Non-radial DEA model: A new approach to evaluation of safety at railway level crossings

Safety Science 103 (2018) 234–246

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Non-radial DEA model: A new approach to evaluation of safety at railway level crossings

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Boban Djordjevića, , Evelin Krmaca, Tomislav Josip Mlinarićb a b

Faculty of Maritime Studies and Transport, University of Ljubljana, Pot pomorscakov 4, SI-6320 Portoroz, Slovenia Faculty of Transport and Traffic Sciences, University of Zagreb, Vukeliceva 4, 10 000 Zagreb, Croatia

A R T I C L E I N F O

A B S T R A C T

Keywords: Non-radial DEA model Evaluation Efficiency Safety Railway level crossing Validation Sensitivity analysis

Railway level crossings (RLCs) are critical points characterized by a large number of accidents per year due to the intersection of railway and roadway infrastructures. The improvement of safety at RLCs is of crucial importance worldwide. This paper proposes a new approach for evaluating safety at RLCs using non-radial DEA (Data Envelopment Analysis) model. The introduced non-radial DEA model is employed for the evaluation of railway efficiency of European countries regarding the level of safety at RLCs through considering desirable and undesirable inputs, as well as desirable and undesirable outputs for the period from 2010 to 2012 and for the year 2014. Due to missing the comprehensive set of data for roadway traffic, the evaluation of safety at railway RLCs has been conducted from railway viewpoint. Through the application of modified non-radial DEA model, results indicate the most efficient countries, as well as countries with a lower efficiency of railway performance in terms of the level of safety at RLCs. In order to check the validity of modified non-radial DEA model, comparison of results and sensitivity analysis were conducted. Despite the validity, results of the sensitivity analysis indicated also some weaknesses of the modified non-radial DEA model related to missing and inaccurate data, the number of variables included, and the selection of inputs and outputs. A modified non-radial DEA model, as proposed in the paper, with overall set of data can be used for evaluating the efficiency of safety improvement at RLCs, efficiency of different countermeasures before and after implementation at RLCs of different.

1. Introduction 1.1. Background Railway level crossings (RLCs) are critical elements of railway networks due to the junction of two of the most frequently used transportation modes – i.e., rail and road transport – where a large number of accidents take place between trains and road users. According to the European Railway Agency (ERA, 2016) within 28 European countries there were 1,14,580 RLCs in 2014, while 47% of them were passive RLCs. On average, there are five RLCs on a 10 km line. However, these data vary between countries. For example, countries such as Sweden, Austria, the Czech Republic and Hungary have the highest level of RLCs per line-kilometre with 75 RLCs per 100 line kilometres, while Bulgaria and Spain have fewer than 25 RLCs per 100 line kilometres. Similar to the number of RLCs, the fatality risk at RLCs also differs between countries, from Ireland with 11 deaths per billion train kilometres representing the lowest fatality risk, to Greece with 550 deaths per billion train kilometres, the highest level of risk (ERA, 2014).

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Although accident rate at RLCs worldwide has decreased by using various technical systems to eliminate hazards, the problem is still present (Wang et al., 2016), and that the number of fatalities has been almost constant (ERA, 2016). Based on the ERA (2014) from all railway accidents on railways of the European Union (EU), RLC accidents and fatalities represent more than one quarter. In Europe, every day one person has been killed or seriously injured in recent years and about 40% were pedestrians. Aside from the tragic outcomes, such accidents have a significant impact on society (ERA, 2014). Khoudour et al. (2009) have estimated that accidents at RLCs entail enormous human and financial costs estimated at a minimum of EUR 110 million annually. Moreover, accidents at RLCs account at least for 90% of passenger and freight train accident costs (Miller et al., 1994). Furthermore, accidents also have a negative impact on the railway sector and its operations regarding significant infrastructure and vehicle damage costs, together with other indirect costs, such as that of traffic disruption (ERA, 2014). In that view, safety at RLCs, as the weak point of railway and road infrastructure, is a worldwide concern which is getting increased

Corresponding author. E-mail addresses: [email protected] (B. Djordjević), [email protected] (E. Krmac), [email protected] (T.J. Mlinarić).

https://doi.org/10.1016/j.ssci.2017.12.001 Received 4 February 2017; Received in revised form 6 November 2017; Accepted 4 December 2017 0925-7535/ © 2017 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/BY-NC-ND/4.0/).

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1.2. Types of railway level crossings

attention from the relevant transport authorities, the rail industry and the public (Tey et al., 2011). Consequently, improving safety at RLCs has become an important field of academic research. Enhancement of the safety level at RLCs is necessary for improving the operation of road and rail transport, safety of people, avoidance of delays in transportation, as well as the reduction in possible release of hazardous materials transported by freight trains into the environment (Eluru et al., 2012; Salmane et al., 2015). Having this in mind and taking into account the fact that the performance regarding the number of RLCs and accidents vary between countries, information about performance regarding efficiency in the improvement of the level of safety at RLCs among countries can be useful in looking for best practices. Therefore, there is a need to provide a potential tool for monitoring, evaluating and benchmarking changes in terms of efficiency in the improvement of the level of safety at RLCs. In the present study, Data Envelopment Analysis (DEA) methodology, as a good tool for measuring efficiency in terms of the level of safety at RLCs at different levels, was considered. The DEA is well-known “non-parametric productive efficiency measurement method for operations with multiple inputs and multiple outputs” (Liu et al., 2012). The DEA method, first popularized by Charnes et al. (1978), combines and transfers multiple inputs and outputs into a single efficiency index, forming the “efficient frontier” with a set of Decision Making Units (DMUs) pointing towards best practices and assigning a level of efficiency to other DMUs that are not on the frontier according to the distance to the efficient frontier (Liu et al., 2012). Today, there are different DEA models for measuring efficiency for different types of measuring requirements (Liu et al., 2012); among them, according to Liu et al. (2010), the most well-known are the CCR model (Charnes et al., 1978), the BCC model (Banker et al., 1984), the Additive model (Charnes et al., 1985) and the Cone Ratio model (Charnes et al., 1989). Within these DEA models, only desirable inputs and outputs are considered, without consideration of undesirable inputs and outputs which can appear in real applications. For example, in evaluating efficiency in terms of improvement of the level of safety at RLCs, different factors can be modeled as undesirable. Therefore, in this study, non-radial DEA models which modeled some inputs and outputs as undesirable were introduced and considered. First, the non-radial DEA model presented by Wu et al. (2015) and here referred to as (M1) was introduced. However, based on the purpose of this study, the model (M1) was improved, forming an new one referred to as (M2). For measuring efficiency related to the improvement of the level of safety at RLCs on the macro level, the improved non-radial DEA model (M2) that includes both desirable and undesirable inputs and outputs has been employed. Improvements to the non-radial DEA model were related to the introduction of efficiency score and weights for desirable inputs in order to improve the discriminating power of the model and to make it suitable for the purpose of efficiency evaluation related to the degree of safety at RLCs on different levels. Based on the data for European countries, the improved non-radial DEA model (M2) was piloted for RLC applications. In the investigation of the improved non-radial DEA model (M2), factors that influence the occurrence of accidents at RLCs were considered as guidelines in the selection of inputs and outputs. Because of the lack of availability of data related to road transport, the non-radial DEA model (M2) was employed only from the railway point of view. Moreover, despite the fact that the DEA approach is fundamentally an economic tool, the costs of safety measures for improvement of the degree of safety at RLCs were not included in the evaluation. Therefore, efficiency evaluation of the degree of safety at RLCs was to a greater extent based on variables related to railway performance. Based on the results and sensitivity analysis, it could be said that the proposed non-radial DEA model (M2) could be applicable for providing information about the state of safety at RLCs for various levels – i.e., micro and macro level. Moreover, it could be used as a support tool for identifying critical RLCs regarding accidents and the planning of upgrading of particular RLCs.

