India’s public transportation system: the repercussions of COVID-19

Abstract

The COVID-19 pandemic has left scars on the Indian public transportation system. In order to regain its original momentum, policymakers will need to assess the barriers hindering the effectiveness of the public transportation sector. In this regard, this article analyzes the various factors affecting the public transportation sector in India and determines their interrelationships. The research is presented in three steps. First, we review the literature to identify the factors that affect the public transportation system in India. Next, we propose an integrated model of grey-DEMATEL and ANP, grey-DANP, to calculate the priority ranking and weight of the factors. The grey-DEMATEL method is used to find the interrelationships among the factors, while ANP determines the local and global weights of the factors to form a priority order. Then, we present the interrelationships in the form of influential relation maps. Furthermore, we provide a sensitivity analysis to enhance the credibility of our study. The paper reveals that governmental regulations are the most influential factors in India's public transportation system. The transportation authorities and policymakers must also focus on improving the financial stability and enhancing the customer’s trust in the public transportation system. The framework provided in this paper can be applied to other countries where similar hindrances in the public transportation system have been caused by COVID-19.

1 Introduction

With 36.5 million passengers daily, the Indian public transportation system (IPTS) is among the most heavily used transportation systems globally. About 18% of the Indian population depends on public modes of transportation for daily commuting (UITP 2020a). Public transportation has been the backbone of the IPTS, with rail, road, air, and water services (World Bank 2014). India’s 5.5 million km road network is the second largest in the world and is used by billions of passengers. According to the National Highway Authority of India (NHAI), 65% of total passenger traffic is carried by road transport. Buses carry 90% of these passengers and form an essential constituent of road transportation. India has about 1.6 million registered buses owned by the government and private operators (Devulapalli 2019). Recent developments include the Bus Rapid Transit (BRT) network in many Indian cities (Pucher et al. 2005; Kathuria et al. 2016). The second highest passenger modal share is that of the Indian Railways (IR). The Indian Rail network is the fourth largest globally, with 74,003 coaches, 12,147 locomotives and 23 million daily passengers (Railway 2019). The railway also runs the suburban rail service in many cities, including coveted metro projects in selected cities. By 2023, the IR is preparing to run India’s first high-speed train between Mumbai and Ahmedabad (Mehta 2020). Regarding air transportation, domestic air traffic continued to grow and stood at 144 million passengers in 2019. Low-cost airlines proved to be more successful than full-service carriers grabbing a major share in passenger traffic while the full-service carriers continued to fall (Indian Express 2020).

With the advent of the pandemic, every transportation mode suffered due to a nationwide lockdown. Kumar and Sankar (2020) analyzed the passenger data before and after the lockdown in India and concluded that for the month of May 2020, the daily passenger suburban rail ridership was reduced from 13 million to zero. The number of people carried by public road transport undertaking buses (per day) also fell from 70 to 4 million. At the same time, the average daily domestic-cum-international passengers reduced from 0.55 million to 0.2 million. Naik (2020) estimated the Indian aviation sector's losses to be USD 3–3.6 billion in the June quarter. Whereas the loss to the IR was estimated to be USD 65,000 million during the lockdown, the toll revenue was also reduced by 6.5–8%. The World Bank survey showed that the bus industry faced a loss of USD 7 billion each month due to inactivity (Gupta 2020).

Over the last 300 years, the world has seen several significant pandemics. The severity of the Spanish flu is attributed to modern transportation modes, due to which it spread very quickly. Infected passengers and crews of ships and trains helped in increasing the area affected by the flu. Air travel has been instrumental in the spreading of viruses like SARS (2002–03) and the Avian Flu (2005) due to its convenience and ubiquity (Luke and Rodrigue 2008). Transportation modes can be considered to be a vector, especially in public transit modes. Once a global pandemic becomes apparent, the first step taken is ceasing international travel through the air, while domestic lockdowns occur at subsequent steps according to the situation at hand (Neister 2019). In this paper, we analyze the various challenges that the IPTS faces during the COVID-19 pandemic by answering the following research questions;

• What are the factors affecting the public transportation system in India during the pandemic?

• What are the impacts of individual factors on public transportation in India during the pandemic?

• How are these factors interrelated, and what is their mutual interdependence?

• What is the significance of individual factors in a consolidated decision framework?

• What are the implications of the proposed study for practitioners and researchers?

Public transportation is a basic need in today’s era and needs to be acknowledged and analyzed correctly. The literature review reveals that studies have been conducted on the effect of COVID-19 on the public transportation systems of various countries. Still, none of them employs an evaluation method to determine the cause-and-effect relations between hindering barriers. Hence, this study aims to fill this research gap and answer the above questions with the help of the following research objectives:

• To identify the factors affecting India's public transportation system during the pandemic and develop an interrelation framework.

• To measure the cause-and-effect influence of each factor and sub-factor.

• To prioritize the factors based on weightage and propose implications based on the inferences drawn by the research.

This study's main contribution is to develop an evaluation structure from multiple users using a methodology that integrates Decision-Making Trial and Evaluation Laboratory (DEMATEL) and Analytic Network Process (ANP) into g-DANP to compute interrelationships among factors and their weights (Shanker et al. 2021b). Many studies in the past have combined these two techniques for evaluating such real-life problems (Büyüközkan and Güleryüz, 2016; Supeekit et al. 2016; Govindan et al. 2018; Shanker et al. 2021a). In g-DANP, the grey-DEMATEL method is used to determine interconnections between the factors, whereas ANP improves the weighing presumptions in the evaluation model (Büyüközkan and Güleryüz, 2016).

The rest of the paper is organized as follows. In Sect. 2, we review the literature on India's public transportation system and the literature on our research methodology. Section 3 discusses the factors that influence the public transportation system in India during the COVID-19 pandemic. These factors assist the decision-makers in formulating the appropriate strategies to tackle the issue. Section 4 discusses the proposed framework for the case illustration, followed by Sect. 5, where we present the results and discussions. Section 6 presents the sensitivity analysis for improving the authenticity of this study. Section 7 discusses the implications for practice and research, and finally, Sect. 8 delivers the concluding remarks.

2 Literature review

2.1 Literature review of Indian public transport

India is among the fastest-growing economies in Asia but faces inadequate transportation infrastructure to meet demands, thus affecting its economic growth and development (Ng and Tongzon 2010). According to World Bank (2010), India’s road density is 0.66 km/km², higher than the USA and China's road density, but these roads are narrow, congested, and of poor service quality. Ng and Gujar (2009) state that the policies should be consistent with political realities for proper growth and development. This is especially hampering the growth of railways. According to Reddy and Balachandra (2011), a major result of an increase in per-capita income is that private car ownership in urban India is much higher than private car ownership in rural India. This is leading to more congested roads. After studying Gurugram, Narain (2009) concluded that a city’s structure and travel demands of private or transit cars highly depend on urban planning. A highly efficient public transportation system encourages the use of public transportation rather than opting for private modes. The main mode of the daily commute in Indian cities, i.e., buses, are overly used due to the lack of a proper rail system and have low standards compared to developed countries. Reddy and Balachandra (2011) identified the key challenges facing transportation in India: urbanization and transportation demand, inefficiency in public transportation systems, externalities like pollution and congestion, institutional weakness, and political instability. Verma et al. (2011) emphasize the need of sustainable transportation as it has a positive impact on the social, economic, and environmental sustainability of the community. It was discovered that inadequate infrastructure, policies, and poor facilities discourage people from using non-motorized transport (NMT) modes in India.

