Routing Optimization of Regional Flexible Transit Under the Mixed Demand Mode

Abstract

To meet trip demands and avoid transit capacity waste or shortages, this study investigates the routing optimization of flexible transit with time windows. We introduce the time penalty costs to accommodate the impacts of early and late vehicle arrivals on passengers’ satisfaction. A routing optimization model is developed to minimize the system operation costs and the costs incurred by passengers' time penalties. The problem is solved by a designed adaptive genetic algorithm that adopts an adaptive mutation strategy to dynamically adjust the mutation probability and mutation operator. The numerical experiments compare the results of the mixed demand model, in which vehicles can pick up and drop off passengers simultaneously, to those of the separate pick-up and delivery modes. Finally, a sensitive analysis is conducted to explore the impact of operational factors (vehicle speed, maximum one-way travel time, and weighting ratios between operating and penalty costs) on the system's performance (total costs, per capita mileage, and average seat occupancy rate). Our results confirm the advantages of the developed adaptive genetic algorithm over traditional ones with respect to the convergence speed and optimality gap. Moreover, the numerical results indicate that the mixed demand operation mode of transit reduces total costs by an average of 2.35% compared to separate pick-up and delivery modes. The results also reveal that an increase in the weights of the operating cost can reduce the total cost. The findings of this work can provide guidance to the operation of regional flexible transit.

Keywords

public transportation bus transit systems optimization

With urbanization, many residential areas have been built on the periphery of cities. This development leads to increased trip demand between urban centers and peripheral areas ( 1 ). Demand for commuting services during peak periods and entertainment trip services during off-peak periods is substantial, causing immense pressures for current transit systems ( 2 ). With the mature use of intelligent transport systems ( 3 , 4 ), the emerging regional flexible transit has become an attractive option. Regional flexible transit is a passenger demand-driven, variable route bus service for a fixed area. This public transport type adopts the “door-to-door” and “one seat for one person” mode ( 5 ) to transfer passengers within the service area to predetermined places or connecting stations, such as central business districts (CBDs), famous tourist attractions, or transfer stations with a large flow of rail transit passengers ( 6 ). It provides higher accessibility ( 7 – 9 ), shorter travel time, and increased mobility ( 10 ) compared to traditional transit. However, because of its flexibility, this public transport type can lead to under capacity or wasted capacity in operations. The different modes of pick-up and drop-off also greatly affect operational efficiency. Therefore, considering both operators and passengers, route optimization studies for regional flexible transit are essential to improve bus operation efficiency and reduce operating costs ( 11 ).

To provide a wide ranging and efficient flexible transit service, scholars have conducted extensive research on the operation strategy of regional flexible transit vehicles. Edwards and Watkins ( 12 ) devised an effective method to determine when and where flexible feeder transit is preferable to traditional transit. Giuffrida et al. ( 1 ) found that flexible transit can satisfy more trip demand with limited cost than other shared mobility services. To verify the accessibility of flexible transit, Alonso-González et al. ( 13 ) proposed an evaluation framework to assess the performance of demand responsive transit (DRT) and empirically analyzed DRT services in the Netherlands. For flexible transit, a larger service area may lead to conflicting service efficiency and coverage. On this basis, to balance the relationship between users and operators, Wang et al. ( 14 ) used a zoning strategy to design the area effectively. The optimal network layout for regional flexible transit was determined. In the current model for flexible transit route optimization, the objectives include minimizing operating costs ( 15 , 16 ) and maximizing service quality ( 17 ), as well as taking two factors into account ( 18 ). To promote urban–rural integration, Li et al. ( 19 ) developed a new scheduling model of variable route transit intended to reduce operating costs and travel time. The model is validated in the northern part of Yongcheng City in China, and its superiority is demonstrated. The operator’s and the passengers’ perspectives do not necessarily conflict ( 20 ). However, many studies have focused on how to balance operating costs and service levels of flexible transit ( 21 , 22 ). Shen et al. ( 7 ) presented a mathematical model of a dynamic DRT vehicle scheduling problem to achieve a balance between operator and passenger costs. Giuffrida et al. ( 1 ) found that route selection strategies are critical to the balance between operator and user costs. Gribkovskaia et al. ( 23 ) explored the carpooling game between passengers and between passengers and operators. These researches also revealed that the selective pick-up and delivery of vehicles affects system efficiency. Therefore, it is worth exploring which is the better transport method to use in the operation.

Vehicle route optimization problems are widely used in reality, with many studies focusing on the algorithmic aspects ( 24 ). The algorithms for optimizing flexible bus routes are also divided into exact ( 25 , 26 ) and heuristic ( 27 , 28 ) algorithms. Shen et al. ( 7 ) presented a mathematical model of a dynamic DRT vehicle routing problem, which they solved using an approach based on approximate dynamic programming (ADP). Stopka et al. ( 29 ) used the nearest neighbor search method to determine the optimal urban waste collection cycle within a pre-designated area of the existing urban road network. However, the exact algorithm is difficult and inefficient for solving routing problems with time windows ( 30 ). Heuristic algorithms seem to be more effective in the study of such problems ( 31 ). The genetic algorithm (GA) has commonly been used to solve complex optimization problems, such as vehicle routing ( 32 , 33 ) and scheduling problems ( 34 , 35 ). Midaoui et al. ( 36 ) proposed a novel intelligent logistics method. They used the GA for path optimization of multiple pharmacies to meet the cargo requirements of hospital pharmacies. Many scholars have used the GA to solve a multi-trip vehicle routing problem with time windows ( 37 , 38 ). The GA can solve many non-deterministic polynomial-time hard (NP-hard) problems and obtain close to optimal solutions ( 39 , 40 ). However, the calculation efficiency and accuracy were low ( 41 , 42 ). In response to the shortcomings of the traditional GA, Cui et al. ( 43 ) designed an adaptive GA to solve the problem to improve the logistics and distribution timeliness. Its superiority was demonstrated compared to the traditional one. Sun et al. ( 44 ) designed an adaptive GA to solve the path problem in air logistics, and the results proved that the algorithm is efficient and accurate. However, adaptive Gas are mostly based on alterations in population fitness, with few mutation improvement strategies on the iteration number.

