Entropy Model of Dynamic Bus Dispatching Based on the Prediction of Back-Station Time

Abstract

In the actual operation of a bus, due to the influences of the passenger flow, traffic conditions and other factors, the vehicle back-station time is often delayed, which brings difficulties in commuting according to a timetable that results in the discontinuity of the bus. This is also the main disadvantage of static bus scheduling. Therefore, the “Entropy model of dynamic bus dispatching based on the prediction of back-station time” is proposed, which can be used for decreasing the passive effect of discontinuity by extending the departure interval of an early bus in advance, and to realize fairness in adjustments of the departure interval by using entropy theory. Finally, the model is validated by two examples, and the results show that the model can match the distribution pattern of the bus departure interval before and after an adjustment and as far as possible, it can reduce bus breaks, balance the occupancy rate and improve the stability of bus operations.

1. Introduction

By 2017, China’s urban bus holdings reached 651,200 vehicles, with 56,786 routes in operation. In 2017, an annual urban passenger volume of 127,215 million people was reached, of which buses commuted 72,287 million people, accounting for 56.82% [1]. In both the terms of the size of the bus system and the amount of passenger transport, bus travel has an irreplaceable position in meeting the travel needs of urban residents; however, most of China’s urban bus operations generally have overlong passenger waiting times, crowded rides and other problems, and the main reasons for this are the low punctuality and poor stability of bus departures. Taking the actual operating data of a bus line in Guangzhou, for example, the average on-time rate of its up and down bus departures is about 70%, and the on-time rate of bus departures during peak hours has dropped by more than 30%. The main reason for the low punctuality of bus departures is that most cities in China mainly use static or empirical bus scheduling methods, which require a high level of road accessibility and bus capacity allocation to face the current, increasingly complex and deteriorating traffic conditions.

With the establishment of the guiding ideology of the priority development of public transportation, many scholars have conducted in-depth research in the theoretical methods of bus scheduling. In static bus scheduling, for example, Yang Xiuhua (2008) [2] and Li Fazhi (2013) [3] predicted the passenger OD distribution based on bus IC card information, and constructed a bus scheduling model matching the passenger flow distribution. Fu and Ali (2008) [4] et al. studied the use of a particle swarm optimization algorithm to solve the bus scheduling model. Yu B (2010) [5] et al. studied the application of genetic algorithms in solving bus scheduling models. Sun Chuanjiao (2008) [6] and Wu Weitiao (2012) [7] proposed a bus scheduling method with different combinations of travel modes, such as intercity buses and large-station express buses, for bus routes with an unbalanced distribution of passenger flow at each station. In dynamic bus scheduling, Zhang Fei Zhou (2002) [8] and Xu Dawei (2007) [9] proposed a dynamic bus scheduling method that uses wireless communication technology to allow drivers to adjust the speed or stop across bus stops to adjust the headway for unexpected situations that may cause deviations between the actual bus operation situations and the travel plan. However, for the static bus scheduling method, its shortcomings are that the vehicle back-station time is often delayed due to the actual operation of the passenger flow, traffic conditions and other factors, making the prepared schedule difficult to implement as planned, resulting in late departures or even a discontinuity of buses; thus, resulting in an imbalance of the full occupancy rate of buses of the former and latter shifts. In addition, in terms of dynamic scheduling, the proposed method of stopping across stations is unfair to passengers waiting for a bus at non-stop stations, and the method of speed control is limited by traffic conditions, neither of which is a fair choice [10].

With the development of intelligent public transportation, bus journey time prediction technology is becoming more and more mature and widely used, making it possible to predict the bus operation status. Shalaby A (2004) [11] et al. proposed a method for bus arrival and departure time prediction using AVL and APC data. Meanwhile, Yu and Bin (2007) [12] investigated the application of support vector machines in bus journey time prediction. Xu Minxia (2007) [13] studied the application of Kalman filtering in bus journey time predictions, while Rajbhandari R (2005) [14] constructed a generalized additive model for transit travel time prediction. Additionally, Kormaksson M (2015) [15] constructed a bus arrival time prediction model based on a stochastic time series and Markov chains.

