1 Introduction

As an important means of transportation for citizens, buses have the advantages of large carrying capacity and environmental protection. Many cities vigorously develop public transportation. In China’s capital of Beijing, there are about 1200 bus lines, with an average of 8.6 million public transport passengers daily. However, with the constant increasement of the number of motor vehicles, the road traffic flow has become increasingly saturated. Traffic congestion is becoming a prominent concern. It also has a serious impact on the normal operation of buses.

Traffic congestion will cause the delay of bus running time. The resulting variations in bus running times can result in some stops experiencing bus bunching, which refers to the case of a bus arriving with a little headway to its predecessor, and an increase in passenger travel time. Based on GPS data from Beijing Transportation Information Center, we take the Yuntong 111 bus line of Beijing as an example, and bus bunching is defined to occur when two buses on the same line arrive at the same stop within one minute of each other. There was an average of 133 cases of bus bunching between 7:00 and 11:00 each day in March 2017. This phenomenon not only seriously undermines the punctuality and regularity of bus schedules but also, more importantly, heavily detracts from the travel experience of passengers, which may affect ridership and revenue (Petit et al. 2019). It is therefore necessary to optimise the bus schedule to improve the quality of a city’s public transport network. As they are providing a public service, bus system operators are responsible for providing a reliable bus service for passengers. Scheduling optimisations must be carried out to alleviate the bus bunching problem.

To reduce bus bunching, a variety of improvement strategies have been proposed in the literature, such as holding strategy (Sánchez-Martínez et al. 2016), stop-skipping strategy (Chen et al. 2015) and boarding limitation strategy (Delgado et al. 2012). These strategies can be very good to alleviate bus bunching, but their implementation is a challenge to the general congestion in big cities. Besides, they are likely to increase the travel time of passengers, thus reducing the satisfaction of passengers (Ibarra-Rojas et al. 2015).

In order to avoid these shortcomings and easier to implement, we design a short-turning strategy to alleviate bus bunching based on the actual needs of bus lines. This study makes three main contributions: (1) for long bus lines with concentrated congestion, this study proposes to solve the problem of bus bunching using a short-turning strategy; (2) in alleviating the bus bunching problem, an optimisation model aiming to minimise schedule deviation is proposed for the first time; and (3) the results obtained from real-world data confirm that the use of big data from public transport can improve operation scheduling.

The remainder of this paper is structured as follows. Section 2 discusses the relevant literature. Section 3 describes the characteristics of bus bunching. Section 4 takes the Yuntong 111 bus line as an example of a bus line with the characteristics that are suited to the alleviation of bus bunching using the short-turning strategy. Section 5 models the short-turning strategy. Section 6 assesses the performance of the short-turning strategy on the Yuntong 111 bus line. Section 7 provides concluding remarks and future research directions.

2 Literature review

In the present work, we review some strategies of relevant studies for alleviating bus bunching, including the holding strategy, stop-skipping strategy, and boarding limitation strategy. At the same time, we also review the application of the short-turning strategy.

One strategy that has been widely studied to alleviate bus bunching is the holding strategy (Newell 1974; Zhao et al. 2016). Due to the complexity of actual road conditions, bus bunching often occurs as part of normal operations (Chen et al. 2009). Bartholdi and Eisenstein (2012) abandoned the idea of a schedule and any a priori target headway and proposed a new method to coordinate buses. In their research, bus holding was only allowed at a predetermined subset of important bus stops to reduce the waiting time variance. Berrebi et al. (2015) presented a global approach to the bus dispatching problem that absorbs large gaps by several or all preceding buses. Hernández et al. (2015) minimised the average travel time of all passengers by developing an optimisation model that is capable of executing a control scheme based on a holding strategy applied to a travel corridor with multiple bus lines. Sánchez-Martínez et al. (2016) formulated a mathematical model for holding control optimisation, which can be used to produce a plan of holding times that accounts not only for the current state of the system, but also for expected changes in travel times and demand.

