Multiple-criteria analysis is crucial for addressing decision-making challenges in practical scenarios (Chen, 2024; Liu, 2024). In the ELECTRE (i.e. ÉLimination Et Choix Traduisant la REalité in French) framework, evaluative criteria serve as standards to assess and prioritize options based on their performance (Akram et al., 2023a, 2023b). However, uncertainty can disrupt the ELECTRE process, leading to inaccurate assessments, inconsistent decisions, and suboptimal choices, undermining decision-makers’ confidence (Ramya et al., 2023; Wu et al., 2023). Insufficient or unreliable data may introduce biases, distort rankings, and increase subjective judgments, compromising reliability (Liu et al., 2023; Yüksel and Dinçer, 2023). This heightened risk complicates the appraisal of options and limits the efficacy of ELECTRE-based analysis (Zhang et al., 2023; Zhou et al., 2022).

To meet the demands of uncertain environments, researchers have expanded the ELECTRE methodology to various fuzzy scenarios. Beyond conventional fuzzy models, recent efforts have focused on higher-order fuzzy frameworks. Akram et al. (2023b) introduced the fuzzy ELECTRE IV method using triangular fuzzy numbers for imprecise data. Wu et al. (2023) developed a triadic decision model combining ELECTRE III with attribute ratio evaluation in a spherical fuzzy setting for customer selection. Wang and Chen (2021) used T-spherical fuzzy score functions and Minkowski distance-dependent indices to design T-spherical fuzzy ELECTRE I and II methods. Yüksel and Dinçer (2023) evaluated sustainability in circular industrialization using quantum spherical fuzzy ELECTRE. Zhang et al. (2023) created an ELECTRE II method with cosine similarity measures to assess financial logistics firms’ efficiency using double hierarchy hesitant fuzzy information. Ramya et al. (2023) proposed a disposal technique selection framework for e-waste using ELECTRE III with wiggly hesitant Pythagorean fuzzy sets. Zhou et al. (2022) developed a Fermatean fuzzy ELECTRE method using Jensen-Shannon divergence and cross-entropy to handle uncertain decision-maker and criteria weights. Akram et al. (2023a) investigated a 2-tuple linguistic Fermatean fuzzy ELECTRE II for group decision-making with linguistic variables. Pinar and Boran (2022) introduced a q-rung picture fuzzy ELECTRE model, integrating the technique for order preference by similarity to ideal solutions (TOPSIS) for group decision analysis. These advancements are a result of continuous work to improve the applicability and efficacy of ELECTRE-based strategies for tackling challenging and ambiguous decision-making situations.

Applying fuzzy extensions of the ELECTRE-based outranking methodology to multi-criteria analysis enhances decision-making, particularly in uncertain environments. These approaches incorporate fuzzy logic to handle uncertainty, offering deeper insights for decision-makers. However, advancements specifically tailored to the emerging circular intuitionistic fuzzy (C-IF) sets remain limited. Intuitionistic fuzzy (IF) sets, introduced by Atanassov (1986), include degrees of hesitancy alongside membership and non-membership, addressing vagueness more effectively than conventional fuzzy sets (Chen, 2024; Liu, 2024). Their integration with various tools enables more precise handling of uncertain information (Liu et al., 2023). Atanassov later extended IF sets into C-IF sets, which represent circles with radii encompassing membership and non-membership components (Atanassov, 2020; Çakır and Taş, 2023). C-IF sets have demonstrated versatility across realistic applications (Alinejad et al., 2024; Ci, 2024; Kong, 2024) and hold promise for future advancements in ELECTRE-based methods.

C-IF sets offer greater flexibility by integrating membership, hesitancy, and non-membership components (Ci, 2024; Jameel et al., 2024). Represented by circular shapes, they capture multiple qualities or attributes, with the circular function enhancing their expressiveness in a triangular distribution space (Kong, 2024; Pratama et al., 2024). When the circle’s radius is set to zero, C-IF sets revert to standard IF sets. C-IF sets excel in handling uncertain, imprecise information, making them suitable for diverse decision-making tasks (Alinejad et al., 2024; Chen, 2023a). Applications include decision assistance and support (Chen, 2023b, 2024), supplier evaluation and selection (Çakır and Taş, 2023; Chen, 2024), sustainable renewable energy management (Jameel et al., 2024), biomass resource strategies (Alinejad et al., 2024), food supply chain monitoring (Alsattar et al., 2024), pattern recognition and medical diagnosis (Khan et al., 2022), and public health risk assessments (Kong, 2024). These examples underscore the adaptability and utility of C-IF sets for addressing uncertainty in practical applications across various domains.

The versatility of C-IF sets makes them valuable for decision-making; however, effective analysis requires reliable methods to interpret and manipulate uncertain information (Pinar and Boran, 2022; Zhang et al., 2023). Chen (2023b) formulated two generalized C-IF distance metrics – three-term and four-term Minkowski-like measures – designed to handle imperfect information. The four-term approach accounts for radius, membership, non-membership, and hesitancy, offering a complete representation of C-IF numbers. In contrast, the three-term model excludes hesitancy. This study adopts the four-term model for its comprehensive nature, providing a solid foundation for developing a C-IF ELECTRE approach to better address research objectives.

