Picture Fuzzy MCDM Approach for Risk Assessment of Railway Infrastructure

To increase the level of safety and prevent significant accidents, it is essential to prioritize risk factors and assess railway infrastructure. The key question is how to identify unsafe railway infrastructure so authorities can undertake safety improvement projects on time. The paper aims to introduce a picture fuzzy group multi-criteria decision-making approach for risk assessment of railway infrastructure. Firstly, picture fuzzy sets are employed for representing and handling risk-related information. Secondly, a picture fuzzy hybrid method based on the direct rating, and Tsallis–Havrda–Charvát entropy is provided to prioritize risk factors. Thirdly, a picture fuzzy measurement of alternatives and ranking according to compromise solution method is developed to rank railway infrastructures. Lastly, the formulated approach is implemented in the Czech Republic context. Two sensitivity analyses verified the high robustness of the formulated approach. The comparative analysis with five state-of-the-art picture fuzzy approaches approved its high reliability. Compared to the state-of-the-art picture fuzzy approaches, the provided three-parametric approach has superior flexibility.


Introduction
Railway infrastructure safety is an important element of urban public safety [1]. The safety of railway infrastructure is vital to secure goods and passengers from departures to destinations [2]. Accidental events in railway transportation cause damages to human health, public property, environment, and the economy [3]. In 2018, there were 1721 significant railway accidents in the European Union, with a total of 853 fatalities and 760 serious injuries. The total cost of railway accidents was approximately 5 billion EUR [4]. Collisions and derailments account for about 200 accidents each year [5].
To increase safety levels of assets, passengers, goods, and employees as well as prevent significant accidents it is essential to prioritize risk factors and assess railway infrastructure. However, the risk assessment of railway infrastructure is an emerging problem for railway planners. It is considered as a multi-criteria decision-making (MCDM) problem, since there are a finite set of available railway infrastructures, which safety levels need to be assessed, and numerous risk factors influencing railway infrastructure (i.e., evaluation criteria). Additionally, to mitigate information loss and prevent erroneous decisions, it is critical to efficiently represent and handle risk-related uncertain information when solving this complex MCDM problem.
Recently, Blagojević et al. [45] coupled the DEA and SAW methods to rank railway line sections in terms of safety. Brumercikova and Sperka [46] used the AHP method to obtain an entrance order of freight carriers in railway stations. Huang et al. [47] utilized two objective criteria weighting methods and historical data to elicit risks of transporting dangerous goods by railway. Kumar and Anbanandam [48] applied the grey decision making trial and evaluation laboratory (DEMATEL) method to quantify relationships of prominent inhibitors to the implementation of railroad freight transport. Li et al. [49] presented a hybrid Pythagorean MULTIMOORA method to order rail transit lines according to the passenger satisfaction level. Majumdar et al. [50] exploited the TOPSIS method to prioritize key factors affecting metro rail infrastructure based on commuter perception. Sangiorgio et al. [51] integrated the AHP method and linear programming approach to estimate the number of accidents and assess the safety of railway networks. Stoilova [52] coupled the AHP and sequential interactive model of urban systems (SIMUS) methods to select railway passenger plans. Stoilova et al. [53] used the SIMUS method to compare railway transport systems of different countries. Vesković et al. [54] combined the pivot pairwise relative criteria importance assessment (PIPRECIA) and evaluation based on distance from average solution (EDAS) methods to rank options for achieving business balance in organizing railway traffic. Wang and Shi [55] applied an interval-valued intuitionistic fuzzy weighted average operator and maximum entropy method to prioritize the traffic service quality of urban railway transit lines. Zhang and Sun [56] formulated a Delphi-DEMATEL-ANP-TOPSIS approach to assess shunting derailment risk response strategies in railway stations. Tavassolirizi et al. [57] used the DEMATEL-ANP approach to prioritize key delay factors in implementing railway development projects and identify their interactions.

Applications and Extensions of the MARCOS Method
The MARCOS method is one of the latest MCDM approaches. Its applications and extensions are summarized in Table 2.
Badi and Pamucar [58] coupled the grey direct rating and MARCOS methods to evaluate suppliers for the iron and steel industry. Ilieva et al. [59] utilized the fuzzy MARCOS method to order cloud storage systems. Chakraborty et al. [60] ranked suppliers for the iron and steel industry based on D numbers and the MARCOS method. Puška et al. [61] used the MARCOS method to rank project management software solutions for facilitating business operations. Stanković et al. [62] developed a fuzzy MARCOS method to identify a road section with the highest traffic risk for all participants. They applied the fuzzy PIPRECIA method to determine the importance of criteria affecting the traffic risk. Stević and Brković [63] combined the FUCOM and MARCOS approach to compare performances of drivers engaged in international road transport. Stević et al. [14] developed the MARCOS method and used it to solve the sustainable supplier selection problem in the healthcare industry. Ulutaş et al. [16] integrated the correlation coefficient and standard deviation (CCSD), indifference threshold-based attribute ratio analysis (ITARA), and MARCOS methods to select manual stacker for performing logistics activities in warehouses.