Subsequent research about the alleviation of bus bunching has combined the holding strategy with other strategies, such as stop-skipping and boarding limitation. The stop-skipping strategy has received significant attention (Liu and Yan 2013; Chen et al. 2015). By combining bus holding and stop-skipping actions, Cortés et al. (2010) presented a hybrid predictive controller to optimise real-time bus operations by taking into account uncertain passenger demand. Bus holding strategy and boarding limitation strategy were also combined by Delgado et al. (2009) and Delgado et al. (2012)). Limiting the number of boarding passengers can effectively increase bus operational speed and reduce the cycle time. Yan et al. (2012) developed a robust optimisation model for schedule design, with the inclusion of a decision variable of slack time for each segment along a real bus route, considering the capacity of bus drivers to affect schedule recovery. Li et al. (2019) proposed a robust dynamic control mechanism for bus transit systems combining the bus holding and operating speed control strategies, which considers variations in congestion delays and passenger demand.

Theoretically, the above control strategies can alleviate the bus bunching problem. However, in actual traffic scenarios, especially during the morning and evening peak times, traffic conditions are too complex for these control strategies to achieve their goals. For example, the forced implementation of the holding strategy seriously affects the normal operation of adjacent buses, while boarding limitations have a serious negative impact on passenger satisfaction. Similarly, there are many obstacles to the implementation of the stop-skipping and speed control strategies. In addition, passenger waiting time and deviation from original schedules increase greatly with the implementation of these control strategies (Petit et al. 2018).

The short-turning strategy has received extensive attention from researchers for dealing with the spatial concentrations of demand. Canca et al. (2016) solved frequent disruption arising from increases in demand by adding a short-turning service, and determined the appropriate turn-back points and service offsets to minimise the difference in passenger loads for both kinds of services in the selected zone. Ghaemi et al. (2016, 2017) considered the optimal scheduling problem for short-turning buses in cases of complete blockage. Leffler et al. (2017) attempted to balance passenger costs and benefits by determining the turnaround station for a short-turning service and when a bus was a short-turning service on a single bidirectional bus line in real time. Zhang et al. (2018) considered scheduling with a short-turning strategy for an urban rail transit line with multiple depots, where short-turning and regular train services were optimised based on a predefined headway obtained by passenger demand analysis. This strategy can solve the problem of a spatial concentration of demand effectively. However, in the applications of the short-turning strategy, a multiple relationship has been assumed between the departure frequency of a regular trip and that of a short-turning trip (Tirachini et al. 2011; Canca et al. 2016). As adopting this assumption means that the final effect has significant limitations, the model proposed in this study can better coordinate the scheduling of regular and short-turning trips.

To the best of our knowledge, the short-turning strategy has not been applied to alleviate bus bunching. Researchers may be searching for a more universal strategy. The short-turning strategy is suitable for long bus lines that are characterised by concentrated congestion. This kind of line is common in bus operations in very large cities, such as Beijing. Compared with other control strategies, the short-turning strategy is easier to implement, even if the road is congested. On the other hand, the short-turning strategy will not increase the total travel time of passengers and thus will not reduce the satisfaction of passengers. Therefore, in this study we explore the use of the short-turning strategy to alleviate bus bunching.

3 Problem description

The design of a bus route schedule is one of the most basic activities in bus system planning (Ceder 2007). In bus operations, strict compliance with the schedule is desirable, and passengers definitely benefit from a reliable bus schedule. The problem is described as follows.

As shown in Fig. 1, there are three buses along a route: A, B (B’) and C. If there is no road congestion and no upsurge in passenger flow, buses A, B and C can maintain the expected headway. Otherwise, once congestion occurs, this ideal state will be broken. For example, if the second section is congested, bus B’ cannot reach the expected location (i.e., the location of bus B), as shown in Fig. 1, which will lead to a long headway between A and B’ and a short headway between B’ and C. When bus B’ arrives at the stop, the number of passengers will have increased significantly due to the long headway. In contrast, when bus C arrives, there will be fewer waiting passengers, and traffic resources will be wasted due to a low occupancy rate. The boarding time for the passengers of bus B’ will also be longer than that of bus C, which will make the situation even worse. The distance between bus A and bus B’ will keep growing, and the distance between bus B’ and bus C will keep shrinking. The imbalance in the headway will thereby become increasingly obvious and will eventually result in bunching. This situation will have a negative impact on bus operators and passengers. For public transport operators, the unstable operation of bus services will lead to delays and affect the next scheduling plan of the bus, thus increasing the cost of operating the services. For passengers, travel time will be less predictable, and expectations for and variance in passenger waiting time will also increase. Furthermore, the late arrival at the terminal will affect the departure time in the opposite direction, thus further affecting the waiting time of additional passengers. It is thus important to design a reasonable schedule, in which the operation of buses can remain close to the original schedule.

