1 Introduction

Since the advent of Bitcoin in 2008, the indispensable folder plan of a blockchain has conventional wide care (Gupta 2017). A blockchain is a circulated archive skill that contains the greatest data consecutively on peer-to-peer systems. Since separately contributors can admittance the complete catalog and important annals (Chen 2018), the dealings container be preserved by a collection of bulges, and thus, a reliable third gathering can be eradicated when standards change (Beck and Muller-Bloch 2017; Birch et al. 2016; Lin et al. 2017). Moreover, as the dispersed archive is of extraordinary clearness, sanctuary, immutability, and decentralization, the blockchain has been practicing in the areas of finances, medication, Internet of Belongings (Li et al. 2017), and reproduction intellect (Zhang et al. 2021), for requests such as economic dealings (Wang et al. 2019; Varma 2019), fitness overhaul data organization (Yaqoob et al. 2021), source cable organization (Liu et al. 2019), and management actions such as translucent elective and numerical autographs. Using this system, one can produce truthful alphanumeric chronicles (Lemieux 2016), condense business budget, confirm data photographs, develop the reimbursement stretch to comprehend efficient methods (Holotiuk et al. 2019), portion data midst members, stop repeated or falsified dealings (Hoy 2017), and inferior the dangers of scheme failure (Ozkan et al. 2019). Individuals encounter their vital needs such as warming and illumination with the assistance of vitality. On the other hand, vitality is considered one of the foremost important crude materials in industrial production. Vitality may be a significant calculate for the maintainable development of the nations.

An object from Harvard Commercial Appraisal declared that a blockchain prepared to set and law companies what the Internet fixed to television (Ito et al. 2017). The monetary facility arena develops an innovator in the examination of substructures and commercial replicas based on blockchains (Beck and Müller-Bloch 2017), and frequent main sets and fiscal organizations have assumed blockchain skill in doings, such as defrayal and defrayal, cross-border payment, numerical promoting, source chain economics, recognition journalism, user individuality confirmation and explanation sympathy (Beck et al. 2016). In addition, numerous blockchain startups have industrialized blockchain technology explanations for doings in major sets and economic organizations. In this situation, the difficulty of the assortment of a fitting blockchain skill provider by a major bank or financial organization, based on several criteria, has appeared. Owing to the intricacy of blockchain technology, which comprises inconsistent and discordant criteria such as the rate and speediness, effectiveness and risk, and deficient perception and understanding of the specific decision-maker, blockchain skill provider variety for chief sets is observed as definitive multi-criteria group decision-making problematic.

Because of the complicacy of blockchain skill and inadequate data and involvement of the DMs, the criteria or substitutes frequently cannot be assessed as finished crisp values. Zadeh (1965) started the fuzzy set principle by applying the membership degree to define indecision. Then, many delays were industrialized. Turksen (1986) protracted this method to an interval-valued fuzzy set that portrayed the membership degree. Atanassov (1986) and Atanassov and Gargov (1989) presented an intuitionistic fuzzy set and an interval-valued intuitionistic fuzzy (IVIF) set, which detained an idea of the nonmembership grade and satisfied the resultant complaint: the synopsis of membership and nonmembership gradations is not developed than 1.

Zhou and Chen (2021) revolutionized fact workers to designate members' risk favorites concerning a blockchain and then prejudiced succeeding rudiments indoors the decision matrix. However, the damage boldness level dislikes animatedly reproduced the DM's thinking in the current methods and the DMs' mental alteration concerning the blockchain program.

Aslam et al. (2022) presented concurrent access control and monitoring of ERP, private permissioned Blockchain using Proof of Elapsed Time consensus is more suitable. The study also investigated the bottleneck issue of transaction processing rates (TPR) of Blockchain consensus, specifically ERP's TPR. Alam et al. (2022) introduced the finally comes up with various implementation requirements in Government, Health, Finance, Economics, and Energy. Qahtan et al. (2022) introduced employed to benchmark blockchain-based IoT healthcare Industry 4.0 systems through the combined gray relational analysis technique for order of preference by similarity to ideal solution (GRA-TOPSIS) and the bald eagle search (BES) optimization method.

Shuaib et al. (2022a) presented avoiding conflict and to support Environmental Sustainability. These traditional land registry applications also lack identity parameters due to weaknesses in identity solutions. A secure and reliable digital identity solution is the need of the hour. Self-sovereign identity (SSI), a new concept, is becoming more popular as a secure and reliable identity solution for users based on identity principles. SSI provides users with a way to control their personal information and consent for it to be used in various ways. In addition, the user's identity details are stored in a decentralized manner, which helps to overcome the problems with digital identity solutions. Shuaib et al. (2022b) presented a design to overcome these limitations and provide a secure, reliable, and efficient identity solution that gives complete control to the users over their personal identity information. Shuaib et al. (2022c) presented examined the current land registry model and its shortcomings. It explains the various blockchain types and their characteristics. It further evaluates the usability of blockchain technology in different aspects of the land registry. Identity management is one of such weaknesses in the blockchain-based land registry model that has been assessed in detail. Identity issues of blockchain-based models have been further evaluated on defined criteria. Rahmani et al. (2022) presented the centralization, huge overhead, trust evidence, less adaptive, and inaccuracy. This systematic review has been performed in six stages: identifying the research question, research methods, screening the related articles, abstract and keyword examination, data retrieval, and mapping processing. Atlas, the software is used to analyze the relevant articles based on keywords. A total of 70 codes and 262 quotations are compiled, and furthermore, these quotations are categorized using manual coding. Bhatia et al. (2022) introduction to train end-to-end segmentation of pixels into vessel and background classes.

This technique reflects both the impartial valuations regarding the consistent criteria and the individual risk outlooks concerning blockchain skills. We describe a set of criteria to lengthily degree the blockchain technology benefactors' performance regarding the cross-border transmittal for chief sets in contract with the available works and stunning topographies of blockchain skill. We assume the TOPSIS, which can precisely define the unclearness and misgivings in rulings due to insufficient assumption and participation related to the blockchain, to acquire the criterion weights with TrFF data. We adventure the TrFF TOPSIS technique to discover the comprehensive sovereignty grade for each claimant, and dynamic the candidates in reducing order seeing the DM's risk desire concerning a blockchain by conclusive the decrease issue of the offended. We appliance compassion training by changing the lessening factor arithmetic value and secondary the consequences with individuals got using the contemporary TrFFPIS and TrFFNIS technique to authenticate its success and suitability of our obtainable mixture method.

The rest of the paper is summarized in Sect. 2, which we offer about the properties of fundamental ideas. Section 3, we exhibit of TrFFNs and operational laws. Section 4, we define the MCDM based on the TrFF TOPSIS technique. Section 5, the request for the developed technique in GDM is a visible example. Section 6, we considered comparative analysis. Finally, a material conclusion is certain in Sect. 7.

1.1 Literature review

This section concisely appraisals the standards and methods for measuring blockchain technologies, stages, crops, and suppliers. This places the basis to create the criteria to choose the suitable blockchain technology supplier for main sets or financial organizations which means relating blockchain skills throughout the procedure of cross-border transmittal.

A blockchain is a type that consists of a growing list of records, called blocks, that are securely linked together using cryptography. Each block contains a cryptographic hash of the previous block, a timestamp, and transaction data (generally represented as a Merkle tree, where data nodes are represented by leaves). The timestamp proves that the transaction data existed when the block was created. Since each block contains information about the block before it, they effectively form a chain (compare linked list data structure), with each additional block linking to the ones before it. Consequently, blockchain transactions are irreversible in that, once they are recorded, the data in any given block cannot be altered retroactively without altering all subsequent blocks.

However, many researchers (50, 51) employed in the arena of smearing digital individuality explanations for blockchain-based property archive schemes inveterate the subject of noncompliance with numerical identity values. So though emerging a unique explanation for a blockchain-based land archive organization, these subjects need attention (52, 53).

Blockchains are typically managed by a computer network for use as a public distributed ledger, where nodes collectively adhere to a consensus algorithm protocol to add and validate new transaction blocks. Although blockchain records are not unalterable, since blockchain forks are possible, blockchains may be considered secure by design and exemplify a distributed computing system with high Byzantine fault tolerance.

