Assessment of Polish Railway Infrastructure and the Use of Artificial Intelligence Methods for Prediction of Its Further Development

Abstract

The aim of this chapter is to analyze economic trends in rail freight volume in Poland, based on the analysis of data from the years 2009–2013, for evaluating decisions on planned investments in railway infrastructure envisioned by Poland and the EU at the time the EU was founded. The theoretical analysis presents a trend of functional Polish railways and its impact on investment decisions. In addition, it shows the long-term plans for railway transport in Poland from both the Polish government and the EU perspectives. An analysis of the current investment to support the development of railways in Poland is also elaborated. The research part of the chapter presents an analysis of statistical data on rail freight. Forecasts are precisely presented of selected transport parameters made by the Bayesian network method and Holt-Winters double exponential smoothing using an artificial immune system to determine parameters and initial conditions.

1 Introduction

A diagnosis of the current state of rail transport in Poland and SWOT [strengths, weaknesses, opportunities, threats] analysis shows that the most significant factor hampering the development of rail transport in Poland is a pronounced deterioration of infrastructure. The most obvious symptom of this degradation is the low maximum speeds on a substantial part of the railway network. These speeds in many cases are smaller than the designed speeds [36] which formerly existed on the individual sections of the line. As a result, speed limits have been introduced more and more often. Reduced maximum speed imposes limitations such that the travel times are being extended considerably in relation to the shorter times achieved in each section in the past. This, in turn, results in lower quality of service regarding transport timeliness.

A systematic decline in the length of railroads, the development of which in the past was not always rationally planned, remains a constant and very unfavorable trend. In 1990, 24.1 thousand km of railway lines were in use, and now only 19.3 thousand km, a decrease of 20%. The last railway line was built in 1987, which, taking into account changes in the technical and technological changes in the global railway industry, puts Poland among the countries with the most neglected and outdated railway infrastructure.

The poor condition of the railway infrastructure is caused by a lack of sufficient financial resources allocated for repair, modernization and maintenance. To counter this trend, an increasing number of projects are being directed to modernization of existing linear and point infrastructure and to infrastructure expansion.

The research goal of this chapter is to verify the rationality and correctness of expenditures on railway infrastructure in relation to future trends in the field of rail freight. The chapter presents an outline of the degradation of the railway infrastructure, as well as recommendations for the appropriate sources and amounts of funding from EU funds for modernization of the railway infrastructure. The rationale for the decisions is verified by the forecast of basic data relating to the quantities of freight carried by rail. Forecasts were made using two methods: exponential smoothing using an artificial immune system for determining the model parameters (modified Holt method), as well as the initial conditions, and by using Bayesian networks method.

2 Prospective Plans for the Railway Transport in Poland

To address unfavorable factors causing the deterioration of the infrastructure, dither Polish State Railways must be transformed or the state-owned enterprise must be delegated to commercial companies. EU directives recommended the separation of railway infrastructure management from transport operations, assuming the coverage of losses as inadmissible passengers profits from freight. The EU Directive pointed to the division of the railway company into three sectors:

• passenger,

• freight,

• infrastructure.

In September 2000 Polish parliament adopted a law on commercialization, re-structuring and privatization of Polish State Railways (Polskie Koleje Państwowe). It established a new company, PKP SA, and four commercial companies:

• PKP Polskie Linie Kolejowe S.A.—infrastructure company,

• PKP Cargo S.A.—freight transport company,

• PKP Intercity Sp. z o.o.—passenger transport company,

• PKP Przewozy Regionalne Sp. z o.o.—regional passenger transport companies.

These statutory solutions, carried out administratively without financial support, did not solve the problem of the railway infrastructure in Poland. Its condition continued to degrade, which caused a further decrease speed throughout the network. The maximum speed has been reduced, and in a number of important connections, journey time was seriously extended, which undermined the competitiveness of rail transport on the market and increased its energy consumption.

2.1 Polish Plans Related to the Development of Rail Transport

Modernization of railway lines became possible only when Poland created substantive conditions for the development and improvement of the situation, producing program documents such as:

• National Development Strategy 2007–2015,

• State Transport Policy for the years 2006–2025,

• Master Plan for Rail Transport until 2030.

The National Development Strategy for the years 2007–2015 is the basic strategic document setting the objectives and priorities of socioeconomic development and the Polish conditions ensuring this development. This strategy directly addresses the national railways [32], where it is stated that:

Increasing the share of rail passenger and freight requires a significant improvement in the quality of rail services, especially in light of the approaching opening of the sector to strong competitive pressure within the Common European Market.

With these assumptions a real support investment was possible, which will result in raising the operational parameters of the main transport routes, including increasing the possible speed of transport and increasing interoperability.

The national strategy envisages support for the construction of a high-speed rail system integrating the Polish metropolises. Investment in railway infrastructure will be primarily aimed at the liquidation of bottlenecks on lines with high traffic, i.e. between the larger destinations, as well as reconstruction activities and modernization of railway lines.

The necessity of modernizing the existing railway lines, caused by years of under-investment in railways, and the lack of any support from public funds, resulted in the need for financing the state budget funds for the tasks associated with repair and maintenance of railway infrastructure from the European Union funds.

Actions taken in the framework presented by the Ministry of Transport “Master Plan for Rail Transport in Poland until 2030” [30] should lead, among other things, to a fundamental improvement of the infrastructure, and consequently to an improvement in the competitive position of rail transport. Thanks to that, the basis for creating a new quality of transport services in the field of transport of passengers and cargo was established.

At the same time directions of development of the railway infrastructure must be consistent with assumptions set forth in the updated Concept of National Spatial Development [23]. Actions on infrastructure included in this section are related to the following planning periods, coinciding with the planning periods of the European Union:

• 2007–2013,

• 2014–2020,

• 2021–2030.

