A random key heuristic method for solving vulnerability envelope of metro network

Abstract

This invention provides a random key heuristic method for solving the vulnerability envelope problem of subway networks. The method includes: constructing a two-layer model of the subway network vulnerability envelope problem; calculating the fitness value of individuals using the two-layer model; assigning continuous real-valued keys ([0, 1]) to each line interval using random key encoding; generating initial key values ​​for each line interval using the betweenness centrality of the line intervals as weights; constructing an initialized population using the initial key values ​​of each line interval; performing neighborhood search optimization on the initialized population to obtain an elite population set; and performing biased crossover and mutation processing on the non-elite population from the previous generation of the elite population set to obtain an output result including the subway network vulnerability envelope curve and combinations of key vulnerable line intervals. This invention can improve the proactive prevention and control capabilities and overall resilience of subway systems, thereby providing direct and quantifiable decision-making basis for planning and construction, emergency response, and resource allocation.

Description

Technical Field

[0001] This invention relates to the field of urban rail transit system operation safety technology, and in particular to a random key heuristic method for solving the vulnerability envelope of subway networks. Background Technology

[0002] A metro network refers to a public rail transit system consisting of multiple metro stations and connecting lines. Its network structure determines passenger flow paths within a city. With the continuous expansion of metro networks, their structure is evolving from a single-line form to multi-line, grid-like, and strongly coupled structures. The presence of numerous transfer nodes, shared line sections, and circular lines gives metro networks significant spatial and functional interconnectivity and transmissibility. When external events cause line disruptions, passenger flow is forced to redistribute within the network, leading to route changes, congestion at transfer links, and even the spread of localized failures to the entire network. Therefore, accurately identifying the potential performance fluctuation range of the rail transit network under different disruption scales is crucial for assessing system resilience, developing contingency plans, and identifying critical weak points.

[0003] Vulnerability envelope: This refers to the function curve representing the range of network performance changes under different scales of outages, used to characterize the boundaries between optimal and worst-case performance. As a tool for characterizing the boundary changes of network performance under optimal and worst-case scenarios, the vulnerability envelope can reflect the network's potential behavior under extreme conditions. However, due to the exponential growth in the number of combinations of multi-line section joint outage scenarios and the strong coupling between outage impact and path reconstruction, the vulnerability envelope calculation problem exhibits typical high-dimensional combinatorial optimization characteristics, making direct and accurate solutions often difficult to achieve. Therefore, efficient intelligent solution algorithms are needed to achieve solution approximation and extreme scenario identification.

[0004] To accurately characterize the performance variation range of a subway network under different outage scales, it is necessary to consider network topology, passenger travel volume, and the mechanism of multi-line section coupling failure simultaneously. Currently, one existing method for assessing subway network vulnerability relies on static topology indicators or single-line / single-node outage analysis, measuring network vulnerability based on topology performance indicators, and equating the complex rail transit network to homogeneous network structural units. The drawbacks of this method include neglecting the impact of multi-line coupling, transfer link connectivity, and passenger travel demand distribution on network performance. This simplification leads to an underestimation of the network performance degradation range caused by simultaneous outages of multiple sections, thus failing to accurately characterize the network's best and worst performance ranges under extreme conditions.

[0005] When dealing with scenarios where multiple sections are interrupted simultaneously, this method often relies on global model solutions or ordinary genetic algorithm strategies, which suffer from problems such as low solution efficiency, slow convergence, and easy getting trapped in local optima, making it difficult to apply to large-scale real subway networks. Summary of the Invention

[0006] This invention provides a random key heuristic method for solving the vulnerability envelope of a subway network, so as to effectively identify the combination of critical vulnerable line sections under different outage scales and accurately reveal the possible fluctuation range of subway network performance under extreme scenarios.

[0007] To achieve the above objectives, the present invention adopts the following technical solution.

[0008] A random key heuristic method for solving the vulnerability envelope of subway networks includes: A two-layer model of the vulnerability envelope problem of subway networks is constructed, which includes an upper-layer model and a lower-layer model. The fitness value of an individual is calculated using the two-layer model. A random key encoding is used to assign a continuous real number key of [0, 1] to each line interval. The initial key value of each line interval is generated with the betweenness centrality of the line interval as the weight. The initial key value of each line interval is used to form an initial population. Based on the initial population and the fitness values ​​of the previous generation individuals, a neighborhood search optimization process is performed to obtain an elite population set; based on the non-elite population of the previous generation of the elite population set, biased crossover and mutation processing is performed to obtain output results including the subway network vulnerability envelope curve and the combination of critical vulnerable line sections.

