A holding strategy to resist bus bunching with dynamic target headway

Computers & Industrial Engineering 140 (2020) 106237

Contents lists available at ScienceDirect

Computers & Industrial Engineering journal homepage: www.elsevier.com/locate/caie

A holding strategy to resist bus bunching with dynamic target headway a,⁎

a,⁎

b

b

a

a

Sheng-Xue He , Shi-Dong Liang , June Dong , Ding Zhang , Jian-Jia He , Peng-Cheng Yuan a b

T

Business School, The University of Shanghai for Science and Technology, Shanghai 200093, China School of Business, SUNY at Oswego, NY 13126, USA

ARTICLE INFO

ABSTRACT

Keywords: Public transit Bus bunching Stability Adaptive control Holding strategy

Bus bunching commonly appears in a high-frequency bus line and will bring the lengthened waiting and riding times to passengers. Among the existing strategies of resisting bus bunching, only a few tried to cope with a highfrequency bus line which has not any pre-specified schedule or headway required to be adhered to. To stabilize a bus line as above, we propose a target-headway-based holding strategy in this paper. The average of the instantaneous time headways is used as the dynamic target headway. Based on the difference between the dynamic target headway and the forward headway of the current bus, the new strategy determines a proper holding time in a dynamic traffic situation. After removing the stochastic factors influencing on the headway, we proved the effectiveness of the new strategy to stabilize an unstable bus line. Different from the existing theoretical analyses which only focus on the single control point situation, the theoretical analysis in this paper takes into account any given number of control points. The numerical experiment demonstrated the effectiveness of the strategy by comparing with other strategies. The following insights were uncovered from the numerical analysis. With the increase of control points in a bus line, the stability performance and the service level of the bus line will increase at the beginning and then change to decrease. With the increase of the stochastic level of travel time, the performance of the bus line will decrease gradually. The new strategy can effectively cope with the non-peak, single-peak and double-peak demand patterns in a similar way.

1. Introduction It is common in a high-frequency bus line that two or more buses serving the same bus line arrive at a bus stop nearly at the same time or are cruising head-to-tail in the road. The above phenomenon is called bus bunching or bus platoon. Bus bunching usually lengthens the average waiting and riding times per passenger. Because many passengers will board the first bus in a bus platoon arriving at a stop, the first bus will become very congested. Bus bunching will give rise to the mismatching of the passenger capacities of buses. The unreliable service of a bus line due to bus bunching will incur more complaints from passengers and reduce the public transit ridership seriously in the long run. Over the past several decades, many effective strategies have been proposed and testified to resist bus bunching. They include the stopskipping strategies (Cortés, Saéz, Milla, Nuñez, & Riquelme, 2010; Fu, Liu, & Calamai, 2003; Liu, Yan, Qu, & Zhang, 2013; Suh, Chon, & Rhee, 2002; Sun & Hickman, 2005), limited-boarding strategies (Barnett, 1974; Delgado, Muñoz, & Giesen, 2012; Delgado, Muñoz, Giesen, & Cipriano, 2009; Newell, 1974; Osuna & Newell, 1972), embeddingslack strategies (Daganzo, 1997, 2009; Xuan, Argote, & Daganzo, 2011; ⁎

Zhao, Dessouky, & Bukkapatnam, 2006), static and dynamic holding strategies (Argote-Cabanero & Daganzo, 2015; Bartholdi & Eisenstein, 2012; Berrebi, Crudden, & Watkins, 2018; Delgado et al., 2012; He, 2015; Liang, Ma, He, Zhang, & Yuan, 2019; Liang, Ma, & He, 2019; Liang, Zhao, Lu, & Ma, 2016; Sun & Hickman, 2008; Xuan et al., 2011), speed adjustment strategies (Daganzo & Pilachowski, 2011; He, 2015; He, Dong, Liang, & Yuan, 2019), transit signal priority strategies (Estrada, Mension, Aymami, & Torres, 2016; Ling & Shalaby, 2004; Liu, Skabardonis, & Zhang, 2003) and bus substitution strategy (Petit, Ouyang, & Lei, 2018). The ideas of various holding strategies to resist bus bunching can be applied to Supply Chain Management (SCM) to improve the effect of resource utilization. Nowadays facing the complicated stochastic environment and various sustainable requirements, researchers in the realm of SCM need to make use of all available ways to determine the optimal lot-sizing, storage time, and the related replenishment policy (Awasthi & Omrani, 2018; Gharaei, Hoseini Shekarabi, & Karimi, 2019; Gharaei, Karimi, & Hoseini Shekarabi, 2019; Gharaei, Karimi, & Shekarabi, 2019; Gharaei, Hoseini Shekarabi, Karimi, Pourjavad, & Amjadian, 2019; Gharaei, Karimi, & Shekarabi, 2018; Giri & Bardhan, 2014; Giri & Masanta, 2018; Hao, Helo, & Shamsuzzoha, 2016; Tsao,

Corresponding authors. E-mail addresses: [email protected] (S.-X. He), [email protected] (S.-D. Liang), [email protected] (J. Dong), [email protected] (D. Zhang).

https://doi.org/10.1016/j.cie.2019.106237 Received 13 May 2019; Received in revised form 5 December 2019; Accepted 17 December 2019 Available online 19 December 2019 0360-8352/ © 2019 Elsevier Ltd. All rights reserved.

Computers & Industrial Engineering 140 (2020) 106237

S.-X. He, et al.

2015). The above decisions can be viewed as complex holding decisions in SCM. Here the raw material, semi-finished products and products replace buses as the control object. A supply chain substitutes for a bus line in the new setting. Interested readers can further refer to the following important studies: Rameshwar, Angappa, and Tripti (2015), Yin, Nishi, and Zhang (2015); Sayyadi and Awasthi (2016), Rabbani, Foroozesh, Mousavi, and Farrokhi-Asl (2017), Reza and Anjali (2018), Duan, Deng, Gharaei, Wu, and Wang (2018), Masoud, Seyed, Amir, and Hamed (2018), Nima, Salwa, Raja, Ehsan, and Simone (2018), Sarkar and Giri (2018), Shah, Chaudhari, and Cárdenas-Barrón (2018) and Hoseini, Gharaei, and Karimi (2019). Most of the existing holding strategies used a predetermined or virtually assumed schedule to estimate the deviation of the actual situation from the schedule. The obtained deviation was then used to construct an optimization objective. If there is no such a schedule, these strategies will be out of work in practice. Only a few of the existing holding strategies focused on a high-frequency bus line which has not any predetermined schedule and any pre-specified headway. We will briefly introduce these few strategies below. Bartholdi and Eisenstein (2012) proposed a strategy to adjust the backward headway of a bus at the control point. Liang et al. (2016) extended the above thought to self-equalizing the headways at control points. They proved the effectiveness of their methods with Markov Chain Theory regarding one control point. Zhang and Lo (2018) further refined the theory of selfequalizing holding strategy. Zhang and Lo (2018) fully investigated the situation with one control point. The above strategies were tested in bus lines with very few control points, generally only one or two. The lack of theory and practice about multiple control points makes the related strategies very vulnerable when many control points are considered. We also note that the above strategies only took into account the local information to determine a holding time. It is very likely to improve the performance of a bus line with some new strategy taking into account the whole bus line information in a dynamic situation. In the paper, we will focus on resisting bus bunching in a bus line without any predetermined schedule or pre-specified headway. We define a critical time point (CTP) as the time point when a bus is about to depart from a stop after finishing its necessary loading and discharging operations. To generate a proper target headway, we estimate the instantaneous time headways of buses at a CTP at first. Then we use the average value of these instantaneous headways as the dynamic target headway with respect to the CTP. The difference between the dynamic target headway and the forward headway of the current bus will be used to adjust the forward headway at the control point. The above headway adjustment is realized by holding the bus for a specified time interval at the control point when the forward headway of the bus is smaller than the target headway at that time. With the evolution of a bus line, the above holding operation can be carried out continuously at all the critical time points. The above holding strategy can not only make use of the dynamic information of the whole bus line, but also resolve the problem of setting a proper criterion which is required to adjust buses’ headways at control points. To strengthen the theoretical basis of related strategies, we will prove the effectiveness of the new strategy when the number of control points is arbitrary. To realize this goal, we will formulate the related control operations into a product of a series of matrices. With the evolution of the bus line under our holding strategy, the product of the ordered matrices will converge to a fixed matrix. The above convergence verifies the effect of our new strategy on stabilizing a bus line. To test our new strategy, a test bus line will be constructed in detail. The passengers’ boarding and alighting processes at stops, the dynamic traffic states in road segments, and the traffic signal control at intersections will all be considered in the construction of the bus line. Through being applied to the test bus line, our new strategy will be compared with other holding strategies. In numerical experiments, we will investigate the influences of the number of control points, the random level of travel time, and the different demand patterns on the

