Improved NSGA-II algorithm-based high-speed rail bus receiving and transporting time table optimization method

Abstract

The invention discloses an improved NSGA-II algorithm-based high-speed rail bus receiving and transporting timetable optimization method, and relates to the technical field of traffic transportation, and the method comprises the following steps: determining basic parameters and precondition assumption of a high-speed rail bus receiving and transporting system; constructing a multi-objective optimization model which aims at minimizing the waiting time of passengers and the operation cost of public transport enterprises and meets the constraints of departure intervals, route lengths and departure train numbers; an improved NSGA-II algorithm fusing championade selection of mixed linear ranking and a uniform multi-parent crossover strategy is adopted, a population is initialized through Gray coding, and a Pareto optimal solution set is obtained through iterative solution of operations such as fitness calculation, selection, crossover, variation, population merging and elite retention; extracting an optimal scheme from the obtained Pareto optimal solution set; according to the optimal scheme, generating a departure time table of each receiving and transporting bus route; the overall coordination and adaptability of the public transportation system are enhanced by constructing a multi-objective optimization model with the goal of minimizing the passenger waiting time and the public transportation enterprise operation cost.

Description

Technical Field

[0001] This invention belongs to the field of transportation technology, and in particular relates to a method for optimizing high-speed rail connecting bus timetables based on an improved NSGA-II algorithm. Background Technology

[0002] High-speed rail operations typically follow strict timetables, ensuring relatively precise arrival times. Under normal circumstances, high-speed trains arrive at their destination stations on time, with an error margin usually within a few minutes, providing passengers with a high degree of time assurance. High-speed rail shuttle buses are a public transportation mode closely connected to high-speed rail stations, facilitating convenient travel between the station and other parts of the city. They are conveniently connected to the high-speed rail station entrances and exits via accessible passageways. Shuttle buses usually have fixed routes and departure times. Vehicles depart from designated starting points on time, travel along predetermined routes, passing several important intermediate stops, and finally arrive at the terminal station before returning. Shuttle buses utilize a regional dispatching model, meaning that after returning to the depot, buses can cross routes to perform the next trip without empty runs. This model improves vehicle utilization efficiency, reduces empty running distances, thereby saving resources and reducing costs.

[0003] The arrival times of high-speed trains are not strictly predictable, occurring throughout various time periods and potentially creating peak passenger flow at any time. Therefore, the efficiency of the connection between high-speed trains and the connecting bus system directly impacts the passenger flow dispersal capacity and the overall travel experience. However, due to the mismatch between high-speed train arrival times and connecting bus departure times, as well as the uneven distribution of high-speed train passenger flow in time and space, the operation of high-speed rail connecting buses faces the following deeper problems:

[0004] (1) Low passenger transfer efficiency: Current bus dispatching schemes mostly adopt fixed departure intervals, ignoring the dynamic changes in high-speed rail arrival time and passenger travel demand. This model results in excessively long waiting times for passengers arriving by high-speed rail during off-peak hours, while during peak hours, there may be insufficient connection capacity, severely restricting transfer efficiency and increasing passengers' travel time costs. The passenger transfer process is as follows: Figure 1 As shown, after arriving at the high-speed rail station, passengers need to walk to the bus stop. Due to the varying walking speeds of passengers, passengers on the same high-speed rail train who need to transfer may take a different bus on the same route.

[0005] (2) Low utilization efficiency of public transport resources: Due to the lack of systematic analysis of high-speed rail arrival times, passenger flow characteristics, and demand at stations along the route, the design of bus schedules failed to fully optimize vehicle dispatching and operation strategies. As a result, the efficiency of passenger flow dispersal at high-speed rail arrival stations was low, and the waiting time for passengers at stations along the route was long. At the same time, the uneven allocation of resources by public transport companies resulted in the dual contradiction of high vehicle idle rate and increased operating costs.

[0006] (3) Complexity of multi-objective coordination: The high-speed rail connecting bus system needs to simultaneously satisfy two relatively contradictory objectives: minimizing passenger waiting time and minimizing bus company operating costs. This multi-objective optimization problem involves the coordinated scheduling of high-speed rail stations and bus routes, the temporal and spatial connection between high-speed rail and buses, and the dynamic distribution of passenger flow along the entire route, which has high complexity and uncertainty.

[0007] Therefore, based on the arrival time and passenger demand characteristics of high-speed rail, a multi-objective optimization model is constructed, and an improved NSGA-II algorithm is proposed to solve the problem. At the same time, the departure timetable and vehicle scheduling plan of high-speed rail station shuttle buses are reasonably formulated, taking into account both enterprise costs and passenger interests, so as to achieve effective connection between high-speed rail station shuttle buses and high-speed rail trains from multiple perspectives. Summary of the Invention

[0008] To solve the above-mentioned technical problems, the present invention is achieved through the following technical solution:

[0009] This invention provides a method for optimizing high-speed rail shuttle bus timetables based on an improved NSGA-II algorithm, comprising the following steps:

[0010] Step S1: Clarify the basic parameters and underlying assumptions of the high-speed rail transfer bus system;

[0011] Step S2: Construct a multi-objective optimization model with the goal of minimizing passenger waiting time and bus company operating costs, while satisfying constraints on departure intervals, route length, and departure frequency;

[0012] Step S3: The improved NSGA-II algorithm, which combines tournament selection with a hybrid linear ranking strategy and uniform multi-parent crossover, is adopted. The population is initialized with Gray encoding, and the Pareto optimal solution set is obtained through iterative operations such as fitness calculation, selection, crossover, mutation, population merging and elite retention.

