A method for robust site selection and configuration of bus transfer standby points in response to subway interruption

Abstract

This invention discloses a two-stage bibliometric optimization method for selecting and configuring backup bus depots to address subway disruptions. It relates to the field of public transportation control technology, and its key technical points are: first, constructing fuzzy sets based on the L1 norm to characterize demand uncertainty; second, establishing a first-stage depot selection and reserve decision model; then, constructing a state-space network to transform the cyclic evacuation problem into a hierarchical network flow problem; further, establishing a second-stage vehicle scheduling model; and finally, transforming the two-layer structure into a directly solvable single-layer mixed-integer linear programming problem through strong dual transformation. The method introduces a two-stage bibliometric optimization approach, constructing a state-space network with the number of evacuations as the state dimension, and then constructing a hierarchical network flow model. Combined with fuzzy sets based on the L1 norm and strong dual transformation techniques, it achieves joint optimization of backup depot location, vehicle reserve quantity, and evacuation routes, balancing the robustness of location decisions and model solution efficiency under uncertain demand conditions.

Description

Technical Field

[0001] This invention relates to the field of public transportation control technology, and more specifically, to a method for the location and configuration of backup bus depots for connecting buses in response to subway interruptions. Background Technology

[0002] With the continuous expansion of urban rail transit networks, subways have become a backbone component of public transportation systems in large and medium-sized cities. However, subway systems inevitably encounter emergencies such as equipment failures, natural disasters, and safety incidents during operation, leading to line interruptions and train stoppages. In such sudden interruption scenarios, a large number of stranded passengers urgently need to be evacuated to nearby stations via emergency bus shuttle services to continue their journeys. Therefore, how to scientifically determine the spatial layout and quantity allocation of bus shuttle resources before an interruption occurs—that is, where to set up backup points and how many vehicles should be reserved at each backup point—to enable rapid response and efficient evacuation in the event of an interruption, has become one of the urgent problems to be solved in the field of urban rail transit emergency management.

[0003] From a practical application perspective, effectively addressing the issues of bus connection backup point location and vehicle reserve will directly contribute to: ① Rapid response and reduced passenger confinement time during emergencies can be achieved through scientifically planning backup points and rationally allocating vehicles; ② Improved resource utilization efficiency and reduced reserve costs can be achieved by optimizing evacuation routes and matching transport capacity.

[0004] In the field of bus connection site selection under subway disruption scenarios, existing research methods are mainly based on different modeling theories and technical means, and can be divided into the following categories:

[0005] 1. Uncertainty-based optimization methods based on stochastic programming

[0006] Stochastic programming methods characterize demand uncertainty by introducing random variables and seek optimal location solutions under the expectation of demand. These methods typically assume that demand follows a known probability distribution (such as a normal or Poisson distribution), constructing a finite set of scenarios through Monte Carlo simulations or scenario generation techniques to solve the expected cost minimization problem. Stochastic programming can, to some extent, account for the impact of demand fluctuations on location decisions. However, in the problem of selecting backup bus stop locations, subway disruptions are sporadic and heterogeneous. The number and distribution of stranded passengers are affected by multiple factors such as the duration of the disruption, the location of the section, and passenger flow patterns. Historical disruption data is scarce, making it difficult to obtain sufficient statistical samples to accurately estimate the probability distribution of demand. This situation limits the applicability of stochastic programming methods based on the exact distribution assumption in the backup bus stop location problem.

[0007] 2. Conservative Decision-Making Method Based on Robust Optimization

[0008] Robust optimization methods do not rely on probability distribution assumptions but instead construct an uncertainty set encompassing all possible demand fulfillment scenarios to seek the optimal location solution in the worst-case scenario. By setting upper and lower bounds or fluctuation ranges for demand, this method ensures that the resulting solution is feasible under any possible demand fulfillment, exhibiting strong resilience. However, in the problem of selecting locations for bus shuttle backup points, the probabilities of different interruption scenarios vary significantly. Determining location and reserve solutions solely based on a single worst-case scenario ignores the probabilistic information of demand fulfillment, often resulting in solutions with excessive redundant vehicles, leading to high reserve costs and low resource utilization. This situation deviates from the practical requirement of "balancing economy and safety" in emergency management.

[0009] 3. A method based on bilobal bar optimization

[0010] Partial Brussels Bar (MBB) optimization is a class of uncertainty optimization methods that has emerged in recent years. Its core idea is to construct a fuzzy set containing all reasonable probability distributions and seek the optimal decision under the worst-case distribution. Compared to traditional stochastic programming, MBB optimization does not require precise knowledge of the probability distribution, only needing to characterize some features of the distribution (such as moment information and distance constraints). Compared to traditional robust optimization, MBB optimization utilizes multi-scenario information through a probability weighting mechanism, avoiding over-conservatism. However, in the current state of public transport connection site selection problems, existing MBB optimization methods have different choices in fuzzy set construction: fuzzy sets based on moment constraints usually introduce a semidefinite programming structure, limiting the solution scale; fuzzy sets based on Wasserstein distance introduce nonlinear coupling when combined with integer decision variables in the site selection problem (such as 0-1 variables for whether a backup point is used), increasing the solution difficulty. Furthermore, there is still a lack of concise and effective methods for efficiently modeling the cyclic evacuation process of vehicles in the site selection planning model to evaluate the evacuation effectiveness of the site selection scheme.