Research Gaps
According to the performed comprehensive literature review, the research gaps are as follows: (i) No previous research has applied a PFS-based MCDM approach for railway transport. Hence, railway planners are unable to naturally express their preferences by voting; (ii) there is no risk assessment framework that can handle ambiguous, uncertain, and vague information, since only deterministic numbers and type-1 fuzzy sets have been used in the available studies; (iii) the direct rating and Tsallis-Havrda-Charvát entropy methods have not been hybridized before to determine the importance of risk factors; and (iv) the MARCOS method has not been extended previously using picture fuzzy sets.

Preliminaries
This section provides some definitions of picture fuzzy sets.
Definition 1 [6,7]. Let PFS A on a universe X is an object in the form of : where µÂ(x) ∈ [0, 1] is called the degree of positive membership of x inÂ; ηÂ(x) ∈ [0, 1] is the degree of neutral membership of x inÂ; υÂ ∈ [0, 1] is the degree of negative membership of x inÂ; and µÂ(x), ηÂ(x), and υÂ(x) satisfy the following condition: The degree of refusal membership of x in the PFS A can be calculated as follows: If X has only one element, For convenience, a PFN is denoted byÂ =< µÂ, ηÂ, υÂ > [64,65].

Picture Fuzzy Group MCDM Approach for Risk Assessment
This section presents the developed picture fuzzy group MCDM approach for risk assessment of railway infrastructure (Figure 1). The approach has three phases. In the first phase, linguistic importance evaluations towards railway infrastructures and risk factors are collected from invited experts and expressed as PFNs. In the second phase, subjective, objective, and hybrid importance of risk factors are computed. This phase represents the formulated picture fuzzy hybrid method for risk factor prioritization. Firstly, the picture fuzzy direct rating method is utilized to compute the subjective importance of risk factors. Secondly, the picture fuzzy Tsallis-Havrda-Charvát entropy method is applied to calculate the objective importance of risk factors. Thirdly, the subjective and objective importance are hybridized. In the last phase, the formulated picture fuzzy MARCOS method is used to rank railway infrastructures.

Picture Fuzzy Group MCDM Approach for Risk Assessment
This section presents the developed picture fuzzy group MCDM approach for risk assessment of railway infrastructure ( Figure 1). The approach has three phases. In the first phase, linguistic importance evaluations towards railway infrastructures and risk factors are collected from invited experts and expressed as PFNs. In the second phase, subjective, objective, and hybrid importance of risk factors are computed. This phase represents the formulated picture fuzzy hybrid method for risk factor prioritization. Firstly, the picture fuzzy direct rating method is utilized to compute the subjective importance of risk factors. Secondly, the picture fuzzy Tsallis-Havrda-Charvát entropy method is applied to calculate the objective importance of risk factors. Thirdly, the subjective and objective importance are hybridized. In the last phase, the formulated picture fuzzy MARCOS method is used to rank railway infrastructures.
Construct the picture fuzzy extended evaluation matrix

3.1
Determine the picture fuzzy additive relative importance of each railway infra.

3.2
Calculate utility degrees to the anti-ideal and ideal solutions STEP

3.3
Calculate utility functions to the anti-ideal and ideal solutions of each railway infra.

3.4
Calculate the utility function of each railway infrastructure  Step 1.1. Construct the linguistic evaluation matrices where m, k, and n are the number of railway infrastructures, invited experts, and risk factors, respectively; γ i ej (i = 1, . . . , m; e = 1, . . . , k; j = 1, . . . , n) is the linguistic evaluation (i.e., yes, abstain, no, or refusal) given by the invited expert D e towards the railway infrastructure A i with respect to the risk factor C j .
Step 1.2. Determine the picture fuzzy evaluation matrixẐ = [ẑ ij ] m×n : whereẑ ij =< µẑ ij , ηẑ ij , υẑ ij > is a PFN, which represents an evaluation of the railway infrastructure A i with respect to the risk factor C j given by the invited experts. Importance evaluations are represented as PFNs by computing the share of each item (i.e., yes, abstain, no, or refusal) in the related voting results of the invited experts.
Step 1.3. Construct the linguistic risk factor importance matrices : where ψ e j is the linguistic importance evaluation (i.e., yes, abstain, no, or refusal) given by the invited expert D e towards the risk factor C j .
Step 1.4. Determine the picture fuzzy risk factor importance matrixV = [v j ] n×1 : whereυ j < µυ j , ηυ j , υυ j > is a PFN, which represents the importance evaluation of the risk factor C j given by the invited experts. It is computed as the share of each item in the corresponding voting results of the invited experts.

