Skip to content
BY 4.0 license Open Access Published by De Gruyter Open Access January 31, 2020

Mathematical approaches for improving the efficiency of railway transport

Martin Vojtek EMAIL logo , Martin Kendra , Vladislav Zitrický and Jaromír Široký
From the journal Open Engineering


Current business trends push transport companies to improve the efficiency of their entrepreneurial activities, what is closely related to costs reduction and revenues increase. Many transport companies achieve it empirically, but some mathematical algorithms, mainly from operational research, can be used to improve the efficiency of railway transport by finding the optimal solution of specific problems. The paper is focused on using of selected methods from applied mathematics in railway transport. These methods are used mainly for improving the operational efficiency of the transport company, what tends to decrease overall operating costs. Major impact is done by efficient use of railway vehicles and train staff. It can be achieved by optimization as a flow network problem or assignment problem, what is further described in the paper. Mathematical approaches can also positively influence the whole logistic process.

1 Introduction

Rapid evolution of production and expansion of economic activities mean that managerial decision-making is more complex and responsible. Main reasons are:

  • more possibilities to deal with the situation,

  • increasing division of work

  • deeper division of work also on strategic level

  • wider division of work causes that consequences from incorrect decisions are more noticeable [1, 2]

Systematic management of all activities in the company is currently associated with the use of modern information technologies related to well-established information system. Creating a suitable information system requires quantification of economic problems, what relates to the use of mathematical and statistical models and methods. The result of all activities in the field of systematic approaches and especially quantitative approaches for development and application of methods designed to support managerial decisions is a new scientific discipline known as operational research or operational analysis [3].

Mathematical models used to describe economic processes can be divided according to different criteria:

  • coincidence

  • deterministic

  • stochastic

  • time factor

    1. static

    2. dynamic

  • usage of mathematical medium

    1. linear

    2. nonlinear

Operational research can be characterized as a scientific discipline that focus on the analysis of several types of decision-making problems [4].

Mathematical approaches from operational research have these advantages:

  • it is possible to find out necessary information when there is not any conclusion directly from surveyed object or in case that it does not already exist;

  • accelerate the decision-making process by shortening all processes that currently take a long time;

  • facilitate and rationalize management by making the model clear, concise and it provides logically correct descriptions of reality;

  • • allow variant solutions with possibility to select the most suitable one according to various circumstances;

  • to prevent possible losses due to incorrect decisions [5].

Application of mathematical approaches in the field of transport is described below.

2 Current research on this field of transport

There are not many similar researches focusing on issues that deal with mathematical approaches which are applied to railway transport. Some studies are done with problems of the bus transport within the town. Topics of these researches are vehicle schedule and also bus-driver schedule. The solution based on applied mathematical approaches of bus-driver schedule is a primary research of Stanislav Paluch’s (University of Zilina). The author describes some mathematical algorithms that are used to optimize of bus-driver schedule and gives some model examples. Some basic applications of flow-network problems are the content of his scientific and educational publications [6, 7, 8].

Other author Jaroslav Kleprlik (University of Pardubice) deals with schedules in general. In his publications, there he suggests using of the Hungarian method as a solution of assignment problem and its practical use in transport optimization. These mathematical approaches that were mentioned above are the content of the paper and its application in railway transport is also there [9].

Some authors, for example Matus Dlugos, Peter Zuber, Rudolf Dzurus, Jozef Staffen, Jozef Bujna (all from University of Zilina) and Robert Geci (University of Presov) deal with similar issues in their bachelor or diploma thesis [10, 11, 12, 13, 14, 15].

3 Background of railway transport operation from mathematical point of view

Operational part of the management of railway transport consists of train timetable, vehicle circulation, run of train staff and operating-economic evaluation. Technical things such as maintenance or infrastructure are not direct part of it, therefore the main emphasis is put on operational issues which are mentioned above. Train timetable is the basic schedule of train operation, which is also available for the traveling public [16, 17].