[0011] Although existing technologies have achieved certain results in the field of subway interruption bus connection site selection and planning, there are still several key technical difficulties in practical applications, which restrict their further promotion and use. Specifically, they have the following shortcomings:

[0012] ① Modeling the cyclic evacuation process is difficult. Existing technologies use the traditional VRP (Vehicle Routing Problem) model framework to model the multi-cycle evacuation process of vehicles. The decision variables increase explosively with the number of stations and evacuation times, resulting in low solution efficiency.

[0013] ② The uncertainty of demand is not accurately characterized. Stochastic programming relies on the assumption of exact distribution, robust optimization is overly conservative, the fuzzy set construction of existing DRO methods is incompatible with integer decision variables, and there is currently a lack of uncertainty modeling techniques suitable for location problems.

[0014] ③ The two-stage optimization model has low solution efficiency. Under the Bruker framework, a nested min-max double-layer structure is formed, and the optimal value of the inner layer is an implicit function of the outer layer, which cannot be directly processed by the standard solver. Summary of the Invention

[0015] The purpose of this invention is to provide a method for the selection and configuration of backup bus depots for connecting buses in response to subway interruptions. It introduces a two-stage split-bulk optimization method, constructs a state space network with the number of evacuation trips as the state dimension, and then constructs a hierarchical network flow model. Combined with fuzzy set and strong dual transformation techniques based on L1 norm, it achieves joint optimization of backup bus depot location, vehicle reserve quantity and evacuation route, and balances the robustness of location decision and model solution efficiency under uncertain demand environment.

[0016] The above-mentioned technical objective of the present invention is achieved through the following technical solution: a method for the location selection and configuration of backup bus depots for responding to subway interruptions, the method comprising two stages, wherein the first stage is before the subway interruption occurs and the second stage is after the subway interruption occurs, specifically comprising the following five steps:

[0017] S1. Construct fuzzy sets based on the L1 norm to represent the uncertainty of demand;

[0018] S2. Establish the first-stage optimization model to determine the location of the standby vehicle point and vehicle reserve decisions;

[0019] S3. Construct a state-space network to transform the vehicle circulatory evacuation problem into a hierarchical network flow problem;

[0020] S4. Establish the second-stage optimization model to determine vehicle scheduling routes and demand satisfaction schemes;

[0021] S5. Based on the strong duality theory, the model equivalent transformation transforms the first and second stage bi-level optimization model into a solvable single-level mixed integer linear programming problem, and solves it using a linear programming solver.

[0022] The present invention is further configured such that the specific steps of S1 are as follows:

[0023] set up A collection of discrete scenarios generated from historical data or simulations, each scenario Corresponding to a set of known demand for stranded passengers Based on historical frequency or the assumption of equal probability, a nominal probability is assigned to each scenario. The nominal probability is taken ;

[0024] However, nominal probability is only an estimate of the true distribution, and the true probability distribution may deviate from it; to characterize this deviation, we define the true probability distribution. The feasible region is defined by the nominal probability distribution. Centered on, with a 1-norm distance not exceeding The set of:

[0025]

[0026] in, ensure This is a valid probability distribution; while The L1 norm distance (i.e., Manhattan distance) between the true probability distribution and the nominal probability distribution is limited to a threshold. ;parameter This is called the robustness coefficient, and its physical meaning is the maximum total deviation that the true probability vector is allowed to deviate from the predicted probability vector; when When = 0, the model degenerates into a deterministic stochastic programming model based on the nominal probability distribution; when When the coefficient is 2, the model degenerates into a robust optimization that only considers the worst-case scenario; therefore... The value of determines the balance between the degree of conservatism in decision-making and optimization efficiency.

[0027] The present invention is further configured such that the specific steps of S2 are as follows:

[0028] The first phase is the planning phase, which involves determining the activation plan for the backup vehicle point before the demand is confirmed. and the number of vehicles in reserve at each standby point The objective function for the first stage is to minimize the sum of the standby point activation cost, vehicle inventory cost, and the worst-case expected operating cost for the second stage.

[0029]

[0030] in, Indicates the activation of the backup vehicle point Fixed costs; This indicates the inventory cost per unit of vehicle; For a given reserve scheme With demand scenarios The optimal operating cost; For fuzzy sets The expected operating cost of the second stage under the worst-case probability distribution; the constraints of the first stage are imposed from the aspects of resource feasibility and service coverage.

[0031] Capacity reserve constraints require each standby point The number of vehicles in reserve must not exceed the product of their maximum capacity and their operational status.

[0032]

[0033] in, Indicates the vehicle preparation point Maximum capacity;

[0034] Coverage constraints guarantee each interruption station and turnaround station It is covered by at least one active backup vehicle location, thus ensuring the accessibility of evacuation services:

[0035]

[0036] in, This indicates that it can cover subway stations. The vehicle preparation point assembly;

[0037] In addition, variable constraints require the location variable to be selected. A binary integer with a value of 0 or 1, used as a storage variable. The value should be a non-negative integer to reflect the discrete nature of actual decision-making.

[0038] .

[0039] The present invention is further configured such that the specific steps of S3 are as follows:

[0040] 1) Define the physical topology of the transportation network:

[0041] Let the set of traffic network nodes be defined. including subway station collection Meet at the alternative pick-up point Among them, subway stations are the collection points. Further divided into sets of interrupted stations Assembly at the turnaround station Meet at the transfer station For each site Define the set of backup vehicle points that can cover the station. ;

[0042] 2) Define the vehicle's travel path:

[0043] To meet the needs of origin-destination (OD) passengers, two types of route sets were constructed: evacuation route sets. and reverse empty driving path set ;

[0044] The evacuation route set includes: ① routes from the interruption station to the turnaround station, ② routes from the interruption station to the transfer station, ③ routes from the turnaround station to the turnaround station on the opposite side, and ④ routes from the turnaround station to the transfer station on the opposite side.