Fig. 1. A circular bus line with 5 buses and 9 stops.

performance of our strategy. The remainder of this paper is organized as follows. In Section 2, we will clarify some important conceptions associated with holding control. Especially, the target headway and the stability index will be defined in this section. In Section 3, we will first introduce the operational rule of the target-headway-based holding strategy and then explain how to implement our strategy. In Section 4, we will prove the effectiveness of our new strategy theoretically regarding multiply control points. In Section 5, a test bus line is constructed to assess the performance of our new strategy. In the end, we will summarize our study and point out several directions for future research. 2. Describe the system 2.1. The circular bus line We will consider a circular bus line, such as the one shown in Fig. 1, to analyze and demonstrate our strategy. The circular bus line depicted in Fig. 1 has 5 buses and 9 stops. The stop with serial number 1 is specified as the reference point. The reference point will be used later to update the serial numbers of buses. The bus line to be focused on in this paper has the following three features. The first one is the high-frequency characteristic. This means when buses are evenly dispersed along the bus route, the time headway between any two successive buses is generally limited to a range from 3 min to 15 min. The second feature is the varying arrival rates of passengers. The traffic demand at a stop will change with time. We will consider the non-peak, single-peak and double-peak demand patterns in this study. The third feature is the changing travel time in a road segment. The expected travel time in a road segment will change with the whole traffic state in the road segment. The travel time in the peak hour of traffic will be longer than in the off-peak hour. The high-frequency feature, the varying demand, and the changing travel time are important factors which may induce bus bunching. To stabilize a bus line with the above three features, we need to design a control strategy which can make use of the whole dynamic information of the bus line. 2.2. Where, when and what to control In the following, we will answer the questions including where to apply the control, when to start to control, and what to control. Terminal station holding strategy (TSHS) uses terminal stations as control points. Other strategies usually choose some stops as control points. For example, He (2015) and Daganzo (2009) use all the stops in a bus line as control points. Bartholdi and Eisenstein (2012), Liang et al. 2

Computers & Industrial Engineering 140 (2020) 106237

S.-X. He, et al.

(2016), Liang, Ma, He, Zhang, et al. (2019) and Liang, Ma, and He (2019) only use one or two stops as control points. Because we plan to analyze the impact of the number of control points on the control effect, the set of control points will be adjusted accordingly in our study. When choosing a control point, we obey the following principle: limiting control points to the stops which are exclusively used by the bus line in question and have relatively low arrival rates of passengers. Obeying the above principle can reduce not only the influence of holding operation on the other bus lines, but also the number of passengers who will be directly influenced by the holding operation. A holding operation is to hold a bus for a while at a control point. It starts when the bus finishes the necessary loading and discharging operations at the control point. Due to various stochastic factors, it is very hard to know the exact start time of a holding operation in advance. In view of the above situation, an adaptive holding strategy needs to determine the exact holding time in a very short time. In practice, the above decision time should be shorter than one second. In another word, a practical strategy must be time-saving with respect to collecting necessary information, calculating the holding time and transferring the control instruction. In this paper, we will only consider using the headway-based holding operation as control means. So the headway will be our control subject. The time headway between two successive buses is the travel time required for the following bus traveling from its current location to the current location of the leading bus. To the following bus, the above headway is called its forward headway. To the leading bus, the above headway is its backward headway. Different strategies usually consider adjusting different headways. For example, the terminal station holding strategy adjusts the forward headway of a bus. Bartholdi and Eisenstein (2012) suggested adjusting the backward headway after a bus passes through the control points. Liang et al. (2016) and Zhang and Lo (2018) proposed to equalize the forward and backward headways for a bus. Most of holding strategies use forward headway as their control subject when a bus finishes the loading and discharging process at the control points. In this paper, we will use forward headway as our control subject.

time point when a bus finishes the necessary loading and discharging operations at a stop as a Critical Time Point (CTP). If a stop is not a control point, a CTP related to this stop is corresponding to the departure time for a bus from this common stop. If a stop is a control point, a bus may be held there when it finishes the necessary loading and discharging operations. So if a bus is held at a control point, the related CTP is the start time of holding. If a bus is not held at a control point, the related CTP is still corresponding to the departure time of the bus from the control point. A CTP related to a control point is defined as Actual Control Time Point (ACTP). Let TCTP be the set of all the CTPs included in an observation period and tc be a typical CTP. Let TM be the set of all the ACTPs included in an observation period and tm be a typical ACTP. Let nCTP and nTM be the sizes of TCTP and TM , respectively. We can define the stability index with respect to the whole observation period as follows:

¯H =

SI

hb (t )/ nB

=

b B

[hb (t )

tc TCTP

[ (tc )

¯H ]2 / nCTP

(4)

3. Realize the control 3.1. Rule to determine the holding time The control rule of our strategy is clarified as follows. When a bus finishes the necessary loading and discharging operations at a control point, the average instantaneous headway, i.e. DAH, will be calculated. If the forward headway of the current bus is equal to or bigger than the DAH at this ACTP, the bus should depart from the control point at once. If the forward headway of the current bus is smaller than the DAH at this ACTP, we will hold the bus at this control point for a time interval. The actual holding time should be smaller than or equal to the absolute difference between the forward headway of the current bus and the DAH. The above absolute difference will be referred to as the maximal allowable holding time. Why should the actual holding time be equal to or smaller than the maximal allowable holding time? If the maximal allowable holding time is too big, to use it as the actual holding time will seriously influence the backward headway of the bus and sometimes may lead to bunching from the behind. A large holding time may also incur many complaints from passengers. The difference between the forward and backward headways of a bus may be big at the beginning of an observation period. In this case, the calculated maximal allowable holding time will be correspondingly big. To alleviate the negative influence of a relatively big holding time at the beginning, we should adopt a relative small actual holding time in this situation. Later when the bus line runs relatively smooth due to the proper holding control, the calculated maximal allowable holding time will become relatively small. In this situation, the actual holding time could be equal to the maximal allowable value. Let atm denote the holding time with respect to the ACTP tm TM . Assume that b is the current bus in question at tm . The rule to determine the holding time is given as follows:

(1)

H (t )]2 / nB

=

In the following, will be used as the stability index of a bus line and will be used to reflect the reliability of ¯H as the estimate of the whole stability of a bus line during an observation period.

where B is the set of all the cruising buses and nB is the number of buses. If all the instantaneous headways at time t are equal to the DAH at time t , we can say the bus line system is perfectly stable at this time. So the deviation of all the instantaneous headways from the corresponding DAH can be used to evaluate the operational stability of the bus line at a given time. The instantaneous stability index at time t is defined as follows: H (t )

(3)

SI

The goal of holding control is to make buses dispersed in the bus route as even as possible. The effectiveness of a strategy should be assessed not only by the classical measurements, such as the average waiting and riding times per passenger, but also some measurements reflecting the operational stability and reliability of a bus line. In the following, we will define a stability index which can be used to evaluate the operational stability of a bus line under all the strategies. Let hb (t ) be the instantaneous forward headway of bus b at time t . We define the average of headways of all buses at time t as the Dynamic Average Headway (DAH) which will be the target headway used in our strategy. DAH can be calculated as follows: b B

H (tc )/ nCTP

This average of instantaneous stability indices over a whole observation period can capture the whole operational stability of a bus line under different strategies. To know the variance of ¯H during an observation period, we define the standard deviation of ¯H as follows:

2.3. Target headway and stability index

H (t ) =

tc TCTP

(2)

To evaluate the operational stability of a bus line during a whole observation period, we need to specify some time points at which the corresponding instantaneous stability index H (t ) can be calculated and stored to be used later. In view of the above requirement, we define the

atm = Here 3

[H (tm )

hb (tm )], if hb (tm ) < H (tm); 0, or else.

