evaluation value to output the optimal sequence of project schedule indicators.

[0007] Furthermore, when constructing a multi-dimensional schedule indicator system for railway engineering, it includes: Collect raw data on factors affecting the construction period in railway engineering, including geological condition level, climate window period, critical line length, number of construction teams, equipment input intensity, and complexity of process connection; The above factors are mapped to discretized or interval-based attribute variables, forming an attribute set A = {a1, a2, ..., a...} n}; Define the schedule control scheme or indicator to be evaluated as the set of decision objects U={x1,x2,...,x}. m}; Establish an initial decision table T, where T = (U, A, V, f). Where: V is the attribute range, and f is the information function.

[0008] Furthermore, when constructing a hesitant fuzzy decision matrix based on the evaluation information of each decision-maker for each decision object under each attribute using hesitant fuzzy sets, it includes: Obtain the set of decision objects and attribute set in the railway engineering schedule indicator decision problem. The decision objects are the schedule indicators to be sorted, and the attributes are multiple factors that affect the schedule evaluation. These serve as the input objects for constructing the decision matrix. For each decision object under each attribute, the membership degree evaluation values ​​independently provided by multiple experts are collected. The membership degree evaluation values ​​are used to characterize the degree to which the decision object conforms to the evaluation target under the corresponding attribute. The membership degree evaluation values ​​given by multiple experts for the same decision object under the same attribute are aggregated to form a membership degree set. The membership degree set is not averaged, weighted or other aggregation operations are performed to retain the hesitation and uncertainty information in the evaluation results. Using the set of membership degrees corresponding to each decision object under each attribute as matrix elements, construct the hesitant fuzzy decision matrix: H=[h ij ] m×n In which each element h in the matrix ij This represents the set of hesitant fuzzy membership degrees of the i-th decision object under the j-th attribute, serving as the input data basis for subsequent indicator screening and ranking.

[0009] Furthermore, when converting the hesitant fuzzy decision matrix into a covering approximate space, the process includes: For each attribute a j ∈A, calculate the hesitant fuzzy distance d(x) between any two objects based on its hesitant fuzzy values ​​across all decision objects. i x k ); Set a distance threshold δ, if d(x) i x k If x ≤ δ, then x is considered to be... i With x k In attribute a j The following cannot be distinguished; For each object x i , with its attribute a j All indistinguishable objects below constitute an overlay block C. j (x i ); All covered blocks C j (x i ) Constituting attribute a j Corresponding coverage C j The entire set of {C1,C2,...,C} is covered. n} constitutes a covering approximate space, Where, x i For the i-th decision object currently being referenced, x k For x i The k-th decision object to be compared.

[0010] Furthermore, when constructing the β-coverage relationship based on parameter β, it includes: For any two project schedule indicator decision objects, under the constraint of the selected attribute subset, the average hesitant fuzzy similarity of the two decision objects on the attribute subset is calculated based on the hesitant fuzzy decision matrix; When the average hesitant fuzzy similarity is not less than a preset similarity threshold, it is determined that one of the decision objects belongs to the threshold neighborhood of the other decision object under the attribute subset. Using the threshold neighborhood formed by each decision object under the attribute subset as the basic information granularity, a covering approximate space based on threshold constraints is constructed. The similarity threshold is a positive number between zero and one, and is preset according to the engineering management objectives or construction phase requirements of the railway project.

[0011] Furthermore, the hesitant fuzzy distance d(x) i ,x k When using the modified Euclidean distance formula, the following is included: For each attribute dimension, pair up all possible membership values ​​in the set of hesitant fuzzy evaluation values ​​of the two decision objects under that attribute, and obtain the squared difference of each pair of membership values. Sum and average all squared differences under the same attribute, then average all attribute dimensions, and finally take the square root of the average to obtain the modified Euclidean distance between the two decision objects, which is used as their hesitant fuzzy distance. Based on hesitant fuzzy distance, the hesitant fuzzy similarity of any two project schedule indicator decision objects under the constraints of a selected attribute subset is obtained, and the hesitant fuzzy similarity is used to construct the neighborhood relationship of the decision objects. The similarity is obtained by subtracting the hesitant fuzzy distance obtained under the attribute subset from one. The numerical range is between zero and one. The closer the similarity value is to one, the higher the similarity between the two decision objects under the specified attribute subset.

[0012] Furthermore, when obtaining the dependence of each attribute on the decision classification based on the upper approximation set and the lower approximation set, it includes: Select any subset of attributes from the attribute set. Based on the discriminative power of the attribute subset for decision classification, determine the set of decision objects to which the attribute subset can uniquely determine the decision classification. Construct the positive domain corresponding to the attribute subset based on the set of decision objects. The ratio of the number of decision objects contained in the positive domain to the total number of decision objects is used as the degree of dependence of the attribute subset on the decision classification. In the attribute subset, remove individual attributes one by one, and re-acquire the dependency level after removal. If the dependency level remains unchanged before and after removing an attribute, the attribute is determined to be a redundant attribute. All attributes deemed non-redundant are retained, and a minimal attribute reduction set is constructed for screening and ranking project schedule indicators.

[0013] Furthermore, when taking the comprehensive evaluation value of each decision-making object, it includes: For each attribute in the reduced attribute set, the hesitant fuzzy entropy of the hesitant fuzzy information corresponding to the attribute is obtained based on the hesitant fuzzy information corresponding to the attribute. The complement of the hesitant fuzzy entropy corresponding to each attribute is used as the objective weight of that attribute. Among them, the lower the uncertainty of an attribute, the higher its weight in the comprehensive evaluation. For each decision object, the mean value of its hesitant fuzzy evaluation value under each attribute is processed to obtain the representative evaluation value of the decision object in each attribute dimension; The objective weights of each attribute are weighted and summed with their corresponding representative evaluation values ​​to obtain the comprehensive evaluation value of the decision-making object.

[0014] Furthermore, when ranking all decision objects, this includes: Construct a complete sorting list or sequence of all project schedule decision objects after sorting by comprehensive evaluation values ​​and comparing hesitant fuzzy variance. The ranking list is arranged from high to low according to the comprehensive evaluation value of the decision objects, and in the case of equal comprehensive evaluation values, the ranking is determined according to the variance of the hesitant fuzzy evaluation value from small to large. The sorted list is used as a preferred sequence of time indicators to guide the optimization of railway engineering construction plans and resource allocation.

