An integrated fuzzy model for evaluation and selection of mobile banking (m-banking) applications using new fuzzy-BWM and fuzzy-TOPSIS

Banking is an essential service in human life. Nowadays, customers can interact with banks through various channels, including branch visits, automated teller machines (ATM), phone calls, internet banking, and m-banking [18]. However, m-banking is the most efficient and convenient way to access banking services. The concept of m-banking is relatively new in management literature, and the definition of m-banking varies significantly across published articles. According to Barnes and Corbitt [19], m-banking is “a channel through which a customer interacts with a bank via a mobile device, such as a mobile phone or personal digital assistant (PDA).” According to Clarke [20], m-banking is a "subset of e-banking or online banking that refers to the shift of conducting financial transactions from wired networks to wireless networks." M-banking is regarded as a subset of m-commerce.

On the other hand, m-commerce is considered a subset of e-commerce [21]. Electronic banking was first implemented using a landline telephone in the 1980s [22], and the concept of m-banking evolved from there. The Bank of Scotland in the United Kingdom was the first to offer e-banking services in 1983. Paybox, on the other hand, launched its m-banking service in early 1999 with the assistance of Deutsche Bank [1]. Initially, m-banking applications were only available in European countries [1]. These applications have percolated to various developing and underdeveloped countries. A few examples of "m-banking" are mobile payments, mobile finance, and mobile transfers [23]. Amin et al. [24] used pocket banking terminology to describe m-banking. There exists a significant difference between e-banking and m-banking. E-banking employs online channels such as computers or laptops, whereas m-banking typically uses a mobile device [25].

Although mobile banking and internet banking offer similar services, banks frequently encourage customers to use m-banking due to its flexibility and low operating costs [26]. Mobile phones have now outnumbered personal computers and laptop computers. As a result, m-banking may be a preferable tool compared to e-banking for both customers and financial institutions. It also contributes to developing stronger relationships between financial institutions and their customers [25]. Customers can use M-banking to perform a variety of banking tasks remotely, such as opening accounts and conducting cashless commercial transactions [27]. The traditional brick-and-mortar banking approach has certain drawbacks, like high operating costs, strict timing of operations, queuing issues, waste of time, and non-availability of services on holidays. On the other hand, M-banking can solve these issues in a cost-effective and real-time manner [28]. The adoption of m-banking is beneficial not only for financial institutions but also for increasing household savings. According to the findings of Loaba [29], m-banking services can increase formal and informal household savings by 2–3%. Ouma et al. [30] also discovered that m-Banking has a similar saving potential. According to Zhu et al. [31], m-banking has enormous potential in rural development.

According to the World Bank 2020 report, m-banking offers millions of people a convenient and secure alternative channel for transactions. It also aids in the growth of cashless financial transactions. M-banking has enormous potential to contribute to the development of a cashless society. Banks are currently projecting m-banking as a regular service provider due to its significant growth potential [32]. According to a KPMG report [33], cashless transactions have increased significantly around the world. Digital transactions grew at a compound annual growth rate of 12.7% between 2016 and 21. A developing country's growth rate is substantially higher than that of a developed country. Even though m-banking has grown in popularity over the years, its adoption rate has been low. The majority of users only use m-banking to check their balances [34]. According to the Global Findex report, until 2017, approximately 52% of account holders worldwide accessed digital banking services [35]. Despite the benefits, selecting appropriate m-banking applications is a challenge for decision-makers.

The selection of alternatives against multiple criteria can be considered as a multi-criteria decision-making (MCDM) problem. A typical MCDM problem is often classified into two categories: MADM (multi-attribute decision making) and MODM (multi-objective decision making) [36]. The study of MCDM began in the 1950s and 1960s. After the 1990s, this field grew at an exponential rate [37]. A complex problem is frequently broken down into smaller pieces in MCDM. These variables are then used to create a decision hierarchy. After calculating the weight of each component, the individual components should be combined to reach a decision [38]. Because of their adaptability to complex problems, MCDM techniques have been used widely [39]. Various researchers over the years have proposed several MCDM techniques. A few examples are the analytic hierarchy process (AHP) [40], analytic network process (ANP) [41], BWM [17], Preference Ranking Organization Method for Enrichment of Evaluations (PROMETHEE) [39], TODIM (an interactive multi-criteria decision making), TOPSIS [42], VlseKriterijumska Optimizacija I Kompromisno Resenje (VIKOR) [43]. Although MCDM is a widely used method for dealing with complex systems, its efficacy is frequently called into question. Consistency is an essential aspect of the model's efficacy. The decision matrix must be consistent with increasing the matrix size during the decision-making process [17]. This section contains studies related to the use of MCDM in m-banking selection.

