In this study, we extend the classical PROBID (Preference Ranking on the Basis of Optimal–Mean Distance) method into the Pythagorean fuzzy environment and propose Pythagorean Fuzzy PROBID (PyF-PROBID) method. The proposed method combines the structural robustness of PROBID with the high representational capability of Pythagorean fuzzy sets to better capture uncertainty and hesitation in real life human judgments. Unlike conventional distance-based MCDM methods, PyF-PROBID evaluates alternatives not only in terms of positive and negative optimal solutions but also with respect to sequential optimal and mean reference solutions, thus reducing rank reversal and inconsistency problems. A numerical example, the evaluation of domestic airline service quality, is presented to demonstrate the applicability of the model. A comparison section is provided with the results of existing PyF-TOPSIS and PyF-MABAC methods. The comparative analysis shows that PyF-PROBID produces stable and consistent rankings, aligning well with alternative Pythagorean fuzzy decision-making schemes. Therefore, the proposed method provides a more flexible, coherent, and reliable framework for multi-criteria decision analysis under uncertainty.

Facing the increasing bicycle providers in the market, it becomes a focus problem for companies to improve differentiated service strategies in this fierce market. Prior research has successively carried out service quality evaluation of bike-sharing providers using various quantitative methods. However, lots of methods cannot accurately adjust the representation range of uncertain information. Besides, service quality is involved of various quantifiable and non-quantifiable criteria. Moreover, considering the fact that different people hold different perceptions, it may be difficult for them to make accurate judgments under fuzzy environment. To address the aforementioned issues, this study places a specific research regarding the bike-sharing enterprise service level from four main dimensions, including perceptibility, availability, dependability and sustainability, and a novel group decision making approach is developed to evaluate bike-sharing service quality using extended Techniques for Order Preferences by Similarity to Ideal Solution (TOPSIS) under interval-valued Pythagorean fuzzy environment. The extended group decision making approach is characterized by the interval-valued Pythagorean fuzzy set (IVPFS), which has a good ability to deal with uncertainty, ambiguity and imprecision in the decision-making process, and enables decision makers to make more accurate and reliable judgments with considering the subjective nature of human judgments under fuzzy conditions. Finally, experiments are carried out to verify the viability and effectiveness of the proposed approach, and the results show that the proposed approach is effective and efficient to help decision makers to select optimal bicycle providers.

As a generalized extension of fuzzy sets, Pythagorean fuzzy sets (PFSs) provide a sophisticated framework for modeling subjective uncertainty inherent in human cognition. Real-world decision-making systems frequently exhibit concurrent uncertainties characterized by incompleteness and fuzziness. This work synthesizes grey system theory with PFSs to establish grey Pythagorean fuzzy sets (GPFSs), a novel uncertainty formalism capable of simultaneously representing vague and incomplete information. GPFSs achieve a unified characterization of both subjective and objective uncertainty. Initially, we express PFS membership and non-membership degrees as generalized grey numbers, thereby defining the degree of greyness for PFSs. Subsequently, fundamental set-theoretic operations for GPFSs are formalized—including whitening, subset relations, complementation, union, intersection, merging, and meet—along with a distance measure between arbitrary GPFSs. Finally, we develop an extended TOPSIS methodology for multi-criteria decision-making (MCDM) under grey Pythagorean fuzzy uncertainty. The proposed MCDM framework is empirically validated through a failure mode and effects analysis case study.