Fig. 1
figure 1

An illustration of bus bunching

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Most studies aimed at alleviating bus bunching have used a control strategy. Although many control strategies, such as static and dynamic holding strategies, have been implemented to mitigate the effects of unreliable bus schedules, most would impose longer waiting time on passengers (Petit et al. 2018). In general, bus bunching can be alleviated using control strategies, such as the holding strategy or operating speed control strategy. However, it is not easy to adjust the speed of buses based on real-time traffic conditions because of traffic congestion and high vehicle density, especially in peak periods. Meanwhile, passenger waiting time and schedule deviation increase greatly because of the prolonged trips.

Here we provide a motivating example, in which, we consider converting some regular trips into short-turning trips. In Fig. 2, we assume that a bus line has five stops. There are two terminal stops, which are stops 1 and 5, and stops 2 and 3 are regular stops with frequent traffic congestion. The travel time with smooth traffic conditions along the route is also presented in Fig. 2.

Fig. 2
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Operation time between adjacent stops

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There are five trips that can be scheduled during the period in this scenario. The scheduled arrival times at each stop are shown in Table 1, and the times in brackets are the actual arrival times. The bus bunching occurs between trips 2 and 3 at stop 3.

In this example, we optimise bus operation by using the short-turning strategy. To alleviate bus bunching, some trips are converted into short-turning trips. The selection of short-turning trips and their departure times are determined by the interactions between adjacent trips.

Table 1 Scheduled arrival time and actual arrival time
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At stop 3, trips 2 and 3 occur bus bunching, and the waiting time of passengers on trips 1 and 2 is longer compared with the other trips. If buses are scarce, this will seriously affect the departure times in the opposite direction, thus affecting the waiting time of passengers. As mentioned before, in Fig. 2 we can see that the road between stop 2 and 3 experiences frequent congestion, so to avoid the traffic congestion on short-turning trips, we set stop 3 as the starting stop of short-turning trips. Table 2 shows the schedule combining short-turning and regular trips. If trip 2 is converted to a short-turning trip, the schedule of stops 3, 4 and 5 will be consistent with the original schedule. As trip 2 is now a short-turning trip, only trips 1 and 3 serve stops 1 and 2, so the waiting time of passengers at stops 1 and 2 increases. However, because the headway of stop 3 and downstream stops are restored to the average, passenger waiting time at these stops are reduced. Because of the short-turning strategy, each trip can arrive at stop 5 on time and bus scheduled in the opposite direction will also operate normally, so passengers travelling in the opposite direction also benefit.

If the holding strategy is used to alleviate the bus bunching between trips 2 and 3, the departure time of trip 3 at stop 3 will be delayed. Although the phenomenon of bus bunching can be alleviated in this manner, the running time of trip 3 will be increased. Therefore, in this case, the holding strategy alleviates the problem of bus bunching, but it does not benefit the passengers or operators, and may even damage their interests.

Table 2 Scheduled arrival time considering the short-turning strategy
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The example shows that bus bunching can be significantly alleviated using a short-turning strategy. A starting stop for the short-turning trips should be selected that avoids the congested road section, allowing for better schedule adjustment. According to the above analysis, when we design our optimisation model, there are two key decision variables that need to be determined: one is the selection of short-turning trips, and the other is the departure time of these trips.

In the next section, we describe in detail the characteristics of an actual bus line on which bus bunching can be alleviated by implementing the short-turning strategy.

4 Data analysis

The Yuntong 111 line is a very popular bus line in Beijing, which has the characteristics of a long line with multiple stops and concentrated congestion. This bus line runs from the Laiguangying North to Dougezhuang South townships,

with 49 stops over a 33.8 km route. Figure 3 shows the bus route map of Yuntong 111. From Fig. 3 we can see that there are many intersections between stops 4 and 15. These create serious traffic jams, especially during peak times. Therefore, this road section is more prone to bus bunching than other sections.