[0009] Preferably, the two-layer model for constructing the vulnerability envelope problem of subway networks includes an upper-layer model and a lower-layer model. The fitness value of an individual is calculated using the two-layer model, including: The subway network is set up with a certain number of line sections and transfer edges. The two ends of the line sections are different stations, and the two ends of the transfer edges are the same station. The lower-level model uses the shortest path algorithm to calculate the minimum travel cost of all OD pairs under the current interruption situation. The upper-level model substitutes the travel cost calculated by the lower-level model into the upper-level model to calculate the network performance index, including the fitness value of individuals, under the current interruption combination. In the genetic algorithm, each individual represents a specific combination of line section interruption schemes. The length of the individual's gene sequence is equal to the total number of line sections in the subway network. Each position in the sequence corresponds to a specific line section. The random key stored at that position determines whether the line section is interrupted. The population is a collection of multiple individuals. In each generation of evolution, a population containing multiple individuals means that the algorithm is simultaneously evaluating multiple different combinations of line section interruption schemes. The formulas for the two-layer model are shown in equations (1)-(8): Upper-level model: (1)

[0010] st (2) (3) (4) (5) Lower-level model: (6)

[0011] st (7) (8)

[0012] The objective function in formula (1) aims to maximize or minimize the residual connectivity of the urban rail transit network, that is, the number of passengers who can still complete their journeys even when some sections of the line are interrupted. Formula (2) imposes a cardinal constraint on the total number of interrupted sections. Formula (3) establishes the connection between the upper and lower level models. Decision variables These are auxiliary variables for travel costs obtained from the lower-level model. Given the interruption scenario, the lower-level model solves for the minimum travel cost of each origin-to-endpoint OD pair, calculates the route interval running time and transfer penalty, forming a causal chain of "interruption-accessibility-travel cost". Formula (6) aims to minimize the total travel cost of the selected edges in each OD pair, and Formula (7) is the conservation constraint for node inflow and outflow. Indicates the starting point. To indicate the endpoint, formula (8) specifies that the decision variable is a binary variable: when connecting edges When used for the shortest path, ;otherwise, .

[0013] The fitness value of an individual is obtained by joint calculation using a two-layer model. The random key values ​​of the route intervals contained in the individual are sorted, and the route intervals corresponding to the top n key values ​​are marked as "interrupted". This interrupted state is substituted into the lower-layer model, and the shortest path algorithm is used to calculate the latest minimum travel cost from all origins to destinations at this time. The minimum travel cost of each OD pair obtained by the lower-layer model is substituted into the upper-layer model. The upper-layer model determines whether each OD pair remains connected based on the passenger tolerance coefficient, and calculates the total travel demand of all connected OD pairs, which is the fitness value of the individual under the current interrupted combination.

[0014] Preferably, the step of assigning a continuous real key of [0, 1] to each line interval using random key encoding, generating initial key values ​​for each line interval using the betweenness centrality of the line interval as weight, and using the initial key values ​​of each line interval to form an initialized population includes: The number of interrupted line sections in the subway network is set as follows: A random key encoding is used to assign consecutive real number keys [0, 1] to each line interval. The key values ​​are sorted and the first few are selected. n One item is designated as the "interruption set," and the rest are designated as the "normal set." Calculate the betweenness centrality of the metro network: Let the metro network be... ,in For a set of nodes, For any line interval, the set of connected edges is defined. Its betweenness centrality The calculation method is as follows: (9) Calculate each line segment in the network Normalized betweenness centrality For each line section Generate a uniformly distributed random number in the range [0, 1]. Normalized betweenness centrality With random numbers Multiply them to obtain the initial key value of the line interval, and use the initial key values ​​of each line interval to form an initialized population.

[0015] Preferably, the step of performing neighborhood search optimization based on the initialized population and the fitness values ​​of individuals from the previous generation to obtain an elite population set includes: The fitness value of the previous generation population is calculated based on the fitness value of the previous generation individuals. The initialized population is then sorted according to the fitness value of the previous generation population to select elite individuals. For each selected elite individual, one of three operators is randomly selected to perform local perturbation to generate a new neighborhood solution. The fitness of the new neighborhood solution is calculated. If the fitness of the new solution is better than that of the original elite individual, the new neighborhood solution is used to replace the original elite individual; otherwise, the original elite individual is retained. After neighborhood search optimization processing of all elite individuals, the elite population set is obtained. The three operators include: insertion operator: replacing the original key with a new key value in a random interval, and rearranging to obtain a new interruption combination; exchange operator: randomly exchanging two key values ​​to cause the critical path interval to enter or leave the "first n" set; and reversal operator: reversing a key sequence to generate a combination perturbation.