[0013] Step S4: Extract the optimal solution from the obtained Pareto optimal solution set;

[0014] Step S5: Generate the departure timetable for each connecting bus route based on the optimal solution.

[0015] Further, in step S1, the basic parameters include high-speed rail arrival time, passenger flow for each line transfer, distance between the connecting bus stop and the high-speed rail platform, distance of the connecting bus line, number of bus departures, unit energy cost of buses, unit labor cost, upper and lower limits of bus departure intervals, maximum number of bus departures, average walking speed of passengers, and so on. The first high-speed rail line transfer The number of passengers on the connecting bus routes, specifically based on the following assumptions:

[0016] Step S11: Buses strictly follow the established routes, with arrival time deviations controlled within ±1 minute, and no instances of buses crossing routes. Buses travel at a constant speed within the dedicated bus lanes, unaffected by traffic lights at intersections or other traffic interference.

[0017] Step S12: All buses used for transportation are of the same model, have a fixed passenger capacity, and have consistent vehicle performance and energy consumption characteristics. Cross-line dispatching can be achieved within the station. After completing the current task, the vehicle is immediately put into operation for the next trip, with no empty running intervals.

[0018] Step S13: Vehicles will only stop at designated stations on each route and are prohibited from picking up or dropping off passengers en route.

[0019] Step S14: All passengers arriving by high-speed rail walk to the bus stop through a dedicated passage. The walking speed follows a normal distribution. Passengers choose to take the first bus available and do not abandon their journey to transfer.

[0020] Step S15: The high-speed rail arrival timetable and corresponding transfer passenger flow are known and fixed values, with no sudden delays or passenger flow fluctuations.

[0021] Furthermore, in the multi-objective optimization model of step S2, the objective function for minimizing passenger waiting time is constructed as follows:

[0022] Step S21: Calculation of the time it takes for passengers to walk from the high-speed rail platform to the bus stop: ,in This refers to the time it takes for passengers to walk from the high-speed rail platform to the bus stop. The distance between bus stops and high-speed rail platforms is considered. The average walking speed of passengers;

[0023] Step S22: Calculation of waiting time for a single passenger: ,in For waiting time, For the first The arrival time of the shuttle bus at the station. For the first The arrival time of the high-speed train at the platform, The indicator function has the following formula: ;

[0024] Step S23, Objective function for total passenger waiting time: ,in The total waiting time for passengers. This refers to the number of high-speed rail lines. The number of bus routes that provide transportation. To determine the number of bus departures for transportation, For the first The first high-speed rail line transfer The number of passengers on each connecting bus route.

[0025] Furthermore, in step S3, the objective function of the multi-objective optimization model, which minimizes the operating costs of the public transport company, is constructed as follows:

[0026] Step S31, Bus operating costs: ,in For driving costs, For the first The distance of the connecting bus routes, The unit energy cost of buses;

[0027] Step S32, Fixed depreciation cost of buses: ,in For depreciation expenses, This refers to the unit depreciation cost of buses.

[0028] Step S33, Bus driver's wages and expenses: ,in For labor costs, This refers to the unit cost of labor.

[0029] Step S34, Objective function for total operating cost of public transport company: ,in This represents the total operating cost of the public transportation company.

[0030] Furthermore, in step S4, the constraints are specifically as follows:

[0031] Step S41, Departure Interval Constraints: ,in For the first The departure interval of each connecting bus route. To preset the minimum bus departure interval, To preset the maximum bus departure interval;

[0032] Step S42, Line length constraint: ,in For the first The distance of the connecting bus routes, and These are the upper and lower limits of the line length;

[0033] Step S43, Departure Number Constraints: ,in This is the preset maximum number of bus departures.

[0034] Furthermore, in step S5, the solution steps of the improved NSGA-II algorithm specifically include:

[0035] Step S51: Set algorithm parameters, including population size, maximum number of iterations, crossover probability, mutation probability, mixed selection probability, and tournament competition size;

[0036] Step S52: Initialize the population: Randomly generate an initial population in the solution space, and use Gray coding to map the departure interval parameter of the shuttle bus to a chromosome gene sequence, with the departure time of the first bus as the initial scheduling benchmark.

[0037] Step S53, Fitness Calculation and Multi-Objective Evaluation: Calculate the objective function value for each individual. and Non-dominated sorting and crowding calculation are performed based on the target value;

[0038] Step S54, Tournament selection operation with mixed linear ranking: Perform linear ranking selection with mixed selection probability, and the remaining individuals are selected through tournament selection until the parent population size reaches the preset size.

[0039] Step S55, Uniform multi-parent crossover operation: Randomly select multiple individuals from the parent population as parents, generate a masking sequence with the same length as the chromosome, and select gene loci from different parents according to the masking sequence to generate new offspring;

[0040] Step S56, Uniform mutation operation: Randomly reset the gene values ​​of offspring individuals according to the mutation probability, and the reset values ​​are within the feasible region;

[0041] Step S57, Merging Populations and Elite Preservation: Merge the parent and offspring populations into a temporary population, perform non-dominated sorting and crowding calculation on the temporary population, and select the top few individuals to form a new generation population.