According to Mani et al. (2012), safety is an essential factor that decides public transportation usage. The WHO (2013) deemed India’s roads dangerous, with the most significant number of road fatalities globally at 866 per million vehicles. The National Urban Transport Policy 2006 is an essential step in ensuring a safe and efficient transportation system. According to Tiwari (2011), better bus services, improved intermediate public transportation systems, and proper enforcement of road speed limits are the three most important factors in improving public transportation systems in India. The author also advises that road and traffic standards should be renewed every two years, and segregated lanes for non-motorized vehicles (NMV) should be constructed. Auto rickshaws can play a vital role in developing an efficient transportation system as they provide first and last-mile and door-to-door connectivity.

Nevertheless, the absence of an organization and the emissions of harmful particulate matter are the primary concerns to be solved (Mani et al. 2012). The average journey speed on Indian roads is very low, especially in high vehicle ownership cities. The average speed in most metro cities was below 20 km/h (Nama et al. 2016). Road congestion is still a complex problem even after sub-urban rail connectivity in metro cities. Policies for easing congestion include better integrated urban planning, the promotion of public transportation, and an intelligent transportation system promotion.

Long delays and red-tapism in transportation infrastructure projects in India are a big concern. The delays decrease the project's financial stability by increasing the total costs and decreasing the customer’s trust (Patil 2013). Despite having a tremendous scope, Rahul and Verma (2013) found that the NMT is not showing any growth in India due to a lack of infrastructural provisions. NMTs can go a long way in reducing congestion costs, decreasing the death rates on Indian roads, and ensuring a safer environment. Suppose 1% of the population shifts to NMT modes for trip distances less than 5 km. In that case, the expected benefits can be about USD 3.4 thousand per day, including social costs, infrastructural costs, congestion costs, health costs, etc. Apart from the government-run public transportation modes, there are also lots of privately run or ‘informal’ public transportation modes consisting of auto-rickshaws, Tata Magics, and Vikram's that cater to a large share of the population (Kumar et al. 2016b). These private modes of transport also include the likes of taxis and cab services, two-wheelers, cars, and jeeps. Of the ‘informal’ modes of transport, the greatest in use for work travel are two-wheelers at 33% and cars at 14% (Eregowda et al. 2021). Other modes of private transport include the hailing of auto-rickshaws and booking of cabs from online services. These informal modes are still active due to government-run public transportation modes' low capacity and inefficiency (Jaiswal et al. 2021). According to Ponodath et al. (2018), the government considers these informal modes unsafe and excludes them from transportation policies. The Indian government should acknowledge these modes' value and help them grow to ensure an efficient transportation system. Paladugula et al. (2018) project the transportation sector's energy intake to increase by 4.1–6.5% every year till 2050, increasing India’s dependency on oil. Thus, the authors suggest that public transportation and NMT modes should be encouraged by the government. Saripalle (2018) claims that the introduction of a Goods and Services Tax (GST) by India’s government will help the transportation sector move from labor-intensive processes to capital-intensive and automated processes. Ahmad and Chang (2020) studied all the transportation policies of India and concluded that the Indian transportation system's biggest problem is the lack of implementation of policies.

The IPTS, which was already facing problems, including inefficiency and vulnerability, was hit hard by the unforeseeable COVID-19 pandemic. Considering the pandemic's economic impacts on the Indian transportation sector, Singh and Neog (2020) conclude that lockdowns in the country have contributed to the shutting down of both public and private transportation modes affecting the demand and supply in the economy. National and private airlines, which are already running in losses, will face severe survival issues. Decreased revenue generation from road and rail transportation will also affect the government treasury. Harikumar (2020) wrote that India's energy demand fell by 11% in March due to the cancellation of flights and other transportation bookings. Owing to an increased risk of being infected, public transportation modes' demand and occupancy have become very low. Risks associated with the spreading of the virus deter more people from traveling through public modes. Further dissuading the public from using transport is an abhorrent behavior of the staff, and in the case of Indian rail transport, a poor refund process and delay in operational activities (Mishra and Panda 2022). The effect on informal public transportation modes is expected to be much more severe than that on organized modes. UITP (2020b) provided an overview of bus operators' challenges during the lockdown. These challenges were segregated into four key areas: operational and service delivery, financial management, crew management-related, and fleet management-related. Government guidelines have also increased the operating costs for operators who are already facing low occupancies. The biggest challenges during the post-COVID era will be operators' financial constraints to sustain operations, low patronage due to fear of safety, lower service availability, and an increase in fares (Mishra and Panda 2022). According to Gupta (2020), the COVID-19 pandemic is an excellent opportunity for India to promote sustainable mobility.

Similar research has focused on the impact of COVID-19 in other nations, such as the survey-based study conducted by Munawar et al. (2021). This manuscript highlighted three main sectors of transport, i.e., air, freight, and public transport, in the Australian transportation sector. They declared that public transport was reduced by over 80% and made it clear that the transport sector's recovery depends on the relaxation of COVID-19 containment policies and financial support from the government. Further studies on reducing the challenges faced by public transport due to COVID-19 have heavily implied the necessity of tailored government regulations (Tirachini and Cats 2020). The authors have suggested specifying restrictions based on the risk of contagion to a specific community in order to increase safety and decrease the deterrence of users. Furthermore, studies spanning multiple countries, such as the USA, members of the EU, China, and India, draw attention to the perception of public transport (Abu-Rayash and Dincer 2020). This argument is of utmost importance because, if public transportation is perceived as unsafe by a majority of the population, it will not be able to fulfill its role of being an accessible mode of transport (Sikarwar et al. 2021). One of the largest factors that may help with public perception is the removal of government-mandated lockdowns. This will help assure the public that adequate measures have been taken and may improve public relations.

2.2 Literature review of g-DEMATEL and ANP methodology

The analysis of factors affecting public transportation in India during the COVID-19 pandemic is a complex decision-making process. The problem itself involves many factors, each requiring quantitative and qualitative assessment. It is observed that real-world scenarios, such as the situation considered for this study, consist of multiple conflicting problems. Also, there is never enough time, staffing, or resources to deal with all barriers. Therefore, it is necessary to use a decision-making technique to analyze the factors identified in the literature review. Multi-criteria decision-making (MCDM) techniques are well known for situations that include conflicting objectives, the presence of numerous factors, and a hierarchical structure (Demirel et al. 2010). In many real-life cases, specialists feel that a single technique is not enough to reach a logical decision and recommend a mix of MCDM techniques (Govindan et al. 2018; Singh and Bhanot 2020; Li and He 2021). The use of the MCDM technique helps in two ways: it decreases the problem's multi-faceted nature and helps in an in-depth analysis of the issue (Kumar and Anbanandam 2020). This study uses a hybrid MCDM technique to discover the nature of interdependencies, interrelationships, and prioritization of the factors influencing public transportation in India during the pandemic.

The DEMATEL strategy is utilized to discover the cause-effect interdependencies of the factors in terms of relation values, while the relative significance is determined by the prominence value of variables (Gölcük and Baykasoğlu 2016). DEMATEL cannot be used to dictate the priority weight of variables; however, ANP helps in processing the structural reliability of the problem (Büyüközkan et al. 2017). The ANP technique demands a pairwise correlation of criteria and also checks the pairwise decision matrix for consistency (Govindan et al. 2018). The mixture of DEMATEL and ANP decreases the complexity of the problem and assists in identifying the global impact of factors (Büyüközkan and Güleryüz 2016; Kumar and Anbanandam 2020). As DEMATEL does not need pairwise examinations, the multi-faceted nature of the decision variables is reduced (Tadić et al. 2014). Thus, it is preferred to feature the interrelationship among the identified factors (Govindan et al. 2018; Li et al. 2020). The above-mentioned advantages of combining DEMATEL and ANP improve computational investigation (Abikova 2020).