(5)

[0, 1] is the parameter used to reflect that the actual holding

Computers & Industrial Engineering 140 (2020) 106237

S.-X. He, et al.

time is usually smaller than the maximal allowable holding time. According to the above analysis, the value of should be relatively small at the beginning. Later the value of can be increased near to 1. For simplicity, we will fix the value of to 0.5 in the numerical experiment of Section 5. The main differences between our strategy and the terminal station holding strategy are summarized as follows. Firstly, in addition to terminal stations, our strategy also uses some common stops as control points. This will obviously improve the effectiveness of our strategy. Secondly, although both our strategy and TSHS adjust the forward headways of buses with some target headways at control points, the target headway used in TSHS is commonly a predetermined static constant, but the target headway adopted in our strategy is dynamically estimated with the real-time information of the bus line. In a dynamically changing traffic circumstance, a dynamic target headway makes our strategy more resilient than a static constant headway does TSHS. Using a dynamic target headway in our strategy requires more computational time than using a constant headway in TSHS. In the following, we will introduce how to calculate an instantaneous forward headway. The calculation of forward headway is the most important computational part of our strategy.

the set of all the road segments. According to the above assumption, we will only consider the expected average travel time in a road segment when we need to calculate the instantaneous headway. In this paper, we assume that passengers use different doors to board and alight from a bus. With this assumption, the expected dwell time of a bus b at a stop e will be the bigger one between the expected boarding time and the expected alighting time of passengers at stop e with respect to b. If there is no passenger who will board or alight from bus b at stop e , a fixed time interval that equals the time of opening and closing the doors will be assigned to the expected dwell time. Here we have taken the following two assumptions. One is that when a bus arrives at a stop, the bus must stop even if no passenger will board or alight from the bus at this stop. Note that to stop at all the bus stops is the common practice in real life since it’s usually hard for a driver to know in advance whether there are some passengers who need to alight from or board the bus at the incoming stop. The other assumption is that when a bus needs to load or discharge passengers at a stop, the time for opening and closing the doors will be overlooked in estimating the dwell time for simplicity. The expected alighting time of a bus b at a stop e can be easily estimated by the product of the average alighting time per passenger and the number of on-board passengers who has e as their destination stop. Note that the expected alighting time only needs to be estimated at the stops which locate between the current bus and its leading bus. Since the number of the above mentioned stops is usually small, there will not be many passengers who will board and also need to alight at these stops. For simplicity, we will overlook the related alighting times of these passengers. So only the on-board passengers at current time will be considered to estimate the expected alighting time. In real life, the above simplicity is reasonable since the number of stops to be traversed by a bus from its current location to the current location of its leading bus in a bus line is usually small, especially when a high-frequency bus line is considered. The expected boarding time of a bus b at a stop e can be estimated as follows. Assume that the arrival time of b at e is tbArr , e and the departure 1 represents the time of the leading bus of b from e is tbDep1, e where b leading bus of b. So during the time interval from tbDep1, e to tbArr , e , the expected number of passengers arriving at stop e is equal to re (tbArr tbDep1, e ) . Here we assume that the arrival rate of passengers at e ,e is a constant re . The total boarding time of these passengers is t¯Boa re (tbArr tbDep1, e ) . Here t¯Boa denotes the average boarding time per ,e tbDep1, e ) , passenger. Note that during the above boarding time t¯Boa re (tbArr ,e e new passengers may arrive at stop and need to board the bus b . The total boarding time of the above newly arrived passengers can be estbDep1, e ) . Since the above deducing logic can timated by (t¯Boa re )2 (tbArr ,e continue, we can obtain a series of boarding times. Finally, all the boarding times can be summed up as follows:

3.2. To estimate the forward headway To obtain the instantaneous forward headway of a bus at a given time, we need to know the expected travel time from a given location to another location in a bus route. Since the expected travel time may include the delays at some intersections, the expected travel times of some related road segments, and the expected dwell times at the stops to be encountered in the journal, we will consider them one by one in the following. Firstly, let us consider an intersection i I where I is the set of all the intersections. We assume that the traffic signal control schemes at all the intersections are the two-phase pre-timing traffic signal control. The amber phase has been included in the other phased properly. tiRed and tiGreen are the lengths of the red and green phases of intersection i , respectively. The expected delay t¯iD at intersection i can be estimated as follows: 1 t¯iD = 2 (tiRed) 2/(tiRed + tiGreen)

(6)

Here we assume that the arrival time of a bus follows the uniform distribution over a signal control cycle. So the probability of encountering a red phase is tiRed/(tiRed + tiGreen) for a bus. For the bus encountering a red phase, its average delay will be the half of the red phase. With the above assumption, the expected delay for a bus at a signal control intersection should equal the product of tiRed/(tiRed + tiGreen) and tiRed/2 . In this paper, we define the part of a bus line between two successive stops as a bus line segment. The stops and intersections are viewed as critical locations in a bus route. The part of a bus route connecting two successive critical locations is called a road segment. The travel time in a road segment mainly depends on the average cruising speed of buses and the length of the road segment. The travel time in a road segment is also influenced by many other stochastic factors, such as the surrounding traffic and the driving habits of bus drivers. To reflect the above observation, we assume that the travel time in a road segment consists of two parts. One is the expected average travel time. This expected average travel time is changing with time responding to the general traffic situation in the road segment. We assume that the value of the expected average travel time can be estimated in advance and out of our control system. The other is the random part obeying the normal distribution with zero mean and ld standard variance. Here is a given positive parameter and ld is the length of the road segment d D . D is

t¯Boa re (tbArr ,e

tbDep1, e )[1 + t¯Boa re + (t¯Boa re )2 +…+ (t¯Boa re) k + …] with k (7)

In real life, the value of t¯Boa re is commonly less than 1. So we can replace the above sum with the limit of (7), i.e. (tbArr tbDep1, e ) t¯Boa re /(1 t¯Boa re) . The above limit can be used as the esti,e mate of the expected boarding time with respect to the bus b and the stop e . Note that if the arrival rate of passengers varies with time, the above calculation needs to be modified with integral. In this situation, we can add up a finite number of the partial boarding times to estimate the expected boarding time. At the end, we can use the bigger one between the expected alighting and boarding times as the expected dwell time. 3.3. Application process of the new strategy The process of implementing the new strategy is summarized as

4

Computers & Industrial Engineering 140 (2020) 106237

S.-X. He, et al.

follows:

follows:

Step 1: Identify the bus to be activated which just finishes the necessary loading and discharging operation at a control point. Step 2: Calculate the target headway with Eq. (1). Step 3: Comparing the forward headway of the bus to be activated with the target headway, we determine whether we need to carry out a holding operation. If a holding operation is required, we should compute the specific holding time with Eq. (5). Step 4: According to the judgment in Step 3, we choose to carry out the necessary holding operation or to dispatch the bus from the control point at once.

1 0 0 1

MHA ( , i) = 0 0

0 0 1

0 0 0 0 (i, i )

0

.Here

is a non-negative

0 0 0 1 0 0 0 0 0 1 number included in the interval [0, 1]. The second parameter i {1, 2, …, nB} in MHA ( , i) indicates the position of the headway in h which will be adjusted. For example, the inner product of the initial h , i.e. (h1, h2, …, hnB , h)T , and MHA ( , i) , is equal to the column vector (h i h ) (h1, h2, …, (1 ) hi + h, …, h)T . (1 ) hi + h is equal to hi which means to adjust the value of hi and make the resulted value near to the target headway h . The specific value of depends on the maximal allowed holding time at the related control point. Obviously, (hi h) is equal to the specified holding time.

4. Theoretical proof of the effectiveness 4.1. The operational matrices

4.2. Theoretical results about effectiveness

When a bus is cruising in a bus line, it will accept two types of control operations. One is renumbering its serial number when any bus passes through the reference point. The other is adjusting its forward headway by holding it for a specified time interval at a control point. We will formulate these two operations into two types of matrices, respectively, below. The rule to assign a serial number to a bus is as follows. At any time, if the location of a bus is downstream of and nearest to the reference point, we should assign the serial number 1 to this bus. The bus downstream of and nearest to the bus with serial number 1 should be assigned serial number 2. The serial numbers of other buses can be assigned in a similar way. Let hi be the forward headway of the bus with initial serial number i .

