Multi-attribute decision making method based on hesitant fuzzy rough set and VIKOR

Abstract

The present application relates to the technical field of railway engineering construction period decision, especially to a multi-attribute decision-making method based on hesitant fuzzy rough set and VIKOR, which comprises constructing a decision matrix containing qualitative and quantitative information, reserving the uncertainty of expert evaluation through hesitant fuzzy processing, eliminating redundancy by using hesitant fuzzy rough set for attribute reduction, combining VIKOR method to calculate group benefit and individual regret, generating compromise ranking value and iterative optimization until the scheme ranking is stable. The present application can effectively improve the scientificity, flexibility and calculation efficiency of decision-making.

Description

Technical Field

[0001] This invention relates to the field of railway engineering schedule decision-making technology, and more specifically, to a multi-attribute decision-making method based on hesitant fuzzy rough sets and VIKOR. Background Technology

[0002] Railway construction typically involves multiple processes and complex schedule management, requiring comprehensive consideration of various objectives and attributes during the decision-making process. However, schedule decision-making involves a large amount of intertwined qualitative and quantitative information, and the subjective preferences of decision-makers can influence the evaluation results, leading to redundant raw decision information and low processing efficiency.

[0003] Currently, traditional methods largely rely on experience or single criteria for evaluation, making it difficult to systematically handle multi-attribute, multi-objective, and criterion-conflicting problems, resulting in decisions lacking scientific rigor and rationality. To address information uncertainty, fuzzy set theory has been introduced into schedule indicator decision-making. By effectively characterizing uncertain, fuzzy, or missing data, it can improve decision-making efficiency while preserving information accuracy. With the development of research, extended fuzzy set theories such as intuitionistic fuzzy sets, hesitant fuzzy sets, and interval-valued fuzzy sets have emerged. These methods can compensate for the limitations of traditional fuzzy sets in handling uncertain information from different dimensions, providing more accurate tools for multi-attribute decision-making. However, simply using fuzzy set methods still struggles to simultaneously consider group benefits and individual shortcomings, resulting in problems of insufficient rationality and flexibility in the ranking of complex schedule indicators.

[0004] Therefore, how to efficiently integrate multi-attribute information, scientifically handle decision-makers' preferences and information uncertainty, and construct a reasonable compromise decision-making model in railway engineering schedule indicator decision-making is an urgent technical problem to be solved. Summary of the Invention

[0005] In view of this, the present invention proposes a multi-attribute decision-making method based on hesitant fuzzy rough set and VIKOR, which aims to solve the problems that existing multi-attribute schedule index decision-making methods in the current technology are unable to systematically handle uncertain information and multiple criterion conflicts, and that simple fuzzy set methods cannot take into account both group benefits and individual regrets, resulting in a lack of scientificity, rationality and flexibility in decision-making results.

[0006] This invention proposes a multi-attribute decision-making method based on hesitant fuzzy rough sets and VIKOR, including: A multi-attribute decision matrix for railway engineering schedule indicators is constructed, wherein the decision matrix contains multiple candidate schemes and evaluation values ​​corresponding to each schedule attribute; The evaluation values ​​in the decision matrix are subjected to hesitant fuzzification processing to form a hesitant fuzzy decision information table; Based on the hesitant fuzzy rough set theory, the hesitant fuzzy decision information table is reduced in terms of attributes. Redundant attributes are removed and key discriminative features are retained to obtain the reduced attribute set. Based on the reduced attribute set, calculate the positive and negative ideal solutions for each candidate scheme under each attribute; The group benefit value, individual regret value, and compromise ranking value of each candidate scheme were calculated based on the VIKOR method. All candidate solutions are ranked according to the compromise ranking value to determine the optimal project duration.

[0007] Furthermore, the construction of the multi-attribute decision matrix for railway engineering schedule indicators includes: Collect key process nodes involved in railway engineering, and determine the attribute set used to evaluate the rationality of the construction period. The attribute set includes qualitative attributes and quantitative attributes. For each candidate schedule, obtain linguistic or numerical evaluation results for each attribute. All evaluation results are entered into a structured table to form an initial multi-attribute decision matrix.

[0008] Furthermore, the process of performing hesitant fuzzification on the evaluation values ​​in the decision matrix includes: For qualitative attributes, language evaluation is converted into corresponding hesitant fuzzy elements, each of which consists of several membership values. For quantitative attributes, they are normalized and mapped to hesitant fuzzy numbers, and multiple possible membership degrees are generated by setting a threshold range. The evaluation values ​​of all attributes are uniformly represented as hesitant fuzzy numbers to construct a complete hesitant fuzzy decision information table.

[0009] Furthermore, the attribute reduction based on hesitant fuzzy rough sets includes: Define hesitant fuzzy similarity relations and construct equivalence classes based on the distance metric between hesitant fuzzy numbers; The coverage capability of each attribute subset to the decision class is obtained based on the upper and lower approximation operators; Based on a heuristic search strategy, attributes whose contribution to classification is lower than a preset threshold are gradually eliminated to obtain the minimum attribute reduction set; Verify whether the reduced attribute set still retains the discriminative power of the original decision information, where: If not, a heuristic search strategy is used, and the preset threshold is adjusted to obtain the minimum attribute reduction set a second time.

[0010] Furthermore, verifying whether the reduced attribute set still retains the discriminative power of the original decision information also includes determining whether the attribute reduction converges, wherein: If the attribute sets obtained from two consecutive attribute reductions are exactly the same, or if the improvement in classification accuracy from the addition of an attribute is less than the set tolerance, then the attribute reduction process is considered to have converged.