In recent years, the emergence of transportation network companies (TNCs) has provided an innovative strategy for shared mobility ( 45 ). TNCs also offer solutions to routing and matching problems for flexible transit. Lotfi and Abdelghany ( 46 ) illustrated a new method of integrating matching and routing for on-demand mobility services. The results show that TNCs could serve more willing passengers and earn higher profits. Noruzoliaee and Zou ( 47 ) investigated the matching problem of shared autonomous vehicles considering network equilibrium. Optimal routing and allocation decisions for TNCs were developed. It was demonstrated that shared travel may lead to significant system benefits. Masoud and Jayakrishnan ( 48 ) proposed a dynamic programming algorithm to solve the matching problem in flexible transit systems. It is demonstrated that the algorithm is effective in solving the matching problem for TNCs.

Because of the above analysis, this study has three contributions. Firstly, this study considers the operation mode of mixed pick-up and delivery demands. Each demand point corresponds to different time window constraints, the time window penalty mechanism is introduced, and the enterprise operation cost is considered. The optimization model is established to minimize operation and time penalty costs. This study compares and analyzes the optimization results under different demand models. Secondly, in the algorithm design, an improved adaptive GA is proposed against the drawbacks of the traditional GA, including easy premature maturation and low search efficiency in the late evolutionary stages. In this paper, the mutation probability and the mutation operator in the traditional GA are modified to dynamically adjust with the number of evolutionary generations. This will contribute to reducing the probability of premature maturation and increasing the speed of evolution. The effectiveness and feasibility of the improved algorithm are verified through a comparative analysis with the GA. Thirdly, a sensitivity analysis is conducted to explore the impact of vehicle speed, maximum one-way travel time, and weighting ratios between operating and penalty costs on regional flexible transit route optimization. Our results also provide new insights into operation planning for urban flexible transit.

The remainder of this paper is organized as follows. The second section describes the problem setting. The third section presents key model assumptions and formally introduces the routing optimization model of flexible transit. A solution algorithm is developed to solve the optimal routes. The fourth section analyzes the feasibility of the proposed algorithm and conducts a sensitivity analysis to identify key factors affecting the optimal decisions. Finally, the paper concludes with the main findings and future research directions.

Problem Description

We consider the routing optimization of flexible transit under the mixed demand mode. The travel plan described in this paper is completely customized to passengers’ reservation time and pick-up and delivery locations. Flexible transit services start and end at the same point, called the terminal. Passengers are divided into two categories based on their travel directions. Class A passengers take the bus from the service area to the terminal, and Class B passengers take the bus from the terminal to the service area. The boarding and alighting stations for both types of passengers are evenly distributed in the area and can be interspersed with one another. Inside the service area, routes and stations are not fixed. The vehicle starts from a terminal, and each vehicle circulates in its own designated area. The vehicle can transport the alighting passengers generated from the terminal to the corresponding demand points, while also transporting the boarding passenger from the demand points to the terminal. According to the scheme issued by the dispatch center, the vehicle visits the demand point locations reserved by passengers in turn. Then, it returns to the terminal after completing the task. Depending on how strict or not the time constraints are, they are categorized as a hard time window or a soft time window. In this study, a soft time window is used, and passengers are relatively relaxed about time requirements. Passenger demand times are divided into expected and tolerable times. No penalty is incurred when the vehicle serves within the expected time window, but a penalty is assessed when it arrives within the tolerable time window. The penalty is increased when the vehicle arrives outside the tolerable time window. Penalties vary according to the length of the breach. All passengers must submit a reservation application in advance. The system obtains the following key information from such applications: (1) the departure place, the expected departure time, and the time of arrival at the terminal of Class A passengers and (2) the destination, the expected departure time, and arrival time of Class B passengers.

Dynamic demand in real-time is not taken into account. Once the route has been successfully planned, the subsequent reservation will be considered in the next route planning process. For flexible transit, the spatial and temporal distribution of passenger flows have no obvious pattern and passengers have high requirements for time windows. A fixed headway interval can result in low passenger satisfaction or wasted capacity. Thus, this study adopts a variable departure interval. When the route optimization is completed, the system will notify each passenger. The passenger who has successfully booked will receive vehicle information, including the license plate number, seat number, estimated arrival time of the vehicle, and fare list. Passengers who fail to book will receive an invitation to re-book.

Methodology

Model Assumptions

The following assumptions are made for establishing the model.

All vehicles have the same capacity.

All vehicles run at a constant speed, regardless of factors such as traffic congestion and traffic accidents.

Flexible transit has pre-determined schedules and routes based on passengers’ information before departure to ensure the shortest length of each route.

All reservations are serviced, and each demand can only be serviced once by one vehicle. Vehicles arriving early or late at the demand point will produce corresponding time penalty costs.

The service time of passengers getting on and off is constant.

Notations

The objective function is to minimize the total system cost, which includes operating cost and penalty cost. Subscripts $i$ , $j$ correspond to passenger demand, and superscript $k$ represents vehicle routes. The sets, variables, and parameters are summarized in Table 1.

Table 1.

