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Optimizing Minimum Headway Time and Its Corresponding Train Timetable for a Line on a Sparse Railway Network

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Optimizing Minimum Headway Time and Its Corresponding Train Timetable for a Line on a Sparse Railway Network S S symmetry Article Optimizing Minimum Headway Time and Its Corresponding Train Timetable for a Line on a Sparse Railway Network Weining Hao 1 , Lingyun Meng 2 and Yuyan Tan 2,* 1 State Key Laboratory of Rail Traffic Control and Safety, Beijing Jiaotong University, Beijing 100044, China; [email protected] 2 School of Traffic and Transportation, Beijing Jiaotong University, Beijing 100044, China; [email protected] * Correspondence: [email protected] Received: 29 June 2020; Accepted: 23 July 2020; Published: 26 July 2020 Abstract: In western China, railway lines are sparse, and there is a large fluctuation in transport demand under different transportation scenarios, which yields severe difficulties in setting up a signaling system and making a train timetable. To meet the fluctuating transport demand effectively and provide efficient train services in the changing multimodal transportation market, a new Chinese train control system based on flexible minimum headway time (FCTCS) is introduced and going to be implemented. Considering that the cost of implementing signaling systems corresponding to different minimum headway times varies significantly, it is necessary to find an optimal minimum headway time and design its corresponding train timetable. In this paper, we propose a mixed integer linear programming model for selecting an optimal minimum headway time, with which a satisfactory train timetable is generated with the consideration of symmetry transport demand. The objective function is to maximize the total profit of train operation. We further develop genetic algorithm with an integer and binary coding method for searching for the solution. Finally, a set of numerical tests based on a railway line in a sparse railway network in western China is used to demonstrate the validity and effectiveness of the proposed model. Keywords: optimal minimum headway time; train timetabling; transport demand; genetic algorithm 1. Introduction Compared with the complex railway network in eastern China, the railway lines in western China are sparse. In a normal transportation scenario, the transport demand in western area is generally low; however, during some special periods, the transport demand increases sharply, which brings a great challenge for train operation on these sparse lines. When higher demand occurs, the provided signaling resource may not adequately meet the given demand. Although we can operate trains with a longer headway—compared with the minimum one based on the existing signaling system, where the minimum headway time remains the same—it will also lead to a waste of signaling resources as implementing excessive signaling resources generates costs, especially in such areas with fluctuating demand and not so good operational conditions. Therefore, a new Chinese train control system based on flexible minimum headway time (FCTCS) is introduced and going to be implemented to attempt to meet the fluctuating transport demand for sparse railway lines. Using the new signaling system, it is more flexible to operate trains corresponding to the generated demand and makes it possible to save more operational costs, due to the flexible minimum headway. To explain clearly about the flexible minimum headway in FCTCS, we illustrate it in Figure1. Symmetry 2020, 12, 1223; doi:10.3390/sym12081223 www.mdpi.com/journal/symmetry Symmetry 2020, 12, 1223 2 of 23 Symmetry 2020, 12, 1223 2 of 24 meet demand Transport demand fluctuation Minimum FCTCS Train timetable adjustment headway time t input t=t±1 Flexible minimum headway time Figure 1. Explanation of flexible flexible minimum headway time. Based on the new proposed FCTCS, train timetablingtimetabling is therefore required toto optimize.optimize. In view of the characteristic of of flexible flexible minimum minimum headway headway in in FCTCS, FCTCS, we we consider consider the the corresponding corresponding cost cost of ofimplementing implementing signaling signaling systems systems in train in train timetabling, timetabling, and andmake make efforts eff ortsto maximize to maximize the profit the profit of train of trainoperation. operation. In addition, In addition, transport transport demand demand is rega is regardedrded as asan an essential essential element element in in solving solving a a train timetabling problem.problem. Generally, Generally, after after a demand a demand analysis, analysis, the train the timetable train timeta is constructedble is constructed to determine to thedetermine departure the anddeparture arrival and time arrival of each time train of ateach each train station, at each which station, is a which sequential is a sequential