Informatica

Realistic decision-making activities are usually highly sophisticated and poorly structured, and many classical decision models cannot directly deal with these complicated problems (Alipour et al., 2021; Garg, 2021b; Garg and Rani, 2021; Ullah et al., 2021b). Numerous classical decision models manage crisp assessment data, which means that the subjective judgment offered by the decision maker is expressed as a precise number. Nevertheless, in considerable down-to-earth situations, the decision information may be inaccurate and/or imprecise (Alipour et al., 2021; Garg, 2021a; Ullah et al., 2021a; Wang and Chen, 2021). Moreover, the decision maker may be unable to explicitly give accurate numerical values for subjective evaluations in uncertain circumstances (Gao and Deng, 2021; Garg, 2021b; Garg and Rani, 2021). In particular, certain judgment criteria are qualitative or equivocal in nature, making it difficult for the decision maker to exploit precise values to externalize subjective assessments and preferences (Alipour et al., 2021; Garg and Rani, 2021; Ullah et al., 2021b). Convoluted decision-making issues usually involve inaccuracy, ambiguity, and indefiniteness, resulting in traditional decision models and relevant canonical techniques often ineffective when manipulating subjective assessment information (Garg, 2021a, 2021b; Wang and Chen, 2021). The aforementioned difficulties and considerations make the notion of fuzzy sets flourish in decision theory (Alipour et al., 2021; Gao and Deng, 2021; Garg, 2021b).

There are different general variants of the fuzzy models that delineate an object’s membership in a fuzzy set in various formats (Smarandache, 2019; Ullah et al., 2020a, 2020b). The notion of ordinary fuzzy sets generalized classical sets and permits a gradual appraisal about an object’s membership in a set. The real world would be full of indeterminism, vagueness, and limited knowledge; thus, the complexities associated with the high-order fuzziness are dependent in a sophisticated way on their uncertainty (Chen, 2021; Donyatalab et al., 2020; Farrokhizadeh et al., 2021; Gül, 2021). In such considerations, it is evidently meaningful to constitute non-standard fuzzy configurations for modelling imprecision and murkiness in recent decades, such as advanced fuzzy models regarding intuitionistic fuzziness (Atanassov, 1986), Pythagorean fuzziness (Yager, 2013), Fermatean fuzziness (Senapati and Yager, 2019a, 2019b), q-rung orthopair fuzziness (Yager, 2017), picture fuzziness (Cuong, 2014), spherical fuzziness (Gündoğdu and Kahraman, 2019), and T-spherical fuzziness (Mahmood et al., 2019). As an efficacious means to expound ambiguous and equivocal information, the generalizations of fuzzy sets exhibit a mathematical strength to expatiate on the uncertainty in subjective thinking and cognitive processes for the reasoning behind intricate decisions in a logical and sensible way (Garg, 2021a; Senapati and Yager, 2019b; Shahzadi et al., 2021; Ullah et al., 2020a). After introducing the generalizations of fuzzy sets, the ordinary fuzzy version of decision models and techniques also began to attain full developments via these higher-order fuzzy sets. Especially, the configuration involving T-spherical fuzziness is a recent advancement in fuzzy theory and has a magnificent capability of tackling decision-making in multiple criteria choice issues (Chen et al., 2021; Garg et al., 2021; Wang and Chen, 2021; Zeng et al., 2020).

The conceptual framework of T-spherical fuzzy (T-SF) sets, initially introduced by Mahmood et al. (2019), is a significant tool for decision makers with multiple criteria evaluation and assessment problems under complicated uncertain circumstances. The uncertain sets of intuitionistic, Pythagorean, Fermatean, and q-rung orthopair fuzziness are elucidated by virtue of belonging and non-belonging functions. More precisely, a total sum of both functions takes a value in a real unit interval from 0 to 1 in the intuitionistic fuzzy model (Atanassov, 1986); the square sum of both functions is valued in $[0,1]$ in the Pythagorean fuzzy model (Yager and Abbasov, 2013); the cube sum of both functions is valued in $[0,1]$ in the Fermatean fuzzy model (Senapati and Yager, 2019a, 2019b); the sum of both functions to the q-th power (q is a positive integer) is valued in $[0,1]$ predicated on the q-rung orthopair fuzzy model (Khan et al., 2021a; Yager, 2017). Herein, the residual term (i.e. the length of the real unit interval excluding the belonging and non-belonging parts) is considered as the grade of indeterminacy. Especially, the conception of q-rung orthopair fuzziness can be deliberated as a broad-ranging kind of non-standard fuzzy models because it becomes the intuitionistic, Pythagorean, and Fermatean fuzzy models when $q=1,2,3$, respectively (Akram et al., 2021a, 2021b; Khan et al., 2021a). The conceptions of picture fuzziness and spherical fuzziness, as well as T-SF sets, are delineated by way of three functions: the first is membership; the second is abstinence; the third is non-membership. Specifically, the sum of three functions takes a value in the interval $[0,1]$ in the picture fuzzy model (Cuong, 2014); the square sum of three functions is valued in $[0,1]$ in the spherical fuzzy model (Gündoğdu and Kahraman, 2019); the summation about three functions to the t-th power (t is a positive integer) is valued in $[0,1]$ in the T-SF model (Mahmood et al., 2019). The residual term (i.e. the length of the real unit interval excluding the belonging, abstinence, and non-belonging parts) is regarded as the grade of indeterminacy. The T-SF configuration reduces to the picture fuzzy and spherical fuzzy models with the conditions that $t=1$ and $t=2$, respectively. In consideration of the null degree of abstinence, T-SF sets are mathematically equivalent to q-rung orthopair fuzzy sets. Taking these discussions into consideration, the concept of T-SF sets is an all-encompassing structure of the before-mentioned non-standard fuzzy models (Chen et al., 2021; Garg et al., 2021; Ullah et al., 2020a, 2020b; Zeng et al., 2020).

