To design an efficient and economical timetable for a heavily congested urban rail corridor, a scheduling model is proposed in this paper. The objective of the proposed model is to find the departure time of trains at the start terminal to minimize the system cost, which includes passenger waiting cost and operating cost. To evaluate the performance of the timetable, a simulation model is developed to simulate the detailed movements of passengers and trains with strict constraints of station and train capacities. It assumes that passengers who arrive early will have more chances to access a station and board a train. The accessing and boarding processes of passengers are all based on a firstcomefirstserve basis. When a station is full, passengers unable to access must wait outside until the number of waiting passengers at platform falls below a given value. When a train is full, passengers unable to board must wait at the platform for the next train to arrive. Then, based on the simulation results, a twostage genetic algorithm is introduced to find the best timetable. Finally, a numerical example is given to demonstrate the effectiveness of the proposed model and solution method.
As the economy develops rapidly, deepening urbanization and the increasing urban population lead to a large demand for urban rail transit in many cities. In 2013, over 3.2 billion and 2.5 billion passenger rides were delivered by the Beijing and Shanghai subway systems, respectively; the average passenger rides on weekdays are more than 10 million and 7 million, respectively [
Timetable design problem (TDP) aims at determining a preoperational schedule for a set of trains and follows some train operational requirements [
Although there are some differences between urban public transportation and railway transportation, the mathematical programming method has still been introduced to create efficient timetables for urban public transportation to reduce passenger waiting time. Cury et al. [
In order to provide a more efficient timetable for passengers, vehicle capacity has been widely considered. Ceder [
Although there is a comprehensive body of literature on TDP, few studies have drawn attention to the limitation of station capacity; also, the outsidestation waiting time (OSWT) caused by being unable to get into a crowding station has always been neglected. In fact, waiting outside the station is a common phenomenon in overcrowded urban rail transit systems, such as subway systems in China. In order to provide a safe and efficient movement in stations, especially in an underground station with more enclosed and very limited internal space, operators will routinely restrict the number of passengers in stations. That is to say, some passengers will be required to wait outside the station when the number of passengers in stations exceeds the safevalue. For example, in Beijing, 63 urban rail stations mainly along Line 1, Line 5, Line 6, Line 13, Batong Line, and Changping Line instituted these restrictions since July 8, 2014, during a.m. peak hours. And OSWT in some stations, such as ShaHe, an intermediate station of Changping Line, is more than 20 minutes.
In addition, many studies in the area of TDP aim to minimize the waiting time of passengers [
This paper fully considers the constraints of station and train capacities. Thus, the passenger waiting time can be divided into three parts, namely,
A simulation model is developed to simulate the movements of passengers and trains with constraints of capacities, such as train maximum capacity and station safety capacity.
The total waiting time of passengers, which includes IPWT, EPWT, and OSWT, is measured based on the outputs of the simulation model.
An optimization model is proposed to minimize the operational and passenger waiting costs; a twostage simulationbased genetic algorithm is developed to solve the model.
The remainder of the paper is as follows. Section
This paper focuses on the TDP of an urban rail line with
The representation of an urban transit rail line.
Let
Other assumptions made throughout the paper are explained as follows.
The distribution of passenger demand is given and is steady during the study period.
Whether a station is under an overcrowding situation is decided by the number of waiting passengers in station. That is, passengers cannot enter a station when the number of waiting passengers at station is larger than the safevalue. In fact, alighting passengers will also lead to a crowding situation at station. However, it is difficult for operators to forecast the number of alighting passengers of each train. For simplification, we only forecast the maximum possible number of alighting passengers. Then, the safe capacity for waiting passengers is equal to design capacity minus this maximum number.
In general, passengers who arrive early will have more chances to access a station or board a train. In order to facilitate simplification, the paper assumes that all passengers accessing a station or boarding a train obey the FCFS principle. The passengers who fail to access an overcrowding station or fail to board a full train must wait for the next chance.
The proposed model focuses on reducing passenger waiting time. The accessing walking time of passengers at a station is not considered in this model. Here, we assume that the accessing walking time of passenger is assumed to be fixed and equal to 0.
The following notations and parameters are used throughout this paper.
Sets are as follows:
set of time intervals,
set of stations,
set of trains.
Indices are as follows:
index of stations,
index of passengers,
index of trains,
index of modeling time intervals,
index of travel direction; let
Parameters are as follows:
number of passengers who arrive at station
passenger destination probability, the probability of potential destination station
total number of passengers who arrive at station
dwelling time at station
running time between stations
recovery times at start terminal and return terminal,
maximum number of trains
prespecified fleet size,
number of time intervals,
train operating cost per vehicle per unit time,
waiting time cost per passenger per unit time,
maximum service headway,
minimum service headway,
maximum capacity of trains,
design capacity of station
maximum possible number of passengers alighting at station
maximum capacity for waiting passengers at station
threshold which is used to judge whether passengers can access the overcrowding station
let
Intermediate variables in simulation modelare as follows:
number of boarded passengers on train
number of passengers waiting outside station
number of passengers waiting at time
number of passengers alighting at station
departure time of train
arrival time of train
departure time of the first train for the direction of the
time by the end of the recovery operation after the train
arrival time of the
accessing time of the
boarding time of the
number of idle train unit at time
let
Decision variable is as follows:
let
The TDP in this paper aims at determining the departure time of each service at start terminal. The objective is to minimize the total cost of the transit system
In this formulation, the objective function (
Constraint (
For our case, the movement of passengers is restricted by a series of constraints, and it is difficult to use mathematical modeling approach to describe the detailed movement of passengers or calculate passenger waiting time. As mentioned in [
In this section, a simulation model of urban rail line is presented to evaluate the performance of timetables. It is characterized by a discreteevent and synchronous simulation to model dynamic processes. The simulation process is illustrated in Figure
Framework of simulation model.
As shown in Figure
Flowchart of passenger arrival event.
Flowchart of passenger accessing event.
Flowchart of passenger boarding event.
Flowchart of passenger alighting event.
Flowchart of train arrival event.
Flowchart of train departure event.
Note that the maximum number of alighting passengers (
The TDP in this paper belongs to the NPhard class (see [
The genetic algorithm (GA) is a stochastic search method that is inspired by the natural evolution of species. Due to its extensive generality, strong robustness, high efficiency, and practical applicability, GA has become increasingly popular in solving complex optimization problems since the seminal work of Holland [
As the decision variable
Optimization procedure integrating simulation and GA.
An initial solution pool with
Mutation operation I.
Mutation operation II.
The procedure for GA includes the following.
Other detailed steps or approaches of GA, such as selection and crossover processes, are similar to the standard GA, and interested readers are referred to the related references (e.g., Gen and Cheng [
In order to show applications of the proposed model and solution algorithm, an urban rail line with seven stations, which is shown in Figure
An urban rail line with 7 stations.
In this example, we aim to determine the departure time of each service at station 1 during the morning peak period
Value of demand related parameters.
 