Mishra and Singh [44] proposed an MCDM framework based on AHP to select appropriate electronic banking channels (EBC) in the Indian context. The researchers identified twenty-six factors in four categories: demographics, technicality, serviceability, and intention. Ecer [45] suggested a model to evaluate Turkey's m-banking services. The author used hybrid MCDM with fuzzy-AHP and additive ratio evaluation for selecting m-banking. The m-banking services were assessed based on three criteria, such as perceived security features, usefulness, and ease of use. Kou et al. [46] used a combination of three MCDM methods to rank the clustering algorithms: VIKOR, TOPSIS, and data envelopment analysis (DEA). The authors ranked the algorithms by taking eleven performance measures into account. Liang et al. [47] conducted a study on the quality of internet banking websites in Ghana. Product quality, responsiveness, ease of use, security, and privacy were used to evaluate the websites. To calculate the weight of the identified factors, the authors used the Pythagorean fuzzy VIKOR technique. Further, TODIM was used to rank the websites. Gbongli et al. [48] developed a model for ranking m-banking operating systems based on various factors such as financial, performance, time, and privacy. The authors quantified the risks of m-banking and suggested areas where managers should focus their efforts. In Poland, Chmielarz and Zborowski [49] proposed a model for selecting the best e-banking channel using the COMET (characteristics objects method). The e-banking sites were evaluated based on twenty criteria and experts’ opinions.

Chou and Cheng [50] proposed a hybrid system to rank the websites of Taiwan's top four accounting firms combining the VIKOR and ANP in a fuzzy setting. The website's qualities were evaluated against various information and service quality factors in their study. Fuzzy-ANP was used in their research to calculate the criteria weight, and fuzzy-VIKOR was used to prioritise the websites. Kaya and Kahraman [51] used an integrated methodology based on MCDM to evaluate e-banking websites using eight evaluation criteria. Their identified criteria were product quality, competence, reliability, receptiveness, access, ease of use, information, and security. The weights of the criteria were calculated using a combination of fuzzy-AHP and ELECTRE in the study. The research was carried out in the context of the Turkish banking industry.

The selection of appropriate m-banking is a complex decision. Different factors can influence the decision maker's choice for selecting a preferred m-banking channel. The decision to choose an m-banking application is frequently affected by system information, products, services, and customer-related factors. Several studies [52‐54] have identified various factors that can influence user acceptance of m-banking. Various researchers propose multiple theories in the literature to explain the adoption behaviour of new technology. Innovation diffusion theory (IDT), technology acceptance model (TAM), and unified theory of acceptance and usage of technology (UTAUT) are a few examples [55, 56]. The majority of studies have found that users frequently adopt technology if they perceive clear benefits regarding ease of use and usefulness [52]. Szopiski [57] identified thirteen criteria related to e-banking adoption in Poland. The author found that trust in banks, mobile phones, and internet service providers was a deciding factor in online banking’s adoption. Furthermore, the author found convenience, usefulness, and safety as critical factors in technology adoption.

Shankar and Rishi [12] identified various aspects of m-banking adoption. According to their findings, access and transaction convenience were critical considerations when deciding on m-banking. According to Oliveira et al. [27], behavioural and technological factors frequently play a crucial role in adopting m-banking applications. Among technical considerations, technology characteristics and performance expectancy play an essential role in m-banking selection. According to some studies, satisfaction and loyalty are essential factors in m-banking adoption [26]. Satisfaction has a direct impact on the decision to use a specific m-banking application. Furthermore, the application's system, information, and service quality influence m-banking adoption [18]. According to Gu et al. [58], the usefulness and ease of using technology are the determining factors for mobile payment. According to Al-Saedi et al. [59], trust, efficiency, expectancy, and social influence play a significant role in m-payment adoption. Technology becomes user-friendly when the process is simple. In this context, the two most important criteria for technology adoption are ease of use and usefulness [53]. Another factor that influences the use of online banking is trust [4, 57]. Mehri et al. [60] discovered that a user's habits, trust, system privacy, and security influence their intention to use m-banking services. In their study, application performance anticipation emerged as a significant factor.

