Optimization Model and Solution Algorithm for Rural Customized Bus Route Operation under Multiple Constraints

Abstract

In order to improve the operational efficiency of public transportation systems in rural areas, we investigated the demand-responsive rural customized bus vehicle route optimization problem. First, a two-stage planning model describing the problem in the reservation phase and real-time phase was constructed with the objectives of minimizing the operating cost of the operator and the travel time cost of the passenger, and the passenger time window, vehicle characteristics, rated passenger capacity and the running time of the route were considered in the constraints. Second, a hybrid algorithm solution model combining bat algorithm and adaptive particle swarm algorithm was designed to obtain a more optimal solution. Finally, the effectiveness of the hybrid algorithm on the optimization model was verified by using the actual road network in some townships of Jing’an County, Jiangxi Province, China, and the obtained objective function value was reduced by 5.5%. The results show that the optimization model and hybrid algorithm designed in this paper can be used to provide theoretical references for opening demand-responsive customized bus route operation schemes in rural areas.

1. Introduction

Rural areas are generally defined as areas outside of towns and cities, where the inhabitants are engaged in agriculture as their main source of livelihood. In terms of living functions, rural areas are home to a large number of people. Although many residents of rural areas elected urban employment and living during the urbanization process, nearly half of the population still lives in rural areas year-round, so there is a need to ensure basic living conditions and basic public services such as housing, transportation and environment for residents of rural areas. Due to the large geographical area, dispersed population, low mobility, low travel demand and dispersed travel locations in some rural areas of China, there is less demand for public transport, making it difficult to maintain traditional public transport services with fixed routes and frequencies, often resulting in loss of operation for operators, reduced service quality and substantial government subsidies. Therefore, for the development of public transport in rural areas and the sustainable development of social resources, there is an urgent need to develop a rural customized bus (CB) service model. This paper aims to provide a high-quality customized bus (CB) service in response to the personalized needs of passengers commuting between the county and rural areas.

This study aims to reduce the cost of both operating companies and passengers, construct a vehicle path model for both reservation and real-time stages, and design a reasonable and efficient customized bus route planning method, which has important theoretical and practical significance for operating demand-responsive customized buses in rural areas in the future.

2. Literature Review

2.1. Demand-Responsive Customized Bus Research

Customized bus (CB) is a form of public transport based on passenger demand response, providing transport service for passengers who make reservations for travel, with operators determining the route, transit stops and specific timetables of vehicles based on the actual travel needs of the passengers themselves. Cole [1] first proposed research on demand-responsive transport systems, which combines the characteristics of conventional buses and taxis. Derek [2] compared fixed-route transportation systems and demand-responsive feeder transit systems, indicating that demand-responsive transportation can be used to improve transit service levels in low-demand areas. Han [3] solved real-time response CB hierarchical scheduling by modifying the genetic algorithm (GA) and non-dominated sorting GA. Montenegro [4] developed a large neighborhood search heuristic in order to optimize a demand-responsive feeder service. Practical results from Italy show that demand response strategies are better adapted to low-density areas than fixed-route public transport services and perform better in terms of cost and performance [5]. In Germany, especially in sparsely populated rural areas, demand-responsive transport systems can provide flexible public transport services on demand within given time and space constraints [6].

2.2. Vehicle Route Optimization Research

In the process of route optimization, researchers have considered various influencing factors from different perspectives to construct the objective function and constraints of the model. From the passengers’ perspective, the main objective in a single-objective optimization model is to minimize travel time [7,8]; from the perspective of the operators, the main objective in a single-objective optimization model is to minimize the operating cost [9,10]. In a multi-objective optimization model, the objective function is mostly to optimize the total cost of both passengers and operators [11,12,13,14,15]. In addition, Huang [16] formulated a two-stage model to optimize the subscription bus services network design (SBSND) using the minimum total service distance and the number of vehicles as a single target objective function. Sun [17] proposed a mixed-integer linear programming model for demand-responsive feeder transport services with the objective of minimizing operating costs and maximizing passenger satisfaction.

With the continuous research on CB, in view of the dynamic period of the vehicle operation process, Huan [18] designed a dynamic genetic algorithm based on simulated annealing to generate the static initial travel path, and kept the dynamic travel paths updated to satisfy the real-time demand. Wang [19] proposed a two-step coordinated optimization model for mixed demand conditions with multi-vehicle responsive bus routes and built a rule-based insertion method. Park [20] presented a more economically efficient route segmentation method that could provide theoretical support for a future policy of replacing conventional bus routes with demand-responsive buses.

2.3. Vehicle Route Algorithm Research

In recent years, many scholars have proposed many different mathematical models for different types of vehicle scheduling problems, and many algorithms to obtain optimal or suboptimal solutions have been proposed. The route optimization problem of customized bus is actually a special kind of vehicle path optimization problem, which belongs to the non-deterministic polynomial problem, and the solution methods are usually divided into two types: exact algorithm and heuristic algorithm.

Exact algorithms are algorithms that use mathematical methods to process the model to find the optimal solution to the problem, which can be obtained in acceptable time when the problem is small in size, but generally face difficulty in finding the optimal solution when the problem is computationally complex and large in scale. When simple analysis is performed for a very small number of requests, the exact algorithm is usually used to solve the problem. Tong [11] provided a solution algorithm based on Lagrangian decomposition framework and also optimized routes and timetables with detailed space-time shortest path searching. Huang [21] established an exact solution method based on branch and bound to optimize the path of the static phase. Guo [22] solved a mixed integer model of customized bus routes considering space and time using branch and cut methods.

Considering that these exact solution algorithms usually require strict constraints and the computation time grows rapidly with the problem size, heuristic intelligent algorithms are generally chosen for solving practical problems with large-scale requirements. Shen [23] proposed a route optimization method based on the reliability shortest circuit, which uses a forbidden search algorithm to solve the vehicle route model. Peng [24] designed a hybrid simulated annealing algorithm to solve the optimization problem of customized feeder bus routes for intercity rail transportation. Both Lei and Liu [25,26] chose the improved ant colony algorithm to optimize the customized bus route network. Genetic algorithms improved by different methods have also been used extensively to solve customized bus routes, thus improving the operational efficiency of vehicles [27,28,29,30,31,32,33,34]. Hu [35] solved the linear model for bus line network optimization using particle swarm algorithm. Yu [36] established a line optimization algorithm based on hybridized particle swarm algorithm, which draws on the concept of hybridization in genetic algorithm to hybridize individuals and optimal individuals or population optimal particles to complete the information exchange between particles and improve the ability to find the best solution. Gao [37] improved the particle swarm algorithm by redefining the iterative formulas for the velocity and position of the particles. In this paper, the vehicle route model constructed by the demand-responsive custom bus connection in rural areas takes into account the costs of operating companies and passengers, and the related parameters and constraints are numerous and complex to solve. The exact algorithm is difficult to solve in the actual vehicle operation process, and the application scenario belongs to the large-scale situation, so it is suitable to use a heuristic algorithm to solve the vehicle route problem.

