Circular Intuitionistic Fuzzy Median Ranking Model with a Novel Scoring Mechanism for Multiple Criteria Decision Analytics

ABSTRACT

This study aims to pioneer an innovative circular intuitionistic fuzzy (C-IF) scoring-mediated median ranking model designed for multiple criteria decision analytics. The primary goal is to establish a comprehensive precedence ranking for competing alternatives, effectively addressing the inherent uncertainties present in decision-analytic challenges within the C-IF environment. The core content delves into the creation of an original scoring mechanism tailored to navigate the complexities of C-IF uncertainties. Moreover, the research introduces a specialized C-IF median ranking model for decision analytics, leveraging the foundational concept of the C-IF scoring mechanism. A significant contribution is made through the formulation of a robust implementation procedure, specifically tailored for the seamless operation of the C-IF scoring-mediated median ranking model within the framework of C-IF information. Drawing from the suggested C-IF scoring mechanism, this research introduces novel concepts related to comprehensive C-IF scoring functions and comprehensive disagreement metrics. Subsequently, a comprehensive disagreement matrix is formulated, with its entries quantifying the extent of disagreement in assigning specific ranks to each alternative across all criterion-wise precedence relationships. This paves the way for the development of a new C-IF scoring-mediated median ranking model, offering decision analysts a tool to navigate intricate C-IF information and derive dependable decision-analytic outcomes.

Introduction

Fuzzy set theory holds significance in the domain of artificial intelligence (AI), particularly playing a crucial role in intelligent decision support systems (Kahraman et al. 2023; Mert 2023). The conceptual framework of fuzzy sets, built upon the principles of fuzzy logic, is specifically designed to tackle the challenges associated with uncertainty and vagueness in down-to-earth circumstances; these challenges often prove difficult to address using traditional precise mathematics and logic (Kokkinos et al. 2024; Zhang et al. 2024). In real-world scenarios, information is often fuzzy, not just binary true or false. Fuzzy set theory empowers decision analysts to represent and manage blurred information, thereby incorporating uncertainty into the decision-analytic process, which is vital for intelligent decision support systems (Bouraima et al. 2024; Mahmood and Rehman 2023). Fuzzy set theory provides an effective reasoning approach to handle fuzzy rules and facts, enabling systems to make decisions when faced with incomplete or ambiguous information and better simulate human decision-making processes (Kahraman et al. 2023; Yin et al. 2023). Fuzzy sets can be used to model complex real-world situations, such as subjective evaluation, satisfaction assessment, organizational performance, risk analysis, safe and sustainable management, strategic location analysis, and more (Kokkinos et al. 2024; Önden et al. 2024; Wu and Monfort 2023), which aids intelligent decision support systems in understanding and addressing these situations more effectively (Bouraima et al. 2024; Mert 2023). However, as societal challenges become increasingly intricate and scientific research delves deeper, the conventional framework of fuzzy set theory encounters growing limitations, complexities, and practical challenges (Mahmood and Rehman 2023; Yin et al. 2023). These issues stem from the fundamental incapacity of the framework to comprehensively encapsulate all pertinent uncertain information associated with the problems under investigation (Kahraman et al. 2023; Zhang et al. 2024). In response to these constraints, Atanassov (1986) spearheaded a significant advancement of classical fuzzy set theory by introducing the groundbreaking conception of intuitionistic fuzzy configurations.

As a higher-ordered evolution of conventional fuzzy counterparts, intuitionistic fuzzy sets encompass a three-fold structure comprising the membership constituent, the non-membership constituents, and the hesitation or intuition degree function (Patel, Jana, and Mahanta 2023). This tripartite framework collectively delineates the intricate relationship between elements and the set (Balaji et al. 2023; Gogoi, Gohain, and Chutia 2023). More precisely, an element’s membership function quantifies how much it belongs to the set, whereas an element’s non-membership function measures how much it does not (Senapati et al. 2023). Furthermore, the hesitation function (i.e., indeterminacy) adds an essential dimension of intuitive uncertainty, which is positioned between the membership and non-membership functions (Acharya et al. 2024; Patel, Jana, and Mahanta 2023). This innovative approach enables a finer-grained representation of decision-related information, simplifying the accurate modeling of uncertainty and vagueness within multiplex problem domains (He, Dong, and Hu 2023). Consequently, intuitionistic fuzzy sets have garnered widespread practical applications and are extensively utilized across various domains, such as phosphorus levels in lake water sediments (Acharya et al. 2024), cancer detection systems (Balaji et al. 2023), hydraulic drive systems (He, Dong, and Hu 2023), supplier selection (Liu, Lin, and Xu 2023), landslide treatment (Liu et al. 2024), transportation problems (Nishad and Abhishekh 2023), and offshore wind farm location (Önden et al. 2024). This increased utilization is attributed to their enhanced capability to address the multifaceted challenges presented by real-world scenarios. This advancement has empowered decision-makers, researchers, and practitioners, equipping them with a versatile means to manipulate the intricate complexities of contemporary issues and decision-making processes (Gogoi, Gohain, and Chutia 2023; Liu et al. 2024).

Expanding upon the intuitionistic fuzzy framework, Atanassov (2020) introduced a groundbreaking extension known as circular intuitionistic fuzzy (C-IF) sets. More explicitly, C-IF sets encompass an adaptable circular structure that enhances the intuitionistic fuzzy configuration through the incorporation of three fundamental functions: the membership constituent, the non-membership constituent, and the hesitation or intuition degree function, along with an innovative built-in circular function (Wang and Chen 2024; Yusoff et al. 2023). These interconnected functions collectively illuminate the intricate relationship between elements and the set under consideration (Ashraf et al. 2024; Çakır and Taş 2023). In particular, the radii related to the built-in circular functions affect the membership function, which counts the extent to which an element is part of the set, as well as the non-membership function, which quantifies the extent to which an element is not part of the set (Alreshidi, Shah, and Khan 2024; Çaloğlu Büyükselçuk and Sarı 2023). Furthermore, the radii of the inbuilt circular functions impact the hesitation function, which quantifies the level of hesitation or intuitive uncertainty oscillating between the domains of membership and non-membership (Chen 2023a; Otay et al. 2023). In essence, C-IF sets meticulously incorporate and synthesize the circular-structured dimensions of membership, non-membership, and hesitation, providing decision analysts with a significantly more nuanced and comprehensive information set (Fetanat and Tayebi 2023; Pratama, Yusoff, and Abdullah 2024).

What distinguishes the C-IF framework from the intuitionistic fuzzy counterpart is its ability to simultaneously incorporate circular-structured membership, non-membership, and hesitation parameters, along with its enhanced adjustability and conceivability in navigating the realms of ambiguity and equivocation (Wang and Chen 2024). This multi-dimensional approach with inbuilt circular functions improves the provision of more comprehensive and nuanced information compared to conventional intuitionistic fuzzy sets (Xu and Wen 2023). It equips decision analysts to represent decision-relevant data in finer detail (Alsattar et al. 2023). This adaptive circular structure further empowers them to accurately capture and model the intricacies of uncertainty and vagueness within complex problems (Pratama, Yusoff, and Abdullah 2024). In particular, the utility of C-IF sets has aided in the development of significant research techniques related to multiple criteria decision modeling (Yusoff et al. 2023). These methodologies include the extended analytic hierarchy process (AHP) in C-IF settings (Çakır and Taş 2022; Otay and Kahraman 2022; Otay et al. 2023), C-IF evaluation methods based on distances from average solutions (EDAS) (Chen 2023b; Garg et al. 2023), C-IF TODIM (an acronym in Portuguese for the “interactive approach for multiple criteria decision making”) (Ashraf et al. 2023), C-IF techniques for order preference by similarity to ideal solutions (TOPSIS) (Alkan and Kahraman 2022; Chen 2023c; Kahraman and Alkan 2021), C-IF inferior ratio techniques predicated on similarity and entropy measurements (Alreshidi, Shah, and Khan 2024), C-IF VlseKriterijumska Optimizacija I Kompromisno Resenje (VIKOR) methods (Çaloğlu Büyükselçuk and Sarı 2023; Chen 2023a; Kahraman and Irem 2022; Khan, Kumam, and Alreshidi 2022), C-IF elimination and choice expressing reality (ELECTRE) III model (Yusoff et al. 2023), C-IF combinative distance-based assessment (CODAS) approach (Fetanat and Tayebi 2023), and the C-IF time series forecasting technique (Ashraf et al. 2024). Over and above that, the versatility of C-IF sets extends across a wide spectrum of practical applications, such as digital transformation (Otay et al. 2023), epidemic hospital site selection (Alreshidi, Shah, and Khan 2024; Chen 2023c), healthcare waste disposal management (Garg et al. 2023), material evaluation (Ashraf et al. 2023), stock change index prediction (Ashraf et al. 2024), supplier assessment (Çakır and Taş 2023), sustainable water resources management (Fetanat and Tayebi 2023), and whey protein supplement selection (Çaloğlu Büyükselçuk and Sarı 2023). C-IF theory enhances decision modeling, especially in intelligent decision support systems, by accommodating uncertainty and bolstering confidence, thereby improving the resolution of intricate decision-analytic challenges.

Research Gaps and Study Motivations

In the broader landscape of decision analysis and applications of C-IF sets, a notable research gap demands both attention and investigation. This gap specifically relates to the creation of decision analysis methods tailored for C-IF environments, with a primary focus on implementing the median ranking methodology. Despite the considerable body of work that has enriched the field of C-IF-based decision analysis, there exists a conspicuous void concerning C-IF decision modeling anchored in the concept of a median ranking model. The median ranking model, introduced by Yoon and Hwang (1995) and based on the distance measure between rankings devised by Cook and Seiford (1978), offers a potent and widely recognized approach for handling situations involving multiple criteria assessment and judgment (Zhang et al. 2018). It excels in its ability to handle both quantitative and rank (ordinal) data, including qualitative data (Zhou and Chen 2020). Yoon and Hwang’s model essentially seeks to identify a median ranking that best aligns with the set of criterion-wise rankings, providing a structured framework for evaluating and ranking alternatives based on predefined criteria. This feature makes it an advantageous means for tackling intricate decision issues (Cui, Yuwen, and Jiang 2021). However, within the context of C-IF environments, characterized by their inherent circular-structured membership, non-membership, and hesitation parameters, there is an absence of a decision analysis methodology grounded in the median ranking model that can effectively navigate the complexities and subtleties intrinsic to C-IF sets. Consequently, this research gap stands as the foundational motivation behind the current study, signifying an opportunity to address and contribute to the advancement of decision analysis methodologies tailored for C-IF environments, particularly those rooted in the median ranking model.

On the flip side, real-life decision scenarios often exhibit criterion dependencies, encompassing aspects like value dependency and preferential dependency (Chen 2022; Wang and Chen 2017). These dependencies can complicate the determination of the overall performance of competing alternatives, sometimes manifesting in quasi-additive or nonlinear formats (Wang and Chen 2017). To tackle this complexity, the median ranking model can offer an elegant and intuitive solution. The distinctive feature of the median ranking model lies in its ability to circumvent the need for synthesizing performance values across multiple (even conflicting) criteria. This distinctive trait makes it particularly adept at mitigating concerns associated with criterion dependencies. The median ranking methodology utilizes a novel approach to tackle challenges in decision evaluation and judgment. More specifically, it strives to produce a ranking that minimizes discrepancies compared to individual rankings based on each criterion. This resulting ranking can be considered as achieving a consensus or the optimal compromise from the various constituent rankings. Utilizing the median ranking model furnishes an effective means to bypass complex issues, including criterion dependencies, thus presenting the second motivation for this paper.

In the practical context of decision-making involving C-IF information, scoring functions serve as fundamental tools for comparing and organizing C-IF performance ratings to establish a prioritization structure. Previous research has introduced various scoring functions tailored for the C-IF state of affairs, as exemplified by the relevant works of Çakır and Taş (2023), Cakir, Ali Tas, and Ulukan (2021, 2022), Chen (2023a), Garg et al. (2023), Kahraman and Irem (2022), and Xu and Wen (2023). However, despite these advancements, they come with notable limitations and practical challenges. The first critical issue revolves around the imperative to enhance the depth and comprehensiveness of C-IF scoring functions. While innovative methods have been proposed, they fall short of fully encompassing the intricate spectrum of uncertainty within C-IF data, as seen in the approaches of Çakır and Taş (2023), Cakir, Ali Tas, and Ulukan (2021, 2022), and Xu and Wen (2023). The second critical issue pertains to circumstances in which the radii corresponding to C-IF numbers equal zero – a scenario not effectively addressed by current approaches propounded by Kahraman and Irem (2022) along with Otay and Kahraman (2022). Notably, the inability to handle this case restricts dimensionality reduction within the intuitionistic fuzzy context, emphasizing a pressing need to develop solutions for C-IF data with zero radii. The third critical issue revolves around instability and data dependency in both a relative C-IF scoring mechanism revealed in Kahraman and Irem (2022) and Otay and Kahraman (2022), as well as in an enhanced C-IF scoring mechanism initiated by Chen (2023a). These methods rely on specific C-IF datasets, leading to result variability and affecting the decision-maker’s confidence. In summary, these critical issues underscore the demand for more comprehensive and effective C-IF scoring functions capable of better handling the complexities of uncertainty and decision-making in real-world scenarios. Therefore, addressing these critical concerns constitutes the third research motivation.

In summary, this study emphasizes three key research motivations that stem from the research gaps previously discussed:

Absence of C-IF Median Ranking Methodology Development

One conspicuous research gap in the realm of multiple criteria assessment within C-IF contexts is the dearth of decision analysis methods designed for C-IF sets utilizing the median ranking model. While numerous valuable C-IF decision-analytic approaches have been developed previously, the creation of decision-analytic methods customized for the C-IF environment and utilizing the median ranking model has been noticeably absent. This gap serves as the central impetus behind the present study.

Complexity of Criterion Dependencies in Decision Analysis

The issue of criterion dependencies may manifest in quasi-additive or nonlinear formats, creating complexity and challenges when assessing the overall performance of competing alternatives. There is a gap in research focusing on harnessing the power of the median ranking model to mitigate these criterion dependencies. The median ranking model provides an elegant and intuitive solution that minimizes the need for synthesizing performance values across multiple, potentially conflicting criteria. This unique trait makes it well-suited for addressing concerns related to criterion dependencies. By circumventing these intricate challenges, this study seeks to bridge the current research gap and offer a fresh approach to conquering criterion dependencies through the utilization of the median ranking model. This gap serves as the second impetus behind the present study.

Enhancement of C-IF Scoring Functions for Addressing Critical Limitations

In the decision-analytic realm involving circular intuitionistic fuzziness, score functions play a pivotal role in comparing and prioritizing C-IF data. Previous research has introduced several scoring functions tailored to C-IF, but they come with limitations and practical issues. This study identifies three key critical issues resulting from these limitations, including enhancing comprehensiveness of C-IF scoring functions, addressing the challenge of dealing with situations where the C-IF radius is zero, and handling the instability and dependence on specific C-IF datasets within relative and enhanced C-IF scoring mechanisms. This gap serves as the third impetus behind the present study.

Research Objectives and Contributions

The ability to manage compounded uncertainties is crucial toward the efficacy of intelligent decision support systems, and the introduction of C-IF sets contributes to enhancing the capabilities of such systems in navigating and processing complex information. The core objective of this paper is to construct a groundbreaking C-IF scoring-mediated median ranking model specifically tailored for multiple criteria decision analytics in complicated and multifaceted scenarios. In order to accommodate the inherent complexities and uncertainties involved in decision-analytic situations, this innovative model seeks to demonstrate a thorough framework for constructing precedence rankings among competing alternatives. The focus of the research is on developing a robust and dependable methodology that leverages an original C-IF scoring mechanism to enhance decision-making processes in situations characterized by multiple evaluative criteria. The envisioned model aims to contribute significantly to the field of decision analytics by offering a systematic and efficacious approach to handling C-IF information, ultimately facilitating more informed and precise decision outcomes.

This research initiates a novel C-IF scoring function aimed at improving data assessment within the domain of circular intuitionistic fuzziness. The study incorporates concepts related to interval-valued estimates and relative significance parameters for both lower and upper estimates, significantly contributing to the formulation of a valuable C-IF scoring approach. The inclusion of theorems pertaining to interval-valued estimates and the newly proposed C-IF scoring mechanism offers a comprehensive understanding of their properties within the context of C-IF circumstances. The research delves into the development of an efficacious C-IF scoring mechanism, emphasizing its theoretical foundations and presenting innovative ideas and theorems in the field of decision analytics.

This research introduces a distinctive median ranking model mediated by C-IF scoring functions, specifically designed to address uncertainty in decision-analytic problems within the realm of C-IF information. The developed median ranking model endeavors to establish a comprehensive precedence ranking for multiple competing options by integrating a specialized scoring mechanism tailored for C-IF uncertainties. In particular, the model utilizes comprehensive disagreement metrics and an innovative C-IF scoring mechanism to assess performance against predetermined criteria. This approach can significantly assist decision-analytic processes by effectively handling equivocation and imprecision. The recommended approach entails using C-IF sets to formulate the problem, producing interval-valued estimates, and determining precedence ranks that are particular to a certain criterion. A matrix capturing the degree of disagreement in precedence connections among competing alternatives is formed in part by the comprehensive C-IF scoring functions and disagreement metrics. Moreover, the proposed model applies scalar multiplication to compute weighted C-IF performance ratings, resulting in the creation of an all-encompassing C-IF scoring methodology. Subsequently, a C-IF scoring-mediated median ranking model is established, employing linear programming to optimize the comprehensive ranking of alternatives. The methodology is presented in four phases, emphasizing problem formulation, C-IF scoring implementation, derivation of comprehensive scoring functions, and modeling with comprehensive disagreement metrics. This comprehensive approach enhances decision-analytic efficiency within implicit and convoluted environments.

Next, this research provides a concrete illustration to explicate the pragmatic significance of the proposed C-IF scoring-mediated median ranking model. Specifically, the research immerses itself in a distinct instance of multiple-expert supplier appraisement, showcasing the concrete application of the suggested model in a real-world scenario. Additionally, the research incorporates a comprehensive comparative analysis and investigations to underscore the merits and advantages inherent in the proposed methodology, rooted in the innovative C-IF scoring technique. Through real-world exemplification and meticulous comparisons, the present investigation underscores the model’s effectiveness and strengths in tackling intricate decision challenges characterized by the uncertainty and ambiguity encapsulated within the C-IF framework.

In a nutshell, this work underscores three significant research contributions, each aimed at rectifying the identified research gaps in C-IF decision analysis. These contributions are poised to serve as the foundation for both practical applications and theoretical advancements, underlining the importance of this study. Then key contributions encompass:

Creating a New Median Ranking Model Tailored for C-IF Environments

The contribution of this research involves the development of a specialized median ranking model designed to operate within the context of C-IF information. This unique model is meticulously crafted to address the distinct challenges and complexities associated with C-IF environments, where circular-structured membership, non-membership, and hesitation parameters play a pivotal role. One of the model’s key strengths is its capacity to efficaciously generate a comprehensive (or consensus) ranking, specifically minimizing disparities among individual rankings based on each criterion. This streamlining effect greatly enhances the decision-making process within C-IF settings, elevating its overall precision. This contribution is particularly significant as it fills a critical void in existing literature. Prior to this research, there was a notable absence of decision-analytic methodologies tailored specifically for C-IF environments employing the median ranking model. Consequently, the model empowers decision-makers and analysts with a potent means to navigate the intricate landscape of decision analytics in scenarios characterized by circular intuitionistic fuzziness. Ultimately, this research advances the cause of more robust and informed decision-making practices.

Mitigating Criterion Dependencies in C-IF Decision Environments for Consensus Generation

This research is driven by the imperative need for practical resolutions to tackle issues about criterion dependencies in complicated and uncertain decision-analytic scenarios. These dependencies, frequently appearing as value and preferential dependencies, pose substantial challenges in evaluating competing alternatives, emphasizing the requirement for more efficacious methodologies. The research contributes to the introduction of a C-IF scoring-mediated median ranking model, which serves as a valuable tool for addressing these intricate issues and providing an intuitive approach to decision support in circumstances involving circular intuitionistic fuzziness. By generating a comprehensive (or consensus) ranking that minimizes disparities among individual rankings for each criterion, this model not only streamlines the decision-aiding process but also enhances its precision. The research endeavors to bridge the existing gap in the literature and provide a novel approach to addressing criterion dependencies in C-IF contexts, thereby contributing to more robust and informed decision-analytic practices.

Innovating a Novel Scoring Mechanism to Enhance Data Assessment in C-IF Contexts

By presenting a groundbreaking C-IF scoring function that is designed to rectify critical restrictions observed in previous C-IF scoring techniques, this research makes a significant contribution within C-IF decision-analytic contexts. This novel scoring function offers an inventive and more thorough approach with regard to C-IF performance rating data. Moreover, it performs exceptionally well in capturing the complex range of uncertainties in C-IF information, effectively managing scenarios where the C-IF radius is zero, and mitigating the instability and data reliance that are present in current C-IF scoring mechanisms. The new scoring function is a powerful yet approachable instrument for decision analysts, researchers, and practitioners to undertake accurate and dependable assessments of C-IF performance rating data because of its exceptional simplicity and user-friendliness.

Collectively, the C-IF scoring mechanism plays a pivotal role in the decision-analytic issue involving multiple evaluative criteria, especially in the development of median ranking modeling systems. The incorporation of the C-IF scoring mechanism into the median ranking model contributes to the progress of intelligent decision-making methodologies, enhancing their resilience and ability to manipulate the complexities inherent in real-world scenarios. These contributions underscore the significance of this study, offering the potential for practical applications and theoretical advancements within the field of C-IF decision analytics.

Paper Organization

The study’s overall structure is laid out as follows: Section 2 provides fundamental explanations about C-IF sets to enhance comprehension of the background. Section 3 establishes a groundbreaking C-IF scoring mechanism, delving into its intriguing and valuable properties tailored for enhancing data assessment in situations characterized by C-IF uncertainties. Section 4 formulates a pioneering C-IF scoring-mediated median ranking model to establish a comprehensive precedence ranking for competing alternatives, tackling uncertainties inherent in decision-analytic issues within the C-IF environment. Section 5 explores a practical case study on multiple-expert supplier appraisement, aiming to demonstrate the actual application of the model. This section also conducts a thorough comparative study with a parametric analysis. In the end, Section 6 consolidates the key findings and suggests potential directions for future exploration.

