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A Branch-And-Price Approach for Trip Sequence Planning of High-Speed R Train Units

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A Branch-And-Price Approach for Trip Sequence Planning of High-Speed R Train Units ARTICLE IN PRESS JID: OME [m5G; November 12, 2019;15:14 ] Omega xxx (xxxx) xxx Contents lists available at ScienceDirect Omega journal homepage: www.elsevier.com/locate/omega A branch-and-price approach for trip sequence planning of high-speed R train units ∗ Yuan Gao a, , Marie Schmidt b, Lixing Yang c, Ziyou Gao c a School of Management and Economics, Beijing Institute of Technology, Beijing, 10 0 081, China b Rotterdam School of Management, Erasmus University Rotterdam, Rotterdam, 30 0 0 DR, The Netherlands c State Key Laboratory of Rail Traffic Control and Safety, Beijing Jiaotong University, Beijing, 10 0 044, China a r t i c l e i n f o a b s t r a c t Article history: In high-speed railway operations, a trip sequence plan is made once the timetable is determined, and Received 24 February 2019 serves as a reference in the subsequent operations of train units scheduling. In light of the maintenance Accepted 27 October 2019 requirements of train units and periodicity characteristics of trip sequences, we introduce a trip sequence Available online xxx graph to describe the train units’ movement and coupling/splitting in a railway network. Based on the Keywords: trip sequence graph, two integer linear programming models are then formulated, namely a path-based Train units scheduling model and an arc-based model. Integrated with the characteristics of the trip sequence graph, a cus- Trip sequence planning tomized branch-and-price algorithm is developed to solve the path-based model. The two models are Maintenance applied to the high-speed railway network in eastern China, and through numerical experiments, the High-speed railway effectiveness and applicability of the models are discussed. Branch-and-price algorithm ©2019 Elsevier Ltd. All rights reserved. 1. Introduction plan is made, which determines a series of trip sequences to cover all trips. In train unit assignment planning , the trip sequence plan Due to the system complexity, the methods of multi-stage plan- serves as a reference. Specifically, based on a given trip sequence ning are widely applied in railway systems, and a typical exam- plan and the maintenance state of the train units in the beginning ple is the scheduling of train units in a high-speed railway net- of a week, a train unit assignment plan assigns individual train work. The process of train units scheduling relates to line plan- units to trip sequences on each day of the week or arranges some ning, train timetabling, trip sequence planning and train unit as- train units to undergo maintenances on certain days of the week. signment planning. The former three belong to the tactical level, Given a timetable, trip sequence planning and train unit assign- the horizons of which are several months or years, while the last ment planning together determine the movement of train units one, which is usually a weekly plan, belongs to the operational in a high-speed railway network, including carrying out trips and level. undergoing maintenances. A main reason that trip sequence plan- Considering network condition and passenger demand, line ning and train unit assignment planning are separated is due to planning determines the origin and destination stations, the route the complex maintenance regulations. For example, in China, there and the stop plan of a trip. Besides, the type of train unit car- are five levels of maintenance, two short-term and three long- rying out the trip is also determined in line planning. The out- term ones. A train unit must undergo maintenance of a certain put of line planning is the input of train timetabling , which gives level before the accumulative travel distance or elapsed time since the specific departure and arrival times at each station of all trips. the last maintenance of this level exceeds a predetermined value, Next is trip sequence planning , which is also referred to as train whichever comes first. The duration of maintenances of differ- unit routing. A trip sequence consists of a series of trips satisfying ent levels and the locations where they can be executed differ. some restrictions, such as shunting, coupling/splitting and main- Since level I maintenance is executed every day or every two days, tenance. Instead of a trip, a trip sequence is the smallest task for when making a trip sequence plan, restrictions of level I mainte- train units. In the stage of trip sequence planning, a trip sequence nance are taken into account. That is, the total travel distance and elapsed time of a trip sequence are respectively less than the max- imum travel distance and elapsed time between two level I main- R This manuscript was processed by