Owed to the existence of developing technology, a methodical evaluative criteria scheme for the presentation assessment of blockchain skill suppliers has not been recognized, though academics have browbeaten sure standards to assess blockchain technologies, crops, or facilities. Tang et al. (2019) invented a usual assessment scheschementing 3 main criteria and 11 sub-club-criteria selected schemes for a suitable community blockchain, in which skill, credit, and applied action remained the key issues. Özkan et al. (2019) created a usual assessment to assess the blockchain skill risk. The safety matter was the greatest dangerous risk at the primary equal, and cyberattacks, confidentiality subjects, and exercise prices were the greatest dangerous dangers at the extra equal. Jin et al. (2019) assessed sustainable blockchain skiskillseing the price, generation, concert, and after-sales facility. Farshidi et al. (2020) obtainable an order of standards, counting functionality, flexibility, and compatibility with obtainable software foodstuffs, to choose a blockchain stage. Colak et al. (2020) offered a multi-criteria evaluative style to measure blockchain skills used in a stock hawser system. Yoon et al. (2020) struggled to discover the issues manipulating the alteration of blockchain technology inside the nautical logistics arena and recognized pertinent alteration variables in footings of ecological, structural, economic, and practical issues. Ozdemir et al. (2019) proposed rudimentary criteria to measure a blockchain request in tourism, which comprised the blockchain supremacy form, stages, agreement kind, operation of cryptocurrency, keen agreements, and marks.

As designated in Table 1, price (85.71%), ability (78.57%), creation and facility excellence (64.29%), and safety (64.29%) exhibition high rank. Though, binary limits are originating in the existing criteria for blockchain agendas. Pertinent danger issues are not devoted to rank in existing poetry. Meanwhile, cross-border payment is an important procedure in financial organizations, the supplies for speed, care, and correctness are severe. Blockchain founded on confusion processes and encryption skills can protect the safety of the transmittal. However, the whole staff must contribute to the control to attain an agreement, which leads to little competence and a minor amount of dealings per additional (Tang et al. 2019). Hence, it is crucial to exploit the assessment of blockchain suppliers, seeing the high presentation and little apparent danger. Care and jeopardy switch are binary features of one driver, so we container shape the chief criteria called refuge risk. Technological equality, creation and facility excellence, fiscal issues, and safety danger are stared as the chief criteria and a crucial essential to finding pertinent sub-criteria to measure the presentation of blockchain skill suppliers relating to the available criteria and exclusive features of cross-border transmittals in financial establishments. From this assessment, this item proposes to shape graded evaluative criteria for chief series or economic foundations that propose to relate blockchain technology in cross-border settlement slog. The criteria for the scientific smooth can be divided into fundamental skill, scalability, interoperability, and dispensation rapidity. The criteria for the creation and facility excellence can be divided into the generation of the creation, rudimentary facility, after deals facility, and additional service. The fiscal influence can be segmented into Table 1.

Table 1 Evaluate criteria pertaining to block chain technologies

First, incomplete devices use fuzzy data to measure blockchain presentation. Among the current works regarding blockchain knowledge valuation, Çolak et al. (2020), Bai and Sarkis (2020), and Karaşan et al. (2021) used hesitant fuzzy data. Owing to the absence of information and skills concerning blockchain skills, rulings relating to the criteria might not be precise, and the crisp numbers might not exactly reflect the DMs' rulings (Tables 2, 3, 4, 5, 6 and 7).

Table 2 Block chain is different technique
Table 3 TrFF decision matrix
Table 4 TrFF decision matrix
Table 5 TrEFFWA operator
Table 6 Normalized TrFF TOPSIS decision matrix
Table 7 Weight normalized TrFF decision matrix

1.2 Novelties

In this article, we mark an effort to integrate and speech the following ideas:

  1. i.

    To describe innovative operational law for trapezoidal fermatean fuzzy data which is a valuable complement of standing operational law and examined their arithmetical possessions.

  2. ii.

    To present new operators like trapezoidal fermatean fuzzy data fuzzy aggregation operators.

  3. iii.

    Proposition of MCGDM strategy in TrFF environment.

  4. iv.

    To demonstrate the proposed method, we solved a numerical problem based on a real-life problem.

  5. v.

    A sensitivity analysis is performed to show the utility and efficiency of the designed method.

1.3 Contributions and structure

This research has contributed to the exploration of MAGDM under uncertainty in the following aspects:

  • The TrFF is introdTFS as a new generalization in TrFS theory to tackle the complexities in numerical data by combing the TrFN terms and FFN.

  • The TrFFE operator, TrFF TOPSIS technique are proposed by the integration MAGDM.

  • A MAGDM innovation for blockchain assessment in TrFF based on the TrFF TOPSIS technique is established.

  • An assessment framework of the blockchain selection scheme using the proposed MAGDM innovation is constructed.

    The essential innovations of this object associated with the current blockchain evaluative standards and approaches container be drawn as surveys:

  • A sequence of regular standards is created to choose the appropriate blockchain skill supplier for main sets or financial organizations, which can contribute to the DMs smearing the blockchain to the cross-border transmittal action.

  • The TrFF sets are practical in the presentation assessment of the blockchain skill workers, which can reproduce additional unstated and clear rulings and favorites of the DMs and exhibition a better scope compared to the additional trapezoidal formation fuzzy sets to precisely get the decision-making marks.

  • TrFF TOPSIS technique is applied to evaluate the blockchain skill workers sighted the DMs' risk favorite completed the blockchain skill and seeing the decrease limit near damage, which things the supremacy results of one additional, and to precisely get the best supplier.

  • A TrFF TOPSIS technique is recognized to language MCGDM problems and delivers an arranged and reliable program to language blockchain skill supplier variety problems.

1.4 Motivation

  1. i.

    TrFFSs in TOPSIS (with MCDM concept) have been implemented in a real-life blockchain assessment problem for manufacturing in the literature. Considering the blockchain width, this study will lead future studies in this field.

  2. ii.

    For TrFF-TOPSIS computations, a new linguistic scale under Fermatean fuzzy documentation has also been developed for experts to disclose their judgments easily. The scope of information uncertainty covered by trapezoidal Fermatean fuzzy is broader than that of conventional trapezoidal fuzzy number, and Fermatean fuzzy number, which is instrumental in avoiding potential information missed in qualitative conversion to quantification. Scholars may benefit from this scale in future studies. In addition, TrFF-TOPSIS uses PIS and NIS as reference points for distance calculation, which can obtain more potential information.

  3. iii.

    Practitioners in the industry can adapt the case application presented in this study for their risk assessment processes. From this point, good practice is demonstrated with detailed steps from expert judgments to risk preventive measure suggestions.

  4. iv.

    A comparative analysis is provided to test the solidity of the proposed approach. To do this, crisp, intuitionistic fuzzy, and FFs are applied to the problem. Analysis results are not exactly the same as TrFF-TOPSIS. Since the TrFFs measure both the membership and non-membership, but crisp, fuzzy, and gray theories only consider the membership.

To initiate with of all, there are frequent considers within the script on source cable management. Predominantly in advanced an extended time, susceptibility in decisions-making procedures for as well as many criteria has extended. This condition completes a feasible decision-making grip certainly more worrying. Then, there's a mounting necessitate for additional complete study plans for this make. In other arguments, seeing MCDM policies lengthily with characteristic fuzzy numbers might donate to intensifying the skill in this concoct. The main position is connected to the scheming of the TrFF-TOPSIS feelings of the joining blockchain. For this reason, selected criteria are measured by four varied authorities. In this organization, assumed standards are assessed to entire the joining outlines by seeing the blockchain feelings (Figs. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 and 11).