From an alternative perspective, the concept of neutrosophic sets, incipiently propounded by Smarandache (2005a, 2005b), is worthy of note as well. The conception of neutrosophic sets can be deemed to be a generalized conformation of unification toward intuitionistic fuzzy logic (Jana et al., 2021; Nabeeh et al., 2021; Pamucar et al., 2020). The primary scheme behind neutrosophic logic is to describe the features of a three-dimensional neutrosophic space (Pamucar et al., 2020; Smarandache, 2019). Specifically, three dimensions of the space characterize the grade of truth-membership, the grade of falsehood-membership, and the grade of indeterminacy-membership that are equivalent to positive, negative, and refusal memberships, respectively, under uncertainty (Chen, 2021; Qin and Wang, 2020). In particular, these grades of membership are independent of each other in essence (Karaaslan and Hunu, 2020; Şahin and Liu, 2017). Notably, the most obvious differentiation between intuitionistic fuzzy logic and neutrosophic logic lies in whether the three membership degrees are independent or dependent (Karaaslan and Hunu, 2020; Pamucar et al., 2020; Smarandache, 2019). In view of the dependent components in intuitionistic fuzzy logic, when one membership degree changes, the other membership degrees need to be changed accordingly to meet the restriction for keeping their total sum being up to 1 (Atanassov, 1986; Smarandache, 2019). On the contrary, making allowance for the independent components in neutrosophic logic, when one membership degree becomes different, the other membership degrees are unnecessary to change accordingly, wherein the summation of three membership degrees is always up to 3 (Smarandache, 2005a, 2005b, 2019). When the decision maker evaluates the choice options, there is bound to be a certain degree of relevance in the assessments of the advantages and disadvantages of the options with respect to specific judgment criteria. In other words, it is impossible for the favourable and unfavourable evaluations of the same subject matter to be unrelated. Therefore, in neutrosophic logic, the assumption that the grades of positive membership and negative membership meet independence is not appropriate for decision-making problems. More precisely, the mechanism involving independent components is not suitable for practical multiple-criteria choice problems, for the reason that the interactions surrounded by three grades of membership have objective reality included in human appraisals and judgments to a great extent (Chen, 2021). The T-SF framework provides an all-encompassing model including intuitionistic fuzzy sets along with certain non-standard fuzzy sets; thus, it would be more appropriate than the neutrosophic framework to portray assessment information for multiple-criteria choice analysis in uncertain circumstances.

Due to the comprehensiveness of T-SF sets, the T-SF configuration serves an important tool to manipulate convoluted uncertainties for formulating and solving multiple-criteria choice problems. By way of illustration, Ali et al. (2020) put forward aggregation operators in complex T-SF settings for the sake of managing multiple-criteria evaluation affairs. With the assistance of the generalized parameter contained in T-SF sets, Chen et al. (2021) carried forward certain useful geometric aggregation operators with the aim of multiple-criteria choice analysis. Garg et al. (2021) launched several beneficial T-SF power aggregation operators to make headway for multiple-criteria evaluation and appraisement. Guleria and Bajaj (2021) progressed aggregation operators for T-SF soft sets to tackle decision-making issues. Ju et al. (2021) gave impetus to a T-SF TODIM (i.e. interactive and multiple-criteria decision making in Portuguese) technique for facilitating group decision-making tasks under incomplete preference information. Khan et al. (2021b) set forth a fresh evaluation approach using the agency of T-SF Schweizer-Sklar power Heronian aggregation operators for uncertain decisions. Liu et al. (2021a) promoted new and creative Muirhead mean aggregation operations via a complex 2-tuple linguistic structure in T-SF circumstances to treat decision analysis issues. Liu et al. (2021b) brought forward Maclaurin symmetric aggregation operators with normal T-SF numbers in the support of uncertain decision making. Munir et al. (2020) evolved T-SF Einstein hybrid aggregating operations to support the determination of choice options. Munir et al. (2021) propounded a T-SF decision-aiding algorithm on grounds of interactive geometric aggregation operations along with associated immediate probabilities. Özlü and Karaaslan (2021) delivered correlation coefficients concerning T-SF type-2 hesitant information and exploited these notions for solving an issue involving clustering the choice options. Wang and Chen (2021) propounded a T-SF ELECTRE (i.e. ELimination Et Choice Translating REality) outranking model for decision analysis with multiple criteria. On account of the advancement of T-SF decision models and techniques, this paper intends to develop an innovative multiple-criteria choice analysis approach using the agency of the T-SF theory for determining the predominance ranks of choice options or alternatives.

The REGIME method, incipiently propounded by Hinloopen et al. (1983a, 1983b), is a well-established technique concerning multiple-criteria choice analysis, especially for qualitative information (Alinezhad and Khalili, 2019; Oztaysi et al., 2021; Tsigdinos and Vlastos, 2021). Based on an efficient and convenient-to-use approach of paired comparisons for choice options, the REGIME method manipulates qualitative information (such as ordinal data) in a mathematically reasonable way (Oztaysi et al., 2022). Of course, the REGIME method accepts both qualitative and quantitative data, and the implementation procedure of this method is simple and easy to understand (Esangbedo et al., 2021; Kamran et al., 2017; Oztaysi et al., 2021). The characteristic of the classical REGIME framework is the formation of a REGIME matrix that collects outcomes about paired comparisons of choice options in an impact matrix (Esangbedo et al., 2021). Herein, the impact matrix elucidates the effect measurements of choice options on an amalgamation of quantitative and qualitative judgment criteria manifested in ordinal values (Alinezhad and Khalili, 2019; Tsigdinos and Vlastos, 2021). REGIME can conclusively generate a complete list of ranking for choice options via comparing the pairs with selected judgment criteria (Esangbedo et al., 2021; Kamran et al., 2017).

