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An Uncertain Multiple-Criteria Choice Method on Grounds of T-Spherical Fuzzy Data-Driven Correlation Measures

Article type: Research Article

Authors: Wang, Jih-Chang1 | Chen, Ting-Yu2; ∗

Affiliations: [1] Department of Information Management, College of Management, Chang Gung University, No. 259, Wenhua 1st Road, Guishan District, Taoyuan City 33302, Taiwan | [2] Department of Industrial and Business Management, Graduate Institute of Management, College of Management, Chang Gung University, No. 259, Wenhua 1st Rd., Guishan District, Taoyuan City 33302, Taiwan

Correspondence: [∗] Corresponding author.

Keywords: T-spherical fuzzy (T-SF) set, multiple-criteria choice method, correlation measure, T-SF comprehensive correlation index, location selection

DOI: 10.15388/22-INFOR500

Journal: Informatica, vol. 33, no. 4, pp. 857-899, 2022

Received January 2022
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Accepted November 2022
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Published: 2022
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Abstract

T-spherical fuzzy (T-SF) sets furnish a constructive and flexible manner to manifest higher-order fuzzy information in realistic decision-making contexts. The objective of this research article is to deliver an original multiple-criteria choice method that utilizes a correlation-focused approach toward computational intelligence in uncertain decision-making activities with T-spherical fuzziness. This study introduces the notion of T-SF data-driven correlation measures that are predicated on two types of the square root function and the maximum function. The purpose of these measures is to exhibit the overall desirability of choice options across all performance criteria using T-SF comprehensive correlation indices within T-SF decision environments. This study executes an application for location selection and demonstrates the effectiveness and suitability of the developed techniques in T-SF uncertain conditions. The comparative analysis and outcomes substantiate the justifiability and the strengths of the propounded methodology in pragmatic situations under T-SF uncertainties.

1Introduction

Multiple-criteria choice modelling under uncertainty forms part of the intelligent decision support system and can be applied to explore an innovative advancement of intelligent decision-making approaches and models (Fernández-Martínez and Sánchez-Lozano, 2021; Jing et al., 2021; Menekse and Camgoz-Akdag, 2022; Riaz et al., 2021). Numerous multiple-criteria assessment models have flourished to evaluate predetermined choice options ascertained from (conflicting) performance criteria for finding the most suitable option (Al-Quran, 2021; Erdogan et al., 2021; Kovač et al., 2021; Naeem et al., 2022). However, it is often troublesome and difficult to manipulate indistinct determinations and blurred assessments for quantifying performance ratings of the choice options in decision analysis processes within involuted and multiplex real-life environments (Al-Quran, 2021; Alsalem et al., 2021; Liu et al., 2021b; Menekse and Camgoz-Akdag, 2022; Oztaysi et al., 2022). When there is intricate uncertain information in the assessment and evaluation processes of choice options, the current decision-making approaches may be challenging to ascertain the performance ratings of choice options on performance criteria, which can result in an unreliable and unacceptable evaluation outcome concerning the most desirable scheme (Chinram et al., 2020; Cihat Onat, 2022; Jing et al., 2021; Liu et al., 2021a; Naeem et al., 2022).

To overcome these types of difficulties, fuzzy sets are capable of providing a supportable representation of imprecise information both beneficially and efficiently (Kovač et al., 2021; Liu et al., 2021a; Wang, 2021; Wang et al., 2021). In numerous realistic fields, fuzzy set theory has been generally accepted and recognized to conduct information modelling issues under uncertainty (Liu et al., 2021b; Wang et al., 2021). Nevertheless, ordinary fuzzy sets possess merely one membership function, which may be inadequate to fully expound the extent of uncertainty in the human cognition of things (Olugu et al., 2021; Wang, 2021). As a result, several high-order fuzzy sets, such as uncertain sets involving intuitionistic, Pythagorean, q-rung orthopair, picture, spherical, and T-spherical fuzziness, have been successively advanced to appropriately manifest human subjective uncertainties in practice (Chen, 2022a, 2022b; Liu et al., 2021c). In particular, the idea of T-spherical fuzzy (T-SF) sets, incipiently presented by Mahmood et al. (2019), can help bring the theoretical development and revolutionary implications according to its strengths of broadening the uncertain space via four parameters of impreciseness, thus composing favourable, neutral (so-called abstinence), unfavourable, and refusal evaluations (Alsalem et al., 2021; Chen, 2022c; Wang and Zhang, 2022; Yang and Pang, 2022).

1.1T-SF Theory in Uncertain Decision Contexts

T-SF sets generalize two uncertain sets on the grounds of the picture fuzzy configuration and the spherical fuzzy (SF) configuration. Picture fuzzy sets and SF sets were advocated by Cuong (2014) and Kahraman and Kutlu Gündoğdu (2018), respectively, and they are high-order mathematical constructions that are more general than ordinary fuzzy sets. Nonetheless, their membership functions are special types of membership functions of the T-SF structure. An illustration in Fig. 1 manifests some general variants of fuzzy sets involving four parameters. Herein, these parameters externalize four-dimensional membership functions consisting of a positive component (μ) for favourable evaluations, neutral component (η) for abstinence, negative component (ν) for unfavourable evaluations, and refusal component (γ) for refusal evaluations. The sum of μ, η, ν, and γ is equal to 1, which behaves as a prerequisite for the picture fuzzy configuration. The sum of μ2, η2, ν2, and γ2 is equal to 1, which indicates a prerequisite for the SF configuration. A positive integer q is placed where q∈Z+. The sum of μq, ηq, νq, and γq is equal to 1, which demonstrates a prerequisite for the T-SF configuration. When q=1 and q=2, the T-SF configuration transforms into the picture fuzzy configuration and the SF configuration, respectively, which provides substance to the generalization of T-SF theory (Chen, 2022a, 2022b). Moreover, in the event that η=0, the T-SF configuration transforms into the intuitionistic, Pythagorean, and q-rung orthopair fuzzy configurations when q=1,2, and q∈Z+. By expounding the membership functions in a much wider range, T-SF sets can give expression to ambiguity and hesitation contained in human opinions in an efficacious manner (Mahnaz et al., 2022; Nasir et al., 2021; Wang, 2021). Moreover, the parameters μ, η, ν, and γ are adequate and appropriate for managing human determinations and assessments and elucidating complicated uncertainties within a changeable and unpredictable decision-making environment.

Fig. 1

General variants of fuzzy sets involving four parameters.

General variants of fuzzy sets involving four parameters.

As of the advancement of T-SF theory in uncertain decision circumstances, a variety of valuable multiple-criteria assessment approaches and evaluation techniques have been constructed for facilitating intelligent decision support and aiding. By way of illustration, Abid et al. (2022) presented improved T-SF similarity measures to suggest an approach to decision-making and pattern recognition. Akram et al. (2022) analysed and addressed threats on social media platforms by employing an uncertain set of the complex cubic T-SF model and put forward a risk-assessing method for cyber-security and social media. By way of the interval-valued complex T-SF relation, Alothaim et al. (2022) identified Hasse diagrams in conformity with T-spherical partial orders to assess cybersecurity. Alsalem et al. (2021) expanded an opinion score-based technique and a fuzzy zero-inconsistency approach to T-SF contexts for implementing distribution decisions of the COVID-19 vaccine. Chen (2022a) instituted new notions of a superiority identifier and a guide index and propounded a T-SF regime prioritization procedure. Chen (2022b) advanced T-SF point operations to derive T-SF informational lower and upper estimations and propounded a point operator-driven method to treat complex assessment and evaluation tasks. By advocating a fresh distance measure with the Minkowski type, Chen (2022c) constructed Gaussian preference functions for conducting an evolved T-SF regime analysis. Nasir et al. (2021) investigated complex T-SF relations for depicting a global market’s time-related interdependence in international trades. Ullah et al. (2021) advanced a new Dijkstra algorithm within the environment of T-SF graphs for addressing the shortest path issue. Wang et al. (2022) launched similarity measures and relations in interval-valued T-SF contexts and investigated an approach to medical diagnostic issues. To execute image segmentation, Xian et al. (2021) based on bias correction to establish a spatial T-SF C-means model.

Table 1

State-of-the-art review of multiple-criteria assessment approaches in T-SF contexts.

ReferenceFuzzy modelMain proposed methodCore concept (or technique)
Abid et al. (2022)T-SF setApproach to decision-making and pattern recognitionSimilarity measure
Improved T-SF similarity measure
Akram and Martino (2022)T-SF soft rough setGroup decision-making approachT-SF soft rough average aggregation operation
Parameterized fuzzy modelling
Akram et al. (2022)Complex cubic T-SF setRisk-assessing method for cyber-security and social mediaCartesian product
Complex cubic T-SF relation
Threat-solving for a social media platform
Alothaim et al. (2022)Interval-valued complex T-SF setMethod of assessing cybersecurityInterval-valued complex T-SF relation
Hasse diagram of interval-valued complex T-spherical partial orders
Al-Quran (2021)T-spherical hesitant fuzzy setMultiple attribute decision-making methodOperational laws of T-spherical hesitant fuzzy information
Weighted (geometric) averaging operation
Alsalem et al. (2021)T-SF setFuzzy decision by opinion score methodFuzzy-weighted zero-inconsistency approach
Distribution decisions of COVID-19 vaccine
Chen (2022a)T-SF setT-SF regime I and II methodsSuperiority identifier
Guide index
Chen (2022b)T-SF setPoint operator-driven approachT-SF point operation for upper and lower estimations
Continuous ordered weighted average operation
Chen (2022c)T-SF setT-SF regime methodologyGaussian preference function
Minkowski-type distance measure
Joint generalized index
Chen et al. (2021)T-SF setGeneralized and group-generalized T-SF aggregation method(Group-)generalized T-SF geometric aggregation operation
Weighted, ordered weighted, and hybrid geometric operations
Gurmani et al. (2022)T-spherical hesitant fuzzy setBorder approximation area comparison approachT-spherical hesitant fuzzy structure with probability
Aggregation method in probabilistic T-spherical hesitant fuzzy settings
Hussain et al. (2022a)Interval-valued T-SF setMethod of assessing business proposalsFrank aggregation operation
Interval-valued T-SF Frank weighted averaging and geometric operations
Hussain et al. (2022b)T-SF setT-SF Aczel-Alsina aggregation methodAczel-Alsina t-(co)norm
T-SF Aczel-Alsina weighted average geometric operation
Karaaslan and Al-Husseinawi (2022)Hesitant T-SF setHesitant T-SF Dombi operation-based methodAggregation approach by way of Dombi operation
Hesitant T-spherical Dombi fuzzy aggregation operation
Khan et al. (2022)Complex T-SF setPerformance measurement methodPower aggregation operation
Complex T-SF power-weighted averaging and geometric operation
Liu et al. (2021c)Normal T-SF numberNormal T-spherical fuzzy aggregation methodMaclaurin symmetric (weighted) mean operation
Mahnaz et al. (2022)T-SF setT-SF Frank aggregation methodFrank t-(co)norm
Frank aggregation operation
T-SF entropy measure
Nasir et al. (2021)Complex T-SF setComplex T-SF relation methodTime-related interdependence of global markets
Interdependence of international trade
Ullah et al. (2021)T-SF setShortest path problem-solving methodDijkstra algorithm
Shortest path in T-SF network
Wang (2021)T-SF rough numberInteractive power Heronian mean operator approachInteraction operational law
Heronian mean operation
Power average operation
Wang and Zhang (2022)T-SF setInteraction power Heronian aggregation methodT-SF interaction power Heronian mean operation
Power averaging operation
Wang et al. (2022)Interval-valued T-SF setApproach to medical diagnosisInterval-valued T-SF relation
Similarity measure
Information measure
Xian et al. (2021)T-SF setSpatial T-SF C-means methodT-spherical fuzzification technology
T-SF C-means model with bias correction
Yang and Pang (2022)T-SF setMultiple attribute decision-making methodT-SF Dombi Bonferroni mean operation
T-SF entropy measure
Symmetric T-SF cross-entropy
Yang et al. (2021)T-SF setAssessment index system for digital transformation solutionsT-SF cloud
T-SF cloud (weighted) Heronian mean operations
Zedam et al. (2022)Complex T-SF setCleaner production evaluation methodComplex T-SF Hamacher weighted averaging operation
Complex T-SF Hamacher weighted geometric operation
Zeng et al. (2021)Complex T-spherical dual hesitant uncertain linguistic setMuirhead mean-based approach to enterprise informatization level evaluationLinguistic Muirhead mean operation
Uncertain linguistic weighted (dual) Muirhead mean operations in complex T-spherical dual hesitant settings