[0016] Preferably, the step of obtaining the output results, including the subway network vulnerability envelope curve and the combination of critical vulnerable line sections, based on the non-elite population of the previous generation of the elite population set after biased crossover and mutation processing, includes: Based on the current generation of elite population and the previous generation of non-elite population, the next generation of non-elite individuals is generated through biased crossover. A certain number of new individuals are randomly generated through mutation. Some non-elite individuals are replaced by these new individuals. All non-elite individuals are used to form a non-elite population. The current generation of elite population is then merged with the newly generated non-elite population to obtain output results including the subway network vulnerability envelope curve and the combination of critical vulnerable line sections. The subway network vulnerability envelope curve corresponds to the upper and lower bounds of network performance indicators under different interruption scales. The combination of critical vulnerable line sections corresponds to the specific line section interruption set at the lower bound of each envelope.

[0017] As can be seen from the technical solutions provided by the embodiments of the present invention described above, the present invention, through precise calculation of the upper and lower bounds of the vulnerability envelope of the subway network, reveals the critical stage of the subway network performance change with the scale of the interruption, the location of vulnerable structures, and the potential risk range of the subway network. This provides a direct and quantifiable basis for planning and construction, operation and maintenance scheduling, emergency response, and resource investment, thereby improving the proactive prevention and control capabilities and overall resilience of the subway system.

[0018] Additional aspects and advantages of the invention will be set forth in part in the description which follows, and will become apparent from the description or may be learned by practice of the invention. Attached Figure Description

[0019] To more clearly illustrate the technical solutions of the embodiments of the present invention, the drawings used in the description of the embodiments will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0020] Figure 1 A schematic diagram illustrating the implementation principle of a random key heuristic method for solving the vulnerability envelope of a subway network, provided in an embodiment of the present invention. Figure 2 A flowchart illustrating a random key heuristic method for solving the vulnerability envelope of a subway network, provided in an embodiment of the present invention; Figure 3 This is a schematic diagram of three random variable neighborhood search operators provided in an embodiment of the present invention; Figure 4 This is a schematic diagram of a biased crossover operator provided in an embodiment of the present invention; Figure 5 A schematic diagram of the Hong Kong MTR topology network provided as an embodiment of the present invention; Figure 6 This invention provides a vulnerability envelope and interval range for the Hong Kong MTR network. Figure 7 This is a schematic diagram illustrating the average improvement of the objective function by three operators in the first 100 generations, as provided in an embodiment of the present invention. Detailed Implementation

[0021] Embodiments of the present invention are described in detail below, examples of which are shown in the accompanying drawings, wherein the same or similar reference numerals denote the same or similar elements or elements having the same or similar functions throughout. The embodiments described below with reference to the accompanying drawings are exemplary and are only used to explain the present invention, and should not be construed as limiting the present invention.

[0022] Those skilled in the art will understand that, unless specifically stated otherwise, the singular forms “a,” “an,” “the,” and “the” used herein may also include the plural forms. It should be further understood that the term “comprising” as used in this specification means the presence of the stated features, integers, steps, operations, elements, and / or components, but does not exclude the presence or addition of one or more other features, integers, steps, operations, elements, components, and / or groups thereof. It should be understood that when we say an element is “connected” or “coupled” to another element, it can be directly connected or coupled to the other element, or there may be intermediate elements. Furthermore, “connected” or “coupled” as used herein can include wireless connections or couplings. The term “and / or” as used herein includes any and all combinations of one or more of the associated listed items.

[0023] It will be understood by those skilled in the art that, unless otherwise defined, all terms used herein (including technical and scientific terms) have the same meaning as commonly understood by one of ordinary skill in the art to which this invention pertains. It should also be understood that terms such as those defined in general dictionaries should be understood to have the same meaning as in the context of the prior art, and should not be interpreted in an idealized or overly formal sense unless defined as herein.

[0024] To facilitate understanding of the embodiments of the present invention, the following will provide further explanation and description with reference to the accompanying drawings and several specific embodiments. These embodiments do not constitute a limitation on the embodiments of the present invention.

[0025] Based on the existing two-layer vulnerability envelope modeling framework, this invention uses an efficient solution strategy at the algorithm level to quickly and stably obtain the upper and lower bounds of network performance under different outage scales, thereby truly reflecting the potential performance fluctuation range of the network under extreme events and providing a more reliable quantitative basis for resilience assessment.

[0026] This invention introduces a random key initialization strategy guided by betweenness centrality and a random neighborhood search strategy for elite solutions. This strengthens the search priority of critical link combinations during the evolution process and provides an improved path for elite solutions to escape local optima. As a result, it can accurately approximate the interruption combinations that lead to the best or worst network performance, and significantly improve the solution efficiency and stability.

