Circular intuitionistic fuzzy sets (C-IFSs) offer an expressive mathematical structure for representing uncertainty, extending traditional intuitionistic fuzzy sets by introducing a geometric interpretation through circular regions. Knowledge measures, which quantify information from the perspective of certainty, serve as a critical tool for capturing informational content and clarity. This study systematically extends the notion of knowledge measures to the framework of C-IFSs by introducing a suite of general construction techniques. These methods employ foundational mathematical constructs, including t-norms, t-conorms, automorphisms, and aggregation operators. The article ensures theoretical soundness by formally deriving expressions and proving mathematical results. Importantly, the proposed framework demonstrates that knowledge measures can be constructed not only from individual t-norms and t-conorms but also through their integration with automorphisms and aggregation strategies. The formulation of arguments within these operators plays a pivotal role, and the inherent non-uniqueness of such structures invites further scholarly exploration. Building on these insights, this study develops a novel score function that enables a refined ranking mechanism balancing both information content and clarity. Experts can adjust decision weights dynamically through tunable parameters. Additionally, the article proposes a new weight generation method based on knowledge-theoretic principles. To illustrate the practical efficacy of the proposed approach, this study introduces a comprehensive multi-attribute group decision-making framework and applies it to a real-world supplier selection problem in the manufacturing sector. The results demonstrate strong robustness and reliability of the proposed model, with an average Spearman rank correlation coefficient of 0.90 compared to existing approaches.

With the rapid expansion of battery-powered electric vehicles, the transportation industry is progressively moving away from fossil fuel-based combustion engines. Lithium-ion batteries have become the dominant energy storage solution due to their efficiency and reliability. However, challenges such as high production costs, scarce raw materials, and limited life cycles have intensified the need for effective recovery and recycling strategies. Determining the optimal location for a recovery center for end-of-life lithium-ion batteries presents a complex multi-criteria decision-making challenge, shaped by technical, environmental, economic, and social considerations. To address this, we introduce a hybrid decision-making framework that integrates the interval rough regime technique. The proposed methodology is structured into three key stages; first leveraging interval rough numbers to manage uncertainty and evaluate alternative performance, then the second stage is dedicated to introduce an interval rough-based logarithmic percentage change-driven as objective weighting method to determine the relative importance of criteria, and final stage presents an outranking-based interval rough regime approach to conduct pairwise comparisons and establish a comprehensive ranking of alternatives. A case study conducted in Istanbul assesses six potential locations for a lithium-ion battery recovery center, demonstrating the applicability and effectiveness of the proposed framework. The results highlight Tuzla as the most favorable location, attributed to its proximity to suppliers, well-developed transportation infrastructure, and strategic positioning. Additionally, a comparative evaluation with existing decision-making methods confirms the robustness and reliability of the findings. The proposed model offers a structured and adaptable decision-support tool for addressing complex location selection challenges in sustainable energy management.

The concept of the Z-number offers a powerful framework for representing fuzzy information, enabling more accurate and reliable decision-making under uncertainty. Building upon this foundation, the Pythagorean fuzzy Z-number further enhances the capacity to model vagueness, accommodating a broader and more nuanced range of data, thus aligning more closely with human reasoning. To enhance this capability, this paper introduces continuous Pythagorean fuzzy Z-numbers, which allow for greater flexibility in expressing uncertainty and better reflect human judgment when dealing with vague or ambiguous data. Also, it proposes a comprehensive arithmetic framework to operate within this context. A probability measure for Pythagorean fuzzy events is established and utilized to assess the results of unary and binary operations involving Pythagorean fuzzy Z-numbers, thereby promoting the advancement of improved arithmetic operations. A computational approach for arithmetic operations of Pythagorean fuzzy Z-numbers has been formulated to determine Pythagorean fuzzy Z-numbers in a more efficient and practical manner, ensuring suitability for real-world applications. To facilitate practical application, a distance measure and a ranking function are developed, enabling more effective comparison and analysis. The study also extends the Weighted Aggregated Sum Product Assessment (WASPAS) method to the Pythagorean fuzzy Z-number environment and demonstrates its utility through a real-world case study. Comparative results and sensitivity analysis are presented to validate the robustness and effectiveness of the proposed approach. The proposed decision-making technique based on Pythagorean fuzzy Z-numbers provides enhanced reliability by jointly modeling uncertainty and expert hesitancy. The consistency of its results with established techniques underscores its superiority and real-world applicability.