In the DANP, DEMATEL is used for gathering information, finding the interrelationships, and measuring the interdependencies, while ANP is used for deriving the unweighted super-matrix, weighted super-matrix, and the limiting matrix to process the influential weight of individual components (Abikova 2020). Grey theory can be combined with various MCDM techniques to improve the accuracy of the decision-making process (Xia et al. 2015; Chithambaranathan et al. 2015). The MCDM based on the grey system depends on the fact that it would provide accurate outcomes even with limited information and provide a state of fuzziness (Rahimnia et al. 2011). The benefit of using a grey system over fuzzy methodology is that it does not require any robust fuzzy enrolment function (Xia et al. 2015). In the past, multiple MCDM techniques have been used by researchers to evaluate the multiple conflicting criteria. Techniques such as Analytic Hierarchy Process (AHP), the Technique for Order Preference by Similarity to Ideal Solution (TOPSIS), and Elimination and Choice Translating Reality (ELECTRE) have been used extensively to analyze the relationship between factors (Supeekit et al. 2016). However, unlike ANP, these methodologies are not apt to deal with the interdependencies between the factors. According to Raj and Sah (2019), including DEMATEL in the study enables easy visualization of causal relationships through causal diagrams. The proposed g-DANP methodology is more accurate and decreases the computational complexity of the decision-problem. Grey systems theory is preferred over fuzzy theory as it does not require any fuzzy membership function (Kumar and Anbanandam 2020). The methodology works in two stages (Appendix 1) for computing the interrelationships and prioritization of the factors affecting public transportation in India during the pandemic.

3 Factors influencing the public transportation system in India during the Pandemic

Analyzing the relationships between the multiple criteria and proposing a proper model is a significant task (Tran et al. 2020). The development of the model begins with a literature review, which leads to differentiating the factors based on their influence. For this purpose, data for factors and their sub-factors are collected from literature reviews, reports, and expert opinions. The sub-factors were chosen from the literature review due to the frequency of manuscripts that mentioned them or the high rating imposed on them by other studies. Then, the factors are categorized under five principal dimensions: Operational, Infrastructural, Financial, Behavioral, and Governmental Regulations. A detailed discussion of the factors and their sub-factors is presented in this section.

3.1 Operational factors (O)

The factors that affect the day-to-day performance of public transportation are known as operational factors (Tang et al. 2018). Operational factors are more concerned with running, scheduling, and managing the system and are aimed at creating an efficient transportation system (Parkan 2002; Yeboah et al. 2019). Table 1(a) gives a description of operational factors that affect the public transportation system.

Table 1 (a) Operational Factors Challenging Indian Transport during COVID-19, (b) Infrastructural Factors Challenging Indian Transport during COVID-19, (c) Financial Factors Challenging Indian Transport during COVID-19, (d) Behavioral Factors Challenging Indian Transport during COVID-19, (e) Governmental Regulation Factors Challenging Indian Transport during COVID-19

3.2 Infrastructural factors (I)

These factors deal with the structures and systems necessary for the operation of transportation systems (Hossain 2019). Transport infrastructure is different for different modes. Roads, bus fleets, and terminals constitute the road infrastructure, while loco fleets, rail tracks, and railway stations constitute the rail infrastructure. Transit stations and ticketing infrastructure are also important components (Skorobogatova and Kuzmina-Merlino 2017). Table 1(b) talks about the impact of infrastructural factors on the public transportation system.

3.3 Financial factors (F)

Financial factors consider many money-related decisions and results, like how much profit the company makes and how much it costs to run the system (Xu et al. 2018; Arora et al. 2022). Achieving financial stability is important for any transportation system, and it leads to more research and development (R&D) in the sector (Efthymiou et al. 2018). Table 1(c) discusses the financial factors that affect the public transportation system.

3.4 Behavioral factors (B)

The factors affecting the transportation system that depend on the passengers, i.e., the service consumer, are behavioral factors. (Kaewkluengklom et al. 2017). These depend upon the perception of the services in their eyes. Behavioral factors differ from person to person and prove to be a major factor influencing the use of public transportation services (Yeboah et al. 2019). Table 1(d) talks about various behavioral factors and their effect on the public transportation system.

3.5 Governmental regulations (G)

The government has laid down several rules and regulations for the proper functioning of the transportation system during this pandemic (Tirachini and Cats 2020). Apart from the nationwide lockdown, several other temporary lockdowns were also imposed at various levels to check the spread of the virus. These regulations have brought losses to this sector, leading to ceased growth (Mayo and Taboada 2020). Table 1(e) discusses the effect of governmental regulations on the public transportation system.

4 The proposed framework

We have selected the case of factors influencing public transportation in India during the COVID-19 pandemic to validate the framework we have prepared. In this study, primary data and a literature review were used to collect information. Primary data was collected through interviews with academic researchers and industry stakeholders. The interviews were conducted between December 2020 and January 2021. Each interview with the experts detailed in Appendix 2 lasted for approximately 50–80 min. The chosen factors were explained to the experts along with the provided linguistic scale. These experts' responses were collected after an explanation of the sub-factors in the study. In the successive section, we describe the process of data assortment and analysis. Further details on the methodology can be found in Appendix 1.

4.1 Determining the interrelationships among the factors and sub-factors affecting public transport

This segment depicts the interrelationships of the factors and sub-factors mentioned in Sect. 3. The steps of grey-DEMATEL are as follows:

Step 1: Formulation of direct relationship and crisp relation matrix ‘A’.

The initial matrix is generated by gathering the linguistic data from the twelve selected experts (refer to Appendix 2). The decision-makers were chosen from academia and industry based on their years of experience in the related fields. These ratings are then converted on the greyscale (Table 2). Thus, twelve individual 22x22 matrices are obtained. The crisp relationship matrix is then prepared using Eq. (7).

Table 2 Linguistic to grey scale

Step 2: Obtain normalized crisp relationship matrix ‘B’

We normalize the initial crisp relationship matrix using Eq. (9) and Eq. (10).

Step 3: Calculate total influence matrix ‘T’

The total influence matrix is calculated from the normalized matrix ‘B’ using Eq. (11). The matrix ‘T’ is given in Table 3.

Table 3 Total Influence Matrix

Step 4: Formulation of the sum of rows ‘${R}_{i}$’ and the sum of columns ‘${D}_{j}$’.

Using Eq. (12), we calculate the total influence given and received by each factor, as shown in Table 4.

Table 4 Influence Among Factors’ Dimensions and Sub Criteria

Step 5: Plot the ‘cause’ and ‘effect’ digraph

Plot cause and effect digraph based on the influence received and given by the dimensions and sub-components. An influence map is made for each of the main factors, which shows the interrelationships between its subfactors. The influential diagrams of dimensions and sub-components are depicted in Figs. 1, 2, 3, 4, 5 and 6.

Fig. 1

Influence relation map of main dimensions

Fig. 2

Influence Relation Map of Operational Factors

Fig. 3

Influence Relation Map of Infrastructural Factors

Fig. 4

Influence Relation Map of Financial Factors

Fig. 5

Influence Relation Map of Behavioral Factors

Fig. 6

Influence Relation Map of Governmental Regulation Factors

4.2 Weight of dimensions and criteria through g-DANP

In continuation to the g-DANP method, the following steps were used to find the priorities of factors.

Step 6: Calculation of unweighted super-matrix (W).