Notations Used in the Model

Sets
$C$	Set of passengers $C = {1, 2, . . ., n}$
$V$	Set of passenger demand points $V = {0, 1, 2, . . ., n}$ , $i, j \in V$ , $i \neq j$
$K$	Set of vehicle routes $k \in K$
$G$	Set of vehicles $G = {1, 2, . . ., m}$
Variables
$C$	Total cost of flexible transit system
$C_{1}$	Operating cost
$C_{2}$	Penalty cost
$F_{1}$	Fuel consumption cost of vehicles
$F_{2}$	Departure cost of the vehicle
$F_{3}$	Revenue from tickets
$F_{4}$	Penalty cost because of early arrival of vehicles
$F_{5}$	Penalty cost because of late arrival of vehicles
$x_{ij}^{k}$	Binary variable; 1 if the vehicle on route $k$ travels from $i$ to $j$ , 0 otherwise
$y_{i}^{k}$	Binary variable; 1 if passenger $i$ is served by the vehicle on route $k$ , 0 otherwise
$P_{k}$	Binary variable; 1 if the vehicle on route $k$ is sent from the terminal, 0 otherwise
$G^{k}$	Departure time of the vehicle on route $k$ from the start point
$A_{i}^{k}$	Vehicles on route $k$ arrival time at demand point $i$
$B_{i}^{k}$	Vehicles on route $k$ departure time from demand point $i$
$U_{i}^{k}$	Passenger $i$ arrival time at the station
$M_{i}^{k}$	Number of passengers onboard at the demand point $i$
$θ_{i}^{k}$	Binary variable; 1 if passenger $i$ takes the vehicle of route $k$ to the terminal, −1 otherwise
Parameters
$γ_{1}$	Weighting of operating costs
$γ_{2}$	Weighting of time penalty costs
$τ_{1}$	Fuel consumption cost per unit distance
$τ_{2}$	Departure cost per unit vehicle
$τ_{3}$	Ticket price per person
$Q$	Maximum passenger capacity of each vehicle
$q^{k}$	Number of passengers onboard when it leaves the start point
$v_{ij}$	Speed between two demand points $i$ and $j$
$S_{ij}$	Distance between two demand points $i$ and $j$
$T_{max}$	Maximum single travel time of each vehicle
$t_{u}$	Dwell time at each demand point
$R$	A finite large number
$f_{1} (t)$	Time penalty function for early arrival of vehicles
$f_{2} (t)$	Time penalty function for late arrival of vehicles
$[E T_{i}, L T_{i}]$	Tolerable service time window at demand point $i$
$α_{1}$	Time penalty factor for early arrival of vehicles within the tolerant time window
$α_{2}$	Time penalty factor for late arrival of vehicles within the tolerant time window
$β_{1}$	Time penalty factor for early arrival of vehicles within the expected time window
$β_{2}$	Time penalty factor for late arrival of vehicles within the expected time window

Model

The key to the implementation of a flexible transit system is the operation mode design, which must satisfy passengers’ needs for convenience and reduce operating costs. This paper transforms the multi-objective optimization problem into a single one. Thus, the objective function can be changed into the minimum value including operating cost and passenger dissatisfaction. Using the time penalty function, the measurement of passenger satisfaction is transformed into the calculation of the travel time value. In this way, uniformity is achieved between passengers and operators. The time window represents the level of passenger satisfaction with the service. Within the constraint of a soft time window, the vehicle is allowed to exceed passengers’ reservation time windows and arrive early or late within the range accepted by passengers. The loss caused by the resulting time difference for passengers is expressed by a penalty function. If a vehicle arrives at the demand point earlier than the time window, then the vehicle must wait until the time window is available; if a vehicle comes late, then the passengers at the demand point will continue to wait until it shows up. In this study, passengers also have to wait in cases where the vehicles fail to arrive in the time window, despite generating significant penalties.

The operating cost considers the fuel consumption cost, departure cost, and ticket price of vehicles. The time penalty cost considers the impact of early and late arrivals on passenger satisfaction. Taking the total cost of a flexible transit system, that is, the sum of operating and penalty costs, as the objective function, this study constructs a flexible route optimization model, as shown in Equation 1:

\begin{matrix} min C = γ_{1} C_{1} + γ_{2} C_{2} \\ = γ_{1} (F_{1} + F_{2} - F_{3}) + γ_{2} (F_{4} + F_{5}) (1) \end{matrix}

(1)

The fuel consumption of the vehicle is related to its driving distance, whereby the corresponding fuel consumption cost $F_{1}$ can be expressed by the product of the driving mileage and the fuel consumption cost per unit mileage, as presented in Equation 2:

F_{1} = τ_{1} \sum_{i} \sum_{j} \sum_{k} S_{ij} x_{ij}^{k}

(2)

The departure cost $F_{2}$ is the depreciation of vehicles owing to each operation. As long as the vehicle starts from the starting point, it generates a fixed departure cost. Therefore, the departure cost is related to the total departure times, as shown in Equation 3:

F_{2} = τ_{2} \sum_{k} P_{k}

(3)

A fare will be charged each time a passenger is served by a vehicle. Ticket prices affect enterprise incomes. The higher the income, the smaller the net operating cost required by enterprises. The revenue from fare $F_{3}$ is given in Equation 4:

F_{3} = τ_{3} \sum_{i} \sum_{k} y_{i}^{k}

(4)

The early arrival of vehicles only results in a wait for passengers onboard. There is no impact on passengers at the demand point. In this case, vehicles are required to wait until passengers arrive. Thus, passenger satisfaction with the service is reduced, and a certain penalty cost is incurred. The penalty cost of early vehicle arrival is related to the time when the vehicle reaches the demand point, the time window of the passengers at the demand point, the number of passengers onboard when reaching the demand point, and the type of passengers at the demand point, as shown in Equation 5:

F_{4} = \sum_{i} \sum_{k} f_{1} (t) M_{i}^{k} y_{i}^{k} θ_{i}^{k}

(5)

Similarly, late arrival of vehicles leads to a wait for passengers at the demand point, whose satisfaction is reduced. Therefore, a corresponding penalty cost is generated. If the vehicle arrives earlier than the passengers, no penalty is incurred. The penalty cost of late vehicle arrival is related to the time when the vehicle arrives at the demand point, the time window, and the type of passengers at the demand point, as shown in Equation 6:

F_{5} = \sum_{i} \sum_{k} f_{2} (t) y_{i}^{k} θ_{i}^{k}

(6)