In summary, it can be seen that the common intelligent heuristic algorithms are forbidden search algorithm, genetic algorithm, simulated annealing algorithm, ant colony algorithm and particle swarm algorithm, all of which are suitable for dealing with large-scale problems. Each algorithm has its own advantages, disadvantages and applicable environment, as shown in

Table 1. Comparison of algorithms for common intelligent heuristics.

Algorithm	Convergence Speed	Operation Mechanism	Features
Taboo search algorithm	Slow	Search for the optimal solution in the domain of the initial solution	Get rid of the local optimum by the solution in the forbidden table, but there is strong dependence on the initial solution and insufficient diversity
Simulated annealing algorithm	Slow	Simulate the laws of physics and chemistry to find the global optimal solution of the objective function randomly	Easy to implement, can effectively avoid falling into local minima and eventually converge to the global optimum; the optimization process is longer
Ant colony algorithm	Fast	Use positive feedback mechanism to solve the optimal solution gradually	With global parallelism, high operational efficiency, but easy to fall into local optimum, slow convergence speed
Genetic algorithm	Slow	Simulate the phenomenon of gene mutation in natural genetic process and find the optimal solution	Simple process, globally searchable, scalable, but prone to premature convergence
Particle swarm algorithm	Faster	Find the optimal solution through collaboration and information sharing among individuals in the group	Simple and easy to implement, but weak global search capability

.

We built a hybrid algorithm through the comparative analysis of the above common intelligent heuristic algorithms, combined with the demand-responsive rural custom bus route operation research problem studied in this paper, with a certain degree of complexity, and the actual use of large-scale and real-time scenarios. The algorithm’s global search capability and convergence speed requirements were high, with comprehensive consideration of the particle swarm algorithm, the adaptive particle swarm algorithm and the bat algorithm to build a hybrid algorithm. The bat algorithm has fast convergence speed and strong global search ability in the early stage, but low convergence effect and easily falls into the local optimum in the late stage. Combining the BA algorithm with other intelligent algorithms can effectively avoid the immature convergence in the process. The particle swarm algorithm has high search accuracy, strong optimization ability and is easy to calculate, but the global search ability is weak. For the advantages and disadvantages of each of the two algorithms, a hybrid algorithm combining the adaptive particle swarm algorithm and the bat algorithm is proposed to solve the demand-responsive rural custom bus route problem.

This paper focuses on mixed travel demand in rural areas, i.e., having both reservation demand and real-time demand, and constructs a CB route optimization model that combines the operating costs of operators and the travel time costs of passengers, and considers multiple vehicle types. This avoids the problem of traditional fixed-point and fixed-route buses failing to efficiently meet the travel needs of people living in areas of weak passenger flow and rural areas. With flexible operational routes generated according to passenger pre-orders and real-time demand, CB offers a higher degree of freedom and a more flexible service, which helps to improve the quality of bus services in rural areas.

3. Definition of the Decision Problem

This paper proposes a demand-responsive rural CB service between the county and rural areas, with the county as a fixed stop and the administrative villages under the jurisdiction of some towns as dynamic stops, and one fixed stop and multiple dynamic stops in the entire road network. The demand-responsive rural bus routes examined in this paper have multiple flexible stops along the way, unlike the conventional concept of one-stop direct buses. Different from the fixed routes of conventional buses, CB routes are dynamically adjusted according to the travel information submitted by passengers, and passengers can make reservations for CB in advance according to the expected departure time. The system judges the submitted information, and if the information meets the requirements, it generates information such as service vehicle, service time and expected arrival time and sends this information to the passengers.

Considering that most bus parks in rural areas are set up in the center of the county, the rural CB needs to serve passengers from the county station to the rural station and passengers from the rural station to the county station, eventually parked in the county passenger station, and the same rural CB must finish serving passengers leaving the county station before it can serve passengers going from the rural station to the county station, as shown by the green line in Figure 1.

The paper takes into account both the bus operation perspective and the passenger perspective, and aims to minimize operating costs and passenger costs by implementing a two-stage demand-responsive custom bus route optimization model that includes constraints such as time windows and passenger capacity, and divides the CB route optimization design process into two stages: dynamic update and static system optimization. In the static stage, passengers make reservations for CB trips and submit their desired pick-up and drop-off stops and times to the operator; in the dynamic stage, passengers make real-time demands for rides before or during the operation of the vehicle, and the demand is inserted into the existing operating routes if the spatio-temporal constraints are satisfied. For demands that cannot be inserted into the existing routes, they are converted into reservation demands and are served by the next bus response, thus obtaining multiple sets of feasible CB transfer solutions.

4. Model Construction

4.1. Basic Hypothesis

Our basic hypotheses are as follows:

(1)

A county station and multiple rural stations exist in the study area, with vehicles all departing from the county station and returning to the county station after completing their transportation tasks.

(2)

Distance between stations is known, vehicles operate at uniform speeds, and there are sufficient vehicles.

(3)

The travel demand is randomly distributed, the reservation demand made by the passenger must be responded to, and the travel information of the passenger (station, time, number of passengers, etc.) is known.

(4)

To simplify the model, the time taken by passengers to get on and off the bus at the stops is not considered.

4.2. Vehicle Route Model for Reservation Demand

In this paper, we need to consider a bidirectional time-windowed vehicle path optimization problem for both the return and departure trips, by differentiating the stations to transform the above problem into a one-way time-windowed vehicle path optimization problem with 2n rural stations, and 2 county stations. The bus from the county town to the rural station $i\left(i=1,2,\cdots ,n\right)$ only provides drop-off service, and the number of boardings is 0; the bus from the rural station $j\left(j=n+1,n+2,\cdots ,2n\right)$ to the county town only provides boarding service, and the number of drop-offs is 0. Together they combine to form a set D with $2n+2$ stations, where the location of stations $1,2,\cdots ,n$ is the same as the location of station $n+1,n+2,\cdots ,2n$. Taking the reservation demand generated in the static stage as the research object, according to the passenger’s reservation station and the time window requirements, the total system cost is minimized by optimizing the coordination of each CB route under the constraints of meeting the passenger’s requirements for vehicle arrival time and vehicle capacity.

