A bus allocation method under full congestion

Abstract

The application discloses a bus distribution method under complete congestion, comprising the following steps: introducing a super-path structure into a bus network to express a travel route selection strategy of passengers; reconstructing a super-path flow loading problem into a monotone VI sequence along a super-path topology node sequence to obtain a super-path flow loading scheme; constructing a super-path flow loading algorithm based on a gradient projection to execute the super-path flow loading, so that the corresponding arc flow distribution, arc use probability and node use probability of each node of each super-path are obtained. Thus, for a problem without capacity constraint, a column generation algorithm is introduced based on the super-path framework to realize local adjustment and rapid equilibrium of the super-path flow in the super-path set of each OD pair. For a problem with capacity constraint, the super-path framework and the augmented Lagrange method are combined to obtain a high-precision feasible solution under the condition of satisfying the capacity constraint.

Description

Technical Field

[0001] This invention relates to the field of public transport network design technology, and in particular to a method for bus allocation under conditions of complete congestion. Background Technology

[0002] The public transport equilibrium allocation model is a core module in public transport network design and operation analysis. It is used to predict passenger route selection behavior and estimate passenger flow distribution across different route segments. In public transport network design, the equilibrium allocation problem for completely congested public transport aims to determine passenger flow distribution within the network based on route service frequency (used to predict passenger route selection behavior and estimate passenger flow distribution across different route segments). Passenger route selection is fundamentally influenced by congestion effects at nodes (i.e., bus stops) and boarding arcs (i.e., route segments) within the transport network. Passengers experience waiting times at stations, and additional queuing delays occur when demand exceeds vehicle capacity. Simultaneously, passengers on the bus bear the costs of congestion due to decreased comfort and extended travel times. The equilibrium state is such that no passenger can reduce their generalized travel costs by unilaterally changing their route selection.

[0003] Therefore, the balanced distribution of fully congested public transport is often regarded as a core component of long-term public transport network planning models, providing key support for strategic decisions such as service design, infrastructure investment, and network optimization.

[0004] In the research and development of balanced allocation of fully congested public transport, DeCea (1993) first proposed the concept of effective departure frequency to characterize the decrease in line frequency caused by boarding congestion, and attempted to solve it by diagonalization method. However, this method is only applicable to partially congested cases that depend on nominal frequency and does not consider strict capacity constraints, so it may produce solutions that exceed the line's carrying capacity.

[0005] Cominetti (2001) further utilized queuing theory to describe the station's flow-based waiting process, modeling the fully congested TEAP as a destination-based fixed-point problem of boarding arc flow, and proved the existence of a solution, but did not propose a specific algorithm. Cepeda et al. (2006) reconstructed this model into a non-convex, non-differentiable gap function optimization problem, and for the first time used the successive mean averaging (MSA) method for heuristic solution, proposing that when the flow exceeds the capacity, the effective frequency should approach zero to prevent oversaturation. Codina (2013) further transformed this model into an equivalent variational inequality (VI) problem in the boarding arc flow space, and proved the existence of an equilibrium solution. Recently, Ren (2021) and Tian (2021) extended the fully congested TEAP to the path segment framework, expanding the research direction for the model's application. However, existing solutions are still mainly frequency-driven models based on MSA. Although these models are simple in form and easy to implement, they have significant limitations at the algorithmic level: on the one hand, the pre-defined monotonically decreasing step size can lead to update vanishing in the later stages of iteration, resulting in slow convergence (Bar-Gera, 2006; Mounce, 2015); on the other hand, in large-scale congested networks, MSA often requires a large number of iterations to achieve acceptable accuracy (Sheffi, 1985). Therefore, solving the problem of balanced bus allocation in fully congested networks still faces considerable challenges in terms of balancing capacity constraints and computational efficiency.

[0006] It is evident that existing public transport equilibrium models generally employ MSA-type algorithms, whose convergence mechanism relies on decreasing step size averaging when solving for equilibrium solutions. This approach fails to incorporate structural information about the public transport network in its design and cannot effectively correct the direction using dual gradient information under capacity constraints. Consequently, as the network size increases, the algorithm's convergence efficiency significantly decreases, the number of iterations grows exponentially, computation time increases, and the accuracy of the solution decreases accordingly. Furthermore, because this type of algorithm updates only based on arc segment flow, it ignores passenger path selection behavior at the policy level, thus failing to retain complete hyperpath information and unable to explicitly output the actual path strategies adopted by passengers.