With the promotion and application of bus travel technology, it has become possible to grasp the dynamic operation information of public transport. The main methods of bus trip prediction include historical trend method, Kalman filter, artificial neural network, support vector machine, etc., which have been widely used in the electronic bus stop announcement, providing more reliable and demand-responsible decision-making basis for passengers to choose travel methods while waiting [16,17,18,19,20,21,22,23,24,25,26,27,28]. However, this application only provides a more reliable basis for passengers to make decisions, and cannot solve the current problems of poor bus operation stability and low service level. There are few relevant studies on using the predicted bus back-station time for dynamic bus scheduling to solve the above problems.

In summary, in view of the characteristics of actual bus operations and the shortcomings of the existing literature studies, this paper proposes a dynamic bus scheduling entropy model based on the predicted back-station time, which reduces the adverse effects of late buses and avoids the discontinuity of buses by extending the previous departure intervals in advance, while the entropy theory is used to achieve the fairness of the adjustment of each departure interval in the previous period. The overall idea of this paper is as follows: first, it compares the departure time of each bus shift corresponding to the expected bus departure schedule with a dynamically predicted bus back-station time, and it evaluates the feasibility of the next multiple corresponding buses to depart at the expected departure time, before then evaluating the feasibility of the bus schedule. Further, it constructs a dynamic bus scheduling entropy model for the infeasible expected bus schedule, it adjusts the bus schedule according to the predicted bus back-station time, and then makes the bus schedules both before and after the adjustment as consistent as possible. Finally, the validity of the model is verified through an example analysis, which shows that the model can effectively reduce the adverse effects caused by late buses and avoid the discontinuity of buses, while also balancing the full occupancy rate of buses of the former and latter shifts and improving the stability of bus operations.

2. Analysis of Bus Departure Delay

The delay of a bus departure means that the corresponding bus schedule fails to be met for the departure time specified in that bus schedule. In order to meet the travel needs of passengers and take into account the operating costs, public transport enterprises usually prepare a bus schedule in advance; however, in actual operation, due to the influence of many factors such as the road traffic conditions, station entrances and exits, and the vehicles themselves, bus vehicles cannot return to a station on time, and most of them will lead to a delay in further bus departures. Based on the two-month operation data of a bus line in Guangzhou, the following will analyze bus departure delays from the perspective of the delay proportion, duration and impact.

2.1. Delay Proportion Time Distribution

At present, the proportion of delays in departures is relatively high, with an average delay rate of 26.94%. Further analysis shows that the delays are mainly concentrated in the morning and evening peak hours and noon hours, of which the average delay rate in the weekday morning and evening peak hours is as high as 74.24%. On the whole, the delay rate in the morning and evening peak hours is higher than that on weekends, especially in the morning and evening peak hours (see Figure 1 for details). Through communication and analysis with the public transport enterprises, the main reasons for the high proportion of delays in the morning and evening rush hours are: (1) during those periods, the demand for passenger travel is large, the frequency of departures is high, and the corresponding interval of departures is short; therefore, delays in departure are easy to occur; (2) during this period, road traffic congestion occurs frequently, and bused cannot return to stations on time, resulting in a delay in departures. On the other hand, due to the fact that the drivers’ shift handovers and lunches are mainly concentrated at noon, there are fewer vehicles available at this time; therefore, the proportion of delayed departures is also high.

2.2. Delay Time Distribution

From the perspective of the delay duration, at present, the maximum delay duration is 600 s, and the duration is mainly concentrated in 60~300 s, accounting for 85%. The distribution of workday and weekend times is basically the same, as shown in Figure 2. Delays with a large duration are mainly distributed in the mornings and evening peak and noon hours, with workday delays being significantly higher than on weekends, especially in the evening peak, as shown in Figure 3.

2.3. Analysis on the Fairness of Delay Impact

A delay of departure will make the waiting time of passengers on a particular route longer, and the load factor will increase, resulting in poor ride comfort. If it is assumed that the arrival of passengers is uniform and the departure interval is 10 min, then the total waiting time of this route will increase by 10% for each minute of delay, and the full load rate will also increase by 10%. From the perspective of fairness, the goal of formulating a bus timetable is to ensure consistency of the waiting times and comfort of passengers in each shift, that is, fairness. Therefore, with the continuous improvement of the bus travel time prediction method, this paper proposes a dynamic bus scheduling method considering the perspective of fairness on the premise that the predicted bus return time is known.