There was something unique about the theory of the REGIME methodology, which constitutes a distinctive outranking-based model for multiple-criteria evaluation and choice analysis. As is well known, the ELECTRE and the PROMETHEE (i.e. Preference Ranking Organization METHod for Enrichment of Evaluations) are widely employed outranking-based models. Compared with these two outranking-based models, the REGIME method exploits a mixed approach with combining the logit analysis and Kendall’s paired comparisons based on ordinal data (Asgharizadeh et al., 2014; Aspen et al., 2015; Hinloopen et al., 1983a). In this regard, a favourable feature possessed by the REGIME is its ability to utilize hybrid qualitative and quantitative data without requiring to convert the qualitative information into quantitative values (Esangbedo et al., 2021; Kamran et al., 2017; Oztaysi et al., 2022). Moreover, the REGIME is capable of conducting an adaptive analysis (Kamran et al., 2017) because it can render a complete predominance ranking for choice options or alternatives supported by paired comparisons with selected criteria in miscellaneous decision scenarios (Briamonte et al., 2021; Wątróbski et al., 2019). Especially, utilizing the REGIME method can generate undisputed consequences, so the dominant choice will be identified for the most cases (Hinloopen and Nijkamp, 1990; Wątróbski et al., 2019). Over and above that, the REGIME method has been proved as an efficacious technique to resolve various evaluation and decision-making affairs (Frank, 2014; Oztaysi et al., 2021; Tsigdinos and Vlastos, 2021). For these reasons, this research makes an effort at choosing the REGIME as the basic framework for extending to T-SF decision environments and conducting specialized decision-making tasks.

The REGIME has been smoothly exploited in the treatment of multiple-criteria evaluation problems (Hinloopen and Nijkamp, 1990), such as assessment and prioritization of coastal areas (Hinloopen et al., 1983b), regional sustainable resource policy (Akgün et al., 2012), assessment of alternative wind park locations (Stratigea and Grammatikogiannis, 2012), environmental management on wastewater from agriculture (Massei et al., 2014), economic-ecological sustainability for rural development (Akgün et al., 2015), sawability related to ornamental and building stones (Kamran et al., 2017), and strategic road network of a metropolitan region (Tsigdinos and Vlastos, 2021). On the flip-side, the REGIME framework has been generalized to fuzzy environments, such as the directly extended REGIME methods involving Pythagorean fuzziness (Oztaysi et al., 2021) and spherical fuzziness (Oztaysi et al., 2022). These fuzzy extensions of the REGIME method would form a basis for further advancement under T-SF uncertainties. Even though the usefulness and effectiveness of the REGIME technique have been demonstrated in the aforesaid literature, the advancement of the REGIME methodology has not been investigated yet in T-SF decision contexts. In view of this, it is critically important in the establishment of a T-SF REGIME approach as the increasing complexity and high-order fuzziness in realistic decision-making processes.

In terms of fuzzy community, the theory of T-spherical fuzziness contains a very comprehensive account of several non-standard fuzzy configurations. The advancements in the study of the fuzzy decision-making field are crucial for the subject of multiple-criteria choice analysis because of the necessity of managing uncertain information in real decisions. According to the investigations regarding the aforementioned literature, the research gaps and motivations for this paper are threefold:

From this basis, this paper attempts to making the REGIME accommodate to the realities of intricate uncertain circumstances. Moreover, this paper contrives of a novel T-SF REGIME method using several beneficial notions for conducting multiple-criteria choice analysis under T-SF uncertainty.

The primary purpose of this study is to specify a suitable measurement system for complex T-SF information and launch an innovative T-SF REGIME method for addressing multiple criteria evaluation and choice issues. It is worth mentioning that this paper intends to exploit the main structure of REGIME to adapt to T-SF decision environments. Because this paper desires to manipulate complex T-SF uncertain information, the evolved REGIME method should have differing adaptive notions corresponding to the differing phases of the core REGIME procedure. First, this paper constitutes a T-SF multiple-criteria choice problem involving judgment criteria and choice options, along with relevant T-SF evaluation values embedded in each T-SF characteristic. Next, in order to differentiate such T-SF assessment information, this study exploits a beneficial score function that signifies grades of satisfaction, neutral satisfaction, dissatisfaction, and refusal membership for utilizing T-SF uncertain information adequately. In accordance with score functions and accuracy values, this study identifies the superiority criteria supported by paired predominance relationships. Soon afterward, this study amalgamates the discrepancy between score functions with the total weights of superiority criteria for enriching the notion of superiority indices. Then, this paper originates an efficacious superiority identifier capable of measuring the relative attractiveness between T-SF characteristics. And following the superiority identifier, this paper presents the notions of REGIME identifiers and REGIME vectors to lay the foundations of a REGIME matrix and establish a guide index capable of measuring the relative fittingness for T-SF characteristics. This study conceives two beneficial procedures for the T-SF REGIME I and II prioritization to yield the eventual partial- and complete-preference rankings, respectively, for available options. By way of the Boolean matrices based on superiority identifiers and guide indices, this paper confirms valuable outranking relationships for generating the T-SF REGIME I predominance ranks for choice options. Moreover, this paper evolves the T-SF REGIME II predominance ranks of options on the basis of the net superiority identifier and the net guide index. To demonstrate the conceivability and advantages of the evolved T-SF REGIME methodology in pragmatic decisions, this paper explores a realistic problem concerning the company selection for plant-building. The effectiveness and constructiveness of the developed techniques can be illustrated through the agency of a comparative analysis.

The remainder of this research paper is exhibited systematically along these lines. Section 2 provides an introductory description of some rudimental notions about T-SF sets. Section 3 puts forward a novel T-SF REGIME method through the utility of the score function-based superiority identifiers and guide indices. Section 4 executes the initiated technique to handle the selection problem of companies for setting up food processing plants and then carry out a comparative analysis with the T-SF versions of other decision-making methods. Section 5 concludes this research work with certain concluding remarks, academic contributions, limitations, and future research directions.

The investigated selection problem of companies for erecting food processing plants was originated from the case study in Garg et al. (2018), as outlined in Fig. 2. The Jharkhand government in India attempts to establish essential agricultural-focused industries in rural regions. The authorities constituted the global investor summit and encouraged companies and enterprises for the investment in the surrounding countryside. Moreover, the authorities made a formal public statement about adequate facilities that were exploitable to construct food processing plants in the countryside.