Fig. 1
figure 1


Fig. 2
figure 2

Model fuzzy way

Fig. 3
figure 3

Procedure flowchart of the offered TrFF-AHP-TOPSIS metho for MCDM Difficulties

Fig. 4
figure 4

Blockchain of hirarechical evaluative

Fig. 5
figure 5

Case 1

Fig. 6
figure 6

Case 2

Fig. 7
figure 7

Case 3

Fig. 8
figure 8

Case 4

Fig. 9
figure 9

Score function of IFWA operator

Fig. 10
figure 10

Score function of FFWG operator

Fig. 11
figure 11

Whole paper

2 Preliminaries

Definition 2.1

Zadeh (1965) Let \(H\) be a universe of discourse. Then the fuzzy set can be defined as: \(J = \{ h,\,\mu_{J} (h)|h \in H\} .\) A fuzzy set in a set \(H\) is denoted by \(\mu_{J} \;\;:\;\;H \to I\), where \(I = [0,\,1]\). The function \(\mu_{J(h)}\) denotes the degree of membership of the element \(h\) to the set \(H\). The collection of all fuzzy subsets of \(H\) is denoted by \(I^{H}\). Define a relation on \(I^{H}\) as follows: \((\forall \mu ,\,\eta \in I^{H} )(\mu \le \eta \Leftrightarrow (\forall h \in H)(\mu (h) \le \eta (h))).\)

2.1 FFN and operational laws

Definition 2.1.1

The fixed set \(C\) and the FFN \(A\) is defined in \(A = \left\{ {\begin{array}{*{20}c} {\langle \varsigma_{A} (x),} \\ {\chi_{A} (x)\rangle } \\ {:\;\;x \in C} \\ \end{array} } \right\},\) where \(\varsigma_{A} (x)\) and \(\chi_{A} (x)\) represent the MED and NOMED, and \(\varsigma_{A} (x) \in [0,\,1]\), \(\chi_{A} (x) \in [0,\,1]\) and \(0 \le \varsigma_{A} (x)^{3} + \chi_{A} (x)^{3} \le 1.\) The degree of indeterminacy is defined as \(\pi_{A} (x) = \sqrt[3]{{(\varsigma_{A} (x)^{3} + \chi_{A} (x)^{3} - \varsigma_{A} (x)^{3} \chi_{A} (x)^{3} )}}.\) The FFN is denoted as \(A = \langle \varsigma_{A} ,\,\chi_{A} \rangle .\)

Definition 2.1.2

Let \(a_{1} = \left\{ {\varsigma_{1} ,\,\chi_{1} } \right\}\) and \(a_{2} = \left\{ {\varsigma_{2} ,\,\chi_{2} } \right\}\) be two FFNs, \(\lambda > 0\), then.

$$ a_{1} \oplus a_{2} = \left[ {\sqrt[3]{{(\varsigma_{1}^{3} + \varsigma_{2}^{3} - \varsigma_{1}^{3} \varsigma_{2}^{3} )}},\,\chi_{1} \chi_{2} } \right]; $$
$$ a_{1} \otimes a_{2} = \left[ {\varsigma_{1} \varsigma_{2} ,\,\sqrt[3]{{(\chi_{1}^{3} + \chi_{2}^{3} - \chi_{1}^{3} \chi_{2}^{3} )}}} \right]; $$
$$ \lambda a_{1} = \left[ {\sqrt[3]{{1 - (1 - \varsigma_{1}^{3} )^{\lambda } }},\,\chi ,\,\chi_{1}^{\lambda } } \right]; $$
$$ a_{1}^{\lambda } = \left[ {\varsigma_{1}^{\lambda } ,\,\sqrt[3]{{1 - (1 - \chi_{1}^{3} )^{\lambda } }}} \right]. $$

Definition 2.1.3

Let \(a = \left\{ {\varsigma ,\,\chi } \right\}\) be the FFNs, then the score function is \(a = \varsigma_{\alpha }^{3} - \chi_{\alpha }^{3} .\)

Definition 2.1.4

Let \(a = \left\{ {\varsigma ,\,\chi } \right\}\) be the FFNs, then the accuracy function is \(a = \varsigma_{\alpha }^{3} + \chi_{\alpha }^{3} .\)

2.2 FF TOPSIS technique

Step 1: Describe the FF decision matrix.

Step 2: Describe the FEFWA operator and \(\xi = \left( {\xi_{1} ,\,\xi_{2} ,...,\,\xi_{n} } \right).\)

$$ {\text{FEFWA}}\,\,(a_{1} ,\,a_{2} ,...,\,a_{n} )\; = \,\,\left[ {\left( {\tfrac{{\mathop \prod \limits_{j = 1}^{n} (1 + G_{j}^{3} )^{{\xi_{j} }} - \mathop \prod \limits_{j = 1}^{n} (1 - G_{j}^{3} )^{{\xi_{j} }} }}{{\mathop \prod \limits_{j = 1}^{n} (1 + G_{j}^{3} )^{{\xi_{j} }} + \mathop \prod \limits_{j = 1}^{n} (1 - G_{j}^{3} )^{{\xi_{j} }} }}} \right)^{{\tfrac{1}{3}}} ,\,\tfrac{{2\mathop \prod \limits_{j = 1}^{n} (F_{j} )^{{\xi_{j} }} }}{{\mathop \prod \limits_{j = 1}^{n} (2 + F_{j} )^{{\xi_{j} }} + \mathop \prod \limits_{j = 1}^{n} (F_{j} )^{{\xi_{j} }} }}} \right] $$

Step 3: To construct a normalized FF TOPSIS decision matrix. The normalized value \(\beta_{ij}\) is calculated as follows:

$$ \beta = \left[ {\tfrac{{G_{j}^{3} }}{{^{3} \sqrt {\mathop \sum \limits_{i = 1}^{n} (G_{j}^{3} )^{2} } }},\,\tfrac{{F_{j}^{3} }}{{^{3} \sqrt {\mathop \sum \limits_{i = 1}^{n} (F_{j}^{3} )^{2} } }}} \right]. $$

Step 4: To construct the weighted normalized FF TOPSIS decision matrix by multiplying the normalized FF TOPSIS decision matrix by its associated weights. The weighted normalized value is given by \(v_{ij} w_{j} = B_{j}\).

Step 5: To find the + ve FF ideal solution and the -veFF ideal solution. It is shown as under:

$$ \alpha_{i}^{ + } = [\max_{i} G_{j} ,\,\min_{i} F_{j} ] $$
$$ \alpha_{i}^{ - } = [\min_{i} G_{j} ,\,\max_{i} F_{j} ] $$

Step 6: Separation of each candidate from the positive FF ideal solution that is given as under:

$$ q_{i}^{ + } \langle [B^{ - } ,\,B^{ + } ],\,\eta \rangle = \langle \tfrac{1}{2}\left( {\left| {G_{ij} - G_{j} } \right| + \left| {F_{ij} - F_{j} } \right|} \right)\,\rangle . $$

Separation of each candidate from the negative FF ideal solution that is given as under:

$$ q_{i}^{ - } \langle [B^{ - } ,\,B^{ + } ],\,\eta \rangle = \langle \tfrac{1}{2}\left( {\left| {G_{ij} - G_{j} } \right| + \left| {F_{ij} - F_{j} } \right|} \right)\,\rangle . $$

Step 7: To compute closeness relative to the ideal solution. Relative closeness to the ideal solution is comprehended by the Equation.

$$ Z_{i} = \tfrac{{q_{i}^{ - } \langle [B^{ - } ,\,B^{ + } ],\,\eta \rangle }}{{q_{i}^{ - } \langle [B^{ - } ,\,B^{ + } ],\,\eta \rangle + q_{i}^{ + } \langle [B^{ - } ,\,B^{ + } ],\,\eta \rangle }}. $$

3 TrFFN and operational laws

Definition 3.1

Let \(a_{1} = \left\{ {\begin{array}{*{20}c} {[\eta_{1} ,\,\kappa_{1} ,\,\varsigma_{1} ,\,\delta_{1} ],} \\ {[\Lambda_{1} ,\,\Upsilon_{1} ,\,\vartheta_{1} ,\,\Theta_{1} ]} \\ \end{array} } \right\}\) and \(a_{2} = \left\{ {\begin{array}{*{20}c} {[\eta_{2} ,\,\kappa_{2} ,\,\varsigma_{2} ,\,\delta_{2} ],} \\ {[\Lambda_{2} ,\,\Upsilon_{2} ,\,\vartheta_{2} ,\,\Theta_{2} ]} \\ \end{array} } \right\}\) be two TrFFNs and \(\lambda > 0,\) then.

