An integrated optimization model and algorithm for subway crew routing and scheduling

Abstract

The present invention discloses an integrated optimization model and algorithm for subway crew route switching and scheduling. Each crew member is assigned to a train duty according to a designated duty plan, ensuring that each train is assigned to a crew member for duty. The IOCSRP model is constructed with the goal of minimizing the total crew operating cost and the meal penalty cost. The present invention achieves the overall optimization of the crew route switching plan and the scheduling plan by modeling and solving the subway crew plan compilation optimization problem.

Description

Technical Field

[0001] The invention relates to the field of urban transportation technology, and in particular to an integrated optimization model and algorithm for subway crew route switching and scheduling. Background Art

[0002] Due to its advantages of high capacity, high speed and few delays, subways play an important role in urban passenger transportation. As subway operation plans become more complex (such as routes, stops, etc.), the difficulty of manually compiling service plans increases. It is particularly important to use automated technology to compile large-scale subway service plans, and it can improve operational efficiency under the condition of limited crew resources. Existing studies usually solve the crew route planning and scheduling sub-problems of the crew planning optimization problem sequentially to reduce operating costs. The optimization models used include set partitioning model, set covering model and multi-commodity network flow model. In addition, some literatures construct integer linear programming models based on set partitioning / network flow models, trying to overcome the problem that traditional path-based modeling methods are difficult to optimize the crew plan and other stage plans in an integrated manner.

[0003] The purpose of the crew route planning problem is to find a feasible crew route plan to cover all trains under the premise of a given car bottom utilization plan, while the crew shift planning problem is to prepare a specific shift plan for each crew member based on the given crew route plan. In addition, the crew route planning optimization problem is to generate a series of duty paths, which include a series of duty sections and meet the working time rules and other related constraints. The crew shift planning problem is to prepare a schedule for each crew member according to the specified scheduling rules and crew preferences, so as to ensure that each crew member can perform one or several duty tasks within the maximum cumulative working time. When the crew member's working time reaches the maximum continuous working time, the crew member needs to get off the train to rest or have a meal before continuing to work. Summary of the invention

[0004] In order to solve the problems existing in the prior art, the purpose of the present invention is to provide an integrated optimization model and algorithm for subway crew route and scheduling plans. The present invention models and solves the subway crew plan compilation optimization problem to achieve overall optimization of crew route plans and scheduling plans.

[0005] To achieve the above purpose, the technical solution adopted by the present invention is: an integrated optimization model for subway crew route and scheduling, characterized in that each crew member is assigned to a train according to a specified duty plan to ensure that each train is assigned a crew member on duty; and the model IOCSRP is constructed with the goal of minimizing the total crew operating cost and the meal penalty cost:

[0006]

[0007] The constraints are:

[0008]

[0009]

[0010]

[0011]

[0012]

[0013] In the objective function (1), c d (i,j,s,e,w,w′) is the travel cost of crew member d on the space-time state arc (i,j,s,e,w,w′), so the first term is the sum of the time costs of the crew member on the arc, and α 1 is the weight of the target item; the second item is the meal penalty fee. If the crew member's non-working arc time period is within the meal time window, if the meal arc is not executed, then the corresponding penalty will be imposed, T 0 Indicates the meal start time, α 2 is the weight of the target item; constraint (2) ensures that an arc starting from the source point is used for any passenger d; constraint (3) is a flow balance constraint, which ensures that the inflow flow of each point in the spatiotemporal state network except the source point and the sink point is equal to the outflow flow; constraint (4) ensures that an arc reaching the sink point is used for any passenger d; constraint (5) ensures that for any value multiplied by the arc (i, j, s, e) ∈ A du , must be assigned a crew member to serve it; constraint (6) is the constraint on the value of the decision variable.