Over and above that, Akram and Martino (2022) delivered T-SF soft rough average aggregation operations and further put forward a proficient group decision-making approach. To attain considerable accuracy in expounding fuzziness and indeterminate data, Al-Quran (2021) brought about weighted (geometric) averaging operators within T-spherical hesitant fuzzy environments for decision aiding. Chen et al. (2021) unfolded generalized and group-generalized T-SF geometric aggregation operations (including (ordered) weighted and hybrid geometric operations) to support multiple-criteria assessments. Next, in the circumstances of probabilistic T-spherical hesitant ambiguity, Gurmani et al. (2022) initiated aggregation operators and advanced an extended approach for boundary approximation region comparison in treating group decision issues. In interval-valued T-SF circumstances, Hussain et al. (2022a) utilized Frank aggregation operators to propose a method of assessing business proposals. Hussain et al. (2022b) exploited Aczel-Alsina t-norms and t-conorms to evolve Aczel-Alsina weighted average and geometric operation in T-SF settings for resolving decision-making issues. Karaaslan and Al-Husseinawi (2022) presented arithmetic and geometric averaging operations in hesitant T-spherical Dombi fuzzy settings for group decision-making. Khan et al. (2022) employed power-weighted averaging and geometric operations in complex T-SF settings to suggest a performance measurement method under uncertainties. Liu et al. (2021c) explored Maclaurin symmetric (weighted) mean operators for normal T-SF numbers and utilized such operators for multiple-criteria decision assistance. Mahnaz et al. (2022) put forward T-SF Frank aggregation operators and utilized them to decide on an unknown preference structure. Wang (2021) came up with T-SF rough numbers for consideration to deliver interaction power Heronian mean operations to carry out collective decision analysis. Wang and Zhang (2022) propounded an interaction power Heronian aggregation method to handle T-SF decision information for decision aiding. Yang and Pang (2022) exploited T-SF entropy and symmetric T-SF cross-entropy measures for weight assessing and advocated T-SF Dombi Bonferroni mean operations for tackling multiple attribute decisions. Yang et al. (2021) launched T-SF cloud weighted Heronian mean operators to fuse evaluation information for digital transformation solutions. Zedam et al. (2022) advocated complex T-SF Hamacher weighted averaging and geometric operations and delivered an approach to cleaner production evaluation. Zeng et al. (2021) explored linguistic Muirhead mean operators to form an intricate decision involving complex T-spherical dual hesitant uncertainties.

Table 1 summarizes a recent review of multiple-criteria assessment and related literature, including specific fuzzy models in the T-SF and extended T-SF setting, the main proposed methods, and the core concepts (or techniques) of these studies. The aforementioned literature manipulates uncertain information in the T-SF configuration from various perspectives to support multiple-criteria assessment tasks. These studies also confirm that handling uncertain information in decision-making environments with the T-SF configuration is a correct and effective way to build a multiple-criteria evaluation method framework.

In particular, based on Table 1, it can be easily observed that many researchers discussed the modularization of multiple-criteria choice methods in the context of T-SF sets with aggregation operations or averaging (i.e. mean) operations, such as Akram and Martino (2022), Al-Quran (2021), Chen et al. (2021), Gurmani et al. (2022), Hussain et al. (2022a, 2022b), Karaaslan and Al-Husseinawi (2022), Khan et al. (2022), Liu et al. (2021c), Mahnaz et al. (2022), Wang (2021), Wang and Zhang (2022), Yang and Pang (2022), Yang et al. (2021), Zedam et al. (2022), and Zeng et al. (2021). That is, many of the above works of literature focus on models of aggregating or averaging operations, which belong to a measurement of the central tendency of a finite set of T-SF information. Nonetheless, they are still unable to reflect the relationship or correlation between T-SF characteristics performed by two available alternatives from the statistical point of view. Moreover, such models and methods may ignore the interrelationships between the two T-SF sets, and cannot precisely measure the degree of relationship or correlation between the two T-SF sets.

1.2Research Gap and Motivations

With the establishment of T-SF theory, the correlation coefficients for T-SF information attempt a solid grounding of multiple-criteria evaluation issues in the fields of decision analysis (Guleria and Bajaj, 2021; Ullah et al., 2020a). A correlation coefficient is one of the most commonly-used statistical notions to estimate linear relationships between quantitative objects (Özlü and Karaaslan, 2022; Riaz et al., 2021), and it is often used in statistical analysis or machine learning. Correlation coefficients in statistics can be negative or positive contingent upon the direction of two objects’ relationship and their values lie between −1 and 1. To expand the applicability of correlation coefficients, an extended definition can be carried out under SF and T-SF conditions (Guleria and Bajaj, 2021; Mahmood et al., 2021). However, in intricate uncertain circumstances, extracting a proper correlation coefficient between two T-SF sets (or SF sets) is nontrivial.

Ullah et al. (2020b) indicated that the correlation coefficients in the intuitionistic fuzzy framework and the picture fuzzy framework do not apply to some practical issues. Because of this, they propounded an innovative notion of correlation coefficients in T-SF settings that range from 0 to 1; moreover, they discussed the fitness of this new measurement in T-SF contexts. Due to its generality, Ullah et al. (2020b) brought forward a clustering algorithm and a multiple attribute evaluation algorithm in T-SF uncertain conditions. In what follows, Guleria and Bajaj (2021) propounded the notion concerning correlation coefficients between T-SF sets and explored their useful properties to analyse the practicality in uncertain real-world conditions. With two applications in pattern recognition and medically diagnostic cases, Guleria and Bajaj gave substance to the effectuality of their evolved correlation coefficients. Riaz et al. (2021) exploited the statistical notions of covariances and variances to evolve a new correlation coefficient for hybrid SF and m-polar fuzzy information. Mahmood et al. (2021) initiated SF cosine similarity measures and (weighted) correlation co-efficient of SF sets for tackling pattern recognition and medical diagnostic issues. Fan et al. (2022) exploited an approach via correlation coefficients and standard deviations to generate the attribute weights and then initiated a T-SF complex proportional assessment method. Liu and Wang (2022) employed an inter-criteria correlation approach to generate objective weights and then combined the subjective weights using a minimum total deviation method for supporting decision analysis. In a T-SF framework, Özlü and Karaaslan (2022) coped with T-spherical type-2 hesitant fuzzy uncertain data to investigate an extended version of correlation coefficients. The aforementioned literature shows the usefulness and practical value of correlation coefficients in managing T-SF uncertain assessment issues with multiple-criteria analysis.

Published findings in support of the advantage of correlation coefficients under SF and T-SF conditions have focused on the usefulness of managing uncertainty contained in compounded and complicated problems efficaciously. However, there are some motivational considerations in advocating the widespread development of correlation coefficients with the help of apposite multiple-criteria analysis in T-SF settings.

  • (1) Few studies have focused on advancing efficient and easy-to-use T-SF correlation measures for differentiating the prioritization relations of available choice options, which is the foremost motivation of this research.

  • (2) Relatively less exploration of correlation-focused measurements as a concept to directly exploit T-SF correlation coefficients when dealing with intricately uncertain information is the second motivation for this research.

  • (3) In the existing T-SF literature predicated on correlation coefficients, the anchored comparisons relative to the universal T-SF set and the null T-SF set were not incorporated into the specification of T-SF correlation-focused measurements, which serves as the third motivation of this research.

  • (4) Comparing T-SF characteristics with universal T-SF sets and null T-SF sets based on existing T-SF correlation measures should be helpful for promoting the construction of an effective and beneficial multiple-criteria selection model, which is the last motivation of our research.

1.3Research Objective and Contributions

The foremost purpose of this research is to construct a practical multiple-criteria choice method by virtue of a correlation-focused approach for facilitating computational intelligence in an uncertain decision analysis involving T-spherical fuzziness. This paper provides novel concepts of T-SF data-driven correlation measures for T-SF performance ratings based on statistical notions of weighted correlation coefficients in T-SF settings. An efficacious algorithmic procedure based on T-SF data-driven correlation measures and an advanced multiple-criteria choice model is propounded to prioritize available choice options for ascertaining the overall desirability of the performance criteria. The initiated approach is to use T-SF weighted informational energies and correlation functions to exactly establish the T-SF weighted correlation coefficients predicated on the “square root function” type and the “maximum function” type. This approach can model empirical data involving imprecision and ambiguity, which facilitates managing T-SF performance ratings in a befitting and effectual manner. Next, by aiming to receive the overall desirability across the criteria, this paper contributes the T-SF comprehensive correlation indices supported by two types of the square root function and the maximum function to identify the relative prioritization of choice options and decide on the most appropriate scheme. Furthermore, a real problem about location selection is demonstrated to illustrate befitting applications of the propounded methodology for verification. Depending on the investigation outcomes, the evolved methodology proves to be efficacious compared with other approaches.

This study makes some interesting contributions to intelligent decision-making practice. The principal contributions of this study are as follows:

  • (1) Through the development of new notions grounded in T-SF correlation coefficients, the evolved T-SF data-driven correlation measures mark a new phase in the advancement of current multiple-criteria choice methods.

  • (2) Based on the square root or maximum functions, a practical measurement of T-SF weighted correlation coefficients is presented to serve as a basis for multiple-criteria choice modelling.

  • (3) Considering anchored comparisons relative to the universal and null T-SF sets, this study delivers advantageous T-SF comprehensive correlation indices for prioritizing competing choice options.

  • (4) This research provides a practical application contribution in delineating a convenient-to-use procedural algorithm to facilitate intelligent decision support in uncertain circumstances. By exploiting realistic applications and comparisons, propounded techniques are considerably more robust and flexible as multiple-criteria tools than comparative approaches.

1.4Paper Organization

In the present work, Section 2 depicts several fundamental notions concerned with T-SF theory. Section 3 advocates some beneficial T-SF data-driven correlation measures and then propounds an efficacious multiple-criteria choice method for treating intricate decision information involving T-spherical fuzziness. Section 4 exploits the initiated techniques to manipulate a location selection issue for a construction company and then puts into effect a comparative study with other approaches. In the end, Section 5 finishes this research work with the main results, limitations, and future research avenues.

2Preliminary Definitions

This part presents an introductory description of T-SF sets and clarifies the relationships among picture fuzzy, SF, and T-SF sets. Throughout the article, the symbols μ, η, ν, and γ will denote four components of positive-, neutral- (i.e. so-called abstinence-membership), negative-, and refusal-membership, respectively, of a part or aspect in an initial universe to a fuzzy configuration.

Definition 1

Definition 1(Cuong, 2014; Kahraman and Kutlu Gündoğdu, 2018; Mahmood et al., 2019).

The symbol U signifies a universal set that is a finite nonempty set. Place three mappings μT,ηT,νT:U→[0,1]. Let T={⟨u,(μT(u),ηT(u),νT(u))⟩|u∈U} and q represent a generalized form of fuzzy sets and a positive integer, respectively; T is named:

  • 1. A picture fuzzy set in U if 0⩽μT(u)+ηT(u)+νT(u)⩽1 for each u;

  • 2. An SF set in U if 0⩽(μT(u))2+(ηT(u))2+(νT(u))2⩽1 for each u;

  • 3. A T-SF set in U if 0⩽(μT(u))q+(ηT(u))q+(νT(u))q⩽1 for each u.

Definition 2

Definition 2(Garg et al., 2018; Ullah et al., 2018).

Place a T-SF set T taking a single positive-integer parameter q in the universal set U. Let t(u) expound a triplet composed of μT(u), ηT(u), and νT(u), namely, t(u)=(μT(u),ηT(u),νT(u)). The triplet t(u) signifies a picture fuzzy number, an SF number, and a T-SF number when q=1, q=2, and q∈Z+, respectively, wherein Z+ represents a collection of positive integers.

Definition 3

Definition 3(Ullah et al., 2018; Mahmood et al., 2019).

Consider a T- SF number t(u)=(μT(u),ηT(u),νT(u)) contained in the T-SF set T. The degrees of refusal-membership γT(u) having relevance for t(u) are exactly delineated by 1−μT(u)−ηT(u)−νT(u), 1−(μT(u))2−(ηT(u))2−(νT(u))2, and 1−(μT(u))q−(ηT(u))q−(νT(u))qq when q=1, q=2, and q∈Z+, respectively.

Definition 4

Definition 4(Modified from Güner and Aygün (2022)).

Let T-SF(U) depict a collection of all T-SF sets delineated in a universal set U. Place T+∈T-SF(U) and T−∈T-SF(U), where T+={⟨u,(μT+(u),ηT+(u),νT+(u))⟩|u∈U} and T−={⟨u,(μT−(u),ηT−(u),νT−(u))⟩|u∈U}.

  • 1. T+ is named a universal T-SF set if T+={⟨u,(1,0,0)⟩|u∈U};

  • 2. T− is named a null T-SF set if T−={⟨u,(0,0,1)⟩|u∈U}.

Definition 5

Definition 5(Garg et al., 2018; Liu et al., 2019; Mahmood et al., 2019).

