In the operational planning process of public transport, the time a passenger spends on waiting is a very critical element for judging passenger service. Schedule synchronization is a useful strategy for reducing bus waiting time and improving service connectivity. This paper develops an extended vehicle scheduling model, taking into account the interests of passengers and operators in attaining optimization of timetable synchronization integrated with vehicle scheduling and considering the passenger waiting cost. Deficit functions at terminals are formulated. Deadheading (DH), shifting departure time (SDT), and network flow technique are used for vehicle scheduling with the consideration of passenger waiting times. An experimental study in Beijing is conducted and three important bus lines are selected as a regional bus network to demonstrate the methodology developed. Results show that both the fleet size of bus operators and the waiting cost of passengers are minimized. For example, the minimum fleet size can be reduced from 28 vehicles to 24 ones while the passenger times are less than 20 minutes in this multidepot network.
In the past 40 years, urbanization in China increased in speed. By the end of 2017, 58.52% of the total population lived in urban areas, a dramatic increase from 17.92% in 1978. Urban population grew rapidly from 170 million to 810 million, and the number of cities increased from 193 to 657. Meanwhile, levels of car ownership in China have risen significantly. By the end of 2018, car ownership reached 240 million, with more than 3 million in eight cities, 2 million in 27 cities, and 1 million in 61 cities [
To alleviate traffic congestion, urban public transport (PT) priority has become a strategic choice for China. The policy of vigorously developing PT has been listed in The 13th Fiveyear Plan for National Economy and Social Development and issued by the State Council [
Crowding at the bus stop. (a) Queuing for a bus line. (b) Climbing by windows.
In the operational planning process of public transport, timetable synchronization is an important issue to reduce transfer waiting time and improve service connectivity. Minimizing waiting times can improve customer satisfaction, which in turn leads to increases in ridership and revenues. However, most of the studies on PT timetable synchronization design have treated the problem independently of other operations planning activities and have focused only on minimizing transfer waiting time. In addition, the impact of schedule changes on PT users’ route/trip choice behavior has not been well investigated yet [
Vehicle scheduling is a crucial step of the PT planning process since it is desirable to minimize the number of vehicles used and operational cost. The purpose of the paper is to address a multidepot vehicle scheduling problem considering the passenger waiting cost. This paper develops an extended vehicle scheduling model together with the deficit function (DF) as follows: First, a practical timetable compromising the passenger waiting cost and the bus operating cost is proposed. Secondly, deadheading (DH), shifting departure time (SDT), and network flow technique are used for vehicle scheduling with the consideration of passenger waiting times. Finally, an experimental study in Beijing is conducted and three important bus lines are selected as a regional bus network to demonstrate the methodology developed. Results show that both the fleet size of bus operators and the waiting cost of passengers are minimized.
Therefore, the contributions of this research are threefold: (a) An extended vehicle scheduling model considering the interests of both passengers and operators is proposed; (b) Timetable synchronization integrated with vehicle scheduling and passenger waiting cost is attained. (c) A detailed numerical example is provided to illustrate the performance of the mathematical model and solution method developed, with a discussion on some promising future research directions.
The rest of the paper is organized as follows. In the next section, a literature review of vehicle scheduling problem (VSP) is provided. Deficit function and an extended vehicle scheduling model are presented in Section
In the last decades, a fruitful development of models and solution techniques were addressed in bus transport systems. The global PT problem is computationally intractable and can hardly be tackled at once [
Guihaire and Hao (2008) presented a global review of the crucial strategic and tactical steps of transit planning: the design and scheduling of the network [
PT operational planning can be considered a multistep process. Because of its complexity, each step is normally conducted separately and sequentially fed into the other. The process includes (1) network route design, (2) timetable development, (3) vehicle scheduling, and (4) crew scheduling [
Vehicle scheduling refers to determining the optimal allocation of vehicles in a given transportation schedule based on the execution of all trips [
Another version of the VSP is the multidepot vehicle scheduling problem, where vehicles can depart from different locations. This assumption leads to complex formulations, such as multicommodity network flow problems. The problem is intractable since it is nondeterministic polynomial hard (NPhard) (proved by Bertossi et al., 1987) [
However, exact approaches towards solving large instances of the multidepot VSP have been presented in recent studies (see reviews by IbarraRojas et al., 2015) [
The deficit function (DF) with information graphics technology features, which represented the deficit number of vehicles required at a specific terminal in a multidepot transit network, was proposed to the multidepot VSP problem (see reviews by Liu and Ceder, 2017) [
Linear programming models were also commonly used in the study of vehicle scheduling. IbarraRojas et al. (2014) discussed the tradeoff between the level of service and the operating cost, which involved the timetabling and the VSP problems at the operational level. They presented two integer linear programming models combining a biobjective integrated model [
Solving the multidepot VSP in an integrated manner can better reflect the operational planning process. Schöbel (2017) described that PT operation system was a multiobjective problem including line planning, timetabling, and vehicle scheduling [
Table
Literature review for the multidepot vehicle scheduling problem.
Authors (year)  Objective  Characteristics  Solution method 