1  2  3  4  5  6  7  

19800  18000  12600  3000  18000  15600  10200 

1680  1680  1800  900  1800  2100  1800 

2700  2700  2100  4200  3300  3600  3600 
Another important parameter related to demand, passenger destination probability
Passenger destination probability
To  

From 
 
1  2  3  4  5  6  7  

1  —  0.05  0.05  0.05  0.2  0.4  0.25 
2  0.05  —  0.05  0.1  0.2  0.3  0.3  
3  0.1  0.05  —  0.05  0.1  0.35  0.35  
4  0.05  0.1  0.05  —  0.25  0.2  0.35  
5  0.1  0.1  0.05  0.05  —  0.3  0.4  
6  0.05  0.05  0.05  0.05  0.4  —  0.4  
7  0.2  0.2  0.05  0.15  0.2  0.2  — 
Other necessary parameters used in the simulation model and GA are summarized in Table
Parameters used in numerical example.
Parameters 



CT 






Maxgeneration 




Value  40  15 min  120 s  1680  640 USD/h  1 USD/h  1  1  40  260  Up to 70  0.9  0.2 
All experiments in this paper are tested on a personal computer with an Inter Celeron G1620 2.7 GHz and 2 GB RAM. The simulationbased optimization model is coded in MATLAB 7.11.
In this section, we solve the optimization problem with different time intervals, namely,
Comparison between different optimal results.




CPU (s) 


2810.59  12693.33  15503.92 (85.2%)  5834.57 

2841.84  12693.33  15535.17 (85.4%)  2768.51 

3136.25  12693.33  15829.58 (87.0%)  929.09 
Evenheadway  3259.35  14933.33  18192.69 (100%)  281.36 
Optimization process of GA.
Best solution: departure times of trains at the start terminal.
Train  Departure time 

1  7:04:55 
2  7:09:30 
3  7:14:20 
4  7:18:20 
5  7:21:40 
6  7:25:10 
7  7:29:10 
8  7:33:10 
9  7:37:10 
10  7:41:30 
11  7:47:30 
12  7:51:30 
13  7:56:35 
14  8:02:25 
15  8:08:50 
16  8:16:45 
17  8:30:00 
Station capacity is an important constraint in our model. In order to analyze the impact of station capacity
Optimal values with different station capacities.
As we can see, when
These results indicate that a limited station capacity will have a great influence on the service quality and operation plan of urban rail transit. And the smaller the station capacity is the higher system cost will be. However, on the other side, a too large capacity for a station will contribute little to the improving of transportation system but may bring a high infrastructure cost. Therefore, it is necessary to preestimate the system cost in the operation stage before building the station.
In this paper, we present a scheduling approach for a heavily congested urban rail line. It aims to create an efficient timetable with minimal passenger waiting cost and operational cost. In order to evaluate the performance of the created timetable, a simulation model is proposed with strict constraints on train and station capacities. Then, based on the simulation results, a twostage GA is designed to find the best timetable. Finally, the feasibility of the solution method is demonstrated through a numerical example.
Although only a small case is discussed in this paper, it is shown that the strict constraint of station capacity is essential for TDP; the timetable designed by the proposed model will have a better balance between passengers and operators.
In addition, the passenger demand in our work is assumed to be steady. However, in real work, timetables will have a feedback on the distribution of passenger demand. In our future research, we will try to create a new timetable considering this feedback.
The authors declare that there is no conflict of interests regarding the publication of this paper.
This research was funded by the Nation Basic Research Program of China (no. 2012CB725406) and the National Natural Science Foundation of China (nos. 71131001 and 71201007). The authors also thank the anonymous reviewers and the editor for their suggestions to improve this paper.