[0027] This invention provides a method for solving the vulnerability envelope of a metro network under the scenario of simultaneous interruption of multiple lines in a metro network. The method is based on random key encoding to realize the interruption combination expression, and combines betweenness-weighted hot start and elite solution neighborhood search strategy, so as to approximate the optimal and worst boundary of network performance under different interruption scales.

[0028] The processing flow of a random key heuristic method for solving the vulnerability envelope of a subway network, provided in this embodiment of the invention, is as follows: Figure 1 As shown, the processing steps include the following: Step S10: Construct a two-layer model of the vulnerability envelope problem of the subway network, which includes an upper-layer model and a lower-layer model.

[0029] The subway network consists of a certain number of line sections and transfer edges. The endpoints of a line section are different stations, while the endpoints of a transfer edge are the same station. Only line sections can be interrupted, and 'n' represents the number of interrupted line sections. The lower-level model uses a shortest path algorithm to calculate the minimum travel cost for all origin-destination (OD) pairs under the current interruption scenario. The upper-level model then substitutes the travel cost calculated by the lower-level model into its own calculations to determine the network performance metrics, including the fitness values ​​of individuals, under the current interruption combination, thereby selecting elite solutions.

[0030] An individual is the basic solution unit in a genetic algorithm. In this invention, each individual represents a specific combination of line section interruptions. The length of an individual's gene sequence is equal to the total number of line sections in the subway network. Each position in the sequence corresponds to a specific line section, and the stored random key value determines whether that line section is interrupted. Transfer edges do not participate in the individual's gene encoding (i.e., it is assumed that transfer edges themselves do not experience interruptions), but they, together with the line sections, constitute the complete subway network topology. The population is a collection of multiple individuals. In each generation of evolution, a population containing multiple individuals means that the algorithm is simultaneously evaluating multiple different combinations of line section interruptions.

[0031] The vulnerability envelope problem aims to determine the combination of line section interruptions that maximizes or minimizes the performance of an urban metro network, given a number of interrupted edges. A two-layer model for the vulnerability envelope problem of metro networks is constructed, consisting of an upper-layer model and a lower-layer model. The upper-layer model focuses on optimizing the interrupted edge variables to affect network connectivity, while the lower-layer model optimizes the path search variables to minimize travel costs. The specific models are shown in equations (1) - (8).

[0032] Upper-level model: (1)

[0033] st (2) (3) (4) (5) Lower-level model: (6)

[0034] st (7) (8)

[0035] The upper-level model aims to "maximize / minimize the demand for travel that can be completed within a given interruption scale," with the decision variable being the set of interrupted line sections. The upper and lower bounds of the vulnerability envelope can be obtained by changing the objective direction. The objective function in formula (1) aims to maximize or minimize the residual connectivity of the urban rail transit network, i.e., the number of passengers who can still complete their journeys even when some line sections are interrupted. Formula (2) imposes a cardinal constraint on the total number of interrupted line sections, ensuring that exactly a certain number of interrupted line sections are included. The connections are affected. The more connections are interrupted simultaneously, the more severe the network damage. Formula (3) establishes the connection between the upper and lower layer models. Here, the decision variables... These are auxiliary variables for travel costs obtained from the lower-level model. The lower-level model, under a given interruption scenario, solves for the minimum travel cost of each OD (Origin-Destination) pair, calculates the route interval travel time and transfer penalty, forming a causal chain of "interruption—accessibility—travel cost". Formula (6) aims to minimize the total travel cost of the selected edges for each OD pair. Formula (7) is the conservation constraint for node inflow and outflow, where Indicates the starting point. Indicates the endpoint. Formula (8) specifies that the decision variable is a binary variable: when connecting edges When used for the shortest path, ;otherwise, .

[0036] In the subsequent population iteration process of step S30 (elite retention and random variable neighborhood search), the two-layer model is used to calculate the fitness value of an individual.

[0037] The fitness value of an individual is obtained through a joint calculation using a two-layer model: First, the random key values ​​of the route intervals contained in the individual are sorted, and the route intervals corresponding to the top n key values ​​are marked as "interrupted". Next, this interrupted state is substituted into the lower-layer model, and the shortest path algorithm is used to calculate the latest minimum travel cost from all origins to destinations at this time. Finally, the minimum travel cost of each OD pair obtained by the lower-layer model is substituted into the upper-layer model. The upper-layer model determines whether each OD pair remains connected based on the passenger tolerance coefficient, and calculates the total travel demand of all connected OD pairs, which is the fitness value of the individual under the current interrupted combination.

