Stochastics and Statistics
Modelling and optimization of average travel time for a metro line by simulation and response surface methodology

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Abstract

This research presents a modelling and solution approach based on discrete-event simulation and response surface methodology for dealing with average passenger travel time optimization problem inherent to the metro planning process. The objective is to find the headways optimizing passenger average travel time with a satisfactory rate of carriage fullness. Due to some physical constraints, traffic safety and legal requirements, vehicle speeds cannot be raised any further to decrease travel time. But travel time can be optimized by arranging headways (i.e. the time period between the departure times of two consecutive transportation vehicles) in a timetable. In the presented approach, simulation metamodels that best fit the data collected from the simulated experiments are constructed to describe the relationship between the responses (average travel time and rate of carriage fullness) and input factors (headways). Then, the Derringer–Suich multi-response optimization procedure is used to determine the optimal settings of the input factors that produce the minimum value of the average travel time by providing a proper rate of carriage fullness. This methodology is applied for a real metro line, and good quality solutions are obtained with reduced number of experiments that needed to provide sufficient information for statistically acceptable results.

Introduction

The problem of optimization in the urban public transportation area has received a great deal of attention in the literature. This problem has been intensively studied from cost minimization or profit maximization viewpoints. In recent years, passenger travel time has also emerged as an important performance measure to be optimized.

In the current paper, we aim to minimize average passenger travel time (average travel time for short). It is obvious that in recent years more regard has been paid to improving passenger travel time. In a passenger flow study, Li (2000) build a simulation model for a train station that includes processes, equipment and passenger queues that affect the total travel time. The objective is to minimize the total travel time and to increase service quality. Assis and Milani (2004) present a methodology for the computation of optimal train schedules in metro lines using a linear-programming-based model predictive control formulation. The train traffic model with passenger demands varying in time comprises dynamic equations describing the evolution of train headways and train passenger loads along the metro line. The authors use a weighted sum of convex piecewise-linear functions for directly or indirectly modelling the waiting time of the passengers at stations, comfort of onboard passengers, train trip duration and number of trains. Gentile et al. (2005) develop a general framework for investigating passengers’ route choices in transit networks when online information about carriers’ arrival times at stops is available. Based on the assumption that the ultimate passenger objective is to minimize his/her total travel time to the destination, a new stops model is proposed for determining the probability of boarding each line available at a given stop and the corresponding expected waiting time when headways show independent general distributions. Hu et al. (2005) firstly propose a transit network optimization model to maximize the non-stop passenger flow, and then put forward the optimization model of headways for all the transit routes in the optimized network. Ant colony algorithm and the genetic algorithm are employed to solve the optimization problem. Guan et al. (2006) develop an integrated approach in which the transit line planning and the passenger transferring process in a general mass transit railway network are jointly modelled, and thus transit line configuration and passenger line assignment are simultaneously optimized. The authors describe sub-problems separately, and then integrate them into a single linear binary integer program so that they can be solved by the standard branch and bound method. Zhao and Zeng (2006) use a combined genetic algorithm and simulated annealing search method for transit network optimization. The objective is to find optimal route network layout and route headways that minimize the overall cost of providing transit services, including both the user cost function based on k-or-less transfer trips, and the corresponding travel time and operator costs. In their later study, in Zhao and Zeng (2007), the authors develop a combined simulated annealing, tabu and greedy search method to find optimal route network design, vehicle headway, and timetable assignment minimizing passenger cost for transit networks. Flamini and Pacciarelli (2007) address a scheduling problem arising in the real time management of a metro rail terminus. The goal is to achieve optimal headways which minimize tardiness and earliness.

Above, a brief review of existing research related to improving passenger service quality in terms of travel time has been presented. While analytical results were obtained by exact algorithms, simulation models and meta-heuristics with approximate outcomes were also employed. Simulation models are flexible and solve real problems without making too many restricting assumptions as in most analytical models. The major benefit of using response surface methodology (RSM) in post-simulation analysis is the significant reduction in the number of simulation runs needed (Myers and Montgomery, 1995). The use of RSM allows some generalizations to be made about the output of the simulation. However, these generalizations have to be within the pre-defined boundaries of the problem. Sensitivity analysis on model parameters can also be easily carried out without re-running costly simulation programs.

Although a comprehensive review for a better understanding of the ways of optimizing travel time has been made, there appears to be no earlier study on the topic using RSM as a solution tool. This observation has been the motivation for the present work on travel time (a measure of the service quality) minimization problem of a metro line with a certain rate of carriage fullness. The modelling and solution approach proposed is the integration of RSM in the simulation.

In Section 2, we present the proposed methodology, which utilizes discrete-event simulation and RSM, and in Section 3 an illustrative example to explain how this methodology can be used to assist public transport providers. Conclusions are pointed out in Section 4.

Section snippets

The proposed procedure

In the proposed procedure, a general metro line simulation model for evaluating the effects of headways on travel time is presented. The simulation model is intended for use in a two-phase approach to build a metamodel that is based on simulation experiments and response surface analysis.

Metamodels are used economically to learn about how the response surface would behave over various regions of input-factor space and thus to estimate how the response would change at a particular point with a

Case study

First, a real metro line is described in detail and its simulation model is built. Then simulation experiments are designed and metamodels for each problem to be solved are developed by data obtained from simulation experiments. Finally, optimal headway settings are found via the Derringer–Suich multi-response optimization procedure.

Conclusion

This study proposed a modelling and solution approach that uses RSM integration with simulation for a complex problem that arises in the planning process of most metro lines. The simulation model built in this study is a generic model that can be easily changed to adapt the changes in headways, rate of carriage fullness, and the length of time periods. By running the model, several what-if questions of a metro company can be replied to make revisions.

Main contribution of the study is to solve

Acknowledgements

The authors would gratefully like to extend their appreciation to the anonymous referee whose valuable suggestions lead to an improved organization of this paper.

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