In terms of public transit, a run denotes the process that a bus sets off from a reference point and travels in the bus line until coming back to the reference point. In a run, a bus will encounter all the control points and has a chance to adjust its forward headway at each of these control points. So we can assume that total n times of headway-adjusting operations will be applied to a bus in a run. n equals the total number of control points. Though we know the value of n , we do not know the specified parameters and i in MHA ( , i) in advance. When a bus line runs smoothly without the appearance of any bunching or overtaking, we can safely assume that all the buses will finish or nearly finish a run when one of buses finishes a run with respect to a given start time. So in a run with respect to a given bus and a given start time, we can say all the buses have the chance to adjust their foreword headways n times.

Put all the forward headways of buses in a column vector h . The initial

h is equal to (h1, h2, …, hi, …, hnB )T . We will use the position of hi in h to indicate the serial number of the bus with initial serial number i . For

Proposition 1. Assume that the adjusting rates in one run for a bus are given as 1, 2 , 3 , …, n and at the beginning of the run the bus has a given forward headway hk . Omitting the influence of various stochastic factors during the run on the headways, the difference between the original headway n h) at the end of the hk and the target headway will be i = 1 (1 i )(hk run.

example, if hi is the second element in h , it means the bus with initial serial number i is assigned the serial number 2 now. Use h to stand for the target headway. In the following analysis, we assume that the value of h is fixed. We define an enlarged vector h consisting of forward headways and h . The initial h is equal to (h1, h2, …, hnB , h)T . We assume

that h always occupies the last position of h . Just as in h , the position of any headway hi in h indicates the serial number of hi . According to the rule of assigning serial numbers to buses mentioned above, when a bus passes through the reference point, the serial numbers of all the buses need to be renumbered. In another word, when a bus passes through the reference point, all the headways of buses need

Proof. After the first adjustment, the original headway will be (1 1) hk + 1 h . The difference between this renewed headway and h) . After the second adjustment, the target headway h is (1 1)(hk (1 (1 the headway becomes 2) 1) hk + 1 h [(1 The difference between the renewed 1) hk + 1 h] + 2 h . h) . In the same way as headway and h becomes (1 2)(1 1)(hk above, we can obtain the difference between the headway resulted from the given one run and the target headway h that is (1 h) . n )…(1 2 )(1 1)(hk

to change their positions in the vectors h and h accordingly. Note that the bus which is about to pass through the reference point always occupies the position next to the last one in h . After the bus passes through the reference point, the position of its headway in h will be the first one and the positions of other headways will move down one position in h . The above position-updating operation can be expressed with a position-updating matrix with respect to h as follows: 0 0 0 1 0 1 0 0 0 0 0 1 0 0 0 MPU = .MPU is an (nB + 1) × (nB + 1) matrix. As

Since is in the interval [0, 1], we can see the above difference becomes smaller and smaller with the evolution of the bus line. In other words, no matter what are the specific values of the original headways, they will all converge to the target headway in the situation where no stochastic factor influences the headways. Proposition 2. Under the operations of position-updating and headwayadjusting, the headway vector h with the initial value (h1, h2, …, hnB , h)T will finally converge to (h, h, …, h)T if the stochastic factors influencing on the headway are removed.

0 0 1 0 0 0 0 0 0 1 an example, the inner product of MPU and (h1, h2, …, hnB , h)T equals (hnB , h1, …, hnB 1, h)T . The above inner product means that the initial serial numbers of buses are changed after the bus with initial serial number nB passes through the reference point. In addition to renumbering the serial numbers of buses, the forward headway of a bus will be adjusted by holding the bus for a specified time interval at a control point. So we need to define a typical headwayadjusting matrix MHA which is also an (nB + 1) × (nB + 1) matrix as

Proof. As we have mentioned if a bus is held for a time interval at a control point when the bus has finished its necessary loading and discharging operations, its forward headway will be adjusted. The above operation can be captured by the inner product between a headway-adjusting matrix MHA ( , i) and the current enlarged headway vector. When a bus passes through the reference point, the positions of 5

Computers & Industrial Engineering 140 (2020) 106237

S.-X. He, et al.

headways in the enlarged headway vector will be updated that can be described with the inner product between MPU and the current enlarged headway vector. According to the above analysis, the value of h will continuously be changed by multiplying the current h by a headwayadjusting matrix or a position-updating matrix. If the statement of this proposition is true, the product resulted from multiplying the headwayadjusting and position-updating matrices should converge to a matrix as follows:

0 0 0 0

Table 2 The relations between bus line segment and road segment.

0 0 1 0 0 1

. 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 Obviously, we have M h = (h, h, …, h)T . From Proposition 1, we know any given value of the original headway of a bus will finally converge to the target headway h . Since the value of the original headway to some extent can be arbitrary for any bus, all the elements except the ones in the last column must be 0. Or else, the statement of Proposition 1 will be violated. This conclusion can be easily explained by an example. For example, let mi, j be the element of M at the i th row and j th column. The original h times the first row of M as follows:

M =

BLS

RS

Length (m)

BLS

RS

Length (m)

1 2 3 4 5 6 7 8 9 10 11 12 13 14

1,2 3 4 5,6 7 8 9 10,11 12 13,14 15 16,17 18,19 20

200,400 500 600 260,350 530 560 600 300,500 600 300,350 600 300,400 300,320 500

15 16 17 18 19 20 21 22 23 24 25 26 27 28

21 22,23,24 25 26 27 28,29 30,31 32 33 34,35 36 37,38 39,40 41,42

450 200,250,100 570 610 600 300,350 200,400 500 600 260,350 530 280,280 260,340 300,500

Note. ‘BLS’ and ‘RS’ stand for bus line segment and road segment, respectively.

reached, the bus will stop loading any passenger until unoccupied space is available at the next stop. The ‘First Stop’ is the stop that a bus is dwelling at or heading to at the beginning of the observation period. The time for a bus to travel to its ‘First Stop’ is presented in the last row. The measurement units of ‘pax’ and ‘s’ stand for passenger and second, respectively. The inclusion relationships between bus line segment (BLS) and road segment (RS) are given in Table 2. The lengths of road segments are listed in the column titled by ‘Length (m)’. The measurement unit of length is one meter (m). Let d denote a typical road segment. ld is the length of d . v¯d (t ) is the average cruising speed for a bus entering the road segment d at time t . The expected average travel time in d equals the result of ld divided by v¯d (t ) . As we have mentioned earlier, the random part of the travel time of a road segment follows a normal distribution with the mean zero and the standard variance ld . Here is a positive parameter with seconds per kilometer as its measurement unit. In the standard variance ld , ld has one kilometer as its measurement unit. In the following analyses, equals 5 if not otherwise specified. We assume that v¯d (t ) will obey the trend shown in Fig. 2. Five stages constitute the trend. Among the stages, the value of v¯d (t ) will remain the same in the first, third and fifth stages which span time intervals [0, 3600], [5400, 9000] and [10800, 14400], respectively. This partition reflects the surrounding traffic in the bus route during the observation period. We assume that at the beginning, traffic is relatively light and the speeds of buses are relatively high. When traffic becomes heavy, the speeds of vehicles will decrease gradually. When the heavy traffic remains, the expected average speed of buses will remain at a relatively low level. Later when traffic becomes light, the cruising speeds of buses will increase accordingly. It is possible to assign different values of v¯dk for k = 1, 2, 3 to different road segments. But for simplicity, we assume that all the road segments have the same values of v¯dk for k = 1, 2, 3. In the following analysis, v¯d1, v¯d2 , and v¯d3 are 35 km/hr, 28 km/hr, and

m1,1 h1 + m1,2 h2 + …+ m1, nB hnB + m1, nB + 1 h Since the values of h1, h2 , …, hnB could be arbitrary, only if m1, j = 0 for all j {1, 2, …, nB} and m1, nB + 1 = 1, the above product will equal the target headway h . In the above analysis, we omit the stochastic factors influencing the headway in order to prove that the new strategy can stabilize a bus line with arbitrary initial headways. Though the stochastic factors, such as the uncertainty of travel time in a road segment, will influence the actual values of headways resulted from the holding operation, their negative effect can be reduced by the holding operation in the one hand. In the other hand, the influences of stochastic factors will cancel each other out with the running of the bus line. In view of the above observation, we will obtain a stable bus line at last with headways near to the target headway but not strictly equal to the target headway. The above deduction will be verified by a simulation bus line in Section 5. In the above analysis, we also assume that the target headway is a given constant. This assumption is fit for the holding strategies with a fixed target headway, such as the TSHS. When the target headway varies continuously to reflect the real-time traffic state, the above analysis can be viewed as a demonstration of the holding effectiveness of the related strategies. The holding operation with a target headway shows the power to limit the variance of headways and keep them around the dynamic target headway. This will also be verified in Section 5. 5. Numerical experiment 5.1. The test bus line A test bus line with 9 cruising buses and 28 stops will be considered in this section. The length of the bus line is 16.7 km. The observation period spans 4 h. Table 1 presents the basic information about buses. The passenger capacity of a bus is the maximal number of passengers including standees that the bus can carry. If the passenger capacity is Table 1 The basic data about the buses. No. of Bus