A key element of the Master Plan for Rail Transport 2030 is to develop a plan and a timetable for the modernization, rehabilitation and expansion of rail infrastructure. Objectives of infrastructure investments provide a comprehensive subset of the specific objectives set for the entire Master Plan, as contained in the introduction to this study, and they must contribute to achieving the main objectives of the Master Plan. They are as follows:

• improving the transport of passengers and freight in the corridors of the trans-European transport network (TEN-T)—the fulfilment of international obligations in the field of standards and bandwidth on modernized lines;

• increasing the efficiency of the rail system, as a result of its reconstruction (including stopping the infrastructure degradation), taking into account the technical standards for interoperability and environmental standards;

• enabling the widest possible use of existing rail infrastructure, especially in two prospective sectors:

  • passenger market: between large urban areas and within large agglomerations,

  • freight market: mass transport of large cargo volumes and intermodal transport;

• facilitating mobility with the use of different modes of transport, in particular for passengers with reduced mobility: rail links to airports, linking with road transport and railway integration with public transport, with particular emphasis on urban agglomerations;

• improving standards of passenger service at railway stations and bus stops, including the adjustment to the needs of people with limited mobility.

The Master Plan framework provides three levels of investment activities, differing in material scope, level of costs and implementation period:

• construction of a new railway infrastructure of a high standard,

• modernization of the existing railway infrastructure, with particular emphasis on lines belonging to the TEN-T,

• investments restoring normal parameters of the railway infrastructure on the lines considered relevant (replacement investments).

A separate group of measures envisaged in the framework of the modernization are investments, including the construction of control systems on lines with small and medium traffic load, with the task of operating the automation and reduction of operating costs of these lines. Investments in infrastructure systems to improve the management of passenger and freight transport are also expected.

The widest range of investment activities will apply to projects relating to the construction of a new railway infrastructure. These investments can be divided into the following groups:

• building connections complementary to significant gaps in the railway network,

• building connections between the centers of large cities and airports supporting these metropolitan areas.

Modernization investments implemented on the Polish railways since the first half of the nineties mainly related to lines in the pan-European transport corridors. These lines are now part of the TEN-T.

However, the new EU perspective for the years 2007–2013 changed the priority of the modernization investments with the support of assistance programs such as the “Infrastructure and Environment” OPI&E and the “Regional Operational Programme” (ROP). The process of modernization of railway lines in these years accelerated, and the scope of modernization diversified and began to depend on the final destination of the line. In the framework of railway line modernization, all railway junctions were reconstructed, especially those that were bottlenecks on the railway network. Modernization included both the expansion of existing track and construction of new control systems and traffic management.

Due to the very poor condition of the rail infrastructure, long overdue for repairs and maintenance, replacement investments were the main measures needed to restore railway network operating parameters to their normal scale, both in terms of the speed of scheduled services and the pressure of the rolling stock axles. Figure 1 shows the values in the range of speeds [31], and Fig. 2 shows the value of the maximum pressure of the rolling stock on the track [36].

Fig. 1

The maximum scheduled speeds on the tracks of important lines in Poland

Fig. 2

The maximum axle loads on the track line of importance lines in Poland

The geographical scope of these investments will be far larger of all groups of investment activities. It is understood that the technical scope of replacement investments will concentrate on railway tracks and, in typical cases, will include the repair of main roads (replacement of individual components) or repair the current extended range of the existing surface. At the same time reconstruction of line should be comprehensive, meaning that reconstruction will include repair of drains and weak spots in the subgrade. It will include reconstruction of damaged or worn-out civil engineering and exploited crossover stations. Reconstruction will also include other work, such as building of automatic vehicular traffic controls at level crossings, which can be introduced as a result of the liquidation of operation speed limits. The aim should be to achieve significant improvement in operational performance without the risk of reducing speed limits.

2.2 European Plans Related to the Development of Railway Transport in Poland

The objectives of the development of railway transport in Poland can be achieved only through investment, including the construction of new sections of lines, and the modernization of existing lines of infrastructure that will improve the operational capacity of the railways and, consequently, help to increase the speed on the tracks. This will also improve traffic safety, increase throughput at hubs, reduction of travel time, and, consequently, improve the competitiveness of rail transport in relation to other modes of transport [37]. Increased railway transport efficiency depends on stable financing of infrastructure and effective management of all its components and systems.

Extensive plans are not feasible without a stable long-term financial plan. Funds for numerous investments come from several sources. First of all, thanks to the Polish accession to the European Union, railways may use funds such as ISPA (until 2004.), the TEN-T Fund, the Cohesion Fund and the European Regional Development Fund. An important role is also played by funds from the European Investment Bank (EIB), which distributes deals with National Economy Bank (NEB) in our country. They allow us to cover insufficiencies in our own matching contribution, which is required to obtain funding from the European Union.

2.2.1 ISPA Funds

The ISPA (Structural Pre-Accession Instrument) was one of three pre-accession instruments (along with PHARE and SAPARD) of the Union’s 10 candidate countries. The ISPA Fund was created by the Council of the European Union under the Decree 1267/1999 of 21 June 1999. Its main objective was to support economic and social cohesion through co-financing of large investment projects in the environment and transport. Principles of ISPA measures referred to the working of the Community Cohesion Fund.

The budget of the program was scheduled for 1.04 billion euros a year in the 2000–2006 period, of which Poland accounted for 30 to 37% of this amount, or an average of about 348 million euros. The program has been used to achieve the objectives set out in the “Partnership” (document prepared by the European Commission) and the priorities identified in the National Programme for the Adoption (as a reply from the Polish side to the EU document).

Financial support for the fund in the area of transport included promoting sustainable transport, in particular those projects that included the creation of connections between national networks and trans-European networks, and allowed for the unification of the conditions of use of these networks.