process. However, process. inHowever, many real-world in many cases,real-world the transport cases, the demand transp fluctuatesort demand randomly, fluctuates and randomly, the obtained and train the scheduleobtained intrain the schedule latter step in mightthe latter not step satisfy might the not demand satisfy in the such demand a sequential in such process.a sequential As aprocess. result, embeddingAs a result, transportembedding demand transport into demand the train into timetable the train phase timetabl has graduallye phase has attracted gradually more attracted attention. more attention. In thisthis paper,paper, basedbased onon flexibleflexible minimumminimum headwayheadway in FCTCS,FCTCS, wewe focusfocus onon findingfinding anan optimaloptimal minimum headway time with which a satisfactory train timetable is designed to well satisfy the fluctuatingfluctuating transport demand.demand. Although Although the the prob problemlem of of optimizing train timetables was often investigated, the optimization of minimum headway time to make the traintrain timetabletimetable designdesign moremore adaptive to the fluctuatingfluctuating transport demand was paid less attention, and can still be improved. Here, Here, we give a method of of how how to to select select the the optimal optimal mi minimumnimum headway headway in intrain train timetabling, timetabling, which which can can be beused used to tosupport support the the planning planning decisions. decisions. Furthe Furthermore,rmore, innovated innovated by by the the flexibility flexibility of minimumminimum headway time, the train timetable can still be well-designed with the cost of implementing signaling system corresponding to different different minimum headways considered in the objective function, because, for traintrain operators, operators, the the greatest greatest concern concern is how is how to maximize to maximize the profit the of profit train operationof train operation after satisfying after demand.satisfying The demand. cost of The implementing cost of implementing signaling system signaling has system a great impacthas a great on train impact operational on train operational costs. Thus, acosts. methodology Thus, a methodology is proposed tois selectproposed an optimal to select minimum an optimal headway minimum with head the costway ofwith implementing the cost of signalingimplementing system signaling corresponding system tocorresponding different minimum to different headways minimum and is headways considered and in is this considered study to fill in inthis the study gap ofto optimizingfill in the gap minimizing of optimizing headway minimizing time in headway train timetabling. time in train timetabling. To solvesolve thisthis problem,problem, aa mixedmixed integerinteger linearlinear programmingprogramming modelmodel isis formulated,formulated, in whichwhich the transport demanddemand corresponding corresponding to to di ffdifferenterent origin origin and and destination destination stations, stations, train train stop stop planning, planning, train routingtrain routing and train and timetablingtrain timetabling are included are included in the modelin the formulation.model formulation. In this trainIn this timetabling train timetabling process, firstprocess, of all, first the of transport all, the demandtransport and demand minimum and headwayminimum time headway are regarded time are as inputsregarded into as the inputs proposed into trainthe proposed timetabling train model, timetabling to obtain model, an initial to obtain train schedulean initial withtrain an schedule objective with function an objective of maximizing function the of profitmaximizing of rail transportationthe profit of revenuerail transportation and train routing revenue cost. and Then, train we routing vary the cost. minimum Then, headwaywe vary time the tminimumto evaluate headway the initial time train t to timetable evaluate by the maximizing initial train the timetable total operational by maximizing profit ofthe rail total transportation operational revenue,profit of rail train transportation routing cost revenue, and the train cost ofrouting implementing cost and the signaling cost of impl system.ementing Finally, signaling we obtain system. the optimalFinally, we minimum obtain the headway optimal time minimum and its headway corresponding time and train its timetable. corresponding The wholetrain timetable. optimization The processwhole optimization in this paper process is shown in inthis Figure paper2. is shown in Figure 2. The remainder of this paper is organized as follows. Section2 provides a literature review on three areas: train timetabling, train timetabling considering transport demand, as well as optimizing headway for train timetable. In Section3, conditions of the model are given, followed by a mathematical model that formulates the optimization problem. Next, a genetic algorithm is introduced
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