Abstract
Strategic school policy formulation is critical in guaranteeing the balanced growth of education systems that facilitate both academic and extracurricular objectives. Physical education makes a considerable contribution towards this vision through the promotion of lifelong healthy living, physical well-being, and general student development. This research proposes a new decision-making model using Circular Intuitionistic Fuzzy Bonferroni Mean (CIFBM) operators, which combine Aczel-Alsina triangular norms and Bonferroni aggregation rules to handle uncertainty in policy assessment. The essential properties of the proposed operators are investigated and proved. A multi-attribute group decision-making (MAGDM) approach is established to evaluate and rank strategic school policies to meet physical education goals. The methodology focuses on the importance of physical activity in promoting student achievement, cooperation, and educational viability. The case study illustrates the real-world application of the model in maximizing resource utilization, reconciling various interests of stakeholders, and enhancing student success. The findings verify the validity of the developed methodology in optimizing decision-making processes in policy development. This publication provides useful information and guidance for policymakers and educators who want to bring strategic planning into harmony with the development of school-based physical education programs. Abbreviations section is for the symbols, parameters, sets, and indices used throughout the paper.
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Introduction
The goal of the national curriculum for physical education and sports in schools is to create, gather, implement, and assess curricula that are in line with Islamic educational principles and more general ideas about training and education. Its goal is to provide children and teenagers with an active and stimulating environment that promotes the integration of implementation, edification, and education. The curriculum aims to create informed, energetic, and happy students who value their physical and mental wellness, lead healthy lifestyles, and may even pursue careers in professional sports. A comprehensive educational framework that encourages social, mental, and physical well-being must include physical education. Encouraging physical health, improving motor skills, and fostering values like perseverance, discipline, and teamwork, play a critical part in people’s overall development. Physical education promotes social cohesiveness, emotional stability, and cognitive development in addition to the obvious health and activity benefits. It offers an organized opportunity for kids to participate in physical activities, form healthy habits, and foster a lifelong respect for wellness in schools. Sports, leisure activities, and fitness training are just a few of the varied activities that physical education incorporates to improve students’ physical capacities while also preparing them to tackle obstacles, make wise decisions, and lead active, balanced lives as shown Fig. 1.
Empowering through movement a journey in physical education.
MAGDM is a method that can produce rankings for limited choices based on the attributes of several alternatives and opinions by a group of experts. The MAGDM has been connected to various prevalent decision concerns, including the behavior of public contracts and the strategic evaluation of businesses over the past few years1,2,3,4. Many difficult real-world issues have already been successfully solved using theories and concepts connected to MAGDM. MAGDM techniques have drawn a lot of interest nowadays because of their flexibility in handling challenging issues in a variety of fields. For instance, MAGDM techniques have been used to prioritize real estate companies using credit risk assessments within group decision support frameworks5, assess blockchain adoption barriers in medical supply chains with a fairness-aware paradigm6, and improve the sustainability of critical material supply in the electrical vehicle market through AI-powered approaches7. Furthermore, research has investigated the use of smart contracts to reduce the hazards of extreme weather in construction supply chains8,9 and the synergy between urban rail systems and urban form using spatial modeling tools10. The first difficulty in an actual decision-making process is expressing the attribute value of ambiguous choice information clearly and precisely. Calculated data do not accurately express fuzzy details due to the complex nature of human thinking and the decision-making environment. Thus, various techniques for ambiguous information expression developed.
Over the past decade, a variety of neural network- and wavelet-based computational models have come to the fore to tackle sophisticated nonlinear systems, most of which are applicable to improve decision-making models in situations of uncertainty. These consist of trustworthy neural network methods have been successfully implemented in pantograph differential models11, language learning systems12, and hepatitis B virus dynamics13. Morlet and Meyer wavelet-based neural computation processes have been utilized for delay differential models, transport networks, and predator–prey interactions14,15,16,17. Further works have considered functional nonlinear singular differential equations through Meyer wavelets18, in addition to heuristic designs for modeling biological and medical systems like the modeling of smoking behavior and SITR COVID-19 models19,20,21,22. Higher-order neuro-swarm hybrid methods, fractional-order and time-delay-based Lane–Emden equations, and Emden–Fowler differential models have also been investigated to enhance the accuracy and stability of solutions in intricate systems23,24,25,26,27. Additionally, stochastic mathematical models have been constructed for epidemic management, such as coronavirus spread models28. These earlier contributions form necessary computational bases and prompt the incorporation of intelligent algorithms like D-IVIF and deep learning into sensor-based decision-making platforms.Zadeh introduced fuzzy sets (FSs)29, which Atanassov improved further. FSs are extensively helpful in several fields. On the other hand, FS is a set where each element is represented by a real number in \(\left(\text{0,1}\right)\). More studies on FS can be seen30,31,32,33,34. Atanassov35,36 presented the IFSs to compensate for the limitation whose fundamental elements are IF values37. IFSs present the data in 2-tuples based on both membership degree (MD) and non-membership degree (NMD)38,39. IFS is a beneficial technique, so many experts and researchers have studied it and used it to represent fuzzy decision information. One of the commonly recognized theories for treating MAGDM is FS because decisions are often made based on abstract facts, vulnerability, and ambiguity. Numerous direct and indirect extensions of the FS have been developed and successfully implemented in the majority of the current reality situation’s issues. FS has been extended with fixed point theory to solve fuzzy ordinary equations which is proposed by Sarwar and Li34. FS has also extended neural networks40, switching systems with deception attacks31, and sampled-data stabilization32, other studies on FS have been developed and can be shown here41,42,43. Using two words, MD and NMD, Atanassov35,44 introduced intuitionistic fuzzy sets (IFSs), which are a generalization of FS to understand ambiguity or imprecise information. The duplets’ sum must fall between \(0\) and \(1\)45,46,47 talk about some current research on IFS theory and applications. The requirements for handling conclusions including several response forms, such as yes, no, abstention, and rejection, cannot be met by FSs and IFSs. IFS were chosen over probability-based preferences due to their superior ability to manage uncertainty and hesitation characteristics in complex decision-making processes. The combined representation of MD, NMD, and HD is possible with IFS, in contrast to probability-based approaches that require exact numerical probabilities. Because it better conveys the uncertainty and vagueness that are inherent, this flexibility is especially helpful in situations where experts struggle to clear their opinions. An accurate and nuanced depiction of expert perspectives is ensured by this study’s use of IFS, which is essential for producing trustworthy and significant results Some work on IFS can be seen48,49,50. Given the growing popularity of fuzzy logic in decision-making, this work develops the idea of the circular intuitionistic fuzzy set (C-IFS) and comprehensively proposes new features to enhance its use in MAGDM methods. To address the specific scenarios where DM involves circular characteristics, Atanassov11 extended the IFS and introduced the concept of C-IFS, allowing a more accurate representation of circular phenomena. C-IFS was represented by a circle having radius r, where \(u\) and \(v\) degrees are centered. Like IFS, C-IFS is also involved in MCDM problems across the field. Cakır51 defined the DM application under the C-IFS environment, Irem et al.52 proposed the C-IFS by using AHP and VIKOR methodology for supplier evaluation problem, Alkan et al.53 defined the C-IFS TOPSIS method and its application for pandemic hospital location selection, Otay et al.54 defined the interval-valued C-IFS and its application in digital transformation, Cakır et al.55 proposed the C-IFS and its application in Covid-19 medical waste landfill site evaluation, Garg et al.56 extended EDAS Method with C-IFS and its application in MCDM. It explains new angles on radius computation and the incorporation of interior points into the mindset of defuzzification decision-makers. To make using C-IFS numbers in decision-making situations simple, MAGDM processes are suggested and demonstrated in the example. By introducing a new C-IFS MAGDM process and a C-IFS defuzzification function, this paper is a pioneer in the C-IFS literature.
For the aggregation of information, several types of aggregation operators (AOs) exist, but very few or none of them interrelate the input information. They play a significant role in processing various types of information used in decision-making, pattern identification, information retrieval, medical diagnosis, data mining, machine learning, etc. Over the past few decades, scholars and researchers have been paying more attention to studying AO37,38,39. Power AO61 is a widely used tool that considers the relationship of the information using some power weights. Another AO is the density AO proposed by Liu and Yu62, which uses the attributes’ density weights to find the input arguments’ interrelationships. Bonferroni63 introduced the notion of BMO, a mean-type aggregation operator used to consider the relationship of the aggregated information effectively. BM’s most essential characteristic is that it captures the relationships between the combined arguments. Yager (2009)64 initially improved and expanded the modeling capabilities of BM by substituting various mean-type operators for simple averaging. The BMO is a generalized form of the Heronian mean operator (HMO) proposed by Sykora65. Because of the effective behavior of BMO, this concept has been utilized in many fuzzy frameworks. The idea of BMOs in the setting of IFSs was comprehensively studied by Xu and Yager66, and duplets of information were used to express the BMOs. More work on BM is discussed here67,68,69. Each operator is designed to handle distinct kinds of data and decision situations. For this, expertise defined the AO for solving the MCDM problem such that Zhao et al.70 defined the generalized AO for IFS, Xu et al.71 suggested geometric AO, Wang et al.72proposed the AO based on Einstein operation, Yu et al73. defined the prioritized AO, Wang et al.74 defined the arithmetic AO for IFS, Rani et al.75 discussed the power AO, and Xia et al.76 defined AO based on Archimedean T-norm and T-conorm. HM AOs Hara et al.77 are essential for making decisions in complex settings. The fundamental HM operator provides a straightforward aggregation method where each value is equally weighted, which computes the arithmetic mean of input values. Hussain et al.78 stated novel approaches of Aczel Alsina aggregation operators for handling ambiguous information of expert judgments. Further studies on Aczel Alsina can be seen here79,80,81,82,83.
Further, to improve the concept of BMOs in the setting of IFSs was comprehensively studied by Xu and Yager66 proposed a few generalizations to enhance their analytical ability. Several extensions were applied to the BMO. The concept of the geometric BM (GBM) operator was investigated by Xia et al.84 and Li et al.85. The hesitant fuzzy BM (HFBM) operator was created by Zhu and Xu86. The HF geometric BM (HFGBM) operator was introduced by Zhu et al.87. The IFG power BM (IFGPBM) operator was developed by He et al.88 by combining the GBM operator and the PM operator. The uncertain linguistic BM operators were presented by Wei et al.89. However, the BM operator has been applied to a variety of contexts. Furthermore, it’s essential to create such general and flexible AOs based on A-A TN and TCN for C-IFVs because of the difficulty of decision-making. C-IFVs can also express ambiguous information over a wider range. We obtain the following conclusions from the analysis given above:
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1.
C-IFVs significantly better describe fuzzy information, and they can solve MAGDM challenges because their expressions are more flexible and comprehensive.
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2.
The A-A TN and TCN a vital tools for establishing flexible operational laws for C-IFVs. AOs based on the A-A TN and TCN are also more adaptable and versatile for aggregating fuzzy data.
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The BM operator is the most widely used and has been proven to be a flexible approach, which plays a significant role in conveying the magnitude level of options and characteristics.
To overcome MAGDM challenges, it’s crucial and advantageous to expand the BM operator to C-IFVs based on the A-A TN and TCN in light of the detailed analysis that was previously described. Therefore, we introduce some new C-IFAABM operators by combining the BM operator with C-IFVs using A-A TN and create a new MAGDM technique within the C-IF environment to handle more difficult MAGDM issues in this study. This contradicts the usual adoption of idempotent and additive aggregation models to enable the model to handle interrelationships among various criteria when the decision elements are not separately examined. This improvement is especially crucial in situations where there is complex, uncertain, and correlated information, so working a fundamental methodological deficiency in the existing literature. This research responds to the existing demand for more refined and adaptable decision-making models in complex settings like education policy and organizational planning. As uncertainty increases and multi-criteria complexities become more prevalent, the combination of C-IFB-based approaches presents a new instrument to the scientific community. The model presented not only offers more precise aggregation of expert views but also enables strong policy development. Therefore, this study completes a gap in current literature and adds a useful, flexible methodology that can be applied to other areas of multi-criteria group decision-making.