[0042] Step S58, Iteration Termination Judgment: If the current iteration count reaches the maximum iteration count, output the Pareto optimal solution set; otherwise, return to step S3 to continue iterating.

[0043] Furthermore, the initial population uses Gray encoding, whose encoding rules ensure that the binary mapping of adjacent integer values ​​has only a single bit difference. When the line operation interval is adjusted, the chromosome only needs to trigger a single bit flip.

[0044] Furthermore, in the tournament selection operation of the hybrid linear ranking, the selection probability of an individual is linearly related to its ranking in the population, with individuals with better fitness having a higher selection probability.

[0045] Furthermore, in the uniform multi-parent crossover operation, the masking sequence is a randomly generated 0-1 sequence, where 1 indicates that the gene locus is selected from the corresponding parent, and 0 indicates that the corresponding gene locus of the parent is not selected; the crowding degree is calculated as follows: for each individual in the non-dominated layer, the distance between it and its neighboring individuals in the target space is calculated, and this distance is used to measure the crowding degree of the individual.

[0046] Furthermore, the method for extracting the optimal solution from the Pareto optimal solution set is as follows: based on actual operational needs, select the Nash equilibrium solution from the solution set that balances passenger waiting time with enterprise operating costs.

[0047] The present invention has the following beneficial effects:

[0048] 1. This invention constructs a multi-objective optimization model aimed at minimizing passenger waiting time and public transport company operating costs, which can systematically coordinate the contradiction between passenger travel experience and enterprise resource investment. It uses an improved NSGA-II algorithm to solve the problem, which can simultaneously obtain multiple Pareto optimal solutions in one optimization process, forming a series of timetable schemes that take into account different preferences. Decision-makers can flexibly choose the optimal scheduling strategy that focuses on improving service efficiency or controlling costs according to actual operational needs, thereby enhancing the overall coordination and adaptability of the public transport system.

[0049] 2. This invention introduces a tournament selection strategy with hybrid linear ranking and a uniform multi-parent crossover strategy, effectively overcoming the problems of premature convergence and decreased population diversity in traditional genetic algorithms. The improved algorithm maintains a wider exploration range in the early stages of the search, avoiding getting trapped in local optima; in the later stages of optimization, it accelerates convergence to a high-quality solution set by enhancing local exploitation capabilities. This mechanism improves the search efficiency and stability of the algorithm in complex solution spaces, making the timetable optimization results more reliable.

[0050] 3. This invention not only generates a single optimal solution, but also outputs a series of Pareto non-dominated solutions, forming a set of timetable schemes covering different optimization tendencies; each solution presents a clear trade-off between passenger waiting time and enterprise operating costs, providing decision-making options for operation managers; this makes the method practically applicable, able to adapt to the dynamic changes of different passenger flow characteristics, resource conditions and operational goals, support multi-scenario and multi-stage bus scheduling planning, and improve the bus system's ability to cope with complex operating environments.

[0051] Of course, any product implementing this invention does not necessarily need to achieve all of the advantages described above at the same time. Attached Figure Description

[0052] To more clearly illustrate the technical solutions of the embodiments of the present invention, the accompanying drawings used in the description of the embodiments will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0053] Figure 1 This is a passenger transfer connection diagram for the present invention;

[0054] Figure 2 This is the encoding method of the present invention;

[0055] Figure 3 This is a cross-operation of the present invention;

[0056] Figure 4 This invention improves the NSGA-II algorithm flow.

[0057] Figure 5 The three algorithms of this invention are used to solve the approximate Pareto front obtained from the ZDT test set;

[0058] Figure 6 This is a flowchart illustrating the high-speed rail connecting bus timetable optimization method based on the improved NSGA-II algorithm of the present invention. Detailed Implementation

[0059] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0060] Please see Figure 1-6 As shown, this invention is a method for optimizing high-speed rail shuttle bus timetables based on an improved NSGA-II algorithm, comprising the following steps:

[0061] Step S1: Clarify the basic parameters and underlying assumptions of the high-speed rail transfer bus system;

[0062] Step S2: Construct a multi-objective optimization model with the goal of minimizing passenger waiting time and bus company operating costs, while satisfying constraints on departure intervals, route length, and departure frequency;

[0063] Step S3: The improved NSGA-II algorithm, which combines tournament selection with a hybrid linear ranking strategy and uniform multi-parent crossover, is adopted. The population is initialized with Gray encoding, and the Pareto optimal solution set is obtained through iterative operations such as fitness calculation, selection, crossover, mutation, population merging and elite retention.

[0064] Step S4: Extract the optimal solution from the obtained Pareto optimal solution set;

[0065] Step S5: Generate the departure timetable for each connecting bus route based on the optimal solution.

[0066] In step S1, the basic parameters include high-speed rail arrival time, passenger flow for each line transfer, distance between the connecting bus stop and the high-speed rail platform, distance of the connecting bus route, number of bus departures, unit energy cost of buses, unit labor cost, upper and lower limits of bus departure intervals, maximum number of bus departures, average walking speed of passengers, and the first... The first high-speed rail line transfer The number of passengers on the connecting bus route is based on the following assumptions:

[0067] Step S11: Buses strictly follow the established routes, with arrival time deviations controlled within ±1 minute, and no instances of buses crossing routes. Buses travel at a constant speed within the dedicated bus lanes, unaffected by traffic lights at intersections or other traffic interference.