For the convenience of model expression, the boarding or alighting demand of each passenger in the service area is set as a demand point. The passenger demand is numbered. Assuming the demand of passengers is $n$ , we set the start and end terminal as 0, $V = {0, 1, 2, . . ., n}$ is a collection of all demand points, passenger demand $i, j \in V$ , $i \neq j$ , the total demand points $N = n + 1$ , and $K$ is the collection of bus operation lines in the area. For the establishment of the model, the following constraints must be met:

0 < q^{k} \leq Q, \forall k \in K

(7)

q^{k} + \sum_{i \in V, i \neq 0} θ_{i}^{k} y_{i}^{k} \leq Q, \forall k \in K

(8)

Equations 7 and 8 express the constraints of vehicle carrying capacity, that is, passengers getting on and off the bus. As well, the number of passengers onboard should not exceed the vehicle’s capacity when it leaves the terminal:

\sum_{i} \sum_{j} x_{ij}^{k} \frac{S_{ij}}{v_{ij}} + \sum_{i} y_{i}^{k} t_{u} \leq T_{max}, \forall k \in K

(9)

Equation 9 expresses the one-way travel time limit of the vehicle, that is, each line has a maximum travel time:

\sum_{k} y_{i}^{k} = 1, \forall i \in V, i \neq 0

(10)

Equation 10 indicates that passengers at each demand point are served and only served by one vehicle:

A_{i}^{k} = (G^{k} + \frac{S_{0 i}}{v_{0 i}}) x_{0 i}^{k}, \forall i \in V 且 i \neq 0, \forall k \in K

(11)

B_{i}^{k} = max {A_{i}^{k}, U_{i}^{k}} + t_{u}, \forall i \in V 且 i \neq 0, \forall k \in K

(12)

B_{i}^{k} + \frac{S_{ij}}{v_{ij}^{k}} + (1 - x_{ij}^{k}) \cdot R \geq A_{j}^{k}, \forall i, j \in V, k \in K

(13)

B_{i}^{k} + \frac{S_{ij}}{v_{ij}^{k}} - (1 - x_{ij}^{k}) \cdot R \leq A_{j}^{k}, \forall i, j \in V, k \in K

(14)

Equations 11–14 are time constraints and represent the relationship among vehicle arrival time, departure time, and service time at the station:

f_{1} (t) = {\begin{array}{l} α_{1} (E T_{i} - A_{i}^{k}) + β_{1} (E T_{i}^{d} - E T_{i}) & A_{i}^{k} < E T_{i}, θ_{i}^{k} = 1 \\ β_{1} (E T_{i}^{d} - A_{i}^{k}) & E T_{i} \leq A_{i}^{k} \leq E T_{i}^{d}, θ_{i}^{k} = 1, \\ 0 & A_{i}^{k} > E T_{i}^{d} or θ_{i}^{k} = - 1 \end{array} \forall i \in V, k \in K

(15)

f_{2} (t) = {\begin{array}{l} 0 & A_{i}^{k} < L T_{i}^{d} or θ_{i}^{k} = - 1 \\ β_{2} (A_{i}^{k} - L T_{i}^{d}) & L T_{i}^{d} \leq A_{i}^{k} \leq L T_{i}, θ_{i}^{k} = 1, \forall i \in V, k \in K \\ α_{2} (A_{i}^{k} - L T_{i}) + β_{2} (L T_{i} - L T_{i}^{d}) & A_{i}^{k} > L T_{i}, θ_{i}^{k} = 1 \end{array}

(16)

Equation 15 is the penalty function for vehicles that arrive early and is set as the soft time window. Early vehicle arrival affects all passengers in the vehicle. Equation 16 is the penalty function for vehicles that are late, and passengers at the demand point must wait until the vehicle arrives:

\sum_{j \in V, j \neq 0} x_{0 j}^{k} = \sum_{i \in V, i \neq 0} x_{i 0}^{k}, \forall k \in K

(17)

\sum_{i} \sum_{j} x_{ij}^{k} = \sum_{i} \sum_{j} x_{ji}^{k}, \forall k \in K

(18)

\sum_{k} P_{k} \geq m

(19)

Equation 17 indicates that the start and end points are the same depot, that is, each line must start from and end at the terminal station. Equation 18 is the conservation constraint of station flow, which posits that the vehicle must leave after arriving and serving at a station. Equation 19 indicates that the total number of departure lines is greater than the number of vehicles.

Adaptive Genetic Algorithm

The GA is a stochastic optimization method based on a parallel search mechanism. Compared to branch and bound, tabu search, and simulated annealing, the GA has higher efficiency ( 36 ). In particular, it has good performance in solving NP-hard problems ( 49 ). The traditional idea is, to begin with, an initial population, and one selects individuals using the natural law of survival of the fittest. Then, through crossover and mutation, a brand-new generation of populations is born. In the process, the population evolves generation by generation until it satisfies its objectives. The GA comprises three operators, namely selection, crossover, and mutation. They usually proceed randomly according to certain selection, crossover, and mutation probabilities, which cause differences between children and parents. However, the traditional GA has poor local search capabilities. This leads to relatively time-consuming solutions and less efficient searches in the later stages of evolution. In practice, the GA is prone to the problem of premature convergence. The choice of crossover and mutation probabilities can also seriously affect the quality of the solution. In the traditional GA, the values of crossover probability and mutation probability are fixed. Consequently, the efficiency of the algorithm may be affected and the results may easily fall into a local optimum. Besides, it is very unfair between superior and inferior individuals. Therefore, the introduction of an adaptive mutation strategy could be a viable solution to the above problems, wherein the mutation operator and probability are dynamically adjusted. The specific description is as follows.

(1) Code. The solution to the passenger transfer problem is encoded into chromosomes. The start and end points of this area are encoded as 0, and the demand points in between are represented as $1, 2, \dots, n$ . When the vehicle on path $k$ completes a transfer and returns to the starting point, the chromosome length of a distribution route is encoded as $n + k - 1$ .

As a limitation of the coding length, the initial population considers the capacity of the pick-up vehicles and the maximum travel time of one way. Passenger requests across different paths are connected with 0 to form an initial chromosome.

(2) Set the initial population.

Step 1: A fixed departure time from the terminal is given.

