Mathematical Analysis and an Exact Solution Combined with Preprocessing Method for Resynchronizing of Bus Timetable Problem

Bus timetabling is a subproblem of bus network planning, and it determines departure time of each trip of lines to make vehicles from different lines synchronously arrive at transfer stations. Due to the well-designed coordination of bus timetables, passengers can make a smooth transfer without waiting a long time for connecting buses. This paper addresses the planning level of resynchronizing of bus timetable problem allowing modifications to initial timetable. Timetable modifications consist of shifts in the departure times and headways. A single-objectivemixed-integer programmingmodel is proposed for this problem tomaximize the number of total transferring passengers benefiting from smooth transfers.We analyze themathematical properties of thismodel, and then a preprocessing method is designed to reduce the solution space of the proposed model. The numerical results show that the reduced model is effectively solved by branch and bound algorithm, and the preprocessing method has the potential to be applied for large-scale bus networks.


Introduction
Bus network planning has been known for long as a complex problem including several subproblems, i.e., line planning, timetable design, vehicle scheduling, and crew scheduling [1][2][3].The bus timetabling is a principal stage, and it determines the departure time of each trip of all lines in the bus network in order to maximize the service quality of bus transit system [4].
With a synchronized bus timetable, passengers can immediately transfer to another line after getting down from the feeder line at stations.However, if the bus network timetable is not well synchronized, passengers have to wait longer for the connecting bus when they get down from the feeder line [5].A well-designed bus timetable with good coordination of bus departure times at transfer stations is very important for the bus operation attracting more travelers choosing transit service.In China, line-by-line timetabling approach is often used to generate bus network timetables.This method does not consider the synchronization of bus departure times among different lines at the same transfer station.In order to provide bus schedule coordination service, schedulers intend to synchronize timetables of different lines at the transfer nodes to facilitate passengers transferring by slightly modifying the departure times and headways of initial bus timetable.Regarding this situation, this paper addresses the planning level of BSC problem through small timetable modifications to given an initial timetable.Timetable modifications consist of shifts in initial departure times of vehicles from the depot.Headway-sensitive passenger demand is also considered in this problem.
In the literature, several studies have proposed solutions to the synchronization of bus timetabling (SBT) problem to maximize the number of synchronized bus arrivals at transfer nodes [6][7][8].For instance, Ibarra-Rojas et al. [8] extended the work of Ceder et al. [6] and Eranki [7] to address a flexible SBT problem with oriented synchronization, almost evenly spaced departures and preventing bus bunching.Shafahi and Khani [9], and Cevallos and Zhao [10] focused on constant headways in the SBT problem to minimize the total transfer waiting time experienced by passengers.Mollanejad et al. [11] developed a MIP model for the SBT problem with uneven headways and trips before and after a planning horizon.Parbo et al. [12] studied a bilevel timetable optimization problem, aimed at minimizing the weighted transfer waiting time.Wu Y. et al. [13] studied a stochastic version of SBT.Slack time was inserted into the timetable to mitigate the randomness of travel times to reduce the rate of transfer failures.A recent paper studied by Wu et al. [14] used integrated real-time holding control and slacks to improve the reliability and efficiency of schedule coordination.Tilahun and Ong [15] formulated bus timetabling just for single frequency routes not multiple trips for each route.A fuzzy multiobjective optimization model was formulated and solved using preference-based genetic algorithm by assigning appropriate fuzzy preference to the need of the customers.
Timetable coordination of railway network was also studied to minimize the waiting time cost (or reduce the chances of missing the connecting trains) of passengers.Wong et al. [16] investigated the synchronization of railway timetabling problem to minimize the total transfer waiting time by adjusting train's run times and dwell times at stations.In the case of high-speed rail lines under given time-dependent origin-destination demand, Niu et al. [17] proposed a synchronization approach for the train schedule.Wang et al. [18] proposed an event-driven model for the train-scheduling problem with the consideration of timevarying passenger demand.Kang et al. [19][20][21] focused on the coordination of last trains aimed at reducing the number of cases in which passengers miss the connecting trains within one headway time.Meanwhile, Kang et al. [22] and Guo et al. [23] proposed the timetable coordination of first trains for the railway network of Beijing.A multiperiod timetabling problem for metro transit networks was studied by Guo et al. [24], in which passenger travel demand varies in the transitional period.
The above-mentioned papers of timetable synchronization issues mainly focused on designing a new timetable for the transit network studied.However, large deviations from the initial timetable destroy regular timetable service and following vehicle and crew scheduling plan [3].Recently, Wu Y. et al. [25] studied the biobjective resynchronizing of bus timetable problem with the consideration of impact on the initial timetable and proposed a nondominated sorting genetic based algorithm to solve this problem, not an exact solution.
Compared to the above papers, the main contributions of this paper are as follows: (1) A resynchronizing of bus timetable (SRSBT) problem that allowing modifications to the initial timetable in tactical planning level is proposed.Timetable modifications consist of shifts in the departure times and headways.And, headway-sensitive passenger demand was also considered in the proposed problem.(2) We formulate this problem as a single-objective mixedinteger programming model maximizing the number of total transferring passengers benefiting from smooth transfers.The mathematical properties of this model were analyzed.(3) Based on these properties of the model, we design a preprocessing method to reduce the solution space of the proposed model, which means that large-scale SRSBT problems can be solved by an exact solution combined with the preprocessing method.For small-cases study, the preprocessing method is effective and reduces the solution spaces of SRSBT model to at least 56%.
The remainder of the paper is organized as follows.Section 2 formulates the SRSBT problem and explores several properties of the single-objective SRSBT model.Based on the properties of the SRSBT, Section 3 describes the preprocessing method to reduce the solution space.In Section 4, the effectiveness of the preprocessing method is evaluated through small instances.Finally, Section 5 concludes this paper.