As a rule, there are two types of RLCs, public and private. Public RLCs are open to the public and are maintained by a specific public authority (Evans, 2013), while private RLCs are not maintained by a public authority and represent “wild” crossings (Haleem, 2016). Further, Khoudour et al. (2009) have highlighted that the ERA has classified RLCs into two groups: active and passive RLCs. According to Evans (2013), active RLCs were further classified into railway-controlled and automatic. Nowadays, RLCs are commonly interlocked with railway signaling, so the proper functioning of crossing impacts the forward movement of the train. Automatic crossings are activated by the passage of the trains over the train detection devices and consist of the combination of flashing lights, audible warnings and barriers which operate only when a train is approaching or is at the crossing. The automatic half barrier (AHB) is the most common type of automatic crossing, consisting of flashing lights and half barriers (Silmon and Roberts, 2010). Railway-controlled RLCs represent control of the crossing operation by a member of the staff – i.e., a signaler or crossingkeeper. Passive RLCs include footpath crossings and have no “active” warning devices for informing the road users about the approaching train. They can consist of a static array of signs that remain unchanged all the time, or fixed warning signs (typically a St Andrew’s Cross or “crossbucks”) (Evans, 2013; Wigglesworth, 2001). 1.3. The aim and the scope of the paper Numerous studies have already assessed multiple factors associated with RLCs accidents (Gruyter and Currie, 2016), investigating the effectiveness of countermeasures and different equipment at RLCs, which can be seen in the following section. However, there are no examples and applications of non-radial DEA model in the field of safety at RLCs. Additionally, Decision Makers (DMs) can require knowing the degree of safety for a particular RLC or RLCs, as well as an overall picture of the safety level in various countries. Consequently, the aim of this paper is the introduction and consideration of the applicability of non-radial DEA models (M1) and (M2), as a non-parametric approach able to benchmark different entities on different levels, for monitoring trends, evaluating and benchmarking changes regarding efficiency in improvement of safety at RLCs. The fact that rail companies cannot control actions of road vehicle drivers or pedestrians at RLCs is a weakness in terms of accident risk for railways (Khoudour et al., 2009). According to Forsberg et al. (2014) accidents at RLCs are the most common cause of train accidents, despite the efforts to improve the existing level crossings and avoid building new ones. Furthermore, infrastructure managers have always been responsible for reducing the risk of accidents at RLCs. There are several reasons for this. Firstly, accidents may derail the train and thereby cause deaths or injuries to passengers and train staff. Secondly, railways operate under strict rules and regulations, and therefore they are easier to control than roads (Silmon and Roberts, 2010). Despite the lack of data for roadways, this new approach is tested through an empirical study which has considered European countries from the railway point of view. The introduced non-radial DEA model (M2) was improved and tested through empirical study, so the evaluation of efficiency in terms of improving safety at RLCs was realized using the improved non-radial DEA model. The main contributions of this paper are the following: (i) newer traditional thorough literature review in the field of evaluation, assessment, appraisal and analysis of RLCs, (ii) new approach/methodology in analyzing safety at RLCs at different levels – i.e., at individual RLC group of RLCs for particular rail line, at national or international levels, (iii) improving and adjusting the non-radial DEA model for RLCs applications, representing the approach through evaluation and comparison of railway efficiency related to the degree of safety at RLCs for EU member states. This new approach and the findings of this paper are 235

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related to the most common factors which cause accidents. Understanding the factors that have an impact on accidents and their significance to injury severity is essential for introducing measures to reduce accidents and the number of deaths and injuries at RLCs (Haleem and Gan, 2015). Examination of accidents and factors could be found in Austin and Carson (2002), Hu et al. (2010), Haleem and Gan (2015) and Oh et al. (2006). Davey et al. (2008) have explored factors which have impact on accidents between heavy vehicles and trains. Eluru et al. (2012) have examined different factors that influence injury severity of drivers involved in vehicle-train accident. Factors of injury severity of truck drivers in accidents at RLCs in the U.S were presented by Hao et al. (2016a). Factors of driver injury severity under various control measures and their differences have been pointed out by Hao and Daniel (2014) and Hao et al. (2015), respectively. Authors Lu and Tolliver (2016) have investigated accidents and contributing factors through comparison of various alternate RLC accident frequency models. Haleem (2016) has identified significant factors of traffic casualties (including both injuries and fatalities) at private RLCs in US.

expected to make a significant contribution to the research community in general, especially to the railway and roadway communities, including decision makers and stakeholders. Expected new insights in railways could be in finding good practice in terms of ways of upgrading/improving safety at RLCs through the application of the nonradial DEA model on a macro level. Also, new insights in railways could be related to the possibility of application of the presented approach on a micro level, and consideration of the influence of safety measures for increasing safety at RLCs. The following section presents the review of previous papers in the field of RLCs. Section 3 describes the methodology of the non-radial DEA model, as well as improvement and application for our purpose. The data used, Decision Making Units (DMUs) selection, desirable and undesirable inputs and desirable and undesirable outputs, as well as weights for them, for EU countries are selected and described in the second part of this section. Section 4 presents the results of empirical study through the application of the improved non-radial DEA (M2) model on data from 2010 to 2012 and during 2014, and the sensitivity analysis of model (M2). Finally, the discussion and conclusions, comprising the summary of the study and future research directions are presented in Sections 5 and 6, respectively.

2.1.1. Review of models used in evaluation Models developed earlier and the most commonly used statistical models for evaluating the relationship between factors and accidents at RLCs, such as the Peabody Dimmick formula, the New Hampshire index, the NCHRP hazard index and the US-DOT accident prediction model were presented and discussed by Austin and Carson (2002). They pointed out the need for a consistent accident prediction method and highlighted that multiple linear regression, Poisson regression methods and binomial regression methods were used for investigating factors that affect accidents at RLCs. Several other models were also used, such as the “zero-inflated” Poisson model and the Gamma probability model (Oh et al., 2006); a multinomial logit model within a systematic Bayesian framework (Miranda-Moreno et al., 2007); a “zero-inflated” binomial, hierarchical tree-based regression (HTBR) model (Yan et al., 2010); generalized logit model (Hu et al., 2010); an ordered logit model (Eluru et al., 2012); ordered probit model (Hao and Daniel, 2014; Hao et al., 2015; Hao et al., 2016b; Hao et al., 2016a); mixed logit model (Haleem and Gan, 2015); the Conway–Maxwell–Poisson (CMP) model, the Bernoulli distribution model and the hurdle Poisson model (Lu and Tolliver, 2016). According to Miranda-Moreno and Fu (2006) in the case of the analysis of accident data a heterogeneous negative binomial model was presented as a more flexible option than the traditional negative binomial model and a zero-inflated negative binomial model. Considering the analysis of drivers‘ injury severities in train-motor vehicle crashes reported at RLCs, among ordered probit, multinomial logit and random parameter logit, random parameter logit was indicated as the most suitable in (Zhao and Khattak, 2015).