[0014] As a further improvement of the present invention, the Lagrange multiplier λ is introduced into the objective function i,j,s,e , relax constraint (5) into the objective function, and then obtain the new relaxed model LRIOCSRP. Equation (7) is the objective function Z of the model LRIOCSRP 2 , the Lagrange multiplier λ i,j,s,e The value is not restricted, as shown in constraint (8):

[0015] Objective function:

[0016]

[0017] st constraints (2), (3), (4) and (6)

[0018]

[0019] As a further improvement of the present invention, the space-time state arc selection variable x of different crew members d (i, j, s, e, w, w′) is only associated in the value multiplication arc coverage constraint (5). After relaxing this constraint, LRIOCSRP is equivalent to the shortest path subproblem for each crew member in the spatiotemporal state network; thus, the objective function of the d subproblem for a single crew member is As shown in equation (9), the cost of different types of space-time state arcs As shown in equation (10):

[0020]

[0021]

[0022] As a further improvement of the present invention, equations (11) and (12) give The value of the Lagrange multiplier in at the kth iteration and step length θ k ; In equation (12), Z ub (k) represents the optimal upper bound value in the first k iterations, Z lb (k) represents the lower bound value at the kth iteration; parameter β k ∈[0,2] will change with the increase of the number of iterations. When the optimal upper bound value does not change within the preset number of iterations, then β k The value is half of the original value. In addition, it can be inferred from equation (12) that Z ub (k)-Z lb The difference in (k) determines the change in the iteration step size:

[0023]

[0024]

[0025] As a further improvement of the present invention, the updating rules of crew working hours are as follows:

[0026]

[0027]

[0028]

[0029]

[0030]

[0031]

[0032] Equations (13) and (14) give the updating methods of continuous working time and cumulative working time; constraints (15), (16) and (17) limit the time of each point to not exceed the specified time standard; constraint (18) gives the time limit of the conversion arc. When the conversion arc is enabled, the crew rest time constraint must be met. Indicates the shift φ i,s,w The crew members waiting at the source gather.

[0033] The present invention also discloses an integrated optimization algorithm for subway crew route switching and scheduling, which is implemented based on the above-mentioned model and includes the following steps:

[0034] Step 1: Initialize the problem parameters, including the number of crew members, the number of crew sections, and the location of the crew duty stations;

[0035] Step 2, introduce the value of the Lagrangian multiplier relaxation model IOCSRP and multiply the arc coverage constraint (5) to construct the Lagrangian relaxation model LRIOCSRP;

[0036] Step 3: Use the labeling method to calculate the shortest path with resource constraints for each crew member in the spatiotemporal state network;

[0037] Step 4: If the current upper and lower bounds are better, update the optimal upper and lower bounds respectively and calculate the optimal Gap;

[0038] Step 5: Determine whether the algorithm termination condition is met. If not, return to step 3.

[0039] The beneficial effects of the present invention are:

[0040] The present invention proposes an optimization method for the preparation of urban rail transit crew plans. On the basis of summarizing previous studies, the characteristics of domestic crew plan preparation are taken into account, and the research period is divided based on the shift, so that the crew plan preparation results are more practical. The present invention takes the minimum weighted sum of crew operating costs and meal time penalty costs as the optimization goal, considers restrictions such as working time rules, and constructs a multi-commodity network flow model with value multiplication arc covering sub-constraints. Based on the characteristics of the problem, the model is dually decomposed to transform the problem into a series of sub-problems that are easier to solve, and each sub-problem can be efficiently solved by searching for the shortest path with resource constraints for each crew member, thereby obtaining a high-quality feasible solution or even the optimal solution of the model. The example test results show that the dual decomposition algorithm based on Lagrangian relaxation can obtain high-quality crew plans within a reasonable time range. By comparing with the on-site experience plan, it is found that the scheme can save cost investment and reduce the scale of personnel, improve the efficiency of duty and the level of on-site operation management, and can be used to assist on-site crew plan preparation. BRIEF DESCRIPTION OF THE DRAWINGS

[0041] Figure 1 This is an example diagram of the relationship between crew planning and inter-shift transfers in an embodiment of the present invention;

[0042] Figure 2 : is the space-time state path of crew member d and the projection diagram of the path on three two-dimensional planes in an embodiment of the present invention;

[0043] Figure 3 It is a Lagrangian relaxation heuristic algorithm solution framework diagram in an embodiment of the present invention;

[0044] Figure 4 This is a schematic diagram of the layout of the Ninggao line in an embodiment of the present invention;

[0045] Figure 5 This is a schematic diagram of the change of the optimal upper and lower bounds with the solution time in an embodiment of the present invention. DETAILED DESCRIPTION

[0046] The embodiments of the present invention are described in detail below with reference to the accompanying drawings.