[0015] Compared with existing technologies, the beneficial effects of this invention are as follows: By constructing a multi-dimensional schedule indicator system, factors affecting the schedule of railway projects, such as natural conditions, construction organization, resource allocation, technical solutions, and management levels, are systematically categorized into attribute sets. Candidate schedule plans or indicators are used as decision-making objects, providing a structured and standardized data foundation for schedule evaluation. This method can take into account the multi-dimensional characteristics of schedule decision-making, enabling project managers to comprehensively and systematically analyze the role and interrelationships of various schedule indicators. Secondly, by introducing hesitant fuzzy set theory, the evaluation information of each decision-maker under different attributes is modeled to form a hesitant fuzzy decision matrix, effectively characterizing the uncertainty, hesitation, and subjective preferences in the evaluation information. This design improves the compatibility and expression accuracy for multiple decision-makers and diverse evaluation information, reflecting the true evaluation state of schedule indicators under various attributes without losing the original information characteristics. Furthermore, by converting the hesitant fuzzy decision matrix into a covering approximate space and constructing β-coverage relationships and upper and lower approximate sets based on parameter β, this invention can scientifically divide the similarity neighborhood of decision objects, achieving effective screening of schedule indicators. The process calculates attribute dependencies based on approximate upper and lower sets, and then reduces the attribute set by eliminating redundant schedule indicators, making the indicator system more concise and discriminative, while also reducing the complexity of subsequent calculations. Finally, a hesitant fuzzy decision matrix is ​​reconstructed based on the reduced attribute set to obtain the comprehensive evaluation value of the decision objects, which is then ranked to generate an optimal schedule indicator sequence. This process not only improves the scientific validity and stability of schedule indicator ranking but also assists project managers in making optimization decisions, enabling the rational formulation of construction plans, the optimization of resource allocation, and the precise control of project progress.

[0016] On the other hand, this application also provides a railway engineering schedule index ranking system, including: The indicator modeling module is used to construct a multi-dimensional schedule indicator system for railway engineering, model influencing factors as a set of attributes, and define candidate schedule options as decision objects. The hesitant fuzzy processing module is electrically connected to the indicator modeling module. The hesitant fuzzy processing module is used to receive evaluations of each decision object by multiple experts under various attributes and form a hesitant fuzzy decision matrix. The covering approximation construction module is electrically connected to the hesitant fuzzy processing module. The covering approximation construction module is used to convert the hesitant fuzzy decision matrix into a β-covering approximation space and generate upper and lower approximation sets. The attribute reduction module is electrically connected to the coverage approximation construction module. The attribute reduction module is used to calculate and remove redundant attributes based on dependency and output the minimum attribute reduction set. The evaluation value calculation module is electrically connected to the attribute reduction module. The evaluation value calculation module is used to calculate the comprehensive evaluation value of each decision object based on the reduced attributes. The sorting output module is electrically connected to the evaluation value calculation module. The sorting output module is used to sort the decision objects according to the comprehensive evaluation value and output the optimal sequence of construction period indicators.

[0017] It is understood that the method and system for ranking railway engineering schedule indicators in the above embodiments of the present invention have the same beneficial effects, and will not be described in detail here. Attached Figure Description

[0018] Various other advantages and benefits will become apparent to those skilled in the art upon reading the following detailed description of preferred embodiments. The accompanying drawings are for illustrative purposes only and are not intended to limit the invention. Furthermore, the same reference numerals denote the same parts throughout the drawings. In the drawings: Figure 1 A flowchart illustrating a method for ranking railway engineering schedule indicators, provided in an embodiment of the present invention; Figure 2 A flowchart illustrating a method for ranking railway engineering schedule indicators according to an embodiment of the present invention; Figure 3 This is a functional block diagram of a railway engineering schedule index ranking system provided in an embodiment of the present invention. Detailed Implementation

[0019] Exemplary embodiments of the present disclosure will now be described in more detail with reference to the accompanying drawings. While exemplary embodiments of the present disclosure are shown in the drawings, it should be understood that the present disclosure may be implemented in various forms and should not be limited to the embodiments set forth herein. Rather, these embodiments are provided to enable a more thorough understanding of the present disclosure and to fully convey the scope of the disclosure to those skilled in the art. It should be noted that, unless otherwise specified, embodiments and features in the embodiments of the present invention can be combined with each other. The present invention will now be described in detail with reference to the accompanying drawings and embodiments.

[0020] like Figures 1-2 As shown in some embodiments of this application, this embodiment provides a method for ranking railway engineering schedule indicators, including: Step S100: Construct a multi-dimensional schedule indicator system for railway engineering. Establish an attribute set based on factors affecting the schedule, such as natural conditions, construction organization, resource allocation, technical solutions, and management level. Establish the candidate schedule control schemes or indicators as decision-making objects.

[0021] Specifically, constructing a multi-dimensional schedule indicator system for railway engineering includes: collecting raw data on factors affecting the schedule of railway engineering projects, including geological condition levels, climate windows, critical line lengths, number of construction teams, equipment investment intensity, and complexity of work processes; mapping these factors into discrete or interval-based attribute variables to form an attribute set A={a1,a2,...,a...} n The schedule control scheme or indicator to be evaluated is defined as the set of decision objects U={x1,x2,...,x}. m Establish an initial decision table T, where T=(U, A, V, f), where V is the attribute range and f is the information function.

[0022] Understandably, by structuring the factors influencing the construction period into a set of attributes and using candidate construction period control schemes or indicators as decision-making objects, a scientific evaluation and ranking of construction period schemes can be achieved. Specifically, firstly, various factors affecting the construction period of railway projects are identified, including natural conditions (such as geological condition level and climate window), construction organization factors (such as the number of construction teams and the arrangement of construction shifts), resource allocation factors (such as equipment investment intensity and material supply guarantee), technical solution factors (such as the complexity of construction process connections and the selection of construction technology), and management level factors (such as project management experience and the perfection of coordination mechanisms). Secondly, these factors are quantified and formed into attribute variables through discretization or interval mapping, thus constituting a complete set of attributes. Each attribute variable has a clearly defined value range and quantitative standard, and can simultaneously accommodate qualitative and quantitative information. Subsequently, the construction period control schemes or indicators to be evaluated are used as decision-making objects, and an initial decision table is constructed, recording the attribute values ​​of each decision object under each attribute in a unified format. In this way, the complex, multi-sourced project schedule evaluation information with uncertainties and subjective judgment components is structured in a unified manner, providing a standardized data foundation for subsequent fuzzification processing based on hesitant fuzzy sets, coverage approximation analysis, attribute reduction, and comprehensive evaluation ranking, thereby enabling the scientific screening, ranking, and optimization of project schedule plans.