With the help of Eqs. (14) and (17), we obtain the unweighted super-matrix $W$($i.e.,W=({{T}_{c}^{\alpha })}^{^{\prime}}$) from the total influence matrix (T). Then, we normalize the total influence matrix with the help of Eqs. (19) and (20).

Step 7: Formulate the weighted super-matrix $({W}^{\alpha })$

The weight of the super-matrix (i.e., ${W}^{\alpha }={T}_{D}^{\alpha } \times W$) is calculated by using Eq. (21).

Step 8: Limit the weighted super-matrix

The final weights are calculated by limiting the power of the weighted super-matrix $\underset{k\to \infty }{\mathrm{lim}}{{(W}^{\alpha })}^{k}$, where $k$ should have a high value. The stable matrix of g-DANP is depicted in Table 5. Each row represents the weight of individual factors. The proposed framework gives the global importance of factors, as summarized in Table 6.

Table 5 Influential Weights in the Stable Matrix with the Power Limit $\underset{k\to \infty }{\mathrm{lim}}{{(W}^{\alpha })}^{k}$

Table 6 Importance of Weight Computation of Factors through g-DANP

5 Results and discussion

The case-based result analysis of the dimensions and their sub-components are presented at the combined and individual dimensional levels.

Tables 4 and 7 show the influence analysis of main factors and report that Operational factors (O), Governmental regulations (G) and Behavioral factors (B) fall in the cause category. In contrast, Infrastructural factors (I) and Financial factors (F) fall into the effect category. This implies that ‘F’ and ‘I’ are the highly influenced dimensions while ‘O’, ‘B’ and ‘G’ are influencing dimensions. Figure 1 represents the graphical relationship between the main factors. As ‘G’ is the main factor with both the highest prominence and relation value, it can be understood that Government regulations are the priority barrier hindering the public transportation system in India during the pandemic. The figure reveals that by solving the issues related to Government regulations, not only will the largest cause of disturbance for Indian public transport be solved, other main factors will be affected, too.

Table 7 Total Influence Matrix of Main Factors (Dimensions)

The prominence (${R}_{i}$+${D}_{i}$) values from Table 4 show that Governmental Regulations (G) obtained the highest score (0.562). The score depicts that it has a very strong influence on public transportation in India during the pandemic. Thus, ‘G’ should be considered the most crucial factor among all the factors affecting public transportation. From (${R}_{i}$-${D}_{i}$) values, ten sub-factors have been classified into the ‘Cause’ group and 12 sub-factors into the ‘Effect’ group. The ranking for the cause group is G1 > G2 > G3 > B2 > B4 > B1 > F3 > B3 > O8 > I2. This result suggests that ‘Nationwide lockdown (G1)’ is the most causal factor which affects the public transportation system in India. The ranking of ‘Effect group’ is F1 > O2 > O1 > I1 > F2 > O4 > O5 > O3 > O6 > I3 > F4 > O7. Thus, decreased profits, a reduced number of trips, and low occupancy are the main effects of the pandemic on the public transportation system in India.

5.1 Operational factors

From Table 4, we can see that all the sub-factors except O8 fall in the effect category. Their relative ranking is Reduced number of trips (O2), Low seat occupancy (O1), Increased waiting time at transit stations (O4), Reduced interchange connectivity (O5), Shortage of workforce (O3), Reduced accessibility (O6), Lesser driver necessities (O7) with ratings of 2.256, 2,151, 1.486, 1.485, 1.46, and 1.419, respectively. As evident by its placement above the x-axis, Increased operational costs (O8) is the only cause factor among these sub-factors (refer to Fig. 2). Also, the factors on the far right of the figure, i.e., Reduced number of trips and Low seat occupancy, are among the major affected operational factors due to their high prominence values. Other studies have come to a similar conclusion regarding effectively managing a limited capacity. Munawar et al. (2021) suggest implementing a queuing system to compensate for the reduced accessibility. However, this solution would be limited to scenarios where digital billing is already available. Furthermore, Bandyopadhyay (2020) suggests public transport should limit itself to use by essential workers and medical staff, thus mitigating some of the social equity ramifications. This suggestion also coincides with our results regarding factor O3. Similarly, policymakers will need to explore new ways to recruit and train additional workers and plan for absenteeism, accounting for new problems as they arise.

5.2 Infrastructural factors

Of the three infrastructural sub-factors, two are counted into the ‘effect’ category, while the third one belongs to the ‘cause’ category (Table 4). Lesser active fleet (I1) and Reduced last mile connectivity (I3) belong to the effect category with ratings of 2.104 and 1.373. Their placement in the effect category is evident from their position below the x-axis in Figure 3. Lack of e-ticketing and e-payment infrastructure (I2) is the only cause sub-factor. This shows that the improvement of e-ticketing and payment infrastructure can help in improving the public transportation infrastructure in India. This conclusion is further supported by the results of Tirachini and Cats (2020), discussing the implications of COVID-19 on public transportation. One major factor towards revitalizing public transport is e-booking and queuing systems. This will be a boon in managing service capabilities and ensuring physical distancing requirements. Beck et al. (2021) also support the results that reduced last-mile connectivity will lead citizens towards employing private transportation methods. The operations of many public transportation modes need to be updated while maintaining the safety of their workers. Another major concern stems from factor I1. A lesser active fleet size is bound to see a quicker deterioration of public transport facilities. Our findings also match Mogaji (2020), who suggests an updated management system for maintaining appropriate services and standards.

5.3 Financial factors

From Fig. 4, we can see that three of the four financial factors, which fall below the x-axis, come under the effect category, which are Declined profits (F1), Increased market risk (F2) and Increased unplanned expenses (F4) with values of 2.365, 1.779, and 1.35, respectively. Increased expenditure on safety procedures and equipment (F3) is the only cause sub-factor with a value of 1.734. Thus, if we find a way to reduce the expenditure, the profits can increase, making the transportation system financially stable. Similar studies have also suggested using government grants to maintain transport supply and observing vehicles' social distancing measures (Munawar et al. 2021). They have concluded that without government intervention, the financial risks faced by the public transportation sector will continue post-COVID-19. Mogaji (2020) also suggests that governments should work with transport unions to curb an increase in post-pandemic fares. The increased market risk of public transport also incentivizes more people to opt for private transportation. Middle-income citizens will begin to prefer motorcycles and auto-rickshaws over standard means of public transport. These results imply that travelers will make their decisions based on their sense of security in public transport and prefer to take larger financial risks for safety.

5.4 Behavioral factors

In our analysis, we can see that all the behavioral factors fall into the cause category (refer to Fig. 5). The relative order of their prominence is Reduced reliability on Public transportation (B3) > Fear of negligence of social distancing norms (B2) > Lack of passenger’s confidence in the hygiene of public transportation (B1) > Lesser need for travel (B4), with values 2.521, 1.901, 1.822, and 1.606, respectively. This shows that behavioral factors contribute a lot to public transportation usage by people. The population's decision to adopt public transportation depends heavily on their perception of its safety. This effect is consistent with the findings of previous studies (Dong et al. 2021). Furthermore, as government regulations remain in effect, most people's priority will be to avoid public spaces for fear of others neglecting social norms. Similar studies describe a state of anxiety that can negatively affect a passenger’s perception of safety (Lau et al. 2006). Investing in proper sanitation and ensuring adequate social distancing norms can rebuild people's trust in the transportation system. Further, Mogaji (2020) suggests properly managing existing infrastructure as that increases the reputation of public transport in the eyes of the public. As all these factors are causal in nature, they must be dealt with first in order to calm the public (refer to Fig. 5).