C-IF sets are highly effective for managing complex and ambiguous information. However, there remains a significant gap in developing ELECTRE-based methods tailored for C-IF decision environments. While existing ELECTRE approaches have shown success in handling uncertainty across various fuzzy frameworks – including spherical fuzzy (Wu et al., 2023), T-spherical fuzzy (Wang and Chen, 2021), quantum spherical fuzzy (Yüksel and Dinçer, 2023), double hierarchy hesitant fuzzy (Zhang et al., 2023), normal wiggly Pythagorean hesitant fuzzy (Ramya et al., 2023), Fermatean fuzzy (Zhou et al., 2022), 2-tuple linguistic Fermatean fuzzy (Akram et al., 2023a), and q-rung picture fuzzy (Pinar and Boran, 2022) – their application within C-IF contexts remains limited. This study addresses this gap with the following motivations: This research aims to develop the C-IF ELECTRE, a multiple-criteria decision model for discrete decisions involving conflicting or incomparable criteria. It integrates circular intuitionistic fuzziness with a joint generalized scoring function, based on Hezam et al.’s (2023) natural exponential function, to address uncertainty. The concepts of aggressive and cautious IF estimates (inspired by Chen, 2023a) are employed to provide upper and lower estimations within the C-IF context, with several theorems outlining their properties and relationships. The C-IF ELECTRE approach involves establishing concordance and discordance sets to determine when one alternative is superior, equal, or inferior to another. C-IF ELECTRE I uses consistency and inconsistency indicators to build a dominance graph for partial-priority rankings. C-IF ELECTRE II introduces consistency-dependent outflow, inconsistency-dependent inflow, and net flow values for complete-priority rankings. Applied to supplier evaluations, the method aligns with comparative results and highlights the impact of parameter settings, such as distance and divergence measures, on ranking outcomes.

This study offers key advancements in ELECTRE-based decision-making: This article is structured as follows: Section 1 highlights research motivations and the need for innovations in ELECTRE-based methodologies for C-IF contexts. Section 2 covers fundamental mathematical notations for IF and C-IF configurations. Section 3 develops a joint generalized scoring function to address C-IF uncertainty. Section 4 introduces the C-IF ELECTRE model for complex decision-making in uncertain contexts. Section 5 demonstrates the application of C-IF ELECTRE I and II techniques through a multi-expert supplier evaluation case. Section 6 analyses the impact of inclination parameter settings on results, highlighting the approach’s advantages. Section 7 provides conclusions and suggests directions for future research.

A standard fuzzy set describes membership, while an IF set adds flexibility by allowing for hesitation between membership and non-membership. Building on this, the C-IF set introduces a circular structure, capturing more complex, ambiguous characteristics. Unlike IF sets, C-IF sets represent uncertainty more precisely using their circular form. Figure 1 compares IF and C-IF sets within a triangular space defined by the vertices $(0,0)$, $(1,1)$, and $(0,1)$, illustrating their overlap and differences. In this space, the C-IF number $c(\chi )$ is represented by the centre $({u_{C}}(\chi ),{v_{C}}(\chi ))$ and radius ${r_{C}}(\chi )$ of the circular function ${\mathfrak{O}_{r}}$, while the IF number $i(\chi )$ corresponds to the point $({u_{I}}(\chi ),{v_{I}}(\chi ))$. Figure 1 shows five ways to depict ${\mathfrak{O}_{r}}$, demonstrating how it constrains ${\mathfrak{O}_{r}}({u_{C}}(\chi ),{v_{C}}(\chi ))$ within the L-fuzzy set ${\mathcal{L}^{\ast }}$. When the radius ${r_{C}}(\chi )=0$ for all elements in the domain ℵ, the C-IF set C reduces to an IF set I, and the C-IF number $c(\chi )$ becomes equivalent to the IF number $i(\chi )$.

Chen (2023b) introduced two generalized C-IF distance metrics to overcome the limitations of traditional measures and improve adaptability. These metrics use triadic and quadripartite representations of C-IF Minkowski-like distances. The triadic model includes radius, membership, and non-membership, while the quadripartite version adds hesitancy, providing a more comprehensive assessment of C-IF dimensionality.

These distance measures quantify the dissimilarity between C-IF numbers, varying by approach (three-term or four-term) and the chosen metric parameter ξ. The triadic model omits hesitancy, while the quaternary model offers a more exhaustive representation. This research adopts the four-term Minkowski-like distance as the foundation for the C-IF ELECTRE methodology due to its comprehensive coverage of all relevant dimensions, ensuring alignment with the study’s objectives.

This section aims to develop a joint generalized scoring function that addresses the C-IF uncertainty. Utilizing a natural exponential-based scoring mechanism (Hezam et al., 2023), it effectively handles IF scenarios. Building on Chen’s (2023a) concepts of aggressive and conservative estimates, this section introduces practical notions of aggressive and cautious IF estimates. It also explores their fundamental properties, highlighting their role in representing upper and lower bounds within C-IF information.

When the inclination parameter φ exceeds 0.5, it indicates that the decision-maker places greater importance on the aggressive IF estimate, reflecting an aggressive stance. Conversely, if φ is less than 0.5, the decision-maker favours the cautious IF estimate, demonstrating a cautious attitude. When φ equals 0.5, both aggressive and cautious perspectives are given equal weight, reflecting a neutral disposition.

The inclination parameter φ in the joint generalized scoring function ${S^{\varphi }}(c(\chi ))$ represents the decision-maker’s psychological predisposition toward aggressive, neutral, or cautious attitudes. Assigning a specific value to φ reflects the decision-maker’s personal tendencies or preferences. Within ${S^{\varphi }}(c(\chi ))$, φ indicates a neutral tendency or preference for either aggressive or cautious outcomes.

This section introduces the C-IF ELECTRE decision-making model, designed to address complex discrete choice situations with multiple, incommensurable, and contradictory criteria. Proposed within the C-IF uncertain context, this model offers a novel approach for tackling these intricate decision-making challenges.

This section delves into the practical quandary of multi-expert supplier appraisal, as posed by Otay and Kahraman (2022), through the application of the proposed C-IF ELECTRE framework. Moreover, the operationalization of the advocated C-IF ELECTRE I and II methodologies is elucidated, highlighting their prowess in both efficacy and efficiency.

The C-IF ELECTRE approach is applied to evaluate and select suppliers in an engineering company, focusing on a specific component from various supplied options. Initially, potential suppliers are listed as choice options. Environmental considerations, particularly pollution control and ISO standards, are prioritized in the assessment. Suppliers failing to meet these criteria are eliminated from further analysis. The remaining options—Supplier Options #1 to #3—are evaluated across three key dimensions: cost, service, and technology/quality. These dimensions are assessed through nine evaluative criteria, including payment terms, on-time delivery, and quality management systems. Figure 3 illustrates the hierarchical structure of the multi-expert supplier evaluation, detailing the relationships among dimensions, evaluative criteria, and supplier options.

