Decision support model for prioritizing railway level crossings for safety improvements: Application of the adaptive neuro-fuzzy system

Expert Systems with Applications 40 (2013) 2208–2223

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Expert Systems with Applications journal homepage: www.elsevier.com/locate/eswa

Decision support model for prioritizing railway level crossings for safety improvements: Application of the adaptive neuro-fuzzy system Goran C´irovic´ a,1, Dragan Pamucˇar b,⇑ a b

The Belgrade University College of Civil Engineering and Geodesy, Belgrade, Serbia University of defence in Belgrade, Department of logistic, Serbia

a r t i c l e

i n f o

Keywords: Railway level crossings Railway accidents Neuro-fuzzy model Safety improvements

a b s t r a c t Every year, more than 400 people are killed in over 1.200 accidents at road-rail level crossings in the European Union (European Railway Agency, 2011). Together with tunnels and specific road black spots, level crossings have been identified as being a particular weak point in road infrastructure, seriously jeopardizing road safety. In the case of railway transport, level crossings can represent as much as 29% of all fatalities caused by railway operations. In Serbia there are approximately 2.350 public railway level crossings (RLC) across the country, protected either passively (64%) or by active systems (25%). Passive crossings provide only a stationary sign warning of the possibility of trains crossing. Active systems, by contrast, activate automatic warning devices (i.e., flashing lights, bells, barriers, etc.) as a train approaches. Securing a level crossing (whether it has an active or passive system of protection) is a material expenditure, and having in mind that Serbian Railways is a public company directly financed from the budget of the Republic of Serbia, it cannot be expected that all unsecured level crossings be part of a programme of securing them. The most common choice of which level crossings to secure is based on media and society pressure, and on the possible consequences of a rise in the number of traffic accidents at the level crossings. The process of selecting a level crossing where safety equipment will be installed is accompanied by a greater or lesser degree of uncertainty of the essential criteria for making a relevant decision. In order to exploit these uncertainties and ambiguities, fuzzy logic is used in this paper. Here also, modeling of the Adaptive Neuro Fuzzy Inference System (ANFIS) is presented, which supports the process of selecting which level crossings should receive an investment of safety equipment. The ANFIS model is a trained set of data which is obtained using a method of fuzzy multi-criteria decision making and fuzzy clustering techniques. 20 experts in road and rail traffic safety at railway level crossings took part in the study. The ANFIS model was trained with the experiential knowledge of these experts and tested on a selection of rail crossings in the Belgrade area regarding an investment of safety equipment. The ANFIS model was tested on 88 level crossings and a comparison was made between the data set it produced and the data set obtained on the basis of predictions made by experts. Ó 2012 Elsevier Ltd. All rights reserved.

1. Introduction It is estimated that each day an average of 1308 people in the world lose their lives in traffic accidents (European Railway Agency, 2011). Out of approximately 54 million people a year who die in the world, the death toll in road accidents amounts to 1.17 million (2.17%). This makes up a third of all victims of all types of injuries in the world and is two times greater than victims of

⇑ Corresponding author. Address: Pavla Jurisica Sturma 33, 11000 Belgrade, Serbia. Tel.: +381 642377908; fax: +381 113603187. E-mail addresses: [email protected] (G. C´irovic´), [email protected] (D. Pamucˇar). 1 Address: Hajduk Stankova 2, 11000 Belgrade, Serbia. Tel.: +381 112422178; fax: +381 112422178. 0957-4174/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.eswa.2012.10.041

war, and almost twice as many as all victims of murder and violence. Among the leading causes of death, traffic accidents are in eighth place in the group of developed countries and in tenth place in the group of developing countries (Bureau of Transport, 2010). However, the absolute number of deaths is seven times higher in the group of developing countries. When comparing geographiceconomic regions, in accordance with the divisions given by the World Health Organization, it can be seen that most deaths are in India (217,000), followed by China (179,000), where the number of those killed in the last decade has increased by almost 30% (Li, Xia, Li, & Li, 2002), and then Africa (170,000) (The World Health Report, 1999). In Europe, around 20,000 die annually in traffic accidents, while around 2.5 million suffer from serious or minor injuries (WHO, 2003).

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One third of fatal traffic accidents are related to persons younger than 25 years of age, of whom 89% are young men between 15 and 24 years (MacNab, 2003). Each year around 9000 children and young people under 19 are killed in traffic accidents, with approximately 355,000 being more or less severely injured (WHO, 2003). Railways remain one of the safest modes of transport in the European Union (EU). Yet, some 1,400 people still die on EU railways each year. Most of the fatalities are unauthorised persons and level-crossing users. Trends derived from the common safety indicators (CSIs) indicate an overall improvement in railway safety since 2006. Both the number of people killed and the number of seriously injured persons fell in 2009. According to the CSI data provided by the national safety authorities (NSAs) to the European Railway Agency, 1391 people were killed and 1114 people were seriously injured in 3073 railway accidents in 2009 (European Railway Agency, 2011). These figures are by far the lowest figures recorded since 2006. Member States reported 1,284 level-crossing users killed in a total of 3063 level-crossing accidents during the three years 2007–2009 (Table 1). There are about 124,000 level crossings in the EU, so that on average there are 4 level crossings in each 10 km section of track. Only 41% are equipped with either manual or automatic protection systems. Level crossings are amongst the most complex of road safety issues, due to the addition of road vehicles with rail infrastructure, trains and train operations. The contributory factors at crossings can be difficult to determine and there are generally several factors for a particular incident. Nevertheless, in Europe, 95% of level crossing accidents are caused by road users (Woods, 2010). Table 2 shows the number of traffic accidents at level crossings in EU countries for the period 2006–2009 (European level crossing forum, 2011). In Serbia, amongst the major causes of collision are adverse weather or road conditions (13%), unintended motor vehicle driver error (46%), alcohol/drug use by a motor vehicle driver (9%), excessive speed (of motor vehicle driver) (7%), collisions involving fatigue (of motor vehicle driver) (3%) and other risk taking (of vehicle driver) (3%) (Serbian Transport Safety Agency, 2010). From these statistics, it is clear that human factors are the major cause of these accidents (total 68%). Railroads and highways are the two primary networks of surface transportation serving the entire nation. Both systems are essential to the public interest. However, exposure to potential collisions between trains and motor vehicles at some 2350 RLC throughout the Serbia has created a serious problem with regard to the convenience and safety of highway travel (Serbian Transport Council, 2010). Accidents at level crossings continue to be the largest single cause of fatalities from rail activity in Serbia (Bureau of Transport, 2008). There are approximately 85 incidents at Serbian crossings every year and these incidents result in the death of an average of 90 people (Serbian Transport Council, 2010). This problem has grown tremendously during the past few decades because of the rapid growth in vehicle-miles of travel. During the four-year period of 2007 through 2010, nationwide statistics indicated that fatalities due to RLC accidents increased

Table 1 Number of fatalities for different categories of railway user (2007–2009). Categories of railway users

Passengers Employees Level-crossing uesrs Unauthorised persons Others

Year 2007

2008

2009

70 38 504 857 50

89 38 380 929 44

37 29 400 852 73

2209

Table 2 Number of traffic accidents at level crossings in the European Union (2006–2009). Year

2006 2007 2008 2009

European Union country

Total

Western Europe

Central Europe

Eastern Europe

135 172 138 116

452 348 305 180

725 676 591 537

1312 1196 1034 833

by 36% in rural areas and 20% in urban areas in Serbia (Bureau of Transport, 2008). In Serbia, the death rate for motor vehicle–train accidents has followed a similar pattern. Between the years 2008 and 2009, traffic deaths in Serbia increased by seven percent, but deaths resulting from motor vehicle collisions with railroad trains increased by 24%. The severity of motor vehicle–train accidents is demonstrated by the fact that the 102 people killed in this type of accident in Serbia during 2009 represented 6.8% of the total highway fatalities while motor vehicle–train accidents accounted for only 2.4% of the total number of traffic accidents (Bureau of Transport, 2008). It is usually difficult to assign a particular cause to RLC accidents. Accidents may be caused by an error in perception, judgment, or action by the motor vehicle driver (European level crossing forum, 2011). Such factors as weather conditions, distractions, obstructions, railroad and highway traffic and operational features, geometry of the railroad, roadway and grade crossing, and type of protective device may be related to the causes of an accident. Possible solutions to the grade crossing problem have included better enforcement of laws and regulations which apply to motor vehicle drivers at grade crossings, improvement of the level of grade crossing protection and construction of grade separations. Application of the latter two alternatives by highway and traffic engineers is economically limited. It is estimated that $ 86 billion would be required to separate all grade crossings in the United States (US Department of transportation, 2008). Even the installation of automatic protective devices at all crossings would cost a minimum of $ 1.8 billion with annual maintenance costs averaging about $ 200 million per year. The total application of either alternative would not only be prohibitive in cost, but economically unjustified. Based upon engineering principles, a feasible solution is to develop some type of priority rating system for the improvement of the level of grade crossing protection. Public crossings in Serbia (approximately 2350) are protected either passively (64%) or actively by automated systems/devices (28%) (Bureau of Transport, 2008). Passive crossings provide only stationary signs without train information. Drivers have to look for the presence of a train before clearing the crossing. An active warning system, by contrast, activates automatic warning devices (i.e., flashing lights, continuous bell, etc.) as a train approaches. In Australia, records show a reduction in accidents following the installation of active warning systems (Ford & Matthews, 2002; Wigglesworth & Uber, 1991). However, improving safety at RLC is costly. The cost of an active level crossing protection system is generally accepted at approximately 500,000 dollars per crossing (Graham & Hogan, 2008). The cost of installing conventional active systems at all passive crossings in Serbia would therefore be as high as $ 1.17 billion. In addition, on-going maintenance costs would be considerable in view of the remote locations of many current passive crossings (Serbian Transport Council, 2010). There has been considerable research and innovation in some countries regarding the development of low cost RLC warning systems at crossings, on trains, or in vehicles. A recent comprehensive literature review identified approximately 50 different systems (Tey,

