Financial management policies exist as strategic architecture to facilitate organizations and governments in their efforts to obtain the best possible benefit from financial resources, minimize risk, and attain sustainable growth. These policies cover various decisions of investment planning, capital structure, liquidity, and risk assessments. The establishment of a well-structured financial policy would allow an organization to achieve financial stability and foster profitability while maintaining resilience to uncertainties of the economy. Moreover, an effective financial management system balances acute requirements for liquidity with long-term policies for mutual fund investments so that resources can be appropriately assigned toward the achievement of organizational objectives and the minimization of financial risk. Financial risk assessment and mitigation are other key aspects of financial management policies. Financial risks can hamper an organization in areas where market conditions are volatile and fluctuate, interest rates sway up and down, or there remain looming credit risks. Such afflictions, therefore, impose a lot of relevance in the contexts of financial encroachments upon existing chances. MODM has been integrated into the financial decision-making process, so that investment alternatives can be considered, with financial stability and risk management mechanisms being a basis upon which such considerations can be resolved. Because of methods that may incorporate fuzzy logic and uncertainty modeling of such types as picture fuzzy sets (PFSs), decision-makers identify managing the determined connections of financial data and ambiguity in the market more effectively. Consequently, financial policies become improved through the efficient employment of advanced decision algorithms in a manner that enhances economic performance, strengthens support for strategic planning, and improves investment decisions.

In the fast-changing financial landscape of today, organizations are being pushed more and more to make high-stakes decisions in the face of uncertainty, imprecision, and missing information. Conventional decision-making models frequently neglect the intricacies of financial systems, especially when expert opinion is based on doubt or conflicting judgments. Practical situations like corporate investment choices during economic crises or policy assessment in times of crises like the COVID-19 pandemic express the necessity for more stringent models that can deal with ambiguity and interdependence between multiple criteria. Notwithstanding increasing studies on ]MCDM methods, there is still a large gap in approaches that manage effectively the subtleties of uncertainty and expert reluctance in financial policy assessment. To fill this gap, in this research, a new C-PFHM operator is proposed, which captures positive, neutral, and negative expert views in a more mathematically coherent and flexible way. This is intended to enhance the accuracy and reliability of financial policy analysis under complex decision-making situations.

Decision-making is an integral and basic part of our daily life since it allows us to realize the problems from a variety of views and successfully tackle them. DM is a technique of choosing the most apt and best possible alternative out of the various possibilities against the numerous criteria to achieve given goals and objectives. Thus, when we deal with decisions involving many criteria that must be weighted, MCDM helps us methodically examine these alternatives and pick the best alternative since it enables the decision-maker to assign weight to such criteria and order the alternatives on the basis of them. For the MCDM problem, traditionally two methods have been followed, i.e., conventional methods such as TOPSIS1, VIKOR2, ELECTRE3, etc., and the other is AO. The former ones only provide us with the Ranking of alternatives. However, the latter ones are more effective and efficient to process the information pertaining to the alternatives and provide us with full information pertaining to other options and ranking values. In the DM process, these methods cannot handle ambiguity and imprecision in linguistic terms and need numerical values. Therefore, keeping this in mind, Zadeh4 proposed the idea of fuzzy set (FS), which only represents the elements by its membership value μ and its complement, i.e., non-membership ν. But the limitation involved in the sets is not our requirement that includes MCDM, Thus, an interval-valued fuzzy set (IVFS)5, an extension of FS that gives greater detail of \(\mu\) over the interval, has been proposed. Next, to address the hesitancy value, Atanassov6 proposed IFS, which, in comparison to classical FS, gives a more sophisticated explanation of ambiguity and uncertainty. In many real-world situations, decision-makers not only had to handle uncertain data but also struggled to make precise decisions. A structure to better represent this vagueness is provided by IFSs, and through them, decision-makers can better specify the degree of membership (DOM) and non-membership (DONM) and refusal associated with each element. We can utilize classical FS to improve AI-based decision-making and for hard transportation problems7,8. IFS offers a wider view of uncertainty, which allows more accurate modeling of complicated decision-making situations, and it is further extended to an interval-valued intuitionistic fuzzy set (IVIFS)9,10. Further research on FS and IFSs can be observed from10,11,12,13. Furthermore, this idea was further extended to Pythagorean FS (PyFS)14 by Yager, picture fuzzy set (PFS)15 by Cuong, and q-rung FS (q-RFS)16 by Yager. picture fuzzy sets (PFSs), introduced as a generalization of Intuitionistic Fuzzy Sets (IFSs), provide a more flexible approach to handling uncertainty and vagueness occurring in decision-making issues. PFSs incorporated another level, referred to as neutrality, for hesitation or indifference to a specific evaluation. A PFS is characterized on the basis of three membership functions: DOM (\(\mu\)), DONM (\(v\)), and degree of Abstinence (\(\pi\)), so that \(0\le \mu +

u +\pi \le 1\). This extra flexibility proves very useful in intricate decision-making issues involving finance, medical usage, and risk analysis, where specialists are likely to disagree partially or hesitate on a certain criterion. The capability to express a broad range of opinions also further increases the usability of PFSs in real-world applications with a view to providing reliable decision results. Later on, a neutrosophic set (NS)17 was introduced, which addresses vagueness, imprecision, and indeterminacy more thoroughly.

