Train operation scheme compiling method and device based on distributed robust chance constraint model

Abstract

The application provides a train operation scheme compiling method and device based on a distributed robust chance constraint model, and the method comprises the following steps: taking the number of trains in a direct process as a decision variable, and constructing a distributed robust chance constraint model for train operation scheme compilation with the target of minimizing operation cost; the distributed robust chance constraint model is converted into a mixed integer linear programming model through linearization; the optimal value under the planning of a deterministic problem is calculated according to the passenger demand sample under the worst distribution, and an initial feasible solution is obtained according to the optimal value, wherein the initial feasible solution is used for representing the upper bound of the feasible region of the mixed integer linear programming model; the mixed integer linear programming model is solved according to the initial feasible solution, and a train operation scheme is obtained. In the case that the passenger demand cannot be accurately estimated, the method can still meet all passenger demands with a given probability, and realizes the fast and accurate compilation of a train operation scheme meeting the passenger demand.

Description

Technical Field

[0001] This application relates to the field of railway scheduling and resource optimization technology, and in particular to a method and apparatus for compiling train operation plans based on a distributed robust chance constraint model. Background Technology

[0002] Railway systems play a vital role in modern transportation, and effective railway line planning is particularly crucial in addressing demand uncertainty. Railway planning typically involves several key stages, including travel demand estimation, operation plan development, timetable creation, platform allocation, and passenger scheduling. Among these stages, operation plan development is a critical issue, requiring the matching of estimated travel demand with the determination of each line / train's origin, destination, stopping patterns, frequency, and capacity. Because operation plan development is usually conducted weeks or months in advance of operation, and passenger demand is subject to significant uncertainty, accurate forecasting becomes difficult.

[0003] Currently, robust optimization and lightly robust methods have been proposed to address the problem of uncertain travel demand. Robust optimization methods assume that uncertain demand belongs to a certain uncertainty set and seek the optimal solution under the worst-case scenario within this set. Lightly robust methods achieve robustness by maximizing demand satisfaction while relaxing the objective value to a certain extent. Although these methods have advantages in computational efficiency, they may lead to overly conservative train operation plans due to a lack of full utilization of the probability distribution information of passenger demand, resulting in a waste of transport capacity resources.

[0004] With the widespread availability of data, various techniques are used to estimate the probability distribution of demand, such as empirical distributions based on historical data or parameter learning of probability distributions. In this regard, stochastic programming methods have been applied to the problem of developing operational plans, optimizing the objective either by minimizing the expected cost or by minimizing the cost of satisfying a specific demand quantile. Considering the error between the estimated probability distribution function and the true probability distribution function, the Partial Brussels Bar (DR) optimization method has been proposed to further enhance the robustness of the model. The DR method aims to find the worst distribution in a set containing all possible true distributions and minimize the loss under this worst distribution. However, existing probability distribution-based optimization methods suffer from computational efficiency issues.

[0005] Therefore, how to quickly and accurately formulate train operation plans that meet the travel needs of passengers is an urgent technical problem to be solved. Summary of the Invention

[0006] In view of the above problems, this application provides a method and apparatus for train operation scheme preparation based on a distributed robust chance constraint model, so as to overcome the above problems or at least partially solve the above problems.

[0007] A first aspect of this application discloses a method for developing train operation plans based on a distributed robust chance constraint model, the method comprising:

[0008] Using the number of trains in the direct route as the decision variable and minimizing operating costs as the objective, a multi-bar chance constraint model for train operation planning is constructed.

[0009] The sub-Bruker chance constraint model is converted into a mixed integer linear programming model by linearization.

[0010] Based on the passenger demand sample under the worst-case distribution, the optimal value under deterministic problem programming is calculated, and an initial feasible solution is obtained based on the optimal value. The initial feasible solution is used to characterize the upper bound of the feasible region of the mixed integer linear programming model.

[0011] Based on the initial feasible solution, the mixed-integer linear programming model is solved to obtain the train operation scheme.

[0012] Optionally, the method further includes:

[0013] Construct an effective inequality, which is used to narrow the range of the feasible region of the mixed-integer linear programming model;

[0014] Based on the initial feasible solution, the mixed-integer linear programming model is solved to obtain the train operation scheme, including:

[0015] Using the effective inequalities as constraints, and based on the initial feasible solution, the mixed-integer linear programming model is solved to obtain the train operation scheme.

[0016] Optionally, the valid inequality includes at least one of the following:

[0017] The first effective inequality is used to narrow the feasible region of train operation schemes in the mixed integer linear programming model.

[0018] The second effective inequality is used to narrow the feasible domain of the passenger demand satisfaction decision variable in the mixed integer linear programming model. The passenger demand satisfaction decision variable is used to characterize whether the passenger demand under the passenger demand sample can be satisfied by the train operation plan.

[0019] Optionally, using the number of trains in the direct route as the decision variable and minimizing operating costs as the objective, a multi-bar chance constraint model for train operation planning is constructed, including:

[0020] Construct an objective function that aims to minimize the operating cost;

[0021] Construct a sub-Bruker chance constraint condition, which is used to constrain the train operation scheme to meet the passenger demand under the worst distribution with a given probability.

[0022] Optionally, the objective function is constructed, including:

[0023] Based on the number of trains in the direct route and the cost of trains traveling through the direct route, a first cost is constructed;

[0024] A second cost is constructed based on the cost of originating at a station and the number of trains originating at the station, wherein the number of trains originating at a station is determined by the non-negative portion of the difference between the number of trains departing from the station and the number of trains arriving at the station.

[0025] A third cost is constructed based on the cost of terminating at a station and the number of trains terminating at the station, wherein the number of trains terminating at the station is determined by the non-negative portion of the difference between the number of trains arriving at the station and the number of trains departing from the station.

[0026] The objective function is obtained based on the first cost, the second cost, and the third cost.

[0027] Optionally, construct the partial bar chance constraints, including:

[0028] Construct constraints for the deterministic problem, and construct the probability distribution of passenger demand.

[0029] Based on the constraints of the deterministic problem, determine the set of train operation schemes that can satisfy passenger flow.

[0030] Based on the set of passenger flows, construct opportunity constraints that satisfy the probability distribution of demand;

[0031] The worst-case probability distribution under the aforementioned demand probability distribution is used as the split-bar chance constraint.

[0032] Optionally, deterministic problem constraints are constructed, including:

[0033] A first constraint is established, which is used to constrain the number of passengers in the direct process to not exceed the train capacity in the direct process, wherein the train capacity in the direct process is determined based on the number of trains passing through the direct process and the capacity of each train.