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Ferreira, & Dia, 2009). Although many of these systems have been invented, their effect on safety and driver acceptance is unknown. There are opportunities for immediate application of some lowcost innovative systems for RLC available worldwide, subject to their effectiveness and adaptation to Serbian conditions. The effectiveness of these alternative systems needs to be assessed to reflect safety improvements at crossings. However, to date, there has been no systematic approach available to evaluate these systems for implementation in Serbian conditions other than before-and-after implementation studies. This paper describes modeling of the Adaptive Neuro Fuzzy Inference System (ANFIS) and defines the criteria which influence the choice of level crossings for installing safety equipment in order to increase their level of safety in Serbia. In this paper is a study in which the criteria were identified for describing the safety parameters at railway crossings and which directly influence the selection of level crossings for an investment in safety equipment. After defining the criteria and relative weight using the Delphi method, modeling of the adaptive neuro-fuzzy network was carried out which has the ability to reproduce the decisions of experts. The ANFIS model was trained with a data set obtained using the method of fuzzy multi-criteria decision making (fuzzy TOPSIS and fuzzy AHP) and the fuzzy clustering technique. The entry parameters in the adaptive neuro-fuzzy network are the criteria which influence the choice of level crossing, and indicators that describe the given criteria are described in linguistic variables represented by membership functions. 20 experts from the field of road and rail traffic safety at level crossings took part in this study. The experiential knowledge of the experts was mapped into the database of rules for the ANFIS model, forming a unique base of knowledge used to make a selection of which level crossing to improve safety at. The value of a criterium function was obtained as output from the system for each level crossing observed. On the basis of the criterium function values obtained, ranking of the level crossings was carried out. The adaptive neuro-fuzzy network was tested on a selection of railway crossings in the Belgrade area. The adaptive neuro-fuzzy network was tested on 88 level crossings and the data set obtained was compared with the data set obtained on the basis of the predictions made by experts. The study is organized as follows. Section 1 presents some of the most important models for the prioritization of level crossings in the world. Modeling of an adaptive neuro-fuzzy network is described in the second section. The process of mapping the fuzzy system in the adaptive neural network is specifically described in the modeling process. In addition, the process of obtaining the numerical set of data is described, which is later used for training the adaptive neuro-fuzzy network. In the third section of the paper is the testing of the ANFIS model on the prioritization of 88 level crossings in the Belgrade area. When testing the model, the output parameters of the ANFIS model are compared with the desired data (training data set).

2. Models for the prioritization of railway crossings The first mathematical models which were developed in the mid 20th century carried out the evaluation and ranking of railway crossings on the basis of the predicted number of accidents which could happen on them. In the next section, an overview will be given of the mathematical model for the evaluation and ranking of railway crossings based on the predicted number of traffic accidents. Accident prediction equations are formulated to estimate the number of accidents that might occur at a particular location over a given period of time. The resulting accident frequency predictions have been used in the determination of priorities for the improvement of grade crossing protection (McEachem, 1960).

Accident prediction equations are expressed in terms of a relative weighting of various influencing variables. These variables are initially selected on the basis of their possible correlation with accident occurrence. Among those variables included in previous research investigations are: average daily traffic volume, average daily train volume, type of protection, daylight or darkness, number of tracks, train speeds, vehicle speeds, type of highway, geometries of the crossing (sight distance, crossing alinement, etc.), pavement width and number of lanes, type of highway surface, distractive influences, visibility, illumination, and vehicle and driver characteristics. An initial study to develop a prediction equation for the number of grade crossing accidents was the previously mentioned investigation by Peabody and Dimmick (McEachem, 1960). A correlation analysis was used to develop the following equation:

I ¼ 1:28 

H0:170  T 0:151 P0:171

þK

ð1Þ

where I- probable number of accidents in a five-year period, H-average daily highway traffic, T- number of trains per day, P-protection coefficient, and K-special variable to be calculated from data in the report. The Oregon State Highway Department completed a study concerned with measuring the relative hazards of railroad grade crossings located on state and federal-aid highway systems (Hays, 1964). The majority of the 400 railway level crossings considered were located in incorporated areas. Using accident data for the five-year period from 1946 to 1950, accidents were correlated with possible combinations of four influencing variables: Vehicle volume (v), Train volume (t), Darkness factor (d) and Protection factor (p). The following curvilinear accident prediction curve provided a 0.72 index of correlation:

a2 ¼ 0:40 þ 7:53  105 V  8:72  105 V 2

ð2Þ

where a2- predicted number of accidents for a five-year period, and V = vtdp. To compensate for the effects of possible influencing variables that were not considered, the ratio of actual accidents (a1) to predicted accidents (a2) for a previous five-year period was used as an adjustment factor in the final equation for measuring relative hazard:

IH ¼ VA

ð3Þ

where IH-index of hazard, V = vtdp, and A = a1/a2. The Armour Research Foundation has conducted two grade crossing accident studies for the Association of American Railroads. The results of an analysis of 2291 grade crossings in the State of Iowa were reported in 1958 (Crecink, 1958). Regression analysis techniques were utilized to develop risk factors (the expected accident rates at grade crossings over a 16-year period) as a function of type of protection, highway traffic volume, number of tracks, and measure of visibility. However, the regression model lacked consistency with accepted a priori assumptions concerning the relationships between the study variables. The second study performed by the Armour Research Foundation was an investigation of the relationships between accidents and nine grade crossing characteristics at 7416 locations in the State of Ohio (Crecink, 1958). A regression analysis routine was used to develop models predicting a 10-year expected accident rate. Equations were developed for four separate types of protection: painted crossbucks, reflectorized crossbucks, flashers, and gates. The predictors used in the models were: Average visibility, Highway grade, Rail traffic volume, Rail traffic speed, Highway traffic volume and Number of tracks and spurs. D.G. Newnan also developed accident prediction equations in conjunction with an engineering economic analysis of grade cross-

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ing protection improvements (Newnan, 1958). By analyzing 617 rail crossings on the California state highway system and collecting accident data for an 18-year period, weighted two-year accident rates were linearly related to the following characteristics: number of tracks, weather (visibility), number of trains, crossing angle, approach grade and corner visibility. The next part of this paper presents a list of models used in different states in USA to prioritize rail-highway grade crossings. The source of this list is a report produced by the University of Illinois in September 2000 (Elzohairy & Benekohal, 2000). The Department of Transportation accident prediction formula (USDOT Accident Prediction Model) combines three calculations to produce an accident prediction value. The expected number of accidents at a crossing is calculated using the following formulas: a formula that contains geometric and traffic factors from the inventory file, a formula that involves crash history and a formula that incorporates the effect of the existing warning devices. The basic formula provides an initial prediction of accidents on the basis of crossing characteristics. It can be expressed as a series of factors that, when multiplied together, yield an initial prediction of the number of accidents per year at a crossing. Each factor in the formula represents a characteristic of the crossing. The formula is:

a ¼ K  EI  DT  MS  HP  HL  HT

ð4Þ

where a-un-normalized initial crash prediction, in crashes per year at the crossing, K-formula constant, EI- factor for exposure index based on the product of highway and train traffic, DT- factor for number of through trains per day during daylight, MS- factor for maximum timetable speed, MT- factor for number of main tracks, HP- factor for highway paved (yes or no), HL-factor for number of highway lanes. The second formula, which is the general DOT accident prediction model, is expressed as follows:

  T o ðaÞ T N B¼ þ ðT o þ TÞ ðT o þ TÞ T 

ð5Þ

where N – observed crashes in T years at the crossing, T – number of years of recorded crash data, To – formula weighting factor 1.0/ (0.05 + a). The formula provided is most accurate if all the accident history data available are used. However, the extent of improvement is minimal if data for more than five years are used. California uses the hazard rating formula (California’s Hazard Rating Formula), which uses four factors: number of vehicles, number of trains, crossing protection type and crash history as input to the model. This formula uses a 10-year crash history as input. This formula does not compute the number of crashes but rather produces a hazard index as a surrogate for the number of crashes. The highest priority is assigned to the crossing with the highest calculated index. The hazard index is calculated as:

HI ¼

V  T  PF þ AH 1000

ð6Þ

where V – number of vehicles, T – number of trains, PF – protection factor and H – crash history (total number of crashes within the last 10 years). Connecticut uses a hazard rating formula (Connecticut’s Hazard Rating Formula) that is very similar to that of California. The only difference between the two is the crash history period. Connecticut uses a 10-year crash history while California uses a five-year history. The Connecticut formula uses four factors: number of vehicles, number of trains, crossing protection type and crash history as input to the model. The Connecticut formula does not compute the number of crashes but rather produces a hazard index as a surrogate for the number of crashes. The highest priority is assigned to the crossing with the highest calculated index. The Hazard Index is calculated from the following formula:

HI ¼

ðT þ 1ÞðA þ 1Þ  AADT  PF 100

2211

ð7Þ

where T – train movements per day, A – number of vehicle/train crashes in the last 5 years, AADT – annual average daily traffic, PF – protection factor from. The original New Hampshire formula uses three factors: number of vehicles per day, number of trains per day and a protection factor based on the type of crossing. However, the modified New Hampshire model does not account for sight distance. The Modified New Hampshire model does not compute the number of crashes but rather produces a hazard index as a surrogate for the number of crashes. The highest priority is assigned to the crossing with the highest calculated index. The Modified New Hampshire Formula is as follows:

HI ¼

TrainADT  HighwayADT  PF  SDf  T s  AHf 100

ð8Þ

where PF – protection factor, SDf – sight distance factor, Ts – train speed (mph), AH – five year crash history factor. Kansas uses the Design Hazard Rating Formula (Kansas’s Design Hazard Rating Formula), which uses five factors: number of vehicles, number of fast trains, number of slow trains, angle of intersection between the road and the track (0–90° range), and sight distances of all the four quadrants (Mohammad et al., 2003). If the computed Hazard Rating is less than 0, it is set to 0. The highest priority is assigned to the crossing with the highest calculated index. The Design Hazard Rating Formula is:

HR ¼

A  ðB þ C þ DÞ 4

ð9Þ

where

A¼

HT  ð2  NFT þ NSSTÞ 400

ð10Þ

where HT – highway traffic, NFT – number of fast trains, NST – number of slow trains, D-constant,

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8000 3 B¼2 sum of max sight distance 4ways sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 90 C¼ Angle of intersection

ð11Þ ð12Þ

Mohammad et al. (2003) modified the Kansas Design Hazard Rating model for the Missouri Department of Transportation. The Mohammad et al. (2003) formula uses eight factors: annual average daily traffic, number of passenger trains, stopping sight distance, approach sight distance, speed of train, total number of trains, speed of highway traffic, number of quadrants sight is restricted from and clearance time. The highest priority is assigned to the crossing with the highest calculated index. The revised formula is given by:

HR ¼

A  ðB þ C þ DÞ 4

ð13Þ

where D is a constant,

A ¼ ðVM  VSÞ½ðFM  FSÞ þ ðPM  PSÞ þ ðSM  10Þ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8000 3 B¼2 sum of max sight distance 4ways sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 90 C¼ Angle of intersection