$$ a_{1} \oplus a_{2} = \left[ {\begin{array}{*{20}c} {\left( {\sqrt[3]{{\tfrac{{\eta_{1}^{3} + \eta_{2}^{3} }}{{1 + \eta_{1}^{3} \eta_{2}^{3} }}}},\,\sqrt[3]{{\tfrac{{\kappa_{1}^{3} + \kappa_{2}^{3} }}{{1 + \kappa_{1}^{3} \kappa_{2}^{3} }}}},\,\sqrt[3]{{\tfrac{{\varsigma_{1}^{3} + \varsigma_{2}^{3} }}{{1 + \varsigma_{1}^{3} \varsigma_{2}^{3} }}}},\,\sqrt[3]{{\tfrac{{\delta_{1}^{3} + \delta_{2}^{3} }}{{1 + \delta_{1}^{3} \delta_{2}^{3} }}}}} \right),} \\ {\left( {\begin{array}{*{20}c} {\tfrac{{\Lambda_{1} \Lambda_{2} }}{{1 + (1 - \Lambda_{1} )(1 - \Lambda_{2} )}},\,\tfrac{{\Upsilon_{1} \Upsilon_{2} }}{{1 + (1 - \Upsilon_{1} )(1 - \Upsilon_{2} )}},} \\ {\tfrac{{\vartheta_{1} \vartheta_{2} }}{{1 + (1 - \vartheta_{1} )(1 - \vartheta_{2} )}},\,\tfrac{{\Theta_{1} \Theta_{2} }}{{1 + (1 - \Theta_{1} )(1 - \Theta_{2} )}}} \\ \end{array} } \right)} \\ \end{array} } \right]; $$
$$ a_{1} \otimes a_{2} = \left[ {\begin{array}{*{20}c} {\left( {\begin{array}{*{20}c} {\tfrac{{\eta_{1} \eta_{2} }}{{1 + (1 - \eta_{1} )(1 - \eta_{2} )}},\,\tfrac{{\kappa_{1} \kappa_{2} }}{{1 + (1 - \kappa_{1} )(1 - \kappa_{2} )}},} \\ {\tfrac{{\varsigma_{1} \varsigma_{2} }}{{1 + (1 - \varsigma_{1} )(1 - \varsigma_{2} )}},\,\tfrac{{\delta_{1} \delta_{2} }}{{1 + (1 - \delta_{1} )(1 - \delta_{2} )}}} \\ \end{array} } \right),} \\ {\left( {\sqrt[3]{{\tfrac{{\Lambda_{1}^{3} + \Lambda_{2}^{3} }}{{1 + \Lambda_{1}^{3} \Lambda_{2}^{3} }}}},\,\sqrt[3]{{\tfrac{{\Upsilon_{1}^{3} + \Upsilon_{2}^{3} }}{{1 + \Upsilon_{1}^{3} \Upsilon_{2}^{3} }}}},\,\sqrt[3]{{\tfrac{{\vartheta_{1}^{3} + \vartheta_{2}^{3} }}{{1 + \vartheta_{1}^{3} \vartheta_{2}^{3} }}}},\,\sqrt[3]{{\tfrac{{\Theta_{1}^{3} + \Theta_{2}^{3} }}{{1 + \Theta_{1}^{3} \Theta_{2}^{3} }}}}} \right)} \\ \end{array} } \right]; $$
$$ \lambda a_{1} = \left[ {\begin{array}{*{20}c} {\left( {\begin{array}{*{20}c} {\sqrt[3]{{\tfrac{{(1 + \eta_{1}^{3} )^{{\lambda_{j} }} - (1 - \eta_{1}^{3} )^{{\lambda_{j} }} }}{{(1 + \eta_{1}^{3} )^{{\lambda_{j} }} + (1 - \eta_{1}^{3} )^{{\lambda_{j} }} }}}},\,\sqrt[3]{{\tfrac{{(1 + \kappa_{1}^{3} )^{{\lambda_{j} }} - (1 - \kappa_{1}^{3} )^{{\lambda_{j} }} }}{{(1 + \kappa_{1}^{3} )^{{\lambda_{j} }} + (1 - \kappa_{1}^{3} )^{{\lambda_{j} }} }}}},} \\ {\sqrt[3]{{\tfrac{{(1 + \varsigma_{1}^{3} )^{{\lambda_{j} }} - (1 - \varsigma_{1}^{3} )^{{\lambda_{j} }} }}{{(1 + \varsigma_{1}^{3} )^{{\lambda_{j} }} + (1 - \varsigma_{1}^{3} )^{{\lambda_{j} }} }}}},\,\sqrt[3]{{\tfrac{{(1 + \delta_{1}^{3} )^{{\lambda_{j} }} - (1 - \delta_{1}^{3} )^{{\lambda_{j} }} }}{{(1 + \delta_{1}^{3} )^{{\lambda_{j} }} + (1 - \delta_{1}^{3} )^{{\lambda_{j} }} }}}}} \\ \end{array} } \right),} \\ {\left( {\begin{array}{*{20}c} {\tfrac{{2(\Lambda_{1} )^{{\lambda_{j} }} }}{{(2 + \Lambda_{1} )^{{\lambda_{j} }} + (\Lambda_{1} )^{{\lambda_{j} }} }},\,\tfrac{{2(\Upsilon_{1} )^{{\lambda_{j} }} }}{{(2 + \Upsilon_{1} )^{{\lambda_{j} }} + (\Upsilon_{1} )^{{\lambda_{j} }} }},} \\ {\tfrac{{2(\vartheta_{1} )^{{\lambda_{j} }} }}{{(2 + \vartheta_{1} )^{{\lambda_{j} }} + (\vartheta_{1} )^{{\lambda_{j} }} }},\,\tfrac{{2(\Theta_{1} )^{{\lambda_{j} }} }}{{(2 + \Theta_{1} )^{{\lambda_{j} }} + (\Theta_{1} )^{{\lambda_{j} }} }}} \\ \end{array} } \right)} \\ \end{array} } \right]; $$
$$ a_{1}^{\lambda } = \left[ {\begin{array}{*{20}c} {\left( {\begin{array}{*{20}c} {\tfrac{{2(\eta_{1} )^{{\lambda_{j} }} }}{{(2 + \eta_{1} )^{{\lambda_{j} }} + (\eta_{1} )^{{\lambda_{j} }} }},\,\tfrac{{2(\kappa_{1} )^{{\lambda_{j} }} }}{{(2 + \kappa_{1} )^{{\lambda_{j} }} + (\kappa_{1} )^{{\lambda_{j} }} }},} \\ {\tfrac{{2(\varsigma_{1} )^{{\lambda_{j} }} }}{{(2 + \varsigma_{1} )^{{\lambda_{j} }} + (\varsigma_{1} )^{{\lambda_{j} }} }},\,\tfrac{{2(\delta_{1} )^{{\lambda_{j} }} }}{{(2 + \delta_{1} )^{{\lambda_{j} }} + (\delta_{1} )^{{\lambda_{j} }} }}} \\ \end{array} } \right),} \\ {\left( {\begin{array}{*{20}c} {\sqrt[3]{{\tfrac{{(1 + \Lambda_{1}^{3} )^{{\lambda_{j} }} - (1 - \Lambda_{1}^{3} )^{{\lambda_{j} }} }}{{(1 + \Lambda_{1}^{3} )^{{\lambda_{j} }} + (1 - \Lambda_{1}^{3} )^{{\lambda_{j} }} }}}},\,\sqrt[3]{{\tfrac{{(1 + \Upsilon_{1}^{3} )^{{\lambda_{j} }} - (1 - \Upsilon_{1}^{3} )^{{\lambda_{j} }} }}{{(1 + \Upsilon_{1}^{3} )^{{\lambda_{j} }} + (1 - \Upsilon_{1}^{3} )^{{\lambda_{j} }} }}}},} \\ {\sqrt[3]{{\tfrac{{(1 + \vartheta_{1}^{3} )^{{\lambda_{j} }} - (1 - \vartheta_{1}^{3} )^{{\lambda_{j} }} }}{{(1 + \vartheta_{1}^{3} )^{{\lambda_{j} }} + (1 - \vartheta_{1}^{3} )^{{\lambda_{j} }} }}}},\,\sqrt[3]{{\tfrac{{(1 + \Theta_{1}^{3} )^{{\lambda_{j} }} - (1 - \Theta_{1}^{3} )^{{\lambda_{j} }} }}{{(1 + \Theta_{1}^{3} )^{{\lambda_{j} }} + (1 - \Theta_{1}^{3} )^{{\lambda_{j} }} }}}}} \\ \end{array} } \right)} \\ \end{array} } \right]. $$