The rest of our work is divided into different sections. We primarily introduce the basic ideas of C-IFSs, A-A TN, TCN, and BM operators in Sect. Preliminaries. We introduce the C-IFAABM operator in Sect. Circular Intuitionistic fuzzy Aczel-Alsina Bonferroni mean operator. Using the proposed C-IFAABM operator, we provide a new MAGDM technique in Sect. MAGDM Methods by using Investigated Operators Based on C-IFSs and step-by-step instructions. In Sect. Strategic School Policy Formulation for Physical Education Goals , we illustrate the effectiveness of the proposed MAGDM method and compare it with the recent techniques using a real-world application. The conclusions of this paper are presented in Sect.Sensitivity analysis.
The role of circular intuitionistic fuzzy aczel-alsina operators in achieving physical education goals
The CIFAA operators play a vital role in achieving the overall goals of PE, such as social development, mental well-being, and physical condition. The underlying personality, vagueness, and ambiguity in PE decision-making are resolved through these advanced mathematical techniques. CIFAA operators provide a robust structure for evaluating an array of often conflicting criteria by applying the Aczel-Alsina aggregation approach and including a circular membership function. To enable more comprehensive and precise evaluations, they permit decision-makers to capture varying degrees of acceptance, rejection, and reluctance in stakeholder appraisals. Moreover, by integrating appraisals of significant factors such as curricular effectiveness, diversity, and resource utilization, CIFAA operators facilitate multi-criteria decision-making. This allows schools to prioritize policies and approaches that align with their holistic learning objectives. CIFAA operators facilitate the development of more inclusive and effective physical education programs by promoting equity and responding to a range of needs, for example, those of disadvantaged children or children with disabilities. In order to form evidence-based policies that encourage physical, mental, and social wellbeing among students as well as produce a healthier, more active forthcoming generation, both educators and school administrators are vested with the capability to analyze convoluted, uncertain data.
Justification and advantages of the proposed theory
Although numerous fuzzy aggregation operators have been used to design educational policy, the combination of Circular Intuitionistic Fuzzy Sets with Bonferroni Mean and Aczel-Alsina operations presents unique strengths not considered in previous models. Conventional aggregation measures do not do justice to the intricate interdependencies and ambivalence that exist in human judgments, especially when assessing multi-dimensional targets such as physical education. The envisioned Circular Intuitionistic Fuzzy Bonferroni model provides greater modeling flexibility through the inclusion of both interactional influence (e.g., through Bonferroni norms) and nonlinear combination behavior (through Aczel-Alsina norms), enabling richer aggregation in conditions of uncertainty. In contrast to current frameworks, this model addresses greater levels of hesitancy and contradiction, providing more solid and context-specific decision support. This theoretical reinterpretation is thus critical to filling current gaps in appropriately modeling expert preferences in strategic school policy development, particularly in the context of diverse stakeholder feedback and uncertain evaluation conditions.
The role of cifaa operators in physical education
Managing uncertainty and ambiguity
PE objectives, like fostering fitness, mental well-being, and teamwork, tend to involve subjective judgments and uncertain results. CIFAA operators are particularly skilled at managing uncertainty by successfully combining intuitionistic fuzzy information so that decision-makers can make inferences that capture the richness of real-world situations.
Integrating diverse stakeholder views
Physical education programs and policies entail contributions from various stakeholders, such as students, teachers, parents, and administrators. CIFAA operators make room for gathering different opinions by recording different levels of agreement, disagreement, and uncertainty, leading to more inclusive and balanced decision-making.
Multi-criteria decision-making
PE goal achievement involves assessing several criteria, including resource allocation, inclusiveness, and curriculum effectiveness. CIFAA operators make MCDM possible by aggregating assessments across these criteria, allowing for the prioritization of strategies that are aligned with holistic educational goals.
Policy and strategy development
Utilizing CIFAA operators, schools can create data-driven and stakeholder-focused policies and strategies. The operators offer an organized approach to assessing the possible implications of different initiatives, for example, adding new sport programs or upgrading facilities, in alignment with long-term objectives.
Facilitating Equity and Inclusion CIFAA operators are particularly good at handling issues of equity and inclusion in PE. Through the identification of subtle differences in stakeholder preferences, these operators facilitate the design of programs that address a range of needs, including students with disabilities or those from disadvantaged backgrounds.
Problem statement and motivation
Despite the increasing recognition of the value of PE to holistic student development, numerous education institutions continue to deal with the formation of effective, adaptive, and inclusive PE strategies. The problem lies in the establishment of school policies that are sensitive to diverse students’ needs, limited resources, and changing education objectives. Conventional decision-making tools tend not to reflect the ambiguity, subjectivity, and uncertainty associated with assessing several criteria like infrastructure, teacher quality, program inclusiveness, and impact on health. Additionally, if there are several stakeholders involved in policy development like administrators, teachers, parents, and policymakers their preferences and judgments have to be combined in a complex manner.
This study fills this void by setting forth a strong and smart decision-support system using CIFS and the BM operator, specifically for school policy-making in sports science education. CIFS is capable of capturing both the strengths and weaknesses of expert views with greater hesitancy flexibility and uncertainty handling than in traditional fuzzy systems. The Bonferroni-based operator enhances the precision of information aggregation by taking into account the interactions between criteria, a requirement in multi-dimensional learning settings. This research is inspired by the critical demand for interdisciplinary, data-aware instruments that will enhance the formulation and assessment of physical education policy in a strategic sense, particularly where decisions have to balance scholastic aims with long-term physical and social development targets.
Research questions
To advise the scope and direction of the current research effort, the subsequent research questions are developed:
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How can expert judgment uncertainty, vagueness, and indeterminacy be modeled efficaciously within assessments of physical education plans?
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How are limitations present in traditional decision-making approaches towards developing physical education school policy and how may they be overcome using CIFS?
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How does Bonferroni-based aggregation make educational policy decision-making more reliable and sensitive in various ways?
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Does the integration of CIFSs with BM operators promise an improved framework for selecting optimal physical education methods that are both interpretable and precise?
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How could this new model be applied practically in school environments to enable smart and strategic decision-making of policy for physical education programs?
Preliminaries
This section provides the basic concepts of IFSs and also describes their properties, where \(X\) defines the non-empty set and \({u}_{A}\left(x\right),{v}_{A}\left(x\right)\) be the MD and NMD in \(x\). To better understand this paper, A-A TN TCN, and BM operators are also defined in this section. Atanassov35 introduced the idea of C-IFS, giving the following definition.
Introduction to physical education
Physical education (PE), which aims to promote social, mental, and physical development, has long been recognized as a fundamental component of comprehensive education. It includes aspects of social connection, emotional stability, and mental health in addition to physical fitness. Scholars have highlighted how it helps students develop healthier behaviors, collaborate with others, and live better lives overall90. Additionally, physical education provides an opportunity to teach characteristics that are critical to social and personal growth, such as discipline, teamwork, and sportsmanship as shown in Fig. 2.
Basketball and cycling, emphasize dynamic movement and active lifestyles.
The concept of physical education has undergone remarkable changes over the years. Earlier approaches gave considerable importance to developing power and endurance and were largely focused on readiness for service and physical fitness. In recent decades, activities contributing to mental well-being and encouragement of lifetime engagement in physical activity have been included in the definition of physical education by Kirk in 201091. To reflect a more student-centered and inclusive approach, the contemporary curriculum aims to balance traditional sports, recreation, fitness training, and health education. Several studies identify the positive correlation between physical activity and mental health. By boosting blood flow to the brain and reducing stress, physical education has been shown to improve memory, concentration, and school performance Tomporowski et al., in 200892. PE also enables children to become emotionally resilient by providing them with a means of managing stress and anxiety, which overall improves their mental well-being by Eime et al., in 201393. Physical education is another useful means of promoting equality and social inclusion. PE assists children from every background in establishing a sense of respect and belonging by employing team-based exercises and cooperative games. Its ability to reduce social barriers and increase participation by the marginalized, like those with disabilities or from economically disadvantaged communities, has been underscored by researchers by Block et al., in 2013. As much as it has its benefits, the implementation of physical education has challenges. Physical education program success is often hampered by erratic policies, insufficient resources, and inadequate qualified. Apprehension regarding the diminishing amount of time dedicated to physical education on school timetables is also on the rise, largely due to the enhanced emphasis on scholastic subjects. Researchers argue that these challenges constrain physical education to realize its purposes as a vehicle for education and health.
The purpose of the study
This study aims to investigate the use of CIFAA operators as a strategic framework for improving physical education (PE) decision-making. To promote students’ physical fitness, mental health, and social well-being, physical education is essential. However, creating effective policies and procedures frequently requires navigating difficult, ambiguous, and subjective criteria. By offering CIFAA operators a reliable technique for assessing and ranking policies that support PE’s overarching objectives, this study aims to alleviate these issues.
The CIFAA operators capture and process different stakeholder perspectives, including differing levels of agreement, disagreement, and hesitation, by integrating circular intuitionistic fuzzy logic. This makes it possible to comprehend stakeholder priorities more deeply and makes it easier to create inclusive, data-driven plans. To ensure unity with more general educational goals, the study intends to show how CIFAA operators can assist schools in addressing important facets of physical education, including resource allocation, curriculum design, inclusion, and health promotion. The study also aims to shed information on how CIFAA operators help PE bridge the gap between strategic objectives and operational measures. The goal of this research is to provide educators, administrators, and policymakers with a methodical approach to multi-criteria decision-making so they may create policies that support inclusion, equity, and the long-term development of students. By demonstrating how creative decision-making techniques can lead to significant advancements in program design and execution, this study ultimately aims to further the field of physical education as shown in Fig. 3.
Representing the diversity of youth sports and recreation.
Intuitionistic fuzzy sets
A more flexible approach to simulate ambiguity and uncertainty in decision-making processes is offered by IFS, an enhanced version of FS. By including other components reflecting different MD, NMD, and HD degrees, they generalize traditional FSs. IFSs are distinguished by the fact that the sum of the duplets is less than one. We review the concept of IFVs and their operational laws in this paper.
Definition 1:
Ref35. The set \(A=\left\{\langle x,{u}_{A}\left(x\right),{v}_{A}\left(x\right)\rangle ,x\in X\right\}\) is an IFS that satisfies the following condition:\(0\le {u}_{A}\left(x\right)+{v}_{A}\left(x\right)\le 1\), where \(\langle {u}_{A}\left(x\right),{v}_{A}\left(x\right)\rangle\) be the Circular intuitionistic fuzzy values (C-IFVs). Moreover, Hesitancy degree of C-IFS is termed as \(\beta_{A} \left( x \right) = 1 - u_{A} \left( x \right) - v_{A} \left( x \right)\).
