Variable and adaptive neighbourhood search algorithms for rail rapid transit timetabling problem
Introduction
An important aspect of the public transportation systems is the passenger waiting times. Specifically, inefficient transportation systems incur additional waiting time to the passengers. Since it is impossible to avoid waiting time completely, it is an essential concern in the daily operation of a public transit system to minimize the passenger waiting times. On the other hands, managing the growing travel demand for public transport is another challenge that urban rail transit systems currently face. To deal with these challenges, the railway companies should try to optimize the rail operations. The optimization of the urban public transportation systems has been intensively studied from both cost minimization and profit maximization viewpoints. In recent researches, passenger waiting time has appeared as an important performance measure to be minimized by optimization methods [50].
Train timetabling problem (TTP) in an urban rail system refers to the procedure of determining the optimal departure times of transit vehicles according to the dynamic behavior of the travel demand in order to minimize the operator and passenger performance measures [31]. In this regard, prediction models aim to analyze the main influencing factors of passenger waiting time. Here, as in early studies in this area, Welding [47], and Osuna and Newell [34] explored the characteristics of passenger waiting time and developed an estimation model for average waiting time per passenger (AWT) at station stops. They defined the expected waiting time as a function of the average and variance of the headway.
The rail managers generally offer a limited number of vehicle services during the daily normal operation, due to the operational and maintenance costs. In this regard, the train headways are affected by the total number of services, fleet size, and vehicle capacity and therefore, better distribution of the train services result in decreasing the travel time and waiting time of passengers. However, urban rail transit is a complex dynamic traffic system, making it challenging to be optimized [51]. Furth and Muller [21] presented methods for generating the waiting time distribution from headway data in order to measure the service reliability. The problem of estimating passenger waiting times and measuring the headway variance at bus stops with incomplete input data was studied by Mcleod [30]. Usually, the establishment of the regular timetables results in minimum waiting time at stops. However, from the application point of view, the even-headway or regular schedules suffer from the lack of enough agility to handle variations of travel demand over the study period [13]. Moreover, the total waiting time during periods of peak demand is expected to be affected by fleet size and train capacity constraints. Very few academic efforts have been devoted to assessment of these factors on waiting times. Islam et al. [24] analyzed the effect of the headway variability and vehicle capacity on service performance and reliability of a high-frequency bus transit system through a Markov chain model. Amin-Naseri and Baradaran [6] extended the Welding׳s formulations and developed a more accurate formula to estimate the AWT at a bus stop with independent headways. A discrete-event simulation method has been applied to evaluate the accuracy of the proposed formulations. Ceder and Philibert [12] presented a method for the generation of timetables in order to eliminate the overcrowding through adjustment of departure times. The developed method produces schedule for vehicles reaching to an even maximum load. The results of this scheduling policy demonstrate the significant improvement over even-headway schedules or with headway plan constructed based on hourly even maximum loads.
Although there is a rigorous literature dealing with train scheduling in the context of off-line planning, there is still a lack of efficient solution methods that can efficiently solve the demand-oriented TTP. Furthermore, a limited number of studies have been found that addressed urban train timetabling problem under dynamic demand and train capacity constraint. The motivation for this study is the development of effective meta-heuristic algorithms for providing solutions to large-sized instances of the demand-oriented TTP. Also, this study aims to contribute to establish new solution encoding–decoding methods for the urban rail transit scheduling.
The outline of the study is organized as follows. A review of the related literature is presented in the next section. The research objectives and contributions are provided in Section 3. The train timetabling models are described and mathematically formulated in Section 4. The proposed neighbourhood search meta-heuristics are presented in Section 5. The specification of the real case is given in Section 6. Computational results and the corresponding discussion are presented in Section 7. Finally, in Section 8 conclusions are drawn and recommendations for further research are given.
Section snippets
Literature review
The optimization approach to timetables design aims to maintain the desired level of service to passengers. Moreover, the minimization of the passenger waiting time in rail transport services is a challenging scientific field that is devoted to optimization models and solution methods. The studies reviewed in this article are in this context a more effective way of improving the quality of service compared with the prediction models. The first optimization model for underground rail transit
Research objectives and contributions
This paper aims to achieve two specific objectives by proposing effective neighbourhood search algorithms. First, the proposed approach improves on current solution methods of the demand-oriented train scheduling problem by developing different efficient neighbourhood search algorithms that can generate train timetables with minimum passenger waiting time in a reasonable computational time. The current study aims to find an optimized daily train timetable for a metro system through minimizing
Problem statement and formulation
Demand for rail travel in both time and space is variable. In a subway system, train services required to operate within the specified safety time and the separation time between two successive departures of the trains known as headway. In this study, the aim is to optimize the operation of train services through adjustment of the headway times in accordance with the dynamic passenger demand. As stated, the passenger waiting times depend on the number of trains, fleet capacity, operational
Neighbourhood search variants
The demand-oriented train timetabling is a NP-hard combinatorial optimization problem which is particularly challenging to be solved optimally for practical cases [39]. Consequently, heuristic and meta-heuristic methods are decidedly worthwhile in practice. This section presents different variants of neighbourhood search algorithm to solve train timetabling problem. First, a general summary of the method and the framework on which the neighbourhood search heuristic is applied will be provided.
Application to real cases
Tehran urban and suburban railway is a rapid system transit rail with five operating lines serving Tehran. Currently, the Tehran metro network is 122 km long and on average two million passengers use it daily. Tehran–Karaj subway line is investigated that connect two major stations (Golshahr and Tehran). It is a 41.5 km long route starting from Tehran terminus and ending at Golshahr. In public transportation systems, the travel demand is characterized by peak and off-peak fluctuations. In the
Results and discussion
For a better comparison of the solution quality, a simple lower bound of the average waiting time per passenger is obtained. The basic idea is designing a regular timetable. Welding [47] estimated the expected passenger waiting time AWT where both the vehicle׳s and passenger׳s arrivals at stop are random. The proposed formulation comprises of the average headway and the headway variance :
From the above equation, it can be concluded that the average passengers waiting time
Conclusion
In railway networks with high rate of congestion, the minimization of the passenger waiting time becomes more and more important. In order to gain higher efficiency and better utilization of the infrastructure and fleet capacity in railway traffic systems, it is required to use optimization approaches. In this study, the focus was on the comfort of the passengers through presenting effective solution methods for minimizing the total waiting time in metro system. Firstly, a mixed integer
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