[0038] Step S20: Use random key encoding to assign a continuous real number key of [0, 1] to each line interval, generate the initial key value of each line interval with the betweenness centrality of the line interval as the weight, and use the initial key value of each line interval to form an initialized population.

[0039] Random key heuristic algorithm: A class of heuristic algorithms that use continuous real numbers from 0 to 1 as chromosome genes, sorting the gene vectors and mapping them to specific discrete combinatorial solutions. The number of interrupted line sections in the subway network is set to... To strictly satisfy the "just enough" condition in evolutionary algorithms n To address the cardinality constraint of "interrupted line intervals," this invention employs random key encoding: assigning a continuous real number key of [0, 1] to each line interval, sorting the key values, and selecting the first... n One item is designated as the "interruption set," and the rest are designated as the "normal set."

[0040] To improve early search efficiency and hit rate in extreme scenarios, a "between-centrality weighted" initialization strategy is introduced: first, the betweenness centrality of the line intervals is calculated and normalized. The betweenness centrality is calculated as the percentage of paths passing through that interval in the shortest paths between all node pairs. Let the subway network be... ,in For a set of nodes, This is the set of connected edges. For any line interval... Its betweenness centrality The calculation method is as follows: (9) The initial key values ​​for each line interval are then generated using the betweenness centrality of the line interval as weights, rather than being purely random. The specific steps are as follows: First, calculate the initial key value for each line interval in the network. Normalized betweenness centrality Then, for each line section Generate a uniformly distributed random number in the range [0, 1]. Finally, the normalized betweenness centrality is multiplied by the random number to obtain the initial key value of the line interval, and the initial key value of each line interval is used to form an initialized population.

[0041] The initial key values ​​generated for each line interval are primarily used in the decoding process to determine the specific line interruption state. Specifically, when selecting the set of interruption intervals, the key value vectors of all line intervals contained in the chromosome are first sorted in descending or ascending order. Then, based on the preset interruption scale, the sorted key values ​​are selected and placed at the top. This operation makes it easier to include high (or low) betweenness line intervals in the interrupt set when determining the lower (or upper) bound.

[0042] Step S30: Elite preservation and random variable neighborhood search.

[0043] The input data for this step consists of the population initialized in step S20 and the fitness of the previous generation individuals calculated using the model in step S10. The output data is the elite population set optimized by neighborhood search. The overall processing procedure is as follows: First, the population initialized in step S20 is sorted according to the fitness values, and elite individuals are selected. For each selected elite individual, one of the three operators is randomly selected to perform local perturbation to generate a new neighborhood solution. Finally, the fitness of the new neighborhood solution is calculated. If the new solution is better than the original elite solution, the original solution is replaced; otherwise, the original solution is retained, resulting in the elite population set.

[0044] The elite population set comprises a certain proportion of individuals with the best fitness performance in the current generation of the population. In addition, after local perturbation optimization, the elite population set also includes a new generation of neighborhood solutions with higher fitness values ​​found through insertion, swap, or inversion operators.

[0045] The acquisition of the non-elite population is divided into two stages: The first stage is the division of the current generation's non-elite population. After calculating the fitness values ​​of the complete initial population and sorting them by quality, the top-ranking elite individuals are excluded, and the remaining individuals directly constitute the current generation's non-elite population. The second stage is the generation of the next generation's non-elite population, mainly achieved through biased crossover and mutation operations. The offspring generated by biased crossover are merged with the newly generated individuals from mutation to obtain the complete non-elite population for the next iteration.

[0046] Figure 3 This is a schematic diagram of three random variable neighborhood search operators provided in an embodiment of the present invention.

[0047] To avoid evolutionary stagnation and accelerate the approach to extreme solutions, this invention selects several elite solutions in each generation and performs a randomized variable neighborhood search on them. Specifically, this includes three types of local perturbation operators: Insertion: Replace the original key with the new key value within a random interval, and rearrange to obtain a new interrupt combination; Swap: Randomly swap two key values ​​to cause the critical path interval to enter or leave the "first n" set; Reversal: Reversing a key sequence to produce a large-scale combined perturbation.

[0048] Each elite individual randomly selects a neighborhood operation. If the objective is improved, the operation is accepted; otherwise, the original solution is retained. This strategy allows for continuous trial and error and acceptance within the neighborhood of the elite solution at a low cost, ensuring the continuation of the elite while maintaining necessary exploration, embodying a "preserve excellence + exit" temporal mechanism.

[0049] Step S40: Biased crossover and mutation.