[0034] A second constraint is constructed to ensure that the number of inbound passengers at intermediate stations of passenger travel demand equals the number of outbound passengers, where passenger travel demand includes origin stations and destination stations.

[0035] Optionally, based on the passenger demand sample under the worst-case distribution, the optimal value under deterministic problem programming is calculated, and an initial feasible solution is obtained based on the optimal value, including:

[0036] Construct a deterministic model corresponding to the deterministic problem planning, wherein the deterministic model is used to characterize minimizing operating costs under deterministic passenger demand;

[0037] The deterministic model is solved based on multiple passenger demand samples under the worst-case distribution to obtain multiple optimal values.

[0038] Optionally, obtaining an initial feasible solution based on the optimal value includes:

[0039] The passenger demand sample under the worst distribution corresponding to the best value that is greater than the cost threshold among the multiple optimal values ​​is determined as the first passenger demand sample that cannot be satisfied by the train operation plan.

[0040] The passenger demand sample corresponding to the worst distribution of the optimal value among the multiple optimal values ​​that is not greater than the cost threshold is determined as the second passenger demand sample that can be satisfied by the train operation plan.

[0041] Set the value of the passenger demand satisfaction decision variable corresponding to the first passenger demand sample to not satisfy, and set the value of the passenger demand satisfaction decision variable corresponding to the second passenger demand sample to satisfy.

[0042] An initial feasible solution is obtained based on the passenger demand satisfaction decision variables corresponding to each of the first passenger demand samples and each of the second passenger demand samples.

[0043] A second aspect of this application discloses a train operation plan compilation device based on a distributed robust chance constraint model, the device comprising:

[0044] The first construction module uses the number of trains in the direct process as the decision variable and aims to minimize operating costs to construct a multi-bar chance constraint model for train operation scheme preparation.

[0045] The first transformation module is used to convert the split-bar chance constraint model into a mixed-integer linear programming model through linearization.

[0046] The first calculation module is used to calculate the optimal value under deterministic problem programming based on the passenger demand sample under the worst distribution, and to obtain an initial feasible solution based on the optimal value. The initial feasible solution is used to characterize the upper bound of the feasible region of the mixed integer linear programming model.

[0047] The first solution module is used to solve the mixed integer linear programming model based on the initial feasible solution to obtain the train operation scheme.

[0048] The embodiments of this application have the following advantages:

[0049] In this embodiment, the number of trains in the direct route is used as the decision variable, and the goal is to minimize operating costs. A partial Brussels chance constraint model for train operation planning is constructed. The partial Brussels chance constraint model is converted into a mixed-integer linear programming model through linearization. Based on the passenger demand sample under the worst-case distribution, the optimal value under deterministic problem programming is calculated, and an initial feasible solution is obtained based on the optimal value. The initial feasible solution is used to characterize the upper bound of the feasible region of the mixed-integer linear programming model. Based on the initial feasible solution, the mixed-integer linear programming model is solved to obtain the train operation plan.

[0050] This method addresses the problem of train operation planning under fluctuating passenger demand. It proposes a partially Bruker chance constraint model, which can still satisfy all passenger demands with a given probability even when passenger demand cannot be accurately estimated. To improve the solution, the partially Bruker chance constraint model is transformed into a mixed-integer linear programming model, allowing train operation plans to be obtained by solving this model. Furthermore, by utilizing passenger demand samples under the worst-case distribution, an initial feasible solution is introduced. When solving the mixed-integer linear programming model, the search can directly start from this initial feasible solution, reducing the search range of the feasible region and thus finding the optimal solution, i.e., the train operation plan, more quickly. In some embodiments, effective inequalities are added to further accelerate the solution of the mixed-integer linear programming model. This enables the rapid and accurate planning of train operation plans that meet passenger demands. Attached Figure Description

[0051] To more clearly illustrate the technical solutions of the embodiments of this application, the drawings used in the description of the embodiments of this application will be briefly introduced below. Obviously, the drawings described below are only some embodiments of this application. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0052] Figure 1 This is a flowchart illustrating the steps of a train operation scheme development method based on a distributed robust chance constraint model, as provided in an embodiment of this application.

[0053] Figure 2 This is a graphical and network representation of a train operation scheme development problem provided in an embodiment of this application;

[0054] Figure 3 This is a performance comparison chart of different train operation plan compilation methods provided in the embodiments of this application;

[0055] Figure 4 This is a schematic diagram of a train operation scheme compilation device based on a distributed robust chance constraint model provided in an embodiment of this application. Detailed Implementation

[0056] To make the above-mentioned objectives, features, and advantages of this application more apparent and understandable, the technical solutions in the embodiments of this application will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of this application, not all embodiments. Based on the embodiments in this application, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of this application.

[0057] To achieve rapid and accurate train operation planning that meets passenger travel needs, this application provides a train operation planning method based on a distributed robust chance constraint model. This method considers the robust train operation planning problem on a single passenger line, assuming all trains have the same passenger capacity. Passenger travel demand is treated as a random vector, determining the origin, destination, stops, and number of departures for each train, ensuring the planned operation can satisfy the travel needs of all passengers on the line with a given probability. This method obtains train operation plans based on a distributed robust chance constraint model. To improve the solution, the distributed robust chance constraint model is converted into a mixed-integer linear programming model, allowing the train operation plan to be obtained by solving the mixed-integer linear programming model. Furthermore, by utilizing passenger demand samples under the worst-case distribution, an initial feasible solution is introduced. When solving the mixed-integer linear programming model, the search can be performed directly based on the initial feasible solution, reducing the search range of the feasible region and thus finding the optimal solution more quickly, i.e., obtaining the train operation plan. This achieves rapid and accurate planning of train operation that meets passenger needs.

[0058] The following description, in conjunction with the accompanying drawings, details the train operation scheme compilation method based on a distributed robust chance constraint model provided in the embodiments of this application.

[0059] Reference Figure 1 As shown, Figure 1 This is a flowchart illustrating the steps of a train operation plan development method based on a distributed robust chance constraint model, as provided in an embodiment of this application. Figure 1 As shown, the train operation plan formulation method based on the distributed robust chance constraint model may include steps S110 to S140:

[0060] Step S110: Using the number of trains in the direct route as the decision variable and minimizing operating costs as the objective, construct a multi-bar chance constraint model for train operation planning.