All projects had to be large enough so that their implementation had a significant impact in priority areas. Thus, the overall cost of the project from the outset could not be less than 5 million euros (derogations from this condition were possible only in exceptional and duly justified cases). Self-government, self-government organizations and other public entities could apply for the grant.

Since 01 May 2004, after Polish accession to the European Union, ISPA ceased its operation in our country. In Poland, in accordance with Annex II of the Accession Treaty, all projects that have received the possibility of funding under the ISPA, and which have not been completed, were pursued within the framework of the Cohesion Fund, operating on similar principles.

2.2.2 TEN-T Fund

The aim of the fund was to support projects implemented by Member States, which are called “common interest.” These projects have been identified in the Community guidelines for the development of the trans-European transport network. The network is aimed at increasing the efficiency of the functioning of the common market. It was also supposed to fully enable citizens of the Union, economic operators and regional and local communities to benefit from the establishment of an area without internal borders. The European Union aims to extend national transport networks through the development of intermodal transport. The aim is also to ensure access from remote regions or islands to the central regions of the EU and to reduce the high transport costs in these regions.

Trans-European networks include, among others, transport network (TEN-T), for which the EU has allocated a separate pool of funds in the EU budget. For the development of trans-European networks of the 4600 million euros was addressed in the years 2000–2006. For 2007–2013, in order to further develop itself, the TEN-T budget has been established in the amount of 8013 million euro, including targets for all EU Member States.

The beneficiaries of the TEN-T budget include both state actors and private entities operating in the area of public services. The Fund also supports projects conducted in public-private partnership. Projects co-financed from the TEN-T projects were those that lay in the interest of all Member States, i.e. those that:

• contributed to the sustainable development of the transport network across the European Union;

• ensured the coherence of the TEN-T and access to it;

• integrated all modes of transport;

• contributed to the protection of the environment and increased safety standards.

Cross-border projects, especially, were supported. Of key importance were also projects related to environmentally friendly transport (rail, sea, inland waterways) and those carried out by more than one member state. A list of priority investments was adopted in the framework of the fund which placed, among other railway infrastructure projects running through Polish territory:

• Railway axis Gdańsk–Warsaw–Brno/Bratislava–Vienna,

• Railway axis “Rail Baltica” Warsaw–Kaunas–Riga–Tallinn–Helsinki.

At the beginning of 2014 the last competition of the TEN-T was settled. Implementation of the projects will be completed in 2015. After 2014, the TEN-T Fund has been replaced by the CEF (Connecting Europe Facility) instrument. The European Union has allocated a separate pool of funds in its budget for this purpose. These funds will be used in 2014–2020 for investments in the construction and modernization of infrastructure in the fields of transport, energy and telecommunications.

2.2.3 Cohesion Fund

The Cohesion Fund was established in 1993 by the Treaty of Maastricht; the decision to create it had been made a year before at the European Council in Edinburgh. The motivation for the creation of the fund was eliminating disparities in the development of the economically weakest members. At the same time, the newly created Cohesion Fund had to compensate for those countries that could bear potential losses associated with the launch of “single market” economies, and make much more competitive goods available to the rest of the Community.

Unlike the Structural Funds, the Cohesion Funds are allocated to states, not to individual regions. The list of countries eligible for aid shall be determined by the decision of the European Commission, on the basis of gross national product. In order for a country to apply for funding from the Cohesion Fund for the implementation of infrastructure investments in the field of environment or TEN-T transport networks, its gross national income must be less than 90% of the average Gross National Income of the European Union. Another condition is that the beneficiary country Cohesion Fund program must lead to the fulfilment of the conditions of economic convergence. This is called the principle of conditionality. Its failure leads to the suspension of aid, but the state is not obliged to return funds already received.

In the years 2007–2013, Poland received 22.2 billion under the Cohesion Fund, which constitutes 33% of the awarded allocation for this period. As already indicated, the Cohesion Fund finances only infrastructure projects in the field of environment.

2.2.4 European Regional Development Fund

The purpose of the European Regional Development Fund is to increase economic and social cohesion in the European Union, eliminating inequalities between regions. In short, the ERDF finances:

• infrastructure related to research and innovation, telecommunications, environment, energy and transport;

• financial instruments (venture capital funds, local development funds, etc.) to support regional and local development and to foster cooperation between cities and regions.

2.2.5 Railway Fund

The Railway Fund is a targeted element of the government’s system-wide approach to the development of railway transport in Poland, the aim of which is to implement transport policy in accordance with sustainable development. The Railway Fund was created by the NEB under the Act of 16 December 2005 on Railway Fund, and started to operate on 9 February 2006.

The task of the Railway Fund is to collect funds for the preparation and implementation of construction and reconstruction of railway lines, repairs and maintenance of railway lines and the elimination of redundant railway lines.

Basic financial sources of the Fund are:

• constant revenues from the fuel tax on motor fuels and gas for motor vehicles; the Railway Fund supplies 20% of the proceeds of this account,

• interest rates available on the Fund’s account in NEB and income from bank deposits, free funds and their investment in securities issued or guaranteed or underwritten by the State Treasury,

• income from shares transferred by the State Treasury and the revenue from their sale.

2.2.6 Infrastructure and Environment Operational Programme

The Partnership Agreement developed by the European Commission defines the main directions of support under the Cohesion Policy, which also includes the Operational Programme Infrastructure and Environment (OPI&E). This program (OPI&E), strives for sustainable economic development and competitiveness, which will be possible by supporting the development of technical infrastructure in Poland.

Existing infrastructure upgrades were based primarily on the OPI&E and the ROP. In 2006, the European Commission granted to Poland, from the Structural Funds and the Cohesion Fund, about 35 billion euros for infrastructure improvements to be spent in 2007–2013. These were measures increased funding, though were still insufficient. The current investments in the railway infrastructure are executed in accordance with the “Long-term Railway Investment Programme for the year 2015” adopted November 5, 2013 by the Council of Ministers. Figure 3 [31] shows a list of infrastructure investments for the years 2011–2013. The program includes 140 investment tasks under the current financial perspective (2007–2013). Of these, 62 investments are made within the OPI&E and 29 under the ROP.