Step I.1 was conducted based on the problem setting from Otay and Kahraman (2022) and the hierarchical structure shown in Fig. 3. Three choice options were evaluated: Supplier Options #1, #2, and #3, represented as $O=\{{O_{1}},{O_{2}},{O_{3}}\}$. These suppliers were assessed across three dimensions: (1) the cost dimension, consisting of price (${E_{1}}$), terms of payments (${E_{2}}$), and handling and transportation (${E_{3}}$); and (2) the service dimension, encompassing flexibility (${E_{4}}$), on-time delivery (${E_{5}}$), and past performance $({E_{6}})$; and (3) the technology and quality dimension, including quality management systems (${E_{7}}$), technological capability (${E_{8}}$), and R&D studies (${E_{9}}$). The set of evaluative criteria was defined as $E=\{{E_{1}},{E_{2}},\dots ,{E_{9}}\}$.

Step I.2 divides the set E into two subsets: ${E_{B}}$ (beneficial criteria) and ${E_{N}}$ (non-beneficial criteria). Beneficial criteria reflect attributes where higher values are preferred, while non-beneficial criteria favour lower values. For instance, price (${E_{1}}$) and terms of payments (${E_{2}}$) are non-beneficial, as lower values are preferred. In contrast, quality management systems (${E_{7}}$) and technological capability (${E_{8}}$) are beneficial, with higher values indicating desirable traits. Although the nine criteria ${E_{1}},{E_{2}},\dots ,{E_{9}}$ include both types, they are standardized as beneficial following Otay and Kahraman (2022). The sets of beneficial and non-beneficial criteria were defined as ${E_{B}}=\{{E_{1}},{E_{2}},\dots ,{E_{9}}\}$ and ${E_{N}}=\varnothing $.

The data creation stage focuses on establishing the C-IF weight ${w_{j}}$ and C-IF evaluation value ${c_{kj}}$ as outlined in Steps I.3 and I.4. This study follows a structured nine-point linguistic rating scale, as proposed by Chen (2024), involving the following steps: (1) Select a Linguistic Scale: Use a nine-point scale to streamline evaluations; (2) Collect Evaluations: Gather semantic evaluations from decision-makers regarding choice options and criterion importance; (3) Convert to IF Numbers: Transform the semantic evaluations into corresponding IF numbers; (4) Calculate C-IF Evaluation Values: For each decision-maker’s assessment, define an IF number. Then, average these IF numbers to find the centre and determine the radius based on maximum deviation; and (5) Establish C-IF Weights: Similarly, calculate C-IF weights by averaging the importance weights to find the center and radius. These steps synthesize insights from multiple decision-makers into a cohesive C-IF framework for evaluation values and weights.

In the context of multi-expert supplier evaluation, the data from Otay and Kahraman (2022) were used, consolidating evaluations from three procurement specialists to derive collective C-IF weights. In Step I.3, the C-IF weight ${w_{j}}$ is presented as a C-IF number (${\omega _{j}},{\varpi _{j}};{r_{j}}$) in the second column of Table 1. Using Eqs. (10) and (11), the weight characteristic was established as: $W=\{\langle {E_{j}},{\omega _{j}},{\varpi _{j}};{r_{j}}\rangle |{E_{j}}\in \{{E_{1}},{E_{2}},\dots ,{E_{9}}\}\}=\{\langle {E_{j}},{\mathfrak{O}_{r}}({\omega _{j}},{\varpi _{j}})\rangle |{E_{j}}\in \{{E_{1}},{E_{2}},\dots ,{E_{9}}\}\}$, where ${\mathfrak{O}_{r}}({\omega _{j}},{\varpi _{j}})=\{\langle \ell ,{\ell ^{\prime }}\rangle \mid \ell ,{\ell ^{\prime }}\in [0,1],{[{({\omega _{j}}-\ell )^{2}}+{({\varpi _{j}}-{\ell ^{\prime }})^{2}}]^{0.5}}\leqslant {r_{j}},\hspace{2.5pt}\text{and}\hspace{2.5pt}\ell +{\ell ^{\prime }}\leqslant 1\}$ for $j\in \{1,2,\dots ,9\}$.

In Step I.4, the C-IF evaluation value ${c_{kj}}$, expressed as a C-IF number (${u_{kj}},{v_{kj}};{r_{kj}}$), was derived from the aggregated evaluation data in Otay and Kahraman’s (2022) study and is shown in the third to fifth columns of Table 1. The C-IF characteristic for each ${O_{k}}$ was established using Eqs. (12) and (13) as follows: ${C_{k}}=\{\langle {E_{j}},{u_{kj}},{v_{kj}};{r_{kj}}\rangle |{E_{j}}\in \{{E_{1}},{E_{2}},\dots ,{E_{9}}\}\}=\{\langle {E_{j}},{\mathfrak{O}_{r}}({u_{kj}},{v_{kj}})\rangle |{E_{j}}\in \{{E_{1}},{E_{2}},\dots ,{E_{9}}\}\}$, where ${\mathfrak{O}_{r}}({u_{kj}},{v_{kj}})=\{\langle \ell ,{\ell ^{\prime }}\rangle \mid \ell ,{\ell ^{\prime }}\in [0,1],{[{({u_{kj}}-\ell )^{2}}+{({v_{kj}}-{\ell ^{\prime }})^{2}}]^{0.5}}\leqslant {r_{kj}},\hspace{2.5pt}\text{and}\hspace{2.5pt}\ell +{\ell ^{\prime }}\leqslant 1\}$ for $k\in \{1,2,3\}$ and $j\in \{1,2,\dots ,9\}$.