Precise crop type classification and agricultural suitability analysis are critical for optimizing land use, improving productivity, and ensuring food security under climate variability. Traditional models often struggle with overlapping class boundaries, high-dimensional input spaces, and lack of interpretability, limiting their practical deployment in decision-support systems. This study proposes a hybrid fuzzy-machine learning framework that integrates Principal Component Analysis for dimensionality reduction, Fuzzy C-Means for uncertainty-aware clustering, and a Random Forest classifier for robust prediction. The framework was trained and validated on a real-world dataset of soil and climatic parameters (soil pH, Nitrogen, Phosphorus, Potassium, temperature, humidity, and wind speed) covering seven major crops, with class-wise stratified five-fold cross-validation ensuring reliability. Principal Component Analysis reduced redundancy while retaining over 87% of total variance, and Fuzzy C-Means membership values captured uncertainty in overlapping crop suitability patterns. The combined Principal Component Analysis, Fuzzy C-Means clustering, and Random Forest model achieved an average accuracy of 93.8%, precision of 94.3%, recall of 94.1 %, and F1-score of 94.1 %, representing a 6% gain in F1-score over baseline Random Forest. Crops such as Cotton, Sugarcane, and Soybean consistently showed high classification performance, while errors were concentrated in pairs with overlapping requirements ( Potato vs. Soybean and Sugarcane vs. Rice). SHapley Additive Explanations-based analysis revealed that The first principal component and fuzzy membership features contributed most to decision-making, aligning with agronomic knowledge and providing interpretability. These results confirm that the proposed framework improves accuracy, enhances transparency, and offers practical decision-support value for sustainable agricultural planning.

The foundation of a nation’s future lies in the performance of its students. Their academic success shapes not only their personal growth but also the strength and progress of the entire country. High-achieving students go on to become skilled professionals, innovators, and leaders who drive economic development and social change. In this way, student performance plays a vital role in building a knowledgeable and competitive society. Investing in education today ensures a stronger, brighter nation tomorrow. Student performance depends on various factors such as the quality of teaching, availability of learning resources, parental support, and a conducive learning environment. Motivation, mental health, and socio-economic background also significantly influence how well a student performs academically. Although previous studies have applied either clustering techniques or machine learning algorithms independently to evaluate student performance, they often lacked the ability to capture the uncertainty and overlap in student characteristics. This research addresses that gap by combining fuzzy c-means clustering, which allows for soft classification, with a machine learning algorithm. The hybrid model enhances the accuracy and interpretability of performance evaluation by leveraging the strengths of both methods. This integrated approach provides a more nuanced and data-driven understanding of student outcomes. In addition to this, explainable artificial intelligence technique is employed to provide a transparent and interpretable summary of how the proposed hybrid model functions. By integrating this technique, the research not only improves prediction accuracy but also ensures that the decision-making process is understandable to educators and stakeholders. Furthermore, feature distribution graphs and correlation heatmap are drawn to have visual understanding of the related features. At the last, comparison with existing techniques, limitations and future directions are being discussed.

This paper presents a unified framework for constructing cosine-based similarity measures for intuitionistic fuzzy sets and demonstrates their effectiveness in decision-making and attribute reduction. A generator-based approach is introduced to derive a broad family of similarity measures from existing ones, complemented by alternative constructions using dissimilarity measures and functions. A general criterion for transforming similarity measures into entropy measures is established, with counterexamples provided to delineate its limitations. To validate the proposed methods, the similarity measures are integrated into the TOPSIS framework with entropy-based weighting and demonstrated through a renewable energy project selection example. In addition, an attribute reduction algorithm, enhanced through TOPSIS and Spearman’s correlation, is developed to improve computational efficiency and decision quality. A systematic framework for comparing similarity measures across multiple evaluation metrics, demonstrated on a specific case study, is also proposed. Numerical experiments confirm the robustness and practical value of the approach.