(1)

The total system cost $Z$ consists of two parts, the operating cost $F_1$ of the operator and the passenger travel time cost $F_2$, as follows:

$$Z={\omega }_1{F}_1+{\omega }_2{F}_2$$

(1)

where ${\omega }_1$ and ${\omega }_2$ are the weight factors for the operating cost of the operator and the cost of passenger travel time, respectively, and ${\omega }_1+{\omega }_2=1$.

(2)

The operating cost $F_1$ of the operator includes vehicle driving cost $f_1$ and vehicle usage cost $f_2$, and the relevant calculation formula is shown in Equations (2)–(4).

$$F_1=f_1+f_2$$

(2)

$$f_1={\displaystyle \sum _{i\in D}{\displaystyle \sum _{j\in D}{\displaystyle \sum _{k\in K}{\displaystyle \sum _{s\in S}{\alpha }_s\cdot x_{ij}^k\cdot d_{ij}}}}}$$

(3)

$$f_2={\displaystyle \sum _{s\in S}{\displaystyle \sum _{k\in K}{x}_s^k\cdot F_s^{Veh}}}$$

(4)

where $\alpha$ is the transport cost per unit distance of different types of vehicles and $d_{ij}$ is the shortest distance between site $i$ to site $j$. Additionally $x_{ij}^k\in \left\{0,1\right\}$, and if vehicle $k\left(k=1,2,\cdots ,K\right)$ travels from site $i$ to site $j$ one after another, then ${\mathrm{x}}_{ij}^k$ is 1; otherwise, it is 0. Moreover, $x_s^k\in \left\{0,1\right\}$, and if different types $s\left(s=1,2,\cdots ,S\right)$ of vehicles $k$ are used, then ${\mathrm{x}}_s^k$ is 1; otherwise, it is 0. $F_s^{Veh}$ is the cost of using different types of vehicles, mainly including driver wages, insurance and maintenance costs.

(3)

Passenger travel time cost $F_2$ is the passenger ride time cost, and the relevant calculation formula is shown in Equation (5).

$$F_2=\beta \cdot {\displaystyle \sum _{i\in D}{\displaystyle \sum _{j\in D}{\displaystyle \sum _{q\in Q}{t}_{ij}\cdot q_{ij}}}}$$

(5)

where $Q$ is the set of passengers in the study area who make a reservation for a trip, $q_{ij}$ is the number of passengers in the vehicle arriving at station $j$ from station $i$, $t_{ij}$ is the time from station $i$ to station $j$, and $\beta$ is the unit time cost factor of the passenger’s ride.

(4)

Vehicle path model for reservation demand.

The objective function of the vehicle path model for reservation demand is $\min (Z)$ and the constraints are as follows:

$${\displaystyle \sum _{j\in D}{x}_{ij}^k}=y_i^k,\forall i\in D,i\ne 0\mathrm{and}i\ne 2n+1;k\in K$$

(6)

$${\displaystyle \sum _{j\in D}{x}_{j(2n+1)}^k}=y_j^k,\forall j\in D,j\ne 0\mathrm{and}j\ne 2n+1;k\in K$$

(7)

$${\displaystyle \sum _{j\in D,j\ne 2n+1}{x}_{(2n+1)j}^k}=0$$

(8)

$${\displaystyle \sum _{j\in D,j\ne 0}{x}_{j0}^k}=0$$

(9)

$${\displaystyle \sum _{j\in D,j\ne 2n+1}{x}_{j(2n+1)}^k}=1$$

(10)

$${\displaystyle \sum _{j\in D,j\ne 0}{x}_{0j}^k}=1$$

(11)

$${\displaystyle \sum _{k\in K}{y}_i^k}={\displaystyle \sum _{k\in K}{y}_j^k}=1,\forall i\in D,i,j\ne 0\mathrm{and}i,j\ne 2n+1$$

(12)

$${{\displaystyle \sum _{i\in D}q}}_i^{-}{y}_i^k\le R_s,\forall k\in K$$

(13)

$${{\displaystyle \sum _{j\in D}q}}_j^{+}{y}_j^k\le R_s,\forall k\in K$$

(14)

$$u_i^k-u_j^k+(2n+2)x_{ij}^k\le 2n+1,\forall i,j\in D;k\in K$$

(15)

$$e_{q_j}\le t_j^k\le l_{q_j},\forall j\in D,k\in K$$

(16)

$$x_{ij}^k(t_i^k+t_{ij}-t_j^k)\le 0,\forall i,j\in D,j\ne 0\mathrm{and}j\ne i;k\in K$$

(17)

$$t_i^k-\left(1-y_i^k\right)M\le t_j^k+\left(1-y_j^k\right)M,\forall i,j\in D,i\le n\mathrm{and}j\ge n+1;k\in K$$

(18)

$$t_{\min }\le {\displaystyle \sum _{i\in D}{\displaystyle \sum _{j\in D}{x}_{ij}^k{t}_{ij}\le t_{\max }}},\forall k\in K$$

(19)

$$x_{ij}^k=\left\{0,1\right\};x_s^k=\left\{0,1\right\};y_{q_i}^k=\left\{0,1\right\};y_i^k=\left\{0,1\right\};q_i^k\ge 0,\forall i,j\in D;k\in K$$

(20)

where $y_i^k\in \left\{0,1\right\}$, and if station $i$ is served by vehicle $k$, then $y_i^k$ is 1; otherwise, 0. Moreover, $q_i^{-}$ is the number of drop-offs at station $i$; $q_j^{+}$ is the number of boardings at station $j$; $R_s$ is the rated capacity of different types of vehicles; $u_i^k$ is an auxiliary variable representing the order of station $i$ in the route of vehicle $k$; $t_i^k$ is the moment when vehicle $k$ arrives at station $i$; $[e_{q_i},l_{q_i}]$ is the desired boarding time window for passenger $q_i$, which is a soft time window; $e_{q_i}$ is the lower bound of the desired boarding time window for passenger $q_i$; $l_{q_i}$ is the upper bound of the desired boarding time window and the lower bound of the tolerable boarding time window for passenger $q_i$; $M$ is an infinite positive integer; $t_{\min }$ and $t_{\max }$ are the minimum and maximum time, respectively, that the vehicle is allowed to run; $q_i^k$ is the number of passengers in the vehicle when vehicle $k$ passes through station $i$; and $y_{q_i}^k\in \left\{0,1\right\}$, which is 1 if passenger $q_i$ is served by vehicle $k$ and 0 otherwise.