[0068] Step S12: All buses used for transportation are of the same model, have a fixed passenger capacity, and have consistent vehicle performance and energy consumption characteristics. Cross-line dispatching can be achieved within the station. After completing the current task, the vehicle is immediately put into operation for the next trip, with no empty running intervals.

[0069] Step S13: Vehicles will only stop at designated stations on each route and are prohibited from picking up or dropping off passengers en route.

[0070] Step S14: All passengers arriving by high-speed rail walk to the bus stop through a dedicated passage. The walking speed follows a normal distribution. Passengers choose to take the first bus available and do not abandon their journey to transfer.

[0071] Step S15: The high-speed rail arrival timetable and corresponding transfer passenger flow are known and fixed values, with no sudden delays or passenger flow fluctuations.

[0072] In step S2 of the multi-objective optimization model, the objective function for minimizing passenger waiting time is constructed as follows:

[0073] Step S21: Calculation of the time it takes for passengers to walk from the high-speed rail platform to the bus stop: ,in This refers to the time it takes for passengers to walk from the high-speed rail platform to the bus stop. The distance between bus stops and high-speed rail platforms is considered. The average walking speed of passengers;

[0074] Step S22: Calculation of waiting time for a single passenger: ,in For waiting time, For the first The arrival time of the shuttle bus at the station. For the first The arrival time of the high-speed train at the platform, The indicator function has the following formula: ;

[0075] Step S23, Objective function for total passenger waiting time: ,in The total waiting time for passengers. This refers to the number of high-speed rail lines. The number of bus routes that provide transportation. To determine the number of bus departures for transportation, For the first The first high-speed rail line transfer The number of passengers on each connecting bus route.

[0076] In step S3, the objective function of the multi-objective optimization model, which minimizes the operating costs of the public transport company, is constructed as follows:

[0077] Step S31, Bus operating costs: ,in For driving costs, For the first The distance of the connecting bus routes, The unit energy cost of buses;

[0078] Step S32, Fixed depreciation cost of buses: ,in For depreciation expenses, This refers to the unit depreciation cost of buses.

[0079] Step S33, Bus driver's wages and expenses: ,in For labor costs, This refers to the unit cost of labor.

[0080] Step S34, Objective function for total operating cost of public transport company: ,in This represents the total operating cost of the public transportation company.

[0081] In step S4, the specific constraints are as follows:

[0082] Step S41, Departure Interval Constraints: ,in For the first The departure interval of each connecting bus route. To preset the minimum bus departure interval, To preset the maximum bus departure interval;

[0083] Step S42, Line length constraint: ,in For the first The distance of the connecting bus routes, and These are the upper and lower limits of the line length;

[0084] Step S43, Departure Number Constraints: ,in This is the preset maximum number of bus departures.

[0085] In step S5, the solution steps of the improved NSGA-II algorithm specifically include:

[0086] Step S51: Set algorithm parameters, including population size, maximum number of iterations, crossover probability, mutation probability, mixed selection probability, and tournament competition size;

[0087] Step S52: Initialize the population: Randomly generate an initial population in the solution space, and use Gray coding to map the departure interval parameter of the shuttle bus to a chromosome gene sequence, with the departure time of the first bus as the initial scheduling benchmark.

[0088] Step S53, Fitness Calculation and Multi-Objective Evaluation: Calculate the objective function value for each individual. and Non-dominated sorting and crowding calculation are performed based on the target value;

[0089] Step S54, Tournament selection operation with mixed linear ranking: Perform linear ranking selection with mixed selection probability, and the remaining individuals are selected through tournament selection until the parent population size reaches the preset size.

[0090] Step S55, Uniform multi-parent crossover operation: Randomly select multiple individuals from the parent population as parents, generate a masking sequence with the same length as the chromosome, and select gene loci from different parents according to the masking sequence to generate new offspring;

[0091] Step S56, Uniform mutation operation: Randomly reset the gene values ​​of offspring individuals according to the mutation probability, and the reset values ​​are within the feasible region;

[0092] Step S57, Merging Populations and Elite Preservation: Merge the parent and offspring populations into a temporary population, perform non-dominated sorting and crowding calculation on the temporary population, and select the top few individuals to form a new generation population.

[0093] Step S58, Iteration Termination Judgment: If the current iteration count reaches the maximum iteration count, output the Pareto optimal solution set; otherwise, return to step S3 to continue iterating.

[0094] The initial population uses Gray encoding, whose encoding rules ensure that the binary mapping of adjacent integer values ​​has only a single bit difference. When the line operation interval is adjusted, the chromosome only needs to trigger a single bit flip.

[0095] In tournament selection operations with mixed linear ranking, the selection probability of an individual is linearly related to its rank in the population, with individuals with better fitness having a higher selection probability.

[0096] In the uniform multi-parent crossover operation, the masking sequence is a randomly generated 0-1 sequence, where 1 indicates that the gene locus is selected from the corresponding parent and 0 indicates that the corresponding gene locus of the parent is not selected. The crowding degree is calculated as follows: for each individual in the non-dominated layer, the distance between it and its neighboring individuals in the target space is calculated. This distance is used to measure the crowding degree of the individual.