2. Literature review The aim of the literature review was to perform an overview of papers related to evaluation, assessment, appraisal and analysis of RLCs and confirm the novelty of the non-radial DEA approach to RLCs. Also, based on the procedure of DEA methodology, a literature review was conducted as a basis for the process of identification of inputs and outputs for the non-radial DEA model (M2), consideration of the application of the non-radial DEA model for different levels of RLC evaluation, as well as discussion of DEA results. Consequently, the literature review was focused on identifying the papers related to evaluation and analysis of safety at RLCs. The search strategy consisted of a literature review of relevant studies published in peer-reviewed journals within scientific sources such as Ebsco, IEEE Xplore, ScienceDirect, Scopus, Springer, and Taylor & Francis without limitation on the time period of publishing. The search, performed on titles, abstracts and keywords for English written full-text free-available scientific journal papers, was performed in October 2016. The application of keywords such as “railway AND level crossings”, “railway AND railroad crossings” or “railway AND grade crossings” resulted in a large number of found papers. In order to reduce this number, the detailed search strings were applied for each database using such keywords as: “evaluation AND railway level crossings”, “assessment AND railway level crossings”, and “appraising AND railway level crossings”, while synonyms like railroad crossings, were also combined in the search strings. Conference papers, projects, periodicals, and working papers related to RLCs were not included in our review, because they went through a less rigorous peer-review process. Based on the results of applied search strings, papers that can support the application of non-radial DEA approach for evaluation of railway efficiency related to the degree of safety at RLCs have been extracted by first reading the abstracts of found papers, and for those relevant the full texts. Finally, 50 relevant papers were selected. The researches aimed through their evaluation consider the factors contributing to, and risks of, accidents occurring at RLCs; as well as to present measures for improving safety and monitoring the trends of these accidents. Therefore, through the review process papers were classified in main areas such as the evaluation of safety, measures for improving safety, and accident analysis at RLCs.

2.1.2. The main factors of compromising and improving safety at RLCs In terms of main contributing factors that lead to a higher probability of accident occurrence at RLCs, Austin and Carson (2002) have highlighted higher train and traffic volumes, number of main track lines and traffic lanes, as well as maximum timetable train speeds as traffic characteristics, while only highway pavement could be found as a road characteristic. Development type, roadway geometry and sight obstructions were not indicated as the most significant factors. As for crossing characteristics, the presence of gates and highway traffic signals represented a factor that significantly reduced RLCs accident frequency, while the presence of stop signs, flashing lights or bells increased the predicted accident frequency. According to Oh et al. (2006) in the proximity of commercial areas, the distance of a train detector from the crossing, the duration of time between the activation of warning signals and the activation of gates probably increased the number of crossing accidents, while the presence of speed bumps decreased the probability of accidents. Beside these factors that affect accidents at RLCs, Hu et al. (2010) found other factors such as obstacle detection devices and crossing approach markings.

2.1. Evaluation of the safety at railway level crossings In terms of literature, evaluation of safety at RLCs is primarily 236

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Consequently, Konur et al. (2013) proposed a mathematical model for resource allocation for RLCs. Based on that, considered upgrades of the model indicated the effectiveness of each countermeasure for different RLCs as follows: gates for passive RLCs were most effective, followed by flashing gates for passive RLCs and gates for RLCs with flashing lights. Moreover, Forgionne (2002) proposed a decision support system in consideration of costs and benefits for investment in RLCs upgrades. Ćirović and Pamučar (2013) defined the criteria for choosing RLCs and represented the modeling of the Adaptive Neuro Fuzzy Inference System (ANFIS) for selection of RLCs that should receive investment(s) in safety equipment. Rezvani et al. (2015) used cost and benefit analysis for identification, selection, and prioritization of safety improvement projects at RLCs.

Davey et al. (2008) pointed out that critical factors of accidents related to heavy vehicles and trains, from the drivers’ point of view, were the design and configuration of RLCs, driver complacency due to high levels of familiarity and willful risk taking as a time-saving measure. On the other hand, from the train driver’s point of view, the size and mass of the vehicles and the behavior of the drivers represented risk factors. Furthermore, factors that affected injury severity were examined by Eluru et al. (2012). They indicated driver age, time of the accident, the presence of snow and/or rain, vehicle role in the accident and motorist action prior to the accident as major risk factors that impacted injury severity. According to age and gender, Hao et al. (2015) concluded that older male and female drivers had higher fatality probabilities when driving in open space or commercial areas under passive control, especially during bad weather conditions. However, younger male drivers were found to be more likely to have severe injuries at rush hours with high vehicle speed passing unpaved RLCs under passive control. Hao et al. (2016b) also confirmed that injury severity for people in vehicles at RLCs was much higher during a.m. and p.m. peaks and p.m. off-peak than in other time periods. However, Hao and Daniel (2014) found that visibility, motor vehicle speed, train speed, area type, traffic volume and highway pavement impacted driver injury severity at both active and passive RLCs. As in previous papers, similar major factors that contributed to accidents at RLCs could be found in Haleem and Gan (2015), Haleem (2016) and Lu and Tolliver (2016). Regarding the strong effect on injury severity level in truckinvolved RLC accidents, Hao et al. (2016b) have indicated driver, environment, and weather as critical characteristics.

2.2.1. Impacts of different systems on the behavior of road users One of the major subjects of much empirical research is the behavior of road users related to different warning devices at RLCs. Driver behavior is one of the major accident factors and is important in determining the use of candidate warning systems at RLCs (Tey et al., 2011). The evaluation of driver behavior towards warning devices in reviewed papers was done for different levels – e.g., micro and macro. Such evaluations and analysis were realized through simulation (Lenné et al., 2011), field video recording and driving simulator in lab (Tey et al., 2011), regression models with data from a driving simulation experiment (Tey et al. 2013a), microsimulation modelling (Tey et al., 2014), and an advanced driving simulator with survey participants (Larue et al., 2015b, 2015a). The effectiveness of flashing light warning devices related to driver behavior at active and passive RLCs in Australia was examined by Wigglesworth (2001). Tey et al. (2011) evaluated driver behavior towards the existing conventional warning devices such as stop sign (passive) and flashing lights and a half boombarrier with flashing lights (active) in relation to driver behavior in Australia. A comparison of driver behavior was also conducted by Lenné et al. (2011) at two RLCs with active controls – e.g., flashing red lights and traffic signals – regarding behavior at the current standard passive level crossing control such as a stop sign. Khattak et al. (2012) assessed changes in drivers’ unsafe maneuvers after removal of a centerline barrier installed at a dual-quadrant gated RLC to improve safety. Further, Tey et al. (2013a, 2014) examined driver responses towards two novel warning devices (rumble strips and in-vehicle audio warning) with two conventional (flashing light and stop sign) ones, where each pair represented active and passive warning devices, respectively. Tey et al. (2013b) studied driver behavior towards two alternative warning devices at crossings (rumble strip a with stop sign and in-vehicle audio warning) compared with current conventional devices (stop sign and flashing light). Larue et al. (2015a) examined the effectiveness of three emerging intelligent transportation systems (ITS) towards the behavior of drivers that the rail industry considered implementing in Australia: a visual in-vehicle warning using a GPS-like system, an audio in-vehicle warning, as well as flashing beacons for on-road intervention. Furthermore, Larue et al. (2016) assessed the impacts of these in terms of possible excessive cognitive load. Results indicated that none of the three technologies resulted in significant changes in cognitive load while approaching crossings. Liu et al. (2016a, 2016b) investigated the use of stop and yield signs as viable alternatives to upgrading a passive RLC to an active RLC. The findings implied that stop signs had the potential to decrease the chance of colliding with a train at passive RLCs and reduce accident severity. The reviewed papers offered quite consistent conclusions: i.e., drivers behaved differently in regard to different warning systems at RLCs, as well as that RLCs equipped with active protection had a higher impact on driver behavior and lowered accident risks in comparison to those with passive protection. Particularly, the papers indicated that RLCs with gates or a combination of flashing lights and gates were significantly more effective at reducing collisions as compared to RLCs with a stop sign, flashing lights, bells or any combination of these. Regarding novel devices, in-vehicle audio