Problem Formulation
In large bus transit systems, passengers often transfer from one bus line to another at a transfer station in order to complete their trips.As an example, Figure 1 shows a transfer opportunity between line  and line , at station .Passengers are assumed to transfer from an arriving trip of a feeder line to the first available trip of their desired line.The transfer opportunity of passengers transferring from trip  of line  to line  depicted in Figure 2 requires a small time, for example, 2 minutes for passengers to disembark the vehicle of trip  and walk to the area for embarking the vehicle of line .In the current timetable, trip  of line  arrives at 9:30 leaving 10 minutes transfer time to the next departing line  trip  + 1 at 9:40, thus resulting in 8 minutes of excess transfer time.
If the departure time of trip  is adjusted from 9:30 to 9:32, the excess transfer time for the trip  of line  would be removed.Figure 3 depicts the change in the departure time of trip .The modification in the departure time of trip  at the transfer station can be realized by adjusting the departure time trip  from the depot.However, such change in the departure time of trip  could also influence the departure times of other trips of this line due to limited range of headway times (interval of time between two consecutive trips on the same line).Alternatively, the required shift in the departure time of trip can be limited if the trip  of line  would be adjusted to arrive earlier at 9:28 (see Figure 4).
The SRSBT problem is how to modify the departure times and headways in the initial timetable with limited range to maximize the number of passengers benefiting from smooth transfers.

Notations.
We introduce notations necessary to formulate the SRSBT problem.
Equation ( 1) is the objective to maximize the total number of passengers whose excess transfer time is limited to [ min  ,  max  ].Constraint (2) ensures that the first trip of each line departs at the beginning of the planning period.Similarly, Constraint (3) forces the last trip of each line to depart at the end of the planning period.Constraint (4) guarantees that the headways between consecutive trips of each line  is in the range of [ℎ min  , ℎ max

