Selection problems arising in higher education admissions and medical waste management are inherently affected by uncertainty, imprecision, and hesitation, which complicate reliable decision analysis. Pythagorean fuzzy sets provide a flexible mathematical framework for modeling such uncertainties, and a variety of Pythagorean fuzzy distance measures have been developed for decision-making applications. Nevertheless, existing distance measures overlook the role of tendency coefficients, which are necessary for characterizing the inclination and relative influence of Pythagorean fuzzy parameters and for enhancing the discriminatory power of distance-based methods. In this paper, a novel Pythagorean fuzzy distance measure incorporating tendency coefficients of the complete Pythagorean fuzzy parameters is proposed. The mathematical properties of the proposed distance measure are established to ensure its validity and consistency. To demonstrate its applicability, the proposed measure is integrated into a multi-criteria decision-making framework and applied to the evaluation of students’ examination scores for higher education admission. In addition, the technique for order of preference by similarity to the ideal solution is employed to address the selection of medical waste management systems under a Pythagorean fuzzy environment. Comparative studies with existing Pythagorean fuzzy distance measures confirm the improved effectiveness, stability, and reliability of the proposed approach in handling uncertainty and supporting robust decision-making.

Wheat remains a main staple crop in Sub-Saharan Africa, yet its cultivation is increasingly endangered by climate unpredictability and environmental uncertainty. The selection of an appropriate farmland for optimal wheat productivity, therefore, involves complex, imprecise, incomplete, and uncertain information. In response to this problem, the proposed study employs the framework of single-valued neutrosophic sets, which has proven effective in modelling imprecision and indeterminacy in decision-making problems. Correlation coefficients under single-valued neutrosophic sets have been deployed to discuss complex selection problems with multi-criteria. Although some single-valued neutrosophic correlation coefficient measures have been established, a number of them lack appropriate statistical justification and reliability. This study introduces new correlation coefficient methods based on Spearman’s approach. The suggested methods are evaluated by comparing them to existing methods, and it is shown that they meet the requirements of the correlation coefficient while providing more accurate and reliable decision outcomes. In particular, based on the numerical experiment in Example 3.1, the two suggested correlation coefficient methods achieve accuracy percentages of 100% and 99.94% in Case I, and 100% and 89.23% in Case II—substantially outperforming the existing methods, which yield notably lower accuracy percentages in both cases. Their applicability is validated through a farmland selection problem aimed at maximizing wheat productivity, using data from a knowledge-based system. The framework is adaptable and can be extended to other complex decision-making scenarios.

This study presents an effective Pythagorean fuzzy similarity method for the analysis of questionnaires on the awareness, usage, and impact of virtual library resources. The questionnaire’s analysis utilizes the multiple criteria decision-making approach. The new Pythagorean fuzzy similarity method explored in this paper is the result of the inefficiency of the existing approaches, which lack credibility in terms of precision and accuracy. Several theoretic properties validate the novel Pythagorean fuzzy similarity method. In addition, comparative studies between existing Pythagorean fuzzy similarity methods and the novel Pythagorean fuzzy similarity method are presented. This study effectively demonstrates the superiority of the novel method in meeting the conditions of the similarity measure. The findings indicate that the novel Pythagorean fuzzy similarity method aligns with the conditions of the similarity measure and produces reasonable results with reliable interpretations when compared to existing Pythagorean fuzzy similarity methods. This application in questionnaire analysis helps to manage any imprecision in data collection and analysis. This research advocates for the use of soft computing approaches, such as the Pythagorean fuzzy similarity measures, to analyze and interpret questionnaires. This recommendation stems from the ability of Pythagorean fuzzy sets to effectively manage imprecision in data.

This article introduces the concept of soluble (or solvable) intuitionistic fuzzy groups as a novel intuitionistic fuzzy algebraic structure. Certain fundamental results concerning intuitionistic fuzzy normal subgroups and quotient intuitionistic fuzzy groups are established. Furthermore, a solvable series for an intuitionistic fuzzy group is constructed and illustrated through a detailed example. It is demonstrated that the family of intuitionistic fuzzy subgroups of a given intuitionistic fuzzy group shares the same support as the group itself. Finally, a biconditional relationship between a solvable intuitionistic fuzzy group and its support is proved.

