Mathematical Problems in Engineering

Volume 2017, Article ID 3047963, 10 pages

https://doi.org/10.1155/2017/3047963

## An Objective Train Timetabling Quality Evaluation Method

School of Transportation and Logistics, National Railway Train Diagram Research and Training Center, National and Local Joint Engineering Laboratory of Comprehensive Intelligent Transportation, 111 North 1 Section, Beierhuan Road, Jinniu District, Chengdu 610031, China

Correspondence should be addressed to Shao-quan Ni; moc.621@nauqoahsin

Received 8 October 2016; Revised 14 December 2016; Accepted 9 May 2017; Published 5 June 2017

Academic Editor: Andrea L. Facci

Copyright © 2017 Feng Jiang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The train timetable dominates the rail traffic organization. The timetabling quality should be evaluated to check the work skill of train timetable managers. The values of existing timetable evaluation indexes vary with infrastructure features and traffic flow; therefore, they are not comparable in fact. Furthermore, subjective inputs like expert scores are involved in evaluation; this will lead to unreliable results because the experts may have different opinions. To overcome these shortages, we propose a relative train path efficiency index by taking the train paths as production units. Each unit consumes some transport resources and produces some feedback outputs. A DEA model is applied to compute the train path efficiency. Two statistical functions of train path efficiency are used to evaluate the timetabling quality. We verify our method with real-world timetables. First, we use the Shibantan-to-Xinqiao line timetable to test the relative feature of the index proposed, and the results show that the train path efficiency value is relative and can reflect whether the stops are evenly distributed or not. Second, we evaluate the timetabling quality of another two timetables of the Qingdao-to-Jinan line with different traffic flows, and the results show that, compared with the 2012 timetable, the timetabling quality decreased in 2013.

#### 1. Introduction

A train timetable defines the departure and arrival times of a train at each station. The Train Timetabling Problem (TTP) is mainly about setting a feasible path for each train by displaying and adjusting the arrival and departure times of the train (see, e.g., Caprara et al. [1]). From this point of view, the evaluation of the train timetabling quality involves evaluating the TTP process mentioned above, particularly the distribution and adjustment of the train paths.

A timetable can be made by a computer or manually. For cases of simple traffic, computer timetabling can provide a good solution, while for large-scale, complex traffic, it is difficult to generate a timetable completely automatically. Therefore, manual adjustments are necessary. In China, where the timetable is nonperiodic and the scale of the rail network is large, at the timetabling stage, a lot of readjustments are required due to the complexity of the problem, so we employ a manual timetabling process. The person in charge of this job is usually known as the timetable manager (TM). As the TTP is NP-hard [1], it is difficult to find the optimal solution for all train paths; thus, different TMs can reach different solutions, and the work skill of the TM will eventually affect the timetable quality. Usually, there are several TMs working on timetabling, each of whom is in charge of different lines. Nowadays, we are facing the problem of how to evaluate the work skill of TMs, in other words, how to find a way to evaluate the train timetabling quality (TTQ).

The TTP is recognized as one of the most important problems in rail traffic organization; a lot of research focuses on this field. In Caprara et al.’s study [1], a single one-way track TTP is studied and an ILP model is used, where the ILP is relaxed in a Lagrangian way and a heuristic method is used to schedule each train while keeping it as close as possible to the ideal timetable. In Brännlund et al.’s study [2], a single track, two-way line case is studied and the Lagrangian relaxation method is used to solve the model. In Zhou and Zhong’s study [3], based on passenger demands, the train plan problem is solved for a double-track high-speed line to minimize both the waiting time for high-speed trains and the total travel time for all trains, and a branch-and-bound algorithm with effective dominance rules is used to solve the bicriteria scheduling problem. Some works focus on periodic TTP, in which for every period (usually 1 hour) the timetable is repeated to assist passengers to remember and transfer between trains. A Periodic Event Scheduling Problem (PESP) model is applied by Liebchen and Möhring [4] and Kroon and Peeters [5]. We refer readers to the review by Cacchiani and Toth [6] of nominal and robust models and algorithms about TTP for further details. After the timetable is created, it should be evaluated to check its quality. In this field, much research focuses on the dynamic performance. In Huisamn and Boucherie’s study [7], the delays caused by fast trains becoming caught behind slower ones are investigated, and a Markov chain is used to build a stochastic model capturing both scheduled and unscheduled train movements; the model allows the analysis of the running times for both fixed periodical timetables and long-term capacity calculation with random trains arriving. In Vromans et al.’s study [8], the reliability of the public railway system is examined by reducing the interdependency between trains and the homogeneity of the train timetable is evaluated. In Goverde’s study [9], a Max-plus method is used to evaluate the stability of a periodic timetable by considering the delay propagation of the initial ones. These studies evaluate the train timetable from the point of view of effects among train paths, some of which call for such a specific condition as no overtaking is allowed (see, e.g., [7]) or a periodic timetable is used (see, e.g., [8]). In China, the evaluation emphasis differs. Usually, evaluation indexes are proposed to evaluate specific features like the passenger service level and infrastructure utilization level, and then subjective scores are given as inputs to describe how good the features are. In Peng et al.’s study [10], an index system including passenger service quality and dynamic performance as well as timetable production efficiency is built, and the timetable is evaluated with a Grey theory based on expert scores. In Zhu et al.’s study [11], the vague sets (see, e.g., Gau and Buehrer [12]) from the fuzzy sets are applied to another index system, which is proposed for the passenger-dedicated line in terms of passenger transport demands, in order to deal with the experts’ scores so that the timetable’s evaluation can be gained.