[0007] In summary, this structural deficiency severely limits the model's ability to characterize path selection behavior, hinders in-depth research on behavioral interpretability and path visualization, and makes it difficult to meet the practical needs of public transportation system analysis and optimization. Summary of the Invention

[0008] The purpose of this invention is to overcome the shortcomings of the prior art and provide a bus allocation method under conditions of complete congestion.

[0009] The objective of this invention is achieved through the following technical solution:

[0010] This application discloses a bus allocation method under full congestion, comprising: introducing a hyperpath structure in the bus network to express the passenger's travel route selection strategy, wherein the hyperpath is a directed acyclic subgraph including a subset of nodes, a subset of arcs, and an arc usage probability vector; reconstructing the hyperpath flow loading problem into a monotonic VI sequence along the topological node order of the hyperpath to obtain a hyperpath flow loading scheme; constructing a hyperpath flow loading algorithm based on gradient projection, executing hyperpath flow loading to obtain the corresponding arc flow distribution, arc usage probability, and node usage probability of each node in each hyperpath; then solving for the hyperpath flow and arc flow under the equilibrium state of the entire network to obtain the complete flow distribution under each hyperpath in the bus network, which is used to update the hyperpath cost, determine whether the current hyperpath flow solution is in an equilibrium state, and obtain the hyperpath flow and arc flow under the equilibrium state of the entire network.

[0011] The beneficial effects of this invention are:

[0012] To address the shortcomings of existing methods in terms of iterative efficiency, path interpretability, and path tracing, this invention achieves three improvements through a modeling approach using hyperpaths as decision variables: First, the hyperpath flow loading problem is reconstructed as a monotonic VI sequence along the hyperpath topological node order, thus proposing an efficient hyperpath flow loading scheme. Second, a hyperpath-based VI representation is established to characterize the fully congested bus equilibrium allocation problem with and without capacity constraints. Third, a hyperpath-based column generation algorithm is proposed to solve the fully congested bus equilibrium allocation problem. Numerical experiments show that column generation significantly outperforms existing MSA-based benchmark algorithms on both large and small networks. Attached Figure Description

[0013] Figure 1 A schematic diagram of the algorithmic framework for the VI problem without considering capacity constraints according to some embodiments of this application;

[0014] Figure 2 This is a schematic diagram of an algorithmic framework for the capacity-constrained VI problem according to some embodiments of this application;

[0015] Figure 3 This is a schematic diagram of a bus stop according to some embodiments of this application. Detailed Implementation

[0016] The technical solution of the present invention will be clearly and completely described below with reference to the embodiments. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0017] See Figures 1-2This invention provides a bus allocation method under complete congestion, which includes introducing a hyperpath structure in the bus network to express the passenger's travel route selection strategy. The hyperpath is a directed acyclic subgraph, including a subset of nodes, a subset of arcs, and an arc usage probability vector.

[0018] For example, in the passenger distribution problem based on effective departure frequency, it can be known that for a bus network's origin O and destination D, a finite set of bus routes... Connected. For line a, It has in-vehicle travel time ; and an effective departure frequency function The effective departure frequency is determined by the number of passengers boarding the bus. The continuous and strictly monotonically decreasing function reaches the saturation flow of the line. It drops to 0.

[0019] Furthermore, according to the bus co-routing theory, we know that when choosing a route, passengers will select a set of non-empty attracting routes s, s∈A, and take the bus that arrives first; set s constitutes a strategy. The passenger's strategy selection must minimize the expected travel time. Minimum. The expected time is defined as:

[0020]

[0021] in, This refers to the waiting time. The probability that a passenger will choose route a for their trip; and This represents the desired in-vehicle travel time.

[0022] To ensure traffic flow conservation, OD (Original Demand) flow (traffic trips) It will be assigned to all possible policies s∈A, and satisfy: ,in This is for the travel flow using strategy s. Each passenger is assigned a probability. Take the route Thus, the total flow of line a is obtained. .

[0023]

[0024] The passenger distribution equation defined by this formula constitutes a fixed-point problem, the solution of which uniquely corresponds to the line flow vector. According to the Wardrop principle (traveler optimality principle), in an equilibrium state, only by making... The strategy with the minimum value will be selected, and the corresponding equilibrium strategy flow vector will be used. satisfy:

[0025]

[0026] In the formula, The flow rate corresponding to strategy s under equilibrium conditions. This is the strategy flow solution under equilibrium conditions. For passengers' shortest expected travel time, and .