Concerning two T-SF sets T1∈T-SF(U) and T2∈T-SF(U) in the universal set U, it is recognized that T1={⟨u,(μT1(u),ηT1(u),νT1(u))⟩|u∈U} and T2={⟨u,(μT2(u),ηT2(u),νT2(u))⟩|u∈U}. Certain fundamental set operations are precisely stated in this manner:

  • 1. T1⊆T2 if μT1(u)⩽μT2(u), ηT1(u)⩽ηT2(u), and νT1(u)⩾νT2(u) for each u;

  • 2. T1=T2 if and only if T1⊆T2 and T2⊆T1;

  • 3. T1∪T2={⟨u,(max{μT1(u),μT2(u)},min{ηT1(u),ηT2(u)},min{νT1(u),νT2(u)})⟩|u∈U};

  • 4. T1∩T2={⟨u,(min{μT1(u),μT2(u)},min{ηT1(u),ηT2(u)},max{νT1(u),νT2(u)})⟩|u∈U};

  • 5. The complement of T1: (T1)c={⟨u,(νT1(u),ηT1(u),μT1(u))⟩|u∈U}.

Definition 6

Definition 6(Ju et al., 2021).

Give consideration to any three T-SF numbers t1(u)=(μT1(u),ηT1(u),νT1(u)), t2(u)=(μT2(u),ηT2(u),νT2(u)), and t(u)=(μT(u),ηT(u),νT(u)) associated with an element u in U. Place a real number α>0. Several operational laws for T-SF numbers are portrayed in this fashion:

  • 1. t1(u)⊕t2(u)=([1−(1−(μT1(u))q)·(1−(μT2(u))q)]1/q,[(1−(μT1(u))q)·(1−(μT2(u))q)−(1−(μT1(u))q−(ηT1(u))q)·(1−(μT2(u))q−(ηT2(u))q)]1/q,[(1−(μT1(u))q−(ηT1(u))q)·(1−(μT2(u))q−(ηT2(u))q)−(1−(μT1(u))q−(ηT1(u))q−(νT1(u))q)·(1−(μT2(u))q−(ηT2(u))q−(νT2(u))q)]1/q);

  • 2. t1(u)⊗t2(u)=([(1−(ηT1(u))q−(νT1(u))q)·(1−(ηT2(u))q−(νT2(u))q)−(1−(μT1(u))q−(ηT1(u))q−(νT1(u))q)·(1−(μT2(u))q−(ηT2(u))q−(νT2(u))q)]1/q,[(1−(νT1(u))q)·(1−(νT2(u))q)−(1−(ηT1(u))q−(νT1(u))q)·(1−(ηT2(u))q−(νT2(u))q)]1/q,[1−(1−(νT1(u))q)·(1−(νT2(u))q)]1/q);

  • 3. α⊙t(u)=([1−(1−(μT(u))q)α]1/q,[(1−(μT(u))q)α−(1−(μT(u))q−(ηT(u))q)α]1/q,[(1−(μT(u))q−(ηT(u))q)α−(1−(μT(u))q−(ηT(u))q−(νT(u))q)α]1/q);

  • 4. (t(u))α=([(1−(ηT(u))q−(νT(u))q)α−(1−(μT(u))q−(ηT(u))q−(νT(u))q)α]1/q,[(1−(νT(u))q)α−(1−(ηT(u))q−(νT(u))q)α]1/q,[1−(1−(νT(u))q)α]1/q).

3Developed Methodology

The purpose of this section is to use effectual T-SF data-driven correlation measures and establish a novel multiple-criteria choice method for manipulating an intricate decision-making issue involving T-spherical fuzziness.

3.1Problem Description

This subsection concerns the formulation regarding a selection problem raised for multiple-criteria assessments and resolutions.

Making allowance for a multiple-criteria choice issue, let A={a1,a2,…,am} and C={c1,c2,…,cn} set forth two limited sets of choice options and performance criteria, respectively, in which the cardinal numbers m,n⩾2. In connection to each performance criterion cj∈C, place the normalized (standardized) weight wj∈[0,1] with the conditioning of weight normalization, i.e. ∑j=1nwj=1. The set C is compartmentalized into the collection of positive (performance) criteria CPo and the collection of negative (performance) criteria CNe. Herein, CPo∩CNe=∅ and CPo∪CNe=C. Positive criteria (such as profit and productivity) refer to the performance attribute cj∈CPo with a positive quality of being desirable from the decision-maker’s viewpoint. More specifically, their higher levels are more favourable from the decision-maker’s position. Negative criteria (such as cost and loss) refer to the performance attribute cj∈CNe with a negative quality of being desirable in line with the decision-maker’s attitude, which indicates that their lower levels are more favourable from the decision-maker’s position.

Multiple-criteria choice models portray decision-makers’ considered evaluations as T-SF numbers of their assessments of the choice options’ prominent features. On grounds of previous experience, knowledge, technical expertise, and appraisal perceptions, the performance ratings related to each choice option about a specific criterion are established after that the decision-maker has established the performance criteria for evaluating the choice options available. Let a T-SF number tij=(μij,ηij,νij) involving a positive-integer exponent q signify a performance rating concerning an alternative ai∈A having relevance for a specified criterion cj∈C (=CPo∪CNe), where the prerequisite 0⩽(μij)q+(ηij)q+(νij)q⩽1 must be fulfilled. In what follows, the degree of refusal-membership is calculated as γij=1−(μij)q−(ηij)q−(νij)qq. By collecting the T-SF performance rating tij of ai across all criteria in C, the T-SF characteristic Ti is formed using this fashion:

(1)
Ti={⟨cj,tij⟩|cj∈C}={⟨cj,(μij,ηij,νij)⟩|cj∈C(=CPo∪CNe)}.

3.2T-SF Data-Driven Correlation Measures

This subsection undertakes several moves to delineate relevant notions of the evolved correlation measures in the T-SF setting and then investigates their valuable features.

Definition 7.

Place the best choice option a+ and the worst choice option a− in a multiple-criteria choice problem. In view of the collections CPo (involving positive criteria) and CNe (involving negative criteria), the T-SF characteristics T+ and T− possessed by a+ and a−, respectively, are represented by way of the concepts of universal T-SF sets and null T-SF sets in this fashion:

  • 1. T+={⟨cj,t+j⟩|cj∈C}={⟨cj,(μ+j,η+j,ν+j)⟩|cj∈C}={⟨cj,(1,0,0)⟩|cj∈CPo,⟨cj,(0,0,1)⟩|cj∈CNe};

  • 2. T−={⟨cj,t−j⟩|cj∈C}={⟨cj,(μ−j,η−j,ν−j)⟩|cj∈C}={⟨cj,(0,0,1)⟩|cj∈CPo,⟨cj,(1,0,0)⟩|cj∈CNe}.

Definition 8.

Considering the normalized (standardized) weight wj and the T-SF characteristic Ti, let TiW state the T-SF weighted characteristic that contains the T-SF weighted performance rating tijw=(μijw,ηijw,νijw). Herein, tijw=(n·wj)⊙tij, where the number of criteria n epitomizes a role of a balancing coefficient. TiW and tijw are elucidated along these lines:

(2)
TiW={⟨cj,tijw⟩|cj∈C}={⟨cj,(μijw,ηijw,νijw)⟩|cj∈C},
(3)
tijw=(n·wj)⊙tij=([1−(1−(μij)q)n·wj]1/q,tijw=[(1−(μij)q)n·wj−(1−(μij)q−(ηij)q)n·wj]1/q,tijw=[(1−(μij)q−(ηij)q)n·wj−(1−(μij)q−(ηij)q−(νij)q)n·wj]1/q).

Theorem 1.

Consider the T-SF characteristic Ti containing the T-SF performance rating tij. When wj=1/n for each performance criterion cj, the T-SF weighted performance rating tijw=tij, and the T-SF weighted characteristic TiW=Ti.

Proof.

With the assistance of Definition 7, it is obtained that tijw=(n·wj)⊙tij=[n·(1/n)]⊙tij=tij, which bring about TiW=Ti straightforwardly. The theorem is proved.  □

Theorem 2.

In consideration of the best choice option a+ and the worst choice option a−, their corresponding T-SF weighted characteristics T+W=T+ and T−W=T− regardless of the values of the weight wj for all performance criteria in C.

Proof.

The T-SF weighted performance rating t+jw connected with the best choice option a+ on a positive criterion cj∈CPo is derived by: t+jw=(n·wj)⊙t+j=([1−(1−1q)n·wj]1/q,[(1−1q)n·wj−(1−1q−0q)n·wj]1/q,[(1−1q−0q)n·wj−(1−1q−0q−0q)n·wj]1/q)=(1,0,0). Next, in what follows, the t+jw of a+ on a negative criterion cj∈CNe is calculated like this: t+jw=([1−(1−0q)n·wj]1/q,[(1−0q)n·wj−(1−0q−0q)n·wj]1/q,[(1−0q−0q)n·wj−(1−0q−0q−1q)n·wj]1/q)=(0,0,1). Therefore, T+W={⟨cj,(1,0,0)⟩|cj∈CPo,⟨cj,(0,0,1)⟩|cj∈CNe⟩}=T+. Analogously, it can be acquired that T−W=T−. The theorem is proved.  □

Ullah et al. (2020a) conquered the non-appositeness limitation of correlation measurements in intuitionistic fuzzy settings or picture fuzzy settings to advance new correlation coefficients within T-SF environments. They put forward the notions of informational energies and correlation functions to exploit new correlation coefficients for T-SF information. By the same token, Guleria and Bajaj (2021) advocated the identical delineation of statistical correlation measurements in T-SF uncertain conditions. In the light of the correlation measures propounded by Guleria and Bajaj (2021) and Ullah et al. (2020b), this paper incorporates the T-SF weighted characteristics TiW, T+W, and T−W into the elucidation of correlation-focused measurements and evolves useful T-SF data-driven correlation measures for facilitating the constitution of an efficacious multiple-criteria choice model.

Definition 9.

In consideration of the T-SF weighted characteristic TiW={⟨cj,(μijw,ηijw,νijw)⟩|cj∈C} (with the refusal-membership γijw=1−(μijw)q−(ηijw)q−(νijw)qq), its T-SF weighted informational energy is expounded such that:

(4)
IE(TiW)=∑j=1n[((μijw)q)2+((ηijw)q)2+((νijw)q)2+((γijw)q)2].

Theorem 3.

The T-SF weighted informational energies IE(TiW), IE(T+W), and IE(T−W) satisfy the following favourable features:

  • 1. 0⩽IE(TiW)⩽n;

  • 2. IE(T+W)=n;

  • 3. IE(T−W)=n.

Proof.

Supported by the axiomatic condition of T-SF sets, it is recognized that (μijw)q+(ηijw)q+(νijw)q+(γijw)q=1, which readily gives rise to 0⩽((μijw)q)2+((ηijw)q)2+((νijw)q)2+((γijw)q)2⩽1. In consequence, the outcome 0⩽IE(TiW)=∑j=1n[((μijw)q)2+((ηijw)q)2+((νijw)q)2+((γijw)q)2]⩽n can be effortlessly confirmed. Next, in conformity with Theorem 2, it is acquainted with T+W=T+ and T−W=T−, which bring about ((μ+jw)q)2+((η+jw)q)2+((ν+jw)q)2+((γ+jw)q)2=1 and ((μ−jw)q)2+((η−jw)q)2+((ν−jw)q)2+((γ−jw)q)2=1, respectively. Under the circumstances, one can corroborate the consequences of IE(T+W)=∑j=1n1=n and IE(T−W)=∑j=1n1=n. The theorem is proved.  □

Definition 10.

Given the T-SF weighted characteristics TiW, T+W, and T−W, the respective T-SF weighted correlation functions of TiW relative to T+W and T−W are elucidated by:

(5)
CF(TiW,T+W)=∑j=1n[(μijw)q·(μ+jw)q+(ηijw)q·(η+jw)q+(νijw)q·(ν+jw)q+(γijw)q·(γ+jw)q]=∑cj∈CPo(μijw)q+∑cj∈CNe(νijw)q,
(6)
CF(TiW,T−W)=∑j=1n[(μijw)q·(μ−jw)q+(ηijw)q·(η−jw)q+(νijw)q·(ν−jw)q+(γijw)q·(γ−jw)q]=∑cj∈CPo(νijw)q+∑cj∈CNe(μijw)q.

Theorem 4.

The T-SF weighted correlation functions CF(TiW,T+W) and CF(TiW,T−W) fulfill the following favourable features:

  • 1. 0⩽CF(TiW,T+W)⩽n and 0⩽CF(TiW,T−W)⩽n;

  • 2. CF(TiW,T+W)=CF(T+W,TiW) and CF(TiW,T−W)=CF(T−W,TiW);

  • 3. CF(T+W,T−W)=CF(T−W,T+W)=0;

  • 4. CF(TiW,TiW)=IE(TiW);

  • 5. CF(T+W,T+W)=n and CF(T−W,T−W)=n.