[0047] Example

[0048] This embodiment transforms the subway crew planning optimization problem into a problem of solving the work shift sequence of each crew member in the study period and the duty task arrangement problem in each work shift by decomposing the study period according to shifts, considering the crew members' working time and related work regulations and constraints. At the same time, by introducing the state dimension in the spatiotemporal network to accurately characterize the changes in the crew members' working status (such as continuous working time, dining status, etc.), the feasibility of the crew plan is guaranteed. In summary, this embodiment models the subway crew planning optimization problem (hereinafter referred to as the original problem) as a multi-commodity network flow model with secondary constraints based on the spatiotemporal state network modeling framework, and introduces a dual decomposition algorithm based on Lagrangian relaxation to efficiently solve the obtained integer linear programming model. Finally, this embodiment designs a series of actual cases based on the Nanjing Metro Ninggao Line to verify the effectiveness of the proposed model and algorithm.

[0049] An integrated optimization model for subway crew route exchange and scheduling, including:

[0050] 1 Problem Description

[0051] Given the vehicle floor utilization plan within the study time, this embodiment generates a feasible crew plan by solving the original problem. Each crew member takes on the train duty according to the specified duty plan, and it is necessary to ensure that each train is assigned a crew member. In addition, the optimization goal of the original problem is to minimize the weighted sum of the total crew operation cost and the penalty cost for violating the dining rules.

[0052] Regarding the original problem during the research period [T s ,T e ] to construct a space-time network G = (V, A), where V and A are the set of space-time nodes and space-time arcs, respectively. For each space-time arc (i, j, s, e) ∈ A, the space-time node (i, s) ∈ V indicates that the crew is at the departure station i at time s, and the space-time node (j, e) ∈ V indicates that the crew is at the arrival station j at time e. In addition, each space-time arc (i, j, s, e) ∈ A corresponds to a train number (i, j), and its travel time is t ij =es, and the travel cost of crew member d∈D through the space-time arc (i, j, s, e)∈A is expressed as c d (i,j,s,e). Figure 1 (a) gives an example of using a spatiotemporal network to describe crew scheduling. Figure 1 (a) is Figure 1 (b) The development of a work shift, i.e., shift 1; Figure 1 (b) shows the shift division during the study period and the transfer relationship of crew members between shifts.

[0053] For the sake of clarity, this embodiment summarizes the definitions of different types of nodes and arcs in the spatiotemporal network as follows:

[0054] (1) Nodes in the spatiotemporal network

[0055] The nodes in the spatiotemporal network include seven types of nodes. Specifically, the source node I Source It is the origin node of the crew, which connects all the source points of the flights through the source arc; the sink I sink It is the final destination node of the crew, which is connected to all the shift convergence points through convergence arcs; the shift source point is the starting node of the train mission included in the shift, and its previous node may be the source point or the shift convergence point; the shift convergence point is the end node of the train mission included in the shift, and its subsequent node may be the convergence point or the shift source point; the duty start point is the starting node of the duty arc; the duty end point is the terminal node of the duty arc.

[0056] (2) Arcs in space-time networks

[0057] The arcs in the space-time network include eleven types of arcs. Specifically, the source arc A so Connect the source point and the shift source point, indicating that the crew members are assigned to the base η d Departure to perform train mission; Convergence Arc A si Connecting the shift meeting point and the meeting point indicates that the crew has finished their duty; the shift source arc A sso Connect the shift source point and the duty start point; shift convergence arc A ssiConnect the shift duty end point and the shift meeting point; duty arc A du Connect the value multiplication start point and the corresponding value multiplication end point, and each value multiplication arc corresponds to a train number; rest arc A re and dining arc A dn Connect the value multiplication end point of a value multiplication arc and the value multiplication start point of another value multiplication arc; empty arc A dh Connect a value multiplication end point and the start node of another rest arc or connect the end node of a rest arc and the start node of another rest arc; directly connect arc A dc Connects the value multiplication end point of a value multiplication arc and the value multiplication start point of another value multiplication arc; shift transfer arc A sh Connect the convergence point of one shift and the source point of the next shift; virtual arc A da Connect the origin and destination of the same flight.