3. Materials and Methods

When dynamically adjusting the bus departure, the following three objectives should be considered: (1) the adjustment amount, that is, the total adjustment amount of each departure interval before and after the adjustment is the minimum required; (2) consistency, that is, the change of the adjustment rate of each shift interval before and after the adjustment is minimal; and (3) fairness, that is, the adjustment of the departure interval before and after the adjustment is fair to the passengers of each trip. Since this paper is carried out on the premise that the predicted value of the bus return time is known, it is fixed that the adjustment amount equals to the length of the departure delay. On the other hand, the goal of formulating a bus timetable is to ensure the consistency of the waiting times and comfort of the passengers on each shift; therefore, consistency and fairness are equivalent. To sum up, the goal of this dynamic shift dispatching method is to achieve a consistency of the adjustment rate of the shift interval before and after the adjustment.

The dynamic adjustment of a bus schedule involves the following: (1) comparing the predicted bus return time with the bus schedule, and evaluating the feasibility of the bus schedule; and (2) when the bus schedule is not feasible, taking the delay time of the departure as the total amount of the adjustment, and the consistency of the adjustment rate of each departure interval before and after the adjustment as the adjustment goal, then adjusting the departure interval of each shift before the delayed shift.

The focus of this study is twofold: the construction of a fairness entropy model for dynamic bus dispatching and an example analysis based on real datasets. A flow chart of the work performed for the study is illustrated in Figure 4.

3.1. Evaluation of Bus Schedule Feasibility

(1) Evaluation of bus shift feasibility

First, the departure time of each bus shift corresponding to the expected bus departure schedule was compared with the dynamically-predicted bus back-station time, to evaluate the feasibility of the corresponding bus departure according to the expected schedule, with the schedule feasibility evaluation function constructed as follows:

$$f\left(t_i,T_i\right)=\left\{\begin{array}{c}1, T_i\le t_i\\ 0, T_i>t_i\end{array}\right.$$

(1)

where $f\left(t_i,T_i\right)$ denotes the feasibility evaluation function for the next bus departure according to the expected bus schedule, $f\left(t_i,T_i\right)=1$ denotes it is feasible, while $f\left(t_i,T_i\right)=0$ denotes it is infeasible; $t_i$ denotes the expected departure time of the next $i^{th}$ bus after the current departure, assuming $t_i$ denotes the optimized expected bus schedule based on the predicted passenger OD distribution; $T_i$ denotes the predicted back-station time of the next $i^{th}$ bus (including the vehicle waiting in a station for departure and the back-station vehicle) after the current departing shift, with the predicted back-station time of the vehicle waiting in station for departure defined as 0; and $i=1,2,\dots ,N, N$ denotes the total number of the vehicles waiting in station for departure and the back-station vehicle.

(2) Evaluation of bus schedule feasibility

To evaluate the feasibility of the next $N$ corresponding bus departures according to the expected bus schedule, the evaluation function of the bus schedule feasibility is constructed as follows:

$$F\left(t_1,\dots ,t_N;T_1,\dots ,T_N\right)=f\left(t_1,T_1\right)·f\left(t_2,T_2\right)\cdots f\left(t_N,T_N\right)$$

(2)

where $F\left(t_1,\dots ,t_N;T_1,\dots ,T_N\right)$ denotes the feasibility of the next $N$ bus departures according to the expected departure schedule, $F\left(t_1,\dots ,t_N;T_1,\dots ,T_N\right)=1$ denotes it is feasible, while $F\left(t_1,\dots ,t_N;T_1,\dots ,T_N\right)=0$ denotes it is infeasible.