ð14Þ ð15Þ ð16Þ

The Missouri crossing improvement program currently uses a calculated Exposure Index (Missouri’s Exposure Index Formula) to prioritize crossings for possible improvements. The Missouri Department of Transportation (MODOT) currently uses an Exposure

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Index (EI) formula to prioritize crossings for safety upgrades at rail-highway crossings. The EI formula was developed in the 1970s and has not changed since then. The EI formula uses nine factors: number and speed of vehicles, number of passenger and freight trains, speed of passenger and freight trains, switching movements and required and actual sight distance (Elzohairy & Benekohal, 2000). All nine factors, or data items, are currently maintained in Missouri’s crossing inventory. The EI is computed differently depending on the type of control at the crossing: (a) When a passive to active upgrade is being considered the EI is

EI ¼ TI þ SDOðTIÞ

ð17Þ

(b) When an Active upgrade is considered EI = TI where TI – traffic index, SDO – sight distance obstruction factor. The traffic index (TI) is the major component of the exposure index. It is determined as follows:

TI ¼

ðVM  VSÞ½ðFM  FSÞ þ ðPM  PSÞ þ ðSM  10Þ 10000

ð18Þ

where TI – traffic index, EI – exposure index, VS – vehicle speed, VM – vehicle movements, PM – passenger train movements, PS – passenger train speed, FM – freight train movements, FS – freight train speed and SM – switching movement. The highest priority is assigned to the crossing with the highest calculated index. In 2000, the Illinois Department of Transportation conducted a study that evaluated the Expected Accident Frequency Formula (Illinois’s modified expected accident frequency formula) used to rank grade crossings (Elzohairy & Benekohal, 2000). The study recommended the model that is being used by this study for evaluation. The model was developed using the non-linear regression analysis procedure on grade crossing accidents in Illinois. The model is as follows:

IHI ¼ 106  A2:59088  B0:09673  C 0:40227  D0:59262  ð15:59  N 5:60977 þ PFÞ

ð19Þ

where A – ln (ADT  NTT), ADT – average daily traffic, NTT – number of total trains per day, B = MTS – maximum timetable speed (mph), C – number of main tracks and other tracks, D – number of highway lanes, N – average number of crashes per year and PF – protection factor. Besides the given model, in many countries worldwide evaluation of railway crossings is carried out using Quantified Risk Analysis. Quantified Risk Analysis (QRA) provides a suitable basis for establishing level crossing improvement priorities. This it does by allowing a ranking of level crossings in terms of their accident risk probability. Those crossings with high accident probabilities would normally qualify for funding allocations (subject to satisfactory cost/benefit results), while those with low accident probabilities would be assigned a low priority for improvement funding. The QRA results were linked to the Level Crossing Inventory Recording System which enables reporting of hazard probabilities against each level crossing. Factors influencing the probability of accident occurrence at level crossings include: rail traffic density (measured in terms of the maximum number of trains passing the crossing within a 24 h period); road traffic density (measured in terms of the maximum number of motor vehicles of all types passing the crossing within a 24 h period); the presence of physical obstructions restricting the visibility of the track; warning signs or signals to road users; absence of a full width barrier protection at level crossings; absence of flashing lights and audible warning devices at level crossings; poor road surface conditions at level crossings (leading to the grounding of low slung road vehicles); and poor alignment and elevation of the road crossing with the track (the road may cross the track at an oblique angle or may approach the crossing on a steeply rising grade). The application of these techniques can be seen in the works of many authors (Anandarao &

Martland, 1998; Bureau of Transport & Regional Economics, 2002; Reiff, Gage, Carroll, & Gordon, 2003; Tey et al., 2009; Woods et al., 2008). In the countries of the European Union, the Traffic movement (TM) model is used for the prioritization of level crossings, which, as the name implies, above all, takes care of the transport flux at the road crossing. TM is a product of the daily number of road vehicles (passenger vehicles, trucks, buses) and the daily number of trains crossing the road. In using the TM model the losses are calculated (or savings, if investment is made in the level crossing) because of waiting time for vehicles per unit of time. Thus, the average waiting time at a road crossing for particular types of motor vehicles is calculated, on the basis of which the cost of passenger and vehicle waiting time can be calculated (fuel, vehicle depreciation, delay in delivery of goods or services etc.). The passenger waiting costs for specific types of journey and vehicle category are obtained on the basis of the estimated vehicle structure by country of origin, the estimated value of a working hour (EUR/hour) in those countries, and on the nature of the journey (Serbian Transport Council, 2010). Roop (2005) and Mendoza (1999) adopted the multi-criteria analysis technique to assess the relative merits of the candidate protection systems and the evaluate RLC. As compared to the conventional cost-benefit approach, multicriteria analysis allows effective comparative evaluation among options and stakeholders over a common set of evaluation objectives. Furthermore, multicriteria analysis could overcome the limitation of cost-benefit analysis whereby all the costs and benefits have to be expressed in monetary terms.

3. ANFIS decision support model for prioritizing railway level crossings for safety improvements in Serbia On the Serbian rail network which is 6974 km long, there are 2354 level crossings, of which 108 are pedestrian crossings. Of the total number of railway crossings in Serbia, 588 railway crossings are secured by automatic or mechanical devices. The remaining railway crossings are secured by road signalization signs. Table 3 shows the safety of railway crossings in the rail network of the Republic of Serbia (STSA, 2011). In Serbia, 77% of railway crossing are not secured according to the Law on Road Traffic and the applicable instructions from Serbian Railways (Serbian Transport Safety Agency, 2010). This means that in Serbia there are a large number of level crossings which have been allowed to operate, and for which basic parameters have not been respected, such as the distance between railway crossings or the visibility of road vehicles when they reach the rail crossing. The safety of rail crossings is a material expenditure for Serbian Railways. However, since Serbian Railways is a public company financed from the state budget, it cannot be expected that all unsecured rail crossings enter a safety programme. For this reason, it is essential that Serbian Railways and the Republic of Serbia have a reliable strategy for the choice of RLC which need to be secured and an implementation plan for investing in their safety. Until now, decisions by Serbian Railways regarding which rail crossings should receive safety intervention have not received enough insight from the state in terms of the choice of RLC which should receive investment. Most commonly, Serbian Railways makes a decision based on public pressure as a result of accidents which have occurred at particular rail crossings. In such situations, the basic criterium considered is that of urgency, which falls into the group of criteria suitable for systems without a strategy. According to statistical data and EU forecasts (European Railway Agency, 2011), the volume of rail traffic in the next 30 years will be doubled, which is a direct indicator of the expected increase

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Table 3 The safety of railway level crossings in the rail network of the Republic of Serbia. Line rank

Rail crossing (SVT)

Pedestrian crossing (BMVT)

Secured with automatic or mechanical devices

Total

MF

LAS

LASH

Pedestrian crossings (LASBM)

Total

International Regional Local Total

382 456 820 1658

35 42 31 108

87 34 11 132

116 13 12 141

176 86 45 307

8 0 0 8

387 133 68 588

804 631 919 2354

⁄ Traffic signs and visibility triangle (SVT), pass-by boundary marker and visibility triangle (BMVT), mechanical fenders (MF), Light-acoustic signals (LAS), Light-acoustic signals and half-fenders (LASH), Light-acoustic signals and pass-by boundary marker (LASBM).

in accidents at all rail crossings on all lines including those of Serbian Railways. An increase in the volume of traffic will result in the necessity for a traffic safety system at rail level crossings. According to the long term plans of the Republic of Serbia, in the period 2010–2025 new railway lines and corridors will not be constructed (Serbian Transport Council, 2010). Economic experts in the field of industrial production forecasts (transportation needs and opportunities and needs of the economy) are of the opinion that the existing corridors and existing tracks will remain unchanged in this part of Europe. It is realistic to expect the reconstruction and modernization of existing systems, for which significant investment is necessary. In this context, the necessity for investing funds in the safety of rail crossings will be defined. Installing safety equipment at RLC is an expensive investment, so when making decisions about investment, management has a great responsibility, because the approved funds must give an adequate effect. In order to make the best possible decisions about investments in rail crossings, a neuro-fuzzy model was developed for use by management. In the following section of the paper, the development of the neuro-fuzzy model for the prioritization of rail crossings is described for making decisions about investing in safety equipment for rail crossings. 3.1. Designing the ANFIS model The problem being considered belongs to the task of allocation (assignment). The problem of allocation belongs to the problem of linear programming. It consists of allocating activities or resources to the agent or the place, at the same time achieving maximum efficiency. In our case, this means that it is necessary to define the objective function i.e. to allocate investments in safety equipment for rail level crossings with limitations and treating the problem as a problem of mathematical programming. The main deficiency of an approach based on mathematical programming is that it is not simple to determine the objective function and to determine the ‘‘hard’’ constraints. In addition, the information available to decision-makers is often imprecise or is given in descriptive form. Furthermore, decision-makers on investments also make their selection of railway crossing based on additional criteria that reflect the real situation on the railway crossing observed. Some of the additional criteria are: – existing safety equipment on rail crossings, – rail crossings found on the lines of higher rank have priority over those of lower rank, – the number of fatal incidents at the rail crossing in the previous six months, – the number of incidents at the rail crossing due to non-compliance with traffic and other regulations, – rank (priority) of road traffic which passes through the rail crossing, – total waiting time of users of the rail crossing for the passage of trains.

Therefore, the conventional approach cannot encompass all the relevant imprecise parameters. In the majority of cases, this phase of the decision-making process comes down to the experiential knowledge of the decision maker. However, a problem arises when a decision on an investment in railway crossings should be made a person who does not have sufficient experiential knowledge. A solution to the above problem is proposed in this paper using the ANFIS model. An integral part of the ANFIS model is a fuzzy reasoning system. The problems that an analyst encounters when developing a fuzzy system are determining a set of linguistic rules used by the expert and the parameters of membership functions of the input/output pairs. Generating the membership function of a fuzzy set and the rules by which decisions are made by experts involves long communication with a large number of experts. Membership functions of fuzzy sets, which describe the same concept, and which have been proposed by experts can differ greatly. For this reason, the characteristics of a developed fuzzy system depend on the number of available experts and their ability to formulate a decision-making strategy. Based on the analysis of these techniques and criteria used for the prioritization of railway crossings (Section 2) eight criteria were identified which influence the selection of railway crossing for installing the equipment necessary for increasing the safety of traffic at that crossing (Table 4). Experts in the field of road and rail traffic safety at railway crossings were involved in choosing the criteria and their indicators. As well as eight input variables the fuzzy system has one output variable, the preference of experts for the choice of railway crossing to receive an investment. The relative importance of criteria and their degree of influence on the preference decided by experts are obtained by normalization of their weight in the following way:

w0i

" n Y

" #1=n n Y kj  g w0ki g ¼ kj wki ! wki ¼ PK ¼ kj wki 0 g j¼1 j¼1 j¼1 kk  wki 8 2" #1=n 391 n
XK j¼1

#1=n

fk 2 ½0; 1k 2 ½0; 1 fk ¼ 1 w w

ð20Þ ð21Þ

where kj is the preference of a decision-maker, i.e. the degree of confidence. The described criteria are listed in Table 4. The composite of Ki(i = 1, 2, . . . , 8) is made of two subsets:  K+, subset of the criteria of the beneficial type, higher values desirable and  K, subset of the criteria of cost type, lower values desirable. The values of some input variables are described by means of linguistic descriptors S = {l1, l2, . . . , li}, i 2 H, H = {1, 2, . . . , T} where T is the overall number of linguistic descriptors. Linguistic variables are presented by a triangle fuzzy number which is defined as (a, b, c).