[0011] Furthermore, the calculation of the positive ideal solution and the negative ideal solution includes: For each attribute, the set of the largest membership degree of the attribute among all candidate solutions is selected as the positive ideal solution, and the set of the smallest membership degree is selected as the negative ideal solution. The attribute weights are determined by the hesitant fuzzy entropy method, reflecting the differences in the amount of information of each attribute in the overall decision-making process.

[0012] Furthermore, in the VIKOR compromise ranking process, each candidate scheme is assigned an independent ranking unit for comprehensive evaluation, wherein: For each candidate solution, construct its corresponding trade-off index vector, which includes a group benefit component and an individual regret component; Based on a preset trade-off coefficient, the two components are linearly combined to generate the final trade-off ranking value; All candidate solutions are ranked in ascending order based on their compromise values, with the solutions ranked higher corresponding to the better schedule indicators.

[0013] Furthermore, regarding the sensitivity adjustment of individual regret values, the compromise coefficient is dynamically adjusted based on the degree of disagreement within the decision-making group.

[0014] Furthermore, during the calculation of the compromise ranking value, the number of attributes involved in the VIKOR operation is determined, and the difference between each scheme and the ideal solution is measured by weighted Euclidean distance.

[0015] Furthermore, the termination condition is that the number of attribute reduction iterations reaches a preset upper limit, or the top three candidate schemes in three consecutive compromise ranking results are completely consistent.

[0016] Compared with existing technologies, the advantages of this invention are as follows: By constructing railway engineering schedule indicators into a multi-attribute decision matrix and applying hesitant fuzzification to the evaluation values, this invention can effectively characterize the uncertainties, fuzziness, and subjective preferences of decision-makers in the decision-making process, retaining key information while reducing information loss, thereby improving the processing accuracy and reliability of complex schedule data. Secondly, attribute reduction based on hesitant fuzzy rough set theory helps to eliminate redundant attributes and retain features with key discriminative capabilities for decision results, achieving information compression and feature extraction. This not only improves the efficiency of decision calculation but also enhances the adaptability and stability of the decision model in multi-attribute, multi-objective environments. Furthermore, by combining the VIKOR method to calculate the group benefit value, individual regret value, and compromise ranking value of the reduced attribute set, a scientific ranking of candidate schemes is achieved while considering overall benefits and individual preferences. This compromise ranking mechanism can reasonably resolve multi-criteria conflicts, making the selection of the final schedule indicator scheme more in line with actual needs and management objectives. Finally, by organically combining attribute reduction and compromise ranking, we not only improved the scientificity, rationality, and flexibility of railway engineering schedule indicator decision-making, but also provided a scalable technical means for multi-objective and multi-attribute decision-making scenarios, which has significant practical value and engineering application potential. Attached Figure Description

[0017] 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 of a multi-attribute decision-making method based on hesitant fuzzy rough set and VIKOR provided for embodiments of the present invention; Figure 2 This is a flowchart illustrating a multi-attribute decision-making method based on hesitant fuzzy rough sets and VIKOR, provided as an embodiment of the present invention. Detailed Implementation

[0018] 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.

[0019] VIKOR (VIKOR) refers to a compromise ranking method used to solve multi-attribute, multi-objective decision-making problems. It is used to balance group benefits and individual regrets to provide a basis for the optimal selection of complex decision-making solutions.

[0020] A multi-attribute decision matrix (MADM) is a matrix form used to represent the evaluation values ​​of each candidate solution on different attributes. It is used to systematically analyze multi-objective, multi-attribute decision problems in order to compare and rank the solutions.

[0021] Hesitant fuzzyization refers to the process of converting evaluation values ​​in a decision matrix into hesitant fuzzy numbers, which are used to characterize the uncertainty and multiple possible preferences of decision-makers during evaluation, thereby improving the accuracy of decision information.

[0022] Attribute reduction refers to the process of eliminating redundant or low-contribution attributes in multi-attribute decision-making, in order to retain attributes that are key to the ability to discriminate the decision results, thereby improving the efficiency of decision computation and the stability of the model.

[0023] A positive ideal solution (PIS) is an ideal solution formed when each attribute takes the optimal evaluation value in multi-attribute decision-making. It is used as a reference benchmark for comparing the merits of candidate solutions to guide ranking and optimization.

[0024] A negative ideal solution (NIS) is an ideal solution formed when each attribute takes the worst evaluation value in a multi-attribute decision-making process. It is used as a reference benchmark for comparing the disadvantages of candidate solutions to assist in compromise evaluation.

[0025] Individual Regret Value (IRV) is an evaluation index calculated based on the difference between a candidate solution and the ideal solution. It is used to reflect the degree of deviation of a solution from its most unfavorable attribute, thereby measuring the risk and dissatisfaction of the solution.

[0026] The Compromise Ranking Value is a comprehensive evaluation index calculated by combining group benefit value and individual regret value. It is used to rank candidate solutions in a compromise manner to balance overall benefits and local biases.

[0027] An equivalence class is a set of attributes or schemes that are partitioned based on hesitant fuzzy similarity relations. It is used to group similar objects in rough set analysis to support reduction and discriminant analysis.

[0028] Coverage ability refers to the degree to which a subset of attributes distinguishes and covers decision-class objects. It is used to evaluate the contribution of a subset of attributes to the ability to maintain decision information and to guide attribute reduction.