Fig. 3
figure 3

Bus route map of Yuntong 111

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The Beijing Transportation Information Center, a branch of the Beijing Municipal Commission of Transport, collects GPS data and IC card data for all 1,020 bus lines and 29,515 vehicles in Beijing. A GPS record consists of the bus number, time, position and direction of travel. We use the vehicle GPS data for the Yuntong 111 line from 1 May 2018 to 31 May 2018. From the GPS data, we derive the actual travel time updated every 5 min between stops. For example, Fig. 4 shows the travel time from 17:35 to 20:25 between stop 4 and stop 15 on 7 May 2018.

Fig. 4
figure 4

Travel time between stops 4 and 15

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To characterise the variations in travel time between stops for a given period, the travel time between two adjacent stops can be expressed by the following formula

$$\begin{aligned} t_{n}(x)=\left\{ \begin{aligned}&t_{n,1}, \quad if\, s\le x<s+\delta ,\\ & t_{n,2}, \quad if\, s+\delta \le x<s+2\delta ,\\ &\,\, \quad\quad\quad \vdots \\ & t_{n,L}, \quad if\, s+(L-1)\delta \le x<s+L\delta ,\\ \end{aligned} \right. \end{aligned}$$
(1)

where s specifies the starting time point of the given period, e.g. 8:00 am, L represents the number of time segments involved in the period, and \(\delta\) represents the length of each time segment, e.g. 5 min. Based on the travel time between stops, for a given departure time from a starting stop, the arrival time of the bus at each stop can be obtained by the following recurrence formula

$$\begin{aligned} O_{k,n}-O_{k,n-1} = t_{n-1}(O_{k,n-1}), \qquad k=1,2,\cdots ,K, n=2,3,\ldots ,N. \end{aligned}$$
(2)

According to the given departure schedule, we can obtain the ideal schedule in Table 3 for each stop. In actual operations, the travel time increases due to

Table 3 Original schedule of Yuntong 111
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Table 4 Actual running schedule of Yuntong 111
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traffic congestion, as shown in Fig. 4. If the operation time is less than 44 min between stops 4 and 15, the driver’s operation experience can guarantee that the actual operation schedule matches the original schedule (as shown in Table 3). When the operation time is more than 44 min, however, the operating bus has no choice but to wait due to the traffic congestion. Bus bunching is then more likely to occur. The actual running schedule (Table 4) can be obtained by Formulas (1) and (2). The phenomenon of bus bunching occurs at stop 15 between trips 10 and 11 and during trips 20, 21 and 22. In this case, the actual operation schedule deviates greatly from the original schedule, which seriously affects the waiting time of passengers and the arrival time at the terminal stop.

As a critical enabler of strategic management, big data analysis has been widely concerned by academics and practitioners (Babar and Arif 2019; Liu et al. 2020; Ramesht et al. 2019; Yan et al. 2019). Through the data analysis of Yuntong 111 bus line, an effective method is found to alleviate bus bunching, and the method is shown in the next section.

5 Problem formulation

Given the characteristics of a long line with concentrated congestion, we design a short-turning strategy to alleviate bus bunching. This section mainly presents the establishment process of the short-turning model and the transformation mechanism from the original nonlinear model to a linear one.

5.1 Preliminary

We consider a bus line with N stops and K trips. For each stop n and each trip k, there is a scheduled arrival time \(a_{k,n}\). We optimise bus operation by proposing a short-turning strategy. There is also an actual arrival time \(O_{k,n}\) of trip k at stop n. The proposed model makes the following assumptions:

  1. (a)

    The passenger demand for each OD and the passenger arrival rate at each stop are known based on the analysis of daily IC card data.

  2. (b)

    There are no transferring passengers between the short-turning trips and the regular trips, which is a reasonable assumption given that a transfer would increase the cost of travel.

  3. (c)

    The capacity of the bus is sufficient to meet passenger demand.

To make this work easier to follow, Table 5 lists the most frequently used notations.

Table 5 Notations
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5.2 Short-turning model development

To minimise the total schedule deviation, we try to optimise the actual running schedule to approximate the original schedule by using the short-turning strategy. In this study, we aim to minimise schedule deviation by developing a short-turning model to alleviate bus bunching.