To solve the particular situations where DoM includes circular attributes, Atanassov18 generalized the IFS and proposed the theory of circular intuitionistic fuzzy sets (C-IFS), which enables a better description of circular phenomena. C-IFS was symbolized by a circle with radius r, whose μ and ν degrees are centered. Similar to IFS, C-IFS is also engaged in MCDM issues throughout the discipline. Cakır18 established the DM application in C-IFS environment, Irem et al.19 established the C-IFS by applying AHP and VIKOR methodology for supplier selection problem, Alkan et al.20 established the C-IFS TOPSIS method and its application in pandemic hospital location selection, Otay et al.21 established the interval-valued C-IFS and its application in digital transformation, Cakır et al.22 introduced the C-IFS and its application in Covid-19 medical waste landfill site evaluation, Garg et al.23 extended EDAS Method with C-IFS and its application in MCDM.

Literature review

In the recent past, decision-making in uncertain financial settings has been studied using fuzzy and hybrid models extensively. For example, introduced hybrid risk assessment techniques integrating FS with failure mode analysis have emerged as eminent for risk prioritization in intricate projects. As an example, the work of Karamoozian and Wu24,25 introduced a hybrid risk prioritization model in construction projects through failure mode and effective analysis that proved more accurate in identifying pivotal risk factors with uncertainty during covid 19. DM methodology was used to choose the best industrial housing building system in Tehran, showing the efficiency of multi-criteria methods for complicated project situations26 and the risk assessment of renewable energy projects and occupational safety in construction project gets using a novel hybrid fuzzy approach27,28 . Though these models enhance the handling of data in more complicated settings, they cannot always maintain interrelationships between several criteria. In addition, most methods use traditional aggregation operators that do not necessarily capture expert opinion hesitation and neutrality. A new fuzzy decision-making method was utilized for green supplier selection for the construction sector, demonstrating the significance of sustainable factors in the evaluation of suppliers when faced with uncertainty29 by Karamoozian et al. More work on Risk Assesment can be seen here30,31, The Table 1 below outlines recent contributions and emphasizes limitations overcome in our research.

Table 1 An overview of related works Emphasising the Research Gaps, Limitations, Application Areas, and Methods the Proposed Study Addresses. Full size table

Recent research has developed decision-making methods for financial risk and performance assessment based on fuzzy and hybrid models. For example32, proposed a hybrid MEREC-RAFSI model based on spherical fuzzy numbers for the determination of banking financial assistance receivers to demonstrate the ease of evolving fuzzy systems in critical financial decisions. The authors of33 explored the effect of financial decision-making authority on the risk-taking behavior of individuals, focusing on the behavioral aspects of financial analysis. An extensive risk analysis process for decentralized finance (DeFi) was put forward in34, unifying multi-criteria decision-making in a blockchain environment. In the same way35, offered an upgraded decision support system through the union of fuzzy logic and machine learning, proving to be more robust in risk assessment. Lastly36, conducted a comparative financial performance analysis using different MCDM methods, further showcasing the benefits of incorporating fuzzy-based models into financial environments. These studies confirm the increasing significance of hybrid fuzzy models but identify a gap in methodology that can cope with picture fuzzy data and interdependent criteria exactly the concern of this research.

Motivation and objectives

In modern financial settings, administrations are challenged with greater complexity and uncertainty in evaluating management policies. Conventional decision-making techniques frequently fail to manage imprecise, vague, or conflicting information. This becomes more critical in risk-sensitive fields like investment analysis, liquidity management, and market volatility evaluation. In addition, due to the emergence of more refined fuzzy models as well as complex aggregation methods, it is now increasingly necessary to create decision tools that are more capable of portraying real-world ambiguity. Inspired by these challenges, this research advances a solid decision algorithm founded upon the PFHM operator in an attempt to improve financial policy assessments of uncertainty.