Fundamental Concepts Pertaining to C-IF Sets

Atanassov (2020) initiated C-IF sets to enhance the representation of uncertainty within single-valued intuitionistic fuzzy elements. The C-IF configuration effectively accounts for variations in membership and non-membership constituents, as perceived by decision-making participants, by encapsulating them within a circular region. This expanded approach provides a robust mechanism for quantifying the decision-maker’s indecision and seamlessly integrating it into the decision-analytic process. The circular representation of a single-valued C-IF set encapsulates the inherent imprecision and ambiguity in its membership and non-membership constituents. The specifics of this unique framework are elaborated in the subsequent definitions.

Definition 1.

(Atanassov 1986) Consider $\mathfrak{Z}$ as a predetermined universe. An intuitionistic fuzzy set $\mathcal{J}$ on $\mathfrak{Z}$ is described as an entity represented by:

(1) $\mathcal{J}=\left\{〈\mathfrak{z},m_{\mathcal{J}}\left(\mathfrak{z}\right),n_{\mathcal{J}}\left(\mathfrak{z}\right)〉|\mathfrak{z}\in \mathit{ℨ}\right\}$(1)

where the functions $m_{\mathcal{J}}\left(\mathfrak{z}\right):\mathit{ℨ}\to \left[0,1\right]$ and $n_{\mathcal{J}}\left(\mathfrak{z}\right):\mathit{ℨ}\to \left[0,1\right]$ correspond to the membership and non-membership degrees of $\mathcal{J}$, respectively. Regarding every $\mathfrak{z}\in \mathfrak{Z}$ within $\mathcal{J}$, it must adhere to the condition $0\le m_{\mathcal{J}}\left(\mathfrak{z}\right)+n_{\mathcal{J}}\left(\mathfrak{z}\right)\le 1$. The equation $h_{\mathcal{J}}\left(\mathfrak{z}\right)=1-m_{\mathcal{J}}\left(\mathfrak{z}\right)-n_{\mathcal{J}}\left(\mathfrak{z}\right)$ calculates the measure of indeterminacy, commonly known as the hesitancy degree, linked to an intuitionistic fuzzy number $I_{\mathcal{J}}\left(\mathfrak{z}\right)=\left(m_{\mathcal{J}}\left(\mathfrak{z}\right),n_{\mathcal{J}}\left(\mathfrak{z}\right)\right)$ within $\mathcal{J}$.

Definition 2.

(Bai 2013; Mishra et al. 2023) Consider $\widetilde{{\mathit{I}}_{\mathcal{I}}}\left(\mathfrak{z}\right)=\left(\left[m_{\mathcal{J}}^{-}\left(\mathfrak{z}\right),m_{\mathcal{J}}^{+}\left(\mathfrak{z}\right)\left],\right[n_{\mathcal{J}}^{-}\left(\mathfrak{z}\right),n_{\mathcal{J}}^{+}\left(\mathfrak{z}\right)\right]\right)$ to be an interval-valued intuitionistic fuzzy number defined over the universe $\mathfrak{Z}$, governed to the constraint $0\le m_{\mathcal{J}}^{+}\left(\mathfrak{z}\right)+n_{\mathcal{J}}^{+}\left(\mathfrak{z}\right)\le 1$. The enhanced scoring function for $\widetilde{{\mathit{I}}_{\mathcal{J}}}\left(\mathfrak{z}\right)$ is calculated using the following approach:

(2) $\mathcal{S}\left(\widetilde{{\mathit{I}}_{\mathcal{J}}}\left(\mathfrak{z}\right)\right)=\frac{m_{\mathcal{J}}^{-}\left(\mathfrak{z}\right)+m_{\mathcal{J}}^{-}\left(\mathfrak{z}\right)\left(1-m_{\mathcal{J}}^{-}\left(\mathfrak{z}\right)-n_{\mathcal{J}}^{-}\left(\mathfrak{z}\right)\right)+m_{\mathcal{J}}^{+}\left(\mathfrak{z}\right)+m_{\mathcal{J}}^{+}\left(\mathfrak{z}\right)\left(1-m_{\mathcal{J}}^{+}\left(\mathfrak{z}\right)-n_{\mathcal{J}}^{+}\left(\mathfrak{z}\right)\right)}{2}$(2)

where $\mathcal{S}\left(\widetilde{{\mathit{I}}_{\mathcal{J}}}\left(\mathfrak{z}\right)\right)\in \left[0,1\right]$.

Definition 3.

(Chen 2023b, 2023b, 2023c) Let $L^{\ast }$ be the set $L^{\ast }=\left\{⟨\mathit{l},l^{\prime }⟩\left|\mathit{l},l^{\prime }\in \left[0,1\right]\mathrm{and}\mathit{l}+l^{\prime }\le 1\right.\right\}$ representing an L-fuzzy set. This arrangement is exploited to define a C-IF set $\mathcal{C}$ within a predefined universe $\mathfrak{Z}$. The C-IF set $\mathcal{C}$ is conceptualized as an entity denoted by:

(3) $\mathcal{C}=\left\{⟨\mathfrak{z},m_{\mathcal{C}}\left(\mathfrak{z}\right),n_{\mathcal{C}}\left(\mathfrak{z}\right);r_{\mathcal{C}}\left(\mathfrak{z}\right)⟩|\mathfrak{z}\in \mathfrak{Z}\right\}=\left\{⟨\mathfrak{z},\text{ti}{O}_R\left(m_{\mathcal{C}}\left(\mathfrak{z}\right),n_{\mathcal{C}}\left(\mathfrak{z}\right)\right)⟩|\mathfrak{z}\in \mathfrak{Z}\right\}$(3)

where the functions $m_{\mathcal{C}}\left(\mathfrak{z}\right):\mathfrak{Z}\to \left[0,1\right]$ and $n_{\mathcal{C}}\left(\mathfrak{z}\right):\mathfrak{Z}\to \left[0,1\right]$ signify the membership and non-membership degrees of $\mathcal{C}$, respectively. Regarding every $\mathfrak{z}\in \mathfrak{Z}$ within $\mathcal{C}$, the condition $0\le m_{\mathcal{C}}\left(\mathfrak{z}\right)+n_{\mathcal{C}}\left(\mathfrak{z}\right)\le 1$ holds true. The inherent circular function $O_R$ associated with $\mathcal{C}$ is expressed with the following formula, representing a circle with a radius $r_{\mathcal{C}}\left(\mathfrak{z}\right)$ over the universe $\mathfrak{Z}$. Here, $r_{\mathcal{C}}\left(\mathfrak{z}\right):\mathfrak{Z}\text{rarrow}\left[0,\sqrt{2}\right]$, and its center is characterized by $\left(m_{\mathcal{C}}\left(\mathfrak{z}\right),n_{\mathcal{C}}\left(\mathfrak{z}\right)\right)$:

$O_R\left(m_{\mathcal{C}}\left(\mathfrak{z}\right),n_{\mathcal{C}}\left(\mathfrak{z}\right)\right)=\left\{〈\mathit{l},l^{\prime }〉|\mathit{l},l^{\prime }\in \left[0,1\right]\text{and}\sqrt{{\left(m_{\mathcal{C}}\left(\mathfrak{z}\right)-\mathit{l}\right)}^2+{\left(n_{\mathcal{C}}\left(\mathfrak{z}\right)-l^{\prime }\right)}^2}\le r_{\mathcal{C}}\left(\mathfrak{z}\right)\right\}\cap L^{*}$

(4) $=\left\{\mathit{l},l^{\prime }|\mathit{l},l^{\prime }\in \left[0,1\right],\sqrt{{\left({\mathit{m}}_{\mathcal{C}}\left(\mathfrak{z}\right)-\mathit{l}\right)}^2+{\left(\mathit{n}\left(\mathfrak{z}\right)-l^{\prime }\right)}^2}\le r_{\mathcal{C}}\left(\mathfrak{z}\right),\mathrm{and}\mathit{l}+l^{\prime }\le 1\right\}.$(4)

Definition 4.

(Chen 2023b, 2023b, 2023c) Take $C_{\mathcal{C}}\left(\mathfrak{z}\right)$ to be a C-IF number associated with an element $\mathfrak{z}$ belonging to a C-IF set $\mathcal{C}$ defined within the specified universe $\mathfrak{Z}$. The C-IF number $C_{\mathcal{C}}\left(\mathfrak{z}\right)$ and the associated measure of indeterminacy, termed the hesitancy degree $h_{\mathcal{C}}\left(\mathfrak{z}\right)$, can be precisely defined as follows:

(5) $C_{\mathcal{C}}\left(\mathfrak{z}\right)=\left(m_{\mathcal{C}}\left(\mathfrak{z}\right),n_{\mathcal{C}}\left(\mathfrak{z}\right);r_{\mathcal{C}}\left(\mathfrak{z}\right)\right),$(5)

(6) $h_{\mathcal{C}}\left(\mathfrak{z}\right)=1-m_{\mathcal{C}}\left(\mathfrak{z}\right)-n_{\mathcal{C}}\left(\mathfrak{z}\right),$(6)

Let $\mathit{\nu }$ represent a non-negative real value ($\mathit{\nu }\ge 0$). Assuming $C_{\mathcal{C}}\left(\mathfrak{z}\right)\ne \left(1,0;r_{\mathcal{C}}\left(\mathfrak{z}\right)\right)$, without losing generality, the definition of the scalar multiplication operation in C-IF circumstances is as follows:

(7) $\mathit{\nu }\odot C_{\mathcal{C}}\left(\mathfrak{z}\right)=\left(1-{\left(1-{\mathit{m}}_{\mathcal{C}}\left(\mathfrak{z}\right)\right)}^{\mathit{\nu }},{\left({\mathit{n}}_{\mathcal{C}}\left(\mathfrak{z}\right)\right)}^{\mathit{\nu }};r_{\mathcal{C}}\left(\mathfrak{z}\right)\right),$(7)

Figure 1 depicts fundamental geometric interpretations of intuitionistic fuzzy and C-IF numbers on the (m, n) axes. Specifically, Figures 1(a)–(b) illustrate the geometric representations of an intuitionistic fuzzy number $I_{\mathcal{I}}\left(\mathfrak{z}\right)=\left(m_{\mathcal{I}}\left(\mathfrak{z}\right),n_{\mathcal{I}}\left(\mathfrak{z}\right)\right)$ within the set $\mathcal{I}$ and a C-IF number $C_{\mathcal{C}}\left(\mathfrak{z}\right)=\left(m_{\mathcal{C}}\left(\mathfrak{z}\right),n_{\mathcal{C}}\left(\mathfrak{z}\right);r_{\mathcal{C}}\left(\mathfrak{z}\right)\right)$ within the set $\mathcal{C}$, respectively. These two diagrams display triangular regions situated in the first quadrant and bounded by vertices (0, 0), (0, 1), and (1, 0). Figures 1(a)–(b) serve to highlight the disparity between intuitionistic fuzzy and C-IF models. Notably, C-IF numbers are endowed with their own unique circular functions characterized by distinct radii within the C-IF framework. To delve deeper, an intuitionistic fuzzy number finds its representation as a point at coordinates $\left(m_{\mathcal{I}}\left(\mathfrak{z}\right),n_{\mathcal{I}}\left(\mathfrak{z}\right)\right)$ within the intuitionistic fuzzy interpretation triangle. On the other hand, a C-IF number is symbolized by a circle referred to as $O_R$, which has a center at $\left(m_{\mathcal{C}}\left(\mathfrak{z}\right),n_{\mathcal{C}}\left(\mathfrak{z}\right)\right)$, and its radius is designated as $r_{\mathcal{C}}\left(\mathfrak{z}\right)$. The circular function $O_R$, defined by $O_R\left(m_{\mathcal{C}}\left(\mathfrak{z}\right),n_{\mathcal{C}}\left(\mathfrak{z}\right)\right)$ in Equation (4), must satisfy the conditions specified in $L^{\ast }$. As a result, this circular function can take on one of five distinct forms, as depicted in Figure 1(b). In the event that $r_{\mathcal{C}}\left(\mathfrak{z}\right)$ equals zero, the C-IF number $C_{\mathcal{C}}\left(\mathfrak{z}\right)$ effectively reverts to an intuitionistic fuzzy number, $I_{\mathcal{I}}\left(\mathfrak{z}\right)$.

Figure 1. Geometrical visualization of intuitionistic fuzzy and C-IF numbers.

Creating a C-IF Scoring Mechanism to Enhance Data Assessment

The current section’s goal is to put forward a brand-new C-IF scoring function dedicated to enhancing data assessment within the context of circular intuitionistic fuzziness. Initially, Bai (2013) introduced an enhanced scoring function tailored for the comparison and assessment of interval-valued intuitionistic fuzzy information, comprehensively addressing uncertainty and indeterminacy. Recognizing the significance of Bai’s approach, Mishra et al. (2023) further established a normalized score value based on enhanced scoring functions. This section builds upon insights from Bai (2013) and Mishra et al. (2023) to innovate a C-IF scoring mechanism, enriching data evaluation tasks involving circular intuitionistic fuzziness. Additionally, the section introduces concepts related to interval-valued estimates and relative significance parameters for both lower and upper estimates, contributing substantially to the formulation of a valuable C-IF scoring approach. The inclusion of theorems related to interval-valued estimates and the new C-IF scoring mechanism provides a thoroughgoing understanding of their properties in C-IF circumstances. Overall, this section explores the development of an effective C-IF scoring mechanism, highlighting its theoretical underpinnings and taking cues from earlier research by Bai (2013) and Mishra et al. (2023) in addition to presenting new ideas and theorems in the decision-analytic area.

To comprehensively account for uncertainty and extract indeterminacy information, Bai (2013) brought forward an enhanced scoring function, specified in Equation (2) within Definition 2, for the purpose of comparing and assessing interval-valued intuitionistic fuzzy numbers. This scoring technique was subsequently employed in the formulation of an extended TOPSIS approach with interval-valued intuitionistic fuzziness, designed to manipulate uncertain decision-analytic scenarios. Recognizing the value of Bai’s improved scoring function, Mishra et al. (2023) established a normalized score value based on a series of enhanced scoring functions. They also created a weight derivation method using a rank sum model within the interval-valued intuitionistic fuzzy framework. This systematic approach for assessing decision experts’ weights through the use of normalized score values and the rank sum model contributes to a reduction in imprecision and biases within the decision- analytic process. This research leverages the formulation presented in (2) to contrive an innovative C-IF scoring mechanism, thereby improving data evaluation within the domain of circular intuitionistic fuzziness. Moreover, it acknowledges the practical significance and utility of the enhanced scoring function $\mathcal{S}\left(\widetilde{{\mathit{I}}_{\mathcal{I}}}\left(\mathfrak{z}\right)\right)$, as suggested by Bai (2013) and Mishra et al. (2023). Furthermore, by introducing concepts related to interval-valued estimates and relative significance parameters concerning lower and upper estimates, this research creates a fresh C-IF scoring mechanism and formulates a valuable C-IF scoring function to enhance data assessment in scenarios involving C-IF uncertainties.

Definition 5.

Consider a C-IF number $C_{\mathcal{C}}\left(\mathfrak{z}\right)=\left(m_{\mathcal{C}}\left(\mathfrak{z}\right),n_{\mathcal{C}}\left(\mathfrak{z}\right);r_{\mathcal{C}}\left(\mathfrak{z}\right)\right)$ within the C-IF set $\mathcal{C}$. The interval-valued estimates corresponding to the membership degree, $m_{\mathcal{C}}\left(\mathfrak{z}\right)$, and the non-membership degree, $n_{\mathcal{C}}\left(\mathfrak{z}\right)$, can be individually specified with these means:

(8) $\left[m_{\mathcal{C}}^{-}\left(\mathfrak{z}\right),m_{\mathcal{C}}^{+}\left(\mathfrak{z}\right)\right]=\left[max\left\{0,m_{\mathcal{C}}\left(\mathfrak{z}\right)-\frac{{\mathit{r}}_{\mathcal{C}}\left(\mathfrak{z}\right)}{\sqrt{2}}\right\},\min \left\{1,m_{\mathcal{C}}\left(\mathfrak{z}\right)+\frac{{\mathit{r}}_{\mathcal{C}}\left(\mathfrak{z}\right)}{\sqrt{2}}\right\}\right]$(8)

(9) $\left[n_{\mathcal{C}}^{-}\left(\mathfrak{z}\right),n_{\mathcal{C}}^{+}\left(\mathfrak{z}\right)\right]=\left[max\left\{0,n_{\mathcal{C}}\left(\mathfrak{z}\right)-\frac{{\mathit{r}}_{\mathcal{C}}\left(\mathfrak{z}\right)}{\sqrt{2}}\right\},\min \left\{1,n_{\mathcal{C}}\left(\mathfrak{z}\right)+\frac{{\mathit{r}}_{\mathcal{C}}\left(\mathfrak{z}\right)}{\sqrt{2}}\right\}\right]$(9)

The interval-valued estimate associated with $C_{\mathcal{C}}\left(\mathfrak{z}\right)$ is denoted using the following approach:

(10) $\widetilde{{\mathit{I}}_{\mathcal{C}}}\left(\mathfrak{z}\right)=\left(\left[m_{\mathcal{C}}^{-}\left(\mathfrak{z}\right),m_{\mathcal{C}}^{+}\left(\mathfrak{z}\right)\right],\left[n_{\mathcal{C}}^{-}\left(\mathfrak{z}\right),n_{\mathcal{C}}^{+}\left(\mathfrak{z}\right)\right]\right)$(10)

Theorem 1.

Concerning the interval-valued estimates for $m_{\mathcal{C}}\left(\mathfrak{z}\right)$, $n_{\mathcal{C}}\left(\mathfrak{z}\right)$, and $C_{\mathcal{C}}\left(\mathfrak{z}\right)$, the subsequent characteristics are applicable:

$0\le m_{\mathcal{C}}^{-}\left(\mathfrak{z}\right)\le m_{\mathcal{C}}\left(\mathfrak{z}\right)\le m_{\mathcal{C}}^{+}\left(\mathfrak{z}\right)\le 1.$

$0\le n_{\mathcal{C}}^{-}\left(\mathfrak{z}\right)\le n_{\mathcal{C}}\left(\mathfrak{z}\right)\le n_{\mathcal{C}}^{+}\left(\mathfrak{z}\right)\le 1.$

(3) The widths of the two intervals $\left[m_{\mathcal{C}}^{-}\left(\mathfrak{z}\right),m_{\mathcal{C}}^{+}\left(\mathfrak{z}\right)\right]$ and $\left[n_{\mathcal{C}}^{-}\left(\mathfrak{z}\right),n_{\mathcal{C}}^{+}\left(\mathfrak{z}\right)\right]$ contained within $C_{\mathcal{C}}\left(\mathfrak{z}\right)$ are $\sqrt{2}\cdot r_{\mathcal{C}}\left(\mathfrak{z}\right)$, provided that $\widetilde{{\mathit{I}}_{\mathcal{C}}}\left(\mathfrak{z}\right)$ is represented by $\left(\left[m_{\mathcal{C}}\left(\mathfrak{z}\right)-r_{\mathcal{C}}\left(\mathfrak{z}\right)/\sqrt{2},m_{\mathcal{C}}\left(\mathfrak{z}\right)+r_{\mathcal{C}}\left(\mathfrak{z}\right)/\sqrt{2}\left],\right[n_{\mathcal{C}}\left(\mathfrak{z}\right)-r_{\mathcal{C}}\left(\mathfrak{z}\right)/\sqrt{2},n_{\mathcal{C}}\left(\mathfrak{z}\right)+\text{break}{r}_{\mathcal{C}}\left(\mathfrak{z}\right)/\sqrt{2}\right]\right)$.

Proof.

When considering Definition 5, it becomes clear that $m_{\mathcal{C}}^{-}\left(\mathfrak{z}\right)\ge 0$ and $m_{\mathcal{C}}^{+}\left(\mathfrak{z}\right)\le 1$ because, correspondingly, $\max \left\{0,m_{\mathcal{C}}\left(\mathfrak{z}\right)-r_{\mathcal{C}}\left(\mathfrak{z}\right)/\sqrt{2}\right\}\ge 0$ and $\min \left\{1,m_{\mathcal{C}}\left(\mathfrak{z}\right)+r_{\mathcal{C}}\left(\mathfrak{z}\right)/\sqrt{2}\right\}\le 1$. Moreover, it is readily apparent that $\max \left\{0,m_{\mathcal{C}}\left(\mathfrak{z}\right)-r_{\mathcal{C}}\left(\mathfrak{z}\right)/\sqrt{2}\right\}\le m_{\mathcal{C}}\left(\mathfrak{z}\right)\le \min \left\{1,m_{\mathcal{C}}\left(\mathfrak{z}\right)+r_{\mathcal{C}}\left(\mathfrak{z}\right)/\sqrt{2}\right\}$, which leads to $m_{\mathcal{C}}^{-}\left(\mathfrak{z}\right)\le m_{\mathcal{C}}\left(\mathfrak{z}\right)\le m_{\mathcal{C}}^{+}\left(\mathfrak{z}\right)$. Thus, property (1) is valid. Similarly, it is recognized that $n_{\mathcal{C}}^{-}\left(\mathfrak{z}\right)\ge 0$ and $n_{\mathcal{C}}^{+}\left(\mathfrak{z}\right)\le 1$ based on Definition 5. Furthermore, one can illustrate that $n_{\mathcal{C}}^{-}\left(\mathfrak{z}\right)\le n_{\mathcal{C}}\left(\mathfrak{z}\right)\le n_{\mathcal{C}}^{+}\left(\mathfrak{z}\right)$ given that $\max \text{break}\left\{0,n_{\mathcal{C}}\left(\mathfrak{z}\right)-r_{\mathcal{C}}\left(\mathfrak{z}\right)/\sqrt{2}\right\}\le n_{\mathcal{C}}\left(\mathfrak{z}\right)\le \min \left\{1,n_{\mathcal{C}}\left(\mathfrak{z}\right)+r_{\mathcal{C}}\left(\mathfrak{z}\right)/\sqrt{2}\right\}$. Accordingly, the property specified in (2) holds true. Finally, assume that $\text{break}\widetilde{{\mathit{I}}_{\mathcal{C}}}\left(\mathfrak{z}\right)$ is represented by $\left(\left[m_{\mathcal{C}}\left(\mathfrak{z}\right)-r_{\mathcal{C}}\left(\mathfrak{z}\right)/\sqrt{2},m_{\mathcal{C}}\left(\mathfrak{z}\right)+r_{\mathcal{C}}\left(\mathfrak{z}\right)/\sqrt{2}\left],\text{break}\right[n_{\mathcal{C}}\left(\mathfrak{z}\right)-r_{\mathcal{C}}\left(\mathfrak{z}\right)/\sqrt{2},n_{\mathcal{C}}\left(\mathfrak{z}\right)+r_{\mathcal{C}}\left(\mathfrak{z}\right)/\sqrt{2}\right]\right)$. In this case, the differences between $m_{\mathcal{C}}^{-}\left(\mathfrak{z}\right)$ and $m_{\mathcal{C}}^{+}\left(\mathfrak{z}\right)$ and between $n_{\mathcal{C}}^{-}\left(\mathfrak{z}\right)$ and $n_{\mathcal{C}}^{+}\left(\mathfrak{z}\right)$ are fixed, i.e., $r_{\mathcal{C}}\left(\mathfrak{z}\right)/\sqrt{2}-\left(-r_{\mathcal{C}}\left(\mathfrak{z}\right)/\sqrt{2}\right)=\sqrt{2}\cdot r_{\mathcal{C}}\left(\mathfrak{z}\right)$. Consequently, the widths of $\left[m_{\mathcal{C}}^{-}\left(\mathfrak{z}\right),m_{\mathcal{C}}^{+}\left(\mathfrak{z}\right)\right]$ and $\left[n_{\mathcal{C}}^{-}\left(\mathfrak{z}\right),n_{\mathcal{C}}^{+}\left(\mathfrak{z}\right)\right]$ are $\sqrt{2}\cdot r_{\mathcal{C}}\left(\mathfrak{z}\right)$, which confirms the property stated in (3).