[0061] This application addresses the problem of developing a daily train operation plan for a single direction (up or down) on a passenger transport route. The decision variables for this problem are the route and frequency. The route decision includes selecting the train's starting station, stops, and destination, while the frequency decision determines the number of trains operating on each route. This application decomposes a route into multiple direct routes. For example, if a route's stop strategy is station 2-station 5-station 6, it can be decomposed into a direct route from station 2 to station 5, and a combination of stations 5 and 6. By decomposing all routes, the original route and frequency decision variables can be changed to the number of trains in each direct route, i.e., using the number of trains in the direct route as the decision variable. After obtaining the appropriate number of trains for each direct route, the direct routes are then combined to obtain the original route and frequency decisions.

[0062] Using the number of trains in the direct route as the decision variable, the train operation planning problem can be viewed as a directed graph network. The edge capacity design problem, such as Figure 2 As shown, V represents the set of nodes, and ε represents the set of directed edges. Each node (e.g., station 0) v∈v in the graph represents a station on a passenger route. A directed edge from A to B represents a direct segment from station A to B without stopping at any other station. If there is no edge connecting two stations, it means that a direct route between these two stations is not allowed (it must stop at other stations). Any directed path in the graph (e.g., station 1-station 3-station 4) can be used to represent a route (i.e., starting from station 1, stopping at station 3, and ending at station 4). The decision variable for the feasible solution problem is the capacity of each edge in the directed graph network. For example, if the capacity of an edge is 3, it means that there are 3 trains on the route with this direct route. Therefore, the decision variable for the distributed chance constraint model is the number of trains in the direct route, i.e., the decision variable is the capacity x of each edge e. e , e∈ε, and require x e It is a non-negative integer.

[0063] In this embodiment of the application, a partial Brussels bar chance constraint model is proposed to address the problem of train operation plan formulation under fluctuating passenger demand. This partial Brussels bar chance constraint model aims to minimize operating costs. Based on this partial Brussels bar chance constraint model, it is possible to meet the needs of all passengers with a given probability even when passenger demand cannot be accurately estimated, while minimizing train operating costs.

[0064] In an optional embodiment, using the number of trains in the direct route as the decision variable and minimizing operating costs as the objective, a multi-bar chance constraint model for train operation planning is constructed, including steps A1 and A2:

[0065] Step A1: Construct an objective function that aims to minimize the operating cost.

[0066] In this embodiment, it is assumed that all trains have identical physical attributes, and their operating costs are only related to the line. The cost of operating a certain line l can be decomposed into fixed costs and variable costs. The fixed costs are related to the originating station s and the destination station t of the line selection, while the variable costs depend on which direct segments the line consists of, i.e., the operating cost cost of a line l. l It can be represented as:

[0067]

[0068] in, This represents the cost of originating from station s on line l. Cost represents the cost of terminating at station t on line l. e This represents the cost of the path passing through edge e.

[0069] With the goal of minimizing cost, if the capacity x of each edge (i.e., each direct process) e Given that, when concatenating the edges into a path (concatenating direct segments into a line), the fixed cost should be minimized as much as possible, that is, the number of lines operated should be minimized, so as to obtain the objective function based on the concatenated path.

[0070] Specifically, constructing the objective function includes: constructing a first cost based on the number of trains in the direct route and the cost of a train traveling through the direct route; constructing a second cost based on the cost of departing from a station and the number of trains departing from that station, wherein the number of trains departing from a station is determined by the non-negative portion of the difference between the number of trains departing from the station and the number of trains arriving at the station; constructing a third cost based on the cost of terminating at a station and the number of trains terminating at that station, wherein the number of trains terminating at the station is determined by the non-negative portion of the difference between the number of trains arriving at the station and the number of trains departing from the station; and obtaining the objective function based on the first cost, the second cost, and the third cost.

[0071] For example, for a station s, the number of trains originating from that station should be equal to the number of trains departing from that station (the capacity of the edge originating from s) minus the number of trains arriving at that station (the capacity of the edge ending at s). Similarly, the number of trains terminating at a station t should be equal to the number of trains arriving at that station minus the number of trains triggered by that station.

[0072] For example, the objective function f(x) is expressed as:

[0073]

[0074] Where ε represents the set of all direct routes; V represents the set of stations on the route; ε + (v) represents the set of all edges leaving node v; ε - (v) represents the set of all edges pointing to v; the function (a) + This means taking the non-negative part of a, i.e., max(a,0); This indicates the cost originating from site v; Cost represents the cost of arriving at station v. e Let f(x) represent the cost of the path passing through edge e; the first term in the objective function f(x) is the first cost, which is the variable cost; the second term in the objective function f(x) is the second cost, which is the cost of starting at each station; the third term in the objective function f(x) is the third cost, which is the cost of ending at each station.

[0075] Step A2: Construct the sub-Bruker chance constraint, which is used to constrain the train operation scheme to meet the passenger demand under the worst distribution with a given probability.

[0076] In this embodiment, the constraint of the Bruker chance constraint model mainly requires that the train operation plan can satisfy the travel needs of all passengers. When solving the operation plan formulation problem, this embodiment does not consider passenger transfer problems (transfer costs), because transfers are time-related, and therefore transfer problems are more suitable to be considered in the subsequent timetable decision-making problem.

[0077] Passenger travel demand is characterized by their origin and destination (hereinafter referred to as OD pairs). Different origins or destinations correspond to different passenger demands, therefore, a triple (s... k ,t k ,d k ) represents the origin / origin station of the OD pair with index k. k Destination / Key Station (t) k ), and passenger demand / number of passengers (d k ); where k belongs to set (This contains the set of all OD pairs). If we consider OD pair k as item k, we can model it as a classic graph and network problem, namely the multi-item flow problem. In directed graph networks... The above can be seen as commodity k with d k The flow of s kThe product flows through the selected path in the diagram to t. k , and then from t k Outflow d k .

[0078] Since the probability distribution of passenger demand cannot be accurately estimated, in order to better meet passenger needs, this application embodiment uses the ability of the train operation plan to meet passenger demand under the worst-case distribution as a Bruker chance constraint. Thus, the Bruker chance constraint model based on this Bruker chance constraint can still meet the needs of all passengers with a given probability, even when passenger demand cannot be accurately estimated.

[0079] In an optional embodiment, the split-bar chance constraint is constructed, including steps A2-1 to A2-4:

[0080] Step A2-1: Construct the deterministic problem constraints, and construct the demand probability distribution that passenger demand follows.

