Informatica

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 $(\mu )$ for favourable evaluations, neutral component $(\eta )$ for abstinence, negative component $(\nu )$ for unfavourable evaluations, and refusal component $(\gamma )$ 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 ${\mu ^{2}}$, ${\eta ^{2}}$, ${\nu ^{2}}$, and ${\gamma ^{2}}$ is equal to 1, which indicates a prerequisite for the SF configuration. A positive integer q is placed where $q\in {Z^{+}}$. The sum of ${\mu ^{q}}$, ${\eta ^{q}}$, ${\nu ^{q}}$, and ${\gamma ^{q}}$ 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 $\eta =0$, the T-SF configuration transforms into the intuitionistic, Pythagorean, and q-rung orthopair fuzzy configurations when $q=1,2$, and $q\in {Z^{+}}$. 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.

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:

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 $A=\{{a_{1}},{a_{2}},\dots ,{a_{m}}\}$ and $C=\{{c_{1}},{c_{2}},\dots ,{c_{n}}\}$ set forth two limited sets of choice options and performance criteria, respectively, in which the cardinal numbers $m,n\geqslant 2$. In connection to each performance criterion ${c_{j}}\in C$, place the normalized (standardized) weight ${w_{j}}\in [0,1]$ with the conditioning of weight normalization, i.e. ${\textstyle\sum _{j=1}^{n}}{w_{j}}=1$. The set C is compartmentalized into the collection of positive (performance) criteria ${C_{\textit{Po}}}$ and the collection of negative (performance) criteria ${C_{\textit{Ne}}}$. Herein, ${C_{\textit{Po}}}\cap {C_{\textit{Ne}}}=\varnothing $ and ${C_{\textit{Po}}}\cup {C_{\textit{Ne}}}=C$. Positive criteria (such as profit and productivity) refer to the performance attribute ${c_{j}}\in {C_{\textit{Po}}}$ 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 ${c_{j}}\in {C_{\textit{Ne}}}$ 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 ${t_{ij}}=({\mu _{ij}},{\eta _{ij}},{\nu _{ij}})$ involving a positive-integer exponent q signify a performance rating concerning an alternative ${a_{i}}\in A$ having relevance for a specified criterion ${c_{j}}\in C$ $(={C_{\textit{Po}}}\cup {C_{\textit{Ne}}})$, where the prerequisite $0\leqslant {({\mu _{ij}})^{q}}+{({\eta _{ij}})^{q}}+{({\nu _{ij}})^{q}}\leqslant 1$ must be fulfilled. In what follows, the degree of refusal-membership is calculated as ${\gamma _{ij}}=\sqrt[q]{1-{({\mu _{ij}})^{q}}-{({\eta _{ij}})^{q}}-{({\nu _{ij}})^{q}}}$. By collecting the T-SF performance rating ${t_{ij}}$ of ${a_{i}}$ across all criteria in C, the T-SF characteristic ${T_{i}}$ is formed using this fashion:

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.

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 ${T_{i}}$ and the normalized (standardized) weight ${w_{j}}$ of an available choice option ${a_{i}}\in A$ and a performance criterion ${c_{j}}\in C$, respectively. Place an anchoring parameter $\xi \in [0,1]$. For each T-SF characteristic, the parameter ξ elucidates the weight of the anchored comparisons relative to universal T-SF sets, while $(1-\xi )$ 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.

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 $A=\{{a_{1}},{a_{2}},\dots ,{a_{m}}\}$ and a limited set of performance criteria $C=\{{c_{1}},{c_{2}},\dots ,{c_{n}}\}$. Separate C into two parts: one is the collection of positive criteria ${C_{\textit{Po}}}$; the other is the collection of negative criteria ${C_{\textit{Ne}}}$.

Step 2. Generate the normalized (standardized) weight ${w_{j}}$ with the conditioning of weight normalization for each performance criterion ${c_{j}}$.

Step 3. Specify a suitable positive-integer exponent q and form a T-SF performance rating ${t_{ij}}$ signified as the T-SF number (${\mu _{ij}},{\eta _{ij}},{\nu _{ij}}$) with the refusal-membership ${\gamma _{ij}}$.

Step 4. Assemble the T-SF characteristic ${T_{i}}$ in Eq. (1) by gathering all T-SF performance rating ${t_{ij}}$ regarding a choice option ${a_{i}}$ across all criteria in C.

Step 5. Employ Eq. (3) to derive the T-SF weighted performance rating ${t_{ij}^{w}}$ (with refusal-membership ${\gamma _{ij}^{w}}$) for the sake of framing the T-SF weighted characteristic ${T_{i}^{W}}$ 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 $\textit{IE}({T_{i}^{W}})$ using Eq. (4) and the T-SF weighted correlation functions $\textit{CF}({T_{i}^{W}},{T_{+}^{W}})$ and $\textit{CF}({T_{i}^{W}},{T_{-}^{W}})$ 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 ${\textit{CC}_{\surd }}({T_{i}^{W}},{T_{+}^{W}})$ and ${\textit{CC}_{\surd }}({T_{i}^{W}},{T_{-}^{W}})$ using Eqs. (7) and (8), respectively.

Step 8-2. Exploit the “maximum function” type to produce the T-SF weighted correlation coefficients ${\textit{CC}_{\wedge }}({T_{i}^{W}},{T_{+}^{W}})$ and ${\textit{CC}_{\wedge }}({T_{i}^{W}},{T_{-}^{W}})$ using Eqs. (9) and (10), respectively.