In today’s digital era, university students’ career-related decision-making has become increasingly complex, shaped not only by personal aspirations and societal expectations but also by the algorithmic recommendations embedded in AI-supported technologies. While such systems provide convenient references, their outputs are often vague, generalized, or contextually limited, which underscores the irreplaceable role of human judgment in situations of uncertainty. Against this backdrop, the present study develops a multi-criteria decision-making (MCDM) framework tailored to the context of student management and career guidance. Specifically, it integrates the compromise ranking strategy CRADIS (Compromise Ranking of Alternatives from Distance to Ideal Solution) with the expressive flexibility of Circular Pythagorean Fuzzy Sets (Cir-PyFS). Cir-PyFS was chosen to reflect the modalities of non-decision, inexactness, and subjective desire in human evaluation that people may have. The linguistic assessments of each alternative were redefined in CPFS representations within the proposed Circular Pythagorean Fuzzy-based CRADIS model (Cir-PyFBCM), which incorporates membership, non-membership, and radius values. The model outperformed traditional fuzzy MCDM approaches in terms of stability, interpretability, and sensitivity to the influence of decision-makers in an empirical exercise using a realistic career-planning example with several alternative courses of action, multiple expert decision-makers, and eight benefit-based evaluation criteria. These comparison and sensitivity analysis results support that the proposed methodology serves as an effective bridge, linking mathematical precision with the human approach to reasoning, and offers a more careful, flexible, and reliable tool for governing high-stakes decisions, especially university students, when making their career decisions in AI-supported realms. More broadly, the study underscores the potential of responsible AI-assisted decision-making in promoting reflective, evidence-based practices within higher education management.Supplementary InformationThe online version contains supplementary material available at 10.1038/s41598-025-22127-7.

The selection of a team in team-based sporting activities and professional situations is a crucial decision that requires an optimal choice from multiple conditions. The solutions currently available do not simplify the issues of the best team selection under different alternatives. In the modern age, the theory of multi-attribute decision-making (MADM) is a well-known approach for assessing decision-making problems. The main objective of this article is to introducing new aggregation operator (AO) called rough Pythagorean Fuzzy Dombi weighted averaging (RPyFDWA) operator, analytic hierarchy process (AHP) for measuring the weights of alternatives and also investigated the organizing preference rankings for enrichment evaluation (PROMETHEE) for ranking of alternatives by imposing a lower and upper approximations, under rough Pythagorean fuzzy set (RPyFS) framework. The RPyFS framework has a superior structure for handling uncertain data compared to existing frameworks. We also establish significant mathematical characteristics of the operator, including idempotency, monotonicity, and boundedness, to determine its flexibility. We have provided a case study based on a basketball team. The proposed theory is applied to the solution of a non-theoretical example related to assessing team selection issues. We can use our proposed AOs to select the best team from the list of four teams based on various attributes like physical fitness, performance criteria, experience, age, and injury prevention. We notice that the team It is the best team among all other considered teams using the proposed RPyFDWA operator and PROMETHEE method. We provided a comparison between other current approaches and established AOs for analyzing the authenticity and validity of the proposed approach. Also, provide a sensitivity analysis of the proposed study to observe the change in input variables that affects the model's or decision's dependability. A solid conclusion is provided at the end.

The following paper presents a new analytical framework for the optimization of player positioning, a methodology with significant practical implications. The method implements the multi-objective optimization by ratio analysis with full multiplicative form (MULTIMOORA) in a decision-making context in which several non-commensurable performance variables have to be combined. The application of Dombi operationalizes the framework by prioritizing weighted aggregation operators coupled with circular q-rung orthopair fuzzy sets (Cq-ROFSs). The Cq-ROFSs allow multidimensional representation of uncertainty, and allow dynamic actions upon the fuzzy parameter q, such that both intuitionistic fuzzy sets and Pythagorean fuzzy sets are subsets. Two Dombi prioritized operators on Cq-ROFSs are thereby devised a Cq-ROFSs Dombi prioritized weighted averaging operator (Cq-ROFSDPWA) and a Cq-ROFSs Dombi prioritized weighted geometric operator (Cq-ROFSDPWG). Results from empirical experiments are reported that demonstrate the performance of the resulting methodology, highlighting its practical relevance. The fundamental properties of these operators are also examined. The proposed aggregation operators are applied within the MULTIMOORA technique to assess their effectiveness. Numerical examples demonstrate that the methods yield logical and consistent results across different decision-making scenarios. Comparative analyses further highlight the advantages of the Cq-ROFSDPWA and Cq-ROFSDPWG operators over existing approaches.Supplementary InformationThe online version contains supplementary material available at 10.1038/s41598-025-18795-0.