[0023] For example, taking a real railway engineering project as an example, suppose we need to evaluate three candidate schedule control schemes, with the goal of selecting the optimal schedule index sequence to guide the construction plan. First, we collect raw data on the factors influencing the project's schedule, including: geological condition level 2, a five-month climate window during the construction period, a critical path length of 20 kilometers, three construction teams, medium equipment input intensity, and high complexity of construction procedures. Then, we map these factors to discretized or interval-based attribute variables. For example, the geological condition level can be mapped to a value range of 1-5, the number of construction teams directly to the integer 3, and the complexity of construction procedures can be mapped to high, medium, and low levels. These attribute variables constitute a complete attribute set. Next, we define the three schedule control schemes as decision object sets, and fill the attribute values ​​of each scheme under each attribute into an initial decision table, forming a unified multi-dimensional schedule index system. For example, the values ​​of the first scheme under attributes such as geological conditions, number of construction teams, and equipment input are clearly recorded in the table; the second and third schemes are similarly recorded. In this way, the originally discrete, qualitative, and quantitative project schedule information is structured and organized into a standardized data format that can be processed by computers and analyzed later. Subsequently, based on hesitant fuzzy sets, each decision object can be evaluated, a covering approximate space can be constructed, attribute dependencies can be calculated, attribute reduction can be performed, and a comprehensive evaluation and ranking can be conducted. This yields a scientific and reliable optimal sequence of project schedule indicators, providing project managers with quantitative decision-making basis and achieving optimization of construction plans and rationalization of resource allocation.

[0024] Step S200: Construct a hesitant fuzzy decision matrix based on the hesitant fuzzy set of evaluation information of each decision-maker for each decision object under each attribute.

[0025] Specifically, when constructing a hesitant fuzzy decision matrix based on the evaluation information of each decision-maker for each decision object under each attribute, the following steps are taken: First, obtain the set of decision objects and the set of attributes in the railway engineering schedule indicator decision problem. The decision objects are the schedule indicators to be ranked, and the attributes are multiple factors affecting the schedule evaluation, which serve as input objects for the subsequent construction of the decision matrix. Second, for each decision object under each attribute, collect the membership degree evaluation values ​​independently given by multiple experts. These membership degree evaluation values ​​characterize the degree to which the decision object conforms to the evaluation objective under the corresponding attribute. Third, aggregate the membership degree evaluation values ​​given by multiple experts for the same decision object under the same attribute to form a membership degree set. This membership degree set is not subjected to averaging, weighting, or other aggregation operations to retain the hesitant and uncertain information in the evaluation results. Fourth, construct the hesitant fuzzy decision matrix using the membership degree sets corresponding to each decision object under each attribute as matrix elements: H = [h...]. ij ] m×n In which each element h in the matrix ijThis represents the set of hesitant fuzzy membership degrees of the i-th decision object under the j-th attribute, serving as the input data basis for subsequent indicator screening and ranking.

[0026] Understandably, this approach treats the schedule indicators to be ranked as decision-making objects, and various factors influencing the schedule (including natural conditions, construction organization, resource input, complexity of construction procedures, and management level) as attributes. For each decision-making object under each attribute, membership evaluation values ​​independently provided by multiple experts are collected. These membership degrees are used to quantify the degree to which the decision-making object meets the schedule target under the corresponding attribute. To preserve the hesitation and uncertainty in expert evaluations, the membership values ​​are not averaged, weighted, or otherwise aggregated; instead, they are directly aggregated to form a membership degree set. Using the membership degree sets of each decision-making object under each attribute as matrix elements, a complete hesitant fuzzy decision matrix is ​​constructed, providing a structured and standardized data foundation for subsequent indicator selection, attribute reduction, comprehensive evaluation, and ranking analysis. Through this technique, when processing information from multiple experts, multiple attributes, and multiple uncertainties, the original judgment information can be preserved to the greatest extent, improving the scientificity and reliability of schedule indicator ranking and optimization decisions, while providing quantifiable and operable data support for project management.

[0027] For example, in a real railway engineering project, suppose three candidate schedule control schemes need to be evaluated, with the goal of determining the optimal sequence of schedule indicators to support the development of the construction plan. First, a set of decision-making objects is established, consisting of the three schedule schemes. Simultaneously, an attribute set is established, including geological conditions, the number of construction teams, equipment input intensity, the complexity of key process connections, and other management and technical factors affecting the schedule. For each schedule scheme, under each attribute, multiple experts provide membership degree evaluation values. For example, for the first scheme, under the geological conditions attribute, experts provide membership degree values ​​of 0.7, 0.8, and 0.9, indicating the degree to which the scheme meets the schedule target under the current geological conditions; under the number of construction teams attribute, experts provide membership degree values ​​of 0.6, 0.65, and 0.7, reflecting the advantages and disadvantages of the scheme in terms of resource input. Similarly, membership degree values ​​from multiple experts are collected for other schemes under each attribute. Subsequently, the membership degree values ​​of each decision-making object under each attribute are directly aggregated to form a membership degree set, maintaining the hesitation and uncertainty in expert evaluation. Finally, these membership sets are used as matrix elements, arranged according to the decision object and attributes, to construct a complete hesitant fuzzy decision matrix. This matrix can not only fully reflect the evaluation information of each scheme in each attribute dimension, but also provide a reliable data foundation for subsequent screening of schedule indicators, attribute reduction, and comprehensive ranking based on the β-coverage approximation space, thereby assisting project managers in making scientific and quantitative decisions on construction plan optimization and resource allocation.

[0028] Step S300: Convert the hesitant fuzzy decision matrix into a covering approximation space, construct a β-covering relation based on parameter β, and establish upper and lower approximation sets.

[0029] Specifically, converting the hesitant fuzzy decision matrix into a covering approximate space includes: for each attribute a j ∈A, calculate the hesitant fuzzy distance d(x) between any two objects based on its hesitant fuzzy values ​​across all decision objects. i x k ); Set a distance threshold δ, if d(x i x k If x ≤ δ, then x is considered to be... i With x k In attribute a j Indistinguishable for each object x; i , with its attribute a j All indistinguishable objects below constitute an overlay block C. j (x i ); All overlay blocks C j (x i ) Constituting attribute a j Corresponding coverage C j The entire set of {C1,C2,...,C} is covered. n} forms a covering approximate space, where x i For the i-th decision object currently being referenced, x k For x i The k-th decision object to be compared.