[0081] In this embodiment of the application, when passenger demand d k When the situation is certain (i.e., assuming the train schedule planners can fully predict passenger demand on the day of train departure), construct constraints for the deterministic problem.

[0082] Specifically, constraints for deterministic problems are constructed, including:

[0083] A first constraint is established, which is used to constrain the number of passengers in the direct process to not exceed the train capacity in the direct process, wherein the train capacity in the direct process is determined based on the number of trains passing through the direct process and the capacity of each train.

[0084] A second constraint is constructed to ensure that the number of inbound passengers at intermediate stations of passenger travel demand equals the number of outbound passengers, where passenger travel demand includes origin stations and destination stations.

[0085] For example, the first constraint (1-1) and the second constraint (1-2) are expressed as follows:

[0086]

[0087] in, It is the introduced decision variable, namely OD with respect to k. A certain number of passengers are assigned to trains that include the direct journey e, that is to say, Let represent the number of passengers traveling on the train with the direct journey e from OD to k; Cap represents the maximum passenger capacity of each train; y represents all decision variables. The set of ; x represents the total number of trains on each direct route, i.e., the train operation plan.

[0088] The first constraint (1-1) states that the number of passengers in the direct process is no greater than the train capacity of the direct process. In other words, the flow through the direct process e cannot exceed the capacity limit of the direct process e. This means the number of passengers allocated to trains containing the direct process e cannot exceed the number of trains containing the direct process e multiplied by the capacity of a single train. The second constraint (1-2) is a flow balance constraint, meaning that the number of inbound passengers at intermediate stations in the passenger travel demand is equal to the number of outbound passengers. This means that the inflow of good k at intermediate nodes is equal to the outflow at departure node s. k The net inflow is d k Upon reaching node t k The net outflow is d k .

[0089] In practical applications, due to passenger demand d k Since passenger demand cannot be accurately predicted, a demand probability distribution is constructed to follow. Specifically, consider demand d. k It follows a probability distribution F, which belongs to a set of distributions D(θ), where D(θ) represents an estimated distribution. The set of all probability distributions with a distance not exceeding θ, the demand probability distribution that passenger demand follows is expressed as:

[0090]

[0091] in, It is supported at N points. The discrete distribution of d, where the probability of d taking each point is {p1,…,p} N}, Each point represents a sample of passenger demand.

[0092] Probability distribution F and estimated distribution The "distance" between them The ∞-Wasserstein distance is used, which is defined as:

[0093]

[0094] Here, Π-ess.sup‖ξ1-ξ2‖1 represents the supremum of ‖ξ1-ξ2‖1.

[0095] Step A2-2: Based on the constraints of the deterministic problem, determine the set of train operation schemes that can satisfy passenger flow.

[0096] In this embodiment of the application, for a given train operation plan x, S(x) represents the set of passenger flows that the train operation plan can satisfy, that is:

[0097]

[0098] In other words, the set of train operation schemes x that can satisfy passenger flow d is determined under the condition that the first constraint (1-1) and the second constraint (1-2) mentioned above are satisfied.

[0099] Step A2-3: Based on the set of passenger flow, construct opportunity constraints that satisfy the probability distribution of demand.

[0100] In this embodiment of the application, the chance constraint condition of the probability distribution requires that the train operation plan can still meet passenger travel demand with a given probability 1-∈ under fluctuating passenger flow, that is:

[0101]

[0102] in, Let ∈ represent the probability function of the distribution, where ∈ represents a number between 0 and 1.

[0103] Step A2-4: Take the worst-case opportunity constraint under the demand probability distribution as the split bar opportunity constraint.

[0104] In this embodiment of the application, since the probability distribution of passenger demand cannot be accurately estimated, a split-bar chance constraint model is adopted, requiring that the chance constraint conditions of the demand probability distribution can still be satisfied even when the passenger demand follows the worst distribution in the demand probability distribution D(θ), that is:

[0105]

[0106] in, This represents the worst-case distribution function under the demand probability distribution D(θ).

[0107] Finally, the multi-bar chance constraint model for train operation planning can be expressed as:

[0108]

[0109] Where (1-4) represents the objective function, and (1-5) and (1-6) represent the random chance constraints.

[0110] Step S120: Convert the split-bar chance constraint model into a mixed-integer linear programming model by linearization.

[0111] In this embodiment of the application, in order to enable the proposed sub-Bruker chance constraint model (1-4)-(1-6) to be solved by a general solver, the sub-Bruker chance constraint model is transformed into an equivalent mixed integer linear programming model (MIP) by linearization.

[0112] For example, the mixed-integer linear programming model is represented as:

[0113]

[0114]

[0115] Among them, u s and u t The auxiliary decision variable z is introduced to linearize the objective function f(x). n (i.e., the passenger demand satisfaction decision variable) is used to represent whether the passenger demand sample n can be satisfied by the train operation plan x, z n =1 indicates that the condition cannot be satisfied, z n =0 indicates that the condition is met; The worst distribution in the data is actually N points. by The probability distribution of z is such that z is a probability distribution of z. n =1 means It cannot be satisfied by x.

[0116] Step S130: Calculate the optimal value under deterministic problem programming based on the passenger demand sample under the worst distribution, and obtain an initial feasible solution based on the optimal value. The initial feasible solution is used to characterize the upper bound of the feasible region of the mixed integer linear programming model.

[0117] In this embodiment, the solver typically employs a branch and bound method to solve the mixed-integer linear programming model. To achieve fast solving, the upper bound of the feasible region of the mixed-integer linear programming model is obtained based on the passenger demand samples under the worst-case distribution. This allows the solver to prune the search tree based on the upper bound of the feasible region, reducing the search range and thus accelerating the solution.

[0118] Specifically, based on multiple passenger demand samples under the worst-case distribution, the optimal value under deterministic problem-solving is calculated for each sample. This is equivalent to transforming the uncertain programming problem into multiple deterministic programming problems for solution. Therefore, solving multiple deterministic programming problems allows for a rapid acquisition of a set of optimal values. Based on these optimal values, an approximately optimal initial feasible solution is determined. Here, the initial feasible solution refers to the initial value of the decision variable z that satisfies the passenger demand. Thus, when subsequently solving the mixed-integer linear programming model, calculations can begin based on the upper bound of the feasible region of the mixed-integer linear programming model given by the initial feasible solution, thereby accelerating the solution process.

[0119] In an optional embodiment, based on the passenger demand sample under the worst-case distribution, the optimal value under deterministic problem programming is calculated, and an initial feasible solution is obtained based on the optimal value, including steps B1 to B2:

[0120] Step B1: Construct a deterministic model corresponding to the deterministic problem planning, wherein the deterministic model is used to characterize minimizing operating costs under deterministic passenger demand.