Step 9. Assign an anchoring parameter ξ to determine the T-SF comprehensive correlation index ${\textit{CI}_{\surd }}({a_{i}})$ (or ${\textit{CI}_{\wedge }}({a_{i}})$) using Eq. (11) (or Eq. (12)).

Step 10. Rank the m choice options in A supported by ${\textit{CI}_{\surd }}({a_{i}})$ (or ${\textit{CI}_{\wedge }}({a_{i}})$) in descending order to identify the prioritization relations “${\succ _{\surd }}$”, “${\sim _{\surd }}$”, and “${\prec _{\surd }}$” (or “${\succ _{\wedge }}$”, “${\sim _{\wedge }}$”, and “${\prec _{\wedge }}$”). Make a final decision for completing the multiple-criteria choice task.

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 (${a_{1}}-{a_{4}}$) for constructing new apartments predicated on the four performance criteria. The performance criteria consist of land cost (${c_{1}}$), surrounding environment (${c_{2}}$), technological capability (${c_{3}}$), and lease value (${c_{4}}$). Fig. 3 provides a profile of the location selection issue under study.

In Step 1, the two limited sets of choice options and performance criteria were designated as $A=\{{a_{1}},{a_{2}},{a_{3}},{a_{4}}\}$ and $C=\{{c_{1}},{c_{2}},{c_{3}},{c_{4}}\}$, respectively. Herein, the set C was separated into two parts: one is the collection of positive criteria ${C_{\textit{Po}}}=\{{c_{2}},{c_{3}}\}$; the other is the collection of negative criteria ${C_{\textit{Ne}}}=\{{c_{1}},{c_{4}}\}$. In Step 2, in conformity with the expert’s professional opinions, the normalized (standardized) weights were given by $({w_{1}},{w_{2}},{w_{3}},{w_{4}})=(0.2,0.1,0.3,0.4)$. 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 ${t_{ij}}=({\mu _{ij}},{\eta _{ij}},{\nu _{ij}})$ and its associated refusal-membership ${\gamma _{ij}}$, where the positive-integer exponent $q=3$ and ${\gamma _{ij}}=\sqrt[3]{1-{({\mu _{ij}})^{3}}-{({\eta _{ij}})^{3}}-{({\nu _{ij}})^{3}}}$. Taking ${t_{13}}=(0.81,0.62,0.11)$ as an illustration, ${\gamma _{13}}=\sqrt[3]{1-{0.81^{3}}-{0.62^{3}}-{0.11^{3}}}=0.61$. In Step 4, the T-SF characteristics were generated by ${T_{i}}=\{\langle {c_{j}},{t_{ij}}\rangle \hspace{0.1667em}|\hspace{0.1667em}{c_{j}}\in C\}=\{\langle {c_{j}},({\mu _{ij}},{\eta _{ij}},{\nu _{ij}})\rangle \hspace{0.1667em}|\hspace{0.1667em}{c_{j}}\in \{{c_{1}},{c_{2}},{c_{3}},{c_{4}}\}\}$ for each location option ${a_{i}}$. For example, ${T_{1}}=\{\langle {c_{1}},(0.43,0.20,0.61)\rangle ,\langle {c_{2}},(0.54,0.35,0.63)\rangle ,\langle {c_{3}},(0.81,0.62,0.11)\rangle ,\langle {c_{4}},(0.18,0.33,0.66)\rangle \}$.

In Step 5, the T-SF weighted performance rating ${t_{ij}^{w}}$ was computed using Eq. (3). To give an instance, ${t_{13}^{w}}=(n\cdot {w_{3}})\odot {t_{13}}=(4\times 0.3)\odot {t_{13}}=({[1-{(1-{0.81^{3}})^{1.2}}]^{1/3}},{[{(1-{0.81^{3}})^{1.2}}-{(1-{0.81^{3}}-{0.62^{3}})^{1.2}}]^{1/3}},{[{(1-{0.81^{3}}-{0.62^{3}})^{1.2}}-{(1-{0.81^{3}}-{0.62^{3}}-{0.11^{3}})^{1.2}}]^{1/3}})=(0.8422,0.6136,0.1060)$, where the refusal-membership ${\gamma _{13}^{w}}=\sqrt[3]{1-{0.8422^{3}}-{0.6136^{3}}-{0.1060^{3}}}=0.5544$. The computed outcomes of ${t_{ij}^{w}}$ and ${\gamma _{ij}^{w}}$ are revealed in the third and fourth columns of Table 3. Moreover, the T-SF weighted characteristics were determined by the use of ${T_{i}^{W}}=\{\langle {c_{j}},{t_{ij}^{w}}\rangle \hspace{0.1667em}|\hspace{0.1667em}{c_{j}}\in C\}=\{\langle {c_{j}},({\mu _{ij}^{w}},{\eta _{ij}^{w}},{\nu _{ij}^{w}})\rangle \hspace{0.1667em}|\hspace{0.1667em}{c_{j}}\in \{{c_{1}},{c_{2}},{c_{3}},{c_{4}}\}\}$ for each ${a_{i}}$. As an illustration, ${T_{1}^{W}}=\{\langle {c_{1}},(0.4003,0.1867,0.5750)\rangle ,\langle {c_{2}},(0.4046,0.2683,0.5031)\rangle ,\langle {c_{3}},(0.8422,0.6136,0.1060)\rangle ,\langle {c_{4}},(0.2104,0.3841,0.7406)\rangle \}$.