2.2. Countermeasures for improving safety at RLCs Analysis of the effectiveness of countermeasures (measures towards increasing safety) could be seen as one of the preliminary steps required for safety upgrades at RLCs. As for the literature, a large number of papers focused on identification and examination of benefits of countermeasures for RLCs. Washington and Oh (2006) illustrated and evaluated safety benefits of 18 countermeasures. The top three performing countermeasures for reducing accidents were in-vehicle warning systems, obstacle detection systems, and constant warning time systems. According to Saccomanno et al. (2007) the strongest countermeasure effects in reducing RLCs were in upgrading the warning device from 2- to 4-quadrant gates and installing photo/video enforcement, while the lowest effect was related to introducing yield signs ahead of RLCs. Millegan et al. (2009) assessed the effectiveness of stop-sign placement on crossing safety before and after upgrades from cross buck only to stop signs, without consideration of other countermeasures. For overall safety improvement at RLCs, Silmon and Roberts (2010) presented the potential benefits of introducing obstacle detection systems on automatic half-barrier level crossings (AHB). In order to reduce accidents at RLCs, Salmane et al. (2014) implemented an intelligent video surveillance system with automatic recognition and evaluation of potentially dangerous situations at RLCs for opened and closed barriers. Besides these different countermeasures, Pattanaik and Yadev (2015) proposed a decision support model using fuzzy logic control, and Sharad et al. (2016) suggested a system for the design and implementation of unmanned RLCs, which are able to avoid accidental fatalities and to eliminate human errors. Even though four quadrant gates could provide a higher level of safety, there is a risk of the vehicle being trapped at the same time. Sensors installed to detect vehicles could avoid this risk. A wider detection area could offer a radar detection system, which was shown by Horne et al. (2016) along with focusing on verification of performance characteristics of the radar device, the interaction of the radar system with warning devices, the influence of site characteristics on crossing conditions and observed driver behavior. Efficient resource allocation and selection of RLCs for safety equipment upgrades had a critical role in reducing accidents at RLCs. 237

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warning systems and external flashing beacons appear to be efficacious accident reduction devices (Larue et al., 2015a; Tey et al., 2014). Moreover, after testing driver acceptance for an in-vehicle visual ITS, an in-vehicle audio ITS, and an on-road valet system designed for RLC safety improvement, Larue et al. (2015b) concluded that drivers accepted all three interventions, with the highest acceptance being found for the valet system at passive crossings.

3.1. Background of the DEA method DEA is a linear programming based method for efficiency measurement based on Farrell’s (1957) original work that was later popularized by Charnes et al. (1978). Proposed by Charnes et al. (1978), DEA has been commonly applied in empirical literature as a nonparametric frontier approach for evaluating the relative efficiency of a comparable set of entities, called DMUs, with multiple inputs and outputs – i.e., DMUs that are able to transform multiple inputs to multiple outputs. This method offers DMs information on efficient (i.e. best practice) and non-efficient DMUs. A major stated advantage of the DEA is that it does not require any prior assumptions on underlying functional relationships between inputs and outputs (Zhou et al., 2008a). Assuming that there are n DMUs, m outputs and s inputs, efficiency score is usually calculated based on the input oriented Charnes, Cooper and Rhodes (CCR) DEA model (Cooper et al., 2006; Zhou et al., 2008a) that can be written as:

2.3. Accident analysis When it comes to literature, many papers have considered the trends of accidents. Additionally, a significant number of the above mentioned papers have studied and compared the accident rates at different types of crossings (active and passive) (Austin and Carson, 2002; Lu and Tolliver, 2016; Oh et al., 2006). For instance, Yan et al. (2010) investigated changes in accident rate at previously passive RLCs after the implementation of stop signs. They found that annual accident rates at crossings controlled by cross bucks only were consistently higher than accident rates where the crossings were controlled by stop signs. Although improvements were made towards a higher level of safety at RLCs, the frequencies of accidents remained high, many resulting in fatalities. The fatal accidents RLCs in Great Britain from 1946 to 2009 were investigated by Evans (2011). The number of fatal accidents and fatalities per year fell between 1946 and 1975, but since then they remained more or less constant at about 11 fatal accidents and 12 fatalities per year. The main reason for the unchanged number of fatal accidents and fatalities in the second half of the period of research was the increase in the number of automatic crossings, replacing the safer railway controlled crossings on some public roads. However, advantages of automatic crossings over the compared crossings were in the reduction of delays for road users and lack of staff. Additionally, the rate per train-kilometer of serious accidents at RLCs remained unchanged in the period 1990–2009 in the EU as compared to fatal train collisions and derailments. The conclusion is that accidents at RLCs represent an increasing proportion of serious accidents (Evans, 2011). The development of railway safety in Finland from 1959 to 2008 was represented in Silla and Kallberg (2012). The number of accidents at RLCs decreased quite steadily from the late 1960s to the mid-1990s, and increased with the rapid growth of the motor vehicle fleet. The number of accidents decreased with the removal of level crossings and the installation of active warning devices, the construction of overpasses or underpasses at crossings with dense traffic where maximum speed was over 140 km/h and on railway sections where dangerous goods were frequently transported, and the improvement of conditions such as visibility at crossings. The fatal train-pedestrian collisions in the Chicago metropolitan area between 2004 and 2012 were analyzed in Savage (2016). The results indicated that unintentional deaths were strongly related to the density of public access points and to the right of way as compared to intentional deaths.

min θ; s. t Xλ ⩽ θx i, Yλ ⩾ yi , λ ⩾ 0 where X, Y and x i , yi are data matrices and vectors of inputs and outputs, respectively, while λ is a vector of variables. θ represents an indicator of technical efficiency where θ ∈ [0,1] and indicates how much an evaluated entity could potentially reduce its input vector while holding the output constant. 3.2. Description of non-radial DEA model The classical DEA model is strongly related and can be presented through production theory, where raw materials and resources are treated as inputs, while products are treated as outputs in the production process. However, in some real applications, production process may also use undesirable inputs and generate undesirable outputs (Liu et al., 2010), like smoke pollution or waste (Liu et al., 2015; Vencheh et al., 2005). Consequently, Amirteimoori et al. (2006) and Seiford and Zhu (2002) have stated that in that production process, however, both desirable and undesirable factors may be presented within the models of DEA methodology. DEA models with undesirable inputs and outputs have been extensively studied (Liu et al., 2010). Some of these papers are summarized below. For instance, Seiford and Zhu (2002) have developed an alternative approach to treating both desirable and undesirable factors differently in the BCC model. Jahanshahloo et al. (2004) have considered the control of changes in input/output levels of a given DMU with the presence of undesirable factors in order to preserve the efficiency index of a DMU. Vencheh et al. (2005) have proposed a model in the framework of DEA for treating undesirable inputs and outputs. Jahanshahloo et al. (2005b) have presented a method of treating both undesirable inputs and outputs simultaneously in non-radial DEA models. Furthermore, Amirteimoori et al. (2006) have extended a CCRDEA model to a DEA-like model able to deal with undesirable inputs and outputs. Liu et al. (2010) have discussed a general approach of deriving DEA models to handle undesirable inputs and outputs without transferring undesirable data, such as in Seiford and Zhu (2002). Including undesirable inputs and outputs, DEA has been extensively represented and used in environmental fields, such as energy and environmental efficiency. Additionally, a large number of papers have focused on the evaluation of transport energy or environmental efficiency (Meng et al., 2016). Considering the level of safety at RLCs, the transportation process could produce undesirable outputs, such as accidents. If inefficiency is present in terms of safety performance at RLCs, the undesirable output should be reduced to improve inefficiency; i.e., the level of safety at RLCs, which means that undesirable and desirable outputs should be treated differently when we evaluate the safety performance at RLCs. Also, transportation movement over RLCs represents the risk for transport participations; therefore RLCs could be treated as undesirable