1

2

3

4

5

6

7

8

9

Passenger Capacity(pax) First Stop Time to First Stop (s)

72 1 20

70 4 0

80 8 40

60 11 30

72 15 50

60 18 10

72 21 30

80 25 36

60 28 24

Fig. 2. The changing trend of v¯d (t ) during the given period. 6

Computers & Industrial Engineering 140 (2020) 106237

S.-X. He, et al.

arrival time of this passenger at the stop. To further determine the destination stop of a passenger, we adopt two series of probabilities as shown in Table 5. The sum of the elements in any series is 1. If a stop uses one of the series to choose the destinations for the passengers generated at this stop, the probability of the n th downstream stop chosen as a destination equals the value of the n th element of the chosen series. For example, if stop e chose the series 1 as its series, 0.027 is the probability of the second downstream stop chosen as a destination for the passengers generated at e . The number of elements in the series chosen by stop e is the maximal number of downstream stops which can be used as destination stops for the passengers generated at e . We assume that series 1 is used by bus stops including 1, 7, 10, 13, 20, 21, 27 and series 2 is used by the other remaining stops. We assign specific values to the following parameters to be used in this section. The average boarding time per passenger is set to 1 s (s). The average alighting time per passenger is set to 0.5 s. Bus stops 1 and 14 will be the terminal stations used as control points in TSHS and the self-equalizing holding strategy. Stop 1 as the reference point is the start stop with zero position in the bus line. To compare with the above two methods, we will also use the terminal stations 1 and 14 as the control points in our new strategy at first. Later taking into account the relative size of the arrival rate of passengers, we will choose stops 1, 6, 8, 14, 18 and 24 as the control points for our new target-headway-based holding strategy. The above control points will remain unless otherwise specified. To justify the subsequent comparisons among various strategies, we will run the bus line simulation system 50 times for each strategy. One time simulation spans the whole observation period of 4 h. The average values over the 50 times simulations will be used in the comparison.

Table 3 The basic data of intersections. No. of Intersection

1

2

3

4

5

tiRed (s)

40

40

40

30

50

30

35

45

tiGreen (s) tiInitial (s) Initial phase Bus line segment

6

7

8

9

10

11

12

13

14

30

40

40

30

30

40

40

40

50

40

30

30

45

35

45

50

30

45

55

35

20

20

10

20

20

20

30

20

20

10

20

10

20

10

2 1

1 4

1 8

2 10

2 12

1 13

2 16

2 16

2 20

2 21

1 24

1 26

1 27

1 28

Fig. 3. The traffic demand with five stages at a stop.

31.5 km/hr, respectively. Between any two successive road segments there is an intersection. There are total 14 intersections in the bus line. We assume that the traffic signal control schemes at all the intersections are the pre-timing two phase traffic signal control. The information about the red and green phases is presented in Table 3. At the beginning of the observation period, intersections may stay in different phases. We use ‘1’ and ‘2’ to indicate the initial red and green phases, respectively. The lengths of the initial phases are listed in the row labeled by ‘tiIniti al ’. The bus line segment which an intersection belongs to is pointed out in the last row of Table 3. There are 28 bus stops in the test bus line. For simplicity, we assume that all the stops have the same traffic demand pattern as shown in Fig. 3. We will call this demand pattern the single-peak pattern. During the 4 h period, the first hour spanning a time interval [0, 3600] in seconds has a constant passenger generation rate, i.e. the arrival rate of passengers, indicated by re1 for stop e . The time intervals [5400, 9000] and [10800, 14400] in seconds have the constant passenger generation rates indicated by re2 and re3 for stop e , respectively. The other two time intervals [3600, 5400] and [9000, 10800] have linear demands specified by the arrival rates of two ends of the time interval . The measurement unit of the passenger generation rate is passengers per minute (pax/min). The specific demand at a stop is characterized by the values of rei , i = 1, 2, 3. Using the triple {re1, re2, re3} to distinguish the traffic demands at stops, the bus stops which have the same traffic demand are summarized in Table 4. The time instant when a passenger is generated at a stop will be the

5.2. Comparison of different strategies The trajectories of buses resulted from a given control strategy can provide us with a direct impression of the operational stability of the bus line system under the control strategy. We draw the resulted trajectories of buses in the subfigures from Fig. 4(a)–(d) corresponding to the non-control scenario, the TSHS, the self-equalizing holding strategy, and our target-headway-based holding strategy, respectively. Note that for our new strategy, stops 1, 6, 8, 14, 18 and 24 are chosen as the control points. The trajectories under the non-control scenario in Fig. 4(a) shows a very strong trend to bus bunching. With the evolution of the bus line system, bus bunching phenomenon shows up and then gradually becomes more and more serious. The above change regarding the relative positions of buses wholly conforms to the field observation and the existing theoretical conclusion about the evolution of the bus bunching phenomenon in a high-frequency bus line. In view of the above observation, we can say the test bus line provides an appropriate subject for us to test our strategy. With stops 1 and 14 as terminal stations, TSHS performs well with respect to preventing the happening of bus bunching during the whole observation period. In Fig. 4(b), we also notice that when the time headway between two successive buses is relatively big, the capacity of the TSHS to reduce the relatively big headways is to some extent limited. The Expected System Headway used here is 218 s. The performance of the self-equalizing holding strategy is similar to the TSHS. With stops 1 and 14 as control points, the effect of equalizing headways is limited. From Fig. 4(c), we can see some headways become very small sometimes, especially in front of the control points. Though adding more control points will bring an improvement of the stability performance, we only use the terminal stations as control points for this strategy in order to reflect the current practice and studies about this strategy. As far as we know, most of the existing studies about this strategy only use one or two control points in their experiments. By the way, the related studies did not supply a sound theoretical proof about the effectiveness of this strategy when more than one control point are

Table 4 Bus stops with given traffic demand patterns. {re1, re2, re3}

Related Stops

{1,1,1} {1,2,1} {1,3,1} {1,3,2} {1,4,1}

6, 8,24 1,2,4,5,7,10,11,14,15,17,18,19,22,23,27 3,12,13,16,21,25 9,20,28 26

Note. The unit of re is passengers per minute in this table. 7

Computers & Industrial Engineering 140 (2020) 106237

S.-X. He, et al.

Table 5 The probabilities for the following downstream stops to be a destination. No.

1

2

3

4

5

6

7

8

9

10

11

12

13

Series 1 Series 2

0.0135 0.0345

0.027 0.0862

0.0541 0.1207

0.0811 0.1552

0.1081 0.1724

0.1351 0.1552

0.1351 0.1207

0.1216 0.0862

0.1216 0.0517

0.0811 0.0172

0.0541 /

0.0405 /

0.0270 /

used. From Fig. 4(d), we can see the trajectories of buses under our targetheadway-based holding strategy is relatively smooth with respect to the distribution of headways. The bus bunching can be prevented effectively. Here we choose 6 stops as control points in our target-headwaybased holding strategy. In the following, we will also consider the impact of different sets of control points on the performance of our new strategy. Before we present the numerical results of different strategies, we need first to define the following notations to facilitate the comparison. Except for the stability index ¯H and its standard deviation SI , we will also use the following quantities to compare these strategies with each other. nP is the number of passengers who finished their transit journals during the observation period. We use tPW , tPR and tPTr to denote the average waiting time, the average onboard riding time and the average travel time per passenger. a¯ is the average holding time for a bus at control points. a¯ is the standard variance of a¯ . Note that except for the passenger number nP with a passenger as its measurement unit and the non-dimensional ACTP number nTM , the other quantities including ¯H , SI a , ¯ , a¯ , tPW , tPR and tPTr all have one second as their measurement unit. In Table 6, we present the results from different control scenarios. The data in Table 6 show that all the control strategies can prevent bus

bunching. Comparing the values of stability index ¯H associated with different strategies, we can see that the self-equalizing holding strategy outperforms the TSHS. The data also show that our target-headwaybased holding strategy outperforms the other two strategies with respect to the stability index. Our new holding strategy has been shown in two cases. In one case, only terminal stations 1 and 14 are used as control points. In the other case, stops 1, 6, 8, 14, 18 and 24 are chosen as the control points. Comparing the above two cases, we can see the case with 6 control points has better performance than with only two control points with respect to the stability index and the average holding time. But the latter with two control points performs a bit of well with respect to the average riding and travel times per passenger. This phenomenon will be further explained in the subsequent subsection. In Fig. 5, we depict the estimated headways of buses at the ACTPs and the target headway in one typical simulation round spanning 4 h. There are total 324 ACTPs when the headways are estimated in Fig. 5. Two observations from this figure are worth mentioning here. One observation is that the changing trend of target headway properly reflects the changing trends of traffic demand and the average bus speed which are shown in Figs. 3 and 2, respectively. The other observation is that the estimated headways at the ACTPs fluctuate around the target