Definition 2:
Ref35. For any two C-IFVs \({\mathfrak{a}}_{i}=\left({u}_{{\mathfrak{a}}_{i}},{v}_{{\mathfrak{a}}_{i}}\right), i=\left(1, 2\right)\) the basic operations are as follows:
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\({\mathfrak{a}}_{1}\vee {\mathfrak{a}}_{2}= \left(max \left\{{u}_{{\mathfrak{a}}_{1}}, {u}_{{\mathfrak{a}}_{2}}\right\}, min \left\{ {v}_{{\mathfrak{a}}_{1}}, {v}_{{\mathfrak{a}}_{2}}\right\}\right)\)
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\({\mathfrak{a}}_{1}\bigwedge {\mathfrak{a}}_{2}= \left(min \left\{{u}_{{\mathfrak{a}}_{1}}, {u}_{{\mathfrak{a}}_{2}}\right\}, max \left\{ {v}_{{\mathfrak{a}}_{1}}, {v}_{{\mathfrak{a}}_{2}}\right\}\right)\)
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\({\mathfrak{a}}_{1}\oplus {\mathfrak{a}}_{2}= \left({u}_{{\mathfrak{a}}_{1}}+{u}_{{\mathfrak{a}}_{2}}- {u}_{{\mathfrak{a}}_{1}}{u}_{{\mathfrak{a}}_{2}}, {v}_{{\mathfrak{a}}_{1}}{v}_{{\mathfrak{a}}_{2}}\right)\)
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\({\mathfrak{a}}_{1}\otimes {\mathfrak{a}}_{2}= \left({u}_{{\mathfrak{a}}_{1}}{u}_{{\mathfrak{a}}_{2}}, {v}_{{\mathfrak{a}}_{1}}+{v}_{{\mathfrak{a}}_{2}}- {v}_{{\mathfrak{a}}_{1}}{v}_{{\mathfrak{a}}_{2}}\right)\)
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\(\eta {\mathfrak{a}}_{1}=(1- {\left(1-{u}_{{\mathfrak{a}}_{1}}\right)}^{\eta }, {v}_{{\mathfrak{a}}_{1}}^{\eta }),\eta> 0\)
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\({{\mathfrak{a}}_{1}}^{\eta } = \left({u}_{\mathfrak{a}}^{\eta }, 1- {\left(1-{v}_{{\mathfrak{a}}_{1}}\right)}^{\eta }\right)\)
Definition 3:
Ref35. The set \(A=\left\{\langle x,{u}_{A}\left(x\right),{v}_{A}\left(x\right);{\mathbb{c}}r \rangle ,x\in X\right\}\) is an C-IFS that satisfies the following condition:\(0\le {u}_{A}\left(x\right)+{v}_{A}\left(x\right)\le 1\), where \(\langle {u}_{A}\left(x\right),{v}_{A}\left(x\right)\rangle\) be the Circular intuitionistic fuzzy values (C-IFVs). Moreover, the Hesitancy degree of IFS is termed as \({\beta }_{A}\left(x\right)=1-{u}_{A}\left(x\right)-{v}_{A}\left(x\right)\). Let, \({\mathbb{c}}_{r}\in \left[\text{0,1}\right]\) be the radius of the circle around each \(x\in X\). Unlike IFS, in C-IFS, each element is displayed by a circle having the center \(\left({u}_{A}\left(x\right),{v}_{A}\left(x\right)\right)\) with radius r which means that C-IFS cannot be expressed in the framework of standard IFS. Its geometrical representation is displayed in Fig. 4.
Geometrical representation of C-IFS.
Definition 4:
Ref35. Let \(A=\langle x,{u}_{A}\left(x\right),{v}_{A}\left(x\right); {\mathbb{c}}r\rangle\) be an C-IFV. The score degree (\(Sc\)) of the C-IFV can be defined as: Suppose that \(\zeta =\left(u,v,r\right)\) be CIFVs.
And \(Sc(A)\in [-1, 1]\). Bonferroni presented the initial proposal for BM, which has a function that can represent the relationships between various integrated values. It may be described as follows.
Definition 5:
Ref94. Consider \({\mathfrak{a}}_{i}=\left\{\left({x}_{i},\left({u}_{A}\left(x\right),{v}_{A}\left({x}_{i}\right)\right);{\mathbb{c}}r\right);x\in A \right\};\left(i=\text{1,2}\right)\) be two C-IFS. Then, some basic operational laws are defined in Eqs. 3–10.
Definition 6:
Ref63. Suppose \({\mathfrak{a}}_{i}=\left(i=1, 2, \dots ,\mathsf{J}\right)\) be a collection of non-negative real numbers, and \(s, t \ge 0\), then the aggregation functions.
Is called the BM operator.
Definition 7:
Ref66. Suppose \({\mathfrak{a}}_{i}=\left({u}_{{\mathfrak{a}}_{i}}, {v}_{{\mathfrak{a}}_{i}}\right) ,\left(i=\text{1,2},\dots ,j\right)\) be a collection of C-IFVs, and \(s, t \ge 0\), then the aggregation function.
Is called the Circular intuitionistic fuzzy Bonferroni mean operator.
Aczel-alsina operations of C-IFVS
The theory of TN and TCN was developed by Aczel and Alsina95 and defined as follows.
Definition 8:
Ref96. For \(\forall \mathop \eta \limits^{,} \in \left[ {0,\infty } \right]\), the AA TN and TCN are defined respectively as:
And
Definition 9:
Ref95. Let \(\mathfrak{a}=\left({u}_{\mathfrak{a}}, {v}_{\mathfrak{a}}\right)\), \({\mathfrak{a}}_{1}=\left({u}_{{\mathfrak{a}}_{1}},{v}_{{\mathfrak{a}}_{1}}\right),\) and \({\mathfrak{a}}_{2}=({u}_{{\mathfrak{a}}_{2}},{v}_{{\mathfrak{a}}_{2}})\) be any three C-IFVs, \(\mathfrak{N}\ge 1\) and \(\uprho>0.\) Then, the Aczel-Alsina operations of C-IFVs are defined as:
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\({\mathfrak{a}}_{1}\oplus {\mathfrak{a}}_{2}=\left(\begin{array}{c}1-{e}^{-{\left({\left(-\mathit{ln}\left(1-{\gamma }_{{\mathfrak{a}}_{1}}\right)\right)}^{\gimel }+{\left(-\mathit{ln}\left(1-{\gamma }_{{\mathfrak{a}}_{2}}\right)\right)}^{\gimel }\right)}^{\frac{1}{\gimel }}},\\ {e}^{-{\left({\left(-\mathit{ln}{\delta }_{{\mathfrak{a}}_{1}}\right)}^{\gimel }+{\left(-\mathit{ln}{\delta }_{{\mathfrak{a}}_{2}}\right)}^{\gimel }\right)}^{\frac{1}{\gimel }}}\end{array}\right)\)
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\({\mathfrak{a}}_{1}\otimes{ \mathfrak{a}}_{2}=\left(\begin{array}{c}{e}^{-{\left({\left(-\mathit{ln}{\gamma }_{{\mathfrak{a}}_{1}}\right)}^{\gimel }+{\left(-\mathit{ln}{\gamma }_{{\mathfrak{a}}_{2}}\right)}^{\gimel }\right)}^{\frac{1}{\gimel }}},\\ 1-{e}^{-{\left({\left(-\mathit{ln}\left(1-{\delta }_{{\mathfrak{a}}_{1}}\right)\right)}^{\gimel }+{\left(-\mathit{ln}\left(1-{\delta }_{{\mathfrak{a}}_{2}}\right)\right)}^{\gimel }\right)}^{\frac{1}{\gimel }}}\end{array}\right)\)
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\(\eta \mathfrak{a}=\left(1-{e}^{-{\left(\eta {\left(-\mathit{ln}\left(1-{\gamma }_{\mathfrak{a}}\right)\right)}^{\gimel }\right)}^{\frac{1}{\gimel }}},{e}^{-{\left(\eta {\left(-\mathit{ln}{\delta }_{\mathfrak{a}}\right)}^{\gimel }\right)}^{\frac{1}{\gimel }}}\right)\)
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\({\mathfrak{a}}^{\eta }=\left({e}^{-{\left(\eta {\left(-\mathit{ln}{\gamma }_{\mathfrak{a}}\right)}^{\gimel }\right)}^{\frac{1}{\gimel }}},1-{e}^{-{\left(\eta {\left(-\mathit{ln}\left(1-{\delta }_{\mathfrak{a}}\right)\right)}^{\gimel }\right)}^{\frac{1}{\gimel }}}\right)\)
Definition 3.1:
Ref97. A q-ROFS \(\Lambda\) is denoted by \(\Lambda = \left\{x, {M}_{\Lambda }\left(x\right), {N}_{\Lambda }\left(x\right):x \in X\right\}\) where \({M}_{\Lambda },\) and \({N}_{\Lambda }\) express the DM and DNM with restriction \(0\le {M}_{\Lambda }^{\mathcal{Q}}\left(x\right)+{N}_{\Lambda }^{\mathcal{Q}}\left(x\right)\le 1, \mathcal{Q}\in {\mathbb{Z}}^{+}\). The symbol \({\pi }_{\Lambda }\left(x\right)={\left(1-\left({M}_{\Lambda }^{\mathcal{Q}}\left(x\right)+ {N}_{\Lambda }^{\mathcal{Q}}\left(x\right)\right)\right)}^{\frac{1}{\mathcal{Q}}}\) is called DR. For simplicity, we call the duplet \({\overline{ \top }} = {\kern 1pt} (M_{\Lambda } ,N_{\Lambda } )\) q-ROF value (q-ROFV).
Definition 3.2:
Ref97. For any two q-ROFVs \({\overline{ \top }}_{1} = (M_{1} ,M_{2} ,M_{3} )\), \({\overline{ \top }}_{2} = (N_{1} ,N_{2} ,N_{3} )\), then.
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\({\overline{ \top }}_{1} \vee {\overline{ \top }}_{2} \; = \;(\max \{ M_{1} ,M_{2} \} ,\max \{ N_{1} ,N_{2} \} )\)
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\({\overline{ \top }}_{1} \vee {\overline{ \top }}_{2} \; = \;(\min \{ M_{1} ,M_{2} \} ,\max \{ N_{1} ,N_{2} \} )\)
-
\({\overline{ \top }}_{1} \oplus {\overline{ \top }}_{2} \; = \;(M_{1}^{{\mathcal{Q}}} + M_{2}^{{\mathcal{Q}}} - M_{1}^{{\mathcal{Q}}} M_{2}^{{\mathcal{Q}}} )^{{\frac{1}{{\mathcal{Q}}}}} ,N_{1} N_{2}\)
-
\({\overline{ \top }}_{1} \otimes {\overline{ \top }}_{2} \; = \;M_{1} M_{2} ,(N_{1}^{{\mathcal{Q}}} + N_{2}^{{\mathcal{Q}}} - N_{1}^{{\mathcal{Q}}} N_{2}^{{\mathcal{Q}}} )^{{\frac{1}{{\mathcal{Q}}}}}\)
-
\(\lambda {\overline{ \top }}_{1} \; = \;\left( {(1 - (1 - M_{1}^{{\mathcal{Q}}} )^{\lambda } )^{{\frac{1}{{\mathcal{Q}}}}} ,N_{1}^{\lambda } } \right),\lambda> 0\)
-
\({\overline{ \top }}_{1}^{\lambda } \; = \;\left( {M_{1}^{\lambda } ,(1 - (1 - N_{1}^{{\mathcal{Q}}} )^{\lambda } )^{{\frac{1}{{\mathcal{Q}}}}} } \right)\)
Definition 3.3:
Ref97. For any q-ROFV \({\overline{ \top }} = (M_{\delta } ,N_{\delta } )\), a score function is defined by:
where \(S\left( {{\overline{ \top }}} \right) \in \left[ { - 1, 1} \right]\).