[0027] Specifically, to describe the flow distribution and equilibrium conditions in the public transport network, combined with Figure 3 The bus network model is defined as follows: the bus network is a directed graph. ,in It is a transfer node Collections and virtual station nodes Union of sets; It is the boarding arc. Disembarkation arc , driving arc and walking arc The union of the two sets. Each bus route that stops at a bus stop corresponds to one set. N ,pass N Constructing the boarding arc and disembarkation arc It depicts the process of passengers getting on and off buses along bus routes and the walking arcs through different transfer nodes. Enables transfers between stations.

[0028] In the public transportation network system of this embodiment, passengers travel strategically, and their routes can be viewed as hyperpaths containing multiple alternative routes. k Hyperpath k for It is a directed graph of a public transport network. Acyclic subgraphs, in which , is a subset of the bus network node N; , is a subset of arc set A; A vector of probabilities used for the arc. Are passengers at the node i Select arc The probability of.

[0029] Due to the acyclic nature of hyperpaths, the usage probability of each transfer node and its corresponding arc can be calculated sequentially using its topological order. Let... , Superpath k Use nodes i Sum of arcs The probability of is then:

[0030]

[0031]

[0032]

[0033] Where o and d are the start and end transfer nodes of hyperpath k, respectively. Let k be the sum of the boarding arc probabilities based on the connection between the previous node i and its j under the superpath k. The total probability of using the starting point o for the superpath k. The total probability of using the endpoint d for the superpath k. The total probability of using the boarding arc (i,j) for the superpath k.

[0034] Based on this, the passenger flow distribution at transfer node i under hyperpath k satisfies a fixed-point problem as shown in the following formula:

[0035]

[0036] in, To enter the transfer node on superpath k Traffic, For the traffic of superpath k, The probability of using node i for superpath k. This refers to the strategy used by hyperpath k at transfer node i. For the reason Assigned to boarding arc The flow rate satisfies the constraints of flow conservation and positive flow. The effective departure frequency is the frequency at which the onboard arc flow is equal to v under the effective departure frequency function.

[0037] when When the solution is the previous formula (i.e., the passenger flow distribution formula), for all transfer points have:

[0038]

[0039] in, Assigning the superpath k to the boarding arc The traffic on the internet, corresponding to For the arc of the vehicle (i,j) n The effective departure frequency; that is, for each transfer node of the superpath k. strategy set Flow rate on each boarding arc and effective departure frequency The ratio is a constant.

[0040] Based on this, the above equation can be equivalently transformed into the following problem VI (variational inequality problem):

[0041] First, define the feasible region. .

[0042] If and only if The solution to the following VI problem is... The condition for satisfying the above formula (i.e., the ratio of each boarding arc to its corresponding effective departure frequency for each transfer node): Find , so that:

[0043]

[0044] in, , It is the flow allocated to the boarding arc (i, j) by the superpath k*. It is the ratio of the flow of the superpath k* through the boarding arc (i, j) to its effective departure frequency, which can be regarded as the waiting cost. .

[0045] Then, based on the above equivalent problem, it is transformed into problem VI, and the GP algorithm is used to efficiently solve the passenger distribution problem. Next, problem VI based on the hyperpath network equilibrium condition is introduced, which includes two cases: considering capacity constraints and not considering capacity constraints.

[0046] Without considering capacity constraints, the equilibrium condition of the superpath network is as follows:

[0047]

[0048] The minimum superpath cost to reach destination d from origin o. The set of hyperpaths from the starting point O to the ending point D (in this article, o and d have the same meaning as O and D). For out-of-path costs, Let W be the flow on hyperpath k, and let W be the set of all Od pairs in the public transport network.

[0049] Out-of-path traffic must satisfy the following traffic conservation and non-negativity constraints:

[0050]

[0051]

[0052] in, Let O be the passenger demand starting at point O and ending at point D; the relationship between arc flow and superpath flow is:

[0053]

[0054] set up This is the hyperpath traffic vector. This is the correlation matrix between hyperpaths and OD pairs. Let this be the OD demand vector. Therefore, we define... And define the set of feasible hyperpath traffic vectors h as It satisfies:

[0055]

[0056] The transformed VI problem is as follows:

[0057] Superpath traffic in equilibrium state if and only if When solving the following hyperpath-based VI problem, To find the hyperpath network equilibrium condition that satisfies the aforementioned conditions. , so that:

[0058]

[0059] in, Indicates a given The hyperpath cost vector below.