By using Geographic Information Systems, satellite imagery from remote sensing techniques provides quantitative and qualitative data about Earth’s natural and human elements. However, the direct use of raw imagery may prevent the accurate identification of the spectral and temporal characteristics of the target objects. To obtain meaningful results from these data, the object and surface features in the image must be classified correctly. In this context, this study develops a new deep learning approach that includes hyperparameter optimization that considers uncertainty factors when classifying satellite imagery. In the proposed approach, a hybrid architecture called CNN-Pythagorean Fuzzy Deep Neural Network (PFDNN) is developed by combining the ability of convolutional neural networks (CNN) to reveal expressive features with the ability of Pythagorean fuzzy set (PFS) theory to predict uncertainty. In addition, to further improve the model’s success, the hyperparameters are automatically optimized using Optuna. In the experiments conducted on the EuroSAT RGB dataset, the CNN+PFDNN+Optuna model achieved 0.9696 ± 0.0037 accuracy and a macro-AUC value of 0.9983, outperforming other methods such as DNN, FDNN, PFDNN and VGG16+PFDNN. Including the Pythagorean fuzzy layer in the system provided about 13.05% higher accuracy than conventional fuzzy systems. These results show that integrating the Pythagorean fuzzy set approach into deep learning models contributes to more effective management of uncertainties in remote sensing data and that hyperparameter optimization significantly impacts model performance.

This study tackles the critical challenge of waste material recycling, which is worsened by the growing global population and the necessity for efficient recycling processes. To address this issue, we introduce a pioneering approach that leverages circular Pythagorean fuzzy set theory, a sophisticated extension of fuzzy and intuitionistic fuzzy information. By formulating Muirhead mean and dual Muirhead mean aggregation operators within this framework, we facilitate structured and intelligent decision-making for assessing waste recycling alternatives. Our methodology and algorithm for multi-criteria group decision-making problems are showcased through a practical example, highlighting the efficacy and dependability of our approach. This research makes a significant contribution to the development of more efficient waste recycling processes and informed decision-making. The proposed approach enables decision-makers to evaluate waste recycling alternatives more comprehensively and systematically, taking into account multiple criteria and stakeholder perspectives. The findings of this study have important implications for policymakers, waste management professionals, and stakeholders seeking to improve waste recycling practices and reduce environmental impacts. By providing a more effective and reliable decision-making framework, this research aims to support the development of sustainable waste management systems. A sensitivity analysis illustrates the effectiveness and reliability of the proposed work. Finally, we adopted the comparative study and highlighted the advantages of defined work.

The concept of Multi Q-Fermatean hesitant fuzzy soft sets (MQFHFSS), derived from the integration of multi-Q fuzzy soft sets and Fermatean hesitant fuzzy sets, can be applied in practice to optimise the resolution of complex multi-criteria decision-making problems. The method exceeds traditional approaches such as Fermatean hesitant fuzzy sets, fuzzy soft sets, and Pythagorean fuzzy sets in enhancing the ability to capture higher levels of uncertainty, hesitation, and symmetry in multi-criteria evaluations, thereby supporting more balanced judgments in complex decision-making situations. In this study, we investigate the novel MQFHFSS concept along with the associated operations. The fundamental characteristics of aggregation operators derived from MQFHFSS have been examined to address some complex decision-making issues. Moreover, we discuss some key algebraic features and their different cases, emphasizing the role of symmetry under the influence of MQFHFSS. Finally, we illustrate some numerical examples and solve the real-world decision-making problem by using the proposed technique.