Phase 2.
Picture fuzzy hybrid method for risk factor prioritization.
Step 2.1. Compute subjective importance of risk factors: whereυ j < µυ j , ηυ j , υυ j > is a PFN, which represents the importance evaluation of the risk factor C j given by the invited experts; and w S = (w S 1 , . . . , w S n ) T represents the subjective importance vector of the risk factors, with w S j ∈ [0, 1] and n j=1 w S j = 1. The subjective importance of risk factors are computed by using the picture fuzzy direct rating method.
Step 2.2. Compute objective importance of risk factors: whereẑ ij =< µẑ ij , ηẑ ij , υẑ ij > is a PFN which represents an evaluation of the railway infrastructure A i with respect to the risk factor C j given by the experts; χ is the information measure parameter; and represents the objective importance vector of the risk factors, with w O j ∈ [0, 1] and n j=1 w O j = 1. The objective importance of risk factors are computed by using the picture fuzzy Tsallis-Havrda-Charvát entropy method.

Phase 3.
Picture fuzzy MARCOS method for railway infrastructure ranking.
Step 3.1. Construct the picture fuzzy extended evaluation matrixΦ = [φ i j ] (m+2)×n : where represents an evaluation of the railway infrastructure A i with respect to the risk factor C j given by the experts: In this step, the picture fuzzy evaluation matrix is extended by adding the anti-ideal and ideal railway infrastructures.
Step 3.2. Determine the picture fuzzy additive relative importance of each railway infrastructure as follows: where ζ is the operational parameter.
Step 3.3. Calculate utility degrees to the anti-ideal and ideal solutions of each railway infrastructure.
(1) Utility degree to the anti-ideal solution: (2) Utility degree to the ideal solution: Step 3.4. Calculate utility functions to the anti-ideal and ideal solutions.
(1) Utility function to the anti-ideal solution: (2) Utility function to the ideal solution: Step 3.5. Calculate the utility function of each railway infrastructure as follows: Step 3.6. Rank railway infrastructures. Rank the railway infrastructures according to the increasing values of utility functions. The lowest value is the safest railway infrastructure.

Case Study
According to the 2020 census, the Czech Republic has 10.7 million inhabitants. More than 50% of the population use railway transportation. A real-life case study of risk assessment of railway infrastructure in the Czech Republic context is presented to evaluate railway infrastructure safety and demonstrate the utility of the formulated approach. The approach is used to prioritize twenty-three relevant risk factors (Table 3) and assess six railway infrastructures ( Figure 2).
The investigated railway infrastructures in the Czech Republic context are: Figure 3a). The railway infrastructure of Pardubice is an important network hub. It is served by international and local trains. This railway infrastructure has connections to Prague, Brno, Bratislava, Budapest, and other large cities. The main railway station, operated by Czech Railways, is vital for passenger traffic and freight transport. Its passenger hall became a place where homeless people concentrate. This railway infrastructure was bombed several times during World War II and many reconstruction projects had been done. Additionally, few significant accidents occurred in the past. • (A 2 ) Hradec Kralove (Figure 3b). It is well-connected with Pardubice. These two railway infrastructures are 21 km away. Hradec Kralove is also an important hub for both passengers and freight. Several significant accidents occurred. Since lots of people pass through its station, the risk of a terroristic attack is possible as well. Figure 3c). This railway infrastructure is located between the capital of the Czech Republic Prague and Pardubice. It is of huge importance for railway transportation in the Czech Republic as a whole, due to a high frequency of trains on the way to Prague. Several years ago, on the railway infrastructure in Kolin, illegal driving was noticed, but luckily there were no injuries and damage declared. (Figure 3d). It is located near the border with Germany. This railway infrastructure counts many departures and arrivals of the local and international trains. Its railway station has been operable for more than sixty years. Due to its location, this railway infrastructure is exposed to various risks such as epidemics, terroristic attacks, and similar accidental events.
In the past, some traffic accidents happened. For instance, a freight train passed a signal prohibiting driving and entered the built train path for a departing passenger train running along the adjacent track in the same direction. The drivers of both trains stopped in front of the switch, with the sides of the trains approaching each other less than a meter. Figure 3e). The railway infrastructure in Chomutov is located in the north-west part of the Czech Republic. It is 86 km away from Prague and 50 km away from Usti nad Labem.
Recently, this railway infrastructure had several significant accidents. In 2010, a freight train knocked down a woman and seriously injured her. In 2016, two people died after being hit by a freight train. The accident occurred at a railway crossing secured by a light warning signal without barriers. According to the information of the Railway Inspectorate, it was in operation at the time of the accident. • (A 6 ) Jaromer (Figure 3f). This railway infrastructure connects Hradec Kralove with Pardubice. Some causes of accidents that happened in the past are walking on the track, passing a signal prohibiting driving, derailment of a train on the ground that was washed up on the track after torrential rain, and track defects.
Step 3.6. Rank railway infrastructures. Rank the railway infrastructures according to the increasing values of utility functions. The lowest value is the safest railway infrastructure.