Equations (6) and (7) ensure the principle of continuity and flow conservation of vehicle k in the route. Equations (8)–(11) ensure that each vehicle’s route must start and end at the county station. Equation (12) ensure that stations with reservation demand must be served and only once. Equations (13) and (14) represent the vehicle capacity constraint, which ensures that the number of people on board the vehicle cannot exceed the rated passenger capacity during the journey. Equation (15) ensures ${\mu }_i^k+1={\mu }_j^k$ when $x_{ij}^k=1$ to avoid sub-circuits in the route. Equation (16) is a time window constraint. Equations (17) and (18) indicate that the vehicle serves passengers from the rural site to the county site later than it serves passengers going from the county site to the rural site, and that the vehicle arrives at the next site $j$ later than it departs from the current site $j$ during the journey. Equation (19) requires that the travel time for each route cannot exceed the maximum time specified nor be less than the minimum time.

4.3. Path Optimization Model for Real-Time Demand

This paper focuses on real-time travel demands made by travelers in the dynamic stage and by increasing the penalty costs incurred for violating hard time window constraints and refusing to respond to passenger demands, thus optimizing demand-responsive rural customized bus vehicle paths to meet passenger travel demands while minimizing operational costs and passenger travel time costs and improving operational efficiency.

The objective function of the path optimization model in the dynamic stage is based on the static stage, adding the penalty cost $f_4$ for vehicles not responding to passengers’ real-time demand to the operating cost $F_1$ of the operator, and adding the passenger waiting penalty cost $f_5$ caused by exceeding the desired boarding time window to the passenger travel time cost $F_2$. The time window constraint of the path optimization model in the dynamic stage is modified on the basis of the static stage, allowing the vehicle to arrive at the station outside the passenger’s desired time window, where the passenger’s desired boarding time constraint is a soft time window and the tolerable boarding time is a hard time window, converted into a time penalty cost, as shown in Equations (21) and (22).

$$f_4=\phi \cdot {\displaystyle \sum _{i\in D}{\displaystyle \sum _{j\in D}{\displaystyle \sum _{k\in K}\left(1-x_{ij}^k\right)\cdot {\displaystyle \sum _{j\in D}{\displaystyle \sum _{q\in Q^{\text{'}}}{q}_j^{+}}}}}}$$

(21)

$$f_5={\displaystyle \sum _{i\in D}{\displaystyle \sum _{k\in K}{\displaystyle \sum _{q\in Q\cup Q^{\text{'}}}{W}_{q_I}^k}}}$$

(22)

$$W_{q_i}^k=\{\begin{array}{ll}0\hfill & e_{q_i}\le t_i^k\le l_{q_i}\hfill \\ y_{q_i}^k\left[p_1\left(t_i^k-l_{q_i}\right)\right]\hfill & l_{q_i}\le t_i^k\le L_{q_i}\hfill \\ y_{q_i}^k\left[p_1\left(L_{q_i}-l_{q_i}\right)+p_2\left(t_i^k-L_{q_i}\right)\right]\hfill & t_i^k>L_{q_i}\hfill \end{array}$$

(23)

where $Q^{\text{'}}$ is the set of passengers in the study area making real-time travel demands; $\phi$ is the penalty cost factor incurred by a vehicle for refusing to respond to a passenger; $W_{q_i}^k$ is the waiting penalty cost for passenger $q_i$; $[l_{q_i},L_{q_i}]$ is the tolerable boarding time window for passenger $q_i$; $L_{q_i}$ is the upper limit of the tolerable boarding time window for passenger $q_i$; and $p_1$ and $p_2$ are the time penalty cost factors for vehicles arriving within and outside the tolerable boarding time window for passengers, respectively.

5. Model Algorithm Solution

The vehicle path model in this paper considers the costs of operators and passengers. It is a multi-objective combined optimization model with many relevant parameters and constraints, and is complex to solve. The exact algorithm is more difficult to solve in the actual vehicle operation process, so it is suitable to use a heuristic algorithm to solve the vehicle path problem. The bat algorithm (BA) has fast convergence speed and strong global search capability in the early stage, but low convergence effect and easily falls into the local optimum in the late stage, whereas the particle swarm optimization (PSO) has high search accuracy, strong optimization capability and is easy to calculate, but weak global search capability. To cope with the advantages and disadvantages of the two algorithms, a hybrid algorithm combining the adaptive particle swarm optimization (APSO) and the bat algorithm (BA) is proposed in this paper.

5.1. Bat Algorithm

The bat algorithm (BA), first proposed by Yang in 2010 [38], is derived from the biological property of bat echolocation, where a number of bats are randomly distributed in the search space to find their respective optimal solutions, and new solutions are generated by continuously adjusting the frequency and modifying the flight speed and position of the bats, which are prompted to move towards the optimal solution by adjusting the pulse incidence and loudness. The algorithm establishes a series of rules to update the position, speed and frequency of the bats as they search for prey, performing the search for the optimal solution until the target stops or each constraint is satisfied, as shown in Equations (24)–(26).

$$f_i=f_{\min }+\left(f_{\max }-f_{\min }\right)\tau$$

(24)

$$v_i^k=v_i^{k-1}+\left(x_i^{k-1}-gbes{t}_i\right)f_i$$

(25)

$$x_i^k=x_i^{k-1}+v_i^k$$

(26)

where $f_i$ is the search frequency of bat $i$ in the $k$ th iteration; v_i is the speed of bat $i$ flying in the $k$ th iteration; x_i is the position of bat $i$ flying in the $k$ th iteration; $f_{\min }$ and $f_{\max }$ are the minimum and maximum values of bat flight frequency, respectively; $\tau$ is a random number with a range of values $[0,1]$; and $gbes{t}_i$ is the global optimal solution in the current population.

For the local search part of the algorithm, once an optimal solution has been selected from the current set of optimal solutions, a new solution is randomly generated in the vicinity of this optimal solution. As the iterations increase, the pulse incidence increases, the loudness decreases, the local search probability decreases, the range narrows and the bat continuously scans to locate the target, eventually searching for the optimal solution, as shown in Equations (27)–(29).

$$x_i^{\mathrm{k}}{}_{new}=x_{best}+ϵA^k$$

(27)

$$A_i^{k+1}=a{A}_i^k$$

(28)

$$r_i^{k+1}=r_i^0[1-\exp (-\gamma k)]$$

(29)

where $ϵ$ is a random number with a range of values $[0,1]$; $A^k$ is the average loudness of all bats in the $k$ th iteration; $A_i^k$ is the loudness of the bat $i$ in the $k$ th iteration; $a$ and $\gamma$ are constants; $r_i^k$ is the pulse emissivity of bat $i$ in the k th iteration; and $r_i^0$ is the initial pulse emissivity of bat $i$.