Proof.

Firstly, let nPo and nNe represent the numbers of criteria in CPo and CNe, respectively, where nPo+nNe=n. It is apparent that 0⩽∑cj∈CPo(μijw)q⩽nPo, 0⩽∑cj∈CNe(νijw)q⩽nNe, 0⩽∑cj∈CPo(νijw)q⩽nPo, and 0⩽∑cj∈CNe(μijw)q⩽nNe. Thus, 0⩽CF(TiW,T+W)⩽nPo+nNe=n, and 0⩽CF(TiW,T−W)⩽nPo+nNe=n. The properties in part 1 are confirmed. The commutative properties in part 2 are straightforward. Next, it demonstrates the correctness of CF(T+W,T−W)=∑cj∈CPo(0)q+∑cj∈CNe(0)q=0, which corroborates the property in part 3. In what follows, it can be effortlessly deduced that CF(TiW,TiW)=IE(TiW) for the reason that CF(TiW,TiW)=∑j=1n[(μijw)q·(μijw)q+(ηijw)q·(ηijw)q+(νijw)q·(νijw)q+(γijw)q·(γijw)q]=∑j=1n[((μijw)q)2+((ηijw)q)2+((νijw)q)2+((γijw)q)2]=IE(TiW); accordingly, it is manifested that CF(T+W,T+W)=IE(T+W)=n and CF(T−W,T−W)=IE(T−W)=n, which demonstrates the truth of the properties in parts 4 and 5. The theorem is proved.  □

Definition 11.

Making allowance for TiW, T+W, and T−W, the respective T-SF weighted correlation coefficients of TiW relative to T+W and T−W based on the “square root function” type are delineated along these lines:

(7)
CC√(TiW,T+W)=CF(TiW,T+W)IE(TiW)·IE(T+W)=∑cj∈CPo(μijw)q+∑cj∈CNe(νijw)qn·∑cj∈C[((μijw)q)2+((ηijw)q)2+((νijw)q)2+((γijw)q)2],
(8)
CC√(TiW,T−W)=CF(TiW,T−W)IE(TiW)·IE(T−W)=∑cj∈CPo(νijw)q+∑cj∈CNe(μijw)qn·∑cj∈C[((μijw)q)2+((ηijw)q)2+((νijw)q)2+((γijw)q)2].

Theorem 5.

Through the utility of the “square root function” type, the T-SF weighted correlation coefficients CC√(TiW,T+W) and CC√(TiW,T−W) fulfill some favourable features:

  • 1. 0⩽CC√(TiW,T+W)⩽1 and 0⩽CC√(TiW,T−W)⩽1;

  • 2. CC√(TiW,T+W)=CC√(T+W,TiW) and CC√(TiW,T−W)=CC√(T−W,TiW);

  • 3. CC√(T+W,T−W)=CC√(T−W,T+W)=0;

  • 4. CC√(TiW,T+W)=1 and CC√(TiW,T−W)=1 if and only if TiW=T+W and TiW=T−W, respectively;

  • 5. CC√(TiW,T+W)=0 and CC√(TiW,T−W)=0ifTiW=T−W and TiW=T+W, respectively.

Proof.

Following Definition 9, the T-SF weighted informational energies of TiW and T+W are given in this fashion: ∑j=1n[((μijw)q)2+((ηijw)q)2+((νijw)q)2+((γijw)q)2] and ∑j=1n[((μ+jw)q)2+((η+jw)q)2+((ν+jw)q)2+((γ+jw)q)2], respectively. The Cauchy–Schwarz inequality is regarded as one of the most celebrated inequalities in mathematics. Its connotative meaning refers to (ι1β1+ι2β2+⋯+ιnβn)2⩽((ι1)2+(ι2)2+⋯+(ιn)2)·((β1)2+(β2)2+⋯+(βn)2) for the real number sequences (ι1,ι2,…,ιn) and (β1,β2,…,βn). Through the utility of the Cauchy–Schwarz inequality, the subsequent consequence can be yielded:

(CF(TiW,T+W))2=(∑j=1n[(μijw)q·(μ+jw)q+(ηijw)q·(η+jw)q+(νijw)q·(ν+jw)q+(γijw)q·(γ+jw)q])2⩽[((μi1w)q)2+((ηi1w)q)2+((νi1w)q)2+((γi1w)q)2+((μi2w)q)2+((ηi2w)q)2+((νi2w)q)2+((γi2w)q)2+⋯+((μinw)q)2+((ηinw)q)2+((νinw)q)2+((γinw)q)2]·[((μ+1w)q)2+((η+1w)q)2+((ν+1w)q)2+((γ+1w)q)2+((μ+2w)q)2+((η+2w)q)2+((ν+2w)q)2+((γ+2w)q)2+⋯+((μ+nw)q)2+((η+nw)q)2+((ν+nw)q)2+((γ+nw)q)2]=∑j=1n[((μijw)q)2+((ηijw)q)2+((νijw)q)2+((γijw)q)2]·∑j=1n[((μ+jw)q)2+((η+jw)q)2+((ν+jw)q)2+((γ+jw)q)2]=IE(TiW)·IE(T+W).
Using this as a basis, we draw the inference that CF(TiW,T+W)⩽IE(TiW)·IE(T+W); thus, CF(TiW,T+W)/IE(TiW)·IE(T+W)⩽1. Because CF(TiW,T+W),IE(TiW),IE(T+W)⩾0, it can be generated that 0⩽CC√(TiW,T+W)⩽1. By the same token, the correctness of 0⩽CC√(TiW,T−W)⩽1 can be proven; thus, the properties in part 1 are validated. The commutative properties in part 2 are straightforward. The property in part 3 is trivially known because CF(T+W,T−W)=CF(T−W,T+W)=0. Regarding the necessity in part 4, the presupposition CC√(TiW,T+W)=1 indicates that (CF(TiW,T+W))2=IE(TiW)·IE(T+W)=IE(TiW)·n, which follows that CF(TiW,T+W)=CF(T+W,T+W)=IE(T+W)=n must be fulfilled. Thus, TiW=T+W. Concerning the sufficiency in part 4, the prerequisite TiW=T+W brings about CC√(T+W,T+W)=CF(T+W,T+W)/IE(T+W)·IE(T+W)=IE(T+W)/IE(T+W)=1. Accordingly, it is received that CC√(TiW,T+W)=1 if and only if TiW=T+W. Analogously, one has CC√(TiW,T−W)=1 if and only if TiW=T−W. Therefore, the properties in part 4 are verified. In part 5, the prerequisite TiW=T−W gives rise to μijw=0 and νijw=0 for cj∈CPo and cj∈CNe, respectively. By virtue of Definition 10, one obtains CF(TiW,T+W)=∑cj∈CPo(μijw)q+∑cj∈CNe(νijw)q=0, which leads to the conclusion that CC√(TiW,T+W)=0/IE(TiW)·IE(T+W)=0. In contrast, the prerequisite TiW=T+W brings about νijw=0 and μijw=0 for cj∈CPo and cj∈CNe, respectively. In the light of Definition 10, one receives CF(TiW,T−W)=∑cj∈CPo(νijw)q+∑cj∈CNe(μijw)q=0, which lets us deduce CC√(TiW,T−W)=0/IE(TiW)·IE(T−W)=0. Thus, one can corroborate that CC√(TiW,T+W)=0 if TiW=T−W; moreover, CC√(TiW,T−W)=0 if TiW=T+W. The theorem is proved.  □

Definition 12.

Making allowance for TiW, T+W, and T−W, the respective T-SF weighted correlation coefficients of TiW relative to T+W and T−W based on the “maximum function” type are delineated along these lines:

(9)
CC∧(TiW,T+W)=CF(TiW,T+W)max{IE(TiW),IE(T+W)}=∑cj∈CPo(μijw)q+∑cj∈CNe(νijw)qmax{∑cj∈C[((μijw)q)2+((ηijw)q)2+((νijw)q)2+((γijw)q)2],n}=∑cj∈CPo(μijw)q+∑cj∈CNe(νijw)qn,
(10)
CC∧(TiW,T−W)=CF(TiW,T−W)max{IE(TiW),IE(T−W)}=∑cj∈CPo(νijw)q+∑cj∈CNe(μijw)qmax{∑cj∈C[((μijw)q)2+((ηijw)q)2+((νijw)q)2+((γijw)q)2],n}=∑cj∈CPo(νijw)q+∑cj∈CNe(μijw)qn.

Theorem 6.

Through the utility of the “maximum function” type, the T-SF weighted correlation coefficients CC∧(TiW,T+W) and CC∧(TiW,T−W) fulfill some favourable features:

  • 1. 0⩽CC∧(TiW,T+W)⩽1 and 0⩽CC∧(TiW,T−W)⩽1;

  • 2. CC∧(TiW,T+W)=CC∧(T+W,TiW) and CC∧(TiW,T−W)=CC∧(T−W,TiW);

  • 3. CC∧(T+W,T−W)=CC∧(T−W,T+W)=0;

  • 4. CC∧(TiW,T+W)=1 and CC∧(TiW,T−W)=1 if and only if TiW=T+W and TiW=T−W, respectively;

  • 5. CC∧(TiW,T+W)=0 and CC∧(TiW,T−W)=0 if TiW=T−W and TiW=T+W, respectively.

Proof.

Firstly, the proofs of parts 2, 3, and 5 are like the proving processes in parts 2, 3, and 5 of Theorem 5. In part 1, as analogous to the proof in Theorem 5, it is recognized that (CF(TiW,T+W))2⩽IE(TiW)·IE(T+W), which gives substance to CF(TiW,T+W)⩽max{IE(TiW),IE(T+W)}. Accordingly, CF(TiW,T+W)/max{IE(TiW),IE(T+W)}⩽1. Because CF(TiW,T+W),IE(TiW),IE(T+W)⩾0, we can state that 0⩽CC∧(TiW,T+W)⩽1. Similarly, one has 0⩽CC∧(TiW,T−W)⩽1. The correctness of the properties in part 1 is confirmed. Concerning the necessity in part 4, the presupposition CC∧(TiW,T+W)=1 implies that CF(TiW,T+W)=max{IE(TiW),IE(T+W)}=max{IE(TiW),n}=n; on account of this, TiW=T+W. For the sufficiency in part 4, the prerequisite TiW=T+W gives rise to CC∧(T+W,T+W)=CF(T+W,T+W)/max{IE(T+W),IE(T+W)}=IE(T+W)/IE(T+W)=1. Therefore, it is acquired that CC∧(TiW,T+W)=1 if and only if TiW=T+W. It is known, just the same, that CC∧(TiW,T−W)=1 if and only if TiW=T−W. As a result, the properties in part 4 are verified. The theorem is proved.  □

3.3Propounded Multiple-Criteria Choice Method in T-SF Settings

This subsection attempts to propound an effective and simple-to-implement approach for tackling an uncertain multiple-criteria evaluation issue predicated on the evolved T-SF data-driven correlation measures.

Consider a multiple-criteria choice task embodying the T-SF characteristic Ti and the normalized (standardized) weight wj of an available choice option ai∈A and a performance criterion cj∈C, respectively. Place an anchoring parameter ξ∈[0,1]. For each T-SF characteristic, the parameter ξ elucidates the weight of the anchored comparisons relative to universal T-SF sets, while (1−ξ) depicts the weight of the anchored comparisons relative to null T-SF sets. In what follows, this study contributes two constructive T-SF comprehensive correlation indices as the measurements of deciding the relative prioritization for available choice options.

Definition 13.

Denote CC_√+=mini′=1mCC√(Ti′W,T+W), CC‾√+=maxi′=1mCC√(Ti′W,T+W), CC_√−=mini′=1mCC√(Ti′W,T−W), and CC‾√−=maxi′=1mCC√(Ti′W,T−W) for the “square root function” type. Denote CC_∧+=mini′=1mCC∧(Ti′W,T+W), CC‾∧+=maxi′=1mCC∧(Ti′W,T+W), CC_∧−=mini′=1mCC∧(Ti′W,T−W), and CC‾∧−=maxi′=1mCC∧(Ti′W,T−W) for the “maximum function” type. By the agency of CC√ and CC∧ on ai, the T-SF comprehensive correlation indices CI√(ai) and CI∧(ai), respectively, are delineated along these lines:

(11)
CI√(ai)=ξ·CC√(TiW,T+W)−CC_√+CC‾√+−CC_√++(1−ξ)·CC‾√−−CC√(TiW,T−W)CC‾√−−CC_√−,
(12)
CI∧(ai)=ξ·CC∧(TiW,T+W)−CC_∧+CC‾∧+−CC_∧++(1−ξ)·CC‾∧−−CC∧(TiW,T−W)CC‾∧−−CC_∧−.