[0045] Reverse empty driving path set This includes: ① lines from turnaround stations to interrupted stations, ② lines from transfer stations to interrupted stations, and ③ lines from transfer stations to turnaround stations.

[0046] 3) Constructing a state-space network :

[0047] Introducing the number of evacuation trips As a state dimension, (where, (For the preset maximum evacuation loop), the cyclical driving process of vehicles in the physical network is unfolded into a hierarchical directed acyclic network; in this state space network, the same physical station corresponds to different state nodes under different evacuation loops. Each time a vehicle completes a passenger evacuation task, its state level automatically increases, thereby eliminating the cyclic structure in the path.

[0048] Specifically, node set It includes three child node sets: virtual source node set Virtual sink set and state node set ;in, and These represent the vehicle's initial standby state at the standby point and the mission termination state, respectively. Each node in the diagram corresponds to a connection task and its real-time status in the physical network, defined as follows: ,in, Indicates physical site, Indicates the number of completed shuttle runs;

[0049] Arc set It contains four types of arc subsets, specifically defined as follows:

[0050] ① Outbound arc subset Vehicle from the standby point The aircraft is dispatched empty to an interruption station or turnaround station, and upon arrival, it is in an initial state where evacuation has not yet been carried out. ), represented as:

[0051]

[0052] ② Sparse arc subset This indicates the process of a vehicle performing an actual passenger-carrying task, from the station... Passengers arrive at the station After completing this transport, the cumulative number of completed laps increased from [previous number]. Increase to , represented as:

[0053]

[0054] ③ Empty Arc Subset : Indicates that the vehicle has completed the first The second evacuation mission has reached the passenger drop-off point. Afterwards, the empty vehicle traveled to the next starting point. The process is represented as:

[0055]

[0056] ④ Task termination arc subset Once a vehicle has transported passengers to the evacuation destination, it ceases service if it does not perform any further tasks.

[0057]

[0058] Define nodes at the same time Outgoing arc set and incoming arc set This is used to express subsequent flow balance constraints.

[0059] The present invention is further configured such that the specific steps of S4 are as follows:

[0060] The second phase is the operational phase, which involves specific demand scenarios. After implementation, based on the vehicle reserve plan of the first phase, determine the specific dispatching routes of the vehicles and the amount of unmet demand; define the arc. Generalized driving costs The specific definition is as follows:

[0061]

[0062] in, For the node arrive Travel time; This is the time coefficient for the outbound stage (without penalty for loops). It is a pre-defined number of cycles A monotonically increasing time coefficient is used; by imposing an increasing penalty on subsequent evacuation trips to force the model to prioritize the use of early capacity, the maximum completion time objective is mathematically minimized, balancing cost optimization and evacuation timeliness; furthermore, it is assumed that... For passenger boarding and alighting times, This represents the maximum number of evacuation trips.

[0063] In addition, definition Representing a scene Middle through arc The number of vehicles, Indicates OD pair Unmet demand; scenario The optimal operating cost is defined as minimizing the sum of the total vehicle operating cost and the penalty for unmet demand:

[0064]

[0065] in, The penalty cost for unmet demand per unit is determined by... Setting it to a larger value forces the model to prioritize passenger demand, allowing some demand to remain unmet only when there is a genuine shortage of capacity.

[0066] The constraints in the second phase ensure the feasibility of the scheduling scheme from three levels: resource coordination, flow conservation, and demand satisfaction.

[0067] Outbound resource constraints require that from each vehicle preparation point The total number of vehicles departing shall not exceed the reserve quantity determined in the first phase. To ensure that vehicle scheduling in the second phase does not exceed the resource supply in the first phase:

[0068]

[0069] in, Indicates from the vehicle preparation point All outbound arcs that have departed;

[0070] The flow balance constraint requires that the number of vehicles flowing into each node equals the number of vehicles flowing out, maintaining the conservation of network flow. That is, vehicles that arrive at a certain physical location and are in a certain evacuation cycle state must all continue to perform subsequent evacuation, empty driving, termination, or other operations from that state, and are not allowed to disappear in an intermediate state or be created out of thin air.

[0071]

[0072] Demand satisfaction constraints express the balance between capacity supply and passenger demand, ensuring that every pair of OD (Original Demand) needs is fully covered by vehicle capacity, while any unmet demand is penalized.

[0073]

[0074] in, This indicates that for each OD pair, from station arrive The evacuation arc; Indicates the passenger capacity of a single vehicle. For the scene Lower OD pair The number of people stranded;

[0075] In addition, all arc flow variables and unmet demand variables must be non-negative integers to ensure physical feasibility:

[0076] .

[0077] The present invention is further configured such that the specific steps of S5 are as follows:

[0078] 1) Linearization process:

[0079] The partial bar term in the objective function of the first stage Its essence is in fuzzy sets Within the defined probability distribution, find the probability distribution that maximizes the expected cost of the second stage, and expand it into a discrete form, which is the probability distribution under the probability normalization constraint. L1 norm constraint and nonnegativity constraints Below, maximize ,in For the scene The corresponding optimal operating cost;

[0080] Since the L1 norm constraint includes absolute value operations, it needs to be linearized first in order to apply duality theory. Therefore, for each scenario... Introduce a pair of auxiliary nonnegative variables and ,make ,but Through this transformation, the L1 norm constraint is converted into a standard linear inequality. The inner problem then becomes a standard linear programming problem;

[0081] 2) Dual transformation:

[0082] To address the linearized inner-layer maximization problem, we take the Lagrange dual and introduce the following dual variables: For free variables, the corresponding probability normalization constraint The duality; For non-negative variables, corresponding to L1 norm constraints The duality; For real-valued variables, they correspond to the dual of the probability definition constraints for each scenario; in the dual derivation process, auxiliary variables... and Nonnegativity constraints, after dual transformation, generate implicit constraints. By the strong duality theorem of linear programming, the optimal values ​​of the primal problem and the dual problem are equal, thus yielding an equivalent dual problem:

[0083]

[0084]

[0085]

[0086]

[0087] in, This reflects the risk hedging costs associated with optimizing the Brussels Bar. This is a weighted summary of cost deviations across different scenarios.