Case Study
According to the 2020 census, the Czech Republic has 10.7 million inhabitants. More than 50% of the population use railway transportation. A real-life case study of risk assessment of railway infrastructure in the Czech Republic context is presented to evaluate railway infrastructure safety and demonstrate the utility of the formulated approach. The approach is used to prioritize twentythree relevant risk factors (Table 3) and assess six railway infrastructures (Figure 2). The investigated railway infrastructures in the Czech Republic context are: (Figure 3a). The railway infrastructure of Pardubice is an important network hub. It is served by international and local trains. This railway infrastructure has connections to Prague, Brno, Bratislava, Budapest, and other large cities. The main railway station, operated by Czech Railways, is vital for passenger traffic and freight transport. Its passenger hall became a place where homeless people concentrate. This railway infrastructure was bombed several times during World War II and many reconstruction projects had been done. Additionally, few significant accidents occurred in the past.  (A2) Hradec Kralove (Figure 3b). It is well-connected with Pardubice. These two railway

Risk Factor Code Description
Prolonged drought C 1 Prolonged drought is an accidental natural phenomenon caused by a deficit of precipitation. It causes significant economic damage and endangers public health, harms forests and crops, and increases the risk of fires.
Extreme temperatures C 2 Extremely high temperatures are a phenomenon that can occur in the Czech Republic in the period from June to August, rarely in late May and early September. The daily maximum air temperature exceeds 30 • C for several days.
Flood C 3 Temporary significant increase in the level of watercourses or other surface waters, during which water floods the area outside the riverbed and can cause damage.
The Czech Republic has a dense hydrographic network with a length of 85.000 km.
Heavy rainfall C 4 The occurrence of intense rainfall in urban areas results in exceeding the capacity of the sewer network, flooding the lower floors of buildings, and technical infrastructure with surface runoff rainwater.
Extreme wind C 5 Wind speed in gusts greater than 30 m/s or in high gusts greater than 25 m/s. Its effects complicate the functionality of critical infrastructure entities. The average annual wind speed in the Czech Republic is between 2 and 4 m/s.
Mass human infection as a result of which, in a local and temporal context, the morbidity of an infectious disease increases above the usual limit. In the Czech Republic, epidemics occur sporadically.
Mass plant disease is a spread of pests at a level that cannot be managed by conventional methods and means of protection. There is a risk of significant societal losses.
Epizootics C 8 Mass animal disease occurs when emergency measures cannot be implemented by veterinary authorities. There is a risk of human infection with zoonoses.
Disrupted food supplies C 9 Large-scale disruption of food supplies. It can be a secondary consequence of other emergencies.
Disrupted communication systems C 10 Disrupted functionality of important electronic communication networks, which consist of broadcasting systems, routing equipment, and other means enabling the transmission of signals. It affects the monitoring systems and information support.
Critical information infrastructure violation C 11 Planned attack or unintentional failure of technology and people, which leads to failure of services provided by information and communication systems, causes security violations of critical information infrastructure.

Special flood C 12
A special flood is caused by a failure or accident through the rupture of a dam of a waterworks swelling or accumulating water or by an emergency solution of a critical situation.
Hazardous chemical leakage C 13 Hazardous chemicals and mixtures are a source of risk for the occurrence of serious accidents. Accidents are manifested by the release of a chemical or mixture under uncontrolled conditions in a stationary installation. Physiochemical effects are fire and explosion. Biological effects are toxicity and ecotoxicity.
Disrupted water supply C 14 Large-scale disruption of drinking water supplies for drinking, cooking, food, and beverage preparation.
Disrupted gas supply C 15 Large-scale disruption of gas supplies. Equipment for gas extraction, transportation, or storage poses a local risk of explosion. The failure of cross-border points does not have an impact on the gas supply of the Czech Republic.
Disrupted oil supplies C 16 A significant and sudden decrease in crude oil and petroleum supplies. This security threat impacts the economy and living conditions.

Radiation accidents C 17
The radiation emergency is unmanageable with the forces and means of staff. It is manifested by the leakage of radioactive substances and requires the introduction of urgent protective measures for the population.
Disrupted electricity supply C 18 Large-scale disruption of electricity supply. The significant and sudden shortage of electricity, which endangers the integrity, safety, and reliability of an electricity system. Transmission and distribution system operators implement measures and activities to prevent a state of emergency.