In Step 6, based on the universal and null T-SF sets, the T-SF characteristic ${T_{+}}=\{\langle {c_{1}},(0,0,1)\rangle ,\langle {c_{2}},(1,0,0)\rangle ,\langle {c_{3}},(1,0,0)\rangle ,\langle {c_{4}},(0,0,1)\rangle \}$ because of ${C_{\textit{Po}}}=\{{c_{2}},{c_{3}}\}$ and ${C_{\textit{Ne}}}=\{{c_{1}},{c_{4}}\}$. Moreover, ${T_{-}}=\{\langle {c_{1}},(1,0,0)\rangle ,\langle {c_{2}},(0,0,1)\rangle ,\langle {c_{3}},(0,0,1)\rangle ,\langle {c_{4}},(1,0,0)\rangle \}$. 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, $\textit{IE}({T_{1}^{W}})={\textstyle\sum _{j=1}^{4}}[{({({\mu _{1j}^{w}})^{q}})^{2}}+{({({\eta _{1j}^{w}})^{q}})^{2}}+{({({\nu _{1j}^{w}})^{q}})^{2}}+{({({\gamma _{1j}^{w}})^{q}})^{2}}$]. The respective computation results of ${({\mu _{ij}^{w}})^{q}}$, ${({\eta _{ij}^{w}})^{q}}$, ${({\nu _{ij}^{w}})^{q}}$, and ${({\gamma _{ij}^{w}})^{q}}$ are shown in the fifth to eighth columns of Table 3. Moreover, their corresponding squared sum, i.e. ${({({\mu _{ij}^{w}})^{q}})^{2}}+{({({\eta _{ij}^{w}})^{q}})^{2}}+{({({\nu _{ij}^{w}})^{q}})^{2}}+{({({\gamma _{ij}^{w}})^{q}})^{2}}$, 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 $\textit{IE}({T_{1}^{W}})=0.5868+0.6405+0.4393+0.4469=2.1135$. In the same fashion, $\textit{IE}({T_{2}^{W}})=2.1830$, $\textit{IE}({T_{3}^{W}})=2.6062$, and $\textit{IE}({T_{4}^{W}})=1.9043$, as shown in the second column of Table 4. Next, the T-SF weighted correlation functions $\textit{CF}({T_{i}^{W}},{T_{+}^{W}})$ and $\textit{CF}({T_{i}^{W}},{T_{-}^{W}})$ were acquired using Eqs. (5) and (6), respectively. To give an example, $\textit{CF}({T_{1}^{W}},{T_{+}^{W}})={\textstyle\sum _{{c_{j}}\in {C_{\textit{Po}}}}}{({\mu _{1j}^{w}})^{q}}+{\textstyle\sum _{{c_{j}}\in {C_{\textit{Ne}}}}}{({\nu _{1j}^{w}})^{q}}=({({\mu _{12}^{w}})^{q}}+{({\mu _{13}^{w}})^{q}})+({({\nu _{11}^{w}})^{q}}+{({\nu _{14}^{w}})^{q}})=(0.0662+0.5974)+(0.1901+0.4062)=1.2599$. The outcomes of $\textit{CF}({T_{i}^{W}},{T_{+}^{W}})$ and $\textit{CF}({T_{i}^{W}},{T_{-}^{W}})$ are displayed in the third and fourth columns of Table 4.

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 ${\textit{CC}_{\surd }}({T_{i}^{W}},{T_{+}^{W}})$ and ${\textit{CC}_{\surd }}({T_{i}^{W}},{T_{-}^{W}})$ using Eqs. (7) and (8), respectively. It was recognized that $\textit{IE}({T_{+}^{W}})=\textit{IE}({T_{-}^{W}})=n=4$ following Theorem 3. To give an instance, ${\textit{CC}_{\surd }}({T_{1}^{W}},{T_{+}^{W}})=\textit{CF}({T_{1}^{W}},{T_{+}^{W}})/\sqrt{\textit{IE}({T_{1}^{W}})\cdot \textit{IE}({T_{+}^{W}})}=1.2599/\sqrt{2.1135\times 4}=0.4333$, and ${\textit{CC}_{\surd }}({T_{1}^{W}},{T_{-}^{W}})=\textit{CF}({T_{1}^{W}},{T_{-}^{W}})/\sqrt{\textit{IE}({T_{1}^{W}})\cdot \textit{IE}({T_{-}^{W}})}=0.2020/\sqrt{2.1135\times 4}=0.0695$. The obtained outcomes of ${\textit{CC}_{\surd }}({T_{i}^{W}},{T_{+}^{W}})$ and ${\textit{CC}_{\surd }}({T_{i}^{W}},{T_{-}^{W}})$ 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 ${\textit{CC}_{\wedge }}({T_{i}^{W}},{T_{+}^{W}})$ and ${\textit{CC}_{\wedge }}({T_{i}^{W}},{T_{-}^{W}})$. The yielded outcomes are manifested in the last two columns of Table 4. For example, ${\textit{CC}_{\wedge }}({T_{i}^{W}},{T_{+}^{W}})=\textit{CF}({T_{1}^{W}},{T_{+}^{W}})/\max \{\textit{IE}({T_{1}^{W}}),\textit{IE}({T_{+}^{W}})\}=1.2599/\max \{2.1135,4\}=0.3150$, and ${\textit{CC}_{\wedge }}({T_{i}^{W}},{T_{-}^{W}})=\textit{CF}({T_{1}^{W}},{T_{-}^{W}})/\max \{\textit{IE}({T_{1}^{W}}),\textit{IE}({T_{-}^{W}})\}=0.2020/\max \{2.1135,4\}=0.0505$.