[0030] Specifically, when constructing a β-coverage relationship based on parameter β, the process includes: for any two project schedule indicator decision objects, under the constraint of a selected attribute subset, calculating the average hesitant fuzzy similarity of the two decision objects on the attribute subset based on the hesitant fuzzy decision matrix; when the average hesitant fuzzy similarity is not less than a pre-set similarity threshold, determining that one decision object belongs to the threshold neighborhood of the other decision object under the attribute subset; using the threshold neighborhood formed by each decision object under the attribute subset as the basic information granularity, constructing a coverage approximation space based on threshold constraints; wherein, the similarity threshold ranges from zero to one positive number, and is pre-set according to the engineering management objectives or construction stage requirements of the railway project.

[0031] Specifically, the hesitant fuzzy distance d(x) i ,x kWhen using the modified Euclidean distance formula, the following steps are taken: For each attribute dimension, pair all possible membership values ​​of the two decision objects in the set of hesitant fuzzy evaluation values ​​under that attribute, and obtain the squared difference of each pair of membership values; sum and average all squared differences under the same attribute, then average over all attribute dimensions, and finally take the square root of the average to obtain the modified Euclidean distance between the two decision objects, which is used as their hesitant fuzzy distance; based on the hesitant fuzzy distance, obtain the hesitant fuzzy similarity of any two project schedule indicator decision objects under the constraints of a selected attribute subset, and use the hesitant fuzzy similarity to construct the neighborhood relationship of the decision objects; wherein, the similarity is obtained by subtracting the hesitant fuzzy distance obtained under the attribute subset from one, where the value ranges from zero to one, and the closer the similarity value is to one, the higher the similarity between the two decision objects under the attribute subset.

[0032] Understandably, by constructing a covering approximation space, the hesitant fuzzy evaluation information of railway engineering schedule indicators is structured into a measurable and comparable form, thereby enabling the scientific screening and ranking of schedule indicators. Specifically, firstly, based on the hesitant fuzzy decision matrix, the pairwise hesitant fuzzy distances of each decision object across each attribute dimension are calculated. These distances are obtained by pairwise pairwise calculation of the squared differences of all membership values ​​of two decision objects under the same attribute, averaging all squared differences under that attribute, summing the averages across all attribute dimensions, and taking the square root to obtain the corrected Euclidean distance, accurately reflecting the similarity and differences between decision objects. This distance calculation method fully preserves the hesitant and uncertain information present in expert evaluations, ensuring that data processing does not lose the original judgment characteristics. Subsequently, based on a set distance threshold, a covering block is formed for each decision object under each attribute: when the hesitant fuzzy distance between two objects under a certain attribute is less than or equal to the threshold, they are determined to be indistinguishable on that attribute, and the set of indistinguishable objects constitutes the covering block. All covering blocks are combined according to attribute dimensions to form a covering approximation space, providing a basic structure for the clustering, similarity analysis, and reduction of decision objects. Furthermore, by introducing the parameter β, the average hesitant fuzzy similarity of the decision objects on a selected subset of attributes is calculated. Objects with a similarity not lower than a pre-set threshold β are identified as the threshold neighborhood of the reference object. This neighborhood is then used as the basic information granularity to construct a β-coverage approximation space based on threshold constraints. This method not only achieves fine-grained division of multi-dimensional project duration indicators but also ensures that, under attribute subset constraints, highly similar and significantly different schemes can be dynamically distinguished, improving the scientific rigor and accuracy of indicator selection, reduction, and ranking.

[0033] For example, taking a real railway engineering project as an example, suppose three schedule control schemes need to be evaluated, considering five key attributes, including geological condition level, number of construction teams, equipment input intensity, complexity of key process connections, and construction management level. First, the membership evaluation values ​​of multiple experts for each scheme under each attribute are collected. For example, for the first scheme, experts give 0.7, 0.8, and 0.9 for the geological condition attribute; 0.6, 0.65, and 0.7 for the number of construction teams; and 0.8, 0.75, and 0.85 for the complexity of key process connections. For each pair of schemes under each attribute, pairwise membership is performed, the squared difference is calculated, and the average of all squared differences under that attribute is calculated. Then, the average of all attributes is calculated and the square root is taken to obtain the corrected Euclidean hesitant fuzzy distance between the three schemes, accurately characterizing the similarity between the schemes. Then, a distance threshold is set, for example, 0.15. When the distance between two schemes under a certain attribute does not exceed the threshold, the two schemes are determined to be indistinguishable on that attribute and included in the coverage block. The sets of cover blocks for each attribute are combined to form a complete cover approximation space. Next, a similarity threshold β is set, for example, 0.8, and the average hesitant fuzzy similarity of each scheme on the selected attribute subset is calculated. When the similarity is not lower than 0.8, the scheme is included in the threshold neighborhood of the reference scheme, and this neighborhood is used as the basic information granularity to construct the β-cover approximation space. Through this process, it is possible to clearly determine which schedule schemes are highly similar in key attributes and which have significant differences, thus providing a reliable basis for subsequent attribute reduction and schedule index ranking. Ultimately, this method not only achieves high-precision similarity classification but also effectively reduces redundant indicators while preserving hesitant information, improving the scientificity and reliability of schedule decisions, and providing quantitative and actionable decision support for project managers to formulate optimized construction plans and rational resource allocation.

[0034] Step S400: Calculate the dependency of each attribute based on the upper and lower approximation sets, reduce the attribute set according to the dependency, and remove redundant schedule indicators.

[0035] Specifically, when obtaining the dependence of each attribute on the decision classification based on the upper approximation set and the lower approximation set, the process includes: selecting any subset of attributes in the attribute set; determining the set of decision objects to which the attribute subset uniquely belongs based on the discriminative power of the attribute subset on the decision classification; constructing a positive domain corresponding to the attribute subset based on the set of decision objects; using the ratio of the number of decision objects contained in the positive domain to the total number of decision objects as the dependence of the attribute subset on the decision classification; sequentially removing individual attributes from the attribute subset and re-obtaining the dependence after removal; determining that the attribute is a redundant attribute when the dependence before and after removing an attribute remains unchanged; retaining all attributes determined to be non-redundant and constructing a minimal attribute reduction set for screening and ranking project schedule indicators.

[0036] Understandably, by selecting any subset of attributes from the attribute set and utilizing its discriminative power to classify decision objects, the set of decision objects uniquely determined by that attribute subset is identified, i.e., the positive domain. The dependence of an attribute subset on the decision classification is represented by the proportion of decision objects contained in the positive domain to the total number of decision objects; a higher value indicates a greater role for the attribute subset in classification. Based on this, individual attributes are sequentially removed from the attribute subset, and the dependence is recalculated. If the dependence remains unchanged after removing an attribute, it indicates that the attribute has no beneficial effect on decision classification and can be considered a redundant attribute. By eliminating all redundant attributes and retaining non-redundant attributes, a minimal attribute reduction set can be constructed, providing an efficient and accurate data foundation for the selection and ranking of schedule indicators, while reducing computational complexity and improving decision-making efficiency. This method can take into account multiple attributes, multiple decision objects, and uncertain information, scientifically quantifying the role of attributes in schedule indicator discrimination, and achieving simplification and optimization of schedule indicators.