Migration wave C 19
Foreigners from third countries coming to the state borders with the aim of legally or illegally enter this territory, transit through to other states, or reside in it. A large-scale migration wave is the arrival of about 750 foreigners/day or 5000 foreigners/month.

Legality violation C 20
Large-scale legality violation is dangerous and intentional behavior in which there is a violation of legislation and security of the Czech Republic to endanger the lives and health of citizens, infrastructure, environment, and democratic system.

Economic destabilization C 21
Large-scale economic destabilization, macroeconomic imbalances, reduction of economic performances, external vulnerability, and economic crisis threats to the financing of social and health expenditure.  A systematic approach is carried out to identify relevant risk factors influencing railway infrastructure in the Czech Republic from the literature [74,75]. Risk factors that do not apply to the Czech Republic context are not listed in Table 3. Each risk is briefly described in Table 3.

Experimental Results
Six assessed railway infrastructures in the Czech Republic context are "Pardubice" (A 1 ), "Hradec Kralove" (A 2 ), "Kolin" (A 3 ), "Usti nad Labem" (A 4 ), "Chomutov" (A 5 ), and "Jaromer" (A 6 ). Twenty-three risk factors influencing railway infrastructure are considered (Table 3). Ten railway planners participated in the case study (Table 4). The online questionnaire approach via Google Forms was utilized to collect evaluations of risk factors and railway infrastructures due to the COVID-19 outbreak. In the first part of the online questionnaire, the invited railway planners are asked to enter their e-mail address, name, occupation, qualifications, total years of experience, and gender. In the second part, the experts are asked to evaluate twenty-three risk factors influencing railway infrastructure in the Czech Republic. In the last part, the participants are requested to provide linguistic evaluations of six investigated railway infrastructures with respect to the risk factors.

Phase 1. Information collection and representation.
Step 1.1. The invited railway planners provided evaluations of six railway infrastructures in the Czech Republic with respect to the risk factors via the online questionnaire. Importance evaluations can be yes, abstain, no, and refusal. Group refusal of voting either is invalid voting papers or does not take the vote. As a result, six linguistic evaluation matrices are constructed with the help of Equation (13). Linguistic evaluations given by 10 relevant railway planners towards the railway infrastructures with respect to twenty-three risk factors are presented in Table 5.
Step 1.2. A picture fuzzy evaluation of railway infrastructure with respect to a risk factor is calculated as the proportion of each out of four items (i.e., yes, abstain, no, or refusal) in the corresponding online voting results of 10 questioned railway planners. The picture fuzzy evaluation matrix is given in Table 6. It is determined based on six linguistic evaluation matrices (Table 5) by using Equation (14).
Step 1.3. Twenty-three risk factors influencing railway infrastructure are evaluated by the relevant experts (Table 7). A sample of the online questionnaire to determine the importance of the risk factors is given in the Appendix A. Ten linguistic risk factor importance matrices are constructed by using Equation 15.
Step 1.4. A picture fuzzy evaluation of a risk factor is computed as the proportion of each out of four items in the corresponding results of the online questionnaire. The picture fuzzy risk factor importance matrix is provided in Table 8. It is determined based on ten linguistic risk factor importance matrices (Table 7) with the help of Equation (16).

Phase 2.
Picture fuzzy hybrid method for risk factor prioritization.
Step 2.1. The subjective importance and ranks of twenty-three risk factors are given in Table 9. The values are determined based on the railway planners' voting on the risk factors influencing railway infrastructure (Table 8) by using the picture fuzzy direct rating method defined in Equation (17).
Step 2.2. The objective importance and ranks of the risk factors are presented in Table 9. The values are calculated based on the railway planners' voting on six railway infrastructures (Table 6) by using the picture fuzzy Tsallis-Havrda-Charvát entropy method defined in Equation (18). In the base case scenario, it is adopted that the information measure parameter χ of the developed picture fuzzy group MCDM approach for risk assessment of railway infrastructure is 1.4.
Step 2.3. The hybrid importance and ranks of the risk factors influencing railway infrastructure are provided in Table 9. The values are computed by using Equation (19). In the base case scenario, it is assumed that the value of the trade-off parameter γ of the formulated picture fuzzy group MCDM approach is 0.5. This value equally appraises the picture fuzzy direct rating method and the picture fuzzy Tsallis-Havrda-Charvát entropy method. Table 5. Linguistic evaluations of the railway infrastructures.   Table 7. Linguistic importance evaluations of the risk factors. Phase 3. Picture fuzzy MARCOS method for railway infrastructure ranking.