[0058] According to the duty system of China's subway system, this embodiment divides the subway operation period into several duty shifts, and each shift has a shift source point and a shift sink point, with φ i,s Indicates the shift to which the spatiotemporal node belongs. If a crew member is assigned to perform a shift, the crew member starts his duty after entering the shift source point and ends his duty after reaching the shift convergence point. In addition, when the crew member completes a series of train tasks within a shift and his cumulative working time does not exceed the maximum cumulative working time, the crew member can Figure 1 The shift transition arc shown in (b) in the figure performs the value multiplication task of the next shift. At the same time, if the source point I Source The cumulative working time of the crew members departing for a certain shift reaches the maximum cumulative working time Or all train missions within the study time are served, then the crew travels directly to the meeting point I sink The other time standards during the working period, except for the duty time, include the sign-in time. Check-out time Lunch and dinner time Note: During a shift, the continuous working time of the crew shall not exceed the maximum continuous working time. The rest and meal time should be and Inside.

[0059] The time and location information of the crew members can be determined through the constructed space-time network, and the continuous working time and cumulative working time of the crew members can be calculated through this information. However, this information is a statistical variable that can only be calculated cumulatively. When it comes to changes in state, it is difficult for the space-time network to characterize the changes in the crew members' working state. For example, the continuous working time of the crew members needs to be reset to zero after passing through the rest arc, and the dining state changes after passing through the dining arc without passing through other dining arcs within the dining time window. Therefore, this embodiment expands the two-dimensional space-time network G into a three-dimensional space-time state network G'=(V',ψ) by introducing a "state" dimension w. In the space-time state network, the state w of the crew members includes work state information such as dining status, continuous working time and cumulative working time. The set of space-time state arcs that the crew members may pass through is denoted by ψ d Indicated by ψ i,j,s,e,d represents the set of space-time state arcs that cover the value multiplied by the arc (i, j, s, e) and that the crew member d may pass through. In addition, the introduced shift source arc, shift conversion arc, and shift sink arc can describe the order in which the crew members perform their work shifts, thereby achieving overall optimization of crew routing and scheduling plans based on the space-time state network constructed in this embodiment. Figure 2 Given the corresponding Figure 1 (a) The space-time state path of crew member d in the crew plan, whose initial state is w 0 , Figure 2 The projection of the space-time state path on three two-dimensional planes is also shown.

[0060] 2 Model construction

[0061] 2.1 Model Assumptions

[0062] (1) In compiling the optimized crew plan, this embodiment assumes that the vehicle bottom operation plan is fixed input data, and the crew sections can be divided according to the plan and the duty stations.

[0063] (2) This embodiment does not consider the attendance rate of each crew member during the study period, that is, there is no absence of crew members.

[0064] (3) During the study period, all trains are required to have at least one attendant assigned to perform crew duties.

[0065] 2.2 Variable Definition

[0066] Table 1 presents the variables involved in the model.

[0067] Table 1 Variable definitions

[0068]

[0069]

[0070] 2.3 Objective Function

[0071] This embodiment aims to minimize the total crew operating costs and meal penalty costs, and constructs the IOCSRP model:

[0072]

[0073] The constraints are:

[0074]

[0075]

[0076]

[0077]

[0078]

[0079] In the objective function (1), c d (i,j,s,e,w,w′) is the travel cost of crew member d on the space-time state arc (i,j,s,e,w,w′), so the first term is the sum of the time costs of the crew member on the arc, and α 1 is the weight of the target item; the second item is the meal penalty fee. If the crew member's non-working arc time period is within the meal time window, if the meal arc is not executed, then the corresponding penalty will be imposed (T 0 indicates the meal start time), α 2 is the weight of the target item. Constraint (2) ensures that an arc from the source is used for any passenger d. Constraint (3) is a flow balance constraint, which ensures that the inflow flow of each point in the spatiotemporal state network except the source and sink is equal to the outflow flow. Constraint (4) ensures that an arc to the sink is used for any passenger d. Constraint (5) ensures that for any value multiplied by the arc (i, j, s, e) ∈ A du , must be assigned a crew member to serve. Constraint (6) is the decision variable value constraint.

