Research on Optimization of Multi-Objective Regional Public Transportation Scheduling

: The optimization of bus scheduling is a key method to improve bus service. So, the purpose of this paper is to address the regional public transportation dispatching problem, while taking into account the association between the departure time of buses and the waiting time of passengers. A bi-objective optimization model for regional public transportation scheduling is established to minimize the total waiting cost of passengers and to maximize the comprehensive service rate of buses. Moreover, a NSGA-II algorithm with adaptive adjusted model for crossover and mutation probability is designed to obtain the Pareto solution set of this problem, and the entropy weight-TOPSIS method is utilized to make a decision. Then the algorithms are compared with examples, and the results show that the model is feasible, and the proposed algorithms are achievable in solving the regional public transportation scheduling problem.


Introduction
With the continuous expansion of urban space scale and population, public transportation has become a travel choice for more and more people. On ordinary weekdays, the number of public transport trips in Beijing is about 6.1 million people/day, accounting for 30% of the city's permanent resident population, and on weekends, it is about 4.4 million people/day. Among them, the proportion of the direct bus travel is only 43.1% [1]. However, the lack of regional public transportation coordinated dispatching tends to result in a consequent worsening of both the magnitude and variability of the average waiting time. This in turn impacts heavily impacts the level of service. Owing to the inconvenience of transfer, public transportation in China is not attractive to travelers. Some studies have shown that transfer is one of the main factors that affect the travel rate of public transportation [2]. The travel time with transfer is obviously longer than the travel time without transfer in Beijing. The average time of a trip in the public transportation system is about 1 h, and the transfer time is more than 10 min [1,3]. So, it is necessary to study the coordinated dispatching problem of regional public transportation to reduce the transfer waiting time of passengers and improve the service level of public transportation.

Literature Review
Compared with single-line public transportation dispatching, regional public transportation dispatching can make public resources more fully utilized [4,5]. Transfer is the main factor of regional public transportation scheduling, and synchronous optimization of regional public transportation timetable can improve the transfer efficiency. Therefore, some researchers have studied the problem of synchronous transfer of regional public transportation scheduling. Liebchen and Stiller [6] considered the bus timetable affected by running late to achieve the optimization of departure interval and fleet size. Liu and Shen [7] established a bilevel programming model between the timetable generation and of easy transfer buses in the process of dispatching, it may lead to ignoring the transfer satisfaction of passengers at other transfer stations. So, the number of passengers who have not realized easy transfer at all the transfer stations should be taken into account so as to improve the service level of public transport.
Some assumptions involved in this paper are as follows: (1) The buses operating on each line are considered to be full-range buses.

Notations
(1) Parameters L = {1, 2, . . . , N} is the bus line, N is the number of lines; s is the station set, s i,q represents station i of line q, and m i is the number of stations on line i; S t = (s i,q , s j,p ) s i,q , s j,p ∈ S, i, j ∈ L is the set of pairs for transfer stations; x i ∈ X implies the bus x uses line i, where X is the bus set; V is the passenger capacity of the bus; p a x i s i,q , p g x i s i,q denote the arrival number and alighting rate of passengers on the bus x i at station s i,q , respectively; f s i,q s j,p is the proportion of passenger transfer from line i to j at transfer-station s i,q , s j,p ; Q a x i s i,q and Q g x i s i,q indicate the number of passengers getting on and off the bus x i at station s i,q of line i; F x i s i,q is the number of passengers who failed to get on the bus x i of line i at station s i,q ; C x i s i,q indicates the number of passengers on the bus x i of line i at station s i,q ; t s i,q−1 s i,q denotes the travel time from station s i,q−1 to station s i,q , when q = 1, t s i,q−1 s i,q represents the time from the parking lot to the departure station; U x i s i,q indicates the dwelling time of bus x i of line i at station s i,q ; δ 1 and δ 2 indicate the time window for easy transfer; t x i s i,q indicates the time when the bus x i of line i arrives at station s i,q ; l x i s i,q denotes the time when the bus x i of line i leaves station s i,q ; T 1 and T 2 are the scheduling simulation time window; I min and I max indicate the minimum and maximum departure interval during the simulation; T s j,p s i,q x j x i 1, δ 1 ≤ t x i s i,q − l x j s j,p ≤ δ 2 0, other indicates whether the bus x j of line j at station s j,q can be easily transferred to the bus x i of station s i,q on line i, where δ 1 , δ 2 are the time window for synchronous transfer; P indicates the proportion of choice other transportation modes for passengers who are failed to get on the bus; Y x i s i,q indicates the number of passengers who failed to get on the bus at the bus x i of line i at station s i,q and chose other transportation modes; indicates whether the two lines can be transferred.
(2) Decision variables A x i indicates the departure time of bus x i of line i from the parking lot.