Theorem 7.

The T-SF comprehensive correlation indices CI√(ai) and CI∧(ai) fulfill the following favoutirable features:

  • 1. 0⩽CI√(ai)⩽1 and 0⩽CI∧(ai)⩽1;

  • 2. CI√(ai)=1 and CI∧(ai)=1 for all ξ∈[0,1] if Ti=T+;

  • 3. CI√(ai)=0 and CI∧(ai)=0 for all ξ∈[0,1] if Ti=T−.

Proof.

Utilizing the foregoing delineation, it is realized that CC√(TiW,T+W)⩽CC‾√+ (=maxi′=1mCC√(Ti′W,T+W)), which follows that CC√(TiW,T+W)−CC_√+⩽CC‾√+−CC_√+, thereby gaining (CC√(TiW,T+W)−CC_√+)/(CC‾√+−CC_√+)⩽1. It is apparent to observe that CC√(TiW,T+W)−CC_√+⩾0 and CC‾√+−CC_√+⩾0 for the reason that CC√(TiW,T+W)⩾mini′=1mCC√(Ti′W,T+W) and maxi′=1mCC√(Ti′W,T+W)⩾mini′=1mCC√(Ti′W,T+W). Accordingly, one has 0⩽(CC√(TiW,T+W)−CC_√+)/(CC‾√+−CC_√+)⩽1. In a similar fashion, 0⩽(CC‾√−−CC√(TiW,T−W))/(CC‾√−−CC_√−)⩽1. Taking 0⩽ξ⩽1 into consideration, it is deduced that 0⩽CI√(ai)⩽1. By the same token, one has 0⩽CI∧(ai)⩽1, which demonstrates the truth of the properties in part 1. Next, it is realized that T+W=T+ and T−W=T− based on Theorem 2. The prerequisite Ti=T+(=T+W) brings about CC√(TiW,T+W)=1 and CC‾√+=1, which indicates that (1−CC_√+)/(1−CC_√+)=1. Moreover, the condition Ti=T+(=T+W) leads to CC√(TiW,T−W)=0 and CC_√−=0, which indicates that (CC‾√−−0)/(CC‾√−−0)=1. From this basis, it is obtained that CI√(ai)=ξ·1+(1−ξ)·1=1 for all ξ∈[0,1]. The correctness of CI∧(ai)=1 is analogously corroborated, which produces proof of part 2. The properties in part 3 are verified similarly. The theorem is proved.  □

Definition 14.

Given two choice options ai and ai′ involving T-SF characteristics Ti and Ti′, respectively, the prioritization procedure of ai and ai′ can be elucidated using the subsequent relations “≻√” (indicating “better than”), “∼√” (indicating “indefinite or indifferent”), and “≺√” (indicating “worse than”) (or “≻∧”, “∼∧”, and “≺∧”), like this:

  • 1. Based on the “square root function” type:

    • a) If CI√(ai)>CI√(ai′), then it is convinced that ai≻√ai′;

    • b) If CI√(ai)=CI√(ai′), then ai∼√ai′;

    • c) If CI√(ai)<CI√(ai′), then it is convinced that ai≺√ai′.

  • 2. Based on the “maximum function” type:

    • a) If CI∧(ai)>CI∧(ai′), then it is convinced that ai≻∧ai′;

      Fig. 2

      The framework of the propounded methodology.

      The framework of the propounded methodology.

    • b) If CI∧(ai)=CI∧(ai′), then ai∼∧ai′;

    • c) If CI∧(ai)<CI∧(ai′), then it is convinced that ai≺∧ai′.

The framework of the propounded multiple-criteria choice method on grounds of T-SF data-driven correlation measures is depicted in Fig. 2. As exhibited in this framework, the evolved methodology comprises four phases, i.e. the organization of a multiple-criteria choice issue in Phase I, the computation of weighted performance information with T-SF sets in Phase II, the generation of T-SF data-driven correlation measures in Phase III, and decision making for treating multiple-criteria choice analysis in Phase IV.

To implement the propounded methodology, this study provides a new algorithm to perform the procedural steps pragmatically in order to facilitate the decision-maker’s multiple-criteria analysis. The following algorithm is expressed using a sequence of simple operations (consisting of Steps 1 and 2 in Phase I, Steps 3–5 in Phase II, Steps 6–8 in Phase III, and Steps 9 and 10 in Phase IV) for conducting the initiated multiple-criteria choice method with T-SF data-driven correlation measures:

Step 1. Place a limited set of choice options A={a1,a2,…,am} and a limited set of performance criteria C={c1,c2,…,cn}. Separate C into two parts: one is the collection of positive criteria CPo; the other is the collection of negative criteria CNe.

Step 2. Generate the normalized (standardized) weight wj with the conditioning of weight normalization for each performance criterion cj.

Step 3. Specify a suitable positive-integer exponent q and form a T-SF performance rating tij signified as the T-SF number (μij,ηij,νij) with the refusal-membership γij.

Step 4. Assemble the T-SF characteristic Ti in Eq. (1) by gathering all T-SF performance rating tij regarding a choice option ai across all criteria in C.

Step 5. Employ Eq. (3) to derive the T-SF weighted performance rating tijw (with refusal-membership γijw) for the sake of framing the T-SF weighted characteristic TiW in Eq. (2).

Step 6. Utilize the universal and null T-SF sets to signify the T-SF characteristics T+ and T− for the best choice option a+ and the worst choice option a−, respectively. Moreover, the T-SF weighted characteristics T+W=T+ and T−W=T−.

Step 7. Derive the T-SF weighted informational energy IE(TiW) using Eq. (4) and the T-SF weighted correlation functions CF(TiW,T+W) and CF(TiW,T−W) using Eqs. (5) and (6), respectively.

Step 8. Proceed to either Step 8-1 or Step 8-2.

Step 8-1. Use the “square root function” type to produce the T-SF weighted correlation coefficients CC√(TiW,T+W) and CC√(TiW,T−W) using Eqs. (7) and (8), respectively.

Step 8-2. Exploit the “maximum function” type to produce the T-SF weighted correlation coefficients CC∧(TiW,T+W) and CC∧(TiW,T−W) using Eqs. (9) and (10), respectively.

Step 9. Assign an anchoring parameter ξ to determine the T-SF comprehensive correlation index CI√(ai) (or CI∧(ai)) using Eq. (11) (or Eq. (12)).

Step 10. Rank the m choice options in A supported by CI√(ai) (or CI∧(ai)) in descending order to identify the prioritization relations “≻√”, “∼√”, and “≺√” (or “≻∧”, “∼∧”, and “≺∧”). Make a final decision for completing the multiple-criteria choice task.

4Practical Application and Comparative Research

This section intends to exemplify the functionality and suitability of the propounded methodology for applications in a location selection issue for a construction company in complex uncertain circumstances. Moreover, this section puts into effect two comparative studies to scrutinize the helpfulness and merits of the current technique.

4.1Realistic Application and Discussions

The multiple-criteria choice case investigated by Chen et al. (2021) focused on the issue of a construction company finding an appropriate location to put up a new apartment. In order to find the most suitable location, the construction company evaluates four location options (a1−a4) for constructing new apartments predicated on the four performance criteria. The performance criteria consist of land cost (c1), surrounding environment (c2), technological capability (c3), and lease value (c4). Fig. 3 provides a profile of the location selection issue under study.

Fig. 3

Profile of the location selection issue of a construction company for building new apartments.

Profile of the location selection issue of a construction company for building new apartments.

In Step 1, the two limited sets of choice options and performance criteria were designated as A={a1,a2,a3,a4} and C={c1,c2,c3,c4}, respectively. Herein, the set C was separated into two parts: one is the collection of positive criteria CPo={c2,c3}; the other is the collection of negative criteria CNe={c1,c4}. In Step 2, in conformity with the expert’s professional opinions, the normalized (standardized) weights were given by (w1,w2,w3,w4)=(0.2,0.1,0.3,0.4). In Step 3, the expert evaluated the location options one by one based on the four performance criteria, and the relevant evaluation data were expressed in terms of T-SF information, as revealed in Table 2. The data fields contain the T-SF performance rating tij=(μij,ηij,νij) and its associated refusal-membership γij, where the positive-integer exponent q=3 and γij=1−(μij)3−(ηij)3−(νij)33. Taking t13=(0.81,0.62,0.11) as an illustration, γ13=1−0.813−0.623−0.1133=0.61. In Step 4, the T-SF characteristics were generated by Ti={⟨cj,tij⟩|cj∈C}={⟨cj,(μij,ηij,νij)⟩|cj∈{c1,c2,c3,c4}} for each location option ai. For example, T1={⟨c1,(0.43,0.20,0.61)⟩,⟨c2,(0.54,0.35,0.63)⟩,⟨c3,(0.81,0.62,0.11)⟩,⟨c4,(0.18,0.33,0.66)⟩}.

Table 2

Data of the T-SF performance rating tij (with the refusal-membership γij) in the location selection problem.

cjt1j=(μ1j,η1j,ν1j)γ1jt2j=(μ2j,η2j,ν2j)γ2jt3j=(μ3j,η3j,ν3j)γ3jt4j=(μ4j,η4j,ν4j)γ4j
c1(0.43,0.20,0.61)0.88(0.14,0.32,0.74)0.82(0.75,0.12,0.41)0.80(0.35,0.44,0.83)0.67
c2(0.54,0.35,0.63)0.82(0.26,0.17,0.26)0.99(0.59,0.29,0.13)0.92(0.91,0.12,0.49)0.50
c3(0.81,0.62,0.11)0.61(0.77,0.23,0.55)0.71(0.56,0.22,0.36)0.92(0.63,0.11,0.27)0.90
c4(0.18,0.33,0.66)0.88(0.61,0.34,0.57)0.82(0.11,0.14,0.45)0.97(0.31,0.36,0.84)0.69

In Step 5, the T-SF weighted performance rating tijw was computed using Eq. (3). To give an instance, t13w=(n·w3)⊙t13=(4×0.3)⊙t13=([1−(1−0.813)1.2]1/3,[(1−0.813)1.2−(1−0.813−0.623)1.2]1/3,[(1−0.813−0.623)1.2−(1−0.813−0.623−0.113)1.2]1/3)=(0.8422,0.6136,0.1060), where the refusal-membership γ13w=1−0.84223−0.61363−0.106033=0.5544. The computed outcomes of tijw and γijw are revealed in the third and fourth columns of Table 3. Moreover, the T-SF weighted characteristics were determined by the use of TiW={⟨cj,tijw⟩|cj∈C}={⟨cj,(μijw,ηijw,νijw)⟩|cj∈{c1,c2,c3,c4}} for each ai. As an illustration, T1W={⟨c1,(0.4003,0.1867,0.5750)⟩,⟨c2,(0.4046,0.2683,0.5031)⟩,⟨c3,(0.8422,0.6136,0.1060)⟩,⟨c4,(0.2104,0.3841,0.7406)⟩}.

Table 3

Outcomes relevant to the T-SF weighted performance rating tijw and the refusal-membership γijw (q=3).

aicjtijw=(μijw,ηijw,νijw)γijw(μijw)q(ηijw)q(νijw)q(γijw)qSquared sum∗
a1c1(0.4003, 0.1867, 0.5750)0.90420.06410.00650.19010.73930.5868
c2(0.4046, 0.2683, 0.5031)0.92330.06620.01930.12740.78710.6405
c3(0.8422, 0.6136, 0.1060)0.55440.59740.23100.00120.17040.4393
c4(0.2104, 0.3841, 0.7406)0.80820.00930.05670.40620.52780.4469
a2c1(0.1300, 0.2974, 0.7002)0.85640.00220.02630.34330.62820.5132
c2(0.1919, 0.1258, 0.1928)0.99460.00710.00200.00720.98380.9679
c3(0.8036, 0.2345, 0.5538)0.66820.51890.01290.16990.29830.3873
c4(0.6963, 0.3758, 0.6098)0.72590.33760.05310.22670.38260.3146
a3c1(0.7080, 0.1156, 0.3965)0.83450.35490.00150.06230.58120.4677
c2(0.4446, 0.2244, 0.1009)0.96540.08790.01130.00100.89980.8175
c3(0.5914, 0.2307, 0.3766)0.89940.20680.01230.05340.72750.5750
c4(0.1286, 0.1637, 0.5210)0.94800.00210.00440.14140.85210.7461
a4c1(0.3254, 0.4109, 0.8012)0.72550.03440.06940.51430.38180.4163
c2(0.7542, 0.1171, 0.5083)0.75950.42890.00160.13130.43810.3932
c3(0.6634, 0.1147, 0.2812)0.88120.29200.00150.02220.68430.5540
c4(0.3615, 0.4165, 0.8922)0.55440.04720.07220.71010.17040.5408

Squared sum∗: ((μijw)q)2+((ηijw)q)2+((νijw)q)2+((γijw)q)2 (q=3).