Haghani and Banihashemi (2002)  Solving the possible fuel dissipation constraints  Route time constraints  Decomposition & Heuristic 
Ceder (2002, 2005)  Minimum size of the fleet  Extending each trip’s arrival time  DF method 
Wagale M (2013)  Optimize the bus scheduling process.  An integrated demand and travel time responsive model  Sensitivity analysis 
IbarraRojas et al. (2014)  Tradeoff between the level of service and operating costs 

Linear programming & composition 
Schöbel (2017)  Optimizing the whole process  Eigenmodel for the design of reoptimization procedures  Heuristic & Iterative approaches 
Liu (2017)  Integrated timetable synchronization and vehicle scheduling problem with transit assignment  A new biobjective, bilevel integer programming model.  Mathematical programming approach & Paretoefficient solutions 
Tang (2018)  Minimum fleet size for singleline  Adjusting variable trips to accommodate different strategies  DF methodology 
Consider a directed graph
The deficit function (DF) was proposed by Ceder and Stern (1981) and Ceder (2016) which is a brief description of a step function method for assigning the least number of vehicles to a given schedule [
The description line segment of
The sum of all DFs over
Given a set of stations
This theorem is also called as the DF fleet size theorem. A formal proof can be found in the study by Ceder (2016) [
As shown in the study by Ceder (2016) [
This formula is based on two assumptions:
Passengers can always leave on the first bus (without overload)
The passenger’s random arrival rate at the terminals is independent of the vehicle’s departure rate, which is a constant within a certain period
In the real traffic, passengers expect more efficient service (e.g., high bus frequency) to reduce the waiting time and improve the invehicle comfort. In contrast, bus operators are unwilling to operate the routes with low ridership. They always expect a longer headway to reduce the operating cost. Yet they also need to accommodate the observed passenger demand as well as possible. This article considers the interests of both the bus operators and the passengers. On the one hand, the total operating cost related to the fleet size should be reduced. On the other hand, the average passenger waiting time should also be reduced.
In the operational system, the operating cost is expressed by the number of vehicles required to complete all trips in
The mathematical model contains three groups of constraints, which are the bundle departure constraints, the DF bound constraints, and the fleet size constraints.
The first group of constraints are the bundle departure constraints as follows:
The number of departures index
The second group of constraints are the vehicle constraints (i.e., the DF bound constraints):
The third group of constraints are the resource constraints (i.e., the fleet size constraints):
Based on the DF, this paper adjusts the departure time and reduces the passenger waiting times to improve the bus service. The total waiting time is expressed by the product of the number of waiting passengers and the waiting times, namely,
In Eq. (
Since objective function components of Eq. (
In model (
For an optimal timetable with proper route offset times, all the trips, including the empty trips, need to be connected to form a vehicle driving plan (the trip chain or the trip group). The construction of a trip chain can follow two rules: the firstin firstout (FIFO) rule or the trip chain extraction process rule [
An example of building a connection in a flat peak time
It should be noted that transfers for passengers are not considered in this paper. That is to say, there are no alternative modes for their trips and all the passengers are captive public transport users. Moreover, only the waiting times spent at the beginning station are considered. Transfer is an important issue in the calculation of the waiting cost; thus more models should be presented in future.