[0037] For example, in a real railway engineering project, suppose we are evaluating three schedule control schemes, involving five key attributes: geological condition level, number of construction teams, equipment input intensity, complexity of key process connections, and construction management level. First, we select a subset of all attributes and calculate the discriminative power of each subset for the classification of the three schemes. This determines the set of schemes that each attribute subset can uniquely distinguish, and the dependency is the proportion of schemes in this set to the total number of schemes. For example, if the positive domain of the attribute subset {geological conditions, number of construction teams} contains all decision objects of the three schemes, its dependency is 1, indicating that this attribute subset can completely distinguish the three schemes. Then, we remove individual attributes one by one. For example, after removing the number of construction teams, we recalculate the dependency. If the dependency is still 1, the number of construction teams is considered a redundant attribute. By performing similar operations on all attributes, redundant attributes are eliminated, and non-redundant attributes are retained, ultimately forming a minimal attribute reduction set, such as {geological conditions, complexity of key process connections}. By utilizing the reduced attribute set, a model for screening and ranking project schedule indicators can be constructed. This model can reduce redundant data and improve the efficiency of indicator screening and ranking while ensuring the accuracy of the judgment, thus providing efficient and scientific decision support for project managers.

[0038] Step S500: Reconstruct the hesitant fuzzy decision matrix based on the reduced attribute set, obtain the comprehensive evaluation value of each decision object, sort all decision objects according to the comprehensive evaluation value, and output the optimal sequence of project schedule indicators.

[0039] Specifically, when obtaining the comprehensive evaluation value of each decision object, the process includes: for each attribute in the reduced attribute set, obtaining the hesitant fuzzy entropy of the hesitant fuzzy information corresponding to that attribute based on the hesitant fuzzy information corresponding to that attribute; using the complement value of the hesitant fuzzy entropy corresponding to each attribute as the objective weight of that attribute, where attributes with lower uncertainty have a higher weight in the comprehensive evaluation; for each decision object, averaging the hesitant fuzzy evaluation values ​​corresponding to each attribute to obtain the representative evaluation value of the decision object in each attribute dimension; and weighting and summing the objective weights of each attribute with the corresponding representative evaluation values ​​to obtain the comprehensive evaluation value of the decision object.

[0040] Specifically, when ranking all decision objects, the process includes: constructing a complete ranking list or sequence of all schedule indicator decision objects after ranking by comprehensive evaluation value and comparison of hesitant fuzzy variance; wherein, the ranking list is arranged from high to low according to the comprehensive evaluation value of the decision objects, and in the case of equal comprehensive evaluation values, the ranking is determined according to the variance of hesitant fuzzy evaluation value from small to large; the ranking list is used as the optimal sequence of schedule indicators to guide the optimization of railway engineering construction plans and resource allocation.

[0041] Understandably, by calculating the hesitant fuzzy entropy based on the hesitant fuzzy evaluation value of each attribute under each decision object after reduction, the uncertainty of the attribute is quantified. Then, the complement of the entropy of each attribute is used as an objective weight, so that attributes with lower uncertainty and stronger discriminative ability occupy higher weights in the comprehensive evaluation, thereby improving the accuracy and reliability of the comprehensive evaluation. For each decision object, its hesitant fuzzy evaluation values ​​under each attribute are averaged to obtain a representative evaluation value for subsequent weighted calculation. The objective weights of each attribute are weighted and summed with their corresponding representative evaluation values ​​to obtain the comprehensive evaluation value of each decision object. Based on this, all decision objects are ranked according to the comprehensive evaluation values ​​to form a complete optimal sequence of project schedule indicators. The ranking method first arranges the decision objects from high to low according to the comprehensive evaluation value. When there are cases where the comprehensive evaluation values ​​are equal, the variance of the hesitant fuzzy evaluation values ​​of the decision objects under each attribute is compared, with the smaller variance given priority, to ensure that the ranking process fully considers uncertainty while preserving the relative advantages of the decision objects. This method can systematically integrate multi-attribute and multi-expert evaluation information, taking into account both objectivity and uncertainty, to achieve scientific screening and ranking of schedule indicators. It provides a quantitative and operable decision-making basis for optimizing railway engineering construction plans and allocating resources, while reducing subjective bias in the decision-making process and improving the accuracy and reliability of schedule indicator selection.

[0042] For example, in a real railway engineering project, suppose there are three schedule control schemes as decision-making objects, and after attribute reduction, three key attributes are selected: geological condition level, complexity of key process connections, and construction management level. First, for each attribute, the hesitant fuzzy evaluation values ​​of multiple experts for each scheme are collected. For example, the membership values ​​of the first scheme under the geological condition attribute are 0.8, 0.85, and 0.9; under the key process connection complexity attribute, they are 0.75, 0.8, and 0.78; and under the construction management level attribute, they are 0.7, 0.72, and 0.75. Then, the hesitant fuzzy entropy of each attribute is calculated. The entropy value reflects the degree of uncertainty of the attribute. For example, the geological condition entropy is low, the key process connection complexity entropy is moderate, and the construction management level entropy is high. Using the entropy complement as the weight, the geological condition has the highest weight, followed by the key process, and the construction management level has the lowest weight. Subsequently, the membership set of each scheme under each attribute is averaged to obtain a representative evaluation value. This value is then weighted and summed with the corresponding attribute weights to calculate the comprehensive evaluation value for each scheme. For example, the comprehensive evaluation value for the first scheme is 0.82, for the second scheme it is 0.78, and for the third scheme it is 0.76. Based on the comprehensive evaluation values, the schemes are ranked from highest to lowest to form an optimal sequence: Scheme 1 > Scheme 2 > Scheme 3. If there are schemes with equal comprehensive evaluation values, the hesitant fuzzy variance of each scheme is further compared, with the schemes having smaller variances ranked higher to ensure that the ranking fully considers uncertainty. Finally, this ranking result serves as the optimal sequence for construction period indicators, which can guide the optimization of construction plans, such as determining priority construction schemes, rationally allocating construction resources, and adjusting the construction sequence of key processes, thereby effectively improving the efficiency and reliability of railway engineering construction.