[0088] 3) Final single-layer DRO model:

[0089] Substituting the dual result into the first stage eliminates the min-max structure, resulting in the final single-layer mixed-integer linear programming model:

[0090]

[0091]

[0092]

[0093]

[0094]

[0095]

[0096]

[0097]

[0098]

[0099]

[0100]

[0101] In summary, the present invention has the following beneficial effects: The present invention decouples the cyclic scheduling problem into a hierarchical network flow problem through a state-space network. Compared with the traditional VRP (vehicle routing problem) modeling method, it eliminates the combinatorial explosion and nonlinear coupling caused by path loops, reduces the problem complexity from exponential to polynomial level, significantly improves the solution efficiency, and can obtain a high-quality solution within the limited time of emergency response.

[0102] This invention employs L1-norm-based fuzzy sets to characterize demand uncertainty. Compared to traditional stochastic programming methods, it does not rely on precise assumptions about the probability distribution of demand and has stronger applicability in sudden interruption scenarios where historical data is scarce. Compared to traditional robust optimization methods, it utilizes multi-scenario information through a probability weighting mechanism, avoiding the overly conservative problem caused by only considering the worst-case scenario. Compared to DRO methods based on moment constraints or Wasserstein distance, L1-norm fuzzy sets have a simple structure and do not introduce semidefinite programming or nonlinear coupling when combined with integer variables, maintaining the standard form of mixed-integer linear programming.

[0103] This invention transforms the min-max bilayer decomposed bar optimization model into a standard mixed integer linear programming problem through strong dual transformation. Compared with existing methods that use iterative solutions or decomposition algorithms, it eliminates the computational overhead caused by nested optimization structures and can directly call mature commercial solvers for efficient solutions, meeting the real-time requirements of emergency scenarios. Attached Figure Description

[0104] Figure 1 This is a technical framework diagram of the method of the present invention.

[0105] Figure 2 This is a schematic diagram of a subway line interruption scenario according to the present invention.

[0106] Figure 3 This invention relates to state-space networks ( (Structure diagram of )

[0107] Figure 4 This is a timing diagram of the shuttle bus dispatching task generated by the present invention in the scenario of subway interruption. Detailed Implementation

[0108] The following is in conjunction with the appendix Figure 1-4 The present invention will be described in further detail below.

[0109] Example: A method for site selection and configuration of backup bus depots to cope with subway interruptions.

[0110] 1. Construct fuzzy sets based on the L1 norm to represent the uncertainty of demand.

[0111] In the event of a sudden subway disruption, the number and distribution of stranded passengers are highly uncertain due to multiple factors, including the duration of the disruption, the location of the affected section, and passenger flow patterns. Traditional deterministic optimization assumes known requirements, and the proposed solution is prone to failure when actual deviations occur. While traditional stochastic programming considers randomness, it relies on precise assumptions about probability distributions (such as a normal distribution). In situations where data is scarce for sudden events, it lacks statistical support, and misclassification of the distribution can lead to systematic distortion of the optimization results.

[0112] To address this, this invention employs a robust optimization framework that does not rely on a precise characterization of the demand distribution. Instead, it constructs a fuzzy set encompassing all reasonable probability distributions to seek the optimal decision under the worst-case distribution. Compared to traditional robust optimization, which only considers a single worst-case scenario, this framework utilizes multi-scenario information through a probability weighting mechanism, maintaining robustness while avoiding excessive conservatism. Regarding the specific construction of the fuzzy set, moment-constrained fuzzy sets typically introduce a semidefinite programming structure, limiting the solution scale. Furthermore, Wasserstein distance-based fuzzy sets introduce nonlinear coupling when combined with integer variables, increasing the solution difficulty. Considering the scarcity of historical data in this problem, the greater suitability of representing uncertainty with finite discrete scenarios, and the fact that the model includes integer variables such as backup point location and needs to maintain the standard form of mixed-integer linear programming, this invention selects an L1-norm-based fuzzy set as the uncertainty characterization tool.

[0113] Specifically, let A collection of discrete scenarios generated from historical data or simulations. Each scenario Corresponding to a set of known demand for stranded passengers Based on historical frequency or the assumption of equal probability, each scenario is assigned a nominal probability. (usually taken) However, nominal probability is only an estimate of the true distribution, and the actual probability distribution may deviate from it. To characterize this deviation, we define the true probability distribution. The feasible region is the nominal distribution Centered on, with a 1-norm distance not exceeding The set of:

[0114]

[0115] in, ensure This is a valid probability distribution; while The L1 norm distance (i.e., Manhattan distance) between the true and nominal distributions is limited to not exceeding a threshold. .parameter This is called the robustness coefficient, and its physical meaning is the maximum total deviation that the true probability vector is allowed to deviate from the predicted probability vector. When When = 0, the model degenerates into a deterministic stochastic programming model based on the nominal distribution; when When = 2, the model degenerates into a robust optimization that only considers the worst-case scenario. Therefore, The value of determines the balance between the degree of conservatism in decision-making and optimization efficiency.