Definition 6.

The C-IF scoring function $S^{\rho }\left(C_{\mathcal{C}}\left(\mathfrak{z}\right)\right)$ related to the C-IF number $C_{\mathcal{C}}\left(\mathfrak{z}\right)$ is determined using the following approach, with $\mathit{\rho }$ and $1-\mathit{\rho }$ representing the relative significance parameters for the lower estimate constituents (i.e., $m_{\mathcal{C}}^{-}\left(\mathfrak{z}\right)$ and $n_{\mathcal{C}}^{-}\left(\mathfrak{z}\right)$) and upper estimate constituents (i.e., $m_{\mathcal{C}}^{+}\left(\mathfrak{z}\right)$ and $n_{\mathcal{C}}^{+}\left(\mathfrak{z}\right)$), respectively, where $\mathit{\rho }\in \left[0,1\right]$, and $\widetilde{{\mathit{I}}_{\mathcal{C}}}\left(\mathfrak{z}\right)=\left(\left[m_{\mathcal{C}}^{-}\left(\mathfrak{z}\right),m_{\mathcal{C}}^{+}\left(\mathfrak{z}\right)\right],\left[n_{\mathcal{C}}^{-}\left(\mathfrak{z}\right),n_{\mathcal{C}}^{+}\left(\mathfrak{z}\right)\right]\right)$ serving as the interval-valued estimate for $C_{\mathcal{C}}\left(\mathfrak{z}\right)$:

(11) $S^{\rho }\left(C_{\mathcal{C}}\left(\mathfrak{z}\right)\right)=\mathit{\rho }\left[m_{\mathcal{C}}^{-}\left(\mathfrak{z}\right)+m_{\mathcal{C}}^{-}\left(\mathfrak{z}\right)\left(1-m_{\mathcal{C}}^{-}\left(\mathfrak{z}\right)-n_{\mathcal{C}}^{-}\left(\mathfrak{z}\right)\right)\right]+\left(1-\mathit{\rho }\right)\left[m_{\mathcal{C}}^{+}\left(\mathfrak{z}\right)+m_{\mathcal{C}}^{+}\left(\mathfrak{z}\right)\left(1-m_{\mathcal{C}}^{+}\left(\mathfrak{z}\right)-n_{\mathcal{C}}^{+}\left(\mathfrak{z}\right)\right)\right]$(11)

Theorem 2.

The boundary constraint related to a C-IF scoring function $S^{\rho }\left(C_{\mathcal{C}}\left(\mathfrak{z}\right)\right)$ (with $0\le \mathit{\rho }\le 1$) applied to a C-IF number $C_{\mathcal{C}}\left(\mathfrak{z}\right)$ is that $0\le S^{\rho }\left(C_{\mathcal{C}}\left(\mathfrak{z}\right)\right)\le 1$.

Proof.

Based on Definition 5, there are four potential numerical outcomes for the interval-valued estimate $\left[m_{\mathcal{C}}^{-}\left(\mathfrak{z}\right),m_{\mathcal{C}}^{+}\left(\mathfrak{z}\right)\right]$ that correspond to $m_{\mathcal{C}}\left(\mathfrak{z}\right)$. These outcomes are as follows:

(m1) $\left[m_{\mathcal{C}}^{-}\left(\mathfrak{z}\right),m_{\mathcal{C}}^{+}\left(\mathfrak{z}\right)\right]=\left[m_{\mathcal{C}}\left(\mathfrak{z}\right)-r_{\mathcal{C}}\left(\mathfrak{z}\right)/\sqrt{2},m_{\mathcal{C}}\left(\mathfrak{z}\right)+r_{\mathcal{C}}\left(\mathfrak{z}\right)/\sqrt{2}\right];$(m1)

(m2) $\left[m_{\mathcal{C}}^{-}\left(\mathfrak{z}\right),m_{\mathcal{C}}^{+}\left(\mathfrak{z}\right)\right]=\left[m_{\mathcal{C}}\left(\mathfrak{z}\right)-r_{\mathcal{C}}\left(\mathfrak{z}\right)/\sqrt{2},1\right];$(m2)

(m3) $\left[m_{\mathcal{C}}^{-}\left(\mathfrak{z}\right),m_{\mathcal{C}}^{+}\left(\mathfrak{z}\right)\right]=\left[0,m_{\mathcal{C}}\left(\mathfrak{z}\right)+r_{\mathcal{C}}\left(\mathfrak{z}\right)/\sqrt{2}\right];$(m3)

(m4) $\left[m_{\mathcal{C}}^{-}\left(\mathfrak{z}\right),m_{\mathcal{C}}^{+}\left(\mathfrak{z}\right)\right]=\left[0,1\right].$(m4)

Likewise, there are four potential numerical outcomes for the interval-valued estimate $\left[n_{\mathcal{C}}^{-}\left(\mathfrak{z}\right),n_{\mathcal{C}}^{+}\left(\mathfrak{z}\right)\right]$ associated with $n_{\mathcal{C}}\left(\mathfrak{z}\right)$, which are as follows:

(n1) $\left[n_{\mathcal{C}}^{-}\left(\mathfrak{z}\right),n_{\mathcal{C}}^{+}\left(\mathfrak{z}\right)\right]=\left[n_{\mathcal{C}}\left(\mathfrak{z}\right)-r_{\mathcal{C}}\left(\mathfrak{z}\right)/\sqrt{2},n_{\mathcal{C}}\left(\mathfrak{z}\right)+r_{\mathcal{C}}\left(\mathfrak{z}\right)/\sqrt{2}\right];$(n1)

(n2) $\left[n_{\mathcal{C}}^{-}\left(\mathfrak{z}\right),n_{\mathcal{C}}^{+}\left(\mathfrak{z}\right)\right]=\left[n_{\mathcal{C}}\left(\mathfrak{z}\right)-r_{\mathcal{C}}\left(\mathfrak{z}\right)/\sqrt{2},1\right];$(n2)

(n3) $\left[n_{\mathcal{C}}^{-}\left(\mathfrak{z}\right),n_{\mathcal{C}}^{+}\left(\mathfrak{z}\right)\right]=\left[0,n_{\mathcal{C}}\left(\mathfrak{z}\right)+r_{\mathcal{C}}\left(\mathfrak{z}\right)/\sqrt{2}\right];$(n3)

(n4) $\left[n_{\mathcal{C}}^{-}\left(\mathfrak{z}\right),n_{\mathcal{C}}^{+}\left(\mathfrak{z}\right)\right]=\left[0,1\right].$(n4)

Based on the previously mentioned potential numerical outcomes, there are 16 ( = 4 × 4) combinations of $\left(\left[m_{\mathcal{C}}^{-}\left(\mathfrak{z}\right),m_{\mathcal{C}}^{+}\left(\mathfrak{z}\right)\right],\left[n_{\mathcal{C}}^{-}\left(\mathfrak{z}\right),n_{\mathcal{C}}^{+}\left(\mathfrak{z}\right)\right]\right)$ that characterize the interval-valued estimate $\widetilde{{\mathit{I}}_{\mathcal{C}}}\left(\mathfrak{z}\right)$ related to $C_{\mathcal{C}}\left(\mathfrak{z}\right)$. To illustrate, let’s consider the most complex combination of (m1) and (n1), that is, $\widetilde{{\mathit{I}}_{\mathcal{C}}}\left(\mathfrak{z}\right)=\left(\left[m_{\mathcal{C}}\left(\mathfrak{z}\right)-r_{\mathcal{C}}\left(\mathfrak{z}\right)/\sqrt{2},m_{\mathcal{C}}\left(\mathfrak{z}\right)+r_{\mathcal{C}}\left(\mathfrak{z}\right)/\sqrt{2}\left],\right[n_{\mathcal{C}}\left(\mathfrak{z}\right)-r_{\mathcal{C}}\left(\mathfrak{z}\right)/\sqrt{2},n_{\mathcal{C}}\left(\mathfrak{z}\right)+\text{break}{r}_{\mathcal{C}}\left(\mathfrak{z}\right)/\sqrt{2}\right]\right)$. It should be noted that, according to Definition 5, the prerequisites for (m1) to hold are as follows: $m_{\mathcal{C}}\left(\mathfrak{z}\right)-r_{\mathcal{C}}\left(\mathfrak{z}\right)/\sqrt{2}>0$ and $m_{\mathcal{C}}\left(\mathfrak{z}\right)+r_{\mathcal{C}}\left(\mathfrak{z}\right)/\sqrt{2}<1$. Similarly, the prerequisites for (n1) to hold are $n_{\mathcal{C}}\left(\mathfrak{z}\right)-r_{\mathcal{C}}\left(\mathfrak{z}\right)/\sqrt{2}>0$ and $n_{\mathcal{C}}\left(\mathfrak{z}\right)+r_{\mathcal{C}}\left(\mathfrak{z}\right)/\sqrt{2}<1$. Based on these inequalities, one can infer the value ranges of $m_{\mathcal{C}}\left(\mathfrak{z}\right)$, $n_{\mathcal{C}}\left(\mathfrak{z}\right)$, and $r_{\mathcal{C}}\left(\mathfrak{z}\right)$ as follows: $r_{\mathcal{C}}\left(\mathfrak{z}\right)/\sqrt{2}<m_{\mathcal{C}}\left(\mathfrak{z}\right)<1-r_{\mathcal{C}}\left(\mathfrak{z}\right)/\sqrt{2}$, $r_{\mathcal{C}}\left(\mathfrak{z}\right)/\sqrt{2}<n_{\mathcal{C}}\left(\mathfrak{z}\right)<1-r_{\mathcal{C}}\left(\mathfrak{z}\right)/\sqrt{2}$, and $0\le r_{\mathcal{C}}\left(\mathfrak{z}\right)<1/\sqrt{2}$. To provide greater detail, the expression for the term $m_{\mathcal{C}}^{-}\left(\mathfrak{z}\right)+m_{\mathcal{C}}^{-}\left(\mathfrak{z}\right)\left(1-m_{\mathcal{C}}^{-}\left(\mathfrak{z}\right)-n_{\mathcal{C}}^{-}\left(\mathfrak{z}\right)\right)$ within Equation (11) can be acquired using the ensuing manner:

$m_{\mathcal{C}}^{-}\left(\mathfrak{z}\right)+m_{\mathcal{C}}^{-}\left(\mathfrak{z}\right)\left(1-m_{\mathcal{C}}^{-}\left(\mathfrak{z}\right)-n_{\mathcal{C}}^{-}\left(\mathfrak{z}\right)\right)$

$=\left(m_{\mathcal{C}}\left(\mathfrak{z}\right)-\frac{{\mathit{r}}_{\mathcal{C}}\left(\mathfrak{z}\right)}{\sqrt{2}}\right)+\left(m_{\mathcal{C}}\left(\mathfrak{z}\right)-\frac{{\mathit{r}}_{\mathcal{C}}\left(\mathfrak{z}\right)}{\sqrt{2}}\right)\left[1-\left(m_{\mathcal{C}}\left(\mathfrak{z}\right)-\frac{{\mathit{r}}_{\mathcal{C}}\left(\mathfrak{z}\right)}{\sqrt{2}}\right)-\left(n_{\mathcal{C}}\left(\mathfrak{z}\right)-\frac{{\mathit{r}}_{\mathcal{C}}\left(\mathfrak{z}\right)}{\sqrt{2}}\right)\right]$

$=\left(m_{\mathcal{C}}\left(\mathfrak{z}\right)-\frac{{\mathit{r}}_{\mathcal{C}}\left(\mathfrak{z}\right)}{\sqrt{2}}\right)\left[1-\left(m_{\mathcal{C}}\left(\mathfrak{z}\right)-\frac{{\mathit{r}}_{\mathcal{C}}\left(\mathfrak{z}\right)}{\sqrt{2}}\right)+1-\left(n_{\mathcal{C}}\left(\mathfrak{z}\right)-\frac{{\mathit{r}}_{\mathcal{C}}\left(\mathfrak{z}\right)}{\sqrt{2}}\right)\right].$

It is observed that $0<m_{\mathcal{C}}\left(\mathfrak{z}\right)-r_{\mathcal{C}}\left(\mathfrak{z}\right)/\sqrt{2}<1$ according to the condition $r_{\mathcal{C}}\left(\mathfrak{z}\right)/\sqrt{2}<m_{\mathcal{C}}\left(\mathfrak{z}\right)<1-r_{\mathcal{C}}\left(\mathfrak{z}\right)/\sqrt{2}$ for (m1) to hold. By incorporating the axiom condition $0\le m_{\mathcal{C}}\left(\mathfrak{z}\right)+n_{\mathcal{C}}\left(\mathfrak{z}\right)\le 1$ and the previously obtained conditions for (m1) and (n1), it can be deduced that $0\le 1-\left(m_{\mathcal{C}}\left(\mathfrak{z}\right)-r_{\mathcal{C}}\left(\mathfrak{z}\right)/\sqrt{2}\right)+1-\left(n_{\mathcal{C}}\left(\mathfrak{z}\right)-r_{\mathcal{C}}\left(\mathfrak{z}\right)/\sqrt{2}\right)\le 1$. Thus, it follows that $0\le m_{\mathcal{C}}^{-}\left(\mathfrak{z}\right)+m_{\mathcal{C}}^{-}\left(\mathfrak{z}\right)\left(1-m_{\mathcal{C}}^{-}\left(\mathfrak{z}\right)-n_{\mathcal{C}}^{-}\left(\mathfrak{z}\right)\right)\le 1$. Subsequently, the expression for the term $m_{\mathcal{C}}^{+}\left(\mathfrak{z}\right)+m_{\mathcal{C}}^{+}\left(\mathfrak{z}\right)\left(1-m_{\mathcal{C}}^{+}\left(\mathfrak{z}\right)-n_{\mathcal{C}}^{+}\left(\mathfrak{z}\right)\right)$ within (11) can be obtained as follows:

$m_{\mathcal{C}}^{+}\left(\mathfrak{z}\right)+m_{\mathcal{C}}^{+}\left(\mathfrak{z}\right)\left(1-m_{\mathcal{C}}^{+}\left(\mathfrak{z}\right)-n_{\mathcal{C}}^{+}\left(\mathfrak{z}\right)\right)$

$=\left(m_{\mathcal{C}}\left(\mathfrak{z}\right)+\frac{{\mathit{r}}_{\mathcal{C}}\left(\mathfrak{z}\right)}{\sqrt{2}}\right)+\left(m_{\mathcal{C}}\left(\mathfrak{z}\right)+\frac{{\mathit{r}}_{\mathcal{C}}\left(\mathfrak{z}\right)}{\sqrt{2}}\right)\left[1-\left(m_{\mathcal{C}}\left(\mathfrak{z}\right)+\frac{{\mathit{r}}_{\mathcal{C}}\left(\mathfrak{z}\right)}{\sqrt{2}}\right)-\left(n_{\mathcal{C}}\left(\mathfrak{z}\right)+\frac{{\mathit{r}}_{\mathcal{C}}\left(\mathfrak{z}\right)}{\sqrt{2}}\right)\right]$

$=\left(m_{\mathcal{C}}\left(\mathfrak{z}\right)+\frac{{\mathit{r}}_{\mathcal{C}}\left(\mathfrak{z}\right)}{\sqrt{2}}\right)\left[1-\left(m_{\mathcal{C}}\left(\mathfrak{z}\right)+\frac{{\mathit{r}}_{\mathcal{C}}\left(\mathfrak{z}\right)}{\sqrt{2}}\right)+1-\left(n_{\mathcal{C}}\left(\mathfrak{z}\right)+\frac{{\mathit{r}}_{\mathcal{C}}\left(\mathfrak{z}\right)}{\sqrt{2}}\right)\right].$

The condition for (m1) to hold establishes that $0<m_{\mathcal{C}}\left(\mathfrak{z}\right)+r_{\mathcal{C}}\left(\mathfrak{z}\right)/\sqrt{2}<1$, as $r_{\mathcal{C}}\left(\mathfrak{z}\right)/\sqrt{2}<m_{\mathcal{C}}\left(\mathfrak{z}\right)<1-r_{\mathcal{C}}\left(\mathfrak{z}\right)/\sqrt{2}$. By combining the axiom condition $0\le m_{\mathcal{C}}\left(\mathfrak{z}\right)+n_{\mathcal{C}}\left(\mathfrak{z}\right)\le 1$ and the previously derived conditions for (m1) and (n1), it draws the inference that $0\le 1-\left(m_{\mathcal{C}}\left(\mathfrak{z}\right)+r_{\mathcal{C}}\left(\mathfrak{z}\right)/\sqrt{2}\right)\text{break}+1-\left(n_{\mathcal{C}}\left(\mathfrak{z}\right)+r_{\mathcal{C}}\left(\mathfrak{z}\right)/\sqrt{2}\right)\le 1$. This finding implies that $0\le m_{\mathcal{C}}^{+}\left(\mathfrak{z}\right)+m_{\mathcal{C}}^{+}\left(\mathfrak{z}\right)\left(1-m_{\mathcal{C}}^{+}\left(\mathfrak{z}\right)-n_{\mathcal{C}}^{+}\left(\mathfrak{z}\right)\right)\le 1$. On the other side, it is crucial to remember that the relative significance parameter falls within the range $0\le \mathit{\rho }\le 1$. Consider the resulting inequalities: $0\le m_{\mathcal{C}}^{-}\left(\mathfrak{z}\right)+m_{\mathcal{C}}^{-}\left(\mathfrak{z}\right)\left(1-m_{\mathcal{C}}^{-}\left(\mathfrak{z}\right)-n_{\mathcal{C}}^{-}\left(\mathfrak{z}\right)\right)\le 1$ and $0\le m_{\mathcal{C}}^{+}\left(\mathfrak{z}\right)+m_{\mathcal{C}}^{+}\left(\mathfrak{z}\right)\text{break}\left(1-m_{\mathcal{C}}^{+}\left(\mathfrak{z}\right)-n_{\mathcal{C}}^{+}\left(\mathfrak{z}\right)\right)\le 1$. This leads to $0\le \mathit{\rho }\left[m_{\mathcal{C}}^{-}\left(\mathfrak{z}\right)+m_{\mathcal{C}}^{-}\left(\mathfrak{z}\right)\left(1-m_{\mathcal{C}}^{\text{break}}-\left(\mathfrak{z}\right)-n_{\mathcal{C}}^{-}\left(\mathfrak{z}\right)\right)\left]+\left(1-\mathit{\rho }\right)\right[m_{\mathcal{C}}^{+}\left(\mathfrak{z}\right)+m_{\mathcal{C}}^{+}\left(\mathfrak{z}\right)\left(1-m_{\mathcal{C}}^{+}\left(\mathfrak{z}\right)-n_{\mathcal{C}}^{+}\left(\mathfrak{z}\right)\right)\right]\le 1$. Consequently, $0\le S^{\rho }\left(C_{\mathcal{C}}\left(\mathfrak{z}\right)\right)\le 1$ is the boundary constraint that governs the C-IF scoring function $S^{\rho }\left(C_{\mathcal{C}}\left(\mathfrak{z}\right)\right)$ within the context of the (m1) and (n1) combination. The same rationale can be applied to the remaining 15 combinations, validating the property outlined in this theorem.

Theorem 3.

The C-IF scoring function $S^{\rho }\left(C_{\mathcal{C}}\left(\mathfrak{z}\right)\right)$, with the radius $r_{\mathcal{C}}\left(\mathfrak{z}\right)$ and the hesitancy degree $h_{\mathcal{C}}\left(\mathfrak{z}\right)$ held constant and for a specific value of the relative significance parameter $\mathit{\rho }$, exhibits a non-decreasing trend as the membership degree $m_{\mathcal{C}}\left(\mathfrak{z}\right)$ increases.

Proof.