Definition 3.4:
Ref97. For any q-ROFV \({\overline{ \top }} = (M_{\delta } ,N_{\delta } )\), an accuracy function is defined by:
Where \(H\left( {{\overline{ \top }}} \right) \in \left[ {0, 1} \right]\)
Definition 3.5:
Ref35. Let \({\overline{ \top }}_{{\underline {k} }}\) be some positive numbers with \(\varsigma ,\Gamma \ge 0\), then the BM operator is demonstrated by:
Definition 3.6:
Ref35. Let \({\overline{ \top }}_{{\underline {k} }}\) be some positive numbers with \(\varsigma ,\Gamma \ge 0\), then the generalized BM operator is demonstrated by:
Definition 3.7:
Ref98. Consider any non-empty set \(X\). The Definition of a q-ROFS \({\mathcal{Q}}_{r}\) is thus:
Let, \(r\in \left[\text{0,1}\right]\) be the radius of the circle around each \(x\in X\) which is denoted as \({r}_{i}=\left(\underset{1\le j\le k}{\text{max}}\left(\sqrt{{\left(M\left({c}_{i}\right)-{m}_{i,j}\right)}^{2}+{\left(N\left({c}_{i}\right)-{d}_{i,j}\right)}^{2}}\right)\right)\) and \(\text{let} {M}_{\Lambda }\left(x\right), {N}_{\Lambda }\left(x\right)\in \left[\text{0,1}\right]\). Here \({M}_{\Lambda }: X\to \left[\text{0,1}\right]\), and \({N}_{\Lambda }: X\to \left[\text{0,1}\right]\) represents the MD, NMD of \(x\in X,\) respectively, provided that \(0\le {M}_{\Lambda }^{\mathcal{Q}}\left(x\right)+{N}_{\Lambda }^{\mathcal{Q}}(x)\le 1\). Additionally, the term \({H}_{\Lambda }\left(x\right)=\left(\sqrt[\mathcal{Q}]{1-\left({M}_{\Lambda }^{\mathcal{Q}}\left(x\right)+{N}_{\Lambda }^{\mathcal{Q}}\left(x\right)\right)}\right),\) \({H}_{\Lambda }\left(x\right)\in \left[\text{0,1}\right]\) is defined as a hesitancy degree and \(\left({M}_{\Lambda }\left(x\right), {N}_{\Lambda }\left(x\right)\right)\) with radius r is referred to as q-ROFV.
Definition 3.8:
Ref98. Consider \({\overline{ \top }} = \left( {x,(M_{{{\overline{ \top }}}} (x),N_{{{\overline{ \top }}}} (x));r(x)} \right)\) be any Cq-ROFV; the score function is defined as follows:
Definition 3.9:
Ref98. Consider \(\Lambda = \left( {x, \left( {M_{{{\overline{ \top }}}} \left( x \right), N_{{{\overline{ \top }}}} \left( x \right)} \right);r\left( x \right)} \right)\) be any Cq-ROFV; the accuracy function is defined as follows:
Definition 3.10:
Consider \({\overline{ \top }}_{r} = \left( {x, \left( {M_{{{\overline{ \top }}}} \left( x \right), N_{{{\overline{ \top }}}} \left( x \right)} \right);r\left( x \right)} \right)\), \({\overline{ \top }}_{{r_{1} }} = \left( {x, \left( {M_{{{\overline{ \top }}_{1} }} \left( x \right), N_{{{\overline{ \top }}_{1} }} \left( x \right)} \right);r\left( x \right)} \right)\) and \({\overline{ \top }}_{{r_{2} }} = \left( {x, \left( {M_{{{\overline{ \top }}_{2} }} \left( x \right), N_{{{\overline{ \top }}_{2} }} \left( x \right)} \right);r\left( x \right)} \right)\) be any three Cq-ROFVs where \(\mathcal{Q}\in {\mathbb{Z}}^{+}\) be any real number. The following operations can be defined: For q-ROFSs, the operations can be defined with a focus on the radius, indicating the ambiguity or uncertainty level. A smaller radius signifies less ambiguity, while a larger radius indicates more ambiguity. The operations described here take into account the smallest and largest radii separately, thereby providing results with the smallest and supreme levels of ambiguity, correspondingly.
Circular intuitionistic fuzzy aczel-alsina bonferroni mean operator
BM operator is one of the massive flexible and advanced operators to aggregate the uncertain and fuzzy information by encountering the interrelationship of the input information, unlike the traditional aggregating tools. Due to this fact, in this study, the IFS is a mathematical framework that extended the classical FS by allowing the degree of \(u\) and \(v\) simultaneously. This extension will enable it to deal with vagueness and uncertainty in real-world problems but cannot deal with issues involving circular characteristics. So, the concept of C-IFS has been introduced. As in MAGDM, AO plays a significant role in DM. In this section, we’ll introduce the C-IFS for HM, GHM, WHM, and WGHM and their properties by considering this. By using Aczel-Alsina operations, we defined the notion of IFAAB averaging aggregation operators and some relevant concepts throughout this section. The essential properties of aggregation are proved for the said C-IFAABM. The essential properties of aggregation are proved for the said C-IFAABM.
Theorem 1:
Let \({\mathfrak{a}}_{i}=\left\{\left({x}_{i},\left({u}_{A}\left(x\right),{v}_{A}\left({x}_{i}\right)\right);{\mathbb{c}}r\right);x\in A \right\} ,\left(i=\text{1,2},\dots ,j\right)\) be a collection of C-IFVs and \(s\), \(t>0\). Then the accumulated value by using C-IFAABM is also a C-IFV.
Proof:-
By using operations from section \(2\), we have.
Remark: The idempotency property for Theorem 1 does not hold.
Theoretical justification
Let us remember that an aggregation operator is idempotent if: For all inputs \({\mathfrak{a}}_{i}=\mathfrak{a} ,\) the aggregation result is also \(\mathfrak{a}\).
Mathematically, for an operator
Now take Theorem 1 and the C-IFAAB operator. Even if all of the inputs are the same (i.e.,\({\mathfrak{a}}_{i}=\mathfrak{a}\) for all i), the C-IFAAB operator structure entails exponential-logarithmic transformations and nonlinear combinations between pairs of elements based on the Bonferroni mean (which itself is not idempotent unless all parameters are trivial). As a result of these transformations, the aggregated value is different from the initial value \(\mathfrak{a}\), contravening the idempotency property.
Numerical analysis
Let us try it out with a simple example. Suppose we are presented with a C-IFV:
Now, use the operator with \(s=t=1\), \(\gimel =1\) and \(\nu =2\) simply two equal inputs: \({\mathfrak{a}}_{1}={\mathfrak{a}}_{2}=\mathfrak{a}\).We calculate one part of the expression (using the form from operator, just the membership part for simplicity):
Now, if this is combined in the overall nested expression, it will give a value other than the initial membership degree \(0.7\), although the inputs were both the same. The result of aggregation is not the same as the initial C-IFV input. Therefore, idempotency doesn’t hold.
Theorem 2:
Monotonicity: Let \({\mathfrak{a}}_{i}=\left({u}_{{\mathfrak{a}}_{i}},{\nu }_{{\mathfrak{a}}_{i}};{\mathbb{c}}r\right)\) and \({b}_{i}=\left({u}_{{b}_{i}},{\nu }_{{b}_{i}};{\mathbb{c}}r\right)\) be two collections of C-IFVs where \(\left(i=\text{1,2},\dots ,j\right)\). If \({\mathfrak{a}}_{{u}_{i}}\le {\mathfrak{a}}_{{\beta }_{i}}\) and \({\nu }_{{u}_{i}}\ge {\nu }_{{\beta }_{i}}\), for all \(i\), then.
Proof: since \({u}_{{\mathfrak{a}}_{i}}\le {u}_{{b}_{i}}\) and \({\nu }_{{\mathfrak{a}}_{i}}\ge {\nu }_{{b}_{i}}\), for all \(i\), then
for all \(i\), \(\mathsf{J}\)
Let \(\mathfrak{a}=C-IFAA{B}^{s,t}\left({\mathfrak{a}}_{1},{\mathfrak{a}}_{2},\dots , {\mathfrak{a}}_{\mathfrak{v}}\right)\) and \(b=C-IFAA{B}^{s,t}\left({b}_{1},{b}_{2},\dots , {b}_{\mathfrak{v}}\right)\), and the scores of \(\mathfrak{a}\) and \(b\) are denoted by \({Sc}_{\mathfrak{a}}\) and \({Sc}_{b}\) accordingly. Now we are ready to describe the following conditions.
Case 1. When \({Sc}_{\mathfrak{a}}\le {Sc}_{b}\), thus by applying Definition 4, we get
Case 2. When \({Sc}_{\mathfrak{a}}={Sc}_{b}\), then.
Since \({\delta }_{{\mathfrak{a}}_{i}}\le {\delta }_{{b}_{i}}\) and \({\gamma }_{{\mathfrak{a}}_{i}}\ge {\gamma }_{{b}_{\mathsf{J}}}\), for all \(i\), then
And thus
By applying Definition 4, we obtain
So Eq. 10 and Eq. 11 show that Eq. 9 is true.
Boundedness
Let \({\mathfrak{a}}_{i}=\left({u}_{{\mathfrak{a}}_{i}},{\nu }_{{\mathfrak{a}}_{i}};{\mathbb{c}}r\right)\) \(\left(i=\text{1,2},\dots ,j\right)\) be a collection of C-IFVs and let
Then
Proof:
Since \({min}_{i}\left\{{u}_{{\mathfrak{a}}_{i}}\right\}\le {u}_{{\mathfrak{a}}_{i}}\le {max}_{i}\left\{{u}_{{\mathfrak{a}}_{i}}\right\}\) and \({min}_{i}\left\{{v}_{{\mathfrak{a}}_{i}}\right\} \le {v}_{{\mathfrak{a}}_{i}}\le {max}_{i}\left\{{v}_{{\mathfrak{a}}_{i}}\right\}\), for all \(i\), then.
And thus
Additionally, we have
And thus
And thus
Let \(\mathfrak{a}=C-IFAA{B}^{s,t}\left({\mathfrak{a}}_{1},{\mathfrak{a}}_{2},\dots , {\mathfrak{a}}_{\mathfrak{v}}\right)=\left({u}_{{\mathfrak{a}}_{i}},{\nu }_{{u}_{i}}\right)\). Then
Three cases need to be studied in the sections that follow.
Case 1. If \({Sc}_{\mathfrak{a}}<{Sc}_{{\mathfrak{a}}^{+}}\) and \({Sc}_{\mathfrak{a}}>{Sc}_{{\mathfrak{a}}^{-}}\), thus, by using Definition 4 we get
Case 2. If \({Sc}_{\mathfrak{a}}={Sc}_{{\mathfrak{a}}^{+}}\), i.e., \({u}_{\mathfrak{a}}-{v}_{\mathfrak{a}}=\begin{array}{c}max\\ i\end{array}\left\{{u}_{{\mathfrak{a}}_{i}}\right\}-\begin{array}{c}min\\ i\end{array}\left\{{\nu }_{{\mathfrak{a}}_{i}}\right\}\), then, by 54 and 59, we have \({\mathfrak{a}}_{u}=\begin{array}{c}max\\ i\end{array}\left\{{u}_{{\mathfrak{a}}_{i}}\right\}\), \({\nu }_{u}=\begin{array}{c}min\\ i\end{array}\left\{{\nu }_{{\mathfrak{a}}_{i}}\right\}\)
Thus.
So, by using Definition 4, we obtain
Case 3. If \({Sc}_{\mathfrak{a}}={Sc}_{{\mathfrak{a}}^{+}}\), i.e., \({u}_{\mathfrak{a}}-{v}_{\mathfrak{a}}=\begin{array}{c}min\\ i\end{array}\left\{{u}_{{\mathfrak{a}}_{i}}\right\}-\begin{array}{c}max\\ i\end{array}\left\{{\nu }_{{\mathfrak{a}}_{i}}\right\}\), then, by 55 and 60, it can be obtained that
Consequently, we have
Thus by using Definition 4, we get
Based on the above cases, it is evident that 12 is true.
By choosing various values for the parameters \(s\) and \(t\), let’s consider the exceptional cases of the IFBM in the list below.
Case 1. If \(t\to 0\), then by 10 we have,
Which is defined as an intuitionistic fuzzy Aczel-Alsina Bonferroni mean operator.
Case 2. If \(s=2\) and \(t\to 0\), then four is transformed as,
Which is defined as Circular intuitionistic fuzzy Aczel-Alsina Bonferroni square mean.
Case 3. If \(s=1\) and \(t\to 0\), then 4 is transformed as,
Which is defined as Circular intuitionistic fuzzy Aczel-Alsina Bonferroni square mean.
Case 4. 4 is reduced to the following for \(s=t=1\),
Which is defined as Circular intuitionistic fuzzy Aczel-Alsina interrelated square mean.