[0116] At this point, the mathematical characterization of demand uncertainty has been completed. The next step will be to establish a two-stage optimization model based on this fuzzy set.

[0117] 2. Establish the first-stage optimization model to determine the location of the standby vehicle point and the vehicle reserve decision.

[0118] The first phase is the planning phase, which involves determining the activation plan for the backup vehicle point before the demand is confirmed. and the number of vehicles in reserve at each standby point The objective function for the first phase is to minimize the sum of the standby point activation cost, vehicle inventory cost, and the worst-case expected operating cost for the second phase.

[0119]

[0120] in, Indicates the activation of the backup vehicle point Fixed costs; This indicates the inventory cost per unit of vehicle; For a given reserve scheme With demand scenarios The optimal operating cost under the given conditions will be defined in detail in step 4 after the state space network is constructed in step 3. For fuzzy sets The expected operating cost for the second stage under the worst-case probability distribution.

[0121] The constraints in the first phase are based on two aspects: resource feasibility and service coverage.

[0122] Capacity reserve constraints require each standby point The number of vehicles in reserve shall not exceed the product of their maximum capacity and their operational status.

[0123]

[0124] in, Indicates the vehicle preparation point Maximum capacity;

[0125] Coverage constraints guarantee each interruption station and turnaround station It is covered by at least one active backup vehicle location, thus ensuring the accessibility of evacuation services.

[0126]

[0127] in, This indicates that it can cover subway stations. The vehicle preparation point is where you gather.

[0128] In addition, variable constraints require the location variable to be selected. A binary integer with a value of 0 or 1, used as a storage variable. The value should be a non-negative integer to reflect the discreteness of actual decision-making.

[0129]

[0130] 3. Construct a state-space network to transform the vehicle circulatory evacuation problem into a hierarchical network flow problem.

[0131] To address the characteristic that bus shuttle vehicles need to perform multiple cyclical evacuations between interruption stations, turnaround stations, and transfer stations when rail transit is interrupted, the number of evacuation trips is introduced as a state dimension. This decouples the complex VRP (Vehicle Routing Problem) into an acyclic hierarchical network flow problem, thus transforming it into an easily solvable minimum cost flow problem.

[0132] 1) Define the physical topology of the transportation network. Let the set of nodes in the transportation network be... including subway station collection Meet at the alternative pick-up point Among them, subway stations are the collection points. Further divided into sets of interrupted stations Assembly at the turnaround station Meet at the transfer station For each site Define the set of backup vehicle points that can cover the station. .

[0133] 2) Define vehicle travel routes. To meet the needs of origin-destination (OD) passengers, two types of route sets were constructed: evacuation route sets. and reverse empty driving path set The evacuation route set includes feasible bus routes that transport passengers from interrupted or turnaround stations to opposite turnaround or transfer stations, enabling them to continue their journeys via emergency bus services. Specifically, these include: ① routes from interrupted stations to turnaround stations, ② routes from interrupted stations to transfer stations, ③ routes from turnaround stations to opposite turnaround stations, and ④ routes from turnaround stations to opposite transfer stations.

[0134] Furthermore, when some stations serve only as originating or terminating points for passenger demand, empty routes need to be designed. Between consecutive evacuation missions, empty routes arise because passengers are not evacuated back to the interruption station, and transfer stations are not used as starting points for subsequent evacuations. Therefore, buses must travel empty to reach the next evacuation starting point. This reverse empty route allows buses to efficiently travel from the evacuation terminus to the new departure station, thus promoting a more flexible circulation of bus resources. Specifically, the empty route set... This includes: ① lines from turnaround stations to interrupted stations, ② lines from transfer stations to interrupted stations, and ③ lines from transfer stations to turnaround stations.

[0135] 3) Constructing a state-space network In subway disruption scenarios, bus shuttles need to make multiple round trips between the disruption station, turnaround station, and transfer station for evacuation. Each bus's route is essentially a closed loop involving multiple alternating periods of passenger transport and empty runs. If modeled directly as a Vehicle Routing Problem (VRP), the model will face combinatorial explosion and nonlinear coupling caused by path loops, resulting in low solution efficiency. Therefore, this invention introduces the number of evacuation trips as a state dimension, unfolding the cyclical travel process of vehicles in the physical network into a hierarchical directed acyclic network. In this state-space network, the same physical station corresponds to different state nodes under different evacuation trips. Each time a vehicle completes a passenger evacuation task, its state level automatically increases, thereby eliminating the cyclical structure in the path.

[0136] Specifically, node set It includes three child node sets: virtual source node set Virtual sink set and state node set .in, and These represent the vehicle's initial standby state at the standby point and the mission termination state, respectively. Each node in the diagram corresponds to a connection task and its real-time status in the physical network, defined as follows: ,in Indicates physical site, This indicates the number of completed shuttle runs.

[0137] Arc set It contains four types of arc subsets, specifically defined as follows:

[0138] ① Outbound arc subset Vehicle from the standby point The vehicle is dispatched empty to the interruption station or turnaround station, and upon arrival, it is in an initial state where evacuation has not yet been carried out. ). Represented as .

[0139] ② Sparse arc subset This indicates the process of a vehicle performing an actual passenger-carrying task, from the station... Passengers arrive at the station After completing this transport, the cumulative number of completed laps increased from [previous number]. Increase to 。 is represented as .

[0140] ③ Empty Arc Subset : Indicates that the vehicle has completed the first The second evacuation mission has reached the passenger drop-off point. Afterwards, the empty vehicle traveled to the next starting point. The process. Represented as... .