There are four potential numerical outcomes, labeled as (m1) to (m4), which correspond to the interval $\left[m_{\mathcal{C}}^{-}\left(\mathfrak{z}\right),m_{\mathcal{C}}^{+}\left(\mathfrak{z}\right)\right]$ concerning $m_{\mathcal{C}}\left(\mathfrak{z}\right)$, as illustrated in the proving process of Theorem 2. Similarly, there are four potential numerical outcomes, designated as (n1) to (n4), associated with $\left[n_{\mathcal{C}}^{-}\left(\mathfrak{z}\right),n_{\mathcal{C}}^{+}\left(\mathfrak{z}\right)\right]$ related to $n_{\mathcal{C}}\left(\mathfrak{z}\right)$. There are a total of 16 combinations of $\widetilde{{\mathit{I}}_{\mathcal{C}}}\left(\mathfrak{z}\right)$. Let’s use the most complex combination, which includes (m1) and (n1), as an example. It is important to note that the conditions for $m_{\mathcal{C}}\left(\mathfrak{z}\right)-r_{\mathcal{C}}\left(\mathfrak{z}\right)/\sqrt{2}>0$ and $m_{\mathcal{C}}\left(\mathfrak{z}\right)+r_{\mathcal{C}}\left(\mathfrak{z}\right)/\sqrt{2}<1$, as well as $n_{\mathcal{C}}\left(\mathfrak{z}\right)-r_{\mathcal{C}}\left(\mathfrak{z}\right)/\sqrt{2}>0$ and $n_{\mathcal{C}}\left(\mathfrak{z}\right)+r_{\mathcal{C}}\left(\mathfrak{z}\right)/\sqrt{2}<1$, are the requirements for (m1) and (n1) to hold, respectively. The first term on the right-hand side of Equation (11) is enunciated along these lines in the context of the combination of (m1) and (n1):

$\mathit{\rho }\left[m_{\mathcal{C}}^{-}\left(\mathfrak{z}\right)+m_{\mathcal{C}}^{-}\left(\mathfrak{z}\right)\left(1-m_{\mathcal{C}}^{-}\left(\mathfrak{z}\right)-n_{\mathcal{C}}^{-}\left(\mathfrak{z}\right)\right)\right]$

$=\mathit{\rho }\left[m_{\mathcal{C}}\left(\mathfrak{z}\right)-\frac{{\mathit{r}}_{\mathcal{C}}\left(\mathfrak{z}\right)}{\sqrt{2}}+\left(m_{\mathcal{C}}\left(\mathfrak{z}\right)-\frac{{\mathit{r}}_{\mathcal{C}}\left(\mathfrak{z}\right)}{\sqrt{2}}\right)\left(1-m_{\mathcal{C}}\left(\mathfrak{z}\right)+\frac{{\mathit{r}}_{\mathcal{C}}\left(\mathfrak{z}\right)}{\sqrt{2}}-n_{\mathcal{C}}\left(\mathfrak{z}\right)+\frac{{\mathit{r}}_{\mathcal{C}}\left(\mathfrak{z}\right)}{\sqrt{2}}\right)\right]$

$=\mathit{\rho }\left[m_{\mathcal{C}}\left(\mathfrak{z}\right)-\frac{{\mathit{r}}_{\mathcal{C}}\left(\mathfrak{z}\right)}{\sqrt{2}}+\left(m_{\mathcal{C}}\left(\mathfrak{z}\right)-\frac{{\mathit{r}}_{\mathcal{C}}\left(\mathfrak{z}\right)}{\sqrt{2}}\right)\left(h_{\mathcal{C}}\left(\mathfrak{z}\right)+\sqrt{2}\cdot r_{\mathcal{C}}\left(\mathfrak{z}\right)\right)\right]$

$=\mathit{\rho }\left[m_{\mathcal{C}}\left(\mathfrak{z}\right)-\frac{{\mathit{r}}_{\mathcal{C}}\left(\mathfrak{z}\right)}{\sqrt{2}}+h_{\mathcal{C}}\left(\mathfrak{z}\right)\cdot m_{\mathcal{C}}\left(\mathfrak{z}\right)+\sqrt{2}\cdot r_{\mathcal{C}}\left(\mathfrak{z}\right)\cdot m_{\mathcal{C}}\left(\mathfrak{z}\right)-\frac{{\mathit{r}}_{\mathcal{C}}\left(\mathfrak{z}\right)}{\sqrt{2}}{h}_{\mathcal{C}}\left(\mathfrak{z}\right)-{\left({\mathit{r}}_{\mathcal{C}}\left(\mathfrak{z}\right)\right)}^2\right]$

$=\mathit{\rho }\left(1+h_{\mathcal{C}}\left(\mathfrak{z}\right)+\sqrt{2}\cdot r_{\mathcal{C}}\left(\mathfrak{z}\right)\right)m_{\mathcal{C}}\left(\mathfrak{z}\right)-\mathit{\rho }\left(\frac{{\mathit{r}}_{\mathcal{C}}\left(\mathfrak{z}\right)}{\sqrt{2}}+{\left({\mathit{r}}_{\mathcal{C}}\left(\mathfrak{z}\right)\right)}^2+\frac{{\mathit{r}}_{\mathcal{C}}\left(\mathfrak{z}\right)}{\sqrt{2}}{h}_{\mathcal{C}}\left(\mathfrak{z}\right)\right).$

The second term on the right-hand side of Equation (11) is enunciated by the following procedure under the combination of (m1) and (n1):

$\left(1-\mathit{\rho }\right)\left[m_{\mathcal{C}}^{+}\left(\mathfrak{z}\right)+m_{\mathcal{C}}^{+}\left(\mathfrak{z}\right)\left(1-m_{\mathcal{C}}^{+}\left(\mathfrak{z}\right)-n_{\mathcal{C}}^{+}\left(\mathfrak{z}\right)\right)\right]$

$=\left(1-\mathit{\rho }\right)\left[m_{\mathcal{C}}\left(\mathfrak{z}\right)+\frac{{\mathit{r}}_{\mathcal{C}}\left(\mathfrak{z}\right)}{\sqrt{2}}+\left(m_{\mathcal{C}}\left(\mathfrak{z}\right)+\frac{{\mathit{r}}_{\mathcal{C}}\left(\mathfrak{z}\right)}{\sqrt{2}}\right)\left(1-m_{\mathcal{C}}\left(\mathfrak{z}\right)-\frac{{\mathit{r}}_{\mathcal{C}}\left(\mathfrak{z}\right)}{\sqrt{2}}-n_{\mathcal{C}}\left(\mathfrak{z}\right)-\frac{{\mathit{r}}_{\mathcal{C}}\left(\mathfrak{z}\right)}{\sqrt{2}}\right)\right]$

$=\left(1-\mathit{\rho }\right)\left[m_{\mathcal{C}}\left(\mathfrak{z}\right)+\frac{{\mathit{r}}_{\mathcal{C}}\left(\mathfrak{z}\right)}{\sqrt{2}}+\left(m_{\mathcal{C}}\left(\mathfrak{z}\right)+\frac{{\mathit{r}}_{\mathcal{C}}\left(\mathfrak{z}\right)}{\sqrt{2}}\right)\left(h_{\mathcal{C}}\left(\mathfrak{z}\right)-\sqrt{2}\cdot r_{\mathcal{C}}\left(\mathfrak{z}\right)\right)\right]$

$=\left(1-\mathit{\rho }\right)\left[m_{\mathcal{C}}\left(\mathfrak{z}\right)+\frac{{\mathit{r}}_{\mathcal{C}}\left(\mathfrak{z}\right)}{\sqrt{2}}+h_{\mathcal{C}}\left(\mathfrak{z}\right)\cdot m_{\mathcal{C}}\left(\mathfrak{z}\right)-\sqrt{2}\cdot r_{\mathcal{C}}\left(\mathfrak{z}\right)\cdot m_{\mathcal{C}}\left(\mathfrak{z}\right)+\frac{{\mathit{r}}_{\mathcal{C}}\left(\mathfrak{z}\right)}{\sqrt{2}}{h}_{\mathcal{C}}\left(\mathfrak{z}\right)-{\left({\mathit{r}}_{\mathcal{C}}\left(\mathfrak{z}\right)\right)}^2\right]$

$=\left(1-\mathit{\rho }\right)\left(1+h_{\mathcal{C}}\left(\mathfrak{z}\right)-\sqrt{2}\cdot r_{\mathcal{C}}\left(\mathfrak{z}\right)\right)m_{\mathcal{C}}\left(\mathfrak{z}\right)+\left(1-\mathit{\rho }\right)\left(\frac{{\mathit{r}}_{\mathcal{C}}\left(\mathfrak{z}\right)}{\sqrt{2}}-{\left({\mathit{r}}_{\mathcal{C}}\left(\mathfrak{z}\right)\right)}^2+\frac{{\mathit{r}}_{\mathcal{C}}\left(\mathfrak{z}\right)}{\sqrt{2}}{h}_{\mathcal{C}}\left(\mathfrak{z}\right)\right).$

These calculation outcomes provide the following expression for the C-IF scoring function:

$S^{\rho }\left(C_{\mathcal{C}}\left(\mathfrak{z}\right)\right)=\mathit{\rho }\left(1+h_{\mathcal{C}}\left(\mathfrak{z}\right)+\sqrt{2}\cdot r_{\mathcal{C}}\left(\mathfrak{z}\right)\right)m_{\mathcal{C}}\left(\mathfrak{z}\right)-\mathit{\rho }\left(\frac{{\mathit{r}}_{\mathcal{C}}\left(\mathfrak{z}\right)}{\sqrt{2}}+{\left({\mathit{r}}_{\mathcal{C}}\left(\mathfrak{z}\right)\right)}^2+\frac{{\mathit{r}}_{\mathcal{C}}\left(\mathfrak{z}\right)}{\sqrt{2}}{h}_{\mathcal{C}}\left(\mathfrak{z}\right)\right)$

$+\left(1-\mathit{\rho }\right)\left(1+h_{\mathcal{C}}\left(\mathfrak{z}\right)-\sqrt{2}\cdot r_{\mathcal{C}}\left(\mathfrak{z}\right)\right)m_{\mathcal{C}}\left(\mathfrak{z}\right)+\left(1-\mathit{\rho }\right)\left(\frac{{\mathit{r}}_{\mathcal{C}}\left(\mathfrak{z}\right)}{\sqrt{2}}-{\left({\mathit{r}}_{\mathcal{C}}\left(\mathfrak{z}\right)\right)}^2+\frac{{\mathit{r}}_{\mathcal{C}}\left(\mathfrak{z}\right)}{\sqrt{2}}{h}_{\mathcal{C}}\left(\mathfrak{z}\right)\right).$

In accordance with the prerequisites for (m1) and (n1) to hold, it is derived that $m_{\mathcal{C}}\left(\mathfrak{z}\right)+r_{\mathcal{C}}\left(\mathfrak{z}\right)/\sqrt{2}<1$ and $n_{\mathcal{C}}\left(\mathfrak{z}\right)+r_{\mathcal{C}}\left(\mathfrak{z}\right)/\sqrt{2}<1$, respectively. These two inequalities separately lead to $1-m_{\mathcal{C}}\left(\mathfrak{z}\right)>r_{\mathcal{C}}\left(\mathfrak{z}\right)/\sqrt{2}$ and $1-n_{\mathcal{C}}\left(\mathfrak{z}\right)>r_{\mathcal{C}}\left(\mathfrak{z}\right)/\sqrt{2}$. Incorporating the aforementioned inequalities and using the relation $h_{\mathcal{C}}\left(\mathfrak{z}\right)=1-m_{\mathcal{C}}\left(\mathfrak{z}\right)-n_{\mathcal{C}}\left(\mathfrak{z}\right)$, while keeping $r_{\mathcal{C}}\left(\mathfrak{z}\right)$ and $h_{\mathcal{C}}\left(\mathfrak{z}\right)$ constant, and considering a specific value of $\mathit{\rho }$, the partial derivative of the function $S^{\rho }\left(C_{\mathcal{C}}\left(\mathfrak{z}\right)\right)$ concerning $m_{\mathcal{C}}\left(\mathfrak{z}\right)$ can be generated as follows:

$\frac{\mathrm{\partial }{S}^{\rho }\left(C_{\mathcal{C}}\left(\mathfrak{z}\right)\right)}{\mathrm{\partial }{m}_{\mathcal{C}}\left(\mathfrak{z}\right)}=\mathit{\rho }\left(1+h_{\mathcal{C}}\left(\mathfrak{z}\right)+\sqrt{2}\cdot r_{\mathcal{C}}\left(\mathfrak{z}\right)\right)+\left(1-\mathit{\rho }\right)\left(1+h_{\mathcal{C}}\left(\mathfrak{z}\right)-\sqrt{2}\cdot r_{\mathcal{C}}\left(\mathfrak{z}\right)\right)$

$=\mathit{\rho }+\mathit{\rho }\cdot h_{\mathcal{C}}\left(\mathfrak{z}\right)+\sqrt{2}\mathit{\rho }\cdot r_{\mathcal{C}}\left(\mathfrak{z}\right)+1+h_{\mathcal{C}}\left(\mathfrak{z}\right)-\sqrt{2}\cdot r_{\mathcal{C}}\left(\mathfrak{z}\right)-\mathit{\rho }-\mathit{\rho }\cdot h_{\mathcal{C}}\left(\mathfrak{z}\right)+\sqrt{2}\mathit{\rho }\cdot r_{\mathcal{C}}\left(\mathfrak{z}\right)$

$=1+h_{\mathcal{C}}\left(\mathfrak{z}\right)+2\sqrt{2}\mathit{\rho }\cdot r_{\mathcal{C}}\left(\mathfrak{z}\right)-\sqrt{2}\cdot r_{\mathcal{C}}\left(\mathfrak{z}\right)$

$=\left(1-m_{\mathcal{C}}\left(\mathfrak{z}\right)\right)+\left(1-n_{\mathcal{C}}\left(\mathfrak{z}\right)\right)+2\sqrt{2}\mathit{\rho }\cdot r_{\mathcal{C}}\left(\mathfrak{z}\right)-\sqrt{2}\cdot r_{\mathcal{C}}\left(\mathfrak{z}\right)$

$>\frac{{\mathit{r}}_{\mathcal{C}}\left(\mathfrak{z}\right)}{\sqrt{2}}+\frac{{\mathit{r}}_{\mathcal{C}}\left(\mathfrak{z}\right)}{\sqrt{2}}+2\sqrt{2}\mathit{\rho }\cdot r_{\mathcal{C}}\left(\mathfrak{z}\right)-\sqrt{2}\cdot r_{\mathcal{C}}\left(\mathfrak{z}\right)=2\sqrt{2}\mathit{\rho }\cdot r_{\mathcal{C}}\left(\mathfrak{z}\right)\ge 0.$

Hence, it can be deduced that the C-IF scoring function $S^{\rho }\left(C_{\mathcal{C}}\left(\mathfrak{z}\right)\right)$ exhibits a non-decreasing pattern as the value of $m_{\mathcal{C}}\left(\mathfrak{z}\right)$ increases within the context of the combination of (m1) and (n1). The remaining 15 combinations can be demonstrated in a similar manner. Therefore, the property mentioned in this theorem holds true.

Theorem 4.

The C-IF scoring function $S^{\rho }\left(C_{\mathcal{C}}\left(\mathfrak{z}\right)\right)$ exhibits a non-increasing trend as the non-membership degree $n_{\mathcal{C}}\left(\mathfrak{z}\right)$ increases, while keeping the radius $r_{\mathcal{C}}\left(\mathfrak{z}\right)$ and the hesitancy degree $h_{\mathcal{C}}\left(\mathfrak{z}\right)$ constant, for a specific value of the relative significance parameter $\mathit{\rho }$.

Proof.

In a manner similar to the proving procedure in Theorem 3, the study will use the most complex combination of (m1) and (n1) as an illustration in this proof. Initially, the C-IF scoring function $S^{\rho }\left(C_{\mathcal{C}}\left(\mathfrak{z}\right)\right)$ is established using a manner described below by combining the derivative results of the first and second terms on the right-hand side of Equation (11) in the context of a combination of (m1) and (n1), as previously calculated:

$S^{\rho }\left(C_{\mathcal{C}}\left(\mathfrak{z}\right)\right)=\mathit{\rho }\left(1+h_{\mathcal{C}}\left(\mathfrak{z}\right)+\sqrt{2}\cdot r_{\mathcal{C}}\left(\mathfrak{z}\right)\right)m_{\mathcal{C}}\left(\mathfrak{z}\right)-\mathit{\rho }\left(\frac{{\mathit{r}}_{\mathcal{C}}\left(\mathfrak{z}\right)}{\sqrt{2}}+{\left({\mathit{r}}_{\mathcal{C}}\left(\mathfrak{z}\right)\right)}^2+\frac{{\mathit{r}}_{\mathcal{C}}\left(\mathfrak{z}\right)}{\sqrt{2}}{h}_{\mathcal{C}}\left(\mathfrak{z}\right)\right)$

$+\left(1-\mathit{\rho }\right)\left(1+h_{\mathcal{C}}\left(\mathfrak{z}\right)-\sqrt{2}\cdot r_{\mathcal{C}}\left(\mathfrak{z}\right)\right)m_{\mathcal{C}}\left(\mathfrak{z}\right)+\left(1-\mathit{\rho }\right)\left(\frac{{\mathit{r}}_{\mathcal{C}}\left(\mathfrak{z}\right)}{\sqrt{2}}-{\left({\mathit{r}}_{\mathcal{C}}\left(\mathfrak{z}\right)\right)}^2+\frac{{\mathit{r}}_{\mathcal{C}}\left(\mathfrak{z}\right)}{\sqrt{2}}{h}_{\mathcal{C}}\left(\mathfrak{z}\right)\right)$

$=\mathit{\rho }\left(1+h_{\mathcal{C}}\left(\mathfrak{z}\right)+\sqrt{2}\cdot r_{\mathcal{C}}\left(\mathfrak{z}\right)\right)\left(1-h_{\mathcal{C}}\left(\mathfrak{z}\right)-n_{\mathcal{C}}\left(\mathfrak{z}\right)\right)-\mathit{\rho }\left(\frac{{\mathit{r}}_{\mathcal{C}}\left(\mathfrak{z}\right)}{\sqrt{2}}+{\left({\mathit{r}}_{\mathcal{C}}\left(\mathfrak{z}\right)\right)}^2+\frac{{\mathit{r}}_{\mathcal{C}}\left(\mathfrak{z}\right)}{\sqrt{2}}{h}_{\mathcal{C}}\left(\mathfrak{z}\right)\right)$

$+\left(1-\mathit{\rho }\right)\left(1+h_{\mathcal{C}}\left(\mathfrak{z}\right)-\sqrt{2}\cdot r_{\mathcal{C}}\left(\mathfrak{z}\right)\right)\left(1-h_{\mathcal{C}}\left(\mathfrak{z}\right)-n_{\mathcal{C}}\left(\mathfrak{z}\right)\right)+\left(1-\mathit{\rho }\right)\left(\frac{{\mathit{r}}_{\mathcal{C}}\left(\mathfrak{z}\right)}{\sqrt{2}}-{\left({\mathit{r}}_{\mathcal{C}}\left(\mathfrak{z}\right)\right)}^2+\frac{{\mathit{r}}_{\mathcal{C}}\left(\mathfrak{z}\right)}{\sqrt{2}}{h}_{\mathcal{C}}\left(\mathfrak{z}\right)\right)$

$=-\mathit{\rho }\left(1+h_{\mathcal{C}}\left(\mathfrak{z}\right)+\sqrt{2}\cdot r_{\mathcal{C}}\left(\mathfrak{z}\right)\right)n_{\mathcal{C}}\left(\mathfrak{z}\right)+\mathit{\rho }\left(1-h_{\mathcal{C}}\left(\mathfrak{z}\right)\right)\left(1+h_{\mathcal{C}}\left(\mathfrak{z}\right)+\sqrt{2}\cdot r_{\mathcal{C}}\left(\mathfrak{z}\right)\right)$

$-\mathit{\rho }\left(\frac{{\mathit{r}}_{\mathcal{C}}\left(\mathfrak{z}\right)}{\sqrt{2}}+{\left({\mathit{r}}_{\mathcal{C}}\left(\mathfrak{z}\right)\right)}^2+\frac{{\mathit{r}}_{\mathcal{C}}\left(\mathfrak{z}\right)}{\sqrt{2}}{h}_{\mathcal{C}}\left(\mathfrak{z}\right)\right)-\left(1-\mathit{\rho }\right)\left(1+h_{\mathcal{C}}\left(\mathfrak{z}\right)-\sqrt{2}\cdot r_{\mathcal{C}}\left(\mathfrak{z}\right)\right)n_{\mathcal{C}}\left(\mathfrak{z}\right)$

$+\left(1-\mathit{\rho }\right)\left(1-h_{\mathcal{C}}\left(\mathfrak{z}\right)\right)\left(1+h_{\mathcal{C}}\left(\mathfrak{z}\right)-\sqrt{2}\cdot r_{\mathcal{C}}\left(\mathfrak{z}\right)\right)+\left(1-\mathit{\rho }\right)\left(\frac{{\mathit{r}}_{\mathcal{C}}\left(\mathfrak{z}\right)}{\sqrt{2}}-{\left({\mathit{r}}_{\mathcal{C}}\left(\mathfrak{z}\right)\right)}^2+\frac{{\mathit{r}}_{\mathcal{C}}\left(\mathfrak{z}\right)}{\sqrt{2}}{h}_{\mathcal{C}}\left(\mathfrak{z}\right)\right).$

Considering the preconditions for (m1) and (n1) to hold, it can be deduced that $m_{\mathcal{C}}\left(\mathfrak{z}\right)+r_{\mathcal{C}}\left(\mathfrak{z}\right)/\sqrt{2}<1$ and $n_{\mathcal{C}}\left(\mathfrak{z}\right)+r_{\mathcal{C}}\left(\mathfrak{z}\right)/\sqrt{2}<1$, respectively. These two inequalities are reformulated as $m_{\mathcal{C}}\left(\mathfrak{z}\right)-1<-r_{\mathcal{C}}\left(\mathfrak{z}\right)/\sqrt{2}$ and $n_{\mathcal{C}}\left(\mathfrak{z}\right)-1<-r_{\mathcal{C}}\left(\mathfrak{z}\right)/\sqrt{2}$. By taking into account these inequalities and utilizing the relation $-h_{\mathcal{C}}\left(\mathfrak{z}\right)=m_{\mathcal{C}}\left(\mathfrak{z}\right)+n_{\mathcal{C}}\left(\mathfrak{z}\right)-1$, while keeping $r_{\mathcal{C}}\left(\mathfrak{z}\right)$ and $h_{\mathcal{C}}\left(\mathfrak{z}\right)$ constant, and for a specific value of $\mathit{\rho }$, the partial derivative of the function $S^{\rho }\left(C_{\mathcal{C}}\left(\mathfrak{z}\right)\right)$ concerning $n_{\mathcal{C}}\left(\mathfrak{z}\right)$ is derived along these lines:

$\frac{\mathrm{\partial }{S}^{\rho }\left(C_{\mathcal{C}}\left(\mathfrak{z}\right)\right)}{\mathrm{\partial }{n}_{\mathcal{C}}\left(\mathfrak{z}\right)}=-\mathit{\rho }\left(1+h_{\mathcal{C}}\left(\mathfrak{z}\right)+\sqrt{2}\cdot r_{\mathcal{C}}\left(\mathfrak{z}\right)\right)-\left(1-\mathit{\rho }\right)\left(1+h_{\mathcal{C}}\left(\mathfrak{z}\right)-\sqrt{2}\cdot r_{\mathcal{C}}\left(\mathfrak{z}\right)\right)$

$=-1-h_{\mathcal{C}}\left(\mathfrak{z}\right)-2\sqrt{2}\mathit{\rho }\cdot r_{\mathcal{C}}\left(\mathfrak{z}\right)+\sqrt{2}\cdot r_{\mathcal{C}}\left(\mathfrak{z}\right)$

$=\left(m_{\mathcal{C}}\left(\mathfrak{z}\right)-1\right)+\left(n_{\mathcal{C}}\left(\mathfrak{z}\right)-1\right)-2\sqrt{2}\mathit{\rho }\cdot r_{\mathcal{C}}\left(\mathfrak{z}\right)+\sqrt{2}\cdot r_{\mathcal{C}}\left(\mathfrak{z}\right)<$

$-\frac{{\mathit{r}}_{\mathcal{C}}\left(\mathfrak{z}\right)}{\sqrt{2}}-\frac{{\mathit{r}}_{\mathcal{C}}\left(\mathfrak{z}\right)}{\sqrt{2}}-2\sqrt{2}\mathit{\rho }\cdot r_{\mathcal{C}}\left(\mathfrak{z}\right)+\sqrt{2}\cdot r_{\mathcal{C}}\left(\mathfrak{z}\right)=-2\sqrt{2}\mathit{\rho }\cdot r_{\mathcal{C}}\left(\mathfrak{z}\right)\le 0.$

The C-IF scoring function $S^{\rho }\left(C_{\mathcal{C}}\left(\mathfrak{z}\right)\right)$ for a given $\mathit{\rho }$ value, it can be argued, exhibits a non-increasing pattern when the value of $n_{\mathcal{C}}\left(\mathfrak{z}\right)$ grows while holding the values of $r_{\mathcal{C}}\left(\mathfrak{z}\right)$ and $h_{\mathcal{C}}\left(\mathfrak{z}\right)$ constant. This conclusion can be applied to all 16 combinations of (m1)–(m4) and (n1)–(n4) using a similar proving approach, thus confirming the property stated in this theorem.