5.5 Governmental regulations

All the Governmental Regulations fall in the cause category (Table 4). The order of importance is Nationwide lockdown (G1) > Frequent temporary shutdowns at various levels (G2) > Sealing of state borders due to lockdown (G3). Their values are 2.024, 1.816 and 1.522, respectively. All these sub-factors belong to the ‘cause’ category because they influence the other factors mentioned above. The government can interfere, regulate, and change the rules, which affects the operational, infrastructural, financial, behavioral, and, thus, public transportation usage. From the above analysis, we recommend giving some ease to the lockdown at various levels for the smooth functioning of the public transportation system. Similar studies have suggested that governments should enact specialized restrictions according to the population's needs (Wielechowski et al. 2020). In order to gain the trust of the public, it is a necessity that government regulations be lessened while maintaining safety precautions. Other researchers also mention that vaccinating the population would solve a majority of these issues; however, this action lies outside the domain of public transport (Beck et al. 2020). In a similar study in China, Shen et al. (2020) found that stricter regulations must be followed at the local level, taking into account individual cases instead of temporarily shutting down service. The diagonal nature of their placement in Fig. 6 gives a clear order for which government regulations sub-factor must be dealt with first. ‘G1’ is not only the most prominent of the three sub-factors, but solving the issues relating to ‘G1’ also helps regarding the barriers ‘G2’ and ‘G3’.

5.6 Prioritization of factors based on g-DANP

Considering the factors affecting the public transportation system in India during the COVID-19 pandemic as multi-dimensional, a single model cannot explain the phenomena fully. To compensate for some disadvantages of DEMATEL, we have used the g-DANP methodology. Table 6 shows the weight of every dimension and its sub-factors. The local importance of each sub-factor is calculated by dividing global weight by dimension weight. From Table 6, Lesser active fleet (I1), Reduced profits (F1), Increased market risk (F2), Reduced reliability on public transportation from the passenger’s perspective (B3), and Reduced last mile connectivity (I3) are the top five sub-factors affecting the public transportation system in India during the pandemic. This analysis can help policymakers and managers work in the right direction and take adequate measures to uplift the public transportation system. Table 8 shows the precise analysis which can be used to develop strategies for specific causes.

Table 8 Classification of performance of factors affecting public transportation in India during COVID-19 pandemic

6 Sensitivity analysis

We face several questions while discussing the results of this paper: How does the selection of decision-makers affect the results? What is the extent of this effect on the results? What is the role of personal bias in the variation of data? Is there any way to determine the credibility of the results and to prove that it is not affected by personal bias? To solve these questions, we performed a sensitivity analysis. According to Shanker et al. (2021a), a sensitivity analysis is a famous practice that determines the stability of the results when the input values are varied marginally. Since human decisions constitute the primary input of this study, a sensitivity analysis can be used to scrutinize the results (Kamat et al. 2022).

The analysis concludes that financial factors hold the highest priority among all the main factors (Table 6). Thus, financial factors can influence other factors as well. To illustrate the influence of financial factors on other sub-factors, we performed a sensitivity analysis. We varied the initial weight of ‘F’ in increments from 0.1 to 0.9 and then redistributed the weights among the other main factors (Table 9). As the weightage of financial factors changed, the weight of sub-factors also changed. In Case 1, when the financial factor weightage is 0.1, ‘I1’ and ‘B3’ hold the first and second positions, respectively, while ‘F3’ stands last (Table 10). ‘I1’ holds the top priority until the financial factors' weight reaches 0.286, which is the normal value. From 0.3 to 0.9, ‘F1’ acquires the top position and the rank trend continues till the last case, i.e., Case 9. Figure 7 illustrates the ranking of sub-factors according to the sensitivity analysis. As shown in Table 10, many sub-factors maintain the same rank over numerous trials. Thus, their lines in Fig. 7 overlap and the ten individual trials cannot be observed. The analysis concludes that the variations in the weight of financial factors lead to a change in the relative ranking of other factors. Thus, policymakers, managers, and operators in the transportation industry need to ensure the sector's financial stability. The results of the sensitivity analysis align well with the normal case in the study. This analysis improves the credibility of the study and also authenticates the results.

Table 9 Weights assigned to main factors during sensitivity analysis

Table 10 Sub-factor rankings for all cases

Fig. 7

Ranking trend of sub-factors during sensitivity analysis

7 Implications for practice and research

The government and other authorities can use the proposed g-DANP evaluation framework for a valid assessment of factors and to develop a strategic plan for uplifting the public transportation sector in India during COVID-19. Grey DEMATEL was integrated with ANP to determine causal relationships and importance weights for developing a framework. The managerial contributions of this study are as follows:

• i. To improve the public transportation sector, policymakers would have to pay more attention to the ‘cause’ factors to reduce the influence of the ‘effect’ barriers.

• ii. This research provides a list of factors with priority rankings. Policymakers and managers can use this list to develop appropriate policies.

• iii. The study reveals that government regulations are the key ‘cause’ factors affecting India's public transportation sector. Relaxation in governmental regulations like easing inter-state connections and lockdown norms can play a crucial role in the rise of the public transportation sector in India.

• iv. By going through the factor interrelationships, we observe that reduced reliability on public transportation from the passengers’ perspective (B3) is the prime cause factor. Transportation operators must take steps to gain passengers’ trust in public transportation as behavioral factors highly influence public transportation.

• v. The study reveals that Reduced profits (F1) is the most affected factor. The result is verified by performing a sensitivity analysis, in which F1 remains a highly rated factor throughout all cases. This is due to increased operational expenses. Decisions taken to reduce overheads and operational costs will play a crucial role in mitigating this externality.

• vi. The study also forms a hierarchical structure that can be used to analyze the various factors affecting India's public transportation system during the COVID-19 pandemic. This will help in setting priorities and working accordingly.

• vii. The paper presents a general framework for all the organizations in the transportation sector. The same framework with modified initial values can be used to provide the ranking list for specific organizations for their specific needs.

8 Concluding remarks

The effect of COVID-19 on the public transportation system in India is a critical research area that provides a more in-depth analysis for the development of appropriate policies. An analysis of the cause and effect factors is needed to improve the condition of the public transportation sector. In complex decision-making and assessment processes like this problem, there are various factors that need subjective and qualitative judgments. Thus, MCDM techniques can adequately process the various interdependencies and relationships between the identified factors. The analysis is done by a coordinated MCDM technique, i.e., a mixture of grey-DEMATEL and ANP.

The study first identified factors through a literature survey, then, using g-DANP, analyzed the results in two phases. First, as the interrelationships between the factors using g-DEMATEL, then as the weights and rankings that we have assigned to each factor through ANP. Twelve of the twenty-two factors were regarded as ‘effect’ factors, while the remaining ten were regarded as ‘cause’ factors. The results of the study indicate that the most hindering factors towards Indian public transport are those caused by government regulations. Public perception is one of public transport's most important critical success factors. If restrictions are kept in place for the use of public transportation, then people will naturally be hesitant towards the safety of such transportation. The study also emphasized the behavioral factors' effect on the public transportation system. Another lesson learned is that financial factors should not be the focus of public transport recovery. The sub-factors listed under the financial category are mostly those that are influenced by the solution of other barriers.

Furthermore, a sensitivity analysis was performed to verify the rankings' consistency and prove that this paper's results are not influenced by bias. Policymakers can use the framework prepared in this paper to take appropriate steps toward improving the public transportation sector. The present study also accentuated that no similar study identifies and analyzes the factors affecting public transportation in India during the pandemic utilizing these procedures. Thus, this paper fills a gap by using g-DANP to find the interrelationships between factors and the significance of their weights.