Definition 3.2

The TrFFNs are \(a = \left\{ {\begin{array}{*{20}c} {[\eta ,\,\kappa ,\,\varsigma ,\,\delta ],} \\ {[\Lambda ,\,\Upsilon ,\,\vartheta ,\,\Theta ]} \\ \end{array} } \right\}\) , then the score function \(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{H}\) is define as: \(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{H} = \tfrac{{\left\{ {[\eta^{3} + \kappa^{3} + \varsigma^{3} + \delta^{3} ] - [\Lambda^{3} + \Upsilon^{3} + \vartheta^{3} + \Theta^{3} ]} \right\}}}{8}.\)

Definition 3.3

The TrFFNs are \(a = \left\{ {\begin{array}{*{20}c} {[\eta ,\,\kappa ,\,\varsigma ,\,\delta ],} \\ {[\Lambda ,\,\Upsilon ,\,\vartheta ,\,\Theta ]} \\ \end{array} } \right\}\) , then the accuracy function \(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{H}\) is define as: \(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{H} = \tfrac{{\left\{ {[\eta^{3} + \kappa^{3} + \varsigma^{3} + \delta^{3} ] + [\Lambda^{3} + \Upsilon^{3} + \vartheta^{3} + \Theta^{3} ]} \right\}}}{8}.\)

Definition 3.4

The gathering of TrFFNs are \(c_{j} = \left\{ {\begin{array}{*{20}c} {[\eta ,\,\kappa ,\,\varsigma ,\,\delta ],} \\ {[\Lambda ,\,\Upsilon ,\,\vartheta ,\,\Theta ]} \\ \end{array} } \right\}\) and the weight vector is \(\Delta = (\Delta_{1} ,\,\Delta_{2} ,...,\,\Delta_{n} )^{T}\) with \(\Delta_{j} \in [0,\,1]\) and \(\mathop \sum \nolimits_{j = 1}^{n} \Delta_{j} = 1\) . Then TrEFFWA \(\left( {c_{1} ,\,c_{2} ,...,\,c_{n} } \right) = \mathop \oplus \nolimits_{j = 1}^{n} \Delta_{j} c_{j}\) is said TrEFFWA operator.

Theorem 3.5

The collection of TrFFNs are \(a_{j} = \left\{ {\begin{array}{*{20}c} {[\mu ,\,S,\,G,\,H],} \\ {[\nu ,\,F,\,M,\,N]} \\ \end{array} } \right\}\) and the weight vector is \(\xi = (\xi_{1} ,\,\xi_{2} ,...,\,\xi_{n} )^{T}\) with \(\xi_{j} \in [0,\,1]\) and \(\mathop \sum \nolimits_{j = 1}^{n} \xi_{j} = 1\) . Then it is said TrEFFWA operator and TrEFFWA \ \(\left( {a_{1} ,\,a_{2} ,...,\,a_{n} } \right) = \left[ {\begin{array}{*{20}c} {\left( {\begin{array}{*{20}c} {\sqrt[3]{{\tfrac{{\mathop \prod \limits_{j = 1}^{n} (1 + \mu_{j}^{3} )^{{\xi_{j} }} - \mathop \prod \limits_{j = 1}^{n} (1 - \mu_{j}^{3} )^{{\xi_{j} }} }}{{\mathop \prod \limits_{j = 1}^{n} (1 + \mu_{j}^{3} )^{{\xi_{j} }} + \mathop \prod \limits_{j = 1}^{n} (1 - \mu_{j}^{3} )^{{\xi_{j} }} }}}},\,\sqrt[3]{{\tfrac{{\mathop \prod \limits_{j = 1}^{n} (1 + S_{j}^{3} )^{{\xi_{j} }} - \mathop \prod \limits_{j = 1}^{n} (1 - \mu_{j}^{3} )^{{\xi_{j} }} }}{{\mathop \prod \limits_{j = 1}^{n} (1 + \mu_{j}^{3} )^{{\xi_{j} }} + \mathop \prod \limits_{j = 1}^{n} (1 - \mu_{j}^{3} )^{{\xi_{j} }} }}}}} \\ {,\,\sqrt[3]{{\tfrac{{\mathop \prod \limits_{j = 1}^{n} (1 + G_{j}^{3} )^{{\xi_{j} }} - \mathop \prod \limits_{j = 1}^{n} (1 - G_{j}^{3} )^{{\xi_{j} }} }}{{\mathop \prod \limits_{j = 1}^{n} (1 + G_{j}^{3} )^{{\xi_{j} }} + \mathop \prod \limits_{j = 1}^{n} (1 - G_{j}^{3} )^{{\xi_{j} }} }}}},\,\sqrt[3]{{\tfrac{{\mathop \prod \limits_{j = 1}^{n} (1 + H_{j}^{3} )^{{\xi_{j} }} - \mathop \prod \limits_{j = 1}^{n} (1 - H_{j}^{3} )^{{\xi_{j} }} }}{{\mathop \prod \limits_{j = 1}^{n} (1 + H_{j}^{3} )^{{\xi_{j} }} + \mathop \prod \limits_{j = 1}^{n} (1 - H_{j}^{3} )^{{\xi_{j} }} }}}}} \\ \end{array} } \right),} \\ {\left( {\begin{array}{*{20}c} {\tfrac{{2\mathop \prod \limits_{j = 1}^{n} (\nu_{j} )^{{\xi_{j} }} }}{{\mathop \prod \limits_{j = 1}^{n} (2 + \nu_{j} )^{{\xi_{j} }} + \mathop \prod \limits_{j = 1}^{n} (\nu_{j} )^{{\xi_{j} }} }},\,\tfrac{{2\mathop \prod \limits_{j = 1}^{n} (F_{j} )^{{\xi_{j} }} }}{{\mathop \prod \limits_{j = 1}^{n} (2 + F_{j} )^{{\xi_{j} }} + \mathop \prod \limits_{j = 1}^{n} (F_{j} )^{{\xi_{j} }} }},} \\ {\tfrac{{2\mathop \prod \limits_{j = 1}^{n} (M_{j} )^{{\xi_{j} }} }}{{\mathop \prod \limits_{j = 1}^{n} (2 + M_{j} )^{{\xi_{j} }} + \mathop \prod \limits_{j = 1}^{n} (M_{j} )^{{\xi_{j} }} }},\,\tfrac{{2\mathop \prod \limits_{j = 1}^{n} (N_{j} )^{{\xi_{j} }} }}{{\mathop \prod \limits_{j = 1}^{n} (2 + N_{j} )^{{\xi_{j} }} + \mathop \prod \limits_{j = 1}^{n} (N_{j} )^{{\xi_{j} }} }}} \\ \end{array} } \right)} \\ \end{array} } \right].\)

4 TrFF-AHP-TOPSIS method for blockchain procedure selection

This section offerings an original mixture technique joining AHP with TOPSIS by using TrFF data to discourse blockchain skill provider collection difficulties for main banks or monetary organizations. First, a hierarchical evaluative criteria scheme is constructed in the opinion of the works appraisal and single features of a blockchain, which can ration together a great presentation and apparent hazards. Following, the AHP is comprehensive to the TrFF setting to fast DMs' doubts of the pairwise assessment finished applicable criteria, and the container is consumed to classify the consistent criterion weights of the blockchain skill suppliers. Then, the TOPSIS system is extended to the TrFF scenario to increase the general supremacy grades of the suppliers seeing the DMs' risk dislike concerning blockchain technology, and the position consequences can rapid both impartial valuations regarding the criteria and DMs' supple personal risk brashness finished the blockchain. Therefore, the planned TrFF-AHP-TOPSIS method can widely reflect material topographies of blockchain skill and DMs' reasoning and emotional performance and more effectually address the blockchain skill worker collection problem. The process flow of the recommended TrFF-AHP-TOPSIS method is accessible in Fig. 3.