Step 2: The nearest uncompleted demand point to the last service demand point is selected. If this demand point satisfies the constraint, it is inserted into the current run path.

Step 3: Repeat Step 2. If the current vehicle capacity limit or maximum run time limit is reached, another departure is added until all demand points are completed, and then the calculation is complete.

(3) Population fitness function. We take $F$ as the fitness value:

F = \frac{1}{C} = \frac{1}{γ_{1} (F_{1} + F_{2} - F_{3}) + γ_{2} (F_{4} + F_{5})}

(20)

The least fit individual is covered by the highest performing one, and the chromosome with the largest fitness is selected as the optimal solution of the system.

(4) Select. We use the roulette method for selecting chromosomes for the next generation. The probability of each individual being selected is directly proportional to his fitness value. The larger the fitness function value, the greater the probability of being chosen.

(5) Crossover. The selection made for a crossover of populations is an analog binary crossover operation. The crossover operator crosses the gene chains of the two selected individuals $x_{1}$ , $x_{2}$ with a certain probability, thus generating two new individuals. Implemented by Equations 21–23, the crossover position is random:

{x_{1, j}}^{'} = 0.5 \times [(1 + γ_{j}) x_{1, j} + (1 - γ_{j}) x_{2, j}]

(21)

{x_{2, j}}^{'} = 0.5 \times [(1 - γ_{j}) x_{1, j} + (1 + γ_{j}) x_{2, j}]

(22)

γ_{j} = {\begin{matrix} {(2 u_{j})}^{1 / (η_{1} + 1)} & u_{j} < 0.5 \\ {(1 / (2 (1 - u_{j})))}^{1 / (η_{1} + 1)} & else \end{matrix}

(23)

where $η_{1}$ takes a value of 2; $j \in [1, m]$ , where $m$ represents the number of optimized variables; and $u_{j} = rand (1)$ , which is a random number among [0,1].

(6) Adaptive mutation. The adaptive GA is an improvement on the traditional GA, where the mutation operation is conducted as follows:

{x_{i, j}}^{''} = {x_{i, j}}^{'} + Δ_{j}, j \in [1, S]

(24)

Δ_{j} = {\begin{matrix} {(2 u_{j})}^{1 / (η_{1} + 1)} & u_{j} < 0.5 \\ {(1 / (2 (1 - u_{j})))}^{1 / (η_{1} + 1)} & else \end{matrix}

(25)

where $η_{1}$ takes a value of 5 and $u_{j} = rand (1)$ , which is a random number among [0,1].

The probability of variation is generally chosen to be within 0.1. If the probability of mutation is too small, the diversity of the population declines too quickly. It tends to cause a rapid loss of effective genes and cannot be easily repaired. If the probability of mutation is too large, the probability of disruption of higher-order patterns increases, even though the diversity of the population is ensured. A fixed mutation probability cannot satisfy the needs of the population evolution process. That is, in the early stages of the iteration, individual performance is slightly poorer and higher variation probabilities are necessary to achieve a rapid search for the optimal solution. Then, as the iteration number increases, smaller mutation probabilities are required to help converge quickly after searching for the optimal solution. Therefore, an adaptive mutation strategy is introduced. By adjusting the mutation probability dynamically, it is related to the number of iterations. This will reduce the possibility of early maturation phenomena. The specific improvements are as follows:

P_{m} = (P_{m_max} - P_{m_min}) \times (1 - \frac{i}{T_{max}}) + P_{m_min}

(26)

where $P_{m_max}$ is the maximum variation probability; $P_{m_min}$ is the minimum variation probability; $i$ is the number of current iterations; and $T_{max}$ is the maximum number of iterations. The mutation operator is also improved in this study, as shown in Equations 27 and 28. A random number related to the number of iterations is introduced. With increasing numbers of iterations, the random number reduces. In this way, the gap between the child and the parent generation becomes narrower and narrower, which helps to ensure its stability:

G_{_child} = {\begin{matrix} G_{_parent} + (U_{_limit} - G_{_parent}) \times fg r_{1} > 0.5 \\ G_{_parent} - (G_{_parent} - L_{_limit}) \times fg r_{1} \leq 0.5 \end{matrix}

(27)

fg = {[r_{2} \times (1 - \frac{i}{T_{max}})]}^{2}

(28)

where $G_{_child}$ stands for the child generation; $G_{_parent}$ stands for the parent generation; $U_{_limit}$ and $L_{_limit}$ are the upper and lower limits, respectively; $fg$ is the mutation random number; and $r_{1}$ and $r_{2}$ are random numbers among [0,1].

The random number $fg$ is changed according to the number of iterations. That is, the larger the number of iterations, the smaller the random number $fg$ generated.

(7) Algorithm termination. Using the maximum number of iterations is proposed to control the maximum running time of the algorithm. In this study, the algorithm terminated after 300 iterations.

Numerical Example

Parameter Setting

This section uses simulation to verify the validity and feasibility of the model and algorithm. A right-angle coordinate system is established with the starting point (0,0) as the origin, and the service area is set to 3 × 3 mi². A total of 100 passenger demand coordinates are randomly generated in the service area. Among these, 80 passengers travel to the terminal from their reserved demand points. Conversely, the remaining 20 passengers travel from the terminal to their reserved demand points. The location information of the demand points includes demand point coordinates, passenger type, and vehicle arrival time (Table 2). The passenger type of 1 indicates that the demand is from the service area to the terminal, and that of −1 means the demand is from the starting point to the service area. The period is set as 7:00–9:00 a.m. According to the actual situation, the parameters are set as $v_{ij}$ = 25 mph, $Q$ = 10 persons/bus, $T_{max}$ = 40 min, $τ_{1}$ = 0.5 $/mile, $τ_{2}$ = 10 $/trip, $τ_{3}$ = 1.5 $/person, $γ_{1}$ = 0.7, $γ_{2}$ = 0.3, and $t_{u}$ = 5 s. The passenger time penalty coefficient is $α_{1}$ = 10 $/h, $β_{1}$ = 2 $/h, $α_{2}$ = 15 $/h, and $β_{2}$ = 3 $/h. To improve passengers’ travel satisfaction, this work adopts a soft time window constraint for vehicles to reach the demand point. That is, the range of the passengers’ expected service time window is 5 min, and the range of the tolerable one is 15 min.