[0060] When considering capacity constraints, the equilibrium condition for a capacity-limited network can be expressed as:

[0061]

[0062] definition ,in Let be the upper limit (saturation capacity) of the loading arc (i,j). Given the superpath cost under this condition, and the feasible superpath flow that satisfies the capacity constraint. Definition of the set:

[0063] The equilibrium conditions described above can be equivalently transformed into the following problem VI based on the following proposition:

[0064] If and only if When solving the following VI problem, Find the equilibrium condition that satisfies the following: , so that: .

[0065] Thus far, all questions concerning the presence or absence of capacity constraints in problem VI have been equivalently transformed using the hyperpath approach.

[0066] In a further embodiment, when using the GP algorithm to solve the passenger distribution problem at transfer node i under hyperpath k, a reference path is introduced to simplify the constraints, thereby improving the computational efficiency of the GP algorithm and overcoming the problems of slow convergence and neglect of network structure in the MSA algorithm; for ease of description, the algorithm is defined as the first algorithm.

[0067] In some embodiments, the boarding arc with the minimum "path cost" at the selected transfer node is chosen. As a reference path, the amount of traffic adjustment from a non-reference path to the reference path is expressed as:

[0068]

[0069] Then calculate the waiting cost of the boarding arc (i,j) and compare it with the reference boarding arc. The difference in waiting costs with respect to the first derivative of the boarding arc (i, j) flow rate It can be viewed as the descent step size:

[0070]

[0071] in , They are respectively and about and The derivative of Assign the superpath k to the boarding arc Traffic, Current boarding arc cost, Let $i$ be the minimum path cost from node $i$ to node $j$. Then, given the current average boarding arc flow... The GP algorithm then updates the non-reference "path" according to the following formula. Traffic:

[0072] .

[0073] Based on the definitions and related information of the above parameters, the specific steps of the solution algorithm (i.e., the first algorithm) for solving the passenger distribution problem under the hyperpath structure of the bus network based on the effective departure frequency in this embodiment of the application include the following:

[0074] S11. Input the strategy used by the superpath k at the transfer node i. The flow of traffic entering transfer node i on superpath k .

[0075] S12. Parameter initialization includes:

[0076] Calculate the initial effective departure frequency Determine the initial boarding arc flow distribution based on passenger flow distribution. Set the target convergence accuracy Current iteration number l =0; Sets the termination condition for the main loop. ,in:

[0077]

[0078] In the formula, For the first In the next iteration, the flow allocated to the boarding arc (i,j) by the hyperpath k, and the corresponding... and For the first l-1 The flow rate allocated to the onboard arc (i,j) by the hyperpath k in the next iteration and the flow rate allocated to the onboard arc (i,j) in the next iteration. l+1 In the next iteration, the superpath k is assigned to the boarding arc. Traffic.

[0079] S13. Perform the main loop, including:

[0080] S131, For sets Each boarding arc in the middle ( i , j Update the path cost for the boarding arc. , During the update process, if the current boarding arc ( i , j If the path cost is smaller than the previously recorded minimum, then the variable will be... Set as the current boarding arc ( i , j The cost is the minimum path cost from node i to node j. .

[0081] S132. On this set, for each non-referenced on-arc arc :

[0082] Calculate update adjustment amount And update That is, the formula mentioned above:

[0083]

[0084] S133. Calculate the flow adjustment amount based on the difference between the current boarding arc cost and the reference boarding arc cost. :

[0085] like but:

[0086]

[0087] like but:

[0088]

[0089] And update the onboard arc flow. , and effective departure frequency .

[0090] Each loop l +1, stop iteration when the set iteration termination condition is met.

[0091] S14, Output the final onboard arc flow rate. and their corresponding effective departure frequencies .

[0092] Therefore, by using the above algorithm to solve the passenger distribution problem corresponding to each transfer node in the topological order of the hyperpath, the boarding arc flow can be obtained. and their corresponding effective departure frequencies .

[0093] In a further embodiment, the complete hyperpath traffic structure of a passenger in the network is obtained through a hyperpath traffic loading algorithm. For ease of description, this is defined as the second algorithm, which specifically includes:

[0094] S21, Input superpath topology Hyperpath traffic .

[0095] S22. Initialize the parameters:

[0096] ; And set the starting point. o corresponding .