´ irovic´, D. Pamucˇar / Expert Systems with Applications 40 (2013) 2208–2223 G. C

2214 Table 4 Criteria for evaluating the RLC. Criterion

Numerical

K1 – Frequency of rail traffic on the monitored rail crossing K2 – The frequency of road traffic on the monitored rail crossing K3 – Number of tracks on the monitored rail crossing K4– Maxim permitted speed of trains for the chainage of the rail crossing K5 - The angle of intersection of track and the roa K6 – The number of incidents at the monitored rail crossing in the past year K7 – Visibility of the monitored rail crossing in terms of road traffic K8 – Investment value of activities related to the width of the rail crossing

     

Linguistic

Min     a 

 

Max

a 



Importance of the criteria 0.12 0.19 0.11 0.08 0.15 0.12 0.14 0.09

a The criterium K5 at the same time falls into the group of beneficial and cost criteria. Criteria values found in the interval 30 6 x1 6 90 are in the group of beneficial criteria, while criteria values found in the interval 90 6 x1 6 175 are the group of cost criteria.

l~li ðxÞ ¼

8 0; x > > > xa < ; a 6x6b ba cx > > cb ; > > : 0;

ð22Þ

b6x6c x>c

In our example, the number of linguistic variables is T = 5 (Fig. 1). Defuzzification of linguistic variables (criteria) K7 iK8 is carried out using the scale shown in Fig. 1. Since linguistic values lki ði ¼ 1; T; k ¼ 1; KÞ are described by fuzzy numbers ~lki ¼ flki ; l~lki g the process of normalization is realized according to the following: (a) for a beneficial criterion k(k 2 K) the process is realized according to the form

ðlki Þn ¼

~l ki max lk

ð23Þ

max

where lk is the maximal value of fuzzy number ~l ðk ¼ 1; 2; . . . ; KÞ, for l~ ðl Þ – 0. ki lki ki (b) for a cost criterion k(k 2 K) the process is realized according to the following

ðlki Þn ¼ 1 

~l  lmin ki k max lk

ð24Þ

min

where lk is the minimal value in the area of fuzzy number ~l ðk ¼ 1; . . . ; KÞ for l~ ðl Þ – 0. ki lki ki Defuzzification of the linguistic descriptors is carried out using the expression (Liou & Wang, 1992):

g a;b ðe LÞ ¼ ½b  fa ðl1 Þ þ ð1  bÞ  fa ðl3 Þ;

0 6 b 6 1; 0 6 a 6 1

ð25Þ

where fa(l3) = l3  (l3  l2)  a is the function which represents the left distribution of the universe of discourse of the fuzzy number e L while fa(l1) = (l2  l1)  a + l1 is the function which represents the right distribution of the universe of discourse of the fuzzy number e L. The value a(0 6 a 6 1) represents the preference of the decision-maker, while the value b(0 6 b 6 1) represents the pessimistic index of the decision-maker. The degree of uncertainty is greatest when the value is a = 0, while on the other hand the value b = 0 represents the optimistic index of the decision-maker. After defining the criteria, the parameters which describe the criteria were identified (Table 5). The parameters were used later in the process of modeling the fuzzy system for defining the universe of discourse of the membership functions (MF) of the input variables. By defining the criteria for the selection and evaluation of rail crossings it is possible on the basis of these criteria to create a database of railway crossings that is unique for all RLC in the network (or defined line) and using the criteria makes it possible for each RLC to be treated in the same way. The main problem faced by the analyst in the development of a fuzzy system is determining the fuzzy rule base and membership function parameters of fuzzy sets that describe the input and output variables. Used as functions in the fuzzy system were Gaussian curves (gmf), S-shaped membership functions (smf) and Z-shaped membership functions (zmf) (Fig. 2). The above membership functions (Table 6) were chosen because they are easy to manipulate while adjusting the fuzzy logic system, because they describe the

Table 5 Crieria and indicators for the evaluation of rail crossings. Criterium

Indicator

K1

K11 K12 K13 K21 K22 K23 K31 K32 K33 K41 K42 K43 K51 K52 K53 K61 K62 K63 K71 K72 K81 K82 K83

K2

VL

1

L

M

H

VH K3

0.8 K4

0.6 K5

0.4 K6

0.2 K7

0

0.1

0.2

0.3

0.4

0.5

0.6

Fig. 1. Linguistic descriptors.

0.7

0.8

0.9

K8

> 100 trains/day 50–100 trains/day 0–49 trains/day > 200 vehicles/hour 80–150 vehicles/hour 1–80 vehicles/hour 1 2 >2 >100 km/h 60–100 km/h 20 59 km/h 30°-80° 80°–100° 100°–175° 0–2 2–4 >4 good bad small medium large

´ irovic´, D. Pamucˇar / Expert Systems with Applications 40 (2013) 2208–2223 G. C

Mala

Srednja

Mala

Velika

Srednja

Velika

Vrlo__velika

Ma li

Sred nji

Veli ki

2215

Mala

0. 8

0. 8

0. 8

0. 8

0. 6

0. 6

0. 6

0. 6

0 4 0

16 0

8 10 12 14 Frekfencija__zeleznickog__saobra 0 0 0 0 caja__na__pp

6 0

mali

srednji

18 0

20 0

veliki

Ma li

15 20 10 0 0 0 Frekfencija__drumskog__saobrac aja__na__pp Sred nji

25 0

30 0

0

Veli ki

1

0. 8

0. 8

0. 8

0. 6

0. 6

0. 6

0. 2

0. 2

0

0

0

2 0

4 0

6 0

8

1 0 0

0 Ugao ukrstanja puta i pruge

1 2 0

1 4 0

1 6 0

1 8 0

0

1

1. 5

2. 2 Broj__koloseka__3 5 na__pp

3. 5

4. 5

4

Dob ra

0. 5

1. 1Broj__nezgoda__na_datom__pp__u_preth 2 5 odnom__periodu

2. 5

3

0

2 0

0

5

Los a

4 0

8 10 12 14 6 0 0 0 0 0 Max__dozvoljene__brzine__na__pruzi__na__pp

Low

16 0

Medium

18 0

20 0

High

1

0.8

0. 4

0. 4

0. 2

0

0. 5

Degree of membership

1

Degree of membership

Degree of membership

1

0. 4

0

0 5 0

0

Degree of membership

2 0

0. 2

0. 2

0. 2

0 0

Degree of membership

Degree of membership

Degree of membership

0. 2

0. 4

0. 4

0. 4

Velika

Degree of membership

1

0. 4

Srednja

1

1

1

0.6

0.4

0.2

0 0. 1

0. 2

0. 0. 0. 0. 0. 0. Preglednost__posmatranog__pp__sa__aspekta__dru 7 8 3 4 5 6 mskog__saobracaja

0. 9

1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Investiciona vrednost aktivnosti u funkciji širine pružnog prelaza

Fig. 2. Membership functions before training the ANFIS model.

Table 6 Parameters of the membership function when training the ANFIS model. Membership function/Input value

MF 1

MF 2

MF 3

MF 4

K1 K2 K3 K4 K5 K6 K7 K8

zmf (40.7, 154.5) zmf (36.9, 183.3) zmf (1, 3.62) gmf (34.3, 18.6, 31.5, 49.8) gmf (18.85, 32.7, 20, 44.7) zmf (1, 1.67) zmf (0.166, 0.784) gmf (0.072, 0.6053)

gmf (33.6, 53,4.3,87.5) gmf (44.2, 64, 48.2, 130.4) gmf (0.94, 2) gmf (35.4, 64.2, 38.2, 93.4) gmf (18.7, 77, 29.2, 104) gmf (0.44, 1.31) smf (0.14, 0.79) gmf (0.139, 0.5092)

smf (7.6, 109) gmf (41.2, 151, 36.9, 184.5) smf (0.38, 2.7) smf (17.31, 113) gmf (63.4, 119.8) smf (0.87, 1.67) – gmf (0.3659, 0.8917)

– smf (65.8, 207) – – – – – –

input variables well and because they ensure a satisfactory level of sensitivity in the system. In addition, adjustment of the above membership functions ensures the smallest error in the output of the ANFIS model. The parameters of the membership functions of the fuzzy system are given in Table 6. For all input variables of the ANFIS model, as well as the membership function, it is also necessary to determine the number of membership functions for each input. Having a greater number of membership functions requires an increase in the number of rules, which can make it difficult to adjust the system. It is therefore recommended, in accordance with the nature of the variables, to begin with the lowest number of membership functions. However, reducing the number of membership functions must not result in an incomplete description of the input variables. Starting from these postulates it is defined that in the ANFIS model, the input variables have at least two linguistic values (input variable K7) and at most four variables (input variable K2). The membership functions of the input variables are shown in Fig. 2. By comparing the output data from the fuzzy system and the desired set of solutions, the designed the fuzzy system did not give satisfactory results. The difference between the expected results and the value of the criteria function obtained at the output of the system was not satisfactory i.e., it was not within the limits of tolerance. An attempt to achieve satisfactory results by changing the type and parameters of the membership functions at the output did not give the expected results. Table 7 shows the values of the criteria functions at the output of the fuzzy system and the expected values of the criteria functions. As well as the criteria functions in Table 7 are also the values of the criteria on the basis of which the prioritization of rail crossings is carried out, which at the same time are the input variables of the fuzzy system. In the example shown in Table 7, a total of 25 railway crossings were processed.

Table 7 Characteristics of twenty-five level crossings.

a

No.

K1

K2 a

K3

K4

K5

K6 a

K7

K8

fexpert

fFIS

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25.