Operation Analysis
(1) Analysis of bus operation process The dwelling time of each bus can be expressed as U x i s i,q = a 0 + max(a 1 Q g x i s i,q , a 2 Q a x i s i,q ), where a 0 is the time of buses to enter and leave the station, a 1 and a 2 indicate the average boarding and alighting time for each passenger, respectively.
The bus arrival time at the departure station can written as t x i s i,1 = A x i + t s i,0 s i,1 , where t s i,0 s i,1 indicates the running time of line i from the parking lot to the departure station, and the bus departure time can be computed as l The total running time of a bus can be calculated by l x i s i,q − A x i .
(2) Passenger number analysis The number of passengers getting off the bus Q g x i s i,q is determined by the number of passengers on the bus, C x i s i,q , and the alighting rate of passengers, p g x i s i,q , which can be calculated using Q g There are two aspects to consider in terms of the number of passengers getting on the bus: the remaining capacity of the bus and the number of passengers the who expect to board the bus which includes the number of people arriving randomly, the number of people who need to transfer, and the number of people who fail to get on the previous bus (do not include those who choose other modes of transportation). So, the number of passengers getting on the bus can be formulated as follows: Then, the number of passengers on the bus at station, C x i s i,q , is the number of passengers on the vehicle, C x i −1s i,q , minus the number of passengers getting off the vehicle, , plus the number of passengers getting on the vehicle, which can be obtained: In addition, the number of passengers who have not achieved easy transfer can be expressed as Q g x j s j,p f s j,p s i,q (1 − T s j,p s i,q x j x i ). Thus, the number of passengers who fail to get on the bus is the number of passengers who need to get on the bus in the current period minus the number of passengers who actually get on the vehicle, which can be computed as follows: (3) Analysis of passenger waiting time The waiting time for transfer passengers is the arrival departure interval of two buses with transfer relationship multiplied by the number of transfer people with easy transfer, which can be calculated as follows: Furthermore, the waiting time for non-transfer passengers includes the average waiting time of random arriving passengers and the extra waiting time of passengers who fail to get on the bus according to Larsen and Sunde [25], which can be calculated by The ratio R of the inter-station operating time of all buses to the total operating time is 1 minus the proportion of stop time, which can reflect the effective operation cost and can be expressed as: Algorithms 2021, 14, 108 5 of 15

The Optimization Model
Based on the above analysis, the first objective of this model is to minimize the total waiting cost of passengers who have not achieved easy transfer and those who have to choose other modes of transportation, and the waiting time cost of non-transfer passengers and transfer passengers. The second objective of this model is to consider the efficiency of public transport operation, including the average full load rate of buses and the ratio of inter-station running time of all buses to the total operation time. Then, the bi-objective optimization model for regional public transport scheduling can be formulated as follows: Formula (8) indicates the total waiting cost of passengers, c 1 , c 2 represent the waiting cost coefficients of passengers who did not realize easy transfer and choose other modes of transportation, and the time value cost of waiting time. Formula (9) indicates the comprehensive service ratio of buses, |X| is the number of elements of the set X. Constraint (10) ensures that the departure time is within the scheduling simulation time window. Constraint (11) is the bus capacity constraint. Constraint (12) indicates the arrival interval limits.

Solution Algorithms
As most of the multi-objective problems have to meet several objectives, it is impossible to obtain a unique solution that satisfies all the objectives simultaneously. Usually, Simulated Annealing (SA) [21], Ant Colony Optimization (ACO) [26], Multi-objective Heuristic Algorithms [27], Discrete Evolutionary Multi-objective Optimization (DEMO) [28], Niched Genetic Algorithm [5], Non-dominated Sorting Genetic Algorithm (NSGA) [29,30] are often chosen to solve multi-objective problems. One of the widely used algorithms proposed to solve multi-objective problems is the Non-dominated Sorting Genetic Algorithm-II (NSGA-II).
The scheduling of different bus lines will affect the transfer efficiency. Moreover, there are many schemes for each bus line and the two objectives conflict with each other. Therefore, it is necessary to design an algorithm that can deal with multi-objective public transportation scheduling quickly. In this section, a two-step algorithm is employed in order to solve this problem. The first step is to obtain the Pareto solutions, and the second step is to obtain a satisfactory scheme from the Pareto solution set on the basis of the entropy weight-TOPSIS method.