This paper introduces a comprehensive mathematical framework that integrates Pythagorean fuzzy sets (PyFS), soft sets (SS), and hypersoft set theory to establish robust methodologies for addressing complex uncertainties in multi-attribute decision-making (MADM). We present two formal approaches: Pythagorean soft sets (PySS) and Pythagorean hypersoft sets (PyHSS), which enhance the conventional soft set and hypersoft set frameworks by incorporating the expressiveness of Pythagorean fuzzy sets (PyFS). The theoretical framework demonstrates that these structures preserve all fundamental set-theoretic operations and facilitate the representation of the membership degree (MD) and non-membership degree (n-MD) with sums not exceeding 1, contingent upon satisfying the Pythagorean condition ψ² + φ² ≤ 1. We illustrate the applicability of our approach through extensive digital implementations in technology selection, cloud-based configuration, and educational recommendation systems. The mathematical functionalities exhibited, including operation closure and generalisation hierarchies, render PySS and PyHSS formidable decision support system instruments for managing imprecise and ambiguous information across several criteria.

Sustainable economies require effective energy planning that goes beyond relying on functioning forecasting models to comprehend energy dynamics, and also provides well-defined decision-making (DM) models that can address risk, ambiguity, and conflicting eco-economic objectives. This type of strategic planning requires an integrated assessment approach that can evaluate forecasting choices in an uncertain and dynamic environment. This paper presents a new and modified methodology for ranking energy forecasting models within a Pythagorean Fuzzy Set (PFS) system by integrating the CRITIC (Criteria Importance Through Inter-Criteria Correlation) weighting framework and the MAIRCA (Multi-Attributive Ideal-Real Comparative Analysis) ranking scheme. In the suggested framework, expert uncertainty and vagueness are represented by the PFS environment. In contrast, some of the leading eco-economic indicators are objectively weighted using CRITIC, and forecasting model alternatives are prioritized based on MAIRCA. A comparative study is conducted on a hypothetical data set that represents realistic energy system capabilities, including adaptability, carbon policy integration, and computing efficiency. The findings suggest that the framework contributes to consistent, interpretable, and uncertainty-aware rankings, and the Deep Q-Network (DQN) model was ranked to be the most effective alternative. The study contributes to the development of more sophisticated decision-support mechanisms to facilitate sustainable energy planning, enabling informed and balanced decisions as the eco-economic climate evolves rapidly.

The exponential rise in the global population, combined with limited food resources, poses a critical challenge to feeding more than 9 billion people. Furthermore, the increasing reliance of people on perishable and refrigerated products has proven that the cold supply chain (CSC) has inherent value in ensuring food quality and global food security. The cold supply chain (CSC) is crucial for maintaining food quality and ensuring global food security, yet it faces significant challenges that lead to quality deterioration and food loss. Therefore, a strategic decision support system is necessary to alleviate the severity of these obstacles and ensure the food quality in the CSC. This study aims to facilitate a novel two-stage decision support model that helps decision-makers identify and prioritize the most critical obstacles affecting CSC performance, and propose effective solutions to overcome these obstacles and ensure the food quality. The study integrates a novel Analytic Hierarchy Process (AHP) and a Multiplicative Multi-Objective Optimization based on Ratio Analysis (MULTIMOORA)-based hybrid MCDM decision-making methodology to establish the practical foundation of the model under interval-valued Pythagorean fuzzy (IVPF) set theory. The IVPF-AHP method is utilized to visualize the severity of the obstacles, while the IVPF-MULTIMOORA analyzes the importance and priorities of the alternatives. The reliability and robustness of the results through the IVPF-based AHP-MULTIMOORA methodology have been supported by conducting a total of thirty-one experiments under different sets of input parameters.

Global trade heavily depends on effective supply chain management and the strategic placement of distributors. Additionally, customer demands are becoming increasingly diverse and dynamic, with each customer expecting prompt and reliable responses from companies or distribution centers. To ensure a fast and efficient delivery process, businesses are striving to make well-informed decisions about the site selection of distribution centers or warehouses. The growth and success of online businesses, such as those on Amazon, largely hinge on the strategic site selection of their warehouses to expedite supply chain operations. This site selection process requires a comprehensive analysis of various factors. However, collecting and processing the relevant data often involves uncertainties and fuzziness. To address these challenges, the proposed research introduces a novel algorithmic approach based on distance measures within the Complex Pythagorean Fuzzy Soft Set (CPFSS) framework. The research presents the formulation of distance measures for the CPFSS, followed by the development of an algorithm. This algorithm is then applied to a real-life case study for the site selection of a warehouse for a company named HCRFT, which specializes in handicrafts. Furthermore, a detailed comparison between the proposed approach and existing models is conducted to validate and demonstrate the effectiveness of the algorithm. Finally, concluding remarks summarize the findings and implications of the study.