Vehicle circulation and the run of train staff are directly dependent on train timetable so there is an enormous potential to optimise the operation. Selected mathematical approaches, which are suitable for this kind of optimisation, are the major content of this paper. The run of train requisites can be described and explained separately. The run means regularly recurring schedule of usage time. Train requisites are subjects or objects, which are necessary to join into one set, which can move independently in the transport process. Simply, it is the train, which may consist of locomotives, different types of wagons, vehicledrivers, conductors, stewards etc. [18, 19].

Basic elements in the transport process are transport hubs, where train requisites can switch from one train into another. In transport hubs, there are also flows of passengers in passenger transport or material flows in freight transport. Directional connection of two transport hubs is called the section. Sequence of transport hubs and sections between these transport hubs is called the route [20].

Almost everything what is related to optimization of the run of train requisites could be described by a graph. The graph is a mathematical object, which consists of ordered pair of sets G = (V, E). First set consists of vertices and other set consists of edges. Basic precondition is that each edge connects two vertices. In case of transport process, vertices might represent trains and edges might represent possible switchings from one train to another. The run of train requisites is specific, because train requisites can switch from one train to another only in one direction therefore the graph must be oriented. Vertices and edges represent values of some attributes very often. In transport process, switching of train requisites might be represented by edges. For example: vehicle-drivers, conductors, stewards etc. can switch from one train to another therefore each edge must have information about duration or cost of that switching. It means that the graph is ordered triplet of sets G = (V, E, d), where d is a function d: H→ R+ which assign the nonnegative real number to each edge and it represents the duration or costs of the switching. These switchings and their values may represent not only duration or costs, because there can be considered many different quantities. The most important things by choosing values is a way of solution, what means that the result should be minimum (duration, costs...) or maximum (efficiency, usage...). In this paper, there is the method for seeking the minimum [21].

Switching of train requisites within each operating day can be solved by flow network. Basic unit in regular railway passenger transport is a train ti. Every train is marked by unique number and its complete characteristics are stated in the train timetable, where are all stations and stops of this train and its temporal position. Majority of trains are short from temporal point of view therefore one train cannot fulfil day-long operation of the railway vehicle or train staff members. This is the reason, why each railway vehicle or train staff member have got the sequence of more trains per one operating day. Train ti foregoes train tj in case, when there is on identical place (station or stop) enough time to tj departure after ti arrival. The difference between tj departure and ti arrival is a temporal dissipation and it is marked as dij with its value in minutes. There is a set of trains T = {t1, t2, ..., tn} and the main task is to schedule these trains into operating days. Every train would be just in one operating day. Number of operating days should be as small as possible because number of railway vehicles and train staff members directly depends on number of operating days. In case of train staff members, higher legal standards, mostly labour Code, must be also considered. It specifies maximum working time, duration of breaks and many other associated criteria. This task can be solved by graph G with set of vertices V1∪ V2 {s, m}, where V1 is the set of all train arrivals from the set T, V2 is the set of all train departures from the set T, s is the fictitious source and m is the fictitious mouth. Symbols vi and wj are signs for vertices of V1 and V2 sets, which represent train ti and tj. Set of oriented edges E in the graph G consists of three types of edges. First type includes oriented edges (s, vi) where vi ∈ V1. Second type includes oriented edges (wj, m) where wj ∈ V2. Third type includes oriented edges (vi, wj) in case when ti foregoes tj with d(vi, wj) [6].

4 Case study of optimization

Finding the optimal solution of this problem by flow network is applied on chosen long-haul railway route represented by shown timetable below. This route was chosen for its relevance as closed system, where vehicles from these trains are used only on this route.

The easiest solution is to use the same number of railway vehicles and train staff as number of trains in the timetable. It is possible to reduce the number of railway vehicles and train staff, when some edge (vi, wj) of the graph G is used. It means that the railway vehicle and train staff members switch from the train ti to the train tj. Every other used edge means saving of another railway vehicle and train staff members. Among used edges, there must not be any edges with the same vertex. It is an assignment of train departures to train arrivals and as many as possible train departures should have assigned some train arrival. This can be done by searching for the maximum flow in the network. In case of train staff, maximum work time and minimum break time must be respected [22].

