Hindawi Mathematical Problems in Engineering Volume 2021, Article ID 3876561, 17 pages https://doi.org/10.1155/2021/3876561 Research Article Time Coordination of Periodic Passenger Train Connections in Conditions of Single-Track Lines Filip Lorenz,1 Vit Janos,2 Dusan Teichmann,2,3 and Michal Dorda 3 1Department of Control Systems and Instrumentation, Faculty of Mechanical Engineering, VSB–Technical University of Ostrava, 17. listopadu 2172/15, 708 00 Ostrava-Poruba, Czech Republic 2Department of Logistics and Management of Transport, Faculty of Transportation Sciences, CTU in Prague, Konviktska 20, 110 00 Prague, Czech Republic 3Institute of Transport, Faculty of Mechanical Engineering, VSB–Technical University of Ostrava, 17. listopadu 2172/15, 708 00 Ostrava-Poruba, Czech Republic Correspondence should be addressed to Michal Dorda; [email protected] Received 30 June 2020; Revised 12 January 2021; Accepted 6 February 2021; Published 9 March 2021 Academic Editor: Luca D'Acierno Copyright © 2021 Filip Lorenz et al. )is 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. )e article addresses creation of a mathematical model for a real problem regarding time coordination of periodic train connections operated on single-track lines. )e individual train connections are dispatched with a predefined tact, and their arrivals at and departures to predefined railway stations (transfer nodes) need to be coordinated one another. In addition, because the train connections are operated on single-track lines, trains that pass each other in a predefined railway stations must be also coordinated. To optimize the process, mathematical programming methods are used. )e presented article includes a math- ematical model of the given task, and the proposed model is tested with real data. )e calculation experiments were implemented using optimization software Xpress-IVE. 1. Introduction: Motivation to Solve the As it is clear from the above-stated formula, one of the Given Problem possibilities of how to shorten the total transport time is to minimize the waiting time for connections at the transfer )e main objective of the transport processes in passenger nodes. transportation is to make them as short as possible. We need )e motivation for solving the problem is to find an to start with an assumption than the moving time, for ex- approach that would allow for a maximal transportation ample, [1], is usually unproductive time. )at is why we fluency by shortening the waiting times at the transfer nodes. insist on making it as short as possible. )is formulation is Time coordination of the transport processes is then expressed by the following formula: based on the principle of proposing such time positions of the given connections that will allow for minimizing time losses of the passengers from the moment they board the first T � t1 + tw + tt + ttr + t2; (1) train on their transport route until the moment they get off the last connection on their way. where T is the total transport time (min), t1 is the walking time from the moving source to the given train stop (min), tw is the waiting time for the train (min), tt is the time spent 2. Analysis of the Current State of the Art on the train (transport time) (min), ttr is the transfer time (including waiting time at the transfer nodes) (min), and t2 is A thorough summary of coordination optimization models the walking time from the train stop to the given moving is provided in [2]. )is relatively extensive work deals with destination (min). multimodal transport networks, individual connections 2 Mathematical Problems in Engineering (lines) of which meet in transfer terminals, which means that accumulation of the transport means at the coordination passengers transfer there from one line to another. )e nodes. In the first part of their study, the authors developed a author claims that the main motivation for solving the issue dynamic management strategy, which is based on the given is the necessity to make public transport more attractive for management theory and thanks to an optimal management passengers by making the transfers among individual strategy which was determined (delays of individual con- connections more efficient. )e author approaches the topic nections at the coordinated nodes). A maximal delay is thus in two ways: determined by the estimates of arrivals of individual con- nections and passengers in real time and by uncertainty of (i) Connection coordination by timetable modifications these estimates. (Schedule Coordination) Gkiotsalitis et al. [6] explore a robust coordination of (ii) Connection coordination in real time (Connection connection times, including public transport, for the pur- Protection) pose of the given network coordination. )e authors focus )e work includes an optimization model for generating on analyzing the robustness of the synchronization results in timetables and minimizing transfer times among individual two limit situations. )e first limit situation occurs under connections. )e objective of the model is to find an optimal standard