[0141] ④ Task termination arc subset Once the vehicle has transported passengers to the evacuation destination, it will cease service if it does not perform any further tasks. This is indicated as... .

[0142] Define nodes at the same time Outgoing arc set and incoming arc set This is used to express subsequent flow balance constraints.

[0143] 4. Establish a second-stage optimization model to determine vehicle scheduling routes and demand fulfillment schemes.

[0144] The second phase is the operational phase, which involves specific demand scenarios. After implementation, based on the vehicle reserve plan of the first phase, the specific dispatching routes of the vehicles and the unmet demand are determined. Define the arc. Generalized driving costs The specific definition is as follows:

[0145]

[0146] in For the node arrive Travel time; This is the time coefficient for the outbound stage (without penalty for loops). It is a pre-defined number of cycles A monotonically increasing time coefficient is used. By imposing an increasing penalty on subsequent evacuation trips to force the model to prioritize the use of early capacity, the model mathematically approximates minimizing the maximum completion time objective, balancing cost optimization and evacuation timeliness. Furthermore, it is assumed that... For passenger boarding and alighting times, This represents the maximum number of evacuation trips.

[0147] In addition, definition Representing a scene Middle through arc The number of vehicles, Indicates OD pair Unmet demand. Scenario The optimal operating cost is defined as minimizing the sum of the total vehicle operating cost and the penalty for unmet demand:

[0148]

[0149] in, The penalty cost for unmet demand per unit is determined by... Setting it to a larger value forces the model to prioritize passenger demand, allowing some demand to remain unmet only when there is a genuine shortage of capacity.

[0150] The constraints in the second phase ensure the feasibility of the scheduling scheme from three levels: resource integration, flow conservation, and demand satisfaction.

[0151] Outbound resource constraints require that from each vehicle preparation point The total number of vehicles departing shall not exceed the reserve quantity determined in the first phase. This ensures that vehicle scheduling in the second phase does not exceed the resource supply in the first phase.

[0152]

[0153] in, Indicates from the vehicle preparation point All outbound arcs that have departed;

[0154] The flow balance constraint requires that the number of vehicles flowing into each node equals the number of vehicles flowing out, maintaining the conservation of network flow. That is, vehicles that arrive at a certain physical location and are in a certain evacuation cycle state must all continue to perform subsequent operations (evacuation, empty driving, or termination) from that state, and are not allowed to disappear or be generated out of thin air in an intermediate state.

[0155]

[0156] Demand fulfillment constraints express the balance between capacity supply and passenger demand. They ensure that every pair of origin-destination (OD) demands is fully covered by vehicle capacity, while any unmet demand is penalized.

[0157]

[0158] in, This indicates that for each OD pair, from station arrive The evacuation arc; Indicates the passenger capacity of a single vehicle. For the scene Lower OD pair The number of people stranded.

[0159] In addition, all arc flow variables and unmet demand variables must be non-negative integers to ensure physical feasibility.

[0160]

[0161] At this point, the two-stage optimization model has been established. However, this model contains a nested min-max structure, which needs to be further transformed into a directly solvable form through dual transformation.

[0162] 5. Based on the model equivalence transformation of strong duality theory, the bi-level optimization model is transformed into a solvable single-level mixed integer linear programming problem, and then solved using a linear programming solver.

[0163] In the two-stage model described above, the objective function of the first stage includes an inner maximization problem, namely, in the fuzzy set... The inner objective seeks the probability distribution that maximizes the expected cost of the second stage, thus forming a nested min-max bi-layer structure of outer minimization and inner maximization. Since the optimal value of the inner problem, as part of the outer objective function, is an implicit function of the decision variables in the first stage, its functional form cannot be directly expanded into an explicit algebraic expression of the decision variables. Therefore, a dual transformation is needed to equivalently replace the inner maximization problem with its dual minimization form and corresponding dual feasibility constraints, thereby reconstructing the original bi-layer model into an equivalent single-layer mixed-integer linear programming problem.

[0164] 1) Linearization processing. The partial bar terms in the objective function of the first stage. Its essence is in fuzzy sets Within the defined probability distribution, we seek the probability distribution that maximizes the expected cost of the second stage. Expanding this into a discrete form, we find the probability distribution under the probability normalization constraint. L1 norm constraint and nonnegativity constraints Below, maximize ,in For the scene The corresponding optimal operating cost.

[0165] Because the L1 norm constraint involves absolute value operations, it needs to be linearized first in order to apply duality theory. Therefore, for each scenario... Introduce a pair of auxiliary nonnegative variables and ,make ,but Through this transformation, the L1 norm constraint is converted into a standard linear inequality. The inner problem then becomes a standard linear programming problem.

[0166] 2) Dual Transformation. For the linearized inner-layer maximization problem, we take the Lagrangian dual and introduce the following dual variables: For free variables, the corresponding probability normalization constraint The duality; For non-negative variables, corresponding to L1 norm constraints The duality; These are real-valued variables, corresponding to the dual of the probability definition constraints for each scenario. During the dual derivation, auxiliary variables... and Nonnegativity constraints, after dual transformation, generate implicit constraints. By the strong duality theorem of linear programming, the optimal values ​​of the primal problem and the dual problem are equal, thus yielding an equivalent dual problem:

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[0171] in, This reflects the risk hedging costs associated with optimizing the Brussels Bar. This is a weighted summary of cost deviations across different scenarios.