[0080] 2.4 Dual decomposition based on Lagrangian relaxation

[0081] In the IOCSRP model, constraint (5) is a difficult constraint, and the Lagrange multiplier λ can be introduced into the objective function. i,j,s,e , relax constraint (5) into the objective function, and then obtain the new relaxed model LRIOCSRP. Equation (7) is the objective function Z of the model LRIOCSRP 2 , the Lagrange multiplier λ i,j,s,e The value is not restricted, as shown in constraint (8).

[0082] Objective function:

[0083]

[0084] st constraints (2), (3), (4) and (6)

[0085]

[0086] Due to the space-time state arc selection variable x of different crew members d (i, j, s, e, w, w′) is only associated in the value multiplication arc coverage constraint (5). After relaxing this constraint, LRIOCSRP is equivalent to the shortest path subproblem for each crew member in the spatiotemporal state network. Based on this, the objective function of the subproblem for a single crew member d is As shown in equation (9). In addition, equation (10) gives the costs of different types of space-time state arcs:

[0087]

[0088]

[0089] Equations (11) and (12) give The value of the Lagrange multiplier in at the kth iteration and step length θ k In equation (12), Z ub (k) represents the optimal upper bound value in the first k iterations, Z lb (k) represents the lower bound value at the kth iteration; parameter β k ∈[0,2] will change with the increase of the number of iterations. When the optimal upper bound value does not change within the preset number of iterations, then β k The value is half of the original value. In addition, it can be inferred from equation (12) that Z ub (k)-Z lb The difference in (k) determines the change in the iteration step size.

[0090]

[0091]

[0092] 3 Algorithm Design

[0093] According to the characteristics of the problem, a Lagrangian relaxation heuristic algorithm is designed. In each iteration, the lower bound problem is to solve the subproblem expressed in equation (9) through the minimum cost path algorithm to find a cost-optimal path for each crew member. Among them, the shortest path algorithm is the labeling method. The cost of the space-time state arc in the subproblem is shown in equation (10). After each iteration, the multiplier in the algorithm is updated according to formulas (11) and (12). In the shortest path update process, due to the restrictions of the working time rules, the state of the crew will change with the change of working hours. Formulas (13)-(18) give the update rules of the crew working time.

[0094]

[0095]

[0096]

[0097]

[0098]

[0099]

[0100] Equations (13) and (14) give the updating methods of continuous working time and cumulative working time; constraints (15), (16) and (17) limit the time of each point to not exceed the specified time standard; constraint (18) gives the time limit of the conversion arc. When the conversion arc is enabled, the crew rest time constraint must be met. Indicates the shift φ i,s,w The set of crew members waiting at the source point. Note that when the crew passes through an arc other than the value multiplication arc, the continuous working time will be reset to zero; when the crew passes through a conversion arc, both the continuous working time and the cumulative working time will be reset to zero. Based on the above shortest path update rules, Figure 3 The solution framework of Lagrangian relaxation heuristic algorithm is given.

[0101] The specific steps of the Lagrangian relaxation heuristic algorithm are as follows:

[0102] Step 1: Initialize the problem parameters, including the number of crew members, the number of crew sections, and the locations of crew duty stations.

[0103] Step 2: Introduce the value of the Lagrangian multiplier relaxation model IOCSRP and multiply the arc coverage constraint (5) to construct the Lagrangian relaxation model LRIOCSRP.

[0104] Step 3: Use the labeling method to calculate the shortest path with resource constraints for each crew member in the spatiotemporal state network.

[0105] Step 4: If the current upper and lower bounds are better, update the optimal upper and lower bounds respectively and calculate the optimal Gap.

[0106] Step 5: Determine whether the algorithm termination condition is met. If not, return to Step 3.

[0107] 4 Case Analysis

[0108] A set of actual cases based on Nanjing Metro Ninggao Line was constructed to verify the effectiveness of the model and algorithm. Figure 4 The line layout of Nanjing Metro Ninggao Line. The Ninggao Line consists of 6 stations and 5 sections, with a total length of 51.47 kilometers. Based on the actual operation diagram of the Ninggao Line, 18 examples are constructed in sequence by reducing the number of operating trains and changing the size of the crew. In addition, the models and algorithms involved in this embodiment are all implemented based on C# language programming, and the test platform is an Intel Core i5-9400F2.9GHz CPU and a 32GB 64-bit computer. The parameters in the model are shown in Table 2.

