Collaborative Optimization Model for the Design and Operation of Feeder Bus Routes Based on Urban Metro

Lai, Yuanwen; Chen, Yansheng

doi:https://doi.org/10.1155/2023/6987787

Journal of Advanced Transportation

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Abstract Introduction Conclusions Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2023 | Article ID 6987787 | https://doi.org/10.1155/2023/6987787

Collaborative Optimization Model for the Design and Operation of Feeder Bus Routes Based on Urban Metro

Yuanwen Lai¹and Yansheng Chen¹

Academic Editor: Rocío de Oña

Received08 Oct 2022

Revised28 Feb 2023

Accepted07 Apr 2023

Published19 Apr 2023

Abstract

The collaborative development of conventional buses and urban metro has become an important research topic for the priority development of urban public transport. The topic of collaborative optimization of feeder bus route design and operation is studied in this study. The objective function is to minimize the total travel time of passengers and the operation cost of feeder buses. The improved particle swarm optimization (PSO) algorithm is used to solve the collaborative optimization model, and the effectiveness of the model and algorithm is verified through the case study. The research shows that it is feasible in model construction and algorithm to carry out collaborative optimization of feeder bus route design and operation. Compared with the multiple-to-one (M to 1) mode, the multiple-to-multiple (M to M) mode can better satisfy the needs of passengers from different places of departure and destinations to achieve a more reasonable and realistic goal. The case study is based on two metro stations and 16 feeder bus stops on Fuzhou Metro line 2 to obtain two bus routes and a corresponding operation scheme. Under the same topology road network, the operation time of the improved PSO algorithm is much shorter than the DFS algorithm, the total cost error of the feeder bus is 0.04%, and the departure frequency error is 4.6%, which is within the reasonable error range. Therefore, the collaborative optimization model proposed in this study is feasible and effective in optimizing the feeder bus routes and operation.

1. Introduction

1.1. Background

In recent years, with the rapid increase in the number of motor vehicles, many cities have run into traffic congestion, including small-, medium-, and large-sized cities [1]. More and more experience all over the world has verified that giving priority to the development of public transit is an effective means to alleviate urban traffic congestion [2]. The development of public transport is a crucial way to satisfy the travel needs of passengers, especially in urban areas. Urban metro is a popular choice for high-capacity public transportation systems in urban areas. It is typically used in urban areas for transporting large numbers of passengers over small distances at high frequencies. It is usually preferred over other transportation modes due to its advantages [3]. Traveling through metro mass transit is 20% cheaper [4]. However, compared with the urban metro, although the conventional bus has a small capacity and slow speed, it has the advantages of low cost, vital accessibility, and comprehensive coverage [5]. Therefore, studying the design and operation optimization of the feeder bus routes can help build a perfect feeder bus service system, give full play to their advantages, expand cooperation with the urban metro, and reduce competition. It can also create convenient conditions for urban residents to travel and attract more urban residents to choose buses or the urban metro. Optimizing the structure of residents’ travel modes and alleviating urban traffic congestion are significant.

1.2. Literature Review

Regarding the design of feeder bus routes, Wei et al. [6] developed a novel bilevel programming model for designing feeder bus routes to simultaneously consider bus stop selection, bus routing, and passenger route/trip choice behavior. Jiang et al. [7] established a model for travel time and accessibility and analyzed the design of bus routes for passengers with different needs. Guo [8] proposed a bus route design method for connecting high-speed railway stations and designed an algorithm to solve the model of the feeder bus routes. Wei et al. [9] proposed a mathematical model for designing a feeder transit service to improve the service quality and accessibility of transportation hubs (such as airports and rail stations). Ma et al. [10] proposed a three-stage hybrid coding method based on the NSGA-II algorithm to optimize customized bus routes under uncertain conditions. Recently, feeder bus routes’ optimization model and algorithm were studied [11–13]. Among them, the leading idea was the heuristic algorithm [11, 14, 15]. To solve the problem of a better connection between the feeder bus and the metro, Zheng et al. [16] built a model to quantify the feeder demand at each bus stop. They proposed an optimization approach based on the tabu search (TS) algorithm. Gao and Shi [17] established a mathematical model for optimizing rail transit bus routes, maximizing feeder bus routes’ passenger turnover as the objective function and solving it using the PSO algorithm. Deng et al. [18] proposed a new split delivery model using the genetic algorithm (GA) for the feeder bus network design based on the transfer network. Taplin and Sun [15] and Li et al. [19] considered the feeder bus route and passenger walking distance and used a GA to solve the optimization model of the feeder bus route.