Based on the literature review, it can be conjectured that insufficient research has been performed on m-banking application selection. In literature, there are studies like selecting e-banking channels [44], m-banking services [45], ranking of the quality of internet banking websites [47], and m-banking operating systems [48]. However, none of the aforementioned research focused on selecting the best m-banking application. Further, none of them evaluated inclusive functionality components and responsiveness criteria, as well as security and reliability concerns for m-banking application selection. This study aims to solve a few research gaps in the field of m-banking adoption. M-banking selection falls under the category of the MCDM problem, and hence, it is critical to use a systematic decision-making approach to solve the problem holistically. Furthermore, a more precise approach needs to be devised for selecting m-banking applications. In addition there exists a research opportunity by applying combined MCDM methodology for solving m-banking selection problem. As per the author's knowledge, no studies have reported combined fuzzy-BWM and fuzzy-TOPSIS methods for m-banking selection. Although combined fuzzy-BWM and fuzzy-TOPSIS method has been applied in other context; but the research adds value to the existing literature in terms of application in the area of m-banking. The research tries to explain how fuzzy-BWM and fuzzy-TOPSIS can be used to determine the criterion's weight, and rank the best m-banking applications. The research also elaborates on how to triangulate the decisions of a panel of experts. The research also demonstrated how the present model can be applied in real world settings.

In this study, BWM is used to determine the weight of the criteria and sub-criteria in a fuzzy environment. Rezaei (2015) proposed BWM, a relatively new MCDM technique [17]. It has successfully solved several MCDM problems, including project site selection [71], evaluating firm performance, airport service quality [61], and others [72]. In comparison to other MCDM techniques such as AHP and ANP, BWM is relatively easy to implement. The technique has advantages in terms of comparison numbers, consistency, and reliability [17]. BWM, for example, requires only (2n − 3) pair-wise comparisons, whereas AHP requires n(n − 1) pair-wise comparisons. AHP results are inconsistent due to many pair-wise comparisons and extensive data set involvement [17]. Rezaei [17, 73] proved that BWM is more consistent than AHP.

In a typical decision-making process, the decision-makers frequently face difficulty to express precise numbers [74, 75]. As a result, decision-makers typically prefer to quantify subjective criteria using interval values [76]. Some researchers [15, 16, 76‐81] have extended the original BWM into the fuzzy environment to deal with such ambiguity in human judgement. For example, Mou et al. [74] developed the traditional BWM into a group decision-based intuitionistic fuzzy multiplicative BWM (IFMBWM). Hafezalkotob and Hafezalkotob [77] proposed GIFBWM or group and Individual Fuzzy BWM to unite the group and individual decisions. To address the prototype design selection problem, Ijadi Maghsoodi et al. [78] proposed HBWFAD, a hierarchical group decision-making method based on BWM and axiomatic design principles in a fuzzy environment. Furthermore, Wan and Dong [79] proposed Intuitionistic Fuzzy-BWM to represent reference comparison using intuitionistic fuzzy values. Wan et al. [80] recently extended BWM to generalised interval-valued trapezoidal fuzzy BWM (GITrF BWM). The study found that GITrF BWM improves overall process consistency. Wan et al. [81] used trapezoidal interval type-2 fuzzy sets to extend the original BWM into a fuzzy environment. The technique can be used in a variety of situations. The BWM was extended into a fuzzy environment by Hafezalkotob and Hafezalkotob [77]. The author demonstrated the group decision-making process by combining decision-makers’ perspectives. Ijadi Maghsoodi et al. [78] used a group decision-making approach to solve prototype design selection problems using fuzzy BWM and the axiomatic design principle.