This work presents the hybrid architecture of C-PFHM, which solves DM problems with circular features while considering the viability and effectiveness of AO. Furthermore, it has proven its theorems and properties for the circular Picture fuzzy weighted heronian mean (C-PFWHM), circular Picture fuzzy geometric heronian mean (C-PFGHM), and circular Picture fuzzy weighted geometric heronian mean (C-PFWGHM). Furthermore, as the literature study explains, we need an air purifier to clean the air because of the rise in airborne contaminants in the environment, which significantly affects the air quality index (AQI). A practical situation was then described based on the C-IFHM and how it operates.

The study has the following key objectives:

1. To devise a decision-making model based on CPFS for evaluating financial policies under uncertain conditions. 2. To examine financial policies against important criteria such as rate of return, liquidity, market volatility, and risk management towards an all-encompassing financial assessment. 3. To clearly illustrate the advantages of CPFS over classical fuzzy models, in particular, its superior treatment of neutrality and hesitation. 4. To explore the applications of an MCDM approach (e.g., CPFS-TOPSIS, CPFS-WASPAS, or CIFHM) to rank financial policies with comparative analysis for validation. 5. To furnish a structured, reliable, and flexible decision support tool for investment planning, policy formulation, and risk assessment for financial institutions and policymakers.

This will act to stimulate developments in financial decision-making models by these objectives, thus assisting strategic planning and risk management amid complex and uncertain financial realms.

Justification of using picture fuzzy sets

PFS further develops IFS by introducing one more dimension-neural (hesitant) membership-along with things normal DoM and DoNM. In any decision-making under real-world situations particularly in environments that are complex like evaluation of air purifiers, financial management, or risk assessment, the decision-makers often face situations in which either neutrality or hesitation, for instance, becomes an important factor. PFS gives a structured approach to represent those uncertain opinions more suitably. Alongside conventional FSs and IFS, which employ the theory of two degrees (memberships and non-memberships), PFS introduces a further “neutral” response, thereby offering greater flexibility in cases where stakeholders or experts may neither fully support nor completely reject an option. This is an exceptionally powerful feature for cases involving environmental assessment, medical diagnosis, or financial policy evaluations, in which experts may have mixed opinions. The key condition in PFS, the sum of the square of the triplets should be between \(0\) and \(1\), serves to ensure that the total uncertainty of the element does not exceed a logical boundary. This helps to reduce inconsistencies in decision-making and ensures better balancing of aggregation of expert opinions. In many real decision-making problems, there are multiple conflicting criteria. PFS offers a good instrument to handle subjective judgments in methods such as TOPSIS, WASPAS, CoCoSo, and aggregation operators. With such a tool, better rankings and evaluations of alternatives can be made in case studies ranging from air quality management to financial policy assessment, healthcare decisions, and industrial optimization. Thus, the application of PFSs is justified because it can model more effectively the uncertainties, neutrality, and expert hesitation in contrast to traditional fuzzy models, making it a potent tool for decision-making under fuzzy and imprecise conditions.

Key contributions of the study

The key contributions of this paper are outlined as follows:

1. Formulation of a new decision-making algorithm under a C-PF setting, which provides for greater representation of uncertainty, hesitancy, and indeterminacy in expert ratings about financial management policies. 2. Introduction of a novel aggregation operator, the C-PFHM, which efficiently captures the interdependencies between decision criteria and enhances the credibility of aggregated expert opinions. 3. Development of an exhaustive computational framework consisting of expert data gathering, normalization, fuzzy aggregation, defuzzification, and ranking of alternatives for financial policy assessment. 4. Implementation of a real-life case study of advanced financial management, proving the practical efficiency and flexibility of the suggested algorithm. 5. Validation by comparative assessment, employing the weighted Spearman’s rank correlation coefficient and confusion matrix visualization, proving the reliability and strength of the suggested approach compared to the existing MCDM methods.

Organization of study

The organization of the paper is as follows: Section “Preliminaries” addresses the basic properties of Complex Pythagorean Fuzzy Sets (C-PFS) and their operations, the heronian mean (HM) and its properties, and the generalized heronian mean (GHM) and its related properties. Section “Proposed aggregation operators” introduces a hybrid framework with the C-PFHM, C-PFWHM, C-PFGHM, and C-PFWGHM operators and their properties and supporting theorems. In Section “Methodology”, an approach based on C-PFS is presented to solve MCDM problems. Section “Case study (decision algorithm with fuzzy framework and evaluation of advanced financial management policy)” presents a practical application in the context of a case study on evaluation of a complex financial management policy. Later, in Section “Advantages and disadvantages”, a numerical example is presented to justify the proposed approach, and an analytical comparison between the proposed and current methods is made. Lastly, Section “Conclusion” concludes the research by providing an overview of the major benefits, limitations, and suggesting possible avenues for future research. A detailed list of the symbols, variables, and indices employed throughout the paper and their descriptions is given in Table 2.