Regarding the optimization of feeder bus operation, Yuan et al. [20] established an optimization model for the timetable and bus dispatching system by considering rigid constraints, such as the maximum number of available buses, the maximum bus departure interval, and the minimum bus departure interval. Li et al. [21] proposed a dynamic optimization model of bus route schedules based on bus integrated circuit card (IC Card) data. A dynamic departure interval optimization method based on an improved GA was designed under different decision preferences. Dou and Meng [22] constructed a mixed integer nonlinear programming (MINLP) model for the feeder bus schedule problem. They developed a hybrid artificial bee colony (ABC) algorithm to solve the model. Xiong et al. [23] took passengers’ minimum travel time delay as the goal. They applied the GA and Frank−Wolfe (FW) algorithms combined with a departure time adjustment algorithm to solve the optimization model of the feeder bus schedule. Verbas and Mahmassani [24] took the minimum waiting time as the objective function and the number of vehicles and headway as constraints. They applied a heuristic algorithm to optimize the departure frequency. Hu et al. [25] constructed a double objective optimization model to maximize the one-way coordination of the local area network, starting from the optimization of the subnetwork, and used the heuristic algorithm to obtain the conventional bus schedule coordinated with the metro line. Martínez et al. [26] proposed using the mixed integer linear programming (MILP) model to optimize the frequency of route departures, using MILP software to solve the exact solution of the small-scale line network problem and the TS algorithm to solve the large-scale line network problem.

Although research studies have conducted some research studies on the design of feeder bus routes and the optimization of feeder bus route operation, most have studied them as independent steps to reduce the model’s complexity and improve the calculation speed [27–29]. There are few studies on the collaborative optimization of feeder bus route design and feeder bus route operation. In addition, the existing research shows that the passenger flow mode is mostly the multiple-to-one (M to 1) mode. Passengers arrive at a given metro station from multiple candidate feeder bus stops. As for the multiple-to-multiple (M to M) mode, each candidate feeder bus stop has an origin-destination (OD) relationship with multiple urban metro stations. There are few relevant studies. The closeness between the model and the actual situation needs to be improved. Given the above gaps, this study contributes to the literature in the following ways:(1)The existing literature studies the design and operation of feeder bus routes as two independent parts, while this study optimizes the design and operation of feeder bus routes simultaneously.(2)The collaborative optimization model proposed in this study will be extended to the multiple-to-multiple model between metro stations and feeder bus stops in the feeder service area. Breaking through the previous optimization model, which primarily considers multiple-to-one mode, makes the optimization model more consistent with the actual passenger flow demand distribution.(3)The transfer coordination constraint is considered in the constraint conditions. The departure time of the feeder bus routes is adjusted by considering the arrival time of the urban metro.

2. Problem Description and Model Assumptions

2.1. Problem Description

The feeder bus routes based on the urban metro have the following characteristics: A bus route serves multiple bus stops, such as residential areas, workplaces, schools, hospitals, transportation hubs, and other major passenger attraction points. The feeder bus route is composed of multiple feeder bus stops that form a route candidate stop set, and several of them need to be selected as feeder bus stops according to the travel demand of passengers. Among the selected feeder bus stops, the closest metro station is the bus transfer station, and passengers can complete the transfer between the urban metro and the feeder bus at this transfer station.

Passengers take the feeder bus to transfer to the urban metro for travel. The time of taking the feeder bus, the transfer time, and how to effectively link the departure time of the feeder bus with the arrival time of the urban metro are the key factors to improving the attractiveness of the entire travel chain. At the same time, the operation cost of feeder bus routes is also a critical factor in stimulating public transport enterprises to improve their services. Therefore, it is necessary to collaboratively optimize the design and operation of feeder bus routes under careful consideration of passenger travel time and operating costs of public transport enterprises.

2.2. Assumptions

For the convenience of modeling and solving, the following model assumptions are made in this study:(i)Each feeder bus stop only passes through one feeder bus route, and each feeder bus route only serves a metro station.(ii)The average speed of feeder buses and the average walking speed of passengers are known.(iii)The passenger flow arriving at each feeder bus stop follows the Poisson distribution.(iv)The departure interval and the number of feeder buses and metros remain unchanged.

2.3. Parameter Definition

The notations used throughout this study are listed in Table 1.

3. Model Development

3.1. Constraints

3.1.1. Integrity Constraints for Feeder Bus Routes

Each feeder bus route shall at least include a metro station and a feeder bus stop (see equation (1)).

3.1.2. Number of Bus Vehicles Available for Feeder Bus Routes

The frequency of the feeder bus route is limited by the total mileage of the bus vehicles within a given range (see equation (2)).