Associate Editor Salazar-Gonzalez. ∗ tenances, and the origin and destination of a trip sequence must Corresponding author. E-mail address: [email protected] (Y. Gao). be the same station where level I maintenance is performed. As a https://doi.org/10.1016/j.omega.2019.102150 0305-0483/© 2019 Elsevier Ltd. All rights reserved. Please cite this article as: Y. Gao, M. Schmidt and L. Yang et al., A branch-and-price approach for trip sequence planning of high-speed train units, Omega, https://doi.org/10.1016/j.omega.2019.102150 ARTICLE IN PRESS JID: OME [m5G; November 12, 2019;15:14 ] 2 Y. Gao, M. Schmidt and L. Yang et al. / Omega xxx (xxxx) xxx result, it is convenient to use trip sequences instead of trips as the satisfied. In the paper, an integer linear programming model based smallest unit in scheduling train units when taking into account on a time-space graph was constructed, without considering main- different levels of maintenance. tenance requirements. The framework of this model, namely an ILP A good trip sequence plan means that the train units assigned model on a directed graph, has been widely used in most subse- to the trip sequences can be fully used before level I maintenance. quent research on train units scheduling problems. From then on, The more the train units are used, the lower operation cost is. In researches on train unit scheduling problem can be divided into this paper, the operation cost of a trip sequence plan includes the two categories. cost of train units, the cost of level I maintenance and the travel In the first category, the researchers focus on the allocation cost. We aim to find a trip sequence plan which leads to the mini- and composition of train units on each trip, under the condition mal operation cost. Note that in different countries, the regulations that the predecessor and successor trips of each trip are given on high-speed railway management and train unit maintenance in advance. Abbink et al. [1] presented an integer programming vary. We propose the trip sequence models on the background of model to allocate train units to different trips, which minimized the Chinese high-speed railway, the length of which occupies more the seat shortages during the morning peak hours. The model was than two thirds of the total length of the high-speed railway in solved by CPLEX directly, and tested on several scenarios based the world. In other countries with similar maintenance regulations, on the 20 01–20 02 timetable of NS Reizigers. Alfieri et al. [2] fo- such as Japan and Korea, similar models can be constructed. cused on optimizing the numbers of train units of different types In order to express the movement and coupling/splitting of together with their efficient circulation on a railway corridor. The high-speed train units, we develop a trip sequence graph in this (un)coupling of train units was taken into account, which largely paper. This graph differs from other graph-based models in the lit- increased the complexity of the problem. A solution approach erature on train unit scheduling or aircraft routing problems, since based on an integer multi-commodity flow model was proposed, they either ignored the difference of shunting times between dif- and applied to a real-life case of NS Reizigers. Peeters and Kroon ferent platforms, or were not able to express coupling/splitting [26] described a model to determine an optimal daily allocation times. In a busy high-speed railway network, in which the train and composition of train units on a railway corridor of the Nether- units circulate rapidly, shunting times and coupling/splitting times lands, in which the solution was evaluated based on service qual- significantly affect the construction of a trip sequence. In the trip ity, robustness and circulation cost. The changes in train composi- sequence graph proposed in this paper, shunting times and cou- tion at origin and destination stations were taken into account, and pling/splitting times are be explicitly considered. a branch-and-price algorithm was developed to solve the model, Based on the trip sequence graph, two ILP models, namely a which outperformed the method of solving an integer program- path-based model and an arc-based model, are formulated for the ming model directly by commercial solvers in a e real-life case trip sequence planning problem. If the problem is of medium size, study in the Netherlands. Fioole et al. [15] extended the model the arc-based model can be solved to optimality by an ILP solver in [26] to a railway line with branches. A complex mixed inte- within acceptable time. If the problem is of large size, only the ger linear programming model was constructed, and was solved by path-based model can be used and a customized branch-and-price CPLEX. Taking the maintenance rules of NS Reizigers in the Nether- algorithm is proposed to solve the model.
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