Circular Intuitionistic Fuzzy ELECTRE Approach: A Novel Multiple-Criteria Choice Model
Abstract
This paper presents a multiple-criteria choice model, the circular intuitionistic fuzzy (C-IF) ELECTRE, designed to resolve C-IF ambiguities through built-in circular functions. Joint generalized scoring functions establish contrast relationships between C-IF evaluation values, facilitating concordance and discordance analyses for option ranking. The efficacy of C-IF ELECTRE I and II—leveraging tools such as the prioritization Boolean matrix, average outflows and inflows, and overall net flow—is validated through a multi-expert supplier evaluation, with outcomes benchmarked against alternative methods. A comparative analysis explores the impact of parameter variations, underscoring how integrating C-IF sets with ELECTRE enhances decision-making in complex, multifaceted environments.
1Introduction
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:
(1) Uncertainty Challenges: Current ELECTRE-based approaches struggle with complex uncertainties, highlighting the need for more robust and enhanced methods.
(2) Limited C-IF Applications: Despite progress in C-IF methods, there has been little integration of ELECTRE in C-IF multiple-criteria analysis, highlighting a key research gap.
(3) Distance Measurement Importance: Accurate measurement of C-IF distances is crucial for distinguishing complex information. This study adopts the four-term Minkowski-like distance model for its completeness in developing the C-IF ELECTRE approach.
This study offers key advancements in ELECTRE-based decision-making:
(1) Joint Generalized Scoring Function: Introduces an inclination parameter to capture the decision-maker’s aggressive, neutral, or cautious tendencies, integrating these attitudes into assessments.
(2) C-IF ELECTRE Framework: Develops a practical model tailored for C-IF contexts with step-by-step algorithms for defining problems, calculating scores, measuring consistency/inconsistency, and generating rankings.
(3) C-IF ELECTRE I and II Techniques: Combines C-IF sets with ELECTRE to manage complex multi-criteria decisions, validated through multi-expert supplier assessments, producing reliable, consistent rankings aligned with comparative methods.
2Foundational Concepts Related to C-IF Sets
Definition 1
Definition 1(Atanassov, 1986).
Assume ℵ is a nonempty set of elements. Let
(1)
Definition 2
Definition 2(Hezam et al., 2023).
Let
(2)
Definition 3
Definition 3(Atanassov, 2020).
Let
(3)
(4)
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
Fig. 1
The visualization of the relationship between standard IF and C-IF constructs.
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.
Definition 4
Definition 4(Chen, 2023b).
Consider two C-IF numbers,
(5)
(6)
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.
3Joint Generalized Scoring Function
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.
Definition 5.
For a C-IF number
(7)
(8)
Theorem 1.
For a C-IF number
Proof.
The conditions for
Theorem 2.
The scoring mechanisms
Proof.
Applying Eq. (2), the scoring mechanisms
Theorem 3.
For the aggressive and cautious IF estimates
Proof.
Consider the scoring mechanism
Definition 6.
Consider
(9)
Theorem 4.
The following characteristics are fulfilled by the joint generalized scoring function
Proof.
From Definition 6, the inclination parameter φ ranges from 0 to 1. As per Theorem 2, the scoring mechanisms for
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
4Proposed C-IF ELECTRE Approach
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.
4.1Suggested Methodology
From a mathematical perspective, a multiple-criteria choice problem involves a set of m choice options, signified as
The significance of a criterion
(10)
(11)
The C-IF evaluation value of a choice option
(12)
(13)
Based on Definition 5, the aggressive IF estimate
Definition 7.
The contrast relationship between the C-IF evaluation values
1. For
2. For
Definition 8.
The concordance set
(14)
(15)
Theorem 5.
The concordance set
Proof.
Properties (1) to (3) are trivially valid. Property (1) states that
This study employs the joint generalized scoring function with W to define consistency indicators between choice pairs. Given the C-IF weight
Definition 9.
When the choice option
(16)
Theorem 6.
Assuming that
Proof.
Applying Theorem 4, we know
Theorem 7.
Assuming
Proof.
As per Definition 8, the concordance sets are derived as:
This study introduces an inconsistency indicator for each choice pair using the joint generalized scoring function
Definition 10.
When choice option
(17)
Theorem 8.
Assuming that
Proof.