Maximum flow in Figure 1 means that there are not any other alternative ways where the flow could be put. Circulation of train staff members may be linked to vehicle circulation, but final number of train staff members should equal or higher than number of vehicles. When the optimal solution of switchings is known, the sequence of these switchings must be done. In the example of the route Kosice – Zvolen, basic temporal period for these trains in the timetable is 24 hours. It means, that each train runs every 24 hours (everyday) repeatedly. Solution for operating days (one operating day = one 24 hours period) can be showed as the permutation P (i → j) what shows switchings of railway vehicles from one train to another.

Figure 1 a) digraph for the route b) graph G with maximum flow. All edges in G have got capacity = 1 Resultant operating days: D1: 930 →933 →936; D2: 931 →934 →937; D3: 932 →935
Figure 1

a) digraph for the route b) graph G with maximum flow. All edges in G have got capacity = 1 Resultant operating days: D1: 930 933 936; D2: 931 934 937; D3: 932 935

Example of permutation


Basic precondition is that every train in the timetable is implemented only once during the temporal period. Stated example of the permutation must be completed with switchings between temporal periods (operating days). This can be solved similarly as solution for one operating day. Then it is necessary to change the permutation P into permutation P*. This adapted permutation is showed in oriented graph (Figure 2), where vertices represent trains and oriented edges represent switchings from one train to another.

Figure 2 Graphical representation of permutation P* (source: authors).
Figure 2

Graphical representation of permutation P* (source: authors).

Example of permutation


Required number of railway vehicles is equal to the number of temporal periods of train switchings. The result can be described in the Table 2.

Table 1

Train timetable on the route Kosice – Zvolen (source: authors).

Table 2

Number of railway vehicles and their schedule (source: authors)

1930 933 936931 934 937932 935
2931 934 937932 935930 933 936
3932 935930 933 936931 934 937

Final solution of this type of assignment problem, for example by “Hungarian method” based on the theorem of Egervary and Konig, is going to be found in the matrix D = (dij)i,j=1,2,...,n of n elements. Each two of these elements are not located in the same column not even in the same row. Sum of the values that represent elements must be as minimal as possible [9].

Assignment problem is defined as the problem that deals with the optimal assignment of elements from the set, which consists of n elements, to the same number of elements from other set. It is assumed that all right-side coefficients values ai, i = 1, 2, . . . , n and bj, j = 1, 2, . . . , n is equal 1. Matrix X consists of variables xij, which value is equal 1 in case, when there is the j-elements assigned to i-element. When it is not true, xij is equal 0. Coefficients dij could be presented as temporal dissipations. Assignment problem can be defined this way:


Practical usage of this formula is shown in the Table 4 below. Appropriate optimization of train requisites schedule in railway passenger transport is an important task of transport technology. Minimizing required number of train requisites and provide maximum efficiency of their usage are main goals [23].

Table 3

Time differences between departures and arrivals of trains.

Table 4

Solution of assignments on the route Kosice – Zvolen.


Many attributes have significant influence to solution of this task. In the Table 3, there are time differences between departures and arrivals of every train on the route Kosice – Zvolen. Each train is signed with its number. Values of time differences are written only for those trains where the arrival station of one train is the same as the departure station of other train.Values of time differences are converted and written in minutes according to formula 2.


where: Td = Time difference converted to minutes [min]

Dij = Departure of the train marked by ij [hh:mm]

Aij = Arrival of the train marked by ij [hh:mm]

In the Table 4, assigned departure of one train to the arrival of the other train is shown with value 1 in the relevant cell of the table. Every value 0 mean that there is not any assignment between those trains. All values in the Table 4 represent the optimal solution of assignment problem. These values are results of calculation according to formula 1 and there are many approaches and algorithms for doing it. In case of many inputs (trains), using of some software is more relevant than manual calculation from temporal point of view. There are many mathematical programs, which can be useful and applicable for this issue. In this paper, there was used Solver with restrictive conditions according to formula 1. Other restrictive condition is that all values must be binary. This issue is linear problem, therefore Simplex LP algorithm is chosen method to solve it.