[0121] Step B2: Solve the deterministic model based on multiple passenger demand samples under the worst-case distribution to obtain multiple optimal values.

[0122] Specifically, in deterministic problem planning, passenger demand is deterministic. The deterministic problem constraints are used as the constraints of the deterministic model, that is, the first constraint (1-1) and the second constraint (1-2) mentioned above are used as the constraints of the deterministic model. The optimization objective of the deterministic model is still to minimize the operating cost. Therefore, the objective function f(x) mentioned above is used as the objective function of the deterministic model.

[0123] For example, the deterministic model is represented as:

[0124]

[0125] st(1-1),(1-2)

[0126]

[0127] For each worst-case distribution of passenger demand sample d k ,make Solving the aforementioned deterministic models separately yields the optimal value OPT. n The optimal value represents the operating cost of the train operation plan under the worst-case distribution of passenger demand sample.

[0128] Further, an initial feasible solution is obtained based on the optimal value, including steps C1 to C4:

[0129] Step C1: The passenger demand sample under the worst distribution corresponding to the optimal value that is greater than the cost threshold among the multiple optimal values ​​is determined as the first passenger demand sample that cannot be satisfied by the train operation plan.

[0130] Step C2: Determine the passenger demand sample under the worst distribution corresponding to the best value among the multiple optimal values ​​that is not greater than the cost threshold as the second passenger demand sample that can be satisfied by the train operation plan.

[0131] Step C3: Set the value of the passenger demand satisfaction decision variable corresponding to the first passenger demand sample to not satisfied, and set the value of the passenger demand satisfaction decision variable corresponding to the second passenger demand sample to satisfied.

[0132] Step C4: Based on the passenger demand satisfaction decision variables corresponding to each of the first passenger demand samples and each of the second passenger demand samples, obtain the initial feasible solution.

[0133] In this embodiment, the optimal value represents the operating cost of the train operation scheme under the passenger demand sample in the worst distribution. In order to achieve the goal of minimizing the operating cost, the optimal value is compared with the cost threshold, and the train operation scheme with the lower operating cost is selected as the initial train operation scheme, that is, a train operation scheme that satisfies the chance constraint under the worst distribution is selected.

[0134] For multiple optimal values We can sort them in descending order, and let the index of the sorted result be denoted as . Right now:

[0135]

[0136] Therefore, based on the sorted optimal values, the optimal values ​​that are greater than the cost threshold and the optimal values ​​that are not greater than the cost threshold are selected, that is:

[0137] Take j max satisfy and

[0138] Let n≤j max The passenger demand sample under the worst distribution is identified as the first passenger demand sample that cannot be satisfied by the train operation plan, where n>j max The worst-case distribution of passenger demand samples is identified as the second passenger demand sample that can be satisfied by the train operation plan. The passenger demand satisfaction decision variable is a 0-1 decision variable; the value of the passenger demand satisfaction decision variable corresponding to the first passenger demand sample is set to "not satisfied," i.e., the passenger demand satisfaction decision variable... Set the value of the passenger demand satisfaction decision variable corresponding to the second passenger demand sample to "satisfied", that is, the passenger demand satisfaction decision variable.

[0139] Finally, the passenger demand satisfaction decision variables corresponding to each first passenger demand sample and each second passenger demand sample are used as the initial feasible solutions.

[0140] Step S140: Solve the mixed integer linear programming model based on the initial feasible solution to obtain the train operation scheme.

[0141] In this embodiment of the application, when solving the mixed-integer linear programming model, the initial feasible solution is used as the initial value of the passenger demand satisfaction decision variable of the mixed-integer linear programming model, so that the solver can prune the search tree according to the initial feasible solution, reduce the search range, and greatly improve the efficiency of solving.

[0142] Through the above implementation process, a partial Bruker chance constraint model is proposed to address the problem of train operation plan formulation under fluctuating passenger demand. This model can still satisfy the needs of all passengers with a given probability, even when passenger demand cannot be accurately estimated. To facilitate better solution, the partial Bruker chance constraint model is transformed into a mixed-integer linear programming model, allowing the train operation plan to be obtained by solving this model. Furthermore, by utilizing passenger demand samples under the worst-case distribution, an initial feasible solution is introduced. When solving the mixed-integer linear programming model, the search can directly start from this initial feasible solution, reducing the search range of the feasible region and thus finding the optimal solution more quickly, i.e., obtaining the train operation plan. This enables the rapid and accurate formulation of train operation plans that meet passenger demand.

[0143] In an optional embodiment, to further accelerate the solution of the mixed-integer linear programming model, the method further includes the following steps:

[0144] Step S150: Construct an effective inequality, which is used to narrow the range of the feasible region of the mixed integer linear programming model.

[0145] Step S160: Using the effective inequality as a constraint, solve the mixed integer linear programming model based on the initial feasible solution to obtain the train operation scheme.

[0146] In this embodiment, considering that in mixed-integer programming problems, effective inequalities can reduce the size of the feasible region after relaxation of integer variables, making the relaxed feasible region closer to the convex hull of the original feasible region, thereby reducing the solution time, this embodiment constructs effective inequalities for the transformed mixed-integer linear programming model to narrow the range of the feasible region of the mixed-integer linear programming model, thereby using effective inequalities as constraints to accelerate the solution of the mixed-integer linear programming model.

[0147] Specifically, for each edge (i.e., a direct path) e∈ε, let w e Let l be any non-negative integer assigned to the edge; k (w) indicates from site s k To station t k The shortest path length, where the length of each edge e is w. e Therefore, h is definedn (w) is:

[0148]

[0149] sorted in descending order Sort the data, and denote the sorted index as τ, i.e.:

[0150]

[0151] Use i max (w) indicates that the condition is satisfied. and Integers.

[0152] Therefore, based on the above positioning h n (w), construct an effective inequality.

[0153] Specifically, the effective inequalities include at least one of the following:

[0154] D-1: The first effective inequality, which is used to narrow the feasible region of train operation schemes in the mixed integer linear programming model;

[0155] D-2: The second effective inequality is used to narrow the feasible domain of the passenger demand satisfaction decision variable in the mixed integer linear programming model. The passenger demand satisfaction decision variable is used to characterize whether the passenger demand under the passenger demand sample can be satisfied by the train operation plan.