In Step 6, based on the universal and null T-SF sets, the T-SF characteristic T+={⟨c1,(0,0,1)⟩,⟨c2,(1,0,0)⟩,⟨c3,(1,0,0)⟩,⟨c4,(0,0,1)⟩} because of CPo={c2,c3} and CNe={c1,c4}. Moreover, T−={⟨c1,(1,0,0)⟩,⟨c2,(0,0,1)⟩,⟨c3,(0,0,1)⟩,⟨c4,(1,0,0)⟩}. According to Theorem 2, it was acquainted with T+W=T+ and T−W=T−. In Step 7, the T-SF weighted informational energies were yielded using Eq. (4). Specifically, IE(T1W)=∑j=14[((μ1jw)q)2+((η1jw)q)2+((ν1jw)q)2+((γ1jw)q)2]. The respective computation results of (μijw)q, (ηijw)q, (νijw)q, and (γijw)q are shown in the fifth to eighth columns of Table 3. Moreover, their corresponding squared sum, i.e. ((μijw)q)2+((ηijw)q)2+((νijw)q)2+((γijw)q)2, can be directly derived, and the results are demonstrated in the last column of Table 3. In conformity with these outcomes, it was derived that IE(T1W)=0.5868+0.6405+0.4393+0.4469=2.1135. In the same fashion, IE(T2W)=2.1830, IE(T3W)=2.6062, and IE(T4W)=1.9043, as shown in the second column of Table 4. Next, the T-SF weighted correlation functions CF(TiW,T+W) and CF(TiW,T−W) were acquired using Eqs. (5) and (6), respectively. To give an example, CF(T1W,T+W)=∑cj∈CPo(μ1jw)q+∑cj∈CNe(ν1jw)q=((μ12w)q+(μ13w)q)+((ν11w)q+(ν14w)q)=(0.0662+0.5974)+(0.1901+0.4062)=1.2599. The outcomes of CF(TiW,T+W) and CF(TiW,T−W) are displayed in the third and fourth columns of Table 4.

Table 4

Outcomes relevant to the T-SF data-driven correlation measures.

aiIE(TiW)CF(TiW,T+W)CF(TiW,T−W)CC√(TiW,T+W)CC√(TiW,T−W)CC∧(TiW,T+W)CC∧(TiW,T−W)
a12.11351.25990.20200.43330.06950.31500.0505
a22.18301.09600.51690.37090.17490.27400.1292
a32.60620.49840.41150.15440.12740.12460.1029
a41.90431.94540.23520.70490.08520.48640.0588

In Step 8, if the “square root function” type was employed, this study would comply with Step 8-1 to determine the T-SF weighted correlation coefficients CC√(TiW,T+W) and CC√(TiW,T−W) using Eqs. (7) and (8), respectively. It was recognized that IE(T+W)=IE(T−W)=n=4 following Theorem 3. To give an instance, CC√(T1W,T+W)=CF(T1W,T+W)/IE(T1W)·IE(T+W)=1.2599/2.1135×4=0.4333, and CC√(T1W,T−W)=CF(T1W,T−W)/IE(T1W)·IE(T−W)=0.2020/2.1135×4=0.0695. The obtained outcomes of CC√(TiW,T+W) and CC√(TiW,T−W) are indicated in the fifth and sixth columns, respectively, of Table 4. On the flip side, if the “maximum function” type was utilized, this study would comply with Step 8-2 to generate the T-SF weighted correlation coefficients CC∧(TiW,T+W) and CC∧(TiW,T−W). The yielded outcomes are manifested in the last two columns of Table 4. For example, CC∧(TiW,T+W)=CF(T1W,T+W)/max{IE(T1W),IE(T+W)}=1.2599/max{2.1135,4}=0.3150, and CC∧(TiW,T−W)=CF(T1W,T−W)/max{IE(T1W),IE(T−W)}=0.2020/max{2.1135,4}=0.0505.

In Step 9, in the light of Definition 13, the following minimal and maximal correlation coefficients were produced as: CC_√+=mini′=14CC√(Ti′W,T+W)=min{0.4333,0.3709,0.1544,0.7049}=0.1544, CC‾√+=maxi′=14CC√(Ti′W,T+W)=0.7049, CC_√−=mini′=14CC√(Ti′W,T−W)=min{0.0695,0.1749,0.1274,0.0852}=0.0695, and CC‾√−=maxi′=14CC√(Ti′W,T−W)=0.1749 for the “square root function” type. In a similar fashion, it was yielded that CC_∧+=min{0.3150,0.2740,0.1246,0.4864}=0.1246, CC‾∧+=0.4864, CC_∧−=0.0505, and CC‾∧−=0.1292. Letting the anchoring parameter ξ=0.5, the T-SF comprehensive correlation indices were calculated using Eq. (11) for the “square root function” type. That is, CI√(a1)=0.6×(CC√(T1W,T+W)−CC_√+)/(CC‾√+−CC_√+)+0.4×(CC‾√−−CC√(T1W,T−W))/(CC‾√−−CC_√−)=0.6×(0.4333−0.1544)/(0.7049−0.1544)+0.4×(0.1749−0.0695)/(0.1749−0.0695)=0.7040,CI√(a2)=0.2360, CI√(a3)=0.1801, and CI√(a4)=0.9402. Next, for the “maximum function” type, the T-SF comprehensive correlation index CI∧(ai) was generated using Eq. (12). Specifically, CI∧(a1)=0.6×(0.3150−0.1246)/(0.4864−0.1246)+0.4×(0.1292−0.0505)/(0.1292−0.0505)=0.7157,CI∧(a2)=0.2478,CI∧(a3)=0.1339,andCI∧(a4)=0.9578.

Finally, in Step 10, the four location options were ranked in descending order of the CI√(ai) values for the “square root function” type, which rendered the prioritization ranking a4≻√a1≻√a2≻√a3. Moreover, the prioritization ranking a4≻∧a1≻∧a2≻∧a3 was yielded in descending order of CI∧(ai) for the “maximum function” type. Regardless of the usage of the square root function and the maximum function, the solution outcomes generated by the current correlation-focused approach are concordant with the final ranking rendered by the technique using T-SF group-generalized hybrid geometric (GGHG) operators in Chen et al. (2021).

The conclusions of the application of the propounded methodology to the pragmatic problem for location selection are consistent with the consequences of the existing literature. The new approach centered on T-SF correlation-focused measurements in this study is not only rigorous in concept but also simple and easy to implement. Findings in practical applications are also consistent with existing literature and expectations.

4.2Comparative Analysis with Other Relevant Approaches

This subsection intends to conduct a comparative analysis to analyse the solution outcomes with those yielded by other T-SF multiple-criteria assessment approaches. As described in the state-of-the-art literature review in Table 1, many studies have explored the modularity of evaluation and decision-making methods involving T-SF information by T-SF averaging aggregation operations. Given the large body of related work that has concentrated on models of aggregated or averaged operations, this comparative analysis will provide a comprehensive discussion of the applied results rendered by some newly-developed aggregating or averaging operations regarding the location selection issue of the construction company to build new apartments. Such comparisons and analyses focus on the process of investigating the solution outcomes with each other and distinguishing their similarities and differences.

The T-SF averaging aggregation operations used for this comparative research cover the T-SF weighted averaging (WA) and T-SF weighted geometric (WG) operators advanced by Ullah et al. (2020a), the T-SF Frank weighted averaging (FWA) and T-SF Frank weighted geometric (FWG) operators initiated by Mahnaz et al. (2022), and the T-SF Aczel-Alsina weighted averaging (AAWA) and T-SF Aczel-Alsina weighted geometric (AAWG) operators advocated by Hussain et al. (2022b). From the arithmetic mean perspective, the technique using T-SF WA operators is a generally recognized T-SF aggregation algorithm. Moreover, the techniques using T-SF FWA or T-SF AAWA operators are rising T-SF aggregation algorithms with great potential. From the geometric mean viewpoint, the technique established on the T-SF WG operator provides a well-known T-SF aggregation algorithm. Furthermore, the techniques using T-SF FWG or T-SF AAWG operators are recently up-and-coming T-SF aggregation algorithms. Next, the mathematical expressions of the aforementioned arithmetic mean operators (i.e. T-SF WA, T-SF FWA, and T-SF AAWA) and the geometric mean operators (i.e. T-SF WG, T-SF FWG, and T-SF AAWG) will be described later.

To perform averaging aggregation operations under T-SF uncertainty, the direction of the negative criteria in the collection CNe should be reversed to be consistent with the direction of the positive criteria in the collection CPo. Let tij′=(μij′,ηij′,νij′) signify the normalized T-SF performance rating associated with tij. Using the means of the complement set operation, the T-SF characteristic Ti can be transformed into the normalized T-SF characteristic Ti′ using the following formula:

(13)
Ti′={⟨cj,tij′⟩|cj∈C}={⟨cj,(μij,ηij,νij)⟩|cj∈CPo,⟨cj,(νij,ηij,μij)⟩|cj∈CNe}.

This comparative study endeavours to aggregate the normalized T-SF performance rating tij′ across all cj∈C concerning each ai into a T-SF comprehensive evaluation value by employing the aggregation operators propounded by Ullah et al. (2020a), Mahnaz et al. (2022), and Hussain et al. (2022b). Let ϕ>1 and Φ⩾1 denote the parameters contained in Mahnaz et al.’s and Hussain et al.’s formulations, respectively. The T-SF comprehensive evaluation value of ti1′,ti2′,…,tin′ using the T-SF WA, T-SF FWA, and T-SF AAWA operators are determined sequentially along these lines:

(14)
WA(ti1′,ti2′,…,tin′)=(1−∏j=1n(1−(μij′)q)wjq,∏j=1n(ηij′)wj,∏j=1n(νij′)wj),
(15)
FWA(ti1′,ti2′,…,tin′)=(1−logϕ(1+∏j=1n(ϕ1−(μij′)q−1)wj)q,logϕ(1+∏j=1n(ϕ(ηij′)q−1)wj)q,logϕ(1+∏j=1n(ϕ(νij′)q−1)wj)q),
(16)
AAWA(ti1′,ti2′,…,tin′)=(1−exp(−{∑j=1nwj[−ln(1−(μij′)q)Φ]}1/Φ)q,exp(−{∑j=1nwj[−ln((ηij′)q)Φ]}1/Φ)q,exp(−{∑j=1nwj[−ln((νij′)q)Φ]}1/Φ)q).

From the geometric mean perspective, the T-SF comprehensive evaluation value of ti1′,ti2′,…,tin′ using the T-SF WG, T-SF FWG, and T-SF AAWG operators are calculated sequentially in the following manner, where ϕ>1 and Φ⩾1:

(17)
WG(ti1′,ti2′,…,tin′)=(∏j=1n(μij′)wj,∏j=1n(ηij′)wj,1−∏j=1n(1−(νij′)q)wjq),
(18)
FWG(ti1′,ti2′,…,tin′)=(logϕ(1+∏j=1n(ϕ(μij′)q−1)wj)q,1−logϕ(1+∏j=1n(ϕ1−(ηij′)q−1)wj)q,1−logϕ(1+∏j=1n(ϕ1−(νij′)q−1)wj)q),
(19)
AAWG(ti1′,ti2′,…,tin′)=(exp(−{∑j=1nwj[−ln((μij′)q)Φ]}1/Φ)q,1−exp(−{∑j=1nwj[−ln(1−(ηij′)q)Φ]}1/Φ)q,1−exp(−{∑j=1nwj[−ln(1−(νij′)q)Φ]}1/Φ)q).

This study exploited a well-grounded score function advanced by Zeng et al. (2019) to help compare the obtained T-SF comprehensive evaluation values. Let ti′=(μi′,ηi′,νi′) signify the T-SF comprehensive evaluation value produced by the T-SF WA, FWA, AAWA, WG, FWG, or AAWG operators, where its degree of refusal-membership γi′=1−(μi′)q−(ηi′)q−(νi′)qq. Following Zeng et al.’s formulation, the aggregated score value of ti′ is elucidated like this:

(20)
AS(ti′)=(μi′)q−(ηi′)q−(νi′)q+(γi′)q(exp((μi′)q−(ηi′)q−(νi′)q)exp((μi′)q−(ηi′)q−(νi′)q)+1−12).

Table 5

Outcomes of the T-SF comprehensive evaluation value ti′ yielded by the comparative approaches.