[0050] The input data for this step consists of the elite population set output from step S30 and the non-elite population from the previous generation. The output data is the updated complete next-generation population (including elites and new offspring). The overall processing procedure is as follows: first, biased crossover is used to generate non-elite individuals for the next generation; then, a small number of entirely new individuals are randomly generated through mutation to replace some offspring individuals; finally, the retained elite population is merged with the newly generated offspring population to obtain the output results, including the subway network vulnerability envelope curve and the combination of critical vulnerable line sections.

[0051] The population update phase uses a combination of biased crossover and a small amount of mutation: (1) Biased crossover: Each time, one parent is selected from both the elite and non-elite groups, and the offspring inherit the elite or non-elite key value for each gene locus according to a given bias probability, so that the superior traits are propagated with a higher probability, while preserving the source of diversity. Figure 4 This is a schematic diagram of a biased crossover operator provided in an embodiment of the present invention.

[0052] (2) Mutation: Randomize some key values ​​of a small number of individuals and inject new combinations with a lower intensity.

[0053] This process maintains stable search progress while balancing the inheritance of excellent patterns and the preservation of diversity, and complements random variable neighborhood search: cross-promotes global reorganization, while neighborhood search refines local structure.

[0054] The random key heuristic method of this invention ultimately outputs a vulnerability envelope curve for the subway network and a combination of critical vulnerable line sections. The former corresponds to the upper bound (optimal performance) and lower bound (worst performance) of the network performance index under different outage scales, respectively; the latter corresponds to the specific set of line section outages at the lower bound of each envelope (i.e., which lines, if simultaneously interrupted, would cause the network performance to drop to its lowest level). Based on this result, line sections that frequently appear in the lower bound solution can be statistically analyzed and identified as critical bottlenecks in the network, allowing for priority allocation of maintenance resources or the addition of redundant facilities. Furthermore, based on the calculated worst-case scenario (lower bound), targeted passenger evacuation and emergency dispatch plans can be developed to ensure that the network maintains a minimum level of service capacity even under extreme disasters.

[0055] Step S50: Verify the effectiveness of the subway network vulnerability envelope solution method of the present invention.

[0056] To verify the effectiveness of the vulnerability envelope solution method for subway networks described in this invention, this scheme selects the representative Hong Kong subway network as the empirical research object. Hong Kong's rail network is densely packed with lines and numerous transfer nodes, exhibiting typical characteristics of a strongly coupled multi-line network, making it suitable as a test platform for vulnerability boundary analysis.

[0057] Figure 5This invention provides a schematic diagram of the Hong Kong MTR topology, featuring 10 operating lines and 98 stations. Passenger flow data for this study was provided by the MTR Academy, using OD data from the morning peak hours (7:00 AM to 9:00 AM) as model input. To analyze the systematic trend of network performance degradation under different outage scales, this invention solves for combinations of line sections with outage scales increasing from 1 to 110, forming a network vulnerability envelope. Figure 6 An embodiment of the present invention provides a vulnerability envelope and interval range for the Hong Kong MTR network, as shown in the attached figure. Figure 6 As shown, it can be observed that as the scale of the outage increases, the upper and lower bounds of network performance gradually separate, and the envelope band gradually widens. This indicates that the network has strong adaptability when the outage scale is small, but when the outage scale is large, the uncertainty of system performance increases significantly, and the network enters a potentially vulnerable state. The inflection point of envelope band expansion can be used as a basis for identifying the critical stage of resilience, providing a direct judgment standard for setting outage risk levels and optimizing resource scheduling in actual operation.

[0058] Meanwhile, to verify the role of the random variable neighborhood search operator in extreme performance approximation, Figure 7 This is a schematic diagram illustrating the average improvement of the objective function by three operators over the first 100 generations, as provided in an embodiment of the present invention. Figure 7 As can be seen, the average improvement effect exhibits a non-monotonic pattern, with the greatest improvement effect occurring when the solution space is large, because more potential adjustment options are available. Differences in improvement effects can be observed among different operators, with insertion and exchange operators being more effective. Overall, the three types of operators can improve the objective function value by approximately 1%-3%, verifying that the randomized variable neighborhood search mechanism can effectively help the algorithm escape local optima. The fundamental reason lies in the diversity of neighborhood structures generated by different randomized variable neighborhood search operators. Insertion introduces new edge combinations, exchange adjusts existing edge connections, and reversal reorganizes edge segments, collectively covering multiple local search directions and enhancing global search capabilities.