In Step 9, in the light of Definition 13, the following minimal and maximal correlation coefficients were produced as: $\underline{\textit{CC}}{\hspace{0.1667em}_{\surd }^{+}}={\min _{{i^{\prime }}=1}^{4}}{\textit{CC}_{\surd }}({T_{{i^{\prime }}}^{W}},{T_{+}^{W}})=\min \{0.4333,0.3709,0.1544,0.7049\}=0.1544$, $\overline{\textit{CC}}{\hspace{0.1667em}_{\surd }^{+}}={\max _{{i^{\prime }}=1}^{4}}{\textit{CC}_{\surd }}({T_{{i^{\prime }}}^{W}},{T_{+}^{W}})=0.7049$, $\underline{\textit{CC}}{\hspace{0.1667em}_{\surd }^{-}}={\min _{{i^{\prime }}=1}^{4}}{\textit{CC}_{\surd }}({T_{{i^{\prime }}}^{W}},{T_{-}^{W}})=\min \{0.0695,0.1749,0.1274,0.0852\}=0.0695$, and $\overline{\textit{CC}}{\hspace{0.1667em}_{\surd }^{-}}={\max _{{i^{\prime }}=1}^{4}}{\textit{CC}_{\surd }}({T_{{i^{\prime }}}^{W}},{T_{-}^{W}})=0.1749$ for the “square root function” type. In a similar fashion, it was yielded that $\underline{\textit{CC}}{\hspace{0.1667em}_{\wedge }^{+}}=\min \{0.3150,0.2740,0.1246,0.4864\}=0.1246$, $\overline{\textit{CC}}{\hspace{0.1667em}_{\wedge }^{+}}=0.4864$, $\underline{\textit{CC}}{\hspace{0.1667em}_{\wedge }^{-}}=0.0505$, and $\overline{\textit{CC}}{\hspace{0.1667em}_{\wedge }^{-}}=0.1292$. Letting the anchoring parameter $\xi =0.5$, the T-SF comprehensive correlation indices were calculated using Eq. (11) for the “square root function” type. That is, ${\textit{CI}_{\surd }}({a_{1}})=0.6\times ({\textit{CC}_{\surd }}({T_{1}^{W}},{T_{+}^{W}})-\underline{\textit{CC}}{\hspace{0.1667em}_{\surd }^{+}})/(\overline{\textit{CC}}{\hspace{0.1667em}_{\surd }^{+}}-\underline{\textit{CC}}{\hspace{0.1667em}_{\surd }^{+}})+0.4\times (\overline{\textit{CC}}{\hspace{0.1667em}_{\surd }^{-}}-{\textit{CC}_{\surd }}({T_{1}^{W}},{T_{-}^{W}}))/\hspace{2.5pt}(\overline{\textit{CC}}{\hspace{0.1667em}_{\surd }^{-}}-\underline{\textit{CC}}{\hspace{0.1667em}_{\surd }^{-}})=0.6\times (0.4333-0.1544)/(0.7049-0.1544)+0.4\times (0.1749-0.0695)/(0.1749-0.0695)=0.7040,{\textit{CI}_{\surd }}({a_{2}})=0.2360$, ${\textit{CI}_{\surd }}({a_{3}})=0.1801$, and ${\textit{CI}_{\surd }}({a_{4}})=0.9402$. Next, for the “maximum function” type, the T-SF comprehensive correlation index ${\textit{CI}_{\wedge }}({a_{i}})$ was generated using Eq. (12). Specifically, ${\textit{CI}_{\wedge }}({a_{1}})=0.6\times (0.3150-0.1246)/(0.4864-0.1246)+0.4\times (0.1292-0.0505)/(0.1292-0.0505)=0.7157,{\textit{CI}_{\wedge }}({a_{2}})=0.2478,{\textit{CI}_{\wedge }}({a_{3}})=0.1339,and{\textit{CI}_{\wedge }}({a_{4}})=0.9578$.

Finally, in Step 10, the four location options were ranked in descending order of the ${\textit{CI}_{\surd }}({a_{i}})$ values for the “square root function” type, which rendered the prioritization ranking ${a_{4}}{\succ _{\surd }}{a_{1}}{\succ _{\surd }}{a_{2}}{\succ _{\surd }}{a_{3}}$. Moreover, the prioritization ranking ${a_{4}}{\succ _{\wedge }}{a_{1}}{\succ _{\wedge }}{a_{2}}{\succ _{\wedge }}{a_{3}}$ was yielded in descending order of ${\textit{CI}_{\wedge }}({a_{i}})$ 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.

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 ${C_{\textit{Ne}}}$ should be reversed to be consistent with the direction of the positive criteria in the collection ${C_{\textit{Po}}}$. Let ${t^{\prime }_{ij}}=({\mu ^{\prime }_{ij}},{\eta ^{\prime }_{ij}},{\nu ^{\prime }_{ij}})$ signify the normalized T-SF performance rating associated with ${t_{ij}}$. Using the means of the complement set operation, the T-SF characteristic ${T_{i}}$ can be transformed into the normalized T-SF characteristic ${T^{\prime }_{i}}$ using the following formula:

This comparative study endeavours to aggregate the normalized T-SF performance rating ${t^{\prime }_{ij}}$ across all ${c_{j}}\in C$ concerning each ${a_{i}}$ 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 $\phi >1$ and $\Phi \geqslant 1$ denote the parameters contained in Mahnaz et al.’s and Hussain et al.’s formulations, respectively. The T-SF comprehensive evaluation value of ${t^{\prime }_{i1}},{t^{\prime }_{i2}},\dots ,{t^{\prime }_{in}}$ using the T-SF WA, T-SF FWA, and T-SF AAWA operators are determined sequentially along these lines:

From the geometric mean perspective, the T-SF comprehensive evaluation value of ${t^{\prime }_{i1}},{t^{\prime }_{i2}},\dots ,{t^{\prime }_{in}}$ using the T-SF WG, T-SF FWG, and T-SF AAWG operators are calculated sequentially in the following manner, where $\phi >1$ and $\Phi \geqslant 1$:

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 ${t^{\prime }_{i}}=({\mu ^{\prime }_{i}},{\eta ^{\prime }_{i}},{\nu ^{\prime }_{i}})$ 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 ${\gamma ^{\prime }_{i}}=\sqrt[q]{1-{({\mu ^{\prime }_{i}})^{q}}-{({\eta ^{\prime }_{i}})^{q}}-{({\nu ^{\prime }_{i}})^{q}}}$. Following Zeng et al.’s formulation, the aggregated score value of ${t^{\prime }_{i}}$ is elucidated like this:

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 ${t^{\prime }_{i}}$, 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 $\phi =2$ in Eqs. (15) and (18) and $\Phi =5$ 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. ${\mu ^{\prime }_{i}}$, ${\eta ^{\prime }_{i}}$, and ${\nu ^{\prime }_{i}}$, respectively) contained in ${t^{\prime }_{i}}$ are sketched in Fig. 4.

Next, this study used Eq. (20) to generate the aggregated score value $AS({t^{\prime }_{i}})$ 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 (${\textit{CI}_{\surd }}({a_{i}})$ and ${\textit{CI}_{\wedge }}({a_{i}})$ 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 ${a_{4}}\succ {a_{1}}\succ {a_{3}}\succ {a_{2}}$. 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 ${a_{4}}\succ {a_{1}}\succ {a_{2}}\succ {a_{3}}$. The use of the technique using the T-SF AAWG operator yielded a particularly different ordering result ${a_{4}}\succ {a_{2}}\succ {a_{1}}\succ {a_{3}}$. 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. ${a_{4}}\succ {a_{1}}\succ {a_{2}}\succ {a_{3}}$) 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 ${a_{4}}\succ {a_{1}}\succ {a_{3}}\succ {a_{2}}$ and ${a_{4}}\succ {a_{2}}\succ {a_{1}}\succ {a_{3}}$ reduces to 0.4.

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 ${a_{4}}$ performed the best among all comparative approaches, while the choice option ${a_{3}}$ 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. ${a_{4}}$ and ${a_{3}}$, respectively) are contrasted in Fig. 5(b) to highlight their juxtaposition.

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. ${a_{4}}\succ {a_{1}}\succ {a_{3}}\succ {a_{2}}$) obtained by the techniques using the T-SF WA, WG, FWA, and FWG operators differs from the benchmark ranking (i.e. ${a_{4}}\succ {a_{1}}\succ {a_{2}}\succ {a_{3}}$) based on the T-SF GGHG operator in the outranking relationship between ${a_{2}}$ and ${a_{3}}$. The difference between the prioritization ranking (i.e. ${a_{4}}\succ {a_{2}}\succ {a_{1}}\succ {a_{3}}$) 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 ${a_{1}}$ and ${a_{2}}$. 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.

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 ${\textit{CI}_{\surd }}({a_{i}})$ (based on the square root function) and ${\textit{CI}_{\wedge }}({a_{i}})$ (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 ${\textit{CI}_{\surd }}({a_{i}})$ and ${\textit{CI}_{\wedge }}({a_{i}})$ for distinct values of ξ are portrayed in Fig. 6(a) and Fig. 6(b), respectively.