3. Methodology of the improved non-radial DEA model application Within the first step of this study, the non-radial DEA model (M1) was introduced. After that, in the second step, for evaluation of railway efficiency related to the level of safety at RLCs through improvement of the model (M1) the non-radial DEA model (M2) was proposed. For employed model (M2) inputs and outputs were defined and then they were classified as desirable and undesirable. This step was followed by a selection of DMUs for evaluation and based on the availability, data for them were collected. The fourth step was the selection of weights for desirable and undesirable inputs and outputs. Finally, validation of the improved model (M2), through comparison of the ranking and sensitivity analysis, was presented in order to check the behavior of the model. Each step is described in detail in the following subsections. 238

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improved in the following way:

inputs in the evaluation of safety performance. Based on the paper Zhou and Ang (2008) regarding energy efficiency, production process, desirable and undesirable outputs are jointly produced by consuming both desirable and undesirable inputs, where x, e, y and u are vectors of energy inputs and non-energy inputs (i.e., here undesirable inputs and desirable inputs), desirable outputs, and undesirable outputs, respectively. Therefore, the joint production process can be represented as:

L 1

minWn N

⎝

L

∑

J

θl +

1 J

l=1

∑ j=1

θj

∑ ∑

λk elk ⩽ θl el0, l = 1,…L

(2)

∑

λk ymk ⩾ ym0 , m = 1,…M

(3)

λk elk ⩽ θl el0, l = 1,…L

(2)

λk ymk ⩾ ym0 , m = 1,…M

(3)

λk ujk = θj uj0, j = 1,…J

(4)

λk ⩾ 0, k = 1,…K .

(5)

(M2)

The improved non-radial DEA model (M2), is able to calculate unified efficiency with additional “simultaneous” reduction of desirable inputs. The reason for this extension is in the fact that the nonradial DEA model (M1) does not provide the result of efficiency with decreasing desirable inputs for a given level of desirable output. The improved non-radial DEA model (M2) was piloted through the introduction of efficiency score θn and weight Wn including for desirable inputs based on the collected data for EU countries. Selected inputs and outputs are described in the next section. 3.4. Selection of inputs and outputs

k=1 K

∑

(1)

k=1

k=1 K

∑

λk x nk ⩽ θn x n0, n = 1,…N

k=1 K

k=1 K

∑

θj

j=1

k=1 K

K

(1)

l=1

∑

k=1 K

⎞ ⎟ ⎠

λk x nk ⩽ x n0, n = 1,…N

J 1

θl + Wj J

K

∑

s. t.

∑

∑

s. t.

In order to overcome the discriminating power of the radial model, following Bian and Jang (2010) and Wang et al. (2013), in Wu et al. (2015) the radial DEA model for energy and environmental efficiency evaluation has been extended to a non-radial model. Assuming that there are K DMUs, and each DMU has n desirable inputs (non-energy inputs) and l undesirable inputs (energy inputs) in order to produce m desirable outputs and j undesirable outputs denoted respectively as x = (x1K ,…,x nK ) , e = (e1l,…,xLK ) , y = (ymK ,…,ymK ) , u = (u1K ,…,uJK ) , the non-radial DEA model named/denoted (M1) is the following: 1 ⎛1 ⎜L

L 1

θn + Wl L

l=1

T = {(x ,e,y,u): (x ,e ) can produce (y,e )}.

min 2

∑

λk ujk = θj uj0, j = 1,…J

In order to analyze the level of safety at RLCs some measures of exposure to the risks giving rise to accidents are required (Evans, 2011). In terms of literature which focused on the evaluation of RLCs, key factors that cause a poor degree of safety were presented. Factors which contributed to the occurrence of the accidents at RLCs were identified and examined; for instance in Eluru et al. (2012), Hao and Daniel (2014), Haleem and Gan (2015), Hu et al. (2010) and Tey et al. (2013a). Furthermore, as the criteria that influenced the selection of RLCs for installing equipment for improvement of safety, Ćirović and Pamučar (2013) indicated frequency of rail traffic on the monitored rail crossing, the frequency of road traffic on the monitored rail crossing, number of tracks on the monitored rail crossing, maximum permitted speed of trains for the change of the rail crossing, the angle of intersection of the track and the road, the number of incidents at the monitored rail crossing in the past year(s), visibility of the monitored rail crossing in terms of road traffic, investment value of activities related to the width of the rail crossing. According to Clarke and Loeb (2005), Horne et al. (2016), Lu and Tolliver (2016), and Wigglesworth (2001) more highway or railway traffic volume increased the likelihood of accidents. Since the selection of variables for DEA methodology was a difficult task, we selected inputs and outputs presented in Table 1, all based on the reviewed literature, available data, as well as the aim of the evaluation – i.e., evaluation on a macro level. Undesirable input (UDI): Since the efficiency evaluation in terms of the level of safety at RLCs was conducted on the macro level, the number of RLCs for each country is introduced as an undesirable input

(4)

k=1

λk ⩾ 0, k = 1,…K .

(5)

(M1)

This non-radial model (M1), could be projected for evaluation of railway efficiency regarding the level of safety at RLCs. When compared to other non-radial models mentioned in the literature, the non-radial model (M1) proportionally decreases the number of undesirable inputs and undesirable outputs as much as possible for the given level of desirable inputs and desirable outputs. The optimal values of unified efficiency are in the interval between 0 and 1. An entity with a higher value of efficiency has better railway efficiency in terms the degree of safety at RLCs as compared to others entities. In the case of the nonradial DEA model (M1), if the entity has objective function equal to 1 it means that the entity is the best, located on the frontier and could not reduce undesirable input and undesirable output. Such a non-radial model (M1) could be applicable for evaluation of the degree of safety at RLCs because it has a relatively strong discriminating power and capability to expanding desirable outputs, while simultaneously reducing undesirable outputs. Additionally, unified efficiency can be calculated through decision maker specified weights assigned to each of these two efficiency scores and depends on the preferences between undesirable inputs utilization and undesirable outputs. In the model proposed by Wu et al. (2015) the weights were set to 1/2. 3.3. Improvement of the non-radial DEA model (M1)

Table 1 Variables employed in non-radial DEA model (M2).