[0059] In summary, this invention, based on the topology and passenger flow distribution characteristics of subway networks, constructs a random key heuristic algorithm that integrates structural feature recognition and neighborhood optimization. This algorithm can efficiently approximate the optimal and worst-case boundaries of network performance under scenarios of joint interruption of multiple lines and sections. This invention can not only identify combinations of critical vulnerable line sections under different interruption scales, but also stably obtain the upper and lower bound curves of the vulnerability envelope, accurately revealing the possible fluctuation range of network performance under extreme scenarios. Furthermore, this invention effectively avoids the algorithm getting trapped in local optima during the search process, improving solution efficiency and accuracy. Therefore, it provides more reliable, refined, and quantifiable decision support for the resilience assessment, resource allocation prioritization, and emergency strategy formulation of rail transit networks.

[0060] The heuristic coding framework based on random key encoding to represent multi-line interval interruptions is one of the core contents of this invention. This coding method transforms the multi-line interval combination interruption problem into a searchable chromosome structure without explicit constraint modeling, providing a basic expressive framework for solving vulnerability envelopes.

[0061] The enhanced evolutionary solution strategy constructed in this invention is one of the core contents of this invention. This strategy combines the initial population construction guided by betweenness centrality with the random neighborhood search mechanism based on elite solutions. During the evolution process, it can not only strengthen the search tendency for key link combinations, but also provide the algorithm with a path to escape local optima, thereby effectively improving the quality of extreme solution approximation and ensuring the stable and efficient solution of the vulnerability envelope.

[0062] Those skilled in the art will understand that the accompanying drawings are merely schematic diagrams of one embodiment, and the modules or processes shown in the drawings are not necessarily essential for implementing the present invention.

[0063] As can be seen from the above description of the embodiments, those skilled in the art can clearly understand that the present invention can be implemented by means of software plus necessary general-purpose hardware platforms. Based on this understanding, the technical solution of the present invention, or the part that contributes to the prior art, can be embodied in the form of a software product. This computer software product can be stored in a storage medium, such as ROM / RAM, magnetic disk, optical disk, etc., and includes several instructions to cause a computer device (which may be a personal computer, server, or network device, etc.) to execute the methods described in various embodiments or some parts of the embodiments of the present invention.

[0064] The various embodiments in this specification are described in a progressive manner. Similar or identical parts between embodiments can be referred to mutually. Each embodiment focuses on describing the differences from other embodiments. In particular, for apparatus or system embodiments, since they are basically similar to method embodiments, the description is relatively simple; relevant parts can be referred to the descriptions in the method embodiments. The apparatus and system embodiments described above are merely illustrative. The units described as separate components may or may not be physically separate. The components shown as units may or may not be physical units; that is, they may be located in one place or distributed across multiple network units. Some or all of the modules can be selected to achieve the purpose of this embodiment according to actual needs. Those skilled in the art can understand and implement this without creative effort.

[0065] The above description is merely a preferred embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any variations or substitutions that can be easily conceived by those skilled in the art within the technical scope disclosed in the present invention should be included within the scope of protection of the present invention. Therefore, the scope of protection of the present invention should be determined by the scope of the claims.

Claims

1. A random key heuristic method for solving the vulnerability envelope of subway networks, characterized in that, include: A two-layer model of the vulnerability envelope problem of subway networks is constructed, which includes an upper-layer model and a lower-layer model. The fitness value of an individual is calculated using the two-layer model. A random key encoding is used to assign a continuous real number key of [0, 1] to each line interval. The initial key value of each line interval is generated with the betweenness centrality of the line interval as the weight. The initial key value of each line interval is used to form an initial population. Based on the initial population and the fitness values ​​of the previous generation individuals, a neighborhood search optimization process is performed to obtain an elite population set; based on the non-elite population of the previous generation of the elite population set, biased crossover and mutation processing is performed to obtain output results including the subway network vulnerability envelope curve and the combination of critical vulnerable line sections.