Table 2.

Location Information of Demand Points

Demand point number	Passenger type	Demand point coordinate	Vehicle arrival time (a.m.)	Demand point number	Passenger type	Demand point coordinate	Vehicle arrival time (a.m.)
1	1	(2.86,1.79)	8:40	51	1	(2.79,1.68)	7:32
2	−1	(2.52,1.33)	7:33	52	1	(0.53,1.85)	7:28
3	1	(2.51,1.56)	8:10	53	1	(1.03,1.08)	7:56
4	1	(0.07,1.13)	8:29	54	1	(1.26,2.38)	7:26
5	−1	(2.70,1.29)	7:28	55	−1	(1.01,2.65)	7:15
6	1	(0.60,0.91)	7:31	56	1	(1.38,0.00)	7:37
7	1	(1.61,2.73)	8:49	57	−1	(2.21,1.28)	7:14
8	1	(1.58,0.92)	8:32	58	1	(0.01,2.90)	7:52
9	−1	(0.10,2.15)	7:25	59	1	(2.01,2.69)	7:21
10	1	(2.31,0.18)	7:40	60	1	(0.61,1.67)	7:23
11	1	(1.88,0.80)	7:12	61	1	(2.13,1.44)	7:13
12	−1	(0.94,1.57)	7:20	62	1	(2.19,1.81)	7:33
13	1	(1.23,2.68)	8:18	63	−1	(2.96,2.71)	7:22
14	1	(1.72,1.70)	7:29	64	1	(1.57,1.23)	7:11
15	1	(1.18,2.07)	7:32	65	1	(0.88,1.10)	7:35
16	1	(1.71,1.03)	8:46	66	1	(0.80,1.19)	8:11
17	−1	(1.78,0.82)	7:49	67	1	(2.68,0.68)	8:07
18	−1	(0.14,2.51)	7:23	68	−1	(1.08,0.52)	7:10
19	1	(0.31,2.18)	7:32	69	1	(1.67,1.21)	8:01
20	1	(1.32,2.99)	7:23	70	−1	(2.93,1.92)	7:17
21	−1	(1.69,1.40)	7:14	71	1	(2.91,0.94)	8:32
22	1	(1.26,0.63)	7:37	72	1	(1.22,2.06)	8:02
23	1	(0.71,0.94)	8:14	73	1	(0.48,2.32)	7:28
24	1	(2.61,0.76)	7:36	74	1	(0.92,0.74)	7:17
25	1	(2.91,0.91)	7:29	75	1	(0.96,2.02)	7:41
26	1	(1.12,1.05)	8:18	76	1	(1.20,2.87)	7:37
27	1	(2.10,1.84)	8:14	77	1	(0.56,1.30)	7:56
28	1	(2.28,2.40)	7:15	78	1	(2.02,1.35)	7:33
29	−1	(0.65,2.68)	7:26	79	−1	(1.72,2.70)	8:42
30	1	(2.70,2.82)	7:37	80	1	(2.53,1.09)	7:14
31	1	(0.82,1.43)	7:27	81	1	(0.84,0.76)	7:18
32	1	(0.76,1.09)	7:35	82	1	(0.73,0.02)	8:34
33	1	(1.58,1.14)	7:37	83	1	(2.41,2.85)	7:18
34	1	(2.47,0.94)	7:52	84	−1	(1.43,1.05)	8:00
35	1	(1.99,1.94)	8:03	85	1	(1.50,0.97)	7:36
36	1	(1.41,2.15)	7:35	86	1	(1.23,1.13)	7:11
37	1	(0.06,1.44)	7:46	87	1	(0.86,0.11)	8:41
38	1	(2.97,2.93)	7:56	88	1	(0.05,2.96)	7:24
39	1	(1.49,0.12)	8:04	89	1	(0.23,0.58)	7:19
40	1	(0.40,2.80)	7:27	90	1	(1.34,0.38)	7:11
41	1	(0.05,0.18)	8:11	91	−1	(2.45,0.91)	7:31
42	1	(1.52,1.13)	7:27	92	1	(0.08,2.86)	8:09
43	−1	(0.31,1.52)	7:24	93	1	(1.93,0.46)	8:23
44	1	(0.69,0.58)	7:15	94	−1	(2.55,0.60)	7:56
45	1	(1.62,2.73)	8:00	95	−1	(2.81,0.05)	7:14
46	1	(1.74,0.62)	8:32	96	1	(1.35,1.15)	8:36
47	1	(2.65,0.86)	7:14	97	1	(1.23,2.73)	7:23
48	1	(0.45,2.93)	8:07	98	1	(2.36,0.39)	7:37
49	1	(0.80,0.43)	7:20	99	1	(1.20,2.46)	8:10
50	−1	(2.57,0.37)	7:52	100	1	(0.59,2.76)	8:21

Algorithm Feasibility Analysis

To verify the advantages of the adaptive GA, it is used together with the traditional GA to solve the regional flexible transit route optimization problem under mixed demand. A comparison of the solution results is illustrated in Figures 1 and 2. When the adaptive GA is used to solve the problem, the population size is set to 100, the maximum number of iterations is 300, the crossover probability is 0.9, the minimum mutation probability is 0.01, and the maximum mutation probability is 0.1. In the traditional GA, the mutation probability is given a set of 10 values between 0.01 and 0.1 to enhance the reliability of the conclusion. The rest of the parameters remain unchanged.

Figure 1.

Iterative curve before improvement.

Figure 2.

Improved iteration curve.

Figures 1 and 2 show that the improved GA has stronger global optimization ability. This is because fewer iterations are required to obtain the optimal solution and the objective function is smaller. The total cost of the improved algorithm is less than that before the improvement. This outcome effectively improves the quality of understanding and verifies the effectiveness of the improved algorithm.