To begin with, the T-SF REGIME I method in Algorithm I would be illustrated with the realistic selection problem of companies for building food processing plants. As disclosed in Fig. 2, there are five judgment criteria (i.e. ${c_{1}}\text{--}{c_{5}}$) and three choice options (i.e. ${a_{1}}\text{--}{a_{3}}$) in this multiple-criteria choice task. In Step I.1 of Phase (i), the collection of choice options $A=\{{a_{1}},{a_{2}},{a_{3}}\}$ and the collection of judgment criteria $C=\{{c_{1}},{c_{2}},\dots ,{c_{5}}\}$. Concerning Step I.2, as stated by Garg et al. (2018), the normalized weights in conjunction with the five criteria were derived by the agency of the normal distribution-based approach. In conformity with the T-SF evaluation value ${t_{ij}}=({\mu _{ij}},{\eta _{ij}},{\nu _{ij}})$ in Garg et al. (2018), this study calculated the grade of refusal membership ${\gamma _{ij}}$ corresponding to ${t_{ij}}$. The foregoing decision data related to each choice option are depicted in Table 1. In Step I.3, the parameter $z=3$ for each ${t_{ij}}$, and the T-SF characteristic ${T_{i}}=\{\langle {c_{1}},{t_{i1}}\rangle ,\langle {c_{2}},{t_{i2}}\rangle ,\dots ,\langle {c_{5}},{t_{i5}}\rangle \}=\{\langle {c_{1}},({\mu _{i1}},{\eta _{i1}},{\nu _{i1}})\rangle ,\langle {c_{2}},({\mu _{i5}},{\eta _{i5}},{\nu _{i5}})\rangle \}$ for each ${a_{i}}\in A$. It is worth pointing out that a weight-assessing approach sprung from normal distributions and launched by Xu (2005) was employed to generate the weights of five judgment criteria that fulfill the normalization requirement. On grounds of the notion of normal distributions, the mean and the standard deviation of the sequence $1,2,\dots ,5$ were derived by $(1+5)/2=3$ and ${\{[{(1-3)^{2}}+{(2-3)^{2}}+{(3-3)^{2}}+{(4-3)^{2}}+{(5-3)^{2}}]/5\}^{0.5}}=1.4142$, respectively. By virtue of Xu’s (2005) approach, the weights were yielded along these lines:

Consider Phase (ii) as an illustration. In Step I.4, the calculation consequences of the score function $\textit{Sf}({t_{ij}})$ and the accuracy value $\textit{Av}({t_{ij}})$ using Eqs. (6) and (5), respectively, are exhibited in the upper part of Table 2. Concerning Step I.5, this study exploited Eq. (8) to produce the collections of superiority criteria for paired T-SF characteristics, as displayed in the lower part of Table 2. Take the juxtaposition of the T-SF characteristics ${T_{2}}$ and ${T_{3}}$ as an example. As stated in Table 2, it was recognized that $\textit{Sf}({t_{22}})>\textit{Sf}({t_{32}})$ and $\textit{Sf}({t_{25}})>\textit{Sf}({t_{35}})$, which leads to $\textit{Cs}({T_{2}},{T_{3}})=\{{c_{2}},{c_{5}}\}$. Additionally, it was realized that $\textit{Sf}({t_{31}})>\textit{Sf}({t_{21}})$, $\textit{Sf}({t_{33}})>\textit{Sf}({t_{23}})$, and $\textit{Sf}({t_{34}})>\textit{Sf}({t_{24}})$, which brings about $\textit{Cs}({T_{3}},{T_{2}})=\{{c_{1}},{c_{3}},{c_{4}}\}$.

Concerning Step I.6 in Phase (iii), the discrepancy between $\textit{Sf}({t_{ij}})$ and $\textit{Sf}({t_{{i^{\prime }}j}})$ for each superiority criterion ${c_{j}}\in \textit{Cs}({T_{i}},{T_{{i^{\prime }}}})$ was received, as demonstrated in the upper part of Table 3. Relating to Step I.7, the weighted discrepancy between $\textit{Sf}({t_{ij}})$ and $\textit{Sf}({t_{{i^{\prime }}j}})$ was aggregated across all ${c_{j}}\in \textit{Cs}({T_{i}},{T_{{i^{\prime }}}})$ to calculate the superiority identifier $\textit{SI}({T_{i}},{T_{{i^{\prime }}}})$ using Eq. (11), and the outcomes are presented in the lowest row of Table 3. Consider the pair $({T_{2}},{T_{1}})$ for instance. Based on $\textit{Cs}({T_{2}},{T_{1}})=\{{c_{1}},{c_{2}},{c_{5}}\}$, it was rendered that $\textit{SI}({T_{2}},{T_{1}})={\textstyle\sum _{{c_{j}}\in \{{c_{1}},{c_{2}},{c_{5}}\}}}{w_{j}}(\textit{Sf}({t_{2j}})-\textit{Sf}({t_{1j}}))=0.1117\times 0.0079+0.2365\times 0.1594+0.1117\times 0.1705=0.0576$.