Guo and Zhao [16] proposed TFN-based fuzzy-BWM. As compared to Rezaei's original BWM [17], the author claimed that their approach provides greater consistency. The study directly fuzzified all BWM parameters. Furthermore, the authors used non-linear programming to arrive at the solution. However, their method may not always be capable of providing a global optimal solution [15]. Dong et al. [15] proposed a new fuzzy-BWM based on TFNs to solve the problem. To find the best solution, the researchers used linear programming. This study employs the fuzzy-BWM proposed by Dong et al. [15] to determine the weight of the criteria. The following sections especially demonstrate fuzzy-BWM and fuzzy-TOPSIS in detail. Supplementary Appendix A contains the fundamental theories of fuzzy set theory.

For applying the fuzzy-BWM method, identification of the factors should be made first. To obtain the weight of criteria, the pair-wise comparison should be carried out using linguistic terms. For multiple experts, the consensus of decisions must be converted into fuzzy numbers. Suppose there are “n” criteria to assess the alternatives. In this condition, total \((2n - 3)\) pair-wise comparisons should be carried out to calculate the weight of each criterion. The pair-wise comparison can be performed as per the fuzzy linguistic scale shown in Table 1.

The steps of the new fuzzy-BWM method, suggested by Dong et al. [15], are presented below:

In this study, the weight of the criteria is obtained from a neutral decision-maker as per the mixed approach-I of Dong et al. [16] by solving the following linear programming model.where \(d_{j}^{t}\) and \(q_{j}^{t}\) \((j = 1,2,...,n; \, t = l,m,u)\) are the values of tolerance parameters. Generally, \(d_{j}^{t}\) and \(q_{j}^{t}\) take any value from1 to 9 to get a unique optimal solution. In this study, for all the \(d_{j}^{t}\) and \(q_{j}^{t}\) have been considered as 1.

The above fuzzy-BWM comparison is treated as fully consistent when \(\tilde{a}_{Bj} \times \tilde{a}_{jW} = \tilde{a}_{BW} {\text{ for all }}j\) However, during judgment, a decision-maker may not remain consistent for every criterion, and inconsistency arises when \(\tilde{a}_{Bj} \times \tilde{a}_{jW} \ne \tilde{a}_{BW} \, for \, all \, j\). Hence, a fuzzy consistency ratio (FCR) needs to be calculated from the derived fuzzy deviation of the comparisons \(\mathop {\upxi ^{*} }\limits^{\sim } = (\mathop {\upxi ^{l} }\limits^{\sim } ,\mathop {\upxi ^{m} }\limits^{\sim } ,\mathop {\upxi ^{u} }\limits^{\sim } )\) to check the pair-wise comparison and the decision's overall consistency.

The pair wise comparisons can be considered consistent if \(R({\text{FCR}}) = < { 1}\), however, if R(FCR) =  > 0.1, then the consistency of comparisons cannot be accepted [15].

TOPSIS is a well-known MCDM technique developed by Hwang and Yoon in 1981 [42], to solve MCDM problems. TOPSIS is used to select the best alternative by comparing it to ideal solutions based on the shortest and farthest distances from positive and negative ideal solutions [82]. A positive ideal solution (PIS) maximises benefits while minimising costs, whereas a negative ideal solution (NIS) minimises benefits while increasing costs. TOPSIS has been used successfully to solve a variety of decision-making problems. Its mathematical simplicity and ease of use compared to other MCDM methods have established its popularity among researchers [70, 83]. Because of its mathematical simplicity and ease of use, the technique has been successfully integrated with other MCDM methods [61, 84‐87]. According to Hsieh et al. [88], if TOPSIS is used after another MCDM method, it can rank the alternatives better than a single MCDM. TOPSIS can also handle a larger number of alternatives than other MCDM methods [89]. However, in real-life decision-making, experts prefer to give interval values to deal with a fuzzy situation. Such evaluations for selecting the best alternative can be adequately explained by the fuzzy TOPSIS technique [90, 91]. Researchers such as Lima Junior et al. [89] have demonstrated that, unlike crisp TOPSIS and other MCDMs, fuzzy TOPSIS does not have a rank reversal problem, and an alternative remains the same with additional alternatives. Furthermore, fuzzy TOPSIS outperforms other popular MCDM, like fuzzy AHP [89], during the decision process. In a recent study, Roy and Shaw [92] applied fuzzy TOPSIS and BWM to develop a sustainable credit scoring technique. The study found that alternatives against subjective criteria are better evaluated using fuzzy-TOPSIS than normal TOPSIS. Until now, various researchers [61, 85, 86] have combined TOPSIS and fuzzy set theory to deal with uncertain scenarios. A few examples of fuzzy-TOPSIS applications include supplier selection [61], m-health application [85], measuring success factors in construction projects [86], and measuring performance [93]. The steps of the fuzzy TOPSIS algorithm are provided below.