Both the C-IF Minkowski-like distance and the joint generalized scoring function possess the non-negativity property. Next, this theorem’s proving procedure follows a resemble approach to that of Theorem 6. □
Theorem 9.
Assuming
Proof.
By Definition 8, the discordance sets are defined as follows:
To construct the C-IF ELECTRE I prioritization procedure, this study first compares the consistency indicator
(18)
(19)
This study compares the inconsistency indicator
(20)
(21)
Using the specified φ, this paper formulates the overall prioritization entry
(22)
(23)
If
This study presents the concepts of consistency-dependent average outflow, inconsistency-dependent average inflow, and overall net flow for ranking choice options in the C-IF ELECTRE II prioritization process. These concepts are inspired by the leaving and entering flows developed by Wang and Chen (2021). Specifically, the consistency-dependent average outflow
(24)
(25)
(26)
The average outflow, average inflow, and overall net flow satisfy these properties: (1)
4.2Suggested Algorithm
Figure 2 presents a systematic framework for applying the C-IF ELECTRE I and II approaches to multiple-criteria choice problems under C-IF uncertainty. The C-IF ELECTRE I method involves several stages: (1) Problem Definition: Identifies choice options and evaluative criteria; (2) Data Creation: Establishes C-IF weights for criteria and generates C-IF evaluation values for options; (3) C-IF Scoring: Computes aggressive and cautious IF estimates, and determines scoring mechanisms based on hesitancy degrees; (4) Consistency/Inconsistency Indexing: Identifies concordance and discordance sets, calculates Minkowski-like distances, and determines consistency/inconsistency indicators; and (5) C-IF ELECTRE I Prioritization: Produces overall prioritization entries and Boolean matrices, and creates a dominance graph for partial-priority ranking. The C-IF ELECTRE II approach follows similar steps but incorporates average outflows and inflows alongside net flows to establish complete-priority rankings via overall net flows.
Fig. 2
The systematic procedure of the C-IF ELECTRE I and II approaches.
The implementation steps of the C-IF ELECTRE I approach are as follows:
Problem definition stage: See Steps I.1 and I.2
Step I.1. Define a multiple-criteria decision problem with evaluative criteria
Step I.2. Split criteria into beneficial
Data creation stage: See Steps I.3 and I.4
Step I.3. Establish C-IF weights
Step I.4. Generate C-IF evaluation values
C-IF scoring stage: See Steps I.5–I.7
Step I.5. Compute the aggressive IF estimates
Step I.6. Derive hesitancy degrees
Step I.7. Designate a value to the inclination parameter
Consistency/inconsistency indexing stage: See Steps I.8–I.10
Step I.8. Employ Eqs. (14) and (15), respectively, to establish the concordance set
Step I.9. Allocate a value for the metric parameter
Step I.10. Determine the consistency indicator
C-IF ELECTRE I prioritization stage: See Steps I.11–I.13
Step I.11. Compare the consistency indicator
Step I.12. Contrast the inconsistency indicator
Step I.13. Produce the overall prioritization entry
The implementation steps of the C-IF ELECTRE II approach include:
Steps II.1 to II.10. Same as Steps I.1 to I.10.
C-IF ELECTRE II prioritization stage: See Steps II.11 and II.12
Step II.11. Calculate the consistency-dependent average outflow
Step II.12. Obtain the overall net flow
5Application to a Problem with Supplier Evaluation
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.
Fig. 3
The hierarchical structure of the multi-expert supplier evaluation issue.
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
Step I.2 divides the set E into two subsets:
The data creation stage focuses on establishing the C-IF weight
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
Table 1
Information regarding the C-IF weight
∗: Refer to Otay and Kahraman (2022).