Developing countries face increasing pressure to balance rising energy demands with the pursuit of sustainable development. This study addresses this challenge by evaluating renewable energy projects using a group decision-making framework that integrates technical, economic, environmental, and social dimensions. To handle incomplete information, expert opinion ambiguity, and inherent variability, a novel dual spherical fuzzy rough number cloud-based approach is proposed. This approach combines spherical fuzzy rough numbers with a probabilistic cloud model, enhancing the representation of uncertainty by capturing both ambiguity and randomness in expert evaluations. The main contributions include introducing this advanced uncertainty-handling approach and developing a hybrid decision-making framework that enhances the reliability and robustness of renewable energy project assessments, supported by real-world validation. An extended entropy-based method is used to determine the weights of evaluation criteria, while a unified utility function integrates the strengths of weighted sum and weighted product methods to rank alternatives. A case study on renewable energy projects in Iran demonstrates the model’s applicability. The results show that solar energy is the most viable renewable option. Validation and sensitivity analyses confirm the robustness of these findings.

In two-sided matching decision problems, the matching objects with different knowledge, experiences, and cultures provide linguistic assessments using diverse or multi-granular sets with a factor that the information provided is hesitant in nature due to different opinions given by experts. In the proposed approach, the hesitant 2-tuple linguistic information is integrated with rough approximations to develop the two novel approaches called hesitant rough numbers and hesitant 2-tuple linguistic rough numbers. The proposed novel approximations are implemented on a two-sided matching optimization model to study hesitant multi-granular uncertainty. Firstly, the matching objects provide their evaluations in the form of hesitant multi-granular terms converted into hesitant 2-tuple linguistic rough numbers. Secondly, certain optimization models based on hesitant 2-tuple linguistic rough approximations are constructed to compute the criteria weights using incomplete information. In hesitant 2-tuple linguistic rough optimization models, the maximizing deviation technique is used to find the distance between two proposed novel coefficients. To maximize the level of satisfaction with matching objects, a hesitant 2-tuple linguistic rough optimization model is developed to evaluate the overall satisfaction degree and stability of matching objects. The significance of the proposed two-sided matching optimization model is illustrated with a case study of matching between the green building technology supply and demand. The out-performance of the proposed model is highlighted by a comparison analysis with existing approaches to analyze that it can provide hesitant multi-granular rough flexibility and deals with incomplete information regarding criterion weights.

Multi-polar fuzzy sets are crucial for capturing and representing diverse opinions or conflicting criteria in decision-making processes with greater flexibility and precision. While, Z-numbers are important for effectively modeling uncertainty by incorporating both the reliability of information and its degree of fuzziness, enhancing decision-making in uncertain environments. To date, no model in the literature exhibits the properties of multi-polar fuzzy sets and Z-numbers. In this article, we introduce a new concept of multi-polar fuzzy Z-number and Hamacher operations for multi-polar fuzzy Z-numbers. Based on the Hamacher operations, we propose aggregation operators for multi-polar fuzzy Z-numbers, namely, multi-polar fuzzy Z-number Hamacher weighted averaging operator, multi-polar fuzzy Z-number Hamacher ordered weighted averaging operator, multi-polar fuzzy Z-number Hamacher weighted geometric operator and multi-polar fuzzy Z-number Hamacher ordered weighted geometric operator. Additionally, we develop a decision-making model based on the proposed Hamacher aggregation operators. Further, we apply the proposed technique to a couple of case studies to check the validity and authenticity of the proposed methodology. Finally, we compare the outcomes of the study with several existing techniques to assess the accuracy of the proposed model.