Fig. 4. The trajectories of buses under different control strategies. 8

Computers & Industrial Engineering 140 (2020) 106237

S.-X. He, et al.

Table 6 The results with respect to different control strategies. Methods

¯H

No Control Terminal Station Holding Self-Equalizing Holding New Strategy with 2 control points New Strategy with 6 control points

198.90 64.90 48.97 43.63 30.27

a¯

SI

113.11 25.94 15.44 13.39 11.17

a¯

/ 55.91 28.60 46.30 19.70

/ 50.13 31.57 42.80 26.84

nTM

tPW

tPR

tPTr

nP

Bunch

/ 110 113 111 328

287.7 159.8 154.4 145.0 145.1

474.5 490.4 478.7 485.6 492.4

762.2 650.2 633.1 630.6 637.5

9168 9129 9169 9225 9183

Yes No No No No

outside interference. We think this is the main reason for the change of the average riding and travel times. The changing trend of stability index can be explained as follows. At the beginning, when the number of control points is relatively small, the holding operation to a bus at a control point can fully release its effect before another holding operation being taken to this bus in the next control point. The newly added control points bring positive influence on the stability of bus line if the total number of control points remains relatively small. With the further increase of the total number of control points, the distances between the successive control points are shortened noticeably. Due to the shortened distances mentioned above, the holding operation to a bus at a control point will not fully release its effect before another holding operation is taken to this bus. The overlapped effects among holding operations will undermine the whole stabilizing effect of our holding strategy. To fully understand the above counterintuitive phenomenon, we think a further study in the future is required.

Fig. 5. The changing headways at the ACTPs under our strategy.

headway. This fluctuation properly reflects the influence of all the stochastic factors on the running of the bus line. Main stochastic factors include the stochastic arrivals of passengers at stops, the random travel times in road segments and the unpredictable delays at intersections.

5.4. The performance under different random levels The random part of the travel time in a road segment d can be indicated by the standard variance ld as mentioned earlier. In Table 8, we present the results regarding the different values of . The data show that the increased random level of the travel times in road segments leads to the gradually decreasing stability of bus line under our strategy. From the data in Table 8, we can see the average waiting, riding and travel times per passenger are roughly increased. The above observation is in accordance with common sense.

5.3. The impact of the number of control points Table 7 shows the results from our target-headway-based holding strategy with respect to different sets of control points. The data show two counterintuitive trends. The first trend is as follows. The values of the stability index ¯H and its standard variance SI decrease at the beginning with the enlarged size of the control point set. But with the further increase of the number of control points, the values of ¯H and SI change to increase. The average holding time a¯ and its standard variance a¯ have the same change trend as ¯H and SI . It is counterintuitive because according to the common sense, more control points should bring better performance. The second counterintuitive trend is that accompanying with the enlarged set of control points, the average riding and travel times gradually increase. It is counterintuitive because we usually think more control points will lead to reduced riding and travel times for passengers. In the following, we will give a brief explanation for the above counterintuitive observations. With the added control points, the frequency of outside interference will be considerably increased. This can be confirmed by the gradually increased value of nTM . The riding time added due to the holding operation will increase with the strengthened

5.5. The performances under different demand patterns In Fig. 3, we have given the single-peak traffic demand pattern. To testify the effectiveness of our strategy further, we will consider two other types of demand patterns shown in Fig. 6. The first type is depicted in the horizontal solid line in Fig. 6. This pattern is characterized by the constant passenger arrival rate indicated by re . If a stop has such a demand pattern, its passenger arrival rate will remain the same during the observation period. The second type is depicted by the dash line in Fig. 6. This pattern is characterized by 5 characteristic values including re1, re2 , re1, re4 , and re5 . For simplicity, we call the first pattern with a constant passenger arrival rate the non-peak pattern, and the second pattern with 5 characteristic values the double-peak pattern. Similar to Table 4, we list the bus stops with given constant passenger arrival rates in Table 9. These data will be used as an example of the non-peak demand pattern. In Table 10, the specified characteristic

Table 7 The results with respect to different sets of control points. Control Points

¯H

{1,14} {1,6,14,18} {1,6,8,14,18,24} {1,6,8,11,14,18,22,24} {1,3,6,8,11,14,18,22,24,27} {1,3,6,8,11,14,16,18,20,22,24,27}

43.63 33.57 30.27 28.56 29.18 31.17

SI

13.39 11.83 11.17 10.08 11.91 12.82

a¯

a¯

46.30 26.59 19.70 17.36 18.30 18.59

42.80 30.46 26.84 22.75 25.81 28.50

9

nTM

tPW

tPR

tPTr

nP

111 219 328 431 529 628

145.0 146.2 145.1 158.8 158.4 158.1

485.6 490.1 492.4 498.3 508.2 510.9

630.6 636.3 637.5 657.1 666.6 669.0

9225 9256 9183 9301 9318 9218

Computers & Industrial Engineering 140 (2020) 106237

S.-X. He, et al.

5.6. Computational times

Table 8 The results with respect to different (seconds/km). ¯H 5 8 10 12 15

30.27 34.91 36.23 38.69 40.05

a¯

SI

11.17 11.05 11.47 12.93 12.84

19.70 23.30 23.85 22.78 25.18

a¯

26.84 30.43 30.04 31.19 33.68

nTM

tPW

tPR

tPTr

nP

328 327 326 327 326

145.1 148.0 155.4 157.7 153.6

492.4 494.1 501.0 499.8 505.0

637.5 642.1 656.4 657.5 658.6

9183 9205 9314 9227 9210

Java 1.8.0_45 is adopted in this paper to realize the simulation and the related strategies. The computer program runs on NetBeans IDE 8.0.2. The computer we used in the numerical analysis has a processor of Intel® Core™ i5-5200U CPU @2.20 GHz and is installed memory (RAM) of 4.00 GB (2.44 GB usable). To simulate the operation of the test bus line over an observation period spanning 4 h, the required computational time is only about 2–3 s. To compute the target headway and to determine a specified holding time during the implementation of our target-headway-based holding strategy, the required computational time is less than one millisecond. The above computational times are very promising for the implementation of our strategy in practice. 6. Conclusions In this paper, we proposed a holding strategy to stabilize an unstable high-frequency bus line which has not any predetermined schedule and any pre-specified headway. To capture the dynamically changing system state which is embodied in traffic demands and bus speeds, we estimate the instantaneous headways at an ACT and then use the average value of the obtained instantaneous headways as a danamic target to adjust the forward headway of the current bus. It is easy to carry out the holding operation in practice. The effectiveness of the new strategy has been proven theoretically in the situation where the number of control points is not limited to one or two. The numerical experiment showed that the time required to supply a control instruction is shorter than one millisecond. The varying traffic demands, the unpredictable delays at intersections due to traffic light, and the varying travel times in road segments were all considered in the numerical experiment. The simulation experiment shows the following findings.