[0097] S23. The main loop includes:

[0098] S231, Calculate the entry node i Traffic ;

[0099] S232, when i When it is a transfer node:

[0100] The first algorithm is used to obtain the boarding arc flow at this node. and their corresponding effective departure frequencies Then calculate the probability of using each boarding arc:

[0101]

[0102] Update the arc usage probability and the next node usage probability:

[0103]

[0104] like i When it is not a transfer node:

[0105] set up Then update the arc usage probability and the next node usage probability:

[0106]

[0107] S233. Finally, output the arc flow distribution of the hyperpath. Arc usage probability vector The probability vector of node usage .

[0108] Therefore, the second algorithm is mainly based on the boarding arc flow obtained from the first algorithm. and their corresponding effective departure frequencies The arc flow distribution of the hyperpath is obtained. Arc usage probability vector The probability vector of node usage This allows us to obtain the complete traffic distribution under each superpath in the public transport network.

[0109] In a further embodiment, combining the aforementioned first and second algorithms and the aforementioned VI problem, for the case where capacity constraints are not considered, refer to Figure 1 To understand, proceed with the following steps:

[0110] S31A. First, give the judgment index function and set the convergence index for a single OD pair. :

[0111]

[0112] And setting relative gap indicators :

[0113]

[0114] S32A, Input public transport network G, traffic demand .

[0115] S33A. Initialization includes: initializing the OD travel route selection set. Initialize traffic loading; update arc traffic, cost, and effective departure frequency according to the first and second algorithms, and set the iteration count n=1, with a target convergence accuracy. .

[0116] The following steps are then repeated to adjust network traffic and achieve traffic balance.

[0117] S34A, the main loop includes:

[0118] S341A, Perform column generation for each OD pair. Search for the minimum cost superpath ;if Add it to This means adding a new column.

[0119] S342A. Construct sub-problems for each OD pair using the Gauss-Seidel decomposition strategy (in this embodiment, there are no capacity constraints).

[0120] S35A, adaptive inner loop includes:

[0121] To solve the OD pair subproblem, for the OD pair subproblem constructed by S341A, while keeping the flow distribution of the remaining OD pairs unchanged, the GP algorithm is used to solve the subproblem, update the hyperpath flow, and update the hyperpath cost using the following formula:

[0122]

[0123] Understandably, since the superpath cost function depends on both arc flow and effective departure frequency, and the relationship between the two is implicitly determined by the passenger distribution problem, it cannot be explicitly expressed as a function of arc flow. Therefore, the flow adjustment amount in the GP algorithm cannot be directly derived. A closed expression.

[0124] The convergence criterion for OD pairs is defined as follows: for a single OD pair... Indicators of convergence status (whether the network is in equilibrium) The corresponding calculation formula, when Update the arc flow, cost, and effective departure frequency; otherwise, continue iterating using the GP algorithm.

[0125] The overall convergence is assessed using the relative gap index, which evaluates the convergence of the overall problem (network equilibrium). As stated in the formula above, when If convergence occurs, the loop ends; otherwise, if the iteration count is not met (n=n+1), return to the main loop and continue the above operations.

[0126] S36A, Output Balanced State Overpath Flow Arc flow .

[0127] Furthermore, in order to determine the optimal transfer amount during the superpath flow adjustment process... A binary search is used to determine the optimal transition amount. Specifically, the third algorithm includes the following steps:

[0128] S41. Input reference superpath and its corresponding traffic Non-reference hyperpath k and its corresponding traffic .

[0129] S42. Initialization includes:

[0130] Based on the hyperpath traffic loading algorithm (i.e., the second algorithm), the hyperpaths 𝑘 and 𝑘 are respectively processed. Perform traffic loading;

[0131] Update the cost and effective departure frequency of all arcs on both hyperpaths, and calculate the cost of each hyperpath. and ;

[0132] like Then set the left endpoint l=0 and the right endpoint l=0. ;

[0133] like Then set .

[0134] Record the initial traffic of the non-reference superpath 𝑘 Initialize traffic transfer volume Historical flow transfer volume And set the target convergence accuracy. .

[0135] S43, Binary Search Main Loop:

[0136] when Execute in a loop:

[0137]

[0138]

[0139] Based on the second algorithm, the hyperpath k and Perform traffic loading and update the superpath k and Calculate the cost and effective departure frequency of all arcs, and calculate the superpath cost. and .

[0140] like Then update ;like Then update .

[0141] After the loop ends, output the optimal transfer amount. .