62 51 64 24 35 39 48 35 72 54 49 47 56 37 41 35 58 46 53 31 49 47 53 34 42

56 80 61 79 67 53 86 41 93 77 69 62 31 55 23 49 56 53 72 57 82 75 98 58 75

1 1 2 2 4 1 4 1 2 2 2 2 1 1 1 1 1 4 4 1 2 2 2 2 2

60 55 60 60 60 60 70 65 65 55 70 70 70 70 65 50 50 50 65 70 50 45 45 50 70

70° 60° 90° 80° 100° 95° 110° 130° 60° 60° 150° 95° 90° 40° 45° 130° 115° 75° 80° 65° 70° 85° 100° 55° 90°

2 1 1 1 2 2 2 4 3 1 4 0 1 3 1 3 0 1 2 2 1 1 0 2 0

L VL M M M M VH VH L VL L M L H H H M M H L L VL H L M

M L H VH M H H H M H L H M M M M H L VH M VL M H M L

0.70 0.72 0.70 0.69 0.75 0.65 0.78 0.68 0.80 0.72 0.87 0.69 0.65 0.58 0.48 0.71 0.69 0.65 0.72 0.65 0.72 0.74 0.75 0.63 0.70

0.48 0.71 0.43 0.67 0.38 0.39 0.45 0.51 0.56 0.37 0.77 0.58 0.59 0.35 0.41 0.40 0.36 0.48 0.64 0.39 0.62 0.59 0.48 0.50 0.60

The values are the average indicators on an annual basis.

By analysing the given data, an average error of 0.325 was obtained. Since the fuzzy system did not achieve the desired results, the fuzzy system was mapped into a five-layered adaptive neural network (ANFIS model), Fig. 3. Mapping of the fuzzy logic system into the adaptive neural network was carried out because the error

´ irovic´, D. Pamucˇar / Expert Systems with Applications 40 (2013) 2208–2223 G. C

2216

Layer 1

Layer 2

Layer 3

Layer 4

Layer 5

A1 K1

x1

A2 A3 B1

K2

x2

B2 B3

µv

K3

) (y

2

C2

x3

(y) µ v1

C1

µv ) (y

3

C3 D1

µv

4 (y

K4

)

D2

x4

D3

µv5(y)

y

Preferential eksperta

E1 K5

x5

y) µ v6(

E2

(y) µ v7

x6

v8

µ

µv

K6

9 (y

F1

)

(y

)

E3

F2 F3 G1

K7

x7

G2 G3 H1

K8

H2

x8

H3 O2i

O1i

O3i

O5i

O4i

Fig. 3. Structure of the ANFIS system.

which occurred at the output of the fuzzy system was unacceptable. As already said, the neuro-fuzzy network consists of five layers. Layer 1. The junctions of the first layer represent verbal categories of input variables that are quantified by fuzzy composites. Each junction of the first layer is an adaptive junction and is described by the function of adherence lxi ðxi Þ; i ¼ 1; . . . ; 4. Functions of adherence are described by the form of Gaussian curves that are featured by two parameters c and r. 1 xc 2

Gaussian ðx; c; rÞ ¼ e2ð r Þ

ð26Þ

Since fuzzy rules are expressed in the form ‘‘IF the condition THEN the consequence’’, the categories of output variables that are quantified by fuzzy composites are shown as adaptive junctions of the first layer. Layer 2. Each junction of this layer counts the minimal value of four input values. The output values of the junction of the second layer are the importance of rules.

O21 ¼ wi ¼ lAi ðx1 Þ  lBi ðx2 Þ  lC i ðx3 Þ  lDi ðx4 Þ

ð27Þ

Layer 3. Every i th node in this layer calculates the ratio of the i th rule’s firing strength to the sum of all rules’ firing strength.

wi O31 ¼ wi P4

i¼1 xi

;

i ¼ 1; . . . ; 4

ð28Þ

Layer 4. The fourth layer has five adaptive junctions which represent the preference of dispatchers that a certain transport requirement serves a certain type of vehicle. Each junction of this layer counts the section of a certain fuzzy composite with maximal value of input importance of rules.

O41 ¼ wi fi

ð29Þ

Layer 5. The only junction of the fifth layer is a fixed junction by which the output result of the fuzzy system is gained. This is a fuzzy composite with certain degrees of adherence of possible preference of dispatchers to direct the transport task to a certain transport route considered. The output value is a real number that is found in the interval of 0–1.

O51 ¼ Ov erall output ¼

P X wi fi wi fi ¼ Pi i wi i

ð30Þ

3.2. Forming a data set for training the ANFIS model ~ ¼ ðyl ðrÞ; yu ðrÞÞ; o 6 r 6 1; ~ ~2X If with ~ x ¼ ðxl ðrÞ; xu ðrÞÞ and y x; y we define the fuzzy numbers which are used to evaluate the observed alternatives in relation to the defined optimization criteria, ~ the following relationships are then for the fuzzy numbers ~ x i y valid:

~x ¼ y ~ ! xl ðrÞ ¼ yl ðrÞ ^ xu ðrÞ ¼ yu ðrÞ; ~x þ y ~ ¼ ðxl ðrÞ þ yl ðrÞ; xu ðrÞ þ yu ðrÞÞ

ð0 6 r 6 1Þ

ð31Þ ð32Þ

´ irovic´, D. Pamucˇar / Expert Systems with Applications 40 (2013) 2208–2223 G. C

( K ~x ¼

l

ðkx ðrÞ; xu ðrÞÞ; k P 0 u

ð33Þ

l

ðkx ðrÞ; x ðrÞÞ; k P 0

Let Ai ði ¼ 1; 2; . . . ; nÞ denote the set of rail crossings evaluated by experts Eg(g = 1, 2, . . . , k) in relation to the observed set of criteria Cj(j = 1, 2, . . . , n). Then the problem of fuzzy multi-criteria decision making can be represented in matrix form as:

2

~x11 6~ 6 x21 6 e ¼ 6 ~x31 D 6 6 4  ~xm1

~x12 ~x22

~x13 ~x23



~x32  ~xm2

~x33  ~xm3

 





3 ~x1n ~x2n 7 7 7 ~x3n 7 7; 7  5 ~xmn

i ¼ 1; 2; . . . ; m;

j ¼ 1; 2; . . . ; n

!

j¼1

j¼1

j ¼ 1; 2; . . . ; n

ð35Þ

j¼1

Normalization of the summarized values in rows is carried out using the following transformation

RSi e S i ¼ Pn ¼ j¼1 RSi

Pn

Pn u l j¼1 xij ðrÞ j¼1 xij ðrÞ Pn Pn l ; Pn Pn u x ðrÞ k¼1 j¼1 ij k¼1 j¼1 xij ðrÞ

¼ 1; 2; . . . ; n

n o eþ ¼ V e ij jj 2 JÞ; ðmin V e þ; V e þ; . . . ; V e þ ¼ fðmax V e ij jj 2 J 0 Þ; i ¼ 1; 2; . . . ngð39Þ A 1 2 n n o e ¼ V e ij jj 2 JÞ; ðmax V e ; V e ; . . . ; V e  ¼ fðmin V e ij jj 2 J 0 Þ; i ¼ 1; 2; . . . ngð40Þ A 1 2 n

J ¼ fj ¼ 1; 2; . . . ; mjj belongs to the criteria which are maximizedg;

where ~xij is the value of the criteria function of the given railway crossing Ai in relation to a criterion Cj. Summarizing the values in e is carried out using the following the rows of the matrix D transformation: n n X X xlij ðrÞ; xuij ðrÞ ;

The next step is to determine the ideal solution from the given set of values of criteria functions. The ideal solution A+ and the negative ideal solution A are obtained using the relation

where

ð34Þ

n X ~xij ¼ RSi ¼

! ;

J 0 ¼ fj ¼ 1; 2; . . . ; mjj belongs to the criteria which are minimizedg; As the best alternatives, those which have the highest value Vij in relation to the criteria which are maximized and the lowest Vij in relation to the criteria which are minimized are chosen. The positive ideal and negative ideal solutions are represented by fuzzy numbers. The following relations describe the ideal posieþÞ tive solution ð A

e þl ðrÞ; A e þu ðrÞÞ; e þ ¼ ðA A

06r61

ð41Þ

where

 l  e þl ¼ V e þ ðrÞ; V e þl ðrÞ; . . . ; V e þl ðrÞ A 1 2 n  u  e þu ¼ V e þ ðrÞ; V e þu ðrÞ; . . . ; V e þu ðrÞ ; A 1 2 n

ð42Þ 06r61

e  is calculated in exactly the same way. The ideal negative solution A The distance between the fuzzy numbers e x and e y is calculated as

i ð36Þ

The weight coefficient of each of these criteria is obtained by formf in which comparison is made in pairs of criteria ing a matrix W based on of decisions made by experts who participated in the study.

ð37Þ

2

ð38Þ

1

ðjxl ðrÞ  yl ðrÞj2 þ jxu ðrÞ  yu ðrÞj2 Þdr

1=2 ð43Þ

0

The next step is to calculate n dimensional Euclidean distances of all observed alternatives for the ideal and the negative ideal solution 00

00 B e d i ¼ @@

e and W f and by using the By multiplying the values of matrices D previously mentioned arithmetical operations, we obtain the final values of the criteria functions which describe the significance of each of the observed railway crossings 2 3 2 3 f1 ~x11 e x 12 e x 13    e x 1n W 6e 7 6f 7 x 22 e x 23    e x 2n 7 6 W 6 x 21 e 7 6 7 6 27 e e f 76W f3 7 e e e e F ¼DW ¼6 x x x    x 31 32 33 3n 6 7 6 7 6 7 6 7 4      5 4  5 fk e x m2 e x m3    e x mn x m1 e W 2 3 f f f fn e x 12  W 2 e x 13  W 3    e x 1n  W x 11  W 1 e 6 f1 e f2 e f3    e fn 7 6e 7 x W x 22  W x 23  W x 2n  W 6 21 7 6 f1 e f2 e f3    e fn 7 ¼6e x 31  W x 32  W x 33  W x 3n  W 7 6 7 4 5      f1 e f2 e f3    e fn e x m2  W x m3  W x mn  W x m1  W