Expert Risk Factor
Step 3.1. Anti-ideal and ideal railway infrastructures are determined by employing Equations (22) and (23), respectively. The picture fuzzy evaluation matrix (Table 6) is extended by adding the anti-ideal and ideal railway infrastructures. The resulting matrix is given in Table 10.
Step 3.2. The picture fuzzy additive relative importance of the anti-ideal and ideal railway infrastructures, as well as six real-life railway infrastructures, are calculated by applying the picture fuzzy Hamacher weighted average operator defined in Equation (24). The obtained values can be found in Table 11. In the base case scenario, it is assumed that the operational parameter ζ of the developed picture fuzzy group MCDM approach for risk assessment of railway infrastructure is 3.
Step 3.3. The utility degrees to the anti-ideal and ideal solutions of railway infrastructures are given in Table 11. They are calculated based on the picture fuzzy additive relative importance of railway infrastructures with the help of Equations (25) and (26).
Step 3.4. According to Equation (27), the utility function to the anti-ideal solution is 0.708. On the other hand, the utility function to the ideal solution is 0.292. This value is obtained by using Equation (28).
Step 3.5. Table 11 provides the utility function for the assessed railway infrastructures. The values are calculated with the help of Equation (29).
Step 3.6. Six railway infrastructures in the Czech Republic context are ranked according to the increasing values of the utility functions (Table 11). The ranking order is A 6 (Jaromer) A 2 (Hradec Kralove) A 5 (Chomutov) A 3 (Kolin) A 1 (Pardubice) A 4 (Usti nad Labem). As a result, the presented picture fuzzy group MCDM approach for risk assessment of railway infrastructure ranked "Jaromer" (A 6 ) as the safest. Additionally, it is found that "Usti nad Labem" (A 4 ) has the highest risk in the Czech Republic context. Table 8. The picture fuzzy risk factor importance matrix.

Risk Factor
Degree of Positive Membership

Degree of Negative Membership
Degree of Refusal Membership

Sensitivity Analyses
The sensitivity analyses to changes in the trade-off and operational parameters of the presented picture fuzzy group MCDM approach are performed to check the robustness of the obtained railway infrastructure risk assessment results.
The first sensitivity analysis thoroughly explores how the trade-off parameter γ of the novel picture fuzzy hybrid method influences the ranking order of the railway infrastructures in the Czech Republic context. This method based on the direct rating and Tsallis-Havrda-Charvát entropy prioritizes the risk factors influencing railway infrastructure. More detailed, the subjective importance of a risk factor is based on importance evaluations of that risk factor by the invited railway planners. Its value is computed by using the picture fuzzy direct rating method. On the other hand, the objective importance of a risk factor is based on the railway planners' voting on six railway infrastructures. Its value is calculated by using the picture fuzzy Tsallis-Havrda-Charvát entropy method. Since different input data are used to obtain the subjective and objective ranking orders as well as the number of considered risk factors is large, significant differences in the ranking order of the risk factors are expected.
The parameter γ allows practitioners to make a trade-off between the picture fuzzy direct rating and Tsallis-Havrda-Charvát entropy methods; i.e., a compromise amongst subjective and objective importance of the risk factors. In this regard, the parameter γ is changed in the range [0, 1] with an increment value of 0.1 (Figure 4). When γ = 1, only the picture fuzzy direct rating method is applied to subjectively prioritize the risk factors. When γ = 0, the picture fuzzy Tsallis-Havrda-Charvát entropy method is solely used to objectively evaluate the risk factors. Therefore, in the base case scenario, γ was set to 0.5 to equally appraise both methods and generate hybrid risk factor importance. According to Figure 4, "Jaromer" (A 6 ) is the best railway infrastructure under all γ values, since its utility function has the lowest value. Moreover, there is no change in the ranks of any railway infrastructure in all 10 new test cases; i.e., the railway infrastructure ranking order is A 6 A 2 A 5 A 3 A 1 A 4 . As a result, it is identified that the railway infrastructure risk assessment results of the investigated real-life context are very stable to changes in the trade-off parameter γ. The second sensitivity analysis is performed to check the robustness of generated solutions to changes in the operational parameter ζ of the developed picture fuzzy group MCDM approach for risk assessment of railway infrastructure ( Figure 5). When ζ = 1, the Hamacher T-norm and T-conorm reduce to probabilistic T-norm and T-conorm, respectively. When ζ = 2, the Hamacher T-norm and T-conorm reduce to Einstein T-norm and T-conorm, respectively. As a result, in the base case scenario, it was assumed that ζ is 3. Ten additional test cases are created to systematically analyze the influence of the operational parameter; i.e., ζ ∈ {0.5, 1, 1.5, 2, 2.5, 3.5, 4, 4.5, 5, 5.5}. In all additional test cases, the ranking order of the railway infrastructures in the Czech Republic context is A6 A2 A5 A3 A1 A4 ( Figure 5). This result was obtained in the base case scenario. It is found that the ranks of the railway infrastructures are very stable to changes in the operational parameter ζ. The second sensitivity analysis is performed to check the robustness of generated solutions to changes in the operational parameter ζ of the developed picture fuzzy group MCDM approach for risk assessment of railway infrastructure ( Figure 5). When ζ = 1, the Hamacher T-norm and T-conorm reduce to probabilistic T-norm and T-conorm, respectively. When ζ = 2, the Hamacher T-norm and T-conorm reduce to Einstein T-norm and T-conorm, respectively. As a result, in the base case scenario, it was assumed that ζ is 3. Ten additional test cases are created to systematically analyze the influence of the operational parameter; i.e., ζ ∈ {0.5, 1, 1.5, 2, 2.5, 3.5, 4, 4.5, 5, 5.5}. In all additional test cases, the ranking order of the railway infrastructures in the Czech Republic context is Figure 5). This result was obtained in the base case scenario. It is found that the ranks of the railway infrastructures are very stable to changes in the operational parameter ζ.
According to the results of both sensitivity analyses, it can be outlined that the formulated picture fuzzy group MCDM approach for risk assessment of railway infrastructure is highly robust.

