Journal of Advanced Transportation

Volume 2018, Article ID 4613468, 21 pages

https://doi.org/10.1155/2018/4613468

## Combinatorial Optimization of Service Order and Overtaking for Demand-Oriented Timetabling in a Single Railway Line

^{1}State Key Lab of Rail Traffic Control & Safety, Beijing Jiaotong University, Beijing 100044, China^{2}School of Traffic and Transportation, Beijing Jiaotong University, Beijing 100044, China

Correspondence should be addressed to Dewei Li; moc.361@utjb.ilwd

Received 24 March 2018; Revised 28 June 2018; Accepted 15 August 2018; Published 12 September 2018

Academic Editor: Luca D'Acierno

Copyright © 2018 Dewei Li 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

Train timetabling is crucial for passenger railway operation. Demand-oriented train timetable optimization by minimizing travel time plays an important role in both theory and practice. Most of the current researches of demand-oriented timetable models assume an idealized situation in which the service order is fixed and in which zero overtaking exists between trains. In order to extend the literature, this paper discusses the combinatorial effect of service order and overtaking by developing four mixed-integer quadratic programming timetabling models with different service order as well as overtaking conditions. With the objective of minimizing passengers’ waiting time and in-vehicle time, the models take five aspects as constraints, namely dwell time, running time, safety interval, overtaking, and capacity. All four models are solved by ILOG CPLEX; and the results, which are based on Shanghai-Hangzhou intercity high-speed rail data, show that either allowing overtaking or changing service order can effectively optimize the quality of timetable with respect to reducing the total passengers’ travel time. Although optimizing train overtaking and service order simultaneously can optimize the timetable more significantly, compared to overtaking, allowing the change of service order can help passengers save total travel time without extending the train travel time. Moreover, considering the computation effort, satisfying both of the conditions in the meantime, when optimizing timetable has not got a good cost benefit.

#### 1. Introduction

Railway timetabling is the basis for train operation, whose quality affects the service level and the efficiency of the whole network. In the form of timetables, the railway operators provide access to their transport products for consumers; passengers then choose the appropriate train according to the schedule. The quality of the train schedule determines whether the passengers are satisfied with the train service, which influences the railway operators’ competitiveness in the passenger transportation market. However, it is quite difficult to design a timetable that can meet the needs of both operators and passengers; thus, the train timetabling problem has attracted much attention from researchers over the past few decades.

##### 1.1. Literature Review

Railway timetabling has been widely addressed by providing a closed analytical form for determining the involved time rates, e.g., travel times, dwell times [1], inversion times, buffer times [2], reserve times, and layover times [3]. Besides, some researchers have studied the problem of train timetable optimization to improve its performance by considering stop-skipping [4–6], overtaking, and service scheduling [7, 8].

Many researchers have studied the problem from the perspective of railway operators; meanwhile, various train schedule optimization models have been established to minimize the total travel time of all trains in a system. Two kinds of timetables have been studied in this literature, namely, cyclic timetables and noncyclic timetables.

Most studies on cyclic train timetables are based on the Periodic Event Scheduling Problem (PESP) model that was put forward by Serafini and Ukovich [9]. In 1993, Voorhoeve [10] first provided a PESP-based model that takes into account the main operating constraints, such as dwell time, running time, and safety headway. Odijk [11] designed a cutting plane algorithm to solve the PESP model. This method can quickly obtain the train timetable of a small railway network. In addition, some researchers [12, 13] transformed the PESP model into the Cycle Periodicity Formulation (CPF) model, which can reduce the number of constraints and variables. A stochastic optimization model was developed by Kroon [14] to allocate the time supplements and buffer times in a given cyclic timetable to enhance the robustness of the timetable. Liebchen [15] integrated many nonstandard requirements and other planning phases into PESP. Caimi [16, 17] developed a timetable model with partial periodicity. Zhou [18] modeled the multiperiodic train timetabling problem to simultaneously optimize operation periods, arrival times, and departure times of various types of trains of all periods. A cyclic train timetable is easier and more convenient for passengers to remember exactly the arrival times and the departure times. However, a cyclic timetable is not sensitive to irregular passengers’ demand, which can result in long waiting time under low frequency and a waste of capacity under high frequency.