[0029] The Minimal Membership Set, in hesitant fuzzy rough set analysis, is a set used to describe the minimum membership degree of an object under a subset of attributes. It is used to assist in determining the importance of attributes and formulating reduction strategies.

[0030] Hesitant fuzzy rough set refers to an analytical method that combines hesitant fuzzy set theory with rough set theory. It is used to perform attribute reduction and discrimination in uncertain and fuzzy information environments in order to extract key features for decision-making.

[0031] The VIKOR-based Multi-Attribute Decision Method is a decision-making strategy that combines the VIKOR method with multi-attribute decision matrices and uncertainty information processing methods. It is used to prioritize candidate solutions by balancing group benefits with individual regrets.

[0032] like Figures 1-2 As shown, in some embodiments of this application, this embodiment provides a multi-attribute decision-making method based on hesitant fuzzy rough sets and VIKOR, including: Step S100: Construct a multi-attribute decision matrix for railway engineering schedule indicators. The decision matrix includes multiple candidate schemes and evaluation values ​​corresponding to each schedule attribute.

[0033] Specifically, when constructing a multi-attribute decision matrix for railway engineering schedule indicators, the process includes: collecting key process nodes involved in railway engineering, determining the attribute set used to evaluate the rationality of the schedule, the attribute set including qualitative and quantitative attributes; for each candidate schedule plan, obtaining linguistic or numerical evaluation results under each attribute; and uniformly entering all evaluation results into a structured table to form an initial multi-attribute decision matrix.

[0034] Understandably, by constructing a multi-attribute decision matrix, key process nodes, schedule-influencing factors, and various attributes involved in railway engineering construction are systematically and quantitatively integrated to form a unified decision-making information carrier. Specifically, qualitative attributes (such as construction difficulty, environmental constraints, and management risks) and quantitative attributes (such as process duration, resource input, and construction frequency) are characterized using a unified method, enabling the decision matrix to simultaneously reflect the precision of quantitative data and the fuzziness of qualitative data. This matrix representation not only facilitates horizontal comparison and vertical analysis of candidate schedule plans but also provides a structured data foundation for subsequent fuzzy processing, attribute reduction, and compromise ranking, achieving systematic management of multi-objective, multi-attribute schedule decisions. Furthermore, by utilizing the multi-attribute decision matrix, the differences in the advantages and disadvantages of different schedule plans under various attributes can be intuitively reflected, providing basic data support for quantifying group benefits and individual shortcomings. This provides an operational mathematical basis for comprehensive evaluation, scientific ranking, and schedule optimization, avoiding the subjectivity and arbitrariness of traditional experience-based judgments.

[0035] For example, for a railway construction project, key process nodes such as tunnel excavation, bridge construction, track laying, and hub platform construction can be used as candidate schedule options for evaluation. For each process node, multi-dimensional attributes can be selected as evaluation indicators, specifically: Quantitative attributes: process duration (days), number of construction personnel (person-days), equipment usage intensity (hours), and material consumption (tons). Qualitative attributes: construction difficulty level (high / medium / low), environmental constraints (strict / moderate / lenient), construction risk level (high / medium / low), and coordination complexity of key nodes (high / medium / low). For each process node, decision-makers can provide verbal evaluations (e.g., "high risk," "medium risk") or numerical evaluations (e.g., 45-day schedule, 200 person-days of manpower), and input all evaluation results into a structured table to form an initial multi-attribute decision matrix. Rows in the matrix represent candidate options (e.g., tunnel excavation option 1, option 2), columns represent attributes (e.g., schedule, risk level), and each cell records the corresponding evaluation value. This matrix allows for a clear observation of the performance of each project schedule under different attributes, providing fundamental information for subsequent hesitant fuzzy number processing: qualitative evaluations can be converted into hesitant fuzzy numbers to quantify the decision-maker's uncertain preferences, while quantitative evaluations can be normalized to eliminate the influence of dimensions. Ultimately, this matrix provides reliable data support for attribute reduction, trade-off ranking, and the selection of the optimal project schedule, making the entire railway project schedule decision-making more scientific, reasonable, and feasible.

[0036] Step S200: Perform hesitant fuzzification processing on the evaluation values ​​in the decision matrix to form a hesitant fuzzy decision information table.

[0037] Specifically, when performing hesitant fuzzification on the evaluation values ​​in the decision matrix, the following steps are taken: for qualitative attributes, the linguistic evaluation is converted into corresponding hesitant fuzzy elements, each of which consists of several membership values; for quantitative attributes, they are normalized and mapped to hesitant fuzzy numbers, and multiple possible memberships are generated by setting a threshold range; the evaluation values ​​under all attributes are uniformly represented in the form of hesitant fuzzy numbers, and a complete hesitant fuzzy decision information table is constructed.

[0038] Understandably, hesitant fuzzification transforms the qualitative and quantitative evaluation values ​​in a multi-attribute decision matrix into hesitant fuzzy numbers, thus characterizing the uncertainty and multiple possible preferences of decision-makers during the evaluation process. For qualitative attributes, fuzzy judgments are quantified by mapping linguistic evaluations to hesitant fuzzy elements composed of membership values. For quantitative attributes, multiple possible membership degrees are generated through normalization and setting threshold intervals, characterizing the uncertainty of numerical information. Ultimately, all attributes are uniformly represented as hesitant fuzzy numbers, forming a complete hesitant fuzzy decision information table, providing a structured and quantifiable decision data foundation for subsequent attribute reduction and compromise ranking.