Fig. 3

Investments in Polish railway infrastructure networks in 2011–2013

2.2.7 Regional Operational Programme

The Regional Operational Programme (ROP) is a planning document defining the areas and sometimes specific actions that government bodies of voivodeships take or intend to take to promote the development of the province or region. As the name suggests, this is a document of an operational nature, so it is more detailed and primarily focused on development strategy.

The legal basis for the functioning of the ROP is the Act of 6th December 2006 on principles of development policy.

In total, the 2007–2013 Operational Programme Infrastructure and Environment for railway investments with OPI&E allocated 4.8 billion euros and spent an amount of 23,314 million zloty. The difference between the amount of aid received in euros and the amount of PLN financing was caused by currency fluctuations. The level of spending of EU funds in the perspective of years 2007–2013 are shown in Fig. 4. The funds were allocated to linear infrastructure projects (construction and maintenance of railway lines) and point infrastructure (railway stations and terminals).

Fig. 4

The amount of funds spent on rail infrastructure in the years 2007–2013

The current investments in railway infrastructure are carried out according to the “Long-term Railway Investment Programme for 2015” adopted by the Council of Ministers on 5th November 2013, the size of which is shown in Table 1 [33].

Table 1 Implementation of projects by funding spent on railways in Poland until 2015

Implementation of EU projects by financing sources, indicating the quantity and value of the projects, are presented in Table 1. The data is in accordance with updates to the Multi-Year Investment Programme Railway 2015 [33], where it was assumed that the implementation would be 142 projects with a total value of 41,304.9 million zł. The locations of these investments are shown in Fig. 5 [33]. PKP PLK in the years 2007–2015 issued a total of 37,157.00 million zł.

Fig. 5

Infrastructure investments covered by the Multi-Year Programme of Railway Investment for the years 2013—2015

2.3 The Current State of Modernization of Railways in Poland

Ongoing and future modernizations of the railway infrastructure in Poland include both national and EU financial funding.

2.3.1 Investments from National Funds

In 2015, PLK announced tenders for the projects of modernization and reconstruction of several important railway lines, to be implemented in 2015–2016. They will be financed from the state budget, the Railway Fund and bonds that support Polish Railway Lines. Their total estimated investment value approximately 1 billion zł. In this group there will be a subtask for endarterectomy detours for those episodes, which will be modernized with EU funds’ new financial perspective.

2.3.2 CEF Investments

In 2014–2020, the EU is launching a new program for co-financing major infrastructure investments (not just rail) called CEF. From the sum of 28 billion euros, Poland was awarded 4 billion euros. Poland applied to this program with measures worth at least 3 billion euros. In addition, the railway tasks involved in interfacing with maritime transport and improving access to ports might be able to qualify for an additional 3.5 billion euros. The European Commission will announce recruitment in three stages, the deadline for the first round of projects ended in February 2015. Contenders were announced in September 2015, 2016 and the closure will take place in December 2016. The first call PLK start with six investment projects. These will be assessed by the European Commission at the end of the third and fourth quarters of 2015. The most advanced project is to complete the modernization of the Wroclaw–Poznan section (to complete work on the Rawicz–Czempin section), worth 1.5 billion zł., and a project to modernize the Sochaczew–Swarzędz line for 2.6 billion zł.

There also will be investment in railway lines peripheral to Warsaw (Warsaw section Gołabki/Warsaw West–Warsaw Gdansk)–Warsaw Wlochy–Grodzisk Mazowiecki (line 447).

In total, in the first competition of the project, Polish CEF investors will report to the European Commission projects worth approximately 10 billion zł, by the probable date of announcement of public tenders to 2016, for the implementation of at least two jobs for more than 4.1 billion zł.

2.3.3 The Remaining Pool of CEF

In addition, PLK predicts that CEF funds for projects funded under the “List of offshore projects” will be available. Three tasks to improve rail access to seaports in Szczecin and Swinoujscie and Gdansk and Gdynia will then be reported. All of these projects will involve a total reconstruction of the station ports, adapting them to the specifics of the projected freight traffic and thus enabling the further development of ports.

2.3.4 Funds from OPI&E

As the financing of railway investment projects are based almost exclusively on the state budget and EU funds, the flow of funds for projects of new perspective 2014–2020 will be initiated after the final approval of the new operational programs. Meanwhile, the negotiations with the European Commission on the final shape of the OPI&E are still in progress. Therefore, in contrast to the railways, road builders can already announce a number of tenders, because the roads are financed from other sources, such as the National Road Fund. Polish Railway Lines must wait for the completion of procedures and complete accounting of financing tasks from the perspective of 2007–2013. Not until 2015 was a new program adopted of co-financing national measures of new tasks for 2014–2020. In total, PKP and PLK expect to receive from the new perspective a total of about 7 to 8 billion euros, which is at least 40 percent more than in the perspective of 2007–2013. Undoubtedly, prospects for the “Master Plan” and “Long-Term Investment Plan” look promising. Part of the work included in both programs is in progress, part completed. Implementation of strategic objectives must overcome a variety of difficulties. First of all, implementation of a project is a multidisciplinary task that requires effective coordination and management. This involves obtaining multiple permits, making a lot of arrangements, and most importantly is connected with the necessity of application of the “Public Procurement Law” Act. All this requires the fulfilment of a number of formalities related to the investment. The whole procedure also extends the requirement to obtain administrative decisions. The problem is determination of the sole criterion for choosing the best contractor, because the commonly used criterion of price does not guarantee the investment in the framework of a negotiated amount.