Regional bus crossline scheduling can be divided into two types: interstation scheduling and intrastation scheduling. To optimize the crossline scheduling, some driving directions can be inserted into the schedule based on DF, and a more scientific scheduling plan can be formulated. The deadheading (DH) is a type of nonpassenger bus which is inserted into the driving plan and runs between two stations. Its insertion can effectively improve the utilization of the operating vehicles. A dispatch with idle time and the empty travel distance being zero can be regarded as DH. However, the reliability of a vehicle may be reduced. Therefore, to ensure smooth transition between the operating vehicles, path and time must be fully considered before inserting the DH trip.
When a DH trip is allowed, the minimum fleet size is determined by the maximum number of the operating vehicles:
More details can be found in the study by Ceder (2016) [
If
Figure
An example of the DH trips.
The small adjustment of the departure times in the driving schedule can make it possible to further reduce the fleet size and the passenger waiting cost. Shifting departure time (SDT) implicates that, for each stop and each two times
However, failure SDT may lead to unbalanced loads, including overcrowding or empty, or bus bunching. Therefore, an experienced operator should be performed very carefully with the help of the DF diagram. See the study by Ceder (2002) for details [
This paper harmonizes the vehicles’ departure times with the passengers’ waiting demands. Let
An example of the SDT process.
The network flow technique is used to estimate the minimum fleet size for a given schedule
Let
Given
An overview of the solution procedure is outlined as follows:
Apply the network flow technique to obtain an initial number of vehicles required for a given timetable
Use the DF method with graphical features to depict the number of vehicles required at each station. Calculate the DF for each
Use the SDT procedure to modify the route offset times.
Insert DH trips to reduce the idle times of vehicles and further reduce the fleet size.
Compare the operating cost and the passenger waiting cost.
City bus in Beijing is the most widely used and affordable amongst all means of public transportation, although it can be very crowded and may meet a traffic jam. In 2016, the entire public bus network has 876 routes with 22688 buses [
Public buses are identified either by numbers or by Chinese characters. Lines 1751 are downtown regular ones and bus routes are mostly in the city’s urban core district. They start running at 05:0005:30 and stop at 22:0023:00.
Public transport along Chang’an Avenue, a major thoroughfare in Beijing, China, was selected for data collection during the evening peak hours (17:0019:30 pm). We recorded the passenger flows generated from these sites. The survey lasted for 2 months from May 2015 to June 2015, and more than 30 undergraduate students were recruited. Direct observation and video tape recording were simultaneously employed. Moreover, bus smart card data were obtained from a bus company. All field works were finished on weekdays with good weather conditions.
Three bus lines including lines 1, 57, and 52 are selected since they all belong to the same bus company. Among them, Line 1 is one of the busiest bus routes in Beijing and is known as “the city boat on Chang’an Street.” Moreover, overlapped or parallel routes help to reschedule vehicles, as shown in Figure
Bus routes of Line1, Line 57, and Line 52.
The schedules
A 20trip schedule with the four terminals.
Trip Number  Route  Departure time  Arrival time 

1  ab  17:03  17:43 


2  ab  17:15  17:55 


3  ab  17:18  17:58 


4  ab  17:22  18:02 


5  ab  17:40  18:20 


6  ab  17:55  18:35 


7  ab  18:03  18:43 


8  ab  18:20  19:00 


9  ab  18:25  19:05 


10  ab  18:47  19:23 


11  ba  17:20  18:00 


12  ba  17:35  18:15 


13  ba  17:50  18:30 


14  ba  18:00  18:40 


15  ba  18:15  18:55 


16  ba  18:32  19:12 


17  ba  18:45  19:25 


18  cb  17:00  18:00 


19  cb  17:13  18:13 


20  cb  17:20  18:20 
Passenger demands at the four terminals.
Time  Terminal  

a  b  c  d  
17:0018:00  294  153  297  72 


18:0019:00  293  115  238  46 


Total  587  268  535  118 
Average DH travel time (minutes) matrix.
DH Trip  ab  bc  cd 