Despite already displaying advantages, of the (M1), it was extended. Since the transportation process cannot be realized without a number of assets of railway or roadway, this input was considered as desirable. However, it also represents primarily a factor of RLC accidents. Therefore, in order to evaluate RLCs at the macro level, we also tried to discover a country that achieves the highest volume of transport with a minimal number of assets. Consequently, in this work in order to allow simultaneous reduction of desirable inputs and get the real picture about results of efficiency, the non-radial DEA model (M1) was 239

Inputs/Outputs

Unit

Category

Number of railway level crossings Number of assets Railway passenger volume Railway freight volume Number of accidents at RLCs

Number per 10 line-km in 2010 and 100 line-km in other Number Billion ptkm Billion tkm Number

UDI DI DO1 DO2 UDO

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although it isn’t highlighted in the literature. The number of RLCs represented the sum of active and passive crossings as one variable. Because the RLCs presented potential accident sites (Lu and Tolliver, 2016), the number of RLCs was classified as an undesirable input. Desirable input (DI): In the literature it was highlighted that a higher number of trains influenced accidents at RLCs (Hu et al., 2010). However, due to the lack of data, we employed a number of assets which represented the total number of locomotives and railcars under assumptions which were used for forming the composition of trains. We examined the behavior – i.e., the results of the model (M2) – once without assets and once with assets based on the data for the year 2010. Different efficiency results gained from the testing’s implied the need for involving/including variable “assets” as necessary in DEA analysis. Without adequate inclusion of necessary variables within the DEA analysis, it was possible that the non-radial DEA model would not provide accurate results of efficiency for DMUs (see Fig. 1). We classified this variable as desirable input because it represented the base of forming transport movement – i.e., base of railway transport realization – although it at the same time indicated a significant factor of the accident. Desirable outputs (DO): The volume of realized passenger and freight transport that represent outcomes of the transportation process and one of the factors that cause RLCs accidents, were recognized in consideration of DMUs and were introduced as desirable outputs. In that way, DMs were able to detect the best practices which had the lowest range of accidents for the given level of volume of transport. Finally, undesirable output (UDO) represented a number of accidents at RLCs for each country. In our case, this variable was involved because it represents a useful indication of the level of safety at RLCs and DMU performance at the macro level. Due to the lack of data the highway traffic volume was not included in the evaluation. The data of accidents at RLCs were collected from Eurostat1 for each country. Data for traffic volume and railway assets were taken from EU statistical pocketbooks (EU, 2014, 2016), while data about RLCs were taken from reports of the European Railway Agency (ERA, 2012, 2013, 2014, 2016).

Fig. 1. Differences of efficiency for different variables.

Table 2 EU countries as DMUs. DMUs-countries Belgium (BE), Bulgaria (BG), Czech Republic (CZ), Denmark (DK), Germany (DE), Estonia (EE), Ireland (IE), Greece (EL), Spain (ES), France (FR), Italy (IT), Latvia (LV), Lithuania (LT), Luxembourg (LU), Hungary (HU), Netherlands (NL), Norway (NO), Austria (AT), Poland (PL), Portugal (PT), Romania (RO), Slovenia (SI), Slovakia (SK), Finland (FI), Sweden (SE), United Kingdom (UK), Croatia (HR), Switzerland (CH)

3.5. Data collection and DMUs selection et al. (2015) set weights to 1/2 in their model (M1). In order to choose the adequate weights for the model (M2), different weights were selected and tested first on the model (M1). Table 3 presents the results of the non-radial DEA model (M1) and the improved non-radial DEA model (M2) for different weights. Hence, selection of weights for the model (M2) was based on the monitoring of the behavior of the nonradial DEA model (M1) and comparison of results of unified efficiency through different cases – i.e., different weights for undesirable input and output on the data of the year 2010.

The general weakness of comprehensive evaluation and analysis of railway safety performance in Europe is a lack of a single source of data and overall necessary data (Evans, 2011). Additionally, not all data are available for each EU member state. Moreover, comprehensive data related to RLCs characteristics for EU member states are missing. Therefore, for the purpose of our work, were combined from various sources such as Eurostat, and Statistical Pocketbooks of European Union, ERA reports. At the time of paper writing, homogenous data were available for the period from 2010 to 2012 and for the year 2014 for particular member states of EU. Each country was defined as DMUs for application of improved non-radial DEA model (M2) (Table 2). The collection of data was conducted in order to meet the requirement that the number of DMUs is at least twice the number of inputs and outputs considered, which consequently ensured that the model was more discriminatory (Golany and Roll, 1989).

3.7. Sensitivity analysis In recent years, the sensitivity analysis of DEA has received great attention from researchers who have addressed the matter from various points of view (Jahanshahloo et al., 2011) because the DEA can be sensitive to measurement changes or errors. In the literature, different approaches of sensitivity analysis of DEA can be found. An important one is the sensitivity analysis of a specific decision-making unit (DMU) which is under evaluation through data variations of inputs and outputs. Another type of sensitivity analysis is based on the super-efficiency DEA approach in which the DMU under evaluation is not included in the reference set (Cooper et al., 2001, 2006; Jahanshahloo et al., 2005a, 2011; Zhu, 2001). In order to determine the sensitivity and behavior of the non-radial DEA model (M2), a sensitivity analysis was conducted and tested on the data for year 2010. Taking into account the novelty of the model (M2), multiplier model approaches presented by Thompson et al. (1996) were considered because they are

3.6. Adoption of weights for inputs and outputs As can be seen, the non-radial DEA model (M1) and the improved model (M2) contain user-specified preference weights. User-specified weights reflect the relative degree of desirability of the adjustments of the current desirable and undesirable inputs and outputs levels. Note that greater weights will produce higher priority of DMU for reduced undesirable inputs and outputs in the non-radial DEA model (M1), and consequently in the model (M2) desirable inputs will be reduced. Wu 1

http://ec.europa.eu/eurostat/data/database.

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Table 3 Comparison of results of efficiency between non-radial DEA models.

Legend:

, /, ” – countries with the same signs imply the change of countries ranking with weights in model (M2)

**

one case, and with PT and PL in the other. Based on this, it can be, concluded that the degree of the weights can have a significant impact on the results. Therefore, in order to avoid subjective bias in the setting of the weights, in the non-radial DEA model (M2) equal weights – i.e., 1/3 – were chosen. The overall conclusion was that efficiency under the non-radial DEA model (M2) was different when compared to the nonradial DEA model (M1) and could provide a better overall estimation of efficiency due to the expansion with efficiency score θn with weight Wn for desirable input. Based on the selection of the weights – i.e., 1/3 and application of the improved non-radial DEA model (M2), Table 4 represents the results of efficiency evaluation of railways for selected European countries regarding the level of safety at RLCs. The green cells represent the best practice – i.e., countries with the best level of safety at RLCs. Red cells represent countries with the lowest level of safety at RLCs. The results of other countries can also be found in Table 4.