3.2. Dynamic Adjustment Method of a Bus Schedule

(1) Evaluate the feasibility of departing at the maximum departure interval based on the bus predicted back-station time

Based on the predicted bus back-station time, determining whether there is a shift that still does not depart on time when buses depart at the maximum departure interval is completed with the following formula:

$$Y=\underset{T_i>t_0+H\times i}{\min }i$$

(3)

where $H$ denotes the maximum departure interval, and if Y exists, then the $1^{st}$ to $Y^{th}$ shifts depart at the time determined by the maximum departure interval or the predicted bus back-station time:

$$t_l^{\prime }=\max \left(t_0+H\times l, T_l\right)$$

(4)

where $l=1,2\dots ,Y$; if Y does not exist, then go to step (2) for an adjustment, starting from the first shift.

(2) Determine the shift to be adjusted

If the expected bus schedule is not feasible, it is necessary to determine the shift $M$ with the longest average delay time that cannot depart according to the expected schedule as the node for adjusting the bus schedule, so as to ensure that the departure interval corresponding to the expected schedule after an adjustment is always extended, in order to achieve the purpose of sharing the delay time in advance. As shown in Equation (5), the departure interval of the $M^{th}$ shift and its former shift is dynamically adjusted to avoid the occurrence of late buses:

$$M=\underset{\left(T_i-t_i\right)/i}{\max }\left(i\right)$$

(5)

(3) Determine the adjustment range

In order to reduce the negative impact caused by a late arrival and to avoid the discontinuity of a bus, the interval between the shifts $1^{st}$ and $M^{th}$ needs to be adjusted dynamically, and the adjustment range is ($T_M-t_M$).

(4) The objectives of adjustment

The following three objectives should be considered when making dynamic adjustments to bus schedules: (1) deviation, i.e., the minimum total difference between the bus departure intervals before and after the adjustment; (2) stability, i.e., an avoidance of a late bus caused by departure intervals exceeding the specified maximum interval; and (3) fairness, that is, the consistency of the rate of change of the bus departure interval before and after the adjustment, assuming that the expected distribution of the departure interval matches the distribution of the passenger demand. The more consistent the rate of change of the bus departure interval before and after the adjustment is, the more consistent it is with the distribution of passenger demand, and the fairer the adjustment then is.

Considering the special nature of this adjustment method, each departure interval can only increase after the adjustment; therefore, the difference between the sum of the intervals before and after the adjustment is the total difference. Additionally, the adjustment range ($T_M-t_M)$ is fixed, i.e., the total difference is fixed; therefore, this method does not need to consider the deviation, but rather only the two objectives of stability and fairness. The stability of the method is mainly reflected in the departure interval that is required to be shorter than the maximum departure interval, while the fairness is mainly reflected in whether the rate of change of each bus departure interval before and after the adjustment is consistent. Entropy, a mathematical abstraction that is widely used in various fields, can be understood as the probability of the appearance of a particular piece of information; the greater the uncertainty of a variable, the greater the entropy. In this paper, the rate of change of each interval is taken as the fairness uncertainty, then the larger the entropy value, the better the fairness. Therefore, this paper uses entropy theory to design the fairness evaluation function of the adjustment method, and then uses it to construct a dynamic bus scheduling entropy model based on the predicted back-station time.

3.3. Entropy Theory

The concept of entropy originated from thermodynamics and is a measure of uncertainty in the state of a system [29,30]. In information theory, Shannon introduced the concept of information entropy as a measure of disorder in a system by introducing the function:

$$H\left(X\right)=-{\displaystyle \sum }_{i=1}^n{p}_i log p_i$$

(6)

where $p_i$ denotes the uncertainty or the occurrence probability of a piece of information, satisfying:

$$0\le p_i\le 1$$

(7)

$${{\displaystyle \sum }}_{i=1}^n{p}_i=1$$

(8)

In the formula, the base of the logarithm has no specified value, and it generally takes 2, e, or 10, while the base numbers are different to measure the information in different units. The smaller the value of the information entropy, the greater the amount of information, and the higher the orderliness of information, while the probability distribution tends to be concentrated. When $\exists p_i=1$, the entropy is at a minimum. The larger the value of the information entropy, the smaller the amount of information, which means the higher the disorder of information whereas the probability distribution tends to be dispersed, and the entropy value is at a maximum only when $p_i=\frac{1}{n}$.