The short-turning strategy is convenient for when there are high demand zones that need to be serviced while other zones experience low demand (Canca et al. 2016). Under this strategy, some trips are converted to serve only a specific segment of the line. This reduces the average waiting time within the short-turning segment, but increases the waiting time on the remainder of the line. Therefore, the selection of the starting stop for short-turning trips must take account into the arrival rate of passengers at each stop. To reduce the cost and complexity of scheduling, this study mainly focuses on the situation in which the terminal stop of short-turning trips is the same as that of regular trips. To better fit the actual situation, we must first ensure that regular trips can meet the passenger demand outside of the short-turning segment. By converting some regular trips to short-turning trips, the schedule from the starting stop of the short-turning trips will remain close to the original schedule.

5.2.1 Objective function

Buses running according to their original schedule provides a good passenger experience and makes bus dispatching convenient. The headway on the original schedule is relatively average, and if buses run close to the schedule, bus bunching will be avoided. Therefore, we aim to minimise the deviation from the original schedule. For each trip, the schedule deviation at stop \(n_1\) depends on the difference between the scheduled departure time and actual departure time. Therefore, we formulate the objective function as follows

$$\begin{aligned} \sum _{k=1}^{K}\sum _{i=1}^{K}\left| x_{k,n_{1}}-a_{i,n_{1}}\right| y_{k,i}. \end{aligned}$$
(3)

The starting stop \(n_1\) of the short-turning trips should not be set in the congested road section; otherwise the schedule cannot be adjusted effectively. Therefore, it is necessary to analyse the traffic data of a given line to determine the optimal location of the starting stop associated with the short-turning trips.

5.2.2 Constraints

Assuming that the number of short-turning trips is \(K^*\), it can be expressed as follows

$$\begin{aligned} \sum _{k=1}^{K}v_{k,n_1}=K^*. \end{aligned}$$
(4)

To reduce passenger waiting time, the interval between two adjacent trips should not be excessive. We can adjust the selection of short-turning trips to limit the maximum departure interval between adjacent trips. Based on an interview with the operator, two adjacent trips cannot be converted to short-turning trips simultaneously, which can be written as

$$\begin{aligned} v_{k,n_1}+v_{k+1,n_1} \leqslant 1, \qquad k=1,2,\cdots ,K-1. \end{aligned}$$
(5)

The kth trip is either a short-turning trip or a regular trip. If it is a short-turning trip, we have \(v_{k,n_1}=1\); otherwise, we have \(v_{k,n_1}=0\), which means that the kth trip follows the original schedule (i.e., \(x_{k,n_1}=O_{k,n_1}\)). When \(v_{k,n_1}=1\), we need to decide the departure time of trip k at stop \(n_1\). We then have

$$\begin{aligned} (O_{k,n_1}-x_{k,n_1})(1-v_{k,n_1})=0, \qquad k=1,2,\cdots ,K. \end{aligned}$$
(6)

The implementation of the short-turning strategy is likely to change the order of some trips (both short-turning and regular trips) at the top \(n_1\), so the order of trips must be adjusted to approximate the original schedule. There is a one-to-one relationship between the original trips and the adjusted trips. In other words, only one trip k is directly transformed into trip i, and trip k can only be directly transformed into unique trip i, which can be rewritten as

$$\begin{aligned}&\sum _{k=1}^{K}y_{k,i}=1, \qquad i=1,2,\cdots ,K, \end{aligned}$$
(7)
$$\begin{aligned}&\sum _{i=1}^{K}y_{k,i}=1, \qquad k=1,2,\cdots ,K. \end{aligned}$$
(8)

The short-turning model is then obtained

$$\begin{aligned} \begin{aligned} \min \qquad&\sum _{k=1}^{K}\sum _{i=1}^{K}\left| x_{k,n_{1}}-a_{i,n_{1}}\right| y_{k,i}\\ \mathrm{s.\,t.}\qquad&Constraints \,\, (4-8),\\&x_{k,n_1}\ge 0, \qquad \quad \,\,\, k=1,2,\cdots ,K, n=1,2,\cdots ,N,\\&v_{k,n_1}\in \{0,1\}, \quad \,\,\,\,\, k=1,2,\cdots ,K, n=1,2,\cdots ,N,\\&y_{k,i}\in \{0,1\}, \qquad k,i=1,2,\cdots ,K. \end{aligned} \end{aligned}$$
(9)

To facilitate the optimisation of the short-turning model, we transform it into a linear model, as shown in next subsection.

5.3 Linear transformation of model

To make the model amenable to computation, the nonlinear parts of the short-turning model are transformed into linear ones.