Table 2 presents the aggressive and cautious IF estimates for each criterion and supplier option. In Step I.5, the aggressive IF estimates ${i_{Wj}^{\alpha }}=({\omega _{j}^{\alpha }},{\varpi _{j}^{\alpha }})$ and ${i_{kj}^{\alpha }}=({u_{kj}^{\alpha }},{v_{kj}^{\alpha }})$ are generated using Eq. (7). For example, with ${w_{1}}=(0.573,0.360;0.158)$ and ${c_{21}}=(0.624,0.254;0.163)$, Eq. (7) yields: ${i_{W1}^{\alpha }}=(\min \{1,0.573+0.158/\sqrt{2}\},\max \{0,0.360-0.158/\sqrt{2}\})=(0.685,0.248)$ and ${i_{21}^{\alpha }}=(\min \{1,0.624+0.163/\sqrt{2}\},\max \{0,0.254-0.163/\sqrt{2}\})=(0.739,0.139)$. These results are shown in the upper portion of Table 2. Similarly, the cautious IF estimates ${i_{Wj}^{\beta }}=({\omega _{j}^{\beta }},{\varpi _{j}^{\beta }})$ and ${i_{kj}^{\beta }}=({u_{kj}^{\beta }},{v_{kj}^{\beta }})$ were generated using Eq. (8) and presented in the bottom half of Table 2.

In Step I.6, the degrees of hesitancy ${\hslash _{j}^{\alpha }}$, ${h_{kj}^{\alpha }}$, ${\hslash _{j}^{\beta }}$, and ${h_{kj}^{\beta }}$ for the estimates ${i_{Wj}^{\alpha }}$, ${i_{kj}^{\alpha }}$, ${i_{Wj}^{\beta }}$, and ${i_{kj}^{\beta }}$ were computed. For instance, for ${i_{W1}^{\alpha }}$, hesitancy was calculated as: ${\hslash _{1}^{\alpha }}=1-{\omega _{1}^{\alpha }}-{\varpi _{1}^{\alpha }}=1-0.685-0.248=0.067$. The scoring mechanisms $M({i_{Wj}^{\alpha }})$, $M({i_{kj}^{\alpha }})$, $M({i_{Wj}^{\beta }})$, and $M({i_{kj}^{\beta }})$ were then derived using the natural exponential function approach from Eq. (2). For example, for ${i_{21}^{\alpha }}=(0.739,0.139)$ with ${h_{21}^{\alpha }}=0.122$: The results, including the scoring mechanisms $M({i_{Wj}^{\alpha }})$ and $M({i_{kj}^{\alpha }})$ for aggressive IF estimates (left part), and $M({i_{Wj}^{\beta }})$ and $M({i_{kj}^{\beta }})$ for cautious IF estimates (right part), are illustrated in Table 3.

In Step I.7, the inclination parameter φ is assigned a value within $[0,1]$ to reflect the decision-maker’s preference between aggressive and cautious IF estimates. Here, setting $\varphi =0.5$ indicates equal importance to both. Next, the joint generalized scoring functions ${S^{0.5}}({w_{j}})$ and ${S^{0.5}}({c_{kj}})$ were computed for each ${w_{j}}$ and ${c_{kj}}$ using Eq. (9). For instance, the calculation for ${S^{0.5}}({c_{21}})$ is: ${S^{0.5}}({c_{21}})=0.5\times 0.809+(1-0.5)\times 0.572=0.691$.

In the example with $\varphi =0.5$, the expressions of Eqs. (14) and (15) were simplified since all criteria were beneficial. The concordance and discordance sets were defined as:

As outlined in Step I.8, Table 4 presents the results for ${\mathbb{C}^{0.5}}({O_{k}}/{O_{l}})$ and ${\mathbb{D}^{0.5}}({O_{k}}/{O_{l}})$ when option ${O_{k}}$ outperforms ${O_{l}}$ ($k\ne l$), along with relevant explanations.

In Step I.9, the metric parameter ξ was set to 1 and 2, reflecting the common use of the Manhattan ($\xi =1$) and Euclidean ($\xi =2$) metrics in practice. The C-IF distances ${D_{\mathfrak{M}}^{1}}({c_{kj}},{c_{lj}})$ and ${D_{\mathfrak{M}}^{2}}({c_{kj}},{c_{lj}})$ were computed using the four-term technique from Eq. (6):

These formulas account for differences in parameters r, u, v, and h between two C-IF evaluation values (${c_{kj}}$ and ${c_{lj}}$) to calculate the distance measures ${D_{\mathfrak{M}}^{1}}$ and ${D_{\mathfrak{M}}^{2}}$. Given ${c_{13}}=(0.346,0.528;0.271)$ and ${c_{23}}=(0.569,0.327;0.112)$ with hesitancy degrees of 0.126 and 0.104, the distances were: ${D_{\mathfrak{M}}^{1}}({c_{13}},{c_{23}})={D_{\mathfrak{M}}^{1}}({c_{23}},{c_{13}})=(1/2)\cdot [(1/\sqrt{2})\cdot |0.271-0.112|+(1/2)\cdot (|0.346-0.569|+|0.528-0.327|+|0.126-0.104|)]=0.168$, and ${D_{\mathfrak{M}}^{2}}({c_{13}},{c_{23}})={D_{\mathfrak{M}}^{2}}({c_{23}},{c_{13}})=(1/2)\cdot \{(1/\sqrt{2})\cdot |0.271-0.112|+{[(1/2)\cdot (|0.346-0.569{|^{2}}+|0.528-0.327{|^{2}}+|0.126-0.104{|^{2}})]^{0.5}}\}=0.163$. These Manhattan- and Euclidean-like distances, detailed in Table 5, quantify the dissimilarity between the C-IF evaluation values, reflecting differences across their parameters.