A modern and efficient rail infrastructure is a prerequisite for the development of rail transport and for the country. Rail transport should be regarded as the most ecological and safest mode of transport. It is attractive both for passengers and for businesses. The existing investment plans should, however, be reliably verified in relation to trends in freight volume forecast.

3 Applied Methods of Forecasting

All the phenomena described in the section related to the operation of shipping are non-periodic investigations, close to linear. Some of the waveform charts are decreasing (Figs. 11, 12, 16, 17, and 18), a few are growing or virtually constant (Figs. 18, 22, 24, 25, 26 and 27). In the case of the variation of electric and diesel multiple locomotives (Figs. 15 and 16), decreases in the number of exhaust units are combined with increasing amounts of electrical components. Both graphs are characterized by clear increments in 2006. Generally, the variable course of the collapse in 2009 can be observed in Figs. 20, 21, 24, and 23.

The trend of changes in the near future can be predicted using the exponential smoothing method, which is easy to use and can be accurate enough. For non-periodic phenomena on the course, a Holt-Winters double exponential smoothing is used, which extracts significant changes in observed rail transport phenomena and reduces the influence of random fluctuations. This method was slightly modified in the monograph. The values of model parameters α and β and the initial values F ₁ and S ₁ are calculated as the optimal values using the clonal selection algorithm. The Bayesian network is second method.

Forecasting, understood as scientific prediction of the development of observed phenomena, plays an important role in planning. Forecasting the volume of transport is of great economic importance for the entire Polish economy.

An interesting approach to the use of forecasting in the supply chain optimization is described in [1]. The article presents the optimization of the supply chain cost using an integer programming method. The data on demand, production and inventory forecasting needed for optimization are obtained by using the exponential smoothing methods.

Mathematical methods are applied to forecasting. A brief history of forecasting is presented in [17]. Early forecasts constituted simple inference from observation; such methods developed especially in the nineteenth century. The situation changed at the beginning of the twentieth century with the proposed treatment of the time series as a realization of a stochastic process in [40]. Brown’s [3], Holt’s in 1957 [25] and Winters’ [38] publications initiated the development of exponential smoothing methods. Viewed state exponential smoothing method for the year 1980 can be found in [21]. Exponential smoothing methods are often used in many fields of science, because they are easy to use and provide forecasts burdened with only small errors.

Smoothing methods continue to develop. The adjustments in the Holt-Winters double exponential smoothing can be read in the article [26]. The choice of model parameters for exponential smoothing by means of empirical performance of two derivative free search methods for solving the problem of minimization is presented in the article [35]. The use of weighted coefficients, moving average and exponential smoothing is written in the [39].

Currently, forecasting methods are often aided by artificial intelligence. For example, to estimate optimal values of coefficients of logarithm support vector regression the immune algorithm was used [29]. Neural networks can be used to reduce the sensitivity to input errors, and the ARIMA method (Autoregressive Integrated Moving Average) was used in [27]. A Bayesian network was used to predict stock price in [20].

3.1 Holt-Winters Double Exponential Smoothing

Looking at the graphs illustrating the course of the various phenomena observed in rail transport, we decided to use the methods of exponential smoothing, and because of the lack of periodic phenomena, we chose the Holt-Winters double exponential smoothings.

The Holt-Winters double exponential smoothing is one of the methods used to smooth the time series of a development trend and random fluctuations [19, 41]. For a time series of length N and the data values sy ₀, y ₁, … y _N−1 the following equation are used:

$$F_{t} = y_{t} + (1 - { \propto })(F_{t - 1} + S_{t - 1} )$$

(1)

$$S_{t} = \beta \left( {F_{t} - F_{t - 1} } \right) + (1 - \beta )S_{t - 1}$$

(2)

where

F _t :

smoothed value of the forecasted variable at time t,

S _t :

growth trend value at the moment t,

t = 0, 1, …, N,

parameters α, β ∈ [0,1].

The equation of prediction for expired periods have the form:

$$y_{t}^{*} = F_{t - 1} + S_{t - 1}$$

(3)

for t = 2, 3, …, N

and for future periods:

$$y_{T}^{*} = F_{N} + (T - N)S_{N}$$

(4)

where

N :

the number of periods in the relevant time series

T :

moment, which forecast, T > N

There are many ways of determining the value of the initial F ₁ and S ₁. Most often it is assumed that F ₁ = y ₁ and S ₁  = Y ₂ − Y ₁. In turn, parameters α, β are determined by error’s minimization of expired forecasts.

In presented calculations some modifications were applied. The parameters α and β and the initial values of F ₁ and S ₁ are determined by minimizing the error MAPE. A similar solution is also used in [34]. In this chapter we will refer to the method as the “modified Holt method”.

MAPE (Mean Absolute Percentage Error) is defined as:

$${\text{MAPE}} = \frac{1}{N}\sum\limits_{t = 1}^{N} {\frac{{\left| {y_{t} - y_{t}^{*} } \right|}}{{y_{t} }}} \cdot 100\%$$

(5)

where

N :

number of observations,

y _t :

value of the time series for a moment or period of time t,

$y_{t}^{*}$ :

predicted value of y for a moment or a period of time t.

The parameters α and β and the initial values of F ₁ and S ₁ have to calculate by minimizing the MAPE error. To assess the quality of forecasts Pearson’s correlation coefficient was calculated for actual data and expired forecasts.

3.1.1 Artificial Immune System

To determine the forecast using the modified Holt method, an artificial immune system was used. One of the methods of artificial intelligence, the artificial immune system mimic the action of the natural immune system of the human body [4, 28].

The defense system of the human body is made up of physical barriers and chemical properties, such as skin, body temperature, saliva, tears, sweat and mucus. If, in spite of these barriers, microorganisms attacking the body penetrate into the interior, the immune system defends. Its operation is quite complex and the defense involves different cell types. Some of them recognize antigens that attack the body [22].