[0097] The optimal solution is extracted from the Pareto optimal solution set by selecting the Nash equilibrium solution that balances passenger waiting time with enterprise operating costs based on actual operational needs.

[0098] One specific application of this embodiment is:

[0099] Premise

[0100] 1. Buses strictly follow the established routes, with arrival time deviations controlled within ±1 minute, and no instances of buses skipping each other; vehicles maintain a constant speed (40 km / h) within the dedicated bus lanes, unaffected by traffic lights at intersections or other traffic interference.

[0101] 2. All buses used for transportation are of the same model, with a passenger capacity of 50 people per vehicle, and consistent vehicle performance and energy consumption characteristics; cross-line dispatching can be realized within the station, and vehicles can be put into operation immediately after completing the current task, with no empty running intervals.

[0102] 3. Vehicles will only stop at designated stops on each route, and passengers are not allowed to get on or off along the way.

[0103] 4. All passengers arriving by high-speed rail walk to the bus stop via a dedicated passage, and their walking speed follows a normal distribution. Passengers choose to take the first bus available and do not abandon their journey to transfer.

[0104] 5. The high-speed rail arrival timetable and corresponding transfer passenger flow are known and fixed values, with no sudden delays or passenger flow fluctuations.

[0105] Model building

[0106] An optimization model for the bus timetable connecting high-speed rail and public transport is constructed with the goal of minimizing passenger waiting time and operating costs for public transport companies.

[0107] Minimize passenger waiting time

[0108] The time it takes for passengers to walk from the high-speed rail station to the pick-up bus stop, i.e.:

[0109]

[0110] Passengers' waiting time, i.e.:

[0111] The total waiting time for passengers is:

[0112] The objective function for minimizing passenger waiting time is:

[0113] Minimize the operating costs of public transport companies

[0114] The operating cost of a bus is calculated by multiplying the route distance by the number of departures by the unit energy consumption cost, i.e.:

[0115] The fixed depreciation cost of a bus is:

[0116] Bus driver wages and expenses, namely:

[0117] The objective function for minimizing the operating costs of public transportation companies is as follows:

[0118] Constraints

[0119] 1. Departure Interval: The departure interval for connecting buses must be controlled within a reasonable range to meet the dual requirements of operational efficiency and passenger service quality. Excessively long departure intervals significantly increase passenger waiting times, reduce the travel experience, and thus weaken the social benefits of public transportation; while excessively short intervals exacerbate scheduling complexity, leading to increased operating costs and affecting the overall stability of the system. The departure interval must meet the following requirements:

[0120] 2. Route Length: The length of the connecting bus routes must meet the following requirements.

[0121] 3. Departure Frequency: The departure frequency of the connecting bus routes is limited by the size of the fleet and vehicle dispatching capacity. Since the bus company has limited vehicle resources, the number of buses on each route is fixed. If the number of departures exceeds the number of vehicles available for that route, the planned departure schedule cannot be achieved. The departure frequency must meet the following requirements: , This represents the maximum number of trains that can depart.

[0122] Solving with the improved NSGA-II algorithm

[0123] The model established in this paper is a multi-objective optimization model. Multi-objective optimization problems are widely found in complex practical applications, characterized by the need to simultaneously optimize two or more conflicting objective functions, such as the trade-off between minimizing passenger waiting time and minimizing the operating costs of public transportation companies. Within the multi-objective optimization framework, there is usually a non-dominant relationship between objective functions, meaning there is no single solution that can simultaneously improve all objectives. The core of multi-objective optimization is finding the Pareto optimal solution set, which was improved by Deb et al. based on the NSGA algorithm. By improving the NSGA-II algorithm and combining tournament selection with mixed linear ranking and uniform multi-parent crossover, the DMPNSGA-II algorithm was proposed.

[0124] Encoding rules

[0125] A timetable optimization model is constructed using Gray coding. Its core mechanism lies in the fact that the binary mapping of adjacent integer values ​​differs by only a single bit, effectively avoiding the Hamming cliff effect of traditional binary coding. The model maps the bus departure interval parameter to a chromosome gene sequence, with the first bus departure time serving as the initial scheduling baseline. Subsequent timetables are dynamically generated through an iterative optimization mechanism. Figure 2 As shown, a single chromosome represents the interval between adjacent trains on a specific route. When the route's operating interval is adjusted from 11 minutes to 12 minutes, the chromosome only needs to trigger a single position flip, improving the neighborhood search efficiency of the genetic operation. This encoding strategy reduces the invalid gene mutation rate, enabling the algorithm to effectively avoid local optimum traps during solution space exploration.

[0126] Selection, crossover, mutation:

[0127] Tournament selection with mixed linear rankings

[0128] Tournament selection method refers to randomly selecting individuals from the population each time. Individuals are compared, and the individual with the best fitness is selected to enter the offspring population. Then, it is returned to the original population, keeping the population size unchanged, and the selected individual is added to the new population. This process is repeated until the new population reaches the original population size. This is called the scale of competition, generally If the value is 2, the tournament is a binary tournament.

[0129] The linear ranking selection method aims to overcome the shortcomings of fitness-based proportional selection strategies. Assume... To determine the population size, in linear ranking selection, individuals in the population are first sorted according to their fitness values ​​from smallest to largest. Right now It is the individual with the lowest fitness value. The individual with the highest fitness value is selected, and then the selection probability of each individual is assigned according to a linear function based on the individual's ranking.