4.1 Depiction of the blockchain skill supplier assortment problem

Blockchain supplier selection, as a characteristic MCDM problem, is applied through a collection of decision matrices. Let \(A = \{ A_{1} ,\,A_{2} ,...,\,A_{n} \}\) represent sure latent blockchain skill providers \(PS = \{ PS_{1} ,\,PS_{2} ,...,\,PS_{n} \}\) denote significant evaluative criteria delivered by numerous experts \(DM = \{ DM_{1} ,\,DM_{2} ,...,\,DM_{n} \} ,\) and the corresponding rank of specialists is denoted \(\Delta = (\Delta_{1} ,\,\Delta_{2} ,...,\,\Delta_{n} )^{T}\) satisfying the environments \(\mathop \sum \nolimits_{j = 1}^{n} \Delta_{j} = 1\). Moreover, the rank of the evaluative criteria is represented by \(W = (W_{1} ,\,W_{2} , \cdots ,\,W_{n} )^{T}\), meeting the following situations: \(\mathop \sum \nolimits_{j = 1}^{n} W_{j} = 1\). Each component in the decision matrix means the applicable candidate's evaluative assessment on a convinced criterion from the DM.

4.2 TrFF-AHP-TOPSIS technique

Step 1: Describe the TrFF decision matrix.

Step 2: Describe the TrEFFWA operator and \(\xi = \left( {\xi_{1} ,\,\xi_{2} ,...,\,\xi_{n} } \right).\)

$$TrEFFWA (a_{1} ,\,a_{2} ,...,\,a_{n} )\; = \left[ {\begin{array}{*{20}c} {\left( {\begin{array}{*{20}c} {\sqrt[3]{{\tfrac{{\mathop \prod \limits_{j = 1}^{n} (1 + \mu_{j}^{3} )^{{\xi_{j} }} - \mathop \prod \limits_{j = 1}^{n} (1 - \mu_{j}^{3} )^{{\xi_{j} }} }}{{\mathop \prod \limits_{j = 1}^{n} (1 + \mu_{j}^{3} )^{{\xi_{j} }} + \mathop \prod \limits_{j = 1}^{n} (1 - \mu_{j}^{3} )^{{\xi_{j} }} }}}},\,\sqrt[3]{{\tfrac{{\mathop \prod \limits_{j = 1}^{n} (1 + S_{j}^{3} )^{{\xi_{j} }} - \mathop \prod \limits_{j = 1}^{n} (1 - \mu_{j}^{3} )^{{\xi_{j} }} }}{{\mathop \prod \limits_{j = 1}^{n} (1 + \mu_{j}^{3} )^{{\xi_{j} }} + \mathop \prod \limits_{j = 1}^{n} (1 - \mu_{j}^{3} )^{{\xi_{j} }} }}}}} \\ {,\,\sqrt[3]{{\tfrac{{\mathop \prod \limits_{j = 1}^{n} (1 + G_{j}^{3} )^{{\xi_{j} }} - \mathop \prod \limits_{j = 1}^{n} (1 - G_{j}^{3} )^{{\xi_{j} }} }}{{\mathop \prod \limits_{j = 1}^{n} (1 + G_{j}^{3} )^{{\xi_{j} }} + \mathop \prod \limits_{j = 1}^{n} (1 - G_{j}^{3} )^{{\xi_{j} }} }}}},\,\sqrt[3]{{\tfrac{{\mathop \prod \limits_{j = 1}^{n} (1 + H_{j}^{3} )^{{\xi_{j} }} - \mathop \prod \limits_{j = 1}^{n} (1 - H_{j}^{3} )^{{\xi_{j} }} }}{{\mathop \prod \limits_{j = 1}^{n} (1 + H_{j}^{3} )^{{\xi_{j} }} + \mathop \prod \limits_{j = 1}^{n} (1 - H_{j}^{3} )^{{\xi_{j} }} }}}}} \\ \end{array} } \right),} \\ {\left( {\begin{array}{*{20}c} {\tfrac{{2\mathop \prod \limits_{j = 1}^{n} (\nu_{j} )^{{\xi_{j} }} }}{{\mathop \prod \limits_{j = 1}^{n} (2 + \nu_{j} )^{{\xi_{j} }} + \mathop \prod \limits_{j = 1}^{n} (\nu_{j} )^{{\xi_{j} }} }},\,\tfrac{{2\mathop \prod \limits_{j = 1}^{n} (F_{j} )^{{\xi_{j} }} }}{{\mathop \prod \limits_{j = 1}^{n} (2 + F_{j} )^{{\xi_{j} }} + \mathop \prod \limits_{j = 1}^{n} (F_{j} )^{{\xi_{j} }} }},} \\ {\tfrac{{2\mathop \prod \limits_{j = 1}^{n} (M_{j} )^{{\xi_{j} }} }}{{\mathop \prod \limits_{j = 1}^{n} (2 + M_{j} )^{{\xi_{j} }} + \mathop \prod \limits_{j = 1}^{n} (M_{j} )^{{\xi_{j} }} }},\,\tfrac{{2\mathop \prod \limits_{j = 1}^{n} (N_{j} )^{{\xi_{j} }} }}{{\mathop \prod \limits_{j = 1}^{n} (2 + N_{j} )^{{\xi_{j} }} + \mathop \prod \limits_{j = 1}^{n} (N_{j} )^{{\xi_{j} }} }}} \\ \end{array} } \right)} \\ \end{array} } \right] $$

Step 3: To construct a normalized TrFF TOPSIS decision matrix\(.\) The normalized value \(\beta_{ij}\) is calculated as: \(\beta = \left\{ {\begin{array}{*{20}c} {\left[ {\tfrac{{\mu_{j}^{3} }}{{^{3} \sqrt {\mathop \sum \limits_{i = 1}^{n} (\mu_{j}^{3} )^{2} } }},\,\tfrac{{S_{j}^{3} }}{{^{3} \sqrt {\mathop \sum \limits_{i = 1}^{n} (S_{j}^{3} )^{2} } }},\,\tfrac{{G_{j}^{3} }}{{^{3} \sqrt {\mathop \sum \limits_{i = 1}^{n} (G_{j}^{3} )^{2} } }},\,\tfrac{{H_{j}^{3} }}{{^{3} \sqrt {\mathop \sum \limits_{i = 1}^{n} (H_{j}^{3} )^{2} } }}} \right],} \\ {\left[ {\tfrac{{\nu_{j}^{3} }}{{^{3} \sqrt {\mathop \sum \limits_{i = 1}^{n} (\nu_{j}^{3} )^{2} } }},\,\tfrac{{F_{j}^{3} }}{{^{3} \sqrt {\mathop \sum \limits_{i = 1}^{n} (F_{j}^{3} )^{2} } }},\,\tfrac{{M_{j}^{3} }}{{^{3} \sqrt {\mathop \sum \limits_{i = 1}^{n} (M_{j}^{3} )^{2} } }},\,\tfrac{{N_{j}^{3} }}{{^{3} \sqrt {\mathop \sum \limits_{i = 1}^{n} (N_{j}^{3} )^{2} } }}} \right]} \\ \end{array} } \right\}\)Step 4: To construct the entropy weights. \(H_{j} = \tfrac{1}{\sqrt 2 - 1}\left\{ {\sin \tfrac{\pi (1 + B - C)}{4} + \sin \tfrac{\pi (1 + B - C)}{4} - 1} \right\}\).

Step 5: To construct the weighted normalized TrFF TOPSIS decision matrix by multiplying the normalized TrFF TOPSIS decision matrix by its associated weights. The weighted normalized value is given by \(v_{ij} w_{j} = H_{j}\).