3.4. Entropy-Based Fairness Evaluation Function

Entropy theory is widely used in the study of disparity and fair distribution. To measure the fairness of the bus departure interval adjustment described in Section 2.2, the fairness evaluation function of the adjustment method is designed by using entropy theory.

Assume that:

$$\Delta t_j^1=k_j\times \Delta t_j$$

(9)

$\mathrm{where} \Delta t_i$ is the bus departure interval between shift $i^{th}$ and shift ${\left(i-1\right)}^{th}$ before an adjustment, i.e., the expected departure interval; $\Delta t_i=t_i-t_{i-1}$, and $t_0$ is the departure moment of the bus of the current departing shift; $\Delta t_j^1$ is the adjusted bus departure interval between shift $j^{th}$ and shift ${\left(j-1\right)}^{th}$; $\Delta t_j^1=t_j^1-t_{j-1}^1$; and $t_i^1$ is the dynamically adjusted bus departure moment of shift $i^{th}$ after the current issued shift; defining $t_0^1=t_0$. $k_j$ is a constant, with $k_j>0$ denoting the rate of change between the bus departure interval before and after an adjustment; and $j$ = 1,2, …, M.

Normalizing $k_j$ yields:

$${\lambda }_j=\frac{k_j}{{{\displaystyle \sum }}_{j=1}^M{k}_j}$$

(10)

Then, when ${\lambda }_j$ tends to be concentrated, the more consistent the distribution of the bus departure intervals before and after an adjustment are, and the more consistent it is with the distribution of the passenger demand, while the fairer the adjustment method, conversely, the more unfair it is. The fairest result is when the adjusted bus departure interval is an equal, proportional extension of the pre-adjustment bus departure interval, i.e.,:

$${\lambda }_j=\frac{1}{M}$$

(11)

At this point, $k_1=k_2=\dots =k_M$, and the distribution of the bus departure interval before and after an adjustment are consistent with the distribution of the passenger OD flow.

Therefore, according to the nature of entropy, ${\lambda }_j$ can be used as an indicator to calculate the entropy value. When the distribution of ${\lambda }_j$ tends to be concentrated, the larger the entropy value is, meaning the more consistent the distribution before and after the bus departure interval adjustment is, and the more consistent it is with the passenger OD distribution. The fairness evaluation function of the adjustment method can, therefore, be expressed as:

$$H\left(\lambda \right)=-{{\displaystyle \sum }}_{j=1}^M{\lambda }_{j}ln{\lambda }_j$$

(12)

4. Results

4.1. Model Construction

In summary, in order to extend each previous departure interval in advance according to the predicted bus back-station time to reduce the adverse effects caused by late arrivals, to avoid the discontinuity of buses, and to achieve the fairness of each previous departure interval by using the entropy theory, a dynamic bus scheduling “entropy” model based on the predicted return time was constructed. The entropy model based on the predicted return time is constructed as follows:

(1) Objective function:

$$\max H\left(\lambda \right)=-{\displaystyle \sum }_{j=1}^M{\lambda }_{j}ln{\lambda }_j$$

(13)

(2) Constraints:

$$T_j-t_j^1\le 0$$

(14)

$$0<\Delta t_j^1\le H$$

(15)

$$t_M^1=T_M$$

(16)

$$j=1,2,\dots ,M$$

(17)

where $H$ is the specified maximum departure interval; and ${\lambda }_j$ is the normalized value of the change rate of the bus departure interval before and after an adjustment, as is defined in Equation (10).

Equation (2) indicates that the adjusted $j^{\mathit{th}}$ bus departure time $t_j^1$ is not earlier than the predicted back-station time of the $j^{\mathit{th}}$ bus $T_j$.

Equation (3) indicates that the adjusted bus departure interval before and after the adjustment is less than or equal to the maximum allowable departure interval $H$ and greater than 0.

Equation (4) indicates that the departure time of the $M^{\mathit{th}}$ bus after an adjustment should be equal to the predicted back-station time of the $M^{\mathit{th}}$ bus, and the subsequent shifts can be executed according to the expected schedule.