3.1.3. Passenger Demand Constraint

The capacity of the feeder bus should meet the passenger travel demand of each feeder bus station (see equation (3)).

3.1.4. Vehicle Carrying Capacity of the Feeder Bus Route

For the feeder bus route, the number of passengers transferring from the feeder bus to the urban metro can be at most the capacity of the feeder bus (see equation (4)).

3.1.5. Service Time of Feeder Bus Route

For the feeder bus route, the service time needs to meet the needs of the connecting service within a certain service level (see equation (5)).

3.1.6. Length of Feeder Bus Route

The feeder bus route’s length shall meet the feeder bus’s actual operation (see equation (6)).

3.1.7. Departure Frequency of Feeder Bus

The feeder bus route’s departure frequency shall meet the feeder bus’s actual operation (see equation (7)).

3.1.8. Transfer Coordination

For the feeder bus route, the time spent by passengers transferring from the feeder bus to the metro should be within the arrival and departure times of the metro train at the metro station (see equation (8)). The time-space diagram of the feeder bus to the metro is shown in Figure 1.

3.2. Analysis of the Total Travel Time of Passengers and the Operation Cost of Feeder Buses

3.2.1. Total Travel Time of Passengers

The total travel time of passengers includes the waiting time of passengers, the boarding and alighting time of passengers at the feeder bus stops, and the travel time between feeder bus stops and the transfer time of passengers from the feeder bus stops to the metro stations.

The waiting time of passengers is shown in equation (9). The boarding and alighting times of passengers at feeder bus stops are shown in equation (10). The travel time between feeder bus stops is shown in equation (11). The transfer time for passengers is shown in equation (12). The total travel time of passengers is shown in equation (13).

3.2.2. The Operation Cost of Feeder Buses

In terms of feeder bus operation, the operating cost of feeder buses is mainly considered. The operating costs of feeder buses include fuel costs, labor costs, and depreciation costs, which are positively related to the operating mileage of each feeder bus route. The operation cost of the feeder bus route is obtained by multiplying the total operating mileage of the feeder bus by the operating cost per unit mileage (see equation (14)).

3.3. Collaborative Optimization Model

In this study, the objective function is to minimize the total travel time of passengers and the operation cost of feeder buses (see equations (15) and (16)).

The objective function (see equations (15) and (16)) and the constraint (see equations (1)–(8) constitute the collaborative optimization model of the feeder bus.

3.4. Transformation of the Double Objective Function

In order to facilitate the solution, the double objective function in the above model is converted into a single objective function, which is divided into two steps:

Step 1. By increasing the average travel time value of passengers , the objective function is transformed into the passenger travel cost , which is unified with the operating cost dimension of the objective function (see equation (17)).

Step 2. The weighted combination method is applied. According to the importance of each influencing factor of the optimization objective, the weights of the objective function and are assigned corresponding weights, respectively. The double objective function is converted into a single objective function. The weight of the objective function, considering passenger travel cost is taken as , and the weight of the feeder route operation cost is taken as (see equation (18)).

4. Improved Particle Swarm Optimization Algorithm

Particle swarm optimization (PSO), derived from complex adaptive system (CAS), is a random search algorithm based on group cooperation developed by simulating the foraging behavior of birds. This algorithm has attracted the attention of academic circles due to its advantages of easy implementation, high accuracy, and fast convergence, and it has shown its advantages in solving practical problems [30–32].

This study uses the improved PSO algorithm to solve the collaborative optimization model for the following reasons. The search for the PSO algorithm is efficient and can be adjusted in the search process, which is conducive to finding the optimal solution under fitness. In addition, it has good universality and is suitable for setting the objective functions and constraints of the model in this study. The specific solution flow chart of the improved PSO of the collaborative optimization model is shown in Figure 2. Step 1: The particle swarm is initialized with the given population size and the number of all candidate feeder bus stops. Then randomly select the position and speed of each particle. The population size is defined as , and the position and speed of each particle are defined as and , respectively. is described as the longitude and latitude coordinates of the candidate feeder bus stops under the feeder bus route, and is described as the dispatching sequence under the feeder bus route. Step 2: Each particle compares the current fitness with the fitness corresponding to the best position it has experienced. The best position of the particle is defined as . If it is better than the latter, update the function value of . Otherwise, it will remain unchanged. Compare the fitness of each particle in this iteration with the fitness of experienced by its subgroup. If it is better than the latter, update the function value of . Otherwise, it will remain unchanged. Finally, compare the fitness of each particle in this iteration with the best fitness experienced by the whole population. If it is better than the latter, update the function value of . Otherwise, it will remain unchanged. Step 3: Updating the particle speed and position of each particle according to equations (19) and (20): Step 4: Satisfying the number of iterations or the minimum error. The optimal solution is output when the maximum number of iterations or the minimum error is met. Otherwise, return to Step 2.