Recognition is done using antibodies that are produced by the body. The antibodies, which fit to the antigen, are cloned. The clones are mutated. After finding the fitted antibody a proliferation follows. The antibodies are rapidly replicated and released into the blood stream.

After suppressing the number of antibodies is reduced. Some of the remaining antibodies form a memory cells. The next time the immune system has a chance to recognize the enemy faster. The process of searching for better and better antibodies is called clonal selection and its numerical model can be used for optimization.

Figure 6 shows the main stages of the clonal selection numerical algorithm. An antygen is optimal solution, which is searched. Antibodies are the solutions, which are proposed. The inverse of the objective function representing the optimization criterion is the measure of the matching.

Fig. 6

The main stages of the clonal selection algorithm

The antibody is a sequence of numbers:

$$[F_{1} ,S_{1} ,\alpha ,\beta ]$$

(6)

where F ₁, S ₁ initial values of Holt model,

α and β are parameters of this model, F ₁, S ₁ ∈ R, α, β ∈ [0,1].

The inverse of the MAPE error is a measure of matching.

For the calculations, we use our own implementation of the algorithm written in the programming language C++.

3.2 Forecasting Using Bayesian Networks

Forecasting the volume of transport and other phenomena in rail transport can also be carried out using a second method, the Bayesian network.

Bayesian network structure corresponds to the relationship of cause and effect in a given set of random variables. As a rule, a Bayesian network topology maps the knowledge of the causes and effects in the area considered. The usefulness of Bayesian networks in practical applications is manifested by the fact that the knowledge of any set of observations (some state variables) allows you to calculate the probability distributions for the remaining unknown variables [2].

The network name is derived from Bayes’s theorem, which postulates a revision of the existing beliefs about the probabilities in the light of new facts. Presentation of the rules of operation requires a reminder of two basic and very important theorems of probability theory: the formula for complete probability (7) and Bayes’s theorem (8).

Let the some event B occur in several mutually ways exclusive of A _i , exhausting all possibilities, and assume that the probabilities $P\left( {A_{i} } \right)$ are known. Then the probability of an event B can be expressed as the complete probability:

$$P(B) = \sum\limits_{i} {P(A_{i} )P(\left. B \right|A_{i} )}$$

(7)

Next, assume that it is known that the event B has just happened and the probability of the above formula was calculated. This knowledge allows us to re-calculate the probability of each event A _k from the original value of $P(A_{k} )$ to new values of $P(\left. {A_{k} } \right|B)$:

$$P(\left. {A_{i} } \right|B) = \frac{{P(A_{k} )P(\left. B \right|A_{k} )}}{P(B)}$$

(8)

Both formulas, despite their simplicity, are powerful tools for inference in each chain based on probabilities. The inference can be conducted in both directions: from causes to effects and from effects to causes; thus a calculated value can propagate throughout the network [16].

3.2.1 The Structure of a Bayesian Network

A Bayesian network is a directed acyclic graph, where the nodes represent random variables and edges correspond to cause/effect relationships between these variables. With each vertex X is associated a conditional probabilities table, which describes the strength of the relationship. This array contains the conditional probabilities $P(\left. X \right|P_{1} ,P_{2} , \ldots )$ of each state of a random variable X with the different states taken by the direct parent P ₁, P ₂, …. For vertices without parents (so-called root causes) conditional probabilities come down to simple probabilities. Figure 7 shows a simple Bayesian network with three binary nodes (receiving only two values: true and false) [18].

Fig. 7

The structure of a simple binary Bayesian network

3.2.2 Construction of a Bayesian Network

Knowledge on Bayesian network is included in the topology and in the tables of conditional probabilities associated with the random variables. The first step is to identify those variables, then the relationships between them must be determined. This task is usually performed by an expert, but in rare cases can be performed automatically on the basis of the available data. Values of conditional probabilities can also be introduced by an expert based on his knowledge and experience, but more often the network is “trained” automatically based on historical data [24]. These data do not have to cover all the states of all the variables. A standard algorithm EM (Expectation/Maximization) can be used. EM is an iterative selection of probability and similarities for incomplete data. The algorithm tries to estimate which of the parameters of such a model, relative to the observed data (data provided learning), were the most likely. Each iteration consists of two stages:

• E step (expectation)—missing data are adjusted taking into account the known and the current parameters of the model,

• M step (maximization)—model parameters are selected so as to maximize the probability on the assumption that the missing data are already known.

3.2.3 Inference in Bayesian Networks

A Bayesian network defining the relationship between random variables allows us to calculate the probabilities of occurrence of events represented by these variables. A practical application of Bayesian networks begins with the information that is currently known. States of some random variables are determined, and then the schedule of probabilities other variables is updated. Possible modes of reasoning are summarized in Table 2.

Table 2 Modes of inference in Bayesian networks

3.2.4 Application of Bayesian Networks for Forecasting

A mode of causal inference is used in forecasting. It is assumed that the future, unknown state results from the state variables corresponding to past moments that are fixed at the time of prognosis. Thus, the Bayesian network provides the probability distribution of future states (Fig. 8), and the expected value can be treated as a sought value forecast (9).

Fig. 8

Forecast as a probability distribution of states

$$w_{\text{out}} = \sum\limits_{i} {p_{i} m_{i} }$$

(9)

where

w _out :

predicted value,

p _i :

probability of ith state,

m _i :

value corresponding to ith state.

An additional state called ‘stx’ can be seen in Fig. 8. This state is used to indicate improper operation of the network. The variable y ₀₊₁ is initialized in a way that assigns this state probability 1, and other states are assigned the value of 0. The ‘stx’ state does not appear anywhere in the training data, thus assigning a non-zero probability in the course of determining the forecast shows an unexpected set of states set for the moments of the past.