A limitation of this study is that studies like these can also be done using intuitionistic fuzzy environments, allowing us to consider further hesitancy. Another limitation is the lack of input from private industry personnel. Including the perspective of experts in the industry from the private sector would have led to a sharper analysis of the situation. This would allow a detailed comparative analysis between public and private transportation scenarios and the effect the COVID-19 pandemic has had on them. Future studies in this topic can be on developing an index for weighing the level of readiness of public transportation through various factors.

Appendices

Appendix 1: A hybrid MCDM model combined with Grey-DEMATEL and ANP

Grey-DEMATEL Method:

The Grey-DEMATEL method is applied through the following steps;

• Step 1: Compute the direct relation matrix

The initial information is obtained from the specialists and presented in the direct relation matrix of size $n\times n$. In this matrix ‘A’, every component ${a}_{ij}$ is indicated as the impact of component $i$ over $j$. Let us consider, $\otimes C_{ij}^{e}$ characterized as the grey number for a specialist $^{\prime}e^{\prime}$, which gives assessment data of factor $i$’s influence on factor $j$. Then, $\overline{ \otimes }C_{ij}^{e}$ and $\underline { \otimes } C_{ij}^{e}$ are the upper and lower estimations of a given number $\otimes C_{ij}^{e}$, i.e.,

$$\otimes C_{ij}^{e} = \left[ {\overline{ \otimes }C_{ij}^{e} { },\underline { \otimes } C_{ij}^{e} } \right]$$

(1)

• Step 2: Calculation of the average grey relation matrix (AGRM).

The AGRM $\otimes \tilde{C}_{ij}^{e}$ is formulated from $Q$ grey relation matrices as,

$$\otimes \tilde{C}_{ij}^{e} = \left[ {\frac{{\mathop \sum \nolimits_{l} \overline{ \otimes }C_{ij}^{e} }}{Q},\frac{{\mathop \sum \nolimits_{l} \underline { \otimes } C_{ij}^{e} { }}}{Q}} \right]$$

(2)

• Step 3: Formulation of crisp relation matrix.

In this formulation, a defuzzification technique is applied to change grey relation to crisp relation by using to convert fuzzy data into the crisp score (CFCS) strategy (Opricovic and Tzeng 2003). The crisp transformation follows a three-stage technique:

• i. i. Normalization:

  $$\overline{ \otimes }\hat{C}_{ij}^{{}} = \frac{{\overline{ \otimes }\tilde{C}_{ij}^{{}} - \min_{j} \overline{ \otimes }\tilde{C}_{ij}^{{}} }}{{\vartriangle_{\min }^{\max } }}$$

  (3)

  where $\overline{ \otimes }\hat{C}_{ij}^{{}}$ is defined as the normalized upper limit of a given grey number $\otimes C_{ij}^{{}}$, and

  $$\underline { \otimes } \hat{C}_{ij}^{{}} = \frac{{\underline { \otimes } \tilde{C}_{ij}^{{}} - \min_{j} \underline { \otimes } \tilde{C}_{ij}^{{}} }}{{\vartriangle_{\min }^{\max } }}$$

  (4)

  where $\underline { \otimes } \hat{C}_{ij}^{{}}$ represents the lower value of the given grey number $\otimes C_{ij}^{{}}$.

The value of $\Delta_{min}^{max}$ is calculated as follows;

$$\vartriangle_{min}^{max} = max_{j} \overline{ \otimes }\tilde{C}_{ij}^{{}} - min_{j} \underline { \otimes } \tilde{C}_{ij}^{{}}$$

(5)

• ii Formulation of the total normalized crisp value;

  $$A_{ij} = \frac{{\left[ {\underline { \otimes } \hat{C}_{ij}^{{}} \left( {1 - \underline { \otimes } \hat{C}_{ij}^{{}} } \right) + \left( {\overline{ \otimes }\hat{C}_{ij}^{{}} \times \overline{ \otimes }\hat{C}_{ij}^{{}} } \right)} \right]}}{{\left( {1 - \underline { \otimes } \hat{C}_{ij}^{{}} + \overline{ \otimes }\hat{C}_{ij}^{{}} } \right)}}$$

  (6)

• iii Calculate the final crisp values;

  $$A_{ij}^{*} = \left[ {min_{j} \underline { \otimes } \tilde{C}_{ij}^{{}} + \left( {A_{ij} \times \vartriangle_{min}^{max} } \right)} \right]$$

  (7)

And,

$$A=[{A}_{ij}^{*}]$$

(8)

• Step 4: Formulate the normalized direct crisp relation matrix

A normalized crisp relation matrix of initial relationship can be processed through the accompanying equation. Let ‘B’ be a normalized crisp relation matrix, then

$$B=A\times U$$

(9)

where $U$ is a normalization factor given by;

$$U=\frac{1}{\begin{array}{c}max\\ 1\le i\le n\end{array}\sum_{j=1}^{n}{A}_{ij}^{*}}$$

(10)

• Step 5: Compute the total relation matrix.

In this step, a total relation matrix (T) is calculated with the given equation,

$$T=\left({t}_{ij}\right)=B\times {(I-B)}^{-1}$$

(11)

where $I$ is the identity matrix and ${t}_{ij}$ are the corresponding values in matrix $T$.

• Step 6: Formulate the prominence and relation of each factor

  $${R}_{i}=\sum_{1\le j\le n}{t}_{ij};\ {D}_{j}=\sum_{1\le i\le n}{t}_{ij}$$

  (12)

where ${R}_{i}$ and ${D}_{j}$ represent the sum of the ith row elements and the sum of the jth column elements in $T,$ respectively.

When $i=j$, $({R}_{i}+{D}_{j})$ is the “Prominence” which shows the importance degree that factor $i$ plays in the entire analysis. Similarly, $({R}_{i}-{D}_{j})$ is called “Relation” and determines whether a factor will be classified in the cause and effect group.

• Step 7: Calculate the total relationship matrix (TRM) threshold and plot the cause & effect graph

The value of the threshold is equal to the sum of the mean matrix value and of one standard deviation of the matrix T. In a digraph, the relationship is drawn with the help of a dataset of $[({R}_{i}+{D}_{j}),({R}_{i}-{D}_{j}) ] \,\forall i=j$.

Integration of Grey-DEMATEL and ANP Methods:

• Step 1: Calculate an unweighted super-matrix ‘W’

${T}_{D}$ and ${T}_{c}$ are the total influence matrix of main factors (Dimensions) and sub-factors (criteria), respectively, which are obtained by the grey-DEMATEL method.

Let ${T}_{c}$ be given by Eq. (13).

(13)

The normalized TRM for criteria is denoted as ${T}_{c}^{\alpha }$, which is obtained from normalizing the total influence matrix of criteria ${T}_{c}$, is given by Eq. (14).