The objective is nonlinear on account of absolute value. For each \(k,i\in \{1,2,\ldots ,K\}\), an auxiliary variable \(z_{ki}\) is introduced to substitute the nonlinear expression \(\left| x_{k,n_{1}}-a_{i,n_{1}}\right| y_{k,i}\). That is, we regard the upper bound of absolute deviation as the optimal objective value for \(k,i=1,2,\cdots ,K\). An additional constraint is then added to the optimisation model

$$\begin{aligned} \left| x_{k,n_{1}}-a_{i,n_{1}}\right| y_{k,i}\le z_{k,i},\qquad k,i=1,2,\ldots ,K. \end{aligned}$$
(10)

The minimum objective value Z is a guaranteed upper bound value for the expected arrival time. Thus, based on variable substitution, the above nonlinear constraint is transformed into a linear constraint

$$\begin{aligned} -z_{k,i}-M_1(1-y_{k,i})\le x_{k,n_{1}}-a_{i,n_{1}}\le z_{k,i}+M_1(1-y_{k,i}),\quad k, i=1,2,\ldots ,K, \end{aligned}$$
(11)

where \(M_1\) is a large constant.

The constraint (6) is nonlinear because it takes the form of the product of two decision variables. By introducing a large constant \(M_2\), the constraint is linearised as follows

$$\begin{aligned} -M_2v_{k,n_1}\le O_{k,n_1}-x_{k,n_1} \le M_2v_{k,n_1},\qquad k=1,2,\ldots ,K. \end{aligned}$$
(12)

To sum up, the linear programming model is obtained by equivalent transformation, as shown in Theorem 1.

Theorem 1

Define a linear programming model as follows

$$\begin{aligned} \begin{aligned} \min \qquad&\sum _{k=1}^{K}\sum _{i=1}^{K}z_{k,i}\\ \mathrm{s.\,t.}\qquad&Constraints \,\,(4), (5), (7),(8), (11), (12),\\&x_{k,n_1}\ge 0, \qquad \quad \,\,\, k=1,2,\ldots ,K,\\&z_{k,i}\ge 0, \qquad \quad \,\,\, k,i=1,2,\ldots ,K,\\&v_{k,n_1}\in \{0,1\}, \quad \,\,\,\,\, k=1,2,\ldots ,K,\\&y_{k,i}\in \{0,1\}, \qquad k,i=1,2,\cdots ,K. \end{aligned} \end{aligned}$$
(13)

If model (13) has an optimal solution, it is also an optimal solution of model (9).

Proof

Suppose that the optimal solution of model (9) is \((X', Y', V')\). According to the constraint (6), we have

$$\begin{aligned} (O-X')(1-V')=0, \end{aligned}$$
(14)

which means that if \(V'=0\) then \(X'=O\); otherwise the constraint is redundant. Thus, the values of \(X'\) and \(V'\) satisfy constraints (12). To prove that the optimal solution of model (9) is a feasible solution of model (13), we introduce the free variable \(Z'\). We assume \(Z'=\left| X'-A\right| Y'\), so \((X', Y', V', Z')\) satisfies constraint (11). Therefore, we conclude that the optimal solution of model (9) is a feasible solution of model (13).

Suppose that the optimal solution of model (13) is \((X'', Y'', V'', Z'')\). According to constraints (11)–(12), we have

$$\begin{aligned} -Z''-M(1-Y'')&\le X''-A\le Z''+M(1-Y''), \end{aligned}$$
(15)
$$\begin{aligned} -MV''\le O-X''&\le MV''. \end{aligned}$$
(16)

It is thus clear that the optimal solution \((X'', Y'', V'')\) of model (13) is a feasible solution of model (9).

To sum up, the optimal solution of model (9) is equal to that of model (13). This completes the proof of the theorem. \(\square\)

By adding the variables Z and substituting constraints (10) and (6) with constraints (11) and (12) respectively, we obtain the linear programming model (13), which can be solved by CPLEX.

6 Numerical experiments

The proposed short-turning strategy is tested for the Yuntong 111 bus line in Beijing. Section 3 includes a detailed description of this bus line, on which bus bunching readily occurs because of traffic congestion. This bunching produces negative effects for bus operators and passengers. Beijing is a metropolis in China with a bus network comprising almost 30,000 operating buses. The originally planned bus lines cover a large geographical area that enables the generation of several interlining lines without requiring long deadheading times. This traffic configuration provides a feasible scenario for the implementation of a short-turning strategy.