Fig. 6. Two types of demand patterns. Table 9 The constant passenger arrival rates and the corresponding stops. {re }

Related Stops

{1} {2} {3} {4}

6, 8,9,14,16,20,24,25,28 1,2,4,5,7,10,11,15,17,18,19,22,23,27 3,12,13,21 26

• Our target-headway-based holding strategy outperforms the TSHS

Table 10 Bus stops and their demands in double-peak demand pattern. {re1, re2, re3, re4, re5}

Related Stops

{1,1,1,2,1} {1,2,1,2,1} {1,3,1,3,1} {1,3,2,3,1} {1,2,1,4,2}

6,8,11,17,24 1,2,4,7,10,14,15,18,19,22,23,27 3,12,13,16,21,25 9,20,28 5,26

• • •

values and the corresponding stops are presented. They will be used as an example of the double-peak demand pattern. In Table 11, we present the results from different control schemes with respect to the two different demand patterns. Though the demand patterns are different, the performances of different control strategies are very similar to the data shown in Table 6. For simiplicity, we omit the detailed explanation about the above results here.

•

and the self-equalizing holding strategy with respect to the stability index and the average waiting time per passenger. With the increase of control points, the operational stability and the average holding time resulted from our strategy will increase at beginning and then change to decrease. With the increase of control points, the average riding and travel times per passenger resulted from our strategy increase continuously. With the rising of the random level of travel times in road segments, the performance of the new strategy will gradually decrease with respect to all the performance indices. Our strategy can effectively cope with the non-peak, signgle-peak, and double-peak demand patterns in the same way.

The above findings can provide the practitioners with the theoretical support for implementing the new strategy and choosing a proper number of control points. To cope with a high-frequency bus line which has a very strong trend to bunching, many issues are worth further

Table 11 The results with respect to two different demand patterns. Demand Pattern

Methods

¯H

Non-peak

No Control Terminal Station Holding Self-Equalizing Holding New Strategy (6 CP) No Control Terminal Station Holding Self-Equalizing Holding New Strategy (6 CP)

213.64 73.15 51.46 30.72 221.81 63.11 54.08 35.33

Double-peak

SI

146.17 27.89 13.19 9.43 126.49 26.76 14.52 11.02

a¯

a¯

/ 48.17 33.11 19.45 / 54.77 27.92 23.38

Note: 6 CP means 6 control points including 1, 6, 8, 14, 18 and 24. 10

/ 48.28 34.90 26.56 / 51.48 29.05 30.45

nTM

tPW

tPR

tPTr

nP

Bunch

/ 108 109 322 / 110 113 325

266.1 166.7 148.5 142.8 260.9 154.3 155.3 140.4

471.9 492.7 478.2 489.9 464.1 481.0 471.8 486.0

738.0 659.4 626.7 632.8 725.0 635.3 627.1 626.4

11,964 12,282 12,271 12,255 9615 9970 9964 9643

Yes No No No Yes No No No

Computers & Industrial Engineering 140 (2020) 106237

S.-X. He, et al.

studying. The possible research directions in the future may include: (a). deal with the situation where more than one bus line jointly use some stops and road segments; (b). determine the optimal locations of control points; (c). combine the new strategy with other control means, such as the public transit signal priority, to further improve the effectiveness of the new strategy; (d). extend the thought of using a dynamic target criterion to supply chain management.

Writing - review & editing. June Dong: Formal analysis, Writing - review & editing. Ding Zhang: Data curation, Funding acquisition. JianJia He: Data curation, Funding acquisition. Peng-Cheng Yuan: Software, Funding acquisition.

CRediT authorship contribution statement

This research was supported in part by National Natural Science Foundation of China (71601118, 71801153, 71871144), the Natural Science Foundation of Shanghai (18ZR1426200) and the Key Climbing Project of USST (SK17PA02).

Acknowledgement

Sheng-Xue He: Project administration, Methodology, Writing original draft, Formal analysis. Shi-Dong Liang: Formal analysis, Appendix A. Estimate the expected system headway

The Expected System Headway (ESH) is used by TSHS as the target headway. In the following, we will introduce the definition of ESH and the way of estimating the value of ESH. In an ideal situation where all of hb (t ), b B at time t are equal to each other and all the stochastic factors are removed from the system, we call the above common headway the ESH of the bus line. If the arrival rate of passengers at any one of stops remains the same and the road traffic has not any vehement fluctuation during the observation period, ESH can be used as a target headway. The value of ESH can be estimated as follows:

H¯ = (X / v¯ +

D ~ te +

t¯iD )/ nB

(8) D ~ D where X is the length of the circular bus line; v¯ is the average bus cruising speed; t e and t¯i are the empirical dwell time at stop e and the expected delay time at intersection i for a bus, respectively. Here I is the set of all the intersections. e stands for a typical stop and E is the set of all the bus D t e comes from the field investigation. The way to calculate the value of t¯iD is given in Eq. (6). When a stops. nB is the size of the bus set B . The value of ~ D t e is unavailable. In this case, we will estimate ESH based on the given arrival rates of simulation bus line is used to demonstrate our strategy, ~ passengers at stops. D t e is unavailable, ESH can be estimated as follows. Firstly, we need to use the average arrival rate r¯e of passengers with respect to stop e to If ~ replace the time-varying traffic demand re (t ) which is the instantaneous arrival rate of passengers at stop e at time t . This can be done easily by dividing the total number of passengers generated at e by the length of the observation time. If we only consider the travel times in road segments and the delays at intersections, the time required for a bus traveling the circular bus line in a round can be calculated by

tcircle = X / v¯ +

e E

i I

i I

t¯iD

(9)

The total boarding time of passengers who arrive at stop e in the time interval tcircle is tcircle r¯e t¯Boa . This boarding time will be shared by nB buses. Each bus will need to stay at e about tcircle r¯e t¯Boa/ nB to load passengers. Just like we deduce Eq. (7) associated with the expected boarding time, we can estimate the necessary boarding time for a bus at stop e with (tcircle r¯e t¯Boa/ nB )[1 + r¯e t¯Boa + (¯re t¯Boa) 2 + …] or tcircle r¯e t¯Boa/[nB (1 r¯e t¯Boa)]. With the above estimate, we can use e E tcircle r¯e t¯Boa/[nB (1 r¯e t¯Boa)] as the necessary boarding time for a bus at all the bus stops. By adding up the necessary boarding time and tcircle , we can obtain the estimate of the total time for a bus finishing a round of the bus line. By dividing the estimate of the total time by the number of buses, we can obtain ESH as follows: ~ H = {tcircle + tcircle r¯e t¯Boa/[nB (1 r¯e t¯Boa )]}/nB (10) e E

Delgado, F., Muñoz, J. C., Giesen, R., & Cipriano, A. (2009). Real-time control of buses in a transit corridor based on vehicle holding and boarding limits. Transportation Research Record, 2090, 59–67. Delgado, F., Muñoz, J. C., & Giesen, R. (2012). How much can holding and/or limiting boarding improve transit performance? Transportation Research Part B, 46, 1202–1217. Duan, C., Deng, C., Gharaei, A., Wu, J., & Wang, B. (2018). Selective maintenance scheduling under stochastic maintenance quality with multiple maintenance actions. International Journal of Production Research, 56(23), 7160–7178. Estrada, M., Mension, J., Aymami, J. M., & Torres, L. (2016). Bus control strategies in corridors with signalized intersections. Transportation Research Part C, 71, 500–520. Fu, L., Liu, Q., & Calamai, P. (2003). Real-time optimization model for dynamic scheduling of transit operations. Transportation Research Record, 1857, 48–55. Gharaei, A., Hoseini Shekarabi, S. A., & Karimi, M. (2019). Modelling and optimal lotsizing of the replenishments in constrained, multi-product and bi-objective EPQ models with defective products: Generalised cross decomposition. International Journal of Systems Science: Operations & Logistics, 2, 1–13. Gharaei, A., Karimi, M., & Hoseini Shekarabi, S. A. (2019). Joint economic lot-sizing in multi-product multi-level integrated supply chains: Generalized benders decomposition. International Journal of Systems Science: Operations & Logistics, 1–17. Gharaei, A., Karimi, M., & Shekarabi, S. A. H. (2019). An integrated multi-product, multibuyer supply chain under penalty, green, and quality control polices and a vendor managed inventory with consignment stock agreement: The outer approximation with equality relaxation and augmented penalty algorithm. Applied Mathematical Modelling, 69, 223–254. Gharaei, A., Hoseini Shekarabi, S. A., Karimi, M., Pourjavad, E., & Amjadian, A. (2019). An integrated stochastic EPQ model under quality and green policies: Generalised