𝑖
]. Constraint (5) defines the maximal value of deviation of departure times, relative to that of initial timetable.Equation ( 6) is the actual number of passengers transferring from the -th trip of line  to line  at node  which depends on the actual headways for the feeder line .Constraints (7) and (8) ensure that    equals one if the difference between the arrivals of two trips at node  is within time window [ min  ,  max  ], in which  is a large constant.Constraint (9) ensures that each passenger makes less than one smooth transfer at one time.Constraints (10) and (11) are the domains of   and    , respectively.Two main differences made by this paper relative to Ibarra-Rojas et al. [8] are emphasized here.On the one hand, in view of the studied problems, this paper studied a resynchronizing of bus timetable problem with the consideration of initial timetable which is very different from the problem proposed in Ibarra-Rojas et al. [8], in which a completely new timetable was generated by their model without considering the initial timetable.In our paper, timetable modifications consist of shifts in the departure times and headways of initial timetable.On the other hand, the objective function in our proposed model is also different from the objective function of Ibarra-Rojas et al. [8].In our model, we considered a uniform-distributed passenger demand for transferring, and the number of transferring passengers reflects the importance of transfer nodes and variation in headways.The passenger demand is sensitive to the realized headways.In Ibarra-Rojas et al. [8] the transferring demand is not considered, which is a special case of our model.

Properties of the SRSBT Model.
We can see that the SRSBT model is a mixed-integer quadratic programming model.To find effective algorithms solving the SRSBT model, we analyze the mathematical model.
From expression (18), it can be seen that   /2 and   /2 + 1 are the watershed lines for discussing the value of |  − x |.  is even or not.We discuss three situations as follows.
where ⌊  /2⌋ is the largest integer not greater than   /2.Therefore, the value of |  − x | is not larger than ⌊  /2⌋  +ℎ  .Based on the discussion above, we can obtain the upper bound of  denoted by  Proof.Using a polynomial reduction from monotone NAE-3SAT, Ibarra-Rojas et al. [8] have proved that the SBT problem is NP-hard.Ibarra-Rojas et al. [8] proposed a MIP model for the SBT problem without considering passenger demand and the limited range of departure times.In other words, the MIP model assumes that the number of transferring passengers for each trip of all lines at each transfer node is one.It is quite clear that the MIP model is a special case of the model consisting of expressions (1)∼( 4), ( 7)∼( 8) and ( 10)∼(11) in Section 2.2.Therefore, the SRSBT model is also NP-hard.

Preprocessing Method
According to Properties 1 and 3, we design a preprocessing method to reduce the solution space of the SRSBT by eliminating useless variables and constraints.Algorithm 1 describes the preprocessing method.

Numerical Examples
In this section, a simple bus network used in Wu Y. H. et al. [25] is adopted to illustrate the effectiveness of the preprocessing method.Figure 6 and Table 1 depict the structure of simple bus network and its initial timetable used.For other operational parameters for these two instances, please refer to Wu Y. H. et al. [25].Under the initial timetable, the average number of passengers for each transferring between different lines is given in Table 2. Other operational parameters are described in Section 2.
We firstly use the preprocessing approach to reduce the solution space of SRSBT model.Then, a branch and bound (B&B) algorithm is used to solve the reduced SRSBT model.The B&B algorithm and preprocessing method used in this section are coded in C# and all tests are executed on an Intel  Core (TM) i3 processor at 2.27 GHz under Windows 7 using 2.00 G of RAM.According to property 2, the maximal value of parameter  is 17 for this case.Under each given value of , there is corresponding solution and one optimal value of objective (i.e., the number of passengers benefiting from smooth transfers).It can be observed from Figure 7 that when  equals zero, nonpassengers can take a smooth transfer.The optimal value of objective increases slightly as  takes value range from 0 to 4. However, the optimal value of objective increases rapidly as  takes value range from 5 to 7. The optimal value of objective reaches its maximal value as  takes 10.When  takes a value larger than 10, the optimal value of  objective remains unchanged.According to Figure 7 we can select appropriate value of  under given service level.Table 3 gives the optimal solutions for the adjusted timetable under  taking the value range from 0 to 10.The optimal solutions for  taking the value from 11 to 17 are the same as  equals 10.We can see that from Table 3 the optimal solutions are sensitive to the departure time modifications.
Table 4 shows the results of solving instances of SRSBT model and reduced SRSBT model using a B&B algorithm.Column one shows different [ℎ min  , ℎ max  ] consisting of different instances.Columns two and four show the relative gap (difference between the best feasible solution and the best upper) for SRSBT and reduced SRSBT, respectively.Columns three and five show the computational time needed by B&B algorithm solving SRSBT model and reduced SRSBT model.The computational time of preprocessing method is neglected since it is less than 0.1 second.Column six shows the percentage of solution space eliminated from SRSBT to obtain the reduced SRSBT model.
From Table 4, we can see that there are high levels of eliminated variables using the preprocessing method.Therefore, the solving times of reduced SRSBT are less than that of SRSBT, which illustrates that the preprocessing method is effective.The variation in headway generates large instances which need more time to be solved.In particular, we see that high percentage levels of eliminated variables for instances with small headway deviation.For the instances, the preprocessing method can reduce the solution spaces of SRSBT model to at least 56%.It will be useful to solve large SRSBT problems using the preprocessing method, which is our following work.