6.1 Implementation of the short-turning strategy

Based on the travel time between adjacent stops, we convert some trips to short-turning trips. To avoid the traffic congestion, we set the starting stop of short-turning trips as \(n_1=15\). We analyse a 5-hour period of the day that is empirically found to exhibit a relatively stable ridership pattern (from 17:25 to 22:25). The number of trips in this period is \(K=22\) in the real scenario of the Yuntong 111 bus line. Under these parameters, the optimal scheduling scheme of the converted trips can be derived by minimising the objective function (3) (see Table 6).

Table 6 Bus running schedule with short-turning strategy
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The following subsections observe the effectiveness of the short-turning strategy in terms of schedule deviation, passenger waiting time and sensitivity analysis.

6.1.1 Schedule deviation

We assume that six trips can be converted to short-turning trips. By solving the proposed model in Sect. 4, trips 7, 9, 11, 13, 20 and 22 are changed to short-turning trips (as shown in Table 6). With the implementation of the short-turning strategy, the headway between trips at stop 15 becomes average, and schedule deviation is 24 min compared with the original schedule. The most obvious deviations caused by the implementation of the short-turning strategy are that the headway of trips 10 and 11 at stop 15 increases from 1 to 8 min, and the average headway of trips 20, 21 and 22 at stop 15 increases from less than 1 minute to 8 min. The schedule of these five trips is thus closer to the original schedule. Compared with the total schedule deviation of 343 min with no control strategy, total schedule deviation decreases by 93%. The reduction of schedule deviation means that the phenomenon of bus bunching can be alleviated to a certain extent. Table 6 shows that bus bunching between trips 10 and 11 and trips 20, 21 and 22 at stop 15 has been effectively alleviated. In summary, we conclude that bus bunching can be alleviated using the proposed short-turning strategy.

Table 7 Bus running schedule with holding strategy
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In alleviating bus bunching, the proposed short-turning strategy also results in less schedule deviation than the alternative holding strategy. To apply the holding strategy, the target headway should be set first to determine the holding time at the control point. The target headway is thus set to the ideal headway of 8 min. To better alleviate bus bunching, each stop is used as a control point. When the headway of two buses is less than 2 min, the holding strategy is implemented for the follower bus, such that the headway with the leader bus is 8 min. The adjusted schedule is shown in Table 7. This shows that the bus bunching in the original schedule has been completely solved. However, the schedule deviation at stop 15 increases to 394 min. Compared with no control strategy, the schedule deviation increases by 14.87%. As a result, the time of the last bus arriving at the terminal stop is delayed, which will complicate the subsequent bus scheduling in the opposite direction.

6.1.2 Passenger waiting time analysis

Given that the bus system is a public service, passenger waiting time should also be considered. Figure 5 shows the impact of the implementation of the short-turning strategy (with six short-turning trips) and holding strategy on the total waiting time of passengers at each stop. In the figure, the horizontal axis represents stops and the two vertical axes represent passenger waiting time and passenger arrival rate, respectively.

Through the implementation of the short-turning strategy, the waiting time of passengers boarding from stop 1 to stop 14 increases by 886 min. However, the waiting time of passengers boarding from stop 15 to stop 48 decreases by 1564 min. Due to the high passenger arrival rate from stop 15 to stop 48, the total waiting time for passengers reduces by 678 (\(1564-886=678\)) min.

Fig. 5
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The total passenger waiting time and the constant passenger arrival rate at stops

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As shown in the previous subsection, the holding strategy is effective in alleviating bus bunching. However, from Fig. 5, we see that the implementation of the holding strategy is not friendly to passengers. Compared with the short-turning strategy, the waiting time of passengers boarding from stop 15 to stop 48 increases by 1669 min, and the waiting time of passengers boarding from stop 1 to stop 14 decreases by 886 min. The total passenger waiting time therefore increases by 783 min. Compared with the holding strategy, the short-turning strategy is beneficial in terms of passenger waiting time.