5.2. Adaptive Particle Swarm Optimization

The particle swarm optimization (PSO) was first proposed by Eberhart and Kennedy [39] in 1995 and originated from the social predation behavior of birds. Each particle in the algorithm moves through the search space over time and changes its position and speed according to the particle’s own optimal solution and the current optimal solution in the whole population, and the n-dimensional vector representing the particle’s position in the search space represents the potential optimization problem solution. During the evolution process, the particle flies through the entire solution space at a certain speed. The particle remembers the optimal solution it encounters during the search process, and the population remembers the optimal solution found among all particles. At each iteration, the velocity and position of particle $i$ are updated, as shown in Equations (30) and (31).

$$v_i^k=w{v}_i^{k-1}+c_1{r}_1\left({\mathrm{pbest}}_i-x_i^{k-1}\right)+c_2{r}_2\left({\mathrm{gbest}}_i-x_i^{k-1}\right)$$

(30)

$$x_i^k=x_i^{k-1}+v_i^k$$

(31)

where $v_i^k$ is the velocity of particle $i$ in the $k$ th iteration; x_i is the position of particle $i$ in the $k$ th iteration; $w$ is the inertia weight, a non-negative number used to regulate the search range for the solution space; $c_1$ and $c_2$ are the learning factors for the own cognitive term and the group cognitive term, respectively; $r_1$ and $r_2$ are random numbers, taking values in the range $[0,1]$, used to increase the search randomness; ${\mathrm{pbest}}_i$ is the historical optimal solution for particle $i$ itself; and ${\mathrm{gbest}}_i$ is the historical optimal solution for the whole group.

Adaptive particle swarm is a particle swarm algorithm that introduces a mutation operation to reinitialize some variables with a certain probability. The mutation operation allows the particles to jump out of the previously searched optimal solution position and search in a larger space, increasing the possibility of the algorithm finding the optimal solution.

5.3. Hybrid Algorithm Combining BA and APSO

The hybrid algorithm based on the combination of BA and APSO aims to converge at a faster computational speed in order to obtain a better optimal solution. In the early stage of the hybrid algorithm, the process of executing BA refers to the particle characteristics of APSO and introduces the inertia weight $w$ in the velocity of the particles into the velocity formula of the bat algorithm, which maintains the early search effect of BA while also regulating the effect of individual inertia on the velocity, and the improved velocity formula is shown in Equation (32). In order to improve the search accuracy in the early process, a larger $w$ value should be chosen, and in the later process, a smaller $w$ value should be chosen for a more accurate local search. The adjustment formula is shown in Equation (33).

$$v_i^k=w{v}_i^{k-1}+\left(x_i^{k-1}-gbes{t}_i\right)f_i$$

(32)

$$w=w_{\min }+\left(w_{\max }-w_{\min }\right)\cdot 2^{-\frac{\zeta \cdot k}{k_{\max }}}$$

(33)

where $w_{\min }$ and $w_{\max }$ are the minimum and maximum values of the inertia weights, respectively; $\zeta$ is the parameter adjustment factor, by which the degree of variation of the $w$ value is adjusted; and $k_{\max }$ is the maximum number of iterations set.

Considering the shortcomings of BA in terms of slow search speed and low convergence accuracy in the later stages, the hybrid algorithm in this paper is improved based on APSO in the later stages, with the speed formula shown in Equation (30). In order to speed up the convergence speed and reduce the search time, its learning factor is adjusted, decreasing and increasing according to the exponential law, as shown in Equations (34) and (35), respectively. In the local search process of the hybrid algorithm, once the constraints are satisfied, the bat will directly explore the location of the current optimal solution, but if the search speed is still maintained at this time, it is easy to miss the region of the optimal solution of the path in the next iteration process, so the decay factor is introduced to reduce the influence of inertia on the bat speed, and the speed formula of the adjusted local search is shown in Equation (36).

$$c_1=c_{1\min }+\left(c_{1\max }-c_{1\min }\right)\cdot 2^{-\frac{\zeta \cdot k}{k_{\max }}}$$

(34)

$$c_2=c_{2\max }-\left(c_{2\max }-c_{2\min }\right)\cdot 2^{-\frac{\zeta \cdot k}{k_{\max }}}$$

(35)

$$v_i^k=\psi w{v}_i^{k-1}+c_1{r}_1\left({\mathrm{pbest}}_i-x_i^{k-1}\right)+c_2{r}_2\left({\mathrm{gbest}}_i-x_i^{k-1}\right)$$

(36)

where $c_{1\min }$ and $c_{1\max }$ are the maximum and minimum values of the learning factor for the own cognitive item, respectively; $c_{2\min }$ and $c_{2\max }$ are the maximum and minimum values of the learning factor for the group cognitive item, respectively; and $\psi$ is the attenuation coefficient, taking a range of values $[0,1]$.

6. Example Analysis

6.1. Description of the Road Network and Passenger Information

Some townships in Jing’an County, Jiangxi Province (Gaohu Town, Shuikou Town, and Shuangxi Town) were selected as the study area. The 20 administrative villages under its jurisdiction are rural stations (black dots), and the urban bus station of Jing’an County is the county station (red five-pointed star), as shown in Figure 2. For the convenience of description, each station is numbered, and the information of reserved passengers is shown in

Table 2. Corresponding number of each station and passenger reservation information.