Step 1: When a set of alternatives \(a_{i} (i = 1, \, 2,.......m)\) needs to be evaluated with respect to identified criteria \(c_{j} (j = 1, \, 2,.......n)\), it is required to form a fuzzy decision matrix \(\tilde{a}_{ij}\).

Step 2: In the current study, the linguistic scale using TFNs (triangular fuzzy numbers), as shown in Table 3, can be used to rate the alternatives [94]. The range of TFN should be between [0, 1] to avoid further normalisation.

Step 3: The decision matrix is multiplied with criteria weight \(w_{i}\) to obtain the weighted normalised fuzzy-decision matrix.

Step 4: From the weighted fuzzy-decision matrix \(\mathop {W_{ij} }\limits^{\sim }\), the fuzzy positive ideal solution or FPIS \(\left( {F^{*} } \right)\) and the fuzzy negative ideal solutions or FNIS \(\left( {F^{ - } } \right)\) are determined.

Here, \(J\) and \(J^{^{\prime}}\) represent positive and negative attributes, respectively.

Step 5: The Euclidian distances from the FPIS (\(S_{ai}^{*}\)) and FNIS (\(S_{ai}^{ - }\)) of the alternatives are determined as under:

Step 6: The closeness index or similarity of alternatives \(C_{ai}^{*}\) relative to the ideal solution is calculated as below:

Step 7: Finally, the alternatives are ranked based on their \(C_{ai}^{*}\) value. An alternative with a maximum \(C_{ai}^{*}\) is ranked as the best alternative, and the minimum \(C_{ai}^{*}\) is ranked as the worst alternative.

This section discusses the case study of the proposed model. M-banking is increasingly becoming popular in both developing and developed countries. The real challenge, however, remains in selecting the appropriate m-banking application. The selection of appropriate m-banking applications is a difficult task that frequently includes multiple criteria. The problem can be formulated as an MCDM problem. This paper proposes an m-banking selection model that combines the recently developed fuzzy-BWM proposed by Dong et al. [15] and fuzzy-TOPSIS. The in-depth case study is discussed below.

It is imperative to identify potential m-banking adoption factors before selecting m-banking applications. The factors were identified by conducting a literature review in the relevant field. Various keywords, such as e-banking, m-banking, Technology Acceptance Model, and e-commerce, were used to search for relevant literature. Furthermore, a few identified factors have been dropped based on expert advice. A total of seven experts from industry and academia were chosen. Four of the seven respondents were from banks. The banking experts had extensive experience in the information technology sections of banks. Furthermore, three academic experts with experience in the field of operation research were identified. Although there were differences in responses initially, a consensus was reached after a thorough discussion [95]. Finally, ten criteria and thirty sub-criteria were agreed upon. Table 4 shows the specific factors.

Following the identification of criteria and sub-criteria, various m-banking applications were identified based on their popularity and market share. A survey was conducted to collect information on the m-banking applications. Supplementary Appendix B contains information on m-banking applications. M-banking applications were treated as alternatives in this study. The main objective of this study is to rank the identified alternatives based on the thirty sub-criteria organised into ten main criteria.

A five-layer model is proposed (shown in Fig. 2). The primary goal of the research is to identify the best m-banking application. In this model, the goal is placed in the first layer. Criteria and sub-criteria are organised into the second, third, and fourth layers, respectively. Finally, on the fifth layer, the alternatives are placed. Fuzzy-BWM was used to determine the weight of factors in this study, and fuzzy TOPSIS was used to rank the applications.