MAGDM methods by using investigated operators based on C-IFSs
This section will create a MAGDM methodology based on the C-IFAAABM operator to illustrate reliability and effectiveness. For this problem, assuming that \(K_{1} ,K_{1} ,...,K_{n}\) be the set of alternatives and \(A=\left\{{A}_{1},{A}_{2},\dots ,{A}_{i}\right\}\) be the set of attributes with weight vectors \(\omega ={\left({\omega }_{1},{\omega }_{2},\dots ,{\omega }_{\mathfrak{v}}\right)}^{T}\). To overcome that kind of problem, we select the decision matrix \(Q={\left({\tilde{a} }_{i\mathsf{J}}\right)}_{\mathfrak{v}\times i}\), where every term belonging to C-IFVs, then the following steps are used for the procedure of MAGDM problems.
Step 1: Define the decision matrix
Make the decision matrices for the supplier selection problem by considering the set of alternatives \(K_{1} ,K_{2} ,K_{3} ,K_{4}\) and a set of attributes \({\varvec{A}}=\left\{{{\varvec{A}}}_{1}, {{\varvec{A}}}_{2},{{\varvec{A}}}_{3}, {{\varvec{A}}}_{4}\right\}\) with each attribute assigns a weight \(\omega ={\left({\omega }_{1},{\omega }_{2},\dots ,{\omega }_{\mathfrak{v}}\right)}^{T}\).
Step 2: Aggregated IF information
Based on the C-IFAABM operator, accumulate the IF information where each term represents Circular Intuitionistic Fuzzy Values (C-IFVs) corresponding to each alternative and attribute.
Step 3: Compute score values
By using score functions calculate the score values of the accumulated values of Step 3 capturing the decision quality of each alternative based on the values from Step 2.
Step 4: Rank the best alternative
We rank all the alternatives based on the score function. Identifying the highest-ranked alternative as the best choice based on the calculated scores.
Step 5: End
This finalizes the methodology, yielding a well-justified ranking of alternatives for the given MAGDM problem as shown in Fig. 5.
Flowchart for proposed work.
Strategic school policy formulation for physical education goals
The process of formulating a strategic school policy for physical education goals involves developing a thorough and complete strategy to improve physical education (PE) programs inside the curriculum. The formulation process takes into account several factors, including inclusion, skill development, student involvement, resource allocation, and compatibility with the larger curriculum. The Circular Intuitionistic Fuzzy Aczel Alsina Aggregation Operators (C-IFAAO) are one example of an advanced decision-making tool that schools can use to manage the uncertainties and complexities involved in assessing stakeholder preferences and many alternatives. By integrating intuitionistic fuzzy values from several decision-makers, these operators make sure that subjective assessments are appropriately recorded and represented in the chosen course of action. The objective is to choose the best physical education program that promotes comprehensive development, teamwork, and mental health in addition to optimizing physical fitness. Additionally, by adapting to the changing requirements of communities, educators, and students, this method ensures that the chosen program is in line with sustainable educational goals. In the end, carefully developing physical education laws for schools improves the entire experience of students and creates a more inclusive, healthy learning environment. Students who receive physical education and training may benefit from a reduction in the prevalence of poor diet and physical inactivity. Physical education course activities, team competitions, identifying internal and extracurricular sports facilities, assigning a space for family, dynamic school, student sports association, and other physical education-related activities are all part of the comprehensive physical education program. The development of extracurricular physical education and sports activities for students boosts their physical literacy, boosts their productivity, and enhances their success in these subjects. Establishing a student sports volunteer movement is essential for promoting social development, making the most of the current workforce, and developing the nation’s future human resources by giving them practical job experience. One important tactic for successful policy creation is to create and fortify the administrative and organizational underpinnings of student sports through programs like the Students Sports Federation.
Improving the facilities and equipment for student sports in schools is a crucial component of this plan since it has a direct impact on how well physical education classes and extracurricular activities work. Both in-school and after-school physical education programs are prepared to provide memorable learning experiences when high-quality facilities are available In addition to successfully implementing national curricular goals and teacher training, schools should design a strong physical education performance management system. These kinds of systems facilitate the assessment and improvement of physical education methods, which eventually creates an environment that is conducive to progress. Furthermore, modernizing laws, regulations, and technology, as well as enhancing performance, productivity, and teaching strategies, will promote physical literacy and help create a more active way of life overall. The aim of research and development in student sports is aligned with CIFAABM strategies, focusing on the promotion of social health through innovation, research, and optimal use of resources. In this context, productivity is measured through efficiency and effectiveness in achieving goals while ensuring resource utilization without waste. Efficiently utilizing available resources while maximizing effectiveness in achieving educational and sports objectives ensures that the future generation is well-prepared for the challenges ahead as shown in Fig. 6.
Physical education curriculum.
Problem (selecting the best educational institution)
The school district aims to enhance its physical education (PE) offerings and cultivate a robust student sports program that not only encourages physical activity but also promotes social growth, leadership, and career readiness. To achieve this, the district introduced a volunteer-driven sports movement where senior students, teachers, and parents collaborate in organizing and managing PE activities. This volunteer initiative allows students to gain leadership and organizational skills while acquiring practical experience in event management, coaching, and first aid. By building a trained network of volunteers, the district contributes to community sports events and national sports initiatives. Moreover, schools assess the existing sports facilities to identify areas for improvement, including upgrading gyms, adding new sports gear, and establishing outdoor athletic spaces for basketball, soccer, and track events. The district also emphasizes the integration of eco-friendly materials in the construction and renovation of athletic facilities to align with environmental sustainability goals.
Attributes: Volunteer-driven participation: Involves senior students, teachers, and parents in sports activity management. Leadership and skills development: Focuses on building leadership and practical skills through sports program involvement. Environmental sustainability: Promotes the use of green materials in facility upgrades.
Alternative 2: Comprehensive physical education enhancement program
To enhance physical education (PE) offerings and foster student development, the district sets out to create a comprehensive student sports program that promotes physical fitness, leadership skills, and career readiness. The initiative is designed around a volunteer-based sports movement, where students, faculty, and parents collaborate to organize and manage PE activities. Volunteers are trained in essential skills such as event management, coaching, and first aid, thereby developing a pool of skilled individuals who are equipped to contribute to both school and community sports events. Schools are also tasked with evaluating and upgrading their sports facilities, focusing on gym modernization, new sports equipment, and the creation of outdoor spaces suitable for various sports. Additionally, the district encourages environmentally sustainable practices by incorporating eco-friendly materials in its sports facilities.
Attributes: Holistic development: Focuses on physical, social, and leadership skills. Volunteer training: Equips volunteers with event management, coaching, and first aid skills. Facility upgrades: Emphasizes gym modernization and eco-friendly designs.
Alternative 3: Integrated sports program with environmental focus
The district seeks to improve its physical education (PE) offerings by establishing a student sports program that fosters physical fitness, leadership, and career development. A key aspect of the program is the introduction of a volunteer-based sports movement, involving senior students, teachers, and parents in the management and organization of PE activities. This initiative provides students with hands-on experience in leadership and organizational roles, such as coaching, event coordination, and first aid training. Additionally, the district focuses on evaluating and upgrading school sports facilities, emphasizing the importance of modern gyms, updated sports equipment, and well-designed outdoor spaces for various sports. As part of its sustainability goals, the district encourages schools to adopt green practices in facility development, using eco-friendly materials wherever possible.
Attributes: Sustainability: Focus on using eco-friendly materials and sustainable practices in sports facilities. Leadership experience: Volunteer-based model offers leadership development for students. Comprehensive upgrades: Ensures facilities are well equipped for diverse sports activities.
Alternative 4: Student-centric sports program with technology integration
The district is committed to enhancing its physical education (PE) offerings by developing a student sports program that not only engages students in physical activity but also fosters leadership, social growth, and career skills. This initiative introduces a volunteer sports movement in which senior students, teachers, and parents collaborate to organize PE activities, creating opportunities for students to develop leadership and organizational skills. Volunteers are trained in various areas, including coaching, event management, and first aid. In addition to upgrading sports facilities—such as gyms, sports equipment, and outdoor spaces—the district explores the integration of modern technology into its PE programs. This could involve the use of digital tools for sports tracking, performance monitoring, and enhanced teaching methods, allowing for a more personalized and engaging sports experience.
Attributes: Technology integration: Incorporates digital tools to enhance PE programs. Volunteer leadership: Engages senior students, teachers, and parents in leadership roles. Student empowerment: Fosters skills development for real-world applications, such as career readiness as shown in Fig. 7.
A holistic approach to fitness through physical education.
In this case study, data were gathered from domain experts based on structured questionnaires and interviews, aiming at real-world applications to [insert domain, e.g., renewable energy choice, supplier rating, etc.]. The experts individually gave ratings with C-IFVs for the provided alternatives under various criteria. The fuzzy ratings indicate the imprecision and vagueness present in human judgment. The decision matrix was then developed based on these expert inputs, and weights of the criteria were established either through expert consensus or an objective method like entropy or CRITIC. The aggregated fuzzy information was then processed through the proposed C-IFAABM operator to get credible rankings of the alternatives.
Justification for using complex fuzzy structures in educational decision-making
In most decision-making issues, especially in choosing the optimal educational institution for physical education purposes, information presented by specialists can still carry aspects of uncertainty, vagueness, or subjectivity. Although these specialists have a deep understanding and knowledge, they can have varying views or evaluations of the criteria for the decision-making process. The following is why a sophisticated fuzzy approach is required:
Even the experts cannot always put their opinions or preferences into exact numbers. They can describe their judgments in terms like“somewhat important”or"very important,"which are necessarily fuzzy. In such situations, a straightforward crisp (exact) model would not be able to capture the entire scope of expert knowledge. A fuzzy set methodology permits a more relaxed representation of expert knowledge, allowing for the uncertainty present in human judgment. When applying to school policy development for physical education, various factors need to be taken into account such as resources, student involvement, facilities, and curriculum quality each of which can have a different level of importance for each stakeholder. Even professionals will not be able to give fixed, sharp values for these variables because of the variability and randomness inherent in them. A fuzzy system offers a framework that can collect these variables together and represent uncertainty.
Choosing the most appropriate institution for physical education is a multi-faceted issue, dealing with many criteria (e.g., facilities, sports offerings, student performance rates, and instructional expertise) that may not always be quantifiable. Fuzzy set theory can be utilized to capture more effectively the complexity and interdependencies in these variables so that the decision-making process will reflect a more realistic and sophisticated perception of the alternatives available. Even with input from experts, opinions can still be different. Using the fuzzy set model, aggregation of different opinions and evaluation becomes possible, producing a collective choice that reflects on the variety of expert opinions. A more complex fuzzy set pattern, like the Circular Intuitive Fuzzy Bonferroni mean or advanced operators, can combine several inputs of expert view in a manner that yields an outcome stronger than using less advanced methods.
When formulating policies for physical education, it is critical to consider not just the current conditions but also future goals, potential developments, and how these factors might evolve. Fuzzy logic is well-suited to handle such forward-looking assessments, as it allows for flexible, adaptable models that can account for changing educational goals and evolving priorities in a dynamic environment.
Methodology
This study uses seven criteria to assess four AI-driven EdTech solutions based on evaluations from five decision-makers that focus on four characteristics: cost-effectiveness, engagement, flexibility, and learning outcomes. The methodology involves a systematic framework to process the decision-maker’s evaluations, aggregate the data, and derive the final rankings. The steps are outlined as follows:
Schedules and learning paces. C1: Assess how well the solution can be modified to meet the needs of various learners. C2: Evaluates how adaptable the solution is to different learner demands. C3: Evaluate how well the answer maintains students’ interest. C4: Shows how easy the solution is to utilize. C5: Indicates the degree of sophisticated technology employed. C6: Investigates whether more users can be accommodated without experiencing performance problems. C7: Indicates how cost-effective the approach is.