[0156] In this embodiment, the first effective inequality is the effective inequality of the transformed mixed-integer linear programming model, that is, it does not change the feasible region. It is added as a constraint to the mixed-integer linear programming model, thereby narrowing the feasible region of the train operation scheme in the mixed-integer linear programming model, without changing the optimal solution.

[0157] For example, the first valid inequality is expressed as:

[0158]

[0159] i = 1, ..., i max (w)

[0160] The first valid inequality allows us to approximate the constraint (2-2) above. Thus, x can be removed e After relaxation is made into a continuous variable, part The value of is adjusted to reduce the feasible region after relaxation, thereby making the feasible region of the relaxed train operation scheme close to the convex hull of the original integer programming problem, thus accelerating the solution.

[0161] The second effective inequality is used to narrow the feasible region for satisfying the decision variables in a mixed-integer linear programming model. For example, let... Satisfy h γ(i) (w)≥h γ(i+1) (w) for i = 1, ..., l; where let...

[0162] The second effective inequality can be expressed as:

[0163]

[0164] The second effective inequality is derived from the first effective inequality. It can further cut off the feasible region after relaxation of the integer variable z (the decision variable for satisfying passenger demand), thereby achieving an acceleration effect.

[0165] When solving a mixed-integer linear programming model, the first and second effective inequalities can be selectively added as constraints to the mixed-integer linear programming model to obtain a new optimization problem. The optimal solution of the new optimization problem is consistent with the optimal solution of the original mixed-integer linear programming model, but the solution speed is faster.

[0166] In this embodiment, the partial Bruker chance constraint model can still satisfy the needs of all passengers with a given probability even when passenger demand cannot be accurately estimated. To improve the solution, the partial Bruker chance constraint model is converted into a mixed-integer linear programming model, so the train operation plan can be obtained by solving the mixed-integer linear programming model. Furthermore, by utilizing passenger demand samples under the worst-case distribution, an initial feasible solution is introduced, and by adding effective inequalities, the search for the mixed-integer linear programming model can directly start from the initial feasible solution, reducing the search range of the feasible region and thus finding the optimal solution more quickly, i.e., obtaining the train operation plan. In this way, a train operation plan that meets passenger needs can be quickly and accurately formulated.

[0167] Furthermore, to illustrate the performance of the train operation plan compilation method based on the distributed robust chance constraint model provided in this application embodiment, the Wuhan-Guangzhou high-speed railway passenger line with 17 stations is used as an example to verify the robustness of the train operation plan generated based on the distributed robust chance constraint model under passenger demand fluctuations, as well as the effectiveness of the two accelerated solution methods proposed in this application embodiment.

[0168] For example, Figure 3 This is a performance comparison chart of different train operation plan compilation methods provided in the embodiments of this application. Figure 3The robustness of the Distributed Robust Chance Constraint (DRCC) model is illustrated under different probability distributions of passenger demand (i.e., empirical distribution, normal distribution, and offset normal distribution). The horizontal axis represents the operating cost of the train operation plan (related to the number of trains operated, total running time, and mileage), and the vertical axis represents the probability that the corresponding operation plan cannot meet the demand of all passengers.

[0169] Generally, as the number of trains increases, operating costs rise, capacity increases, and the probability of passenger demand not being met decreases. Therefore, there is a trade-off between operating costs and the probability of meeting demand. Figure 3 As can be seen, compared with the classic robust optimization method (RO), the train operation scheme generated by the distributed robust chance constraint model (DRCCθ>0) in this application embodiment can meet the travel needs of passengers at a lower cost. Figure 3 The results under the rightmost offset normal distribution show that, compared with the general chance constraint model (θ=0), the distributed robust chance constraint model is still robust when the test data distribution deviates from the training data distribution, that is, as the cost increases, the probability of unmet passenger demand steadily decreases.

[0170] Table 1 shows the solution time (in seconds) for solving the mixed integer linear programming (MIP) model problem (Origin) with data weights of empirical distribution and approximate normal distribution, using the initial feasible solution-based solution method (WS) in the embodiments of this application, and further using the first effective inequality (WS-VI) and the second effective inequality (WS-SVI), as well as the optimization gap (percentage) with the solution time limited to 1 hour.

[0171] The results show that, compared with directly solving the MIP problem, the hot-start method can find feasible solutions quickly, thus resulting in a smaller final optimization gap, especially when the problem size is large. Furthermore, by using effective inequalities, the optimal solution can be obtained within a few minutes for problems that could not be found within one hour, which greatly improves the solution efficiency.

[0172] Table 1 Comparison of solution time results

[0173]

[0174] This application also provides a train operation plan compilation device based on a distributed robust chance constraint model, referring to... Figure 4 As shown, Figure 4This is a schematic diagram of a train operation plan compilation device based on a distributed robust chance constraint model provided in an embodiment of this application. The device includes:

[0175] The first construction module 410 uses the number of trains in the direct process as a decision variable and aims to minimize operating costs to construct a multi-bar chance constraint model for train operation scheme preparation.

[0176] The first conversion module 420 is used to convert the sub-Bruker chance constraint model into a mixed integer linear programming model by linearization.

[0177] The first calculation module 430 is used to calculate the optimal value under deterministic problem programming based on the passenger demand sample under the worst distribution, and to obtain an initial feasible solution based on the optimal value. The initial feasible solution is used to characterize the upper bound of the feasible region of the mixed integer linear programming model.

[0178] The first solution module 440 is used to solve the mixed integer linear programming model based on the initial feasible solution to obtain the train operation scheme.

[0179] In an optional embodiment, the device further includes:

[0180] The second construction module is used to construct effective inequalities, which are used to narrow the range of the feasible region of the mixed integer linear programming model.

[0181] The second solution module is used to solve the mixed integer linear programming model based on the effective inequalities as constraints and the initial feasible solution to obtain the train operation scheme.

[0182] In one alternative embodiment, the valid inequality includes at least one of the following:

[0183] The first effective inequality is used to narrow the feasible region of train operation schemes in the mixed integer linear programming model.

[0184] The second effective inequality is used to narrow the feasible domain of the passenger demand satisfaction decision variable in the mixed integer linear programming model. The passenger demand satisfaction decision variable is used to characterize whether the passenger demand under the passenger demand sample can be satisfied by the train operation plan.

[0185] In one alternative embodiment, the first building module includes:

[0186] A function building module is used to construct an objective function that aims to minimize runtime cost.

[0187] The constraint construction module is used to construct the sub-Bruker chance constraint conditions, which are used to constrain the train operation scheme to meet the passenger demand under the worst distribution with a given probability.