References Argote-Cabanero, J., & Daganzo, C. F. (2015). Dynamic control of complex transit systems. Transportation Research Part B, 81(1), 146–160. Awasthi, A., & Omrani, H. (2018). A goal-oriented approach based on fuzzy axiomatic design for sustainable mobility project selection. International Journal of Systems Science: Operations & Logistics, 1–13. Barnett, A. (1974). On controlling randomness in transit operations. Transportation Science, 8(2), 102–116. Bartholdi, J. J., III, & Eisenstein, D. D. (2012). A self-coördinating bus route to resist bus bunching. Transportation Research Part B, 46(4), 481–491. Berrebi, J. S., Crudden, O. S., & Watkins, E. K. (2018). Translating research to practice: Implementing real-time control on high-frequency transit routes. Transportation Research Part A, 111, 213–226. Cortés, C. E., Saéz, D., Milla, F., Nuñez, A., & Riquelme, M. (2010). Hybrid predictive control for real-time optimization of public transport system’s operations based on evolutionary multi-objective optimization. Transportation Research Part C, 18(5), 757–769. Daganzo, C. F. (1997). Passenger waiting time: Advertised schedules. In C. F. Daganzo (Eds.), Fundamentals of transportation and traffic operations. New York, NY: Elsevier (pp. 291–292). Daganzo, C. F. (2009). A headway-based approach to eliminate bus bunching: Systematic analysis and comparisons. Transportation Research Part B: Methodological, 43(10), 913–921. Daganzo, C. F., & Pilachowski, J. (2011). Reducing bunching with bus-to-bus cooperation. Transportation Research Part B: Methodological, 45(1), 267–277.

11

Computers & Industrial Engineering 140 (2020) 106237

S.-X. He, et al. cross decomposition under the separability approach. International Journal of Systems Science: Operations & Logistics. https://doi.org/10.1080/23302674.2019.1656296. Gharaei, A., Karimi, M., & Shekarabi, S. A. H. (2018). An integrated multi-product multibuyer supply chain under the penalty, green, and VMI-CS policies: OA/ER/AP algorithm. Applied Mathematical Modelling. https://doi.org/10.1016/j.apm.2018.11. 035. Giri, B. C., & Bardhan, S. (2014). Coordinating a supply chain with backup supplier through buyback contract under supply disruption and uncertain demand. International Journal of Systems Science: Operations & Logistics, 1(4), 193–204. Giri, B. C., & Masanta, M. (2018). Developing a closed-loop supply chain model with price and quality dependent demand and learning in production in a stochastic environment. International Journal of Systems Science: Operations & Logistics, 1–17. Hao, Y., Helo, P., & Shamsuzzoha, A. (2016). Virtual factory system design and implementation: Integrated sustainable manufacturing. International Journal of Systems Science: Operations & Logistics, 5(2), 116–132. He, S. X. (2015). An anti-bunching strategy to improve bus schedule and headway reliability by making use of the available accurate information. Computers & Industrial Engineering, 85, 17–32. He, S. X., Dong, J., Liang, S. D., & Yuan, P. (2019). An approach to improve the operational stability of a bus line by adjusting bus speeds on the dedicated bus lanes. Transportation Research Part C, 107(10), 54–69. Hoseini, S. A., Gharaei, A., & Karimi, M. (2019). Modelling and optimal lot-sizing of integrated multi-level multi-wholesaler supply chains under the shortage and limited warehouse space: Generalised outer approximation. International Journal of Systems Science: Operations & Logistics, 6(3), 237–257. Liang, S., Ma, M., He, S. X., Zhang, H., & Yuan, P. (2019). Coordinated control method to self-equalize bus headways: An analytical method. Transportmetrica B: Transport Dynamics, 1–28. Liang, S., Ma, M., & He, S. X. (2019). Multiobjective optimal formulations for bus fleet size of public transit under headway-based holding control. Journal of Advanced Transportation. Liang, S., Zhao, S., Lu, C., & Ma, M. (2016). A self-adaptive method to equalize headways: Numerical analysis and comparison. Transportation Research Part B, 87, 33–43. Ling, K., & Shalaby, A. (2004). Automated transit headway control via adaptive signal priority. Journal of Advanced Transportation, 38(1), 45–67. Liu, H., Skabardonis, A., & Zhang, W. (2003). A dynamic model for adaptive signal priority. In 82nd Transportation Research Board Annual Meeting, Washington, DC. Preprint CD-ROM. Liu, Z., Yan, Y., Qu, X., & Zhang, Y. (2013). Bus stop-skipping scheme with random travel time. Transportation Research Part C, 35, 46–56. Masoud, R., Seyed, A. A. H.-M., Amir, H. O., & Hamed, F.-A. (2018). A hybrid robust possibilistic approach for a sustainable supply chain location-allocation network design. International Journal of Systems Science: Operations & Logistics, 1–16. https:// doi.org/10.1080/23302674.2018.1506061. Newell, G. F. (1974). Control of pairing of vehicles on a public transportation route, two vehicles, one control point. Transportation Science, 8(3), 248–264.

Nima, K., Salwa, H. A.-R., Raja, A. R. G., Ehsan, S., & Simone, Z. (2018). Economic order quantity models for items with imperfect quality and emission considerations. International Journal of Systems Science: Operations & Logistics, 5(2), 99–115. Osuna, E. E., & Newell, G. F. (1972). Control strategies for an idealized bus system. Transportation Science, 6(1), 52–71. Petit, A., Ouyang, Y., & Lei, C. (2018). Dynamic bus substitution strategy for bunching intervention. Transportation Research Part B, 115, 1–16. Rabbani, M., Foroozesh, N., Mousavi, S. M., & Farrokhi-Asl, H. (2017). Sustainable supplier selection by a new decision model based on interval-valued fuzzy sets and possibilistic statistical reference point systems under uncertainty. International Journal of Systems Science: Operations & Logistics, 1–17. Rameshwar, D., Angappa, G. S., & Tripti, S. (2015). Building theory of sustainable manufacturing using total interpretive structural modelling. International Journal of Systems Science: Operations & Logistics, 2(4), 231–247. Reza, S., & Anjali, A. (2018). An integrated approach based on system dynamics and ANP for evaluating sustainable transportation policies. International Journal of Systems Science: Operations & Logistics. https://doi.org/10.1080/23302674.2018.1554168. Sarkar, S., & Giri, B. C. (2018). Stochastic supply chain model with imperfect production and controllable defective rate. International Journal of Systems Science: Operations & Logistics, 1–14. Sayyadi, R., & Awasthi, A. (2016). A simulation-based optimisation approach for identifying key determinants for sustainable transportation planning. International Journal of Systems Science: Operations & Logistics, 5(2), 161–174. Shah, N. H., Chaudhari, U., & Cárdenas-Barrón, L. E. (2018). Integrating credit and replenishment policies for deteriorating items under quadratic demand in a three echelon supply chain. International Journal of Systems Science: Operations & Logistics, 1–12. Suh, W., Chon, K., & Rhee, S. (2002). Effect of skip-stop policy on a Korean subway system. Transportation Research Record, 1793, 33–39. Sun, A., & Hickman, M. (2005). The Real-Time Stop-Skipping problem. Journal of Intelligent Transportation Systems, 9(2), 91–109. Sun, A., & Hickman, M. (2008) The holding problem at multiple holding stations. In M. Hickman, P. Mirchandani, S. Voss (Eds.), Computer-aided systems in public transport. Berlin/Heidelberg: Springer (Vol. 600, pp. 339–359). Tsao, Y. C. (2015). Design of a carbon-efficient supply-chain network under trade credits. International Journal of Systems Science: Operations & Logistics, 2(3), 177–186. Xuan, Y., Argote, J., & Daganzo, C. F. (2011). Dynamic bus holding strategies for schedule reliability: Optimal linear control and performance analysis. Transportation Research Part B, 45, 1831–1845. Yin, S., Nishi, T., & Zhang, G. (2015). A game theoretic model for coordination of single manufacturer and multiple suppliers with quality variations under uncertain demands. International Journal of Systems Science: Operations & Logistics, 3(2), 79–91. Zhao, J., Dessouky, M., & Bukkapatnam, S. (2006). Optimal slack time for schedule-based transit operations. Transportation Science, 40(4), 529–539. Zhang, S., & Lo, H. K. (2018). Two-way-looking self-equalizing headway control for bus operations. Transportation Research Part B, 110, 280–301.

12