(14)

Normalized element ${T}_{c}^{\alpha 11}$ is explicated in detail in Eqs. (15) and (16).

$${d}_{ci}^{11}=\sum_{j=1}^{{m}_{1}}{t}_{cij}^{11}, i=\mathrm{1,2},\dots , {m}_{1}$$

$${{\varvec{T}}}_{{\varvec{c}}}^{\boldsymbol{\alpha }11}=\left[\begin{array}{ccc}\begin{array}{c}\begin{array}{cc}{{\varvec{t}}}_{{\varvec{c}}11}^{11}/{{\varvec{d}}}_{{\varvec{c}}1}^{11}& \cdots \end{array}\\ \begin{array}{cc}\vdots & \end{array}\end{array}& \begin{array}{c}\begin{array}{cc}{{\varvec{t}}}_{{\varvec{c}}1{\varvec{j}}}^{11}/{{\varvec{d}}}_{{\varvec{c}}1}^{11}& \cdots \end{array}\\ \begin{array}{cc}\vdots & \end{array}\end{array}& \begin{array}{c}{{\varvec{t}}}_{{\varvec{c}}1{{\varvec{m}}}_{1}}^{11}/{{\varvec{d}}}_{{\varvec{c}}1}^{11}\\ \vdots \end{array}\\ \begin{array}{c}\begin{array}{cc}{{\varvec{t}}}_{{\varvec{c}}{\varvec{i}}1}^{11}/{{\varvec{d}}}_{{\varvec{c}}{\varvec{i}}}^{11}& \cdots \end{array}\\ \begin{array}{cc}\vdots & \end{array}\end{array}& \begin{array}{c}\begin{array}{cc}{{\varvec{t}}}_{{\varvec{c}}{\varvec{i}}{\varvec{j}}}^{11}/{{\varvec{d}}}_{{\varvec{c}}{\varvec{i}}}^{11}& \cdots \end{array}\\ \begin{array}{cc}\vdots & \end{array}\end{array}& \begin{array}{c}{{\varvec{t}}}_{{\varvec{c}}{\varvec{i}}{{\varvec{m}}}_{1}}^{11}/{{\varvec{d}}}_{{\varvec{c}}{\varvec{i}}}^{11}\\ \vdots \end{array}\\ \begin{array}{cc}{{\varvec{t}}}_{{\varvec{c}}{{\varvec{m}}}_{1}1}^{11}/{{\varvec{d}}}_{{\varvec{c}}{{\varvec{m}}}_{1}}^{11}& \cdots \end{array}& \begin{array}{cc}{{\varvec{t}}}_{{\varvec{c}}{{\varvec{m}}}_{1}{\varvec{j}}}^{11}/{{\varvec{d}}}_{{\varvec{c}}{{\varvec{m}}}_{1}}^{11}& \cdots \end{array}& {{\varvec{t}}}_{{\varvec{c}}{{\varvec{m}}}_{1}{{\varvec{m}}}_{1}}^{11}/{{\varvec{d}}}_{{\varvec{c}}{{\varvec{m}}}_{1}}^{11}\end{array}\right]$$

(15)

$$=\left[\begin{array}{ccc}\begin{array}{c}\begin{array}{cc}{{\varvec{t}}}_{{\varvec{c}}11}^{\boldsymbol{\alpha }11}& \cdots \end{array}\\ \begin{array}{cc}\vdots & \end{array}\end{array}& \begin{array}{c}\begin{array}{cc}{{\varvec{t}}}_{{\varvec{c}}1{\varvec{j}}}^{\boldsymbol{\alpha }11}& \cdots \end{array}\\ \begin{array}{cc}\vdots & \end{array}\end{array}& \begin{array}{c}{{\varvec{t}}}_{{\varvec{c}}1{{\varvec{m}}}_{1}}^{\boldsymbol{\alpha }11}\\ \vdots \end{array}\\ \begin{array}{c}\begin{array}{cc}{{\varvec{t}}}_{{\varvec{c}}{\varvec{i}}1}^{\boldsymbol{\alpha }11}& \cdots \end{array}\\ \begin{array}{cc}\vdots & \end{array}\end{array}& \begin{array}{c}\begin{array}{cc}{{\varvec{t}}}_{{\varvec{c}}{\varvec{i}}{\varvec{j}}}^{11}& \cdots \end{array}\\ \begin{array}{cc}\vdots & \end{array}\end{array}& \begin{array}{c}{{\varvec{t}}}_{{\varvec{c}}{\varvec{i}}{{\varvec{m}}}_{1}}^{\boldsymbol{\alpha }11}\\ \vdots \end{array}\\ \begin{array}{cc}{{\varvec{t}}}_{{\varvec{c}}{{\varvec{m}}}_{1}1}^{\boldsymbol{\alpha }11}& \cdots \end{array}& \begin{array}{cc}{{\varvec{t}}}_{{\varvec{c}}{{\varvec{m}}}_{1}{\varvec{j}}}^{\boldsymbol{\alpha }11}& \cdots \end{array}& {{\varvec{t}}}_{{\varvec{c}}{{\varvec{m}}}_{1}{{\varvec{m}}}_{1}}^{\boldsymbol{\alpha }11}\end{array}\right]$$

(16)

The unweighted super-matrix is formulated by transposing the normalized TRM for criteria as shown in Eq. (17).

(17)

(18)

• Step 2: Calculate the Weighted Super-matrix

To calculate the weighted super-matrix, the TRM for main factors (Dimensions) were considered as shown in Eq. (19).

$${{\varvec{T}}}_{{\varvec{D}}}=\left[\begin{array}{ccc}\begin{array}{c}\begin{array}{cc}{{\varvec{t}}}_{{\varvec{D}}}^{11}& \cdots \end{array}\\ \begin{array}{cc}\vdots & \end{array}\end{array}& \begin{array}{c}\begin{array}{cc}{{\varvec{t}}}_{{\varvec{D}}}^{1{\varvec{j}}}& \cdots \end{array}\\ \begin{array}{cc}\vdots & \end{array}\end{array}& \begin{array}{c}{{\varvec{t}}}_{{\varvec{D}}}^{1{\varvec{n}}}\\ \vdots \end{array}\\ \begin{array}{c}\begin{array}{cc}{{\varvec{t}}}_{{\varvec{D}}}^{{\varvec{i}}1}& \cdots \end{array}\\ \begin{array}{cc}\vdots & \end{array}\end{array}& \begin{array}{c}\begin{array}{cc}{{\varvec{t}}}_{{\varvec{D}}}^{{\varvec{i}}{\varvec{j}}}& \cdots \end{array}\\ \begin{array}{cc}\vdots & \end{array}\end{array}& \begin{array}{c}{{\varvec{t}}}_{{\varvec{D}}}^{{\varvec{i}}{\varvec{n}}}\\ \vdots \end{array}\\ \begin{array}{cc}{{\varvec{t}}}_{{\varvec{D}}}^{{\varvec{n}}1}& \cdots \end{array}& \begin{array}{cc}{{\varvec{t}}}_{{\varvec{D}}}^{{\varvec{n}}{\varvec{j}}}& \cdots \end{array}& {{\varvec{t}}}_{{\varvec{D}}}^{{\varvec{n}}{\varvec{n}}}\end{array}\right]$$

(19)

Then, normalization of TRM ‘${T}_{D}$’ is done and a new matrix ‘${T}_{D}^{\alpha }$’ is obtained as shown in Eq. (20).