[0188] In one alternative embodiment, the function building module includes:

[0189] The first construction submodule is used to construct a first cost based on the number of trains in the direct process and the cost of the trains traveling through the direct process;

[0190] The second construction submodule is used to construct a second cost based on the cost of originating at the station and the number of trains originating at the station, wherein the number of trains originating at the station is determined based on the non-negative part of the difference between the number of trains departing from the station and the number of trains arriving at the station.

[0191] The third construction submodule is used to construct a third cost based on the cost of terminating at the station and the number of trains terminating at the station, wherein the number of trains terminating at the station is determined based on the non-negative part of the difference between the number of trains arriving at the station and the number of trains departing from the station.

[0192] The objective function module is used to obtain the objective function based on the first cost, the second cost, and the third cost.

[0193] In one optional embodiment, the constraint construction module includes:

[0194] The fourth submodule is used to construct constraints for deterministic problems, and to construct the demand probability distribution that passenger demand follows;

[0195] The first determining module is used to determine the set of passenger flow that the train operation plan can satisfy based on the constraints of the deterministic problem;

[0196] The fifth construction submodule is used to construct opportunity constraints that satisfy the demand probability distribution based on the set of passenger flow.

[0197] The constraint module is used to take the worst-case opportunity constraint under the demand probability distribution as the split bar opportunity constraint.

[0198] In one alternative embodiment, the fourth construction submodule includes:

[0199] The first constraint construction submodule is used to construct a first constraint condition, which is used to constrain the number of passengers in the direct process to be no greater than the train capacity in the direct process. The train capacity in the direct process is determined based on the number of trains passing through the direct process and the capacity of each train.

[0200] The second constraint construction submodule is used to construct a second constraint condition, which is used to constrain the number of inbound passengers at intermediate stations of passenger travel demand to be equal to the number of outbound passengers. The passenger travel demand includes the origin station and the destination station.

[0201] In an optional embodiment, the first computing module includes:

[0202] The third construction module is used to construct a deterministic model corresponding to the deterministic problem planning. The deterministic model is used to characterize the minimization of operating costs under deterministic passenger demand.

[0203] The optimal value calculation module is used to solve the deterministic model based on multiple passenger demand samples under the worst-case distribution to obtain multiple optimal values.

[0204] In an optional embodiment, the third building module includes:

[0205] The second determining module is used to determine the passenger demand sample under the worst distribution corresponding to the optimal value that is greater than the cost threshold among the multiple optimal values ​​as the first passenger demand sample that cannot be satisfied by the train operation plan.

[0206] The third determining module is used to determine the passenger demand sample under the worst distribution corresponding to the optimal value that is not greater than the cost threshold among the multiple optimal values ​​as the second passenger demand sample that can be satisfied by the train operation plan.

[0207] The first setting module is used to set the value of the passenger demand satisfaction decision variable corresponding to the first passenger demand sample to not satisfy, and set the value of the passenger demand satisfaction decision variable corresponding to the second passenger demand sample to satisfy.

[0208] The initial feasible solution module is used to obtain an initial feasible solution based on the passenger demand satisfaction decision variables corresponding to each of the first passenger demand samples and the passenger demand satisfaction decision variables corresponding to each of the second passenger demand samples.

[0209] The various embodiments in this specification are described in a progressive manner, with each embodiment focusing on the differences from other embodiments. The same or similar parts between the various embodiments can be referred to each other.

[0210] This application describes embodiments of methods and apparatus according to flowchart illustrations and / or block diagrams. It should be understood that each block of the flowchart illustrations and / or block diagrams, and combinations of blocks in the flowchart illustrations and / or block diagrams, can be implemented by computer program instructions. These computer program instructions can be provided to a processor of a general-purpose computer, special-purpose computer, embedded processor, or other programmable data processing terminal device to produce a machine, such that the instructions, which execute via the processor of the computer or other programmable data processing terminal device, generate instructions for implementing the flowchart. Figure 1 One or more processes and / or boxes Figure 1 A device that provides the functions specified in one or more boxes.

[0211] These computer program instructions may also be stored in a computer-readable storage medium that can direct a computer or other programmable data processing terminal device to operate in a particular manner, such that the instructions stored in the computer-readable storage medium produce an article of manufacture including instruction means, which are implemented in a process Figure 1 One or more processes and / or boxes Figure 1 The function specified in one or more boxes.

[0212] These computer program instructions can also be loaded onto a computer or other programmable data processing terminal equipment, causing a series of operational steps to be performed on the computer or other programmable terminal equipment to produce a computer-implemented process, thereby providing instructions that execute on the computer or other programmable terminal equipment for implementing the process. Figure 1 One or more processes and / or boxes Figure 1 The steps of the function specified in one or more boxes.

[0213] Although preferred embodiments of the present application have been described, those skilled in the art, upon learning the basic inventive concept, can make other changes and modifications to these embodiments. Therefore, the appended claims are intended to be interpreted as including the preferred embodiments as well as all changes and modifications falling within the scope of the embodiments of the present application.

[0214] Finally, it should be noted that in this document, relational terms such as "first" and "second" are used only to distinguish one entity or operation from another, and do not necessarily require or imply any such actual relationship or order between these entities or operations. Furthermore, the terms "comprising," "including," or any other variations thereof are intended to cover non-exclusive inclusion, such that a process, method, article, or terminal device that comprises a list of elements includes not only those elements but also other elements not expressly listed, or elements inherent to such a process, method, article, or terminal device. Without further limitations, an element defined by the phrase "comprising one..." does not exclude the presence of other identical elements in the process, method, article, or terminal device that includes said element.

[0215] The above provides a detailed description of the train operation scheme compilation method and apparatus based on a distributed robust chance constraint model provided in this application. Specific examples have been used to illustrate the principles and implementation methods of this application. The descriptions of the above embodiments are only for the purpose of helping to understand the method and core ideas of this application. At the same time, for those skilled in the art, there will be changes in the specific implementation methods and application scope based on the ideas of this application. Therefore, the content of this specification should not be construed as a limitation of this application.