Station	Number	Number of People on Board	Pick-Up and Drop-Off Stations	Expected Boarding Time Window
Jing’an County	0	7	(0, 2), (0, 3), (0, 10), (0, 11), (0, 12), (0, 13), (0, 18)	[6:25, 6:35]
		7	(0, 1), (0, 6), (0, 14), (0, 15), (0, 19), (0, 19), (0, 20)	[7:00, 7:10]
		5	(0, 2), (0, 3), (0, 5), (0, 7), (0, 17)	[7:10, 7:20]
		8	(0, 4), (0, 5), (0, 7), (0, 13), (0, 13), (0, 13), (0, 14), (0, 18), (0, 18)	[7:30, 7:40]
		10	(0, 2), (0, 4), (0, 6), (0, 6), (0, 6), (0, 10), (0, 11), (0, 12), (0, 12), (0, 15)	[7:55, 8:05]
Daqiao	1	2	(1, 0)	[9:55, 10:05]
Hebei	2	1	(2, 0)	[9:00, 9:10]
		1	(2, 0)	[8:15, 8:25]
Mawei Mountain	3	2	(3, 0)	[8:55, 9:05]
		1	(3, 0)	[8:10, 8:20]
Caoshan	4	3	(4, 0)	[9:50, 10:00]
Guanzhou	5	1	(5, 0)	[9:45, 9:55]
		1	(5, 0)	[8:50, 9:00]
Shuikou	6	2	(6, 0)	[9:45, 9:55]
Shagang	7	1	(7, 0)	[9:40, 9:50]
		1	(7, 0)	[8:40, 8:50]
Laibao	8	1	(8, 0)	[9:35, 9:45]
		2	(8, 0)	[8:30, 8:40]
Zhoujia	9	2	(9, 0)	[8:20, 8:30]
Zhonglun	10	1	(10, 0)	[7:50, 8:00]
Taoyuan	11	1	(11, 0)	[7:45, 7:55]
Qinshan	12	1	(12, 0)	[7:40, 7:50]
Tangdi	13	1	(13, 0)	[7:50, 8:00]
Zhonggang	14	1	(14, 0)	[9:15, 9:25]
		1	(14, 0)	[8:05, 8:15]
Gaohu	15	2	(15, 0)	[9:15, 9:25]
		1	(15, 0)	[8:00, 8:10]
Tengfeng	16	2	(16, 0)	[8:30, 8:40]
Xiaguan	17	1	(17, 0)	[7:55, 8:05]
Shankou	18	1	(18, 0)	[7:10, 7:20]
Yongfeng	19	1	(19, 0)	[9:05, 9:15]
Xitou	20	1	(20, 0)	[9:00, 9:10]

, including the pick-up station, drop-off station and expected boarding time window for each passenger, and the distance between each station is obtained using the map, as shown in

Table 3. Matrix of distances between stations.

Station	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20
0	0	1.4	1.7	3.5	4.5	6.6	10	12	15	17	16	17	20	16	15	18	20	21	24	24	29
1	1.4	0	1.3	3	4	4.9	8.4	9.9	13	15	14	16	18	14	13	16	18	20	22	23	27
2	1.7	1.3	0	2.1	3.1	5	8.6	10	13	15	14	16	18	14	14	16	18	20	22	23	27
3	3.5	3	2.1	0	2.4	4.3	7.9	9.3	13	14	14	15	17	13	13	15	18	19	22	22	26
4	4.5	4	3.1	2.4	0	2.9	6.4	7.9	11	13	12	14	16	12	11	14	16	18	20	21	25
5	6.6	4.9	5	4.3	2.9	0	3.5	4.9	8.1	10	9.1	11	13	8.8	8.5	11	13	15	17	18	22
6	10	8.4	8.6	7.9	6.4	3.5	0	4.1	7.3	9.3	5.7	7.1	9.3	5.3	5	7.6	13	11	14	14	19
7	12	9.9	10	9.3	7.9	4.9	4.1	0	3.4	5.3	9.8	11	14	9.4	9.1	12	8.6	15	18	18	23
8	15	13	13	13	11	8.1	7.3	3.4	0	2.1	13	14	17	13	12	15	5.3	18	21	21	26
9	17	15	15	14	13	10	9.3	5.3	2.1	0	15	16	18	14	14	17	5.8	20	23	23	28
10	16	14	14	14	12	9.1	5.7	9.8	13	15	0	1.6	3.9	2.4	5.5	8.1	18	12	14	12	19
11	17	16	16	15	14	11	7.1	11	14	16	1.6	0	5.5	3.9	7	7.1	20	13	13	16	21
12	20	18	18	17	16	13	9.3	14	17	18	3.9	5.5	0	9.7	9.4	12	22	16	18	19	23
13	16	14	14	13	12	8.8	5.3	9.4	13	14	2.4	3.9	9.7	0	0.7	3.3	18	6.9	9.5	9.8	14
14	15	13	14	13	11	8.5	5	9.1	12	14	5.5	7	9.4	0.7	0	2.6	18	6.2	8.8	9.1	14
15	18	16	16	15	14	11	7.6	12	15	17	8.1	7.1	12	3.3	2.6	0	20	3.6	6.2	5.9	10
16	20	18	18	18	16	13	13	8.6	5.3	5.8	18	20	22	18	18	20	0	24	26	27	31
17	21	20	20	19	18	15	11	15	18	20	12	13	16	6.9	6.2	3.6	24	0	2.6	8.1	13
18	24	22	22	22	20	17	14	18	21	23	14	13	18	9.5	8.8	6.2	26	2.6	0	11	15
19	24	23	23	22	21	18	14	18	21	23	12	16	19	9.8	9.1	5.9	27	8.1	11	0	4.2
20	29	27	27	26	25	22	19	23	26	28	19	21	23	14	14	10	31	13	15	4.2	0

.

6.2. Parameter Setting

There are two types of electric customized buses in the study area, with an average running speed of 45 km/h. The two types of vehicles can accommodate 7 and 15 passengers, with fixed costs of 100 and 300 yuan per vehicle and unit transportation costs of 1.8 yuan/km and 2.2 yuan/km, respectively.

In the objective function of the vehicle path model, the proportions of operating costs and passengers’ travel time costs are 0.4 and 0.6, respectively. The travel time cost of passengers is 0.15 yuan per min, and the minimum time and maximum time allowed to run the vehicle are 30 min and 180 min, respectively. In the dynamic phase, the penalty cost when the operator refuses to respond to the real-time travel request of passengers is 10 yuan per person, the tolerable time window for passengers is within 5 min after the expected time window, and the penalty cost coefficients of the vehicle arriving inside and outside the passenger’s tolerable boarding time window are 0.5 and 1, respectively. The values of the relevant parameters of the vehicle path model are shown in

Table 4. Vehicle path model parameter values.

Parameter	Value	Meaning
${\omega }_1$	0.4	Operating business operating cost weights in the objective function
${\omega }_2$	0.6	Passenger travel time cost weight in objective function
$\beta$	0.15 yuan/min	Unit time cost factor of passenger travel
$t_{\min }$	30 min	The minimum time required for the vehicle to operate
$t_{\min }$	180 min	The maximum time required for the vehicle to operate
$\phi$	10 yuan	Penalty cost factor for vehicle refusal to respond to a passenger in the real-time phase
$L_{q_i}-l_{q_i}$	5 min	The tolerable time window for passengers is 5 min after the expectation time window
$p_1$	0.5	Penalty factor for vehicles arriving at stations within the tolerable boarding time window for passengers
$p_2$	1	Penalty factor for vehicles arriving at stations outside of the tolerable boarding time window for passengers

.