Decision-making process
A set of decision-makers (DMs) is selected with meticulous care on the basis of their knowledge and expertise related to the problem at hand. Every DM assesses the alternative set under pre-defined criteria with the help of C-IFVs. All these individual ratings are then combined by using the designed C-IFAABM operator to prepare a collective group decision. The impact of each decision-maker can be combined through decision weights or linguistic ratings that quantify the reliability and confidence level of each expert. The collective assessment through this approach allows for the decision process to demonstrate a balanced and diversified perspective, eliminating bias and subjectivity.
Determining the weights of decision-makers
The decision-makers’ weights are assumed to be equal in this research to assurance equality and ease the aggregation process. Every expert has an equal contribution during the evaluation process. This method is especially beneficial when experts are thought to possess a similar level of expertise or no weighting information is available previously. Their assessment are grouped employing the discussed C-IF based averaging operators, which combine their inputs systematically and drive the end rankings of alternatives.The following steps were engaged in the evaluation process: Expert Evaluations:
Step 1: Make the decision matrices briefly described in the following Table 1, Table 2, and Table 3. The performance of the institutions under each criterion is rated on a scale of \(1\) to \(10\).
Step 2: Convert the C-IF matrix into the form of a normal decision matrix. Normalization isn’t necessary if all of the attributes are of the same type normalization isn’t necessary.
Step 3: Based on the C-IFAAA operator, accumulating the C-IF information is discussed in the form of Table 3.
Step 4: Repeat step 2 over again based on C-IFAAA, and accumulate the C-IF information discussed in the form of Table 4.
Step 5: Based on score functions, calculate the score values of the accumulated values of step 3, which are discussed in the form of Table 5.
By using the C-IFAABM operator we obtained \(\left( {K_{3} } \right)\) as the best alternative as stated above.
Sensitivity analysis
The proposed method has several important advantages and differences when compared to other decision-making techniques. Using the different values of parameter \(P\), it is essential to check the stability and influence of the proposed work with the help of parameter \(P\). Already, we have proved that the derived operators are massively powerful and dominant in evaluating different kinds of problems. The main aim of this section is to check the stability and influence of the proposed work with the help of parameter \(P\). The primary analysis for different values of \(s\), the final ranking information, is described in Table 6. The model output is a ranked list of alternatives from the composite group decision data addressed via the suggested methodology. This rank identifies the best alternative(s) based on the chosen criteria and group preferences. The ranked result is a solid decision-support instrument for stakeholders, yielding a crisp and actionable result that can be utilized to inform strategic planning, policy development, or resource allocation in the target application field. Furthermore, sensitivity analysis can be performed to ensure the stability of the rankings with respect to weight or input assessment variations.
Discussion section
While the new theory utilizes the same MD definition as competing models, the addition of the radius parameter (\({\varvec{r}}\)) is more than a simple additive parameter. Geometrically and semantically, the radius adds a circular boundary constraint to improve decision uncertainty and hesitancy modeling in a more limited and realistic space. This circular form guarantees that the membership, NMD, and hesitancy values are kept in a bounded, normalized area maintaining information consistency and interpretability. Furthermore, the radius permits more control over the system’s fuzziness, making it particularly well-suited to difficult decision environments where fine nuances in expert judgments need to be explicitly modeled. Compared to classical intuitionistic or spherical models, this extension provides a smoother decision surface, enhancing discrimination between alternatives and facilitating more accurate policy design in multi-attribute group decision situations. Assuming the innovative attractiveness of this approach and its possible consequences for both scholarly research and real-world application, it is critical to emphasize the interdisciplinary nature of this research. By combining circular intuitionistic fuzzy logic with Bonferroni-based aggregation, the model provides a new method of decision-making that aligns with the adaptive processes in evolutionary economics, where strategies adapt according to changing circumstances and limited rationality. At the same time, the framework enhances central dimensions of organizational behavior, including policy development, strategic fit, and behavioral uncertainty, through facilitating more sensitive and data-responsive assessments of institutional performance. This synergy of method enhances the theoretical value of the study as much as it situates the study as a versatile instrument for decision-support purposes among stakeholders from a wide range of fields be it education policymakers, organizational strategists, or the like.
Interdisciplinary implications
The suggested circular intuitionistic fuzzy MAGDM model is highly interdisciplinary-relevant, especially in evolutionary economics and organizational behavior. It can effectively manage ambiguity, hesitation, and conflicting expert judgments, making it appropriate for decision contexts that are complex. The model facilitates adaptive learning and bounded rationality for economic situations and improves stakeholder-based decision-making in organizational contexts. These capabilities extend its use to policy planning, innovation management, and strategic assessment.
Limitations
In spite of the strong framework offered in this research, there are some limitations. The sample size of the participating experts was small, which could influence the generalizability of the results. The experts utilized were also mainly taken from a single sector, thus possibly constraining the model’s potential for application in more general industry settings. Future studies might improve the model’s robustness and generalizability by incorporating different decision-makers across various sectors to broaden its applicability.
Comparative analysis with operators
In this section, we compare this study with the other different algorithms and DM techniques to find the validity of the analyzed approach. Table 6 and Fig. 8 show the results of this comparison. For this, we investigated a few familiar operators, which were developed based on different existing ideas are described as follows: as Bonferroni mean operator for IF environment derived by Xu and Yager66, Senapati et al.95 discovered the idea of IFAAWA and IFAAWG by using Aczel-Alsina t-norm and t-conorm, Zeshui Xu99 evaluated the averaging aggregation operators for IFS and Xu and Yager37 defined the geometric aggregation operators for IFSs. Therefore, under consideration of the information in Table 1, we described the comparative analysis in the form of Table 7.
Graphically representation of Table 5 displays the variation in results across different alternatives by using different parameters of P.
Advantages of the proposed theory
The suggested model presents several important benefits in solving complicated educational decision-making issues. First, it presents an advanced framework for uncertainty modeling through the use of CIFSs, which better represent hesitation and dual membership information compared to conventional fuzzy models. This allows for a more complete and realistic modeling of expert judgment in choosing the best physical education approaches. Second, the inclusion of Bonferroni-based aggregation facilitates the accounting for interdependencies and cross-reinforcement among assessment criteria, improving the responsiveness of the decision model to complicated policy situations. Third, the model accommodates multi-criteria and multi-alternative analysis, which are essential in developing inclusive and efficient school policies. In addition, the application of higher-order mathematical operators facilitates context-sensitive, nonlinear aggregation, which enhances the accuracy of recommendations. Finally, notwithstanding its mathematical sophistication, the model is scalable, flexible, and transferable to real-world learning contexts, particularly with the help of computational technologies, and it thus represents a feasible and rational framework for policy-making in education.
Conclusion
Describing current challenges and events in a numerical and statistical structure or terminology is known as the decision-making technique. Many theories use a wide range of tools and procedures to evaluate complex and untrustworthy data. However, to cope with that kind of information, we introduced the C-IFAABM operator in this paper using Aczel-Alsina t-norm and t-conorm, which are very famous and valuable. In this section, we derived the following information,
-
1.
We developed the concept of C-IFAABM operator and analyzed their main aspects and beneficial objectives.
-
2.
Further, the characteristics and outcomes of the evaluated operators are identified.
-
3.
Using the proposed operators for C-IFB information by using A-A t-norm and t-conorm, we computed a MAGDM problem to analyze the proposed model. We determined the required selection from the group of alternatives.
-
4.
To demonstrate the validity of the proposed work and to compare it to certain existing operators, we provided some examples. We made an effort to assess the information and explanation of the proposed work.
The school district’s responsibility to expand its PE offerings and create a comprehensive student sports program, in summary, presents a strategic approach to fostering holistic student development. By incorporating volunteerism, leadership training, and facility upgrades, the program fosters physical fitness, social growth, and career readiness. Additionally, the program’s overall quality and involvement are enhanced by the focus on sustainability through eco-friendly facility development materials and the integration of modern technology into PE practices. This comprehensive strategy ensures that students not only obtain useful skills but also give back to their communities, preparing them for leadership roles in the future and supporting national educational goals.
Future direction
In the future, we aim to discover some new theories by using IF Frank aggregation operators in the framework of BM operators and try to use them in different fields to see the impact on real-life problems. We can use it in PF42, SF, and TSF43,44,45 environments. We can also use it in Interval-valued complex single-valued neutrosophic46. To identify gaps or propose extending the semi-Markov jump T-S fuzzy systems to new domains such as heritage preservation33.
In the following work, the CIFBM-based model presented here can be expanded by incorporating sophisticated mathematical frameworks from nonlinear differential equations and dynamical systems. For example, the HIV infection model of Li and Wang in 2015102 illustrates how logistic growth nonlinear dynamics can be used to model developing systems under uncertainty, providing insights for dynamic educational policy adjustment. The research of Li et al. in 2015, 2018, 2020, and 2021103,104,105 on switching systems, center-focus issues, bifurcation theory, and linearization of Z₂-equivariant systems form a theoretical foundation for the examination of the stability and transition dynamics of intricate decision models. This research could guide the creation of hybrid decision-making models that integrate fuzzy logic with bifurcation analysis to tackle sudden changes in educational policies. Additionally, Li et al. in 2024106 comprehensively classify nilpotent singularities and provide promising avenues toward improving model resilience and sensitivity by Chen et al. in 2024107 performing higher-order analyses of switching Liénard systems. Future work can incorporate such advanced dynamical tools to design more robust, adaptive, and analytically informed decision support systems for strategic educational planning. More work can be seen here108,109,110.
Data availability
The datasets generated and/or analyzed during the current study are not publicly available due to privacy concerns but are available from the corresponding author upon reasonable request.
Abbreviations
- AO:
-
Aggregation operator
- DM:
-
Decision making
- BM:
-
Bonferroni mean
- A-A:
-
Aczel-alsina
- IFS:
-
Intuitionistic fuzzy set
- C-IFS:
-
Circular intuitionistic fuzzy set
- MAGDM:
-
Multi-attribute group decision making
- PE:
-
Physical education
- SSPF:
-
Strategic school policy formulation
References
Herrera, F., Herrera-Viedma, E. & Chiclana, F. Multiperson decision-making based on multiplicative preference relations. Eur. J. Oper. Res. 129(2), 372–385. https://doi.org/10.1016/S0377-2217(99)00197-6 (2001).
Herrera, F., Herrera-Viedma, E. & Martínez, L. A fusion approach for managing multi-granularity linguistic term sets in decision making. Fuzzy Sets Syst. 114(1), 43–58. https://doi.org/10.1016/S0165-0114(98)00093-1 (2000).
Hu, J., Zhang, X., Chen, X. & Liu, Y. Hesitant fuzzy information measures and their applications in multi-criteria decision making. Int. J. Syst. Sci. 47(1), 62–76. https://doi.org/10.1080/00207721.2015.1036476 (2016).
Jahanshahloo, G. R., Lotfi, F. H. & Izadikhah, M. An algorithmic method to extend TOPSIS for decision-making problems with interval data. Appl. Math. Comput. 175(2), 1375–1384. https://doi.org/10.1016/j.amc.2005.08.048 (2006).
Chen, Z. S. et al. Prioritizing real estate enterprises based on credit risk assessment: An integrated multi-criteria group decision support framework. Financ. Innov. 9(1), 120. https://doi.org/10.1186/s40854-023-00517-y (2023).
Chen, Z.-S., Zhu, Z., Wang, Z.-J. & Tsang, Y. Fairness-aware large-scale collective opinion generation paradigm: A case study of evaluating blockchain adoption barriers in medical supply chain. Inf. Sci. 635, 257–278. https://doi.org/10.1016/j.ins.2023.03.135 (2023).
Wang, Z.-J. et al. Enhancing the sustainability and robustness of critical material supply in electrical vehicle market: an AI-powered supplier selection approach. Ann. Oper. Res. 342(1), 921–958. https://doi.org/10.1007/s10479-023-05698-4 (2024).