The result of assignment problem solution is the same as result of flow network. Assignment problem is better for cases where there are many trains. Flow network solution is friendlier for these users, who prefer graphical approaches and outputs.

5 Conclusions

Mathematical approaches are very suitable for optimization of all problems that are connected to railway transport operation. In the paper, optimization of the run of train requisites in passenger transport is shown. Two possible approaches are used there. Firstly, this problem is solved on the flow network as a simple task of finding the maximum flow. Secondly, optimization task is transformed to assignment problem and the optimal assigns represent also the optimal solution of the whole task. It is necessary to pay more attention to using these and other mathematical methods in a field of transport, because they can make the railway transport operation more efficient. Every software, which can solve these mathematical tasks fast, can be used and applied to field of railway transport. Its application could help in all aspect of transport business either in operational or strategic level of decision-making and the whole managerial issues. Synergic effects are directly influenced by efficiency of train operation. The most relevant impact is that costs would decrease by efficient using of train requisites mainly railway vehicles and train staff members such as vehicle-drivers, conductors and stewards. Mathematical solution is better than empirical solution, because there do not have to be any person who make working schedules since everything is done automatically. There is also the possibility to find many alternative solutions is case that some circumstances are changed. In case when the train is delayed, the software, which use mathematical methods, can easily calculate and remake the plan of working schedule. Then the vehicle from delayed train could be used on another train than it was previously planned. Current trends are very supportive for using automatic procedures and techniques therefore these issues are very actual. They can provide new benefits and different point of views in the field of railway transport.

Conclusions should be summarized:

  1. Operating processes are more efficient and less time-consuming,

  2. Decision-making on operational and strategic level is more flexible,

  3. Costs reduction mostly in direct costs influenced by operation efficiency,

  4. People could focus on creative work because mechanic work is done by software.


The paper is supported by the VEGA Agency by the Project 1/0791/18 “The Assessment of Economic and Technological Aspects in the Provision of Competitive Public Transport Services in Integrated Transport Systems” that is solved at Faculty of Operation and Economics of Transport and Communication, University of Zilina.


[1] Benesova M. Operační analýza: bakalárska práca /Operations research: bachelor thesis/. Olomouc: Palacký University. 2008. 52 p.Search in Google Scholar

[2] Houda M. Operační analýza /Operations research/. Ceské Budejovice: University of South Bohemia. 2012. 156 p.Search in Google Scholar

[3] Brezina I, Ivanicova Z, Pekar J. Operačná analýza /Operations research/. Bratislava: Iura Edition. 2007. 189 p.Search in Google Scholar

[4] Jablonsky J. Operační výzkum – kvantitativní modely pro ekonomické rozhodování /Operations research – quantitative models for economic decision-making/. Prague: Professional Publishing. 2007. 234 p.Search in Google Scholar

[5] Brozova H, Houska M. Základní metody operační analýzy /Basic methods of operations research/. Prague: Czech University of Life Sciences. 2008. 120 p.Search in Google Scholar

[6] Paluch S. Algoritmická teória grafov /Algorithmic graph theory/. Zilina: EDIS. 2008. 198 p.Search in Google Scholar

[7] Paluch S, Majer T. Kastor - A vehicle and crew scheduling system for regular bus passenger transport. Transport Problems: An International Scientific Journal. 2017;12(1): 103-11010.20858/tp.2017.12.1.10Search in Google Scholar

[8] Paluch S,Majer T. The role of independent sets in decomposition of the vehicle scheduling problem, 19th International Carpathian Control Conference (ICCC). 2018;19(1):619-62410.1109/CarpathianCC.2018.8399704Search in Google Scholar

[9] Kleprlik J. Tvorba turnusů náležitostí /Run of train requisites creation/. Perner’s Contacts [Internet]. 2007 Mar [cited 2019 Oct 10];5(1):48-51. Available from: in Google Scholar