[0043] In the above embodiments, by constructing a multi-dimensional schedule indicator system, factors affecting the railway project schedule, such as natural conditions, construction organization, resource allocation, technical solutions, and management level, are systematically categorized into attribute sets. Candidate schedule plans or indicators are then used as decision-making objects, providing a structured and standardized data foundation for schedule evaluation. This method can take into account the multi-dimensional characteristics of schedule decision-making, enabling project managers to comprehensively and systematically analyze the role and interrelationships of various schedule indicators. Secondly, by introducing hesitant fuzzy set theory, the evaluation information of each decision-maker under different attributes is modeled, forming a hesitant fuzzy decision matrix, effectively characterizing the uncertainty, hesitation, and subjective preferences in the evaluation information. This design improves the compatibility and expression accuracy for multiple decision-makers and diverse evaluation information, reflecting the true evaluation state of schedule indicators under various attributes without losing the original information features. Furthermore, by converting the hesitant fuzzy decision matrix into a covering approximate space and constructing β-coverage relationships and upper and lower approximate sets based on parameter β, this invention can scientifically divide the similarity neighborhood of decision objects, achieving effective screening of schedule indicators. The process calculates attribute dependencies based on approximate upper and lower sets, and then reduces the attribute set by eliminating redundant schedule indicators, making the indicator system more concise and discriminative, while also reducing the complexity of subsequent calculations. Finally, a hesitant fuzzy decision matrix is ​​reconstructed based on the reduced attribute set to obtain the comprehensive evaluation value of the decision objects, which is then ranked to generate an optimal schedule indicator sequence. This process not only improves the scientific validity and stability of schedule indicator ranking but also assists project managers in making optimization decisions, enabling the rational formulation of construction plans, the optimization of resource allocation, and the precise control of project progress.

[0044] In another preferred embodiment based on the above embodiments, such as Figure 3 As shown in the figure, this embodiment provides a railway engineering schedule index ranking system, including: an index modeling module, a hesitant fuzzy processing module, a coverage approximation construction module, an attribute reduction module, an evaluation value calculation module, and a ranking output module.

[0045] Specifically, the indicator modeling module is used to construct a multi-dimensional schedule indicator system for railway engineering, modeling influencing factors as attribute sets and defining candidate schedule schemes as decision objects; the hesitant fuzzy processing module is electrically connected to the indicator modeling module, and is used to receive evaluations of each decision object under various attributes from multiple experts, forming a hesitant fuzzy decision matrix; the coverage approximation construction module is electrically connected to the hesitant fuzzy processing module, and is used to convert the hesitant fuzzy decision matrix into a β-coverage approximation space, generating upper and lower approximation sets; the attribute reduction module is electrically connected to the coverage approximation construction module, and is used to eliminate redundant attributes based on dependency calculation, outputting the minimum attribute reduction set; the evaluation value calculation module is electrically connected to the attribute reduction module, and is used to calculate the comprehensive evaluation value of each decision object based on the reduced attributes; the ranking output module is electrically connected to the evaluation value calculation module, and is used to rank the decision objects according to the comprehensive evaluation value and output the optimal schedule indicator sequence.

[0046] To enable those skilled in the art to fully understand and implement this invention, the specific implementation principles of this invention are further supplemented below with a specific application scenario.

[0047] During the route selection process for a high-speed railway in a mountainous area of ​​Southwest China, five representative schedule control schemes (x1 to x5) were identified, each corresponding to different bridge-to-tunnel ratios, construction organization models, and resource allocation strategies. The project management team needed to comprehensively evaluate each scheme from dimensions such as natural conditions, construction organization, resource allocation, technical solutions, and management level to determine the optimal schedule control path. Against this backdrop, the system was launched. First, the index modeling module retrieved engineering geological survey reports, meteorological statistics, construction organization design documents, and expert survey records to extract six core influencing factors: a1 (geological condition level, classified as I–IV), a2 (climate window period, taking the annual effective construction days range), a3 (critical path length, unit km), a4 (number of construction teams), a5 (equipment input intensity, unit / month / km), and a6 (complexity of process connections, discretely assigned values ​​according to levels 1–5). An initial decision table T was constructed, and the five schemes were input into the system as a set of decision objects (U).

[0048] Subsequently, the hesitant fuzzy processing module receives independent evaluations from three senior railway engineering experts on each solution across six attributes. For example, regarding the performance of solution x3 on a6 (process connection complexity), expert A considers its membership degree of "beneficial to shortening the construction period" to be 0.65, expert B gives 0.70, and expert C gives 0.60. Then h... 3 6={0.60, 0.65, 0.70}. All such sets of multiple membership degrees constitute a complete 5×6 hesitant fuzzy decision matrix H, which fully preserves the divergence and hesitation characteristics of expert judgments without any prior aggregation, ensuring that the original uncertainty information is not diluted.

[0049] After receiving matrix H, the coverage approximation construction module first calculates the hesitant fuzzy distance between any two schemes attribute by attribute according to the modified Euclidean distance formula. Taking the distance between x2 and x4 under a1 (geological conditions) as an example, if h 2 1 ={0.25, 0.30}, h 4 1 ={0.28, 0.32, 0.29}, then the mean of the internal differences is calculated by double summation, and the square root is taken to obtain d1(x2,x4). When the average distance of all attributes d(x2,x4) ≤ δ (let δ = 0.15), it is determined that the two are indistinguishable under the current attribute. Therefore, for each attribute a j Generate overlay block C j (x i For example, C2(x1)={x1,x3} indicates that x1 and x3 have similar performance during the climate window period. Based on this, the system sets β=0.8 according to the current stage management objectives and calculates the similarity s_A(x) between any two schemes in the full attribute set. i ,x k )=1-d(x i ,x k If s_A(x1,x3)=0.83≥0.8, then x3∈N_A^0.8(x1), forming a β-neighborhood. All β-neighborhoods together constitute the β-covering approximation space, and based on this, the lower approximation (completely classifiable objects) and the upper approximation (objects that may belong to a certain class) are divided, providing a granular basis for subsequent reduction.

[0050] Based on the aforementioned approximate space, the attribute reduction module first categorizes the five solutions into three decision classes D according to their expected project duration and performance: D1={x4} (Excellent), D2={x2,x1} (Good), and D3={x3,x5} (Average). Then, the dependency γ(A,D) of the entire attribute set A on D is calculated. Testing revealed that after removing a3 (critical path length), the positive domain POS_{A{a3}}(D) remains equal to the original positive domain, i.e., γ(A{a3},D)=γ(A,D), indicating that the classification information provided by a3 has been covered by other attributes; similarly, a5 is also considered redundant. Finally, Red={a1,a2,a4,a6} is retained as the minimal reduction set, effectively eliminating redundant indicators caused by data collinearity or information overlap, and reducing model complexity.