Comparative Analysis
The comparative analysis is performed to check the reliability of the presented results of the developed picture fuzzy group MCDM approach for risk assessment of railway infrastructure. The railway infrastructure risk assessment problem in the Czech Republic context is solved with five available state-of-the-art picture fuzzy approaches: (1) picture fuzzy TOPSIS [76], (2) picture fuzzy EDAS [77], (3) picture fuzzy grey relational analysis [78], (4) picture fuzzy grey relational projection [79], and (5) picture fuzzy cross-entropy [80]. reduce to probabilistic T-norm and T-conorm, respectively. When ζ = 2, the Hamacher T-norm and T-conorm reduce to Einstein T-norm and T-conorm, respectively. As a result, in the base case scenario, it was assumed that ζ is 3. Ten additional test cases are created to systematically analyze the influence of the operational parameter; i.e., ζ ∈ {0.5, 1, 1.5, 2, 2.5, 3.5, 4, 4.5, 5, 5.5}. In all additional test cases, the ranking order of the railway infrastructures in the Czech Republic context is A6 A2 A5 A3 A1 A4 ( Figure 5). This result was obtained in the base case scenario. It is found that the ranks of the railway infrastructures are very stable to changes in the operational parameter ζ. According to the results of both sensitivity analyses, it can be outlined that the formulated picture fuzzy group MCDM approach for risk assessment of railway infrastructure is highly robust. The comparison results are presented in Table 12. The developed picture fuzzy group MCDM approach and picture fuzzy grey relational projection generate the identical ranking order of the railway infrastructures; i.e., A 6 (Jaromer) A 2 (Hradec Kralove) A 5 (Chomutov) A 3 (Kolin) A 1 (Pardubice) A 4 (Usti nad Labem). On the other hand, the picture fuzzy TOPSIS, grey relational analysis, and cross-entropy produce the same ranking order; i.e., A 6 (Jaromer) A 5 (Chomutov) A 2 (Hradec Kralove) A 3 (Kolin) A 1 (Pardubice) A 4 (Usti nad Labem). Six compared approaches assess "Jaromer" (A 6 ) as the safest railway infrastructure of the analyzed real-life context. The proposed approach, picture fuzzy grey relational projection and picture fuzzy EDAS, rank "Hradec Kralove" (A 2 ) as the second-best railway infrastructure. The other three approaches put this railway infrastructure in the third position and evaluate "Chomutov" (A 5 ) as the second-best. "Kolin" (A 3 ) holds the fourth position in five out of six approaches. Only the picture fuzzy EDAS puts this railway infrastructure in the third place. "Pardubice" (A 1 ) is the second-worst railway infrastructure in five out of six approaches. Finally, "Usti nad Labem" (A 4 ) is the worst-ranked in all approaches. Table 12. The comparative analysis of the state-of-the-art picture fuzzy approaches.