Following Step I.10, the consistency indicator ${\mathcal{I}_{\mathbb{C}}^{0.5}}({O_{k}}/{O_{l}})$ and inconsistency indicator ${\mathcal{I}_{\mathbb{D}}^{0.5}}({O_{k}}/{O_{l}})$ (for $\varphi =0.5$) were derived using Eqs. (16) and (17). In the case where ${O_{3}}$ outperforms ${O_{2}}$, Table 4 shows the concordance set ${\mathbb{C}^{0.5}}({O_{3}}/{O_{2}})=\{{E_{2}},{E_{3}},{E_{6}},{E_{7}},{E_{9}}\}$ and the discordance set ${\mathbb{D}^{0.5}}({O_{3}}/{O_{2}})=\{{E_{1}},{E_{4}},{E_{5}},{E_{8}}\}$. The consistency indicator ${\mathcal{I}_{\mathbb{C}}^{0.5}}({O_{3}}/{O_{2}})$ was then computed as: By employing Eq. (17) and utilizing the C-IF Manhattan-like distance ${D_{\mathfrak{M}}^{1}}({c_{kj}},{c_{lj}})$, the inconsistency indicator ${\mathcal{I}_{\mathbb{D}}^{0.5}}({O_{3}}/{O_{2}})$ can be produced in this fashion: Using the C-IF Euclidean-like distance ${D_{\mathfrak{M}}^{2}}({c_{kj}},{c_{lj}})$, the inconsistency indicator ${\mathcal{I}_{\mathbb{D}}^{0.5}}({O_{3}}/{O_{2}})$ equals 0.454. Additional results are provided in Table 6’s third, fifth, and seventh columns.

Following Step I.11, the average consistency indicator ${\overline{\mathcal{I}}_{\mathbb{C}}^{0.5}}$ is 0.5 (Theorem 7). Using Eq. (18), the consistency entry ${\mathcal{B}_{\mathbb{C}}^{0.5}}({O_{k}}/{O_{l}})$ was generated by comparing ${\mathcal{I}_{\mathbb{C}}^{0.5}}({O_{k}}/{O_{l}})$ with ${\overline{\mathcal{I}}_{\mathbb{C}}^{0.5}}$. The fourth column of Table 6 presents these comparisons. The consistency Boolean matrix ${\mathfrak{B}_{\mathbb{C}}^{0.5}}$ was then constructed using Eq. (19):

In Step I.12, the average inconsistency indicator ${\overline{\mathcal{I}}_{\mathbb{D}}^{0.5}}$ was yielded to be 0.5 (Theorem 9). By comparing ${\mathcal{I}_{\mathbb{D}}^{0.5}}({O_{k}}/{O_{l}})$ with ${\overline{\mathcal{I}}_{\mathbb{D}}^{0.5}}$, the inconsistency entry ${\mathcal{B}_{\mathbb{D}}^{0.5}}({O_{k}}/{O_{l}})$ was derived using Eq. (20). Results based on the C-IF Manhattan-like distance ${D_{\mathfrak{M}}^{1}}$ and Euclidean-like distance ${D_{\mathfrak{M}}^{2}}$ are shown in the sixth and eighth columns of Table 6, respectively. Table 6 summarizes the consistency and inconsistency indicators ${\mathcal{I}_{\mathbb{C}}^{0.5}}({O_{k}}/{O_{l}})$ and ${\mathcal{I}_{\mathbb{D}}^{0.5}}({O_{k}}/{O_{l}})$, along with their entries ${\mathcal{B}_{\mathbb{C}}^{0.5}}({O_{k}}/{O_{l}})$ and ${\mathcal{B}_{\mathbb{D}}^{0.5}}({O_{k}}/{O_{l}})$ for $\varphi =0.5$. As noted from Eq. (21), both distance measures yield the same inconsistency Boolean matrix ${\mathfrak{B}_{\mathbb{D}}^{0.5}}$, as shown:

Following Step I.13, the overall prioritization entry ${\mathcal{B}_{\mathbb{O}}^{0.5}}({O_{k}}/{O_{l}})$ was calculated using ${\mathcal{B}_{\mathbb{O}}^{0.5}}({O_{k}}/{O_{l}})={\mathcal{B}_{\mathbb{C}}^{0.5}}({O_{k}}/{O_{l}})\cdot {\mathcal{B}_{\mathbb{D}}^{0.5}}({O_{k}}/{O_{l}})$ from Eq. (22). These results were then used to build the overall prioritization Boolean matrix ${\mathfrak{B}_{\mathbb{O}}^{0.5}}$ via Eq. (23), as shown:

Continuing the procedure in Step I.13, a dominance graph was created to illustrate the partial-prioritization ranking of the three supplier options, as shown in Fig. 4. The overall outranking relationships using the C-IF ELECTRE I techniques were depicted utilizing orange arrows: ${O_{2}}{\succcurlyeq _{\mathbb{O}}^{0.5}}{O_{1}}$ and ${O_{3}}{\succcurlyeq _{\mathbb{O}}^{0.5}}{O_{1}}$. Consistency-based relationships are shown with green arrows: ${O_{2}}{\succcurlyeq _{\mathbb{C}}^{0.5}}{O_{1}}$, ${O_{2}}{\succcurlyeq _{\mathbb{C}}^{0.5}}{O_{3}}$, and ${O_{3}}{\succcurlyeq _{\mathbb{C}}^{0.5}}{O_{1}}$. Inconsistency-based relationships are indicated by yellow arrows: ${O_{2}}{\succcurlyeq _{\mathbb{D}}^{0.5}}{O_{1}}$ and ${O_{3}}{\succcurlyeq _{\mathbb{D}}^{0.5}}{O_{1}}$. The ranking from the C-IF analytic hierarchy process (AHP) and VIKOR (i.e. VlseKriterijumska Optimizacija I Kompromisno Resenje in Serbian) methodology in Otay and Kahraman (2022) was ${O_{2}}\succcurlyeq {O_{3}}\succcurlyeq {O_{1}}$. The outranking relationships from the C-IF ELECTRE I approach with $\varphi =0.5$ and $\xi =1,2$ align closely with the findings of Otay and Kahraman.