[0039] For example, in evaluating the construction period of a railway construction process, the qualitative attribute "construction difficulty" can be mapped to hesitant fuzzy elements with multiple membership values, such as "high," "medium," and "low," for linguistic evaluations. For instance, high difficulty corresponds to {0.8, 0.9, 1.0}. For the quantitative attribute "process duration," the original day data, such as 45 days and 60 days, is normalized to the 0-1 range, and several possible membership values ​​are generated based on a preset threshold range, such as 45 days being mapped to {0.7, 0.8}. By summarizing the hesitant fuzzy numbers for each attribute, a complete hesitant fuzzy decision information table can be formed. This table comprehensively reflects the uncertain evaluation of candidate construction period schemes under multiple attributes, providing accurate data support for subsequent decision-making.

[0040] Step S300: Based on the hesitant fuzzy rough set theory, the hesitant fuzzy decision information table is reduced in terms of attributes. Redundant attributes are removed and key discriminative features are retained to obtain the reduced attribute set.

[0041] Specifically, when performing attribute reduction based on hesitant fuzzy rough sets, the process includes: defining hesitant fuzzy similarity relations and constructing equivalence classes based on the distance metric between hesitant fuzzy numbers; obtaining the coverage capability of each attribute subset for the decision class based on upper and lower approximation operators; gradually eliminating attributes whose contribution to classification is lower than a preset threshold based on a heuristic search strategy to obtain the minimum attribute reduction set; and verifying whether the reduced attribute set can still maintain the discriminative ability of the original decision information. If not, the minimum attribute reduction set is obtained a second time based on a heuristic search strategy and by adjusting the preset threshold.

[0042] Specifically, when verifying whether the reduced attribute set can still maintain the discriminative ability of the original decision information, it also includes judging whether the attribute reduction is converged. If the attribute sets obtained by two consecutive attribute reductions are exactly the same, or the improvement of classification accuracy by the added attribute is less than the set tolerance, then the attribute reduction process is judged to be converged.

[0043] Understandably, by defining hesitant fuzzy similarity relations and based on the distance metric between hesitant fuzzy numbers, decision objects are grouped into equivalence classes, providing a foundation for rough set upper and lower approximation calculations. Subsequently, upper and lower approximation operators are used to calculate the coverage capability of each attribute subset to the decision class, quantifying the contribution of each attribute to the subset classification. Combined with a heuristic search strategy, attributes with a classification contribution below a preset threshold are gradually eliminated, achieving a minimization reduction of the attribute set. Simultaneously, discriminative capability verification ensures that the reduced attribute set retains the classification capability of the original decision information, avoiding information loss. Furthermore, a convergence judgment mechanism (such as two consecutive reductions of the attribute set being identical, or the improvement in classification accuracy from a newly added attribute being less than a set tolerance) controls the stability and reliability of the reduction process, ensuring that the final reduced attribute set is both minimized and possesses high discriminative capability, providing high-quality input data for subsequent VIKOR compromise ranking. It can be seen that, on the one hand, this method uses rough set theory to systematically analyze hesitant and fuzzy information, thereby achieving fine screening and compression of multi-attribute and multi-objective decision-making information; on the other hand, it uses heuristic search and threshold adjustment mechanisms to dynamically control the attribute reduction process, so that the method can balance efficiency and accuracy, and has flexibility and adaptability when dealing with redundant information and attribute conflicts.

[0044] For example, in evaluating the schedule of a railway construction project, candidate options include key processes such as tunnel excavation, bridge construction, and track laying. The attribute set includes process duration, construction difficulty, resource consumption, environmental constraints, and coordination complexity. First, the hesitant fuzzy decision information under each attribute is divided into equivalence classes according to a preset distance metric, grouping schedule options with similar performance into the same class. Then, the coverage capability of each attribute subset in classifying schedule options is calculated using upper and lower approximation operators. For example, it is found that environmental constraints and coordination complexity contribute little to distinguishing options in most cases. Next, a heuristic search strategy is used to remove low-contribution attributes, and the discriminative ability verification results are used to determine whether the reduced attribute set can still correctly distinguish different options. If the discriminative ability decreases, the process reverts to the previous step and adjusts the threshold for recalculation until attribute reduction converges, i.e., two consecutive reduction results are consistent or the improvement in classification accuracy from adding attributes is less than the tolerance. Ultimately, we obtain a minimal set of attributes that retains key discriminative features, such as only retaining the duration of the work process and the difficulty of construction. This provides a stable and efficient data foundation for subsequent calculations of group benefit values, individual regret values, and compromise ranking, ensuring the scientific and rational nature of the schedule indicator decisions.

[0045] Step S400: Based on the reduced attribute set, calculate the positive ideal solution and negative ideal solution of each candidate scheme under each attribute.

[0046] Specifically, the calculation of positive and negative ideal solutions includes: for each attribute, selecting the set of the largest membership degree of the hesitant fuzzy number of the attribute among all candidate solutions as the positive ideal solution and the set of the smallest membership degree as the negative ideal solution; the attribute weight is determined by the hesitant fuzzy entropy method, reflecting the difference in information content of each attribute in the overall decision-making.