Step 6: To find the + ve TrFF ideal solution and the -ve TrFF ideal solution. It is shown as under:

$$ \alpha_{i}^{ + } = \left\{ {\begin{array}{*{20}c} {[\max_{i} \mu_{j} ,\,\max_{i} S_{j} ,\,\max_{i} G_{j} ,\,\max_{i} H_{j} ],} \\ {[\min_{i} \mu_{j} ,\,\min_{i} S_{j} ,\,\min_{i} G_{j} ,\,\min_{i} H_{j} ]} \\ \end{array} } \right\} $$
$$ \alpha_{i}^{ - } = \left\{ {\begin{array}{*{20}c} {[\min_{i} \nu_{j} ,\,\min_{i} F_{j} ,\,\min_{i} M_{j} ,\,\min_{i} N_{j} ],} \\ {[\max_{i} \nu_{j} ,\,\max_{i} F_{j} ,\,\max_{i} M_{j} ,\,\max_{i} N_{j} ]} \\ \end{array} } \right\} $$

Step 7: Separation of each candidate from the positive TrFF ideal solution that is given as under:

$$ q_{i}^{ + } \langle [B^{ - } ,{\mkern 1mu} B^{ + } ],{\mkern 1mu} \eta \rangle = \left\langle {\tfrac{1}{8}\left( {\begin{array}{*{20}c} {\left| {\mu _{{ij}} - \mu _{j} } \right| + \left| {S_{{ij}} - S_{j} } \right| + \left| {G_{{ij}} - G_{j} } \right| + \left| {H_{{ij}} - H_{j} } \right|} \\ {\left| {\upsilon _{{ij}} - \upsilon _{j} } \right| + \left| {F_{{ij}} - F_{j} } \right| + \left| {M_{{ij}} - M_{j} } \right| + \left| {N_{{ij}} - N_{j} } \right|} \\ \end{array} } \right)} \right\rangle . $$

Separation of each candidate from the negative TrFF ideal solution that is given as under:\( q_{i}^{ - } \langle [B^{ - } ,{\mkern 1mu} B^{ + } ],{\mkern 1mu} \eta \rangle = \left\langle {\tfrac{1}{8}\left( {\begin{array}{*{20}c} {\left| {\mu _{{ij}} - \mu _{j} } \right| + \left| {S_{{ij}} - S_{j} } \right| + \left| {G_{{ij}} - G_{j} } \right| + \left| {H_{{ij}} - H_{j} } \right|} \\ {\left| {\upsilon _{{ij}} - \upsilon _{j} } \right| + \left| {F_{{ij}} - F_{j} } \right| + \left| {M_{{ij}} - M_{j} } \right| + \left| {N_{{ij}} - N_{j} } \right|} \\ \end{array} } \right)} \right\rangle . \)

Step 8: The relative closeness to the ideal solution is comprehended by the equation.

$$ Z_{i} = \tfrac{{q_{i}^{ - } \langle [B^{ - } ,\,B^{ + } ],\,\eta \rangle }}{{q_{i}^{ - } \langle [B^{ - } ,\,B^{ + } ],\,\eta \rangle + q_{i}^{ + } \langle [B^{ - } ,\,B^{ + } ],\,\eta \rangle }}. $$

5 Case study

TE-FOOD is a community permissioned nourishment traceability system that enables all supply chain participants blockchain-based farm-to-table and customer trace of the food's data. The FoodChain (TE-FOOD's blockchain) is a community permissioned blockchain that lets supply chain performers and patrons to uphold chief bulges to disperse traceability data. Clienteles of TE-FOOD consume the suppleness to improvement painstaking visions into the nourishment manufacturing's supply chain.

TFD is TE-ERC FOOD's symbolic, which is typically cast-off on the ethereal stage. Its assignment is to offer slide in the nourishment supply cable by nursing the items finished the whole source chain (farm, computers or abattoir, distributor, shop) and given that tools to clienteles, supply chain companies, and administration activities to study around food past and excellence. The TE-FOOD goals to upsurge customer faith and make contact, get better supply chain information to recover working efficiency, obey with spread rubrics, defend their makes from forging, and perform earlier creation memories.

TE-Food system includes of dissimilar fields:

Empathy Gears: It comprises 1D/2D barcodes/RFID ear labels, safety closures and tag stickers. Traceability tackles: It contains of a B2B traceability management moveable app, web app, dominant system, outside interfaces, and journalism tools. Trade and customer gears comprise B2C new harvest past vision moveable app, and web app, trade side nourishment antiquity numerical signage gears. Nationwide cattle organization solutions: It contains of cattle management and implementation schemes. Farm organization gears: These gears are founded upon the category-specific (Inoculation, nourishing, manures, vegetable defense crops, etc.). Food care gears: These gears comprise a Scam organization scheme, Food disorder device gear, and Meat excellence graphic examination scheme. To classify the followed bodily substances (crops, sites, etc.), the scheme gears dissimilar documentations resources: malleable closures with QR ciphers, malleable credentials labels with RFID, and paper-based marker label with QR ciphers).

For traceability, it delivers dissimilar customer requests: moveable app used by B2B and client to image documentation resources and appeal/arrive data, Web App for the client who fixes not usage the moveable app to admission the creation history, IoT API for nourishment businesses that allow to syndicate data conventional from the devices and Exposed Border for supply chain businesses who custom software now to lever invention's information's-FOOD suggestions binary execution representations, remote or formal.

In Isolated application, a scheme is recycled to suggestion their doings though in Official application, establishments or community government, the scheme is rummage-sale to suggestion a collection of businesses (physically or manufacturing group related). TE-FOOD originates with the filled set of gears and requests wanted for the entire supply chain to tool nourishment traceability by endwise working discernibility and procedure switch.

TE-FOOD existence absorbed on nourishment track ability, delivers sole answers in agrarian industry.

It is lone the tractability answer that proposals dissimilar facilities B2B(Business-to-Business), B2C(Business-to-Consumer) and B2A (Business-to-Authorities), helping businesses, customers and establishments.

It shapes customer faith as they are talented to path the source of the nourishment creation counting all dispensation the creation experienced.

Due to good track ability and sensors, the nourishment creation that are dirty can be isolated at initial phase before it spreads to shop, plummeting numerous foodborne disease.

Controlling forms have actual viewpoint of the nourishment marketplace that assistance to recover food care regulatory nursing and implementation.

One of the curb TE-FOOD is that TFD symbolic that innings on the Ethereal net has little deal/second (15 TPS) which is comparatively slow. Likewise TE-FOOD has to expression a straight and unintended rivalry form dissimilar contestants businesses like: AMbrosus, WABI, MOD, WTC. Also TE-FOOD consuming big statistics of customers in Hungary and Vietnam, it is stressed to become contact in global marketplace.

5.1 Numerical application of TrFF TOPSIS method

The Array of HBL in Pakistan is dedicated to applying investment founded on large statistics. To understand suitable, protected, and real cross-border remittances, the Array of HBL in Pakistan means to choice the best blockchain skill supplier. A decision group connecting four DMs is shaped to select a suitable supplier from four before screened suppliers \((PS_{1} ,\,PS_{2} ,\,PS_{3}\) and \(PS_{4} ).\) The hierarchical evaluative criteria scheme to measure the presentation of the blockchain skill providers, which comprises four chief criteria ( \(GBU_{1} :\) technical close \(GBU_{2}\): produce and amenity, \(GBU_{3}\): economic issue, and \(GBU_{4}\): safety jeopardy). To source passable flexibility inside the assessment of the standards of the four criteria of each optional manufacturing, rulers are allowable to exploit TrFFNs.

Step 1: Describe the TrFF decision matrix.

Step 2: Describe the TrEFFWA operator and \(\xi = \left( {0.23,\,0.24,\,0.25,\,0.28} \right).\)

Step 3: To construct a normalized TrFF TOPSIS decision matrix.

Step 4: To construct the entropy weights.\(\begin{gathered} H_{1} = 0.0231,\,H_{2} = 0.3123,\, \hfill \\ H_{3} = 0.0109,\,H_{4} = 0.0088. \hfill \\ \end{gathered}\)

Step 5: To construct the weighted normalized TrFF TOPSIS decision matrix.