With regard to the noncyclic train timetabling problem, Szpigel [19] first studied the optimal train scheduling problem on a single line track and presented a linear programming model originally based on job-shop scheduling problem to minimize the total travel time, while the best crossing and overtaking positions are determined. The problem is proved to be NP-hard by Cai [20] and Caprara [21], so it is difficult to obtain the optimal solution, especially for large-scale cases. In Carey [22], total service running time was the objective, and binary variables were used to describe the precedence between service. Higgins [23] provided a lower bound that allows the branch-and-bound algorithm to find the optimal solution of complex instances in a reasonable time. Lindner [24] developed a timetable model by minimizing the total operation cost. In order to find the suboptimal solution, Zhou and Zhong [25] developed a bi-criteria train scheduling model. With effective dominance rules, utility evaluation rules, and a beam search algorithm, the train scheduling model considered the acceleration and deceleration times and solved by a branch-and-bound algorithm. Caprara [26] designed train timetables. The train timetables took into account several additional constraints, which arose in real-world applications and provided a Lagrangian heuristic algorithm for real-world instances. Zhou and Zhong [27] studied a single-track train timetabling problem, aiming to minimize the total travel time. They proposed lower bound rules and heuristic upper bound construction methods to improve computational performance. Cacchiani [28] proposed a column generation approach to solve train timetabling problem. R.L. Burdett [29] extended a discrete sequencing approach for train scheduling, considering multiple overtaking conflicts and compound moves. Corman [30] described a train dispatching support tool to manage more saturated railway networks by changing dwell times and train orders and routes. Cordone [31] proposed a mixed-integer nonlinear model with a nonconvex continuous relaxation. In the same year, to construct a feasible NWBPMJSS train schedule, satisfying the blocking and no-wait constraints in job-shop environments, Liu [32] proposed a two-stage hybrid heuristic algorithm. Canca [33] proposed a tactical model to determine optimal policies of short-turning and nonstopping at certain stations, considering the objective of minimizing the arrival times of the last shuttle. Kroon [34] dealt with connection problem in cyclic passenger railway timetabling. Fröidh [35] discussed how different dwell times and skip-stop operation affected capacity. To minimize delays after an unexpected event perturbs the operations, Pellegrini et al. [36] proposed a mixed-integer linear model for the real-time railway traffic management problem and represented the infrastructure with fine granularity. Considering the stopping and skipping stations for skip-stop rail operation, Lee et al. [37] proposed a mathematical model; and a Genetic Algorithm is used to solve the model. Chen [38] integrated optimization of train service headways and stop-skipping strategy to improve the operation efficiency and service quality of a BRT system. He also proposed a genetic algorithm to solve the problem. Castillo et al. [39] presented a time partition technique to reduce the complexity of the problem. Liu [40] considered different kinds of headway time; an integer linear programming and a branch and price algorithm are proposed to optimize the timetable.

In recent years, passenger demand for service quality has grown ever higher, while the supply capacity of rail service operators has increased as well. Therefore, some researchers focus on the dynamic passenger demand and construct models to maximize the benefit of passengers.

In [41] and [42], Niu studied the time-dependent characteristics of passenger demand and formulated a timetabling model under oversaturated conditions to minimize passenger waiting times at stations. A local improvement algorithm was presented to find the optimal timetables for individual stations and a genetic algorithm was provided to solve the whole line problem. Barrena [43] discretized the time horizon and assumed that the arrival of passengers in each small interval is subject to a uniform distribution. A nonlinear programming model was formulated and solved by branch-and-bound algorithm. The algorithm is insufficient to solve the large-scale case, so Barrena [44] introduced Riemann’s sum theory to calculate the passenger waiting time and developed a simulated annealing algorithm to solve large instances of the problem within short computation times. Sun [45] proposed the concept of equivalent time to synchronize train operations and passenger demand, then he provided a mixed-integer programming model that allows for train capacity constraints in order to design demand-driven timetables for metro services. The capacitated metro service timetabling problem can simply be solved by CPLEX. Canca [46] presented a nonlinear integer programming model to meet the dynamic passenger demand; this model can also be used to measure the timetable quality and to offer the service provider a trade-off between service quality and operational cost. Niu [5] considered time-dependent demand and skip-stop patterns; a nonlinear integer programming model with quadratic and quasi-quadratic objective functions was proposed to compute the total passenger waiting time under both minute-dependent demand and hour-dependent demand. However, the above studies only considered passenger waiting time at stations; the in-vehicle travel time is ignored. Wang [47, 48] proposed a nonlinear nonconvex model, the objective function of which is total energy consumption and total passenger travel time. A set of possible approaches is used to solve the model. Besides, the iterative convex programming approach is proved to provide a better trade-off between the quality of solution and computational time. Xu [49] proposed a multiobjective timetable optimization approach to minimize the passenger time and energy consumption. Robenek et al. [50, 51] proposed the Elastic Passenger Centric Train Timetabling Problem (EPCTTP) model with the objective of maximizing the train operating company’s revenue. Robenek [52] proposed a hybrid timetable combining the benefits of cyclic and demand-oriented timetables. Hassannayebi et al. [53] applied an adaptive and variable neighborhood search algorithm to optimize the train timetable problem. Yin [54] developed a timetable optimizing model, considering the dynamic passenger demands and energy saving objective. A Lagrangian relaxation-based heuristic algorithm is proposed to solve the model. Zhang [6] considered flexible skip-stop scheme and proposed a mixed-integer nonlinear programming model to minimize the average passenger travel time. Shen [55] and Zhang [56] proposed timetabling model to minimize the passenger travel time under congestion conditions. Liu [57] considered joint routing and scheduling between freight and passenger trains and proposed a model to study the robust passenger train timetable. A branch-and-bound framework with hybrid heuristics is used to solve the model. A systematic comparison among the typical existing studies is shown in Table 1.