[0047] Understandably, for each attribute, by traversing the hesitant fuzzy numbers of all candidate solutions, the maximum set of membership values ​​is selected as the positive ideal solution, representing the optimal performance under that attribute; the minimum set of membership values ​​is selected as the negative ideal solution, representing the worst performance under that attribute, thus constructing an evaluation reference range for each attribute. In this way, the information of uncertain and diverse candidate solutions can be quantified into clear ideal and anti-ideal indicators. Simultaneously, this method combines the hesitant fuzzy entropy method to calculate the weights of each attribute, reflecting the differences in information content and importance of different attributes in the overall decision-making process. The hesitant fuzzy entropy method calculates information entropy based on the membership distribution of each attribute. A larger entropy value indicates that the attribute's information is more dispersed and contributes less to the decision; a smaller entropy value indicates that the attribute's information is more concentrated and contributes more to the decision, thus assigning reasonable weights to attributes. This weight calculation method can objectively reflect the impact of each attribute on the overall scheme ranking in multi-attribute decision-making, providing a scientific basis for subsequent compromise ranking and ensuring that the decision results conform to both overall benefits and local attribute biases. It can be seen that, on the one hand, by constructing positive and negative ideal solutions, the multi-attribute performance of candidate solutions is quantified into standardized reference points, which facilitates systematic comparison among candidate solutions; on the other hand, by calculating entropy weights, the contribution of each attribute in the trade-off ranking is reasonably allocated, thereby improving the scientificity, rationality and operability of the decision-making process.

[0048] For example, when evaluating the construction sequence and schedule of a railway project, let's assume the reduced attribute set consists of "process duration" and "construction difficulty." For the "process duration" attribute, the hesitant fuzzy numbers of candidate schemes such as tunnel excavation, bridge construction, and track laying are compared. The set with the largest membership degree is taken as the positive ideal solution (representing the best value reference for the shortest-term scheme), and the set with the smallest membership degree is taken as the negative ideal solution (representing the worst value reference for the longest-term scheme). Similarly, for the "construction difficulty" attribute, the set with the largest membership degree is taken as the positive ideal solution, and the set with the smallest membership degree is taken as the negative ideal solution, thus forming a complete ideal solution reference system. Next, the hesitant fuzzy entropy method is applied to calculate the weights of each attribute. For example, "process duration" has a lower entropy value and more concentrated information, so it is assigned a higher weight; "construction difficulty" has a higher entropy value and more dispersed information, so it is assigned a relatively lower weight. This weight allocation can reasonably reflect the importance of each attribute to the overall schedule ranking. Finally, by combining the performance and attribute weights of each candidate scheme under positive and negative ideal solutions, accurate and quantifiable basic data are provided for the VIKOR method to calculate the group benefit value, individual regret value, and compromise ranking value, making the process of optimizing the schedule scheme scientific, reasonable, and efficient.

[0049] Step S500: Calculate the group benefit value, individual regret value, and compromise ranking value for each candidate scheme based on the VIKOR method.

[0050] Specifically, in the VIKOR compromise ranking process, each candidate scheme is assigned an independent ranking unit for comprehensive evaluation. This involves: constructing a corresponding compromise index vector for each candidate scheme, which includes a group benefit component and an individual regret component; generating the final compromise ranking value by linearly combining the two components based on a preset compromise coefficient; and sorting the compromise ranking values ​​of all candidate schemes in ascending order, with the scheme ranked higher corresponding to a better schedule index scheme.

[0051] Specifically, for the sensitivity adjustment of individual regret values, the compromise coefficient is dynamically adjusted based on the degree of disagreement in the decision-making group.

[0052] Specifically, the termination condition is that the number of attribute reduction iterations reaches a preset upper limit, or the top three candidate solutions in three consecutive compromise ranking results are completely consistent.

[0053] Understandably, for each candidate solution, a compromise index vector is constructed, containing two components: a group benefit component reflecting the solution's contribution to the overall decision-making group, and an individual regret component reflecting the solution's inferiority relative to the optimal solution in various attributes. By pre-setting a compromise coefficient, the group benefit component and the individual regret component are linearly combined to generate the final compromise ranking value, thereby achieving a comprehensive evaluation of the candidate solutions. Secondly, this method also introduces a sensitivity adjustment mechanism for individual regret values: when there are significant differences or disagreements among members of the decision-making group, the compromise coefficient is dynamically adjusted to increase the weight of individual regret in the final ranking, ensuring that the ranking result reflects both overall benefits and local biases and individual concerns. Furthermore, to ensure the stability of the ranking process and the convergence of the algorithm, iteration termination conditions are introduced, including reaching a preset upper limit for the number of attribute reduction iterations, or the top three candidate solutions being completely consistent in three consecutive compromise ranking results, thereby avoiding excessive iteration and result fluctuations, and improving decision reliability and operational efficiency. It can be seen that, on the one hand, this method quantifies and standardizes conflict information in multi-attribute decision-making by constructing a trade-off index vector between group benefits and individual regrets; on the other hand, it achieves adaptive adjustment and stable output of trade-off ranking through dynamic trade-off coefficients and termination condition control, thus ensuring the scientific, rational and operable nature of the railway construction schedule plan.