The NSGA-II Algorithm
The NSGA-II algorithm is one of the most commonly used multi-objective optimization algorithms, but it has some drawbacks of an uneven convergence of the population and the ease of falling into the local optimal solution [31]. In order to avoid premature convergence of Genetic algorithm, it is necessary to keep the diversity of population. According to the characteristics of this model, an adaptive adjusted model of the crossover and the mutation probability is introduced, and an adaptive crossover operator for judging the similarity of chromosomes and multipoint mutation operator are designed.
(1) Construction of chromosomes Chromosome X = (X 1 , X 2 , · · · , X N ) is designed by a positive integer, where gene X i = (A 1 , A 2 , · · · , A x i ) of the chromosome represents the set of the departure time for each bus of line i. Figure 1 illustrates the specific structure of a randomized chromosome.

The NSGA-II Algorithm
The NSGA-II algorithm is one of the most commonly used multi-objective optimization algorithms, but it has some drawbacks of an uneven convergence of the population and the ease of falling into the local optimal solution [31]. In order to avoid premature convergence of Genetic algorithm, it is necessary to keep the diversity of population. According to the characteristics of this model, an adaptive adjusted model of the crossover and the mutation probability is introduced, and an adaptive crossover operator for judging the similarity of chromosomes and multipoint mutation operator are designed.
(1) Construction of chromosomes Chromosome  of the chromosome represents the set of the departure time for each bus of line i. Figure 1 illustrates the specific structure of a randomized chromosome. The process of generating a chromosome is as follows: Step  The process of generating a chromosome is as follows: Step 1: Let x i = 1 and A 1 = r 0 (I max − I min ) + I min + T 1 (where r 0 is a random number obeying 0-1 uniform distribution).
(2) Select operator The selection of the parental population is using binary tournament method calculate the performing non-dominated sorting and crowding calculation of the parental population.
(3) Adaptive adjusted model for crossover and mutation probability In order to improve the search performance of the algorithm and avoid falling into the local optimal solution, an adjusted model is adopted to adjust the crossover probability and mutation probability.
The crossover probability can be adjusted by the following method.
where P c1 , P c2 are the crossover probability for different generations, β is the adjustment coefficient of crossover and mutation probability in the later stage of evolution. g is the current evolutionary generation, g 1 , g 2 are the dividing points of the early stage and the middle stage, and g max is the total maximum evolution generation. Furthermore, the mutation probability can be adjusted by the following method according to the running phase.
where L c represents the total number of bus departures for all lines in a chromosome, P m1 is the maximum mutation probability.
(4) Crossover operator According to the character of the chromosome structure, the traditional single-point crossover is easy to fall into the local optimal solution. Therefore, an adaptive crossover operator is designed through judging the similarity of chromosomes.
The process of adaptive crossover operator is stated as follows: Step1: Let i = 1, k = 0.
Step2: If i > N, then go to step 4. Otherwise, compare the number of bus departures of a line for the two selected chromosomes, if the departure times are equal, let the departure times of line i be x i and go to Step3, otherwise, let i = i + 1, return to Step2.
Step3: Count the number Z of equal departure time of lines i of the two chromosomes. If Z ≥ x i − 2, then regenerate the departure time of line i for one chromosome and exchange the genes of line i of two chromosomes, and let i = i + 1, k = k + 1, go to Step2.
Step4: If k ≤ 1, randomly generate two different lines and exchange the corresponding genes of two chromosomes.
In the case of Z ≥ x i − 2, the crossover process is shown in Figure 2. Assume that the first genes of parents P 1 , P 2 are highly similar to the N − 1 genes, then randomly select one chromosome from P 1 , P 2 , regenerate the departure time of the high similarity line, and exchange the corresponding genes of P 1 , P 2 (where C P q x i is the regenerated departure time). The crossover process when k ≤ 1 is shown in Figure 3.
Step4: If 1 k ≤ , randomly generate two different lines and exchange the corresponding genes of two chromosomes.
In the case of 2 i Z x ≥ − , the crossover process is shown in Figure 2. Assume that the first genes of parents 1 2 , P P are highly similar to the 1 N − genes, then randomly select one chromosome from 1 2 , P P , regenerate the departure time of the high similarity line, and exchange the corresponding genes of 1 2 , P P (where q i P x C is the regenerated departure time). The crossover process when 1 k ≤ is shown in Figure 3. Figure 2. The crossover process case    Step4: If 1 k ≤ , randomly generate two different lines and exchange the corresponding genes of two chromosomes.
In the case of 2 i Z x ≥ − , the crossover process is shown in Figure 2. Assume that the first genes of parents 1 2 , P P are highly similar to the 1 N − genes, then randomly select one chromosome from 1 2 , P P , regenerate the departure time of the high similarity line, and exchange the corresponding genes of 1 2 , P P (where q i P x C is the regenerated departure time). The crossover process when 1 k ≤ is shown in Figure 3.        Randomly select some places form line gene and regenerate these departure time according to the following conditions (A x i , A x i −1 and A x i +1 denote the departure time of bus x i , x i−1 and x i+1 respectively, and r 0 indicates a random number from uniform distribution u(0, 1)).