Conclusions
In this paper, we have studied a SRSBT problem by adjusting the departure times of initial bus timetable in a limited range.For this problem, we proposed a mixed-integer programming model with the objective of maximizing the number of total transferring passengers benefiting from smooth transfers.By analyzing the mathematical properties of this model, we developed a preprocessing method to reduce the synchronization variables and constraints of the SRSBT model.The preprocessing method proved to be effective in terms of the solving time and reduced solution spaces.For the instances studied in Section 4, the reduced SRSBT has at least 56% fewer solution spaces than that of SRSBT.Obtaining a high quality solutions for SBT problems in a reasonable time is beneficial to large bus transit networks in real life.Combined with B&B algorithm, the preprocessing method proposed in this paper has the potential to solve large SRSBT problems in a reasonable time.
Besides using the preprocessing method with exact algorithms to solve large bus networks, several aspects should be considered for future studies.In real cases, the travel times of bus are often stochastic.Recent studies deal with multiperiod transit timetabling, robust timetable coordination, and integrated timetabling and vehicle-scheduling problems [3,14,24,26].These topics also should be paid more attention

Figure 1 :
Figure 1: Geographic representation of two bus lines, line  and line , and station , where passengers can transfer.

Figure 2 :
Figure 2: Example of passenger transfer from trip  of line  to line  at station  under a given timetable.

Figure 3 :Figure 4 :
Figure 3: Representation of change in the departure time of trip .

1 Figure 5 :
Figure 5: Typical structure of the matrix formed by the synchronization variables    .

≤ 1 ≤ 1 Property 4 . 1 ,
is a valid inequality for trip  of line  synchronizing with line  as ℎ min  means that the maximum number of smooth transfers that passengers make between one trip of line  and all trips of line  is 1.Let us suppose ∑   =1    =  ( > 1) so that trip  of line  can synchronize with trips  1 <  2 < ⋅ ⋅ ⋅ <   of line .According to constraints(7) and (8), the arrival times of trips that synchronize with trip  of line  must be within the time window [  +   +  min  ,   +   +  max  ].Therefore, the difference between    +   and   1 +   must be less than  max  −  min  ; i.e.,    −   1 ≤  max  −  min  .According to constraint (4), we obtain    −   1 ≥ ( − 1)ℎ min  .We then have ℎ min  ≤ ( max  − min  )/(−1) ( > 1), which is in conflict with ℎ min  >  max  −  min  .According to constraints (7) ∼ (9), the matrix formed by the synchronization variables    has following typical structure as shown in Figure5.Proof.According to constraints(7) and(8), when  trip  of line  at most synchronizes with one trip of line .We thus get the typical structure of the matric formed by the synchronization variables    , which is different from that of Ibarra-Rojas et al.[8].Property 5.The SRSBT problem is NP-hard.

Figure 7 :
Figure 7: Optimal value of the objective function in SRSBT with different .

Table 1 :
Initial timetable for the simple bus network used by Wu Y.