In the C-IF ELECTRE II approach, Steps II.1–II.10 mirror Steps I.1–I.10. In Step II.11, Eq. (24) calculates the consistency-dependent average outflow ${\mathcal{A}_{\mathbb{C}}^{0.5}}({O_{k}})$. For example, for ${O_{1}}$:${\mathcal{A}_{\mathbb{C}}^{0.5}}({O_{1}})={\textstyle\sum _{l=1,l\ne 1}^{3}}{\mathcal{I}_{\mathbb{C}}^{0.5}}({O_{1}}/{O_{l}})/(3-1)=[{\mathcal{I}_{\mathbb{C}}^{0.5}}({O_{1}}/{O_{2}})+{\mathcal{I}_{\mathbb{C}}^{0.5}}({O_{1}}/{O_{3}})]/2=(0.000+0.019)/2=0.010$. Moreover, ${\mathcal{A}_{\mathbb{C}}^{0.5}}({O_{2}})=0.771$ and ${\mathcal{A}_{\mathbb{C}}^{0.5}}({O_{3}})=0.720$. In Step II.11, Eq. (25) computes the inconsistency-dependent average inflow ${\mathcal{A}_{\mathbb{D}}^{0.5}}({O_{k}})$. Using the C-IF Manhattan-like distance measure ${D_{\mathfrak{M}}^{1}}:{\mathcal{A}_{\mathbb{D}}^{0.5}}({O_{1}})={\textstyle\sum _{l=1,l\ne 1}^{3}}{\mathcal{I}_{\mathbb{D}}^{0.5}}({O_{1}}/{O_{l}})/(3-1)=[{\mathcal{I}_{\mathbb{D}}^{0.5}}({O_{1}}/{O_{2}})+{\mathcal{I}_{\mathbb{D}}^{0.5}}({O_{1}}/{O_{3}})]/2=(1.000+0.990)/2=0.995$. Additionally, ${\mathcal{A}_{\mathbb{D}}^{0.5}}({O_{2}})=0.275$ and ${\mathcal{A}_{\mathbb{D}}^{0.5}}({O_{3}})=0.231$. Using the C-IF Euclidean-like distance measure ${D_{\mathfrak{M}}^{2}}$:${\mathcal{A}_{\mathbb{D}}^{0.5}}({O_{1}})={\textstyle\sum _{l=1,l\ne 1}^{3}}{\mathcal{I}_{\mathbb{D}}^{0.5}}({O_{1}}/{O_{l}})/(3-1)=[{\mathcal{I}_{\mathbb{D}}^{0.5}}({O_{1}}/{O_{2}})+{\mathcal{I}_{\mathbb{D}}^{0.5}}({O_{1}}/{O_{3}})]/2=(1.000+0.990)/2=0.995$, ${\mathcal{A}_{\mathbb{D}}^{0.5}}({O_{2}})=0.273$, and ${\mathcal{A}_{\mathbb{D}}^{0.5}}({O_{3}})=0.232$.

Finally, Eq. (26) calculates the overall net flow ${\mathcal{N}_{\mathbb{O}}^{0.5}}({O_{k}})$. Using the C-IF Manhattan-like distance measure ${D_{\mathfrak{M}}^{1}}$, the calculations yield: ${\mathcal{N}_{\mathbb{O}}^{0.5}}({O_{1}})={\mathcal{A}_{\mathbb{C}}^{0.5}}({O_{1}})-{\mathcal{A}_{\mathbb{D}}^{0.5}}({O_{1}})=0.010-0.995=-0.985$, ${\mathcal{N}_{\mathbb{O}}^{0.5}}({O_{2}})=0.496$, and ${\mathcal{N}_{\mathbb{O}}^{0.5}}({O_{3}})=0.489$. Using the C-IF Euclidean-like distance ${D_{\mathfrak{M}}^{2}}$:${\mathcal{N}_{\mathbb{O}}^{0.5}}({O_{1}})={\mathcal{A}_{\mathbb{C}}^{0.5}}({O_{1}})-{\mathcal{A}_{\mathbb{D}}^{0.5}}({O_{1}})=0.010-0.995=-0.985$, ${\mathcal{N}_{\mathbb{O}}^{0.5}}({O_{2}})=0.498$, and ${\mathcal{N}_{\mathbb{O}}^{0.5}}({O_{3}})=0.488$. Arranging these in descending order, the complete prioritization ranking is ${O_{2}}{\succcurlyeq _{\mathcal{N}}^{0.5}}{O_{3}}{\succcurlyeq _{\mathcal{N}}^{0.5}}{O_{1}}$ for both measures ${D_{\mathfrak{M}}^{1}}$ and ${D_{\mathfrak{M}}^{2}}$. This aligns with the preference order from the C-IF AHP and VIKOR methodology in Otay and Kahraman (2022), confirming that the outranking relationships from the current C-IF ELECTRE II approach with $\varphi =0.5$ and $\xi =1,2$ are consistent with their findings.

The proposed C-IF ELECTRE I and II techniques have demonstrated feasibility and efficiency in addressing multiple-criteria supplier assessments. By integrating C-IF theory with the established ELECTRE framework, these methodologies provide a comprehensive and reliable decision-making toolset. They effectively consider both beneficial and non-beneficial criteria while accounting for the decision-maker’s attitudes toward aggression and caution. The rankings of supplier options generated from these techniques are coherent and dependable, aligning with the findings from the C-IF AHP and VIKOR approach by Otay and Kahraman (2022). This indicates that the C-IF ELECTRE I and II methods significantly enhance decision-making in complex supplier evaluation scenarios.

The C-IF ELECTRE approach provides valuable insights into how decision-makers evaluate suppliers across multiple criteria. Key dimensions—such as cost, service, technology, and quality—greatly influence supplier rankings, particularly factors like price, on-time delivery, and technological capability. By incorporating both aggressive and cautious estimates, this method addresses uncertainty and adapts to varying market conditions and expert judgments, improving the reliability of recommendations.

Key advantages of the C-IF ELECTRE methodology include: (1) Refined Uncertainty Representation: Circular structures capture uncertainty, offering a balanced view of fluctuating factors like supplier reliability; (2) Flexibility in Risk Management: The inclination parameter allows customization based on the decision-maker’s risk tolerance, accommodating aggressive or cautious evaluations; and (3) Improved Sensitivity to Complex Criteria: This approach effectively manages complex, interdependent criteria, making it ideal for multidimensional supplier evaluations where factors are interconnected.