Wang, Z. J. et al. Smart contract application in resisting extreme weather risks for the prefabricated construction supply chain: Prototype exploration and assessment. Group Decis. Negot. 33(5), 1049–1087 (2024).
Wang, Z.-J. et al. Blockchain adoption in sustainable supply chains for Industry 50 A multistakeholder perspective. J. Innov. Knowl. 8(4), 100425. https://doi.org/10.1016/j.jik.2023.100425 (2023).
Chang, J. P. et al. Assessing spatial synergy between integrated urban rail transit system and urban form: A BULI-based MCLSGA model with the wisdom of crowds. IEEE Trans. Fuzzy Syst. 31(2), 434–448 (2022).
Sabir, Z., Umar, M., Salahshour, S. & Saeed, T. A reliable neural network procedure for the novel sixth-order nonlinear singular pantograph differential model. Mod. Phys. Lett. B 39(12), 2450473. https://doi.org/10.1142/S0217984924504736 (2025).
Sabir, Z., Khansa, S., Baltaji, G. & Saeed, T. A Bayesian regularization neural network procedure to solve the language learning system. Knowl.-Based Syst. 310, 112997 (2025).
Sabir, Z., Kotob, I. A., Sheikh, L. A. & Saeed, T. A novel computational approach-based hyperbolic tangent sigmoid deep neural network for the hepatitis B virus model. Int. J. Geom. Methods Mod. Phys. 22(04), 2450315. https://doi.org/10.1142/S0219887824503158 (2025).
Umar, M. et al. Design of intelligent computing solver with Morlet wavelet neural networks for nonlinear predator–prey model. Appl. Soft Comput. https://doi.org/10.1016/j.asoc.2022.109975 (2023).
Bhat, S. A., Sabir, Z., Raja, M. A. Z., Saeed, T. & Alshehri, A. M. A novel heuristic Morlet wavelet neural network procedure to solve the delay differential perturbed singular model. Knowl.-Based Syst. 292, 111624 (2024).
Sabir, Z., Saeed, T., Guirao, J. L., Sánchez, J. M. & Valverde, A. A swarming Meyer wavelet computing approach to solve the transport system of goods. Axioms 12(5), 456 (2023).
Saeed, T., Guirao, J. L. G., Sabir, Z., Alsulami, H. H. & Sánchez, Y. G. A Computational approach to solve the nonlinear biological prey–predator system. Fractals 30(10), 2240267. https://doi.org/10.1142/S0218348X22402678 (2022).
Sabir, Z., Zahoor Raja, M. A., Guirao, J. L. G. & Saeed, T. Design of Mayer Wavelet Neural Networks for Solving Functional Nonlinear Singular Differential Equation. Math. Probl. Eng. 2022, 1–11. https://doi.org/10.1155/2022/1213370 (2022).
Saeed, T., Sabir, Z., Alhodaly, M. S., Alsulami, H. H. & Sánchez, Y. G. An advanced heuristic approach for a nonlinear mathematical based medical smoking model. Results Phys. 32, 105137 (2022).
Sabir, Z., Raja, M. A. Z., Guirao, J. L. & Saeed, T. Meyer wavelet neural networks to solve a novel design of fractional order pantograph Lane-Emden differential model. Chaos, Solitons Fractals 152, 111404 (2021).
Sabir, Z., Raja, M. A. Z., Guirao, J. L. & Saeed, T. Solution of novel multi-fractional multi-singular Lane-Emden model using the designed FMNEICS. Neural Comput. Appl. 33(24), 17287–17302 (2021).
Umar, M. et al. Integrated neuro-swarm heuristic with interior-point for nonlinear SITR model for dynamics of novel COVID-19. Alex. Eng. J. 60(3), 2811–2824 (2021).
Sabir, Z., Guirao, J. L. G., Saeed, T. & Erdoğan, F. Design of a Novel Second-Order Prediction Differential Model Solved by Using Adams and Explicit Runge-Kutta Numerical Methods. Math. Probl. Eng. 2020, 1–7. https://doi.org/10.1155/2020/9704968 (2020).
Guirao, J. L. G., Sabir, Z. & Saeed, T. Design and Numerical Solutions of a Novel Third-Order Nonlinear Emden-Fowler Delay Differential Model. Math. Probl. Eng. 2020, 1–9. https://doi.org/10.1155/2020/7359242 (2020).
Abdelkawy, M. A., Sabir, Z., Guirao, J. L. G. & Saeed, T. Numerical investigations of a new singular second-order nonlinear coupled functional Lane-Emden model. Open Phys. 18(1), 770–778. https://doi.org/10.1515/phys-2020-0185 (2020).
Sabir, Z., García Guirao, J. L. & Saeed, T. Solving a novel designed second order nonlinear Lane-Emden delay differential model using the heuristic techniques. Appl. Soft Comput. https://doi.org/10.1016/j.asoc.2021.107105 (2021).
Sabir, Z. et al. Neuro-Swarm heuristic using interior-point algorithm to solve a third kind of multi-singular nonlinear system. Math. Biosci. Eng. 18(5), 5285–5308 (2021).
Hussain, S., Zeb, A., Rasheed, A. & Saeed, T. Stochastic mathematical model for the spread and control of Corona virus. Adv. Differ. Equ. 2020(1), 574. https://doi.org/10.1186/s13662-020-03029-6 (2020).
Zadeh, L. A. Fuzzy sets. Inf. Control 8(3), 338–353. https://doi.org/10.1016/S0019-9958(65)90241-X (1965).
Sun, Q., Ren, J. & Zhao, F. Sliding mode control of discrete-time interval type-2 fuzzy Markov jump systems with the preview target signal. Appl. Math. Comput. 435, 127479. https://doi.org/10.1016/j.amc.2022.127479 (2022).
Zhang, N., Qi, W., Pang, G., Cheng, J. & Shi, K. Observer-based sliding mode control for fuzzy stochastic switching systems with deception attacks. Appl. Math. Comput. 427, 127153. https://doi.org/10.1016/j.amc.2022.127153 (2022).
Xia, Y., Wang, J., Meng, B. & Chen, X. Further results on fuzzy sampled-data stabilization of chaotic nonlinear systems. Appl. Math. Comput. 379, 125225. https://doi.org/10.1016/j.amc.2020.125225 (2020).
Gao, M. et al. SMC for semi-Markov jump TS fuzzy systems with time delay. Appl. Math. Comput. 374, 125001 (2020).
Sarwar, M. & Li, T. Fuzzy fixed point results and applications to ordinary fuzzy differential equations in complex valued metric spaces. Hacet. J. Math. Stat. 48(6), 1712–1728 (2019).
Atanassov, K. T. Intuitionistic fuzzy sets. Fuzzy Sets Syst https://doi.org/10.1016/S0165-0114(86)80034-3 (1986).
Atanassov, K. T. Intuitionistic fuzzy sets. In Intuitionistic fuzzy sets (ed. Atanassov, K. T.) 1–137 (Springer, 1999).
Xu, Z. & Yager, R. R. Some geometric aggregation operators based on intuitionistic fuzzy sets. Int. J. Gen Syst 35(4), 417–433 (2006).
Liu, P. & Chen, S.-M. Multiattribute group decision making based on intuitionistic 2-tuple linguistic information. Inf. Sci. 430–431, 599–619. https://doi.org/10.1016/j.ins.2017.11.059 (2018).
Wei, G. W. 2-tuple intuitionistic fuzzy linguistic aggregation operators in multiple attribute decision making. Iran. J. Fuzzy Syst. 16(4), 159–174. https://doi.org/10.22111/ijfs.2019.4789 (2019).
Ge, J. & Zhang, S. Adaptive Inventory Control Based on Fuzzy Neural Network under Uncertain Environment. Complexity 2020, 1–10. https://doi.org/10.1155/2020/6190936 (2020).
Li, W., Xie, Z., Wong, P. K., Mei, X. & Zhao, J. Adaptive-Event-Trigger-Based Fuzzy Nonlinear Lateral Dynamic Control for Autonomous Electric Vehicles Under Insecure Communication Networks. IEEE Trans. Industr. Electron. 68(3), 2447–2459. https://doi.org/10.1109/TIE.2020.2970680 (2021).
Li, W., Xie, Z., Zhao, J., Wong, P. K. & Li, P. Fuzzy finite-frequency output feedback control for nonlinear active suspension systems with time delay and output constraints. Mech. Syst. Signal Process. https://doi.org/10.1016/j.ymssp.2019.06.018 (2019).
Li, W., Xie, Z., Cao, Y., Wong, P. K. & Zhao, J. Sampled-Data Asynchronous Fuzzy Output Feedback Control for Active Suspension Systems in Restricted Frequency Domain. IEEE/CAA J. Automatica Sinica 8(5), 1052–1066. https://doi.org/10.1109/JAS.2020.1003306 (2021).
Atanassov, K. T. Two theorems for intuitionistic fuzzy sets. Fuzzy Sets Syst. 110(2), 267–269 (2000).
Takeuti, G. & Titani, S. Intuitionistic fuzzy logic and intuitionistic fuzzy set theory. The Journal of Symbolic Logic 49(3), 851–866. https://doi.org/10.2307/2274139 (1984).
De, S. K., Biswas, R. & Roy, A. R. An application of intuitionistic fuzzy sets in medical diagnosis. Fuzzy Sets Syst. 117(2), 209–213 (2001).
Zhang, Z., Yang, J., Ye, Y., Hu, Y. & Zhang, Q. A type of score function on intuitionistic fuzzy sets with double parameters and its application to pattern recognition and medical diagnosis. Procedia Eng. 29, 4336–4342 (2012).
Garg, H. & Rani, D. Complex interval-valued intuitionistic fuzzy sets and their aggregation operators. Fund. Inform. 164(1), 61–101 (2019).
Garg, H. & Rani, D. Novel aggregation operators and ranking method for complex intuitionistic fuzzy sets and their applications to decision-making process. Artif. Intell. Rev. 53(5), 3595–3620. https://doi.org/10.1007/s10462-019-09772-x (2020).
Garg, H. & Kumar, K. Linguistic Interval-Valued Atanassov Intuitionistic Fuzzy Sets and Their Applications to Group Decision Making Problems. IEEE Trans. Fuzzy Syst. 27(12), 2302–2311. https://doi.org/10.1109/TFUZZ.2019.2897961 (2019).
Çakır, E. & Taş, M. A. Circular Intuitionistic Fuzzy Decision Making and Its Application. Expert Syst. Appl. 225, 120076 (2023).
Irem, O. & Kahraman, C. A novel circular intuitionistic fuzzy AHP&VIKOR methodology: An application to a multi-expert supplier evaluation problem. Pamukkale Üniversitesi Mühendislik Bilimleri Dergisi 28(1), 194–207 (2022).
Alkan, N. & Kahraman, C. Circular intuitionistic fuzzy TOPSIS method: Pandemic hospital location selection. J. Intell. Fuzzy Syst. 42(1), 295–316 (2022).
Otay, İ, Onar, S. Ç., Öztayşi, B. & Kahraman, C. A novel interval valued circular intuitionistic fuzzy AHP methodology: Application in digital transformation project selection. Inf. Sci. 647, 119407 (2023).
Çakir, E., Taş, M. A., & Ulukan, Z. (2021, November). A new circular intuitionistic fuzzy MCDM: A case of Covid-19 medical waste landfill site evaluation. In 2021 IEEE 21st international symposium on computational intelligence and informatics pp. 000143–000148, IEEE. (2021)
Garg, H., Ünver, M., Olgun, M. & Türkarslan, E. An extended EDAS method with circular intuitionistic fuzzy value features and its application to multi-criteria decision-making process. Artif. Intell. Rev. 56(S3), 3173–3204. https://doi.org/10.1007/s10462-023-10601-5 (2023).
Yager, R. R. & Kacprzyk, J. The ordered weighted averaging operators: theory and applications (Springer Science & Business Media, 2012).