4.2. Model Solving

According to Section 3.2, and the $1^{\mathit{st}}$ to $M^{\mathit{th}}$ shifts when Y does not exist:

$$T_j<t_0+H\times j$$

Therefore, there must be feasible solutions to the model. One of the characters of the entropy function is that $H\left(X\right)$ is a strictly upper convex function of the probability distribution $p_i$. Obviously, the set of solutions consisting of the constraints in Equations (2)–(4) is a convex set. Therefore, this optimization problem is a convex programming problem, and its local optimal solution is its global optimal solution. Lingo is a very classical and comprehensive set of tools for solving linear programming, quadratic programming, integer programming and nonlinear programming optimization problems, with a series of fully built-in solvers and the ability to automatically select the appropriate solver by reading equations. This was applicable to the solution of the model in this paper, and this paper uses the software of Lingo to solve the model.

5. Example Analysis

In order to further explain and test the effectiveness of the model, the following examples, which were based on the historical bus operation data of a bus line in Guangzhou, China, were taken to analyze the two periods that were more prone to bus delays, namely, the morning peak at 08:00–09:00 and the evening peak at 18:00–19:00. The schematic map of the bus line is shown in Figure 5.

5.1. Example 1

Take the example of bus scheduling during the morning rush hour from 08:00 to 09:00, the departure time of the current bus as $t_0$ = 08:00, and the number of buses in the station yard and on the way back to the station as $N=8$. Assuming that the maximum departure interval constraint was $H=15 \min ,$ the expected bus departure time for the next eight shifts was $t_i$, the predicted back-station time for the corresponding shift was $T_i$, the adjusted departure schedules were $t_i^1$, and the actual unadjusted departure time $t_i^2$ are shown in

Table 1. The next 8 bus departure times before and after the adjustment (Example 1).

Shift	the Predicted Back-Station Time  ${\mathit{T}}_{\mathit{i}}$	the Expected Bus Departure Time  ${\mathit{t}}_{\mathit{i}}$	the Adjusted Departure Schedules  ${\mathit{t}}_{\mathit{i}}^1$	the actual Unadjusted Departure Time  ${\mathit{t}}_{\mathit{i}}^2$
1	08:00	08:03	08:04	08:03
2	08:00	08:06	08:08	08:06
3	08:00	08:12	08:15	08:12
4	08:08	08:18	08:22	08:18
5	08:18	08:24	08:29	08:24
6	08:37	08:33	08:40	08:37
7	08:55	08:45	08:55	08:55
8	08:57	08:57	08:57	08:57

.

Observe in Example 1 that (1) if no adjustment were made to the bus schedule, then there may be $\Delta t_7^2=18 \min >H=15 \min ,$ causing an occurrence of late buses and reducing the stability of bus operations, then the dynamically adjusted departure interval could meet the maximum departure interval constraint of $H=15 \min ; \mathrm{and} ($2) as shown in Figure 6, which depicts the distribution of the expected departure interval $t_i,$ the adjusted departure interval $t_i^1$ and the actual departure interval $t_i^2$, obviously $t_i^1$ was more similar to $t_i$ in terms of the departure interval distribution. This is more in line with the distribution of the passenger demand and can effectively reduce the negative impact of late buses, avoid a discontinuity of buses, balance the full occupancy rates of buses of the former and latter shifts and improve the stability of bus operations.

5.2. Example 2

Take the example of bus scheduling during the evening rush hour from 18:00 to 19:00, the current departure time of the bus $t_0$= 18:00, the number of buses in the station yard and on the way back to the station is $N=11$. Assuming the maximum departure interval constraint of $H=15 \min ,$ then the expected bus departure time for the next 11 shifts is $t_i$, the predicted back-station time for the corresponding shift is $T_i$, the adjusted departure schedules are $t_i^1$, and the actual unadjusted departure time is $t_i^2$ are shown in

Table 2. The next 11 bus departure times before and after the adjustment (Example 2).