Step 6: To find positive TrFF ideal solution \(\left[ {\begin{array}{*{20}c} {[0.0099,} \\ {0.0091,} \\ {0.0121,} \\ {0.0130],} \\ {[0.0140,} \\ {0.0141,} \\ {0.0142,} \\ {0.039]} \\ \end{array} } \right],\,\left[ {\begin{array}{*{20}c} {[0.2172,} \\ {0.1248,} \\ {0.1123,} \\ {0.1189],} \\ {[0.1896,} \\ {0.1893,} \\ {0.1892,} \\ {0.0041]} \\ \end{array} } \right],\,\left[ {\begin{array}{*{20}c} {[0.0028,} \\ {0.0044,} \\ {0.0042,} \\ {0.0067],} \\ {[0.0065,} \\ {0.0066,} \\ {0.0066,} \\ {0.0014]} \\ \end{array} } \right],\,\left[ {\begin{array}{*{20}c} {[0.0036,} \\ {0.0037,} \\ {0.0035,} \\ {0.0055],} \\ {[0.0019,} \\ {0.0053,} \\ {0.0053,} \\ {0.0053]} \\ \end{array} } \right]\) To find negative TrFF ideal solution.

\(\left[ {\begin{array}{*{20}c} {[0.0032,} \\ {0.0037,} \\ {0.0035,} \\ {0.0036],} \\ {[0.0143,} \\ {0.0144,} \\ {0.0145,} \\ {0.0147]} \\ \end{array} } \right],\,\left[ {\begin{array}{*{20}c} {[0.0239,} \\ {0.0461,} \\ {0.0405,} \\ {0.0067],} \\ {[0.2151,} \\ {0.1945,} \\ {0.1961,} \\ {0.2141]} \\ \end{array} } \right],\,\left[ {\begin{array}{*{20}c} {[0.0016,} \\ {0.0017,} \\ {0.0019,} \\ {0.0004],} \\ {[0.0067,} \\ {0.0068,} \\ {0.0068,} \\ {0.0067]} \\ \end{array} } \right],\,\left[ {\begin{array}{*{20}c} {[0.0007,} \\ {0.0015,} \\ {0.0017,} \\ {0.0017],} \\ {[0.0055,} \\ {0.0056,} \\ {0.0055,} \\ {0.0056]} \\ \end{array} } \right]\) Step 7: To separate every candidate from + ve TrFF ideal solution. It is as under:\(\begin{gathered} q_{1}^{ + } = {0}{\text{.0655}},\,q_{2}^{ + } = 0.{1685},\, \hfill \\ q_{3}^{ + } = 0.0043,\,q_{4}^{ + } = 0.0047. \hfill \\ \end{gathered}\)

To separation each candidate from -ve TrFF ideal solution. It is as under:

$$ \begin{gathered} q_{1}^{ - } = 0.0053,\,q_{2}^{ - } = 0.{4734},\, \hfill \\ q_{3}^{ - } = 0.0052,\,q_{4}^{ - } = 0.{4843}. \hfill \\ \end{gathered} $$

Step 8: The closeness of solution of the problem under consideration is given by the following equations.

$$ \begin{gathered} Z_{1} = {0}{\text{.2531}},\,Z_{2} = {0}{\text{.5501}},\, \hfill \\ Z_{3} = {0}{\text{.9879}},\,Z_{4} = {0}{\text{.3974}}. \hfill \\ \end{gathered} $$

6 Comparison analysis

To favor and customary up feasibility of the optional plan, its comparison with another plan inferior upon IFSs (Atanassov and Gargov 1989) and FFWG statistics (Varma 2019) is showed with cases. Other plans are strange cases of our method plan that's originated on TrFFN to the similar descriptive situation.

6.1 IFN with existing technique

Step 1: IF decision matrix is Table 8 in given below (Table 9).

Table 8 IF decision
Table 9 IFWA operator

Step 2: Describe the IFWA operator and \((0.1,\,0.2,\,0.3,\,0.4).\)

Now applying given formula.

Step 3: The score function is

$$ \begin{gathered} \eta_{1} = 0.0555,\eta_{2} = - 0.0746, \hfill \\ \eta_{3} = 0.2133,\,\eta_{4} = 0.3765. \hfill \\ \end{gathered} $$

Step 4: Given the ranking \(\eta_{4} > \eta_{3} > \eta_{1} > \eta_{2}\) and the \(\eta_{4}\) is the best.

6.2 FF number with existing technique

Step 1: The FF decision matrix is given in Table 10.

Table 10 The FF decision matrix

Step 2: The FFWG operator and \((0.1,\,0.2,\,0.3,\,0.4).\)

Applying given formula.

Step 3: The score function is

$$ \begin{gathered} \eta_{1} = - 0.9996,\eta_{2} = - 0.9873, \hfill \\ \eta_{3} = - 0.9824,\,\eta_{4} = - 0.9111. \hfill \\ \end{gathered} $$

Step 4: Find the ranking \(\eta_{1} > \eta_{2} > \eta_{3} > \eta_{4}\) and \(\eta_{1}\) is the best.

Moreover, we discuss some experiments study to reinforce our claim of developing an improved framework for TrFF-based MCDA concerns.

In (Tang et al. 2019) and (Senapati and Yager 2020), the alternatives are ranked using the relative closeness coefficient and suitability index, respectively, between the overall value of the alternatives and the ideal alternative. This is not sufficient to conclude how good or bad an alternative is. In the TrFF TOPSIS method, the benefit and the cost criteria are both considered with proposed AOs on TrFFSs which comprise a more precise outcome compared with simply dealing with benefit or cost criteria. In the meantime, it increases the practicality of assessment data and the precision of outcomes as well.

The main benefit of the introduced TrFF model is capable of assessing any MCDA issues with uncertainty through TrFFNs as well as IFs and FFNs (Atanassov and Gargov 1989; Senapati and Yager 2019a) as described in the previous sections.

The proposed TrFF framework, which is utility or scoring degree-based model for MCDA, selects an option with the highest utility degree; therefore, the concern is how to assess the prior multi-criteria utility degree for an appropriate decision setting, whereas the extant models, which are compromise degree models, select an option which is nearest to the ideal solution.

The proposed TrFF technique is one of the robust and novel MCDA utility measuring methods. This framework is a combination of TrFFWA operator. The ranking of TrFF-TOPSIS technique is strengthening than PIS and NIS. The proposed method enables to reach the highest accurateness of assessment for utilizing the proposed approach for optimization of weighted AOs.

All the existing AOs utilize different operations on FFNs and IFS information, it is necessary to propose some neutral AOs about them due to that we are neutral in several issues and need to be treated fairly. Here, we have implemented TrEFFWA aggregation operators to get more reasonable outcomes.

7 Conclusion

The analysis part for valuation of data in a numerous existing extension of FSs is limited, making it hard to manage with the problem of evaluating blockchain technology under a complex condition, which is full of hesitations. To overcome this difficulty, we have employed a new generalization of FSs, called as TrFFNs. We define TrFF data based on decision-making. Define the Einstein aggregation operator and operational laws. We apply the Einstein operator on TrFF-TOPSIS technique, based on the TrFEFWA operator, a general framework has been developed for MAGDM with TrFF-TOPSIS data. The suggested TrFF -TOPSIS decision framework combines TrFF with TOPSIS in the assessment process, to determine the subjective weights of attributes and to evaluate blockchain strategies. To determine the normalized solution of every value, TrFF-TOPSIS uses PIS and NIS as the reference points for distance calculation, which will provide more potential information’s, realistic case study on the assessment of blockchain technology and demonstrate the practicality and validity of the suggested technique.

It is perceived that the method proposed in this work stretches a massive variety for articulating assessment data, which facilitates DEs in evaluating blockchain technology efficiently and flexibly. The experimental findings suggest that using our technique to establish the finalized blockchain technology selection scheme depending on the ranking outcomes is acceptable and adequate. Furthermore, the outcomes of an analysis of different approaches demonstrate that this approach has substantial benefits in terms of exploring multi-layer diverse relationships between attributes and reducing the influence of immense importance of assessment information (Tables 11, 12, 13, 14 and 15).

Table 11 FFWA operator
Table 12 When there are interrelationships among any manifold criteria: The diverse approaches can grip this state
Table 13 Comparison
Table 14 Different existing techniques
Table 15 Comparison table with existing method

This research also link-up a collection of data with a relationship, which indicated that the selection of simulated data is addressed in our suggested research. In future studies, we plan to extend the proposed structure further by introducing new characteristics, including the use of TrFF probabilistic aggregations. Additionally, we will discuss additional decision-making aspects like cluster analysis, performance analysis, sustainable city logistics, risk investment assessment, Wireless Sensor Networks, capital budgeting techniques, home buying process, and other domains under the uncertain environment.