[0130] Assume the individual with the smallest fitness value in the current population The expected quantity after the selection operation is: That is, The individual with the highest fitness value The expected quantity after the selection operation is: That is, .in, , They are respectively , The probability of choosing, , It is a pre-specified constant. The expected number of other individuals is calculated according to an arithmetic sequence, that is, if we let Then the expected number of the i-th individual is:

[0131]

[0132] Therefore, the individual The probability of selection is

[0133]

[0134] in, The probability of the worst individual's choice. The probability of choosing the best individual.

[0135] Combining linear ranking selection with tournament selection, a hybrid linear ranking tournament selection method is proposed. Tournament selection focuses more on the direct competition between individuals and is based on local competition. Each selection only considers a small number of individuals participating in the tournament, lacking full utilization of global information of the entire population. This may cause the algorithm to miss some potential global optima located in other areas of the population during the search process. In contrast, when a "super individual" with fitness much higher than other individuals appears in the population, the probability of that individual being selected increases significantly. This may lead to the super individual quickly dominating the population, eliminating the genes of other individuals, causing a sharp decline in population diversity and premature convergence.

[0136] The tournament selection algorithm, which combines hybrid linear ranking, maintains diversity in the early stages of the search, extensively exploring the solution space, and converges quickly to the vicinity of the optimal solution in the later stages, preventing premature convergence and stagnation. The combination of these two approaches simultaneously considers an individual's global ranking and local competitive advantage, providing a more comprehensive assessment of individual fitness and improving the accuracy and effectiveness of the selection.

[0137] Tournament selection for mixed linear ranking, first with a certain probability Linear ranking selection is performed, choosing a subset of parent individuals from the population according to their linear ranking. Then, for the remaining unselected individuals, a probability-based selection process is used... A tournament selection process is used to supplement parent individuals, thus combining the two selection methods.

[0138] Uniform multi-parent crossover

[0139] Traditional crossover operations typically involve two parent individuals, exchanging partial genes to produce offspring. This two-parent crossover approach can sometimes limit the exploration of the solution space. Multi-parent crossover, on the other hand, utilizes the genetic information of three or more parent individuals simultaneously to generate offspring. This aims to integrate the superior genes of more individuals, increasing population diversity and the search space, allowing offspring to inherit the advantageous characteristics of multiple parents, and improving the algorithm's likelihood of finding a better solution. The designed uniform multi-parent crossover uses a randomly generated 0-1 sequence of the same length as the chromosome as a mask sequence. 1 indicates that the gene locus is selected from the parent, and 0 indicates that it is not selected. The mask sequence is divided into three parts, controlling the crossover between parent 1 and parent 2, parent 1 and parent 3, and parent 2 and parent 3, respectively, as follows: Figure 3 As shown. Compared to single-point crossover and multi-point crossover, it is not limited to exchanging genes at a few crossover points, but considers genes from different parents evenly across the entire chromosome. Therefore, it is more global in searching the solution space and can explore the solution space more extensively.

[0140] Uniform variation

[0141] Uniform mutation effectively enhances population diversity and global exploration capabilities by randomly resetting gene values ​​to any value within the feasible region with a fixed probability. It exhibits broad applicability in both continuous and discrete optimization problems, particularly suitable for multimodal function optimization and high-dimensional search spaces.

[0142] Algorithm Flow

[0143] The steps of the improved NSGA-II algorithm are as follows:

[0144] Step 1: Set parameters. These include: population size, maximum number of iterations, crossover probability, mutation probability, mixed selection probability, tournament competition size, etc.

[0145] Step 2: Initialize the population. Randomly generate an initial population within the solution space.

[0146] Step 3: Fitness Calculation and Multi-Objective Evaluation. Calculate the objective function value for each individual, and perform non-dominated ranking and crowding calculation based on the objective value.

[0147] Step 4: Mixed Selection Operation. A mixed linear ranking tournament selection is used to generate the parent population. First, linear ranking selection is performed with probability P, sorting individuals according to fitness and assigning selection probabilities. The remaining individuals are replenished through tournament selection, randomly selecting k individuals each time, and choosing the best one to join the parent generation.

[0148] Step 5: Uniform Multiple-Parent Crossover. Randomly select 3 individuals from the parent population and perform uniform multiple-parent crossover. Generate a masked sequence (0-1 random sequence) to determine the gene source of the offspring. Select gene loci from different parents according to the masked sequence to generate new offspring.

[0149] Step Six: Mutation Operation. Perform a uniform mutation operation on the offspring individuals.

[0150] Step 7: Merge the population and retain elites; merge the parent and offspring generations into a temporary population of size 2N. Perform non-dominated sorting and crowding calculation on them. Select the top individuals according to sorting level and crowding to form a new generation population of size N.

[0151] Step 8: Iteration termination judgment; if the current iteration count has reached the maximum iteration count, output the Pareto optimal solution set; otherwise, return to step 3 to continue iterating.

[0152] Step 9: Output the results; extract the Pareto front solution set, and the decision-maker selects the optimal timetable scheme according to actual needs; the improved NSGA-II algorithm process is as follows: Figure 4 As shown.