[0142] Furthermore, the complete process of using the GP algorithm to solve the OD pair subproblem, i.e., the fourth algorithm, specifically includes the following steps:

[0143] S51. Input the current superpath solution. .

[0144] S52. Initialization includes:

[0145] Calculate the initial effective departure frequency For each hyperpath implement:

[0146] The second algorithm is used to load traffic onto superpath k; the cost and effective departure frequency of all arcs on superpath k are updated, and the superpath cost is calculated. If the cost of superpath k is smaller, then .

[0147] S53. Perform flow adjustment loop, including:

[0148] For each superpath implement:

[0149] The optimal transfer amount is solved based on the third algorithm. And update the hyperpath traffic:

[0150]

[0151] Based on the second algorithm, the hyperpath k and Perform traffic loading, then update the superpath k and Calculate the cost and effective departure frequency of all arcs, and calculate the superpath cost. and .

[0152] S54. Output the updated hyperpath traffic solution. .

[0153] Therefore, based on the above method, the superpath flow equilibrium solution under the equilibrium condition of the public transport network is obtained. .

[0154] In other embodiments, capacity limitations of bus routes are taken into account, and upper limit constraints are added to all travel arcs belonging to the same route, for example:

[0155]

[0156] in, The total flow rate on the travel arc (x, y) It is the upper limit of the capacity of the travel arc (𝑖,𝑗).

[0157] To reflect the additional delay costs caused by limited vehicle capacity, Lagrange multipliers related to capacity constraints are introduced. This allows us to construct a generalized superpath cost that combines delay costs and travel costs. The definition is as follows:

[0158]

[0159] In this embodiment, penalty parameters are set under the ALM framework. With the Initialization formula and update rules:

[0160]

[0161]

[0162]

[0163]

[0164]

[0165] in, Solution for the initial travel arc flow rate; This represents the number of arcs in the initial solution that do not satisfy the capacity constraint. A multiplier factor greater than 1 A positive parameter less than 1; The degree of unreliability is used to determine whether the current solution satisfies the capacity constraint.

[0166] Based on this, the generalized cost of an arc is further defined as follows:

[0167]

[0168] The degree of convergence of individual OD pairs and the convergence status of the overall problem:

[0169]

[0170]

[0171] like Figure 2 As shown, the solution algorithm combining hyperpath and augmented Lagrange method (ALM) specifically includes:

[0172] S31B, Input public transport network G, traffic demand .

[0173] S32B, Initialization:

[0174] Establish a walking arc for each OD pair to satisfy capacity feasibility;

[0175] Initialize the OD (Operational Destination) route selection set;

[0176] Initialize traffic loading;

[0177] Initialize penalty parameters , ride ;

[0178] Update arc flow, generalized cost, and effective departure frequency;

[0179] Set the number of iterations n=1, and the target convergence accuracy. ;

[0180] Secondly, regarding the setting of parameter m=1 to control the number of self-loop iterations, the convergence accuracy of the subproblem objective is... .

[0181] S33B performs a loop to adjust network traffic to achieve traffic balance, including:

[0182] S331B, Column generation, for each OD pair Search for the minimum generalized cost to reflect the direct impact of capacity constraints on path selection behavior (refer to the aforementioned generalized superpath cost). And the superpath k); if Add it to .

[0183] S332B: Construct subproblems for each OD pair using the Gauss-Seidel decomposition strategy (without capacity constraints).

[0184] S34B: Perform an adaptive inner loop to solve the OD pair problem with capacity constraints, including:

[0185] S341B: Adaptively solves the unconstrained OD pair problem with low convergence accuracy based on the GP algorithm, updating the hyperpath flow and generalized hyperpath cost (refer to the aforementioned generalized hyperpath cost). And the superpath k).

[0186] S342 B. Convergence of the solved individual OD pair subproblems Make a judgment based on the aforementioned formula, if it satisfies If the OD pair converges, update the corresponding arc flow, generalized cost, and effective departure frequency, and delete unused hyperpaths.

[0187] S35 B, Subproblem convergence judgment: Perform a convergence judgment on the subproblems as a whole. If This indicates that the subproblem has converged globally, and parameter updates can be performed: update the penalty parameter. With Lagrange multipliers This leads to a new capacity constraint problem.

[0188] S36B, Capacity constraint test to determine infeasibility. and (Given the relationship of boundary values), if Then the superpath traffic solution If the arc flow rate v is not feasible, update the parameter n=n+1, return to the steps of the above loop (the loop after initialization), and continue the above operation.