5. Case Study

5.1. Case General Situation

This study takes Juyuanzhou station and Jinshan station of Fuzhou Metro line 2 in Fuzhou, China, as research examples to analyze and verify the model’s validity. The study area is 6.67 square kilometers. Within the research scope, there are two metro stations, 16 candidate feeder bus stops, and two bus transfer stations. Baidu Map is a free map tool focusing on Chinese geography, providing the location information of Fuzhou metro stations and feeder bus stops. The topological relationship between metro stations and feeder bus stops is shown in Figure 3. The path length between feeder bus stops is shown in Table 2.

According to the land use types around Juyuanzhou station and Jinshan station, as well as the travel rate indicators for Fuzhou, the travel demand of each feeder bus stop is obtained by using TransCAD software and combining the traffic survey data, as shown in Table 3.

Based on the field survey and the information officially released by Fuzhou Metro Group Co., Ltd, the morning rush hour in Fuzhou is from 8:00 to 9:00. Table 4 shows the arrival times of metro trains at Juyuanzhou station and Jinshan station during the period from 8:00 to 9:00.

According to the existing feeder bus operation standards in Fuzhou, the average speed of feeder bus operation is 20 km/h, and the feeder bus’s operating cost per unit mileage is 30 CNY/km. The salary level report for Fuzhou (2021) shows that the passengers’ average travel time value is 26 CNY/hour. The feeder bus mainly provides travel services for urban residents. Therefore, the weight of the objective function of passenger travel cost is 0.6, and the weight of the operation cost of feeder buses is 0.4 [33]. The values of other relevant parameters in the model are shown in Table 5.

5.2. Model Solving

The improved PSO algorithm is programmed with MATLAB. The initial population size is 50, and the number of iterations is 100. The final result is obtained, as shown in Figure 4. The optimal feeder bus routes are shown in Figure 5.

(a)

(b)

According to the solution results in Figure 5, the feeder bus routes are drawn, and the optimal feeder bus routes are shown in Table 6.

According to the optimal feeder bus routes (Figure 5) and the optimal departure frequency (Table 6), the feeder bus route schedule is obtained, as shown in Table 7.

5.3. Validation

In order to prove the feasibility and accuracy of the collaborative optimization model, the depth-first search (DFS) algorithm is applied to traverse all feasible paths of metro stations (M1 and M2) under the topological road network. It calculates the total cost value and the corresponding departure frequency value of all paths that meet the constraints and selects them to compare with the model solution results, as shown in Table 8.

Table 8 shows the optimal path obtained by the improved PSO and DFS algorithms is the same. The operation time of the improved PSO algorithm is much shorter than the DFS algorithm. The total cost error of the feeder routes is 0.04%, and the departure frequency error is 4.6%, which is within a reasonable range. Therefore, the collaborative optimization model established in this study can obtain the optimal feeder bus routes and feeder bus route operation.

6. Conclusions

The collaborative development of conventional buses and urban metro has become an important research topic for the priority development of urban public transport. This study optimizes the design and operation of feeder bus routes simultaneously, studies the collaborative optimization of feeder bus route design and feeder bus route operation, and proves that the model construction and algorithm are feasible.

With in-depth research on the feeder bus, the multiple-to-one mode can no longer satisfy the travel needs of passengers [34], and the multiple-to-multiple mode emerges as the times require. Therefore, the study of the multiple-to-multiple mode in this study can achieve a more reasonable and realistic purpose of satisfying the needs of passengers from different origins and destinations.

In this study, the transfer coordination constraint is considered. The departure time of the feeder bus routes can be dynamically adjusted according to the actual arrival time of the urban metro, which can realize the synchronous transfer between the feeder bus and the urban metro.

Although the study area selected in the case study in this study is a small area, with few feeder bus stops and metro stations considered, the study’s general rules of theoretical modeling and algorithm design are still applicable to larger and more complex networks. Our research results apply to optimizing existing feeder bus routes and operation schemes and designing new feeder bus routes and operation schemes. In addition, it has good theoretical reference significance for developing and improving the planning theory of the metro-related feeder bus system.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The authors thank the Fuzhou Public Transport Group Co., Ltd., and Fuzhou Metro Group Co., Ltd., for providing some of the required data. They would also like to thank the teachers and graduate students of the traffic engineering department at Fuzhou University for their help.

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Copyright © 2023 Yuanwen Lai and Yansheng Chen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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