Claims

1. A method for developing train operation plans based on a distributed robust chance constraint model, characterized in that, The method includes: Using the number of trains in the direct route as the decision variable and minimizing operating costs as the objective, a partial Bruker chance constraint model for train operation planning is constructed, including: constructing an objective function, wherein the objective function aims to minimize operating costs; and constructing partial Bruker chance constraints, wherein the partial Bruker chance constraints are used to constrain the train operation plan to meet passenger demand under the worst-case distribution with a given probability. The sub-Bruker chance constraint model is converted into a mixed integer linear programming model by linearization. Based on the passenger demand sample under the worst-case distribution, the optimal value under deterministic problem programming is calculated, and an initial feasible solution is obtained based on the optimal value. The initial feasible solution is used to characterize the upper bound of the feasible region of the mixed integer linear programming model. Based on the initial feasible solution, the mixed integer linear programming model is solved to obtain the train operation scheme; The objective function is constructed as follows: A first cost is constructed based on the number of trains in the direct route and the cost of the trains traveling through the direct route; a second cost is constructed based on the cost of trains originating at a station and the number of trains originating at that station, wherein the number of trains originating at a station is determined by the non-negative portion of the difference between the number of trains departing from the station and the number of trains arriving at the station; a third cost is constructed based on the cost of trains terminating at a station and the number of trains terminating at that station, wherein the number of trains terminating at the station is determined by the non-negative portion of the difference between the number of trains arriving at the station and the number of trains departing from the station; and the objective function is obtained based on the first cost, the second cost, and the third cost. Constructing the sub-Bruker chance constraints includes: constructing deterministic problem constraints, and constructing a demand probability distribution that passenger demand follows; determining the set of passenger flows that can be satisfied by the train operation plan based on the deterministic problem constraints; constructing chance constraints that satisfy the demand probability distribution based on the set of passenger flows; and using the chance constraints under the worst distribution of the demand probability distribution as the sub-Bruker chance constraints. The constraints for constructing deterministic problems include: constructing a first constraint condition, which constrains the number of passengers in the direct journey to be no greater than the train capacity in the direct journey, wherein the train capacity in the direct journey is determined based on the number of trains passing through the direct journey and the capacity of each train; and constructing a second constraint condition, which constrains the number of inbound passengers at intermediate stations of passenger travel demand to be equal to the number of outbound passengers, wherein the passenger travel demand includes the origin station and the destination station.

2. The method according to claim 1, characterized in that, The method further includes: Construct an effective inequality, which is used to narrow the range of the feasible region of the mixed-integer linear programming model; Based on the initial feasible solution, the mixed-integer linear programming model is solved to obtain the train operation scheme, including: Using the effective inequalities as constraints, and based on the initial feasible solution, the mixed-integer linear programming model is solved to obtain the train operation scheme.

3. The method according to claim 2, characterized in that, The effective inequalities include at least one of the following: The first effective inequality is used to narrow the feasible region of train operation schemes in the mixed integer linear programming model. The second effective inequality is used to narrow the feasible domain of the passenger demand satisfaction decision variable in the mixed integer linear programming model. The passenger demand satisfaction decision variable is used to characterize whether the passenger demand under the passenger demand sample can be satisfied by the train operation plan.

4. The method according to claim 1, characterized in that, Based on the passenger demand sample under the worst-case distribution, calculate the optimal value under deterministic programming problem, and obtain an initial feasible solution based on the optimal value, including: Construct a deterministic model corresponding to the deterministic problem planning, wherein the deterministic model is used to characterize minimizing operating costs under deterministic passenger demand; The deterministic model is solved based on multiple passenger demand samples under the worst-case distribution to obtain multiple optimal values.

5. The method according to claim 4, characterized in that, The initial feasible solution is obtained based on the optimal value, including: The passenger demand sample under the worst distribution corresponding to the best value that is greater than the cost threshold among the multiple optimal values ​​is determined as the first passenger demand sample that cannot be satisfied by the train operation plan. The passenger demand sample corresponding to the worst distribution of the optimal value among the multiple optimal values ​​that is not greater than the cost threshold is determined as the second passenger demand sample that can be satisfied by the train operation plan. Set the value of the passenger demand satisfaction decision variable corresponding to the first passenger demand sample to not satisfy, and set the value of the passenger demand satisfaction decision variable corresponding to the second passenger demand sample to satisfy. An initial feasible solution is obtained based on the passenger demand satisfaction decision variables corresponding to each of the first passenger demand samples and each of the second passenger demand samples.

6. A train operation scheme compilation device based on a distributed robust chance constraint model, characterized in that, The device includes: The first construction module uses the number of trains in the direct process as the decision variable and aims to minimize operating costs to construct a partial Bruker chance constraint model for train operation schemes. This includes: constructing an objective function, which aims to minimize operating costs; and constructing partial Bruker chance constraints, which are used to constrain the train operation schemes to meet passenger demand under the worst-case distribution with a given probability. The first transformation module is used to convert the split-bar chance constraint model into a mixed-integer linear programming model through linearization. The first calculation module is used to calculate the optimal value under deterministic problem programming based on the passenger demand sample under the worst distribution, and to obtain an initial feasible solution based on the optimal value. The initial feasible solution is used to characterize the upper bound of the feasible region of the mixed integer linear programming model. The first solution module is used to solve the mixed integer linear programming model based on the initial feasible solution to obtain the train operation scheme; The objective function is constructed as follows: A first cost is constructed based on the number of trains in the direct route and the cost of the trains traveling through the direct route; a second cost is constructed based on the cost of trains originating at a station and the number of trains originating at that station, wherein the number of trains originating at a station is determined by the non-negative portion of the difference between the number of trains departing from the station and the number of trains arriving at the station; a third cost is constructed based on the cost of trains terminating at a station and the number of trains terminating at that station, wherein the number of trains terminating at the station is determined by the non-negative portion of the difference between the number of trains arriving at the station and the number of trains departing from the station; and the objective function is obtained based on the first cost, the second cost, and the third cost. Constructing the sub-Bruker chance constraints includes: constructing deterministic problem constraints, and constructing a demand probability distribution that passenger demand follows; determining the set of passenger flows that can be satisfied by the train operation plan based on the deterministic problem constraints; constructing chance constraints that satisfy the demand probability distribution based on the set of passenger flows; and using the chance constraints under the worst distribution of the demand probability distribution as the sub-Bruker chance constraints. The constraints for constructing deterministic problems include: constructing a first constraint condition, which constrains the number of passengers in the direct journey to be no greater than the train capacity in the direct journey, wherein the train capacity in the direct journey is determined based on the number of trains passing through the direct journey and the capacity of each train; and constructing a second constraint condition, which constrains the number of inbound passengers at intermediate stations of passenger travel demand to be equal to the number of outbound passengers, wherein the passenger travel demand includes the origin station and the destination station.

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