Case 1: When
Case 2: When x i = L g , A x i = A x i −1 + (I max − I min )r 0 + I min . Case 3:  Since Case 2 has only one calculation method, we will give examples of Cases 1 an 3 respectively, the shaded part is the point to be mutated, assuming that the departur interval is [3,10] min and the scheduling time window is 30 min.

Obtain Satisfactory Scheme from Pareto Solution Set
The second step is to obtain a satisfactory scheme from the Pareto solution set. Th entropy weight-TOPSIS method is an improvement over the entropy weight method an the TOPSIS method [32]. Moreover, it is a common evaluation method used in mu ti-objective decision-making problems. Consequently, the entropy weight-TOPSI method is used to obtain satisfactory schedule from the Pareto set.
Suppose that the number of Pareto solutions is M. Then, a decision matrix 2 M D × obtained with M Pareto solutions and 2 objectives. The process is stated as follows: Step 1: Obtaining the standardization matrix Step 2: Calculating the information entropy of each objective j: Step 3: Computing the weight of objective j:   Since Case 2 has only one calculation method, we will give examples of Cases 1 an 3 respectively, the shaded part is the point to be mutated, assuming that the departur interval is [3,10] min and the scheduling time window is 30 min.

Obtain Satisfactory Scheme from Pareto Solution Set
The second step is to obtain a satisfactory scheme from the Pareto solution set. Th entropy weight-TOPSIS method is an improvement over the entropy weight method an the TOPSIS method [32]. Moreover, it is a common evaluation method used in mu ti-objective decision-making problems. Consequently, the entropy weight-TOPSI method is used to obtain satisfactory schedule from the Pareto set.
Suppose that the number of Pareto solutions is M. Then, a decision matrix 2 M D × obtained with M Pareto solutions and 2 objectives. The process is stated as follows: Step 1: Obtaining the standardization matrix Step 2: Calculating the information entropy of each objective j: Step 3: Computing the weight of objective j: Since Case 2 has only one calculation method, we will give examples of Cases 1 and 3 respectively, the shaded part is the point to be mutated, assuming that the departure interval is [3,10] min and the scheduling time window is 30 min.

Obtain Satisfactory Scheme from Pareto Solution Set
The second step is to obtain a satisfactory scheme from the Pareto solution set. The entropy weight-TOPSIS method is an improvement over the entropy weight method and the TOPSIS method [32]. Moreover, it is a common evaluation method used in multiobjective decision-making problems. Consequently, the entropy weight-TOPSIS method is used to obtain satisfactory schedule from the Pareto set.
Suppose that the number of Pareto solutions is M. Then, a decision matrix D M×2 is obtained with M Pareto solutions and 2 objectives. The process is stated as follows: Step 1: Obtaining the standardization matrix Step 2: Calculating the information entropy of each objective j: Step 3: Computing the weight of objective j: Step 4: Computing the weighted judgment matrix: Algorithms 2021, 14, 108 9 of 15 Step 5: Calculating the Euclidean distance and closeness for each solution: Moreover, the satisfactory schedule can be obtained according to the closeness F i .