As depicted in Fig. 6(a), the three prioritization rankings ${a_{1}}{\succ _{\surd }}{a_{4}}{\succ _{\surd }}{a_{3}}{\succ _{\surd }}{a_{2}}$, ${a_{4}}{\succ _{\surd }}{a_{1}}{\succ _{\surd }}{a_{3}}{\succ _{\surd }}{a_{2}}$, and ${a_{4}}{\succ _{\surd }}{a_{1}}{\succ _{\surd }}{a_{2}}{\succ _{\surd }}{a_{3}}$ were generated when $\xi =0.0$, 0.1, 0.2, $\xi =0.3$, 0.4, 0.5, and $\xi =0.6,0.7,\dots ,1.0$, respectively. As revealed in Fig. 6(b), the rankings ${a_{1}}{\succ _{\wedge }}{a_{4}}{\succ _{\wedge }}{a_{3}}{\succ _{\wedge }}{a_{2}}$, ${a_{4}}{\succ _{\wedge }}{a_{1}}{\succ _{\wedge }}{a_{3}}{\succ _{\wedge }}{a_{2}}$, and ${a_{4}}{\succ _{\wedge }}{a_{1}}{\succ _{\wedge }}{a_{2}}{\succ _{\wedge }}{a_{3}}$ were produced when $\xi =0.0,0.1$, $\xi =0.2,0.3,0.4$, and $\xi =0.5,0.6,\dots ,1.0$, 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 $\xi =0.2$ and $\xi =0.5$ are inconsistent. On the other hand, it is worth mentioning that the obtained ${\textit{CI}_{\surd }}({a_{i}})$ and ${\textit{CI}_{\wedge }}({a_{i}})$ values gave rise to identical rankings (i.e. the prioritization rankings ${a_{4}}{\succ _{\surd }}{a_{1}}{\succ _{\surd }}{a_{2}}{\succ _{\surd }}{a_{3}}$ and ${a_{4}}{\succ _{\wedge }}{a_{1}}{\succ _{\wedge }}{a_{2}}{\succ _{\wedge }}{a_{3}}$ when $\xi =0.6,0.7,\dots ,1.0$ and $\xi =0.5,0.6,\dots ,1.0$, 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 ${a_{4}}{\succ _{\surd }}{a_{1}}{\succ _{\surd }}{a_{3}}{\succ _{\surd }}{a_{2}}$ (based on the square root function) and ${a_{4}}{\succ _{\wedge }}{a_{1}}{\succ _{\wedge }}{a_{3}}{\succ _{\wedge }}{a_{2}}$ (based on the maximum function) were acquired when $\xi =0.3,0.4,0.5$ and $\xi =0.2,0.3,0.4$, respectively. Nonetheless, different outcomes were yielded in face of the small values of ξ, namely ${a_{1}}{\succ _{\surd }}{a_{4}}{\succ _{\surd }}{a_{3}}{\succ _{\surd }}{a_{2}}$ and ${a_{1}}{\succ _{\wedge }}{a_{4}}{\succ _{\wedge }}{a_{3}}{\succ _{\wedge }}{a_{2}}$ when $\xi =0.0,0.1,0.2$ and $\xi =0.0,0.1$, respectively. Overall, stable and justified consequences can be generated under most settings of ξ. When $\xi =0.0,0.1,0.2$ based on the “square root function” type or $\xi =0.0,0.1$ 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 ${a_{\ast }}$ indicate the positive ideal scheme, where the T-SF characteristic ${T_{\ast }}=\{\langle {c_{j}},{t_{\ast j}}\rangle \hspace{0.1667em}|\hspace{0.1667em}{c_{j}}\in C\}=\{\langle {c_{j}},({\mu _{\ast j}},{\eta _{\ast j}},{\nu _{\ast j}})\rangle \hspace{0.1667em}|\hspace{0.1667em}{c_{j}}\in C\}$. Let ${a_{\mathrm{\# }}}$ signify the negative ideal scheme, where the T-SF characteristic ${T_{\mathrm{\# }}}=\{\langle {c_{j}},{t_{\mathrm{\# }j}}\rangle \hspace{0.1667em}|\hspace{0.1667em}{c_{j}}\in C\}=\{\langle {c_{j}},({\mu _{\mathrm{\# }j}},{\eta _{\mathrm{\# }j}},{\nu _{\mathrm{\# }j}})\rangle \hspace{0.1667em}|\hspace{0.1667em}{c_{j}}\in C\}$. Utilizing the set operations ∪ and ∩, ${T_{\ast }}$ and ${T_{\mathrm{\# }}}$ are delineated in this fashion:

Recall that ${C_{\textit{Po}}}=\{{c_{2}},{c_{3}}\}$ and ${C_{\textit{Ne}}}=\{{c_{1}},{c_{4}}\}$ in the location selection problem. Using the aforesaid manner, the T-SF characteristics of ${a_{\ast }}$ and ${a_{\mathrm{\# }}}$ were identified as follows: ${T_{\ast }}=\{\langle {c_{1}},(0.14,0.12,0.83)\rangle ,\langle {c_{2}},(0.91,0.12,0.13)\rangle ,\langle {c_{3}},(0.81,0.11,0.11)\rangle ,\langle {c_{4}},(0.11,0.14,0.84)\rangle \}$ and ${T_{\mathrm{\# }}}=\{\langle {c_{1}},(0.75,0.12,0.41)\rangle ,\langle {c_{2}},(0.26,0.12,0.63)\rangle ,\langle {c_{3}},(0.56,0.11,0.55)\rangle ,\langle {c_{4}},(0.61,0.14,0.45)\rangle \}$. The corresponding T-SF weighted characteristics were given by: ${T_{\ast }^{W}}=\{\langle {c_{1}},(0.1300,0.1156,0.8012)\rangle ,\langle {c_{2}},(0.7542,0.1171,0.1009)\rangle ,\langle {c_{3}},(0.8422,0.1147,0.1060)\rangle ,\langle {c_{4}},(0.1286,0.1637,0.8922)\rangle \}$ and ${T_{\mathrm{\# }}^{W}}=\{\langle {c_{1}},(0.7080,0.1156,0.3965)\rangle ,\langle {c_{2}},(0.1919,0.1171,0.5083)\rangle ,\langle {c_{3}},(0.5914,0.1147,0.5538)\rangle ,\langle {c_{4}},(0.6963,0.1637,0.5210)\rangle \}$. The T-SF weighted informational energies were derived as: $\textit{IE}({T_{\ast }^{W}})=2.1053$ and $\textit{IE}({T_{\mathrm{\# }}^{W}})=2.0837$. The comparisons of the T-SF weighted correlation functions $\textit{CF}({T_{i}^{W}},{T_{+}^{W}})$, $\textit{CF}({T_{i}^{W}},{T_{\ast }^{W}})$, $\textit{CF}({T_{i}^{W}},{T_{-}^{W}})$, and $\textit{CF}({T_{i}^{W}},{T_{\mathrm{\# }}^{W}})$ are manifested in Fig. 7. Furthermore, the T-SF weighted correlation coefficients ${\textit{CC}_{\surd }}({T_{i}^{W}},{T_{+}^{W}})$, ${\textit{CC}_{\surd }}({T_{i}^{W}},{T_{\ast }^{W}})$, ${\textit{CC}_{\surd }}({T_{i}^{W}},{T_{-}^{W}})$, and ${\textit{CC}_{\surd }}({T_{i}^{W}},{T_{\mathrm{\# }}^{W}})$ are contrasted in Fig. 8(a), while the comparisons of ${\textit{CC}_{\wedge }}({T_{i}^{W}},{T_{+}^{W}})$, ${\textit{CC}_{\wedge }}({T_{i}^{W}},{T_{\ast }^{W}})$, ${\textit{CC}_{\wedge }}({T_{i}^{W}},{T_{-}^{W}})$, and ${\textit{CC}_{\wedge }}({T_{i}^{W}},{T_{\mathrm{\# }}^{W}})$ are exhibited in Fig. 8(b).