Shift	the Predicted Back-Station Time  ${\mathit{T}}_{\mathit{i}}$	the Expected Bus Departure Time  ${\mathit{t}}_{\mathit{i}}$	the adjusted Departure Schedules  ${\mathit{t}}_{\mathit{i}}^1$	the Actual Unadjusted Departure Time  ${\mathit{t}}_{\mathit{i}}^2$
1	18:00	18:04	18:05	18:04
2	18:06	18:08	18:10	18:08
3	18:06	18:12	18:15	18:12
4	18:12	18:16	18:20	18:16
5	18:18	18:20	18:25	18:20
6	18:28	18:26	18:34	18:28
7	18:38	18:32	18:42	18:38
8	18:50	18:38	18:50	18:50
9	18:58	18:48	18:58	18:58
10	19:03	18:58	19:03	19:03
11	19:08	19:08	19:08	19:08

.

Observe in Example 2 that although the departure interval $\Delta t_i^2>H=15 \min$ does not occur without adjustment, there is still a situation where the actual departure interval is significantly different from the expected departure interval because the predicted bus back-station time does not meet the expected departure time. This leads to an imbalance in the full occupancy rate of the buses of the former and latter shifts, such as the expected $\Delta t_7=\Delta t_8=6 \min ,$ the actual $\Delta t_7^2=10 \min$, $\Delta t_8^2=12 \min ,$ and the dynamically adjusted $\Delta t_7^1=\Delta t_7^1=8 \min .$ Figure 7 depicts the distribution of the expected departure interval $t_i$, the adjusted departure interval $t_i^1 \mathrm{and}$ the actual departure interval $t_i^2$. It also shows that the method can effectively reduce the negative impact of late buses and avoid a discontinuity of buses, while balancing the full occupancy rate of the buses of the former and latter shifts; thus, improving the stability of bus operations.

In view of the shortcomings of static public transport scheduling, this paper proposes a dynamic public transport scheduling method based on fair entropy on the basis of the existing technical support. This method is based on the expected timetable obtained from static scheduling optimization, and adjusts the action state according to the actual public transport vehicles available for scheduling. The inadequacy of this paper is that it assumes that the delay or break of a bus departure is affected by factors other than a change in the passenger demand, and although it includes a change in the passenger demand, the distribution trend of the passenger demand before and after a change is consistent, while the change in the actual passenger demand distribution is not fully considered. In addition, the specific passenger OD demand is not considered, but the passenger demand of the whole line is considered. In future research work, we can consider combining the real-time bus operation status with real-time passenger flow data to achieve a dynamic bus dispatching method that is more in line with the real-time passenger demand.

6. Conclusions

With the continuous development of intelligent public transportation, the prediction of passenger flow OD distribution methods based on bus IC card technology and video counting technology provide more accurate passenger flow information support for static bus scheduling optimization. Furthermore, the promotion and application of bus trip prediction technology can further realize the potential lateness or discontinuity of buses by predicting the bus back-station time and comparing it with the expected departure schedule; thus, predicting the potential unpunctuality or discontinuity of buses in advance. This paper innovatively applies entropy theory to dynamic bus scheduling, and builds a dynamic bus scheduling entropy model based on the predicted back-station time and on static bus scheduling optimization for dynamic bus schedules. Finally, the solution method of the model has been given, and an example analysis performed to verify the validity of the model. Consequently, it is feasible to use the dynamic bus scheduling entropy model to adjust a bus schedule based on the predicted bus back-station time, in order to reduce the negative impact of late buses, avoid the discontinuity of buses, to balance the full occupancy rate of the buses of former and latter shifts and to improve the stability of bus operations. It should also be noted that, compared with other models, the advantage of the entropy model is that it does not need to standardize the data when measuring data volatility, which solves the problem that other models seem to have to ensure data consistency when in use.

In the face of the current situation, the problem of poor stability, service levels that are difficult to guarantee in public transport operations, and a growing demand for public transport travel are contradictory, but the research results of this paper can improve the stability of public transport operations and ensure a high service level. Furthermore, they can effectively alleviate overcrowding in buses caused by delays in departure. Consequently, the attraction of public transport travel can be improved, the share rate of public transport travel can be improved, urban traffic pressures can be alleviated, and environmental pollution can be reduced.