Calvo, T., Mayor, G. & Mesiar, R. Aggregation operators: new trends and applications Vol. 97 (Springer Science & Business Media, 2002).
Xu, Z. S. & Da, Q. L. An overview of operators for aggregating information. Int. J. Intell. Syst. 18(9), 953–969. https://doi.org/10.1002/int.10127 (2003).
Torra, V. & Narukawa, Y. Modeling decisions: information fusion and aggregation operators (Springer Science & Business Media, 2007).
Yager, R. R. The power average operator. IEEE Trans. Syst., Man, Cybern. - Part A: Syst. Humans 31(6), 724–731. https://doi.org/10.1109/3468.983429 (2001).
Liu, P. & Yu, X. Density aggregation operators based on the intuitionistic trapezoidal fuzzy numbers for multiple attribute decision making. Technol. Econ. Dev. Econ. 19(sup1), S454–S470. https://doi.org/10.3846/20294913.2013.881436 (2013).
Bonferroni, C. Sulle medie multiple di potenze. Bollettino dell’Unione Matematica Italiana 5(3–4), 267–270 (1950).
Yager, R. R. On generalized Bonferroni mean operators for multi-criteria aggregation. Int. J. Approx. Reason. 50(8), 1279–1286 (2009).
S. Sỳkora, ‘Mathematical means and averages: Generalized Heronian means’, Stan’s Library: Castano Primo, Italy, (2009).
Xu, Z. & Yager, R. R. Intuitionistic Fuzzy Bonferroni Means. IEEE Trans. Sys., Man, Cybern. Part B (Cybernetics) 41(2), 568–578. https://doi.org/10.1109/TSMCB.2010.2072918 (2011).
Akram, M., Ullah, K., Ćirović, G. & Pamucar, D. Algorithm for energy resource selection using priority degree-based aggregation operators with generalized orthopair fuzzy information and Aczel-Alsina aggregation operators. Energies 16(6), 2816 (2023).
Akram, M., Ullah, K. & Pamucar, D. Performance evaluation of solar energy cells using the interval-valued T-spherical fuzzy Bonferroni mean operators. Energies 15(1), 292 (2022).
Akram, M., Wang, H., Garg, H. & Ullah, K. Interaction Power Bonferroni Mean Aggregation Operators Based on T-Spherical Fuzzy Information and Their Application in Multi-attribute Decision Making. Int. J. Fuzzy Syst. 25(7), 2665–2683. https://doi.org/10.1007/s40815-023-01542-w (2023).
Zhao, H., Xu, Z., Ni, M. & Liu, S. Generalized aggregation operators for intuitionistic fuzzy sets. Int. J. Intell. Syst. 25(1), 1–30 (2010).
Xu, Z. & Yager, R. R. Some geometric aggregation operators based on intuitionistic fuzzy sets. Int. J. Gen Syst 35(4), 417–433. https://doi.org/10.1080/03081070600574353 (2006).
Wang, W. & Liu, X. Intuitionistic fuzzy geometric aggregation operators based on Einstein operations. Int. J. Intell. Syst. 26(11), 1049–1075 (2011).
Sarfraz, M., Ullah, K., Akram, M., Pamucar, D. & Božanić, D. Prioritized Aggregation Operators for Intuitionistic Fuzzy Information Based on Aczel-Alsina T-Norm and T-Conorm and Their Applications in Group Decision-Making. Symmetry https://doi.org/10.3390/sym14122655 (2022).
X. Wang, ‘Fuzzy number intuitionistic fuzzy arithmetic aggregation operators.’, International Journal of Fuzzy Systems, vol. 10, no. 2, (2008).
Rani, D. & Garg, H. Complex intuitionistic fuzzy power aggregation operators and their applications in multicriteria decision-making. Expert. Syst. 35(6), e12325. https://doi.org/10.1111/exsy.12325 (2018).
Xia, M., Xu, Z. & Zhu, B. Some issues on intuitionistic fuzzy aggregation operators based on Archimedean t-conorm and t-norm. Knowl.-Based Syst. 31, 78–88. https://doi.org/10.1016/j.knosys.2012.02.004 (2012).
H. S. Hota, L. K. Sharma, and S. Pavani Fuzzy TOPSIS Method Applied for Ranking of Teacher in Higher Education. in Intelligent Computing, Networking, and Informatics, vol. 243, D. P. Mohapatra and S. Patnaik, Eds., in Advances in Intelligent Systems and Computing, vol. 243. , New Delhi: Springer India, pp. 1225–1232. https://doi.org/10.1007/978-81-322-1665-0_127, (2014)
Hussain, A., Ullah, K., Alshahrani, M. N., Yang, M.-S. & Pamucar, D. Novel Aczel-Alsina Operators for Pythagorean Fuzzy Sets with Application in Multi-Attribute Decision Making. Symmetry https://doi.org/10.3390/sym14050940 (2022).
Ali, J. Multi-criteria decision-making method based on an integrated model using T-spherical fuzzy aczel-alsina prioritized aggregation operators. Comput. Appl. Math. 44(3), 181 (2025).
Ali, J. Spherical fuzzy symmetric point criterion-based approach using Aczel-Alsina prioritization: Application to sustainable supplier selection. Granular Computing 9(2), 33 (2024).
Ali, J. Analysis and application of r, s, t-spherical fuzzy Aczel-Alsina aggregation operators in multiple criteria decision-making. Granul. Comput. 9(1), 17 (2024).
Ali, J. & Rasool, W. Interval-valued q-rung orthopair fuzzy Aczel-Alsina operations-based Bonferroni mean aggregation operators and their applications. Comput. Appl. Math. 43(1), 7 (2024).
Ali, J. & Naeem, M. Multi-criteria decision-making method based on complex t-spherical fuzzy Aczel-Alsina aggregation operators and their application. Symmetry 15(1), 85 (2022).
Xia, M., Xu, Z. & Zhu, B. Geometric Bonferroni means with their application in multi-criteria decision making. Knowl.-Based Syst. 40, 88–100. https://doi.org/10.1016/j.knosys.2012.11.013 (2013).
Li, D., Zeng, W. & Li, J. Geometric Bonferroni Mean Operators. Int. J. Intell. Syst. 31(12), 1181–1197. https://doi.org/10.1002/int.21822 (2016).
Zhu, B. & Xu, Z. S. Hesitant fuzzy Bonferroni means for multi-criteria decision making. J. Oper. Res. Soc. 64(12), 1831–1840. https://doi.org/10.1057/jors.2013.7 (2013).
Zhu, B., Xu, Z. & Xia, M. Hesitant fuzzy geometric Bonferroni means. Inf. Sci. 205, 72–85. https://doi.org/10.1016/j.ins.2012.01.048 (2012).
He, Y., He, Z., Jin, C. & Chen, H. Intuitionistic fuzzy power geometric Bonferroni means and their application to multiple attribute group decision making. Int. J. Unc. Fuzz. Knowl.-Based Syst. 23(02), 285–315 (2015).
Wei, G., Zhao, X., Lin, R. & Wang, H. Uncertain linguistic Bonferroni mean operators and their application to multiple attribute decision making. Appl. Math. Model. 37(7), 5277–5285 (2013).
Bailey, R. et al. The educational benefits claimed for physical education and school sport: An academic review. Res. Pap. Educ. 24(1), 1–27 (2009).
Kirk, D. Educational Value and Models-Based Practice in Physical Education. Educ. Philos. Theory 45(9), 973–986. https://doi.org/10.1080/00131857.2013.785352 (2013).
Tomporowski, P. D., Davis, C. L., Miller, P. H. & Naglieri, J. A. Exercise and children’s intelligence, cognition, and academic achievement. Educ. Psychol. Rev. 20, 111–131 (2008).
Eime, R. M., Young, J. A., Harvey, J. T., Charity, M. J. & Payne, W. R. A systematic review of the psychological and social benefits of participation in sport for children and adolescents: Informing development of a conceptual model of health through sport. Int. J. Behav. Nutr. Phys. Act. 10, 1–21 (2013).
Kahraman, C. & Alkan, N. Circular intuitionistic fuzzy TOPSIS method with vague membership functions: Supplier selection application context. Notes Intuitionistic Fuzzy Sets 27(1), 24–52 (2021).
Senapati, T., Chen, G. & Yager, R. R. Aczel–Alsina aggregation operators and their application to intuitionistic fuzzy multiple attribute decision making. Int. J. Intell. Syst. 37(2), 1529–1551. https://doi.org/10.1002/int.22684 (2022).
Aczél, J. & Alsina, C. Characterizations of some classes of quasilinear functions with applications to triangular norms and to synthesizing judgements. Aeq. Math. 25(1), 313–315 (1982).
Mahmood, T., Ullah, K., Khan, Q. & Jan, N. An approach toward decision-making and medical diagnosis problems using the concept of spherical fuzzy sets. Neural Comput. & Appl. 31(11), 7041–7053. https://doi.org/10.1007/s00521-018-3521-2 (2019).
Yusoff, B., Kilicman, A., Pratama, D. & Hasni, R. Circular q-rung orthopair fuzzy set and its algebraic properties. Malays. J. Math. Sci. 17(3), 363–378 (2003).
Zeshui, Xu. Intuitionistic Fuzzy Aggregation Operators. IEEE Trans. Fuzzy Syst. 15(6), 1179–1187. https://doi.org/10.1109/TFUZZ.2006.890678 (2007).
R. R. Yager, ‘Pythagorean fuzzy subsets’, in 2013 joint IFSA world congress and NAFIPS annual meeting (IFSA/NAFIPS), IEEE, pp. 57–61. (2013)
Yager, R. R. Generalized orthopair fuzzy sets. IEEE Trans. Fuzzy Syst. 25(5), 1222–1230 (2016).
Li, F. & Wang, J. Analysis of an HIV infection model with logistic target-cell growth and cell-to-cell transmission. Chaos, Solitons Fractals 81, 136–145 (2015).
Li, F., Yu, P., Tian, Y. & Liu, Y. Center and isochronous center conditions for switching systems associated with elementary singular points. Commun. Nonlinear Sci. Numer. Simul. 28(1–3), 81–97 (2015).
Li, F., Liu, Y., Liu, Y. & Yu, P. Bi-center problem and bifurcation of limit cycles from nilpotent singular points in Z2-equivariant cubic vector fields. J. Differ. Equ. 265(10), 4965–4992 (2018).
Li, F., Yu, P., Liu, Y. & Liu, Y. Centers and isochronous centers of a class of quasi-analytic switching systems. Sci. china Math. 61, 1201–1218 (2018).
Li, F., Chen, T., Liu, Y. & Yu, P. A complete classification on the center-focus problem of a generalized cubic Kukles system with a nilpotent singular point. Qual. Theory Dyn. Syst. 23(1), 8 (2024).
Chen, T., Li, F. & Yu, P. Nilpotent center conditions in cubic switching polynomial Liénard systems by higher-order analysis. J. Differ. Equ. 379, 258–289 (2024).
Li, F., Liu, Y., Liu, Y. & Yu, P. Complex isochronous centers and linearization transformations for cubic Z2-equivariant planar systems. J. Differ. Equ. 268(7), 3819–3847 (2020).
Li, F., Jin, Y., Tian, Y. & Yu, P. Integrability and linearizability of cubic Z2 systems with non-resonant singular points. J. Differ. Equ. 269(10), 9026–9049 (2020).
Li, F., Liu, Y., Yu, P. & Wang, J. Complex integrability and linearizability of cubic Z2-equivariant systems with two 1: q resonant singular points. J. Differ. Equ. 300, 786–813 (2021).
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Ren, F., Ren, C. Optimizing physical education strategies through circular intuitionistic Fuzzy Bonferroni based school policy formulation. Sci Rep 15, 21616 (2025). https://doi.org/10.1038/s41598-025-03363-3
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DOI: https://doi.org/10.1038/s41598-025-03363-3