The parameters in the hybrid algorithm combined with the BA algorithm and the APSO algorithm used in this paper are as follows: the population size is 100; the number of iterations is 200; the minimum and maximum inertia weights in the BA algorithm are 0.2 and 0.9, respectively; the minimum and maximum values of the bat flight frequency are 0 and 100, respectively; the initial loudness of the bat is 0; the initial pulse emissivity is 0.5; the maximum and minimum learning factors of the self-cognitive term in the APSO algorithm are 0.5 and 2.5, respectively; the maximum and minimum learning factors of the group cognition item are 1 and 2.25, respectively; and the values of $r_1$, $r_2$, $\psi$, $\zeta$, $a$, $\gamma$ and $ϵ$ are 0.2, 0.4, 0.5, 0.5, 0.9, 0.9 and 0.5, respectively. The values of the parameters of the hybrid algorithm designed in this paper are shown in

Table 5. Hybrid algorithm parameter values.

Parameter	Value	Meaning
$k_{\max }$	200	Number of iterations
$w$	0.9	Inertia weights for the velocity equation
$w_{\min }$	0.2	Minimum value of inertia weights
$w_{\max }$	0.9	Maximum value of inertia weights
$f_{\min }$	0	Minimum value of flight frequency
$f_{\max }$	100	Maximum value of flight frequency
$c_1$	2	Learning factor for the own cognitive term of the speed formula
$c_{1\min }$	0.5	Minimal value of learning factor for own cognitive term
$c_{1\max }$	2.5	Maximal value of the learning factor of own cognitive term
$c_2$	2	Learning factors for the group cognitive term of the speed formula
$c_{2\min }$	1	Minimal value of learning factor of group cognitive items
$c_{2\max }$	2.25	Maximal value of learning factor of group cognitive items
$r_1$	0.2	Random number of speed formula, general value range [0, 1]
$r_2$	0.4	Random number of speed formula, general value range [0, 1]
$\psi$	0.5	Attenuation coefficient during local search
$\zeta$	0.5	Parameter adjustment coefficient in the early stage
$r_i^0$	0.5	Initial pulse emissivity of bat $i$
$a$, $\gamma$	0.9	Constant in bat algorithm
$ϵ$	0.5	Random number, general value range [0, 1]
$A_i^0$	0	Initial loudness of bat $i$

.

6.3. Analysis of Results

MATLAB was used to construct a demand-responsive rural customized bus path optimization model, and a hybrid algorithm combining BA algorithm and APSO algorithm was used to solve it.

Table 6. Vehicle path scheme for the static phase of the BP-APSO hybrid algorithm.

Vehicle Number	Vehicle Type	Traveling Path	Travel Distance/km	Travel Time/min	Number of Passengers	Passenger Load Factor/%	F1	F2
1	1	0-2-3-10-11-12-13-18-20-19-17-15-12-8-1-0	118.4	221.9	14	100	313.1	58
2	1	0-1-6-14-15-19-20-18-16-7-0	88.5	145.8	11	78.6	260.4	49.5
3	1	0-2-3-5-7-17-20-5-4-3-0	71.8	141	8	57.1	229.2	24.2
4	2	0-4-5-7-13-14-18-20-15-11-4-3-2-1-0	84.5	163.3	15	50	485.9	66
5	2	0-2-4-6-10-11-12-15-20-15-14-13-10-9-8-7-6-5-4-3-0	98.6	180.7	25	83.3	516.9	113.8

provides information on vehicle paths that serve the reservation requirements of the static phase. The optimal solution of the objective function is 2117, using five cars and two models.

The results of the BP algorithm, APSO algorithm and hybrid algorithm to solve the vehicle route model were compared, as can be seen in

Table 7. Comparison of vehicle path schemes by algorithm.

Algorithm	Total Travel Distance/km	Total Travel Time/min	Average Passenger Load Factor/%	Target Value
BP	441.8	732.55	55.0	2224
APSO	469.6	861.9	76.8	2239
BP-APSO	461.8	852.8	73.8	2117

. Compared with the BP algorithm, the average load factor of BP-APSO increased by 18.8%, and the target value decreased by 97. Compared to the APSO algorithm, the total driving distance of BP-APSO was reduced by 8 km and the total travel time reduced by 9 min, and the target value was reduced by 112. BP-APSO makes the target value of the whole model the lowest, followed by BP and APSO algorithms. The specific number of iterations and the objective function values are shown in Figure 3, and it can be seen that the BP-APSO algorithm can obtain better optimal solution values than the BP and APSO algorithms, and it has strong optimization and fast convergence speed.

After the vehicle scheduling arrangement in the static phase starts operation, the operating company receives a certain number of real-time travel requests, and the vehicle path optimization in the dynamic phase begins. A total of eight passengers had real-time travel requests, and the dynamic demand information received is shown in

Table 8. Dynamic phase adds real-time travel demand information.

Number	Number of People on Board	Pick-Up and Drop-Off Stations	Expected Boarding Time Window
0	1	(0, 2)	[7:10, 7:20]
	1	(0, 13)	[7:00, 7:10]
4	1	(4, 0)	[7:55, 8:05]
8	1	(8, 0)	[8:30, 8:40]
8	1	(8, 0)	[8:35, 8:45]
12	1	(12, 0)	[7:40, 7:50]
19	1	(19, 0)	[9:05, 9:15]
20	1	(20, 0)	[8:50, 9:00]

.

We selected the actual road network of some townships in Jing’an County, Jiangxi Province for empirical analysis, applied the static and dynamic two-stage rural customized bus route optimization model and the BP- and APSO-based hybrid algorithm to plan the vehicle driving path, verified the effectiveness of the hybrid algorithm on the optimization model and used MATLAB software to solve the optimization model. The solution time is 114 s. Compared with the BP and APSO algorithms, the hybrid algorithm has better advantages in convergence speed and accuracy. It can be used to solve real-scale studies, reducing the result of the objective function value by 5.5%. The results show that the optimization model and algorithm proposed in this paper can provide a theoretical reference for the improvement of public transportation systems in rural areas.

The vehicle routing scheme in the dynamic phase of the hybrid algorithm is shown in

Table 9. BP-APSO hybrid algorithm for dynamic phase of vehicle path scheme.