Approach Ordering
Picture fuzzy group MCDM approach for risk assessment of railway infrastructure (our approach)

Picture fuzzy TOPSIS [76]
Picture fuzzy grey relational analysis [78] A 6 A 5 A 2 A 3 A 1 A 4 Picture fuzzy grey relational projection [79] A 6 A 2 A 5 A 3 A 1 A 4 Picture fuzzy cross-entropy [80] A According to the results of the comparative analysis, it can be outlined that the developed approach for risk assessment of railway infrastructure is highly reliable.
The ranking similarity between six compared approaches is examined by Spearman's rank correlation coefficient. This quantitative metric reveals the strength of the relationship between compared approaches (Table 13).
The picture fuzzy group MCDM approach for risk assessment of railway infrastructure has 93.2% of ranks matched. Additionally, there is a perfect relationship between the formulated approach and picture fuzzy grey relational projection. Finally, a very strong correlation exists between the introduced approach and picture fuzzy TOPSIS, EDAS, grey relational analysis, and cross-entropy.
According to the results of Spearman's rank correlation analysis (Table 13), it can be outlined that the introduced approach for risk assessment of railway infrastructure produces highly consistent results.

Conclusions
The picture fuzzy group MCDM approach for risk assessment of railway infrastructure is introduced in this paper. Its major contributions are: (i) For the first time, PFSs are employed for representing railway planners' preferences and handling risk-related information; (ii) the novel picture fuzzy hybrid method based on the direct rating and Tsallis-Havrda-Charvát entropy is provided to prioritize risk factors influencing railway infrastructure; (iii) the new picture fuzzy MARCOS method is developed to rank railway infrastructures; and (iv) the formulated approach is implemented in the Czech Republic context.
The merits of the presented real-life case study are: (a) The utility of the introduced approach is demonstrated; (b) the high robustness of the formulated approach is verified by two sensitivity analyses, since the ranks of the railway infrastructures are very stable to changes in the trade-off and operational parameters; (c) the high reliability of the developed approach is approved by the comparative analysis with the picture fuzzy TOPSIS, EDAS, grey relational analysis, grey relational projection, and cross-entropy; and (d) the high consistency with five state-of-the-art picture fuzzy approaches is confirmed by Spearman's rank correlation analysis, since 93.2% of railway infrastructure ranks are matched.
The picture fuzzy group MCDM approach for risk assessment of railway infrastructure generates the following ranking order: A 6 (Jaromer) A 2 (Hradec Kralove) A 5 (Chomutov) A 3 (Kolin) A 1 (Pardubice) A 4 (Usti nad Labem). This novel approach identified "Jaromer" as the safest. On the other hand, the worst-ranked railway infrastructure is "Usti nad Labem". It has the highest risk in the Czech Republic context. As a result, it is strongly recommended to undertake a safety improvement project for this railway infrastructure.
The picture fuzzy grey relational analysis and grey relational projection are one-parametric approaches with build-in distinguishing coefficients. The picture fuzzy TOPSIS, EDAS, and cross -entropy have no parameters. On the other hand, the formulated approach involves three intrinsic parameters, which is highly desirable for solving the complex railway infrastructure risk assessment problem. They are the trade-off parameter γ, the operational parameter ζ, and the information measure parameter χ. As a result, compared to the state-of-the-art picture fuzzy approaches, the provided three-parametric approach has superior flexibility in assessing the risk of railway infrastructure.
Limitations of this paper can indicate its possible extension areas. The limitations are: (1) The risk factors influencing railway infrastructure are not filtered; (2) interrelationships between the risk factors are mainly ignored. A well-known technique for filtering influential factors is the Delphi method. It is traditionally used to obtain a consistent flow of answers through the results of questionnaires. Its major features are anonymous response, iteration and controlled feedback, and finally statistical group response. In a future study, an online multi-round questionnaire approach, in line with the Delphi method, could be performed to collect risk-related information and overcome the first limitation. On the other hand, the DEMATEL method is mostly applied for modeling interrelationships of influential factors. This method has not been extended before into the picture fuzzy environment. One of the future researches may integrate a picture fuzzy DEMATEL method into the introduced methodological framework for risk assessment to handle the second limitation. Additionally, a comparison of the risk factors influencing railway infrastructure in the Czech Republic with risk factors proposed by other researchers can also be seen as an interesting topic that deserves a future research effort. Finally, the introduced methodological approach can be used not only for risk assessment of railway infrastructure but also in macro-level issues. Indicatively, various risks exist in different frames, due to climate change, like fires, floods, droughts, hurricanes, and tornadoes, which affect social and economic life seriously. Acknowledgments: The first author thanks Miloš Milenković for valuable comments on an earlier version of this paper. The authors are grateful for the valuable comments of five anonymous reviewers, who helped to improve the manuscript greatly.

Conflicts of Interest:
The authors declare no conflict of interest.

Appendix A
Sample questionnaire-Importance of risk factors influencing railway infrastructure in the Czech Republic.
Response in each row is not required since a participant can refuse to vote on a particular risk factor. There are three responses/columns to choose from: YES-risk factor is important for railway infrastructure; ABSTAIN-if you are not certain about the importance of a risk factor; NO-risk factor is not important for railway infrastructure.

YES ABSTAIN NO
(