Methodt1′=(μ1′,η1′,ν1′)t2′=(μ2′,η2′,ν2′)t3′=(μ3′,η3′,ν3′)t4′=(μ4′,η4′,ν4′)
The aggregation technique using Ullah et al.’s (2020a) operators
T-SF WA(0.7051, 0.3629, 0.2095)(0.6765, 0.2787, 0.4045)(0.4999, 0.1672, 0.2343)(0.8093, 0.2353, 0.3190)
T-SF WG(0.6771, 0.3629, 0.3607)(0.6076, 0.2787, 0.5287)(0.4846, 0.1672, 0.4894)(0.7749, 0.2353, 0.3379)
The aggregation technique using Mahnaz et al.’s (2022) operators
T-SF FWA(0.7030, 0.3647, 0.2103)(0.6740, 0.2789, 0.4081)(0.4993, 0.1673, 0.2367)(0.8071, 0.2359, 0.3192)
T-SF FWG(0.6790, 0.4574, 0.3583)(0.6124, 0.2981, 0.5272)(0.4851, 0.1921, 0.4825)(0.7780, 0.3321, 0.3375)
The aggregation technique using Hussain et al.’s (2022b) operators
T-SF AAWA(0.7443, 0.3161, 0.1735)(0.7185, 0.2668, 0.2913)(0.5247, 0.1596, 0.1716)(0.8375, 0.1869, 0.3117)
T-SF AAWG(0.6510, 0.5505, 0.5025)(0.5024, 0.3168, 0.5703)(0.4733, 0.2307, 0.6497)(0.7254, 0.3811, 0.3896)
Fig. 4

Juxtaposition of three components of positive-, neutral-, and negative-membership in ti′.

Juxtaposition of three components of positive-, neutral-, and negative-membership in ti′.

In the light of the location selection issue of a construction company for building new apartments, this research exploited Eqs. (14)–(19) to produce the T-SF comprehensive evaluation value ti′, and the determination outcomes are displayed in Table 5. Herein, referring to the specifications by Mahnaz et al. (2022) and Hussain et al. (2022b), the two parameters ϕ and Φ were designated as ϕ=2 in Eqs. (15) and (18) and Φ=5 in Eqs. (16) and (19). To get a general idea of the obtained T-SF comprehensive evaluation values, the juxtaposition results of the three components of positive-, neutral-, and negative-membership (i.e. μi′, ηi′, and νi′, respectively) contained in ti′ are sketched in Fig. 4.

Next, this study used Eq. (20) to generate the aggregated score value AS(ti′) and then identify the corresponding prioritization ranking order, as revealed in Table 6. Over and above that, to conduct a baseline analysis, the technique using the T-SF GGHG operator evolved by Chen et al. (2021) will be exploited to be a beginning point used for comparisons. The aggregated score values generated by the T-SF GGHG operator are exhibited in Table 6. As described in the previous subsection, when employing the propounded methodology in this study, the T-SF comprehensive correlation indices (CI√(ai) and CI∧(ai) based on the square root and maximum functions, respectively) are also displayed in Table 6. Moreover, the numbers in parentheses are the orders of precedence for each choice option. The techniques using the T-SF WA, WG, FWA, and FWG operators generated the identical prioritization ranking a4≻a1≻a3≻a2. The techniques using the T-SF AAWA and GGHG operators and the current multiple-criteria choice method using the square root and maximum functions generated the same prioritization ranking a4≻a1≻a2≻a3. The use of the technique using the T-SF AAWG operator yielded a particularly different ordering result a4≻a2≻a1≻a3. Of all the comparative approaches, only the solution results yielded by the T-SF AAWA operator and the current methodology ranked the same as the benchmark method using the T-SF GGHG operator. The Spearman correlation between the benchmark ranking (i.e. a4≻a1≻a2≻a3) and the solution outcome based on the T-SF WA, WG, FWA, and FWG operators is equal to 0.8. The Spearman correlation between the benchmark ranking and the solution outcome based on the T-SF AAWG operator is also equal to 0.8. It is noted that the Spearman correlation between the two prioritization rankings a4≻a1≻a3≻a2 and a4≻a2≻a1≻a3 reduces to 0.4.

Table 6

The aggregated score value and the T-SF comprehensive correlation index with their rank orders.

Source of methodsComparative approacha1a2a3a4
Ullah et al. (2020b)T-SF WA operator0.7252 (2)0.6520 (4)0.6991 (3)0.8091 (1)
T-SF WG operator0.6397 (2)0.4664 (4)0.5001 (3)0.7768 (1)
Mahnaz et al. (2022)T-SF FWA operator0.7224 (2)0.6472 (4)0.6982 (3)0.8074 (1)
T-SF FWG operator0.5550 (2)0.4624 (4)0.5048 (3)0.7355 (1)
Hussain et al. (2022b)T-SF AAWA operator0.7855 (2)0.7575 (3)0.7245 (4)0.8423 (1)
T-SF AAWG operator0.2675 (3)0.3334 (2)0.1994 (4)0.6316 (1)
Chen et al. (2021)T-SF GGHG operator0.4620 (2)0.3257 (3)0.1951 (4)0.6322 (1)
Current paperSquare root function type0.7040 (2)0.2360 (3)0.1801 (4)0.9402 (1)
Maximum function type0.7157 (2)0.2478 (3)0.1339 (4)0.9578 (1)

The aggregated score values and T-SF comprehensive correlation indices yielded by the T-SF averaging aggregation operations and the evolved multiple-criteria choice method, respectively, are contrasted in Fig. 5. In particular, Fig. 5(a) reveals the comparisons among the four choice options under distinct comparative approaches. Furthermore, consider that the choice option a4 performed the best among all comparative approaches, while the choice option a3 performed the worst among most comparative approaches (i.e. the T-SF AAWA, AAWG, GGHG operators, and the current method based on the square root and maximum functions). The relative performances associated with the best and comparatively worst choice options (i.e. a4 and a3, respectively) are contrasted in Fig. 5(b) to highlight their juxtaposition.

Fig. 5

Comparison results of the aggregated score values/T-SF comprehensive correlation indices.

Comparison results of the aggregated score values/T-SF comprehensive correlation indices.

Going a step further, this study attempts to examine the solution outcomes produced by the comparative approaches with a benchmark ranking by Chen et al. (2021). The prioritization ranking (i.e. a4≻a1≻a3≻a2) obtained by the techniques using the T-SF WA, WG, FWA, and FWG operators differs from the benchmark ranking (i.e. a4≻a1≻a2≻a3) based on the T-SF GGHG operator in the outranking relationship between a2 and a3. The difference between the prioritization ranking (i.e. a4≻a2≻a1≻a3) rendered by the technique using the T-SF AAWG operator and the benchmark ranking based on the T-SF GGHG operator lies in the outranking relationship between a1 and a2. Different from the techniques using the aggregation operators initiated by Ullah et al. (2020a), Mahnaz et al. (2022), and Hussain et al. (2022b), the prioritization rankings yielded by the two approaches based on square root and maximum functions in this study are consistent with the benchmark ranking determined from the T-SF GGHG operator. Therefore, the comparative investigation of the application outcomes supports the superiority of the proposed multiple-criteria choice method grounded in T-SF data-driven correlation measures.

4.3More Comparative Discussion Based on Parametric Analysis

This subsection has the objective of conducting a comprehensive comparative analysis from a problem-oriented point of view. In the first comparative study, different settings of the anchoring parameter are explored and the yielded outcomes of T-SF comprehensive correlation indices under each scenario are discussed holistically. In the second comparative study, the best and worst choice options that are constituted by the universal and null T-SF sets are replaced by the positive and negative ideal schemes, respectively, to be a benchmark for exploring the effects on the T-SF correlation-focused measurements.

The first comparative study gives thought to distinct assigned values of the anchoring parameter ξ and investigates the yielded consequences of T-SF comprehensive correlation indices under various parameter settings. By conducting such a comparative study, the effect of the distinct controlling or deciding of the parameter ξ on the T-SF comprehensive correlation indices CI√(ai) (based on the square root function) and CI∧(ai) (based on the maximum function) can be obtained; moreover, the stability and controllability of the prioritization ranking results can be investigated. In the comparative analysis, the values of the anchoring parameter ξ were set to 0.0, 0.1, …, 1.0. The juxtaposition and comparisons of CI√(ai) and CI∧(ai) for distinct values of ξ are portrayed in Fig. 6(a) and Fig. 6(b), respectively.

Fig. 6

Contrasts of the T-SF comprehensive correlation indices in distinct settings of the anchoring parameter.

Contrasts of the T-SF comprehensive correlation indices in distinct settings of the anchoring parameter.

As depicted in Fig. 6(a), the three prioritization rankings a1≻√a4≻√a3≻√a2, a4≻√a1≻√a3≻√a2, and a4≻√a1≻√a2≻√a3 were generated when ξ=0.0, 0.1, 0.2, ξ=0.3, 0.4, 0.5, and ξ=0.6,0.7,…,1.0, respectively. As revealed in Fig. 6(b), the rankings a1≻∧a4≻∧a3≻∧a2, a4≻∧a1≻∧a3≻∧a2, and a4≻∧a1≻∧a2≻∧a3 were produced when ξ=0.0,0.1, ξ=0.2,0.3,0.4, and ξ=0.5,0.6,…,1.0, respectively. In this respect, the prioritization ranking outcomes using the square root function were not much different from those using the maximum function. The main discrimination was that the ranking outcomes in the case of ξ=0.2 and ξ=0.5 are inconsistent. On the other hand, it is worth mentioning that the obtained CI√(ai) and CI∧(ai) values gave rise to identical rankings (i.e. the prioritization rankings a4≻√a1≻√a2≻√a3 and a4≻∧a1≻∧a2≻∧a3 when ξ=0.6,0.7,…,1.0 and ξ=0.5,0.6,…,1.0, respectively) in comparison to the ranking outcome rendered by Chen et al. (2021). Thus, the efficacy and reasonableness of the proposed methodology can be corroborated because of consistent ranking results in most cases. Furthermore, somewhat different rankings a4≻√a1≻√a3≻√a2 (based on the square root function) and a4≻∧a1≻∧a3≻∧a2 (based on the maximum function) were acquired when ξ=0.3,0.4,0.5 and ξ=0.2,0.3,0.4, respectively. Nonetheless, different outcomes were yielded in face of the small values of ξ, namely a1≻√a4≻√a3≻√a2 and a1≻∧a4≻∧a3≻∧a2 when ξ=0.0,0.1,0.2 and ξ=0.0,0.1, respectively. Overall, stable and justified consequences can be generated under most settings of ξ. When ξ=0.0,0.1,0.2 based on the “square root function” type or ξ=0.0,0.1 based on the “maximum function” type, distinct prioritization ranking outcomes can be rendered to reflect the change of the ξ values, which gives substance to the pliability of the current methods by adjusting the anchoring parameter ξ. The comparison consequence demonstrates that by controlling the parameter values, stable and flexible prioritization rankings can be produced by using the propounded methodology.

In the second comparative study, the best choice option a+ and the worst choice option a− (composed of the universal T-SF set and the null T-SF set) are replaced by the positive and negative ideal schemes, respectively, as an alternate benchmark for calculating the T-SF correlation-focused measurements. To accommodate the change of the reference points, this study would like to yield the corresponding T-SF correlation-focused measurements, so that the subsequent practical data processing and multiple-criteria evaluation procedures can operate smoothly. Through the juxtaposition and comparison of the solution outcomes, the influence of distinct points of reference on the yielded results can be clarified. Moreover, through the side-by-side comparison, the advantages of taking a+ and a− as points of reference can be demonstrated and justified.

The positive and negative ideal schemes would be exploited to replace the best and worst choice options, respectively, to explore the influences of different points of reference on the T-SF correlation-focused measurements and resolution consequences. More specifically, instead of the universal and null T-SF sets, the T-SF characteristics of the ideal schemes would be established using the union and intersection operations. Let a∗ indicate the positive ideal scheme, where the T-SF characteristic T∗={⟨cj,t∗j⟩|cj∈C}={⟨cj,(μ∗j,η∗j,ν∗j)⟩|cj∈C}. Let a# signify the negative ideal scheme, where the T-SF characteristic T#={⟨cj,t#j⟩|cj∈C}={⟨cj,(μ#j,η#j,ν#j)⟩|cj∈C}. Utilizing the set operations ∪ and ∩, T∗ and T# are delineated in this fashion:

  • 1. T∗={⟨cj,(maxi=1mμij,mini=1mηij,mini=1mνij)⟩|cj∈CPo,⟨cj,(mini=1mμij,mini=1mηij,maxi=1mνij)⟩|cj∈CNe};

  • 2. T#={⟨cj,(mini=1mμij,mini=1mηij,maxi=1mνij)⟩|cj∈CPo,⟨cj,(maxi=1mμij,mini=1mηij,mini=1mνij)⟩|cj∈CNe}.