The TTQ impacts the rail transport system directly, but it is difficult to compare the work skills of TMs. The major reason is that when a train path is set, one needs to consider several different constraints, usually including capacity constraints, which involve the running time of a train between stations, headway distances, crossover constraints, and so forth (see, e.g., [1]). Meanwhile, these constraints are caused by traffic flow, such as the number of trains and train stop patterns. Usually, we consider all the relevant factors such as capacity constraints and traffic flow as the background of the timetable. Once the timetable background is changed, the timetable features become incomparable. That is why when we try to evaluate a timetable’s quality, a subjective input like an* expert score* is involved (see, e.g., [10, 11]). However, the result of this method is unstable because it is based on the assumption that the experts are familiar with the timetable background. As mentioned before, when a TM adjusts a train path, since the constraints for the train paths will affect each other, the timetable background changes at the same time. Due to this consideration, there is no guarantee that the experts will give an objective score. For example, supposing we have 2 lines, one consisting of 100 trains with two speeds, 120 km/h and 140 km/h, and the other consisting of 80 trains with four speeds, 160 km/h, 140 km/h, 120 km/h, and 80 km/h, given the different traffic flow and capacity constraints, it is hard to say which timetabling process is better. As the timetable background varies from line to line, the key point in evaluating the TTQ is to find a relative index which is still comparable even when the timetable background changes. Then, it will be possible to abandon the subjective inputs in the evaluation.

This paper introduces a TTQ evaluation method where the infrastructure features and traffic flows are no longer relevant. This allows us to determine the skill of the TMs even if the timetable background is different. We tested the method on different lines and obtained objective evaluation results without any subjective inputs, where the results showed the method to be effective and reasonable. As far as we know, this TTQ evaluation method is new.

The construction of this paper is as follows. In Section 2, we describe the problem and propose a* train path efficiency *index to evaluate the timetabling quality. In Section 3, we analyze the elements of the train path efficiency and give calculation methods. In Section 4, a solution model of train path efficiency based on Data Envelopment Analysis (DEA) is given, and the model is solved by linear programming. The method is verified with real-world timetables in China in Section 5, while in the final section some conclusions are given.

#### 2. Problem Description

In TTP, the shortest running time between two stations is known as the ideal running time (see, e.g., [13]). A timetable with the ideal running time and minimum stop time is called an ideal timetable (see, e.g., [1]). In real-world cases, it is difficult to set an ideal timetable for every train, because some trains will conflict with others and violate the capacity constraints. To make a conflict-free timetable, some trains need to stop at a station longer or to slow down in the rail sector. In China, we allow TMs to modify the stop pattern and stop time of a train to meet the capacity constraints. Once the train type is given, the timetabling quality is affected mainly by the adjustment of the departure and arrival time instants as well as extra stop time.

An example is given to show how a train TM affects the timetabling quality in Figure 1: T and K are train types, where T is a long-distance train with a fixed departure time at station A and K is a short-distance train with a variable departure time at station A, and the overtaking is due to capacity constraints. By modifying the departure time at station A, the improved train paths (expressed with dashed lines) improved the performances; for example, K1′ has a better stop equilibrium because of the shorter average stop distance and K2′ has less total travel time because some unnecessary stops are avoided.