[0054] For example, for the key processes of a railway construction project, including tunnel excavation, bridge construction, and track laying, candidate timeline options are Option A, Option B, and Option C. Under the reduced key attributes, such as process duration and construction difficulty, each option first calculates a group benefit value (e.g., average time reduction, resource utilization optimization) and an individual regret value (e.g., the degree of exceeding the shortest time or lowest difficulty option). Then, the group benefit value and individual regret value are linearly combined according to a preset trade-off coefficient to generate the final trade-off ranking value for each option. For example, if Option A has the lowest trade-off ranking value, then Option A is considered the optimal timeline option under comprehensive evaluation. During the trade-off process, if significant disagreements are found within the decision-making group, such as some members favoring the shortest timeline while others focus more on construction difficulty, the trade-off coefficient is dynamically adjusted to increase the weight of individual regret values, making the ranking result more consistent with the group's overall preference. The algorithm terminates when the top three options are completely consistent in three consecutive rankings, or when the attribute reduction iteration count reaches its upper limit, thus ensuring the stability and reliability of the ranking results. Ultimately, this method can provide a scientific and quantitative basis for optimizing the construction period of railway projects, achieve a reasonable balance between multiple attribute conflicts, and provide a reliable reference for the selection of actual construction schemes.

[0055] Step S600: Sort all candidate schemes according to the compromise ranking value and determine the optimal schedule index scheme.

[0056] Specifically, in the process of calculating the compromise ranking value, the number of attributes involved in the VIKOR operation is determined, and the difference between each scheme and the ideal solution is measured by weighted Euclidean distance.

[0057] Understandably, a compromise ranking value is used to comprehensively quantify and evaluate candidate solutions, thereby determining the optimal project schedule. First, the number of reduced key attributes participating in the VIKOR calculation is determined, and corresponding weights are assigned based on the information content or importance of each attribute in the overall decision-making process to ensure a reasonable contribution of different attributes to the final ranking result. Then, for each candidate solution, the difference between it and the positive ideal solution (the reference value that performs best on each attribute) and the negative ideal solution (the reference value that performs worst on each attribute) is calculated using weighted Euclidean distance, transforming the combined impact of group benefits and individual regrets into a single compromise ranking value. This compromise ranking value quantifies the overall performance of the candidate solution across all attributes while balancing the shortcomings of the solution in local attributes, thus providing a unified evaluation index for scientific decision-making. Second, during the ranking process, the compromise ranking values ​​of all candidate solutions are arranged in ascending order, and the solutions ranked highest are determined as the optimal project schedule. Simultaneously, this method, by dynamically adjusting attribute weights and compromise coefficients, can adaptively correct situations where there are significant disagreements within the decision-making group, improving the rationality and stability of the ranking results. The advantages of this technology are: on the one hand, it enables quantitative comparison of multi-attribute and multi-objective project schedule plans; on the other hand, it takes into account both group benefits and individual regrets, avoids a single indicator dominating the ranking, and ensures that the decision-making results are scientific, reasonable and operable.

[0058] For example, consider three key processes in a railway construction project: tunnel excavation, bridge construction, and track laying. Each process corresponds to candidate schedule options A, B, and C. The reduced key attributes include "process duration" and "construction difficulty." For each option, the compromise index value under the reduced attributes is first calculated using weighted Euclidean distance with the positive ideal solution (shortest process duration and lowest construction difficulty) and the negative ideal solution (longest process time and highest construction difficulty). Assuming option A performs best in process duration and second best in construction difficulty, the calculated compromise ranking value is 0.25; option B is 0.37, and option C is 0.42. By sorting the compromise ranking values ​​in ascending order, option A is ultimately determined as the optimal schedule option. In this process, the attribute weights can be dynamically adjusted based on the differences in opinions among the decision-making group. For example, if some decision-makers are more concerned with construction difficulty than the length of the project, the weight of the "construction difficulty" attribute is increased, and the compromise ranking value is recalculated, thus making the ranking result more consistent with the overall preferences of the group. Meanwhile, this method can monitor the stability of the compromise sorting in real time during the iteration process. When the top three schemes of the sorting results are completely consistent for multiple consecutive sorting results, the calculation is terminated to ensure the reliability and operability of the sorting results, and to provide a scientific and quantifiable decision-making basis for optimizing the construction period of railway projects.

[0059] 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.

[0060] In the practical application of railway engineering schedule optimization, the project management team first identifies key process nodes affecting the overall schedule based on engineering design drawings and construction organization plans, and determines an initial attribute set accordingly. This set includes seven attributes: "subgrade construction period", "bridge erection time", "tunnel breakthrough period", "track laying progress", "joint commissioning and testing duration", "construction organization difficulty", and "geological risk level". The first five are quantitative attributes, and the last two are qualitative attributes. Subsequently, a decision-making group composed of five railway engineering experts with senior professional titles is invited. Each expert independently evaluates three candidate schedule schemes A1, A2, and A3 under the above attributes, forming an initial multi-attribute decision matrix.

[0061] Entering the hesitant fuzzification stage, the system, based on the preset linguistic terminology-membership degree mapping rules, converts the linguistic values ​​such as "high," "medium," and "medium-high" given by experts under "construction organization difficulty" into hesitant fuzzy elements such as {0.8,0.9}, {0.4,0.5,0.6}, and {0.6,0.7,0.8}, respectively. It also performs union and deduplication on the evaluations of multiple experts under the same scheme, forming a single hesitant fuzzy number. For quantitative attributes such as "estimated total construction period in days," the system first uses a maximum normalization formula to map the original number of days to the [0,1] interval, and then generates discrete membership degree points centered on the mean and with ±0.05 as the half-width, thus forming a hesitant fuzzy number. After all attributes are converted, a hesitant fuzzy decision information table is formed, where each cell is a finite set of real numbers, fully preserving the diversity and uncertainty of expert evaluations.