First, consider the contrast outcomes of the T-SF weighted correlation functions concerning the best choice option ${a_{+}}$ and the positive ideal scheme ${a_{\ast }}$, as revealed in Fig. 5. The differences among the $\textit{CF}({T_{i}^{W}},{T_{+}^{W}})$ values of the four choice options (${a_{1}}-{a_{4}}$) were significantly higher than the differences among the $\textit{CF}({T_{i}^{W}},{T_{\ast }^{W}})$ values. In particular, the gap between the maximum value (i.e. $\textit{CF}({T_{4}^{W}},{T_{+}^{W}})$) and the minimum value (i.e. $\textit{CF}({T_{3}^{W}},{T_{+}^{W}})$) was quite pronounced. However, the gap between the maximum value (i.e. $\textit{CF}({T_{4}^{W}},{T_{\ast }^{W}})$) and the minimum value (i.e. $\textit{CF}({T_{3}^{W}},{T_{\ast }^{W}})$) 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 ${a_{\mathrm{\# }}}$, the maximum value of $\textit{CF}({T_{i}^{W}},{T_{-}^{W}})$ and the maximum value of $\textit{CF}({T_{i}^{W}},{T_{\mathrm{\# }}^{W}})$ correspond to different options; the same is true for the minimum value of $\textit{CF}({T_{i}^{W}},{T_{-}^{W}})$ (or $\textit{CF}({T_{i}^{W}},{T_{\mathrm{\# }}^{W}})$). To be precise, the options ${a_{2}}$ and ${a_{1}}$ enjoy the largest and smallest values, respectively, of $\textit{CF}({T_{i}^{W}},{T_{-}^{W}})$; ${a_{3}}$ and ${a_{4}}$ enjoy the largest and smallest values, respectively, of $\textit{CF}({T_{i}^{W}},{T_{\mathrm{\# }}^{W}})$.