Z

Dðe x; e yÞ ¼

B e d þi ¼ @@

3 ef 1 6 7 6 ef 7 6 27 7 e eW f¼6 F¼D 6 ef 3 7 6 7 6 7 45 ef i

2217

Z

l

0

Z 0

0 @

Xh

v

e lij ðrÞ

e þl ðrÞ V j

i2

þ

j2J 0 l

0 @

Xh j2J 0

Xh

v

e uij ðrÞ

j2J

ve uij ðrÞ  Ve j

þu

ðrÞ

i2

þ

Xh j2J

1 11=2 1 i2 C þu e  V j ðrÞ A þ dr A Að44Þ

11=2 1 ve lij ðrÞ  Ve j ðrÞ A þ drA C Að45Þ þl

i2

1

where i = 1, 2, . . . , m For each alternative the relative distance of the coefficients e dþ i i e d is calculated according to the relation i

eþ ¼ Q i

e d i e d þi þ e d i

;

i ¼ 1; 2; . . . ; m

ð46Þ

e þ 6 1 is the alternative Ai closer to an ideal solution if where 0 6 Q i e þ is closer to a value of 1. After obtaining the values of the criteria Q i functions the processed experimental data are accessed using clustering techniques. By cluster we mean a finite number of similar points that can be classified into the same group, by one or more distinctive features. The center of the cluster can be considered as representative of one group of data. It replaces the group of data and is the basis for establishing a code of conduct for the problem studied. In this way, a large number of experimental data is reduced to a smaller number of representative cluster centers and the research can continue with a smaller amount of data. In this research, a fuzzy clustering technique was used (Nasibov & Ulutagay, 2007). Nasibov and Ulutagay developed an iteration procedure based on minimizing a function that represents the geometric distance from any given point to the cluster center, but with an additional weight factor based on the membership function of the analyzed point. The e is calcudistance between two points of the data set of variable v lated as the minimal negative value of similarity

´ irovic´, D. Pamucˇar / Expert Systems with Applications 40 (2013) 2208–2223 G. C

2218

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

200

400

600

800

1000

1200

Fig. 4. The set of criteria functions before and after use of the clustering technique model after training the adaptive neural network.

dXY ¼ minflev g ev

ð47Þ

developed in the Matlab software package for implementing clustering techniques the set F0 was reduced to a total of F00 = 547 values of the criteria functions. A comparative presentation of the set of criteria functions F0 and F00 is shown in Fig. 4. Training of the adaptive neuro-fuzzy network is carried out with set F00 and a base of fuzzy rules is formed. If the initial set of criteria functions (F or F00 ) were used for developing a base of rules and training the neuro-fuzzy system, all data would be treated with the same importance and it would be impossible to construct a base of rules which would, as output from the neuro-fuzzy system, give results which deviate very little from the required values. With this kind of presentation of the system being studied it is possible to generate a neuro-fuzzy system with a minimal number of fuzzy rules. As well as this, the time for training the neuro-fuzzy system is significantly reduced.

The degree of belonging of each point to a cluster is defined as

mik ¼

1 2 P dik q1

ð48Þ

djk

where dik = kuk  cik is the distance of point from the cluster center i. The two points which have the lowest value dXY are considered to be the closest points. By analyzing the total number of optimality criteria and the number of linguistic descriptors which describe the given criteria, we see that the set of criteria functions is about F  a  bK = 1 953 125, where b = 5 is the total number of linguistic P factors, K ¼ 8i¼1 K i ¼ 8 is the total number of optimality criteria, and a = 5 is the total number of parameter values a, b which represents the degree of uncertainty (a) and the optimistic index of the decision-maker (b). Five values of the given parameters were considered in the research 0,0.3, 0.5, 0.7 and 1. Since neuro-fuzzy networks have the ability to generalize the data obtained, the set (F0 ) was identified for the research from F0 = 2200 of the criteria functions. Using the described clustering techniques and the toolbox Srednja

Velika

Mala

Velika

Vrlo__velika

Mali

Srednji

Veliki

Mala 1

0.8

0.8

0.8

0.8

0.4

0.6

0.4

0.6

0.4

0.2

0.2

0.2

0

0

0

20

40 60 80 100 120 140 160 Frekfencija__zeleznickog__saobracaja__na__pp

mali

srednji

180

200

0

veliki

50

Mali

1

100 150 200 250 Frekfencija__drumskog__saobracaja__na__pp

Srednji

300

0.8

Degree of membership

Degree of membership

0.4

0.6

0.4

0

20

40

60

80

100

Ugao ukrstanja puta i pruge

120

140

160 180

1.5

2 2.5 3 Broj__koloseka__na__pp

3.5

4

4.5

0

0.5 1 1.5 2 2.5 Broj__nezgoda__na_datom__pp__u_prethodnom__periodu

3

0.6

0.4

5

0

20

Losa

40 60 80 100 120 140 160 Max__dozvoljene__brzine__na__pruzi__na__pp

Mala

1

1

0.8

0.8

0.6

0.4

180

Srednja

200

Velika

0.6

0.4

0.2

0

0

0

1

0.2

0.2

0.2

0.5

Dobra

Veliki

0.8

0.6

Velika

0 0

1

Srednja

0.2

Degree of membership

0.6

Degree of membership

1

Degree of membership

1

0

Degree of membership

Srednja

The adaptivna neural network is trained using a Backpropagation algorithm (Horikawa, Furuhashi, & Uchikawa, 1992; Shi and Mizumoto, 2000).

1

Degree of membership

Degree of membership

Mala

3.3. Training the ANFIS model

0 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Preglednost__posmatranog__pp__sa__aspekta__drumskog__saobracaja

1

0

Fig. 5. Membership functions of the ANFIS model after training the adaptivne neural network.

1

2 3 4 5 6 7 8 Investiciona__vrednost__aktivnosti__u__funkciji__sirine__pp

9

10

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Table 8 Values of function parameters after training the ANFIS system. MF/Ulazna vrednost

MF 1

MF 2

MF 3

MF 4

K1 K2 K3 K4 K5 K6 K7 K8

zmf (10.3, 155.3) zmf (13.1, 148) zmf (0.36, 2.99) gmf (36.1, 14, 47.51, 36.7) gmf (23.9, 34.2, 24.8, 45.8) zmf (0.1782, 2.079) zmf (0.0807, 0.9272) gmf (2.13, 2.9, 2, 3.664)

gmf (46.9, 79.1, 35.03, 114) gmf (62.5, 56, 75.86, 103.6) gmf (0.9291, 2) gmf (37.6, 72.2, 38.5, 98.2) gmf (30, 95.6, 26.77, 107.3) gmf (0.5287, 1.31) smf (0.091, 0.911) gmf (1.96, 5.63, 2.01, 6.55)

smf (11.4, 188.6) gmf (73.2, 159, 61.75, 175) smf (1.08, 3.552) smf (27.78, 174) gmf (56.71, 161) smf (0.717, 2.647) – gmf (3.85, 9.405)

– smf (54.37, 257) – – – – – –

Table 9 Test results for fitting capability of the ANFIS. No.

Relative error (0.279)

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30.

1.2

Relative error (0.1312) Predicted value

Measured value

Predicted value

Measured value

Predicted value

0.720 0.441 0.812 0.439 0.676 0.565 0.744 0.502 0.839 0.635 0.639 0.695 0.697 0.494 0.681 0.600 0.357 0.570 0.656 0.777 0.817 0.729 0.525 0.750 0.418 0.299 0.687 0.736 0.802 0.991

0.441 0.720 0.533 0.718 0.955 0.844 0.465 0.781 0.560 0.356 0.360 0.974 0.976 0.773 0.402 0.321 0.636 0.849 0.377 0.498 0.538 0.450 0.246 0.471 0.697 0.578 0.408 0.457 0.523 0.712

0.720 0.441 0.812 0.439 0.676 0.565 0.744 0.502 0.839 0.635 0.639 0.695 0.697 0.494 0.681 0.600 0.357 0.570 0.656 0.777 0.817 0.729 0.525 0.750 0.418 0.299 0.687 0.736 0.802 0.991

0.589 0.572 0.681 0.570 0.807 0.696 0.613 0.633 0.708 0.504 0.508 0.826 0.828 0.625 0.550 0.469 0.488 0.701 0.525 0.646 0.686 0.598 0.394 0.619 0.549 0.430 0.556 0.605 0.671 0.860

0.720 0.441 0.812 0.439 0.676 0.565 0.744 0.502 0.839 0.635 0.639 0.695 0.697 0.494 0.681 0.600 0.357 0.570 0.656 0.777 0.817 0.729 0.525 0.750 0.418 0.299 0.687 0.736 0.802 0.991

0.691 0.470 0.783 0.468 0.705 0.594 0.715 0.531 0.810 0.606 0.610 0.724 0.726 0.523 0.652 0.571 0.386 0.599 0.627 0.748 0.788 0.700 0.496 0.721 0.447 0.328 0.658 0.707 0.773 0.962

(a)

1

Relative error (0.0291)

Measured value

fANFIS ftraining

1

(b)

0.9

fANFIS ftraining

0.8

0.8

0.7

0.6

0.6

0.4

0.5 0.4

0.2 0.3

0

0.2

0

20

40

60

Error = 0.279 80 100

120

140

160

180 192

0

20

40

60

Error = 0.1312 120 80 100

140

160

180 192

Fig. 6. Training data (Phase I and II) – ANFIS output.

By training the neural network with numerical examples of decisions made, the initial forms of input/output membership functions of fuzzy sets are adjusted. If there is a difference between the obtained and the expected data, modifications are made to the

connections between the neurons in order to reduce errors, i.e., the membership functions are adjusted into adaptive nodes. Fig. 5 shows the appearance of the membership functions of the ANFIS model after training the adaptive neural network.

´ irovic´, D. Pamucˇar / Expert Systems with Applications 40 (2013) 2208–2223 G. C

2220

1

fANFIS fexpert

0.9

Table 11 Comparative review of expert decisions and ANFIS model. No.

K1 a

K2 a

K3

K4

K5

K6 a

K7

K8

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40.

62 51 64 24 35 39 48 35 72 54 49 47 56 37 41 35 58 46 53 31 49 47 53 34 42 113 41 61 45 66 46 31 36 53 80 57 100 72 95 115

56 80 61 79 67 53 86 41 93 77 69 62 31 55 23 49 56 53 72 57 82 75 98 58 75 86 35 38 71 57 34 59 48 62 72 35 97 38 69 100

1 1 2 2 4 1 4 1 2 2 2 2 1 1 1 1 1 4 4 1 2 2 2 2 2 4 1 4 2 3 4 3 1 3 1 2 1 1 3 4

60 55 60 60 60 60 70 65 65 55 70 70 70 70 65 50 50 50 65 70 50 45 45 50 70 55 61 60 64 59 39 61 53 31 33 33 54 43 58 50

70° 60° 90° 80° 100° 95° 110° 130° 60° 60° 150° 95° 90° 40° 45° 130° 115° 75° 80° 65° 70° 85° 100° 55° 90° 108° 120° 55° 100° 70° 144° 140° 70° 100° 115° 140° 120° 90° 90° 80°

2 1 1 1 2 2 2 4 3 1 4 0 1 3 1 3 0 1 2 2 1 1 0 2 0 1 1 1 0 0 0 0 0 2 1 1 0 1 0 0

L VL M M M M VH VH L VL L M L H H H M M H L L VL H L M L VL M L VL M M L VL M M M L VL M

H M M H L L M M M L M H VH VH H VH M M M L L M H M M L M H H M L H M M H H H H H M

Final rank

0.8 0.7 0.6 0.5 0.4 0.3

Error = 0.0291 0

20

40

60

80

100

120

140

160

180 192

Fig. 7. Training data (Phase III) – ANFIS output.