The idea of Pythagorean fuzzy distance metrics (PFDMs) has been used to discuss sundry selection problems. Existing PFDMs often fail to incorporate all three essential parameters of a Pythagorean fuzzy set (PFS), namely; membership degree (MD), non-membership degree (NMD), and hesitation degree (HD), thereby limiting their precision and effectiveness in real-world decision-making situations. To explore this gap, this research presents a novel three-dimensional (3D) weighted distance metric under the Pythagorean fuzzy framework, which integrates MD, NMD, and HD for a more comprehensive representation of imprecision. The proposed PFDM is theoretically validated and it is shown to fulfill the axioms of a distance function. It is then embedded into the technique for order of preference by similarity to ideal solution (TOPSIS) to enhance multi-criteria decision-making (MCDM), particularly in the context of smartphone selection. A comparative analysis against existing PFDMs demonstrates the superior precision and stability of the new approach. Furthermore, a sensitivity analysis of the novel 3D distance model confirms its robustness with respect to changes in criteria weights. This enhanced 3D distance metric provides a more reliable and interpretable tool for decision-makers in decision making fields.

Glaucoma is one such major cause of irreversible blindness, yet early diagnosis and constant monitoring remain some of the major clinical challenges due to variability in ocular parameters and patient response. In this paper, a Fermatean Fuzzy Distance Model is developed to quantify and interpret glaucomatous neuropathy progression through multi-parametric ophthalmic indicators. Each patient’s profile-given by IOP, CCT, C/D, RNFL thickness, and age-is translated into Fermatean fuzzy triples, thereby encapsulating supporting, contradicting, and uncertain evidence for disease classification. Clinical data and anonymised patient records from the Ophthalmology Department were procured from the Optometry Department of the Abia State Ministry of Health between 2021 and 2024. The dataset contained clinically verified readings collected from confirmed cases of glaucoma and non-glaucoma, under due ethical clearance and according to institutional guidelines for secondary data use. Using established Fermatean Fuzzy Distance Models, a few comparative studies are performed to determine the closeness of various patient conditions to positive and negative ideal states. The results obtained show a consistent ranking scheme according to the physiological progression of the disease-from controlled intraocular pressure with preserved optic nerve function to advanced optic neuropathy. The new distance function based on logarithms is given greater discriminatory power in differentiating between borderline cases of glaucoma and those with advancing glaucoma, providing a smoother transition between disease stages. On the other hand, interpretability analysis uncovered that the proposed Fermatean Fuzzy Distance Model allows a transparent ranking system from a clinical perspective. Therefore, the proposed approach offers an interpretable, mathematically sound framework of glaucoma risk assessment that, in turn, supports ophthalmologists in monitoring diseases on a patient basis and making appropriate decisions.

The theory of fuzzy multigroups is an algebraic structure derivable from the application of fuzzy multisets to groups. Various concepts in group theory have been considered in fuzzy multigroup theory. However, simple group, maximal normal subgroup, normal series, composition series, and the Jordan-Hölder Theorem are open problems in fuzzy multigroup theory. Hence, this paper defines simple fuzzy multigroups, maximal normal fuzzy submultigroups, normal series for fuzzy multigroups, and composition series for fuzzy multigroups with illustrations. In addition, the Jordan-Hölder Theorem in fuzzy multigroup theory is established. We show that each finite fuzzy multigroup of a finite group possesses a composition series, and also prove that two composition series for a finite fuzzy multigroup of a finite group are comparable.

Q-rung orthopair fuzzy sets (Q-ROFS) have been widely employed in decision-making problems due to their strong ability to handle uncertainty, indecision, and imprecision. Consequently, several q-rung orthopair fuzzy correlation measures (Q-ROFCM) have been developed and applied in various decision-making contexts. However, many existing correlation measures exhibit inherent limitations, which reduce their effectiveness in addressing practical, real-world problems. In this study, a novel q-rung orthopair fuzzy correlation coefficient (Q-ROFCC) based on Spearman’s correlation scheme is proposed to overcome the shortcomings of existing approaches. The fundamental mathematical properties of the proposed correlation measure are rigorously analyzed to ensure compliance with the standard axioms of correlation coefficients. Furthermore, the proposed method is incorporated into a multi-attribute decision-making (MADM) framework. The results demonstrate that the proposed Spearman-based Q-ROFCM technique is reliable, effective, and accurate when compared with existing methods. Its applicability is illustrated through a vehicle selection problem, where the most suitable alternative is identified based on optimal performance and user satisfaction. Comparative analysis confirms the superiority of the proposed approach over Pearson-based Q-ROFCM approaches. The proposed Q-ROFCM technique provides a robust and efficient alternative for solving MADM problems under uncertainty. Owing to its improved performance and practical applicability, the method is well suited for real-life decision-making scenarios.