[0189] S36B, Output Balanced State, Superpath Flow Arc flow .

[0190] Therefore, the process of solving capacity-constrained problems described above involves transforming the original problem with capacity constraints into an approximate subproblem without capacity constraints. By continuously solving these approximate subproblems, a solution that ultimately satisfies the capacity constraints is found. In other words, the implementation steps of "column generation" and "solving OD subproblems without capacity constraints" within the ALM framework are essentially consistent with the framework for algorithms considering no capacity constraints. The main difference lies in the fact that the hyperpath flow and cost updates in the ALM framework are based on generalized path costs to reflect the impact of capacity constraints on path selection behavior.

[0191] Furthermore, when the original problem with capacity constraints is transformed into an approximate subproblem without capacity constraints, the third algorithm updates the hyperpath k and ... during initialization. Calculate the cost and effective departure frequency of all arcs, and calculate the superpath cost. and The cost of the arc in the main loop and the corresponding parameters updated in the main loop, as well as the arc cost in the corresponding content of the fourth algorithm (i.e., the main loop and initialization update of the above parameters), should all be calculated according to the formula:

[0192]

[0193] Update to ensure that the superpath cost (corresponding to the generalized superpath cost) is updated in each iteration. It can accurately reflect the impact of capacity constraints on hyperpath selection.

[0194] The above description is merely a preferred embodiment of the present invention. It should be understood that the present invention is not limited to the forms disclosed herein and should not be construed as excluding other embodiments. It can be used in various other combinations, modifications, and environments, and can be altered within the scope of the concept described herein through the above teachings or related technologies or knowledge. Modifications and variations made by those skilled in the art that do not depart from the spirit and scope of the present invention should be within the protection scope of the appended claims.

Claims

1. A public transport allocation method under conditions of complete congestion, characterized in that, include: In a public transport network, a hyperpath structure is introduced to express the travel route selection strategy of passengers. The hyperpath is a directed acyclic subgraph, which includes a subset of nodes, a subset of arcs, and an arc usage probability vector. The hyperpath traffic loading problem is reconstructed as a monotonically varying sequence of inequalities along the order of nodes in the hyperpath topology, resulting in hyperpath traffic loading schemes, including: The passenger flow distribution at transfer node i on superpath k satisfies a fixed-point problem as shown in the following formula: in, To enter the transfer node on superpath k flow, For the traffic of superpath k, Let be the probability that hyperpath k passes through transfer node i. This refers to the strategy used by hyperpath k at transfer node i. For the reason Assigned to boarding arc The flow rate satisfies the constraints of flow conservation and positive flow. The effective departure frequency is the frequency at which the arc flow rate equals v under the effective departure frequency function. when When the solution is the passenger flow distribution formula, for all transfer points have: in, Assigning the superpath k to the boarding arc The traffic on the internet, corresponding to For boarding arc Effective departure frequency; for each node on superpath k strategy set Flow on all arcs and effective departure frequency The ratio is a constant. , is a subset of the bus network node N; Based on this, the above equation can be equivalently transformed into the following variational inequality problem: Define feasible region ; If and only if It is the solution to the following variational inequality problem; turn up , so that: in, , It is the flow allocated to the boarding arc (i, j) by the superpath k*. It is the ratio of the flow of the superpath k* through the boarding arc (i, j) to its effective departure frequency, which can be regarded as the waiting cost. ； A hyperpath traffic loading algorithm is constructed based on gradient projection. The algorithm loads hyperpath traffic to obtain the arc traffic distribution, arc usage probability, and node usage probability of each node in each hyperpath. Then, the hyperpath traffic and arc traffic under the network equilibrium state are solved to obtain the complete traffic distribution under each hyperpath in the public transport network. This is used to update the hyperpath cost and determine whether the current hyperpath traffic solution is in an equilibrium state, thus obtaining the hyperpath traffic and arc traffic under the network equilibrium state.

2. The bus allocation method under complete congestion as described in claim 1, characterized in that, Without considering capacity constraints, the hyperpath network equilibrium condition is: in, The minimum superpath cost to reach destination d from origin o. The set of hyperpaths from the starting point o to the ending point d. For out-of-path costs, Let W be the flow on hyperpath k, and let W be the set of all origin-destination pairs od in the public transport network.