The rise of streaming services and online story-sharing has led to a vast amount of cinema and television content being viewed and reviewed daily by a worldwide audience. It is a unique challenge to grasp the nuanced insights of these reviews, particularly as context, emotion, and specific components like acting, direction, and storyline intertwine extensively. The aim of this study is to address said complexity with a new hybrid Multi Criteria Decision-Making MCDM model that combines the Deck of Cards Method (DoCM) with the Circular Pythagorean Fuzzy Set (CPFS) framework, retaining the symmetry of information. The study is conducted on a simulated dataset to demonstrate the framework and outline the plan for approaching real-world press reviews. We postulate a more informed mechanism of assessing and choosing the most appropriate deep learning assembler, such as the transformer version, the hybrid Convolutional Neural Network CNN-RNN, and the attention-based framework of aspect-based sentiment mapping in film and television reviews. The model leverages both the cognitive ease of the DoCM and the expressive ability of the Pythagorean fuzzy set (PFS) in a circular relationship setting possessing symmetry, and can be applied to various decision-making situations other than the interpretation of media sentiments. This enables decision-makers to intuitively and flexibly compare alternatives based on many sentiment-relevant aspects, including classification accuracy, interpretability, computational efficiency, and generalization. The experiments are based on a hypothetical representation of media review datasets and test whether the model can combine human insight with algorithmic precision. Ultimately, this study presents a sound, structurally clear, and expandable framework of decision support to academicians and industry professionals involved in converging deep learning and opinion mining in entertainment analytics.

In this communication, we construct new multiple attribute decision-making (MADM) problems using the redefined square root interval-valued normal Pythagorean fuzzy set (RSIVNPFS). The interval-valued Pythagorean fuzzy sets (IVPFSs) and square root PFSs are extended by the square RSIVNPFS. We introduce RSIVNPF weighted averaging (RSIVNPFWA), RSIVNPF weighted geometric (RSIVNPFWG), generalized RSIVNPFWA (RSGIVNPFWA), and generalized RSIVNPFWG (RSGIVNPFWG). Idempotence, boundedness, commutativity, and monotonicity in algebraic operations are all satisfied by RSIVNPFSs. We develop an algorithm for dealing with MADM problems using the aggregation operators (AOs). The applications of the Euclidean distance (ED) and the Hamming distance (HD) are described using examples from everyday scenarios. We also compare several suggested and current models to show the validity and applicability of the models. Our objective is to compare expert opinions with the criteria in order to determine the best option and to demonstrate the superiority and validity of the suggested AOs.

Sustainability evaluation in manufacturing industries is increasingly vital for promoting responsible growth and long-term competitiveness amid environmental, social, and economic challenges. Effective decision-making (DM) under uncertainty is crucial for managing multiple, often conflicting sustainability objectives. In this paper, we propose a novel hybrid model, termed Pythagorean fuzzy N-bipolar soft sets (PFNBSSs), which integrates Pythagorean fuzzy sets (PFSs), N-soft sets (NSSs), and bipolar soft sets (BSSs) within a unified multi-criteria decision-making (MCDM) framework. For theoretical purposes, we define basic operations and algebraic properties of PFNBSSs, supported by illustrative examples. To demonstrate practical applicability, the PFNBSS model is applied to assess sustainability practices in manufacturing industries through two numerical examples: one focusing on positive and negative sustainability indicators, and another emphasizing comparative sustainability risk assessment across diverse manufacturing sectors. Detailed interpretations of computational results and their relevance in practical DM are provided. This is followed by a comparative analysis confirming the superior discrimination power and expressive capability of the PFNBSS model over existing alternatives. The paper concludes with a critical evaluation of the model and suggestions for future research.