6.1.3 Sensitivity analysis

To analyse the scheduling effect with different numbers of short-turning trips, we also perform a sensitivity analysis of the bus operation status. We compare the performance of the optimal trips adjustment with the difference in the number of short-turning trips (Num). The results, which are summarised in Table 8, demonstrate that schedule deviation decreases with an increasing number of short-turning trips. The schedule deviation is reduced by at least 43.44% \(((343-194)/343\times 100\%=43.44\%)\), corresponding to one short-turning trip) compared with no control strategy. With an increasing number of short-turning trips, the total waiting time of passengers initially decreases and then increases. The total waiting time of passengers achieves its minimum value with four short-turning trips.

Table 8 The influence of different number of short-turning trips on bus operation
Full size table

The selection of the number of short-turning trips can be adjusted according to the desired effect. Table 8 shows that the schedule deviation decreases as the number of short-turning trips increase. Based on the above sensitivity analysis, however, there is a limit to the increase in the number of short-turning trips. On the one hand, such an increase makes the overall passenger waiting time longer; on the other hand, it decreases the scheduling cost. We take a schedule deviation of less than 100 min as the threshold to select the number of short-turning trips. If we want to schedule as few short-turning trips as possible, the optimal number of regular trips to convert to short-turning trips is three; if we prioritise reducing the waiting time of passengers, the optimal number of short-turning trips is four.

6.2 Discussions

The short-turning strategy makes the actual running schedule of trips at the starting stop of short-turning trips and downstream stops closer to the original schedule. Bus bunching is alleviated and bus scheduling at the terminal stop (the starting stop in the opposite direction) is facilitated. Although the implementation of the short-turning strategy increases the waiting time of passengers outside of the short-turning segment, the total passenger waiting time is reduced due to the reduction in the waiting time of passengers in the short-turning segment. However, a large number of short-turning trips has a negative impact on passengers outside of the short-turning segment, without much additional reduction in the waiting time for passengers in the segment. For example, when the number of short-turning trips increases to eight, the total waiting time begins to increase sharply compared with four short-turning trips. When the optimised schedule at stop 15 is completely consistent with the original schedule, even if the number of short-turning trips increases, the waiting time for passengers boarding from stop 15 to stop 48 will not be reduced, but the waiting time for passengers boarding from stop 1 to stop 14 will increase significantly. There is thus a trade-off between the number of short-turning trips and passenger waiting time.

The holding strategy can alleviate bus bunching by increasing the headway at the control point. However, this extends the overall journey time. As a result, both the passenger waiting time and operation costs will increase. The implementation of the holding strategy increases the deviation of the actual schedule from the original schedule, which will reduce passenger satisfaction. Furthermore, metropolises such as Beijing are characterised by complex traffic networks. With each bus stop being used by many bus lines, the implementation of the holding strategy on one bus line will affect the operation of other lines. Therefore, it is unrealistic to implement the holding strategy under complex traffic conditions. In summary, compared with the holding strategy, the short-turning strategy is easier to implement and can not only alleviate bus bunching but also reduce the total waiting time of passengers.

7 Conclusions

In this study, we propose a short-turning strategy to alleviate bus bunching. We model the short-turning strategy with the objective of minimising schedule deviation. By converting some regular trips to short-turning trips, the schedule at the starting stop of the short-turning trips and at the downstream stops remains close to the original schedule, and bus bunching is alleviated. To facilitate the optimisation of the model, the nonlinear short-turning model with complex objectives and constraints is transformed into a linear programming model. The method is tested to a numerical example in Beijing, China. Compared with the current bus schedule, the results show that the method can relieve bus bunching with a more than 43.44% reduction of schedule deviation and also reduce the total passenger waiting time by up to 8.99%. With an increasing number of short-turning trips, the total waiting time of passengers initially decreases and then increases. This indicates that there is a trade-off between the number of short-turning trips and total passenger waiting time. For long bus lines with concentrated congestion, the holding strategy is not advantageous. The short-turning model presented here is designed for, but not limited to, urban public transport systems. It can be extended and applied to the scheduling for intercity passenger transportation systems and ferry service systems.

There are several promising future research directions. First, because of the complex characteristics of traffic conditions, it would be interesting to apply the proposed short-turning strategy to multiple competing bus lines. Second, the cost of bus operations is also an important concern for a bus operator, so operation cost could be added as an optimisation objective in future research. Finally, considering the advantages and disadvantages of various strategies, scheduling models to alleviate bus bunching that combine the short-turning strategy and other control strategies could be studied in the future.