It was important to evaluate the weight of the factors after selecting the relevant factors and sub-factors. In this study, the importance of factors was calculated using the new fuzzy BWM proposed by Dong et al. [15]. Furthermore, the study was conducted from the perspective of a neutral decision-maker, using the mixed approach-I described by Dong et al. [15]. In this study, the optimal global weight of criteria was computed using pair-wise comparisons at a tolerance parameter of 1.

The detailed procedure for calculating the factor weight is shown below. The panel of experts from banks and academia were asked to choose the Best (most important) and Worst (least important) criteria from each set of main criteria and sub-criteria in this study. Subsequently, the panel was asked to perform fuzzy pair-wise comparisons using the linguistic scale suggested by Dong et al. [15]. The combined responses were used to create the fuzzy Best-to-Others and Others-to-Worst vectors. Similar comparisons were performed at different levels of the model. The pair-wise comparisons are shown in Tables 5, 6, 7, and 8.

After obtaining the best-to-others and others-to-worst vectors for the main criteria, a linear programming model was developed in Lingo software as per Eq. 5. In this study, a tolerance parameter of 1 was adopted to get the unique global optimal solution. The fuzzy BWM algorithm is shown below.

The global optimal fuzzy weights \((\tilde{w}_{1} ,\tilde{w}_{2} ,\tilde{w}_{3} )\) were obtained by solving the linear programming model shown in Eq. 18 in Lingo 11.0. The experiments were carried out in a Dell Core i5 processor with an 8 GB Ram. Further, the crisp weights, i.e., GMIR of fuzzy weights, were calculated using Eq. 10. The fuzzy weight, crisp weight, fuzzy deviations \(\mathop {\upxi ^{*} }\limits^{\sim } = (\mathop {\upxi ^{l} }\limits^{\sim } ,\mathop {\upxi ^{m} }\limits^{\sim } ,\mathop {\upxi ^{u} }\limits^{\sim } )\), and GMIR of the fuzzy consistency ratio, i.e., R(FCR), are shown in Table 9.

The FCR was calculated from the fuzzy deviations \(\tilde{\upxi} *\) (0.038, 0.071, 0.102) and fuzzy consistency index (FCI) (1.31, 1.63, 5.69) shown in Table 2 using Eqs. 9 and 10. The R(FCR) was observed at 0.04, which is very close to zero, satisfying the consistency requirement suggested by Dong et al. [15].

After obtaining the weight of the different dimensions, similar experiments were carried out for sub-criteria. In this study, the global weight was calculated by multiplying the weight of dimensions with the local weight of the main criteria and sub-criteria. The final local and global weight of all the criteria is shown in Table 10.

Table 10 shows that the experts placed the greatest emphasis on the usefulness of m-banking applications (0.5816), followed by ease of use (0.2755) and riskiness (0.1428). The effectiveness of an m-banking application specifies how it can improve the users' financial performance. On the other hand, ease of use refers to how quickly and easily an m-banking application can be learned.

Furthermore, among the main criteria for selecting an m-banking application, functionality (C6) has emerged as the most important, followed by convenience (C1), expectancy (C4), performance quality (C5), and security (C8). The global weight of sub-criteria was calculated by multiplying the local importance of the main criteria by the weight of sub-criteria. Innovativeness (0.1610), transaction convenience (0.1216), performance expectancy (0.096), service quality (0.0704), customisation (0.066), and encryption (0.0538) emerged as significant factors with a weight of greater than 5% in the fuzzy BWM evaluation. Application innovation and transaction convenience are critical considerations when selecting an m-banking application. R(FCR) values were calculated for all pair-wise comparisons using the Dong et al. [15] specification. An effort was made to keep the R(FCR) value below 0.10, which confirmed the decision's consistency. The global weight of various criteria was then used in fuzzy TOPSIS to rank m-banking applications.

The m-banking applications were evaluated against the identified factors after the weights of the criteria were determined. Seven m-banking applications provided by Indian banks were used as alternatives in this study. The same experts were asked to assign a fuzzy preference weight to each application in relation to the identified factors. The linguistic terms (very good, good, medium, poor, and very poor) were used to evaluate the m-banking applications. The scale is shown in Table 3 [100]. If an application's performance in that area was found best, it was rated as very good. On the other hand, an application was rated as very poor when it was the worst in that category.