[0062] The hesitant fuzzy rough set attribute reduction process is then initiated. First, the average Hausdorff distance between any two schemes under each attribute is calculated. For example, the hesitant fuzzy numbers of schemes A1 and A2 under the "geological risk level" attribute are {0.7, 0.8} and {0.5, 0.6}, respectively, and their distance is 0.2. If the preset threshold δ=0.15, the two are determined to be dissimilar under this attribute. Based on all attribute combinations, an equivalence class partition U / P is constructed, and the approximation accuracy α_P is calculated using the lower and upper approximation operators of the hesitant fuzzy rough set. A greedy strategy is used to successively eliminate attributes that minimize the decrease in α_P. When the attribute set is reduced to three items: "tunnel breakthrough period", "geological risk level", and "critical path float time", further elimination of any attribute will cause the decrease in α_P to exceed the tolerance ε=0.02. At this point, the reduced attribute set is output. At the same time, the classification accuracy of this attribute set for the original decision class D is verified. If the error rate is ≤5%, the reduction result is accepted; otherwise, the callback parameters are recalculated.

[0063] Based on the reduced attribute set, the ideal solution calculation stage begins. For the "geological risk level" attribute, the hesitant fuzzy numbers {0.7, 0.8}, {0.5, 0.6}, and {0.8, 0.9} of A1, A2, and A3 are traversed, and the union {0.5, 0.6, 0.7, 0.8, 0.9} is taken. The set of maximum values ​​{0.9} is taken as the positive ideal solution h_a. + The set of minimum values ​​{0.5} is taken as the negative ideal solution h_a - Simultaneously, the hesitant fuzzy entropy E(a) of each attribute is calculated. For example, E(a) = 0.32 for "tunnel breakthrough period", E(a) = 0.41 for "geological risk level", and E(a) = 0.48 for "critical path float time". Based on this, the weights w_a are assigned to 0.4, 0.35, and 0.25 respectively, to ensure that attributes with more information receive higher weights.

[0064] Then, a VIKOR compromise sort is performed. For scheme A2, its hesitant fuzzy number under the "geological risk level" attribute is {0.5, 0.6}, which is consistent with h_a. + ={0.9} distance d(h,h + =√[((0.5-0.9)] 2 +(0.6-0.9) 2 [) / 2]=0.354, and ha - ={0.5} distance d(h,h - =0.071, the normalized distance ratio is 0.354 / (0.354+0.071)=0.833; after weighting, the calculation of S_i and R_i is taken into account, and finally the S_i of A2 is 0.31, R_i is 0.35; combined with v=0.5, the Qi is calculated as 0.5×(0.31-0.31) / (0.45-0.31)+0.5×(0.35-0.35) / (0.42-0.35)=0.38; similarly, the Q values ​​of A1 and A3 are calculated as 0.62 and 0.75 respectively, and the compromise ranking value is generated.

[0065] Then, the convergence judgment phase begins. The current reduced attribute set is recorded as {a3, a6, a4}, and the compromise ranking value is [0.62, 0.38, 0.75]. Compared with the result of the previous iteration, if the attribute set remains unchanged and the newly added attribute improves α_P by <0.01, and the top three Q values ​​for three consecutive iterations are A2, A1, and A3, then the termination condition is met. Otherwise, the label is returned, the δ value is finely adjusted (e.g., from 0.15 to 0.13) to enhance the discriminative power, and the attribute reduction and ranking are re-executed to form a closed-loop feedback.

[0066] Once the termination condition is met, the optimal solution A2 is output, along with its dependent reduced attribute set, hesitant fuzzy evaluation values ​​for each attribute, and compromise ranking values. Throughout the process, the hesitant fuzzy decision information table serves as the data hub, dynamically updated between the attribute reduction module and the VIKOR ranking module. The reduced attribute set serves as the simplified feature set, simultaneously driving the ideal solution calculation and the VIKOR module. The compromise ranking values ​​are fed back to the convergence judgment module as stability criteria. The three components achieve collaborative optimization through structure. The modules pass structure data through memory pointers. Attribute reduction uses adjacency lists to store equivalence classes to reduce space complexity. VIKOR distance calculation uses NumPy vectorized operations. Under the scale of 3 solutions and 3 attributes, the time taken for a single iteration is less than 0.1 seconds, demonstrating practical engineering efficiency.

[0067] In the above embodiments, by constructing railway engineering schedule indicators into a multi-attribute decision matrix and applying hesitant fuzzification to the evaluation values, this invention effectively characterizes the uncertainties, fuzziness, and subjective preferences of decision-makers in the decision-making process, retaining key information while reducing information loss, thereby improving the processing accuracy and reliability of complex schedule data. Secondly, attribute reduction based on hesitant fuzzy rough set theory helps eliminate redundant attributes and retain features with key discriminative capabilities for decision results, achieving information compression and feature extraction. This not only improves decision computation efficiency but also enhances the adaptability and stability of the decision model in multi-attribute, multi-objective environments. Furthermore, by combining the VIKOR method to calculate the group benefit value, individual regret value, and compromise ranking value of the reduced attribute set, a scientific ranking of candidate schemes is achieved while considering overall benefits and individual preferences. This compromise ranking mechanism can reasonably resolve multi-criteria conflicts, making the selection of the final schedule indicator scheme more in line with actual needs and management objectives. Finally, by organically combining attribute reduction and compromise ranking, we not only improved the scientificity, rationality, and flexibility of railway engineering schedule indicator decision-making, but also provided a scalable technical means for multi-objective and multi-attribute decision-making scenarios, which has significant practical value and engineering application potential.