[0109] Table 2 Parameter values ​​in the model

[0110]

[0111] 4.1 Objective function coefficient sensitivity test

[0112] Analyze the relative relationship between the coefficients of the two sub-objective items in the objective function, and determine the coefficient values ​​of the two sub-objective items as the basis for problem analysis. In order to determine the best coefficient combination, this embodiment selects Cases 2 and 16 for sensitivity testing, and sets α 2 The value of is set to 0.1, α 1 The value increases from 1 to 10, and the upper and lower bounds and the optimal Gap are obtained. Table 3 reports the calculation results of the original problem based on the Lagrangian relaxation method. In Table 3, as α 1 As the value of increases, the upper and lower bounds of the test case gradually increase. In Case 2, when α 1 =3, the optimal Gap is 1.12%, and the corresponding upper and lower bounds are 9974 and 9864 respectively; in case 16, the optimal Gap is 4.97%, and α 1 =7. It is easy to see that the weight coefficients for obtaining the optimal Gap in Case 2 and Case 16 are not consistent, but considering that the case scale of Case 16 is much larger than that of Case 2, this embodiment takes α 1 =7 and α 2 =0.1 for subsequent case analysis. At this time, compared with α1 =3, the optimal Gap of Case 2 only increases by 0.26%.

[0113] Table 3 Upper and lower bounds and optimal gaps with α 1 Change value (unit: RMB)

[0114]

[0115] 4.2 Algorithm Performance Test

[0116] Table 4 lists the detailed calculation results of 18 test cases. The optimal Gap can be used to evaluate the quality of the solution, while the solution time reflects the solution efficiency of the Lagrangian relaxation heuristic algorithm. As can be seen from Table 4, the optimal Gap and solution time show a fluctuating increase trend as the problem size increases. Among them, the optimal Gap of Case 4 is the smallest, which is 0.67%; while the optimal Gap of Case 11 is the largest among all cases, which is 8.04%. In terms of solution time, the shortest solution time of Case 1 is 20.56 seconds, while the longest solution time of Case 18 is 394 seconds. In addition, the analysis of the calculation results of the 18 cases shows that the change of the problem size will lead to a polynomial increase in the solution time. In terms of average solution time and Gap, the average solution time of the 18 cases is only 150.28 seconds, and the average Gap is 5.03%. Figure 5 The optimal lower bound value of case 16 shows a trend of gradually increasing in the early stage and then remaining unchanged, and the optimal upper bound value shows a sawtooth change, gradually decreasing and then remaining unchanged, and both show a stable and unchanged trend after 146 seconds. Therefore, in the process of updating the Lagrange multiplier using the subgradient method, the Lagrange relaxation heuristic algorithm can obtain high-quality feasible solutions for the test case in a short time, and can effectively solve the IOCSRP model.

[0117] Table 4 Calculation information of 18 examples (unit: RMB)

[0118]

[0119]

[0120] In order to illustrate the practicality and effectiveness of the algorithm proposed in this embodiment, the Nanjing Metro Ninggao Line is taken as an example, considering the actual scheduling rules of the Ninggao Line, comparing the optimization scheme obtained by the Lagrangian relaxation crew planning method with the empirical scheme used on site, and analyzing the gap between the two schemes in various factors. Table 5 compares the total cost, number of people, duty efficiency, per capita cost and per capita working time of the empirical scheme and the optimization scheme of this embodiment. Among them, the duty efficiency is defined as the total number of duty tasks divided by the total number of crew members used in this embodiment. It can be seen from Table 5 that the optimization scheme is smaller than the empirical scheme in terms of the total cost index, saving 9.98% of the cost investment than the empirical scheme, and the optimization scheme uses fewer crew members, only 36 people; the optimization scheme has the largest increase in per capita working time, with an increase of 39.34 minutes and an increase of 13.00%. The optimization scheme is inferior to the empirical scheme only in the per capita cost index, but the gap between the two is small. Compared with the improvement of other indicators, the increase in this cost is acceptable. In summary, the optimization scheme has a higher crew utilization efficiency than the empirical scheme, saving total costs and crew investment.

