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 μ 2 , η 2 , ν 2 , and γ 2 is equal to 1, which indicates a prerequisite for the SF configuration. A positive integer q is placed where q ∈ Z + . The sum of μ q , η q , ν q , and γ q 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.

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.

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.

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.

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.

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.

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 = { a 1 , a 2 , … , a m } and C = { c 1 , c 2 , … , c n } 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 c j ∈ C, place the normalized (standardized) weight w j ∈ [ 0 , 1 ] with the conditioning of weight normalization, i.e. ∑ j = 1 n w j = 1. The set C is compartmentalized into the collection of positive (performance) criteria C Po and the collection of negative (performance) criteria C Ne . Herein, C Po ∩ C Ne = ∅ and C Po ∪ C Ne = C. Positive criteria (such as profit and productivity) refer to the performance attribute c j ∈ C Po 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 c j ∈ C Ne 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 t i j = ( μ i j , η i j , ν i j ) involving a positive-integer exponent q signify a performance rating concerning an alternative a i ∈ A having relevance for a specified criterion c j ∈ C ( = C Po ∪ C Ne ), where the prerequisite 0 ⩽ ( μ i j ) q + ( η i j ) q + ( ν i j ) q ⩽ 1 must be fulfilled. In what follows, the degree of refusal-membership is calculated as γ i j = 1 − ( μ i j ) q − ( η i j ) q − ( ν i j ) q q . By collecting the T-SF performance rating t i j of a i across all criteria in C, the T-SF characteristic T i is formed using this fashion: (1) T i = { ⟨ c j , t i j ⟩ | c j ∈ C } = { ⟨ c j , ( μ i j , η i j , ν i j ) ⟩ | c j ∈ C ( = C Po ∪ C Ne ) } .

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.

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 T i W , 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 T i W = { ⟨ c j , ( μ i j w , η i j w , ν i j w ) ⟩ | c j ∈ C } (with the refusal-membership γ i j w = 1 − ( μ i j w ) q − ( η i j w ) q − ( ν i j w ) q q ), its T-SF weighted informational energy is expounded such that: (4) IE ( T i W ) = ∑ j = 1 n [ ( ( μ i j w ) q ) 2 + ( ( η i j w ) q ) 2 + ( ( ν i j w ) q ) 2 + ( ( γ i j w ) q ) 2 ] .