[0051] The evaluation value calculation module is based on the reduced set Red, and first calculates the hesitant fuzzy entropy of each attribute. Taking a1 as an example, if the membership degree sets of each scheme under this attribute are generally concentrated (such as most h...), i1 If the differences between elements are small, then E(a1) is lower, and the corresponding weight w1 = 1 - E(a1) is higher, reflecting the objective weighting principle that "the more certain the information, the greater the weight". Then, for each scheme x... i In Red, take the mean membership degree μ for each attribute. ij For example, x4 under a2 h 4 2 If μ = {0.75, 0.80, 0.78}, then μ 4 2 =0.777. Finally, through weighted summation, we obtain S(x4) = w1·μ 4 1 +w2·μ 4 2 +w4·μ 4 4 +w6·μ 4 6 Complete the quantification of the comprehensive evaluation value.

[0052] The sorting output module sorts S(x1) to S(x5) in descending order. If a critical situation arises where S(x2) = S(x3), the variance of the hesitation fuzzy values ​​of the two under the Red attribute is further calculated. For example, x2's evaluation under a6 is {0.68, 0.70, 0.69}, with a small variance, indicating consensus among experts; while x3's evaluation under a6 is {0.60, 0.75, 0.65}, with a large variance, reflecting disagreement. In this case, x2 is prioritized due to its more stable evaluation. The final system outputs the optimal sequence [x4, x2, x1, x3, x5], explicitly recommending x4 as the optimal schedule control scheme. Its advantages stem from its comprehensive performance in geological adaptability, climate utilization efficiency, team configuration rationality, and process coordination. Moreover, this conclusion is based on eliminating redundant indicators, retaining original hesitation information, and introducing dynamic adjustment of the β threshold, significantly improving the robustness and interpretability of the decision.

[0053] It is understood that the method and system for ranking railway engineering schedule indicators in the above embodiments of the present invention have the same beneficial effects, and will not be described in detail here.

[0054] Those skilled in the art will understand that embodiments of this application can be provided as methods, systems, or computer program goods. Therefore, this application can take the form of a completely hardware embodiment, a completely software embodiment, or an embodiment combining software and hardware aspects. Furthermore, this application can take the form of a computer program goods embodied on one or more computer-usable storage media (including, but not limited to, disk storage, CD-ROM, optical storage, etc.) containing computer-usable program code.

[0055] This application is described with reference to flowchart illustrations and / or block diagrams of methods, apparatus (systems), and computer program goods according to embodiments of this application. It will be understood that each block of the flowchart illustrations and / or block diagrams, and combinations of blocks in the flowchart illustrations and / or block diagrams, can be implemented by computer program instructions. These computer program instructions can be provided to a processor of a general-purpose computer, special-purpose computer, embedded processor, or other programmable data processing apparatus to produce a machine, such that the instructions, which execute via the processor of the computer or other programmable data processing apparatus, generate instructions for implementing the flowchart... Figure 1 One or more processes and / or boxes Figure 1 A device that provides the functions specified in one or more boxes.

[0056] These computer program instructions may also be stored in a computer-readable storage medium that can direct a computer or other programmable data processing device to function in a particular manner, such that the instructions stored in the computer-readable storage medium produce an article of manufacture including instruction means, which are implemented in a process Figure 1 One or more processes and / or boxes Figure 1 The function specified in one or more boxes.

[0057] These computer program instructions may also be loaded onto a computer or other programmable data processing equipment to cause a series of operational steps to be performed on the computer or other programmable equipment to produce a computer-implemented process, thereby providing instructions that execute on the computer or other programmable equipment for implementing the process. Figure 1 One or more processes and / or boxes Figure 1 The steps of the function specified in one or more boxes.

[0058] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and not to limit it. Although the present invention has been described in detail with reference to the above embodiments, those skilled in the art should understand that modifications or equivalent substitutions can still be made to the specific implementation of the present invention. Any modifications or equivalent substitutions that do not depart from the spirit and scope of the present invention should be covered within the protection scope of the claims of the present invention.

Claims

1. A method for ranking railway engineering schedule indicators, characterized in that, include: Construct a multi-dimensional schedule indicator system for railway engineering, establish an attribute set based on factors affecting the schedule such as natural conditions, construction organization, resource allocation, technical solutions and management level, and establish the decision-making object based on candidate schedule control schemes or indicators; Based on the hesitant fuzzy set, a hesitant fuzzy decision matrix is ​​constructed for each decision-maker's evaluation information for each decision object under each attribute; The hesitant fuzzy decision matrix is ​​transformed into a covering approximation space, a β-covering relation is constructed based on the parameter β, and upper and lower approximation sets are established. The dependency of each attribute is calculated based on the upper and lower approximation sets. The attribute set is then reduced according to the dependency and redundant schedule indicators are removed. The hesitant fuzzy decision matrix is ​​reconstructed based on the reduced attribute set, the comprehensive evaluation value of each decision object is obtained, and all decision objects are sorted according to the comprehensive evaluation value to output the optimal sequence of project schedule indicators.

2. The method for ranking railway engineering schedule indicators as described in claim 1, characterized in that, When constructing a multi-dimensional schedule indicator system for railway engineering, the following should be included: Collect raw data on factors affecting the construction period in railway engineering, including geological condition level, climate window period, critical line length, number of construction teams, equipment input intensity, and complexity of process connection; The above factors are mapped to discretized or interval-based attribute variables, forming an attribute set A = {a1, a2, ..., a...} n }; Define the schedule control scheme or indicator to be evaluated as the set of decision objects U={x1,x2,...,x}. m }; Establish an initial decision table T, where T = (U, A, V, f). Where: V is the attribute range, and f is the information function.

3. The method for ranking railway engineering schedule indicators as described in claim 2, characterized in that, When constructing a hesitant fuzzy decision matrix based on the evaluation information of each decision-maker for each decision object under each attribute using hesitant fuzzy sets, the following is included: Obtain the set of decision objects and attribute set in the railway engineering schedule indicator decision problem. The decision objects are the schedule indicators to be sorted, and the attributes are multiple factors that affect the schedule evaluation. These serve as the input objects for constructing the decision matrix. For each decision object under each attribute, the membership degree evaluation values ​​independently provided by multiple experts are collected. The membership degree evaluation values ​​are used to characterize the degree to which the decision object conforms to the evaluation target under the corresponding attribute. The membership degree evaluation values ​​given by multiple experts for the same decision object under the same attribute are aggregated to form a membership degree set. The membership degree set is not averaged, weighted or other aggregation operations are performed to retain the hesitation and uncertainty information in the evaluation results. Using the set of membership degrees corresponding to each decision object under each attribute as matrix elements, construct the hesitant fuzzy decision matrix: H=[h ij ] m×n In which each element h in the matrix ij This represents the set of hesitant fuzzy membership degrees of the i-th decision object under the j-th attribute, serving as the input data basis for subsequent indicator screening and ranking.