Table 2
Specifics on the aggressive IF estimates
| Outcomes associated with the aggressive IF estimates | ||||
| Outcomes associated with the cautious IF estimates | ||||
In Step I.4, the C-IF evaluation value
Table 2 presents the aggressive and cautious IF estimates for each criterion and supplier option. In Step I.5, the aggressive IF estimates
In Step I.6, the degrees of hesitancy
Table 3
Specifics on the scoring mechanisms
| Outcomes related to aggressive IF estimates | Outcomes related to cautious IF estimates | |||||||
| 0.722 | 0.604 | 0.809 | 0.748 | 0.494 | 0.208 | 0.572 | 0.602 | |
| 0.447 | 0.603 | 0.705 | 0.878 | 0.286 | 0.397 | 0.705 | 0.603 | |
| 0.607 | 0.604 | 0.705 | 0.769 | 0.379 | 0.208 | 0.543 | 0.603 | |
| 0.612 | 0.603 | 0.812 | 0.809 | 0.362 | 0.315 | 0.705 | 0.549 | |
| 0.729 | 0.705 | 0.852 | 0.748 | 0.479 | 0.564 | 0.705 | 0.602 | |
| 0.498 | 0.748 | 0.913 | 0.967 | 0.265 | 0.602 | 0.701 | 0.705 | |
| 0.416 | 0.705 | 0.809 | 0.872 | 0.310 | 0.564 | 0.549 | 0.706 | |
| 0.663 | 0.894 | 0.707 | 0.705 | 0.557 | 0.396 | 0.603 | 0.543 | |
| 0.492 | 0.792 | 0.705 | 0.707 | 0.492 | 0.396 | 0.543 | 0.603 | |
In Step I.7, the inclination parameter φ is assigned a value within
In the example with
As outlined in Step I.8, Table 4 presents the results for
Table 4
Specifics on the concordance and discordance sets for
| Explanation | ||||
| ∅ | ||||
| ∅ | ||||
In Step I.9, the metric parameter ξ was set to 1 and 2, reflecting the common use of the Manhattan (
These formulas account for differences in parameters r, u, v, and h between two C-IF evaluation values (
Table 5
Specifics on the C-IF Manhattan- and Euclidean-like distances
| Outcome of the C-IF Manhattan-like distances | Outcome of the C-IF Euclidean-like distances | |||||
| 0.177 | 0.196 | 0.034 | 0.176 | 0.192 | 0.033 | |
| 0.156 | 0.135 | 0.089 | 0.151 | 0.134 | 0.086 | |
| 0.168 | 0.197 | 0.031 | 0.163 | 0.192 | 0.031 | |
| 0.192 | 0.114 | 0.078 | 0.190 | 0.114 | 0.076 | |
| 0.072 | 0.021 | 0.052 | 0.072 | 0.020 | 0.051 | |
| 0.084 | 0.126 | 0.041 | 0.080 | 0.120 | 0.040 | |
| 0.053 | 0.082 | 0.079 | 0.050 | 0.081 | 0.076 | |
| 0.109 | 0.098 | 0.030 | 0.108 | 0.097 | 0.029 | |
| 0.077 | 0.105 | 0.030 | 0.074 | 0.102 | 0.029 | |
Following Step I.10, the consistency indicator
Table 6
Specifics on the consistency/inconsistency indicators and entries for
| Use of the | Use of the | ||||||
| 0.000 | 0 | 1.000 | 0 | 1.000 | 0 | ||
| 0.019 | 0 | 0.990 | 0 | 0.990 | 0 | ||
| 1.000 | 1 | 0.000 | 1 | 0.000 | 1 | ||
| 0.542 | 1 | 0.549 | 0 | 0.546 | 0 | ||
| 0.981 | 1 | 0.010 | 1 | 0.010 | 1 | ||
| 0.458 | 0 | 0.451 | 1 | 0.454 | 1 | ||
Following Step I.11, the average consistency indicator
In Step I.12, the average inconsistency indicator
Following Step I.13, the overall prioritization entry
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:
Fig. 4
The dominance graph for the multi-expert supplier evaluation issue.
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
Finally, Eq. (26) calculates the overall net flow
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.
6Comprehensive Comparative Analysis
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 (
The C-IF Chebyshev-like distance between
(27)
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
(28)
(29)
(30)
(31)
(32)
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
Fig. 5
Comparison outcomes of consistency indicators and consistency-dependent average outflows.
In the second comparative analysis, the study examined the combined effects of different inclination parameter values (
Fig. 6
Comparison outcomes of inconsistency indicators for distinct C-IF distance/divergence measures.
The third comparative analysis considered key parameters: inclination
Fig. 7
Comparison outcomes of inconsistency-dependent average inflows.
 |
 |
The fourth comparative study considered the parameters
Fig. 8
Comparison outcomes of overall net flows.
7Conclusions
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.
Acknowledgements
The author thanks the editor and anonymous referees for their insightful feedback, which greatly enhanced the paper’s quality.
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