An effective transportation system is fundamental for socioeconomic development, public safety, and environmental sustainability. A critical component of such a system is the Vehicle Selection Problem (VSP), which is inherently a Complex Decision-Making (CDM) problem due to conflicting and uncertain criteria such as fuel efficiency, purchase cost, maintenance cost, and warranty. To address this complexity, this study develops two novel logarithmic-based distance measures within the q-Rung Orthopair Fuzzy (q-ROF) framework. The proposed distance metrics incorporate membership, non-membership, and hesitation degrees, along with the cardinality of the universe of discuss, ensuring a more comprehensive representation of uncertainty. Their metric properties are rigorously proven, and they are integrated with the Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) to solve a VSP. The case study involving seven vehicle brands and seven evaluation criteria, assessed by domain experts, demonstrates the effectiveness of the proposed approach. Comparative analysis with existing logarithmic-based distance measures shows that the new methods provide superior accuracy, stability, and discrimination ability in ranking alternatives. The findings highlight the practical significance of the proposed q-ROF distance measures, offering a robust decision-support tool for vehicle selection and other CDM scenarios under uncertainty.

A Pythagorean fuzzy correlation coefficient (PFCC) is a reliable approach for eliminating ambiguity during the measure of relationships. Numerous Pythagorean fuzzy correlation coefficient methods (PFCCMs) have been constructed using Pearson's correlation coefficient technique. In this study, anew PFCCM is constructed based on Spearman's correlation coefficient to eliminate all possible uncertainties that may impede decision-makers from making a dependable selection. To validate the construction of anew PFCCM, we examine the existing PFCCMs and pinpoint their inadequacies. Among the extant PFCCMs, one approach was constructed through Spearman's correlation coefficient but it does not takes into cognizance the properties of the PFSs. In addition, it sometimes fails the axiomatic conditions of the PFCC, and yields invalid result for PFSs that are defined on a singleton set. These setbacks justify the construction of anew Spearman's correlation coefficient-like PFCCM, which is shown to overcome the limitations of the extant PFCCMs. Equally, the strength of the new PFCCM is verified by some theoretical results, and it fulfills the conditions of PFCC. Additionally, the use of the novel PFCCM is discussed in the solution of supplier selection problems to eliminate supplier selection ambiguity through the multiple criteria decision-making (MCDM) approach. To unarguably show the intrinsic worth of the new PFCCM, the effectiveness of the new PFCCM is compared with the existing PFCCMs and it is observed that the new PFCCM is reliable, consistent and precise, and in the same way satisfies the axioms of the PFCC. In particular, the existing Spearman's PFCCM yields cc in Example 4, while the new PFCCM produces 0.7603, which justifies the construction of anew Spearman's PFCCM. Finally, it is found that the new approach can suitably handle the hesitancies associated with the art of selection.

The term q-rung orthopair fuzzy set is an essential variant of fuzzy set with the capacity of tackling fuzziness and imprecision in the decision-making process. A fundamental concept in the decision-making process is the idea of correlation coefficient because of its wide applications. The process of decision-making is complex due to imprecisions, and as such the idea of correlation coefficient has been investigated under q-rung orthopair fuzzy setting. Some authors have constructed some techniques of correlation coefficient under q-rung orthopair fuzzy sets with practical applications. However, these existing techniques are defectives with several drawbacks in terms of precision and alignment with the conditions of correlation coefficient. In this work, two new techniques for estimating correlation coefficient under q-rung orthopair fuzzy sets are presented and theoretically discussed. Moreover, we apply the new techniques of correlation coefficient under q-rung orthopair fuzzy sets in disease diagnosis and employment process by using simulated q-rung orthopair fuzzy data based on multi-criteria decision-making approach and recognition principle. Some comparative analyses are provided to ascertain the benefits of the new techniques of correlation coefficient under q-rung orthopair fuzzy sets over the obtainable techniques with regard to reliability and performance rating.

Multigroup theory is the application of multisets to the theory of groups. Many group’s theoretic notions have been studied in multigroup theory, however, the ideas of maximal normal subgroup, simple group, normal series, composition series, and the Jordan-Hölder Theorem are yet to be investigated in multiset context. In this article, we define simple multigroup, maximalnormal submultigroup, normal series for multigroup, and composition series for multigroup with examples. With these concepts, we establish the Jordan-Hölder Theorem in multigroup theory. It is shown that every finite multigroup defined over a finite group has a composition series. In addition, it is established that every finite multigroup defined over a finite group has at least two composition series which are equivalent.

This paper presents intuitionistic fuzzy characteristic subgroups and discusses some of its properties. It is established that an intuitionistic fuzzy subgroup of an intuitionistic fuzzy group is characteristic provided its cuts are characteristic subgroups. In addition, it is proven that every intuitionistic fuzzy characteristic subgroup of an intuitionistic fuzzy group is an intuitionistic fuzzy normal subgroup. Furthermore, the notions of intuitionistic fuzzy maximal subgroups and intuitionistic fuzzy Frattini subgroups are established. It is shown that every intuitionistic fuzzy Frattini subgroup is an intuitionistic fuzzy characteristic subgroup as well as an intuitionistic fuzzy normal subgroup, respectively. Finally, some results on intuitionistic fuzzy Frattini subgroups are presented with regards to the level sets and cuts of an intuitionistic fuzzy groups.