Circular intuitionistic fuzzy sets (C-IFSs) emerge as a powerful extension of fuzzy and intuitionistic fuzzy sets (IFSs) to handle uncertain situations. Moreover, preference relations are widely used for solving group decision-making (GDM) problems. This paper defines circular intuitionistic fuzzy preference relations (C-IFPRs), using C-IFSs and preference relations. We define basic operations and aggregation operators for C-IFPRs. Several entropy measures for C-IFPRs are given and justified theoretically. A method is developed to generate the weights of decision-makers using entropy measures. New similarity measures are constructed to measure the resemblance between two C-IFPRs. On the other hand, an algorithm is proposed to improve the additive consistency of C-IFPRs. This algorithm guarantees the additive consistency of any C-IFPR, and once the consistency is achieved, it will hold even after further manipulations. Furthermore, a new method is developed to achieve an acceptable consensus level for GDM. It is based on similarity measures and aggregation operators. This method will guarantee that any sequence of C-IFPRs can achieve the group consensus. If the sequence is additively consistent, then this method will not disturb their consistency while achieving the group consensus. Additionally, it remains insensitive to similarity measures, and changing the similarity measure only affects the number of iterations to achieve group consensus. A TOPSIS-type selection process is extended to obtain a complete ranking, guaranteeing that the optimal solution is closest to the ideal solution and farthest from the worst option. In summary, this paper presents a GDM method leveraging C-IFPRs, an entropy-based weight generation method, additive consistency, a group consensus-reaching method, and a TOPSIS-type selection process. Numerical examples are provided to justify our developed approaches.

Cadmium (Cd) contamination in various regions of Pakistan has been reported which poses severe threats to the health of local communities through various exposure routes. There is relatively scarce data and information regarding Cd contamination status in the groundwater of Punjab, Pakistan, which is typically used for drinking purposes. The present research work was carried out to assess the concentration of Cd in the drinking water samples collected from the Khanewal district. Drinking water samples (196) were collected from different sources of groundwater (hand and electric pumps, tube wells) at different depths (50-400 feet) in rural and urban areas of four tehsils (Jahanian, Kabirwala, Khanewal, and Mian Channu) of district Khanewal. The collected water samples were evaluated for Cd level and physico-chemical properties such as electrical conductivity, pH, carbonates, cations, anions, and bicarbonates. It was noticed that 90% of collected samples of water were unsafe for drinking purposes as these contained higher levels of Cd compared to the World Health Organization (WHO) permissible limit of Cd (3.0 x 10-3 mg L-1) in drinking water. Cd-induced health risks were also calculated concerning the hazard quotient (HQ), the average daily dose (ADD), and carcinogenic risk (CR) for humans who were reliant on the Cd-mixed water for consumption. Overall, the study found that people in the Khanewal district were at a severe/serious carcinogenic health risk due to Cd contamination in drinking water. This study highlights that management and monitoring steps are necessary for people in study regions, to decrease Cd-induced health issues and build effective remediation methods for Cd-contaminated water.

In order to store and oversee inventory, raw materials, or finished goods in different Konya regions, warehouse building management is a challenging task for the authorities that calls for creative and efficient distribution operations. Choosing the ideal site for warehouse facility is typically a difficult procedure, particularly when taking into account a number of parameters.Due to the costs of purchasing the property and associated improvements, the location selection problem is a long-term investment decision; as such, it should be carefully considered using a reliable method to avoid the negative consequences that follow bad decisions. To handle the inherent complexity and numerous ambiguities in real-world circumstances, the problems underlying warehouse management require the application of appropriate multi-criteria decision-making techniques. The goal of the present study is to develop an integrated spherical fuzzy rough number based -Measurement of Alternatives and Ranking according to Compromise Solution (MARCOS) method. Through the spherical fuzzy rough ideal and anti-ideal solutions, the spherical fuzzy rough-MARCOS technique yields the reference points and allows for a more accurate assessment of alternatives via utility degree. The study’s emphasis on appropriately evaluating the criterion weights utilizing the spherical fuzzy rough -Stepwise Weight Assessment Ratio Analysis (SWARA) algorithm to ascertain the viability of the decisions and reduce the decision-makers’ personal impact. To address ambiguity and imprecision in expert assessment, the spherical fuzzy rough numbers are employed throughout the examination process. To demonstrate how the recommended decision-making methodology may be applied to select the ideal location for warehouses, a case study of Konya is resolved. Our proposed method is compared with two current multi-criteria decision-making techniques, namely the spherical fuzzy rough-TOPSIS method and spherical fuzzy rough-WASPAS, in order to confirm the consistency of results. Sensitivity analysis is applied to verify if the recommended method’s computations are appropriate.