operation conditions and the second limit situation timetable by shifting existing connections to a new time under extreme operation conditions (extreme operation position and/or by delaying a given connection, i.e., waiting conditions are represented by significant disruptions of the for a subsequent connection, thus minimizing the overall planned timetable). )e optimization approach proposed in delay of both connections. )e model considers waiting time their article is based on evolution algorithms. )e authors of the transferring passengers as well as delay times of the execute the corresponding calculation experiments using people who use one of the two connections, but who do not real data obtained from the bus transport system in the city transfer. )e model has been validated by case studies and of Hague, Netherlands. the positive results are demonstrated by concrete time )e objective of [7] was to identify individual bus savings. Furthermore, the author concludes that even if a connections and their mutual relations from the perspective timetable is modified based on the corresponding mathe- of their competitiveness, complementarity, or indepen- matical model output, it is still necessary to pay attention to dency. Next, the authors decided about time changes of the stochastic character of the transport, i.e., occurrence of individual bus connections, which were servicing a given extraordinary situations and delays, which disrupt the unified area based on a calculation of the travel cost of the transfer relations. )at is why the work also included de- passengers, walking time, and waiting time related to each velopment of a CP model (Connection Protection). )e individual bus route. Furthermore, the authors explored objective function of the model considers probability of the passenger behavior when they choose suitable bus con- implementation of the transfer relations and optimizes nections, and they incorporate the given survey results in expenses. )is model is evaluated and compared with their solution. previous models with the objective to demonstrate it as a tool Omar and Yasmin [8] focus on the creation of timetables for improving the efficiency of the transfer relations and for of the bus lines operated by several private bus companies in reducing waiting times of the passengers. the city of Monterrey, Mexico. )e following are the ob- Ciaffi et al. [3] focus on transfer relations between a jectives of the study: secondary public transport network in the cities (primarily (1) Preventing cumulation of bus connections of various bus transport) and the main transport network (metro and bus lines in individual transfer nodes railway). )e objective is to develop a procedure that si- (2) Increasing potential transfer links multaneously generates routes and frequencies (time in- tervals) in the bus transport network, making sure it is Based on the above-stated objectives, the authors de- applicable for solving problems of a size that corresponds to veloped a suitable model, which is able to fulfil the defined the extent of real cities. objectives. Sun et al. [4] address classification of bus routes and their )e study carried out by Almasi et al. [9] looks into the relations to the main transit (railway) transport. )e ob- timetable proposal of individual bus connections and their jective of the proposed solution is to maximize the use of the intensity, with the objective to conveniently interlink them transit (railway) routes by passengers and to optimize the with the railway transit system. )e authors created an existing bus lines from the perspective of their contribution improved metaheuristic approach based on the Imperialist to the given transit (railway) route. )e article also presents Competitive Algorithm and Water Cycle Algorithm. )e an algorithm that is used for defining the relation between optimization criterion here is the cost function, which in- bus lines and main transit routes. cludes expenses of the networks users, expenses related to Authors of study [5] address coordination of the transfer the operation of technical equipment, and also social ex- relations among individual transit routes in real time. )ey penses, such as the cost of the network operation and cost realize the advantages of coordinated transfers, i.e., short- related to the pollution caused by individual vehicles. )e ening of the passengers’ waiting time, while also considering developed metaheuristic approach was tested on an actual the risks of delays of the passengers who are on the given network of the city of Petaling Jaya in Kuala Lumpur, line, passengers who intend to travel on that line, and Malaysia. Mathematical Problems in Engineering 3 Wong et al. [10] present a proposal of a coordination of Authors of [17] explore maximization of the number of public transport timetables, which allow for smooth the current connection arrivals to a transfer hub, with the transfers with minimal delays for all passengers based on objective to minimize transfer waiting times and thus also mixed-integer programming.