[0121] Table 5 Comparison of indicators between empirical scheme and optimized scheme

[0122]

[0123] The above-mentioned embodiments only express the specific implementation of the present invention, and the description thereof is relatively specific and detailed, but it cannot be understood as limiting the scope of the present invention. It should be pointed out that, for ordinary technicians in this field, several variations and improvements can be made without departing from the concept of the present invention, which all belong to the protection scope of the present invention.

Claims

1. A method for constructing an integrated optimization model for subway crew route exchange and scheduling, characterized in that: According to the designated duty plan, each crew member is assigned to perform the train mission, ensuring that each train is assigned a crew member to perform the mission; and with the goal of minimizing the total crew operating cost and meal penalty cost, the model IOCSRP is constructed: The constraints are: In the objective function (1), c d (i,j,s,e,w,w′) is the travel cost of crew member d on the space-time state arc (i,j,s,e,w,w′), so the first item is the sum of the time costs of the crew member's arc, and α1 is the weight of the target item; the second item is the meal penalty cost. If the time period of the crew member's non-working arc is within the meal time window, if the meal arc is not executed, then the corresponding penalty will be imposed. T0 represents the meal start time, and α2 is the weight of the target item; constraint (2) ensures that an arc starting from the source point is used for any crew member d; constraint (3) is a flow balance constraint, which ensures that the inflow flow of each point in the space-time state network except the source point and the sink point is equal to the outflow flow; constraint (4) ensures that an arc to the sink point is used for any crew member d; constraint (5) ensures that for any value multiplied by the arc (i,j,s,e)∈A du , must be assigned a crew member to serve; constraint (6) is the value constraint of the decision variable; Introducing the Lagrange multiplier λ into the objective function i,j,s,e , relax constraint (5) into the objective function, and then obtain the new relaxed model LRIOCSRP. Equation (7) is the objective function Z2 of the model LRIOCSRP, and the Lagrange multiplier λ i,j,s,e The value is not restricted, as shown in constraint (8): Objective function: st constraints (2), (3), (4) and (6) Due to the space-time state arc selection variable x of different crew members d (i, j, s, e, w, w′) is only associated in the value multiplication arc coverage constraint (5). After relaxing this constraint, LRIOCSRP is equivalent to the shortest path subproblem for each crew member in the spatiotemporal state network; thus, the objective function of the d subproblem for a single crew member is As shown in equation (9), the cost of different types of space-time state arcs As shown in equation (10): Equations (11) and (12) give The value of the Lagrange multiplier in at the kth iteration and step length θ k ; In equation (12), Z ub (k) represents the optimal upper bound value in the first k iterations, Z lb (k) represents the lower bound value at the kth iteration; parameter β k ∈[0,2] will change with the increase of the number of iterations. When the optimal upper bound value does not change within the preset number of iterations, then β k The value is half of the original value. In addition, it can be inferred from equation (12) that Z ub (k)-Z lb The difference in (k) determines the change in the iteration step size: The rules for updating crew working hours are as follows: Equations (13) and (14) give the updating methods of continuous working time and cumulative working time; constraints (15), (16) and (17) limit the time of each point to not exceed the specified time standard; constraint (18) gives the time limit of the conversion arc. When the conversion arc is enabled, the crew rest time constraint must be met. Indicates the shift φ i,s,w The crew members waiting at the source gather.

2. A method for using an integrated optimization model for subway crew route switching and scheduling, based on the model constructed by the method described in claim 1, characterized in that: The steps include: Step 1: Initialize the problem parameters, including the number of crew members, the number of crew sections, and the location of the crew duty stations; Step 2, introduce the value of the Lagrangian multiplier relaxation model IOCSRP and multiply the arc coverage constraint (5) to construct the Lagrangian relaxation model LRIOCSRP; Step 3: Use the labeling method to calculate the shortest path with resource constraints for each crew member in the spatiotemporal state network; Step 4: If the current upper and lower bounds are better, update the optimal upper and lower bounds respectively and calculate the optimal Gap; Step 5: Determine whether the algorithm termination condition is met. If not, return to step 3.

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