The theory of intuitionistic fuzzy groups is an algebraic structure derivable from the utilization of groups in intuitionistic fuzzy sets. Many notions in group theory have been presented in intuitionistic fuzzy group theory. However, concepts like simple group, maximal normal subgroup, normal series, composition series, and the Jordan-Hölder Theorem are open problems in intuitionistic fuzzy group theory. Hence, this paper defines simple intuitionistic fuzzy groups, maximal normal intuitionistic fuzzy subgroups, normal series for intuitionistic fuzzy groups, and composition series for intuitionistic fuzzy groups with illustrations. In addition, the Jordan-Hölder Theorem in intuitionistic fuzzy group theory is verified. It is shown that every intuitionistic fuzzy group of a finite group possesses a composition series, and any two composition series for an intuitionistic fuzzy group of a finite group are equivalent.

To effectively evaluate relationships in uncertain environments, such as in decision-making, the idea of Intuitionistic Fuzzy Correlation Coefficient (IFCC) is employed. Several authors have provided many methods of IFCC, but their methods have some setbacks in correctness and accuracy. In this article, some novel methods of IFCC are developed that possess better exactness and reliability compared to the existing IFCC techniques. The new IFCC techniques possess high reliability and precision based on their mathematical formulations and the inclusion of all intuitionistic fuzzy parameters. The setbacks of the existing IFCC techniques are enumerated and verified with some numerical examples. Several of their properties are discussed in order to authenticate the novel IFCC techniques. More so, utilizing the new IFCC schemes in the issue of course allocations in higher institutions is discussed. Finally, the preeminence of the novel IFCC techniques is discussed in terms of precision and consistency with correlation principles by juxtaposing their effectiveness with other IFCC methods.

This work presents a modified Pythagorean fuzzy similarity operator and utilizes its potential in the analysis of questionnaire. Similarity operator is a formidable methodology for decision-making under uncertain domains. Pythagorean fuzzy set is an extended form of intuitionistic fuzzy set with a better accuracy in complex real-world applications. Lots of discussions bordering on the uses of Pythagorean fuzzy sets have been explored based on Pythagorean fuzzy similarity operators. Among the extant Pythagorean fuzzy similarity operators, the work of Zhang et al. is significant but it contains some flaws which need to be corrected/modified to enhance reliable interpretation. To this end, this work explicates the Zhang et al.’s techniques of Pythagorean fuzzy similarity operator by pinpointing their drawbacks to develop an enhanced Pythagorean fuzzy similarity operator, which appropriately satisfies the similarity conditions and yields consistent results in comparison to the Zhang et al.’s techniques. Succinctly speaking, the aim of the work is to correct the flaws in Zhang et al.’s techniques via modifications. To theoretically validate the enhanced Pythagorean fuzzy similarity operator, we discuss it properties and find out that the similarity conditions are well satisfied. In addition, the enhanced PFSO and the Zhang et al.’s PFSOs are compared in the context of precision, and it is verified that the enhanced Pythagorean fuzzy similarity operator can successfully measure the similarity between vastly related but inconsistent PFSs and as well yields a very reasonable results. Furthermore, the enhanced Pythagorean fuzzy similarity operator is applied to the analysis of questionnaire on virtual library to ascertain the extent of awareness and effects of virtual library on students’ academic performance via real data collected from fieldwork. Finally, it is certified that the enhanced Pythagorean fuzzy similarity operator can handle diverse everyday problems more precisely than the Zhang et al.’s Pythagorean fuzzy similarity operators.

The idea of Pythagorean fuzzy sets (PFSs) has been extensively applied in various decision-making scenarios. Many of the applications of PFSs were carried out based on similarity functions. Some methods of similarity functions for PFSs (SFPFSs) cannot be trusted for a reliable interpretations in practical cases due to some of their setbacks. In this work, a new method of SFPFSs is developed with the capacity to outsmart the efficiency of the extant SFPFSs in terms of precise results and appropriately satisfying the rules of SFs. The new method is described with some results to validate the properties of SFs. In terms of practical application, we use the newly developed method of SFPFSs to discuss the relationship between the players of the Liverpool Football Club (FC) in the 2022/2023 English Premier League (EPL) season to assess their performances in their resurgent moments within the season. Using data from BBC Sport analysis (BBCSA) on the players' rating per match in a Pythagorean fuzzy setting, we establish the players' interactions, communications, passing, contributions, and performances to ascertain the high ranking players based on performances. Similarly, a comparative analyses are presented in tables to undoubtedly express the superiority of the newly developed method of SFPFSs. Due to the flexibility of the newly developed method of SFPFSs, it can be used for clustering analysis. In addition, the new method of SFPFSs can be extended to other uncertain environments other than PFSs.