[0153] Algorithm verification

[0154] The ZDT test set is a widely used benchmark problem for multi-objective optimization, used to evaluate and compare the performance of multi-objective optimization algorithms. Three functions, ZDT1, ZDT2, and ZDT3, were selected (see Table 1) to validate the standard non-dominated sorting genetic algorithm (NSGA-II), the adaptive crossover and uniform mutation NSGA-II (DCNSGA-II), and the improved genetic algorithm DMPNSGA-II, respectively. Algorithm parameters were set as follows: population size 100, maximum number of iterations 300, crossover probability 0.9, and mutation probability 0.05.

[0155] Table 1 ZDT Test Set

[0156]

[0157] The ZDT series of test functions, as benchmark problems in the field of multi-objective optimization, possess Pareto front characteristics that are of significant research value: ZDT1 exhibits a continuous convex solution set, ZDT2 displays a concave distribution, and ZDT3 exhibits a discontinuous convex structure. Algorithms are compared by setting the same maximum number of iterations. Figure 5 Experimental data show that, compared with the standard NSGA-II algorithm, the improved DCNSGA-II and DMPNSGA-II algorithms exhibit significant advantages in solution set convergence and distribution uniformity. Specifically, the DCNSGA-II algorithm incorporates an adaptive crossover operator and a uniform mutation strategy, dynamically adjusting genetic operation parameters to improve convergence efficiency while effectively avoiding premature convergence. The DMPNSGA-II algorithm is further optimized with a multi-parent uniform crossover mechanism and integrates a tournament selection strategy based on hybrid linear sorting. This dual improvement strategy significantly enhances population diversity in the early stages of iteration to expand the solution space exploration range, and strengthens local search capabilities in the later stages of optimization, achieving fast and accurate convergence. The results show that the Pareto front obtained by the improved algorithms not only closely approximates the theoretical optimal solution set, but also improves the convergence accuracy and coverage density of its solution set distribution. This fully verifies the effectiveness and applicability of the improved algorithms in solving multi-objective optimization problems.

[0158] In the description of this specification, references to terms such as "an embodiment," "example," "specific example," etc., indicate that a specific feature, structure, material, or characteristic described in connection with that embodiment or example is included in at least one embodiment or example of the invention. In this specification, illustrative expressions of the above terms do not necessarily refer to the same embodiment or example. Furthermore, the specific features, structures, materials, or characteristics described may be combined in any suitable manner in one or more embodiments or examples.

[0159] The preferred embodiments of the present invention disclosed above are merely illustrative of the invention. These preferred embodiments do not exhaustively describe all details, nor do they limit the invention to the specific implementations described. Clearly, many modifications and variations can be made based on the content of this specification. This specification selects and specifically describes these embodiments to better explain the principles and practical applications of the invention, thereby enabling those skilled in the art to better understand and utilize the invention. The invention is limited only by the claims and their full scope and equivalents.

Claims

1. A method for optimizing high-speed rail shuttle bus timetables based on an improved NSGA-II algorithm, characterized by: Includes the following steps: Step S1: Clarify the basic parameters and underlying assumptions of the high-speed rail transfer bus system; Step S2: Construct a multi-objective optimization model with the goal of minimizing passenger waiting time and bus company operating costs, while satisfying constraints on departure intervals, route length, and departure frequency; Step S3: The improved NSGA-II algorithm, which combines tournament selection with a hybrid linear ranking strategy and uniform multi-parent crossover, is adopted. The population is initialized with Gray encoding, and the Pareto optimal solution set is obtained through iterative operations such as fitness calculation, selection, crossover, mutation, population merging and elite retention. Step S4: Extract the optimal solution from the obtained Pareto optimal solution set; Step S5: Generate the departure timetable for each connecting bus route based on the optimal solution.

2. The method for optimizing high-speed rail shuttle bus timetables based on the improved NSGA-II algorithm according to claim 1, characterized in that, In step S1, the basic parameters include high-speed rail arrival time, passenger flow for each line transfer, distance between the connecting bus stop and the high-speed rail platform, distance of the connecting bus line, number of bus departures, unit energy cost of buses, unit labor cost, upper and lower limits of bus departure intervals, maximum number of bus departures, average walking speed of passengers, and so on. The first high-speed rail line transfer The number of passengers on the connecting bus routes, specifically based on the following assumptions: Step S11: Buses strictly follow the established routes, with arrival time deviations controlled within ±1 minute, and no instances of buses crossing routes. Buses travel at a constant speed within the dedicated bus lanes, unaffected by traffic lights at intersections or other traffic interference. Step S12: All buses used for transportation are of the same model, have a fixed passenger capacity, and have consistent vehicle performance and energy consumption characteristics. Cross-line dispatching can be achieved within the station. After completing the current task, the vehicle is immediately put into operation for the next trip, with no empty running intervals. Step S13: Vehicles will only stop at designated stations on each route and are prohibited from picking up or dropping off passengers en route. Step S14: All passengers arriving by high-speed rail walk to the bus stop through a dedicated passage. The walking speed follows a normal distribution. Passengers choose to take the first bus available and do not abandon their journey to transfer. Step S15: The high-speed rail arrival timetable and corresponding transfer passenger flow are known and fixed values, with no sudden delays or passenger flow fluctuations.