One of the simplest Bayesian networks that can be used in a prediction is shown in Fig. 9. The premise is that the value (stocks of random variables) for the present time and the past moment (y ₀₋₁, y ₀₋₂, y ₀₋₃) are known (here, four values), and a probability distribution for the future moment is sought. There is no relationship between the variables corresponding to the past because they are not needed; all of these variables belongs to the past, and are already known facts. This model allows the calculation of forecasts for the single step, a future moment.

Fig. 9

The simplest Bayesian network used for prediction

If one needs to perform following forecasts for two or more moments, a Bayesian network can have the structure shown in Fig. 10. The obtained test results confirm the usefulness of such a model, but a very important condition against the practical application of such a solution for predictions year after year is the limitation of historical data. The learning sequence must contain data treated as past, and two or more values corresponding to the forecast. Therefore, a specified time frame significantly reduces the number of available learning sequences.

Fig. 10

Bayesian network used for two step prediction

In this situation, the predictions for moments more distant than one step first iteratively apply the simple model: the prediction y ₀₊₁ obtained in the first step is treated as the current value y ₀ in the next step. The number of such iterations depends on the required number of predicted values. It is clear that the reliability of such distant predictions decreases with increasing distance in time.

3.2.5 The Choice of the Number of Random Variables States

An important issue in the design of a Bayesian network is to determine the appropriate number of states adopted by random variables. The higher the number, the more accurate the obtained forecasts. On the other hand, a limited number of historical data increases the probability that, during the learning, multiple network states will not be assigned any value. This situation negatively affects the learning processes of the network. To avoid this, tests have shown that the optimal number of states is seven. At the same time, to ensure the accuracy of processing, the following procedure was used. In the place of unique assignment of a value from a set of historical data to one of the conditions, we made a variable decomposition process. A single historical value was expressed as a linear combination of values corresponding to the states for each random variable. In this way, a single string of historical data generated a whole set of records containing clean learner states, to which the distribution coefficients of decomposition provided an answer.

4 Forecasts of Rail Transport in Poland

The forecasts are based on statistical data presented by the Central Statistical Office of Poland in Transport of activity results for years 2000–2014 [5–15]. The predictions included the following data:

• operated railway lines,

• standard gauge rolling stock (electric and diesel locomotives, electric railcars, freight cars and coaches),

• cargo rail transportation (in tons),

• cargo rail transportation (in ton-kilometers),

• rail transportation according to types of consignment (export, import, transit),

• total national and international rail transport of containers.

4.1 Operated Railway Lines’ Length

Predictions of operated railway line lengths in Poland were based on data presented in Table 3.

Table 3 Operated railway line length

As you can see in Fig. 11, the length of railway lines in the coming years decreases slightly. Both forecasts for the next three years provide for a further reduction of railway lines. Using the Bayesian network here is more optimistic. It gives a smaller forecast error MAPE of 0.85%, but the modified Holt’s method gives a Pearson correlation coefficient closer to unity, which shows a better correlation of the predicted values with the actual data.

Fig. 11

Predicted total length of operated railway lines

The aforementioned trend of shortening railways applies in particular to railway standard gauge (Fig. 12). Here, again, the first method predicts a smaller reduction of standard gauge lines. Its MAPE forecast error is 0.88%, but the second method gives a higher Pearson correlation coefficient, which is 0.85.

Fig. 12

Predicted length of standard gauge railway lines

The length of electrified railway lines since 2004 changes slightly and the graph for real data, shown in Fig. 13, differs very little from both forecasts. Both methods show a slight downward trend for the next three years. MAPE error is smaller for the first method at 0.4%. The correlation coefficient is superior to other methods and is equal 0.95, which indicates a strong linear correlation of actual data and forecast results.

Fig. 13

Predicted length of electrified operated railway lines

4.2 Standard Gauge Rolling Stock Number

Predictions of standard gauge rolling stock in Poland were based on predictions of electric and diesel locomotives and electric railcars, which are presented in Table 4, as well as freight cars and coaches, presented in Table 5.

Table 4 Standard gauge rolling stock prediction—part 1

Table 5 Standard gauge rolling stock—part 2

The number of electric locomotives starts to grow slightly (Fig. 14). Forecasting with Holt method in this respect is more optimistic and predicts greater sales growth than prediction by Bayesian networks. However, the first method has a smaller MAPE error so the prediction should be based on this method. In turn, the second method has a higher correlation of actual data and forecast data for the years 2001–2014. The correlation coefficient is 0.67.

Fig. 14

Predicted number of electric locomotives

In accordance with predictive analysis shown in Fig. 15, the number of operating diesel locomotives declines. The Bayes prediction expects the number of operating locomotives to stabilize. It can be seen from the chart that this method is more sensitive to the jumping-off data. The Holt’s method clearly smooths the data in such a situation. For the first method, correlation of actual data and predicted is almost non-existent. For the Holt method it is 0.24, understood also as weak.

Fig. 15

Predicted number of diesel locomotives

Since 2006, a systematic increase has been observed in the number of electric multiple units, as shown in Fig. 16. The method of forecasting based on the Bayesian network provides stability in the next three years. Holt’s method indicates a rapid increase in the number of teams in the coming years. The actual data and the forecasts for both methods are correlated poorly. MAPE error for the first is 4.16%, and 7.67% for the second.

Fig. 16

Predicted number of electric railcars

Analyzing the data contained in Table 5, it can be said that since 2002 the number of freight cars has been steadily decreasing (Fig. 17). The Bayesian network method forecasts a slight further decline in this number in the coming years. Holt’s method predicts a greater decrease in the amount of freight cars. Because the MAPE error for the first method is smaller and is 2.8%, and the correlation of real data and the forecast is as high as 0.94, the forecast for the next few years according to this method seems to be more reliable.