[0068] 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.

[0069] 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.

[0070] 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.

[0071] 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.

[0072] 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 scope of protection of the claims of the present invention.

Claims

1. A multi-attribute decision-making method based on hesitant fuzzy rough sets and VIKOR, characterized in that, include: A multi-attribute decision matrix for railway engineering schedule indicators is constructed, wherein the decision matrix contains multiple candidate schemes and evaluation values ​​corresponding to each schedule attribute; The evaluation values ​​in the decision matrix are subjected to hesitant fuzzification processing to form a hesitant fuzzy decision information table; Based on the hesitant fuzzy rough set theory, the hesitant fuzzy decision information table is reduced in terms of attributes. Redundant attributes are removed and key discriminative features are retained to obtain the reduced attribute set. Based on the reduced attribute set, calculate the positive and negative ideal solutions for each candidate scheme under each attribute; The group benefit value, individual regret value, and compromise ranking value of each candidate scheme were calculated based on the VIKOR method. All candidate solutions are ranked according to the compromise ranking value to determine the optimal project duration.

2. The multi-attribute decision-making method based on hesitant fuzzy rough sets and VIKOR as described in claim 1, characterized in that, The construction of the multi-attribute decision matrix for railway engineering schedule indicators includes: Collect key process nodes involved in railway engineering, and determine the attribute set used to evaluate the rationality of the construction period. The attribute set includes qualitative attributes and quantitative attributes. For each candidate schedule, obtain linguistic or numerical evaluation results for each attribute. All evaluation results are entered into a structured table to form an initial multi-attribute decision matrix.

3. The multi-attribute decision-making method based on hesitant fuzzy rough sets and VIKOR as described in claim 2, characterized in that, When performing hesitant fuzzification on the evaluation values ​​in the decision matrix, the following steps are included: For qualitative attributes, language evaluation is converted into corresponding hesitant fuzzy elements, each of which consists of several membership values. For quantitative attributes, they are normalized and mapped to hesitant fuzzy numbers, and multiple possible membership degrees are generated by setting a threshold range. The evaluation values ​​of all attributes are uniformly represented as hesitant fuzzy numbers to construct a complete hesitant fuzzy decision information table.

4. The multi-attribute decision-making method based on hesitant fuzzy rough sets and VIKOR as described in claim 3, characterized in that, When performing attribute reduction based on hesitant fuzzy rough sets, it includes: Define hesitant fuzzy similarity relations and construct equivalence classes based on the distance metric between hesitant fuzzy numbers; The coverage capability of each attribute subset to the decision class is obtained based on the upper and lower approximation operators; Based on a heuristic search strategy, attributes whose contribution to classification is lower than a preset threshold are gradually eliminated to obtain the minimum attribute reduction set; Verify whether the reduced attribute set still retains the discriminative power of the original decision information, where: If not, a heuristic search strategy is used, and the preset threshold is adjusted to obtain the minimum attribute reduction set a second time.

5. The multi-attribute decision-making method based on hesitant fuzzy rough sets and VIKOR as described in claim 4, characterized in that, Verifying whether the reduced attribute set still retains the discriminative power of the original decision information also includes determining whether the attribute reduction converges, where: If the attribute sets obtained from two consecutive attribute reductions are exactly the same, or if the improvement in classification accuracy from the addition of an attribute is less than the set tolerance, then the attribute reduction process is considered to have converged.

6. The multi-attribute decision-making method based on hesitant fuzzy rough sets and VIKOR as described in claim 1, characterized in that, The calculation of the positive ideal solution and the negative ideal solution includes: For each attribute, the set of the largest membership degree of the attribute among all candidate solutions is selected as the positive ideal solution, and the set of the smallest membership degree is selected as the negative ideal solution. The attribute weights are determined by the hesitant fuzzy entropy method, reflecting the differences in the amount of information of each attribute in the overall decision-making process.

7. The multi-attribute decision-making method based on hesitant fuzzy rough sets and VIKOR as described in claim 1, characterized in that, In the VIKOR compromise ranking process, each candidate solution is assigned an independent ranking unit for comprehensive evaluation, wherein: For each candidate solution, construct its corresponding trade-off index vector, which includes a group benefit component and an individual regret component; Based on a preset trade-off coefficient, the two components are linearly combined to generate the final trade-off ranking value; All candidate solutions are ranked in ascending order based on their compromise values, with the solutions ranked higher corresponding to the better schedule indicators.

8. The multi-attribute decision-making method based on hesitant fuzzy rough sets and VIKOR as described in claim 7, characterized in that, Sensitivity adjustments are made to individual regret values, and the compromise coefficient is dynamically adjusted based on the maximum degree of disagreement within the decision-making group.

9. The multi-attribute decision-making method based on hesitant fuzzy rough sets and VIKOR as described in claim 1, characterized in that, In the process of calculating the compromise ranking value, the number of attributes involved in the VIKOR operation is determined, and the difference between each scheme and the ideal solution is measured by weighted Euclidean distance.

10. The multi-attribute decision-making method based on hesitant fuzzy rough sets and VIKOR as described in claim 8, characterized in that, The termination condition is that the number of attribute reduction iterations reaches a preset upper limit, or the top three candidate schemes in three consecutive compromise ranking results are completely consistent.

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