able to consider cases where DMUs are numerous (Cooper et al., 2001, 2006). Having in mind different types of this approach which allow the data variations to occur at random (Cooper et al., 2001, 2006) and specificity of model (M2), as well as its application in RLCs, the sensitivity analysis was conducted applying “approach II” through “development model” (Cooper et al., 2001) for three cases under certain percentages of perturbation until the status of at least one DMU was changed from inefficient to efficient or vice versa (Cooper et al., 2006). 4. Results for railway efficiency in safety level at RLCs All changes of the results in terms of non-radial DEA models and the results of the improved non-radial DEA model were gained by applying Excel Solver. Table 3 presents the results of unified efficiency of model (M1) and model (M2) based on different weights for undesirable input and output on the data of the year 2010. It should be noted that there are differences of unified efficiency in comparison model (M1) and model (M2) for selected weights. Results of the efficiency of the model (M1) were changed due to various weights set to undesirable output – i.e., a number of accidents at RLCs. The results of efficiency were the lowest for combination of weights UDI = 1/3 and UDO = 1/3. The colored cells, numbers and signs indicated the differences in the position regarding efficiency between countries under the represented non-radial DEA models. Moreover, results indicated that, for example, AT had better efficiency under the non-radial DEA model (M2) and the nonradial DEA model (M1) for the last two combinations of weights as compared to NL, while efficiency results were lower for the combination UDI = 1/3 and UDO = 1/2. Furthermore, for the two first results, FI had better efficiency as compared to PL, while for the last two PL had better efficiency. Moreover, NL with a sign (/) had better efficiency as compared to PT for the first three cases under model (M1), while PT was better under model (M2). It should also be noted that one country could have a different position when compared to several other countries. Such an example was NL which was compared to DK and EE in

4.1. Validation of the non-radial DEA model (M2) 4.1.1. Comparison of ranking of countries In order to check the ranking and validity of the non-radial DEA model (M2), its results of efficiency for year 2012 (see Table 4) with data related to ranked countries in terms of fatality risk – i.e., fatalities per million train-km (EU-28: 2010–2012) from the ERA report (ERA, 2014) – were compared. Note that the NO has not been considered due to lack of data and consequently missing result of efficiency for the year 2012 (see Table 4). As can be seen in Fig. 2, countries with the lowest fatality risk such as UK, DE, SE have the highest rate of efficiency. Countries such as EL, HR, SI, and SK with a higher fatality risk have a lower efficiency. However, it can be also seen that some countries with lower number of fatalities such as IE, DK, LU do not have the highest rate of efficiency, as for example the UK, DE, and SE. Moreover, it should be noted that some countries with a lower fatality risk do not have a significant score of efficiency (compare NL, AT, BE with similar pair for example FR) and vice versa (for example RO, PT, HU in 241

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Table 4 Trends of efficiency of European railways in safety performance at RLCs.

was changed from inefficient to efficient or vice versa (Cooper et al., 2006). Therefore, in our example, three cases were performed. Within Case 1 for efficient countries both desirable and undesirable inputs, as well as undesirable output were increased and simultaneously for inefficient countries were decreased. Then, in order to check the sensitivity of the model (M2), while desirable input was fixed, in Case 2 undesirable input and output were increased for efficient countries and decreased for inefficient countries. Since the data for road transport were missing, in Case 3 the behavior of the model (M2) was considered, under the assumption that data for the road portion will be added in terms of traffic volume and number of vehicles. Therefore, for the efficient countries desirable outputs were decreased and desirable input increased, and vice versa for inefficient countries. After each data

comparison with similar ranked countries such as LT, EE, PL). Also, some countries, for example IE, IT, DK, LU, ES do not have higher relative efficiency despite the minimum values of undesirable input and output, as well as desirable input producing relatively lower desirable outputs in comparison to other countries which have relative efficiency equal to 1. Therefore the sensitivity analysis of the model (M2) was conducted in order to confirm the results of the comparison and to realize a complete validity of the model (M2). 4.1.2. Sensitivity analysis of non-radial DEA model (M2) Table 5 presents the results of sensitivity analysis of non-radial DEA model (M2). Sensitivity analysis is made for certain percentages of perturbation (1%, 2%, 5% and 10%) until the status of at least one DMU

Fig. 2. Comparison of ranking of non-radial DEA model (M2) and European Railway Agency.

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Table 5 Sensitivity of non-radial DEA model (M2) using the data for 2010 year.

Through comparison of the non-radial DEA models (M1) and (M2), the non-radial DEA model (M1) could be applied in efficiency evaluation including only undesirable inputs and outputs. However, our improved non-radial DEA model (M2) is more suitable for efficiency evaluation, especially for our purposes, because desirable and undesirable inputs, as well as undesirable outputs, can be simultaneously reduced assessing all weights by decision makers. Therefore, it should be noted that the non-radial DEA model (M1) minimizes only undesirable inputs and outputs for given levels of desirable inputs and outputs. Besides this minimization, with the proposed improved nonradial DEA model (M2) it is possible to minimize desirable inputs as well. Actually, for example in some future applications, without including the efficiency score θn and weight Wn in terms of desirable inputs, the non-radial DEA model (M1) can give an unreal picture regarding efficiency. With the model (M2) consideration of changes in terms of the level of safety at RLCs on macro or micro levels is more representative and strict, which can provide a more accurate picture of relative efficiency. The paper also indicates the results of the non-radial DEA model (M1) for 2010 under different weights. Therefore, in both models weights for desirable and undesirable inputs and outputs can be used. Based on a selected set of preference weights the relative degree of desirability of the adjustment of the input and output levels can be reflected. Consequently, the greater degree of weight, for example, for undesirable output implies the reduction of that output. Having that in mind, based on the preference of DM and the aim of the evaluation, selection of weights should be carefully carried out. Based on their degree they can influence the results of non-radial DEA models. In order to avoid the subjective bias, in this study all undesirable input and output, as well as desirable input weights were set to 1/3. The picture of efficiency may be inaccurate if all necessary variables are not involved. This paper has shown that efficiency results change both with and without a variable number of assets (see Fig. 1). It should also be noted that for different sets of weights, the efficiency results can vary for both non-radial DEA models (see Table 3). Therefore, in terms

perturbation in each case, Excel Solver was repeated and obtained results are represented in Table 5. From the results of all three cases, it can be concluded that the score of efficiency was improved for all inefficient countries. Within Case 1 with 5% and 10% of decrement in data the firstly inefficient IT and then LT became efficient, while a score of efficiency was changed for efficient FR with 10% of increment in data. Regarding Case 2, the only changes can be seen for inefficient countries (IT, LT), only with 10% decrease in undesirable input and output, while the score of the efficiency for some efficient countries was not changed. In comparison with Case 1, the efficiency of FR was not changed, and the inefficient IT became efficient for a higher rate of perturbation. Based on that, it can be concluded that the results of the model (M2) probably depend on the desirable input due to the unchanged efficiency of FR. Results of sensitivity analysis for Case 3 show that both inefficient countries (IT, LT) became efficient with 5% decrement/increment, while in comparison with Case 1 beside the efficient FR, LV also became inefficient with 10% decrement/increment. This yields the information that the countries are more sensitive to the data of desirable outputs. In order to achieve further changes from inefficient to efficient or vice versa, higher percentages of increment/decrement were necessary. However, it should be noted that the efficiency of some countries, such as BE, DK, ES, NL, NO, and UK are constant at some point of decrement/increment. 5. Discussion The proposed tools – i.e., non-radial models (M1) and (M2) – are both relevant for considering best practices and are potentially good techniques for assessing safety at different levels and analyze for short or long terms. Specifically, these models are able to measure the efficiency of some countermeasures before and after implementation, efficiency in the improvement of safety at particular RLCs, and monitoring efficiency in the improvement of safety at RLCs on national and international levels over time. 243