DH Time (minute)  18  20  25 
It is assumed that passengers will board the first feasible transfer connecting trip (i.e., the vehicle capacity is sufficient). And the waiting weights at the four terminals are assumed to be equal. Namely,
Based on the data in Table
DF values at four terminals.
Sum of deficit functions (of vehicles in service).
Since the passenger waiting time is very critical to judge the bus service, this paper makes endeavor to keep it within 20 minutes. The DH trips are inserted to reduce the fleet size. As shown in Figure
SDT for the twelve trips.
Trip No.  Travel Route  Original Time  Time After SDT  

Departure time  Arrival time  Departure time  Arrival time  
10  ab  18:47  19:23  18:45  19:25 


12  ba  17:35  18:15  17:43  18:23 


31  bc  17:54  18:54  17:55  18:55 


13  ba  17:50  18:30  17:58  18:38 


39  cd  17:46  18:36  18:00  18:50 


24  cb  18:05  19:05  18:10  19:10 


32  bc  18:05  19:05  18:13  19:13 


15  ba  18:15  18:55  18:20  19:00 


41  cd  18:18  19:08  18:25  19:15 


26  cb  18:30  19:30  18:34  19:34 


48  dc  17:47  18:37  17:55  0:00 


49  dc  18:13  19:03  18:15  0:00 
Timetable for inserting DH trips.
DH Trip  Travel Route  Departure time  Arrival time 

DH  ba  18:02  18:20 
In Figure
Sum of deficit functions after inserting DH and SDT.
Before the SDT and the DH processes, the fleet size was 28 and the target function value was 2154.57. Considering the passenger waiting cost, the departure times of some buses are adjusted and one empty bus is inserted. The fleet size reduces to 24, and the objective function value becomes 1793.22, which has saved 16.90%.
This work develops a methodology for the multidepot vehicle scheduling problem considering the passenger waiting cost. The purpose of this methodology is to minimize the fleet size and reduce the passenger waiting times. The deficit function (DF) approach is applied to realize the shifting departure time (SDT) and the deadheading (DH) to minimize the fleet size. An optimization function is introduced to minimize both the bus operating cost and the passenger waiting cost. Finally, this methodology is applied to a PT network in Beijing, China. Results show that the total cost can be reduced by 16.90% compared with the current timetables.
A limitation in this study is need of more accurate calculations of timedependent passenger demands, which have important impacts on the vehicle scheduling optimization. With the development of modern technology, realtime passenger information is easily obtained and collected. The mobile phone can be used as a tracking device, which can be used to query the line, the vehicle arrival times, and the vehicle locations, etc. It can also collect the passenger’s precise origindestination information to evaluate the passenger’s path selection preferences. In addition, the following studies are advised:
(a) Studying more accurate weight factors for passengers
(b) Investigating the effect of different vehicle size to meet the needs of passengers at different departure times
(c) Considering transfers for passengers in their trips
(d) If largesize vehicles are allowed to operate on those trips designed for small vehicles, the fleet size can be reduced. This problem also involves issues such as departure intervals and operating costs, which need to be further discussed in detail in subsequent studies
All data included in this study are available upon request by contact with the corresponding author (
The authors declare that they have no conflicts of interest.
This research was jointly supported by grants from the National Natural Science Foundation of China (71890971, 71371128), the Beijing Natural Science Foundation (8192006), and the Beijing Municipal Education Commission Foundation (SZ201910038021). Thanks are due to Dr. Tao Liu for his kind help.