Useful decisions are made based on reliable information. The concept of Z-number involves the issue of reliability of information. Multipolar information is particularly important in scenarios involving multiple attributes in a decision making process. There does not exist a study in the literature that conveys multipolar information with reliability. In this research article, the concept of multipolar fuzzy Z-Dombi aggregation operators is first introduced. An outranking method based on the proposed multipolar fuzzy Z-Dombi aggregation operators is then developed. The proposed method is applied to a case study related to the selection of the best rehabilitation centre for the treatment of teenage drug users. The proposed method is compared with four existing techniques in multipolar fuzzy and fuzzy environments to validate the approach. A sensitivity analysis is performed to test the credibility of the study. Further, the Spearman coefficient is calculated for ranking lists obtained by different methods to verify the method’s consistency. The study’s findings are presented in graphical illustrations for a clear understanding of the results. The method shows validity through consistent comparison with four established techniques. This alignment supports its robustness and relevance in practical applications. Moreover, a positive Spearman correlation coefficient confirms its reliability by aligning rankings with expected outcomes.

Soft set theory builds on the idea of a parameterized family of subsets of a universal set, where for each pertinent characteristic, any specific member of the universe either satisfies it or not. The concept of an N-soft set sharpens this model with the aid of multinary parameterized descriptions; that is, N-soft sets categorize the options in terms of multiple classifications of the characteristics. The aim of this research is fourfold. First, this research focuses on daily-life decision-making problems that involve both positive and negative attributes that can be naturally distributed among classes. Each comparable group of attributes produces an N-soft set, and we can represent all these N-soft sets using separable N-soft sets. We show that this structure facilitates decision-making in the presence of large numbers of attributes. Second, to develop tools that provide a mechanism for the selection of an alternative in this new model, we first develop a complement operator for N-soft sets to uniformize the data, and then, we propose strategies for taking advantage of the qualities of the attributes. Aggregation operators are employed to aggregate the data into a resultant N-soft set, a fuzzy N-soft set, or a hesitant N-soft set. Several algorithmic procedures are proposed to define these methods. Third, we define the novel notion of a multihesitant N-soft set. This loosely defined concept is helpful for representing data with multiple and repetitive entries while avoiding information loss. Finally, we provide solutions to several real-life decision-making problems to illustrate the versatility of our approaches. We apply this theory to construct a new method for ranking countries participating in the Olympic Games. Our motivation is that the existing lexicographic procedure is unable to distinguish among gold, silver, and bronze medals won at sports with very different characteristics.

Supply chain network design is a strategic framework that optimizes the flow of materials, information, and resources. Designing such networks under circular economy principles requires an integrated approach that embeds economic, environmental, sustainability, and resilience criteria into competitive priorities. To address incomplete data, diverse information, subjective judgments, and cognitive limitations, this study introduces a novel multi-stage circular supplier solution method based on a Fermatean fuzzy rough approach. Unlike conventional models, the proposed framework combines Fermatean fuzzy rough weighted sum and weighted product techniques to simultaneously capture uncertainty, imprecision, and subjectivity in supplier evaluation and network design. A distinctive strength of the method lies in its capacity to handle incomplete information without predefined parameters, thereby reducing distortion and information loss, enabling experts to fine-tune precision, and overcoming constraints in circular supplier selection. The framework further integrates suppliers' sustainability and circular economy ratings directly into supply chain configuration. Empirical validation using expert data from Iran's food, dairy, and beverage sectors, covering raw material suppliers, production facilities, and logistics centers evaluated against sustainability and circularity criteria, demonstrates that embedding circular economy and resilience objectives reshapes supplier prioritization, with sustainability-oriented suppliers outperforming cost-driven ones. Sensitivity analysis confirms robustness, as a +/- 10% variation in the conservatism coefficient results in less than a 5% change in outcomes, while comparative analysis demonstrates the superiority of the proposed model in achieving higher decision validity, consistency, and transparency. Overall, this study advances multi-criteria decision-making under uncertainty and equips managers and policymakers with a rigorous and adaptable tool for developing sustainable and resilient circular supply networks in emerging economies.