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the score of efficiency with a smaller percentage of modifications in inputs and outputs. On the other hand, this was the effect of greater discrimination of the non-radial DEA model (M2). In the case of inaccurate data, the model can provide an unrealistic picture regarding the best DMUs. Another weakness of the model is related to the correctness of the results depending on the number of variables included. The third weakness is associated with the selection of inputs and outputs. Depending on the selected inputs and outputs, results of the efficiency of safety levels at RLCs will be affected. Moreover, in the case of inappropriate and missing data, results also can be different. For example, including roadway data – i.e., a number of assets and traffic volume (Case 3) – it is sure that results of the model (M2) will be different because sensitivity analysis showed instability with a lower data variation. Overall, these weaknesses can impact the results of the nonradial DEA model (M2) and provide a thankless picture about efficiency in terms of the degree of safety at RLCs. However, in the case of accurate data and selection of appropriate inputs and outputs in accordance with the aim of the evaluation, the proposed new methodology could be a good tool for evaluation of safety at RLCs.

of the mentioned issue, consultations with experts are necessary. Moreover, it is necessary to precisely consider the allocations of variables between desirable and undesirable factors. In our case, the degree of safety at RLCs was evaluated on the macro level in the case of European countries which represent DMUs in the period 2010 to 2012 and for 2014. Undesirable input and output, as well as desirable input, were minimized for a given level of desirable outputs. This means that countries with higher relative efficiency realized the highest level of traffic volume with a minimum number of assets, accidents, and RLCs. In that view, among the most relative efficient countries in that period were DE, FR, LV, SE, and the UK (see Table 4). Although the costs of safety interventions were not considered in the study, based on the results it could be said that these countries have implemented the best safety measures in for the protection of RLCs, which entail significant costs of safety interventions. This assumption is justified by the fact that for the given level of desirable outputs they have minimum undesirable input and output, as well as desirable input in comparison with other countries. Since that is the only assumption, the non-radial DEA model could be used for monitoring the degree of safety at RLCs on different levels after upgrading it, while costs of safety measures would be included as the inputs. In that case, since the costs of safety measures should be minimal, such inputs would be modeled within non-radial DEA models as undesirable. It should be noted that our case considered efficiency only from the railway viewpoint due to the lack of data for roadways, primarily roadway volume at RLCs for each selected country. Moreover, there are limited data for some additional variables for railways that can describe the level of safety at RLCs. It is clear that including some other variables, especially road data, the results presented in Table 4 would change. With a comprehensive set of data – for example, in terms of pedestrians and number of road vehicles passing via RLCs or realized roadway volume – the DEA approach could provide a better picture of the real state of affairs and be used for monitoring best practices. Additionally, the non-radial DEA models can be used for different factors and impacts that cause accidents, such as economic impacts, social impacts and environmental impacts of RLCs (Gruyter and Currie, 2016). In order to check the ranking and validity of the non-radial DEA model (M2), its results were compared with data related to ranked countries from the ERA report. As can be seen from the Fig. 2, there is difference in ranking provided by model (M2) and ERA report. The reason for these differences lies in the fact that ERA ranked countries only based on the fatality risk, while in the model (M2) the total number of accidents was involved. Moreover, the advantage of the nonradial DEA model (M2) is the inclusion of multiple desirable and undesirable inputs and outputs in evaluation (see Table 1) which produce different outcomes of ranking. Based on the sensitivity analysis it can be said that the non-radial DEA model (M2) is valid. However, there is a significant sensitivity to data and selected variables for a smaller data variation that cause reduced stability of the model. For example, the efficiency of some countries is constant at some point of decrement/ increment. The reason for that lies in the model (M2) that evaluates countries – i.e., minimizes inputs for a given level of outputs. Therefore, besides decrement in Case 1 and 2, as well as the increment of desirable outputs and decrement of undesirable input in Case 3, from some point of perturbation the efficiency was constant because these countries have a lower level of desirable outputs in comparison with other countries. Comparing Cases 1 and 2 for FR it can be noted that desirable input may have significant influence on the efficiency score of the nonradial DEA model (M2). Moreover, in the case of lack of data, like in our application in terms of road data, based on Case 3 it can be concluded that the model can also give a different picture of efficiency. It should be noted that with a higher percentage of data modifications the model would provide a picture related to more significant changes of a score of efficiency. Therefore, taking into account the results of the cases it can be noted that sensitivity analysis shows some anomalies/weaknesses in the non-radial DEA model (M2). The first is related to the sensitivity of

6. Conclusion This paper focused on the introduction of a new approach in efficiency evaluation of railways regarding the level of safety at RLCs. For that purpose, an improved non-radial DEA model (M2) was proposed and employed. The evaluation of the degree of safety with the improved non-radial DEA model (M2) was conducted by including desirable and undesirable inputs, as well as desirable and undesirable outputs. Improvement of the model (M2) was based on the introduction of an efficiency score and weight for a desirable output, which was missing in the model (M1), presented by Wu et al. (2015). Based on that the improved non-radial DEA model (M2) can offer a more precise picture regarding efficiency evaluation for our purposes and other applications, as well. The evaluation was conducted for European countries in the period 2010 to 2012 and for 2014. The results of the improved nonradial DEA model (M2) showed the country with best efficiency in terms of the degree of safety at RLCs, as well as other countries with lower efficiency. It should be noted that the evaluation of the efficiency of the level of safety at RLCs was conducted from the railway point of view. With the inclusion of data for a roadways the results of efficiency from the improved non-radial DEA model (M2) presented in Table 4 would change. Based on the sensitivity analysis of the model (M2) it can be concluded that the model is valid, but also illustrates reduced stability due to the smaller variation of data. Therefore, properly selected variables and accurate data are necessary. Precisely, Case 3 of the sensitivity analysis showed that by including road data a picture of the efficiency of DMU has started to change with lower data variation. Therefore, with the inclusion of road data, the results of the improved non-radial DEA model would change. Through empirical study it can be shown that the improved nonradial DEA model (M2), employed using a comprehensive and accurate set of data, can provide an evaluation of the efficiency of the degree of safety at RLCs both at the macro and micro levels, as well as benchmarking for different levels. Evaluation of other factors and their impacts on accidents at RLCs is also possible with the model (M2). For example, with the division of RLCs into active and passive, their impacts on a number of accidents could be measured. Moreover, with this model costs of safety measures and improvements of degrees of safety after costs interventions in upgrading RLCs can be considered. In such circumstances – i.e., ideal implementation of the model (M2), the costs of safety measures should be considered as an undesirable input. Furthermore, this model could be a useful tool for identifying a list of the RLCs with a higher number of accidents and for the planning of the upgrading of particular RLCs. The unavailability of data concerning the volume of roadway traffic and therefore the focus on railway characteristics with limited variables 244

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is the main limitation of this paper. However, since the onus has always been on the railway manager to reduce the risk of accidents at RLCs and due to the lack of the mentioned data, the efficiency in the improvement of the degree of safety at RLCs has been considered only from the railway point of view. Therefore, it is not possible to obtain the comprehensive picture about the real state in terms of the safety at RLCs. Based on that, the future work could be focused on efficiency evaluation in terms of improvement of safety at RLCs, including data for roadway characteristics and an expanded set of variables connected to railways. Another limitation is the subjective selection of variables and weights for variables and the allocation of variables on inputs and outputs. In future research, the selection of variables and weights can be based on the opinions of experts through questionnaires and the allocation of variables can be derived through the use of the an additional tool such as the DEMATEL (Decision-Making Trial and Evaluation Laboratory) method. The improved non-radial DEA model (M2) can be used in future work for monitoring of the degree of safety after costs interventions since in the study costs of safety measures were not considered. Moreover, with this model the evaluation of the efficiency of any particular countermeasure, and any individual RLC, could be conducted in the future. Upgrades for the improved non-radial DEA model in terms of objective allocations of inputs and outputs could also be considered.

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