C-IF Scoring-Mediated Median Ranking Model for Decision Analytics

This section strives to develop an innovative C-IF scoring-mediated median ranking model. The general goal is to establish a comprehensive precedence ranking for competing alternatives, addressing uncertainties inherent in decision-analytic challenges within the C-IF environment. The preceding content detailed the creation of an original scoring mechanism in the realm of C-IF uncertainties. The accompanying inventive C-IF scoring function was crafted to enhance data evaluation in situations characterized by circular intuitionistic fuzziness. This systematic method proves valuable in evaluating the performance of alternatives against criteria, contributing to a reduction in imprecision and biases in decision-analytic processes. Furthermore, a specialized C-IF median ranking model for decision analytics can be created using an underlying concept of the C-IF scoring mechanism. In this section, a noteworthy contribution is made through the formulation of a robust implementation procedure specifically tailored for the operation of the C-IF scoring-mediated median ranking model within the framework of C-IF information. Utilizing the suggested C-IF scoring mechanism, this section introduces innovative concepts of comprehensive C-IF scoring functions and comprehensive disagreement metrics. Subsequently, a comprehensive disagreement matrix is established, with its entries quantifying the degree of disagreement regarding the assignment of a specific rank to each alternative in all criterion-wise precedence relationships. Following this, a novel C-IF scoring-mediated median ranking model is formulated, providing decision analysts with a tool to handle intricate C-IF information and derive dependable decision-analytic results.

Figure 2 illustrates the framework of the C-IF scoring-mediated median ranking model, which encompasses three primary objectives aimed at addressing decision-analytic challenges within the C-IF context. The first objective, as depicted in Figure 2, focuses on formulating multiple criteria decision-analytic problems within C-IF scenarios. This includes representing decision-analytic tasks with multiple criteria, generating necessary data for C-IF performance and importance ratings, and forming C-IF sets with associated circular functions. The second objective aims to address C-IF uncertainties inherent in decision-analytic challenges. This involves establishing the C-IF scoring mechanism, which includes creating interval-valued estimates for membership and non-membership constituents, designating relative significance parameters, computing C-IF scoring functions, and identifying criterion-specific precedence ranks. Additionally, constructing a comprehensive C-IF scoring mechanism entails weighting C-IF performance ratings, estimating interval values, and calculating comprehensive scoring functions. The third objective focuses on rendering comprehensive precedence rankings through median ranking modeling. This encompasses determining comprehensive disagreement metrics, initiating permutation matrices, instituting the C-IF scoring-mediated median ranking model, and resolving optimal permutation matrices to derive optimal comprehensive rankings. Through these action plans, the C-IF scoring-mediated median ranking model offers a structured approach to effectively address decision-analytic challenges in complex and uncertain environments.

Figure 2. The framework of the C-IF scoring-mediated median ranking model.

Consider a multiple criteria decision-analytic problem where $\mathit{A}=\left\{{\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_a\right\}$ represents a finite collection of a ($\mathit{a}\ge 2$) competing alternatives, and $\mathfrak{X}=\left\{{ }_1,{ }_2,\cdots ,{ }_z\right\}$ is a finite collection of $\mathit{z}$ ($\mathit{z}\ge 2$) evaluative criteria. The C-IF performance rating of each competing alternative ${\alpha }_i\in \mathit{A}$ concerning an evaluative criterion ${\mathfrak{z}}_j\in \mathfrak{Z}$ is symbolized by a C-IF number $C_{{\mathcal{C}}_i}\left({\mathfrak{z}}_j\right)$, denoted as $\left(m_{{\mathcal{C}}_i}\left({\mathfrak{z}}_j\right),n_{{\mathcal{C}}_i}\left({\mathfrak{z}}_j\right);r_{{\mathcal{C}}_i}\left({\mathfrak{z}}_j\right)\right)$. Henceforth, for the sake of notation simplicity, let $C_{\mathit{ij}}\left(=\left(m_{\mathit{ij}},n_{\mathit{ij}};r_{\mathit{ij}}\right)\right)=C_{{\mathcal{C}}_i}\left({\mathfrak{z}}_j\right)\left(=\left(m_{{\mathcal{C}}_i}\left({\mathfrak{z}}_j\right),n_{{\mathcal{C}}_i}\left({\mathfrak{z}}_j\right);r_{{\mathcal{C}}_i}\left({\mathfrak{z}}_j\right)\right)\right)$. The membership degree $m_{\mathit{ij}}\left(:=m_{{\mathcal{C}}_i}\left({\mathfrak{z}}_j\right)\right):\mathfrak{Z}\to \left[0,1\right]$ and non-membership degree $n_{{\mathcal{C}}_i}\left({\mathfrak{z}}_j\right)):\mathfrak{Z}\to \left[0,1\right]$ embedded in the C-IF performance rating $C_{\mathit{ij}}$ depict how well or poorly ${\alpha }_i$ performs in terms of ${\mathfrak{z}}_j$. It is essential to note that they must adhere to the axiom condition $0\le m_{\mathit{ij}}+n_{\mathit{ij}}\le 1$. The hesitancy degree $h_{\mathit{ij}}\left(:=h_{{\mathcal{C}}_i}\left({ }_j\right)\right)$ is linked to $C_{\mathit{ij}}$, and $h_{\mathit{ij}}=1-m_{\mathit{ij}}-n_{\mathit{ij}}$. Each competing alternative ${\alpha }_i$ is characterized by a C-IF set ${\mathcal{C}}_i$, consisting of $\mathit{z}$ C-IF numbers, represented as:

(12) ${\mathcal{C}}_i=\left\{〈{\mathfrak{z}}_j,m_{\mathit{ij}},n_{\mathit{ij}};r_{\mathit{ij}}〉{|}_j\in \mathfrak{X}\right\}=\left\{〈{\mathfrak{z}}_j,O_{R_i}\left(m_{\mathit{ij}},n_{\mathit{ij}}\right)〉|{\mathfrak{z}}_j\in \mathit{ℨ}\right\}$(12)

where the inherent circular function $O_{R_i}$ with the radius $r_{\mathit{ij}}\left(:=r_{{\mathcal{C}}_i}\left({\mathfrak{z}}_j\right)\right):\mathit{ℨ}\to \left[0,\sqrt{2}\right]$ associated with ${\mathcal{C}}_i$ is enunciated as follows:

(13) $O_{R_i}\left(m_{\mathit{ij}},n_{\mathit{ij}}\right)=\left\{〈\mathit{l},l^{\prime }〉|\mathit{l},l^{\prime }\in \left[0,1\right],\sqrt{{\left({\mathit{m}}_{\mathit{ij}}-\mathit{l}\right)}^2+{\left({\mathit{n}}_{\mathit{ij}}-l^{\prime }\right)}^2}\le r_{\mathit{ij}},\text{and\hspace{0.17em}}\mathit{l}+l^{\prime }\le 1\right\}$(13)

Similarly, the C-IF importance rating for each evaluative criterion ${\mathfrak{z}}_j$ is symbolized by a C-IF number ${\mathcal{W}}_j$, expressed as $\left(w_j,{\omega }_j;r_j\right)$, ensuring adherence to the axiom condition $0\le w_j+{\omega }_j\le 1$. Considering that all evaluative criteria are crucial factors throughout the decision-analytic process, the C-IF importance rating ${\mathcal{W}}_j$ should adhere to the condition $\left(w_j,{\omega }_j;r_j\right)\ne \left(1,0;r_j\right)$ for ${\mathfrak{z}}_j\in \mathfrak{Z}$.

In situations of C-IF uncertainties, the formulated C-IF scoring mechanism unveils specific valuable and beneficial properties, elucidated in Theorems 2 to 4. In this regard, the derived $S^{\rho }$ value serves as an instrument to establish the precedence relationship among C-IF performance ratings across distinct C-IF sets ${\mathcal{C}}_i$ (for ${\alpha }_i\in \mathit{A}$) concerning a specific criterion ${\mathfrak{z}}_j$. When contemplating a C-IF performance rating $C_{\mathit{ij}}=\left(m_{\mathit{ij}},n_{\mathit{ij}};r_{\mathit{ij}}\right)$ within the C-IF set ${\mathcal{C}}_i$, the corresponding interval-valued estimates for $m_{\mathit{ij}}$ and $n_{\mathit{ij}}$ are specifically defined as $\left[m_{ij}^{-},m_{ij}^{+}\left]=\right[\max \left\{0,m_{\mathit{ij}}-r_{\mathit{ij}}/\sqrt{2}\right\},\min \left\{1,m_{\mathit{ij}}+r_{\mathit{ij}}/\sqrt{2}\right\}\right]$ using Equation (8) and $\left[n_{ij}^{-},n_{ij}^{+}\left]=\right[\max \left\{0,n_{\mathit{ij}}-r_{\mathit{ij}}/\sqrt{2}\right\},\min \left\{1,n_{\mathit{ij}}+r_{\mathit{ij}}/\sqrt{2}\right\}\right]$ using Equation (9) in alignment with Definition 5. Following this, the interval-valued estimate linked to $C_{\mathit{ij}}$ is specified as ${\tilde{I}}_{\mathit{ij}}=\left(\left[m_{ij}^{-},m_{ij}^{+}\left],\right[n_{ij}^{-},n_{ij}^{+}\right]\right)$ using Equation (10). Next, as outlined in Definition 6, the selection of an appropriate value for the relative significance parameter $\mathit{\rho }$ enables the formulation of a C-IF scoring strategy for $C_{\mathit{ij}}$. The C-IF scoring outcome is generated using Equation (11) as follows: $S^{\rho }\left(C_{\mathit{ij}}\right)=\mathit{\rho }\left[m_{ij}^{-}+m_{ij}^{-}\left(1-m_{ij}^{-}-n_{ij}^{-}\right)\left]+\left(1-\mathit{\rho }\right)\right[m_{ij}^{+}+m_{ij}^{+}\left(1-m_{ij}^{+}-n_{ij}^{+}\right)\right]$. A larger $S^{\rho }\left(C_{\mathit{ij}}\right)$ value signifies superior performance and a stronger preference for $C_{\mathit{ij}}$. In contrast, when the $S^{\rho }\left(C_{\mathit{ij}}\right)$ value is smaller, it indicates poorer performance and a weaker preference for $C_{\mathit{ij}}$. When two C-IF scores are equal, it implies an indifferent preference. Consequently, organizing the considered alternatives based on their respective $S^{\rho }\left(C_{\mathit{ij}}\right)$ values allows for the accurate generation of criterion-specific precedence rankings.

The ordering of precedence ranks for each criterion can be established by assessing the C-IF scores obtained. Let ${\mathfrak{p}}_{\mathit{ij}}$ signify the criterion-specific precedence rank assigned to an alternative ${\alpha }_i\in \mathit{A}$ concerning criterion ${\mathfrak{z}}_j\in \mathfrak{Z}$, where $\mathit{i}\in \left\{1,2,\cdots ,\mathit{a}\right\}$ and $\mathit{j}\in \left\{1,2,\cdots ,\mathit{z}\right\}$. In cases where there are ties in the criterion-specific precedence ranks, the tied alternatives can be assigned a mean rank. By way of illustration, if three alternatives share the second rank, the precedence rank assigned would be the average of the second, third, and fourth ranks, i.e., ${\mathfrak{p}}_{\mathit{ij}}=\left(2+3+4\right)/3=3$. This approach facilitates the production of the $\mathit{a}\times \mathit{z}$ criterion-specific precedence rank matrix $\mathfrak{P}$ as outlined below:

(14) $\mathfrak{P}={\left[{\mathfrak{p}}_{\mathit{ij}}\right]}_{\mathit{a}\times \mathit{z}}=\left[\begin{array}{c}\begin{array}{cc}{\mathfrak{p}}_{11}& {\mathfrak{p}}_{12}\\ {\mathfrak{p}}_{21}& {\mathfrak{p}}_{22}\\ ⋮& ⋮\end{array}\begin{array}{cc}\cdots & {\mathfrak{p}}_{1\mathit{z}}\\ \cdots & {\mathfrak{p}}_{2\mathit{z}}\\ \ddots & ⋮\end{array}\\ \begin{array}{cc}{\mathfrak{p}}_{\mathit{a}1}& {\mathfrak{p}}_{\mathit{a}2}\end{array}\begin{array}{cc}\cdots & {\mathfrak{p}}_{\mathit{az}}\end{array}\end{array}\right]$(14)

where $1\le {\mathfrak{p}}_{\mathit{ij}}\le \mathit{a}$ and $\sum _{\mathit{i}=1}^a{\mathfrak{p}}_{\mathit{ij}}=\mathit{a}\left(\mathit{a}+1\right)/2$ for all $\mathit{j}\in \left\{1,2,\cdots ,\mathit{z}\right\}$.

Utilizing the C-IF scoring mechanism and the resulting precedence ranks allocated to competing alternatives based on individual criteria, this research advances the formulation of specific concepts for assessing the appositeness of an overall ranking among the competing alternatives. Precisely, the ranking that closely aligns with the rankings derived from precedence relationships, the associated precedence ranks, and the integration of C-IF scoring outcomes (i.e., $S^{\rho }\left(C_{\mathit{ij}}\right)$ values) with the C-IF importance rating ${\mathcal{W}}_j$ is referred to as the comprehensive (or overall) ranking. Employing such a comprehensive precedence ranking as a reference can produce a consensus on all precedence rankings within the criterion-specific precedence rank matrix $\mathfrak{P}$.

To attain the stated purpose, this research exploits the scalar multiplication operation in C-IF circumstances, as represented in (7), to formulate the weighted C-IF performance rating symbolized as ${\mathfrak{C}}_{ij}^{\rho }$ for a C-IF performance rating $C_{\mathit{ij}}$ within each C-IF set ${\mathcal{C}}_i$. Given the criticality of all evaluative criteria in decision-making, the C-IF importance rating ${\mathcal{W}}_j$ should fulfill the condition $\left(w_j,{\omega }_j;r_j\right)\ne \left(1,0;r_j\right)$ for all ${\mathfrak{z}}_j\in \mathfrak{Z}$. It is essential to note that the $S^{\rho }\left(C_{\mathit{ij}}\right)$ value is non-negative, as stipulated in Theorem 2. The application of the scalar multiplication operation, as outlined in Equation (7), generates ${\mathfrak{C}}_{ij}^{\rho }$ through the operation $S^{\rho }\left(C_{\mathit{ij}}\right){\mathcal{W}}_j$, namely $S^{\rho }\left(C_{\mathit{ij}}\right)\left(w_j,{\omega }_j;\mathit{rj}\right)$, in the following manner:

(15) ${\mathfrak{C}}_{ij}^{\rho }=\left(1-{\left(1-w_j\right)}^{S^{\rho }\left(C_{\mathit{ij}}\right)},{\left({\omega }_j\right)}^{S^{\rho }\left(C_{\mathit{ij}}\right)};r_j\right)$(15)

Under a specific setting value of the relative significance parameter $\mathit{\rho }$, let ${\widetilde{{\mathit{I}}_{\mathfrak{C}}}}_{\mathit{ij}}^{\mathit{\rho }}$ represent the interval-valued estimate corresponding to the weighted C-IF performance rating ${\mathfrak{C}}_{ij}^{\rho }$; it is externalized using the following format:

(16) ${\widetilde{{\mathit{I}}_{\mathfrak{C}}}}_{\mathit{ij}}^{\mathit{\rho }}=\left(\left[m_{{\mathfrak{C}}_{ij}^{\rho }}^{-},m_{{\mathfrak{C}}_{ij}^{\rho }}^{+}\right],\left[n_{{\mathfrak{C}}_{ij}^{\rho }}^{-},n_{{\mathfrak{C}}_{ij}^{\rho }}^{+}\right]\right)$(16)

where the interval-valued estimates contained in ${\widetilde{{\mathit{I}}_{\mathfrak{C}}}}_{\mathit{ij}}^{\mathit{\rho }}$ are specified with the ensuing means:

(17) $\left[m_{{\mathfrak{C}}_{ij}^{\rho }}^{-},m_{{\mathfrak{C}}_{ij}^{\rho }}^{+}\right]=\left[max\left\{0,1-{\left(1-w_j\right)}^{S^{\rho }\left(C_{\mathit{ij}}\right)}-\frac{r_j}{\sqrt{2}}\right\},\min \left\{1,1-{\left(1-w_j\right)}^{S^{\rho }\left(C_{\mathit{ij}}\right)}+\frac{r_j}{\sqrt{2}}\right\}\right]$(17)

(18) $\left[n_{{\mathfrak{C}}_{ij}^{\rho }}^{-},n_{{\mathfrak{C}}_{ij}^{\rho }}^{+}\right]=\left[max\left\{0,{\left({\omega }_j\right)}^{S^{\rho }\left(C_{\mathit{ij}}\right)}-\frac{r_j}{\sqrt{2}}\right\},\min \left\{1,{\left({\omega }_j\right)}^{S^{\rho }\left(C_{\mathit{ij}}\right)}+\frac{r_j}{\sqrt{2}}\right\}\right]$(18)

By keeping the relative significance parameter $\mathit{\rho }$ constant, the comprehensive C-IF scoring function $\mathfrak{S}\left({\mathfrak{C}}_{ij}^{\rho }\right)$ for the weighted C-IF performance rating ${\mathfrak{C}}_{ij}^{\rho }$ is formulated as follows:

(19) $\mathfrak{S}({ℭ}_{ij}^{\rho })=\mathit{\rho }\left[m_{{ℭ}_{ij}^{\rho }}^{-}+m_{{ℭ}_{ij}^{\rho }}^{-}\left(1-m_{{ℭ}_{ij}^{\rho }}^{-}-n_{{ℭ}_{ij}^{\rho }}^{-}\right)\right]+\left(1-\mathit{\rho }\right)\left[m_{{ℭ}_{ij}^{\rho }}^{+}+m_{{ℭ}_{ij}^{\rho }}^{+}\left(1-m_{{ℭ}_{ij}^{\rho }}^{+}-n_{{ℭ}_{ij}^{\rho }}^{+}\right)\right]$(19)

Recall that, within the context of two collections of competing alternatives, $\mathit{A}$, and evaluative criteria, $\mathfrak{Z}$, the criterion-specific precedence rank of ${\alpha }_i\in \mathit{A}$ concerning ${\mathfrak{z}}_j\in \mathfrak{Z}$ is represented as ${\mathfrak{p}}_{\mathit{ij}}$, determined by the C-IF scoring outcomes, i.e., $S^{\rho }\left(C_{\mathit{ij}}\right)$ values. The initial disagreement metric ${\bar{d}}_{\mathit{ik}}$, signifying the appropriateness of ranking ${\alpha }_i$ as k-th, is explained as the cumulative sum of absolute discrepancies between k-th rank and ${\mathfrak{p}}_{\mathit{ij}}$:

(20) ${\bar{d}}_{\mathit{ik}}={\sum }_{j=1}^z\left|{\mathfrak{p}}_{\mathit{ij}}-\mathit{k}\right|$(20)

However, the initial disagreement metric ${\bar{d}}_{\mathit{ik}}$ fails to fully encapsulate C-IF decision information due to its omission of factors related to C-IF performance and importance ratings. In an effort to incorporate influential decision elements into the disagreement measurement, this research advocates for a more comprehensive technique that thoroughly leverages the impacts of both C-IF performance and importance ratings. The research proposes a specific substitution, advocating for replacing the original $C_{\mathit{ij}}$ and ${\mathcal{W}}_j$ with the $\mathfrak{S}\left({\mathfrak{C}}_{ij}^{\rho }\right)$ value. This substitution is endorsed because the $\mathfrak{S}\left({\mathfrak{C}}_{ij}^{\rho }\right)$ concept accurately reflects the performance of competing alternatives regarding each evaluative criterion, establishing precedence relationships across various alternatives for each criterion. This adjustment, utilizing the $\mathfrak{S}\left({\mathfrak{C}}_{ij}^{\rho }\right)$ value, promises a more fitting and efficacious determination of a novel disagreement metric. Consequently, this research integrates the $\mathfrak{S}\left({\mathfrak{C}}_{ij}^{\rho }\right)$ value into the disagreement measurement, departing from the initial disagreement metric ${\bar{d}}_{\mathit{ik}}$. Under a specific parameter configuration involving the relative significance parameter $\mathit{\rho }$, the resulting comprehensive disagreement metric $d_{\mathit{ik}}$, which incorporates the $\mathfrak{S}\left({\mathfrak{C}}_{ij}^{\rho }\right)$ value into ${\bar{d}}_{\mathit{ik}}$, is derived by summing the absolute discrepancies cumulatively for ${\alpha }_i$ to achieve the k-th overall rank:

(21) $d_{\mathit{ik}}={\sum }_{j=1}^z\left(\mathfrak{S}\left({\mathfrak{C}}_{ij}^{\rho }\right)\cdot \left|{\mathfrak{p}}_{\mathit{ij}}-\mathit{k}\right|\right)$(21)

The proposed C-IF scoring-mediated median ranking model, utilizing the C-IF scoring mechanism for multiple criteria decision analytics, seeks to establish a comprehensive (or consensus) ranking of competing alternatives, commonly known as the median ranking. The primary objective of designing a median ranking is to minimize disparities from individual criterion-specific ranks as much as possible. Attaining this goal involves the effective creation of a linear assignment model within the C-IF context. Managing intricate uncertain issues in multiple criteria decision analytics often entails considering a group of overall sorting outcome for competing options, which can be effectively addressed through a linear programming formulation. Significantly, when concentrating on an overall ranking, the multiplex nature of multiple-criteria decision analytics in a C-IF state of affairs can be more productively tackled by formulating it as a C-IF assignment problem. To provide further detail, the symbol ${\mathfrak{b}}_{\mathit{ik}}$ is defined as a binary variable, governed to the values of either 0 or 1. Let’s think about $\mathfrak{B}$ as an $\mathit{a}\times \mathit{a}$ permutation matrix, where each entry ${\mathfrak{b}}_{\mathit{ik}}$ equals 1 if alternative ${\alpha }_i$ is designated to a comprehensive precedence rank k, and ${\mathfrak{b}}_{\mathit{ik}}$ equals 0 otherwise. The way to represent the permutation matrix $\mathfrak{B}$ is in this fashion:

(22) $\mathfrak{B}={\left[{\mathfrak{b}}_{\mathit{ik}}\right]}_{\mathit{a}\times \mathit{a}}=\left[\begin{array}{c}\begin{array}{cc}{\mathfrak{b}}_{11}& {\mathfrak{b}}_{12}\\ {\mathfrak{b}}_{21}& {\mathfrak{b}}_{22}\\ ⋮& ⋮\end{array}\begin{array}{cc}\cdots & {\mathfrak{b}}_{1\mathit{a}}\\ \cdots & {\mathfrak{b}}_{2\mathit{a}}\\ \ddots & ⋮\end{array}\\ \begin{array}{cc}{\mathfrak{b}}_{\mathit{a}1}& {\mathfrak{b}}_{\mathit{a}2}\end{array}\begin{array}{cc}\cdots & {\mathfrak{b}}_{\mathit{aa}}\end{array}\end{array}\right]$(22)

In this formulation, the binary variable ${\mathfrak{b}}_{\mathit{ik}}$, subject to the constraints $\sum _{\mathit{k}=1}^a{\mathfrak{b}}_{\mathit{ik}}=1$ and $\sum _{\mathit{i}=1}^a{\mathfrak{b}}_{\mathit{ik}}=1$, serves to guarantee that each competing alternative ${\alpha }_i$ is designated to a singular rank in the comprehensive precedence ranking (as per the former condition) and that each rank k is exclusively assigned to a single alternative (as per the latter condition).