Despite its strengths, the C-IF ELECTRE approach has limitations: (1) Computational Complexity: The integration of C-IF sets and the need for precise calibration of parameters (φ and ξ) increases computational intricacy, potentially burdening decision-makers without advanced tools or expertise; and (2) Limited Sensitivity to Extremes: The approach may not adequately respond to suppliers excelling or underperforming in specific criteria, risking the omission of those with exceptional performance in niche areas but lacking broader competencies.

This section examines the effects of different inclination parameter settings on the C-IF ELECTRE approach’s results. It also incorporates the Chebyshev distance metric ($\xi \to \infty $) within the C-IF Minkowski-like distance measure, alongside the previously used Manhattan and Euclidean metrics. Moreover, it evaluates divergence functions proposed by Khan et al. (2022) to compare the results generated by the C-IF ELECTRE techniques.

The C-IF Chebyshev-like distance between ${c_{kj}}$ and ${c_{lj}}$ is derived by setting the metric parameter ξ in Definition 4 to infinity. This is calculated using the four-term strategy as:

Divergence measures in fuzzy contexts help identify differences between fuzzy sets. For C-IF sets, Khan et al. (2022) developed various divergence functions to address higher-order uncertainties and evaluated their performance. They established five divergence functions based on chi-square and Canberra distances, as well as the exponential function. This study will adopt these five divergence measures to calculate inconsistency indicators. Let ${D_{\mathfrak{E}}^{\epsilon }}({c_{kj}},{c_{lj}})$ represent the divergence measure between ${c_{kj}}$ and ${c_{lj}}$, where ϵ is an identifier parameter from the set $\{1,2,\dots ,5\}$. The formulas for the five divergence measures based on chi-square distances (${D_{\mathfrak{E}}^{1}}$ and ${D_{\mathfrak{E}}^{2}}$), Canberra distances $({D_{\mathfrak{E}}^{3}}\hspace{2.5pt}\text{and}\hspace{2.5pt}{D_{\mathfrak{E}}^{4}})$, and the exponential function (${D_{\mathfrak{E}}^{5}}$) are presented below:

In the initial comparative analysis, the inclination parameter φ was systematically varied from 0 to 1 in increments of 0.1. Using these eleven configurations, the study applied the C-IF ELECTRE to the same supplier evaluation issue. Figure 5 illustrates the results in two parts: Fig. 5(a) shows the distribution of consistency indicators ${\mathcal{I}_{\mathbb{C}}^{\varphi }}({O_{k}}/{O_{l}})$ among option pairs. The distribution remains stable for φ values between 0 and 0.4, but shifts significantly when φ exceeds 0.5. As φ increases, ${\mathcal{I}_{\mathbb{C}}^{\varphi }}({O_{2}}/{O_{1}})$, ${\mathcal{I}_{\mathbb{C}}^{\varphi }}({O_{2}}/{O_{3}})$, and ${\mathcal{I}_{\mathbb{C}}^{\varphi }}({O_{3}}/{O_{1}})$ decrease, while ${\mathcal{I}_{\mathbb{C}}^{\varphi }}({O_{1}}/{O_{2}})$, ${\mathcal{I}_{\mathbb{C}}^{\varphi }}({O_{3}}/{O_{2}})$, and ${\mathcal{I}_{\mathbb{C}}^{\varphi }}({O_{1}}/{O_{3}})$ increase. Figure 5(b) displays the consistency-dependent average outflows ${\mathcal{A}_{\mathbb{C}}^{\varphi }}({O_{k}})$ for each option. It reveals that the average outflow of ${O_{1}}$ is consistently the lowest across all φ values. As φ increases from 0 to 0.6, ${O_{2}}$ has the highest average outflow, followed by ${O_{3}}$. From $\varphi =0.7$ to 1, ${O_{3}}$ surpasses ${O_{2}}$ in average outflow, indicating a shift in advantage. These findings suggest that ${O_{2}}$ is favoured at lower φ values, while ${O_{3}}$ gains an advantage at higher φ values.

In the second comparative analysis, the study examined the combined effects of different inclination parameter values ($\varphi =0,0.1,\dots ,1$) and various C-IF distance and divergence measures on inconsistency indicators. It evaluated the C-IF Minkowski-like distance ${D_{\mathfrak{M}}^{\xi }}$ with metric parameters $\xi =1,2,\infty $, alongside five divergence measures ${D_{\mathfrak{E}}^{\epsilon }}$ based on chi-square, Canberra, and exponential functions. The study tested these eight measures: Manhattan-like ${D_{\mathfrak{M}}^{1}}$, Euclidean-like ${D_{\mathfrak{M}}^{2}}$, Chebyshev-like ${D_{\mathfrak{M}}^{\infty }}$, chi-square ${D_{\mathfrak{E}}^{1}}$ and ${D_{\mathfrak{E}}^{2}}$, Canberra ${D_{\mathfrak{E}}^{3}}\hspace{2.5pt}\text{and}\hspace{2.5pt}{D_{\mathfrak{E}}^{4}}$, and exponential ${D_{\mathfrak{E}}^{5}}$. It compared outcomes for various parameter combinations, as shown in Fig. 6: (1) Similar trends: Inconsistency indicators calculated with ${D_{\mathfrak{M}}^{1}}$, ${D_{\mathfrak{M}}^{2}}$, and ${D_{\mathfrak{M}}^{\infty }}$ showed high consistency. Likewise, results from divergence measures ${D_{\mathfrak{E}}^{1}}$, ${D_{\mathfrak{E}}^{3}}$, and ${D_{\mathfrak{E}}^{5}}$ aligned closely with those from the Minkowski-like distances; (2) Distinct patterns: Line charts in Fig. 6(e) and (g) revealed unique trends, suggesting these cases differ from others; and (3) Stable results: Indicators based on ${D_{\mathfrak{E}}^{2}}$ and ${D_{\mathfrak{E}}^{4}}$ remained relatively consistent across all φ values, contrasting with other measures that captured parameter variations more effectively. These findings highlight the C-IF ELECTRE framework’s robustness in accommodating diverse distance and divergence measures.