Relating to Step I.8 in Phase (iv), the REGIME identifier $\textit{Ri}({t_{ij}},{t_{{i^{\prime }}j}})$ was produced with the aid of $\textit{Sf}({t_{ij}})$ and $\textit{Sf}({t_{{i^{\prime }}j}})$ using Eq. (12) to constitute the REGIME vector $\textit{Rv}({T_{i}},{T_{{i^{\prime }}}})$ for paired T-SF characteristics ${T_{i}}$ and ${T_{{i^{\prime }}}}$. Employing Eq. (14), the REGIME matrix $\textit{Rm}$ was constructed by taking together the REGIME vectors for all ordered pairs of (${a_{i}},{a_{{i^{\prime }}}}$):

Concerning Step I.9, this study exploited Eq. (15) to synthesize the normalized weight ${w_{j}}$ and the REGIME identifier $\textit{Ri}({t_{ij}},{t_{{i^{\prime }}j}})$ for determining the guide index $\textit{GI}({T_{i}},{T_{{i^{\prime }}}})$ by: $\textit{GI}({T_{1}},{T_{2}})={\textstyle\sum _{j=1}^{5}}{w_{j}}\cdot \textit{Ri}({t_{1j}},{t_{2j}})=0.1117\cdot (-1)+0.2365\cdot (-1)+0.3036\cdot 1+0.2365\cdot 1+0.1117\cdot (-1)=0.0802$, $\textit{GI}({T_{1}},{T_{3}})=-0.3928,\textit{GI}({T_{2}},{T_{1}})=-0.0802$, $\textit{GI}({T_{2}},{T_{3}})=-0.3036,\textit{GI}({T_{3}},{T_{1}})=0.3928$, and $\textit{GI}({T_{3}},{T_{2}})=0.3036$.

Considering Steps I.10 and I.11 in Phase (v), this paper made use of Eqs. (16) and (18) to acquire the entries $\textit{Bs}({a_{i}},{a_{{i^{\prime }}}})$ and $\textit{Bg}({a_{i}},{a_{{i^{\prime }}}})$, respectively, for ${a_{i}},{a_{{i^{\prime }}}}\in A$. Moreover, this paper employed Eqs. (17) and (19) to generate the superiority-based Boolean matrix $\textit{Bs}$ and the guide-based Boolean matrix $\textit{Bg}$, respectively. In Step I.12, the entry $\textit{Bc}({a_{i}},{a_{{i^{\prime }}}})$ was delineated as $\textit{Bs}({a_{i}},{a_{{i^{\prime }}}})\cdot \textit{Bg}({a_{i}},{a_{{i^{\prime }}}})$ according to Eq. (20) for organizing the comprehensive Boolean matrix $\textit{Bc}$ in Eq. (21). The three Boolean matrices are as shown:

In Step I.13, the partial-preference predominance rankings ${a_{1}}{\succ ^{\mathrm{REGIME}\mathrm{I}}}{a_{2}}$, ${a_{3}}{\succ ^{\mathrm{REGIME}\mathrm{I}}}{a_{1}}$, and ${a_{3}}{\succ ^{\mathrm{REGIME}\mathrm{I}}}{a_{2}}$ were generated by way of $\textit{Bc}({a_{1}},{a_{2}})=1$, $\textit{Bc}({a_{3}},{a_{1}})=1$, and $\textit{Bc}({a_{3}},{a_{2}})=1$, respectively. On the report of the comprehensive Boolean matrix $\textit{Bc}$, the dominance graph was portrayed in Fig. 3, which follows that ${a_{3}}$ (i.e. Parle Products Ltd.) was recognized as the most beneficial choice option.

On the flip-side, the T-SF REGIME II method in Algorithm II would be utilized to tackle the identical selection problem of companies for food processing plants. As mentioned previously, Steps II.1–II.9 are the same as Steps I.1–I.9. Concerning Step II.10 in Phase (v), this paper employed Eq. (22) to identify the net superiority identifiers in this manner: ${N_{\textit{SI}}}({a_{1}})=(\textit{SI}({T_{1}},{T_{2}})+\textit{SI}({T_{1}},{T_{3}}))-(\textit{SI}({T_{2}},{T_{1}})+\textit{SI}({T_{3}},{T_{1}}))=(0.1826+0.0761)-(0.0576+0.1103)=0.0908$, ${N_{\textit{SI}}}({a_{2}})=-0.2840$, and ${N_{\textit{SI}}}({a_{3}})=0.1933$. The complete ranking ${a_{3}}{\succ ^{\textit{SI}}}{a_{1}}{\succ ^{\textit{SI}}}{a_{2}}$ was received in keeping with ${N_{\textit{SI}}}({a_{3}})>{N_{\textit{SI}}}({a_{1}})>{N_{\textit{SI}}}({a_{2}})$. Next, relating to Step II.11, this study made use of Eq. (23) to attain the net guide indices in this fashion: ${N_{\textit{GI}}}({a_{1}})=(\textit{GI}({T_{1}},{T_{2}})+\textit{GI}({T_{1}},{T_{3}}))-(\textit{GI}({T_{2}},{T_{1}})+\textit{GI}({T_{3}},{T_{1}}))=(0.0802-0.3928)-(-0.0802+0.3928)=-0.6252$, ${N_{\textit{GI}}}({a_{2}})=-0.7676$, and ${N_{\textit{GI}}}({a_{3}})=1.3928$. The complete ranking ${a_{3}}{\succ ^{\textit{GI}}}{a_{1}}{\succ ^{\textit{GI}}}{a_{2}}$ was received in keeping with ${N_{\textit{GI}}}({a_{3}})>{N_{\textit{GI}}}({a_{1}})>{N_{\textit{GI}}}({a_{2}})$.

Finally, in Step II.12, the complete-preference predominance ranking ${a_{3}}{\succ ^{\mathrm{REGIME}\mathrm{II}}}{a_{1}}{\succ ^{\mathrm{REGIME}\mathrm{II}}}{a_{2}}$ were rendered as supported by ${a_{3}}{\succ ^{\textit{SI}}}{a_{1}}{\succ ^{\textit{SI}}}{a_{2}}$ and ${a_{3}}{\succ ^{\textit{GI}}}{a_{1}}{\succ ^{\textit{GI}}}{a_{2}}$. Therefore, in the same vein, the top-ranked option ${a_{3}}$ was selected as the eventual solution. The resulting outcomes yielded by the prioritization procedures of T-SF REGIME I and II were identical to that used in the decision-aiding approach with the aid of geometric aggregation operations in Garg et al. (2018).