Results and Discussion

Mixed Demand Model Versus Separate Delivery and Separate Pick-Up Model

The passenger path information derived from the MATLAB simulation is shown in Table 3 for the mixed mode and presented in Tables 4 and 5 for the single mode.

Table 3.

Passenger Routes Under the Mixed Demand Mode

Vehicle number	Vehicle path	Vehicle departure time (a.m.)	Vehicle arrival time (a.m.)	Path length (miles)	Seat occupancy rate
1	0-47-81-12-97-31-91-65-85-0	7:06	7:41	13.68	80%
2	0-61-21-89-18-54-52-6-24-98-0	7:05	7:43	14.43	90%
2	0-37-34-45-48-66-23-100-0	7:42	8:26	13.75	70%
2	0-46-96-4-87-7-0	8:28	8:55	10.98	50%
3	0-68-90-57-74-20-5-15-56-0	7:05	7:39	12.80	80%
4	0-64-44-63-73-78-30-0	7:04	7:44	16.08	60%
4	0-50-53-72-92-0	7:45	8:16	9.54	40%
5	0-95-59-9-40-2-0	7:08	7:41	13.71	50%
6	0-28-70-88-29-51-62-22-75-0	7:07	7:46	15.68	80%
6	0-38-35-3-41-0	7:47	8:12	9.22	40%
7	0-11-80-83-43-42-14-19-36-76-0	7:06	7:45	15.34	90%
7	0-17-94-69-67-27-26-93-8-1-0	7:46	8:46	15.51	90%
8	0-86-55-49-60-25-32-33-10-0	7:06	7:46	15.62	80%
8	0-58-77-84-39-99-13-0	7:46	8:26	11.95	60%
8	0-71-82-79-16-0	8:26	8:53	10.95	40%

Table 4.

Passenger Routes Under the Separate Pick-Up Mode

Vehicle number	Vehicle path	Vehicle departure time (a.m.)	Vehicle arrival time (a.m.)	Path length (miles)	Seat occupancy rate
1	0-80-40-4-46-10-48-0	7:05	7:45	16.94	60%
1	0-14-22-36-44-86-0	7:46	8:13	10.29	50%
2	0-56-28-47-66-24-59-33-69-71-0	7:07	7:50	18.33	90%
2	0-20-75-45-96-82-0	7:50	8:19	9.90	50%
3	0-51-13-25-15-7-67-0	7:07	7:45	16.49	60%
3	0-89-97-98-27-74-77-37-6-65-0	7:45	8:25	13.38	90%
3	0-38-34-32-0	8:26	8:48	10.26	30%
4	0-93-35-78-88-39-62-61-83-0	7:08	7:50	18.15	80%
4	0-49-30-76-19-11-16-0	7:50	8:27	11.97	60%
4	0-26-53-42-1-0	8:27	8:56	8.00	40%
5	0-54-52-92-41-64-23-0	7:06	7:36	12.31	60%
6	0-100-8-81-0	7:07	7:24	7.79	30%
7	0-99-58-73-72-60-85-31-90-87-0	7:06	7:41	14.96	90%

Table 5.

Passenger Routes Under the Separate Delivery Mode

Vehicle number	Vehicle path	Vehicle departure time (a.m.)	Vehicle arrival time (a.m.)	Path length (miles)	Seat occupancy rate
1	0-84-17-21-55-12-43-0	7:06	7:26	8.47	60%
2	0-9-18-29-68-50-57-2-5-91-0	7:06	7:36	12.30	90%
3	0-95-94-70-63-79-0	7:06	7:33	11.03	50%

Under the same demand and parameters, the solution results of the route optimization model under the mixed and separate demand modes are compared (Table 6). If the vehicle is responsible exclusively for transporting passengers from an area to the terminal, then at least seven vehicles are needed to meet the constraints. By contrast, when the vehicle is only responsible for delivering passengers from the starting point to the area, a minimum of three vehicles is necessary. Compared with the mixed demand counterpart, the single demand mode requires two more vehicles to serve the same passengers. This means that the number of vehicles required increases by 25%, thereby making the cost higher. The per capita driving mileage is 2.01 mi/person in the separate demand mode and 1.99 mi/person in the mixed one. As a result, the average mileage per passenger is reduced by 1.0%. In addition, the total cost is $127.84 for the mixed demand mode and $132.42 for the separate mode. The cost of the mixed demand mode is 3.5% less. For the tolerable time window, the average time for vehicles arriving early or late was 5.08 min for the mixed mode and 4.69 min for the separate one. The fewer the number of departures, the longer the waiting time for passengers, resulting in higher penalty costs. Meanwhile, the decrease in the number of vehicles required and departures made would lead to a reduction in operating costs. This indicates that the effect on total costs because of the early or late arrival of vehicles is minor compared to operating costs.

Table 6.

Comparison of the Two Modes

Mode	Total mileage (miles)	Per capita mileage (miles/person)	Number of vehicles required	Total number of departures	Total cost ($)	Avg. of early or late arrival time of vehicles (min)
Mixed demand model	199.24	1.99	8	15	127.84	5.08
Separate pick-up and delivery model	200.57	2.01	10	16	132.42	4.69

During testing, 10 different sets of demand points were randomly generated for validation. The comparison results are shown in Table A1. The vast majority of results prove the superiority of the mixed demand mode. With respect to the mixed demand model, the average improvement in cost is 2.35%.

Sensitivity Analysis in the Mixed Demand Model

A sensitivity analysis is used to discuss the impact of several factors on operational efficiency. Holding all other conditions constant, only the vehicle speed, the maximum one-way travel time, and the weighting ratios between operating and penalty costs are changed. Following this, the effects of these factors on the total cost, number of trips, miles traveled per capita and average seat utilization are analyzed.