Vehicle Number	Vehicle Type	Traveling Path	Travel Distance/km	Travel Time/min	Number of Passengers	Passenger Load Factor/%	F1	F2	Time Window Penalty Cost	Rejection Response Cost
1	1	0-2-3-10-11-12-13-18-20-16-8-7-5-0	110.3	175.5	14	100	298.5	69.1	27.4	10
2	2	0-1-6-14-15-19-20-19-18-11-5-4-3-2-0	75.8	129.1	17	56.7	466.8	71.4	22.3	0
3	1	0-2-3-5-7-13-17-20-15-14-12-10-8-4-0	96.7	165.1	13	92.9	274.1	46.9	14.7	0
4	2	0-4-5-7-13-14-18-20-17-13-12-6-4-3-2-1-0	98.7	158.1	18	60	517.1	100.8	28.6	0
5	2	0-2-4-6-10-11-12-15-20-19-15-9-8-3-0	91.7	147.6	18	60	501.7	63.5	29.9	0

. The five buses carried a total of 80 passengers, responded to the travel needs of seven passengers in the dynamic phase and refused to respond to the demand of passengers with the expected boarding window [8:30, 8:40] for stop 8. The optimal solution objective function value is 2410, and the maximum delay occurs at the return stop of the fourth bus, exceeding the expected time window by 12.1 min, which was within the acceptable range in ordinary travel. The results of this case study show that the BP-APSO hybrid algorithm can provide a better scheduling effect for the operation of rural customized bus lines in response to demand.

In summary, a total of 81 passengers requested rides, of which 73 were in the reservation phase, and the rural demand-responsive customized bus responded to all of them. Eight passengers were in the real-time phase and the rural demand-responsive customized bus responded to seven of them and rejected the real-time ride request of one passenger. A total of five vehicles were used, including two small vehicles with seven seats and three medium-sized vehicles with 15 seats. The buses traveled a total distance of 473.2 km, and the operating cost of the vehicles was 2058.3 yuan.

Assume that the regular bus in the rural area with a fixed schedule and stops leaves the county station every hour and passes through each rural station, with the shortest one-way route of 1-2-3-4-5-6-7-8-9-16-14-13-10-11-12-15-17-18-19-20 and a mileage of 90.2 km, traveling to the farthest station 20 and then back, finally returning to the county station. According to the survey, the current rural area buses are large- and medium-sized vehicles, and of a single type, so in the comparison process, we chose 15-seat vehicles the conventional bus vehicles.

The two modes of transportation are compared, as shown in

Table 10. BP-APSO hybrid algorithm for dynamic phase of the vehicle path scheme.

Mode	Number of Passengers Serviced	Number of Vehicles Used	Vehicle Type	Average Passenger Load Factor	Travel Distance/km	Travel Time/min	Operating Costs
Demand Response Customized Bus	80	5	7/15	73.9%	473.2	775.4	2058.3
Regular Bus	81	4	15	66.6%	721.6	962.1	2787.5

.

The above comparison shows that demand-responsive customized transit yields better results both in terms of cost and time. In the range of the expected time window of [6:25, 10:05] presented by 81 passengers, the average load factor of demand-responsive customized buses is improved by 7.3%, the distance traveled is reduced by 248.4 km, the travel time is reduced by 186.7 min and the cost is reduced by 2529.2 yuan compared to the regular bus.

7. Results and Discussion

This paper focuses on public transportation services for residents in rural areas. Surveys and interviews targeting many transportation-related government departments have shown that with increasing private car ownership in rural areas with low passenger density, the vehicle capacity is often low or vehicles are even empty, and passengers in rural areas face the problems of infrequent public transportation and long waiting times, while the costs for operating companies remain high.

Therefore, this paper proposes to run demand-responsive customized buses in rural areas, which can effectively avoid empty vehicles, low load factor and high costs. In this study, in order to improve the operational efficiency of public transportation systems in rural areas and take into account the interests of both operators and passengers, a two-stage vehicle route planning model of the reservation phase and the real-time stage was constructed, aiming at minimizing the costs of the operating enterprises and the travel time cost of passengers, considering the passenger time window, different vehicle characteristics, rated passenger capacity and line operation time in the constraints. The novelty of this paper lies in the fact that the two-stage vehicle routing model not only meets the passenger demand at the reservation stage, but also takes into account the real-time demand from passengers during the vehicle operation, serving a larger range of passengers. A variety of constraints are considered, and in order to better facilitate vehicle route planning in different time periods and with different numbers of passengers, this paper provides alternative vehicle types instead of a single type, which is rare in other work.

For the demand-responsive rural customized bus route operation problem studied in this paper, through the analysis and comparison of common vehicle path algorithms, we propose a BA-APSO hybrid algorithm. The process of using the BA algorithm in the early stage considers the particle characteristics of the APSO algorithm. The inertia weight in the velocity of the particle is introduced into the bat algorithm velocity. The middle and late stage based on the APSO algorithm are improved, the learning factor is adjusted and the optimal solution selection mode of the BA algorithm is improved by using the particle adaptive change mode in the APSO algorithm, which can improve the accuracy of the hybrid algorithm, accelerate the convergence speed and reduce the search time. In previous studies, bat algorithms and adaptive algorithms have rarely been combined, mutually improved and applied to vehicle route planning.

Finally, the actual road network of some townships in Jing’an County, Jiangxi Province, China, was selected for the case study, and the reservation and real-time two-stage rural customized bus vehicle route optimization model designed in this paper was applied to plan the vehicle driving path with the hybrid algorithm based on BA and APSO, and the effectiveness of the hybrid algorithm on the optimization model was verified.

The case study shows that the designed reservation and real-time two-stage rural customized bus vehicle route optimization model and the hybrid algorithm of BA and APSO proposed in this paper have certain reference value for planning vehicle driving routes, but due to some objective reasons, this work can still be further improved, specifically considering the following points:

(1)

This research on a demand-responsive custom bus line was carried out under partially known assumptions. We set the vehicle travel speed for a fixed value, but in the actual operation process, due to road conditions, the environment, weather and other factors, the vehicle speed cannot be kept constant, so this paper is based on the ideal state of research. Moreover, calculations of the passenger information were made without actual data support, as the software randomly generated information. In the future, real passenger flow data can be collected for analysis based on actual surveys.

(2)

The demand-responsive rural customized bus route research presented in this paper focuses on passengers. In the future, it can be combined with the integrated development of rural passengers, cargo and mail to develop a demand-responsive customized bus route for passengers, cargo and mail cooperation and explore a new model for the development of rural transportation services, reduce operating costs and improve transport service levels in rural areas.