[10] Dlugos M. Optimalizácia obehu vlakových súprav RegioJet, a.s. na trati Komárno – Dunajská Streda – Bratislava diplomová práca /Optimization of railway vehicle circulation for RegioJet on the route Bratislava – Dunajska Streda – Komarno: diploma thesis/. Zilina: University of Zilina. 2017. 66 p.Search in Google Scholar

[11] Zuber P. Interaktívny system na podporu tvorby a úpravy turnusov vozidiel MHD: diplomová práca /Interactive system for support to create and edit the run of vehicles in city public transport: diploma thesis/. Zilina: University of Zilina. 2013. 49 p.Search in Google Scholar

[12] Dzurus R. Tvorba plánu práce vodičov prímestskej autobusovej dopravy v spoločnosti SAD Prešov a. s. a možnosti jeho optimalizácie: bakalárska práca /Creation of working plan for drivers in suburban bus transport for the company SAD Presov a.s. and possibilities for its optimization: bachelor thesis/. Zilina: University of Zilina. 2017. 42 p.Search in Google Scholar

[13] Staffen J. Plánovanie zachádzok do garáže v turnusoch mestskej hromadnej dopravy: diplomová práca /Planning of using garage in the run for city public transport: diploma thesis/. Zilina: University of Zilina. 2013. 55 p.Search in Google Scholar

[14] Bujna J. Návrh zásad optimalizácie turnusov vlakového personálu: diplomová práca /Proposal for principles of run of train staff optimization: diploma thesis/. Zilina: University of Zilina. 2013. 39 p.Search in Google Scholar

[15] Geci R. Optimalizácia priraďovania vodičov na linky vo vybranom dopravnom podniku: diplomová práca /Optimization of driver assignments on lines in selected company: diploma thesis/. Presov: University of Presov. 2014. 54 p.Search in Google Scholar

[16] Rybicka I, Drozdziel P, Stopka O, Luptak V. Methodology to propose a regional transport organization within specific integrated transport system: A case study. Transport Problems. 2018;13(1):115-12510.20858/tp.2018.13.4.11Search in Google Scholar

[17] Luptak V., Bartuska L., Hanzl J., Assessment of connection quality on transport networks applying the empirical models in traffic planning: A case study. Transport Means – Proceedings of the International Conference. 2018;22(1):236-239.Search in Google Scholar

[18] Kampf R, Stopka O, Kubasakova I, Zitricky V. Macroeconomic Evaluation of Projects Regarding the Traffic Constructions and Equipment. Procedia Engineering. 2016;1(1):13-17.10.1016/j.proeng.2016.08.623Search in Google Scholar

[19] Kampf R, Lorincová S, Hitka M, Stopka O. Generational Differences in the Perception of Corporate Culture in European Transport Enterprises. Sustainability. 2016;9(9):55-61.10.3390/su9091561Search in Google Scholar

[20] Simkova I, Konecny V, Liscák S, Stopka O. Measuring the quality impacts on the performance in transport company. Transport Problems. 2015;10(1):113-124.10.21307/tp-2015-039Search in Google Scholar

[21] Tuzar A. Teorie dopravy /Theory of Transport/. Prague: Czech Technical University. 1997. 239 p.Search in Google Scholar

[22] Gasparik J, Luptak V, Kurenkov PV, Mesko P. Methodology for assessing transport connections on the integrated transport network. Communications – Scientific Letters of the University of Zilina. 2017;19(2):61-67.10.26552/com.C.2017.2.61-67Search in Google Scholar

[23] Luptak V, Drozdziel P, Stopka O, Stopkova M, Rybicka I. Approach methodology for comprehensive assessing the public passenger transport timetable performances at a regional scale. Sustainability. 2017;11(13):45-52.Search in Google Scholar

Received: 2019-10-30
Accepted: 2019-12-18
Published Online: 2020-01-31

© 2020 M. Vojtek et al., published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

Downloaded on 6.12.2022 from
Scroll Up Arrow