4. The method for ranking railway engineering schedule indicators as described in claim 1, characterized in that, Converting the hesitant fuzzy decision matrix into a covering approximate space includes: For each attribute a j ∈A, calculate the hesitant fuzzy distance d(x) between any two objects based on its hesitant fuzzy values ​​across all decision objects. i x k ); Set a distance threshold δ, if d(x) i x k If x ≤ δ, then x is considered to be... i With x k In attribute a j The following cannot be distinguished; For each object x i , with its attribute a j All indistinguishable objects below constitute an overlay block C. j (x i ); All covered blocks C j (x i ) Constituting attribute a j Corresponding coverage C j The entire set of {C1,C2,...,C} is covered. n } constitutes a covering approximate space, Where, x i For the i-th decision object currently being referenced, x k For x i The k-th decision object to be compared.

5. The method for ranking railway engineering schedule indicators according to claim 4, characterized in that, When constructing a β-coverage relationship based on parameter β, the following is included: For any two project schedule indicator decision objects, under the constraint of the selected attribute subset, the average hesitant fuzzy similarity of the two decision objects on the attribute subset is calculated based on the hesitant fuzzy decision matrix; When the average hesitant fuzzy similarity is not less than a preset similarity threshold, it is determined that one of the decision objects belongs to the threshold neighborhood of the other decision object under the attribute subset. Using the threshold neighborhood formed by each decision object under the attribute subset as the basic information granularity, a covering approximate space based on threshold constraints is constructed. The similarity threshold is a positive number between zero and one, and is preset according to the engineering management objectives or construction phase requirements of the railway project.

6. The method for ranking railway engineering schedule indicators as described in claim 5, characterized in that, The hesitant ambiguity distance d(x) i ,x k When using the modified Euclidean distance formula, the following is included: For each attribute dimension, pair up all possible membership values ​​in the set of hesitant fuzzy evaluation values ​​of the two decision objects under that attribute, and obtain the squared difference of each pair of membership values. Sum and average all squared differences under the same attribute, then average all attribute dimensions, and finally take the square root of the average to obtain the modified Euclidean distance between the two decision objects, which is used as their hesitant fuzzy distance. Based on hesitant fuzzy distance, the hesitant fuzzy similarity of any two project schedule indicator decision objects under the constraints of a selected attribute subset is obtained, and the hesitant fuzzy similarity is used to construct the neighborhood relationship of the decision objects. The similarity is obtained by subtracting the hesitant fuzzy distance obtained under the attribute subset from one. The numerical range is between zero and one. The closer the similarity value is to one, the higher the similarity between the two decision objects under the specified attribute subset.

7. The method for ranking railway engineering schedule indicators according to claim 1, characterized in that, When obtaining the dependence of each attribute on the decision classification based on the upper approximation set and the lower approximation set, it includes: Select any subset of attributes from the attribute set. Based on the discriminative power of the attribute subset for decision classification, determine the set of decision objects to which the attribute subset can uniquely determine the decision classification. Construct the positive domain corresponding to the attribute subset based on the set of decision objects. The ratio of the number of decision objects contained in the positive domain to the total number of decision objects is used as the degree of dependence of the attribute subset on the decision classification. In the attribute subset, remove individual attributes one by one, and re-acquire the dependency level after removal. If the dependency level remains unchanged before and after removing an attribute, the attribute is determined to be a redundant attribute. All attributes deemed non-redundant are retained, and a minimal attribute reduction set is constructed for screening and ranking project schedule indicators.

8. The method for ranking railway engineering schedule indicators as described in claim 1, characterized in that, When obtaining the comprehensive evaluation value of each decision-making object, the following should be included: For each attribute in the reduced attribute set, the hesitant fuzzy entropy of the hesitant fuzzy information corresponding to the attribute is obtained based on the hesitant fuzzy information corresponding to the attribute. The complement of the hesitant fuzzy entropy corresponding to each attribute is used as the objective weight of that attribute. Among them, the lower the uncertainty of an attribute, the higher its weight in the comprehensive evaluation. For each decision object, the mean value of its hesitant fuzzy evaluation value under each attribute is processed to obtain the representative evaluation value of the decision object in each attribute dimension; The objective weights of each attribute are weighted and summed with their corresponding representative evaluation values ​​to obtain the comprehensive evaluation value of the decision-making object.

9. The method for ranking railway engineering schedule indicators as described in claim 8, characterized in that, When ranking all decision objects, including: Construct a complete sorted list or sequence of all project schedule decision objects after sorting by comprehensive evaluation values ​​and comparing hesitant fuzzy variance; The ranking list is arranged from high to low according to the comprehensive evaluation value of the decision objects, and in the case of equal comprehensive evaluation values, the ranking is determined according to the variance of the hesitant fuzzy evaluation value from small to large. The sorted list is used as a preferred sequence of time indicators to guide the optimization of railway engineering construction plans and resource allocation.

10. A railway engineering schedule index ranking system, employing the railway engineering schedule index ranking method as described in any one of claims 1-9, characterized in that, include: The indicator modeling module is used to construct a multi-dimensional schedule indicator system for railway engineering, model influencing factors as a set of attributes, and define candidate schedule options as decision objects. The hesitant fuzzy processing module is electrically connected to the indicator modeling module. The hesitant fuzzy processing module is used to receive evaluations of each decision object under various attributes from multiple experts and form a hesitant fuzzy decision matrix. The covering approximation construction module is electrically connected to the hesitant fuzzy processing module. The covering approximation construction module is used to convert the hesitant fuzzy decision matrix into a β-covering approximation space and generate upper and lower approximation sets. The attribute reduction module is electrically connected to the coverage approximation construction module. The attribute reduction module is used to calculate and remove redundant attributes based on dependency and output the minimum attribute reduction set. The evaluation value calculation module is electrically connected to the attribute reduction module. The evaluation value calculation module is used to calculate the comprehensive evaluation value of each decision object based on the reduced attributes. The sorting output module is electrically connected to the evaluation value calculation module. The sorting output module is used to sort the decision objects according to the comprehensive evaluation value and output the optimal sequence of construction period indicators.

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