Table 7 shows the effect of vehicle speed on each optimization result. Given the morning peak hour, 15–35 mph is a reasonable speed for analysis. Table 7 indicates that the total cost first reduces and then increases as the vehicle speed grows, with a minimum objective value of $119.36 at the speed of 30 mph. The number of departures and per capita mileage decrease with increasing speed; however, the average seat occupancy rate increases with the rising speed, reaching 90.9% at the speed of 35 mph. The number of passengers served in a round trip is reduced when the speed is low under the maximum travel time constraint, resulting in a poorer average seat occupancy rate. Consequently, the number of departures needs to be raised to meet demand. The total and per capita mileage are accordingly increased. At the same time, faster or slower speeds can both lead to an increase in the time penalty cost, thus making the total cost larger. A moderate speed must be chosen for flexible transit operation.

Table 7.

Optimal Solution for Vehicle Speed Variations

Vehicle speed (mph)	Total cost ($)	Total number of departures	Total mileage (mile)	Per capita mileage (miles/person)	Average seat occupancy rate
15	281.03	29	238.38	2.38	34.5%
20	139.49	19	206.85	2.07	52.6%
25	127.84	16	199.24	1.99	62.5%
30	119.36	12	194.85	1.95	83.3%
35	132.86	11	185.13	1.85	90.9%

Table 8 reveals that the values of the optimization results vary greatly when the maximum one-way travel time is changed. As it decreases, the total cost of the system increases, with a corresponding rise in the number of departures and per capita mileage, and a reduction in the average seat occupancy rate. With a maximum travel time of 40 min as a basis, when it is reduced to half, the total cost increases by 140.6%, and the mileage per person is 2.67 mi/trip, which is up by 34.2%. In reality, as the maximum one-way travel time of the vehicle becomes smaller, the number of departures is increased to meet passenger demand. Fewer passengers are then carried in a vehicle, leading to low occupancy, which is consistent with the results obtained.

Table 8.

Effect of Maximum One-Way Travel Time on Optimization Results

Maximum one-way travel time (min)	Total cost ($)	Total number of departures	Total mileage (mile)	Per capita mileage (miles/person)	Average seat occupancy rate
20	292.51	36	267.21	2.67	27.8%
25	223.48	28	244.20	2.44	35.7%
30	167.46	23	229.22	2.29	43.5%
35	155.03	19	223.37	2.23	52.6%
40	127.84	16	199.24	1.99	62.5%
45	119.78	13	186.87	1.87	76.9%
50	116.02	12	185.37	1.85	83.3%

The study also explores the impact of different weighting ratios between operating and penalty costs on the results. Table 9 shows that when the weight of operating costs is 0.9, the least total cost is $70.05 and the minimum number of departures is 14. When the weight of operating costs is 0.1, the highest total cost is $201.32 and the highest number of departures is 18. The analysis reveals that as the penalty cost weight is increased, the penalty cost should be made as small as possible to minimize the total cost. Therefore, the number of departures should be increased so that the vehicles can better cater to the time windows of each demand. If the number of departures increases, then the number of passengers carried per vehicle decreases.

Table 9.

Effects of Weighting Ratios on Operations

Weighting ratios between operating and penalty costs	Total cost ($)	Total number of departures	Total mileage (mile)	Per capita mileage (miles/person)	Average seat occupancy rate
9:1	70.05	14	183.02	1.83	71.4%
7:3	127.84	16	199.24	1.99	62.5%
5:5	167.72	17	206.6	2.07	58.8%
3:7	183.61	17	210.13	2.10	58.8%
1:9	201.32	18	210.66	2.11	55.6%

Conclusion and Future Works

This study develops a routing optimization model for regional flexible transit by considering the mixed demand mode and proposes an improved adaptive genetic solution algorithm. We numerically verify the feasibility of the proposed model and the validity of the algorithm. Our results indicate the following.

Firstly, the proposed algorithm can efficiently solve the optimal routes of flexible transit, significantly improves the quality of the solution, and greatly reduces the system cost compared with the traditional GA.

Secondly, using the mixed mode can reduce vehicle departures, mileage, and costs compared with the separate pick-up and delivery modes.

Thirdly, under the same demand conditions, the vehicle speed, maximum one-way travel time limit, and weights of cost affect the total cost and operational efficiency of the flexible transit system to an extent. Choosing a moderate speed is beneficial to efficiency. In addition, the greater the maximum one-way travel time limit within a certain range, the better for vehicle scheduling. Moreover, the proportional setting of the cost components has a significant impact on the operating results, wherein increasing the weight of operating costs can reduce the total cost.

This work has certain guiding significance for the development of regional flexible transit. Further work can be conducted in the following avenues. We only consider static reservation demand, so incorporating dynamic trip demand into route optimization is meaningful. As our insights are derived from hypothetical numerical examples, more empirical studies should be conducted to verify our conclusions.

Supplemental Material

sj-docx-1-trr-10.1177_03611981231162600 – Supplemental material for Routing Optimization of Regional Flexible Transit Under the Mixed Demand Mode

Supplemental material, sj-docx-1-trr-10.1177_03611981231162600 for Routing Optimization of Regional Flexible Transit Under the Mixed Demand Mode by Siqing Wang, Jian Wang, Xiaowei Hu, Tingting Dong and Zhipeng Niu in Transportation Research Record

Footnotes

Author Contributions

The authors confirm contribution to the paper as follows: study conception and design: S. Wang, J. Wang, X. Hu; data collection: S. Wang, Z. Niu; analysis and interpretation of results: S. Wang, T. Dong; draft manuscript preparation: S. Wang, J. Wang, X. Hu, T. Dong, Z. Niu. All authors reviewed the results and approved the final version of the manuscript.

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research was supported by the Major Research Plan of the National Natural Science Foundation of China (91846301), Heilongjiang Philosophy and Social Science Research Planning Project (20GLC204).

ORCID iD

Zhipeng Niu

Data Accessibility Statement

Some or all data, models, or codes that support the findings of this study are available from the corresponding author on reasonable request.

Supplemental Material

Supplemental material for this article is available online.

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