In an attempt to validate the methodological advantages, this paper implements a comparative approach to different score values in the initiated T-SF REGIME methods and techniques. Based on Definition 4, the score value $\textit{Sv}({t_{ij}})$ was exploited to replace the score function $\textit{Sf}({t_{ij}})$. Herein, it was received that $\textit{Sv}({t_{ij}})={({\mu _{ij}})^{3}}-{({\nu _{ij}})^{3}}$ for each T-SF characteristic ${t_{ij}}$ in the selection problem of companies. Table 4 reveals the summary consequences through the agency of the score value $\textit{Sv}({t_{ij}})$.

First, let us examine the results using the score value-based T-SF REGIME I method. As attested by the entry $\textit{Bc}({a_{i}},{a_{{i^{\prime }}}})$, the comparative study produced the partial-preference predominance rankings ${a_{1}}{\succ ^{\mathrm{REGIME}\hspace{0.1667em}\mathrm{I}}}{a_{2}}$, ${a_{1}}{\succ ^{\mathrm{REGIME}\hspace{0.1667em}\mathrm{I}}}{a_{3}}$, and ${a_{3}}{\succ ^{\mathrm{REGIME}\hspace{0.1667em}\mathrm{I}}}{a_{2}}$ owing to $\textit{Bc}({a_{1}},{a_{2}})=1$, $\textit{Bc}({a_{1}},{a_{3}})=1$, and $\textit{Bc}({a_{3}},{a_{2}})=1$, respectively. Next, consider the score value-based T-SF REGIME II method. Supported by ${a_{1}}{\succ ^{\textit{SI}}}{a_{3}}{\succ ^{\textit{SI}}}{a_{2}}$ and ${a_{1}}{\succ ^{\textit{GI}}}{a_{3}}{\succ ^{\textit{GI}}}{a_{2}}$, the comparative study generated the complete-preference predominance ranking ${a_{1}}{\succ ^{\mathrm{REGIME}\hspace{0.1667em}\mathrm{II}}}{a_{3}}{\succ ^{\mathrm{REGIME}\hspace{0.1667em}\mathrm{II}}}{a_{2}}$. Nevertheless, such outcomes were directly conflicting with the ultimate ranking ${a_{3}}\succ {a_{1}}\succ {a_{2}}$ yielded by Garg et al. (2018). Garg et al. (2018) corroborated that the choice option ${a_{3}}$ is superior to ${a_{1}}$; however, the comparative study delivered different consequences, i.e. the partial ordering ${a_{1}}{\succ ^{\mathrm{REGIME}\hspace{0.1667em}\mathrm{I}}}{a_{3}}$ and the complete ordering ${a_{1}}{\succ ^{\mathrm{REGIME}\hspace{0.1667em}\mathrm{II}}}{a_{3}}{\succ ^{\mathrm{REGIME}\hspace{0.1667em}\mathrm{II}}}{a_{2}}$. These conflicting results were attributed to the use of the score value $\textit{Sv}({t_{ij}})$ in the T-SF REGIME I and II series of steps. Thus, the exploitation of the score function $\textit{Sf}({t_{ij}})$ has the ability to achieve more desirable solution consequences. The correctness and effectuality of the evolved methodology have been manifested through the medium of the comparative study and discussions.

The evolved T-SF REGIME methodology possesses an exceptional ability to make more accurate decisions by using a more suitable and efficacious measurement system concerning the characteristics of T-SF information. In particular, the initiated T-SF REGIME I and II prioritization procedures can make more precise decisions. Results are presented in the format of the partial-and complete-preference predominance ranks and can be utilized for the further applications in decision-making support systems.

This subsection attempts to make comparisons with other decision-making approaches carefully and objectively to verify the performance of the evolved T-SF REGIME methodology. As is well known, the TOPSIS and VIKOR have been widely used throughout the compromise decision-making processes and have been renowned as anchor dependent models in multiple criteria analysis. From this basis, this subsection exploits the T-SF versions of TOPSIS and VIKOR to facilitate comparisons.

Choice options that come nearer to the positive-ideal option in separation measures are more favourable than those that come farther away. In contrast, choice options that keep away from the negative-ideal option via separation measures are more favourable than those that come near. The rationale of human choice is to simultaneously come near to the positive-ideal option and keep away from the negative-ideal option to the most possible extent. Such axioms of choice have explicitly assumed that there exist sharply bipolar anchor values of reference. Two types of bipolar anchor values were utilized in the comparative analysis with the T-SF TOPSIS and the T-SF VIKOR: fixed ideals and displaced ideals.

Take into consideration the selection problem of companies for erecting food processing plants. Place the fixed positive-ideal option ${a_{+}}$ and the fixed negative-ideal option ${a_{-}}$; their corresponding T-SF characteristics ${T_{+}}$ and ${T_{-}}$ were expounded in this manner:

Place the displaced positive-ideal option ${a_{\ast }}$ and the displaced negative-ideal option ${a_{\mathrm{\# }}}$. On the grounds of the outcomes of score functions in Table 2, the largest values with respect to five judgment criteria were identified as follows: $\textit{Sf}({t_{31}})=0.6842$, $\textit{Sf}({t_{22}})=0.7629$, $\textit{Sf}({t_{13}})=0.6842$, $\textit{Sf}({t_{34}})=0.7449$, and $\textit{Sf}({t_{25}})=0.5977$ for ${c_{1}},{c_{2}},\dots ,{c_{5}}$, respectively. Additionally, the smallest values of score functions with relevance to five judgment criteria were identified in this manner: $\textit{Sf}({t_{11}})=0.2128$, $\textit{Sf}({t_{12}})=0.6035$, $\textit{Sf}({t_{23}})=0.2864$, $\textit{Sf}({t_{24}})=0.3013$, and $\textit{Sf}({t_{15}})=0.4272$ for ${c_{1}},{c_{2}},\dots ,{c_{5}}$, respectively. As a direct consequence, the T-SF characteristics ${T_{\ast }}=\{\langle {c_{j}},({\mu _{\ast j}},{\eta _{\ast j}},{\nu _{\ast j}})\rangle |\forall {c_{j}}\in C\}$ and ${T_{\mathrm{\# }}}=\{\langle {c_{j}},({\mu _{\mathrm{\# }j}},{\eta _{\mathrm{\# }j}},{\nu _{\mathrm{\# }j}})\rangle \mid \forall {c_{j}}\in C\}$ associated with ${a_{\ast }}$ and ${a_{\mathrm{\# }}}$, respectively, were delineated in this fashion: First, give consideration to the determination outcomes yielded by the T-SF TOPSIS method. Place a distance parameter $\chi \in {Z^{+}}$. By exploiting the Minkowski distance unfolded by Ju et al. (2021), the weighted distance between the T-SF characteristics ${T_{i}}$ and ${T_{{i^{\prime }}}}$ can be derived along these lines:

The closeness coefficients $C{C_{+}^{\chi }}({a_{i}})$ and $C{C_{\ast }^{\chi }}({a_{i}})$ of each choice option ${a_{i}}\in A$ on the grounds of the fixed and displaced ideal options, respectively, can be explicated in this manner:

Based around the data in the selection problem of companies, the computation outcomes of the closeness coefficients $C{C_{+}^{\chi }}({a_{i}})$ and $C{C_{\ast }^{\chi }}({a_{i}})$ are exhibited in Table 5. Moreover, the contrasts of the resulting $C{C_{+}^{\chi }}({a_{i}})$ and $C{C_{\ast }^{\chi }}({a_{i}})$ are sketched in Fig. 4(a) and (b), respectively. It should be noted that the consistent predominance rankings ${a_{3}}{\succ ^{{\mathrm{TOPSIS}_{+}}}}{a_{2}}{\succ ^{{\mathrm{TOPSIS}_{+}}}}{a_{1}}$ and ${a_{3}}{\succ ^{{\mathrm{TOPSIS}_{\ast }}}}{a_{2}}{\succ ^{{\mathrm{TOPSIS}_{\ast }}}}{a_{1}}$ were acquired in cases that $\chi =1,2,\dots ,10$ and $\chi \to \infty $ regardless of the exploitation of fixed ideals and displaced ideals. However, such a ranking (i.e. ${a_{3}}\succ {a_{2}}\succ {a_{1}}$) had a conflicting partial ordering in comparison with the ultimate ranking ${a_{3}}\succ {a_{1}}\succ {a_{2}}$ rendered by Garg et al. (2018), which indicates unreliability and untrustworthiness of the consequences rendered by the T-SF TOPSIS method predicated on the fixed and displaced ideal options. Contrary to the TOPSIS results, the propounded T-SF REGIME methodology can generate reasonable and convincing outcomes in preference predominance rankings of choice options. No matter the partial-preference rankings ${a_{1}}{\succ ^{\mathrm{REGIME}\hspace{0.1667em}\mathrm{I}}}{a_{2}}$, ${a_{3}}{\succ ^{\mathrm{REGIME}\hspace{0.1667em}\mathrm{I}}}{a_{1}}$, and ${a_{3}}{\succ ^{\mathrm{REGIME}\hspace{0.1667em}\mathrm{I}}}{a_{2}}$ or the complete-preference ranking ${a_{3}}{\succ ^{\mathrm{REGIME}\hspace{0.1667em}\mathrm{II}}}{a_{1}}{\succ ^{\mathrm{REGIME}\hspace{0.1667em}\mathrm{II}}}{a_{2}}$ were consistent with the yielded results by Garg et al. (2018). The findings of the comparative study have given substance to the superiority of the T-SF REGIME approach over the T-SF TOPSIS method.

Next, consider the determination outcomes produced by the T-SF VIKOR method. Place any two T-SF evaluation values ${t_{ij}}=({\mu _{ij}},{\eta _{ij}},{\nu _{ij}})$ and ${t_{{i^{\prime }}j}}=({\mu _{{i^{\prime }}j}},{\eta _{{i^{\prime }}j}},{\nu _{{i^{\prime }}j}})$ in a specific multiple-criteria choice issue. The Minkowski distance presented by Ju et al. (2021) can be utilized to delineate two measurements about group utility and individual regret with the intention of forming a joint measure for compromise rankings. The weighted distance between ${t_{ij}}$ and ${t_{{i^{\prime }}j}}$ can be computed in this fashion:

By incorporating the normalized weight ${w_{j}}$ into the relative ratio of the distances ${D^{\chi }}({t_{ij}},{t_{+j}})$ to ${D^{\chi }}({t_{+j}},{t_{-j}})$, the group utility measure $G{u_{+}^{\chi }}({T_{i}})$ and the individual regret measure $I{r_{+}^{\chi }}({T_{i}})$ of the T-SF characteristic ${T_{i}}=\{\langle {c_{j}},({\mu _{ij}},{\eta _{ij}},{\nu _{ij}})\rangle \mid \forall {c_{j}}\in C\}$ based on the fixed ideals are manifested as follows:

In an analogous way, the measures $G{u_{\ast }^{\chi }}({T_{i}})$ and $I{r_{\ast }^{\chi }}({T_{i}})$ based on the displaced ideals are given by:

By virtue of a VIKOR parameter $\iota \in [0,1]$, the joint measures $J{m_{+}^{\chi ,\iota }}({a_{i}})$ and $J{m_{\ast }^{\chi ,\iota }}({a_{i}})$ of each choice option ${a_{i}}$ are elucidated in this manner:

Giving consideration to the selection problem of companies, the determined outcomes of the measures $G{u_{+}^{\chi }}({T_{i}})$, $I{r_{+}^{\chi }}({T_{i}})$, $G{u_{\ast }^{\chi }}({T_{i}})$, and $I{r_{\ast }^{\chi }}({T_{i}})$ are indicated in Table 6. Letting the VIKOR parameter $\chi =0.5$, the resulting joint measures $J{m_{+}^{\chi ,0.5}}({a_{i}})$ and $J{m_{\ast }^{\chi ,0.5}}({a_{i}})$ are also revealed in this table. First, consider the bipolar anchor values using the fixed ideals. Two predominance rankings were produced in compliance with an increasing order of the $J{m_{+}^{\chi ,0.5}}({a_{i}})$ values, namely ${a_{1}}{\succ ^{{\mathrm{VIKOR}_{+}}}}{a_{3}}{\succ ^{{\mathrm{VIKOR}_{+}}}}{a_{2}}$, when $\chi =1$, and ${a_{1}}{\succ ^{{\mathrm{VIKOR}_{+}}}}{a_{2}}{\succ ^{{\mathrm{VIKOR}_{+}}}}{a_{3}}$, when $\chi =2,3,\dots ,10$ and $\chi \to \infty $. Nevertheless, such consequences were strikingly different from the ultimate ranking ${a_{3}}\succ {a_{1}}\succ {a_{2}}$ resolved by Garg et al. (2018) and the proposed T-SF REGIME II algorithm. To be specific, the predominance relationship between ${a_{1}}$ and ${a_{2}}$ (i.e. ${a_{1}}\succ {a_{2}}$) was generally concordant with the consequences in Garg et al. (2018) and the T-SF REGIME methodology. In particular, the choice option ${a_{3}}$ was the best solution in the results yielded by Garg et al. (2018) and the T-SF REGIME I and II algorithms; however, the T-SF VIKOR method pointed out that ${a_{3}}$ was the worst solution in cases that $\chi =2,3,\dots ,10$ and $\chi \to \infty $ from the perspective of the fixed ideals. Thus, based on bipolar anchor values of the fixed ideals, the T-SF VIKOR approach gave rise to disbelieving and unreasonable ranking outcomes. Next, making allowance for the bipolar anchor values using the displaced ideals, a reasonable and consistent ranking ${a_{3}}{\succ ^{{\mathrm{VIKOR}_{\ast }}}}{a_{1}}{\succ ^{{\mathrm{VIKOR}_{\ast }}}}{a_{2}}$ was obtained in keeping with the ascending order of the $J{m_{\ast }^{\chi ,0.5}}({a_{i}})$ values when $\chi =1,2,\dots ,10$ and $\chi \to \infty $. By comparison, the consequence judging by the $J{m_{\ast }^{\chi ,0.5}}({a_{i}})$ values were more believable and creditable than that by the $J{m_{+}^{\chi ,0.5}}({a_{i}})$ values. Even so, the results of applying the T-SF VIKOR method were still disputable and confusing for the decision maker because the obtained ranking outcomes would have great differences. The reason was that the contrasts of the obtained joint measures may be unstable and controversial based on distinct bipolar anchor values. In contrast, the T-SF REGIME methodology can deliver steady and well supported solution results, which demonstrates the strong points of the current approach.

In order to observe the changes of the joint measures $J{m_{+}^{\chi ,\iota }}({a_{i}})$ and $J{m_{\ast }^{\chi ,\iota }}({a_{i}})$ under different settings of the VIKOR parameter ι, this study designated the VIKOR parameter as $\iota =0.0,0.1,\dots ,1.0$. First, make allowance for the bipolar anchor values using the fixed ideals. It should be noted that $J{m_{+}^{\chi ,\iota }}({a_{3}})=1$ in every parameter combination of $\iota =0.0,0.1,\dots ,1.0$ and $\chi =2,3,\dots ,10$ and $\chi \to \infty $ because ${\max _{{i^{\prime }}=1}^{3}}G{u_{+}^{\chi }}({T_{{i^{\prime }}}})=G{u_{+}^{\chi }}({T_{3}})$ and ${\max _{{i^{\prime }}=1}^{3}}I{r_{+}^{\chi }}({T_{{i^{\prime }}}})=I{r_{+}^{\chi }}({T_{3}})$. Thus, it was recognized that $J{m_{+}^{\chi ,\iota }}({a_{3}})=\iota \times 1+(1-\iota )\times 1=1$ in such scenarios. In this regard, it would merely have a look at the changes of the values of $J{m_{+}^{\chi ,\iota }}({a_{1}})$ and $J{m_{+}^{\chi ,\iota }}({a_{2}})$ in various settings, as portrayed in Fig. 5(a) and (b), respectively. On the flip-side, it would give consideration to the bipolar anchor values using the displaced ideals. In an analogous way, it was known that $J{m_{\ast }^{\chi ,\iota }}({a_{2}})=1$ in every parameter combination of $\iota =0.0,0.1,\dots ,1.0$ and $\chi =1,2,\dots ,10$ and $\chi \to \infty $. Over and above that, it was found that $J{m_{\ast }^{\chi ,\iota }}({a_{3}})=0$ because ${\min _{{i^{\prime }}=1}^{3}}G{u_{\ast }^{\chi }}({T_{{i^{\prime }}}})=G{u_{\ast }^{\chi }}({T_{3}})$ and ${\min _{{i^{\prime }}=1}^{3}}I{r_{\ast }^{\chi }}({T_{{i^{\prime }}}})=I{r_{\ast }^{\chi }}({T_{3}})$ in all scenarios. On the grounds of this, $J{m_{\ast }^{\chi ,\iota }}({a_{3}})=\iota \times 0+(1-\iota )\times 0=0$ no matter which combination of parameters ι and χ. From this basis, the change of the $J{m_{\ast }^{\chi ,\iota }}({a_{1}})$ value was explored and depicted in Fig. 5(c). As sketched in Fig. 5, three special cases of $\iota =0.0,0.5,1.0$ were represented using line charts, whereas the other cases were expressed by bar graphs. As revealed in Fig. 5, the values of the joint measures $J{m_{+}^{\chi ,\iota }}({a_{1}})$, $J{m_{+}^{\chi ,\iota }}({a_{2}})$, and $J{m_{\ast }^{\chi ,\iota }}({a_{1}})$ were deeply affected by the parameters ι and χ. Thus, such values were not stable, which may bring about different final rankings of the choice options. In contrast to the T-SF VIKOR technique, the evolved T-SF REGIME methodology can render more reliable and convincing results, which can make the final decision easy to implement.