To derive the optimal comprehensive precedence ranking of competing alternatives with minimal discordance among all criterion-specific precedence ranks, the notion of the comprehensive disagreement metric $d_{\mathit{ik}}$ is utilized to formulate the objective function in the proposed C-IF scoring-mediated median ranking model. Identifying the optimal comprehensive precedence ranking, achieved by minimizing ${\sum }_{i=1}^a{\sum }_{k=1}^a\left(d_{\mathit{ik}}\cdot {\mathfrak{b}}_{\mathit{ik}}\right)$, is outlined in the following assignment model:

$\min {\sum }_{i=1}^a{\sum }_{k=1}^a\left(d_{\mathit{ik}}\cdot {\mathfrak{b}}_{\mathit{ik}}\right)$

(23) $\mathrm{subject}\mathrm{to}{\sum }_{k=1}^a{\mathfrak{b}}_{\mathit{ik}}=1,\mathit{i}=1,2,\cdots ,\mathit{a}$(23)

$\mathit{\hspace{0.17em}}{\sum }_{i=1}^a{\mathfrak{b}}_{\mathit{ik}}=1,\mathit{k}=1,2,\cdots ,\mathit{a},{\mathfrak{b}}_{\mathit{ik}}=0\mathrm{or}1\mathrm{for}\mathit{i},\mathit{k}\in \left\{1,2,\cdots ,\mathit{a}\right\}.$

The proposed C-IF scoring-mediated median ranking model is succinctly formulated by means of a simple zero-one linear programming structure. Employing the Hungarian method facilitates the efficient solution of the model to acquire the optimal comprehensive precedence ranking, minimizing $\sum _{\mathit{i}=1}^a\sum _{\mathit{k}=1}^a\left(d_{\mathit{ik}}\cdot {\mathfrak{b}}_{\mathit{ik}}\right)$. This technique effectually tackles multiple criteria decision-analytic challenges in C-IF circumstances. The optimal comprehensive precedence ranks for the set of competing alternatives (denoted by $\mathit{A}=\left\{{\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_a\right\}$) are produced through a multiplication of the alternative vector $\left({\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_a\right)$ by the optimal permutation matrix $\widehat{\mathfrak{B}}$ as outlined below:

(24) $\mathit{A}\cdot \widehat{\mathfrak{B}}=\left({\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_a\right)\cdot \left[\begin{array}{ccc}{\widehat{\mathfrak{b}}}_{11}& {\widehat{\mathfrak{b}}}_{12}\cdots & {\widehat{\mathfrak{b}}}_{1\mathit{a}}\\ {\widehat{\mathfrak{b}}}_{21}& {\widehat{\mathfrak{b}}}_{22}\cdots & {\widehat{\mathfrak{b}}}_{2\mathit{a}}\\ ⋮& ⋮\ddots & ⋮\\ {\widehat{\mathfrak{b}}}_{\mathit{a}1}& {\widehat{\mathfrak{b}}}_{\mathit{a}2}\cdots & {\widehat{\mathfrak{b}}}_{\mathit{aa}}\end{array}\right]$(24)

In the end, the optimal comprehensive alternative, often referred to as the most favored option, can be discerned among the competing alternatives based on evaluative criteria, thereby facilitating decision analytics.

The C-IF scoring-mediated median ranking methodology involves distinct phases, each crucial in conducting comprehensive scoring assessments and establishing precedence rankings for competing alternatives in multiplex and complicated decision-analytic scenarios. In the initial phase (referring to Steps 1–3), the emphasis is on formulating the decision-analytic task and constructing the necessary data. This entails defining a task with multiple criteria, encompassing finite collections of evaluative criteria and competing alternatives. C-IF numbers are then utilized to articulate both the C-IF performance and importance ratings. Additionally, the decision-maker (or analyst) formulates C-IF sets related to competing alternatives and their inherent circular functions. In the second phase (referring to Steps 4–6), the C-IF scoring mechanism is being implemented as part of the methodology. This entails the generation of interval-valued estimates regarding membership and non-membership constituents. The decision-maker (or analyst) needs to select a suitable value for the relative significance parameter, and this value, along with the interval-valued estimates, is utilized to calculate the C-IF scoring function. Subsequently, criterion-specific precedence ranks are determined based on the descending order of C-IF scores. In instances of ties, mean ranks are assigned, leading to the construction of the criterion-specific precedence rank matrix. Transitioning to the third phase (referring to Steps 7–9), the methodology involves deriving comprehensive C-IF scoring functions. In this phase, the scalar multiplication operation is employed to calculate the weighted C-IF performance ratings. Apart from that, the interval-valued estimates for these ratings are determined, contributing to the establishment of comprehensive C-IF scoring functions. In the concluding phase (referring to Steps 10–12), the methodology focuses on modeling with comprehensive disagreement metrics. During this phase, the comprehensive disagreement metric for each alternative’s specific overall rank is generated. Subsequently, a permutation matrix is formulated to create the C-IF scoring-mediated median ranking model. Solving the model results in the production of the best-fitting permutation matrix, enabling the generation of the optimal comprehensive ranking for competing alternatives.

The C-IF scoring-mediated median ranking algorithm is comprised of the following 12 procedural sequences:

Phase I (Steps 1–3): Defining the Problem and Constructing Data.

Step 1: Formulate a decision-analytic task encompassing multiple criteria, including a finite collection of evaluative criteria ($\mathfrak{X}=\left\{{ }_1,{ }_2,\cdots ,{ }_z\right\}$), and a finite collection of examined competing alternatives ($\mathit{A}=\left\{{\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_a\right\}$).

Step 2: Employ a C-IF number $\left(m_{\mathit{ij}},n_{\mathit{ij}};r_{\mathit{ij}}\right)$ to articulate the C-IF performance rating $C_{\mathit{ij}}$ for each ${\alpha }_i\in \mathit{A}$ in relation to ${\mathfrak{z}}_j\in \mathfrak{Z}$. Simultaneously, utilize a C-IF number $\left(w_j,{\omega }_j;r_j\right)$ to signify the C-IF importance rating ${\mathcal{W}}_j$ for each ${\mathfrak{z}}_j$.

Step 3: Apply Equations (12) and (13) to formulate a C-IF set ${\mathcal{C}}_i$ and its corresponding inherent circular function $O_{R_i}$, respectively.

Phase II (Steps 4–6): Implementing the C-IF Scoring Mechanism.

Step 4: Generate the interval-valued estimates $\left[m_{ij}^{-},m_{ij}^{+}\right]$ for the membership degree $m_{\mathit{ij}}$ and $\left[n_{ij}^{-},n_{ij}^{+}\right]$ for the non-membership degree $n_{\mathit{ij}}$ as per Equations (8) and (9), respectively.

Step 5: Select an appropriate value for the relative significance parameter $\mathit{\rho }$, and employ this value along with the interval-valued estimates to calculate the C-IF scoring function $S^{\rho }\left(C_{\mathit{ij}}\right)$ for $C_{\mathit{ij}}$ as per Equation (11).

Step 6: Identify the criterion-specific precedence rank ${\mathfrak{p}}_{\mathit{ij}}$ by arranging the established C-IF scoring outcomes in descending order. In instances of ties, allocate a mean rank. Build the criterion-specific precedence rank matrix $\mathfrak{P}$ outlined in Equation (14).

Phase III (Steps 7–9): Dering Comprehensive C-IF Scoring Functions.

Step 7: Utilize the scalar multiplication operation to figure out the weighted C-IF performance rating ${\mathfrak{C}}_{ij}^{\rho }$ using Equation (15).

Step 8: Derive the interval-valued estimate ${\widetilde{{\mathit{I}}_{\mathfrak{C}}}}_{\mathit{ij}}^{\mathit{\rho }}$ for ${\mathfrak{C}}_{ij}^{\rho }$ using Eq. (16), which includes $\left[m_{{\mathfrak{C}}_{ij}^{\rho }}^{-},m_{{\mathfrak{C}}_{ij}^{\rho }}^{+}\right]$ and $\left[n_{{\mathfrak{C}}_{ij}^{\rho }}^{-},n_{{\mathfrak{C}}_{ij}^{\rho }}^{+}\right]$ as determined by Equations. (17) and (18), respectively.

Step 9: Determine the comprehensive C-IF scoring function $\mathfrak{S}\left({\mathfrak{C}}_{ij}^{\rho }\right)$ for ${\mathfrak{C}}_{ij}^{\rho }$ as per Eq. (19) with the relative significance parameter $\mathit{\rho }$ unchanged.

Phase IV (Steps 10–12): Modeling with comprehensive disagreement metrics.

Step 10: Calculate the comprehensive disagreement metric $d_{\mathit{ik}}$ for ${\alpha }_i$ to be assigned the k-th overall rank as per Equation (21).

Step 11: Create a permutation matrix $\mathfrak{B}$ using (22) to formulate the C-IF scoring-mediated median ranking model outlined in Equation (23).

Step 12: Solve the C-IF scoring-mediated median ranking model to receive the optimal permutation matrix $\widehat{\mathfrak{B}}$, and subsequently generate the optimal comprehensive ranking for ${\alpha }_i\in \mathit{A}$ as per Equation (24).

Model Application and Validation

This section offers a tangible example to elucidate the practical relevance of the proposed C-IF scoring-mediated median ranking model. The study delves into a particular case of multiple-expert supplier appraisement to exhibit the application of the model in an actual scenario. Furthermore, the section incorporates an in-depth comparison and investigations to highlight the merits and conveniences of the suggested methodology, grounded in the novel C-IF scoring technique. Through the real-world illustration and an examination of comparisons, the current section endeavors to emphasize the model’s efficacy and benefits in addressing complex decision problems marked by C-IF uncertainty and ambiguity.

Validation Through Practical Application

The application of the established C-IF scoring-mediated median ranking model is demonstrated in addressing a practical challenge of assessing and selecting competing suppliers within an engineering company, leveraging the expertise of multiple experts. Building on the framework outlined by Otay and Kahraman (2022), the research focuses on the appraisement of a single component chosen from a range of supplied components. The initial phase of the assessment process involves compiling and listing potential competing suppliers for consideration. The assessments commence by giving thought to environmental factors relevant to the engineering company, resulting in the exclusion of suppliers that do not align with specified environmental requirements, such as pollution control systems and International Organization for Standardization (ISO) standards.

Step 1 involved referencing the problem setting outlined in Otay and Kahraman’s work (2022) and the hierarchical configuration exhibited in Figure 3. In this step, following the elimination process based on environmental concerns, the remaining competing suppliers (competitors #1 to #3) undergo a thorough appraisement. Specifically, three competing alternatives, namely Suppliers #1, #2, and #3 (represented as ${\alpha }_1$, ${\alpha }_2$, and ${\alpha }_3$, respectively), were taken into account. The collection of competing alternatives was signified as $\mathit{A}=\left\{{\alpha }_1,{\alpha }_2,{\alpha }_3\right\}$. This assessment is conducted across three core dimensions: cost, service, and technology/quality. These dimensions undergo a more detailed investigation through nine evaluative criteria, encompassing factors like technological capability, flexibility, and handling and transportation. The hierarchical configuration of the multiple-expert supplier appraisement is illustrated in Figure 3, delineating the relationships and levels among dimensions, evaluative criteria, and competing suppliers throughout the analytical procedure.

Figure 3. Hierarchical configuration and C-IF data for the multiple-expert supplier appraisement.

The three competing suppliers underwent evaluation based on three dimensions: the cost dimension, the service dimension, and the technology/quality dimension. The cost dimension comprised three factors: price (${\mathfrak{z}}_1$), representing the amount of money the supplier requested for goods or services; terms of payments (${\mathfrak{z}}_2$), indicating the deadlines and conditions for payment; and handling and transportation (${\mathfrak{z}}_3$), reflecting the means and effectiveness of handling and moving supplied goods to their final destination. The service dimension included flexibility (${\mathfrak{z}}_4$), demonstrating the supplier’s ability to adapt products or services to the changing needs of the engineering company; on-time delivery (${\mathfrak{z}}_5$), assessing the supplier’s capability to deliver within agreed-upon timeframes; and past performance (${\mathfrak{z}}_6$), evaluating the supplier’s reliability, quality, and commitment to terms. The technology/quality dimension encompassed quality management systems (${\mathfrak{z}}_7$), indicating the systems and processes ensuring product or service quality; technological capability (${\mathfrak{z}}_8$), reflecting the supplier’s level of technological expertise and innovation; and R&D studies (${\mathfrak{z}}_9$), revealing the supplier’s engagement in research and development activities to enhance products or services. The significance of ${\mathfrak{z}}_1$–${\mathfrak{z}}_3$ in the multiple-expert supplier appraisal case is clarified as follows: (1) Price plays a pivotal role in supplier evaluation, directly influencing the overall cost structure of the engineering company; (2) Terms of payments have an impact on the financial stability and cash flow management of the engineering company; and (3) Efficient handling and transportation can enhance timely deliveries and reduce the risk of damages. The significance of ${\mathfrak{z}}_4$–${\mathfrak{z}}_6$ is elucidated as follows: (1) Flexibility is vital for adapting to dynamic market conditions and meeting the specific requirements of the engineering company; (2) On-time delivery is essential to avoid disruptions in the supply chain and uphold production schedules; and (3) Past performance serves as an indicator of the supplier’s track record and reliability. The significance of ${\mathfrak{z}}_7$–${\mathfrak{z}}_9$ is detailed below: (1) Quality management systems are essential for preserving and enhancing the overall quality of the supplied goods; (2) Technological capability impacts the competitiveness and future relevance of the supplied products; and (3) R&D studies signify the supplier’s dedication to innovation, potentially resulting in enhanced and advanced offerings over time. These criteria collectively contribute to a comprehensive evaluation of the three competing suppliers across different dimensions. The collection of evaluative criteria was specified as $\mathfrak{Z}=\left\{{\mathfrak{z}}_1,{\mathfrak{z}}_2,\cdots ,{\mathfrak{z}}_9\right\}$.

In Step 2, aligning with the methodology outlined by Otay and Kahraman (2022), the evaluation opinions of three experts were consolidated for the competing suppliers across nine evaluative criteria. To maintain consistency with the direction of beneficial criteria, the orientation of non-beneficial criteria (such as price ${\mathfrak{z}}_1$) was adjusted, following the approach of Otay and Kahraman. The individual C-IF performance ratings were then amalgamated, considering the same orientation, where higher values indicate superior performance. Using the aggregated outcomes of the assessment information from Otay and Kahraman’s (2022) research, the C-IF performance rating $C_{\mathit{ij}}$ and importance rating ${\mathcal{W}}_j$, enunciated as C-IF numbers $\left(m_{\mathit{ij}},n_{\mathit{ij}};r_{\mathit{ij}}\right)$ and $\left({ }_j,{\omega }_j;{ }_j\right)$ respectively, were illustrated in Figure 3. In Step 3, the C-IF set ${\mathcal{C}}_i$ associated with each ${\alpha }_i\in \mathit{A}$ is determined as per Equations (12) and (13): ${\mathcal{C}}_i=\left\{{\mathfrak{z}}_j,m_{\mathit{ij}},n_{\mathit{ij}};r_{\mathit{ij}}\left|{\mathfrak{z}}_j\in \left\{{\mathfrak{z}}_1,{\mathfrak{z}}_2,\cdots ,{\mathfrak{z}}_9\right\}\right.\right\}=\left\{{\mathfrak{z}}_j,O_{R_i}\left(m_{\mathit{ij}},n_{\mathit{ij}}\right)\left|{\mathfrak{z}}_j\in \left\{{\mathfrak{z}}_1,{\mathfrak{z}}_2,\cdots ,\text{break}{\mathfrak{z}}_9\right\}\right.\right\}$. Here, the circular function $O_{R_i}\left(m_{\mathit{ij}},n_{\mathit{ij}}\right)=\left\{⟨\mathit{l},l^{\prime }⟩\left|\mathit{l},l^{\prime }\in \left[0,1\right],\text{break}\sqrt{{\left({\mathit{m}}_{\mathit{ij}}-\mathit{l}\right)}^2+{\left({\mathit{n}}_{\mathit{ij}}-\mathit{l}{ }^{\mathrm{\prime }}\right)}^2}\le r_{\mathit{ij}},\mathrm{and}\mathit{l}+\mathit{l}{ }^{\mathrm{\prime }}\le 1\right.\right\}$ for $\mathit{i}\in \left\{1,2,3\right\}$ and $\mathit{j}\in \left\{1,2,\cdots ,9\right\}$.

In Step 4, the application of Equations. (8) and (9) yields interval-valued estimates $\left[m_{ij}^{-},m_{ij}^{+}\right]$ for $m_{\mathit{ij}}$ and $\left[n_{ij}^{-},n_{ij}^{+}\right]$ for $n_{\mathit{ij}}$, respectively. The detailed outcomes are showcased in Table 1. Moving to Step 5, the relative significance parameter $\mathit{\rho }$ was allocated a median value of 0.5. Subsequently, for each C-IF performance rating $C_{\mathit{ij}}$, the C-IF scoring function $S^{\rho }\left(C_{\mathit{ij}}\right)$ was calculated using Equation (11). The C-IF scoring results are also displayed in Table 1.

Table 1. Detailed outcomes of interval-valued estimates and C-IF scoring functions ($\mathit{\rho }=0.5$).