3. The bus allocation method under complete congestion as described in claim 2, characterized in that, The superpath flow must satisfy the following flow conservation and non-negativity constraints: in, Let O be the traffic demand originating from point O and ending at point D; the relationship between arc flow and superpath flow is: in, Let A be the probability that the superpath k passes through the boarding arc (i, j), and let A be the set of arcs. set up This is the hyperpath traffic vector. This is the correlation matrix between hyperpaths and OD pairs. This represents the OD demand vector. definition And define the set of feasible hyperpath traffic vectors h as It satisfies: The transformed variational inequality problem is: superpath flow in equilibrium state. To find the hyperpath network equilibrium condition that satisfies the aforementioned conditions. , so that: in, Indicates a given The hyperpath cost vector below.

4. The bus allocation method under complete congestion as described in claim 3, characterized in that, When considering capacity constraints, the hyperpath network equilibrium condition is: in, The minimum superpath cost to reach destination d from origin o. The set of hyperpaths from the starting point o to the ending point d. For out-of-path costs, Let W be the flow on hyperpath k, and let W be the set of all Od pairs in the public transport network.

5. The bus allocation method under complete congestion as described in claim 4, characterized in that, definition in Given the saturation capacity of the onboard arc (i, j), and the feasible superpath flow that satisfies the capacity constraint. h Definition of the set: The equilibrium conditions described above are equivalently transformed into the following variational inequality problem: Superpath traffic The equilibrium condition is met: turn up , so that: .

6. The bus allocation method under complete congestion as described in claim 5, characterized in that, The gradient projection-based hyperpath traffic loading algorithm, which performs hyperpath traffic loading, includes: Select the boarding arc with the minimum path cost at the transfer node. As a reference path, the amount of traffic adjustment from a non-reference path to the reference path is expressed as: Then calculate the waiting cost of the boarding arc (i,j) and compare it with the reference boarding arc. The difference in waiting costs with respect to the first derivative of the boarding arc (i, j) flow rate It can be viewed as the descent step size: in , They are respectively and about and The derivative of Assigning the superpath k to the boarding arc Traffic, Current arc cost, The minimum path cost from node i to node j; At the current arc flow The gradient projection algorithm updates the non-reference path according to the following formula. Traffic: 。 7. The bus allocation method under complete congestion as described in claim 1, characterized in that, The process involves obtaining the arc flow distribution, arc usage probability, and node usage probability for each node along each hyperpath, then solving for the hyperpath flow and arc flow under the network-wide equilibrium state. This yields the complete flow distribution for each hyperpath in the public transport network, which is used to update the hyperpath cost, determine whether the current hyperpath flow solution is in an equilibrium state, and obtain the hyperpath flow and arc flow under the network-wide equilibrium state, including: Based on the topological order of the hyperpath, starting from the hyperpath origin, calculate the usage probability of each transfer node in sequence; For transfer nodes, the usage probability of each arc leaving the node is calculated based on the arc flow and effective departure frequency output by the gradient projection algorithm. Based on the usage probability of the node and the usage probability of the arc, the flow distribution of each arc is calculated.

8. The bus allocation method under complete congestion as described in claim 7, characterized in that, The step of calculating the usage probability of each node sequentially, starting from the origin of the hyperpath, according to the topological order of the hyperpath includes: A vector of probabilities used for the arc. Are passengers at the node i Select arc The probability of; set up , Superpath k Use nodes i Sum of arcs The probability of is then: Where o and d are the start and end transfer nodes of hyperpath k, respectively. Let k be the sum of the arc probabilities connected to the previous node i and its j along the superpath k. Let be the probability that the superpath k passes through the starting point o. The probability of using the endpoint d for the superpath k. Let be the probability that hyperpath k uses arc (i,j) at node i. , is a subset of arc set A.

9. The bus allocation method under complete congestion as described in claim 1, characterized in that, For transfer nodes, the calculation of the usage probability of each arc leaving the node based on the arc flow and effective departure frequency output by the gradient projection algorithm includes: Based on the onboard arc flow at this node and their corresponding effective departure frequencies Calculate the probability of using all arcs: ； in Let k be the set of strategies at node i for the hyperpath k. The calculation of the flow distribution of each arc based on the usage probability of the node and the usage probability of the arc includes: Update the arc usage probability and the next node usage probability: like i When it is not a transfer node: set up Then update the arc usage probability and the next node usage probability: Finally, output the complete arc flow distribution of the hyperpath. Arc usage probability vector The probability vector of node usage Among them, hyperpath k for It is a directed graph of a public transport network. Acyclic subgraphs.

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