$${{\varvec{T}}}_{{\varvec{D}}}^{\boldsymbol{\alpha }}=\left[\begin{array}{ccc}\begin{array}{c}\begin{array}{cc}{{\varvec{t}}}_{{\varvec{D}}}^{11}/{{\varvec{d}}}_{1}& \cdots \end{array}\\ \begin{array}{cc}\vdots & \end{array}\end{array}& \begin{array}{c}\begin{array}{cc}{{\varvec{t}}}_{{\varvec{D}}}^{1{\varvec{j}}}/{{\varvec{d}}}_{1}& \cdots \end{array}\\ \begin{array}{cc}\vdots & \end{array}\end{array}& \begin{array}{c}{{\varvec{t}}}_{{\varvec{D}}}^{1{\varvec{n}}}/{{\varvec{d}}}_{1}\\ \vdots \end{array}\\ \begin{array}{c}\begin{array}{cc}{{\varvec{t}}}_{{\varvec{D}}}^{{\varvec{i}}1}/{{\varvec{d}}}_{{\varvec{i}}}& \cdots \end{array}\\ \begin{array}{cc}\vdots & \end{array}\end{array}& \begin{array}{c}\begin{array}{cc}{{\varvec{t}}}_{{\varvec{D}}}^{{\varvec{i}}{\varvec{j}}}/{{\varvec{d}}}_{{\varvec{i}}}& \cdots \end{array}\\ \begin{array}{cc}\vdots & \end{array}\end{array}& \begin{array}{c}{{\varvec{t}}}_{{\varvec{D}}}^{{\varvec{i}}{\varvec{n}}}/{{\varvec{d}}}_{{\varvec{i}}}\\ \vdots \end{array}\\ \begin{array}{cc}{{\varvec{t}}}_{{\varvec{D}}}^{{\varvec{n}}1}/{{\varvec{d}}}_{{\varvec{n}}}& \cdots \end{array}& \begin{array}{cc}{{\varvec{t}}}_{{\varvec{D}}}^{{\varvec{n}}{\varvec{j}}}/{{\varvec{d}}}_{{\varvec{n}}}& \cdots \end{array}& {{\varvec{t}}}_{{\varvec{D}}}^{{\varvec{n}}{\varvec{n}}}/{{\varvec{d}}}_{{\varvec{n}}}\end{array}\right]=\left[\begin{array}{ccc}\begin{array}{c}\begin{array}{cc}{{\varvec{t}}}_{{\varvec{D}}}^{\boldsymbol{\alpha }11}& \cdots \end{array}\\ \begin{array}{cc}\vdots & \end{array}\end{array}& \begin{array}{c}\begin{array}{cc}{{\varvec{t}}}_{{\varvec{D}}}^{\boldsymbol{\alpha }1{\varvec{j}}}& \cdots \end{array}\\ \begin{array}{cc}\vdots & \end{array}\end{array}& \begin{array}{c}{{\varvec{t}}}_{{\varvec{D}}}^{\boldsymbol{\alpha }1{\varvec{n}}}\\ \vdots \end{array}\\ \begin{array}{c}\begin{array}{cc}{{\varvec{t}}}_{{\varvec{D}}}^{\boldsymbol{\alpha }{\varvec{i}}1}& \cdots \end{array}\\ \begin{array}{cc}\vdots & \end{array}\end{array}& \begin{array}{c}\begin{array}{cc}{{\varvec{t}}}_{{\varvec{D}}}^{\boldsymbol{\alpha }{\varvec{i}}{\varvec{j}}}& \cdots \end{array}\\ \begin{array}{cc}\vdots & \end{array}\end{array}& \begin{array}{c}{{\varvec{t}}}_{{\varvec{D}}}^{\boldsymbol{\alpha }{\varvec{i}}{\varvec{n}}}\\ \vdots \end{array}\\ \begin{array}{cc}{{\varvec{t}}}_{{\varvec{D}}}^{\boldsymbol{\alpha }{\varvec{n}}1}& \cdots \end{array}& \begin{array}{cc}{{\varvec{t}}}_{{\varvec{D}}}^{\boldsymbol{\alpha }{\varvec{n}}{\varvec{j}}}& \cdots \end{array}& {{\varvec{t}}}_{{\varvec{D}}}^{\boldsymbol{\alpha }{\varvec{n}}{\varvec{n}}}\end{array}\right]$$

(20)

The weighted super-matrix (${W}^{\alpha }$) is obtained with the help of Eq. (21);

$${{\varvec{W}}}^{\boldsymbol{\alpha }}={{\varvec{T}}}_{{\varvec{D}}}^{\boldsymbol{\alpha }}\boldsymbol{ }\times {\varvec{W}}=\left[\begin{array}{ccc}\begin{array}{c}\begin{array}{cc}{{\varvec{t}}}_{{\varvec{D}}}^{\boldsymbol{\alpha }11}\times {{\varvec{W}}}^{11}& \cdots \end{array}\\ \begin{array}{cc}\vdots & \end{array}\end{array}& \begin{array}{c}\begin{array}{cc}{{\varvec{t}}}_{{\varvec{D}}}^{\boldsymbol{\alpha }{\varvec{i}}1}\times {{\varvec{W}}}^{{\varvec{i}}1}& \cdots \end{array}\\ \begin{array}{cc}\vdots & \end{array}\end{array}& \begin{array}{c}{{\varvec{t}}}_{{\varvec{D}}}^{\boldsymbol{\alpha }{\varvec{n}}1}\times {{\varvec{W}}}^{{\varvec{n}}1}\\ \vdots \end{array}\\ \begin{array}{c}\begin{array}{cc}{{\varvec{t}}}_{{\varvec{D}}}^{\boldsymbol{\alpha }1{\varvec{j}}}\times {{\varvec{W}}}^{1{\varvec{j}}}& \cdots \end{array}\\ \begin{array}{cc}\vdots & \end{array}\end{array}& \begin{array}{c}\begin{array}{cc}{{\varvec{t}}}_{{\varvec{D}}}^{\boldsymbol{\alpha }{\varvec{i}}{\varvec{j}}}\times {{\varvec{W}}}^{{\varvec{i}}{\varvec{j}}}& \cdots \end{array}\\ \begin{array}{cc}\vdots & \end{array}\end{array}& \begin{array}{c}{{\varvec{t}}}_{{\varvec{D}}}^{\boldsymbol{\alpha }{\varvec{n}}{\varvec{j}}}\times {{\varvec{W}}}^{{\varvec{n}}{\varvec{j}}}\\ \vdots \end{array}\\ \begin{array}{cc}{{\varvec{t}}}_{{\varvec{D}}}^{\boldsymbol{\alpha }1{\varvec{n}}}\times {{\varvec{W}}}^{1{\varvec{n}}}& \cdots \end{array}& \begin{array}{cc}{{\varvec{t}}}_{{\varvec{D}}}^{\boldsymbol{\alpha }{\varvec{i}}{\varvec{n}}}\times {{\varvec{W}}}^{{\varvec{i}}{\varvec{n}}}& \cdots \end{array}& {{\varvec{t}}}_{{\varvec{D}}}^{\boldsymbol{\alpha }{\varvec{n}}{\varvec{n}}}\times {{\varvec{W}}}^{{\varvec{n}}{\varvec{n}}}\end{array}\right]$$

(21)

• Step 3: Limit the weighted super-matrix.

To acquire a stable and converged matrix, the weighted super-matrix which we got in the last step is to be limited by expanding to a satisfactorily large power k to fulfil the above prerequisites, which will finally result in global priority scores, which are called grey-DANP (grey-DEMATEL based ANP) influential weights, such as $\underset{k\to \infty }{\mathrm{lim}}{{(W}^{\alpha })}^{k}$, where $k$ represents any number of power (Govindan et al. 2018).

Appendix 2: Detailed Information about Experts

Expert S. No	Experts’ Affiliation	Years of Experience	Designation
1	Indian Ministry of Road Transport and Highways	9	Junior Manager
2		11	Technical Consultant
3		21	Deputy Director
4		15	Deputy General Manager
5	Indian Public Transport Researchers	11	Assistant Professor
6		12	Associate Professor
7		10	Assistant Professor
8		13	Associate Professor
9		12	Associate Professor
10	Indian Ministry of Railways	12	Technical Consultant
11		14	Deputy General Manager
12		10	General Manager