Humanitarian supply chain management plays a crucial role in effectively delivering aid and allocating resources during crises. It involves coordinated logistics, inventory control, and collaboration among stakeholders to ensure timely and ethical support. The integration of artificial intelligence with human judgment enhances logistics efficiency, resource allocation, and adaptability. Key enablers such as advanced technology, robust infrastructure, and efficient communication systems support seamless operations across the supply chain. Structured around the phases of preparedness, response, and recovery, these processes guide effective resource mobilization. This study introduces a multi-criteria group decision-making outranking method to determine the most suitable phase for strengthening enablers in the humanitarian supply chain. The approach involves normalizing the decision matrix for comparability, computing the border approximation area by assessing distances from ideal and anti-ideal solutions, and ranking alternatives based on their relative closeness to the ideal. The methodology is validated through two case studies, selecting the optimal phase for improving supply chain enablers and opting the best logistic strategy in supply chain for e-commerce retailer. The phase F1 and logistic strategy J6 are chosen as the best options in the considered case studies as they have highest border approximation area relative to their scenarios. To validate the credibility of the proposed technique, the suggested method is compared with existing methods and the selection of same optimal choice assures the integrity of the proposed method. At the last, limitations and future directions are being discussed to address the pros and cons of the proposed method.

These days, infectious illness mathematical modeling is a major global trend. With the use of current data, mathematical models enable us to predict the occurrence of disease outbreaks in the future. In this work, we use a fractal fractional operator with two fractal and fractional orders to solve a system of fractional differential equations using a Caputo Fabrizio type kernel. A six chambered model with a single source of chlamydia is studied using the concept of fractal fractional derivatives with nonsingular and nonlocal fading memory. The fractal fractional model of the Chlamydia system can be solved by using the characteristics of a non-decreasing and compact mapping. Initially, we calculate the system’s equilibrium points and fundamental reproduction number R0. We then look at the system’s stability at the equilibrium point. Through the application of the Picard Lindelof methodology, we establish the existence of a unique solution for the given fractional CF-system of the hearing loss model and use fixed point theory to examine the stability of the iterative process. By taking into account the therapy as a control technique to lower the number of infected individuals, the system’s optimal control is established. Calculating the estimated solution of the system involves applying the Euler technique for the fractional order Caputo Fabrizio derivative. In two scenarios, R0 < 1 and R0 > 1, we provide a numerical simulation of the disease’s spread with regard to the basic reproduction number and the transmission rate. We compute the results for various fractional order derivatives and compare the findings in order to examine the impact of the fractional order derivative on the behavior and value of each variable in the model. Additionally, we examine the sensitivity of R0 with regard to each model parameter and ascertain the influence of each parameter, taking into account the significance of reproduction number in the persistence of disease transmission. Finally, it can be said that once more, fractional operator mathematical models can help in making better decisions on how to manage financially turbulent situations.

The circular intuitionistic fuzzy set (C-IFS) is a recent extension of the intuitionistic fuzzy set, whose elements are represented as circles instead of specific orthopairs. Entropy measures provide us with a quantitative measure of uncertainty. This research addresses the gap in entropy measures for C-IFSs by introducing innovative construction methods. It establishes robust theoretical frameworks for entropy measures, leveraging established mathematical concepts including t-norms, t-conorms, automorphisms, and aggregation operators. It provides several proven mathematical results and expressions for entropy measures. Moreover, it is worth mentioning that entropy measures can be generated not only using t-norms and t-conorms alone but also when coupled with automorphisms and aggregation operators. Additionally, the argument formulation of t-norms and t-conorms plays a significant role in entropy measure generation, and these formulations are not unique, thereby paving the way for future research avenues. In addition to these contributions, a novel transformation method from entropy to a similarity measure is proposed. This transformation method fully incorporates all aspects of C-IFSs. Lastly, the technique for order preference by similarity to ideal solution (TOPSIS) is extended for C-IFSs to deal with multi-criteria decision-making problems and overcome its previous extensions' limitations. This method consists of a novel criteria weight generation method that is entropy-based and the modified relative closeness index, adding further depth to the approach. Various numerical examples are given to elaborate on our results.