	Results for Supplier #1 (${\alpha }_1$):			Results for Supplier #2 (${\alpha }_2$):			Results for Supplier #3 (${\alpha }_3$):
${\mathfrak{z}}_j$	$\left[m_{1j}^{-},m_{1j}^{+}\right]$	$\left[n_{1j}^{-},n_{1j}^{+}\right]$	$S^{\rho }\left(C_{1\mathit{j}}\right)$	$\left[m_{2j}^{-},m_{2j}^{+}\right]$	$\left[n_{2j}^{-},n_{2j}^{+}\right]$	$S^{\rho }\left(C_{2\mathit{j}}\right)$	$\left[m_{3j}^{-},m_{3j}^{+}\right]$	$\left[n_{3j}^{-},n_{3j}^{+}\right]$	$S^{\rho }\left(C_{3\mathit{j}}\right)$
${\mathfrak{z}}_1$	[0.1544, 0.5376]	[0.3364, 0.7196]	0.3162	[0.5087, 0.7393]	[0.1387, 0.3693]	0.6736	[0.5473, 0.6887]	[0.2063, 0.3477]	0.6729
${\mathfrak{z}}_2$	[0.3386, 0.5394]	[0.3386, 0.5394]	0.4724	[0.6500, 0.6500]	[0.2500, 0.2500]	0.7150	[0.5406, 0.8094]	[0.0726, 0.3414]	0.7186
${\mathfrak{z}}_3$	[0.1544, 0.5376]	[0.3364, 0.7196]	0.3162	[0.4898, 0.6482]	[0.2478, 0.4062]	0.6156	[0.5484, 0.7096]	[0.1864, 0.3476]	0.6814
${\mathfrak{z}}_4$	[0.2610, 0.5410]	[0.3400, 0.6200]	0.4095	[0.6457, 0.7503]	[0.1417, 0.2463]	0.7679	[0.4891, 0.7409]	[0.1411, 0.3929]	0.6559
${\mathfrak{z}}_5$	[0.5087, 0.6473]	[0.2467, 0.3853]	0.6297	[0.6459, 0.7901]	[0.1029, 0.2471]	0.7844	[0.5473, 0.6887]	[0.2063, 0.3477]	0.6729
${\mathfrak{z}}_6$	[0.5473, 0.6887]	[0.2063, 0.3477]	0.6729	[0.6345, 0.8395]	[0.0375, 0.2425]	0.8066	[0.6353, 0.9167]	[0.0000, 0.2367]	0.8215
${\mathfrak{z}}_7$	[0.5087, 0.6473]	[0.2467, 0.3853]	0.6297	[0.4891, 0.7409]	[0.1411, 0.3929]	0.6559	[0.6477, 0.8103]	[0.0847, 0.2473]	0.7923
${\mathfrak{z}}_8$	[0.3329, 0.8151]	[0.0519, 0.5341]	0.5341	[0.5471, 0.6489]	[0.2451, 0.3469]	0.6562	[0.4898, 0.6482]	[0.2478, 0.4062]	0.6156
${\mathfrak{z}}_9$	[0.3364, 0.7196]	[0.1544, 0.5376]	0.5211	[0.4898, 0.6482]	[0.2478, 0.4062]	0.6156	[0.5471, 0.6489]	[0.2451, 0.3469]	0.6562

In Step 6, the criterion-specific precedence rank ${\mathfrak{p}}_{\mathit{ij}}$ was generated by organizing the obtained C-IF scoring outcomes in descending order. To illustrate this procedure, let’s take the evaluative criterion ${\mathfrak{z}}_5$. The C-IF scoring outcomes for this criterion were $S^{0.5}\left(C_{15}\right)=0.6297$, $S^{0.5}\left(C_{25}\right)=0.7844$, and $S^{0.5}\left(C_{35}\right)=0.6729$. Based on these scores, it was recognized that ${\alpha }_2$ outperforms ${\alpha }_3$ and subsequently outperforms ${\alpha }_1$ in relation to criterion ${\mathfrak{z}}_5$. Accordingly, the precedence ranks were assigned on this wise: $\mathit{Accessisdenied}$, ${\mathfrak{p}}_{25}=1$, and ${\mathfrak{p}}_{35}=2$. These outcomes stipulate the precedence orders of ${\alpha }_1,{\alpha }_2,$ and ${\alpha }_3$ concerning ${\mathfrak{z}}_5$. It is acknowledged that $\mathit{a}=3$ and $\mathit{z}=9$ in this case study. Following Equation (14), a $3\times 9$ criterion-specific precedence rank matrix $\mathfrak{P}$ in the situation where $\mathit{\rho }=0.5$ was constructed by this action:

(25) $\mathfrak{B}={\left[\mathfrak{p}{\mathit{ }}_{\mathit{ij}}\right]}_{3\times 9}=\left[\begin{array}{ccc}3& 3& 3\\ 1& 2& 2\\ 2& 1& 1\end{array}\begin{array}{ccc}3& 3& 3\\ 1& 1& 2\\ 2& 2& 1\end{array}\begin{array}{ccc}3& 3& 3\\ 2& 1& 2\\ 1& 2& 1\end{array}\right]$(25)

In Step 7, the weighted C-IF performance rating ${\mathfrak{C}}_{ij}^{0.5}$ ($\mathit{\rho }=0.5$) for each C-IF performance rating $C_{\mathit{ij}}$ was calculated using the scalar multiplication operation, as described in Equation (15). In Step 8, (16) was employed to generate the interval-valued estimate ${\widetilde{{\mathit{I}}_{\mathfrak{C}}}}_{\mathit{ij}}^{0.5}$ associated with each $C_{\mathit{ij}}$, defined as ${\widetilde{{\mathit{I}}_{\mathfrak{C}}}}_{\mathit{ij}}^{0.5}=\left(\left[m_{{\mathfrak{C}}_{ij}^{0.5}}^{-},m_{{\mathfrak{C}}_{ij}^{0.5}}^{+}\left],\right[n_{{\mathfrak{C}}_{ij}^{0.5}}^{-},n_{{\mathfrak{C}}_{ij}^{0.5}}^{+}\right]\right)$. Specifically, Equations (17) and (18) were used to calculate $\left[m_{{\mathfrak{C}}_{ij}^{0.5}}^{-},m_{{\mathfrak{C}}_{ij}^{0.5}}^{+}\right]$ and $\left[n_{{\mathfrak{C}}_{ij}^{0.5}}^{-},n_{{\mathfrak{C}}_{ij}^{0.5}}^{+}\right]$, respectively. In Step 9, Equation (19) was utilized to determine the comprehensive C-IF scoring function $\mathfrak{S}\left({\mathfrak{C}}_{ij}^{0.5}\right)$ corresponding to ${\mathfrak{C}}_{ij}^{0.5}$. The third, fourth, and final columns of Table 2 display the computed values of ${\mathfrak{C}}_{ij}^{0.5}$, ${\widetilde{{\mathit{I}}_{\mathfrak{C}}}}_{\mathit{ij}}^{0.5}$, and $\mathfrak{S}\left({\mathfrak{C}}_{ij}^{0.5}\right)$, respectively.

Table 2. Calculation outcomes related to comprehensive C-IF scoring functions.

${\alpha }_i$	${\mathfrak{z}}_j$	${\mathfrak{C}}_{ij}^{0.5}$	${\widetilde{{\mathit{I}}_{\mathfrak{C}}}}_{\mathit{ij}}^{0.5}$	$\mathfrak{S}\left({\mathfrak{C}}_{ij}^{0.5}\right)$
${\alpha }_1$	${\mathfrak{z}}_1$	(0.2359, 0.7240; 0.1580)	([0.1242, 0.3476], [0.6123, 0.8357])	0.2204
	${\mathfrak{z}}_2$	(0.1706, 0.7788; 0.1120)	([0.0914, 0.2498], [0.6996, 0.8580])	0.1667
	${\mathfrak{z}}_3$	(0.1751, 0.7871; 0.1580)	([0.0634, 0.2868], [0.6754, 0.8988])	0.1567
	${\mathfrak{z}}_4$	(0.2195, 0.7404; 0.1740)	([0.0965, 0.3425], [0.6174, 0.8634])	0.1980
	${\mathfrak{z}}_5$	(0.4131, 0.5311; 0.1740)	([0.2901, 0.5361], [0.4080, 0.6541])	0.4059
	${\mathfrak{z}}_6$	(0.2455, 0.6883; 0.1610)	([0.1316, 0.3593], [0.5745, 0.8021])	0.2358
	${\mathfrak{z}}_7$	(0.2185, 0.7189; 0.0740)	([0.1662, 0.2708], [0.6665, 0.7712])	0.2267
	${\mathfrak{z}}_8$	(0.3684, 0.5795; 0.0740)	([0.3161, 0.4207], [0.5272, 0.6318])	0.3821
	${\mathfrak{z}}_9$	(0.2732, 0.6770; 0.0000)	([0.2732, 0.2732], [0.6770, 0.6770])	0.2868
${\alpha }_2$	${\mathfrak{z}}_1$	(0.4363, 0.5025; 0.1580)	([0.3245, 0.5480], [0.3908, 0.6142])	0.4380
	${\mathfrak{z}}_2$	(0.2466, 0.6849; 0.1120)	([0.1674, 0.3258], [0.6057, 0.7641])	0.2509
	${\mathfrak{z}}_3$	(0.3126, 0.6274; 0.1580)	([0.2009, 0.4243], [0.5157, 0.7392])	0.3064
	${\mathfrak{z}}_4$	(0.3717, 0.5691; 0.1740)	([0.2486, 0.4947], [0.4461, 0.6922])	0.3634
	${\mathfrak{z}}_5$	(0.4851, 0.4546; 0.1740)	([0.3621, 0.6082], [0.3315, 0.5776])	0.4841
	${\mathfrak{z}}_6$	(0.2865, 0.6390; 0.1610)	([0.1727, 0.4004], [0.5252, 0.7529])	0.2819
	${\mathfrak{z}}_7$	(0.2265, 0.7090; 0.0740)	([0.1742, 0.2788], [0.6567, 0.7614])	0.2356
	${\mathfrak{z}}_8$	(0.4314, 0.5115; 0.0740)	([0.3791, 0.4837], [0.4592, 0.5638])	0.4506
	${\mathfrak{z}}_9$	(0.3141, 0.6307; 0.0000)	([0.3141, 0.3141], [0.6307, 0.6307])	0.3315
${\alpha }_3$	${\mathfrak{z}}_1$	(0.4359, 0.5029; 0.1580)	([0.3242, 0.5477], [0.3911, 0.6146])	0.4377
	${\mathfrak{z}}_2$	(0.2476, 0.6836; 0.1120)	([0.1685, 0.3268], [0.6044, 0.7628])	0.2521
	${\mathfrak{z}}_3$	(0.3396, 0.5969; 0.1580)	([0.2278, 0.4513], [0.4852, 0.7087])	0.3362
	${\mathfrak{z}}_4$	(0.3276, 0.6179; 0.1740)	([0.2046, 0.4506], [0.4949, 0.7410])	0.3152
	${\mathfrak{z}}_5$	(0.4342, 0.5085; 0.1740)	([0.3111, 0.5572], [0.3854, 0.6315])	0.4288
	${\mathfrak{z}}_6$	(0.2910, 0.6338; 0.1610)	([0.1771, 0.4048], [0.5199, 0.7476])	0.2869
	${\mathfrak{z}}_7$	(0.2667, 0.6601; 0.0740)	([0.2144, 0.3191], [0.6078, 0.7124])	0.2808
	${\mathfrak{z}}_8$	(0.4112, 0.5331; 0.0740)	([0.3589, 0.4635], [0.4808, 0.5855])	0.4286
	${\mathfrak{z}}_9$	(0.3310, 0.6118; 0.0000)	([0.3310, 0.3310], [0.6118, 0.6118])	0.3499

In Step 10, Equation (21) was utilized to derive the comprehensive disagreement metric $d_{\mathit{ik}}$, which evaluates the ranking of ${\alpha }_i$ in the k-th position within the comprehensive precedence ranking, using the parameter configuration $\mathit{\rho }=0.5$. Using the $\mathfrak{S}\left({\mathfrak{C}}_{2j}^{0.5}\right)$ values from Table 2 and the precedence ranks ${\mathfrak{p}}_{2\mathit{j}}$ within the matrix $\mathfrak{P}$ in Equation (25), the resulting values for the comprehensive disagreement metrics are as follows: $d_{11}=4.5585$, $d_{12}=2.2792$, $d_{13}=0$, $d_{21}=1.4063$, $d_{22}=1.7361$, $d_{23}=4.8785$, $d_{31}=1.6102$, $d_{32}=1.5059$, and $d_{33}=4.6220$. Moving to Step 11, the resulting $3\times 3$ permutation matrix $\mathfrak{B}$ was constructed using Equation (22), wherein the binary variable ${\mathfrak{b}}_{\mathit{ik}}$ takes on values of either 0 or 1 for $\mathit{i},\mathit{k}\in \left\{1,2,3\right\}$:

$\mathfrak{B}={\left[{\mathfrak{b}}_{\mathit{ik}}\right]}_{3\times 3}=\left[\begin{array}{ccc}{\mathfrak{b}}_{11}& {\mathfrak{b}}_{12}& {\mathfrak{b}}_{13}\\ {\mathfrak{b}}_{21}& {\mathfrak{b}}_{22}& {\mathfrak{b}}_{23}\\ {\mathfrak{b}}_{31}& {\mathfrak{b}}_{32}& {\mathfrak{b}}_{33}\end{array}\right].$

Following that, Equation (23) was utilized to articulate the C-IF scoring-mediated median ranking model by this action:

$\min {\sum }_{i=1}^3{\sum }_{k=1}^3\left({{\mathit{d}}_{\mathit{ik}}\cdot \mathfrak{b}}_{\mathit{ik}}\right)=4.5585\cdot {\mathfrak{b}}_{11}+2.2792\cdot {\mathfrak{b}}_{12}+0\cdot {\mathfrak{b}}_{13}+1.4063\cdot {\mathfrak{b}}_{21}$

$+1.7361\cdot {\mathfrak{b}}_{22}+4.8785\cdot {\mathfrak{b}}_{23}+1.6102\cdot {\mathfrak{b}}_{31}+1.5059\cdot {\mathfrak{b}}_{32}+4.6220\cdot {\mathfrak{b}}_{33}$

$\mathrm{subject}\mathrm{to}{\sum }_{k=1}^3{\mathfrak{b}}_{\mathit{ik}}=1,\mathit{i}=1,2,3,{\sum }_{i=1}^3{\mathfrak{b}}_{\mathit{ik}}=1,\mathit{k}=1,2,3,{\mathfrak{b}}_{\mathit{ik}}=0\mathrm{or}1\mathrm{for}\mathit{i},\mathit{k}\in \left\{1,2,3\right\}.$

Finally, in Step 12, the previously formulated model was solved to yield the optimal solution: ${\stackrel{\mathfrak{ˆ}}{\mathfrak{b}}}_{13}={\stackrel{\mathfrak{ˆ}}{\mathfrak{b}}}_{21}={\stackrel{\mathfrak{ˆ}}{\mathfrak{b}}}_{32}=1$, with the other variable values being 0. The corresponding optimal permutation matrix $\stackrel{\mathfrak{ˆ}}{\mathfrak{B}}$ is presented below:

$\stackrel{\mathfrak{ˆ}}{\mathfrak{B}}=\left[\begin{array}{ccc}{\stackrel{\mathfrak{ˆ}}{\mathfrak{b}}}_{11}& {\stackrel{\mathfrak{ˆ}}{\mathfrak{b}}}_{12}& {\stackrel{\mathfrak{ˆ}}{\mathfrak{b}}}_{13}\\ {\stackrel{\mathfrak{ˆ}}{\mathfrak{b}}}_{21}& {\stackrel{\mathfrak{ˆ}}{\mathfrak{b}}}_{22}& {\stackrel{\mathfrak{ˆ}}{\mathfrak{b}}}_{23}\\ {\stackrel{\mathfrak{ˆ}}{\mathfrak{b}}}_{31}& {\stackrel{\mathfrak{ˆ}}{\mathfrak{b}}}_{32}& {\stackrel{\mathfrak{ˆ}}{\mathfrak{b}}}_{33}\end{array}\right]=\left[\begin{array}{ccc}0& 0& 1\\ 1& 0& 0\\ 0& 1& 0\end{array}\right].$

Afterward, Equation (24) was utilized to generate the optimal comprehensive ranks for the three competing alternatives. This was done in the following manner:

$\mathit{A}\cdot \stackrel{\mathfrak{ˆ}}{\mathfrak{B}}=\left({\alpha }_1,{\alpha }_2,{\alpha }_3\right)\cdot \left[\begin{array}{ccc}0& 0& 1\\ 1& 0& 0\\ 0& 1& 0\end{array}\right]=\left({\alpha }_2,{\alpha }_3,{\alpha }_1\right).$

Based on the obtained $\left({\alpha }_2,{\alpha }_3,{\alpha }_1\right)$, the optimal comprehensive ranking indicated that ${\alpha }_2$ surpasses ${\alpha }_3$ and ${\alpha }_1$, consistent with the findings from Otay and Kahraman’s (2022) study. Furthermore, among the three contenders, Supplier #2 (${\alpha }_2$) emerged as the optimal comprehensive alternative. In essence, the utilization of the C-IF scoring-mediated median ranking model in the multi-expert supplier appraisal case underscores its practical utility in real-world decision-making contexts.

Comparative Evaluation and Discussion

The current subsection engages in comparative research, stability, and adaptability testing, aiming to thoroughly investigate the effects of varying the relative significance parameter. Ensuring consistent yet adaptable outcomes in parametric examinations is crucial for achieving reliable and accurate decision-making. Parametric analyses involve adjusting the values of the relative significance parameter from 0 to 1, exploring their impact on the (comprehensive) C-IF scoring functions and comprehensive disagreement metrics generated by the C-IF scoring-mediated median ranking model in a real-world multiple-expert supplier appraisement case. To illustrate and validate the effectiveness and advantages of the C-IF scoring-mediated median ranking methodology, the results obtained through this approach are compared with those from Otay and Kahraman (2022). This comparative analysis serves to emphasize the performance and strengths of the established methodology in comparison to existing research endeavors.

The primary focus of the initial comparison lies in assessing the influence of the relative significance parameter $\mathit{\rho }$ on the C-IF scoring function $S^{\rho }\left(C_{\mathit{ij}}\right)$. More precisely, for this investigation, a range of $\mathit{\rho }$ values – that is, $0,0.1,\cdots ,1$—was considered. Decision analysts can evaluate how variations in $\mathit{\rho }$ impact the determination of C-IF scoring outcomes through this methodical investigation, which offers advantageous information on the function of the parameter $\mathit{\rho }$ in in the C-IF scoring mechanism within the median ranking modeling framework.

Figure 4 illustrates comparisons of the C-IF scoring function $S^{\rho }\left(C_{\mathit{ij}}\right)$ across various $\mathit{\rho }$ values in the multiple-expert supplier appraisement case. The figure comprises three line charts labeled (a) to (c), corresponding to the three competing alternatives from ${\alpha }_1$ to ${\alpha }_3$, respectively. These charts reveal different patterns in C-IF performance ratings. Upon examining the comparative patterns depicted in Figures 4(a)–(c) for the three competing alternatives, it becomes evident that, overall, for varied $\mathit{\rho }$ settings, the C-IF scoring outcomes linked to the C-IF performance ratings of ${\alpha }_2$ and ${\alpha }_3$ are considerably greater than those linked to ${\alpha }_1$. To provide greater detail, the average C-IF scoring functions in connection with ${\alpha }_2$ (i.e., $\sum _{\rho }\sum _j{S}^{\rho }\left(C_{2\mathit{j}}\right)/\left(9\times 11\right)$) and ${\alpha }_3$ (i.e., $\sum _{\rho }\sum _j{S}^{\rho }\left(C_{3\mathit{j}}\right)/\left(9\times 11\right)$) were 0.6990 and 0.6986, respectively. On the contrary, the average C-IF scoring function related to ${\alpha }_1$ (i.e., $\sum _{\rho }\sum _j{S}^{\rho }\left(C_{1\mathit{j}}\right)/\left(9\times 11\right)$) was 0.5002. To delve into more detail, the minimal and maximal C-IF scoring functions (i.e., $\mathrm{mi}{\mathrm{n}}_{\rho }\mathrm{mi}{\mathrm{n}}_j{S}^{\rho }\left(C_{2\mathit{j}}\right)$ and $\mathrm{ma}{\mathrm{x}}_{\rho }\mathrm{ma}{\mathrm{x}}_j{S}^{\rho }\left(C_{2\mathit{j}}\right)$, respectively) associated with ${\alpha }_2$ were 0.6129 and 0.8426, respectively. Meanwhile, the minimal and maximal C-IF scoring functions (i.e., $\mathrm{mi}{\mathrm{n}}_{\rho }\mathrm{mi}{\mathrm{n}}_j{S}^{\rho }\left(C_{3\mathit{j}}\right)$ and $\mathrm{ma}{\mathrm{x}}_{\rho }\mathrm{ma}{\mathrm{x}}_j{S}^{\rho }\left(C_{3\mathit{j}}\right)$, respectively) corresponding to ${\alpha }_3$ were 0.6129 and 0.8670, respectively. In contrast, the minimal and maximal C-IF scoring functions (i.e., $\mathrm{mi}{\mathrm{n}}_{\rho }\mathrm{mi}{\mathrm{n}}_j{S}^{\rho }\left(C_{1\mathit{j}}\right)$ and $\mathrm{ma}{\mathrm{x}}_{\rho }\mathrm{ma}{\mathrm{x}}_j{S}^{\rho }\left(C_{1\mathit{j}}\right)$, respectively) linked to ${\alpha }_1$ were 0.2330 and 0.6822, respectively. Based on these contrasting outcomes, it is recognized that the C-IF scoring functions $S^{\rho }\left(C_{2\mathit{j}}\right)$ and $S^{\rho }\left(C_{3\mathit{j}}\right)$ related to ${\alpha }_2$ and ${\alpha }_3$, respectively, were notably higher than the C-IF scoring function $S^{\rho }\left(C_{1\mathit{j}}\right)$ for ${\alpha }_1$ across various $\mathit{\rho }$ values. These findings signify that the C-IF performance ratings of ${\alpha }_2$ and ${\alpha }_3$ concerning the nine evaluative criteria were generally superior to those of ${\alpha }_1$ for the eleven setting values of the relative significance parameter $\mathit{\rho }=0,0.1,\cdots ,1$.

Figure 4. Comparative patterns of C-IF scoring outcomes across various $\mathit{\rho }$ values.

With increasing $\mathit{\rho }$ values, the C-IF scoring function $S^{\rho }\left(C_{1\mathit{j}}\right)$ for ${\alpha }_1$, in relation to criteria ${\mathfrak{z}}_1$, ${\mathfrak{z}}_2$, ${\mathfrak{z}}_3$, ${\mathfrak{z}}_4$, and ${\mathfrak{z}}_9$, gradually declines as portrayed in Figure 4(a). Particularly noteworthy is the substantial decrease in the $S^{\rho }\left(C_{13}\right)$ values. Moving on to Figure 4(b), it illustrates that as the $\mathit{\rho }$ values increase, the C-IF scoring function $S^{\rho }\left(C_{2\mathit{j}}\right)$ for ${\alpha }_2$ across criteria ${\mathfrak{z}}_1$, ${\mathfrak{z}}_3$, ${\mathfrak{z}}_4,\cdots {\mathfrak{z}}_9$ experiences a gradual rise. Remarkably, there is a notable increase in the values of $S^{\rho }\left(C_{24}\right)$, $S^{\rho }\left(C_{25}\right)$, and $S^{\rho }\left(C_{26}\right)$. Interestingly, the C-IF scoring function $S^{\rho }\left(C_{22}\right)$ yields values that don’t change when the $\mathit{\rho }$ value rises. In Figure 4(c), it is observed that with an increase in the $\mathit{\rho }$ value, the C-IF scoring function $S^{\rho }\left(C_{3\mathit{j}}\right)$ for ${\alpha }_3$ across all nine criteria shows a gradual increase. Particularly noteworthy is the significant increase in the values of $S^{\rho }\left(C_{32}\right)$, $S^{\rho }\left(C_{36}\right)$, and $S^{\rho }\left(C_{37}\right)$.