Table 10 Railway accidents in the Belgrade area for the period 2001–2011. Year

Total railway accidents

Accidents at RLCs

Share of RLC accidents(%)

2001–02 2002–03 2003–04 2004–05 2005–06 2006–07 2007–08 2008–09 2009–10 2010–11 P

97 63 73 57 93 77 64 87 81 88 780

21 13 14 12 25 22 19 28 33 41 228

21.65 20.10 19.17 21.05 26.88 28.57 29.69 32.18 40.07 46.59 29.23

The values of the function parameters after training the neurofuzzy system are shown in Table 8. The proposed neural network is trained on 547 expert decisions. Appendix A (Table 12) shows a set of 192 railway crossings with which the adaptive neural network was trained. While training the ANFIS model the data from the training set xk, k = 1, 2, . . . , n, where is the total number of input values in the ANFIS model, were periodically passed through the network. A comparative view of the values of the criteria functions of the ANFIS model (fANFIS) and the criteria functions from the training set (ftraining) is shown in Table 9. Fig. 6 shows the deviation values of the function ftraining (Measured value) and fANFIS (Predicted value) which are presented above. Training of the ANFIS model was carried out in three phases which lasted a total of 250 epochs. The first phase of training the ANFIS model was completed after 90 epochs. After completion of the first phase an output error of 0.279 was obtained (Fig. 6a). In the following phase, after 180 epochs, the output error was 0.1312 (Fig. 6b), which compared to the previous phase is a reduction in error of 52.97%. In the third and final phase, which was completed after 250 epochs, at the output from the model the error was 0.0291 (Fig. 7), which compared to the second phase is an error reduction of 77.82%. Upon completion of the third phase, it was concluded that the error obtained at the output of the ANFIS model was acceptable. In addition, it was concluded that the neuro-fuzzy network was trained and able to generalize new input data that has not been trained. The five-layered adaptive network was tested on 40 expert decisions. The values of the criteria which describe the given railway crossing were put through the neuro-fuzzy system, where specific values of the expert preferences were obtained. The selec-

a

0.54 0.58 0.58 0.56 0.64 0.56 0.66 0.55 0.66 0.60 0.74 0.55 0.47 0.47 0.39 0.55 0.54 0.55 0.60 0.50 0.57 0.63 0.63 0.52 0.56 0.73 0.54 0.46 0.61 0.63 0.62 0.61 0.47 0.61 0.68 0.59 0.71 0.55 0.66 0.73

0.50 0.54 0.64 0.59 0.61 0.58 0.67 0.58 0.64 0.62 0.70 0.58 0.47 0.47 0.41 0.54 0.54 0.53 0.62 0.48 0.58 0.62 0.62 0.55 0.59 0.72 0.56 0.48 0.61 0.63 0.66 0.60 0.45 0.59 0.65 0.59 0.68 0.59 0.63 0.70

34 26 14 21 13 25 6 27 7 16 1 28 36 37 40 29 32 30 15 35 24 11 12 33 22 2 31 38 17 10 9 18 39 19 5 20 4 23 8 3

Values represent the average indicators on an annual basis.

0.85 fANFIS fexpert

0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35

0

10

20

30

40

50

60

70

80

90

Fig. 8. A comparison of the preferences of the ANFIS model and the experts.

tion process for the installation of railway crossing safety equipment is based on the expression:

fRi ¼ maxðfRi Þ;

i ¼ 1; . . . ; 8

ð49Þ

4. Results and discussion Testing of the described ANFIS model was carried out on the prioritization of railway level crossings in the Belgrade area. The

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Table 12 Characteristics of 192 level crossings (training pairs). No.

K1

K2

K3

K4

K5

K6

K7

K8

ftraining

No.

K1

K2

K3

K4

K5

K6

K7

K8

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59. 60. 61. 62. 63. 64. 65. 66. 67. 68. 69. 70. 71. 72. 97. 98. 99. 100. 101. 102. 103. 104. 105. 106. 107. 108. 109. 110. 111. 112. 113. 114. 115. 116. 117. 118. 119. 120. 145. 146.

119 91 45 107 45 74 91 61 44 36 52 63 36 77 86 119 104 80 119 62 111 55 99 110 104 59 115 67 44 64 50 109 96 40 33 92 39 77 60 80 113 31 42 94 93 80 111 79 110 50 77 96 66 43 42 45 43 48 79 58 32 112 84 106 40 88 104 91 51 36 57 101 62 45

133 33 83 26 216 181 163 226 218 220 76 35 235 123 33 78 230 214 25 239 128 155 122 180 77 209 27 157 97 189 44 83 17 201 85 209 85 116 62 19 110 64 35 56 12 38 133 15 134 175 196 96 221 169 12 115 119 209 221 174 21 44 101 177 179 222 180 215 16 119 194 159 98 76

2 2 3 2 3 2 1 3 3 2 4 2 3 2 3 1 2 2 2 1 4 3 2 4 3 3 3 4 4 2 3 1 3 4 3 3 3 2 1 3 3 2 3 2 1 4 3 3 1 3 4 3 1 2 4 1 1 4 3 1 2 1 3 4 3 2 2 2 2 4 3 3 2 3

64 60 72 49 61 57 42 67 58 82 86 52 56 52 55 51 86 48 45 75 76 81 82 55 83 79 58 48 43 82 79 83 48 59 69 40 80 51 63 78 65 57 64 61 61 49 78 59 50 54 74 81 79 79 68 88 47 52 44 60 74 55 49 87 73 54 58 52 63 70 75 49 53 50

62° 155° 72° 143° 83° 87° 128° 129° 41° 141° 86° 128° 118° 152° 147° 136° 124° 130° 58° 55° 102° 129° 112° 113° 66° 105° 102° 121° 91° 113° 145° 168° 111° 32° 58° 101° 135° 48° 43° 36° 61° 107° 167° 152° 31° 153° 130° 56° 125° 87° 51° 70° 86° 90° 65° 106° 39° 147° 75° 66° 167° 89° 88° 130° 106° 169° 76° 71° 147° 51° 31° 35° 168° 54°

3 4 6 1 2 2 4 7 6 6 5 2 5 5 6 1 7 1 8 3 3 7 3 2 7 7 8 7 7 7 4 5 4 2 8 4 4 2 0 3 2 3 4 5 8 2 8 2 3 3 1 4 4 8 1 7 7 5 5 6 8 3 2 6 0 2 3 5 1 1 4 4 2 1

0.43 0.77 0.83 0.83 0.6 0.83 0.6 0.43 0.77 0.6 0.43 0.43 0.43 0.26 0.6 0.43 0.6 0.77 0.77 0.6 0.77 0.83 0.26 0.77 0.43 0.83 0.6 0.83 0.6 0.43 0.77 0.6 0.43 0.43 0.43 0.26 0.6 0.43 0.6 0.77 0.77 0.43 0.43 0.77 0.83 0.43 0.6 0.26 0.26 0.83 0.83 0.83 0.83 0.77 0.77 0.83 0.83 0.83 0.43 0.6 0.26 0.26 0.83 0.83 0.83 0.83 0.83 0.43 0.6 0.26 0.26 0.83 0.26 0.26

0.6 0.43 0.43 0.43 0.43 0.43 0.6 0.77 0.6 0.6 0.6 0.83 0.43 0.83 0.43 0.6 0.77 0.83 0.43 0.26 0.26 0.26 0.77 0.77 0.6 0.43 0.43 0.26 0.26 0.77 0.83 0.26 0.43 0.6 0.26 0.83 0.77 0.77 0.83 0.43 0.26 0.26 0.26 0.26 0.6 0.6 0.43 0.6 0.83 0.77 0.77 0.83 0.83 0.26 0.43 0.6 0.26 0.6 0.43 0.83 0.6 0.83 0.6 0.43 0.77 0.6 0.43 0.43 0.83 0.6 0.83 0.6 0.43 0.43

0.68 0.80 0.86 0.85 0.61 0.67 0.57 0.42 0.65 0.48 0.80 0.84 0.47 0.56 0.77 0.69 0.39 0.54 0.87 0.58 0.68 0.62 0.62 0.58 0.75 0.55 0.79 0.61 0.77 0.51 0.84 0.62 0.84 0.70 0.82 0.47 0.77 0.79 0.97 1.03 0.79 0.84 0.78 0.74 0.98 0.78 0.55 0.95 0.57 0.70 0.70 0.81 0.58 0.64 1.07 0.74 0.84 0.52 0.51 0.65 0.75 0.79 0.81 0.53 0.71 0.45 0.64 0.52 0.90 0.85 0.64 0.73 0.64 0.89

25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 73. 74. 75. 76. 77. 78. 79. 80. 81. 82. 83. 84. 85. 86. 87. 88. 89. 90. 91. 92. 93. 94. 95. 96. 121. 122. 123. 124. 125. 126. 127. 128. 129. 130. 131. 132. 133. 134. 135. 136. 137. 138. 139. 140. 141. 142. 143. 144. 169. 170.