Chi-square Distance Measure of Pythagorean Fuzzy Sets Based on Expected Boundary and its Applications

Quality education and instructor training are foundational to the development of any society, as they directly influence the effectiveness of learning and the overall performance of educational systems. Decision-making is crucial in enhancing quality education and instructor training by ensuring teaching policies. This article modifies some robust mathematical methodologies of the fuzzy framework and decision-making techniques to aggregate an authentic ranking of preferences. We also explore a novel approach to the circular pythagorean fuzzy set (Cr-PyFS) that is used to manage uncertainty and vagueness in complicated real-life applications. Some flexible operations of Frank t-norm and t-conorm are also formulated under the system of circular pythagorean fuzzy (Cr-PyF) information. Furthermore, we derive a list of mathematical aggregation operators such as circular pythagorean fuzzy frank weighted average (Cr-PyFFWA) and circular pythagorean fuzzy frank weighted geometric (Cr- PyFFWG) operators with prominent properties. Additionally, a decision-making approach to the criteria importance through the intercriteria correlation (CRITIC) method is adopted to investigate the weight of criteria by incorporating the Cr-PyF situations. Moreover, an optimization technique of the weighted aggregated sum product assessment (WASPAS) method is established to rank alternatives under different conflicting criteria. A numerical example is constructed to examine suitable training institutes to improve instructor skills and expertise. The comparative study is also established to demonstrate the superiority and validation of pioneering approaches with existing terminologies. Finally, a summary of the article is presented at the end of the manuscript.

The q-rung orthopair fuzzy set (q-ROFS) has been developed as an extension of the Pythagorean fuzzy set (PFS) to address ambiguity in various decision-making contexts. Group theory is a significant area of mathematics with numerous applications across various scientific fields. This paper examines q-rung orthopair fuzzy group theory, emphasizing the importance of q-ROFS and group theory. The concept of a q-rung orthopair fuzzy subgroup (q-ROFSG) is introduced, and its various algebraic properties are examined. A comprehensive investigation into q-rung orthopair fuzzy cosets (q-ROFCs) and q-rung orthopair fuzzy normal subgroups (q-ROFNSGs) has been conducted. The definitions of q-rung orthopair fuzzy homomorphism and isomorphism are presented. We extend the concept of the quotient group of a classical group V in relation to its normal subgroup U by introducing a q-ROFSG of V⁄U. The q-rung orthopair fuzzy variant of the three fundamental isomorphism theorems has been demonstrated.

In this paper, we develop an edge detection model based on Pythagorean fuzzy sets (PFSs) for dealing with uncertainty in images before edge detection. In this regard, novel Pythagorean fuzzy (PF)-entropy and discrimination measures are developed to enumerate the degree of difference between objects or images, which can evade the limitations of existing PF-discrimination measures. Moreover, several elegant properties of the developed PF-discrimination and entropy measures are presented in the context of PFSs. Numerical examples are presented to illustrate the efficacy of the developed PF-entropy and discrimination measures over the existing PF-discrimination measures. Further, an edge detection model is presented with the developed PF-discrimination measures and entropy and its application in edge detection. To show the usefulness of the developed edge detection approach on PFSs, the computational outcomes are compared with existing edge detection models. Finally, some measurement parameters such as mean square error (MSE), peak signal-to-noise ratio (PSNR) and structural similarity index measure (SSIM) are determined. The findings of this study conclude that the PSNR and SSIM values of the developed model are continuously greater than or equal to the PSNR and SSIM values of various extant edge detection models, respectively. The obtained results on the medical images have shown their applicability to detect edges appropriately in the occurrence of uncertainty and in the presence of noise. Experimental results on several images validate the theoretical results and demonstrate good performance of the developed model.