3. The method for optimizing high-speed rail shuttle bus timetables based on the improved NSGA-II algorithm according to claim 1, characterized in that, In the multi-objective optimization model of step S2, the objective function for minimizing passenger waiting time is constructed as follows: Step S21: Calculation of the time it takes for passengers to walk from the high-speed rail platform to the bus stop: ,in This refers to the time it takes for passengers to walk from the high-speed rail platform to the bus stop. The distance between bus stops and high-speed rail platforms is considered. The average walking speed of passengers; Step S22: Calculation of waiting time for a single passenger: ,in For waiting time, For the first The arrival time of the shuttle bus at the station. For the first The arrival time of the high-speed train at the platform, The indicator function has the following formula: ; Step S23, Objective function for total passenger waiting time: ,in The total waiting time for passengers. This refers to the number of high-speed rail lines. The number of bus routes that provide transportation. To determine the number of bus departures for transportation, For the first The first high-speed rail line transfer The number of passengers on each connecting bus route.

4. The method for optimizing high-speed rail shuttle bus timetables based on the improved NSGA-II algorithm according to claim 1, characterized in that, In step S3, the objective function of the multi-objective optimization model, which minimizes the operating costs of the public transport company, is constructed as follows: Step S31, Bus operating cost: ,in For driving costs, For the first The distance of the connecting bus routes, The unit energy cost of buses; Step S32, Fixed depreciation cost of buses: ,in For depreciation expenses, This refers to the unit depreciation cost of buses. Step S33, Bus driver's wages and expenses: ,in For labor costs, This refers to the unit cost of labor. Step S34, Objective function for total operating cost of public transport company: ,in This represents the total operating cost of the public transportation company.

5. The method for optimizing high-speed rail shuttle bus timetables based on the improved NSGA-II algorithm according to claim 1, characterized in that, In step S4, the specific constraints are as follows: Step S41, Departure Interval Constraints: ,in For the first The departure interval of each connecting bus route. To preset the minimum bus departure interval, To preset the maximum bus departure interval; Step S42, Line length constraint: ,in For the first The distance of the connecting bus routes, and These are the upper and lower limits of the line length; Step S43, Departure Number Constraints: ,in This is the preset maximum number of bus departures.

6. The method for optimizing high-speed rail shuttle bus timetables based on the improved NSGA-II algorithm according to claim 1, characterized in that, In step S5, the solution steps of the improved NSGA-II algorithm specifically include: Step S51: Set algorithm parameters, including population size, maximum number of iterations, crossover probability, mutation probability, mixed selection probability, and tournament competition size; Step S52: Initialize the population: Randomly generate an initial population in the solution space, and use Gray coding to map the departure interval parameter of the shuttle bus to a chromosome gene sequence, with the departure time of the first bus as the initial scheduling benchmark. Step S53, Fitness Calculation and Multi-Objective Evaluation: Calculate the objective function value for each individual. and Non-dominated sorting and crowding calculation are performed based on the target value; Step S54, Tournament selection operation with mixed linear ranking: Perform linear ranking selection with mixed selection probability, and the remaining individuals are selected through tournament selection until the parent population size reaches the preset size. Step S55, Uniform multi-parent crossover operation: Randomly select multiple individuals from the parent population as parents, generate a masking sequence with the same length as the chromosome, and select gene loci from different parents according to the masking sequence to generate new offspring; Step S56, Uniform mutation operation: Randomly reset the gene values ​​of offspring individuals according to the mutation probability, and the reset values ​​are within the feasible region; Step S57, Merging Populations and Elite Preservation: Merge the parent and offspring populations into a temporary population, perform non-dominated sorting and crowding calculation on the temporary population, and select the top few individuals to form a new generation population. Step S58, Iteration Termination Judgment: If the current iteration count reaches the maximum iteration count, output the Pareto optimal solution set; otherwise, return to step S3 to continue iterating.

7. The method for optimizing high-speed rail shuttle bus timetables based on the improved NSGA-II algorithm according to claim 6, characterized in that, The initial population uses Gray encoding, whose encoding rules ensure that the binary mapping of adjacent integer values ​​has only a single bit difference. When the line operation interval is adjusted, the chromosome only needs to trigger a single bit flip.

8. The method for optimizing high-speed rail shuttle bus timetables based on the improved NSGA-II algorithm according to claim 6, characterized in that, In the tournament selection operation of the hybrid linear ranking, the selection probability of an individual is linearly related to its ranking in the population, with individuals with better fitness having a higher selection probability.

9. The method for optimizing high-speed rail shuttle bus timetables based on the improved NSGA-II algorithm according to claim 6, characterized in that, In the uniform multi-parent crossover operation, the masking sequence is a randomly generated 0-1 sequence, where 1 indicates that the gene locus is selected from the corresponding parent and 0 indicates that the corresponding gene locus of the parent is not selected; the crowding degree is calculated as follows: for each individual in the non-dominated layer, the distance between it and its neighboring individuals in the target space is calculated, and this distance is used to measure the crowding degree of the individual.

10. The method for optimizing high-speed rail shuttle bus timetables based on the improved NSGA-II algorithm according to claim 6, characterized in that, The method for extracting the optimal solution from the Pareto optimal solution set is as follows: based on actual operational needs, select the Nash equilibrium solution from the solution set that balances passenger waiting time with enterprise operating costs.

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