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 ${\textit{CC}_{\surd }}({T_{i}^{W}},{T_{\ast }^{W}})$ values of the four choice options were significantly higher than the ${\textit{CC}_{\surd }}({T_{i}^{W}},{T_{+}^{W}})$ values; this phenomenon was also found in the comparisons of the values of ${\textit{CC}_{\surd }}({T_{i}^{W}},{T_{\mathrm{\# }}^{W}})$ and ${\textit{CC}_{\surd }}({T_{i}^{W}},{T_{-}^{W}})$. The higher the value of ${\textit{CC}_{\surd }}({T_{i}^{W}},{T_{+}^{W}})$ (or ${\textit{CC}_{\surd }}({T_{i}^{W}},{T_{\ast }^{W}})$), the higher the correlation between the corresponding option ${a_{i}}$ and ${a_{+}}$ (or ${a_{\ast }}$). 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 ${\textit{CC}_{\surd }}({T_{i}^{W}},{T_{-}^{W}})$ (or ${\textit{CC}_{\surd }}({T_{i}^{W}},{T_{\mathrm{\# }}^{W}})$), the lower the correlation between ${a_{i}}$ and ${a_{-}}$ (or ${a_{\mathrm{\# }}}$). 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 ${a_{\ast }}$ were ${\textit{CC}_{\surd }}({T_{4}^{W}},{T_{+}^{W}})>{\textit{CC}_{\surd }}({T_{1}^{W}},{T_{+}^{W}})>{\textit{CC}_{\surd }}({T_{2}^{W}},{T_{+}^{W}})>{\textit{CC}_{\surd }}({T_{3}^{W}},{T_{+}^{W}})$ and ${\textit{CC}_{\surd }}({T_{4}^{W}},{T_{\ast }^{W}})>{\textit{CC}_{\surd }}({T_{1}^{W}},{T_{\ast }^{W}})>{\textit{CC}_{\surd }}({T_{2}^{W}},{T_{\ast }^{W}})>{\textit{CC}_{\surd }}({T_{3}^{W}},{T_{\ast }^{W}})$, respectively. Different from the identical ranking orders above, the numerical orders of the T-SF weighted correlation coefficients for mutual relationships with ${a_{-}}$ and ${a_{\mathrm{\# }}}$ were ${\textit{CC}_{\surd }}({T_{1}^{W}},{T_{-}^{W}})<{\textit{CC}_{\surd }}({T_{4}^{W}},{T_{-}^{W}})<{\textit{CC}_{\surd }}({T_{3}^{W}},{T_{-}^{W}})<{\textit{CC}_{\surd }}({T_{2}^{W}},{T_{-}^{W}})$ and ${\textit{CC}_{\surd }}({T_{4}^{W}},{T_{\mathrm{\# }}^{W}})<{\textit{CC}_{\surd }}({T_{1}^{W}},{T_{\mathrm{\# }}^{W}})<{\textit{CC}_{\surd }}({T_{2}^{W}},{T_{\mathrm{\# }}^{W}})<{\textit{CC}_{\surd }}({T_{3}^{W}},{T_{\mathrm{\# }}^{W}})$, 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 ${\textit{CC}_{\wedge }}({T_{i}^{W}},{T_{\mathrm{\# }}^{W}})$. Specifically, the numerical orders of ${\textit{CC}_{\wedge }}({T_{i}^{W}},{T_{\mathrm{\# }}^{W}})$ were given by ${\textit{CC}_{\wedge }}({T_{4}^{W}},{T_{\mathrm{\# }}^{W}})<{\textit{CC}_{\wedge }}({T_{1}^{W}},{T_{\mathrm{\# }}^{W}})<{\textit{CC}_{\wedge }}({T_{3}^{W}},{T_{\mathrm{\# }}^{W}})<{\textit{CC}_{\wedge }}({T_{2}^{W}},{T_{\mathrm{\# }}^{W}})$. In the matter of the numerical orders of the T-SF weighted correlation coefficients among the four options, the two ranking outcomes employing ${\textit{CC}_{\surd }}({T_{i}^{W}},{T_{+}^{W}})$ and ${\textit{CC}_{\surd }}({T_{i}^{W}},{T_{\ast }^{W}})$ were consistent; however, the two ranking outcomes via ${\textit{CC}_{\surd }}({T_{i}^{W}},{T_{-}^{W}})$ and ${\textit{CC}_{\surd }}({T_{i}^{W}},{T_{\mathrm{\# }}^{W}})$ 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 $\xi =0.0,0.1,\dots ,1.0$. Let $\underline{\textit{CC}}{\hspace{0.1667em}_{\surd }^{\ast }}={\min _{{i^{\prime }}=1}^{4}}{\textit{CC}_{\surd }}({T_{{i^{\prime }}}^{W}},{T_{\ast }^{W}})$, $\overline{\textit{CC}}{\hspace{0.1667em}_{\surd }^{\ast }}={\max _{{i^{\prime }}=1}^{4}}{\textit{CC}_{\surd }}({T_{{i^{\prime }}}^{W}},{T_{\ast }^{W}}$), $\underline{\textit{CC}}{\hspace{0.1667em}_{\surd }^{\mathrm{\# }}}={\min _{{i^{\prime }}=1}^{4}}{\textit{CC}_{\surd }}({T_{{i^{\prime }}}^{W}},{T_{\mathrm{\# }}^{W}})$, and $\overline{\textit{CC}}{\hspace{0.1667em}_{\surd }^{\mathrm{\# }}}={\max _{{i^{\prime }}=1}^{4}}\hspace{-0.1667em}{\textit{CC}_{\surd }}({T_{{i^{\prime }}}^{W}},{T_{\mathrm{\# }}^{W}})$ for the “square root function” type. Let $\underline{\textit{CC}}{\hspace{0.1667em}_{\wedge }^{\ast }}\hspace{-0.1667em}=\hspace{-0.1667em}{\min _{{i^{\prime }}=1}^{4}}{\textit{CC}_{\wedge }}({T_{{i^{\prime }}}^{W}},{T_{\ast }^{W}})$, $\overline{\textit{CC}}{\hspace{0.1667em}_{\wedge }^{\ast }}={\max _{{i^{\prime }}=1}^{4}}{\textit{CC}_{\wedge }}({T_{{i^{\prime }}}^{W}},{T_{\ast }^{W}})$, $\underline{\textit{CC}}{\hspace{0.1667em}_{\wedge }^{\mathrm{\# }}}={\min _{{i^{\prime }}=1}^{4}}{\textit{CC}_{\wedge }}({T_{{i^{\prime }}}^{W}},{T_{\mathrm{\# }}^{W}})$, and $\overline{\textit{CC}}{\hspace{0.1667em}_{\wedge }^{\mathrm{\# }}}={\max _{{i^{\prime }}=1}^{4}}{\textit{CC}_{\wedge }}({T_{{i^{\prime }}}^{W}},{T_{\mathrm{\# }}^{W}})$ for the “maximum function” type. On the grounds of the ideal schemes ${a_{\ast }}$ and ${a_{\mathrm{\# }}}$, the T-SF comprehensive correlation indices ${\textit{CI}^{\prime }_{\surd }}({a_{i}})$ and ${\textit{CI}^{\prime }_{\wedge }}({a_{i}})$ are elucidated in this fashion:

Regarding the “square root function” type, the juxtaposition of the T-SF comprehensive correlation indices ${\textit{CI}_{\surd }}({a_{i}})$ and ${\textit{CI}^{\prime }_{\surd }}({a_{i}})$ 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 ${a_{\ast }}$ and ${a_{\mathrm{\# }}}$. For the “maximum function” type, the juxtaposition of the T-SF comprehensive correlation indices ${\textit{CI}_{\surd }}({a_{i}})$ and ${\textit{CI}^{\prime }_{\surd }}({a_{i}})$ 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 ${a_{\ast }}$ and ${a_{\mathrm{\# }}}$.

On the grounds of the reference points of the ideal schemes ${a_{\ast }}$ and ${a_{\mathrm{\# }}}$, 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. ${\textit{CI}^{\prime }_{\surd }}({a_{3}})=0$ and ${\textit{CI}^{\prime }_{\surd }}({a_{4}})=1$ for all $\xi =0.0,0.1,\dots ,1.0$, as displayed in Fig. 9(b). Additionally, it was received that ${\textit{CI}^{\prime }_{\wedge }}({a_{4}})=1$ predicated on the “maximum function” type for all $\xi =0.0,0.1,\dots ,1.0$, as displayed in Fig. 10(b). Regardless of how the value of the anchoring parameter ξ changed, the indices ${\textit{CI}^{\prime }_{\surd }}({a_{3}})$, ${\textit{CI}^{\prime }_{\surd }}({a_{4}})$, and ${\textit{CI}^{\prime }_{\wedge }}({a_{4}})$ 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.