The third comparative analysis considered key parameters: inclination $(\varphi =0,0.1,\dots ,1)$, metric ($\xi =1,2,\infty $), and identifier $(\varepsilon =1,2,\dots ,5)$. Figure 7 contrasts outcomes of inconsistency-dependent average inflows across various C-IF distance/divergence measures, including Manhattan, Euclidean, and Chebyshev distances (${D_{\mathfrak{M}}^{1}}$, ${D_{\mathfrak{M}}^{2}}$, and ${D_{\mathfrak{M}}^{\infty }}$) from Chen (2023b) and five divergence measures (${D_{\mathfrak{E}}^{1}}$–${D_{\mathfrak{E}}^{5}}$) from Khan et al. (2022). Key findings include: (1) Consistent trends: Radar charts in Fig. 7(a)–(d), (f), and (h) show similar patterns across distance measures ${D_{\mathfrak{M}}^{1}}$, ${D_{\mathfrak{M}}^{2}}$, and ${D_{\mathfrak{M}}^{\infty }}$ and divergence measures ${D_{\mathfrak{E}}^{1}}$, ${D_{\mathfrak{E}}^{3}}$, and ${D_{\mathfrak{E}}^{5}}$; (2) Distinct cases: Fig. 7(e) (${D_{\mathfrak{E}}^{2}}$) and Fig. 7(g) (${D_{\mathfrak{E}}^{4}}$) reveal unique patterns, indicating these scenarios differ from others; and (3) Disadvantage patterns: Across all φ values, option ${O_{1}}$ shows the highest average inflow (disadvantage). For φ from 0 to 0.4, ${O_{2}}$ has the lowest inflow, followed by ${O_{3}}$. When φ exceeds 0.5, ${O_{3}}$ becomes the least disadvantaged, followed by ${O_{2}}$. These results confirm that the C-IF ELECTRE techniques produce consistent and reliable findings, regardless of the measurement method used.

The fourth comparative study considered the parameters $\varphi =0,0.1,\dots ,1$, $\xi =1,2,\infty $, and $\varepsilon =1,2,\dots ,5$. Figure 8 illustrates the overall net flow ${\mathcal{N}_{\mathbb{O}}^{\varphi }}({O_{k}})$ for each option, reflecting both the advantages from consistency indicators and disadvantages from inconsistency indicators. When using C-IF Manhattan, Euclidean, and Chebyshev distances (${D_{\mathfrak{M}}^{1}}$, ${D_{\mathfrak{M}}^{2}}$, and ${D_{\mathfrak{M}}^{\infty }}$), or divergence measures based on Canberra distances (${D_{\mathfrak{E}}^{3}},{D_{\mathfrak{E}}^{4}}$) and the exponential function $({D_{\mathfrak{E}}^{5}})$, the rankings were ${O_{2}}{\succcurlyeq _{\mathcal{N}}^{\varphi }}{O_{3}}{\succcurlyeq _{\mathcal{N}}^{\varphi }}{O_{1}}$ for φ values from 0 to 0.5, and shifted to ${O_{3}}{\succcurlyeq _{\mathcal{N}}^{\varphi }}{O_{2}}{\succcurlyeq _{\mathcal{N}}^{\varphi }}{O_{1}}$ for φ values from 0.6 to 1. With the chi-square-based divergence measure ${D_{\mathfrak{E}}^{1}}$, the rankings remained ${O_{2}}{\succcurlyeq _{\mathcal{N}}^{\varphi }}{O_{3}}{\succcurlyeq _{\mathcal{N}}^{\varphi }}{O_{1}}$ for φ between 0 and 0.4, but shifted to ${O_{3}}{\succcurlyeq _{\mathcal{N}}^{\varphi }}{O_{2}}{\succcurlyeq _{\mathcal{N}}^{\varphi }}{O_{1}}$ for φ between 0.5 and 1. Using ${D_{\mathfrak{E}}^{2}}$, the shift occurred later, with rankings changing from ${O_{2}}{\succcurlyeq _{\mathcal{N}}^{\varphi }}{O_{3}}{\succcurlyeq _{\mathcal{N}}^{\varphi }}{O_{1}}$ ($\varphi =0$ to 0.6) to ${O_{3}}{\succcurlyeq _{\mathcal{N}}^{\varphi }}{O_{2}}{\succcurlyeq _{\mathcal{N}}^{\varphi }}{O_{1}}$ ($\varphi =0.7$ to 1). These findings reveal how rankings vary with φ values under different distance and divergence measures, offering insights into the options’ performance across varying conditions.

The proposed C-IF ELECTRE I and II approaches effectively handle uncertainties, hesitancies, and imprecise data through C-IF theory. They integrate a scoring mechanism, consistency and inconsistency indexing, and prioritization steps. Applied to supplier evaluation tasks, these methods demonstrate practical utility and reliability. The key contributions include: introducing the joint generalized scoring function, capturing decision-maker preferences, developing the C-IF ELECTRE framework, and validating its benefits through multi-expert supplier assessments.

The C-IF ELECTRE methodology, while effective for handling uncertainty in supplier evaluations or other decision-making issues, requires careful parameter tuning, may be complex for non-experts, and could face limitations in other fuzzy contexts or decision-making scenarios, requiring further validation and adaptation for broader applicability. Future research could broaden the application of the C-IF ELECTRE approach and strengthen its validity through comparisons with other methods and model quality metrics. While this study focused on supplier evaluation, future work should apply the approach to diverse fields—such as healthcare, project management, and environmental sustainability—to assess its versatility. Comparative analyses with other decision-making models, including traditional and fuzzy ELECTRE methods, would highlight the unique strengths of the C-IF ELECTRE. Additionally, evaluating its stability and reliability through metrics like accuracy and consistency would provide valuable quantitative insights, further validating its effectiveness and practical value.