[0172] 3) Final Single-Layer DRO Model. Substituting the dual result into the first stage, the min-max structure is eliminated, resulting in the final single-layer mixed-integer linear programming model:

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[0184] The final model described above is a standard mixed-integer linear programming problem, which can be solved efficiently by directly calling commercial solvers (such as CPLEX and Gurobi).

[0185] As can be seen from the above technical solution, this invention decouples the cyclic scheduling problem into a hierarchical network flow problem through a state-space network. Combining an L1-norm-based sub-Bruker optimization framework with strong duality transformation, it transforms the min-max bi-level sub-Bruker optimization model into a standard mixed-integer linear programming problem, achieving joint optimization of standby point location, vehicle reserves, and scheduling routes. This method balances decision robustness and computational feasibility, and has practical application value for improving the response efficiency and resource utilization of emergency bus connections in scenarios of sudden subway disruptions.

[0186] This specific embodiment is merely an explanation of the present invention and is not intended to limit the invention. After reading this specification, those skilled in the art can make modifications to this embodiment without contributing any inventive step, but such modifications are protected by patent law as long as they are within the scope of the claims of the present invention.

Claims

1. A method for the location and configuration of backup bus depots to cope with subway interruptions, characterized by: The method comprises two phases: the first phase occurs before the subway disruption, and the second phase occurs after the subway disruption. Specifically, it includes the following five steps: S1. Construct fuzzy sets based on the L1 norm to represent the uncertainty of demand; S2. Establish the first-stage optimization model to determine the location of the standby vehicle point and vehicle reserve decisions; S3. Construct a state-space network to transform the vehicle circulatory evacuation problem into a hierarchical network flow problem; S4. Establish the second-stage optimization model to determine vehicle scheduling routes and demand satisfaction schemes; S5. Based on the strong duality theory, the model equivalent transformation transforms the first and second stage bi-level optimization model into a solvable single-level mixed integer linear programming problem, and solves it using a linear programming solver.

2. The method for site selection and configuration of backup bus depots for responding to subway interruptions as described in claim 1, characterized in that: The specific steps of S1 are as follows: set up A collection of discrete scenarios generated from historical data or simulations, each scenario Corresponding to a set of known demand for stranded passengers Based on historical frequency or the assumption of equal probability, a nominal probability is assigned to each scenario. The nominal probability is taken as ; However, nominal probability is only an estimate of the true distribution, and the true probability distribution may deviate from it; to characterize this deviation, we define the true probability distribution. The feasible region is defined by the nominal probability distribution. Centered on, with a 1-norm distance not exceeding The set of: in, ensure This is a valid probability distribution; while The L1 norm distance between the true probability distribution and the nominal probability distribution is limited to a threshold. ;parameter This is called the robustness coefficient, and its physical meaning is the maximum total deviation that the true probability vector is allowed to deviate from the predicted probability vector; when When = 0, the model degenerates into a deterministic stochastic programming model based on the nominal probability distribution; when When the coefficient is 2, the model degenerates into a robust optimization that only considers the worst-case scenario; therefore... The value of determines the balance between the degree of conservatism in decision-making and optimization efficiency.

3. The method for site selection and configuration of backup bus depots for responding to subway interruptions as described in claim 2, characterized in that: The specific steps of S2 are as follows: The first phase is the planning phase, which involves determining the activation plan for the backup vehicle point before the demand is confirmed. and the number of vehicles in reserve at each standby point The objective function for the first stage is to minimize the sum of the standby point activation cost, vehicle inventory cost, and the worst-case expected operating cost for the second stage. in, Indicates the activation of the backup vehicle point Fixed costs; This indicates the inventory cost per unit of vehicle; For a given reserve scheme With demand scenarios The optimal operating cost; For fuzzy sets The expected operating cost of the second stage under the worst-case probability distribution; the constraints of the first stage are imposed from the aspects of resource feasibility and service coverage. Capacity reserve constraints require each standby point The number of vehicles in reserve must not exceed the product of their maximum capacity and their operational status. in, Indicates the vehicle preparation point Maximum capacity; Coverage constraints guarantee each interruption station and turnaround station It is covered by at least one active backup vehicle location, thus ensuring the accessibility of evacuation services: in, Indicates coverage of subway stations The vehicle preparation point assembly; In addition, variable constraints require the location variable to be selected. A binary integer with a value of 0 or 1, used as a storage variable. The value should be a non-negative integer to reflect the discreteness of actual decision-making. 。 4. The method for site selection and configuration of backup bus depots for responding to subway interruptions as described in claim 1, characterized in that: The specific steps of S3 are as follows: 1) Define the physical topology of the transportation network: Let the set of traffic network nodes be defined. including subway station collection Meet at the alternative pick-up point Among them, subway stations are the collection points. Further divided into sets of interrupted stations Assembly at the turnaround station Meet at the transfer station For each site Define the set of backup vehicle points that can cover the station. ; 2) Define the vehicle's travel path: To meet the needs of origin-destination (OD) passengers, two types of route sets were constructed: evacuation route sets. and reverse empty driving path set ; The evacuation route set includes: ① routes from the interruption station to the turnaround station, ② routes from the interruption station to the transfer station, ③ routes from the turnaround station to the turnaround station on the opposite side, and ④ routes from the turnaround station to the transfer station on the opposite side. Reverse empty driving path set This includes: ① lines from turnaround stations to interrupted stations, ② lines from transfer stations to interrupted stations, and ③ lines from transfer stations to turnaround stations. 3) Constructing a state-space network : Introducing the number of evacuation trips As a state dimension, where To determine the maximum number of evacuation loops, the cyclical driving process of vehicles in the physical network is unfolded into a hierarchical directed acyclic network. In this state space network, the same physical station corresponds to different state nodes under different evacuation loops. Each time a vehicle completes a passenger evacuation task, its state level automatically increases, thereby eliminating the cyclical structure in the path. Specifically, node set It includes three child node sets: virtual source node set Virtual sink set and state node set ;in, and These represent the vehicle's initial standby state at the standby point and the mission termination state, respectively. Each node in the diagram corresponds to a connection task and its real-time status in the physical network, defined as follows: ,in, Indicates physical site, Indicates the number of completed shuttle runs; Arc set It contains four types of arc subsets, specifically defined as follows: ① Outbound arc subset Vehicle from the standby point The aircraft is dispatched empty to an interruption station or turnaround station, and upon arrival, it is in an initial state where evacuation has not yet been carried out. ), represented as: ② Sparse arc subset This indicates the process of a vehicle performing an actual passenger-carrying task, from the station... Passengers arrive at the station After completing this transport, the cumulative number of completed laps increased from [previous number]. Increase to , is represented as: ③ Empty Arc Subset : Indicates that the vehicle has completed the first The second evacuation mission has reached the passenger drop-off point. Afterwards, the empty vehicle traveled to the next starting point. The process is represented as: ④ Task termination arc subset Once a vehicle has transported passengers to the evacuation destination, it ceases service if it does not perform any further tasks. Define nodes at the same time Outgoing arc set and incoming arc set This is used to express subsequent flow balance constraints.