Table 11 depicts the linguistic comparative decision matrix.

After evaluating m-banking applications using fuzzy linguistic terms, the ratings were transformed into fuzzy numbers as per the linguistic scale shown in Table 3. The fuzzy evaluation matrix is shown in Table 12.

In the next step, the fuzzy evaluation matrix was multiplied with the corresponding sub-criteria weights shown in Table 10 derived by fuzzy-BWM. The weighted fuzzy decision matrix is depicted in Table 13.

A fuzzy positive ideal solution (FPIS) and a fuzzy negative ideal solution (FNIS) were identified from the matrix. The best and worst solutions in this study are denoted as very good (VG) and very poor (VP), respectively. The weighted matrix was used to calculate the FPIS and FNIS values for each factor. Equations 15 and 16 were used to calculate the Euclidean distances from the FPIS and FNIS, respectively. Finally, the m-banking applications were ranked based on the closeness index \(C_{ai}^{*}\) derived from \(S_{ai}^{*}\) and \(S_{ai}^{ - }\) using Eq. 17. M-banking application with the highest \(C_{ai}^{*}\) was considered as best while the lowest \(C_{ai}^{*}\) was regarded as the worst application.

The importance of various criteria primarily determines the selection of m-banking applications. As a result, it was intriguing to conduct the sensitivity analysis of the model. The primary goal of sensitivity analysis was to assess the impact of individual factors on the final decision. The sensitivity analysis was carried out by varying the weights of the factors. The weights of factors (ease of user − E ± ; usefulness − U ± and riskiness − R ±) were changed by ± 10%, ± 20%, and ± 30% once at a time. Changing weight once at a time ensures observing the effects of the factor on the overall decision. For example, how do usefulness, riskiness, and ease of use affect the final selection of m-banking applications? A total of eighteen experiments (six for each dimension) were carried out, and the results are presented in Fig. 3.

When the base scenario is compared to the E + 30 scenario, it can be seen that there is a significant difference in the ranking. For example, in the base scenario, the m-banking application's preference is found in the following order: A2–A6–A1–A5–A9–A8–A4–A3–A7. For the E + 30 scenario, the preference order has been changed to A2–A6–A1–A5–A9–A4–A8–A3–A7. If usability is prioritised, the A4 outperforms the A8. However, if the risk is prioritised in the R + 20 scenario, A1 ranks higher than A6 compared to the base scenario. If the risk factor (such as R + 20) is given higher importance, the sequence of preference of m-banking applications should be A2–A1–A6–A5–A9–A8–A4–A3–A7.

A similar variation is observed when changing the weight of usefulness. In this study, the m-banking application A2 was found to be the most efficient application based on the importance of identified factors and sub-factors. In this study, no significant ranking changes were observed for the A7 and A9 m-banking applications. Based on the sensitivity analysis, it can be assumed that the weights of the factors have a significant impact on the ranking of m-banking applications. A faulty or biased judgement can lead to the incorrect selection of m-banking applications.

Following the sensitivity analyses, a comparison with the existing literature was performed. Although a few studies [7, 45, 65, 102] addressed the selection of m-banking technologies, none of them addressed human judgement of uncertainty when making a decision. The proposed fuzzy model, which employs fuzzy-BWM and fuzzy-TOPSIS to select m-banking applications, is unique in that it takes into account human uncertainty when selecting m-banking applications. The current study identified thirty sub-criteria and classified them into ten main criteria. None of the existing literature compiled all of these factors into a single report. Our findings differ significantly from those of Zarifopoulos and Economides [102]. For example, the Zarifopoulos and Economides [102] framework did not consider the user's security and trust when selecting m-banking applications. In terms of methodology, the current study differs from Komlan et al. [7]. Komlan et al. [7] make no mention of the uncertainty of human judgement. This issue, however, has been addressed in the current study. Furthermore, the analysis differs significantly in terms of the number of factors used to evaluate m-banking applications.