79 43 113 80 118 85 112 76 50 75 39 81 100 119 85 30 92 41 38 107 58 112 82 115 117 70 52 31 87 52 111 86 106 100 59 104 31 41 64 56 35 109 31 91 41 72 57 97 34 67 73 109 83 58 93 48 101 97 103 110 62 99 80 94 36 44 83 34 111 65 73 57 71 85

135 30 175 60 92 62 70 64 155 32 181 27 66 99 224 126 195 143 217 238 121 134 196 147 34 22 110 22 104 202 39 161 95 125 30 201 175 71 146 145 235 148 233 217 26 46 43 67 165 102 39 175 160 199 150 84 90 16 89 130 221 132 34 160 143 109 151 167 200 194 180 216 134 157

1 4 2 1 3 3 3 3 4 2 2 1 2 2 1 1 1 3 2 2 3 3 3 3 2 4 2 2 3 3 1 3 2 3 3 1 1 2 2 2 2 3 4 4 3 3 3 4 4 3 4 1 1 3 2 1 4 2 3 4 3 2 1 1 3 3 2 3 2 2 1 2 4 3

89 41 74 40 77 75 79 85 70 83 62 83 67 51 75 55 62 82 59 63 53 48 76 43 80 66 86 40 65 74 85 78 83 63 80 45 40 52 62 84 84 84 62 71 56 75 82 76 86 58 55 60 45 42 50 46 55 60 73 58 51 73 69 67 46 43 79 64 74 63 46 44 74 57

72° 67° 163° 107° 106° 149° 83° 59° 96° 82° 170° 81° 57° 50° 94° 58° 101° 155° 67° 169° 100° 125° 88° 107° 43° 63° 154° 40° 118° 32° 34° 107° 155° 108° 99° 159° 91° 86° 136° 72° 32° 102° 99° 61° 76° 139° 112° 164° 43° 147° 33° 62° 51° 54° 143° 65° 40° 38° 49° 40° 47° 38° 61° 136° 112° 109° 82° 128° 135° 49° 147° 40° 143° 34°

4 7 3 3 7 3 5 6 6 5 5 5 3 6 0 3 5 8 6 7 1 6 7 7 8 5 1 3 2 3 4 7 3 6 8 1 4 7 2 4 5 7 1 7 5 4 0 4 6 1 6 1 5 1 0 7 3 7 4 5 3 6 4 3 0 2 2 7 7 2 7 7 7 5

0.6 0.83 0.6 0.83 0.6 0.83 0.6 0.43 0.77 0.6 0.43 0.43 0.43 0.26 0.6 0.43 0.83 0.6 0.83 0.6 0.43 0.77 0.6 0.43 0.26 0.83 0.83 0.83 0.83 0.77 0.77 0.83 0.83 0.6 0.43 0.43 0.43 0.6 0.77 0.6 0.6 0.6 0.83 0.43 0.83 0.43 0.6 0.26 0.83 0.83 0.83 0.77 0.77 0.83 0.83 0.6 0.43 0.43 0.26 0.26 0.77 0.83 0.26 0.43 0.6 0.26 0.83 0.77 0.77 0.83 0.43 0.26 0.26 0.26

0.83 0.77 0.83 0.43 0.26 0.26 0.26 0.83 0.83 0.77 0.83 0.43 0.26 0.26 0.26 0.26 0.83 0.83 0.83 0.83 0.77 0.77 0.83 0.83 0.77 0.83 0.43 0.43 0.83 0.26 0.83 0.26 0.43 0.6 0.26 0.83 0.77 0.77 0.83 0.43 0.26 0.26 0.26 0.83 0.26 0.43 0.6 0.26 0.43 0.77 0.6 0.43 0.43 0.43 0.26 0.6 0.43 0.6 0.77 0.77 0.43 0.43 0.77 0.83 0.43 0.6 0.26 0.26 0.83 0.83 0.6 0.43 0.26 0.6

0.72 0.89 0.48 0.85 0.66 0.78 0.77 0.83 0.68 0.89 0.49 0.87 0.83 0.70 0.56 0.80 0.56 0.57 0.63 0.31 0.74 0.60 0.54 0.54 0.81 0.98 0.74 0.94 0.75 0.72 0.94 0.60 0.68 0.63 0.83 0.44 0.65 0.83 0.67 0.73 0.63 0.58 0.62 0.50 0.98 0.77 0.91 0.64 0.77 0.75 0.97 0.68 0.71 0.72 0.65 0.83 0.81 0.91 0.76 0.68 0.65 0.76 0.88 0.54 0.74 0.73 0.72 0.61 0.45 0.71 0.47 0.55 0.54 0.67

(continued on next page)

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2222

Table 12 (continued) No.

K1

K2

K3

K4

K5

K6

K7

K8

ftraining

No.

K1

K2

K3

K4

K5

K6

K7

K8

147. 148. 149. 150. 151. 152. 153. 154. 155. 156. 157. 158. 159. 160. 161. 162. 163. 164. 165. 166. 167. 168.

62 65 111 113 71 83 108 47 120 86 67 45 66 50 115 87 78 81 45 49 85 115

23 152 137 184 231 193 161 69 173 114 200 201 116 87 112 44 45 185 57 30 159 176

1 4 4 1 2 1 1 3 2 3 2 3 3 3 3 2 4 4 2 3 2 1

69 74 64 50 80 63 75 54 43 56 82 59 50 85 90 80 49 81 42 47 59 63

37° 123° 71° 124° 140° 76° 48° 108° 93° 80° 51° 120° 79° 119° 132° 163° 68° 148° 44° 96° 168° 65°

1 7 8 4 7 1 5 7 8 7 4 1 3 7 6 0 1 6 4 6 1 6

0.26 0.6 0.6 0.43 0.6 0.77 0.83 0.43 0.43 0.83 0.26 0.83 0.26 0.43 0.6 0.26 0.83 0.77 0.77 0.83 0.43 0.26

0.26 0.6 0.43 0.83 0.6 0.83 0.6 0.43 0.77 0.6 0.43 0.43 0.83 0.6 0.83 0.6 0.43 0.77 0.6 0.43 0.43 0.43

0.99 0.59 0.63 0.48 0.41 0.66 0.69 0.77 0.49 0.73 0.59 0.63 0.72 0.71 0.60 0.74 0.96 0.51 0.97 0.92 0.53 0.53

171. 172. 173. 174. 175. 176. 177. 178. 179. 180. 181. 182. 183. 184. 185. 186. 187. 188. 189. 190. 191. 192.

116 82 117 42 44 97 60 31 74 58 118 69 62 92 43 104 116 52 60 74 92 107

130 148 117 17 71 90 201 206 121 75 128 156 58 42 74 215 57 215 100 21 41 135

2 3 4 2 3 3 3 3 3 3 1 3 2 2 2 2 4 1 3 2 3 1

65 67 62 45 77 63 55 52 52 61 79 59 53 70 56 75 80 52 81 46 89 40

83° 139° 102° 69° 122° 57° 142° 88° 49° 98° 122° 63° 121° 137° 94° 155° 107° 132° 103° 134° 105° 149°

3 3 2 1 3 1 7 5 6 8 4 2 7 3 0 7 7 2 4 3 6 7

0.83 0.26 0.43 0.6 0.26 0.83 0.77 0.77 0.83 0.83 0.26 0.43 0.6 0.26 0.83 0.77 0.77 0.83 0.43 0.26 0.26 0.26

0.43 0.6 0.77 0.77 0.43 0.43 0.77 0.83 0.43 0.6 0.26 0.26 0.83 0.83 0.83 0.83 0.77 0.77 0.83 0.83 0.6 0.43

railway crossings in Belgrade were chosen because of their high frequency of road and rail traffic, and because of the worrying statistics regarding the number of accidents which occur at them (Serbian Transport Safety Agency, 2010). Table 10 shows the number of railway accidents in the Belgrade area for the period 2001–2011. As seen in Table 10, the total number of accidents on railways in the Belgrade area for the period 2001–2011 does not have a trend of growth. However, the share of accidents at level crossings in the total number of accidents has risen since 2003. For the period 2001– 2011 a total of 767 accidents occurred, and of that number, 214 were at railway level crossings. These data indicate the fact that it is necessary to improve the safety of road users on railway crossings. In 2012, the Czech Development Agency donated 2.7 million euro to Serbian Railways with the aim of improving the safety equipment on railway crossings. Serbian Railways have decided to invest 85comparative review of the results of expert prioritization and prioritization using the ANFIS model is shown in Table 11. The parameters shown in Table 11 were obtained by recording traffic parameters at the given level crossings in the period 2009–2011. By adding together the ranks of the railway crossings obtained by the ANFIS model and by expert assessments, synthetic ranks of the given rail crossings were achieved. By arranging the synthetic ranks according to their increasing totals, a final rank (prioritization) of each level crossings was reached. As seen in Table 11, the ANFIS model has a high degree of generalization for the preferences of the experts. By comparing the output preferences of the ANFIS model and the experts, an average error of 0.029 is obtained. In Fig. 8 is a comparison of the output values for the criteria functions (preferences) of the ANFIS model and the experts for all 88 railway level crossings in the Belgrade area. On the basis of the data presented in Fig. 8 and Table 11, we can conclude that the ANFIS model successfully simulates the preferences of the experts. The experiential knowledge of experts was successfully mapped into the base of rules for the neuro-fuzzy system, and a unique base of knowledge was formed by means of which a selection can be made regarding which level crossings should receive safety improvements.

5. Conclusion The ANFIS model developed in this paper enables the quantification of criteria and selection of the best alternative from a set

0.71 0.55 0.66 1.03 0.77 0.87 0.50 0.63 0.79 0.80 0.56 0.70 0.77 0.74 0.92 0.40 0.76 0.56 0.74 0.81 0.76 0.48

of alternatives. The presented model makes it possible to evaluate the proposed railway crossings and select the best alternative from the set of those offered, which are described by means of criteria which can be benefit or cost related. One of the main criteria for the evaluation of the quality of new methodologies in soft computing is their usefulness in the analysis of real data. It has been shown that the ANFIS model for the prioritization of railway level crossings, as well as the output data are equal to the experts’ assessments. The development of the ANFIS model has made it possible for the strategy of choosing a railway level crossing in which to make an investment of safety equipment to be transformed into an automatic control strategy. As a result of the research it has been shown that the system developed has the ability to learn and can emulate the expert estimates and demonstrate a comparable level of expertise. Looking at the performance of the ANFIS model and the results obtained we can conclude that the ANFIS model can reproduce the decisions of experts with great accuracy. This makes it possible to prioritize which railway level crossings should receive an investment of safety equipment in a straightforward way without the use of the complex statistical and mathematical transformations which have been used so far (Berg, 1966; Mohammad et al., 2003). Furthermore, the ANFIS model contributes to a reduction in subjective influences and a saving in the amount of time required for making the decision. Acknowledgements The work reported in this paper is a part of the investigation within the research project TR 36017 supported by the Ministry for Science and Technology, Republic of Serbia. This support is gratefully acknowledged. Appendix A See Table 12. References Anandarao, S., & Martland, C. D. (1998). Level crossing safety on east Japan company: Application of probabilistic risk assessment techniques. Transportation, 25(3), 265–286. Berg, W. D. (1966). Evaluation of safety at railroad-highway grade crossings in urban areas, Joint Highway Research Project, Indiana Department of Transportation and Purdue University, West Lafayette, Indiana.

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