Fig. 17

Predicted number of freight car

Figure 18 shows the number of passenger coaches, which since 2000 decreases linearly. Both methods of forecasting predict a further linear drop in the number of freight cars. The correlation coefficients of both methods show a very strong linear relationship between the actual data and the forecast. Forecast errors are also small.

Fig. 18

Predicted number of passenger coaches

4.3 Transportation of Cargo in Tons

Forecasts for freight transport and goods by rail in tons are based on data contained in Table 6.

Table 6 Cargo transportation by mode of transport prediction

There is clearly a growing trend in Poland in transportation of cargo by all means of transport, which is shown in Fig. 19. As for the forecast for the next three years, a modified Holt method predicts a slight decline in traffic, while the Bayesian network method shows growth. The forecast with the Holt method in this case has a smaller MAPE error, equal to 2.5%. Also, Pearson’s correlation coefficient is high, close to unity, indicating a strong correlation of actual data and forecasts. The forecasting method based on Bayesian networks gives much worse results, so it seems safer to assume that there may be drop in freight volume.

Fig. 19

Predicted total transportation of cargo

According to the diagram in Fig. 20, cargo transportation by rail in the past fifteen years has large fluctuations. Holt’s method in this case gives a forecast burdened with a smaller MAPE error, equal to 5.9%. It also has a correlation coefficient that is almost twice as large, which is about 60% of the linear correlation of actual data and forecasts. Both methods of forecasting predict a fall in freight cargo transportation amount.

Fig. 20

Predicted amount of freight cargo transportation

4.4 Transportation of Cargo in Ton-Kilometers

Forecasts for rail freight transport denominated in ton-kilometers are based on data contained in Table 7.

Table 7 Cargo transportation by rail

Transportation of cargo by rail in Poland, expressed in ton-km, have a clear growing trend as shown in Fig. 21. The correlation coefficient of the modified Holt method is two times higher than the correlation coefficient of the Bayesian network. It is amounting to 0.34, what does not indicate for very strong correlation of observed data and forecasts. In turn, the forecast error MAPE for Holt’s method here is lower and amounts to 5.75%. Despite the complicated course of the graph in Fig. 21, Holt’s method shows the greatest smoothing of data in 2009.

Fig. 21

Predicted amount of total cargo transportation

4.5 Rail Transportation According to Types of Consignment

Forecasts of rail transportation according to types of consignment prediction, by export, import and transit are based on data contained in Table 8. Rail transportation by different types of consignment in the years 2003–2014 is characterized with great variability in the amount of cargo.

Table 8 Rail transportation by types of consignment

Export of goods by rail hit bottom in 2009. In the whole period 2003–2014, high variability of the freight volume can be observed, see Fig. 22. Data and forecasts are weakly correlated, and the results are burdened with a high MAPE error. The future of export goods by rail is also projected by two different methods. Modified Holt’s method predicts the future stabilization of export volume by rail, while the Bayesian network method predicts the decline of transport.

Fig. 22

Predicted export by rail transportation

The import of goods by rail is a constant upward trend. In Fig. 23 a strong linear correlation data and forecasts can be seen at the same time. For the next three years a steady growth in freight volume is expected.

Fig. 23

Predicted imports by rail

Railroad transportation of goods by transit through Poland is characterized with very large fluctuations in the years 2003–2014, as shown in Fig. 24. The modified Holt method has smoothed the graph. For subsequent years, both methods predict a slight increase in the volume of cargo in transit. The resulting predictions are affected by significant MAPE error. The correlation coefficient indicates a weak link between data and forecasts.

Fig. 24

Predicted transit by rail transportation

4.6 Rail Transportation of Containers

Utilization of railway for the transport of containers has a clear upward trend; the forecasts are based on data contained in Table 9.

Table 9 Rail transportation of containers

Utilization of railway for the transport of containers has a clear growing trend as shown in Fig. 25. The two forecasting methods predict a further increase in this type of transport. Data and forecasts are strongly correlated linearly. The results are, unfortunately, burdened with high MAPE errors.

Fig. 25

Predicted total transport of containers by rail

Transportation of containers in the country is also characterized by an upward trend, as shown in Fig. 26. Both forecasting methods predict a further increase in this type of transport. Data and forecasts are strongly correlated linearly. The results, unfortunately, are subject to very large MAPE errors.

Fig. 26

Predicted national transport of containers by rail

Transportation of containers in international transport are also characterized by an upward trend, see Fig. 27. Both forecasting methods predict a further increase in this type of transport. Data and forecasts are strongly correlated linearly. The results are, unfortunately, burdened with high MAPE errors. The Holt method has the lower MAPE error and, as the chart shows, smooths fluctuations in freight volume.

Fig. 27

Predicted international transport of containers by rail

4.6.1 A Summary of the Calculations

The results and future predictions look believable. The two methods used, modified Holt method and Bayesian networks, give comparable results. The values of MAPE errors and the coefficients of corelations are respectively similar for both methods. In the case of the Bayesian network, the expired forecasts depict on the graphs of results the slight shift to the right. The maxima and minima of the original time series were preserved. The modified Holt method clearly smooths the random jumps of the original time series.

5 Conclusion

On the basis of the calculation results, it is obvious that if nothing changes in the policy of rail transport, it will continue to see reduced length of the railway line, electrified lines and locomotives (although for electrical forecasts predict stagnation or even a slight increase). The number of freight cars and passenger cars is also predicted to be continuously reduced. No wonder that in such a situation, rail services have large fluctuations in the coming years and are predicted to continue to show a slight decrease in traffic, which both methods confirm in extremely similar fashion (Fig. 18). This is alarming because, as shown in Figs. 20, 21, and 22, interest is growing in rail transport. The railway has large share of export, import and transit. An increase in container traffic is observed, which may be the greatest argument in the context of the modernization of infrastructure and investment programs analyzed in the chapter.