Example Analysis
A case study is carried out to test the dispatch approach and algorithm on a regional public transportation network with four bus lines, as shown in Figure 6. The scheduling simulation time window T 1 and T 2 were set to be 7:00, 7:30, respectively. The minimum and maximum departure interval, I min and I max , were set to be 3 min, 10 min, and the time window for easy transfer, δ 1 and δ 2 , were 1 min and 5 min, respectively. Let the waiting cost coefficient c 1 = 1.5, c 2 = 0.18. The passenger capacity of the bus V = 100.
Step 4: Computing the weighted judgment matrix: (17 Step 5: Calculating the Euclidean distance and closeness for each solution:

Example Analysis
A case study is carried out to test the dispatch approach and algorithm on a regiona public transportation network with four bus lines, as shown in Figure 6. The scheduling simulation time window 1 T and 2 T were set to be 7:00, 7:30, respectively. The mini mum and maximum departure interval, min Based on the investigation of bus transfer stations in line 7, line 15 and line 131 of Lanzhou City, it is found that the more the number of passengers who failed to get on the bus, the higher the proportion of passengers who choose other modes of transportation. The corresponding proportion of choice other transportation modes for those passengers is as follows: The size of population is 50, the maximum number of iterations is 150, the maximum mutation rate P m1 = 0.3, and P c1 = 0.6, P c2 = 0.4. The main transfer stations and transfer The Pareto solutions obtained and the weighted objective value of them are shown in Table 3, and the Euclidean distance between solutions and positive and negative ideal points and the closeness of solutions are listed in Table 4 after using the entropy-weight TOPSIS method. It can be seen from Table 4 that solution 7 is the most satisfactory operation schedule, and its operation schedule is given in Table 5.
By comparing solution 7 with the average value of schedules at a fixed departure interval of 3~10 min (as shown in Table 6), it can be seen that the total cost of loss for passengers is reduced by 31.42%, the comprehensive service ratio of buses is increased by 0.04%, and the number of passengers who not achieved easy transfer decreased by 74.29%.
For comparing the performance of NSGA-II with the fixed crossover probability and mutation probability (FNSGA-II) and NSGA-II with adaptive adjusted model for crossover and mutation probability (ANSGA-II), the two algorithms solve the test experiment for 10 times. The Pareto solutions obtained of two algorithms after running 10 times are shown in Figure 7, and the number of nondominated set with different iterations of each algorithm is shown in Figure 8. The results show that the obtained solution set by ANSGA is superior to that obtained by FNSGA-II.

Conclusions
A bi-objective optimization model for regional public transpo lished considering the total cost of passengers and the efficiency o eration. Furthermore, a two-step algorithm is designed to solve th step, the NSGA-II algorithm with adaptive adjusted model for cr probability is used to obtain the Pareto solutions. In the seco weight-TOPSIS method is implemented to find a satisfactory sche public transportation network is collected to test the dispatch app The results show that the scheduling scheme obtained in this pap than the schemes with fixed departure interval, and the obtained so is superior to that obtained by FNSGA-II.
In this paper, we assumed that the running time between stat ity, the running time may be uncertain due to delay. Therefore, fu sider the followings: (1) Considering the uncertainty of the running time to design transport scheduling model and algorithm. (2) Considering other factors, such as multiple types of buses, depots capacities, to the regional bus scheduling problem is s the regional public transportation scheduling.

Conclusions
A bi-objective optimization model for regional public transport scheduling is established considering the total cost of passengers and the efficiency of public transport operation. Furthermore, a two-step algorithm is designed to solve this problem. In the first step, the NSGA-II algorithm with adaptive adjusted model for crossover and mutation probability is used to obtain the Pareto solutions. In the second step, the entropy weight-TOPSIS method is implemented to find a satisfactory scheme. At last, a regional public transportation network is collected to test the dispatch approach and algorithm. The results show that the scheduling scheme obtained in this paper is obviously better than the schemes with fixed departure interval, and the obtained solution set by ANSGA is superior to that obtained by FNSGA-II.
In this paper, we assumed that the running time between stations is known. In reality, the running time may be uncertain due to delay. Therefore, future research can consider the followings: (1) Considering the uncertainty of the running time to design the regional public transport scheduling model and algorithm. (2) Considering other factors, such as multiple types of buses, energy consumption, depots capacities, to the regional bus scheduling problem is studied, to research on the regional public transportation scheduling.  Data Availability Statement: Data sharing is not applicable to this article.