Next, the contrasted results of the comprehensive C-IF scoring function $\mathfrak{S}\left({\mathfrak{C}}_{ij}^{\rho }\right)$ across different $\mathit{\rho }$ configurations are displayed in Figure 5. The figure consists of three radar charts labeled (a) to (c), in connection with the three competing alternatives from ${\alpha }_1$ to ${\alpha }_3$, respectively. Upon examining the comparative patterns depicted in Figures 5(a)–(c) for the three competing alternatives, the radar chart in Figure 5(a) reveals evidently inconsistent pattern of $\mathfrak{S}\left({\mathfrak{C}}_{ij}^{\rho }\right)$ related to each weighted C-IF performance rating ${\mathfrak{C}}_{ij}^{\rho }$. The comprehensive C-IF scoring outcomes, as portrayed in the radar charts of Figures 5(b)–(c) for ${\alpha }_2$ and ${\alpha }_3$, respectively, exhibit similar comparative patterns, indicating the generally consistent change trends of $\mathfrak{S}\left({\mathfrak{C}}_{2j}^{\rho }\right)$ and $\mathfrak{S}\left({\mathfrak{C}}_{3j}^{\rho }\right)$ in relation to ${\mathfrak{z}}_j\in \mathfrak{Z}$ across deviating $\mathit{\rho }$ settings.

Figure 5. Comparative patterns of comprehensive C-IF scoring outcomes across various $\mathit{\rho }$ values.

The trends in comprehensive C-IF scoring outcomes show distinct changes when the C-IF importance rating ${\mathcal{W}}_j$ was used to weight the C-IF performance rating $C_{\mathit{ij}}$. As depicted in Figure 5, the comprehensive C-IF scoring functions $\mathfrak{S}\left({\mathfrak{C}}_{2j}^{\rho }\right)$ and $\mathfrak{S}\left({\mathfrak{C}}_{3j}^{\rho }\right)$ are markedly higher than the $\mathfrak{S}\left({\mathfrak{C}}_{1j}^{\rho }\right)$ values across different $\mathit{\rho }$ settings. To be precise, the average comprehensive C-IF scoring functions for ${\alpha }_2$ (i.e., $\sum _{\rho }\sum _j\mathfrak{S}\left({\mathfrak{C}}_{2j}^{\rho }\right)/\left(9\times 11\right)$) and ${\alpha }_3$ (i.e., $\sum _{\rho }\sum _j\mathfrak{S}\left({\mathfrak{C}}_{3j}^{\rho }\right)/\left(9\times 11\right)$) were 0.3495 and 0.3466, respectively. Conversely, the average comprehensive C-IF scoring function regarding ${\alpha }_1$ (i.e., $\sum _{\rho }\sum _j\mathfrak{S}\left({\mathfrak{C}}_{1j}^{\rho }\right)/\left(9\times 11\right)$) was 0.2520. Furthermore, the minimal and maximal comprehensive C-IF scoring functions (i.e., $\mathrm{mi}{\mathrm{n}}_{\rho }\mathrm{mi}{\mathrm{n}}_j\mathfrak{S}\left({\mathfrak{C}}_{2j}^{\rho }\right)$ and $\mathrm{ma}{\mathrm{x}}_{\rho }\mathrm{ma}{\mathrm{x}}_j\mathfrak{S}\left({\mathfrak{C}}_{3j}^{\rho }\right)$, respectively) for ${\alpha }_2$ were 0.2054 and 0.4866, respectively. Meanwhile, the minimal and maximal comprehensive C-IF scoring functions (i.e., $\mathrm{mi}{\mathrm{n}}_{\rho }\mathrm{mi}{\mathrm{n}}_j\mathfrak{S}\left({\mathfrak{C}}_{3j}^{\rho }\right)$ and $\mathrm{ma}{\mathrm{x}}_{\rho }\mathrm{ma}{\mathrm{x}}_j\mathfrak{S}\left({\mathfrak{C}}_{3j}^{\rho }\right)$, respectively) for ${\alpha }_3$ were 0.2184 and 0.4549, respectively. Conversely, the minimal and maximal comprehensive C-IF scoring functions (i.e., $\mathrm{mi}{\mathrm{n}}_{\rho }\mathrm{mi}{\mathrm{n}}_j\mathfrak{S}\left({\mathfrak{C}}_{1j}^{\rho }\right)$ and $\mathrm{ma}{\mathrm{x}}_{\rho }\mathrm{ma}{\mathrm{x}}_j\mathfrak{S}\left({\mathfrak{C}}_{1j}^{\rho }\right)$, respectively) for ${\alpha }_1$ were 0.0257 and 0.4327, respectively. In general, the comprehensive C-IF scoring functions indicate significantly higher values for ${\alpha }_2$ and ${\alpha }_3$ compared to ${\alpha }_1$ across various $\mathit{\rho }$ configurations, substantiating the superiority of ${\alpha }_2$ and ${\alpha }_3$ over ${\alpha }_1$.

Figure 6 illustrates the variations in the comprehensive disagreement metric $d_{\mathit{ik}}$ for each competing alternative ${\alpha }_i$ when assigned the k-th overall rank, across diverse $\mathit{\rho }$ configurations in the supplier appraisement instance. The most noteworthy observation, as exhibited in Figure 6, is that the resulting $d_{13}$ is consistently zero under all $\mathit{\rho }$ settings, indicating that ${\alpha }_1$ fully fits the third place in the overall ranking. This outcome aligns with the earlier comparative analyses, where both $S^{\rho }\left(C_{\mathit{ij}}\right)$ and $\mathfrak{S}\left({\mathfrak{C}}_{ij}^{\rho }\right)$ for ${\alpha }_2$ and ${\alpha }_3$ were significantly superior to those for ${\alpha }_1$. It is also worth noting that, on average, the lowest comprehensive disagreement metrics for ${\alpha }_2$ and ${\alpha }_3$ occurred when they were placed in the first position in the overall ranking. This suggests that, under all $\mathit{\rho }$ settings, ${\alpha }_2$ and ${\alpha }_3$ were highly suitable for securing the top spot within the optimal comprehensive ranking. This finding underscores an intense competition between ${\alpha }_2$ and ${\alpha }_3$, both performing exceptionally well.

Figure 6. Comparisons of the comprehensive disagreement metric $d_{\mathit{ik}}$ across various $\mathit{\rho }$ values.

For a more detailed analysis of Figure 6, the mean comprehensive disagreement metrics corresponding to the first, second, and third overall ranks assigned to ${\alpha }_1$ were 4.5367, 2.2683, and 0.0000, respectively. Subsequently, the mean comprehensive disagreement metrics for assigning the first, second, and third overall ranks to ${\alpha }_2$ were 1.4840, 1.6617, and 4.8073, respectively. Furthermore, the mean comprehensive disagreement metrics for ${\alpha }_3$ to obtain the first, second, and third overall ranks were 1.5324, 1.5873, and 4.7071, respectively. Notice that the competing alternative ${\alpha }_2$ attained the minimum comprehensive disagreement metric when the $\mathit{\rho }$ values ranged from $\left\{0,0.1,\cdots ,0.4\right\}$, favoring ${\alpha }_2$ for securing the second comprehensive ranking. Similarly, when the $\mathit{\rho }$ values were within the range $\left\{0.5,0.6,\cdots ,1\right\}$, the support shifted toward ${\alpha }_2$ to secure the first comprehensive ranking. Apart from that, the competing alternative ${\alpha }_3$ received the minimum comprehensive disagreement metric in the range of $\mathit{\rho }$ values $\left\{0,0.1,\cdots ,0.4\right\}$, indicating support for ${\alpha }_3$ to secure the first comprehensive ranking. Similarly, within the range of $\mathit{\rho }$ values $\left\{0.5,0.6,\cdots ,1\right\}$, ${\alpha }_3$ received backing for attaining the second comprehensive ranking. Based on these findings, the rivalry between ${\alpha }_2$ and ${\alpha }_3$ appears highly competitive. In scenarios with $\mathit{\rho }$ values within the range $\left\{0,0.1,\cdots ,0.4\right\}$, ${\alpha }_3$ secured the first rank, while ${\alpha }_2$ claimed the second position. Conversely, within the range $\left\{0.5,0.6,\cdots ,1\right\}$, ${\alpha }_2$ took the lead, with ${\alpha }_3$ following in the second spot.

The final comparison aims to investigate disparities in priority ranking outcomes between established decision-analytic methods and the optimal comprehensive rankings derived from the proposed C-IF scoring-mediated median ranking model across different $\mathit{\rho }$ configurations. This analysis serves to authenticate the effectiveness and dependability of the recommended methodology, which incorporates the innovative C-IF scoring mechanism, and to assess practical implications concerning the developed C-IF scoring technique. Table 3 provides a thorough examination of the ultimate sorting results produced by various techniques in the supplier appraisement instance. This table compares key aspects such as the fundamental framework of each technique, the indicators utilized for supplier sorting, the approach to parameter setting, decision-makers’ preferences (optimism, pessimism, or comprehensiveness), and the ultimate supplier rankings.

Table 3. Assessment of the final sorting outcomes generated by different methods.

Solution outcome derived from Otay and Kahraman’s (2022) approach			Ultimate supplier rank order
Technique	Indicator	Inclination	${\alpha }_1$	${\alpha }_2$	${\alpha }_3$
Integrated C-IF AHP	Group utility measure	Optimism	1st	2nd	3rd
& VIKOR approach		Pessimism	3rd	1st	2nd
		Comprehensiveness	3rd	1st	2nd
	Individual regret measure	Optimism	1st	2nd	3rd
		Pessimism	3rd	1st	2nd
		Comprehensiveness	3rd	1st	2nd
	Trade-off index	Optimism	1st	2nd	3rd
		Pessimism	3rd	1st	2nd
		Comprehensiveness	3rd	1st	2nd
Solution outcome derived from the developed approach in the current paper			Ultimate supplier rank order
Technique	Indicator	Parameter setting	${\alpha }_1$	${\alpha }_2$	${\alpha }_3$
C-IF scoring-mediated	Comprehensive C-IF scoring	$\mathit{\rho }\in \left\{0,0.1,\cdots ,0.4\right\}$	3rd	2nd	1st
median ranking model	function/disagreement metric	$\mathit{\rho }\in \left\{0.5,0.6,\cdots ,1\right\}$	3rd	1st	2nd

As indicated in Table 3, the attitudinal inclinations of the decision-maker, encompassing optimistic, pessimistic, and comprehensive perspectives, were considered when applying the integrated C-IF AHP & VIKOR strategy that was suggested by Otay and Kahraman (2022). The application of this methodology to the multiple-expert supplier appraisement case study demonstrated that the outcomes of the solution varied based on the adopted perspectives. Under comprehensive and pessimistic views, the preferred supplier was ${\alpha }_2$, followed by ${\alpha }_3$ and ${\alpha }_1$. Nevertheless, under an optimistic view, the preferred supplier became ${\alpha }_1$, followed by ${\alpha }_2$ and ${\alpha }_3$. Consistent ranking orders were observed across different sorting indicators, including group utility measurement, individual regret measurement, and trade-off indices. These findings reveal the incongruity between optimistic, pessimistic, and comprehensive views.

Significantly, the earlier findings from the comparative analyses conducted in this research indicate that, irrespective of the assessment indicators such as (comprehensive) C-IF scoring functions and comprehensive disagreement metrics, the overall performance of competing suppliers ${\alpha }_2$ and ${\alpha }_3$ surpasses that of ${\alpha }_1$ remarkably. Consequently, it can be deduced that a high-quality decision-analytic method should be capable of prioritizing options ${\alpha }_2$ and ${\alpha }_3$ over ${\alpha }_1$. In terms of the performance contrast between ${\alpha }_2$ and ${\alpha }_3$, their proximity in performance is noteworthy. Accordingly, employing different decision-analytic methods may yield disparate conclusions regarding the priority between these two options.

The C-IF scoring-mediated median ranking model produced two distinct sorting outcomes: ${\alpha }_3$ leading, followed by ${\alpha }_2$, and trailed by $\mathit{AccessisdeniedAccessisdenied}$ for $\mathit{\rho }$ values in the range $\left\{0,0.1,\cdots ,0.4\right\}$; and ${\alpha }_2$ taking the lead, followed by ${\alpha }_3$, and trailed by ${\alpha }_1$ for $\mathit{\rho }$ values in the range $\left\{0.5,0.6,\cdots ,1\right\}$. These results underscore the superiority of ${\alpha }_2$ and ${\alpha }_3$ over ${\alpha }_1$, affirming the soundness of the suggested approach. On the other hand, given an optimistic perspective, the integrated C-IF AHP & VIKOR methodology identified ${\alpha }_1$ as the preferred supplier, followed by ${\alpha }_2$, and then by ${\alpha }_3$. This discrepancy leads to illogical outcomes and a confounding conclusion for decision support.

In conclusion, this research applied the proposed C-IF scoring-mediated median ranking model to the multiple-expert supplier appraisement case, demonstrating its superior performance through comprehensive comparative analyses. These comparisons, which evaluated (comprehensive) C-IF scoring functions, comprehensive disagreement metrics, and priority ranking outcomes across different $\mathit{\rho }$ configurations, highlight the efficacy and versatility of our suggested methodology. The specific outcomes generated by the C-IF scoring-mediated median ranking model, where ${\alpha }_2$ leads in one scenario ($\mathit{\rho }\in \left\{0.5,0.6,\cdots ,1\right\}$) and ${\alpha }_3$ leads in another ($\mathit{\rho }\in \left\{0,0.1,\cdots ,0.4\right\}$), further substantiate the model’s superiority by illustrating its adaptability to varying $\mathit{\rho }$ values. In contrast, the C-IF AHP & VIKOR approach’s designation of ${\alpha }_1$ as the preferred supplier under an optimistic view raises concerns about the methodology’s logical coherence and suitability for decision assistance. Overall, the proposed C-IF scoring-mediated median ranking model demonstrates its robustness and effectiveness in addressing complex decision issues associated with supplier appraisement, emphasizing its practical significance and potential for enhancing decision-analytic processes.

Conclusions

Faced with progressively intricate societal challenges and the expanding horizons of scientific exploration, both conventional and intuitionistic fuzzy theories have confronted mounting limitations and hurdles in practical applications. These limitations are primarily attributed to their inherent inability to fully encompass the comprehensive spectrum of uncertain information that is relevant to the issues under scrutiny. To address these challenges, the development of C-IF sets involves a thorough consideration and synthesis of the circular-structured aspects of membership, non-membership, and hesitation. This comprehensive approach equips decision analysts with a substantially more intricate and detailed information set.

The research contributes to creating an innovative C-IF scoring function dedicated to improving data assessment in the context of circular intuitionistic fuzziness. Bai (2013) was credited for introducing an improved scoring function tailored for the comparison and evaluation of interval-valued intuitionistic fuzzy numbers. This scoring approach was specifically crafted to systematically address uncertainty and indeterminacy. The study also emphasizes Mishra et al. (2023) recognition of the significance of Bai’s scoring function. Going beyond, Mishra et al. established a normalized score value, incorporating enhancements to the scoring functions. This study has extended upon the insights gleaned from the prior research of Bai (2013) and Mishra et al. (2023). The central purpose was to forge an innovative C-IF scoring mechanism, enhancing data assessment in uncertain contexts with circular intuitionistic fuzziness. In fortifying the development of this scoring function, the research introduced pivotal concepts, including interval-valued estimates and relative significance parameters for both lower and upper estimates. These additions significantly contribute to formulating a valuable C-IF scoring technique. Furthermore, this study has presented several theorems pertaining to interval-valued estimates, the C-IF scoring mechanism, and associated boundary conditions. These theorems elucidate the properties and behaviors of the scoring function in C-IF circumstances, furnishing a robust theoretical foundation for the proposed mechanism. In summary, this research has provided valuable insights into the creation of a C-IF scoring mechanism, emphasizing its theoretical foundations. The emphasis lies in crafting a scoring mechanism proficient in evaluating data amidst circular intuitionistic fuzziness. This involves leveraging insights from previous research while introducing novel concepts and theorems to advance the field.

This research has propounded an innovative C-IF scoring-mediated median ranking model crafted to tackle uncertainties intrinsic to decision-analytic challenges within the C-IF environment. The main objective of this model is to establish a comprehensive precedence ranking for competing alternatives, aiming to minimize imprecision and biases in decision-analytic processes. The developed C-IF scoring mechanism has been utilized to formulate comprehensive C-IF scoring functions and disagreement metrics. The process involves scalar multiplication to calculate weighted C-IF performance ratings, generating interval-valued estimates. A novel C-IF scoring-mediated median ranking model has been formulated, utilizing a linear programming structure to minimize disagreement metrics and achieve optimal comprehensive precedence rankings. Specifically, the methodology, comprising four phases (Steps 1–3, Steps 4–6, Steps 7–9, and Steps 10–12), systematically progresses from defining the decision-analytic task and constructing data to modeling with comprehensive disagreement metrics, ultimately providing decision analysts with a tool for handling complex C-IF information and deriving dependable decision-analytic outcomes.

To illustrate the application and authentication of the developed approach, this study practically employed the proposed C-IF scoring-mediated median ranking model in assessing and selecting competing suppliers within an engineering company. The case study entailed evaluating three suppliers based on cost, service, and technology/quality dimensions, utilizing nine evaluative criteria. Each dimension comprised three criteria crucial for supplier appraisal, with their significance explained in the context of the engineering company’s needs and goals. The methodology involved applying C-IF scoring techniques to generate interval-valued estimates for performance ratings. The relative significance parameter $\mathit{\rho }$ was introduced to account for varying preference opinions. C-IF scoring outcomes for each supplier on each criterion were calculated, and a C-IF scoring-mediated median ranking methodology was created through the use of comprehensive disagreement metrics. Furthermore, the validation process included comparing the implemented outcomes of the suggested model with those of a prior study (Otay and Kahraman 2022) and conducting a parametric analysis to examine the model’s sensitivity to changes in the relative significance parameter $\mathit{\rho }$. The findings suggest that the model effectively delivers consistent and reliable rankings across various scenarios. The comparative evaluation and discussions delved deeper into the effects of $\mathit{\rho }$ on (comprehensive) C-IF scoring functions, comprehensive disagreement metrics, and overall rankings. The results underscored the model’s robustness and its capacity to navigate uncertainties in decision-making. The analytical findings highlight the practical utility of the C-IF scoring-mediated median ranking model in real-world contexts.

Overall, the paper presents a comprehensive and structured approach – the C-IF scoring-mediated median ranking model – to addressing decision-analytic problems in real-world contexts. It incorporates advanced scoring techniques and tackles the challenges associated with complex decision-making under intricate uncertainty. The validation through a practical case study adds credibility to the proposed model, showcasing its applicability in complex decision problems. The comparative analysis strengthens the argument for the model’s superiority over existing methodologies. The final emphasis is on the empowering aspect of C-IF theory, enabling decision analysts to undertake more refined and precise fuzzy modeling endeavors. This is particularly pertinent in the context of intelligent decision support systems, where the utilization of C-IF information with inbuilt circular functions can significantly enhance the resolution of intricate decision-analytic challenges. Furthermore, the C-IF configuration accomplishes this by accommodating both the degree of uncertainty and the confidence associated with such uncertainty, which, in turn, elevates the intelligence of these systems.

While this study contributes to the field of C-IF decision analytics, several research limitations persist in the developed model. These limitations include:

1. Limitation on the C-IF scoring mechanism with subjectivity in parameter selection: The introduction of the relative significance parameter $\mathit{\rho }$ aims to accommodate varying preference opinions. However, alterations to this parameter could influence both the C-IF scores and the outcomes of the C-IF scoring-mediated median ranking model. Additionally, seeking to refine or introduce additional factors to enhance the parameter-based C-IF scoring mechanism may introduce complexity, potentially complicating the model’s implementation and interpretation. Consequently, the selection of the relative significance parameter or refinements for the scoring mechanism may involve subjective judgments, potentially resulting in biases or inconsistencies in model outcomes.

2. Limitation on the selection of alternative models and contextual relevance: The selection of alternative decision-making models for comparative studies may introduce bias, as different models possess inherent strengths and weaknesses that could impact comparison results. Furthermore, the applicability of alternative models in diverse decision-making contexts may vary, restricting the extent to which comparative studies can offer insights into the superiority of the proposed C-IF scoring-mediated median ranking model.

3. Limitation on industry-specific nuances for model application: Each industry presents unique characteristics and requirements, which may pose challenges in extrapolating the effectiveness and adaptability of the C-IF scoring-mediated median ranking model across diverse industry contexts. The successful deployment of the model in various industry settings could hinge on factors like resource limitations, stakeholder buy-in, and cultural influences, potentially impacting its uptake and efficacy.

These limitations highlight areas for future research and refinement of the developed approach in the realm of C-IF decision analytics. Moving forward, potential research directions could include:

1. Enhancement of the C-IF scoring mechanism: To minimize bias and bolster the model’s robustness, consider implementing additional exploration of objective methods for parameter selection. Further enhancements to the C-IF scoring mechanism should be explored, taking into account additional factors or refinements to enhance accuracy and applicability across diverse decision-analytic contexts.

2. Comparative studies with alternative models: To offer a more thorough assessment of the model’s performance, conducting additional comparative studies with a broader range of alternative decision-making models is recommended. This will enable the validation and benchmarking of the proposed C-IF scoring-mediated median ranking model against existing methodologies, thus providing a deeper insight into its advantages.

3. Application in various industry contexts: Conducting investigations into industry-specific adaptations of the model can address unique challenges and requirements in different industry contexts. Specifically, expanding the employment of the C-IF scoring-mediated median ranking model to industries beyond engineering can assess its effectiveness and adaptability across diverse decision-making domains.

Author Contribution

Ting-Yu Chen: Conceptualization, Methodology, Validation, Formal analysis, Data curation, Writing – original draft, Writing – review & editing, Visualization, Funding acquisition.

Human and Animal Rights

This article does not contain any studies with human participants or animals that were performed by any of the authors.

Acknowledgements

The author acknowledges the assistance of the respected editor and the anonymous referees for their insightful and constructive comments, which helped to improve the overall quality of the paper. The author would like to acknowledge the financial support of the National Science and Technology Council, Taiwan (NSTC 111-2410-H-182-012-MY3) and the Fundamental Research Funds from Chang Gung Memorial Hospital, Linkou, Taiwan (BMRP 574) during the completion of this study.

Data availability statement

The datasets generated during and/or analyzed during the current study are available from the corresponding author upon reasonable request.

Disclosure statement

The author declares that she has no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Supplementary Materials

Supplemental data for this article can be accessed online at https://doi.org/10.1080/08839514.2024.2335416

Additional information

Funding

The work was supported by the Chang Gung Memorial Hospital, Linkou [BMRP 574]; National Science and Technology Council [NSTC 111-2410-H-182-012-MY3].