2. The method according to claim 1, characterized in that, The aforementioned two-layer model for constructing the vulnerability envelope problem of subway networks includes an upper-layer model and a lower-layer model. The fitness value of an individual is calculated using this two-layer model, including: The subway network is set up with a certain number of line sections and transfer edges. The two ends of the line sections are different stations, and the two ends of the transfer edges are the same station. The lower-level model uses the shortest path algorithm to calculate the minimum travel cost of all OD pairs under the current interruption situation. The upper-level model substitutes the travel cost calculated by the lower-level model into the upper-level model to calculate the network performance index, including the fitness value of individuals, under the current interruption combination. In the genetic algorithm, each individual represents a specific combination of line section interruption schemes. The length of the individual's gene sequence is equal to the total number of line sections in the subway network. Each position in the sequence corresponds to a specific line section. The random key stored at that position determines whether the line section is interrupted. The population is a collection of multiple individuals. In each generation of evolution, a population containing multiple individuals means that the algorithm is simultaneously evaluating multiple different combinations of line section interruption schemes. The formulas for the two-layer model are shown in equations (1)-(8): Upper-level model: (1) st (2) (3) (4) (5) Lower-level model: (6) st (7) (8) The objective function in formula (1) aims to maximize or minimize the residual connectivity of the urban rail transit network, that is, the number of passengers who can still complete their journeys even when some sections of the line are interrupted. Formula (2) imposes a cardinal constraint on the total number of interrupted sections. Formula (3) establishes the connection between the upper and lower level models. Decision variables These are auxiliary variables for travel costs obtained from the lower-level model. Given the interruption scenario, the lower-level model solves for the minimum travel cost of each origin-to-endpoint OD pair, calculates the route interval running time and transfer penalty, forming a causal chain of "interruption-accessibility-travel cost". Formula (6) aims to minimize the total travel cost of the selected edges in each OD pair, and Formula (7) is the conservation constraint for node inflow and outflow. Indicates the starting point. To indicate the endpoint, formula (8) specifies that the decision variable is a binary variable: when connecting edges When used for shortest path, ;otherwise, ; The fitness value of an individual is obtained by joint calculation using a two-layer model. The random key values ​​of the route intervals contained in the individual are sorted, and the route intervals corresponding to the top n key values ​​are marked as "interrupted". This interrupted state is substituted into the lower-layer model, and the shortest path algorithm is used to calculate the latest minimum travel cost from all origins to destinations at this time. The minimum travel cost of each OD pair obtained by the lower-layer model is substituted into the upper-layer model. The upper-layer model determines whether each OD pair remains connected based on the passenger tolerance coefficient, and calculates the total travel demand of all connected OD pairs, which is the fitness value of the individual under the current interrupted combination.

3. The method according to claim 2, characterized in that, The method of assigning a continuous real number key of [0, 1] to each line interval using random key encoding, generating initial key values ​​for each line interval using the betweenness centrality of the line interval as weight, and using the initial key values ​​of each line interval to form an initialized population includes: The number of interrupted line sections in the subway network is set as follows: A random key encoding is used to assign consecutive real number keys [0, 1] to each line interval. The key values ​​are sorted and the first few are selected. n One item is designated as the "interrupt set," and the rest are designated as the "normal set." Calculate the betweenness centrality of the metro network: Let the metro network be... ,in For a set of nodes, For any line interval, the set of connected edges is defined. Its betweenness centrality The calculation method is as follows: (9) Calculate each line segment in the network Normalized betweenness centrality For each line section Generate a uniformly distributed random number in the range [0, 1]. Normalized betweenness centrality With random numbers Multiply them to obtain the initial key value of the line interval, and use the initial key values ​​of each line interval to form an initialized population.

4. The method according to claim 3, characterized in that, The process of performing neighborhood search optimization based on the initialized population and the fitness values ​​of individuals from the previous generation to obtain an elite population set includes: The fitness value of the previous generation population is calculated based on the fitness value of the previous generation individuals. The initialized population is then sorted according to the fitness value of the previous generation population to select elite individuals. For each selected elite individual, one of three operators is randomly selected to perform local perturbation to generate a new neighborhood solution. The fitness of the new neighborhood solution is calculated. If the fitness of the new solution is better than that of the original elite individual, the new neighborhood solution is used to replace the original elite individual; otherwise, the original elite individual is retained. After neighborhood search optimization processing of all elite individuals, the elite population set is obtained. The three operators include: insertion operator: replacing the original key with a new key value in a random interval, and rearranging to obtain a new interruption combination; exchange operator: randomly exchanging two key values ​​to cause the critical path interval to enter or leave the "first n" set; and reversal operator: reversing a key sequence to generate a combination perturbation.

5. The method according to claim 4, characterized in that, The aforementioned process, which involves biased crossover and mutation of the previous generation's non-elite population within the elite population set, yields output results including the subway network vulnerability envelope curve and combinations of critical vulnerable line sections. Based on the current generation of elite population and the previous generation of non-elite population, the next generation of non-elite individuals is generated through biased crossover. A certain number of new individuals are randomly generated through mutation. Some non-elite individuals are replaced by these new individuals. All non-elite individuals are used to form a non-elite population. The current generation of elite population is then merged with the newly generated non-elite population to obtain output results including the subway network vulnerability envelope curve and the combination of critical vulnerable line sections. The subway network vulnerability envelope curve corresponds to the upper and lower bounds of network performance indicators under different interruption scales. The combination of critical vulnerable line sections corresponds to the specific line section interruption set at the lower bound of each envelope.

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