Recall that CPo={c2,c3} and CNe={c1,c4} in the location selection problem. Using the aforesaid manner, the T-SF characteristics of a∗ and a# were identified as follows: T∗={⟨c1,(0.14,0.12,0.83)⟩,⟨c2,(0.91,0.12,0.13)⟩,⟨c3,(0.81,0.11,0.11)⟩,⟨c4,(0.11,0.14,0.84)⟩} and T#={⟨c1,(0.75,0.12,0.41)⟩,⟨c2,(0.26,0.12,0.63)⟩,⟨c3,(0.56,0.11,0.55)⟩,⟨c4,(0.61,0.14,0.45)⟩}. The corresponding T-SF weighted characteristics were given by: T∗W={⟨c1,(0.1300,0.1156,0.8012)⟩,⟨c2,(0.7542,0.1171,0.1009)⟩,⟨c3,(0.8422,0.1147,0.1060)⟩,⟨c4,(0.1286,0.1637,0.8922)⟩} and T#W={⟨c1,(0.7080,0.1156,0.3965)⟩,⟨c2,(0.1919,0.1171,0.5083)⟩,⟨c3,(0.5914,0.1147,0.5538)⟩,⟨c4,(0.6963,0.1637,0.5210)⟩}. The T-SF weighted informational energies were derived as: IE(T∗W)=2.1053 and IE(T#W)=2.0837. The comparisons of the T-SF weighted correlation functions CF(TiW,T+W), CF(TiW,T∗W), CF(TiW,T−W), and CF(TiW,T#W) are manifested in Fig. 7. Furthermore, the T-SF weighted correlation coefficients CC√(TiW,T+W), CC√(TiW,T∗W), CC√(TiW,T−W), and CC√(TiW,T#W) are contrasted in Fig. 8(a), while the comparisons of CC∧(TiW,T+W), CC∧(TiW,T∗W), CC∧(TiW,T−W), and CC∧(TiW,T#W) are exhibited in Fig. 8(b).

Fig. 7

Contrast outcomes of the T-SF weighted correlation functions concerning distinct points of reference.

Contrast outcomes of the T-SF weighted correlation functions concerning distinct points of reference.
Fig. 8

Contrast outcomes of the T-SF weighted correlation coefficients concerning distinct points of reference. (a) Outcomes using the “square root function” type. (b) Outcomes using the “maximum function” type.

Contrast outcomes of the T-SF weighted correlation coefficients concerning distinct points of reference. (a) Outcomes using the “square root function” type. (b) Outcomes using the “maximum function” type.

First, consider the contrast outcomes of the T-SF weighted correlation functions concerning the best choice option a+ and the positive ideal scheme a∗, as revealed in Fig. 5. The differences among the CF(TiW,T+W) values of the four choice options (a1−a4) were significantly higher than the differences among the CF(TiW,T∗W) values. In particular, the gap between the maximum value (i.e. CF(T4W,T+W)) and the minimum value (i.e. CF(T3W,T+W)) was quite pronounced. However, the gap between the maximum value (i.e. CF(T4W,T∗W)) and the minimum value (i.e. CF(T3W,T∗W)) did not show a particularly significant difference. Next, concerning the T-SF weighted correlation functions toward the worst choice option a− and the negative ideal scheme a#, the maximum value of CF(TiW,T−W) and the maximum value of CF(TiW,T#W) correspond to different options; the same is true for the minimum value of CF(TiW,T−W) (or CF(TiW,T#W)). To be precise, the options a2 and a1 enjoy the largest and smallest values, respectively, of CF(TiW,T−W); a3 and a4 enjoy the largest and smallest values, respectively, of CF(TiW,T#W).

Next, consider the comparisons of the T-SF weighted correlation coefficients with relevance to two types of points of reference (i.e. one type for the best and worst choice options and the other type for the positive and negative ideal schemes). Let us investigate the contrast outcomes in Fig. 8(a) using the “square root function” type. The CC√(TiW,T∗W) values of the four choice options were significantly higher than the CC√(TiW,T+W) values; this phenomenon was also found in the comparisons of the values of CC√(TiW,T#W) and CC√(TiW,T−W). The higher the value of CC√(TiW,T+W) (or CC√(TiW,T∗W)), the higher the correlation between the corresponding option ai and a+ (or a∗). Accordingly, the decision-maker expects to choose the option that is highly correlated with the best choice option (or the positive ideal scheme). The lower the value of CC√(TiW,T−W) (or CC√(TiW,T#W)), the lower the correlation between ai and a− (or a#). In this regard, the decision-maker expects to choose the option that lowly correlates with the worst choice option (or the negative ideal scheme). In Fig. 8(a), the numerical orders of the T-SF weighted correlation coefficients for mutual relationships with a+ and a∗ were CC√(T4W,T+W)>CC√(T1W,T+W)>CC√(T2W,T+W)>CC√(T3W,T+W) and CC√(T4W,T∗W)>CC√(T1W,T∗W)>CC√(T2W,T∗W)>CC√(T3W,T∗W), respectively. Different from the identical ranking orders above, the numerical orders of the T-SF weighted correlation coefficients for mutual relationships with a− and a# were CC√(T1W,T−W)<CC√(T4W,T−W)<CC√(T3W,T−W)<CC√(T2W,T−W) and CC√(T4W,T#W)<CC√(T1W,T#W)<CC√(T2W,T#W)<CC√(T3W,T#W), respectively. In Fig. 8(b), the findings concerning the contrast outcomes using the “maximum function” type were about the same as those using the “square root function” type except for the results of CC∧(TiW,T#W). Specifically, the numerical orders of CC∧(TiW,T#W) were given by CC∧(T4W,T#W)<CC∧(T1W,T#W)<CC∧(T3W,T#W)<CC∧(T2W,T#W). In the matter of the numerical orders of the T-SF weighted correlation coefficients among the four options, the two ranking outcomes employing CC√(TiW,T+W) and CC√(TiW,T∗W) were consistent; however, the two ranking outcomes via CC√(TiW,T−W) and CC√(TiW,T#W) were somewhat different. To unify the inconsistent numerical orders, this study continued to calculate the T-SF comprehensive correlation indices for the final decision.

To facilitate discussing the effects of the anchoring parameter ξ regarding the solution consequences, this study set eleven different values for the parameter to calculate the T-SF comprehensive correlation indices and identify the ultimate ranking outcome under various scenarios. This study designated the anchoring parameter ξ ranging from 0 to 1, wherein ξ=0.0,0.1,…,1.0. Let CC_√∗=mini′=14CC√(Ti′W,T∗W), CC‾√∗=maxi′=14CC√(Ti′W,T∗W), CC_√#=mini′=14CC√(Ti′W,T#W), and CC‾√#=maxi′=14CC√(Ti′W,T#W) for the “square root function” type. Let CC_∧∗=mini′=14CC∧(Ti′W,T∗W), CC‾∧∗=maxi′=14CC∧(Ti′W,T∗W), CC_∧#=mini′=14CC∧(Ti′W,T#W), and CC‾∧#=maxi′=14CC∧(Ti′W,T#W) for the “maximum function” type. On the grounds of the ideal schemes a∗ and a#, the T-SF comprehensive correlation indices CI√′(ai) and CI∧′(ai) are elucidated in this fashion:

(21)
CI√′(ai)=ξ·CC√(TiW,T∗W)−CC_√∗CC‾√∗−CC_√∗+(1−ξ)·CC‾√#−CC√(TiW,T#W)CC‾√#−CC_√#,
(22)
CI∧′(ai)=ξ·CC∧(TiW,T∗W)−CC_∧∗CC‾∧∗−CC_∧∗+(1−ξ)·CC‾∧#−CC∧(TiW,T#W)CC‾∧#−CC_∧#.

Fig. 9

Contrast outcomes of CI√(ai) and CI√′(ai) for various values of ξ.

Contrast outcomes of CI√(ai) and CI√′(ai) for various values of ξ.

Regarding the “square root function” type, the juxtaposition of the T-SF comprehensive correlation indices CI√(ai) and CI√′(ai) for various values of ξ are displayed in Fig. 9. Specifically, Fig. 9(a) reveals the contrast outcomes concerning the points of reference a+ and a−, while Fig. 9(b) demonstrates the comparisons in connection with the points of reference a∗ and a#. For the “maximum function” type, the juxtaposition of the T-SF comprehensive correlation indices CI√(ai) and CI√′(ai) for various values of ξ are sketched in Fig. 10. Herein, Fig. 10(a) shows the comparison consequence on the grounds of a+ and a−, while Fig. 10(b) exemplifies the contrasts based on a∗ and a#.

Fig. 10

Contrast outcomes of CI∧(ai) and CI∧′(ai) for various values of ξ.

Contrast outcomes of CI∧(ai) and CI∧′(ai) for various values of ξ.

On the grounds of the reference points of the ideal schemes a∗ and a#, the contrast outcomes of the T-SF comprehensive correlation indices among the four options presented moderately unreasonable patterns; moreover, these results may be difficult to be accepted by the decision-maker. To be specific, the unusual consequences were produced using the “square root function” type, i.e. CI√′(a3)=0 and CI√′(a4)=1 for all ξ=0.0,0.1,…,1.0, as displayed in Fig. 9(b). Additionally, it was received that CI∧′(a4)=1 predicated on the “maximum function” type for all ξ=0.0,0.1,…,1.0, as displayed in Fig. 10(b). Regardless of how the value of the anchoring parameter ξ changed, the indices CI√′(a3), CI√′(a4), and CI∧′(a4) were fixed at 0, 1, and 1, respectively. These findings revealed that the multiple-criteria analysis approach that exploited the positive and negative ideal schemes as a benchmark for reference was not sensitive enough to reflect changes in various ξ values. On the contrary, the propounded methodology predicated on the best and worst choice options (i.e. a+ and a−) that were established on the universal and null T-SF sets generated reasonable and desirable consequences.

5Conclusions and Future Research Avenues

The framework based on T-spherical fuzziness provides an important tool for overcoming complex uncertainties in multiple-criteria choice issues by manipulating the four membership degrees involving positive, neutral, negative, and refusal components. One of the recent developments of multiple-criteria analysis techniques under T-SF conditions, the notion of correlation coefficients, has an increased uncertainty modelling capacity for decision-making. This paper creates some valuable concepts of T-SF data-driven correlation measures predicated on correlation coefficients in T-SF settings. Furthermore, this paper formulates a beneficial multiple-criteria choice method through a correlation-focused approach, which assists with computational intelligence in uncertain decision analysis. Following the anchored comparisons relative to the universal T-SF set and the null T-SF set, this paper constructs the T-SF weighted correlation coefficients using the types of square root and maximum functions. This paper also institutes the T-SF comprehensive correlation indices to determine the relative prioritization of all competing options and decide on the most appropriate scheme.

The evolved methodology is applied to a location selection issue to support a construction company in constructing new apartments. In addition to an applicable illustration, two types of comparative analyses (by changing anchoring parameters and reference T-SF sets) are performed to examine the robustness and merits of the developed techniques. It was found that the obtained T-SF comprehensive correlation indices render relatively stable but adjustable ranking results in different scenarios of anchoring parameters. Moreover, comparative analyses demonstrate that the T-SF comprehensive correlation indices predicated on the universal and null T-SF sets are more reasonable and justifiable than the yielded outcomes on the other reference T-SF sets.

Although the advantages of the multiple-criteria choice methodology are demonstrated through practical applications and comparative studies, the current methods still struggle with research limitations and disadvantages. The core concept of schematizing the evolved methodology is the notion of the T-SF data-driven correlation measures, which are derived from the T-SF correlation coefficients. The T-SF correlation coefficients can be utilized in statistical analysis or machine learning, and they mainly measure the degree of linear correlation between two T-SF sets. In other words, the T-SF correlation coefficients can explore whether there is a linear relationship between two T-SF sets (or multiple T-SF sets). However, if the relationship between two T-SF sets is nonlinear, the T-SF correlation coefficients may not precisely represent the relationship between them. This limitation may reduce the accuracy or sensitivity of the T-SF data-driven correlation measures in distinguishing between superior and inferior T-SF (weighted) characteristics.

Future research avenues can be improved and extended in two aspects. First, the proposed methodology and techniques can be exploited for other relevant high-order fuzzy configurations, such as uncertain sets of interval-valued spherical fuzziness, (complex) T-spherical fuzziness, (complex) q-rung orthopair fuzziness, T-spherical hesitant fuzziness, and neutrosophic fuzziness. Second, colloquially referred to as a normalized measurement of the covariance using correlation coefficients in applied statistics, the initiated T-SF data-driven correlation measures can also be employed for a variety of tasks in data analysis, decision aiding, engineering, intelligence sciences, and other areas.

Acknowledgements

The authors acknowledge the assistance of the respected editor and the anonymous referees for their insightful and constructive comments, which helped improve the overall quality of the paper.

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