5. The method for site selection and configuration of backup bus depots for responding to subway interruptions as described in claim 1, characterized in that: The specific steps of S4 are as follows: The second phase is the operation phase, which is implemented in specific demand scenarios. After implementation, based on the vehicle reserve plan of the first phase, determine the specific dispatching routes of the vehicles and the unmet demand; define the arc. Generalized driving costs The specific definition is as follows: in, For the node arrive Travel time; This is the time coefficient for the outbound stage (without cycle penalty). It is a pre-defined number of cycles A monotonically increasing time coefficient is used; by imposing an increasing penalty on subsequent evacuation trips to force the model to prioritize the use of early capacity, the maximum completion time objective is mathematically minimized, balancing cost optimization and evacuation timeliness; furthermore, it is assumed that... For passenger boarding and alighting times, This represents the maximum number of evacuation trips. In addition, definition Representing a scene Middle through arc The number of vehicles, Indicates OD pair Unmet demand; scenario The optimal operating cost is defined as minimizing the sum of the total vehicle operating cost and the penalty for unmet demand: in, The penalty cost for unmet demand per unit is determined by... Setting it to a larger value forces the model to prioritize passenger demand, allowing some demand to remain unmet only when there is a genuine shortage of capacity. The constraints in the second phase ensure the feasibility of the scheduling scheme from three levels: resource integration, flow conservation, and demand satisfaction. Outbound resource constraints require that from each vehicle preparation point The total number of vehicles departing shall not exceed the reserve quantity determined in the first phase. To ensure that vehicle scheduling in the second phase does not exceed the resource supply in the first phase: in, Indicates from the vehicle preparation point All outbound arcs that have departed; The flow balance constraint requires that the number of vehicles flowing into each node equals the number of vehicles flowing out, maintaining the conservation of network flow. That is, vehicles that arrive at a certain physical location and are in a certain evacuation cycle state must all continue to perform subsequent evacuation, empty driving, termination, or other operations from that state, and are not allowed to disappear in an intermediate state or be created out of thin air. Demand satisfaction constraints express the balance between capacity supply and passenger demand, ensuring that every pair of origin-destination (OD) demand is fully covered by vehicle capacity, while any unmet demand is penalized. in, This indicates that for each OD pair, from station arrive The evacuation arc; Indicates the passenger capacity of a single vehicle. For the scene Lower OD pair The number of people stranded; In addition, all arc flow variables and unmet demand variables must be non-negative integers to ensure physical feasibility: 。 6. The method for site selection and configuration of backup bus depots for responding to subway interruptions as described in claim 1, characterized in that: The specific steps of S5 are as follows: 1) Linearization process: The partial bar term in the objective function of the first stage Its essence is in fuzzy sets Within the defined probability distribution, find the probability distribution that maximizes the expected cost of the second stage, and expand it into a discrete form, which is the probability normalization constraint. L1 norm constraint and nonnegativity constraints Below, maximize ,in For the scene The corresponding optimal operating cost; Since the L1 norm constraint includes absolute value operations, it needs to be linearized first in order to apply duality theory. Therefore, for each scenario... Introduce a pair of auxiliary nonnegative variables and ,make ,but ; Through this transformation, the L1 norm constraint is converted into a standard linear inequality. The inner problem then becomes a standard linear programming problem; 2) Dual transformation: To address the linearized inner-layer maximization problem, we take the Lagrange dual and introduce the following dual variables: For free variables, the corresponding probability normalization constraint The duality; For non-negative variables, corresponding to L1 norm constraints The duality; For real-valued variables, they correspond to the dual of the probability definition constraints for each scenario; in the dual derivation process, auxiliary variables... and Nonnegativity constraints, after dual transformation, generate implicit constraints. By the strong duality theorem of linear programming, the optimal values ​​of the primal problem and the dual problem are equal, thus yielding an equivalent dual problem: in, This reflects the risk hedging costs associated with optimizing the Brussels Bar. This is a weighted summary of cost deviations across different scenarios. 3) Final single-layer DRO model: Substituting the dual result into the first stage eliminates the min-max structure, resulting in the final single-layer mixed-integer linear programming model: 。

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